NEEP 536: Feasibility of Controlled Fusion Energy | LECTURE PRESENTATION SLIDES

NEEP 536: Feasibility of Controlled Fusion Energy LECTURE PRESENTATION SLIDES

Overview: Fusion Energy • Why? – To meet growing world energy demands using a safe, clean method of producing electricity.

• What?

– Extract net energy from controlled thermonuclear fusion of light elements

• Who? – International research teams of engineers and physicists.

• Where?

– Research is worldwide (including UW-Madison)….eventual energy source has few geographical restrictions.

• When?

– Many feasibility issues resolved….demonstration in next decades

• How? – This course

Course outline 1.

2.

3.

4.

5.

6.

Fusion reactor physics – Fusion reactions, fuels, Coulomb collisions, power balance Magnetic confinement concept – Configurations, MHD, pressure limits, power exhaust – Burning plasma experiments (ITER) Inertial confinement concept – Ignition physics, Rayleigh-Taylor, drivers – National Ignition Facility Electrostatic confinement concept Technology of fusion – Neutronics, tritium, activation – MFE: plasma-surface interactions, magnets – ICF: drivers, chambers Economics and social aspects of fusion – Cost of electricity estimates, scalings, safety

Expected work for course • 5 homework assignments (30%) • Written final exam: Friday May 16, 7:25 p.m. (45%) • Paper on related topic (25%) – Written report will be due 1 week before….

– Brief oral presentation (~15 minutes) to the class near end of semester – Before starting…please let me advise you on topics

Fusion & fission: binding energy, E = ∆m c2

Exothermic fusion reactions of interest •

Hydrogenic species definitions (A=1): –

Hydrogen or p (M=1), Deuterium (M=2), Tritium (M=3)

D + T --> 4He + n + 17.6 MeV D + D --> (50%) T + p + 4 MeV or (50%) 3He + n + 3.3 MeV D + 3He --> 4He + p + 18.3 MeV T + T --> 4He + 2 n + 11.3 MeV

Exothermic fusion reactions of interest Fusion reaction cross sections 10

barns (1e-28 m2)

1 D+D = T + p D+D = 3He + n D +T D + 3He T+T

0.1

0.01 0.001

0.0001 1

10

100

Particle energy (keV)

1000

Estimating fusion reaction rates • Fundamental competition: nuclear binding force vs. repulsive Coulomb force

U

rnucleus repulsive Coulomb force

r attracting nuclear force

Estimating fusion reaction rates • Repulsive force acting on two particles with same charge (ions here),

F=

Z1Z2e

U

2 ε, dielectric const

rnucleus repulsive Coulomb force

~8.9x10-12 F/m

4 πεr

r attracting nuclear force

2 r

Z1Z2e2

Z1Z2e2 dr = U ≡ ∫ F(r)dr = − ∫ 2 4πεr ∞ 4πεr So for protons (taking U=0 at infinity), rnucleus ~ 4x10-15m, e=1.6x10-19 C, we get U~5.5x10-14 J

Or U ~ 350 keV

Some useful units & conversions • • •

1 eV = 1.6 x 10-19 J (particle energy) 1 eV ~ 104 K (describing temperatures) 10 keV ~ 108 K ~ 100 million K (typical fusion T)

• •

1 unit ~ 5x1011 J (per capita energy consumption, U.S.) 16 kW/per capita power consumption --> ~ 5 TW

•

1 barn = 10-28 m2 (cross section)

•

Vth ~ 1.4x104 (T / M)1/2 m/s {T in eV, M in amu}

If Ucoulomb~350 keV, how do we get fusion at even ~10 keV? •

Two important effects 1.

2.

Quantum-mechanical tunneling provides finite cross-sections below Ucoulomb Fusion achieved in thermalized population distribution with “significant” population at E > T (will derive next class)

Quantum-tunneling provides finite crosssection for E < UCoulomb Esimate of tunneling effect

100

barns (1e-20 m2)

10 1 0.1

D +T: expt

0.01

prop to exp(-1/sqrt(E)) 0.001 0.0001 1

10

100

Particle energy (keV)

1000

D-T is obviously the easiest reaction to achieve…what about D-T as fuel?

– D + n --> T + γ (e.g. CANDU reactor)

• …and for fusion we have 6Li + n --> T + 4He + 4.8 MeV 7Li + n --> T + 4He + n - 2.3 MeV

Fusion reaction cross sections 10 1

barns (1e-28 m2)

• Deuterium is so abundant on earth…fD/H ~ 1.5x10-4 …as to make an infinite supply for fusion.

• Tritium does not occur naturally due to its short half-life, T --> 3He + e- 1/2 life = 12.3 years.

• So T must be bred

D+D = T + p D+D = 3He + n D +T D + 3He T+T

0.1 0.01 0.001 0.0001 1

10

100

Particle energy (keV)

1000

Fusion by tunneling Quantum

Classical

Proton-proton potential energy

Classical turning point

7 February 2002

Whyte 2003-01-22 10:31:52

1 2

Probability density mv2

-------------------------------------------7 February 2002 Astronomy 142, Spring 2002 0 1

10−13 cm

Distance

Distance Nonzero probability!

Astronomy 142, Spring 2002

10

D-T is obviously the easiest reaction to achieve…what about D-T as fuel?

Fusion reaction cross sections 10 1

barns (1e-28 m2)

• Deuterium is so abundant on earth…fD/H ~ 1.5x10-4 …as to make an infinite supply for fusion.

• Tritium does not occur naturally due to its short half-life, T --> 3He + e- 1/2 life = 12.3 years.

• So T must be bred

D+D = T + p D+D = 3He + n D +T D + 3He T+T

0.1 0.01 0.001 0.0001 1

10

100

1000

Particle energy (keV)

– D + n --> T + γ (e.g. CANDU reactor)

• …and for fusion we have 6Li + n --> T + 4He + 4.8 MeV 7Li + n --> T + 4He + n - 2.3 MeV

NEEP 536, Spring 2003, D. Whyte

With Li blanket/breeder We can produce closed T fuel cycle and 6Li + D = 24He + 22.4 MeV Is actual fusion fuel

Fusion & Fission exhibit important differences in energetic and neutronics Fusion

Fission

Fuel

D, (T) (1 neutron)

235U (~2.5 neutrons)

Yield

~20 MeV

~200 MeV

Energy / kg Units / kg # Neutrons/ MeV Eneutrons /unit energy

3.4x1014 J / ~700

8.8x1013 / ~180

0.05

~0.01

~1

~0.02

NEEP 536, Spring 2003, D. Whyte

Fusion & Fission exhibit important differences in energetic and neutronics Fusion

Fission

Fuel

D abundant, T bred

235U, breeding

Ash

He

Sn, Rb, Cs, etc.

Energy source

Neutron and radiation Volumetric capture through heating surface Temperature > 100 Million K ~ Thermal

NEEP 536, Spring 2003, D. Whyte

D-T is obviously the easiest reaction to achieve…what about D-T as fuel?

– D + n --> T + γ (e.g. CANDU reactor)

• …and for fusion we have 6Li + n --> T + 4He + 4.8 MeV 7Li + n --> T + 4He + n - 2.3 MeV

NEEP 536, Spring 2003, D. Whyte

Fusion reaction cross sections 10 1

barns (1e-28 m2)

• Deuterium is so abundant on earth…fD/H ~ 1.5x10-4 …as to make an infinite supply for fusion.

• Tritium does not occur naturally due to its short half-life, T --> 3He + e- 1/2 life = 12.3 years.

• So T must be bred

D+D = T + p D+D = 3He + n D +T D + 3He T+T

0.1 0.01 0.001 0.0001 1

10

100

Particle energy (keV)

1000

D-T is obviously the easiest reaction to achieve…what about D-T as fuel?

– D + n --> T + γ (e.g. CANDU reactor)

• …and for fusion we have 6Li + n --> T + 4He + 4.8 MeV 7Li + n --> T + 4He + n - 2.3 MeV (with n~14 MeV, fbreed>1!!)

NEEP 536, Spring 2003, D. Whyte

Fusion reaction cross sections 10 1

barns (1e-28 m2)

• Deuterium is so abundant on earth…fD/H ~ 1.5x10-4 …as to make an infinite supply for fusion.

• Tritium does not occur naturally due to its short half-life, T --> 3He + e- 1/2 life = 12.3 years.

• So T must be bred

D+D = T + p D+D = 3He + n D +T D + 3He T+T

0.1 0.01

0.001 0.0001 1

10

100

1000

Particle energy (keV)

With Li blanket/breeder We can produce closed T fuel cycle and 6Li + D = 24He + 22.4 MeV Is actual fusion fuel

Fusion & Fission exhibit important differences in energetic and neutronics Fusion

Fission

Fuel

D, (T) (1 neutron)

235U (~2.5 neutrons)

Yield

~20 MeV

~200 MeV

Energy / kg Units / kg # Neutrons/ MeV Eneutrons /unit energy

3.4x1014 J / ~700

8.8x1013 / ~180

0.05

~0.01

~1

~0.02

NEEP 536, Spring 2003, D. Whyte

Fusion & Fission exhibit important differences in energetic and neutronics Fusion

Fission

Fuel

D abundant, T bred

235U, breeding

Ash

He

Sn, Rb, Cs, etc.

Energy source

Neutron and radiation Volumetric capture through heating surface Temperature > 100 Million K ~ Thermal

NEEP 536, Spring 2003, D. Whyte

Reaction rates in thermalized plasma: Kinetic theory & intro to plasmas Fundamental: f(v) --> probability distribution function (essentially density of particles in velocity phase space)

Knowing f(v) tells us what we need to know about the plasma (collection of energetic particles/ions for the moment)

Density (# / m3): n = ∫ f dvx dvy dvz = ∫ f d3v Mean velocity: <v> = ∫ f v d3v / ∫ f d3v Velocity vector: v Particle flux: Γ = ∫ f v d3v NEEP 536, Spring 2003, D. Whyte

Reaction rates are calculated by integrating cross section through velocity space of thermalized population Volumetric reaction rate for N atoms: N Γ σ = R So R = N ∫ f(v) v σ(v) d3v for a stationary target

How about two “intercepting” populations (denoted 1 and 2)?

Relative velocity of particles u = | v1 - v2| So R = ∫ f1(v) f2(v) u σ(u) d3v1 d3v2 And we define: n1 n2 <σv> = R….volumetric reaction rate (I.e. fusion reactions / s / m3) NEEP 536, Spring 2003, D. Whyte

v N

Reaction rates are calculated by integrating cross section through velocity space of thermalized population <σv> = ∫ f1(v) f2(v) u σ(u) d3v1 d3v2 / (n1 n2 ) in m3/s So now we take the standard assumption that the population follows the Maxwell-Boltzmann distribution Namely, f1 (v) = n1 (m/2πkT)3/2 exp(-mv2/2kT) vth = (2kT/m)1/2 f1 (v) = n1 π−3/2 (vth)-3 exp(-(v/vth)2) Maxwell-Boltzman dist'n

vx 2 ∫−∞ exp(− vth2 )dvx = π ⋅ vth

f(v)

∞

∞

∫ f1(v)d3v = n1

−∞ NEEP 536, Spring 2003, D. Whyte

-4

-3

-2

-1

0

v / vth

1

2

3

4

Reaction rates are calculated by integrating cross section through velocity space of thermalized population So let’s examine probability for tunneling and speed v: the product (taking hydrogenic ions): mv 2 2 e P ≡ exp − − 2ε o hv 2kT Since the two terms have opposite v dependence, it is informative to look at where we might expect highest rate dP

e2 − mv 2 ⋅ e2 − mv 2 hv 2kT 2 ε hv kT = 0 = exp − o dv 2ε o e2 2ε o

NEEP 536, Spring 2003, D. Whyte

hv 2

e2kT

mv =

kT

⇒v=

2ε o hm

1/ 3

Reaction rates are calculated by integrating cross section through velocity space of thermalized population mv e = ⇒v= 2 2εohv kT

2

e kT 2εohm

1/ 3 Maxwell-Boltzman dist'n

f(v)

2

At T=10 keV --> v ~ 1.7x106 m/s Recall: vth ~ 1x106 m/s

-4

-3

-2

-1

0

1

2

3

4

v / vth Fusion reaction cross sections 10

and vth ~ 2.2x106 m/s

1

barns (1e-28 m2)

At T=50 keV --> v ~ 3x106m/s

D+D = T + p D+D = 3He + n D +T D + 3He T+T

0.1 0.01 0.001

0.0001 1

10 100 Particle energy (keV)

1000

So it’s the 1-10% percent of ions in the E/Eth > 2-3 distribution that are giving us the “substantial” fusion rates in the10’s keV temperature range NEEP 536, Spring 2003, D. Whyte

Fusion rate coefficient (cm3/s)

Maxwellian-averaged fusion rate coefficients: Pfusion = nDnT<σv> QD-T (W/m3) 1.E-15

1.E-16

1.E-17

D-D D-T

1.E-18

D-3He

1.E-19 1

10

100 T_ion (keV)

NEEP 536, Spring 2003, D. Whyte

1000

So we have introduced (exploited) the properties of a Maxwellian fion(v) with T~10 keV: What does this mean?

Gas: (hydrogenic) is completely ionized (stripped of electrons) with T ~ 10 keV ….ionization potential of H =13.6 eV Needed density of ions ~ 1020 m-3 (derived later on) With these conditions, we are dealing with a plasma Plasmas exhibit unique behavior that greatly affect fusion -ne ~ ni , despite being composed of charged particles, plasmas are actually quasi-neutral, but good conductors - Coulomb collisions dominate physics - radiation power losses - “action at a distance” --> differs from gas kinetics --> “collective behaviour”

NEEP 536, Spring 2003, D. Whyte

For fusion plasmas bremsstrahlung radiation is an unavoidable source of power loss Bremsstrahlung: (German for “braking radiation”) is a result of acceleration of plasma electrons during collisions with ions, the acceleration causes the electron to radiate.

Bremsstrahlung energy radiated

q = 2πbvf ⋅ db ⋅ d 3v ⋅ ∆E

e-

# electrons colliding with ion per unit time

b +

In spherical coordinates: f d3v = 4 π v2 f dv

q = 8π 2bv 3 f ⋅ db ⋅ dv ⋅ ∆E a

≈

Ze22 4πε ⋅ b m

S≈

e

Acceleration of eNEEP 536, Spring 2003, D. Whyte

Pbrenss for given acceleration

1 e2 6πε c

3

a

2

b ∆t ≈ v Time spent undergoing acceleration

For fusion plasmas bremsstrahlung radiation is an unavoidable source of power loss Substituting we get: ∞

1

2

Z 2e6 ∆E ≡ ∫ Sdt ≈ (4πε)3 3 m2e c 3b3v −∞ We also can estimate on the frequency of the radiation: ν = v / b So db ~ b2 d ν / v And q = 8π 2bv 3 f ⋅ db ⋅ dv ⋅ can be written as

q

∆E

≈

Z 2e6vf

12πε m c 3

2 3 e

dv ⋅ dν

So the total power dissipate in bremsstrahlung is obtained by integration: ∞

∞

0

vmin

P = ∫ dν ∫ dv NEEP 536, Spring 2003, D. Whyte

niZ 2e6vf 12πε3m2ec 3

Where integration limit is set by quantum relationship: 1/2 m v2 min = h ν

For fusion plasmas bremsstrahlung radiation is an unavoidable source of power loss ∞

∞

P = ∫ dν ∫ dv vmin

0

n i Z 2e6vf

ni is the ion density f is Maxwell-Boltzmann --> exp(mv2/2kT)

12πε3m2ec 3

After integrating through velocity and frequency space we get

P=

1 24 πε

3

kTe

2πme

1/ 2

Z 2 e6 3

mec h

neni

Pbremsstralung= 5.35x10-37 Z2 ne ni Te1/2 W / m3 (Te in keV)

NEEP 536, Spring 2003, D. Whyte

Fusion Ignition: when self-heating (from hot alphas in D-T case) wins over radiative losses Pfusion= nD nT <σv> Qα,,D-T > 5.35x10-37 Z2 ne ni Te1/2 = Pbrems.

But in a plasma 2nD=2nT = ni = ne (quasi-neutrality constraint) Take a pure D-T plasma (Z=1) with Te=Ti

(density independent!)

So if considering an Infinitely large plasma we Will reach ignition at T~ 4 keV (note plasma is transparent To Pbremsstahlug) NEEP 536, Spring 2003, D. Whyte

Power efficiency (W m3)

So <σv>ignition ~ 5.35x10-37 ne2 Te1/2 / (Q* 0.25 * ne2 )= 4 x 5.35x10-37 Te1/2 / Q 2E-35 1.8E-35 1.6E-35 1.4E-35 1.2E-35 1E-35

D-T fusion

8E-36

bremsstralung

6E-36 4E-36 2E-36 0 0

5

10 T (keV)

15

20

Lawson criterion: the realistic limit for reaching ignition in a fusion system Of course it is physically nonsense to take an infinitely large plasma system that undergoes no energy losses….

plasma energy

So let’s parameterize the energy loss of our plasma by the concept of an energy confinement time, τe, which is essentially the e-folding time Of the energy containment ‘efficiency” of the system

-5

0

τe time

Response of system To energy input at time=0 5

10

Stored energy: 3 n kT (Ti=Te=T) partition theorem

For the moment we ignore the physical explanation for energy Confinement time (convection,conduction, etc.)

NEEP 536, Spring 2003, D. Whyte

Lawson criterion: the realistic limit for reaching ignition in a fusion system 3nkT – 3nkT 1 2 d3nkT −37 n 2 T 1/ 2 − = n σ v E – 5.35 ×10 = 0 = Pα – PB α τe 4 dt τe So we can rewrite the criterion for achieving heat gains and losses that keep T constant

nτe =

12kT

σv Eα − 2.14 ×10

−36

1/ 2

T

1E+22

τe (s / m3)

Lawson criterion

Ignition

1E+21

n

The Lawson criterion tells us that we must provide a plasma/fusion system that Obtains high T and adequate product of density and confinement.

1E+20 1

10 T (keV)

NEEP 536, Spring 2003, D. Whyte

100

Quasi-neutrality: Debye shielding I previously claimed that we can macroscopically treat the plasma as a neutral.

How is this justified?

Let’s consider the potential in a plasma that arises from non-uniform density ne ≠ ni e(ne − n) 2ϕ(r) = − ∇ From Poisson’s equation

ε0

And the potential determines the Local ne thru Maxwell-Bolzmann dist’n 2 So combining these results we get:

∇ ϕ(r) =

ne (r) = n exp

ne2

ε0

kTe

1

ϕ(r) 2 ϕ (r) ≡ kT λ e

With the solution in spherical coordinates: ϕ (r) =

NEEP 536, Spring 2003, D. Whyte

eϕ

e

e 4πεor

exp

−r λe

≅ n 1+

eϕ kTe

Quasi-neutrality, Debye shielding define the criterion for plasmas So with r --> 0 we just recover the usual potential for a point charge, …the exponential term informs us that past λe, the potential is shielded out.

ϕ(r) =

e −r exp 4πεor λe

λ is called the Debye length and from our solution : e

1/ 2

In meters

ne2

The number of electrons within the Debye length

4

4

ε kT

Ndebye = n πλe3 = n π o e ne2 3 3

3/ 2

8.E-05 Electrostatic potential (V)

λ =

εokTe

exp(-r/lambdae)/r

6.E-05

1/r

4.E-05 2.E-05 0.E+00 0

0.0001

0.0002 radius (m)

NEEP 536, Spring 2003, D. Whyte

0.0003

0.0004

Quasi-neutrality, Debye shielding define the criterion for plasmas So with r --> 0 we just recover the usual potential for a point charge, …the exponential term informs us that past λe, the potential is shielded out. λ is called the Debye length and from our solution :

λe = εo kT e 2 ne

1/ 2

ϕ(r) =

e −r exp 4πεor λe

Example case: T=10 keV, n=1x1020m-3 λe ~ 70 x 10-6 m << Lplasma > 1 m

The number of electrons within the Debye length

4 Ndebye = n

4

ε kT

πλe3 = n π o e ne2 3 3

NEEP 536, Spring 2003, D. Whyte

3/ 2

Ndebye ~ 170 million electrons >> 1 Conditions for plasma easily met.

Another useful example on the behavior of plasmas: Restoring force from suddenly displaced electrons in the plasma Initial - distribution

dE dx

=

perturbed distribution

ρ εo x1

x

xo

From Poisson’s equation this charge imbalance will produce an electric field that will act to restore charge balance ..

2 n noe oe ma e = me x = eE(x 1 ) = E(x 1 ) = −∇ϕ = εo (x1 − x o ) εo (x1 − x o ) So substituting ξ for (x1-xo) we find the solution for the motion which is oscillatory 2 1/ 2

ξ ≡ Cexp(−iω pe t) NEEP 536, Spring 2003, D. Whyte

ω pe =

ne

m eεo

Plasma frequency: electrons & ions ω pe =

ne2

Example fusion case: n=1x1020m-3

1/ 2

fpe = ωpe/2π ~ 90 GHz

m e εo

ω pi = ω pe m mi

e

1/ 2

~ ω pe /50

fpi ~ 2 GHz

So one can see that the plasma (ions & electrons) “rings” at very high frequencies to charge perturbation.

NEEP 536, Spring 2003, D. Whyte

Let’s examine another consequence of charge neutrality in plasma fusion systems: Helium Ash impurity Let’s consider a D-T fusion plasma: as power is being produced by “burning” the D-T fuel we are creating “ash” ion, namely the resultant, non-fusing 4He ion From D+ + T+ --> 4He ++ + n (the neutron escapes the plasma constraints since it is has no charge) What will this ash do and how much might we expect to tolerate?

First: the He ash dilutes the fuel (D,T) density since the charge density Of electrons and ions must be the same, i.e. ne= ∑i Zi ni = nD + nT + 2nHe (Zi is the charge of each ion) We typically then reference densities to the ne Impurity fraction of He: fHe = nHe++/ne And the hydrogenic fraction/density is diluted by nD+nT = nH = ne (1- 2 fHe) NEEP 536, Spring 2003, D. Whyte

Let’s examine another consequence of charge neutrality in plasma fusion systems: Helium Ash impurity Remember our derivation of the bremsstrahlung radiation power:

P = ∫ dν

∞

0

vmin

∞

We considered the collisions of the electron population with a given ion species of charge Z. So we can see that for multiple ion species we just sum over the Various ion species to obtain total bremsstrahlung power.

Pbremsstralung= 5.35x10-37 ne Te1/2 ∑ i n i Z i2

n Z 2e6vf

∫ dv 12

W / m3 (Te in keV)

We now define a “effective” ion charge (typically related to Coulomb collisions)

∑n Z 2 ∑n Z 2

Z eff

=

i

i

i

∑niZ

=

i

i

ne

i

Zeff = 1+ ∑ f i Z i (Z i −1) i

i

So Pbremsstralung= 5.35x10-37 ne2 Te1/2 Zeff NEEP 536, Spring 2003, D. Whyte

W / m3 (Te in keV)

i

i

3

2 3

πε m e c

The He ash is doubly detrimental to fusion: its increases fuel dilution and increases radiative power loss Remember our power balance for the Lawson criterion

3nkT – 3nkT 1 2 d3nkT −37 n 2 T 1/ 2 − = n σ v E – 5.35 ×10 = 0 = Pα – PB α τe 4 dt τe The alpha power is now reduced:

1 1 P = n 2 σv E = ne 2(1− 2 f He )2 σv Eα α α 4 H 4

And bremsstahlung losses increase: PB = 5.35 ×10−37 ne2Z eff = 5.35 ×10−37 n 2e(1+ 2 ⋅ f He )

12kT neτ e = 2 36 1/ 2 (1− 2 f He ) σv Eα – (1+ 2 f He )2.14 ×10− T

Lawson criterion: Effect of He

fHe=1 : nτ --> ∞ No fusion!

NEEP 536, Spring 2003, D. Whyte

The He ash is doubly detrimental to fusion: yet another criterion is set for fusion…keep the fuel clean! neτ e =

12kT 2

(1− 2 f He ) σv Eα – (1+ 2 f He )2.14 ×10−36T1/ 2

ne τe (s /m3)

1E+22 fHe = 0 fHe = 10% fHe = 20% fHe=30% 1E+21

1E+20 1

10 Te (keV)

NEEP 536, Spring 2003, D. Whyte

100

Having set the criteria for fusion ignition, let’s examine the confinement and size scale needed First, fusion T~104 keV~30,000 Tmelt for tungsten So fusion and direct solid material containment are incompatible.

Secondly, let’s imagine a uniform plasma volume at fusion conditions:

τe ~ W / (dW/dt) W is the stored energy of the plasma volume, V (e- and ions): 2 * (3/2) n k T V (Ti=Te)

plasma L NEEP 536, Spring 2003, D. Whyte

And the rate of loss of W is given by power loss thru The “sides” of our container with area A P ~ Γ 3 k T A where Γ is the particle flux From kinetic theory:

Γ=

1

4

nv ~

1

4

nvth,ion

Having set the criteria for fusion ignition, let’s examine the confinement and size scale needed

plasma L

4 3 3nkT πL W L 3 nv ~ v τe = ≡ 1 nv 2 th,ion th,ion 3kT 4πL dW / dt 4 So back to our usual example T=10 keV --> Lawson criterion: neτe~4x1020 s m-3

Taking ne~1020m-3 --> Why? Pfusion ~ 0.25 n2 <σv> QD-T ~ 1 MW/m3 “reasonable” τe ~ 4 seconds L ~ τe vth,ion ~ (4 s) 1.4x104 (T/M)1/2 ~ 4 x 106 m !!!

Obviously the proper confinement scheme will be necessary for fusion.

NEEP 536, Spring 2003, D. Whyte

How do we obtain confinement?

• Magnetic confinement: – Exploit confining properties of strong magnetic fields on charge particles / plasmas.

• Inertial confinement: –

Obtain ignition conditions so quickly that the inertia of the D-T fuel does not allow them to escape

• Electrostatic confinement –

Exploit electric potential effects on plasmas.

NEEP 536, Spring 2003, D. Whyte

The He ash is doubly detrimental to fusion: its increases fuel dilution and increases radiative power loss

Remember our power balance for the Lawson criterion

3nkT – 3nkT 1 2 d3nkT −37 n 2 T 1/ 2 − = n σ v E – 5.35 ×10 = 0 = Pα – PB α τe 4 dt τe The alpha power is now reduced:

1 1 P = n 2 σv E = ne 2(1− 2 f He )2 σv Eα α α 4 H 4

And bremsstahlung losses increase: PB = 5.35 ×10−37 ne2Z eff = 5.35 ×10−37 n 2e(1+ 2 ⋅ f He )

12kT neτ e = 2 36 1/ 2 (1− 2 f He ) σv Eα – (1+ 2 f He )2.14 ×10− T

Lawson criterion: Effect of He

As fHe--> 0.5, no ignition in pure He plasma Since only energy losses!

NEEP 536, Spring 2003, D. Whyte

1

Another useful example on the behavior of plasmas: Restoring force from suddenly displaced electrons in the plasma & stationary ions dE dx

=

Initial - distribution

perturbed distribution

ρ εo

x1

x

xo

From Poisson’s equation this charge imbalance will produce an electric field that will act to restore charge balance .. noe noe2 ma e = me x = eE(x 1 ) = E(x 1 ) = −∇ϕ = εo (x1 − x o ) εo (x1 − x o ) So substituting ξ for (x1-xo) we find the solution for the motion which is oscillatory 2 1/ 2

ξ ≡ Cexp(−iω pe t) NEEP 536, Spring 2003, D. Whyte

ω pe =

ne

m eεo 2

Plasma frequency: electrons & ions

ω pe =

ne2

Example fusion case: n=1x1020m-3

1/ 2

m eεo

ω pi = ω pe m mi

e

fpe = ωpe/2π ~ 90 GHz

1/ 2

~ ω pe /50

fpi ~ 2 GHz

So one can see that the plasma (ions & electrons) “rings” at very high frequencies to charge perturbation. On fusion confinement time scales (~ 0.1 - 1 s) the plasma is quasi-neutral.

NEEP 536, Spring 2003, D. Whyte

3

Derive simple dispersion relation for electrostatic oscillations in plasma Force

Current density J ⋅

..

me x = eE

J = neev = n e ex

Solve for plane wave solution:

−k Eoe

i( kx−ωt )

2

2

xˆ + εµω Eoe

−k + εµω = εµ

NEEP 536, Spring 2003, D. Whyte

2

i( kx−ωt )

dt

..

= n e ex =

2

ne E e

me

E = E o exp(i(kx − wt)) ⋅ x̂

Plug this into wave equation from Maxwell’s equation: 2

dJ

∇ 2 E − εµ

nee 2 E oe i( kx−ωt ) xˆ xˆ = µ me

∂ 2E ∂t 2

=µ

nee2

Electron plasma frequency: ω 2

εm

pe

e

dJ dt

4

Derive simple dispersion relation for electrostatic oscillations in plasma −k 2 + εµω 2 = εµωpe2

So reaction to EM waves: ω < ωpe then k is imaginary and The wave damps in the plasma ekx

2 k2 ω pe ) = εµ(1− 2 ω ω2

k2

=

1

εµ

ω pe 2

1−

ω ~ ωpe k-->0, cutoff / reflection

ω2

E = E o exp(i(kx − wt)) ⋅ xˆ

NEEP 536, Spring 2003, D. Whyte

−1

angular frequency

v 2phase =

ω2

ω > ωpe then the wave propagates

ωpe wavenumber

5

Implications of plasma-wave interaction example •

Plasma exhibits complex EM properties (we only scratched the surface with our simple example here)

•

Affects radio frequencies (RF) used for plasma heating – Frequencies above plasma frequency “penetrate” to reach other resonances (in particular electron cyclotron resonance)…but this penetration is density dependent.

– Frequencies much less than plasma frequency --> “plasma wave”

•

Bremsstrahlung radiation losses: – Plasma cutoff wavelength: λ ~ c/ν ~ c / 90GHz ~ 3mm – Radiation frequency / wavelength: hc/λ ~ kT --> λ ~ 0.12 nm is therefore more than 6 orders of magnitude above plasma frequency --> plasma is completely transparent in this sense.

E = E o exp(i(kx − wt)) ⋅ xˆ NEEP 536, Spring 2003, D. Whyte

6

Implications of plasma-wave interaction example •

Plasma exhibits complex EM properties (we only scratched the surface with our simple example here)

•

Affects radio frequencies (RF) used for plasma heating – Frequencies above plasma frequency “penetrate” to reach other resonances (in particular electron cyclotron resonance?)…but this penetration is density dependent.

– Frequencies much less than plasma frequency --> “plasma wave”

•

Bremsstrahlung radiation losses: – Plasma cutoff wavelength: λ ~ c/ν ~ c / 90GHz ~ 3mm – Radiation frequency / wavelength: hc/λ ~ kT --> λ ~ 0.12 nm is therefore more than 6 orders of magnitude above plasma frequency --> plasma is completely transparent in this sense.

NEEP 536, Spring 2003, D. Whyte

7

Fundamentals of the magnetic confinement method •

As previously states we need T~100 million K and this is incompatible with direct material containment – Stars get confinement through size and gravity

•

…so we need a confinement scheme.

•

Since the plasma consists of charged particles (ions and electrons) we exploit the fact that a charged particle will not cross a magnetic field but instead will rotate around the field lines….this is the essence of using magnetic fields as a non-material “bottle” to contain the plasma

NEEP 536, Spring 2003, D. Whyte

8

Cyclotron motion •

Investigate the motion of charge particles in electro-magnetic (EM) fields.

– Charged particles in motion induce EM fields, which in turn affect the motion of the charged particles --> complexity.

•

Start with a single charged particle of mass m and charge q in a magnetic field (B in Tesla) is given by the Lorentz force

m

dv dt

= q( r r r E + v × B)

v is the velocity of the particle

So in the absence of electric field, E, the particle undergoes acceleration in a direction perpendicular to its motion and the magnetic field.

NEEP 536, Spring 2003, D. Whyte

9

Cyclotron motion •

dv/dt ~ (qB/m) v --> which leads to a cyclic motion around the field line at frequency: ωc = qB/m called the cyclotron frequency.

– Note that the field does not affect the particle by doing any work on it…it’s kinetic energy 1/2 m v2 remains the same.

•

And the Lorentz force is providing centripetal acceleration, thus

v ⊥2 m = qv ⊥ B R

NEEP 536, Spring 2003, D. Whyte

mv⊥ R= qB

Is known as the Larmor radius

10

Cyclotron motion B

B

•

Ions and electrons rotate in different directions.

•

Electrons will have Rlarmor,e << Rlarmor,ions because of their mass difference.

•

Motion // to B unaffected --> no intrinsic confinement // to B along “guiding center”

-

+

Typical fusion plasma T~104 K: vth,i ~ 106 m/s ~ 1/50 vth,e

m 2kT 1/ 2

mv R=

⊥

qB

=

NEEP 536, Spring 2003, D. Whyte

m qB

e- cyclotron frequency: ~ 28 B in GHz (B in T)

ω = qB/m

Ion cyclotron frequency: ~ 14 B / M in MHz

c

Ion gyro-radius: 10-2 M1/2 / B

in m

11

Confinement example shows that a simple linear magnetic field is not practical for fusion •

B

B

Long cylinder with Bz along the axis: with losses out ends only.

•

Take B ~ 2 T --> MRI superconductor >> Bearth (0.5 Gauss) {104 Gauss = 1 Tesla}

•

So with Rlarmor ~ 1 cm let’s make the cylinder R ~ 1 m >> Rlarmor

-

+

•

Recalling our derivation from the Lawson criterion on the need size scale completely unconfined plasma: but Vcylinder ~ π R2 L

R=

mv⊥ qB

=

2kT m m qB

4 ×1020 sm−3

1/ 2

τe ~

ωc = qB/m •

ne

3nkTπR2L

W =

dW dt

= 1

v th n3kT2 πR2

4 So n~1020 m-3 and vth,ion ~ 106 m/ give us

2L

=v

th

L ~ 2x106 m ~ 2000 km!

NEEP 536, Spring 2003, D. Whyte

12

Quick summary of Maxwell’s EM equations • Gauss’ law, Poisson equation for electric fields and potential caused by charge density • No magnetic monopole • Faraday’s induction law • Ampere’s law of magnetomotive force

∇ ⋅ E = −∇2ϕ = ρ /ε ∇⋅B =0 ∇×E

=−

∇ × B = µJ + ε

∂B ∂t

∂E ∂t

NEEP 536, Spring 2003, D. Whyte

13

Examples of applying Maxwell’s equations •

Already derived plasma potential from Poisson’s equation

•

Let’s look at producing magnetic fields through Ampere’s law (dE/dt = 0 --> Biot-Savart law) µ

B=

∇ × B = µJ

µJz Bθ = 2π r

J J

r Poloidal magnetic Field from Axial current

NEEP 536, Spring 2003, D. Whyte

Idl × ∆r 4π ∆r 3

B r

Bz|z= 0,r= 0 =

µJθ 2r Axial magnetic Field from Poloidal current 14

Magnetic moment & flux surfaces •

Notice that the charged particle circulating around the magnetic field produces a current ~ qvperp/(2πR) and thus produces a magnetic moment, µ ~ I π R2 which is a constant.

–

Note from picture that m is opposed to the magnetic field --> diamagnetic

KE = •

Conservation of kinetic energy since the Lorentz force does no work on the particle.

1

2

m[v 2 + v 2 ] = constan t //

B

⊥ 2

•

mRLarmorv⊥ ∝ v⊥ B

We also have conservation of angular momentum

1 µm ≡ 2

mv 2 ⊥

= constant

•

The magnetic moment is defined:

•

Furthermore we can see that the charged particle is restrained to a magnetic flux “tube” 2v 2 m 2πm µ ⊥ 2 Φ = BzπR = πB = m L

NEEP 536, Spring 2003, D. Whyte

2

qB

2

2

q

B

+

= constant 15

Magnetic mirror: a confinement method •

So you might have noticed that we had two constraints on vperp, therefore the free parameter when combining them is v//, namely

1

mv 2 = KE − µ B m

//

2 •

So as B increases, at B=KE/µm v// --> 0 and the particle is mirrored or trapped.

•

So let’s consider a cylindrical magnetic topology with smoothly varying B(z) increasing-->

•

As the particle approaches the location of higher B, it’s gyro-orbit becomes smaller and its orbital velocity increases, but it is actually being accelerated toward regions of lower B

•

B Z

So along Z we can see that` conservation of energy`````````` 1 2 1 2 1 1 2 1 1 mv (z) = mv (z ) + mv 2 (z ) − µ (z )B(z) = mv (z ) + mv 2 (z ) − mv 2 (z

2

//

NEEP 536, Spring 2003, D. Whyte

2

//

o

2

⊥

o

m

o

2

//

o

2

⊥

o

2

⊥

) o

B(z) Bmin 16

Magnetic mirror: a confinement method •

So by solving for vperp we see that there is a constraint on the minimum ratio of perp. to parallel velocity known as the mirror ratio necesssary for trapping.

v ⊥ (zo ) = ± Bmax −1 Bmin •

−1/ 2

v // (z o )

This essentially depends on how efficiently you can converge the magnetic field.

Bmin Bmax

•

Now we a have a ”simple” scheme to confine the plasma charged particles by bouncing them back and forth between two magnetic mirrors (rather than just letting them leak out the ends as before).

NEEP 536, Spring 2003, D. Whyte

17

Magnetic mirror: a confinement method •

So what stops us from building a fusion reactor based on this simple mirror?

– – –

•

Throw energy at plasma to heat it up to T ~ 104 K Lose small fraction of hot plasma ~ (Bmin/Bmax) that don’t meet mirror criterion (high v//) Remaining particles confined infinitely long!

The reasons this doesn’t work

as stated are illustrative of many magnetic fusion issues

–

The collisions between the plasma particles: will tend to re-thermalize the velocity distribution, continually sending plasma into the loss region.

–

The collisions between the plasma particles: will make them “jump” across field lines, degrading our perfect magnetic confinement for a single particle.

–

MHD (Magneto Hydro Dynamics): the shape of the magnetic field and considerations about the plasma and magnetic pressure lead to plasma instability.

NEEP 536, Spring 2003, D. Whyte

18

Issues we left to consider for magnetic fusion from the last lecture •

•

Remember that we had used a simple description of single particle motion around and along a field line to develop a reactor based on a magnetic mirror confinement scheme.

But this picture is too simple…in particular we need to deal with – Since we obviously want our reactor to be densely filled with fusion ions we need to consider the effects of multiple particle scattering --> back to Coulomb collisions! (short-range interactions within the Debye length) – In a related manner the scattering collision events which cause the particles to cross field lines and thus degrade confinement.

– The collective movement of the plasma ions and electrons under the influence of various topologies of B and E (drifts) – The magneto hydrodynamic (MHD) stability: plasma pressure vs. magnetic pressure.

NEEP 536, Spring 2003, D. Whyte

1

We re-examine Coulomb scattering – Consider two charged particles acting r on each other via Coulomb potential

mm

q1,m1

q1q2r

˙r˙= mr˙r˙ = 3 m1 + m2 4πεor 1

2

x r

θ

r

q2,m2

– Where mr is the reduced mass of the colliding particles – The solution for the equation of motion in the center-of-mass frame yields the deflection angle

tan

θ 2

=

q1q2 4πεo x ⋅ mrv

2

– So the deflection becomes larger as • The impact parameter becomes smaller • The relative velocity of the particles , v, decreases.

NEEP 536, Spring 2003, D. Whyte

2

We re-examine Coulomb scattering – Consider two charged particles acting on each other via Coulomb potential r

mm

q1,m1

q1q2r

˙r˙= mr˙r˙ = 3 m1 + m2 4πεor 1

2

x r

θ

r

q2,m2 – Where mr is the reduced mass of the colliding particles – The solution for the equation of motion in the center-of-mass frame yields the deflection angle

tan

θ 2

=

q1q2 4πεo x ⋅ mrv

2

– So the deflection becomes larger as • The impact parameter becomes smaller • The relative velocity of the particles , v, decreases.

NEEP 536, Spring 2003, D. Whyte

Special case (e- on ion) m1 << m2 --> mr ~ m1 And θ∼0 small 2 q1q2 U = Coulomb θ= Ekinetic 4πεo x ⋅ mrv 2

3

Consider the cause of large-angle scattering of particle •

Probability of direct, single interaction causing θ > π/2 ~ 90 degrees is determined by the impact parameter. (q q ) 2 qq

x π /2 =

1 2

4πεo ⋅ mrv 2

σ (θ > π /2) = πx 2π /2 =

1 2

π (4εo mrv 2 ) 2 • Now let’s consider the accumulative effect of many small angle scattering, events of a particle going length L through a plasma with density n. xmax qq 2 2 1 2 ∆θ = (∆θ ) = nL ∫ (∆θ ) 2πxdx 2πεo x ⋅ mrv 2 xmin •

For the limits of the integration we take xmin ~ xπ/2 (where large angle calculation started) and for xmax we take the Debye length

NEEP 536, Spring 2003, D. Whyte

4

Consider the cause of large-angle scattering of particle •

• •

So integrating through x we get 2 nL q q 2 ln Λ 1 2 (∆θ ) = 2 2π εomrv

kT) 3/ 2 nq4q2

o

1/ 2

2 1

Where ln Λ is known as the Coulomb logarithm which is ~ constant for fusion plasmas with values ~15.

So we can determine the effective cross section to get enough small deflections add up to ~ 90 degrees deflections to ∆θ=π/2~1

σ≡ •

Λ ≈ 8π

(ε

1 nLπ /2

=

1

2

qq 1 2

2 π ε o mrv

2

ln Λ

σ (θ > π /2) = πx 2

π /2

(q q )

2

= π (4ε1 m2 v 2 ) 2 o

r

So comparing the two we see that the effective cross section is 8 x ln Λ ~ 100 times larger for accumulative small-angle scatters to dominate the deflection of the plasma particles.

NEEP 536, Spring 2003, D. Whyte

5

Our previous exercise also informs us about the relative equilibration times for species in the plasma •

It’s reasonable to use the 90 degree deflection as the benchmark for characteristic time when particles lose their momentum identity – Recall our concern about deflection of particle into the mirror loss cone.

Lπ /2 = τm ≡ v •

2π

(q q 2 ⋅ 1 2 ) ln Λ

1 mr ⋅

n

⋅ (3kT)

3/ 2

So as the plasma becomes – Less collisional as the temperature increases.

– More collisional as the density increases

•

Electrons will transfer their momentum equally fast to other electrons and to ions since mr~me (~10-4 s)

•

Ions will lose their momentum (mi/me)1/2 times slower than electrons (~10-2 s)

•

In a similar vein, the energy equilibration time is fastest between electrons, then ions-ions and finally ions - electrons.

NEEP 536, Spring 2003, D. Whyte

6

Plasma resistivity •

•

Reconsider electron motion with applied electric field, E, where the 90 degree scattering event(s) are destroying the momentum order. So we r can approximate the equation of motion r dv = −eE − mev me ei τ m dt andrthe current density, j=e r n v, so r

r

r

me dj = E − ηj 2 e n dt

NEEP 536, Spring 2003, D. Whyte

E = ηj In steady state

eie me ln Λ η≡ 6 3 πε 2 (kT) 3/ 2

7

Plasma resistivity •

Re-introducing the Zeff as before for multiple ion species + Coulomb collisions we find the resistivty (in ohm-m) to be (T in eV)

η •

≈ 5 ×10−5

Zeff ln Λ

So for T~1 keV, η~2x10-8 ohm-m ~ ηcopper

T 3/ 2

Plasma is an extremely good conductor

Plasma resistivity does not depend on the plasma density (except extremely weakly through the Coulomb logarithm) – Charge carrying electrons and scattering-causing collisions cancel out.

•

The plasma resistivity is highly dependent on the plasma temperature and its purity.

NEEP 536, Spring 2003, D. Whyte

8

Application of E perpendicular to B: ExB drift • •

Let’s rewrite the Lorentz force equation

m

dv = q( r

+ r × r) E v B

dt

And suppose we have an applied E which is solely perpendicular to B What do we expect for the motion of the ions electrons? r E qEy qv × Bz + v˙= m m

Let’s re-arrange this in terms of the cycoltron E Frequency qB/m v˙ = ω y + ω c v c Bz

B

This form of the differential equation suggests the solution for v r r r r r r r dvo r ExB ExBxB m v = vExB + vo = B2 + vo = −E 2 = q(vo × B) B Back into Lorentz formula NEEP 536, Spring 2003, D. Whyte

dt

We recover our original cyclic motion Around B with no E

9

Application of E perpendicular to B causes a plasma drift in the ExB direction

v=

r r ExB B2

E

ions

+ vcyclic⊥B electrons

B •

Note that the ExB drift velocity has some very interesting properties – The velocity is independent of charge --> the electron and ions travel in the same direction (although undergoing different rotation around B) – The velocity is also independent of mass!

– So the entire plasma (ions and electrons) move uniformly in the ExB direction at velocity E/B. • No charge separation or currents arise in the plasma.

NEEP 536, Spring 2003, D. Whyte

10

Let’s now consider a non-uniform magnetic field: ∇B drift •

The magnetic field gradient produces a force on the charge particles toward regions of lower B (recall our solution for the magnetic mirror)

r 1 mv 2 ⊥ F =− ∇B 2 B •

dt

r r r = q(v × B) + F∇B

Again we separate the cyclic motion: v = vcyclic + vgrad-B

r r r 0 = q(v∇B × B) + F∇B

•

m

dv

The electrons and ions drift in opposite directions, producing charge separation and net current

NEEP 536, Spring 2003, D. Whyte

∇B

r r mv r B × ∇B v∇B = 2q 2 ⊥

ions

B

electrons 11

Let’s now consider a magnetic field that is curved: curvature drift •

The particle following the curved field line will undergo a centrifugal force in the direction normal to curvature

r Fcentrifugal = −

rˆ // n

mv 2 R

r r r dv m = q(v × B) + Fcentrifugal dt

B R

electrons

•

ions

Again we separate the cyclic motion: v = vcyclic + vcurv B

r 0 = q( × r ) + Fr v∇B B centrifugal

•

rˆ n

r rˆ r B×n mv // vcurvB = − 2 B qR 2

The electrons and ions drift in opposite directions, producing charge separation and net current.

NEEP 536, Spring 2003, D. Whyte

12

Plasma: electrically conducting medium with magnetic forces. So now let’s start putting this together into an equilibrium • We re-write the force balance equation in terms of a unit volume of plasma with kinetic pressure p

r r F = q(v × B)

• And Ampere’s law

r r ∇p = j × B

(N / m3)

r µo j = ∇ × B

• So we notice that current and B lie in isobaric surfaces (I.e. ∇p = 0)….these are our flux surfaces again.

– But current and B are not parallel unless there is no pressure gradient anywhere.

NEEP 536, Spring 2003, D. Whyte

13

Simple equilibrium illustrates generic features of MHD •

Consider a cylindrical plasma of radius a with uniform current density j. This simple geometry --> dp

∇p = r × r j B µ j =∇× r B o

•

• •

=jB

j

1∂ dr µ j= (rB ) z θ

o

r ∂r

θ

Recall the Biot-Savart law for poloidal magnetic field caused by a straight wire ---> tells us 1 2 2 Back into µ balance p(r) = µj z ( a – r 2 ) Bθ (r) = j z r Force And integrating thru r 4 2

Bθ

So isobaric surfaces are r=constant “rings” --> flux surfaces 2 And removing j we find a term that looks B p(r) = 2 θ like magnetic pressure ∝ B

µ

NEEP 536, Spring 2003, D. Whyte

a

2

−1

r 14

Summary of drifts: Occur perpendicular to B, ExB moves whole plasma, curvature and ∇B drift cause plasma charge separation --> E --current Consider example of our simple mirror in cylindrical coordinate (r, θ z)

•

r nˆ Curvature of B

Biot-Savart Law: Coils with poloidal current --> Bz~I/a

∇B

•

Because of the axi-symmetry, there are no gradients in θ so for the drifts we find 2

mv // 1 mv ⊥2 ∇B vθ ≅ + eR B 2e B2

So the plasma will “spin” in the poloidal direction, but since there is no Radial motion, the mirror is stable to drifts.

NEEP 536, Spring 2003, D. Whyte

13

Plasma: electrically conducting medium with magnetic forces. So now let’s start putting this together into an equilibrium • We re-write the force balance equation in terms of a unit volume of plasma with kinetic pressure p

r r F = q(v × B)

• And Ampere’s law

r r ∇p = j × B

(N / m3)

r µo j = ∇ × B

• So we notice that current and B lie in isobaric surfaces (I.e. for ∇p = 0)….these are our flux surfaces again.

– But current and B are not parallel unless there is no pressure gradient anywhere in the system.

NEEP 536, Spring 2003, D. Whyte

14

Simple equilibrium illustrates generic features of MHD •

Consider a cylindrical plasma of radius a with uniform current density j. This simple geometry --> dp

∇p = r × r j B µ j =∇× r B o

•

• •

=jB

j

1∂ dr µ j= (rB ) z θ

o

r ∂r

a

θ

Recall the Biot-Savart law for poloidal magnetic field caused by a straight wire ---> tells us 1 2 2 Back into µ balance p(r) = µj z ( a – r 2 ) Bθ (r) = j z r Force And integrating thru r 4 2

Bθ

So isobaric surfaces are r=constant “rings” --> flux surfaces 2 And removing j we find a term2that looks B p(r) = θ like “magnetic pressure” ∝ B

µ

NEEP 536, Spring 2003, D. Whyte

a

2

−1

r 15

Magnetic equilibrium construction •

We can examine this formally by combining the force balance and Ampere’s law r 1 r B2

r r ∇p = j × B r µo j = ∇ × B

∇ p+

2µ o

Kinetic Pressure ~ nT

=

µo (B ⋅ ∇)B

Magnetic pressure

•

In most of the magnetic topologies we consider, the RHS of this equation is very small since B is straight (or nearly straight). So the sum of the kinetic pressure and magnetic pressure are ~ constant

•

Let’s consider our cylinder with j distributed evenly through… a

p(r) =

p Bθ NEEP 536, Spring 2003, D. Whyte

j

1

µj 2 ( a 2 − r 2 ) 4 z

Bθ (r) =

µ 2

jzr

µj a z r 2

2

Bθ (r) =

For r>a

Bθ 16

Ideal MHD: Plasma is treated as a “fluid” with no resistivity •

The fluid obeys – Maxwell’s equations – Force balance

r r ∇p = j × B dn

– Continuity (sources and sinks must balance)

dt

– Ohm’s law (vxB) arises from MHD motion of plasma in B

r + ∇( nv) =0 r r r ηj = E + v × B

and in “ideal” MHD the plasma resistivity is taken as 0 so

•

Useful illustration: put ideal Ohm’s law back into Faraday’s induction law r r r r ∂Φ r Integrate over

∂B = ∇ × E = ∇ × r × v B dt

• •

r r E +v×B=0

= dS

A fixed surface

dt

∫

· ∇ × v × B = ∫ B ⋅ (dl × v )

So the rate of change of flux is geometrically linked to the MHD plasma “movement”.

I.e. the flux lines move along with the plasma….they are frozen into the fluid

MHD equilibrium: Balance between kinetic and magnetic driven forces ∇B

B2/2µ

B2 p+ 2µ o

≈ constant

p • •

The outward radial force caused by the internal kinetic pressure is balanced by an inward magnetic pressure in equilibrium.

Perturbation stability analysis on mirror: What if the flux surface deforms slightly outward (dashed line)?

– – – –

•

The surface moves into a region of lower B --lower magnetic pressure.

The plasma continues to exert same outward pressure (~isobaric) The deformation continues to grow Confinement is destroyed.

Plasmas + magnetic topology --> MHD instability

MHD stability plays a defining role in the possibilities for magnetic confinement fusion • • • •

What we are interested in is a magnetic bottle that can provide the containment for the fusion What we really need is plasma kinetic pressure: Near 10-20 keV for D-T: Fusion power density ~ n2 <σv> QD-T ~ n2 T2 ~ p2 But we will have to provide magnetic pressure to contain this kinetic pressure…Pmag ~ B2/2µ So we define one of the key parameters in MFE: β, the ratio of kinetic pressure to magnetic pressure, nkT

β≡

•

B2 2µ o

And the relative success or failure of the MFE concept will be highly influenced by how high this parameter can be pushed.

– How far can you push β before the bottle breaks?

MHD stability plays a defining role in the possibilities for magnetic confinement fusion •

Consider a “theta pinch” fusion device.

– Apply current through plasma, Bθ forms.

– Bθ applies magnetic compression, constricting the plasma – As the plasma constricts, j increases

j

Bθ

– Bθ increases ----> compression …….

•

However the theta pinch is unstable to a “sausage” type instability – Where the plasma column bulges in the movement is toward pmag > pkinetic – Where the plasma column bulges out the the movement is toward pmag < pkinetic – So the deformation tends to grow --> instability

plasma

I a

Bθ

rplasma

Stability is typically gained by adding in an “axial” field that provides stabilization for the “confining field” • • • •

In the case of the pinch, we add in an axial magnetic field embedded in the plasma. Φ= πr2 B ~ constant plasma z So in ideal MHD the flux is conserved: So at the constriction, the pressure associated with the axial magnetic field will exert an increasing outward pressure ~ B z2/2µ which will stabilize against the deformation….ditto at the bulge.

The axial magnetic field provides stiffness to the plasma flux surfaces to stabilize it.

Bz

plasma

I a

Bθ

rplasma

The kiink is another form of instability.

•

A kinking of the plasma column will produce regions of weaker and stronger poloidal field – Poloidal magnetic flux, Bθ dr dl(θ) depends only on I. – Similar to sausage instability, where the stronger field Bθ appears tends to increase the deformation of the column and vice versa.

– Again an axial magnetic field will stabilize the kink…the tension in the axial field resists the kinking.

I

dl

Strong Bθ

Kink Deformation

Weak Bθ

Our final consideration of ‘real’ plasmas is the deconfining effect of collisions • • •

We have only considered particle moving along flux surfaces to now …but particles can “jump” across flux surfaces due to collisions.

Let’s consider a diffusion-like transport resulting from a random-walk process as collisions push particles in and out with a mean perpendicular displacement ~ Larmor radius….and the accompanying “radial” flux 2 RLarmor D= τ π /2

Fick’s law m2 / s

Adjacent Flux surfaces

Γr ≅ −D∇n 90 degree deflection

•

So plugging in the formulae for Larmor radius and collision time we obtain – Simplifying T~constant tells us that collisions will produce a net flux “outward”…I.e. in the opposite direction of the pressure gradient.

– So the higher the axial magnetic field, the smaller the Larmor radius, the better the confinement is based on classical transport.

– And τ global ~ aplasma /D ~ a x B2 – The bigger the plasma and magnetic field, the better the confinement.

D∝

n T1/ 2Bz 2

Our considerations of the real plasma have shown us the way to magnetic fusion • • • •

Bigger size scale is better for confinement ---> $$$ Bigger axial magnetic field is better for confinement ---> $$$ Bigger axial magnetic field is better for stability ---> $$$ Higher temperature --> result of confinement -->fusion competes better against Coulomb collisions.

•

“ay, there's the rub” Hamlet – As an engineering and economic enterprise magnetic fusion tends to largescale, expensive devices.

– This has necessarily slowed down the practical development of MFE.

MHD stability plays a defining role in the possibilities for magnetic confinement fusion •

Consider a “theta pinch” fusion device.

– Apply current through plasma, Bθ forms.

– Bθ applies magnetic compression, constricting the plasma – As the plasma constricts, j increases

j

Bθ

– Bθ increases ----> compression …….

•

However the theta pinch is unstable to a “sausage” type instability – Where the plasma column bulges in the movement is toward pmag > pkinetic – Where the plasma column bulges out the the movement is toward pmag < pkinetic – So the deformation tends to grow --> instability

plasma

I a

rplasma

Bθ 1

Stability is typically gained by adding in an “axial” field that provides stabilization for the “confining field” • • • •

In the case of the pinch, we add in an axial magnetic field embedded in the plasma. Φ= πr2 B ~ constant plasma z So in ideal MHD the flux is conserved: So at the constriction, the pressure associated with the axial magnetic field will exert an increasing outward pressure ~ B z2/2µ which will stabilize against the deformation….ditto at the bulge.

The axial magnetic field provides stiffness to the plasma flux surfaces to stabilize it.

Bz

plasma

I a

rplasma

Bθ 2

Stabilizing influence of Bz can be seen by looking at drift and current response of plasma •

Look at the deformation at the “top of the cylinder, – Vector normal for curvature has radial component perp. To B

•

Bz

n

n

Jpoloidal Ⓒ

plasma

vcurvature,ions ◉ Jpoloidal

I

So the ions will drift in the -B x nr poloidal direction…the electron in the opposite direction.

–

The charge separation can be viewed as a poloidal current.

•

The Jpoloidal x Bz term(s) pushes the constriction (bulge) outward (inward)…I.e. it tries to restore the original shape of the flux surface and heal the perturbation.

•

Note this is independent of which way we point Bz

Jpoloidal x Bz restoring forces

3

The kiink is another form of instability.

•

A kinking of the plasma column will produce regions of weaker and stronger poloidal field – Poloidal magnetic flux, Bθ dr dl(θ) depends only on I. – Similar to sausage instability, where the stronger field Bθ appears tends to increase the deformation of the column and vice versa.

– Again an axial magnetic field will stabilize the kink…the tension in the axial field resists the kinking.

I

dl

Strong Bθ

Weak Bθ

Kink Deformation

4

Our final consideration of ‘real’ plasmas is the deconfining effect of collisions • • •

We have only considered particle moving along flux surfaces to now …but particles can “jump” across flux surfaces due to collisions.

Let’s consider a diffusion-like transport resulting from a random-walk process as collisions push particles in and out with a mean perpendicular displacement ~ Larmor radius….and the accompanying “radial” flux 2 RLarmor D= τ π /2

Fick’s law m2 / s

Adjacent Flux surfaces

Γ r ≅ −D∇n 90 degree deflection

•

So plugging in the formulae for Larmor radius and collision time we obtain – Simplifying T~constant tells us that collisions will produce a net flux “outward”…I.e. in the opposite direction of the pressure gradient.

– So the higher the axial magnetic field, the smaller the Larmor radius, the better the confinement is based on classical transport.

– And τ global ~ aplasma /D ~ a x B2 – The bigger the plasma and magnetic field, the better the confinement.

D∝

n T1/ 2Bz 2 5

Our considerations of the real plasma have shown us the way to magnetic fusion • • • •

Bigger size scale is better for confinement ---> $$$ Bigger axial magnetic field is better for confinement ---> $$$ Bigger axial magnetic field is better for stability ---> $$$ Higher temperature --> result of confinement -->fusion competes better against Coulomb collisions.

•

“ay, there's the rub” Hamlet – As an engineering and economic enterprise magnetic fusion tends to largescale, expensive devices.

– This has necessarily slowed down the practical development of MFE.

6

Plasma instabilities continued: Instability with axial magnetic field in simple mirror • • • •

•

The simple mirror is subject to the “flute” or “interchange” instability At the point in minimum Bz we get a B gradient toward the plasma center The ions and electrons drift in opposite poloidal direction from Bx ∇B.

e-

◉

i+

B

∇B

If the surface is deformed (a “flute” shape) then the “sides” of the deformation will build up charge

◉

e- i+

The resultant electric field will make the flute expand due to Er x Bz drift

B

∇B -

+ +

-

E

ExB

7

In general we need to avoid situations where the field has “bad” topology in order to avoid flute like instability.

•

•

•

In essence, we want the accelerating forces (like (-)gradB, curvature vector) to point into the plasma.

acurvature

Good curvature

B

plasma Bad curvature

The good configurations will tend to stabilize deformation.

For the mirror case this leads one to re-arranging the magnets to create a 3-D magnetic well such that ∇B is always point outward – The “mininum |B|” solution.

Where the field increases away from plasma.

plasma

Bad ∇B

acurvature

e- i+

◉

∇B Good ∇B

e- i+

◉

∇B 8

Extensive research was carried out on minimum-B mirror configurations in the 60, 70 and early 80’s •

•

An optimal magnet shape, which look like the seams on a baseball (also called ‘yinyang’ magnets, which are much too complicated for me to draw in Powerpoint), was successfully used to make min.-B mirrors that suppressed the flute instability.

However other problems arise…mostly arising from the parallel loss channel – Remember that a defined population of the velocity distribution is lost out the ends because they don’t meet the mirror criterion.

– The “hole” left in velocity space gets filled in by collisions…leading to further parallel losses.

– The attempted relaxation of the velocity distribution to a Maxwellian prompts particledriven instability.

– Possible remedy for this is external injection of energetic particles with v// >> vperp --> power – Collisions also lead to cross-field transport (obviously not related to end losses)

•

Optimistic calculations showed that the maximum Q expected from min.-B configurations is Q ~ 1-2…which is insufficient for fusion energy.

9

What is Q? •

This is the dimensionless number given to the ratio of how much fusion power we get compared to the external heating power we have to provide to heat the plasma.

It is the essential measure on the feasibility of an MFE concept – Q < 1 no net gain….all experiments to-date – Q~1 scientific breakeven – Q > 10 reactor engineering feasibility --> electricity on the grid – Q = ∞ ignition…completely self-sustained fusion reactions

Pfusion Q= Pinput

10

The tandem mirror is a variation on the standard mirror configuration • •

In a simple mirror the electrons scatter into the loss cone faster than the ions so the “central” cell becomes positively charged.

This effect of mirrors is used to intentionally create an electrostatic confining well for the ions: the mirrors are added in tandem with other simple mirrors.

– The use of this configuration greatly improved confinement.

•

•

However without any special tricks (which continue to make the design more complicated) the field strength necessary for fusion at the end coils is ~20 Tesla…this is restrictively high.

The tandem mirror research program was cancelled in the U.S., although some research continues in Japan (Gamma-10).

– The latest paper from Gamma-10, B=3T with T~10 keV: n τe ~ 1018 m3s-1.

•

The mirror configurations were out-performed by toroidal systems.

– Although mirrors have engineering advantages (paper topic?)

11

Toroidal magnetic configurations eliminate end losses by bending the plasma cylinder into a torus •

•

The configuration is essentially cylindrical with the “axial” direction now denoted as “toroidal”.

Another important geometric factor is now the ratio of R, the major radius as compared to a, the plasma minor radius.

A=R/a

Top view

– R >> a --> skinny donut – R ~ a --> spherical

•

toroidal

As before we want to apply a Bz -> Btoroidal to provide – Plasma confinement (particles pinned to B by Larmor effects) – Stability.

r,a

R

poloidal

12

Toroidal magnetic configurations eliminate end losses by bending the plasma cylinder into a torus •

The toroidal magnetic field is produced by a ring of current carrying field coils.

•

The geometric concentration of the current in the center of the donut “hole” gives rise to a toroidal magnetic field strength that decreases with major radius R, in the plasma namely BT ( R)= Bo (Ro/R) – Bo is the magnetic field at Ro --> r=0, or the magnetic axis.

– Therefore there exists an inevitable B gradient pointed toward lower R.

NEEP 536, Spring 2003, D. Whyte

Top view

Toroidal Field coils

toroidal

r,a

Icoil R

poloidal BT 13

Because of the toroidal geometry and plasma drifts, a purely toroidal B cannot confine plasma • • •

Curvature drift and ∇B drift both lead to the same vertical separation of ions and electrons.

The charge separation causes a vertical electric field.

The resulting ExB drift quickly sends the plasma outward to the wall

Btoroidal

ncurv

eions

◉ B

– No confinement!

•

∇B

Estimate confinement: n ~ 1020 m-3 and T ~ 1 eV, R=1m, B=10 T – vz,curv ~ kT/eBR ~ 10-1 m/s – So in 10-6 s, E ~ e n 10-7 / εο ~ 105 V/m – vExB ~ E --> plasma is essentially “instantly” deconfined and smashes into the wall.

------

ExB / B2

E ++++ wall 14

The addition of a poloidal field is necessary in order to ”short” out the deconfining drifts.

• • •

•

Curvature drift and ∇B drift still apply.

But (e.g. the ions) must now follow the poloidal field around (see picture) So as the ions orbit on different halves (upper and lower) they, along with electrons that are performing similarly, will provide both inward and outward radial drift that cancel each other.

In other words the poloidal field averages out the drift effects.

– This is called “rotational transform” and is necessary for confinement in a toroidal device.

•

B

Btoroidal

Bpoloidal

Poloidal field line

There are two fundamentally different ways to produce the poloidal field: – A toroidal current through the plasma ---> tokamak.

– Add toroidal “twist” to the field coils --> stellarator, heliotron, etc.

15

The Tokamak has been the most successful MFE configuration and will be the primary focus of our study •

Q ~ 1 has been obtained.

•

Fusion power of > 10 MW with D-T plasma in two experiments (TFTR at PPPL and JET in UK)

•

Confinement quality n τ ~ 6x1019 m3 s achieved at T=10 keV.

•

First MFE burning plasma will be a tokamak (e.g. ITER)

•

However, tokamaks have severe engineering drawbacks and may not be the “final” MFE fusion solution.

16

The tokamak relies on toroidal plasma current to provide poloidal field for rotational transform •

Faraday’s induction law

How does one apply this current to

−

the plasma?

•

The originating and standard design of the tokamak (a Russian acronym for toroid -kamera - magnit - katushka meaning

dB/dt

∂B =∇× E ∂t

Etoroidal

“toroidal chamber and magnets) uses

induction to drive the toroidal current.

•

dB/dt

Essentially the tokamak plasma acts like the secondary (single) winding on a transformer.

Itoroidal ≡

Etoroidal

η

Isolenoid 17

Inductive current drive is not steady-state •

•

The induced emf relies on the rate of change of the Bprimary ∝ Isolenoid We ignore the impedance of the plasma compared to the solenoid coil, and we find that plasma toroidal current ∝ (L/R) dIsolenoid/dt –

•

R plasma resistance here, not major radius.

So for a constant plasma resistivity (I.e. constant T) we must drive continually changing current through solenoid – –

•

Bvertical,solenoid ~

h

Itoroidal ≡

∂B

hZ ,solenoid

∂t

∂I

NR E∝

µoNI

µo

Vloop Rplasma

=

loop

Vloop

ηa

Nµo ∂I hsolenoid ∂t ∂I

N2πR2

V

∂t

∝

2 plasma

≡ E 2πR ∝

∝

h

L

∂Isolenoid

Rplasma

∂t

µo

∂t

∂I ∝L

∂t

inductance

Isolenoid dB/dt

Practical limitation to V-s capability.

This is why we use transformers on AC!

Iplasma

We’ll come back to inductance in a more formal manner when we examine the coil sets in more detail.

Central solenoid coil with N turns, length h

Itoroidal ≡

Isolenoid

Etoroidal

t

η

18

The Tokamak has been the most successful MFE configuration and will be the primary focus of our study •

Q ~ 1 has been obtained.

•

Fusion power of > 10 MW with D-T plasma in two experiments (TFTR at PPPL and JET in UK)

•

Confinement quality n τ ~ 6x1019 m3 s achieved at T=10 keV.

•

First MFE burning plasma will be a tokamak (e.g. ITER)

•

However, tokamaks have severe engineering drawbacks and may not be the “final” MFE fusion solution.

NEEP 536, Spring 2003, D. Whyte

16

The tokamak relies on toroidal plasma current to provide poloidal field for rotational transform •

Faraday’s induction law

How does one apply this current to

−

the plasma?

•

The originating and standard design of the tokamak (a Russian acronym for toroid -kamera - magnit - katushka meaning

dB/dt

∂B =∇× E ∂t

Etoroidal

“toroidal chamber and magnets) uses

induction to drive the toroidal current.

•

dB/dt

Essentially the tokamak plasma acts like the secondary (single) winding on a transformer.

Eθ jtoroidal ≡

E

toroidal

η

Isolenoid NEEP 536, Spring 2003, D. Whyte

17

“Inductive” current drive is not steady-state • •

•

The induced emf relies on the rate of change of the Bprimary ∝ Isolenoid We ignore the impedance of the plasma compared to the solenoid coil, and we find that plasma toroidal current ∝ (L/η) dIsolenoid/dt So for a constant plasma resistivity (I.e. constant T) we must drive continually changing current through solenoid – –

•

E∝

NR h

µoNI

∂B

hZ ,solenoid

∂t

∂I

µ o

∂t

Vloop ≡ E 2πR ∝

∝

Nµo ∂I hsolenoid ∂t ∂I

N2πR2 h

L ∂Isolenoid V ∝ I toroidal ≡ loop R η ∂t η 2 a Isolenoid

µo

∂t

∂I ∝L

∂t

inductance

dB/dt Iplasma

Normal tokamak --> pulsed operation.

–

•

Practical limitation to V-s capability.

This is why we use transformers on AC!

Bvertical,solenoid ~

Implications for practicality as energy source.

We’ll come back to inductance in a more formal manner when we examine the coil sets in detail.

NEEP 536, Spring 2003, D. Whyte

t Central solenoid coil with N turns, length h

Itoroidal ≡

Isolenoid

Etoroidal η

18

The geometry of the magnetic field in the tokamak • •

It is convenient to work in a toroidal coordinate system as shown The toroidal magnetic field is given by:

θ or poloidal

r,a

Bo Bo = Btoroidal = r 1+ cosθ 1+ ε cosθ R – –

•

Notice this only a function of R. In limit of inverse aspect ratio ε--> 0 we recover cylindrical geometry.

φ or Btoroidal Magnetic or minor axis

r

rBθ = ∫ µo j(r) r dr

∇ × B = µo j

In toroidal coordinates ∂B R ∂ o [(1+ ε cosθ)B (r,θ)] + φ = ∇ ⋅ B = 0 Whyte

R Major axis

The toroidal current density j will produce a poloidal magnetic field through Ampere’s law

r ∂θ

toroidal

θ

∂φ

θ

Bo axisymmetry

B (r,θ) ≡ Bθ (r,θ = π /2) θ

1+ ε cosθ

Another result Of toroidal geometry

19

The concentration of toroidal and poloidal field towards lower R causes force imbalance which must be counteracted by a vertical stabilizing field.

•

In general we can write the total field as Bo B(r,θ) = 1+ ε cosθ

•

So comparing the inboard to the outboard side of a flux surface (isobaric) we have B Bo B(θ = π ) = o > 1+ ε = B(θ = 0) 1− ε

•

•

And through magnetic pressure (∝B2) we have a net outward radial force on plasma

Iplasma

The higher field at low R pushes the plasma out.

Therefore a vertical magnetic field must be applied in order to balance this radial force.

–

Whyte

◉

Applied vertical field

This can understood through the IplasmaxBθ force as shown –

•

Bθ

Iplasmax Bvertical is inward 20

In a similar manner one can find vertical movement instability caused by plasma flux surface vertical displacement from the midplane.

•

A vertical displacement will tend to increase poloidal B in on the side opposite to displacement.

•

The JxB force imbalance will tend to further vertically displace the plasma.

•

As with the radial force imbalance, this is controlled by applying an external field to counteract the instability.

–

Whyte

The radial components of the concave field will tend to oppose the vertical displacement.

Bθ

◉I

plasma

Applied concave field

21

High plasma current is desired for confinement yet we are limited in the current we can apply by stability • • • • •

Let’s consider the simple sausage instability in cylindrical geometry.

We saw that an axial magnetic field was required to suppress the instability….but what Bz is required?

Remember the instability was being driven by the pinch effect through Bpoloidal.

We evaluate stability limit where the restoring pressure from Bz “wins” over the instability-driving Bpoloidal.

This tells us that Bz must be ~ sqrt(2) times larger than the poloidal magnetic field.

πr2B

B =

z = const.

∇p ≡ ∂

θ

B2z

∂r 2µ o

−

Bθ2

µI 2π r

>0

2µ o

2B2 > B 2 z

θ

Bz

plasma

I a

rplasma

Bθ

Whyte

22

High plasma current is desired for confinement yet we are limited in the current we can apply by stability •

The kink is also driven the the poloidal magnetic field –

•

•

Bθ dr dl = I= const

Since poloidal field flux is fixed by the plasma current, Bpoloidal increases on the inside of the kink (where dl decreases) and decreases on the outside of the kink (dl increases).

pBΖ

The magnetic pressures can be visualized as pushing and pulling on a disk surrounding the ends of the column.

a

–

The poloidal field pressure tries to kink the column

–

further.

The axial field pressure straightens the column.

B2

R

1

L /2 µI2

B2(a)

∫

yields stabilization needed at boundary, a, of pBθ ~ dr = 2 plasma. π a a 4 πr

B 2 > 2B 2(a)ln z

Whyte

L

pBz ~ µz 2 o

For the kink instability in cylindrical geometry

Bz >~ 2Bθ (a)

Bz

θ

L 2a

θ

µ

L /2 ln

≈ 2B 2(a)ln θ

Bpoloidal pBθ

a

πR a

tokamak

23

Another important instability in a tokamak is the ballooning instability •

• •

As we showed before the curvature acts like an accelerating or gravitational force directed outboard.

At the outboard side of the plasma the plasma pressure gradient is inward --> bad curvature.

The plasma kinetic pressure tends to “pop-out” at the weakest point in the bottle…like a tire bursting --> “ballooning”…at the outboard or low-field side surface (as shown), while the inboard side is stable (good curvature).

– This is essentially the Rayleigh-Taylor instability (heavy fluids don’t rest stably on lighter fluids in a gravity field) with curvature.

a p βθ ≡ ≤~1 Ro ( Bθ2 /2 µ) A Ballooning stability Whyte

24

We introduce the dimensionless parameter, q, which describes both the helicity and stability of the tokamak field configuration.

•

The q parameter describes how many times a field line at r will go around the torus (the long way) until it completes one poloidal rotation.

– Smaller q --> more twisted.

– Higher q --> less twisted.

• • •

•

The minimum stability criterion against the internal kink on axis is that q ~ q(0) > 1 And at the surface (r=a) the requirement is typically q(a) ≥ 2-3 The locations of “rational” number q (q=1,2,3/2) are the most dangerous for instability since their periodicity helps “amplify” perturbations.

Looking at the function for q we can start to realize some trade-offs in tokamak design

r q= Ro

Bφ

B φ 2 π r2 = Bθ (r) µo I(r)Ro

– E.g. lower aspect ratio allows for lower Btoroidal.

Whyte

25

High plasma current is desired for confinement yet we are limited in the current we can apply by stability • • • • •

Let’s consider the simple sausage instability in cylindrical geometry.

We saw that an axial magnetic field was required to suppress the instability….but what Bz is required?

Remember the instability was being driven by the pinch effect through Bpoloidal.

We evaluate stability limit where the restoring pressure from Bz “wins” over the instability-driving Bpoloidal.

This tells us that Bz must be ~ sqrt(2) times larger than the poloidal magnetic field.

πr2B

z

B =

= const.

∇p ≡ ∂

θ

B2z

∂r 2 µ o

−

Bθ2

µI 2π r

>0

2µ o

2B2 > B2 z

θ

Bz plasma

I a

rplasma

Bθ

1

High plasma current is desired for confinement yet we are limited in the current we can apply by stability •

The kink is also driven by the the poloidal magnetic field –

•

•

Bθ dr dl = I= const

Since poloidal field flux is fixed by the plasma current, Bpoloidal increases on the inside of the kink (where dl decreases) and decreases on the outside of the kink (dl increases).

pBΖ

The magnetic pressures can be visualized as pushing and pulling on a disk surrounding the ends of the column.

a

–

The poloidal field pressure tries to kink the column

–

further.

The axial field pressure straightens the column.

B2

R

L

pBz ~ µz 2 o

For the kink instability in cylindrical geometry

1

L /2 µI2

B2(a)

∫

yields stabilization needed at boundary, a, of pBθ ~ dr = 2 plasma. π a a 4 πr

Bz >~ 2Bθ (a)

Bz

B 2 > 2B 2(a)ln z

θ

L 2a

θ

µ

ln

≈ 2B 2(a)ln θ

Bpoloidal

L /2 pBθ

a

πR a

tokamak

2

We introduce the dimensionless parameter, q, which describes both the helicity and stability of the tokamak field configuration.

•

The q parameter describes how many times a field line at r will go around the torus (the long way) until it completes one poloidal rotation.

– Smaller q --> more twisted.

– Higher q --> less twisted.

• • •

•

The minimum stability criterion against the internal kink on axis is that q ~ q(0) > 1 And at the surface (r=a) the requirement is typically q95~q(a) ≥ 2-3 The locations of “rational” number q (q=1,2,3/2) are the most dangerous for instability since their periodicity helps “amplify” perturbations.

Looking at the function for q we can start to realize some trade-offs in tokamak design – E.g. lower aspect ratio allows for lower Btoroidal.

r q= Ro

Bφ Bθ (r)

=

Bφ 2πr2

µoI(r)Ro

q=1

q=2

Field line poloidal location at a Fixed toroidal angle

q= irrational number 3

Another important instability in a tokamak is the ballooning instability •

• •

As we showed before the curvature acts like an accelerating or gravitational force directed outboard.

At the outboard side of the plasma the plasma pressure gradient is inward --> bad curvature.

The plasma kinetic pressure tends to “pop-out” at the weakest point in the bottle…like a tire bursting --> “ballooning”…at the outboard or low-field side surface (as shown), while the inboard side is stable (good curvature).

– This is essentially the Rayleigh-Taylor instability (heavy fluids don’t rest stably on lighter fluids in a gravity field) with curvature.

a p βθ ≡ ≤~1 Ro ( Bθ2 /2 µ) A Ballooning stability

4

The kink stability limit + “Beta” limit from ballooning determine to a large extent MFE efficiency.

•

•

"Because the cost of a reactor is strongly influenced by the strength of the magnetic field that must be provided, Beta values are directly related to the economics of fusion power production.”

The beta limit for a conventional tokamak was derived semi-empirically and is known as the Troyon limit. It is characterized by a “normalized Beta” (βN) value which takes into account the plasma current, field and size.

–

–

•

This is sometimes a “soft limit” in that attempts to push past the βN ~3.5 result in rapidly degraded confinement from ballooning like instability.

The q limit (q(a) ~ q95 < 2-3) meanwhile usually provides a hard limit…that is the resulting kink mode will rapidly lead to a global instability --> “disruption”

One can conveniently re-write Beta in terms of the normalized Beta, torus shape (inverse aspect ratio ε = a/R) and q.

–

A conventional tokamak (ε∼1/3) achieves β ~ 2-3%.

βt =

p Bt2 2 µo

W / m 3 ∝~ p2 ≡ β 2B 4

P fusion

[

]

βN ≡ β

aB ≤ 3.5 Ip

t

β in %, a in m, B in T, I in MA

β ⋅ε β ≡5 N q95 5

Non-circular “shaping” of the plasma is used to increase the available β.

•

•

Remember how we decided that the outboard midplane would be the “weakest link”.

κ~2

–

κ=1

Because of the low B and max.curvature at this region.

The idea of shaped plasmas is to make the trajectory of field lines avoid this region as much as possible.

This is achieved by – –

Elongating the plasma --> shaping factor κ=“elongation” κ=1 circle.

Squeezing the outside midplane, making the plasma look like a triangle --> shaping factor δ = “triangularity” δ = 0 circle.

•

E.g. for the ballooning limit, the allowed pressure gradient scales as ~ ( 1 + κ2(1 + 5 δ2 ))

•

There are some practical limits to the use of these factors….

–

•

Elongation

E.g. κ>2 difficult to maintain vertical stability

However the use of shaping has allowed the tokamak configuration to achieve β ~ 10%.

– Low aspect ratio tokamaks β > 20%.

Triangulariy

β ≡ 2.5(1+ κ 2(1+ 5δ 2))

βN ⋅ ε qcyl,95 6

Non-circular “shaping” of the plasma is used to increase the available β.

•

Remember how we decided that the outboard midplane would be the “weakest link”.

–

•

Because of the low B and max.curvature at this region.

The idea of shaped plasmas is to make the trajectory of field lines avoid this region as much as possible.

This is achieved by – –

Elongating the plasma --> shaping factor κ=“elongation” κ=1 circle.

Squeezing the outside midplane, making the plasma look like a triangle --> shaping factor δ = “triangularity ” δ = 0 circle.

•

E.g. for the ballooning limit, the allowed pressure gradient scales as ~ ( 1 + κ2(1 + 5 δ2 ))

•

There are some practical limits to the use of these factors….

–

•

“D-shaped” plasma for accessing higher Beta

E.g. κ>2 difficult to maintain vertical stability

However the use of shaping has allowed the tokamak configuration to achieve β ~ 10%.

– Low aspect ratio tokamaks β > 20%.

7

Besides stability, the other criterion for a successful fusion system is energy confinement.

– Collision make the ions jump from one flux tube to the next by a Larmor radius.

– Expected diffusion coefficient D {m2 /s} ∝ R L2 ∝ 1/B2.

Adjacent Flux surfaces

plasma energy

Response of system to energy input at time=0 (W=3nkT)

-5

0

τe

5

10

time

n τe =

12kT

σv E α − 2.14 ×10−36T1/ 2

1E+22 Lawson criterion

τe (s / m3)

•

The Lawson criterion requires that our plasma system confines the energy (externally provided and internally from fusion) sufficiently well so as to maintain T necessary for fusion.

Classical transport: Coulomb collisions lead to cross-field transport.

1E+21

n

•

1E+20 1

10 T (keV)

100

90 degree deflection

8

The global particle and energy confinement times can be characterized by the plasma size and the transport coefficients Γr ≅ −D∇n

Particle flux density

Τp ≅

q ≡ −χnk∇T

Heat flux density

τe ≅

a2

∇n, ∇T

D a2

T, n

χ

r

a

• Simple geometry considerations immediately inform us that “bigger is better”.

• Volumetric power source with diffusion/flux losses through surface.

• Low D or χ is most desired.

• In fact in plasmas, the classical scaling is rare. “Bohm” diffusion is an empirical scaling that is more typical of highly turbulent plasmas.

• Bohm represents the “worst-case” scenario for transport

DBohm ∝

T B

Dclassical ∝

n 1/ 2

T B

2

9

Neo-classical confinement is the “ideal” level of transport available in a toroidal geometry…but is rarely achieved.

•

– –

•

•

•

∇B

Neo-classical transport takes into account unique aspects of toroidal geometry

Banana orbit

Particle drift orbits.

Portion of particle exhibit “banana” orbits which occur due to the mirror effect in a toroidal geometry as particles approach inboard high magnetic fields.

The theoretical basis of neo-classical transport was worked out in 70’s…too complicated to go into in detail.

Bottom line: We expect an enhancement over classical transport with factors related to aspect ratio and the safety factor, q.

Negative aspect of neo-classical transport: Preferential peaking of impurity density due to Z dependences in Coulomb collisions – –

ε −3/2 R 2Lq 2 D,χ ∝ τ π /2 τ e,neo−classical ≈

Enhances fuel dilution and radiative losses from most reactive portion of plasma Impurity peaking Illustrates important differentiation between energy confinement and particle confinement. nZ (r) ni (r)

nz (a)

=

ε 3/ 2 a 2 τ π /2 R2L q 2

Z

n i (a) 10

Empirical confinement scaling has been constructed from 25+ years of experiments τ e = 0.0562 ⋅ H ⋅ I p Bt ne M 0.93

• •

0.15

0.4

0.19

1.97 0.58

R ε κ

0.78

P

−0.69

(MA, T, n in 1e19 m-3, m, P in MW

Wide variety of tokamak sizes, field strength, shape, etc. used.

No physics basis whatsoever --strictly regression analysis.

– But works over factor of 100!

•

•

•

Related back to stability factor, q, reveals quasi Bohm-like process and strong geometry( R) and shape (ε,κ,δ) dependence …and degradation with power (P)!!?

What is this?

What is the H factor?

Τ e ∝ H ⋅ Bt

n M 0.19R2.97ε2.58κ 0.78P−0.69 f (κ,δ) /q 95,cyl

1.15 0.4 e

11

The degradation of confinement time with heating: Not completely understood but is generally a result of drift-wave turbulent transport driven by T gradients • •

•

The plasma exhibits electrostatic turbulence that causes cross-field transport.

Intensive computation and theoretical work is currently underway to provide first principles based predictive capability for confinement.

In general result is that there exists a critical gradient for the ion temperature gradient

Computation of electro-static turbulence In DIII-D tokamak core (Waltz et al.) (looking at cross-section of minor radius) Notice the radial coherent structures on outboard side --> transport

– Central T thus given by critical gradient and size scale…”stiff” profiles.

Tcentral ≈ ∇Tcritical ⋅ a 12

An operational mode of high confinement (H-mode: H=1) was discovered in the 80’s that features reduced turbulence.

This mode is the baseline for near-term fusion MFE. • •

Again no first principles theory exists to predict the onset and cause of H-mode Empirical work shows that the transition is related to the creation of strongly sheared Er xB flows that break apart the turbulent eddies

Example transition into H-mode on the DIII-D tokamak The input power is constant ~ 2.5 MW At 2200 ms the plasma spontaneously transitions into higher Confinement as witnessed by increase in plasma energy.

– Complex ‘self-organization’ – ExB shear mostly present in periphery of plasma --> transport barrier – …and leads to local edge instabilities..Edge Localized Modes, ELM which periodically relax the edge gradient--these have a major impact on heat removal

•

Empirical scaling shows that there exist a critical power across boundary of plasma is needed to initiate H-mode, PL-H (MW).

– Also sensitive to plasma shape and the presence of a “divertor” and vacuum cleanliness.

ELMs

L-mode

H-mode

time

PL– H = 2.84 ⋅ Bt0.82 n e0.58 M −1Ra 0.81 13

Besides stability, the other criterion for a successful fusion system is energy confinement.

– Collision make the ions jump from one flux tube to the next by a Larmor radius.

– Expected diffusion coefficient D {m2 /s} ∝ R L2 ∝ 1/B2.

Adjacent Flux surfaces

plasma energy

Response of system to energy input at time=0 (W=3nkT)

-5

0

τe

5

10

time

nτe =

12kT

σv Eα − 2.14 ×10−36T1/ 2

1E+22 Lawson criterion

τe (s / m3)

•

The Lawson criterion requires that our plasma system confines the energy (externally provided and internally from fusion) sufficiently well so as to maintain T necessary for fusion.

Classical transport: Coulomb collisions lead to cross-field transport.

1E+21

n

•

1E+20 1

10 T (keV)

100

90 degree deflection NEEP 536, Spring 2003, D. Whyte

1

The global particle and energy confinement times can be characterized by the plasma size and the transport coefficients Γr ≅ −D∇n

Particle flux density

q ≡ −χnk∇T

Heat flux density

τp ≅ τe ≅

a2

∇n, ∇T

D a2

T, n

χ

r

a

• Simple geometry considerations immediately inform us that “bigger is better”.

• Volumetric power source with diffusion/flux losses through surface.

• Low D or χ is most desired.

• In fact in plasmas, the classical scaling is rare. “Bohm” diffusion is an empirical scaling that is more typical of highly turbulent plasmas.

• Bohm represents the “worst-case” scenario for transport

DBohm ∝

NEEP 536, Spring 2003, D. Whyte

T B

Dclassical ∝

n T1/ 2 B 2 2

Neo-classical confinement is the “ideal” level of transport available in a toroidal geometry…but is rarely achieved.

•

– –

•

•

•

∇B

Neo-classical transport takes into account unique aspects of toroidal geometry

Banana orbit

Particle drift orbits.

Portion of particle exhibit “banana” orbits which occur due to the mirror effect in a toroidal geometry as particles approach inboard high magnetic fields.

The theoretical basis of neo-classical transport was worked out in 70’s…too complicated to go into in detail.

Bottom line: We expect an enhancement over classical transport with factors related to aspect ratio and the safety factor, q.

Negative aspect of neo-classical transport: Preferential peaking of impurity density due to Z dependences in Coulomb collisions – –

ε −3/2 RL2 q 2 D,χ ∝ τ π /2 τ e,neo−classical ≈

Enhances fuel dilution and radiative losses from most reactive portion of plasma Impurity peaking Illustrates important differentiation between energy confinement and particle confinement. nZ (r) ni (r)

nz (a)

NEEP 536, Spring 2003, D. Whyte

=

ε 3/ 2 a 2 τ π /2 R2L q 2

Z

n i (a) 3

Empirical confinement scaling has been constructed from 25+ years of experiments τ e = 0.0562 ⋅ H ⋅ I p Bt ne M 0.93

• •

0.15

0.4

0.19

1.97 0.58

R ε κ

0.78

P

−0.69

(MA, T, n in 1e19 m-3, m, P in MW

Wide variety of tokamak sizes, field strength, shape, etc. used.

No physics basis whatsoever --strictly regression analysis.

– But works over factor of 100!

•

•

•

Related back to stability factor, q, reveals quasi Bohm-like process and strong geometry( R) and shape (ε,κ,δ) dependence …and degradation with power (P)!!?

What is this?

What is the H factor?

NEEP 536, Spring 2003, D. Whyte

Τ e ∝ H ⋅ Bt

n M 0.19R2.97ε2.58κ 0.78P−0.69 f (κ,δ) /q 95,cyl

1.15 0.4 e

4

The degradation of confinement time with heating: Not completely understood but is generally a result of drift-wave turbulent transport driven by ion T gradients •

•

•

The plasma exhibits electrostatic turbulence that causes cross-field transport.

Intensive computation and theoretical work is currently underway to provide first principles based predictive capability for confinement.

In general result is that there exists a critical gradient for the ion temperature gradient

Computation of electro-static turbulence In DIII-D tokamak core (Waltz et al.) (looking at cross-section of minor radius) Notice the radial coherent structures on outboard side --> transport

– Central T thus given by critical gradient and size scale…”stiff” profiles.

Tcentral ≈ ∇Tcritical ⋅ a

NEEP 536, Spring 2003, D. Whyte

5

An operational mode of high confinement (H-mode: H=1) was discovered in the 80’s that features reduced turbulence.

This mode is the baseline for near-term fusion MFE. • •

Again no first principles theory exists to predict the onset and cause of H-mode Empirical work shows that the transition is related to the creation of strongly sheared Er xB flows that break apart the turbulent eddies

Example transition into H-mode on the DIII-D tokamak The input power is constant ~ 2.5 MW At 2200 ms the plasma spontaneously transitions into higher Confinement as witnessed by increase in plasma energy.

– Complex ‘self-organization’ – ExB shear mostly present in periphery of plasma --> transport barrier – …and leads to local edge instabilities..Edge Localized Modes, ELM which periodically relax the edge gradient--these have a major impact on heat removal

•

Empirical scaling shows that there exist a critical power across boundary of plasma is needed to initiate H-mode, PL-H (MW).

– Also sensitive to plasma shape and the presence of a “divertor” and vacuum cleanliness.

NEEP 536, Spring 2003, D. Whyte

ELMs

L-mode

H-mode

time

PL– H = 2.84 ⋅ Bt0.82 n e0.58 M −1Ra 0.81 6

The increased understanding and control of the ion thermal energy loss channel has been one of the great successes of MFE fusion research over the last 10-20 years.

•

Discovery of H-mode.

•

The paradigm of ExB velocity shear stabilization of turbulent transport.

•

The use of experimental tools to reduce heat transport --> transport barriers.

– E.g. low magnetic shear (dq/dr).

– Combinations of mutliple transport barriers (central + edge)

•

These successes are a key in our confidence in obtaining fusion conditions in future large-scale experiments.

– But caution is necessary since we have not obtained true predictive capability.

– Other key transport issues remain unresolved: e.g. particle transport.

NEEP 536, Spring 2003, D. Whyte

Experimental Obtainment of Neo-classical Ion heat transport

7

A topic related to ion confinement was/is the concern about how the 3.5 MeV alpha ions from D-T fusion behaved and heated the plasma --> ignition •

From a treatment based on the

τ −1 ≡ ν α−ions ~ 5 ×10−13 n ⋅ ln Λ ⋅ E −3/2

Coulomb collisions the classical slowing down time for the 3.5 MeV alphas is known in the limit that the alpha energy, Eα >> Tplasma.

2 τ −1 ≡ νsα−electrons ~ 10−9 n e ⋅ ln Λ ⋅ E −3/ α

– E in eV, n in m-3.

– First we see that the alphas slow down on the electrons --> therefore alphas directly heat the electrons only.

– For our prototypical fusion plasma, n~1020 m-3 we find alpha slowing down time ~ 10 ms << τe.

•

What about the confinement of the hot alpha ion?

s

i

α

1/ 2 1/ 2 mv E −4 M RLarmor = ~ 10 qB Z⋅B

R in m, E in eV, B in T

– For a 5 T field the alpha has a Larmor radius ~ 3.5 cm << a.

NEEP 536, Spring 2003, D. Whyte

8

A topic related to ion confinement was/is the concern about how the 3.5 MeV alpha ions from D-T fusion heated the plasma --> ignition

•

The classical slowing down of the fusion alpha particles was confirmed on the TFTR tokamak using a D-T plasma.

– Slowing down time << confinement time.

•

A sensitive diagnostic designed by Prof. Fonck measured alpha particle density in the plasma and showed signs that MHD could re-distribute the alphas – Current topic of research for burning plasmas is the worry that energetic particle driven modes will instability (TAE modes).

NEEP 536, Spring 2003, D. Whyte

9

•

Since MHD fixes the pressure (nT) allowed in the plasma then their exists a most efficient T --> 1/n for the fusion power – Pfusion ~ n2 <σv>(T)

•

•

Are there any restriction on the density we can obtain?

fusion power rate

One further important limit for the tokamak MFE concept: The density limit

0

10

20

30

T (keV) ~

40 1/n

50

60

Empirically the answer is yes on for tokamaks (with similar limits for other concepts) – If one tries to raise density arbitrarily high, a hard, disruptive limit is found

NEEP 536, Spring 2003, D. Whyte

10

One further important limit for the tokamak MFE concept: The density limit •

The empirical scaling for this density limit is known as the Greenwald limit:

nGreenwald =

Ip

πa 2

– With n in 1020 m-3, Ip in MA and a in m.

•

•

This scaling works convincingly well over a wide variety of plasma shapes, etc.

There remains no first-principles physics principle to explain this scaling.

NEEP 536, Spring 2003, D. Whyte

Ip

B ∝ nGreenwald = πa2 qcyl R

Compact high-field Tokamak (e.g. Alactor) Achieve high densities 11

We have now laid out the criteria for our MFE fusion reactor: stability, confinement, alpha heating and density limit. So what will this thing look like? One example is ITER

NEEP 536, Spring 2003, D. Whyte

12

The increased understanding and control of the ion thermal energy loss channel has been one of the great successes of MFE fusion research over the last 10-20 years.

•

Discovery of H-mode.

•

The paradigm of ExB velocity shear stabilization of turbulent transport.

•

The use of experimental tools to reduce heat transport --> transport barriers.

– E.g. low magnetic shear (dq/dr).

– Combinations of mutliple transport barriers (central + edge)

•

These successes are a key in our confidence in obtaining fusion conditions in future large-scale experiments.

– But caution is necessary since we have not obtained true predictive capability.

– Other key transport issues remain unresolved: e.g. particle transport.

NEEP 536, Spring 2003, D. Whyte

Experimental Obtainment of Neo-classical Ion heat transport

7

A topic related to ion confinement was/is the concern about how the 3.5 MeV alpha ions from D-T fusion behaved and heated the plasma --> ignition •

From a treatment based on the

τ −1 ≡ ν α−ions ~ 5 ×10−13 n ⋅ ln Λ ⋅ E −3/2

Coulomb collisions the classical slowing down time for the 3.5 MeV alphas is known in the limit that the alpha energy, Eα >> Tplasma.

2 τ −1 ≡ νsα−electrons ~ 10−9 n e ⋅ ln Λ ⋅ E −3/ α

– E in eV, n in m-3.

– First we see that the alphas slow down on the electrons --> therefore alphas directly heat the electrons only.

– For our prototypical fusion plasma, n~1020 m-3 we find alpha slowing down time ~ 10 ms << τe.

•

What about the confinement of the hot alpha ion?

s

i

α

1/ 2 1/ 2 mv E −4 M RLarmor = ~ 10 qB Z⋅B

R in m, E in eV, B in T

– For a 5 T field the alpha has a Larmor radius ~ 3.5 cm << a.

NEEP 536, Spring 2003, D. Whyte

8

A topic related to ion confinement was/is the concern about how the 3.5 MeV alpha ions from D-T fusion heated the plasma --> ignition

•

The classical slowing down of the fusion alpha particles was confirmed on the TFTR tokamak using a D-T plasma.

– Slowing down time << confinement time.

•

A sensitive diagnostic designed by Prof. Fonck measured alpha particle density in the plasma and showed signs that MHD could re-distribute the alphas – Current topic of research for burning plasmas is the worry that energetic particle driven modes will instability (TAE modes).

NEEP 536, Spring 2003, D. Whyte

9

Remember from first lectures: The He ash is doubly detrimental to fusion: yet another criterion is set for fusion…keep the fuel clean! neτ e =

12kT 2

(1− 2 f He ) σv E α – (1+ 2 f He )2.14 ×10−36T1/ 2

ne τe (s /m3)

1E+22 fHe = 0 fHe = 10% fHe = 20% fHe=30% 1E+21

1E+20 1

10

100

Te (keV) NEEP 536, Spring 2003, D. Whyte

10

The helium ash, which also provides fusion heating, must be exhausted efficiently •

Consider that the creation rate of the helium is directly linked to the fusion reaction itself.

He /s / m-3

* nHe ~ nD nT σv fusion · τ He

– So we assign an effective ash particle confinement time: τHe*.

•

Of course the fusion rate and the energy confinement time, τe, are directly linked to our ability to heat the plasma through He++ slowing down on electrons and maintain DT burn conditions.

s

τ

*He

=

n He n D n T σv fusion

n Dn T σv fusion ⋅ kEα ~

3nkT τe

– Ignoring bremsstrahlung losses.

• •

So we can see that to not jeopardize the ignition criterion too much, fHe<10%, the ratio of τHe*/ τe ~< 10.

This means that we will have to worry about “pumping” away helium at some point.

NEEP 536, Spring 2003, D. Whyte

n E E * τ He ~ He α ~ f He α ~ f He100 3T n 3T τe * τ He < 15 τe

Reactor requirement 11

•

Since MHD fixes the pressure (nT) allowed in the plasma then their exists a most efficient T --> 1/n for the fusion power – Pfusion ~ n2 <σv>(T)

•

•

Are there any restriction on the density we can obtain?

fusion power rate

One further important limit for the tokamak MFE concept: The density limit

0

10

20

30

T (keV) ~

40 1/n

50

60

Empirically the answer is yes on for tokamaks (with similar limits for other concepts) – If one tries to raise density arbitrarily high, a hard, disruptive limit is found

NEEP 536, Spring 2003, D. Whyte

12

One further important limit for the tokamak MFE concept: The density limit •

The empirical scaling for this density limit is known as the Greenwald limit:

nGreenwald =

Ip

πa 2

– With n in 1020 m-3, Ip in MA and a in m.

•

•

This scaling works convincingly well over a wide variety of plasma shapes, etc.

There remains no first-principles physics principle to explain this scaling.

NEEP 536, Spring 2003, D. Whyte

Ip

B ∝ nGreenwald = πa2 qcyl R

Compact high-field Tokamak (e.g. Alactor) Achieve high densities 13

We have now laid out the criteria for our MFE fusion reactor: stability, confinement, alpha heating and density limit. So what will this thing look like? One example is ITER

NEEP 536, Spring 2003, D. Whyte

14

We now look at techniques that can be used to heat up the plasma to T~10 keV… ”lighting the fusion match” •

Inductive or ohmic heating – Use finite resistance and large currents to heat plasma.

•

Energetic neutral beams – High energy neutrals will penetrate through plasma without feeling effect of magnetic field.

•

Radio-frequency (RF) heating – Exploit EM properties of the plasma and launch RF waves into them for heating.

– Wide range of frequencies due to multiple resonances in plasma.

NEEP 536, Spring 2003, D. Whyte

15

Ohmic heating is insufficient to heat standard tokamaks to fusion temperatures.

•

Our derivation of the Coulomb scattering gave us the plasma resistivity

•

So for a given current density (Ip/πa2) through the plasma cross section we get “ohmic” resistive heating of the plasma.

–

–

•

Re-arranging this in terms of the stability limit, q, shows that ohmic heating saturates at high Te. ..and ohmic heating is most efficient in high field, small tokamaks (e.g. Alcator C-Mod at MIT).

However the T-3/2 scaling makes ohmic heating impractical for fusion temperatures – –

Example of ITER D-T burn…ohmic heating x50 too low.

General rule of thumb is that ohmic heating only efficient up to ~ 1 keV •

With RFP and high-field compact tokamaks somewhat of exception

NEEP 536, Spring 2003, D. Whyte

η ≈ 5 ×10−5

W/m3

Zeff ln Λ

Ohm m

T 3/ 2

Ip in A, a in m, Te in keV

Pohmic ~ η ⋅ j 2 ≅

2 −9 3 ×10 Z eff I p

Z Pohmic ∝ eff 1 T e3/ 2 q95

a4T e3/ 2 2

B 2 T R

ITER MJ

3nkT 0.4 m−3 kW ~ ~ 100 −3 Pohmic > Ptransport ≡ m 3.5s τe

kW 3 ×10−9 (1.5)(15 ×106 ) ~ 2.5 Pohmic ~ 248.53 / 2 m−3 2

16

Energetic neutral beam heating: basic principles •

•

•

•

•

Proven technology for making ion beams with E>100 kV….but these cannot “penetrate” through magnetic field to plasma.

Solution: create an energetic, hydrogenic neutral beam of that is not affected by the magnetic field.

The energetic H neutral will eventually ionize through collisions with the plasma ions and electrons.

The resultant “hot” ion will now be confined to the magnetic field and heat the plasma through collisions.

What we need:

B ◉

Energetic neutral beam ( E >50 keV)

plasma

B ◉

Pbeam ~ 2-5 MW

Ionized beam particle Is confined by magnetic Field and heats plasma Via collisions

plasma

– Penetration of the beam into the central portion of the plasma.

NEEP 536, Spring 2003, D. Whyte

17

Energetic neutral beam heating: hardware

NEEP 536, Spring 2003, D. Whyte

•

Neutral beams have been the principal means of heating tokamak plasmas.

•

However their complexity, size and cost rival or surpass that of the actual tokamak plasma system.

18

Neutral beams are limited in their penetration into the plasma by ionization and charge-exchange losses •

As the neutral beam, Ho, encounter plasma ions it undergoes –

•

Charge exchange: Ho + H+ --> H+ + Ho Impact (and e-) ionization: Ho + H+ --> H+ + H+ + e-

So the beam flux, Ibeam decreases as

dIb ~ n(σCX + σ ion )Ib dr

I b (r) ≡ I b(a) ⋅ exp −

λ≡

(r − a) λ

Charge exchange

6.E-16 5.E-16 4.E-16

proton impact ionization

3.E-16

total

2.E-16 1.E-16 0.E+00

0.0E+00

5.0E+04

1.0E+05

1.5E+05

2.0E+05

Beam neutral H energy (eV)

1 n ⋅ σ (E)

Energetic neutral beam penetration into ITER 1.2 1

Including the enhanced stopping effect of impurities we get the the beam penetration length, λ, to be 17 –

So penetration into λ= ITER core is challenging for Eb<200 keV.

5 ×10 Ebeam (keV) M ⋅ n ⋅ Z eff

λ/a

•

Cross section (cm2)

–

Ionization processes for energetic H beams

0.8 0.6 0.4 0.2 0 0

200

400

600

800

1000

Beam energy (keV)

NEEP 536, Spring 2003, D. Whyte

19

A recent technological advance has been the development of “negative” ion sources for neutral beam heating.

•

Positive ions (H+, H +, H +) made in 2

Neutralization efficiency vs. beam energy

3

standard plasma sources experience a sharp decline in their neutralization efficiency for E ~> 100 keV – Ionization dominates over CX in the neutralizer cell (see previous slide) – Neutralization efficiency is ~0 at 1 MeV.

•

This is avoided by producing negative H- ions and sending them through the neutralizer gas.

– The weakly bound extra electron (Eb,H~ 0.5 eV) is easily removed by collisions to make a neutral.

•

The technology of producing ~10 A, MeV H- is quite demanding, but progress is being made.

NEEP 536, Spring 2003, D. Whyte

20

Some example of negative-ion source development for ITER. IPP-Germany

JAERI- Japan

NEEP 536, Spring 2003, D. Whyte

21

We have now laid out the criteria for our MFE fusion reactor: stability, confinement, alpha heating and density limit. So what will this thing look like? One example is ITER

NEEP 536, Spring 2003, D. Whyte

1

We now look at techniques that can be used to heat up the plasma to T~10 keV… ”lighting the fusion match” •

Inductive or ohmic heating – Use finite resistance and large currents to heat plasma.

•

Energetic neutral beams – High energy neutrals will penetrate through plasma without feeling effect of magnetic field.

•

Radio-frequency (RF) heating – Exploit EM properties of the plasma and launch RF waves into them for heating.

– Wide range of frequencies due to multiple resonances in plasma.

NEEP 536, Spring 2003, D. Whyte

2

Ohmic heating is insufficient to heat standard tokamaks to fusion temperatures.

•

Our derivation of the Coulomb scattering gave us the plasma resistivity

•

So for a given current density (Ip/πa2) through the plasma cross section we get “ohmic” resistive heating of the plasma.

–

–

•

Re-arranging this in terms of the stability limit, q, shows that ohmic heating saturates at high Te. ..and ohmic heating is most efficient in high field, small tokamaks (e.g. Alcator C-Mod at MIT).

However the T-3/2 scaling makes ohmic heating impractical for fusion temperatures – –

Example of ITER D-T burn…ohmic heating x50 too low.

General rule of thumb is that ohmic heating only efficient up to ~ 1 keV •

With RFP and high-field compact tokamaks somewhat of exception

NEEP 536, Spring 2003, D. Whyte

η ≈ 5 ×10−5

W/m3

Zeff ln Λ

Ohm m

T 3/ 2

Ip in A, a in m, Te in keV

Pohmic ~ η ⋅ j 2 ≅

2 −9 3 ×10 Z eff I p

Z Pohmic ∝ eff 1 T e3/ 2 q95

a4T e3/ 2 2

B 2 T R

ITER MJ

3nkT 0.4 m−3 kW ~ ~ 100 −3 Pohmic > Ptransport ≡ m 3.5s τe

kW 3 ×10−9 (1.5)(15 ×106 ) ~ 2.5 Pohmic ~ 248.53 / 2 m−3 2

3

Energetic neutral beam heating: basic principles •

•

•

•

•

Proven technology for making ion beams with E>100 kV….but these cannot “penetrate” through magnetic field to plasma.

Solution: create an energetic, hydrogenic neutral beam of that is not affected by the magnetic field.

The energetic H neutral will eventually ionize through collisions with the plasma ions and electrons.

The resultant “hot” ion will now be confined to the magnetic field and heat the plasma through collisions.

What we need:

B ◉

Energetic neutral beam ( E >50 keV)

plasma

B ◉

Pbeam ~ 2-5 MW

Ionized beam particle Is confined by magnetic Field and heats plasma Via collisions

plasma

– Penetration of the beam into the central portion of the plasma.

NEEP 536, Spring 2003, D. Whyte

4

Energetic neutral beam heating: hardware

NEEP 536, Spring 2003, D. Whyte

•

Neutral beams have been the principal means of heating tokamak plasmas.

•

However their complexity, size and cost rival or surpass that of the actual tokamak plasma system.

5

Neutral beams are limited in their penetration into the plasma by ionization and charge-exchange losses •

As the neutral beam, Ho, encounter plasma ions it undergoes –

•

Charge exchange: Ho + H+ --> H+ + Ho Impact (and e-) ionization: Ho + H+ --> H+ + H+ + e-

So the beam flux, Ibeam decreases as

dIb ~ n(σCX + σ ion )Ib dr

I b (r) ≡ I b (a) ⋅ exp −

λ≡

(r − a) λ

Charge exchange

6.E-16 5.E-16 4.E-16

proton impact ionization

3.E-16

total

2.E-16 1.E-16 0.E+00

0.0E+00

5.0E+04

1.0E+05

1.5E+05

2.0E+05

Beam neutral H energy (eV)

1 n ⋅ σ (E)

Energetic neutral beam penetration into ITER 1.2 1

Including the enhanced stopping effect of impurities we get the the beam penetration length, λ, to be 17 –

So penetration into λ= ITER core is challenging for Eb<200 keV.

5 ×10 Ebeam (keV) M ⋅ n ⋅ Z eff

λ/a

•

Cross section (cm2)

–

Ionization processes for energetic H beams

0.8 0.6 0.4 0.2 0 0

200

400

600

800

1000

Beam energy (keV)

NEEP 536, Spring 2003, D. Whyte

6

A recent technological advance has been the development of “negative” ion sources for neutral beam heating.

•

Positive ions (H+, H +, H +) made in 2

Neutralization efficiency vs. beam energy

3

standard plasma sources experience a sharp decline in their neutralization efficiency for E ~> 100 keV – Ionization dominates over CX in the neutralizer cell (see previous slide) – Neutralization efficiency is ~0 at 1 MeV.

•

This is avoided by producing negative H- ions and sending them through the neutralizer gas.

– The weakly bound extra electron (Eb,H~ 0.5 eV) is easily removed by collisions to make a neutral.

•

The technology of producing ~10 A, MeV H- is quite demanding, but progress is being made.

NEEP 536, Spring 2003, D. Whyte

7

Some example of negative-ion source development for ITER. IPP-Germany

JAERI- Japan

NEEP 536, Spring 2003, D. Whyte

8

So how much heating power do we need? For an ITER scale device ~ 50-100 MW external power •

The power input has to offset the power losses from transport and radiation.

•

Based on the confinement scaling we know Ptransport ~ 100 kW/m3.

•

We previously derived the formula for bremsstrahlung radiation, which is the minimum radiative loss: Pbremsstranlung ~ 20 kW/m3.

•

So P ~> 120 kW/m3 --> total power ~ 100 MW needed (without fusion alpha heating) –

•

~ 100 Amperes of 1 MeV neutral beam!

Pneutral−beam > Ptransport + Pbremsstrahlung

Pbremsstrahlung (W / m3 ) = 5.3 ×10 −37 ne2 Zeff Te1/ 2 (keV )

E ~ 3nk(∆T) ⋅Vplasma ~ 400MJ

How about the energy needed to raise the plasma from 0 --> 10 keV? –

Not so large…e.g. with extra 50 MW injected power will take 8 seconds to raise plasma to 10 keV.

NEEP 536, Spring 2003, D. Whyte

9

RF heating: Review of EM dispersion •

•

–

r r r r ∂v nm = ne( E + v × B) ∂t r j ≡ e(nivi − n e ve )

Parallel to B NEEP 536, Spring 2003, D. Whyte

x:

z:

For electrons and ions

r r r x(r,t) = xˆ ⋅ exp(i(k ⋅ r − ωt)) ne2 2 ωp ≡ εo m

k : propagation vector wavenumber ω: traveling wave frequency.

Perp. To B

r r ∂B ∇× E =− ∂t

r r r 1 ∂E ∇× B = µo j + 2 c ∂t

We previously derived a simple dispersion relationship regarding EM wave propagation / cutoff near the plasma frequency.

In general for EM dispersion a perturbed solution for a cold, plasma in equilibrium (Eo~jo~vo~0 Bz=constant) can be arrived at by linearizing Faraday’s and Ampere’s law, momentum balance and seeking a solution to the plane wave:

Plasma frequency

eBz,0 Ω ≡ m 2

2

Cyclotron frequency

2 2 2 2 ω pi 2Ω iω ω pi 2Ω eω r ω ω pe ω r pi ω 2 2 2 Ex + i 2 Ey = 0 + + c k −ω + 2 ω − Ω 2i ω 2 − Ω2e ω − Ω2i ω 2 − Ω2e

r (−ω + ω + ω ) E z = 0 2

2 pi

2 pe

10

RF heating: Review of EM dispersion •

•

•

•

As before we see the plasma can resonate at near the electron plasma frequency with no constraint on k.

More interesting solutions are arrived at by making the determinant of the solutions in x and y vanish: this is written as. (for waves propagating // to B) So when the frequency approaches the ion or electron cyclotron frequency the propagation vector k goes to infinity --> resonance!!

I.e. externally generation EM waves with these frequencies readily drive plasma motion --> heating!!

ω ≡ p2

ne2

εo m 2 pi

Cyclotron frequency

2

r (−ω + ω + ω ) E z = 0 2

ω2 = ω2 + ω2 ≈ ω2

2 pe

pi

2

pe

pe

2

ω ω ω ω c k − ω + pi + pe =0 ω ± Ω i ω ±Ω e 2 2

2

ITER: ω p,e ≡ Ωe≡

ne2

εom

1/2

1020(1.6e −19)2 ~ 9e −12 ⋅ 9e − 31

1/2

~ 5.6 ×1011rad / s ~ 100 GHz

eB 1.6e −19 ⋅ 5.3(Tesla) ~ ~ 9.42 ×10 11 rad / s ~ 150 GHz me 9e − 31 Ωi ≡ Ωi

NEEP 536, Spring 2003, D. Whyte

eB z,0 2 Ω ≡ m

Plasma frequency

me ~ 40 MHz mi 11

The general goal of RF heating is to exploit the various resonances in the plasma for heating •

The three main resonances of interest are

ω 2 = ΩeΩi

– electron cyclotron heating (ECH) f ~ 100 GHz – ion cyclotron heating (ICRF) f ~ 50 MHz – Lower hybrid (LH) f ~ 2 GHz

•

Lower Hybrid Frequency

building

RF has several attractive features: – Variety of resonances and frequencies to choose from with various advantages to each.

– Some technology commercially developed.

– Remote RF generator can bring heating to plasma by ~ loss-free waveguides – Modular design.

NEEP 536, Spring 2003, D. Whyte

waveguide plasma

RF generator

Antannae or launcher Vacuum vessel 12

ECH provides a pinpoint means of heating the plasma due to its sharp resonance and it can propagate easily through plasma above plasma frequency •

Can use fundamental or harmonics of the electron cyclotron frequency.

•

Quasi-optical waveguides sufficient.

•

Since the waves propagate in vacuum, antannae are simple copper mirrors and can be far from the plasma.

•

Steerable launcher moves flux surface location of resonance and thus provides means to control profile of heating.

NEEP 536, Spring 2003, D. Whyte

13

Great challenge in ECH is development of reliable RF generators in 100 GHz range

•

RF generator in 100 GHz range: gyrotron – Exploits e- resonance in magnetic field + resonant cavity.

– Superconducing magnet technology necessary since Bgyrotron= n Btokamak.

– Arcing, non-resonant modes and power handling a challenge.

NEEP 536, Spring 2003, D. Whyte

14

Recent technological advance has bee the development of pure diamond windows for interface between gryotron and waveguides •

CVD (chemical vapor deposition) diamond windows

•

< 0.2% power absorption in 100 GHZ range.

•

High thermal conductivity.

•

Transmission of ~1 MW --> 100 MW/m2 in steady state through windows recently demonstrated.

NEEP 536, Spring 2003, D. Whyte

15

Several tokamaks are currently expanding their ECH heating systems to demonstrate technology and science needed for ITER & beyond.

Schematic of DIII-D tokamak ECH system NEEP 536, Spring 2003, D. Whyte

16

RF heating: Review of EM dispersion •

•

•

•

As before we see the plasma can resonate at near the electron plasma frequency with no constraint on k.

More interesting solutions are arrived at by making the determinant of the solutions in x and y vanish: this is written as. (for waves propagating // to B) So when the frequency approaches the ion or electron cyclotron frequency the propagation vector k goes to infinity --> resonance!!

I.e. externally generation EM waves with these frequencies readily drive plasma motion --> heating!!

ω ≡ p2

ne2

εo m

2

r (−ω + ω + ω ) E z = 0 2

2 pi

eB z,0 m

2

Cyclotron frequency

ω2 = ω2 + ω2 ≈ ω2

2 pe

pi

2

pe

pe

2

ω ω ω ω c k − ω + pi + pe =0 ω ±Ωi ω ±Ωe 2 2

2

ITER: ω p,e ≡ Ωe≡

ne2

εom

1/2

~

1020(1.6e −19)2 9e −12 ⋅ 9e − 31

1/2

~ 5.6 ×1011rad / s ~ 100 GHz

eB 1.6e −19 ⋅ 5.3(Tesla) ~ ~ 9.42 ×10 11 rad / s ~ 150 GHz me 9e − 31 Ωi ≡ Ωi

NEEP 536, Spring 2003, D. Whyte

Ω ≡

Plasma frequency

me ~ 40 MHz mi 1

The general goal of RF heating is to exploit the various resonances in the plasma for heating •

The three main resonances of interest are

ω 2 = ΩeΩi

– electron cyclotron heating (ECH) f ~ 100 GHz – ion cyclotron heating (ICRF) f ~ 50 MHz – Lower hybrid (LH) f ~ 2 GHz

•

Lower Hybrid Frequency

building

RF has several attractive features: – Variety of resonances and frequencies to choose from with various advantages to each.

– Some technology commercially developed.

– Remote RF generator can bring heating to plasma by ~ loss-free waveguides – Modular design.

NEEP 536, Spring 2003, D. Whyte

waveguide plasma

RF generator

Antannae or launcher

Vacuum vessel 2

ECH provides a pinpoint means of heating the plasma due to its sharp resonance and it can propagate easily through plasma above plasma frequency •

Can use fundamental or harmonics of the electron cyclotron frequency.

•

Quasi-optical waveguides sufficient.

•

Since the waves propagate in vacuum, antannae are simple copper mirrors and can be far from the plasma.

•

Steerable launcher moves flux surface location of resonance and thus provides means to control profile of heating.

NEEP 536, Spring 2003, D. Whyte

3

Here is an example from the DIII-D tokamak of ECH being used for controlled localized heating at a specific radial location

Also notice How the heat initially Deposited in a single location Diffuses radially Through the plasma --> Energy confinement!

NEEP 536, Spring 2003, D. Whyte

4

Great challenge in ECH is development of reliable RF generators in 100 GHz range

•

RF generator in 100 GHz range: gyrotron – Exploits e- resonance in magnetic field + resonant cavity.

– Superconducing magnet technology necessary since Bgyrotron= n Btokamak.

– Arcing, non-resonant modes and power handling a challenge.

NEEP 536, Spring 2003, D. Whyte

5

Recent technological advance has bee the development of pure diamond windows for interface between gryotron and waveguides •

CVD (chemical vapor deposition) diamond windows

•

< 0.2% power absorption in 100 GHZ range.

•

High thermal conductivity.

•

Transmission of ~1 MW --> 100 MW/m2 in steady state through windows recently demonstrated.

NEEP 536, Spring 2003, D. Whyte

6

Several tokamaks are currently expanding their ECH heating systems to demonstrate technology and science needed for ITER & beyond.

Schematic of DIII-D tokamak ECH system NEEP 536, Spring 2003, D. Whyte

7

Ion cyclotron heating: simpler RF generation, more difficult wave propagation.

•

RF frequencies in the 10’s MHz range (at harmonics of the ion cyclotron frequency).

– More standard cw, RF technology --> MW range / generator commercially available.

– Coaxial transmission lines

•

Wavelength of the order or larger than the plasma minor radius.

– Vacuum cutoff occurs in periphery of plasma….therefore complicated propagation and mode conversions happen for these waves as they travel through plasma – Also implies more complicated antennae structure that must couple to the edge plasma--> reliability issues with thermonuclear plasma

NEEP 536, Spring 2003, D. Whyte

More direct control of ion thermal distribution: Important since ions are the fusing particles

8

Propagation and absorption of ICRF is extremely complicated but is now handled by large-scale computing.

NEEP 536, Spring 2003, D. Whyte

9

ICRF technology

NEEP 536, Spring 2003, D. Whyte

10

ICRF typically relies on 2nd harmonic heating or minority species ion heating

NEEP 536, Spring 2003, D. Whyte

11

Successful examples of ICRF heating in tokamaks

NEEP 536, Spring 2003, D. Whyte

12

Lower hybrid heating •

Intermediate frequency between electron and ion cyclotron frequencies.

– Less demanding source technology than ECH.

•

Like ICRF, waves must propagate through plasmas – More complicated propagation scheme than ECH. – Further involves plasma-antannae coupling issues.

•

Affects electrons: – Typically looked to for as a means of current drive without ultra-high frequencies of ECH.

NEEP 536, Spring 2003, D. Whyte

13

LH hardware

Antennae NEEP 536, Spring 2003, D. Whyte

RF generator: ~ 1 MW, cw 14

Example from current tokamaks of antannae structure, wave propagation and heating.

NEEP 536, Spring 2003, D. Whyte

15

Steady-state magnetic confinement fusion •

•

•

We have now examined the “global” system issues of fusion: stability, confinement and heating…we now turn to specifics regarding the spatial and temporal profiles.

Easily the biggest technological and physics “step” to be made from current experiments to realistic burning plasmas --> reactor is the issue of plasma/burn duration.

Compare: >> to typical plasma timescales (ITER) – – – –

•

Ideal MHD: τ ~ 10-5 s Energy confinement: τ ~ 101 s Current redistribution (a.k.a. skin time: ~ 200 s.

L/R time ~10 x skin time

Therefore a reactor will be in a steadystate never truly obtained by today’s experiments.

Reactor

1 year 1 week 1 hr

ITER

1s 1 ms 1960 η ≈ 5 ×10−5

2000

Zeff ln Λ ≈ 3 ×10−8 ohm − m at T = 103eV T 3/ 2

Ltorus ~ µoR ⋅ ln(8 Rresistan ce ~ η

NEEP 536, Spring 2003, D. Whyte

1980

R − 2) ~ 4 ×10−6 ⋅ R Henry a

2

ε⋅a

Ohm 1

Main issues regarding steady-state operation •

Maintaining confining magnetic fields – High B required (> 5 T) --> high coil current --> pulse limit for normal conducting coil due to resistive heating.

– Answer: cw super-conducting coils (covered later in technology)

•

Maintain plasma current – For tokamak configuration inductive current limited by flux swing.

– Answer(s) • Develop means to sustain current without central solenoid --> “non-inductive” current drive • Drop tokamaks and use stellarator configuration that does not require toroidal current for plasma confinement.

•

Particle and heat exhaust.

– – – –

Endure neutron and plasma heat flux ~ 5-10 MW/m2 for > 1 year.

Material erosion.

Helium exhaust.

Partial Answer: • Divertor magnetic geometry for He pumping….others??

NEEP 536, Spring 2003, D. Whyte

2

Examine pulse duration possible using solenoid and inductive current drive.

• •

Take case of ITER (I=15 MA) Flux swing denotes the total (Webers: V-s) capacity of the solenoid to inductively drive current in the plasma (limited by maximum I --> B in the superconducting coil) –

•

Gives duration for a given loop voltage in the plasma

Plasma e.m.f. Vloop~ Ip R –

– –

R ∝ T-3/2 so the majority of the flux swing is consumed in starting the (cold) plasma and ramping up the current to 15 MA At 15 MA, heated condition (Q~10), Vloop ~ 50 mV Inductive current duration at D-T burn: ~ 30 V-s / 0.05 V ~ 600 s.

NEEP 536, Spring 2003, D. Whyte

3

Development of ITER central solenoid

NEEP 536, Spring 2003, D. Whyte

4

The scientific and technological basis for the use of cw noninductive current drive is being developed for tokamaks •

Note that the power needed to sustain the current is small: – ~ IV ~ 107 A x 0.1 V ~ 1 MW << Pfusion.

– Remember current negligible for heating.

•

Two principle means of non-inductive current sustainment.

1.

Apply external power / heating that drives particle distribution anisotropy --> toroidal current.

2.

Exploit aspect of neo-classical transport that allows the plasma profiles to self-generate or “bootstrap” its own current.

NEEP 536, Spring 2003, D. Whyte

In general, this progression toward directly controlling profiles for obtaining desired effects is called “Advanced Tokamak” or AT

5

Neo-classical transport: trapped particle orbits, “banana” orbits •

•

“Mirror” point

As particles follow field lines poloidally around they undergo a // acceleration towards region of lower B -> outboard.

Conservation of magnetic moment, total kinetic energy and toroidal angular momentum.

1 2 µB + mv// = K.E. = const.

2

µB ≡ const.

R

R( mv toroidal + eAtoroidal ) = p = const.

• • •

Where A is the magnetic vector potential linked to poloidal magnetic flux: So using the usual conventions for the B and flux structure the conservations can be written So when the particle reflects and has “negative” // direction, it is moved to an inner surface. In the limit of this width << a we find:

NEEP 536, Spring 2003, D. Whyte

∆b Orbit width

r r Bθ = ∇ × Atoroidal

r2Bo Ψθ = −RAtoroidal ≡ 2q

1

Ωr2

µB

2

o

1+ ε cosθ + mv// = const.

2

8q2 v ∆b = ε Ω

v// − 2qRo = const.

6

The trapped particle orbits resulting from neoclassical transport lead to self-generated “bootstrap” current • •

•

•

We examine projection of trapped electron orbits in a common test volume.

Because there are more electrons radially inboard (density highest at centre) there are more trapped epassing through // to current than against it.

This net current appears as a distortion in the distribution function which couples into passing particles and amplifies the effect.

The full theory shows that the bootstrap current is driven by the radial pressure gradient

(∇p) r = jbs × Bθ NEEP 536, Spring 2003, D. Whyte

7

Tokamaks have demonstrated high fraction bootstrap current discharges •

The existence of bootstrap occurred when sufficient heating made large pressure gradient in TFTR and diagnostics advanced to extract accurately the contribution of the inductive and energetic-beam driven current.

NEEP 536, Spring 2003, D. Whyte

8

The JT-60U tokamak in Japan recently demonstrated stationary 80% bootstrap current discharges

NEEP 536, Spring 2003, D. Whyte

9

Energetic neutral beams can drive current •

Beams directed in a toroidal direction tend to have a directed flow of fast ions.

– –

•

In general the efficiency of the neutral beam current drive increases/decreases with plasma temperature/density is proportional to: ICD/Pnb∝ (Te/ne ) ∝ (tslowing down/v) –

•

Opposed by e- collisions with background plasma.

Effect of neo-classical trapped electrons allows net current drive for H beams.

Although this saturates at high T for fixed beam energy .

Higher energy beams (negative-ion) needed for CD efficiency at ITER temperatures have been verified.

NEEP 536, Spring 2003, D. Whyte

ηCD =

ICD nR Current drive efficiency P

10

An example of full non-inductive current drive from JT-60U in Japan

•

Note the very large amount of beam power (~ 20 MW) needed in order to obtain ~ 1 MA. –

For ITER: CD efficiency --> ICD (A) ~ 0.04 P (W) I ~ 15 MA --> 300 MW beam power ~ Pfusion

NEEP 536, Spring 2003, D. Whyte

11

Electron cyclotron heating can also be used to drive current --> great advantage is localization

NEEP 536, Spring 2003, D. Whyte

12

EC Current drive demonstrated

Like NB current drive the efficiency is proportional to (T/n) Looking to ITER: IECCD (A) ~ 1% PECCD (W)

NEEP 536, Spring 2003, D. Whyte

13

Lower hybrid RF is also a means of producing non-inductive current •

Lower hybrid has the highest current drive efficiency of the RF techniques.

•

For ITER: ~ 0.3 (1020 A/m2/W)

•

…but like other techniques is too “power” inefficient to drive feasible current by itself.

NEEP 536, Spring 2003, D. Whyte

14

In the end the key to successfully sustaining noninductive current will be profile control •

As we have seen, the externally driven non-inductive current techniques are simply too inefficient by themselves: Must rely on bootstrap current.

•

Since the bootstrap current is driven by local pressure gradients – Depends strongly on ability to create pressure gradients through heating and confinement.

– But the current profile itself strongly affects confinement.

•

So we have a case where we must try to optimize complex, non-linear set of profiles (q, p, Ψ) – This is at the heart of current Advanced Tokamak research • Detectors: real-time q, flux equilibrium, pressure • Actuators: spatially controlled heating and CD (ECH, LH, etc.)

NEEP 536, Spring 2003, D. Whyte

15

An example of full non-inductive current drive from JT-60U in Japan

•

Note the very large amount of beam power (~ 20 MW) needed in order to obtain ~ 1 MA. –

For ITER: CD efficiency --> ICD (A) ~ 0.04 P (W) I ~ 15 MA --> 300 MW beam power ~ Pfusion

NEEP 536, Spring 2003, D. Whyte

11

Electron cyclotron heating can also be used to drive current --> great advantage is localization

NEEP 536, Spring 2003, D. Whyte

12

EC Current drive demonstrated

Like NB current drive the efficiency is proportional to (T/n) Looking to ITER: IECCD (A) ~ 1% PECCD (W)

NEEP 536, Spring 2003, D. Whyte

13

Lower hybrid RF is also a means of producing non-inductive current •

Lower hybrid has the highest current drive efficiency of the RF techniques.

•

For ITER: ~ 0.3 (1020 A/m2/W)

•

…but like other techniques is too “power” inefficient to drive feasible current by itself.

NEEP 536, Spring 2003, D. Whyte

14

In the end the key to successfully sustaining noninductive current will be profile control •

As we have seen, the externally driven non-inductive current techniques are simply too inefficient by themselves: Must rely on bootstrap current.

•

Since the bootstrap current is driven by local pressure gradients – Depends strongly on ability to create pressure gradients through heating and confinement.

– But the current profile itself strongly affects confinement.

•

So we have a case where we must try to optimize complex, non-linear set of profiles (q, p, Ψ) – This is at the heart of current Advanced Tokamak research • Detectors: real-time q, flux equilibrium, pressure • Actuators: spatially controlled heating and CD (ECH, LH, etc.)

NEEP 536, Spring 2003, D. Whyte

15

The fusion plasma “core” must reside within a vacuum chamber •

Let’s say we want to start ITER up: – Start with thermal D-T gas which will be ionized and formed into plasma.

– The final desired particle density: 2 xnD2 ~ ne ~ 1020 m-3.

– Air density at STP: n = 2.5x1025 m-3.

– So the starting gas pressure in the vessel: ~ 10-6 atmospheres.

•

In reality we will need P < 10-10 atmospheres since air (N, O) would constitute avoidable, undesirable impurities in the plasma.

– These pressures are obtainable with UHV technology (stainless steel vacuum vessel, turbo or cryo pumping) – But a tedious task for a 1000 m3 plasma volume!

– Looking to a reactor --> vacuum integrity must be maintained ~ 1 year.

NEEP 536, Spring 2003, D. Whyte

16

The boundary of the plasma is defined by the presence of solid surfaces Vacuum vessel

•

What determines the plasma minor radius in the simplest configuration?

The vacuum vessel Why?: Because solid surfaces are sinks for a plasma.

plasma a

– Even most energetic fusion particle will only penetrate through 1 micron of solid material – Plasma (e- and ions) rapidly recombine in the solid 2(e- + D+) --> D2 …plasma is lost. e-

D+

Solid surface

NEEP 536, Spring 2003, D. Whyte

D2

D2

17

In practice, the vital vacuum vessel does not undergo direct plasma contact --> limiter Vacuum vessel

•

•

The limiter is specially designed surface deliberately inserted to keep the very hot plasma away from the vacuum vessel.

Because the limiter is a plasma sink – No rotational transform outside the limiter touching flux surface --> lacks confinement.

– Limiter directly intercepts field lines carrying plasma particles and heat.

– Plasma is “drained away” by the limiter.

• Small effective area surfaces can easily deplete cw plasma transported energy / power.

NEEP 536, Spring 2003, D. Whyte

Confined plasma a

Scrape-off layer

limiter 1 1 20 −3 5 m ~ 1025 m−2s−1 Γ ≡ nvi ~ 10 m ~ 4 ×10 4 4 s T =1keV

q ≡ 3kT ⋅ Γ ~ 3(1.6 ×10−19)103eV ⋅1025 ~ 5 ×109 //

W m2

18

Equilibrium and shaping coils are used to stabilize and define the plasma shape •

X

We already worked out that tokamaks need a stabilizing vertical field – This balances out the radial force imbalance caused by the interaction of the poloidal field with the current that generates the field.

– We must produce this field with a set of toroidally extended current carrying coils as shown, which will produce an ~ vertical BE • Produces equilibrating inward force on plasma.

Ip ◉

BE

Equilibrium Field coils

X

– These are called “poloidal field” coils since they produce poloidal magnetic fields (no toroidal field) NEEP 536, Spring 2003, D. Whyte

19

Equilibrium and shaping coils are used to stabilize and define the plasma shape •

•

•

We had previously discussed the benefits of shaping the plasma away from a circle in order to gain stability.

An example of how to obtain elongated plasma is shown. The toroidal current in the coils acts to decrease the poloidal magnetic field at the top and bottom of the plasma column.

Since: Flux ~ Bpol. * Area, the flux surface (isobaric) elongates at the top and bottom.

– Can also view this as a Ip x Br,coil force acting on each half of plasma.

NEEP 536, Spring 2003, D. Whyte

◉

Ip ◉

Poloidal Magnetic fields from Plasma and coil

Plasma Shaping Coils

◉

Essentially the shaping coils push and pull The flux surfaces around (depending on Sign of coil to Ip 20

Equilibrium and shaping coils are used to stabilize and define the plasma shape •

In a similar fashion we can construct a triangular or D-shaped plasma by placing shaping coils on the inboard and outboard side of the plasma.

Plasma Shaping Coils

X Ip ◉

◉

X

NEEP 536, Spring 2003, D. Whyte

21

With sufficient coil current we can create a poloidal field null --> diverted magnetic geometry •

If the coil current becomes large enough then there will be a location of zero poloidal field. The flux surface diverge at this position: X-point.

– Note that the coil current can be much less than the plasma current if the position of the coil is closer to the Xpoint.

•

•

There also now exists a “Last-Closed Flux Surface” defined by the X-point.

This separates regions with and without rotational transform and is called the ‘separatrix’.

The limiter is now replaced by the regions where the separatrix touches the wall. This region is called the divertor.

NEEP 536, Spring 2003, D. Whyte

◉

Ip ◉

X-points: Locations of Zero poloidal field

Plasma Shaping Coils

◉

Essentially the shaping coils push and pull The flux surfaces around (depending on Sign of coil to Ip 22

Example of shaped plasma from DIII-D •

Note the position of the vacuum vessel and shaping coils (values indicate A/turn in coils).

•

Plasma is elongated and slightly triangular, with an X-point on one side only.

•

The plasma flux lines are determined by a large non-linear fitter that solves the poloidal flux equilibrium based on force balance and Ampere’s law – Fit is based on the various magnetic and flux loop diagnostics (blue squares, purple line) placed around the vessel

NEEP 536, Spring 2003, D. Whyte

23

Inductive loops positioned around the plasma in various geometries provide real-time diagnosis of flux equilibrium.

•

Example of typical coils: – Rogowski coil measures Vemf ~ dI/dt through the conductor.

– Poloidal flux loop: measured loop voltage at region outside plasma.

•

Rogowski coil

All of these measurements rely on analog integrators – Rogowski: Ip = ∫ Vemf dt – Pol. flux loop: ψ = ∫ Vloop dt

NEEP 536, Spring 2003, D. Whyte

Surface loop voltage / poloidal flux loop

24

Sophisticated non-linear control systems and realtime equilibrium constructions allow for extremely accurate real-time control of the plasma shape.

•

Long-term stability /operational viablity of shaping/equilibrium coils and diagnostic coil will be vital for reactor tokamak operations.

NEEP 536, Spring 2003, D. Whyte

25

In practice, the vital vacuum vessel does not undergo direct plasma contact --> limiter Vacuum vessel

•

•

The limiter is specially designed surface deliberately inserted to keep the very hot plasma away from the vacuum vessel.

Because the limiter is a plasma sink – No rotational transform outside the limiter touching flux surface --> lacks confinement.

– Limiter directly intercepts field lines carrying plasma particles and heat.

– Plasma is “drained away” by the limiter.

• Small effective area surfaces can easily deplete cw plasma transported energy / power.

NEEP 536, Spring 2003, D. Whyte

Confined plasma a

Scrape-off layer

limiter 1 1 20 −3 5 m ~ 1025 m−2s−1 Γ ≡ nvi ~ 10 m ~ 4 ×10 4 4 s T =1keV

q ≡ 3kT ⋅ Γ ~ 3(1.6 ×10−19)103eV ⋅1025 ~ 5 ×109 //

W m2

18

Equilibrium and shaping coils are used to stabilize and define the plasma shape •

X

We already worked out that tokamaks need a stabilizing vertical field – This balances out the radial force imbalance caused by the interaction of the poloidal field with the current that generates the field.

– We must produce this field with a set of toroidally extended current carrying coils as shown, which will produce an ~ vertical BE • Produces equilibrating inward force on plasma.

Ip ◉

BE

Equilibrium Field coils

X

– These are called “poloidal field” coils since they produce poloidal magnetic fields (no toroidal field) NEEP 536, Spring 2003, D. Whyte

19

Equilibrium and shaping coils are used to stabilize and define the plasma shape •

•

•

We had previously discussed the benefits of shaping the plasma away from a circle in order to gain stability.

An example of how to obtain elongated plasma is shown. The toroidal current in the coils acts to decrease the poloidal magnetic field at the top and bottom of the plasma column.

Since: Flux ~ Bpol. * Area, the flux surface (isobaric) elongates at the top and bottom.

– Can also view this as a Ip x Br,coil force acting on each half of plasma.

NEEP 536, Spring 2003, D. Whyte

◉

Ip ◉

Poloidal Magnetic fields from Plasma and coil

Plasma Shaping Coils

◉

Essentially the shaping coils push and pull The flux surfaces around (depending on Sign of coil to Ip 20

Equilibrium and shaping coils are used to stabilize and define the plasma shape •

In a similar fashion we can construct a triangular or D-shaped plasma by placing shaping coils on the inboard and outboard side of the plasma.

Plasma Shaping Coils

X Ip ◉

◉

X

NEEP 536, Spring 2003, D. Whyte

21

With sufficient coil current we can create a poloidal field null --> diverted magnetic geometry •

If the coil current becomes large enough then there will be a location of zero poloidal field. The flux surface diverge at this position: X-point.

– Note that the coil current can be much less than the plasma current if the position of the coil is closer to the Xpoint.

•

•

There also now exists a “Last-Closed Flux Surface” defined by the X-point.

This separates regions with and without rotational transform and is called the ‘separatrix’.

The limiter is now replaced by the regions where the separatrix touches the wall. This region is called the divertor.

NEEP 536, Spring 2003, D. Whyte

◉

Ip ◉

X-points: Locations of Zero poloidal field

Plasma Shaping Coils

◉

Essentially the shaping coils push and pull The flux surfaces around (depending on Sign of coil to Ip 22

Example of shaped plasma from DIII-D •

Note the position of the vacuum vessel and shaping coils (values indicate A/turn in coils).

•

Plasma is elongated and slightly triangular, with an X-point on one side only.

•

The plasma flux lines are determined by a large non-linear fitter that solves the poloidal flux equilibrium based on force balance and Ampere’s law – Fit is based on the various magnetic and flux loop diagnostics (blue squares, purple line) placed around the vessel

NEEP 536, Spring 2003, D. Whyte

23

Inductive loops positioned around the plasma in various geometries provide real-time diagnosis of flux equilibrium.

•

Example of typical coils: – Rogowski coil measures Vemf ~ dI/dt through the conductor.

– Poloidal flux loop: measured loop voltage at region outside plasma.

– Diamagnetic coil: Measures small changes in toroidal flux which is linked to diamagnetic response of plasma from gryomotion --> m ∝ Eperp.

•

Rogowski coil

Surface loop voltage / poloidal flux loop

All of these measurements rely on analog integrators – Rogowski: Ip = ∫ Vemf dt – Pol. flux loop: ψ = ∫ Vloop dt

NEEP 536, Spring 2003, D. Whyte

24

Sophisticated non-linear control systems and realtime equilibrium constructions allow for extremely accurate real-time control of the plasma shape.

•

Long-term stability /operational viablity of shaping/equilibrium coils and diagnostic coil will be vital for reactor tokamak operations.

NEEP 536, Spring 2003, D. Whyte

25

The poloidal divertor magnetic configuration is the leading topology for magnetic fusion •

Access to H-mode confinement was discovered (accidentally) with the application of a divertor at ASDEX. Access is much easier with this topology – If the ion grad-B drift points toward the X-point???

•

Provides convenient engineering location for – Helium exhaust due to large buildup of neutral pressure.

– Dedicated location for power exhaust.

– Helps to isolate interaction of plasmas & surfaces away from fusion core.

•

Drawbacks: – Power exhaust is concentrated in single area.

– Takes up valuable “real-estate” inside the toroidal field coil volume.

NEEP 536, Spring 2003, D. Whyte

26

Two fundamental concepts control the interaction of the plasma with the solid-material: Recycling & Sheaths •

Recycling: What happens to the plasma at the wall?

– – –

It recombines in the surface.

The near-surface (implantation range ~ 10 nm) rapidly saturates with the gas (~1021 m-2) and releases back into plasma.

The plasma ionizes the gas, creating more plasma, closing the loop --> recycling •

–

plasma e-

e-

eD2

D+ recombination Solid surface

D+

ionization D2

Note: this needs energy input.

Therefore concentration of plasma flux --> neutral buildup --> He exhaust Ion undergoes acceleration

•

Sheath: The electrons arrive at the surface at a rate ~ 50 times that of the ions.

–

– –

This imbalance creates an electrostatic potential near the surface that attempts to equilibrate the ion and electron flux to the surface --> ambipolarity.

The sheath therefore controls the energy of ions striking the material --> sputtering.

The sheath controls how the plasma loses its energy to the wall.

NEEP 536, Spring 2003, D. Whyte

Electron are decelerated

Solid surface

Only highest energy Electrons get to surface Potential caused by Velocity difference in Electrons and ions 27

The poloidal divertor magnetic configuration is the leading topology for magnetic fusion •

Access to H-mode confinement was discovered (accidentally) with the application of a divertor at ASDEX. Access is much easier with this topology – If the ion grad-B drift points toward the X-point???

•

Provides convenient engineering location for – Helium exhaust due to large buildup of neutral pressure.

– Dedicated location for power exhaust.

– Helps to isolate interaction of plasmas & surfaces away from fusion core.

•

Drawbacks: – Power exhaust is concentrated in single area.

– Takes up valuable “real-estate” inside the toroidal field coil volume.

NEEP 536, Spring 2003, D. Whyte

1

Two fundamental concepts control the interaction of the plasma with the solid-material: Recycling & Sheaths •

Recycling: What happens to the plasma at the wall?

– – –

It recombines in the surface.

The near-surface (implantation range ~ 10 nm) rapidly saturates with the gas (~1021 m-2) and releases back into plasma.

The plasma ionizes the gas, creating more plasma, closing the loop --> recycling •

–

plasma e-

e-

eD2

D+ recombination Solid surface

D+

ionization D2

Note: this needs energy input.

Therefore concentration of plasma flux --> neutral buildup --> He exhaust Ion undergoes acceleration

•

Sheath: The electrons arrive at the surface at a rate ~ 50 times that of the ions.

–

– –

This imbalance creates an electrostatic potential near the surface that attempts to equilibrate the ion and electron flux to the surface --> ambipolarity.

The sheath therefore controls the energy of ions striking the material --> sputtering.

The sheath controls how the plasma loses its energy to the wall.

NEEP 536, Spring 2003, D. Whyte

Electron are decelerated

Solid surface

Only highest energy Electrons get to surface Potential caused by Velocity difference in Electrons and ions 2

It is useful to examine the 1-D, parallel to B, “fluid” properties of the SOL plasma •

Plasma as fluid with density n, bulk velocity, v.

• Particle conservation – Sp --> ionization source, plasma creation\

•

Momentum conservation – Dynamic (mnv2) and static pressure.

– Drag caused by collision of plasma with neutrals.

•

d (nv) = S p (z) dz

Energy conservation (for electrons): – LHS: Heat convection and conduction – RHS: electric field, equilibration with ions, resistive heating, ionization losses.

NEEP 536, Spring 2003, D. Whyte

d (minv 2 + p) = −mυσvi−neutral n ⋅ nneutral dz

d d 3 dT ( p// e v e ) + (−κ0Te5/ 2 e ) = dz dz dz 2 −enve E − Qeq // + QR + Qionization

3

Recycling from the wall --> in the edge, plasma and neutral gas co-exist Ionization is the basic process that creates plasma – Inelastic collision between a plasma electron and a neutral atom/molecule – Calculating the ionization MFP for an example 10 eV, 1020 m-3 plasma informs us that the plasma is extremely efficient at shielding neutrals.

– Ionization and the radiation associated with it are a fundamental energy sink for plasma electrons

•

H + e --> H+ + e + e Electron impact ionization of H,D 1E-13

ionization rate coefficient (m3/s)

•

1E-14 1E-15 1E-16 1E-17 1

10

100

Te (eV)

Momentum transfer – Not governed by Coulomb collision physics since one species neutral.

NEEP 536, Spring 2003, D. Whyte

λionization ~

103 m / s vD ~ 20 ~1 mm −14 10 10 ne σv

4

Plasmas exhibit a very strong T dependence for heat conduction •

Similar to principles for ideal kinetic gas – Major difference is that MFP for self-collisions, λs, in plasma given by Coulomb interaction -> T3/2 – // plasma heat conduction: • Strongly T dependent!

• Electrons dominate

•

•

Example: A 100 eV plasma:

“one-way” particle and heat flux for Maxwellian +∞

Γ x ~ ∫ vx f

Maxwellian

0

(v)dvx ∫ dvy ∫ dv z = −∞

−∞

1 8kT n 1/2 4 πm

=

1 nc 4

∞ ∞ 1 2 Maxwellian Max qx ~ ∫ mv vx f (v)dvx ∫ dv y ∫ dvz = 2kTΓx 0 2 −∞ −∞

“Net” heat flux r with Temperature gradient From “cold”side

From “hot” side

dT dT 1 − (To + λs )) nc dx dx 4 dkT qcond ≡ −K K ≡ ncλs dx

qnet = 2k(To − λs

First approx. of SOL heat flux:

NEEP 536, Spring 2003, D. Whyte

∞

+∞

– Ke ~ 20,000 W/m/K – >> Kcopper ~ 400 W/m/K – ITER: Qcond. ~ 100 MW – Let Te,SOL~100 eV to T << Te,sol at target ~ 100 m // length q// ~ 0.2 GW/m2 Atarget~ Q / (q// (Bp/BT)) ~ 5 m2

∞

λs =

c

K ∝n

νs

K = κ T 5/ 2 κ o

kT mν s

νs ∝

n m1/ 2T 3/ 2

~ 2000(W / m /eV 7/ 2 ) ~ 50 × κ o,e

o,i

5

Magnetic geometry plays a key role in the behavior of the SOL •

A fundamental aspect of the B geometry is that BT > Bp

– Take typical qedge ~ 3..since q = (a/R) *(Bt/Bp) then typically Bp/Bt ~ 10 at midplane.

•

In the divertor region the proximity of the poloidal field null leads to Bt/Bp ~ 30-50.

midplane

– This “oblique incidence” of the field lines helps to increase the effective wetted area and reduce target heat loading.

•

The parallel connection length is related to q: L// ~ 3 R q

– So for ITER we get L// ~ 50 m.

– Indicates necessary compromise in obtaining confinement (low q) vs. easing of target heat loads (high q)

NEEP 536, Spring 2003, D. Whyte

target

6

SOL are characterized by rapid transport parallel to B and much slower transport perpendicular to B. • • •

Estimate typical SOL times: In parallel direction the plasma can move at ~ its sound speed.

However, in the direction perpendicular to B, some sort of collision effect must occur to bump the particle to next field line.

v// ~ cs ~ 104 T m / s

τ SOL ~ (L /c s ) Γ ≡ D⊥∇n

– Characterize this with diffusion coefficient in presence of density gradient.

– Empirically D ~ 1 m2/s > Dclassical

•

So we can make a simple estimate for the perpendicular scale size of the SOL – L ~ 100 m, cs ~ 105 m/s --> λSOL ~ 3 cm – So we can see that the SOL is extremely narrow compared to its length.

– Simple estimate confirms approximate target area << Awall ~ 1000 m2.

λSOL ~ (D⊥τ SOL )

1/ 2

1/ 2

ITER

ASOL,t arg et ~ 2πR ⋅ 2 ⋅ λSOL ⋅ 2 ~ 4.7 m 2 Flux expansion

NEEP 536, Spring 2003, D. Whyte

~ ( D ⊥L /c s )

# divertor legs 7

From these limited physical models we already see the key aspects of SOL •

For a given geometry, the (electron) temperature actually has very weak dependences on other parameters – T is “pinned” by ~T7/2 dependence on heat conduction.

– Te,SOL ~ 100 eV is ~ universal.

•

Power and particles will be strongly exhausted in the // direction.

– Most intense local heat and particle flux occurs at divertor strikepoints.

– Can now estimate “received” surface heat flux in ITER. – Particle flux estimate: Received flux ~ 50 kA/m2 – This ion flux must be ~ balanced by the recycle flux of neutrals from the target since the target particle “sink” is filled in ~ 1021 m-2/ 1023 s-1m-2 ~ 10 ms.

NEEP 536, Spring 2003, D. Whyte

qsurface ~ Q/ ASOL,t arget ~ 100MW /5m 2 = 20MW / m 2

Γt arg et ~ ncs

Bp Bt

~ 1020 m−3 ⋅105 m / s ⋅ 0.03 ~ 3 ×1023 m−2s−1 t arg et

8

From these limited physical models we already see the key aspects of SOL •

• • •

Core plasma

SOL

– Lperp ~ 10-4 L//

q//

Because the surface quickly saturates, we obtain a recycle flux of H neutrals which must ~ incident flux of ions.

In cases where the electron conduction is “attached” to the plate, the T ~ 20 eV and the H is quickly ionized (MFP ~ 1 cm).

These newly formed ions return along B to the plate --> recycling –

•

Qtransport

The “shape” of the SOL is governed by the vastly different time-scales associated with parallel vs. perpendicular transport.

Is “fed” primarily by q//…needed to provide energy necessary for H ionization.

Γcore

surface

Estimate recycle time constants: – – – –

τneutral ~ λion / v ~ 10-2/103 ~ 10 µs τrecycled,ion ~ λion (Bt/Bp) / cs ~ 10 µs So there is a recycle transport loop at the surfaces with τ << τenergy Local recycling particle flux > flux from core….”flux amplification”.

NEEP 536, Spring 2003, D. Whyte

Γion,e

qtarget, Γtarget ionization

ΓH

9

The helium ash, which also provides fusion plasma heating, must be exhausted efficiently •

Consider that the creation rate of the helium is directly linked to the fusion reaction itself.

He /s / m-3

* nHe ~ nD nT σv fusion ⋅ τ He

– So we assign an effective ash particle confinement time: τHe*.

•

Of course the fusion rate and the energy confinement time, τe, are directly linked to our ability to heat the plasma through He++ slowing down on electrons and maintain DT burn conditions.

s

τ

*He

=

n He n D n T σv fusion

n Dn T σv fusion ⋅ kEα ~

3nkT τe

– Ignoring bremsstrahlung losses.

• •

So we can see that to not jeopardize the ignition criterion too much, fHe<10%, the ratio of τHe*/ τe ~< 10.

This means that we will have to worry about “pumping” away helium at some point.

NEEP 536, Spring 2003, D. Whyte

n E E * τ He ~ He α ~ f He α ~ f He100 3T n 3T τe * τ He < 15 τe

Reactor requirement 10

Divertors concentrate particle flux and recycling, making practical He pumping exhaust possible •

•

We can break the He transport times into two essentially independent portions – Core (from fusion) time, τcore – Edge recycle time, τedge

ionization

Γion,e

Reff ΓH,He

It’s easy to find a solution for the situation that at t=0 we somehow “turnoff” the fusion, He production, that

Baffle structure

pump

(1-Reff) ΓH,He

Reff

* = τ core + τ edge τ He (1− Reff )

•

ITER

This informs us that we must provide sufficient “geometry access” to the recycled He such that Reff ~< 1 – Reff much smaller than unity becomes inefficient since τedge << τcore

NEEP 536, Spring 2003, D. Whyte

11

Divertors concentrate particle flux and recycling, making practical He and H pumping exhaust possible --> particle control

NEEP 536, Spring 2003, D. Whyte

12

Two fundamental concepts control the interaction of the plasma with the solid-material: Recycling & Sheaths •

Recycling: What happens to the plasma at the wall?

– – –

It recombines in the surface.

The near-surface (implantation range ~ 10 nm) rapidly saturates with the gas (~1021 m-2) and releases back into plasma.

The plasma ionizes the gas, creating more plasma, closing the loop --> recycling •

–

plasma e-

e-

eD2

D+ recombination Solid surface

D+

ionization D2

Note: this needs energy input.

Therefore concentration of plasma flux --> neutral buildup --> He exhaust Ion undergoes acceleration

•

Sheath: The electrons arrive at the surface at a rate ~ 50 times that of the ions.

–

– –

This imbalance creates an electrostatic potential near the surface that attempts to equilibrate the ion and electron flux to the surface --> ambipolarity.

The sheath therefore controls the energy of ions striking the material --> sputtering.

The sheath controls how the plasma loses its energy to the wall.

NEEP 536, Spring 2003, D. Whyte

Electron are decelerated

Solid surface

Only highest energy Electrons get to surface Potential caused by Velocity difference in Electrons and ions 1

From these limited physical models we already see the key aspects of SOL •

• • •

Core plasma

SOL

– Lperp ~ 10-4 L//

q//

Because the surface quickly saturates, we obtain a recycle flux of H neutrals which must ~ incident flux of ions.

In cases where the electron conduction is “attached” to the plate, the T ~ 20 eV and the H is quickly ionized (MFP ~ 1 cm).

These newly formed ions return along B to the plate --> recycling –

•

Qtransport

The “shape” of the SOL is governed by the vastly different time-scales associated with parallel vs. perpendicular transport.

Is “fed” primarily by q//…needed to provide energy necessary for H ionization.

Γcore

surface

Estimate recycle time constants: – – – –

τneutral ~ λion / v ~ 10-2/103 ~ 10 µs τrecycled,ion ~ λion (Bt/Bp) / cs ~ 10 µs So there is a recycle transport loop at the surfaces with τ << τenergy Local recycling particle flux > flux from core….”flux amplification”.

NEEP 536, Spring 2003, D. Whyte

Γion,e

qtarget, Γtarget ionization

ΓH

2

The helium ash, which also provides fusion plasma heating, must be exhausted efficiently •

Consider that the creation rate of the helium is directly linked to the fusion reaction itself.

He /s / m-3

* n He ~ n D n T σv fusion ⋅ τ He

– So we assign an effective ash particle confinement time: τHe*.

•

Of course the fusion rate and the energy confinement time, τe, are directly linked to our ability to heat the plasma through He++ slowing down on electrons and maintain DT burn conditions.

s

τ

*He

=

n He n D n T σv fusion

n Dn T σv fusion ⋅ kEα ~

3nkT τe

– Ignoring bremsstrahlung losses.

• •

So we can see that to not jeopardize the ignition criterion too much, fHe<10%, the ratio of τHe*/ τe ~< 10.

This means that we will have to worry about “pumping” away helium at some point.

NEEP 536, Spring 2003, D. Whyte

nHe E α Eα * τ He ~ ~ f He ~ f He100 3T n 3T τe

τ *He < 15 τe

Reactor requirement 3

Divertors concentrate particle flux and recycling, making practical He pumping exhaust possible •

•

We can break the He transport times into two essentially independent portions – Core (from fusion) time, τcore – Edge recycle time, τedge

Γion,e

Reff ΓH,He

It’s easy to find a solution for the situation that at t=0 we somehow “turnoff” the fusion, He production, that

τ •

ionization

* He

= τ core +

Reff (1− Reff )

τ edge

Baffle structure

pump

(1-Reff) ΓH,He ITER

This informs us that we must provide sufficient “geometry access” to the recycled He such that Reff ~< 1 – Reff much smaller than unity becomes inefficient since τedge << τcore

NEEP 536, Spring 2003, D. Whyte

4

Divertors concentrate particle flux and recycling, making practical He and H pumping exhaust possible --> particle control

NEEP 536, Spring 2003, D. Whyte

5

ITER will used cyropumping to exhaust both H and He gas •

Liquid He cooled cryopanels – Charcoal activated.

– Must be regenerated for the surface to continue pumping --> more difficult in cw conditions.

•

Pumping speed for Helium: 60 m3/s – He/alpha production rate: ~ 1.4e20 s-1 – Necessary pressure such that the core production rate is matched by divertor exhaust rate? ~ 0.01 Pa helium pressure.

NEEP 536, Spring 2003, D. Whyte

6

Sheaths: The Bohm criterion •

The Boltzmann factor relation is used to “modify” the edistribution in the presence of the retarding potential field near the surface

Ion undergoes acceleration

– Loss rate of electrons << kinetic velocity.

– Velocity distribution is ~ Maxwellian even in the sheath – Te remains constant.

Sheath edge

Electron are repelled Solid surface

Potential caused by Velocity difference in Electrons and ions

Plasma flow velocity

f ≡ f MaxwellianeeV / kT

e

In the sheath

ne (x) = nse exp[e(V −Vse )/ kTe ] Se: Sheath edge

NEEP 536, Spring 2003, D. Whyte

7

Sheaths: The Bohm criterion (simple derivation with Ti=0) 1 miv 2 = −eV 2

Conservation of Energy for ions

niv ≡ const

Particle conservation

n i = n se (Vse /V )1/ 2

Ion undergoes acceleration Electron are repelled Solid surface

Potential caused by Velocity difference in Electrons and ions

Sheath edge

Poisson’s equation in sheath (ne<ni) e d 2V ∇ ⋅ E ≡ 2 = − ( ni − ne ) εo dx

d2V e = − nse dx 2 εo

NEEP 536, Spring 2003, D. Whyte

V se V

1/ 2

– exp(e(V −Vse )/ kTe )

Plasma flow velocity

x

ne (x) = nse exp[e(V −Vse )/ kTe ] Se: Sheath edge

8

Sheaths: The Bohm criterion (simple derivation with Ti=0) d2V e nse = − dx 2 εo

Vse V

1/ 2

– exp(e(V −Vse )/ kTe )

Ion undergoes acceleration Electron are repelled Solid surface

∆ ≡ Vse −V (> 0) Vse V

1/ 2

≈1− 1 ∆ 2 Vse

exp[. ..... ] ≈ 1 −

e∆ kTe

Solved through expansion

e 1 d2∆ e ≈ −∆ n se − dx 2 kTe 2Vse εo

NEEP 536, Spring 2003, D. Whyte

Potential caused by Velocity difference in Electrons and ions

Sheath edge

Plasma flow velocity

x

ne (x) = nse exp[e(V −Vse )/ kTe ] Se: Sheath edge

9

Sheaths: The Bohm criterion (simple derivation with Ti=0) e e 1 d2∆ − ≈ ∆ n se dx 2 kTe 2V se εo •

In order for this 2nd order diff.

Equation to have a non-oscillatory solution the the bracketed term must be positive –

An oscillating sheath has never been observed.

e 1 > kTe 2Vse miv 2se ≥ kTe

1 mi vse2 = −eVse 2 vse ≥ cs

The Bohm criterion: The plasma “exit” velocity into The sheath must be at least at the Ion sound speed: flux = n cs NEEP 536, Spring 2003, D. Whyte

Ion undergoes acceleration Electron are repelled Solid surface

Potential caused by Velocity difference in Electrons and ions

Sheath edge

Plasma flow velocity

x

Generalized solution with Ti non-zero

v se ≥ c s = [ (kTe + kTi )/ m i ]

1/ 2

10

Sheaths: The Bohm criterion (simple derivation with Ti=0) Ion undergoes acceleration

e e 1 d2∆ − ≈ ∆ n se dx 2 kTe 2V se εo •

In order for this 2nd order diff.

Equation to have a non-oscillatory solution the the bracketed term must be positive –

An oscillating sheath has never been observed.

e 1 > kTe 2Vse miv 2se ≥ kTe

1 mi vse2 = −eVse 2

Solid surface

Potential caused by Velocity difference in Electrons and ions

Sheath edge

Plasma flow velocity

vse ≥ cs

The Bohm criterion: The plasma “exit” velocity into The sheath must be at least at the Ion sound speed: flux = n cs NEEP 536, Spring 2003, D. Whyte

Electron are repelled

∆ L

2 sheath

≈

en∆ e εo kTe

x e Lsheath ≈ εokT 2 en

1/ 2

≡ λDebye

..& the physical dimension of the sheath is (not surprisingly) the plasma Debye length (~10 microns)

11

Sheath potential drop •

•

We derive the “floating” (ambipolarity: e-flux = ion flux) potential drop between the plasma and the wall.

So the relative potential drop is

IONS

Γi = Γse = nsecs ELECTRONS

[

Γ = ΓMax exp eV

linked/normalized to the electron temperature and the ion species – For deuterium plasma with Ti = Te the electrostatic sheath potential drop ~ 3 Te. – Therefore ions will be accelerated toward the wall an reach a final an energy ~ Ti + V + Vpresheath ~ 5 Te.

•

c s = [ (kTe + kTi )/ m i ]

1/ 2

e

e

wall / kTe

]

1 1 8kT nv = n Max e e e Γ =4 4 πme

1/ 2

1 m T eV / kTe = ln 2π e 1 + i mi 2 Te

Most important consequence: Ion induced material sputtering.

NEEP 536, Spring 2003, D. Whyte

12

Power transmission through the sheath •

•

The sheath acts as an energy transfer system between the electron and ions through the presence of the electrostatic sheath.

It is most convenient to relate the power entering at the sheath entrance by a sheath transmission coefficient.

– Can assume no particle sources in sheath due to small size.

– Ion contribution rough estimate (ions are non-Maxwellian near sheath).

•

qse ≡ γ e kTeΓse γ e ≈ 2 + eVsf / kTe + eVpresheath / kTe ≈ 5.5 “one-way” E flux component

γi ≈

“potential” energy Into sheath

3 1 kTi + mi c s2 / kTe ≈ 2 2 2

What q,// expect for T=100 eV, n~1020 m-3?

– Must keep surfaces well-aligned an not intercept // heat flux.

NEEP 536, Spring 2003, D. Whyte

qse ~ γkTeΓse ~ 8kTeΓse 13

Sheath potential drop •

•

We derive the “floating” (ambipolarity: e-flux = ion flux) potential drop between the plasma and the wall.

So the relative potential drop is

IONS

Γi = Γse = nsecs ELECTRONS

[

Γ = ΓMax exp eV e

linked/normalized to the electron temperature and the ion species – For deuterium plasma with Ti = Te the electrostatic sheath potential drop ~ 3 Te. – Therefore ions will be accelerated toward the wall an reach a final an energy ~ Ti + V + Vpresheath ~ 5 Te.

•

c s = [ (kTe + kTi )/ m i ]

1/ 2

e

wall / kTe

]

1 1 8kT nv = n Max e e e Γ =4 4 πme

1/ 2

1 m T eV / kTe = ln 2π e 1 + i mi 2 Te

Most important consequence: Ion induced material sputtering.

NEEP 536, Spring 2003, D. Whyte

12

Power transmission through the sheath •

•

The sheath acts as an energy transfer system between the electron and ions through the presence of the electrostatic sheath.

It is most convenient to relate the power entering at the sheath entrance by a sheath transmission coefficient.

– Can assume no particle sources in sheath due to small size.

– Ion contribution rough estimate (ions are non-Maxwellian near sheath).

•

qse ≡ γe kTeΓse γ e ≈ 2 + eVsf / kTe + eVpresheath / kTe ≈ 5.5 “one-way” E flux component

γi ≈

“potential” energy Into sheath

3 1 2 kTi + mi cs / kTe ≈ 2 2 2

What q,// expect for T=100 eV, n~1020 m-3?

– Must keep surfaces well-aligned and not intercept // heat flux.

NEEP 536, Spring 2003, D. Whyte

qse ~ γkTeΓse ~ 8kTeΓse 13

Divertor power heat loading is one of the major concerns for MFE reactors • • • •

Energy is necessary aspect of fusion energy production, so heat removal seems inevitable.

What can we reasonably expect from materials for heat removal?

Power extraction capability given by heat conduction of material and thickness.

Further limit imposed by thermal stress on materials –

•

q plasma

Heat flux extraction

q(W / m 2 ) = k(W / m / k)∇T = k

Large stresses buildup across the material interface due to the large temperature gradient.

NEEP 536, Spring 2003, D. Whyte

dtile

Thermal stress (pressure) across material α: coefficient thermal expansion, E: Young’s modulus ν: Poisson’s ratio

Coolant abilities (water pressure --> temperature) Qefm is material’s “figure-ofmerit” for stress tolerance to heating

(Tsurface − Tcool )

k, Heat conductivity

Further limits imposed by melting point of material.

–

d

coolant

Coolant substrate

σ =[ σ yield σ

=

αE ]⋅ q ⋅ d 2k(1− ν )

σ yield 2k(1− ν ) 1 1 =Q efm q⋅ d αE q⋅ d

14

The extreme demands on divertor heat removal limit our choice of candidate materials • Material must have both high thermal conductivity, high yield strength and high melting point – This limits us primarily to refractory metals like Molybdenum and Tungsten.

– In addition, high thermal conductivity CFC (carbonfibre composites: k ~ 300 W/m/K) meet these criteria • Additional bonus that C sublimes…no melting!

NEEP 536, Spring 2003, D. Whyte

15

The extreme demands on divertor heat removal limit the allowable tile thickness • •

Let’s take two examples: copper and W. If we need to exhaust ~ 20 MW/m2 what maximum thickness can we use?

– This is a an overly pessimistic strain limit.

dtile = k

(Tmelt − Tcool )

k (W/m/K)

~130

~320

Qemf

~45

~50

(MW mm/m2)

Tmelt

3700

1200

1 q

dmelt

~2 cm

~1.5 cm

dstrain

~2 mm

~2.5 mm

So with even the best heat conducting materials the tiles surfaces are limited to ~ cm thickness.

NEEP 536, Spring 2003, D. Whyte

Copper

q(W / m 2 )

dtile ≅ Qefm

•

Tungsten

16

Transient heating also sets a limit on the material --> high Tmelt •

What if the plasma loses its energy on a fast time-scale

Tungsten

Carbon (CFC)

Copper

k (W/m/K)

~130

~300

~320

Cv (J/m3/K)

2.5e6

1.9e6

3.4e6

The plasma loses it energy to the wall surface on MHD timescales (~ 0.1ms).

Dheat (cm2/s)

0.7

1.5

0.9

Skin depth in 0.1ms

~ 0.09mm

0.11 mm

~0.09 mm

– So penetration “skin” depth of heat is ~ sqrt(D*t) << dtile.

– Surface heating:

Fig. Of merit (MJ/m2/s1/2)

62

71

26

– Edge-Localized Mode instability associated with Hmode.

– Global stability limit?

•

∆T =

2 π

1 Q k ⋅ Cv t1/ 2

Transient heating Figure-of-merit

π k ⋅ Cv ⋅ Tmelt 2 NEEP 536, Spring 2003, D. Whyte

Copper is out!

17

The wall material/element is a critical choice for the fusion reactor design • 1.

2.

3.

Several fundamental limits arise from the interaction of the plasma with the material The plasma bombardment can cause the wall material to be released (sputtered) into the plasma --> wall material acts as an impurity--> fuel dilution, radiation losses --> loss of ignition margin.

The plate material can be eroded away by the interaction --> limited thickness (~1 cm) based on heat conduction.

The hydrogenic fuel, in particular the tritium, can become trapped in regions where the plasma net deposits impurity material.

NEEP 536, Spring 2003, D. Whyte

18

Sputtering: Removal of wall material from plasma interaction •

• •

When energetic plasma ions or neutrals strike a solid surface, sufficient momentum / energy can be transferred to an atom in the material lattice such that is ejected.

The (more-or-less) straightforward means of sputtering can be understood through classical treatment of the colliding species.

A common sputtering event: – – –

Light incident ion reflects off target back toward surface Collides with surface atoms which is ejected Clear threshold exists set by surface binding energy.

Maximum energy fraction that can be transferred In the collision

γ=

Maximum energy fraction that can be transferred Into sputtered atom

Eejected = Eoγ(1− γ ) − Ebinding

4 Mincident M t arg et (Mincident + M t arg et )2

NEEP 536, Spring 2003, D. Whyte

E threshold =

Ebinding

γ(1− γ ) 19

Deuterium ion physical sputtering for candidate materials

•

•

Binding energies, M: 0.1 Be: 3.4 eV / 9 C: 7.4 eV / 12 Beryllium Mo: 6.9 eV / 96 0.01 W: 8.8 eV / 184 Carbon What sets the incident 0.001 Molybdenum deuterium ion energy?

The sheath ~ 5 Te Tungsten As one would expect the 0.0001 10 100 1000 10000 sputtering efficiency Incident deuterium ion energy (eV) decreases as the target particles M >> M • Carbon has unique property that it undergoes chemical plasma. erosion: volatile hydrocarbon molecules are formed with Sputter yield

•

free H species – Weak energy dependence: yield ~ 1%.

NEEP 536, Spring 2003, D. Whyte

20

Self-sputtering is the most efficient form of sputtering

•

•

Momentum-energy transfer is the most efficient for species with same mass.

For this reason ‘selfsputtering” can be very dangerous.

Potential of runaway effects in sputtering if Yss > 1.

10

Sputter yield

•

Beryllium

1

Carbon

Molybdenum

0.1

Tungsten

0.01 10

100

1000

10000

Incident self ion energy (eV)

NEEP 536, Spring 2003, D. Whyte

21

Uncontrolled sputtering can easily limit the plate lifetime

• • • • • •

What is the ~ gross rate at which our divertor plasma could remove the plate material through sputtering?

Flux: n cs (Bp/BT) ~ 1024 m-2s-1 Yield: ~ 0.001 to 0.01 Efflux of material: ~ yield * flux ~ 1021 - 1022 s-1 m-2 Atomic density of materials: ~ 1029 m-3.

Gross erosion rate: 10-100 nm/s Time to erode through 1 cm plate?

107 nm / G.E.R. ~ 105-106 s ~ 1 day - 2 weeks.

NEEP 536, Spring 2003, D. Whyte

0.1

Sputter yield

•

Beryllium

0.01

Carbon

0.001

Molybdenum

Tungsten

0.0001 10

100

1000

10000

Incident deuterium ion energy (eV)

Overly pessimistic?

Ionization of material near Surface should limit erosion

22

In reality the interplay between the plasma and wall material is complicated by the recycling •

•

•

•

Experiments and modeling find very complex interaction between the plasma and material The rapid ionization and return of the eroded material tends to help limit the net erosion. “ “ tends to increase selfsputtering at the plate and enhance total yield (1-Yss)-1 The “answer” for net erosion depends on a complex interaction of yield, self-yield, sheath physics and plasma transport.

NEEP 536, Spring 2003, D. Whyte

Carbon gross and net erosion on DIII-D 23

The impurities released from the wall affect fusion performance through radiation and fuel dilution Remember rule for H-like ionization energy: E ∝ 13.6 eV * Z2 – Therefore low-Z materials (C, Be) will be fully ionized in core plasma Te ~ 10 keV. – However high-Z refractory metals like Mo (A=42), W (A=74) are only partially ionized --. Bound electrons.

•

90000

H-like ionization energy (eV)

•

60000

30000

0 0

20

40

60

80

Atomic number

Fuel dilution: nD-T/ne ~ 1 - fz x Z

– Small fractions of high-Z impurities quickly dilute hydrogenic fuel.

NEEP 536, Spring 2003, D. Whyte

24

The impurities released from the wall affect fusion performance through radiation and fuel dilution •

•

Radiation efficiency is a complicated function of the atomic details of each impurity (ionization levels, valence electrons, etc.) and the electron temperature.

Line radiation efficiency increases strongly with atomic number of impurity (~ Z4) in fusion temperature range.

–

•

Line radiation efficiency, L P = fz ne2 L

Impurities are not ionized through --> line radiation from bound electrons.

Line radiation is in addition to the increased losses through continuum/bremsstrahlung radiation (L ~ T1/2 Z2 )

NEEP 536, Spring 2003, D. Whyte

Mo W m3 Si C Be

Te (keV)

25

The fusion core plasma has very little tolerance to high Z impurities

•

•

•

The effects of the impurity (line) radiation can have a very significant impact on the global power balance.

Going through an exercise along the lines of the Lawson criterion we can calculate the “fatal fraction” for a given impurity based on its radiation properties (At a given Te) The plasma cannot ignite if it holds impurity fraction larger than this value.

Fatal fraction for Te ~ 10 keV 0.1

fatal fraction

•

0.01 0.001 0.0001

One see that the plasma has very small tolerance to the presence of any high-Z impurities like Mo or W.

0

20

40

60

80

Impurity atomic number

So while high-Z wall materials has lower erosion yields, we must weigh this against the plasma’s intolerance for Prad from high-Z impurities.

NEEP 536, Spring 2003, D. Whyte

26

In reality the interplay between the plasma and wall material is complicated by the recycling •

•

•

•

Experiments and modeling find very complex interaction between the plasma and material The rapid ionization and return of the eroded material tends to help limit the net erosion. “ “ tends to increase selfsputtering at the plate and enhance total yield (1-Yss)-1 The “answer” for net erosion depends on a complex interaction of yield, self-yield, sheath physics and plasma transport.

NEEP 536, Spring 2003, D. Whyte

Carbon gross and net erosion on DIII-D 1

The impurities released from the wall affect fusion performance through radiation and fuel dilution Remember rule for H-like ionization energy: E  13.6 eV * Z2 – Therefore low-Z materials (C, Be) will be fully ionized in core plasma Te ~ 10 keV. – However high-Z refractory metals like Mo (A=42), W (A=74) are only partially ionized --. Bound electrons.

•

90000

H-like ionization energy (eV)

•

60000

30000

0 0

20

40

60

80

Atomic number

Fuel dilution: nD-T/ne ~ 1 - fz x Z

– Small fractions of high-Z impurities quickly dilute hydrogenic fuel.

NEEP 536, Spring 2003, D. Whyte

2

The impurities released from the wall affect fusion performance through radiation and fuel dilution •

•

Radiation efficiency is a complicated function of the atomic details of each impurity (ionization levels, valence electrons, etc.) and the electron temperature.

Line radiation efficiency increases strongly with atomic number of impurity (~ Z4) in fusion temperature range.

–

•

Line radiation efficiency, L P = fz ne2 L

Impurities are not ionized through --> line radiation from bound electrons.

Line radiation is in addition to the increased losses through continuum/bremsstrahlung radiation (L ~ T1/2 Z2 )

NEEP 536, Spring 2003, D. Whyte

Mo W m3 Si C Be

Te (keV)

3

The fusion core plasma has very little tolerance to high Z impurities

•

•

•

The effects of the impurity (line) radiation can have a very significant impact on the global power balance.

Going through an exercise along the lines of the Lawson criterion we can calculate the “fatal fraction” for a given impurity based on its radiation properties (At a given Te) The plasma cannot ignite if it holds impurity fraction larger than this value.

Fatal fraction for Te ~ 10 keV 0.1

fatal fraction

•

0.01

0.001

0.0001

One see that the plasma has very small tolerance to the presence of any high-Z impurities like Mo or W.

0

20

40

60

80

Impurity atomic number

So while high-Z wall materials has lower erosion yields, we must weigh this against the plasma’s intolerance for Prad from high-Z impurities.

NEEP 536, Spring 2003, D. Whyte

4

The “gas-box” divertor was envisioned as a means of controlling energy and particle removal.

•

Magnetic fusion has an acute need for controlling both the plasma energy release and the sputtering of impurities from the wall.

It was discovered early in the exploration of plasmas that a substantial gas pressure or gas “buffer” could “extinguish” the plasma.

A basic result of this work was an understanding that ion-neutral collisions reduce power and particle flux to the solid surfaces

•

•

Heat flux

Competition between ionization and neutral damping

∇

d 2 n + ngas σv ion dz2 Damb

Sion

Damb  ion−

neutral f (

NEEP 536, Spring 2003, D. Whyte

Te Tgas

)

0

Main result

wall  ion− neutral Lz 5

The use of divertor plasma “detachment” has proven successful in controlling heat flux in divertors •

•

Two-point model (Stangeby) uses energy and momentum balance to simply describe relationship between upstream and target plasma parameters.

Basic phenomenology – With fixed power the plasma density is increased (through gas injection) – The divertor plasma density progressively increases and bifurcates into a situation where the Te becomes too low to sustain heat conduction.

– Plasma flux continues to increase & through recycling continues to increase the local divertor neutral pressure.

NEEP 536, Spring 2003, D. Whyte

Two-point model Upstream: u Target: t

2ntTt

nuTu

Tu7/ 2

Tt7/ 2 +

q//

nt kTtcs,t

7q// L 2κ0,e

Tu  q//2/ 7 / 7 −4 / 7 −2 Tt  q10 L nu //

n t  n u3q−//8/ 7 L6/ 7 t  n u2q−//3/ 7 L4 / 7

6

The use of divertor plasma “detachment” has proven successful in controlling heat flux in divertors •

Detachment finds substantial increase in peak heat flux

DIII-D

Alcator C-Mod

– Parallel heat conduction reduced or eliminated.

– Impurities (from wall or seeded) provide radiative dissipation further upstream where T > 5 eV (Prad  n2) – Ion-neutral momentum extraction, recombination strongest near separatrix!

NEEP 536, Spring 2003, D. Whyte

7

At sufficiently high upstream density the divertor plasma cools to the point where it becomes dominated by plasma volume recombination rather than ionization.

NEEP 536, Spring 2003, D. Whyte

8

The uniformly low Te from plasma detachment eliminates physical sputtering at the divertor plates. This controls the erosion rate of the divertor target 50 10

Divertor Plate: Attached vs.

Detached

Te

Attached

(eV)

Yphys < 10-6

Detached

1

1

i (1023 m-2 s-1)

0

q 1 (MW m-2) 0 -0.1

private flux region

0

0

15 0 -15 -0.04 NEEP 536, Spring 2003, D. Whyte

Outer divertor

Inner divertor private flux exposure

0.2

ROSP (m)

RISP (m) Carbon Net Erosion Rate 30/ burn-year) (cm

0.1

erosion deposition

0

.04

RISP (m)

0

.04

ROSP (m)

.08 9

For the purpose of heat flux and erosion control, detached divertor plasma operation is the basis for ITER: Modeled by 2-D plasma “fluid” coes

NEEP 536, Spring 2003, D. Whyte

10

Inertial Confinement Fusion •

This approach is based on scaling down a thermonuclear explosion to a sufficiently small size that it can be readily utilized for power production.

•

The invention of the laser in 1960’s with its potential for producing very high intensities initiated the ICF research.

– Experiments based on lasers and charge-particle drivers.

•

Inertial confinement means that the inertia of the mass of exploding fuel provides a sufficient confinement time that the thermonuclear reaction can take place before the mass blows apart.

•

ICF seeks to compress a tiny amount of fusion fuel to high density and temperature to ignite the fusion burn.

NEEP 536, Spring 2003, D. Whyte

1

Inertial Confinement Fusion works…it is the principle behind the “H bomb”

• 15 M-ton D-T fusion bomb test at Bikini atoll (Hiroshima/Nagasaki fission bombs ~ 20 k-ton).

• The D-T is bred (T from Li) and rapidly heated by a fission explosion.

• While energy production is eventually hoped for from ICF, the primary raison d’etre of present ICF research is to support the “Stockpile Stewardship” program…i.e. the nuclear weapons program.

NEEP 536, Spring 2003, D. Whyte

2

Inertial Fusion Energy (IFE): conceptual power plant

• •

Modular design: target , driver and fusion chamber separately operated Driver/target explosion at repetition rate of few Hz to produce necessary time-averaged electrical power.

NEEP 536, Spring 2003, D. Whyte

3

The grand challenges of IFE energy production • High gain (Efusion / Edriver@target) targets – Gains ~ 50-100 required.

• Energy efficient (10-30%) high repetition rate (~5 Hz) driver.

– Both ion beams and lasers being considered

• Low cost target fabrication at high production rate (~100 million targets / year) – Target cost ~ 25-50 ¢/target.

• Long life fusion chamber /w low radioactivity.

NEEP 536, Spring 2003, D. Whyte

4

IFE in comparison to MFE

•

•

Both use D-T since it is the easiest fuel combination to achieve fusion power (also why it is used in bombs)

•

D-T plasma thermal temperature must be increased to ~ 10 keV for both.

Inertial confinement: we can guess that the reaction must occur very quickly since thermal velocity of 10 keV plasmas ~ 106 m/s.

•

Since IFE relies on a scaled-down thermonuclear explosion we immediately know this a pulsed power source. MFE is inherently steady-state.

•

Therefore we have to reformulate the equivalent of the Lawson criterion for IFE.

NEEP 536, Spring 2003, D. Whyte

5

IFE fundamentals • • • • • • • •

•

Spherical target.

D-T fuel is confined in a microballoon The driver deposits energy on the surface layer of the target, the ablator.

This layer becomes hot, generating high pressure.

The surface layer ablates (solid --> gas/plasma) outward at high speed.

This rocket forces acts to compress and heat the fuel.

A large part of the input energy is conveyed to the ablator.

Some part of this does work in compressing and heating the D-T fuel.

The ratio of the energy supplied to the D-T fuel at the instant the fuel is compressed to highest density to the total beam energy is the EDT hydrodynamic efficiency. = Edriver The typical efficiency for an ablation-driven rocket is ~10%

NEEP 536, Spring 2003, D. Whyte

6

IFE fundamentals

NEEP 536, Spring 2003, D. Whyte

7

IFE fundamentals • • •

The rho x r expression replace n x tau from MFE. The burn efficiency of the pellet must be ~ 3 g/cm2 to achieve 30% burn efficiency.

The equivalent Lawson criterion is not actually that different between IFE and MFE….but the absolute values are very different.

– Density is ~ 1011 higher in IFE (n~1031 m-3)…compression needed – Confinement time is ~ 1011 smaller in IFE…

•

Compression of the fuel center to a “hot spot” is necessary for ignition..otherwise the energy of the driver and the resulting released energy become too large for technological reasons.

NEEP 536, Spring 2003, D. Whyte

8

IFE fundamentals • • • • • • • •

•

Spherical target.

D-T fuel is confined in a microballoon The driver deposits energy on the surface layer of the target, the ablator.

This layer becomes hot, generating high pressure.

The surface layer ablates (solid --> gas/plasma) outward at high speed.

This rocket forces acts to compress and heat the fuel.

A large part of the input energy is conveyed to the ablator.

Some part of this does work in compressing and heating the D-T fuel.

The ratio of the energy supplied to the D-T fuel at the instant the fuel is compressed to highest density to the total beam energy is the EDT hydrodynamic efficiency. = Edriver The typical efficiency for an ablation-driven rocket is ~10%

NEEP 536, Spring 2003, D. Whyte

6

IFE fundamentals

NEEP 536, Spring 2003, D. Whyte

7

Fuel and burn considerations

•

• •

A high burn fraction is necessary in IFE due to the low hydrodynamic efficiency of the drivers.

In IFE mass density, rho = m n, takes the place of density and r ~ tau / cs, takes the place of confinement time.

So for a 1/3 burn fraction at T~20 keV we require r ~ 3 g /cm2 This is equivalent to n τ ~ 2x1015 s/cm3 ~ 2x1021 s/m3.

r r ~ 8mics r + 6 g /cm 2 r+ σv D T

fburn

6 mi cs /sigma_v

•

1.00E+02

1.00E+01

1.00E+00 0

10

20

30

40

50

Temperature (keV)

NEEP 536, Spring 2003, D. Whyte

8

Ignition conditions •

Energy balance from self-heating gives us rough equivalent of Lawson criterion in IFE.

Lawson criterion

– Efusion,alphas > Edriver

 r ~ 1.6 - 2 g/cm2 – ~ 1-2x1021 s/m3 in 10-20 keV range.

–

1E+21

τ

For hydrodynamic efficiency of 10%

n e (s / m3)

•

1E+22

1E+20 1

•

Therefore ignition criterion is comparable to MFE.

NEEP 536, Spring 2003, D. Whyte

10 T (keV)

100

9

Compression / size considerations •

From consideration of total electrical plant efficiency the energy gain must be ~ 100 --> linked to fburn -->  r

•

The fuel mass and driver energy necessary scales as 1/ 

•

At normal density, ~ 0.2 g/cm3 this would require 2500 g of DT, EDT ~ 5x1012 J and result in Efusion ~ 3x1014 J – ~ 70 kT explosion ~ Hiroshima.

•

Therefore we must rely on substantial compression (>x1000) – M ~ 2 mg, yield ~ 300 MJ, Edriver ~ MJ.

•

So if the fuel is heated to T~ 10 keV and is compressed to n ~ 5e31 m3 the pressured of the fuel is P ~ n k T ~ 1017 Pa…cannot be applied directly.

NEEP 536, Spring 2003, D. Whyte

10

Compression / size considerations •

Compression is energetically attractive.

•

Here we show that if the DT remains Fermi degenerate during the compressions process that the compression requires much less energy density than direct ignition.

NEEP 536, Spring 2003, D. Whyte

11

Compression / ignition issues •

• •

•

•

Although compression is energetically attractive and reduces the driver size required for ignition, high gain also requires hot-spot ignition.

5 mg target requires ~60-90 MJ driver if all heated to 10 keV. However if the target can ignited from a central hot spot containing about 2% of the total mass then the energy required to heat and compress this mass is ~ 2.5 MJ --> gain ~ 100.

The hot spot forms during compression from material the center of the fuel where its implosion is suddenly halted Further requirement is that the heating alphas deposit their energy into spot to propagate burn throughout the fuel mass.

NEEP 536, Spring 2003, D. Whyte

12

Uniformity •

• •

Convergence ratio ratio of initial outer radius to final compressed radius of the hot spot.

Typical convergence ratio is ~ 30 for high-gain target.

If target surface with average acceleration g has a location on it surface with acceleration perturbation g then we can estimate that the requirement for compression uniformity is ~ 1%.

NEEP 536, Spring 2003, D. Whyte

13

Ignition conditions for IFE • • •

The rho x r expression replace n x tau from MFE. The burn efficiency of the pellet must be ~ 3 g/cm2 to achieve 30% burn efficiency.

The equivalent Lawson criterion is not actually that different between IFE and MFE….but the absolute values are very different.

– Density is ~ 1011 higher in IFE (n~1031 m-3)…compression needed – Confinement time is ~ 1011 smaller in IFE…

•

•

Large compression of the fuel center to a “hot spot” is necessary for ignition..otherwise the energy of the driver and the resulting released energy become too large for technological reasons.

Uniform compression velocity / acceleration is required --> must avoid hydrodynamic instabilities.

NEEP 536, Spring 2003, D. Whyte

14

Inertial Confinement Fusion •

This approach is based on scaling down a thermonuclear explosion to a sufficiently small size that it can be readily utilized for power production.

•

The invention of the laser in 1960’s with its potential for producing very high intensities initiated the ICF research.

– Experiments based on lasers and charge-particle drivers.

•

Inertial confinement means that the inertia of the mass of exploding fuel provides a sufficient confinement time that the thermonuclear reaction can take place before the mass blows apart.

•

ICF seeks to compress a tiny amount of fusion fuel to high density and temperature to ignite the fusion burn.

NEEP 536, Spring 2003, D. Whyte

1

Inertial Confinement Fusion works…it is the principle behind the “H bomb”

• 15 M-ton D-T fusion bomb test at Bikini atoll (Hiroshima/Nagasaki fission bombs ~ 20 k-ton).

• The D-T is bred (T from Li) and rapidly heated by a fission explosion.

• While energy production is eventually hoped for from ICF, the primary raison d’etre of present ICF research is to support the “Stockpile Stewardship” program…i.e. the nuclear weapons program.

NEEP 536, Spring 2003, D. Whyte

2

Inertial Fusion Energy (IFE): conceptual power plant

• •

Modular design: target , driver and fusion chamber separately operated Driver/target explosion at repetition rate of few Hz to produce necessary time-averaged electrical power.

NEEP 536, Spring 2003, D. Whyte

3

The grand challenges of IFE energy production • High gain (Efusion / Edriver@target) targets – Gains ~ 50-100 required.

• Energy efficient (10-30%) high repetition rate (~5 Hz) driver.

– Both ion beams and lasers being considered

• Low cost target fabrication at high production rate (~100 million targets / year) – Target cost ~ 25-50 ¢/target.

• Long life fusion chamber /w low radioactivity.

NEEP 536, Spring 2003, D. Whyte

4

IFE in comparison to MFE

•

•

Both use D-T since it is the easiest fuel combination to achieve fusion power (also why it is used in bombs)

•

D-T plasma thermal temperature must be increased to ~ 10 keV for both.

Inertial confinement: we can guess that the reaction must occur very quickly since thermal velocity of 10 keV plasmas ~ 106 m/s.

•

Since IFE relies on a scaled-down thermonuclear explosion we immediately know this a pulsed power source. MFE is inherently steady-state.

•

Therefore we have to reformulate the equivalent of the Lawson criterion for IFE.

NEEP 536, Spring 2003, D. Whyte

5

IFE fundamentals • • • • • • • •

•

Spherical target.

D-T fuel is confined in a microballoon The driver deposits energy on the surface layer of the target, the ablator.

This layer becomes hot, generating high pressure.

The surface layer ablates (solid --> gas/plasma) outward at high speed.

This rocket forces acts to compress and heat the fuel.

A large part of the input energy is conveyed to the ablator.

Some part of this does work in compressing and heating the D-T fuel.

The ratio of the energy supplied to the D-T fuel at the instant the fuel is compressed to highest density to the total beam energy is the EDT hydrodynamic efficiency. = Edriver The typical efficiency for an ablation-driven rocket is ~10%

NEEP 536, Spring 2003, D. Whyte

6

IFE fundamentals

NEEP 536, Spring 2003, D. Whyte

7

Fuel and burn considerations

•

• •

A high burn fraction is necessary in IFE due to the low hydrodynamic efficiency of the drivers.

In IFE mass density, rho = m n, takes the place of density and r ~ tau / cs, takes the place of confinement time.

So for a 1/3 burn fraction at T~20 keV we require r ~ 3 g /cm2 This is equivalent to n τ ~ 2x1015 s/cm3 ~ 2x1021 s/m3.

r r ~ 8mics r + 6 g /cm 2 r+ σv D T

fburn

6 mi cs /sigma_v

•

1.00E+02

1.00E+01

1.00E+00 0

10

20

30

40

50

Temperature (keV)

NEEP 536, Spring 2003, D. Whyte

8

Ignition conditions •

Energy balance from self-heating gives us rough equivalent of Lawson criterion in IFE.

MFE D-T Lawson criterion Lawson criterion

– Efusion,alphas > Edriver

 r ~ 1.6 - 2 g/cm2 – ~ 1-2x1021 s/m3 in 10-20 keV range.

–

1E+21

τ

For hydrodynamic efficiency of 10%

n e (s / m3)

•

1E+22

1E+20 1

•

Therefore ignition criterion is comparable to MFE.

NEEP 536, Spring 2003, D. Whyte

10 T (keV)

100

9

Compression / size considerations •

From consideration of total electrical plant efficiency the energy gain per pellet must be ~ 100 --> linked to fburn -->  r

•

The fuel mass and driver energy necessary scales as 1/ 

•

At normal density, ~ 0.2 g/cm3 this would require 2500 g of DT, EDT ~ 5x1012 J and result in Efusion ~ 3x1014 J – ~ 70 kT explosion ~ Hiroshima.

•

Therefore we must rely on substantial compression (>x1000) in order to make the ignition energetically feasible and “controllable”.

– M ~ 2 mg, yield ~ 300 MJ, Edriver ~ MJ.

•

So if the fuel is heated to T~ 20 keV and is compressed to n ~ 5e31 m3 the pressure of the fuel is P ~ n k T ~ 2x1017 Pa…cannot be applied directly.

NEEP 536, Spring 2003, D. Whyte

10

The implosion of the ICF capsule is described by a rocket equation

=

V shell

•

•

Pa

m  m  o ln   V exhaust ln o  ṁ  m f   mf 

The energy efficiency of an ablator rocket is is typically a factor of 4 or more smaller than that of an ideal rocket because the exhaust is continually being heated by the incident flux driving the implosion.

For this reason the hydrodynamic efficiency in IFE is limited to ~ 10%….this is what drives the need for high pellet gain.

NEEP 536, Spring 2003, D. Whyte

11

Compression / size considerations •

Compression is energetically attractive.

•

Here we show that if the DT remains Fermi degenerate during the compressions process that the compression requires much less energy density than direct ignition.

NEEP 536, Spring 2003, D. Whyte

12

Compression / ignition issues •

• •

•

•

Although compression is energetically attractive and reduces the driver size required for ignition, high gain also requires hot-spot ignition.

5 mg target requires ~60-90 MJ driver if all heated to 10 keV. However if the target can ignited from a central hot spot containing about 2% of the total mass then the energy required to heat and compress this mass is ~ 1 MJ --> gain ~ 100.

The hot spot forms during compression from material the center of the fuel where its implosion is suddenly halted Further requirement is that the heating alphas deposit their energy into spot to propagate burn throughout the fuel mass.

NEEP 536, Spring 2003, D. Whyte

13

Implosion velocity • We can quickly estimate the implosion velocity from requirements for compression and expected driver parameters.

– E ~ MJ – Tpulse ~ 30 ns 2 – Ppulse ~ (no/n) * Phot-spot – Pellet radius ~ 4-5 mm --> Aspect ratio?

• Implosion velocity ~ 5-8 x 107 cm/s

NEEP 536, Spring 2003, D. Whyte

14

Uniformity •

Convergence ratio of initial outer radius to final compressed radius of the hot spot.

•

Typical convergence ratio is ~ 30 for high-gain target.

•

If target surface with average acceleration g has a location on it surface with acceleration perturbation g then we can estimate that the requirement for compression uniformity is ~ 1%.

NEEP 536, Spring 2003, D. Whyte

15

Ignition conditions for IFE • • •

The r expression replace n τ from MFE. The burn efficiency of the pellet must be ~ 3 g/cm2 to achieve 30% burn efficiency.

The equivalent Lawson criterion is not actually that different between IFE and MFE….but the absolute values are very different.

– Density is ~ 1011 higher in IFE (n~1031 m-3)…compression needed – Confinement time is ~ 1011 smaller in IFE…

•

•

Large compression of the fuel center to a “hot spot” is necessary for ignition..otherwise the energy of the driver and the resulting released energy become too large for technological reasons.

Uniform compression velocity / acceleration is required --> must avoid hydrodynamic instabilities.

NEEP 536, Spring 2003, D. Whyte

16

Ignition Physics: Simple energy balance model.

NEEP 536, Spring 2003, D. Whyte

17

Ignition Physics: Complete numerical simulation.

NEEP 536, Spring 2003, D. Whyte

18

Ignition conditions for IFE • • •

The r expression replace n τ from MFE. The burn efficiency of the pellet must be ~ 3 g/cm2 to achieve 30% burn efficiency.

The equivalent Lawson criterion is not actually that different between IFE and MFE….but the absolute values are very different.

– Density is ~ 1011 higher in IFE (n~1031 m-3)…compression needed – Confinement time is ~ 1011 smaller in IFE…

•

•

Large compression of the fuel center to a “hot spot” is necessary for ignition..otherwise the energy of the driver and the resulting released energy become too large for technological reasons.

Uniform compression velocity / acceleration is required --> must avoid hydrodynamic instabilities.

NEEP 536, Spring 2003, D. Whyte

1

Ignition Physics: Simple energy balance model.

NEEP 536, Spring 2003, D. Whyte

2

Ignition Physics: Complete numerical simulation.

NEEP 536, Spring 2003, D. Whyte

3

Numerical simulations have verified burn propagation

NEEP 536, Spring 2003, D. Whyte

4

D-T isentropes and shock dynamics Initial conditions R = 0.25 g/cm3 T=11.6 K S (108 J/g/keV)

Typical implosion P ~ 100 Mbar

Adiabatic compressibility Is weak function of s for s < 4 but is exponentially Dependent on s for s>4.

NEEP 536, Spring 2003, D. Whyte

5

Multiple-shocks (~3-4) are adequate to maintain ~isentropic compression --> Pulse shaping is necessary

NEEP 536, Spring 2003, D. Whyte

6

Pulse shaping experiment verified its effectiveness in compressibility

NEEP 536, Spring 2003, D. Whyte

7

Leading candidate for IFE is indirect drive

NEEP 536, Spring 2003, D. Whyte

8

A multitude of pellet designs have been experimented with: e.g. multiple pusher layers used for implosion velocity multiplication

NEEP 536, Spring 2003, D. Whyte

9

Experiments with high-pressure D-D gaseous target validated the Tr dependence for implosion --> yield.

NEEP 536, Spring 2003, D. Whyte

10

Energy ignition threshold is hyper-sensitive to the implosion velocity

−5 1  0.05  3/ 2  V Eth (MJ) =      2  hydro   3 10 7 m / s

NEEP 536, Spring 2003, D. Whyte

11

Demonstration of sharpness of the ignition cliff.

2 <  < 8.5

NEEP 536, Spring 2003, D. Whyte

12

Implosion velocity dynamics differ greatly between indirect and direct drive Indirect drive

Direct drive

I15 2/ 3 P(Mbar) = 40( ) 

7/ 8 P(Mbar) = 3Tr3.5 = 170I15

Vimplosion (cm / s) = 10

7

Tr

ln

mo

= 1.8e7 I1/ 8 ln

m

Radiation temperature important

NEEP 536, Spring 2003, D. Whyte

15

mo m

Vimplosion (cm / s) = 1.5e8(I2)1/ 3 ln

mo  V

exhaust

m

ln

mo m

Laser wavelength important

13

Gain vs. laser energy designed for NIF

NEEP 536, Spring 2003, D. Whyte

14

Leading candidate for IFE is indirect drive

NEEP 536, Spring 2003, D. Whyte

8

A multitude of pellet designs have been experimented with: e.g. multiple pusher layers used for implosion velocity multiplication

NEEP 536, Spring 2003, D. Whyte

9

Energy ignition threshold is hyper-sensitive to the implosion velocity

 1  0.05  3/ 2  V Eth (MJ) =     2 hydro   3 10 7 m / s

−5

NEEP 536, Spring 2003, D. Whyte

10

Demonstration of sharpness of the ignition cliff.

2 <  < 8.5

NEEP 536, Spring 2003, D. Whyte

11

Implosion and ablation velocity dynamics differ greatly between indirect and direct drive Indirect drive

Direct drive

I P(Mbar) = 40( 15 ) 2/ 3

P(Mbar) = 3T 3.5 = 170I 7/ 8 r

Vimplosion (cm / s) = 10

7

Tr

ln



15

mo

= 1.8e7 I

m

1/ 8

ln

15

m

Vablation (cm / s) = 2e5 3/ 5T 9 /10 = 6e5 3/ 5 I 9/ 40 r

mo

15

Vimplosion (cm / s) = 1.5e8(I )

2 1/ 3

ln

mo  V

mo

m

m

exhaust ln

Vablation (cm / s) = 4.3e4 3/ 5I−1/15−14 /15 15

Radiation temperature important

Laser wavelength important

High ablation rate, low exhaust velocity, High ablator mass burnup

High exhaust velocity

NEEP 536, Spring 2003, D. Whyte

12

Experiments with high-pressure D-D gaseous target validated the Tr dependence for implosion --> yield.

NEEP 536, Spring 2003, D. Whyte

13

The shell In-Flight Aspect Ratio (IFAR) is examined for purposes of understanding hydrodynamic stability t1

mo R = V ln dt  exhasut 0 vdt = 2 m 0

• IFAR is defined as the ratio of the shell radius.

• Is related to implosion, exhaust and ablation velocities.

• Assumes shell is accelerated over half of it radius

m1 mo

1 Vablatet ∆R

ln

m1 Vimplosion = mo Vexhaust

V R V 2 exhaust f ( implosion ) Vablation Vexhaust ∆R 1.2 1 0.8

f(x)

– Most of the work done through ∫vol PdV is performed early.

t1

1

0.6 0.4 0.2 0 0

2

4

6

X NEEP 536, Spring 2003, D. Whyte

14

IFAR dynamics differ significantly between indirect and direct drive Indirect drive

Direct drive

Vimplosion > Vexhaust --> f ~ linear in x

4 1 2 P πR3 MVimp 3 2 R 3 V2 or ∆R 2 P PV

V V R ≈ 0.56 implosion , for 4 > imp > 0.8 ∆R Vablation Vex R

∆R

R 21(Vimplosion / 3e7) ≈ 3/ 5 9/ 40 ∆R I15

≈ 0.7

2 Vimplosion

VexhaustVablation

80(Vimplosion / 3e7) R ≈ 3/ 5 (I15 / )4 /15 ∆R

2

1 2 4πR2∆R  Vimp 2

, for 0.8 >

Vimp Vex

> 0.1

60(Vimplosion / 3e7)

2

3/ 5 4 /15 15

I

Taking 1/3 micron laser wavelength Taking same intensity, indirect drive at v=3e7 cm/s has ~x3 less IFAR than direct drive…this is direct Consequence of the effective shorter wavelengths being used in radiation / indirect drive

I  Tr4 NEEP 536, Spring 2003, D. Whyte

  T−1

Vimplosion

R 0.9 Tr ∆R

15

Rayleigh-Taylor is the most important/detrimental instability for ICF • •

Driving force --> acceleration Damping force --> ablation velocity

•

Original (ca 1972) estimates for RT growth rates.

Too optimistic for laser absorption and RT --> led to predictions of ignition with ~ 1 kJ driver Through its links to implosion velocity through shell geometry (RT is essentially a surface pheonomenon) during the compression the RT instability growth sets the minimum driver size…here n-5 !!

• •

NEEP 536, Spring 2003, D. Whyte

Growth amplitude

e te ( kx + kV t )  e(

e te kx →

2

abl

ka(1 k∆R)

kVa )t kx

e

Max. e-foldings for RT

n max 2 R 80(Vimplosion / 3e7) ≈ 3/ 5 ∆R (I15 / )4 /15

max dt ≈

1 R 2 ∆R

60(Vimplosion / 3e7)2 3/ 5 4 /15 I 15

−5 1  0.05  3/ 2  V −5 −2/3  n I Eth (MJ) =      2  hydro   3 107 m / s 16

Experimental and theoretical assessment of Rayleigh-Taylor instability

ka 1+ kL

k̃V

ablation

L is density gradient scale length across boundary  is a constant ~ 1 for indirect drive and ~ 3 for direct drive

NEEP 536, Spring 2003, D. Whyte

17

RT dynamics differ significantly between indirect and direct drive Indirect drive

Direct drive

Vimplosion > Vexhaust

l

n≈

1+ 0.2l

∆R R

0.8l

nmax ≈ 8.5

∆R R

≈

E driver

2/ 5

nmax 0.5 8.5

Find permitted IFAR ~ 30 if we allow n=6 Maximum growth for l~100

V 3e7

1.4

I151/15

25 / 7 1/14

I 5/ 21 15

Taking 1/3 micron laser wavelength and 5% thermodynamic efficiency

Large R is also found for indirect drive.

Complete simulations indicate much lower RT growth rates for indirect drive.

NEEP 536, Spring 2003, D. Whyte

18

3-D radiation hydrodynamic models have been developed to predict RT.

NEEP 536, Spring 2003, D. Whyte

19

Experiments using intentionally “bumpy” capsule surfaces have verified RT growth models

NEEP 536, Spring 2003, D. Whyte

20

Models and experiments indicate surface roughness must be limited to 10 nm range

NEEP 536, Spring 2003, D. Whyte

21

Gain vs. laser energy designed for NIF

NEEP 536, Spring 2003, D. Whyte

22

The National Ignition Facility is being constructed at LLNL

NEEP 536, Spring 2003, D. Whyte

22

The National Ignition Facility is being constructed at LLNL Approximately same Scale and cost as ITER But very large difference in Fusion energy output One ITER pulse: ~ 100 GJ, several /day One NIF pulse: ~10 MJ, one / day

NEEP 536, Spring 2003, D. Whyte

23

NIF is designed with glass-lasers for indirect drive

NEEP 536, Spring 2003, D. Whyte

24

NIF laser and target chamber

NEEP 536, Spring 2003, D. Whyte

25

NIF target, laser and hohlraum needs

NEEP 536, Spring 2003, D. Whyte

26

Gain vs. laser energy designed for NIF

NEEP 536, Spring 2003, D. Whyte

27

Magnets for fusion •

The success of MFE depends greatly on our ability to create and maintain high magnetic fields.

•

Magnet construction tends to be the single largest component cost in the construction of the fusion “core”.

•

For energy production we require steady-state magnetic fields – Confinement --> toroidal B2 – Shaping and equilibrium control

NEEP 536, Spring 2003, D. Whyte

1

Normal conducting magnets do not seem feasible for tokamak reactors The simple power loss from ohmic Dissipation in the toroidal field Coils would be too large for Cu magnets on a conventional tokamak

CW Cooling requirements for the magnet heat Dissipation disallow very large current Densities --> fields

Pdissipation > Pfusion

NEEP 536, Spring 2003, D. Whyte

2

Superconductors are the design choice for almost all MFE reactor designs • Zero dc electrical resistivity below critical temperature (~ 10’s K) • Meissner effect: Excludes magnetic flux (perfect diamagnet).

• Arises from ordered pairing of conduction electrons in metals (Cooper pair).

• SC state is destroyed by a sufficiently strong magnetic field --> limited current can be passed through.

NEEP 536, Spring 2003, D. Whyte

3

Type-II superconductor materials

NEEP 536, Spring 2003, D. Whyte

4

Flux penetration, Lorentz force and flux pinning determine the maximum B and I for the SC coil

Liquid helium cooling: T=4.2 K

NEEP 536, Spring 2003, D. Whyte

5

SC magnetic coil design must ensure stability against going normal and mechanical stresses

SS casing provides Mechanical Support against Large JxB forces

NEEP 536, Spring 2003, D. Whyte

Cu provides Thermal stability To transients

6

ITER superconducting magnet design

NEEP 536, Spring 2003, D. Whyte

7

Design parameters for the ITER coil systems

NEEP 536, Spring 2003, D. Whyte

8

The scale of the ITER magnets is impressive/scary

NEEP 536, Spring 2003, D. Whyte

9

An additional design feature needed is a crysostat to ensure thermal isolation Volume: 14,000 m3 Evacuated 400 penetrations for torus access Heat shields placed between warm vacuum vessel and magnets.

NEEP 536, Spring 2003, D. Whyte

10

Magnets for fusion •

The success of MFE depends greatly on our ability to create and maintain high magnetic fields.

•

Magnet construction tends to be the single largest component cost in the construction of the fusion “core”.

•

For energy production we require steady-state magnetic fields – Confinement --> toroidal B2 – Shaping and equilibrium control

NEEP 536, Spring 2003, D. Whyte

1

Normal conducting magnets do not seem feasible for tokamak reactors The simple power loss from ohmic Dissipation in the toroidal field Coils would be too large for Cu magnets on a conventional tokamak

CW Cooling requirements for the magnet heat Dissipation disallow very large current Densities --> fields

Pdissipation > Pfusion

NEEP 536, Spring 2003, D. Whyte

2

Superconductors are the design choice for almost all MFE reactor designs • Zero dc electrical resistivity below critical temperature (~ 10’s K) • Meissner effect: Excludes magnetic flux (perfect diamagnet).

• Arises from ordered pairing of conduction electrons in metals (Cooper pair).

• SC state is destroyed by a sufficiently strong magnetic field --> limited current can be passed through.

NEEP 536, Spring 2003, D. Whyte

3

Type-II superconductor materials

NEEP 536, Spring 2003, D. Whyte

4

Flux penetration, Lorentz force and flux pinning determine the maximum B and I for the SC coil

Liquid helium cooling: T=4.2 K

NEEP 536, Spring 2003, D. Whyte

5

SC magnetic coil design must ensure stability against going normal and mechanical stresses

SS casing provides Mechanical Support against Large JxB forces

NEEP 536, Spring 2003, D. Whyte

Cu provides Thermal stability To transients

6

ITER superconducting magnet design

NEEP 536, Spring 2003, D. Whyte

7

Design parameters for the ITER coil systems

NEEP 536, Spring 2003, D. Whyte

8

The scale of the ITER magnets is impressive/scary

NEEP 536, Spring 2003, D. Whyte

9

An additional design feature needed is a crysostat to ensure thermal isolation Volume: 14, 000 m3 Evacuated 400 penetrations for torus access Heat shields placed between warm vacuum vessel and magnets.

NEEP 536, Spring 2003, D. Whyte

10

Schematic of fusion power plant

NEEP 536, Spring 2003, D. Whyte

11

Neutrons in fusion reactors are both useful and harmful: 80% of DT fusion energy • Breeds tritium to close fuel cycle (n + Li) • Generates nuclear heat • Produces radiation damage in materials • Activates reactor material via transmutations (generates radioactive waste).

• Must stop high-energy neutron damage to SC coils – Necessitates blanket as a “shield” NEEP 536, Spring 2003, D. Whyte

12

14.1 MeV neutron history • Immediately leaves the fusion plasma (no charge).

• Penetrates with ~10 cm 1/e folding length into most materials --> blanket • Loses energy to collisions with blanket material: – Lithium --> must produce > 1 T per neutron (TBR: tritium breeding ratio) --> lithium is majority of blanket material.

– Consider using neutron multipliers to increase tritium breeding ratio.

– Neutron slows/moderates through interactions with solid --> displaces atoms within lattice (DPA) – Nuclear reactions /w non Li --> transmutations.

NEEP 536, Spring 2003, D. Whyte

13

Lithium-neutron tritium cross-sections: blanket must moderate neutrons

NEEP 536, Spring 2003, D. Whyte

14

Typical neutron energy spectrum at first wall: Normalized to 1 MW/m2 wall loading

NEEP 536, Spring 2003, D. Whyte

15

Schematic of fusion power plant

NEEP 536, Spring 2003, D. Whyte

1

Neutrons in fusion reactors are both useful and harmful: 80% of DT fusion energy • Breeds tritium to close fuel cycle (n + Li) • Generates nuclear heat • Produces radiation damage in materials • Activates reactor material via transmutations (generates radioactive waste).

• Must stop high-energy neutron damage to SC coils – Necessitates blanket as a “shield” NEEP 536, Spring 2003, D. Whyte

2

Typical blanket design components

NEEP 536, Spring 2003, D. Whyte

3

Lithium-neutron tritium cross-sections: blanket must moderate neutrons

NEEP 536, Spring 2003, D. Whyte

4

Complicated neutron interactions with blanket materials are solved by numerical simulations

NEEP 536, Spring 2003, D. Whyte

5

14.1 MeV neutron history • Immediately leaves the fusion plasma (no charge).

• Penetrates with ~10 cm 1/e folding length into most materials --> blanket • Loses energy to collisions with blanket material: – Lithium --> must produce > 1 T per neutron (TBR: tritium breeding ratio) --> lithium is majority of blanket material.

– Consider using neutron multipliers to increase tritium breeding ratio.

– Neutron slows/moderates through interactions with solid --> displaces atoms within lattice (DPA) – Nuclear reactions /w non Li --> transmutations.

NEEP 536, Spring 2003, D. Whyte

6

Neutron can react in many ways with the solid blanket materials

NEEP 536, Spring 2003, D. Whyte

7

Some important neutron reactions

Energy releasing nuclear reactions

Neutron multiplication

E14.1 MeV Neuton ~ 17-24 MeV

Helps in maintaining acceptable Tritium breeding ratios

NEEP 536, Spring 2003, D. Whyte

8

Neutrons and auxiliary gamma-rays lead to heating in near-surface of blanket

NEEP 536, Spring 2003, D. Whyte

9

Primary effect of neutrons: Atomic displacement and He/H production

Dpa per 1e21 14 MeV neutrons

NEEP 536, Spring 2003, D. Whyte

Helium production Rate per 1e21 14 MeV neutrons

10

The magnitude of direct neutron effects on materials

NEEP 536, Spring 2003, D. Whyte

11

Neutron radiation degrades material viability Swelling for Ti-modified austentic SS At 300 C

Reduction in tensile elongation limits for 316 SS caused by neutron irradiation.

Highlights needs for materials Test facility with high fusion Neutron fluence NEEP 536, Spring 2003, D. Whyte

12

Neutron fluence also degrades thermal fatigue limits

316 SS @430 C

NEEP 536, Spring 2003, D. Whyte

13

Neutron damage is very harmful to Nb3Sn superconductor ability

NEEP 536, Spring 2003, D. Whyte

14

The resistivity of typical (normal) conductors is greatly enhanced by atom displacement.

NEEP 536, Spring 2003, D. Whyte

15

Neutron shields must surround the blanket to attenuate/modify/capture neutrons

NEEP 536, Spring 2003, D. Whyte

16

Neutron capture transmutation of blanket materials leads to substantial activation --> radioactive waste for fusion Activation level: Curies / kWth

NEEP 536, Spring 2003, D. Whyte

17

DT fusion has somewhat lower waste disposal issues --> but large volume of waste

NEEP 536, Spring 2003, D. Whyte

18

Neutron capture transmutation of blanket materials leads to substantial activation --> radioactive waste for fusion Activation level: Curies / kWth

NEEP 536, Spring 2003, D. Whyte

16

DT fusion has somewhat lower waste disposal issues --> but large volume of waste

NEEP 536, Spring 2003, D. Whyte

17

Coolants He gas Benefits: Operates at high T --> better thermal efficiency Inert gas Detriments: Poor thermal conductivity, high T incompatible with many materials, impurity corrosion Water Benefits: Large experience database, good heat removal Detriments: Requires high pressure (~ 140 bar), Tmax~ 320 C low outlet temperature --> thermal inefficiency Safety (lithium explosions)

Liquid metal Benefits: Heat removal & breeder rolled into one.

Detriments: MHD, corrosion of structural material high melting point (slugged breeder), chemical reactivity

NEEP 536, Spring 2003, D. Whyte

18

Activation implies that all maintenance on fusion core is robotic --> $$$

NEEP 536, Spring 2003, D. Whyte

19

Material’s lifetime: limited by neutron fluence and temperature

NEEP 536, Spring 2003, D. Whyte

20

Lithium-neutron tritium cross-sections: blanket must moderate neutrons Lithium isotope Abundance: 6Li: 7.5% 7Li: 92.5 %

NEEP 536, Spring 2003, D. Whyte

21

Tritium breeding from lithium: breeding yield and safety are primary concerns Candidate breeding materials Chemical reactivity

NEEP 536, Spring 2003, D. Whyte

22

Tritium breeding ratio MUST be > 1

Thicker blanket helps but leads to greater cost (larger magnets) Benefit of neutron multipliers obvious

NEEP 536, Spring 2003, D. Whyte

23

Realistic TBR ~ 1.1-1.2 because of finite structural losses and blanket geometry

NEEP 536, Spring 2003, D. Whyte

24

Tritium recovery through the vessel must be very efficient

Since tritium breeding ratio is about 1.1 tritons/neutron, almost all tritons introduced into the vacuum vessel must be burned.

Tritium self-sufficiency requires that: Tritium recovery must be very efficient ITER as example: 4 10 1000s shots burns 25kg T , while safety limit is 0.3 kg T in vessel.

NT triton flow – through rate = = 1 helium pumping rate 2 cHe N He

≈ 5 ~ 25

cHe = nHe / (n D + nT) = core plasma helium concentration ≈ 0.1

 =

n o He 2 no, 2 D + T

n D + nT = helium (de)enrichment ~ 0.2 nHe

Rate of tritium deposition in wall = NT,wall ≤

NHe N ~ T 10 250

Present experiments: T retention rate ~20% !!

Two orders of magnitude too large.

NEEP 536, Spring 2003, D. Whyte

25

Tritium inventory and safety Because of the low burn efficiency in fusion, this requires large T inventories On-site to ensure power production: 1 Mwe plant: ~ 40 kg Tritium inventory --> 400 M Ci

Exposure safety limit: 0.1 micro Ci / ml of tritiated water At these levels biological hazard results from OH radical formation during HTO beta decay rather than from beta radiation itself.

Biological half life ~ 10 days.

NEEP 536, Spring 2003, D. Whyte

26

Tritium breeding ratio MUST be > 1

Thicker blanket helps but leads to greater cost (larger magnets) Benefit of neutron multipliers obvious

NEEP 536, Spring 2003, D. Whyte

23

Realistic TBR ~ 1.1-1.2 because of finite structural losses and blanket geometry

NEEP 536, Spring 2003, D. Whyte

24

Tritium recovery through the vessel must be very efficient

Since tritium breeding ratio is about 1.1 tritons/neutron, almost all tritons introduced into the vacuum vessel must be burned.

Tritium self-sufficiency requires that: Tritium recovery must be very efficient ITER as example: 4 10 1000s shots burns 25kg T , while safety limit is 0.3 kg T in vessel.

NT triton flow – through rate = = 1 helium pumping rate 2 cHe N He

≈ 5 ~ 25

cHe = nHe / (n D + nT) = core plasma helium concentration ≈ 0.1

 =

n o He 2 no, 2 D + T

n D + nT = helium (de)enrichment ~ 0.2 nHe

Rate of tritium deposition in wall = NT,wall ≤

NHe N ~ T 10 250

Present experiments: T retention rate ~20% !!

Two orders of magnitude too large.

NEEP 536, Spring 2003, D. Whyte

25

Tritium inventory and safety Because of the low burn efficiency in fusion, this requires large T inventories On-site to ensure power production: 1 Mwe plant: ~ 40 kg Tritium inventory --> 400 M Ci

Exposure safety limit: 0.1 micro Ci / ml of tritiated water At these levels biological hazard results from OH radical formation during HTO beta decay rather than from beta radiation itself.

Biological half life ~ 10 days.

NEEP 536, Spring 2003, D. Whyte

26

MFE Reactor studies • Bring all the pieces of the puzzle into a large systems code.

• Takes into account capital and operating costs per standard techniques.

• Strict safety criteria set for site.

• Not surprisingly, if one picks more “aggressive” physics (smaller size, higher beta, etc) the C.O.E. decreases --> but with greater risk of outright failure.

• The operation of interim devices like ITER will allow us to better assess those risks.

NEEP 536, Spring 2003, D. Whyte

27

Conceptual 1000 MWe power plants are based on the latest physics advances from actual tokamak experiments

NEEP 536, Spring 2003, D. Whyte

28

Individual advances on several fronts help improve the attractiveness of fusion 80

ARIES-RS

COE, mill/kWe-h

AT physics, PbLi/SiC

70 →59%

60

LSA=1 HTSC A=80%

50 ARIES-AT

Improvements NEEP 536, Spring 2003, D. Whyte

29

Current drive in ARIES-AT: 91% bootstrap current

NEEP 536, Spring 2003, D. Whyte

30

ARIES-AT device parameters: notice that is nearly identical to ITER but with about 4 times the fusion power

NEEP 536, Spring 2003, D. Whyte

31

Sample calculation: Outer divertor target heat load

NEEP 536, Spring 2003, D. Whyte

32

Reactor design effects on total waste

NEEP 536, Spring 2003, D. Whyte

33

Our Vision of Magnetic Fusion Power Systems Has Improved Dramatically in the Last Decade, and Is Directly Tied to Advances in Fusion Science & Technology

Major radius (m) Estimated Cost of Electricity (¢/kWh)

10

9

14 12

8 7

10 8 6 4

6 5 4

3

2 0

2 1

Mid 80's Physics

Early 90's Physics

Late 90's Physics

Advance Technology

0

Mid 80's Pulsar

ARIES-AT parameters: Major radius: 5.2 m Toroidal : 9.2% Wall Loading: 4.75 MW/m2 NEEP 536, Spring 2003, D. Whyte

Early 90's ARIES-I

Fusion Power Net Electric COE

Late 90's ARIES-RS

2000 ARIES-AT

1,720 MW 1,000 MW 5.5 ¢/kWh 34