Traping Boundary of Turbulent Magnetic Field Lines

Trapping Boundary of Turbulent Magnetic Field Lines

Mr. Jakapan Meechai

Senior Project submitted to the Department of Physics, Chulalongkorn University in partial fulfillment of the requirements for the degree of Bachelor of Science Academic Year 2003

Project Title: Name: Project Advisor: Project Co-Advisor: Department: Academic Year:

Trapping Boundary of Turbulent Magnetic Field Lines Mr. Jakapan Meechai Asst. Prof. Dr. Udomsilp Pinsook Assoc. Prof. Dr. David Ruffolo Physics 2003

Abstract We use a computer program to simulate turbulent magnetic field lines in the 2D+slab model of turbulence, which has been shown to provide a good description of solar wind turbulence. In this model, the magnetic field lines seem to be temporarily trapped inside or outside of certain boundaries. We develop a mathematical definition of local trapping boundaries, which are found to coincide with the edges of the trapping regions of these turbulent magnetic field lines in computer simulations.

i

Acknowledgements I am grateful to my Co-Advisor, Assoc. Prof. Dr. David Ruffolo, for many guidance and suggestions, for giving me a chance to use the scientific side of my mind (by this project) and correcting my English. I am thankful to Asst. Prof. Dr. Udomsilp Pinsook for being my advisor and giving me some recommendations. I am also thankful to Asst. Prof. Manit Rujiwarodom, and Dr. Thiti Bovornratanaraks for being my senior project examiners.

I would like to thank Miss Piyanate Chuychai, Mr. Peera Pongkitiwanichkul and everybody in the “Astrophysics Laboratory” for their kind instructions and recommendations. This work could not be finished without them.

I also would like to thank everybody at the Department of Physics, Chulalongkorn University, everybody at the Gr.3, everybody at the “BaanDork”, and all other friends from other faculties for everything in my university life. Many thanks to everybody at the “Music Club”, Faculty of Science, Chulalongkorn University. They always brighten my days up humorously.

Finally, I wish to thank my family for their love. Special thanks to someONE for her existence in this world.

ii

Table of Contents

Abstract

i

Abstract in Thai

ii

Acknowledgements

iii

Table of Contents

iv

List of Figures

vi

1 Introduction

1

2 Turbulent Magnetic Field Model and Diffusion Theory

5

2.1

Turbulent Magnetic Field Model . . . . . . . . . . . . . . . . . . .

5

2.2

Diffusion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.3

Temporary Trapping . . . . . . . . . . . . . . . . . . . . . . . . .

10

3 Simulation of Turbulent Magnetic Field Lines 3.1

3.2

15

Generating the Magnetic Field . . . . . . . . . . . . . . . . . . . .

15

3.1.1

Generating the Slab Field . . . . . . . . . . . . . . . . . .

16

3.1.2

Generating the 2D Field . . . . . . . . . . . . . . . . . . .

18

Tracing the Magnetic Field Lines . . . . . . . . . . . . . . . . . .

19

3.2.1

19

Tracing Slab Trajectories . . . . . . . . . . . . . . . . . . .

iii

3.3

3.2.2

Tracing 2D Trajectories . . . . . . . . . . . . . . . . . . .

21

3.2.3

Tracing 2D+slab Trajectories . . . . . . . . . . . . . . . .

22

Local Trapping Boundaries . . . . . . . . . . . . . . . . . . . . . .

23

4 Results and Discussion

25

5 Conclusions

33

References

34

Appendices

35

A Numerical Methods

35

A.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

A.1.1 Linear Interpolation . . . . . . . . . . . . . . . . . . . . .

35

A.1.2 Bilinear Interpolation . . . . . . . . . . . . . . . . . . . . .

36

A.2 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . .

37

B Source Code

38

iv

List of Figures 1.1

Interplanetary magnetic field . . . . . . . . . . . . . . . . . . . . .

1.2

Energy of H-Fe ions (in units of MeV nucleon−1 ) vs. arrival time

2

at 1 AU for impulsive flare events of 1999 January 9-10 (Mazur et al. 2000). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.1

Pure slab magnetic field lines . . . . . . . . . . . . . . . . . . . .

7

2.2

Contours of a magnetic potential function for 2D turbulence . . .

8

2.3

A pure 2D magnetic field line . . . . . . . . . . . . . . . . . . . .

9

2.4

A 2D+slab (80:20) magnetic field line . . . . . . . . . . . . . . . .

10

2.5

Interplanetary magnetic field lines (Ruffolo et al. 2003) . . . . . .

12

2.6

Contours of a magnetic potential function with O-points and Xpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.7

Scatter plot of magnetic field lines with dropouts . . . . . . . . .

14

3.1

Kolmogorov power spectrum . . . . . . . . . . . . . . . . . . . . .

15

4.1

Contours of a magnetic potential function . . . . . . . . . . . . .

26

4.2

Scatter plot of initial magnetic field line locations at z = 0. . . . .

27

4.3

Scatter plot of locations of magnetic field lines at about 0.25 AU from the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

27

Scatter plot of locations of magnetic field lines at about 0.50 AU from the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

28

4.5

Scatter plot of locations of magnetic field lines at about 0.75 AU from the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6

Scatter plot of locations of magnetic field lines at about 1.00 AU from the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

29

Scatter plot of locations of magnetic field lines at about 1.50 AU from the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.9

29

Scatter plot of locations of magnetic field lines at about 1.25 AU from the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

28

30

Scatter plot of locations of magnetic field lines at about 1.75 AU from the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

4.10 Scatter plot of locations of magnetic field lines at about 2.00 AU from the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

4.11 Scatter plot of locations of magnetic field lines at about 2.25 AU from the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

4.12 Contour plot of b22D . . . . . . . . . . . . . . . . . . . . . . . . . .

32

A.1 Linear interpolation at point P between two values F (xi ) and F (xi+1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

A.2 Bilinear interpolation at point S . . . . . . . . . . . . . . . . . . .

36

vi

Chapter 1 Introduction The solar wind is a stream of ionized gas flowing outward from the Sun in all directions at speeds of about 400 km/s (about 1 million miles per hour). It is caused by the heat of the outermost region of the Sun which is called the corona. The corona can be seen with the naked eyes during a solar eclipse. The temperature of the corona is very high, so hot that the coronal gas can expand and escape gravitational attraction. The solar wind also drags the solar magnetic field outward from the Sun to be the interplanetary magnetic field (IMF). Although the solar wind moves out almost radially from the Sun, the rotation of the Sun gives the magnetic field a spiral form (garden hose effect).

Turbulence is an irregular flow of fluids. Most flows occurring in nature are turbulent. The turbulent flows must have these following characteristics.

First, the turbulent flows must have irregularity or randomness which makes it impossible to use a deterministic approach to turbulence problems. Instead, we rely on statistical methods. Second, the turbulent flows must have diffusivity which causes a rapid rate of momentum, heat, and mass transfer. Third, turbulent flows occur at high Reynolds number1 . Fourth, turbulent flows are always dissipative. The large scale must feed energy to the smaller scale with some losses, 1

The fluid velocity times length scale divided by kinematic viscosity, R =

1

uL ν .

2

Figure 1.1: Interplanetary magnetic field so turbulence will decay rapidly if there is no continuous supply of energy.

The solar wind is one example of a turbulent flow. It also drags out the interplanetary magnetic field, so this is a turbulent magnetic field. As we know that charged particles generally gyrate and travel along magnetic field lines, this interplanetary magnetic field will control the path of accelerated charged particles from solar events (such as solar flares and coronal mass ejections), which are called solar energetic particles (SEP). The transport of solar energetic particles can be divided into two different types. One is the transport along the mean magnetic field. It is called parallel transport. Another is perpendicular transport, the transport along magnetic field lines as they separate from the mean magnetic field. Solar energetic particles diffuse away from the mean field by this kind of transport.

3 The Advanced Composition Explorer (ACE) observed the transport of solar energetic particles in many solar events. In a large number of impulsive solar events, “dropouts” can be seen. Dropouts refer to discontinuous data where the intensity of solar energetic particles repeatedly disappears and reappears (see Figure 1.2). This observation seems to indicate that solar energetic particles are

Figure 1.2: Energy of H-Fe ions (in units of MeV nucleon−1 ) vs. arrival time at 1 AU for impulsive flare events of 1999 January 9-10 (Mazur et al. 2000). confined to filaments in space, and therefore have a slow diffusion rate. However, observations from the IMP-8 and Ulysses spacecraft while they were on opposite sides of the Sun showed very similar intensity profiles, indicating a fast diffusion rate of solar energetic particles. The contradictory observations have been explained recently by the idea that the solar energetic particles are trapped by the small-scale topology of solar wind turbulence [1]. The dropouts reflect that particles are temporarily trapped in three-dimensional filaments corresponding to the shapes of magnetic potential islands in the two perpendicular directions.

These are irregular, non-geometric shapes.

This undergraduate senior project will examine the boundary of trapping by using the turbulence model called the “2D+slab Model” [3], and will study how the magnetic field lines and the boundary are related. The explanation of the

4 model will appear in the next chapter. Chapter 3 will explain the simulation of magnetic field lines and the method of finding the trapping boundary of turbulent magnetic field lines. The fourth and fifth chapters will show the results and conclusions from the simulations, respectively.

Chapter 2 Turbulent Magnetic Field Model and Diffusion Theory 2.1

Turbulent Magnetic Field Model

This senior project uses the 2D+slab model of turbulence, which is suited to the actual solar wind (Matthaeus, Goldstein, and Roberts 1990) and can explain the parallel transport of solar energetic particles in the solar wind (Bieber et al.

1994). In this model, we assume that the total magnetic field can be separated into two terms: B = B0 + b(x, y, z).

(2.1)

The mean field B0 is constant, not depending on the position coordinates x, y and z. The mean field also has the constant direction ˆ z, the direction along the spiral curve of the interplanetary magnetic field (Figure 1.1). Another term is b, the fluctuating field which is perpendicular to the mean field and depends on x, y and z: B0 = B0ˆ z,

b⊥ˆ z.

(2.2)

We model the fluctuating field by two terms. One is a term depending on x and y. Another one depends on only z. They are called the 2D term and slab term,

5

6 respectively: b = b2D (x, y) + bslab (z).

(2.3)

From Maxwell’s equations, we know that ∇ · B = 0,

(2.4)

∇ · B = ∇ · B0 + ∇ · b2D (x, y) + ∇ · bslab (z) = 0.

(2.5)

so

B0 does not depend on the position coordinates x, y and z, so ∇ · B0 = 0.

(2.6)

∂bslab ∂bslab y (z) x (z) + = 0, ∂x ∂y

(2.7)

In general, we also know that ∇ · bslab (z) =

slab because bslab x (z) and by (z) do not depend on x and y. Therefore we must have

∇ · b2D (x, y) = 0.

(2.8)

b2D (x, y) = ∇ × [a(x, y)ˆ z],

(2.9)

This implies that

where aˆ z can be interpreted as a vector potential for the 2D component of turbulence. Here a(x, y) is called the magnetic potential function.

Both 2D and slab terms are random functions that depend on the Kolmogorov power spectrum. This spectrum is the natural spectrum of turbulence phenomena. We generate the power spectrum first, and then transform to the magnetic field by an inverse Fourier transform. The explanation of the power spectrum and method of generating field lines will appear in the next chapter.

The 2D and slab fluctuating fields are random functions that average to zero.

7 This makes sense because it leaves the first term in Equation 2.1 as the mean field. If there is only a slab fluctuating field, the magnetic field lines undergo a simple random walk in x and y. The shapes of the magnetic field lines are similar if they have same initial z position (beginning on the same x-y plane). Figure 2.1 shows an example of magnetic field lines with only a slab fluctuating field.

Figure 2.1: Pure slab magnetic field lines For 2D fluctuations at every z, we have the same contours of the magnetic potential function. Figure 2.2 is an example showing contours of the magnetic potential function. The color shows the magnitude of the magnetic potential function. White coloring is for highly positive values, black coloring is for highly negative values, and grey coloring is for positive or negative values between those for white and black. If there is only a 2D fluctuating field, the magnetic field lines follow equipotential lines of the magnetic potential function (which have the same color in the contour plot). We must combine slab and 2D components to complete the turbulence model. Figure 2.4 shows an example of magnetic field

8

Figure 2.2: Contours of a magnetic potential function for 2D turbulence lines with both 2D and slab fluctuating fields.

2.2

Diffusion Theory

By using the 2D+slab model of turbulence, we can explain the magnetic field line trajectory as a diffusive random walk. The ensemble average statistics of the field line random walk were calculated by Matthaeus et al. (1995). A diffusion coefficient, D, is defined by h∆x2 i = 2D∆z,

(2.10)

where ∆x is the change in a perpendicular coordinate and ∆z is the distance along the mean field. The diffusion coefficient of only the slab fluctuating field is Dslab ∼

hb2x i lc , B02

(2.11)

9

Figure 2.3: A pure 2D magnetic field line where lc is the correlation range R∞ lc ≡

0

hbx (z0 )bx (z0 + ∆z)id∆z . hb2x i

(2.12)

Dslab is very small (≈ 5 × 10−4 AU) under the normal solar wind conditions, so the magnetic field lines diffuse very slowly if we have only the slab fluctuating field. The diffusion coefficient of the 2D fluctuating field alone is p e hb2 i λ D2D ∼ √ , 2 B0 e is referred to as the “ultrascale” (Matthaeus et al. 1995), where λ s a2 e= λ . hb2 i2D

(2.13)

(2.14)

If we have only a 2D fluctuating field, the magnetic field lines are trapped in some equipotential lines permanently, so we must combine the two types of fluctuating fields. The overall value of the diffusion coefficient is " # 21 2 Dslab Dslab D= + + (D2D )2 .

2 2

(2.15)

10

Figure 2.4: A 2D+slab (80:20) magnetic field line Under normal solar wind conditions, the 2D component of turbulence dominates (Matthaeus et al. 1995), and D2D is much greater than Dslab (Ruffolo, Matthaeus, and Chuychai 2003). Because of the high diffusion coefficient, the magnetic field lines can diffuse far from the mean field at long distances from the Sun.

2.3

Temporary Trapping

Ensemble average statistics for the 2D+slab model give a diffusion coefficient in the long-distance limit. Over shorter distances, the diffusion of a magnetic field line in the 2D+slab model of solar wind turbulence depends on its starting point.

We call these conditional statistics. There are two interesting types of points in the contour plot of the magnetic potential function. First are the local maxima or minima of the magnetic potential function, which are called the O-points.

At these points, 2D fluctuating field is very small, so the slab fluctuating field

11 will dominate. The magnetic field lines that start near the O-points will diffuse slowly due to the slab fluctuating field and then may be temporarily trapped within “islands� of 2D turbulence, if the islands are not smaller than the mean free path of slab trajectories. By an island we mean the enclosed area in which magnetic fields lines are found to be temporarily trapped. The island boundary separates the slow and fast diffusion zones. Other points of interest are saddle points. They are called X-points. The magnetic field lines starting near the X-points rapidly diffuse away from the initial position.

These two types of diffusion cause dropouts. As described in the introduction, dropouts refer to discontinuous data where the intensity of solar energetic particles repeatedly disappears and reappears. When solar energetic particles undergo transport along the interplanetary magnetic field lines (as we show in Figure 2.5) which can be temporarily trapped in the islands, they are also trapped in the islands for some distance. This phenomenon is illustrated in Figure 2.6 which shows O and X-Points, and Figure 2.7, which shows a distribution of field lines at various z values. We can see that the intensity of solar energetic particles in the islands is much higher than that outside, so we can observe the dropouts.

This mechanism implies that the size of dropouts corresponds to the size of the islands and the empty space between the islands. The shapes of dropouts are similar to the shapes of islands because the magnetic field lines outside the islands have a faster diffusion rate. This leaves the magnetic field lines inside the islands, which have a slow diffusion rate. The trapping of field lines inside the islands is temporary.

This senior project introduces the local trapping boundary (LTB) for describing the edge of trapping. At a local trapping boundary, there is a locally maximal intensity of the 2D field on that contour, so the 2D field has a strong

12

Figure 2.5: Interplanetary magnetic field lines (Ruffolo et al. 2003) effect on this boundary. The magnetic field lines are forced to go mainly along the boundary. The mathematical definition and method of finding the local trapping boundaries will be presented in the next chapter.

13

Figure 2.6: Contours of a magnetic potential function with O-points and X-points

14

Figure 2.7: Scatter plot of magnetic field lines with dropouts

Chapter 3 Simulation of Turbulent Magnetic Field Lines This chapter will explain about the method of simulating the magnetic field lines and about the local trapping boundary, which we use to describe the edge of magnetic field line trapping.

3.1

Generating the Magnetic Field

As we explained in the previous chapter, we must generate a random function in the wavenumber domain first and then transform this to the real space domain by an inverse Fourier transform. We use the Kolmogorov power spectrum which is the natural spectrum of turbulent phenomena. Figure 3.1 shows the Kollog P (k) 6

5

P (k) � k − 3 -

k0

log k

Figure 3.1: Kolmogorov power spectrum 15

16 mogorov power spectrum in a log-log scale. We can see that the function at small wavenumbers (very large scales) is constant and after the scale k0 the spectrum has the slope (power law index) of -5/3. This spectrum arises when large scale fluctuations feed energy to the smaller scales with some losses. Such spectra are observed for all turbulent phenomena and can be explained theoretically.

3.1.1

Generating the Slab Field

The slab fluctuating field is a random function for which hbslab x i = 0,

hbslab y i = 0,

(3.1)

and their characteristics depend on the Kolmogorov power spectrum, so we use these following steps to generate them. Note that the power spectrum is defined as the Fourier transform of the correlation function: Z ∞ 1 Rxx (r) exp[i(r · k)]dn r Pxx (k) ≡ n (2π) 2 −∞ Z ∞ 1 Pyy (k) ≡ Ryy (r) exp[i(r · k)]dn r, n 2 (2π) −∞

(3.2) (3.3)

where n is the dimensionality of the fluctuations (slab: n = 1, 2D: n = 2) and the correlation function is defined by Rxx (r) ≡ hbx (r0 )bx (r0 + r)i

(3.4)

Ryy (r) ≡ hby (r0 )by (r0 + r)i.

(3.5)

Note also that the power spectrum is related to the amplitude of the Fourier transform of b(x): Pxx (k) ∝ |bx (k)|2

(3.6)

Pyy (k) ∝ |by (k)|2 .

(3.7)

17 slab slab ). Since or Pyy In the first step, we generate the power spectrum (Pxx

bslab depends only on z, these can be written as a function of the wavenumber kz , e.g., slab (kz ) = Pxx

C1 5

[1 + (kz lz )2 ] 6

,

(3.8)

where lz is a characteristic length scale related to lc in Equation 2.12, and C1 is a constant. For the y component, we also have slab (kz ) = Pyy

C1 5

[1 + (kz lz )2 ] 6

.

(3.9)

2 slab slab (kz ) are simply |bslab (kz ) and Pyy With the proper normalization Pxx x (kz )| and 2 |bslab y (kz )| , respectively. After that, we generate a random phase to split them

into two components of the complex number (the real part and imaginary part): p slab (k ) exp(iφ) Pxx z p p slab (k ) cos φ + i slab (k ) sin φ, = Pxx Pxx z z

bslab x (kz ) =

where i is

√

(3.10)

−1 and φ is a random phase for each kz (varying from 0 to 2π). We

also have q slab (k ) exp(iφ) Pyy z q q slab slab (k ) sin φ. = Pyy (kz ) cos φ + i Pyy z

bslab y (kz ) =

(3.11)

Finally, when we already have both the x-components and y-components of the magnetic field in the wavenumber domain, we transform them to the real space domain by the inverse Fourier transform. We can write the two slab fluctuating components as Z ∞ 1 = √ bslab x (kz ) exp(−ikz z)dkz , 2π −∞ Z ∞ 1 slab by (z) = √ bslab y (kz ) exp(−ikz z)dkz , 2π −∞ bslab (z) = bslab x + bslab y. x (z)ˆ y (z)ˆ bslab x (z)

(3.12) (3.13) (3.14)

18 By the way, the numerical method that we use for the inverse Fourier transform is the “fast Fourier transform” (Press et al. 1992). In the numerical method, we divided the total length of z into 2n grid points, where we used n = 22.

3.1.2

Generating the 2D Field

The 2D fluctuating field is a function of x and y. It must also have the characteristics of a Kolmogorov power spectrum and the condition in Equation 2.8, so we use the following steps to generate them. First, we generate the power spectrum.

A(k⊥ ) =

1 7

[1 + (k⊥ l⊥ )2 ] 3

,

(3.15)

where A(k⊥ ), the power spectrum of a(x, y), is defined as the Fourier transform of correlation function ha(r0 )a(r0 + r)i, k⊥ =

q kx2 + ky2 ,

(3.16)

e in Equation 2.14. The reason for using the exponent 7/3 and l⊥ is related to λ will be given below.

First, we assign a random phase to a(k⊥ ): a(k⊥ ) =

p

A(k⊥ ) exp(iφ),

(3.17)

Now recall Equation 2.9, b2D (x, y) = ∇ × [a(x, y)ˆ z].

(3.18)

After Fourier transforms to the wavenumber domain, we have b2D (kx , ky ) = −ik × [a(kx , ky )ˆ z].

(3.19)

2D Pxx (kx , ky ) = ky2 A(k⊥ )

(3.20)

2D Pyy (kx , ky ) = kx2 A(k⊥ ),

(3.21)

Then we have

19 and 2D 2D 2 Pxx + Pyy = k⊥ A.

(3.22)

The power spectrum which we use for the 2D fluctuating field must be summed over all directions to obtain the omnidirectional power spectrum (OPS), which 2D should vary as k −5/3 (following the Kolmogorov law). This gives OPS ∝ k⊥ (Pxx + 2D 3 Pyy ) = k⊥ A ∝ k −5/3 . This is why we use the exponent 7/3 in Equation 3.15.

Now, we can find b2D as a function of wavenumber (kx , ky ). Finally, we perform an inverse Fourier transform of b2D to the space domain: Z ∞Z ∞ 1 2D b2D (kx , ky ) exp[−i(r · k)]dkx dky , bx (x, y) = 2π −∞ −∞ x Z ∞Z ∞ 1 2D by (x, y) = b2D (kx , ky ) exp[−i(r · k)]dkx dky , 2π −∞ −∞ y

(3.23) (3.24)

As for the slab field, we divide the total length of x and y into 2n × 2m grid points (n and m are integers), and we use periodic boundary conditions at the edges of the 2D plane.

3.2

Tracing the Magnetic Field Lines

This section is based on a computer program by Mr. Peera Pongkitiwanichkul.

The author thanks him for kindly providing the program and instructions. After we generate the magnetic field, we trace the magnetic field lines using the equation dy dz dx = = .

Bx By Bz

(3.25)

In this model Bz is always the constant B0 , so we can calculate the trajectories of fluctuating field lines by the following method.

3.2.1

Tracing Slab Trajectories

We create the slab magnetic field lines from the equation dx bslab = x , dz B0

bslab dy y = . dz B0

(3.26)

20 and bslab For the grid points of the Fourier transform, we know bslab y . At intermex diate z values, bslab or bslab is calculated using linear interpolation (see Appendix x y A): bslab x (z) =

z − iδz slab (i + 1)δz − z slab (bx )i+1 + (bx )i , δz δz

(3.27)

where δz is the total length of z divided by the number of grid points (Lz /Nz ), and i and i + 1 label the grid points surrounding z. We can rewrite that as bslab x (z) = mx z + cx ,

(3.28)

where mx =

slab (bslab x )i+1 − (bx )i , δz

(3.29)

and slab cx = (bslab x )i (i + 1) − (bx )i+1 i,

(3.30)

and bslab We can see that cx has the same dimensions as bslab x . Similarly, for the x y component, bslab y (z) = my z + cy ,

(3.31)

where my =

slab (bslab y )i+1 − (by )i , δz

(3.32)

and slab cy = (bslab y )i (i + 1) − (by )i+1 i.

(3.33)

Now, we have the equations dx mx z + cx = dz B0 dy my z + cy = . dz B0

(3.34) (3.35)

21 The solutions of these equations are 1 B0 1 = B0

x − x0 =

mx (z 2 − z02 ) + cx (z − z0 )

(3.36)

y − y0

my (z 2 − z02 ) + cy (z − z0 ) .

(3.37)

These solutions are valid if there is only a slab fluctuating field.

3.2.2

Tracing 2D Trajectories

We use bilinear interpolation to find the potential function at any position a(x, y): a(x, y) =

(x − iδx)(y − jδy) [(i + 1)δx − x](y − jδy) ai+1,j+1 + ai,j+1 (3.38) δx δy δx δy [(i + 1)δx − x][(j + 1)δy − y] (x − iδx)[(j + 1)δy − y] ai+1,j + ai,j , + δx δy δx δy

or we can write this as a(x, y) = a1 xy + a2 x + a3 y + a4

(3.39)

From Chapter 2, we know that b2D (x, y) = ∇ × [a(x, y)ˆ z].

(3.40)

This gives b2D x =

∂a = a1 x + a3 , ∂y

(3.41)

b2D y =

∂a = a1 y + a2 . ∂x

(3.42)

and

From Equation 3.25, we get dx a1 x + a3 = dz B0 dy a1 y + a2 = . dz B0

(3.43) (3.44)

22 The solutions of these equations are

z − z0 1 a1 x + a3 = ln B0 a1 a x + a3 1 0

1 a1 y + a2 z − z0 = ln .

B0 a1 a1 y 0 + a2

(3.45) (3.46)

These solutions are valid if there is only a 2D fluctuating field.

3.2.3

Tracing 2D+slab Trajectories

From Equations 3.25, 3.34 and 3.43, we have dx mx z + a1 x + Cx = , dz B0

(3.47)

Cx = cx + a3 .

(3.48)

my z + a1 x + Cy dy = , dz B0

(3.49)

Cy = cy + a2 .

(3.50)

where

For the y component, we have

with

The solutions of these 2D+slab trajectories are

mx mx B0 + Cx a1 a1 x = Ax exp z − z− B0 a1 a21

a1 my my B0 + Cy a1 y = Ay exp z − z− , B0 a1 a21

(3.51) (3.52)

where Ax and Ay are integration constants that are determined by the initial location of the field line.

All the above solutions are only valid in one cell (the cubic space surrounded by 3D grid points of the 2D+slab field which we already generated in the last section). When a field line reaches the edge of a cell, we have to find the

23 position where the magnetic field line leaves the cell. We take the exit position of the previous cell as the initial position in the new cell. By this method, we can trace entire magnetic field lines.

3.3

Local Trapping Boundaries

As can be seen in the scatter plot of magnetic field lines in Figure 2.7, there seem to be sharp boundaries that can temporarily trap the magnetic field lines inside.

Dropouts occur with these boundaries because there are a lot of magnetic field lines inside the boundaries and few magnetic field lines outside the boundaries.

In this senior project, we develop the concept of the local trapping boundary (LTB), which is a mathematical concept and can be interpreted as the boundary of trapping.

We define the local trapping boundary as an equipotential contour that has a maximum average 2D fluctuation energy compared with neighboring contours. We know that the 2D magnetic energy at each position depends on the square of the magnetic field: E(x, y) � |b(x, y)|2 ,

(3.53)

so we can view the average value of |b|2 along the equipotential line as a average value of the energy.

As we explained in Chapter 2, there are two special types of points in the contour plot of the 2D magnetic field. There is slow diffusion (almost slab) near the O-points and there is fast diffusion (both 2D and slab) near the X-points. At both O-points and X-points we have b = 0. The local trapping boundary contour has a maximum hb2 i, so it could be the edge that separates near-O-point zones and near-X-point zones.

24 The magnetic field lines that start inside the local trapping boundary are always temporarily trapped inside. The magnetic field lines that have starting points outside the local trapping boundary diffuse away over a short distance in the z direction. We can find the equipotential contours by tracing the 2D magnetic field lines, without slab turbulence. While tracing the pure 2D magnetic field lines we can collect the values of |b(x, y)|2 and calculate their average value: R |b(x, y)|2 dl , (3.54) |b|2av = L where dl is a small step along the equipotential contour (in the x-y plane) and L is the total length of the equipotential line. Another method of finding the average value is integration along the z direction: R |b(x, y)|2 dz 2 |b|av = .

Lz

(3.55)

(In practice, we observe little difference in the results for the two definitions.) We use a computer for all the steps listed above.

When we already have the average value of the energy along an equipotential line, we print out and compare it with neighboring equipotential contours by eye. If its average value of energy is greater than those of contours both inside and outside, it is called a local trapping boundary.

Chapter 4 Results and Discussion After simulating the magnetic field lines by the methods that we described in Chapters 2 and 3, and using appropriate parameters for solar wind conditions (see Appendix A), we obtain the results presented in this chapter.

We set the magnetic field lines to start from random initial locations inside a circle at z = 0. We can see in Figures 4.2-4.11 that the magnetic field lines that start outside a local trapping boundary rapidly diffuse away from the initial position. The magnetic field lines that start inside the local trapping boundary are temporarily trapped for some distance. This means that inside the local trapping boundary is the near-O-Point-zone and outside the local trapping boundary is the near-X-Point-zone. The local trapping boundary not only traps the magnetic field lines inside, but also inhibits the magnetic field lines from going inside, as seen for the small island in the upper left corner of these figures.

Generally, regions of high b22D coincide with sharp gradients in the density of field lines.

We can also see that LTBs, which are equipotential contours that have a maximum value of hb22D i compared with neighboring contours, nearly coincide with sharp gradients in the scatter plots of magnetic field lines, so it can be interpreted as an boundary of trapping which we draw from the simulation. This

25

26

Figure 4.1: Contours of a magnetic potential function seems to tell us that trapping of turbulent magnetic field lines depends on the value of b22D . It seems that the magnetic field lines rarely go through an area or a contour that has a high amount of 2D fluctuating field. We can see a contour plot of b22D in Figure 4.12.

From these new results, we know that now we can mathematically predict the trapping edges of turbulent magnetic field lines, including where to find solar energetic particles when dropouts occur (as can be seen in Ruffolo, Matthaeus, and Chuychai 2003).

27

Figure 4.2: Scatter plot of initial magnetic field line locations at z = 0.

28

Figure 4.3: Scatter plot of locations of magnetic field lines at about 0.25 AU from the Sun

29

Figure 4.4: Scatter plot of locations of magnetic field lines at about 0.50 AU from the Sun

30

Figure 4.5: Scatter plot of locations of magnetic field lines at about 0.75 AU from the Sun

31

Figure 4.6: Scatter plot of locations of magnetic field lines at about 1.00 AU from the Sun

32

Figure 4.7: Scatter plot of locations of magnetic field lines at about 1.25 AU from the Sun

33

Figure 4.8: Scatter plot of locations of magnetic field lines at about 1.50 AU from the Sun

34

Figure 4.9: Scatter plot of locations of magnetic field lines at about 1.75 AU from the Sun

35

Figure 4.10: Scatter plot of locations of magnetic field lines at about 2.00 AU from the Sun

36

Figure 4.11: Scatter plot of locations of magnetic field lines at about 2.25 AU from the Sun

37

Figure 4.12: Contour plot of b22D

Chapter 5 Conclusions We have simulated the turbulent magnetic field lines by the 2D+slab model, and we also mathematically defined a local trapping boundary(LTB), which is an equipotential contour that has a maximum average 2D fluctuation energy compared with neighboring contours. We have proven that the LTB really functions as a boundary of the trapping region. That is, the magnetic field lines that start inside the local trapping boundary are temporarily trapped inside the local trapping boundary for some distance. The magnetic field lines that start outside the local trapping boundary rarely go inside the local trapping boundary. We can conclude that the mathematically defined local trapping boundary can truly be interpreted as a trapping boundary of turbulent magnetic field lines. This gives us a way to mathematically express where trapping will take place, based on the characteristics of the 2D turbulence alone.

38

References [1] Ruffolo, D., Matthaeus, W. H., and Chuychai, P., Trapping of Solar Energetic Particles by The Small-Scale Topology of Solar Wind Turbulence (2003), Astrophys. J. 597, L169.

[2] Matthaeus, W. H., Goldstein, M. L., and Roberts, D. A. (1990) J. Geophys.

Res. 95, 20,673.

[3] Matthaeus, W. H., Gray, P. C., Pontius, D. H. Jr., and Bieber, J. W. (1995), Phys. Rev. Lett. 75, 2136.

[4] Ruffolo, D., Matthaeus, W. H., Separation of Magnetic Field Line in TwoComponent Turbulence, submitted by Astrophys. J..

[5] Mazur, J. E., Mason, G. M., Giacolone, J., Jokipii, J. R., and Stone, E. C. (2000), Astrophys. J. 532, L79.

[6] Bieber, J. W., Matthaeus, W. H., Smith C. W., Wanner, W., Kallenrode, M.-B., and Wibberenz, G., (1994), Astrophys. J., 420, 294.

[7] Tennekes, H., and Lumley, J. L., A First Course in Turbulence (1994), The MIT Press, Messachusetts, United States of America.

[8] Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery B. P., Numerical Recipes in C (1992), Cambridge University Press, Cambridge, England.

39

Appendix A Numerical Methods A.1

Interpolation

Often a function, say F, is known for specific values of the independent variable(s), call “grid points.” When we have to find the value of F at a point between two 1D grid points or at a point surrounded by four 2D grid points, we use the interpolation method. There are two types of interpolation in this senior project.

They are called linear interpolation and bilinear interpolation which are use for 1D and 2D grid points, respectively.

A.1.1

Linear Interpolation

We use the linear interpolation for slab grid points. The linear interpolation between two points is illustrated in Figure A.1 below.

• Fxi

FP (x) ×

• Fxi+1

Figure A.1: Linear interpolation at point P between two values F (xi ) and F (xi+1 ) The formula for F at point P is Fp (x) =

xi+1 − x x − xi F (xi ) + F (xi+1 ) xi+1 − xi xi+1 − xi 40

(A.1)

41 = (1 − fx )F (xi ) + fx F (xi+1 ), where fx ≡

A.1.2

x − xi . xi+1 − xi

(A.2)

Bilinear Interpolation

We use linear interpolation in two dimensions between 2D grid points. We can calculate a function value Fs at S by using bilinear interpolation as follows: At point P :

(A.3)

FP (x, yj ) = (1 − fx )F (xi , yj ) + fx F (xi+1 , yj ).

At point Q :

(A.4)

FQ (x, yj+1 ) = (1 − fx )F (xi , yj+1 ) + fx F (xi+1 , yj+1 ) At point S :

(A.5)

FS (x, y) = .(1 − fy )FP (x, yj ) + fy FQ (x, yj+1 ), where fy ≡

y − yj . yj+1 − yj

F (xi , yj+1 ) Q • ×

(A.6)

F (xi+1 , yj+1 ) •

S × FS (x, y)

• F (xi , yj )

× P

• F (xi+1 , yj )

Figure A.2: Bilinear interpolation at point S

42 Now, we can calculate the function value FS (x, y) as follows: FS (x, y) = (1 − fy )(1 − fx )F (xi , yj ) + (1 − fy )fx F (xi+1 , yj )

(A.7)

+(1 − fx )fy F (xi , yj+1 ) + fx fy F (xi+1 , yj+1 ).

A.2

Simulation Parameters

We use simulation parameters similar to the solar wind conditions [1]. The parameters are as follow:

mean field

B0 = 1

parallel scale length

λk = 1

perpendicular scale length

λ⊥ = 4

turbulent to mean magnetic field ratio

1 b = B0 2

energy ratio (2D:slab) grid points

80 : 20 Nx = 2048 Ny = 2048 Nz = 4194304

total box lengths

Lx = 100 Ly = 100 Lz = 1000

random seed numbers

iseed 1(for slab) = 43330176 iseed 2(for slab) = 43330038 iseed 3(for 2D) = 223346

Appendix B Source Code c-------------------------------------------------------------------program fline call inputflds call makeline end c-------------------------------------------------------------------subroutine makeline include ’parm.h’ integer nx,ny,nz integer step,datasize parameter (step=419430) parameter (datasize=10) parameter (twopi=2*3.14159265358979323846d0) real*8 lx,ly,lz common /nbox/ nx,ny,nz common /lbox/ lx,ly,lz real*8 eslab,lambda_par,alpha1,phase,rrandom common /slab/ eslab,lambda_par,alpha1 real*8 bx1,by1,a2d common /bslab/ bx1(nzmax+2),by1(nzmax+2) common /aa/ a2d(nxmax,nymax) real*8 xp,yp,zp,radcir common /xp/ xp(20001),yp(20001),zp(20001) real*8 h,ran1 integer noline,check,check2,ii real*8 xx,yy,zz iseed_4=1627190184 iseed_5=708969183 iseed_6=13894221 call genslab() call gen2d() open(2,FILE=’VXtest5000.dat’) write(2,*)’ ’ close(2) radcir=9. iseed_4=-iseed_4 iseed_5=-iseed_5 iseed_6=-iseed_6 do j=1,10000 rrandom=sqrt((radcir**2)*dble(ran1(iseed_5))) phase=dble(twopi*ran1(iseed_4)) xp(1)=(12.5)+(rrandom*cos(phase)) yp(1)=(12.5)+(rrandom*sin(phase))

43

44 zp(1)= 0. if (eslab.ne.0) then call trace(step,datasize) else call trace2d(step,datasize) endif open(2,FILE=’VXtest5000.dat’,access=’append’) do s=1,datasize write(2,*) xp(s),yp(s) enddo close(2) enddo end c-------------------------------------------------------------------subroutine inputflds integer nx,ny,nz real*8 lx,ly,lz common /lbox/ lx,ly,lz common /nbox/ nx,ny,nz namelist /box/ lx,ly,lz,nx,ny,nz,nn real*8 b0,dbob common /field/ b0,dbob namelist /field/ b0,dbob real*8 eslab,lambda_par,alpha1 common /slab/ eslab,lambda_par,alpha1 namelist /slab/ eslab,lambda_par,alpha1 real*8 e2d,lambda_per,alpha2 common /twod/ e2d,lambda_per,alpha2 namelist /twod/ e2d,lambda_per,alpha2 integer iseed_1,iseed_2,iseed_3 common /seeds/ iseed_1,iseed_2,iseed_3 namelist /seeds/ iseed_1,iseed_2,iseed_3 character*256 flnm integer fldfile flnm=’input.nml’ fldfile = 9 open(fldfile,file=flnm) read(fldfile,nml=box) read(fldfile,nml=field) read(fldfile,nml=slab) read(fldfile,nml=twod) read(fldfile,nml=seeds) close(fldfile) return end c-------------------------------------------------------------------subroutine genslab() include ’parm.h’ real*8 dbob,alpha1 real*8 b0,eslab,lambda_par parameter (twopi=2*3.14159265358979323846d0) real*8 ssp1,sp,ssp2,enorm,ewant real*8 phase1,phase2 real*8 ran1,deltax,deltay,deltaz integer iseed_1,iseed_2,iseed_3 real*8 rkx,rky,kz integer loc,kx,ky integer anz(1) real*8 lx,ly,lz integer nx,ny,nz common /lbox/ lx,ly,lz common /nbox/ nx,ny,nz

45 common /field/ b0,dbob common /slab/ eslab,lambda_par,alpha1 common /seeds/ iseed_1,iseed_2,iseed_3 real*8 bx1,by1 common /bslab/ bx1(nzmax+2),by1(nzmax+2) deltax=twopi/lx deltay=twopi/ly deltaz=twopi/lz anz(1)=nz ewant=(dbob**2.)*eslab iseed_1=-iseed_1 iseed_2=-iseed_2 ssp1=0.d0 c********************************* c generate magnetic field c********************************* bx1(1)=0 bx1(2)=0 by1(1)=0 by1(2)=0 do k=3,nz+2,2 kz=deltaz*(k-1)/2 sp=dble(1+dble(kz*lambda_par)**2.)**(-alpha1/4.) phase1=twopi*dble(ran1(iseed_1)) phase2=twopi*dble(ran1(iseed_2)) bx1(k)=dsin(phase1)*sp bx1(k+1)=dcos(phase1)*sp by1(k)=dsin(phase2)*sp by1(k+1)=dcos(phase2)*sp ssp1=ssp1+bx1(k)**2+by1(k)**2 & +bx1(k+1)**2+by1(k+1)**2 enddo enorm=dsqrt(ewant/(2*ssp1)) do k=1,nz+2 bx1(k)=bx1(k)*enorm by1(k)=by1(k)*enorm enddo call four2(bx1,anz,1,-1,-1) call four2(by1,anz,1,-1,-1) ssp1=0.d0 do i=1,nz bx1(i)=bx1(i)*2.d0 ssp1=ssp1+(bx1(i)**2.) by1(i)=by1(i)*2.d0 enddo return end c-------------------------------------------------------------------subroutine gen2d() include ’parm.h’ real*8 dbob,alpha2 real*8 b0,e2d,lambda_per parameter (twopi=2*3.14159265358979323846d0) real*8 ssp1,sp,ssp2,enorm,ewant real*8 phase3 real*8 ran1,deltax,deltay,deltaz integer iseed_1,iseed_2,iseed_3 real*8 rkx,rky integer loc,kx,ky,i,j,k integer n(2) real*8 lx,ly,lz integer nx,ny,nz,nn

46 common /lbox/ lx,ly,lz common /nbox/ nx,ny,nz common /field/ b0,dbob common /twod/ e2d,lambda_per,alpha2 common /seeds/ iseed_1,iseed_2,iseed_3 real*8 a((nxmax+2)*nymax) real*8 a2d,a1,a2,a3,a4 common /aa/ a2d(nxmax,nymax) nn=(nx+2)*ny n(1)=nx n(2)=ny deltax=twopi/lx deltay=twopi/ly deltaz=twopi/lz do i=1,nn a(i)=0.d0 enddo ewant=e2d*(dbob**2.) ssp1=0.d0 iseed_3=-iseed_3 do i=4,nn,2 phase3=twopi*dble(ran1(iseed_3)) call kloc(kx,ky,n,i) rkx=kx*deltax rky=ky*deltay sp=1/(1+dble((rkx**2+rky**2)*(lambda_per**2)))**(alpha2/4.d0) a(i-1)=dcos(phase3)*sp a(i)=dsin(phase3)*sp ssp1=ssp1+((rky**2.+rkx**2.)*(sp**2.)) if (kx.eq.0) then call lloc(0,-ky,n,loc) a(loc-1)=a(i-1) a(loc)=-a(i) endif enddo enorm=dsqrt(ewant/(2*ssp1)) do k=1,nn a(k)=a(k)*enorm enddo call four2(a,n,2,-1,-1) k=1 open(59,FILE=’a.dat’) do j=1,ny do i=1,nx a(k)=a(k)*2.d0 write(59,*)a(k) a2d(i,j)=a(k) k=k+1 enddo enddo close(59) return end c-------------------------------------------------------------------function ran1(idum) integer idum,ia,im,iq,ir,ntab,ndiv double precision ran1,am,eps,rnmx parameter (ia=16807,im=2147483647,am=1./im,iq=127773, & ir=2836,ntab=32,ndiv=1+(im-1)/ntab,eps=1.2e-7, & rnmx=1.-eps) integer j,k,iv(ntab),iy/0/ SAVE iv,iy

47 DATA iv /ntab*0/,iy/0/ if(idum.le.0.or.iy.eq.0) then idum=max(-idum,1) do j=ntab+8,1,-1 k=idum/IQ idum=ia*(idum-k*iq)-ir*k if(idum.lt.0) idum=idum+im if(j.le.ntab) iv(j)=idum enddo iy=iv(1) endif k=idum/iq idum=ia*(idum-k*iq)-ir*k if(idum.lt.0) idum=idum+im j=1+iy/ndiv iy=iv(j) iv(j)=idum ran1=min(am*iy,rnmx) return end c-------------------------------------------------------------------subroutine kloc(KX,KY,N,loc) dimension N(2) NINROW=N(1)+2 NTOP=NINROW*(N(2)/2+1) IF (LOC.GT.(NINROW*N(2)))GO TO 220 IF (LOC.GT.NTOP) GO TO 210 KY=(LOC-1)/NINROW KX=(LOC-KY*NINROW-1)/2 RETURN 210 KY=1-N(2)/2+(LOC-NTOP-1)/NINROW KX=(LOC-NTOP-(KY+N(2)/2-1)*NINROW-1)/2 RETURN 220 WRITE (6,230)LOC 230 FORMAT(’0’,’ERROR IN KLOC,LOC TOO LARGE’,2X,I10) RETURN END c-------------------------------------------------------------------subroutine lloc(kx,ky,n,loc) integer n(2) nadd=0 ninrow=n(1)+2 if(ky.lt.0) nadd=n(2) loc=ninrow*(nadd+ky)+2*(kx+1) return end c-------------------------------------------------------------------subroutine trace2d(step,datasize) include ’parm.h’ real*8 hz,bz integer step,datasize parameter (twopi=2*3.14159265358979323846d0) real*8 x,y,z,h,zmax,zz real*8 px1d,py1d,pz real*8 ztoy,ztox,app1,app2 integer check,p integer d,x1,x2,y1,y2 real*8 px,py,bx,by,ppx,ppy real*8 exceedz integer ox,oy real*8 xzmax,yzmax,zzmax,dz,zround real*8 gbx,gby,xpeak,ypeak real*8 trx,try,trz

48 real*8 xp,yp,zp common /xp/ xp(20001),yp(20001),zp(20001) integer nx,ny,nz,check2 real*8 lx,ly,lz common /lbox/ lx,ly,lz common /nbox/ nx,ny,nz real*8 a2d,bx1,by1 real*8 x3,y3,z3,px3,py3 common /bslab/ bx1(nzmax+2),by1(nzmax+2) common /aa/ a2d(nxmax,nymax) integer kstep kstep=int(step/datasize) zmax=2.d0 bz=1.d0 trx=lx/dble(nx) try=ly/dble(ny) trz=lz/dble(nz) hz=1.d0 c ************************ c start gen field c ************************ p=2 x=xp(1)/trx y=yp(1)/try z=zp(1)/trz px=x py=y ppx=x ppy=y exceedz=0.d0 ox=0 oy=0 gbx=bx(x,y,px,py) gby=by(x,y,px,py) if (gbx.lt.0) then px=x+0.1 ppx=px else px=x-0.1 ppx=px endif if (gby.lt.0) then py=y+0.1 ppy=py else py=y-0.1 ppy=py endif h=hz z=zp(1)/trz zz=dble(int(z/100.d0)) z=z-(100.d0*zz) c******************************* c one line c******************************* do i=1,step check=0 if (z.gt.100) then z=z-100. zz=zz+1. endif call locate(x,y,ox,oy,check) 51 if (abs(x).lt.1.d-11) x=0. if (abs(y).lt.1.d-11) y=0. xpeak=x ypeak=y

49 px3=px py3=py x3=x y3=y z3=z px1d=x py1d=y pz=z check2=0 if ((int(x).ne.0).and.(int(y).ne.0)) then if (z.lt.zmax) goto 71 endif 61 call edge(x3,y3,z3,zmax,bz,px3,py3) check2=1 71 px=x py=y app1=app(px,py) z=z+h if (check2.eq.1) then if (z.gt.zmax) then exceedz=z-zmax z=zmax else exceedz=0.d0 endif endif x=ztox(px,py,pz,z,bz,ppx,ppy) y=ztoy(px,py,pz,z,bz,ppx,ppy) if (integ(x).ne.integ(px)) then if (abs(x-integ(x)).gt.1.d-8) then if (abs(px-integ(px)).gt.1.d-8) then x=px y=py px=px3 py=py3 z=z3 goto 61 endif endif endif if (integ(y).ne.integ(py)) then if (abs(y-integ(y)).gt.1.d-8) then if (abs(py-integ(py)).gt.1.d-8) then x=px y=py px=px3 py=py3 z=z3 goto 61 endif endif endif app2=app(x,y) if (i.lt.10) then if (app1.ne.app2) then x=xpeak y=ypeak z=pz+h endif endif ppx=px ppy=py c******************************************* c Small steps c******************************************* if ((exceedz.gt.1.d-10).and.(app1.eq.app2)) then h=exceedz

50

&

check=check+1 if (check.eq.1) then xzmax=x yzmax=y zzmax=z endif if (check.ge.5) then check=0 if((abs(xzmax-x).lt.1.d-8) .and.(abs(yzmax-y).lt.1.d-8))then dz=z-zzmax zround=exceedz/dz z=z+int(zround)*dz exceedz=exceedz-(dz*int(zround)) if (dz.lt.1.d-8) then z=z+exceedz exceedz=0.d0 endif h=exceedz endif endif goto 51

else h=hz endif x=x+(dble(nx)*ox) y=y+(dble(ny)*oy) if(i-kstep*int(i/kstep).eq.0) then xp(p)=trx*x yp(p)=try*y zp(p)=trz*(z+(100.0*zz)) p=p+1 endif enddo return end c----------------------------------------------------------------subroutine trace(step,datasize) include ’parm.h’ real*8 hz,bz integer step integer datasize parameter (twopi=2*3.14159265358979323846d0) real*8 xp,yp,zp common /xp/ xp(20001),yp(20001),zp(20001) real*8 x,y,z integer nx,ny,nz real*8 lx,ly,lz common /lbox/ lx,ly,lz common /nbox/ nx,ny,nz real*8 a2d,bx1,by1 common /bslab/ bx1(nzmax+2),by1(nzmax+2) common /aa/ a2d(nxmax,nymax) real*8 mx,my,cx,cy real*8 a1,a2,a3,a4,betax,betay integer s,i,j,x1,x2,y1,y2 integer l,check,zo,integ,ox,oy real*8 zz,findzx,xx,yy,findzy,x3,y3 real*8 bx,by,bxx,byy,px,py,pz,zi,zz1,zz2 real*8 eslab,lambda_par,alpha1 common /slab/ eslab,lambda_par,alpha1 integer checkx,checky,checkz,kstep,p,w w=69092 p=1 kstep=int(step/datasize)

51

&

1

&

hz=1.d0 trx=lx/dble(nx) try=ly/dble(ny) trz=lz/dble(nz) bz=trx/trz x=xp(1)/trx y=yp(1)/try z=zp(1)/trz zo=int(z/10.d0) z=z-(zo*10) pz=z ox=int(x/dble(nx)) oy=int(y/dble(ny)) if (x.lt.0) ox=ox-1 if (y.lt.0) oy=oy-1 x=x-(ox*dble(nx)) y=y-(oy*dble(ny)) if ((int(x).eq.x).and.(int(y).eq.y) .and.(eslab.eq.0)) then i=int(x) j=int(y) if (a2d(i,j).gt.a2d(i-1,j)) then if (a2d(i,j).gt.a2d(i+1,j)) then if (a2d(i,j).gt.a2d(i,j-1)) then if (a2d(i,j).gt.a2d(i,j+1)) then goto 100 endif endif endif endif endif px=x py=y bxx=bx(x,y,px,py) byy=by(x,y,px,py) if (bxx.gt.0) then px=x-0.0001 else px=x+0.0001 endif if (byy.gt.0) then py=y-0.0001 else py=y+0.0001 endif do s=1,step dz=0.1 if (z.ge.11.d0) then z=z-10.d0 zo=zo+1 endif zi=z pz=z call parameters(betax,betay,mx,cx,my, cy,a1,a2,a3,bz,x,y,z,zo,px,py) px=x py=y z=dble(int(z))+1 checkx=0 checky=0 checkz=0 if (abs(a1).lt.1.d-17) then call mfl1d(x,y,pz+(zo*10.d0),z+(zo*10.d0),bz,a2,a3) goto 95 endif

52 5 & &

&

& &

& &

x=(betax*dexp(a1*z/bz))-(mx*z/a1) -((((cx+a3)/bz)+(mx/a1))*bz/a1) y=(betay*dexp(-a1*z/bz))+(my*z/a1) +((((cy-a2)/bz)-(my/a1))*bz/a1) if (checkz.eq.1) then if((int(x).eq.int(px)).and.(int(y).eq.int(py))) then call parameters(betax,betay,mx,cx,my, cy,a1,a2,a3,bz,x,y,z,zo,px,py) px=x py=y else zi=zi-dz dz=dz*0.9 endif endif edgex=0 edgey=0 if ((integ(x).ne.integ(px)) .and.(abs(x-dble(int(x))).gt.1.d-8) .and.(abs(px-dble(int(px))).gt.1.d-8)) then edgex=1 else if (abs(x-px).gt.1.d0) then edgex=1 endif endif if ((integ(y).ne.integ(py)) .and.(abs(y-dble(int(y))).gt.1.d-8) .and.(abs(py-dble(int(py))).gt.1.d-8)) then edgey=1 else if (abs(y-py).gt.1.d0) edgey=1 endif if ((edgex.eq.1).or.(edgey.eq.1)) then zz1=0.d0 zz2=0.d0 if (edgex.eq.1) then checkx=checkx+1 if (x.gt.px) then if ((x-px).lt.(nx-10)) then xx=dble(int(px))+1.d0 else xx=0.d0 endif else if ((x-px).lt.-(nx-10)) then xx=dble(nx) else xx=dble(int(px)) endif endif zz1=findzx(a1,a3,bz,betax,mx,cx,pz,xx) if (zz1.le.pz) zz1=pz-1.d0 checkz=0 endif if (edgey.eq.1) then checky=checky+1 if (y.gt.py) then if ((y-py).lt.(ny-10)) then yy=dble(int(py))+1.d0 else yy=0.d0 endif else if ((y-py).lt.-(ny-10)) then

53 yy=dble(ny) else

& &

& &

&

&

&

&

& &

&

& & &

yy=dble(int(py)) endif endif zz2=findzy(a1,a2,bz,betay,my,cy,pz,yy) if (zz2.le.pz) zz2=pz-1.d0 checkz=0 endif if (edgex.eq.edgey) then if ((zz1.gt.int(pz)) .and.(zz1.lt.(dble(int(pz))+1.d0))) then if ((zz2.gt.int(pz)) .and.(zz2.lt.(dble(int(pz))+1.d0))) then if (zz1.gt.zz2) then z=zz2 edgex=0 x=(betax*dexp(a1*z/bz))-(mx*z/a1) -((((cx+a3)/bz)+(mx/a1))*bz/a1) y=yy if ((integ(x).ne.integ(px)) .and.(px.ne.dble(int(px)))) z=pz-1.d0 else z=zz1 y=(betay*dexp(-a1*z/bz))+(my*z/a1) +((((cy-a2)/bz)-(my/a1))*bz/a1) x=xx edgey=0 if ((integ(y).ne.integ(py)).and. (py.ne.dble(int(py)))) z=pz-1.d0 endif else z=zz1 y=(betay*dexp(-a1*z/bz))+(my*z/a1) +((((cy-a2)/bz)-(my/a1))*bz/a1) x=xx edgey=0 if ((integ(y).ne.integ(py)) .and.(py.ne.dble(int(py)))) z=pz-1.d0 endif else if ((zz2.gt.int(pz)) .and.(zz2.lt.dble(int(pz+1.d0)))) then z=zz2 x=(betax*dexp(a1*z/bz))-(mx*z/a1) -((((cx+a3)/bz)+(mx/a1))*bz/a1) y=yy edgex=0 if ((integ(x).ne.integ(px)) .and.(px.ne.dble(int(px)))) z=pz-1.d0 else z=pz-1.d0 endif endif else if (edgex.eq.1) then if ((zz1.gt.int(pz)) .and.(zz1.lt.dble(int(pz+1.d0)))) then z=zz1 y=(betay*dexp(-a1*z/bz)) +(my*z/a1)+((((cy-a2)/bz)-(my/a1))*bz/a1) x=xx if ((integ(y).ne.integ(py)) .and.(py.ne.dble(int(py)))) z=pz-1.d0

54 else

z=pz-1.d0 endif else if ((zz2.gt.int(pz)) .and.(zz2.lt.dble(int(pz+1.d0)))) then z=zz2 x=(betax*dexp(a1*z/bz)) & -(mx*z/a1)-((((cx+a3)/bz)+(mx/a1))*bz/a1) y=yy if ((integ(x).ne.integ(px)) & .and.(px.ne.dble(int(px)))) z=pz-1.d0 else z=pz-1.d0 endif endif endif checkz=0 if ((z.le.pz).or.(z.gt.(pz+1.d0))) then if (dz.lt.1.d-10) then if (edgex.eq.1) then x=dble(int(x+0.01)) y=py endif if (edgey.eq.1) then y=dble(int(y+0.01)) x=px endif else zi=zi+dz z=zi checkz=1 goto 5 endif endif endif oox=ox ooy=oy if ((x.ge.nx).and.(x.gt.px)) then x=x-dble(nx) oox=ox+1 else if ((x.lt.-1.d-10).and.(x.lt.px)) then x=x+dble(nx) oox=ox-1 endif endif if ((y.ge.ny).and.(y.gt.py)) then y=y-dble(ny) ooy=oy+1 else if ((y.lt.-1.d-10).and.(y.lt.py)) then y=y+dble(ny) ooy=oy-1 endif endif ox=oox oy=ooy 95 if (z.ne.int(z)) goto 1 if ((s-kstep*int(s/kstep)).eq.0) then p=p+1 xp(p)=(x+(ox*nx))*trx yp(p)=(y+(oy*ny))*try zp(p)=(z+(zo*10.d0))*trz endif enddo 100 return end c---------------------------------------------------------------

55

subroutine parameters(betax,betay,mx,cx,my,cy,a1,a2,a3,bz,x,y, & z,zo,px,py) include ’parm.h’ real*8 mx,cx,my,cy,a1,a2,a3,x,y,px,py,x3,y3,betax,betay,bz real*8 z integer x1,x2,y1,y2 integer nx,ny,nz real*8 lx,ly,lz common /lbox/ lx,ly,lz common /nbox/ nx,ny,nz real*8 a2d,bx1,by1 common /bslab/ bx1(nzmax+2),by1(nzmax+2) common /aa/ a2d(nxmax,nymax) integer i,j,k,zo,integ,ki,k1,k2 x3=x y3=y i=integ(x) j=integ(y) if (abs(x-integ(x)).lt.1.d-12) then if (x.lt.px) then x3=x3-1 i=i-1 endif endif if (abs(y-integ(y)).lt.1.d-12) then if (y.lt.py) then y3=y3-1 j=j-1 endif endif call pgrid(x1,x2,y1,y2,x3,y3) ki=integ(z) k1=ki+(zo*10.d0)+3 if (k1.gt.nz) k1=k1-nz k2=k1+1 if (k1.eq.nz) k2=1 mx=bx1(k2)-bx1(k1) my=by1(k2)-by1(k1) cx=((ki+1)*bx1(k1))-(ki*bx1(k2)) cy=((ki+1)*by1(k1))-(ki*by1(k2)) a1=a2d(x2,y2)-a2d(x1,y2)-a2d(x2,y1)+a2d(x1,y1) a1=a1*nx/lx a2=(j*(a2d(x1,y2)-a2d(x2,y2))) & +((j+1)*(a2d(x2,y1)-a2d(x1,y1))) a2=a2*ny/ly a3=(i*(a2d(x2,y1)-a2d(x2,y2))) & +((i+1)*(a2d(x1,y2)-a2d(x1,y1))) a3=a3*nx/lx betax=(x+(mx*z/a1)+((((cx+a3)/bz) & +(mx/a1))*bz/a1))*dexp(-a1*z/bz) betay=(y-(my*z/a1)-((((cy-a2)/bz) & -(my/a1))*bz/a1))*dexp(a1*z/bz) return end c--------------------------------------------------------------function findzx(a1,a3,bz,betax,mx,cx,zz,xx) integer l,counts real*8 a1,a3,bz,betax,mx,cx,zz,xx,findzx real*8 fx,dfx,zoo,fx2 fx2=100.d0 zoo=zz

56 counts=0 counts=counts+1 fx=(a1*bz*betax*dexp(a1*zoo/bz))-(mx*zoo*bz) & -((((cx+a3)*a1)+(mx*bz))*(bz/a1))-(xx*bz*a1) dfx=(betax*a1*a1*dexp(a1*zoo/bz))-(mx*bz) zoo=zoo-(fx/dfx) if (abs(zoo-zz).lt.100.d0) then if ((abs(fx/dfx).lt.1.d-10).and.(abs(fx).lt.1.d-10)) then if (abs(fx).le.abs(fx2)) findzx=zoo goto 110 else if (abs(fx).le.abs(fx2)) then findzx=zoo fx2=fx endif if (counts.lt.100) goto 105 endif else zoo=zz-1 endif 110 return end c--------------------------------------------------------------function findzy(a1,a2,bz,betay,my,cy,zz,yy) integer l,counts real*8 a1,a2,bz,betay,my,cy,zz,yy real*8 fy,dfy,findzy,zoo,fy2 fy2=100.d0 zoo=zz counts=0 115 counts=counts+1 fy=((-a1)*bz*betay*dexp(-a1*zoo/bz))-(my*zoo*bz) & -((((cy-a2)*(-a1))+(my*bz))*(-bz/a1))-(yy*bz*(-a1)) dfy=(betay*a1*a1*dexp(-a1*zoo/bz))-(my*bz) zoo=zoo-(fy/dfy) if (abs(zoo-zz).lt.100.d0) then & if ((abs(fy/dfy).lt.1.d-10) .and.(abs(fy).lt.1.d-10)) then if (abs(fy).lt.abs(fy2)) findzy=zoo goto 120 else if (abs(fy).lt.abs(fy2)) then findzy=zoo fy2=fy endif if (counts.lt.100) goto 115 endif else findzy=zz-1 endif 120 return end c--------------------------------------------------------------function app(x,y) include ’parm.h’ integer nx,ny,nz common /nbox/ nx,ny,nz real*8 a2d common /aa/ a2d(nxmax,nymax) real*8 s1,s2,x,y real*8 appx,appx2,appx1,app integer t1,t2 real*8 xx,yy xx=x-nx*int(x/(1.*nx)) if (xx.le.1) xx=xx+nx 105

57 yy=y-ny*int(y/(1.*ny)) if (yy.lt.1) yy=yy+ny i=integ(xx) j=integ(yy) s1=xx-integ(xx) s2=yy-integ(yy) if(i.ge.nx+1)i=i-nx if(j.ge.ny+1)j=j-ny t1=i+1 t2=j+1 if(i.eq.nx)t1=1 if(j.eq.ny)t2=1 if ((i.eq.0).or.(i.eq.nx)) then i=nx t1=1 endif if ((j.eq.0).or.(j.eq.ny)) then j=ny t2=1 endif appx1=a2d(i,j)*(1-s1)+a2d(t1,j)*s1 appx2=a2d(i,t2)*(1-s1)+a2d(t1,t2)*s1 appx=appx1*(1-s2)+appx2*s2 app=appx return end c--------------------------------------------------------------subroutine mfl1d(x,y,pz,z,bz,a2,a3) include ’parm.h’ integer i,j,nx,ny,nz common /nbox/ nx,ny,nz real*8 bx1,by1 common /bslab/ bx1(nzmax+2),by1(nzmax+2) real*8 x,y,z,h,dh,dz,bz real*8 lx,ly,lz,a3,a2,dx,dy,pz common /lbox/ lx,ly,lz dx=0.d0 dy=0.d0 h=z-pz j=integ(h) dh=h-j do i=0,j if (i.eq.j) then dz=dh else dz=1. endif call xy1d(dx,dy,pz,dz,bz,a2,a3) x=x+dx y=y+dy enddo return end c--------------------------------------------------------------subroutine xy1d(dx,dy,pz,h,bz,a2,a3) include ’parm.h’ integer nz,i,jj,ii,nx,ny common /nbox/ nx,ny,nz real*8 bx1,by1 common /bslab/ bx1(nzmax+2),by1(nzmax+2) real*8 zz,h,lz,pz,a2,a3 real*8 dx,dy,bz,over,abc

58 pz=pz zz=pz-nz*dble(int(pz/nz)) over=0. i=integ(zz) ii=i-int(dble(i)/dble(nz))+3 if (ii.eq.0) ii=nz jj=ii+1 if (ii.eq.nz) jj=1 z2=zz+h if (z2.gt.i+1) then z2=(i+1.) over=zz+h-i-1. endif dx=(bx1(jj)-bx1(ii))*((z2)**2-zz**2)/2. dx=dx+(((bx1(ii)*(i+1))-(bx1(jj)*i)+a3)*(z2-zz)) dx=dx/bz dy=(by1(jj)-by1(ii))*((z2)**2-zz**2)/2. dy=dy+(((by1(ii)*(i+1))-(by1(jj)*i)-a2)*(z2-zz)) dy=dy/bz zz=z2 if (over.ne.0) then ii=ii+1 if (ii.eq.0) ii=nz jj=ii+1 if (ii.eq.nz) jj=1 z2=z2+over dx=(bx1(jj)-bx1(ii))*((z2)**2-zz**2)/2. dx=dx+(((bx1(ii)*(i+1))-(bx1(jj)*i)+a3)*(z2-zz)) dx=dx/bz dy=(by1(jj)-by1(ii))*((z2)**2-zz**2)/2. dy=dy+(((by1(ii)*(i+1))-(by1(jj)*i)-a2)*(z2-zz)) dy=dy/bz endif

return end c--------------------------------------------------------------subroutine pgrid(x1,x2,y1,y2,x,y) integer i,j,x1,x2,y1,y2,integ integer nx,ny,nz real*8 x,y,ox,oy,xx,yy,bx,by common /nbox/ nx,ny,nz if (x.gt.0) then ox=1.*dble(int(x/nx)) else ox=-1.+1.*dble(int(x/nx)) endif xx=x-dble(nx)*ox if (y.gt.0) then oy=1.*int(y/ny) else oy=-1.+1.*dble(int(y/ny)) endif yy=y-dble(ny)*oy i=integ(xx) j=integ(yy) x1=i-(nx*int(i*1./nx)) y1=j-(ny*int(j*1./ny)) if (x1.le.0) x1=x1+nx if (y1.le.0) y1=y1+nx x2=x1+1 y2=y1+1 if ((x1.eq.0).or.(x1.eq.nx)) then

59 x1=nx x2=1 endif if ((y1.eq.0).or.(y1.eq.ny)) then y1=ny y2=1 endif

return end c--------------------------------------------------------------function bx(x,y,px,py) include ’parm.h’ integer i,j,x1,y1,x2,y2,integ integer nx,ny,nz common /nbox/ nx,ny,nz real*8 a2d,x,y,c1,c2 common /aa/ a2d(nxmax,nymax) real*8 ox,oy,bx,px,py,x3,y3,ty real*8 lx,ly,lz common /lbox/ lx,ly,lz ty=dble(ny)/ly x3=x y3=y i=integ(x3) j=integ(y3) if (abs(x3-integ(x3)).lt.(1.d-8)) then if (((x3-px).lt.0).or.((x3-px).gt.500))then if ((x3-px).gt.(-500)) then if (i.ne.0) then i=i-1 else i=nx-1 x3=dble(nx) endif endif endif endif if (abs(y3-integ(y3)).lt.(1.d-8)) then if (((y3-py).lt.0).or.((y3-py).gt.500))then if ((y3-py).gt.(-500)) then if (j.ne.0) then j=j-1 else j=ny-1 y3=dble(ny) endif endif endif endif x1=i-(nx*int(i*1./nx)) y1=j-(ny*int(j*1./ny)) if (x1.le.0) x1=x1+nx if (y1.le.0) y1=y1+nx x2=x1+1 y2=y1+1 if ((x1.eq.0).or.(x1.eq.nx)) then x1=nx x2=1 endif if ((y1.eq.0).or.(y1.eq.ny)) then y1=ny y2=1 endif c1=((i+1.)*(a2d(x1,y2)-a2d(x1,y1)))+(i*(a2d(x2,y1)-a2d(x2,y2))) c2=a2d(x1,y1)+a2d(x2,y2)-a2d(x2,y1)-a2d(x1,y2)

60 bx=(c1+(c2*x3))*ty return end c--------------------------------------------------------------function by(x,y,px,py) include ’parm.h’ integer i,j,x1,y1,x2,y2,integ integer nx,ny,nz common /nbox/ nx,ny,nz real*8 a2d,x,y,c1,c2,x3,y3 common /aa/ a2d(nxmax,nymax) real*8 xx,yy,ox,oy,by,px,py,tx real*8 lx,ly,lz common /lbox/ lx,ly,lz tx=dble(nx)/lx x3=x y3=y i=integ(x3) j=integ(y3) if (abs(x3-integ(x3)).lt.(1.d-8)) then if (((x3-px).lt.0).or.((x3-px).gt.500))then if ((x3-px).gt.(-500)) then if (i.ne.0) then i=i-1 else i=nx-1 x3=dble(nx) endif endif endif endif if (abs(y3-integ(y3)).lt.(1.d-8)) then if (((y3-py).lt.0).or.((y3-py).gt.500))then if ((y3-py).gt.(-500)) then if (j.ne.0) then j=j-1 else j=ny-1 y3=dble(ny) endif endif endif endif x1=i-(nx*int(i*1./nx)) y1=j-(ny*int(j*1./ny)) if (x1.le.0) x1=x1+nx if (y1.le.0) y1=y1+nx x2=x1+1 y2=y1+1 if ((x1.eq.0).or.(x1.eq.nx)) then x1=nx x2=1 endif if ((y1.eq.0).or.(y1.eq.ny)) then y1=ny y2=1 endif c1=(j+1.)*(a2d(x1,y1)-a2d(x2,y1))+j*(a2d(x2,y2)-a2d(x1,y2)) c2=a2d(x2,y1)+a2d(x1,y2)-a2d(x1,y1)-a2d(x2,y2) by=(c1+(c2*y3))*tx return end c--------------------------------------------------------------function integ(x)

61 integer integ real*8 x integ=int(x) if (x.lt.0) integ=integ-1 if (abs(1.d0+dble(integ)-x).lt.(1.d-8)) then integ=integ+1 endif return end c--------------------------------------------------------------subroutine dxy(xp,ddz,dstep,dx2) integer ddz,i,dstep real*8 xp(ddz) real*8 ndata,dx2,results,sumdx2 i=1 j=0 sumdx2=0.d0 ndata=0.d0 25 results=xp(i+dstep)-xp(i) sumdx2=sumdx2+(results**2.) i=i+1 ndata=ndata+1.d0 c if (ndata.eq.20000) goto 26 if ((i+dstep).lt.ddz) goto 25 26 dx2=sumdx2/ndata return end c--------------------------------------------------------------subroutine edge(x,y,z,zmax,bz,px,py) integer nx,ny,si,nz integer y1,y2 common /nbox/ nx,ny,nz real*8 x,y,z,dz,bx,by,x3,y3,bz real*8 edx,edy,zmax,ztox,ztoy,app1,app real*8 bxcheck,bycheck,px,py x3=x y3=y if (abs(x-integ(x)).lt.(1.d-8)) then bxcheck=bx(x,y,px,py) if (bxcheck.lt.0) then x3=x-1. endif endif if (abs(y-integ(y)).lt.(1.d-8)) then bycheck=by(x,y,px,py) if (bycheck.lt.0) then y3=y-1. endif endif y1=integ(y3) y2=y1+1 call zmaxx(x,y,z,edx,zmax,bz,px,py) edy=ztoy(x,y,z,zmax,bz,px,py) if (zmax.le.z) then edy=dble(y2)+1. endif if ((edy.ge.dble(y2)).or.(edy.le.dble(y1))) then call zmaxy(x,y,z,edy,zmax,bz,px,py) edx=ztox(x,y,z,zmax,bz,px,py) endif return end c---------------------------------------------------------------

62

subroutine zmaxx(x,y,z,edx,zmax,bz,px,py) include ’parm.h’ integer nx,ny,x1,x2,y1,y2 integer i,j,integ,nz common /nbox/ nx,ny,nz real*8 a2d,x,y,z,c1,c2,ztox,dz common /aa/ a2d(nxmax,nymax) real*8 z1,z2,zmax,edx,x3,y3,by,bx,bz,xxx real*8 bxcheck,bycheck,xx2,xx1,px,py,ratio real*8 lx,ly,lz common /lbox/ lx,ly,lz ratio=(nz*lx)/(nx*lz) bxcheck=bx(x,y,px,py) i=integ(x) j=integ(y) if (i.lt.0) i=i-1 if (j.lt.0) j=j-1 x3=x y3=y if (abs(x-integ(x)).lt.1.d-8) then if (bxcheck.lt.0) then x3=x-1. i=i-1 endif endif if (abs(y-integ(y)).lt.1.d-8) then bycheck=by(x,y,px,py) if (bycheck.lt.0) then y3=y-1. j=j-1 endif endif call pgrid(x1,x2,y1,y2,x3,y3) c1=(i+1.)*(a2d(x1,y2)-a2d(x1,y1)) & +(i)*(a2d(x2,y1)-a2d(x2,y2)) c2=a2d(x1,y1)+a2d(x2,y2)-a2d(x1,y2)-a2d(x2,y1) c1=c1*nx/lx c2=c2*nx/lx if (bxcheck.gt.0) then edx=dble(i+1) else edx=dble(i) endif if (c2.ne.0) then dz=((bz/c2)*dlog(abs((c1+(c2*edx))/(c1+(c2*x)))))*ratio else dz=(edx-x)/c1 endif if (dz.lt.1.d-8) then bxcheck=-bxcheck if (bxcheck.gt.0) then edx=dble(i+1) else edx=dble(i) endif if (c2.ne.0) then dz=((bz/c2)*dlog(abs((c1+(c2*edx))/(c1+(c2*x)))))*ratio else dz=(edx-x)/c1 endif endif zmax=z+dz xxx=ztox(x,y,z,zmax,bz,px,py) if ((abs(xxx-edx).gt.1.d-8).or.(abs(xxx-x).lt.1.d-8)) then

63 zmax=z-1. endif

return end c--------------------------------------------------------------subroutine zmaxy(x,y,z,edy,zmax,bz,px,py) include ’parm.h’ integer nx,ny,x1,x2,y1,y2 integer i,j,integ,nz common /nbox/ nx,ny,nz real*8 a2d,x,y,z,c1,c2,ztoy,dz common /aa/ a2d(nxmax,nymax) real*8 z1,z2,zmax,edy,x3,y3,by,bx,bz,yyy real*8 bycheck,bxcheck,yy1,yy2,px,py,ratio real*8 lx,ly,lz common /lbox/ lx,ly,lz ratio=(nz*lx)/(lz*nx) bycheck=by(x,y,px,py) i=integ(x) j=integ(y) if (i.lt.0) i=i-1 if (j.lt.0) j=j-1 x3=x y3=y if (abs(x-integ(x)).lt.1.d-8) then bxcheck=bx(x,y,px,py) if (bxcheck.lt.0) then x3=x-1. i=i-1 endif endif if (abs(y-integ(y)).lt.1.d-8) then if (bycheck.lt.0) then y3=y-1. j=j-1 endif endif call pgrid(x1,x2,y1,y2,x3,y3) c1=(j+1.)*(a2d(x1,y1)-a2d(x2,y1))+j*(a2d(x2,y2)-a2d(x1,y2)) c2=a2d(x2,y1)+a2d(x1,y2)-a2d(x2,y2)-a2d(x1,y1) c1=c1*ny/ly c2=c2*ny/ly if (bycheck.gt.0) then edy=dble(j+1) else edy=dble(j) endif if (c2.ne.0) then dz=((bz/c2)*dlog(abs((c1+(c2*edy))/(c1+(c2*y)))))*ratio else dz=(edy-y)/c1 endif if (dz.lt.1.d-8) then bycheck=-bycheck if (bycheck.gt.0) then edy=dble(j+1) else edy=dble(j) endif if (c2.ne.0) then dz=((bz/c2)*dlog(abs((c1+(c2*edy)) & /(c1+(c2*y)))))*ratio else

64 dz=(edy-y)/c1 endif endif zmax=z+dz

return end c--------------------------------------------------------------function ztoy(x,y,pz,z,bz,px,py) include ’parm.h’ integer nx,ny,x1,x2,y1,y2 integer i,j,integ,nz common /nbox/ nx,ny,nz real*8 a2d,x,y,z,c1,c2,pz common /aa/ a2d(nxmax,nymax) real*8 ztoy,x3,y3,bx,by,bz,dx,dy real*8 bycheck,bxcheck,px,py,ratio real*8 lx,ly,lz common /lbox/ lx,ly,lz ratio=(lz*nx)/(nz*lx) bycheck=by(x,y,px,py) i=integ(x) j=integ(y) if (i.lt.0) i=i-1 if (j.lt.0) j=j-1 x3=x y3=y if (abs(x-integ(x)).lt.(1.d-8)) then bxcheck=bx(x,y,px,py) if (bxcheck.lt.0) then x3=x-1. i=i-1 else x3=dble(integ(x)) endif endif if (abs(y-integ(y)).lt.(1.d-8)) then if (bycheck.lt.0) then y3=y-1. j=j-1 else y3=dble(integ(y)) endif endif call pgrid(x1,x2,y1,y2,x3,y3) c1=(j+1.)*(a2d(x1,y1)-a2d(x2,y1))+(j*(a2d(x2,y2)-a2d(x1,y2))) c1=c1*ny/ly c2=a2d(x2,y1)+a2d(x1,y2)-a2d(x1,y1)-a2d(x2,y2) c2=c2*ny/ly if (c2.ne.0) then ztoy=(((c1+(c2*y))*dexp((c2/bz)*ratio*(z-pz)))-c1)/c2 else ztoy=c1*(z-pz)+y endif return end c--------------------------------------------------------------function ztox(x,y,pz,z,bz,px,py) include ’parm.h’ integer nx,ny,x1,x2,y1,y2,nz integer i,j,integ common /nbox/ nx,ny,nz real*8 a2d,x,y,z,c1,c2,pz common /aa/ a2d(nxmax,nymax)

65 real*8 ztox,x3,y3,by,bx,bz,dx,dy real*8 bxcheck,bycheck,px,py,ratio real*8 lx,ly,lz common /lbox/ lx,ly,lz ratio=lz*nx/(lx*nz) bxcheck=bx(x,y,px,py) i=integ(x) j=integ(y) if (i.lt.0) i=i-1 if (j.lt.0) j=j-1 x3=x y3=y if (abs(x-integ(x)).lt.(1.d-8)) then if (bxcheck.lt.0) then x3=x-1. i=i-1 else x3=dble(integ(x)) endif endif if (abs(y-integ(y)).lt.(1.d-8)) then bycheck=by(x,y,px,py) if (bycheck.lt.0) then y3=y-1. j=j-1 else y3=dble(integ(y)) endif endif call pgrid(x1,x2,y1,y2,x3,y3) c1=((i+1.)*(a2d(x1,y2)-a2d(x1,y1))) & +(i*(a2d(x2,y1)-a2d(x2,y2))) c2=a2d(x1,y1)+a2d(x2,y2)-a2d(x2,y1)-a2d(x1,y2) c1=c1*nx/lx c2=c2*nx/lx if (c2.ne.0) then ztox=(((c1+(c2*x))*dexp((c2/bz)*ratio*(z-pz)))-c1)/c2 else ztox=c1*(z-pz)+x endif return end c--------------------------------------------------------------subroutine locate(x,y,ox,oy,check) integer i,j,integ,ox,oy,check integer oox,ooy,nx,ny,nz real*8 x,y,xx,yy,bx,by common /nbox/ nx,ny,nz if (x.gt.1.d-11) then oox=1*dble(integ(x/nx)) else oox=-1+1*dble(integ(x/nx)) endif ox=oox x=x-dble(nx)*oox if (y.gt.1.d-11) then ooy=1*integ(y/ny) else ooy=-1+1*dble(integ(y/ny)) endif oy=ooy y=y-dble(ny)*ooy return end