Vertical and horizontal components of the electric background field at the sea bottom Endre Håland∗ , Eirik G. Flekkøy∗
†
and Knut Jørgen Måløy
†∗
ABSTRACT The natural E-field variations are measured at the sea bottom, and the magnitude of the different field components are compared and discussed in the light of the theory for induction caused by ocean surface waves. At shallow sea depths of 107-122 meters only the vertical component carries an observable effect of ocean waves, while the horizontal field is dominated by the larger magnetotelluric noise. This agrees well with theoretical predictions.
INTRODUCTION Understanding the electromagnetic noise that reaches the bottom of the sea is of fundamental interest in a range of contexts. In magnetotellurics the natural electromagnetic field variations is used to detect resistive or conductive bodies in the subsurface. Marine controlled source electromagnetic measurements (CSEM) rely on the resolution of a signal which must be stronger than the background noise. In many circumstances the averaging techniques applied to reduce the noise in the signal have problems in the frequency range of the ocean waves. It is therefore of both fundamental, and technological, interest to quantify the effect of ocean waves. The effects of the air-sea boundary give qualitatively different noise spectra in the different E-field components, and it is the purpose of the present paper to compare the two. The sea bottom measurements, which are obtained by the recently developed equipment of PetroMarker, include the vertical field component, which contain the effect of ocean waves, while this effect does not appear to be observable in strictly horizontal components. The reasons are that the wave-induced effects are much weaker, and the overall magnetotelluric noise much larger in the horizontal components. Only recently has it been possible to measure the relatively weak vertical component with any accuracy, as even minute deviations from the vertical antenna orientation will cause contamination of the signal. Our measurements are accurate both in terms of signal-to-noise ratio, resolving field strengths down to 0.5 nV/m, and in terms of verticality of the receiver antenna, as it is aligned with the direction of gravity to within 0.1o . This allows for a separation between horizontal and truly vertical field components, which, to our knowledge, is unprecedented. E-field variations due to ocean waves have been studied by a number of authors Podney (1975); T.B.Sanford (1971); Petersen and Poehls (1982); Davey and Barwes (1985); Ochadlik (1989); Cox et al. (1978); Chave (1984); Longuett-Higgins (1950); Weaver (1965), where Podney Podney (1975) in particular has provided a general formalism for calculating the ∗
e-mail: eirik.flekkoy@petromarker.com
2
magnetic field due to ocean surface waves over a layered earth. However, for clarity and simplicity we here follow a more direct, or pedestrian route, generalising Weaver (1965) solution for an infinite sea, to a 3-layer, air-sea-bottom model. This solution differs qualitatively from the Weaver solution and the solution quoted by Cox et al. (1978) through the fact that there are no horizontal field components in our solution. Our prediction is well confirmed by our measurements in the ocean wave range of frequencies, where indeed, it is observed that the weaker vertical E-field component carries a clear effect of ocean waves.
MEASUREMENT TECHNIQUE
Figure 1: (a) Tripod antenna structure, and (b) tripod with screening fabric. The measurements are carried out by 10 meter high tripods positioned on the sea bottom from a ship at water depths of d=106-122 m and d=313 m, Holten et al. (2009). The vertical tripod antenna is suspended like a pendulum with lead-chloride electrodes at the extreme of this pendulum arm. This is shown in Fig. 1 a. Figure 1 b shows the screened pendulum with its fabric coating as it is ready to be placed in the sea. The screening shields the antenna from hydrodynamic forces, and is a key factor in ensuring both accuracy and verticality. The horizontal antennas are located in the base of the structure and the 3 sets of horizontal electrodes are seen as white cylinders on the base poles. The vertical and horizontal E-field components are measured in periods of half an hour and their power spectra calculated. The sampling rate is 1000 Hz. Through independent GPS measurements the heave motion of the ship is recorded and the corresponding power-spectrum calculated. These heave measurements are challenging as the GPS must resolve displacements which are in the range less than the amplitude of the ships vertical movement. This amplitude may be as low as half a meter. The eigenfrequency f of the boats heave-motion may be estimated by the boats weight to be roughly ω = 2πf = 2 Hz, which lies above the relevant
3
wave frequencies. So, although this frequency is observable in the heave-spectrum, it lies above the wave frequencies and in a frequency range that does not affect the sea bottom E-fields.
POWER SPECTRUM IN THE DOMAIN OF THE OCEAN WAVE FREQUENCIES h(ω) ei ω t
λ
x v v d z
Figure 2: Sketch of the wave of wavelength λ in a sea of depth d with coordinate system. The positive y-direction points into the plane. Magnetotelluric noise that comes from electromagnetic activity in the ionosphere, lighting and tornadoes around the earth etc., appears largely in the horizontal E-field components. This follows directly from Maxwells equations and the large scale-separation between the thickness and the horizontal dimensions of the atmosphere. However, the noise induced by ocean waves enters mainly in the vertical field component, when measurements are made on the sea bottom which is generally the case in CSEM surveys. The reason for this is the combination of electric and hydrodynamic boundary conditions: There can be no electric current through the sea surface, and there is no water flow through the sea bottom. When conductive sea water is set in motion by the surface waves, the Lorentz force created by the water velocity and the earths magnetic background field, will create time-dependent electric currents. Thanks to induction these currents will spread out diffusively with a diffusivity D = 1/(σµ0 ), where σ is the sea water conductivity and µ0 the magnetic permittivity, see Weaver (1965). It is instructive to compare the characteristic diffusion time 1/(Dk2 ) to the wave period 2π/ω, where ω and k are the angular frequency and wave number of the ocean waves. The wave number k is related to the frequency through the shallow water dispersion relation, as given by Landau and Lifshitz (1959) ω 2 = gk tanh(kd)
(1)
where g is the acceleration of gravity. The ratio of the diffusion time to the wave period β=
µ0 σω 2k2
(2)
is generally small, as may be seen from Fig. 3. At least this is the case for the frequency range ω > 0.1 of primary interest relative to the ocean waves. This means that the inductive currents comes close to their steady state values where diffusion has had the time to relax. This observations allows us to estimate the electric fields.
log10(k)
log10 (β)
4
0
-2
-4
-6 -3
-2
-1
0
log10(ω/Hz)
Figure 3: The dispersion relation k = k(ω) (red curves) and the parameter β = β(ω) (green curves), which estimates the relative magnitude of the diffusion time to the wave period. B. The stapled curves show the d =300 m results, and the full curves the d =122 m results.
Assuming that the sea water is neutral, so that the charge density vanishes, Ohms law gives the following current density in the sea water j = σ(E + µ0 u × H)
(3)
where E is the electric field, u the water velocity, H the magnetic field, and all fields are measured relative to the sea bottom frame of reference. In a frame moving with the sea water the electric field would be measured as E + µ0 u × H and the electric current density would be the same, j. Now, assuming a layered earth model where there are no horizontal variations in charge density, a stationary current will flow freely in all horizontal directions and there can be no horizontal electric field. On the other hand, there may well be charge density variations in the vertical direction, both on the surface and the bottom of the sea. These charge densities will in particular cause the sea surface boundary condition jz = 0 to hold. The stationary solution of the diffusion equation, is a solution of the Laplace equation. With the boundary condition jz = 0 on the sea surface, the solution is simply jz = 0 everywhere, and therefore Eq. (3) predicts Ez = −µ0 (u × H)z = ux By (4) at the sea bottom. We consider a plane ocean wave of frequency ω with a horizontal wave vector k in the x-direction, so that u ∝ ei(ωt+k·x) . An inviscid and incompressible velocity field from waves of small amplitude compared to their wavelength will satisfy the Laplace equation. In particular this is true for the vertical component of the field ∇2 uz = 0. If we add the boundary condition of the surface velocity at z = 0, and uz (d) = 0 at the bottom, it may be shown (Landau and Lifshitz (1959)) that uz (x, t) = −h0 ω
sinh(k(z − d)) i(ωt+k·x) e sinh(kd)
(5)
5
where k = |k| and h0 is the wave height, and k is taken to point in the x-direction. We have chosen coordinates with z = 0 at the sea surface and z positive in the downwards direction. The incompressibility condition ∇ · u = 0 gives the other non-zero velocity component ux = −∂z uz /(ik), where ∂z denotes the derivative with respect to z, so that u(x, t) =
−h0 ω (ex i cosh(k(z − d)) + ez sinh(k(z − d)))ei(ωt+kx) sinh(kd)
(6)
Using this velocity field we can calculate Ez in response to the wave of heighth0 and frequency ω. However, our theory is entirely linear, so it will apply to a superposition of waves of different frequencies as well. In that case we need only to consider each seperate frequency, and this is simply done by allowing h0 to depend on frequency, h0 → h(ω). The power spectrum of the electric field is given as |E(d, ω)|2 ≈ |Ez (d, ω)|2 =
|h(ω)|2 ω 2 By2 sinh2 (kd)
(7)
where|h(ω)|2 is now the power spectrum of the waves and the horizontal E-field components vanish. Note, by the way, that this solution does not depend on the sea bottom conductivity. The above approximate analysis may be done exactly, and with more rigor, by generalizing Weaver’s classical solution for the induced magnetic field generated by linear waves in an infinitely deep sea. This generalization involves the introduction of the sea bottom with both its electromagnetic and hydrodynamic boundary condition and is carried out in the Appendix .
Figure 4: Shallow sea (d =107 m) powerspectra |Ez (f )|2 , |Ex (f )|2 and their ratio, |Ez (f )/Ex (f )|2 , taken at different times. In the following we compare measurements and theory. Figure 4 shows the power spectra of the vertical and horizontal field components taken at different times and at the relatively
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shallow depth of 107 meters. The spectrum of Ez has a very clear peak at the dominant wave frequency, while the Ex spectrum is more smeared out. It should be noted that in the course of half a day the strength of the Ez peak decreases by more than an order of magnitude, corresponding to variations in the wave conditions. The same is true for the low frequency part of the Ex spectrum. 2
|E (ω)|2
|E (ω)| z
(a)
y 2
z
−12
10 2
2
∝ 1/ω
10
−14
|E (ω)| y
B2|h(ω)|2 Theory
−12
10
2
∝ 1/ω
−14
10
i
10
2
(b)
−10
B |h(ω)| Theory
i
E (ω)2[sV2/m2]
10
|E (ω)|2
E (ω)2[sV2/m2]
−10
−16
−16
10
10
−18
10
−18
−1
10
10
0
−1
10
ω[1/s]
0
10
ω[1/s]
10
2
|E (ω)| z
−10
|E (ω)|2
(c)
y 2
2
B |h(ω)| Theory
−12
2
∝ 1/ω
10
−14
10
i
E (ω)2[sV2/m2]
10
−16
10
−18
10
−1
10
0
ω[1/s]
10
Figure 5: Shallow sea measurements and theoretical predictions of the sea bottom electric fields at different locations. The black, thick curves show the power spectra of Ez , and the magenta curves the associated theory given above. The noisy stapled curve shows the power spectrum of a horizontal field component, and the stapled blue curve has a slope of -2. The red curves show the ships heave power spectrum multiplied by a squared B-field of 15 µT to put it on the same scale. The three different figures correspond to the different depths (a) 107 m, (b) 116 m and (c) 122 m. In figure 5 it is seen that the vertical E-field component indeed confirms to the theoretical prediction for angular frequencies below 0.8 Hz, corresponding to wave periods roughly above 10 s. The magnetic field component By is taken to be a reasonable 15 µT in all three shallow sea figures, as well as in the d = 313 m figure. The theory predicts a very sharp decay in the spectrum with frequencies, as the sinh-factor in Eq. (7) behaves as 2
sinh−2 (kd) ≈ e−2kd ≈ e−2dω /g (8) p in the deep water limit. Frequencies above g/(2d) will therefore cause a quick decay, above which no effect of the waves are observed. In figure 6 the cutoff sets in at ω = 0.3 Hz, and in figure 5 at ω = 0.5 Hz, which is in nice agreement with the above expression. In Fig. 5 a the dominating wave frequency is a little too high for it to be measured as a peak in the bottom field, while in Fig. 5 b and 5 c the main wave peak is clearly seen in the spectrum. In all three figures the crossover from the theoretically predicted wave-signal to some othe type of
7
2
|E (ω)| z
2
|E (ω)|
−10
y 2
2
B |h(ω)| Theory
−12
10
2
∝ 1/ω
2
2
2
Ei(ω) [sV /m ]
10
−14
10
−16
10
−18
10
−1
0
10
ω[1/s]
10
Figure 6: Deeper sea measurements and theoretical predictions of the sea bottom electric fields at depth d=313 m. The black, thick curve shows the power spectrum of the vertical E-field component, and the magenta curve the associated theory given above. The noisy stapled curve shows the power spectrum of a horizontal field component, and the stapled blue curve has a slope of -2. The red curve shows the ships heave power spectrum multiplied by a squared B-field of 15 µT.
1/ω 2 noise is quite sharp since the wave signal has such a pronounced decay. Also, in Figs. 5 a and 5 b there is significant discrepancies between the low frequency measurements and predictions. This is most likely due to noise in the heave spectrum. Any kind of random drift in the GPS altitude measurements would contribute an 1/ω 2 noise (if the steps in the drift are uncorrelated) We also see some low frequency oscillations in the Ex spectrum of Fig. 5 b which are not presently understood.
log10(Ey(ω)/Ez(ω))
-1.5 -2 -2.5 -3 -3.5 -4 -3
-2
-1 log10(ω/Hz)
0
Figure 7: The ratio of the horizontal- to the vertical field components that are generated by ocean waves at the sea bottom, calculated from Eq. (A-19). The sea depths are d=122 meters (full curve) and 300 meters (stapled curve). The ratio Ey /Ez , which is calculated exactly, for finite β in Appendix , is shown in Fig. 7, and it seen that the horizontal component of the wave induced field is more than
8
two orders of magnitude smaller than the vertical one. This prediction sharply contrasts the measurements, where the horizontal component exceeds the vertical component by two orders of magnitude outside the dominating frequency of the waves. This means that it must have other causes than the waves, and that the wave motion does not dominate as a source in these horizontal E-field measurements. In fact, the measurement of the horizontal component is consistent with the standard magnetotelluric background noise. This may be concluded from separate longer time, deeper sea measurements.
CONCLUSIONS AND DISCUSSION A general conclusion of our measurements is the electromagnetic noise around the ocean wave frequencies is significantly larger in the horizontal, than in the vertical E-field components, and the difference grows with sea depth. At depths around 100 m the power spectrum of the horizontal components lies 1-2 orders of magnitude above the vertical spectrum, except at the peak corresponding to the maximum of the surface wave spectrum. At the depth of 313 meters, the wave effect is screened out, and the difference is 2-3 orders of magnitude over the whole range of frequencies. The fact that we observe agreement between the field measurements and the prediction of the shallow water wave theory for the vertical- but not the horizontal field has several immediate conclusions: First, in the frequency range of the waves, the noise in vertical field component measurements are indeed dominated by wave-generated field, at least at shallow water. In our d =122 m case the dominant wavelength is roughly 200 m, which is somewhat longer than the depth. In deeper waters it is likely that we will see similar effects when the wavelength-to-depth-ratio is the same. Second, the noise in the horizontal field components must have other causes, most likely the standard magnetotelluric ones, since the frequency dependence of the spectra are consistent with those. The fact that the vertical prediction and measurement are significantly weaker than the horizontal measurements also means that the vertical signal is not contaminated by the horizontal one. This is interesting, not only from the technological perspective, as it was a priori conceivable that 3D effects of bathymetry could cause the horizontal signal to pollute the vertical one. Third, we see no traces in the E-field measurements of a peak at the double frequency of the main peak of the wave spectrum. Such a peak in the horizontal E-field measurements was pointed out by Cox et al. and attributed to the standing wave effect described by Longuett-Higgins (1950). The absence of such a peak in our data could be due to an absence of significantly strong wave trains in the opposite direction of the main ones. If so, the effect of standing waves would be weak. The first conclusion also implies that a number of noise sources in the vertical channel can be ruled out. For instance, the tidal currents around the receiver station will generally cause Karman streets that produce a hydrodynamic force with a characteristic frequency. Also, small scale turbulence around the receiver station will cause an independent source of noise. While these effects are most likely present, the measurements show that they do not dominate, at least not in the wave frequency range. Other noise sources are likely to dominate outside the given frequency range. In the high frequency range the white electronic noise, with the Nyquist noise as a lower threshold, will dominate. In the low frequency range electrode drift is one source which is likely to dominate. From a practical point of view however, the extreme low and extreme high frequency ranges
9
are not problematic when a measurement of a controlled source signal is performed. The reason for this is that the high frequency noise may be averaged away by time window binning, while the low frequency noise may be removed by subtracting signals measured at different times. This is done in some CSEM techniques where responses to a subsequent signals of opposite polarity are measured. The effect of subtracting responses is to remove any noise components that vary insignificantly over the time between the two measurements. These times must be smaller than the time window of interest for the measurement and the correlation time of the low frequency noise. The frequency range observed in figure 2 is a problematic range in this context, and therefore one that it is important to understand and control. Turbulence around the receiver stations will have a velocity field that is not curl-free, and therefore provide an independent source of noise that we have not taken into account in the present treatment. However, this noise is likely to have a very different frequency spectrum from that of the surfaces waves.
APPENDIX: THEORY FOR THE SEA BOTTOM EFFECT OF OCEAN SURFACE WAVES
Air
σ =0
Sea
σ0
Infinite bottom layer
σ1
Figure A-1: The geological model When the conductive sea water moves due to waves, the Lorentz force created by the earths magnetic background field will cause an electric current. This current subsequently sets up a magnetic field that adds to the background field. This magnetic field was calculated already by Weaver (1965) for an infinitely deep sea, though to our knowledge, the shallow water, equivalent of Weavers calculations has not been presented in detail.
Governing equations When displacement currents are neglected and the last of Maxwell equations takes the form ∇×H=j
(A-1)
10
the electric field measured at a stationary receiver is from Eq. (3) 1 E = ( ∇ × H − µ0 u × H) . σ
(A-2)
Using Eq. (5) and Eq. (1) as the hydrodynamic starting points, the magnetic field follows from Maxwells equations ∇ · H = 0 and ∇ × H = σ(E + µ0 u × H). If ∇× is applied to the last equation and Amperes law ∇ × E = −µ0 ∂H/∂t is used to replace the E-field, then ∂H − σµ0 ∇ × (u × H) ∂t ∂H = µ0 σ + σµ0 (u · ∇H − H · ∇u) ∂t ∂H ≈ mu0 σ − σµ0 H0 · ∇u (A-3) ∂t where H0 is the background magnetic field, and we have used the fact that the perturbation in the field δH = H − H0 is much smaller than the background field. Introducing the notation δH = hei(ωt+kx) and u = vei(ωt+kx) (A-4) ∇ 2 H = µ0 σ
the above equation takes the form (∂z2 − k2 )h = iµ0 σωh − µ0 σH0 · ∇v.
(A-5)
Solution of the linear three-layer problem Since u has no y-component, it may be observed that the last term of Eq. (A-3) with the earth magnetic background field acts as a source term with no y-component. This means that if the H-field is created from zero by gradually turning on the velocity field, Hy = 0 at all times. Taking the z-component of Eq. (A-5) we get (∂z2 − k2 )hz = iµ0 σωhz − µ0 σH0 · ∇vz
(A-6)
h0 kω (iH0x sinh(k(z − d)) + H0z cosh(k(z − d))) sinh(kd)
(A-7)
where H0 · ∇vz = − so that (∂z2
h0 k (H0x sinh(k(z − d)) − iH0z cosh(k(z − d)) . (A-8) − k )hz = iµ0 σω hz + sinh(kd) 2
The homogeneous equation (∂z2 −k2 )hz = iµ0 σωhz has solutions e±qz where q 2 = k2 +iµ0 σω, while the particular solution may be found by equating the right hand side to 0. We will use the notation q0 and q1 when σ = σ0 and σ = σ1 respectively. In the air z < 0 where σ = 0, (∂z2 − k2 )hz = 0, and hz ∝ ekz . Below the bottom there is no water flow, so vz = 0, so hz satisfies the homogeneous equation with q = q1 . This means that we can write the solution as Qekz when z < 0 h0 k P1 cosh q0 (z − d) + P2 sinh q0 (z − d)− (A-9) hz (z) = H sinh k(z − d) + iH0z cosh k(z − d) when d > z > 0 sinh(kd) 0x−q1 z Re when z > d
11
Now, at the sea surface and bottom boundaries, both hz and ∂z hz are continuous. The first condition follows from integrating ∇ · H = 0 over a thin volume enclosing the boundary, while the second follows from the same equation and the fact that ikH0x is continuous across the surface. These conditions take the form Q = P1 cosh(q0 d) − P2 sinh(q0 d) + (H0x sinh(kd) + iH0z cosh(kd))
(A-10)
kQ = −P1 q0 sinh(q0 d) + P2 q0 cosh(q0 d) − k(H0x cosh(kd) + iH0z sinh(kd))(A-11) R = P1 + iH0z
(A-12)
−q1 R = q0 P2 − kH0x
(A-13)
These equations are readily solved. The two first may be rearranged to give a relation between P1 and P2 , and the last two another relation between P1 and P2 . Together these results yield P1 =
(k sinh(q0 d) + q0 cosh(q0 d))(kH0x − iq1 H0z ) − kq0 (H0x + iH0z )ekd q0 (k + q1 ) cosh(q0 d) + (q02 + kq1 ) sinh(q0 d)
P2 =
(k cosh(q0 d) + q0 sinh(q0 d))(kH0x − iq1 H0z ) + kq1 (H0x + iH0z )ekd . (A-14) q0 (k + q1 ) cosh(q0 d) + (q02 + kq1 ) sinh(q0 d)
By a bit of straightforward algebra the above equations may be used to verify that, when the conductivity goes to zero, σ → 0 and q0 → k, there is no current and no induced magnetic field hz . However, this does not imply that there is no electric field in the σ → 0 limit, as will be demonstrated below. Now, in order to evaluate Eq. (A-2) we note that ∇×H=
iey 2 ∂z − k2 hz ei(ωt+kx) k
(A-15)
so that the cosh k(z − d) and sinh k(z − d) terms above vanish when the curl operator is applied. In the sea layer we are left only with the e−q0 z terms. At the sea bottom z = d the P2 term vanishes and we get ∇ × H = −ey
σ0 µ0 h0 ω P1 ei(ωt+kx) sinh(kd)
(A-16)
for the current density j at the sea bottom. The expressions in Eq. (A-16) and Eq. (A-14) give the exact solutions for the magnetic field and current density in an air-sea-earth model. For the other term of Eq. (A-2) we must calculate u × H ≈ u × H0 = [−uz Hy , uz H0x − ux H0z , ux Hy ] ,
(A-17)
which on the bottom, where uz = 0, becomes u × H ≈ u × H0 = ux (d)(ez H0y − ey H0z ) =
ih0 ω (ey H0z − ez H0y ) . sinh(kd)
(A-18)
Inserting the above in Eq. (A-2) gives E=−
µ0 h0 ω ((P1 + iH0z )ey + iH0y ez ) sinh(kd)
(A-19)
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for the sea bottom E-field. Note that the vertical component only contains the Lorentz force contribution and does not contain any effect of the inductive currents. It is the above equation that is used in plotting Fig.¨7. In most applications q0 is quite close to k. Taylor expanding to first order q0 ≈ k(1 + iβ) where µ0 σ0 f λ2 µ 0 σ0 ω ≈ 0.03 . (A-20) = β= 2k2 4π Taking a wavelength of λ = 300 m and a corresponding frequency from Eq. (1) f = 0.1 Hz, β =0.003. Since the β → 0 limit is equivalent to the σ0 → 0, or q0 → k limit we need the limit lim P1 = −iH0z . (A-21) q0 →k
Combining the above results we obtain E(d) = ez
iµ0 h0 ω H0y ei(ωt+kx) . sinh(kd)
(A-22)
where the lack of a horizontal field component comes from the fact that the Lorentz term u × H term cancels the induction term ∇ × H in this limit.
The infinite depth limit It is instructive to compare the above results with the case of an infinitely deep sea, which is described by the d → ∞ limit. In this case the velocity field of the water, which follows from Eq. (6) is u∞ (x, t) = h0 ω(ez − iex )ei(ωt+kx)−kz (A-23) where the superscript on u∞ denotes the d → ∞ limit. Taking the infinite depth limit d → ∞ of Eqs. (A-14) it may be shown that 2k −q0 z ∞ −kz e . (A-24) hz (z) → hz (z) = h0 k(H0x + iH0z ) e − k + q0 Note again that when the conductivity goes to zero, σ0 → 0 and q0 → k, there is no current and no induced magnetic field hz . Using the above velocity field and Eq. (A-24) in Eq. (A-2) we get the electric field E∞ (z) = µ0 h0 ω (ex H0y + ez H0y ) ei(ωt+kx)−kz ,
(A-25)
which has no y-component. This, however is only true in the β = 0 limit. If we keep a finite, but small β-value and Taylor expand Eq. (A-24) to first order in β, then the use of Eq. (A-2) with Eq. (A-23) gives Ey∞ (z) = βh0 ωµ0 (H0z − iH0x )(kz + 1/2)e−kz
(A-26)
as obtained by Cox et al. (1978). The smallness of β, though, will make Ey∞ ≪ Ez∞ , or Ex∞ . In order to compare the infinite sea depth- and the bottom values of the fields, we take the large kd limit of Eq. (A-22), which, by the approximation sinh(kd) ≈ ekd /2, becomes E(d) = ez 2iµ0 h0 ωH0y ei(ωt+kx)−kd .
(A-27)
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which, apart from not having an x-component, has twice the magnitude of the infinite depth field Ez∞ (d). The fact that the bottom field has no x-component simply reflects the fact that the water does not flow through the sea floor, so that bottom velocity field has no z component. The doubling of the magnitude of the bottom field, compared to the deep-water field at the same depth z = d is attributed to an increase in the horizontal water velocity due to the presence of the bottom. It is easily seen from the large kd limit of Eq. (6) that the bottom velocity ux (z = d) is twice the velocity u∞ x (z = d) calculated from Eq. (A-23).
Power spectra Now, if h0 is taken to be only of many frequency components in the ocean wave spectrum h(ω), we can use that fact that all the above results are linear in h0 to generalize the above expressions to hold for an arbitrary wave spectrum h(ω). This gives the following expressions for the sea bottom power spectrum of the vertical field component |E(d, ω)|2 ≈ |Ez (d, ω)|2 =
|h(ω)|2 ω 2 By2 sinh2 (kd)
(A-28)
where we have used B = µ0 H0 , and where the frequency and wave number is related through the dispersion relation of Eq. (1). The approximation |E(d, ω)|2 ≈ |Ez (d, ω)|2 holds to order β ∼ 10−3 , as is shown above.
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