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Induction in a model ocean
328
Physics of the Earth and Planetary Interiors, 53 (1989) 328—336
Elsevier Science Publishers By., Amsterdam
—
Printed in The Netherlands
Induction in a model ocean D.E. Winch Department of Applied Mathematics, University of Sydney, Sydney, N.S. W (Australia)
(Received January 29, 1987; revision accepted August 20, 1987)
Winch, D.E., 1989. Induction in a model ocean. Phys. Earth Planet. Inter., 53: 328—336. The theory of Rikitake for the induction of electric currents in a thin uniformly conducting spherical shell representing the ocean, surrounding a non-conducting shell and uniformly conducting core, is extended to deal with a non-uniformly conducting shell to represent the land—ocean distribution. A method by which conductivity rather than resistivity of the land—ocean distribution could be used is also suggested.
1. Introduction Analyses of solar and lunar magnetic variations and magnetic storms have provided spherical harmonic coefficients for external (inducing) and internal (induced) parts of the magnetic potential. Up until 1960, the results, expressed as amplitude ratios and phase angle differences between the internal and external parts, provided the only input for the modelling of electrical conductivity of the Earth on a global scale. The potential function for magnetic storms used only the one surface spherical harmonic, P10, and the results were used to discriminate between the various models determined from the potential of solar daily magnetic variations at 1, 2, 3 and 4 cycles per day based on the surface spherical harmonics P~,P~,P~and P~,respectively. Seasonal variations also gave usable potential coefficients for the sectorial surface spherical harmonics P~and P~. The very marked variability of conductivity models based on daily magnetic variations (e.g., Parkinson, 1974), led to a preference for models based on the Pf potential for the continuum of frequency associated with the 27-day recurrence tendency of magnetic activity (Banks and Bullard, 1966). Knowledge of the P 1°response over a range of frequencies has made conductivity modelling 0031-9201/89/$03.50
© 1989 Elsevier Science Publishers B.V.
possible with data from a single observatory only. When used with the Backus—Gilbert inversion theory (e.g., Parker and Whaler, 1981), the conductivity models are found to vary significantly from those based on the daily magnetic variations and magnetic storms, which give only a single sharp increase in conductivity between 400 and 800 km. Those based on inversion theory show a minimum between 400 and 600 km (e.g., Parker and Whaler, 1981). In this respect, the warning given by Price (1973) is still very relevant: “...the first earth models, Schuster’s and Chapman’s, necessarily incorporated a sudden rise in conductivity for purely mathematical simplicity and convenience. Some investigators apparently claim that the conductivity profiles that they have deduced from geomagnetic induction studies establish the existence of a phase change at a certain depth, but, personally, while I think it probable that a phase change occurs somewhere, I do not believe that the geomagnetic evidence at present available can alone establish this fact incontestably”. Price was referring to the estimation of two-parameter models of the conductivity distribution, corresponding to the radius, q (given as a dimensionless fraction of the entire Earth, denoted a km) of a uniformly conducting core, and its conductivity, a~S,based on
329
two input parameters, namely the amplitude ratio and the phase angle difference between the internal and external potential coefficients. The depth of the sudden rise in conductivity referred to the depth (1 q) am of the uniformly conducting core below the surface of the Earth. The electrical conductivity of the oceans is able to influence the estimate of coefficients in the potential of the magnetic daily variations (both regular and irregular) by two means. One is owing to the magnetic potential of electrical currents induced in the ocean by the westward-moving daily magnetic variations fields, and the second is owing to the magnetic potential of electrical currents flowing in response to the electric fields set up by the tidal movements of the ocean acting as a dynamo in the presence of the Earth’s main magnetic field. For the Sq magnetic variation at 1 cycle per day and the lunar magnetic variation at 2 cycles per lunar day, the ocean dynamo effect is significant, perhaps more significant than the effect of induction in the ocean by the Sq or L fields. For lunar magnetic tides (L), Maim (1970) devised a simple calculation to give an estimate of the lunar semi-diurnal ocean dynamo effect, based on the assumption that there are no ionospheric current systems at local midnight. However, the problem that has attracted most attention is that of induction of electric currents in the ocean by the westward movement of the magnetic daily variation fields. No results appear to be available for effects associated with the transient phases of magnetic storms, although the calculation would, in principle, be very similar to that for the magnetic daily variation fields, A fundamental paper by Chapman and Whitehead (1923) developed the earlier theory of Schuster into the theory of induction in a uniformly conducting shell surrounding a concentric uniformly conducting spherical core, and the determination of what might be referred to as (q, a~) models of global conductivity began with the Chapman—Whitehead paper. The work emphasized the determination of (q, a~)parameters and expressed little interest in the corresponding current function for the system of electric currents induced in the thin shell, The Chapman—Whitehead theory was then de—
veloped further by Price (1949) to deal with the induction of electric currents in non-uniform thin sheets and shells. Price was concerned more with induction in the ionosphere than in the ocean, and he chose a model in which the non-uniform resistivity was proportional to (1 ± cos 9), where 9 is the colatitude, to represent the increase in resistivity that occurs in the ionosphere during the nighttime hours. The theory concentrated on the determination of the magnetic potential e~’for the currents in the shell from the coefficients E~ for the potential of the external inducing field, and suggested two iterative schemes for the solution of the problem. Bullard and Parker (1970) then took up the work in an important paper on the induction of electric currents specifically in a model ocean, using numerical techniques based on a 3 X 3° grid representation of the ocean bathymetry. Electrical conductivity of the deep oceans was shown to be proportional to the ocean depth to quite a good level of accuracy. The theory also included the effect of an infinitely conducting inner core by use of the method of images. Bullard and Parker preferred to work in terms of the current function ‘I’( 9, ~), so that the requirement that no electric current should cross a land—ocean boundary became a requirement that the land—ocean boundary should be a constant contour of the current function. The current function contour value around islands, e.g., Australia or New Zealand, would be different for different islands, and solutions for this problem have only recently been obtained. Beamish et a!. (1980) show contours for the current function around Antarctica which do not appear in the corresponding contours given by Builard and Parker (1970). The work of Builard and Parker (1970) drew attention for the first time to difficulties associated with the convergence of the iterative procedure suggested by Price (1949). Basically, Price’s method converged for the 24-h variation, but not for the 12-, 8- or 6-h terms. A discussion of the convergence problem is given in the summary by Kendall and Quinney (1983). Hewson-Browne (1973) indicated some preliminary results of ocean dynamo effects using studies of tidal flow in channels. Rikitake (1961) presented the theory for induc-
330
tion of electric currents in an ocean represented by a hemispherical shell surrounding a non-conducting layer and a conducting core of finite conductivity. His work gave results in terms of the magnetic potential rather than the current function and, estimated an effect of the order of only 2 nT. The results are also given diagrammatically in terms of the current function. The Rikitake theory ignores derivatives of conductivity in the model, and these will of course be indeterminate at the boundary of the hemispherical conductor. The Rikitake solution is therefore valid only in regions away from the boundary, and is what Kendall and Quinney (1983) would refer to as the ‘outer solution’ corresponding to the solution away from the edge of the hemispherical shell. The early Chapman and Whitehead (1923) paper made use of the amplitude ratio and phase angle difference derived from the external and internal parts of the P2’ diurnal Sq term, but the authors expressed some reservations about this particular term. The vital point is the phase angle difference between the internal (induced) and external (inducing) terms. On the basis of causality, the internal (induced) terms should lag behind the external (inducing) term, and in the same way that sin(t + 30°) lags behind sin(t ± 10°), the phase angle difference, internal minus external, should be positive, and in the case of the P~diurnal term should be 20°.However, recent analysis of the diurnal term, using all observatory data available (e.g., Parkinson, 1971; Maim, 1973; Winch, 1981), have all shown that the phase angle difference is very close to 0° and possibly negative and that therefore the Chapman—Whitehead theory cannot be used to derive (q, a~)models of the Earth’s electrical conductivity as it requires a positive phase angle difference before one can start. This argument can be taken further: the observed negative phase angle difference cannot be a consequence of induction. It could, however, be associated with an ocean dynamo effect whose phase is in advance of the induced internal Sq dynamo field. Recent analyses by Beamish et al. (1980) have included results for the amplitudes of magnetic field variations associated with electrical currents induced in the ocean, but the information given is —
not sufficient to determine if the influence on the phase angle difference is favourable for the theory, i.e., likely to increase the phase angle difference between internal and external parts of P~term. Takeda (1985), in an analysis at 2-hourly intervals of the Sq current system, sought an ocean influence by averaging with respect to longitude to determine the purely local time field. He then graphed the residual terms, hoping to show that such non-local time residuals would be related to the land—ocean distribution. The residuals showed only a little correlation with the land—ocean distribution. It can easily be shown, however, that currents induced in the ocean include terms which depend solely on local time. All in all, one is left with the view that there is likely to be a significant ocean dynamo contribution to the P~ term in the Sq potential and also to the P~ term in the lunar magnetic variation. The Rikitake (1961) paper provided a very useful benchmark against which computer programs for non-uniformly conducting shells could be checked (e.g., Hobbs, 1971). The Rikitake analysis gave results directly in terms of a magnetic potential by solving sets of equations directly, without making use of the iterative procedure suggested by Price (1949). It is significant therefore that Rikitake (1961) gave results for both the 24- and 12-h variations and did not find the difficulty with convergence at 12 h that was found later by other authors. For this reason, it seemed worthwhile to show how the Rikitake theory could be extended to deal with a non-uniformly conducting shell used to represent the conductivity of the oceans, with conductivity proportional to the ocean depth. The purpose is to provide a spectral representation of the induction equation along the lines used with such success in the Bullard-Gellman theory of the geomagnetic dynamo, using conductivity, rather than resistivity which would be very great over non-conducting regions, e.g., the land, and not amenable to representation as a linear combination of spherical harmonics. The theory will treat the oceans as a thin shell, and the induced current in them as a surface current represented as a toroidal current system (producing a poloidal magnetic field), or to put it another way, repre-
331
sented by an equivalent stream function. The model also includes a concentric uniformly conducting core. In the sense that one is using a current function as a depth integrated model of
1 in the region r a m above 9, ~) Acurrent m sheet is given by the0(r, spherical H ~ 0(r,
the current systems (much as is done for ionospheric the dynamo theory), of it is simple matter to include conductivity thea sub-oceanic lithosphere. To deal with a thick spherical shell and to treat the sub-oceanic conducting lithosphere in the presence of poloidal and toroidal inducing fields requires the inclusion of poloidal and toroidal fields in the thick spherical shell and would be more complex than the simpler analysis for induction by a toroidal field analysis for a thin shell, Poloidal current systems, with non-zero radial components can be used to represent leakage from the system induced in the ocean. However, the boundary conditions and orthogonality of poloidal and toroidal fields over the surface of a sphere indicate that such current systems cannot be induced in a spherical shell even with a non-uniform distribution of conductivity by the toroidal current systems that constitute the inducing external daily magnetic variation fields. This argument does not apply to regions with non-spherical boundaries, and it might therefore appear that one electromagnetic mode has been eliminated in regions with spherical boundaries, but the nature of the inducing field, together with the boundary conditions over the surface of a sphere, do not require it. The critical nature of the assumption is dealt with by Ranganyaki and Madden (1980) and Coxetal.(1986).
0,
~)
=
a
~
[(~)n+t(J0)~
m(o,
n.m
ç,
x
~
where
(~)~(E0):}
+
4)) e””~
(1)
(E
0)~ A m~ are complex numbers giving the potential of the so-called internal and external parts, respectively, of the magnetic variation field corresponding to parts which originate either within the sphere r a m or outside it, e.g., in the ionosphere. The sum over n is from 1 to some finite number N and the sum over m from —n to n. The surface spherical harmonics )‘~(0, 4)) are normalized so that the r.m.s. value over the surface of a unit sphere is unity. The corresponding magnetic field strength H0(r, 9, 4)) A m in the region r a m above the spherical current sheet is given by —v~0A m~, and hence in spherical polar co-ordinates at the level r a, its components have the form =
=
(H0),(a, 9, 4)) =
—
~
(n
[—
+
1)(I~)’
n.m
+
(H0)9(a, 0, =
~
—
~)
[(ic~)’
n.m
(H0)q,(a, 0,
n(Eo)1 Y~m(9,~) e~
I
+
aY,n ~
(E0)’ —~—e
(2)
~) m
=
~ [10:±E0:
—
1
n.m
8Y
It is convenient to denote the magnetic potential 2. Theory A thin spherical shell of radius a m of non-uniform electric conductivity corresponding to the land—ocean distribution carries a surface density of electric current, i.e., is a spherical current sheet, and surrounds a non-conducting spherical shell of
of the magnetic field strength in the current free (non-conducting) region r a within the spherical current sheet, ~‘2.(r,9, 4)) A, where ~.
(
,~,
9, 4))
=
a
~{ (~)
+
(I,)
n.m
+
(~)~(E 1)’]Y~(9, 4)) e’~’
thickness (1 — q)a m and a concentric uniformly conducting core of radius qa m and conductivity a S. Colatitude and east longitude will be potential denoted by 9 and 4) throughout. The magnetic ~2 0(r, 9, 4)) A of the magnetic field strength
(3) 1 relate to current systems where the A sphere rn outside and(E~ on)~‘ the r a and (I,)~ A m’ =
to current systems within the conducting core. The
332
corresponding spherical polar components of the magnetic field strength H1(r, 9, 4)) A m’ within the spherical current sheet are given by
~)
0,
(~)r(a,
=
[-(n
~
-
±i)(i~)
1
(H,)
~
0(a, ~ 4))
~
—
=
n ,m
=
4)) e~w
m
‘ + (E, ) J
[(ii)
lines of current flow. It follows from eqns. (7) and (9), that m n+1 (E0)~ 2n + 1 ~“ + (E1) (11) ~ 2n±1 ~ ±(i,)~ (12) =
n,m
±n(Ej)]
tween them so that contours of ‘I’(O, 4)) are the
e””
ay
(4)
In the conducting core r qa, the magnetic flux density being flux solenoidal is made of poloidal and toroidal densities. If theup magnetic meability is constant, the same applies to perthe
(H,) =
4,(a, 9 4)) ~ [(i~)’ —
+ (E,)n 1
n ,m
magnetic field strength. The boundary conditions daily that apply variation to induction poloidal field by the imply external that nomagnetic toroidal field is generated, although it could exist as a
m —a—’ sin9 04) e’~” 1 OY
The boundary conditions across the spherical current sheet are given by ~ [B 0(a, 0, e., x [H0(a,
~)
B1(a, 0,
—
9, 4))
H,(a, 9,
—
0
(5)
~)}
K(0, 4)) (6)
4))1
=
=
freely decaying current system. The poloidal field components are of the form Hr(r, 9, 4)) = ~
n(n ~ s(r)Y~m(0, 4)) e’~’ mr OYm ds ~ Cm__~___~?_eI~l (13) dr 89
n,m
H 9(r, 9, 4))
=
n, m
where K(0, 4)) A m’ is the surface current density in the shell r= a. If the magnetic permeability is the same on both sides of the spherical current sheet, then the boundary condition (5) gives ±i)(I~)’ + n(E0)’ =
—
(n ±i)(i,)’
(7)
+ n(E,)’
K,~=
aiç,
1
—~—e”~
m
—
‘i’,~”aY
~
—~-~--
~
n ,m
ds,” 1 C,,”—~---—
8ym
—‘——i —~—e”~
where the coefficients are constants and the dimensions of s~(r)C,~” are Ametres. The functions
strength H(r, 9, 4)) A m’~satisfies OH V X (v X H) + ,aoac~’~- 0 (14) where a~ S is the conductivity of the core. For =
tm
8= ~ ~
=
s’(r) must be chosen so that the magnetic field
and the boundary condition (6) gives K
H4,(r, 9, 4))
(8) e””
time dependence of the form e”, the radial cornponent of eqn. (14) leads to the Ricatti—Bessel equation with solutions of the form snm(r)
n,m
where the coefficients ‘I’,~”A m
—
are given by
where
=
rjn(Ar)
A2
=
—
i,.L
(9)
0coa~ m -2, so that the argument Ar of the spherical Bessel function j~(Ar) is seen
We write K(0, 4)) aV x [e,’I’(O,~)] A m’ for the surface current the orfunction 1 is thedensity currentwhere function stream ‘I’(O, 4)) A m by function given
to be dimensionless. Chapman and Bartels (1940) make good use of the functions e’~p~(kr)in place of the Besselpn(kr) functions. For integer values of nspherical the function is a polynomial (kr) 1 fl (n ±1)
=
(Ia)’
±(E0)~”’—
(i,)’
—
(E,)
=
‘~p(9 ‘1))
=
~
4..rnym(9
p)
e~’~
(10)
—
—
2k,’
n ,m
The difference between values of ‘I’(9, 4)) at different points gives the total current flowing be-
(2kr)2
(n—1)n(n±1)(n+2) ±
2!(2kr)3
—
(is)
333
which is similar to the Legendre polynomial F,, ( ~t) expanded as a power series in (1 ~t)/2
Numerical evaluation of the real and imaginary parts of the bracketed expression on the right of
P,,(
eqn. (22)theis oceanic most easily done thesub-oceanic functions 2 of layer, withwith the )~~(1~T) of Chapman and Bartels (1940). If the depth-integrated resistivity (or surface resistivity) p~, lithosphere if required, is a function of colatitude 0 and east longitude 4), and if K(0, 4)) A m’ is the surface current density, then the correspond-
—
(1 ‘~ n(n ±1) ±(1 ~ )2 2’2 x (n — 1)n(n ±1)(n + 2) (16) Assuming the magnetic permeability to be the same on both sides of the conducting core surface, r qa, the continuity of the normal component of magnetic induction Br( r, 0, 4)) T gives the following result from eqns. (3) and (13) 1it)
=
1
—
—
—
...
=
ing surface electric field E( 0, current sheet is given by
~)
V m1 in the
(n ±1)q”2(1 1)’
—
E(0, 4))
nq”’(E1)’
1
(17) and, in the absence of a surface current at the surface of the conducting core, the continuity of the tangential component of magnetic field strength, using eqns. (3) and (13) gives =n(n+1)C,,m__j,(Xqa)
—
q~—1(E,)~
1
=cnmi{jn(xqa)±xqa(dJ~(r)~ / r—xqaj ____
=
p(0,
4))K(0, 4))
(23)
and on taking the curl of eqn. (23), one obtains 2’I’) (24)
vXE=—e~(vp.v”I’+pv
Hence the radial component of the Faraday equation V x E ±8 B/8t 0 reduces to the fundamental equation for induction in non-uniformly conducting current sheets =
8t
(18)
—
vpv4’±pv2’I~
(25)
Solving eqns. (17), (18) for the coefficients (I, )~‘,(E, )‘ A m1 of the potential field in the non-conducting region, and making use of the recurrence relations for spherical Bessel functions, one obtains 2(1)’ nC~mj,,÷i(Aqa) (19) (2n ±1)q~
(e.g., Price, 1949; Zhdanov, 1980). Because of the continuity of the normal (i.e., radial) component of magnetic flux density across the current sheet at r a, the normal component of the magnetic flux density there can be expressed in terms of (I,),,~’and (E ent to choose1)~’ (I,)’ and (E (E0)~’.It is convenior (1~)’and 1)’, so that, from eqn.
(2n±1)q~~(E1)= —(n±1)C,~j,,_1(Xqa)
(4)
=
(20) 1 can also be written The as eqn. (19) for (I,)’ A m
~
~
N.M
8t +
N(E,)NMI Y~(0, 4)) e”
(26)
tm 2n ±1 —j,,_
(2n + 1)q~2(I 1)
=
nC
1(Xqa)
Xqa
(21) tm A from eqns.
and the constant C,, (20) on andeliminating (21), we have m
=
in which the sum over N, M means only the terms for the chosen inducing field. For example, the diurnal analysis (1940) uses M terms 1 andonly N of1, Benkova’s 2, 3, 4, 5. Substituting for (I,)~ from eqn. (22) into eqn. (26) gives =
=
fl
8Br
=
x[1
—
2n+1 j,,(Aqa) Aqa j,,_
1
1(Aqa)j~’~
=
—
~
[A~(Xqa)a±iB~(Xqa)]
N.M
(22)
XN(E1)~Y~(9,4))e~wt
(27)
334
where, following Rikitake (1961), we have made use of the dimensionless variables A~(Aqa), B~(Aqa), defined by
M’) are defined by I(p, n, N’, m, M’) 2~r
~f0 fp(94))Ytm(94))~M’smn0d9d4) 1
A~(Xqa)±iB~(Aqa)
L
]
2N±1 j~(Xqa) (28) Xqa J~_J(Aqa) which can be tabulated for given q, A and which for an infinitely conducting core reduces to the value 1— q2N+i, Using eqn. (25), and using eqn. (11) to replace (E 1)~in eqn. (27), the equation for induction in a non-uniformly conducting thin sheet in the presence of a uniformly conducting core becomes =
1 _q2N+iF1
(31)
m
—
1
Op 81T~n
1 1 jo2ir fo‘~f8p 4~r (,89 Oiç,” 89
=
1
8~
_‘hi~~(o,
84) 84) j
±~
4)) sin9 dO d4)
N
(32) which can be evaluated numerically, once and for all, for any given land—ocean distribution. Price (1949) considered the nature of such solutions for the surface resistivity p 1 + cos 0, for induction without a conducting inner core. If the surface =
~ I[Op 89 8}~, 80 + sin2O 84) 84) n,m
resistivity is given as a sum of surface spherical —
=
J(p ~ N’, m, M’)
n(n
±1)p)~,tm(O,4))]4~’”
harmonics, the required integrals can be obtained in terms of Wigner 3 —j coefficients 1
~ i~owa(AN+iBN)N[~7~~I’M
_j
N
f
2,r
N,
4))
(29)
Y~(0,4))Ym(0
=(-1)M~(2L+1)(2n+1)(2N’+1) IL
This equation is satisfied over the whole surface of the sphere. Hence, on multiplying eqn. (29) throughout by Y~.’(0,4)) and integrating over the surface of the sphere, one obtains an infinite set of
4))Y~’ sin0d0d4)
1
n
~
N’\1L
o)U
n m
N’\ -M)
(33)
2~ ~ 8y~8Y,”
j j
~—
1
equations
+
i~tOc~~a(AN + iBN)N(EO)~~~’aM’ N ~‘M
=
~
(~
—~-
—~--
8y~8ym —a-- I~’(9, 4)) sin 9 dO d4) 84) 84) ) N
(_1)M~[L(L ±1) + n(n + 1)
N(N± 1)q,M =i,tO~a(AN±iBN) 2N±1 — ~ [J(p, n, N’, M, M’)
6N~~~M~ N
N
M
(30)
x
N(N± 1)] I(2L+ 1)(2n ±1)(2N’+ 1) tL n N’ (,~ 0 0 / m —M) (34) n N’\
n.m
—n(n
±1)I(p, n, N’, M, M’)] 4’,~
where I(p, n, N’, m, M’) and J(p, n, N’, m,
where L + n + n’ must be even and the values for L, n, n’ must be such that it is possible to form a triangle with them (e.g., Winch, 1974). If one wishes to make use of surface conductivity, a, in
335
place of surface resistivity, p, then with the substitution p 1/a, eqn. (29) becomes ~[8a81~,m 1 Oa8Y,m [~ 89 + ~ ~
where K(n, N’, m, M’) is here defined by
=
K(n, N’, m, iivi’) 1 =
n.m
±n(n+1)aY,,m(O,
=
— ~
4))]~~n
i~t~wa2a(A,~ ±iBN)N[
N,
N+ 1 2N+1N
(35)
(Eo)~I Y~(O,4))
—
which is valid over the entire surface of the 9’sphere. 4)) On multiplying eqn. (35) throughout ~N~ and integrating over the surface of thebysphere, one obtains the following set of equations i,iOwa(AN + =
iBN)N(Eo)1(a2,
n,
N’,
m,
M’)
N(N±1) 2N + 1
±~
[J(a,
‘4’NI(a,
n,
N’, m, M’)
(39) The solution consists of a set of parameters { ‘I’,~} for the current function, for a given set of inducing terms {(E 0)’), under whatever assumptions are appropriate concerning the distribution of resistivity or conductivity in the spherical shell. A great of effort has gone into finding contours of thedeal current function ‘I’ which have the continental shelf as a contour, but what is required now is numerical results for those ‘I’,” which correspond to the principal input (E 0)~’ so that the terms (I~)~’ can be corrected for the inductive
n, N’, M, M’)
n.m
+
4)) sin 9 dO d4)
(as opposed to the dynamo) influence of the ocean. Equation (38) can be solved directly if recurrence methods fail.
4tOwa(AN + iBN) X
j,cean~~tm(9~4))Y~(O,
3. Time dependence
n(n
±1)I(a, n, N’, M, M’)]’4’,”
(36)
There is a need to consider the explicit
which are very similar in form to eqn. (30). If eqn. (29) is to be applied in ocean areas only and the surface resistivity is taken to be a constant, then it reduces to
mathematical form of terms in the magnetic potential of the daily variations arising from the inductive influence of the oceans. It is a simple matter to show that such terms are not only UT dependent, but also include local time terms as well. Consider the magnetic potential associated with the current function induced in the oceans by
n(n
— ~
+ 1)pY,tm(O,
a)’I’,”
n.m
=
[N+1 ip0wa(A~,,,±iBN)N[ 2N ±1
~ N.M
the diurnal magnetic daily variation: it consists of a ‘real’ or ‘in-phase’ part
..JfM N
(37)
a~A~1cos(k4)+a~)I~(cosO)
(40)
j,k
and on multiplying throughout by ~N~’ and integrating over the ocean area only, it follows that
and an ‘imaginary’ or ‘in-quadrature’ part a>BJ~, cos(k4)+/3j’)I~’(cos 0)
— ~ n(n + 1)p’I’,~”K(n,N’, m, M’)
which are combined together as
n ,m
[N+1
=
(41)
J,k
~ iftowa(Afl±iBN)N[2N
1
~[A~~costcos(k4)-i-4)
N
N, j,k —
(Eo)] K(N, N’, M, M’)
(38)
±B~1 sin t smn(k4) ±$1k)1 Pk(cos 0)
(42)
336
and then written in the form
tion in the oceans. In: E.C. Bullard and J.L. Worzel (Edi-
tors), The Sea, Vol. IV. Wiley, New York, pp. 695—730. Chapman, S. and Bartels, J., 1940~Geomagnetism, Vol. II, Oxford University Press, Oxford, p. 739. Chapman, S. and Whitehead, T.T., 1923. The influence of
[Ci,cos(k4) + t ±
~ j,k
±D~, cos(k4) — t + ~~)]
pk(C
05
0)
(43)
In these equations t is the UT and the local time t* t ± 4), hence with k 1 there is clearly one term in eqn. (43) with diurnal local time dependence. =
=
4. Summary The paper has discussed at some length the theory needed for the study of induction in a non-urnformly conducting ocean, surrounding a concentric conducting shell, and the need to determine the contribution to the principal spherical harmonic terms in order to correct them for the inductive influence of the oceans. The ocean also acts as a dynamo, and there is evidence for a strong contribution from the ocean dynamo to the coefficient of F2’ in the internal field potential at one cycle per day. Now that tidal movements in the deep oceans have been so well determined, it should be possible to determine, in the first instance, current function contours associated with the ocean dynamo current system, and then to correct further the principal terms in the internal magnetic field potential used for conductivity modelling of the Earth.
References Banks, R.J. and Bullard, E.C., 1966. The annual and 27 day magnetic variations. Earth Planet. Sci. Lett., 1: 118—120. Beamish, D., Hewson-Browne, R.C., Kendall, P.C., Main, S.R.C. and Quinney, D.A., 1980. Induction in arbitrarily shaped oceans—V. The circulation of Sq induced currents around land masses. Geophys. J. R. Astron. Soc., 61: 479—488. Benkova, N.P., 1940. Spherical harmonic analysis of the Sq variations, May—August 1933. Terr. Magn. Atmos. Elect., 45: 425—432. Bullard, E.C. and Parker, R.L., 1970. Electromagnetic induc-
electrically conducting material within the earth on various phenomena of terrestrial magnetism. Trans. Camb. Phil. Soc., 22: 463—482. Cox, CS., Constable, S.C., Chave, A.D. and Webb, S.C., 1986. Controlled-source electromagnetic sounding of the oceanic lithosphere. Nature, 320: 52—54. Hewson-Browne, R.C., 1973. Magnetic effects of sea tides. Phys. Earth Planet. Inter., 7: 161—166. Hobbs, BA., 1971. Calculation of geophysical induction effects using surface integrals. Geophys. J. R. Astron. Soc., 25: 481—509. Kendall, P.C. and Quinney, DA. 1983. Induction in the occans. Geophys. J. R. Astron. Soc., 74: 239—255. . . Malin, S.R.C., 1970. Separation of lunar daily geomagnetic variations into parts of ionospheric and oceanic origin. Geophys. J. R. Astron. Soc., 21: 447—455. Malin, S.R.C., 1973. Worldwide distribution of geomagnetic tides. Phil. Trans. R. Soc., London, 5cr. A, 274: 551—594. Parker, R.L. and Whaler, K.A., 1981. Numerical methods for . establishing solutions to the inverse electromagnetic induction. J. Geophys. Res., 86problem (BlO): of 9574—9584. Parkinson, W.D., 1971. An analysis of the geomagnetic diurnal variation during the International Geophysical Year. Beitr. Geophys., 80: 199—932. Parkinson, W.D., 1974. The reliability of conductivity derived from diurnal vanations. J. Geomagn. Geoelectr., 26: 281284. Price, AT., 1949. The induction of electric currents in non-urnform thin sheets and shells. Quart. J. Mech. Appl. Math., 2: 283—310. Price, A.T. 1973. The theory of geomagnetic induction. Phys. Earth Planet. Inter., 7: 227—233. Ranganayaki, R.P. and Madden, T.R., 1980. Generalised thin sheet analysis in magnetotellurics: an extension of Price’s analysis. Geophys. J. R. Astron. Soc., 60: 435—444. Riltitake, T., 1961. Sq and Ocean. J. Geophys. Res., 66: 3245—3254. Takeda, M., 1985. UT variation of internal Sq currents and the oceanic effect during March 1—18. Geophys. J. R. Astron. Soc., 80: 649—659. Winch, D.E., 1974. Evaluation of geomagnetic dynamo integrals. J. Geomagn. Geoelectr., 26: 87—94. Winch, D.E., 1981. Spherical harmonic analysis of geomagnetic tides, 1964—1965. Phil. Trans. R. Soc. London, Ser. A, 303: 1—101. Zhdanov, MS., 1980. Allowing for the effect of the world ocean on the variable geomagnetic field by means of a spherical shell of finite thickness. Geomag. Aeron. (English edition), 20: 355—358.
Physics of the Earth and Planetary Interiors, 53 (1989) 328—336
Elsevier Science Publishers By., Amsterdam
—
Printed in The Netherlands
Induction in a model ocean D.E. Winch Department of Applied Mathematics, University of Sydney, Sydney, N.S. W (Australia)
(Received January 29, 1987; revision accepted August 20, 1987)
Winch, D.E., 1989. Induction in a model ocean. Phys. Earth Planet. Inter., 53: 328—336. The theory of Rikitake for the induction of electric currents in a thin uniformly conducting spherical shell representing the ocean, surrounding a non-conducting shell and uniformly conducting core, is extended to deal with a non-uniformly conducting shell to represent the land—ocean distribution. A method by which conductivity rather than resistivity of the land—ocean distribution could be used is also suggested.
1. Introduction Analyses of solar and lunar magnetic variations and magnetic storms have provided spherical harmonic coefficients for external (inducing) and internal (induced) parts of the magnetic potential. Up until 1960, the results, expressed as amplitude ratios and phase angle differences between the internal and external parts, provided the only input for the modelling of electrical conductivity of the Earth on a global scale. The potential function for magnetic storms used only the one surface spherical harmonic, P10, and the results were used to discriminate between the various models determined from the potential of solar daily magnetic variations at 1, 2, 3 and 4 cycles per day based on the surface spherical harmonics P~,P~,P~and P~,respectively. Seasonal variations also gave usable potential coefficients for the sectorial surface spherical harmonics P~and P~. The very marked variability of conductivity models based on daily magnetic variations (e.g., Parkinson, 1974), led to a preference for models based on the Pf potential for the continuum of frequency associated with the 27-day recurrence tendency of magnetic activity (Banks and Bullard, 1966). Knowledge of the P 1°response over a range of frequencies has made conductivity modelling 0031-9201/89/$03.50
© 1989 Elsevier Science Publishers B.V.
possible with data from a single observatory only. When used with the Backus—Gilbert inversion theory (e.g., Parker and Whaler, 1981), the conductivity models are found to vary significantly from those based on the daily magnetic variations and magnetic storms, which give only a single sharp increase in conductivity between 400 and 800 km. Those based on inversion theory show a minimum between 400 and 600 km (e.g., Parker and Whaler, 1981). In this respect, the warning given by Price (1973) is still very relevant: “...the first earth models, Schuster’s and Chapman’s, necessarily incorporated a sudden rise in conductivity for purely mathematical simplicity and convenience. Some investigators apparently claim that the conductivity profiles that they have deduced from geomagnetic induction studies establish the existence of a phase change at a certain depth, but, personally, while I think it probable that a phase change occurs somewhere, I do not believe that the geomagnetic evidence at present available can alone establish this fact incontestably”. Price was referring to the estimation of two-parameter models of the conductivity distribution, corresponding to the radius, q (given as a dimensionless fraction of the entire Earth, denoted a km) of a uniformly conducting core, and its conductivity, a~S,based on
329
two input parameters, namely the amplitude ratio and the phase angle difference between the internal and external potential coefficients. The depth of the sudden rise in conductivity referred to the depth (1 q) am of the uniformly conducting core below the surface of the Earth. The electrical conductivity of the oceans is able to influence the estimate of coefficients in the potential of the magnetic daily variations (both regular and irregular) by two means. One is owing to the magnetic potential of electrical currents induced in the ocean by the westward-moving daily magnetic variations fields, and the second is owing to the magnetic potential of electrical currents flowing in response to the electric fields set up by the tidal movements of the ocean acting as a dynamo in the presence of the Earth’s main magnetic field. For the Sq magnetic variation at 1 cycle per day and the lunar magnetic variation at 2 cycles per lunar day, the ocean dynamo effect is significant, perhaps more significant than the effect of induction in the ocean by the Sq or L fields. For lunar magnetic tides (L), Maim (1970) devised a simple calculation to give an estimate of the lunar semi-diurnal ocean dynamo effect, based on the assumption that there are no ionospheric current systems at local midnight. However, the problem that has attracted most attention is that of induction of electric currents in the ocean by the westward movement of the magnetic daily variation fields. No results appear to be available for effects associated with the transient phases of magnetic storms, although the calculation would, in principle, be very similar to that for the magnetic daily variation fields, A fundamental paper by Chapman and Whitehead (1923) developed the earlier theory of Schuster into the theory of induction in a uniformly conducting shell surrounding a concentric uniformly conducting spherical core, and the determination of what might be referred to as (q, a~) models of global conductivity began with the Chapman—Whitehead paper. The work emphasized the determination of (q, a~)parameters and expressed little interest in the corresponding current function for the system of electric currents induced in the thin shell, The Chapman—Whitehead theory was then de—
veloped further by Price (1949) to deal with the induction of electric currents in non-uniform thin sheets and shells. Price was concerned more with induction in the ionosphere than in the ocean, and he chose a model in which the non-uniform resistivity was proportional to (1 ± cos 9), where 9 is the colatitude, to represent the increase in resistivity that occurs in the ionosphere during the nighttime hours. The theory concentrated on the determination of the magnetic potential e~’for the currents in the shell from the coefficients E~ for the potential of the external inducing field, and suggested two iterative schemes for the solution of the problem. Bullard and Parker (1970) then took up the work in an important paper on the induction of electric currents specifically in a model ocean, using numerical techniques based on a 3 X 3° grid representation of the ocean bathymetry. Electrical conductivity of the deep oceans was shown to be proportional to the ocean depth to quite a good level of accuracy. The theory also included the effect of an infinitely conducting inner core by use of the method of images. Bullard and Parker preferred to work in terms of the current function ‘I’( 9, ~), so that the requirement that no electric current should cross a land—ocean boundary became a requirement that the land—ocean boundary should be a constant contour of the current function. The current function contour value around islands, e.g., Australia or New Zealand, would be different for different islands, and solutions for this problem have only recently been obtained. Beamish et a!. (1980) show contours for the current function around Antarctica which do not appear in the corresponding contours given by Builard and Parker (1970). The work of Builard and Parker (1970) drew attention for the first time to difficulties associated with the convergence of the iterative procedure suggested by Price (1949). Basically, Price’s method converged for the 24-h variation, but not for the 12-, 8- or 6-h terms. A discussion of the convergence problem is given in the summary by Kendall and Quinney (1983). Hewson-Browne (1973) indicated some preliminary results of ocean dynamo effects using studies of tidal flow in channels. Rikitake (1961) presented the theory for induc-
330
tion of electric currents in an ocean represented by a hemispherical shell surrounding a non-conducting layer and a conducting core of finite conductivity. His work gave results in terms of the magnetic potential rather than the current function and, estimated an effect of the order of only 2 nT. The results are also given diagrammatically in terms of the current function. The Rikitake theory ignores derivatives of conductivity in the model, and these will of course be indeterminate at the boundary of the hemispherical conductor. The Rikitake solution is therefore valid only in regions away from the boundary, and is what Kendall and Quinney (1983) would refer to as the ‘outer solution’ corresponding to the solution away from the edge of the hemispherical shell. The early Chapman and Whitehead (1923) paper made use of the amplitude ratio and phase angle difference derived from the external and internal parts of the P2’ diurnal Sq term, but the authors expressed some reservations about this particular term. The vital point is the phase angle difference between the internal (induced) and external (inducing) terms. On the basis of causality, the internal (induced) terms should lag behind the external (inducing) term, and in the same way that sin(t + 30°) lags behind sin(t ± 10°), the phase angle difference, internal minus external, should be positive, and in the case of the P~diurnal term should be 20°.However, recent analysis of the diurnal term, using all observatory data available (e.g., Parkinson, 1971; Maim, 1973; Winch, 1981), have all shown that the phase angle difference is very close to 0° and possibly negative and that therefore the Chapman—Whitehead theory cannot be used to derive (q, a~)models of the Earth’s electrical conductivity as it requires a positive phase angle difference before one can start. This argument can be taken further: the observed negative phase angle difference cannot be a consequence of induction. It could, however, be associated with an ocean dynamo effect whose phase is in advance of the induced internal Sq dynamo field. Recent analyses by Beamish et al. (1980) have included results for the amplitudes of magnetic field variations associated with electrical currents induced in the ocean, but the information given is —
not sufficient to determine if the influence on the phase angle difference is favourable for the theory, i.e., likely to increase the phase angle difference between internal and external parts of P~term. Takeda (1985), in an analysis at 2-hourly intervals of the Sq current system, sought an ocean influence by averaging with respect to longitude to determine the purely local time field. He then graphed the residual terms, hoping to show that such non-local time residuals would be related to the land—ocean distribution. The residuals showed only a little correlation with the land—ocean distribution. It can easily be shown, however, that currents induced in the ocean include terms which depend solely on local time. All in all, one is left with the view that there is likely to be a significant ocean dynamo contribution to the P~ term in the Sq potential and also to the P~ term in the lunar magnetic variation. The Rikitake (1961) paper provided a very useful benchmark against which computer programs for non-uniformly conducting shells could be checked (e.g., Hobbs, 1971). The Rikitake analysis gave results directly in terms of a magnetic potential by solving sets of equations directly, without making use of the iterative procedure suggested by Price (1949). It is significant therefore that Rikitake (1961) gave results for both the 24- and 12-h variations and did not find the difficulty with convergence at 12 h that was found later by other authors. For this reason, it seemed worthwhile to show how the Rikitake theory could be extended to deal with a non-uniformly conducting shell used to represent the conductivity of the oceans, with conductivity proportional to the ocean depth. The purpose is to provide a spectral representation of the induction equation along the lines used with such success in the Bullard-Gellman theory of the geomagnetic dynamo, using conductivity, rather than resistivity which would be very great over non-conducting regions, e.g., the land, and not amenable to representation as a linear combination of spherical harmonics. The theory will treat the oceans as a thin shell, and the induced current in them as a surface current represented as a toroidal current system (producing a poloidal magnetic field), or to put it another way, repre-
331
sented by an equivalent stream function. The model also includes a concentric uniformly conducting core. In the sense that one is using a current function as a depth integrated model of
1 in the region r a m above 9, ~) Acurrent m sheet is given by the0(r, spherical H ~ 0(r,
the current systems (much as is done for ionospheric the dynamo theory), of it is simple matter to include conductivity thea sub-oceanic lithosphere. To deal with a thick spherical shell and to treat the sub-oceanic conducting lithosphere in the presence of poloidal and toroidal inducing fields requires the inclusion of poloidal and toroidal fields in the thick spherical shell and would be more complex than the simpler analysis for induction by a toroidal field analysis for a thin shell, Poloidal current systems, with non-zero radial components can be used to represent leakage from the system induced in the ocean. However, the boundary conditions and orthogonality of poloidal and toroidal fields over the surface of a sphere indicate that such current systems cannot be induced in a spherical shell even with a non-uniform distribution of conductivity by the toroidal current systems that constitute the inducing external daily magnetic variation fields. This argument does not apply to regions with non-spherical boundaries, and it might therefore appear that one electromagnetic mode has been eliminated in regions with spherical boundaries, but the nature of the inducing field, together with the boundary conditions over the surface of a sphere, do not require it. The critical nature of the assumption is dealt with by Ranganyaki and Madden (1980) and Coxetal.(1986).
0,
~)
=
a
~
[(~)n+t(J0)~
m(o,
n.m
ç,
x
~
where
(~)~(E0):}
+
4)) e””~
(1)
(E
0)~ A m~ are complex numbers giving the potential of the so-called internal and external parts, respectively, of the magnetic variation field corresponding to parts which originate either within the sphere r a m or outside it, e.g., in the ionosphere. The sum over n is from 1 to some finite number N and the sum over m from —n to n. The surface spherical harmonics )‘~(0, 4)) are normalized so that the r.m.s. value over the surface of a unit sphere is unity. The corresponding magnetic field strength H0(r, 9, 4)) A m in the region r a m above the spherical current sheet is given by —v~0A m~, and hence in spherical polar co-ordinates at the level r a, its components have the form =
=
(H0),(a, 9, 4)) =
—
~
(n
[—
+
1)(I~)’
n.m
+
(H0)9(a, 0, =
~
—
~)
[(ic~)’
n.m
(H0)q,(a, 0,
n(Eo)1 Y~m(9,~) e~
I
+
aY,n ~
(E0)’ —~—e
(2)
~) m
=
~ [10:±E0:
—
1
n.m
8Y
It is convenient to denote the magnetic potential 2. Theory A thin spherical shell of radius a m of non-uniform electric conductivity corresponding to the land—ocean distribution carries a surface density of electric current, i.e., is a spherical current sheet, and surrounds a non-conducting spherical shell of
of the magnetic field strength in the current free (non-conducting) region r a within the spherical current sheet, ~‘2.(r,9, 4)) A, where ~.
(
,~,
9, 4))
=
a
~{ (~)
+
(I,)
n.m
+
(~)~(E 1)’]Y~(9, 4)) e’~’
thickness (1 — q)a m and a concentric uniformly conducting core of radius qa m and conductivity a S. Colatitude and east longitude will be potential denoted by 9 and 4) throughout. The magnetic ~2 0(r, 9, 4)) A of the magnetic field strength
(3) 1 relate to current systems where the A sphere rn outside and(E~ on)~‘ the r a and (I,)~ A m’ =
to current systems within the conducting core. The
332
corresponding spherical polar components of the magnetic field strength H1(r, 9, 4)) A m’ within the spherical current sheet are given by
~)
0,
(~)r(a,
=
[-(n
~
-
±i)(i~)
1
(H,)
~
0(a, ~ 4))
~
—
=
n ,m
=
4)) e~w
m
‘ + (E, ) J
[(ii)
lines of current flow. It follows from eqns. (7) and (9), that m n+1 (E0)~ 2n + 1 ~“ + (E1) (11) ~ 2n±1 ~ ±(i,)~ (12) =
n,m
±n(Ej)]
tween them so that contours of ‘I’(O, 4)) are the
e””
ay
(4)
In the conducting core r qa, the magnetic flux density being flux solenoidal is made of poloidal and toroidal densities. If theup magnetic meability is constant, the same applies to perthe
(H,) =
4,(a, 9 4)) ~ [(i~)’ —
+ (E,)n 1
n ,m
magnetic field strength. The boundary conditions daily that apply variation to induction poloidal field by the imply external that nomagnetic toroidal field is generated, although it could exist as a
m —a—’ sin9 04) e’~” 1 OY
The boundary conditions across the spherical current sheet are given by ~ [B 0(a, 0, e., x [H0(a,
~)
B1(a, 0,
—
9, 4))
H,(a, 9,
—
0
(5)
~)}
K(0, 4)) (6)
4))1
=
=
freely decaying current system. The poloidal field components are of the form Hr(r, 9, 4)) = ~
n(n ~ s(r)Y~m(0, 4)) e’~’ mr OYm ds ~ Cm__~___~?_eI~l (13) dr 89
n,m
H 9(r, 9, 4))
=
n, m
where K(0, 4)) A m’ is the surface current density in the shell r= a. If the magnetic permeability is the same on both sides of the spherical current sheet, then the boundary condition (5) gives ±i)(I~)’ + n(E0)’ =
—
(n ±i)(i,)’
(7)
+ n(E,)’
K,~=
aiç,
1
—~—e”~
m
—
‘i’,~”aY
~
—~-~--
~
n ,m
ds,” 1 C,,”—~---—
8ym
—‘——i —~—e”~
where the coefficients are constants and the dimensions of s~(r)C,~” are Ametres. The functions
strength H(r, 9, 4)) A m’~satisfies OH V X (v X H) + ,aoac~’~- 0 (14) where a~ S is the conductivity of the core. For =
tm
8= ~ ~
=
s’(r) must be chosen so that the magnetic field
and the boundary condition (6) gives K
H4,(r, 9, 4))
(8) e””
time dependence of the form e”, the radial cornponent of eqn. (14) leads to the Ricatti—Bessel equation with solutions of the form snm(r)
n,m
where the coefficients ‘I’,~”A m
—
are given by
where
=
rjn(Ar)
A2
=
—
i,.L
(9)
0coa~ m -2, so that the argument Ar of the spherical Bessel function j~(Ar) is seen
We write K(0, 4)) aV x [e,’I’(O,~)] A m’ for the surface current the orfunction 1 is thedensity currentwhere function stream ‘I’(O, 4)) A m by function given
to be dimensionless. Chapman and Bartels (1940) make good use of the functions e’~p~(kr)in place of the Besselpn(kr) functions. For integer values of nspherical the function is a polynomial (kr) 1 fl (n ±1)
=
(Ia)’
±(E0)~”’—
(i,)’
—
(E,)
=
‘~p(9 ‘1))
=
~
4..rnym(9
p)
e~’~
(10)
—
—
2k,’
n ,m
The difference between values of ‘I’(9, 4)) at different points gives the total current flowing be-
(2kr)2
(n—1)n(n±1)(n+2) ±
2!(2kr)3
—
(is)
333
which is similar to the Legendre polynomial F,, ( ~t) expanded as a power series in (1 ~t)/2
Numerical evaluation of the real and imaginary parts of the bracketed expression on the right of
P,,(
eqn. (22)theis oceanic most easily done thesub-oceanic functions 2 of layer, withwith the )~~(1~T) of Chapman and Bartels (1940). If the depth-integrated resistivity (or surface resistivity) p~, lithosphere if required, is a function of colatitude 0 and east longitude 4), and if K(0, 4)) A m’ is the surface current density, then the correspond-
—
(1 ‘~ n(n ±1) ±(1 ~ )2 2’2 x (n — 1)n(n ±1)(n + 2) (16) Assuming the magnetic permeability to be the same on both sides of the conducting core surface, r qa, the continuity of the normal component of magnetic induction Br( r, 0, 4)) T gives the following result from eqns. (3) and (13) 1it)
=
1
—
—
—
...
=
ing surface electric field E( 0, current sheet is given by
~)
V m1 in the
(n ±1)q”2(1 1)’
—
E(0, 4))
nq”’(E1)’
1
(17) and, in the absence of a surface current at the surface of the conducting core, the continuity of the tangential component of magnetic field strength, using eqns. (3) and (13) gives =n(n+1)C,,m__j,(Xqa)
—
q~—1(E,)~
1
=cnmi{jn(xqa)±xqa(dJ~(r)~ / r—xqaj ____
=
p(0,
4))K(0, 4))
(23)
and on taking the curl of eqn. (23), one obtains 2’I’) (24)
vXE=—e~(vp.v”I’+pv
Hence the radial component of the Faraday equation V x E ±8 B/8t 0 reduces to the fundamental equation for induction in non-uniformly conducting current sheets =
8t
(18)
—
vpv4’±pv2’I~
(25)
Solving eqns. (17), (18) for the coefficients (I, )~‘,(E, )‘ A m1 of the potential field in the non-conducting region, and making use of the recurrence relations for spherical Bessel functions, one obtains 2(1)’ nC~mj,,÷i(Aqa) (19) (2n ±1)q~
(e.g., Price, 1949; Zhdanov, 1980). Because of the continuity of the normal (i.e., radial) component of magnetic flux density across the current sheet at r a, the normal component of the magnetic flux density there can be expressed in terms of (I,),,~’and (E ent to choose1)~’ (I,)’ and (E (E0)~’.It is convenior (1~)’and 1)’, so that, from eqn.
(2n±1)q~~(E1)= —(n±1)C,~j,,_1(Xqa)
(4)
=
(20) 1 can also be written The as eqn. (19) for (I,)’ A m
~
~
N.M
8t +
N(E,)NMI Y~(0, 4)) e”
(26)
tm 2n ±1 —j,,_
(2n + 1)q~2(I 1)
=
nC
1(Xqa)
Xqa
(21) tm A from eqns.
and the constant C,, (20) on andeliminating (21), we have m
=
in which the sum over N, M means only the terms for the chosen inducing field. For example, the diurnal analysis (1940) uses M terms 1 andonly N of1, Benkova’s 2, 3, 4, 5. Substituting for (I,)~ from eqn. (22) into eqn. (26) gives =
=
fl
8Br
=
x[1
—
2n+1 j,,(Aqa) Aqa j,,_
1
1(Aqa)j~’~
=
—
~
[A~(Xqa)a±iB~(Xqa)]
N.M
(22)
XN(E1)~Y~(9,4))e~wt
(27)
334
where, following Rikitake (1961), we have made use of the dimensionless variables A~(Aqa), B~(Aqa), defined by
M’) are defined by I(p, n, N’, m, M’) 2~r
~f0 fp(94))Ytm(94))~M’smn0d9d4) 1
A~(Xqa)±iB~(Aqa)
L
]
2N±1 j~(Xqa) (28) Xqa J~_J(Aqa) which can be tabulated for given q, A and which for an infinitely conducting core reduces to the value 1— q2N+i, Using eqn. (25), and using eqn. (11) to replace (E 1)~in eqn. (27), the equation for induction in a non-uniformly conducting thin sheet in the presence of a uniformly conducting core becomes =
1 _q2N+iF1
(31)
m
—
1
Op 81T~n
1 1 jo2ir fo‘~f8p 4~r (,89 Oiç,” 89
=
1
8~
_‘hi~~(o,
84) 84) j
±~
4)) sin9 dO d4)
N
(32) which can be evaluated numerically, once and for all, for any given land—ocean distribution. Price (1949) considered the nature of such solutions for the surface resistivity p 1 + cos 0, for induction without a conducting inner core. If the surface =
~ I[Op 89 8}~, 80 + sin2O 84) 84) n,m
resistivity is given as a sum of surface spherical —
=
J(p ~ N’, m, M’)
n(n
±1)p)~,tm(O,4))]4~’”
harmonics, the required integrals can be obtained in terms of Wigner 3 —j coefficients 1
~ i~owa(AN+iBN)N[~7~~I’M
_j
N
f
2,r
N,
4))
(29)
Y~(0,4))Ym(0
=(-1)M~(2L+1)(2n+1)(2N’+1) IL
This equation is satisfied over the whole surface of the sphere. Hence, on multiplying eqn. (29) throughout by Y~.’(0,4)) and integrating over the surface of the sphere, one obtains an infinite set of
4))Y~’ sin0d0d4)
1
n
~
N’\1L
o)U
n m
N’\ -M)
(33)
2~ ~ 8y~8Y,”
j j
~—
1
equations
+
i~tOc~~a(AN + iBN)N(EO)~~~’aM’ N ~‘M
=
~
(~
—~-
—~--
8y~8ym —a-- I~’(9, 4)) sin 9 dO d4) 84) 84) ) N
(_1)M~[L(L ±1) + n(n + 1)
N(N± 1)q,M =i,tO~a(AN±iBN) 2N±1 — ~ [J(p, n, N’, M, M’)
6N~~~M~ N
N
M
(30)
x
N(N± 1)] I(2L+ 1)(2n ±1)(2N’+ 1) tL n N’ (,~ 0 0 / m —M) (34) n N’\
n.m
—n(n
±1)I(p, n, N’, M, M’)] 4’,~
where I(p, n, N’, m, M’) and J(p, n, N’, m,
where L + n + n’ must be even and the values for L, n, n’ must be such that it is possible to form a triangle with them (e.g., Winch, 1974). If one wishes to make use of surface conductivity, a, in
335
place of surface resistivity, p, then with the substitution p 1/a, eqn. (29) becomes ~[8a81~,m 1 Oa8Y,m [~ 89 + ~ ~
where K(n, N’, m, M’) is here defined by
=
K(n, N’, m, iivi’) 1 =
n.m
±n(n+1)aY,,m(O,
=
— ~
4))]~~n
i~t~wa2a(A,~ ±iBN)N[
N,
N+ 1 2N+1N
(35)
(Eo)~I Y~(O,4))
—
which is valid over the entire surface of the 9’sphere. 4)) On multiplying eqn. (35) throughout ~N~ and integrating over the surface of thebysphere, one obtains the following set of equations i,iOwa(AN + =
iBN)N(Eo)1(a2,
n,
N’,
m,
M’)
N(N±1) 2N + 1
±~
[J(a,
‘4’NI(a,
n,
N’, m, M’)
(39) The solution consists of a set of parameters { ‘I’,~} for the current function, for a given set of inducing terms {(E 0)’), under whatever assumptions are appropriate concerning the distribution of resistivity or conductivity in the spherical shell. A great of effort has gone into finding contours of thedeal current function ‘I’ which have the continental shelf as a contour, but what is required now is numerical results for those ‘I’,” which correspond to the principal input (E 0)~’ so that the terms (I~)~’ can be corrected for the inductive
n, N’, M, M’)
n.m
+
4)) sin 9 dO d4)
(as opposed to the dynamo) influence of the ocean. Equation (38) can be solved directly if recurrence methods fail.
4tOwa(AN + iBN) X
j,cean~~tm(9~4))Y~(O,
3. Time dependence
n(n
±1)I(a, n, N’, M, M’)]’4’,”
(36)
There is a need to consider the explicit
which are very similar in form to eqn. (30). If eqn. (29) is to be applied in ocean areas only and the surface resistivity is taken to be a constant, then it reduces to
mathematical form of terms in the magnetic potential of the daily variations arising from the inductive influence of the oceans. It is a simple matter to show that such terms are not only UT dependent, but also include local time terms as well. Consider the magnetic potential associated with the current function induced in the oceans by
n(n
— ~
+ 1)pY,tm(O,
a)’I’,”
n.m
=
[N+1 ip0wa(A~,,,±iBN)N[ 2N ±1
~ N.M
the diurnal magnetic daily variation: it consists of a ‘real’ or ‘in-phase’ part
..JfM N
(37)
a~A~1cos(k4)+a~)I~(cosO)
(40)
j,k
and on multiplying throughout by ~N~’ and integrating over the ocean area only, it follows that
and an ‘imaginary’ or ‘in-quadrature’ part a>BJ~, cos(k4)+/3j’)I~’(cos 0)
— ~ n(n + 1)p’I’,~”K(n,N’, m, M’)
which are combined together as
n ,m
[N+1
=
(41)
J,k
~ iftowa(Afl±iBN)N[2N
1
~[A~~costcos(k4)-i-4)
N
N, j,k —
(Eo)] K(N, N’, M, M’)
(38)
±B~1 sin t smn(k4) ±$1k)1 Pk(cos 0)
(42)
336
and then written in the form
tion in the oceans. In: E.C. Bullard and J.L. Worzel (Edi-
tors), The Sea, Vol. IV. Wiley, New York, pp. 695—730. Chapman, S. and Bartels, J., 1940~Geomagnetism, Vol. II, Oxford University Press, Oxford, p. 739. Chapman, S. and Whitehead, T.T., 1923. The influence of
[Ci,cos(k4) + t ±
~ j,k
±D~, cos(k4) — t + ~~)]
pk(C
05
0)
(43)
In these equations t is the UT and the local time t* t ± 4), hence with k 1 there is clearly one term in eqn. (43) with diurnal local time dependence. =
=
4. Summary The paper has discussed at some length the theory needed for the study of induction in a non-urnformly conducting ocean, surrounding a concentric conducting shell, and the need to determine the contribution to the principal spherical harmonic terms in order to correct them for the inductive influence of the oceans. The ocean also acts as a dynamo, and there is evidence for a strong contribution from the ocean dynamo to the coefficient of F2’ in the internal field potential at one cycle per day. Now that tidal movements in the deep oceans have been so well determined, it should be possible to determine, in the first instance, current function contours associated with the ocean dynamo current system, and then to correct further the principal terms in the internal magnetic field potential used for conductivity modelling of the Earth.
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