May 15, 1990 - 1954] and the development of the geomagnetic electrokineto- ..... tions of the induced currents with the vertically varying con- ductivity, while the ...
JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 95, NO. C5, PAGES 7185-7200, MAY 15, 1990
Low-Frequency,Motionally InducedElectromagneticFields in the Ocean 1. Theory ALAN D. CHAVE
AT& T Bell Laboratories,Murray Hill, New Jersey
DOUGLAS S. LUTHER
ScrippsInstitutionof Oceanography,La Jolla, California
The theoryof electromagnetic inductionby motionalsourcesin the oceanis examinedfrom a first principles point of view. The electromagnetic field is expandedmathematicallyin poloidaland toroidalmagneticmodes basedon the Helmholtz decomposition.After derivinga set of Greenfunctionsfor the modesin an unbounded oceanof constantdepthand conductivityunderlainby an arbitraryone-dimensionalconductingearth, a set of exactintegralequationsare obtainedwhichdescribethe inductionprocessin an oceanof verticallyvaryingconductivity. Approximatesolutionsare constructedfor the low-frequency(subinertial)limit wherethe horizontal lengthscaleof the flow is largecomparedto the waterdepth,the effect of self inductionis weak, and the vertical velocityis negligible,explicitly yieldingcomplexrelationshipsbetweenthe vertically-integrated, conductivityweightedhorizontalwatervelocity andthe horizontalelectricand threecomponentmagneticfields and accounting for interactionswith the conductiveearth. After introducinggeophysicallyreasonablemodelsfor the conductivitystructures of the oceanandearth,thesereduceto a spatiallysmoothedproportionalitybetweenthe electromagneticfield components andthe vertically-integrated, conductivity-weighted horizontalwatervelocity. An upperboundof a few times the water depthfor the lateral averagingscaleof the horizontalelectric field is derived,andits constantof proportionality is shownto be nearly 1 for mostof the deepoceanbasedon geophysicalarguments.The magneticfield is shownto have a similar form but is a relatively weak, larger-scaleaverage of the velocityfield. Becauseverticalvariationsin the conductivityof seawaterlargelyreflect its thermal structureand are weak beneaththe thermocline,the horizontalelectricfield is a spatiallyfiltered versionof the true water velocitywhich stronglyattenuatesthe influenceof baroclinicityand accentuates the barotropiccomponent. This is quantifiedusingconductivityprofilesandvelocityinformationfrom a varietyof locations.
INTRODUCTION
Natural electromagnetic fields in the oceansare inducedby both external,ionosphericand magnetospheric, electriccurrent systemsandby the dynamointeractionof oceanwater currents with the earth'smagneticfield. The former have been fairly well
characterized
at the surface of the earth and are often
whileit haslongbeenrecognized thatmotional horizontal
measuredduring geophysicalinvestigationsof its electrical structure.
due to Sanford [1971]. This work has formed the basisfor the successfulinterpretationof submarinecable measurementsin terms of transport[Sanford, 1982' Larsen and Sanford, 1985] and spawnedsomenovel oceanographicinstrumentation[Sanford et al., 1985]. Despitethe obviousutility of the theoretical studies,the resultsare formally deficientin two respects.First,
While the existence of motional electric currents in
the oceanwasfirstpostulated by Faraday [1832], oceanicelectromagneticfields remainlesswell understood, primarilydue to the difficultyof collectingin situobservations. Theoreticalstudiesof motionalinductiondivide naturally at the inertialfrequencydueto changesin theirphysicalbehavior. Only the subinertialcasewill be considered in this paper,and the oceantides are specificallyexcludedfrom treatment. of long-periodmotionalelectromagnetic fields post-dateWorld War II andbeganwith the theoreticalwork of Stommel[1948], Longuet-Higgins [1949], Malkus and Stern [1952], and Longuet-Higginset al. [1954]. These investigationsled to someearly measurements on submarinecables[e.g., Wertheim, 1954] and the developmentof the geomagneticelectrokinetograph(GEK) [vonArx, 1950]. The mostcompletetheoretical investigation of low-frequency motionalinductionavailableis
Copyright1990by the AmericanGeophysicalUnion.
electric and magneticfields are proportionalto the verticallyintegrated, conductivity-weightedhorizontal water velocity, most of the investigationsfail to demonstratethis explicitly.
Furthermore,the importanceof conductivityweightingin oceanographic applications has received relatively little emphasis. Second, while it is known that electromagnetic interactions
with the conductive
material
beneath the seafloor
may be significant,due to analyticalcomplexityit is common to model the earthin a simpleandprobablyunrealisticmanner. Because of these limitations and increasing oceanographic experimentalactivity using electromagnetictools [e.g., Sanford, 1986; Luther et al., 1987], it is appropriateto re-examine the motionalinductionproblemin its entirety: In this paper, the relationshipsbetween all six components of the electromagneticfield and the water velocity are derived from first principlesin the low-frequencylimit where the horizontal lengthscaleof the flow is greaterthan the water depth and the effect of self-inductionis weak. The electromagnetic fields with a motionalsourceare decomposedinto poloidaland toroidal magnetic modes using a formalism suggestedby Backus [1986]. This approach allows an arbitrary onedimensional
electrical
structure for the earth beneath the sea to
Papernumber90JC00189.
be modeledin a straightforwardmanner. Using a set of Green
0148-0227/90/90JC-00189505.00
functions 7185
valid for an unbounded
ocean of constant conduc-
7186
CHAVE ANDLUTHER:MOTIONALEM FIELDSIN THEOCEAN. 1, THEORY
tivity and depth, an exact set of integral equationsare derived for the primary and secondaryelectromagneticfields in an ocean of vertically varying conductivity. Approximate solutions for the integralequationscan be constructedthat explicitly yield the expected relationship with the verticallyintegrated,conductivity-weighted horizontal water velocity in the low-frequencylimit. The effect of the earth's electrical structureon the electromagneticfields is then examined in detail. While a complexrelationshipbetweenthe frequencywave number dependenceof the water velocity field and the earth's electricalstructureis formally required,it is shownthat a geophysicallyreasonablemodel for the structureunder the deep ocean floor leads only to weak coupling, and a simple proportionalitybetween the horizontalelectric field and the vertically-integrated,conductivity-weightedhorizontal water velocity smoothedover a horizontal scale of no more than a few times the water depth obtains. The constantof proportionality is approximately1, reflecting the minimal extent of current leakage into the geophysicaldeep seafloor. For the magneticfield, the conclusionsare similar, althoughthe horizontal averagingdistanceis larger and the fields are very weak. Becausethe depthdependence of seawaterconductivity largely reflects its thermal structureand is nearly constant beneaththe main thermocline,the vertical integrationproperty of the electromagnetic field dramaticallyreducesthe influence of the baroclinic (depth-dependent) field, yielding a filtered versionof the velocitythat is usuallydominatedby the barotropic (depth-averaged)component. The next four sectionsof the paperand the four appendices containthe completemathematicaldevelopmentof the theory of low-frequencymotionalinduction. Readersinterestedin the key resultsand oceanographicapplicationsmay wish to skim these sectionsand concentrateon the last four parts of the paper. As an aid to the reader,the symbolsare summarizedin the notationsectionat the end of the paper togetherwith the numberof the equationwhereeachquantityis defined. Vector quantitiesare alwaysindicatedby boldfacetype. In a separatepaper (D.S. Luther et al., Low-frequency, motionally-inducedelectromagneticfields in the ocean, 2, electric field and Eulerian currentcomparisonfrom BEMPEX, submitted to Journal of Geophysical Research, 1990; hereinafterdesignatedLutheret al., 1990) the theoryis verified observationallyby comparingthe water velocity as measured directly by a mechanicalcurrent meter mooring with contemporaneousrecordsof the seafloorelectricandmagneticfields.
Since the induced magnetic induction is very weak compared to the earth'smain field, the Lorentzforce on waterparticlesis many orders of magnitudesmaller than hydrodynamicones. As a result, the total magnetic inductionB in (4) may be approximatedby that of the earth F that is assumedto be both sourcelessand static. The electric current (4) is then the sum of an inducedpart I = (sE and a sourceterm (5(vx F). Note that the electric field in (2)-(4) is the value for an observerfixed to the earth;for a referenceframe movingat a relativevelocity v,. and neglecting relativistic effects, the electric field is
E'=E+ v,.x F, whilethemagnetic induction B isunchanged. It is essentialto understandthe role which electriccharge playsunderthe quasi-staticapproximation.Neglectof the displacementcurrent in (3) requiresthat the electric current density J be divergence-free,and the time rate of changeof the chargedensityis zero to the samelevel of approximation.In fact, the electric currentswhich distributeelectric charges occuron a time scaleof O(v/(5) (wherev is the electricalpermittivity) that is instantaneous comparedto the time scaleof interest,so that they do not produceany significanttimevaryingmagneticfields. However,the electricfieldsproduced by chargeswhich move into place with seemingly infinite speedare quite importantand are not removedby the approximation. The quasi-staticscaling is a singular perturbation problemwith someinterestingconsequences at breakdown;see Backus [1982] for a discussion. The pre-Maxwell equations will be solved in terms of
poloidaland toroidalmagnetic(PM and TM) modesbasedon the Helmholtz representationof a vector field. The modal form is usefulbecauseit reducesthephysicsinto two relatively simple and independentparts. The PM mode is marked by electric currentsflowing in horizontalplanes that couple by induction,and has no vertical electric field component. The TM mode is associatedwith electriccurrentsflowing in planes containingthe vertical,andhasno verticalmagneticfield component. The TM mode magneticfield vanishesat the Earth's surface,and is not observableoutsideof it. Electric charges associatedwith conductivitygradientsaffect only the TM mode. This physicalapproachshouldbe contrastedwith one basedeither on the magneticvectorpotentialor directly on the fields in which interactionswith conductingmaterialare difficult to handle,necessitating simplificationsthat are often geophysicallyuntenable. Using the Helmholtz representationtheorem defined in AppendixA and (1), the magneticinductionmay be written
B = •xVhI] + Vh0:tP-V}tP•
GOVERNINGEQUATIONS
For the lengthand time scalesof interestin motionalinduction in the subinertialrange, the quasi-staticor pre-Maxwell approximationto the full Maxwell equationsis sufficient. This limit involves neglect of displacementcurrent, polarization current,and advectedchargein comparisonto the conduction current. For a movingmediumand with the magneticpermeability la that of free spaceeverywhere,the resultingequations are
V.B=0
(1)
VxE+0, B=0
(2)
VxB-laJ =0
(3)
where the electriccurrentdensityis given by J=o(E+vxB)
(4)
(5)
where FI and tp are scalarfunctionswhich representthe TM and PM modes. Let the sourcecurrentin (4) be expandedin a similarway so that
O(VXF) = E• + VhT+ •XVhY
(6)
Under the conditionthat the Earth's conductivityvaries only verticallyand that the magneticpermeabilityhasthe free space value everywhere,it is proved in Appendix A that the modal scalarssatisfythe differentialequations
V2t?- lao•)t t?-- lay
(7)
V•l-I + oO:(O:l-I/o) - laoOtFl = laE- lao0:(T/o)
(8)
and
where the electric field is
CHAVEANDLUTHER:MOTIONALEM FIELDSIN THEOCEAN,1, THEORY
E: 5XVh•,•- Vh(•_-l-I+gT)/go + (V•l-I-!a•,)/g(•^ (9)
0 ~ PM
For most of this paper, it will be more convenientto work with the horizontal
Fourier transfo•s
(z): -o•[tk •zx•)I dz'g•(z,z')'•(z) -H
0
of the modal differential
(z), - imp I dz'Xw(z,z') AC•(z') •_]PM
equations(7) and (8). The time dependencewill be taken as e -'m' and the horizontal coordinates transfo•
7187
(17)
-H
as
This is a second kind integral equation for the spatially Fourier-transformedelectric field. The first term on the right •(B,•):I I dxdy et(n'+•) )f (x,y) sideis the primaryor driving electricfield neglectinginteractions of the inducedcurrentswith the vertically varying conductivity, while the next term is the secondaryelectric field due to scatteringfrom the depth-varyingconductivity.EquaLet• and• beunitvectors in thevertical andpropagationtion (17) is similar to those derived for two- or threedimensionalelectromagnetic modelingin explorationgeophydirections,respectively.The expressions for the magneticand sics[e.g.,Hohmann,1983]. electric fields (5) and (9) become Applying Ampere'slaw, the PM mode horizontalmagnetic
12.... I I dnd• e-'m'+•"' ?(n,•) (10) f (x,y)= (2•)
field follows from (17) 0
and
~ PM
,
,
Bh(z): -igk• I dz'O:g•,(z,z )'•(z)
• = -ok(zx•:) - (k2•-I +[tz-)/bto ^
-H
0
(12)
_ gxIdz'O.g.(z,z _ ,.)a.(z')•_•?M ,. (z) -H
while the differentialequations(7) and (8) for the PM and TM modes transform
to
(18)
while the verticalmagneticfield satisfies 0
0• - [k2-imgo(z)l• = g•
(13)
•_-(z) =gk21dz'gw(z,z') '•(z') -H
and
d-[[k2^ •XI &'gw(z,z') AC•(z') •PM(3'
o•(•:fi/o)- [k2-imgo(z)]fi = g• - go•:(•/o) (14)
(19)
-H
wherek=(q2+•2) • is themagnitude of th• hori•onta) wave
The PM mode magnetic field may be regardedas a quantity derived from the electric field, and the interpretationof the (0:•+g_•)[go is•quired athorizontal interfaces. Thesource right-handsides of (18) and (19) in terms of primary and te•s Y, E, and T in (13) and (14) are discussedin the next secondarycomponentsis similarto that for (17). section. Proceedingas for the PM mode, the TM mode counterparts Solutionsof (13) and (14) are soughtfor an oceanwith vertof (17)-(19) are a pair of integralequationsfor the magnetic ically varying conductivity o(z) and an arbitrary onefield and inducedelectriccurrentdensity
number. In addition, continuity of •,
O:•,
dimensional electrical structure beneath the seafloor.
fo•
H, and
Closed
0
solutionis not tractable,and the approachused in this
paperis approximate.Greenfunctionsolutionsto (13) and (14) are constructed for an unboundedoceanof depthH and fixedconductivityequalto thatat theseafloorwith an arbitrary one-dimensional conductingmediumbeneaththe seafloor,as detailedin AppendixB. The Greenfunctionscontaininformation on the structureof the earth throughcomplexreflection coefficientswhosecomputationfor a layeredearth model is outlinedin AppendixC. Thesewill be usefullaterin assessing the importance of electromagnetic interactions with the earth. The Greenfunctionsmay thenbe usedto convertthe differential expressions (13) and (14) includingvertically varying oceanconductivityinto integralequationswhich completely describethe motionalinductionproblem. To illustrate,rea•angethetermsin (13) to give
-H
0
-H
0
-
_ log (z)' g•xI &'gz(z,z') a-' o(z') ihTM -H
0
- img I &'g
(20)
-H
and 0
Ih (z): ik• I dz'O:gr•(z,z') 2(z') -H
0
-H
where A,(z)=o(z)-,(-H)
and
• = •k2-iwgff(-H)
(16)
o ,
-H
The PM mode Green function satisfies the left-hand side of
(15) with delta functionforcing and the relevantboundaryconditions,so that (15) may be rewritten as an integral equation. Using the definitionsin (12), the result may be expressedin te•s
of the PM mode electric field
~TM
+ I dz'O:g•(z,z') •_-' log o(z) Ih (z') 0
- io•xI dz'O:Xr•(z,z')Ac•(z')•r•U(z') (21) -H
~ TM
~ TM
whereI• =oEt, . The first two termson the right-handsides
7188
CHAVEANDLUTHER:MOTIONALEM FIELDSIN THEOCEAN,1, THEORY
of (20) and (21) are the primary field terms,reflectingthe presence of two source terms in (14). The last two terms in (20)
and (21) are the secondaryfield termsdue to scatteringby the depth-varyingseawaterconductivity. Finally, usingFaraday's law, an expressionfor the vertical electricfield is
satisfythe conditionon (24) without difficulty. However,for large-scalemotionsat relatively high frequencies,breakdown may occur. For example,in the basin-wideoscillationdetected by Luther [1982] with T=O(5 d), L=O(10000 km) the induction number
is about 75.
Note
also that the tides violate
the
induction number criterion; a diurnal coastal Kelvin wave has
•o(z)
o(z)
The set of equations(17)-(22) are exact. The form of the integral equationssuggeststhat the motionalelectromagnetic field is a weightedspatialaverageof the velocity field with the weighting dependingon the Green functions,and hence the conductivitystructurebelow the seafloor,and the vertical distribution of seawaterconductivity. In principle, they can be solvedfor an arbitraryvelocity field and conductivitystructure by numerical methods using the complete Green function expressionsof Appendix B, although such an approach is unlikely to be enlighteningin termsof the relevantphysics.
œ=800,soinductioneffectsare quitelarge. It is alsonecessaryto examinethe behaviorof the reflection coefficientsto simplify the Green functionsin (17)-(22). Consider a simplemodel consistingof an oceanlayer of conductivity {•(-H) overlying a half-spaceof conductivity{•l. The PM mode sea surface and seafloor
T.
In the integral equations(17)-(22), kH enters through the exponentialterms in the Green functionsgiven in Appendix B.
These terms contain argumentsof the form [•z,-H_z)
(33)
CHAVEAND LUTHER:MOTIONALEM FIELDSIN THEOCEAN. 1, THEORY
i)_i)_'g•(z,z )= g•(z,z')-iS(z-z) -- h•-15(z-z')
7189
(34)
where
13(2-t• TM)
h,;= t•rl 4_2R•r14[• H
(35)
and 15is the Dirac delta function. The expressions (29)-(35)
arevalidtoterms ofO( I [•HI ) andhavebeenverified numerically. The use of theseapproximationsin (17)-(22) formally requiresthat the principal contributionto the inverseFourier transform in (10) be from small wave numbers where the velo-
city is significantand is equivalentto replacingthe infinite integrationlimits by a cutoff wave numberk, which is not well defined,but O(1/H). Note that (29) and (34), which appearin the expressions for the horizontalelectricand verticalmagnetic fields, are depth-independent,while (30) and (33), which appear in the expressionsfor the horizontal magnetic field, undergoa step changewithin the water column. The former are consistentwith intuition about low-frequencyinductionin the ocean, but the integral equationexpressionsin (17)-(22) are actually more complicatedthan this due to the secondary
Pm MODE
SIDE
VIEW
I I I
I I
I I
fieldtermsanddepthzva_rying conductivity. The sourcetermsY, T, and E in (13)-(14) and (17)-(22) are given by the Fourier transformsof Poissonequationsderived in Appendix A and the vertical component of the source currentdensity. To simplify theseequations,some properties of the real geomagneticfield mustbe considered.In Appendix D, an inclined geocentricdipole model for the geomagnetic field is used to get an approximationvalid for mid-latitudes under the assumptionthat the vertical velocity is small compared to the horizontalones. Taking Fourier transformsand neglectingtermsinvolvingthe spatialvariationof the geomagnetic field gives
,, I
I I I
z
TM •(z) = lJ(z)[7½h(Z ) XFh]
(36)
•'(z)= i•J(z) [•xZ%(z)l.F,•,, k
(37)
•(z)= - i•(z__•_) [•.Z%(z)]F,[ k
(38)
where F ø is the geomagneticinductionat a referencelatitude
ko and longitudeq•o. Note that (37) and (38) containthe Fourier transformsof the vertical componentof relative vorticity and the horizontaldivergenceof the horizontalwater velocity, and drive the TM and PM modes, respectively, in (17)-(22). For a graphicalview of this association,seeFigure
MODE
SIDE VIEW
Fig. 1. Cartoon showing the modal source mechanismsfor lowfrequency,large-scaleoceanicflows. The dashedline showsthe vertical componentof the geomagneticfield, while the solid line denotesthe watervelocity,andthe doubleline is the sourceemf. The top partof the figure depictsa sideview of a velocityfield with horizontaldivergence, suchas occursin gravity waveswith displacementof the sea surface. The sourcecurrentsflow in horizontalloopsin alternatedirectionsalong peaksand troughsof the wave, closingat infinity. The bottompart of the figure showsa plan view of a velocity field with relative vorticity. The induced electric currents are horizontally divergent, cannot penetratethe sea surfacebecausethe atmosphereis insulating,and so dive down into the earthover horizontaldistancescomparableto that of the velocity field.
1.
and* is the vertically-integrated, conductivity-weighted
APPROXIMATE PM MODE SOLUTION
horizontalwater velocity 0
The integralequationfor the PM mode electric field (17) is combined with the approximateGreen function (29) and the sourceterm (38) to give 0 ~/>14
,
(z ') E (z)+imgf• vI dzA•(z)•/>14 -H
(k') -- imgHF'__'f,v" ~.
(39)
E
1 + iwg(-•J(-H))Hf•,
0
1_IHd z,,(zß)
The definition in (41) reducesto the depth-averaged(barottopic) one if the velocity field or conductivity is depthindependent. Expression(39) is a simple Fredholm integral equation with a degeneratekernel and can be reduced to an algebraicequationby standardmethods.The solutionis
~p14 imgHF•-'f•, (•:.*) = -
where is the depth-averaged conductivityof seawater
= H
~ , 1 _.[HdZ,(j(Z,)•h(Z, ) (41)
= H
(40)
(k') : icogHF'•f•," ~*
(42)
7190
CHAVEANDLUTHER:MOTIONALEM FIELDSIN THEOCEAN.1, THEORY
where the secondform is obtainedby neglectingsmall terms and is just the primary field in (17). The PM mode electric field is in the horizontalplane and orthogonalto the direction of propagationfor the velocityfield. The PM mode horizontalmagneticfield follows from (18) with the approximateGreen function (30), the sourceterm (38), and(42). Neglectingsmallterms,the resultis
Bh (z)=-gF'_.'•c ß wH +(2+H)_-H (43) where
--:n = (z +H)dzo(z)Vh(Z )
(44)
The right side of (47) is an unknownconstant,and a Neumann seriessolutionto the remainingVolterra integralequationin terms of this constantcan easily be constructed.The remaining problemreducesto the solutionof an algebraicequation, yielding ~TM
~r•4 /h.(i)(Z) =
The PM mode
horizontalmagneticfield is orientedin the directionof propagationfor the velocityfield. The vertical magneticfield is depth-independent and follows from a similarcalculationwith (19), giving
B_= -igkHF•_' f • k' ^~ * APPROXIMATE TM
(48)
o(-H) + hnH(-o(-H))
Note that the electric field correspondingto (48) is depthindependentwhile that for the primary term (46) is not, and that the new term obtainedby solving(47) is not small. The lowest-orderapproximationfor the TM modemagnetic field from (20) is givenby the sumof the first threetermsafter applying (31), (33), (35)-(37), and (48). As for the induced
SOthat_0 n = *.Expression (43) is depth-dependent,electric current (46),
and the second term vanishes at the seafloor.
l h,(o)O(z)
it can be shown that the contribution due
to the vertical sourcecurrent is negligible comparedto that from the horizontally divergent part of the source current, yielding
(0)(z)=laF•_. '/cx o(-H)(1-hnH)
(45) -H
MODE SOLUTION
The integralequationsfor the TM mode horizontalelectric currentdensityand magneticfields (20) and (21) are a coupled set and should be solved simultaneously. To simplify and decouplethem, the motional magneticfield will be assumed small enoughthat its contributionto (21) is negligibleand a solutionfor the inducedelectriccurrentdensityat lowestorder will be obtained. This will be usedin (20) to get the motional magneticfield, and substitutedback into (21) to get a better estimateof the currentdensity. This iterativeprocedureends when the new terms are small in the usual sense.
*/ [(•(-H) +hnH(c•(-H)) 1 - (z+H)_ZH] '•
(49)
Note that this vanishesat z =0, as expectedfor the TM mode magneticfield. The next orderapproximationfor the magneticfield follows by substitutingthe Green function (31) into the last term in (20), yieldingthe integralequation
It is first necessary to simplifythe primarypart of (21) since there are two sources,the vertical source current (36) and the
scalarrepresenting the horizontallydivergentpart of the horizontal sourcecurrent (37). Substitutingthe Green functions (32) and (34) and the sourceterms(36) and (37) givesa sumof three terms. H61der'sinequality may be used to bound and comparetheir magnitudes.Such an analysisshowsthat the two terms from the vertical source current (36) are similar in
magnitude butsimultaneously O(kHIF•ø./F•-'l) compared to that from (37). The horizontalcurrentterm is dominantexcept
near the magneticequatorwhere Ft__ ' vanishes.That vertical sourcecurrentsare not efficient at inducingelectriccurrentsin the oceanis not surprising;vertical sourceshave a scalelike that of the water column, while horizontal sourceshave a scale
comparableto that of the hydrodynamicfield and henceare better able to induce horizontal currents in the ocean and earth.
At middle to high latitudes,the primary inducedelectric currentdensityis well approximatedby ~TM
li,,(O) = -h•HF'__' ?kX*)-z ^ (46) The next orderapproximationto the electriccurrentdensity satisfiesthe integralequation
~TM i dz'•:'1og{J(z,)Ih.(l)(z ~TM ,)
Ih.(•)(z) -
-H
0
~TM
,
~TM
=lh,{O ) +hnldz'z•-'10g(•(z')lh.{•)(z') (47) -H
~TM
i ' '
, ~TM
BtlI (z)+ icog(1-h•H) dz (z-z )A(•(z )B(•)(z) -H
o
~ TM
,
,
~ TM
,
=B{0 )(z)-icog(1-hnH) I dz'zAo(z )B{•) (z)
(50)
-H
As in (47), the right-handside of (50) is an unknownconstant, and the Volterra part of the integralequationmay be solvedby constructinga Neumann series for the equivalent resolvent equation. This gives a convergentserieswhose entriesare
eachO[(kH)2] compared totheprevious one,andthefirstterm alone is a satisfactoryapproximation.Using this, the second
termin (50)is seentobeO[(kH)2] compared to (49)andmay be neglected. In a similar way the last term in (50) is also found to be small, and (49) is a consistentexpressionfor the motionalmagneticfield. Returningto the inducedelectric current integralequation (21), it is now possibleto evaluatethe contributionfrom the secondarymagneticfield givenby the last term with (49). This
termis O[(kH)2] compared to (48), sothecoupling between the integral equations(20) and (21) is weak. Similarly, the third term in (20) does not contribute significantlyto the motionalmagneticfield, and the approximateprocedureused to get a solutionis justified. Finally, the vertical electric field follows from (22) with (36) and (49). Applying H61der'sinequalityonce again,the ratio
of
O(kHI
the
first
to
the
second
term
is
seen
to
be
I IF'__' I / I F>', I ). Thisistypically O(kH)away
CHAVEANDLUTHER:MOTIONALEM FIELDSIN THEOCEAN,1, THEORY
from the poles,so the verticalelectricfield is very nearlyequal to
•: = --(ZVh(Z )X•",',)'•
(51)
This result is the wave number domain counterpart to that given by Harvey [1974]. The vertical electricfield is independent of the electrical
structure
of the earth because vertical
electric currentscannotreadily penetratethe relatively poorly conductingearthandnearlyinsulatingatmosphere. THE ELECTROMAGNETIC FIELDS IN THE WAVE NUMBER DOMAIN
While the PM and TM modesprovideboth a physicalview of the inductionprocessand a substantialmathematicalsimplification, they cannot be measuredindependently,and the separateexpressionsfor the horizontalelectric and magnetic fields
must be combined.
For the horizontal
electric
com-
ponent,the relevant expressionsare given by (42) and the inducedcurrentdensity(48) dividedby the in situ conductivity
of seawater.The resultcanb•e•implifiedusingthe identity • [kx ^ ~ , ]'• • Zzx ~ , +(•xk)k' ~ , . Neglecting thePM mode term that is O(kH) comparedto the TM mode contributions,the horizontalelectricfield is given by
where
weak in the presenceof resistivematerial. Equations(52) and (54), togetherwith (45) and (51), are wave number domain expressionsvalid in the low-frequency,large-scalelimit when inductionis not large. INTERACTIONS OF MOTIONAL
FIELDS WITH THE EARTH
Interactionsof motionalelectromagneticfields with the conductingoceanand earth are describedby the term (53) which appearsin both (52) and (54) and by (30). Understandingthe influenceof theseregionsrequiresa geophysicallyreasonable model for the electricalconductivitystructuresof both the oceans and earth.
Precise empirical relationshipsbetween seawater conductivity and its temperature,salinity, and pressureexist [e.g., Fofonoffand Millard, 1983] and may be usedto computeseawater conductivityusingtemperatureand salinity profile data. The dominant effect is due to temperature,although rising pressuretypically resultsin a small increasein conductivity near the seafloor. Figure 2 shows such a calculation using historic 5¸ zonal averagesof potentialtemperature(converted to in situ temperature)and salinity at 42.5øN in the Pacific taken from Levitus[1982]. Most of the changein conductivity occurs above the main thermocline,and the conductivityis nearly constantbelow that point. The depth-averagedconductivity for this profile is 3.18 S/m, about 2% smaller than the seafloor value o(-H) of 3.25 S/m.
f,= o(-H)
h •H
+ h •H( - o(-H))
(53)
Conductivity
is a dimensionlessfunction. The electric field is depthindependentand consistsof two parts. The first term is due to direct forcing by the velocity field, while the secondone is relatedto large-scalehorizontalvelocitydivergenceandwill be discussedlater. The electric field is proportionalto a factor (53) which is dependenton wave number,frequency,and the electrical structuresof the ocean and earth, and contains all of
the essentialinformationon the leakageof electriccurrentinto the conductingearththat determinesthe size of (52). This term is nearlyreal and has a magnitudewhich lies between0 and 1 as the earthrangesfrom a good conductorto a perfectinsulator. Note that (53) is a generalizationof simplercalculations given by Sanford[ 1971] andCox [ 1981]. The horizontalmagneticfield is given by the sum of (43) and (49). Unlike the horizontal electric field, the motional magneticfield is markedly depth-dependent.Becausevector magneticfield sensorsare extremelysensitiveto motionin the geomagneticfield, it is not feasibleto measuremagneticcomponentsexcept at the seafloor,and only that locationwill be consideredfurther. The resultcan be simplifiedusingthe idenA
7191
^
^
ß
^ ^
~
2.8
3.2
3.6 I
o ttt
lOOO ,
/' I!
,
[
2000
'
3000
4OO0
ß
tity(zxk)[kx•]-• = _k(k.) togive
fih(-H) =gHF? I(1-•-') * --•(1--•+ hv)k'
5000
(54)
The balanceof termsin (54) is not nearly as straightforwardas in (52) and requirescareful examinationof (30) and (53) as a function of frequencyand wave number for realistic earth models. However, the sensitivityof the magneticfield to the
earth'sconductivity has the opposite• senseto (52) sincea poorly conductingearth resultsin l-F--0. This is intuitively reasonablebecauseimage currentsin the earth result in reinforcement of the seafloor horizontal magnetic field and are
Fig. 2. Profile of the electricalconductivityof seawatercomputedfrom the empiricalformulaegiven by Fofonoffand Millard [1983] using5¸ zonal averagesof the potentialtemperature(convertedto in situ temperature) and salinity from Levitus [1982]. The short-dashedline includesthe temperaturedependenceonly and assumesa constantsalinity of 33.4 ppt and zero pressure.The long-dashedline addsthe real salinity dependenceto the computation.The solid line is the full treatment includingthe pressuredependenceof conductivity.
7192 A reference
CHAVE ANDLUTHER: MOTIONAL EM FIELDS INTHEOCEAN, 1,THEORY model
for the electrical
structure
beneath
the
deep seafloor has recently been proposedby Chave et al. [1990] basedon a variety of geophysicalandgeochemicaldata. It canbe dividedinto threemain regions:sediments, crust,and mantle. The thicknessand conductivity of the sedimentary region is highly variable in space. It nearly vanishesin much of the Pacific
and can be 1 km or more thick in the Atlantic
Ocean, so the model is subdivided into two types. In the Pacific model, no sediment is included, while in the Atlantic
-2 . 5
model, a 0.5 km thick sedimentaryzone of conductivity0.3 S/m is added. The oceanic crust is about 6.5 km thick and con-
sistsof a 0.6 km upperzone of conductivity=0.03 S/m overlying a 5.9 km lower region of conductivity =0.003 S/m. Beneaththis point, the conductivityfalls sharplyto a value of
=10-5 S/movera distance of 30km;thepresence of a resistive
-3.0
-3.5
zone beneaththe crust is supportedby controlledsourceelectromagneticsoundingsreportedby Cox'et al. [1986]. Below this region,the conductivitymustrise rapidly due to increasing temperature. This is simulatedby a 40 km thick zone of conductivity0.003 S/m overlyinga 0.1 S/m half-space.The reference model is believed to be generallyvalid away from coastlines and mid-oceanspreadingcenters;for a justification see Chave et al. [1990]. Figure 3 summarizesthe Pacific and Atlantic conductivitymodels. Figure 4 showsa contourplot of (53) as a function of frequencyin cyclesper hour and inversewavelengthin cyclesper kilometer using the Pacific referencemodel and the seawater conductivityprofile of Figure 2. The frequencyscale covers
therangeof 10-4-10 -• cph(10hoursto 1 yearperiod), whilea usefulrangefor the inversewavelength is 5)0.99) across a 300km by 300 km arrayfrom a periodof a few minutesto about10 days (the longestperiodthat could be studiedin a statistical
sense)duringEMSLAB,implyingthatthemagnetic fieldwas dominatedby externalsources,while the seafloorelectricfield was incoherentexcept for a few narrow band featuresat
periods longerthana fewdays.Similarseafloor magnetic field coherences across distances to 1000km andoutto periodsof several months have been observed in the BEMPEX data. The
reasonsthat motionalmagneticfields are weak relativeto the
externalpartaretwofold.First,thediscussion aroundequation (58) indicatesthat motionalmagneticfieldsare large-scale averagesof the velocity field with a net contributionthat is
correspondingly smallandarefurtherreducedby theweakness of electromagnetic interactions withtheearth. Second, power spectraof the horizontalmagneticfield are similarat seafloor
andterrestrial sitesanddisplay roughly f-I behavior atperiods longerthana day at mid-latitudes. The increasing external field with increasingperiodeffectivelymasksany weak motionalcomponent.Evenif theyproveddetectable, it is not
TABLE 1. Modal Expansionof ConductivityProfiles Mode
12.5øN Pacific
32.5øN Atlantic
42.5øN Pacific
57.5øS Atlantic
1 2 3 4 5
3.302 -0.334 0.116 0.002 0.029
3.522 0.419 -0.051 -0.043 0.029
3.181 0.053 0.067 -0.007 0.025
3.036 -0.012 -0.029 -0.013 -0.003
clearthatmotionalmagnetic fieldshavereal oceanographic applications because of thelargescalenatureof thevelocity field averagingand the ambiguityof the relativesizesof the termsin (58). However,the electricfield appearsto be dominatedby localmotionalsources at periodslongerthana few days.Thisis duein partto thefactthattheexternally induced electricfield actuallydecreases with increasing periodbelow
CHAVE AND LUTHER:MOTIONAL EM FIELDS IN THE OCEAN, 1, THEORY
about a day' see Chave et al. [1989] and Luther et al. (1990) for elaboration.
7197
horizontal componentsand an independentrelation for the vertical one. This resultsin the equation u2 +iv2 =f2(x+iy)
APPENDIXA: DERIVATIONOF THE MODAL EQUATIONS
(A8)
where
Using the Helmholtz representationtheorem given by Backus[ 1986], any vector field on the plane may be written in terms of its vertical component,a consoidalvector field, and a toroidal
vector field
T = r • + Vhs+ • x Vht
E = c•^+ Vh[3+ • x VhT
(A2)
The sourcecurrent(5(vxF) may be expressedin a similar way using (6). The scalar E is just the vertical componentof (5(vxF), while the consoidal and toroidal parts of the source
v2: [tot + [tY - V2•
(A10)
with f2 an analyticfunction,and
Vh2H - [toot- [tE=0
Since&l] is arbitrary(this is just the choiceof gaugefrom
the two modal scalarfunctions(7) and (8) and the electric field (9). The usual boundaryconditionson the horizontal componentsof E and B and the vertical componentsof B and J must be satisfiedat horizontal interfaces. These require con-
tinuityof •, •:•, l-I, and (•:I]+[tT)/[t(5. Sincethe boundary conditionsare not coupled,the PM and TM modesrepresented by solutionsof (7) and (8) are independent. APPENDIX B: GREEN FUNCTIONS FOR A CONSTANT CONDUCTIVITY
Vh2T = Vh'[(5(vx F)]
(A3)
(All)
(5)), it is permissibleto set u2 in (A8) to zero. This requires that v2 in (A10) be constant,but (6) is unchangedif that constantis absorbedinto Y, and v 2 may alsobe setto zero. Combiningtheseconditionsgives differential equationsfor
current in (6) are solutionsof
OCEAN
Let the Cartesiancoordinatesystembe the usual oceano-
graphic onewith•' east,) north,and• positive upwards, sothat
and
V•Y = (• x Vh)'[(5(v x F)]' •
(A4)
Further simplificationof (A3) and (A4) for realistic ocean flows and models of the geomagneticfield are discussedin AppendixD. Substituting(5) and (A2) into (2) gives three equations when expressedin Cartesiancomponentform. The two horizontal componentsare identifiable as the Cauchy-Riemann conditions
(A9)
and
(A1)
where s and t are scalarfunctionswhich are uniqueup to arbitrary additive constants. Uniquenessconsiderationsusually require that the constantsbe known; Backus [1986] suggests requiring the averagevalue of s and t to vanish on a ball of specifiedradiuswhen working with a sphericalgeometry. In this paper, a plane geometrywill be usedexclusively,and all functionswill be assumedto have a spatial Fourier transform representation,so that uniquenessconditionsare automatically satisfied. Combining (A1) and (1), the magnetic induction may be written in terms of scalarfunctionsas in (5). Using (A1), the electric field is written as
u2 = O:l-I+ [t(513 + [tT
the water column covers-H_
