H.K. Moffatt and R.F. Dillon- The Correlation Between Gravitational and Geomagnetic Fields Caused by Interaction of the Core Fluid Motion with a Bumpy Core-Mantle Interface

8/3/2019 H.K. Moffatt and R.F. Dillon- The Correlation Between Gravitational and Geomagnetic Fields Caused by Interaction

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Physics of the Earth and Planetary Interiors, 1 3 ( 1 9 7 6 ) 6 7 - 7 80 lsevier Scicntific Publishing Comp any , Amsterda m - Printed in The Nether lands

THE CORRELATION BETWEEN GRAVITATIONAL A ND GEOM AGNETIC FIELDS CAU SED BYINTERACTION OF THE CORE FLUID MOTION WITH A BUMPY CORE-MANTLE INTERFACEH.K. MOFFATT and R.F. DILLON *

Department of Applied Mathematics and Theoretical Physics, Cambridge (Great Brifa in)(Received December 19 , 1975 ; revised and accepted Febru ary 2 7, 19 76)

Moffat t , H.K. and Dillon, R.F. , 1976 . The co rre la t ion between gravi ta t ional and g eomagnet ic f ields caused by inter-act ion of the core f luid mot ion w i th a bum py core-m ant le interface . Phys. Earth Planet . Inter . , 1 3: 67-78.

The corre la t ion discovered by Hide and Mal in between the var iable par ts of the Earths gravi ta t ional f ield andmagnetic field (suitably displaced in longitude) was tentatively and qualitatively explained by them in terms ofthe inf luence on b ot h f ields of i r regular i ties (or surface bum ps) a t the core-mantle interface . In this paper , a quan-t i ta t ive analysis of this phen omen on is developed, through study of an ideal ised problem in which conduct ing f luidoccupying the region z < q ( x ) flows over the surface z = q(x ) in th e presence of a magnetic field (Bo,O,O), he wholesystem rotating with angular velocity (O,O,n) . It is assumed that Iq(x)l > Iql ) as a funct ion of xo /d and the curves obta ined are qual i ta t ively similar to that based OI I theobserved data ; the maxim um corre la t ion obta ined var ies between 0.67 and 1, depending on values of the parametersof the problem , and is about 0.7 2 for reasonable est imates of these parameters in the geophysical co ntext .

1. IntroductionIt has been suggested (Hide 19 70 ; Hide and Malin 1 970 , 1972) that the observed correlation betwe en thevariable parts of the Earths surface gravitational and geomagnetic fields may be explained on the assumptionthat bumps (or inverted m ountain s) on the core-mantle interface are responsible directly or indirectly forthe observed perturba tions in b oth fields. Clearly an y density jum p a t the interface will lead directly to gravita-tional perturbation s vertically above the bum ps. Moreover any flow of the electrically co nductin g fluid in thecore region across the Earths magnetic field and over the bumps will generate electric currents in the core andso magnetic field perturbations on the Earths surface that are in some way determined by the bum p struc tureor statistics.

magnetic field at the same location, but with the geomagnetic field shifted in longitude thro ugh an angle cpo(t)towards the e ast, where cpo(t) ncreases linearly with time a t a rate of about 0.27/year, a manifestation of thewell-known westward d rift of the field. The data for the year 1965 [see Fig. 2 B on p . 76 , copied from Hideand Malin (19 70) l gave tpo = 160 , and the value of the correla tion corresponding t o this shift was 0.84. Back-ward extrapo lation of the available data suggested tha t p0 was zero abou t 500-600 years ago, and it wastentatively conc luded by Hide and Malin that some kind of occurrence at the core-mantle interface might

An important feature of the observed co rrelation is that the gravitational field is correlated no t w ith the geo-

6 1

* Present address: Shell Australia Ltd ., P.O. B ox 872K, Melbourne, Victor ia 3001, Aust ral ia.

www.moffatt.tc


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68

II z = d

- XFLUID COREc ( inviscid .mag netic diffus ivity Adensity p c > v,)0

0Fig. 1. The geometry of the model.

then have been responsible for th e generation of the co rrelation th at is now observed. Such an interp reta tion isperha ps open to ques tion; and yet the c orrelation itself is a striking pheno me non , for which no alternative e x-planatio n has yet been offered.The ob ject of this paper is to explore the nature of the magn etohydrodynamic interaction between a bum pyinterface and a flow over it influenced both by rotation and by the magnetic field of the Ea rth; and to derivethe correlatio n at the Earths surface befw een the resulting variable parts of the gravity and geomagnetic potentialfields. In order to make progress, certain drastic idealisations are required. The m odel studie d is dep icted in Fig. 1 .Firstly, we replace the Earths spherical geometry by a Cartesian geometry w ith coord inate z vertically upwards.The Earths surface is taken to be z = d, and the bum py core-mantle interface to be z = q(x),whe re, for all x,) q ( x )


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69

poloidal field). Such an at tem pt a t verisimilitude would however lead to severe mathe matica l difficulties; and it isbelieved that the simpler model a dop ted here, althou gh by n o means perfe ct, does not drastically misrepresentthe essential physics of the situation. Th e unifo rm field Bo may be imagined as an applied field extendingthrougho ut all space, bu t its effects will in fact be confined to a neighbourhood of the interfa ce in the core region,as in the real spherical situation.terms of scalar potentials @(x ,z) and *(x,z):

In the insulating region z >q(x), the gravitational field g(x) and magnetic field B ( x ) can be represe nted ing = - V@ , (popm)-l/(B ~ Bo ) = - v 9 ( 3 )We shall seek to d etermin e the steady form of @ and 9, nd hence to evaluate the cross-correlation:

where (.;) is an appropriate ly defined scalar prod uct. It will be clear tha t, in restricting at ten tio n to the steadyrelative t o the @-field can not be obtained from this primitive m odel. Nevertheless, it seems reasonable to co n-centrate a t this stage on gaining an unders tanding of the steady s ituati on, and on isolating those param eters ofthe problem on which the function R(xo ) (and in particular its maxim um value) depends.

situation, we can only derive a steady correlation function R(x o) , and that the westward drift of the *-field

2. General considerations concerning the correlation fun ction R ( x o )Le t @(x) @(x,d) nd $(x) = 9(x,d) and suppose that @ and $ have Fourier representations:

m m

@(x)= i ( k ) xp(ikx)dk, $(x) = 1 ( k ) xp(ikx)dk (5 )- m - mwhere 6 nd & may be generalised fu nc tio ns (Lighthill, 1959)(to include the possibilities that the spectra may bediscrete, or that @ and $ may be stationary random functions o fx ). These satisfy the reality con ditions:&-k) = & k) , &k) = & * ( k ) (6 )where the star indicates a complex co njugate, and we may therefore rewrite eq. 5 in the form:@(x)= 2 R e / $(k) exp(ikx)dk,m $(x) = 2 R e 9 (k ) xp(ikx)dk (7 )

0 0I

It is conve nient to be able to regard th e wavenumber k as always positive, and we shall therefore use the form (7)throughout.If @ and $ have well-behaved Fourier transforms, then the natural definition of the scalar product is:m m

[@(x),$(x - xO)] = @(x)Jl(x- X O ) ~ 47rReJ ?(k)$(k) exp(-ikxo)dk (8)- m 0

and the correlation function (4) becomes:


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70This expression can also be a dop ted as the definition of the correlation function fo r the case of stationary randomfields, or fields with discrete sp ectra .If @ and $ are sinusoidal with the same wavelength 2 n / k l , so that:

R (x o) = R e [ @ h x P( -ik lx o) l/l @ ll I $ 1 l = cos(k1xo - rg@1/$1) (11) .and it follows th at, as would be expe cted from the most elemen tary considerations, R(x o) = 1 when klx o =arg(G1/$]); i.e. there is perfect correlation between the two sine curves if one is shifted relative to the other sothat their maxima coincide.ingredients, since clearly the sh ift x (k l) tha t would maximise th e correlation if the wave number k l alonecon tributed need n ot be the same as the shift x(k 2) at another wavenumber k 2 . To be specific, let:

0his simple property obviously does not exten d to the case when @(x) and $(x) contain tw o or more spectral0k) = @16(k- k i ) + @ d ( k- kz), $(k) = $i6(k - k1) + $z6(k - kz) (12)then i t is readily verified tha t:

where (0 < t


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2t of filtering o ut high-frequency [0 ( !2 ) ] inertial-typ e waves from the system (20)-(21). The approxim a-

not essential in the presen t co nte xt (Dillon, 1975 ); bu t it greatly simplifies the analysis, and it m ay be veri-6, the effects of inertia are

If we now put U = U , + u(x),H = H o t h(x)and linearise (20)-(21) (with inertia terms in eq. 20 dropp ed) we= - V p + Ho Vh (23)

, . Vh= Ho.V u + hV2h (24)

p is the perturba tion in P.These equations admit solutions of the form :, , P ) = 2 R e

mA . . . .

( u , h , p ) xp(im .x)dk0

m = k(l,O,y) and possible values of y are to be determ ined; clearly these m ust satisfy Im y < 0 since themust vanish as z -+- -W. Sub stitut ion in eqs. 23-25 gives:= -imp + i H o k i

= iHok; - M2 (1 t y2 ) i(27)( 2 8 )

n terms of t he dimensionless numbers:= U o / H o ,

= H o k l n ,q. 28 becomes:

A A

Q = 2!2hl@

U = 2 [Q(1 + y2)+ tiA~-' -'(iU/K)G

27 becomes:u ; t 2n A; = -imp

uct (twice) wit h m gives:m A i - 2ky;=0, -urn2;-2kymA;=0

(32)

(33)

(34)elimination of; and mA now gives:

= -4y2( 1 t y2)-l

1 t 7') -+ y2 [Q( 1 t y2) + 2iA K - ~(35)

(36)eqs. 31 a nd 35, we obtain a cubic equation for yz

= 0e roots of this cubic be y i ( n = 1, 2 , 3) , with Imy,


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7 3Eq. 33, together with m . h = 0 now determ ine the ratios and relative phases of the velocity components in eachmode; for n = 1, 2 and 3 , we have:

in a,(k)( 1, --2 U;' , 7 ), &*= (i0n /K)& (37)where a,(k) are amplitude s as yet u ndete rmin ed. The general solution of eqs. 23-25 vanishing at z = -00 thentakes the form:

3 wU = 2Re 1 ,(k)(I, -2u;', -7;') xp[ik(x + y n z ) ]dkn = l

3 wh = 2R e -!,(k)(u,, -2 , -u,y;') exp[ik(x + y n z ) ]dkn = l " K (39)

5. Application of the boundary conditions at the core-mantle interface

The amplitudesa,(k) and $ o ( k )must now be determined in terms of { ( k ) hrough the boundary conditionsat z = ~ (x ) . hese conditions are that the normal compo nent of U must vanish and all three components of Bmust be con tinuou s. The linearised form of these co ndition s is:uz = Uoaq/ax , h = - (Pm/Pc) ' ' z v\k at z = O (40)With U , an d \k given by eqs. 38 , 39 and 17 respectively, these conditions may be expressed in matrix notation:

(41)0 3

1 1

where r = (Pm/pc )1 / 2CLHo . he determinant A of th e 4 X 4 matrix may be evaluated in the fo rm:= y;' A , + y;'AZ + y i 1 A 3

where:

and inversion of (41 ) gives the am plitude s an in the forma, = -(An/A)ikUoq,Finally, $o is given by :$ o = -nr-' T(K;Q, ) ; ( / c )where r = iAAO/KA,and:

. ( n = 1, 2 , 3 )

0 1 0 2 (53Y1 Y2 Y3A o = - ( ~ 3 - 0 2 ) + - ( 0 1 - ~ 3 ) + - ( ~ 2 - 0 1 )

(44)

(45)


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746 . Asym ptotic evaluation of the correlation R(xO)

On dimensional grounds, it is appropriate to express the Fourier transform of q(x) in the form:&)= 770LHkL) (47) I

l;j(k)12 = q ; L 2 I{(kL)12 = q&L(kL)

q(x) = 4 ~ ~ q ~ ( 12 x 2 / L 2 ) x p ( - x 2 / L 2 ) = 2 n 2 q o L 2 d 2 xp ( - x 2 /L 2 ) /d x 2

where, by virtue of the condition ( l) , {(O) = 0; here, qo is a measure of the a mp litude of the surface irregularities.The b um p spec trum is given by:(48)

say, and in general F ( t ) = O ( E 2 ) as t -+ 0. If the bum ps are symmetric about x = 0, then { ( E ) = O ( t 2 ) and F ( { ) =O( t 4 ) as t -+ 0. A simple prototyp e symm etric bum p is that given by: 0(49)Substitution of eqs. 16, 1 8, 4 5 and 48 in eq . 9 now gives R( xo ) as a functio n of the dimensionless variableX = x o / d and the parameters A , Q an d

P = L / d , S = H o / f l d (51)

R(X; f i , Qr A ,S)= I ( X ; , Q j A ,S)/ ~ G ( f i ) ~ M @ , ,A ,S)I 12 (52)in the form:

where:

I = Re j - r ( S K ;Q , A ) F @ K ) xp(-2K t iKX)dKIG = 1 2 F @ K ) xp(-2K)dK

m

0m

0

and:ZM = IT (SK;Q,A ) J 2 F @ K )xp(-2K)dK

0

(53)(54),

(5 5 )0Th e gravitational integral IG may be simply evaluated for specific choices of F(E),e.g. tha t given by e q. 50.Similarly, I and I M could in principle be evaluated numerically for specific values of X, P, Q, A an d S. owever,in the geophysical context, Q, A and S are ail small (see Table I). and asym ptotic evaluation of Z nd I M , akingadvantage of this fac t, is possible.Guided by the estimates of Table I , we let:

Q = @ , S = S E (56)where 0 < E


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7s7 ( s K ;Q,A ) only for K = O(1). T o do this, we must first return to the cubic (36) which with the su bstitutions(56) and K = S K becomes:( I t r2) 4 y 2 [@(I t 7') t e Z - ' l / s ~ 1 2 o (57)The leading terms of the asy mpto tic expansions of the three roots 7; or small E are:1 . - 17;- -1 , 7; h iq - ' e - p , y; -- E-pan d so:

Correspondingly, from eq. 31:- ~ K P , u2 -2 4 u3 -2 iand, from eqs. 4 2 ,4 3 and 46, we then have:

1 - -4 , A, - sKep-' , A 3 - - sKefi-'and: A --.4Kq11 &'J2+p-2-4ix. 4 b K ~ ' " - ~ ,Hence, in particular, from eq. 45 : (62) L Sr ( S K ;Q,A )- i t K/Ko)- ' (63)

It is now evident that the situation p + i p = 2 is critical in the sense tha t on ly th en is K O = O( 1). If p f $ p < 2 ,then K O + 0 as E -+ 0 and rTable I give K O = 4 and dearly therefore the condit ionother relation would be incom patible with the geophysical estimates) then the ch oice:

K o / K ,while if p t p > 2, then KO + CO as E -+ 0 and r - i. The estimates of+ i p = 2 is appro priate. If moreover p = p (and any(65)4,=,=3would appear to provide the best description in the geophysical conte xt. Note that this choice leads to the

following estimates of magnetic Reynolds nu mbe r R , and Rossby number R o :R , = U o d / h= A Q - ' S - ' = y - ' s - ' ~ - ~ / ~Ro = Uo/LtL= P-'AS = SP-'E'~/~ (67)It is this small estimate of Rossby n um ber w hich ensures th at ine rtia effects are negligible, as anticipated inSection 4.

Subst i tut ion of eq. 63 in eqs. 53 and 55 now gives:m

Z(X) - s K-I [ 1 + (K/Ko) ' ] - ' [ (K/Ko) osKX t sinKX] exp(-2K)F(PK)dK1~- s 1 + (K/KO)')-' exp(-2K)F(PK)dK = K d ( 0 )

(68)0and:m

(69)0

Note that the depende nce of these integrals on Q , A and S now appears only via the particular combin ationK~ =AS-' Q - ~ / z .


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76If /3 is small (and again this is suggested by the estimates of Table I) , then it is reasonable to evaluate th e ex-

pressions 5 4 ,6 8 and 69 by replacing F(F'(PK) y th e leading term of its expansion fo r small OK. If F(F'(PK)= 0 [(F'(PK)2n]as OK + 0, then eq. 52 reduces to:21/2J;KZn-1 [ l + (K/Ko)']-' [(K/Ko) cosK4 + sinKX] exp(-2K)dK[J;KZn (1 t (K/KO)'}-' exp(--2K)dK] 1 /2R n ( X ,KO) = (70) '

with limiting form s, corresponding to the possibilities n = 1, 2:1

and:R2(X,O) = (1 ix2)31 -1P 3'/'X(1 - - i X 2 )R2(X,") = - (72)4 01 t $X2)4The func t ionsRZ(X,K o)are sketche d in Fig. 2 for four values of K O (0 ,0 .42 5 ,4 .25 an dm ) . No te tha t a sK oincreases, the m aximum value of R2(X,Ko)decreases from 1 to 0.67, and th e value X, of X at which thismaximum occurs decreases from 0 to -0.65. In the geophysical cont ext this would corre spon d to a longitudeshift of approximately 20". Note that, since X,


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77

7 . The physical character of the three wave mod esTh e values of y 2 and y3 given by eq . 59 indicate that the corresponding velocity and magnetic perturbationshave a bound ary layer struc ture , the thickness of th e layer being O(6) where:

6 =q1/2&'/2k-l = Q l/ zk -' (74)Since, from eq. 6 3, A2 = -A3 in the limit conside red, the energy densities of these tnode s are equa l; in fact:

A 8 S 2 U i d 2 k 4;I21i212 lu312= exp(kz/Q1/2)~ ~ 1 ~ 1 ~ (75)

Note that, for K = 0(1), the magnetic energy de nsity is a factor O(S-2) greater than the kinetic energy density.3) is given by :Both mo des are highly helical; in fact, from eq . 34 , the helicity density (Moffatt, 1970) in either mode ( n = 2 orX , = Re(;: . &A = 2kli,I2 Re(iy,/a,) =+k l ; n 1 2 q - 1 / 2 e - P / 2+k l&12Q - 1 / 2 (77)This positive helicity in bo th m odes is associated with the flux of energy aw ay fro m the boun dary in the direc-tion -a.Moffat t, 1970, 5 2 ; Soward, 1975). A region of concen trated helicity near the boun dary is of potentialimpo rtance in the co nte xt of dyn am o regeneration of the large-scale magnetic field.magnetic field in this mode are given by:The m ode corresponding to suffix n = 1, with y1 = -i has a totally differen t charac ter. The velocity and

and are irrotational to leading order (i.e. neglecting the small y-co mp one nts). Th e mode is therefore almo st un-affected by ohmic dissipation. The kinetic energy density I 1 is smaller by a facto r O(e4,Y2) than that in modes2 and 3 ; it is however d istributed through out a layer of thickness O(k-') in the fluid. Th e magnetic energ ydensity Ih l 1 is of the same order of magnitude as l @ 1 2 and lL3l2. f & a n d M , ( n = 1, 2 ,3 ) a re the to ta lkinetic and magnetic energies in the three mod es (integrated over z) then, in the case p = y = $ .M~ = = 0 ( e - 1 0 1 3 ) ~ 2 , 3o ( ~ - ~ ) E ~ (79)It may readily be verified that th e mod e n = 1 is non-helical in the lim it conside red.The term U , . Vh in eq. 24 introduces a distinction between the positive and negative x-directions, and isresponsible for the final asymm etry in the correlation functions sketch ed in Fig. 2. In the n = 1 mod e, in whichV2h= 0 at lowest order in e (since yt = -l), ihisrconvection term is of dom inant imp ortance. In the other tw omodes, the relative importance of U , . Oh and XV2h in eq . 24 is given by:

so that , since y < 2, diffusion d om inates in these mo des (due to the small vertical length scale) and con vection isunimportant. If U , is gradually increased from small values (so that K O ncreases in proportio n) the gradual shiftin the correlation pattern from symme tric to antisymm etric form (Fig. 2A ) is therefore entirely due t o th e in-fluence of convection in the n = 1 mo de; the dom inant magnetic energy in this mode (eq. 79) ensures that struc-tural changes in the mode are transmitted via the bo undary condition s at the interface into the m antle region.It seems in fact as if the n = 1 mode plays the do mina nt role in determining the correlation pattern; the n = 2and 3 modes play a passive role, driven by the n = 1 mode via its interaction with the interface.


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7 8Referencrs

Dillon, R.I:., 19 75 . Gravity and mag netic field cor rela tio n an drelated g eoniagn etic topics. P1i.D. Thesis, Cam bridge Univer-sity, Cambridge, 1 83 pp.rotating fluid layer. Proc. R. Soc. London, Ser. A, 326:229-254.Eltayeb, I. A. , 1975. Overstable hydrom agnetic convection in aiota ting fluid layer. .I. luid Mech., 71: 161-179.Hide, R., 1970 . On the Earths core-m antle interface. Q.J.R.Meteorol. S oc., 96 : 579-59 0.riidz, K. and Malin, S.R.C., 1970 . Novel correlation s betw eenglobal featiires of the Earths gravitational and m agneticfields. M u r e (London) , 225: 605-609 .

Il ide, R. arid Malin, S.R.C., 197 2. Correla tions betwe en th e

Hta yeb , I.A., 1972 . Hydromagnetic convection in a rapidly

gravitational and geomagnetic potentials: recent develop-ments. Int. Conf. on The Core-Mantle Interfac e, Trans.Am. Geophys. Union, Rep., 53: 610.Hide, R. and Rob erts , P.H., 19 61. The origin of the m a in geo-magnetic field. In: L.H. Ahrens, S.K. Runcorn and H.C.Urey (Editors), Physics and Chemistry of the Earth, Vol. 4 ,Pergamon, London, pp. 25-98.Generalised Functions. Cambridge U niversity Press, Lond on,

aLighthill, M.J., 1959 . Introd ucti on to F ourier Analysis and

80 PP . IMoffatt , H.K., 1970. Dy namo action associated with randominertial waves in a rotating c ondu cting fluid. J. F luid Mech.,44: 705-719.Rob erts, P.H. and Soward, A.M., 197 2. Ma gneto hyd rodyn amic sof the Earths core. Annu. Rev. Fluid Mech., 4: 117-1 53.Soward, A.M., 1975 . Ran dom waves and dy na mo action. J.Fluid Mech., 69: 145-177.

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H.K. Moffatt and R.F. Dillon- The Correlation Between Gravitational and Geomagnetic Fields Caused by Interaction of the Core Fluid Motion with a Bumpy Core-Mantle Interface