H.K. Moffatt Fixed Points of Turbulent Dynamical Systems and Suppresion of Nonlinearity
H.K. Moffatt Topographic Coupling at the CoreMantle Interface
Transcript of H.K. Moffatt Topographic Coupling at the CoreMantle Interface

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Topographic Coupling a t theCore Mantle I nterfacetH. K. MOFFATTSchool of Mathematics, University of Bristol, Bristol BS8 1TW, England(ReceiccdApril 26. 1 9 7 7 )
l h c niodcl of Anufriyev and Ijraginskii (1975) and Moffatt and Dillon (1076) is further studied,with ;I view t o calculating ( i ) the mean tangential strcv 01 1 thc corem antle interface. and ( i i ) thelevel of the acfTect duc t o flow over the surkice bumps. l h c stress c;ilculalcd agrecs in ordcr ofmagnitude with the stress of0.04N mhat is inferred from the dccadc variatio ns in the length ofthe day.
1. INTRODUCTIONIt has been suggested by Hide (1969, 1977) t h a t an important. and perhapsdominant, contribution to the tangential stress transmitted from the Earthsliquid core to the solid mahtlc m a y be acwciated uith thc presence ofundulations, or bumps. on the corcmantle inlcrfacc. I t i\ knoun fromstudies of t h e decade flticttiiitions i n Ilic Icngtliofday that the total meanlangcntial slrcbs at t tic intcrfiicc ISapprouimatcly
Fs0.04N m W 2 . ( 1 1This stress may be decomposed into viscous, elcctromagnctic and topographicingredients:
F = F , + F , F,. ( 2 )The viscous ingredient F , is almost ccrtainly negligible, duc t o the smallestimated value of the kinematic viscosity in the corc of thc Earth, Theelectromagrictic ingredient FEarises from the lcakagc ofelec!ric currzil ts fromthe core into the lower mantle and is also estimated to be insufficient to_
tPaper prcsented at Britlsh Geophysical Assembly. Edinburgh, April 1977.219
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2x0 H. K M O F F A I Iaccount for the total stress (Rochester, 1973; Jacobs, 1975, $4.8). n this paper,we evaluate Fr on the basis of an idealised model previously treated byAnufriyev and Braginskii (1975) and by Moffatt and Dillon (1976, hereafterreferred to as I ) and we show that under reasonable estimates of the variousphysical parameters in the core of the E arth, topograp hic effects may indeedprovide a con tribution of the required or de r of magnitude.The stress F , has been expressed (Hide, 1977) in the form
FT =C$PcUoh, ( 3 )where SZ is the Earths angular velocity, p E he core density,? U. the relativevelocity between core an d mantle, h a m easure of the bum p am plitude, an d C ra drag coefficient, of o rder unity or less. C, may however depend in a nontrivial way on the various dimensionless parameters controlling the flow overthe bumps, and in particular on the magnetic Reynolds number.
Rn,T = u o h f ~ ~ ~ (41based on bu m p amplitude (1. being the magnetic diffusivity in the core), and i tis important to determine this dependence before a result of the form (3) can beused with confidence.M any of the results to be presented here are already implicit in the paper ofAnufriyev an d Braginskii (1975). It is hoped however tha t the discussion to begiven here will clarify the various physical mechanisms involved .2. ESSENTIAL FEATURES OF THE M O D E LConsider the problem depicted in Figure 1. Th e interface z = ( x ) separates anonconducting solid from a liquid of uniform density pc and magnetic+
oC t 1Ia 0CLX..FIGURE 1. Sketch of the configuration considered.
~
tWe shall use a sufix c to denote liquid core quantities, and a suffix rn to dcnote mantlequantities.

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1 () i < ) (J \ a magnet ic tield 11 \I hiclitends to the uniform field Bo= ( BO , ,O ) far from the interfacc T he liquid flowsteadily relative t o the b ou nd ar y, with velocity U tending to U , = ( U o , ,O)farfrom the interface. T he whole configuration (including the m agnetic field)rotates with angu lar velocity R = O,O,Q). We shall suppose tha t l ( ( x ) l 41 sothat all perturbations associated with the bumps are small. Let L be thehorizontal scale of the bumps, i.e. I[/(\ =O(L).There are three independent dimensionless numbers that arise i n theanalysis of this situa tion, viz.
A = Uo /H o , Q =RA/H& R , I= U,/RL, ( 5 )where Ho= (pop , )  /BO is the Alfven velocity associated with the field B,.With the estimates (see I )
Lr,z104ni/s, I I , , sO.4ni /s . Q = 7 . 3 10 \ I , i : 3 i n 2 , s , L = IOhin ( 6 )in the pmphysical context, wc liavc
A z 3 . 1 0 4 , Q s 2 . 1 0 , K , , : 1 5 . 1 0  , ( 7 )an d it is reasonable to base th e analysis on th e awu mp t i o n s j
4 , 6 1, Q41 . (8 )Thc conditioi: R , 6 1 allows us to neglect the incrtia term DUIDt in thecquation of motion relative to the Coriolis term ~ R A. Under steadyconditions, the govern ing equ at ion s in the fluid region a re then
v . i= v f I =o, (1 1 )where H =(pup1)B. and
P = p; p , +p0 B 2 ) . (12)p r is the fluid pressure including modifications associated with gravity andcentrifugal acceleration. Note that the stress tensor (including the Maxwell
?The importance of the particular conibination A z / R o is emphasised by Anulriyev andBraginskii (1975).

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282 H. K. MOFFATTelectromagnetic ingredient) is given by
cij=  p / 6 i j +C lo1 (B iB j ~B2S i I )=p , (  Pd i j+HiH, ) . (13)Writing
the linearised form of Eqs. (9)( 11) becomes2 R b . U =  V p + H o * V h ,U. * Vh =Ho * VU+LV'h,
V * u = V . h = O .In the solid region z > (x), the magnetic field is a potentia l field, i.e.
an d evidently $,+O as z+m. Moreoverh,u+O as z+m, (19)
an d we have also the m atching cond itions on the interface
so th at the linearised form o f (20) isu , =Vo t ; ' ( x ) , [h]=O on z=O. (22)
The problem as specified above has been treated exactly in I, and it emergesthat under the conditions (8), the disturbance modes available are of twotypes: (i) a currentfree irro tat iona l mo de, and (ii) a m ode having a boundarylayer structure, the boundary layer thickness being of order Q112. In thefollowing two sections, we discuss furthe r the structure of these modes and theman ner in which their amplitudes an d phases are determined by the boundaryconditions (22).

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TOPOGRAPHIC COREMANTLE COUPLING 283
3. THE CURRENTFREE IRROTATIONAL MODEWe shall suppo se for simplicity that
[(x) = Re (Ceikx), k >0, I k q < 1 . (231The linearity of the problem allows more general interface profiles to betreated by Fourier synthesis.In the formal limit A0 (or Q0) Eq. (16) becomes
(H , .V)(ul  A h , ) = O , i.e. u1 = A hl (24)(where we use the suffix 1 to denote this dissipationless type ofdisturbance), andEq. (15) then becomes
( 2 6 )so that, under the condition A 2 G R o (see (8) above), Eq. (25) takes theapproximate form
Ho . V h , = V p , , (27)
(28)The field h, (and so U, from (24)) is therefore irrotational. The normalcomponent of this field must match the no rmal com ponent of V$,,, across z=[;since both $< nd $,,,are harmonic fields, they may then be easily obta inedin the form
or equivalcntlyh , x ( i k H , )  ' V p , = V$c, say.
(29)< = " i k X + k z , II/, = ,Jpk =where is a complex amplitude to be determined. Note th at
There is a corresponding current sheet on z =0, the surface current beingJ, = pc/po)"z(O, ik$etkX, ). (31)

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284 H . K. MOFFATTTh e struc ture of this cur ren t sheet can be resolved on ly by restoring the effectsof magnetic diffusion.
4. THE M A G N E T IC O lF F U S lO N L A Y E RLet us now focus attention on solutions of Eqs. (15)(17) having a scale ofvariation 6 in the z direction small co m pa red with L ( w k  ). As mentionedabove, the scale th at emerges from a n exact analysis is ~ 5 = 0 ( Q ~ ) k  .henQ O an d H o > O ; ifeither Q or H o is negative, then 101 and lHoi
must be used.

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TOIOCiKAPtllC C O R E  M A N T L E COUP L ING 285the relevant solu tions of (37), vanishing as 2 3  3, are
x + s + e i k x e ( l + i ) z / b , 2 c e s _ e i k x e ( l  i P / a I (39)and we then have
The cond itions of continu ity of the x and y com ponen ts of h = h, + h, acrossz=O can now be satisfied. Since hl,=O, the condition [h,.] = O gives simplyYt+ =.% .Thecond i t ion Eh,] then givesik$+i(.%?+ + .*  )= i k$ . Hence
Y + .% =2ik$, (411and so, determining /I , ,rom the condition V .h, = 0, we obtain
Similarly, from (33), and V .u ,=O,
( f ; Ho k 2$eikxez/d[s in i , cos ,iQ cos sin ~ . (43) R 6Note that l / id2)=O(Q)lh,,l on z =0, so tha t to leading ordcr the condition ofcontinuity of h .n across the interface is unaffected by the boundary layerillode.It remains to satisfy the condition
U, = u1 +udz =V,C (x )= kU , [ on z = 0. (44 )O n z = 0, we have
Hence (44)gives
(46)
(471or
 Ho

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286 H. . MOFFATTwhere
ko = (Uo/Ho)(fi/A)1'2.Note that ul,/uaz=  ikok on z=O,and that the ratio k/ko is crucial in determ ining the phase shift of $ relative toe. If k +k,, then the bo un da ry layer mode makes negligible contribution in thecondition (44), while if k + ko i t makes a dominant contribution. In thegeophysical context, k; '=700 km, and values of k/ko of order unity are ofparticular interest; in this situation the boundarylayer and irrotationalmodes make com parable contributions in (44).5. T H E M E A N STR ESS ON T H E B O U N D A R YFrom the expression (13) for the stress tensor, it follows that the mean xcomponen t of stress on z = ( s
F T = ( o l j n j ) = p c (  P n , +H,H vn). (50)Now, P = P o + p l +pd, and from (27)
P1 =Hob*,, (51 )while, from (15), within the boundary layer,
~4 =O(aHokhb:)=O(Q P I (52)Hence, t o o rder 12, nd neglecting the O(Q) ontribution from pa, we havefrom (50)
F T p t ( H o h i X '  f:C'+ HxHr)r=o. ( 5 3 )Now both H , and 1, are continu ous across z =0, and H , and H, re exactlyout of phase? on the mantle side (z=O+); hence ( H , H , > , , o = O . With H,=Ho h,, ( 5 3 ) hen gives
?If finite mantle conductivity is taken into account, this result no longer holds, and the resultingcontribution is just the term F E eferred to in the introduction.

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TOPOGRAPHIC COREMANTLE COUPLING 287and , using (47) , his reduces to
A similar, though not identical, expression has been obtained by Anufriyevand Braginskii (1975).T he expression ( 5 5 ) s maximal as a function of k /ko when k / k , = 1, and then
It is interesting that this maximal stress, although entirely of magnetic origin,does not depend on the field strength H , , [although ofcourse ( 5 6 ) s valid o n lyi f H o is sufficiently large for (8 ) to bc satisfied]. W ith the values of R, U. and 1given in ( 6 ) bove, and with
pcx104kg/m3, 1t1x5.103m, ( 57 )(56)gives
F,,,x0.045 Nm, ( 5 8 )which is of the right order of magn itude to account ior the total stress given by
I f C(x) is a stationary ran do m function of x with spectrum function W ( k ) ,(1).then the approp ria te generalisation of ( 5 5 ) s clearly
The approp riate cxprcssion for C , may be inferred from (3), and in generalthere appears to be no guarantee that th is will bt: of order unity.6. T H E a E F F E C T A S S O C IA T E D WITH T H EB O U N D A R Y LA YER F L O WAs observed in I, the bou ndary layer mode is strongly helical in cha racter, andastrongaeffect is therefore to beexpected (see,forexample, M oFdtt, 1976).Themean electromotive force 8 associated with the pe rtu rbatio n fields is given by
(popc)12B (U A h) = (U, A h,)+ (U, A h , ) + U, A h,). ( 60 )

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288 H. K. MOFFATTOf particular interest is the com po ne nt 8, n the direction of H o , given by
Now ul r and hay are exactly ou t o f phase so tha t (ulrhdy)=O; on the otherhand uay nd hl, are exactly in phase, an d
Fu rthe r, from (42) and (43),
T he expression (62) clearly provides the do m ina nt contribution, an d we havein fact 8, ctB, where (since k g6  )
Again i t must bc emphasised that this result holds only when Ho is strongenough for (8) to be satisficd; the dependence of a on H , is indicative of thestro ng dynamic interactions taking place. When H o + CO, k , + O and a(z) endst o the satu ra tio n level givcn by
ReferencesAnufriycv,A. P. and fhpinskii, S. I . Effect of a magnetic field on the stream ofrotatingliquid at arough surkicc, MCCR.i idrod. 4, 62 68 (1975). [h.loRfteto/i~rlrt,dynclmics, I , 461467. 19761.Hide, R . Interaction bclwccn ihc Earths liquid core and solid mantle, Nuture 222, 10551056
(1969 .Hide. R. Towards a theory of irregular variations in the lcngth of the day and coremantlecoupling, P l i i l . Tr