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On the dynamics of topographical core-mantle coupling
PHYSICS O F T H E EARTH ANDPLANETARY INTERIORS
ELSEVIER
Physics of the Earth and Planetary Interiors99 (1997) 289-294
On the dynamics of topographical core-mantle coupling Weijia Kuang, Jeremy Bloxham * Department of Earth and Planetary Sciences, Harvard University. Cambridge, MA 02138, USA
Received 19 August 1995; revised 24 May 1996; accepted6 June 1996
Abstract
We examine the dynamics of topographical core-mantle coupling, the process by which angular momentum is exchanged between the Earth's fluid outer core and solid mantle through the action of pressure forces acting on topography on the core-mantle boundary. We show that methods for calculating the torque directly from observations based on the assumption of tangentially geostrophic flow at the core surface should yield a zero torque if the mantle is an electrical insulator and the gravitational acceleration is radial. We discuss the effect of relaxing these assumptions and argue that such methods remain incapable of determining the torque on the core, either due to violation of the underlying assumption of tangential geostrophy or the admission of larger non-topographical torques. 1. Introduction
Topographical core-mantle coupling, first proposed by Hide (1969), is one of the leading candidates for the mechanism of the exchange of angular momentum between the solid mantle and fluid core which results in decadal variations in the length-ofday. The topographical, or pressure, torque F arises from the pressure p associated with flow in the outer core acting on asymmetrical topography on the core-mantle boundary (CMB)
F= fcmBr×pdS=fvc~r ×
VpdV
(1)
where the CMB is given by r = r0[1 + 8 H ( 0 , ~b)]
(2)
with r 0 the mean radius of the CMB, H(O, q~) ~ O(1) and t~ .~z 1.
* Corresponding author.
Previous studies aimed at evaluating topographical coupling have adopted one of two approaches. In the first, a forward modelling approach, simple models of the basic fluid flow and magnetic field are assumed, and the torque that results from the perturbation to the basic state by small amplitude boundary topography (2) is calculated (for example, Anufriev and Braginsky, 1975a,b, 1977a,b, Moffatt and Dillon, 1976, Moffatt, 1978, Kuang and Bloxham, 1993). The pressure torque calculated by this approach is proportional to 8 2 (for long wavelength topography). In the other approach, an inverse modelling approach proposed by Hide (1986), observations are used to deduce the fluid pressure at the CMB. The tangentially geostrophic part of the fluid flow at the core-mantle boundary is determined from the geomagnetic secular variation; then, using the tangentially geostrophic approximation, the horizontal pressure gradient at the CMB is calculated; next, a map of core-mantle boundary topography is determined using seismic observations; and finally, with both the pressure gradient and the CMB topography known,
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W. Kuang, J. Bloxham / Physics of the Earth and Planetary Interiors 99 (1997) 289-294
the torque is calculated. The pressure torque from this approach is proportional to 6 and is thus much larger than that predicted by the forward modelling. In this paper, we examine these two approaches and demonstrate how they give rise to such seemingly inconsistent conclusions. This is a matter of some importance, not only in studies of the lengthof-day, but also in studies of the dynamics of the lowermost mantle through the constraints that topographical coupling may place on the amplitude ,~ of CMB topography. First we consider a simplified problem. In order to separate topographical coupling from other coupling mechanisms, we assume that the mantle is a perfect electrical insulator, thereby eliminating electromagnetic core-mantle coupling, and we assume that the gravitational acceleration in the outer core is radial, thereby eliminating gravitational core-mantle coupling. Later, we discuss the effect of relaxing these assumptions. We consider a reference frame co-rotating with the solid mantle, with angular velocity g~. Then, the fluid flow u and magnetic field B in the outer core are described by the following equations
(0 u
)
3t + V. puu + 2 ~ X p u = - Vp + j X n + p g + vpV2u
aB at
= VX ( u X B ) + ~IVZB
V'(pu)= 1 j= --VXB /x
V-B=0
(3) (4) (5) (6)
in which p is the density, "q the magnetic diffusivity, the kinematic viscosity, g the gravitational acceleration, j the current density and /x the magnetic permeability. We have assumed that v, "q and /x are constants. The anelastic approximation in Eq. (5) eliminates compressionai waves which are not important in this context. We have also neglected the Poincar~ term in the momentum Eq. (3). This approximation does not introduce qualitative changes because the moment of inertia of the Earth's outer core is about an order of magnitude smaller than that of the solid mantle (Jault and LeMoufil, 1989).
At the CMB we have the boundary condition ~.u=0
(7)
where ~ is a unit vector normal to the CMB.
2. Contributions to the topographical torque The topographical or pressure torque arising from p is given by Eq. (1). Using Eq. (3) we write rX VpdV
F= f
J, Vcore
= -
ir ( °°u v,o,~
3t
)
+ V. puu dV
-2f Vcorep r X ( g ~ X u ) d V +f
rX(JXB)dV gcorc
+f
rXpgdV-Ft+Fc+FB+F
G
Vcore
(8) where the superscript I represents the contribution from the inertial force, c the contribution from the Coriolis force, B the contribution from the Lorentz force and G the contribution from the buoyancy force. We have neglected the contribution from the viscous term which is small. We can see immediately that F C = 0 since we have assumed g is parallel to r. We denote by F. c the axial component of F c i.e.
IC= -2£. fv
pr X ( g~X u) dU
=-ear
p ~ ' r X ( ~ × u ) dV Vcore
= -- 2 , Q f
p(~,Xr).(~,Xu)dV
V~ore
= -2x f
p r s i n O ~ b . ( ~ . × u ) dV
Vcore
= -2, f
psu~dV
(9)
gt ore
where u, is the radial component of the velocity in the cylindrical coordinate system (s, ,;b, z).
W. Kuang, J. Bloxham / Physics of the E.arth and Planetary Interiors 99 (1997) 289-294
CA~
291
Similarly,
CAPT
fvjpOUs dV s 8
:-f,:'sds[-fa:oPou,dA+fcA,sPo"dS] =-f
Sm
sdsf V'(pu) dV=O
SO
(12)
V2 s
where s,. is the maximum distance CAPB
,. =
Fig. 1. Division of the CMB into the surfaces CAPT, CAPB and CAPS by the cylinder with radius s.
Taylor (1963) showed that Fz c vanishes if the CMB is spherical. Here, we extend his result for the case of a nonspherical boundary. We use A s to represent the cylindrical surface within the core at distance s from the rotation axis, and use V~s to represent the region of the outer core inside this cylinder and V2, to represent the region of the outer core outside this cylinder. In particular, we may choose one such cylinder, say As0, which divides the outer core into two regions V l and V2. Then,
rzc= - 2 o f
z)]
Note in the above, that since the CMB is asymmetric, sm varies with longitude th and may not occur precisely at z = 0. In deriving Eq. (I 1) and Eq. (12), we have applied the boundary condition (7), resulting in vanishing surface integrals on CAPT, CAPB and CAPS. Thus we have shown that Fz c = 0. Next, to show that U n = 0, we write
F a = f,11 r × ( j × n ) d V = f rX V.'rndV Vcor, J'vco~
(13) where ¢ n is the Maxwell stress tensor. Then, using Gauss's law and the symmetry of cn, we have
SpousdV Vcore
1" B = - f
r×¢B.dS
(14)
"C MB
Then, applying Gauss's law a second time, We denote by CAPT and CAPB the parts of the CMB enclosed by the intersection of Aso with the CMB, and we denote by CAPS the remainder of the CMB (see Fig. 1). Then,
since j = 0 in the mantle.
fvspou,dV= fS°sdsfA sPou,dA
3. Discussion
l
0
= fo °Sd,[f Oou, dA [ Aso
=foS°sdsf V'(PoU)dV=O Vis
(11)
U n= - f
r×(j×B)dV=O
(15)
Vmantle
We have shown that if the mantle is an electrical insulator and the gravitational acceleration is radial, the only contribution to the axial topographical torque F z comes from the inertial force. Physically, this is not surprising (as has been noted previously by Bloxham and Kuang (1995)) since from Newton's Second Law the torque must be associated with a
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W. Kuang, J. Bloxham/ Physics of the Earth and Planetary Interiors 99 (1997) 289-294
change in angular momentum, i.e. with the inertial term. It is apparent from this result, if not from these basic physical considerations, that Hide's method (Hide, 1986) for computing the topographical torque should, in the absence of errors in the determination of the flow from the geomagnetic secular variation, yield a zero torque. Jault and LeMouS1 (1989) have also addressed this issue. They use a perturbation approach, and examine the perturbation to the flow due to boundary topography. They expand the flow V in the core V = V0 + o
(16)
in terms of a leading order flow V0 and a perturbation o (or u 2 in their notation). Their leading order flow is tangentially geostrophic and is that which is derived from the geomagnetic secular variation; it satisfies 2p(OX
V0)n = - V n P
~. V0 = O,
(17)
atr=r 0
The perturbation 1, represents the response of Vo to the boundary topography; it satisfies
0(p ) -
-
at
+2aX
= - Vp
(pv)
(19)
atr=r o
~.o=roa(VH).Vo=-g(s,~b,z+_)
(20) where z _ denotes the upper (i.e. northern hemisphere) and lower (i.e. southern hemisphere) parts of the CMB, respectively. They show that the part of o corresponding to the zonal symmetrical part of g (i.e. g = g ( s ) ) is timevarying, and is responsible for the time-variations of the axial angular momentum of the outer core aM= Ot
e fvc
e(p,,) r×
ore
3t
dV=-I"
z
(21) ' ~
where Fz. ~ is, in their terminology, the fictitious Coriolis torque of the flow v arising from the nonhomogeneous boundary condition (20). Since the total flow V satisfies the impenetrable boundary condition h . V = 0 at the CMB (see conditions (18) and (20)), the total torque F z is zero,
r ,vo +
0
where FZ.Vo is the pressure torque of the leading
order flow (17). /'Z.Vo is the pressure torque derived using Hide's method outlined above. Jault and LeMouifl (1989) conclude that the pressure torque Fz,vo indeed changes the axial angular momentum M z of the outer core, since dM. dt
Fz'v = Fz'v°
(22)
How do we explain this apparent contradiction between their analysis and that which we have presented above? In our analysis, the flow is required to satisfy the boundary condition ~ . u at the CMB exactly, in which case the geostrophic contribution to the torque vanishes. Jault and LeMoui~l, on the other hand, consider the boundary condition P- V0 = 0 at r = r 0 for the leading order flow, in which case the geostrophic contribution to the topographical torque does not vanish. As discussed, they then examined the perturbation to the flow by the topography and showed that the perturbation l, (again, u 2 in their notation) results in an equal and opposite torque; this equal and opposite torque arises in their analysis precisely so that the full flow V = V0 + o, which does satisfy the boundary condition ~ • V = 0 at the CMB), gives rise to a zero torque. The torque must be calculated from a flow which satisfies the full boundary condition; no physical significance can be attached to the torque FZ.Vo calculated by Jault and LeMouSl since it is calculated using a flow that does not satisfy the boundary condition. This matter of boundary conditions is the critical factor explaining the difference in these results. A further difficulty in their approach is that the leading order flow is assumed to be steady, but, unlike the steady basic states in the forward modelling approaches referred to earlier, the leading order flow V0 in Eq. (16) varies on a decadal time scale and contributes to the time-variations of the angular momentum variation of the core. Indeed, good agreement is found between the time-variations of the flow V0 determined from the geomagnetic secular variation and the angular momentum of the core inferred from time-variations in the length-ofday (Jault et al., 1988, Jackson et al., 1993). If we estimate that 6 ~ d~'(10-3), which is not unreasonable, then 3pVo/Ot is comparable to the Coriolis force 2 g ~ × p~, and so should be included at this
W. Kuang, J. Bloxham / Physics of the Earth and Planetary Interiors 99 (1997) 289-294
order. Thus Eq. (19) is inadequate for analysing the perturbed flow l,, and hence Eq. (21) is inadequate for analysing the angular momentum of the outer core. It should be noted that this shortcoming in their analysis pertains even under the restrictive set of conditions (electrically insulating mantle and radial gravity) which we have considered so far. In other words, the analysis of Jault and LeMou~l (1989) would incorrectly predict a non-zero torque even if the mantle were an insulator and gravity radial. Let us now consider the effects of relaxing these conditions. First we consider the effect of a non-radial component of the gravitational acceleration. If the gravitational acceleration is not entirely radial then the flow at the core surface will not be tangentially geostrophic, in violation of the underlying assumption for calculating the pressure torque. Tangential geostrophy is violated because pg, which need not be entirely irrotational, will then have a horizontal component which enters the tangential balance. In other words, the tangential balance is no longer entirely irrotational, a result of the fact that Vp × g will not necessarily vanish. Note also that the term which arises, the gravitational body torque I~C= f
r×pgdV
(23)
Vcore
which was first identified by Jault and LeMouSl (1989) represents the effect of the non-radial part of the gravitational acceleration in the core (primarily due to density variations in the mantle) acting on density variations in the core. Separating the body coupling from the topographical coupling due to this term is not straightforward. Next we consider the effect of relaxing the requirement that the mantle is an electrical insulator. We need to separate two contributions: a tangential stress at the CMB (electromagnetic core-mantle coupling in the conventional sense) and a normal stress (magnetotopographical core-mantle coupling resulting from the magnetic pressure). The second of these, since it arises from a pressure gradient, can contribute to the tangentially geostrophic balance at the CMB, and so is of direct concern to this discussion.
293
Comparing these two electromagnetic contributions, we have normal contribution tangential contribution B2
B 23ro/2 Iz o r o BhBr/tX o
t~
----
-I-
~.
BhB , 2 (24)
where B is the typical field strength, B h is the typical horizontal field strength and B r the typical radial field strength at the CMB. Now B h >~ Br, SO the ratio becomes normalcontribution
Bh 8 ~
tangentialcontfibution
B, 2
(25)
C h o o s i n g B r -- 5 x 10-4 T and 8 -,, 10-3, the ratio is small for B h ~ 1 T. Thus the normal stress is
insignificant compared to the tangential stress for reasonable values of B h and the CMB topography amplitude. Thus, if we relax the assumptions of a perfectly insulating mantle and radial gravitational acceleration, then the underlying assumption of tangential geostrophy is violated and the tangentially geostrophic contribution to pressure coupling is small compared to the electromagnetic tangential stress coupling. Finally, in order again to isolate topographical coupling from body coupling and tangential stress coupling, we return to our original set of assumptions; then ask, what are the implications of these results for the mechanism of angular momentum exchange at the CMB? Estimating the part of the pressure gradient Vpt due to the inertial force, i.e. the part that contributes under the set of assumptions introduced earlier, we have IVp, I ~
pU T
(26)
where U is a typical velocity and ~- the characteristic timescale on which the flow varies. Taking U -- 5 × 10 -4 ms- i and ~-~ 30 yr ~ 109 s we have [Vpt[ ~ 5 × 10 -9 Nm -3. With topography 8 ~ 10 -3, the resultant topographical torque is about 200 times smaller than the typical values of 10 Ig Nm required
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W. Kuang, J. Bloxham / Physics of the Earth and Planetary Interiors 99 (1997) 289-294
to explain the observed decadal changes in the length-of-day. One final matter is worthy of some consideration. The demonstration that decadal period variations in the length-of-day are due to the exchange of angular momentum between the core and mantle demands that the inertia of the core is not inconsequential at these periods. This is not inconsistent with the above result which concerns merely whether the pressure gradients associated with the inertial term are large enough to contribute significantly to the required torque. The manner in which the inertia of the core is important is a function of timescale. At the longperiod limit, the core behaves rigidly and is coupled to the mantle. For example, in studies of the exchange of angular momentum between the Earth and the Moon, the moment of inertia of the core is included in the calculation of the moment of inertia of the Earth. On decadal timescales, the core behaves quasi-rigidly, with the angular momentum carried by zonal flows of the form u6 = u6(s); the inertia of each cylindrical annulus must be considered. At very short periods, for example for variations in the length-of-day on diurnal timescales, the core is decoupled from the mantle, and its moment of inertia can be neglected. Between the decadal timescale and the diurnal timescale, the behaviour of the core is more complex, with the zonal flow assuming a z-dependence as angular momentum is transferred from the CMB to points at depth within the core (or vice-versa). A complete theory of the decadal variations in the length-of-day must include the process by which the flow at depth adjusts to maintain, on decadal timescales, the form u6 = u~(s). Whatever the mechanism of core-mantle coupling, the issue of transferring angular momentum from the CMB to points at depth within the core must be addressed. It is not clear that this can be accomplished unless the Lorentz force is significant near the CMB.
4. Conclusions We have shown that Hide's method for calculating the topographical contribution to core-mantle coupling should yield zero if the mantle is an electrical insulator and the gravitational acceleration is
radial. Furthermore, the pressure torque would, if calculated correctly using a more complete theory, prove too small to play a significant role in the exchange of angular momentum between the core and mantle on decadal timescales. Relaxing the requirement that the mantle is an electrical insulator permits electromagnetic normal and tangential stresses at the CMB. However, the tangential stress is dominant, so core-mantle coupling would then be predominantly electromagnetic (in the usual sense) rather than topographic in origin. Relaxing the requirement that the gravitational acceleration is radial violates the assumption of tangentially geostrophic flow at the core surface.
Acknowledgements JB is supported by the Packard Foundation and by an NSF Presidential Young Investigator Award (EAR-9158298). This work was also supported by NSF Award EAR-.
References Anufriev, A. and Braginsky, S. (1975a). Geomag. Aeron., I1: 461-467. Anufriev, A. and Braginsky, S. (1975b). Geomag. Aeron., 15: 754-757. Anufriev, A. and Braginsky, S. (1977a). Geomag. Aeron., 17: 492-496. Anufriev, A. and Braginsky, S. (1977b). Geomag. Aeron., 17: 78-82. Bloxham, J. and Kuang, W. (1995). Geophys. Res. Lett., 22: 3561-3562. Hide, R. (1969). Nature, 222: 1055-1056. Hide, R. (1986). Quart. J. Roy. Astron. Soc., 278: 3-14. Jackson, A., Bloxham, J. and Gubbins, D. (1993), in J.-L. LeMou~l, D. Smylie, and T. Herring (Eds.) Dynamics of the Earth's deep interior and Earth rotation, volume 72 of Geophys. Monog. American Geophysical Union. Jault, D. and LeMoui~l, J.-L. (1989). Geophys. and Astrophys. Fluid Dyn., 48: 273-296. Jault, D., Gire, C. and LeMou~l, J.-L. (1988). Nature, 333: 353-356. Kuang, W. and Bloxham, J. (1993). Geophys. and Astrophys. Fluid Dyn., 72: 161-195. Moffatt, H. (1978). Geophys. and Astrophys. Fluid Dyn., 9: 278-288. Moffatt, H. and Dillon, R. (1976). Phys. Earth Planet Inter., 13: 67-78. Taylor, J. (1963). Proc. Roy. Soc. Lond., A274: 274-283.
ELSEVIER
Physics of the Earth and Planetary Interiors99 (1997) 289-294
On the dynamics of topographical core-mantle coupling Weijia Kuang, Jeremy Bloxham * Department of Earth and Planetary Sciences, Harvard University. Cambridge, MA 02138, USA
Received 19 August 1995; revised 24 May 1996; accepted6 June 1996
Abstract
We examine the dynamics of topographical core-mantle coupling, the process by which angular momentum is exchanged between the Earth's fluid outer core and solid mantle through the action of pressure forces acting on topography on the core-mantle boundary. We show that methods for calculating the torque directly from observations based on the assumption of tangentially geostrophic flow at the core surface should yield a zero torque if the mantle is an electrical insulator and the gravitational acceleration is radial. We discuss the effect of relaxing these assumptions and argue that such methods remain incapable of determining the torque on the core, either due to violation of the underlying assumption of tangential geostrophy or the admission of larger non-topographical torques. 1. Introduction
Topographical core-mantle coupling, first proposed by Hide (1969), is one of the leading candidates for the mechanism of the exchange of angular momentum between the solid mantle and fluid core which results in decadal variations in the length-ofday. The topographical, or pressure, torque F arises from the pressure p associated with flow in the outer core acting on asymmetrical topography on the core-mantle boundary (CMB)
F= fcmBr×pdS=fvc~r ×
VpdV
(1)
where the CMB is given by r = r0[1 + 8 H ( 0 , ~b)]
(2)
with r 0 the mean radius of the CMB, H(O, q~) ~ O(1) and t~ .~z 1.
* Corresponding author.
Previous studies aimed at evaluating topographical coupling have adopted one of two approaches. In the first, a forward modelling approach, simple models of the basic fluid flow and magnetic field are assumed, and the torque that results from the perturbation to the basic state by small amplitude boundary topography (2) is calculated (for example, Anufriev and Braginsky, 1975a,b, 1977a,b, Moffatt and Dillon, 1976, Moffatt, 1978, Kuang and Bloxham, 1993). The pressure torque calculated by this approach is proportional to 8 2 (for long wavelength topography). In the other approach, an inverse modelling approach proposed by Hide (1986), observations are used to deduce the fluid pressure at the CMB. The tangentially geostrophic part of the fluid flow at the core-mantle boundary is determined from the geomagnetic secular variation; then, using the tangentially geostrophic approximation, the horizontal pressure gradient at the CMB is calculated; next, a map of core-mantle boundary topography is determined using seismic observations; and finally, with both the pressure gradient and the CMB topography known,
0031-9201/97/$17.00 Copyright © 1997 Elsevier Science B.V. All fights reserved. PI! S0031-9201 (96)03203-7
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W. Kuang, J. Bloxham / Physics of the Earth and Planetary Interiors 99 (1997) 289-294
the torque is calculated. The pressure torque from this approach is proportional to 6 and is thus much larger than that predicted by the forward modelling. In this paper, we examine these two approaches and demonstrate how they give rise to such seemingly inconsistent conclusions. This is a matter of some importance, not only in studies of the lengthof-day, but also in studies of the dynamics of the lowermost mantle through the constraints that topographical coupling may place on the amplitude ,~ of CMB topography. First we consider a simplified problem. In order to separate topographical coupling from other coupling mechanisms, we assume that the mantle is a perfect electrical insulator, thereby eliminating electromagnetic core-mantle coupling, and we assume that the gravitational acceleration in the outer core is radial, thereby eliminating gravitational core-mantle coupling. Later, we discuss the effect of relaxing these assumptions. We consider a reference frame co-rotating with the solid mantle, with angular velocity g~. Then, the fluid flow u and magnetic field B in the outer core are described by the following equations
(0 u
)
3t + V. puu + 2 ~ X p u = - Vp + j X n + p g + vpV2u
aB at
= VX ( u X B ) + ~IVZB
V'(pu)= 1 j= --VXB /x
V-B=0
(3) (4) (5) (6)
in which p is the density, "q the magnetic diffusivity, the kinematic viscosity, g the gravitational acceleration, j the current density and /x the magnetic permeability. We have assumed that v, "q and /x are constants. The anelastic approximation in Eq. (5) eliminates compressionai waves which are not important in this context. We have also neglected the Poincar~ term in the momentum Eq. (3). This approximation does not introduce qualitative changes because the moment of inertia of the Earth's outer core is about an order of magnitude smaller than that of the solid mantle (Jault and LeMoufil, 1989).
At the CMB we have the boundary condition ~.u=0
(7)
where ~ is a unit vector normal to the CMB.
2. Contributions to the topographical torque The topographical or pressure torque arising from p is given by Eq. (1). Using Eq. (3) we write rX VpdV
F= f
J, Vcore
= -
ir ( °°u v,o,~
3t
)
+ V. puu dV
-2f Vcorep r X ( g ~ X u ) d V +f
rX(JXB)dV gcorc
+f
rXpgdV-Ft+Fc+FB+F
G
Vcore
(8) where the superscript I represents the contribution from the inertial force, c the contribution from the Coriolis force, B the contribution from the Lorentz force and G the contribution from the buoyancy force. We have neglected the contribution from the viscous term which is small. We can see immediately that F C = 0 since we have assumed g is parallel to r. We denote by F. c the axial component of F c i.e.
IC= -2£. fv
pr X ( g~X u) dU
=-ear
p ~ ' r X ( ~ × u ) dV Vcore
= -- 2 , Q f
p(~,Xr).(~,Xu)dV
V~ore
= -2x f
p r s i n O ~ b . ( ~ . × u ) dV
Vcore
= -2, f
psu~dV
(9)
gt ore
where u, is the radial component of the velocity in the cylindrical coordinate system (s, ,;b, z).
W. Kuang, J. Bloxham / Physics of the E.arth and Planetary Interiors 99 (1997) 289-294
CA~
291
Similarly,
CAPT
fvjpOUs dV s 8
:-f,:'sds[-fa:oPou,dA+fcA,sPo"dS] =-f
Sm
sdsf V'(pu) dV=O
SO
(12)
V2 s
where s,. is the maximum distance CAPB
,. =
Fig. 1. Division of the CMB into the surfaces CAPT, CAPB and CAPS by the cylinder with radius s.
Taylor (1963) showed that Fz c vanishes if the CMB is spherical. Here, we extend his result for the case of a nonspherical boundary. We use A s to represent the cylindrical surface within the core at distance s from the rotation axis, and use V~s to represent the region of the outer core inside this cylinder and V2, to represent the region of the outer core outside this cylinder. In particular, we may choose one such cylinder, say As0, which divides the outer core into two regions V l and V2. Then,
rzc= - 2 o f
z)]
Note in the above, that since the CMB is asymmetric, sm varies with longitude th and may not occur precisely at z = 0. In deriving Eq. (I 1) and Eq. (12), we have applied the boundary condition (7), resulting in vanishing surface integrals on CAPT, CAPB and CAPS. Thus we have shown that Fz c = 0. Next, to show that U n = 0, we write
F a = f,11 r × ( j × n ) d V = f rX V.'rndV Vcor, J'vco~
(13) where ¢ n is the Maxwell stress tensor. Then, using Gauss's law and the symmetry of cn, we have
SpousdV Vcore
1" B = - f
r×¢B.dS
(14)
"C MB
Then, applying Gauss's law a second time, We denote by CAPT and CAPB the parts of the CMB enclosed by the intersection of Aso with the CMB, and we denote by CAPS the remainder of the CMB (see Fig. 1). Then,
since j = 0 in the mantle.
fvspou,dV= fS°sdsfA sPou,dA
3. Discussion
l
0
= fo °Sd,[f Oou, dA [ Aso
=foS°sdsf V'(PoU)dV=O Vis
(11)
U n= - f
r×(j×B)dV=O
(15)
Vmantle
We have shown that if the mantle is an electrical insulator and the gravitational acceleration is radial, the only contribution to the axial topographical torque F z comes from the inertial force. Physically, this is not surprising (as has been noted previously by Bloxham and Kuang (1995)) since from Newton's Second Law the torque must be associated with a
292
W. Kuang, J. Bloxham/ Physics of the Earth and Planetary Interiors 99 (1997) 289-294
change in angular momentum, i.e. with the inertial term. It is apparent from this result, if not from these basic physical considerations, that Hide's method (Hide, 1986) for computing the topographical torque should, in the absence of errors in the determination of the flow from the geomagnetic secular variation, yield a zero torque. Jault and LeMouS1 (1989) have also addressed this issue. They use a perturbation approach, and examine the perturbation to the flow due to boundary topography. They expand the flow V in the core V = V0 + o
(16)
in terms of a leading order flow V0 and a perturbation o (or u 2 in their notation). Their leading order flow is tangentially geostrophic and is that which is derived from the geomagnetic secular variation; it satisfies 2p(OX
V0)n = - V n P
~. V0 = O,
(17)
atr=r 0
The perturbation 1, represents the response of Vo to the boundary topography; it satisfies
0(p ) -
-
at
+2aX
= - Vp
(pv)
(19)
atr=r o
~.o=roa(VH).Vo=-g(s,~b,z+_)
(20) where z _ denotes the upper (i.e. northern hemisphere) and lower (i.e. southern hemisphere) parts of the CMB, respectively. They show that the part of o corresponding to the zonal symmetrical part of g (i.e. g = g ( s ) ) is timevarying, and is responsible for the time-variations of the axial angular momentum of the outer core aM= Ot
e fvc
e(p,,) r×
ore
3t
dV=-I"
z
(21) ' ~
where Fz. ~ is, in their terminology, the fictitious Coriolis torque of the flow v arising from the nonhomogeneous boundary condition (20). Since the total flow V satisfies the impenetrable boundary condition h . V = 0 at the CMB (see conditions (18) and (20)), the total torque F z is zero,
r ,vo +
0
where FZ.Vo is the pressure torque of the leading
order flow (17). /'Z.Vo is the pressure torque derived using Hide's method outlined above. Jault and LeMouifl (1989) conclude that the pressure torque Fz,vo indeed changes the axial angular momentum M z of the outer core, since dM. dt
Fz'v = Fz'v°
(22)
How do we explain this apparent contradiction between their analysis and that which we have presented above? In our analysis, the flow is required to satisfy the boundary condition ~ . u at the CMB exactly, in which case the geostrophic contribution to the torque vanishes. Jault and LeMoui~l, on the other hand, consider the boundary condition P- V0 = 0 at r = r 0 for the leading order flow, in which case the geostrophic contribution to the topographical torque does not vanish. As discussed, they then examined the perturbation to the flow by the topography and showed that the perturbation l, (again, u 2 in their notation) results in an equal and opposite torque; this equal and opposite torque arises in their analysis precisely so that the full flow V = V0 + o, which does satisfy the boundary condition ~ • V = 0 at the CMB), gives rise to a zero torque. The torque must be calculated from a flow which satisfies the full boundary condition; no physical significance can be attached to the torque FZ.Vo calculated by Jault and LeMouSl since it is calculated using a flow that does not satisfy the boundary condition. This matter of boundary conditions is the critical factor explaining the difference in these results. A further difficulty in their approach is that the leading order flow is assumed to be steady, but, unlike the steady basic states in the forward modelling approaches referred to earlier, the leading order flow V0 in Eq. (16) varies on a decadal time scale and contributes to the time-variations of the angular momentum variation of the core. Indeed, good agreement is found between the time-variations of the flow V0 determined from the geomagnetic secular variation and the angular momentum of the core inferred from time-variations in the length-ofday (Jault et al., 1988, Jackson et al., 1993). If we estimate that 6 ~ d~'(10-3), which is not unreasonable, then 3pVo/Ot is comparable to the Coriolis force 2 g ~ × p~, and so should be included at this
W. Kuang, J. Bloxham / Physics of the Earth and Planetary Interiors 99 (1997) 289-294
order. Thus Eq. (19) is inadequate for analysing the perturbed flow l,, and hence Eq. (21) is inadequate for analysing the angular momentum of the outer core. It should be noted that this shortcoming in their analysis pertains even under the restrictive set of conditions (electrically insulating mantle and radial gravity) which we have considered so far. In other words, the analysis of Jault and LeMou~l (1989) would incorrectly predict a non-zero torque even if the mantle were an insulator and gravity radial. Let us now consider the effects of relaxing these conditions. First we consider the effect of a non-radial component of the gravitational acceleration. If the gravitational acceleration is not entirely radial then the flow at the core surface will not be tangentially geostrophic, in violation of the underlying assumption for calculating the pressure torque. Tangential geostrophy is violated because pg, which need not be entirely irrotational, will then have a horizontal component which enters the tangential balance. In other words, the tangential balance is no longer entirely irrotational, a result of the fact that Vp × g will not necessarily vanish. Note also that the term which arises, the gravitational body torque I~C= f
r×pgdV
(23)
Vcore
which was first identified by Jault and LeMouSl (1989) represents the effect of the non-radial part of the gravitational acceleration in the core (primarily due to density variations in the mantle) acting on density variations in the core. Separating the body coupling from the topographical coupling due to this term is not straightforward. Next we consider the effect of relaxing the requirement that the mantle is an electrical insulator. We need to separate two contributions: a tangential stress at the CMB (electromagnetic core-mantle coupling in the conventional sense) and a normal stress (magnetotopographical core-mantle coupling resulting from the magnetic pressure). The second of these, since it arises from a pressure gradient, can contribute to the tangentially geostrophic balance at the CMB, and so is of direct concern to this discussion.
293
Comparing these two electromagnetic contributions, we have normal contribution tangential contribution B2
B 23ro/2 Iz o r o BhBr/tX o
t~
----
-I-
~.
BhB , 2 (24)
where B is the typical field strength, B h is the typical horizontal field strength and B r the typical radial field strength at the CMB. Now B h >~ Br, SO the ratio becomes normalcontribution
Bh 8 ~
tangentialcontfibution
B, 2
(25)
C h o o s i n g B r -- 5 x 10-4 T and 8 -,, 10-3, the ratio is small for B h ~ 1 T. Thus the normal stress is
insignificant compared to the tangential stress for reasonable values of B h and the CMB topography amplitude. Thus, if we relax the assumptions of a perfectly insulating mantle and radial gravitational acceleration, then the underlying assumption of tangential geostrophy is violated and the tangentially geostrophic contribution to pressure coupling is small compared to the electromagnetic tangential stress coupling. Finally, in order again to isolate topographical coupling from body coupling and tangential stress coupling, we return to our original set of assumptions; then ask, what are the implications of these results for the mechanism of angular momentum exchange at the CMB? Estimating the part of the pressure gradient Vpt due to the inertial force, i.e. the part that contributes under the set of assumptions introduced earlier, we have IVp, I ~
pU T
(26)
where U is a typical velocity and ~- the characteristic timescale on which the flow varies. Taking U -- 5 × 10 -4 ms- i and ~-~ 30 yr ~ 109 s we have [Vpt[ ~ 5 × 10 -9 Nm -3. With topography 8 ~ 10 -3, the resultant topographical torque is about 200 times smaller than the typical values of 10 Ig Nm required
294
W. Kuang, J. Bloxham / Physics of the Earth and Planetary Interiors 99 (1997) 289-294
to explain the observed decadal changes in the length-of-day. One final matter is worthy of some consideration. The demonstration that decadal period variations in the length-of-day are due to the exchange of angular momentum between the core and mantle demands that the inertia of the core is not inconsequential at these periods. This is not inconsistent with the above result which concerns merely whether the pressure gradients associated with the inertial term are large enough to contribute significantly to the required torque. The manner in which the inertia of the core is important is a function of timescale. At the longperiod limit, the core behaves rigidly and is coupled to the mantle. For example, in studies of the exchange of angular momentum between the Earth and the Moon, the moment of inertia of the core is included in the calculation of the moment of inertia of the Earth. On decadal timescales, the core behaves quasi-rigidly, with the angular momentum carried by zonal flows of the form u6 = u6(s); the inertia of each cylindrical annulus must be considered. At very short periods, for example for variations in the length-of-day on diurnal timescales, the core is decoupled from the mantle, and its moment of inertia can be neglected. Between the decadal timescale and the diurnal timescale, the behaviour of the core is more complex, with the zonal flow assuming a z-dependence as angular momentum is transferred from the CMB to points at depth within the core (or vice-versa). A complete theory of the decadal variations in the length-of-day must include the process by which the flow at depth adjusts to maintain, on decadal timescales, the form u6 = u~(s). Whatever the mechanism of core-mantle coupling, the issue of transferring angular momentum from the CMB to points at depth within the core must be addressed. It is not clear that this can be accomplished unless the Lorentz force is significant near the CMB.
4. Conclusions We have shown that Hide's method for calculating the topographical contribution to core-mantle coupling should yield zero if the mantle is an electrical insulator and the gravitational acceleration is
radial. Furthermore, the pressure torque would, if calculated correctly using a more complete theory, prove too small to play a significant role in the exchange of angular momentum between the core and mantle on decadal timescales. Relaxing the requirement that the mantle is an electrical insulator permits electromagnetic normal and tangential stresses at the CMB. However, the tangential stress is dominant, so core-mantle coupling would then be predominantly electromagnetic (in the usual sense) rather than topographic in origin. Relaxing the requirement that the gravitational acceleration is radial violates the assumption of tangentially geostrophic flow at the core surface.
Acknowledgements JB is supported by the Packard Foundation and by an NSF Presidential Young Investigator Award (EAR-9158298). This work was also supported by NSF Award EAR-.
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