The MAGSAT-program has added significantly to our knowledge of planetary magnetism. The accuracy of observations has been improved such that a reliable extrapolation of the magnetic field to the core surface is now much more feasible than it has been before, and the prospect of further MAGSAT missions raises the expectation that the time dependence of the geomagnetic field will be known with similar accuracy in the future. In the research support it has been attempted to develop dynamo theory with these applications in mind

Busse, F. H.

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I,M
FINAL TECHNICAL REPORT
"Planetary Dynamos"
Grant No. NSG-7484
(BaSA-CIA-176001) ELIMPTIBY ITNHOS Final 	 NE5-32f57
Technical Bel-crt (Califcrzia Oriv.) 	 12 p
HC ► 02/MF I01	 CSCL 20C
Ucc la, E
G3/70 21683
Friedrich H. Busse
Principal Investigator
Julv 19, 1985
M AGo
bj
t
TABLE OF CONTENTS
PaSe
Foreword
	
6
1. Generation of magnetic fields by convection in a
rotating sphere
	
7
2. Dvnamo oscillations
	 8
3. A model of nonlinear convection in rapidly rotating
spheres
	 9
4. A semi-analytical model for planetary dynamos
	
10
5. Paper; resulting from research supported by the Grant
	
13
6. Other references
	 14
7. Students supported by the Grant
	
15
6
IV
B
FCC REWARD
The discovery of the magnetic fields of the planets and the detailed
measurement: of the geomagnetic field by space probes has introduced a new
era in the development of dynamo theory.
Only a decade ago, it seemed rather unlikely that the problem of the
origin of geomagnetism could ever be solved with sufficient accuracy to
permit quantitative comparison with the observations. The uncertainty about
the energy source and the genera ' , ck of knowledge about the physical
conditions in the earth's core tended to prohibit a unique determination of
the mechanism of the geodynamo. The discovery of the magtetic fields of
other planets and the recognition of their general similarity with the
geomagnetic field have changed this unsatisfactory state of the theory.
With	 respect to some parameters relevant to the dynamo process, the cores
of the major planets are actually better known than the earth's core. And
the constraints offered by the various realizations of pla^etary magnetism
further restrict the possibilities of different dynamo models. Thus the
theory c;f geomagnetism has become a part of the general theory of planetary
magnetism. The MAGSAT-program has added significantly to our knowledge of
planetary magnetism. The accuracy of observations has been improved such
that a reliable extrapolation of the magnetic field to the core surface is
now much more feasible than it has been before, and the prospect of further
MAGSAT missions raises the expectation that the time dependence of the
geomagnetic field will be known with similar accuracy in the future. In
the research support under NASA Grant NSG 7484 it has been attempted to
develop dynamo theory with these applications in mind.
Because of the three-dimensional nonlinear magnetohydrodynamic equations
which must be solved in spherical geometries and because of the complex
feedback provided by the Lorentz force in the strongly nonlinear regime,
PRECEDTNC PAGE,
	 NOT ^TUTT'7
^I
1
v
2
the development of realistic models of planetary dynamos is anything but 	 1
straightforward. Much of the recent work had to be devoted towards an
exploration of the parameter space in order to distinguish between important
and less important influences and to gain th-! experience necessary for the
construction of more flexible and more cost effective dynamo computer codes.
But the progress made in the past year and the promising results obtained so
far have encouraged us to continue these efforts. In the following we
discuss those areas where we have made progress during the pa_t few years.
1. Generation of magnetic fields by convecticn in a rotating sphere
A major step in understanding planetary dynamos depends on the analysis
of magnetic field generation by convective motions in rotating spheres and
spherical shells. This problem has been attacked in collaboration with
Dr. Cuong and more recently with Dr. Bolton.
Major results presented in the paper by Cuong and Busse (1981) are:
(i) For small amplitudes of convection the magnetic field has quadrupolar
symmetry. For larger amplitudes axisymmetric circulations and differential
r
rotation become important and a field of dipolar symmetry becomes preferred.
The close competition of fields of dipolar and quadrupolar symmetry is in
agreement with the analysis of apherical a 2 -dynamos by Proctor (1977).
(ii) Since convection is most vigorous outside the cylindrical surface
I
touching the equator of an inner spherical boundary, the poloidal field
strength is lower at the poles and higher at the equator than a purely
dipolar field. (iii) A sufficiently high azimuthal wave number (m > 6) of
the convection motion is required for dynamo action. Dynamos appear to
function best with a clear separation between large and small scale
components of the magnetic field. (iv) The increase of magnetic field
strength with increasing Rayleigh number suggests a stable equilibrium
8state for the parameter range that has been investigated.
The work of Cuong and Busse (1981) has been extended, at least as far
as the convection part is concerned, in the Ph.D. thesis of Bolton (1985).
The axisymmetric interaction approximation of the earlier work has baen
dropped, since larger computer memories have become available. Accordingly,
the higher harmonics in the azimuthal direction have been taken into account.
The comparison with the earlier work indicates that the axisymmetric
interaction approximation works well at low amplitudes of convection (as is
to be expected) and at higher values of the dominant azimuthal wavenumber m.
Unfortunately the limited time available for the completion of the
thesis has not allowed Dr. Bolton to complete the solution of the
nonlinear dynamo problem. The attempts to generate equilibrium solutions
for the complete nonlinear magnetohydr.odynamic problem of the generation of
magnetic fields by convection have failed so far. The forward integration 	
.a-
in time, which has also been attempted, has indicated a divergence of the
axisymmetric component of the magnetic field. Further testing of the 	 IA+
magnetic part of the computer code of Bolton appears to be necessary before
it can be decided whether a programming error or numerical instabilities
or physical ins ability of the equilibrium solution are the cause of
convergence failure. This work is currently in progress.
2. Dynamo oscillations
Compared to the problem of thermal convection, the problem of the
magnetohydrodynamic dynamo driven by convection offers even more possibilities
for time dependent states because of the additional time derivative in the
equation of induction. It is thus not surprising that stable steady
equilibrium states are exceptional in dynamo theory even though they are
most accessible to calculations.
grow
I
9
To elucidate some typical features of nonlinear dynamo oscillations the
stability of two cases of magnetic equilibrium states has been studied. The
investigation of the model of magnetic field generation by convection rolls
in the presence of a shear (Busse, 1973) revealed an instability which is
caused by the reduction of the convection amplitude with increasing magnetic
field strength in combination with the property that the efficiency of
dynamo action decreases for high magnetic Reynolds numbers because of flux
expulsion. Thus, an incidental increase of the magnetic field strength
reduces the amplituae of convection which increases the efficiency of the
dynamo process and leads to a further growth of the magnetic field strength. 	 j
The numerical integration of the time-dependent equations for the problem
exhibits nonlinear relaxation oscillations with short-lived peak amplitudes
of the magnetic field, sharp decays, and long periods of slowly growing
magnetic energy in the neighborhood of the equilibrium value (Busse, 1982).
It is not unlikely that reversals could occur if small additional disturbances
F
are introduced.
The other example of an unstable equilibrium state occurs in the annulus
model of a convection driven dynamo if the enhancement of dynamo action owing
to a particular component of the Lorentz force is taken into account (Busse,
1977). Fr r a particular parameter range t: yo unstable steady magnetic states
exist which gives rise to nonlinear oscillations between these states (Busse,
1982).
3. A model of nonlinear convection in rapidly rotating spheres
Numerical computations of convection in rotating spheres are limited
to relative low values of the l'aylor number which is the dimensionless
parameter incorporating the rotation rate. Because of the small scale in
the direction perpendicular to the axis of rotation assumed by the
10
convection columns it is difficult to obtain sufficiently high resolution as
the Taylor number is increased. To overcome this limitation analytical
expressions for zonvection columns have been developed recently (Busse, 1983a,b)
using an expansion of the basic _quations in powers of the Prandtl number.
This expansion is particularly appropriate for planetary cores where the
Prandtl number is certainly less than unity aTLd probably small compared to
',	 r
unity.
In the present form the analysis has first been applied to the atmospheres
of the major planets with the goal of explaining the structure of the
observed mean zonal flows. The success of the model in describing the major 	
1
features of zonal flows on Jupiter and Saturn suggests that it is likely
to give a good description of convection in ilanetary cores if magnetic
effects are taken into account. This is the basic idea of a new approach
to model dynamos in rapidly rotating planetary cores which is discussed in 	
o	 i
more detail in the following section,
4. A semi-analytical model for planetary dynamos 	 t
The success of the analytical solution for convection in a rapidly
rotating internally heated sphere (Busse, 1983a,b, which is referred to in
the following by B83) has motivated us to develop a dynamo model on the basis
of that solution. This model describes explicitly in the numerical scheme
only the large scale magnetic field and the large scale motions, while
analytical expressions are used for the small scale convection velocity field
and the small scale magnetic field. Expressions for the latter have been
obtained from the velocity field derived in B83 since boundary conditions
for the small scale magnetic field can be neglected. At the same time the
small scale component of the Lorentz force acting on the conv-velocity field
has been taken into account as a perturbation in analogy to the buoyancy
%I
r^
11
and viscous forces treated in B83.
The objective of the semianalytical model is to complement the fully
numerical model discussed. Because of the limite,: aarameter space accessible
to the latter and its high cost of computations, the semi-analytical model
offers considerable advantages in exploring the parameter domain relevant
for planetary dynamos. On the other hand the assumptions underlying the
analytical part of the model can be tested by comparison with the fully
numerical model. Such a test will be of special importance when the two
scales are not widely separated and an influence of boundary conditions for
the small scale magnetic field may become noticeable.
In its kinematic aspects the semi-analytical model corresponds to
the method of mean field electrodynamics developed by Steenbeck and Krause
and Radler (1966). But instead of introducing assumptions about the small
scale velocity field, the latter is obtained directly as a solution of the
equations of motion. This method permits the introduction of the Lorentz
force without additional complications. The large scalt :omponent of the
Lorentz force may even exceed the large scale Coriolis force. Only on the
small scale the Lorentz force is taken into account as perturbation which is
realistic in terms of planetary applications. Because of the analytical
i
expression for the small scale velocity field, the small scale Lorentz
force becomes a parameter of the problem. The main advantage of the nea
formulation is that the a ik tensor, which is assumed to be isotropic in
the earlier work, can be calculated from first principles.
In other respects the new model can be regarded as an extension of the
,analytical annulus model of Busse (1975). Owing to the replacement of the
annulus geometry by the realistic configuration of a spherical shell the
magnetohydrodynamic part of the problem becomes much more interesting. The
large scale Lorentz force is no longer automatically balanced by the pressure
P	 12
gradient and the consequences of Taylor's (1963) constraint must be faced
directly. Taylor has pointed out that the component of large scale flow that
corresponds to a differential rotation w(s) k x r is driven most effectively
by the Lorentz force. There k denotes the unit vector in the direction of
the axis of rotation and w(s) is the angular velocity which depends only on
the distance from the axis. The Lorentz force can be balanced only by
viscous forces or Reynolds stresses for this particular component, while
the Coriolis force is available for other compon:nts of large scale flow.
While the problem pointed out by Taylor (1963) is usually regarded as a
severe difficulty of nonlinear dynamo theory (see, for example, Malkus and
Proctor, 1975; Ierley, 1982), the Taylor shear has turned out to be essential
in our new dynamo model. Because the a ik tensor is highly anisotropic in
the asymptotic case of large rotation rates, no steady dynamos have been
found in spherical shells (Busse and Miin, 1979). By taking into account the
Taylor shear, however, Busse and Zhang (1985) have found steady dynamos.
Presently the parameter dependence of these solutions is investigated and
attempts are being made to derive a realistic spatial dependence of the
a ik tensor. This new type of dynamo has been called a2w-dynamo.
v
1
L:4
13
5. Papers resulting from research supported under NASA Grant NSG 7484
Busse, F.H., Some new results on spherical dynamos. Phs. Earth Planet. Int.
20, 152-157, 1979.
Busse, F.H., and S.W. Miin, Spherical dynamos with anisotropic a-effect.
Geophys. Astrophys. Fluid Dyn. 14, 167-181, 1979.
Busse, F.H., Nonlinear dynamo oscillations, Geophys. Astrophys., Fluid Dyn.
21, 129-141, 1982.
Busse, F.H., A model of mean zonal flows in the major planets. Geophys.
As t rophys. Fluid Dyn. 23, 1-3-174, 1983a.	 111
Busse, F.H., Convection Driven Zonal Flows in the M p.jor Planets. PAGEOPH
121, 375-390, 1983b.
Busse, F.H., Recent developments in the dynamo theory of planetary magnetism.
Ann. Rev. Earth Plan. Sci. 11, 241-268, 1983c.
Busse, F.H., Generation of magnetic Yields by thermal Rossby waves in rotating
spherical shells. Invited lecture at Mathematical Geophysics Symposium,
Loen, Norway. Terra Cognita 4, 235, 1984.
Busse, F.H., Motions in the earth's core and core-mantle coupling, p. 377-378
in Landolt-Bornstein, vol. V2. Geophysics of the solid earth, the moon
and the planets. Springer, 1984.
{
Busse, F.H. and K.-K. Zhang, a 2w-dynamo in the Earth's core. Abstract
submitted for presentation. Symposium I 16, IAGA meeting in Prague,
August, 1985.
Cuong, P.G., Thermal convection and magnetic field generation in rotating
spherical shells. Dissertation, University of California, Los Angeles,
1979.
Cuong, P.G. and Busse, F.H., Generation of magnetic field by convection in a
rotating sphere I.	 Phys. Earth Planet. Int. 24, 272-283, 1981.
t
14
6. Other references mentioned in this report
Busse, F.H., J. Fluid Mech. 57, 529-44, 1973.
Busse, F.h., Geophys. J. R. Astr. Soc. 42, 437-459, 1975.
Busse, F.H., J. of Geo phys. 43, 441-452, 1977.
Malkus, W.V. and M.R.E. Proctor, J. Fluid Mech. 67, 417-443, 1975.
Ierley, G.R., Bull. Amer. Phys. Soc. 27, 549, 1982.
Proctor, M.R.E., Geophys. Astrophys. Fluid Dyn. 8, 311-324, 1977.
Taylor, J.B., Proc. Roy. Soc. London A274, 274-283, 1963.
Steenbeck. M., I. Krause, and N.-H. Radler, Z. Naturforschunq 21a. 369-376. 1966
i-1
IM
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15
T. Students Supported by the Grant
P.C. Cuong
	
	 Ph.D. thesis - "Thermal Convection and Magnetic Field
Generation in Rotating Spherical Geometries", UCLA,
1979. Some post doctoral work of Dr. P.C. Cuong
was also supported.
E.W. Bolton
	
	
Ph.D. thesis - "Problems in Nonlinear Convection it
Planar and Spherical Geometries," UCLA, 1985.
vli
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Planetary Dynamos

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