Background and Motivation
Interest in the problem of the generation of the geomagnetic field
has increased significantly in recent years (see for example Fearn, 1995),
with several groups embarked upon, or planning, full numerical attacks
on the problem. A UK collaboration has taken a complementary 2D approach
(Jones et al 1995). The earlier results of Glatzmaier & Roberts
(1995a,b) illustrate the limitations of fully three-dimensional highly
nonlinear calculations. Their generous allocation of Cray time permitted
just one integration (which was completed in sections, spread over a year),
covering only one diffusion time. With such a numerically intensive problem,
there is no opportunity to gain insight into the problem by exploring its
dependence on the controlling parameters. One means of understanding of
the extremely complicated dynamo process is by investigating simpler, more
tractable problems. One such problem is that of magnetic stability, where
the mechanism generating the Earth's field is neglected to concentrate
on the evolution of the field itself. As well as helping to understand
the results of full dynamo calculations, such work should have a direct
bearing on observations since it has been suggested (McFadden & Merrill
1993) that magnetic instabilities are responsible for the observed secular
variation (see for example Courtillot and Le Mouel 1988, Bloxham and Jackson
1992) and reversals of the geomagnetic field (see for example Hoffman 1992).
During the past few years there have been important advances in our knowledge
of the conditions under which magnetic fields become unstable and significant
progress has been made in investigating how such instabilities evolve nonlinearly.
A review of progress is given, for example, in Fearn (1993), and below.
The problem of magnetic field stability is complementary to the (much more widely studied) kinematic dynamo problem The dynamo mechanism converts the mechanical energy of convection into magnetic energy. If the field generated is unstable then the instability extracts energy from the field and might be expected to limit the strength of field attainable. This is an important consideration since the toroidal part of the field in the Earth's core is shielded from us by the insulating mantle and we do not know its strength. [Measurements of the DC electrical potential near the top of the mantle have been extrapolated downward to estimate the toroidal field at the core-mantle boundary (CMB), see for example Lanzerotti et al (1993). This gives a toroidal field strength of a few gauss ( at the CMB), but with a weakly conducting mantle, a low value is to be expected and this result is not inconsistent with a strong toroidal field in the interior of the core (Levy and Pearce, 1991).] There are good reasons for believing the toroidal field may be significantly stronger than the observed (poloidal) part of the field. If so, theoretical considerations may be the only way we have of determining the strength of the major component of the field. Linear theory (Zhang and Fearn 1994, 1995) has shown that a broad class of fields, characteristic of the Earth's field, are unstable when their maximum strength exceeds a value of the order of 60 gauss which is a factor 10 greater than the observed poloidal field but is in line with many estimates of the hidden toroidal field. By contrast, the maximum field strength generated in Glatzmaier and Roberts (1995b) (highly super-critical) calculation was over 500 gauss. To explain the observed features of the geomagnetic field, it is clearly important to understand how a magnetic instability evolves. Below, we give some details of the development of the linear theory and review more recent nonlinear work.
Brief Review of Progress
The importance of differential rotation in the cores of rotating planets
and the consequent omega-effect suggests that planetary dynamos may be
of 'strong field' type, with the azimuthal component of the field
strong compared with the meridional component (see for example Soward 1991).
This motivates the choice of B=B(s,z) 1f
[where (s, f, z) are cylindrical
polar coordinates and 1f is
the unit vector in the f-direction] as a reasonable
model (as simple as possible while retaining the essential physics) for
a planetary magnetic field. Early linear studies of magnetic field stability
in rapidly rotating systems (Acheson, 1972, 1973) considered B =
B(s)
and showed non-axisymmetric perturbations to be more unstable than axisymmetric
ones (for example Michael, 1954). Additionally, the importance of considering
a range of choices of B(s) was emphasised. Earlier work (for
example Malkus 1967) had used B proportional to s for simplicity
but this choice is atypically stable.
Acheson's (1972, 1973) analytic work was non-diffusive and established local conditions on B(s) necessary for ideal instability. Fearn (1983, 1984, 1985, 1988) followed this with a series of numerical studies using a cylindrical geometry. This was consistent with the choice of field, and its relative simplicity permitted the stability problem to be reduced to solving a one-dimensional linear eigenvalue problem. Requiring only modest computing resources, this model has proven to be invaluable in exploring parameter space and investigating many aspects of the problem of magnetic stability (see also Fearn and Weiglhofer 1992a, Fearn and Kuang 1994, McLean and Fearn 1995).
In planetary cores, we are not in the almost perfectly conducting regime familiar in astrophysical applications but rather in a regime where the ohmic decay timescale th is comparable with the natural (slow hydromagnetic) timescale ts of hydromagnetic waves in a rapidly rotating system. (The ratio of these gives the Elsasser number L = th / ts which is the controlling parameter in magnetic stability calculations.) In the perfectly conducting limit (L tends to infinity), two important classes of instability can be identified: ideal (the mode found by Acheson) and resistive. The former exists when diffusive effects are entirely absent (L = infinity) but the latter depends on ohmic diffusion (which gives the system greater freedom by permitting field lines to move relative to the fluid and to reconnect). In both cases, as L is decreased, ohmic diffusion acts more and more strongly to damp any perturbation, until at some L = Lc the system is marginally stable. For L < Lc the system is (linearly) stable. When L = O(Lc ) the there is not a clear distinction between ideal and resistive instabilities (McLean and Fearn 1996).
To establish the relevance of this work to the planetary magnetic field
problem, it was important to determine Lc
and compare this with estimated values of L.
The cylindrical model gave useful qualitative information but the constraint
of the spherical geometry is important in determining realistic values
of Lc . Additionally, the
z-dependence
of the field is known to be destabilising (Fearn and Proctor 1983). Fearn
and Weiglhofer (1991a,b, 1992b) have addressed this, and more recent work
(Zhang and Fearn 1993, 1994, 1995) has firmly established that Lc
= O(10). In the Earth this corresponds to a field strength of about 60
gauss, which is comparable with estimates of the azimuthal field strength.
With the relevance of magnetic instability studies confirmed, a nonlinear study is required to determine how such instabilities influence the field and, in particular, what are the observable consequences that could provide the essential link between theory and observation. Specifically we have in mind the major features of the geomagnetic field ; the secular variation and reversals. We conclude with some details of the nonlinear progress made so far.
In the magnetostrophic limit (Ekman number E tends to 0), the most important nonlinear effect is the differential rotation VG(s) 1f called the geostropic flow. A study currently in progress (Fearn et al 1997) has shown that differential rotation can decrease Lc significantly. This requires that dW/ds (where W = VG/s) is largely negative. When VG is determined dynamically (see for example Fearn 1994), it is found that in some cases it has this property; the nonlinear effect hence reduces Lc giving a subcritical instability.
When E is finite, other nonlinear effects (which are found to have a stabilising effect) are important. Hutcheson and Fearn (1995a,b, 1996, 1997) have integrated the governing equations forward in time in a cylindrical geometry for E=10-3 - 10-4. (Lower values would require considerably more computational resources.) They find no evidence of subcritical instability of the most unstable mode. Geophysical values of E are certainly smaller than 10-4, so at present it is unclear whether subcritical instability can be expected in the core. Hutcheson and Fearn's study has shown an interesting series of bifurcations as Lambda is increased above Lc .
Subcritical instability and transitions between different nonlinear states due to small changes in L may be relevant to changes in the observed field. Future work will pursue this idea using a spherical geometry. Paul Fotheringham is currently working on a study in a spherical geometry where, instead of the magnetic field being prescribed, it is instead generated by prescribed a and w-effects, this allowing a study of the feedback of magnetic instability on the strength of the 'imposed' field.
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