Implication of kinematic dynamo studies for the geodynamo

Abstract

SUMMARY In the kinematic dynamo problem Maxwell's equations are solved for the magnetic field given a prescribed fluid velocity. Although no dynamic equations are involved, it does provide an accurate link between the magnetic field and fluid velocity and can therefore be used to infer something about the flow underlying the observed geomagnetic field. In this sense it complements the commonly used frozen‐flux theory for inverting secular variation for core flow, in which electrical diffusion is neglected, and can be used to show up some of the strengths and weaknesses of the frozen‐flux approximation. It might be thought that kinematic models have been superceded by dynamic models that include the momentum and heat equations, but this is not the case. Even the biggest numerical simulation cannot approach the correct parameters for the Earth's core, and the classes of flows that result are, in fact, quite restricted as well as being too complex for simple physical interpretation. A variety of simple flows have been studied for dynamo action; of particular interest here are a broad class of flows, based loosely on extensions of the early simple choice of Bullard & Gellman and to some extent representative of what might be generated by convection in the Earth's rapidly rotating core: this paper reviews the implications of the solutions for geomagnetism. The non‐axisymmetric flows mimic convection rolls in a rotating sphere, the axisymmetric poloidal flows describe meridional circulation, a likely secondary flow, and the axisymmetric toroidal flow is a simple differential rotation. Helicity, which seems to be important for dynamo action, is related to spiralling of the rolls. Dipole, rather than quadrupole, fields are preferred when spiralling is eastward and differential rotation westward at the surface. Magnetic flux tends to be concentrated at stagnation points of the flow, and dynamo action fails when this concentration becomes so intense that steep gradients develop so as to enhance energy loss by diffusion. On the surface these stagnation points are centres of downwelling that concentrate vertical flux. Flux concentration plays an important role in determining the magnetic field's symmetry: for example, concentration of vertical flux in high latitudes favours dipolar fields while concentration near the equator, where dipole symmetry requires a change of sign, favours quadrupole symmetry. Steady fluid flow can only produce steady or oscillatory solutions to the kinematic problem at onset of instability. Steady solutions are preferred in three dimensions, in contrast to predictions of axisymmetric mean‐field dynamo theory, where oscillatory solutions are the most common. Oscillatory solutions are preferred when poloidal and toroidal field coincide, which is unlikely to happen in the Earth because poloidal field is concentrated at high latitudes around the tangent cylinder while the toroidal field is probably strongest in mid‐latitudes. Geomagnetic reversals are not oscillatory in nature and, therefore, require t...

A dynamo driven by zonal jets at the upper surface: Applications to giant planets

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The optimal kinematic dynamo driven by steady flows in a sphere

“…Helicity is an important measure in fluid dynamics and its significance for dynamo action has been discussed in many places, e.g. review by Moffatt (1983); Gubbins (2008), as well as specific examples given by Livermore et al (2007). While our optimal dynamo does have large helicity for some dominant components, helicity alone cannot explain all the optimal structures we have.…”

“…Early numericists found flow fields that are capable of dynamo action in a sphere with electrically insulating boundary conditions (Backus 1958;Pekeris et al 1973;Kumar & Roberts 1975;Gubbins 1973;Dudley & James 1989). However, there is no universal recipe on how to obtain such flow fields; known dynamo solutions do not necessarily share similar spatial structures (Gubbins 2008).…”

Optimized boundary driven flows for dynamos in a sphere

“…The problem of flow optimization for the kinematic dynamo in spherical geometry was addressed by many researchers [28][29][30][31][32]. The necessity of balancing the relative amplitudes of the toroidal and poloidal flow components for the dynamo action was originally noticed for the Kumar and Roberts flow (KR flow) [3] and later for the Dudley and James flows (DJ flows) [4].…”