Exact evaluation of the effect of an arbitrary mean flow in kinematic dynamo theory

Fluid particles can have a mean motion in time, even when the Eulerian mean flow disappears everywhere in space. In the present article, we demonstrate that this phenomenon, known as the Stokes drift, plays an essential role in the problem of magnetic field generation by fluctuation flows (kinematic dynamo) at high Rm. At leading order, the dynamo is generated by the Stokes drift that acts as if it were a mean flow. This result is derived from a mean-field dynamo theory in terms of time averages, which reveals how classical expressions for alpha and beta tensors actually recombine into a single Stokes drift contribution. In a test case, we find fluctuation flows that have a G. O. Roberts flow as Stokes drift and this allows to confront our model to direct integration of the induction equation. We find an excellent quantitative agreement between the prediction of our model and the results of our simulations. We finally apply our Stokes drift model to prove that a broad class of inertial waves in rapidly rotating flows cannot drive a dynamo.

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Kinematic dynamo models

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1972.0074 ◽

1972 ◽

Vol 272 (1230) ◽

pp. 663-698 ◽

Cited By ~ 226

Keyword(s):

Magnetic Fields ◽

Northern Hemisphere ◽

Mean Field ◽

Magnetic Reynolds Number ◽

The State ◽

Ekman Pumping ◽

Mean Flow ◽

Kinematic Dynamo ◽

The Mean ◽

Dynamo Wave

Spherical kinematic dynamo models with axisymmetric magnetic fields are examined, which arise from the mean field electrodynamics of Steenbeck and Krause, and also from the nearly axisymmetric limit of Braginskii. Four main cases are considered: (i) there is no mean flow, but the dynamo is maintained by microscale motions which create a mean electromotive force, ( E ) , proportional to the mean magnetic field, B (the α effect); (ii) in addition to an α effect which creates poloidal mean field from toroidal, a mean toroidal shearing flow (angular velocity w ) is present which creates toroidal mean field from poloidal more efficiently than by the α effect; (iii) in addition to the processes operative in (ii), a mean meridional circulation, m , is present; (iv) ( E ) is produced by a second order inductive process first isolated by Radler. When these processes are sufficiently strong, they can maintain magnetic fields. The resulting situations are known as (i) α 2 dynamos, (ii) αω dynamos, (iii) αω dynamos with meridional circulations, and (iv) Rädler dynamos. Models of each type are considered below, but cases (ii) and (iii) give rise to particularly interesting results. If | m | is sufficiently small, or zero [case (ii)], the most easily excited dynamo is oscillatory and is of dipole type if α ω ′ < 0 in the northern hemisphere (and negative in the southern); here ω ′ denotes the outward gradient of ω . The oscillation resembles a Parker dynamo wave, generated at the poles, absorbed at the equator and always moving towards lower latitudes, as for the butterfly diagrams of sunspots. If α ω ′ > 0 in the northern hemisphere, the direction of wave motion is reversed, and also the quadrupolar solution is more readily excited than the dipolar. If | m | is sufficiently large, and of the right magnitude and sense (which is model dependent), it is found that the dynamo which regenerates most easily is steady. It is of dipolar form if α ω ′ > 0 but quadrupolar if α ω ′ < 0 . These models appear to be relevant to the Earth, where meridional circulations might be provided by, for example, Ekman pumping. Evidence for a remarkable symmetry property is adduced. If m and α ω ′ are reversed everywhere in the state in which the dipole (say) is most readily excited, it is found that the state in which a quadrupole is most easily regenerated is recovered, almost precisely. Moreover, the critical magnetic Reynolds number for each is closely similar. As a corollary, the critical Reynolds numbers for dipolar and quadrupolar solutions of opposite α ω ′ are nearly identical for the α ω dynamo (m = 0).

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