At the end of my Phd, I wanted to find out if inertial waves in rotating flows could drive dynamos. In his book on dynamo theory, K.Moffatt provides a statistical description of mean field action by random superpositions of inertial waves, but the kinematic dynamo action by simple inertial waves had not been studied in much detail before (a short contribution in 2000 by M. Rieutord in Dynamo and Dynamics: a Mathematical Challenge, NATO Science Series, II (26)).
With analytical expressions for inertial wave profiles available in several fluid domains (periodic boxes, plane layers of fluid, cylinders and spheres) all what is required is a numerical code that solves the induction equation. This defined the subject for my post-doctoral stay (CNES grant) at LERMA-ENS Paris, in E. Dormy’s group.
During this stay, I programmed two solvers for kinematic dynamo problems in plane rotating fluid layers and in cylinders. Plugging in inertial wave profiles, the verdict was simple : no kinematic dynamo action with simple waves up to high magnetic Reynolds numbers. This negative result motivated me to try to understand what made these dynamos so bad.
Inertial waves are very rapidly oscillating, typically at frequencies of the order of the background rotation frequency. This time-scale is much faster than the dynamo time-scales at which we expect magnetic fields to vary or grow. We speak of a time-scale separation and this information can be used to formulate a trustworthy mean field dynamo model based on time-averages alone. Spatial averages are no option here, since there is no reason to suspect a space-scale separation.
For simple wave-like flows, it is possible to analytically compute the spatially dependent alpha and beta tensors in the high magnetic Reynolds number limit. This allowed me to understand why inertial wave dynamos were not efficient: decisive couplings were lacking in the mean field tensors.
But in making the model, I also came across a reduction of alpha and beta tensors, that I really did not expect. It turned out that alpha and beta effects recombined into a single physically more appealing effect: rapid waves can drive dynamos through their Stokes drift
Stokes drift
The Stokes drift phenomenon is very classical and typically first encountered by students when analyzing gravity waves on the surface of a liquid. Under a propagative surface wave (black), the flow (in red) decays exponentially in amplitude
Particle tracers released in this oscillating flow, follow gyring paths: a rapid elliptical rotation combined with a slow mean displacement. This due to the fact that along an elliptical path, a particle sees a varying wave-amplitude. On each turn, this results in a mean displacement, a mean drift velocity in the direction of propagation of the wave (blue).
We can expect the Stokes drift phenomenon in any wave-like flow that has spatial inhomogeneity in its amplitude or phase and it is the perfect illustration of how Eulerian averages are different from Lagrangian averages.
Stokes drift dynamos
The Stokes drift associated with a wave can itself be spatially inhomogeneous. Two particles, initially nearby can then drift as in the following picture
An inhomogeneous drift will not only displace particles, but can also separate them. Material lines will be displaced and stretched on average just as indicated by the Stokes drift.
With this information we are not so far from understanding the physical mechanism that is operating in dynamos driven by fast waves. Fast waves are only fast when they vary so quickly that magnetic fields cannot change significantly change on that short timescale. The magnetic field then has no time to diffuse and so it remains glued to the moving fluid particles as in the well known frozen flux limit. Initial magnetic fields will then by displaced and stretched (amplified) just as much as the line-elements are in the picture above.
More precisely speaking, the kinematic dynamo problem for rapid waves
can be reasonably well approximated by the kinematic dynamo problem
in which the wave-like flow is replaced by the stationary Stokes drift flow calculated as
Here we use the spatial profiles of the waves u and omega is the frequency of the waves. To apply the model, it is necessary to verify that the waves are indeed fast, the period, the advective timescale and the diffusive timescale need order as
for the model to have a chance. Here U is the typical dimensional wave-magnitude, L the typical wavelength and eta the diffusivity.
The model extends to superpositions of waves and to waves accompanied by small mean flows. We have tested the model by designing artificial wave-like flows that had the famous G.O.Roberts flow as Stokes drift. Magnetic fields produced by the rapid waves were successfully compared with magnetic fields produced by the G.O.Roberts dynamo.
On the left, you can see that a particle displaced by the wave-like flow indeed drifts around Robert’s flow streamlines. In the middle, we see typical magnetic energy density plots in the plane for a G.O.Roberts dynamo. On the right, we see the magnetic eigenmode that was driven by the waves. For the growth rates of the kinematic dynamos we get excellent agreement.
The Stokes drift model was applied to the case of simple inertial waves. We find that the drift associated with such waves is absent or has a too simple spatial structure (azimuthal circulation in cylinders and spheres) and so it cannot act as a Stokes drift dynamo due to the toroidal anti-dynamo theorem. This result does not exclude dynamo action by simple inertial waves, but it does tell us that large growth rates and low dynamo thresholds should not be expected.
