In 2012, A. Willis (U. Sheffield) publishes a lovely PRL paper ‘‘The optimized magnetic dynamo’’. Using a variational optimization method, Willis finds the flow that is the most efficient kinematic dynamo within the a huge class of possible candidates, the only restriction being that flow is stationary, incompressible and that it fits in a periodical box. This parameter space is infinitely dimensional, but typically is ~100000 dimensional in pratice, due to the finite numerical resolution that is used.
I was very impressed by this work and I tried to code the algorithm. This worked out fine as I reproduced Willis’ result. During an iGDR Dynamo-meeting at CFS Monte Verita, I proposed A. Jackson (ETH Zurich) to work together on this subject of optimized dynamos. His group has expertise and numerical tools to tackle variational data-assimilation problems in spheres. During L. Chen’s Phd (2014-2017), we have collaborated to find optimal dynamos in confined fluid systems, in cubes and spheres.
The optimization method can also be used to do minimal defect studies. Shear flows can never by kinematic dynamos on their own, but small flow perturbations can trigger a kinematic dynamo. It is possible to measure the smallest possible flow perturbations that can trigger dynamo.
Optimized dynamos in cubes
Using a Taylor-Green basis we found optimal dynamos in impermeable cubes with idealized magnetic boundary conditions (perfect conductor (tangential) or pseudovacuum (normal). Dynamo thresholds were different from Willis’ case and could even be lower when comparing boxes of the same size. Here is a plot that shows the optimal flow streamlines (left) and the magnetic field lines (right) for one particular boundary condition set-up
The flow (left) mainly is composed of a dominant roll around the y-axis but there also is a poloidal flow that allows an inflow. The center of the cube is a stagnation point and the flow is inversion symmetric around this point. The magnetic field (right) is perpendicular to the x and y plates and tangential to the z plates (NNT = Normal, Normal, Tangential) and clearly has a large spatial structure. The critical magnetic Reynolds number for this optimal dynamo is
In a cubical domain, no other stationary flow can have a lower critical magnetic Reynolds number. Note that the magnetic Reynolds number is here defined in an unusual way, with S being the rms shear, L the dimensional box-size and eta the diffusivity. For the more conventional magnetic Reynolds number we measure
with U being the rms speed. This study was published in 2015 in JFM. A preprint may be downloaded here
Optimized dynamos in spheres
In a second study, L. Chen found optimal dynamos in spheres. The use of realistic boundary conditions (no-slip for flow, conitnuity for magnetic field) makes the optimization problem more difficult: variational derivatives used to update flows do not necessarily satisfy the right boundary conditions. The optimization algorithm is implemented using a specifically designed spectral basis functions designed by P. Livermore (U. Leeds) and starting from K. Li’s (ETH Zurich) variational data-assimilation code. Here are the optimal flow (left) and the magnetic eigenmode (right)
The optimal flow (left) has a rotational symmetry (rotation by pi) around an axis and is intense and helical near the center of the sphere. The magnetic field (right) mainly has the structure of an equatorial dipole when seen from the outside. Inside the sphere, the fieldlines wind up along a torus. The critical magnetic Reynolds number for this optimal dynamo is
and L being the radius of the sphere. Using the more convential Rm-definition, we can get as low as
Such low values should not be a real surprise since according M. Proctor (2015) there is no lower bound on this rms-velocity based magnetic Reynolds number. Comparing to all other known dynamos in spheres, our optimal dynamo reaches its onset at critical magnetic Reynolds number that is at least 3 times lower. This work is published in JFM. A preprint may be downloaded here.
Minimal perturbations that trigger dynamos in shear flows
In dynamo theory, we have a multitude of anti-dynamo theorems, forbidding kinematic dynamo action by entire classes of too symmetrical flows. For example, Zel’dovich’s antidynamo theorem rules out kinematic dynamo action by all 1D shear flows. In the kinematic approach, magnetic fields B can then grow transiently but they will always utlimately decay on long times
But how surreal is the hypothesis of a perfectly 1D flow ? Any realistic flow is susceptible to being imperfected by finite amplitude perturbations (small red arrows below) and we can expect that some of them can trigger a sustained growth of the magnetic field at long times.
In this study, I numerically measure the minimal magnitude of dynamo triggering perturbations. This required small modifications of the variational optimization method of Willis. To be able to use the periodic box code, I consider Kolmogorov’s sine flow as a archetypical shear flow:
in non-dimensional form. The magnetic Reynolds number in this study is defined as
with U is the dimensional Kolmogorov flow magnitude, L the periodicity length (or boxsize) and eta the diffusivity. From Zel’dovichs antidynamo theorem, we know that Kolmogorov flow is not a kinematic dynamo for all values of Rm. Using the optimization method, I then find the smallest stationary and solenoidal flow perturbation that can trigger dynamo, measuring its magnitude with
This norm is better adapted than the rms speed norm, since we want to avoid perturbations with infinite shear. We can ilustrate the idea of this study graphically as follows
In the horizontal s-Rm plane, the diagonally barred part is the region where flow perturbations that trigger kinematic dynamo exist. The green arrow illustrates how a magnetic field will be able to grow by a supercritical bifurcation (kinematic dynamo), towards a nonlinear dynamo state that is out of reach in the kinematic approach. Due to the anti-dynamo theorem, the diagonally barred region is disconnected from the s=0 axis. In this minimal perturbation study, I measure the location of the thick red line, in other words how the minimal perturbation magnitude varies as a function of Rm.
Many optimisations later, I have found a scaling law for this minimal pertrubation magnitude
The optimal configurations had surprisingly simple spatial structures (left: mean flow, middle: perturbation flow rms speed, right: field lines for perturbation flow and magnetic field lines).
The perturbation flow (red) mainly is a jet-like structure, transverse to the mean flow. The magnetic field is a ‘‘mean’’ magnetic field, that searches to align with the base flow (left) and that localizes on planes of maximal mean flow shear.
Scaling laws as the one I found here quantify the fragility of the anti-dynamo theorem. To be sure about the affirmation that a flow is not a kinematic dynamo at some large value of magnetic Reynolds number Rm, you need to be
sure that your flow indeed is what you assume it is. To me this means that anti-dynamo theorems are beautiful maths but not relevant physics.
The article was published in 2016 in JFM. A preprint is available here.
Minimal perturbations that trigger mean field dynamos in shear flows
A suprising outcome of the minimal perturbation study, was that the minimal perturbation flow u and the magnetic field B had very simple structures along x (streamwise) and z (spanwise) at high Rm, mainly
for the flow perturbation and
for the magnetic field. This observation motivated me to recast the minimal perturbation study in a reduced functional space. Rather than optimizing the full 3D-structure of the flow and initial magnetic field, I optimize only the y-structure of the 7 fields that appear in the previous truncation. The magnetic field evolution is not constrained by the induction equation but by a second order mean field field dynamo model. Running optimizations in this new model is much more economical. It allows us to confirm the scaling law observed in Herreman (2016)
up to much higher magnetic Reynolds number Rm ~ 1000. This scaling law is clearly visible in the left panel of the following figure
and the new data points align very well with previous ones, suggesting that the mean field model is rather well adapted to capture these optimal dynamos. In the right panel, we replot the data in terms of a new perturbation flow magnetic Reynolds number
where s_d is the dimensional rms enstrophy of the perturbation flow. This suggest a simpler reformulation of the result. At high Rm, it seems that
defines the lower bound for kinematic dynamo action in Kolmogorov flows, perturbed by stationary flow perturbations. Mainly Rm_s decides whether there can be a kinematic dynamo or not in the perturbed Kolmogorov, the precise value of the original Rm is less important, provided that it is high enough.
Since high values of Rm could be reached in the optimizations, it became possible to access the asymptotical, high Rm-structure of the optimal perturbation flow and the magnetic field mode. For the perturbation flow we find the simple expression
Only the magnitude decreases with Rm, but the spatial structure remains mainly the same. The mean and fluctuating parts of the magnetic field mode are
and
with
typical for a critical layer rescaling. All y-profiles can be found in the article. We have succeeded to explain these scalings by analyzing the optimal dynamo mechanism at threshold. It turns out that our optimal dynamos are of alpha-omega type. The feedback loop is organized as follows
A dominant streamwise mean magnetic field interacts with the perturbation flow to generate magnetic fluctuations localized on critical layers. Interaction of these fluctuations with the flow regenerates a small normal mean magnetic field. This field is amplified and rotated by the dominant shear in the Kolmogorov flow, to recreate the original streamwise mean field.
Anyone familar with the problem of subcritical transition to turbulence in shear flows, will notice the striking similarilty between the identified optimal dynamo mechanism and the self-sustaining process. This likely is not a coincidence and is briefly discussed in the article.
Optimising kinematic dynamos in geophysically relevant classes of flow
During the summer of 2019, I realized together with a student (T. Novaes Borges Da Cunha, ENSTA) a new optimised dynamo study. Rather than finding optimal flows within the class of all stationary solenoidal and normalizable (with enstrophy or energy) flow fields, we search for the most effecient general superposition of stationnary geostrophic flows and vertical jets and time-dependent inertial waves.
Physically speaking, inertial waves and stationary geostrophic and vertical flows are much more geophysically relevant than stationary flows, as they exist as a family of weakly interacting linear modes in rapidly rotating flows.
First results of these optimisations suggest that inertial waves never come out as the most efficient dynamos. The optimizer naturally points stationary flows as being way more efficient to generate magnetic fields. I hope to report on this soon enough.
Ideas for future work
I have loved working on this topic and there are many follow-up studies to imagine. Here are a few ideas.
1. We are trying to find out what flow characteristics make the optimal dynamos so efficient. This is not an easy question. We try to manipulate the Euler-Lagrange equations to isolate flow-properties that may be interesting.
2. It would be interesting to extend the minimal perturbation study to the subcritical dynamo problem. What is the smallest initial flow or magnetic field perturbation that can trigger a subcritical dynamo in let’s say shear flows. This question would be of astrophysical interest since it would quantify the fragility of linear stability based arguments that are too often seen in accretion disk context.
3. If we can parametrize particular classes of forcing of flows (impellers, boundary forcing), we may be able to optimize flow forcing rather than the flow. This would then allow to answer the following question: ‘' What is the optimal forcing of a flow that allow to maximize magnetic energy at time T with a normalized kinetic energy input.’’ In other words, how should we drive a flow to get the quickest magnetic field amplification.
