bpbrown/kinematic_dynamos

Kinematic dynamos for Roberts flows

Scripts in this repository solve kinematic dynamo problems from Roberts (1972). Here we use the pseudospectral Dedalus framework and search for peak growth rates via solutions to sparse eigenvalue problems (evps).

The scripts in this repo are:

• roberts_flow_peak.py: solve for peak growth rate at fixed lambda. Reproduces growth rates in Roberts (1972)
• roberts_flow_scan_lambda.py: solve for growth curves, scanning in lambda and kx. Reproduces Figure 7 of Roberts (1972)

Also included are:

• roberts_flow.py: solve for fastest growing mode at fixed kx and lambda
• roberts_flow_scan_kx.py: solve for growth curves at fixed lambda, scanning in kx
• roberts_flow_3D_evp.py: solve full 3-D evp; very slow

Usage

Note: currently scripts in this repo must be run on the d3-half-dimensions branch of dedalus.

Growth curves

To compute growth curves at a variety of lambda, run

python roberts_flow_scan_lambda.py

which produces the following figure:

This figure agrees very well, by eye, with the results in Roberts (1972), Figure 7.

Peak growth rates

To compute peak growth rates at fixed lambda, but searching over kx, run:

python roberts_flow_peak.py

To reproduce all reported values in Roberts (1972), run these:

python roberts_flow_peak.py --flow=1
python roberts_flow_peak.py --flow=2
python roberts_flow_peak.py --flow=3
python roberts_flow_peak.py --flow=4 --lambda=1/64 --kx=1.5 --N=32

These yield the folowing values.

• flow 1, lambda=1/8
  • ω_R = 0.17153310520117904, ω_I = 2.445162511133008e-16, at kx = 0.5384618674254258
• flow 2, lambda=1/8
  • ω_R = 0.02489507608928554, ω_I = 0.5090183158721694, at kx = 0.279865912211303
• flow 3, lambda=1/8
  • ω_R = 0.08972691904475676, ω_I = 0.4467167319782522, at kx = 0.27248575594328633
• flow 4, lambda=1/64, N=32
  • ω_R = 0.10714809611885308, ω_I = 3.062963590975353e-17, at kx = 1.6476979951898494

The literature values from Roberts (1972) are (their p => ω, j => kx):

• flow 1, lambda=1/8
  • p = 0.173, j = 0.55
• flow 2, lambda=1/8
  • Re p = 0.025, j = 0.28
• flow 3, lambda=1/8
  • Re p = 0.09, j = 0.27
• flow 4, lambda=1/64, N=32
  • p = 0.10, j = 1.6

All of these agree well, except for flow 1.

There are reasons to suspect that the Roberts (1972) flow 1 peak growth rate is not fully converged in kx, and indeed another source (David Hughes, private correspondence) finds

• flow 1
  • p = 0.17154, kx = 0.54

which agrees well with the values produced by python roberts_flow_peak.py --flow=1.

References

Roberts, G.O., 1972, ``Dynamo action of fluid motions with two-dimensional periodicity'', Philosophical Transactions of the Royal Society of London