benchmark03_rotating_convective_MHD
We increase the complexity of our benchmark by considering the case of magnetic field generation by the flow of an electrically conducting fluid. We employ a double potential decompositions for the (incompressible) velocity field and the magnetic field, which we are both solenoidal. This benchmark thus provides an example for magneto-hydrodynamics (MHD) equations. Results are compared against published results by Stellmach and Hansen, Phys. Rev. E, vol. 70 (2004), which have been recently reproduced by Cooper, Bushby, and Guervilly, Phys. Rev. Fluids vol. 5 (2020). Thereafter, we denote this publication CBG20
Input files for this benchmark are located in the {CORAL_ROOT}/etc/benchmarks/03_convective_dynamo, together with a python routine for the computation of the magnetic Reynolds number and of the Elsasser number, which provide dimensionless measures of the kinetic and magnetic energies, respectively. These diagnostic quantities are defined in equations (8) and (10) in CBG20. A potential formulation of their governing equations (1-5) will be uploaded shortly as a jupyter notebook. Control parameters are choosen identical to run I1 of CBG20.
In this benchmark, we compute the fluid instability that develops when a layer of an electrically conducting fluid is heated from below and cooled from the top in presence of rotation. As a result of this (linear) thermal instability, a turbulent flow appears with a magnetic Reynolds number approaching 100. This value is slightly above the onset for the generation of a magnetic field through the dynamo effect. This secondary instability explains the (comparatively slow) growth of a magnetic field. During the first stage, this magnetic field has a small intensity and does not modify the velocity field substantially, hence the name "kinematic dynamo". Eventually, the magnetic field saturates at a value (measured by the Elsasser number) such that the flow is modified: in this "saturated dynamo" stage, the magnetic Reynolds number is increased compared to the value in absence of magnetic field.
These findings are illustrated in details in CBG20. Here we reproduce the timeseries on their Figure 2, which corresponds to their run I1 in their Table 1. We also illustrate the difference between the kinematic and the saturated phase (their Figure 3).
Prepare a directory for this benchmark, where you copy the content of this folder. The input files are setup for reproducing their run I1.
The coral.paramaters.in file defines the numbers of alias-free modes to (NX,NY,NZ)=(128,128,128), identical to the resolution used in CBG20. This corresponds to a grid-size (NXAA, NYAA, NZAA)=(192,192,192). For reference, the code uses 14Gb of memory on a Haswell workstation, when ran with 20 processes. For long time series, a workstation may be limiting for this benchmark, and a small cluster is perhaps better suited.
The output file coral.timeseries is set to record time series of volume averaged kinetic and magnetic energy. These time series will be dialed into the magnetic Reynolds number and the Elsasser number.
A posteriori, you can compute the Nusselt number using compute_Elsasser_Rm.py. Time series of kinetic and magnetic energy are displayed below. For reference, CBG report Rm = 131 ± 20 and Elsasser = 0.20 ± 0.04 in the saturated phase for kinetic and magnetic energy, respectively (see their table 1). For comparison, these mean values (blue dashed line) and the standard deviations (blue-shaded area) are superimposed to the time series obtained with Coral:
We also illustrate the difference between the kinematic and the saturated phase by representing a horizontal slice of the vertical velocity during the kinematic phase:
In the saturated phase, the flow is much less uniform in space (vertical velocity and magnetic field modulus):
