Convection test
→ Working material: Convection/,
Convection.tar.gz
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Aim of this test: to reproduce the critical Rayleigh number Ra_crit in
Gough et al. (1976) for a given density ratio
Z+1 = rho_bot/rho_top=11.
and m=1. Assume dynamical viscosity mu=rho*nu and heat conductivity K to be
constant.
For these values, Gough et al. find a critical Rayleigh number
Ra_crit and critical wave number k_crit of
Ra_crit = 1189, k_crit*Lz = 2.42 ,
where Ra is defined as
[dT/dz - (dT/dz)_ad)]*g*Lz^4
Ra = ------------------------------ ,
T*nu*chi
K
chi = --------- ,
c_p*rho
with respect to the values in the mid-layer (z=-0.6 in our box which
ranges from -1.1 to -0.1). The setup in this directory is such that the
lowest horizontal mode corresponds to the above value of the critical
wave number, and the values of nu and Kbot correspond to a Rayleigh
number Ra = 1189.28.
Resolution nx×nz | Critical Ra |
20×51 | 1130.1 |
20×101 | 1168.6 |
20×201 | 1189.2 |
Note that we don't need more points in x, since the individual Fourier
modes evolve independently as long as the perturbations are small (and the
governing equations for perturbations are linear).
For diagnostics, open e.g. a ipython shell (by typing 'ipython
-pylab') and use the following python subroutines [you can also run
the python subroutines outside an ipython shell by typing directly
'python python/ra.py' etc...]
In [1]: run python/ra # compute the Rayleigh number of the polytrope
In [2]: run python/init # plot the initial setup (density and entropy fields)
In [3]: ts=read_ts(plot_data=False) # read the data/time_series.dat file
In [4]: plot(ts.t, ts.urms) # plot the evolution of urms
In [5]: run python/flux # plot the radiative, convective and kinetic fluxes
In [6]: run python/pvid # animation purpose (velocity field superimposed to the vorticity one)
Reference: D. O. Gough, D. R. Moore, E. A. Spiegel, and N. O. Weiss:
"Convective instability in a compressible atmosphere. II."
Ap.J. 206, 536--542 (1976).
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$Date: 2009-09-25 08:18:36 $, $Author: dintrans $, $Revision: 1.4 $