Convection test

→ Working material: Convection/, Convection.tar.gz [untar this file by typing tar zxf Convection.tar.gz]

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). [PDF]

Numerical Experiments homepage

$Date: 2009/09/25 08:18:36 $, $Author: dintrans $, $Revision: 1.4 $