Directory:
    ${PENCIL_HOME}/samples/2d-tests/GMSW1976
SVN Id:
    $Id$
Maintainer:
Added:
    13-Apr-2004
Status:
    succeeds # One of [succeeds|failing since <date>|obsolete| ...]
Recommended resolution:
    128x128x128 (nu=eta=1.3e-3)
Comments:
    Reproduce critical Rayleigh number for 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 .

    At resolution nx×nz=20×201, we find Ra_crit ~ 1189.2 ;
    at            nx×nz=20×101,         Ra_crit ~ 1168.6 ;
    at            nx×nz=20× 51,         Ra_crit ~ 1130.1 .

    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, use the idL command sequence
      idL>  .r ts
      idL>  plot, ts.t, ts.uzrms   ; plot rms of vertical velocity over time
      idL>  plot, ts.t, deriv(ts.t,alog(ts.uzrms))  ; plot growth rate
    for time series, and
      idL>  .r start
      idL>  .r r
      idL>  .r thermo
      idL>  .r pvert        ; vertical profiles
      idL>  .r plot_uu+ss   ; visualize velocity and entropy
      idL>  .r Ra           ; print Rayleigh number
    for looking at the latest snapshot.
      

    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).