Penetrative convection - Parametrized expression for the growth rates

The equations determining the linear growth rate omega characterizing a convectively unstable fluid with Rayleigh number R(u) bounded below by an impenetrable free boundary and above by a convectively stable fluid with Rayleigh number R(s), are solved numerically. Using the analytical Rayleigh-Benard growth rate omega (RB) as a convenient functional form, it is possible to fit the numerical values for omega if the vertical wave number k(z) = n(pi) and the Rayleigh number R(RB) are taken to be functions of R(s), R(u), and the horizontal wave number k-perpendicular rather than n = integer as in the Rayleigh-Benard case. In addition, contrary to Rayleigh-Benard convection, in which the critical Rayleigh number is fixed, it is found that R super (cr) sub u is variable in the presence of a stable layer, (i.e., it depends on R(s)).