On linear growth rates in thermohaline convection with viscosity variations

In the present paper it is proved that the complex growth rate (where and are real and imaginary parts of p) of an arbitrary oscillatory motions of growing amplitude, neutral or unstable, for thermohaline convection configuration of Veronis type (Veronis, G., J. Mar. Res., 23(1965)1), with the viscosity variations must lie inside a semicircle in the right half of the prpiplane whose centre is at the origin and radius equals. A similar theorem is also proved for thermohaline convection of Stern type (Stern, M.E., Tellus 12(1960)172). Furthermore the above results are uniformly valid for all combinations of rigid and free bounding surfaces. The results obtain herein, in particular, also yield sufficient conditions for the validity of the ‘principle of the exchange of the stabilities’ for the respective configurations.