Upper Bounds for the Complex Growth Rate of a Disturbance in Ferrothermohaline Convection

It is proved analytically that the complex growth rate σ= σ_r+iσ_i (σ_r and σ_i are the real and imaginary parts of σ, respectively) of an arbitrary oscillatory motion of neutral or growing amplitude in ferrothermohaline convection in a ferrofluid layer for the case of free boundaries is located inside a semicircle in the right half of the σ_rσ_i-plane, whose center is at the origin and radius = Rs[1−M1′(1−1M5)]Pr′, {\rm{radius}}\, = \,\sqrt {{{{R_s}\left[{1 - M_1^{'}\left({1 - {1 \over {{M_5}}}} \right)} \right]} \over {P_r^{'}}}}, where R_s is the concentration Rayleigh number, P_r^′ is the solutal Prandtl number, M₁^′ is the ratio of magnetic flux due to concentration fluctuation to the gravitational force, and M₅ is the ratio of concentration effect on magnetic field to pyromagnetic coefficient. Further, bounds for the case of rigid boundaries are also derived separately.