Governing equations

The non-dimensional governing equations are those of a two-dimensional Boussinesq fluid with the topographic b effect caused by the inclined boundaries (Busse,1986).

$\displaystyle\Dinv{P}\DP{}{t} \Dlapla_2 \psi + \Dinv{P}J(\psi, \Dlapla_2 \psi) - \eta \DP{\psi}{x} = R \DP{\theta}{x} + \Dlapla_2 \Dlapla_2 \psi,$ (1)

$\displaystyle\DP{\theta}{t} + J(\psi, \theta) - \DP{\psi}{x} = \Dlapla_2 \theta.$ (2)

y and q are stream function and temperature disturbance. Non-dimensional numbers appearing in the system are the Rayleigh number R, the Prandtl number P, and the parameter of topographic b effect h. Further explanation of these equations is given in Appendix A.

The fixed heat flux condition is given as the thermal boundary condition.

$\begin{displaymath} \DP{\theta}{z} = 0 , \qquad \mbox{at} \quad z=0,1. \end{displaymath}$ (3)

The kinematic and dynamic boundary conditions are impermeable and free-slip.

$\begin{displaymath} \psi = \DP[2]{\psi}{z} = 0, \qquad\mbox{at}\quad z=0,1. \end{displaymath}$ (4)

Governing equations

$\displaystyle\Dinv{P}\DP{}{t} \Dlapla_2 \psi + \Dinv{P}J(\psi, \Dlapla_2 \psi) - \eta \DP{\psi}{x} = R \DP{\theta}{x} + \Dlapla_2 \Dlapla_2 \psi,$	(1)
$\displaystyle\DP{\theta}{t} + J(\psi, \theta) - \DP{\psi}{x} = \Dlapla_2 \theta.$	(2)