Appendix A: Governing equations

A Boussinesq fluid is governed by the equations of heat, motion and continuity (e.g. Kimura, 1983):



   $\displaystyle\DP{T}{t}+(\Dvect{v}\cdot\Dgrad)T = \kappa\Dlapla T,$ (14)

   $\displaystyle\DP{\Dvect{v}}{t}+(\Dvect{v}\cdot\Dgrad)\Dvect{v}= - \Dinv{\rho}\Dgrad P + \alpha T \Dvect{g}+ \nu\Dlapla\Dvect{v},$ (15)

   $\displaystyle\Ddiv \Dvect{v}=0.$ (16)

where, T is temperature, $\Dvect{v}=(u,w)$ is velocity, $\rho$ is density, P is pressure, $\alpha$ is the thermal expansion coefficient, $\kappa$ is thermal diffusivity, $\nu$ is kinematic viscosity, $\Dvect{g}=(0,-g)$ is acceleration of gravity.
We introduce the stream function $\psi$ as
$u \equiv -\DP{\psi}{z}, \ \ w \equiv \DP{\psi}{x},$ (17)

Subsituting this into (14) and (15), we obtain



   $\displaystyle\DP{T}{t}+J(\psi,T)=\kappa \Dlapla T,$ (18)

   $\displaystyle\DP{(\Dlapla \psi)}{t}+J(\psi,\Dlapla \psi) = \alpha \DP{T}{x} \Dvect{g}+ \nu\Dlapla (\Dlapla \psi),$ (19)

where J(A,B) is the Jacobian

$J(A,B)=\DP{A}{x}\DP{B}{z}-\DP{A}{z}\DP{B}{x}.$ (20)

The governing equations are non-dimensionalized using the thickness of the fluid layer b as the length scale, the thermal diffusion time $\Ddsty \frac{b^{2}}{\kappa}$ as the time scale, and the temperature difference between the boundaries $\Delta T =T_{1}-T_{0}$ as the temperature scale, yielding,

$& & x=b \tilde{x}, \ z=b \tilde{z},\ \psi = \frac{\kappa}{b^...\ & & t=\frac{b^{2}}{\kappa}\tilde{t},\ T=\Delta T \tilde{T}.$
Here $\tilde{\ }$ denotes a non-dimensional variable. Equations (18) and (19) then become



   $\displaystyle\DP{T}{t} + J(\psi,T)= \Dlapla T,$ (21)

   $\displaystyle\DP{\zeta}{t} + J(\psi,\zeta)= Pr \cdot Ra \DP{T}{x} + Pr \Dlapla \zeta,$ (22)

where, $\zeta = \Dlapla \psi$ is vorticity, $\displaystyle Pr=\frac{\nu}{\kappa}$ is the Prandtl number, and $\displaystyle Ra=\frac{\alpha g \Delta T b^{3}}{\nu \kappa}$ is the Rayleigh number. The subscirpts $\tilde{\ }$ denoting a non-dimensional variable are omitted for simplicity.
A fixed temperature condition is specified at the top and the bottom boundaries:

$T = 0 \ \ {\rm at} \ \ z=0,1.$ (23)

The kinematic and dynamic boundary conditions are impermeable, and free-slip, such that

$w = \tau_{zx} = 0, \ {\rm at}\ \left\{\begin{array}{ll...\ (6+L) \leq x \leq 12 & (流体層上面)\ ,\end{array}\right.$ (24)

$u = \tau_{xz} = 0 \ {\rm at} \ (1-d) \leq z \leq 1 \ ,\ x=6,\ (6+L). \ \ \ (プレート側面)$ (25)

Horizontal boundaries are periodic.
These kinematic and dynamic conditions are described by the voriticity and the stream function. The impermeable condition becomes

$\displaystyle\psi = {\rm constant.}$    (26)

Since the initial condtion is one of no motion and $\psi=0$ , stream function always vanishes at the boundaries,

$\psi=0$ (27)

Since w=0 at the horizontal plane of the boundaries, the free-slip condition becomes
$\tau_{zx} = \DP{u}{z}+\DP{w}{x} = \Ddsty \DP{u}{z}= 0,$
that is,

$\DP[2]{\psi}{z}=0.$ (28)

Further, since
$\DP[2]{\psi}{x}=0,$
at the horizontal plane of the boundaries, we have,

$\zeta = 0.$ (29)

By following the similar procedure for vertical plane of the boundaries, we obtain (29).

		$\displaystyle\DP{T}{t}+(\Dvect{v}\cdot\Dgrad)T = \kappa\Dlapla T,$	(14)
		$\displaystyle\DP{\Dvect{v}}{t}+(\Dvect{v}\cdot\Dgrad)\Dvect{v}= - \Dinv{\rho}\Dgrad P + \alpha T \Dvect{g}+ \nu\Dlapla\Dvect{v},$	(15)
		$\displaystyle\Ddiv \Dvect{v}=0.$	(16)

		$\displaystyle\DP{T}{t}+J(\psi,T)=\kappa \Dlapla T,$	(18)
		$\displaystyle\DP{(\Dlapla \psi)}{t}+J(\psi,\Dlapla \psi) = \alpha \DP{T}{x} \Dvect{g}+ \nu\Dlapla (\Dlapla \psi),$	(19)

		$\displaystyle\DP{T}{t} + J(\psi,T)= \Dlapla T,$	(21)
		$\displaystyle\DP{\zeta}{t} + J(\psi,\zeta)= Pr \cdot Ra \DP{T}{x} + Pr \Dlapla \zeta,$	(22)