Scaling of the equations

We use $d$ as length scale, $d^2/k$ as time scale and $T_b-T_u=\beta d$ as the scale for temperature. Let $\boldsymbol {u},p,\theta$ are the nondimensional variables. The linear stability equations become

$$\nabla\cdot \boldsymbol {u}=0$$ (10)

$$\frac{\ensuremath{\partial}\boldsymbol {u}}{\ensuremath{\partial}t} = -\nabla p + R_a P_r \theta \hat{k} + P_r \Delta\boldsymbol {u}$$ (11)

$$\frac{\ensuremath{\partial}\theta}{\ensuremath{\partial}t} - w = \Delta\theta$$ (12)

Here the nondimensional Rayleigh number and Prandtl number are given by

$$R_a = \frac{g \alpha \beta d^4}{\kappa \nu},\quad P_r=\frac{\nu}{k}$$ (13)

The boundary conditions are to be applied on the boundary at $z=0$ and $z=1$. Note that Rayleigh number is positive when $\beta > 0$ and is the ratio of buoyancy force and viscous force. On the other hand, Prandtl number depends on the properties of the fluid: ratio of molecular diffusion due to momentum and heat.

We now eliminate all the dependent variables except $w=u_3$. The curl of equation (11) gives

$$\frac{\ensuremath{\partial}\boldsymbol {\omega}}{\ensuremath{\par... ..._r (\nabla\theta \boldsymbol {\times}\hat{k}) + P_r \Delta\boldsymbol {\omega},$$ (14)

where we $\boldsymbol {\omega}=\nabla\boldsymbol {\times}\boldsymbol {u}$ is the vorticity of the flow. In the above equation we have used the vector identity

$$\nabla\boldsymbol {\times}(\phi \boldsymbol {A}) = \phi (\nabla\b... ...bol {\times}\boldsymbol {A}) + \nabla\phi \boldsymbol {\times} \boldsymbol {A}$$

Taking the curl of equation (14) again and using the vector identities

$$\nabla\boldsymbol {\times}(\nabla\boldsymbol {\times}\boldsymbol {A}) = \nabla(\nabla\boldsymbol {\cdot}\boldsymbol {A}) - \Delta\boldsymbol {A}$$

$$\nabla\boldsymbol {\times}(\boldsymbol {A}\boldsymbol {\times}\bo... ... {A}(\nabla\cdot\boldsymbol {B}) - \boldsymbol {B}(\nabla\cdot\boldsymbol {A})$$

we get

$$\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}\Delta\boldsy... ...{\partial}\theta}{\ensuremath{\partial}z}\right) + P_r \Delta^2\boldsymbol {u},$$ (15)

In particular for the vertical velocity $w$ we have

$$\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}\Delta w = R_a P_r \Delta_1\theta+ P_r \Delta^2 w,$$ (16)

where $\Delta_1=\ensuremath{\partial}^2/\ensuremath{\partial}x^2 + \ensuremath{\partial}^2/\ensuremath{\partial}y^2$ is the horizontal laplacian. This equation can be written is

$$\left(\frac{1}{P_r}\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}-\Delta\right)\Delta w = R_a \Delta_1 \theta$$

Similarly the heat equation (12) can be written as

$$\left(\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}-\Delta\right)\theta = w$$

If we eliminate $\theta$ then we get

$$\left(\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}-\Delta... ...remath{\partial}}{\ensuremath{\partial}t}-\Delta\right)\Delta w = R_a\Delta_1 w$$ (17)

If we eliminate $w$ then we get same equation (17) satisfied by $\theta$.

From the vertical component of $\boldsymbol {\omega}$ we have

$$\frac{\ensuremath{\partial}\omega_3}{\ensuremath{\partial}t}=P_r \Delta \omega_3$$ (18)

from which we can determine the vertical vorticity component. From the definition of $\omega_3=\ensuremath{\partial}v/\ensuremath{\partial}x - \ensuremath{\partial}u/\ensuremath{\partial}y$ and the equation of continuity, we get

$$\Delta_1 u = -\frac{\ensuremath{\partial}^2 w}{\ensuremath{\parti... ...remath{\partial}z}-\frac{\ensuremath{\partial}\omega_3}{\ensuremath{\partial}y}$$ (19)

$$\Delta_1 v = -\frac{\ensuremath{\partial}^2 w}{\ensuremath{\parti... ...remath{\partial}z}+\frac{\ensuremath{\partial}\omega_3}{\ensuremath{\partial}x}$$ (20)

and these equation can be used to find $u$ and $v$.