Normal modes

Rayleigh was able to take normal mode of the following form (in view of the symmetry of the (16), (17) and the boundary conditions (26),27)

$$w(x,y,z,t) = W(z) f(x,y)e^{\sigma t},\quad \theta(x,y,z,t)=\Theta(z) f(x,y) e^{\sigma t}$$ (28)

with the requirement that

$$\Delta_1 f + a^2 f=0$$ (29)

Here $a$ is called the horizontal wavenumber. In general, a disturbance excites components for each real value of $a$. Now equations (12), (16) and (17) becomes

$$(D^2-a^2-\sigma)\Theta=-W$$ (30)

$$(D^2-a^2)(D^2-a^2-\frac{\sigma}{P_r})W = a^2 R_a \Theta$$ (31)

$$(D^2-a^2)(D^2-a^2-\sigma)(D^2-a^2-\frac{\sigma}{P_r})W = - a^2 R_a W$$ (32)

subject to boundary conditions

$$\left. \begin{array}{c} W = DW = \Theta = 0 \quad \mbox{(rigid)}\\ W=D^2W=\Theta=0\quad \mbox{(free)} \end{array} \right\} z=0,1$$ (33)

which can be written as using (31)

$$\left. \begin{array}{llr} W& = DW = D^4W-\left(2a^2+\frac{\sigma}... ...\quad &\mbox{(rigid)}\\ W&=D^2W=D^4W=0\quad &\mbox{(free)} \end{array} \right\} z=0,1$$ (34)

This gives three conditions at each of the end point for the sixth order equation (32) to determine the countable infinity of eigen values $\sigma_j(a,R_a,P_r)$ and associated eigen functions $W_j (j=1,2,\cdots)$. Let $\sigma=\sigma_r + i \sigma_i$. For a given values of $a,R_a$ and $P_r$, a complete set of solutions $W_j$ satisfying the boundary conditions (BCs) is needed to represnt an arbitrary initial disturbance. These $W_j$ are called modes of the solution. For given $R_a,P_r$, the flow is unstable if $\sigma_r > 0$ for any mode with any real value of $a$ and stable if $\sigma_r \le 0$ for all modes. Hence the critical value of $R_a$ denoted by $R_c$ is such that $\sigma_r(a,R_a,P_r)>0$ for some $a$ whenever $R_a > R_c$ and $\sigma_r(a,R_a,P_r)\le 0$ for all $a$ whenever $R\le R_c$.