Derivation of scaled Richardson number and speed of internal gravity waves for density stratified fluid

Question

My questions relate to page 29 of this document. In particular, on page 29, two expressions for the bulk Richardson number are given; one is said to be 'before scaling'. How is the first expression for the bulk Richardson number $Ri_0 = gh/\rho_0$ derived from the unscaled $Ri_0 = g\Delta\rho h/(\rho_0 \Delta U^2)$? I can't seem to find a way of linking the two. Also, how is the phase speed of the internal gravity wave derived as $c = \sqrt{Ri_0/\alpha}$? I can't seem to figure out how this was obtained. Any help with the above questions is much appreciated.

Answer 1

I am not sure but there might be a small error in this derivation that compensates itself. It has been quite some time since I have seen a linear stability analysis so take the following thoughts with a pinch of salt.

The Richardson number is defined as

$$ Ri := \frac{g}{\rho} \frac{\frac{\partial \rho}{\partial z}}{\left( \frac{\partial u}{\partial z} \right)^2} \phantom{spacespace} \frac{\text{buoyancy}}{\text{flow shear}}. \tag{1}\label{1}$$

When approximating the derivatives in \eqref{1} as finite differences (bulk Richardson number) this leads to

$$ Ri = \frac{g}{\rho} \frac{\frac{\Delta \rho}{\Delta z}}{\left( \frac{\Delta u}{\Delta z} \right)^2} = \frac{g}{\rho} \frac{\Delta \rho \Delta z}{\Delta u^2}. \tag{2}\label{2}$$

We transform the velocity to $\overline{U} = \frac{U_1 + U_2}{2}$ and for convenience introduce $\Delta z = h$ (see page 26) for the parameter for velocity transition between 1 and 2 which is assumed linear and of finite thinkness. Thus \eqref{2} can also be written for the point $z = 0$ (thus the index $0$) when considering the intervals I and II as

$$ Ri_0 = \frac{g}{\rho_0} \frac{\Delta \rho h}{\Delta \overline{U}^2}. \tag{3}\label{3} $$

Now we consider a scaled system where the all differences are normalised to the range $\left[ -1, 1 \right]$ such as in the figure on page 29 and therefore all the changes are $1 - (-1) = 2$ ($\Delta U = 2$ and $\Delta \rho = 2$). This allows us to simplify \eqref{3} to

$$ Ri_0 = \frac{g h}{2 \rho_0} \tag{4}\label{4} $$

which is referred to as "after scaling". Contrary to the document I still got a factor of two in there but it eliminates itself later on.

The phase speed is given as (page 5)

$$ c_p = \frac{\omega}{k} \tag{5}\label{5} $$

and the wave number by (page 5)

$$ \omega^2 = \frac{g ( \rho_2 - \rho_1) k}{\rho_1 + \rho_2} \tag{6}\label{6} $$

When transforming \eqref{6} to the scaled system $\rho_{1,2} = \rho_0 \pm 1$ this results in

$$ \omega^2 = \frac{g \overbrace{[\rho_0 + 1 - (\rho_0 - 1)]}^{2} k}{\underbrace{ \rho_0 + 1 + \rho_0 - 1}_{2 \rho_0} } = \frac{g k}{\rho_0} \tag{7}\label{7}$$

and therefore combining \eqref{6} and \eqref{7} $c_p$ is given by

$$ c_p = \frac{1}{k^2} \sqrt{\frac{g k}{\rho_0}} = \sqrt{\underbrace{\frac{g h}{2 \rho_0}}_{Ri_0} \underbrace{\frac{2}{k h}}_{\frac{1}{\alpha}}} = \sqrt{\frac{Ri_0}{\alpha}} \tag{8}\label{8} $$

as (page 26) $ \alpha = \frac{k h}{2}$ holds.