📂Fluid Mechanics

Matsuno-Gill Model

Model^{1 2 3}

Assume an ocean in the vicinity of the equator. Consider a layer of incompressible fluid with mean height $H$ above the surface. Let $\eta$ be the displacement from that height, and let the zonal and meridional flows be $u$ and $v$ respectively. Then the governing equations for the flow velocity can be approximated as follows. $$ \begin{align*} {\frac{ \partial h }{ \partial t }} + H \left( {\frac{ \partial u }{ \partial x }} + {\frac{ \partial v }{ \partial y }} \right) =& 0 \\ {\frac{ \partial u }{ \partial t }} + u {\frac{ \partial u }{ \partial x }} + v {\frac{ \partial u }{ \partial y }} - \beta y v =& -g {\frac{ \partial h }{ \partial x }} \\ {\frac{ \partial v }{ \partial t }} + u {\frac{ \partial v }{ \partial x }} + v {\frac{ \partial v }{ \partial y }} + \beta y u =& -g {\frac{ \partial h }{ \partial y }} \end{align*} $$ Here $g$ is the gravitational acceleration, and $\beta$ is a constant called the Rossby parameter.

Explanation¹

In the shallow water equations, the Coriolis frequency $f$ can be expressed in terms of the latitude $\theta$ as $f = 2 \Omega \sin \theta$. For the radius $R$ of the celestial body and a sufficiently small latitude $\theta$, $y$ can be expressed as $$ y = R \sin \theta \approx R \theta $$ and the point is that $\theta$ can be linearized with respect to $y$. Meanwhile, by the Taylor expansion of the sine function, $\sin \theta$ admits the following approximation for some $f_{0}$ and $\beta$. $$ f \approx f_{0} + \beta y $$

What should be emphasized once again here is that the latitude $\theta$ is small, in other words, that we are assuming the vicinity of the equator. This approximation is also called the $\beta$-plane approximation. Exactly on the equator the Coriolis force is $0$, so theoretically $f_{0} = 0$ and the expression reduces neatly to $f = \beta y$.