This section describes the mathematically simplest situation where topographic Rossby waves form: a uniform bottom slope.
Shallow water equations A coordinate system is defined with x in eastward direction, y in northward direction and z as the distance from the earth's surface. The coordinates are measured from a certain reference coordinate on the earth's surface with a reference latitude \varphi_0 and a mean reference layer thickness H_0. The derivation begins with the
shallow water equations: \begin{align} {\partial u\over\partial t}&+u{\partial u\over\partial x}+v{\partial u\over\partial y}-f_0v = -g{\partial \eta\over\partial x}\\[3pt] {\partial v\over\partial t}&+u{\partial v\over\partial x}+v{\partial v\over\partial y}+f_0u = -g{\partial \eta\over\partial y}\\[3pt] {\partial \eta\over\partial t}&+{\partial \over\partial x}hu + {\partial \over\partial y}hv=0, \end{align} where In the equation above, friction (
viscous drag and
kinematic viscosity) is neglected. Furthermore, a constant Coriolis parameter is assumed ("f-plane approximation"). The first and the second equation of the shallow water equations are respectively called the zonal and meridional momentum equations, and the third equation is the
continuity equation. The shallow water equations assume a
homogeneous and
barotropic fluid.
Linearization For simplicity, the system is limited by means of a weak and uniform bottom slope that is aligned with the y-axis, which in turn enables a better comparison to the results with planetary Rossby waves. The mean layer thickness H for an undisturbed fluid is then defined as H = H_0 + \alpha_0y \qquad \text{with} \qquad \alpha ={\left\vert \alpha_{0} \right\vert L \over H_0}\ll1, where \alpha_0 is the slope of bottom, \alpha the topographic parameter and L the horizontal
length scale of the motion. The restriction on the topographic parameter guarantees that there is a weak bottom irregularity. The local and instantaneous fluid thickness h can be written as h(x,y,t)=H_0+\alpha_0 y+\eta (x,y,t). Utilizing this expression in the continuity equation of the shallow water equations yields {\partial \eta\over\partial t}+\left(u{\partial \eta \over\partial x}+ v{\partial \eta \over\partial y}\right)+\eta\left({\partial u \over\partial x}+{\partial v \over\partial y}\right)+ (H_0+\alpha_0y)\left({\partial u \over\partial x}+{\partial v \over\partial y}\right)+\alpha_0v=0. The set of equations is made linear to obtain a set of equations that is easier to solve analytically. This is done by assuming a
Rossby number Ro (=
advection / Coriolis force), which is much smaller than the temporal Rossby number RoT (= inertia / Coriolis force). Furthermore, the length scale of \eta \Delta H is assumed to be much smaller than the thickness of the fluid H. Finally, the condition on the topographic parameter is used and the following set of linear equations is obtained: \begin{align} {\partial u\over\partial t}&-f_0v = -g{\partial \eta\over\partial x}\\[3pt] {\partial v\over\partial t}&+f_0u = -g{\partial \eta\over\partial y}\\[3pt] {\partial \eta\over\partial t}&+H_0\left({\partial u \over\partial x} + {\partial v \over\partial y}\right)+\alpha_0v=0. \end{align}
Quasi-geostrophic approximation Next, the quasi-geostrophic approximation Ro, RoT \ll 1 is made, such that \begin{align} u = \bar{u}+\tilde{u} \qquad &\text{with} \qquad \bar{u}=-{g \over f_0}{\partial \eta \over \partial y}\\[3pt] v = \bar{v}+\tilde{v} \qquad &\text{with} \qquad \bar{v}={g \over f_0}{\partial \eta \over \partial x},\\[3pt] \end{align} where \bar{u} and \bar{v} are the geostrophic flow components and \tilde{u} and \tilde{v} are the
ageostrophic flow components with \tilde{u}\ll\bar{u} and \tilde{v}\ll\bar{v}. Substituting these expressions for u and v in the previously acquired set of equations, yields: \begin{align} &-{g \over f_0}{\partial^2 \eta \over \partial y \partial t}+{\partial \tilde{u} \over\partial t}-f_0\tilde{v} = 0\\[3pt] &{g \over f_0}{\partial^2 \eta \over \partial x \partial t}+{\partial \tilde{v} \over\partial t}+f_0\tilde{u} = 0\\[3pt] &{\partial \eta\over\partial t}+H_0\left({\partial \tilde{u} \over\partial x} + {\partial \tilde{v} \over\partial y}\right)+\alpha_0 {g \over f_0}{\partial \eta \over \partial x}+ \alpha_0\tilde{v}=0. \end{align} Neglecting terms where small component terms (\tilde{u}, \tilde{v}, {\partial\over\partial t} and \alpha_0) are multiplied, the expressions obtained are: \begin{align} &\tilde{v} = -{g \over f_0^2}{\partial^2 \eta \over \partial y \partial t}\\[3pt] &\tilde{u} = -{g \over f_0^2}{\partial^2 \eta \over \partial x \partial t}\\[3pt] &{\partial \eta\over\partial t}+H_0\left({\partial \tilde{u} \over\partial x} + {\partial \tilde{v} \over\partial y}\right)+\alpha_0 {g \over f_0}{\partial \eta \over \partial x}=0. \end{align} Substituting the components of the ageostrophic velocity in the continuity equation the following result is obtained: {\partial \eta \over\partial t}-R^2{\partial\over\partial t}\nabla^2 \eta + \alpha_0{g \over f_0}{\partial \eta \over \partial x}=0, in which R, the
Rossby radius of deformation, is defined as R = {\sqrt{gH_0}\over f_0}.
Dispersion relation Taking for \eta a
plane monochromatic wave of the form \eta = A \cos(k_xx+k_yy-\omega t+\phi), with A the
amplitude, k_x and k_y the
wavenumber in x- and y- direction respectively, \omega the
angular frequency of the wave, and \phi a
phase factor, the following
dispersion relation for topographic Rossby waves is obtained: \omega = {\alpha_0 g \over f_0}{k_x \over 1+R^2(k_x^2+k_y^2)}. If there is no bottom slope (\alpha_{0}=0), the expression above yields no waves, but a steady and geostrophic flow. This is the reason why these waves are called topographic Rossby waves. The maximum frequency of the topographic Rossby waves is \left\vert \omega \right\vert _{max} ={ \left\vert \alpha_0 \right\vert g \over 2\left\vert f_0 \right\vert R }, which is attained for k_x = R^{-1} and k_y = 0. If the forcing creates waves with frequencies above this threshold, no Rossby waves are generated. This situation rarely happens, unless \alpha_0 is very small. In all other cases \left\vert \omega \right\vert_{max} exceeds \left\vert f_0 \right\vert and the theory breaks down. The reason for this is that the assumed conditions: \alpha \ll 1 and RoT \ll 1 are no longer valid. The shallow water equations used as a starting point also allow for other types of waves such as
Kelvin waves and inertia-gravity waves (
Poincaré waves). However, these do not appear in the obtained results because of the quasi-geostrophic assumption which is used to obtain this result. In wave dynamics this is called filtering.
Phase speed The
phase speed of the waves along the
isobaths (lines of equal depth, here the x-direction) is c_x = {\omega \over k_x}={\alpha_0g\over f_0}{1 \over 1 + R^2(k_x^2+k_y^2)}, which means that on the northern hemisphere the waves propagate with the shallow side at their right and on the southern hemisphere with the shallow side at their left. The equation of c_x shows that the phase speed varies with wavenumber so the waves are dispersive. The maximum of c_x is \left\vert c_x \right\vert_{max} = {\alpha_0g \over f_0}, which is the speed of very long waves (k_x^2+k_y^2\rightarrow 0). The phase speed in the y-direction is c_y = {\omega \over k_y}={k_x \over k_y}c_x, which means that c_y can have any sign. The phase speed is given by c = {\omega \over k} = {k_x \over k}c_x, from which it can be seen that \left\vert c \right\vert \leq \left\vert c_x \right\vert as \left\vert k \right\vert = \sqrt{k_x^2+k_y^2} \geq \left\vert k_x \right\vert. This implies that the maximum of \left\vert c_x \right\vert is the maximum of \left\vert c \right\vert. == Analogy between topographic and planetary Rossby waves ==