Topographic Rossby waves

Figure 1: Animation of propagating Rossby wave.

Topographic Rossby waves are one of the two types of Rossby waves. Rossby waves can be due to the changing Coriolis parameter over the earth, which are called planetary Rossby waves, or due to bottom irregularities, which are topographic Rossby waves. Bottom irregularities can exist on the oceanic floor such as the mid-ocean ridge for ocean dynamics. For atmospheric dynamics bottom irregularities on land can exist in the form of, for example, mountains. Rossby waves are quasi-geostrophic, dispersive waves (Figure 1). This means that not only the Coriolis force and the pressure-gradient force influence the flow as in geostrophic flow, but also inertia.

Physical derivation

We define a coordinate system with x in eastward direction, y in northward direction and z the distance from the earth 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 $H_{0}$. We start the derivation with the shallow water equations:

${\begin{aligned}{\partial u \over \partial t}&+u{\partial u \over \partial x}+v{\partial u \over \partial y}-f_{0}v=-g{\partial \eta \over \partial x}\\[3pt]{\partial v \over \partial t}&+u{\partial v \over \partial x}+v{\partial v \over \partial y}+f_{0}u=-g{\partial \eta \over \partial y}\\[3pt]{\partial \eta \over \partial t}&+{\partial \over \partial x}hu+{\partial \over \partial y}hv=0\end{aligned}}$

where

$u$	is the velocity in the x direction, or zonal velocity
$v$	is the velocity in the y direction, or meridional velocity
$h$	is the local and instantaneous fluid layer thickness
$\eta$	is the height deviation of the fluid from its mean height
$g$	is the acceleration due to gravity
$f_{0}$	is the Coriolis parameter at the reference coordinate with $f_{0}=2\Omega \sin(\varphi _{0})$, where $\Omega$ is the angular frequency of the Earth and $\varphi _{0}$ is the reference latitude

and where friction (viscous drag and kinematic viscosity) is neglected and a constant Coriolis parameter is assumed. 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.

Uniform bottom slope example

For simplicity, the system is limited by means of a weak and uniform bottom slope which we align 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

Figure 2: A layer of homogeneous fluid over a sloping bottom and the attending notation.

$H=H_{0}+\alpha _{0}y\qquad {\text{with}}\qquad \alpha ={\left\vert \alpha _{0}\right\vert L \over H_{0}}\ll 1$

where $\alpha _{0}$ is the slope of bottom, $\alpha$ the topographic parameter and $L$ the horizontal length scale of the motion. The condition on the topographic parameter guarantees that there is a weak bottom irregularity, and the local and instantaneous fluid thickness h can be written as (Figure 2)

$h(x,y,t)=H_{0}+\alpha _{0}y+\eta (x,y,t)$.

Utilizing this expression in the continuity equation yields

${\partial \eta \over \partial t}+\left(u{\partial \eta \over \partial x}+v{\partial \eta \over \partial y}\right)+\eta \left({\partial u \over \partial x}+{\partial v \over \partial y}\right)+(H_{0}+\alpha _{0}y)\left({\partial u \over \partial x}+{\partial v \over \partial y}\right)+\alpha _{0}v=0$.

By neglecting the nonlinear terms through assuming a Rossby number Ro (= advection / Coriolis force), which is much smaller than the temporal Rossby number Ro_T (= inertia / Coriolis force), assuming that the length scale of η ΔH $\ll$ H, and finally using the condition on the topographic parameter, the following set of linear equations is obtained:

${\begin{aligned}{\partial u \over \partial t}&-f_{0}v=-g{\partial \eta \over \partial x}\\[3pt]{\partial v \over \partial t}&+f_{0}u=-g{\partial \eta \over \partial y}\\[3pt]{\partial \eta \over \partial t}&+H_{0}\left({\partial u \over \partial x}+{\partial v \over \partial y}\right)+\alpha _{0}v=0\end{aligned}}$

Now taking the quasi-geostrophic approximation: Ro, Ro_T $\ll$ 1 such that

${\begin{aligned}u={\bar {u}}+{\tilde {u}}\qquad &{\text{with}}\qquad {\bar {u}}=-{g \over f_{0}}{\partial \eta \over \partial y}\\[3pt]v={\bar {v}}+{\tilde {v}}\qquad &{\text{with}}\qquad {\bar {v}}={g \over f_{0}}{\partial \eta \over \partial x}\\[3pt]\end{aligned}}$

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 ${\tilde {v}}\ll {\bar {v}}$, we substitute these expressions for $u$ and $v$ to obtain

${\begin{aligned}&-{g \over f_{0}}{\partial ^{2}\eta \over \partial y\partial t}+{\partial {\tilde {u}} \over \partial t}-f_{0}{\tilde {v}}=0\\[3pt]&{g \over f_{0}}{\partial ^{2}\eta \over \partial x\partial t}+{\partial {\tilde {v}} \over \partial t}+f_{0}{\tilde {u}}=0\\[3pt]&{\partial \eta \over \partial t}+H_{0}\left({\partial {\tilde {u}} \over \partial x}+{\partial {\tilde {v}} \over \partial y}\right)+\alpha _{0}{g \over f_{0}}{\partial \eta \over \partial x}+\alpha _{0}{\tilde {v}}=0.\end{aligned}}$

Neglecting terms where small component terms are multiplied, the expressions obtained are:

${\begin{aligned}&{\tilde {v}}=-{g \over f_{0}^{2}}{\partial ^{2}\eta \over \partial y\partial t}\\[3pt]&{\tilde {u}}=-{g \over f_{0}^{2}}{\partial ^{2}\eta \over \partial x\partial t}\\[3pt]&{\partial \eta \over \partial t}+H_{0}\left({\partial {\tilde {u}} \over \partial x}+{\partial {\tilde {v}} \over \partial y}\right)+\alpha _{0}{g \over f_{0}}{\partial \eta \over \partial x}=0\end{aligned}}$

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

$R={{\sqrt {gH_{0}}} \over f_{0}}$.

Figure 3: Dispersion relation of topographic Rossby waves.

Taking for $\eta$ a plane monochromatic wave of the form

$\eta =A\cos(k_{x}x+k_{y}y-\omega t+\phi )$

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 (Figure 3):

$\omega ={\alpha _{0}g \over f_{0}}{k_{x} \over 1+R^{2}(k_{x}^{2}+k_{y}^{2})}$.

If there is no bottom slope ($\alpha _{0}=0$) this yields no waves but a steady and geostrophic flow, which 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, because $\left\vert \omega \right\vert _{max}$ exceeds $\left\vert f_{0}\right\vert$ and the theory breaks down because the assumed conditions: $\alpha \ll 1$ and Ro_T $\ll 1$ 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), but these do not appear here because of the quasi-geostrophic assumption which is used to obtain this result. In wave dynamics this is called filtering.

Figure 4: The propagation speed of topographic Rossby waves in x-direction on the northern hemisphere.

The phase speed of the waves along the isobaths (lines of equal depth, here the x-direction) is

$c_{x}={\omega \over k_{x}}={\alpha _{0}g \over f_{0}}{1 \over 1+R^{2}(k_{x}^{2}+k_{y}^{2})}$

which means that on the northern hemisphere the waves propagate with the shallow side at their right (Figure 4) and on the southern hemisphere with the shallow side at their left. We can see in the formula of $c_{x}$that it varies with wavenumber so the waves are dispersive. The maximum of $c_{x}$ is

$\left\vert c_{x}\right\vert _{max}={\alpha _{0}g \over f_{0}}$,

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

Planetary and topographic Rossby waves are the same in the sense that, if you exchange ${\alpha _{0}g/f_{0}}$ for $-\beta _{0}R^{2}$ in the expressions above, where $\beta _{0}$ is the beta-parameter or Rossby parameter, the expression of planetary Rossby waves is obtained. This means that if we want to do experiments to research geophysical fluid dynamics we can simulate the beta effect, the changing of the Coriolis parameter over the earth, with a tank with a sloping bottom. The reason for this similarity is that for the nonlinear shallow water equations for a frictionless, homogeneous flow the potential vorticity q is conserved:

${dq \over dt}=0\qquad {\text{with}}\qquad q={\zeta +f \over h}$

with $\zeta$ the relative vorticity, which is twice the rotation speed of fluid elements about the z-axis, and is mathematically defined as

$\zeta ={\partial v \over \partial x}-{\partial u \over \partial y}$,

with $\zeta >0$ an anticlockwise rotation about the z-axis. On a beta-plane and for a linearly sloping bottom in the meridional direction, the potential vorticity becomes

$q={f_{0}+\beta _{0}y+\zeta \over H_{0}+\alpha _{0}y+\eta }$.

In the derivations above it was assumed that

${\begin{aligned}\beta _{0}L\ll \left\vert f_{0}\right\vert &{\text{ : small planetary number }}\beta \\[3pt][\zeta ]\ll \left\vert f_{0}\right\vert &{\text{ : small Rossby number Ro}}\\[3pt]\alpha _{0}L\ll H_{0}&{\text{ : small topographic parameter }}\alpha \\[3pt]\Delta H\ll H&{\text{ : linear dynamics}},\end{aligned}}$

so $q={f_{0}\left(1+{\beta _{0}y \over f_{0}}+{\zeta \over f_{0}}\right) \over H_{0}\left(1+{\alpha _{0}y \over H_{0}}+{\eta \over H_{0}}\right)}={f_{0} \over H_{0}}\left(1+{\beta _{0}y \over f_{0}}+{\zeta \over f_{0}}\right)\left(1-{\alpha _{0}y \over H_{0}}+{\eta \over H_{0}}-\ldots \right)$ (where a Taylor expansion was used on the denominator). Only keeping the largest terms and neglecting the rest, the following result is obtained:

$q\approx {f_{0} \over H_{0}}\left(1+{\beta _{0}y \over f_{0}}+{\zeta \over f_{0}}-{\alpha _{0}y \over H_{0}}-{\eta \over H_{0}}\right).$

So the analogy that appears in potential vorticity is that $\beta _{0}/f_{0}$ and $-\alpha _{0}/H_{0}$ play the same role in the potential vorticity equation. Rewriting these terms a bit differently, this boils down to the earlier seen $-\beta _{0}R^{2}$ and ${\alpha _{0}g/f_{0}}$, which demonstrates the link between planetary and topographic Rossby waves. In the equation for potential vorticity we see that planetary and topographic Rossby waves exsist because of a background gradient in potential vorticity.

Conceptual explanation

Figure 5: The physical mechanisms that propel topographic waves. Displaced fluid parcels react to their new location by developing either clockwise or counterclockwise vorticity. Intermediate parcels are entrained by neighboring vortices, and the wave progresses forward.

The waves are formed because potential vorticity must be conserved. The conservation of potential vorticity is very similar to the conservation of angular momentum. An example of the conservation of angular momentum is a figure skater who makes a pirouette. When the figure skater pulls his/her arms in, the figure skater will get a higher angular velocity. The opposite is also true: extending the arms slows the figure skater down.

When the surface has a slope, the thickness of the fluid layer h is not constant. The conservation of the potential vorticity makes that the relative vorticity ζ or the Coriolis parameter f should change too. Because the Coriolis parameter is constant at a given latitude, the relative vorticity must change. In figure 5 you can see that when a fluid moves to a shallower environment, where h is smaller, the fluid forms a crest. Because the height is smaller, the relative vorticity must also be smaller. In the figure this becomes a negative relative vorticity (on the northern hemisphere a clockwise spin) shown with the rounded arrows. On the southern hemisphere this is an anticlockwise spin, because the Coriolis parameter is negative on the southern hemisphere. If a fluid moves to a deeper environment, the opposite is true. The fluid parcel on the original depth is sandwiched between two fluid parcels with one of them having a positive relative vorticity and the other one a negative relative vorticity. This causes a movement of the fluid parcel to left in the figure. In general the displacement causes a wave pattern that propagates with the shallower side on the right on the northern hemisphere and to the left on the southern hemisphere.

Measurements of topographic Rossby waves on earth

From 1 jan 1965 till 1 jan 1968, the The Buoy Project at the Woods Hole Oceanographic Institution let in buoys on the western side of the Northern Atlantic to measure the velocities. The data has several gaps because some of the buoys went missing. Still they managed to measure topographic Rossby waves at 500 meters deep . Several other research projects have confirmed that there are indeed topographic Rossby waves in the Northern Atlantic .

In 1988, barotropic planetary Rossby waves were found in the Northwest Pacific basin . In 2017, more research was done and they concluded that the Rossby waves are no planetary Rossby waves, but topographic Rossby waves .

In 2021 research in the South China Sea confirmed that there are topographic Rossby waves .