Topographic Rossby waves

Topographic Rossby waves are

.

Physical derivation

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 ${\displaystyle \varphi _{0}}$ and a mean reference layer thickness ${\displaystyle H_{0}}$. The derivation begins with the shallow water equations:

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

where

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

In the equation above, friction (

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 ${\displaystyle H}$ for an undisturbed fluid is then defined as

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

where ${\displaystyle \alpha _{0}}$ is the slope of bottom, ${\displaystyle \alpha }$ the topographic parameter and ${\displaystyle L}$ 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 ${\displaystyle h}$ can be written as

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

Utilizing this expression in the continuity equation of the shallow water equations yields

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

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 Ro_T (= inertia / Coriolis force). Furthermore, the length scale of ${\displaystyle \eta }$ ${\displaystyle \Delta H}$ is assumed to be much smaller than the thickness of the fluid ${\displaystyle H}$. Finally, the condition on the topographic parameter is used and the following set of linear equations is obtained:

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

Quasi-geostrophic approximation

Next, the quasi-geostrophic approximation Ro, Ro_T ${\displaystyle \ll }$ 1 is made, such that

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

where ${\displaystyle {\bar {u}}}$ and ${\displaystyle {\bar {v}}}$ are the geostrophic flow components and ${\displaystyle {\tilde {u}}}$ and ${\displaystyle {\tilde {v}}}$ are the ageostrophic flow components with ${\displaystyle {\tilde {u}}\ll {\bar {u}}}$ and ${\displaystyle {\tilde {v}}\ll {\bar {v}}}$. Substituting these expressions for ${\displaystyle u}$ and ${\displaystyle v}$ in the previously acquired set of equations, yields:

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

Neglecting terms where small component terms (${\displaystyle {\tilde {u}},{\tilde {v}},{\partial \over \partial t}}$ and ${\displaystyle \alpha _{0}}$) are multiplied, the expressions obtained are:

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

Substituting the components of the ageostrophic velocity in the continuity equation the following result is obtained:

${\displaystyle {\partial \eta \over \partial t}-R^{2}{\partial \over \partial t}\nabla ^{2}\eta +\alpha _{0}{g \over f_{0}}{\partial \eta \over \partial x}=0,}$

in which R, the Rossby radius of deformation, is defined as

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

Dispersion relation

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

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

with ${\displaystyle A}$ the amplitude, ${\displaystyle k_{x}}$ and ${\displaystyle k_{y}}$ the wavenumber in x- and y- direction respectively, ${\displaystyle \omega }$ the angular frequency of the wave, and ${\displaystyle \phi }$ a phase factor, the following dispersion relation for topographic Rossby waves is obtained:

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

If there is no bottom slope (${\displaystyle \alpha _{0}=0}$), 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

${\displaystyle \left\vert \omega \right\vert _{max}={\left\vert \alpha _{0}\right\vert g \over 2\left\vert f_{0}\right\vert R},}$

which is attained for ${\displaystyle k_{x}=R^{-1}}$ and ${\displaystyle k_{y}=0}$. If the forcing creates waves with frequencies above this threshold, no Rossby waves are generated. This situation rarely happens, unless ${\displaystyle \alpha _{0}}$ is very small. In all other cases ${\displaystyle \left\vert \omega \right\vert _{max}}$ exceeds ${\displaystyle \left\vert f_{0}\right\vert }$ and the theory breaks down. The reason for this is that the assumed conditions: ${\displaystyle \alpha \ll 1}$ and Ro_T ${\displaystyle \ll 1}$ are no longer valid. The shallow water equations used as a starting point also allow for other types of waves such as

). 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

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

which means that on the northern hemisphere the waves propagate with the shallow side at their right and on the southern hemisphere with the shallow side at their left. The equation of ${\displaystyle c_{x}}$ shows that the phase speed varies with wavenumber so the waves are dispersive. The maximum of ${\displaystyle c_{x}}$ is

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

which is the speed of very long waves (${\displaystyle k_{x}^{2}+k_{y}^{2}\rightarrow 0}$). The phase speed in the y-direction is

${\displaystyle c_{y}={\omega \over k_{y}}={k_{x} \over k_{y}}c_{x},}$

which means that ${\displaystyle c_{y}}$ can have any sign. The phase speed is given by

${\displaystyle c={\omega \over k}={k_{x} \over k}c_{x},}$

from which it can be seen that ${\displaystyle \left\vert c\right\vert \leq \left\vert c_{x}\right\vert }$ as ${\displaystyle \left\vert k\right\vert ={\sqrt {k_{x}^{2}+k_{y}^{2}}}\geq \left\vert k_{x}\right\vert }$. This implies that the maximum of ${\displaystyle \left\vert c_{x}\right\vert }$ is the maximum of ${\displaystyle \left\vert c\right\vert }$.^[1]

Analogy between topographic and planetary Rossby waves

Planetary and topographic Rossby waves are the same in the sense that, if the term ${\displaystyle {\alpha _{0}g/f_{0}}}$ is exchanged for ${\displaystyle -\beta _{0}R^{2}}$ in the expressions above, where ${\displaystyle \beta _{0}}$ is the beta-parameter or Rossby parameter, the expression of planetary Rossby waves is obtained. The reason for this similarity is that for the nonlinear shallow water equations for a frictionless, homogeneous flow the potential vorticity q is conserved:

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

with ${\displaystyle \zeta }$ being the

, which is twice the rotation speed of fluid elements about the z-axis, and is mathematically defined as

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

with ${\displaystyle \zeta >0}$ 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

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

In the derivations above it was assumed that

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

so

${\displaystyle q={f_{0}\left(1+{\beta _{0}y \over f_{0}}+{\zeta \over f_{0}}\right) \over H_{0}\left(1+{\alpha _{0}y \over H_{0}}+{\eta \over H_{0}}\right)}={f_{0} \over H_{0}}\left(1+{\beta _{0}y \over f_{0}}+{\zeta \over f_{0}}\right)\left(1-{\alpha _{0}y \over H_{0}}-{\eta \over H_{0}}+\ldots \right),}$

where a

was used on the denominator and the dots indicate higher order terms. Only keeping the largest terms and neglecting the rest, the following result is obtained:

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

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

The analogy between planetary and topographic Rossby waves is exploited in laboratory experiments that study geophysical flows to include the beta effect which is the change of the

Conceptual explanation

As shown in the last section, Rossby waves are formed because potential vorticity must be conserved. When the surface has a slope, the thickness of the fluid layer ${\displaystyle h}$ is not constant. The conservation of the potential vorticity forces the relative vorticity ${\displaystyle \zeta }$ or the Coriolis parameter ${\displaystyle f}$ to change. Since the Coriolis parameter is constant at a given latitude, the relative vorticity must change. In the figure a fluid moves to a shallower environment, where ${\displaystyle h}$ is smaller, causing the fluid to form a crest. When 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 the left in the figure. In general, the displacement causes a wave pattern that propagates with the shallower side to the right on the northern hemisphere and to the left on the southern hemisphere.[1]

Measurements of topographic Rossby waves on earth

From 1 January 1965 till 1 January 1968, The Buoy Project at the Woods Hole Oceanographic Institution dropped 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 depth.^[2] Several other research projects have confirmed that there are indeed topographic Rossby waves in the Northern Atlantic.^[3][4][5]

In 1988, barotropic planetary Rossby waves were found in the Northwest Pacific basin.^[6] Further research done in 2017 concluded that the Rossby waves are no planetary Rossby waves, but topographic Rossby waves.^[7]

In 2021, research in the South China Sea confirmed that topographic Rossby waves exist.^[8][9]

In 2016, research in the East Mediterranean showed that topographic Rossby Waves are generated south of Crete due to lateral shifts of a mesoscale circulation structure over the sloping bottom at 4000 m (https://doi.org/10.1016/j.dsr2.2019.07.008).

References