Editing Rossby wave (section)

===Free barotropic Rossby waves under a zonal flow with linearized vorticity equation===

To start with, a zonal mean flow, ''U'', can be considered to be perturbed where ''U'' is constant in time and space. Let <math>\vec{u} = \langle u, v\rangle</math> be the total horizontal wind field, where ''u'' and ''v'' are the components of the wind in the ''x''- and ''y''- directions, respectively.  The total wind field can be written as a mean flow, ''U'', with a small superimposed perturbation, ''u&prime;'' and ''v&prime;''.
<math display="block"> u = U + u'(t,x,y)\!</math>
<math display="block"> v = v'(t,x,y)\!</math>

The perturbation is assumed to be much smaller than the mean zonal flow.
<math display="block"> U \gg u',v'\!</math>

The relative vorticity <math>\eta</math> and the perturbations <math>u'</math> and <math>v'</math> can be written in terms of the [[stream function]] <math>\psi</math> (assuming non-divergent flow, for which the stream function completely describes the flow):
<math display="block">
\begin{align}
u' & =  \frac{\partial \psi}{\partial y} \\[5pt]
v' & =  -\frac{\partial \psi}{\partial x} \\[5pt]
\eta & = \nabla \times (u' \mathbf{\hat{\boldsymbol{\imath}}}  + v' \mathbf{\hat{\boldsymbol{\jmath}}}) = -\nabla^2 \psi
\end{align}
</math>

Considering a parcel of air that has no relative vorticity before perturbation (uniform ''U'' has no vorticity) but with planetary vorticity ''f'' as a function of the latitude, perturbation will lead to a slight change of latitude, so the perturbed relative vorticity must change in order to conserve [[potential vorticity]]. Also the above approximation ''U'' >> ''u''' ensures that the perturbation flow does not advect relative vorticity.

<math display="block">\frac{d (\eta + f) }{dt} = 0 = \frac{\partial \eta}{\partial t} + U \frac{\partial \eta}{\partial x} + \beta v'</math>
with <math>\beta = \frac{\partial f}{\partial y} </math>.  Plug in the definition of stream function to obtain:
<math display="block"> 0 = \frac{\partial \nabla^2 \psi}{\partial t} + U \frac{\partial \nabla^2 \psi}{\partial x} + \beta \frac{\partial \psi}{\partial x}</math>

Using the [[method of undetermined coefficients]] one can consider a traveling wave solution with [[zonal and meridional]] [[wavenumbers]] ''k'' and ''ℓ'', respectively, and frequency <math>\omega</math>:
<math display="block">\psi = \psi_0 e^{i(kx+\ell y-\omega t)}\!</math>

This yields the [[dispersion relation]]:
<math display="block"> \omega = Uk - \beta \frac k {k^2+\ell^2}</math>

The zonal (''x''-direction) [[phase speed]] and [[group velocity]] of the Rossby wave are then given by
<math display="block">
\begin{align}
c & \equiv \frac \omega k = U - \frac \beta {k^2+\ell^2}, \\[5pt]
c_g & \equiv \frac{\partial \omega}{\partial k}\ = U - \frac{\beta (\ell^2-k^2)}{(k^2+\ell^2)^2},
\end{align}
</math>
where ''c'' is the phase speed, ''c''<sub>''g''</sub> is the group speed, ''U'' is the mean westerly flow, <math>\beta</math> is the [[Rossby parameter]], ''k'' is the [[Zonal and meridional|zonal]] wavenumber, and ''ℓ'' is the [[Zonal and meridional|meridional]] wavenumber. It is noted that the zonal phase speed of Rossby waves is always westward (traveling east to west) relative to mean flow ''U'', but the zonal group speed of Rossby waves can be eastward or westward depending on wavenumber.