Rossby wave

A Rossby wave is a low-frequency wave restored by the spatial variation of planetary or background potential vorticity. On a beta plane its phase commonly propagates westward.

Near a north pole, $f=2\Omega -\gamma r^2/2$ with $\gamma =2\Omega /a^2$. If a quasi-geostrophic streamfunction satisfies ${\nabla }_h^2\psi -\psi /L_d^2=-{\alpha }^2\psi$, then $Q=f-{\alpha }^2\psi$. Its nonlinear self-advection cancels because $J(\psi ,\psi )=0$, leaving $-{\alpha }^2{\psi }_t+\gamma {\psi }_{\varphi }=0$. An azimuthal mode $\cos (\omega t-n\varphi )$ therefore has $\omega =-n\gamma /{\alpha }^2$ at any amplitude consistent with the underlying balanced model.

For a polar gamma-plane Rossby wave, the drift fixes ${\alpha }^2=K^2+L_d^{-2}$. It does not separately fix radial wavenumber $K$ and Rossby deformation radius $L_d$. Scaling an eigenfunction changes its amplitude without changing either wavenumber, so a meander amplitude cannot supply the missing radial wavelength. A radial boundary condition or a separately justified envelope approximation is needed to determine $L_d$.

Linearizing shallow-water quasi-geostrophic potential vorticity about a zonal current $U(y)$ gives $Q_y=\beta -U^{\prime \prime }+U/R_D^2$. A normal mode $\hat{\varphi }(y)e^{i(kx-\omega t)}$ obeys $(U-\omega /k)[{\hat{\varphi }}^{\prime \prime }-(k^2+R_D^{-2})\hat{\varphi }]+Q_y\hat{\varphi }=0$. A global transverse plane wave is generally unavailable for an arbitrary nonuniform jet; boundary conditions or a local short-wavelength approximation are required.

When a zonal current varies slowly compared with the wavelength, freeze its velocity and background potential vorticity gradient locally in the Rossby-wave equation for a sheared zonal current. The resulting local dispersion relation includes curvature $U^{\prime \prime }$ and, for a free surface, the basic surface-slope contribution $U/R_D^2$. In the nondivergent limit it becomes $\omega =kU-k(\beta -U^{\prime \prime })/(k^2+l^2)$.

For a sinusoidal mean jet on an $f$ plane, its QG potential-vorticity gradient is $({\pi }^2/L^2+k_d^2)U$. Ground-stationary modes satisfy the displayed circle and can solve the full stationary shear equation even though their scale does not meet a local short-wave assumption. The local stationary group vector is $2Uk(k,l)/(k^2+l^2+k_d^2)$, reducing to a zonal scalar formula only for $l=0$.

For a mode with streamfunction $\varphi$, deformation radius $L_D$, and constant planetary-vorticity gradient $\beta$, the linear Rossby-wave equation isIn the long-wave limit it becomes a first-order westward transport equation.

For a resting midlatitude layer, the slow Rossby wave equation is $({\nabla }_h^2-a^2){\eta }_t=-\beta {\eta }_x$, where $a=f_0/\sqrt{gH}$. With the plane wave convention $e^{i(kx+ly-\omega t)}$,The zonal phase velocity is westward for $\beta >0$, but the zonal group velocity changes sign at $k^2=l^2+a^2$.

A stationary straight meridional wall preserves frequency and tangential wavenumber $l$. The two normal wavenumbers obeyIncidence and reflection are distinguished by the normal group velocity. In the leading quasi-geostrophic approximation, a nonzero $l$ and the impermeability condition give reflected height amplitude $A_r=-A_i$. The $l=0$ no-normal-flow condition is degenerate in that reduced model; exact ageostrophic transport generally changes the amplitude ratio.

The wavevectors of a nonzero-frequency Rossby wave in a resting shallow layer satisfyThis follows by completing the square in the shallow-water Rossby-wave dispersion relation. A real circle requires $|\omega |\le |\beta |/(2|a|)$. Its two intersections with a given tangential wavenumber provide the incident and reflected normal wavenumbers at a meridional wall.

In a one-layer quasi-geostrophic fluid with effective meridional PV gradient ${\beta }_{\mathrm{eff}}$,Bottom slope contributes to ${\beta }_{\mathrm{eff}}$ through vortex stretching.

In a square with impermeable walls and background topographic potential-vorticity gradient $\beta$, the linear shallow-water quasi-geostrophic potential vorticity equation is ${\partial }_t({\nabla }^2-L_D^{-2})\psi +\beta {\psi }_x=0$. For a nonzero-frequency normal mode, setting $\hat{\psi }=\psi e^{i\beta x/(2\omega )}$ gives a Helmholtz equation with homogeneous wall data. The eigenfunctions are $\hat{\psi }=\sin (m\pi x/L)\sin (n\pi y/L)$, with $m,n\ge 1$, and the displayed frequencies. The positive and negative frequency representations are complex conjugates for a real wave.

A depth discontinuity between two constant-depth rotating layers supports an exponentially trapped Rossby wave. Continuity of surface height and cross-step volume flux gives $\omega (H_{+}{\kappa }_{+}+H_{-}{\kappa }_{-})=fk(H_{+}-H_{-})$, where ${\kappa }_{\pm }^2=k^2+(f^2-{\omega }^2)/(g{H}_{\pm })$ and positive roots give decay. For $H_{\pm }=H_0\pm d$ with $|d|\ll H_0$, the low-frequency branch has $\omega \simeq (fd/H_0)k/\sqrt{k^2+R_D^{-2}}$. In the Northern Hemisphere it propagates with the shallow side on its right.

A long disturbance trapped on a depth step is sometimes called a double Kelvin wave. With $H_{+}>H_{-}$, $f>0$ and $c_{\pm }=\sqrt{g{H}_{\pm }}$, cross-step geostrophic balance and the along-step acceleration yield the transport equation shown for a background height jump ${\eta }_0\mathrm{sgn}(x)$. Its zero-initial-amplitude solution is $A={\eta }_0[\mathrm{sgn}(x)-\mathrm{sgn}(x-(c_{+}-c_{-})t)]$. The adjusted region diverts cross-step volume flux along the step; the propagating front still carries flux. This outer approximation does not resolve fast adjustment at the fronts.

The long step-trapped topographic Rossby wave has equal phase velocity and group velocity, $c_T=fd{R}_D/H_0$. For an outer surface profile $\eta ={\eta }_0\mathrm{sgn}(x)-A(x,t)e^{-|y|/R_D}$, cross-step matching gives $A_t+c_T{A}_x=2c_T{\eta }_0\delta (x)$. With $A(x,0)=0$, its weak solution is $A={\eta }_0[\mathrm{sgn}(x)-\mathrm{sgn}(x-c_{T}t)]$. This describes the slowly varying outer response; the fast adjustment region at the origin is unresolved.

A barotropic Rossby wave has no vertical shear. For horizontal wavenumbers $k,l$, its quasi-geostrophic frequency is $\omega =-\beta k/(k^2+l^2)$.

For a stationary $n=2$, $l=0$ wave in a uniform zonal current, a perturbation $n>2$ has phase speed $U(1-4/n^2)$ and zonal group speed $U(1+4/n^2)$. Phase speed relative to the air is westward; packet propagation relative to the ground is faster downstream.

A wave packet can transfer energy downstream faster than individual phase features move, allowing successive downstream disturbances to develop while upstream ones weaken. In the simple uniform-current barotropic model with zero meridional wavenumber, the zonal group speed is $U+\beta /k^2$ while phase speed is $U-\beta /k^2$. Linear superposition and interference are not direct nonlinear eddy scattering. Real weather development also involves baroclinic energy conversion and wave–mean-flow feedback, which require additional dynamics.

In the uniform-current beta-plane model, $U=\beta /(k_2^2+l^2)$ with $k_2=2/(a\cos \varphi )$. This is a model-dependent balance between eastward advection and westward intrinsic propagation, not a unique speed set by zonal wavenumber alone.

For a uniform eastward current $U>0$ on a beta plane with $\beta >0$, stationary nondivergent Rossby waves have $k^2+l^2=\beta /U$. Their ground-frame group velocity is $2U(k^2,kl)/(k^2+l^2)$, tracing a circle centered at $(U,0)$ with radius $U$. Packets emitted by a localized source throughout $0<s<t$ occupy the disk shown in geometrical ray theory. This envelope is distinct from the full balanced near field. A westward current has no nontrivial stationary propagating branch in this model.

A baroclinic Rossby wave has nontrivial vertical structure. Constant stratification and rigid boundaries add a positive vertical-mode contribution to the denominator of its dispersion relation.

For fixed meridional wavenumber and deformation radius, differentiating $\omega =Uk-\beta k/(k^2+l^2+L_d^{-2})$ gives $c_{gx}=U+\beta (k^2-l^2-L_d^{-2})/(k^2+l^2+L_d^{-2}{)}^2$. A localized packet uses this velocity rather than $\omega /k$. For $l=0$ its intrinsic sign changes at $|k|L_d=1$.

For constant buoyancy frequency and uniform zonal flow, the three-dimensional quasi-geostrophic potential vorticity equation gives $\omega =Uk-\beta k/[k^2+l^2+(f_0^2/N^2)m^2]$. Set $\omega =0$ with $k\ne 0$ to obtain the displayed formula. For $\beta >0$, oscillatory vertical structure requires $0<U<\beta /(k^2+l^2)$. Differentiation gives $c_{g,z}=2\beta k(f_0^2/N^2)m/[k^2+l^2+(f_0^2/N^2)m^2{]}^2$, so an upward outgoing radiation condition selects $km>0$. The Coriolis parameter enters squared: the criterion does not reverse between hemispheres. A density-scale-height correction belongs to a different, compressible log-pressure quasi-geostrophic potential vorticity model.

For dispersion relation $\omega =Uk-\beta k/(k^2+{\alpha }^2)$ with fixed ${\alpha }^2\ge 0$, nonzero zonal wavenumber and stationary phase imply $U=\beta /(k^2+{\alpha }^2)$. Differentiating at fixed $\alpha$ yields $c_{g,x}=U+\beta (k^2-{\alpha }^2)/(k^2+{\alpha }^2{)}^2$, which reduces to the displayed positive value for $\beta >0$. Thus a stationary Rossby wave can transport energy eastward even though its crests stay fixed.

For constant buoyancy frequency $N$, the log-pressure quasi-geostrophic potential vorticity equation has modes $\psi =e^{z/(2H)}{e}^{i(kx+mz-\omega t)}$ withThe prefactor compensates the density decrease in the energy weight. For $\beta ,k>0$, vertical propagation occurs for $-\beta k/[k^2+f_0^2/(4N^2{H}^2)]<\omega <0$. An outgoing radiation condition selects the root with positive vertical group velocity; bounded weighted amplitude alone does not select between two propagating roots.