Air flowing across a broad mountain range does not necessarily return to a straight path as soon as it passes the summit. Under westerly flow, the disturbance can extend far downstream as a stationary Rossby wave train. Under easterly flow, the response is typically confined to the vicinity of the mountain in the simplest barotropic model. Why does reversing the background wind produce such a different result?
The explanation begins with potential vorticity conservation. As an air column stretches or contracts while crossing a mountain, its relative vorticity changes. The planetary vorticity gradient, or β effect, then couples that curvature to north–south displacement. We will derive the governing stationary-wave equation, examine its solutions, and interpret them using an interactive parcel visualization.
Try the interactive model: Use Hide visualization to clear the overlay and Show visualization to restore it. The settings panel lets you change wind direction, wind speed, mountain height and width, β, and playback speed. You can hide the visualization while reading the derivation.
1. Potential vorticity conservation
Consider an idealized, inviscid fluid column bounded by two material surfaces. Let its thickness be \(H\), its vertical component of relative vorticity be \(\zeta\), and the Coriolis parameter be \(f\). In the shallow-column approximation, potential vorticity is
\boxed{q=\frac{\zeta+f}{H},\qquad\frac{Dq}{Dt}=0.}
\]
The material derivative follows an air parcel:
\frac{D}{Dt}=\frac{\partial}{\partial t}+u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}.
\]
We choose \(x\) positive eastward and \(y\) positive northward, with
\zeta=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y},\qquad f(y)=f_0+\beta y.
\]
The parameter \(\beta=\partial f/\partial y\) represents the meridional variation of planetary vorticity. On a Northern Hemisphere midlatitude β plane, \(\beta>0\).
2. How mountain crossing changes relative vorticity
Let \(h(x)\) denote mountain elevation and \(\eta_{\mathrm{top}}(x)\) the vertical displacement of the column’s upper boundary. If \(H_0\) is its undisturbed thickness, then
\boxed{H(x)=H_0+\eta_{\mathrm{top}}(x)-h(x),\qquad\delta H=\eta_{\mathrm{top}}-h.}
\]
Near the mountain, the lower boundary rises. The upper boundary can rise over a broader region, so the column may begin to stretch upstream of the actual slope. As it climbs the mountain, the rising lower boundary may cause the column to contract.
Suppose a parcel began at \(y=y_0\) with \(\zeta=0\) and \(H=H_0\). Potential vorticity conservation gives
\frac{\zeta+f_0+\beta y}{H_0+\delta H}
=\frac{f_0+\beta y_0}{H_0}.
\]
For small fractional thickness variations and to leading order about a reference latitude,
\boxed{\zeta\simeq\frac{f_0}{H_0}\delta H-\beta(y-y_0).}
\]
This relation separates two contributions. Column stretching tends to increase cyclonic relative vorticity when \(\delta H>0\). Meridional displacement modifies the required relative vorticity because planetary vorticity changes with latitude. As the column starts climbing the slope and contracts, its relative vorticity tends to become more anticyclonic. The net sign depends on both effects.
Consequently, a parcel need not wait until it reaches the mountain summit to begin curving. The disturbance can appear over the windward foothills or even farther upstream if the upper boundary has already been displaced.
3. Linearizing the barotropic potential-vorticity equation
We now derive a tractable stationary model. Assume a uniform zonal background wind
(\overline u,\overline v)=(U,0),\qquad U\ne 0,
\]
and weak perturbations about a reference state of thickness \(H_0\). Define the effective topography
\boxed{h_{\mathrm{eff}}(x)\equiv h(x)-\eta_{\mathrm{top}}(x),\qquad H=H_0-h_{\mathrm{eff}}.}
\]
When \(|h_{\mathrm{eff}}|/H_0\ll1\), the first-order expansion of potential vorticity is
\begin{aligned}
q&=\frac{f_0+\beta y+\zeta’}{H_0-h_{\mathrm{eff}}}\\[3pt]
&\simeq\frac{f_0}{H_0}+\frac{1}{H_0}
\left[\zeta’+\beta y+\frac{f_0}{H_0}h_{\mathrm{eff}}\right].
\end{aligned}
\]
Here products of perturbations and the small correction proportional to \(\beta y\,h_{\mathrm{eff}}/H_0\) have been neglected. Substitution into \(Dq/Dt=0\), retaining the advection by the background wind and the meridional advection of planetary vorticity by \(v’\), yields
\boxed{\frac{\partial\zeta’}{\partial t}
+U\frac{\partial\zeta’}{\partial x}
+\beta v’
=-\frac{f_0U}{H_0}\frac{d h_{\mathrm{eff}}}{dx}.}
\]
The right-hand side is the stationary forcing associated with changes in column thickness. Setting \(\partial\zeta’/\partial t=0\) gives the steady equation.
Show the linearization step by step
To first order, the dynamically relevant part of the potential vorticity is proportional to
\zeta’+\beta y+\frac{f_0}{H_0}h_{\mathrm{eff}}(x).
\]
Its material derivative retains the terms
\left(\partial_t+U\partial_x\right)\zeta’
+v’\partial_y(\beta y)
+U\frac{f_0}{H_0}\partial_x h_{\mathrm{eff}}=0,
\]
because the remaining products of perturbation quantities are second order.
4. Deriving the forced stationary-wave equation
For a horizontally nondivergent flow, introduce a streamfunction
\Psi(x,y,t)=-Uy+\psi'(x,y,t),\quad
u=-\frac{\partial\Psi}{\partial y},\quad
v=\frac{\partial\Psi}{\partial x}.
\]
Then \(v’=\partial_x\psi’\) and \(\zeta’=\nabla^2\psi’\). In steady flow, the linearized equation becomes
U\frac{\partial}{\partial x}\nabla^2\psi’
+\beta\frac{\partial\psi’}{\partial x}
=-\frac{f_0U}{H_0}\frac{d h_{\mathrm{eff}}}{dx}.
\]
Integrating with respect to \(x\) and selecting the zero perturbation reference upstream gives
\boxed{\nabla^2\psi’+\frac{\beta}{U}\psi’
=-\frac{f_0}{H_0}h_{\mathrm{eff}}(x).}
\]
Our idealized mountain ridge is uniform in the north–south direction. Choosing the corresponding meridionally uniform disturbance \(\psi’=\psi'(x)\) reduces the equation to
\boxed{\frac{d^2\psi’}{dx^2}+\frac{\beta}{U}\psi’
=-\frac{f_0}{H_0}h_{\mathrm{eff}}(x).}
\]
This is a one-dimensional forced stationary Rossby-wave equation. The ridge is cropped in the animation for visual clarity, but the mathematical forcing remains independent of \(y\).
5. Why westerlies and easterlies behave differently
Far from the mountain, \(h_{\mathrm{eff}}\to0\). The homogeneous equation is
\frac{d^2\psi’}{dx^2}+\frac{\beta}{U}\psi’=0.
\]
5.1 Westerly flow: \(U>0\)
For \(\beta>0\) and \(U>0\), define
\boxed{k_s=\sqrt{\frac{\beta}{U}},\qquad
\lambda_s=2\pi\sqrt{\frac{U}{\beta}}.}
\]
The homogeneous response is oscillatory. Imposing a downstream-radiation condition (with no incoming free wave from the upstream boundary) gives the forced solution
\boxed{\psi'(x)=-\frac{f_0}{H_0 k_s}
\int_{-\infty}^{x}\sin\!\left[k_s(x-\xi)\right]
h_{\mathrm{eff}}(\xi)\,d\xi.}
\]
Once the parcel has crossed the mountain, the free oscillatory component persists: a stationary lee-wave train forms downstream. Increasing \(U\) lengthens its wavelength, while increasing \(\beta\) shortens it.
5.2 Easterly flow: \(U<0\)
For the same positive β but \(U<0\), write
\kappa=\sqrt{-\frac{\beta}{U}},\qquad
\frac{d^2\psi’}{dx^2}-\kappa^2\psi’
=-\frac{f_0}{H_0}h_{\mathrm{eff}}.
\]
Requiring the response to decay at both distant boundaries gives
\boxed{\psi'(x)=\frac{f_0}{2\kappa H_0}
\int_{-\infty}^{\infty}e^{-\kappa|x-\xi|}
h_{\mathrm{eff}}(\xi)\,d\xi.}
\]
The solution is not spatially oscillatory far from the ridge. Instead, the disturbance decays over a characteristic distance \(\kappa^{-1}=\sqrt{-U/\beta}\). Thus, in this idealized model, reversing the background flow changes the mathematical type of the stationary response.
5.3 The same result from the Rossby-wave dispersion relation
Without topographic forcing, a plane-wave disturbance \(e^{i(kx+ly-\omega t)}\) satisfies
\omega=Uk-\frac{\beta k}{k^2+l^2}.
\]
For nonzero \(k\), the stationarity condition \(\omega=0\) requires \(k^2+l^2=\beta/U\). With \(l=0\), this reproduces \(k_s^2=\beta/U\); a real stationary wavenumber exists only when \(U\) and \(\beta\) have the same sign. This is the planetary β effect, not a separate topographic β parameter.
6. Terrain geometry and the prescribed upper boundary
To produce a visible slope and a finite mountain width, the simulation uses the following ridge profile:
h(x)=\begin{cases}
h_0\sqrt{1-(x/a)^2},& |x|<a,\\
0,&|x|\ge a.
\end{cases}
\]
Here \(h_0\) is the peak elevation and \(a\) is the east–west half-width. The upper boundary is prescribed, not computed from a full three-dimensional vertical-motion model. To make the upstream stretching and slope contraction visible, we use
\begin{aligned}
\eta_{\mathrm{top}}(x)=h_0\bigg[&0.32\exp\!\left(-\frac{x^2}{2(1.75a)^2}\right)\\
&+0.45\exp\!\left(-\frac{(x+1.30sa)^2}{2(0.70a)^2}\right)\bigg],\\
&s=\operatorname{sgn}(U).
\end{aligned}
\]
The second term is displaced toward the upstream side and reverses location when the wind direction changes. The numerical coefficients are visualization assumptions; they are not a universal property of mountain waves. The model uses \(f_0=10^{-4}\,\mathrm{s^{-1}}\) and \(H_0=10^4\,\mathrm{m}\).
At the initial settings—\(U=10\,\mathrm{m\,s^{-1}}\), \(\beta=2.5\times10^{-11}\,\mathrm{m^{-1}s^{-1}}\), \(h_0=450\,\mathrm{m}\), and \(a=950\,\mathrm{km}\)—the theoretical stationary wavelength is about 3,974 km. The plot spans approximately 2.5 wavelengths from the mountain center to its downstream edge for the meridionally uniform component.
7. Reading the parcel animation
Because \(\psi’=\psi'(x)\), the velocity and relative vorticity are
u=U,\qquad v=\frac{d\psi’}{dx},\qquad
\zeta’=\frac{d^2\psi’}{dx^2}.
\]
A parcel moves through this time-independent field according to
\boxed{\frac{dx_p}{dt}=U,\qquad
\frac{dy_p}{dt}=\left.\frac{d\psi’}{dx}\right|_{x=x_p}.}
\]
Every 0.4 seconds of display time, a vertical line of seven parcels is released from the upstream edge. The release positions span \(\pm0.5a\) in the north–south direction. By observing successive columns, you can see that the velocity field is stationary even though the parcels are moving.
Parcels are colored by the relative vorticity at their current location: blue for negative (anticyclonic in the Northern Hemisphere), off-white for zero, and red for positive (cyclonic). Each segment of a parcel trail retains the color corresponding to the vorticity at the time that location was crossed.
When a parameter is changed, the stationary field is recalculated immediately and all existing parcels continue from their current positions under the updated field. This is an interactive visualization technique, not a physical time-dependent adjustment from one equilibrium to another. Trails produced across a parameter change are not streamlines of any one fixed field.
8. Assumptions and limitations
This is a linear, idealized, horizontally nondivergent barotropic model for a meridionally uniform ridge. It does not explicitly solve three-dimensional vertical motion, atmospheric stratification, friction, nonlinear wave–mean-flow interaction, or mountain blocking. The shaped upper-boundary displacement is prescribed. Very large mountain heights or perturbation velocities may violate the assumptions underlying the linear approximation.
The visualization represents horizontal north–south meandering on a plan view; it is not a vertical cross-section showing air flowing up and down a mountain. The terrain is drawn over a finite north–south segment for legibility, although the governing equation assumes an indefinitely long ridge.
Reference
Holton, J. R. (2004). An Introduction to Dynamic Meteorology (4th ed.). Elsevier Academic Press. Chapter 4, §4.3, pp. 97–100.