### Broken Symmetry and Atmospheric Waves, 2

Part II: Stationary Mountains and Travelling Waves

Atmospheric flow over mountains can generate large-scale waves that propagate upwards. Although the mountains are stationary(!), the waves may have a component that propagates towards the west. In this post, we look at a simple model that explains this curious asymmetry.

Earth’s Rotation and Symmetry Breaking

If the Earth were not spinning, there would be a perfect symmetry between east and west. However, the angular velocity vector ${\boldsymbol{\Omega}}$, pointing towards Polaris, is associated with a breaking of this symmetry. To model the atmospheric flow, we often use the quasi-geostrophic equations on a beta-plane. The beta parameter is ${\beta = 2\Omega \cos\phi/ a}$ where ${\Omega}$ is the Earth’s angular velocity. The rotation of the Earth that breaks the east-west symmetry enters the quasi-geostrophic equations via the beta term, which strongly dominates the dynamics. Together with the differential solar heating, this brings about the westerly mean zonal flow. If the rotation were reversed, the mean flow would be easterly.

Rossby Waves and the Stratospheric Window

The circulation in the winter stratosphere comprises large planetary-scale waves superimposed on a westerly zonal flow known as the polar-night jet. The large-scale waves in the troposphere (lower atmosphere) move westward relative to the mean flow. The phase speed derived by Carl Rossby for these waves is $\displaystyle c = \bar u - \frac{\beta L^2}{4\pi^2} \,.$

For the planetary waves (large wavelength ${L}$) the second rhs term dominates, so ${c<0}$ and the waves propagate towards the west. Mean circulation at 50hPa. Left: Winter Northern Hemisphere. Right: Summer Northern Hemisphere [source NCEP/NCAR Reanalysis].

In the winter, the mean stratospheric flow is westerly, and comprises a circumpolar vortex with large amplitude wave-like disturbances superimposed on it (Figure above, left panel). In the Summer, the winds in the stratosphere are easterly ( ${\bar u < 0}$) and the planetary waves are all trapped and the dominant flow is an axially symmetric vortex (Figure, right panel). Since ${L^2}$ is largest for planetary-scale waves, these are the ones found in the winter stratosphere.

Jule Charney and Philip Drazin (1961), showed how waves originating in the troposphere can propagate vertically into the stratosphere only when the mean zonal wind is weak and blowing from the west. For vertical propagation, forced stationary waves are found only when $\displaystyle 0 < \bar u < U_c, \qquad\mbox{where}\qquad U_c = \frac{\beta}{(k^2+\ell^2) + f^2/4 N^2H^2} \,.$

That is, the waves can propagate upwards only when the mean zonal wind ${\bar u}$ is positive (westerly) and less than the critical velocity ${U_c}$ .

The amplitude of the waves, as well as their vertical structure, depends on the value of the index of refraction ${n^2}$. More detail is found in the previous blog post [here] and in Charney and Drazin (1961).

Hirota’s Model

The analysis of the vertical propagation of wave energy was continued in several studies. One especially interesting study was that of Isamu Hirota (1971). To enable an analytical treatment, various factors like the sphericity of the Earth, horizontal shear of the zonal wind, critical line absorption and damping due to Newtonian cooling were omitted. Vertical structure of mean zonal flow ${\bar u}$. Shaded area shows range of variation.

Observational studies of the winter stratospheric circulation strongly suggested that the quasi-periodic variation of planetary waves in the stratosphere is caused by the time change of mean zonal westerlies below 30 km. Since the range of oscillation in ${\bar u}$ is maximum in the lower atmosphere, the excitation of traveling waves takes place there, and the wave energy propagates upward through the zonal westerlies.

Hirota examined the response of planetary Rossby waves to periodic forcing in the lower atmosphere. His analysis started from the linearised QG vorticity equation and thermodynamic equation. For the lower boundary condition, he assumed that the stream function was of sinusoidal form and varied periodically with time. For the upper boundary condition, he used a radiation condition, requiring the direction of energy propagation to be upwards.

Hirota defined the zonal flow to have a constant component ${u_0}$ and a part oscillating in time with amplitude maximum at the surface and vanishing above 60km (Figure above). We assume the mountains can be represented by a sinusoidal function: ${h = h_0\exp(ikx)}$. Suppose the zonal wind near the surface has a constant part and a part oscillating in time: $\displaystyle \bar u(x,t) = [ u_0 + u^\prime \cos\omega t ]\exp(ikx) \,.$

This standing wave pattern can be expressed as $\displaystyle \begin{array}{rcl} \bar u &=& \left[ u_0 + u^\prime\left(\frac{\exp(i\omega t)+\exp(-i\omega t)}{2}\right)\right]\exp(ikx) \\ &=& u_0 \exp(ikx) + u^\prime\exp[ ikx+\omega t ] + u^\prime \exp[ ikx-\omega t] \\ &=& u_0 \exp(ikx) + u^\prime\exp[ik(x+c t)] + u^\prime\exp[ik(x-c t)] \\ &=& \qquad \mathbf{A} \hspace{3cm} \mathbf{B} \hspace{3cm} \mathbf{C} \end{array}$

which is the sum of a steady component ${\mathbf{A}}$ of amplitude ${u_0}$ and two travelling waves, ${\mathbf{B}}$ travelling westward with phase speed ${c=-\omega/k< 0}$ and ${\mathbf{C}}$ travelling eastward with phase speed ${c}$. Variations of amplitudes and phase at 60 km for the transient wave and composite (transient plus steady waves) in Hirota’s model (Hirota, 1971).

Since the equations have been linearised, the three components can be treated separately and then combined. The Figure above shows the time-dependent behaviour of the composite solution (transient plus steady waves) and the transient part at 60km, high in the stratosphere. The dashed line near the top of the figure shows the phase of the transient part steadily increasing; this indicates a westward propagation.

We can see that, component ${\mathbf{C}}$ of the forcing, representing a wave travelling eastward, is trapped in the lower atmosphere: this is confirmed by the negative index of refraction ${n^2<0}$. On the other hand, ${n^2}$ is positive for the westward-travelling component ${\mathbf{B}}$, which has an oscillating vertical structure and propagates unhindered into the stratosphere.

In summary, a standing-wave forcing at the surface results in a westward-travelling response in the stratosphere. This is rather remarkable, and is a consequence of the symmetry-breaking effect of the Earth’s rotation.

Sources ${\bullet}$ Charney, Jule and Philip Drazin, 1961: Propagation of planetary-scale disturbances from the lower into the upper atmosphere. J. Geophys. Res., 66(1), 83–109. ${\bullet}$ Hirota, Isamu, 1971: Excitation of planetary Rossby waves in the winter stratosphere by periodic forcing. J. Met. Soc. Japan, Ser. II 49.6, 439–449. ${\bullet}$ Holton, James, 1975: The Dynamic Meteorology of the Stratosphere and Mesosphere. Met. Monographs Vol. 15 No. 37., Amer. Met. Soc. Reprinted by Springer, 2016. ${\bullet}$ Pieter Thyssen and Arnout Ceulemans, 2017: Shattered Symmetry. Oxford Univ. Press. 498pp. ISBN: 978-0-19061-139-2.