### Broken Symmetry and Atmospheric Waves, 1

Part I: Vertically propagating Waves and the Stratospheric Window

Symmetry is a powerful organising principle in physics. It is a central concept in both classical and quantum mechanics and has a key role in the standard model. When symmetry is violated, interesting things happen. The book Shattered Symmetry by Pieter Thyssen and Arnout Ceulemans discusses many aspects and examples of broken symmetry.

In this article (and the following one) we look at some consequences of broken symmetry in atmospheric dynamics. In particular, we see how mountains (which are stationary!) can generate waves in the atmosphere that propagate towards the west. We will look at this unexpected breaking of symmetry and try to explain it.

The Stratospheric Wind Field

The winds in the stratosphere are easterly in summer and westerly in winter (see Figure). This has a strong influence on the propagation of energy upward from the troposphere (lowest 12 km) to the stratosphere. The circulation in the winter stratosphere comprises large planetary-scale waves superimposed on a westerly zonal flow known as the polar-night jet. The figures below show the pressure and wind fields at 250 hPa (around 10km high). The waves have quasi-stationary components but also components that propagate towards the west.

The Stratospheric Window

The origin of these patterns was first studied by Jule Charney and Philip Drazin, whose conclusions were published in 1961. Charney and Drazin showed how waves originating in the troposphere can propagate vertically only when the mean zonal wind is weak, and blowing from the west.

Charney and Drazin used the quasi-geostrophic potential vorticity equation on a beta-plane with a mean zonal wind independent of latitude but varying with height. They sought a solution in terms of a stream function

$\displaystyle \psi^\prime = \Psi(z) \exp [ i(kx+\ell y-kct)+z/2H] \,.$

For constant zonal flow ${\bar u}$, the equation reduces to

$\displaystyle \frac{\mathrm{d}^2\Psi}{\mathrm{d}z^2} + n^2 \Psi = 0 \ \ \ \ \ (1)$

where

$\displaystyle n^2 = \frac{N^2}{f^2}\left[ \frac{\beta}{\bar u-c} - (k^2+\ell^2) \right] - \frac{1}{4H^2} \,.$

(all notation is standard in the meteorological literature) with ${n}$ being, essentially, an index of refraction. For vertical propagation, we require ${n^2 > 0}$. Thus, forced stationary waves are found only when

$\displaystyle 0 < \bar u < \frac{\beta}{(k^2+\ell^2) + f^2/4 N^2H^2} \equiv U_c \,.$

The quantity ${U_c}$ is called the Rossby critical velocity. This means that vertically propagating stationary waves can occur only if the wind is westerly (${\bar u > 0}$) and weaker than the critical value (${\bar u < U_c}$).

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. In the winter, the mean flow is westerly, and comprises a circumpolar vortex with large amplitude wave-like disturbances superimposed on it. Since ${K^2+\ell^2}$ is least for the largest waves, these are the ones found in the winter stratosphere.

The amplitude of the waves, as well as their vertical structure, depends on the value of ${n^2}$. Much more detail is found in the paper of Charney and Drazin and in Holton’s (1975) book.

The analysis of the vertical propagation of wave energy was continued in several studies. One especially interesting study, by Isamu Hirota, will be reviewed in Part II of this post.

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.