= Solution
Apply the intended, corrected <dispersion relation> in <5/c>. A uniform mean zonal wind gives $\omega_{\rm ground}=\widehat\omega+kU$. Stationarity with $U=-20\,\mathrm{m\,s^{-1}}$ therefore requires eastward intrinsic <Phase velocity>[phase speed] $\widehat\omega/k=-U=20\,\mathrm{m\,s^{-1}}$. With $k=1/a$,
$$
|m|=\left[\frac{N^2}{20^2}-\frac1{a^2}-\frac1{4H^2}\right]^{1/2}
\simeq4.975\times10^{-4}\,\mathrm{m^{-1}},
\qquad\boxed{\lambda_z=\frac{2\pi}{|m|}\simeq12.6\,\mathrm{km}.}
$$
The period determines $\Omega=2\pi/(24.4\times3600\,\mathrm s)$ and $\beta=2\Omega/a\simeq4.21\times10^{-11}\,\mathrm{m^{-1}\,s^{-1}}$. Hence
$$
\boxed{L_D=\sqrt{20/\beta}\simeq6.89\times10^5\,\mathrm m,\qquad
L_D/a\simeq0.203\,\mathrm{rad}\simeq11.6^\circ.}
$$
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2017-martian-kelvin-wave.png]
{title=Equatorial trapping and vertical phase of the stationary Martian Kelvin-wave example}
{description=The original calculation uses an eastward intrinsic speed of 20 metres per second. The Gaussian amplitude has a latitude scale of 11.6 degrees; the density-weighted vertical velocity has a vertical wavelength of 12.6 kilometres. The plotted negative vertical wavenumber corresponds to upward intrinsic energy propagation.}
A storm within roughly $12^\circ$ of the equator has substantial projection on the <Gaussian distribution>[Gaussian] mode. There is no sharp <latitude> cutoff: $|y|=L_D$ gives amplitude $e^{-1/2}$ and the $1/e$ amplitude <latitude> is about $16.4^\circ$. Forcing outside that core has progressively weaker overlap.
The intrinsic wave is eastward even though its observed ground pattern is stationary; westward advection cancels its <Phase velocity>[phase propagation]. Its intrinsic period is about $12.4$ <Earth> days. The inferred $|m|\gg1/(2H)$ puts it in the short-vertical-wavelength branch, well below the long-wave speed $2NH=200\,\mathrm{m\,s^{-1}}$. For upward <energy> radiation, $\partial\widehat\omega/\partial m$ must be positive, which selects $m<0$ for positive $k,\widehat\omega$: vertical <Phase velocity>[phase propagation] is then downward. Dust <absorption> supplies <Diabatic heating>[thermal forcing] with a zonally asymmetric component; the response does not imply that dust itself moves eastward at the intrinsic <Phase velocity>[phase speed].
Southern spring to summer, the warmer perihelion-side dust season, is a reasonable climatological window for such a regional storm. This seasonal inference uses observed dust climatology, not the <dispersion relation>: the specified wind and wave parameters alone do not determine a unique season, and individual regional storms can occur outside that window.<NASA Mars Orbiters Reveal Seasonal Dust Storm Pattern>[2016 report, discussion of the warmer southern spring and summer, perihelion near the end of southern spring, and the observed regional-storm sequence]{ref}
Back to article page
