Stability of atmospheric flows

Absolute and convective instability

Figure 1: Classification of observed and numerically simulated cases of orographic cyclogenesis [1]
and a possible transition threshold (dashed line).

Secondary instability of a finite amplitude Eady wave displays convective (Mode 1) or absolute (Mode 2) character according to the amplitude of the primary wave. The unstable wave packet can be induced by the encounter of the large-scale wave with an orographic feature, and therefore be related to orographic modifications of the atmospheric flow. [2]

Instability of finite amplitude lower-neutral Eady waves

Figure 2: Contours of the non-dimensional growth rate of the Mode 2 of
Fantini and Davolio (2001) as a function of non-dimensional wavelength
and maximum meridional velocity of the primary Eady wave. Thick line is
the value 0.31, the maximum Eady growth rate (see [3] for details).

From an approximate analytic solution of the stability problem proposed by Fantini and Davolio (2001) a simple condition for the appearance of meridionally structured unstable secondary modes (Mode 2) is obtained. The condition is most simply expressed in terms of the relative vorticity of the primary wave:

for absolute instability to develop.

Work in progress

Further research is needed to relate the observational/case-studies evidence (Figure 1) to the analytic/idealized-modeling results (Figure 2). A partial list of what we are working on:

Extract from observed orographic cyclogenesis cases the parameters of mean flow and "primary" wave needed to combine the "empirical" classification of Fig.1 with the theory of Fig.2.
Extend the domain of the theoretical resuls of Figure 2 from two-dimensional neutral Eady waves to flow configurations closer to the observed atmosphere (spherical domain, deepening large-scale wave, frontal structure)
Prove the "absolute" behavior of packets of Mode 2 waves from the truncated spectral series solution of [3].

References

[1] Illari, Malguzzi and Speranza, 1980: On breakdowns of the westerlies, Geophys. Astrophys. Fluid Dyn., 17, 27-49
[2] Fantini and Davolio, 2001: Instability of neutral Eady waves and orography, J. Atmos, Sci., 58, 1146-1154
[3] Fantini, 2006: Instability of finite amplitude lower-neutral Eady waves. Quart. J. Royal Met. Soc., 132, 2157-2170.