Normal modes in the symmetric stability problem in a vertically bounded domain

ISSN： 0309-1929

Source： Geophysical and Astrophysical Fluid Dynamics, Vol.102, Iss.4, 2008-08, pp. : 333-348

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

The stability/instability properties of symmetric normal mode disturbances in a vertically bounded domain are thoroughly examined for both constant and non-constant Brunt-Vaisala frequency (N2) cases. Sufficient conditions for stability and instability are derived. The impact of model parameters on the growth rate is investigated. The results are compared with those derived for a vertically bounded domain. The comparison shows that normal modes have very different spatial structures in the bounded and unbounded domains. Their instability properties in the two domains are also very different. (a) In the bounded domain case, the growth rate is symmetric with respect to horizontal wavenumber k and - k of normal modes. For a given vertical wavenumber, disturbances are stable when k is small and unstable when k is large. However, in the unbounded domain case, the growth rate is not symmetric in k and - k. Disturbances are unstable in a narrow range of small k and stable when k is large. (b) When the basic flow is inertially unstable all normal modes are unstable in the bounded domain case, but normal modes with large k or negative k are still stable in the unbounded domain case. (c) In the bounded domain case, the growth rate is symmetric with respect to positive and negative vertical shear of the basic flow ( [image omitted]), and a strong [image omitted] makes all normal modes more unstable. But in the unbounded domain case it is not symmetric, and a strong [image omitted] makes some normal modes more unstable and others more stable. (d) When the Richardson number is less than the ratio of the Coriolis parameter to absolute vorticity, there exists a horizontal wavenumber cut-off for stability in the bounded domain case but there is no such wavenumber cut-off in the unbounded domain case.