Nonmodal and unstable normal mode baroclinic growth as a function of horizontal scale

Abstract

Linear, quasi-geostrophic, Cartesian, spectral models based on Grotjahn (1980) are solved as initial-value problems. The basic-state wind flow includes realistic vertical shear in the form of an upper-level jet but no horizontal shear. Two archetype initial vertical structures are selected. One structure, labeled “connected”, develops strong nonmodal growth (NG). The other structure, labeled “separated”, is intended to approximate better conditions prior to observed cyclogenesis. NG is deduced from growth rates of common growth measures: amplitude, total energy, potential enstrophy, and their components. Significant NG may occur, usually early on, before a solution asymptotes to the most unstable normal mode. This study focuses on how the relative amounts of NG and unstable normal mode growth vary for different scales in both horizontal dimensions. The peak NG in most growth measures is greatest for wavelengths much smaller than the most unstable normal mode wavelength. The peak NG occurs earlier as wavelength decreases consistent with relative phase speed and distance arguments applied to constituent eigenmodes coming into favorable superposition. The peak NG is much less at all wavelengths for a separated trough than a connected initial condition (IC), except for the boundary contribution to potential enstrophy. Also, the connected IC has peak NG at shorter wavelengths than the separated IC. The peak NG occurs at a shorter wavelength for amplitude than for total energy. The connected and separated ICs are shown with the horizontal structure of a square wave and for a wave having initially localized structure along the meridional axis but allowed to evolve in that dimension. The main differences are initially localized waves develop larger meridional scale and commensurately larger growth rates. When the meridional structure is allowed to evolve, transient horizontal tilts appear leading most commonly to zonal mean convergence of eddy momentum. Phase speed differences between the main eigenmodes comprising the total solution primarily explain this result.