1JUNE1999 1579SMITH AND VALLISq 1999 American Meteorological SocietyLinear Baroclinic Instability in Extended Regime Geostrophic ModelsK. SHAFERSMITHDepartment of Physics, University of California, Santa Cruz, Santa Cruz, CaliforniaGEOFFREYK. VALLIS*Department of Ocean Science, University of California, Santa Cruz, Santa Cruz, California(Manuscript received 16 September 1997, in final form 29 May 1998)ABSTRACTThe linear wave and baroclinic instability properties of various geostrophic models valid when the Rossbynumber is small are investigated. The models are the ‘‘L1’’ dynamics, the ‘‘geostrophic potential vorticity’’equations, and the more familiar quasigeostrophic and planetary geostrophic equations. Multilayer shallow waterequations are used as a control. The goal is to determine whether these models accurately portray linear baroclinicinstability properties in various geophysically relevant parameter regimes, in a highly idealized and limited setof cases. The L1and geostrophic potential vorticity models are properly balanced (devoid of inertio-gravitywaves, except possibly at solid boundaries), valid on thebplane, and contain both quasigeostrophy and planetarygeostrophy as limits in different parameter regimes; hence, they are appropriate models for phenomena that spanthe deformation and planetary scales of motion. The L1model also includes the ‘‘frontal geostrophic’’ equationsas a third limit. In fact, the choice to investigate such relatively unfamiliar models is motivated precisely bytheir applicability to multiple scales of motion.The models are cast in multilayer form, and the dispersion properties and eigenfunctions of wave modes andbaroclinic instabilities produced are found numerically. It is found that both the L1and geostrophic potentialvorticity models have sensible linear stability properties with no artifactual instabilities or divergences. Theirgrowth rates are very close to those of the shallow water equations in both quasigeostrophic and planetarygeostrophic parameter regimes. The growth rate of baroclinic instability in the planetary geostrophic equationsis shown to be generally less than the growth rate of the other models near the deformation radius. The growthrate of the planetary geostrophic equations diverges at high wavenumbers, but it is shown how this is amelioratedby the presence of the relative vorticity term in the geostrophic potential vorticity equations.1. IntroductionThe large-scale circulation of the mid- and high-lat-itude atmosphere and ocean is characterized by a smallRossby number and velocities close to geostrophic bal-ance. Although it is true that the primitive equations,which do not explicitly employ such a balance, are morecommonly used for forecasting and climate studies,much of our conceptual understanding of the circulationhas been attained by exploiting the simplifications thatcan then be made in the equations of motion.The two classic simplified sets of equations that havebeen most commonly used for theoretical and concep-tual studies are the quasigeostrophic (QG) and the plan-* Current affiliation: Program in Atmospheric and Oceanic Sci-ences, Princeton University, Princeton, New Jersey.Corresponding author address: K. Shafer Smith, Department ofPhysics, University of California, Santa Cruz, Santa Cruz, CA 95064.E-mail: [email protected] geostrophic (PG) equations. Both are valid for lowRossby number flow. The former requires scales nearthe deformation radius and simultaneously much smallerthan the planetary scale, while the latter is valid forscales that are large compared to the deformation radiusand on the order of the planetary scale. Typically, forthe atmosphere, the QG equations are valid for scalesof order one to a few thousand kilometers, and the PGequations are valid for nearly global scales (excludingthe equatorial region where the Rossby number may notbe small). In the ocean the QG equations are valid forscales of order tens to hundreds of kilometers, and thePG equations again are valid for much larger scales.Furthermore, the large separation in spatial scale be-tween the deformation radius and the radius of the planetin the ocean yields an additional parameter regime,namely, the so-called frontal geostrophic (FG) regime.In this regime large variations in the height field (orstratification) are allowed, but the Coriolis parameter isnot allowed to vary significantly. A balanced set ofequations valid in this regime was asymptotically de-rived by Cushman-Roisin (1986). Whether this regime1580 VOLUME56JOURNAL OF THE ATMOSPHERIC SCIENCESexists or is important in the atmosphere is less likely,due to the lack of a significant scale separation betweenthe deformation radius and the planetary radius.Many of the important circulation patterns in the at-mosphere or ocean span these parameter regimes. Forexample, although baroclinic instability may preferen-tially occur near the deformation scale, there may be asignificant instability at larger scales [e.g., the Greenmodes, found by Green (1960)], which might moreproperly be described with a model that is valid in thePG regime. In any case, the nonlinear interactions ofeddies at the deformation scale (leading to a cascade ofenergy to larger scales), and eddy–mean flow interactioncertainly span the parameter regime from deformationscale to planetary scale, although the flow is in neargeostrophic balance at all scales. An ideal model forconceptual studies of the circulation would contain boththe QG and PG (and possibly FG) regimes, while ex-ploiting the smallness of the Rossby number. While for-mal accuracy with respect to the primitive equations (orshallow water equations, in an idealized setting) shouldbe roughly maintained over the parameter range of in-terest (as a function of the small parameter exploited inthe approximation), it is (we believe) more importantthat it be valid over a broad parameter regime than thatthe model have high-order accuracy with respect to thatsmall parameter.Two ‘‘geostrophic’’ models (by geostrophic model wemean merely that it is based on the smallness of theRossby number) have been proposed that (we explicitlyshow) do in fact span both QG and PG regimes. Theseare the L1model (Salmon 1983) and the simpler geo-strophic potential vorticity (GPV) equations (Vallis1996; see also Bleck 1973). That is, both models includeboth the QG and the PG equations in the appropriatelimit in parameter space. Each model