Linear Baroclinic Instability in Extended (15 pages)

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Linear Baroclinic Instability in Extended



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1 JUNE 1999 SMITH AND VALLIS 1579 Linear Baroclinic Instability in Extended Regime Geostrophic Models K SHAFER SMITH Department of Physics University of California Santa Cruz Santa Cruz California GEOFFREY K VALLIS Department of Ocean Science University of California Santa Cruz Santa Cruz California Manuscript received 16 September 1997 in final form 29 May 1998 ABSTRACT The linear wave and baroclinic instability properties of various geostrophic models valid when the Rossby number 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 water equations are used as a control The goal is to determine whether these models accurately portray linear baroclinic instability properties in various geophysically relevant parameter regimes in a highly idealized and limited set of cases The L1 and geostrophic potential vorticity models are properly balanced devoid of inertio gravity waves except possibly at solid boundaries valid on the b plane and contain both quasigeostrophy and planetary geostrophy as limits in different parameter regimes hence they are appropriate models for phenomena that span the deformation and planetary scales of motion The L1 model also includes the frontal geostrophic equations as a third limit In fact the choice to investigate such relatively unfamiliar models is motivated precisely by their applicability to multiple scales of motion The models are cast in multilayer form and the dispersion properties and eigenfunctions of wave modes and baroclinic instabilities produced are found numerically It is found that both the L1 and geostrophic potential vorticity models have sensible linear stability properties with no artifactual instabilities or divergences Their growth rates are very close to those of the shallow water equations in both quasigeostrophic and planetary geostrophic parameter regimes The growth rate of baroclinic instability in the planetary geostrophic equations is shown to be generally less than the growth rate of the other models near the deformation radius The growth rate of the planetary geostrophic equations diverges at high wavenumbers but it is shown how this is ameliorated by the presence of the relative vorticity term in the geostrophic potential vorticity equations 1 Introduction The large scale circulation of the mid and high latitude atmosphere and ocean is characterized by a small Rossby number and velocities close to geostrophic balance Although it is true that the primitive equations which do not explicitly employ such a balance are more commonly used for forecasting and climate studies much of our conceptual understanding of the circulation has been attained by exploiting the simplifications that can then be made in the equations of motion The two classic simplified sets of equations that have been most commonly used for theoretical and conceptual studies are the quasigeostrophic QG and the plan Current affiliation Program in Atmospheric and Oceanic Sciences Princeton University Princeton New Jersey Corresponding author address K Shafer Smith Department of Physics University of California Santa Cruz Santa Cruz CA 95064 E mail shafer physics ucsc edu q 1999 American Meteorological Society etary geostrophic PG equations Both are valid for low Rossby number flow The former requires scales near the deformation radius and simultaneously much smaller than the planetary scale while the latter is valid for scales that are large compared to the deformation radius and on the order of the planetary scale Typically for the atmosphere the QG equations are valid for scales of order one to a few thousand kilometers and the PG equations are valid for nearly global scales excluding the equatorial region where the Rossby number may not be small In the ocean the QG equations are valid for scales of order tens to hundreds of kilometers and the PG equations again are valid for much larger scales Furthermore the large separation in spatial scale between the deformation radius and the radius of the planet in the ocean yields an additional parameter regime namely the so called frontal geostrophic FG regime In this regime large variations in the height field or stratification are allowed but the Coriolis parameter is not allowed to vary significantly A balanced set of equations valid in this regime was asymptotically derived by Cushman Roisin 1986 Whether this regime 1580 JOURNAL OF THE ATMOSPHERIC SCIENCES exists or is important in the atmosphere is less likely due to the lack of a significant scale separation between the deformation radius and the planetary radius Many of the important circulation patterns in the atmosphere or ocean span these parameter regimes For example although baroclinic instability may preferentially occur near the deformation scale there may be a significant instability at larger scales e g the Green modes found by Green 1960 which might more properly be described with a model that is valid in the PG regime In any case the nonlinear interactions of eddies at the deformation scale leading to a cascade of energy to larger scales and eddy mean flow interaction certainly span the parameter regime from deformation scale to planetary scale although the flow is in near geostrophic balance at all scales An ideal model for conceptual studies of the circulation would contain both the QG and PG and possibly FG regimes while exploiting the smallness of the Rossby number While formal accuracy with respect to the primitive equations or shallow water equations in an idealized setting should be roughly maintained over the parameter range of interest as a function of the small parameter exploited in the approximation it is we believe more important that it be valid over a broad parameter regime than that the model have high order accuracy with respect to that small parameter Two geostrophic models by geostrophic model we mean merely that it is based on the smallness of the Rossby number have been proposed that we explicitly show do in fact span both QG and PG regimes These are the L1 model Salmon 1983 and the simpler geostrophic potential vorticity GPV equations Vallis 1996 see also Bleck 1973 That is both models include both the QG and the PG equations in the appropriate limit in parameter space Each model is thus valid for O 1 variations in the layer thickness provided the variations occur on a sufficiently


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