Origins and Evolution of Imbalance in Synoptic-Scale Baroclinic Wave Life Cycles

Andrew B. G. Bush Department of Physics, University of Toronto, Toronto, Ontario, Canada

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James C. McWilliams Department of Physics, University of Toronto, Toronto, Ontario, Canada

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W. Richard Peltier Department of Physics, University of Toronto, Toronto, Ontario, Canada

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Abstract

A set of balance equations is derived that is appropriate for analysis of the three-dimensional anelastic system and is based on expansions in Rossby and Froude number similar to those employed in the study of the shallow-water equations by Spall and McWilliams. Terms that constitute the usual balance equations are formally retained here in addition to non-Boussinesq terms of the same order arising from the vertical variation of the background density field. The authors apply the derived set of equations diagnostically to the analysis of three-dimensional, anelastic numerical simulations of a synoptic-scale baroclinic wave. Of particular interest in this analysis is the degree to which and the time at which the flow becomes appreciably unbalanced, as well as the form of the imbalance itself. Unbalanced motions are here defined as departures from solutions of the balance equations. Application of this analysis procedure allows us to identify two classes of unbalanced motion, respectively: 1) unbalanced motion that is slaved to the balanced motion and is therefore characterized by the same time and length scales as the balanced motion (i.e., higher-order corrections on the “slow” manifold) and 2) unbalanced motion that is on a faster timescale than the large-scale balanced motion but is nevertheless forced by these same balanced motions (e.g., forced internal gravity waves). It will he shown in the analysis that both forms of imbalance arise in the frontal zones generated during the numerical simulation, but that the gravity wave generation is probably a numerical artifact of insufficient vertical resolution as the slope of the surface front decreases below the threshold required for consistent horizontal and vertical resolution. The total unbalanced motion field is dominated by the slower advective motion, but the numerically generated gravity waves nevertheless reach a peak amplitude comparable to that of the slower unbalanced motion. Whether internal wave radiation would persist, or perhaps become more intense, with increased spatial resolution is an issue that is left unresolved in the present analysis.

Abstract

A set of balance equations is derived that is appropriate for analysis of the three-dimensional anelastic system and is based on expansions in Rossby and Froude number similar to those employed in the study of the shallow-water equations by Spall and McWilliams. Terms that constitute the usual balance equations are formally retained here in addition to non-Boussinesq terms of the same order arising from the vertical variation of the background density field. The authors apply the derived set of equations diagnostically to the analysis of three-dimensional, anelastic numerical simulations of a synoptic-scale baroclinic wave. Of particular interest in this analysis is the degree to which and the time at which the flow becomes appreciably unbalanced, as well as the form of the imbalance itself. Unbalanced motions are here defined as departures from solutions of the balance equations. Application of this analysis procedure allows us to identify two classes of unbalanced motion, respectively: 1) unbalanced motion that is slaved to the balanced motion and is therefore characterized by the same time and length scales as the balanced motion (i.e., higher-order corrections on the “slow” manifold) and 2) unbalanced motion that is on a faster timescale than the large-scale balanced motion but is nevertheless forced by these same balanced motions (e.g., forced internal gravity waves). It will he shown in the analysis that both forms of imbalance arise in the frontal zones generated during the numerical simulation, but that the gravity wave generation is probably a numerical artifact of insufficient vertical resolution as the slope of the surface front decreases below the threshold required for consistent horizontal and vertical resolution. The total unbalanced motion field is dominated by the slower advective motion, but the numerically generated gravity waves nevertheless reach a peak amplitude comparable to that of the slower unbalanced motion. Whether internal wave radiation would persist, or perhaps become more intense, with increased spatial resolution is an issue that is left unresolved in the present analysis.

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