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On the Conservation and Impermeability Theorems for Potential Vorticity

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  • 1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom
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Abstract

If the partial analogy between the behavior of Rossby-Ertel potential vorticity (PV) and the behavior of chemical tracers is to be correctly used in the general case of diabatic, frictional motion, then certain fundamental differences, as well as similarities, between the behavior of PV and that of chemical tracers must be recognized. These differences stem from the well-known kinematical relationship between PV and isentropic circulation (via Stokes' theorem), which has no counterpart for chemical substances.

One way of stating the analogy while recognizing the differences is to say first that PV behaves like the mixing ratio of a peculiar chemical “substance” that has zero source; i.e., is exactly conserved, away from boundaries (conserved not in the material or Lagrangian sense, but in the general sense associated with the idea of an indestructible chemical substance), and second that isentropic surfaces behave exactly as if they were impermeable to this “PV-substance” or “PVS,” even when diabatic heating or cooling, including that associated with turbulent mixing, makes them permeable to mass and chemical substances. In this respect isentropic surfaces can be said to act like semipermeable membranes. The PV itself can of course change, as can the mixing ratio of an exactly conserved chemical substance or decay-corrected radioactive tracer. For instance, all these mixing ratios can change by dilution when cumulonimbus clouds penetrate isentropic surfaces in a tropopause fold.

The net flux or transport of PVS along isentropic surfaces can be either up or down any pre-existing isentropic gradient of PV. For instance the typical effect of the small-scale turbulence due to breaking internal gravity waves is to transport PVS along isentropes in a gradient-independent sense, while transporting chemical substances across isentropes in a downgradient sense. It is the turbulent transport of PVS along isentropes that gives rise to the phenomenon of gravity-wave drag. Such a transport is absent from the formulation given in Danielsen (1990), which supposes that PV always behaves like the mixing ratio of a chemical even in three-dimensionally turbulent flow. The latter supposition is demonstrably incorrect.

Abstract

If the partial analogy between the behavior of Rossby-Ertel potential vorticity (PV) and the behavior of chemical tracers is to be correctly used in the general case of diabatic, frictional motion, then certain fundamental differences, as well as similarities, between the behavior of PV and that of chemical tracers must be recognized. These differences stem from the well-known kinematical relationship between PV and isentropic circulation (via Stokes' theorem), which has no counterpart for chemical substances.

One way of stating the analogy while recognizing the differences is to say first that PV behaves like the mixing ratio of a peculiar chemical “substance” that has zero source; i.e., is exactly conserved, away from boundaries (conserved not in the material or Lagrangian sense, but in the general sense associated with the idea of an indestructible chemical substance), and second that isentropic surfaces behave exactly as if they were impermeable to this “PV-substance” or “PVS,” even when diabatic heating or cooling, including that associated with turbulent mixing, makes them permeable to mass and chemical substances. In this respect isentropic surfaces can be said to act like semipermeable membranes. The PV itself can of course change, as can the mixing ratio of an exactly conserved chemical substance or decay-corrected radioactive tracer. For instance, all these mixing ratios can change by dilution when cumulonimbus clouds penetrate isentropic surfaces in a tropopause fold.

The net flux or transport of PVS along isentropic surfaces can be either up or down any pre-existing isentropic gradient of PV. For instance the typical effect of the small-scale turbulence due to breaking internal gravity waves is to transport PVS along isentropes in a gradient-independent sense, while transporting chemical substances across isentropes in a downgradient sense. It is the turbulent transport of PVS along isentropes that gives rise to the phenomenon of gravity-wave drag. Such a transport is absent from the formulation given in Danielsen (1990), which supposes that PV always behaves like the mixing ratio of a chemical even in three-dimensionally turbulent flow. The latter supposition is demonstrably incorrect.

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