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Boundary Effects in Potential Vorticity Dynamics

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Boundary Effects in Potential Vorticity Dynamics 1024 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 Boundary Effects in Potential Vorticity Dynamics TAPIO SCHNEIDER* Courant Institute of Mathematical Sciences and Center for Atmosphere±Ocean Science, New York University, New York, New York ISAAC M. HELD AND STEPHEN T. G ARNER NOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey (Manuscript received 19 December 2001, in ®nal form 30 August 2002) ABSTRACT Many aspects of geophysical ¯ows can be described compactly in terms of potential vorticity dynamics. Since potential temperature can ¯uctuate at boundaries, however, the boundary conditions for potential vorticity dy- namics are inhomogeneous, which complicates considerations of potential vorticity dynamics when boundary effects are dynamically signi®cant. A formulation of potential vorticity dynamics is presented that encompasses boundary effects. It is shown that, for arbitrary ¯ows, the generalization of the potential vorticity concept to a sum of the conventional interior potential vorticity and a singular surface potential vorticity allows one to replace the inhomogeneous boundary conditions for potential vorticity dynamics by simpler homogeneous boundary conditions (of constant potential temperature). Functional forms of the surface potential vorticity are derived from ®eld equations in which the potential vorticity and a potential vorticity ¯ux appear as sources of ¯ow quantities in the same way in which an electric charge and an electric current appear as sources of ®elds in electrodynamics. For the generalized potential vorticity of ¯ows that need be neither balanced nor hydrostatic and that can be in¯uenced by diabatic processes and friction, a conservation law holds that is similar to the conservation law for the conventional interior potential vorticity. The conservation law for generalized potential vorticity contains, in the quasigeo- strophic limit, the well-known dual relationship between ¯uctuations of potential temperature at boundaries and ¯uctuations of potential vorticity in the interior of quasigeostrophic ¯ows. A nongeostrophic effect described by the conservation law is the induction of generalized potential vorticity by baroclinicity at boundaries, an effect that plays a role, for example, in mesoscale ¯ows past topographic obstacles. Based on the generalized potential vorticity concept, a theory is outlined of how a wake with lee vortices can form in weakly dissipative ¯ows past a mountain. Theoretical considerations and an analysis of a simulation show that a wake with lee vortices can form by separation of a generalized potential vorticity sheet from the mountain surface, similar to the separation of a friction-induced vorticity sheet from an obstacle, except that the generalized potential vorticity sheet can be induced by baroclinicity at the surface. 1. Introduction potential vorticity dynamics. And in situations in which the quasigeostrophic approximation is inadequate, such Since potential vorticity is materially conserved in as for planetary-scale ¯ows, considerations of potential adiabatic and frictionless ¯ows, and since it contains all vorticity dynamics on isentropes (or on isopycnals in relevant information about balanced ¯ows in a single the ocean) have proven fruitful (see, e.g., Tung 1986; scalar ®eld, many aspects of geophysical ¯ows can be Rhines and Young 1982). Since potential temperature described compactly in terms of potential vorticity dy- can ¯uctuate at boundaries, however, the boundary con- namics. For example, the propagation of Rossby waves ditions for potential vorticity dynamics are inhomoge- and the development of baroclinic instability have tra- neous, which complicates considerations of potential ditionally been described in terms of quasigeostrophic vorticity dynamics when boundary effects are dynam- ically signi®cant. Within quasigeostrophic theory, * Current af®liation: Division of Geological and Planetary Sciences Bretherton (1966) has shown that the inhomogeneous and Division of Engineering and Applied Sciences, California Insti- boundary condition implied by a ¯uctuating potential tute of Technology, Pasadena, California. temperature at a boundary can be replaced by a ho- mogeneous boundary condition of constant potential temperature if a singular surface potential vorticity pro- Corresponding author address: Tapio Schneider, California Insti- tute of Technology, Mail Code 100-23, 1200 E. California Blvd., portional to the surface potential temperature ¯uctua- Pasadena, CA 91125. tions is included in the quasigeostrophic potential vor- E-mail: [email protected] ticity. Extending Bretherton's argumentation, Rhines q 2003 American Meteorological Society 15 APRIL 2003 SCHNEIDER ET AL. 1025 (1979) has shown that not only surface potential tem- in coordinate-independent form in section 3, is expand- perature ¯uctuations, but also the topography of a ed in isentropic coordinates. boundary can be taken into account in a quasigeostroph- Section 5 discusses the baroclinic induction of gen- ic surface potential vorticity. Bretherton's and Rhines's eralized potential vorticity at boundaries, a nongeo- generalization of the quasigeostrophic potential vorticity strophic effect. An analysis of a simulated Boussinesq concept has been used to describe the interaction be- ¯ow demonstrates that the formation of a wake with lee tween quasigeostrophic potential vorticity ¯uctuations vortices in ¯ows past a mountain with a free-slip surface in the interior of a ¯ow on the one hand and surface can be described in terms of generalized potential vor- potential temperature ¯uctuations and/or topographic ticity dynamics and the baroclinic induction of gener- slopes on the other hand, for example, in unstable bar- alized potential vorticity at the mountain surface. A oclinic waves [see Hoskins et al. (1985) and Hallberg wake with lee vortices can form by separation of a gen- and Rhines (2000) for reviews]. Here we present a sim- eralized potential vorticity sheet from the mountain sur- ilar generalization of the potential vorticity concept that face, similar to the separation of a friction-induced vor- allows for the inclusion of boundary effects in the po- ticity sheet from an obstacle, except that the generalized tential vorticity dynamics of arbitrary nongeostrophic potential vorticity sheet can be induced by baroclinicity ¯ows. at the surface. In arbitrary ¯ows as in quasigeostrophic ¯ows, the Section 6 summarizes the conclusions. The appendix inhomogeneous boundary conditions for potential vor- lists the notation and symbols used in this paper. ticity dynamics can be replaced by homogeneous bound- The analyses presume an ideal-gas atmosphere with ary conditions if the simpli®cation of the boundary con- the planet's surface as the only dynamically relevant ditions is compensated by a generalization of the po- boundary. However, the concepts and mathematical tential vorticity concept to a sum of the conventional techniques presented are easily adaptable, for example, interior potential vorticity and a singular surface poten- to ocean ¯ows with lateral boundaries and with a more tial vorticity. In addition to the contributions from sur- complex thermal equation of state, irrespective of the face potential temperature ¯uctuations that are contained fact that the more complex thermal equation of state of in the quasigeostrophic surface potential vorticity, the seawater implies that potential vorticity is not neces- surface potential vorticity generally also contains con- sarily materially conserved in adiabatic and frictionless tributions from surface vorticity ¯uctuations, which are ocean ¯ows (cf. McDougall 1988). neglected in the quasigeostrophic approximation. We derive functional forms of the generalized potential vor- ticity and of its conservation law and discuss nongeo- 2. Potential vorticity and potential vorticity ¯ux as strophic effects described by the generalized potential sources of ¯ow quantities vorticity conservation law. As an illustration of how the a. Field equations generalized potential vorticity concept can be used to describe ¯ows for which the quasigeostrophic approx- The potential vorticity is the pseudoscalar function imation is inadequate, we demonstrate that this concept va ´ =u can form a basis of theories of lee vortex formation in P 5 (1) r mesoscale ¯ows past a mountain. Sections 2 through 4 set up the formal framework of of absolute vorticity va, potential temperature u, and den- generalized potential vorticity dynamics. Section 2 casts sity r. The absolute vorticity is the curl va 5 = 3 ua the momentum equation with the help of the thermo- of the three-dimensional absolute velocity ua 5 u 1 dynamic equation in the form of ®eld equations in which V 3 r, or the sum va 5 vr 1 2V of the relative the potential vorticity and the potential vorticity ¯ux vorticity vr 5 = 3 u and the vorticity = 3 (V 3 r) appear as sources of ¯ow quantities in the same way in 5 2V of a planetary rotation with constant angular which an electric charge and an electric current appear velocity V. The potential vorticity P is a conserved as sources of ®elds in electrodynamics. Section 3 de- quantity with a conservation law of the ¯ux form rives, by means of techniques from electrodynamics, the ( P) ´( J) 0, (2) functional forms of the generalized potential vorticity ]t r 1 = r 5 and of the generalized potential vorticity ¯ux that re- with a potential vorticity ¯ux1 place the conventional interior potential vorticity and 21 21 potential vorticity ¯ux when the
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