Boundary Effects in Potential Vorticity Dynamics

1024 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60

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 dynamics 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 quasigeostrophic 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 dynamically 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 homogeneous 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

᭧ 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 vorticity 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 ١␪ ´ ␻a can form a basis of theories of lee vortex formation in P ϭ (1) ␳ mesoscale ¯ows past a mountain.

Sections 2 through 4 set up the formal framework of of absolute vorticity ␻a, potential temperature ␪, and den- ϫ ua ١ generalized potential vorticity dynamics. Section 2 casts sity ␳. The absolute vorticity is the curl ␻a ϭ the momentum equation with the help of the thermo- of the three-dimensional absolute velocity ua ϭ u ϩ dynamic equation in the form of ®eld equations in which ⍀ ϫ r, or the sum ␻a ϭ ␻r ϩ 2⍀ of the relative (ϫ (⍀ ϫ r ١ ϫ u and the vorticity ١ the potential vorticity and the potential vorticity ¯ux vorticity ␻r ϭ appear as sources of ¯ow quantities in the same way in ϭ 2⍀ of a planetary rotation with constant angular which an electric charge and an electric current appear velocity ⍀. 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) ␳ ϭ ١ t ␳ ϩץ functional forms of the generalized potential vorticity and of the generalized potential vorticity ¯ux that re- with a potential vorticity ¯ux1 place the conventional interior potential vorticity and Ϫ1 Ϫ1 (١␪ ϫ F (3 potential vorticity ¯ux when the inhomogeneous bound- J ϭ uP Ϫ ␳ Q␻a ϩ ␳ ary conditions for potential vorticity dynamics are re- in which Q ϭ D␪/Dt is a diabatic heating rate and F a placed by homogeneous boundary conditions. For the generalized potential vorticity, a conservation law holds that reduces, in the quasigeostrophic limit, to the con- 1 This potential vorticity ¯ux J differs from the quantity ␳J that Haynes and McIntyre (1987) call potential vorticity ¯ux by a density servation law for Bretherton's (1966) generalized qua- factor ␳. We refer to the quantity ␳J as the potential vorticity ¯ux sigeostrophic potential vorticity. In section 4, the con- density. For a uniqueness property of the potential vorticity ¯ux (3), servation law for generalized potential vorticity, derived see Bretherton and SchaÈr (1993). 1026 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60

frictional force per unit mass (Haynes and McIntyre H uniquely. The Maxwell equations (4) are invariant 1987). Diabatic heating Q and frictional forces F con- under gauge transformations of the form tribute to the potential vorticity ¯ux J and redistribute ϫ A ١ D ← D ϩ potential vorticity within a ¯ow, but in the interior of (5) ← ,١␺ tA ϩץthe ¯ow, they do not create or destroy potential vorticity H H ϩ (Haynes and McIntyre 1987, 1990). where A is a vector ®eld and ␺ a scalar ®eld. Given a Since the divergence of the absolute vorticity van- potential vorticity P, a potential vorticity ¯ux J, and a ١␪ of density ´ ␻ ϭ 0, the product ␳P ϭ ␻ ´ ١ ,ishes a a density ␳, the ®elds D and H are only determined up D of ´ ١ and potential vorticity is the divergence ␳P ϭ to such gauge transformations. a vector ®eld D. The density and potential vorticity The freedom in the de®nition of the ®elds D and H determine the vector ®eld D up to a nondivergent com- can be exploited to ®nd gauges of the ®elds D and H ponent. For example, one might choose D ϭ ␪␻ ,orD a that are amenable to physical interpretation or are con- ١␪. The difference between these two choices ϭ u ϫ a venient in speci®c contexts. ϫ (␪u a). For ١ for D is the nondivergent vector ®eld both choices for D, the product ␳P is the divergence, and hence the source, of D. c. Gauge I That the product ␳P is both a conserved quantity and With the choice the divergence of a vector ®eld can be expressed through

(١␪, (6 eld equations that make some properties of potential D ϭ ua ϫ® vorticity manifest and that will be convenient in analyz- -D ϭ ␳P is the diver ´ ١ the potential vorticity density ing the role of boundaries in potential vorticity dynam- gence of a ®eld that indicates, up to a minus sign, the ,D ´ ١ ics. Upon substitution of the divergence ␳P ϭ absolute angular momentum of the ¯ow along isentro- the conservation law (2) for potential vorticity becomes pes.3 D ϩ ␳J) ϭ 0 (cf. Bretherton and SchaÈr 1993). It ץ)´ ١ t A ®eld H for this choice for D can be found from follows that the potential vorticity ¯ux density ␳J has the Maxwell equation (4b) by expanding the time de- the form -t␪) and by subץ)١ ١␪ ϩ ua ϫ tua ϫץtD ϭץ rivative ,ϫ H ١ D ϩץ␳J ϭϪ tu from theץtua ϭץ t stituting for the velocity derivative tD ϩ ␳J. momentum equationץ where H is the vector potential of the sum The facts that ␳P is both a conserved quantity and the 11 2 ١⌽ϩF ١࿣u࿣ Ϫ ١p Ϫ tau ϩ ␻ ϫ u ϭϪץ divergence of a vector ®eld D can thus be expressed through the ®rst two Maxwell equations: ␳ 2 t␪ from theץ D ϭ ␳P, (4a) and for the potential temperature derivative ´ ١ thermodynamic equation (tD ϭ ␳J. (4bץϫ H Ϫ ١ (١␪ ϭ Q. (7 ´ ␪ ϩ u ץ The quantity ␳P corresponds to the charge density in t electrodynamics, the ®eld D to the electric displacement Using the differential dhˆ ϭ ␳Ϫ1dp ϩ TdsÃ of the speci®c

-log␪ be ١sÃ ϭ cp١ eld, the ®eld H to the magnetic ®eld, and the potential enthalpy hˆ ϭ cpT and the relation® vorticity ¯ux density ␳J to the current density. We refer tween gradients of speci®c entropy sÃ and gradients of to the quantity ␳P as the potential vorticity density.2 In potential temperature ␪, one can cast the momentum the Maxwell equations (4), the potential vorticity P and equation in the form (cf. Batchelor 1967, chapter 3.5; the potential vorticity ¯ux J appear as sources of the SchaÈr 1993) ®elds D and H, just as in electrodynamics charges and u u cT log B F, (8) ϩ ١ ␪ Ϫ taϩ ␻ ϫ ϭ p١ץ currents appear as sources of the electric displacement ®eld and of the magnetic ®eld. The conservation law with Bernoulli function (2) for potential vorticity follows from the Maxwell 1 equations by adding the time derivative of Eq. (4a) to 2 B ϭ ࿣u࿣ ϩ cTp ϩ⌽. (9) the divergence of Eq. (4b). 2 Taking the cross product of the momentum equation (8) b. Gauge invariance

3 ١␪ in isentropic The potential vorticity P, the potential vorticity ¯ux More precisely, if one expands the ®eld D ϭ ua ϫ J, and the density ␳ do not determine the ®elds D and coordinates, using the hydrostatic approximation and the notation and Ϫ1 ␪ techniques that follow in section 4, one obtains D ϭ h (␷a, Ϫua,0). Within the approximations of the primitive equations (in which the 2 We prefer the term ``potential vorticity density'' to Haynes and distance from any point in the atmosphere to the center of the planet McIntyre's (1990) term ``amount per unit volume of potential vor- is taken to be equal to the planet radius a), this ®eld D is related to ␪ ticity substance.'' The word substance connotes an independence of the absolute angular momentum density ␳L␪ ϭ ␳a(Ϫ␷a, ua,0) of the Ϫ1 other quantities and a subsistence in itself that are not characteristic ¯ow along isentropes by ␳␪D ϭϪa ␳L␪. The divergence of the ®eld Ϫ1 .(yuaץx␷a Ϫץ) D ϭ h ´ ١ of the potential vorticity. D becomes in isentropic coordinates 15 APRIL 2003 SCHNEIDER ET AL. 1027

١␪ leads to the d. Gauge II and the potential temperature gradient Maxwell equation (4b) with potential vorticity ¯ux (3) For the analysis of boundary conditions in section 3, and ®eld the choice

(ta␪)u . (10ץ) H ϭϪB١␪ Ϫ D ϭ ␪␻a (11) The explicit time derivative in this expression for t␪ is convenient. A ®eld H for this choice for D can beץ the ®eld H could be expanded by substitution from the found, as above, from the Maxwell equation (4b) by thermodynamic equation (7); however, with the explicit ( ␻ ץ)␪)␻ ϩ ␪ ץ) D ϭ ץ expanding the time derivative time derivative , it is evident that the second term in t t a t a -t␪ and by substituting for the potential temperature derivץ the expression for the ®eld H does not appear in a rep- ␪ from the thermodynamic equation (7) and for ץ ative resentation of the ®eld in isentropic coordinates. In is- t t␻r from the vorticityץt␻a ϭץ the vorticity derivative entropic coordinates, the ®eld H is proportional to the equation Bernoulli function times the cross-isentropic basis vec-

(log␪ ϩ F) (12 ϫ (u ϫ ␻ aϩ cT p١ ١ tr␻ ϭץ ϫ H involves only derivatives ١ ١␪, and the curl tor 4 of the Bernoulli function along isentropes. belonging to the momentum equation (8). Similar al- In a steady state, the Maxwell equation (4b) reduces gebra to the above leads to the potential vorticity ¯ux ϫ H, which becomes, in this gauge, ␳J ϭ (3) and to the ®eld5 ١ to ␳J ϭ ١B. That is, as noted by SchaÈr ١␪ ϫ ϫ B١␪ ϭ Ϫ١ (the potential vorticity ¯ux is directed along lines H ϭ u ϫ ␪␻apϩ cT١␪ ϩ ␪F. (13 ,(1993) of intersection between surfaces of constant potential This functional form of the ®eld H does not seem to temperature ␪ (isentropes) and surfaces of constant Ber- have a simple physical interpretation. noulli function B. The Bernoulli function B is the Irrespective of the gauge of the ®elds D and H, the streamfunction of the potential vorticity ¯ux J along Maxwell equations (4) are a way of arranging the mo- isentropes. Since the Bernoulli function is a measure of mentum equation with the help of the thermodynamic the speci®c energy of the moving ¯uid, along-stream equation such that the existence of a conservation law gradients of the Bernoulli function imply energy dis- (2) for potential vorticity is immediately evident. At the ١B be- same time, the Maxwell equations represent differential ١␪ ϫ sipation. According to the relation ␳J ϭ tween potential vorticity ¯uxes and Bernoulli function, equations that link the potential vorticity and the po- energy dissipation indicated by gradients of the Ber- tential vorticity ¯ux to the velocity (or vorticity) and noulli function along streamlines on isentropes are as- the potential temperatureÐa link that is necessary in sociated, in a steady state, with across-stream potential any formulation of dynamics in which the conservation vorticity ¯uxes. Some practical implications of this gen- law for potential vorticity is taken as a fundamental eralization of Bernoulli's theorem are discussed by equation. Together with boundary conditions and with SchaÈr (1993), SchaÈr and Durran (1997), and, in an oce- other dynamical equations, the Maxwell equations de- anic context, by Marshall et al. (2001). termine the primitive variables given the potential vor- The Maxwell equations (4) in the gauge given by Eqs. ticity and its ¯ux. The Maxwell equations provide a (6) and (10) generalize to time-dependent ¯ows the re- framework within which one can analyze the role of lation between potential vorticity ¯uxes and Bernoulli boundaries in the potential vorticity dynamics of ¯ows function that SchaÈr (1993) noted for steady ¯ows. In that need be neither balanced nor hydrostatic and that this gauge, the potential vorticity P and the potential can be in¯uenced by diabatic processes and friction. vorticity ¯ux J appear as sources of orthogonal ®elds D and H that indicate, up to minus signs, the absolute angular momentum and the energy of the ¯ow along 3. Generalized potential vorticity in coordinate- isentropes. With the potential vorticity ¯ux J understood independent form as source (or, because of the minus sign, as sink) of a a. Boundary conditions and boundary sources ®eld that indicates the energy of the ¯ow along isentropes, it is not as surprising as it might seem that a Since potential temperature can ¯uctuate at the sur- nonconservative force F and the diabatic heating rate face, the boundary conditions for the ®elds D and H in Q contribute to a quantity J that is called a ¯ux. the Maxwell equations (4) are generally inhomogeneous. These inhomogeneous boundary conditions at

4 t␪)uaץ) More precisely, if one expands the ®eld H ϭϪB١␪ Ϫ in isentropic coordinates, using the hydrostatic approximation and 5 Alternatively, the frictional term in the potential vorticity ¯ux J ϫ F, in which case the ®eld H the notation and techniques that follow in section 4, one obtains H could have been written as Ϫ␳Ϫ1␪١ Ϫ1 ␪ ␪ Ϫ1 ␪ y z, h ) is the contravariant basis would not contain a frictional term. However, an advantage of theץx z, ϪץϭϪh Be , where e ϭ (Ϫ ١␪ nondimensionalized by multiplication by the scale factor functional forms (3) and (13) of the potential vorticity ¯ux J and vector h, and derivatives are to be understood as derivatives along isentropes ®eld H is that the frictional term ␳Ϫ1١␪ ϫ F in the potential vorticity (cf. Arfken 1985, chapter 3.8). So the ®eld H is related to the Ber- ¯ux has no cross-isentropic component and is therefore easier to ␪ noulli density ␳B by ␳␪H ϭϪ␳Be . The curl of the ®eld H becomes expand in isentropic coordinates than a frictional term of the form Ϫ1 ␪ Ϫ1 .(ϫ F (cf. section 4 x B,0). Ϫ␳ ␪١ ץy B, Ϫ ץ) ϫ H ϭϪh ١ in isentropic coordinates 1028 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 the surface, or ``immediately above'' it, can be replaced tion (4a) a source with a surface density of potential by homogeneous boundary conditions ``inside'' the sur- vorticity equal to n ´(Ds Ϫ Db) ϭ n ´ Ds. One can think face by inclusion of suitable boundary sources in the of this surface density of potential vorticity as the Maxwell equations (see, e.g., Morse and Feshbach 1953, across-surface integral of the potential vorticity density chapter 7). Since the potential vorticity P and the po- ␳S that belongs to the singular surface potential vorticity tential vorticity ¯ux J are sources in the Maxwell equa- n ´ D tions, the boundary sources can be viewed as boundary s S ϭ ␦(z Ϫ zs). contributions to the potential vorticity and to the po- ␳ tential vorticity ¯ux. The surface potential vorticity S is concentrated in a For the ®elds D and H in Gauge II, we specify the delta-function potential vorticity sheet on the surface at homogeneous boundary conditions Db ϭ 0 and Hb ϭ 0 z ϭ zs(x, y). The Gauge II expression D ϭ ␪␻a leads inside the surface. The subscript b denotes quantities to the surface potential vorticity inside the surface, and the subscript s will denote quantities immediately above the surface. Taking the ®elds ␻a ´ n S ϭ ␪␦(z Ϫ zs). (14) Db and Hb to be zero inside the surface can be viewed ␳ as a consequence of taking the potential temperature ␪b and with it, for consistency, all other ¯ow quantities in The divergence equation (4a) with inhomogeneous the thermodynamic equation (7) and momentum equa- boundary condition n ´ D ϭ n ´ Ds and potential vortic- tion (8) to be zero inside the surface.6 Additionally, we ity P is equivalent to a divergence equation (4a) with specify the potential vorticity density ␳P and the po- homogeneous boundary condition n ´ D ϭ n ´ Db ϭ 0 tential vorticity ¯ux density ␳J to be zero inside the and generalized potential vorticity surface. The homogeneous boundary conditions for the P ϭ P ϩ S. (15) ®elds D and H must be compensated by boundary con- g tributions to the potential vorticity and to the potential Replacing the inhomogeneous boundary condition for vorticity ¯ux. the ®eld D by a homogeneous boundary condition is compensated by putting an idealized potential vorticity sheet with surface potential vorticity S on the surface. b. Boundary contributions to the potential vorticity

At the interface between two media, the normal com- c. Boundary contributions to the potential vorticity ponent of the electric displacement ®eld has a discon- ¯ux tinuity proportional to a surface charge density at the interface (Jackson 1975, section I.5). Analogously, the At the interface between two media, the tangential normal component of the ®eld D has, at the surface, a component of the magnetic ®eld has a discontinuity pro- discontinuity proportional to a surface density of po- portional to a current density at the interface (Jackson tential vorticity. 1975, section I.5). Analogously, the tangential component of the ®eld H has, at the surface, a discontinuity Given the ®elds Db inside the surface and Ds immediately above the surface with upward normal n, one proportional to a surface density of potential vorticity can compute the surface potential vorticity that is re- ¯ux. quired to force the normal component of the ®eld D The surface ¯ux of potential vorticity that is required to force the tangential component n ϫ H of the ®eld from n ´ D ϭ n ´ Db ϭ 0 inside the surface to n ´ D ϭ H from n ϫ H ϭ n ϫ H ϭ 0 inside the surface to n ´ Ds immediately above the surface. Integrating the b divergence equation (4a) over an in®nitesimally small n ϫ H ϭ n ϫ Hs immediately above the surface can volume enclosing the surface and using Gauss's theo- be found by integrating the second Maxwell equation t D ϭ ␳J over an area perpendicular to andץϫ H Ϫ ١ -rem, one ®nds that the homogeneous boundary condi including the surface. In the limit of an in®nitesimally tion n ´ D ϭ n ´ Db ϭ 0 must be compensated by in- t Dץ cluding on the right-hand side of the divergence equa- small area, the area integral of the time derivative t D is ®nite at the surface. Usingץ vanishes because Stokes's theorem, one ®nds that the homogeneous 6 One could also specify a boundary condition of constant potential boundary condition n ϫ H ϭ n ϫ Hb ϭ 0 must be temperature ␪0 ± 0 inside the surface, leading to ®elds Db ϭ ␪0␻a compensated by including on the right-hand side of the ١␪ ϩ ␪ F inside the surface. Such(and H ϭ u ϫ ␪ ␻ ϩ c T(␪ /␪ b 0 a p 0 0 Maxwell equation (4b) a source with a surface density an alternative boundary condition corresponds to specifying homo- of potential vorticity ¯ux equal to n ϫ (Hs Ϫ Hb) ϭ geneous boundary conditions DbbЈ ϭ 0 and HЈ ϭ 0 for the ®elds DЈ ϭ (␪Ј/␪)D and HЈϭ(␪Ј/␪)H, obtained by replacing the potential n ϫ Hs. Analogous to the line of reasoning that led to temperature ␪ by the ¯uctuation ␪Јϭ␪ Ϫ␪0 about the constant po- the surface potential vorticity, one can think of the tential temperature ␪0. Equivalently, the ®elds DЈ and HЈ can be surface density of potential vorticity ¯ux as the obtained from the ®elds D and H by a gauge transformation of the across-surface integral of the potential vorticity ¯ux form (5) with A ϭϪ␪0ua. The ambiguities due to the invariance of the Maxwell equations under such gauge transformations are dis- density ␳K that belongs to the singular potential vor- cussed in section 3e. ticity ¯ux 15 APRIL 2003 SCHNEIDER ET AL. 1029

n ϫ H d. Conservation of generalized potential vorticity K ϭ s ␦(z Ϫ z ). ␳ s The generalized potential vorticity Pg and ¯ux Jg con- This surface potential vorticity ¯ux is concentrated on tain the boundary contributions to the potential vorticity the surface and has only components tangential to the balance. The original Maxwell equations (4) with in- surface. Substituting for the ®eld H from the Gauge II homogeneous boundary conditions at the surface and expression (13) and using the fact that, at the surface, with the interior potential vorticity P and ¯ux J as sourc- the normal component n ´ u of the velocity u vanishes, es are equivalent to Maxwell equations with homoge- one obtains the surface potential vorticity ¯ux neous boundary conditions inside the surface and with the generalized potential vorticity Pg and ¯ux Jg as

K ϭ uS ϩ Kbc ϩ KF (16a) sources. Adding the time derivative of the ®rst Maxwell equation (4a) to the divergence of the second equation with baroclinic component (4b) yields the conservation law K Ϫ1cT(n ) (z z ) (16b) (␳J g) ϭ 0 (18)´ ١ tg(␳P ) ϩץ ١␪ ␦ Ϫ bc ϭ ␳ psϫ and frictional component for the generalized potential vorticity Pg. The conservation law (18) for generalized potential Ϫ1 KFsϭ ␳␪(n ϫ F)␦(z Ϫ z ). (16c) vorticity is similar to the conservation law (2) for in- The surface potential vorticity ¯ux K consists of an terior potential vorticity. In contrast to interior potential vorticity, however, generalized potential vorticity is not, advective component uS, of a baroclinic component Kbc that is directed along lines of intersection between is- in general, materially conserved in adiabatic and fric- entropes and the surface, and of a frictional component tionless ¯ows. The baroclinic component (16b) of the 7 surface potential vorticity ¯ux can redistribute gener- KF. The baroclinic component Kbc has its origin in the -١p to alized potential vorticity nonadvectively along the sur contribution of the baroclinicity vector ␳Ϫ2١␳ ϫ the vector potential H. With a no-slip boundary con- face, even in adiabatic and frictionless ¯ows. dition at the surface, the velocity u along the surface Nevertheless, the integral of the generalized potential vanishes, and the surface potential vorticity ¯ux K con- vorticity density ␳Pg over the volume of the atmosphere is conserved. Integrating the conservation law (18) over sists only of the nonadvective components Kbc and KF. The advective component of the surface potential vor- the volume V of the atmosphere and using Gauss's the- ticity ¯ux might, nevertheless, be of practical relevance. orem with the boundary condition n ´(␳Jg) ϭ 0 inside For example, numerical atmosphere models typically the surface and with the assumption that the ¯ux density use not a no-slip boundary condition, but a drag-law ␳Jg vanish at the top of the atmosphere, one obtains boundary condition, so the advective component of the ,␳Pdx ϭ 0 ץ surface potential vorticity ¯ux might not be negligible t ͵ g in a generalized potential vorticity budget of such a V model. The advective component of the surface poten- or, equivalently, tial vorticity ¯ux is proportional to the surface heat ¯ux, and horizontal heat ¯uxes are signi®cant down to the .␳Sdx ץ␳Pdx ϭϪ ץ lowest levels of typical general circulation models and t ͵ t ͵ down to the lowest atmospheric levels for which ob- VV servational data are available (cf. Held and Schneider Any increase in the volume-integrated interior potential 1999). vorticity density ␳P is compensated by a decrease in the The Maxwell equation (4b) with inhomogeneous volume-integrated surface potential vorticity density ␳S, boundary condition n ϫ H ϭ n ϫ Hs and potential and vice versa. The conservation law (2) for interior vorticity ¯ux J is equivalent to a Maxwell equation (4b) potential vorticity implies that the volume-integrated in- with homogeneous boundary condition n ϫ H ϭ n ϫ Hb terior potential vorticity density ␳P can change only if ϭ 0 and generalized potential vorticity ¯ux the integral of the normal ¯ux component

(١␪ ϫ F)´ Jg ϭ J ϩ K. (17) n ´(␳J) ϭϪQ(n ´ ␻a) ϩ n Replacing the inhomogeneous boundary condition for over the surface area A is nonzero, the ®eld H by a homogeneous boundary condition is ١␪ ϫ F)] dA)´ ␳Pdx ϭϪ [Q(n ´ ␻ ) Ϫ n ץ -compensated by the added ¯ux K in the idealized po tential vorticity sheet on the surface. t ͵͵a VA

7 (t ␳Sdx. (19ץIf the frictional term in the interior potential vorticity ¯ux J would ϭϪ ͵ ϫ F, the ®eld H and, with it, the surface have been written as Ϫ␳Ϫ1␪١ V potential vorticity ¯ux K would not contain frictional components (cf. footnote 5). Since the normal component of the velocity vanishes at 1030 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 the surface, only diabatic and frictional processes at the generalized potential vorticity does not depend on the surface can effect changes in the volume-integrated in- gauge chosen. The gauge invariance of the Maxwell terior potential vorticity density ␳P. Given that changes equations (4) translates into gauge invariance of the con- in the volume-integrated interior potential vorticity den- servation law (18) for generalized potential vorticity. sity ␳P are compensated by opposing changes in volume- integrated surface potential vorticity density ␳S, diabatic f. Adiabatic and frictionless Boussinesq ¯ows and frictional processes at the surface can be viewed as converting surface potential vorticity into interior poten- The surface potential vorticity S and the surface po- tial vorticity, and vice versa. The surface integral of the tential vorticity ¯ux K for adiabatic and frictionless normal ¯ux component n ´(␳Jg) ϭ n ´(␳J) indicates the Boussinesq ¯ows can be derived in a similar manner as conversion rate. in the general case. In the Boussinesq approximation, the density ␳ in the potential vorticity (1) is taken to be equal to a constant reference density ␳ , and the potential e. Alternative generalized potential vorticity 0 vorticity is de®ned with the potential temperature ¯uc- functionals tuation ␪Јϭ␪ Ϫ ␪0 about a constant reference potential The functional forms of the generalized potential vor- temperature ␪0 in place of the absolute potential tem- ticity Pg ϭ P ϩ S and of the generalized potential vor- perature ␪: ticity ¯ux J ϭ J ϩ K are not unique because the func- ١␪Ј ´ g ␻ tional forms of the surface potential vorticity S and of P ϭ a . (20) the surface potential vorticity ¯ux K depend on the ␳ 0 gauge of the ®elds D and H. Correspondingly, the surface potential vorticity be- We chose Gauge II because, in this gauge, the func- comes tional forms of the surface potential vorticity (14) and of the surface potential vorticity ¯ux (16) resemble the ␻a ´ n S ϭ ␪Ј␦(z Ϫ zs). (21) functional forms of the interior potential vorticity (1) ␳ 0 and of the interior potential vorticity ¯ux (3). For example, the surface potential vorticity (14) is proportional The surface potential vorticity ¯ux for adiabatic and frictionless Boussinesq ¯ows consists of the advective to the absolute vorticity component ␻a ´ n normal to the surface, while the interior potential vorticity (1) is pro- component uS and of a baroclinic component Kbc. Which form the baroclinic component Kbc takes can be ١␪ ´ portional to the absolute vorticity component ␻a normal to isentropes. And like the interior potential vor- seen by going back to the derivation of the surface po- ticity ¯ux (3), the surface potential vorticity ¯ux (16) tential vorticity ¯ux in the general case. The baroclinic contains an advective component, which legitimizes its component has its origin in the baroclinicity vector Ϫ1 ١p. The baroclinic component resulted from ϫ ␳ interpretation as a ¯ux. Ϫ١ Ϫ1 ١p) in the expansion of ϫ ␳ ١)Alternatively, however, we could have chosen a gauge writing the term Ϫ␪ t D as the curl of the vector ®eldץ the time derivative in which, for example, a ®eld DЈϭ␪Ј␻a is de®ned with .the potential temperature ¯uctuation ␪Јϭ␪ Ϫ ␪ about cpT١␪, making this ®eld cpT١␪ part of the ®eld H [Eq 0 (13)], and taking the tangential component n ϫ H to a constant reference potential temperature ␪0 in place of the absolute potential temperature ␪ (cf. footnote 6). determine the surface potential vorticity ¯ux. In the Such a gauge suggests itself when a reference potential Boussinesq approximation, the baroclinicity vector is ϫ (g␪Ј/␪0)k, with vertical unit vector k, and the ١ temperature is given, such as is the case in Boussinesq Ϫ1 ١p) in the general case becomes ϫ ␳ ١)ows (see below). One obtains the alternative gauge DЈ term Ϫ␪¯ 2 ,ϫ (g␪Ј /2␪0)k (cf. Salmon 1998 ١ ϫ (g␪Ј/␪0)k) ϭ ١)and HЈ from Gauge II by a transformation of the form ␪Ј chapter 2.16). Consequently, the baroclinic component (5) with A ϭϪ␪0ua. Replacing inhomogeneous with homogeneous boundary conditions in this alternative gauge of the surface potential vorticity ¯ux becomes results in a surface potential vorticity SЈϭ(␪Ј/␪)S and a 1 g␪Ј2 surface potential vorticity ¯ux KЈϭ(␪Ј/␪)K. All of the Kbc ϭ (n ϫ k)␦(z Ϫ zs). (22) ␳ 2␪ above statements about the conservation of generalized 00 potential vorticity and about the structure of the surface The baroclinic component of the surface potential vor- potential vorticity and the surface potential vorticity ¯ux ticity ¯ux is quadratic in potential temperature ¯uctu- remain valid if the surface potential vorticity S and the ations ␪Ј and hence would not appear in a linearized ¯ux K are replaced by the alternative functionals SЈ and Boussinesq system. Instead of being directed along lines KЈ. This arbitrariness in the de®nition of the surface of intersection between isentropes and the surface, as potential vorticity and of the surface potential vorticity the baroclinic component (16b) in the general case, the ¯ux means that their absolute values by themselves car- baroclinic component of the Boussinesq surface poten- ry no dynamical signi®cance. tial vorticity ¯ux is directed along lines of constant sur- Other gauges are possible and may be convenient in face elevation. At a ¯at surface (n ϭ k), the baroclinic some contexts. However, the conservation law (18) for component vanishes, and the surface potential vorticity 15 APRIL 2003 SCHNEIDER ET AL. 1031

¯ux reduces to the advective ¯ux K ϭ uS. Since the We will determine the generalized potential vorticity Boussinesq approximation is often an adequate approx- and the components of the generalized potential vortic- imation for atmospheric ¯ows near the surfaceÐsay, ity ¯ux in isentropic coordinates by expanding the co- within the planetary boundary layerÐthe vanishing of ordinate-independent expressions of sections 2 and 3 in the baroclinic component Kbc of the surface potential isentropic coordinates. Expressions for the interior po- vorticity ¯ux for Boussinesq ¯ows over a ¯at surface tential vorticity (1) and for the interior potential vorticity suggests that this component is only important if to- ¯ux (3) in isentropic coordinates are well known; they pography exerts a signi®cant in¯uence on the ¯ow. The are usually derived from the equations of motion in baroclinic component Kbc of the surface potential vor- isentropic coordinates [see, e.g., Salmon (1998, chapter ticity ¯ux and topographic effects will be discussed in 2.18); Andrews et al. (1987, chapter 3.8)]. The tech- more detail in section 5. nique of expanding the coordinate-independent expressions in isentropic coordinates has the advantage of be- g. Quasigeostrophic Boussinesq ¯ows over ¯at ing applicable to the singular surface potential vorticity surface and its ¯ux, without it being necessary to go back to the equations of motion to deduce the representation of Taking the quasigeostrophic limit of the Maxwell these quantities in isentropic coordinates. equations for Boussinesq ¯ows, one ®nds that, for qua- Isentropic coordinates are nonorthogonal, so contra- sigeostrophic Boussinesq ¯ows over a ¯at surface (zs ϭ variant and covariant vector components must be dis- 0), the normal component ␻a ´ n of the absolute vorticity tinguished (see, e.g., Arfken 1985, chapter 3). We use in the surface potential vorticity (21) must be approx- the notation (ax, ay, a␪)␪ for the contravariant compo- imated by a constant reference value f0 of the Coriolis nents of a vector a in isentropic coordinates. The con- parameter f, so that the surface potential vorticity be- travariant horizontal components ax ϭ a ´ i and a y ϭ comes a ´ j are equal to the local Cartesian components of the f vector a, the local Cartesian unit vectors i and j being S ϭ 0 ␪Ј␦(z). directed eastward and northward.8 The contravariant ١␪ is the scalar ´ ␳ 0 cross-isentropic component a␪ ϭ a This surface potential vorticity corresponds to that product of the vector a and the potential temperature .١␪ boundary contribution to the quasigeostrophic potential gradient vorticity with which Bretherton (1966) replaced an inhomogeneous thermodynamic boundary condition. a. Generalized potential vorticity The baroclinic component (22) of the surface potential vorticity ¯ux vanishes for Boussinesq ¯ows over a The generalized potential vorticity takes a particularly ¯at surface, and, for adiabatic and frictionless quasi- simple form in isentropic coordinates. In the primitive geostrophic ¯ows, the surface potential vorticity ¯ux equations, the planetary vorticity 2⍀ is approximated reduces to the advective ¯ux by its local vertical component f k, and in the hydrostatic approximation, horizontal derivatives of the ver- f0 ϫ u are ١ K ϭ uggS ϭ u ␪Ј␦(z), tical velocity in the relative vorticity ␻r ϭ ␳ 0 neglected compared with vertical derivatives of the hor- where the advecting velocity ug is the geostrophic ve- izontal velocity. The absolute vorticity hence becomes ϫ v of the planetary vorticity ١ locity. For quasigeostrophic Boussinesq ¯ows over a ¯at the sum ␻a ϭ f k ϩ ϫ v of the horizontal ١ surface, the surface potential vorticity ¯ux is propor- f k and the relative vorticity tional to the geostrophic surface heat ¯ux. This surface ¯ow v ϭ (u, ␷, 0). The relative vorticity of the horizontal potential vorticity ¯ux corresponds to the boundary con- ¯ow can be represented in isentropic coordinates as (u),␪ (23ץ␷ Ϫ ץ ,u ץ ,␷ ץϫ v ϭ hϪ1(Ϫ ١ tribution to the quasigeostrophic potential vorticity ¯ux discussed by Bretherton (1966). ␪␪xy

-y are to be unץ x andץ where the horizontal derivatives derstood as derivatives along isentropes, and the scale 4. Generalized potential vorticity in isentropic factor coordinates h z, (24) ␪ץWe assume that for each time t and at each point in ϭ the (x, y) plane, the potential temperature ␪ is a strictly an inverse measure of static stability, is the Jacobian h -x, y, ␪) of the transformation from Car)ץ/(x, y, z)ץmonotonic function of height z, so that the instanta- ϭ neous thermal strati®cation is everywhere statically tesian coordinates to isentropic coordinates. By the rep- z␪ Ͼ 0), and the potential temperature can beץ) stable used as the vertical coordinate in an isentropic coor- 8 As horizontal coordinates, we use local Cartesian coordinates in dinate system. We adopt the hydrostatic approximation what follows. The transformation of the horizontal coordinates from and carry out the analysis within the framework of the local Cartesian coordinates to spherical coordinates is straightfor- primitive equations. ward. 1032 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60

ϫ v with the help of the expression ١ resentation (23) of the relative vorticity in isentropic relative vorticity coordinates, the interior potential vorticity density ␳P (23); the frictional force per unit mass F is assumed to x y -١␪Ðthe contravariant cross-isentropic component have only horizontal components F and F ; and prod ´ ϭ ␻a Ϫ1 ١␪ ϭ h ( f ϩ ␨␪), ucts of the scale factor h and the density ␳ are combined ´ of the absolute vorticityÐis ␻a -yu is the relative vorticity of the to the isentropic density ␳␪. The cross-isentropic comץx␷ Ϫץwhere ␨␪ ϭ horizontal ¯ow v along isentropes. In the hydrostatic ponents of the advective ¯ux uP and of the diabatic Ϫ1 Ϫ1 z p, and the term ␳ Q␻a cancel because, by the thermodynamicץ approximation, the density is ␳ ϭϪg product of density ␳ and scale factor h is the isentropic equation (7), the contravariant cross-isentropic com- ١␪ of the velocity is the heating rate Q, and ´ density ponent u ١␪ of ´ the contravariant cross-isentropic component ␻ p. a ץ␳ ϭ ␳h ϭϪgϪ1 ␪␪ the absolute vorticity is the potential vorticity density Combining the density ␳ and the scale factor h in the ␳P. Combining all terms, one ®nds the well-known in- Ϫ1 ١␪ ϭ h ( f ϩ ␨␪) yields terior potential vorticity ¯ux (cf. Haynes and McIntyre ´ potential vorticity density ␻a the well-known interior potential vorticity 1987)

␪ f ϩ ␨␪ J ϭ (u, ␷,0)P ϩ J ϩ J , (27a) P ϭ H(␪ Ϫ ␪ ). (25) QF ␳ s ␪ with diabatic ¯ux

The step function Ϫ1 ␪ (␪u,0)H(␪ Ϫ ␪s) (27b ץ␷, Ϫ ץ)JQ ϭ ␳␪␪Q 1if␪ Ͼ ␪s and frictional ¯ux H(␪ Ϫ ␪s) ϭ Ά0if␪ Ͻ ␪s Ϫ1 yx ␪ JF ϭ ␳␪ (ϪF , F ,0)H(␪ Ϫ ␪s). (27c) indicates that, according to our conventions, the interior potential vorticity P contributes to the generalized po- Even in the presence of diabatic heating and friction, the interior potential vorticity ¯ux has no cross-isentro- tential vorticity Pg only above the surface, on isentropes with potential temperature ␪ greater than the surface pic component. Therefore, the impermeability theorem holds: interior potential vorticity can only be redistrib- potential temperature ␪s(x, y, t). An isentropic-coordinate representation of the bound- uted along isentropes but cannot be transferred across ary contribution S to the generalized potential vorticity isentropes (Haynes and McIntyre 1987). In order to represent the surface potential vorticity Pg ϭ P ϩ S can be found in a similar way. Under the ¯ux (16) in isentropic coordinates, we use for the unit assumption of static stability, the delta function ␦(z Ϫ zs) Ϫ1 normal vector at the surface z ϭ z (x, y) the explicit transforms according to ␦(z Ϫ zs) ϭ h ␦(␪ Ϫ ␪s), where s the scale factor h is to be evaluated immediately above representation ,( ١z z Ϫ z ) ϭ ␮(k Ϫ)the surface (since the surface potential vorticity and its n ϭ ␮١ ¯ux only contain quantities immediately above the sur- ss face; cf. sections 3b and 3c). Combining the density ␳ with normalization factor

in the surface potential vorticity (14) with the scale factor 2 Ϫ1/2 .( h from the transformation of the delta function yields the ␮ ϭ (1 ϩ ࿣١zs࿣ isentropic-coordinate expression The hydrostatic approximation is only justi®able if the

horizontal scale of the topography zs(x, y) is much great- ␻a ´ n S ϭ ␪␦(␪ Ϫ ␪s) (26) er than the vertical scale, such that ␮ ഠ 1. For consis- ␳ ␪ tency with the hydrostatic approximation, we should set for the surface potential vorticity. the normalization factor ␮ equal to one. But with the Within the approximations of the primitive equations understanding that the hydrostatic approximation would and under the assumption of static stability, the gen- be inappropriate if the normalization factor ␮ were sig- eralized potential vorticity (15) is the sum of the interior ni®cantly less than one, we retain the normalization fac- potential vorticity (25) and the surface potential vortic- tor ␮ in the following equations as a marker of where ity (26) in isentropic coordinates. topographic effects can play a role. With the explicit representation of the normal vector n, the surface potential vorticity ¯ux can be expanded in b. Generalized potential vorticity ¯ux isentropic coordinates term-by-term. The delta functions The generalized potential vorticity ¯ux in isentropic in the surface potential vorticity ¯ux are transformed in Ϫ1 coordinates can likewise be found by expanding the the same way as above: ␦(z Ϫ zs) ϭ h ␦(␪ Ϫ ␪s). And vectors and differential operators of the coordinate-in- horizontal derivatives of the potential temperature ␪ at dependent interior potential vorticity ¯ux (3) and sur- constant height z are transformed into horizontal deriva- face potential vorticity ¯ux (16). The isentropic-coor- tives of the height z at constant potential temperature Ϫ1 Ϫ1 z | ␪ for xi ϭ ץ xxii␪ | z ϭϪhץ dinate representation of the term ␳ Q␻a in the interior ␪ by means of the relation potential vorticity ¯ux (3) follows by expanding the x, y. One obtains the surface potential vorticity ¯ux 15 APRIL 2003 SCHNEIDER ET AL. 1033

␪ K ϭ (u, ␷, Q) S ϩ Kbc ϩ KF, (28a) 5. Baroclinic induction of generalized potential vorticity with baroclinic component In adiabatic and frictionless ¯ows, the surface poten- ␮E yx ␪ ␪ Kbc ϭ (Ϫ␪ ss, ␪ ,0)␪␦(␪ Ϫ ␪ s) (28b) tial vorticity ¯ux (28) is the sum K ϭ (u, ␷,0) S ϩ Kbc ␳␪ of the advective component (u, ␷,0)␪ S and the baroclinic and frictional component component Kbc. The presence of the nonadvective baroclinic component K of the surface potential vorticity ␮ bc yxxy yx␪ ¯ux implies that surface potential vorticity, and through K Fsssϭ (ϪF , F , F ␪ Ϫ F ␪ ) ␪␦(␪ Ϫ ␪ ), (28c) ␳␪ it generalized potential vorticity, can be induced baroclinically. where (z Ϫ z ) (29) ץz Ϫ z ) and ␪ y ϭϪhϪ1) ץ␪ x ϭϪhϪ1 sxssys a. Origin of baroclinic component of surface are the derivatives of the surface potential temperature potential vorticity ¯ux ␪ (x, y, t) with respect to x and y. In the baroclinic s The baroclinic component K of the surface potential component (28b), the speci®c enthalpy c T has been bc p vorticity ¯ux arises because of differences between the written as the product c T ϭ ␪E of potential temperature p interior potential vorticity (f ϩ ␨ )/␳ and the quantity ␪ and Exner function E ϭ c (p/p )␬, p being a constant ␪ ␪ p 0 0 (␻ ´ n)/␳ to which the surface potential vorticity (26) reference pressure. a ␪ Within the approximations of the primitive equations is proportional. Denoting the relative vorticity component perpendicular to the surface by ␨ ϭ ␻ ´ n, one and under the assumption of static stability, the gen- ␴ r can write (␻ ´ n)/␳ ϭ (␮f ϩ ␨ )/␳ . In the hydrostatic eralized potential vorticity ¯ux (17) is the sum of the a ␪ ␴ ␪ interior potential vorticity ¯ux (27) and the surface po- approximation, the normalization factor ␮ is equal to one, so that the interior potential vorticity (f ϩ ␨ )/␳ tential vorticity ¯ux (28) in isentropic coordinates. Since ␪ ␪ and the quantity (␻ ´ n)/␳ ϭ (f ϩ ␨ )/␳ differ only by both the advective component (28a) and the frictional a ␪ ␴ ␪ component (28c) of the surface potential vorticity ¯ux the relative vorticity factors: the interior potential vor- have cross-isentropic components, the impermeability ticity theorem does not hold for the generalized potential vor- f ϩ ␨ P ϭ ␪ H(␪ Ϫ ␪ ) ticity ¯ux in Gauge II. The generalized potential vor- ␳ s ticity in Gauge II can be transferred across isentropes ␪ by diabatic heating and friction at the surface. However, contains the relative vorticity ␨␪ of the ¯ow along is- whether the generalized potential vorticity ¯ux has a entropes; the surface potential vorticity cross-isentropic component is gauge-dependent; for ex- f ϩ ␨␴ ample, the generalized potential vorticity ¯ux has no S ϭ ␪␦(␪ Ϫ ␪s) cross-isentropic component in Gauge I. ␳␪

contains the relative vorticity ␨␴ of the ¯ow along the c. Conservation of generalized potential vorticity surface. Because of this difference in the relative vorticity factors, the surface potential vorticity ¯ux has a The conservation law (18) for generalized potential nonadvective baroclinic component in adiabatic and vorticity becomes in isentropic coordinates frictionless ¯ows, while the interior potential vorticity .␳ J ␪ ) ϭ 0, (30) ¯ux is purely advective in such ¯ows)ץ␳ J ) ϩ)ץ␳ J xy) ϩ)ץ␳ P ) ϩ) ץ t ␪ gx␪ gy␪ g ␪␪g The regular part of the conservation law (30) de- xy␪ ␪ where (JJJggg , , ) are the contravariant components of scribes the time evolution of the isentropic relative vor- 9 the generalized potential vorticity ¯ux Jg. ticity ␨␪. In the absence of diabatic heating and friction, the interior potential vorticity ¯ux (27) reduces to an 9 The conservation law (30) results from the representation advective ¯ux along isentropes, so the interior potential vorticity (f ϩ ␨ )/␳ is materially conserved and is, in 1 ␪ ␪ xy␪ -␪(ha )] particular, materially conserved in adiabatic and fricץha ) ϩ)ץxy(ha ) ϩץ] a ϭ ´ ١ h tionless ¯ows along isentropes at or immediately above of the divergence of a vector a ϭ (ax, ay, a␪)␪ in isentropic coordinates the surface. The singular part of the conservation law (cf. Arfken 1985, chapter 3.9). Since the scale factor h can be viewed (30) describes the time evolution of the relative vorticity x, y, ␪, t) of the four-dimensional)ץ/(x, y, z, t)ץas the Jacobian h ϭ transformation from (x, y, z, t)-coordinates to (x, y, ␪, t)-coordinates, ␨␴ on isentropes ␪ ϭ ␪s(x, y, t) at the surface. Even in and since it is the Jacobian h of the transformation from Cartesian the absence of diabatic heating and friction, the surface to isentropic coordinates that appears in the representation of the potential vorticity ¯ux (28) contains the nonadvective divergence in isentropic coordinates, the explicit time derivative baroclinic component Kbc, so the quantity (f ϩ ␨␴)/␳␪ t(␳P) can be viewed as being part of a four-dimensional divergenceץ Ϫ1 is not, in general, materially conserved in adiabatic and (t(h␳Pץ operator and can, like the space derivatives, be written as h Ϫ1 frictionless ¯ows along isentropes at the surface. -t(␳␪ P); hence the conservation law (30) in isentropic coordiץ ϭ h nates. The quantity (f ϩ ␨␴)/␳␪ is not materially conserved 1034 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 because baroclinicity at the surface can affect the rel- quasigeostrophic scaling. In quasigeostrophic scaling, ative vorticity ␨␴ of the surface ¯ow. The contribution the baroclinic component of the surface potential vor- .␳Kbc) of the baroclinic component (16b) to the di- ticity ¯ux is negligible, irrespective of topography)´ ١ vergence of the surface potential vorticity ¯ux density Hence, in large-scale atmospheric ¯ows, the baroclinic is equal, up to factors that are constant along isentropes component of the surface potential vorticity ¯ux will at the surface, to the downward normal component often be negligible. cT ١␪ p n ϫ ´ ١ log␪) ϭ ϫ (cT١ ١ ´ Ϫn p ΂΃␪ c. Example: Wake formation in ¯ows past a mountain of the baroclinicity vector One example of a ¯ow in which nongeostrophic effects and the baroclinic component of the surface po- Ϫ1 -log␪. tential vorticity ¯ux can play a signi®cant role is strat ١ ϫ cTp ١ ١p ϭ ϫ ␳ Ϫ١ The component of the baroclinicity vector normal to the i®ed ¯ow past a mountain. Smolarkiewicz and Rotunno surface does not generally vanish but affects the surface (1989) have demonstrated with numerical simulations that, in a strati®ed ¯ow that is irrotational upstream of potential vorticity S via the relative vorticity ␨␴ of the surface ¯ow. In contrast, the component of the baro- an isolated mountain, a wake with a pair of lee vortices clinicity vector normal to isentropes vanishes, so the can form downstream of the mountain even when the boundary condition at the mountain is a free-slip con- isentropic relative vorticity ␨␪ is not affected by baroclinicity and the ¯ux of interior potential vorticity is dition. A free-slip condition at the surface implies that purely advective in adiabatic and frictionless ¯ows, in- there cannot be a usual frictional boundary layer from cluding ¯ows along isentropes at or immediately above which vorticity could be transferred into the wake in the surface. the interior of the ¯ow. In place of frictional processes, The baroclinic component of the surface potential baroclinic effects have been linked to the induction of vorticity ¯ux, then, is due to the difference between the wake vorticity in ¯ows past a mountain with a free-slip surface (Smolarkiewicz and Rotunno 1989; Rotunno et relative vorticities ␨␪ and ␨␴. al. 1999; Epifanio and Durran 2002a,b). Even in the presence of baroclinicity, however, interior potential b. Scale analysis for small Rossby numbers vorticity is materially conserved in adiabatic and fric- For hydrostatic ¯ows with small Rossby numbers, the tionless ¯ows, and so baroclinic effects alone cannot be ratio of the difference ␨␪ Ϫ ␨␴ between the relative vor- responsible for the induction of potential vorticity in a ticities to the relative vorticities ␨␪ and ␨␴ themselves wake; the interior potential vorticity would have to re- scales like the ratio Fr2/Ro of squared Froude number main zero throughout adiabatic and frictionless ¯ows Fr ϭ U/(NH) to Rossby number Ro ϭ U/(fL), where past a mountain if it is zero upstream of the mountain. U is a velocity scale, N the Brunt±VaÈisaÈlaÈ frequency, Yet, in simulations with weak frictional and thermal H a height scale, and L a length scale. Since the bar- dissipation, a ¯ow with zero interior potential vorticity oclinic component Kbc of the surface potential vorticity upstream of a mountain with a free-slip surface can

¯ux is due to the difference ␨␪ Ϫ ␨␴ between the relative develop a wake with nonzero interior potential vorticity vorticities, it is of order O(Fr2/Ro) compared with the downstream of the mountain (see, e.g., SchaÈr and Dur- advective component (u, ␷, Q)␪ S. That is, for ¯ows with ran 1997; Rotunno et al. 1999; Epifanio and Durran small Rossby numbers, only if Fr2 տ Ro can surface 2002b). The nonzero interior potential vorticity in the baroclinicity lead to signi®cant deviations from material wake implies that dissipative processes, however weak, conservation of generalized potential vorticity. must be active somewhere in the ¯ow. But the extent For quasigeostrophic ¯ows on the scale of the Rossby to which baroclinic effects and dissipative processes radius LRo ϭ NH/f, the Froude number Fr is of the same play a role in the wake formation has been the subject order as the Rossby number Ro, and the baroclinic com- of controversy (see, e.g., Smith 1989; SchaÈr and Durran ponent Kbc of the surface potential vorticity ¯ux is of 1997; Rotunno et al. 1999; Epifanio and Durran order O(Ro) compared with the advective component 2002a,b). The generalized potential vorticity concept (u, ␷, Q)␪ S. The baroclinic component of the surface allows for a scenario of how a wake can form in weakly potential vorticity ¯ux hence is negligible in quasigeo- dissipative ¯ows past a mountain. strophic scaling.10 Figure 1 shows the time evolution of the surface po- As shown above (section 3f), the baroclinic com- tential vorticity S and of the interior potential vorticity ponent of the surface potential vorticity ¯ux vanishes P on two isentropes during the spinup of a mountain for Boussinesq ¯ows over a ¯at surface, irrespective of wake from a potential-¯ow initial condition. The ®gure is based on a simulation by Rotunno et al. (1999) of a Boussinesq ¯ow past an isolated mountain. In the sim- 10 x x(␳␪ K bc)|␪ץ␳K bc) ϭ)´ ١ Strictly speaking, it is the divergence ±␳ K y )| that is negligible in quasigeostrophic scaling. The bar- ulation, the planetary vorticity is zero, and the Brunt)ץϩ y ␪ bc ␪ VaÈisaÈlaÈ frequency N, the velocity u (U, 0, 0), and oclinic component K bc may have a nondivergent component that is ϭ irrelevant for the transfer of surface potential vorticity. the surface potential temperature ␪s are uniform far up- 15 APRIL 2003 SCHNEIDER ET AL. 1035

FIG. 1. Generalized potential vorticity in simulated Boussinesq ¯ow past a mountain. The ¯ow impinges along the x axis (from the left) upon a radially symmetric mountain at the coordinate origin. (a), (b), (c) Colored contours indicate the interior potential vorticity (20) on the isentropes ␪Јϭ0.8 and ␪Јϭ2.0 and the surface potential vorticity (21) at the mountain surface for three different times t after the start of the simulation from a potential-¯ow initial condition. (d) Colored contours indicate, projected onto the (x, y) plane, the convergence

␳0Kbc) of the baroclinic component (22) of the surface potential vorticity ¯ux. Vectors indicate the magnitude and direction of the)´ Ϫ١ advective interior potential vorticity ¯ux (u, ␷,0)␪P along the isentropes and of the advective surface potential vorticity ¯ux uS. Quantities are given in units of the scales listed in Table 1. The delta function ␦(z Ϫ zs) in the surface potential vorticity was replaced by the inverse height scale 1/H, so that the surface potential vorticity is ®nite and of magnitude comparable with the interior potential vorticity. 1036 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60

TABLE 1. Scales used for nondimensionalization of quantities in Figs. 1 and 2. The fundamental scales used for the nondimensionalization are the horizontal scale L of the mountain and the velocity U and Brunt±VaÈisaÈlaÈ frequency N of the ¯ow far upstream of the mountain. Quantity Scale t L/U x, y, z H ϭ U/N 2 ␪Ј ⌰ϭ␪0(N H/g)

P, S N⌰/(␳0L)

J, K UN⌰/(␳0L) 2 ␳0K) UN⌰/L)´ ١ .FIG. 2. Isentropes in the y ϭ 0 symmetry plane at time t ϭ 14.4 The thin lines represent isentropes (␪Јϭconst) with a contour interval of ⌬␪Јϭ0.5. The thick lines represent the surface and the isentropes ␪Јϭ0.8 and ␪Јϭ2.0 on which the generalized potential vorticity state has been reached in the simulation (see Table 1 is shown in Fig. 1. Potential temperature ¯uctuations ␪Ј are given in 2 for the scales used for nondimensionalization of the units of the scale ⌰ϭ␪0 (N H/g) (cf. Table 1), so that a unit potential temperature ¯uctuation ␪Јϭ1 corresponds to a downward displace- time and other quantities). As Fig. 2 shows, the grav- ment of an isentrope by one height scale H. ity waves in the simulation are not overly steep and do not break, whence adiabatic and inviscid linear theory can account qualitatively for the modi®cation stream of the mountain, so that the interior potential of the thermal strati®cation above a boundary layer. vorticity P, the surface potential vorticity S, and the For example, for a mountain that is suf®ciently high generalized potential vorticity P ϭ P ϩ S are zero there. g (h տ H), such as the mountain in the simulation The ¯ow impinges along the x axis upon a radially M considered here (h ϭ 1.25H), adiabatic and inviscid symmetric Gaussian mountain at the coordinate origin. M linear theory predicts that, in the lee of the mountain, The mountain is of height h ϭ 1.25H, where H ϭ U/N M isentropes are de¯ected downward and collapse onto is a height scale, and its slopes are gentle in that the the mountain surface (Smith 1980, 1988). The down- horizontal scale L ϭ 10H of the mountain is consid- ward de¯ection of isentropes in the lee of the moun- erably greater than the mountain height h . The only M tain is evident in Fig. 2. However, adiabatic and in- dissipative processes in the simulation are viscous mo- viscid ¯ows generally do not satisfy the free-slip mentum dissipation and thermal diffusion with constant conditions that the tangential stress and the normal viscosity ␯ and constant thermal diffusivity ␬ ϭ ␯ . e e e heat ¯ux vanish. The prediction of linear theory that The Reynolds number Re ϭ UL/␯ ϭ 500 is chosen e isentropes collapse onto the leeward slope of the such that the simulated ¯ow is only weakly dissipative, mountain (which linear theory constrains to be an yet remains laminar. The boundary condition at the isentrope except at stagnation points) must be mod- mountain is a free-slip condition, namely, the vanishing i®ed in a boundary layer. In the boundary layer, dis- u and of the normal(١ ´ of the tangential stress Ϫ␯ (n e sipative processes can become important even at -␪Ј. [See Rotunno et al. (1999, sec(١ ´ heat ¯ux Ϫ␬ (n e high Reynolds numbers because the inhibition of tion 4) for a detailed description of the simulation.] motion normal to the surface allows the buildup of In the simulation, the formation of a wake with non- large tangential gradients in vorticity and potential zero interior potential vorticity is the result of ®ve pro- temperature. Figure 2 shows that at the leeward slope cesses: 1) modi®cation of the thermal strati®cation in of the mountain, where linear theory predicts a re- the vicinity of the mountain by gravity waves, 2) bar- gion of collapsed isentropes, a region of high and oclinic induction of a surface potential vorticity dipole relatively uniform surface potential temperature at the leeward slope of the mountain, 3) downslope forms, bounded downslope (and what cannot be in- advection of the surface potential vorticity dipole and ferred from the ®gure: also laterally) by a region of accumulation of surface potential vorticity in a region large gradients in surface potential temperature. Fig- of large gradients in surface potential temperature, 4) ures 1a±c each show one isentrope (␪Јϭ2.0) that dissipative conversion of the surface potential vorticity does not intersect the mountain surface and one is- dipole into an interior potential vorticity dipole, and 5) entrope (␪Јϭ0.8) that intersects the leeward slope separation of the interior potential vorticity dipole from of the mountain in the region of large gradients in the surface and advection into the wake along isentropes surface potential temperature. The essential character- that intersect the surface. istics of the thermal strati®cation in the vicinity of 1) The way in which gravity waves modify the thermal the mountain are established at time t ϭ 1.8 (Fig. 1a) strati®cation in the vicinity of the mountain above a and evolve only slightly during the further spinup of boundary layer is qualitatively well described by adi- the mountain wake (Fig. 1b, c). abatic and inviscid linear theory. Figure 2 shows 2) The thermal strati®cation in the vicinity of the moun- isentropes in the y ϭ 0 symmetry plane of the ¯ow tain implies that, in the region of high and relatively at time t ϭ 14.4, a time at which a nearly steady uniform surface potential temperature at the leeward 15 APRIL 2003 SCHNEIDER ET AL. 1037

slope, a surface potential vorticity dipole is induced gradients in surface potential temperature near the lee- baroclinically. Since the baroclinic component (22) ward foot of the mountain. Since a ¯ow with weak of the surface potential vorticity ¯ux is quadratic in thermal diffusion as the only diabatic process effec- potential temperature ¯uctuations ␪Ј, the baroclinic tively cannot cross isentropes at the surface, the ad- induction of surface potential vorticity is not taken vective surface potential vorticity ¯ux converges and into account in linear theories [cf. the analysis of surface potential vorticity accumulates in the region Smolarkiewicz and Rotunno (1989), which shows of large surface potential temperature gradients. In that the baroclinic induction of wake vorticity is an Fig. 1, the convergence of the advective surface po- effect of second order in perturbation amplitude]. tential vorticity ¯ux can be inferred from the vectors The across-stream component along the surface, which indicate the magnitude and direction of the advective surface potential vorticity ␮ g␪Ј2 y ¯ux. The accumulation of surface potential vorticity (xsz )␦(z Ϫ z sץ) K bc ϭ ␳ 002␪ in the region of large surface potential temperature gradients near the leeward foot of the mountain is dominates the baroclinic component of the surface clearly recognizable in the succession of Figs. 1a±c. potential vorticity ¯ux at the leeward slope and trans- 4) As surface potential vorticity accumulates in the re- fers surface potential vorticity from the left (facing gion of large surface potential temperature gradients downstream) of the y ϭ 0 symmetry plane to the near the leeward foot of the mountain, the magnitude right (since K y Ͻ 0 at the leeward slope). Figure 1d bc of the conversion rate (19) between surface potential shows, at time t ϭ 14.4 and projected onto the vorticity and interior potential vorticity increases on ␳ K ) of the)´ x, y) plane, the convergence Ϫ١) 0 bc both sides of the y 0 symmetry plane. In a steady baroclinic component of the surface potential vor- ϭ state, a balance is established between the conver- ticity ¯ux. Since the baroclinic component K de- bc gence of surface potential vorticity ¯ux and the pends only on the topography and on the surface conversion of surface potential vorticity into inte- potential temperature ®eld ␪Ј, which is already close rior potential vorticity. This balance is a conse- to its steady state at time t ϭ 1.8, the structure of quence of the fact that, in a steady state, the di- ␳ K ) at the earlier times t)´ the convergence Ϫ١ -␳ J ϩ ␳ K) of the gen)´ ١ ␳ J ) ϭ)´ ١ bc vergence 0 ϭ 1.8 and 7.2 is similar to the shown convergence 0 g 0 0 eralized potential vorticity ¯ux vanishes, whence, at time t ϭ 14.4. The convergence of the diabatic by integration over a volume V that encloses a sur- and frictional components of the generalized poten- face patch of area A and that has in®nitesimally tial vorticity ¯ux is, in the region of the extrema of small faces normal to the surface, it follows that the baroclinic component, an order of magnitude smaller than the convergence of the baroclinic com- (␳ K) dx ϭ n ´(␳ J) dA. (31)´ ١ Ϫ 11 ponent. The convergence of the baroclinic com- ͵ 0 ͵ 0 ponent of the surface potential vorticity ¯ux at the VA leeward slope of the mountain leads to a surface In the region of large surface potential temperature potential vorticity dipole with negative surface po- gradients near the leeward foot of the mountain, the tential vorticity to the left of the y ϭ 0 symmetry 2 baroclinic component Kbc ϰ ␪Ј (n ϫ k) is negligible plane and positive surface potential vorticity to the in the convergence of the surface potential vorticity right. ¯ux (cf. Fig. 1d) because the normal vector n is 3) The baroclinically induced surface potential vorticity nearly parallel to the vertical unit vector k. The bal- dipole is advected downslope into the region of large ance (31) between the convergence of surface potential vorticity ¯ux and the conversion of surface 11 Scale analysis gives an indication of the relative magnitudes of the potential vorticity into interior potential vorticity can baroclinic and dissipative contributions to the convergence of the gen- therefore be approximated as

(␳0Jg)´ eralized potential vorticity ¯ux. Integrating the convergence Ϫ١ of the generalized potential vorticity ¯ux over a volume V that encloses ␳ uS) dx)´ ١ Ϫ a surface patch of area A and that has in®nitesimally small faces normal ͵ 0 to the surface, one ®nds that, at about half-height at the leeward slope, V

␳0Kbc) dx to the integral scales like)´ ١ the baroclinic contribution Ϫ#V 2 2 ␳0KF) dx)´ ١ g/␪0)(⌰ hM/L )A, whereas the dissipative contributions Ϫ#V) ,ϫ F)] dA ١)´ ϫ F)] dA scale like (␯ ⌰U/H2L)A ഠ Ϫ [Q␨␴ ϩ ␪Јn ١)´ Ϫ # n ´(␳ J) dA ϭ # [Q␨ ϩ ␪Јn A 0 A ␴ e ͵ (cf. the scales in Table 1). The ratio of the baroclinic contribution to the A 2 3 2 dissipative contributions is therefore N H hM/(␯eUL) ϭ HhM/L ´Reഠ 6

(since Re ϭ 500 and L ϭ 8hM ϭ 10H in the simulation considered here). where we have combined the frictional components This scaling estimate underestimates the actual ratio of the baroclinic KF and JF of the surface potential vorticity ¯ux and contribution to the dissipative contributionsÐlargely, it appears, because of the interior potential vorticity ¯ux in the surface the scaling estimate ␨␴ ϳ U/L for the vorticity in the dissipative term overestimates the vorticity in the region at the leeward slope where the integral on the right-hand side (cf. footnotes 5 and baroclinic contribution to the convergence of the generalized potential 7). Since the region of large surface potential tem- vorticity ¯ux is largest. perature gradients near the leeward foot of the moun- 1038 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60

tain is anomalously warm (cf. Fig. 2), the diffusive is L/U. Within that time, potential temperature and diabatic heating rate Q tends to be negative there, vorticity anomalies that give rise to a potential

so that, since the surface potential temperature anom- vorticity anomaly can diffuse a distance ␦0 ϭ aly ␪Ј is positive at the leeward slope, the diabatic Ϫ1/2 ͙␯eeL/U ϭϭ͙␬ L/U Re L away from the sur- conversion ϪQ␨ damps surface potential vorticity ␴ face. This distance ␦0 is an estimate of the thickness anomalies S ϰ ␪Ј␨␴ by converting positive surface of the boundary layer at the free-slip surface; it co- potential vorticity into positive interior potential vor- incides with the thickness of a boundary layer at a ticity and negative surface potential vorticity into no-slip surface (cf. Lighthill 1963). For the simu- negative interior potential vorticity. The magnitude lation considered here, with Re ϭ 500 and L ϭ 10H, of the conversion rate ϪQ␨ increases with increas- ␴ the estimate of the boundary layer thickness is ␦0 ϭ ing magnitude of the surface potential vorticity H/͙5 , which is roughly consistent with the thickness anomaly S ϰ ␪Ј␨␴. The frictional conversion of the region where Fig. 1 suggests signi®cant ab- -ϫ F) has a similar effect: to the extent solute values of the divergence of the interior po ١)´ Ϫ␪Јn that extrema in the relative vorticity ␨␴ ϭ n ´ ␻r are tential vorticity ¯ux along isentropes. [The thick- approximately concomitant with the extrema in sur- nesses of the regions where Fig. 6c of Rotunno et face potential vorticity, the frictional conversion rate al. (1999) and Fig. 10 of Epifanio and Durran 2 -ϫ F) ϭϪ␯e␪Јn ´(ٌ ␻r) is positive at the (2002b) indicate signi®cant viscous and thermal dis ١)´ Ϫ␪Јn surface potential vorticity maximum near the lee- sipation are likewise consistent with the scaling es- ward foot of the mountain and negative at the min- timate ␦0 of the boundary layer thickness.] imum. Dissipative conversion of surface potential vorticity into interior potential vorticity thus coun- In this scenario, a wake with a pair of counterrotating teracts the accumulation of surface potential vorticity lee vortices forms by conversion of a baroclinically in- due to the convergence of the advective surface po- duced surface potential vorticity dipole into an interior tential vorticity ¯ux. potential vorticity dipole. As in the analysis of Rotunno 5) Since the large along-stream gradients in surface po- et al. (1999), baroclinicity is posited as fundamental for tential temperature imply convergence of the surface the formation of a wake with nonzero interior potential ¯ow, the region of large surface potential tempera- vorticity. The generalized potential vorticity perspective ture gradients is not only a region of signi®cant dis- emphasizes the baroclinicity near the surface of the sipation, but also a region in which streamlines break mountain and shows that the principal role of dissipative away from the surface (cf. Rotunno et al. 1999, Figs. processes is to convert the surface potential vorticity 3 and 6). The ¯ow separates from the surface and dipole that develops at the leeward slope of the mountain advects with it the interior potential vorticity dipole into an interior potential vorticity dipole. The interior that originated in the dissipative conversion of the potential vorticity dipole subsequently separates from surface potential vorticity dipole (Fig. 1). Since the the surface. The description from the perspective of gen- ¯ow is only weakly dissipative, the interior potential eralized potential vorticity resembles the classical de- vorticity dipole is advected predominantly along is- scriptions of the separation of a vorticity sheet induced entropesÐalong isentropes that intersect the surface by frictional processes at a no-slip surface (cf. Lighthill near the leeward foot of the mountain. On isentropes 1963), with two main differences: whereas the surface that do not intersect the surface, such as the upper vorticity sheet at a no-slip surface is induced by friction, isentropes in Fig. 1, interior potential vorticity can- a surface potential vorticity sheet can be induced by not be induced by conversion from surface potential baroclinicity; and whereas the surface vorticity sheet at vorticity, but arises by vertical diffusion of potential a no-slip surface involves vorticity components tangen- temperature and/or vorticity anomalies from the sur- tial to the surface, a surface potential vorticity sheet at face and from lower-lying isentropes that intersect a free-slip surface involves the vorticity component nor- the surface. The interior potential vorticity anomalies mal to the surface. are therefore weaker and appear later on isentropes that do not intersect the surface than on isentropes 6. Summary that do intersect the surface (Fig. 1). The thickness of the boundary layer in which dis- We have presented a formulation of potential vorticity sipative processes are signi®cant for converting the dynamics that encompasses boundary effects. For ar- surface potential vorticity dipole into an interior po- bitrary ¯ows, the generalization of the potential vorticity tential vorticity dipole and for separating the interior concept to a sum of the conventional interior potential potential vorticity dipole from the surface can be vorticity and a singular surface potential vorticity allows estimated from the timescale of potential vorticity one to replace the inhomogeneous boundary conditions advection in the wake. To the extent that in the wake for potential vorticity dynamics by simpler homoge- downstream of the mountain, the upstream velocity neous boundary conditions. For the generalized poten- scale U and the length scale L are adequate ¯ow tial vorticity, a conservation law holds that is similar to scales, the timescale of potential vorticity advection the well-known conservation law for the interior po- 15 APRIL 2003 SCHNEIDER ET AL. 1039

tential vorticity. The generalized potential vorticity re- ventional potential vorticity dynamics carries over to duces in the quasigeostrophic limit to Bretherton's generalized potential vorticity dynamics: like the con- (1966) generalized quasigeostrophic potential vorticity, ventional potential vorticity combined with typically in- which includes a surface potential vorticity that is pro- homogeneous boundary conditions, the generalized po- portional to surface potential temperature ¯uctuations. tential vorticity combined with simpler homogeneous Not being limited to quasigeostrophic ¯ows, however, boundary conditions contains all relevant information the generalized potential vorticity concept can be used about ¯ows that satisfy fairly general balance conditions to describe ¯ows such as mesoscale or planetary-scale (cf. Hoskins et al. 1985; McIntyre and Norton 2000). ¯ows, for which the quasigeostrophic approximation is Thus, as Bretherton's extension of the quasigeostrophic inadequate. potential vorticity has proven fruitful in the analysis of The formal framework of generalized potential vor- boundary effects in quasigeostrophic ¯ows, the gener- ticity dynamics issues from ®eld equations in which the alized potential vorticity concept could prove fruitful in potential vorticity and the potential vorticity ¯ux appear the analysis of boundary effects in more general bal- as sources of ¯ow quantities in the same way in which anced ¯ows. And irrespective of whether a ¯ow is bal- an electric charge and an electric current appear as anced and the generalized potential vorticity contains sources of ®elds in electrodynamics. The boundary all relevant information about the ¯ow, the conservation sourcesÐthe surface potential vorticity and the surface law for generalized potential vorticity describes how potential vorticity ¯uxÐthat must be included in the potential vorticity ¯uctuations in the interior of the ¯ow ®eld equations if the usual inhomogeneous boundary interact with boundaries and how the interior and sur- conditions for potential vorticity dynamics are replaced face potential vorticities evolve in time. by simpler homogeneous boundary conditions were determined with techniques from electrodynamics. We de- Acknowledgments. We thank Richard Rotunno, Vanda rived functional forms of the surface potential vorticity GrubisÏicÂ, and Piotr Smolarkiewicz for their willingness and of its ¯ux, pointed out ambiguities in these func- to share their simulation data; Piotr Smolarkiewicz for tional forms, and discussed the conservation law for making the data available in a format convenient for us; generalized potential vorticity. Robert Hallberg, Peter Haynes, David Marshall, Olivier In an example, we demonstrated how the generalized Pauluis, and Richard Rotunno for helpful comments on potential vorticity and its conservation law can be used drafts of this paper; and Heidi Swanson for editing the to analyze the dynamical role of boundaries in ¯ows for manuscript. Parts of this paper where drafted while T. which the quasigeostrophic approximation is inade- Schneider was with the Atmospheric and Oceanic Sci- quate. We outlined a theory of how a wake with lee ences Program at Princeton University, where he was vortices can form in mesoscale ¯ows past a mountain supported by a NASA Earth System Science Fellowship. with a free-slip surface. Even in adiabatic and frictionless ¯ows, generalized potential vorticity is not, in gen- APPENDIX eral, materially conserved but can be induced by baroclinicity at a boundary. In strati®ed ¯ows past a moun- Notation and Symbols tain, generalized potential vorticity can be induced by baroclinicity at the leeward slope of the mountain. As Most symbols follow standard meteorological and illustrated in a simulation of a strati®ed Boussinesq ¯ow, mathematical conventions. Listed here are only those weak dissipative processes in a boundary layer suf®ce symbols that are used repeatedly in different sections to convert a baroclinically induced surface potential vor- of this paper. ticity dipole that develops at the leeward slope of a mountain into an interior potential vorticity dipole,

␰ Partial derivative with respect to space or timeץ which separates from the surface and is advected into the wake along isentropes that intersect the surface. coordinate ␰

-fol ١ ´ t ϩ uץThus a wake with a pair of counterrotating lee vortices D/Dt Material derivative D/Dt ϭ can form by separation of a baroclinically induced gen- lowing the three-dimensional ¯ow u

eralized potential vorticity sheet from the surface of a cp Speci®c heat at constant pressure ␬ mountain, even when frictional processes in the bound- E Exner function E ϭ cp(p/p0) ary layer are so weak that no friction-induced vorticity f Coriolis parameter f ϭ 2࿣⍀࿣ sin␾, where ␾ can be transferred into the wake in the interior of the is latitude ¯ow. F Frictional force per unit mass This example illustrates how the generalized potential g Gravitational acceleration Ϫ1 -z␪) of isentropic coorץ) vorticity concept extends the conventional potential vor- h Scale factor h ϭ ticity concept to encompass boundary effects. The gen- dinates eralized potential vorticity concept is applicable to ar- H(´) Heaviside step function bitrary ¯ows, even to ¯ows that are not balanced. For i, j, k Local Cartesian unit vectors (eastward, north- balanced ¯ows, the inversion principle known from con- ward, upward) 1040 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60

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