Zonal Momentum Balance, Potential Vorticity Dynamics, and Mass Fluxes on Near-Surface Isentropes

Transcription

1 1884 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 Zonal Momentum Balance, Potential Vorticity Dynamic, and Ma Fluxe on Near-Surface Ientrope TAPIO SCHNEIDER California Intitute of Technology, Paadena, California (Manucript received 2 December 2003, in final form 13 October 2004) ABSTRACT While it ha been recognized for ome time that ientropic coordinate provide a convenient framework for theorie of the global circulation of the atmophere, the role of boundary effect in the zonal momentum balance and in potential vorticity dynamic on ientrope that interect the urface ha remained unclear. Here, a balance equation i derived that decribe the temporal and zonal mean balance of zonal momentum and of potential vorticity on ientrope, including the near-urface ientrope that ometime interect the urface. Integrated vertically, the mean zonal momentum or potential vorticity balance lead to a balance condition that relate the mean meridional ma flux along ientrope to eddy fluxe of potential vorticity and urface potential temperature. The ientropic-coordinate balance condition formally reemble balance condition well known in quaigeotrophic theory, but on near-urface ientrope it generally differ from the quaigeotrophic balance condition. Not taking the interection of ientrope with the urface into account, quaigeotrophic theory doe not adequately repreent the potential vorticity dynamic and ma fluxe on near-urface ientrope a hortcoming that call into quetion the relevance of quaigeotrophic theorie for the macroturbulence and global circulation of the atmophere. 1. Introduction Potential vorticity i one of the principal conerved quantitie in adiabatic and invicid atmopheric flow. Conideration of potential vorticity dynamic therefore form a good bai for decription of the interaction between a mean flow and rapid, nearly adiabatic and invicid fluctuation uch a occur in baroclinic eddie. Since the potential vorticity flux ha only component along ientrope, but not acro ientrope (Hayne and McIntyre 1987, 1990), it i convenient to conider the interaction between a mean flow and eddie in ientropic coordinate. For example, on ientrope in the interior of the extratropical atmophere, the mean meridional ma flux, via the zonal momentum balance, i primarily aociated with the meridional eddy flux of potential vorticity (Tung 1986; Yang et al. 1990). For a theory of the mean meridional ma flux along ientrope, or, equivalently, for a theory of the mean meridional entropy flux, one therefore need a theory of the meridional eddy flux of potential vorticity along ientrope. No concluive theory ha yet been propoed. But a theory of the meridional eddy flux of Correponding author addre: Tapio Schneider, California Intitute of Technology, Mail Code , 1200 E. California Blvd., Paadena, CA tapio@caltech.edu potential vorticity along ientrope a calar flux of a quantity that i approximately conerved in baroclinic eddie eem to be eaier to develop than a theory, for intance, of the flux of entropy (or potential temperature), which would necearily involve vectorvalued eddy fluxe. Since the Eulerian mean ma flux along ientrope approximate the Lagrangian mean ma flux and ince it, in concert with radiative procee, determine uch baic climatic feature a the tropopaue height and the thermal tratification of the atmophere, it ha been peculated that a theory of the global circulation of the atmophere may be baed on conideration of potential vorticity dynamic in ientropic coordinate (ee, e.g., Hokin 1991). However, ince potential temperature fluctuate at the urface, implying that ientrope ometime lie above the urface, ometime inide the urface, the role of boundary effect in the zonal momentum balance and in potential vorticity dynamic on near-urface ientrope ha remained unclear. Since an undertanding of how potential vorticity fluxe along ientrope in the interior atmophere are related to the zonal momentum balance and to potential vorticity dynamic on nearurface ientrope ha thu far been lacking, it ha been impoible to contruct a theory of the global circulation of the atmophere baed on conideration of potential vorticity dynamic. The preent paper dicue the temporal and zonal mean balance of zonal momentum and potential vor American Meteorological Society

2 JUNE 2005 S C H N E I D E R 1885 ticity on ientrope, including the near-urface ientrope that ometime interect the urface. A form of the mean zonal momentum balance i derived that hold throughout the flow domain in ientropic coordinate and that i equivalent to the mean potential vorticity balance (ection 2). 1 Integrated vertically, the mean zonal momentum or potential vorticity balance lead to a balance condition that, at each latitude, relate the mean meridional ma flux along ientrope to eddy fluxe of potential vorticity and urface potential temperature (ection 3). Thi balance condition reemble balance condition well known in quaigeotrophic theory, but on near-urface ientrope it generally differ from the quaigeotrophic balance condition (ection 4). The development of thi paper preume a hydrotatic ideal-ga atmophere with tationary flow tatitic and with the planet urface a the only dynamically relevant boundary. The boundary condition at the urface i aumed to be a no-lip condition. The urface can have arbitrary topography, o long a the hydrotatic approximation remain adequate. Appendix A decribe an idealized GCM ued to illutrate theoretical development. Appendix B lit the notation and ymbol. 2. Mean balance of zonal momentum in ientropic coordinate a. Zonal momentum equation The zonal momentum equation in ientropic coordinate can be formulated uch that it hold both on ientrope above the urface and on ientrope inide the urface, that i, both on ientrope with potential temperature greater than or equal to the intantaneou urface potential temperature (x, y, t) and on ientrope with potential temperature le than the intantaneou urface potential temperature (x, y, t). The flow domain in ientropic coordinate can formally be extended to ientrope inide the urface by introducing male ientropic layer inide the urface (cf. Lorenz 1955). In the definition of the ientropic denity (g 1 p)h( ) the ma denity in (x, y, ) pace one include the Heaviide tep function H( )to indicate that the ientropic denity vanihe on ientrope inide the urface. With the ientropic denity et to zero on ientrope inide the urface, the continuity equation (cf. Andrew et al. 1987, chapter 3.8) t x u y Q 0, with diabatic heating rate Q D/Dt, hold on ientrope both above and inide the urface. (Horizontal 1 A form of the mean zonal momentum balance that i eentially equivalent to that preented in ection 2 ha been independently derived by Koh and Plumb (2004). 1 and time derivative here are to be undertood a derivative at contant potential temperature.) Similarly, the zonal momentum equation in flux form, t u f x u 2 y u uq x M F x, 2 with Montgomery treamfunction M c p T gz and with zonal frictional force per unit ma F x, hold on ientrope both above and inide the urface (Johnon 1980; Gallimore and Johnon 1981). Combining the continuity equation (1) and the flux-form zonal momentum equation (2), one obtain the form t u f x B Q u F x 3 of the zonal momentum equation, where x y u i the relative vorticity component normal to ientrope and B 1 2 u2 2 M i the Bernoulli function. On ientrope above the urface, the ientropic denity i greater than zero (provided that the atmophere i tably tratified, which we aume) and can be dropped from the zonal momentum equation (3). The zonal momentum equation reduce to one of it tandard form (cf. Andrew et al. 1987, appendix 3A). On ientrope inide the urface, the ientropic denity vanihe and cannot be dropped from the zonal momentum equation (3). The zonal momentum equation reduce to the trivial tatement 0 0. Although we cannot generally drop the ientropic denity from the zonal momentum equation (3), we only need to retain the tep function factor to obtain a form of the zonal momentum equation that hold on ientrope both above and inide the urface, t u f H x B Q u F x H. 4 We will average thi form of the zonal momentum equation to derive a mean balance of zonal momentum on ientrope. b. Temporal and zonal mean When conidering mean flow in ientropic coordinate, one mut ditinguih two region of the flow domain (cf. Jucke et al. 1994; Held and Schneider 1999). The urface layer at a given latitude i that region of the flow domain in which potential temperature lie within the range of urface potential temperature that typically occur at the latitude. The above-lying interior atmophere i that region of the flow domain in which potential temperature lie above the range of urface potential temperature that typically occur at the latitude. In the interior atmophere, the tep function factor

3 1886 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 H( ) i alway equal to one, o the mean acceleration term t uh( ) and the mean Bernoulli gradient x BH( ) appearing in the ientropic temporal and zonal mean ( ) of the zonal momentum equation (4) vanih by tatitical tationarity and periodicity, repectively. In the urface layer, the mean acceleration term t uh( ) i the zonal mean of a normalized um of difference in zonal velocity between pair of time at which ientrope attach to and detach from the urface. Since the velocity vanihe at a no-lip boundary, the mean acceleration term vanihe alo in the urface layer. Similarly, the mean Bernoulli gradient x BH( ) in the urface layer i the temporal mean of a normalized um of difference in Bernoulli function between pair of point at which ientrope enter and leave the urface. Integrating by part, uing the fact that the ditributional derivative of the tep function i the Dirac delta function, H() (), and uing the no-lip boundary condition to et the Bernoulli function B (u 2 2 )/2 M at the urface equal to the Montgomery treamfunction M, one can write the mean Bernoulli gradient a x B H M x. Thi term generally doe not vanih; a we will ee, it i eential for the mean zonal momentum balance of the urface layer. The mean zonal momentum balance take on a uggetive form if one write the abolute vorticity f P a the product of ientropic denity and potential vorticity P (f )/. Introducing the diabatic component J Q y 1 Q u H and the frictional component J F y 1 F x H 5a 5b of the meridional potential vorticity flux (cf. Hayne and McIntyre 1987; Schneider et al. 2003) and averaging the zonal momentum equation (4) lead to where P * J Q y * J F y * Mx, 6 * denote the denity-weighted mean aociated with the temporal and zonal mean along ientrope. The explicit tep function in the advective potential vorticity flux P * wa omitted with the undertanding that the velocity i taken to vanih on ientrope inide the urface. The left-hand ide of the mean zonal momentum balance (6) i generally nonzero both in the interior atmophere and in the urface layer. The right-hand ide i nonzero only in the urface layer. It i well known that the mean zonal momentum balance (6) on interior ientrope i a tatement of mean potential vorticity balance. On interior ientrope, the mean zonal momentum balance can alternatively be derived by averaging the potential vorticity balance (cf. Hayne and McIntyre 1987; Andrew et al. 1987, chapter 3.9). Similarly, the mean zonal momentum balance (6), which hold both on interior and urface layer ientrope, can alternatively be derived by averaging a generalized potential vorticity balance a potential vorticity balance in which boundary effect are taken into account by generalizing the potential vorticity concept to a um of the conventional interior potential vorticity and a ingular urface potential vorticity (Schneider et al. 2003). 2 From a generalized potential vorticity perpective, the urface term on the right-hand ide of the mean zonal momentum balance (6) arie a the average of a ingular urface potential vorticity flux. The mean zonal momentum balance (6), then, can be viewed a a tatement of mean potential vorticity balance, not only on interior ientrope, but throughout the flow domain in ientropic coordinate. c. Vertically integrated mean zonal momentum balance How the contribution of the urface term M x ( ) to the mean zonal momentum or potential vorticity balance (6) i to be interpreted become clearer upon vertical integration. Integrating from ome nominal lower boundary of the domain at a potential temperature b le than or equal to the lowet potential temperature that occur at the urface to ome ientrope i in the interior atmophere yield b i P * J Q y * J F y * d M x. Vertically integrated over the urface layer, an ientropic mean ( )( ) that include a urface delta function become a mean ( ) of urface quantitie (marked by the ubcript ). Since the mean x ( ) of a zonal derivative vanihe, only fluctuation ( )( ) ( ) about the urface mean contribute to the urface term M x M x. Integrating the urface term by part and introducing a balanced meridional eddy velocity at the urface by f 1 x M c p lead to the vertically integrated mean zonal momentum balance 2 For the derivation of the mean zonal momentum balance (6) from the generalized potential vorticity balance, it i convenient to adopt Gauge I of Schneider et al. (2003). The derivation lead to the ame reult a the direct derivation from the zonal momentum balance given here. 7

4 JUNE 2005 S C H N E I D E R 1887 b i P * J Q y * J F y * d f. If fluctuation of urface preure, temperature, potential temperature, and denity about contant reference value p r, T r, r T r, and r p r /(RT r ) are mall, one can approximate urface temperature fluctuation a T p, T r p r r and fluctuation of the Montgomery treamfunction at the urface a M c p T gz 1 r p gz c p. The balanced eddy velocity (7) at the urface then i approximately equal to the geotrophic eddy velocity, f 1 x r 1 p gz f r 1 x p zz. So to the extent that fluctuation of thermodynamic variable about contant reference value are mall near the urface, the contribution of the urface term to the vertically integrated mean zonal momentum balance (8) i approximately equal to the geotrophic eddy flux of urface potential temperature, multiplied by the Corioli parameter. The balance condition (8) reemble a well-known quaigeotrophic balance condition, 0 g q z 0 F 0 x z g z dz f 0 0, 9 z 0z0 obtained by integrating the quaigeotrophic tranformed Eulerian mean of the zonal momentum equation in the vertical and taking into account ma conervation [cf. Andrew and McIntyre (1976); Edmon et al. (1980); Andrew et al. (1987, chapter 3.5)]. Here, q f 0 0 y g f 0 z 0 0 z i the quaigeotrophic potential vorticity, 0 g the geotrophic meridional velocity, and g the relative vorticity of the geotrophic flow; the ubcript 0 mark reference value and profile; ( ) z denote the temporal and zonal mean along horizontal plane, and ( ) ( ) ( ) z denote fluctuation about thi mean. The quaigeotrophic balance condition (9) tate that, vertically integrated over an atmopheric column, the flux of quaigeotrophic potential vorticity i balanced by zonal frictional force and by the geotrophic eddy flux of urface potential temperature, with the momentum flux convergence contained in the quaigeotrophic potential vorticity flux balancing the frictional force (Green 1970; Held and Hokin 1985). The diabatic potential vorticity flux J y * Q Q u H( ) appearing in the ientropic-coordinate balance condition (8) ha no counterpart in the quaigeotrophic balance condition (9). It repreent the cro-ientropic advection of zonal 8 momentum, whoe counterpart the vertical advection of zonal momentum i neglected in quaigeotrophic theory. d. Phyical interpretation of urface term The urface term M x ( ) x MH( ) in the mean zonal momentum balance (6) arie from averaging the gradient x M of the Montgomery treamfunction along ientrope that interect the urface. Since the gradient of the Montgomery treamfunction x M repreent minu the zonal preure force per unit ma in ientropic coordinate, the urface term x MH( ) repreent a mean zonal preure drag per unit ma, imilar to the preure drag at mountain appearing in the mean momentum balance in preure or height coordinate (cf. Peixoto and Oort 1992, chapter 11). Although thi mean preure drag can contain topographic contribution, it doe not require topography; it i the mean zonal preure drag per unit ma that the flow along ientrope experience at interection of ientrope with the urface, whether at topographic obtacle or at a flat urface. Hence, in the vertically integrated zonal momentum balance (8), the term f involving the balanced eddy flux of urface potential temperature i the integral acro ientrope of the mean zonal preure drag per unit ma that the flow along ientrope experience at the urface. Since thi term appear in the vertically integrated zonal momentum balance even if the boundary condition i a no-lip condition, o that the actual eddy flux of urface potential temperature vanihe, the balanced eddy flux of urface potential temperature i not necearily to be undertood a an approximation of the actual eddy flux of urface potential temperature. Even without a mall Roby number approximation, it i the balanced and not the actual eddy flux of urface potential temperature that appear in the vertically integrated zonal momentum balance (8). Thi ugget that, to the extent that quaigeotrophic potential vorticity dynamic can be interpreted a repreenting potential vorticity dynamic on ientrope (Charney and Stern 1962), the term involving the geotrophic eddy flux of urface potential temperature in the quaigeotrophic balance condition (9) can likewie be interpreted a a vertical integral of a mean urface preure drag. e. Other formulation of mean zonal momentum balance Andrew (1983) and Koh and Plumb (2004) have offered imilar formulation of the mean zonal momentum balance on ientrope, including ientrope that ometime interect the urface. Andrew (1983) formulation differ from the preent formulation. Andrew introduce fictitiou nonzero velocitie on ientrope inide the urface and define the Montgomery treamfunction and it gradient uch that

5 1888 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 they are continuou at the interection of ientrope with the urface. While thee convention allowed him to prove a finite-amplitude Eliaen Palm theorem, they make it neceary to include in the momentum equation additional force on ientrope inide the urface. Thi approach left unclear the relation between the mean zonal momentum balance and the mean potential vorticity balance, and between the vertically integrated potential vorticity flux and the balanced eddy flux of urface potential temperature. Koh and Plumb (2004) have independently derived a formulation of the mean zonal momentum balance that i eentially equivalent to that preented here, although the equivalence of the formulation may not be obviou becaue of difference in mathematical technique, notation, and convention. Koh and Plumb mean zonal momentum balance alo contain the term that arie when the velocity at the urface doe not vanih in the preent formulation, an acceleration term t uh( ) and a kinetic energy term 1 2 x (u 2 2 )H( ) due to the difference between the Bernoulli function and the Montgomery treamfunction at the urface. If the acceleration t u at the urface cale advectively a U 2 /L, with a velocity cale U and a length cale L, the acceleration term and the kinetic energy term are of the ame order of magnitude and maller than the preure drag term x MH( ) by a factor of the order of the urface Roby number U/(fL). Hence, to the extent that the urface Roby number i mall, the acceleration term and the kinetic energy term are negligible, for example, in the mean zonal momentum balance of an atmophere model that ue a drag-law boundary condition in place of the realitic no-lip boundary condition. 3. Eddy fluxe and the mean ma flux along ientrope a. Derivation of balance condition A relationhip between the mean ma flux along ientrope and eddy fluxe of potential vorticity and of urface potential temperature can be obtained from the mean zonal momentum or potential vorticity balance (6) if the advective potential vorticity flux P * *P * ˆPˆ * i decompoed into a mean advective component *P * and an eddy component ˆPˆ *. [Hat ( ˆ) ( ) ( ) * denote fluctuation about the denityweighted mean along ientrope.] Dividing the mean zonal momentum balance (6) by the mean potential vorticity P * and rearranging term yield * 1 P * ˆPˆ * M x y J * Q y J F *. 10 Thi form of the mean zonal momentum balance hold where the mean potential vorticity i nonzero. It repreent a balance equation that relate the mean ma flux * along ientrope to the eddy flux of potential vorticity, to the urface preure drag, and to the diabatic and frictional component of the potential vorticity flux. To obtain a balance condition that relate the mean ma flux along ientrope to eddy fluxe of potential vorticity and urface potential temperature, we integrate the balance equation (10) vertically from the lower boundary of the domain at potential temperature b to ome ientrope i in the interior atmophere. The integrated contribution of the urface preure drag can be implified by uing the expanion P *1 P *1 P *1 for the invere mean potential vorticity in the urface layer. Keeping only the zeroth-order term and uing the relation that led to the vertically integrated zonal momentum balance (8) give i M x P * d 1 i P * M x d b b f P *. 11 At next order, a term involving the flux 2 of urface potential temperature variance would appear. Thi term and all other term involving fluxe of even moment of urface potential temperature fluctuation vanih to leading order for weakly nonlinear quaigeotrophic eddie. For uch eddie, the leading-order term of fluxe of even moment 2n have a zonal phae () dependence of the form in() co 2n () (n 1, 2,...) and o have vanihing phae average. Hence, although eddie cannot generally be aumed to be quaigeotrophic and weakly nonlinear, the higherorder correction to the integral (11) are generally maller than caling argument would ugget. Neglecting thee higher-order correction, we obtain i i ˆPˆ * y J * F * d b P * d f P *. b 12 The diabatic component J y Q * of the potential vorticity flux, or the cro-ientropic advection of zonal momentum, wa neglected becaue it i maller than the eddy flux ˆPˆ * by a factor of order Roby number (ee, e.g., Tung 1986; Hayne and McIntyre 1987). The balance condition (12) between the integrated mean ma flux along ientrope, on the one hand, and friction and eddy fluxe of potential vorticity and urface potential temperature, on the other hand, hold in the extratropic, where the Roby number i mall and the mean potential vorticity typically i nonzero. The frictional component J y F * / P * F x H( )/ P * of the ma

6 JUNE 2005 S C H N E I D E R 1889 flux i an Ekman ma flux. If the top of the atmophere i taken a the upper limit i of the integral, the lefthand ide of the balance condition (12) i zero by ma conervation, o the balance condition (12) become a balance condition between friction and eddy fluxe of potential vorticity and urface potential temperature. Uing more retrictive aumption, Held and Schneider (1999) argued that, in the extratropic, the mean ma flux along ientrope integrated over the urface layer contain a component proportional to the geotrophic eddy flux of urface potential temperature. The balance condition (12) how in more detail that not only the balanced eddy flux of urface potential temperature and the frictional component J y F * of the potential vorticity flux, but alo the eddy flux ˆPˆ * of potential vorticity contribute to the mean ma flux integrated over the urface layer. b. Convention for potential vorticity in urface layer The contribution of ientrope inide the urface to the mean potential vorticity P * [ (f )/ ]/ involve the indefinite expreion (f ) /. The total potential vorticity flux P * i independent of the convention one adopt to aign value to thi indefinite expreion. But the mean potential vorticity P * in the urface layer, and thu the eddy flux of potential vorticity ˆPˆ * P * *P *, depend on convention. The magnitude of the Ekman ma flux and of the ma fluxe aociated with the eddy fluxe of potential vorticity and urface potential temperature in the balance condition (12) likewie depend on convention. 1) CONVENTION I Under convention I, the abolute vorticity on ientrope inide the urface equal the Corioli parameter, P f f. One contrual of thi convention i that, on ientrope inide the urface, the Corioli parameter f take on it uual value, the relative vorticity vanihe, and the indefinite expreion / equal one. Convention I i convenient for cloure of the eddy flux of potential vorticity in the urface layer becaue, under convention I, the mean potential vorticity gradient in the urface layer i uually well defined and varie on large cale. Under convention I, the mean potential vorticity throughout the atmophere, including the urface layer, can be written a P * (f )/,and, if the relative vorticity gradient can be neglected, the mean potential vorticity gradient can be approximated by y P * f y The firt term on the right-hand ide i poitive both in the interior atmophere and in the urface layer. The econd term typically change ign near the top of the FIG. 1. Mean potential vorticity gradient y P * (10 6 PVU m 1, with 1 PVU 10 6 m 2 1 Kkg 1 ) under convention I. The contouring i logarithmic, with contour level at (4 3,4 2,..., 4 4 ) 10 6 PVU m 1. Here and in ubequent figure, the dotted line repreent the 10%, 50%, and 90% ioline of the cumulative ditribution of urface potential temperature. urface layer: it i uually poitive in the interior tropophere, where the mean ientropic denity decreae poleward along ientrope; it i uually negative in the urface layer, where the mean ientropic denity increae poleward along ientrope. The mean ientropic denity increae poleward along urface-layer ientrope primarily becaue a pole-to-equator urface potential temperature gradient implie that the relative frequency with which an ientrope lie above the urface increae poleward along the ientrope. (If variation of the denity and tatic tability near the urface are ignored, the mean ientropic denity H( ) in the urface layer i proportional to the cumulative ditribution of urface potential temperature (y, ) H( ), or to the relative frequency with which an ientrope lie above the urface.) If the variance and higher moment of urface potential temperature fluctuation vary on much larger meridional cale than the mean urface potential temperature, the ientropic gradient y of the cumulative ditribution of urface potential temperature i directly related to the mean urface potential temperature gradient, which i uually well defined and varie on large cale. To the extent that the term involving i mall, thi implie that the gradient (13) of the mean potential vorticity in the urface layer i uually negative and likewie varie on large cale. Figure 1 how the mean potential vorticity gradient under convention I in a imulation with an idealized GCM (decribed in appendix A). Included in Fig. 1 are the 10%, 50%, and 90% ioline of the cumulative ditribution of urface potential temperature. The 50% ioline, the median, approximate the mean urface potential temperature, and the 10% and 90% ioline can be taken a demarcating the urface layer. The figure how that the mean potential vorticity gradient i indeed poitive in the interior tropophere and negative in the urface layer and that it varie on large cale.

7 1890 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 If the mean potential vorticity gradient varie on meridional cale that are large in comparion with eddy length cale an acceptable albeit not very accurate aumption in the urface layer term involving third and higher derivative in an expanion of the eddy flux of potential vorticity ˆPˆ * in term of meridional derivative of the mean potential vorticity can be neglected (term involving even derivative vanih by ymmetry). The eddy flux can then be modeled uing a diffuive cloure ˆPˆ * D y P * with an eddy diffuivity D (Corrin 1974). In the imulation hown here and in other imulation with the idealized GCM, the empirical eddy diffuivitie ˆPˆ */ y P * in the urface layer indeed vary on large cale under convention I and are poitive, except near the top of the urface layer, where the mean potential vorticity gradient change ign and eddy diffuivitie are not defined. The approximation (11) of the integrated urface preure drag contribution to the ma flux along ientrope appear to be very accurate under convention I. In the imulation with the idealized GCM, it entail error of 5% 10% in midlatitude. In other imulation with the idealized GCM, panning a wide range of climate, the approximation (11) conitently entail midlatitude error of about 10% and often le. 3 Uing convention I and the approximation of mall Roby number, P * f f 0, 14 where 0 ( ) denote the ientropic denity at the mean urface potential temperature (y), one can write the balance condition (12) a b i * d b i ˆPˆ * J F y * P * d Schneider (2004) take thi balance condition a the point of departure for the derivation of a contraint on the tropopaue height and the thermal tratification of the tropophere. 3 To the extent that eddie are weakly nonlinear and quaigeotrophic and that the ditribution of urface potential temperature fluctuation i ymmetric about the mean urface potential temperature, heuritic argument may account for the accuracy of the approximation (11) under convention I. For weakly nonlinear quaigeotrophic eddie, the fluxe of even moment of urface potential temperature fluctuation vanih to leading order, and for a ymmetric ditribution of urface potential temperature fluctuation, the econd derivative of the cumulative ditribution function vanihe at the mean urface potential temperature. So if the Roby number i mall (P * f / ) and if the mean ientropic denity in the urface layer can be approximated by, the firt correction to the zeroth-order term in the integral (11) may only be the fourth-order term involving the flux of the fifth moment of urface potential temperature fluctuation. FIG. 2. Mean potential vorticity gradient y P * (10 6 PVU m 1 ) under convention II. The contouring i logarithmic, with contour level at (4 3,4 2,...,4 4 ) 10 6 PVU m 1. 2) CONVENTION II Under convention II, the abolute vorticity on ientrope inide the urface equal zero, P f 0. One contrual of thi convention i that the indefinite expreion / vanihe on ientrope inide the urface. Schneider et al. (2003) and Koh and Plumb (2004) adopted thi convention in their dicuion of the potential vorticity budget on ientrope. Convention II i le convenient for cloure of the eddy flux of potential vorticity in the urface layer than convention I becaue, under convention II, the mean potential vorticity gradient in the urface layer doe not necearily have a definite ign and can vary on relatively mall cale. Under convention II, the mean potential vorticity can be written a P * (f )/, with the undertanding that the relative vorticity vanihe on ientrope inide the urface. If the relative vorticity gradient can be neglected, the mean potential vorticity gradient y P * f y f y involve a third term proportional to the gradient of the cumulative ditribution of urface potential temperature. If the mean ientropic denity in the urface layer i proportional to the cumulative ditribution function,, thi third term cancel the econd term on the right-hand ide. But to the extent that the cancellation i incomplete and that the term involving i mall, the mean potential vorticity gradient under convention II doe not necearily have a definite ign and can vary on relatively mall cale. Figure 2 how the mean potential vorticity gradient under convention II in the imulation with the idealized GCM. The mean potential vorticity gradient in the urface layer varie on maller cale under convention II than under convention I. Correpondingly, in the imulation hown here and in other imulation with the 2

8 JUNE 2005 S C H N E I D E R 1891 idealized GCM, empirical eddy diffuivitie ˆPˆ */ y P * for potential vorticity in the urface layer vary on maller cale under convention II than under convention I and even are, in ome intance, negative under convention II. If the mean urface potential temperature i approximately equal to the median urface potential temperature, the mean potential vorticity at the mean urface potential temperature i about a factor of 2 maller under convention II than under convention I: P * ( ) f ( )/ 0 0.5f / 0. Therefore, the ma flux aociated with the eddy flux of urface potential temperature on the right-hand ide of the balance condition (12) i about a factor of 2 greater under convention II than under convention I. The other term on the right-hand ide of the balance condition (12) have correpondingly changed magnitude under convention II, and the ma flux aociated with the eddy flux of potential vorticity in the urface layer can have a different ign under convention II than under convention I. The approximation (11) of the integrated urface preure drag contribution to the ma flux along ientrope alo appear to be accurate under convention II, albeit le o than under convention I. In the imulation with the idealized GCM, the approximation (11) entail error of 10% 30% in midlatitude. Since the ma flux aociated with the integrated urface preure drag i about a factor of 2 greater under convention II, the fact that the relative error of the approximation (11) i about a factor of 2 3 greater tranlate into an abolute error that i about a factor of 4 6 greater than under convention I. In other imulation with the idealized GCM, the abolute error of the approximation (11) i conitently larger under convention II than under convention I. Convention I and II are two poible convention for the mean potential vorticity in the urface layer. Other convention, aigning other value to the abolute vorticity P on ientrope inide the urface, are poible in principle. The analye here will be focued on convention I, ince it i convenient for eddy flux cloure and ince it will allow u to draw analogie between the ientropic-coordinate urface layer and the lower layer of quaigeotrophic two-layer model (ection 4b). However, ince the quaigeotrophic limit of the balance condition (12) i obtained under convention II (ection 4a), ome reult under convention II that are pertinent to the quaigeotrophic limit will alo be dicued. c. Geotrophic limit of mean zonal momentum balance To interpret the balance condition (15) in term of the mean zonal momentum balance on ientrope and to make connection with quaigeotrophic theory, it i helpful to conider the geotrophic limit. Introducing the geotrophic meridional velocity in ientropic coordinate by g f 1 x MH, with the horthand H H( ) for the tep function, we can write the denity-weighted mean of the geotrophic balance equation a f * g x M. In thi form, geotrophic balance tate that the Corioli force on the geotrophic ma flux along an ientrope balance the zonal preure gradient force, which, in turn, i equal to the form drag exerted on the ientrope (Andrew 1983; Held and Schneider 1999; Koh and Plumb 2004). The form drag include the urface preure drag that the flow along ientrope experience at interection of ientrope with the urface. Decompoing the geotrophic ma flux and the zonal preure gradient force into mean and eddy component, f g x MH and f g x MH, with prime here denoting fluctuation ( ) ( ) ( ) about the ientropic mean, one can alternatively write the geotrophic balance equation a * g g g f 1 x M. In the urface layer, unlike in the interior atmophere, the mean g of the geotrophic velocity i generally nonzero. In the geotrophic limit, the vertical integral of the eddy component g and of the mean component g of the geotrophic ma flux can be identified with the two eddy flux term on the right-hand ide of the balance condition (15). To conider thi limit, we neglect relative vorticity contribution to the potential vorticity, writing P g f /, and we replace velocitie by geotrophic velocitie. (i) In the geotrophic limit, potential vorticity fluctuation about the denity-weighted ientropic mean can be written a Pˆ g f( 1 1 ). Upon multiplication by the ientropic denity, it follow that Pˆ g (f/ ). Since, for ientropic mean of quadratic term, the relation ( ˆ)( ) ( )( ) hold (cf. Tung 1986), the ma flux aociated with the eddy flux of potential vorticity i, in the geotrophic limit, equal to the eddy ma flux, * ˆPˆ g P *, 16a g or, upon ubtitution of geotrophic velocitie, * ˆgPˆ g P * g. 16b g Thu, the ma flux aociated with the eddy flux of potential vorticity that appear in the balance condition (15) i, in the geotrophic limit, equal to the geotrophic eddy ma flux. From the perpective of the zonal momentum balance, thi implie that, in the geotrophic limit, the Corioli force on the

9 1892 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 ma flux aociated with the eddy flux of potential vorticity balance an eddy form drag. The relation (16b) between geotrophic eddy fluxe of potential vorticity and ma ha a counterpart in quaigeotrophic theory. If the contribution of the relative vorticity to the quaigeotrophic potential vorticity i neglected, the ma flux aociated with the quaigeotrophic potential vorticity flux i likewie equal to the geotrophic eddy ma flux (cf. Rhine and Holland 1979). Unlike in quaigeotrophic theory, however, in ientropic coordinate it i only neceary to aume that the contribution of the relative vorticity to the potential vorticity i mall; it i not neceary to aume that fluctuation of the ientropic denity are mall. (ii) In the geotrophic limit, the term x MH / P * M x ( )/P * in the zonal momentum balance (10) become x MH / f g. The approximate vertical integral (11) of thi term i what gave rie to the ma flux aociated with the balanced eddy flux of urface potential temperature, FIG. 3. Ma flux treamfunction (10 9 kg 1 ) (olid line: counterclockwie rotation; dahed line: clockwie rotation). between the ma flux aociated with the eddy flux of potential vorticity and a geotrophic eddy ma flux. The mean geotrophic ma flux i related to the balanced eddy flux of urface potential temperature by b i g d b i g d, 19 Thu, upon vertical integration over the urface layer, the geotrophic mean ma flux i approximately equal to the ma flux aociated with the balanced eddy flux of urface potential temperature that appear in the balance condition (15). From the perpective of the zonal momentum balance, thi implie that, in the geotrophic limit and upon vertical integration, the Corioli force on the ma flux aociated with the balanced eddy flux of urface potential temperature approximately balance a mean form drag, or a mean urface preure drag due to the interection of ientrope with the urface. Thee relation between geotrophic eddy and mean ma fluxe and eddy fluxe of potential vorticity and urface potential temperature hold under convention I for the potential vorticity. Under convention II, one can derive imilar relation between geotrophic eddy and mean ma fluxe and eddy fluxe of potential vorticity and urface potential temperature if one introduce a mean H that i normalized by the relative frequency with which an ientrope lie above the urface and if one define eddy field ( ˇ) ( ) ( ) a fluctuation about thi mean. Reaoning analogou to the above then lead to the relation * ˆgPˆ g P * ˇˇg g 18 where ( ) 0 /( ) 2 0 i the mean ientropic denity at the mean urface potential temperature, normalized by the value of the cumulative ditribution function ( ) 0.5. In the geotrophic limit under convention II, the vertical integral of the eddy component ˇ ˇg and of the mean component g of the geotrophic ma flux * g ( g ˇ ˇg ) can be identified with the two eddy flux term on the righthand ide of the balance condition (12). Thu, in the geotrophic limit under convention II a under convention I, the Corioli force on the ma fluxe aociated with the eddy fluxe of potential vorticity and of urface potential temperature balance eddy and mean component of the form drag. Convention I and II differ in the way in which the form drag on ientrope i partitioned into mean and eddy component. d. Validity of balance condition in GCM imulation Figure 3 how the ma flux treamfunction, 2a * d co b in the idealized GCM imulation. The mean meridional ma flux repreented by the treamfunction i primarily compoed of the ma flux aociated with the eddy flux of potential vorticity (Fig. 4) and the ma flux aociated with the urface preure drag (Fig. 5). The vertical integral (11) of the ma flux aociated with the urface preure drag i what give rie to the ma flux aociated with the balanced eddy flux of urface potential temperature.

10 JUNE 2005 S C H N E I D E R 1893 FIG. 4. Ma flux ˆPˆ */P * aociated with eddy flux of potential vorticity under convention I (kg m 1 K 1 1 ). Under convention I, which we adopt for the remainder of thi ection, the ma flux aociated with the eddy flux of potential vorticity change ign near the top of the urface layer, from equatorward flux in the urface layer to poleward flux in the interior atmophere (Fig. 4). The ma flux aociated with the urface preure drag i equatorward in the urface layer and vanihe in the interior atmophere (Fig. 5). Taken together, thee ma flux component reult in poleward ma flux in the interior tropophere and equatorward ma flux near the urface, approximately within the urface layer (Fig. 3). Figure 6 diplay term in the balance condition (15) with an approximate top of the urface layer a the upper limit i ( ) of the integration. The ma fluxe diplayed thu are ma fluxe integrated over the urface layer. The approximate top of the urface layer i taken to be the 90% ioline of the cumulative ditribution of urface potential temperature (the uppermot dotted line in Fig. 3 5). Figure 6a how to what extent the balance condition (15) i quantitatively accurate at the approximate top of the urface layer. The integrated ma fluxe in Fig. 6a, repreenting the left-hand and right-hand ide of the balance condition (15), differ by 10% 20%. Thi difference i due to four factor: (i) neglecting the diabatic component of the potential vorticity flux (5% 10% error in midlatitude; 10% 15% error in ubtropic); (ii) taking the ma flux 0 a repreenting, in place of the integral (11) of the ma flux aociated with the urface preure drag up to the top of the urface layer, the integral up to the approximate top of the urface layer (10% error); (iii) dicretizing and interpolating to ientropic coordinate (5% 10% error); (iv) approximating the integral (11) of the urface preure drag term (in addition to the error due to the approximation of the upper limit of integration; 5% 10% error). Figure 6b how individual contribution to the ma flux integrated over the urface layer. The balanced eddy flux of urface potential temperature and the eddy flux of potential vorticity in the urface layer are aociated with equatorward ma fluxe of imilar magnitude, a i alo evident from Fig. 4 and 5. [Since the ma flux aociated with the eddy flux of potential vorticity change ign below the 90% ioline of the cumulative ditribution of urface potential temperature (Fig. 4), ome poleward ma flux contribute to the urface layer integral hown in Fig. 6b, thu reducing the magnitude of the equatorward ma flux hown there.] The Ekman ma flux i directed poleward in region of urface weterlie and equatorward in region of urface eaterlie. In the extratropic, the Ekman ma flux i coniderably weaker than the ma fluxe aociated with the eddy fluxe of potential vorticity and urface potential temperature. e. Relative magnitude of ma fluxe aociated with eddy fluxe Scaling etimate how that, under convention I, the ma fluxe aociated with the eddy fluxe of urface potential temperature and potential vorticity in the urface layer are of the ame order of magnitude not only in the imulation with the idealized GCM, but in general. If the contribution of the relative vorticity to the potential vorticity i negligible (a good approximation in the urface layer), the ma flux aociated with the eddy flux of potential vorticity i approximately equal to the eddy ma flux (16a). To obtain an etimate of the magnitude of thi eddy ma flux integrated from the bottom b to the top I of the urface layer, we ue the relation (g 1 p)h( ) and ignore correlation between the preure and the vertical hear of the meridional velocity on ientrope above the urface, I ˆPˆ * I b P * d d b FIG. 5. Ma flux x MH / P * f g /P * g aociated with urface preure drag under convention I (kg m 1 K 1 1 ). g 1 b I ph d g 1 p I p. 20

11 1894 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 FIG. 6. Mean ma flux along ientrope integrated over urface layer (up to the 90% ioline of the cumulative ditribution of urface potential temperature). (a) Actual ma flux * d (olid line) and ma flux aociated with eddy fluxe and friction [dahed line; right-hand ide of balance condition (15)]. (b) Individual contribution to ma flux under convention I: eddy flux of potential vorticity ˆPˆ */P * d (olid line); balanced eddy flux of urface potential temperature 0 (dahed line); Ekman ma flux J y F * /P * d (dah dotted line). The ma fluxe are multiplied by the length 2a co() of latitude circle, o that they are comparable with the value of the treamfunction in Fig. 3. In the lat line, we ubtituted the balanced eddy velocity at the urface a an etimate of a repreentative meridional eddy velocity on ientrope between the urface (potential temperature ) and the top of the urface layer (potential temperature I ). Approximating the preure at the top of the urface layer by expanding about the urface preure and neglecting fluctuation of the ientropic denity at the urface, one obtain for the intantaneou ma per unit area of the urface layer (cf. Jucke et al. 1994; Held and Schneider 1999) g 1 p I p g 1 p I I, 21 where i the ientropic denity averaged along the urface. Combining the etimate (20) and (21) and uing the fact that the potential temperature I at the top of the urface layer i fixed, one find the etimate I ˆPˆ * b P * d for the ma flux aociated with the eddy flux of potential vorticity in the urface layer. The urface mean of the ientropic denity and the ientropic mean 0 at the mean urface potential temperature are of imilar magnitude (the ientropic mean 0 i typically about a factor of 2 maller than the urface mean ). So the ma flux aociated with the eddy flux of potential vorticity in the urface layer i of the ame direction and order of magnitude a the ma flux aociated with the balanced eddy flux of urface potential temperature. By analye of National Center for Environmental Prediction National Center for Atmopheric Reearch (NCEP NCAR) reanalyi data (Kalnay et al. 1996) and analye of idealized GCM imulation, including thoe decribed by Schneider (2004), it wa verified that, in midlatitude, the ma fluxe aociated with the eddy fluxe of urface potential temperature and of potential vorticity in the urface layer indeed are generally of the ame direction and order of magnitude. The analyzed imulation pan a wide range of climate, from weakly baroclinic flow with pole-to-equator urface potential temperature difference of about 10 K to trongly baroclinic flow with pole-to-equator urface potential temperature difference of about 150 K. 4 f. Qualitative account of ma flux along ientrope The ma flux treamfunction in ientropic coordinate i characterized by an overturning cell in each hemiphere, with equatorward ma flux in the urface layer and poleward ma flux aloft (Johnon 1989). With the imulation with the idealized GCM, we illutrate that qualitative apect of the ma flux treamfunction in the extratropical tropophere can be undertood by auming that eddie tend to homogenize quantitie that are materially conerved in adiabatic and invicid flow (cf. Held and Schneider 1999). In the interior of the extratropical tropophere, the ma flux along ientrope i primarily aociated with the eddy flux of potential vorticity (Tung 1986). To the extent that eddie tend to homogenize potential vorticity on interior ientrope, the eddy flux ˆPˆ * i 4 A dicued in ection 3b, the relative magnitude of the ma fluxe aociated with the eddy fluxe of urface potential temperature and of potential vorticity in the urface layer depend on the convention for the potential vorticity on urface-layer ientrope. Under convention II, it appear to be difficult to make general tatement about the ign of the ma flux aociated with the eddy flux of potential vorticity in the urface layer. Analye of NCEP NCAR reanalyi data and of imulation with the idealized GCM how that, under convention II, the ma flux aociated with the eddy flux of potential vorticity in the urface layer can be equatorward or poleward.

12 JUNE 2005 S C H N E I D E R 1895 generally directed outhward, ince the mean potential vorticity gradient i generally poitive in the interior tropophere (Fig. 1). According to the balance condition (15), a outhward eddy flux of potential vorticity i aociated with a poleward ma flux, conitent with the ma flux hown in Fig. 4. In the urface layer of the extratropical atmophere, the ma flux along ientrope i compoed of the Ekman ma flux and the ma fluxe aociated with the eddy fluxe of urface potential temperature and of potential vorticity in the urface layer. (i) The mean urface potential temperature generally increae equatorward. To the extent that eddie tend to homogenize potential temperature along the urface, the eddy flux i therefore directed poleward. According to the balance condition (15), a poleward eddy flux of urface potential temperature i aociated with an equatorward ma flux in the urface layer, conitent with the ma fluxe hown in Fig. 5 and 6b (cf. Held and Schneider 1999). (ii) The mean potential vorticity gradient under convention I i uually negative in the urface layer (Fig. 1). To the extent that eddie tend to homogenize potential vorticity in the urface layer, the eddy flux ˆPˆ * i therefore directed northward. According to the balance condition (15), a northward eddy flux of potential vorticity implie an equatorward ma flux, conitent with the ma fluxe hown in Fig. 4 and 6b. (iii) The Ekman ma flux i directed poleward in the region of the extratropical urface weterlie, hence i directed oppoite to the net ma flux in the urface layer, and thu i weaker than the um of the other component of the urface-layer ma flux. Cloer inpection of Fig. 3 and 4 reveal that the total ma flux and the ma flux aociated with the eddy flux of potential vorticity in the urface layer are not ditributed ymmetrically about the mean urface potential temperature but are kewed to lower potential temperature. Thi kewed ditribution of ma fluxe appear to be due to the preence of a mixed layer near the urface, which implie a delta-function ingularity of the ientropic denity at the urface (Held and Schneider 1999). The ditribution of ma fluxe in imulation with an idealized GCM that doe not have a urface mixed layer i ymmetric about the mean urface potential temperature (ee Schneider 2004, Fig. 2). 4. Comparion with quaigeotrophic theory a. Continuouly tratified model Rearranging the quaigeotrophic balance condition (9), one obtain the quaigeotrophic counterpart of the ientropic-coordinate balance condition (12), 0 g q z 0 F x z 0 dz 0 0 f 0 z g zz The eddy fluxe of quaigeotrophic potential vorticity along horizontal plane can be viewed a repreenting eddy fluxe of potential vorticity along ientrope (Charney and Stern 1962). Formally, the quaigeotrophic balance condition (22) thu reemble the ientropic-coordinate balance condition (12) if the integral in the latter are likewie taken to extend to the top of the atmophere. Near the urface, however, the two balance condition generally differ. In quaigeotrophic model of a continuouly tratified atmophere, potential vorticity gradient are uually poitive throughout the atmophere, down to immediately above the urface (ee, e.g., Solomon and Stone 2001a, b). The eddy flux of quaigeotrophic potential vorticity in a tatitically tationary tate i directed downgradient in the mean, hence i negative, and i aociated with a ma flux 0 g q z /f 0 that i directed poleward down to immediately above the urface. The equatorward ma flux ( 0 / z 0 ) g z z0 aociated with the eddy flux of urface potential temperature cloe the quaigeotrophic tranformed Eulerian mean ma circulation in an infiniteimally thin heet at the urface. In contrat, the equatorward ma flux in ientropic coordinate i ditributed over a urface layer that comprie up to one third of the ma of the Earth tropophere. (According to NCEP NCAR reanalyi data, the mean preure at the 90% ioline of the cumulative ditribution of urface potential temperature varie between about 850 and 725 hpa in the extratropic.) A we have een, the Corioli force on the ma flux aociated with the eddy flux of potential vorticity in the urface layer balance, in the geotrophic limit, an eddy form drag. Irrepective of how eddy and mean field are defined, thi eddy form drag and the ma flux aociated with the eddy flux of potential vorticity in the urface layer have no counterpart in quaigeotrophic model of a continuouly tratified atmophere becaue the horizontal plane to which the quaigeotrophic potential vorticity flux i confined do not interect the urface. The quaigeotrophic balance condition (22) can only be adequate if the ma of the urface layer i negligible. If vertical variation of denity and tatic tability are negligible, the ratio of the ma of the urface layer to the ma of the tropophere cale a ( / z z )/H t, where denote a typical urface potential temperature fluctuation and H t the tropopaue height. With urface potential temperature fluctuation that cale a y L, with a near-urface eddy length cale L, the condition that the ma of the urface layer i mall i tantamount to the condition that the lope of ientrope near the urface i mall,