VOL. 61, NO.12 JOURNAL OF THE ATMOSPHERIC SCIENCES 15 JUNE 2004

The Tropopause and the Thermal Strati®cation in the Extratropics of a Dry Atmosphere

TAPIO SCHNEIDER California Institute of Technology, Pasadena, California

(Manuscript received 8 October 2002, in ®nal form 5 December 2003)

ABSTRACT A dynamical constraint on the extratropical tropopause height and thermal strati®cation is derived by con- siderations of entropy ¯uxes, or isentropic mass ¯uxes, and their different magnitudes in the troposphere and stratosphere. The dynamical constraint is based on a relation between isentropic mass ¯uxes and eddy ¯uxes of potential vorticity and surface potential temperature and on diffusive eddy ¯ux closures. It takes baroclinic eddy ¯uxes as central for determining the extratropical tropopause height and thermal strati®cation and relates the tropopause potential temperature approximately linearly to the surface potential temperature and its gradient. Simulations with an idealized GCM point to the possibility of an extratropical climate in which baroclinic eddy ¯uxes maintain a statically stable thermal strati®cation and, in interaction with large-scale diabatic processes, lead to the formation of a sharp tropopause. The simulations show that the extratropical tropopause height and thermal strati®cation are set locally by extratropical processes and do not depend on tropical processes and that, across a wide range of atmospheric circulations, the dynamical constraint describes the relation between tro- popause and surface potential temperatures well. An analysis of observational data shows that the dynamical constraint, derived for an idealized dry atmosphere, can account for interannual variations of the tropopause height and thermal strati®cation in the extratropics of the earth's atmosphere. The dynamical constraint implies that if baroclinic eddies determine the tropopause height and thermal strat- i®cation, an atmosphere organizes itself into a state in which nonlinear interactions among eddies are inhibited. The inhibition of nonlinear eddy±eddy interactions offers an explanation for the historic successes of linear and weakly nonlinear models of large-scale extratropical dynamics.

1. Introduction tropopause height and thermal strati®cation into two parts: a radiative constraint and a dynamical constraint. The troposphere is the atmospheric layer within The radiative constraint takes for each latitude a measure which the circulation redistributes the bulk of the en- of the tropospheric thermal strati®cation (e.g., the tem- tropy (heat) that the atmosphere receives by the heating perature lapse rate) and a lower boundary condition at the surface. In ¯uid dynamical parlance, the tropo- (e.g., the temperature or potential temperature at the sphere is the caloric boundary layer of the atmosphere, surface) as given and determines the tropopause height with the tropopause as the upper boundary of this layer. by radiative transfer considerations. The tropopause The height of the tropopause and the thermal strati®- height at each latitude is determined as the minimum cation of the troposphere are determined by a dynamical height at which the tropospheric temperature pro®le that equilibrium between radiative processes and dynamic is consistent with the measure of the thermal strati®- entropy ¯uxes. For the Tropics, the fundamental sig- cation and with the lower boundary condition matches ni®cance of moist convection for the entropy transport a radiative equilibrium temperature pro®le that implies and for determining the tropopause height and thermal the same amount of upwelling longwave radiation at the strati®cation are well established. But despite decades tropopause as the tropospheric temperature pro®le. That of experimentation with general circulation models, the is, to determine the tropopause height, one assumes the dynamics that determine the tropopause height and ther- stratosphere to be approximately in radiative equilib- mal strati®cation in the extratropics have remained ob- rium. Thuburn and Craig (1997, 2000) have shown in scure. a series of GCM simulations that, if one takes a rep- Held (1982) suggested subdividing theories of the resentative latitude-dependent temperature lapse rate and the surface temperature as given, radiative con- straints of this kind can approximately account for the Corresponding author address: Tapio Schneider, California Insti- tute of Technology, Mail Code 100-23, 1200 E. California Blvd., tropopause height. Pasadena, CA 91125. The dynamical constraint provides the measure of the E-mail: [email protected] tropospheric thermal strati®cation that is taken as given

᭧ 2004 American Meteorological Society 1317 1318 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 in the radiative constraint. For the Tropics, the fact that the effects of extratropical moist convection may not moist convection maintains the temperature lapse rate always be negligible, I regard it as foundational for any close to the moist pseudoadiabatic lapse rate (Xu and future general circulation theory to study how baroclinic Emanuel 1989) provides such a dynamical constraint. eddies, in concert with radiative processes but in iso- The dif®culty in accounting for the extratropical tro- lation from convection and other complicating factors popause height and thermal strati®cation has its roots such as surface topography and moisture effects, can in the lack of a dynamical constraint of comparable determine the tropopause height and thermal strati®- simplicity for the extratropics. If a dynamical constraint cation in the extratropics. The theoretical developments for the extratropics were available, the temperature dis- and analyses of simulations in this paper therefore focus tribution in the extratropical troposphere and the tro- on the mean state of a dry ideal-gas atmosphere with popause height could be computed from a radiative con- stationary and axisymmetric circulation statistics. straint of the kind discussed by Thuburn and Craig, Section 2 derives a dynamical constraint on the ex- provided the temperature or potential temperature at the tratropical tropopause height and thermal strati®cation surface is taken as given or can be inferred from energy by considerations of entropy ¯uxes, or isentropic mass balance considerations. ¯uxes, and their relation to baroclinic eddy ¯uxes of Baroclinic eddies constitute central elements in the potential vorticity and surface potential temperature. dynamical constraints on the extratropical tropopause The tropopause is taken as the upper boundary up to height and thermal strati®cation that have been pro- which signi®cant isentropic mass ¯uxes originating at posed. For example, according to baroclinic adjustment the surface extend, a view that leads to a balance con- hypotheses, baroclinic eddies maintain the thermal strat- dition for baroclinic eddy ¯uxes and, with the help of i®cation of the extratropical troposphere in a state that diffusive eddy ¯ux closures, to the dynamical constraint. is neutral with respect to linear baroclinic instability Section 3 describes an idealized GCM and a reference (see, e.g., Stone 1972, 1978; Lindzen and Farrell 1980; simulation with the idealized GCM, pointing to the pos- Lindzen 1993). A priori, however, there is no reason to sibility of an extratropical climate in which baroclinic expect that baroclinic eddies would maintain the extra- eddy ¯uxes maintain a statically stable thermal strati- tropical atmosphere in a state that is neutral with respect ®cation and, in interaction with large-scale diabatic pro- to linear baroclinic instability; baroclinic instability can cesses, lead to the formation of a sharp tropopause. Sec- equilibrate nonlinearly, which can result in a dynamical tion 4 summarizes a series of simulations with the ide- equilibrium that is linearly unstable (see, e.g., Vallis alized GCM that show that, across a wide range of at- 1988). As an alternative to baroclinic adjustment hy- mospheric circulations, the proposed dynamical potheses, Juckes (2000) has proposed a dynamical con- constraint describes the relation between tropopause and straint that takes the sporadic moist convection in warm surface potential temperatures well. Section 5 shows sectors of surface cyclones as the determinant of the that the dynamical constraint, derived for an idealized tropospheric thermal strati®cation. In contrast to baro- dry atmosphere, can also account for observed inter- clinic adjustment hypotheses, Juckes's dynamical con- annual variations of the tropopause height and thermal straint takes convective in¯uences on the extratropical strati®cation in the extratropics of the earth's atmo- thermal strati®cation into account and does not presup- sphere. Section 6 discusses an implication of the dy- pose the equilibration of baroclinic eddies to be weakly namical constraint, namely, that if baroclinic eddies de- nonlinear (as the equilibration would have to be to result termine the tropopause height and thermal strati®cation, in a baroclinically neutral dynamical equilibrium). A an atmosphere organizes itself into a state in which non- similar proposal in which slantwise moist convection linear interactions among the eddies are inhibited. Sec- plays a central role in determining the extratropical tro- tion 7 summarizes the conclusions. The appendix lists popause height and thermal strati®cation has been made the notation and symbols used in this paper. by Emanuel (1988, 2002). In this paper, a dynamical constraint is proposed that, like baroclinic adjustment hypotheses, is based on the 2. A dynamical constraint on the extratropical premise that baroclinic eddies are central for determin- tropopause height and thermal strati®cation ing the extratropical tropopause height and thermal strat- a. De®ning features of troposphere and tropopause i®cation. The dynamical constraint relates the tropo- pause potential temperature via a diffusive closure of To develop a dynamical constraint on the extratropical baroclinic eddy ¯uxes to the surface potential temper- tropopause height and thermal strati®cation, one needs ature and its gradient, which are taken as given. Unlike a dynamically based de®nition of what constitutes the baroclinic adjustment hypotheses, the proposed dynam- troposphere and tropopause. The existing ad hoc con- ical constraint does not presuppose the equilibration of ventions of determining the tropopause, such as the baroclinic eddies to be weakly nonlinear. And unlike World Meteorological Organization's convention or the Juckes's and Emanuel's dynamical constraints, the pro- convention of identifying the extratropical tropopause posed dynamical constraint neglects convective in¯u- with an isoline of potential vorticity (Holton et al. 1995), ences on the extratropical thermal strati®cation. Though provide adequate characterizations of the tropopause in 15 JUNE 2004 SCHNEIDER 1319

y(␳␷␪ *) of isentropic mass ¯uxesץ the present-day climate, but they may be inadequate in of the divergence a changed climate. One needs a quantitative de®nition originating at the surface implies a large rate at which of the tropopauseÐor, equivalently, a quantitative dis- air masses in isentropic layers are replaced with surface tinction between troposphere and stratosphereÐthat is air. Dividing typical values of the isentropic density Ϫ2 Ϫ1 independent of the speci®c climatic state of the atmo- (␳␳␪ ϳ 100 kg m K in the troposphere and ␪ ϳ 20 sphere and, preferably, is applicable to the Tropics and kg mϪ2 KϪ1 in the lower stratosphere) by the typical y(␳␷␪ *) and assuming thatץ subtropics as well as to the extratropics. values of the divergence Based on the view that the troposphere is the atmo- the divergences of isentropic mass ¯uxes in the tropo- spheric layer within which the entropy received at the sphere and in the lower stratosphere are due to entropy surface is redistributed, a quantitative distinction be- ¯uxes originating at the surface, one obtains typical val- tween troposphere and stratosphere can be derived from ues of the mean replacement times of air masses with the entropy conservation law. In isentropic coordinates, surface air: on the order of 10 days in the troposphere entropy conservation is equivalent to mass conserva- and on the order of 1 year in the lower stratosphere, tion, consistent with estimates of the ``mean age'' of air de- rived from tracer measurements (cf. Hall and Plumb (␳ Q) ϭ 0, (1)ץ␳␷) ϩ)ץ␳ u) ϩ)ץ␳ ϩ ץ t ␪ x ␪ y ␪␪␪ 1994; Andrews et al. 2001). That is, as is well known, where horizontal derivatives are understood as deriva- air masses in isentropic layers in the troposphere are tives along isentropes and Q ϭ D␪/Dt denotes the ma- replaced much more ef®ciently with surface air than are terial derivative of potential temperature ␪.1 The isen- air masses in isentropic layers in the stratosphere. Ϫ1 ␪p)H(␪ Ϫ ␪s) is the density Making no reference to speci®c dynamic mechanismsץ tropic density ␳␪ ϭϪ(g in (x, y, ␪) space (␳␪ dx dy d␪ ϭ ␳ dx dy dz is a mass of entropy redistribution, the distinction of troposphere element), and the Heaviside step function H(´) indicates and stratosphere in terms of the divergence of isentropic that the isentropic density vanishes on isentropes ``in- mass ¯uxes allows for a consistent characterization of side'' the surface, that is, on isentropes with potential the troposphere and tropopause in the Tropics, subtrop- temperature ␪ less than the instantaneous surface po- ics, and extratropics. tential temperature ␪s(x, y, t). Averaging the conser- To the extent that the entropy received at the surface vation law (1) yields and any additional entropy received in the interior tro- * posphere are entirely redistributed within the tropo- (␳ Q ) ϭ 0, (2)ץy(␳␷␪␪␪*) ϩץ * sphere and do not reach the stratosphere, the mass cir- where (´)ϭ (␳␪ ´)/␳ ␪ denotes the density-weighted culation formed by the mass ¯ux along isentropes mean associated with the temporal and zonal mean (´) * ␳␷␪ * and the mass ¯ux across isentropes ␳␪ Q closes along isentropes. The conservation law (2) can be re- within the troposphere. That is, at each latitude, the mass garded as the mean conservation law for mass or, if ¯ux ␳␷* along isentropes integrated from a potential multiplied by the speci®c entropy, as the mean conser- ␪ temperature ␪b(y) less than the lowest potential tem- vation law for entropy. perature that occurs at the latitude to the temporal and In terms of the conservation law (2), the view of the zonal mean potential temperature ␪ t(y) of the tropo- troposphere as the atmospheric layer within which the pause approximately vanishes: entropy received at the surface is redistributed means ␳␷*) of isentropic mass ¯uxes ␪ t) ץ that the divergence y ␪ ␳␷* d␪ 0. (3) originating at the surface is of much larger magnitude ͵ ␪ ഠ in the troposphere than in overlying atmospheric layers. ␪b Both in the extratropics and in the Tropics and sub- This constraint on the mass ¯ux along isentropes de®nes tropics of the earth's atmosphere, the absolute value of the tropopause potential temperature ␪ t as the potential y(␳␷␪ *) of the annual and zonal mean temperature at which the streamfunction of the massץ the divergence mass ¯ux along isentropes drops by about two orders ¯uxes along and across isentropes closes approximately. of magnitude across the tropopause, from values up to (A quantitative version of this de®nition is given in 10Ϫ4 kg KϪ1 mϪ2 sϪ1 in the troposphere to values on section 4b.) For the present-day earth atmosphere, the the order of 10Ϫ6 kg KϪ1 mϪ2 sϪ1 in the lower strato- tropopause potential temperature thus determined sphere.2 Since the divergence of a mass ¯ux indicates roughly agrees with the conventionally determined tro- a rate of replacement of air masses, a large magnitude popause potential temperature (cf. Bartels et al. 1998, their Fig. 1). 1 For simplicity of notation, local Cartesian coordinates x and y are To derive a dynamical constraint on the extratropical used as horizontal coordinates. Where numerical values of derivatives tropopause height and thermal strati®cation from the are given, however, they have been computed in spherical coordinates. mass ¯ux constraint (3), one needs to relate the extra- 2 The values are computed from National Centers for Environ- tropical mass ¯ux along isentropes to eddy ¯uxes of mental Prediction±National Center for Atmospheric Research (NCEP±NCAR) reanalysis data (Kalnay et al. 1996). They are con- adiabatically conserved quantities, and these eddy ¯ux- sistent with the data, for example, of Yang et al. (1990, their Figs. es, via a turbulence closure, to the thermal strati®cation 6 and 10). of the troposphere. 1320 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 b. Extratropical eddy ¯uxes and the mass ¯ux along ¯ux in the balance condition (4) is tantamount to ne- isentropes glecting the contribution of the eddy momentum ¯ux convergence to the eddy ¯ux of potential vorticity. In the extratropics, where the Rossby number is small, The assumption that eddies tend to homogenize quan- * the isentropic mass ¯ux ␳␷␪ at each latitude is related tities that are materially conserved in adiabatic and fric- to eddy ¯uxes of potential vorticity P and of surface tionless ¯ows leads to a qualitative account of the ex- potential temperature ␪s via the approximate balance tratropical mass ¯ux along isentropes. Isentropes in two condition [Schneider (2003); see also Held and Schnei- separate layers must be distinguished (Held and Schnei- der (1999) and Tung (1986)]: der 1999; Schneider 2003). First, the interior atmosphere at a latitude comprises ␪␪ii* Ã s ␳␷␪ Ã P 0 isentropes ␪ Ն max(␪ ) that, at that latitude, typically ␳␷* d␪ ഠ Ϫ d␪ Ϫ ␳␷Ä Ј␪Ј . (4) s ͵ ␪ ͵ * ␪ ss do not intersect the surface. The mass ¯ux along is- ␪␪P bb entropes in the interior atmosphere is associated with * The upper limit of the integrations is an arbitrary po- the eddy ¯ux ␷ÃPÃ of potential vorticity. The meridional P* of potential vorticity along isentropes is ץ tential temperature ␪i greater than the largest surface gradient potential temperature that typically occurs at the latitude y 0 usually positive in the interior troposphereÐbecause the under consideration. The symbol ␳␳␪␪ ϭ ␪(y, s) denotes planetary vorticity gradient ␤ is positive, and because the isentropic density at the temporal and zonal mean the isentropic density ␳ ␪ usually decreases poleward surface potential temperature ␪ s, and along interior isentropes, such that, neglecting relative MЈϪc ␪Ј), vorticity gradients, both factors contribute to a positive) ץ␷Ä Јϭ f Ϫ1 sxsps potential vorticity gradient with Montgomery streamfunction M ϭ cpT ϩ gz and y ␳␪ץ ␤ with the subscript s marking surface quantities, denotes * f (ϭϪf 2 . (5ץ yyP ഠץ a balanced meridional velocity at the surface (approx- ΂΃␳␳␪␪ ␳ ␪ imately the geostrophic velocity). The temporal s Downgradient mixing of potential vorticity hence leads and zonal mean (´) along the surface appears in ad- * to a southward eddy ¯ux ␷ÃPÃ , which implies a poleward (´) dition to the temporal and zonal mean along isen- contribution to the mass ¯ux (4). tropes and the associated density-weighted mean (´)*. * Second, the surface layer at a latitude comprises is- Hats denote ¯uctuations (ˆ´ ) ϭ (´) Ϫ (´) about the den- entropes ␪ Յ ␪ Յ max(␪ ) that, at that latitude, some- sity-weighted isentropic mean, and primes denote ¯uc- b s s times intersect the surface. The mass ¯ux along isen- tuations (´)Јϭ(´) Ϫ (´) about the surface mean. On tropes in the surface layer is associated with the bal- s isentropes inside the surface, the isentropic density ␳␪, anced eddy ¯ux ␷Ä Јss␪Ј of surface potential temperature the velocities u and ␷, and the relative vorticity com- Ã * ponent ␨ normal to isentropes vanish. Since the isen- and with the eddy ¯ux ␷ÃP of potential vorticity. Mixing ␪ of surface potential temperature down the meridional tropic density vanishes, the contribution of isentropes s ys␪␷leads to a poleward eddy ¯ux Ä Јss␪Ј , whichץ inside the surface, for example, to the mean potential gradient vorticity P* ϭ [␳ ( f ϩ ␨ )/␳ ]/␳ involves the indef- implies an equatorward contribution to the mass ¯ux ␪␪␪␪ (4). This equatorward mass ¯ux can be augmented by inite expression ( f ϩ ␨␪)␳␪/␳␪. Implicit in the balance * condition (4) is the convention that, inside the surface, a mass ¯ux associated with the eddy ¯ux ␷ÃPÃ of po- ( f ϩ ␨ )␳ /␳ ϭ f ϩ ␨ ϭ f, such that the mean potential tential vorticity in the surface layer. Within the surface ␪ ␪ ␪ ␪ * vorticity can be written as P ϭ ( f ϩ ␨␳␪)/ ␪, even on layer, the isentropic density ␳␳␪ ϭ ␪ H(␪ Ϫ ␪s) increases isentropes that sometimes intersect the surface [see poleward along isentropes, primarily because the fre- Schneider (2003) for details]. quency with which the potential temperature ␪ of an The balance condition (4) derives from the zonal mo- isentrope is greater than the instantaneous surface po- mentum balance of isentropes. It states that the vertically tential temperature ␪s (i.e., the frequency with which integrated mass ¯ux ␳␷␪ * along isentropes is approxi- the isentrope is above the surface) increases as one mately composed of a mass ¯ux associated with the moves poleward along the isentrope. The isentropic- * -y ␳ ␪ thus provides a negative contriץ eddy ¯ux ␷ÃPÃ of potential vorticity along isentropes, density gradient 2 y ␳␳␪/␪ , to the potential vorticity gradientץand a mass ¯ux associated with the balanced eddy ¯ux bution, Ϫ f s (5). As one moves downward through the surface layer ␷ÄЈss␪Ј of surface potential temperature. Neglected in the balance condition (4) are the Ekman mass ¯ux due to at constant latitude, this negative contribution to the surface friction, terms of higher order in Rossby number, potential vorticity gradient eventually dominates the s 2 positive contribution ␤/␳ ␪ of the planetary vorticity gra- and terms containing eddy ¯uxes such as ␷Ä Јss␪Ј that involve higher moments of surface potential tempera- dient ␤ because the isentropic density ␳ ␪ decreases to ture ¯uctuations. Since the Ekman mass ¯ux is primarily zero at the bottom of the surface layer. Downgradient mixing of potential vorticity hence can lead to a north- balanced by the convergence of eddy momentum ¯uxes * in the upper troposphere, neglecting the Ekman mass ward eddy ¯ux ␷ÃPÃ , which, similar to a poleward eddy 15 JUNE 2004 SCHNEIDER 1321

s ¯ux ␷Ä Јs ␪Ј of surface potential temperature, implies an is negligible, surface potential temperature and potential equatorward contribution to the mass ¯ux (4).3 vorticity are nearly materially conserved on the time The constraint (3) on the mass ¯ux along isentropes scales of extratropical eddies; and typical meridional implies that the equatorward mass ¯ux in the surface length scales of the energy-containing baroclinic eddies layer and the poleward mass ¯ux in the interior atmo- in the earth's atmosphere (about 3500 km) are smaller sphere are in balance within the troposphere. Combining than the planetary scales over which the mean surface the mass ¯ux constraint (3) with the balance condition potential temperature and the mean potential vorticity (4) relating mass ¯uxes and eddy ¯uxes, one obtains a vary (about 10 000 km).4 However, since the eddy constraint on extratropical eddy ¯uxes: length scales are not much smaller than the mean-¯ow length scalesÐin particular near the top of the surface ␪ t * Ã s ␳␷␪ Ã P 0 layer, where the meridional potential vorticity gradient d␪ ഠ Ϫ␳␷␪ Ä Јss␪Ј . (6) changes signÐthe accuracy that can be expected of dif- ͵ P * ␪b fusive closures is limited [see Corrsin (1974) for esti- At each latitude, the vertically integrated mass ¯ux as- mates of how scale separation relates to the accuracy sociated with the eddy ¯ux of potential vorticity ap- of diffusive closures]. proximately balances the vertically integrated mass ¯ux associated with the balanced eddy ¯ux of surface po- d. Diffusive eddy ¯ux closure and tropopause tential temperature. A similar constraint holds for qua- properties sigeostrophic eddy ¯uxes: if one neglects dissipation and the eddy momentum ¯ux convergence (or, equiv- With the diffusive closure (7b) of the eddy ¯ux of alently, the meridional wave activity ¯ux divergence), potential vorticity as the point of departure, one can the vertically integrated eddy ¯ux of quasigeostrophic distinguish how different factors in¯uence properties of potential vorticity is proportional to the geostrophic the extratropical tropopause. If the troposphere is clearly eddy ¯ux of potential temperature at the surface (see, distinguished from the stratosphere in that the bulk of e.g., Treguier et al. 1997). However, the mass ¯ux as- the entropy received at the surface is redistributed within sociated with the eddy ¯ux of potential vorticity in the the troposphere but does not reach the stratosphere, the surface layer has no counterpart in quasigeostrophic mass ¯ux along isentropes in the interior atmosphere models of a continuously strati®ed atmosphere. (cf. Tung 1986), * *P ץ ␳␷Ã PÃ ␳ D c. Diffusive closure of extratropical eddy ¯uxes ␪␪i y ␳␷␪ * ഠ Ϫ ഠ , (8) PP** Going beyond the assumption that the meridional eddy ¯uxes of potential temperature along the surface must decrease strongly across a sharp tropopause. Ob- and of potential vorticity along isentropes are directed servational data show that kinematic eddy diffusivities downgradient, one can model the eddy ¯uxes as dif- generally do not vary strongly across the extratrop- fusive ¯uxes, ical tropopause. The empirical eddy diffusivity s Ä Ã * * y implied by the observed eddy ¯uxץ/␪ and (7a) Di ϭϪ␷ÃPP ץ␷Ä Јss␪Ј ഠ ϪD sys and mean gradient of potential vorticity does not gen- * P *, (7b) erally exhibit sharp variations across the extratropical ץ ␷Ã PÃ ഠ ϪD iy tropopause, with the exception of stronger variations with nonnegative eddy diffusivities Ds and Di for sur- in the winter hemisphere in the vicinity of localized face potential temperature and for potential vorticity on regions of strong baroclinicity, where the potential vor- isentropes. The eddy diffusivities can, and usually do, ticity gradient is close to zero (Bartels et al. 1998). depend on the mean thermal strati®cation of the at- And Nakamura's (1996) effective diffusivity, a La- mosphere and may vary with latitude on length scales grangian eddy diffusivity that quanti®es mixing ef®- that are large compared with typical eddy length scales. ciency, exhibits a minimum near the subtropical tro- However, the precise form of the eddy diffusivities is popause, in the core of the subtropical jet, but it does irrelevant for the considerations that follow. not exhibit sharp variations across the extratropical tro- Modeling the meridional eddy ¯uxes in the extra- popause (Haynes and Shuckburgh 2000). Observation- tropics as diffusive ¯uxes is justi®able because two con- al data thus suggest that, unlike the subtropical tro- ditions are approximately met (cf. Rhines and Holland popause, the extratropical tropopause cannot, as is 1979; Held 1999): to the extent that convective heating

4 Additionally, the distinction between the balanced (approximately 3 See Held and Schneider (1999) and Schneider (2003) for more geostrophic) meridional eddy ¯ux of surface potential temperature detailed discussions of the relation between isentropic mass ¯uxes appearing in the balance condition (6) and the actual meridional eddy and eddy ¯uxes of potential vorticity and surface potential temper- ¯ux of potential temperature near the surface can be neglected be- ature, including a discussion of how the presence of a mixed layer cause meridional velocity ¯uctuations in baroclinic eddies are ap- at the surface modi®es the distribution of isentropic mass ¯uxes. proximately geostrophic. 1322 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 sometimes stated, be viewed as a mixing barrier, at be brought into a form that relates the tropopause po- least not in the sense that strong variations of kinematic tential temperature to external parameters and to the mixing properties would in general de®ne the tropo- surface potential temperature. pause. The strong decrease of the isentropic mass ¯ux To the extent that the baroclinic eddies that effect across the extratropical tropopause, and the concomi- the eddy ¯uxes of potential vorticity and surface po- tant large gradients of tracers such as nitrous oxide tential temperature are vertically coherent within the (see, e.g., Mahlman 1997), are not primarily due to a troposphere, one can expect that, at each latitude, the strong decrease of kinematic eddy diffusivities. eddy diffusivity D i for potential vorticity exhibits no If the kinematic eddy diffusivity Di does not vary essential vertical structure within the troposphere (ex- strongly across the extratropical tropopause, the is- cept in regions in which the potential vorticity gra- entropic density ␳ , and with it the ``isentropic dy- * y P is close to zero); moreover, one can expect ץ ␪ dient namic eddy diffusivity'' ␳ D , must decrease strongly ␪ i a vertical mean of the eddy diffusivity D i for potential for the isentropic mass ¯ux to decrease strongly across vorticity to be approximately equal to the eddy dif- the tropopause. The only other way for the isentropic fusivity D s for surface potential temperature. This ex- mass ¯ux (8) to decrease strongly across the tropopause pectation of essentially barotropic mixing appears to would be through a strong decrease of the logarithmic * * be a plausible heuristic, whether the mixing is effected -y PP/ Ða decrease that by waves or turbulence. For Rossby waves, the meץ potential vorticity gradient is not observed in the earth's atmosphere. In the ob- ridional group velocity is largest for barotropic servational data shown by Bartels et al. (1998, their waves; for fully developed turbulence, one would ex- Fig. 4) and in the simulations described in section 4, pect an inverse energy cascade to lead to barotropi- the empirical eddy diffusivity for potential vorticity zation of the energy-containing scales of the ¯ow. To typically decreases by about a factor of 2 across the be sure, in quasigeostrophic theories, in which the extratropical tropopause, except in localized regions of thermal strati®cation and the tropopause height (or strong baroclinicity, where the decrease can be greater. their correlatives) are ®xed and unaffected by eddy In contrast, the isentropic density typically decreases ¯uxes, it is generally impossible to satisfy eddy-¯ux by about a factor of 4±5 across the tropopause, not constraints such as the one [Eq. (9)] above with an only in localized regions of strong baroclinicity, but eddy diffusivity for potential vorticity that is verti- throughout the extratropics. The decrease of the is- cally constant and equal to the eddy diffusivity for 2 entropic density ␳ ␪ ϰ ␳/N by about a factor of 4±5 surface potential temperature, unless the planetary is consistent with the observed increase of the Brunt± vorticity gradient ␤ can be neglected (Treguier et al. VaÈisaÈlaÈ frequency N by about a factor of 2 (Peixoto 1997; see also footnote 6); rather, given the tropo- and Oort 1992, chapter 7). The maximum of the po- pause height and thermal strati®cation, the vertical tential vorticity gradient (5) at the tropopause implied structure of the eddy diffusivities must be adjusted by the strong decrease of the isentropic density is as- such that the eddy-¯ux constraints are satis®ed (see, sociated with a maximum of the eddy ¯ux of potential e.g., Smith and Vallis 2002). But if the tropopause vorticity at the tropopause (Bartels et al. 1998, their height and thermal strati®cation, instead of being Fig. 3), consistent with the relatively weak variation ®xed, are determined by eddy ¯uxes, a different ad- of the eddy diffusivity across the tropopause. None- justment process is possible. The tropopause height theless, the isentropic mass ¯ux streamfunctions that and thermal strati®cation can adjust themselves such Johnson (1989) and Bartels et al. (1998) obtained from that the eddy ¯ux constraint (9) is satis®ed with an observational data show that the mass ¯ux associated eddy diffusivity D i for potential vorticity that exhibits with the maximum of the potential vorticity gradient no essential vertical structure within the troposphere at the tropopause is small compared with the mass ¯ux and that, vertically averaged, is approximately equal in the interior troposphere. These data show, then, that to the eddy diffusivity D s for surface potential tem- it is the increase of the static stability that primarily perature. This adjustment process is posited as deter- accounts for the decrease of the isentropic mass ¯ux mining the extratropical tropopause height and ther- across the extratropical tropopause. mal strati®cation: if no vertical inhomogeneities are imposed, the tropopause height and thermal strati®- e. A dynamical constraint on the extratropical cation adjust such that they are consistent with eddy tropopause height and thermal strati®cation ¯uxes whose kinematic mixing properties, re¯ected by the eddy diffusivities, exhibit no essential vertical Substituting the diffusive closure (7) into the con- inhomogeneities within the troposphere. The left- straint (6) on the extratropical eddy ¯uxes yields hand side of the constraint (9) can then be approxi- mated by ␪ t * y P 0ץ ␳␪Di (␪ s. (9 ץd␪ ഠ Ϫ␳ ␪ Dsy * ␪␪tt** P ץ yyP ␳ץ ␳ D ͵ P i ␪b ␪ d␪ Dd␪ ␪, (10) ͵ **ഠ s ͵ By a series of rough approximations, this constraint can ␪␪bbPP 15 JUNE 2004 SCHNEIDER 1323 and the constraint (9) becomes independent of the eddy at the center of the surface layer, where the mean pres- diffusivities. sure is smaller than the mean surface pressure, one ex- The integral (10) can be calculated using the small pects the approximation (12) to overestimate the isen- 0 Rossby number approximation of the potential vorticity tropic density␳ ␪ . However, the relation between the * Ϫ1 0 ␪ p between isentropic density␳ ␪ and the discrete approximationץ P ഠ f/␳␳␪ and the relation ␪ ϭϪg Ϫ1 isentropic density and pressure, with Lorenz's (1955) g (pps Ϫ t)/(␪␪t Ϫ s) is complex. The variations with 0 convention that the instantaneous pressure on isentropes latitude of the extratropical isentropic density␳ ␪ at the inside the surface is equal to the surface pressure. One mean surface potential temperature are considerably ®nds greater than those of the discrete approximation Ϫ1 g (pps Ϫ t)/(␪␪t Ϫ s). In wintertime polar inversions, ␪␪tt* ±y P ␤ according to NCEP±NCAR reanalysis data, the Bruntץ ␳␪ ␳ d␪ץd␪ ഠ ␳ Ϫ ͵͵* f ␪ y ␪ VaÈisaÈlaÈ frequency N near the surface can be about a ␪␪P ΂΃ bb factor of 2 greater than in the midtroposphere, resulting ␤ p Ϫ p in an isentropic density ␳ 0 ϰ N Ϫ2 at the mean surface ഠ st, (11) ␪ fg potential temperature that is about a factor of 4 smaller than the discrete approximation (12) would suggest. where pps ϭ (␪b) and ppt ϭ (␪ t) denote the temporal Nonetheless, when multiplied by the surface potential and zonal mean pressure at the surface and at the tro- ys␪ , as on the right-hand side ofץ temperature gradient -p | ␪ t of the integra ץpopause. The boundary term gϪ1 y ␪b the constraint (9), and averaged over the extratropics, y ␳ ␪ has been the approximation (12) entails an error of 20%±45%ץ tion of the isentropic-density gradient neglected. In the earth's atmosphere, the surface according to NCEP±NCAR reanalysis data. The simu- y (␪b) is much smaller in lations analyzed in section 4e moreover indicate that inץysppϭץ pressure gradient absolute value than the scaled pressure difference the presence of strong surface inversions, what are de- (␤/ f )(pps Ϫ t) between surface and tropopause; the noted as surface quantities on the right-hand side of the y p(␪ t) along isentropes at the tro- constraint (9) should be interpreted as quantities at theץ pressure gradient popause (a measure of the slope of isentropes) is about 0 top of the boundary layer. If the isentropic density ␳ ␪ an order of magnitude smaller in absolute value than is interpreted as a quantity at the top of the boundary 5 the scaled pressure difference (␤/ f )(pps Ϫ t). Neglect- layer, the approximation (12) appears to be relatively ing this boundary term means that the part of the po- good. For a dynamical constraint on the tropopause tential vorticity gradient that is associated with the is- height and thermal strati®cation averaged over the ex- entropic-density gradient does not contribute to the in- tratropics, then, the approximation (12) appears to be tegral (11). In terms of isentropic mass ¯uxes, neglect- justi®able, at least as a scaling estimate. ing the boundary term amounts to the approximation With the approximations (9)±(12), one obtains a dy- that, if the eddy diffusivity Di for potential vorticity namical constraint that relates the tropopause potential varies only weakly in the vertical, the mass ¯uxes as- temperature to external parameters and to the surface sociated with the contributions of the isentropic-density potential temperature and its gradient: gradient to the potential vorticity gradient (which are of opposite sign in the surface layer and interior atmo- f (␪ . (13ץ␪ Ϫ ␪ ഠ Ϫ sphere) cancel upon vertical integration over the tro- ts␤ ys posphere. A further simpli®cation of the constraint (9) results This dynamical constraint states that the potential tem- with the rough approximation perature difference ␪␪t Ϫ s between tropopause and 1 p Ϫ p surface is approximately equal to minus the surface po- ␳ 0 ഠ st (12) tential temperature gradient multiplied by f/␤ ϭ a tan(␾), ␪ g ␪ Ϫ ␪ ts where a is the planet radius and ␾ is latitude. If the for the isentropic density at the mean surface potential surface potential temperature gradient in midlatitudes temperature. This approximation generally is not very [where tan(␾) ഠ 1] multiplied by the planet radius a 0 accurate. Since the isentropic density␳ ␪ at the mean can be taken as a measure of the equator-to-pole surface surface potential temperature is the isentropic density potential temperature difference, the dynamical con- straint (13) states that the potential temperature differ-

5 Ϫ1 2 2 2 ence between tropopause and surface in midlatitudes is yT | p and valuesץ( y␪ | p ϭ pg /(RN Tץ (p␪ץ)y p | ␪ ϭϪץ Using typical of the midlatitude tropopause [p ഠ 250 hPa, T ഠ 220 K, approximately equal to the surface potential temperature N ഠ 1.9 ϫ 10Ϫ2 sϪ1 (corresponding to a temperature lapse rate of difference between equator and pole, as is the case in Ϫ1 Ϫ6 Ϫ1 .yT | p ഠ Ϯ3 ϫ 10 Km (see Peixoto and Oort the earth's atmosphereץ 2Kkm ), and 1992, chapter 7)], one ®nds for the scaled isentropic pressure gradient Since the dynamical constraint is based on diffusive p(␪ )| 90 hPa. In midlatitudes, the scaled ץ | p(␪ )| a ץ(␤/f)| y t ഠ y t ഠ closures of eddy ¯uxes, and given the roughness of the y p(␪ t) along isentropes at the tropopause isץ(␤/pressure gradient ( f thus about an order of magnitude smaller in absolute value than the approximations made, the dynamical constraint (13) can pressure difference pps Ϫ t. only be expected to hold on meridional scales that are 1324 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 large compared with typical eddy length scales. In an diffusivity for surface potential temperature.6 Consid- earth-like atmosphere in which eddy length scales are erations of the dynamics of isentropic layers combine not much smaller than the planetary scale, the dynamical elements of quasigeostrophic layer models and contin- constraint (13) can therefore only be expected to hold uously strati®ed models and were essential for the der- for extratropical averages. ivation of the dynamical constraint (13)Ðsimilar to qua- The dynamical constraint derived from the baroclinic sigeostrophic theory, with the approximation that the adjustment hypothesis that the quasigeostrophic poten- Rossby number vanishes, but different from quasigeo- tial vorticity gradient vanishes in the interior atmosphere strophic theory, without the approximation that ¯uctu- has a form similar to that of the present constraint (13), ations of the static stability (or isentropic density) about with the potential temperature gradient along pressure a ®xed reference pro®le be small, an approximation that surfaces in the midtroposphere in place of the surface would be inadequate for surface-layer isentropes. potential temperature gradient (cf. Lindzen and Farrell The dynamical constraint (13) can be combined with 1980; Held 1982). Considerations of the dynamics of a radiative constraint to obtain a closed theory of the quasigeostrophic models, however, would not lead to tropopause height and thermal strati®cation in the ex- the constraint (13). In quasigeostrophic theory, the po- tratropics. In addition to the postulate that the eddy dif- tential vorticity ¯ux along isentropes is represented as fusivity Di for potential vorticity exhibits no essential a quasigeostrophic potential vorticity ¯ux along ®xed vertical structure within the troposphere and is, verti- horizontal planes (Charney and Stern 1962). But since cally averaged, approximately equal to the eddy diffu- the horizontal planes in quasigeostrophic theory are sivity Ds for surface potential temperature, the dynam- ®xed, whereas a surface-layer isentrope at a given lat- ical constraint is based on several premises: (i) baro- itude lies sometimes inside the surface, sometimes clinic eddies dominate the entropy transport and trans- above the surface, the quasigeostrophic potential vor- port entropy ef®ciently enough to maintain the static ticity balance averaged along near-surface horizontal stability of the extratropics, while convective entropy planes differs fundamentally from the potential vorticity transport can be neglected; (ii) turbulent ¯uxes due to balance averaged along surface-layer isentropes. Layer baroclinic eddies, in interaction with large-scale diabatic models such as Phillips's (1954) two-layer model can processes, can lead to the formation of a distinct tro- represent the potential vorticity ¯uxes and associated posphere separated from a stratosphere by a tropopause; mass ¯uxes in isentropic layers (or in the surface layer and (iii) the extratropical tropopause height and thermal and interior atmosphere), but since they lack a repre- strati®cation are set locally by extratropical processes sentation of the fact that surface-layer isentropes some- and do not depend on tropical processes. By means of times lie inside the surface, they cannot represent the simulations with an idealized GCM, it was investigated isentropic mass ¯ux associated with the balanced eddy whether these premises are plausible and to what degree ¯ux of surface potential temperatureÐthe contribution to the isentropic mass ¯ux that gives rise to the ap- 6 An argument put forth by I. Held (2002, personal communication) pearance of the surface potential temperature gradient shows in more detail how quasigeostrophic dynamical constraints on in the constraint (13). Continuously strati®ed models the thermal strati®cation differ from the present constraint. Neglecting such as Charney's (1947) model can represent the po- eddy momentum ¯uxes and taking the upper boundary at in®nity, the tential vorticity ¯uxes along interior isentropes, the bal- quasigeostrophic counterpart of the balance condition (6) is ϱ zz anced eddy ¯ux of surface potential temperature, and ␳␷0 ЈggqЈ ␷Ј␪Ј dz ഠ Ϫ␳ 0 , z ␪0ץ the associated mass ¯uxes, but since they lack a rep- ͵ f0 0 Ηzϭ0 resentation of the fact that surface-layer isentropes where ␷g is the geostrophic meridional velocity, q is the quasigeo- sometimes lie above the surface, they cannot represent strophic potential vorticity, the subscript 0 marks reference values z the isentropic mass ¯ux associated with the eddy ¯ux and pro®les, (´) denotes a temporal and zonal mean along horizontal z of potential vorticity in the surface layer. In the mean planes, and (´)Јϭ(´) Ϫ (´) denotes ¯uctuations about this mean. quasigeostrophic potential vorticity balance near the Substituting diffusive eddy ¯ux closures with an eddy diffusivity D(z) for potential vorticity that is, at the surface (z ϭ 0), equal to surface, there appears either the geostrophic eddy ¯ux the eddy diffusivity for surface potential temperature (cf. Treguier et of surface potential temperature (in continuously strat- al. 1997), neglecting the relative vorticity gradient in the potential z z Ϫ1 z␪0), and integratingץ/ yץz(␳0ץ y q ഠ ␤ ϩ f 0 ␳␪0ץ i®ed models), or an eddy ¯ux of potential vorticity (in vorticity gradient layer models) that can be interpreted as a ¯ux in the the left-hand side by parts yields ϱ z y ␪ץ ␤ -isentropic surface layer; however, both eddy ¯ux com .zDdzഠ 0ץ␳ 0 D Ϫ z ␪0ץ ponents do not appear simultaneously in quasigeo- ͵ f0 0 ΂΃ strophic models [see Schneider (2003) for details]. Yet Unlike in the more general case (9), the boundary term of the inte- it is the simultaneous appearance of both eddy ¯ux com- gration by parts cancels the term involving the surface potential tem- ponents in the mean potential vorticity balance of sur- perature gradient on the right-hand side. This relation shows that the face-layer isentropes that makes it possible to postulate surface potential temperature gradient generally does not appear in quasigeostrophic constraints on the thermal strati®cation and that the that the eddy diffusivity for potential vorticity exhibits eddy diffusivity for quasigeostrophic potential vorticity can only be no essential vertical structure within the troposphere and ␤ zD ϭ 0) if the planetary vorticity gradientץ) constant with height is, vertically averaged, approximately equal to the eddy is negligible. 15 JUNE 2004 SCHNEIDER 1325 the dynamical constraint (13) holds if circulation pa- TABLE 1. Parameters of the reference simulation. rameters such as the planetary angular velocity and the Planet and ¯uid differential heating of the surface are varied. Planet radius a ϭ 6.371 ϫ 106 m Planetary angular velocity ⍀ϭ7.292 ϫ 10Ϫ5 sϪ1 Ϫ1 Ϫ1 Speci®c heat at constant pressure cp ϭ 1004.6 J kg K 3. An idealized general circulation model Gas constant R ϭ 286.9 J kgϪ1 K Ϫ1 Mean surface pressure p0 ϭ 1000 hPa The idealized GCM used is a primitive-equation mod- Surface drag 0 Ϫ5 Ϫ1 el in which the spherical lower boundary has no to- Frictional wavenumber kd ϭ 0.7 ϫ 10 m pography, quadratic damping of near-surface winds Extent of ``planetary boundary layer'' ␴b ϭ 0.85 mimics turbulent dissipation in the planetary boundary Diabatic processes e layer, and Newtonian relaxation of temperatures toward Mean surface temperature T s ϭ 300 K a radiative equilibrium state represents diabatic pro- Equator-to-pole temperature difference ⌬h ϭ 60 K Skin temperature Te ϭ 200 K cesses. Vertical diffusion and convective adjustment are t Scale height ratio Hp /Ha ␣ ϭ 3.5 not taken into account in the model, with the exception Rescaling factor of pseudoadiabatic that, in a narrow region around the equator, relaxation lapse rate ␥ ϭ 1 of temperatures toward a moist pseudoadiabat mimics ``Radiative'' relaxation time in interior ␶ i ϭ 50 days dynamic heating due to moist convection. What distin- ``Radiative'' relaxation time near surace ␶ s ϭ 7 days ``Convective'' relaxation time ␶ ϭ 4 days guishes this model from similar idealized GCMs with c Newtonian temperature relaxation is that the radiative equilibrium state toward which temperatures are relaxed b. Surface drag is, in the extratropics, statically unstable, in contrast to the statically stable radiative±convective equilibrium Quadratic damping states used in other idealized GCMs (e.g., James and vץ Gray 1986; Held and Suarez 1994; Haynes et al. 2001). ϭ ´´´Ϫ kd(␴)࿣v࿣v (14) tץ Since the model does not contain a representation of convective adjustment, only the turbulent entropy trans- of the horizontal winds v near the surface mimics tur- port due to baroclinic eddies (possibly augmented by bulent dissipation in the planetary boundary layer. The convective entropy transport on the grid scale) can frictional wavenumber maintain the static stability of the model extratropics. Hence, this idealized GCM is well suited for studying ␴ Ϫ ␴ 0 b to what extent and how turbulent ¯uxes due to baroclinic kdd(␴) ϭ k max 0, ΂΃1 Ϫ ␴b eddies, in interaction with diabatic processes, can main- tain the thermal strati®cation of a dry atmosphere and is nonzero near the surface, in ␴ levels with ␴ Ͼ ␴b ϭ can contribute to the formation of an extratropical tro- 0.85. The frictional wavenumber decreases linearly in 0 popause. ␴ from kd ϭ kd at the surface (␴ ϭ 1) to kd ϭ 0in␴ 0 levels above ␴ ϭ ␴b. The frictional wavenumberkd at the surface is one of the parameters that were varied to a. Dynamical core investigate their in¯uence on the extratropical tropo- pause height and thermal strati®cation; in the simulation The dynamical core of the idealized GCM is a hy- referred to as the reference simulation (see Table 1), it 0 Ϫ5 Ϫ1 drostatic spectral transform model in Bourke's (1974) was set to kd ϭ 0.7 ϫ 10 m . -vorticity±divergence form, with semi-implicit time dif- Aside from the ٌ 8 hyperdiffusion representing sub ferencing and a vertical ␴ coordinate (Durran 1999, grid-scale dissipation, the quadratic damping of near- chapter 7.6). The spectral truncation is triangular, and surface winds is the only frictional process in the ide- most simulations used T42 resolution (for an exception, alized GCM. see section 4a). The vertical discretization is based on a centered difference scheme, with 30 unequally spaced c. Diabatic processes ␴ levels affording a relatively high vertical resolution throughout the troposphere that forms in the model Diabatic processes are represented by Newtonian re- (6 ␴ levels below ␴ ϭ 0.8 and 17 ␴ levels below ␴ ϭ laxation of temperatures T toward a zonally symmetric Subgrid-scale dissipation is represented by ٌ 8 hy- radiative equilibrium temperature T e and, in a narrow .(0.2 perdiffusion in the vorticity, divergence, and tempera- region around the equator, toward a moist pseudoadiabat ture equations. The dynamical core of the idealized T m: (TTϪ T em(␾, p) T Ϫ T (pץ -GCM is identical with the spectral dynamical core de scribed by Held and Suarez (1994). Further numerical ϭ ´´´ϪϪw(␾) . (15) t ␶ (␾, ␴) ␶ץ speci®cations can be taken from Held and Suarez's de- rc scription. The radiative equilibrium temperature T e(␾,p), a func- 1326 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

FIG. 1. (a) Temperature (K) and (b) potential temperature (K) in radiative equilibrium of the reference simulation. tion of latitude ␾ and pressure p, is the radiative equi- for the function that controls the latitude dependence of librium temperature of a semigray atmosphere (Weaver the radiative equilibrium temperature (16). and Ramanathan 1995). Temperatures are relaxed to- Figure 1 shows the radiative equilibrium temperature ward this radiative equilibrium temperature on a ``ra- and the associated potential temperature for the param- diative'' time scale ␶r(␾, ␴) that depends on latitude ␾ eters of the reference simulation. It can be seen that the and ␴ level. In a narrow region around the equator in temperature gradient is largest at the surface and de- which the weight function w(␾) has nonzero amplitude, creases to zero at the top of the atmosphere. The strat- temperatures are relaxed toward the temperature T m(p) i®cation of the radiative equilibrium state is statically of a moist pseudoadiabat on the ``convective'' time scale unstable below ␴ ഠ 0.4 and statically stable above that

␶c. level, with a smooth transition between static stability and static instability. The statically unstable radiative equilibrium strati®cation in this model contrasts with 1) RELAXATION TOWARD RADIATIVE EQUILIBRIUM the statically stable radiative equilibrium strati®cation OF SEMIGRAY ATMOSPHERE used in other idealized GCMs, such as those used by The surface air temperature in radiative equilibrium Held and Suarez (1994) and Haynes et al. (2001). is speci®ed as a function of latitude ␾ as The temperature T e is the radiative equilibrium tem- perature of a semigray atmosphere, that is, of an at- e 1 mosphere transparent to solar radiation and gray for T e(␾) ϭ T ϩ⌬ Ϫsin2␾ , ssh3 infrared radiation (Weaver and Ramanathan 1995). Sup- ΂΃ pose the optical depth d of a semigray atmosphere de- e where T s ϭ 300 K is the global-mean surface temper- creases exponentially with altitude z so that d ϭ ature. The equator-to-pole temperature difference ⌬h in d 0 exp(Ϫz/Ha), where d 0 is the optical thickness of the radiative equilibrium is one of the parameters that were atmosphere and Ha is the partial-pressure scale height varied; in the reference simulation, it was set to ⌬h ϭ of the principal infrared absorber (water vapor in the 60 K. earth's atmosphere). If pressure decreases likewise ex-

The latitude±pressure dependence of the radiative ponentially with altitude, p ϭ p 0 exp(Ϫz/Hp) with pres- equilibrium temperature is given by sure scale height Hp, the optical depth can be written

1/4 as a function of pressure as p ␣ ee T (␾, p) ϭ T 1 ϩ d (␾) . (16) ␣ t 0 p p []΂΃0 d ϭ d0 , e ΂΃p0 The skin temperatureT t at the top of the atmosphere (p ϭ 0) is one of the parameters that were varied; in where ␣ ϭ H /H is the ratio of the pressure scale height e p a the reference simulation, it was set to T t ϭ 200 K. The to the partial-pressure scale height of the infrared ab- constant p 0 ϭ 1000 hPa is a reference surface pressure, ␣ sorber. The function d(␾,p) ϭ d 0(␾)(p/p 0) in the ra- and the exponent ␣ ϭ 3.5 controls the lapse rate of the diative equilibrium temperature (16) can thus be viewed radiative equilibrium state. The requirement that for as an optical depth that depends on latitude and pressure. e p ϭ p 0 the radiative equilibrium temperature T (␾,p0) Indeed, the radiative equilibrium temperature (16) is, up e match the radiative equilibrium temperatureT s (␾) at the to constants, an equilibrium solution of the two-stream surface yields equations for a semigray atmosphere, with the function 4 d (␾)(p/p )␣ playing the role of an optical depth (cf. Te(␾) 0 0 d (␾) ϭϪs 1 (17) Goody and Yung 1989, chapter 9.2). The choice of the 0 Te []t exponent ␣ ϭ 3.5 in the radiative equilibrium temper- 15 JUNE 2004 SCHNEIDER 1327 ature (16) is motivated by this correspondence between simulation and was varied to examine possible depen- the radiative equilibrium temperature (16) and the ra- dencies of the extratropical tropopause height and thermal diative equilibrium temperature of a semigray atmo- strati®cation on the tropical thermal strati®cation. The sphere: in the earth's atmosphere, the ratio of the pres- instantaneous surface temperature Ts and surface pressure sure scale height to the scale height of water vapor ps provide the lower boundary conditions for the vertical pressure is roughly Hp/Ha ഠ 7 km/2 km ϭ 3.5. However, integration of the pressure lapse rate. Above the level of the correspondence between radiative properties of the neutral buoyancy, at which the temperature T m(p) of the earth's atmosphere on the one hand and the radiative moist pseudoadiabat is equal to the instantaneous tem- equilibrium temperatures of the idealized GCM and of perature T, the temperature of the moist pseudoadiabat a semigray atmosphere on the other hand is incomplete. is set equal to the instantaneous temperature, so that the m For example, the latitude dependence of the optical ``convective'' heating Ϫ [T Ϫ T (p)]/␶c vanishes. thickness (17) is not motivated by the actual distribution Equatorial temperatures are relaxed toward the moist m of infrared absorbers in the earth's atmosphere, but by pseudoadiabat T (p) on the ``convective'' time scale ␶c the desire to specify in radiative equilibrium both a sur- ϭ 4 days. face air temperature with nonzero meridional gradient and a skin temperature with zero meridional gradient. d. Climate of a reference simulation Temperatures are relaxed toward the radiative equi- librium temperature T e on the ``radiative'' time scale As a baseline for comparisons of simulations with

␶r(␾, ␴), which is a function of latitude ␾ and ␴ level different parameters, a reference simulation was per- with formed in which the parameters were chosen such that the simulated large-scale circulation resembles that of ␴ Ϫ ␴b the earth's atmosphere. Table 1 lists the values of the ␶ Ϫ1(␾, ␴) ϭ ␶ Ϫ1 ϩ (␶ Ϫ1 Ϫ ␶ Ϫ1)max 0, cos 8 ␾. risi parameters in the reference simulation. The planet radius ΂΃1 Ϫ ␴b and the angular velocity are those of the earth, and the (18) thermodynamic properties of the simulated ¯uid are those of dry air. In ␴ levels above ␴ ϭ ␴b ϭ 0.85, the relaxation time scale is ␶ (␾, ␴ Յ ␴ ) ϭ ␶ ϭ 50 days. In ␴ levels below The simulation was started from an isothermal rest r b i state with superimposed small random perturbations. ␴ ϭ ␴b, the relaxation time scale decreases toward the surface and, along the surface, toward the equator, tak- After a spinup time of 800 simulated days, circulation ing the value ␶ (␾ ϭ 0, ␴ ϭ 1) ϭ ␶ Յ ␶ at the equator. statistics were computed from another 1200 simulated r s i days in which ¯ow ®elds were sampled 5 times per day. The near-surface relaxation time scale ␶s is one of the parameters that were varied; in the reference simulation, Since the forcing and the dissipation in the idealized it was set to ␶ ϭ 7 days. GCM are hemispherically symmetric, the circulation s statistics are, up to sampling error, likewise hemispher- ically symmetric. To reduce the sampling error in the 2) RELAXATION TOWARD MOIST PSEUDOADIABAT circulation statistics, the simulated statistics of the two The weight function hemispheres were averaged, thereby enforcing sym- metry of the plotted ®elds. sin2␾ Figure 2 shows the streamfunction ⌿ of the mean w(␾) ϭ 1 Ϫ tanh , ␾ ϭ 6Њ, mass ¯ux (␳␷*, ␳Q*) in the dynamical equilibrium sin2 t ␪ ␪ ΂΃␾t that develops in the reference simulation. The mass ¯ux- determines the meridional extent of the region in which es along isentropes and across isentropes are the deriv- temperatures are relaxed toward a moist pseudoadiabat atives m T (p). The weight function decays steeply from w(0) 1 ⌿andץϭ 1 at the equator to zero away from the equator. In ␳␷␪␪* ϭ the narrow region around the equator ( | ␾ | Շ 6Њ)in 2␲a cos(␾) which the weight function has signi®cant amplitude, 1 ⌿ץ ␳ Q* ϭϪ temperatures are relaxed toward the moist pseudoadi- ␪ 2␲a cos(␾) y abatic temperature of the streamfunction ⌿(y, ␪). Figure 2 shows that, as p discussed in section 2b, equatorward mass ¯ux along T m(p) ϭ T ϩ ␥ ⌫ (pЈ) dpЈ (19) s ͵ m isentropes predominates in the surface layer, demarcated p s approximately by the 5% and 95% isolines of the cu- obtained by columnwise vertical integration of the moist mulative distribution of surface potential temperatures; pseudoadiabatic lapse rate ⌫m(p) with respect to pressure poleward mass ¯ux along isentropes predominates in decreases (cf. Emanuel 1994, chapter 4.7). The dimen- the interior troposphere. Upward and downward mass sionless factor ␥ rescales the pseudoadiabatic lapse rate ¯uxes across isentropes, corresponding to diabatic heat-

⌫m; the rescaling factor was set to ␥ ϭ 1 in the reference ing and cooling, close the mass circulation. Figure 2 1328 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

10Ϫ4 kg KϪ1 mϪ2 sϪ1, and a stratosphere in which the absolute value of this divergence is signi®cantly smaller, on the order of or less than 10Ϫ6 kg KϪ1 mϪ2 sϪ1. The transition between troposphere and stratosphere is sharp in the extratropics and more gradual in the Tropics (cf. Fig. 5 below). Figure 3 shows the simulated mean temperature and mean potential temperature. In the dynamical equilib- rium that develops in the simulation, baroclinic eddies transport entropy poleward and upward, such that the mean meridional gradient of the surface temperature is reduced compared with the radiative equilibrium gra- dient (cf. Fig. 1). At the surface near the equator, a FIG. 2. Mass ¯ux streamfunction ␺ (109 kg sϪ1) in the reference shallow statically unstable layer forms; this unrealistic simulation (solid lines, counterclockwise rotation; dashed lines, feature of the simulation is unlikely to have a signi®cant clockwise rotation). The dotted lines represent the 5%, 50%, and 95% isolines of the cumulative distribution of surface potential tempera- impact on the examined properties of the large-scale tures. The thick line marks the tropopause, determined as the potential circulation. In the extratropics, although the radiative temperature ␪ t(y) in the interior atmosphere at which the absolute equilibrium state toward which temperatures are relaxed value of the streamfunction | ⌿(y, ␪) | amounts to 10% of the max- is statically unstable (Fig. 1b), the turbulent entropy imum value max | ⌿(y, ␪)|. ␪ transport due to baroclinic eddies is ef®cient enough to maintain a dynamical equilibrium that is in the mean, shows that the atmosphere of the idealized GCM is heat- though not at every instant, statically stable (Fig. 3b). ed near the surface and in the Tropics, in the region in Less than 5% of the vertical entropy ¯ux in the extra- which temperatures are relaxed toward a moist pseu- tropics occurs at zonal wavenumbers greater than 20, doadiabat; elsewhere, the atmosphere is cooled in the which means that, if convective entropy transport on mean. There are no signi®cant entropy sources in the the grid scale contributes to the entropy transport at all, interior atmosphere of the idealized GCM except in the it must be coupled to the large-scale eddies and so is Tropics, and so the statement that the troposphere is the dif®cult to distinguish from them. With the caveat that layer of the atmosphere within which the entropy re- grid-scale convective entropy ¯uxes coupled to baro- ceived at the surface is redistributed means that the cir- clinic eddies may contribute to the entropy transport, culation of mass along and across isentropes closes the facts that the idealized GCM does not contain a within the troposphere. Provided ⌿ϭ0 is chosen as representation of extratropical convective adjustment the lower boundary condition on the streamfunction ⌿, and that the extratropics are statically stable imply that, the constraint (3) on the mass ¯ux along isentropes im- in this simulation, baroclinic eddies transport entropy plies that the mean tropopause potential tempera- ef®ciently enough to maintain the static stability of the extratropics. ture ␪ t(y) is a streamfunction envelope along which ⌿(y, ␪ t) ഠ 0. The tropopause plotted in Fig. 2 is de- At each latitude, the tropopause plotted in Fig. 3b termined as such an envelope of the streamfunction (see indicates the mean ␴ level of the isentrope with potential section 4b for details). As in the earth's atmosphere, the temperature equal to the tropopause potential temper- tropopause marks a transition between a troposphere in ature ␪ t(y) plotted in Fig. 2. Figure 3b shows that, in

-y(␳␷␪ *)of accordance with the reasoning of section 2d, the extraץ which the absolute value of the divergence the mean mass ¯ux along isentropes is on the order of tropical tropopause marks a transition between a tro-

FIG. 3. (a) Mean temperature (K) and (b) potential temperature (K) in the reference simulation. The thick line in (b) marks the tropopause. 15 JUNE 2004 SCHNEIDER 1329 posphere with high isentropic density (low static sta- bility) and a stratosphere with low isentropic density (high static stability). The Brunt±VaÈisaÈlaÈ frequency in- creases from 1.0 ϫ 10Ϫ2 sϪ1 in the interior of the ex- tratropical troposphere to 2.2 ϫ 10Ϫ2 sϪ1 in the lower stratosphereÐan increase similar to that observed in the earth's atmosphere. The extratropical tropopause is also recognizable in instantaneous ¯ow ®elds. Figure 4 shows a typical in- stantaneous potential vorticity ®eld on the 298-K is- entrope in the dynamical equilibrium of the reference simulation. A sharp increase of the static stability or decrease of the isentropic density ␳␪ shows up in the potential vorticity ®eld P ϭ ( f ϩ ␨␪)/␳␪ on isentropes FIG. 4. An instantaneous potential vorticity ®eld (contour interval that intersect the tropopause as a region of large me- is 1 PVU) on the 298-K isentrope. ridional gradients. The tropopause, roughly coinciding with the Ϯ2 PVU isolines7 of potential vorticity (cf. Holton et al. 1995), is clearly recognizable as the un- cation turned out to be statically unstable. A statically dulating region of large meridional potential vorticity stable strati®cation for ⍀ϭ4⍀* was obtained by raising e gradients. Breaking of potential vorticity contours is the skin temperature to T t ϭ 225 K and increasing the seen on the entire isentrope, with ®lamentation of con- equator-to-pole temperature difference to ⌬h ϭ 120 K, tours most clearly seen in the subtropics, where Rossby thereby increasing the baroclinicity of the thermal forc- waves encounter their critical layers. The potential vor- ing. ticity mixing that these processes entail is what is mod- All simulations were run with the same vertical res- eled with the diffusive eddy ¯ux closure (7b). olution of 30 ␴ levels. The horizontal resolution was That in the idealized GCM a well-de®ned extratrop- T42 for all simulations except for the one with quadru- ical tropopause forms without convective adjustment pled angular velocity ⍀ϭ4⍀*, for which the horizontal indicates that, in this model as in the model of Haynes resolution was increased to T63 to ensure that the small- et al. (2001), the extratropical tropopause forms through er baroclinic eddies were resolved. Each simulation was an interaction between large-scale diabatic processes spun up for at least 200 days from a circulation state and turbulent ¯uxes due to baroclinic eddies. The ref- for a point nearby in parameter space (a day referring erence simulation points to the possibility of a climate to 86 400s, independently of the angular velocity ⍀). in which premises (i) and (ii) of section 2e, assigning Circulation statistics were sampled 5 times per day over central importance to baroclinic eddy ¯uxes, hold. another 400 simulated days. For the higher-resolution simulation (T63) with quadrupled angular velocity ⍀ϭ 4⍀*, the circulation statistics were sampled 5 times per 4. Test of dynamical constraint with idealized day over 300 simulated days. GCM a. Variation of parameters b. Determination of tropopause The dynamical constraint (13) was tested in simula- The mean tropopause potential temperature ␪ t(y) was tions with the idealized GCM with different values of determined as the line in the interior atmosphere along some of the model parameters that in¯uence the extra- which tropical tropopause height and thermal strati®cation. |⌿(y, ␪)| The model parameters that were varied are listed in the ϭ 0.1. (20) ®rst column of Table 2. max |⌿(y, ␪)| Simulations were carried out in which one parameter ␪ at a time was set to a different value than in the reference The maximum absolute value max␪ | ⌿(y, ␪) | of the simulation. Only in the simulation with angular velocity streamfunction at each latitude occurs near the top of ⍀ϭ4⍀* (the asterisk denotes the reference parameter the surface layer and gives the magnitude of the ver- value listed in Table 1) were the angular velocity ⍀, the tically integrated mean equatorward mass ¯ux ␳␷␪ * (cf. e skin temperatureT t , and the equator-to-pole temperature Fig. 2). Determining the tropopause potential temper- difference ⌬h varied simultaneously, for with an in- ature ␪ t(y) at each latitude as the potential temperature creased angular velocity ⍀ϭ4⍀* and with reference in the interior atmosphere at which the absolute value values for the other parameters, the simulated strati®- of the streamfunction | ⌿(y, ␪) | amounts to 10% of the

maximum value max␪ | ⌿(y, ␪) | implies that 90% of 7 Potential vorticity is measured in potential vorticity units with the equatorward mass ¯ux along surface-layer isentro- 1 PVU ϭ 10Ϫ6 Km2 kgϪ1 sϪ1 (Hoskins et al. 1985). pes are returned as a poleward mass ¯ux along interior 1330 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

TABLE 2. Circulation statistics in simulations with idealized GCM. Extratropical mean values ͗´͘ are mean values over latitudes poleward ys␪ ͘ expected accordingץ(␤/of␾ˆ (see footnote 8). The last column gives the relative deviations of the potential temperature differences Ϫ͗( f to the dynamical constraint (13) from the actual potential temperature differences ͗␪␪t Ϫ s͘. The asterisk denotes the reference parameter values listed in Table 1. y␪ s͘ (K) Errorץ(␤/␾ˆ ͗p t͘ (hPa) ͗T t͘ (K) ͗␪ t͘ (K) ͗␪ s͘ (K) ͗␪ t Ϫ ␪ s͘ (K) Ϫ͗( f Reference 29.4Њ 302.7 211.3 297.5 269.3 28.2 24.6 Ϫ0.13 ␥ ϭ 0.6 26.6Њ 314.3 215.6 301.2 271.5 29.6 23.1 Ϫ0.22 0.7 26.6Њ 315.2 214.8 299.5 271.1 28.4 23.2 Ϫ0.18 0.8 26.6Њ 312.4 214.1 299.0 271.0 27.9 23.7 Ϫ0.15 0.9 26.6Њ 304.9 212.5 298.8 270.6 28.3 23.6 Ϫ0.16 1.1 29.4Њ 299.5 210.3 297.1 268.5 28.6 23.6 Ϫ0.17 1.2 29.4Њ 305.9 210.7 296.1 268.1 28.0 22.5 Ϫ0.20 1.3 29.4Њ 311.3 210.6 294.6 267.7 27.0 21.5 Ϫ0.20 1.4 29.2Њ 313.7 210.0 293.1 267.5 25.6 21.5 Ϫ0.16

⌬h ϭ 30 K 26.6Њ 314.1 211.7 295.2 277.6 17.6 15.0 Ϫ0.15 45 K 29.2Њ 309.5 210.6 294.8 272.8 22.0 18.1 Ϫ0.18 75 K 29.4Њ 296.1 211.5 299.6 265.3 34.3 30.8 Ϫ0.10 90 K 29.4Њ 290.2 212.1 302.2 261.2 41.0 37.2 Ϫ0.09 105 K 31.8Њ 281.7 211.9 304.0 256.4 47.6 44.1 Ϫ0.08 120 K 32.0Њ 281.2 212.5 305.3 251.6 53.7 50.6 Ϫ0.06 135 K 32.2Њ 275.4 212.9 307.8 247.2 60.6 57.0 Ϫ0.06 150 K 32.2Њ 272.3 213.9 310.1 242.2 67.9 65.6 Ϫ0.03 165 K 32.2Њ 267.3 214.2 312.4 239.6 72.8 72.8 0.00 180 K 34.6Њ 266.9 213.8 312.0 232.7 79.3 80.2 0.01

e Tt ϭ 160 K 32.2Њ 199.5 174.2 277.2 253.8 23.4 18.6 Ϫ0.20 170 K 32.0Њ 222.1 183.3 282.6 257.8 24.7 19.9 Ϫ0.19 180 K 32.0Њ 246.8 192.5 287.6 261.6 25.9 21.0 Ϫ0.19 190 K 29.4Њ 273.6 202.4 293.3 266.1 27.2 23.0 Ϫ0.16 210 K 29.2Њ 334.6 219.9 300.9 271.7 29.2 26.4 Ϫ0.09 220 K 26.4Њ 372.2 229.4 304.7 274.4 30.4 27.6 Ϫ0.09 230 K 23.8Њ 408.4 238.8 309.0 276.9 32.1 29.4 Ϫ0.08 240 K 23.8Њ 441.3 247.6 313.5 278.4 35.1 32.3 Ϫ0.08

␶ s ϭ 4 days 29.4Њ 296.4 214.5 303.9 272.4 31.5 28.5 Ϫ0.10 15 days 29.2Њ 325.9 210.0 289.5 266.7 22.8 19.4 Ϫ0.15 30 days 26.6Њ 347.0 209.7 284.4 266.2 18.3 16.0 Ϫ0.12 40 days 26.6Њ 355.7 209.6 282.3 265.7 16.5 15.6 Ϫ0.06 50 days 26.4Њ 358.9 209.6 281.5 265.6 15.9 15.1 Ϫ0.05

0 Ϫ6 0* kd ϭ 2 kd 29.4Њ 323.2 214.9 297.1 270.4 26.6 22.6 Ϫ0.15 Ϫ5 0* 2 kd 32.0Њ 330.4 214.8 295.1 269.5 25.6 20.7 Ϫ0.19 Ϫ4 0* 2 kd 32.0Њ 329.6 214.6 295.1 270.2 24.9 21.7 Ϫ0.13 Ϫ3 0* 2 kd 29.4Њ 321.0 212.5 294.3 269.4 24.9 20.5 Ϫ0.18 Ϫ2 0* 2 kd 29.4Њ 313.5 211.6 294.9 269.8 25.1 21.0 Ϫ0.17 Ϫ1 0* 2 kd 29.4Њ 307.8 211.4 296.2 269.5 26.7 22.2 Ϫ0.17 1 0* 2 kd 32.0Њ 296.3 211.6 299.3 269.1 30.3 28.9 Ϫ0.05 2 0* 2 kd 29.4Њ 289.2 212.2 302.6 269.9 32.7 31.0 Ϫ0.05 ⍀ϭ0.5 ⍀* 40.6Њ 307.9 212.5 297.9 270.9 26.9 19.9 Ϫ0.26 2Ϫ1/2 ⍀* 34.8Њ 304.5 211.6 297.3 269.7 27.6 24.3 Ϫ0.12 21/2 ⍀* 23.8Њ 311.8 212.3 296.7 270.5 26.2 23.5 Ϫ0.10 2 ⍀* 21.0Њ 312.5 213.7 298.5 274.9 23.6 22.5 Ϫ0.05 ⍀ϭ4 ⍀*

⌬h ϭ 120 K 14.0Њ 333.9 235.4 322.8 277.7 45.1 52.1 0.16 e Tt ϭ 225 K isentropes with potential temperature less than the tro- tropical latitudes in the reference simulation. As in the popause potential temperature ␪ t(y). This criterion for reference simulation, a well-de®ned extratropical tro- determining the tropopause potential temperature is con- popause forms in all simulations presented in Table 2. sistent with the mass ¯ux constraint (3) from which the In all simulations, the extratropical tropopause deter- dynamical constraint (13) was derived; it is applicable mined by the mass ¯ux criterion (20) marks a sharp to the extratropics and to the Tropics and subtropics. transition between a troposphere and a stratosphere dis- Figure 5 shows the mass ¯ux fraction on the left-hand tinguishable by typical values of the isentropic mass y(␳␷␪ *) or of the isentropic densityץ side of the criterion (20) at typical tropical and extra- ¯ux divergence 15 JUNE 2004 SCHNEIDER 1331

scaling factor ␥ of the pseudoadiabatic lapse rate was set to 0.6 and 1.4. Temperatures are relaxed toward the rescaled pseudoadiabatic temperature (19) in a narrow region around the equator. In the dynamical equilibria that develop in the simulations, the thermal strati®- cation throughout the Tropics changes as the rescaling factor ␥ is varied. At a midtropospheric level (␴ ഠ 0.5) in the Tropics, the Brunt±VaÈisaÈlaÈ frequency changes from 1.5 ϫ 10Ϫ2 s Ϫ1 for ␥ ϭ 0.6 (Fig. 6a) over 1.1 ϫ 10Ϫ2 s Ϫ1 for ␥ ϭ 1 (Fig. 3b) to 0.6 ϫ 10Ϫ2 s Ϫ1 for ␥ ϭ 1.4 (Fig. 6b). The extratropical thermal strati®cation, however, is hardly affected by the chang- es of the tropical thermal strati®cation. For all values of the rescaling factor ␥, the Brunt±VaÈisaÈlaÈ frequency FIG. 5. Mass ¯ux fraction | ⌿(y, ␪) | /max␪ | ⌿(y, ␪) | in the ref- erence simulation at latitudes 5Њ (dashed line) and 50Њ (solid line). at a midtropospheric level in midlatitudes is approxi- The dotted line marks the critical mass ¯ux fraction (0.1) used to mately 1.0 ϫ 10Ϫ2 s Ϫ1 . Correspondingly, the tropical determine the tropopause. The maxima of the mass ¯ux fraction lie tropopause height changes signi®cantly as the rescaling near the top of the surface layer. factor ␥ is varied, while the tropopause height in mid- latitudes, suf®ciently far away from the subtropics,

␳ ␪. Because of the sharpness of this transition, the po- changes only slightly (see Fig. 6 and Table 2). tential temperature ␪ t(y) of the extratropical tropopause The simulations for various values of the rescaling determined by the mass ¯ux criterion (20) is not very factor ␥ demonstrate that the extratropical tropopause sensitive to the chosen value of the mass ¯ux ratio. Only height and thermal strati®cation do not depend on the in the polar regions of some simulations is the transition tropical thermal strati®cation. The fact that, in the earth's between troposphere and stratosphere not very sharp atmosphere, the tropical and the extratropical Brunt± (see footnote 8). VaÈisaÈlaÈ frequencies are almost equal cannot be taken as The tropical tropopause that forms in the simulations evidence for a dynamical link between the extratropical is less clearly de®ned than is the extratropical tropo- and tropical thermal strati®cations. The extratropical pause (cf. Fig. 5). The transition in typical values of the tropopause height and thermal strati®cation are set lo-

-y(␳␷␪ *) or of the is- cally by extratropical processes, as was assumed in deץ isentropic mass ¯ux divergence entropic density ␳ ␪ from the troposphere to the strato- riving the dynamical constraint (13). sphere is less sharp in the Tropics than in the extra- tropics, and the tropical tropopause potential tempera- d. Empirical eddy diffusivities ture ␪ t(y) determined by the mass ¯ux criterion (20) is more sensitive to the chosen value of the mass ¯ux ratio. The derivation of the dynamical constraint (13) rests

However, here we focus on the extratropical tropopause on the postulate that the eddy diffusivity Di for potential and thermal strati®cation. vorticity exhibits no essential vertical structure within the troposphere and is, vertically averaged, approximately

equal to the eddy diffusivity Ds for surface potential tem- c. Independence of extratropical and tropical thermal perature. Figure 7 shows empirical eddy diffusivities in strati®cations the simulation in which the near-surface thermal relax-

Figure 6 shows the mean potential temperature and ation time ␶s was equal to the thermal relaxation time the tropopause in two simulations in which the re- ␶i ϭ 50 days in the interior atmosphere. Plotted are the

FIG. 6. Mean potential temperature (K) in simulations with a rescaled pseudoadiabatic lapse rate near equator and rescaling factors of (a) ␥ ϭ 0.6 and (b) ␥ ϭ 1.4. The thick lines mark the tropopauses. 1332 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

* y P changes sign from negative valuesץ ticity gradient near the surface to positive values in the interior at- mosphere moves upward, toward potential temperatures signi®cantly above the 95% isoline of the cumulative distribution of surface potential temperatures. As a con- sequence, diffusivities become noisy and/or ill-de®ned in extended regions of the interior atmosphere. In these simulations, it is unclear to what extent the vertical structure of the diffusivity for potential vorticity is neg- ligible. Overall, however, it appears that although the postulate regarding the eddy diffusivities is not quan- titatively accurate latitude by latitude, it is a plausible starting point for understanding large-scale features of FIG. 7. Empirical eddy diffusivities in the simulation with near- the turbulent mixing in the extratropical troposphere. surface relaxation time ␶ ϭ 50 days. Eddy diffusivity DÄ ϭ s s s -ys␪ for surface potential temperature (solid line); eddy difץ/ Ϫ␷Ä Јss␪Ј Ä Ã * * e. Validity of dynamical constraint y for potential vorticity in the surface layerץ/fusivity Di ϭϪ␷ÃPP (along the mean surface potential temperature, dashed line) and in the interior troposphere (along the P* ϭ 0.5 PVU isoline of po- The circulation statistics listed in Table 2 indicate tential vorticity, dotted line). the degree to which the dynamical constraint (13) holds in the simulations with the idealized GCM. The quan- tities in Table 2 are extratropical mean values, denoted Ä s -ys␪ for surface by angle brackets ͗´͘.8 Besides the potential temperaץ/ empirical eddy diffusivity Ds ϭϪ␷ÄЈss␪Ј potential temperature and the empirical eddy diffusivity tures of the tropopause ͗␪␪t ͘ and of the surface ͗ s ͘ Ä Ã * * y for potential vorticity in the surface and the scaled surface potential temperature gradientץ/Di ϭϪ␷ÃPP ys␪ ͘ appearing in the dynamical constraintץ(␤/ layer (along the mean surface potential temperature) and Ϫ͗( f in the interior troposphere (along a potential vorticity (13), Table 2 contains the pressure ͗p t ͘ and the tem- isoline that, in midlatitudes, lies roughly half way be- perature ͗T t ͘ of the extratropical tropopause. The table tween the tropopause and the 95% isoline of the cu- shows that, in the idealized GCM, the tropopause tem- mulative distribution of surface potential temperatures). perature ͗T t ͘ is largely determined by the skin tem- e The potential temperature at the lowest model level (␴ peratureT t in radiative equilibrium. Given the skin e ϭ 0.989) is taken to represent the surface potential tem- temperatureT t , therefore, one can estimate how the perature. Figure 7 shows that, at least in a large-scale tropopause pressure ͗p t ͘ changes as the tropopause mean, these eddy diffusivities are indeed approximately potential temperature ͗␪ ͘ changes, without the explicit Ä t equal. The empirical eddy diffusivity Di for potential radiative transfer calculation that would ordinarily be vorticity does not exhibit pronounced vertical structure necessary to determine the tropopause pressure from within the troposphere, except near the top of the surface a radiative constraint. * -yP changes The deviations of the tropopause and surface potenץ layer, where the potential vorticity gradient sign and the diffusivity is ill-de®ned. tial temperatures from exact consistency with the dy- In the other simulations, the empirical eddy diffusiv- namical constraint (13) are within the accuracy that ity for surface potential temperature and that for poten- can be expected given the approximations made in the tial vorticity in the surface layer are generally approx- derivation. The median absolute value of the relative imately equal. The diffusivity for potential vorticity in deviation of the expected potential temperature differ-

-ys␪ ͘ from the actual potential temperץ(␤/ the interior atmosphere in the subtropics and in lower ence Ϫ͗( f extratropical latitudes tends to be smaller than the near- ature difference ͗␪␪t Ϫ s ͘ amounts to 13% and does surface diffusivities when the near-surface thermal re- laxation time ␶ is smaller than the interior relaxation s 8 The extratropical mean values ͗´͘ in Table 2 are mean values over time ␶i. [The near-surface diffusivities increase with de- (␾␾ ␪ t(␾ץ latitudes poleward of the latitude␾ˆ at which the curvature creasing thermal relaxation time near the surface, and of the temporal and zonal mean tropopause potential temperature the thermal relaxation time (18) near the surface ap- ␪␾t(␾) is greatest. The latitudeˆ of greatest curvature of the mean tropopause potential temperature ␪ (␾) is approximately the latitude proaches the interior relaxation time ␶i with increasing t latitude.] In localized strongly baroclinic regions, the of the core of the subtropical jet. As ¯ow parameters are varied, the latitude␾ˆ changes with the changing width of the Hadley cell (see diffusivity for potential vorticity in the interior atmo- Table 2). For the simulations with angular velocity ⍀ Ն 2⍀*, with sphere often exceeds the near-surface diffusivities, usu- equator-to-pole temperature difference ⌬h ϭ 30 K, or with frictional 00*Ϫ4 ally in the vicinity of regions in which the potential wavenumber kkddՅ 2 , the mean values ͗´͘ are mean values over vorticity gradient is so close to zero that it is dif®cult the extratropics poleward of␾ˆ up to | ␾ | ϭ 60Њ; in these simulations, the tropopause is not clearly de®ned and/or the mean thermal strat- to estimate the diffusivities. As the baroclinicity is in- i®cation of the lower troposphere is statically unstable poleward of creased by increasing the equator-to-pole temperature | ␾ | ϭ 60Њ. For all other simulations, the mean values ͗´͘ are mean difference ⌬h, the line along which the potential vor- values over the extratropics poleward of␾ˆ up to | ␾ | ϭ 85Њ. 15 JUNE 2004 SCHNEIDER 1333

FIG. 9. Extratropical mean values of the tropopause potential FIG. 8. Extratropical mean values of the potential temperature dif- temperature ͗ ␪ t ͘ vs the expected tropopause potential temperature ference ͗␪␪t Ϫ s͘ vs the expected potential temperature difference ,ys͘ in simulations with idealized GCM. Filled star ץ(␤/ ␪␪s Ϫ ( f ͗ -ys␪ ͘ in simulations with idealized GCM. Filled star, referץ(␤/Ϫ͗( f ence simulation; circles, variation of equator-to-pole surface tem- reference simulation; open stars, variation of skin temperature Tke; triangles, variation of frictional wavenumber0 ; and crosses, perature difference ⌬ ; squares, variation of angular velocity ⍀ (with t d h variation of rescaling factor . 0.5⍀* Յ ⍀ Յ 2⍀*); triangles, variation of near-surface relaxation ␥ time ␶s; and diamond, angular velocity ⍀ϭ4⍀*, equator-to-pole tem- e perature difference ⌬h ϭ 120 K, and skin temperature T t ϭ 225 K. all latitudes, the inversions being most pronounced in the simulations with large equator-to-pole tem- not exceed 26% in any simulation (Table 2, last col- perature difference ⌬h (cf. Held and Schneider 1999, umn). Given the roughness of the approximations made section 4). Concomitant with surface inversions, in deriving the dynamical constraint (13), one might there forms a statically very stable boundary layer have expected the constraint to hold only as a scaling with a sharp top that lies in all simulations below estimate, but it appears to hold as an approximate the 925-hPa pressure level. Within this boundary lay- equality. er, the Brunt±VaÈisaÈlaÈ frequency is up to a factor of Figures 8 and 9 are scatterplots of quantities in Table 2±3 greater than at the top of the boundary layer, 2 that, according to the dynamical constraint (13), should resulting in low isentropic densities at the mean sur- be pairwise equal. Figure 8 shows the potential temper- face potential temperature and in the approximation ature differences plotted against the expected ͗␪␪t Ϫ s͘ (12) overestimating the actual isentropic density at potential temperature differences ( f/ ) for the ys␪ ͘ the mean surface potential temperature by up to aץ ␤ Ϫ͗ simulations in which the equator-to-pole surface tem- factor of 6 in the extratropical mean (for ⌬ ϭ 180 K). perature difference , the angular velocity , and the h ⌬h ⍀ It appears that the limit in which the isentropic near-surface thermal relaxation time were varied. If ␶s density and the thickness of the boundary layer ap- the remaining simulations in Table 2 were included in proach zero is relevant for the simulations with Fig. 8, they would correspond to points scattered in the strong surface inversions. In this limit, the top of the vicinity of the point that corresponds to the reference boundary layer plays the role of the surface in the simulation (Fig. 8, ®lled star). To bring out more clearly arguments of section 2. The approximation (12) the quantities that change in the simulations in which the seems to provide a good approximation of the is- rescaling factor ␥ of the pseudoadiabatic lapse rate, the entropic density at the potential temperatures that, skin temperatureTke , and the frictional wavenumber 0 tdin the mean, correspond to the top of the boundary were varied, Fig. 9 shows the tropopause potential tem- layer, such that the dynamical constraint (13) is es- peratures plotted against the expected tropopause ͗␪ t͘ sentially unaffected by the error of the approximation .ys͘. The approximate (12) at the actual mean surface potential temperatureץ(␤/potential temperatures ͗␪␪s Ϫ ( f consistency of the tropopause and surface potential tem- Consistent with this reasoning, the bias of the dy- peratures with the dynamical constraint (13) is evident namical constraint (13) is reduced if the surface po- from Figs. 8 and 9, as is a bias of the expected potential tential temperature gradient is replaced by the po- -ys␪ ͘ toward values less tential temperature gradient at the top of the boundץ(␤/temperature difference Ϫ͗( f than the actual potential temperature difference ͗␪␪t Ϫ s͘. ary layer. For example, replacing the surface poten- The following simulation results are notable: tial temperature gradient by the potential temperature 1) A possible source of the bias of the dynamical con- gradient at the ␴ ϭ 0.925 level reduces the me- straint (13) is the approximation (12) for the isen- dian absolute value of the relative deviation of tropic density at the mean surface potential temper- the expected potential temperature difference ys␪ ͘ from the actual potential temperatureץ(␤/ature. Surface inversions similar to the wintertime Ϫ͗( f

inversions that form in the earth's atmosphere at high difference ͗␪␪t Ϫ s͘ from 13% to 5%, with increased latitudes form in most of the present simulations at absolute values of the relative deviations primarily 1334 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

for those simulations in which no clear surface in- 1986). Nevertheless, as the frictional wavenumber 0 versions form. The bias of the dynamical constraint kd is varied by a factor of 256, the changes in tro- (the mean of the relative deviations) reduces from popause height and tropopause potential temperature Ϫ12% to Ϫ1%. However, given the roughness of are relatively small and are consistent with the dy- the approximations made in the derivation of the namical constraint (13) (Fig. 9, triangles), irrespec- dynamical constraint, this close agreement may not tive of the fact that the considerable changes of the be completely explained by the arguments of section barotropic shear of the zonal ¯ow do affect the ex- 2 and may be fortuitous. tratropical eddy ¯uxes. 2) According to the dynamical constraint (13), the po- 5) Although the dynamical constraint derived from the tential temperature difference between tropopause baroclinic adjustment hypothesis that the quasigeo- and surface does not change as the angular velocity strophic potential vorticity gradient vanishes in the ⍀ of the planetary rotation is varied, provided the interior atmosphere is similar to the constraint (13), surface potential temperature gradient does not with the potential temperature gradient along pressure change. As the angular velocity ⍀ is varied in the surfaces in the midtroposphere in place of the surface simulations, at constant equator-to-pole surface tem- potential temperature gradient, the baroclinic adjust-

perature difference ⌬h in radiative equilibrium, the ment constraint does not account for the simulated potential temperature difference ͗␪␪t Ϫ s͘ between changes of the tropopause height and thermal strati- tropopause and surface and the scaled surface po- ®cation (cf. Thuburn and Craig 1997).9 For example,

-ys␪ ͘ indeed do as the equator-to-pole temperature difference ⌬h is inץ(␤/tential temperature gradient Ϫ͗( f not change signi®cantly (Fig. 8, squares). As in Thu- creased, the potential temperature difference ͗␪␪t Ϫ s͘ burn and Craig's (1997) simulations with a compre- between tropopause and surface increases more strong- hensive GCM, the tropopause height likewise does ly than linearly with the scaled potential temperature p y ␪ | pϭpm͘ along a midtroposphericץ(␤/not change signi®cantly as the angular velocity ⍀ is gradient Ϫ͗( f variedÐdespite signi®cant changes in the atmo- pressure surface (e.g., for pm ϭ 500 or 600 hPa). spheric circulations. For example, the length scale Nonetheless, the similarity between dynamical con- of the energy-containing extratropical eddies de- straints derived from baroclinic adjustment hypoth- creases with increasing angular velocity ⍀, and for eses and the dynamical constraint (13) may explain ⍀ϭ2⍀*, each hemisphere contains two belts of the relative success with which baroclinic adjustment surface westerlies that are clearly separated from constraints, despite their de®ciencies (see, e.g., Barry each other by a belt of easterlies. In the simulation et al. 2000), account for some aspects of the thermal with angular velocity ⍀ϭ4⍀*, the tropopause strati®cation of the troposphere [see, e.g., Stone and height changes because the equator-to-pole surface Nemet (1996) or, for modi®cations due to moisture,

temperature difference ⌬h and the skin temperature Juckes (2000)]. e T t in radiative equilibrium were also set to values different from the reference values. Nevertheless, the The simulations span a wide range of atmospheric circulations, exhibiting, for example, multiple belts of potential temperature difference ͗␪␪t Ϫ s͘ between tropopause and surface is within 16% of the expected surface westerlies in each hemisphere and surface tem- perature gradients 3 times as large as those on earth. In -ys␪ ͘ (Fig. 8, diamond). In this simץ(␤/value Ϫ͗( f ulation, the extratropics poleward of a narrow Hadley all these simulations, the dynamical constraint (13) ap- cell contain three alternating belts of surface west- pears to hold to within the degree of accuracy that could erlies and easterlies in each hemisphere and three be expected, or perhaps with greater accuracy than could associated upper-tropospheric jet cores, the jet cores, be expected, given the approximations made in the der- however, not being separated from each other by ivation. lines of zero wind. e 3) With increasing skin temperatureT t in radiative f. Limits of validity of dynamical constraint equilibrium, the tropopause temperature in dynam- ical equilibrium increases, and the tropopause height The dynamical constraint (13) is only valid if baro- decreases (Table 2). At the same time, the tropopause clinic eddies dominate the entropy transport in the ex- tratropics and transport entropy ef®ciently enough to potential temperature ͗␪ t͘ increases in proportion to the increase of the expected tropopause potential maintain a statically stable thermal strati®cation. If con- ys͘, although the bias ofץ(␤/temperature ͗␪␪s Ϫ ( f the expected tropopause potential temperature is ev- 9 Among the technical differences between the study of Thuburn ident (Fig. 9, stars). and Craig (1997) and the present study is the way in which the 0 4) With decreasing frictional wavenumberkd , the zonal tropopause potential temperature and height are determined. Thuburn winds become increasingly barotropic, and the mag- and Craig determine the tropopause according to the World Meteo- rological Organization's lapse rate criterion. If one determines the nitudes of the extratropical eddy ¯uxes of potential tropopause potential temperature in the present simulations according vorticity along isentropes and of potential temper- to the lapse rate criterion, the results summarized here only change ature along the surface decrease (cf. James and Gray in quantitative details, but not qualitatively. 15 JUNE 2004 SCHNEIDER 1335 vective or other dynamic ¯uxes dominate the entropy transport, the dynamical constraint (13) cannot be ex- pected to be valid. For example, the dynamical con- straint (13) is probably not valid for the Venus atmo- sphere, since the baroclinicity of the Venus atmosphere appears to be too weak for baroclinic eddies to be able to dominate the entropy transport. Since the idealized GCM does not contain a repre- sentation of extratropical convective adjustment, the simulated extratropical thermal strati®cations are stati- cally unstable if the entropy transport due to baroclinic eddies does not suf®ce to maintain a statically stable thermal strati®cation. Extratropical thermal strati®ca- tions that are, in the mean, statically unstable develop, FIG. 10. Variations of annual mean tropopause potential temper- for example, if the equator-to-pole surface temperature ature ͗␪ t͘ (solid line, left axis), expected tropopause potential tem- ys͘ (dashed line, left axis), and tropopauseץ(␤/perature ͗␪␪s Ϫ ( f difference ⌬h in radiative equilibrium is reduced to less pressure ͗p t͘ (dashed±dotted line, right axis) about the mean values than 30 K, or if the angular velocity ⍀ of the planetary of the years 1960±97. Axis scales are chosen such that a potential rotation is reduced to less than 0.5 ⍀*. (The instanta- temperature increment of 1 K corresponds to a pressure decrement neous thermal strati®cations in simulations with greater of 2.7 hPa (cf. footnote 11). baroclinicity are occasionally statically unstable, in par- ticular near the surface, but in the mean, they are stat- ically stable.) The fact that simulations with low bar- tions of the water vapor content of the atmosphere can oclinicity are statically unstable in the mean limited the be neglected and that baroclinic eddies dominate the range of circulation regimes explored with the idealized entropy transport in the extratropics, the dynamical GCM. constraint (13) should account for interannual varia- In addition to circulation regimes in which convective tions of the extratropical tropopause. Figure 10 shows or other dynamic ¯uxes dominate the entropy transport, interannual variations of the tropopause potential tem- the dynamical constraint (13) also cannot be valid in perature ͗␪ t ͘ and of the expected tropopause potential ys͘ in the extratropics of theץ(␤/ circulation regimes in which the planetary vorticity gra- temperature ͗␪␪s Ϫ ( f dient ␤ does not affect baroclinic eddies. It is unclear Northern Hemisphere. The plotted variations are low- how the tropopause height and thermal strati®cation pass-®ltered variations of annual mean values about would adjust if the energy-containing baroclinic eddies the mean values of the years 1960±97.10 The variations were so small compared with the planet radius that the of the tropopause potential temperature ͗␪ t ͘ follow the planetary vorticity gradient ␤ could effectively be set variations of the expected tropopause potential tem- to zero.

10 Figure 10 is based on NCEP±NCAR reanalysis data (Kalnay et 5. Relevance of dynamical constraint for earth's al. 1996) provided by the NOAA±CIRES Climate Diagnostics Center. atmosphere The potential temperature at 850 hPa was taken as representing the surface potential temperature ␪s. The tropopause potential tempera-

The dynamical constraint (13) offers an explanation ture ␪t and pressure pt, determined according to the World Meteo- for the fact that in the earth's atmosphere, the equator- rological Organization's convention, were provided by the Climate to-pole surface potential temperature difference is ap- Diagnostics Center. The extratropical mean values ͗´͘ are annual mean values of the Northern Hemisphere, computed from monthly mean proximately equal to the potential temperature differ- values over latitudes poleward of the latitude␾ˆ at which the curvature

-␾␾ ␪ t(␾) of the monthly and zonal mean tropopause potential temץ ence between tropopause and surface in midlatitudes (i.e., the lowest isentrope that crosses the tropopause perature ␪ t(␾) is greatest. (The overbar(´) here denotes a monthly near the poles grazes the surface in the subtropics or and zonal mean, and angle brackets ͗´͘ denote an annual and extra- tropical mean.) Latitudes up to 72.5ЊN are included in the extratrop- Tropics). However, to account quantitatively, for ex- ical mean values ͗´͘. The variations about the mean values of the ample, for seasonal variations of the extratropical tro- years 1960±97 were low-pass ®ltered by convolution of the time popause, a dynamical constraint on the tropopause series with a Gaussian smoothing kernel of standard deviation 2.5 height and thermal strati®cation would need to take at yr. Although the reanalysis includes upper-tropospheric observations least the thermodynamics of water vapor near the sur- from 1958 onward, data only from 1960 onward were used, since ys␪ ͘,aץ(␤/the scaled surface potential temperature gradient Ϫ͗( f face into account. For example, it may be possible to noisy time series, takes a value for 1959 that appears anomalously extend the theoretical developments of section 2 by con- small. Because of the low-pass ®ltering, this anomalously small value, sidering mass ¯uxes along surfaces of constant moist if included, would affect the analysis for later years. Results for the entropy. It may also be necessary to take the entropy years before 1979, prior to the availability of satellite data, are to be interpreted cautiously because NCEP±NCAR reanalysis data for that transport due to extratropical moist convection into ac- period for the lower stratosphere are known to be biased (Santer et count, as, for example, in the theory of Juckes (2000). al. 1999). See Santer et al. (2003a,b) for further analyses of tropo- Nevertheless, to the extent that interannual varia- pause data and for comparisons with simulations. 1336 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

-ys͘, with an rms deviation of The dynamical constraint (13), then, accounts quanץ(␤/ perature ͗␪␪s Ϫ ( f 0.2 K. titatively for observed interannual variations of the ex-

Variations of the surface potential temperature ͗␪ s͘ tratropical tropopause and makes it possible to predict and of the scaled surface potential temperature gradient how the tropopause potential temperature and pressure

-ys␪ ͘ both contribute to the variations of change as the climate changes. However, convective enץ(␤/Ϫ͗( f the expected tropopause potential temperature ͗␪ s Ϫ tropy ¯uxes, which were neglected in the derivation of ys␪␪͘. The surface potential temperature ͗ s͘ de- the dynamical constraint (13), may still play a role inץ(␤/f ) creased from 1960 to the early 1970s by about 0.3 K the maintenance of the extratropical thermal strati®ca- and increased from the early 1970s to the 1990s by about tion. With the reanalysis data, for example, Juckes's 1.0 K (cf. Jones 1994). The scaled surface potential (2000) dynamical constraint cannot be distinguished

-ys␪ ͘ decreased from 1960 from the dynamical constraint (13), provided an unץ(␤/temperature gradient Ϫ͗( f to the early 1970s by about 1.6 K and increased from speci®ed constant of proportionality in Juckes's dynam- the early 1970s to the 1990s by about 0.9 K, thus con- ical constraint is adjusted to ®t the data. tributing to the variations of the expected tropopause ys͘ with the sameץ(␤/potential temperature ͗␪␪s Ϫ ( f sign as the variations of the surface potential temper- 6. Implications for a turbulent energy cascade and for linear models of extratropical dynamics ature ͗␪ s͘. Included in Fig. 10 are the variations of the extratropical The dynamical constraint (13) has implications for tropopause pressure ͗p t͘ about the mean value of the years two characteristic length scales of geostrophic turbu- 1/␬ 1960±97. Linearizing the relation p ϭ p0(T/␪) between lence, the Rossby radius and the Rhines scale. Baro- pressure, temperature, and potential temperature, one ex- clinic energy generated by the differential heating of the pects small changes ␦␪t and ␦Tt of the tropopause potential atmosphere is converted into barotropic energy at the temperature ␪ and of the tropopause temperature T to lead t t Rossby radius LR ϭ NHt/ f. From the Rossby radius, to changes of the tropopause pressure pt of approximately barotropic energy cascades upscale, up to a scale at which the inverse energy cascade is halted (see, e.g., Salmon 1998, chapter 6). Unless friction or the limited ptt␦T ␦␪ t size of the planet halt the inverse energy cascade at a ␦pt ഠ Ϫ . ␬ ΂΃Ttt␪ smaller scale, barotropic energy cascades up to the Rhi- 1/2 nes scale L␤ ϭ (UЈ/␤) , with turbulent velocity scale For the years 1960±97, the rms deviation (0.18 K) of UЈ. At the Rhines scale, energy is channeled into zonal the low-pass-®ltered annual mean tropopause temper- jets and Rossby waves (Rhines 1975). Taking the an- ature ͗T t͘ from the long-term mean (217.6 K) was a isotropy of the Rossby wave dispersion relation into factor of 3.5 smaller than the rms deviation (0.62 K) of account, one expects the Rhines scale L␤ to approximate the low-pass-®ltered annual mean tropopause potential the meridional length scale of the energy-containing ed- temperature ͗␪ t͘ from its long-term mean (324.5 K). dies (Vallis and Maltrud 1993). Therefore, variations of the tropopause pressure ͗p t͘ To express the Rhines scale in terms of mean ¯ow about the long-term mean (250.6 hPa) were dominated quantities, let us assume that the turbulent velocity scale by variations of the tropopause potential temperature UЈ is related to the velocity scale U of the mean ¯ow ͗␪ ͘, with ␦͗p ͘ Ϫ2.7 hPa KϪ1␦͗␪ ͘.11 Figure 10 r t t ഠ t in the upper troposphere by UЈϳ(L␤/LR) U, where the shows that variations of the tropopause pressure ͗p ͘ t ratio L␤/LR of Rhines scale to Rossby radius is a measure and of the tropopause potential temperature ͗␪ t͘ indeed of the nonlinearity, or supercriticality, of the ¯ow and follow this relation approximately, to within the error r is a scaling exponent. Held and Larichev (1996) sug- that is to be expected if one neglects the contribution gest r ϭ 1 for the scaling exponent. Neglecting surface of tropopause temperature variations ␦͗T t͘ to pressure topography and using discrete approximations of the variations ␦͗p t͘. Since variations of the surface potential thermal wind relation and of the squared Brunt±VaÈisaÈlaÈ temperature and of its gradient approximately deter- frequency, mined variations of the tropopause potential tempera- gH g ␪ Ϫ ␪ ture, which, in turn, approximately determined varia- tts2 (ys␪ and N ഠ , (21ץU ഠ Ϫ tions of the tropopause pressure, a large part of the f ␪␪00Ht interannual variations of the extratropical tropopause in the years 1960±97 appears to be dynamically linked to one ®nds the ratio of Rhines scale to Rossby radius: (␪ 1/(2Ϫr ץvariations of the surface climate. Lf ␤ ϳ y s . (22) LRtsΗΗ␤␪ Ϫ ␪ 11 The constant of proportionality between variations of the tro- popause pressure ͗p ͘ and variations of the tropopause potential tem- t For arbitrary values of the scaling exponent r, it follows perature ͗␪ t͘ is the derivative of the tropopause pressure with respect Ϫ1 that inasmuch as the dynamical constraint (13) on the -␪ pt ϭϪ␬ (pt /␪t), evaluated at the longץ ,to potential temperature term mean: Ϫ␬Ϫ1 (250.6 hPa/324.5 K) ഠ Ϫ2.7 hPa KϪ1. tropopause height and thermal strati®cation holds, the 15 JUNE 2004 SCHNEIDER 1337

Rhines scale and the Rossby radius are of the same order troposphere was taken as de®ning the tropopause. By 12 of magnitude, L␤/LR ϳ 1. means of a relation between isentropic mass ¯uxes and The coincidence of Rossby radius and Rhines scale eddy ¯uxes, this condition on isentropic mass ¯uxes was implies that barotropic energy cannot go through a sig- transformed into a balance condition for eddy ¯uxes: ni®cant inverse cascade. The thermal strati®cation of upon vertical integration over the troposphere, the is- the troposphere is such that the nonlinear eddy±eddy entropic mass ¯ux associated with the eddy ¯ux of po- interactions that would give rise to an inverse energy tential vorticity approximately balances the isentropic cascade are inhibited. The reasoning that led to the dy- mass ¯ux associated with the balanced eddy ¯ux of namical constraint (13) did not presuppose nonlinear surface potential temperature. A diffusive closure of the eddy±eddy interactions to be weak; hence, it offers an eddy ¯uxes combined with the postulate that the eddy explanation for the historic successes of linear or weakly diffusivity for potential vorticity exhibits no essential nonlinear models of large-scale extratropical dynamics vertical structure within the troposphere and is, verti- and for the apparent absence of a signi®cant inverse cally averaged, approximately equal to the eddy diffu- energy cascade in the troposphere (see, e.g., Boer and sivity for surface potential temperature led to the dy- Shepherd 1983; Randel and Held 1991; Welch and Tung .ys ␪ץ(␤/namical constraint ␪␪t Ϫ s ഠ Ϫ( f 1998). A linear relation between the tropopause potential For example, one could not have expected a priori temperature and the surface potential temperature and that typical length scales and spatial structures of the its gradient, the dynamical constraint has a very simple most unstable baroclinic waves in linear models such formÐa form that could not have resulted from similar as Eady's (1949) model or Charney's (1947) model re- considerations within quasigeostrophic theory. The semble typical length scales and spatial structures of the mean potential vorticity balance of near-surface isen- energy-containing baroclinic eddies observed in the tropes differs fundamentally from the mean potential earth's atmosphere. In an atmosphere with strong non- vorticity balance of near-surface horizontal planes in linear eddy±eddy interactions and with a signi®cant in- quasigeostrophic theory. The postulated structure of verse energy cascade, one would expect that typical eddy diffusivities that led to the present dynamical con- length scales of the energy-containing eddies would be straint is impossible in quasigeostrophic models. greater than those of the linearly most unstable waves, A series of simulations with an idealized GCM and there would be no necessary resemblance between showed that, across a wide range of atmospheric cir- the spatial structure of the energy-containing eddies and culations, the dynamical constraint describes the re- that of the linearly most unstable waves. The successes lation between tropopause and surface potential tem- of linear and weakly nonlinear models in accounting for the structure of observed large-scale eddies in the ex- peratures well. The simulations pointed to the possi- tratropics may be a consequence of the atmospheric cir- bility of an earth-like extratropical climate in which culation organizing itself into a state in which nonlinear baroclinic eddy ¯uxes maintain a statically stable ther- eddy±eddy interactions are weak. mal strati®cation and, in interaction with large-scale diabatic processes, lead to the formation of a sharp tropopause. In the simulations as in observational data, 7. Summary the empirical eddy diffusivity for potential vorticity A dynamical constraint on the extratropical tropo- varied only weakly across the extratropical tropopause, pause height and thermal strati®cation has been derived implying that the extratropical tropopause cannot be by considerations of entropy ¯uxes, or isentropic mass viewed as a kinematic mixing barrier. That a well- ¯uxes. The condition that the circulation of mass along de®ned tropopause formed although the radiative equi- and across isentropes approximately closes within the librium temperature in the simulations decreases smoothly and monotonically with height means that, as the simulations by Thuburn and Craig (1997) in- 12 In the simulations of section 4, the Rossby radius NHt / f [with dicated, the radiative heating of the earth's stratosphere N estimated according to Eq. (21)], the Rhines scale (UЈ/␤)1/2 (with due to absorption of solar radiation by ozone is only UЈ estimated from the vertically averaged eddy kinetic energy), and s 2 1/2 of secondary signi®cance for the existence of a tro- ys␪ | for surface potential temperatureץ|/ ( the mixing length (␪Јs indeed covary and are of the same order of magnitude. How the three popause, although it can modify the temperature and length scales vary with latitude depends on the method of averaging height of the tropopause. The simulations also showed employed, but averaged over the extratropics, they differ by at most that the extratropical tropopause height and thermal a factor of 2, despite variations of the mixing length by up to a factor of 6 across the simulations. In the simulations of Barry et al. (2002), strati®cation are set locally by extratropical processes the Rossby radius and Rhines scale likewise appear to be of the same and do not depend on tropical processes. The postulate order of magnitude, though in some of their simulations, the differ- that the eddy diffusivity for potential vorticity exhibits ence between Rossby radius and Rhines scale appears to be larger no essential vertical structure within the troposphere than in the present simulations. The reasons for the discrepancy be- tween the simulation results are unclear; they may lie in different and is, vertically averaged, approximately equal to the ways of averaging the quantities relevant for the computation of the eddy diffusivity for surface potential temperature, al- Rossby radius and Rhines scale. though not quantitatively accurate latitude by latitude, 1338 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 was seen to be a plausible starting point for under- APPENDIX standing large-scale features of the turbulent mixing in the extratropical troposphere. Notation and Symbols An analysis of observational data showed that the ␰ Partial derivative with respect to coordinateץ -dynamical constraint, derived for an idealized dry at mosphere with stationary and axisymmetric circulation ␰ (unless otherwise noted, with horizontal de- statistics, can also account for interannual variations of rivatives understood as derivatives along is- the tropopause height and thermal strati®cation in the entropes if the argument depends on a vertical extratropics of the earth's atmosphere. The agreement coordinate) of the dynamical constraint with the observational data (´) Temporal and zonal mean (along isentropes means that the dynamical constraint can be combined if the argument depends on a vertical coor- with a radiative constraint to compute the variations of dinate) (´)* Isentropic mean( ´) / weighted by isen- the tropopause height and thermal strati®cation that are ␳␪ ␳ ␪ tropic density ␳ associated with interannual variations of the surface po- ␪ ()ˆ´ Fluctuation (ˆ´ ) (´) (´)* about density- tential temperature. ϭ Ϫ weighted isentropic mean (´)* The dynamical constraint implies that if baroclinic s (´) Temporal and zonal mean along surface eddies determine the tropopause height and thermal s (´)Ј Fluctuation (´)Јϭ(´) Ϫ (´) about surface strati®cation, an atmosphere organizes itself into a s mean (´) state in which nonlinear interactions among eddies ͗´͘ Extratropical mean are inhibited. Given that there is no reason a priori a Planet radius to suppose that nonlinear eddy±eddy interactions are c Speci®c heat at constant pressure weak, the inhibition of nonlinear eddy±eddy inter- p Ds, Di Eddy diffusivities for surface potential tem- actions offers an explanation for the historic successes perature and for potential vorticity on isen- of linear and weakly nonlinear models of large-scale tropes extratropical dynamicsÐan explanation that rests on f Coriolis parameter f ϭ 2⍀ sin(␾) the postulate that the kinematic mixing properties of g Gravitational acceleration the baroclinic eddies exhibit no essential vertical Ht Tropopause height structure. The fact that the dynamical constraint im- H(´) Heaviside step function plies that the atmosphere organizes itself into a critical M Montgomery streamfunction M ϭ cpT ϩ gz state with weak nonlinear eddy±eddy interactions in- N Brunt±VaÈisaÈlaÈ frequency timates that there may exist a fundamental variational p, p0 Pressure, constant reference pressure principle that would justify the postulate on eddy dif- P Potential vorticity P ϭ ( f ϩ ␨␪)/␳␪ fusivities and/or the dynamical constraint itself. What Q Material derivative of potential temperature form such a variational principle might take remains Q ϭ D␪/Dt to be investigated. R Gas constant t Time Acknowledgments. My thanks go to Isaac Held for T Temperature advice on the research on which this paper is based and u, ␷ Horizontal velocity components (eastward, on constructing the idealized GCM, and for many dis- northward) cussions over several years that helped to clarify, among U, UЈ Mean velocity scale, turbulent velocity scale other things, the commonalities and differences between v Horizontal velocity [v ϭ (u, ␷, 0) in local Cartesian coordinates] the theoretical developments of this paper and quasi- ␷Ä Ј Balanced meridional eddy velocity at the sur- geostrophic theory. The simulations described in section s face 4c were prompted by discussions with Kerry Emanuel, x, y, z Local Cartesian coordinates (eastward, north- Richard Lindzen, and Alan Plumb. I also thank Vladimir ward, upward) Gryanik, Peter Haynes, Paul Kushner, Shafer Smith, Ka- ␤ Meridional derivative ␤ ϭ 2⍀aϪ1 cos(␾)of Kit Tung, and a reviewer for helpful discussions and Coriolis parameter comments on the paper, and Heidi Swanson for editing ␨␪ Relative vorticity of horizontal ¯ow along is- the manuscript. Parts of the research on which this paper entropes is based were carried out while I was with the Courant ␬ ␪, ␪ 0 Potential temperature ␪ ϭ T(p 0/p) , constant Institute of Mathematical Sciences at New York Uni- reference potential temperature versity (supported partially by NSF Grant DMS- ␪b A potential temperature ␪b ϭ ␪b(y) less than 9972865 and ONR Grant N00014-96-1-0043) and with the lowest potential temperature that occurs the Atmospheric and Oceanic Sciences Program at at a given latitude

Princeton University (supported by a NASA Earth Sys- ␪t Tropopause potential temperature tem Science Fellowship). ␬ Adiabatic exponent ␬ ϭ R/cp 15 JUNE 2004 SCHNEIDER 1339

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