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 q 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