## 1. Introduction

[2] Several studies have reported an increase in the meridional extent of the Hadley circulation (HC) in both recent observational data [e.g., *Hu and Fu*, 2007; *Seidel and Randel*, 2007; *Seidel et al.*, 2008] and simulations of 21st-century climate [e.g., *Lu et al.*, 2007; *Seager et al.*, 2007]. Such shifts can dramatically alter climate on regional scales, underscoring the need to understand the dynamical mechanisms responsible. While it has long been suspected that the HC extent may be related to baroclinic eddy activity, an expression for its meridional extent built on a dynamically consistent foundation has not been developed.

[3] Axisymmetric HCs for Earth-like planets are baroclinically unstable, which has led to formulations for HC extent that combine baroclinic instability measures with expressions for the flow within HCs. For example, combining the zonal wind obtained by assuming angular momentum conservation of the meridional flow in the upper troposphere [*Held and Hou*, 1980] with the critical shear for baroclinic instability in the quasigeostrophic two-layer model [*Phillips*, 1954] yields an expression for the HC terminus as the latitude at which the HC would become baroclinically unstable in the two-layer model [*Held*, 2000]. However, the critical shear is an artifact of the vertical truncation of the two-layer model; in a continuously stratified atmosphere, there is none. While some other baroclinic instability measure could be used in its place (e.g., the growth rate from the Eady or Charney model), an additional problem is that upper-tropospheric flows in HCs across a range of climates including Earth's deviate substantially from angular momentum conservation [*Walker and Schneider*, 2006; *Schneider*, 2006], rendering the physical basis for this formulation dubious. Existing theories do not capture the quantitative dependence of HC extent on the static stability and other mean-flow quantities [*Walker and Schneider*, 2006; *Schneider*, 2006].

[4] Here we propose a new formulation built on a foundation that neither requires angular momentum conservation of the tropical upper-tropospheric flow nor uses an expression for baroclinic instability that is predicated on the architecture of a particular model. We take as the defining characteristic of the HC's subtropical terminus that there the divergence of meridional eddy angular momentum fluxes in the upper troposphere changes sign. Upper-tropospheric divergence of eddy angular momentum fluxes near the terminus and poleward thereof is balanced primarily by the Coriolis torque on (Eulerian) mean meridional mass fluxes because the Rossby number there is small [e.g., *Walker and Schneider*, 2006; *Bordoni and Schneider*, 2008]. Where there is eddy angular momentum flux divergence (in the Hadley cells), there is poleward mass flux; where there is convergence (in the Ferrel cells), there is equatorward mass flux. Convergence of meridional eddy angular momentum fluxes is tantamount to divergence of meridional wave activity fluxes, and the wave activity diverging poleward of the HC terminus is brought into the upper troposphere by vertical fluxes, which are related to meridional eddy entropy fluxes [e.g., *Edmon et al.*, 1980]. Therefore, the HC terminus can be interpreted as the latitude poleward of which eddy entropy fluxes are sufficiently deep to reach the upper troposphere, leading to wave activity flux divergence and thus to eddy angular momentum flux convergence and a meridional mass flux that opposes that of the HCs.

[5] The preceding discussion suggests that the HC terminus may be characterized by a critical *O*(1) value of the supercriticality

which is a nondimensional measure of the pressure range over which eddy entropy fluxes in dry atmospheres extend [*Schneider and Walker*, 2006; *Schneider*, 2007]. The supercriticality generalizes a similar measure of the depth of eddy entropy fluxes proposed in the context of quasigeostrophic theory by *Held* [1978]. The fields entering (1) are temporal and zonal means: the surface or near-surface potential temperature _{s}; the pressures at the surface (_{s}), at the tropopause (_{t}), and at the level up to which eddy entropy fluxes extend (_{e}); and the bulk stability

which depends on the static stability near the surface. Although the derivation of *S*_{c} as a measure of the depth of eddy entropy fluxes relative to the tropopause height is based on diffusive eddy flux closures, and therefore may be expected only to hold on length scales large compared with those of eddies [*Schneider and Walker*, 2006], here we evaluate *S*_{c} locally in latitude, using local values of *f*, *β*, and the mean-flow quantities. Formulated this way, the latitude poleward of which *S*_{c} first exceeds a critical *O*(1) value may be that at which eddy entropy fluxes first reach the upper troposphere and, therefore, where the eddy angular momentum flux divergence in the upper troposphere changes sign.

[6] We use simulations with an idealized dry GCM that span a wide range of climates to investigate how the HC extent depends on mean-flow quantities and how the specific formulation proposed here performs. The dry GCM allows us to test theories based on scaling laws for baroclinic eddies in dry atmospheres, postponing questions of how they may be generalized to moist atmospheres. We show that *S*_{c} indeed assumes an approximately constant value at the HC's terminus in all but a few simulations.