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Chapter 1
Hadley cell dynamics
Adam P. Showman March 6 2009
1.1 Introduction
Perhaps the simplest possible idealization of a circulation that transports heat from equator to poles is an axisymmetric circulation — that is, a circulation that is independent of longitude — where hot air rises at the equator, moves poleward aloft, cools, sinks at high latitudes, and returns equatorward at depth (near the surface on a terrestrial planet). Such a circulation is termed a Hadley cell, and was first envisioned by Hadley in 1735 to explain Earth’s trade winds. Most planetary atmospheres in our Solar System, including those of Venus, Earth, Mars, Titan, and possibly the giant planets, exhibit Hadley circulations. Hadley circulations on real planets are of course not truly axisymmetric; on the terrestrial planets, longitudinal variations in topography and ther- mal properties (e.g., associated with continent-ocean contrasts) induce asymmetry in longitude. Nevertheless, the fundamental idea is that the longitudinal variations are not crucial for driving the circulation. This differs from the circulation in midlatitudes, whose longitudinally aver- aged properties are fundamentally controlled by the existence of three- dimensional eddies. Planetary rotation generally prevents Hadley circulations from extend- ing all the way to the poles. Because of planetary rotation, equatorial air contains considerable angular momentum about the planetary rotation axis; to conserve angular momentum, equatorial air would accelerate to unrealistically high speeds as it approached the pole, a phenomenon which is dynamically inhibited. To illustrate, the specific angular momentum about the rotation axis on a spherical planet is M = (Ωa cos φ + u)a cos φ,
1
Figure 1.1: Schematic of Earth’s three-cell structure. The cell closest to the equator is the Hadley cell.
Figure 1.3: Global mosaic of Earth without clouds or sea ice, illustrating the effect of the Hadley cell. Equatorial regions (within ± 20 ◦^ of equator) receive abundant rainfall and show up green; this is the rising branch of the cell. Subtropical regions at ∼ 20 – 30 ◦^ latitude receive little rainfall and show up brown; this is the descending branch of the cell.
where the first and second terms represent angular momentum due to planetary rotation and winds, respectively. If u = 0 at the equator, then M = Ωa^2 , and an angular-momentum conserving circulation would then exhibit winds of
u = Ωa
sin^2 φ cos φ
Given Earth’s radius and rotation rate, this equation implies zonal-wind speeds of 134 m sec−^1 at 30◦^ latitude, 700 m sec−^1 at 60◦^ latitude, and 2 .7 km sec−^1 at 80◦^ latitude. Such high-latitude wind speeds are unrealisti- cally high and would furthermore be violently unstable to 3D instabilities. On Earth, the actual Hadley circulations extend to ∼ 30 ◦^ latitude. The Hadley circulation exerts strong control over the wind structure, latitudinal temperature contrast, and climate. Hadley circulations trans- port thermal energy by the most efficient means possible, namely straight- forward advection of air from one latitude to another. As a result, the latitudinal temperature contrast across a Hadley circulation tends to be modest; the equator-to-pole temperature contrast on a planet will there- fore depend strongly on the width of the Hadley cell. Moreover, on plan- ets with condensable gases, Hadley cells exert control over the patterns of cloudiness and rainfall. On Earth, the rising branch of the Hadley
Figure 1.4: Polar region of Venus in the UV (top; blue) and IR (bottom; red). Polar vortex is visible in the IR data and may represent descending branch of the Hadley cell; ascending branch would then cover most of the rest of the planet.
(a) Vertically and zonally averaged temperature struc- ture. Top shows mean temperature, middle shows vari- ance due to traveling eddies and bottom shows variance due to stationary eddies. The temperature structure near the equator is nearly isothermal and eddy-free, indicating the region of dominance of the Hadley cell. From Peixoto and Oort (1992, Fig. 7.9).
(b) Vertically and zonally averaged zonal-wind structure. Top shows mean zonal wind, mid- dle shows variance due to traveling eddies and bottom shows variance due to stationary eddies. Near the equator, eddies are weak and the wind is westward but increases with latitude, showing the region of dominance of the Hadley cell. From Peixoto and Oort (1992, Fig. 7.20).
Figure 1.6: Signature of the Hadley cell is visible in zonal-mean precipitation, which peaks at the equator but has local minima at ∼ 20 – 30 ◦^ latitude. From Peixoto and Oort (1992, Fig. 7.25).
circulation leads to cloud formation and abundant rainfall near the equa- tor, helping to explain for example the prevalence of tropical rainforests in Southeast Asia/Indonesia, Brazil, and central Africa.^1 On the other hand, because condensation and rainout dehydrates the rising air, the de- scending branch of the Hadley cell is relatively dry, which explains the abundances of arid climates on Earth at 20–30◦^ latitude, including the deserts of the African Sahara, South Africa, Australia, central Asia, and the southwestern United States. The Hadley cell can also influence the mean cloudiness, hence albedo and thereby the mean surface tempera- ture. Venus’s slow rotation rate leads to a global equator-to-pole Hadley cell, with the descending branch confined to small polar vortices and the ascending branch covering most of the planet. Coupled with the presence of trace condensable gases, this near-global ascent leads to a near-global cloud layer that helps generate Venus’ high bond albedo of 0.75. Dif- ferent Hadley cell patterns would presumably cause different cloudiness patterns, different albedos, and therefore different global-mean surface temperatures. A variety of studies have been carried out using fully nonlinear, global 3D numerical circulation models to determine the sensitivity of the Hadley cell to the planetary rotation rate and other parameters (e.g. Hunt, 1979; Williams and Holloway, 1982; Williams, 1988a,b; del Genio and Suozzo, 1987; Navarra and Boccaletti, 2002; Walker and Schneider, 2005, 2006). These studies show that as the rotation rate is decreased the width of the Hadley cell increases, the equator-to-pole heat flux increases, and the equator-to-pole temperature contrast decreases. Figs. 1.9–1.10 illustrates examples from Navarra and Boccaletti (2002) and del Genio and Suozzo (1987). For Earth parameters, the circulation exhibits mid-latitude east- ward jet streams that peak in the upper troposphere (∼ 200 mbar pres- sure), with weaker wind at the equator (Fig. 1.9). The Hadley cells extend from the equator to the equatorward flanks of the mid-latitude jets. As the rotation rate decreases, the Hadley cells widen and the jets shift poleward. At first, the jet speeds increase with decreasing rotation rate, which re- sults from the fact that as the Hadley cells extend poleward (i.e., closer to the rotation axis) the air can spin-up faster (cf Eq. 1.1). Eventually, once the Hadley cells extend almost to the pole (at rotation periods exceeding ∼ 5–10 days for Earth radius, gravity, and vertical thermal structure), further decreases in rotation rate reduce the mid-latitude jet speed. Perhaps more interestingly for planetary observations, these changes in the Hadley cell significantly influence the planetary temperature structure. This is illustrated in Fig. 1.10 from a series of simulations by del Genio
(^1) Regional circulations, such as monsoons, also contribute.
Figure 1.9: Zonal-mean circulation versus latitude (abscissa) and pressure (ordinate) in a series of Earth-based GCM experiments from Navarra and Boccaletti (2002) where the rotation period is varied from 18 hours (top) to 360 hours (bottom). Greyscale and thin grey contours depict zonal-mean zonal wind, u, and thick black contours denote streamfunction of the circulation in the latitude-height plane, with solid being clockwise and dashed being counterclockwise. The two cells closest to the equator correspond to the Hadley cell. As rotation period increases, the jets move poleward and the Hadley cell widens, becoming nearly global at the longest rotation periods.
Figure 1.11: Latitudinal temperature profiles for simplified Held-Hou-type axisymmetric models. Shown is the radiative equilibrium temperature profile and several solutions. Γ = 0 corresponds to the inviscid Held-Hou solution where angular momentum is conserved in the upper branch of the Hadley cell. Combined with thermal-wind balance, this leads to a T ∝ φ^4 dependence (Eq. 1.3). A constant (θequator from Eq. 1.3) is added to ensure that the Hadley cell is thermally closed, which requires that the areas between the solution and the radiative-equilibrium profile sum to zero. This generally means that the solutions strike the radiative-equilibrium profile twice: once in the subtropics (equatorward of which heating occurs and poleward of which cooling occurs) and once closer to the pole; the latter defines the latitudinal extent of the Hadley cell. The Γ > 0 curves correspond to axisymmetric solutions where drag has been added to the upper branch of the Hadley cell. At low latitudes, these models produce profiles of temperature, heating, and zonal wind that agree better with Earth data than the Γ = 0 Held-Hou model. However, the Γ > 0 models shown here have Hadley cells that extend all the way to the pole. The fact that the Hadley cell does not extend to the pole hints that the Held-Hou model does not provide the correct explanation for the width of Earth’s Hadley cell. From Farrell (1990).
of sunlight and loss of heat to space generate a latitudinal temperature contrast that drives the circulation; for concreteness, let us parameter- ize the radiation as a relaxation toward a radiative-equilibrium potential temperature profile that varies with latitude as θrad = θ 0 − ∆θrad sin^2 φ, where θ 0 is the radiative-equilibrium potential temperature at the equator and ∆θrad is the equator-to-pole difference in radiative-equilibrium poten- tial temperature. If we make the small-angle approximation for simplicity (valid for a Hadley cell that is confined to low latitudes), we can express this as θrad = θ 0 − ∆θradφ^2.
In the lower layer, we assume that friction against the ground keeps the wind speeds low; in the upper layer, assumed to occur at an altitude H, the flow conserves angular momentum. The upper layer flow is then
specified by Eq. 1.1, which is just u = Ωaφ^2 in the small-angle limit. We expect that the upper-layer wind will be in thermal-wind balance with the latitudinal temperature contrast^2 :
f
∂u ∂z
= f
u H
g θ 0
∂θ ∂y
where ∂u/∂z is simply given by u/H in this two-layer model. Inserting u = Ωaφ^2 into Eq. 1.2 and integrating, we obtain a temperature that varies with latitude as
θ = θequator −
Ω^2 θ 0 2 ga^2 H
y^4 (1.3)
where θequator is a constant to be determined. At this point, we introduce two constraints. First, Held and Hou (1980) assumed the circulation is energetically closed, i.e. that no net exchange of mass or thermal energy occurs between the Hadley cell and higher latitude circulations. Given an energy equation with radiation parameterized using Newtonian cooling, dθ/dt = (θrad − θ)/τrad, where τrad is a radiative time constant, the assumption that the circulation is steady and closed requires that (^) ∫ (^) φ H
0
θ dy =
∫ (^) φH
0
θraddy (1.4)
where we are integrating from the equator to the poleward edge of the Hadley cell, at latitude φH. Second, temperature must be continuous with latitude at the poleward edge of the Hadley cell. In the axisymmetric model, baroclinic instabilities are suppressed, and the regions poleward of the Hadley cells reside in a state of radiative equilibrium. Thus, θ must equal θrad at the poleward edge of the cell. Inserting our expressions for θ and θrad into these two constraints yields a system of two equations for φH and θequator. The solution yields a Hadley cell with a latitudinal half-width of
φH =
5∆θradgH 3Ω^2 a^2 θ 0
in radians. This solution suggests that the width of the Hadley cell scales as the square root of the fractional equator-to-pole radiative-equilibrium temperature difference, the square root of the gravity, the square root of the height of the cell, and inversely with the rotation rate. Inserting Earth annual-mean values (∆θrad ≈ 70 K, θ = 260 K, g = 9.8 m sec−^2 , H = 15 km, a = 6400 km, and Ω = 7. 2 × 10 −^5 sec−^1 ) yields ∼ 32 ◦. (^2) This form differs slightly from Eq. ?? because Eq. 1.2 adopts a constant basic-state density (the so-called “Boussinesq” approximation) whereas Eq. ?? adopts the compressible ideal-gas equation of state.
pendence of θrad. The highest latitude to which the cell can extend without encountering this problem is simply given by Eq. 1.4.
The model can be generalized to consider a more realistic treatment of radiation than the simplified Newtonian cooling/heating scheme employed by Held and Hou (1980). Caballero et al. (2008) reworked the scheme using a two-stream, non-grey representation of the radiative transfer with parameters appropriate for Earth and Mars. This leads to a prediction for the width of the Hadley cell that differs from Eq. 1.5 by a numerical constant of order unity. Although the prediction of Held-Hou-type models for φH provides im- portant insight, several failures of these models exist. First, the model underpredicts the strength of the Earth’s Hadley cell (e.g., as character- ized by the magnitude of the north-south wind) by about an order of magnitude. This seems to result from the lack of turbulent eddies in axisymmetric models; several studies have shown that turbulent three- dimensional eddies exert stresses that act to strengthen the Hadley cells beyond the predictions of axisymmetric models (e.g. Kim and Lee, 2001; Walker and Schneider, 2005, 2006; Schneider, 2006).^3 Second, the Hadley cells on Earth and probably Mars are not energetically closed; rather, mid-latitude baroclinic eddies transport thermal energy out of the Hadley cell into the polar regions. As a result, net heating occurs throughout the latitudinal extent of Earth’s Hadley cell (see Fig. 1.13) — rather than heating in the equatorial branch and cooling in the poleward branch as postulated by Held and Hou (1980). Third, the poleward-moving upper tropospheric branches of the Hadley cells do not conserve angular mo- mentum — although the zonal wind does become eastward as one moves poleward across the cell, for Earth this increase is a factor of ∼ 2–3 less than predicted by Eq. 1.1. The Hadley cell also experiences a net torque with the surface (Fig. 1.14), which can only be balanced by exchange of momentum between the Hadley cell and higher-latitude circulations. Overcoming these failings requires the inclusion of three-dimensional ed- dies.
Several studies have shown that turbulent eddies in the mid- to high- latitudes — which are neglected in the Held-Hou and other axisymmetric models — can affect the width of the Hadley circulation and alter the parameter dependences suggested by Eq. 1.5 (e.g., del Genio and Suozzo, 1987; Walker and Schneider, 2005, 2006). Turbulence can produce an acceleration/deceleration of the zonal-mean zonal wind, which breaks the
(^3) Held and Hou (1980)’s original model neglected the seasonal cycle, and it has been suggested that generalization of the Held-Hou axisymmetric model to include seasonal effects could alleviate this failing (Lindzen and Hou, 1988). Although this improves the agreement with Earth’s observed annual-mean Hadley-cell strength, it predicts large solstice/equinox oscillations in Hadley-cell strength that are lacking in the observed Hadley circulation (Dima and Wallace, 2003).
Figure 1.13: Zonal-mean solar absobed flux, emitted IR flux, and net (solar – IR) heat- ing/cooling versus latitude from Earth observations. The net heating rate is positive from the equator all the way to 30– 40 ◦^ latitude, implying that heating occurs throughout the entire width of the Hadley cell. This is inconsistent with the Held-Hou model, which predicts that heating occurs in the equatorial part of the Hadley cell and that cooling occurs in the subtropical part of the cell. The implication is that, contrary to the assumption of the Held-Hou model, the Hadley cell is not energetically closed but transports net thermal energy into midlatitudes. From Peixoto and Oort (1992, Fig. 6.14.)
angular-momentum conservation constraint in the upper-level wind, caus- ing u to deviate from Eq. 1.1. With a different u(φ) profile, the latitudinal dependence of temperature will change (via Eq. 1.2), and hence so will the latitudinal extent of the Hadley cell required to satisfy Eq. 1.4. In- deed, within the context of axisymmetric models, the addition of strong drag into the upper-layer flow (parameterizing turbulent mixing with the slower-moving surface air, for example) can lead Eq. 1.4 to predict that the Hadley cell should extend to the poles even for Earth’s rotation rate (e.g., see Fig. 1.11; Farrell, 1990). It could thus be the case that the width of the Hadley cell is funda- mentally controlled by eddies. For example, in the midlatitudes of Earth and Mars, baroclinic eddies generally accelerate the zonal flow eastward in the upper troposphere; in steady state, this is generally counteracted by a westward Coriolis acceleration, which requires an equatorward up- per tropospheric flow — backwards from the flow direction in the Hadley cell. Such eddy effects can thereby terminate the Hadley cell, forcing its confinement to low latitudes. Based on this idea, Held (2000) suggested that the Hadley cell width is determined by the latitude beyond which the troposphere first becomes baroclinically unstable (requiring isentrope slopes to exceed a latitude-dependent critical value). Adopting the hor- izontal thermal gradient implied by the angular-momentum conserving wind (Eq. 1.3), making the small-angle approximation, and utilizing a common two-layer model of baroclinic instability, this yields (Held, 2000)
φH ≈
gH∆θv Ω^2 a^2 θ 0
in radians, where ∆θv is the vertical difference in potential temperature from the surface to the top of the Hadley cell. Note that the predicted dependence of φH on planetary radius, gravity, rotation rate, and height of the Hadley cell is weaker than predicted by the Held-Hou model. Earth- based GCM simulations suggest that Eq. 1.6 may provide a better repre- sentation of the parameter dependences (Frierson et al., 2007; Lu et al., 2007; Korty and Schneider, 2008). Nevertheless, even discrepancies with Eq. 1.6 are expected since the actual zonal wind does not follow the angular-momentum conserving profile (implying that the actual thermal gradient will deviate from Eq. 1.3). Substantially more work is needed to generalize these ideas to the full range of conditions relevant for other planets.
