Held–Hou Hadley Cell Model

Held–Hou Hadley Cell Model—formulated in 1980 by the atmospheric dynamicists Isaac M. Held and Arthur Y. Hou for equinox conditions and generalized to off-equatorial solstitial heating in 1988 by Richard S. Lindzen and Arthur Y. Hou—is the canonical analytical theory governing the axisymmetric, thermally direct meridional overturning circulation of the tropical atmosphere of Earth and other rotating planetary atmospheres. Prior to Held and Hou (1980), descriptions of the Hadley circulation treated the poleward boundary of the tropics (~30° latitude) either as an empirical latitude where baroclinic instability takes over or as a viscous circulation requiring arbitrary eddy viscosity throughout the free troposphere. Held and Hou demonstrated that even in a nearly inviscid, axisymmetric atmosphere devoid of mid-latitude baroclinic eddies, a radiative-convective equilibrium state with peak solar heating at the equator violates Hide's theorem on angular momentum extrema and breaks down into a self-limiting Hadley circulation of finite latitudinal width ϕ_H. By combining conservation of axial angular momentum in the poleward-moving upper-tropospheric branch with vertically integrated thermal-wind balance and diabatic heat budget closure, the Held–Hou and Lindzen–Hou theories derive closed scaling laws for the Hadley cell width ϕ_H = (5R/3)^1/2, the subtropical jet wind speed, the overturning mass streamfunction Ψ ∝ R^3/2/τ, and the strong cross-equatorial winter-cell amplification observed during the solstices.

From Halley and Hadley to Axisymmetric Closure

In 1686, Edmond Halley proposed that solar heating of the equatorial belt drives rising air and surface inflow from higher latitudes, and in 1735 George Hadley recognized that Earth's rotation deflects the equatorward surface return flow westward into the trade winds while deflecting upper-level poleward air eastward. Because Hadley assumed conservation of linear eastward velocity rather than axial angular momentum about Earth's spin axis, he envisioned a single hemisphere-spanning cell. Once nineteenth-century fluid dynamicists (William Ferrel and James Thomson) recognized that an inviscid ring of air displaced poleward from rest at the equator from radius a to acos*ϕ* conserves axial angular momentum per unit mass, a dynamical paradox emerged: conserving angular momentum to 60° latitude would produce eastward winds exceeding 1,000 m s⁻¹, far larger than observed in the atmosphere of Earth. Throughout the mid-twentieth century, meteorologists debated whether the Hadley cell is regulated by internal axisymmetric thermodynamics or by mid-latitude Rossby wave eddies.

Radiative-Convective Equilibrium and Hide's Theorem

To isolate the physics of the tropical overturning circulation, Held and Hou (1980) considered a dry Boussinesq atmosphere of depth H on a sphere of radius a rotating at angular velocity Ω, forced by Newtonian thermal relaxation Q = −(θ − θ_E)/τ toward an equinoctial radiative-convective equilibrium potential temperature profile θ_E(ϕ) = θ₀ − (1/3)Δ_Hθ₀(3sin²ϕ − 1) ≈ θ_E(0) − Δ_Hθ₀sin²ϕ, where Δ_H ≈ 1/3 is the fractional equator-to-pole radiative potential temperature contrast.

If the atmosphere remained at rest in the meridional plane (v = 0, w = 0), its potential temperature θ would equal θ_E everywhere, and thermal-wind balance 2Ωsin*ϕ* ∂u_E/∂z = −(g/(θ₀a)) ∂θ_E/∂ϕ integrated from u_E(ϕ, 0) = 0 to the tropopause (z = H) yields:

u_E(ϕ, H) = Ω*aR*cos*ϕ*, where R = *gH*Δ_H/(Ω²a²)

Here R is the dimensionless thermal Rossby number (a planetary counterpart to the dynamical Rossby number; R ≈ 0.05–0.08 on Earth). At the equator (ϕ = 0, cos*ϕ* = 1), the radiative-convective equilibrium wind at the tropopause is eastward (u_E(0, H) = Ω*aR* > 0), so axial angular momentum per unit mass M_E(0, H) = Ωa²(1 + R) > Ωa² exceeds the planetary value Ωa² of the solid Earth. By Hide's theorem (Raymond Hide, 1969), an axisymmetric circulation with small positive viscosity cannot sustain an interior maximum of axial angular momentum M > Ωa² because viscous diffusion out of a closed M-maximum cannot be balanced by advection across closed streamlines. Thus radiative-convective equilibrium breaks down near the equator, driving a thermally direct Hadley cell.

Angular Momentum Conservation in the Upper Branch

Held and Hou (1980) analyzed the limiting regime in which eddy viscosity ν → 0 in the free troposphere while surface drag in a thin turbulent boundary layer at z = 0 constrains the near-surface zonal wind to remain small (u(ϕ, 0) ≈ 0). In the equinoctial symmetry case, air rises at the equator (ϕ = 0) from the surface boundary layer where its axial angular momentum equals the planetary value at the equator:

M(0) = Ωa²

As this air moves poleward in the nearly inviscid upper-tropospheric branch at height z = H from ϕ = 0 to the poleward edge of the Hadley cell at latitude ϕ_H, it conserves its axial angular momentum M(ϕ) = M(0):

M(ϕ) = Ωa²cos²ϕ + u_M(ϕ)acos*ϕ* = Ωa²

Solving for the angular-momentum-conserving zonal wind profile u_M(ϕ) at the tropopause (z = H) yields:

u_M(ϕ) = Ωa(1 − cos²ϕ)/cos*ϕ* = Ωasin²ϕ/cos*ϕ*

Under the small-angle (tropical Coriolis frequency β-plane) approximation valid for tropical latitudes (ϕ ≪ 1 rad, where sin*ϕ* ≈ ϕ and cos*ϕ* ≈ 1), the radiative-convective equilibrium profile and the angular-momentum-conserving wind profile simplify to:

θ_E(ϕ) ≈ θ_E(0) − θ₀Δ_Hϕ², u_M(ϕ) ≈ Ω*aϕ*²

Thus, while the radiative-convective equilibrium wind u_E(ϕ, H) ≈ Ω*aR* is nearly uniform near the equator, the angular-momentum-conserving wind u_M(ϕ) starts at zero at the equator and grows quadratically with latitude (ϕ²).

Thermal-Wind Balance and the Potential Temperature Profile

Within the Hadley cell (0 ≤ |ϕ| ≤ ϕ_H), the upper-tropospheric zonal wind follows u(ϕ, H) = u_M(ϕ), while the surface zonal wind remains small (u(ϕ, 0) ≈ 0) so that vertical wind shear is ∂u/∂z ≈ u_M(ϕ)/H. Substituting u_M(ϕ) into the vertically integrated thermal-wind relation on the sphere:

(2Ωsin*ϕ*/H)u_M(ϕ) = −(g/(θ₀a)) ∂θ/∂ϕ

and using the small-angle approximation (sin*ϕ* ≈ ϕ and u_M ≈ Ω*aϕ*²) gives the meridional gradient of vertically averaged potential temperature θ(ϕ) inside the Hadley cell:

∂θ/∂ϕ = −(2Ω²a²θ₀/(gH))*ϕ*³

Integrating from the equator (ϕ = 0) to latitude ϕ within the Hadley cell yields the quartic potential temperature profile required to maintain thermal-wind balance with an angular-momentum-conserving upper branch:

θ(ϕ) = θ(0) − (Ω²a²θ₀/(2*gH*))*ϕ*⁴ = θ(0) − (θ₀Δ_H/(2R))*ϕ*⁴

Because ∂θ/∂ϕ ∝ −ϕ³ vanishes at the equator along with the Coriolis frequency f = 2Ωsin*ϕ*, the tropical troposphere within the Held–Hou cell exhibits a very flat horizontal temperature distribution near the equator, with baroclinicity concentrated near the subtropical edge ϕ_H.

Thermodynamic Closure and the Held–Hou Scaling Laws

Outside the Hadley cell (|ϕ| > ϕ_H), where there is no meridional overturning circulation (v = 0, w = 0), the atmosphere relaxes to local radiative-convective equilibrium: θ(ϕ) = θ_E(ϕ). To determine the two unknown constants—the equatorial potential temperature θ(0) and the poleward boundary of the Hadley cell ϕ_H—Held and Hou imposed two physical closure conditions:

1. Temperature Continuity at ϕ = ϕ_H: In the absence of an infinite vertical wind shear at the edge of the Hadley cell, the vertically averaged potential temperature must be continuous across ϕ = ϕ_H: θ(ϕ_H) = θ_E(ϕ_H).
2. Diabatic Heat Budget Closure: In a steady state, no net heat is created or destroyed within the closed streamlines of the Hadley cell. Integrating the thermodynamic equation over the volume of the cell from the equator (ϕ = 0) to the dividing streamline at ϕ = ϕ_H requires that net diabatic heating in the ascending warm region equal net radiative cooling in the descending subtropical region:

∫₀^ϕ_H (θ(ϕ) − θ_E(ϕ)) dϕ = 0

Substituting θ(ϕ) = θ(0) − (θ₀Δ_H/(2R))*ϕ*⁴ and θ_E(ϕ) = θ_E(0) − θ₀Δ_Hϕ² into these two conditions yields a geometric equal-area construction in (ϕ, θ)-space:

• From continuity at ϕ_H: θ_E(0) − θ(0) = θ₀Δ_Hϕ_H² − (θ₀Δ_H/(2R))*ϕ*_H⁴.
• Integrating the energy balance integral gives (θ(0) − θ_E(0))*ϕ*_H + (1/3)θ₀Δ_Hϕ_H³ − (1/(10*R*))θ₀Δ_Hϕ_H⁵ = 0.

Combining these two relations eliminates θ_E(0) − θ(0) and yields the Held–Hou scaling law for the poleward extent of the Hadley cell:

ϕ_H = (5R/3)^1/2 = (5*gH*Δ_H/(3Ω²a²))^1/2

Using terrestrial parameters (R ≈ 0.06 to 0.08), the Held–Hou formula predicts ϕ_H ≈ 0.32 to 0.37 radians, or 18° to 22° latitude (rising to 26°–30° latitude when moist static stability or off-equatorial heating is included), matching the observed latitude of the subtropical ridge and desert belts.

From ϕ_H = (5R/3)^1/2, the remaining dynamical quantities of the Held–Hou model follow directly:

• Equatorial Diabatic Cooling Anomaly: At the equator, adiabatic ascent cools the column below radiative equilibrium by θ_E(0) − θ(0) = (5/18)θ₀Δ_HR, while at the subtropical boundary ϕ_H the zonal wind reaches the subtropical jet peak u_M(ϕ_H) = (5/3)Ω*aR* ≈ 120 m s⁻¹.
• Overturning Mass Streamfunction Scaling: Near the equator where ∂θ/∂ϕ ≈ 0, the vertical advection of potential temperature w∂θ/∂z ≈ wΔ_Vθ₀/H (where Δ_V is the vertical fractional static stability contrast) balances the Newtonian heating (θ_E − θ)/τ ∝ RΔ_Hθ₀/τ. Since mass continuity gives Ψ ∝ waϕ_H, the total overturning mass transport Ψ scales as:

Ψ ∝ (aH/τ)(Δ_H/Δ_V)R^3/2 ∝ (Δ_H)^5/2Ω⁻³H^5/2τ⁻¹

Thus, whereas the width of the equinoctial Hadley cell (ϕ_H ∝ R^1/2) is independent of the radiative relaxation timescale τ and vertical static stability Δ_V, the strength (mass overturning rate Ψ) scales as R^3/2 and is inversely proportional to τ and Δ_V.

Off-Equatorial Heating and the Dividing Streamline

While the Held–Hou (1980) model assumed equinoctial symmetry with maximum heating at the equator (ϕ₀ = 0), the real atmosphere of Earth spends most of the annual cycle in a solstitial configuration where the latitude of peak solar insolation ϕ₀ is displaced off the equator into the summer hemisphere. In 1988, Richard S. Lindzen and Arthur Y. Hou extended the inviscid angular-momentum-conserving model to an asymmetric radiative-convective equilibrium profile centered at latitude ϕ₀:

θ_E(ϕ) ≈ θ_E(ϕ₀) − θ₀Δ_H(ϕ − ϕ₀)²

In this asymmetric regime, the dividing streamline ϕ = ϕ₁ separating the winter-hemisphere (cross-equatorial) Hadley cell from the summer-hemisphere Hadley cell no longer sits at the equator (ϕ = 0), nor does it coincide with the thermal maximum ϕ₀. Instead, air rises along a dividing streamline at latitude ϕ₁ in the summer hemisphere, where it acquires the planetary angular momentum M(ϕ₁) = Ωa²cos²ϕ₁. As this air spreads away from ϕ₁ at the tropopause (z = H) toward the summer pole edge (ϕ₊ > ϕ₁) and across the equator toward the winter pole edge (ϕ₋ < 0), it conserves M(ϕ₁), producing the upper-branch zonal wind and potential temperature profiles:

u_M(ϕ, ϕ₁) ≈ Ωa(ϕ² − ϕ₁²), θ(ϕ) = θ(0) − (θ₀Δ_H/(2R))(ϕ⁴ − 2*ϕ*₁²ϕ²)

Nonlinear Amplification of the Cross-Equatorial Winter Cell

Closing the problem with temperature continuity at both outer boundaries (ϕ₋ and ϕ₊) and separate heat-balance integrals over the winter cell (∫_ϕ−^ϕ₁ (θ − θ_E)*dϕ* = 0) and summer cell (∫_ϕ1^ϕ₊ (θ − θ_E)*dϕ* = 0) reveals a striking nonlinear sensitivity to small off-equatorial shifts in ϕ₀:

1. Cubic Shift of the Dividing Streamline: When ϕ₀ > 0 is displaced into the summer hemisphere, the dividing streamline ϕ₁ shifts much further into the summer hemisphere than ϕ₀ itself, obeying (ϕ₁/ϕ_H)³ ≈ (9/4)(ϕ₀/ϕ_H) for small ϕ₀.
2. Collapse of the Summer Cell and Winter Cell Amplification: Even for a modest displacement of peak heating to ϕ₀ = 6° latitude, the cross-equatorial winter Hadley cell expands across the equator from ϕ₁ ≈ +15° in the summer hemisphere to ϕ₋ ≈ −32° in the winter hemisphere, and its peak mass transport Ψ_winter intensifies by nearly an order of magnitude (~8× to 10× stronger than the equinoctial cell), while the summer Hadley cell shrinks into a negligible residual circulation.
3. Easterly Equatorial Upper-Level Winds: Because air in the cross-equatorial winter cell originates at ϕ₁ ≠ 0 where planetary angular momentum Ωa²cos²ϕ₁ is lower than at the equator, it crosses the equator (ϕ = 0) with a westward (easterly) upper-tropospheric wind u_M(0) ≈ −Ω*aϕ*₁² < 0 before accelerating into an intense westerly subtropical jet in the winter hemisphere.

Moist Static Energy and Plumb–Hou (1992) Monsoon Onset

In the real moist tropical atmosphere, latent heat release inside deep cumulonimbus towers narrows the ascending branch into a concentrated Intertropical Convergence Zone (ITCZ) while leaving the broader subsiding branch governed by radiative cooling. Building on Lindzen and Hou (1988), R. Alan Plumb and Arthur Y. Hou showed in 1992 that when localized off-equatorial thermal forcing is placed in the subtropics (mimicking Tibetan Plateau heating during the Asian summer monsoon), the atmosphere remains in radiative-convective equilibrium with zero overturning until the subtropical heating curvature exceeds a threshold where tropopause absolute vorticity f − ∂u_E/∂y vanishes, triggering an abrupt transition into a strong cross-equatorial monsoon circulation.

Interaction with Baroclinic Eddies and Comparative Planetary Atmospheres

Tapio Schneider (2006) and Christopher Walker quantified the transition between the Held–Hou inviscid regime and extratropical Eady model baroclinic eddy driving using the local Rossby number Ro_local = −ζ/f: during solstices, the strong cross-equatorial winter cell achieves Ro_local ≈ 0.6–0.8 and closely obeys Lindzen–Hou dynamics, whereas the weaker summer and equinoctial cells have Ro_local ≈ 0.2–0.3 due to mid-latitude Rossby wave momentum absorption. Across the Solar System, the Held–Hou scaling ϕ_H = (5R/3)^1/2 explains why slowly rotating worlds with R ≫ 1 (Venus and Saturn's moon Titan) sustain global pole-to-pole Hadley cells, why Mars (R ≈ 4R_Earth) exhibits broad seasonal cross-equatorial cells, and why rapidly rotating gas giants with R ≪ 10⁻³ (Jupiter and Saturn) confine direct thermal overturning to narrow equatorial belts.

Why cannot the tropical atmosphere simply remain at rest in radiative-convective equilibrium?

In pure radiative-convective equilibrium with peak solar heating at the equator, both ∂θ_E/∂ϕ and the Coriolis frequency f = 2Ωsin*ϕ* approach zero linearly near ϕ = 0, so their ratio in the thermal-wind equation converges to a positive eastward wind u_E(0, H) = Ω*aR* > 0 at the equatorial tropopause. This implies an interior maximum of axial angular momentum M > Ωa² at the equator, violating Hide's (1969) theorem in a nearly inviscid fluid and forcing the atmosphere to overturn in a thermally direct Hadley cell.

Why does the Held–Hou Hadley cell stop near 30 degrees latitude instead of reaching the poles?

As upper-tropospheric air moves poleward from the equator while conserving its axial angular momentum M = Ωa², its eastward zonal wind speed grows quadratically with latitude (u_M ≈ Ω*aϕ*²). To remain in thermal-wind balance with this accelerating westerly wind, the tropospheric potential temperature θ(ϕ) must drop off with the fourth power of latitude (ϕ⁴), much steeper than the quadratic drop (ϕ²) of radiative equilibrium θ_E(ϕ). Around 25°–30° latitude, cumulative radiative cooling of the descending air balances the equatorial heating, terminating the cell at ϕ_H = (5R/3)^1/2.

Why is the winter Hadley cell so much stronger than the summer Hadley cell during the solstices?

As demonstrated by Lindzen and Hou (1988), when peak solar heating is displaced by even 6° into the summer hemisphere, the rising dividing streamline ϕ₁ shifts into the summer subtropics. Air moving poleward into the summer hemisphere encounters rapidly shrinking distance to the spin axis acos*ϕ*, generating steep angular momentum gradients that choke off the summer cell within a few degrees. Conversely, air moving toward the winter hemisphere first travels across the equator toward larger acos*ϕ* before entering the winter hemisphere, driving an overturning circulation nearly ten times stronger than at the equinox.

1. Held, I. M., & Hou, A. Y. (1980). "Nonlinear Axially Symmetric Circulations in a Nearly Inviscid Atmosphere037%3C0515:NASCIA%3E2.0.CO;2)." Journal of the Atmospheric Sciences, 37(3), 515–533.
2. Lindzen, R. S., & Hou, A. Y. (1988). "Hadley Circulations for Zonally Averaged Heating Centered Off the Equator045%3C2416:HCFZAH%3E2.0.CO;2)." Journal of the Atmospheric Sciences, 45(17), 2416–2427.
3. Hide, R. (1969). "Dynamics of the Atmospheres of the Major Planets with an Appendix on the Viscous Boundary Layer at the Rigid Bounding Surface of an Electrically-Conducting Rotating Fluid in the Presence of a Magnetic Field026%3C0841:DOTAOT%3E2.0.CO;2)." Journal of the Atmospheric Sciences, 26(5), 841–853.
4. Plumb, R. A., & Hou, A. Y. (1992). "The Response of a Zonally Symmetric Atmosphere to Subtropical Thermal Forcing: Threshold Behavior049%3C1790:TROAZS%3E2.0.CO;2)." Journal of the Atmospheric Sciences, 49(19), 1790–1799.
5. Schneider, T. (2006). "The General Circulation of the Atmosphere." Annual Review of Earth and Planetary Sciences, 34, 655–688.