Eddington-Sweet Circulation

Eddington-Sweet circulation (also termed meridional circulation) is the large-scale, axisymmetric quadrupolar fluid flow that develops in the stably stratified radiative zones of rotating stars to restore thermal equilibrium when centrifugal forces distort spherical isobaric surfaces into oblate spheroids. First proposed qualitatively by Arthur Stanley Eddington in 1925 and quantified in 1950 by Peter Alan Sweet following Heinrich Vogt's independent 1925 analysis, the circulation resolves the thermal paradox of the von Zeipel theorem: in a uniformly rotating radiative star without internal nuclear energy generation in its envelope, the radiative heat flux vector F_rad has a non-zero divergence (∇ · F_rad ≠ 0) along equipotential surfaces, causing the rotational poles to heat up relative to the equator. Within astronomy, Eddington-Sweet circulation provides the primary advective driver of rotational mixing and angular momentum redistribution in intermediate- and high-mass stars, interacting with chemical composition barriers (Mestel μ-currents) and shear turbulence across the main sequence.

Hydrostatic Equilibrium on Oblate Equipotentials

Consider a star rotating with angular velocity Ω about the z-axis. In a barotropic star where Ω is constant or depends only on cylindrical distance ϖ = r sin θ from the rotation axis, the sum of the gravitational potential Φ_g and the centrifugal potential Φ_rot = −(1/2) Ω² r² sin²θ defines a total effective potential Ψ(r, θ):

g_eff = −∇Ψ = −∇Φ_g + Ω² r sin θ e_ϖ

Hydrostatic equilibrium (∇P = ρ g_eff) requires that pressure P, density ρ, and—for a chemically homogeneous ideal gas—temperature T be constant on surfaces of constant total potential Ψ = constant (isobars). Because centrifugal force acts outward perpendicular to the rotation axis, each equipotential surface is compressed at the poles (θ = 0, π) and bulges outward at the equator (θ = π/2). Consequently, the spacing between adjacent equipotential shells is narrower at the poles than at the equator, so the magnitude of effective gravity g_eff = |∇Ψ| is greater at the poles than at the equator.

Breakdown of Local Radiative Equilibrium

In a radiative zone, photon diffusion carries an outward energy flux given by Fourier's radiative law:

F_rad = −[ (4 a c T³) / (3 κ ρ) ] ∇T = −χ_rad(P, T) ( dT / dΨ ) ∇Ψ

As proved by the von Zeipel theorem (1924), because P, T, ρ, and opacity κ (governed by Kramers' opacity law or electron scattering) are constant across any equipotential surface Ψ = constant, the radiative flux magnitude is directly proportional to the local effective gravity:

F_rad(Ψ, θ) ∝ g_eff(Ψ, θ)

However, if the star is to remain in strict local radiative thermal equilibrium without fluid motion, the divergence of the radiative flux must match the local nuclear energy generation rate per unit volume (∇ · F_rad = ρ ε_nuc). Evaluating ∇ · F_rad using Poisson's equation (∇²Φ_g = 4π G ρ) for a uniformly rotating star yields:

∇ · F_rad = (d / dΨ)[ χ_rad (dT / dΨ) ] g_eff² + χ_rad (dT / dΨ) (4π G ρ − 2 Ω²)

Because g_eff²(r, θ) varies with colatitude θ across an oblate equipotential while ρ and Ω are constant on that surface, ∇ · F_rad cannot vanish simultaneously at both the poles and the equator (unless Ω varies follow a special non-physical law). Instead, radiative flux divergence is negative near the poles (net radiative heating) and positive near the equator (net radiative cooling).

Conservation of Entropy and Meridional Velocity

As Eddington (1925) and Sweet (1950) demonstrated, this latitudinal heating imbalance cannot persist in a fluid star: the polar regions become slightly warmer and more buoyant than the equatorial regions on the same isobar, driving a continuous meridional circulation cell U = (U_r, U_θ, 0) in the (r, θ) plane. In a steady state, the entropy advection by the vertical circulation velocity U_r balances the non-zero divergence of the radiative flux:

ρ T (U · ∇S) = ρ c_P T [ (∇_ad − ∇) / H_P ] U_r = − ∇ · F_rad + ρ ε_nuc

Expanding the perturbation to first order in the dimensionless rotational oblateness parameter (ratio of centrifugal to gravitational acceleration):

χ_rot(r) ≡ (Ω² r³) / (G m(r))

the radial component of the meridional circulation velocity has a quadrupolar Legendre polynomial dependence P₂(cos θ) = (1/2)(3 cos²θ − 1):

U_r(r, θ) = U₂(r) P₂(cos θ)

where the characteristic velocity amplitude U_ES in a chemically homogeneous radiative envelope is:

U_ES(r) ≈ [ ∇_ad / (∇_ad − ∇) ] [ L(r) / (m(r) g(r)) ] [ R / r ]² ( Ω² r³ / [G m(r)] )

In classical Sweet (1950) solid-body flow, fluid rises along the rotational poles (θ = 0, π, where P₂(cos θ) = +1) and sinks at the equator (θ = π/2, where P₂(cos θ) = −1/2), connected by horizontal return currents U_θ.

The Eddington–Sweet Timescale (t_ES)

Dividing the stellar radius R by the characteristic circulation speed U_ES defines the Eddington–Sweet timescale t_ES, which equals the global Kelvin-Helmholtz thermal timescale t_KH = G M² / (R L) divided by the surface rotational oblateness factor:

t_ES ≈ t_KH / ( Ω² R³ / [G M] ) = t_KH ( v_crit / v_eq )²

where v_eq = Ω R is the equatorial rotation speed and v_crit = (G M / R)^1/2 is the Keplerian breakup velocity.

Compositional Barriers and Mestel's Choking Criterion

Sweet's original 1950 formula ignores gradients in mean molecular weight μ. Because t_ES in rapidly rotating O and B stars is much shorter than the nuclear main-sequence lifetime t_MS, unrestricted Sweet circulation would homogenize every moderately rotating star. In 1953, Leon Mestel resolved this discrepancy by showing that as nuclear burning builds a positive helium gradient (∇_μ > 0, satisfying the Ledoux criterion) outside the convective core, rising circulation currents carry heavier helium-rich fluid upward at the poles while sinking currents carry lighter hydrogen-rich fluid downward at the equator.

This nonspherical molecular weight perturbation Δμ(r, θ) = Λ(r) P₂(cos θ) generates an opposing baroclinic buoyancy current—termed a Mestel μ-current (U_μ)—that opposes the thermal Eddington–Sweet velocity (U_Ω):

U₂(r) = U_Ω(r) + U_μ(r)

When the stabilizing composition gradient exceeds a threshold value (∇_μ ≳ [δ / φ] ∇_ad [Ω² r³ / G m]), the Mestel μ-current chokes off circulation across the core boundary, creating a stagnant barrier unless eroded by horizontal shear turbulence.

Zahn's Advection–Diffusion Balance and Gratton–Öpik Term

In modern stellar evolution theory (Zahn, 1992; Maeder & Zahn, 1998), the assumption of uniform rotation is relaxed in favor of shellular rotation Ω(r). In this baroclinic state, the radial velocity amplitude U₂(r) depends on both the local differential rotation dΩ / dr and the Gratton–Öpik density term (1 − [Ω² / 2π G ρ]), which increases strongly in low-density outer stellar envelopes where ρ → 0. Near the stellar surface, the Gratton–Öpik term reverses the sense of the circulation, creating an outer cell that rises at the equator and sinks at the poles while the deeper interior cell rises at the poles and sinks at the equator, transporting angular momentum radially until balanced by shear diffusion and surface magnetic braking or mass loss.

Why is Eddington-Sweet circulation negligible in the present-day Sun?

The present-day Sun has spun down via magnetic braking to a slow equatorial velocity of 2 km s⁻¹ (P_rot ≈ 25.4 days), giving a centrifugal-to-gravitational ratio Ω² R_☉³ / (G M_☉) ≈ 2 × 10⁻⁵. Consequently, the solar Eddington–Sweet timescale is ~1.5 × 10¹² years—more than 100 times the Sun's entire 10-gigayear main-sequence lifetime—resulting in circulation velocities of only ~10⁻¹⁰ cm s⁻¹ in the solar radiative interior.

Does Eddington-Sweet circulation transport angular momentum like a diffusion process?

No. Unlike turbulent shear eddies, which always diffuse angular momentum down an angular velocity gradient (toward uniform rotation Ω = constant), Eddington-Sweet circulation is a large-scale advective flow. Depending on the direction of the meridional loop (U₂ > 0 or U₂ < 0), it can transport angular momentum inward toward the stellar core even when the core already rotates faster than the envelope, maintaining differential rotation inside radiative zones.

1. Eddington, A. S. (1925). "Circulating currents in rotating stars." The Observatory, 48, 73–75. https://ui.adsabs.harvard.edu/abs/1925Obs....48...73E
2. Sweet, P. A. (1950). "The importance of rotation in stellar evolution." Monthly Notices of the Royal Astronomical Society, 110, 548–558. https://doi.org/10.1093/mnras/110.6.548
3. Mestel, L. (1953). "Rotation and stellar evolution." Monthly Notices of the Royal Astronomical Society, 113, 716–745. https://doi.org/10.1093/mnras/113.6.716
4. Maeder, A., & Zahn, J.-P. (1998). "Stellar evolution with rotation. III. Meridional circulation with μ-gradients and non-stationarity." Astronomy and Astrophysics, 334, 1000–1006. https://ui.adsabs.harvard.edu/abs/1998A%26A...334.1000M