Marshall–Radko Residual-Mean Theory of the Antarctic Circumpolar Current

Marshall–Radko Residual-Mean Theory of the Antarctic Circumpolar Current is an analytical dynamical and thermodynamic model formulated in 2003 by John Marshall and Timour Radko that explains how surface westerly winds, air–sea buoyancy fluxes, and mesoscale baroclinic eddies jointly determine the isopycnal stratification, zonal volume transport, and meridional overturning circulation (MOC) of the Southern Ocean. Across the circumpolar belt of Drake Passage (roughly 55°S to 62°S), the absence of continental boundaries prevents a zonally averaged east–west pressure gradient from developing above the depth of the highest submarine ridges (~1,500–2,000 m), prohibiting any net time-mean geostrophic meridional flow in the upper ocean. Consequently, the strong northward surface Ekman transport driven by Southern Hemisphere subpolar westerly winds cannot return southward geostrophically above the sill depth; instead, it steepens isopycnal surfaces until baroclinic instability generates an opposing southward eddy-induced overturning circulation. By solving the transformed Eulerian-mean (residual-mean) buoyancy equation along characteristic isopycnal coordinates, Marshall and Radko demonstrated that the residual overturning streamfunction Ψ_res is governed at the sea surface by air–sea buoyancy fluxes and drops to zero across the adiabatic interior of the upper thermocline, establishing the quantitative basis for understanding eddy saturation and eddy compensation in the global climate system.

Absence of Zonal Geostrophic Pressure Gradients in Drake Passage

In ocean basins bounded by eastern and western continents—such as the North Atlantic and North Pacific—zonal wind stress builds up a basin-wide east–west sea-surface tilt and subsurface pressure difference Δp_E−W that supports meridional geostrophic flow through the Coriolis effect: ρ₀ f v_g = ∂p/∂x. In the latitude band of Drake Passage between South America and the Antarctic Peninsula, however, the Southern Ocean forms a zonally periodic re-entrant channel uninterrupted by land above the crests of the Scotia Arc, Kerguelen Plateau, and Pacific–Antarctic Ridge.

Integrating the zonal geostrophic relation around a closed latitude circle above the submarine ridges yields zero net meridional geostrophic velocity:

⟨v_g⟩ = (1 / (ρ₀ f L_x)) ∮ (∂p / ∂x) dx = 0

where ⟨·⟩ denotes the zonal average over the circumpolar channel length L_x and f < 0 is the Southern Hemisphere Coriolis parameter. When subpolar westerly winds apply an eastward surface wind stress τ_x > 0, they drive an equatorward (northward) surface Ekman transport integrated over the surface Ekman layer of depth δ_Ek:

Ψ_mean(y) = − τ_x(y) / (ρ₀ f) > 0

where Ψ_mean is the Eulerian-mean overturning streamfunction (conventionally positive for a clockwise cell looking westward, corresponding to northward surface flow and deep southward return flow). Because this northward surface Ekman flux diverges south of the westerly wind maximum and converges to the north, it drives upwelling of deep water around Antarctica and Ekman downwelling in the subantarctic zone—a wind-driven Eulerian-mean vertical loop historically termed the Deacon Cell.

Cancellation Between Eulerian-Mean and Eddy-Induced Streamfunctions

If the Eulerian-mean Deacon Cell represented the true net lagrangian transport of water parcels, the roughly 25–35 Sverdrups (1 Sv = 10⁶ m³ s⁻¹) of northward surface Ekman flow would have to sink across steeply tilted density surfaces in the subantarctic zone, requiring unrealistically large diapycnal mixing rates in the stratified thermocline. Marshall and Radko (2003) resolved this paradox by adapting the transformed Eulerian-mean (TEM) formalism of Andrews and McIntyre (1976) and the Gent–McWilliams isopycnal eddy parameterization to the zonally averaged Southern Ocean.

In a two-dimensional meridional–vertical plane (y, z), the net residual overturning streamfunction Ψ_res(y, z)—which governs the time-mean trajectory of buoyancy and biogeochemical tracers—is the sum of the wind-driven Eulerian-mean streamfunction Ψ_mean and the mesoscale eddy-induced streamfunction Ψ^*:

Ψ_res(y, z) = Ψ_mean(y, z) + Ψ^*(y, z)

such that the meridional and vertical residual velocities are given by v_res = −∂Ψ_res/∂z and w_res = ∂Ψ_res/∂y. Below the surface Ekman layer and above the topographic form-drag layer at the seafloor, the Eulerian-mean streamfunction is vertically uniform and set directly by the surface wind stress: Ψ_mean = −τ_x / (ρ₀ f). Conversely, the eddy-induced streamfunction arises from the correlation between eddy velocity and buoyancy perturbations, which closes via the Gent–McWilliams thickness diffusivity κ_GM as proportional to the mean meridional isopycnal slope s_b:

Ψ^*(y, z) = (⟨v′ b′⟩) / (∂b / ∂z) = κ_GM s_b

where b = −g (ρ − ρ₀) / ρ₀ is mean buoyancy and s_b = −(∂b/∂y) / (∂b/∂z) > 0 is the positive northward-sloping isopycnal tilt across the Antarctic Circumpolar Current (since cold, dense polar waters lie to the south, ∂b/∂y > 0, and stable stratification requires ∂b/∂z > 0). Thus, in the Southern Ocean interior, the interior residual streamfunction takes the explicit diagnostic form:

Ψ_res(y, z) = − τ_x(y) / (ρ₀ f) + κ_GM s_b(y, z)

Under the standard Southern Hemisphere sign convention where f < 0, τ_x > 0, and isopycnals slope downward toward the north (s_b = −(∂b/∂y)/(∂b/∂z) < 0), the two terms have opposite signs:

Ψ_res(y, z) = |τ_x / (ρ₀ f)| − κ_GM |s_b(y, z)|

Northward wind-driven Ekman overturning steepens the isopycnals, whereas baroclinic mesoscale eddies slump the isopycnals via a counter-rotating anticlockwise bolus cell that opposes the Deacon Cell.

Adiabatic Interior Regime Versus Diabatic Surface Mixed Layer

In a steady state, the zonally averaged buoyancy budget in residual-mean form equates the advection of mean buoyancy by the residual circulation to the vertical divergence of diabatic buoyancy fluxes B(y, z) generated by air–sea heat and freshwater exchange within the surface mixed layer of depth h_m:

J(Ψ_res, b) = v_res (∂b / ∂y) + w_res (∂b / ∂z) = ∂B / ∂z

where J(Ψ_res, b) = (∂Ψ_res/∂y)(∂b/∂z) − (∂Ψ_res/∂z)(∂b/∂y) is the Jacobian operator in the (y, z) plane. Below the base of the diabatic surface mixed layer (z < −h_m), diapycnal mixing in the upper thermocline is weak (∂B/∂z ≈ 0), reducing the steady buoyancy equation to:

J(Ψ_res, b) = 0   ⇒   Ψ_res = Ψ_res(b)

This fundamental relationship states that throughout the adiabatic interior of the Southern Ocean, streamlines of the residual overturning circulation coincide with surfaces of constant buoyancy (isopycnals); the magnitude of Ψ_res is constant along each isopycnal characteristic curve z = z(y, b).

Setting Ψ_res from Surface Buoyancy Forcing B₀(y)

Integrating the buoyancy budget vertically across the diabatic surface mixed layer from z = −h_m to the sea surface z = 0 where the air–sea buoyancy flux is B₀(y) (positive for surface warming or freshening, negative for cooling or brine rejection during sea-ice formation), Marshall and Radko derived the surface boundary condition that links the residual overturning directly to air–sea buoyancy transformations:

Ψ_res(y, z = −h_m) = B₀(y) / (∂b_s / ∂y)

where b_s(y) is the meridional distribution of surface buoyancy across the Southern Ocean outcrop zone. Once the value of Ψ_res is set at the base of the mixed layer where an isopycnal of buoyancy b outcrops at latitude y_out(b), that value of Ψ_res(b) propagates unaltered down along the isopycnal into the ocean interior. Using the method of characteristics, the differential equation for the depth z(y, b) of each isopycnal characteristic follows directly from its local slope s_b = ∂z/∂y|_b:

(∂z / ∂y)_b = s_b = (1 / κ_GM) [ τ_x(y) / (ρ₀ f) + Ψ_res(b) ]

In the upper thermocline where isopycnals do not undergo net diabatic water-mass transformation (Ψ_res = 0), the Eulerian-mean Ekman cell and the eddy-induced bolus cell cancel exactly, yielding the classical Deacon-Cell cancellation slope:

s_b = − τ_x / (ρ₀ f κ_GM)

Combining this isopycnal slope with thermal-wind balance (f ∂u/∂z = −∂b/∂y = s_b ∂b/∂z) allows direct analytical integration for the vertical shear and baroclinic zonal transport of the Antarctic Circumpolar Current, connecting Southern Ocean stratification seamlessly to the pycnocline depth of the Atlantic Meridional Overturning Circulation and the global thermohaline circulation.

Over the late twentieth and early twenty-first centuries, anthropogenic stratospheric ozone depletion and rising greenhouse gas concentrations shifted the Southern Annular Mode toward its positive phase, strengthening and contracting the Southern Hemisphere westerly winds. The Marshall–Radko residual-mean framework clarifies two distinct nonlinear dynamical responses of the Southern Ocean when eddy diffusivity κ_GM is allowed to vary with baroclinic eddy kinetic energy rather than remaining constant:

1. Eddy Saturation (Insensitivity of Isopycnal Slope and ACC Zonal Transport to Wind Stress):

In the adiabatic balance s_b ≈ |τ_x / (ρ₀ f)| / κ_GM, a constant κ_GM would imply that a 20% increase in westerly wind stress τ_x steepens the isopycnal slope s_b by 20% and increases the baroclinic volume transport of the ACC proportionally. However, high-resolution eddy-resolving models and theories of two-layer baroclinic turbulence show that strengthened wind forcing is channeled rapidly into mesoscale eddy kinetic energy, causing the effective eddy diffusivity κ_GM to scale nearly linearly with wind stress (κ_GM ∝ |τ_x|). As a result, the ratio |τ_x| / κ_GM—and therefore the mean isopycnal slope s_b and total baroclinic ACC volume transport (~130–170 Sv through Drake Passage)—remains remarkably invariant to multi-decadal increases in wind forcing, a regime termed eddy saturation (Straub, 1993; Hallberg and Gnanadesikan, 2006).

1. Eddy Compensation (Partial Buffering of the Residual Overturning Ψ_res):

As westerly winds intensify, the northward Eulerian-mean Ekman transport Ψ_mean = |τ_x / (ρ₀ f)| increases linearly with τ_x. Because enhanced baroclinic instability simultaneously amplifies the opposing southward eddy-induced overturning Ψ = κ_GM |s_b|, the net change in the residual overturning circulation ΔΨ_res = ΔΨ_mean − Δ|Ψ| is substantially smaller than the change in the Ekman transport alone. This partial cancellation of wind-driven overturning anomalies by an intensified eddy bolus circulation is termed eddy compensation and regulates the rate at which carbon-rich Circumpolar Deep Water is ventilated to the atmosphere.

1. Marshall, J., & Radko, T. (2003). Residual-mean solutions for the Antarctic Circumpolar Current and its associated overturning circulation. Journal of Physical Oceanography, 33(11), 2341–2354. https://doi.org/10.1175/1520-0485(2003)033<2341:RSFTAC>2.0.CO;2033<2341:RSFTAC>2.0.CO;2)
2. Marshall, J., & Radko, T. (2006). A model of the dynamical hysteresis of the Antarctic Circumpolar Current. Geophysical Research Letters, 33(17), L17603. https://doi.org/10.1029/2006GL026629
3. Karsten, R., Jones, H., & Marshall, J. (2002). The role of eddy transfer in setting the stratification and transport of a circumpolar current. Journal of Physical Oceanography, 32(1), 39–54. https://doi.org/10.1175/1520-0485(2002)032<0039:TROETI>2.0.CO;2032<0039:TROETI>2.0.CO;2)
4. Hallberg, R., & Gnanadesikan, A. (2006). The role of eddies in determining the structure and response of the wind-driven Southern Hemisphere overturning: Results from the Modeling Eddies in the Southern Ocean (MESO) project. Journal of Physical Oceanography, 36(12), 2232–2252. https://doi.org/10.1175/JPO2980.1
5. Marshall, J., & Speer, K. (2012). Closure of the meridional overturning circulation through Southern Ocean upwelling. Nature Geoscience, 5(3), 171–180. https://doi.org/10.1038/ngeo1391