Eliassen–Palm Flux

Eliassen–Palm Flux (abbreviated EP flux, denoted F = [F^(y), F^(z)] or [F_ϕ, F_z], and extended to three dimensions as the Plumb wave-activity flux by R. Alan Plumb in 1985) is a fundamental vector diagnostic in atmospheric and oceanic dynamics that unifies the meridional eddy transport of momentum and heat into a single measure of wave propagation and wave–mean-flow interaction. Originating in the 1961 work of the Norwegian meteorologists Arnt Eliassen and Enok Palm on energy and momentum fluxes in mountain gravity waves and stationary planetary waves, and cast into its modern Transformed Eulerian Mean (TEM) formulation by David G. Andrews and Michael E. McIntyre in 1976 (Journal of the Atmospheric Sciences, 33, 2031–2048), the Eliassen–Palm flux vector combines the poleward eddy momentum flux (−u′v′) in its horizontal component with the upward eddy buoyancy/form-drag flux (f₀v′θ′/θ̄_z) in its vertical component. Within quasi-geostrophic theory for the atmosphere of Earth, the EP flux satisfies three powerful dynamical properties simultaneously: (1) its direction in the latitude–height (y, z) plane is parallel to the meridional group velocity c_g of Rossby waves (F = c_gA, where A is wave-activity pseudomomentum density); (2) its divergence ∇ · F equals the northward eddy flux of quasi-geostrophic potential vorticity (ρ₀a cos ϕ v′q′) and acts as the sole net eddy forcing driving accelerations of the zonal-mean wind ∂ū/∂t and the diabatic Brewer–Dobson circulation (v̄^*, w̄^*); and (3) under the Generalized Eliassen–Palm Theorem (∂A/∂t + ∇ · F = D), steady, linear, conservative waves satisfy ∇ · F = 0 and induce zero net acceleration of the zonal-mean flow (Charney–Drazin Non-Acceleration Theorem, 1961).

Near-Cancellation in the Conventional Eulerian Zonal Mean

In classical meteorology (Lorenz, 1967), atmospheric variables are decomposed into a zonal average around a latitude circle, (¯) = (2π)⁻¹∫₀^2π(·)*dλ*, and an eddy departure (′) from that zonal average. In log-pressure vertical coordinates z = −H ln(p/p_s) with reference density profile ρ₀(z) = ρ_s exp(−z/H) and Coriolis frequency f₀ on a mid-latitude β-plane, the conventional Eulerian-mean quasi-geostrophic zonal momentum, thermodynamic (potential temperature θ), and continuity equations are:

1. Eulerian Zonal Momentum Equation: ∂ū/∂t − f₀v̄ = −∂(u′v′)/∂y + X̄
2. Eulerian Thermodynamic Equation: ∂θ̄/∂t + w̄(∂θ̄/∂z) = −∂(v′θ′)/∂y + Q̄
3. Eulerian Continuity Equation: ∂v̄/∂y + ρ₀⁻¹∂(ρ₀w̄)/∂z = 0

Although mathematically rigorous, the conventional Eulerian-mean decomposition suffers from a major physical obscurity in the extratropical troposphere and winter stratosphere: the eddy heat flux divergence −∂(v′θ′)/∂y and the adiabatic vertical motion cooling w̄(∂θ̄/∂z) are nearly equal in magnitude and opposite in sign, canceling each other to over 85% to 95%. Growing baroclinic Rossby waves in mid-latitudes transport warm air poleward (v′θ′ > 0), converging heat in high latitudes (−∂(v′θ′)/∂y > 0) and driving a thermally indirect Eulerian Ferrel cell with rising motion (w̄ > 0) at subpolar latitudes and sinking motion (w̄ < 0) in the subtropics. Because w̄(∂θ̄/∂z) nearly cancels −∂(v′θ′)/∂y, the Eulerian mean vertical velocity w̄ does not represent the net Lagrangian cross-isentropic mass transport of air parcels or chemical tracers such as ozone and water vapor.

The Residual Meridional Circulation (v̄^*, w̄^*)

To eliminate this artificial cancellation between eddy heat fluxes and the Eulerian Ferrel cell, David G. Andrews and Michael E. McIntyre (1976) introduced the Transformed Eulerian Mean (TEM) circulation by defining the residual meridional and vertical velocities (v̄^*, w̄^*):

v̄^* ≡ v̄ − ρ₀⁻¹∂[ρ₀v′θ′ / (∂θ̄/∂z)]/∂z, w̄^* ≡ w̄ + ∂[v′θ′ / (∂θ̄/∂z)]/∂y

By construction, the Stokes drift correction Ψ_eddy = ρ₀(v′θ′)/(∂θ̄/∂z) is non-divergent, so (v̄^*, w̄^*) satisfies the same mass continuity equation ∂v̄^*/∂y + ρ₀⁻¹∂(ρ₀w̄^*)/∂z = 0. When (v̄, w̄) are eliminated in favor of (v̄^*, w̄^*), the eddy heat flux divergence disappears from the thermodynamic equation, which simplifies to:

∂θ̄/∂t + w̄^*(∂θ̄/∂z) = Q̄

For steady-state conditions (∂θ̄/∂t ≈ 0), w̄^* ≈ Q̄/(∂θ̄/∂z) directly measures the diabatic (Lagrangian-mean) mass circulation—demonstrating a single thermally direct pole-to-pole Brewer–Dobson cell in the stratosphere (rising in the radiatively heated tropics Q̄ > 0 and sinking over the radiatively cooled winter pole Q̄ < 0).

The TEM Momentum Equation and EP Flux Components

Substituting v̄ = v̄^* + ρ₀⁻¹∂[ρ₀(v′θ′)/(θ̄_z)]/∂z into the Eulerian zonal momentum equation collects all eddy momentum and heat flux terms into the divergence of a single two-dimensional vector F = (F^(y), F^(z)) in the meridional (y, z) plane:

∂ū/∂t − f₀v̄^* = ρ₀⁻¹(∇ · F) + X̄

where the quasi-geostrophic Eliassen–Palm flux vector on the β-plane is defined as:

F = (F^(y), F^(z)) ≡ ρ₀(z) ( −u′v′ , f₀[v′θ′ / (∂θ̄/∂z)] )

Each component of F has a clear physical meaning:

1. Horizontal (Meridional) Component F^(y) = −ρ₀(u′v′): Proportional to the equatorward flux of eastward eddy momentum (the negative of the poleward Reynolds stress u′v′). When a Rossby wave propagates equatorward (F^(y) < 0), its troughs and ridges tilt southwest-to-northeast in the Northern Hemisphere, transporting eastward momentum poleward (u′v′ > 0) into the mid-latitude eddy-driven jet.
2. Vertical Component F^(z) = +ρ₀f₀(v′θ′)/(∂θ̄/∂z): Proportional to the poleward eddy heat flux v′θ′ divided by static stability ∂θ̄/∂z, which by hydrostatic balance equals the upward interfacial pressure form drag exerted by undulating corrugated isentropic surfaces on the overlying fluid layer. When a baroclinic or planetary Rossby wave propagates upward (F^(z) > 0), its troughs and ridges tilt westward with height, transporting sensible heat poleward (v′θ′ > 0).

Equivalence of EP Flux Divergence and Northward Potential Vorticity Flux

A fundamental mathematical identity links the divergence of the Eliassen–Palm flux to the linearized quasi-geostrophic potential vorticity perturbation q′ = ∂v′/∂x − ∂u′/∂y + ρ₀⁻¹∂[ρ₀f₀θ′/(∂θ̄/∂z)]/∂z. Multiplying q′ by ρ₀v′ and taking the zonal average yields the Taylor–Bretherton identity (G. I. Taylor, 1915; F. P. Bretherton, 1966):

ρ₀⁻¹(∇ · F) = ρ₀⁻¹(∂F^(y)/∂y + ∂F^(z)/∂z) = v′q′

Thus, the divergence of the Eliassen–Palm flux per unit mass equals the northward eddy flux of quasi-geostrophic potential vorticity (v′q′). Eddies exert a net torque on the zonal-mean wind ū if and only if they accomplish an irreversible meridional transport of potential vorticity across latitude circles.

Wave-Activity Pseudomomentum Conservation Law

Consider the linearized quasi-geostrophic potential vorticity equation for waves on a zonal basic state U(y, z) with basic-state potential vorticity gradient ∂q̄/∂y = β − U_yy − ρ₀⁻¹∂(ρ₀f₀²N⁻²U_z)/∂z:

(∂/∂t + U∂/∂x)q′ + v′(∂q̄/∂y) = S′

where S′ represents non-conservative diabatic heating and frictional dissipation perturbations. Defining the meridional particle displacement η′ such that (∂/∂t + U∂/∂x)η′ = v′ (so q′ = −η′[∂q̄/∂y] for conservative waves), multiplying the linearized PV equation by ρ₀q′/(∂q̄/∂y), and taking the zonal average yields the Generalized Eliassen–Palm Wave-Activity Conservation Relation (Andrews & McIntyre, 1976):

∂A/∂t + ∇ · F = D

where A ≡ ½ρ₀[q′²/(∂q̄/∂y)] = ½ρ₀(∂q̄/∂y)η′² is the Wave-Activity Density (Pseudomomentum / Enstrophy Density)—which is positive-definite whenever ∂q̄/∂y > 0 (violated only when the flow satisfies the Charney–Stern–Pedlosky Criterion or Rayleigh–Kuo Inflection Point Theorem for normal-mode instability)—and D = ρ₀(q′S′)/(∂q̄/∂y) is the non-conservative wave damping/source term. For a slowly varying WKB wave packet with group velocity c_g = (∂ω/∂l, ∂ω/∂m), the EP flux satisfies F = c_gA, so arrows of F plotted in a latitude–height (y, z) cross-section point along the rays of Rossby wave packet propagation.

The Charney–Drazin Non-Acceleration Theorem (1961)

Substituting ∇ · F = −∂A/∂t + D into the TEM zonal momentum equation gives ∂ū/∂t − f₀v̄^* = ρ₀⁻¹(−∂A/∂t + D) + X̄. If a wave field is (1) steady (∂A/∂t = 0), (2) linear (no wave-breaking cascades or singular zero-wind absorption at U(y, z) = c), and (3) conservative/adiabatic/inviscid (D = 0, Q̄ = 0, X̄ = 0), then:

∇ · F = ρ₀(v′q′) = 0 ⇒ ∂ū/∂t = 0, v̄^* = w̄^* = 0

This result is the Charney–Drazin Non-Acceleration Theorem (derived by Jule G. Charney and Philip G. Drazin in 1961): steady, linear, conservative waves pass through a stratified shear flow without exerting any net acceleration on the zonal-mean wind or driving any residual meridional circulation, even when their Eulerian eddy fluxes −∂(u′v′)/∂y and −∂(v′θ′)/∂y are large. Net wave driving of the mean flow occurs only via wave transience (∂A/∂t ≠ 0), radiative/frictional damping (D ≠ 0), or nonlinear wave breaking.

The Edmon, Hoskins, and McIntyre (1980) Tropospheric Storm-Track Diagram

In 1980, Harold J. Edmon Jr., Brian J. Hoskins, and Michael E. McIntyre (Journal of the Atmospheric Sciences, 37, 2600–2616) introduced "Eliassen–Palm cross-sections"—plotting arrows of F superimposed on contours of ∇ · F—to synthesize the life cycle of mid-latitude baroclinic cyclones:

1. Stage I (Baroclinic Growth at Low Levels): Near the lower boundary (z = 0 to 3 km) between 35° and 55° latitude, poleward eddy heat fluxes (v′θ′ > 0) generate an upward EP flux (F^(z) > 0) and low-level EP flux divergence (∇ · F** > 0)**, accelerating the surface westerlies against planetary boundary-layer Ekman pumping drag.
2. Stage II (Upward and Equatorward Refraction): As the baroclinic wave packets propagate upward into the upper troposphere (7 to 12 km / 400 to 200 hPa), the spherical refractive index curves their group-velocity rays equatorward (F^(y) < 0, corresponding to poleward momentum flux u′v′ > 0).
3. Stage III (Subtropical Barotropic Decay and Wave-Breaking Absorption): Approaching the subtropical zero-wind line where U(y, z) = c_r, the waves break nonlinearly, producing a broad zone of upper-tropospheric EP flux convergence (∇ · F** < 0)** that decelerates the upper flanks while concentrating westerly momentum into the barotropic eddy-driven jet core and homogenizing potential vorticity (Rhines–Young potential vorticity homogenization).

Planetary Wave Driving of the Brewer–Dobson Circulation and Sudden Stratospheric Warmings

As established by Jule G. Charney and Philip G. Drazin (1961), stationary planetary Rossby waves (c = 0, zonal wavenumbers k = 1 and k = 2 forced by continents and large mountain ranges such as the Himalayas and Rockies) can propagate vertically from the troposphere into the stratosphere (F^(z) > 0) only when the stratospheric zonal wind ū is westerly and weaker than the Charney–Drazin cutoff velocity:

0 < ū < U_c ≡ β / [ k² + l² + ( f₀² / [4 N² H²] ) ]

Because U_c decreases with total horizontal wavenumber k² + l², synoptic-scale weather waves (k ≥ 4) are trapped in the troposphere, whereas ultralong planetary waves (k = 1, 2) penetrate deep into the winter hemisphere westerly polar night vortex (while the summer hemisphere stratospheric easterlies ū < 0 block upward Rossby wave propagation).

As planetary waves ascend into the thinning air of the middle stratosphere (ρ₀(z) ∝ exp(−z/H)), their velocity amplitudes grow as ρ₀^−1/2 until the waves overturn and break irreversibly in the mid-latitude stratospheric "surf zone" (McIntyre & Palmer, 1983), producing persistent Eliassen–Palm flux convergence (∇ · F** < 0). Through the downward-control principle (Haynes et al., 1991), this steady westward wave torque −f₀v̄^* = ρ₀⁻¹∇ · F** < 0 acts as a mechanical "wave-driven pump" forcing poleward residual drift (v̄^* > 0 in the Northern Hemisphere) and polar downwelling (w̄^* < 0)—the stratospheric Brewer–Dobson circulation.

When an abnormally intense pulse of upward-propagating k = 1 or k = 2 planetary wave activity (F^(z) ≫ 0) enters the winter stratosphere and converges rapidly (∇ · F ≪ 0), it triggers a Major Sudden Stratospheric Warming (SSW), first explained dynamically by Taroh Matsuno (1971): the westward EP flux torque decelerates the polar night jet at rates exceeding 10 m·s⁻¹ per day, reversing the 60° N, 10-hPa (~30 km) zonal-mean wind from westerly (ū > +40 m·s⁻¹) to easterly (ū < 0). Once an easterly zero-wind level (ū = 0) forms, all subsequent upward-propagating stationary planetary waves are absorbed at the zero-wind layer, concentrating ∇ · F < 0 downward and driving rapid adiabatic descent (w̄^* < 0) over the pole that warms the polar stratosphere by 30 to 60 K within 5 to 7 days.

Plumb's (1985) Three-Dimensional Wave-Activity Flux

Because the classical Eliassen–Palm flux F = (F^(y), F^(z)) is defined via zonal averages around a full latitude circle, it cannot diagnose the longitudinal (x, y, z) source regions and downstream Rossby wave trains forced by localized mountains or El Niño tropical heating anomalies. In 1985 (Journal of the Atmospheric Sciences, 42, 217–229), R. Alan Plumb (and subsequently Takaya & Nakamura, 2001, for migratory eddies on a zonally varying basic state) generalized the EP flux to a three-dimensional conservation law ∂A_3D/∂t + ∇_3D · F_Plumb = D_3D, where the Plumb 3D wave-activity flux vector F_Plumb = (F^(x), F^(y), F^(z)) is parallel to the three-dimensional group velocity of stationary Rossby wave trains and reduces upon zonal averaging to the classical Eliassen–Palm flux (F_EP = ⟨F_Plumb⟩_x).

Why is the Transformed Eulerian Mean (TEM) framework preferred over the conventional Eulerian zonal mean?

In the conventional Eulerian zonal-mean equations, the mid-latitude eddy heat flux divergence (−∂(v′θ′)/∂y) and the adiabatic cooling of the thermally indirect Eulerian Ferrel cell (*w̄*∂θ̄/∂z) cancel each other by 85% to 95%, obscuring both the net wave forcing of the zonal wind and the true Lagrangian transport of air parcels. The Transformed Eulerian Mean (TEM) absorbs the eddy heat flux into a residual meridional circulation (v̄^*, w̄^*) (which directly approximates the diabatic Brewer–Dobson circulation) and collects all eddy forcing of the mean flow into the single divergence term ρ₀⁻¹∇ · F.

What do the arrows and divergence contours of an Eliassen–Palm (EP) flux cross-section show?

In a latitude–height (y, z) cross-section: (1) Arrows of the EP flux vector F point in the direction of wave-activity propagation (group velocity)—upward arrows (F^(z) > 0) indicate poleward eddy heat flux (v′θ′ > 0) and upward wave radiation, while equatorward arrows (F^(y) < 0) indicate poleward eddy momentum flux (u′v′ > 0); and (2) Contours of EP flux divergence (∇ · F = ρ₀v′q′) show where waves are generated (∇ · F > 0, accelerating eastward mean wind) vs. where waves break or dissipate (∇ · F < 0, decelerating eastward mean wind).

What is the Charney–Drazin Non-Acceleration Theorem?

Derived by Jule G. Charney and Philip G. Drazin in 1961 and generalized via the Eliassen–Palm theorem (∂A/∂t + ∇ · F = D), the Non-Acceleration Theorem states that if atmospheric waves are steady (∂A/∂t = 0), linear (no wave breaking or zero-wind absorption levels), and conservative (D = 0, no radiative damping or friction), then the divergence of the Eliassen–Palm flux vanishes identically (∇ · F = 0) and the waves exert zero net acceleration on the zonal-mean wind (∂ū/∂t = 0).

1. Eliassen, A., & Palm, E. (1961). "On the transfer of energy in stationary mountain waves." Geofysiske Publikasjoner, 22(3), 1–23. https://books.google.com/books?id=O0I_AAAAIAAJ
2. Charney, J. G., & Drazin, P. G. (1961). "Propagation of planetary-scale disturbances from the lower into the upper atmosphere." Journal of Geophysical Research, 66(1), 83–109. https://doi.org/10.1029/JZ066i001p00083
3. Andrews, D. G., & McIntyre, M. E. (1976). "Planetary waves in horizontal and vertical shear: The generalized Eliassen–Palm relation and the mean zonal acceleration." Journal of the Atmospheric Sciences, 33(11), 2031–2048. https://doi.org/10.1175/1520-0469(1976)033%3C2031:PWIHAV%3E2.0.CO;2033%3C2031:PWIHAV%3E2.0.CO;2)
4. Edmon, H. J., Jr., Hoskins, B. J., & McIntyre, M. E. (1980). "Eliassen–Palm cross sections for the troposphere." Journal of the Atmospheric Sciences, 37(12), 2600–2616. https://doi.org/10.1175/1520-0469(1980)037%3C2600:EPCSFT%3E2.0.CO;2037%3C2600:EPCSFT%3E2.0.CO;2)
5. Plumb, R. A. (1985). "On the three-dimensional propagation of stationary waves." Journal of the Atmospheric Sciences, 42(3), 217–229. https://doi.org/10.1175/1520-0469(1985)042%3C0217:OTTDPO%3E2.0.CO;2042%3C0217:OTTDPO%3E2.0.CO;2)