Charney–DeVore Blocking Model

Charney–DeVore Blocking Model—formulated in 1979 by the American meteorologists Jule G. Charney and John G. DeVore at the Massachusetts Institute of Technology—is the landmark nonlinear dynamical theory explaining why the extratropical circulation of the atmosphere of Earth can reside in two distinct, self-sustaining weather regimes under identical external solar and thermal forcing: a fast, weakly meandering high-index (zonal) jet state and a slow, high-amplitude low-index (meridional blocking) state characterized by persistent stationary anticyclones that deflect migratory storm tracks for weeks at a time. By applying a spectral Galerkin truncation to the quasi-geostrophic barotropic potential vorticity equation in a mid-latitude β-plane channel with sinusoidal bottom topography, Charney and DeVore demonstrated that the nonlinear topographic form-drag torque exerted by stationary Rossby wave disturbances back onto the zonal-mean wind yields a cubic steady-state equilibrium equation. When the external momentum forcing exceeds a bifurcation threshold, this cubic balance admits three stationary solutions: two stable equilibria (the super-resonant zonal state and the near-resonant blocked state) separated by an intermediate unstable saddle state susceptible to orographic instability.

High-Index Zonal Flow Versus Low-Index Blocking Anticyclones

Since the pioneering synoptic analyses of Carl-Gustaf Rossby, Hurd Willett, and Daniel Rex between 1939 and 1950, mid-latitude dynamicists had observed that the Northern Hemisphere westerly jet stream alternates irregularly between two recurring planetary-scale circulation patterns:

1. High-Index (Zonal) Circulation: Strong, predominantly west-to-east (zonal) winds blow across the mid-latitude oceans and continents with relatively small stationary wave amplitudes, steering transient baroclinic cyclones rapidly eastward across the storm tracks.
2. Low-Index (Blocked) Circulation: The westerly jet splits or buckles into a high-amplitude quasi-stationary planetary ridge—typically anchored over the northeastern Atlantic/Scandinavia or the northeastern Pacific/Alaska—often paired with an equatorward cut-off cyclonic low (a Rex block or Omega block). While transient baroclinic eddies have synoptic lifetimes of only 3 to 5 days, blocking episodes persist for 10 to 30 days, causing prolonged winter cold waves or summer heatwaves and droughts before transitioning back to zonal flow.

Linear stationary wave theory (formulated by Charney and Arnt Eliassen in 1949) showed that flow of uniform speed U over planetary mountain ranges (such as the Rocky Mountains and the Tibetan Plateau) excites stationary Rossby wave trains whose amplitude becomes resonant when U matches the stationary wave phase speed. However, linear models treated the zonal-mean wind U as an externally prescribed constant, leaving open why the atmosphere locks into resonant blocking episodes for weeks and why transitions between zonal and blocked states occur abruptly without changes in solar insolation.

Quasi-Geostrophic Barotropic Potential Vorticity Equation over Topography

Charney and DeVore (1979) examined a homogeneous, incompressible fluid of mean depth H in a mid-latitude β-plane channel of meridional width πL bounded by rigid walls at y = 0 and y = πL, rotating with Coriolis frequency f = f₀ + βy at small Rossby number. The flow is governed by the quasi-geostrophic barotropic potential vorticity equation with bottom topography h(x, y), surface Ekman pumping friction, and Newtonian zonal momentum driving toward an externally forced radiative-thermal wind profile ψ*(y):

∂(∇²ψ)/∂t + J(ψ, ∇²ψ + βy + f₀h/H) = −k_E∇²(ψ − ψ*)

where ψ(x, y, t) is the geostrophic streamfunction (with zonal velocity u = −∂ψ/∂y and meridional velocity v = ∂ψ/∂x), J(A, B) = (∂A/∂x)(∂B/∂y) − (∂A/∂y)(∂B/∂x) is the horizontal Jacobian advection operator, f₀h(x, y)/H represents vortex-tube stretching and compression over the mountain profile h(x, y), and k_E = f₀d_E/(2H) is the spin-down rate due to the turbulent Ekman boundary layer of thickness d_E (typically giving a damping timescale k_E⁻¹ ≈ 5 to 10 days). When the planetary Rossby deformation radius is finite, an additional −ψ/L_D² term appears in the vorticity operator, shifting the effective total wavenumber without altering the mathematical structure.

Spectral Galerkin Truncation in the β-Plane Channel

To capture the nonlinear feedback between the zonal-mean flow and the topographically forced stationary wave in analytical form, Charney and DeVore expanded the streamfunction ψ(x, y, t) and the bottom topography h(x, y) in a truncated set of orthonormal eigenfunctions of the horizontal Laplacian ∇²F_i = −a_i²F_i satisfying no-normal-flow boundary conditions (∂ψ/∂x = 0) at the channel walls y = 0, πL:

• Zonal-Mean Flow Mode: F_A(y) = 2^1/2cos(y/L), representing a broad mid-channel westerly jet with zonal velocity u_A(y, t) = U(t)2^1/2sin(y/L).
• Topographically Forced Wave Modes: F_K(x, y) = 2cos(nx/L)sin(y/L) and F_L(x, y) = 2sin(nx/L)sin(y/L), representing a planetary wave of zonal wavenumber k = n/L and meridional wavenumber l = 1/L (with total squared wavenumber K² = k² + l² = (n² + 1)/L²).

When the bottom mountain profile lies in the cosine wave phase, h(x, y) = h₀cos(nx/L)sin(y/L), projecting the potential vorticity equation onto the three spectral amplitudes—the zonal-mean flow speed U(t), the in-phase (ridge-over-mountain) wave amplitude X(t), and the out-of-phase (trough-upslope) wave amplitude Y(t)—yields a closed three-variable autonomous system of ordinary differential equations:

1. In-Phase Wave Amplitude (X): dX/dt = −k(U − U_R)Y − k_EX
2. Out-of-Phase Wave Amplitude (Y): dY/dt = k(U − U_R)X − k_EY + αh₀U
3. Zonal-Mean Wind Speed (U): dU/dt = −k_E(U − U*) − γh₀Y

where U* is the externally forced radiative-equilibrium zonal wind speed that would prevail in the absence of mountains (h₀ = 0), α and γ are positive geometric coupling constants arising from the Jacobian projection integrals, and U_R is the linear stationary Rossby wave resonance speed:

U_R = β/(k² + l²)

At U = U_R, the intrinsic westward phase propagation of a free Rossby wave (c_x = U − β/(k² + l²)) cancels the eastward advection by the zonal mean flow U, rendering the free wave stationary with respect to the underlying mountain range. In this three-mode truncation, the nonlinear wave–wave self-advection Jacobian J(ψ_wave, ∇²ψ_wave) vanishes identically because all wave modes share the same total wavenumber K²; the sole nonlinearity is the wave–mean flow interaction coupling U with X and Y.

Stationary Wave Response for a Given Zonal Wind U

Setting the time derivatives dX/dt = dY/dt = 0 in the two wave equations gives the steady-state stationary wave amplitudes (X_s, Y_s) as functions of the zonal-mean wind speed U:

X_s(U) = −(αh₀kU(U − U_R))/(k²(U − U_R)² + k_E²)

Y_s(U) = (αh₀k_EU)/(k²(U − U_R)² + k_E²)

Here, Y_s(U) is the component of the stationary pressure perturbation shifted by a quarter wavelength (90°) relative to the mountain crest h(x, y). Physically, when high geostrophic pressure sits on the windward (westward) slope of a mountain range and low pressure sits on the leeward (eastward) slope (Y_s > 0), the east–west pressure difference across the mountain range exerts a net westward topographic form drag −γh₀Y_s(U) that extracts eastward momentum from the atmosphere and transfers it to the solid Earth.

Cubic Steady-State Torque Balance and Resonance Fold Catastrophe

Substituting the out-of-phase wave amplitude Y_s(U) into the steady-state zonal momentum equation (dU/dt = 0) yields the fundamental Charney–DeVore steady-state balance:

k_E(U* − U) = D_form(U) = (αγh₀²k_EU)/(k²(U − U_R)² + k_E²)

Multiplying both sides by the denominator (k²(U − U_R)² + k_E²) and dividing by k_E shows that the equilibrium zonal wind speed U satisfies a cubic polynomial equation:

(U − U*) [k²(U − U_R)² + k_E²] + αγh₀²U = 0

The physical origin of multiple equilibria can be visualized by plotting the two sides of the torque balance as functions of U:

• The left-hand side, F_drive(U) = k_E(U* − U), is a straight line with negative slope −k_E that crosses zero at the unperturbed thermal wind U = U*.
• The right-hand side, D_form(U), is a bell-shaped Lorentzian resonance curve centered at the stationary Rossby resonance speed U = U_R, with peak topographic drag proportional to h₀²/k_E and half-width ΔU = k_E/k.

When the mountain height h₀ is sufficiently large and Ekman damping k_E is weak, the straight driving line k_E(U* − U) for a super-resonant forcing U* > U_R intersects the Lorentzian drag curve D_form(U) at three distinct equilibrium points (U₁ < U₂ < U₃):

Orographic Instability of the Intermediate Equilibrium

Linearizing the three-mode dynamical system around each stationary point reveals why the intermediate root U₂ is dynamically unstable while U₁ and U₃ are stable. On the super-resonant falling flank of the Lorentzian resonance curve (U > U_R), increasing the zonal wind U moves the flow further away from stationary Rossby resonance (U − U_R increases), which reduces the topographic form drag (∂D_form/∂U < 0).

At the intermediate root U₂, this negative drag-slope exceeds the stabilizing Ekman damping slope (−∂D_form/∂U > k_E). Consequently, a small positive perturbation δU > 0 weakens the mountain wave drag more than Ekman friction increases, causing the flow to accelerate into the high-index zonal state (U₃). Conversely, a small negative perturbation δU < 0 brings the zonal wind closer to resonance U_R, amplifying the stationary wave and its form drag until the flow locks onto the low-index blocking state (U₁). This positive feedback between stationary wave amplitude and zonal deceleration is termed Charney–DeVore orographic instability.

Six-Mode Channel Model and Wave–Wave Triad Interactions

In the second half of their 1979 paper, Charney and DeVore expanded their spectral truncation from 3 modes to 6 modes by adding a second meridional gravest mode (y-wavenumber 2/L, including zonal shear mode F_C(y) = 2^1/2cos(2y/L) and corresponding wave modes F_M, F_N). In the 6-mode system, nonlinear wave–wave triad interactions J(ψ_wave, ∇²ψ_wave) no longer vanish, allowing the primary topographically forced wave to exchange energy with free Rossby modes through topographic barotropic instability. Even in the presence of these wave–wave interactions, the bistability between a high-index zonal regime and a low-index topographically locked blocking regime persists.

Charney–Straus (1980) Baroclinic Extension and Eddy-Driven Transitions

Because the real mid-latitude atmosphere is driven by meridional solar heating gradients rather than direct mechanical momentum forcing U*, Jule Charney and David M. Straus extended the multiple-equilibria model in 1980 to a two-layer baroclinic quasi-geostrophic atmosphere on a β-plane (combining the Eady model and Phillips two-layer baroclinic instability with sinusoidal bottom topography). Charney and Straus demonstrated that multiple stationary and wavy equilibria arise when the meridional radiative temperature gradient exceeds the baroclinic instability threshold, with baroclinic conversion energizing the resonant blocking ridge.

Subsequent theoretical and observational studies in the 1980s and 1990s (by Klaus Fraedrich, Roberto Benzi, Rein Haarsma, and Brian Hoskins) refined the original Charney–DeVore framework in two major directions:

1. Hemispheric vs. Regional Localization of Blocks: Because the 3-mode Charney–DeVore model used a single global sinusoidal wavenumber, its blocked state U₁ represented a hemisphere-wide standing wave amplification. Pierre Malguzzi and Paola Malanotte-Rizzoli (1984) and Judith Berner and Tim Palmer showed that when localized topography and transient synoptic eddies are included, the blocking ridge becomes regionally localized over the Euro-Atlantic or Pacific sectors while the rest of the hemisphere remains zonal.
2. Stochastic Noise and Transient Eddy Vorticity Forcing: Using the Kramers escape-rate theory of stochastic differential equations, Roberto Benzi, Alfonso Sutera, and Angelo Vulpiani showed that high-frequency synoptic cyclone eddies act as both an upscaling potential vorticity pump (which helps maintain the low-potential-vorticity anticyclonic anomaly inside the block against dissipation) and a stochastic noise forcing that triggers rapid regime transitions across the unstable saddle barrier U₂ between the high-index and blocked attractors.

What is the physical difference between the high-index and low-index equilibria in the Charney–DeVore model?

In the high-index (zonal) equilibrium (U₃), the westerly zonal wind blows much faster than the stationary Rossby wave resonance speed (U₃ ≫ U_R), so the flow passes over planetary mountain ranges out of resonance, generating only small-amplitude stationary waves and weak mountain form drag. In the low-index (blocking) equilibrium (U₁ ≈ U_R), the zonal wind is decelerated close to the resonance speed U_R = β/(k² + l²), exciting a high-amplitude stationary ridge whose large westward topographic form drag holds the zonal jet in its slow, near-resonant state.

Why are there three mathematical equilibria when the equations only contain quadratic nonlinearities?

Although the quasi-geostrophic advection term J(ψ, ∇²ψ + f₀h/H) is quadratically nonlinear in the state variables (U, X, Y), eliminating the two linear wave amplitudes (X_s, Y_s) in terms of U introduces the resonant Lorentzian denominator k²(U − U_R)² + k_E². Multiplying the linear Ekman driving term k_E(U* − U) by this quadratic denominator produces a cubic polynomial in U, which has three real roots whenever the mountain height h₀ and external forcing U* exceed the fold-bifurcation threshold.

Why is the middle equilibrium state never observed as a persistent weather pattern?

The middle root (U₂) lies on the super-resonant side of the mountain drag peak where ∂D_form/∂U < −k_E. Any slight strengthening of the westerly wind moves the atmosphere further away from resonance and reduces the mountain drag, causing the wind to accelerate into the high-index zonal state; any slight weakening moves the flow closer to resonance and increases the mountain drag, collapsing the wind into the blocked state. Thus U₂ is an unstable saddle point in phase space that acts as a watershed boundary separating the basins of attraction of the two stable weather regimes.

1. Charney, J. G., & DeVore, J. G. (1979). "Multiple Flow Equilibria in the Atmosphere and Blocking036<1205:MFEITA>2.0.CO;2)." Journal of the Atmospheric Sciences, 36(7), 1205–1216.
2. Charney, J. G., & Straus, D. M. (1980). "Form-Drag Instability, Multiple Equilibria and Propagating Planetary Waves in Baroclinic, Orographically Forced, Planetary Wave Systems037<1157:FDIMEA>2.0.CO;2)." Journal of the Atmospheric Sciences, 37(6), 1157–1176.
3. Charney, J. G., & Eliassen, A. (1949). "A Numerical Method for Predicting the Perturbations of the Middle Latitude Westerlies." Tellus, 1(2), 38–54.
4. Rex, D. F. (1950). "Blocking Action in the Middle Troposphere and its Effect upon Regional Climate: I. An Aerological Study of Blocking Action." Tellus, 2(3), 196–211.
5. Benzi, R., Malguzzi, P., Speranza, A., & Sutera, A. (1986). "The Theory of Multiple Equilibria Revisited." Quarterly Journal of the Royal Meteorological Society, 112(473), 661–675.