Charney–Stern Instability Criterion

The Charney–Stern instability criterion is a necessary condition for the linear instability of a parallel zonal flow in quasigeostrophic (QG) theory, derived by Jule Charney and Melvin Stern in 1962. It states that a basic-state zonal flow can support growing disturbances only if the meridional gradient of quasigeostrophic potential vorticity (PV) changes sign somewhere in the domain, or if the interior PV gradient has the opposite sign to the temperature gradient on a horizontal boundary. The result unifies two older theorems for purely barotropic flow, the Rayleigh–Kuo and Fjørtoft theorems, and embeds the classical Eady and Charney models of baroclinic instability as special cases. Because it is a necessary but not sufficient condition, it is used primarily to prove stability: flows that fail the criterion cannot amplify small perturbations, whatever their strength.

Lord Rayleigh showed in 1880 that an inviscid, incompressible, two-dimensional shear flow can be unstable only if its vorticity profile has an inflection point, meaning that the basic-state absolute vorticity gradient vanishes somewhere. Hsiao-Lan Kuo adapted this argument in 1949 to the barotropic vorticity equation on a rotating sphere, giving a criterion that includes the planetary vorticity gradient β (see Kuo barotropic instability theorem). Rolf Fjørtoft in 1950 added a sharper necessary condition involving the product of the vorticity gradient and the velocity difference from the inflection-point velocity.

Meanwhile, Jule Charney (1947) and Eric Eady (1949) explained mid-latitude cyclones as baroclinic instability of westerly flow with vertical shear, a different mechanism relying on release of available potential energy (see Eady model). The two lines of theory were unified in the 1962 paper by Charney and Stern, "On the stability of internal baroclinic jets in a rotating atmosphere," published in the Journal of the Atmospheric Sciences. They showed that a single integral argument based on conservation of wave activity captures both barotropic and baroclinic cases.

Consider a zonal flow U(y, z) in a stratified, rotating fluid on a β-plane with the QG approximation. The basic-state QG potential vorticity gradient is

∂q/∂y = β − U_yy − (f₀²/ρ₀) ∂/∂z (ρ₀ U_z / N²) ,

where f₀ is the Coriolis parameter, N the buoyancy frequency, and ρ₀ a reference density profile. On rigid horizontal boundaries at z = 0 and z = H, the boundary condition links vertical velocity to the advection of potential temperature, so the boundary acts like an additional PV sheet with effective gradient proportional to −U_z (the meridional temperature gradient through thermal wind balance).

Linearizing the QG PV equation about U and multiplying by the complex conjugate of the streamfunction perturbation yields, for a growing normal mode with complex phase speed c = c_r + i c_i and c_i > 0, an integral identity whose imaginary part reads

c_i ∫ (∂q/∂y) |ψ|² / |U − c|² dV + boundary terms = 0 .

For this to hold with c_i ≠ 0, the weighted integral cannot have a single sign. Thus a necessary condition for instability is one of:

1. ∂q/∂y takes both signs within the interior.
2. ∂q/∂y has the same sign throughout, but the boundary gradient has the opposite sign at the upper boundary, or at the lower boundary with the same sign convention reversed.
3. The boundary gradients at the two boundaries have opposing signs.

When the flow is barotropic (no vertical shear, N → ∞ in the stretching term), ∂q/∂y reduces to β − U_yy and the criterion becomes the Rayleigh–Kuo theorem: the absolute vorticity gradient must vanish somewhere. A refinement due to Fjørtoft states that an additional necessary condition is that U_yy(U − U_s) be positive somewhere, where U_s is the velocity at the point of vanishing gradient. The Charney–Stern criterion therefore generalizes the barotropic theorems to three-dimensional QG flow and supplies analogous Fjørtoft-type extensions that involve the wave activity budget.

Jets stable under the criterion include broad Gaussian profiles with small curvature at ocean basin scales, while sharp mid-latitude jets with narrow cores usually satisfy it through a sign change in the PV gradient on the flanks. The criterion is thus an efficient diagnostic for checking, from the mean zonal wind and stratification alone, whether an observed flow can develop eddies.

In the Eady model, interior PV is uniform (∂q/∂y = 0 because β = 0 and U_z is constant), so the interior gradient has no sign. Instability arises solely from the boundaries, where the temperature gradients have opposite signs on the lower and upper lids. The criterion is satisfied by the opposition of boundary gradients, matching Eady's result that Rossby edge waves at the two boundaries phase-lock and grow.

In the Charney model, β is nonzero and shear is uniform in a semi-infinite atmosphere. The interior PV gradient is positive (β − the stretching term involving U_z is negative for sufficiently strong shear), while the surface temperature gradient carries the opposite sign, again satisfying condition 2. The model reveals that the planetary gradient β is a stabilizing influence on the shortest waves and that long waves are suppressed by β as well, selecting the fastest-growing wavelength near the Rossby radius of deformation.

In both cases, the physical interpretation is a resonant interaction between two counter-propagating Rossby waves: one riding on the interior or upper PV gradient, another on the lower boundary, which phase-lock and extract energy from the mean shear.

An elegant derivation follows from the Eliassen–Palm flux relation. The pseudomomentum density of a small-amplitude perturbation is proportional to q'²/(∂q/∂y) in the interior and to θ'²/(−U_z) on boundaries. Conservation of the total wave activity then implies that growth of one component requires a negative-activity component elsewhere, which is possible only when the PV gradient (or boundary temperature gradient) changes sign. The viewpoint links the criterion to PV-based dynamical reasoning described in Hoskins–McIntyre–Robertson potential vorticity thinking and to the conservation laws in the vorticity equation.

The theorem is restricted to QG dynamics, linear perturbations, and inviscid flow; it does not guarantee instability when satisfied and cannot predict growth rates. Extensions include:

• Primitive-equation versions, such as the work of Pedlosky and Held, that restore ageostrophic effects.
• Generalization to spherical geometry and non-zonal flow through Andrews-style wave-activity conservation laws.
• Finite-amplitude versions based on conserved quantities, such as the Arnold theorems discussed in Arnold nonlinear stability theorems for geophysical flows.
• Applications to the Antarctic Circumpolar Current, where the sign of the PV gradient near fronts determines eddy generation, a topic connected to Held–Larichev baroclinic eddy diffusivity and Salmon's two-layer baroclinic turbulence.

The criterion remains a staple of dynamic meteorology and physical oceanography courses because it connects the shape of the mean flow to the existence of eddies without solving an eigenvalue problem.

1. Charney, J. G., & Stern, M. E. (1962). On the stability of internal baroclinic jets in a rotating atmosphere. Journal of the Atmospheric Sciences, 19(2), 159–172. https://doi.org/10.1175/1520-0469(1962)019<0159:OTSOIB>2.0.CO;2019%3C0159:OTSOIB%3E2.0.CO;2)
2. Charney, J. G. (1947). The dynamics of long waves in a baroclinic westerly current. Journal of Meteorology, 4(5), 135–162. https://doi.org/10.1175/1520-0469(1947)004<0136:TDOLWI>2.0.CO;2004%3C0136:TDOLWI%3E2.0.CO;2)
3. Eady, E. T. (1949). Long waves and cyclone waves. Tellus, 1(3), 33–52. https://doi.org/10.3402/tellusa.v1i3.8507
4. Vallis, G. K. (2017). Atmospheric and Oceanic Fluid Dynamics (2nd ed.). Cambridge University Press. https://doi.org/10.1017/9781107588417