Charney–Stern–Pedlosky Criterion

Charney–Stern–Pedlosky Criterion (also known as the Charney–Stern theorem or Charney–Stern–Pedlosky necessary condition for instability) is a fundamental stability theorem in atmospheric and oceanic dynamics derived by Jule G. Charney and Melvin E. Stern in 1962 and generalized to non-isothermal horizontal boundaries by Joseph Pedlosky in 1964. Extending Lord Rayleigh's 1880 inflection-point theorem for homogeneous parallel shear flows and H. L. Kuo's 1949 barotropic β-plane stability theorem to three-dimensional rotating, stratified fluids, the theorem establishes a necessary integral condition for an inviscid, adiabatic quasi-geostrophic zonal jet U(y, z) to support exponentially growing normal-mode disturbances (combined barotropic and baroclinic instability).

Specifically, for a zonal flow U(y, z) confined laterally by walls or decay conditions at y = y₁, y₂ and vertically by rigid horizontal boundaries at z = 0 and z = H, the imaginary part of the complex phase speed c_i of any normal-mode wave exp[ik(x − ct)] can be non-zero (c_i > 0, instability) only if the integral identity:

∫₀^H ∫_y1^y₂ ρ₀(z) (∂q̄ / ∂y) [ |Φ|² / |U − c|² ] dy dz + ∫_y1^y₂ [ (f₀² / N²) ρ₀(z) (∂U / ∂z) ( |Φ|² / |U − c|² ) ]_z=0^z=H dy = 0

is satisfied, where ∂q̄/∂y is the basic-state meridional gradient of quasi-geostrophic potential vorticity and ∂U/∂z is proportional by thermal wind balance to the negative meridional buoyancy/temperature gradient (−∂B̄/∂y / f₀) at the lower (z = 0) and upper (z = H) boundaries.

Linearized Quasi-Geostrophic Potential Vorticity Equation

Consider a Boussinesq or anelastic quasi-geostrophic fluid on a β-plane (f = f₀ + βy) with reference density profile ρ₀(z) and buoyancy frequency N(z), containing a steady zonal basic state U(y, z) = −∂Ψ̄/∂y. The basic-state meridional gradient of quasi-geostrophic potential vorticity q̄ is:

∂q̄ / ∂y = β − ∂²U / ∂y² − (1 / ρ₀) ∂ / ∂z [ (ρ₀ f₀² / N²) (∂U / ∂z) ]

Here, the three terms represent the planetary vorticity gradient (β), the curvature of the horizontal shear (−∂²U/∂y², governing barotropic instability), and the vertical derivative of stretching vorticity (governing baroclinic instability).

Introducing a normal-mode perturbation streamfunction ψ′(x, y, z, t) = Re{ Φ(y, z) exp[ik(x − ct)] } with complex phase speed c = c_r + ic_i (k > 0) into the linearized conservation of potential vorticity (∂/∂t + U ∂/∂x)q′ + v′(∂q̄/∂y) = 0 yields the elliptic eigenvalue problem inside 0 < z < H, y₁ < y < y₂:

(U − c) [ ∂²Φ / ∂y² − k²Φ + (1 / ρ₀) ∂ / ∂z ( [ρ₀ f₀² / N²] ∂Φ / ∂z ) ] + (∂q̄ / ∂y) Φ = 0

subject to lateralimpermeability Φ(y₁, z) = Φ(y₂, z) = 0 and rigid-lid thermodynamic boundary conditions (w = 0) at z = 0 and z = H:

(U − c) (∂Φ / ∂z) − (∂U / ∂z) Φ = 0 at z = 0 and z = H

Multiplication by Φ* / (U − c) and Integration by Parts

Assuming an unstable mode exists (c_i > 0, so the critical-layer denominator U(y, z) − c never vanishes anywhere in the domain), divide the interior equation by (U − c), multiply by ρ₀(z) times the complex conjugate Φ*(y, z), integrate over the cross-sectional area (y, z) ∈ [y₁, y₂] × [0, H], and integrate the second-derivative terms by parts using the boundary conditions at z = 0 and z = H:

∫₀^H ∫_y1^y₂ ρ₀ [ |∂Φ / ∂y|² + k²|Φ|² + (f₀² / N²)|∂Φ / ∂z|² ] dy dz = ∫₀^H ∫_y1^y₂ ρ₀ (∂q̄ / ∂y) [ Φ |Φ|² / ( Φ [U − c] ) ] dy dz + [ ∫_y1^y₂ (ρ₀ f₀² / N²) (∂U / ∂z) ( |Φ|² / [U − c] ) dy ]_z=0^z=H

Because the left-hand side is a strictly real, positive-definite quadratic integral, taking the imaginary part of both sides and using Im{ 1 / (U − c) } = c_i / |U − c|² yields the fundamental Charney–Stern–Pedlosky identity:

c_i · { ∫₀^H ∫_y1^y₂ ρ₀ (∂q̄ / ∂y) [ |Φ|² / |U − c|² ] dy dz + ∫_y1^y₂ [ (ρ₀ f₀² / N²) (∂U / ∂z) ( |Φ|² / |U − c|² ) ]_z=H dy − ∫_y1^y₂ [ (ρ₀ f₀² / N²) (∂U / ∂z) ( |Φ|² / |U − c|² ) ]_z=0 dy } = 0

Bretherton's Generalized Potential Vorticity Sheet Interpretation

For an unstable wave (c_i ≠ 0) to exist, the curly-bracketed sum of positive-weighted integrals (+ |Φ|² / |U − c|² > 0) must sum to zero. As Francis P. Bretherton showed in 1966, the upper and lower boundary terms are mathematically equivalent to incorporating Dirac delta-function sheets of potential vorticity into a generalized potential vorticity gradient ∂q̄_gen/∂y at z = 0 and z = H:

∂q̄_gen / ∂y = ∂q̄ / ∂y + (f₀² / N²)(∂U / ∂z) δ(z − H⁻) − (f₀² / N²)(∂U / ∂z) δ(z − 0⁺)

Therefore, normal-mode instability requires ∂q̄_gen/∂y to take both positive and negative values across the domain [0, H] × [y₁, y₂], which can occur through four distinct physical configurations:

Reduction to Rayleigh's Inflection-Point and Kuo's Barotropic Theorems

In the special case of a purely horizontal, barotropic shear flow U = U(y) independent of height z (so ∂U/∂z = 0 at both boundaries and in the interior), the boundary integrals vanish and the interior potential vorticity gradient reduces to ∂q̄/∂y = β − d²U/dy²:

• Rayleigh's Inflection-Point Theorem (1880): On a non-rotating plane (β = 0), instability requires d²U/dy² = 0 somewhere in the channel (y₁, y₂).
• Kuo's Theorem (1949): On a β-plane (β > 0), barotropic instability requires β − d²U/dy² = 0 somewhere in the channel. Because β > 0 is positive everywhere, an eastward jet (U > 0) with a sharp core has −d²U/dy² > 0 at its peak (reinforcing β and stabilizing the jet core, while β − d²U/dy² can turn negative only in the flanks if d²U/dy² > β), whereas a westward jet (U < 0) has −d²U/dy² < 0 right at its core, making westward jets far more susceptible to barotropic instability than eastward jets (as seen in the Rhines scale PV staircase).

Wave-Activity Conservation and Pseudomomentum Interpretation

From the perspective of Eliassen–Palm wave-activity conservation, the integrand ρ₀(∂q̄/∂y) |Φ|² / |U − c|² is proportional to the negative pseudomomentum density A = −ρ₀ ⟨q′²⟩ / (2 ∂q̄/∂y) = −(ρ₀/2) (∂q̄/∂y) ⟨η²⟩ of a Rossby wave with meridional parcel displacement η = Φ / (U − c). Because total domain-integrated pseudomomentum must remain constant (zero for a mode growing exponentially from infinitesimal amplitude without external forcing), an unstable normal mode must consist of at least two coupled counter-propagating Rossby waves carrying opposite signs of pseudomomentum—which is possible only when ∂q̄_gen/∂y takes both positive and negative signs across the fluid domain.

Is the Charney–Stern–Pedlosky criterion a sufficient condition for instability or only a necessary condition?

It is a necessary condition, not a sufficient condition. Even if ∂q̄_gen/∂y changes sign across the domain (for instance, in the Eady model, where the top and bottom boundary terms have opposite signs for every positive shear Λ > 0), individual wavelengths can still be stable—such as short waves above the Eady cutoff μ > 2.3994—if the vertical or horizontal separation between the opposite-signed PV regions exceeds the Rossby deformation radius so the two counter-propagating waves cannot phase-lock.

Why did Charney and Stern's original 1962 theorem appear to exclude the Eady and Charney models?

Charney and Stern (1962) derived their theorem for an internal jet in which potential temperature (and thus ∂U/∂z) is constant along the horizontal boundaries (z = 0 and z = H), causing the boundary integrals to vanish and requiring ∂q̄/∂y to change sign inside the fluid interior. In 1964, Joseph Pedlosky restored the boundary integrals at z = 0 and z = H, proving that surface temperature gradients ∂θ̄/∂y ∝ −∂U/∂z act just like delta-function sheets of interior potential vorticity gradient.

Why are midlatitude easterly (westward) surface winds stable when interior ∂q̄/∂y is positive?

In the midlatitude troposphere with no rigid lid at the top (z → ∞), the interior potential vorticity gradient ∂q̄/∂y ≈ β > 0 is positive. At the Earth's surface (z = 0), the boundary contribution to the Charney–Stern–Pedlosky integral has the sign of −∂U/∂z(0). For normal westerly vertical shear (∂U/∂z(0) > 0, cold pole and warm equator), the surface term is negative and opposes the positive interior ∂q̄/∂y, allowing Charney baroclinic instability. For easterly vertical shear (∂U/∂z(0) < 0) with positive interior ∂q̄/∂y > 0, both the interior and surface terms are positive, so their sum cannot vanish and normal-mode instability is impossible unless ∂q̄/∂y reverses sign aloft (as occurs in the African Easterly Jet over the Sahara–Sahel gradient).

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<0159:OTSOIB>2.0.CO;2)
2. Pedlosky, J. (1964). "The stability of currents in the atmosphere and the ocean: Part I." Journal of the Atmospheric Sciences, 21(2), 201–219. https://doi.org/10.1175/1520-0469(1964)021<0201:TSOCIT>2.0.CO;2021<0201:TSOCIT>2.0.CO;2)
3. Bretherton, F. P. (1966). "Critical layer instability in baroclinic flows." Quarterly Journal of the Royal Meteorological Society, 92(393), 325–334. https://doi.org/10.1002/qj.49709239302
4. Kuo, H.-L. (1949). "Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere." Journal of Meteorology, 6(2), 105–122. https://doi.org/10.1175/1520-0469(1949)006<0105:DIOTDN>2.0.CO;2006<0105:DIOTDN>2.0.CO;2)
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