Eady Model

Eady Model is a linear, continuous-stratification mathematical model of baroclinic instability formulated by the British meteorologist Eric Thomas Eady in 1949 (alongside Jule G. Charney's 1947 β-plane model) that explains the spontaneous growth of midlatitude synoptic weather systems (extratropical cyclones and anticyclones) and oceanic mesoscale eddies from a vertically sheared geostrophic zonal flow. To isolate the pure thermodynamic mechanism by which available potential energy (APE) stored in horizontal temperature gradients is converted into eddy kinetic energy, Eady idealized the midlatitude troposphere (or ocean thermocline) as an inviscid, Boussinesq fluid on an f-plane (β = df/dy = 0) with constant Coriolis parameter f₀, constant Brunt–Väisälä frequency N, and uniform vertical shear U(z) = Λz confined between two rigid horizontal boundaries at the Earth's surface (z = 0) and the tropopause (z = H).

Because both β = 0 and the vertical curvature d²U/dz² = 0 vanish identically in the Eady setup, the interior gradient of quasi-geostrophic potential vorticity is zero throughout the fluid depth (∂q̄/∂y = 0). Instability arises purely through the mutual phase-locking and constructive interaction of two counter-propagating thermal Rossby edge waves trapped at the lower boundary (z = 0) and upper boundary (z = H), yielding an explicit analytical dispersion relation with a sharp short-wave cutoff at dimensionless wavenumber μ_c ≈ 2.3994 and a maximum growth rate σ_max ≈ 0.3098 f₀ Λ / N at the scale of the internal Rossby radius of deformation L_D = NH / f₀.

Thermal Wind Balance and Vanishing Interior PV Gradient

Consider a Boussinesq fluid of depth H on an f-plane (f = f₀ = constant) with constant static stability N² = (g/θ₀)(∂θ̄/∂z) > 0. The basic-state zonal flow U(z) increases linearly with altitude z ∈ [0, H] with constant vertical shear Λ > 0:

U(z) = Λ z , U(0) = 0 , U(H) = Λ H = U_T

By hydrostatic and geostrophic thermal wind balance, the constant vertical shear ∂U/∂z = Λ is supported by a uniform equator-to-pole horizontal buoyancy gradient ∂B̄/∂y = −f₀Λ (where buoyancy b = gθ′/θ₀ = f₀ ∂ψ/∂z), so warm air lies equatorward (y < 0) and cold air lies poleward (y > 0):

B̄(y, z) = N² z − f₀ Λ y

In the quasi-geostrophic approximation, the basic-state meridional potential vorticity gradient is:

∂q̄ / ∂y = β − ∂²U / ∂y² − (f₀² / N²) (∂²U / ∂z²) = 0

Consequently, linearized perturbations ψ′(x, y, z, t) satisfy zero interior perturbation potential vorticity (q′ = 0) throughout 0 < z < H:

q′ = ∂²ψ′ / ∂x² + ∂²ψ′ / ∂y² + (f₀² / N²) (∂²ψ′ / ∂z²) = 0

All dynamical time-dependence enters through the thermodynamic buoyancy equations at the two rigid horizontal boundaries (w = 0 at z = 0 and z = H):

(∂ / ∂t + U(z) ∂ / ∂x) (∂ψ′ / ∂z) − Λ (∂ψ′ / ∂x) = 0 at z = 0 and z = H

Comparison of the Eady, Charney, and Phillips Baroclinic Instability Models

Edge Waves and the Eady Dispersion Relation

Substituting a normal-mode plane wave ψ′(x, y, z, t) = Re{ Φ(z) exp[i(kx + ly − kc t)] } into the interior Laplace-like PV equation q′ = 0 gives the vertical structure equation:

d²Φ / dz² − (K² N² / f₀²) Φ = 0 ⇒ Φ(z) = A cosh(μz / H) + B sinh(μz / H)

where K = (k² + l²)^1/2 is the total horizontal wavenumber and μ is the dimensionless scaled wavenumber (measuring the ratio of the internal Rossby radius of deformation L_D = NH/f₀ to the horizontal wavelength):

μ = K L_D = (K N H) / f₀

Applying the linearized thermodynamic boundary conditions at z = 0 (where U = 0) and z = H (where U = ΛH) yields a 2 × 2 homogeneous linear algebraic system for the amplitudes (A, B), whose determinant vanishes when the complex phase speed c = c_r + ic_i satisfies the Eady quadratic dispersion relation:

c = (Λ H / 2) ± (Λ H / μ) [ (μ / 2 − coth[μ / 2]) (μ / 2 − tanh[μ / 2]) ]^1/2

Short-Wave Cutoff, Fastest-Growing Mode, and Energetics

Because tanh(μ/2) < μ/2 holds for all μ > 0, the second factor (μ/2 − tanh[μ/2]) under the square root is always positive. Thus, the sign of the discriminant is controlled by the first factor (μ/2 − coth[μ/2]):

1. Short-Wave Stability (μ ≥ μ_c ≈ 2.3994): At short wavelengths where μ/2 ≥ coth(μ/2) (occurring above the root of (μ_c/2) tanh(μ_c/2) = 1, i.e., μ_c ≈ 2.3994, or wavelengths λ < 2πL_D / 2.3994 ≈ 2.62 L_D), both factors under the radical are positive. The two roots c₁ and c₂ are purely real, representing two neutral, vertically trapped edge waves that decay exponentially away from z = 0 and z = H with trapping depth H/μ too shallow for the upper and lower waves to feel and lock onto one another across the depth H.
2. Baroclinic Instability Window (0 < μ < μ_c ≈ 2.3994): For longer wavelengths where 0 < μ < 2.3994, the vertical penetration scale H/μ spans the entire fluid depth H. The upper boundary wave (propagating retrograde against the fast upper flow ΛH) and the lower boundary wave (propagating prograde relative to the surface U(0) = 0) match their real phase speed at the mid-level steering velocity:

c_r = Λ H / 2 = U(H / 2) and lock into a mutually amplifying pair with exponential growth rate σ(k, l) = k c_i: σ(k, l) = (k Λ H / μ) [ (coth[μ / 2] − μ / 2) (μ / 2 − tanh[μ / 2]) ]^1/2

1. Fastest-Growing Mode (μ_max ≈ 1.6062): For a given zonal wavenumber k, the growth rate σ(k, l) is maximized for two-dimensional waves with zero meridional wavenumber (l = 0, so K = k and μ = kL_D). Differentiating σ(μ) with respect to μ shows that the maximum growth rate occurs at:

μ_max = k_max L_D ≈ 1.6062 (wavelength λ_max = [2π / 1.6062] L_D ≈ 3.91 L_D) with peak exponential growth rate: σ_max ≈ 0.3098 (f₀ Λ / N) = 0.3098 (f₀ / Ri^1/2) where Ri = N² / Λ² is the basic-state Richardson number.

For typical terrestrial midlatitude tropospheric values (H = 10 km, N = 10⁻² s⁻¹, f₀ = 10⁻⁴ s⁻¹, and Λ = 3 × 10⁻³ s⁻¹ corresponding to a 30 m/s upper-tropospheric jet), the Rossby deformation radius is L_D = NH/f₀ = 1,000 km, the fastest-growing wavelength is λ_max ≈ 3,910 km, the steering phase speed is c_r = 15 m/s eastward, and the e-folding growth rate is σ_max ≈ 0.93 × 10⁻⁵ s⁻¹ (an e-folding doubling time of ~1.25 days), matching observed extratropical cyclogenesis.

Westward Phase Tilt and Slantwise Convection

Physically, the growing Eady normal mode extracts available potential energy from the basic state by tilting westward with height for the geopotential/pressure trough axis (by an angle Δϕ ≈ π/2 at small μ decreasing to ~67° at μ_max = 1.61 and 0° at the short-wave cutoff μ_c = 2.40) while the temperature wave tilts eastward with height. This quadrature phase shift places warm anomalies (θ′ > 0) in the poleward-moving (v′ > 0) ascending (w′ > 0) sector ahead of the surface trough, and cold anomalies (θ′ < 0) in the equatorward-moving (v′ < 0) descending (w′ < 0) sector behind the trough. Parcel trajectories follow slantwise convection along surfaces tilted at half the slope of the mean isentropes (s_parcel ≈ (1/2) s_isentrope = (1/2)(f₀Λ/N²)), releasing maximum gravitational potential energy into eddy kinetic energy (w′b′ > 0 and v′θ′ > 0).

How can the Eady model be baroclinically unstable if the interior potential vorticity gradient ∂q̄/∂y is zero everywhere?

Under the Charney–Stern–Pedlosky criterion, normal-mode instability requires the generalized potential vorticity gradient—which includes Dirac delta-function sheets of potential temperature at the top and bottom boundaries (Bretherton's 1966 boundary PV equivalence)—to change sign. In the Eady model, the interior gradient ∂q̄/∂y vanishes, but the meridional buoyancy gradient ∂B̄/∂y = −f₀Λ < 0 has the same sign at both z = 0 and z = H, which corresponds to opposite-signed effective boundary PV gradients (+∂B̄/∂y at the bottom boundary and −∂B̄/∂y at the top lid), enabling counter-propagating edge waves at z = 0 and z = H to phase-lock and amplify.

Why are very short wavelengths (μ > 2.3994) stable in the Eady model?

In the quasi-geostrophic interior equation ∂²Φ/∂z² = (K²N²/f₀²)Φ, the vertical e-folding decay depth (the Prandtl–Rossby height scale) of a surface thermal wave of horizontal wavenumber K is h_R = f₀/(KN) = H/μ. When the horizontal wavelength is short (μ > 2.3994), h_R < 0.42*H*: the upper and lower edge waves decay exponentially before reaching the opposite boundary, so they cannot interact strongly enough to overcome the differential advective shearing ΛH between the top and bottom lids.

Why are midlatitude oceanic mesoscale eddies ~10 to 30 times smaller in diameter than atmospheric cyclones?

In the Eady model, the wavelength of maximum baroclinic growth scales directly with the internal Rossby radius of deformation L_D = NH/f₀ (λ_max ≈ 3.91*L*_D). In the midlatitude troposphere (H ≈ 10 km, N ≈ 10⁻² s⁻¹), L_D ≈ 1,000 km, giving cyclone wavelengths of ~4,000 km; in the midlatitude ocean thermocline (H ≈ 1–4 km, smaller vertical buoyancy contrast), the first baroclinic Rossby radius is L_D ≈ 30–50 km, giving mesoscale eddy wavelengths of ~120–200 km.

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