Salmon Two-Layer Baroclinic Turbulence Model

The Salmon two-layer baroclinic turbulence model is a foundational theoretical framework in geophysical fluid dynamics—formulated by Rick Salmon in 1978 and 1980 building on Jule G. Charney's (1971) theory of three-dimensional geostrophic turbulence—that explains the spectral transfer of energy and potential enstrophy across horizontal and vertical scales in rapidly rotating, stably stratified planetary atmospheres and oceans. By decomposing the Phillips two-layer quasigeostrophic (QG) potential vorticity equations into a vertically averaged barotropic mode ψ = (ψ₁ + ψ₂)/2 and a vertically sheared baroclinic (thermal) mode τ = (ψ₁ − ψ₂)/2 separated by the internal Rossby deformation wavenumber k_R = L_D⁻¹ = √(2f₀² / (g′ H)), Salmon demonstrated that conservation of total mechanical energy and layer-wise potential enstrophy in nonlinear wave-triad interactions enforces a dual-branch cascade known throughout atmospheric circulation and physical oceanography as the Salmon Diagram.

At horizontal scales much larger than the Rossby deformation radius (k ≪ k_R), planetary radiative or wind forcing injects baroclinic available potential energy (APE), which cascades downscale (forward) toward k_R because the barotropic velocity field advects the baroclinic thermal streamfunction τ like a passive scalar. Near the deformation wavenumber (k ≈ k_R), Eady and Phillips baroclinic instability converts this baroclinic APE into vertically coherent barotropic eddy velocity variance—a vertical modal homogenization process termed barotropization. Once transferred into the barotropic mode at k ≈ k_R, barotropic eddy energy undergoes a two-dimensional Kraichnan upscale (inverse) energy cascade with a wavenumber spectrum E_BT(k) ∝ k^−5/3 toward smaller wavenumbers (k < k_R) until arrested by planetary Rossby wave dispersion at the Rhines wavenumber k_β = √(β / (2U_rms)) or by bottom Ekman transport drag, while total potential enstrophy cascades downscale at k > k_R with a Charney k⁻³ spectrum toward small-scale dissipation.

Layer-Wise Potential Vorticity Equations

To capture the interaction between horizontal rotation—governed by the Coriolis effect with Coriolis parameter f = f₀ + βy—and vertical density stratification, Norman A. Phillips (1956) and Rick Salmon (1978, 1980) modeled a rotating fluid as two immiscible layers of equal undisturbed thickness H/2 and densities ρ₁ < ρ₂ (reduced gravity g′ = g(ρ₂ − ρ₁)/ρ₀). In the quasigeostrophic limit (Ro = U / (f₀L) ≪ 1), the horizontal geostrophic velocity in layer i ∈ {1, 2} is u_i = (−∂ψ_i/∂y, ∂ψ_i/∂x) in terms of the layer streamfunctions ψ₁(x, y, t) and ψ₂(x, y, t).

Material conservation of quasigeostrophic potential vorticity q₁ and q₂, subject to thermal forcing F₁, F₂, bottom Ekman friction −r_E∇²ψ₂, and hyperviscous dissipation D_i, takes the form:

• Upper Layer (i = 1): ∂q₁/∂t + J(ψ₁, q₁) = F₁ + D₁, where q₁ = ∇²ψ₁ + βy − ½ k_R² (ψ₁ − ψ₂)
• Lower Layer (i = 2): ∂q₂/∂t + J(ψ₂, q₂) = F₂ − r_E∇²ψ₂ + D₂, where q₂ = ∇²ψ₂ + βy + ½ k_R² (ψ₁ − ψ₂)

Here J(A, B) = (∂A/∂x)(∂B/∂y) − (∂A/∂y)(∂B/∂x) is the Jacobian operator, and k_R = L_D⁻¹ = √(2f₀² / (g′ H)) is the internal Rossby deformation wavenumber (L_D ≈ 700–1,000 km in the midlatitude troposphere and 20–50 km in the ocean thermocline).

Decomposition into Barotropic (ψ) and Baroclinic (τ) Vertical Modes

Following Salmon (1978, 1980), the coupled layer equations are diagonalized vertically by introducing the barotropic streamfunction ψ (vertically averaged horizontal motion) and the baroclinic (thermal) streamfunction τ (proportional to the vertical geostrophic wind shear u₁ − u₂ and interface displacement):

ψ = ½ (ψ₁ + ψ₂) and τ = ½ (ψ₁ − ψ₂)

Adding and subtracting the upper- and lower-layer vorticity equation balances yields the evolution equations for barotropic vorticity ∇²ψ and baroclinic potential vorticity (∇² − k_R²)τ:

1. Barotropic Mode Evolution:

∂(∇²ψ)/∂t + J(ψ, ∇²ψ) + J(τ, ∇²τ) + β ∂ψ/∂x = −½ r_E ∇²(ψ − τ) + D_ψ

1. Baroclinic Mode Evolution:

∂(∇²τ − k_R²τ)/∂t + J(ψ, ∇²τ − k_R²τ) + J(τ, ∇²ψ) + β ∂τ/∂x = F_τ + ½ r_E ∇²(ψ − τ) + D_τ

Inspection of the nonlinear Jacobian terms reveals a key selection rule: every nonlinear triad interaction involves either three barotropic modes (ψ–ψ–ψ) via J(ψ, ∇²ψ) or one barotropic mode and two baroclinic modes (ψ–τ–τ) via J(τ, ∇²τ), J(ψ, ∇²τ − k_R²τ), and J(τ, ∇²ψ). For layers of equal depth (H₁ = H₂ = H/2), there is zero pure baroclinic self-advection triad (τ–τ–τ).

Conservation of Total Mechanical Energy and Potential Enstrophy

In the inviscid, unforced limit (F_τ = r_E = D = 0), expanding ψ(x, y, t) and τ(x, y, t) in spatial Fourier modes with horizontal wavenumber k = |k| = √(k_x² + k_y²) shows that the two-layer QG system conserves two quadratic invariants simultaneously:

1. Total Mechanical Energy (E): The sum of barotropic eddy energy (E_BT), baroclinic shear flow energy (K_BC), and available potential energy (P_APE) stored in the tilted density interface:

E = E_BT + E_BC = ½ ∑_k k² |ψ̂_k|² + ½ ∑_k (k² + k_R²) |τ̂_k|²

1. Total Potential Enstrophy (Z): Half the domain-integrated mean-squared potential vorticity summed across both layers (Z = ¼ ∬ (q₁² + q₂²) dx dy), decomposing into barotropic enstrophy (Z_BT) and baroclinic potential enstrophy (Z_BC):

Z = Z_BT + Z_BC = ½ ∑_k k⁴ |ψ̂_k|² + ½ ∑_k (k² + k_R²)² |τ̂_k|²

In every Fourier component, the ratio of potential enstrophy to mechanical energy defines an effective squared total wavenumber K_eff²:

• For a barotropic mode ψ̂_k at horizontal wavenumber k: Z_BT(k) / E_BT(k) = k²
• For a baroclinic mode τ̂_k at horizontal wavenumber k: Z_BC(k) / E_BC(k) = k² + k_R²

Fjørtoft–Salmon Triad Geometry Across the Deformation Radius

Because a baroclinic mode of horizontal wavenumber k carries the effective total wavenumber √(k² + k_R²) in the potential enstrophy invariant, Ragnar Fjørtoft's (1953) spectral theorem acquires a three-dimensional generalization when applied to ψ–ψ–ψ and ψ–τ–τ triads (Salmon, 1978, 1980):

1. Pure Barotropic Triads (ψ_p–ψ_q–ψ_k at All Scales): Among three barotropic modes, the effective squared wavenumbers are p², q², and k². Simultaneous conservation of ∑ E_BT and ∑ k²E_BT forces barotropic eddy energy to cascade upscale toward k → 0 (inverse energy cascade) while barotropic enstrophy cascades downscale toward k → ∞.
2. Large-Scale Baroclinic–Barotropic Triads (ψ_p–τ_q–τ_k at p, q, k ≪ k_R): At horizontal scales much larger than the Rossby deformation radius (k ≪ k_R), baroclinic energy is almost purely available potential energy (E_BC(k) ≈ ½ k_R² |τ̂_k|²) and the baroclinic effective squared wavenumber is nearly constant (q² + k_R² ≈ k_R²). The dominant nonlinear term in the baroclinic equation is −k_R² J(ψ, τ), meaning that the large-scale barotropic velocity u_BT = ∇^⊥ψ advects the baroclinic thermal field τ like a passive scalar, stretching thermal contours into finer filaments and driving a forward (downscale) cascade of baroclinic available potential energy from the planetary forcing wavenumber k₀ toward k_R.
3. Deformation-Scale Barotropization Triads (ψ_p–τ_q–τ_k with q, k ≈ k_R and p < k_R): Once baroclinic APE reaches wavenumbers near k_R (q, k ≈ k_R), the baroclinic effective squared wavenumbers are q² + k_R² ≈ 2k_R², whereas a barotropic mode at p ≤ k_R has a smaller effective squared wavenumber p² ≤ k_R² < 2k_R². Consequently, a triad combining two baroclinic modes near k_R with a barotropic mode at p ≲ k_R transfers energy into the barotropic mode ψ_p while transferring potential enstrophy into the higher-wavenumber baroclinic mode—corresponding in physical space to baroclinic instability (Eady, 1949; Charney, 1947).

Tracing the Three Regimes of the Salmon Cascade

Using Markovian eddy-damped quasi-normal (EDQNM) closures (1978) and direct numerical integrations (1980), Rick Salmon synthesized these triad selection rules into the Salmon Diagram:

1. Branch I — Downscale Baroclinic APE Cascade (k₀ < k < k_R): Differential solar heating (atmosphere) or wind-driven Ekman pumping (ocean) injects baroclinic available potential energy at the planetary scale k₀ ≪ k_R. Advection of interface anomalies τ by large-scale barotropic eddies cascades baroclinic APE downscale (from k₀ toward k_R) at constant spectral flux Π_APE > 0.
2. Branch II — Barotropization Conversion Zone (k ≈ k_R): At horizontal scales matching the Rossby deformation radius (k ≈ k_R), baroclinic instability reaches its peak growth rate (σ_max ≈ 0.31 f₀ |∂u/∂z| / N), releasing available potential energy through vertical secondary circulations (governed by the quasigeostrophic omega equation) and converting E_BC into barotropic velocity variance E_BT.
3. Branch III — Upscale Barotropic Inverse Cascade (k_β < k < k_R) and Downscale Charney Enstrophy Cascade (k > k_R): Barotropic eddy energy injected at k_R cascades upscale toward smaller wavenumbers (k → k_β) with a Kraichnan spectrum E_BT(k) = C_K ε^2/3 k^−5/3, energizing synoptic storm tracks and ocean mesoscale eddies larger than L_D. Simultaneously, at sub-deformation scales (k ≫ k_R), upper and lower layers decouple vertically and undergo a forward potential enstrophy cascade with Charney's (1971) spectrum: E_BT(k) ≈ E_BC(k) ∝ η^2/3 k⁻³.

Rhines Beta-Arrest and Zonal Jet Striation (k_β)

On a rotating spherical planet where the Coriolis parameter varies meridionally with gradient β = ∂f/∂y = 2Ω cos φ / R_p, Peter B. Rhines (1975) demonstrated that as eddies grow larger via the inverse barotropic cascade, their nonlinear turnover frequency ω_turb ∼ U_rms k decreases while the linear Rossby wave frequency ω_Rossby = −β k_x / k² increases, crossing at the Rhines wavenumber k_β = √( β / (2 U_rms) ) (Rhines scale L_β = k_β⁻¹). Near k_β, isotropic barotropic eddies radiate Rossby wave packets and channel upscale energy into zonal wavenumbers (k_x → 0, k_y ≈ k_β), forming alternating eastward and westward barotropic zonal jets—as observed in midlatitude atmospheric jet streams, Southern Ocean fronts, and the banded winds of Jupiter and Saturn.

Bottom Ekman Friction and Held–Larichev Eddy Diffusivity Scaling

In parallel with β-arrest, bottom Ekman drag (−r_E∇²ψ₂) removes barotropic eddy energy at large scales, setting a frictional arrest wavenumber k_drag ∼ (r_E³ / ε)^1/2. As shown by Isaac M. Held and V. D. Larichev (1996) and Brian K. Arbic and Glenn R. Flierl (2004), the supercriticality parameter ξ = k_R / k_β ≈ U_BC / (β L_D²) governs the regime of the Salmon cascade:

1. Weakly Supercritical Regime (Earth's Midlatitude Troposphere, k_R / k_β ≈ 1.5–2.5): Because the atmospheric deformation radius (L_D ≈ 800 km) is close to the Rhines scale (L_β ≈ 2,000 km), the inverse cascade is arrested within half a decade of wavenumbers, producing a synoptic upper-tropospheric spectrum that follows k⁻³ at wavelengths of 600–3,000 km before transitioning to k^−5/3 at mesoscales (< 500 km).
2. Strongly Supercritical Regime (Ocean Mesoscale Eddies & Gas Giants, k_R / k_β ≈ 3–10): In the midlatitude ocean (L_D ≈ 25–45 km, L_β ≈ 120–250 km), satellite altimetry confirms that baroclinic instability near L_D fuels barotropization and an upscale cascade to peak eddy scales of 150–250 km, regulated by bottom drag and abyssal topography.
3. Held–Larichev (1996) Baroclinic Eddy Diffusivity Scaling: Because the inverse barotropic cascade from k_R to k_β sets the barotropic velocity U_BT ∼ β L_β² that advects temperature anomalies downscale, the turbulent meridional heat diffusivity κ_eddy scales as κ_eddy ∼ U_BT L_β ∼ ξ³ β L_D³ ∼ U_BC³ / (β² L_D³).

What is "barotropization" in the Salmon two-layer turbulence model?

Barotropization is the nonlinear transfer of energy from the vertically sheared baroclinic (thermal) mode τ = (ψ₁ − ψ₂)/2 into the depth-independent barotropic mode ψ = (ψ₁ + ψ₂)/2 near the internal Rossby deformation wavenumber k_R = L_D⁻¹. Through baroclinic instability and ψ–τ–τ wave-triad interactions, tilted density surfaces flatten while upper- and lower-layer pressure anomalies lock in phase vertically, converting available potential energy into vertically coherent barotropic eddy circulation.

Why does baroclinic available potential energy cascade downscale while barotropic energy cascades upscale?

At horizontal scales much larger than the Rossby deformation radius (k ≪ k_R), a baroclinic mode has a nearly constant effective squared wavenumber (k² + k_R² ≈ k_R²) and is advected by the barotropic velocity field like a passive temperature tracer (J(ψ, τ)), driving a forward (downscale) cascade of available potential energy toward k_R. Once converted into the barotropic mode ψ at k ≈ k_R, simultaneous conservation of barotropic energy (∑ k²|ψ̂|²) and enstrophy (∑ k⁴|ψ̂|²) forces barotropic energy to transfer upscale (inverse cascade) toward k < k_R.

How does the Salmon model explain why ocean mesoscale eddies are larger than the Rossby radius of deformation?

In the midlatitude ocean, linear baroclinic instability is most unstable near the first baroclinic Rossby deformation radius (L_D ≈ 20–50 km), yet satellite altimetry observes dominant mesoscale eddies with diameters of 100–250 km. The Salmon two-layer cascade explains this discrepancy: baroclinic eddies generated near L_D undergo barotropization and merge upscale via the inverse barotropic energy cascade until arrested near the Rhines scale L_β = √(2U_rms/β) or by bottom Ekman drag.

How does Charney's (1971) geostrophic turbulence theory relate to Salmon's (1978, 1980) two-layer model?

Jule G. Charney (1971) showed that for three-dimensional continuously stratified quasigeostrophic flow away from horizontal boundaries, conservation of 3D energy and potential enstrophy yields a forward potential enstrophy cascade with a k⁻³ spectrum. Salmon's (1978, 1980) two-layer model incorporated the vertical confinement of the tropopause and seafloor, capturing the transition at k_R = L_D⁻¹ between the large-scale (k < k_R) baroclinic-to-barotropic conversion regime and the sub-deformation (k > k_R) Charney k⁻³ enstrophy cascade.

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