Rhines Scale

Rhines Scale is a fundamental horizontal length scale L_R (and associated wavenumber k_R) in geophysical fluid dynamics, derived by the American oceanographer Peter B. Rhines in 1975, that marks the transition between isotropic two-dimensional geostrophic turbulence at smaller scales and dispersive Rossby wave dynamics at larger scales on a rotating sphere or β-plane. In two-dimensional and quasi-geostrophic flows, the simultaneous conservation of turbulent kinetic energy and potential enstrophy drives an inverse energy cascade that transfers eddy kinetic energy from small injection scales toward progressively larger horizontal eddies (k → 0). On a rotating planet where the meridional gradient of the Coriolis parameter is β = df/dy = (2Ω / a) cos φ, expanding eddies eventually reach a horizontal scale where the linear Rossby wave phase speed c_R ~ β/k² matches the root-mean-square turbulent fluid velocity U_rms. Equating the turbulent eddy turnover frequency ω_turb = U_rmsk to the Rossby wave frequency ω_Rossby = β/k defines the Rhines wavenumber k_R and Rhines scale L_R:

k_R = (β / [2U_rms])^1/2 (or k_β = (β / U_rms)^1/2) , L_R = k_R⁻¹ = (2U_rms / β)^1/2

At horizontal scales approaching L_R (often expressed as the half-wavelength πL_R ≈ π(2U_rms/β)^1/2), the isotropic inverse cascade is deflected anisotropically toward zonal wavenumbers (k_x → 0), organizing the flow into alternating east–west zonal jets such as the banded belts and zones of Jupiter and Saturn and the multiple jet cores of the Antarctic Circumpolar Current.

Competition Between Nonlinear Advection and Planetary β-Dispersion

The barotropic vorticity equation for inviscid, horizontally nondivergent two-dimensional flow u = −∂ψ/∂y, v = ∂ψ/∂x on a midlatitude β-plane is:

∂(∇²ψ) / ∂t + J(ψ, ∇²ψ) + β ∂ψ / ∂x = 0

where ζ = ∇²ψ is the relative vorticity and J(ψ, ζ) = u(∂ζ/∂x) + v(∂ζ/∂y) is the nonlinear vorticity advection operator. Consider an eddy field with characteristic root-mean-square horizontal velocity U_rms = ⟨u² + v²⟩^1/2 and isotropic horizontal wavenumber k = (k_x² + k_y²)^1/2:

1. Nonlinear Turbulent Advection Term (J(ψ, ∇²ψ)): With streamfunction amplitude ψ ~ U_rms/k and relative vorticity ζ ~ kU_rms, the advective rate of change of vorticity scales as U_rmsk · (kU_rms) = k²U_rms², corresponding to an eddy turnover rate (inverse advective timescale):

ω_turb(k) = k U_rms

1. Linear Planetary Dispersion Term (β ∂ψ/∂x): Substituting a plane Rossby wave exp[i(k_xx + k_yy − ωt)] into the linearized vorticity equation yields the anisotropic Rossby dispersion relation:

ω_Rossby(k_x, k_y) = − (β k_x) / (k_x² + k_y²) = − (β cos θ) / k where θ = arctan(k_y/k_x) is the wavevector angle in Fourier space.

At large wavenumbers (k ≫ k_R, small length scales), ω_turb ≫ |ω_Rossby|: eddies deform and shear one another within a fraction of a Rossby wave period, and Fjørtoft's (1953) and Kraichnan's (1967) two-dimensional turbulence theorem governs an isotropic k^−5/3 inverse cascade of eddy kinetic energy toward smaller k. As energy cascades to smaller wavenumbers and k decreases down to k_R = (β/[2U_rms])^1/2 (where the factor of 2 arises from Rhines's 1975 spectral centroid identity ⟨k²⟩ = Z/E = ζ_rms²/U_rms² for waves with phase speed c = β/[2k²] = U_rms), the Rossby wave period becomes shorter than the eddy turnover time and linear wave radiation takes over.

Anisotropic "Dumbbell" Arrest and Zonal Jet Formation

Crucially, the Rossby frequency ω_Rossby = −βk_x / (k_x² + k_y²) is anisotropic in the (k_x, k_y) Fourier plane. Setting the turbulent strain rate U_rms(k_x² + k_y²)^1/2 equal to the directional Rossby frequency |βk_x|/(k_x² + k_y²) defines a two-lobed figure-eight (dumbbell) arrest boundary described by Geoffrey K. Vallis and Michael E. Maltrud (1993):

(k_x² + k_y²)^3/2 = (β / U_rms) |k_x| ⇒ k(θ) = (β |cos θ| / U_rms)^1/2

Inside this figure-eight boundary (k < k(θ)), Rossby wave frequencies are too rapid for resonant triad interactions among eddies to transfer energy efficiently across wavevectors. However, along the meridional axis (k_x = 0, θ = ±π/2, corresponding to purely zonal flows u(y) that do not vary with longitude x), the Rossby wave frequency vanishes identically (ω_Rossby(0, k_y) = 0). Consequently, the inverse energy cascade is blocked from entering the equatorial lobes of the figure-eight (k_x ≠ 0) and is funneled into the narrow "channel" along the k_y axis (k_x → 0) near k_y ≈ k_R. In physical space, energy accumulating at (k_x ≈ 0, k_y ≈ k_R) manifests as alternating eastward and westward zonal jets separated in latitude by half a wavelength:

L_jet ≈ π / k_R = π (2U_rms / β)^1/2

Potential Vorticity Homogenization vs. Sharp Eastward Jet Barriers

In physical space, Michael E. McIntyre (1982, 2008) and David G. Dritschel explained the emergence of Rhines-scale jets through the mixing of barotropic potential vorticity q(x, y) = ζ + f₀ + βy. Vigorous Rossby-wave breaking within meridional bands of width L_jet ~ π(2U_rms/β)^1/2 homogenizes q toward constant plateau values (∂q̄/∂y ≈ 0), while concentrating the planetary potential vorticity gradient into narrow, high-gradient interfaces (∂q̄/∂y ≫ β).

Inverting the zonally averaged relation ∂q̄/∂y = β − ∂²ū/∂y² over this potential vorticity staircase produces broad westward flows (∂²ū/∂y² = β > 0, parabolic retrograde troughs) inside the mixed zones and narrow, cusp-like eastward (prograde) jets (∂²ū/∂y² ≪ 0) along the sharp PV steps, where the concentrated PV gradient acts as an elastic waveguide barrier that inhibits meridional eddy transport and stabilizes the eastward jet under the Charney–Stern–Pedlosky criterion.

Comparison of Rhines Scales Across Planetary Atmospheres and Terrestrial Oceans

In forced–dissipative turbulence with large-scale bottom drag −rζ and energy injection rate ε, Boris Galperin and Sukumar Sukhatme defined the dimensionless zonostrophy index R_β = k_R / k_fr ≈ 0.5 (β³ / [ε r⁵])^1/20 (where k_fr = (r³/ε)^1/2 is the frictional arrest wavenumber). Terrestrial oceans and gas giant atmospheres operate in the zonostrophic regime (R_β ≳ 2.5), where the Rhines arrest occurs well before frictional dissipation and develops an anisotropic E_zonal(k_y) ≈ C_Z β² k_y⁻⁵ spectrum. Earth's troposphere has R_β ≈ 1.5 (the friction-dominated or macroturbulent regime), where the Rhines scale L_R (~1,200 km) is close to both the baroclinic Rossby radius of deformation L_D (~1,000 km) and the planetary radius, permitting only one eddy-driven polar jet and one thermally driven subtropical jet per hemisphere.

Why does two-dimensional turbulence transfer energy to larger scales rather than smaller scales?

In three-dimensional turbulence, vortex tubes are stretched by velocity gradients, cascading turbulent kinetic energy downscale to viscous dissipation scales. In two-dimensional or quasi-geostrophic flow, vortex stretching is absent or constrained by stratification, so both eddy kinetic energy E = (1/2)∫ U² dA and enstrophy Z = (1/2)∫ k²U² dA are conserved simultaneously. By Fjørtoft's theorem, any spectral redistribution that conserves both E and Z must send enstrophy to high wavenumbers (k → ∞) and eddy kinetic energy to low wavenumbers (k → 0).

Why does the Rhines arrest produce east–west (zonal) jets instead of north–south (meridional) jets?

On a rotating sphere, the Coriolis parameter f = 2Ω sin φ varies with latitude (y) but is constant with longitude (x). Therefore, the linear Rossby wave restoring frequency ω = −βk_x/(k_x² + k_y²) vanishes for zonal flows (k_x = 0, east–west jets) but reaches its maximum for meridional flows (k_y = 0, north–south currents). North–south currents immediately radiate their energy away as dispersive Rossby wave packets, leaving only east–west zonal jets stable against linear wave dispersion.

Why does Jupiter have many narrow zonal bands whereas Uranus and Neptune have only two broad jets?

The meridional spacing of zonal jets scales with the Rhines wavelength πL_R = π(2U_rms/β)^1/2. Jupiter combines a large planetary radius (a = 69,911 km) and fast 9.9-hour rotation (giving a large planetary scale relative to L_R ≈ 4,500 km), so roughly a dozen jets fit from pole to pole. Uranus and Neptune have smaller radii (a ≈ 25,000 km) and very high wind speeds (U_rms ~ 100–150 m/s), making their Rhines wavelength (~30,000–40,000 km) comparable to the entire hemisphere.

1. Rhines, P. B. (1975). "Waves and turbulence on a beta-plane." Journal of Fluid Mechanics, 69(3), 417–443. https://doi.org/10.1017/S0022112075001504
2. Vallis, G. K., & Maltrud, M. E. (1993). "Generation of mean flows and jets on a beta plane and over topography." Journal of Physical Oceanography, 23(7), 1346–1362. https://doi.org/10.1175/1520-0485(1993)023<1346:GOMFAJ>2.0.CO;2023<1346:GOMFAJ>2.0.CO;2)
3. Dritschel, D. G., & McIntyre, M. E. (2008). "Multiple jets as PV staircases: The Phillips effect and the resilience of eddy-transport barriers." Journal of the Atmospheric Sciences, 65(3), 855–874. https://doi.org/10.1175/2007JAS2227.1
4. Sukoriansky, S., Dikovskaya, N., & Galperin, B. (2007). "On the arrest of inverse energy cascade and the Rhines scale." Journal of the Atmospheric Sciences, 64(9), 3312–3327. https://doi.org/10.1175/JAS4013.1
5. Galperin, B., & Read, P. L. (Eds.). (2019). Zonal Jets: Phenomenology, Genesis, and Physics. Cambridge University Press. https://doi.org/10.1017/9781107358225