Two-Dimensional $\beta$-plane Turbulence: Dual Cascade and Zonal Jets

Technical Summary: Two-Dimensional $\beta$-plane Turbulence: Dual Cascade and Zonal Jets

Problem Statement
The paper investigates the statistical properties of two-dimensional, incompressible, forced-dissipative Navier-Stokes equations subject to the Coriolis force under the $\beta$-plane approximation. This system models geophysical flows (such as oceanic currents and atmospheric jets) where planetary rotation varies linearly with the meridional coordinate ($f(x) = f_0 + \beta x_2$). The authors aim to rigorously characterize two primary phenomena in this regime:

1. Geostrophic Balance: The statistical relationship between the Coriolis force and the pressure gradient at large scales.
2. Dual Cascade: The behavior of energy and enstrophy transfer across scales in the presence of rotation, specifically addressing whether the Coriolis force alters the universal laws of two-dimensional turbulence (inverse energy cascade and direct enstrophy cascade) derived for non-rotating fluids.

The mathematical setting involves a periodic channel $D = \mathbb{T} \times [a, b]$ with no-slip boundary conditions in the meridional direction, driven by stochastic forcing and subject to large-scale Ekman damping ($\alpha$) and small-scale viscosity ($\nu$).

Methodology
The authors employ a rigorous probabilistic framework based on the Kármán–Howarth–Monin (KHM) relations, adapted for the stochastic Navier-Stokes equations on a $\beta$-plane. Key methodological components include:

• Stationary Measure and Regularity: The analysis relies on the existence of a stationary measure for the stochastic system, established in prior work [11]. The authors assume sufficient regularity of the noise to ensure the stationary measure has $H^2$ regularity, allowing for the manipulation of two-point correlation functions.
• Weak KHM Relations: Instead of assuming spatial homogeneity or isotropy, the authors utilize weak formulations of the KHM equations involving test functions and spatial averaging over the domain and the unit sphere.
• Dual Projections:
  1. Trace (Symmetric) Projection: To analyze the dual cascade, the authors take the trace of the tensor KHM relation. This recovers scalar balance equations for energy and enstrophy.
  2. Antisymmetric Projection: To analyze geostrophic balance and zonal jets, the authors introduce a novel antisymmetric test function (a weighted off-diagonal projection). This specific projection isolates terms that vanish in standard isotropic analyses, specifically the Coriolis and pressure gradient contributions.
• Weak Anomalous Dissipation (WAD): For the cascade results, the authors adopt the WAD hypothesis introduced by Bedrossian et al. [5]. This assumption posits that in the inviscid limit ($\nu, \alpha \to 0$), injected energy is dissipated entirely by large-scale friction and injected enstrophy by viscosity, without requiring uniform bounds on vorticity moments that are stronger than necessary.

Key Contributions and Results

1. Vanishing of Spherically Averaged Coriolis Terms (The Dual Cascade)
The paper proves that the spherically averaged contributions of the Coriolis force to the energy and enstrophy fluxes vanish identically at all scales.

• Theorem 3.1: It is shown that the trace of the velocity Coriolis correlation tensor, $\Theta(l)$, and the enstrophy Coriolis correlation, $Q(l)$, are exactly zero for any separation scale $l > 0$.
• Theorem 3.2 & 3.3 (Dual Cascade Laws): Under the WAD hypothesis, the authors derive the asymptotic behavior of the averaged third-order structure functions.
  • Direct Enstrophy Cascade: At small scales ($l_\nu \ll l \ll l_I$), the mixed velocity-vorticity structure function follows the law $S^{(1)}_3(l) \sim -2\eta l$, identical to the non-rotating case.
  • Inverse Energy Cascade: At large scales ($l_I \ll l \ll l_\alpha$), the velocity structure function follows $S^{(3)}_3(l) \sim 2\varepsilon l$.
• Significance: These results demonstrate that the net rate of energy and enstrophy transfer in the isotropic, spherically averaged sense is unaffected by the Coriolis force. The Coriolis force does not inject or dissipate energy/enstrophy in the isotropic average; it acts purely as a geometric constraint.

2. Exact Geostrophic Balance and Zonal Jet Formation
The paper derives a novel, exact statistical identity connecting the Coriolis force to the pressure gradient via a two-point correlation function.

• Theorem 3.4 (Geostrophic KHM Balance): By applying the antisymmetric projection to the KHM relation, the authors derive the identity:
  $$\beta l \mathbb{E}\langle (n_2)^2 (u(x) \cdot u(x+ln)) \rangle = \mathbb{E}\langle n_2 (\nabla p(x) \cdot u^\perp(x+ln) - u^\perp(x) \cdot \nabla p(x+ln)) \rangle$$
  where $n$ is the direction of the separation vector.
• Interpretation:
  • At small scales, the $\beta l$ term is negligible, recovering local isotropy.
  • At large scales, the $\beta l$ term grows unbounded. To maintain the balance, the pressure gradient correlation must grow proportionally.
  • This identity provides a rigorous mathematical signature of how the Coriolis force induces anisotropy. The balance forces a reorganization of the flow where meridional velocity fluctuations are suppressed, leading to the formation of zonal jets (bands of east-west flow).
• Rhines Scale: The authors show via dimensional analysis (Appendix A) that this identity recovers the Rhines scale $l_R \sim (\varepsilon/\beta^3)^{1/5}$, the transition scale where the inverse cascade is arrested by the $\beta$-effect.

3. Limitations on Longitudinal Laws
The paper notes that while the spherically averaged (isotropic) cascade laws are unaffected, the authors cannot currently prove the longitudinal third-order structure function laws (e.g., the 4/5-law analogues) in the presence of the Coriolis force. The Coriolis contribution to the longitudinal projection does not vanish and introduces error terms that diverge in the inviscid limit under standard WAD assumptions.

Significance and Claims
The authors position this work as providing a "clear picture" of the Coriolis force's role in $\beta$-plane turbulence by reconciling two seemingly contradictory physical observations:

1. Conservation of Cascade Rates: The Coriolis force does not alter the magnitude of the dual cascade (energy and enstrophy transfer rates) when averaged isotropically.
2. Induction of Anisotropy: The Coriolis force fundamentally alters the spatial distribution of energy, driving the formation of large-scale zonal structures.

The paper claims to provide the first exact statistical balance for $\beta$-plane turbulence that captures the emergence of zonal jets without relying on macroscopic approximations or statistical closures. The derived identity (Theorem 3.4) serves as a rigorous, fully nonlinear, two-point mathematical signature of geostrophic balance, linking the Coriolis parameter directly to pressure-velocity correlations.

The work complements previous results on small scales [10] by extending the rigorous understanding to large scales and the transition regime, confirming that while the mechanism of the cascade (transfer rates) remains robust to rotation, the structure of the flow is profoundly reorganized by it.