Quantitative theory of the split energy cascade in turbulent thin fluid layers: flux partition scaling, 3D-mode saturation, and rotating/stratified extensions

When a fluid layer of thickness $L_z$ smaller than the forcing scale $L_f$ is turbulently forced at an intermediate horizontal scale, the injected energy splits into an inverse cascade of two-dimensional (2D) modes toward large scales and a residual direct cascade of three-dimensional (3D) modes toward viscous scales. Musacchio and Boffetta (Phys. Fluids 29, 111106, 2017) established this split-cascade phenomenology in high-resolution simulations and left four quantitative questions open. The present objective addresses three of these open questions: (i) the dimensional prediction that the residual direct flux scales as $\Pi(k_z)\sim\varepsilon_f S^2$ with $S=L_z/L_f$ has been verified only in shell models and, per two independent 2023-2024 reviews, has never been tested in fully resolved 3D simulations; (ii) no predictive closure exists for the saturated 3D-mode energy $E^{3D}(L_z,\nu,\eta_f)$ and hence for the reduced inverse-cascade growth rate $\varepsilon_\alpha$ as a function of $L_z/L_\nu$; (iii) the 2D/3D mode-decomposed energy and enstrophy diagnostics, including the enstrophy-conserving intermediate range $L_z<\ell<L_f$, have not been extended to rotating and stably stratified thin layers.

Background

In two-dimensional (2D) turbulence the Navier-Stokes equations conserve, in the absence of forcing and dissipation, two quadratic invariants: the kinetic energy $E=\tfrac12\langle|\mathbf{u}|^2\rangle$ and the enstrophy $Z=\tfrac12\langle|\boldsymbol\omega|^2\rangle$, half the mean-square vorticity (Kraichnan 1967). This forces a double cascade: energy flows to large scales (inverse cascade, spectrum $E(k)\simeq\varepsilon_f^{2/3}k^{-5/3}$) while enstrophy cascades to small scales ($E(k)\simeq\eta_f^{2/3}k^{-3}$), where $\varepsilon_f$ and $\eta_f$ are the energy and enstrophy injection rates. In three dimensions (3D) the vortex-stretching term $\boldsymbol\omega\cdot\nabla\mathbf{u}$ destroys enstrophy conservation and energy cascades downscale. Many geophysical and laboratory flows (the atmospheric weather layer, oceans, soap films, electrolytic layers) are thin: nearly 2D at large scales while retaining 3D motions at small scales.

For a layer of thickness $L_z$ forced at horizontal scale $L_f$ with $L_\nu<L_z<L_f$ ($L_\nu$ the Kolmogorov viscous scale, i.e. the scale at which the local Reynolds number equals one), the control parameter is $S=L_z/L_f$ (throughout, $\nu$ denotes the kinematic viscosity and $k_z=2\pi/L_z$ the smallest nonzero vertical wavenumber). For $S\ll1$ the flow is quasi-2D; increasing $S$ past a critical value produces 3D turbulence with a purely direct cascade (Smith, Chasnov and Waleffe 1996; Celani, Musacchio and Vincenzi 2010). A useful decomposition splits the velocity into a 2D mode $\mathbf{u}^{2D}$, the average of the horizontal velocity over the vertical coordinate (the Fourier modes with vertical wavenumber $k_3=0$), and the 3D remainder $\mathbf{u}^{3D}=\mathbf{u}-\mathbf{u}^{2D}$, so that $E=E^{2D}+E^{3D}$.

Musacchio and Boffetta (2017) simulated such a layer at resolution $4096^2\times64$ with $S=1/4$, purely 2D forcing, hyperviscous dissipation (a dissipation operator $\nu_p\Delta^p$ with $p=4$, used to push the dissipative range to smaller scales) and no large-scale friction (no linear drag removing energy at the largest scales). They observed the split (bidirectional) cascade: $E^{3D}$ saturates to a statistically steady value while $E^{2D}$ grows at a reduced rate $\varepsilon_\alpha<\varepsilon_f$; in the intermediate range $L_z<\ell<L_f$ the enstrophy flux is approximately constant and the enstrophy production by vortex stretching is negligible, so enstrophy acts as a quasi-invariant supporting the upscale energy transfer; and, unexpectedly, truncating the 2D modes and treating them as a standalone 2D flow fails to reproduce the measured fluxes, showing that the 3D modes are dynamically necessary for the constancy of the 2D energy and enstrophy fluxes (unlike a passive two-dimensional three-component scenario, i.e. a flow depending on only two spatial coordinates but with three velocity components, in which the vertical velocity is simply transported by the horizontal flow). A dimensional argument then predicts the residual direct energy flux at the layer scale, $\Pi(k_z)\sim\varepsilon_f S^2$, from the residual energy flux $\Pi(k)\sim\varepsilon_f(k_f/k)^2$ of the constant-enstrophy-flux cascade evaluated at $k_z=2\pi/L_z$; this scaling had been verified only in a shell model for quasi-2D turbulence — a severely truncated dynamical model of turbulence in which each octave of wavenumbers is represented by a single complex velocity amplitude (Boffetta, De Lillo and Musacchio 2011) — not in 3D simulations. Their Appendix A derives a two-level reduced model (retaining only the $k_3=0,\pm k_z$ vertical modes) in which the first vertical modes feel an effective linear friction $\alpha=\nu k_z^2$ and the 2D mode grows at $\varepsilon_\alpha=\varepsilon_f-2\alpha E^{3D}$; the model supplies no quantitative estimate of $E^{3D}$, so the dependence of $\varepsilon_\alpha$ on $L_z/L_\nu$ remains undetermined.

Later work placed these questions in a broader phase diagram of thin-layer turbulence: there is an upper critical height $H_{3D}$ above which the cascade is purely direct, and a lower critical height $H_{2D}\sim\ell Re^{-1/2}$ (or $\ell Re^{-3/4}$ when a condensate, i.e. a system-scale coherent structure fed by the inverse cascade, is present) below which the flow becomes exactly 2D (Benavides and Alexakis 2017; van Kan and Alexakis 2019). Related rotating and stratified thin-layer flows were studied before the source paper by Deusebio et al. (2014) and Sozza et al. (2015), establishing that rotation promotes two-dimensionalization while stable stratification converts kinetic into potential energy that is promptly dissipated at small scales. The condensate-formation and dissipation-model questions raised in the source paper (behavior with and without large-scale friction, hyperviscosity versus ordinary viscosity) have been substantially clarified by the 2019-2022 simulations cited below (flux-loop condensates, eddy-viscosity saturation of the condensate, bistability).

Problem Statement

Obtain a quantitative, predictive account of the split energy cascade in a turbulent thin fluid layer — that is, of how the injected power $\varepsilon_f$ divides between the inverse cascade of 2D-mode energy toward large scales (rate $\varepsilon_\alpha$) and the residual direct cascade carried by 3D modes toward viscous scales (flux $\Pi(k_z)$) — as posed by Musacchio and Boffetta (2017) and left open by the subsequent literature. Concretely, the problem comprises: (i) a systematic, fully resolved three-dimensional numerical (or experimental) verification of the dimensional prediction $\Pi(k_z)\sim\varepsilon_f S^2$ with $S=L_z/L_f$, establishing its range of validity in $S$ and the corrections from finite Reynolds number, forcing dimensionality (whether the force injects horizontal-only or fully three-dimensional velocity components), and dissipation model, each of which is a required element of (i); (ii) a closure or first-principles prediction of the saturated 3D-mode energy $E^{3D}(L_z,\nu,\eta_f)$ — equivalently, of the reduced large-scale growth rate $\varepsilon_\alpha=\varepsilon_f-2\alpha E^{3D}$ in the two-level reduction with $\alpha=\nu k_z^2$ — that determines the scaling of $\varepsilon_\alpha$ with $L_z/L_\nu$ and is consistent with the measured near-critical behavior of the 3D-mode energy $\mathcal{E}_{3D}$ (the notation of van Kan and Alexakis 2019 for $E^{3D}$); and (iii) the extension of the 2D/3D mode-decomposed energy and enstrophy flux diagnostics to rotating and stably stratified thin layers — rotation quantified by the Rossby number $Ro$, the ratio of the nonlinear turnover time to the rotation period, and stratification by the Froude number $Fr$, the ratio of the turnover time to the buoyancy period — determining how rotation and stratification modify the coupling between 2D vortical and 3D modes and the existence and extent of the enstrophy-conserving intermediate range $L_z<\ell<L_f$ that underlies the inverse cascade. Parts (i), (ii) and (iii) are independent open sub-problems of this objective: each can be studied on its own, but a complete resolution of the split-cascade problem requires all three, so a result that addresses only one or two of them is a partial contribution and does not by itself resolve the task. The condensate-formation and dissipation-model questions raised in the same paper (behavior with and without large-scale friction, hyperviscosity versus ordinary viscosity) are not required for (i)-(iii); they need to be addressed only insofar as the friction and dissipation-model choices fix the flux balance in (i)-(ii).

References

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