Deterministic surface quasi-geostrophic (SQG) flow, simulated at small but finite viscosity from smooth, few-Fourier-mode initial data, develops a turbulent regime exhibiting three hallmark anomalies of three-dimensional turbulence: a dissipative anomaly for the surface kinetic energy, multifractal scaling, and superdiffusive forward-and-backward Lagrangian particle separation (spontaneous stochasticity) - yet the stochasticity is tempered, with non-universal limiting statistics (Valade-Thalabard-Bec, Ann. Henri Poincare 2024). The open problem identified in that work is to determine the inviscid-limit statistical content of deterministic SQG turbulence: (1) whether the observed nonzero dissipation rate of the surface kinetic energy persists in the strict viscosity-to-zero limit or is an artifact of finite viscosity and of the limiting protocol; (2) which mechanisms (the deterministic large-scale flow, the conserving Hamiltonian, or Lagrangian time correlations) temper the spontaneous stochasticity and produce its non-universality; (3) how sensitive the limiting dispersion laws and transition probabilities are to the initial condition, the joint viscosity-and-noise limiting protocol, and finite-domain effects; (4) the construction of an analytic framework (extending Kraichnan-type zero-mode descriptions to this non-Markovian, nonlinear setting) for the limiting transition measures; and (5) whether SQG admits Eulerian spontaneous stochasticity - intrinsic probabilistic inviscid solutions tied to non-uniqueness at the Hölder threshold h≤1/3h\le 1/3 - and how it relates to the tempered Lagrangian stochasticity. All five components remain open: the Onsager conjecture for the SQG Hamiltonian was resolved in 2024, but that concerns the inverse-cascade Hamiltonian, not the surface kinetic energy, and does not touch the spontaneous-stochasticity questions.

Background

Surface quasi-geostrophy (SQG) is a two-dimensional model of the active transport of a surface temperature (buoyancy) field in a rapidly rotating, strongly stratified environment, introduced in the 1990s in the geophysical literature (Held-Pierrehumbert-Garner-Swanson) and as a mathematical testbed by Constantin-Majda-Tabak. The dynamics reads ∂tθ+v⋅∇θ=νΔθ\partial_t\theta+\mathbf{v}\cdot\nabla\theta=\nu\Delta\theta with v=R⊥θ\mathbf{v}=\mathcal{R}^\perp\theta, where R⊥\mathcal{R}^\perp is a rotation of the Riesz transforms - Fourier multipliers homogeneous of degree zero - so that the advecting velocity has the same spatial regularity and physical dimension as the scalar itself. An active scalar, unlike a passive one, determines its own advecting velocity. This makes SQG the canonical two-dimensional analogue of three-dimensional turbulence: like the 3D energy cascade, the cascade of the SQG scalar requires a rough, non-differentiable velocity field, while the model remains far more tractable numerically and analytically. Inviscid SQG conserves two quadratic quantities: the surface kinetic energy E=12⟨θ2⟩=12⟨∣v∣2⟩\mathcal{E}=\frac12\langle\theta^2\rangle=\frac12\langle|\mathbf{v}|^2\rangle (angular brackets denote spatial averages over the periodic domain), which cascades to small scales, and the Hamiltonian H=12⟨θΨ⟩\mathcal{H}=\frac12\langle\theta\Psi\rangle with θ=ΛΨ\theta=\Lambda\Psi - equivalently the homogeneous Sobolev H˙−1/2\dot H^{-1/2} norm of θ\theta - which undergoes an inverse cascade towards large scales. A dissipative anomaly is the persistence of a nonzero dissipation rate, here εθ=ν⟨∣∇θ∣2⟩\varepsilon_\theta=\nu\langle|\nabla\theta|^2\rangle for E\mathcal{E}, in the limit ν→0\nu\to0. Onsager's 1949 conjecture for 3D Euler - conservation of energy for weak solutions with velocity increments δℓu∼ℓh\delta_\ell u\sim\ell^h, h>1/3h>1/3, and possible dissipation for h≤1/3h\le1/3 - was proved by Isett (2018) and is the prototype for such threshold statements. For SQG the analogous Onsager statement for the Hamiltonian (conservation for h>0h>0, dissipation possible for h≤0h\le0; conservation above threshold by Isett-Vicol 2015, explicit dissipative examples for h<−1/5h<-1/5 by Buckmaster-Shkoller-Vicol 2019) was settled in 2024 by Isett-Looi and, independently, Dai-Giri-Radu. For the surface kinetic energy, however, only the conservation side (h>1/3h>1/3) is known (Akramov-Wiedemann 2019): whether weak solutions of inviscid SQG can dissipate E\mathcal{E} at h≤1/3h\le1/3, and whether a vanishing-viscosity limit from smooth data realizes such dissipation, is open. Spontaneous stochasticity is the persistence of intrinsic randomness under a singular limit: two tracer particles released together with infinitesimally small noise and initial separation fail to coalesce as the noise and the flow regularisation are removed, and their trajectories are described by a nontrivial probability law. The phenomenon was first understood rigorously in the Kraichnan model, a Gaussian, white-in-time random velocity ensemble, where zero modes of the n-particle forward generators simultaneously determine the anomalous scaling of correlation functions (intermittency) and the statistics of particle separation; for Burgers equation, Eyink-Drivas proved that Lagrangian spontaneous stochasticity and anomalous dissipation are two facets of the same phenomenon. Eulerian spontaneous stochasticity is the Eulerian counterpart in which the inviscid limit of the field itself - not only of its trajectories - is random, the rigorous counterpart of Lorenz's 1969 'real butterfly effect', in which perturbations at infinitesimal scales are amplified by a direct cascade of errors up to the largest scales. The source paper (Valade-Thalabard-Bec, Ann. Henri Poincare 25, 1261-1283, 2024; arXiv:2210.12366, dedicated to the memory of K. Gawedzki) simulated deterministic viscous SQG on the doubly periodic domain, starting from smooth initial data consisting of a few large-scale Fourier modes. At finite but small ν\nu a turbulent regime appears in finite time featuring (i) a dissipative anomaly for E\mathcal{E} - dissipation rates at different ν\nu collapse onto a ν\nu-independent curve at late times, while the dissipation of H\mathcal{H} decreases proportionally to ν\nu - (ii) multifractal scaling of temperature increments, and (iii) superdiffusive, approximately Richardson-like (R2∝t3R^2\propto t^3) forward and backward pair separation, signalling spontaneous stochasticity of the Lagrangian flow. However, the observed stochasticity is tempered: the limiting dispersion laws and transition probabilities are non-universal, with prefactors depending on the initial separation and on the small-scale regularisation, and the Eulerian error-energy statistics likewise fail to collapse onto a universal regime. Inviscid SQG is locally well-posed for smooth data, but whether finite-time singularities form from smooth data is a famous open problem, weak solutions are known to be non-unique (Buckmaster-Shkoller-Vicol 2019), and for forced critical dissipative SQG anomalous dissipation is absent in a long-time-averaged sense (Constantin-Tarfulea-Vicol 2013) - a forced setting distinct from the decaying, smooth-data problem considered here.

Problem Statement

The common scientific objective is to determine the statistical content of the inviscid limit of deterministic SQG turbulence evolving from smooth initial data, connecting the finite-viscosity phenomenology (dissipative anomaly, tempered Lagrangian spontaneous stochasticity) to provable inviscid-limit statements. For the SQG equation on the doubly periodic domain with smooth large-scale initial data, as formulated in Valade-Thalabard-Bec (Ann. Henri Poincare 2024), the problem has five linked components: (1) Dissipative-anomaly status: decide whether the observed finite, non-zero dissipation rate of the surface kinetic energy E\mathcal{E} persists in the strict ν→0\nu\to0 limit of the exact inviscid SQG dynamics from smooth initial data - i.e. whether lim⁡ν→0ν⟨∣∇θν∣2⟩>0\lim_{\nu\to0}\nu\langle|\nabla\theta^\nu|^2\rangle>0 - or whether it is an artifact of the finite-ν\nu dynamics and of the specific manner of taking the limit (for instance of the order in which the long-time and inviscid limits are taken); a resolution may prove persistence, prove vanishing, or exhibit regularization-dependent selection. (2) Tempering mechanisms: identify which mechanism(s) - candidates include the deterministic (non-random) large-scale flow, the conserving Hamiltonian H\mathcal{H} favouring large-scale organisation, and non-trivial Lagrangian time correlations - temper the spontaneous stochasticity observed numerically, i.e. control the dependence of the limiting Lagrangian transition probabilities and dispersion laws on the small-scale regularisation. (3) Generality and protocols: determine how sensitive the limiting dispersion laws and transition probabilities are to the choice of initial condition, to the joint protocol for taking ν→0\nu\to0 and the noise/perturbation amplitude to zero, and to finite-domain effects (including possible condensation of energy at large scales); in particular, decide whether the observed non-universality is a robust feature of SQG turbulence or a consequence of the specific numerical setup. (4) Analytic framework: construct an analytic framework capable of computing or characterising the limiting transition measures of deterministic SQG, extending to this non-Markovian, nonlinear setting the zero-mode-based descriptions that link intermittency and Lagrangian spontaneous stochasticity in Markovian Gaussian models. (5) Eulerian spontaneous stochasticity: decide whether there is a regime of SQG in which inviscid non-uniqueness - related to the Hölder-regularity threshold h≤1/3h\le 1/3 for the dissipation of E\mathcal{E}, and to the now-settled Onsager threshold for H\mathcal{H} - leads to intrinsic probabilistic Eulerian solutions in the vanishing-viscosity limit, and how such Eulerian spontaneous stochasticity relates to the tempered Lagrangian stochasticity observed. The five components are facets of one question - whether the inviscid limit of deterministic SQG turbulence from smooth data is dissipative and probabilistic - and answers reinforce one another; for example a selection or mechanism theory (components 2 and 4) directly informs components 1 and 5. A resolution of any one of the five components, as stated, resolves this task. A claimed resolution of any component must address that component as stated, without substituting a proxy or a narrowed target (e.g. replacing E\mathcal{E} by H\mathcal{H}, or the deterministic inviscid limit by a randomly forced system).

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