--- title: Intermittent Obukhov-Corrsin Regime url: https://www.emergentmind.com/topics/intermittent-obukhov-corrsin-regime type: topic --- # Intermittent Obukhov-Corrsin Regime The intermittent Obukhov–Corrsin regime is a family of passive-scalar turbulence regimes in which the classical Obukhov–Corrsin inertial-convective picture is retained at second order or in an averaged sense, while intermittency appears through anomalous higher-order scaling, non-Gaussian small-scale statistics, anomalous dissipation, or codimension-sensitive concentration of dissipation. In the classical homogeneous isotropic setting, the baseline prediction is the Obukhov–Corrsin law \(E_\theta(k)\sim C\,\chi\,\epsilon^{-1/3}k^{-5/3}\) together with the dimensional estimate \(S_p(r)\propto r^{p/3}\). In recent rigorous PDE work, the same phrase is used in a different but related sense: a supercritical regularity regime for transport by rough divergence-free flows, in which uniform regularity and anomalous dissipation coexist. By contrast, some recent mathematical derivations of the Obukhov–Corrsin spectrum explicitly prove annularly averaged second-order spectral scaling without claiming intermittency itself [1202.4113, 2207.06833, 2508.00115, 2512.02853]. ## 1. Classical Obukhov–Corrsin framework In incompressible homogeneous isotropic turbulence with a passive scalar, the basic object is the normalized two-point scalar correlation \[ f_\theta(r,t)=\frac{\langle \vartheta \vartheta'\rangle}{\theta^2}, \] whose evolution is governed by the Corrsin equation \[ \theta^2 \frac{\partial f_\theta}{\partial t} + f_\theta \frac{d \theta^2}{d t} -G -2 \chi \theta^2 \left( \frac{\partial^2 f_\theta}{\partial r^2} + \frac{2}{r} \frac{\partial f_\theta}{\partial r} \right) = 0, \] with transfer term \[ G = - \frac{\partial}{\partial r_k} \langle \vartheta \vartheta' (u_k'-u_k) \rangle. \] In the same setting, the scalar spectrum is defined by a sine transform of \(f_\theta\), and the inertial-convective regime is identified with \[ \Theta(\kappa)\sim C_\theta \epsilon^{-1/3}\epsilon_\theta\,\kappa^{-5/3}. \] The same closed theory also recovers the Batchelor viscous-convective regime \(\Theta(\kappa)\sim \kappa^{-1}\) at high \(Pr\), and a steeper low-\(Pr\) range with exponent \(n\) satisfying \(-17/30. \] The proof uses the stochastic flow \[ dX_t^\kappa = u(t,X_t^\kappa)\,dt+\sqrt{2\kappa}\,dW_t \] and the Feynman–Kac representation \[ \vartheta^\kappa(t,x)=\mathbb E\bigl[\vartheta_{\rm in}(X_{0,t}^\kappa(x))\bigr]. \] The same work also proves lack of selection by vanishing diffusivity and by convolution mollification: for every \(\alpha\in[0,1)\), there are \(C^\alpha\) divergence-free velocities for which distinct inviscid weak limit points coexist, one conserving \(L^2\) and another dissipating it [2207.06833]. A more explicitly intermittent formulation refines the regularity threshold by the dimension of the dissipation support. If \[ u \in L^p([0,1],B^{\sigma,p}_\infty(T^2)),\qquad \theta \in L^q([0,1], B^{\beta,q}_\infty(T^2)), \] and \(\theta\) solves \[ \partial_t \theta + \nabla \cdot (u \theta) =0, \] then the dissipation distribution \[ D := \partial_t \theta^2 + \nabla \cdot (u \theta^2) \] is assumed to be a Radon measure supported on a set \(S\) of Hausdorff dimension \(\gamma\in[0,3]\). If \(D\neq 0\), then one must have \[ \frac{2\beta}{1-\sigma} \leq 1 - \Big(1 - \frac{2}{q} - \frac{1}{p}\Big)(3-\gamma). \] This is the codimension-sensitive intermittent Obukhov–Corrsin constraint. Its novelty is specific to \(\gamma<3\), where dissipation lies on a positive-codimension set. The same paper constructs an explicit divergence-free field \(V\in L^\infty([0,1],C^\alpha(T^2))\) whose dissipation measure is supported on countably many timeslices, and proves sharpness at the endpoint \(p=\infty\), \(\gamma=2\): whenever \[ \frac{2\beta}{1-\sigma} < \frac{2}{q}, \] there exist \(u\) and \(\theta\) with non-trivial negative Radon dissipation supported on \[ S=\{1/2\}\times T^2. \] The same construction is also tied to asymptotic total dissipation, enhanced dissipation, Richardson dispersion, anomalous regularization, and spatial intermittency [2508.00115]. ## 4. Anomalous regularization and annular spectral laws A distinct rigorous direction proves the Obukhov–Corrsin spectrum in rough passive-scalar models without claiming an intermittent regime in the higher-moment sense. For passive scalar advection by a white-in-time Kraichnan-type transport noise with spatial Hölder regularity \(C_x^\alpha\), \(0<\alpha<1\), the classical rough-flow prediction \[ \mathbb E_\mu |\theta(k)|^2 \approx |k|^{-d-(1-\alpha)} \] is shifted by the white-in-time scaling to \[ \mathbb E_\mu |\theta(k)|^2 \approx |k|^{-d-2(1-\alpha)}, \] corresponding to essentially \(H_x^{1-\alpha}\) regularity in equilibrium. The rigorous theorem does not establish a pointwise lower bound for each Fourier mode. It proves the predicted scaling only after summing over geometric annuli, and only up to logarithmic factors and slightly super-geometric shell widths. In the main Kraichnan examples, \(m=2\), and the total shell mass is bounded above and below by terms of order \(r^{2(1-\alpha)}\|F\|_{L^2}^2\) times logarithmic corrections [2512.02853]. The mechanism is anomalous regularization. For the free-decay problem \[ \dot \theta_t^\kappa - \kappa \Delta \theta_t^\kappa + \circ du_t\cdot\nabla \theta_t^\kappa =0, \] one has the uniform decay estimate \[ \mathbb E \|\theta_t^\kappa\|_{L^2}^2 \le C e^{-c t}\|\theta_0^\kappa\|_{L^2}^2, \] together with the time-integrated gain of roughly \(1-\alpha\) derivatives, \[ \mathbb E\int_0^1 \sum_{k\neq 0}\frac{|k|^{2(1-\alpha)}{(\log|k|+1)^4}\,|\theta_t^\kappa(k)|^2\,dt \le C\|\theta_0^\kappa\|_{L^2}^2. \] This regularization is transferred to the invariant measure of the forced problem through \[ \mathbb E_{\mu^\kappa}|\theta^\kappa(k)|^2 = \int_0^\infty \mathbb E|\theta_t^\kappa(k)|^2\,dt. \] The analytic core is a Fourier-space \(\ell^p\) energy identity and a weighted lattice Poincaré inequality. Writing \(a_k(t)=\mathbb E|\theta_t^\kappa(k)|^2\), the energy identity takes the form \[ \frac{d}{dt}\sum_k a_k^p = -8\pi^2\kappa p\sum_k |k|^2 a_k^p -2\pi^2p\sum_{k,j}w_j^2|\Pi_{j^\perp}k|^2 \big(a_k-a_{k-j}\big)\big(a_k^{p-1}-a_{k-j}^{p-1}\big), \] and the weighted lattice estimate gives control of weighted Fourier mass by directional lattice differences. The admissible Kraichnan-type models include an isotropic choice \[ w_k \sim |k|^{-d/2-\alpha}(\log|k|+1)^{-1} \] and an anisotropic shear model supported on coordinate axes, \[ w_k \sim |k|^{-1/2-\alpha}(\log|k|+1)^{-1} \quad\text{when }k_3=\cdots=k_d=0, \] subject to structural conditions on \[ S(r)=\sum_{|k|\le r}|k|^{1+\alpha}w_k^2 \] that exclude purely one-dimensional shears and ensure sufficient directional mixing. The paper explicitly states that intermittency corrections are expected only for moments \(>2\) and are not its focus; its contribution is a rigorous verification of the Obukhov–Corrsin regime with annular averaging and logarithmic losses, not a derivation of an intermittent spectrum [2512.02853]. ## 5. Nonstationary and buoyancy-dominated generalizations The Obukhov–Corrsin framework has also been generalized to nonstationary settings in which the scalar is globally active but effectively passive at inertial scales. In a turbulent puff evolving under the Oberbeck–Boussinesq equations, the strong-buoyancy regime occurs for \[ t\gg t_b \equiv \frac{u_0}{\beta g T_0}, \] and the bulk scales obey \[ L(t)\sim L_0 (t_0/t_b)^{1/4} (t/t_0)^{1/2},\qquad u_L \sim u_0 (t_0/t_b)^{1/4} (t/t_0)^{-1/2}, \] \[ T_L\sim T_0 (t_b/t_0)^{3/4} (t/t_0)^{-3/2}. \] The underlying assumption is an adiabaticity hypothesis: small-scale fluctuations relax rapidly to the slowly evolving large-scale background. Under this hypothesis, the inertial range remains K41/OC-like, with time-dependent fluxes inherited from the puff dynamics [2107.13178]. The generalized Obukhov–Corrsin prediction for temperature increments in the buoyancy-dominated puff is \[ \delta_r T (t)\sim \varepsilon_0^{1/2}\epsilon_0^{-1/6} r^{1/3} \left ( \frac{t_b}{t_0}\right )^{5/6} \left ( \frac{t}{t_0}\right )^{-5/3}, \] while the viscous-range scalar increments satisfy \[ \delta_r T (t) \sim r \left ( \frac{\varepsilon_0}{\nu}\right )^{1/2} \left ( \frac{t_0}{t_b}\right )^{-3/4} \left ( \frac{t}{t_0}\right )^{-2}. \] Intermittency is then incorporated through anomalous exponents imported from stationary turbulence, specifically \[ \sigma_4=0.06,\qquad \sigma_6=0.27,\qquad \xi_4=0.24,\qquad \xi_6=0.37. \] The DNS evidence reported in that work includes collapse of second-, fourth-, and sixth-order structure functions after applying the predicted temporal prefactors, ESS slopes consistent with the intermittency-corrected theory, and clear disagreement between the data and the no-intermittency lines. The same paper explicitly contrasts this with a Bolgiano-type scenario and states that such a scale-by-scale buoyancy balance is not observed in the simulations. The resulting interpretation is that buoyancy modifies the large-scale amplitudes and time dependence, but the inertial-range scalar cascade remains in a generalized intermittent Obukhov–Corrsin class [2107.13178]. ## 6. Scope, ambiguities, and recurrent misconceptions Across the literature, the phrase “intermittent Obukhov–Corrsin regime” is used in several related but non-identical senses. In homogeneous isotropic turbulence and in statistical analogues, it usually means that the second-order Obukhov–Corrsin law remains approximately valid while higher-order structure functions, flatness, hyperflatness, or increment PDFs display anomalous scaling and non-Gaussianity [1202.4113, 2501.09907]. In rigorous transport theory, it refers instead to supercritical regularity regimes, codimension-localized dissipation measures, and integrability-dependent constraints on the admissible pair \((u,\theta)\) [2207.06833, 2508.00115]. A recurrent misconception is to identify every rigorous derivation of an Obukhov–Corrsin spectrum with an intermittency result. The Kraichnan-type annular spectrum theorem does not define or prove an intermittent Obukhov–Corrsin regime in the turbulence-physics sense; it derives the expected second-order spectrum on geometric annuli, modulo logarithmic losses, from anomalous regularization. Its own discussion states that intermittency corrections are expected only for moments \(>2\) and are not being addressed [2512.02853]. Another ambiguity concerns dynamical versus statistical evidence. Image-based analyses may show spectra, structure functions, and anisotropy tests that are statistically consistent with passive-scalar intermittency, but such evidence is not a dynamical derivation from transport equations. Conversely, closure theories based on Lagrangian separation or finite-scale Lyapunov analysis can supply dynamical mechanisms for cascade and finite-time development without, by themselves, fixing anomalous multiscaling exponents. The intermittent Obukhov–Corrsin regime is therefore best understood as a technically structured extension of the Obukhov–Corrsin framework, not as a single universally standardized model. Source: https://www.emergentmind.com/topics/intermittent-obukhov-corrsin-regime