Self-similar solutions to the hypoviscous Burgers and SQG equations at criticality

After reviewing the source-type solution of the Burgers equation with standard dissipativity, we study the hypoviscous counterpart of the Burgers equation. 1) We determine an equation that governs the near-identity transformation underlying its self-similar solution. 2) We develop its approximation scheme and construct the first-order approximation. 3) We obtain the source-type solution numerically by the Newton-Raphson iteration scheme and find it to agree well with the first-order approximation. Implications of the source-type solution are given, regarding the possibility of linearisation of the hypoviscous Burgers equation. Finally we address the problems of the incompressible fluid equations in two dimensions, centering on the surface quasi-geostrophic equation with standard and hypoviscous dissipativity

Remarks on the principles of statistical fluid mechanics

This is an idiosyncratic survey of statistical fluid mechanics centering on the Hopf functional differential equation. Using the Burgers equation for illustration, we review several functional integration approaches to the theory of turbulence. We note in particular that some important contributions have been brought about by researchers working on wave propagation in random media, among which Uriel Frisch is not an exception. We also discuss a particular finite-dimensional approximation for the Burgers equation

Oscillatory damping in long-time evolution of the\ud surface quasi-geostrophic equations with generalised\ud viscosity: a numerical study

We study numerically the long-time evolution of the surface quasi-geostrophic\ud equation with generalised viscosity of the form $(-¥Delta)^¥alpha$, where global regularity has\ud been proved mathematically for the subcritical parameter range $¥alpha ¥geq ¥frac{1}{2}$. Even in\ud the supercritical range, we have found numerically that smooth evolution persists, but\ud with a very slow and oscillatory damping in the long run. A subtle balance between\ud nonlinear and dissipative terms is observed therein. Notably, qualitative behaviours\ud of the analytic properties of the solution do not change in the super and subcritical\ud ranges, suggesting the current theoretical boundary $¥alpha =¥frac{1}{2}$ is of technical nature

Growth rate analysis of scalar gradients in generalized surface quasigeostrophic equations of ideal fluids

The growth rates of scalar gradients are studied numerically and analytically in a family of two-dimensional (2D) incompressible fluid equations related to the surface quasigeostrophic (SQG) equation. The active scalar is related to the stream function ψ by θ=(−△)α/2ψ (0⩽α⩽2). A notable difference is observed in a comparison of the instantaneous growth rates in Lp and in L∞ norms, depending on the stage of the time evolution. The crux is the phase-shift effect of singular integral operators, which displaces the peak location of the scalar gradient from that of the strain rate. On this basis, a method of detecting such a dislocation is proposed in view of the importance of their coalescence needed for a possible blow-up. Moreover, it is found in the long-time evolution that a solution of the SQG equation (whose regularity is not known) is less singular than that of the 2D Euler equations (known to be regular) on the time interval covered by this computation. This consistently expands an earlier observation by Majda and Tabak [Physica D 98, 515 (1996).] in some detail. A 1D model problem is discussed to illustrate the present method, and extensions to the 3D case are also are briefly discussed

Intermittency and local Reynolds number in Navier-Stokes turbulence: A cross-over scale in the Caffarelli-Kohn-Nirenberg integral

We study space-time integrals, which appear in the Caffarelli-Kohn-Nirenberg (CKN) theory for the Navier-Stokes equations analytically and numerically. The key quantity is written in standard notations δ(r)=1/(νr)∫Qr(∇,u)2dxdtδ(r)=1/(νr)∫Qr∇u2dxdt, which can be regarded as a local Reynolds number over a parabolic cylinder Q r . First, by re-examining the CKN integral, we identify a cross-over scale r∗∝L(∥∇u∥2L2¯¯¯¯¯¯¯¯¯¯¯¯∥∇u∥2L∞)1/3,r*∝L‖∇u‖L22¯‖∇u‖L∞21/3, at which the CKN Reynolds number δ(r) changes its scaling behavior. This reproduces a result on the minimum scale r min in turbulence:r2min∥∇u∥∞∝ν,rmin2‖∇u‖∞∝ν, consistent with a result of Henshaw et al. [“On the smallest scale for the incompressible Navier-Stokes equations,” Theor. Comput. Fluid Dyn.1, 65 (1989)10.1007/BF00272138]. For the energy spectrum E(k) ∝ k −q   (1 < q < 3), we show that r * ∝ ν a with a=43(3−q)−1a=43(3−q)−1. Parametric representations are then obtained as ∥∇u∥∞∝ν−(1+3a)/2‖∇u‖∞∝ν−(1+3a)/2 and r min ∝ ν3(a+1)/4. By the assumptions of the regularity and finite energy dissipation rate in the inviscid limit, we derive limp→∞ζpp=1−ζ2limp→∞ζpp=1−ζ2 for any phenomenological models on intermittency, where ζ p is the exponent of pth order (longitudinal) velocity structure function. It follows that ζ p ⩽ (1 − ζ2)(p − 3) + 1 for any p ⩾ 3 without invoking fractal energy cascade. Second, we determine the scaling behavior of δ(r) in direct numerical simulations of the Navier-Stokes equations. In isotropic turbulence around R λ ≈ 100 starting from random initial conditions, we have found that δ(r) ∝ r 4 throughout the inertial range. This can be explained by the smallness of a ≈ 0.26,with a result that r * is in the energy-containing range. If the β-model is perfectly correct, the intermittency parameter a must be related to the dissipation correlation exponent μ as μ=4a1+a≈0.8,μ=4a1+a≈0.8, which is larger than the observed μ ≈ 0.20. Furthermore, corresponding integrals are studied using the Burgers vortex and the Burgers equation. In those single-scale phenomena, the cross-over scale lies in the dissipative range.The scale r * offers a practical method of quantifying intermittency. This paper also sorts out a number of existing mathematical bounds and phenomenological models on the basis of the CKN Reynolds number

Asymptotics and numerics of a family of two-dimensional generalized surface quasi-geostrophic equations

We study the generalised 2D surface quasi-geostrophic (SQG) equation, where the active scalar is given by a fractional power α of Laplacian applied to the stream function. This includes the 2D SQG and Euler equations as special cases. Using Poincaré’s successive approximation to higher α-derivatives of the active scalar, we derive a variational equation for describing perturbations in the generalized SQG equation. In particular, in the limit α → 0, an asymptotic equation is derived on a stretched time variable τ = αt, which unifies equations in the family near α = 0. The successive approximation is also discussed at the other extreme of the 2D Euler limit α = 2–0. Numerical experiments are presented for both limits. We consider whether the solution behaves in a more singular fashion, with more effective nonlinearity, when α is increased. Two competing effects are identified: the regularizing effect of a fractional inverse Laplacian (control by conservation) and cancellation by symmetry (nonlinearity depletion). Near α = 0 (complete depletion), the solution behaves in a more singular fashion as α increases. Near α = 2 (maximal control by conservation), the solution behave in a more singular fashion, as α decreases, suggesting that there may be some α in [0, 2] at which the solution behaves in the most singular manner. We also present some numerical results of the family for α = 0.5, 1, and 1.5. On the original time t, the H 1 norm of θ generally grows more rapidly with increasing α. However, on the new time τ, this order is reversed. On the other hand, contour patterns for different α appear to be similar at fixed τ, even though the norms are markedly different in magnitude. Finally, point-vortex systems for the generalized SQG family are discussed to shed light on the above problems of time scale

Study of the 3D Euler equations using Clebsch potentials: dual mechanisms for geometric depletion (vol 31, pg R25, 2018)

After the publication of [1], it has come to the author’s attention that a class of Clebsch potentials for the Kida-Pelz flow, similar to what was derived in Appendix B of [1], has been studied in detail in [2]. We also note that there are typos in the formulas for one such example in [3], and these are corrected in [1]