Takahiro Iwayama

Unified Scaling Theory for Local and Non-local Transfers in Generalized Two-dimensional Turbulence

The enstrophy inertial range of a family of two-dimensional turbulent flows, so-called α-turbulence, is investigated theoretically and numerically. Introducing the large-scale correction into Kraichnan–Leith–Batchelor theory, we derive a unified form of the enstrophy spectrum for the local and non-local transfers in the enstrophy inertial range of α-turbulence. An asymptotic scaling behavior of the derived enstrophy spectrum precisely explains the transition between the local and non-local transfers at α= 2 observed in the recent numerical studies by Pierrehumbert et al. [Chaos, Solitons & Fractals 4 (1994) 1111] and Schorghofer [Phys. Rev. E 61 (2000) 6572]. This behavior is comprehensively tested by performing direct numerical simulations of α-turbulence. It is also numerically examined the validity of the phenomenological expression of the enstrophy transfer flux responsible for the derivation of the transition of scaling behavior. Furthermore, it is found that the physical space structure for the loca...

Differential geometric structures of stream functions: incompressible two-dimensional flow and curvatures

The Okubo–Weiss field, frequently used for partitioning incompressible two-dimensional (2D) fluids into coherent and incoherent regions, corresponds to the Gaussian curvature of the stream function. Therefore, we consider the differential geometric structures of stream functions and calculate the Gaussian curvatures of some basic flows. We find the following. (I) The vorticity corresponds to the mean curvature of the stream function. Thus, the stream-function surface for an irrotational flow and that for a parallel shear flow correspond to the minimal surface and a developable surface, respectively. (II) The relationship between the coherency and the magnitude of the vorticity is interpreted by the curvatures. (III) Using the Gaussian curvature, stability of single and double point vortex streets is analyzed. The results of this analysis are compared with the well-known linear stability analysis. (IV) Conformal mapping in fluid mechanics is the physical expression of the geometric fact that the sign of the Gaussian curvature does not change in conformal mapping. These findings suggest that the curvatures of stream functions are useful for understanding the geometric structure of an incompressible 2D flow.

Universal spectrum in the infrared range of two-dimensional turbulent flows

The low-wavenumber behavior of decaying turbulence governed by the generalized two-dimensional (2D) fluid system, the so-called α-turbulence system, is investigated theoretically and through direct numerical simulation. This system is governed by the nonlinear advection equation for an advected scalar q and is characterized by the relationship between q and the stream function ψ: q=−(−∇2)α/2ψ. Here, the parameter α is a real number that does not exceed 3. The enstrophy transfer function in the infrared range (k → 0) is theoretically derived to be TαQ(k→0)∼k5 using a quasi-normal Markovianized model of the generalized 2D fluid system. This leads to three canonical cases of the infrared enstrophy spectrum, which depend on the initial conditions: Qα(k → 0) ∼ Jk, Qα(k → 0) ∼ Lk3, and Qα(k → 0) ∼ Ik5, where J, L, and I are various integral moments of two-point correlation for q. The prefactors J and L are shown to be invariants of the system, while I is an increasing function of time. The evolution from a narr...

Linear stability analysis of parallel shear flows for an inviscid generalized two-dimensional fluid system

The linear stability of parallel shear flows for an inviscid generalized two-dimensional (2D) fluid system, the so-called ? turbulence system, is studied. This system is characterized by the relation q = ?( ? ?)?/2? between the advected scalar q and the stream function ?. Here, ? is a real number not exceeding 3 and q is referred to as the generalized vorticity. In this study, a sufficient condition for linear stability of parallel shear flows is derived using the conservation of wave activity. A stability analysis is then performed for a sheet vortex that violates the stability condition. The instability of a sheet vortex in the 2D Euler system (? = 2) is referred to as a Kelvin?Helmholtz (KH) instability; such an instability for the generalized 2D fluid system is investigated for 0 < ? < 3. The sheet vortex is unstable in the sense that a sinusoidal perturbation applied to it grows exponentially with time. The growth rate is finite and depends on the wavenumber of the perturbation as k3 ? ? for 1 < ? < 3, where k is the wavenumber of the perturbation. In contrast, for 0 < ? ? 1, the growth rate is infinite. In other words, a transition of the growth rate of the perturbation occurs at ? = 1. A physical model for KH instability in the generalized 2D fluid system, which can explain the transition of the growth rate of the perturbation at ? = 1, is proposed.

Anomalous eddy viscosity for two-dimensional turbulence

We study eddy viscosity for generalized two-dimensional (2D) fluids. The governing equation for generalized 2D fluids is the advection equation of an active scalar q by an incompressible velocity. The relation between q and the stream function ψ is given by q = − (−∇2)α/2ψ. Here, α is a real number. When the evolution equation for the generalized enstrophy spectrum Qα(k) is truncated at a wavenumber kc, the effect of the truncation of modes with larger wavenumbers than kc on the dynamics of the generalized enstrophy spectrum with smaller wavenumbers than kc is investigated. Here, we refer to the effect of the truncation on the dynamics of Qα(k) with k < kc as eddy viscosity. Our motivation is to examine whether the eddy viscosity can be represented by normal diffusion. Using an asymptotic analysis of an eddy-damped quasi-normal Markovian (EDQNM) closure approximation equation for the enstrophy spectrum, we show that even if the wavenumbers of interest k are sufficiently smaller than kc, the eddy viscosity...

SELF-SIMILARITY OF VORTICITY DYNAMICS IN DECAYING TWO-DIMENSIONAL TURBULENCE

A new similarity theory is proposed for decaying two-dimensional Navier–Stokes turbulence, including the viscous range, which encompasses all Reynolds numbers and various degrees of hyperviscosity. In the high Reynolds number limit where the energy E is invariant, the theory predicts the enstrophy decay law Q ∼ t−1/p, where t is time and p is the degree of hyperviscosity (p = 1 is the usual Laplacian viscosity). This is at variance with the vortex scaling theory of Carnevale et al. (1991). However it is consistent with previously published numerical simulations using the usual viscosity. That enstrophy decay in the high Reynolds number limit may depend on the degree of hyperviscosity suggests that the inviscid limit is singular. Indeed, our similarity theory based on the inviscid equations predicts an upscale energy flux for all wavenumbers, in violation of basic physical constraints. This may be part of the reason for the failure of Batchelor’s (1969) decay law E ∼ t0, Q ∼ t−2.

Self-Similarity of Decaying Two-Dimensional Turbulence governed by the Charney—Hasegawa—Mima Equation

In decaying two-dimensional Navier—Stokes turbulence, Batchelor’s simi larity hypothesis fails due to the existence of coherent vortices. However, it has recently been shown that in decaying two-dimensional turbulence governed by the Charney—Hasegawa—Mima (CHM) equation

A Comprehensive Analysis of Nonlinear Corrections to the Classical Ekman Pumping

\frac{\partial }{{\partial t}}\left( {{\nabla ^2}\varphi - {\lambda ^2}\varphi } \right) + J\left( {\varphi ,{\nabla ^2}\varphi } \right) = D