In decaying two-dimensional Navier–Stokes turbulence, Batchelor's similarity hypothesis fails due to the existence of coherent vortices. However, it is shown that decaying two-dimensional turbulence governed by the Charney–Hasegawa–Mima (CHM) equation(∂/∂t)(∇2φ−λ2φ) +J(φ, ∇2φ) = D,where D is a damping, is described well by Batchelor's similarity hypothesis for wave numbers k [Lt ] λ (the so-called AM regime). It is argued that CHM turbulence in the AM regime is a more ‘ideal’ form of two-dimensional turbulence than is Navier–Stokes turbulence itself.
A Green's function for a generalized two-dimensional (2D) fluid in an unbounded domain (the so-called α turbulence system) is discussed. The generalized 2D fluid is characterized by a relationship between an advected quantity q and the stream function ψ : namely, q=-(-Δ){α/2}ψ . Here, α is a real number and q is referred to as the vorticity. In this study, the Green's function refers to the stream function produced by a delta-functional distribution of q , i.e., a point vortex with unit strength. The Green's function has the form G{(α)}(r)∝r{α-2} , except when α is an even number, where r is the distance from the point vortex. This functional form is known as the Riesz potential. When α is a positive even number, the logarithmic correction to the Riesz potential has the form G(r){(α)}∝r{α-2} ln r . In contrast, when α is a negative even number, G{(α)} is given by the higher-order Laplacian of the delta function. The transition of the small-scale behavior of q at α=2 , a well-known property of forced and dissipative α turbulence, is explained in terms of the Green's function. Moreover, the azimuthal velocity around the point vortex is derived from the Green's function. The functional form of the azimuthal velocity indicates that physically realizable systems for the generalized 2D fluid exist only when α≤3 . The Green's function and physically realizable systems for an anisotropic generalized 2D fluid are presented as an application of the present study.
The Charney-Hasegawa-Mima equation, with random forcing at the narrow band wave-number region, which is set to be slightly larger than the characteristic wave number , evaluating the inverse ion Larmor radius in plasma, is numerically studied. It is shown that the Fourier spectrum of the potential vorticity fluctuation in the development of turbulence with an initial condition of quiescent state obeys a dynamic scaling law for kӶ. The dimensional analysis with the assumption that the energy transfer rate ⑀ in the inverse cascade is constant with time leads to the scaling form S(k,t) ϭ 1/2 ⑀ 5/4 t 7/4 F"k/k(t)…͓k(t)ϳ 3/4 ⑀ Ϫ1/8 t Ϫ3/8 ͔ with a scaling function F(x), which turns out to be in good agreement with numerical experiments. ͓S1063-651X͑97͒08205-6͔
The enstrophy inertial range of a family of two-dimensional turbulent flows, so-called -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 -turbulence. An asymptotic scaling behavior of the derived enstrophy spectrum precisely explains the transition between the local and non-local transfers at ¼ 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 -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 local transfer is dominated by the small scale vortical structure, while it for the non-local transfer is done by the smooth and thin striped structures caused by the random straining motions.
The local and nonlocal characteristics of triad enstrophy transfer in the enstrophy inertial range of generalized two-dimensional turbulence, so-called alpha turbulence, are investigated using direct numerical simulations, with a special emphasis on alpha=1 , 2, and 3. The enstrophy transfer via nonlocal triad interactions dominates the transfer dynamics in the enstrophy inertial range, irrespective of alpha . However, the contributions from more local interactions to the total enstrophy transfer increase as alpha decreases. The results are discussed in connection with the local and nonlocal transition of the enstrophy transfer at alpha=2 expected from the phenomenological scaling theory. The specific nature of the enstrophy transfer in surface quasigeostrophic turbulence (alpha=1) is also discussed.
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