Tsutomu Imamura

Stationary Spectrum of Pseudo-Three-Dimensional Electrostatic Turbulence in Magnetized Plasmas

We present a stationary k -spectrum of pseudo-three-dimensional electrostatic plasma turbulence in a uniform, loss-less, magnetized plasma based on the model equation of Hasegawa and Mima; \(\partial/\partial t(\phi-\mbi{\nabla}^{2}\phi)+[\mbi{\nabla}\phi\times\hat{z}{\cdot}\mbi{\nabla}]\mbi{\nabla}^{2}\phi{=}0\). The k -spectrum for a potential φ k is given by =(1+ k 2 ) -1 (α+β k 2 ) -1 , where α and β are constants. The existence of the two constants, α and β, in the spectrum is the consequence of the two rugged constants of motion. The spectrum is obtained using three different methods, namely, the method of Gibbs distribution, of Hopfs equation and of Wiener Hermite expansion. Because of the similarity of the equation, the obtained spectrum should also apply to the hydrodynamic turbulence of geostrophic vortices.

Turbulent Flows near Flat Plates

The method to study the effect of the plate moving in the homogeneous or isotropic turbulence is presented. The crucial point of this method is to solve the Orr-Sommerfeld like equation, which is satisfied by the kernel of the Wiener-Hermite expansion of the velocity field, under the inhomogeneous boundary condition. In the special case of constant mean flow, our method gives the same result as that of Hunt and Graham and succeeds in explaining the experimental results of Thomas and Hancock. The method is also applied to the case of non-uniform mean flow, where the shear effect comes up.

An Exact Gaussian Solution for Two-Dimensional Incompressible Inviscid Turbulent Flow

An exact Gaussian solution for two-dimensional incompressible inviscid turbulence is obtained by using the Wiener-Hermite expansion with the time-dependent ideal random function. The one-time characteristic functional corresponding to this solution is found to agree with the one obtained by Cook and Taylor, but the present method gives more information about the time dependence of the ideal random function, which is useful for dealing with the many-time correlation and extending to the viscous turbulence.

Relative Diffusion of a Pair of Fluid Particles in Turbulence

The effective Hamiltonian method is used to find the distribution function describing the positions as a function of time of a given pair of fluid particles subject to convection in steady, incompressible, statistically isotropic turbulent flow. By assuming a joint Gaussian distribution for the velocity field, a reasonable time dependence is obtained for the statistical average of the distance between the pair of fluid particles. The basis for Richardsons four-thirds law is indicated and the proportionality constant C * is shown to be related to the Kolmogorov constant C by C * =3 1/3 C 1/2 /2 1/2 .

Two-Time Correlation Function of the Time-Dependent Ideal Random Function

Two-time correlation function of the time-dependent ideal random function is obtained as exp [-ω k 2 ( t 1 - t 2 ) 2 /2] where ω k is determined by means of a function which governs the time development of the ideal random function. The result compares well with the width of the frequency spectrum of the pseudo-three dimensional electrostatic plasma turbulence in a uniform, loss-less, magnetized plasma, and also with a numerical simulation for a plasma diffusion normally across a magnetic field

A Gaussian Solution for Drift Wave Turbulence and for Rossby Wave Turbulence

An exact Gaussian solution is obtained for the probability distribution of a turbulent field given by a nonlinear differential equation, which governs electrostatic potential for the drift wave in a pseudo-three dimensional plasma and depth of the air for the Rossby wave turbulence in the atmospheric pressure system. In the weakly turbulent case, it is obtained that the two-time correlation function of the solution shows the exponential decay. Anisotropy of the solution does not appear in the one-time correlation function but appears in the two-time correlation function.

Mechanism in Leading to Richardson's Four-Thirds Law

For the relative diffusion of a pair particles in turbulent flow, the mechanism in leading to Richardsons four-third law is studied analytically and numerically by using the effective Hamiltonian method. With this analysis, in connection with the Kolmogorovs spectrum and the scaling law, it is shown that the main contribution to the Richardsons law comes from a relatively small region of the inertial range, and there is a considerably large contribution from the vortexes whose diameters are larger than the relative distance between a pair of particles. It is also reported that the range of Richardsons law in the smaller scales in the turbulence with a multi-range spectrum is extended toward large scales beyond a typical length in the smaller scales.

The Lagrangian auto-velocity correlation function

An effective Hamiltonian method is developed to evaluate the distribution function of \(\bm{r}_{\alpha}\) (α=1, 2), which are the positions of the α-th particle at the time t α . By using this method the Lagrangian auto-velocity correlation function is expressed in terms of the Eulerian one. The spectrum of the Lagrangian one clearly shows ω -2 form under a specific model Eulerian correlation function which realizes the Kolmogorov spectrum.

Evaluation of Functional Integral

A new method of approximate evaluation of functional integral by means of recurrence formula is proposed. Effectiveness of this method is checked by using evaluation of Feynman kernel in the case of harmonic oscillator in one dimension. Application is made to the problem of 3-mode model. This method gives satisfactory results to this model.

An Exact Gaussian Solution for the Two-Dimensional Ideal Incompressible Magnetohydrodynamic Turbulence

An exact solution associated with a stationary Gaussian distribution is obtained for two-dimensional ideal incompressible magnetohydrodynamic turbulence by using the Wiener-Hermite expansion with a time-dependent ideal random function. The spectra are given by \(\langle\tilde{\phi}(\mbi{p}:t)\tilde{\phi}(\mbi{q}:t)\rangle{\propto}q^{-2}\delta(\mbi{p}+\mbi{q})\) and \(\langle\tilde{A}(\mbi{p}:t)\tilde{A}(\mbi{q}:t)\rangle{\propto}(\mbi{q}^{2}+\gamma)^{-1}\delta(\mbi{p}+\mbi{q})\) where \(\tilde{\phi}\) and \(\tilde{A}\) are the Fourier transforms of the stream function and the electro-magnetic potential respectively. A remarkable result is that the two-time correlation function \(\langle\tilde{\phi}(\mbi{k}:t)\tilde{A}(\mbi{p}:0)\rangle{=}0\) for an arbitrary t , but this does not necessarily mean the statistically independence of \(\tilde{\phi}\) and \(\tilde{A}\).