Energy Cascade Process Research Articles

Cascade model of turbulence

A simple low-order dynamic system is proposed to simulate the energy cascade process of a fully developed hydrodynamic turbulence, and a series of numerical experiments is conducted. The main structural properties of this dynamic system are the same as those of the normalized Navier–Stokes equation. A quick estimation of its first Lyapunov exponent λ1 is made by calculating the chaos index at sampling points on trajectories. The large positive values (20–22) of λ1 indicate that the system is highly chaotic and that it has a strange attractor. After the transient period the dynamic system moves along its strange attractor, and a very wide inertial range is observed, in which the energy flux across the wavenumbers is a constant and equal to the energy dissipation rate. Moreover, the Kolmogorov − (5)/(3) spectrum is obtained as a statistical property of chaotic trajectories along the strange attractor.

Simple multifractal cascade model for fully developed turbulence.

A simple model is presented for the energy-cascading process in the inertial range that fits remarkably well the entire spectrum of scaling exponents for the dissipation field in fully developed turbulence. The scheme is a special case of weighted curdling and its one-dimensional version is a simple generalized two-scale Cantor set with equal scales but unequal weights (with ratio\ensuremath{\sim}(7/3). This set displays all the measured multifractal properties of one-dimensional sections of the dissipation field.

The power spectrum of interplanetary Alfvénic fluctuations: Derivation of the governing equation and its solution

A model is presented to explain the radial evolution of the power spectra of interplanetary Alfvénic fluctuations found by Bavassano et al. (1982a) based on the magnetic field data of Helios 1 and 2. It is assumed in this model that Alfvénic fluctuations represent an asymmetric state of MHD turbulence in which most of the fluctuations are in the Alfvénic wave mode propagating outward from the sun, with a small part of the fluctuations in the wave mode propagating inward. There is weak nonlinear interaction between the two modes. It is also assumed that the turbulence will not become dissociated from the sources that create the waves propagating inward and that the weak nonlinear interaction will remain between 0.3 AU and 1 AU. The nonlinear interaction results in an energy cascading process. Both the effects of the cascading process and the effects of the slow variation of the solar wind on the outward propagating waves determine the radial evolution of the power spectra. Starting with the magnetohydrodynamic equations and deriving the equations governing fluctuations and correlation moments, we finally get an equation which describes the power spectrum. We present an analytic solution. The radial evolution of the power spectra of the fluctuations given by the analytical solution, from 0.29 AU to 0.87 AU, is in agreement with the observation of Helios 1 and 2 in the following aspects: (1) The spectral slope increases (in absolute value) for the frequency range lower than 2.5×10−3Hz while the slope remains almost unchanged for the frequency range higher than 10−1.5Hz. (2) The radial gradient of the power spectrum densities increases with increasing frequency. The dissipation length remains nearly unchanged for the frequency rangef≥ 10−2Hz. (3) The radial variation of 〈b²〉 is approximatelyr−3.56. (4) The ratio of 〈b²〉/Bo² is nearly a constant.

Radar observations of two‐dimensional turbulence in the equatorial electrojet, 3. Nighttime observations of Type 1 waves

The large vertically directed 50‐MHz radar at Jicamarca has sufficient sensitivity to study nighttime echoes from the equatorial electrojet in detail with good resolution. Here we concentrate on type 1 (‘two‐stream’) echoes. We find that (1) these echoes sometimes dominate the spectrum (which is never the case in daytime for a vertically directed radar), (2) they are observed over a much wider range of altitudes than they are during daytime, (3) they show an asymmetry which reverses from day to night (downgoing waves are more common at night), (4) the magnitude of the mean Doppler shift appears to increase somewhat with altitude, and (5) there is evidence of large‐scale (kilometers) structure as well as intermediate (tens or hundreds of meters) and small (the wavelengths of meters or less which are directly responsible for radar echoes). We describe a model which qualitatively accounts for all the important characteristics of the observations. Long and intermediate waves with roughly horizontal k are initially unstable, and a complicated two‐dimensional energy cascade process develops, with energy transfers probably going in both directions in k space. The asymmetry in the echoes can be explained if either the density depletions at the intermediate wavelengths are stronger than the enhancements or the positive density gradients are steeper than the negative.