Numerical Simulation

We have studied the evolution of a random initial realization of the stream function. The Euler equations in the stream function-vorticity formultion were solved by the fully-dealiased, Spectral method of Orzag [13]. The program to implement this method was developed by Prof. R. Salmon of Scripps Institute of Oceanography. For this particular simulation a cutoff wavenumber of $32$ and an eddy viscosity term $-\sigma\Delta^2C$, where $\sigma=1.E-6$ were employed. Starting from an initial stream function with amplitude proportional to $k^2/(1+k^6)$ and a random phase, the method employs a leap-frog time differencing with smoothing every $20$ time steps to eliminate the spurious computational mode. Since the energy is normalized to be one and a time step is $0.01$, an eddy turnover time consists of nearly $100$ time steps. Every $10$ time steps the stream function was output onto disk. Thus in the figures a label Record 17 indicates a time of $1.7$. From the stream function we have produced contour plots for the vorticity, magnitude of strain, the quantity $\lambda\overline{\lambda}-C^2$, the magnitude of $\frac{d\lambda}{dt}$ and the magnitude of the gradients of vorticity. In addition, we have produced graphs of the spectra for the vorticity (=rate of strain) and the gradients of the vorticity.

Since it is necessary to use an eddy-viscosity term to prevent reflection of energy at the cutoff wavenumber, the total enstrophy is decaying with time. See Figure 1. Nevertheless, the qualitative features of the solution are quite interesting.

The random initial gradients of vorticity have evolved by frame $11$ into a tightly localized patterm that is related to the central hyperbolic region. There are two hyperbolic regions. The stronger one located at the center of the frame and a weaker one located in the upper left corner, with label on the left. These hyperbolic regions may be identified with reference to either the vorticity or ${\rm tr}A^2$ contours.

The strain appears to be confined almost solely to the hyperbolic region. The magnitude of $\frac{d\lambda}{dt}$ is highly intermittent; being confined to several small regions that appear to migrate from left to right during the evolution.

A curious event occurs in frames $26$ through $37$. In these frames the vorticity gradients streaming from the hyperbolic centers are observed to interact and intensify. This phenomenon is strikingly similar to the interaction between hyperbolic (unstable) fixed points observed for general area preserving maps [3]. This process is associated with the first peak in the value of the total vorticity gradients; the following decrease being associated with the increased dissipation caused by the transfer of enstrophy to the higher wavenumbers. The further evolution of the system appears to involve the wrapping of vorticity gradients about the hyperbolic centers, by folding and stretching of the fluid in these regions.

Further numerical studies of the phenomena discussed in this paper are presented by M. Brachet et. al. in reference [16].