Vortex Structure in Planetary Turbulence

In a series of calculations aimed at understanding some essential features of rotating flows, McWilliams and his colleagues posed an idealized problem. They neglected the influence of outside forces, such as the planet's geometry and its global circulation, on overall turbulence. "We believe," McWilliams says, "that the behavior exhibited in such idealized situations underlies many of the particular phenomena that occur in planetary flows."

Availability of the C90 made it possible to take a different numerical approach than prior computations of planetary turbulence. McWilliams used a three-dimensional grid with higher resolution than prior calculations and with an equal number of grid points (320) in each direction. To accommodate computing limitations, most prior calculations used a reduced number of points in one dimension. McWilliams used all 16 C90 processors for hundreds of hours at a rate of 6.5 billion calculations a second.

Parallel processing, he says, was a natural approach for these calculations because a multi-dimensional grid was used. The same equations were solved at each processor, and "we simply gave different parts of the domain to different processors." This algorithmic approach and the C90's capabilities, he says, produced the fast sustained speed, one of the fastest that has run at the Pittsburgh Supercomputing Center.

    [Image One]         [Image Two]         [Image Three]           [Image Four]

This sequence, from McWilliams' simulation of planetary turbulence shows how the flow changes from random and disoriented to coherent, organized structure. The images represent a property called "potential vorticity," an indicator of turbulence. Color shows negative vorticity (red, orange, yellow) and positive vorticity (green, blue, purple). The sequence from left to right shows dimensionless time increasing (5.0, 10.0, 25.6, and 72.1).

As time proceeds, explains McWilliams, small centers of vorticity interact and aggregate and empty out the space between them. "They become fewer, sparser, larger, less deformed." The centers also cluster in the vertical dimension until they reach an approximate end state: two columns of vorticity, one positive and one negative. "All of these little elemental vortices are captured into two long, grainy columns. This stationary stable state is a global attractor that this whole system is moving toward."

These results, says McWilliams, disprove the classical beliefs about turbulence. "What we established from disordered conditions was that the vortices eventually become ordered. During the entire process, the turbulence goes from random and disordered to structured and complex to structured and simple."

These findings confirm ideas that have emerged during the last 10 years: that turbulence is far more ordered, or coherent, than previously believed. "This point of view -- order out of chaos -- certainly finds support from modern dynamical systems theory," says McWilliams. "The implication is that scientists will increasingly try to interpret what happens in turbulence in terms of what the coherent vortices are doing. This is revolutionary, although this revolution has been underway for a decade or more."

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