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Oceans in Motion: The Emergence of Coherent Structures


n climatology, the world is considered as several worlds at once: the atmosphere, the lithosphere (referring to landmasses and their contents), the cryosphere (polar caps, ocean-borne ice, and glaciers), and the hydrosphere (rivers and oceans). The last of these, the oceans, cover 71 percent of the Earth's surface, and while scientists have known for hundreds of years some basic features of ocean circulation-such long-lived currents as the warm Gulf Stream and the Northern Pacific "Kuroshio" near Japan-the details of smaller-scale currents and flows have yet to be represented adequately in computational simulations and hence in climate models. "We need extraordinary computational power to improve our models," said geophysicist Jeffrey B. Weiss, "and our experiments at SDSC have taken advantage of that power. We've found some important clues to the way in which relatively small-scale, energetic flows-ocean eddies and vortices-form, give way to instabilities, and re-form."

Figure 1. The Motion of the Ocean

These figures show potential vorticity in the upper layer of the simulated ocean, with positive and negative extrema represented by red and blue, respectively, at increasing horizontal resolution-from left, 25 km, 12.5 km, 6.25 km, and 3.13 km.

Weiss is a professor in the University of Colorado's Program in Atmospheric and Oceanic Sciences (PAOS). In his work at SDSC, he is leading one of the projects within the research program of geophysicist James C. McWilliams, who is the Louis B. Slichter Professor of Earth Sciences in the Department of Atmospheric Sciences, UCLA, and head of UCLA's Center for Earth Systems Research.

"Professor Weiss's project is an important one within our program of investigations of the detailed physics of ocean circulation," McWilliams said. "Properly diagnosing oceanic transport is crucial for the development of accurate climate models." Because the oceans cover more than 70 percent of the Earth's surface and absorb most of the heat coming to Earth from the Sun, oceanic heat transport largely governs the planet's climate on the scale of decades to centuries. In general, heat is transferred poleward from the equator, but the details of heat and other transports are not well realized in oceanic general circulation models (OGCMs).

The computer time used is part of a grant to McWilliams made by the PACI program's National Resource Allocation Committee, whose mission is to ensure that researchers of national stature have adequate computational support from a bundle of national resources. McWilliams is widely known as among the first to call the attention of the oceanographic community to such phenomena as the long-lived, deep Atlantic vortices called Gulf Stream rings, and he has long studied the formation of coherent structures within turbulent fluid flows in the oceans, atmosphere, and solar system. (A canonical example of such a structure is Jupiter's Great Red Spot, a planetary storm that has persisted for several hundred years.)

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"The very large-scale currents, like the Gulf Stream and the Kuroshio, run for thousands of kilometers across their respective basins," Weiss notes. "But coexisting with these currents are numerous highly energetic vortex motions at relatively small scales-tens to hundreds of kilometers. These constitute the medium or 'mesoscale' motions, which contribute substantially to fluxes of material properties." But that contribution has been difficult to model in OGCMs, for example, because of limitations in computational resources that restrict the problem size and force scientists to use ad hoc representations of energy transfer on small scales.

Using the Cray T3E and Blue Horizon supercomputers at SDSC, Weiss and colleagues have carried out a series of numerical experiments, each at a higher degree of horizontal resolution. In all the experiments, the ocean basin was represented as a box of 3200 ¤ 3200 ¤ 5 km-dimensions about equivalent to the Atlantic ocean basin. The results are to be published in Geophysical Research Letters.

The only other difference among the simulations was the value of a quantity called the Reynolds number (after British physicist and engineer Osborne Reynolds, 1842-1912). The Reynolds number (Re) is a ratio that shows the effect of viscosity in a fluid flow. For a smooth flow, Re may be quite low; the higher Re goes, the more likely the flow is to contain eddies and vortices and ultimately to become fully turbulent.

Because the computational cost increases rapidly with resolution, the scientists chose to address the problem in the context of simplified "quasigeostrophic" (nearly two-dimensional) equations in the geometrically idealized box domain. They were thus able to conduct simulations over a range of previously unexplored grid resolutions and for multiyear time intervals required to understand statistical trends. "More detailed analyses of these idealized solutions can be used to guide the development of subgrid parameterizations for more realistic models," Weiss pointed out.

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Figure 2. Oceans in Motion

Potential vorticity of the simulated ocean at the highest horizontal resolution-1.56 km.

Some results are shown in Figures 1 and 2. In the first image, coherent vortices are beginning to emerge. "When we examine this flow as time-dependent, using computer animation," Weiss said, "we find that the vortices originate as 'rings,' evolving from jet meanders and becoming pinched-off, elliptical structures. They have diameters on the order of 100 km and lifetimes of several months, which is inconsistent with oceanic observations." The Gulf Stream rings have been shown to have lifetimes of several years, for example. They are characterized, in fact, by their unusual persistence and ability to maintain temperature and chemical properties distinct from the surrounding ocean.

"When we raise Re and increase the horizontal resolution, the flows are brought more in line with observations," Weiss said. In Figure 2, vortex structures covering a wide range of scales now populate almost the entire region south of the main jet. Some small vortices endure for as long as a year in these simulations. Vortex-vortex mergers are also frequent and conspicuous in the flow animations. Ultimately, Weiss and colleagues speculated, this process limits the growth of the vortex population and the whole may tend asymptotically toward a nearly constant state.

"The explosion of small and mesoscale structure at higher Re is significant for developing appropriate ways to represent this activity in oceanic general circulation models," Weiss said. Calculations of such quantities as the eddy kinetic energy and its distribution over the basin, poleward fluxes of fluid elements, and other basin-averaged quantities yield important insight into the behavior of mesoscale processes in the ocean.

"While the time-mean kinetic energy is relatively independent of Re," Weiss said, "the emergence of vortices contribute to the increase of eddy kinetic energy and meridional vorticity flux as Re increases." The rate of increase slows slightly at the highest Re, he noted, indicating the possibility of a regime in which eddy variability decouples from further increases in Re.

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"The abundance and variety of coherent ocean vortices challenges us to develop better modeling skill and theoretical understanding of their dynamics and roles in the general circulation," McWilliams said. "These simulations represent contributions to our understanding" of the processes that control the rates of change in vortex conformations and influences. In particular, he noted, they exhibit recurrent, spatially localized patterns; coherent structures that are close to stationary or evolve with considerably self-similarity; unusual longevity; and spatial isolation from other coherent structures.

If these are the essential characteristics of actual ocean eddies and vortices, Weiss added, the incorporation of this behavior into larger models should produce more realistic outcomes and better predictions of climatic variables. Plans are to continue using Blue Horizon to verify what appears to happen as Re and horizontal resolution are increased. "This painstaking modeling is increasingly important to geophysicists, particularly in an epoch when they and policy makers who depend upon them are considering rapid environmental changes," McWilliams said. -MM

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Project Leader
Jeffrey B. Weiss
University of Colorado, Boulder

James C. McWilliams,
Pavel S. Berloff

Brett DiFrischia
University of Colorado, Boulder

Andrew Siegel
University of Chicago

Juri Toomre
Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder

Irad Yavneh
Israel Institute of Technology, Haifa