High gradient phenomena in two-dimensional vortex interactions

H. B. Yao*, N. J. Zabusky, D. G. Dritschel

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

34 Citations (Scopus)


Hyperdiffusion, a simple linear eddy diffusivity scheme, is commonly used in atmospheric and oceanic simulations because it increases the range of inertially behaving spatial scales for a given model resolution. Compared with molecular diffusion (which is utterly negligible in the atmosphere and oceans), hyperdiffusion more sharply confines the dissipation to the smallest scales of the numerical model. But is this all that hyperdiffusion does? In this paper, the inelastic interaction of two distributed vortices of unequal size is examined. Contour surgery (CS) simulations are compared with pseudospectral (PS) simulations employing hyperdiffusion or molecular diffusion. The example illustrates what is believed to be the most fundamental characteristic of two-dimensional (2-D) (and layerwise-2-D) vortex dynamics, namely, the formation of exceedingly high vorticity gradients. There is an excellent agreement between the hyperdiffusive PS and CS calculations at early times (i.e., for a few vortex rotation periods). Thereafter, significant discrepancies develop, beginning abruptly from the time when vorticity-gradient intensification is arrested by diffusion. A rapid inward erosion of the smaller of the two vortices then takes place. This erosion takes place under the joint action of (hyper) diffusion and stripping (the peeling of the vortex periphery by the external flow). With hyperdiffusion, the erosion is accompanied by a serious numerical artifact: a climb in the peak vorticity by 30% in this example. Eventually, the erosion reaches the vortex center and the vortex is sheared into a filament. In the CS calculation, there is no erosion, no climb in peak vorticity, and the vortex appears to last indefinitely. In the PS calculations, the viscosity or hyperdiffusion is adjusted according to the resolution to give the largest possible inertial range while ensuring numerical stability. It is found that vortices that are spanned by fewer than 10-20 grid points are eroded away in only a few vortex rotation periods (a time scale that is very much shorter than one would estimate from pure viscous decay). These findings bring into question the results of many 2-D turbulence simulations using hyperdiffusion, for hyperdiffusion simulates neither inviscid dynamics nor molecular-diffusive dynamics.

Original languageEnglish
Pages (from-to)539-548
Number of pages10
JournalPhysics of Fluids
Issue number3
Publication statusPublished - 1 Jan 1995


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