Scale dependence of divergence and vorticity of near-surface flows in the sea

Abstract

An attempt is made to find relations of ensemble averages ofQ ²/2 andζ ²/2 to length scale,L, whereQ andζ are the area-averaged horizontal divergence and vertical component of vorticity, respectively. To calculateQ andζ the polygon, loop and crossing methods have been used in Part 1. HereL is the square root of an area of a polygon connecting drifters or of a domain enclosed by a looping trajectory or by a closed integration curve. In the light of an extended −5/3 power law for energy spectrum, the ensemble averages ofQ ²/2 andζ ²/2 are shown to be proportional toL ^−4/3 not only in the three-dimensionally isotropic range but also in the mesoscale range (3 m≲L≲30 km). They approach constants at microscales (L≲1 cm), and become proportional toL ⁻² at macroscales (L≳100 km). By the polygon method, unbiased random samples ofQ ²/2 andζ ²/2 are liable to be drawn from a population. By the loop and crossing methods, the same is true ofQ ²/2, but samples ofζ ²/2 much greater than the average are to be drawn for the following reason. The loop and crossing methods are intentionally applied to vortices of various scales from a tidal vortex to the Antarctic Circumpolar Gyre. Since vorticity is locally concentrated within vortices and shear zones distributed intermittently, the loop and crossing methods always catch greatest values of vorticity but the polygon method does not. Values ofQ ²/2 andζ ²/2 at 30 m depth are reduced, at the lowest, to one hundredth of those at the surface. Those around the Kuroshio are, at the highest, one hundred times those in the eastern North Pacific.