Geostrophic Scatter Diagrams and Potential Vorticity Dynamics

Abstract

A scatter diagram may be constructed by choosing an appropriate closed or open horizontal curve in physical space and plotting the value of any scalr quantity q against the geostrophic streamfunction ? for each data point on the curve. The area enclosed on the scatter diagram is equal to the net geostrophic advective flux of q across the chosen curve in physical space. When q is the (quasi-geostrophic) potential vorticity Q, and suitable normalizations are adopted, this result may he exploited to derive measures of departure from free-mode form Q)= Q(?) along the curve in physical space. For a certain class of open space curves, an appropriate measure is the width-to-length ratio of the circuit in (?, Q) space. Most scatter diagrams that have appeared in the literature included the (?, Q) points corresponding to all the data or grid points within a given horizontal domain. The significance of the area enclosed on these diagrams is less clear, but the spread about some curve Q) = Q(?) is evidently a qualitative measure of the extent to which the flow deviates from free-mode form. For steady or time-averaged flows which are approximately of this form, the gradient dQ/d? of the scatter diagram may be used to infer some properties of the forcing and dissipative processes acting. When dissipation is principally due to Qtransfer by transient eddy motion (or viscosity), the key diagnostic relation iswhere S is the potential vorticity forcing, K the lateral eddy (or viscous) v the horizontal velocity, and the integrals are taken over and around any region enclosed by a mean streamline. Hence dQ/d?is often negative. corresponding to two common properties of quasi-geostrophic circulations: that the eddy motion (or viscosity) transport Q down its mean gradient (K > 0) and that the circulation integral have the same sign as the potential vorticity forcing. Two sets of examples, both involving (Q,?) scatter diagrams constructed from numerically simulated data, are presented. One relates to steady baroclinic wave motion in a rotating annulus system, and the other to the time-averaged circulation in an ocean basin.