## 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 ψ 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(ψ) 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 (ψ, *Q*) space. Most scatter diagrams that have appeared in the literature included the (ψ, *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(ψ) 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*ψ of the scatter diagram may be used to infer some properties of the forcing and dissipative processes acting. When dissipation is principally due to *Q*transfer 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*ψis 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*,ψ) 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.