Contraction rate and its relationship to frontogenesis, the Lyapunov exponent, fluid trapping, and airstream boundaries

Although a kinematic framework for diagnosing frontogenesis exists in the form of the Petterssen frontogenesis function and its vector generalization, a similar framework for diagnosing airstream boundaries (e.g., drylines, lee troughs) has not been constructed. This paper presents such a framework, beginning with a kinematic expression for the rate of change of the separation vector between two adjacent air parcels. The maximum growth rate of the separation vector is called the instantaneous dilatation rate and its orientation is called the axis of dilatation. Similarly, a maximum decay rate is called the instantaneous contraction rate and its orientation is called the axis of contraction. These expressions are related to the vector frontogenesis function, in that the growth rate of the separation vector corresponds with the scalar frontogenesis function, and the rotation rate of the separation vector corresponds with the rotational component of the vector frontogenesis function. Because vorticity can rotate air-parcel pairs out of regions of deformation, the instantaneous dilatation and contraction rates and axes may not be appropriate diagnostics of airstream boundaries for fluid flows in general. Rather, the growth rate and orientation of an airstream boundary may correspond better to the so-called asymptotic contraction rate and the asymptotic axis of dilatation, respectively. Expressions for the asymptotic dilatation and contraction rates, as well as their orientations, the asymptotic dilatation and contraction axes, are derived. The asymptotic dilatation rate is related to the Lyapunov exponent for the flow. In addition, a fluid-trapping diagnostic is derived to distinguish among adjacent parcels being pulled apart, being pushed together, or trapped in an eddy. Finally, these diagnostics are applied to simple, idealized, steady-state flows and a nonsteady idealized vortex in nondivergent, diffluent flow to show their utility for determining the character of air-parcel trajectories and airstream boundaries.