Dynamics of “critical” trajectories

Two diagnostic dynamic models for flow in hyperbolic and elliptic regions of a geophysical fluid are developed and compared. As the main interest here is in local dynamical processes, these models are used to study trajectories near stagnation points in the flow field. The simplest model presumes a balance between the Coriolis and geopotential accelerations. This model is equivalent to the classic approach that characterizes these regimes by the quadratic equation for the eigenvalues of the velocity gradient. However, since that model imposes geostrophic dynamics, the eigenvalues of the velocity gradient can be replaced by the local curvature or Hessian of the geopotential scaled by Coriolis. The general model adds both local and inertial accelerations to the dynamical balance. In contrast to the classic result the consequent frequency equation is a quartic that involves both the Hessian of the geopotential field, the components of the velocity gradient, and Coriolis. Roots of this equation give two distinct time scales, which are interpreted as Lagrangian time scales. Motion of the geopotential field produces a third Eulerian time scale. Critical trajectories are those whose initial positions and velocities are such that they are independent of the Lagrangian time scales. These simple models establish that within hyperbolic and elliptic regions of the geopotential field there may be trajectories whose time scales differ radically from even their nearest neighbors.A characteristic of critical trajectories in the ocean is that they often are found near stagnation points. These may be hard to identify even in model simulations, but a similar quantity, the null in the geopotential gradient, might be easier to obtain. To analyze the relation between the critical trajectories, stagnation points, and gradient null, evolution models for the later two objects are proposed. For a steady geopotential all three coincide. However with a time varying geopotential, they are distinct even though all have the same time scale. The analysis provides a metric for the separation of all three objects.