Date Approved

7-6-2026

Embargo Period

7-6-2026

Document Type

Thesis

Degree Name

M.A. Mathematics

Department

Mathematics

College

College of Science & Mathematics

Advisor

Helga S. Huntley, Ph.D.

Committee Member 1

A.D. Kirwan Jr., Ph.D.

Committee Member 2

Robert Freeman, Ph.D.

Committee Member 3

Thanh Nguyen, Ph.D.

Keywords

analytic solutions;differential equations;dynamical systems;kinematic property equations;ODE;physical oceanography

Disciplines

Mathematics | Physical Sciences and Mathematics

Abstract

With the recent advent of large submesoscale drifter deployments, interpreting the timeseries of kinematic properties (KP) - divergence, vorticity, shear, and normal strain rates observed therein is a pressing issue. The evolution equations for the four KP are derived from the two-dimensional momentum conservation equations. The resulting equations, a nonlinear coupled system of ordinary differential equations, capture the submesoscale motion of drifters along fixed ocean surfaces, with time and space scales on the order of days and kilometers. This thesis builds theory around KP timeseries behavior by solving for the analytic solutions in the particular cases that one KP is constant with respect to time, known as a partially steady state, as well as for the general analytic solutions in the case of zero forcing and only divergence forcing. We examine how solution stability depends on the initial conditions and forcing terms for the partially steady, zero-forced, and divergence-forced solutions and investigate the behavior (periodicity, limiting behavior, etc.) of the aforementioned solutions. In the unforced case, solutions are stable if and only if the total strain rate is less than the absolute vorticity, with stable solutions being periodic and tracing an ellipse in phase space. In the divergence-forced case, solutions are stable if the total strain rate is less than the total vorticity, however the contrapositive does not hold, as there exist stable solutions with total strain rate greater than total vorticity, so long as the initial total strain rate is not too large relative to the initial vorticity and the initial divergence is not too large relative to the divergence forcing. Thus, for the divergence-forced case, we see that greater absolute vorticity magnitude is stabilizing, and larger magnitude of total strain rate and negative divergence are destabilizing. We found partially steady solutions for each of the four KP, and we found partially steady solutions with steady total deformation rate as well. These solutions are varied in their behavior, being periodic or limiting to an equilibrium if they are stable, and unstable solutions can be unstable in only one or both directions of time, with varied stability criteria. However, the general trend observed in the literature of negative divergence being destabilizing and greater magnitude of strain rate being destabilizing was present in all partially steady solutions.

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