#### Title

Classical chaos in mesoscale ocean dynamics: Lateral stirring and geometric acoustics

#### Date of Award

1991

#### Availability

Article

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Applied Marine Physics/Ocean Engineering

#### First Committee Member

Michael G. Brown, Committee Chair

#### Abstract

Chaos is unpredictable behavior in a deterministic low-order dynamical system. Two dynamical systems which arise naturally in ocean physics are examined here, midocean fluid motion and underwater acoustic ray propagation. Both have Hamiltonian form with one degree of freedom. Chaotic solutions appear, in general, when the Hamiltonian is explicitly time-dependent and the canonical equations of motion are nonintegrable. When the Hamiltonian is not explicitly time-dependent, the equations of motion are integrable, and trajectories are regular and predictable for all times. Neighboring trajectories are found to diverge rapidly (exponentially, on average) or slowly (according to a power law, on average) when the motion is chaotic or regular, respectively.Midocean fluid particle trajectories are assumed to obey Lagrangian equations of motion with Hamiltonian form. The presence of chaos is observed to stir passive tracers efficiently, enhancing diffusive processes. To determine if observed behavior in the midocean is chaotic, power spectra of SOFAR float trajectory data are computed and found to contain structure on all resolvable scales. Attempts to directly estimate Lyapunov exponents, a measure of the exponential divergence, from a reconstructed streamfunction are unsuccessful. The Kolmogorov entropy (here equivalent to the Lyapunov exponent) is estimated to be $\sim$(140 day)$\sp{-1}$. These results suggest the presence of chaos. Furthermore, analysis of SOFAR float trajectories suggests, albeit ambiguously, that the underlying dynamics are those of a low-order system. The fractal dimension of the trajectories is estimated to be $\sim$1.2. A possible rationale for this value, and the associated implications for anomalous diffusion, are addressed.Underwater acoustic rays obey equations of Hamiltonian form where range plays the role of the time-like variable. The appearance of chaos implies a limited ability to predict eigenrays at long range. Power spectra calculations and construction of Poincare' sections consistently predict the appearance of chaos in simple, periodically range-dependent models of the sound speed profile. When a realistic model of oceanic mesoscale sound speed perturbations is introduced, consisting of a randomly phased superposition of baroclinic Rossby wave modes, numerical integration of the ray equations yields Lyapunov exponent estimates for near-axial rays of $\sim$(400 km)$\sp{-1}$ when perturbation strengths are consistent with ocean observations. This implies that prediction of near-axial underwater acoustic rays is not feasible at ranges beyond one or two thousand km.

#### Keywords

Physical Oceanography; Physics, Acoustics

#### Recommended Citation

Smith, Kevin Barkley, "Classical chaos in mesoscale ocean dynamics: Lateral stirring and geometric acoustics" (1991). *Dissertations from ProQuest*. 3004.

https://scholarlyrepository.miami.edu/dissertations/3004

## Link to Full Text

http://access.library.miami.edu/login?url=http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9214837