Topological order in the noisy rotating nonlinear shallow water equations

The linear rotating shallow water equations are known to support topologically-protected edge modes and bulk bands with nonzero Chern numbers. Previous work has shown that the topological nature of the system can be also quantified by a winding number of a gauge-invariant two-point correlation function. However, in real systems such as the oceans and atmosphere, noise and nonlinearities are important. Conventional methods for understanding topological order in quantum band insulators rely upon the linearity of the Schrodinger equation. Therefore, new methods for studying topological order in nonlinear systems are needed. Here we use the Martin-Siggia-Rose-Janssen-De Dominicis formalism to extend the winding number method to the noisy rotating nonlinear shallow water system and study the effects of the new terms on the topological order present in the system. For large wave amplitudes, turbulence due to strong nonlinearities may be expected to destroy the signature of non-trivial topology.