Internal wave energy flux from density perturbations in nonlinear stratifications

Abstract

Internal gravity wave energy contributes significantly to the energy budget of the oceans, affecting mixing and the thermohaline circulation. Hence it is important to determine the internal wave energy flux $\boldsymbol{J}=p\,\boldsymbol{v}$, where $p$ is the pressure perturbation field and $\boldsymbol{v}$ is the velocity perturbation field. However, the pressure perturbation field is not directly accessible in laboratory or field observations. Previously, a Green’s function based method was developed to calculate the instantaneous energy flux field from a measured density perturbation field $\unicode[STIX]{x1D70C}(x,z,t)$, given a constant buoyancy frequency $N$. Here we present methods for computing the instantaneous energy flux $\boldsymbol{J}(x,z,t)$ for an internal wave field with vertically varying background $N(z)$, as in the oceans where $N(z)$ typically decreases by two orders of magnitude from the pycnocline to the deep ocean. Analytic methods are presented for computing $\boldsymbol{J}(x,z,t)$ from a density perturbation field for $N(z)$ varying linearly with $z$ and for $N^{2}(z)$ varying as $\tanh (z)$. To generalize this approach to arbitrary $N(z)$, we present a computational method for obtaining $\boldsymbol{J}(x,z,t)$. The results for $\boldsymbol{J}(x,z,t)$ for the different cases agree well with results from direct numerical simulations of the Navier–Stokes equations. Our computational method can be applied to any density perturbation data using the MATLAB graphical user interface ‘EnergyFlux’.