Determination of the energy flux of internal gravity waves
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Internal gravity waves are traveling disturbances that propagate within a fluid whose density varies with depth, and two prominent examples where these occur are the atmosphere and the ocean. In the latter case, which is the focus of this work, the tidal forcing by the moon creates internal gravity waves (oftentimes referred to simply as "internal waves") that originate from the ocean bottom topography. The energy generated in the internal waves by this mechanism contributes significantly to the energy budget of the ocean. Hence it is important to determine the energy flux in the internal waves. However, it is not possible to obtain the energy flux J = p v directly because the pressure and velocity perturbation fields, p and v, cannot be simultaneously measured at the present time. The two primary methods for measuring internal waves in the laboratory are particle image velocimetry (PIV), which gives velocity perturbation fields v(x,z,t), and synthetic schlieren, which gives density perturbation fields [rho](x,z,t). We present one method for obtaining the time-averaged energy flux (J) from PIV data by calculating the stream function [psi](x,z,t), whose results agree to within 0.5% when compared with direct numerical simulations of the Navier-Stokes equations. The method was also applied to laboratory data, and again using direct numerical simulations, the agreement was found to be very good. A MATLAB code was developed with a graphical user interface that can be used to compute the energy flux and power from any two-dimensional velocity field data. Another method, using a Green's function approach, was developed to obtain the instantaneous energy flux J(x,z,t) from density perturbation data [rho](x,z,t) such as that from synthetic schlieren. This was done for a uniform, tanh, and linear buoyancy frequency N(z). Additionally, a finite-difference method was developed for the case of arbitrary N(z). The results for J(x,z,t) are found to agree with results from direct numerical simulations, typically to within 6%. These methods can be applied to any density perturbation data using the MATLAB graphical user interface EnergyFlux.