On the Calculation of Available Potential Energy in Internal Wave Fields

A comparison of three common formulations for calculating the available potential energy (APE) in internal wave fields is presented. The formulations are the perturbation APE (APE(1)), the exact local APE (APE(2)), and its approximation for linear stratification (APE(3)). The relationship among these formulations is illustrated through a graphical interpretation and a derivation of the energy conservation laws. Numerical simulations are carried out to quantitatively assess the performance of each APE formulation under the influence of different nonlinear and nonhydrostatic effects. The results show that APE(2) is the most attractive in evaluating the local APE, especially for nonlinear internal waves, since use of APE(2) introduces the smallest errors when computing the energy conservation laws. Larger errors arise when using APE(1) because of the large disparity in magnitude between the kinetic energy and APE(1). It is shown that the disparity in the tendency of APE(1) is compensated by a large flux arising from the reference pressure and density fields. Because the tendency of the kinetic energy is close to that of APE(3), computational errors arise when using APE(3) only in the presence of nonlinear stratification, and these errors increase for stronger flow nonlinearity.