Analytical Solutions of Vertically Propagating Gravity Waves—from the Perspective of Buoyancy-Vorticity Thinking

Abstract

On the basis of buoyancy-vorticity (BV) formulation of Harnik et al. (2008), the initial value problem of vertically propagating gravity waves is analytically solved in a zonal-vertical two-dimensional system. The analytical solutions provide an example of the visualization of BV thinking. Further, the analytical solutions enable a qualitative understanding of the growth of gravity waves in a vertically sheared zonal flow (so-called shear instability of gravity waves) by BV thinking. To this end, the basic buoyancy (i.e., basic potential temperature) is assumed to be piecewise uniform in the vertical direction, and the Green function method is employed. The obtained analytical solutions show the following. In a vertically uniform basic zonal flow, the gravity wave, which is initially at the lowest level, propagates vertically upwards, gets reflected from the highest level back to the lowest level and again from the lowest level to the highest level, and so on. In a vertically sheared basic zonal flow, the behavior of the gravity waves depends on the horizontal wave number. This is caused by the dependence of horizontal propagation velocity on the horizontal wave number. Here, horizontal propagation is defined relative to the fluid. If the horizontal propagation and advection by the basic zonal flow are successfully balanced so that the lower and upper phase velocities are nearly equal, then the gravity wave propagates vertically, and the upper and lower disturbances are phase-locked to each other; this results in an effective interaction between them and in growth as an exponential function of time. On the other hand, if the horizontal propagation and advection by the basic zonal flow are out of balance so that the lower and upper phase velocities are different from each other, then the gravity wave hardly propagates vertically, and the upper and lower disturbances horizontally flow away from each other resulting in an absence of interaction between them and in the oscillation (i.e., no growth). At the marginal points between oscillation and growth, the gravity wave grows as a linear function of time. The behavior of analytical solutions can be qualitatively explained by the BV thinking of Harnik et al. (2008).