Beyond ray tracing for internal waves. I. Small-amplitude anelastic waves

Abstract

We compute the transmission of small-amplitude two-dimensional anelastic internal waves in nonrotating inviscid fluid having arbitrarily specified stratification and background velocity. Whereas stability analyses of the flow involve solving an eigenvalue problem that relates the frequency to horizontal wavenumber, we focus upon the evolution of waves incident from below with independently specified frequency and horizontal wavenumber. A numerical method is developed to ensure that the wave field in the upper domain corresponds only to upward-propagating transmitted waves. Two particular applications are discussed. First, internal waves incident upon a piecewise-linear shear layer are examined and their transmission is computed as a function of the bulk Richardson number Rib and the ratio of the density scale height relative to the depth of the shear layer. The waves transmit partially across critical levels if they coincide with heights where the gradient Richardson number is less than 1/4. Transmission is larger if Rib is smaller. Decreasing the density scale height reduces the frequency and wavenumber range over which internal waves propagate, but this does not significantly affect the magnitude of transmission. Second, internal waves generated by flow over Jan Mayen island are examined. Although the waves are ducted, the waves are found to transmit partially through the top of the duct. The results are used to interpret the discrepancy between predictions of the Fourier-ray tracing model and fully nonlinear numerical simulations of Eckermann et al. [Mon. Weather Rev. 134, 2830 (2006)].