It is shown that information about the intrinsic frequency of internal inertial gravity waves, together with their vector wavenumber and their steepness spectrum, may be derived using data from a meter recording current and temperature just above a uniform sloping boundary. The analysis assumes that, over the periods of time in which estimates are made, the local mean buoyancy frequency, N, and temperature gradient are uniform, and the bottom slope, α, is constant over the scale of the observed waves and is known. It is further assumed that, during short periods of analysis, the internal wave field is dominated by waves having a single wavenumber vector at each frequency. Measurements of the temperature and up-slope current spectra can be used to determine the intrinsic frequency and the along-slope wavenumber, l, of the waves provided the mean along-slope current, V, is non-zero. If the internal wave field near the sloping boundary is composed of perfectly reflecting wave rays, it is found that the ratio, R, of the potential energy density of the waves of given frequency to that of their kinetic energy depends on α and on the ratio, l/k, of waves in the reflecting wave field, where k is the up-slope wavenumber. Since l is already determined if V ≠ 0, k may be estimated from the measured R. The third component of wavenumber, m, normal to the slope is found from the dispersion relation. The estimated values may be used to test assumptions of small wave steepness and the uniformity of buoyancy frequency over the vertical scale of the waves. Problems of application are revealed by using the analysis on data collected from a mooring located within the hyperlimnion on the sloping side of Lake Geneva. Whilst the along-slope wavenumbers and corresponding intrinsic frequencies can be found, it proves impossible to derive estimates of wave directions for waves in the subcritical frequency range of group velocities having angles to the horizontal, β, less than α, and supercritical waves (β > α) have inferred wavelength which violate assumptions of the vertical uniformity of N. It is inferred that in the lake the wave propagation is dominated by waves with a modal structure rather than rays, even near the buoyancy frequency. The methodology should however be applicable in the deep ocean, subject to the various assumptions which are made. Fundamental questions are raised about the directional characteristics and persistence of internal waves near sloping boundaries which deserve further investigation in view of their possible relation to mixing.
Thorpe, S. A., and L. Umlauf. 2002. "Internal gravity wave frequencies and wavenumbers from single point measurements over a slope." Journal of Marine Research 60, (5). https://elischolar.library.yale.edu/journal_of_marine_research/2449