Spontaneous Gravity Wave Radiation in a Shallow Water System on a Rotating Sphere

Abstract

The effect of the earth’s rotation on spontaneous inertial gravity wave radiation from an unsteady rotational flow is investigated numerically using a shallow water system on a rotating sphere, changing three parameters, the Rossby number (1 ≤ Ro ≤ 30), the Froude number 0.1 ≤ Fr ≤ 0.7), and the latitudinal positions of the jet (11.25°N ≤θ0 ≤ 78.75 °N). The spectral-like three-point combined compact difference (sp-CCD) scheme is used for numerical calculation to estimate the amplitude of gravity waves with high accuracy and resolution similar to the spherical harmonics model. The jet, which is initially balanced but evolves with time because of barotropic instability, is maintained by relaxation forcing. The time variations of nearly balanced rotational flows continuously radiate gravity waves. While the amount of gravity wave flux for large Ro (≥ 10) in all positions of the jet is almost constant, that for relatively small Ro (< 10) depends considerably on the positions of the jet and the observational latitude. First, there is almost no gravity wave radiation from the jet positioned at a high latitude. Second, even when gravity waves are radiated from the jet positioned at low latitude, they rarely propagate toward a higher latitude. To discuss the results, the source of gravity waves is introduced as an analogy to the aeroacoustic sound wave radiation theory (Lighthill theory) with an f-plane approximation at the position of the jet. Spectral analysis of the sources reasonably explains the waves. Because the effect of the earth’s rotation varies with the latitude, gravity waves are radiated only from an unsteady source with higher frequency than the inertial cut-off frequency at the position of the jet. In a similar way, gravity waves that propagate toward high latitudes should have a higher frequency than the inertial cut-off frequency at that latitude.