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Spontaneous gravity wave radiation from an unsteady rotational flow is investigated numerically in an f-plane shallow water system. Unlike the classical Rossby adjustment problem, where free development of an initially unbalanced state is investigated, we consider development of a barotropically unstable zonal flow which is initially balanced but maintained by zonal mean forcing. Gravity waves are continuously radiated from a nearly balanced rotational flow region even when the Froude number is so small that balance dynamics is thought to be a good approximation for the full system. The source of gravity waves is discussed by analogy with the theory of aero-acoustic sound wave radiation (the Lighthill theory). It is shown that the source regions correspond to regions of strong rotational flow. The gradual change of rotational flow causes gravity wave radiation. We propose an approximation for these strong sources on the assumption that the dominant flow in the jet region is non-divergent rotational flow. In addition, we calculate the zonally symmetric component of gravity waves far from the source regions, solving the Lighthill equation. Using scaling analyses for perturbations, these gravity waves can be calculated with only one approximated source term that is related to the latitudinal gradient of the fluid depth and the latitudinal mass flux. In spite of its simplicity, this approximation not only explains the physical cause of gravity wave radiation, but gives an amount of source close to that obtained by classical approximation derived from vortical motion.

To study a limit of validity of balanced models, stability of a zonal jet is investigated both linearly and nonlinearly in an f-plane shallow water system for a wide range of parameter. It is shown that quasi-geostrophic approximation gives not only a good estimation of maximum growth rate for high Rossby number, Ro, but also is valid even in the nonlinear phase of instability for high Ro as long as Froude number, Fr, is low. While the maximum growth rate of unstable modes is well estimated by the quasi-geostrophic approximation, dominant balance is different between high and low Ro. In the low Ro regime (Ro 5), cyclostrophic balance with basic shear is dominant, ||||/||ψ||∝Fr2. Considering that the barotropic instability is caused by the resonance of neutral Rossby wave modes, we can explain the difference of the ratio in each regime. Using the ratio ||||/||ψ|| being small, different approximation of the linear shallow water equations for each regime is deduced. Properties of the linear unstable modes are explained with these approximations.

Inertia–gravity wave (IGW) radiation from the elliptical vortex is investigated in the f-plane shallow water system. The far field of IGW is analytically derived for the case of an almost circular Kirchhoff vortex with a small aspect ratio. Cyclone–anticyclone asymmetry appears at finite values of the Rossby number (Ro) caused by the source originating in the Coriolis acceleration. While the intensity of IGWs from the cyclone monotonically decreases as f increases, that from the anticyclone increases as f increases for relatively smaller f and has a local maximum at intermediate f. A numerical experiment is conducted on a model using a spectral method in an unbounded domain. The numerical results agree quite well with the analytical ones for elliptical vortices with small aspect ratios, implying that the derived analytical forms are useful for the verification of the numerical model. For elliptical vortices with larger aspect ratios, however, significant deviation from the analytical estimates appears. The intensity of IGWs radiated in the numerical simulation is larger than that estimated analytically. The reason is that the source of IGWs is amplified during the time evolution because the shape of the vortex changes from ideal ellipse to elongated with filaments. Nevertheless, cyclone–anticyclone asymmetry similar to the analytical estimate appears in all the range of aspect ratios, suggesting that this asymmetry is a robust feature.