The spontaneous generation of inertia–gravity waves (IGWs) in the idealized simulation of vortical flows is investigated using the isentropic two-layer model on a sphere. The contour-advective semi-Lagrangian (CASL) algorithm is applied to solve the primitive equations using the potential vorticity (PV), velocity divergence, and acceleration divergence as the prognostic variables. The CASL algorithm consists of a Lagrangian part and an Eulerian part. While the Lagrangian part is attributed to PV equation and solved by contour advection, the Eulerian part includes the prognostic and diagnostic equations for the grid-based variables of velocity divergence, acceleration divergence and the depth field. The fourth-order compact differencing and spectral transform are used, respectively, in latitudinal and longitudinal directions and time stepping is carried out using a three-time-level semi-implicit scheme.
The model is set up using 256 grid points in both latitudinal and longitudinal directions, the upper- and lower-layer potential temperatures of, respectively, 280 K and 310 K, and the same horizontal mean depth of 5 km for the two layers. A balanced, zonal jet is used as the initial state and a very small perturbation is added to it as a trigger instability. In order to determine the IGWs more accurately, the Bolin–Charney balance relations are used to decompose the flow into a balanced part controlled by PV and an unbalanced part representing free inertia–gravity waves.
In this study, the analysis of the velocity divergence, acceleration divergence and PV points to breakdown of balance and generation of two wave packets of IGWs where sharp PV gradients appear to contribute significantly to the generation and organization of the wavepackets of IGWs. Application of the CASL algorithm helps to capture fine-scale structures in PV and to quantify IGWs generated by vortical flows more accurately. With regard to balanced initial conditions used, these gravity waves are spontaneously generated from the breakdown of the balance. The first wave packet is found on the downstream side of the trough similar to the mesoscale waves described by Zhang in 2004. The second wave packet is identified on the upstream side of the trough similar to the wave packet described by Plougonven and Snyder in 2007 in idealized simulations of a baroclinic life cycle dominated by cyclonic behavior. It seems that the propagation direction of the first wave packet is the same as that of the Zhang, but the second wave packet propagates perpendicular to the wave packet of Plougonven and Snyder. By determining the characteristics of the waves, the magnitude of the intrinsic frequency of both wave packets are found to be larger than those of the previous studies and is near to the results of Wang and Zhang in 2007 for their low static stability experiment.
Further, a short-time Fourier transform is used to determine the dominant absolute frequency and verify frequency characteristics of the packets obtained by the dispersion relation. In this method, the time–frequency analysis for each signal is provided by applying a moving time window to the signal and taking the fast Fourier transform. The more accurate results obtained by this method for the intrinsic frequency confirm the estimates based on the local dispersion relation.