By using the renormalization group (RG) method, the interaction between balanced flows and Doppler-shifted inertia-gravity waves (GWs) is formulated for the hydrostatic Boussinesq equations on the f plane. The derived time-evolution equations [RG equations (RGEs)] describe the spontaneous GW radiation from the components slaved to the vortical flow through the quasi resonance, together with the GW radiation reaction on the large-scale flow. The quasi resonance occurs when the space-time scales of GWs are partially comparable to those of slaved components. This theory treats a coexistence system with slow time scales composed of GWs significantly Doppler-shifted by the vortical flow and the balanced flow that interact with each other. The theory includes five dependent variables having slow time scales: one slow variable (linear potential vorticity), two Doppler-shifted fast ones (GW components), and two diagnostic fast ones. Each fast component consists of horizontal divergence and ageostrophic vorticity. The spontaneously radiated GWs are regarded as superpositions of the GW components obtained as low-frequency eigenmodes of the fast variables in a given vortical flow. Slowly varying nonlinear terms of the fast variables are included as the diagnostic components, which are the sum of the slaved components and the GW radiation reactions. A comparison of the balanced adjustment equation (BAE) by Plougonven and Zhang with the linearized RGE shows that the RGE is formally reduced to the BAE by ignoring the GW radiation reaction, although the interpretation on the GW radiation mechanism is significantly different; GWs are radiated through the quasi resonance with a balanced flow because of the time-scale matching.
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