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For understanding the dissipation in a rotating flow when resonance occurs,we study the rotating flow driven by the harmonic force in a periodic box. Boththe linear and nonlinear regimes are studied. The various parameters such asthe force amplitude $a$, the force frequency $\omega$, the force wavenumber$k$, and the Ekman number $E$ are investigated. In the linear regime, thedissipation at the resonant frequency scales as $E^{-1}k^{-2}$, and it is muchstronger than the dissipation at the non-resonant frequencies on the largescales and at the low Ekman numbers. In the nonlinear regime, at the resonantfrequency the effective dissipation (dissipation normalised with the square offorce amplitude) is lower than in the linear regime and it decreases with theincreasing force amplitude. This nonlinear suppression effect is significantnear the resonant frequency but negligible far away from the resonantfrequency. Opposite to the linear regime, in the nonlinear regime at theresonant frequency the lower Ekman number leads to the lower dissipationbecause of the stronger nonlinear effect. This work implies that the previouslinear calculations overestimated the tidal dissipation, which is important forunderstanding the tides in stars and giant planets.

Linear and nonlinear responses to harmonic force in rotating flow