Spectrum of shallow water gravity waves generated by confined two-dimensional turbulence

We apply Lighthill's theory of aeroacoustic sound generation to shallow water gravity waves generated by spatially confined two-dimensional turbulence. We show that the frequency spectrum of surface waves at large distances from the source of turbulence is, under suitable conditions, proportional to the spatiotemporal spectrum of the energy momentum tensor associated with the turbulent fields acting as the wave source and hence that it follows a power-law behavior. We compute the exponent for shallow water waves generated by isotropic two-dimensional turbulence and show that the integrated power radiated scales as ${\omega }^{-3}$ when the turbulent fluctuations arise from an inverse energy cascade and as ${\omega }^{-7}$ when they arise from the enstrophy cascade.

Figure 1

A bounded two-dimensional domain $\mathcal{A}$ displays a turbulent energy-momentum tensor $T^{ij}({\mathbf{r}}^{\prime },t)$ which is the source of shallow gravity water waves measured at a point $\mathbf{r}$ within the circumference $\mathcal{C}$, with $\left|\mathbf{r}\right|=r\gg L={\mathcal{A}}^{1/2}$.