Scattering of surface shallow water waves on a draining bathtub vortex

In the linear approximation, we study long-wave scattering on an axially symmetric flow in a shallow water basin with a drain in the center. Besides of academic interest, this problem is applicable to the interpretation of recent laboratory experiments with draining bathtub vortices and description of wave scattering in natural basins, and also can be considered as the hydrodynamic analog of scalar wave scattering on a rotating black hole in general relativity. The analytic solutions are derived in the low-frequency limit to describe both pure potential perturbations (surface gravity waves) and perturbations with nonzero potential vorticity. For the moderate frequencies, the solutions are obtained numerically and illustrated graphically. It is shown that there are two processes governing the dynamics of surface perturbations, the scattering of incident gravity water waves by a central vortex, and emission of gravity water waves stimulated by a potential vorticity. Some aspects of their synergetic actions are discussed.

Figure 1

Left frame (a) shows a sketch of the wave pattern in the neighborhood of a rotating black hole model: black spot in the center shows a drainage orifice mimicking a black hole; black circle bounding the dark disk shows the event horizon; the red circle bounding the pink disk shows the ergosphere. Right frame (b) is a photo of a natural whirlpool “Old Sow” of about 80 m diameter that has existed for hundreds years off the coast of Maine, USA (image by Jim Lowe: https://io9.gizmodo.com/a-250-foot-whirlpool-that-has-existedfor-hundreds-of-y-5501419?IR=T).

Figure 2

(a) Reflection coefficient of gravity waves with the azimuthal number $m=1$ as the function of normalized frequency $\omega /\mathrm{\Omega }$ for $\mathrm{\Omega }=0.5$ (line 1), $\mathrm{\Omega }=1$ (line 2), $\mathrm{\Omega }=1.5$ (line 3), and $\mathrm{\Omega }=2$ (line 4). (b) Reflection coefficient of gravity waves as the function of normalized frequency $\omega /\left(m\mathrm{\Omega }\right)$ for $\mathrm{\Omega }=2$ and few values of azimuthal number $m$.

Figure 3

Dependences of maximal reflection coefficient (a) and corresponding wave frequency ${\omega }_{*}$ (b) on $\mathrm{\Omega }$ for few several values of azimuthal number $m=1,2,3$.

Figure 4

Dependence of normalized amplitude of PV-stimulated gravity wave on the normalized frequency for $\mathrm{\Omega }=1$ and various numbers $m$. Frame (a) shows a whole dependence, and frame (b) shows a fragment for $\omega \ll m\mathrm{\Omega }$. Dashed lines in frame (b) show the analytical dependences $\left|A\right|\sim {\omega }^{m/2}$ for $\omega /\left(m\mathrm{\Omega }\right)\ll 1$.

Figure 5

Maximal amplitude of PV-stimulated gravity wave as the function of $\mathrm{\Omega }$ for various values of $m$. Frame (a) shows this dependence in the wide range of $\mathrm{\Omega }\le 20$, and frame (b) shows a fragment for $\mathrm{\Omega }\le 0.5$.