Viscous reflection of internal waves from a slope

Motivated by the laboratory experiments of Rodenborn et al. [Phys. Fluids 23, 026601 (2011)], a weakly nonlinear model is developed that accounts for viscous dissipation in the reflection of a finite-width internal wave beam from a uniform slope. Asymptotically, at high Reynolds number, viscous effects come into play predominantly in the immediate vicinity of the critical slope angle equal to the propagation angle to the horizontal of the incident wave beam. However, in the experiments of Rodenborn et al. where the Reynolds number is moderately large, it turns out that viscosity is important throughout the slope range considered, which explains the overall poor agreement of their observations with earlier inviscid models. The predictions of the proposed model, by contrast, are in good qualitative, and in some respects quantitative, agreement with these experiments, and they also compare favorably against numerical Navier-Stokes simulations.

Figure 1

Geometry of internal wave beam reflection from a slope of angle $\alpha$, for the case when the incident beam hits the slope at an angle $\theta <\alpha$, to the horizontal. The primary reflected beam is also inclined to the horizontal by $\theta$, while the radiated second-harmonic beam stretches along ${\theta }_2$ to the horizontal, where $sin{\theta }_2=2sin\theta <1$.

Figure 2

Plots of normalized amplitude of primary-harmonic reflected beam as function of the slope angle $0<\alpha <{30}^{\circ }$, for angle of incidence $\theta =22.7^{\circ }$ and two values of Re: solid line $(\mathrm{Re}=103)$ and dashed-dotted line $(\mathrm{Re}=1030)$. The dotted line corresponds to the inviscid $(\mathrm{Re}=\infty )$ response, which features an infinite peak at $\alpha ={\alpha }_{\mathrm{crit}}=\theta$. Symbols denote results of numerical simulations for certain values of Re and $\varepsilon$: $\times (\mathrm{Re}=103,\varepsilon =0.2),◯(\mathrm{Re}=103,\varepsilon =0.04)$, and $▵(\mathrm{Re}=1030,\varepsilon =0.04)$.

Figure 3

Plots of the slope angle ${\alpha }_{max}$ at which the primary-harmonic reflected beam attains the maximum amplitude as function of the incidence angle $5^{\circ }<\theta <{30}^{\circ }$, for two values of Re: solid line $(\mathrm{Re}=103)$ and dashed-dotted line $(\mathrm{Re}=1030)$. The dotted line indicates the inviscid limit $(\mathrm{Re}=\infty )$, where ${\alpha }_{max}=\theta$. Symbols denote results of numerical simulations for certain values of Re and $\varepsilon$: $\times (\mathrm{Re}=103,\varepsilon =0.2),◯(\mathrm{Re}=103,\varepsilon =0.008)$, and $▵(\mathrm{Re}=1030,\varepsilon =0.008)$.

Figure 4

Plots of normalized amplitude of second-harmonic radiated beam as function of the slope angle $0<\alpha <{30}^{\circ }$, for angle of incidence $\theta =22.7^{\circ }$ and two values of Re: solid line $(\mathrm{Re}=103)$ and dashed-dotted line $(\mathrm{Re}=1030)$. The dotted line corresponds to the inviscid $(\mathrm{Re}=\infty )$ response of Ref. [14], which features an infinite peak at $\alpha ={\alpha }_{\mathrm{crit}}=\theta$. Symbols denote results of numerical simulations for certain values of Re and $\varepsilon$: $\times (\mathrm{Re}=103,\varepsilon =0.2),◯(\mathrm{Re}=103,\varepsilon =0.04)$, and $▵(\mathrm{Re}=1030,\varepsilon =0.04)$.

Figure 5

Plots of the slope angle ${\alpha }_{max}^{2\mathrm{nd}}$ at which the second-harmonic radiated beam attains the maximum amplitude as function of the incidence angle $5^{\circ }<\theta <{30}^{\circ }$, for two values of Re: solid line $(\mathrm{Re}=103)$ and dashed-dotted line $(\mathrm{Re}=1030)$. The dotted line indicates the inviscid limit $(\mathrm{Re}=\infty )$, where ${\alpha }_{max}^{2\mathrm{nd}}=\theta$. Symbols denote results of numerical simulations for certain values of Re and $\varepsilon$: $\times (\mathrm{Re}=103,\varepsilon =0.2),◯(\mathrm{Re}=103,\varepsilon =0.008)$, and $▵(\mathrm{Re}=1030,\varepsilon =0.008)$.

Figure 6

Intensity of radiated second-harmonic beam as function of slope angle $0<\alpha <{30}^{\circ }$ for incidence angle $\theta =22.7^{\circ }$, under the flow conditions of RKZS $(\mathrm{Re}=103$ and $\varepsilon =0.2)$. The solid line corresponds to the theoretical predictions. The crosses are results of numerical simulations. The open triangles denote the experimental results in Fig. 7 of RKZS after being multiplied by the constant factor 0.1 such that the peak value at $\alpha =13.2^{\circ }$ matches the theoretical intensity estimate at this $\alpha$.