The nonlinear reflection of a finite-width plane internal gravity wave incident onto a uniform slope is addressed, relying on the inviscid theory of Thorpe ( J. Fluid Mech. , vol. 178, 1987, pp. 279–302) for pure plane waves. The aim of this theory is to determine the conditions under which the incident and the reflected waves form a resonant triad with the second-harmonic wave resulting from their interaction. Thorpe’s theory leads to an indeterminacy of the second-harmonic wave amplitude at resonance. In waiving this indeterminacy, we show that the latter amplitude has a finite behaviour at resonance, increasing linearly from the slope. We investigate the influence of background rotation and find similar results with a weaker growth rate. We then adapt the theory to the case of an incident plane wave of finite width. In this case, nonlinear interactions are confined to the area where the incident and reflected finite-width waves superpose, implying that the amplitude of the second-harmonic wave is bounded at resonance. We find good agreement with the results of numerical simulations in a vertical plane as long as the dissipated power of the incident and reflected waves remain smaller than the power transferred to the second-harmonic wave. This is the case for small slope angles. As the slope angle increases, the focusing of the reflected wave enhances viscous effects and dissipation eventually dominates over nonlinear transfer. We finally discuss the relevance of laboratory experiments to assess the validity of the theoretical results.

中文翻译：

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二维有限宽度内重力波在斜坡上的非线性反射

根据 Thorpe 的无粘理论（J. Fluid Mech., vol. 178, 1987, pp. 279–302），针对纯平面波，解决了有限宽度平面内部重力波入射到均匀斜坡上的非线性反射. 该理论的目的是确定入射波和反射波与由它们相互作用产生的二次谐波形成共振三元组的条件。索普的理论导致谐振时二次谐波振幅的不确定性。在放弃这种不确定性时，我们表明后者的振幅在共振时具有有限的行为，从斜率线性增加。我们研究了背景旋转的影响，并发现了类似的结果，但增长率较弱。然后我们将理论应用于有限宽度的入射平面波的情况。在这种情况下，非线性相互作用仅限于入射和反射有限宽度波叠加的区域，这意味着二次谐波的振幅以共振为界。只要入射波和反射波的耗散功率保持小于传递到二次谐波的功率，我们就会发现与垂直平面中数值模拟的结果非常吻合。这是小倾斜角的情况。随着倾斜角的增加，反射波的聚焦增强了粘性效应，耗散最终超过了非线性传输。我们最后讨论了实验室实验的相关性，以评估理论结果的有效性。这意味着二次谐波的振幅以共振为界。只要入射波和反射波的耗散功率保持小于传递到二次谐波的功率，我们就会发现与垂直平面中数值模拟的结果非常吻合。这是小倾斜角的情况。随着倾斜角的增加，反射波的聚焦增强了粘性效应，耗散最终超过了非线性传输。我们最后讨论了实验室实验的相关性，以评估理论结果的有效性。这意味着二次谐波的振幅以共振为界。只要入射波和反射波的耗散功率保持小于传递到二次谐波的功率，我们就会发现与垂直平面中数值模拟的结果非常吻合。这是小倾斜角的情况。随着倾斜角的增加，反射波的聚焦增强了粘性效应，耗散最终超过了非线性传输。我们最后讨论了实验室实验的相关性，以评估理论结果的有效性。只要入射波和反射波的耗散功率保持小于传递到二次谐波的功率，我们就会发现与垂直平面中数值模拟的结果非常吻合。这是小倾斜角的情况。随着倾斜角的增加，反射波的聚焦增强了粘性效应，耗散最终超过了非线性传输。我们最后讨论了实验室实验的相关性，以评估理论结果的有效性。只要入射波和反射波的耗散功率保持小于传递到二次谐波的功率，我们就会发现与垂直平面中数值模拟的结果非常吻合。这是小倾斜角的情况。随着倾斜角的增加，反射波的聚焦增强了粘性效应，耗散最终超过了非线性传输。我们最后讨论了实验室实验的相关性，以评估理论结果的有效性。