Triadic resonance is one mechanism via which internal waves dissipate their energy, often at locations away from their generation sites. In this paper, we perform a combined theoretical and numerical study of triadic resonance in internal wave modes in a finite-depth ocean with background rotation and an arbitrary stratification profile. The spatial evolution of the modal amplitudes within a resonant triad are first derived based on the requirement that the nonlinear solution at leading order cannot diverge. The amplitude evolution equations are then numerically solved for two different cases: (i) modes 1 and 2 at a specific frequency ({\omega_0}) in triadic resonance with the mode-1 superharmonic wave (2{\omega_0}) in a uniform stratification, and (ii) a self-interacting mode-3 at a specific frequency {\omega_0} in triadic resonance with the mode-2 (2{\omega_0}) in a nonuniform stratification representative of the ocean. Quantitative estimates of energy transfer rates within the resonant triad show that superharmonic wave generation resulting from modal interactions should be an important consideration alongside other triadic resonances like parametric subharmonic instability (PSI). Remarkably, in case (ii), the amplitude evolution equations suggest that any initial energy in mode-3 at {\omega_0} would get permanently transferred to mode-2 at frequency 2{\omega_0}. Direct numerical simulations (DNS) are then performed to show the spontaneous generation of superharmonic internal waves resulting from modal interactions in the aforementioned two cases, and quantitatively validate the initial spatial evolution of the wave field predicted by the amplitude evolution equations. DNS at off-resonant frequencies are used to identify the range of primary wave frequencies (around the resonant frequency) over which spontaneous superharmonic wave generation occurs.

中文翻译：

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均匀和非均匀分层中模态相互作用的自发超谐波内波激发

三元共振是内波耗散能量的一种机制，通常在远离其产生地点的位置。在本文中，我们对具有背景旋转和任意分层剖面的有限深度海洋中的内波模式的三重共振进行了理论和数值组合研究。共振三元组内模态振幅的空间演化首先基于领先阶的非线性解不能发散的要求推导出来。然后针对两种不同情况对振幅演化方程进行数值求解：(i) 模式 1 和模式 2 在特定频率 ({\omega_0}) 与模式 1 超谐波 (2{\omega_0}) 的三重共振分层，(ii) 特定频率 {\omega_0} 下的自相互作用模式 3，与代表海洋的非均匀分层中的模式 2 (2{\omega_0}) 进行三重共振。共振三元组内能量转移率的定量估计表明，与参数次谐波不稳定性 (PSI) 等其他三元共振一起，由模态相互作用产生的超谐波应该是一个重要的考虑因素。值得注意的是，在情况 (ii) 中，振幅演化方程表明 {\omega_0} 模式 3 中的任何初始能量将永久转移到频率 2{\omega_0} 的模式 2。然后执行直接数值模拟 (DNS) 以显示在上述两种情况下由模态相互作用引起的超谐波内波的自发产生，并对振幅演化方程预测的波场初始空间演化进行了定量验证。偏共振频率的 DNS 用于识别发生自发超谐波产生的主波频率范围（共振频率附近）。