Nonlinear internal gravity wave beams
Nonlinear internal gravity wave beams
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非线性内重力波梁
DOI：
10.1017/s0022112003003902
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发表时间：
2003
影响因子：
3.7
通讯作者：
T. Akylas
中科院分区：
文献类型：
--
作者：
A. Tabaei;T. Akylas
基于线性无粘理论，在均匀分层（恒定Brunt-Väisälä频率$N_{0}$）的Boussinesq流体中以频率$\omega_{0}$振荡的二维源诱导出稳态波模式，也称为圣安德鲁十字，其特征是四个直波束以相对于垂直方向的$\pm\cos^{-1}(\omega_{0}/N_{0})$角度从源向外径向伸展。类似的波束是由地形上的振荡分层流产生的，也出现在大气中雷暴产生的重力波的模拟中。无限延伸的均匀平面波实际上是非线性无粘运动方程的精确解，本文利用这一性质来研究考虑弱粘性和折射效应的有限振幅波束的传播。首先考虑斜光束（$\omega_{0}\,{<}\,N_{0}$），并推导出沿光束方向缓慢调制的振幅演化方程。值得注意的是，在这个演化方程中，前阶非线性项被消去了，因此，Thomas ＆ Stevenson（1972）关于线性粘性梁的稳态相似解在非线性方程组中也是有效的。此外，由于同样的原因，对于在Boussinesq流体中几乎垂直传播的二维和轴对称光束（$\omega_{0}\,{\approx}\,N_{0}$），发现非线性效应相对不重要。然而，由于群速度在$\omega_{0}\,{=}\,N_{0}$时消失，近垂直光束的瞬态演化在较慢的时间尺度上发生；这被证明可以解释Gordon和Stevenson（1972）的稳态相似解与他们的实验观察结果之间的差异。最后，利用渐近理论研究了近垂直非线性光束在背景剪切和Brunt-Väisälä频率变化情况下的折射。
Based on linear inviscid theory, a two-dimensional source oscillating with frequency $\omega_{0}$ in a uniformly stratified (constant Brunt–Väisälä frequency $N_{0}$) Boussinesq fluid induces a steady-state wave pattern, also known as St Andrew's Cross, that features four straight wave beams stretching radially outwards from the source at angles $\pm\cos^{-1}(\omega_{0}/N_{0})$ relative to the vertical. Similar wave beams are generated by oscillatory stratified flow over topography and also appear in simulations of thunderstorm-generated gravity waves in the atmosphere. Uniform plane-wave beams of infinite extent are in fact exact solutions of the nonlinear inviscid equations of motion, and this property is used here to study the propagation of finite-amplitude wave beams taking into account weak viscous and refraction effects. Oblique beams ($\omega_{0}\,{<}\,N_{0}$) are considered first and an amplitude-evolution equation is derived assuming slow modulations along the beam direction. Remarkably, the leading-order nonlinear terms cancel out in this evolution equation and, as a result, the steady-state similarity solution of Thomas & Stevenson (1972) for linear viscous beams is also valid in the nonlinear régime. Moreover, for the same reason, nonlinear effects are found to be relatively unimportant for two-dimensional and axisymmetric beams that propagate nearly vertically ($\omega_{0}\,{\approx}\,N_{0}$) in a Boussinesq fluid. Owing to the fact that the group velocity vanishes when $\omega_{0}\,{=}\,N_{0}$, however, the transient evolution of nearly vertical beams takes place on a slower time scale than that of oblique beams; this is shown to account for the discrepancies between the steady-state similarity solution of Gordon & Stevenson (1972) and their experimental observations. Finally, the present asymptotic theory is used to study the refraction of nearly vertical nonlinear beams in the presence of background shear and variations in the Brunt–Väisälä frequency.