An inertia-gravity wave (IGW) propagating in a vertically sheared, rotating stratified fluid interacts with the pair of inertial levels that surround the critical level. An exact expression for the form of the IGW is derived here in the case of a linear shear and used to examine this interaction in detail. This expression recovers the classical values of the transmission and reflection coefficients |T|=exp(−πμ) and |R|=0 , where μ 2 =J(1+ν 2 )−1/4 , J is the Richardson number and ν the ratio between the horizontal transverse and along-shear wavenumbers. For large J , a WKB analysis provides an interpretation of this result in term of tunnelling: an IGW incident to the lower inertial level becomes evanescent between the inertial levels, returning to an oscillatory behaviour above the upper inertial level. The amplitude of the transmitted wave is directly related to the decay of the evanescent solution between the inertial levels. In the immediate vicinity of the critical level, the evanescent IGW is well represented by the quasi-geostrophic approximation, so that the process can be interpreted as resulting from the coupling between balanced and unbalanced motion. The exact and WKB solutions describe the so-called valve effect, a dependence of the behaviour in the region between the inertial levels on the direction of wave propagation. For J<1 this is shown to lead to an amplification of the wave between the inertial levels. Since the flow is inertially unstable for J<1 , this establishes a correspondence between the inertial-level interaction and the condition for inertial instability.
展开▼
机译:在垂直剪切的旋转分层流体中传播的惯性重力波(IGW)与围绕临界水平面的一对惯性水平面相互作用。在线性剪切的情况下,可以得出IGW形式的精确表达式,并用于详细检查这种相互作用。该表达式恢复了透射和反射系数的经典值| T | = exp(-πμ)和| R | = 0,其中μ2 = J(1 +ν2)-1 / 4,J是理查森数, ν水平横向波数与沿剪切波数之比。对于较大的J,WKB分析用隧道效应来解释该结果:入射到较低惯性级的IGW在惯性级之间逐渐消失,返回到高于惯性级的振荡行为。传输波的振幅与惯性能级之间的the逝解的衰减直接相关。在临界水平附近,the逝的IGW由准地转近似近似表示,因此该过程可以解释为平衡运动与不平衡运动之间的耦合所导致。精确和WKB解决方案描述了所谓的阀效应,即惯性能级之间的区域中的行为对波传播方向的依赖性。对于J <1,这表明导致惯性能级之间的波放大。由于对于J <1,流动是惯性不稳定的,因此在惯性级相互作用和惯性不稳定性的条件之间建立了对应关系。
展开▼