Stable and unstable shear modes of rotating parallel flows in shallow water
Stable and unstable shear modes of rotating parallel flows in shallow water
复制标题
浅水中旋转平行流的稳定和不稳定剪切模式
DOI：
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发表时间：
1987
影响因子：
3.7
通讯作者：
W. Young
中科院分区：
文献类型：
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作者：
Y. Hayashi;W. Young
本文考虑赤道β平面上旋转的浅水切变流动的不稳定性。由于自由面的存在，运动是水平发散的，能量密度在场变量中是立方的(即，在标准记数法中，动能密度为1/2小时(u2+v2))。Marinone&Ripa(1984)观察到，结果是波能不再是正定的(存在交叉项Uh‘u’)。具有负波能的波可以通过将能量转移到平均流来增长。当然，总能量(平均加波能量)在这个过程中是守恒的。此外，当基本态具有恒定的位涡时，我们证明了在增长波和平均流之间不存在能量和动量交换。因此，当基本态没有位涡梯度时，不稳定波的波能为零，平均流被修正，使其能量不变。这一结果引人注目地表明，增长波和平均流之间的能量和动量交换通常不是不稳定的特征，也不是不稳定的本质。理解这些违反直觉的结果的一个有用的概念工具是剪切模式的扰动能量(或赝能)。这是当模式被激发时流体中的能量减去未被扰动的介质中的能量。等价地，扰动能量是波能和修正平均流中的波能之和。类似地定义了扰动动量(或伪矩)。对于无外源增长的不稳定模，扰动能量必须为零。另一方面，波能可以增加到正无穷大，保持为零，或者减小到负无穷大。因此，对不稳定性进行了三方分类。我们认为，这三种情况的一个共同特征是，不稳定的剪切模式大致是共振剪切模式的线性组合，如果另一种模式被以某种方式抑制，则每一种模式都是稳定的。两个共振成分必须具有相反符号的扰动能量，才能使不稳定联盟的扰动能量为零。不稳定性是扰动能量从具有负扰动能量的成员向具有正扰动能量的成员的转移。
This article considers the instabilities of rotating, shallow-water, shear flows on an equatorial β-plane. Because of the free surface, the motion is horizontally divergent and the energy density is cubic in the field variables (i.e. in standard notation the kinetic energy density is ½ h(u2 + v2)). Marinone & Ripa (1984) observed that as a consequence of this the wave energy is no longer positive definite (there is a cross-term Uh′u′). A wave with negative wave energy can grow by transferring energy to the mean flow. Of course total (mean plus wave) energy is conserved in this process. Further, when the basic state has constant potential vorticity, we show that there are no exchanges of energy and momentum between a growing wave and the mean flow. Consequently when the basic state has no potential vorticity gradients an unstable wave has zero wave energy and the mean flow is modified so that its energy is unchanged. This result strikingly shows that energy and momentum exchanges between a growing wave and the mean flow are not generally characteristic of, or essential to, instability. A useful conceptual tool in understanding these counterintuitive results is that of disturbance energy (or pseudoenergy) of a shear mode. This is the amount of energy in the fluid when the mode is excited minus the amount in the unperturbed medium. Equivalently, the disturbance energy is the sum of the wave energy and that in the modified mean flow. The disturbance momentum (or pseudomomentum) is defined analogously. For an unstable mode, which grows without external sources, the disturbance energy must be zero. On the other hand the wave energy may increase to plus infinity, remain zero, or decrease to minus infinity. Thus there is a tripartite classification of instabilities. We suggest that one common feature in all three cases is that the unstable shear mode is roughly a linear combination of resonating shear modes each of which would be stable if the other were somehow suppressed. The two resonating constituents must have opposite-signed disturbance energies in order that the unstable alliance has zero disturbance energy. The instability is a transfer of disturbance energy from the member with negative disturbance energy to the one with positive disturbance energy.