An asymptotic study of instability to Langmuir circulation in shallow layers

作者：Hayes D T; Phillips W R C^*

来源：Geophysical and Astrophysical Fluid Dynamics, 2016, 110(4): 295-316.

DOI：10.1080/03091929.2016.1173207

• Instability theory
• shallow water waves
• wave-mean flow interactions
• boundary conditions
• Langmuir circulation

The role of boundary conditions applied to perturbation equations describing the instability of wavy shear flows to counter-rotating vortical structures aligned with the flow in shallow water layers is considered. In the ocean context the structures are described by the Craik-Leibovich equations and are known as Langmuir circulation (LC); they arise via an instability mechanism known as CL2, which requires the presence of shear and differential drift of the same sense and that a threshold Rayleigh number be exceeded. While Neumann or stress-free boundary conditions applied at the free surface in deep water act to yield a physically plausible spanwise wavenumber l at onset, that is not the case in shallow layers, where non-zero l at onset is ensured only when mixed boundary conditions are applied. This paper questions why that is so and goes on to determine how onset is affected by distributions of . In order to proceed analytically, results are computed by a small-l asymptotic approximation for which an algorithm is generated enabling computation to, essentially, any order. It is found that the extra stress induced by the perturbed motion, which is reflected in mixed but not Neumann boundary conditions, acts to suppress the growth rate of the instability over all wavenumbers, thereby restricting instability to non-zero l; further details are seen to depend on the distributions of and . Specifically, while the critical wavenumber is not overly sensitive to precise distributions of , the critical Rayleigh number is sensitive. Minor variations in can also lead to multiple rather than single layers of LC and even contrive to ensure non-zero l at onset with Neumann boundary conditions. Finally, although crafted as an asymptotic approximation for , the results concur closely with numerical calculations at much largerl.

• 出版日期2016