Within the context of the well-known interpretation in terms of the wave interaction [P. G. Baines and H. Mitsudera, J. Fluid Mech. 276, 327 (1994); J. R. Carpenter et al., Phys. Fluids 22, 054104 (2010)], instability of sharply stratified (so that the vertical scale ℓ of density variation is much smaller than the scale Λ of velocity shear) flows with inflection-free velocity profiles should be treated as Holmboe’s instability. In such flows with a relatively weak stratification (when the bulk Richardson number J < (ℓ/Λ)3/2), eigenoscillations (i.e., Holmboe waves) have much the same phase velocities in a broad spectral range. This creates favorable conditions for a wide variety of three-wave interactions, in contrast to the homogeneous boundary layers where subharmonic resonance is the only effective three-wave process. In the paper, evolution equations are derived which describe three-wave interactions of Holmboe waves and have the form of nonlinear integral equations. Analytical and numerical methods are both used to find their solutions in different cases, and it is shown that at the nonlinear stage disturbances increase, as a rule, explosively. Some possible relations of the results obtained with those of numerical simulations and laboratory experiments are briefly discussed.
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November 2011
Research Article|
November 09 2011
Resonant three–wave interaction of Holmboe waves in a sharply stratified shear flow with an inflection–free velocity profile
S. M. Churilov
S. M. Churilov
Institute of Solar–Terrestrial Physics SB RAS, 126a Lermontov Street, Irkutsk 664033,
Russia
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Physics of Fluids 23, 114101 (2011)
Article history
Received:
April 28 2011
Accepted:
October 08 2011
Citation
S. M. Churilov; Resonant three–wave interaction of Holmboe waves in a sharply stratified shear flow with an inflection–free velocity profile. Physics of Fluids 1 November 2011; 23 (11): 114101. https://doi.org/10.1063/1.3657093
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