TY - JOUR
T1 - Shallow water models with constant vorticity
AU - Hur, Vera Mikyoung
N1 - The author is supported by the National Science Foundation under the Faculty Early Career Development (CAREER) Award DMS-1352597 , an Alfred P. Sloan Research Fellowship, a Simons Fellowship in Mathematics, and by the U niversity of Illinois at Urbana-Champaign under the Arnold O. Beckman Research Awards Nos. RB14100 and RB16227 . She is grateful to the Department of Mathematics at Brown University and for its generous hospitality.
The author is supported by the National Science Foundation under the Faculty Early Career Development (CAREER) Award DMS-1352597, an Alfred P. Sloan Research Fellowship, a Simons Fellowship in Mathematics, and by the U niversity of Illinois at Urbana-Champaign under the Arnold O. Beckman Research Awards Nos. RB14100 and RB16227. She is grateful to the Department of Mathematics at Brown University and for its generous hospitality.
PY - 2019/1/1
Y1 - 2019/1/1
N2 - We modify the nonlinear shallow water equations, the Korteweg–de Vries equation, and the Whitham equation, to permit constant vorticity, and examine wave breaking, or the lack thereof. By wave breaking, we mean that the solution remains bounded but its slope becomes unbounded in finite time. We show that a solution of the vorticity-modified shallow water equations breaks if it carries an increase of elevation; the breaking time decreases to zero as the size of vorticity increases. We propose a full-dispersion shallow water model, which combines the dispersion relation of water waves and the nonlinear shallow water equations in the constant vorticity setting, and which extends the Whitham equation to permit bidirectional propagation. We show that its small amplitude and periodic traveling wave is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin–Feir instability in the irrotational setting; the critical wave number grows unboundedly large with the size of vorticity. The result agrees with that from a multiple scale expansion of the physical problem. We show that vorticity considerably alters the modulational stability and instability in the presence of the effects of surface tension.
AB - We modify the nonlinear shallow water equations, the Korteweg–de Vries equation, and the Whitham equation, to permit constant vorticity, and examine wave breaking, or the lack thereof. By wave breaking, we mean that the solution remains bounded but its slope becomes unbounded in finite time. We show that a solution of the vorticity-modified shallow water equations breaks if it carries an increase of elevation; the breaking time decreases to zero as the size of vorticity increases. We propose a full-dispersion shallow water model, which combines the dispersion relation of water waves and the nonlinear shallow water equations in the constant vorticity setting, and which extends the Whitham equation to permit bidirectional propagation. We show that its small amplitude and periodic traveling wave is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin–Feir instability in the irrotational setting; the critical wave number grows unboundedly large with the size of vorticity. The result agrees with that from a multiple scale expansion of the physical problem. We show that vorticity considerably alters the modulational stability and instability in the presence of the effects of surface tension.
KW - Breaking
KW - Constant vorticity
KW - Modulational instability
KW - Shallow water
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U2 - 10.1016/j.euromechflu.2017.06.001
DO - 10.1016/j.euromechflu.2017.06.001
M3 - Article
AN - SCOPUS:85024829540
SN - 0997-7546
VL - 73
SP - 170
EP - 179
JO - European Journal of Mechanics, B/Fluids
JF - European Journal of Mechanics, B/Fluids
ER -