TY - JOUR

T1 - Experimental Study of Dispersion and Modulational Instability of Surface Gravity Waves on Constant Vorticity Currents

AU - Steer, James

AU - Borthwick, Alistair

AU - Stagonas, Dimitris

AU - Buldakov, Eugeny

AU - van den Bremer, Ton

PY - 2020/2/10

Y1 - 2020/2/10

N2 - This paper examines experimentally the dispersion and stability of weakly nonlinear waves on opposing linearly vertically sheared current proﬁles (with constant vorticity). Measurements are compared against predictions from the 1D+1 constant vorticity nonlin-ear Schr¨odinger equation (the vor-NLSE) derived by Thomas, Kharif & Manna [Physics of Fluids, 24, 127102 (2012)]. The shear rate is negative in opposing currents when the magnitude of the current in the laboratory reference frame is negative (i.e. opposing the direction of wave propagation) and reduces with depth, as is most commonly encountered in nature. Compared to a uniform current with the same surface velocity, negative shear has the eﬀect of increasing wavelength and enhancing stability. In experiments with a regular low-steepness wave, the dispersion relationship between wavelength and frequency is examined on ﬁve opposing current proﬁles with shear rates from 0 s−1 to −0.87 s−1. For all current proﬁles, the linear constant vorticity dispersion relation predicts the wavenumber to within the 95% conﬁdence bounds associated with estimates of shear rate and surface current velocity. The eﬀect of shear on modulational instability was determined by the spectral evolution of a carrier wave seeded with spectral sidebands on opposing current proﬁles with shear rates between 0 s−1 and −0.48 s−1. Numerical solutions of the vor-NLSE are consistently found to predict sideband growth to within two standard deviations across repeated experiments, performing considerably better than its uniform-current NLSE counterpart. Similarly, the ampliﬁcation of experimental wave envelopes is predicted well by numerical solutions of the vor-NLSE, and signiﬁcantly over-predicted by the uniform-current NLSE.

AB - This paper examines experimentally the dispersion and stability of weakly nonlinear waves on opposing linearly vertically sheared current proﬁles (with constant vorticity). Measurements are compared against predictions from the 1D+1 constant vorticity nonlin-ear Schr¨odinger equation (the vor-NLSE) derived by Thomas, Kharif & Manna [Physics of Fluids, 24, 127102 (2012)]. The shear rate is negative in opposing currents when the magnitude of the current in the laboratory reference frame is negative (i.e. opposing the direction of wave propagation) and reduces with depth, as is most commonly encountered in nature. Compared to a uniform current with the same surface velocity, negative shear has the eﬀect of increasing wavelength and enhancing stability. In experiments with a regular low-steepness wave, the dispersion relationship between wavelength and frequency is examined on ﬁve opposing current proﬁles with shear rates from 0 s−1 to −0.87 s−1. For all current proﬁles, the linear constant vorticity dispersion relation predicts the wavenumber to within the 95% conﬁdence bounds associated with estimates of shear rate and surface current velocity. The eﬀect of shear on modulational instability was determined by the spectral evolution of a carrier wave seeded with spectral sidebands on opposing current proﬁles with shear rates between 0 s−1 and −0.48 s−1. Numerical solutions of the vor-NLSE are consistently found to predict sideband growth to within two standard deviations across repeated experiments, performing considerably better than its uniform-current NLSE counterpart. Similarly, the ampliﬁcation of experimental wave envelopes is predicted well by numerical solutions of the vor-NLSE, and signiﬁcantly over-predicted by the uniform-current NLSE.

U2 - 10.1017/jfm.2019.951

DO - 10.1017/jfm.2019.951

M3 - Article

VL - 884

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -