TY - JOUR

T1 - A comparison of frequency downshift models of wave trains on deep water

AU - Carter, John D.

AU - Henderson, Diane

AU - Butterfield, Isabelle

N1 - Funding Information:
We thank Andrea Armaroli, Bernard Deconinck, Debbie Eeltink, Paul Milewski, and Harvey Segur for helpful discussions. This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-1716120 and DMS-1716159.
Publisher Copyright:
© 2019 Author(s).

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Frequency downshift (FD) in wave trains on deep water occurs when a measure of the frequency, typically the spectral peak or the spectral mean, decreases as the waves evolve. Many FD models rely on wind or wave breaking. We consider seven models that do not include these effects and compare their predictions with four sets of experiments that also do not include these effects. The models are the (i) nonlinear Schrödinger equation (NLS), (ii) dissipative NLS equation (dNLS), (iii) Dysthe equation, (iv) viscous Dysthe equation (vDysthe), (v) Gordon equation (Gordon), which has a free parameter, (vi) Islas-Schober equation (IS), which has a free parameter, and (vii) a new model, the dissipative Gramstad-Trulsen (dGT) equation. The dGT equation has no free parameters and addresses some of the difficulties associated with the vDysthe equation. We compare a measure of overall error and the evolution of the spectral amplitudes, means, and peaks. We find the following: (i) The NLS and Dysthe equations do not accurately predict the spectral amplitudes. (ii) The Gordon equation, which is a successful model of FD in optics, does not accurately model FD in water waves, regardless of the choice of free parameter. (iii) The dNLS, vDysthe, dGT, and IS (with optimized free parameter) models do a reasonable job predicting the measured spectral amplitudes, but none captures all spectral evolutions. (iv) The vDysthe, dGT, and IS models most accurately predict the observed evolution of the spectral peak and the spectral mean. (v) The IS and vDysthe models have the smallest overall errors.

AB - Frequency downshift (FD) in wave trains on deep water occurs when a measure of the frequency, typically the spectral peak or the spectral mean, decreases as the waves evolve. Many FD models rely on wind or wave breaking. We consider seven models that do not include these effects and compare their predictions with four sets of experiments that also do not include these effects. The models are the (i) nonlinear Schrödinger equation (NLS), (ii) dissipative NLS equation (dNLS), (iii) Dysthe equation, (iv) viscous Dysthe equation (vDysthe), (v) Gordon equation (Gordon), which has a free parameter, (vi) Islas-Schober equation (IS), which has a free parameter, and (vii) a new model, the dissipative Gramstad-Trulsen (dGT) equation. The dGT equation has no free parameters and addresses some of the difficulties associated with the vDysthe equation. We compare a measure of overall error and the evolution of the spectral amplitudes, means, and peaks. We find the following: (i) The NLS and Dysthe equations do not accurately predict the spectral amplitudes. (ii) The Gordon equation, which is a successful model of FD in optics, does not accurately model FD in water waves, regardless of the choice of free parameter. (iii) The dNLS, vDysthe, dGT, and IS (with optimized free parameter) models do a reasonable job predicting the measured spectral amplitudes, but none captures all spectral evolutions. (iv) The vDysthe, dGT, and IS models most accurately predict the observed evolution of the spectral peak and the spectral mean. (v) The IS and vDysthe models have the smallest overall errors.

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U2 - 10.1063/1.5063016

DO - 10.1063/1.5063016

M3 - Article

AN - SCOPUS:85059817377

SN - 1070-6631

VL - 31

JO - Physics of Fluids

JF - Physics of Fluids

IS - 1

M1 - 013103

ER -