TY - JOUR

T1 - Asymptotics for the long-time evolution of kurtosis of narrow-band ocean waves

AU - Janssen, Peter A.E.M.

AU - Janssen, Augustus J.E.M.

PY - 2019/1/25

Y1 - 2019/1/25

N2 - In this paper we highlight that extreme events such as freak waves are a transient phenomenon in keeping with the old fisherman tale that these extreme events seem to appear out of nowhere. Janssen (J. Phys. Oceanogr., vol. 33, 2003, pp. 863-884) obtained an evolution equation for the ensemble average of the excess kurtosis, which is a measure for the deviation from normality and an indicator for nonlinear focusing resulting in extreme events. In the limit of a narrow-band wave train, whose dynamics is governed by the two-dimensional nonlinear Schrödinger (NLS) equation, the excess kurtosis is under certain conditions seen to grow to a maximum after which it decays to zero for large times. This follows from a numerical solution of the problem and also from an analytical solution presented by Fedele (J. Fluid Mech., vol. 782, 2015, pp. 25-36). The analytical solution is not explicit because it involves an integral from initial time to actual time. We therefore study a number of properties of the integral expression in order to better understand some interesting features of the time-dependent excess kurtosis and the generation of extreme events.

AB - In this paper we highlight that extreme events such as freak waves are a transient phenomenon in keeping with the old fisherman tale that these extreme events seem to appear out of nowhere. Janssen (J. Phys. Oceanogr., vol. 33, 2003, pp. 863-884) obtained an evolution equation for the ensemble average of the excess kurtosis, which is a measure for the deviation from normality and an indicator for nonlinear focusing resulting in extreme events. In the limit of a narrow-band wave train, whose dynamics is governed by the two-dimensional nonlinear Schrödinger (NLS) equation, the excess kurtosis is under certain conditions seen to grow to a maximum after which it decays to zero for large times. This follows from a numerical solution of the problem and also from an analytical solution presented by Fedele (J. Fluid Mech., vol. 782, 2015, pp. 25-36). The analytical solution is not explicit because it involves an integral from initial time to actual time. We therefore study a number of properties of the integral expression in order to better understand some interesting features of the time-dependent excess kurtosis and the generation of extreme events.

KW - surface gravity waves

KW - waves/free-surface flows

UR - http://www.scopus.com/inward/record.url?scp=85057374937&partnerID=8YFLogxK

U2 - 10.1017/jfm.2018.844

DO - 10.1017/jfm.2018.844

M3 - Article

AN - SCOPUS:85057374937

SN - 0022-1120

VL - 859

SP - 790

EP - 818

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -