# New kinetic equations and their modelling for wind wave forecasting.

Lead Research Organisation:
Keele University

Department Name: Institute Env Physical Sci & App Maths

### Abstract

Wind waves are a key link in the feedback loop of ocean/atmosphere interaction. The quality of wind wave modelling directly affects the quality of weather forecasting and the accuracy of description of all processes at the air/water interface, since waves control fluxes of momentum, gas and heat exchange. Better wind wave modelling, especially of 'rogue' waves, is literally vital for reliability of growing shipping and offshore activities. At present all wave forecasting and modelling is based on numerical integration of the kinetic equation (KE), which is also referred to as the Hasselmann equation. The equation takes into account wind input, dissipation and interaction between waves of different scales and directions and describes the slow evolution of wind wave spectra in time and space. The interaction term, usually denoted as Snl, is dominant for energy carrying waves. The expression for Snl has not changed for half a century. Now the improved quality of observations and data input for wave modelling made it impossible to ignore the situations where the discrepancy between the KE based models and observations could not be bridged by any means. The following fundamental shortcomings of the modelling based upon the existing kinetic equation became apparent: (i) By construction, the KE cannot describe reaction of wave fields to rapid perturbations (e.g. abrupt changes of wind, wind gusts, sharp boundaries, etc). (ii) The KE based models can predict evolution of wave spectra only, while it is highly desirable to model the evolution of the wave height probability density function, or, more precisely, its departure from the Gaussian distribution. This, in particular, is badly needed for forecasting rogue waves, but also for assessing the validity of the KE based models. The KE itself 'does not know' when it ceases to be applicable. To address these shortcomings, the following radical ideas have been put forward very recently by the authors. New generalised kinetic equations have been derived from first principles by lifting two most restrictive assumptions: proximity of the wave field to equilibrium, and the oversimplification in taking into account the wave field departure from Gaussianity. Crucially, it was also discovered by the authors that for a typical wind waves it is possible to reconstruct evolution of probability density function once the spectrum is found. The proposal aims at creating a new numerical and conceptual framework for the study of wind wave evolution, by elaborating the above ideas and developing a new way of wind wave modelling based on the novel expression for Snl, derived from first principles. The purpose of this proposal is to create a numerical tool (robust parallel code) for solving the new kinetic equations and then to use this tool to address outstanding questions of wave field evolution. In developing the code, we will be greatly helped by the direct numerical simulation (DNS) algorithm developed by the authors for the simulation of long time evolution of wave field. The algorithm will be used for the validation of the new code. With the help of the new tool, we will delineate the situations where the standard KE is indeed valid and where it ceases to be applicable, and will investigate the regimes of wave evolution well beyond the limits of its applicability (e.g. gusty wind). We will model the evolution of wave field departure from Gaussianity, which, in particular, will make possible to assess probability of freak waves. We will formulate 'practical' parameterisations and recommendations for wave forecasting. This project aims to revolutionise wind wave modelling and forecasting. The new approach we propose is better suited to parallelisation of simulations and increase of power of computers, eventually it will replace the existing algorithms of calculating Snl. It will produce not only a more accurate description of reality, it has the potential to do it faster.

### Organisations

## People |
## ORCID iD |

Victor Shrira (Principal Investigator) | |

Sergei Annenkov (Researcher) |

### Publications

Shrira V
(2013)

*Advances in Wave Turbulence*
Annenkov SY
(2022)

*Effects of finite non-Gaussianity on evolution of a random wind wave field.*in Physical review. E
Annenkov SY
(2013)

*Evaluation of kurtosis of JONSWAP spectra*
Shrira V
(2014)

*Evaluation of Skewness and Kurtosis of Wind Waves Parameterized by JONSWAP Spectra*in Journal of Physical Oceanography
Annenkov SY
(2018)

*Evolution of water wave spectra under a sharp increase of wind*
Annenkov Sergei
(2018)

*Evolution of water wave spectra under a sharp increase of wind*in EGU General Assembly Conference Abstracts
Annenkov SY
(2011)

*Evolution of wave turbulence under "gusty" forcing.*in Physical review lettersDescription | (i) The problem with the current generation of the wave models is that they are based upon a number of restrictive assumptions often violated in nature. In particular, winds are presumed to vary only slowly. The main overall achievement of the project is in the creation of a new conceptual and numerical framework for wind wave modelling based on novel wave nonlinear interaction term Snl derived by the authors from first principles. The generalized kinetic equation is free from the restrictive assumptions and can be forced by the wind fields with high temporal resolution which are becoming available now. (ii) The main technical achievement is an effective algorithm for simulation of the new nonlinear term in the generalized kinetic equation. The algorithm has been implemented in a robust parallel code for wave modelling. Now it has become possible to model evolution of wave fields forced by rapidly varying winds. (iii) Employing the developed code it has become possible to model a class of hitherto inaccessible important geophysical situations with rapidly varying winds, such as, for example, squall. The first of its kind study of wave evolution under squall wind has been carried out. The most surprising finding with significant practical implications is that the limits of applicability of the classical kinetic equation proved to be much wider than anticipated. However, after the end of the squall the generalized kinetic equation shows that a transient spectrum is formed, which has a considerably narrower peak than the classical kinetic equation predicts. (iv) There has been a breakthrough in modelling the evolution of probabilistic characteristics of wind waves, such as kurtosis and other higher moments of wave height distribution. It has been found that under constant wind conditions these moments evolve in a particular self-similar way and the evolution can be described analytically. For the wave fields parameterized by the JONSWAP spectra of the kurtosis and skewness has been simulated for a wide range of parameters and on this basis a simple parameterization has been suggested. The key grants objectives have been met. Further implementation of the findings in the large weather prediction centres such as ECMWF or UK MetOffice requires considerable time and effort and is determined by their own priorities. The findings regarding the evolution of wave height distributions are easier to implement and even small companies working in this area could do this. A number of collaborations partly capitalizing on the results of this project have been initiated: (i) EU FP7 2014-18: Air-Sea Interaction under Stormy and Hurricane Conditions: Physical Models and Applications to Remote Sensing (The Coordinator) (ii) MetOfice/ECMWF 2011-2014: "Modelling of new kinetic equations". Access to the ECMWF supercomputer for 3 years (PI) (iii) EU FP7 2010-12, "Deterministic Forecasting of Rogue Waves in the Ocean" (PI) (iv) EPSRC 2015-18. Network "Living with Environmental Change (LWEC)" |

Exploitation Route | The findings have the potential to revolutionize wind wave modelling and forecasting. The new code for simulation of the generalized kinetic equation can be implemented in the wave modelling routines in the large weather forecasting centers (ECMWF, MetOffice). The expected improvement of wave modelling and forecasting will benefit all marine related industries. The findings on the evolution of wave height distributions are easier to implement and can be taken forward by small companies and consultancies interested in risk in marine industries. Creation of an efficient algorithm for solving the generalized kinetic equation shows a way how such equations could be handled in other branches of science, e.g. for light propagation in optical fibres. |

Sectors | Aerospace, Defence and Marine,Environment,Financial Services, and Management Consultancy,Transport,Other |

URL | http://www.nl-wave.com |

Description | It is too early to expect the findings to be used in earnest, the project suggests a radical change in the practice of wave modelling and forecasting. Such changes do not happen overnight. Implementation of the key findings depends primarily on large weather prediction centres such as ECMWF and MetOffice, which is determined by their internal priorities. At the moment the findings are now being scrutinized. |

First Year Of Impact | 2014 |

Sector | Other |

Title | gKE numerical tank |

Description | A new approach to water waves modelling has been created, based on the generalised kinetic equation. The efficient algorithm, parallelised and adapted for the use in a modern supercomputing environment, allows for the first time to model transient wave states, arising from rapid changes of wind forcing. The algorithm allows to trace, along with the spectra, the evolution of higher statistical moments of a wave field, quantifying the changes in the likelihood of freak waves. |

Type Of Material | Computer model/algorithm |

Provided To Others? | No |

Impact | Numerical simulations performed with the model allowed to identified transient sea states with increased probability of freak waves. |