Results of
extensive simulations of swell evolution within the duration-limited setup
for the kinetic Hasselmann equation
for long durations of up to

Ocean swell is an important constituent of the field of surface gravity waves in the sea and, more generally, of the sea environment as a whole. Swell is usually defined as a fraction of a wave field that does not depend (or depends slightly) on local wind. Being generated in confined stormy areas, these waves can propagate long distances of many thousands of miles, thus influencing vast ocean stretches. For example, swell from the Roaring Forties in the Southern Ocean can traverse the Pacific and reach distant shores of California and Kamchatka. Predicting swell as a part of sea wave forecast remains a burning problem for maritime safety and marine engineering.

Pioneering works by

The authors are thankful to Gerbrant van Vledder, Delft University of Technology, for this reference.

Nonlinear wave–wave interactions have been sketched by

It should be stressed that a number of theoretical and numerical models,
including those mentioned above, treat swell as a quasi-monochromatic wave
and, thus, ignore nonlinear interactions of the swell harmonics themselves
and the swell coupling with locally generated wind waves. The latter effect
can be essential, as observations and simulations clearly show

Synthetic aperture radar (SAR) allows for a spatial resolution of up to tens
of meters

Meanwhile, the linear treatment remains quite restrictive and cannot explain
important features of swell. The observed swell spectra exhibit frequency
downshift which is not predicted by deterministic linear or weakly nonlinear
models of narrow-banded wave guide evolution

In this paper we present results of extensive simulations of ocean swell
within the Hasselmann equation for deep water waves. The simplest
duration-limited setup has been chosen to obtain numerical solutions for the
duration of up to

We analyze the simulation results within the framework of the theory of weak
turbulence

We demonstrate the fact that is usually ignored: the power-law swell
attenuation within the conservative kinetic equation. We show that it does
not contradict the observations mentioned above. We also reveal a remarkable
feature of collapsing the swell spectra onto an angular distribution that
depends weakly on initial angular spreading. Such universality can be of
great value for modeling swell and
developing methods for its monitoring

We conclude this paper with a discussion of how to apply this model.
Evidently, the setup of duration-limited evolution is quite restrictive and
does not reflect essential features of ocean swell when wave dispersion and
spatial divergence play a key role. At the same time, wave–wave interactions
remain of importance independently of the setup. The weakening of swell
evolution is not directly related to abatement of wave–wave interactions
which are able to effectively restore perturbations of these quasi-stationary
states

In this section we reproduce previously reported theoretical results on the
evolution of swell as a random field of weakly interacting wave harmonics. We
apply the statistical theory of wind-driven seas

The right-hand term

An approximate weakly anisotropic Kolmogorov–Zakharov solution has been
obtained by

Originally, solutions
(Eqs.

Swell solutions evolve slowly with time and, thus, give a good opportunity
for discussing features of the KZ solutions (or, alternatively, the KZ
solutions can be used as a reference case for the swell studies). One of the
key points of this discussion is the question of uniqueness or universality
of the swell solutions that can be treated in the context of general KZ
solutions (Eq.

The homogeneity property Eq. (

The shape functions

Two-lobe patterns can be observed beyond the spectral peak as local maxima at
oblique directions or as “shoulders” in wave frequency spectra. Their
appearance within the kinetic equation approach is generally associated with
wave generation by wind

An essential approximation which is widely used both for experimentally
observed and simulated wave spectra is generally treated as an important
property of spectral shape invariance

The self-similar solution for swell is just a member of a family of solutions
(Eqs.

The swell solution manifests another general feature of evolving spectra: the
downshifting of the spectral peak frequency (or other characteristic
frequency), i.e.,

The theoretical background presented above is used below for analysis of results of simulations.

Simulations of ocean swell require special care. First of all, calculations
for quite long periods of time (up to

Standard angular resolution

Initial conditions were similar in all series of simulations: spectral
density of action in wavenumber space was almost constant in a box of the
wavenumber modulo and angles. A slight modulation (

Dissipation was absent in the runs. Free boundary conditions were applied at
the high-frequency end of the domain of calculations: generally, short-term
oscillations of the spectrum tail do not lead to instability; i.e., the
resulting solutions can be regarded as ones corresponding to conditions of
decay at infinitely small scales (

Calculations with a hyper-viscosity

Initial parameters of the simulation series.

In contrast to wind-driven waves where wind speed is an essential physical
parameter that gives a useful physical scale, the swell evolution is
determined by initial conditions only, i.e., by

Below we focus on the series of Table

Frequency spectra of energy at different times (legend, in hours)
for case sw330 (

Evolution of swell spectra with time is shown in Fig.

Evolution of wave parameters for runs of Table

The dependence of key wave parameters on time is shown in Fig.

A simple quantitative estimate of the “degree of anisotropy” is given in
Fig.

Evolution of the left-hand side of the invariant (Eq.

Similar dispersion of runs depending on anisotropy of initial distributions
is seen in Fig.

Despite a significant difference in the runs in integral characteristics of
the swell anisotropy (e.g., Fig.

Normalized frequency spectra for direction

The only physical mechanism of the mode selection in the swell problem is
nonlinear relaxation to an inherent state due to four-wave resonant
interactions. This relaxation generally occurs at essentially shorter
timescales than ones of wind pumping and wave dissipation

Normalized sections of spectra at the peak frequency

Normalized dependence of swell energy spectra on angle at peak
frequency

Evolution of directional spreading in time is shown in absolute values in
Fig.

Angular spreading of the swell spectra at different times (in hours;
see the legend). Left column – wave spectra at peak frequency; right –
integral of wave spectra in frequency as a function of direction.

The effect of sharpening of angular distributions of the run sw330 in
Fig.

Isolines of spreading functions for different runs (see Table 1):

Bi-modality of directional spreading of ocean swell is widely discussed for
experimental data as a possible result of swell evolution

The patterns similar to ones of Fig.

An important issue of agreement of our results and findings by

Generally, the phenomenon of side-lobe occurrence is associated with a joint
effect of wave–wave interactions and wave generation by wind

The very slow evolution of swell in our simulations provides a chance to
check the relevance of the classic Kolmogorov–Zakharov
solutions (Eqs.

Two runs of Table

Top row – spectral fluxes
of energy for series sw030

The domain of quasi-constant fluxes

The first and second Kolmogorov constants can easily be estimated for the
approximate solution (Eq.

Left – estimates of the first Kolmogorov constant

The analytic estimate gives the very close result

While the estimates of the Kolmogorov's constants for the swell look
consistent, the numerical solutions differ essentially from the approximate
weakly anisotropic KZ solution (Eq.

The estimates of

Results of our simulations showed their fairly good correspondence to findings of the theory of wave (weak) turbulence. The relevance of these results to experimental facts seems to be a logical conclusion of this work. The issue of relevance is 2-fold. First, our results can help in explaining effects whose interpretation in terms of alternative approaches (mostly, within linear theory) is questionable. Secondly, one can formulate, or, at least, sketch cases where our approach becomes invalid or requires an extension. Both aspects are considered in the final section.

Attenuation in course of long-term swell evolution is an appealing problem of
the swell monitoring. We show that contribution of wave–wave interactions to
this process can be important mostly at initial stages of swell evolution.
The observed rates of swell attenuation in an open ocean cannot be treated
within our approach for a number of reasons. First of all, the
duration-limited setup of our simulations do not account for important
mechanisms of frequency dispersion and spatial divergence due to sphericity
of the Earth. These mechanisms can both contribute into swell attenuation
together with wave–wave interactions and essentially contaminate results of
observations. The intrinsic swell attenuation is, generally, small as
compared to the effect of reduction (or amplification at large fetches)

Ocean swell for long times (fetches) likely becomes an important constituent of the ocean environment which can be heavily affected by relatively short wind-driven waves. We discuss the effect of swell amplification at rather low wind speeds and give tentative estimates based on the approach of this paper.

Top – dependence of significant wave height

Dependence of wave height on time is shown in upper panel of
Fig.

For comparison with other models, and available observations, the
duration-limited simulations have been recasted into dependencies of fetch
through the simplest time-to-fetch transformation

It should be noted that our model describes attenuation of the ocean swell “on its own” due to wave–wave interactions without any external effects. Thus, the effect of an abrupt drop in wave amplitude at short times (fetch) should be taken into consideration above all others when discussing the possible application of our results to swell observations and physical interpretation of the experimental results.

Extremely weak attenuation of swell due to wave–wave interactions provokes a
question on robustness of this effect. A variety of physical mechanisms in
the ocean environment can change the swell evolution qualitatively. The above
discussion of swell attenuation presents a remarkable example of such
transformation when dissipation becomes dominant. Tracking of swell events
from space gives an alternative scenario of transformation when swell appears
to be growing. Satellite tracks can comprise up to

As noted and shown above, evolution of swell can occur at different
timescales for different physical quantities. Integrals of motion (energy,
action, momentum) evolve at relatively large scales: frequency downshift and
energy follows power-law dependencies

Oppositely, spectral shaping is evolving due to excursions from an “inherent
state” at much shorter scales that can be estimated following

A similar effect can be realized in the mixed sea when background of
relatively short wind-driven waves feeds the swell. Total energy flux of the
swell is decaying as rapidly as

Simple estimates of the possibility of the effect can be made in terms of
balancing of two fluxes: direct cascade of swell and inverse cascade of
wind-driven fraction. The swell energy leakage can be estimated from the
weakly turbulent law

The simple estimate (Eq.

We presented results of sea swell simulations within the framework of the
kinetic equation for water waves (the Hasselmann equation) and treated these
properties within the paradigm of the theory of weak turbulence. A series of
numerical experiments (duration-limited setup, WRT algorithm) has been
carried out in order to outline features of wave spectra in a range of scales
usually associated with ocean swell, i.e., wavelengths larger than

Key results of the study are the following.

A strong tendency for self-similar asymptotics is demonstrated. These asymptotics are shown to be insensitive to initial
conditions in terms of evolution of integral quantities (wave energy, momentum). Moreover, universal angular distributions of
wave spectra at large times have been obtained for both narrow (initial angular spreading

The classic Kolmogorov–Zakharov (KZ) isotropic and weakly anisotropic solutions for direct and inverse cascades are shown to be relevant to slowly evolving sea swell solutions. Estimates of the corresponding KZ constants are found to agree well with previous analytical, numerical and experimental results. Thus, features of KZ solutions can be used as a reference for advanced approaches in the swell studies.

We show that an inherent peculiarity of the Hasselmann equation, energy and momentum leakage, can also be considered
a mechanism of the sea swell attenuation. Today's models of sea swell are
unlikely to account for this effect. Possible
problems of the models are sketched in Sect. 3.1 when different options of simulation of the “conservative dissipation” are
discussed. All these options require sufficiently large high-frequency range where the short-term oscillations in absence of
dissipation or hyper-viscosity can mimic the energy leakage at

Long-term evolution of swell is associated with rather slow frequency downshift (

The last conclusion uncovers deficiency of the duration-limited setup for the phenomenon of swell. An
alternative setup of fetch-limited evolution (

No data sets were used in this article.

The authors declare that they have no conflict of interest.

The authors are thankful for the support of Russian Science Foundation grant no. 14-22-00174. The authors are indebted to Victor Shrira and Vladimir Geogjaev for discussions and valuable comments. The authors are also grateful to Andrei Pushkarev for his assistance in simulations. The authors appreciate critical consideration of the paper by reviewers Gerbrant van Vledder and Sergei Annenkov. Their constructive feedback led to substantial revision of Sects. 3 and 4. Edited by: V. Shrira Reviewed by: G. van Vledder, S. Y. Annenkov and two anonymous referees