The new Zakharov–Resio–Pushkarev (ZRP) wind input source term

The scientific description of wind-driven wave
seas, inspired by
solid state physics statistical ideas (see, for instance,

Since Hasselmann's work, Eq. (

The HE is a specific example of the kinetic equation for
quasi-particles, widely used in different areas of
theoretical physics. There are standard methods for its
derivation. In the considered case, two forms of
the

Equation (

To understand the relation and difference between the Boltzmann
equation and the HE, one should recall the above-mentioned

Purely cubical (applicable to classical waves, not to classical
particles) systems are relatively new objects in physics. Such
equations describe the simplest case of the wave turbulence by
the theory called weak turbulence theory (WTT)

It is clear now that the WTT can be used for the description of
a very broad class of physical phenomena, including waves in
magneto-hydrodynamics

It is unfortunate that the discussion of the HE in the context of WTT has been overlooked by a major part of the oceanographic community for many years now. The community accepts, nevertheless, the HE as the basis for the operational wave forecasting models, thereby believing de facto in WTT without fully appreciating its ramifications.

The WTT essentially differs from the kinetic theory of classical
particles and quantum quasi-particles. In the “traditional” gas
kinetics (both classical and quantum) the basic solutions are
thermodynamic equilibrium spectra, such as Boltzmann and
Planck
distributions. In the WTT such solutions, though formally
existing, play no role – they are non-physical. The physically
essential solutions are the non-equilibrium Kolmogorov–Zakharov spectra (or
KZ spectra,

The accuracy advantage of knowing the analytical expression
for the

In contrast to

Similar to the wind input term, there is little consensus on the
parameterization of the source dissipation term

The next chapters present a balanced set of wind energy input
and wave energy dissipation source terms, based on WTT and
experimental data analysis. Further, they are numerically
checked to comply with WTT predictions and experimental observations. As
mentioned above, contrary to previous attempts to build the detailed-balance
source terms, the current approach is based neither on the development of
a rigorous analytic theory of turbulent atmospheric boundary layers nor on
reliable and repeatable air to ocean momentum measurements.
The new

Section

Section

Such a situation is similar to one realized in incompressible liquid turbulence for large Reynolds numbers, where the energy distribution is given by the famous Kolmogorov spectrum, transferring the energy from large to small scales, where the energy dissipation is realized due to viscosity, but the viscosity coefficients, i.e., the dissipation details, are not included in the final Kolmogorov spectrum expression. The ZF spectrum and its further KZ generalizations are in this sense the ideological Kolmogorov spectrum counterparts, having the significant difference that the Kolmogorov spectrum is a plausible conjecture, while KZ spectra are the exact solutions of the wave kinetic equation.

Since the current research is application oriented, it is important to understand why this formally academic approach is connected with reality. In this context, there is no such thing as the dissipation at infinitely small waves in nature: however, it is clear that the existence of an absorption at sufficiently high finite frequencies provides a wave scale in real applications that still preserves the KZ solutions, found from the HE equation in infinite space.

As was mentioned before, this statement was confirmed in
a different physical context with radically different inertial ranges (the
wave-number band between characteristic energy
input and characteristic wave energy dissipation), showing KZ
solutions with different corresponding indices. As for the
considered case of gravity waves on the surface of a deep
fluid, KZ spectra have been routinely observed in multiple
experiments. The results, published before 1985, are summarized
by

The assumed close relation of the HE in the infinite space and
finite domain, bounded by high-frequency dissipation, also has
a much deeper meaning, consisting in the fact that

The indices, corresponding to self-similar solutions, allow one to wrap up
Sect.

Section

There is considerable freedom in choosing a specific analytic form of such a
high-frequency dissipation term, given the lack of
a generally accepted rigorous derivation for this
mechanism. Consequently, one can choose a preferred one and
possibly justify it, but any particular choice will be
questioned since it will remain somewhat artificial. Because of
that, our motivation was that at the current stage of
development, we considered simplicity as a primary motivating
factor. Instead of following the previous path of
time-consuming numerical and empirical formulations based on
field experiments, the authors decided to continue the spectrum
from some specific frequency point, well above the spectral
peak, with the Phillips law

The spectrum amplitude at the junction frequency

At this point, all that remains for source-term closure in the
HE model is the coefficient in front of the wind input term, since it is not
well defined experimentally. If we carry out the
numerical simulation with some arbitrary chosen coefficient, we
could obtain a range of spectral energies but would retain the qualitative
properties of the HE, like the

To solve this, we choose the wind source coefficient to reproduce the same wave energy growth as was observed in field experiments. The value of this coefficient, found from the comparison with field observations of wave energy growth, is equal to 0.05. This step completes the construction of the HE model.

In the next sections we proceed with numerical simulations
based on the HE model described above. Section

Here we examine the empirical evidence from around the
world, which has been utilized to quantify energy levels
within the equilibrium spectral range by

The notations in Eqs. (

Correlation of the equilibrium range coefficient

Here

Self-similar solutions consistent with the conservative kinetic
equation

One should note that this equation does not contain any explicit
wave dissipation term; the role of dissipation is played by the existence of
the energy sink at infinitely high wave numbers, in the spirit of the WTT; see

For our purposes, it is sufficient to simply use the dimensional
estimate for

For many years, the assumption has been that there could be a net
input or dissipation within the equilibrium range; however,

Another important theoretical relationship that can be derived
from joint consideration of Eqs. (

At the end of the section, we present the summary of important relationships.

Wave action

The self-similar relations for the duration limited case are given by

The same sort of self-similar analysis gives self-similar
relations for the fetch limited case:

Now that the construction of the ZRP wind input term with the unknown coefficient has been accomplished in the spirit of WTT in the previous chapter, the HE model, suitable for numerical simulation, still misses the dissipation term localized at finite wave numbers – there is no such thing as the infinite phase volume in reality: the real ocean Fourier space is confined by a characteristic wave number corresponding to the start of the dissipation effects caused by the wave-breaking events.

There is a lot of freedom in choosing the dissipation term. Since there is no current interpretation of the wave-breaking dissipation mechanism, one can choose it in whatever shape it is preferred, but any particular choice will be questioned since it is an artificial one.

Because of that, the motivation consisted in the fact that at
the current “proof of concept” stage one needs to know the
effective sink with the simplest structure. Continuation of the
spectrum from

Specifically, the coefficient

The whole set of the input and dissipation terms is
accomplished now with one uncertainty: the explained approach
leaves one parameter arbitrary – the constant in front of the
wind input term. We choose it to be equal to 0.05 from the condition
of the reproduction of the field observations of wave energy
growth along the fetches, analyzed in

To check the self-similar hypothesis posed in
Eq. (

All simulations used the WRT (Webb–Resio–Tracy) method (see

There is a balance between the number of nodes of the grid and the volume of the calculation to be performed. The particular version of the WRT model has been tuned to the minimum grid number of nodes to solve realistic physical problems, but is still fast enough to simulate them over a reasonable time span. The correctness of this statement is confirmed by the multiple numerical experiments cited above, reproducing mathematical properties of the Hasselmann equation.

For convenience, we present the pseudo-code used for the main cycle of the
described model.

Calculate

Overwrite

Update

Solve analytically

Return to step 1.

The same ZRP wind input term Eq. (

The above-described “implicit dissipation” term

The duration limited simulation has been performed for a wind
speed of

Dimensionless energy

Figure

Energy local power function index

Dimensionless mean frequency

One should specifically elaborate on the local index

The relatively small systematic deviation from self-similar
behavior, visible in Fig.

Mean frequency local power function index

“Magic number”

First, the transition process in the beginning of the simulation, when the wave system behavior is far from a self-similar one. The self-similar solution is a pure power function, which does not take into account the initial transition process, and which causes the systematic difference. This systematic difference could be diminished via a parallel shift, which would take into account the initial transition process. Such a parallel shift is equivalent to starting the simulation from different initial conditions.

The second fact is the asymptotic nature of the self-similar
solution, producing an evolution of the simulated wave system
toward self-similar behavior with increasing time. As seen in
Fig.

The dependence of the mean frequency on time, shown in
Fig.

Decimal logarithm of the angle averaged spectrum as the function of the
decimal logarithm of the frequency for the wind speed

Compensated spectrum as the function of linear frequency

The systematic deviation of two lines in Fig.

Typical, angle averaged, wind input function density

Angular spectrum for the wind speed

A check of the consistency with the “magic number”

One should note that indices

Experimental, theoretical and numerical evidence of the dependence of

Figure

the spectral peak region,

the inertial (equilibrium) range

a Phillips high-frequency tail

The compensated spectrum

One can see a plateau-like region responsible for

The angular spectral distribution of energy, presented in
Fig.

To compare the duration limited numerical simulation results with
the experimental analysis by

The limited fetch simulation was performed in the framework of the
stationary version of Eq. (

The stationarity in Eq. (

Dimensionless energy

Energy local power function index

Since the wind forcing index

Figure

The wave evolution for the wind speed

Dimensionless mean frequency as the function of the dimensionless fetch,
calculated as

Local mean frequency exponent

The dependence of the mean frequency
on the fetch, shown in
Fig.

“Magic number”

Decimal logarithm of the angle averaged spectrum as the function of the
decimal logarithm of the frequency for the wind speed

The local values of indices

Typical, angle averaged, wind input function density

Relative wave energy distribution

The reasons for the

The check on the consistency of the calculated “magic number”

Compensated spectrum as the function of linear frequency

Angular spectrum for the wind speed

One should note that indices

Experimental, theoretical and numerical evidence of the dependence of

The solid line (pointed to by arrows) presents non-dimensional total
energy from the limited fetch numerical experiments, superimposed onto
Fig. 5.4, which is adapted from

The solid line (pointed to by arrows) presents non-dimensional average
frequency as the function of the fetch for limited fetch numerical
experiments, superimposed on Fig. 5.5 adapted from

Figure

the spectral peak region,

the inertial (equilibrium) range

a Phillips high-frequency tail

The compensated spectrum

The angular spectral distribution of energy, presented in
Fig.

The excess spectral energy at very oblique angles is a numerical
artifact connected with the specifics of how the limited fetch statement is
simulated here, i.e., the above-mentioned singularity presence on the
left-hand side of Eq. (

The detailed structure of angular spreading for both the duration and limited
fetch cases is given in Fig.

It is clearly seen that the “blobs” in the limited fetch case
contain no more than

To compare the limited fetch numerical simulation results with
the experimental analysis by

A comparison of limited fetch and duration limited simulations with
the experimental results by

The dependencies of the dimensionless energy and the frequency on
the dimensionless fetch for the limited fetch simulation,
superimposed on the experimental observations collected by

We have analyzed the new ZRP form for wind input, proposed in

The numerical simulations for both the duration limited and fetch
limited cases, using the ZRP wind input term, XNL nonlinear term

The proposed model is the proof of the concept, providing strong
support for simplified assumptions, such as discontinuity and the
fixed frequency transition point of the source terms. The
influence of these effects will be studied, in particular, using a more
sophisticated approach by

Although the integral parameters of the model have been verified against the experimental observations, the verification of the spectral details, such as angular spreading, requires additional studies.

Observed oscillations of self-similar indices are interpreted as the effects of the discreteness of the model, which suggests that a study of the influence of the grid resolution on such oscillations is desirable in future research.

A test of the model invariance with respect to wind speed change
from

At the moment of submission of the manuscript, the main technical obstacle to effective development of a new generation of physically based HE models was insufficiently fast calculation of the exact nonlinear interaction. The transition to the 2-D case requires a radical increase in the calculation speed. We hope that such improvements will be made in the near future.

The authors hope that this new framework will offer additional guidance for the source terms in operational models.

The research presented in Sect.

The authors gratefully acknowledge the support of these foundations. Edited by: Victor Shrira Reviewed by: two anonymous referees