NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-581-2017Balanced source terms for wave generation within the Hasselmann equationZakharovVladimirResioDonaldPushkarevAndreidr.push@gmail.comDepartment of Mathematics, University of Arizona, Tucson, AZ 85721,
USALebedev Physical Institute RAS, Leninsky 53, Moscow 119991,
RussiaNovosibirsk State University, Novosibirsk, 630090, RussiaWaves and Solitons LLC, 1719 W. Marlette Ave., Phoenix, AZ 85015,
USATaylor Engineering Research Institute, University of North
Florida, Jacksonville, FL, USAAndrei Pushkarev (dr.push@gmail.com)9October201724458159722November201616August20177August20175December2016This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/24/581/2017/npg-24-581-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/581/2017/npg-24-581-2017.pdf
The new Zakharov–Resio–Pushkarev (ZRP) wind input source term is examined for its
theoretical consistency via numerical simulation of the Hasselmann
equation. The results are compared to field experimental data,
collected at different sites around the world, and theoretical
predictions based on self-similarity analysis. Consistent results are
obtained for both limited fetch and duration limited statements.
Introduction
The scientific description of wind-driven wave
seas, inspired by
solid state physics statistical ideas (see, for instance,
), was proposed by in the form of
the Hasselmann equation (hereafter HE), also known as the kinetic equation
for waves:
∂ε∂t+∂ωk∂k∂ε∂r=Snl+Sin+Sdiss,
where ε=ε(ωk,θ,r,t) is the
wave energy spectrum as a function of wave dispersion ωk=ω(k), angle θ, two-dimensional real space coordinate
r=(x,y) and time t. Snl, Sin and
Sdiss are the nonlinear, wind input and wave-breaking
dissipation source terms, respectively. Hereafter, only the deep
water case ω=gk is considered, where g is the
gravity acceleration and k=|k| is the absolute value of the
vector wave number k=(kx,ky).
Since Hasselmann's work, Eq. () has become the basis of
operational wave forecasting models such as WAM, SWAN and Wavewatch
III . While the physical oceanography community
consents on the general applicability of Eq. (), there is
no consensus on universal parameterizations of the source
terms Snl, Sin and Sdiss.
The Snl term and weak turbulence theory
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 Snl term were derived by different methods from
the Euler equations for free surface incompressible
potential flow of a liquid by and
. showed that they are identical on
the resonant surface:
ωk1+ωk2=ωk3+ωk4,k1+k2=k3+k4.
The Snl term is the complex nonlinear operator acting
on εk, concealing hidden symmetries
and cubic with respect to the spectrum ε.
Equation () sometimes is called “the Boltzmann
equation” by the oceanographic community, although this is
a misconception. The Boltzmann equation, derived in the
nineteenth century for description of gas kinetics, is quadratic,
rather than cubic, with respect to the distribution function.
To understand the relation and difference between the Boltzmann
equation and the HE, one should recall the above-mentioned
equation. This equation, applicable to quantum quasi-particles,
contains both quadratic and cubic terms. Hence, the Boltzmann
equation and the HE present opposite limiting cases of a general
quantum kinetic equation.
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 , waves in nonlinear optics
, gravitational waves in the universe ,
plasma waves , capillary waves
, and Kelvin waves in super-fluid helium
.
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, ), which are the solutions of
the corresponding kinetic equation
Snl=0.
The simplest one is the Zakharov–Filonenko (hereafter ZF) solution
, which is the sub-class of KZ solutions:
ε≃P1/3ω4,
where P is the energy flux toward high wave numbers.
The accuracy advantage of knowing the analytical expression
for the Snl term, also known in physical
oceanography as the XNL, is overshadowed by its computational
complexity. Today, none of the operational wave forecasting
models can afford to perform XNL computations in real
time. Instead, the operational approximation, known as DIA and
its derivatives, is used to replace this source term. The
implication of such simplification is the inclusion of
a tuning coefficient in front of the nonlinear term; however,
several publications have shown that DIA does not provide
a good approximation of the actual XNL form. The paradigm of
replacement of the XNL by DIA and its variations leads to
even more grave consequences: other source terms must be
adjusted to allow the model Eq. () to produce
desirable results. In other words, deformations suffered by
the XNL model due to the replacement of Snl by its surrogates
need to be compensated for by non-physical
modification of other source terms to achieve reasonable model
behavior in any specific case, leading to a loss of physical
universality in the HE model.
Operational formulations for the wind energy input Sin and wave energy dissipation Sdiss terms
In contrast to Snl, the knowledge of
Sin and Sdiss source terms is poor;
furthermore, both include many heuristic factors and
coefficients. The creation of a reliable, well-justified
theory of Sin has been hindered by strong
turbulent fluctuations, uncorrelated with the wave motions, in
the boundary layer over the sea surface. Even one of the most
crucial elements of this theory, the vertical distribution of
horizontal wind velocity in the region closest to the ocean
surface, where wave motions strongly interact with atmospheric
motions, is still the subject of debate. The history of the
development of different wind input forms is full of heuristic
assumptions, which fundamentally restrict the magnitude and
directional distribution of this term. As a result, the values
of different wind input terms scatter by a factor of
300–500 %. For example, experimental
determination of Sin, as provided by direct
measurements of the momentum flux from the air to the water,
cannot be rigorously performed in a laboratory due to gravity
wave dispersion dependence on the water depth, as well as
problems with scale effects in laboratory winds. Additional
information on the detailed analysis of the current state of the
art of wind input terms can be found in .
Similar to the wind input term, there is little consensus on the
parameterization of the source dissipation term
Sdiss. The physical dissipation mechanism, which
most physical oceanographers agree on, is the effect of wave
energy loss due to wave breaking, while there are also other
dubious ad hoc “long wave” dissipation source terms with
heuristically justified physical explanations. Currently,
there is not even an agreement on the location of wave
breaking events in Fourier space. The approach currently
utilized in operational wave forecasting models mostly relies
on the dissipation localized in the vicinity of the spectral
energy peak. Recent numerical experiments show
, however, that such an approach does not pass
most of the tests associated with the essentially nonlinear
nature of the HE Eq. ().
Road map for the construction and verification of balanced source terms
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 Sin is constructed in the artificial way
realizing, in a sense, “the poor man approach”, based on the
finding of a two-parameter family of HE self-similar solutions
and their restriction to the single-parameter one with the help
of comparison with the data of experimental observations,
accumulated for several decades.
Section presents experimental evidence of wave
energy spectrum characteristics in the form of
a specific
regression line, found by . The analytic form of
this regression line will play a crucial role in narrowing the
circle of possible outcomes, obtained with WTT analysis.
Section studies self-similar solutions of the HE – a kinetic
equation for surface ocean waves, starting with the
analysis of the behavior of the dissipationless HE in infinite
space, containing the wind source term in power function
form. This approach is similar in spirit to one realized by
for finding the solution of the equation
Snl=0 in the infinite Fourier domain, which derived
the ZF spectrum ε∼P1/3ω4, where
P is the energy flux toward high wave numbers. The Fourier
domain in both situations does not contain any dissipation
function: its role is played by infinite phase volume as the
effective energy sink at infinitely high wave numbers.
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 . Thereafter, they were observed and discussed by
. A complete survey of all measurements requires
a separate comprehensive paper, which is in our plans for the
future.
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
Snl is the leading term of the HE . This
allows further use of the solutions found from
the “zero-dissipation” HE Eq. () in infinite
space for “practical” Fourier domains with the dissipation
localized at finite high enough wave numbers. They take the form of a
two-parameter family of self-similar solutions, which
can be further restricted to the single-parameter one using
experimental regression dependence, presented in
Sect. . These self-similar solutions present
realistic HE solutions and describe a broad class of wave-energy spectra
observed in ocean and wave-tank experiments.
The indices, corresponding to self-similar solutions, allow one to wrap up
Sect. with the specific form of the wind input term in
infinite phase space, called the ZRP wind input term , with an
arbitrary coefficient in front of
it. Now, the theoretical part of the wind source term
Sin construction is finished, but the obtained
model is not suitable yet for numerical simulation, since to
perform in finite phase space, it has to be augmented with the
wave-breaking dissipation term.
Section explains the dissipation function used in the presented
model. The wave-breaking dissipation, also
known as “white-capping dissipation”, is an important
physical phenomenon not properly studied yet for reasons
of mathematical and technological complexity.
achieved important results, but did not accomplish the theory
completely. studied the wave-breaking of short
waves, “squeezed” by surface currents, caused by longer
waves, and showed that they become steep and unstable. Our
explanation is simpler but has the same consequences: the
“wedge” formation, preceding the wave breaking, causes the
“fat tail” appearance in Fourier space. Subsequent smoothing
of the tip of the wedge is equivalent to a “chopping off” of
the developed high-frequency tail in Fourier space – a sort of
natural low-pass filtering – leading to the loss of the wave
energy. Both scenarios lead to smoothing of the wave
surface, and are indirectly confirmed by the numerical
experiments presented in the current study.
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 ∼ω-5, which decays
faster than the equilibrium spectrum ω-4 and therefore
corresponds to a net wave energy absorption. Although a version
of this concept was incorporated by , detailed forms
of this source term have not been developed to date, other than
that the spectrum at high frequencies appears to consistently tend
toward an ∼ω-5 form as noted by
. Additional evidence for a transition from ∼ω-4 to ∼ω-5 at frequencies above the
equilibrium range comes from analysis of multiple data sets by
. In that paper the transition from ∼ω-4 to ∼ω-5 occurs approximately at
fd=1.1Hz; i.e., the physical spectrum has to be
continued from this point by ∼ω-5.
The spectrum amplitude at the junction frequency
fd is dynamically changing in time. It is
important that this analytic continuation contributes to
a differential in inverse action, which also affects
frequencies lower than fd, since the nonlinear
interaction term Snl is calculated over both
“dynamic” and fixed Phillips areas. Therefore, the Phillips
part of the spectrum “sends” the information about the presence
of the dissipation above fd to the rest of the
spectrum.
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 ∼ω-4 spectrum,
spectral peak down-shift and peak frequency behavior in
accordance with self-similar laws.
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 discusses the
details of the numerical model setup. Section describes
the duration limited
numerical simulation, which is the subject of more academic
than applied interest, targeted at self-similarity concept
support, while the limited fetch numerical simulation results,
described in Sect. , besides academic
interest, are the subject of comparison with the field
experiments. A check of the compliance of numerical results
with field experimental measurements is presented in
Sect. .
Experimental evidence
Here we examine the empirical evidence from around the
world, which has been utilized to quantify energy levels
within the equilibrium spectral range by . For
convenience, we shall also use the same notation used by
in their study, for the angular averaged
spectral energy densities in frequency and wave-number
spaces:
E4(f)=2πα4Vg(2πf)4,F4(k)=βk-5/2,
where f=ω2π, α4 is the constant, V is
some characteristic velocity and β=12α4Vg-1/2. These notations are based on relation of spectral
densities E(f) and F(k) in frequency f=ω2π
and wave-number k bases:
F(k)=cg2πE(f),
where cg=dωdk=12⋅2πgf is the group velocity.
The notations in Eqs. () and () are
connected with the spectral energy density
ϵ(ω,θ) through
E(f)=2π∫02πϵ(ω,θ)dθ.
The analysis showed that experimental energy spectra
F(k), estimated through averaging 〈k5/2F(k)〉,
can be approximated by a linear regression line as the function of
(uλ2cp)1/3g-1/2. Figure shows
that the regression line
β=12α4(uλ2cp)1/3-u0g-1/2,
indeed, seems to be a reasonable approximation of these
observations.
Correlation of the equilibrium range coefficient β with (uλ2cp)1/3/g1/2 based on data from six disparate sources.
Adapted from .
Here α4=0.00553, u0=1.93ms-1, cp is
the spectral peak phase speed and uλ is the wind speed
at the elevation equal to a fixed fraction λ=0.065 of the
spectral peak wavelength 2π/kp, where kp is the spectral
peak wave number. It is important to emphasize that the
experiments show that parameter β increases with
development of the wind-driven sea, when fp decreases and
Cp increases. This observation is consistent with the weak
turbulent theory, where β∼P1/3; here P
is the wave energy flux toward small scales.
assumed that the near-surface boundary layer can be
treated as neutral and thus follows a conventional logarithmic
profile
uλ=u*κlnzz0
with a Von Karman coefficient κ=0.41, where z=λ⋅2π/kp is the elevation equal to a fixed
fraction λ=0.065 of the spectral peak wavelength
2π/kp, where kp is the spectral peak
wave number, and z0=αCu*2/g is subject to
surface roughness with αC=0.015.
Theoretical considerations
Self-similar solutions consistent with the conservative kinetic
equation
∂ϵ(ω,θ)∂t=Snl
were studied in and . In this section we study
self-similar solutions of the forced kinetic equation
∂ϵ(ω,θ)∂t=Snl+γ(ω,θ)ϵ(ω,θ)
where
ϵ(ω,θ)=2ω4gN(k,θ) is
the energy spectrum.
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
and .
For our purposes, it is sufficient to simply use the dimensional
estimate for Snl,
Snl≃ωω5ϵg22ϵ.
Eq. () has a self-similar solution if
γ(ω,θ)=αω1+sf(θ)
where s is a constant. Looking for a self-similar solution in the
form
ϵ(ω,t)=tp+qF(ωtq),
we find
q=1s+1,p=9q-12=8-s2(s+1).
The function F(ξ) has a maximum at ξ∼ξp; thus, the
frequency of the spectral peak
is
ωp≃ξpt-q.
The phase velocity at the spectral peak is
cp=gωp=gξptq=gξpt1s+1.
According to experimental data, the main energy input into the
spectrum occurs in the vicinity of the spectral peak, i.e., at
ω≃ωp. For ω≫ωp, the spectrum is described by
the Zakharov–Filonenko tail
ϵ(ω)∼P1/3ω-4.
Here
P=∫0∞∫02πγ(ω,θ)ϵ(ω,θ)dωdθ.
This integral converges if s<2. For large ω,
ϵ(ω,t)≃tp-3qω4≃t2-s2(s+1)ω4.
More accurately,
ϵ(ω,t)≃μgω4u1-ηcpηg(θ),η=2-s2.
Now, supposing s=4/3 and γ≃ω7/3, we get η=1/3,
which is exactly the experimental regression line prediction. Because it is
known from the regression line in
Fig. that ξ=1/3, we immediately get s=4/3 and
the wind input term
Swind≃ω7/3ϵ.
For many years, the assumption has been that there could be a net
input or dissipation within the equilibrium range; however,
recently used extensive data from Ocean Station Papa
to show that there was minimal wind input into the wave spectrum
in the equilibrium range. suggest that the existence
of significant net energy input or dissipation within the
frequency range would tend to force the spectrum away from an
f-4 form, contrary to the pattern found in field
measurements. If we assume that the wind source is primarily
centered on the spectral peak, the only missing component in our
numerical solution is an unknown coefficient in front of it, which
will be defined later from the comparison with total energy growth
in experimental observations.
Another important theoretical relationship that can be derived
from joint consideration of Eqs. (), ()
and () is
1000β=λ(u2cp)1/3g1/2,
which shows a theoretical equivalence to the experimental
regression, where λ is an unknown constant, defined
experimentally.
At the end of the section, we present the summary of important
relationships.
Wave action N, energy E and momentum M in frequency-angle
presentation are
N=2g2∫0∞∫02πω3ndωdϕ,E=2g2∫0∞∫02πω4ndωdϕ,M=2g3∫0∞∫02πω5ncosϕdωdϕ.
The self-similar relations for the duration limited case are given by
ϵ=tp+qF(ωtq),9q-2p=1,p=10/7,q=3/7,s=4/3,N∼tp+q,E∼tp,M∼tp-q,〈ω〉∼t-q.
The same sort of self-similar analysis gives self-similar
relations for the fetch limited case:
ϵ=χp+qF(ωχq),10q-2p=1,p=1,q=3/10,s=4/3,N∼χp+q,E∼χp,M∼χp-q,〈ω〉∼χ-q.
The details of “implicit” dissipation
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 ωd with the Phillips law
A(ωd)⋅ω-5 (see ), decaying faster than
the equilibrium spectrum ω-4, will get high-frequency
dissipation. The corresponding analytic parameterization of
this dissipation term will be unknown, while not in principle
impossible to figure out in some way. One should note that this
method of dissipation is not our invention: it is described in
.
Specifically, the coefficient A(ωd) in front of
ω-5 is unknown but is not required to be defined
in an explicit form. Instead, it is dynamically determined from
the continuity condition of the spectrum, at frequency
ωd, on every time step. In other words, the starting
point of the Phillips spectrum coincides with the last point of
the dynamically changing spectrum, at the frequency point
ωd=2πfd, where fd≃1.1Hz, as
per . This is the way the high-frequency
“implicit” damping is incorporated into the alternative
computational framework of the HE. The question of the finer
details of the high-frequency “implicit” damping structure is
of secondary importance, at the current “proof of
concept” stage.
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 .
Numerical validation of the relationship
To check the self-similar hypothesis posed in
Eq. (), we performed a series of numerical
simulations of Eq. () in the spatially homogeneous
duration limited ∂N∂r=0 and
spatially inhomogeneous fetch limited ∂N∂t=0 situations.
All simulations used the WRT (Webb–Resio–Tracy) method (see
), which calculates the nonlinear interaction term in
the exact form. The presented numerical simulation utilized the
version of the WRT method, previously used in , ,
, , , , , and
, and used the grid of
71 logarithmically spaced points in the frequency range from
0.1 to 2.0Hz and 36 equidistant points in the angle
domain. The constant time step in the range between 1 and
2 s has been used for explicit first-order accuracy
integration in time.
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 Snl(ε(f,θ)).
Overwrite ε(f,θ) to f-5 for f>1.1Hz.
Update ε(f,θ)=ε(f,θ)+dt⋅Snl(f,θ).
Solve analytically ∂(f,θ)∂t=Swind(f,θ)ε(f,θ) for time dt.
Return to step 1.
All numerical simulations discussed in the current paper have been
started from a uniform noise energy distribution in Fourier space
ε(ω,θ)=10-6m4, corresponding to a small
initial wave height with an effectively negligible nonlinearity
level. The constant wind of speed 10 ms-1 was assumed
to blow away from the shoreline, along the fetch. The assumption of constant
wind speed is a necessary simplification due to
the fact that the numerical simulation is being compared to
various data from field experiments, and the considered setup is the simplest
physical situation which can be modeled.
The same ZRP wind input term Eq. () has been
used in both cases as
Sin(ω,θ)=γ(ω,θ)⋅ε(ω,θ),γ(ω,θ)=0.05ρairρwaterωωω04/3q(θ)forfmin≤f≤fd,ω=2πf0 otherwise,q(θ)=cos2θ for -π/4≤θ≤π/40 otherwise,ω0=gU,ρairρwater=1.3×10-3,
where U is the wind speed at the reference level of 10 m,
and ρair and ρwater are the air and water
density, respectively. It is conceivable to use a more
sophisticated expression for q(θ), for instance q(θ)=q(θ)-q(0). To make direct comparison with experimental
results of , we used the relation u*≃U/28
(see ) in Eq. (). Frequencies
fmin and fd depend on the wind speed and
should be found empirically. In current numerical experiments for
U=10 and U=5ms-1, fmin=0.1Hz and
fd=1.1Hz. This choice is justified by the
obtained numerical results.
The above-described “implicit dissipation” term Sdiss has
played the dual role of a direct energy cascade flux sink due to wave
breaking as well as a numerical scheme stabilization factor at high wave
numbers.
Duration limited numerical simulation
The duration limited simulation has been performed for a wind
speed of U=10ms-1.
Dimensionless energy Eg2/U4 vs. dimensionless time tg/U for the wind
speed U=10ms-1 duration limited case – solid line.
Self-similar solution with the empirical coefficient in front of it: 1.3×10-9(tg/U)10/7 – dashed line.
Figure shows the total energy growth as the
function of time, consistent with the self-similar prediction
Eq. () for index p=10/7, supplied with the
empirical coefficient in front of it; see
Fig. .
Energy local power function index p=dlnEdlnt as a function of dimensionless time tg/U for the wind
speed U=10ms-1 duration limited case – solid line.
Theoretical value of the self-similar index p=10/7 – thick horizontal
dashed line.
Dimensionless mean frequency 〈f〉⋅U/g=E/N⋅U/g
(solid line) vs. dimensionless time tg/U for the wind speed
U=10ms-1 duration limited case – solid line, self-similar
solution with the empirical coefficient in front of it: 16.0⋅(tg/U)-3/7 – dashed line.
One should specifically elaborate on the local index p
numerical calculation procedure for
Fig. . First, the total energy function was
smoothed via a moving average, then the corresponding derivative is
estimated numerically via finite differences, and finally
a moving average is used to obtain the time-varying index value.
The relatively small systematic deviation from self-similar
behavior, visible in Fig. , is connected with
the following two facts.
Mean frequency local power function index -q=dln〈ω〉dlnt as the function of dimensionless time
tg/U for the wind speed U=10ms-1 duration limited case
(solid line). Theoretical value of self-similar exponent q=-3/7 – thick
horizontal dashed line.
“Magic number” 9q-2p as the function of dimensionless time tg/U for the
wind speed U=10ms-1 duration limited case – solid line. The
target value 1 for the self-similar relation Eq. ()
is represented by the horizontal dashed line.
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 numerical value of the local
exponent converges to the theoretical value p=10/7, reaching
approximately 6% accuracy for a sufficient
dimensionless time 3×104.
The dependence of the mean frequency on time, shown in
Fig. , is consistent with the self-similar
dependence found in Eq. () for q=3/7,
supplied with the empirical coefficient in front of it: see
Fig. .
Decimal logarithm of the angle averaged spectrum as the function of the
decimal logarithm of the frequency for the wind speed
U=10ms-1 duration limited case – solid line. Spectrum ∼f-4 – dashed line; spectrum ∼f-5 – dash-dotted line.
Compensated spectrum as the function of linear frequency f for the wind
speed U=10ms-1 duration limited case.
The systematic deviation of two lines in Fig. remains
within 3% of the target value q=3/7 for the same reasons as for
wave-energy
behavior – the transition process in the beginning of the
simulation and the asymptotic nature of the self-similar solution.
Typical, angle averaged, wind input function density 〈Sin〉=12π∫γ(ω,θ)ε(ω,θ)dθ (dotted line) and angle averaged
spectrum 〈ε〉=12π∫ε(ω,θ)dθ (solid line) as the functions of
the frequency f=ω2π for the wind speed
U=10ms-1 duration limited case.
Angular spectrum for the wind speed U=10ms-1 duration limited
case.
A check of the consistency with the “magic number” 9q-2p=1
(see Eq. ) is presented in Fig. .
The reason for systematic deviation
from the target value 1 is obviously connected with the reasons
for the systematic deviations of p and q, as the “magic
number” is calculated as their linear combination, reaching the
accuracy of approximately 10% for a long enough
dimensionless time of 3×104.
One should note that indices p and q and the “magic relation”
9q-2p exhibit asymptotic convergence to the corresponding
target values.
Experimental, theoretical and numerical evidence of the dependence of 1000β on (uλ2cp)1/3/g1/2. Dashed line –
theoretical prediction Eq. () for λ=2.74; dotted line –
experimental regression line from and . Line connected
diamonds – results of numerical calculations for the wind speed
U=10ms-1 duration limited case. Being parameterized by
dimensionless time tg/U, the numerical simulation trajectory evolves from
the left to the right on the graph, covering a time span from tg/U=0 to
tg/U≃3.5×105.
Figure presents an angle-integrated energy spectrum as the
function of frequency, in logarithmic coordinates. One can see that it
consists of the segments of
the spectral peak region,
the inertial (equilibrium) range ω-4 spanning from the spectral peak to the beginning of the “implicit dissipation”
fd=1.1Hz, and
a Phillips high-frequency tail ω-5 starting approximately from
fd=1.1Hz.
The compensated spectrum F(k)⋅k5/2 is presented in
Fig. .
One can see a plateau-like region responsible for k-5/2 behavior,
equivalent to the ∼f-4 tail in Fig. . This shape of
the spectrum is similar to that
observed by . This exact solution of
Eq. (), known as the KZ spectrum, was found by
. The universality of f-4 asymptotic for the
“inertial” (also known as “equilibrium” in oceanography)
range between spectral peak energy input and high-frequency
energy dissipation areas has been observed in multiple
experimental field observations and is accepted by the
oceanographic community after the seminal work of
. One should note that most of the energy flux into
the system comes in the vicinity of the spectral peak, as shown
in Fig. , providing a significant inertial
interval for the KZ spectrum.
The angular spectral distribution of energy, presented in
Fig. , is consistent with the results of
experimental observations that show a broadening of
the angular spreading in both directions away from the spectral
peak frequency.
To compare the duration limited numerical simulation results with
the experimental analysis by , presented in
Fig. , Fig. shows the function
β=F(k)⋅k5/2 as the function of (uλ2Cp)1/3/g1/2 for wind speed
U=10ms-1, along with the regression line from
and the theoretical prediction Eq. () for
λ=2.74. The numerical results and theoretical prediction
line fall within a very small rms deviation (r2=0.939; see
Fig. ) from the regression line. One should note
asymptotic convergence of the numerical simulation results to the
theoretical line.
Limited fetch numerical simulation
The limited fetch simulation was performed in the framework of the
stationary version of Eq. ():
12gcosθω∂ϵ∂x=Snl(ϵ)+Swind+Sdiss,
where x is chosen as the coordinate axis orthogonal to the
shore and θ is the angle between the individual wave number
k and the axis x. To find the dependence on the
wind speed, directed off the shore, two numerical simulations for
wind speeds of U=5ms-1 and U=10ms-1 have been
performed.
The stationarity in Eq. () is somewhat difficult for
numerical simulation, since it contains a singularity in the form
of cosθ in front of ∂ϵ∂x. This problem was overcome by zeroing one-half of the Fourier
space of the system for the waves propagating toward the
shore. Since the energy in such waves is small with respect to
waves propagating in the offshore direction, such an approximation
is quite reasonable for our purposes.
Dimensionless energy Eg2/U4 vs. dimensionless fetch xg/U2 for the
fetch limited case: wind speed U=10ms-1 – solid line; wind
speed U=5ms-1 – dash-dotted line. Self-similar solution with
the empirical coefficient in front of it: 2.9×10-7xg/U2 –
dashed line.
Energy local power function index p=dlnEdlnx as the function of dimensionless fetch xg/U2 for
the fetch limited case: wind speed U=10ms-1 – solid line;
wind speed U=5ms-1 – dash-dotted line. Theoretical value of
self-similar index p=1 – thick horizontal dashed line.
Since the wind forcing index s in the fetch limited case is
similar to that in the duration limited case, the numerical
simulation of Eq. () has been performed for the same
input functions as in the duration limited case with the same
low-level energy noise initial conditions in Fourier space.
Figure presents total energy growth as a function
of fetch, consistent with the self-similar solution
Eq. () for index p=1, and its appropriate
empirical coefficient. The corresponding values of indices p
along the fetch are presented in Fig. . The small
amplitude oscillations observed in the index behavior can be
attributed to the finite grid resolution used in the simulation.
The wave evolution for the wind speed
U=5ms-1 case is expected to be slower than for the
U=10ms-1 case due to the weaker nonlinear interaction
term. One can see, indeed, slower asymptotic convergence of the
calculated total energy local power index to the target value p=1
for the U=5ms-1 case compared to the U=10ms-1
case. The deviation of results from the
U=10ms-1 case relative to the target value does not
exceed an error of about 5%, while for the
U=5ms-1 case the error does not exceed 20%. The
role of the relatively short (in time)
non-self-similar development of the wave system at the very
beginning of the fetch should be noted as well as the factor
contributing to the deviation from the target value of index p=1:
the wave system obviously needs some time to evolve into a fully
self-similar mode.
Dimensionless mean frequency as the function of the dimensionless fetch,
calculated as 〈f〉=12π∫ωndωdθ∫ndωdθ, where
n(ω,θ)=ε(ω,θ)ω is the wave
action spectrum, for wind speed 10ms-1 (solid line) and
5ms-1 (dashed line). The dash-dotted line is the self-similar
dependence 3.4⋅(xgU2)-0.3 with the empirical
coefficient in front of it.
Local mean frequency exponent -q=dln〈ω〉dlnx as the function of dimensionless fetch
xg/U2 for the limited fetch case. Wind speed U=10ms-1 –
solid line; wind speed U=5ms-1 – dashed line. Horizontal
dashed line – target value of the self-similar exponent q=0.3.
The dependence of the mean frequency
on the fetch, shown in
Fig. , is consistent with the self-similar
dependence Eq. () for index q=0.3, supplied
with the empirical coefficient in front of it. The small amplitude
oscillations observed in index behavior can be attributed to the
finite grid resolution used in the simulation, since the spectral
peak moves continuously between discrete frequencies in a manner
that cannot be matched in these discretized simulations.
“Magic number” 10q-2p as a function of dimensionless fetch xg/U2 for
the limited fetch case. Wind speed U=10ms-1 – solid line;
wind speed U=5ms-1 – dashed line. Horizontal dashed line –
self-similar target value 10q-2p=1.
Decimal logarithm of the angle averaged spectrum as the function of the
decimal logarithm of the frequency for the wind speed
U=10ms-1 limited fetch case – solid line. Spectrum ∼f-4 – dashed line; spectrum ∼f-5 – dash-dotted line.
The local values of indices q for two different wind speed
amplitudes are presented in Fig. along with the target
value of the self-similar index q=0.3. After sufficient
fetch one can see only about 14% deviation from the
target value for the U=10ms-1 case and about 2.5%
for the U=5ms-1 case.
Typical, angle averaged, wind input function density 〈Sin〉=12π∫γ(ω,θ)ε(ω,θ)dθ (dotted line) and angle averaged
spectrum 〈ε〉=12π∫ε(ω,θ)dθ (solid line) as the functions of
the frequency f=ω2π for the wind speed
U=10ms-1 limited fetch case.
Relative wave energy distribution E(θ)/Etot=∫fminfdε(ω,θ)dω/∫fminfd∫02πε(ω,θ)dωdθ as the function of angle θ for the duration limited (solid line) and limited fetch (dotted line) cases.
The reasons for the 10% systematic deviation from the
self-similar solutions observed in the lines in
Fig. , corresponding to the wind speeds of U=5ms-1
and U=10ms-1, are the same as noted previously for
wave energy behavior – the transition process in the beginning of
the simulation and the asymptotic nature of the self-similar solution.
The check on the consistency of the calculated “magic number”
(10q-2p) (see Eq. ) is presented in
Fig. . The reason for systematic deviation from
the target value 1 is obviously connected with the systematic
deviations of p and q, as the “magic number” is calculated as
their linear combination, reaching the accuracy of approximately
10% for fetches 3×104. As noted previously,
the small amplitude oscillations observed in the indices' behavior
can be attributed to the finite grid resolution used in the
simulation.
Compensated spectrum as the function of linear frequency f for the wind
speed 10ms-1 limited fetch case.
Angular spectrum for the wind speed U=10ms-1 limited fetch
case.
One should note that indices p and q and the “magic relation”
10q-2p exhibit asymptotic convergence to the corresponding target
values.
Experimental, theoretical and numerical evidence of the dependence of 1000β on (uλ2cp)1/3/g1/2. Dashed line –
theoretical prediction Eq. () for λ=2.11; dotted line –
experimental regression line from and . Line connected
diamonds – the results of numerical calculations for the wind speed
U=10ms-1 limited fetch case. Being parameterized by the
dimensionless fetch coordinate χ=xgU2, the numerical
simulation trajectory evolves from the left to the right on the graph,
covering a fetch span from χ=0. to χ≃3.0×104.
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 original caption is
“A composite of data from variety of studies showing the development of the
non-dimensional energy, ε as a function of non-dimensional fetch,
χ. The original JONSWAP study (Hasselmann et al., 1973) used the data
marked, JONSWAP, together with that of Burling (1959) and Mitsuyasu (1968).
Also shown are a number of growth curves obtained from the various data sets.
These include JONSWAP Eq. (5.27), Donelan et al. (1985) Eq. (5.33) and
Dobson et al. (1989) Eq. (5.38).”
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 . The original
caption is “A composite of data from a variety of studies showing the
development of the non-dimensional peak frequency, ν as a function of
non-dimensional fetch, χ. The original JONSWAP study (Hasselmann et al.,
1973) used all the data shown with the exception of that marked Donelan
et al. (1985) and Dobson et al. (1989). Also shown are a number of growth
curves obtained from the various data sets. These include JONSWAP
Eq. (5.28), Donelan et al. (1985) Eq. (5.34) and Dobson et al. (1989)
Eq. (5.39).”
Figure presents an angle-integrated energy
spectrum, as the function of frequency, in logarithmic
coordinates. As could be seen in the duration
limited case, one can
see that it consists of three process-related segments:
the spectral peak region,
the inertial (equilibrium) range ω-4 spanning from the spectral peak to the beginning of the “implicit dissipation”
fd=1.1Hz, and
a Phillips high-frequency tail ω-5 starting approximately at
fd=1.1Hz.
The compensated spectrum F(k)⋅k5/2 is presented in
Fig. . One can see a plateau-like region responsible for
k-5/2 behavior, equivalent to the ω-4 tail in
Fig. and similar to that observed by
. As in the duration limited case, the KZ solution
also holds for the fetch limited case, and most of the
energy flux into the system comes in the vicinity of the spectral
peak as well, as shown in Fig. ,
providing a significant inertial (equilibrium) range for the KZ
spectrum between spectral peak energy input and high-frequency
energy dissipation areas.
The angular spectral distribution of energy, presented in
Fig. , as in the duration limited case, is
consistent with the results of experimental observations by
that show a broadening of the angular spreading in
both directions away from the spectral peak frequency.
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. () at
θ=±π/2.
The detailed structure of angular spreading for both the duration and limited
fetch cases is given in Fig. . The time that
would be required to produce such a pattern is far in excess of
the time for this excess energy to be removed from the
equilibrium range by the nonlinear flux and can be shown to
vanish when a time–space simulation is used instead of the
stationary solution assumed here.
It is clearly seen that the “blobs” in the limited fetch case
contain no more than 5% of the total energy of the
corresponding spectrum and could be neglected for the purposes of
the presented research.
To compare the limited fetch numerical simulation results with
the experimental analysis by , presented in
Fig. , Fig. shows the function
β=F(k)⋅k5/2 as a function of (uλ2Cp)1/3/g1/2 for wind speed
U=10.0ms-1, along with the regression line from
and its theoretical prediction Eq. () for
λ=2.11. The numerical results and theoretical prediction
line fall within the rms deviation (r2=0.939; see
Fig. ) from the regression line. One should note
asymptotic convergence of the numerical simulation results to the
theoretical line. Being parameterized by the fetch coordinate, the
numerical simulation results evolve from the left to the right on
the graph, from the dimensionless fetch equal to 0, to 3.0×104.
Comparison with the experiments
A comparison of limited fetch and duration limited simulations with
the experimental results by and the theoretical
prediction based on Eq. () is presented in
Figs. and . One should note that the
numerical results and theoretical prediction line with
corresponding values of λ fall into the rms deviation (r2=0.939;
see Fig. ) relative to the experimental
regression line Eq. ().
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
, are presented in Figs. and ,
showing good consistency of the presented numerical results and
the experimental observations.
Conclusions
We have analyzed the new ZRP form for wind input, proposed in
in terms of both fetch limited and duration limited
wave growth. The approach proposed here for the development of
a set of balanced source terms uses only two empirical
coefficients: one in the magnitude of the wind source term and
the second in the location of the transition from the ∼ω-4 to ∼ω-5 spectrum. This approach
focuses on the combination of the theoretical finding of the
self-similar solutions and the extraction of the relevant one
through the comparison with the field experimental data.
The numerical simulations for both the duration limited and fetch
limited cases, using the ZRP wind input term, XNL nonlinear term
Snl and “implicit” high-frequency dissipation, show
remarkable consistency with predicted self-similar properties of
the HE and with the regression line from field studies, relating
energy levels in the equilibrium range to wind speed by
and .
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 in
the future.
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 5 to 10ms-1 has already been performed,
but further study of the effects of a wider range of wind speed
variation on self-similar properties of the model is desired in
the future.
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.
Acknowledgements
The research presented in Sect. has been accomplished due
to the support of the grant “Wave turbulence: the theory, mathematical
modeling and experiment” of the Russian Scientific Foundation
no. 14-22-00174. The research set forth in Sect. was funded by the
program of the presidium of RAS: “Nonlinear dynamics in mathematical and
physical sciences”. The research presented in other chapters was supported
by ONR grant N00014-10-1-0991.
The authors gratefully acknowledge the support
of these foundations. Edited by: Victor
Shrira Reviewed by: two anonymous referees
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