Dmitry Chalikov*
Department of Oceanology, University of Melbourne, Australia
*Corresponding author: Dmitry Chalikov, Department of Oceanology, University of Melbourne, Australia
Submission: February 07, 2021;Published: February 26, 2021
ISSN 2578-031X Volume4 Issue1
A new approach to the three-dimensional modeling based on the analysis of three-dimensional equations of potential waves in the periodic domain is developed. Instead of the 3-D equation for the velocity potential, the 2-D Poisson equation on a surface is used. This equation contains both the first and second vertical derivatives of the potential. It is suggested that the equations be closed by introducing the connection between these derivatives. Finally, the model for 3-D waves includes only surface equations, which dramatically simplifies the modeling as well as provides a substantial acceleration of the calculations and reduces a volume of the memory used. The examples of integration of the equation over long periods, taking into account the input energy and dissipation, proved that such a simplified approach gives quite realistic results. It is typical that a new model runs faster by around two orders than the 3-D model
Keywords: Phase resolving; Wave modeling; Wave development; Wave spectrum; Wind input; Dissipation
Several different 3-D numerical methods for wave processes investigation have been developed over the past two decades [1]. The most advanced approaches are Boundary Integral Method [2], the finite-difference model designed at Technical University of Denmark [3], High-Order Spectral (HOS) model developed at Ecole Centrale Nantes, LHEEA Laboratory [4], Full Wave Model (FWM), [5-7].
A common disadvantage of the 3-D models is their low performance because all of them somehow or other resolve the vertical structure of wave field on the basis of 3-D equation for the velocity potential. It is well known that most of the computer time for 3-D model is spent on solution of Poisson equation for the velocity potential. However, the 3-D solution is used for calculation of the vertical velocity w on a surface only. A new method is based on separation of the velocity potential ϕ into the linear ϕ and nonlinear ϕ components and on the considering of Poisson equation for the nonlinear component on a surface. This exact equation contains both the first (ϕζ =w ) and second (ϕζζ = wζ ) derivatives, which means that the formulation of the entire problem is not closed. The multiple numerical calculations with an accurate 3-D model [8] show that w and wζ are well connected. This suggestion was found to be reliable enough to formulate the entire model which can realistically simulate wave dynamics. We call this model Heuristic Wave Model (HWM). The comparison of similar runs with the models supplied by identical physics shows that the new model is by two decimal orders faster than FM.
The model is supplied by a developed system of the run-time processing and recording of the results. The volume of data generated by the model typically exceeds dozens of gigabytes. Note that the given paper gives but a brief description of a new approach. A detailed description of the results will be given in the forthcoming papers.
The equations are written in the non-stationary surface-following non-orthogonal coordinate system:
ξ = x,ϑ= y,ζ = z− η (ξ ,ϑ,τ ),τ = t→(1)
where η (x, y,t) =η (ξ ,ϑ,ζ ) is a moving periodic wave surface?
The surface conditions for potential waves in the system of coordinates (1) at ζ = 0 take the following form:
Where ϕ is the velocity potential? The Laplace equation for ϕ at ζ ≤ 0 turns into Poisson equation
The equations (2)-(5) are written in a non-dimensional form
introduced formally by assuming that the gravity acceleration is
equal to one.
The equations (2-5) are the basis of the three-dimensional
FWM. The method of solution combines a 2-D Fourier transform
method in the ‘horizontal surfaces’ and a second-order finitedifference
approximation on the stretched staggered grid that
provides high accuracy of a finite-difference approximation in the
vicinity of surface. An equation (4) is solved as Poisson equations
with the iterations over the right-hand side with the prescribed
accuracy.
In was suggested in [5] that it would be convenient to represent
the velocity potential as a sum of two components such as the
analytical (‘linear’) component ϕ and the arbitrary (‘non-linear’)
component ϕ
ϕ= ϕ+ ϕ˜→(6)
The analytical component ϕ satisfies Laplace equation:
ϕξξ + ϕϑϑ + ϕζζ = 0→(7)
(7) with a known solution that satisfies the following boundary conditions:
The nonlinear component satisfies an equation:
ϕξξ + ϕϑϑ + ϕζζ = ϒ(ϕ) →(9)
Note that the potential ϕ0 on a surface is full. So, the nonlinear component ϕ˜ on a surface is equal to zero. Eq. (9) is solved with the boundary conditions:
Hence, on a surface ζ = 0 an equation (9) takes the form:
ϕζζ = ϒ (ϕ˜) + ϒ(ϕ)→(11)
The derivatives of the linear component ϕ in (11) are calculated analytically. Since ϕ˜(0) = 0, the equation (11) for velocity potential at ζ = 0 takes the form:
→(10)
(13) that is calculated analytically using Fourier presentation. The presence of w˜ζ in (12) makes the system of equations unclosed. Since the relation (13) follows formally from Poisson equation, the agreement between these values characterizes accuracy of the entire solution. The calculations of w˜ζ on the basis of an equation (4) and the calculations by surface condition (13) showed a very good agreement shown in Figure 1; [8].
Figure 1:Evolution of integral characteristics of
simulate developing wave field: curves 1-4 are the
rates of energy transitions, multiplied by:
A. 1-Input energy.
B. 2-Dissipation due to breaking.
C. 3-Dissipation in spectral tail.
D. 4-Balance of energy.
E. 5-Evolution of peak frequency weighted by
spectrum divided by 1000.
F. 6-Evolution of potential energy multiplied by
Thin straight line shows that transformation of energy
in adiabatic part of model is equal to zero.
For evaluation of the connection between w˜ and w˜ζ several thousand short-term numerical experiments with FWM were performed. The calculations were performed at large variations of integral parameters. The problem is reduced to finding dependence
→(14)
Where is the dispersion of elevation and is the dispersion of horizontal Laplacian Λ = Δη while t s is the averaged steepness of elevation? Finally, the best results were obtained in a form
A= σ F(μ )→(15)
Where μ is a parameter
μ = σσL →(16)
and function F is approximated by the formula
Where d0= 0.535, d1= 0.0414, d2= 0.00321
The two-dimensional model which is presumably able to replace the full 3-D model (2-5) includes the following three surface equations:
Note that the right side of Eq. (20) contains full vertical velocity w = w+ w˜ as well as the linear component of the vertical velocity wζ. The equation is represented in a form convenient for iterations. The iterations were being performed until the following condition was reached.
max |w˜i− Ri−1| < 10−7→(21)
Where R is the right-hand side of (20) while i is the number
of iterations?
The comparison of function F calculated in the course of
simulation by FWM with the results of the calculations by formula
(17) showed that the agreement between those functions was
quite satisfactory. A correlation coefficient between the values F
calculated with FWM and those calculated by Eq. (17) is 0.989. The
root mean square error is 0.014, which is by two orders smaller
than the typical value of F .
The main idea of the current approach is to replace the
cumbersome bloc for calculation of the vertical velocity equation
for ϕ by a simple 2-D equation (20). Such trick reduces the total
3-D problem to a 2-D one which spares us the trouble of countless
problems associated with the numerical solution of Poisson
equations. The question is how well these simplified equations
reproduce wave dynamics. The most obvious way of validation of
the simplified model is performing runs with the identical setting
and same initial conditions.
The data on evolution of the integral characteristics for both
models were represented in [8]; (Figure 2) which confirmed that
HFM could reproduce the integral characteristics quite similar to
the results obtained with the full model. The similarity of the results
was also illustrated by comparison of the spectra for different
statistical characteristics of solution.
Figure 2:Probability of elevation η normalized by significant wave height Hs
Further calculations were carried out with HWM model with initiated algorithms for the input and dissipation of energy. The algorithms for calculation of those terms are described in detail in [1-7]. The last minor modification of the algorithm for breaking was done in [7]. Those algorithms are not described in the current paper.
The model used 257 × 257 Fourier modes, or 1024 × 512 knots. The number of degrees of freedom (the minimum number of Fourier coefficients for elevation ƞ and the surface velocity potential ϕ is 264,196. In the initial conditions JONSWAP spectrum (Hasselmann et al, 1973) with the peak wave number kp =100 and an angle distribution proportional to (sch (ψ))256 was assigned, which corresponds to nearly unidirected waves with a very small energy. The model was integrated with the time step equal to 0.01 over the period t = 3000 As seen, all the characteristics undergo fast transformation during the first period of length t = 2000 After that the total input of energy (curve 4) starts to decrease and the evolution of energy approaches an equilibrium level but does not reach it by the end. The frequency of wave peak ωp (curve 5) decreases as t−1/2 which corresponds to the dependence on fetch ωp f−1/3 which follows from JONSWAP approximation.
An important characteristic is the probability distribution for elevation (Figure 2), which is in good agreement with the results obtained earlier with FWM. For calculations 3.1⋅108 points were used. Note that the results represented in Figure 2 are quite similar to those with the probability results obtained earlier. If we refer freak waves to those with the height exceeding 1.3Hs the total probability of such waves is higher than 10-6. As seen in Figure 2, wave height can exceed the value Hs = 2with the probability equal to 109. It was not noticed before because no one had ever used such a big volume of data. Obviously, the dependence of probability on wave height is maintained for larger wave height. This again confirms that big waves in a statistically uniform wave field are not generated by some special mechanism, but their physics does not differ (in anything other than size) from the physics of smaller waves.
The paper suggests a new approach to the phase-resolving
modeling of two-dimensional periodic wave fields. The only, though
an important, shortcoming of the adiabatic 3-D model is quite a low
performance connected with solution of 3-D Poisson equation at
each time step with iterations over the right side. The runs of such
models, even with a medium resolution, take many days or months
(depending on the speed of a computer).
In our opinion, the most important result of the current
investigation is introducing an exact surface condition (11)
connecting the first and second vertical derivatives of the velocity
potential. This equation might be useless, however, there are
some considerations indicating that these variables could be well
connected. The analysis of many thousands of the velocity potential
profiles generated by 3-D FWM model proves that this hypothesis
does work [8]; (Figure 1). The formulas (15-17) obtained can be
inserted into the new surface condition (11) that gives the third
surface equation (20) and closes the problem (Eqs. (18-20)). The
structure of governing equations for both models is the same;
hence, the HWM can be considered as a perturbed FWM. The
figures demonstrated in the paper prove that the results generated
by the models are at least quite satisfactory.
An obvious advantage of the new scheme is a dramatic increase
of the simulation speed. The HWM runs 50-100 times faster than
FWM. This ratio can depend on parameterization algorithms as
well as on the content and frequency of the results recorded and
on the setting of models. At any rate, this ratio remains high. It is
obvious that the approach developed here is simplified. It cannot be
applied to the individual cases with a small number of modes, for
example, for simulation of the steep Stokes wave. The peculiarity
of the HWM is that it works well for simulation of the statistically
uniform in space wave field with a very large number of modes. It
is easy to transform the scheme into a finite-difference form and
use the model for investigation of the wave regime in a specific
geographical environment. It is important that the model, opposite
to FWM, does not contain any global operations, which makes it
convenient for parallel computers.
HWM, due to its high performance, can be used as a component
of the wave prediction model (like WAM or WAVEWATCH) for
interpretation of spectral information in terms of the phaseresolved
wave field. The spectra calculated in the spectral model in
the selected areas are used for generation of the initial conditions
for HWM; then the calculations can be performed up to the
stationary statistical regime.
Author would like to thank Mrs. O. Chalikova for her assistance in preparation of the manuscript. This research was performed in the framework of the state assignment of Russian Academy of Science (Theme 0128-2021-0014).
© 2021 Dmitry Chalikov. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work non-commercially.