3.2 Formulation of the Wave Propagation Problem
51
The existence of this problem was mentioned by many scientists (Ocean
Wave Modeling (SWAMP group), 1985; Booij &Holtuijsen, 1987; The WAM
model, 1988; Ryabinin, 1991a,b; Tolman, 1992), but no sufficiently simple
solution was found. It appears that the most natural way to solve this problem is to increase the spectral resolution. According to the estimations of
Booij & Holtuijsen (1987), a typical width of the frequency (~w) and the angular (~(3) bands for wave calculation in the North Atlantic area should be
~w = 0.03w and ~(3 = 1.5°. However, the use of such a fine resolution is
unlikely to be advisable in practical applications.
The second solution of the "garden-sprinkler" problem was proposed by
Booij & Holtuijsen (1987) for the case of wave propagation in a plane surface. They suggest adding two terms to the wave energy balance equation,
to correct the effects connected with the finite width of the frequency and
angular bands. The inclusion of the correction terms into the numerical wave
propagation schemes requires the solution of a more complex equation than
the traditional wave energy balance equation. This includes additional terms
with second-order partial derivatives and the solution of an additional equation for estimating the wave age, which is not determined locally. The solution
of this problem requires additional computational time.
That is why it is necessary to develop an alternative method, which
would be no worse than the finite difference method in accuracy and requires
less computational time. This approach can be elaborated by combining the
method of characteristics and a simple polynomial interpolation allowing us
to solve (1.84) at the grid points on a sphere.
Thus, an attempt is undertaken to solve the well-known "philosophical
problem" of the wave description~ whether waves are considered to be a field
description or particle propagation. The method of characteristics considers
the propagation of wave packets (particles), and at the same time the finitedifference and interpolation methods appear to describe the field. In our
opinion, the proposed numerical method is different from ordinary finitedifference methods because it describes more precisely the correspondence
with wave propagation physics.
It should be noted that the alternative approach proposed below removes
the "garden-sprinkler" effect without a noticeable increase in computation
time. Moreover, application of this approach in the semi-Lagrangian numerical method (hereinafter referred to as the interpolation-ray or the INTERPOL method) allows the use of much larger integration time steps in comparison with the Courant condition ( CLF) without losing numerical accuracy.
3.2 Formulation of the Wave Propagation Problem
Main equation. The evolution of a two-dimensional sea wave spectrum
S ( w, (3, cp, ' 19, t), being a function of the frequency w, direction (3 (measured
counterclockwise from the parallel), latitude cp, longitude ' 19 and time t is
51
The existence of this problem was mentioned by many scientists (Ocean
Wave Modeling (SWAMP group), 1985; Booij &Holtuijsen, 1987; The WAM
model, 1988; Ryabinin, 1991a,b; Tolman, 1992), but no sufficiently simple
solution was found. It appears that the most natural way to solve this problem is to increase the spectral resolution. According to the estimations of
Booij & Holtuijsen (1987), a typical width of the frequency (~w) and the angular (~(3) bands for wave calculation in the North Atlantic area should be
~w = 0.03w and ~(3 = 1.5°. However, the use of such a fine resolution is
unlikely to be advisable in practical applications.
The second solution of the "garden-sprinkler" problem was proposed by
Booij & Holtuijsen (1987) for the case of wave propagation in a plane surface. They suggest adding two terms to the wave energy balance equation,
to correct the effects connected with the finite width of the frequency and
angular bands. The inclusion of the correction terms into the numerical wave
propagation schemes requires the solution of a more complex equation than
the traditional wave energy balance equation. This includes additional terms
with second-order partial derivatives and the solution of an additional equation for estimating the wave age, which is not determined locally. The solution
of this problem requires additional computational time.
That is why it is necessary to develop an alternative method, which
would be no worse than the finite difference method in accuracy and requires
less computational time. This approach can be elaborated by combining the
method of characteristics and a simple polynomial interpolation allowing us
to solve (1.84) at the grid points on a sphere.
Thus, an attempt is undertaken to solve the well-known "philosophical
problem" of the wave description~ whether waves are considered to be a field
description or particle propagation. The method of characteristics considers
the propagation of wave packets (particles), and at the same time the finitedifference and interpolation methods appear to describe the field. In our
opinion, the proposed numerical method is different from ordinary finitedifference methods because it describes more precisely the correspondence
with wave propagation physics.
It should be noted that the alternative approach proposed below removes
the "garden-sprinkler" effect without a noticeable increase in computation
time. Moreover, application of this approach in the semi-Lagrangian numerical method (hereinafter referred to as the interpolation-ray or the INTERPOL method) allows the use of much larger integration time steps in comparison with the Courant condition ( CLF) without losing numerical accuracy.
3.2 Formulation of the Wave Propagation Problem
Main equation. The evolution of a two-dimensional sea wave spectrum
S ( w, (3, cp, ' 19, t), being a function of the frequency w, direction (3 (measured
counterclockwise from the parallel), latitude cp, longitude ' 19 and time t is