38
2 Mathematical Simulation of Wave Propagation at Global Distances
is changed. Actually, there is no turn at all. The effect is created by the geometry of the spherical surface, on which the wave packet propagates along
the arc of the Earth's great circle.
Thus, the problem of wind wave energy evolution under oceanic surface
sphericity results in integrating the spectral equation (2.1) with the characteristics (2.6)-(2.8) for the prescribed boundary or initial conditions.
2.2 Correspondence
of Local and Global Coordinate Systems
The transformation of the spectral energy balance equation (2.1), written
with the use of the spherical variables r.p, {), {3, to a local plane rectangular coordinate system { x, y} is now considered. The local coordinates
{x, y} around the point { r.p0 , 1J0 } (Fig. 2.1) are introduced as follows (Kamenkovich & Monin, 1978):
x = Rcos(r.po)(1J- 1Jo); y = R(r.p- r.po).
(2.9)
Then
dx
d{J
dt = Rcos(r.p)dt ,
dy = Rdr.p.
dt
dt
Substituting the variables (2.9) into (2.1) gives
(2.10)
The equation (2.10) coincides with the spectral density equation, written
in the usual problem formulation in the plane rectangular coordinate system {x,y}, with the exception of the fourth term in its left part. Using the
expression (2.8), this can be written as:
as df3
9
( y
)
as
- - = - - t a n - + r.po cos(f3)-.
a{3 dt
2a
R
a{3
(2.11)
The expression (2.11) can be interpreted as a local error arising in the
energy spectral density estimation using the rectangular coordinate system
{x, y} due to the fact that the Earth's surface sphericity is not taken into
account. The value of the expression (2.11) is considerably less than the other
items in the left side of (2.10) for regional spatial-temporal scales. However,
in the case of IY / R + r.p0 I -+ n/2, the expression (2.11) becomes sufficiently
large, and it cannot be neglected. Since the local coordinate system (2.9) is
introduced for a small area around the point {r.po,1Jo}, (i.e. at y/R « 1), the
term (2.11) is sharply increased close to the poles (r.po = ±n/2). In this case
the largest errors occur in (2.10), due to neglecting the expression (2.11). The
2 Mathematical Simulation of Wave Propagation at Global Distances
is changed. Actually, there is no turn at all. The effect is created by the geometry of the spherical surface, on which the wave packet propagates along
the arc of the Earth's great circle.
Thus, the problem of wind wave energy evolution under oceanic surface
sphericity results in integrating the spectral equation (2.1) with the characteristics (2.6)-(2.8) for the prescribed boundary or initial conditions.
2.2 Correspondence
of Local and Global Coordinate Systems
The transformation of the spectral energy balance equation (2.1), written
with the use of the spherical variables r.p, {), {3, to a local plane rectangular coordinate system { x, y} is now considered. The local coordinates
{x, y} around the point { r.p0 , 1J0 } (Fig. 2.1) are introduced as follows (Kamenkovich & Monin, 1978):
x = Rcos(r.po)(1J- 1Jo); y = R(r.p- r.po).
(2.9)
Then
dx
d{J
dt = Rcos(r.p)dt ,
dy = Rdr.p.
dt
dt
Substituting the variables (2.9) into (2.1) gives
(2.10)
The equation (2.10) coincides with the spectral density equation, written
in the usual problem formulation in the plane rectangular coordinate system {x,y}, with the exception of the fourth term in its left part. Using the
expression (2.8), this can be written as:
as df3
9
( y
)
as
- - = - - t a n - + r.po cos(f3)-.
a{3 dt
2a
R
a{3
(2.11)
The expression (2.11) can be interpreted as a local error arising in the
energy spectral density estimation using the rectangular coordinate system
{x, y} due to the fact that the Earth's surface sphericity is not taken into
account. The value of the expression (2.11) is considerably less than the other
items in the left side of (2.10) for regional spatial-temporal scales. However,
in the case of IY / R + r.p0 I -+ n/2, the expression (2.11) becomes sufficiently
large, and it cannot be neglected. Since the local coordinate system (2.9) is
introduced for a small area around the point {r.po,1Jo}, (i.e. at y/R « 1), the
term (2.11) is sharply increased close to the poles (r.po = ±n/2). In this case
the largest errors occur in (2.10), due to neglecting the expression (2.11). The