7.4 Non-Linear Transfer in the Spectrum for Waves with Ice Cakes
333
The algorithm for calculating the integral (7.69) described in Chap. 4
for the case of deep water without ice. The situation is more complicated
for finite depth water with ice. Due to the difficulty in using the accurate
dispersion ratio determined by the formula (7.9) for waves in water with ice,
its approximate expression can be used as follows:
(7.71)
where w6 = g / H is the typical frequency defined by water depth, and the
frequency 0' is found using the formula:
It should be noted that (7.71) is valid for every frequency and wave number with an error of no more than 10 per cent.
There are two parameters: w 0 and wb in this problem. In other respects
the algorithm for calculating the integral (7.69) coincides completely with
that used in Chap. 4.
Estimation results of non-linear energy transfer in the wave spectrum.
According to the above mentioned algorithm the calculations of
non-linear energy transfer in the wave spectrum are fulfilled, taking into account water depth and the presence of ice cakes. The wave spectrum is used
in the form of the JONSWAP approximation, and the angle distribution function is accepted as the cosine. The calculations are made for different values
of the peakedness parameter equal to "/ = 1.0 and 3.3, and the cosine power
in the angle distribution function is varied from 2 to 8. The water depth is
accepted to be equal to 10m and the ice cover thickness is 0.3, 0.5 and 0.8 m,
correspondingly.
An example of such a calculation for the peakedness parameter equal to
1.0 is shown in Fig. 7.3. The value of the non-linear transfer is given in the
normalized form:
For comparison, the results are shown in the case of infinitely deep water
without ice cover. The results show that the character of the non-linear energy
transfer is qualitatively the same as in the traditional case of deep water
without ice. The energy flow is directed from the central spectrum area to
the high and low frequency ranges. The depth decrease results in increasing
the non-linear energy transfer, otherwise, the increase of ice cover thickness
diminishes the intensity of the non-linear energy transfer.
It can be concluded from the aforesaid that the presence of ice causes
narrowing of the frequency range of the wave spectrum. The positive lowfrequency and negative areas are shifted to low frequencies in comparison
with the traditional case without ice. This shows the intensive spectrum shift
333
The algorithm for calculating the integral (7.69) described in Chap. 4
for the case of deep water without ice. The situation is more complicated
for finite depth water with ice. Due to the difficulty in using the accurate
dispersion ratio determined by the formula (7.9) for waves in water with ice,
its approximate expression can be used as follows:
(7.71)
where w6 = g / H is the typical frequency defined by water depth, and the
frequency 0' is found using the formula:
It should be noted that (7.71) is valid for every frequency and wave number with an error of no more than 10 per cent.
There are two parameters: w 0 and wb in this problem. In other respects
the algorithm for calculating the integral (7.69) coincides completely with
that used in Chap. 4.
Estimation results of non-linear energy transfer in the wave spectrum.
According to the above mentioned algorithm the calculations of
non-linear energy transfer in the wave spectrum are fulfilled, taking into account water depth and the presence of ice cakes. The wave spectrum is used
in the form of the JONSWAP approximation, and the angle distribution function is accepted as the cosine. The calculations are made for different values
of the peakedness parameter equal to "/ = 1.0 and 3.3, and the cosine power
in the angle distribution function is varied from 2 to 8. The water depth is
accepted to be equal to 10m and the ice cover thickness is 0.3, 0.5 and 0.8 m,
correspondingly.
An example of such a calculation for the peakedness parameter equal to
1.0 is shown in Fig. 7.3. The value of the non-linear transfer is given in the
normalized form:
For comparison, the results are shown in the case of infinitely deep water
without ice cover. The results show that the character of the non-linear energy
transfer is qualitatively the same as in the traditional case of deep water
without ice. The energy flow is directed from the central spectrum area to
the high and low frequency ranges. The depth decrease results in increasing
the non-linear energy transfer, otherwise, the increase of ice cover thickness
diminishes the intensity of the non-linear energy transfer.
It can be concluded from the aforesaid that the presence of ice causes
narrowing of the frequency range of the wave spectrum. The positive lowfrequency and negative areas are shifted to low frequencies in comparison
with the traditional case without ice. This shows the intensive spectrum shift