WIND STRESS
11
Introducing the wind steadiness s,
VuW 2 +VW2
S = --'-----===--WW
(14)
the vector mean pseudo stress can be expressed as a function of squared wind speed and wind
steadiness.
- T -
- -
--=sWW
(15)
pGD
Both factors on the right hand side are
liable to observational errors.
Obviously, the observation error ~ W of
the wind speed has to be taken into account,
because stress depends on W 2 , so that error effects are not compensated by averaging. The mean stress is overestimated by
the factor
~W~W
1+ WW
(16)
where ~ W ~ W denotes the error variance.
Not only the observational errors of the
wind speed affect the mean stress values,
but also errors of the wind direction. These
errors, denoted here as ~d, cause a spurious decrease of wind steadiness, consequently the mean wind stress is underestimated. It is easy to show that this effect
can be quantified by the factor cos ~d.
Thus, the ratio of calculated and true wind
stress is
1.0
0.9
"§' 0.8
I
,...,
e
~ 0.7
()
0.6
0.5
0
+
aO = 0.841
al = -0.49 pro Mm
+
+
+
+
45N to 50N
lOW to 20W
error = arccas(sqrt(aO)) = 23.5 degr
100
200
distance / km
300
Figure 13: Example for concluding the error of
the wind direction. The mean value of cosD1 -
D2 where Dl and D2 denote tewo individual simultaneous ship observations of the wind direction, is plotted against the ship distance s. A
linear fit with ao + al s yields a value for s = 0
-
= 1 +
cos
Tobs
(
~W~W)
~d
T
WW
(17)
To determine the error effects, the mean value of cos ~d and the relative error variance of
Ware evaluated for different monthly 5°x 10°_ fields in the North Atlantic. For this purpose
pairs of simultaneous wind reports were extracted from COADS. Their differences in the observed wind speed were computed and sorted into groups of ship distance (fig.13). In this way
it is possible to extrapolate lineary to a hypothetical value at zero distance, where only observational errors take effect. If (1) more than 100,000 pairs of observations were found and
(2) the linear fit explained more than 95% of the variance, the relative error ~ W ~ W /WW
is plotted as a function of WW. A relative error of about 7% results, decreasing with higher
wind speeds (fig. 14). An analogous procedure is applied to determinate the mean directional
11
Introducing the wind steadiness s,
VuW 2 +VW2
S = --'-----===--WW
(14)
the vector mean pseudo stress can be expressed as a function of squared wind speed and wind
steadiness.
- T -
- -
--=sWW
(15)
pGD
Both factors on the right hand side are
liable to observational errors.
Obviously, the observation error ~ W of
the wind speed has to be taken into account,
because stress depends on W 2 , so that error effects are not compensated by averaging. The mean stress is overestimated by
the factor
~W~W
1+ WW
(16)
where ~ W ~ W denotes the error variance.
Not only the observational errors of the
wind speed affect the mean stress values,
but also errors of the wind direction. These
errors, denoted here as ~d, cause a spurious decrease of wind steadiness, consequently the mean wind stress is underestimated. It is easy to show that this effect
can be quantified by the factor cos ~d.
Thus, the ratio of calculated and true wind
stress is
1.0
0.9
"§' 0.8
I
,...,
e
~ 0.7
()
0.6
0.5
0
+
aO = 0.841
al = -0.49 pro Mm
+
+
+
+
45N to 50N
lOW to 20W
error = arccas(sqrt(aO)) = 23.5 degr
100
200
distance / km
300
Figure 13: Example for concluding the error of
the wind direction. The mean value of cosD1 -
D2 where Dl and D2 denote tewo individual simultaneous ship observations of the wind direction, is plotted against the ship distance s. A
linear fit with ao + al s yields a value for s = 0
-
= 1 +
cos
Tobs
(
~W~W)
~d
T
WW
(17)
To determine the error effects, the mean value of cos ~d and the relative error variance of
Ware evaluated for different monthly 5°x 10°_ fields in the North Atlantic. For this purpose
pairs of simultaneous wind reports were extracted from COADS. Their differences in the observed wind speed were computed and sorted into groups of ship distance (fig.13). In this way
it is possible to extrapolate lineary to a hypothetical value at zero distance, where only observational errors take effect. If (1) more than 100,000 pairs of observations were found and
(2) the linear fit explained more than 95% of the variance, the relative error ~ W ~ W /WW
is plotted as a function of WW. A relative error of about 7% results, decreasing with higher
wind speeds (fig. 14). An analogous procedure is applied to determinate the mean directional