Geoscience Reference
In-Depth Information
Note that we have assumed that the variance of the phase uncertainty is small
compared with the characteristic length scale of the disturbance such that the
truncation of the Taylor-series to the quadratic term is sensible. Physically, the
reason the distribution of
is approximately chi-square at this location arises
because the values of pressure at this location can never be below
p
p
0
but can vary
as high as the far-field values of pressure allow. This type of hard lower limit
characterizing the distribution of pressures is a characteristic of distributions like
that of the chi-square distribution and always leads to significant third moments.
Similarly, we may perform a Taylor-expansion around the inflection point
(
x
D
x
1
), which lies along the edge of the storm (See Fig.
7.1
):
p.x
1
I
'/
D
p.x
1
/
C
˛.x
1
'/
C
:::;
(7.37)
where
ˇ
ˇ
ˇ
ˇ
x
D
x
1
:
dp
dx
˛
D
(7.38)
Note that at the inflection point of the TC the second derivative vanishes while the
first derivative (
) is large. Hence, the distribution in the vicinity of the inflection
point is nearly Gaussian and therefore has vanishingly small third moment, which
can be seen in the values of its moments:
˛
h
p
i D
p
1
C
˛x
1
;
(7.39)
D
.p
h
p
i
/
2
E
D
˛
2
2
;
(7.40)
D
.p
h
p
i
/
3
E
D
0:
(7.41)
Therefore, we have so far seen that at the mean position of the phase distribution
the structure of the pressure distribution is strongly non-Gaussian, but as we make
our way away from the center of the phase distribution we find that the pressure
distribution appears to become approximately Gaussian.
This variation in the structure of the scalar distributions at each grid point along
the disturbance can be summarized by eliminating the Taylor-expansion and simply
randomly sampling from a known distribution. We define a function that is vaguely
similar to the pressure field of a tropical cyclone,
exp
"
2
#
.x
'/
4
p.x
I
'/
D
1000
25
;
(7.42)
and proceed to randomly sample this function by drawing values of
.
This function represents a Gaussian-shaped depression that is positioned at random
locations and is in fact the function plotted in Fig.
7.1
a. By calculating the scalar
moments of this distribution at each grid point we find a pattern as in Fig.
7.1
b.
'
N.0;2/
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