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sea

1m

100

micro

turbulence

1 cm

10-2

1 year

1 day 10 10 100

1 second 1 hour

1 minute 1000

1

0 2 3 4 56791

80

1

0 2 3 456789

1

0

10-2 100 102 104 106 108 1010

Characteristic time scale [s]

Figure 3.1: Length and time scales in the atmosphere and ocean. After [390].

3: Distributions of Climate Variables

54

â€¢ The statistics of the amount of precipitation

depend on the accumulation time, as demon-

strated in Figure 3.2. The curves, which

are empirical distribution functions of accu-

mulated precipitation, are plotted so that a

normal distribution appears as a straight line.

For shorter accumulation times, such as days

and weeks, the curves are markedly concave

with medians (at probability 0.5) that are less

than the mean (normalized precipitation = 1),

indicating that these accumulations are not

normally distributed. For the annual accumu-

lation, the probability plot is a perfect straight

line with coinciding mean and median. Thus,

for long accumulation times the distribution

Figure 3.2: Empirical distribution functions of

is normal. Figure 3.2 is a practical demonstra-

the amount of precipitation, summed over a

tion of the Central Limit Theorem.

day, a week, a month, or a year, at West

Glacier, Montana, USA. The amounts have been

â€¢ The number of rainy days per month is often

normalized by the respective means, and are

independent of the amount of precipitation.

plotted on a probability scale so that a normal

distribution appears as a straight line. For further

â€¢ The time between any two rainfall events, or

explanations see [3.1.3]. From Lettenmaier [252].

between two rainy days, is the interarrival

time.

3.1.2 Precipitation. Precipitation, in the form

Lettenmaier [252] deals with the distribution

of rain or snow, is an extremely important

aspects of precipitation and offers many references

climate variable: for the atmosphere, precipitation

to relevant publications.

indicates the release of latent heat somewhere

in the air column; for the ocean, precipitation

represents a source of fresh water; on land, 3.1.3 Probability Plotsâ€”a Diversion. Dia-

precipitation is the source of the hydrological grams such as Figure 3.2 are called probability

cycle; for ecology, precipitation represents an plots, a type of display we discuss in more detail

important external controlling factor. here.

There are two different dynamical processes that The diagram is a plot of the empirical

yield precipitation. One is convection, which is distribution function, rotated so the possible

the means by which the atmosphere deals with outcomes y lie on the vertical axis, and the

estimated cumulative probabilities p(y) = F Y (y)

vertically unstable conditions. Thus, convection

lie on the horizontal axis.7 Alternatively, if

depends mostly on the local thermodynamic

conditions. Convective rain is often connected we consider p the independent variable on the

âˆ’1

with short durations and high rain rates. The horizontal axis, then y = F Y ( p) is scaled by

other precipitation producing process is large- the vertical axis. For reasons outlined below, the

âˆ’1

scale uplift of the air, which is associated with variable p is re-scaled by x = FX ( p) with some

the large-scale circulation of the troposphere. chosen distribution function FX . The horizontal

Large-scale rain takes place over longer periods axis is then plotted with a linear scale in x. The

but is generally less intense than convective p-labels (which are given on a nonlinear scale)

rain. Sansom and Thomson [338] and Bell are retained. Thus, Figure 3.2 shows the function

and Suhasini [40] have proposed interesting âˆ’1

x â†’ F Y [FX (x)]. If F Y = FX , the function is

approaches for the representation of rain-rate

the identity and the graph is the straight line (x, x).

distributions, or the duration of rain-events, as

The probability plot is therefore a handy visual

a sum of two distributions: one representing

tool that can be used to check whether the observed

the large-scale rain and the other the convective

random variable Y has the postulated distribution

rain.

FX .

There are a number of relevant parameters

that characterize the precipitation statistics at a 7 The â€˜hatâ€™ notation, as in F (y), is used throughout this

Y

location. book to identify functions and parameters that are estimated.

3.1: Atmospheric Variables 55

apparently occur over a broad range (caus-

ing the long negative tail) whereas warm

extremes are more tightly clustered.

â€¢ The distribution has two marked maxima at

35 â—¦ F and at 75 â—¦ F. This bimodality might

be due to the interference of the annual

cycle: summer and winter conditions are

more stationary than the â€˜transientâ€™ spring

and autumn seasons, so the two peaks may

represent the summer and winter modes.

The summer peak is taller than the winter

peak because summer weather is less variable

than winter weather. Also, the peak near the

Maximum Temperature

freezing point of 33 â—¦ F might reï¬‚ect energy

absorption by melting snow.

Figure 3.3: Frequency distribution of daily

â€¢ There is a marked preference for temperatures

maximum temperature in â—¦ F at Napoleon (North

ending with the digits 0 and 5. Nese

Dakota, USA) derived from daily observations

[291] also found that the digits 2 and 8

from 1900 to 1986. From Nese [291].

were overrepresented. This is an example of

psychology interfering with science.

When the observed and postulated random

variables both belong to a â€˜location-scaleâ€™ family

Averages of daily mean air temperature8 are also

of distributions, such as the normal family, a

sometimes markedly non-normal, as is the case

straight line is also obtained when Y and X have

in Hamburg (Germany) in January and February.

different means and variances. In particular, if a

Weather in Hamburg is usually controlled by

random variable X has zero mean and unit variance

a westerly wind regime, which advects clouds

such that FY (y) = FX ( yâˆ’Âµ ), then

Ïƒ and maritime air from the Atlantic Ocean. In

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