by the model.

climatologists. The eddy and transient components

Barker [17] describes a radiative transfer

of a ¬eld are indicated by superscripts ˜*™ and

parameterization that requires the mean („ ) and

˜ ™ respectively. The time mean (equivalent to the

standard deviation (σ„ ) of cloud optical depth „

sample mean in this context) is denoted by an over-

within the grid box as input.1 In contrast, the cloud

bar, and square brackets denote the zonal average.

1 Optical depth is a measure of opacity. With this notation the meridional transient eddy

8.2: Correlation 147

Figure 8.2: Optical depth inferred from three 0.83 µm Landsat images. The brightest pixels in these

images correspond to an optical depth of about 20. From Barker et al. [18].

Left: Scene A3. Overcast stratocumulus, Ac = 1.000 and „ = 11.868.

Middle: Scene B2. Broken stratocumulus, Ac = 0.644 and „ = 3.438.

Right: Scene C14. Scattered cumulus, Ac = 0.291 and „ = 3.741.

Mean Log Optical Depth

Mean Log Optical Depth

•• • • • •

3

•

3

•

•• • •

• • • •• •

•

•• •

• ••

2

•••

2

•• •

•

•• • •••

•• •

••

•

•

• •• • •

••

•

1

••

1

• •• •

•••• •

•• •

•• • •

• •

•• •

•• •

0

• ••

0

5 10 15 20 0.2 0.4 0.6 0.8 1.0

Mean Optical Depth Fractional Cloud Cover

Figure 8.3: Left: Mean cloud optical depth („ ) versus mean log cloud optical depth (ln „ ) for 45 Landsat

scenes.

Right: Fractional cloud cover (Ac ) versus mean log cloud optical depth (ln „ ) for the same scenes. Data

courtesy of H. Barker.

transport of zonal momentum is formally given by transport of zonal momentum. Poleward transport

u — v — , where u and v represent the zonal and of zonal momentum in the Southern Hemisphere

is indicated by negative covariances. Figure 8.4

meridional wind components.2

illustrates that the transient eddies are a powerful

When u — v — > 0 in the Northern agent for exporting zonal momentum from the

Hemisphere, as in Figure 8.4, then easterly u tropical and subtropical latitudes polewards in both

anomalies (u — > 0) are usually connected hemispheres.

with northerly v anomalies (v — > 0), and

westerly anomalies (u — < 0) with southerly 8.2.2 The Correlation Coef¬cient. The corre-

anomalies (v — < 0). The distribution in lation coef¬cient is given by

Figure 8.4 represents a northward (poleward)

E((X ’ µ X )(Y ’ µY ))

ρ XY = ,

σX σY

2 The complete decomposition of the total transport is

√

[uv] = [u — v — ] + [u — v — ] + [u] [v] + [u v]. The ¬rst two terms

¯¯ ¯¯

where σX = Var(X) and σY is de¬ned

represent the transport by transient and stationary eddies, and

the last two terms the transports by the transient and stationary analogously. Note that ρ XY takes values in the

cells. For maps and further details, see Peixoto and Oort [311]. range [’1, 1].

8: Regression

148

when using the monthly mean SST anomaly to

specify the SOI, or the root mean square error is

76% of the standard deviation. This is in general

agreement with the level of scatter displayed in

Figure 8.1.

Note that the mean squared error is zero when

σ

ρ XY = 1; that is, Y = µY + σY (X ’ µx )

X

with probability 1 when ρ XY = 1. Also, note

that zero correlation is generally not the same as

independence (except when X and Y are normally

distributed, then X and Y are independent if and

only if ρ XY = 0; see [2.8.14]).

8.2.3 Making Inferences about Correlations.

When the sample {(Xi , Yi )T : i = 1, . . . , n}

Figure 8.4: Zonally averaged covariance between consists of independent, identically distributed

the ˜transient™ eddy components of the zonal and random vectors of length two, a good estimator of

meridional wind = ˜meridional transient eddy the correlation coef¬cient ρ XY is

transport of zonally averaged zonal momentum™

n

i=1(Xi ’ X)(Yi ’ Y)

during DJF simulated by a GCM. Units: m2 /s2 .

ρXY = . (8.4)