Section 13.2: Expansion into a Few Guess Patterns
231
13.2.2 Example:
Temperature Distribution In the
Mediterranean Sea
As an example 3 , we present here an expansion (13.1) of a time-dependent
3-dimensional temperature field of the Mediterranean Sea. The output of a
9-year run of an OGCM, forced by monthly mean atmospheric conditions as
analysed by the US-NMC for the years 1980 to 1988, was decomposed such
that
T(r, z, t) = L O!k(Z, t)p;
(13.11)
k
with rrepresenting the horizontal coordinates, Z the vertical coordinate and t
the time. The temperature field is given on a (r, z)-grid with better resolution
in the upper levels. In the representation (13.11), coefficients O!k depend on
depth and time. The orthogonal patterns pk depend only on the horizontal
distribution and are independent of the depth z and of the time t. The
deeomposition was determined by a "Singular Value Decomposition" but the
technical aspects are not relevant for the present discussion.
Prior to the analysis, the data have been processed. For eaeh depth z
the horizontal spatial mean and standard de:viation have been calculated.
Then, the temperature values at each depth are normalized by subtracting
the (spatial) mean and dividing by the (spatial) standard deviation of the
respective depth. The annual eyde is not subtracted so that the time series
are not stationary (but cydostationary) with an annual cyde of the time
mean and of the temporal standard deviation.
The first two patterns are shown in Figure 13.1. The first one, which
represents 57% of the total second moment of the normalized temperature,
exhibits a dipole, with about half of the basin being warmer than average and
the other half being cooler than average. The second mode, which represents
32% of the second moment, is relatively uniform throughout the Mediterranean Sea. The relative importance of the two mo des for different layers
of the ocean is described by the amount of the 2nd moment accounted for
by the two mo des (Figure 13.2). The second mode explains most of the 2nd
moment above 100 m, whereas the first mode dominates below the top 4
layers. An inspection of the time series O!k(Z, t) for different depths z reveals that the two modes represent different aspects of the dimatology of the
Mediterranean Sea. The time series O!l(Z, t) is always positive with irregular
variations superimposed. Such a behaviour is indicative that the first mode
describes mostly the overall mean and its interannual variability. The time
series 0!2(Z, t) describe a regular annual cyde (with a negative minimum in
winter so that 0!2PJ is a negative distribution; and a positive maximum in
3This material was presented by Gerrasimos Korres in a "student paper" during the
Autumn School. It will be available as a regular paper coauthored by Korres and Pinardi
in 1994.
231
13.2.2 Example:
Temperature Distribution In the
Mediterranean Sea
As an example 3 , we present here an expansion (13.1) of a time-dependent
3-dimensional temperature field of the Mediterranean Sea. The output of a
9-year run of an OGCM, forced by monthly mean atmospheric conditions as
analysed by the US-NMC for the years 1980 to 1988, was decomposed such
that
T(r, z, t) = L O!k(Z, t)p;
(13.11)
k
with rrepresenting the horizontal coordinates, Z the vertical coordinate and t
the time. The temperature field is given on a (r, z)-grid with better resolution
in the upper levels. In the representation (13.11), coefficients O!k depend on
depth and time. The orthogonal patterns pk depend only on the horizontal
distribution and are independent of the depth z and of the time t. The
deeomposition was determined by a "Singular Value Decomposition" but the
technical aspects are not relevant for the present discussion.
Prior to the analysis, the data have been processed. For eaeh depth z
the horizontal spatial mean and standard de:viation have been calculated.
Then, the temperature values at each depth are normalized by subtracting
the (spatial) mean and dividing by the (spatial) standard deviation of the
respective depth. The annual eyde is not subtracted so that the time series
are not stationary (but cydostationary) with an annual cyde of the time
mean and of the temporal standard deviation.
The first two patterns are shown in Figure 13.1. The first one, which
represents 57% of the total second moment of the normalized temperature,
exhibits a dipole, with about half of the basin being warmer than average and
the other half being cooler than average. The second mode, which represents
32% of the second moment, is relatively uniform throughout the Mediterranean Sea. The relative importance of the two mo des for different layers
of the ocean is described by the amount of the 2nd moment accounted for
by the two mo des (Figure 13.2). The second mode explains most of the 2nd
moment above 100 m, whereas the first mode dominates below the top 4
layers. An inspection of the time series O!k(Z, t) for different depths z reveals that the two modes represent different aspects of the dimatology of the
Mediterranean Sea. The time series O!l(Z, t) is always positive with irregular
variations superimposed. Such a behaviour is indicative that the first mode
describes mostly the overall mean and its interannual variability. The time
series 0!2(Z, t) describe a regular annual cyde (with a negative minimum in
winter so that 0!2PJ is a negative distribution; and a positive maximum in
3This material was presented by Gerrasimos Korres in a "student paper" during the
Autumn School. It will be available as a regular paper coauthored by Korres and Pinardi
in 1994.