The contribution of a wide range of time scales to the long time average of atmospheric variances and covariances, and in particular on the contribution of the annual cycle is discussed. The interannual variability of seasonally averaged fluxes is analyzed in the same context. For definiteness, numerical results will be shown for the eddy momentum flux at 200 mb as obtained from seven years of NMC analyses (March 8, 1970 - March 10, 1977). The role of the seasonal cycle is considered

Straus, D. M.

English

NASA Technical Reports Server

I S3WOSf3 
•/' 
David M. Straus 
Laboratory for Atmospheric Sciences

Modeling and Simulation Facility

NASA/Goddard Space Flight Center

Greenbelt, MD 20771 
P882-219791 
The Contribution of a Wide Range of Space and Time Scales

to the Northward Flux of Westerly Momentum 
•	 The focus of this talk will be on the contribution of a wide range of time 
scales to the long-time average of atmospheric variances and covariances, and 
in particular on the contribution of the annual cycle. In addition, the 
interannual variability of seasonally averaged fluxes will be analyzed in. the 
same context.	 For definiteness, numerical results will be shown for the eddy 
-	 momentum flux at-200 mb as obtained from seven years of NMC analyses (March 8, 
1970 - March 10,	 1977).
Any discussion of variability encompassing time scales of longer than One 
month should address the question of the role of the seasonal cycle. One way of 
defining the seasonal cycle is as the average value of any atmospheric variable 
as a function of the time of year. The seasonal cycle represents the completely 
predictable component of the general circulation, predictable on the basis of 
past data alone .
. What is its contribution to the long-term averagnd eddy 
momentum flux?	 .	 .	 ..	 . 
The very existence of the seasonal cycle renders atmo
. heric time series 
non-stationary, and this causes the traditional separation between time mean 
and departures therefrom to be quite misleading when applied to records whose. 
length is one season or greater. For instance, consider a hypothetical time 
series whose seasonal cycle resembles a parabola over the course of one season 
(see Figure 1). If we define the transients as departures from this smooth 
seasonal cycle, they represent fluctuations about the completely predictable 
component. Comparison of this definition of transients with the traditional 
one of departures from the time mean (Figure 1) shows that the traditional 
definition includes contributions due to the seasonal cycle, contributions 
which ought to be separated from the less predictable fluctuations. 
In mathematical terms, let A(t) and 8(t) be any two atmospheric time 
series, '(t) and (t) their seasonal cycle components, and A'(t) and B I (t) the 
fluctuations about the seasonal cycle components: A I (t) = A(t) - A(t), B I (t) = 
B(t) -t). Letting an overbar denote a time averageover t
	 length of the 
record, the time averaged product AB is AB
	 + A'R' +	 B' + A'. In 
order to unambiguously ascribe a certain fraction of the total covariance to 
the seasonal cycle and a certain fraction to the transients, it is necessary to 
guarantee the orthogonality of these two components over the record length, so 
that the cross terms vanish. (This orthogonality holds automatically if one 
uses the traditional definition of transients, namely A I (t) = A(t) - A , for 
then AB A'S' + AB.) 
• When the record length T is an integer number of years (here T = 7 years), 
there is a natural solution to the orthogonality problem. This involves the 
use of Fourier analysis:
 
A(t)= A +	 (A cos(29m/T) + Bm sin(2utm/T)). 
247 
https://ntrs.nasa.gov/search.jsp?R=19830005477 2020-03-21T05:21:28+00:00Z
The mth term in the series describes fluctuations with a period of Tim. Based 
on this decomposition, it is possible to divide A(t) in four pieces: 
A(t) = A3( t ) + AULF(t) + A(t) + A(t) 
ASC(t) is the seasonal cycle component, and consists of the time mean A plus 
the annual (m=7) and semiannual (m=14) Fourier components. AU (t) is the 
ultra-low frequency component, and consists of fluctuations with periods longer 
than one year (m<7). ALF(t) is the low frequency component, and consists of 
fluctuations with periods less than one year but greater than 7.20 days, 
excluding the semi-annual period (8<m<355, u14). A(t) is the medium frequency 
component, and consists of fluctuations with periods between 7.20 days and 2.0 
days, the Nyquist period (356<m2560). Because of the orthogonality of the 
Fourier components over the seven year record length,. 
A(t) B(t) = A (t) B (t) + AF(t) B(t) 
+ A(t) BF(t) + A(t)BM2(t) 
The seven year averaged eddy momentum flux is thus a sum of separate fluxes, 
one for each term, with no cross terms. . A comparison of the magnitude of the 
four terms (Figure 2) shows that the seasonal cycle and low frequency components 
together contribute roughly 70%-80% of the total. The seasonal cycle component 
alone is significant. 
These ideas are now applied to the interannual variability of the fluxes. 
If we want to retain the high degree of time scale separation afforded by the 
Fourier basis set, and yet.be
 able to examine seasonal (rather than long-term) 
averages, we lose the orthogonality property and must deal with non-zero-cross 
terms. Combining the ultra-low and low frequency categories into a new low 
frequency category, and denoting a time average over a single season with a 
double overbar, we. have: 
A(t) B(t) = ASC BSC + ALF BLF + AMF B + 
(ASC BLF + ALF B ) + (ASC B + Bsc A) + (ALF B + A. BLF). 
The cross terms arise because the different components are, not orthogonal over 
a given season. 
The seasonal eddy momentum flux and its contributing terms were examined 
for the seven winter seasons, and the results shown in Figures 3-10. The seasonal 
cycle contribution is not shown on these g-raphs, since it is the same every 
winter and hence cannot contribute to interannual variability. The interaction 
terms-between the ,
 medium frequency component and the other components were very 
small, and are not shown. The interaction between the seasonal cycle and low 
frequency components is, however, quite important. On the average it seems to 
explain more of the year to year variation than any of the other components. 
This suggests.that the interannual variability of the winter eddy momentum 
flux at 200 mb is due to distortions of the climatologicalflow which, although 
they may not transport a significant amount of momentum themselves, interact 
with the normal (r climatological) component to transport anomalous momentum. 
248	 .	 .
I'
Figure Captions 
Figure 1. Hypothetical time series (solid line), the seasonal. cycle based on 
that time series (dashed line), and the seasonal mean (4tt	 line). 
Figure 2. Contributions to long time averaged 200 mb eddy momentum flux as a 
function of latitude. The seasonal cycle ontribution is the heavy - 
solid line, the low frequency contribution J s the heavy dashed line, 
the medium frequency contribution is the li;ht solid line, and the 
ultra-low frequency contribution is the light dashed line. 
Figure 3. Contributions to the winter averaged 200mb eddy momentum flux at 
62N as a function of year. The total winter flux is shown by the 
ieavy solid line, the contribution due to interactions between low 
frequency and seasonal cycle components by the heavy dashed line, 
the contribution due to low frequency components alone by the. light 
solid-line, and the contribution due to medium frequency components 
alone by the light dashed line. 
Figure 4. Contributions to the winter averaged 200 nib eddy momentum flux at 
56 0 N as a function of year. As in Figure 2. 
Figure 5. Contributions to the winter averaged 200 mb eddy momentum flux.at 
500 N as a function of year. As in Figure 2. 
Figure 6. Contributions to the winter averaged 200 mb eddy momentum flux at 
44 0N as a function of year.. As in Figure 2. 
Figure 7. Contributions to the winter averaged 200 mb eddy momentum flux at 
380 N as a function of year. As in Figure 2.
Figure 8. Contributions to the winter averaged 200 mb eddy momentum flux at 
32°N as a function of year. As in Figure 2. 
Figure 9. Contributions to the winter averaged 200 mb eddy momentum flux at 
26°N as a function of year. As in Figure 2. 
Figure 10. Contributions to the winter averaged 200 mb eddy momentum flux at 
20°N as a function of year. As in Figure 2. The contribution due 
to interactions between low frequency and seasonal cycle components 
has had 10 (rn/sec) 2 added to it. 
249 
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Figure 1 
Time Serlas 
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254


The Contribution of Wide Range of Space and Time Scales to the Northward Flux of Westerly Momentum

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