Spatial variability

A plot of P(s, n) is shown in Figure 1-4, along with the frequency spectrum (P(s) =

P

nP(s, n)) and the spatial spectrum (P(n) =^P_sP(s, n)) of variability. As expected, the dominant feature ofP(s, n) is a ridge at annual periods with a maximum at degree number n = 1. Ridges are also apparent at the higher frequency harmonics of the annual cycle, i.e. 2,3,4... cycles/year, and each of these ridges appear as peaks in the frequency spectrum, P(s). As observed in the earlier atmospheric temperature records [Figure 1-1], away from the peaks, the frequency spectrum is characterized by a power law process with q = −0.4. The degree number spectrum also shows greater energy towards longer spatial scales, but with a broad peak surrounding ann of roughly five.

Because the SAT response to the annual cycle is so large, it obscures the behavior of the background continuum. Removing mean monthly temperatures, as calculated at each grid point, suppresses the energy associated with the annual cycle and its higher harmonics. Results are shown in Figure 1-5. The background continuum

asso-3Because the NCEP/NCAR reanalysis uses a spectral model, it should be possible to find the spherical harmonic loadings without ever transforming into the gridded domain.

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Figure 1-4: top The spectrum of spherical harmonic coefficients for NCEP tempera-ture variability plotted as a function of spatial degree number (n) and frequency (s) measured in cycles/year. Spectral estimates are made using the multi-taper method with three windows, and the resulting n by s field is smoothed using a tapered 5x5 window. bottom A summation over frequency yields the degreen spatial spectrum (left), while summation over spatial scales yields the frequency spectrum (right). The degreenspectrum increases towards larger spatial scales with q= 2 up ton = 7, and then increases more weakly with q = 1, as indicated by the dashed red-lines. In the frequency spectrum, the concentration of energy at the annual cycle and its higher harmonics is evident, and the background variability has q = 0.4.

ciated with the frequency spectra of the filtered and unfiltered monthly temperature estimates is nearly the same, but after filtering the degree number spectrum now shows enhanced variability at n = 5 to 7, with power rolling off remarkably steadily with q = .9 for n < 5 and q = −2 for n > 7. Thus it appears that temperature variability, at monthly to decadal timescales, predominantly occurs at spherical

har-−8

Figure 1-5: Similar to Figure 1-4 but for NCEP temperature variations with the monthly averages removed. Most notable is the peak of energy at degree n = 6 and 7, with energy diminishing at an exponential rate towards the larger and smaller scales.

monic degree number 6. Note that taking the monthly average of SAT effectively filters out the synoptic scales of variability; these shorter scales were the focus of the study by Trenberth and Solomon [1993]. At periods longer than a month, the most active spatial-scales of temperature variability are on the order of continents and oceans, and are probably associated with the land-sea temperature contrast, as well as hemispheric meridional temperature gradients. Connecting these results with the synoptic scale variability could prove useful, but is not further pursued here.

Figure 1-6 shows the first moment of the spatial scale as a function of frequency

and weighted by the fraction of temperature variance, M1(s) =

P

nnP(s, n)

P

nP(s, n) . (1.1)

Small values of M1 indicate relatively more energy at large spatial scales, and will be interpreted as greaterorganization of the temperature variability. Away from the annual cycles and its higher-harmonics, M1 hovers around eleven. At the annual cycles M1 drops to four because the annual cycle is spatially organized. At two cy-cles per year, a weak organizing effect is evident, while at higher harmonics slightly greater disorganization is observed — an unexpected result. For temperatures with their monthly means removed, M1 remains close to 11 showing only minor variability at the annual cycle and its harmonics. The consistency of spatial organization at timescales ranging from months to decades suggests that the spatial description of climate variability is no simpler at long timescales than it is at the monthly timescales.

This result is apparently at odds with Mitchell’s [1976] suggestion of larger spatial scales of variability at longer timescales, and is in some sense surprising. One might expect that dissipative systems such as the atmosphere and ocean would not main-tain strong gradients over long timescales. But perhaps the persistence of features such as the atmospheric jets, western boundary currents, land/ocean configuration, mountains, ice-sheets, vegetation, etc. is more telling. Suffice it to say that the long-term behavior of fluids on a rough, heterogeneous, and rotating planet is not easily intuited, particular when dynamical interactions with the cryosphere, geosphere, and biosphere come into play.

The spectra discussed in conjunction with Figures 1-4 and 1-5 contain a lot of information but are rather abstract. To provide a more tangible example, the cross-correlations between temperature at a single location with temperature at every other point (the one-point correlation) is shown in Figure 1-7. To focus on the inter-annual timescales of interest, all timeseries were first filtered to remove the energy at the annual and higher frequencies. First, the GISP2 site in Greenland is considered.

There is a strong local correlation extending over parts of the Arctic, Northeastern Canada, and Siberia. Interestingly, there also exists weak positive correlation with the Atlantic and Antarctic. But the overall result is patchy. Given only observations from Greenland, it would be difficult to infer inter-annual temperature variability outside of the Northern N. Atlantic.

Another record which has aroused attention comes from Devils Hole in Nevada

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