How does the spatial scale of natural climate fluctuations vary across timescales?

What is the spatial scale of natural climate fluctuations, and how does it depend on timescale? To answer this question, we characterize the spatio-temporal correlation structure of global surface temperature fields by estimating frequency spectra of the effective spatial degrees of freedom (ESDOF), which can be interpreted as the effective number of independent spatial samples on the globe at each frequency. ESDOF spectra are estimated from the HadCRUT4 global gridded, monthly mean temperature anomaly dataset, based exclusively on instrumental measurements, covering the period 1850 to near-present. Because this dataset includes gaps (due to a lack of observations in certain months and regions on the globe), we employ a newly developed method that allows for bias-free spectral estimation from gappy data without interpolation across gaps. To correct for the anthropogenic warming trend, the data is detrended prior to the analysis, by subtracting the linear response to the anthropogenic global mean log(CO2-equivalent) forcing time series. The resulting ESDOF spectra reveal a reduction of the ESDOF value by a factor of 10, from about 130 (±15%) at sub-annual timescales to about 13 (±50%) at multi-decadal time scales. Uncertainties are estimated by applying the same analysis to a CMIP6 climate model ensemble, with HadCRUT4 data gaps imposed. To test for the possible impact of the data gaps, the ESDOF analysis is applied to global temperature fields with and without gaps, taken from both the climate model ensemble and from the NOAA 20th Century Reanalysis dataset. Results suggest slightly higher ESDOF values for complete fields, with the increase being negligible at sub-annual timescales and of the order of 15-20% at multi-decadal timescales. Overall, the results indicate that natural temperature variability at multi-decadal timescales is characterised by an ESDOF value between 10 and 20. Since it is unlikely, due to physical constraints, that the ESDOF value increases towards timescales longer than those resolved by the instrumental record, the above multi-decadal ESDOF estimate can be taken as an upper limit for centennial and longer timescales. This may have important implications in the context of paleo-climate reconstructions and their comparison with model simulations.