The power spectrum of climate change

Abstract

Both global, intermediate and local scales of Climate Change have been studied extensively, but a unified diagnostic framework for examining all spatial scales concurrently has remained elusive. Here, we present a new tool-set using spherical harmonics to examine climate change through surface temperature anomalies from 1850 to 2021 on spatial scales ranging from planetary to 50 km scales. We show that the observed temperature anomalies are accurately decomposed in spherical harmonics typically within 0.05 K. This decomposition displays a remarkably simple dependence on spatial scale with a universal shape across seasons and decades. The decomposition separates two distinct regimes by a characteristic turnover-length of approximately 3000 km. The largest scales confirm established trends, while local fluctuations are consistent with 2-dimensional turbulence. We observe a downward cascade, from which it follows that climate change feeds increasing volatility on all spatial scales from 2.000 to 50 km. This increase is primarily driven by growing volatility along longitudes.

Appendix

1.1 A: Power spectral analysis of ERA5 dataset

While the above analysis centres on The Berkeley Earth Land/Ocean Temperature Record—other coincident catalogues tell a similar story. In Fig. 6, the power spectrum for each decade from temperature anomalies in the ERA5 reanalysis (from 1979 to present) and the preliminary ERA5 reanalysis (from 1950 to 1978) is illustrated. The anomalies are defined relative to the locally inferred average temperature for that month over the period January 1951–December 1980 (i.e. similar to the Berkeley Earth Land/Ocean Temperature Record). Crucially, this shows an identical functional form of the power spectrum with a similar normalisation, turnover length-scale and powerlaw-decline (see Fig. 3). Additionally, the evolution with time displays a coherent growth of power for modes within and above the turbulent cascade.

Note, the analysis of the 30 km spatial resolution of ERA5 allows an examination of small modes ($90<l<360$). The previous observed powerlaw-regime with $\alpha =-3$ is observed within the entire range from 2000 to 50 km. The scale-height of the atmosphere (order 10 km) still remains significantly smaller than the horizontal resolution, so at first glance one might believe we are still well above the scales of 3-dimensional turbulence, but several previous analysis (especially of wind spectra within the Global Assimilation and Prognosis System) indicate $\alpha =-5/3$ regime up to several hundreds of kilometres [10, 32]. Thus, the singular powerlaw extending over several orders of magnitude for temperature anomalies on the earths surface is seemingly in contrast with some atmospheric fields at higher altitudes. Nevertheless, we emphasise that ERA5 reanalysis is not an empirical field—it is obtained through variational data assimilation techniques, which require smoothing over small spatial scales. Thus, the spectral power on the smallest scales remains ill-constrained is subject to the potential systematics effects of smoothing (see Appendix E).

1.2 B: Power spectrum of average temperature

Temperature anomalies are reported relative to the baseline average temperature for each month from January 1951 to December 1980. The power spectrum for this baseline temperature is seen in Fig. 7. Firstly, the power of all modes is larger than observed for temperature anomalies; this is expected as absolute temperature varies far more drastically—ranging from $-70^{\circ }\hbox {C}$ to $40^{\circ }\hbox {C}$ with elevation, climate and latitude. For instance, the systematic variation with elevation will increase the amplitude of temperature fluctuations on spatial scales ranging from continents to mountains. Secondly, the shape of the power-spectrum is different than temperature anomalies, but still consistent with a powerlaw for large degrees. The powerlaw slope $\alpha$ is for all months within the range $-3.1$ to $-3.4$, which remains significantly steeper than for temperature anomalies. The cause of the increased steepness remains (as of yet) unresolved. Thirdly, the spread from the powerlaw has increased with large systematic bias observed across months for specific degrees (i.e. $C_3$ is remarkably small). Such rugged power-spectra are commonly observed when examining altitude [33], which illustrates the apparent systematic bias in decomposing absolute temperature at the surface.

Spherical harmonics have not previously been used as a standard descriptor of climate change, because structures like elevation, latitude and differences between land and ocean dominate the variation of absolute temperature. However, by limiting the analysis to temperature anomalies we focus only on the underlying fluctuations due to atmospheric processes. This effectively decouples the effect of climate change from the larger irrelevant structures.

Fig. 7

Power spectrum of average temperature for each month from January 1951 to December 1980. The power for all modes is larger than for temperature anomalies

1.3 C: Power spectrum characteristics are robust to limiting coverage

While the specific numerical value of the power for any mode is dependent on the coverage, the characteristic features of the power-spectrum (discussed in Sects. 3.2 and 3.3) are unchanged when limiting the coverage to that 1850 [See Fig. 8]. The turnover-length remains at $\approx 3000$ km, the powerlaw slope remains consistent with $\alpha =-3$ for the entire time-series and the relative growth of small to large modes is unchanged.

Fig. 8

Power spectrum of temperature distribution from 1850s (in black) to 2010s (in red) in log-log plot limiting coverage to that of 1850. The power for modes $l \ge 10$ is increasing with time, especially from 1960 to 2020. For comparison see the full coverage in Fig. 3

1.4 D: Increasing resolution

From 1850 to 2020 the number of surface temperature measurements over land and sea increased drastically. In Fig. 9 we show the evolution in number of active temperature stations over land (reported in [16]) and number of “superobservations” of temperatures over sea. Super-observations are the Winsorised mean of temperature anomalies used in HadSST [17].

We emphasise that most of the evolution of the power-spectrum is from 1960 to 2020, where no significant change in number of either land stations nor super-observations over sea takes place. On the other hand, the methodology yields consistent power-spectra from 1850 to 1960 even though the number-of-observations increased by orders of magnitude. Thus, increasing the number of measurements will improve the statistical uncertainty on the monthly temperature-estimates, but improving accuracy does not correlate with a bias in the spherical harmonics decomposition.

Fig. 9

Top: power of degree l relative to power in 1850. Bottom: Number of active temperature stations over land (brown line) and number of “superobservations” of temperatures over sea (blue dots) from 1850 to 2021. Grey shaded region indicates the period 1960–2021. The significant growth in power-spectrum does not correlate with an increasing number of observing stations

1.5 E: Interpolation bias

Figure 10 shows the underestimation of power when generated temperature anomalies are interpolated over a characteristic length-scale, L. As expected larger interpolation length results in a larger bias. For interpolation over $0.5^{\circ }$, $1^{\circ }$ and $2^{\circ }$ latitude and longitude the estimated power remains accurate within, respectively, 1%, 3% and 10% up to degree $l=50$. Therefore, the typical bias remains significantly smaller, than the factor 2 growth in power of these modes from 1960 to 2020. Furthermore, the bias has a clear dependence on angular scale. In contrast, from 1960 to 2020 all power within the cascade grew coherently maintaining the powerlaw-slope, $\alpha =-3$ (see Fig. 11).

Over land the typical scale of interpolation from 1960 to 2020 is less than 100 km so any potential bias would remain negligible for modes $l \le 90$ [16]. Note, the growth observed in the power-spectrum from 1960 to 2020 is also detected in a when only examines temperature anomalies over land using a multi-tapered power-spectrum.

The HadSST grid-cells are separated by up to 500 km, which is re-interpolated into a $1^{\circ } \times 1^{\circ }$ lat-long grid [3]. Thus, interpolation lengths are typically of order 250 km (which suggests underestimation of around 20% for degree $l=50$). Thus, sparse sea coverage may bias the power-spectrum on small scales at all times, but neither the magnitude nor the functional shape of the bias matches the growing power from 1960 to 2020.

Fig. 10

Inferred power for smoothed temperature anomalies (linearly interpolated over $0.5^{\circ }$, $1^{\circ }$, $2^{\circ }$ and $5^{\circ }$ latitude and longitude) relative to power of unsmoothed temperature anomalies. The smoothed power is underestimated with most suppression at small angular scales. Larger interpolation lengths result in smoothing structure at larger scales

Fig. 11

Inferred power for smoothed temperature anomalies (linearly interpolated over $2^{\circ } \times 2^{\circ }$ lat-lon) relative to power of unsmoothed temperature anomalies (red line). Observed power in 1960 to 1970 relative to power in 2010 to 2020 (blue error-bars). The angular dependence of any interpolation bias contrast the coherent growth of modes $20 \le l \le 90$

1.6 F: Difference between land and sea

The analysis presented thus far has averaged over the entire globe, thereby neglecting any kinematic or dynamical differences between land and sea. However, if we limit the spatial coverage, we can determine the contribution of different parts of the globe. In general, the procedure of obtaining a multitaper power spectrum estimate is well established. [19, 33] As seen in Fig. 12, the contribution of earth and sea temperatures are not interchangeable with slightly different functional shapes and powerlaw slopes. While the typical temperature fluctuations over land and sea have a similar turnover length, the fluctuation on most length scales has larger amplitudes on land than at sea. For instance, the dipole power, $C_1$, is substantially larger over land than sea—as extensively observed the seasonal difference is larger over land.

We again emphasise that limiting coverage will introduce a bias for degrees close to the spectral bandwidth of the aliased structure [18, 19]. This is a direct consequence of the signal from low degrees leaking into high degrees over the spectral bandwidth of the excluded regions. However, previous studies specifically dividing land and sea have highlighted (1) that any leakage of signal is small between these domains and (2) any bias remains small for large degrees [19]. Thus, we conclude that the spectral differences of land and sea is not merely caused by aliasing the spatial structure. Furthermore, Fig. 12 clearly indicates that the land-surface temperature field remains the dominant contribution to the observed power spectrum. The overall power, the turnover length and the powerlaw decline for the entire globe traces the power spectrum from land.

1.7 G: The flow of temperature fluctuations

To determine whether the observed cascade is downwards (from large spatial scales to smaller) or inverse (from small to large) we compute the Pearson Correlation Coefficient between the power of each mode;

$$\begin{aligned} r_{C_l(t),C_l(t+\Delta t)} = \frac{Cov(C_l(t),C_l(t+\Delta t))}{\sigma _{C_l(t)} \sigma _{C_l(t+\Delta t)}} \end{aligned}$$

(1)

For $\Delta t = 0$, we can determine the correlation between the power of different modes. In Fig. 13 we see that all modes within the cascade ($l \ge 10$) show large internal correlation with $r > 0.9$. The large-scale spatial structures are far more uncorrelated with $r<0.6$.

Fig. 12

Power spectrum from 2000 to 2020 when only calculated from Land or Sea temperatures. Importantly, the temperature fluctuations are not consistent with identical within $5\sigma$. The power spectrum over land is similar to the entire surface (indicated by grey shading) with similar powerlaw slopes, turnover length-scale and overall power on large scales. Sea temperatures display fewer large-scale temperature fluctuations, but with a shallower powerlaw decline than land

Fig. 13

Correlation coefficient between power of modes. Typically modes of similar spatial scales are more correlated while all modes within the cascade show large correlation

For $\Delta t = 1$ Month, we can determine the correlation between the power of different modes in neighbouring months, see Fig. 14 (left). Over time correlations in general decay, as the former temperature fluctuations diffuse and dissipate. Nevertheless, the relative distribution of correlations remain unchanged with for instance the highest correlation remaining within the cascade. However, the correlation is no longer symmetrical; for instance $C_{20}(t + \Delta t)$ is more correlated with the power of higher modes in subsequent months, than the correlation between $C_{20}(t)$ and higher modes of prior months. This implies, that increasing power on large spatial scales is often followed by increased power on smaller spatial scales, but the inverse does not hold. Thus, the direction of the cascade of temperature oscillations is downwards.

In Fig. 14 (right) this contrast is emphasised by taking the difference in correlation between previous and following month. All modes from $10 \le l \le 40$ the flow is from large to smaller spatial scales (i.e. the correlations from large to small are forward in time). Ultimately, this resolution across time yields observational evidence, that temperature oscillations follow a downward cascade from large to small spatial scales. For modes $l < 8$ the inverse relationship seems to hold, smaller spatial scales precede oscillations on larger scales. This tentatively suggests, that temperature oscillations are generally driven around the characteristic transition within the power-spectrum (i.e. $l \approx 8$), which then feeds temperature fluctuations on both smaller and larger spatial scales.

Fig. 14

Correlation coefficient between power of modes in neighbouring months (left) and residual correlation coefficient [i.e. difference in correlation between power in prior and subsequent months] (right). We still see that modes of similar spatial scales are more correlated, while all modes within the cascade show large correlation. However, the correlation between powers across subsequent months is not symmetric. There is more correlation between lower modes in previous months with higher modes in subsequent months then between higher modes in previous months with lower modes in ensuing months

Fig. 15

Left: the turnover length scale as a function of months for the northern and southern hemisphere [averaged over the period 1950–2020]. Right: the turnover length scale from 1950 to 2020 for entire globe. We see the smallest turnover-lengths in summer months, with the larger turnover-length in winter. Furthermore, in recent decades the turnover length has increased

1.8 H: The temporal evolution of turnover length

The emphasis on the coherent temporal evolution of the power within the cascade underlines the growing weather volatility. However, another interesting temporal trend is the changing spatial scale for the transition from flat to cascade. We can determine the intersection between the powerlaw (fitted from modes $10 \le l \le 50$) with the average power of modes in the flat regime ($2 \le l \le 8$), which yields an estimate of the turnover length. As seen in Fig. 15 this definition proves that the characteristic length scale is dynamically evolving. A noteworthy feature is that the turnover length has increased from $2960 \pm 30$ km (for 1950–1960) to $3230 \pm 60$ km (for 2010–2020). The discrepancy in the length scales for these two periods is more than $3\sigma$ significant.

Fig. 16

Left panels show simulated temperature field composed from 250 isotropically distributed Gaussian temperature fluctuations with right panels indicating corresponding power spectrum for any spherical harmonic $Y_l^m$. Top illustrates equal Gaussian variance along longitudes and latitudes, with bottom showing 5 times larger variance along longitudes than latitudes. Evidently anisotropies of the scale of temperature fluctuations cause an asymmetry in power across m

1.9 I: Simulated anisotropic temperature fields

To simulate a global temperature field with a prescribed scale of fluctuations, we randomly generate 250 points sampled isotropically across the sphere. Each point represents the centre of a Gaussian with the variance setting the scale of fluctuations. Importantly, the variance along latitudes and longitudes can be set independently. The centre and spread of the i’th Gaussian is denoted, respectively ($\phi _i$, $\theta _i$) and ($\sigma _\phi , \sigma _\theta$). Thus, the total temperature field is determined as a superposition of all individual fluctuations:

$$\begin{aligned} T(\theta ,\phi ) = \sum _i{\frac{1}{2 \pi \sqrt{\sigma _\phi \sigma _\theta }} \exp { \left( -\frac{(\phi -\phi _i)^2 \sin (\theta )^2}{2 \sigma _{\phi }^2} - \frac{(\theta -\theta _i)^2}{2 \sigma _{\theta }^2}\right) } } \end{aligned}$$

(2)

As seen in Fig. 16 varying the scale of fluctuations between longitudes and latitudes varies the power across m at fixed l.