Paleoclimate proxies, extreme excursions, and persistence in the. climate continuum. Gerard H. Roe and Marcia B. Baker

Transcription

1 Paleoclimate proxies, extreme excursions, and persistence in the climate continuum Gerard H. Roe and Marcia B. Baker Department of Earth and Space Sciences, University of Washington, Seattle, WA. February 22, 212 1

2 1 Abstract In this study we explore the impact of interannual persistence in climate variability on the natural fluctuations of glacier length that occur even without a climate change. We focus on climate persistence whose power spectrum is characterized by a power-law of the form P (f) 1/f ν. Such spectra have been shown to apply for long paleoclimate 6 records, and are consistent with the basic physics of ocean heat uptake. The auto correlation, or memory, in a power-law process decays more slowly with time than exponentially and hence it is also referred to as a long-memory process. This small chance that the climate forcing has the same sign for several years in succession drives length fluctuations that are much large than would occur if there were no memory in the climate forcing. Using a simple glacier model we show that even for ν =.25, a degree of persistence so small that is hard to identify in century-long instrumental records, the variance of glacier length fluctuations is increased by seventy percent over that for memoryless forcing, and that this causes a dramatic reduction in the expected return time of large advances. The basic behavior applies to anything that act as an integrator of climate forcing, and so the results presented here generalize to a variety of other paleoclimate proxies. 2

3 18 1 Introduction Knowledge of Earth s climate prior to the availability of instrumental records depends on inferences made from elements of the Earth System that are influenced by climate, and whose histories can be recovered from the geologic record. Such paleoclimate proxies are affected by both the natural variability that is intrinsic to a constant climate and by the trends and shifts that constitute actual climate change. A major goal and challenge in paleoclimatology is to identify when and where these proxies provide compelling evidence of a change in the forcing or dynamics of the climate system. In turn, when clearly established, such evidence provides a challenge to the climate dynamics community to understand the cause of the changes Expressed in a different way, it is a classic problem in the detection of signal versus noise. This immediately raises the question of how to define noise. In a climate system without a clear separation of timescales, this definition will always have a degree of arbitrariness. And although it is also always going to be true that one man s noise is another man s signal (attributed to Edward Ng, New York Times, 199), one natural definition of noise is that it is the variability that occurs even in a constant climate. In other words, the signal is the climate change that is worth studying. However, this merely begs the question, for it leaves unresolved what is meant by constant. The World Meteorological Organization defines climate as the statistics of the atmosphere averaged over a 3-yr period (WMO, 1989), in which case the noise would be the variability described by those statistics. A slightly different definition is that noise is the natural variability that accompanies a constant underlying generating process. In other words, it is the variability that accompanies a fixed set of parameters in the governing climate equations. As will be described, because of the 3

4 39 4 very long timescales associated with ocean heat uptake, such a definition would imply that even without climate change, the 3-yr running-mean statistics might vary Regardless of the nuances of the definition, a body of recent work has demonstrated that natural climate variability alone can generate large, persistent fluctuations in proxy climate records. Care should be taken not to misinterpret such fluctuations as requiring a climate change. For example, Oerlemans (2) and Reichert et al. (22) find that, for two glaciers in Europe, Little Ice Agescale advances should be expected every few centuries, even without a climate change, but that the observed modern retreats exceed the natural variability. These results are corroborated by Roe and O Neal (29) and Roe (211) who make similar findings for glaciers in the Pacific Northwest of the North America. Relatedly, Huybers and Roe (29) characterize the spatial extent over which glacier fluctuations are coherent in a constant climate, and Huybers et al. (212) demonstrates that lake levels exhibit similarly persistent fluctuations in response to interannual climate variability A central issue is whether a prolonged excursion of a given climate proxy reflects (i) a change in climate, (ii) a persistent climate fluctuation in a constant climate, (iii) the proxy s dynamical response, or (iv) the time averaging that might have occurred in obtaining or processing the record. It is obviously important to distinguish between records that show an unusual event that can only be explained in terms of a change in climate dynamics or climate forcing, and those that would occur in the ordinary scheme of things driven by internal climate variability The purpose of this study is to explore how paleoclimate proxies should be expected to respond to climate persistence. We define the term climate persistence mathematically below; for the present discussion we define it loosely to mean finite autocorrelation of climate variables at long time scales. 4

5 Although the problem is a general one, the particular focus here is on centennial and millennial time-scales, and on glacier-length variations that act as low pass filters of climate variability. These choices are relevant for the interpretation of Holocene records, where excursions of such proxies are typically interpreted in terms of a climate change, and often associated with, for example, climate events such as a Little Ice Age or a Mediaeval Warm Period We find that even for a degree of climate persistence that is so small it is hard to formally establish from century-length instrumental records, the effect of the persistence is to substantially increase the likelihood of large glacier excursions, and to broaden the zone over which moraines unrelated to climate change might be formed on the landscape. We also demonstrate there are some metrics of glacier fluctuations that are sensitive discriminants between the effect of climate persistence and the effect of a climate trend. Although we focus on glaciers as a particular paleoclimate proxy, our analysis applies to any proxy that acts as a filter of climate variability, the implications of which are broached in the Discussion Climate persistence and climate spectra We begin by contrasting two common representations of climatic persistence. The first, an autore- gressive process, is widely used as a model for climate variability. The second, a power-law, or long-memory, process has also been extensively studied, although it is less often applied in mod- 77 ern and paleoclimate studies. The power-law process can represent relatively small amounts of 78 persistence that may be present for long periods. Because of this property, it is the power-law 5

6 process that we use to model the climate variability driving glacier fluctuations. There are also strong physical grounds, as well as observational evidence, that when a wide range of timescales is being considered, the power-law process is a better representation of nature. The remainder of the section presents some idealized representations of power-law processes and then some examples from long-term instrumental records The autoregressive process A straightforward and common way of characterizing persistence in climate data is to represent it as an autoregressive process (e.g., Jenkins and Watts, 1968; vonstorch and Zwiers, 1999). Let the data, y t, be evenly spaced in increments of t. A p th -order autoregressive model ( AR(p)) of these data is y t = a u t + a 1 y t t + a 2 y t 2 t a p y t p t, (1) where u t is a residual of uncorrelated, normally distributed random noise. The data can be modeled as an AR(p) process by finding the set of a i s that minimize the size of the residual term. The optimal order of the model, p, is chosen based on a minimization criteria that penalizes higher order models because of their extra degrees of freedom (e.g., von Storch and Zwiers, 1999) An advantage of eq. (1) is that it directly represents how the data at one time depends on its values at previous times. Moreover an AR(p) model is the discrete form of a p th -order differential equation and, as such, can be cleanly interpreted in terms of the dynamical equations that generated the 6

7 data. Depending on the values of the a i s, a general AR(p) process represents a combination of oscillations and decaying exponentials. The autocorrelation function at lag k t, defined as ρ(k), can be calculated in terms of the a i s: ρ(k) = a 1 ρ(k 1) a p ρ(k p), (2) 99 for k 1, and ρ() = 1 (Jenkins and Watts, 1968) The simplest example, AR(1), represents a first-order differential equation with an exponentially decaying autocorrelation function: ρ(k) = exp( k t/τ), (3) where τ = t/(1 a 1 ). Hassleman (1976) demonstrated that this AR(1) process was an effective representation of midlatitude sea-surface temperature variability for decadal-length records, where the response time, τ, was due to the thermal inertia of the mixed layer. Many other subsequent studies have used it, or closely related processes, to characterize climate variability (e.g., Barsugli and Battisti 1998, Newman et al., 23, Roe and Steig, 24). An assumption of AR(1) is almost always used as the null hypothesis for natural climate variability, against which any possible significant trends or spectral peaks are evaluated. It is the basis, for example, of the trend tests in the IPCC 27 report that declared global warming to be unequivocal (Trenberth et al., 27). 11 A notable disadvantage of modeling climate variability as an AR(p) process in practice is that 7

8 low-order models are preferred on grounds of parsimony and that the a i s tend to be influenced by the autocorrelations at short lags. The effect is that, if there is persistence at long time lags in the data, it may not be captured in the AR(p) model The power-law process An alternative perspective on persistence comes from the power spectrum of the data. The link is through the Wiener-Khinchin theorem, which states that the autocovariance function (i.e., the autocorrelation function multiplied by the variance) is the Fourier transform of the power spectral density (e.g., Jenkins and Watts, 1968). For an AR(p) process with finite p, the power spectrum always asymptotes to a constant value as the frequency tends to zero. However, observations are not consistent with this. In an important study that extended the earlier work of Pelletier (1998), Huybers and Curry (26) compiled a wide variety of instrumental and paleoclimate records to present spectra of surface temperature variability across a wide range of frequencies, spanning hourly observations at the high end and isotope variations from ocean sediment cores at the low end. Among their results was that spectral power increased towards low frequencies across the full range considered (i.e., from 1 1 yrs 1 to 1 5 yrs 1 ) There are also clear physical explanations this behavior. At lower and lower frequencies, more and more of the deep ocean becomes involved in the energy budget, and the effective thermal inertia of the climate system increases. This physical behavior is adequately approximated by an upwelling diffusion model (e.g., Hoffert et al., 198; Hansen et al., 1985; Pelletier, 27; Fraedrich et al., 23; and MacMynowski et al., 211, among others). All demonstrate that one should expect 8

9 spectra whose amplitude increases towards low frequencies. Although this is an oversimplification of the actual ocean dynamics (e.g., Gregory, 2), such models produce realistic vertical profiles of ocean temperature, and are able to emulate the behavior of more complex climate models (e.g., MacMynowksi et al., 211) Eventually at low-enough frequencies, the finite volume of the ocean means that the effective thermal inertia cannot continue to increase without limit. At frequencies below ω min ξ/h 2, where ξ is effective diffusivity and H is ocean depth, we expect the spectrum to flatten (for H = 4 km, ξ 1 4 m 2 s 1, ω min yr 1. However observations nonetheless suggest that the sloped amplitude of the power spectrum continues to even lower frequencies. Huybers and Curry (26) identified a transition in the spectral slope between centennial and millennial frequencies, which they speculated as resulting from power in orbital bands affecting the background climate system The important point for the present study is that such power-law spectra imply the presence of some amount of interannual persistence in climate variability. The goal here is to explore the effect of such persistence on the fluctuations of climate proxies, and in particular of glaciers. Therefore, consider climate variability that is characterized by a power-law spectrum of the form: ( ωmax ) ν S F (ω, ν) = P, (4) ω where ω( 2πf) is the angular frequency; ω max and P are constants; and the exponent ν is the slope of the power spectrum on log-log axes. The subscript F denotes climate forcing. We will 9

10 149 refer to a model process whose spectrum has the form of eq. (4) as a power-law process. 15 From the Wiener-Khinchin theorem, the autocorrelation function (ACF) at time lag is given by ρ F F ( ) = S F (ω, ν)cos( ω)dω. (5) S F (ω, ν)dω As illustrated in the next section, a function of this form declines to zero with increasing lag less rapidly than the exponential decay of an AR(1) process, meaning it can represent greater persistence in the data at long lags. This is the reason that models of the form described by eqs. (4) and (5) can be also deemed long-memory models We note that a power-law process can be emulated by an AR(p) process with an infinite numbers of terms (e.g., Beran, 1994). Also, as well as AR(p), there is a generalized class of models that exist for explaining long-memory behavior in time series, known as autoregressive, fractionally-integrated, moving average (ARFIMA) models (e.g., Beran, 1994). We restrict our attention to power-law and AR(1) models here, as the physical grounds for proposing them are clear, and each involve only two parameters Idealized examples of a power-law process 162 Time series can be generated for a power-law process following the algorithm given in, for example, 163 Percival et al. (21). The time series is reconstructed from the Fourier transform of a power 164 spectrum of frequencies whose amplitudes are governed by eq. (4), and whose phases are chosen at 1

11 random from a normal distribution. We consider values for ν of,.25,.5, and.75, which spans the range found in observations. To aid the comparison, the same set of random numbers were used for the phases for each time series. We note that eq. (4) cannot characterize the full frequency range (i.e., ω ) because the variance of such a process would be infinite. Hence we make the simplifying approximation that P (ω) = for ω > 2π yr 1 (i.e., ω max 2π yr 1 ), corresponding to a maximum frequency, f max = 1 yr 1. Alternative choices have little impact on the shape of the ACF for lags at interannual time scales. For ν = (no persistence) the variance of this process is then σf 2 = (P ω max /2π). The analytical expression for the ACF of such a power spectrum is given in Appendix A Figure 1a shows 2 yr realizations of time series generated by idealized power-law processes for four different ν. The increased variance for higher values of ν is evident by eye. It is also clear that higher ν creates more persistence in the time series excursions away from the mean are more prolonged. These visual impressions are confirmed by the power spectra and the ACF shown in panels (b) and (c). Thus, these idealized climate time series are well suited for exploring how the climatic persistence affects natural fluctuations of glacier length For the curves shown in Figure 1 we have, for simplicity, fixed the value of P in eq. (4) and varied only ν. P is related to the variance of the climate forcing via σ 2 F = (P ω max /2π)/(1 ν) (Equation A-2). In practice, P would typically be estimated only after detrending the instrumental 183 data to remove anthropogenic influences. If there were significant persistence in the data, this detrending would remove some the variance that should be attributed to the persistence. In other words, calculating variance from short time series may alias the apparent value of P. For typical 11

12 year instrumental records and temperature trends, we estimate that this bias is about 2% for ν =.25, and about 2% for ν =.75. Thus it is only a secondary effect for the analyses presented here If ν =.25 in Nature, it is very hard to detect from instrumental data The standard measure for establishing persistence in a time series is that the lag-1 autocorrelation exceeds 2/ N, where N is the number of data points in the data. This rejects the null hypothesis that there is no memory at 2σ, or better than 95% confidence (e.g., Jenkins and Watts, 1968). Therefore, for a typical 1-yr instrumental record, the lag-1 autocorrelation of the data would have to exceed.2 to pass this test (see, e.g., Fig 2c). However for a power-law process governed by ν =.25 the lag-1 autocorrelation is in fact only.17 (Figure 1, and eq. (A-5)), By this test then, it would require N 15 yrs to identify such a level of persistence in nature, if it was in fact present Finally we briefly note that a common practice in paleoproxy studies that focus on decadal, or longer, time scales is to apply some kind of multi-year smoothing filter. This is dangerous if not interpreted correctly. It can create a greatly exaggerated visual impression of persistent fluctuations, and preclude the analysis of any true persistence that is present in the data. 12

13 A few examples from observational data In this section we present a few examples of long instrumental records of observed climate variability, and calculate the best-fit parameters for both AR(1) and power-law representations of the variability. We focus particularly on variables that are relevant for glacier fluctuations. In all cases the datasets are linearly detrended before analyzing. The rationale for doing so, and the possible (small) bias that might be imparted, is discussed in the previous section. 28 The first record we analyze is the standard index of the Pacific Decadal Oscillation (e.g., Mantua 29 et al., 1997), shown in Figure 2a. It is a 111-yr record of the dominant pattern of sea surface temperatures (SSTs) in the North Pacific. Some significant persistence is clear, in that there is more power at lower frequencies than at higher frequencies (Fig. 2a). Fitting an AR(1) process to this data gives a decorrelation time scale, τ in eq. (3), of 1.6 ±.8 yrs (2σ uncertainties used throughout). Fitting a power-law process to the data gives a best slope, ν in eq. (4), of.6 ±.2 (Fig. 2b,c) Our analysis of the PDO closely follows the study of Percival et al. (21) who analyzed a related measure of Pacific variability, the North Pacific index (NPI), which tracks the strength of the wintertime Aleutian Low. That study found values for τ and ν of.7 ±.3 yrs and.3 ±.2, respectively, suggesting the presence of some small amount of persistence, though one would be hesitant to conclude too much from a τ shorter than one year. Percival et al. (21) further concluded that AR(1) and power-law processes were both consistent, and equally good models of the NPI and that, furthermore, one would need several centuries more data in order to be 13

14 able to discriminate between them. Note that greater persistence is indicated in the sea surface temperatures of the PDO index than for an overlying atmospheric variable, the sea level pressure of the NPI index We also analyzed another commonly discussed index of SST variability, the Atlantic Multidecadal Oscillation, or AMO (results not shown). The AMO index is a time series of annual-mean North 227 Atlantic SSTs averaged between and 7N (e.g., Enfield et al., 21). We found values for τ and ν of 1.8 ±.7 yrs and.7 ±.2, respectively. The value for ν in particular suggests quite a high degree of persistence (see for example, Fig. 2c), although such a result should be interpreted cautiously because it is controversial whether the AMO is really a mode of natural variability, or is an artifact of non-monotonic anthropogenic forcing, particularly the fluctuating production of industrial aerosols from North America (e.g., Zhang, 28 vs. Shindell and Faluvegi, 29) The longest instrumental climate record in existence is the Central England Temperature series (e.g., Parker and Horton, 25). We analyze the 352 yrs of summertime (JJAS) near-surface air temperature and Figures 2d,e,f presents the results. We find τ =.6 ±.2 yrs, and ν =.5 ±.1. Thus some persistence in summertime temperatures is indicated. However, the analysis is consistent with the results already cited about Pacific variability, atmospheric variables generally show less persistence than oceanic ones. The longest available precipitation record we are aware of is the monthly England and Wales precipitation series (e.g., Alexander and Jones, 21). For 245 years of annual-mean precipitation measurements we find τ =.3 ±.4 yrs and ν =.1 ±.2 (Figs. 2g,h,i). In this case then, there is no evidence of persistence. 242 Finally, the longest continuous record of glacier mass balance is from Clarinden glacier, Switzerland, 14

15 extending a remarkable 97 years (Bauder and Ryser, 211). Our analysis is shown in Figs. 2j,k,l. For this time series we find τ =.3±.5 yrs and ν =.1±.2. For this one annual-mean mass balance record, then, persistence is not established with confidence. Using autoregression modeling only, Burke and Roe (212) find indications of some significant persistence in summertime temperatures in southern Europe, but only hints of persistence in the several shorter records of glacier mass balance they analyzed. It may be that factors local to the glacier confound any persistence in the overlying climate. It s also the case that it is very hard to establish persistence from short records that must be detrended in an attempt to remove anthropogenic influence. As we show in the next section, even levels of climate persistence that may be unidentifiable in the instrumental record have an importance impact on the magnitude of natural glacier fluctuations In summary, the four time series presented in Figure 2 are a brief tour of the climate persistence that can be established from instrumental records. They are illustrative of the broader results from other studies: significant multi-year persistence that is well characterized by a power-law process can be demonstrated for ocean variables; a diminished echo of that persistence can be identified in atmospheric temperature and pressure; significant interannual persistence in precipitation is not established in the instrumental record. While we ve focussed on instrumental records here, in Appendix B we present an analysis of the spatial pattern of persistence from a 5-yr integration of a coupled climate model, which supports these general results. 15

16 261 3 The response of a glacier to climatic persistence The rest of the study explores what effect climatic persistence has on the expected statistics of glacier excursions. As noted in the introduction the essential results of the analysis extend to any climate proxy that has an approximately linear relationship to the climate it reflects. A simple linear model of a glacier s response to climate variability is dl(t) dt + L(t) τ g = αt (t) + βp (t) = F (t), (6) where L(t) are the anomalous length fluctuations over time (i.e., the departure from the equilibrium value). τ g is the response time, and depends on glacier geometry and mass balance parameters. Climate forcing occurs in the form of fluctuations in melt-season temperature, T, and annual accumulation, P. The coefficients α and β are functions of glacier geometry. The particular form of eq. (6) is derived in Roe and O Neal (29), though there are a whole class of similar models (e.g., Jóhanneson et al., 1989; Raper et al., 2). Our purpose here is to explore the principle of how adding climatic persistence affects the glacier s response, and so using a simple model is appropriate. Roe (211) makes a detailed comparison of eq. (6) to a fully dynamical flowline glacier model Although the atmospheric controls on T and P are separate and, as we ve seen, the νs for each can be different, it is sufficient for our purposes to amalgamate the climate forcing into a single variable F (t). We assume that at each time t F (t) is normally distributed, with variance σf 2. If there is no persistence in the climate fluctuations, then this is equal to the average over the time 16

17 series of F (t)f (t). Calibrating the model to the geometry and climate of the glaciers around Mt. Baker in Washington State, Roe and O Neal (29) took τ = 7 yrs and σ F = 2 m yr 1. When quantitative results are presented, it is these values we use Although we focus on presenting results for glacier fluctuations, eq. (6) is the simplest one-parameter relationship between forcing and response, and thus the essential results of the analysis extend to 284 any climate proxy that has an approximately linear relationship to the climate it reflects. For example lake-level fluctuations can be described by an equation identical to eq. (6) (e.g., Mason et al., 1994, Huybers et al., 212), where the parameters are instead functions of the lake geometry and the atmospheric factors that control precipitation and evaporation The variance of glacier length, σl 2, for a glacier forced by general power-law climate variability can be solved analytically from eq. (6). The derivation is presented in Appendix A, and the solution is a relatively simple expression: σ 2 L = π ( πν ) (ω max τ g ) ν τ g sec σ 2 ω max 2 F. (7) By substituting our standard glacier parameters and a range of ν into eq. (7), we see that there is a striking sensitivity of σ L to ν. For ν =.,.25,.5, and.75, and constant σ F, we calculate σ L = 37 m, 62 m, 1.1 km, and 2.5 km, respectively. Even the relatively small amount of climate persistence implied by ν =.25 leads to a nearly 7% increase in σ L, compared to that for ν =. This is a remarkable increase in variance when it is recalled that the lag-1 autocorrelation coefficient for ν =.25 is only.17. The glacier undergoes dramatically larger fluctuations because of the 17

18 297 small but prolonged tendency for the climate forcing in successive years have the same sign For climate variability governed by ν =.75, there is a more than six-fold increase in σ L. In the limit of ν 1, we see that σ L due to the secant dependence, reflecting that the variance of the climate forcing becomes unbounded An illustration of the effect of power-law climate variability on glacier length is shown in Figure 3, which was created by integrating eq. (6) forward in time, for ν = and ν =.5. Figure 3a clearly shows the increased variance of glacier length for the ν =.5 case. This is consistent with greater power at lower frequencies (Figure 3b). Higher values of ν also lead to higher autocorrelations in the glacier response, which is particularly evident at short lags (Figure 5c). This effect might be important when decadal-scale glacier records are used to estimate the response time (e.g., Oerlemans, 25): care might be needed to separate out the autocorrelations due to the glacier response, and that due to climatic persistence. An exact expression for the ACF in glacier length can be derived from the model equations, and is given in Appendix A Our results can be compared to two other studies. Reicher et al. (22) investigated the impact of climate variability on Nigardsbreen glacier in Norway and Rhonegletscher in the Alps, and modeled local climate persistence as an AR(3) process (i.e., see eq. 1). Compared to a white-noise climate, they found glacier variance was enhanced by about 35% for Nigardsbreen and about 5% for Rhonegletscher (whose climate had very little persistence). For Nigardsbreen, the coefficients of the AR(3) process reflect two exponentially decaying timescales of 1.7 and 1.1 yrs. A second study, Huybers and Roe (29) provide formulae from which we can calculate that for our case of τ g = 7 yrs, and a climate governed by an AR(1) process with a decorrelation time of one year (i.e., 18

19 a lag-1 autocorrelation of e 1 =.38 in eq. 3) the variance in glacier length would be increased by 38%. The much larger increase of variance that we find here even for the ν =.25 case illustrates the impact of the long memory associated with the power-law process Statistics of glacier excursions How likely is it for a given glacier excursion to have occurred in a given period of time, even in the absence of a climate change? We now use extreme-value statistics first developed by Rice (1948) (see also Vanmarcke, 1983), to characterize the likelihoods the glacier advancing past a given point (i.e., an up-crossing ). This extends the analyses of Roe (211) who considered only the case of ν = (i.e., white noise climate forcing). Here, we show the presence of persistence in the climate variability enhances the likelihood of large glacier fluctuations. Rice (1948) showed that the expected rate of an advance past L is given by λ(l ) = 1 σ L e 2π σ L 1 L 2 σ L 2, (8) 329 where σ L is the standard deviation of dl/dt, which from eq. (6) can be written as σ 2 L = σf 2 σ2 L τg 2, (9) assuming L (t) L (t) >=. The average return time of a glacier advance beyond L is equal to λ 1 (L ). Equations (7), (8), and (9) can be combined to derive an expression for the return time 19

20 332 as a function of ν, and the results are shown in Figure 4. The analytically derived curves are compared with a direct numerical determination of return times from 1 6 yr integrations of eq. (6), and the two methods agree well. Where there are differences it is due to the fact that the analytical solution solves the continuous equation, whereas the numerical model solves the discrete version The return time of a given advance is an acutely sensitive function of ν. For example, while an advance past 15 m would occur only every 5, yrs in a climate with ν =, it occurs every 3 yrs for ν =.25, every 6 yrs for ν =.5, and every 3 yrs for ν =.75. Thus the addition of even a small amount of climate persistence can dramatically affect the return time of large advances. As can be seen in Figure 3, the larger the value of ν, the larger the glacier variance, the more time it spends away from equilibrium, and the more often the glacier will cross a given point of advance. This is also the reason why small advances become less frequent with increasing ν A metric that has perhaps has more practical relevance for paleo-glaciological reconstructions is the likelihood of a total glacier excursion occurring in a given period of time. For example, for any given glacier reconstruction the question to be asked is: how likely is it that, just by chance in a statistically constant climate, the glacier would have advanced down-valley as far as that particular moraine, and also retreated as far back up-valley as we now see it? Let L be the total excursion of a glacier (i.e., its maximum extent minus its minimum extent). For the case of ν =, Roe (211) derived the statistics governing the likelihood of finding a given L in a given time by assuming that maximum and minimum excursions could be treated as Poisson processes, which is to say that they are independent events which occur at particular average rate, and that the chance of simultaneous events occurring is vanishingly small (e.g., von Storch and 2

21 Zwiers, 1999). In this study the inclusion of climate persistence means that excursions occurring at different times cannot be treated independently, but must account for the finite autocorrelation of the glacier, even at large lags. In other words, the ability of the glacier to reach a given minimum extent after a given maximum depends on the persistence of the climate variability. In Appendix A we outline an approximate modification of the statistical analysis to include this effect. The excursion statistics can also be derived directly from long simulations of eq. (6) Figure 5 shows the effect of changing the value of ν on the excursions probabilities in a 1 yr period. Larger values of ν cause a strong increase in the likelihood of seeing large excursions, to the extent that there is almost no overlap in the curves for the values of ν we have considered. For example, if one assumed that climate variability was governed by ν =, one would conclude from Figure 5 that a total excursion of 3 km was virtually impossible in a 1 yr period, and that an observation of such an excursion would be proof that a climate change must have occurred. However if the natural variability was, in fact, governed by ν =.25, such an excursion would be virtually certain to occur in a constant climate. This highlights the importance of knowing the underlying climate persistence. The challenge this creates is that the difference between ν = and ν =.25 is practically impossible to distinguish from even century-length instrumental records (e.g., Section 2.3.1; Percival et al., 21) Can we distinguish between climate trends and climate persistence? We live on a planet where most regions are warming because of anthropogenic emissions (e.g., IPCC, 27). We also live in a period where instrumental records have not been kept long enough to clearly 21

22 discern the degree of climate persistence present in natural variability. Since both climate trends and climate persistence affect glacier (and other proxy) behavior, what is their relative importance, and are there aspects of a proxy record that would potentially allow us to discriminate between the two? Standing at the modern glacier front, the presence of either a warming trend or climate persistence (ν =.25, say) would increase the chance of the glacier front having, in the past, extended further down the valley than would be the case for ν = and no climate trend. This is illustrated in Figure 6a, which shows the probability density functions (PDFs) of the glacier terminus position for the cases of ν = and ν =.25 with no climate trend, and for ν = plus an added warming trend of 1 o C century 1, which is typical for the midlatitudes. The PDFs were generated from a normal distribution using eq. (7) for the no-trend cases, and from 1, realizations of 1 yrlong simulations with normally-distributed climate forcing, for warming-trend case. The PDFs are centered around the long-term mean for the no-trend cases, and for the mode of the distribution at the end of the 1 yrs for the warming-trend case, the latter being the most likely position for the glacier front in the present day (i.e., after 1 years of warming has elapsed) As expected Figure 6a shows that, compared to the ν = case, both ν =.25 and the warming trend increase the likelihood of the glacier having being 1 to 2 km down valley in the past century. However the PDF for ν =.25 is quite broad, whereas the warming-trend PDF remains narrow but is translated down-slope. In other words, for the warming-trend case, it is likely that the glacier will have spent most of its time slightly down valley of its present position in the past century. 393 A second big difference is that the statistics for the ν =.25 case are stationary whereas those of 22

23 the warming-trend case are not. For this reason, there is no simple analytic treatment adequate for this case and we rely exclusively on the numerical simulations. The effect of a trend is particularly pronounced for the expected return time of an advance past a given position. Figure 4 showed that the effect of having ν > is to dramatically reduce the return time of large excursions. However, in the warming-trend case, the mean position of the glacier front is retreating, the chances of having a large down-valley excursion are decreasing extremely rapidly with time. This difference is shown in Figure 6b, and can be understood from eq. (8): L is the position of a point on the landscape relative to the average position of the glacier front, and so from the time the warming trend begins, the exponent is changing as the square of time Although the return time is potentially a very sensitive metric of the difference between a climate trend and climate persistence, in practice it would require detailed histories of terminus position for a population of glaciers that can be regarded as independently forced. However, as detailed by Huybers and Roe (29), glaciers within a region experience essentially the same climate and 47 so are not independent. More importantly, since the modern warming trend is already clearly established from thermometers, the more useful challenge is the inference of past climate changes from paleoreconstructions of glacier extent. And in many instances, the predominant evidence is moraines that mark the furthest extent of glacier fluctuations that have not been subsequently overridden. Neglecting all the complications of how moraines get created, these glacier maxima can treated as potential moraine locations and can be diagnosed from model simulations using eq. (6). Figure 7 presents the statistics of such moraine locations for 1,-member Monte Carlo simulations of the same three cases presented in Figure 6. For consistency with Figure 6, we consider the same 1-yr climate trend. Figure 7a shows, for instance, that for the ν =, no-trend case it 23

24 is most likely to find a moraine about 5 m down valley from the modern position, and Figure 7b shows that the expected age of that moraine is about 4 years. It becomes less-and-less likely that moraines would be found further down-valley (within the 1 yr interval we have allowed) Both the ν =.25 case and the warming-trend case increase the likelihood of finding moraines further down-valley from the present day position (Fig. 7a), and the two PDFs lie nearly on top of each other. However a significant difference between these two cases is the average age of those moraines (Fig. 7b), particularly for those that are found more than 1 km down valley. For the ν =.25 case the expected age of the moraine stays around 6 yrs, whereas for the warming-trend case the expected age is older - between 7 and 9 yrs. In other words, when a climate trend is present, down-valley moraines will be older than would be expected if they were due to climate persistence Obviously, there are many caveats to the above analyses, and they are presented tentatively. Firstly, the processes and timescales for moraine formation remain poorly characterized (e.g., Mathews et al., 1995; Winkler and Nesje, 1999; Winkler and Mathews, 21), and obviously are much more complex than simply reflecting glacier maxima. Secondly, the linear glacier model produces too many high-frequency terminus fluctuations compared to a model that represents the physics of glacier flow (e.g., Roe, 211). These preliminary analyses should be redone with such a flow-line model, and also target a specific paleoclimate question such as the statistics of moraine formation during the late Holocene (e.g., Schaefer et al., 29). The challenge will be to tease apart the relative importance of climate trends and climate persistence, and to establish whether the glacier modeling can be accurate enough, and the local climate variability known well enough, to distinguish between 24

25 437 the two Summary and discussion A wide variety of instrumental data, paleoclimate proxy data, and climate model output all suggest that Earth s climate variability is characterized by a continuum background spectrum. The few significant spectral peaks that do exist are generally associated with external forcing that has to do with the planet s orbit (i.e., diurnal, seasonal, annual, and orbital forcing), with El-Nino being perhaps the singular case where it is clearly established that a significant spectral peak arises from internal dynamics (though the presence of other peaks is perpetually speculated, e.g., Chylek, 211). In general, this continuum spectrum has increasing power towards lower frequencies, although the slope of the spectrum depends on the climate variable and the location in question. The observational evidence is strongly supported by basic thermodynamics and climate physics that predicts that a quasi-diffusive ocean heat uptake exchanging heat with the atmosphere should 449 produce such surface-temperature spectra. We analyzed the persistence present in several long instrumental records of climate. The main purpose of this study was to explore what the effect of such persistence is on the natural variability of paleoproxy climate records. Even with no climate persistence, paleoclimate proxies such as glaciers and lakes have a dynamical response time, and will produce large fluctuations on long timescales because they act as low pass filters of climate variability. 455 The effect of adding even a small amount of climatic persistence is to greatly increase the variance 25

26 of the proxy fluctuations. Even a small chance that the climate forcing will have the same sign for several successive years makes a big difference. For a power spectrum variability with a slope of.25 a value so low that it is hard to detect even in century-length instrumental records of 459 climate change glacier variance increases by almost 7%. This increased variance produces a 46 spectacular reduction in the expected return time of large advances or retreats. For instance, for our glacier parameters, we demonstrated that by adding this small amount of persistence the average return time of a 15 m advance plummets from once every 5, years to once every 3 years. The statistics of total glacier excursions (accounting for both advances and retreats) is similarly impacted. Typical results for this are summarized in Figure Generalization of this Analysis For the linear model, the variance, autocorrelation, return time, and excursion statistics can all be expressed as analytical functions of the parameters governing the model and the climate. This is important because the sensitivity to changes in these parameters can be clearly understood, and rapidly calculated. The effect of climatic persistence depends on the geographic location of the proxy, and on what climate variables it is sensitive to. Two big simplifications were made in assuming 1) a single slope to the power-law spectra, and 2) a linear model for the climate-proxy 472 relationship. These assumptions can be relaxed to allow for a general function for the climate spectrum, and allow for the paleoclimate proxy to act as a more complicated (but still linear) filter of the climate forcing. In the Appendix, section A-5 presents these generalized expressions. 475 Eq. (6) is the simplest one-parameter relationship between forcing and response, and thus the 26

27 essential results of the analysis extend to any climate proxy that has an approximately linear relationship to the climate it reflects. For example lake-level fluctuations can be described by an equation identical to eq. (6) (e.g., Mason et al., 1994, Huybers et al., 212), where the parameters are instead functions of the lake geometry and the atmospheric factors that control precipitation and evaporation. Further examples of proxies acting as filters of climate variability include tree ring growth, bioturbation in sediment cores, isotope diffusion in ice cores, and carbonate formation 482 in paleosols or speleothems. The functional relationship between these proxies and the climate they experience is perhaps more complicated than for glacier extents and lake levels. The analyses could also be repeated with some nonlinearities included, such as diffusive ice flow in the case of glaciers, though we are confident that the basic relationships between climate spectra, climate persistence, and the effect on the variance of the paleoclimate proxy are robust. It is essential to understand that relationship well before being able to determine what fraction of the persistence in the proxy record reflects climate persistence and what fraction simply reflects the proxy s behavior as a low-pass filter of climate information Holocene climate variability How much of late Holocene climate variability can be explained by the natural climate variability that occurs in a constant climate with a quasi-diffusive ocean heat uptake? Is there, in fact, a need to invoke any external climate forcing such as solar variability and volcanic eruptions to explain, for instance, the putative Little Ice Age (LIA) and Mediaeval Warm Period (MWP)? A null hypothesis of natural variability that has a power-law spectrum is appealing because it is 27

28 supported by long-term instrumental and proxy data, is grounded in basic physics, and invokes the fewest factors. For climate variability over the last millennium, the issue appears finely balanced. Combining multiple high-resolution paleoclimate records (mainly trees), Osborn and Briffa (26) find indications of widespread warm and cool episodes during the intervals commonly associated with the LIA and MWP. Using a nearly identical dataset, Tingley and Huybers (211) identify a MWP that was both significantly warmer and significantly more variable. A useful extension of these studies would be to characterize the climatic persistence present in the data, and to address whether any filtering of the climate signal occurred because of the way the proxies record climate. Using a simple climate model of the last millennium, Crowley (2) finds that volcanic and solar forcing can explain approximately half of the smoothed pre-anthropogenic global-mean temperature variations, though there are wide bounds due to uncertainty in the assumed climate forcing For Holocene variability beyond the last millennium, the scarcity of high-resolution datasets, and uncertainties in the dating (and cross-dating) of proxies such as glaciers and lake levels, may preclude definitively answering the question of whether a climate change beyond that driven by 51 the slow progression of the orbital cycles is required to explain their fluctuations. However it is also certain that the fundamental nature of climate variability has not changed: year-to-year climate variability and, depending on the location and climate variable, some degree of interannual persistence, have been combining to drive fluctuations in climate proxies throughout the entire interval. The task, clearly, is to evaluate whether the proxy records show fluctuations which are larger, more frequent, or more spatially extensive, than would be predicted to occur from natural variability alone. 28

29 517 Acknowledgements 29

30 518 Appendix A: Analytical solutions for glacier statistics 519 A1 Autocorrelation function for climate forcing 52 The power-law climate spectrum is described by ( ωmax ) ν S F (ω, ν) = P, (A-1) ω 521 for < ω ω max and P (ω) = for ω > ω max. We define P 2πσ2 F ω max (A-2) As we now show, σ 2 F is equal to the average of F (t) 2 if ν = but it is smaller than that average if ν >. 524 The Wiener-Khinchin Theorem tells us that F (t) 2 (ν) = ωmax S F (ω, ν)dω = σ2 F (1 ν). (A-3) This is the variance that would be deduced from a time series of instrumental climate observations. We have defined σ 2 F (ν) < F (t) 2 > (ν), leading to the relationship σf 2 (ν) = σ2 F (ν = ) (1 ν). (A-4) 3

31 527 The ACF of the climate forcing is F (t)f (t ± ) (ν) σ 2 F (ν) ρ F F (, ν) = cos(ω )S F (ω, ν)dω S F (ω, ν)dω = 1 F 2 [{ 1 2 ν 2 }, { 1 2, 3 2 ν 2 ; (A-5) }, 1 4 (ω max ) 2 ]. (A-6) 528 where 1 F 2 (a, b, c) is a generalized hypergeometric function. 529 A2 Standard deviation of glacier response 53 We begin by forming the Fourier transforms of climate forcing F (t) and glacier length L(t): ˆF (ω) = ˆL(ω) = exp(iωt)f (t)dt exp(iωt)l(t)dt, (A-7) 531 where the prime notation have been dropped. The spectrum of forcing is S F (ω) ˆF (ω) Substituting these into the glacier model equation (6) gives ) (iω + 1τg ˆL(ω) = ˆF (ω). (A-8) 533 so the spectrum of glacier length variations is S L (ω) ˆL(ω) 2 = S F (ω)r L (ω) (A-9) 31