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, 2012

1

Abstract1

In this study we explore the impact of interannual persistence in climate variability on2

the natural fluctuations of glacier length that occur even without a climate change. We3

focus on climate persistence whose power spectrum is characterized by a power-law4

of the form P (f) ∼ 1/fν. Such spectra have been shown to apply for long paleoclimate5

records, and are consistent with the basic physics of ocean heat uptake. The auto-6

correlation, or memory, in a power-law process decays more slowly with time than7

exponentially and hence it is also referred to as a long-memory process. This small8

chance that the climate forcing has the same sign for several years in succession drives9

length fluctuations that are much large than would occur if there were no memory10

in the climate forcing. Using a simple glacier model we show that even for ν = 0.25,11

a degree of persistence so small that is hard to identify in century-long instrumental12

records, the variance of glacier length fluctuations is increased by seventy percent over13

that for memoryless forcing, and that this causes a dramatic reduction in the expected14

return time of large advances. The basic behavior applies to anything that act as an15

integrator of climate forcing, and so the results presented here generalize to a variety16

of other paleoclimate proxies.17

2

1 Introduction18

Knowledge of Earth’s climate prior to the availability of instrumental records depends on inferences19

made from elements of the Earth System that are influenced by climate, and whose histories can20

be recovered from the geologic record. Such paleoclimate proxies are affected by both the natural21

variability that is intrinsic to a constant climate and by the trends and shifts that constitute actual22

climate change. A major goal and challenge in paleoclimatology is to identify when and where these23

proxies provide compelling evidence of a change in the forcing or dynamics of the climate system.24

In turn, when clearly established, such evidence provides a challenge to the climate dynamics25

community to understand the cause of the changes.26

Expressed in a different way, it is a classic problem in the detection of signal versus noise. This27

immediately raises the question of how to define noise. In a climate system without a clear separa-28

tion of timescales, this definition will always have a degree of arbitrariness. And although it is also29

always going to be true that “one man’s noise is another man’s signal” (attributed to Edward Ng,30

New York Times, 1990), one natural definition of ‘noise’ is that it is the variability that occurs even31

in a constant climate. In other words, the ‘signal’ is the climate change that is worth studying.32

However, this merely begs the question, for it leaves unresolved what is meant by constant. The33

World Meteorological Organization defines climate as the statistics of the atmosphere averaged over34

a 30-yr period (WMO, 1989), in which case the noise would be the variability described by those35

statistics. A slightly different definition is that noise is the natural variability that accompanies36

a constant underlying generating process. In other words, it is the variability that accompanies a37

fixed set of parameters in the governing climate equations. As will be described, because of the38

3

very long timescales associated with ocean heat uptake, such a definition would imply that even39

without climate change, the 30-yr running-mean statistics might vary.40

Regardless of the nuances of the definition, a body of recent work has demonstrated that natural41

climate variability alone can generate large, persistent fluctuations in proxy climate records. Care42

should be taken not to misinterpret such fluctuations as requiring a climate change. For example,43

Oerlemans (2000) and Reichert et al. (2002) find that, for two glaciers in Europe, Little Ice Age-44

scale advances should be expected every few centuries, even without a climate change, but that the45

observed modern retreats exceed the natural variability. These results are corroborated by Roe and46

O’Neal (2009) and Roe (2011) who make similar findings for glaciers in the Pacific Northwest of47

the North America. Relatedly, Huybers and Roe (2009) characterize the spatial extent over which48

glacier fluctuations are coherent in a constant climate, and Huybers et al. (2012) demonstrates that49

lake levels exhibit similarly persistent fluctuations in response to interannual climate variability.50

A central issue is whether a prolonged excursion of a given climate proxy reflects (i) a change51

in climate, (ii) a persistent climate fluctuation in a constant climate, (iii) the proxy’s dynamical52

response, or (iv) the time averaging that might have occurred in obtaining or processing the record.53

It is obviously important to distinguish between records that show an unusual event that can only54

be explained in terms of a change in climate dynamics or climate forcing, and those that would55

occur in the ordinary scheme of things driven by internal climate variability.56

The purpose of this study is to explore how paleoclimate proxies should be expected to respond to57

climate persistence. We define the term ‘climate persistence’ mathematically below; for the present58

discussion we define it loosely to mean finite autocorrelation of climate variables at long time scales.59

4

Although the problem is a general one, the particular focus here is on centennial and millennial60

time-scales, and on glacier-length variations that act as low pass filters of climate variability. These61

choices are relevant for the interpretation of Holocene records, where excursions of such proxies are62

typically interpreted in terms of a climate change, and often associated with, for example, climate63

events such as a ‘Little Ice Age’ or a ‘Mediaeval Warm Period’.64

We find that even for a degree of climate persistence that is so small it is hard to formally establish65

from century-length instrumental records, the effect of the persistence is to substantially increase66

the likelihood of large glacier excursions, and to broaden the zone over which moraines unrelated67

to climate change might be formed on the landscape. We also demonstrate there are some metrics68

of glacier fluctuations that are sensitive discriminants between the effect of climate persistence and69

the effect of a climate trend. Although we focus on glaciers as a particular paleoclimate proxy, our70

analysis applies to any proxy that acts as a filter of climate variability, the implications of which71

are broached in the Discussion.72

2 Climate persistence and climate spectra73

We begin by contrasting two common representations of climatic persistence. The first, an autore-74

gressive process, is widely used as a model for climate variability. The second, a power-law, or75

long-memory, process has also been extensively studied, although it is less often applied in mod-76

ern and paleoclimate studies. The power-law process can represent relatively small amounts of77

persistence that may be present for long periods. Because of this property, it is the power-law78

5

process that we use to model the climate variability driving glacier fluctuations. There are also79

strong physical grounds, as well as observational evidence, that when a wide range of timescales80

is being considered, the power-law process is a better representation of nature. The remainder of81

the section presents some idealized representations of power-law processes and then some examples82

from long-term instrumental records.83

2.1 The autoregressive process84

A straightforward and common way of characterizing persistence in climate data is to represent85

it as an autoregressive process (e.g., Jenkins and Watts, 1968; vonStorch and Zwiers, 1999). Let86

the data, yt, be evenly spaced in increments of ∆t. A pth-order autoregressive model (≡ AR(p)) of87

these data is88

yt = a0ut + a1yt−∆t + a2yt−2∆t + ...+ apyt−p∆t, (1)

where ut is a residual of uncorrelated, normally distributed random noise. The data can be modeled89

as an AR(p) process by finding the set of ais that minimize the size of the residual term. The optimal90

order of the model, p, is chosen based on a minimization criteria that penalizes higher order models91

because of their extra degrees of freedom (e.g., von Storch and Zwiers, 1999).92

An advantage of eq. (1) is that it directly represents how the data at one time depends on its values93

at previous times. Moreover an AR(p) model is the discrete form of a pth-order differential equation94

and, as such, can be cleanly interpreted in terms of the dynamical equations that generated the95

6

data. Depending on the values of the ais, a general AR(p) process represents a combination of96

oscillations and decaying exponentials. The autocorrelation function at lag k∆t, defined as ρ(k),97

can be calculated in terms of the ais:98

ρ(k) = a1ρ(k − 1) + ...+ apρ(k − p), (2)

for k ≥ 1, and ρ(0) = 1 (Jenkins and Watts, 1968).99

The simplest example, AR(1), represents a first-order differential equation with an exponentially100

decaying autocorrelation function:101

ρ(k) = exp(−k∆t/τ), (3)

where τ = ∆t/(1 − a1). Hassleman (1976) demonstrated that this AR(1) process was an effective102

representation of midlatitude sea-surface temperature variability for decadal-length records, where103

the response time, τ , was due to the thermal inertia of the mixed layer. Many other subsequent104

studies have used it, or closely related processes, to characterize climate variability (e.g., Barsugli105

and Battisti 1998, Newman et al., 2003, Roe and Steig, 2004). An assumption of AR(1) is almost106

always used as the null hypothesis for natural climate variability, against which any possible signif-107

icant trends or spectral peaks are evaluated. It is the basis, for example, of the trend tests in the108

IPCC 2007 report that declared global warming to be ‘unequivocal’ (Trenberth et al., 2007).109

A notable disadvantage of modeling climate variability as an AR(p) process in practice is that110

7

low-order models are preferred on grounds of parsimony and that the ais tend to be influenced by111

the autocorrelations at short lags. The effect is that, if there is persistence at long time lags in the112

data, it may not be captured in the AR(p) model.113

2.2 The power-law process114

An alternative perspective on persistence comes from the power spectrum of the data. The link115

is through the Wiener-Khinchin theorem, which states that the autocovariance function (i.e., the116

autocorrelation function multiplied by the variance) is the Fourier transform of the power spectral117

density (e.g., Jenkins and Watts, 1968). For an AR(p) process with finite p, the power spectrum118

always asymptotes to a constant value as the frequency tends to zero. However, observations are119

not consistent with this. In an important study that extended the earlier work of Pelletier (1998),120

Huybers and Curry (2006) compiled a wide variety of instrumental and paleoclimate records to121

present spectra of surface temperature variability across a wide range of frequencies, spanning122

hourly observations at the high end and isotope variations from ocean sediment cores at the low123

end. Among their results was that spectral power increased towards low frequencies across the full124

range considered (i.e., from 101 yrs−1 to 10−5 yrs−1).125

There are also clear physical explanations this behavior. At lower and lower frequencies, more and126

more of the deep ocean becomes involved in the energy budget, and the effective thermal inertia of127

the climate system increases. This physical behavior is adequately approximated by an upwelling128

diffusion model (e.g., Hoffert et al., 1980; Hansen et al., 1985; Pelletier, 2007; Fraedrich et al.,129

2003; and MacMynowski et al., 2011, among others). All demonstrate that one should expect130

8

spectra whose amplitude increases towards low frequencies. Although this is an oversimplification131

of the actual ocean dynamics (e.g., Gregory, 2000), such models produce realistic vertical profiles132

of ocean temperature, and are able to emulate the behavior of more complex climate models (e.g.,133

MacMynowksi et al., 2011).134

Eventually at low-enough frequencies, the finite volume of the ocean means that the effective135

thermal inertia cannot continue to increase without limit. At frequencies below ωmin ≈ ξ/H2,136

where ξ is effective diffusivity and H is ocean depth, we expect the spectrum to flatten (for H =137

4 km, ξ ≈ 10−4m2s−1, ωmin ≈ 0.2× 10−3 yr−1. However observations nonetheless suggest that the138

sloped amplitude of the power spectrum continues to even lower frequencies. Huybers and Curry139

(2006) identified a transition in the spectral slope between centennial and millennial frequencies,140

which they speculated as resulting from power in orbital bands affecting the background climate141

system.142

The important point for the present study is that such power-law spectra imply the presence of143

some amount of interannual persistence in climate variability. The goal here is to explore the effect144

of such persistence on the fluctuations of climate proxies, and in particular of glaciers. Therefore,145

consider climate variability that is characterized by a power-law spectrum of the form:146

SF (ω, ν) = P0

(ωmaxω

)ν, (4)

where ω(≡ 2πf) is the angular frequency; ωmax and P0 are constants; and the exponent ν is the147

slope of the power spectrum on log-log axes. The subscript F denotes“climate forcing”. We will148

9

refer to a model process whose spectrum has the form of eq. (4) as a ‘power-law’ process.149

From the Wiener-Khinchin theorem, the autocorrelation function (ACF) at time lag ∆ is given by150

ρFF (∆) =

∫∞0 SF (ω, ν)cos(∆ω)dω∫∞

0 SF (ω, ν)dω. (5)

As illustrated in the next section, a function of this form declines to zero with increasing lag less151

rapidly than the exponential decay of an AR(1) process, meaning it can represent greater persistence152

in the data at long lags. This is the reason that models of the form described by eqs. (4) and (5)153

can be also deemed ‘long-memory’ models.154

We note that a power-law process can be emulated by an AR(p) process with an infinite numbers of155

terms (e.g., Beran, 1994). Also, as well as AR(p), there is a generalized class of models that exist for156

explaining long-memory behavior in time series, known as autoregressive, fractionally-integrated,157

moving average (ARFIMA) models (e.g., Beran, 1994). We restrict our attention to power-law and158

AR(1) models here, as the physical grounds for proposing them are clear, and each involve only159

two parameters.160

2.3 Idealized examples of a power-law process161

Time series can be generated for a power-law process following the algorithm given in, for example,162

Percival et al. (2001). The time series is reconstructed from the Fourier transform of a power163

spectrum of frequencies whose amplitudes are governed by eq. (4), and whose phases are chosen at164

10

random from a normal distribution. We consider values for ν of 0, 0.25, 0.5, and 0.75, which spans165

the range found in observations. To aid the comparison, the same set of random numbers were166

used for the phases for each time series. We note that eq. (4) cannot characterize the full frequency167

range (i.e., ω → ∞) because the variance of such a process would be infinite. Hence we make the168

simplifying approximation that P (ω) = 0 for ω > 2π yr−1 (i.e., ωmax ≡ 2π yr−1), corresponding to169

a maximum frequency, fmax = 1 yr−1. Alternative choices have little impact on the shape of the170

ACF for lags at interannual time scales. For ν = 0 (no persistence) the variance of this process is171

then σ2F = (P0ωmax/2π). The analytical expression for the ACF of such a power spectrum is given172

in Appendix A.173

Figure 1a shows 200 yr realizations of time series generated by idealized power-law processes for174

four different ν. The increased variance for higher values of ν is evident by eye. It is also clear175

that higher ν creates more persistence in the time series – excursions away from the mean are more176

prolonged. These visual impressions are confirmed by the power spectra and the ACF shown in177

panels (b) and (c). Thus, these idealized climate time series are well suited for exploring how the178

climatic persistence affects natural fluctuations of glacier length.179

For the curves shown in Figure 1 we have, for simplicity, fixed the value of P0 in eq. (4) and180

varied only ν. P0 is related to the variance of the climate forcing via σ2F = (P0ωmax/2π)/(1 − ν)181

(Equation A-2). In practice, P0 would typically be estimated only after detrending the instrumental182

data to remove anthropogenic influences. If there were significant persistence in the data, this183

detrending would remove some the variance that should be attributed to the persistence. In other184

words, calculating variance from short time series may alias the apparent value of P0. For typical185

11

100 year instrumental records and temperature trends, we estimate that this bias is about 2% for186

ν = 0.25, and about 20% for ν = 0.75. Thus it is only a secondary effect for the analyses presented187

here.188

2.3.1 If ν = 0.25 in Nature, it is very hard to detect from instrumental data189

The standard measure for establishing persistence in a time series is that the lag-1 autocorrelation190

exceeds 2/√N , where N is the number of data points in the data. This rejects the null hypothesis191

— that there is no memory — at 2σ, or better than 95% confidence (e.g., Jenkins and Watts,192

1968). Therefore, for a typical 100-yr instrumental record, the lag-1 autocorrelation of the data193

would have to exceed 0.2 to pass this test (see, e.g., Fig 2c). However for a power-law process194

governed by ν = 0.25 the lag-1 autocorrelation is in fact only 0.17 (Figure 1, and eq. (A-5)), By195

this test then, it would require N ≈ 150 yrs to identify such a level of persistence in nature, if it196

was in fact present.197

Finally we briefly note that a common practice in paleoproxy studies that focus on decadal, or198

longer, time scales is to apply some kind of multi-year smoothing filter. This is dangerous if not199

interpreted correctly. It can create a greatly exaggerated visual impression of persistent fluctuations,200

and preclude the analysis of any true persistence that is present in the data.201

12

2.4 A few examples from observational data202

In this section we present a few examples of long instrumental records of observed climate vari-203

ability, and calculate the best-fit parameters for both AR(1) and power-law representations of the204

variability. We focus particularly on variables that are relevant for glacier fluctuations. In all cases205

the datasets are linearly detrended before analyzing. The rationale for doing so, and the possible206

(small) bias that might be imparted, is discussed in the previous section.207

The first record we analyze is the standard index of the Pacific Decadal Oscillation (e.g., Mantua208

et al., 1997), shown in Figure 2a. It is a 111-yr record of the dominant pattern of sea surface209

temperatures (SSTs) in the North Pacific. Some significant persistence is clear, in that there is210

more power at lower frequencies than at higher frequencies (Fig. 2a). Fitting an AR(1) process211

to this data gives a decorrelation time scale, τ in eq. (3), of 1.6 ± 0.8 yrs (2σ uncertainties used212

throughout). Fitting a power-law process to the data gives a best slope, ν in eq. (4), of 0.6 ± 0.2213

(Fig. 2b,c).214

Our analysis of the PDO closely follows the study of Percival et al. (2001) who analyzed a related215

measure of Pacific variability, the North Pacific index (NPI), which tracks the strength of the216

wintertime Aleutian Low. That study found values for τ and ν of 0.7 ± 0.3 yrs and 0.3 ± 0.2,217

respectively, suggesting the presence of some small amount of persistence, though one would be218

hesitant to conclude too much from a τ shorter than one year. Percival et al. (2001) further219

concluded that AR(1) and power-law processes were both consistent, and equally good models220

of the NPI and that, furthermore, one would need several centuries more data in order to be221

13

able to discriminate between them. Note that greater persistence is indicated in the sea surface222

temperatures of the PDO index than for an overlying atmospheric variable, the sea level pressure223

of the NPI index.224

We also analyzed another commonly discussed index of SST variability, the Atlantic Multidecadal225

Oscillation, or AMO (results not shown). The AMO index is a time series of annual-mean North226

Atlantic SSTs averaged between 0 and 70N (e.g., Enfield et al., 2001). We found values for τ227

and ν of 1.8 ± 0.7 yrs and 0.7 ± 0.2, respectively. The value for ν in particular suggests quite a228

high degree of persistence (see for example, Fig. 2c), although such a result should be interpreted229

cautiously because it is controversial whether the AMO is really a mode of natural variability, or230

is an artifact of non-monotonic anthropogenic forcing, particularly the fluctuating production of231

industrial aerosols from North America (e.g., Zhang, 2008 vs. Shindell and Faluvegi, 2009).232

The longest instrumental climate record in existence is the Central England Temperature series233

(e.g., Parker and Horton, 2005). We analyze the 352 yrs of summertime (JJAS) near-surface air234

temperature and Figures 2d,e,f presents the results. We find τ = 0.6± 0.2 yrs, and ν = 0.5± 0.1.235

Thus some persistence in summertime temperatures is indicated. However, the analysis is consistent236

with the results already cited about Pacific variability, atmospheric variables generally show less237

persistence than oceanic ones. The longest available precipitation record we are aware of is the238

monthly England and Wales precipitation series (e.g., Alexander and Jones, 2001). For 245 years of239

annual-mean precipitation measurements we find τ = 0.3± 0.4 yrs and ν = 0.1± 0.2 (Figs. 2g,h,i).240

In this case then, there is no evidence of persistence.241

Finally, the longest continuous record of glacier mass balance is from Clarinden glacier, Switzerland,242

14

extending a remarkable 97 years (Bauder and Ryser, 2011). Our analysis is shown in Figs. 2j,k,l. For243

this time series we find τ = 0.3±0.5 yrs and ν = −0.1±0.2. For this one annual-mean mass balance244

record, then, persistence is not established with confidence. Using autoregression modeling only,245

Burke and Roe (2012) find indications of some significant persistence in summertime temperatures246

in southern Europe, but only hints of persistence in the several shorter records of glacier mass247

balance they analyzed. It may be that factors local to the glacier confound any persistence in the248

overlying climate. It’s also the case that it is very hard to establish persistence from short records249

that must be detrended in an attempt to remove anthropogenic influence. As we show in the next250

section, even levels of climate persistence that may be unidentifiable in the instrumental record251

have an importance impact on the magnitude of natural glacier fluctuations.252

In summary, the four time series presented in Figure 2 are a brief tour of the climate persistence253

that can be established from instrumental records. They are illustrative of the broader results from254

other studies: significant multi-year persistence that is well characterized by a power-law process255

can be demonstrated for ocean variables; a diminished echo of that persistence can be identified256

in atmospheric temperature and pressure; significant interannual persistence in precipitation is257

not established in the instrumental record. While we’ve focussed on instrumental records here, in258

Appendix B we present an analysis of the spatial pattern of persistence from a 500-yr integration259

of a coupled climate model, which supports these general results.260

15

3 The response of a glacier to climatic persistence261

The rest of the study explores what effect climatic persistence has on the expected statistics of262

glacier excursions. As noted in the introduction the essential results of the analysis extend to any263

climate proxy that has an approximately linear relationship to the climate it reflects. A simple264

linear model of a glacier’s response to climate variability is265

dL(t)dt

+L(t)τg

= αT ′(t) + βP ′(t) = F (t), (6)

where L(t) are the anomalous length fluctuations over time (i.e., the departure from the equilibrium266

value). τg is the response time, and depends on glacier geometry and mass balance parameters.267

Climate forcing occurs in the form of fluctuations in melt-season temperature, T ′, and annual268

accumulation, P ′. The coefficients α and β are functions of glacier geometry. The particular form269

of eq. (6) is derived in Roe and O’Neal (2009), though there are a whole class of similar models270

(e.g., Johanneson et al., 1989; Raper et al., 2000). Our purpose here is to explore the principle271

of how adding climatic persistence affects the glacier’s response, and so using a simple model is272

appropriate. Roe (2011) makes a detailed comparison of eq. (6) to a fully dynamical flowline glacier273

model.274

Although the atmospheric controls on T ′ and P ′ are separate and, as we’ve seen, the νs for each275

can be different, it is sufficient for our purposes to amalgamate the climate forcing into a single276

variable F (t). We assume that at each time t F ′(t) is normally distributed, with variance σ2F . If277

there is no persistence in the climate fluctuations, then this is equal to the average over the time278

16

series of 〈|F ′(t)F ′(t)|〉. Calibrating the model to the geometry and climate of the glaciers around279

Mt. Baker in Washington State, Roe and O’Neal (2009) took τ = 7 yrs and σF = 200 m yr−1.280

When quantitative results are presented, it is these values we use.281

Although we focus on presenting results for glacier fluctuations, eq. (6) is the simplest one-parameter282

relationship between forcing and response, and thus the essential results of the analysis extend to283

any climate proxy that has an approximately linear relationship to the climate it reflects. For284

example lake-level fluctuations can be described by an equation identical to eq. (6) (e.g., Mason et285

al., 1994, Huybers et al., 2012), where the parameters are instead functions of the lake geometry286

and the atmospheric factors that control precipitation and evaporation.287

The variance of glacier length, σ2L, for a glacier forced by general power-law climate variability can288

be solved analytically from eq. (6). The derivation is presented in Appendix A, and the solution is289

a relatively simple expression:290

σ2L =

π

ωmax(ωmaxτg)ντgsec

(πν2

)σ2

F. (7)

By substituting our standard glacier parameters and a range of ν into eq. (7), we see that there is291

a striking sensitivity of σL to ν. For ν = 0.0, 0.25, 0.5, and 0.75, and constant σF , we calculate292

σL = 370 m, 620 m, 1.1 km, and 2.5 km, respectively. Even the relatively small amount of climate293

persistence implied by ν = 0.25 leads to a nearly 70% increase in σL, compared to that for ν = 0.294

This is a remarkable increase in variance when it is recalled that the lag-1 autocorrelation coefficient295

for ν = 0.25 is only 0.17. The glacier undergoes dramatically larger fluctuations because of the296

17

small but prolonged tendency for the climate forcing in successive years have the same sign.297

For climate variability governed by ν = 0.75, there is a more than six-fold increase in σL. In the298

limit of ν → 1, we see that σL → ∞ due to the secant dependence, reflecting that the variance of299

the climate forcing becomes unbounded.300

An illustration of the effect of power-law climate variability on glacier length is shown in Figure 3,301

which was created by integrating eq. (6) forward in time, for ν = 0 and ν = 0.5. Figure 3a clearly302

shows the increased variance of glacier length for the ν = 0.5 case. This is consistent with greater303

power at lower frequencies (Figure 3b). Higher values of ν also lead to higher autocorrelations in304

the glacier response, which is particularly evident at short lags (Figure 5c). This effect might be305

important when decadal-scale glacier records are used to estimate the response time (e.g., Oerle-306

mans, 2005): care might be needed to separate out the autocorrelations due to the glacier response,307

and that due to climatic persistence. An exact expression for the ACF in glacier length can be308

derived from the model equations, and is given in Appendix A.309

Our results can be compared to two other studies. Reicher et al. (2002) investigated the impact310

of climate variability on Nigardsbreen glacier in Norway and Rhonegletscher in the Alps, and311

modeled local climate persistence as an AR(3) process (i.e., see eq. 1). Compared to a white-noise312

climate, they found glacier variance was enhanced by about 35% for Nigardsbreen and about 5%313

for Rhonegletscher (whose climate had very little persistence). For Nigardsbreen, the coefficients314

of the AR(3) process reflect two exponentially decaying timescales of 1.7 and 1.1 yrs. A second315

study, Huybers and Roe (2009) provide formulae from which we can calculate that for our case of316

τg = 7 yrs, and a climate governed by an AR(1) process with a decorrelation time of one year (i.e.,317

18

a lag-1 autocorrelation of e−1 = 0.38 in eq. 3) the variance in glacier length would be increased by318

38%. The much larger increase of variance that we find here even for the ν = 0.25 case illustrates319

the impact of the long memory associated with the power-law process.320

3.1 Statistics of glacier excursions321

How likely is it for a given glacier excursion to have occurred in a given period of time, even322

in the absence of a climate change? We now use extreme-value statistics first developed by Rice323

(1948) (see also Vanmarcke, 1983), to characterize the likelihoods the glacier advancing past a given324

point (i.e., an “up-crossing”). This extends the analyses of Roe (2011) who considered only the325

case of ν = 0 (i.e., white noise climate forcing). Here, we show the presence of persistence in the326

climate variability enhances the likelihood of large glacier fluctuations. Rice (1948) showed that327

the expected rate of an advance past L0 is given by328

λ(L0) =1

2πσLσLe− 1

2

“L0σL

”2

, (8)

where σL is the standard deviation of dL/dt, which from eq. (6) can be written as329

σ2L

= σ2F −

σ2L

τ2g

, (9)

assuming 〈L′(t)L′(t) >= 0〉. The average return time of a glacier advance beyond L0 is equal to330

λ−1(L0). Equations (7), (8), and (9) can be combined to derive an expression for the return time331

19

as a function of ν, and the results are shown in Figure 4. The analytically derived curves are332

compared with a direct numerical determination of return times from 106 yr integrations of eq. (6),333

and the two methods agree well. Where there are differences it is due to the fact that the analytical334

solution solves the continuous equation, whereas the numerical model solves the discrete version.335

The return time of a given advance is an acutely sensitive function of ν. For example, while an336

advance past 1500 m would occur only every 50,000 yrs in a climate with ν = 0, it occurs every337

300 yrs for ν = 0.25, every 60 yrs for ν = 0.5, and every 30 yrs for ν = 0.75. Thus the addition338

of even a small amount of climate persistence can dramatically affect the return time of large339

advances. As can be seen in Figure 3, the larger the value of ν, the larger the glacier variance, the340

more time it spends away from equilibrium, and the more often the glacier will cross a given point341

of advance. This is also the reason why small advances become less frequent with increasing ν.342

A metric that has perhaps has more practical relevance for paleo-glaciological reconstructions is343

the likelihood of a total glacier excursion occurring in a given period of time. For example, for any344

given glacier reconstruction the question to be asked is: how likely is it that, just by chance in a345

statistically constant climate, the glacier would have advanced down-valley as far as that particular346

moraine, and also retreated as far back up-valley as we now see it?347

Let ∆L be the total excursion of a glacier (i.e., its maximum extent minus its minimum extent). For348

the case of ν = 0, Roe (2011) derived the statistics governing the likelihood of finding a given ∆L349

in a given time by assuming that maximum and minimum excursions could be treated as Poisson350

processes, which is to say that they are independent events which occur at particular average rate,351

and that the chance of simultaneous events occurring is vanishingly small (e.g., von Storch and352

20

Zwiers, 1999). In this study the inclusion of climate persistence means that excursions occurring at353

different times cannot be treated independently, but must account for the finite autocorrelation of354

the glacier, even at large lags. In other words, the ability of the glacier to reach a given minimum355

extent after a given maximum depends on the persistence of the climate variability. In Appendix356

A we outline an approximate modification of the statistical analysis to include this effect. The357

excursion statistics can also be derived directly from long simulations of eq. (6).358

Figure 5 shows the effect of changing the value of ν on the excursions probabilities in a 1000 yr359

period. Larger values of ν cause a strong increase in the likelihood of seeing large excursions, to360

the extent that there is almost no overlap in the curves for the values of ν we have considered.361

For example, if one assumed that climate variability was governed by ν = 0, one would conclude362

from Figure 5 that a total excursion of 3 km was virtually impossible in a 1000 yr period, and363

that an observation of such an excursion would be proof that a climate change must have occurred.364

However if the natural variability was, in fact, governed by ν = 0.25, such an excursion would be365

virtually certain to occur in a constant climate. This highlights the importance of knowing the366

underlying climate persistence. The challenge this creates is that the difference between ν = 0367

and ν = 0.25 is practically impossible to distinguish from even century-length instrumental records368

(e.g., Section 2.3.1; Percival et al., 2001).369

3.2 Can we distinguish between climate trends and climate persistence?370

We live on a planet where most regions are warming because of anthropogenic emissions (e.g., IPCC,371

2007). We also live in a period where instrumental records have not been kept long enough to clearly372

21

discern the degree of climate persistence present in natural variability. Since both climate trends373

and climate persistence affect glacier (and other proxy) behavior, what is their relative importance,374

and are there aspects of a proxy record that would potentially allow us to discriminate between375

the two?376

Standing at the modern glacier front, the presence of either a warming trend or climate persistence377

(ν = 0.25, say) would increase the chance of the glacier front having, in the past, extended further378

down the valley than would be the case for ν = 0 and no climate trend. This is illustrated in379

Figure 6a, which shows the probability density functions (PDFs) of the glacier terminus position380

for the cases of ν = 0 and ν = 0.25 with no climate trend, and for ν = 0 plus an added warming381

trend of 1oC century−1, which is typical for the midlatitudes. The PDFs were generated from a382

normal distribution using eq. (7) for the no-trend cases, and from 10,000 realizations of 100 yr-383

long simulations with normally-distributed climate forcing, for warming-trend case. The PDFs are384

centered around the long-term mean for the no-trend cases, and for the mode of the distribution385

at the end of the 100 yrs for the warming-trend case, the latter being the most likely position for386

the glacier front in the present day (i.e., after ∼100 years of warming has elapsed).387

As expected Figure 6a shows that, compared to the ν = 0 case, both ν = 0.25 and the warming388

trend increase the likelihood of the glacier having being 1 to 2 km down valley in the past century.389

However the PDF for ν = 0.25 is quite broad, whereas the warming-trend PDF remains narrow but390

is translated down-slope. In other words, for the warming-trend case, it is likely that the glacier391

will have spent most of its time slightly down valley of its present position in the past century.392

A second big difference is that the statistics for the ν = 0.25 case are stationary whereas those of393

22

the warming-trend case are not. For this reason, there is no simple analytic treatment adequate for394

this case and we rely exclusively on the numerical simulations. The effect of a trend is particularly395

pronounced for the expected return time of an advance past a given position. Figure 4 showed that396

the effect of having ν > 0 is to dramatically reduce the return time of large excursions. However, in397

the warming-trend case, the mean position of the glacier front is retreating, the chances of having398

a large down-valley excursion are decreasing extremely rapidly with time. This difference is shown399

in Figure 6b, and can be understood from eq. (8): L0 is the position of a point on the landscape400

relative to the average position of the glacier front, and so from the time the warming trend begins,401

the exponent is changing as the square of time.402

Although the return time is potentially a very sensitive metric of the difference between a climate403

trend and climate persistence, in practice it would require detailed histories of terminus position404

for a population of glaciers that can be regarded as independently forced. However, as detailed405

by Huybers and Roe (2009), glaciers within a region experience essentially the same climate and406

so are not independent. More importantly, since the modern warming trend is already clearly407

established from thermometers, the more useful challenge is the inference of past climate changes408

from paleoreconstructions of glacier extent. And in many instances, the predominant evidence is409

moraines that mark the furthest extent of glacier fluctuations that have not been subsequently410

overridden. Neglecting all the complications of how moraines get created, these glacier maxima411

can treated as ‘potential moraine locations’ and can be diagnosed from model simulations using412

eq. (6). Figure 7 presents the statistics of such moraine locations for 10,000-member Monte Carlo413

simulations of the same three cases presented in Figure 6. For consistency with Figure 6, we consider414

the same 100-yr climate trend. Figure 7a shows, for instance, that for the ν = 0, no-trend case it415

23

is most likely to find a moraine about 500 m down valley from the modern position, and Figure 7b416

shows that the expected age of that moraine is about 40 years. It becomes less-and-less likely that417

moraines would be found further down-valley (within the 100 yr interval we have allowed).418

Both the ν = 0.25 case and the warming-trend case increase the likelihood of finding moraines419

further down-valley from the present day position (Fig. 7a), and the two PDFs lie nearly on top420

of each other. However a significant difference between these two cases is the average age of those421

moraines (Fig. 7b), particularly for those that are found more than 1 km down valley. For the422

ν = 0.25 case the expected age of the moraine stays around 60 yrs, whereas for the warming-trend423

case the expected age is older - between 70 and 90 yrs. In other words, when a climate trend is424

present, down-valley moraines will be older than would be expected if they were due to climate425

persistence.426

Obviously, there are many caveats to the above analyses, and they are presented tentatively. Firstly,427

the processes and timescales for moraine formation remain poorly characterized (e.g., Mathews et428

al., 1995; Winkler and Nesje, 1999; Winkler and Mathews, 2010), and obviously are much more429

complex than simply reflecting glacier maxima. Secondly, the linear glacier model produces too430

many high-frequency terminus fluctuations compared to a model that represents the physics of431

glacier flow (e.g., Roe, 2011). These preliminary analyses should be redone with such a flow-line432

model, and also target a specific paleoclimate question such as the statistics of moraine formation433

during the late Holocene (e.g., Schaefer et al., 2009). The challenge will be to tease apart the relative434

importance of climate trends and climate persistence, and to establish whether the glacier modeling435

can be accurate enough, and the local climate variability known well enough, to distinguish between436

24

the two.437

4 Summary and discussion438

A wide variety of instrumental data, paleoclimate proxy data, and climate model output all suggest439

that Earth’s climate variability is characterized by a continuum background spectrum. The few440

significant spectral peaks that do exist are generally associated with external forcing that has441

to do with the planet’s orbit (i.e., diurnal, seasonal, annual, and orbital forcing), with El-Nino442

being perhaps the singular case where it is clearly established that a significant spectral peak443

arises from internal dynamics (though the presence of other peaks is perpetually speculated, e.g.,444

Chylek, 2011). In general, this continuum spectrum has increasing power towards lower frequencies,445

although the slope of the spectrum depends on the climate variable and the location in question.446

The observational evidence is strongly supported by basic thermodynamics and climate physics447

that predicts that a quasi-diffusive ocean heat uptake exchanging heat with the atmosphere should448

produce such surface-temperature spectra. We analyzed the persistence present in several long449

instrumental records of climate. The main purpose of this study was to explore what the effect of450

such persistence is on the natural variability of paleoproxy climate records. Even with no climate451

persistence, paleoclimate proxies such as glaciers and lakes have a dynamical response time, and452

will produce large fluctuations on long timescales because they act as low pass filters of climate453

variability.454

The effect of adding even a small amount of climatic persistence is to greatly increase the variance455

25

of the proxy fluctuations. Even a small chance that the climate forcing will have the same sign456

for several successive years makes a big difference. For a power spectrum variability with a slope457

of 0.25 – a value so low that it is hard to detect even in century-length instrumental records of458

climate change – glacier variance increases by almost 70%. This increased variance produces a459

spectacular reduction in the expected return time of large advances or retreats. For instance,460

for our glacier parameters, we demonstrated that by adding this small amount of persistence the461

average return time of a 1500 m advance plummets from once every 50,000 years to once every462

300 years. The statistics of total glacier excursions (accounting for both advances and retreats) is463

similarly impacted. Typical results for this are summarized in Figure 5.464

4.1 Generalization of this Analysis465

For the linear model, the variance, autocorrelation, return time, and excursion statistics can all466

be expressed as analytical functions of the parameters governing the model and the climate. This467

is important because the sensitivity to changes in these parameters can be clearly understood,468

and rapidly calculated. The effect of climatic persistence depends on the geographic location of469

the proxy, and on what climate variables it is sensitive to. Two big simplifications were made in470

assuming 1) a single slope to the power-law spectra, and 2) a linear model for the climate-proxy471

relationship. These assumptions can be relaxed to allow for a general function for the climate472

spectrum, and allow for the paleoclimate proxy to act as a more complicated (but still linear) filter473

of the climate forcing. In the Appendix, section A-5 presents these generalized expressions.474

Eq. (6) is the simplest one-parameter relationship between forcing and response, and thus the475

26

essential results of the analysis extend to any climate proxy that has an approximately linear476

relationship to the climate it reflects. For example lake-level fluctuations can be described by an477

equation identical to eq. (6) (e.g., Mason et al., 1994, Huybers et al., 2012), where the parameters478

are instead functions of the lake geometry and the atmospheric factors that control precipitation479

and evaporation. Further examples of proxies acting as filters of climate variability include tree480

ring growth, bioturbation in sediment cores, isotope diffusion in ice cores, and carbonate formation481

in paleosols or speleothems. The functional relationship between these proxies and the climate482

they experience is perhaps more complicated than for glacier extents and lake levels. The analyses483

could also be repeated with some nonlinearities included, such as diffusive ice flow in the case484

of glaciers, though we are confident that the basic relationships between climate spectra, climate485

persistence, and the effect on the variance of the paleoclimate proxy are robust. It is essential to486

understand that relationship well before being able to determine what fraction of the persistence in487

the proxy record reflects climate persistence and what fraction simply reflects the proxy’s behavior488

as a low-pass filter of climate information.489

4.2 Holocene climate variability490

How much of late Holocene climate variability can be explained by the natural climate variability491

that occurs in a constant climate with a quasi-diffusive ocean heat uptake? Is there, in fact, a492

need to invoke any external climate forcing such as solar variability and volcanic eruptions to493

explain, for instance, the putative Little Ice Age (LIA) and Mediaeval Warm Period (MWP)? A494

null hypothesis of natural variability that has a power-law spectrum is appealing because it is495

27

supported by long-term instrumental and proxy data, is grounded in basic physics, and invokes the496

fewest factors. For climate variability over the last millennium, the issue appears finely balanced.497

Combining multiple high-resolution paleoclimate records (mainly trees), Osborn and Briffa (2006)498

find indications of widespread warm and cool episodes during the intervals commonly associated499

with the LIA and MWP. Using a nearly identical dataset, Tingley and Huybers (2011) identify a500

MWP that was both significantly warmer and significantly more variable. A useful extension of501

these studies would be to characterize the climatic persistence present in the data, and to address502

whether any filtering of the climate signal occurred because of the way the proxies record climate.503

Using a simple climate model of the last millennium, Crowley (2000) finds that volcanic and solar504

forcing can explain approximately half of the smoothed pre-anthropogenic global-mean temperature505

variations, though there are wide bounds due to uncertainty in the assumed climate forcing.506

For Holocene variability beyond the last millennium, the scarcity of high-resolution datasets, and507

uncertainties in the dating (and cross-dating) of proxies such as glaciers and lake levels, may508

preclude definitively answering the question of whether a climate change beyond that driven by509

the slow progression of the orbital cycles is required to explain their fluctuations. However it510

is also certain that the fundamental nature of climate variability has not changed: year-to-year511

climate variability and, depending on the location and climate variable, some degree of interannual512

persistence, have been combining to drive fluctuations in climate proxies throughout the entire513

interval. The task, clearly, is to evaluate whether the proxy records show fluctuations which are514

larger, more frequent, or more spatially extensive, than would be predicted to occur from natural515

variability alone.516

28

Acknowledgements517

29

Appendix A: Analytical solutions for glacier statistics518

A1 Autocorrelation function for climate forcing519

The power-law climate spectrum is described by520

SF (ω, ν) = P0

(ωmaxω

)ν, (A-1)

for 0 < ω ≤ ωmax and P (ω) = 0 for ω > ωmax. We define521

P0 ≡2πσ2

F

ωmax(A-2)

As we now show, σ2F is equal to the average of |F ′(t)|2 if ν = 0 but it is smaller than that average522

if ν > 0.523

The Wiener-Khinchin Theorem tells us that524

〈|F ′(t)|2〉(ν) =∫ ωmax

0SF (ω, ν)dω =

σ2F

(1− ν). (A-3)

This is the variance that would be deduced from a time series of instrumental climate observations.525

We have defined σ2F (ν) ≡< |F ′(t)|2 > (ν), leading to the relationship526

σ2F (ν) =

σ2F (ν = 0)(1− ν)

. (A-4)

30

The ACF of the climate forcing is527

〈F ′(t)F ′(t±∆)〉(ν)σ2F (ν)

≡ ρFF (∆, ν) =

∫∞0 cos(ω∆)SF (ω, ν)dω∫∞

0 SF (ω, ν)dω; (A-5)

= 1F2

[{12− ν

2

},

{12,32− ν

2

},−1

4(ωmax∆)2

]. (A-6)

where 1F2(a, b, c) is a generalized hypergeometric function.528

A2 Standard deviation of glacier response529

We begin by forming the Fourier transforms of climate forcing F (t) and glacier length L(t):530

F (ω) =∫ ∞

0exp(iωt)F (t)dt (A-7)

L(ω) =∫ ∞

0exp(iωt)L(t)dt,

where the prime notation have been dropped. The spectrum of forcing is SF (ω) ≡ |F (ω)|2.531

Substituting these into the glacier model equation (6) gives532

(iω +

1τg

)L(ω) = F (ω). (A-8)

so the spectrum of glacier length variations is533

SL(ω) ≡ |L(ω)|2 = SF (ω)RL(ω) (A-9)

31

where the “response function” of glacier length fluctuations is RL(ω) ≡ 1(ω2+1/τ2

g ). For a power law534

climate forcing spectrum SF (ω) = SF (ω, ν) (Equation (4)), and535

SL(ω) = SF (ω, ν)RL(ω) (A-10)

=σ2Fω

ν−1max

ων [ω2 + 1/τ2g ]. (A-11)

and the standard deviation of the glacier response, σL, is536

σ2L =

σ2F

πων−1max

∫ ∞0

dω

ων(ω2 + 1/τ2g )

=π

ωmaxσ2F (ωmaxτg)ντgsec

(πν2

). (A-12)

We have taken the upper limit of the integral to be ∞. This approximation is justified because537

we are interested in glaciers with response times of several years and more, which means that high538

frequencies (ω > 2π× 1yr−1) are strongly damped by the glacier dynamics, and also by inspection;539

for reasonable values of the parameters, extending the upper limit does not affect the result.540

A3 Autocorrelation of glacier response541

The autocorrelation of the glacier response is given by542

ρLL(∆) =1πσ2

L

∫ ∞0

cos(ω∆)SL(ω)dω. (A-13)

Solving by inspection:543

32

ρLL(∆) =12

cosh(∆/τg)−1π

(∆τg

)1+ν

Γ(−1− ν) 1F2

[1,{

1 +ν

2,32

+ν

2

},14

(∆τg

)2]

sin(πν),

(A-14)

where Γ(z) is a gamma function, and 1F2(a,b, c) is a generalized hypergeometric function.544

A4 Excursion probabilities545

Let the expected rate of up-crossings past a given point be given by λ(L). In any interval (t1, t2),546

this can be written as547

λ(L) =1

(t2 − t1)〈n(L, t2 − t1)〉 , (A-15)

where 〈n(L, t2 − t1)〉 is the expected numbers of crossings of L in (t2− t1). In general we can write548

that the number of crossings is equal to549

n(L, t2 − t1) =∫ t2

t1

δ(t′ − tcross)dt′ =∫ t2

t1

δ(x(t)′ − L)|x|dt′, (A-16)

where tcross denotes up-crossing times, and x represents dx/dt. The expected number of up-550

crossings can now be found by integrating over the respective probability distributions:551

〈n(L, t2 − t1)〉 =∫ t2

t1

dt′∫ ∞

0|x|fx(x)dx

∫ ∞−∞

δ(x(t′)− L)fx(x(t′))dx, (A-17)

33

where we have assumed the probability distributions f(x) andf(x) are independent, as demon-552

strated in Vanmarke (1983). The δ-function in the third integral picks out just fx(x = L). We553

are only interested in up-crossings so the second integral has a lower bound of 0, and for a normal554

distribution, the integral is equal to σL/√

(2π). Upon substitution, eq. (A-15) becomes eq. (8).555

We now briefly review the derivation of glacier excursion statistics given in Roe (2011). Let (t2−t1)556

be the interval of interest, and let the occurrence of maxima and minima be governed by Poisson557

statistics (e.g., vonStorch and Zwiers, 1999). Then the probability of seeing at least one maximum558

event of magnitude L1 (≡ p(L1)) is given by559

p(L1) = 1− exp[−(t2 − t1)λ(L1)] (A-18)

A similar expression can be written for the probability of seeing one minimum event of magnitude,560

L2, p(L2). We are interested in total glacier excursion, ∆L. In other words L1 and L2 are linked561

via L2 = L1 −∆L.562

Roe (2011) showed that the probability of at least one event with a total excursion exceeding ∆L563

is given by:564

p(∆L) =∫ ∞

0

dp(L1)dL1

p(L2 = L1 −∆L)dL1. (A-19)

The remaining challenge is to determine the correct value of λ(L2). Roe (2011) assumes that565

p(L1) and p(L2) were independent of each other. However for non-zero values of ν, the climatic566

34

persistence means this can no longer be assumed. Instead, the probability that the glacier reaches567

L2 is conditional on L1. Here we give a very heuristic derivation of the excursion probabilities for568

this case. Frankly, no one was more surprised than we were that it actually seemed to work, at569

least for some cases.570

We can write down the joint probability distribution that L lies between L1 and L1 + dL at time t,571

and also lies between L2 and L2+dL2 at some other time t±∆. This depends on the autocorrelation572

of L:573

h(L1, t, L2, t±∆) =1

2πσ2L(1− ρLL(∆))

exp[−(L2

1 − 2L1L2ρLL(∆) + L22)

2σ2L(1− ρ2

LL(∆))

]. (A-20)

Hence the chance that L lies between L2 and L2 + dL2 given that L was in fact between L1 and574

L1 + dL at time t is:575

h1(L2, t±∆|L1, t) =h(L1, t, L2, t±∆)

1√2πσL

exp[− L2

1

2σ2L

] . (A-21)

Next, for a given t that lies within the desired interval, we integrate over all possible leads and lags576

within this interval at which L = L2 might occur:577

h2(L2|L1, t) =1

(t2 − t1)

∫ (t2−t1)−t

−th1(L2, t+ ∆|L1, t)d∆. (A-22)

Finally, we integrate over all possible ts within the desired interval. This gives the probability of578

35

L = L2 occurring within the desired interval that is contingent on L = L1 also having occurred579

somewhere in that same interval:580

h3(L2|L1) =1

(t2 − t1)

∫ (t2−t1)

0h2(L2|L1, t)dt. (A-23)

It is this probability distribution that we use in calculating the expected rate of crossing L2, given581

an L1 has occurred:582

λ(L2|L1) =σL√2πh3(L2|L1). (A-24)

This is the λ used in the calculation of p(L2), and the rest of the calculation is analogous to Roe583

(2011).584

A5 Generalized expressions585

Let S(ω) be a general form for the power spectrum of climate variability, and let RL(ω) be a general586

filter response function due to glacier dynamics (e.g., for our model RL(ω) = R0/(ω2 +1/τ2g ), where587

R0 is a constant that is a combination of model parameters). The variance of the proxy response588

is given by589

|σ2L| =

∫ ∞0

S(ω) ·RL(ω)dω ≡∫ ∞

0HL(ω)dω (A-25)

36

From eq. A-7 we have that590

dL

dt=∫ ∞

0iω exp(iωt)L(ω)dω, (A-26)

from which591

|σ2L| =

∫ ∞0

ω2S(ω) ·RL(ω)dω ≡∫ ∞

0ω2HL(ω)dω. (A-27)

The derivation of eq. (8) still applies for these general expressions. This means that the variance,592

return times, and excursion statistics can all be expressed in terms of the zeroth and second moments593

of the spectrum of HL(ω) (e.g., vanMarcke, 1983). As a result of the integral, all frequencies594

contribute to these statistics, and hence they are very insensitive to the presence of any narrow595

peaks in the climate spectrum (e.g., Wunsch, 2006).596

Moreover, this general formalism holds for any proxy variable x (lake sediment depth, tree ring597

width, etc.) that depends linearly on the climate forcing. If Rx(ω) is the proxy response function,598

then599

Sx(ω) = S(ω)Rx(ω) (A-28)

and the standard deviation, ACF and other statistical characteristics of x can be derived just as we600

derived those of glacier extent and excursions. Thus the effect of climate persistence on the proxy601

variables can be quantitatively examined and insight can be gained as to the relative importance602

of climate shifts and climate persistence in the proxy records.603

37

Appendix B: Persistence in a long integration of a climate model604

Instrumental records extending beyond 100 yrs are relatively uncommon and of course apply605

only to a single location, so we turn to climate models for an estimate of how ν varies spa-606

tially. We analyze the spatial pattern of the best-fit ν from a 500-yr long integration of the607

GFDL coupled ocean-atmosphere climate model (the GFDL CM2.0 model, output available at608

http://nomads.gfdl.noaa.gov), for summertime temperature and annual-mean precipitation vari-609

ability in the extratropics (Figure B1). For summertime temperatures, the value of ν (and there-610

fore the persistence) is substantially greater over ocean than over land, with largest values in the611

high North Atlantic, the North Pacific, and near the Weddell Sea in the Southern Ocean. These612

are all regions of strong coupling between the atmosphere and ocean. Consistent with the instru-613

mental observations in the previous section, the values of ν for precipitation are generally lower,614

indicating less persistence than for summertime temperature. There is a suggestion of some persis-615

tence in precipitation in the Barents and Weddell Seas, presumably influenced by the persistence616

in temperature there.617

38

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44

Figures720

45

20 40 60 80 100 120 140 160 180 200−20

−15

−10

−5

0

5

Year

Clim

ate

forc

ing

10−2

10−1

100

100

102

104

106

frequency (yr−1)

spec

tral p

ower

0 5 10 15 200

0.2

0.4

0.6

0.8

1

lag (yr)

auto

corre

latio

n

ν = 0ν = 0.25ν = 0.5ν = 0.75

(a)

(b) (c)

Figure 1: Realizations of time series with different amounts of persistence, generated from

eq. (4) for νs of 0, 0.25, 05, and 0.75. Also shown is their spectra (panel b) and theoretical

autocorrelation function (panel c, and see Appendix A). The same random noise process

was used to generate each time series. In panels a) and b) the curves have been offset for

clarity. Larger values of ν produce time series with greater variance and persistence.

46

1650 1700 1750 1800 1850 1900 1950 2000

-2

-1

0

1

2

year

PDO

inde

x

5-yr running mean

10-3 10-2 10-1 100

10-2

100

102

frequency (yr-1)

spec

tral p

ower

PDO datalong mem.AR(1)

0 5 10 15 20-0.2

00.20.40.60.8

lag (yr)

auto

corre

latio

n

PDO datalong mem.AR(1)

1650 1700 1750 1800 1850 1900 1950 2000

-2

-1

0

1

2

year

Cen

t. En

g. T

JJA

5-yr running mean

10-3 10-2 10-1 100

10-2

100

102

frequency (yr-1)

spec

tral p

ower

CET datalong mem.AR(1)

0 5 10 15 20-0.2

00.20.40.60.8

lag (yr)

auto

corre

latio

n

CET datalong mem.AR(1)

1650 1700 1750 1800 1850 1900 1950 2000

-2

-1

0

1

2

year

Eng.

Wal

es P

reci

p.

5-yr running mean

10-3 10-2 10-1 100

10-2

100

102

frequency (yr-1)

spec

tral p

ower

Precip datalong mem.AR(1)

0 5 10 15 20-0.2

00.20.40.60.8

lag (yr)

auto

corre

latio

n

Precip datalong mem.AR(1)

1650 1700 1750 1800 1850 1900 1950 2000

-2

-1

0

1

2

year

Cla

rinde

n M

ass

Bal

5-yr running mean

10-3 10-2 10-1 100

10-2

100

102

frequency (yr-1)

spec

tral p

ower

Clarinden Mass Bal datalong mem.AR(1)

0 5 10 15 20-0.2

00.20.40.60.8

lag (yr)

auto

corre

latio

n

Clarinden Mass Bal datalong mem.AR(1)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figure 2: Examples of power-law climate variability from long instrumental data sets. Top

row: (a) The Pacific Decadal Oscillation (PDO) index - 110 years; (b) The solid line is the

power spectrum of the PDO index. The dashed red line shows the spectrum of the best-fit

power-law slope, the dotted green line shows the spectrum of the best-fit AR(1) process; (c)

The diamond symbols show the lag autocorrelation function (ACF) of the PDO index. The

dashed red line shows the ACF of the best-fit power-law process, and the green dotted line

shows the ACF of the best-fit power-law process. The thin dashed horizontal lines indicate

95% confidence bounds for significant autocorrelations. Second row: as for top row, but for

the summertime temperatures the Central England Temperature record - 352 years. Third

row: as for top row, but for annual precipitation in England and Wales - 245 years. Bottom

row: as for top row, but for the mass balance measured at upper Clarinden glacier - 97

years.

47

0 100 200 300 400 500 600 700 800 900 1000−2000

−1000

0

1000

2000

Time (yr)

Dis

tanc

e (m

)

ν = 0.5ν = 0

10−2 100102

104

106

108

1010

frequency (yr−1)

spec

tral

pow

er

0 20 40 60 80 100−0.2

0

0.2

0.4

0.6

0.8

lag (yr)

auto

corr

elat

ion

Figure 3: A 1000 yr realization of the response of an idealized glacier to climate variability

with ν = 0 and ν = 0.5: (a) length fluctuations, (b) power spectra, and (c) ACF. The ACF

is calculated from the 1000 yr realization. As in Figure 1, the same set of random numbers

was used to generate both climate time series.

48

0 500 1000 1500 2000 2500 3000101

102

103

104

105

106

Glacier advance, L0 (m)

Ave

. ret

urn

time

(yrs

)

ν = 0ν = 0.25ν = 0.5ν = 0.75

Figure 4: The return time of a given glacier advance as a function of the value of ν governing

the climate variability. The lines result from solving eqs. (7), (8), and (9), and the plotted

symbols are diagnosed from 106 year simulations using eq. (6).

49

2000 4000 6000 8000 10000 12000 140000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ν = 0

ν = 0.25

ν = 0.5

ν = 0.75

Total Excursion length (m)

Pro

babi

lity

of e

xcee

ding

Figure 5: The probability of exceeding a given total glacier-length excursion in any 1000 yr

period. The thicker curves show direct calculations from long simulations of the linear

glacier model (eq. 6), as a function of ν. The symbols show calculations from the formulae

in section A-4, using the σL and σL from the model simulation. The thin grey lines, where

visible, show the impact of neglecting to account for the co-dependence of between maximum

and minimum excursions, which matters most for large ν.

50

-2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

1

x 10-3

Down valley distance (m)

Prob

abilit

y de

ns. (

m-1

)

ν = 0, no trendν = 0.25, no trendν = 0, 1C century-1

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

Distance of down-valley advance (m)

Ave.

retu

rn ti

me

(yrs

)

no trend = nu = 0no trend, nu = 0.25

(a) (b)

trend, t = 0 yrstrend, t = 50 yrstrend, t = 100 yrs

Figure 6: A comparison of the glacier response to three cases of different climate forcing: (1)

ν = 0, no trend; (2) ν = 0.25, no trend; (3) ν = 0 and a warming trend of 1oC century−1.

Panel (a): the probability density function (PDF) of the glacier terminus position. For cases

1 and 2 the PDFs are centered on the long-term mean terminus location. For case 3 the

PDF is centered on the mode of the PDF after 100 years of warming has occurred (roughly

corresponding to the most likely position in the present day. Panel (b): the likelihood of

seeing an advance past a given point on the landscape, expressed as an expected return time.

For case 3 (red lines) the likelihood changes rapidly with time as the warming progresses.

51

0 500 1000 1500 2000 2500 30000

0.5

1

1.5x 10-3

Down-valley distance (m)

Mor

rain

e Pr

ob (m

-1)

0 500 1000 1500 2000 2500 30000

20

40

60

80

100

Ave.

Mor

rain

e Ag

e (y

r)

ν = 0, no trendν = 0.25, no trend

(a) (b)

ν = 0, +1oC 100 yr-1

Down-valley distance (m)

Figure 7: As for Fig. 6 but for potential moraines left on the landscape by glacier maxima

that were not overridden by a subsequent advance. The plotted curves were diagnosed from

10,000 member Monte Carlo simulations of the linear model. a) The probability density of

finding a moraine that was created within the last 100 years on the landscape, as a function

of down-valley distance from the modern glacier front. b) The average age of the moraines,

as a function of down-valley distance. For large down-valley distances, the average age of

a moraine becomes noisy because of the small number of moraines reaching that far in the

Monte Carlo simulations.

52

−180 −90 0 90 18020

40

60

80νT, JJA Temperature

20

40

60

80νP, Annual precipitation

−80

−60

−40

−20νT, DJF temperature

−0.2

0

0.2

0.4

0.6

0.8

−80

−60

−40

−20νP, Annual precipitation

(a) (b)(a) (b)

(c) (d)

−180 −90 0 90 180

−180 −90 0 90 180 −180 −90 0 90 180

Figure 8: Figure B1. Best-fit power-law slopes in the extratropics for variability in summer-

time temperature and annual-mean precipitation, from a 500-yr integration of the GFDL

coupled ocean-atmosphere climate model. Values of ν that were not significantly different

from zero (at 95% confidence, based on a two-sided t-test) are displayed as zero.

53