paleoclimate proxies, extreme excursions, and persistence...
TRANSCRIPT
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
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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
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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
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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
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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
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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
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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
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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
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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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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