The response of glaciers to intrinsic climate variability: observations and models of late-Holocene variations in the Pacific Northwest

The response of glaciers to intrinsic climate variabilityobservations and models of late-Holocene variations in the

Pacific Northwest

Gerard H ROE1 Michael A OrsquoNEAL2

1Department of Earth and Space Sciences University of Washington Seattle Washington 98195-1310 USAE-mail gerardesswashingtonedu

2Department of Geography University of Delaware Newark Delaware 19716 USA

ABSTRACT Discriminating between glacier variations due to natural climate variability and those dueto true climate change is crucial for the interpretation and attribution of past glacier changes and forthe expectations of future changes We explore this issue for the well-documented glaciers of MountBaker in the Cascades Mountains of Washington State USA using glacier histories glacier modelingweather data and numerical weather model output We find that natural variability alone is capable ofproducing kilometer-scale excursions in glacier length on multi-decadal and centennial timescales Suchchanges are similar in magnitude to those attributed to a global Little Ice Age The null hypothesis thatno climate change is required to explain the glacier fluctuations in this setting cannot be rejected Theseresults for Mount Baker glaciers are also consistent with an earlier study analyzing individual glaciers inScandinavia and the Alps The principle that long-timescale fluctuations of glacier length can be drivenby short-timescale fluctuations in climate reflects a robust and fundamental property of stochasticallyforced physical systems with memory It is very likely that this principle also applies to other Alpineglaciers and that it therefore complicates interpretations of the relationship between glacier and climatehistory However the amplitude and timescale of the length fluctuations depends on the details of theparticular glacier geometry and climatic setting and this remains largely unevaluated for most glaciers

1 INTRODUCTIONThe existence of mountain glaciers hinges on a sensitivebalance between mass accumulation via snowfall and masswastage (ie ablation) via melting evaporation sublimationand calving All of these processes are ultimately controlledby climate While climate changes will obviously tend todrive glacier variations not all glacier variations should beinterpreted as being caused by climate changes Climate isthe statistics of weather averaged over some time period ofinterest The World Meteorological Organization takes thistime period as 30 years but it can be any interval relevantfor the question at hand By definition then a constantclimate means that the statistical distributions of atmos-pheric variables do not change with time Thereforevariability as characterized by the standard deviation andhigher-order statistical moments is in fact intrinsic to aconstant (ie stationary) climate Glaciers reflect thisvariability The characteristic response time (ie inertia orlsquomemoryrsquo) of a glacier ranges from years to centuries (egJohannesson and others 1989 Harrison and others 2001Oerlemans 2001 Pelto and Hedlund 2001) and any givenglacier will reflect an integrated climate history on thosetimescales Thus we arrive at a key question in interpretingrecords of changes in glacier geometry is the reconstruc-tion of past glacier variations significantly different (in astatistical sense) from what would be expected as a naturalresponse to intrinsic variability in a stationary climateOnly when this significance has been demonstrated can arecorded glacier advance or retreat be confidently inter-preted as reflecting an actual change in climate

Using glacier histories to infer past climate changescenters on a classic case of understanding the signal-to-noise ratio Glaciers respond to the lsquonoisersquo of interannual

variability as well as the lsquosignalrsquo of actual climate changesWithout understanding the response of a glacier to inter-annual variability there is a potential to conclude thatchanges in climate forcing must have occurred and ofsearching for a climate mechanism (eg volcanoes or solarvariability) where none may in fact be needed Theimportance of the issue can be understood by a directanalogy to the detection and attribution of anthropogenicinfluence on global mean temperature The Intergovern-mental Panel on Climate Change (IPCC Solomon andothers 2007) concluded that it was not until the late 20thcentury that the signal of anthropogenic warming emergedfrom the background of the natural interannual variabilityUp to that point there was not enough information and thesignal was too small so the effect remained undemon-strated Crucially this exercise required the use of climatemodels calibrated primarily to instrumental climate recordsto determine the magnitude of the natural variability inglobal mean temperature in the absence of any climatechange In using glaciers as lsquopast climate-change detectorsrsquothe same issues exist and the challenge therefore is tocharacterize the timescales and magnitudes of the glaciervariability that are driven by the natural interannual climatevariability in the absence of any real climate change

In this paper we adapt a linear glacier model to includean explicit and separate treatment of precipitation and melt-season temperature We use reconstructed geometrieshistorical climate data and numerical model output fromlocalities on and near Mount Baker in the Cascade Range ofwestern Washington USA (Fig 1) in order to determine theglacier response to intrinsic climate variability in this region

Although the examples used in this study are based on thegeometries of typical valley glaciers in this setting the goal

Journal of Glaciology Vol 55 No 193 2009 839

in this paper is not to create a simulation of the evolution ofany observed glacier Rather our approach is to employ thesimplest modeling framework in which the basic behaviorcan be demonstrated We use a combination of observationsand reconstructions of climate and glaciers to calibrate andevaluate a simple model for the glaciers on Mount BakerWe emphasize here that the model adequately reproducesthe observed characteristic variations of glacier length and isalso consistent with the observed trends over the lastcentury The analyses lead to some important results againstwhich to interpret glaciers in natural settings

Our approach mirrors that of Reichert and others (2002)who used a downscaled global climate model (GCM) outputand a dynamical glacier model for two European glaciers(Nigardsbreen Norway and Rhonegletscher Switzerland)They concluded that the present retreat exceeded naturalvariability but that Little Ice Age (LIA) advances did notThus a climate change (at least within their GCMglaciermodel system) was not required to explain LIA-like ad-vances Here our use of a linear model is a trade-off thelevel of sophistication of the glacier model is less but itssimplicity allows us to derive some simple expressions thatmake clear the dependencies of the system response

2 THE GLACIER MODELGlaciers are dynamic physical systems wherein ice deformsand flows in response to hydrostatic pressure gradientscaused by sloping ice surfaces There are other importantfactors to glacier motion among which are ice flow istemperature-dependent glaciers can slide over their base ifsubglacial water pressures are sufficient glaciers interactwith their constraining side-walls and glacier mass balancecan be sensitive to complicated mountain environments(eg Nye 1952 Pelto and Riedel 2001 Anderson andothers 2004) Despite these somewhat daunting complica-tions a series of papers have shown that simple linearmodels based on basic mass-balance considerations can beextremely effective in characterizing glacier response toclimate change (eg Huybrechts and others 1989Johannesson and others 1989 Raper and others 2000Harrison and others 2001 Oerlemans 2001 2005 Klok2003)

The model that we use includes an explicit and separaterepresentation of melt-season temperature and annual meanprecipitation in the mass balance A schematic of the modelis depicted in Figure 2 and a derivation of the modelequations is presented in the Appendix The model operates

Fig 1 Major Mount Baker glaciers (elevation contour interval 250m) Glaciers are shown at their LIA maxima 1930 and present positions

Roe and OrsquoNeal Late-Holocene variations in the Pacific Northwest840

on three key assumptions that are typical of such modelsThe first assumption is that a fixed characteristic glacierdepth and a fixed width of the glacier tongue can representthe glacier geometry The second assumption is that glacierdynamics can be essentially neglected All accumulationand ablation anomalies act immediately to change thelength of the glacier The third assumption is that lengthvariations are departures from some equilibrium value andare small enough that the system can be made linear Thesethree assumptions together with a constraint of massconservation allow for prescribed climate variations inthe form of accumulation and temperature anomalies to betranslated directly into length changes of the glacier Insection 52 the validity of these assumptions is evaluated bycomparing the results from the linear model with those froma non-linear dynamical flowline model

The geometry of the glacier model is depicted in Figure 2the total glacier area is Atot there is a melt zone where themelt-season temperature is above zero with an area ATgt0and there is an ablation zone where the annual melt exceedsthe annual accumulation with an area Aabl In other wordsthe ablation zone is the region of the glacier below theequilibrium-line altitude (ELA) This definition of ablationzone is in accord with for example Paterson (1994) but it isimportant to note that the net mass loss above the ablationzone also plays a role in the glacier dynamics as can beseen in the derivation of the model equation in theAppendix The glacier has a protruding tongue with acharacteristic width w has uniform thickness H and restson a bed with a constant slope angle rsquo The center-linelength is assumed to represent the total glacier length LFurther refinements of the simple model are possible It

would be possible to incorporate a feedback betweenthickness and glacier length (eg Oerlemans 2001) or atime-lag between climate anomaly and terminus response(eg Harrison and others 2003) with some incrementalincrease in model complexity However the agreementbetween the linear model and a dynamic flowline modeldemonstrated in section 52 is sufficient to justify the use ofthe current model for the question posed

Climate is prescribed in terms of a spatially uniformaccumulation rate P and a temperature-dependent ablationrate T where T is the mean melt-season temperature and is an empirical coefficient or melt factor In effect thisablation parameterization is a simplified form of the morefrequently used positive-degree-day model (eg Braithwaiteand Olesen 1989) Percolation of meltwater and freezing ofrainfall are neglected A simplified treatment of ablation isadequate for the purpose of this paper which is tocharacterize the general magnitude of the glacier responserather than to accurately capture the details

In the Appendix it is shown that when the model islinearized the evolution of the terminus position is governedby

dL0ethtTHORNdt

thorn Aabl tanrsquowH

L0ethtTHORN frac14 Atot

wHP 0ethtTHORN ATgt0

wHT 0ethtTHORN eth1THORN

where the prime denotes perturbations from the equilibriumstate and all other variables are their climatological (ie timemean) values is the atmospheric lapse rate (the decreaseof temperature with elevation) and P0 and T0 are annualanomalies of respectively the average annual accumulationon the glacier and the average melt-season temperature onthe glacierrsquos melt zone

Fig 2 Idealized geometry of the linear glacier model based on Johannesson and others (1989) Precipitation falls over the entire surface ofthe glacier Atot while melt occurs only on the melt-zone area ATgt0 The ablation zone Aabl is the region below the ELA Melt is linearlyproportional to the temperature which in turn decreases linearly as the tongue of the glacier recedes up the linear slope tanrsquo and increasesas the glacier advances downslope The height H of the glacier and the width of the ablation area w remain constant Figure courtesy ofK Huybers

Roe and OrsquoNeal Late-Holocene variations in the Pacific Northwest 841

In sections 4 and 5 Equation (1) is applied to the glaciersof Mount Baker We use climate forcing that is consistentwith the observed mass balance and show that Equation (1)adequately reproduces the observed variability and trends ofglacier length First though we discuss the physics repre-sented by Equation (1) as there are important behaviors tohighlight

3 DISCUSSION OF MODEL PHYSICSAs noted above the model is similar in spirit to that ofJohannesson and others (1989) and Harrison and others(2001) and the essence of the dynamics is identicalHowever the model does differ in some important respectsFirstly in the model presented here the climate forcingreflects the separate contributions of accumulation and melt-season temperature Secondly ablation in this model iscalculated from the melt occurring over the whole ablationzone rather than being calculated from the net mass balanceat the terminus These differences in model formulationreveal that the glacier response has some importancedependencies on the glacier geometry and climatic settingso a discussion of the model physics is warranted

In the absence of a climate perturbation (P0 frac14 T0 frac14 0)Equation (1) can be written as

dL0ethtTHORNdt

thorn L0ethtTHORNfrac14 0 eth2THORN

where

frac14 wH tanrsquoAabl

eth3THORN

Equation (2) represents a system that if perturbed relaxesasymptotically back to equilibrium (L0 frac14 0) with a char-acteristic timescale which is a function of the glaciergeometry and the sensitivity of ablation to temperature

In the presence of climate variability another interpret-ation of is that it is the timescale over which the glacierintegrates the mass-balance anomalies In other words can be thought of as the length of the glacierrsquos lsquomemoryrsquo ofprevious climate states

Without loss of generality the numerator and denomi-nator in Equation (3) can both be multiplied by L0 andwrittenas

frac14 wHL0

L0 tanrsquoAabl

Recall that the model linearizes the glacier into an equi-librium component and an anomalous departure from thatequilibrium (see Appendix) wHL0 is the anomalous volumeof ice associated with anomalous length L0 (Fig 2) L0 tanrsquois the temperature anomaly resulting from a displacement ofthe glacier terminus Therefore L0 tanrsquoAabl is the anom-alous melt rate summed over the ablation zone (m3 andash1) Thus is equivalent to a volume divided by a volume rate and canbe interpreted as the time taken to melt (or build) ananomalous volume of ice due to the anomalously warm (orcold) temperature conditions experienced in the ablationzone due to an advance (or retreat) of the terminus

The ways that the various parameters affect have clearphysical explanations increasing the value of or tanrsquoaffects the local melt rate (ie m andash1) increasing Aabl

increases the ability of the glacier terminus to accommodatean increase in the mass balance conversely increasing H

results in a greater amount of mass that must be removed fora given climate change In the model of Johannesson andothers (1989) the equivalent timescale is given by H= _bwhere _b is the net mass balance at the terminus Thedenominator in Equation (2) plays the equivalent role of _b inthis model

31 Equilibrium response to changes in forcingWe first consider the steady-state response of the glaciersystem to a step-function change in climate The second andthird terms on the right-hand side of Equation (1) representthe climatic forcing separated into precipitation and tem-perature respectively Equation (1) can be rearranged andused to calculate the steady-state response of the glacierterminus L to a change in annual accumulation P ormelt-season temperature T using the fact that dL0dt frac14 0in steady state In response to a change in melt-seasontemperature T the response of the terminus is given by

LT frac14 ATgt0T tanrsquoAabl

eth4THORN

In response to a negative anomaly in temperature theglacier will advance downslope until ablation comes backinto balance with the new climate A steeper basal slope orlapse rate (ie a larger value of tanrsquo or ) means the glacierwill not have to advance as far to find temperature warmenough to establish mass balance The ratio of areas appearsin Equation (4) because as the glacier terminus advances italso expands the area over which accumulation occurs andthis partially offsets the increased melting that occurs atlower elevations

In response to a step change in annual accumulation PEquation (1) can be rearranged to give

LP frac14 AtotP tanrsquoAabl

eth5THORN

Equation (5) is analogous to Equation (4) both the imposedgeometry of the glacier and the melt rate at the terminus arerequired to account for the accumulation and the areaadded to the glacier tongue Looking at the terms inEquation (5) Aabl is the area of the ablation zoneLP tanrsquois the temperature change of the terminus due to the changein length and P is the change in the total accumulationEquation (5) is therefore a perturbation mass-balance equa-tion it gives the change in the length of the glacier such thatthe change in the total ablation rate balances the prescribedchange in the total accumulation rate

Another useful property of the linear model is that it isstraightforward to evaluate the relative sensitivity of theglacier length to accumulation and melt-season tempera-ture Let R equal the ratio of length changes due totemperature LT to the length change due to precipitationLP From Equations (4) and (5)

R frac14 LTLP

frac14 ATgt0

Atot

TP

eth6THORN

Thus R is equal to the ratio of the ablation and melt-zoneareas multiplied by the ratio of the ablation-rate (ie T)and accumulation-rate changes

32 Response to climate variabilityFor particular accumulation and melt-season temperaturerecords and for parameter choices appropriate to specificglaciers Equation (1) can be numerically integrated forward


in time to calculate the glacier response to a given climateforcing However to begin with we first derive generalexpressions for the glacier length variations expected in aconstant climate As discussed in section 1 a constantclimate means the climate has a constant mean and aconstant standard deviation

If we assume that the accumulation and temperatureanomalies are described by normally distributed randomnoise then Equation (1) is formally equivalent to a first-orderautoregressive process or AR(1) (eg von Storch and Zwiers1999) It represents a physical system with memory subjectto stochastic forcing A robust and fundamental property ofsuch systems is that even when driven by forcing with nopersistence (ie white noise) they respond with persistentfluctuations (ie red noise) that can have surprisingly longtimescales as shown below (eg Jenkins and Watts 1968von Storch and Zwiers 1999 Roe 2009)

Assuming a normal distribution of accumulation cannotbe of course strictly correct because negative precipitationis not physical but provided the standard deviation issmall compared to the mean this approximation is stillinstructive to make We further assume that neitheraccumulation nor melt-season temperature is correlated intime and also that they are not correlated with each otherHuybers and Roe (in press) showed that these assumptionsare appropriate for glaciers in the Pacific NorthwestAlthough there is some interannual memory in precipi-tation it is not very strong (eg Huybers and Roe in press)and much shorter than characteristic glacier responsetimescales Moreover 80 of annual precipitation inthe region falls in the winter half-year so correlationsbetween annual precipitation and melt-season temperatureare not significant A more detailed treatment of the massbalance might include the temperature dependence of thewintertime snowpack Further possible model refinementsare discussed in section 7

Let T be the standard deviation of melt-season tempera-ture let P be the standard deviation of annual accumu-lation and let t and t be independent normally distributedrandom processes Then using finite differences to discretizethe equation into time increments of t frac14 1 year Equa-tion (1) can be written as

L0tthornt frac14 L0t 1t

thorn Atott

wHPt thorn ATgt0t

wHTt eth7THORN

where subscript t denotes the year We first calculateexpressions for the standard deviation of glacier length dueto temperature and precipitation variations separately Lethxi represent the statistical expectation value of x Thefollowing relationships hold htti = ht Lti = ht Lti = 0hLtLti= hLtthornt Ltthornti and the expectation value of both sidesof Equation (7) must be the same Firstly let P =0 in whichcase it follows from Equation (7) that

LT frac14ffiffiffiffiffiffiffiffiffiffiffiffiLtLth i

pfrac14 T

ATgt0

wH

ffiffiffiffiffiffiffiffiffit2

r eth8THORN

and similarly for LP

LP frac14 PAtot

wH


r eth9THORN

As might be expected the relative sensitivity of the glacier toprecipitation and temperature variations is similar to that for

a step-change

R frac14 LT

LP

frac14 ATgt0

Atot

T

P eth10THORN

Note that this ratio describes the relative importance ofaccumulation and melt-season temperature for glacierlength It is different of course from the relative importanceof accumulation and temperature for local mass balancewhich is simply given by T=P

Since the model is linear and the climate variations areuncorrelated the standard deviation of glacier length whenboth temperature and precipitation are varying can bewritten

L frac14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2LT thorn 2

LP

q eth11THORN

Thus for specified glacier geometry and parameters we candirectly calculate the expected response of the glacier torandom year-to-year fluctuations in the meteorologicalvariables In section 4 we apply and evaluate this modelfor typical conditions of Mount Baker glaciers and theclimate of the Pacific Northwest of the United States Weemphasize that Equations (7ndash11) follow directly from thegoverning Equation (1) and involve no additional assump-tions

4 CALIBRATION OF MODEL FOR MOUNT BAKERGLACIERS AND CASCADE CLIMATE41 Climate parametersMost of the model parameters are readily determined oravailable from published literature The value of the meltrate at the terminus per 8C of melt-season temperature isassumed to range from 05 to 084m 8Cndash1 andash1 (eg Paterson1994) We take to be 658Ckmndash1 (eg Wallace and Hobbs2004) In practice varies somewhat as a function oflocation and season (JR Minder and others unpublishedinformation) Our choices here produce vertical mass-balance gradients that are consistent with profiles obtainedfor glaciers in the region (eg Meier and others 1971)

42 Glacier geometryFor our rectangular slab-shaped model glacier the meanablation area Aabl is calculated using the accumulationndasharea ratio (AAR) method which assumes that the accumu-lation area of the glacier is a fixed portion of the total glacierarea (eg Meier and Post 1962 Porter 1977) Although themethod does not account for the distribution of glacier areaover its altitudinal range or hypsometry it is appropriate forthe model since we are trying to generalize to large tabularvalley glaciers with similar shapes Porter (1977) indicatesthat for mid-latitude glaciers like the large valley glaciers ofthe Cascade Range a steady-state AAR is generally in therange 06ndash08

A range of areas for the ablation zone Aabl for our modelis readily determined from the area of glaciers and theircharacteristic tongue widths using 750 US GeologicalSurvey topographic maps and past glacier-geometry datafrom Burke (1972) Fuller (1980) Harper (1992) Thomas(1997) and OrsquoNeal (2005) For the major glaciers on MountBaker this information is compiled in Table 1 The largevalley glaciers around Mount Baker are all similar geo-metrically so we also choose a representative set ofparameters which we use for analyses in section 5 (Table 1)


Substituting this characteristic set of parameters into Equa-tion (2) and accounting for the range of uncertainties in and the AAR varies between 7 and 24 years with a mid-range value of 12 years This range of timescales is consistentwith those identified by Harper (1992) for these glaciers

43 Climate dataWe take the melt season to be JunendashSeptember (denotedJJAS) We also use annual mean precipitation as a proxy forannual mean accumulation of snow In this region of thePacific Northwest about 80 of the precipitation occursduring the OctoberndashMarch winter half-year so we assumethat it falls as snow at high elevation This also means thatannual precipitation and melt-season temperature in theregion are not significantly correlated and therefore can beassumed independent of each other Since we are seeking afirst-order characterization of the glacier response to climatevariability these are appropriate approximations For wantof a satisfactory treatment of these processes we alsoneglect mass input to the glacier from avalanching andblowing snow

The nearest long-term meteorological record is fromDiablo Dam (488300N 1218090W 271masl) about60 km from Mount Baker (488460N 1218490W 3285masl) and extends back about 75 years The observations ofannual precipitation and melt-season temperature areshown in Figure 3a and b

It is quite common in the glaciological literature to finddecadal climate variability invoked as the cause of glaciervariability on these timescales (eg Hodge and others 1998Moore and Demuth 2001 Kovanen 2003 Nesje and Dahl2003 Pederson and others 2004 Lillquist and Walker2006) In particular for the Pacific Northwest much is madeof the Pacific Decadal Oscillation (PDO) The PDO is theleading empirical orthogonal function of sea-surface tem-peratures in the North Pacific and exerts an importantinfluence on climate patterns in the Pacific Northwest (egMantua and others 1997) In fact gridded datasets for theatmospheric variables that control glacier variability showvery little persistence there is no significant interannualmemory in melt-season temperature (Huybers and Roe inpress) and only weak interannual memory in the annualprecipitation (the 1 year lag autocorrelation is 02ndash03 andso explains lt10 of the variance Huybers and Roe inpress) The interannual memory that does exist in NorthPacific sea-surface temperatures comes from re-entrainmentof ocean heat anomalies into the following winterrsquos mixedlayer (Deser and others 2003 Newman and others 2003)

and is consistent with longer-term proxy records foraccumulation (eg Rupper and others 2004) The appear-ance of decadal variability in time series of the PDO is oftenartificially exaggerated by the application of a several-yearrunning mean through the data (eg Roe 2009)

At Diablo Dam and for this length of record a standardstatistical test for autoregression (a stepwise least-squaresestimator for the significance of autocorrelations usingSchwarzrsquos Bayesian criterion as implemented in ARfit VonStorch and Zwiers 1999 Schneider and Neumaier 2001)does not allow the conclusion that there is any statisticallysignificant interannual persistence in either the annualprecipitation or the melt-season temperature (after linearlydetrending the time series) ARfit uses the modified Li-McLeod portmanteau statistic (eg Schneider and Neumaier2001) to test for the presence of significant autocorrelationsFor Diablo Dam the test has a p value of 51 A value oflt5 would be necessary to demonstrate statistically signifi-cant persistence in the time series In other words one yearbears no relation to the next The oft-used technique ofputting a 5 year running mean through the data gives adeceptive appearance of regimes lasting many years or evendecades that are wetterdrier or warmercolder but this isartificial A graphic demonstration of this is given inFigure 3c and e which show random realizations of whitenoise with the same mean and standard deviation as theannual precipitation Figure 3e and f show the same thingbut for melt-season temperature There is a large amount ofyear-to-year variability and just by chance there are intervalsof a few years of above- or below-normal conditions This ishighly exaggerated by the application of the running meanThe similarity of the general characteristics of these randomtime series to the data (visible in Fig 3) reflects the fact thatthe linearly detrended annual-mean accumulation and melt-season temperature are statistically indistinguishable fromnormally distributed white noise Thus for the atmosphericfields that are relevant for forcing glaciers there is little or noevidence of any interannual persistence of climate regimesin this setting

It is very important to stress this result A large fraction ofthe glaciological literature has interpreted the recent (butpre-anthropogenic) glacier history of this region and else-where in terms of the decadal-scale persistence of climateregimes For the Pacific Northwest this is simply notsupported by the available weather data Five other long-term weather-station records near Mount Baker were alsoanalyzed Upper Baker Dam (488390N 1218420W 2103masl 44 years) Bellingham airport (488480N 1228320W

Table 1 Parameters for the major Mount Baker glaciers obtained from a variety of sources See Figure 2 for a schematic illustration of themodel parameters and the text for details Aabl is shown for several different choices of the AAR Also given is a choice of a set of typicalparameters used in the glacier model

Boulder Deming Coleman Easton Rainbow lsquoTypicalrsquo

Atot (km2) 430 540 215 360 270 40

Aabl 08 (km2) 086 108 043 072 054 080Aabl 07 (km2) 129 162 064 108 081 120Aabl 06 (km2) 172 216 086 144 108 160tanrsquo 047 036 047 034 032 040w (m) 550 450 650 550 300 500H (m) 50 50 39 51 47 50


454masl 46 years) Clearbrook (488580N 1228200W195m asl 76 years) Concrete (488320N 1218450W594m asl 76 years) and Sedro Wooley (488300N1228140W 183masl 76 years) None of these stationsshows any statistically significant autocorrelations of annualprecipitation or melt-season temperature either The lack ofstrong evidence for dominant decadal-scale regimes (iesignificant autocorrelations on decadal timescales) of atmos-pheric circulation patterns is widely appreciated in theclimate literature (eg Frankignoul and Hasselmann 1977Frankignoul and others 1997 Barsugli and Battisti 1998Wunsch 1999 Bretherton and Battisti 2000 Stouffer andothers 2000 Deser and others 2003 Newman and others2003) At best the interannual persistence that does exist forsome atmospheric variables only accounts for a smallfraction of their total variance (for simple cases the varianceexplained by the interannual persistence is equal to thesquare of the 1 year lag autocorrelation)

While we have analyzed only local records here Stoufferand others (2000) addressed this issue on a global scaleAlthough they did not study the atmospheric fields mostdirectly relevant for glacier mass balance they analyzed thepersistence of anomalies in annual-mean surface tempera-ture using the longest available datasets from observationsextending back as far as 1854 Quoting from theirconclusions lsquothe annual mean SAT [=surface air tempera-ture] variance is very close to 5 times the variance obtained

from the 5 [year] time series in most continental areas itappears the SAT time series is nearly white noise on thesetimescales This suggests according to Hasselmannrsquos [1976]theory that most of the surface temperature variance isdominated by atmospheric white noise [emphasis added]rsquoStouffer and others (2000) do find weak interannual persist-ence (perhaps 1ndash2 year decorrelation times) for some mari-time climates as well as those in the vicinity of the sea-iceedge or centers of deep-ocean convection (ie well-mixeddeep water columns) As emphasized below and in thecontext of glacier variability it is the memory intrinsic to theglacier itself and not persistence in the climate system thatdrives the long-timescale variations

For the specific climate fields used for the model we useoutput from a high-resolution (4 km) numerical weather-prediction model the Pennsylvania State UniversityndashUSNational Center for Atmospheric Research Mesoscale Modelversion 5 (MM5 Grell and others 1995) MM5 has been inoperational use over the region for the past 8 years at theUniversity of Washington It provides a unique opportunityto obtain information about small-scale patterns of atmos-pheric variables in mountainous terrain that begins to extendtowards climatological timescales a series of studies in theregion has shown persistent patterns in orographic precipi-tation at scales of a few kilometers (Colle and others 2000Anders and others 2007 Garvert and others 2007 Minderand others 2008)

Fig 3 (a) Annual mean precipitation recorded at Diablo Dam near Mount Baker over the past 75 years equal to 189 036 (1)m andash1(b) melt-season (JJAS) temperature at the same site equal to 168 078 (1)8C These atmospheric variables at this site are statisticallyuncorrelated and both are indistinguishable from normally distributed white noise with the same mean and variance The application of a5 year running mean imparts the artificial appearance of multi-year regimes (cndashf) Random realizations of white noise shown for annualmean accumulation (c e) and for melt-season temperature (d f) Note the general visual similarity of the random realizations and theobservations


Although 8 years is a short interval for obtaining robuststatistics the output from MM5 at the gridpoint nearestDiablo Dam agrees quite well with the observations thereFrom 75 years of observations at Diablo Dam the meanannual accumulation is 189 036mandash1 (1 hereafterunless stated otherwise) whereas the output from MM5 atthe nearest gridpoint to Diablo Dam is 23041mandash1 Formelt-season temperatures the values in observations andMM5 are 1680788C and 127 0938C respectively Thenearest meteorological observation to Mount Baker comesfrom the Elbow Lake SNOTEL site (US Natural ResourcesConservation Service httpwwwwccnrcsusdagovgisin-dexhtml 488410N 1218540W 985masl) about 15 kmaway For 11 years of data the observed annual accumu-lation is 37077mandash1 compared with 47080mandash1 inthe MM5 output For melt-season temperatures the valuesare 133 128C and 117088C respectively It is thestandard deviations that matter for driving glacier variationsso we consider this agreement sufficient to proceed withusing the MM5 output For Mount Baker MM5 output givesan annual accumulation of 55 10mandash1 and a melt-season temperature of 93088C

Spatial correlations in the interannual variability of meanannual precipitation and melt-season temperature in thevicinity of Mount Baker are high (gt08) (OrsquoNeal 2005 Pelto2006 Huybers and Roe in press) Therefore when theglacier model is evaluated against the glacier history of thepast 75 years we use the time series of observations atDiablo Dam scaled to match the standard deviation of theMM5 output at Mount Baker

44 Parameter and data uncertaintiesThe combined uncertainty in AAR and is nearly a factor offour These dominate over other sources of uncertainty sowe focus on their effects in the analyses that follow Both ofthese factors have their biggest proportional effects on theablation side of the mass balance (for the melt factorexclusively so) Thus as we find for Mount Baker glacierswhile it may be that a glacier is most responsive toaccumulation variations the uncertainty in that responsive-ness is dominated by uncertainty in the factors controllingablation In this paper we seek a general picture of glacierresponse to climate so we explore this full range ofuncertainty However for a specific glacier of interest it ispossible to better constrain both the AAR and the melt factor

by careful measurements at which point it may be thatother sources of uncertainty need to be more carefullyaccounted for

45 Comparison with available mass-balance dataHow well does the climate forcing applied to the glaciermodel agree with observations Although a detailed mass-balance comparison is hampered by lack of data some veryuseful guidance exists As part of the North Cascade GlacierClimate Project (httpwwwnicholsedudepartmentsglacier) Pelto (2000) reports above-snowline accumulationof 50223mwe andash1 (1) between 1990 and 1999 forEaston Glacier This is an average over two elevations (2300and 2450m) and measured in late summer For RainbowGlacier the value is 42156mandash1 averaged over 1984ndash99and measured at 1900m again in late summer Obviouslyhaving only accumulation rates at high elevations is notideal and they cannot be directly applied over the wholeglacier Recall that it is the standard deviation of interannualfluctuations that is important for characterizing the naturalglacier variability using the MM5 value of 1mandash1 thusappears reasonable and is perhaps conservative Ablationmeasurements are even sparser Pelto (httpwwwnicholsedudepartmentsGlacierbibhtm) reports ablation ratesaveraging 44mandash1 for Easton Glacier in the 2002 and2003 seasons (averaged over the whole glacier) Thiscompares quite favorably with the 37mandash1 for the linearmodel calculated from the mid-range set of parameters inTable 1 The low and high end of parameter choices give 19and 59mandash1 suggesting that our mid-range values areappropriate in this setting

5 RESULTSWe first use the parameters of the typical Mount Bakerglacier (Table 2) and use Equations (8) and (9) to calculateLT and LP the variations in the model glacierrsquos terminusdue to characteristic melt-season temperature (T) andprecipitation (P) variability respectively The range in LTis 128ndash180m with a mid-range value of 150m Themagnitude of LP is significantly larger 301ndash554m with amid-range value of 386m Assuming the melt-seasontemperature and annual precipitation are uncorrelated thecombination of the two climate forcings gives a terminusrange of 344ndash569m with a mid-range estimate of 415mdominated by precipitation variability Note that this rangeof glacier length response is not found by simply combiningthe ranges of the separate responses to precipitation andmelt-season temperature since consistent combinations ofparameters have to be chosen

The relative importance of precipitation and melt-seasontemperature for the local mass balance is equal to T=P For Mount Baker then the model suggests that precipitationis more important than melt-season temperature by a factorof 15ndash2 depending on the chosen value of This range isconsistent with for example the conclusions of Bitz andBattisti (1999) as to the cause of mass-balance variations forseveral glaciers in this region

Because of the differing accumulation and ablation areasthe relative importance of precipitation and temperature onthe glacier length is different than for local mass balanceUsing Equation (10) the ratio R varies from 025 to 056with a mid-range value of 038 In other words the model

Table 2 Minimum mean and maximum estimates of standarddeviations in glacier lengths for various glacier properties for thetypical Mount Baker glacier defined in Table 1 and driven byclimate variability determined from the MM5 model output Therange of values here is generated from the range of uncertainties inthe melt factor and the AAR Note that each row has to becalculated separately from other rows because of mutuallyincompatible parameter combinations

Min Mean Max

(years) 7 12 24LT (m) 128 150 180LP (m) 301 386 554L frac14

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2LT thorn 2LP

q(m) 344 415 569

Sensitivity ratio R frac14 LT =LP 025 038 0566L (m) 2064 2490 3414


suggests that taking the characteristic local climate vari-ability into account the length of the average Mount Bakerglacier is two to four times more sensitive to precipitationthan to temperature variations This is due to the very largeinterannual variability in precipitation Thus we concludethat variability in Mount Baker glaciers is predominantlydriven by precipitation variability

A key point to appreciate about Equation (10) is that therelative importance of precipitation and melt-season tem-perature for a glacier length depends on the characteristicmagnitude of the climate variability and so depends onlocation as well as glacier geometry Huybers and Roe (inpress) use regional datasets of climate variability to explorehow R varies around the Pacific Northwest Maritimeclimates tend to have high precipitation rates and highprecipitation variability but muted temperature variabilityThe reverse is the case farther inland where in morecontinental climates temperature variability becomes moreimportant for driving glacier variations

51 Historical fluctuations of Mount Baker glaciersHistorical maps photos and reports of Mount Baker glaciersindicate that they retreated rapidly from 1931 to 1940paused then began readvancing between 1947 and 1952(eg Long 1955 Fuller 1980 Harper 1992) This advancecontinued until 1980 when they again began to retreatAlthough Rainbow and Deming Glaciers began to advanceabout 1947 earlier than the other Mount Baker glaciers theterminal movement between 1947 and 1980 was 600ndash700mfor eachMount Baker glacier underscoring the similar lengthresponses of these glaciers over this period

We use the 75 year long record of precipitation and melt-season temperature from the Diablo Dam weather-station

data scaled to have variance equal to the MM5 output atMount Baker and integrate Equation (1) for the typical MountBaker glaciers from 1931 to 2006 and for the range of modeluncertainties given in Table 2 (Fig 4) The initial condition forthe glacier model terminus is a free parameter ChoosingL0 frac14 600m produces the best agreement with the observedrecord Maximum changes in glacier length are on the orderof 1000m similar to the observed data for this period andapproximately 50 of the observedmagnitude of the glacier-length changes over the last 200 years There are somediscrepancies between the model and the historical recordthe model appears to respond a little quicker than the actualglaciers probably due to the neglect of glacier dynamics inthe model However we emphasize that the point is not tohave the model be a simulation of the historical recordcorrect in all details Rather the point is to establish that thecharacteristic magnitude and the approximate timescale ofglacier variations are adequately captured by the model

Are the century-scale trends in glacier length which arecaptured by the linear model in Figure 4 consistent withobserved climate trends For timescales much longer than Equations (4) and (5) can be used to calculate the glacierlength response to climate trends so this provides anotherway to evaluate whether the sensitivity of the linear model isreasonable For mid-range parameters Equation (4) predictsthat a 18C temperature change in melt-season temperaturecauses a 920m retreat of the glacier Mote (2003) finds a0838C increase in annual mean temperature in the centralCascades (statistically significant at gt95) and a 068Cincrease in summertime temperature over the course of the20th century Thus the observed retreat of around 500m onMount Baker is indeed consistent with the glacier modelsensitivity and observed temperature trends

Fig 4 Model glacier length variations from 1931 to 2006 using the time series of annual precipitation and melt-season temperature fromDiablo Dam scaled by the MM5 output for Mount Baker The grey shading shows the zone of the range of model uncertainties given inTable 2 Also shown are the historical glacier fluctuation records from Harper (1992) OrsquoNeal (2005) and Pelto (2006) Negative numbersindicate retreat The initial perturbation length at 1931 for the glacier model is a free parameter and was chosen to be 600m to produce thebest fit with the historical record


Establishing trends in glacier accumulation is harder thanfor temperature for several reasons the absence of longrecords the high degree of interannual variability in regionalprecipitation and snowpack the opposing tendencies ofincreasing precipitation (eg Mote 2003) and decreasingmountain snowpack (eg Mote and others 2005) in awarming climate and the complicated elevation depend-ence of changes in snowpack Equation (5) predicts that a05mandash1 decrease in accumulation causes a 1 km retreatof the glacier terminus

52 Glacier variations over longer timescalesThe success at simulating glacier length variations usinghistorical climate data for the last 75 years suggests the modelprovides a credible means for estimating characteristiclength-scale variations on longer timescales Table 2 givesthe range of estimates of the standard deviation of glacierfluctuations in response to the observed interannual climatevariability If forced with normally distributed interannualfluctuations Equation (1) will produce normally distributedglacier length fluctuations Thus by definition of the standarddeviation the glacier will spend 30 of its time outside1 5 of its time outside 2 and 03 of its timeoutside 3 Therefore the statistical expectation is that forthree years out of every thousand the maximum andminimum lengths of the glacier during that time will be

separated by at least 6 Table 2 shows that this range ofparameter uncertainty 6 varies between 2064 and 3414mwith a mid-range estimate of 2316m We emphasize thismillennial-scale variability must be expected of a glaciereven in a constant climate as a direct result of the simpleintegrative physics of a glacierrsquos response time or memory

To convey a sense of what this means in practiceFigure 5a shows a 5000 year integration of the linear modelwith geometry parameters equal to our typical Mount Bakerglacier The glacier model was driven by normally distrib-uted random temperature and precipitation variations withstandard deviations given by the MM5 output for MountBaker By eye it can be seen that there is substantialcentennial-scale variability with amplitude 2ndash3 km Alsoshown in Figure 4a are maximum terminus advances that arenot subsequently overridden Therefore these suggest occa-sions when moraines might be left preserved on thelandscape (though the precise mechanisms of morainedeposition and conditions for preservation remain uncer-tain) Just by the statistics of chance the further back in timeone goes the likelier it is that moderate advances andretreats have been overridden so moraines become morewidely separated in time (eg Gibbons and others 1984)Again we emphasize that none of the centennial andmillennial variability in our modeled glacier terminus arisesbecause of climate change To infer a true climate change

Fig 5 (a) 5000 year integration of the linear glacier model using parameters similar to the typical Mount Baker glacier (Table 1) and drivenby random realizations of interannual melt-season temperature and precipitation variations consistent with statistics for Mount Baker fromthe MM5 model output The black curve shows the time series for the mid-range estimate of parameters The green curve is a 100 yearrunning average The dots denote maximum terminus advances that are not subsequently overridden and so are possible times for moraineformation (b) The black curve is the power spectral density (PSD) estimate of the time series generated using the mid-range parameters Thegray curve is the theoretical red-noise spectrum (solid) together with its 95 confidence band (dashed) (AR(1) first-order autoregressiveprocess) Spectra were calculated using a periodogram with a 1000 year Hanning window The arrow shows the frequency corresponding to1 so the spectrum emphasizes that much of the variability in the glacier time series occurs at periods much longer than the physicalresponse time of the glacier


from a single glacier reconstruction the glacier change mustexceed at some statistical level of confidence the variabilityexpected in a constant climate

Figure 5b shows the power-spectral estimate of theterminus variations in Figure 4a together with the theoret-ical spectrum for a statistical process given by Equation (8)(eg Jenkins and Watts 1968 von Storch and Zwiers 1999)It can be shown that half the variance in the power spectrumoccurs at periods at least 2 times longer than the physicaltimescale of the system in this case frac14 12 years (eg Roe2009) Thus there is centennial and even millennialvariability in the spectrum all fundamentally driven by thesimple integrative physics of a process with a perhapssurprisingly short timescale and forced by simple stochasticyear-to-year variations in climate

53 Comparison with a dynamic glacier modelWe next briefly evaluate the behavior of the linear glaciermodel compared to that of a dynamic flowline model (egPaterson 1994) The model incorporates the dynamicalresponse of a single glacier flowline based on the shallow-ice approximation (eg Paterson 1994) Assuming a no-sliplower boundary condition the flowline obeys a non-lineardiffusion equation governed by Glenrsquos flow law (egPaterson 1994)

hethxTHORNt

thorn

x2AethgTHORNnhethxTHORNnthorn2 dzsethxTHORN

dx

n frac14 _bethxTHORN eth12THORN

where x is the horizontal coordinate h(x) is the thickness ofthe glacier zs is the surface slope _bethxTHORN is the net mass

balance at a point on the glacier is ice density and g isgravitational acceleration The exponent relating appliedstress to strain rates n is set equal to 3 and A is a softnessfactor taken to be 50 10ndash24 Pandash3 sndash1 The flowline model isa reasonable representation of ice flow and also includesthe non-linearities of the mass-balance perturbation Thus acomparison between models is a useful evaluation of thevalidity of the assumptions made in deriving the linearmodel

Equation (12) can be solved using standard numericalmethods Figure 6a shows the length history from anintegration of the dynamical flowline model for climaticand geometrical conditions identical to the linear modelcalculations shown in Figure 5 and for mid-range values ofparameters shown in Table 1 Figure 6bndashd show a smallersegment of the integration together with the melt-seasontemperature and accumulation variations that force theglacier We emphasize again that the climate just reflects therandom interannual fluctuations inherent to a constantclimate The figure demonstrates that large and persistentclimate fluctuations are not necessary to drive large andpersistent glacier fluctuations

The dynamical model produces a glacier length recordwith a standard deviation of 370m compared with 420m forthe linear model Thus the magnitudes of variations producedby the two models are similar though there is a suggestionthat the linear model may overestimate the glacier responseby about 15 Obviously a much fuller exploration of theglacier dynamics is possible no glacier sliding has beenincluded nor has there been any account of the confininglateral stresses of the side-walls Computational constraints

Fig 6 (a) Glacier length variations from a 5000 year integration of a dynamical flowline glacier model in a constant climate using the sameparameters as the linear model and as described in the text Note the slightly reduced length variations compared to the linear model (b) A500 year long segment of glacier-length variations from (a) (c d) Melt-season temperature (c) and accumulation (d) variations that drive thelength variations A 30 year running mean is also shown As in Figure 5 the climate forcing is white noise scaled by the characteristicamplitude of variability near Mount Baker reflecting the random interannual fluctuations inherent in a constant climate The figuredemonstrates that large persistent climate fluctuations are not necessary to produce large persistent fluctuations in glacier length


limit the model resolution to 75m and the pattern of theclimatic perturbations has been assumed uniform Such anexploration is important to undertake and is the subject ofongoing work However the purpose in the present study isto establish that the linear model produces a reasonablemagnitude of glacier variability compared with a modelincorporating ice-flow dynamics and without linearization ofthe mass balance

6 SUMMARYA simple linear model has been presented for estimating theresponse of typical mid-latitude glaciers to climate forcingwith a particular focus on the interannual variability inaccumulation and melt-season temperature that is inherenteven in a constant climate With one free matchingparameter allowed and otherwise using standard physicalgeometrical and climatic parameters pertaining to the regionthe model produces a reasonable simulation of the observedvariations of glaciers on Mount Baker in the Cascade Rangeof Washington State over the past 75 years The magnitude ofvariability in the simple model also approximates that seen ina more complicated model incorporating ice-flow dynamics

Mount Baker glacier lengths are more sensitive toaccumulation than to melt-season temperature by a factorof 2ndash4 The maritime climate and mountainous terrain ofthe region produces large interannual accumulation vari-ability (1mandash1) and muted melt-season temperaturevariability By contrast calculations using the same modelfor glaciers in contintental climates show the reversesensitivity (Huybers and Roe in press) The expressiongiven in Equation (10) is a simple and robust indicator ofthe relative importance of melt-season temperature andaccumulation variability for a glacier The uncertainty factorof 2 is principally due to uncertainty in the melt factor andthe AAR Both of these can be much more tightlyconstrained for specific glaciers by careful observationsand several lines of evidence support the mid-rangeparameter values as being appropriate for this setting

Within the bounds of the observed natural variability inclimate expressed by instrumental observations between1931 and 1990 and the range of model parameters that weconsider to be reasonable the 13ndash25 km length fluctua-tions on Mount Baker attributed to the LIA can be accountedfor by the model without recourse to changes in climate Avariety of external climate forcings are commonly invoked toexplain glacier length fluctuations on centennial to millen-nial scales (eg changes in the strength of the atmosphericcirculation (eg OrsquoBrien and others 1995) atmosphericdust from volcanic eruptions (eg Robock and Free 1996)and variations in sunspot activity (eg Soon and Baliunas2003)) However the model results indicate that at least inthis setting kilometer-scale fluctuations of the glacierterminus do not require a substantial change in temperatureor precipitation and should be expected simply from naturalyear-to-year variability in weather

It is worth being clear about the logic of the argumentbeing made Nothing we have done is a definitivedemonstration that no climate change occurred in thePacific Northwest However in order to demonstrate thatglacier histories are indeed a response to climate change wemust first falsify the null hypothesis of no climate change Inparticular to attribute the nested sequences of late-Holocene moraines on Mount Baker to a distinct climate

change we require that changes in glacier length werelarger or of longer duration than that expected to be drivenby the observed instrumental record of interannual vari-ability Our results suggest that by itself the most recent twocenturies of glacier history at Mount Baker do not falsify thenull hypothesis

7 DISCUSSION

71 Model frameworkThe linear glacier model required several important assump-tions about ice dynamics and also neglected non-linearitiesin the glacier mass balance We discuss here what theconsequences of doing this might be Comparing with anon-linear dynamical flowline glacier model we find thatthe linear model produces variability about 15 larger thanthe dynamical model This is adequate agreement weemphasize that for the purpose of this paper the magnitudeof glacier variability only needs to be reasonably representedand therefore that the linear model is an adequate tool forthe question posed Indeed in view of other uncertainties inthe problem such as the details of the glacier mass balanceit is unclear what value would be added by employing amore complex glacier model Nonetheless further explora-tion of the reasons for the differences between the twoglacier models would be interesting and enlightening

Note also that we focused on a single characteristicMount Baker glacier but one should expect some sizeabledifferences in the magnitude of glacier variability evenamong glaciers as close together as those around MountBaker because of differences in geometry For example Atot

has a considerable influence on glacier variability as we seefor example from Equations (8) or (9) and Atot varies by afactor of 2 among Mount Baker glaciers (Table 1)

Our approach to the relationship between climate andglacier mass balance was crude A distinction between snowand rain might be more carefully made Based on thefraction of annual precipitation in winter we estimate thismight perhaps have a 20 effect on our answers Inaddition we assumed a simple proportionality betweenablation and temperature of a melt season of fixed length Atreatment based on positive degree days could easily besubstituted (eg Braithwaite and Olesen 1989) However itis not temperature per se but rather heat that causesablation A full surface energy-balance model is necessary toaccount for the separate influence of radiative and turbulentfluxes albedo variations cloudiness and aspect ratio of theglacier surface on steep and shaded mountain sides (egRupper and Roe 2008 Rupper and others in press) It ishard to single out any of these effects as more important thanthe others To pursue all of them in a self-consistentframework would require a detailed surface energy-balanceand snowpack model including the infiltration percolationand refreeze of meltwater The resulting system would becomplicated and it is not clear that with all its attendantuncertainties it would produce a higher-quality or moremeaningful answer than our first-order approach

Several other factors that we have not incorporatedprobably act to enhance glacier variability over and abovewhat we have calculated The interannual variability weassumed was less than that implied by limited wintertimemass-balance observations perhaps due to our havingneglected mass sources due to avalanching and wind-blown


snow both of which increase the effective area over which aglacier captures precipitation We have assumed the glaciersurface slope is linear The characteristically convex-upprofile of a real glacier acts to enhance the glacier sensitivitysince the ablation area as well as the ablation rate increaseswith increasing melt-season temperature (eg Roe andLindzen 2001) Including a feedback between glacierthickness and length increases the timescale and magnitudeof the glacier fluctuations (eg Harrison and others 2001Oerlemans 2001) Finally while no statistically significantinterannual memory in annual precipitation could bedemonstrated from the available weather-station recordsnear Mount Baker other gridded atmospheric reanalysisdatasets do suggest some weak interannual persistenceAlthough annual mean precipitation varies spatially withinthe region Huybers and Roe (in press) find some 1 year lagcorrelations in anomalies of around 02ndash03 and Stoufferand others (2000) find similar values for mean annualtemperature in some maritime settings A small autocorrela-tion like this makes it slightly more likely that the next yearrsquosprecipitation anomaly will have the same sign as this yearrsquosand so act to re-enforce it Huybers and Roe (in press) showthat a 1 year climate autocorrelation of r is enough toamplify the glacier variability in Equations (8) and (9) by afactor of

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1thorn 2reth1t=THORN=eth1 rt=THORNp

or about 35 forr frac14 025 similar to that found by Reichert and others(2002) For addressing the persistence of climate anomalieselsewhere in the world it is important to extend the analysesof Stouffer and others (2000) to melt-season temperaturesand annual precipitation which are fields of more directrelevance to glacier mass balance This work is in progressOn the basis of all the arguments given above we haveevery reason to think that our estimate of glacier response tonatural climate variability in this setting errs on theconservative side

72 ImplicationsOne lesson from our analyses is that small-scale patterns inclimate forcing inevitable in mountainous terrain aretremendously important for adequately capturing the glacierresponse Had we used the nearest long-term record fromthe weather station at Diablo Dam we would haveunderestimated the glacier variability by 65 The lapserate that the glacier surface experiences during the meltseason has an important effect on the glacier response ascan be seen from Equations (2) (8) and (9) The relevantlapse rate is likely not simply a typical free-air value asassumed here but rather some complicated dependence onlocal setting and mountain meteorology (PW Mote andothers unpublished information) The archive of high-resolution MM5 output provides an invaluable resource forthe investigation of such effects and will be the focus offuture investigations

It is also possible to take advantage of spatial patterns ofglacier variability in interpreting climate Huybers and Roe(in press) show that spatial patterns of melt-season tempera-ture and annual precipitation are coherent across large tractsof western North America though not always of the samesign (eg there is an anticorrelation of precipitation betweenAlaska and the Pacific Northwest) On spatial scales forwhich patterns of natural climate variability are coherentcoherent glacier variability must also be expected tightlyclustered glaciers provide only one independent piece of

information about climate Huybers and Roe (in press) useEquation (1) to evaluate how patterns of melt-seasontemperature and annual accumulation are convolved byglacier dynamics into regional-scale patterns of glacierresponse

Patterns of climate variability that are spatially coherentand also account for a large fraction of the local variance areregional in scale so the current worldwide retreat consti-tutes a powerful suggestion of global climate change (egOerlemans 2005) However a formal attribution of stat-istical significance requires (1) propagation of the prob-ability distribution of uncertainties in model parametersthrough the glacier model (2) accounting for the relativeimportance of melt-season temperature and precipitation indifferent climate settings (3) evaluation of how muchindependent information is represented by clustered glacierrecords and (4) determination of whether the trend risesabove the expected background variability We anticipatethat the global glacier record will probably pass such asignificance test but performing it will add greatly to thecredibility of the claim The model presented here provides atool for such a test

The long-term kilometer-scale fluctuations predicted bythe model provide the opportunity to suggest alternativeinterpretations or scenarios for moraine ages which are oftenattributed to poorly dated glacier advances between the 12thand 20th centuries Many moraines at Mount Baker and inother Cascade glacier forelands with similar physiographicsettings and glacier geometries have been dated bydendrochronology using tree species that are at the limit oftheir lifespan The range of glacier fluctuations produced bythe model combined with these poor constraints in theactual landform ages suggests that these moraines may beproducts of even earlier advances not necessarily synchro-nous with each other and certainly not necessarily part of aglobal pattern of climate fluctuations Random climaticfluctuations over the past 1000 years may have been ampleenough to produce large changes in glacier length and untilquantitative dating techniques can be used to reliablycorrelate widely separated advances from this interval theseadvances cannot be used as the main evidence for asynchronous signal of regional or global climate change

The primary purpose of this paper was to explore the ideathat substantial long-timescale glacier variability occurseven in a constant climate We have focused our analyseson the Pacific Northwest and found that the null hypothesisthat no climate change is required to explain the 19th-century advances on Mount Baker cannot be ruled outReichert and others (2002) came to the same conclusion forNigardsbreen and Rhoneglestcher Given the importance ofthe question it is crucial to extend these calculations toother glacier settings both considering variations in glaciergeometry and evaluating the interannual persistence ofclimate variables most relevant to glaciers Such work iscurrently under way

The results raise the possibility that variations recorded inmany glacier histories may have been misattributed toclimate change The effect on glacier length of the inter-annual variations inherent in a constant climate should notbe ruled out as a factor in explaining glacier historieswithout a careful analysis being made Although glacierrecords form the primary descriptor of climate history inmany parts of the world those records are in generalfragmentary and provide only a filtered glimpse of the


magnitude of individual glacier advances and retreats andof the regional or global extent of the coherent patterns ofglacier variations

The formal evaluation of whether the magnitude orregional coherence of glacier variability does or does notexceed that expected in a constant climate is a detailed andcomplicated exercise At a minimum it involves knowing(1) small-scale patterns of climate forcing and their vari-ability (2) the relationship between those variables and theglacier mass balance and (3) that the glacier dynamics arebeing adequately captured Regional- or global-scalepatterns of past glacier variability are also useful but sufferfrom difficulties in accurately cross-dating the histories Ourresults demonstrate however that such an evaluation mustbe performed before glacier changes can confidently beascribed to climate changes Given the very few exampleswhere this has been done at the necessary level of detail asubstantial re-evaluation of the late-Holocene glacier recordmay be called for

ACKNOWLEDGEMENTSWe thank K Huybers S Rupper E Steig B HansenC Wunsch E Waddington and C Raymond for insightfulconversations and comments We are enormously grateful toJ Minder and N Johnson for heroic efforts to compile theMM5 output from archives We also appreciate detailedcomments from T Johannesson and two anonymousreviewers that were instrumental in focusing the manuscriptGHR acknowledges support from US National ScienceFoundation (NSF) Continental Dynamics grant No 0409884

REFERENCESAnders AM GH Roe DR Durran and JR Minder 2007 Small-

scale spatial gradients in climatological precipitation on theOlympic Peninsula J Hydromet 8(5) 1068ndash1081

Anderson RS and 6 others 2004 Strong feedbacks betweenhydrology and sliding of a small alpine glacier J Geophys Res109(F3) F03005 (1010292004JF000120)

Barsugli JJ and DS Battisti 1998 The basic effects of atmos-pherendashocean thermal coupling on midlatitude variabilityJ Atmos Sci 55(4) 477ndash493

Bitz CC and DS Battisti 1999 Interannual to decadal variabilityin climate and the glacier mass balance in Washington westernCanada and Alaska J Climate 12(11) 3181ndash3196

Braithwaite RJ and OB Olesen 1989 Calculation of glacierablation from air temperature West Greenland In OerlemansJ ed Glacier fluctuations and climatic change DordrechtKluwer Academic Publishers 219ndash233

Bretherton CS and DS Battisti 2000 An interpretation of theresults from atmospheric general circulation models forced bythe time history of the observed sea surface temperaturedistribution Geophys Res Lett 27(6) 767ndash770

Burke R 1972 Neoglaciation of Boulder Valley Mt BakerWashington (MS thesis Western Washington University)

Colle BA CF Mass and KJ Westrick 2000 MM5 precipitationverification over the Pacific Northwest during the 1997ndash99 coolseasons Weather Forecast 15(6) 730ndash744

Denton GH and SC Porter 1970 Neoglaciation Sci Am222(6) 100ndash110

Deser C MA Alexander and MS Timlin 2003 Understandingthe persistence of sea surface temperature anomalies inmidlatitudes J Climate 16(1) 57ndash72

Frankignoul C and K Hasselmann 1977 Stochastic climatemodels Part 2 Application to sea-surface temperature anoma-lies and thermocline variability Tellus 29(4) 289ndash305

Frankignoul C P Muller and E Zorita 1997 A simple model ofthe decadal response of the ocean to stochastic wind forcingJ Phys Oceanogr 27(8) 1533ndash46

Fuller SR 1980 Neoglaciation of Avalanche Gorge and theMiddle Fork Nooksack River valley Mt Baker Washington (MSthesis University of Western Washington)

Garvert MF B Smull and C Mass 2007 Multiscale mountainwaves influencing a major orographic precipitation eventJ Atmos Sci 64(3) 711ndash737

Gibbons AB JD Megeath and KL Pierce 1984 Probability ofmoraine survival in a succession of glacial advances Geology12(6) 327ndash330

Grell GA J Dudhia and DR Stauffer 1995 A description of thefifth-generation Penn StateNCAR mesoscale model (MM5)Boulder CO National Center for Atmospheric Research (NCARTech Note TN-398+STR)

Harper JT 1992 The dynamic response of glacier termini toclimatic variation during the period 1940ndash1990 on Mount BakerWashington USA (MS thesis Western Washington University)

Harrison WD DH Elsberg KA Echelmeyer and RM Krimmel2001 On the characterization of glacier response by a singletime-scale J Glaciol 47(159) 659ndash664

Harrison WD CF Raymond KA Echelmeyer andRM Krimmel 2003 A macroscopic approach to glacierdynamics J Glaciol 49(164) 13ndash21

Hasselmann K 1976 Stochastic climate models Part 1 TheoryTellus 28 473ndash483

Hodge SM DC Trabant RM Krimmel TA HeinrichsRS March and EG Josberger 1998 Climate variations andchanges in mass of three glaciers in western North AmericaJ Climate 11(9) 2161ndash2179

Huybers K and GH Roe In press Spatial patterns of glaciers inresponse to spatial patterns in regional climate J Climate

Huybrechts P P de Nooze and H Decleir 1989 Numericalmodelling of Glacier drsquoArgentiere and its historic front vari-ations In Oerlemans J ed Glacier fluctuations and climaticchange Dordrecht etc Kluwer Academic Publishers 373ndash389

Jenkins GM and DG Watts 1968 Spectral analysis and itsapplications San Francisco CA Holden-Day

Johannesson T C Raymond and E Waddington 1989 Time-scalefor adjustment of glaciers to changes in mass balanceJ Glaciol 35(121) 355ndash369

Klok EJ 2003 The response of glaciers to climate change (PhDthesis University of Utrecht)

Kovanen DJ 2003 Decadal variability in climate and glacierfluctuations on Mt Baker Washington USA Geogr Ann85A(1) 43ndash55

Lillquist K and K Walker 2006 Historical glacier and climatefluctuations at Mount Hood Oregon Arct Antarct Alp Res38(3) 399ndash412

Long WA 1955 Whatrsquos happening to our glaciers Sci Mon81(2) 57ndash64

Mantua NJ SR Hare Y Zhang JM Wallace and RC Francis1997 A Pacific interdecadal climate oscillation with impacts onsalmon production Bull Am Meteorol Soc 78(6) 1069ndash1079

Meier MF and AS Post 1962 Recent variations in mass netbudgets of glaciers in western North America IASH Publ 58(Symposium at Obergurgl 1962 ndash Variations of the Regime ofExisting Glaciers) 63ndash77

Meier MF WV Tangborn LR Mayo and A Post 1971Combined ice and water balances of Gulkana and WolverineGlaciers Alaska and South Cascade Glacier Washington 1965and 1966 hydrologic years USGS Prof Pap 715-A

Minder JR DR Durran GH Roe and AM Anders 2008 Theclimatology of small-scale orographic precipitation over theOlympic Mountains patterns and processes Q J R MeteorolSoc 134(633) 817ndash839

Moore RD and MN Demuth 2001 Mass balance and stream-flow variability at Place Glacier Canada in relation to recentclimate fluctuations Hydrol Process 15(18) 3472ndash3486


Mote PW 2003 Trends in snow water equivalent in the PacificNorthwest and their climatic causes Geophys Res Lett30(12) 1601 (1010292003GL017258)

Mote PW AF Hamlet MP Clark and DP Lettenmaier 2005Declining mountain snowpack in western North America BullAm Geogr Soc 86(1) 39ndash49

Nesje A and SO Dahl 2003 The lsquoLittle Ice Agersquo ndash onlytemperature Holocene 13(1) 139ndash145

Newman M GP Compo and MA Alexander 2003 ENSO-forced variability of the Pacific decadal oscillation J Climate16(23) 3853ndash3857

Nye JF 1952 The mechanics of glacier flow J Glaciol 2(12)82ndash93

OrsquoBrien SR PA Mayewski LD Meeker DA MeeseMS Twickler and SI Whitlow 1995 Complexity of Holoceneclimate as reconstructed from a Greenland ice core Science270(5244) 1962ndash1964

OrsquoNeal MA 2005 Late Little Ice Age glacier fluctuations in theCascade Range of Washington and northern Oregon (PhDthesis University of Washington)

Oerlemans J 2001 Glaciers and climate change Lisse etcAA Balkema

Oerlemans J 2005 Extracting a climate signal from 169 glacierrecords Science 308(5722) 675ndash677

Ohmura A M Wild and L Bengtsson 1996 A possible change inmass balance of Greenland and Antarctic ice sheets in thecoming century J Climate 9(9) 2124ndash2135

Paterson WSB 1994 The physics of glaciers Third editionOxford etc Elsevier

Pederson GT DB Fagre ST Gray and LJ Graumlich 2004Decadal-scale climate drivers for glacial dynamics in GlacierNational Park Montana USA Geophys Res Lett 31(12)L12203 (1010292004GL019770)

Pelto MS 2000 Summer snowpack variations with altitude onMount Baker Washington 1990ndash1999 a comparison withrecord 19981999 snowfall In Hardy JP and J Pomeroy edsProceedings of the 57th Annual Eastern Snow Conference17ndash19 May 2000 Syracuse New York Hanover NH USArmy Cold Regions Research and Engineering Laboratory79ndash84

Pelto MS 2006 The current disequilibrium of North Cascadeglaciers Hydrol Process 20(4) 769ndash779

Pelto MS and C Hedlund 2001 Terminus behavior and responsetime of North Cascade glaciers Washington USA J Glaciol47(158) 497ndash506

Pelto MS and J Riedel 2001 Spatial and temporal variations inannual balance of North Cascade glaciers Washington 1984ndash2000 Hydrol Process 15(18) 3461ndash3472

Pollard D 1980 A simple parameterization for ice sheet ablationrate Tellus 32(4) 384ndash388

Porter SC 1977 Present and past glaciation threshold in theCascade Range Washington USA topographic and climaticcontrols and paleoclimatic implications J Glaciol 18(78)101ndash116

Raper SCB O Brown and RJ Braithwaite 2000 A geometricglacier model for sea-level change calculations J Glaciol46(154) 357ndash368

Reichert BK L Bengtsson and J Oerlemans 2002 Recent glacierretreat exceeds internal variability J Climate 15(21)3069ndash3081

Robock A and MP Free 1996 The volcanic record in ice coresfor the past 2000 years In Jones PD RS Bradley and J Jouzeleds Climatic variations and forcing mechanisms of the last 2000years Berlin Springer-Verlag (NATO ASI Series 1 GlobalEnvironmental Change 41)

Roe G 2009 Feedbacks timescales and seeing red Annu RevEarth Planet Sci 37 93ndash15

Roe GH and RS Lindzen 2001 A one-dimensional model forthe interaction between continental-scale ice-sheets and atmos-pheric stationary waves Climate Dyn 17(5ndash6) 479ndash487

Rupper S and G Roe 2008 Glacier changes and regional climatea mass and energy balance approach J Climate 21(20)5384ndash5401

Rupper S EJ Steig and G Roe 2004 The relationship betweensnow accumulation at Mt Logan Yukon Canada and climatevariability in the North Pacific J Climate 17(24) 4724ndash4739

Rupper SB GH Roe and A Gillespie In press Spatial patterns ofglacier advance and retreat in Central Asia in the HoloceneQuat Res

Schneider T and A Neumaier 2001 Algorithm 808 ARfit ndash aMatlab package for the estimation of parameters and eigen-modes of multivariate autoregressive models ACM Trans MathSoftw 27(1) 58ndash65

Solomon S and 7 others eds 2007 Climate change 2007 thephysical science basis Contribution of Working Group I to theFourth Assessment Report of the Intergovernmental Panel onClimate Change Cambridge etc Cambridge University Press

Soon W and S Baliunas 2003 Proxy climatic and environmentalchanges of the past 1000 years Climate Res 23(2) 89ndash110

Stouffer RJ G Hegerl and S Tett 2000 A comparison of surfaceair temperature variability in three 1000-yr coupled oceanndashatmosphere model integrations J Climate 13(3) 513ndash537

Thomas PA 1997 Late Quaternary glaciation and volcanism onthe south flank of Mt Baker Washington (MS thesis WesternWashington University)

Von Storch H and FW Zwiers 1999 Statistical analysis in climateresearch Cambridge etc Cambridge University Press

Wunsch C 1999 The interpretation of short climate records withcomments on the North Atlantic and Southern Oscillations BullAm Meteorol Soc 80(2) 245ndash255

APPENDIX

DERIVATION OF LINEAR GLACIER MODELEQUATIONSHere we derive the equations used in the linear glaciermodel using the geometry shown in Figure 2 FollowingJohannesson and others (1989) the model considers onlyconservation of mass The rate of change of glacier volumeV can be written as

dVdtfrac14 accumulation ablation ethA1THORN

We assume the ablation rate is T where T is the melt-season temperature and is the melt factor A constantmight be added to the ablation rate as in Pollard (1980) orOhmura and others (1996) However since the modelequations will be linearized about the equilibrium state theconstant would not enter into the first-order terms We alsoassume that all melting ends up as run-off

Let the temperature at any point on the glacier be givenby

T frac14 Tp thorn x tanrsquo ethA2THORNwhere x is the distance from the head of the glacier Tp is themelt-season temperature at the head of the glacier is theatmospheric lapse rate and rsquo is the slope of the glaciersurface (assumed parallel to the bed) The temperature at thetoe of the glacier of length L is then equal to

T ethx frac14 LTHORN frac14 Tp thorn L tanrsquo ethA3THORNThere is some melting wherever the melt-season temperatureexceeds 08C and the area of the melting zone is given byATgt0 frac14 w L xTfrac140eth THORN Since temperature decreases linearlywith elevation and the local melt rate is proportional totemperature the total melting that occurs is equal to


05 times the total melt area multiplied by the melt rate atthe toe of the glacier (this is simply the area under thetriangle of the melt-ratendashdistance graph) or in other words

ablation frac14 12ATgt0Txfrac14L ethA4THORN

Note that melting above the ELA is included in thiscalculation The total accumulation is just the product ofthe precipitation rate P (assumed uniform over the glacier)and the total glacier area Atot

We now linearize the glacier system about someequilibrium climate state denoted by an overbarP P thorn P 0 Tp Tp thorn T 0 and L Lthorn L0 where theclimate anomalies are uniform over the glacier Similar toprevious studies H and w are assumed constant (egJohannesson and others 1989 though this assumption canbe relaxed (eg Oerlemans 2001)) which meansV 0 frac14 wHL0 We also note that the melt-season freezing linexTfrac140 can be found from Equation (A3)

Tp thorn T 0 frac14 xTfrac140 tanrsquo ethA5THORNSo in the perturbed state and using Equations (A2) and (A3)Equation (A4) becomes

ablation frac14 12w ethLthorn L0THORN xTfrac140

Tp thorn T 0 thorn ethLthorn L0THORN tanrsquo

ethA6THORNSubstituting xTfrac140 from Equation (A5) and rearranging a littlegives

ablation frac14 w2 tanrsquo

ethLthorn L0THORN tanrsquothorn Tp thorn T 0 2

ethA7THORN

which using Equation (A5) can be written as


ethL xTfrac140THORN tanrsquothorn L0 tanrsquothorn T 0 2

ethA8THORN

Note that wethL xTfrac140THORN ATgt0 is the area over whichmelting occurs For accumulation we can write

accumulation frac14 ethP thorn P 0THORNethAtot thornwL0THORN ethA9THORNTaking only first-order terms and combining Equations (A1)(A8) and (A9) we obtain

wHdL0

dtfrac14 Pw ATgt0 tanrsquo

L0 thorn AtotP 0 ATgt0T 0

ethA10THORNOne last simplification is possible At the climatologicalELA

P frac14 T ela

and so

wP frac14 wT ela frac14 wethTp thorn xela tanrsquoTHORNfrac14 w tanrsquoethxela xTfrac140THORN ethA11THORN

where Equations (A2) and (A5) have been used Defining theablation zone in the sense of for example Paterson (1994)as the reg ion below the ELA we can wri teAabl frac14 ATgt0 wethxTfrac140 xelaTHORN Finally then Equation (A10)becomes

dL0

dtthorn Aabl tanrsquo

wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN

where the overbars have been dropped for convenienceEquation (A12) is the governing equation of the glaciermodel used in this study

Lastly using Equation (A11) the relationship between themelt-zone area and the ablation-zone area is given by

ATgt0 frac14 Aabl thornwethxela xTfrac140THORN frac14 Aabl thorn Pw tanrsquo

ethA13THORN

For the standard model parameters given in Table 1 ATgt0

adds another 167 km2 to Aabl

MS received 3 May 2009 and accepted in revised form 7 June 2009


in this paper is not to create a simulation of the evolution ofany observed glacier Rather our approach is to employ thesimplest modeling framework in which the basic behaviorcan be demonstrated We use a combination of observationsand reconstructions of climate and glaciers to calibrate andevaluate a simple model for the glaciers on Mount BakerWe emphasize here that the model adequately reproducesthe observed characteristic variations of glacier length and isalso consistent with the observed trends over the lastcentury The analyses lead to some important results againstwhich to interpret glaciers in natural settings

Our approach mirrors that of Reichert and others (2002)who used a downscaled global climate model (GCM) outputand a dynamical glacier model for two European glaciers(Nigardsbreen Norway and Rhonegletscher Switzerland)They concluded that the present retreat exceeded naturalvariability but that Little Ice Age (LIA) advances did notThus a climate change (at least within their GCMglaciermodel system) was not required to explain LIA-like ad-vances Here our use of a linear model is a trade-off thelevel of sophistication of the glacier model is less but itssimplicity allows us to derive some simple expressions thatmake clear the dependencies of the system response

2 THE GLACIER MODELGlaciers are dynamic physical systems wherein ice deformsand flows in response to hydrostatic pressure gradientscaused by sloping ice surfaces There are other importantfactors to glacier motion among which are ice flow istemperature-dependent glaciers can slide over their base ifsubglacial water pressures are sufficient glaciers interactwith their constraining side-walls and glacier mass balancecan be sensitive to complicated mountain environments(eg Nye 1952 Pelto and Riedel 2001 Anderson andothers 2004) Despite these somewhat daunting complica-tions a series of papers have shown that simple linearmodels based on basic mass-balance considerations can beextremely effective in characterizing glacier response toclimate change (eg Huybrechts and others 1989Johannesson and others 1989 Raper and others 2000Harrison and others 2001 Oerlemans 2001 2005 Klok2003)

The model that we use includes an explicit and separaterepresentation of melt-season temperature and annual meanprecipitation in the mass balance A schematic of the modelis depicted in Figure 2 and a derivation of the modelequations is presented in the Appendix The model operates

Fig 1 Major Mount Baker glaciers (elevation contour interval 250m) Glaciers are shown at their LIA maxima 1930 and present positions







dL0ethtTHORNdt











dL0ethtTHORNdt


where


eth3THORN




frac14 wHL0

L0 tanrsquoAabl







eth4THORN




eth5THORN



R frac14 LTLP

frac14 ATgt0

Atot

TP

eth6THORN









thorn Atott

wHPt thorn ATgt0t

wHTt eth7THORN



pfrac14 T

ATgt0

wH


r eth8THORN


LP frac14 PAtot

wH


r eth9THORN


a step-change

R frac14 LT

LP

frac14 ATgt0

Atot

T

P eth10THORN




LP

q eth11THORN















Atot (km2) 430 540 215 360 270 40


















Min Mean Max



q(m) 344 415 569




















hethxTHORNt

thorn


dx














7 DISCUSSION


























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN










dL0ethtTHORNdt











dL0ethtTHORNdt


where


eth3THORN




frac14 wHL0

L0 tanrsquoAabl







eth4THORN




eth5THORN



R frac14 LTLP

frac14 ATgt0

Atot

TP

eth6THORN









thorn Atott

wHPt thorn ATgt0t

wHTt eth7THORN



pfrac14 T

ATgt0

wH


r eth8THORN


LP frac14 PAtot

wH


r eth9THORN


a step-change

R frac14 LT

LP

frac14 ATgt0

Atot

T

P eth10THORN




LP

q eth11THORN















Atot (km2) 430 540 215 360 270 40


















Min Mean Max



q(m) 344 415 569




















hethxTHORNt

thorn


dx














7 DISCUSSION


























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN








dL0ethtTHORNdt


where


eth3THORN




frac14 wHL0

L0 tanrsquoAabl







eth4THORN




eth5THORN



R frac14 LTLP

frac14 ATgt0

Atot

TP

eth6THORN









thorn Atott

wHPt thorn ATgt0t

wHTt eth7THORN



pfrac14 T

ATgt0

wH


r eth8THORN


LP frac14 PAtot

wH


r eth9THORN


a step-change

R frac14 LT

LP

frac14 ATgt0

Atot

T

P eth10THORN




LP

q eth11THORN















Atot (km2) 430 540 215 360 270 40


















Min Mean Max



q(m) 344 415 569




















hethxTHORNt

thorn


dx














7 DISCUSSION


























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN










thorn Atott

wHPt thorn ATgt0t

wHTt eth7THORN



pfrac14 T

ATgt0

wH


r eth8THORN


LP frac14 PAtot

wH


r eth9THORN


a step-change

R frac14 LT

LP

frac14 ATgt0

Atot

T

P eth10THORN




LP

q eth11THORN















Atot (km2) 430 540 215 360 270 40


















Min Mean Max



q(m) 344 415 569




















hethxTHORNt

thorn


dx














7 DISCUSSION


























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN














Atot (km2) 430 540 215 360 270 40


















Min Mean Max



q(m) 344 415 569




















hethxTHORNt

thorn


dx














7 DISCUSSION


























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN




















Min Mean Max



q(m) 344 415 569




















hethxTHORNt

thorn


dx














7 DISCUSSION


























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN














Min Mean Max



q(m) 344 415 569




















hethxTHORNt

thorn


dx














7 DISCUSSION


























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN






















hethxTHORNt

thorn


dx














7 DISCUSSION


























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN














hethxTHORNt

thorn


dx














7 DISCUSSION


























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN








hethxTHORNt

thorn


dx














7 DISCUSSION


























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN











7 DISCUSSION


























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN
























































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN













































































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN






































APPENDIX


















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN















ethA7THORN




ethA8THORN



wHdL0




P frac14 T ela

and so



dL0


wHL0 frac14 Atot

wHP 0 ATgt0

wHT 0 ethA12THORN




ethA13THORN





The response of glaciers to intrinsic climate variability: observations and models of late-Holocene variations in the Pacific Northwest

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