VOLUME 31 JOURNAL OF CLIMATE 1JANUARY 2018

A Robust Null Hypothesis for the Potential Causes of Megadrought in Western North America

a b c d c,e,f TOBY R. AULT, SCOTT ST.GEORGE, JASON E. SMERDON, SLOAN COATS, JUSTIN S. MANKIN, a e,c g CARLOS M. CARRILLO, BENJAMIN I. COOK, AND SAMANTHA STEVENSON a Department of Earth and Atmospheric Science, Cornell University, Ithaca, New York b Department of Geography, University of Minnesota, Twin Cities, Minneapolis, Minnesota c Division of Ocean and Climate Physics, Lamont–Doherty Earth Observatory, Columbia University, Palisades, New York d Climate and Large Scale Dynamics Laboratory, National Center for Atmospheric Research, Boulder, Colorado e NASA Goddard Institute for Space Studies, New York, New York f Department of Geography, Dartmouth College, Hanover, New Hampshire g Bren School of Environmental Science and Management, University of California, Santa Barbara, Santa Barbara, California

(Manuscript received 7 March 2017, in final form 11 September 2017)

ABSTRACT

The western United States was affected by several megadroughts during the last 1200 years, most prominently during the Medieval Climate Anomaly (MCA; 800 to 1300 CE). A null hypothesis is developed to test the possibility that, given a sufficiently long period of time, these events are inevitable and occur purely as a con- sequence of internal climate variability. The null distribution of this hypothesis is populated by a linear inverse model (LIM) constructed from global sea surface temperature anomalies and self-calibrated Palmer drought severity index data for North America. Despite being trained only on seasonal data from the late twentieth century, the LIM produces megadroughts that are comparable in their duration, spatial scale, and magnitude to the most severe events of the last 12 centuries. The null hypothesis therefore cannot be rejected with much confidence when considering these features of megadrought, meaning that similar events are possible today, even without any changes to boundary conditions. In contrast, the observed clustering of megadroughts in the MCA, as well as the change in mean hydroclimate between the MCA and the 1500–2000 period, are more likely to have been caused by either external forcing or by internal climate variability not well sampled during the latter half of the twentieth century. Finally, the results demonstrate that the LIM is a viable tool for determining whether paleoclimate reconstructions events should be ascribed to external forcings or to ‘‘out of sample’’ climate mechanisms, or if they are consistent with the variability observed during the recent period.

1. Introduction and dynamic circulation changes (Seager et al. 2007b; Woodhouse et al. 2010; Ault et al. 2014, 2016; Cook et al. Megadroughts are prolonged periods of aridity typically 2015, 2016; Kelley et al. 2015). If a megadrought were to defined by their multidecadal duration (Woodhouse and unfold in the coming decades, it would require water Overpeck 1998; Ault et al. 2014; Cook et al. 2016)and management decisions to be made on scales that have no have been linked to the demise of several preindustrial historical precedent. Understanding the risks of these civilizations (Benson et al. 2002, 2003; Hodell et al. 1995; events is therefore essential for supporting regional ad- Buckley et al. 2010; Stahle and Dean 2011). Mounting aptation strategies in the face of a changing climate. evidence suggests that the risk of these events is in- Furthermore, quantitative assessments of megadrought creasing throughout the southwestern United States and risk require us to disentangle the influences of external other subtropical dry zones due to rising temperatures and internal climate variability in the past because future likelihoods will be governed by both of these factors Denotes content that is immediately available upon publica- (Ault et al. 2014, 2016; Cook et al. 2015). tion as open access. The last millennium affords some insights into the range of forced and unforced preindustrial climate var- Corresponding author: Toby Ault, [email protected] iability, yet it is only one realization of many plausible

DOI: 10.1175/JCLI-D-17-0154.1 Ó 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). 3 Unauthenticated | Downloaded 10/03/21 08:07 PM UTC 4 JOURNAL OF CLIMATE VOLUME 31 alternative climate trajectories that could have occurred contrast, suggests that megadroughts could occur as a during that same time period. If a given climate forcing consequence of internal climate variability, if given a (e.g., enhanced solar output) made megadroughts more sufficiently long period of time. For example, Hunt probable in the past, but they failed to occur due to in- (2006) argued that megadroughts should be interpreted ternal climate variability, then one might erroneously as normal events in the unforced statistics of a long conclude that the forcing was not important to the sta- (10 000 yr) control run. Likewise, Meehl and Hu (2006) tistics of megadrought on the basis of the one realization linked megadroughts in the U.S. Southwest to low- being examined. Alternatively, if megadroughts occur frequency modes of climate variability using a millennial- without any clear external forcing, yet are nonetheless scale integration of a moderate-resolution GCM. More more probable during certain climate regimes, one recently, Coats et al. (2013b, 2015a,b, 2016a,b)and might correctly conclude that megadroughts do not re- Stevenson et al. (2015) argued that megadroughts occur quire exogenous changes to climate boundary condi- as a result of internal variability, and hence they do tions, but would miss the critical piece of information not require an exogenous mechanism. Furthermore, that such forcings can make megadroughts more likely. fully coupled GCMs forced with realistic, time-evolving These inherent constraints of having only one re- boundary conditions of the last millennium do not alization of reality have largely prevented a consensus universally simulate clusters of megadroughts during view from emerging on whether or not megadroughts the MCA (Cook et al. 2016; Coats et al. 2016a,b). This are forced [see the review by Cook et al. (2016)]. result could be interpreted as evidence that models fail A number of studies have suggested that mega- to simulate forced megadroughts, or alternatively that droughts in the western United States, and especially in these events occurred more frequently during the MCA the southwestern United States, require external forcing due to chance (Coats et al. 2016b). mechanisms to alter regional and global circulation The GCM-based studies above are inherently limited patterns in ways that have not been observed over the by the rarity of megadroughts even in long control historical period (e.g., Clement et al. 1996; Graham et al. simulations, and by structural biases in GCM moisture 2007, 2011; Mann et al. 2009; Seager et al. 2007a). Evi- fields (Ault et al. 2012; Flato et al. 2013) and tele- dence for this hypothesis stems in part from the ob- connections (Coats et al. 2013a). Even the largest en- servation that the Medieval Climate Anomaly (MCA; sembles (e.g., Kay et al. 2014; Otto-Bliesner et al. 2016) 801–1300 CE) occurred during a period of slightly en- only include a few dozen realizations. If megadroughts hanced solar activity (e.g., Lean 2000; Vieira and tend to occur at a rate of 1–2 events per millennium Solanki 2010) and saw at least three multidecadal in- [as originally proposed by Woodhouse and Overpeck tervals of aridity with drought area index (DAI) values (1998) and supported analytically by Ault et al. (2014)], above the worst droughts of the twentieth century then independent millennial-scale realizations still (Cook et al. 2004, 2016; Coats et al. 2016b). Idealized yield a relatively small sample size to assess mega- climate model experiments support this interpretation drought statistics [as does a 10 000-yr control run as in to varying degrees: when the Cane–Zebiak model Hunt (2006)]. Likewise, GCMs are known to have sys- (Zebiak and Cane 1987) of tropical Pacific Ocean dy- tematic biases both in their mean precipitation (Flato namics is forced with increased solar activity, it et al. 2013), as well as their tropical Pacific variability simulates a ‘‘La Niña–like’’ mean state (Clement et al. (Guilyardi et al. 2009; Ault et al. 2013b). Together, these 1996). Prolonged La Niña–like sea surface temperatures biases imply that megadrought statistics estimated from (SSTs), in turn, favor relatively arid conditions across GCMs are an imperfect approximation of the ‘‘true’’ the southwestern quadrant of the United States (Seager limiting behavior of the system. et al. 2007a; Burgman et al. 2010). In addition to external Here we develop an empirical approach for ruling out forcing factors, there is also evidence that regional the possibility that megadroughts could arise from un- feedbacks related to dust and land surface dynamics forced, stationary statistics. Specifically, we use a linear could make megadroughts more probable when the inverse model (LIM) framework to emulate the ob- mean state of the western U.S. hydroclimate is drier servable spatiotemporal patterns inherent to the climate than today (e.g., Cook et al. 2011, 2016). system over the instrumental period, when ocean– Under the interpretations above, prolonged and atmosphere data are widespread and of high quality. widespread megadroughts arise from climate regimes In climate studies, LIMs have been used to characterize that are inherently out-of-sample realizations of forced whether or not the recent clustering of central Pacific hydroclimate states not observed during the relatively El Niño events (also called ‘‘Modokis’’) is anomalous in short instrumental era. Analysis of unforced fully cou- the context of El Niño–Southern Oscillation (ENSO) pled general circulation model (GCM) simulations, in dynamics as they evolve naturally over the course of

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FIG. 1. Key characteristics of WUS megadrought from 801 to 2000 CE. (a) DAI35 computed as the fraction of grid points with 35-yr average PDSI values below 21 (red dot indicates the most widespread event). (b) Smoothed (35-yr running mean) PDSI from the SWUS (328–418N, 1248–1058W). The gray vertical bars highlight five MCA droughts (35-yr averages below 20.5 PDSI units), with the lowest 35-yr averaged PDSI value (21.04) circled in red (1159 CE). The long-term means of PDSI are plotted in red for MCA (20.19 PDSI units) and 1500-present (0.26 PDSI units). (c) Average PDSI across the WUS domain during the most widespread event (1130–1164 CE). The map of 35-yr averaged PDSI during the most negative (worst) SWUS drought in (b) is very similar due to the map of the most widespread drought because the two periods overlap in time. centuries (Newman et al. 2011). It has also been ap- 3) A ‘‘cluster’’ comprising five megadroughts occurred plied to test the significance of the substantial low- during a 500-yr period corresponding to the MCA frequency (decadal to centennial) variability in the tropi- (gray vertical bars in Fig. 1; Coats et al. 2016b; Cook cal Pacific (Ault et al. 2013b) and to evaluate significance et al. 2016). of ‘‘peaks’’ in the power spectrum of Pacific decadal os- 4) And, finally, the mean hydroclimatic state of the cillation (PDO) reconstructions (Newman et al. 2016). Southwest was drier on average during the MCA Our null hypothesis is that megadrought characteris- than any other subsequent period of time (red line in tics in the western United States are consistent with those Fig. 1b; Cook et al. 2004, 2016; Graham et al. 2007; of a stochastically forced, linearly damped system with Coats et al. 2016b). stationary statistics. That is, over any given millennium, The list above serves as a basis for developing mega- the interannual variability in global SSTs and western drought test statistics, the significance of which can then U.S. (WUS) hydroclimate, combined with spatial and be evaluated against the LIM’s distribution of these temporal sources of autocorrelation, will produce meg- same quantities. For example, if LIM realizations adrought statistics that are indistinguishable from re- regularly reproduce the spatial scale, magnitude, and constructions. We use an LIM to populate the null clustering of megadroughts, as well as changes in the distribution associated with this hypothesis, which in long-term mean values of reconstructed PDSI, then we turn permits us to test the significance of the following would not be able to reject the null hypothesis. Failing to characteristics of WUS megadrought (Fig. 1). reject this null hypothesis would imply that mega- 1) Events were widespread, encompassing between drought occurrence does not require external forcing 20% and 40% of the domain at their peak (Fig. 1a; mechanisms or climate behaviors that are any differ- Cook et al. 2004). ent from those observed during the twentieth century 2) The worst event in the southwestern United States (e.g., a multicentury warm phase of the Atlantic multi- occurred during the late 12th century and was as decadal oscillation; Feng et al. 2008; Oglesby et al. 2012; severe as the interannual droughts of the twenty-first Coats et al. 2016b; Wang et al. 2017). Rejecting the null century, but persisted for decades (Cook et al. 2004, hypothesis, on the other hand, would imply that the 2016). This event can be clearly identified in the 35-yr processes responsible for generating multidecadal pe- running mean of southwestern U.S. Palmer drought riods of aridity in the western U.S. hydroclimate origi- severity index (PDSI) in Fig. 1b (red circle). nate from either external forcings (e.g., solar variability,

Unauthenticated | Downloaded 10/03/21 08:07 PM UTC 6 JOURNAL OF CLIMATE VOLUME 31 explosive volcanism, or slow-varying orbital changes), an AR(1) model to evaluate the significance of behavior from low-frequency climate processes that are not well in a scalar time series (e.g., Bartlett 1978). The key dif- sampled over the twentieth century, or from nonlinear ference is that the AR(1) model applies only to a uni- coupled processes and feedbacks at multidecadal and variate setting (xt), with variations governed by an longer time scales. Accordingly, applying our null hy- autocorrelation term (r) and stochastic white noise (h), pothesis to the list of western U.S. hydroclimate char- such that xt 5 rxt21 1 h; it therefore cannot incorporate acteristics from the last 1200 years will help clarify which information about both the spatial and temporal scales aspects of megadrought are fundamentally unusual, ver- of autocorrelation. An LIM, in contrast, is appropriate sus merely unexpected. for simulating the variability produced by any multi- Finally, for clarity and consistency with previous variate system that evolves according to the following studies, we make a distinction between hydroclimate of Langevin equation for an Ornstein–Uhlenbeck process the entire western United States (WUS; 328–508N, 1258– (Penland and Sardeshmukh 1995): 1058W,), and the southwestern (SWUS) portion of that dX domain (328–418N, 1248–1058W). The dual foci of this 5 LX 1 z, (1) study therefore permits analysis of both the spatial scale dt of megadroughts across the WUS and their severity and where X is an m-dimensional state vector and z is white- duration in the SWUS, the latter features having re- noise forcing term, which itself may have spatial (or ceived somewhat more attention in the literature than temporal) structure, but whose realizations are un- the former. Moreover, among the leading modes of correlated from one time step to the next. The term L is variability in WUS hydroclimate is a ‘‘dipole’’ pattern an m 3 m linear deterministic feedback matrix, which with dry and wet anomalies of opposite sign between the includes information about how stochastic excitations Pacific Northwest and the SWUS (e.g., Ropelewski and propagate (linearly) through the system. If the system is Halpert 1986; Cook et al. 2017), which further motivates indeed approximately linear, the eigenvalues of L must analysis of both domains because megadroughts condi- have negative real parts; otherwise, it would contain tions in the SWUS are not necessarily guaranteed to be modes that grow exponentially and violate our as- representative of WUS-wide conditions. sumptions about stable linear statistics. The deterministic linear feedback matrix, L, can be 2. Linear inverse modeling estimated empirically from data as L 5 21 C C21 A linear inverse model presents a simple method to n lnf n 0 g. (2) emulate the statistical characteristics of certain dynamical C systems (Penland and Matrosova 1994; Penland and Here, the term n is a time lag parameter, n is the lag C Sardeshmukh 1995). In addition to the examples cited in covariance matrix (at lag n), and 0 is the instantaneous the previous section, the LIM approach has been used covariance matrix. To the extent that the system is ap- to make seasonal predictions of ENSO (Penland and proximately linear, the choice of n does not critically L Sardeshmukh 1995; Newman et al. 2011), to quantify alter the structure of because, by definition, the most ^ 1 predictability of the PDO (Newman 2007; Alexander probable state vector (X) at some lead time (t n)is et al. 2008; Newman et al. 2016), to characterize tropical given by Penland and Sardeshmukh (1995) as Pacific ocean–atmosphere coupling across time scales X^ 5 exp(Ln)X . (3) (Newman et al. 2009), and to evaluate decadal forecasts t1n t made by fully coupled global climate models (Newman In a typical application of an LIM, the noise term (z)is 2013). This suite of studies demonstrates that seasonal to not known, but instead its covariance matrix (Q 5 hzzTidt) decadal SST variability is well approximated by an LIM is estimated as a residual using the fluctuation–dissipation (Penland and Sardeshmukh 1995; Newman 2007, 2013; theorem (Penland and Matrosova 1994): Alexander et al. 2008; Smirnov et al. 2014). Because SSTs, dC and especially tropical Pacific SSTs, exert a major influ- 0 5 LC 1 C LT 1 Q 0 0 . (4) ence on WUS hydroclimate (Ropelewski and Halpert dt 1986, 1996; Redmond and Koch 1991; Brown and Comrie Further information on the assumptions, limitations, 2004), we extend the LIM approach to emulate drought and details of implementing an LIM for any generalized statistics in the WUS and the influence of SSTs on this system with approximately linear dynamics can be region across seasonal to centennial time scales. found in Penland and Sardeshmukh (1995) and Penland Using an LIM to generate null distributions for the and Matrosova (1994). Importantly, LIMs partition vari- statistics of widespread megadrought is similar to using ability into an approximately linear part, represented

Unauthenticated | Downloaded 10/03/21 08:07 PM UTC 1JANUARY 2018 A U L T E T A L . 7 by L, and a noise forcing part (Q). The variables contrib- PDSI (termed scPDSIpm) was generated using the uting to the noise term (Q) might exhibit nonlinear be- Penman–Montieth method for calculating potential havior at short time scales, and so an LIM approach is most evapotranspiration, and the University of East Anglia’s appropriate for situations where there is tempo- (UEA) Climate Research Unit’s (CRU) TS3.10 pre- ral averaging inherent to the system of interest (X). cipitation data product as input, as well as near-surface Additionally, the time series included in X are usually pressure, humidity, and radiation fields derived from re- smoothed to emphasize the approximately linear sources of analysis and remote sensing data (Sheffield et al. 2006). variability in the system (Penland and Sardeshmukh 1995). For comparison of the widespread multidecadal mega- This averaging (and smoothing), in turn, causes multiple drought intervals identified by Cook et al. (2004),were- realizations of the nonlinear phenomena to be integrated stricted our use of this scPDSIpm database to the WUS into each time step, and hence exhibit Gaussian statistics region (328–508N; 1258–1058W). Again, the period from via the central limit theorem (Yuval and Hsieh 2002). This January 1950 through December 2000 was used to train assumption is demonstrably valid for monthly SST anom- the LIM. Sensitivities of our LIM to the underlying data alies, as shown by many previous studies across weekly to are presented at the end of the section 4. multidecadal time scales (e.g., Penland and Sardeshmukh b. Data preparation 1995; Newman et al. 2009, 2011; Newman 2013). As with SST, the drought indices of interest here integrate monthly Following prior LIM studies (Penland and Sardeshmukh moisture supply and demand, and hence their variations 1995; Newman et al. 2011), principal components (PCs) are well suited to be simulated by an LIM approach. were used to reduce the dimensionality of each field, Given the similarity between Eq. (1) and the equation and a 3-month moving average was applied to smooth for univariate red noise, LIM output can be thought of as each time series prior to computing the PCs. The first 11 ‘‘multivariate red noise’’ (Newman et al. 2011; Ault et al. PCs were retained from global SST [as in Newman 2013b). Unlike univariate red noise, which smoothly (2013)] and jointly account for 60% of the total variance increases in amplitude from short to long time scales in that field, although the RMSE between the filtered and (Bartlett 1978), multivariate red noise can be used to raw data, as well as the total fraction of shared variability, generate time series characterized by strong spectral can be locally much higher (Fig. 2). While the smoothing peaks, as is the case in the tropical Pacific. These peaks has the benefit of stabilizing our estimate of L (i.e., arise from the linear dynamics of the system, even when making it less sensitive to noise), an obvious limitation it is forced by stochastic white noise, and hence yield an is that the LIM cannot reproduce 100% of the variance important baseline for variations generated in regions at all locations (Fig. 2). This limitation is not a serious affected by phenomena dominated by quasi-periodic one for our applications because, locally, at least 90% of variability (e.g., ENSO). the variability in the tropical Pacific and the WUS is captured by the LIM. In other applications, however, 3. A linear inverse model of western U.S. drought such as seasonal prediction, this limitation could present problems if we needed high-accuracy LIM forecasts of a. Data future hydroclimate states. We use global SST anomalies and PDSI values to The SST portion of the LIM preserves information about construct an LIM of the surface ocean and WUS the growth and decay structures of tropical Pacific ENSO hydroclimate. Monthly SST anomalies from 608Sto variability. Hence, it produces realistic interannual vari- 758N originate from the Kaplan et al. (1998) dataset. ability and Niño-3.4 power spectra, both of which are crit- This product is generated at a native spatial resolution of ical for emulating hydroclimate statistics in the WUS. The 58358 (500 km) lat/lon using a statistical procedure that characteristics of the LIM were not particularly sensitive to estimates missing data through time from spatial em- this choice, and a somewhat larger or smaller number of pirical orthogonal functions (EOFs), statistical inter- PCs could have been retained to yield similar results. polation, and Kalman filtering (Kaplan et al. 1998). In selecting the number of PCs to retain from the ei- Although data are available as far back as 1856, we only genvector decomposition of scPDSIpm, we required use data from 1950–2000 for training the LIM. It is also the total amount of variance in the field be comparable detrended to minimize the contribution of anthropo- to the full range of the raw dataset (otherwise, any in- genic forcing over that period, and smoothed prior to ferences about the statistics of megadrought in the LIM estimating L (see section 3b). would have limited relevance to paleoclimate records of Hydroclimate data for the WUS are obtained from these events). We therefore retained the leading 22 PCs the ‘‘self-calibrating’’ PDSI dataset developed and from scPDSIpm, which accounted for 90.8% of the raw published by Sheffield et al. (2006). This version of the (unsmoothed) variance of the WUS domain.

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FIG. 2. (top) Root-mean-square error (RMSE) and (bottom) percent variance accounted for from smoothing and EOF filtering the raw (left) SST and (right) PDSI. White patches over the ocean are areas of missing data.

T The final step in constructing X entails normalizing instantaneous covariance matrix: C0 5 hX Xi, where the PCs from SST and scPDSIpm. We required that the angled brackets indicate averaging. The matrix jjSSTjj jjscPDSIjj so that variability in the scPDSIpm C0 will have diagonal sections that are the covariances L 5 T 5 would not appreciably affect the structure of . Physi- of the individual fields (i.e., CT0 hT0 T0i, and CD cally, this would be interpreted to mean that SST vari- hDTDi), and off-diagonal sections that are the co- ability and regional sources of atmospheric noise can variances between SST and scPDSIpm fields: force scPDSIpm in the LIM, but scPDSIpm cannot af- 2 5 5 3 CT CT ,D fect SST. By requiring SST 1 and jjscPDSIjj 10 , C 5 0 0 . (6) we found that the spatiotemporal structure of the SST 0 C C D,T0 D patterns in L were insensitive to scPDSIpm. Our state vector is therefore defined as Finally, there is a well-known trend in global SSTs   over the twentieth century, and possible (albeit more T uncertain) trends in PDSI over the same period (e.g., X 5 0 , (5) D Sheffield and Wood 2008; Sheffield et al. 2012; Dai 2011). Trends can influence the structure of L because where T0 and D are the filtered SST and scPDSIpm they will appear as weakly damped modes with decor- sets of PCs, respectively. The state vector defines the relation time scales proportional to the length of the

Unauthenticated | Downloaded 10/03/21 08:07 PM UTC 1JANUARY 2018 A U L T E T A L . 9 training interval (1950 to 2000 in this case). This NADA, which was recently updated to 0.58 latitude/ component, in turn, could impart ultralow-frequency longitude resolution (Cook et al. 2014) and incorpo- variability on the dynamical system that would be un- rates a greater number of predictor chronologies. For representative of its unforced statistics. We took several comparison with the observational data (Sheffield steps to guard against this possibility. First, SST and et al. 2006), we regridded the high-resolution NADA scPDSIpm fields were both linearly detrended prior to to the same 18 lat/lon grid as the scPDSIpm product analysis. Second, patterns and time series of the PCs derived from observations. obtained from both fields were examined to ensure they Given our focus on multidecadal time scales, we refer to did not include any ‘‘residual’’ trend. Third, the eigen- the 35-yr running average of PDSI as PDSI35. The specific values of L were inspected to make sure their decay time choice of 35 years for the time averaging is arbitrary; we scales were relatively short (seasonal to interannual). adopt it here following earlier studies because it is simple And fourth, by focusing on the interval of 1950–2000, and reliably identifies megadroughts as 20:5s anomalies and especially by calibrating our LIM on the instanta- in PDSI35 preindustrial times in the WUS (Ault et al. neous and lag 3-month covariance matrices, we mini- 2014, 2016; Cook et al. 2016). It also satisfies earlier, more mize the effects of secular and long-term variations qualitative definitions of megadroughts as being periods because seasonal and interannual sources of variability of aridity as severe as the worst droughts of the twentieth dominate the fluctuations at these time scales. century, but much longer lasting (e.g., Woodhouse and The LIM was run for 12000 months (1000 yr) a total of Overpeck 1998; Acuna-Soto et al. 2002), and it is ame- 1000 times [using the formula provided by Penland and nable to analytical treatment [see the supplementary

Sardeshmukh (1995), Eqs. (15a) and (15b) therein]. The material of Ault et al. (2016)]. This quantity (PDSI35)is PC time series generated from each of these stochastic computed from reconstructed JJA PDSI in the NADA realizations were projected onto the original subset of and compared to the LIM output of JJA PDSI. spatial loading vectors to generate spatially complete fields of each variable. Drought and tropical Pacific d. Megadrought test statistics statistics (e.g., the DAI, spatial correlations, and power We identify megadroughts in the NADA and in the spectra) were then computed from this output. Al- LIM based on the four features of these events described though output from the LIM is available at monthly in the introduction (and summarized in Table 1). Meg- resolution, the PDSI field was averaged over June– adroughts have been widespread, of comparable mag- August (JJA) for comparison with paleoclimate re- nitude to the twentieth century in their severity, and constructions of this same variable over the same season clustered during the MCA. There has also been a shift in (see next section). mean hydroclimatic state of 0.47 PDSI units since the c. Reconstructed PDSI data MCA (Coats et al. 2016b), from relatively more arid to more humid modern conditions. Each of these charac- We evaluated the synthetic droughts produced by the teristics are expressed as the quantitative test statistics LIM by comparing them against the North American described below: Drought Atlas (NADA), which is a gridded set of mean JJA PDSI reconstructions derived from tree rings (Cook d Spatial scale. We generate a multidecadal DAI from et al. 2004, 2014). The NADA uses point-by-point reconstructions and LIM output by calculating the principal component regression to estimate JJA PDSI fraction of WUS grid points with PDSI35 at or for the hundreds of years before instrumental data are below 21 PDSI units. To distinguish this quantity available using nearby tree-ring data, drawing first upon from the DAI employed by Cook et al. (2004), which those chronologies within a 450-km search radius of a first computed the fraction of the domain below 21 given grid point, and then expanding the search by 50 km PDSI units, then smoothed the resulting time series, until at least five predictors have entered the model we term the DAI employed here ‘‘DAI35’’ to empha- (Cook et al. 1999, 2007). As part of the reconstruction size its multidecadal dependence. Standard deviations procedure built into the NADA, the tree-ring data are of the reconstructed PDSI are close to 2 across much prewhitened to remove temporal autocorrelation be- of the WUS, so that the threshold used to identify lieved to be due to biological memory and forest dy- events (21) is generally consistent with earlier studies namics. After the reconstructions are generated, the that used a 20:5s threshold to identify events in time estimates are ‘‘re-reddened’’ by adjusting the autocor- series normalized to unit variance over the historical relation of the proxy drought series to match the mem- period (e.g., Cook et al. 2004, 2016; Ault et al. 2014, ory observed in instrumental PDSI (Cook et al. 1999, 2016). The DAI35 may therefore be interpreted as the 2007). Here, we used a high-resolution version of the fraction of the WUS experiencing megadrought lasting

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TABLE 1. Summary of key acronyms and quantities used in this study.

Term Meaning Valid range WUS Western United States 1258–1058W, 328–508N SWUS Southwestern United States 1248–1058W, 328–418N NADA North American Drought Atlas 614 PDSI units PDSI35 35-yr running average of PDSI 62 PDSI units DAI35 Drought area index computed from PDSI35 0, 100% f * Test statistic for extent of megadrought 0, 1 m* Test statistic for magnitude of worst megadrought 21.5, 20.3 c* Test statistic for clustering of megadroughts 0, 6 Dms Test statistic for 500-yr mean PDSI 60.47 L Linear deterministic feedback matrix — X State vector — Q Noise covariance matrix —

35 years or longer at a given time. It could theoretically Empirical p values defined in this way will be close to vary between zero (no grid points in megadrought) and zero for megadrought characteristics that are unlikely to 100% (the entire WUS in megadrought). LIM output is have been drawn from the null distribution of the LIM, used to compute the DAI35, which is then compared and larger (closer to unity) when associated with events against the NADA’s DAI35 (see below). for which the null hypothesis cannot be rejected. d Magnitude. The significance of multidecadal mega- For example, the most widespread event from 801 to

drought magnitude is assessed by comparing LIM 2000 CE reached peak spatial extent (maximum DAI35) output to the worst event in the mean (domainwide) over the period from 1130 to 1164 CE (Fig. 1a), with SWUS PDSI35 between 801 and 2000 CE. This event 40.8% of the domain experiencing megadrought con- reached a peak amplitude from 1124 to 1159 CE ditions (PDSI35 below 21.0 PDSI units). We use the of 21.04 PDSI units on average (red circle in Fig. 1b). LIM to test if events of this scale are significant by d Clustering. Five events of relatively low amplitude computing a ‘‘fraction of time’’ test statistic (f *) over (20.5 PDSI units, or approximately 20.25s) and long which megadrought conditions prevailed across some duration (35 yr or more) occurred during a 500-yr portion of the WUS domain. In the NADA, a total of period of the MCA (Fig. 1b). We tally the total 3 years (1/400 of all the years from 801 to 2000 CE) saw 2 : s number of events of equivalent amplitude ( 0 25 )in DAI35 values above 40%. The test statistic characteriz- each possible consecutive 500-yr interval of each LIM ing such an event at the DAI35 40% level would there- realization to evaluate the significance of the cluster- fore be f * 5 1/400. The values of f * can likewise be

ing observed in the MCA (applying this same criterion calculated for any other DAI35 threshold. to the reconstructed PDSI identifies the five low- Output from the 1000 LIM realizations is used to amplitude megadroughts of the MCA; Fig. 1). generate the null distribution of f *. For each run, the d Mean shift. Mean PDSI values of the MCA were 0.47 fraction of years above various DAI35 values is com- PDSI units lower than they were during 1500–2000 puted, yielding an empirical probability density func- CE. Differences between the first and last 500 years of tion (PDF) of f * based on the entire LIM ensemble. each LIM realization are compared against this value. NADA-based values of f * are then ranked against this e. Evaluating significance empirical distribution to establish p values, and hence confidence limits for rejecting the null hypothesis. For We generate empirical p values for each test statistic example, if a reconstructed value of f * falls outside the above by comparing its reconstructed value against the range of 95% of the LIM-generated data for that com- distribution of those same quantities generated by the bination of area and fraction of time, we would reject the LIM. The p value is therefore the empirical probability null hypothesis at the 95% confidence limit. that a given megadrought test statistic was drawn from As with our significance testing of NADA-based the LIM’s distribution, and is defined as values of f * for the entire WUS domain, we evaluate   each of the three other test statistics for the SWUS: the LIM events less severe than those in NADA p 5 12 . magnitude of SWUS PDSI35 minima (m*), clustering of Total number of LIM realization (low-amplitude) megadroughts during a 500-yr period (7) (c*), and the change in mean PDSI (Dm*). Because the

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FIG. 3. Correlation maps between boreal winter (JFM) Niño-3.4 and summer (JJA) PDSI calculated from observations (shown at top left) and the first 11 LIM realizations. The iteration number of each LIM realization is shown at the bottom left of each panel.

value of m* is negative (20.62s), we evaluate signifi- hydroclimate (Fig. 3). Correlations patterns between cance by comparing the distribution of the LIM with the January–March (JFM) Niño-3.4 time series (average minimum PDSI35 values below this threshold. In con- SSTs over 58S–58N, 1708–1208W) and June–August (JJA) trast, the number of low-amplitude events (20.25s scPDSIpm (Sheffield et al. 2006) are very similar in both anomalies in PDSI35) that occur during a given 500-yr the observations and the LIM output (Fig. 3). period (c*) is a positive integer. Significance is therefore Power spectra of the Niño-3.4 time series generated assigned by tallying the number of LIM realizations with by the LIM exhibit a broad spectral peak over the in- as many or more low-amplitude events over any con- terannual time scales associated with ENSO (gray tinuous 500-yr period. Finally, if the shift in the 500-yr shading, Fig. 4). Indeed, the range of LIM power spectra mean of the MCA was generated merely by chance encompasses three different observational versions of (following our null hypothesis), then its sign is neces- Niño-3.4 (Fig. 4): Kaplan SST (Kaplan et al. 1998), sarily arbitrary. Significance is therefore evaluated ERSSTv3b (Smith et al. 2008), and HadSST1 (Rayner against the absolute value of all differences between the et al. 2003), which verifies that this version of the LIM first and last 500 years of all realizations. generates realistic interannual variability [a result also discussed in Newman (2013) and Penland and Sardeshmukh (1995), among others]. Although one 4. Results observational product (ERSSTv3b) exhibits spectral Using an LIM that represents global SST variability and densities above the LIM at near-centennial time hori- drought in the western United States, we generated 1000 zons, this low-frequency variability probably reflects synthetic PDSI and SST realizations that each span one long-term trends in the dataset that are not robust across millennium. The LIM reproduces realistic spatial corre- the tropical Pacific (e.g., Solomon and Newman 2011; lation patterns between the tropical Pacific and WUS Emile-Geay et al. 2013a,b).

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FIG. 4. Power spectra computed from Niño-3.4 time series. The gray shading encompasses the lower 2.5% and 97.5% quantiles of Niño-3.4 time series simulated by the LIM (e.g., 95% of the distribution of all realizations at all frequencies). The three thin lines are spectra computed from Niño-3.4 time series derived from the following gridded observational data products: Kaplan SST (Kaplan et al. 1998), ERSSTv3v (Smith et al. 2008), and HadSST1 (Rayner et al. 2003). Vertical dashed lines bracket the 2–7-yr ENSO band.

To help visualize the output generated by our LIM of are found throughout most of the entire Southwest, and global SSTs and western hydroclimate, we illustrate the most especially in the central part of the domain. Modest timing and spatial structure of megadroughts in just one positive values are present across the northern regions LIM realization (Figs. 5 and 6). In this run, the maxi- of the WUS (Fig. 6c). mum WUS DAI35 and minimum SWUS PDSI35 values DAI35 time series from individual LIM realizations coincide, but this does not occur in all realizations, and encompass a similar range as the tree-ring-based re- high DAI35 values are not always driven by the same constructions (Fig. 7). Maps of the most widespread SST patterns as low SWUS PDSI35 values. During the events (maximum DAI35) likewise depict prolonged first widespread event (culminating at year 54), DAI35 events commensurate with observations in their spatial values reached 27.1% (Fig. 5a). Other comparable pe- scale and severity (Fig. 8). Several of these realizations riods are also present in this run (e.g., years 230–270, have at least one event with DAI35 values greater than 310–390, and 950–970). Here we focus on the year 54 or equal to the maximum reconstructed value (40.8% of event (vertical dashed line) to distinguish its SST pattern WUS in the NADA). There are also instances of wide- from that of the most severe event (Fig. 6). SST anom- spread megadroughts that are even more spatially ex- alies were cool almost everywhere, and most notably in tensive than have been observed (e.g., run 2). In total, the northern Atlantic (Fig. 5b). Negative values are 35.4% of the runs included at least one event as wide- apparent in much of the central portion of the domain, spread as the 1130–1164 megadrought in the NADA. with wet anomalies occurring in the Northwest and Average SST anomalies from the 35 years associated along much of the Pacific coast (Fig. 5c). The most se- with each run’s maximum DAI35 are reminiscent of the vere event in the Southwest reached a value of 20:72s canonical patterns associated with La Niña and the (red circle in Fig. 6; LIM year 946). Average SST ‘‘cold phase’’ of the PDO with cool SSTs in the Indian anomalies in the LIM during that time were close to Ocean, equatorial Pacific, and along the western edges neutral in the Indian Ocean, cool in the tropical Pacific, of North and South America (Fig. 9). Warm SST anom- and warm in the western Pacific and North Atlantic alies are found in the northwestern Pacific and North (Fig. 6b). Negative PDSI values (between 21 and 22) Atlantic. The amplitudes of the averaged anomalies,

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FIG. 5. Example output from one LIM realization. (a) DAI35 from this run, with the vertical dashed line marking the first widespread event during the realization (year 54). Also mapped are average (b) SST and (c) PDSI con- ditions for this event. however, are much weaker than during La Niñaor PDF identifies the portion of LIM realizations that PDO cold phases (individual events can have stronger have a lower fraction of years than the NADA at each anomalies relative to composites). The westward ex- threshold. For example, in the NADA 1072 out of 1200 tent of the cold anomaly along the equator is also less years saw DAI35 values of at least 5% (f * 5 0.893). Of zonal than it is during normal La Niña events. the 1000 LIM realizations, 81.3% simulated a smaller Global average SST anomalies associated with the fraction of time below this threshold than in the NADA, lowest values of PDSI35 from each run (Fig. 10) differ meaning that the probability (p value) that the observed from those connected with maximum DAI35 (Fig. 9). In value at least as large as f * was drawn from the null particular, cool SST anomalies along the equatorial distribution is 0.187 (p 5 1 2 813/1000).

Pacific extend less westward in the maximum DAI35 The shape of the distributions for f * is clearly more pattern than they do at times of minimum PDSI35 (cf. skewed for higher DAI35 thresholds because most of the Figs. 9 and 10). Moreover, SST anomalies associated realizations only simulate a small fraction of years with with SWUS megadrought have a more distinctively La DAI35 values greater than 35% (Fig. 11). The LIM also Niña–like pattern, as compared with the anomalies as- produces a small number of events (not shown) with sociated with widespread drought. even greater values (i.e., the DAI35 thresholds of 45% Distributions of our test statistic for widespread WUS and 50% both include a few nonzero values when con- megadrought (f *) from LIM output establish confi- sidering the full 1000-member ensemble). dence limits for various reconstructed DAI35 thresholds None of the reconstructed DAI35 values are signifi- (Fig. 11). In this figure, each panel shows the LIM-based cant at the 95% confidence limit (Fig. 12). In this case,

PDF of time (f *) spent at or above each DAI35 we applied a one-sided significance test because values threshold in 5% increments. The vertical dashed line of f * can only be positive, and we are interested in the marks the fraction of time in the NADA spent at or fraction of LIM realizations with more time spent below above these thresholds; the gray shaded area of each various DAI35 thresholds. A significance level value of

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FIG. 6. Example output from the same LIM realization used in Fig. 5. (a) Southwestern U.S. PDSI35 from this run, with the red circle marking the most severe event (year 929) during the realization. Also mapped are average (b) SST and (c) PDSI anomalies during this drought.

0.3 (y axis of Fig. 12) means that 3/10 of all LIM re- during most 500-yr periods of the LIM (Fig. 14). In alizations simulated less time at a given DAI35 level total, 2.1% of LIM runs had 500-yr periods with as than in the NADA, while 7/10 spent more time at that many or more megadroughts, suggesting that the null threshold. Even the most widespread reconstructed hypothesis for megadrought clusters (c*) can be event (1130–1164 CE), when approximately 40.8% of rejected at the 90% confidence level (p value 5 0.021). the WUS was affected by megadrought, is not significant d Mean shift. The long-term difference in the MCA at the 90% confidence limit (p value 5 0.354). mean versus the 1500–2000 CE mean (0.47 PDSI We assign significance to the other test statistics units) is highly significant against the null hypothesis (m*, c*, and Dm*) as follows: (Fig. 15). Out of the 1000 LIM realizations, none had higher magnitude 500-yr mean shifts, hence the re- d Magnitude. During the worst SWUS drought, tree-ring constructed values are significant at the 99.9% confi- reconstructed PDSI reached 20.62s (m* 520.62s). 35 dence limit or higher (p value , 0.001). Out of the 1000 LIM realizations, 34.8% of the runs included events as severe or worse (more negative) Our results were robust with respect to a number of than this period (Fig. 13), while the other runs did not other alternative assumptions made as part of con- include such high magnitude events. The null hy- structing of the LIM, identifying megadrought, and pothesis can only be rejected at about the 65% confi- evaluating significance. For example, other calibration dence limit (p value 5 0.348) for the magnitude of periods (e.g., 1980–2010) yielded structurally similar the worst megadroughts in the SWUS over the last linear deterministic matrices, meaning that estimating 1200 years. L is not particularly sensitive to the 1950–2000 interval d Clustering. Five low-amplitude (PDSI35 # 20:25s) used here. Likewise, the scPDSIpm product we included megadroughts occurred during the MCA (Fig. 1 in in the LIM uses precipitation from the CRU TS3.10 data Coats et al. 2016b), which is more than occurred product [for details, see Sheffield et al. (2006)]. There

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FIG. 7. DAI35 calculated from a subset of 19 LIM realizations. The top left panel shows values computed from the NADA (Cook et al. 2004), while the remaining panels are each computed

from a single 1000-yr-long realization. The dashed line on each panel marks the highest DAI35 value in the NADA for reference. The realization number is given on the upper left of each panel. are 15 alternative PDSI products, however, including variability. The limitation of detrending is, of course, ones with different underlying precipitation inputs and that there is some possibility that we have inadvertently approaches to estimating evapotranspiration (Sheffield removed unforced low-frequency climate variability. If et al. 2006). Use of the other 15 versions of PDSI did not this is the case, rejecting the null hypothesis would be appreciably change the structure of the LIM or the even more difficult, as it would raise the confidence confidence limits we established. Furthermore, the limits even further. Our results should therefore be in- scPDSIpmdiffers from its predecessor, the PDSI, in that terpreted as being slightly ‘‘conservative’’ because of the local surface storage coefficient is estimated empir- our use of detrended data in constructing the LIM. Note, ically from data, rather than prescribed as a spatially however, that the tree-ring data are also detrended, so invariant constant. Use of the original PDSI did not there is also a possibility that the true amplitude of cli- critically alter the nature of variability in the LIM, nor mate fluctuations is greater in reality than inferred from did use of alternative SST datasets (e.g., ERSSTv3b; the reconstruction (see discussion). In either case, the Smith et al. 2008). results presented here are at least consistent, in that both There was not any strong dependence of the LIM on the LIM and the tree-ring data have had long-term the details of the detrending; use of a spline fit, high-pass trends removed. filter, or linear fit all yielded structurally similar LIMs. We did find, however, that failing to detrend the data 5. Discussion had a modest impact on the significance levels, in that it made more prolonged and more widespread events Several key features of Pacific SST variability and its somewhat more common in the LIM. The likely expla- impact on WUS hydroclimate can be reproduced by the nation for this particular sensitivity is that the global relatively simple, low-order linear dynamical system warming pattern becomes included in the LIM as a represented by the LIM. For example, the correlation ‘‘least damped’’ mode that grows and decays on multi- patterns between the Niño-3.4 time series and JJA PDSI decadal time scales, imparting additional low-frequency are nearly identical between observations and the LIM

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FIG. 8. Reconstructed PDSI during the 1130–1164 CE period (shown at top left) shown with average PDSI from a subset of 19 LIM realizations (the same runs as in Fig. 7) over the 35-yr

period leading to maximum DAI35 values during each run.

(Fig. 3). Likewise, the LIM simulates spectral densities investigated. This finding could be further used as a basis in the central Pacific Ocean that encompass the range of for predicting an optimal forcing pattern (or combina- observational estimates (Fig. 4). Both of these features tions of patterns) that would elevate the risk of wide- are important for verifying that the LIM’s underlying spread megadrought if it were to appear across ocean statistics in space and time are representative of obser- basins this century. vations over the short instrumental period, and hence We used the LIM to evaluate our null hypothesis, relevant to the longer time scales of megadrought. namely that widespread WUS megadroughts occur fun- The LIM generates synthetic megadroughts, but this damentally from stochastic forcing. To test the signifi- outcome was not guaranteed from its formulation be- cance of reconstructed events against this null hypothesis, cause we used only seasonal data from the late twentieth we compared the fraction of years spent below each century (an interval when such events have not been DAI35 threshold (f *; see section 3e) in the NADA to observed). We also emphasize the LIM did not include the LIM’s distribution at those same levels. This com- any information from tree-ring reconstructions of past parison, in turn, permits us to establish confidence limits droughts in the western United States. Megadroughts in for rejecting the null hypothesis across reconstructed the LIM are therefore to be regarded as an emergent DAI35 thresholds. For instance, the maximum DAI35 feature of the low-order dynamical system we have value of the NADA is 40.8%, and just 3 years over the

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FIG. 9. Average SST anomalies from all LIM realizations during maximum DAI35 values from each run. last 1200 years were spent at or above the 40% thresh- normal occurrences in the hydroclimatic statistics of the old. The fraction of time megadrought conditions pre- region on millennial time scales. vailed in the NADA at this level is therefore 3/1200 Two features of SWUS megadrought are more sig- (0.0025). While 64.6% of the LIM realizations did not nificant than the scale (f *) and severity (m*) of past yield DAI35 values this high, 35.4% of the runs include events. First, the clustering (c*) of low-amplitude meg- megadroughts that are equally (or more) widespread adroughts during the MCA is unlikely to have occurred (Fig. 12). We may therefore only reject our null hy- from chance alone (p value 5 0.021), and hence we re- pothesis at the nominal 64% confidence level. Similar ject the null hypothesis for this characteristic of SWUS considerations apply to other lower thresholds, none hydroclimate with fairly high (.95%) confidence. We of which may be said to be significant at the 95% can be even more confident in rejecting the null hy- confidence limit. pothesis for the shift in 500-yr mean reconstructed As with the spatial scale of reconstructed WUS PDSI values (p value , 0.001), implying that such long- droughts, the null hypothesis cannot be rejected with term changes are extremely unlikely to be driven by very much confidence for the magnitude (m*) of the chance alone. worst droughts in the SWUS. That is, about a third What does accepting the null hypothesis for two of (34.8%) of the LIM realizations generated events as the four megadrought test statistics, and rejecting it for the severe as, or worse than, the worst drought in the SWUS. other two, mean for the nature of hydroclimate in the Hence, such events should be considered relatively WUS and SWUS? First and foremost, our results help

FIG. 10. Average SST anomalies from all LIM realizations during the most severe megadroughts (35-yr average PDSI values below 20:5s) from each run.

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FIG. 11. LIM-based empirical distributions of widespread megadrought for various DAI35 thresholds. In each panel, the ordinate is the fraction of all years spent below a given DAI35 threshold (marked at the top of each panel in bold). The shaded portion of the PDF is the fraction of LIM realizations with a proportionally lower number of years spent below each threshold than the NADA, while the white shading is the remaining portion of the PDF. The

vertical dotted line is fraction of time (f) in the NADA spent at or below each DAI35 threshold. The y axis is normalized so that the number of realizations (N) at each value of f is divided by the total number of realizations (r). clarify what aspects of megadrought occurrence in the proposed by Bjerknes (1969) or the influences of ex- past are truly exceptional given what we can observe in ternal forcings [including ‘‘quiet’’ periods of relatively the modern system. Clusters of megadroughts, and es- low variability in forcings as described by Bradley et al. pecially changes in long-term means, are unlikely to be (2016)]. It is clear that a smaller fraction of total SST generated by stationary statistics and stochastic forcing. variance is captured by the LIM in the Atlantic than in Their existence in the NADA therefore requires further the Pacific (Fig. 2). Consequently we could be under- examination. One possibility is that their odds of oc- sampling AMO variability in construction of the LIM currence were externally forced during the MCA [e.g., due to the short time scale of the observational record. If as discussed extensively in Cook et al. (2016)], either by the AMO is responsible for megadrought clustering slightly enhanced summer insolation (Lean 2000)orby during the MCA, then we cannot completely rule out the coupled feedbacks between the land surface, vegetation, possibility of megadroughts being internally generated and atmosphere (Cook et al. 2011). Alternatively, a because our LIM does not fully characterize variability multicentury warm phase of the Atlantic multidecadal in the North Atlantic. oscillation (AMO) (Coats et al. 2016b; Wang et al. 2017) The change in the long-term mean of reconstructed could have shifted the mean state of hydroclimate in the PDSI is the most statistically significant feature of SWUS and increased megadrought risk (Feng et al. SWUS hydroclimate. If it is climatic in origin (Cook 2008; Oglesby et al. 2012). Such persistence itself would et al. 2004; Coats et al. 2016b), it too likely arises from be unusual in the context of seasonal dynamics of global either nonlinear or externally forced sources of variability. SST over the late twentieth century, and could re- If it is not driven by climate, then it could arise from quire either ocean–atmosphere coupling as originally phenomena related to the proxy itself (e.g., biological

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FIG. 12. Empirical confidence limits for rejecting the null hypothesis that widespread megadroughts of the last millennium are caused by SST variability, combined with spatial and

temporal autocorrelation. For each DAI35 threshold (abscissa), the fraction of years in the NADA found at that value is compared against the total fraction of years in all LIM re- alizations. The line with open circles traces the significance limits for the amount of time spent

at each DAI35 threshold. The 50% (black), 66% (dashed), 90% (dash-dotted), and 95% (dotted) confidence limits are also shown for reference. growth trends not completely removed through de- WUS reconstructions to compare against the LIM. Fur- trending; Fritts 1976) or the methodology used to de- ther work should therefore test ensembles of drought velop the reconstruction. For example, reconstructed atlas reconstructions against our LIM-based significance PDSI values of the NADA are associated with calibra- levels, although we have focused here exclusively on tion and measurement uncertainties, and the effect of the available expected value reconstructions over the these uncertainties on our rejection or acceptance of the NADA domain. null hypothesis cannot be directly addressed using the Notwithstanding the inherent methodological limita- NADA. In contrast, more recent drought atlases (Cook tions of reconstructing past climates, the differences in et al. 2010, 2015) have been constructed using methods the multicentury mean PDSI values between the mod- that explicitly generate an ensemble of probable re- ern period and the MCA are extremely unlikely to have constructions over the respective domains, and a similar arisen from stationary statistics. The LIM therefore approach could be incorporated to derive an ensemble of provides important insight into the details of the data

FIG. 13. PDF of normalized minimum PDSI35 (m*), averaged over the SWUS. Black vertical bars indicate realizations with runs that produced minimum values below the NADA (20.63s).

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FIG. 14. Distribution of maximum number of low-amplitude megadroughts (c*) to have occurred over a 500-yr period in each LIM realization. Black vertical bars identify the fraction of runs with 500 year periods that include an equal or greater number of events than occurred during the MCA (801–1300 CE), as inferred from tree-ring reconstructions. that require further investigation even if they are related that exogenous factors may have played a role in making to the proxy or reconstruction methodology. megadroughts more probable during that time. Under the changing climates of the present day, it is Our results suggest that the LIM is a viable tool to often stated that individual extreme weather events establish baselines (or expectations) when evaluating cannot be unambiguously attributed to climate change, long model simulations or suites of paleoclimate re- but that the occurrence of more frequent extremes re- constructions. For example, it is not entirely clear if flects the net effects of anthropogenic activity on the GCMs simulate megadroughts at realistic rates with or statistics of the climate system (e.g., Trenberth et al. without externally varying boundary conditions (e.g., 2015). A similar principle applies to megadroughts Ault et al. 2014; Coats et al. 2013b). A LIM could be during the MCA: such events should occur naturally trained on model output, and the PDF of megadrought even without any changes to the boundary conditions occurrence derived from it could be compared to the (e.g., solar, orbital, or land cover) of WUS hydroclimate. observation-based LIM we used. Such an analysis would However, their frequency from 801 to 1300 CE suggests therefore not be limited by the size of millennial-scale

FIG. 15. PDF of SWUS shifts in 500-yr means, expressed in PDSI unit differences (Dm*). The bars are white because all runs had smaller amplitude differences than inferred from the NADA for the SWUS between 801–1300 CE and 1500–2000 CE (this difference is marked by an asterisk on the x axis).

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GCM ensembles because the LIM would empirically fill states that have not been observed during the twentieth out the PDF of megadrought from a limited set of runs. century [e.g., a multicentury warm phase of the AMO, as Likewise, an LIM could be adapted to simulate the risk discussed by Coats et al. (2016b)]. These results help of widespread megadrought, building on the work of clarify which aspects of megadrought are unusual but Ault et al. (2014) and Cook et al. (2015), both of which unremarkable in the context of the observable system focused on single time series methods of estimating (widespread intervals and high magnitude events), and risk. Finally, and perhaps most importantly, this effort what characteristics are truly unusual and require fur- should motivate wider adoption of the LIM framework ther investigation (clustering and the lower mean of in paleoclimate significance testing. In high-resolution the MCA). paleoclimatology, researchers frequently observe low- In summary: frequency variability in time series or networks of pa- d The most widespread megadrought in the last mil- leoclimate archives, and it is often unclear whether such lennium occurred between 1130 and 1164 CE and fluctuations require external forcings (e.g., volcanic, encompassed 40% of the WUS domain. About 35% of solar, or land surface processes), ‘‘out of sample’’ cli- the LIM output simulates megadroughts that are mate mechanisms, or if they are simply consistent with equally (or more) widespread (Fig. 12), meaning we the variability observed during the late twentieth cen- cannot reject the null hypothesis of internal variability tury. Adapting an LIM to these settings would shed new as causative at the 90% or 95% confidence limit light on the importance of internal variability and forced (p value 5 0.646). climate change across a broad range of reconstructed d The worst drought in the reconstructed SWUS PDSI phenomena. average was 20.62s. Because 34.8% of LIM runs had

PDSI35 values below this level, we cannot reject the 6. Conclusions and implications null hypothesis for the severity of the worst mega- drought in the NADA (p value 5 0.652). We constructed a simple, low-order empirical model d During the MCA five low-amplitude megadroughts (an LIM) of SST and WUS hydroclimate to evaluate (PDSI # 20.25s) occurred over a 500-yr period; the odds that megadroughts occur by chance—that is, 35 only 2.1% of LIM runs had 500-yr periods with as arising from observable spatiotemporal modes of late- many or more megadroughts, suggesting that we can twentieth-century climate variability, stochastic forcing, reject the null hypothesis at the 95% confidence limit and sufficient time. Despite being trained only on sea- (p value 5 0.021). sonal data from the late twentieth century, the LIM d Mean PDSI values from 801 to 1300 CE were 0.47 successfully reproduces realistic power spectra in the PDSI units lower than from 1500 to 2000 CE. All of tropical Pacific and correlation patterns between the the LIM runs had smaller mean shifts between their Niño-3.4 region and WUS hydroclimate. It also pro- first and last 500 years, meaning we can reject the duces megadroughts that are comparable in their dura- null hypothesis with a high degree of confidence tion, spatial scale, and magnitude as the most severe (p value , 0.001). events of the last 12 centuries (801–2000 CE). For ex- ample, the most widespread megadrought in the NADA Finally, our results hold a number of key implications lasted from between 1130 and 1164 CE and reached for stakeholder and research communities alike. First,

DAI35 values of approximately 40%. About 35% of events as arid as and even more arid than the most LIM realizations included events of this scale or worse. widespread drought of the last 1200 years are possible, Because the LIM is an empirical model of dynamics with even without any changes to the boundary conditions. fundamentally stationary statistics, we cannot reject the Given their spatial extent, these events would require null hypothesis that even the most widespread mega- large-scale efforts to manage water across much of the droughts are internally generated. Similar consider- western United States, as well as to protect ecosystems ations apply to other smaller DAI35 thresholds, and to and agriculture. A LIM could be used to support water the magnitude of the most severe SWUS megadroughts. resource management strategies because it permits us to In contrast to the spatial scale and magnitude of generate a large number of statistically plausible out- megadroughts, the clustering of these anomalies and comes even without climate change. These scenarios the change in the 500-yr mean since the MCA are could, in turn, be used by policy makers and water re- nominally, and highly, significant against the null hy- source managers to prepare for possible future water pothesis, respectively. Accordingly, these characteristics stress. Second, given that the LIM simulates events that of the NADA may require nonlinear processes, external are slightly worse than those in the NADA, and given forcing mechanisms, or exceptionally persistent climate that the NADA is based on tree-ring data that are

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