# Influence of Stationary Waves on Precipitation Change in North American Summer during the Last Glacial Maximum

Hung-I Lee, Jonathan Mitchell, Juan Lora, Aradhna Tripati

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Hung-I Lee, Jonathan Mitchell, Juan Lora, Aradhna Tripati. Influence of Stationary Waves on Precipitation Change in North American Summer during the Last Glacial Maximum. Journal of Climate, 2023, 36 (10), pp.3165-3182. 10.1175/JCLI-D-21-0886.1. hal-04218162

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# Influence of Stationary Waves on Precipitation Change in North American Summer during the Last Glacial Maximum

HUNG-I LEE<sup>a,b</sup> JONATHAN L. MITCHELL,<sup>c,b</sup> JUAN M. LORA,<sup>d</sup> AND ARADHNA TRIPATI<sup>c,b,e,f</sup>

<sup>a</sup> Department of Geophysical Sciences, University of Chicago, Chicago, Illinois

<sup>b</sup> Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California

<sup>c</sup> Department of Earth, Planetary and Space Sciences, University of California, Los Angeles, Los Angeles, California

<sup>d</sup> Department of Earth and Planetary Sciences, Yale University, New Haven, Connecticut

<sup>e</sup> Institute of the Environment and Sustainability, University of California, Los Angeles, Los Angeles, California

<sup>f</sup> European Institute of Marine Sciences (IUEM), Université de Brest, Plouzané, France

(Manuscript received 29 November 2021, in final form 21 September 2022)

**ABSTRACT:** Paleo-proxy reconstructions reveal a moistening of the American Southwest during the Last Glacial Maximum (LGM; 21 ka). However, the primary mechanisms driving the moistening trend are still debated, with relatively few studies focused on hypotheses related to synoptic changes in precipitation. Analysis of the Paleoclimate Intercomparison Model Project (PMIP3) simulations shows enhancement of precipitation in the southwest and south-central United States during the winter and summer. Here, we suggest that summertime eastward phase shifting of stationary waves at the LGM enhanced pre-cipitation in the south-central United States and dried the southeastern United States. Mechanism denial experiments performed with version 3 of the Hadley Centre Coupled Model (HadCM3) indicate that the thermodynamic effect of the Laurentide Ice Sheet forced eastward phase shifting of stationary waves. By comparing a synthesis of LGM proxies to the PMIP3 ensemble, we find models that compare more favorably to the reconstructions simulate a weaker Laurentide ice thermodynamic effect, smaller eastward phase shifting of stationary waves, and weaker jet stream anomalies.

**SIGNIFICANCE STATEMENT:** Our study is motivated by the impact of climate change on hydroclimate in North America, especially in the semiarid areas near the southwestern United States, and the persistent problem of significant disagreements among CMIP model projections. Yet modern models and observations alone cannot tell us the likeliest climate change outcome in this region. Here we analyze precipitation during the LGM reconstructed from proxies and as simulated by the PMIP3, which reveal a precipitation trend with wetting in the southwestern United States and drying in the southeastern United States. Our interpretation of PMIP3 simulations involves application of simple theories for midlatitude stationary waves, and indicates that the LGM precipitation is generated by the phase shift of stationary waves from the enhancement of jet speeds. Our model–data comparison also suggests that in the PMIP3 simulations, weak LGM jet stream anomalies, due to weak polar amplification, compare more favorably to the proxies.

**KEYWORDS:** Large-scale motions; Orographic effects; Stationary waves; Ice age; Precipitation; Paleoclimate

## 1. Introduction

It is anticipated that many regions in North America will experience changes in the hydrological cycle over the coming century based on global warming scenarios from simulations of Climate Model Intercomparison Project phases 5 (CMIP5) (Stocker et al. 2013) and 6 (CMIP6) (Tebaldi et al. 2021). However, disagreements between CMIP5 models make assessing the exact signs, magnitudes, and distributions of precipitation changes a challenge in many regions, such as the marginal semiarid areas of western North America. One way to evaluate the ability of climate models to capture the dynamics of moisture transport under different climate conditions is to run simulations for past climate intervals, such as the Last Glacial Maximum (LGM) 21 000 years ago (21 ka), and compare simulations to geological records to investigate changes in moisture transport.

Proxy data on hydroclimate during the LGM, including the distribution of pluvial lake shorelines (Munroe and Laabs 2013; Ibarra et al. 2014, 2018; Santi et al. 2019), speleothem datasets (Oster et al. 2009, 2015b), and pollen-based reconstructions of precipitation (Bartlein et al. 2011), indicate enhanced net precipitation in most of southwestern North America, including the Great Basin, and decreasing precipitation in neighboring areas. This spatial pattern of the paleo-precipitation wetting and drying has been hypothesized to reflect a moisture “dipole” (Swain et al. 2016; Oster et al. 2015b; Oster and Kelley 2016). Several mechanisms have been proposed to explain the moistening trend in the Great Basin and the American Southwest during the LGM. First, increased winter precipitation in western North America has been attributed to the southward displacement of jet streams, as storm tracks were deflected by the Laurentide Ice Sheet (COHMAP Members 1988; Asmerom et al. 2010). The intensification of the Aleutian low during the LGM could also induce the south-westward displacement of atmospheric rivers, which constitute a major source of moisture for winter precipitation in California

Corresponding author: Hung-I Lee, hungi@uchicago.edu

DOI: 10.1175/JCLI-D-21-0886.1

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).
(Lora et al. 2017). A challenge in identifying the primary driver of LGM moistening in the Great Basin and the southwestern United States is that the mechanisms listed above can also interact with each other. For instance, a weakening of the North American summer monsoon could be induced by cold air advected from the southward displacement of the jet stream (Bhattacharya et al. 2018) and increased Laurentide Ice Sheet albedo (Bhattacharya et al. 2017).

This study explores the potential controls and impact of synoptic changes in summer precipitation since the LGM in the United States to complement these other investigations that have focused on wintertime changes. We explore mechanisms that govern North American precipitation change during the LGM using model analysis, and then we discuss possible sources of disagreement between climate model simulations. We analyze the Paleoclimate Modeling Intercomparison Project phase 3 (PMIP3) simulations that consist of nine models of phase 5 of the Coupled Model Intercomparison Project (CMIP5) running under LGM and preindustrial (PI; pre-1850) conditions (Braconnot et al. 2011). Our analysis has been built on several theoretical approaches for studying synoptic change in precipitation that apply a moisture budget analysis, which quantifies the roles of precipitation, evaporation, and transport in moisture conservation. The transport of moisture can be further decomposed into stationary and transient (Seager et al. 2014; Lora 2018; Simpson et al. 2016). The pattern of summertime moisture change during the LGM in the United States has been shown to be similar to moisture transport by stationary processes (Lora 2018). A number of model simulations also indicate a strong influence of stationary waves on the North American hydrological cycle during the LGM (Li and Battisti 2008; Löfverström et al. 2014; Löfverström and Lora 2017; Löfverström et al. 2016).

The PMIP3 climatological (averaged at least 100 model years) data with nine models are compared to LGM-minus-PI precipitation anomalies derived from pollen data for 29 sites in the United States (Bartlein et al. 2011; Scheff et al. 2017). Pollen data of precipitation anomalies and model setup are described in the introduction. In section 2, a pollen-based proxy synthesis of continental precipitation of North America is compared to PMIP3 simulations, which allows an objective determination of the statistical significance of model simulations and model skill. Models are then identified that perform well against the proxies, and these simulations are used to diagnose mechanisms responsible for the LGM-minus-PI precipitation anomalies in sections 4 and 5.

## 2. North American precipitation changes since the LGM

Figure 1a displays the annual mean precipitation (abbreviated *P*) of the PMIP3 model ensemble average for the PI climate, and is further divided to show wintertime (November–April) precipitation (Fig. 1b) and summertime (May–October) precipitation (Fig. 1c). The sources of North American precipitation roughly divide into two types: storm tracks, which provide moisture to northwestern and eastern North America (marked by red rectangles respectively in Fig. 1a), and the North American summer monsoon (NASM), which delivers

<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Annual mean PMIP3 ensemble-mean PI precipitation with regions influenced by extratropical storms (marked by two red rectangles) and by North American summer monsoon (NASM; marked by the yellow rectangle). (b) As in (a), but for winter (November–April). (c) As in (a), but for summer (May–October).</figcaption></figure>

moisture to the south-central United States and Mexico (marked by the yellow rectangle in Fig. 1a). During the wintertime, North American precipitation is mainly governed by storms (Fig. 1b); during summertime, storms become weaker although remain the dominant contribution to precipitation in the northwestern and the eastern parts of North America, while the NASM is the main source of moisture to the south-central United States and Mexico.

The ratios of winter- and summer-mean precipitation to the annual-mean precipitation, as shown in Fig. 2, indicate that winter storms contribute the majority of precipitation along the west coast of North America (Lora and Ibarra 2019). Figure 3a shows the annual-mean, LGM-minus-PI precipitation anomalies for the PMIP3 model ensemble average in LGM-minus-PI ($\delta P$, with $\delta$ referring to LGM minus PI). A significance test is applied to each grid cell by assessing whether the magnitude of $\delta P$ can be differentiated from the standard deviation of PMIP3 model predictions; if the magnitude of the anomaly is greater than twice the standard deviation of model estimates, then it is defined as being significant.
<figure><img src="image_3.png" /><figcaption>FIG. 2. The ratio of total PMIP3 ensemble winter precipitation to total PMIP3 ensemble annual precipitation (colored shading). The thick black contours show the U.S. regions, including the southwest, the south-central, and the southeast regions.</figcaption></figure>

Note that $\delta P$ (colored shading in Fig. 3a) shows a significant enhancement of precipitation during the LGM in the western part of the United States. By dividing $\delta P$ into wintertime and summertime averages (Figs. 3b,c), we further show that enhanced $\delta P$ during wintertime occurs near the West Coast of the United States, while the summertime average of $\delta P$ increases more inland, close to the conventional NASM regions (see discussion in section 3). Also, the enhancement of $\delta P$ is south of winter P in PI conditions (green shading in Fig. 3b), indicating the southward displacement of winter storms during the LGM.

To explore how the enhancement of $\delta P$ in North America is a consequence of different mechanisms such as changing winter storms or the NASM during the LGM, we divide $\delta P$ into three areas (as shown in Fig. 2): the southwest United States, with a positive $\delta P$ and the ratio of winter/annual precipitation being more than 50%; the south-central United States, with positive $\delta P$ and the ratio of summer/annual precipitation being more than 50%; and the southeastern United States, with negative $\delta P$ and the ratio of summer/annual precipitation being more than 50%. We note that the definitions of the three regions rely on the patterns of precipitation, and do not necessarily match the conventional geographic divisions of the United States. Also, we take a closer look at the southwestern and the south-central United States since both places experience higher precipitation during the LGM.

Figure 3a displays the annual mean precipitation anomalies from the PMIP3 model ensemble average and from pollen and diatom reconstructions (LGM minus PI; i.e., $\delta P$). The annual mean $\delta P$ in the PMIP3 model ensemble shows a significant wetting in the southwestern and south-central United States. We divide $\delta P$ into winter (Fig. 3b) and summer (Fig. 3c) averages that clearly exhibit different precipitation patterns. The significant enhancement of $\delta P$ during wintertime mostly occurs in the southwestern United States, while the increase of $\delta P$ during summertime is more prominent in the south-central United States. The summertime $\delta P$ shows a wetting in the south-central region and a drying in the southeastern region. The significance test (stippling in Fig. 3c) indicates that this pattern is a robust feature of the model simulations.

The PMIP3 ensemble mean $\delta P$ generally shows the same spatial patterns from the LGM-minus-PI as observed in reconstructions of precipitation from pollen for most regions in the continental United States, except for three proxy locations

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>FIG. 3. (a) Annual mean PMIP3 ensemble-mean LGM-minus-PI precipitation (colored shading), pollen reconstruction of precipitation anomalies (colored circles), diatom reconstruction of precipitation anomalies (colored square), and significant changes of precipitation (stippling). (b) As in (a), but for winter. (c) As in (a), but for summer. (d) Spatial correlations (with their uncertainty) of annual mean LGM-minus-PI precipitation between PMIP3 models and precipitation proxies. PMIP3 models are divided into group 1 and group 2 with correlation coefficients above and below 0.5.</figcaption></figure>
that show opposite trends (one in Arizona and two in Florida; Fig. 3a). Also, since the region with increasing summertime precipitation (the south-central United States in Fig. 2) extends to Mexico, we include the diatom-based reconstruction of precipitation anomalies (Caballero et al. 2019) with detailed descriptions in section A1, and the diatom-based reconstruction shows a slight increase in precipitation during the LGM (Fig. 3). We calculate the correlation coefficients between all the proxy data and the corresponding data from each of the PMIP3 model simulations and their ensemble mean (Fig. 3d), and those show a range of coefficients from ~-0.05 to 0.7.

A simple bootstrapping method is applied to quantify the uncertainty of correlations. The technique involves developing 100 datasets from both the proxies and models by randomly selecting 30 data points while allowing for a single site to be selected more than once. We then calculate the correlation coefficients between the proxy and model data of the 100 datasets. The uncertainty of correlations is obtained from the distributions of correlations from the 100 datasets, showing a range from ~0.05 to 0.2 in Fig. 3d.

Other proxies (Oster et al. 2015a), including glaciers, lakes, vegetation, soils, speleothems, and groundwater, provide a quantitative reconstruction of annual mean $\delta(P - E)$ during the LGM, where $E$ stands for local evaporation. The reconstruction of $\delta(P - E)$ (not shown) reveals a moistening trend in almost the entire United States except a drier northwest region. The overall moistening trend over the United States during the LGM is likely due to the reduction of the local evaporation rate. In this paper, we focus on exploring the spatial pattern of $\delta P$ which highlights patterns in the LGM winter storm tracks and the NASM, and leave the discussion of $\delta E$ for future work.

### 3. Patterns of LGM precipitation anomalies in the southern United States during summertime

In this section, we discuss the enhancement of LGM precipitation ($\delta P$) in the southern United States. (Fig. 3). The increase of wintertime $\delta P$ in southwestern United States (Fig. 3b) is consistent with recent studies (COHMAP Members 1988; Asmerom et al. 2010; Lora et al. 2017; Lora 2018). Therefore, in this study, we mainly focus on exploring the primary drivers of the summertime $\delta P$ in the south-central United States. We note that the model-data constraint is mainly accomplished with the pollen data from the eastern and western United States, and there are few pollen sites [marked with a dagger (†) in Table A1] and a diatom site in the south-central United States. Based on the correlation of pollen-reconstructed precipitation anomalies and $\delta P$ in the PMIP3 models, we classify each PMIP3 model into one of two groups (Fig. 3d). Group 1 contains the five models that have correlation coefficients with the pollen reconstructions higher than 0.5, while the remaining four models are in group 2. The PMIP3 ensemble mean exhibits a better correlation with the reconstructions than most individual models, which we discuss later in section 5.

#### a. Summertime precipitation anomalies correlate with anomalies of meridional wind and wind divergence in the southern United States

Figures 4a and 4b show positive $\delta P$ in the south-central United States and negative $\delta P$ in the southeastern United States for groups 1 and 2. Both groups of models simulate negative-positive-negative 300-mb LGM-minus-PI meridional wind, $\delta V$, approximately from the southwestern, the south-central, and the southeastern United States (Figs. 4e,f). Positive $\delta V$ indicates anomalous southerly winds during the LGM. The northerly-southerly-northerly wind anomaly (negative-positive-negative $\delta V$) in Figs. 4e and 4f is consistent with negative-positive-negative $\delta P$ patterns from the southwestern to the southeastern United States. Apparently, models in groups 1 and 2 exhibit weak and strong stationary wave responses during the LGM respectively. The north-south-north stationary wave phase lines also tilt northeast-southwest, and those in group 2 tilt less than those in group 1. In both groups, upper level (300 mb) LGM-minus-PI wind divergence ($\delta D$, where $D$ stands for wind divergence; Figs. 4c,d) shows a convergence-divergence-convergence anomaly (negative-positive-negative $\delta D$) pattern across the southwestern, the south-central, and the southeastern United States. The high spatial correlations, by evaluating the correlation coefficients (Table 1) between $\delta P$, $\delta V$, and $\delta D$ at each grid in the south-central and southeastern regions, indicate strong relations of $\delta P$, $\delta V$, and $\delta D$. Moderate spatial correlations are also found when the domains include the southwestern United States or parts of the Laurentide Ice Sheet in the United States.

Vertical profiles of tropospheric $\delta V$ and $\delta D$ during summertime in the southern United States (averaged over the sector of 21°–35°N; Fig. 5) for both groups reveal the same 300-mb north-south-north (N-S-N) pattern as seen in Figs. 4e and 4f. The $\delta V$ for group 2 (Fig. 5b) is barotropic, meaning that the N-S-N pattern is aligned from the surface to the top of the troposphere. The $\delta V$ for group 1 (Fig. 5a) is less barotropic than for group 2, but still shows the N-S-N pattern below 300 mb. Upper-level $\delta D$ in Figs. 5c and 5d shows convergence-divergence-convergence, which is consistent with Figs. 4c,d and also in-phase with N-S-N patterns with $\delta V$ in Figs. 4a and 4b. The $\delta D$ (Figs. 5c,d) in the south-central United States turns to negative values below 500 mb, showing the baroclinic structures with lower-level wind convergence and upper-level wind divergence.

The moisture transport anomalies by the meridional wind ($\delta Vq$, where $q$ stands for specific humidity) in Figs. 5a and 5b is consistent with $\delta V$ below 500 mb, indicating the increasing moist air transport by the southerly wind anomalies in the south-central United States during the LGM summer. From the perspective of the moisture budget (Trenberth and Guillemot 1995; Seager et al. 2014; Lora 2018), precipitation is the sum of evaporation ($E$) and the vertically averaged convergence of moisture transport. Therefore, the enhancement of precipitation could be a consequence of increasing convergence of moisture transport [negative $\delta \nabla \cdot (qV)$, where $V$ is the vector of horizontal wind velocity], or increasing evaporation (positive $\delta E$) that increases specific humidity in the air. However, evaporation decreases since the air temperature is generally cooler in the LGM climate, and if this were the only
<figure><img src="image_8.png" /><figcaption>FIG. 4. (a) Summertime $\delta P$ in group 1. (b) Summertime $\delta P$ in group 2. (c) Summertime wind divergence anomalies at 300 mb in group 1. (d) As in (c), but in group 2. (e) Summertime meridional wind anomalies (V) at 300 mb in Group 1. (f) As in (e), but in group 2.</figcaption></figure>

change we would expect less LGM precipitation due to the reduction of evaporation. Hence, we expect the positive anomalies of convergence of moisture transport [$\delta\nabla \cdot (q\mathbf{V})$] if precipitation increases during the LGM. The vertical profiles of $\delta\nabla \cdot (q\mathbf{V})$ in Figs. 5c and 5d are positive in the lower troposphere around the central United States (110°–100°W for group 1 and 105°–90°W for group 2), and are consistent with increasing low-level wind convergence (negative $\delta D$). The reduction of $\delta\nabla \cdot (q\mathbf{V})$ occurring in the southeastern United States also matches with the increasing low-level wind divergence (positive $\delta D$).

b. The meridional wind and wind divergence anomalies are a manifestation of stationary waves

We have so far shown that the summer precipitation anomalies ($\delta P$) match with barotropic wind anomalies ($\delta V$) and the

baroclinic structure of lower-level wind convergence and upper-level wind divergence (positive $\delta D$) in Figs. 4 and 5, as supported by the spatial correlations of $\delta P$ with $\delta V$ and $\delta D$ at 300 mb (Table 1). A consistent interpretation is that southerly wind anomalies bring moisture into the south-central United States, whereas northerly wind anomalies bring cool, dry air into the southwestern and southeastern United States. The N-S-N patterns of $\delta V$ in the southern United States during the LGM summer can be conceived as stationary waves traveling from the southwestern United States to the southeastern United States. We emphasize that the relationship between the LGM summer stationary waves and precipitation anomalies mainly relies on the spatial correlations; the mechanism that causes $\delta D$ to match with $\delta V$ remains unclear, and we leave it for future study.

<table><caption>TABLE 1. The spatial correlations of summertime $\delta P$, 300-mb $\delta V$, and 300-mb $\delta D$ in the south-central and the southeastern United States (15°–35°N, 110°–75°W), the southern United States (15°–35°N, 118°–75°W), and the entire United States (15°–45°N, 118°–75°W), where angle brackets represent spatial correlations.</caption><thead><tr><th>Domains</th><th>⟨δP|δV⟩</th><th>⟨δP|δD⟩</th><th>⟨δV|δD⟩</th></tr></thead><tbody><tr><td>South-central and southeastern United States</td><td>0.79</td><td>0.74</td><td>0.83</td></tr><tr><td>Southern United States</td><td>0.71</td><td>0.61</td><td>0.81</td></tr><tr><td>Entire United States</td><td>0.48</td><td>0.61</td><td>0.64</td></tr></tbody></table>
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>FIG. 5. (a) The meridional average (21°–35°N) of summertime LGM-minus-PI meridional wind ($\delta V$; colored shading) and moisture transport by meridional wind ($10^{-6}\delta Vq$; black contours) for group 1. (b) As in (a), but for group 2. (c) The meridional average (21°–35°N) of summertime LGM-minus-PI wind divergence ($\delta D$; colored shading) and convergence of moisture transport [$-10^{-10}\delta\nabla \cdot (qV)$; black contours] for group 1. (d) As in (c), but for group 2.</figcaption></figure>

The shifting locations of NASM during the LGM are related to this strengthening moisture and heat transport in the south-central United States during summer (Bhattacharya et al. 2018; Lyle et al. 2012; Oglesby et al. 2012), which we now demonstrate in the PMIP simulations. We apply two NASM indices, the precipitation-based IPCC index and the moist static energy (MSE)-based index, to determine the boundaries of the NASM (see details of the indices in section A2). Figure 6 shows the NASM regions for the LGM and the PI based on the two indices. From this it is evident that PMIP3 models simulate a southward displacement of the NASM areas during the LGM based on the MSE monsoon index. Since the NASM during both the PI and the LGM mismatches $\delta P$, it is less likely that the NASM during the LGM contributes to the enhancement of precipitation during the LGM summer in the south-central United States. The NASM domains from the IPCC index are farther equatorward and beyond the south-central United States, where we are interested for this study. We emphasize that the moisture sources of increasing LGM summer precipitation in the south-central United States are mainly related to the stationary waves; therefore, we will focus more on the dynamics of stationary waves in the following section.

## 4. Mechanisms of the stationary waves

To identify the sources of the stationary wave anomalies that drive $\delta V$ in the southern United States, we use the wave activity flux (WAF) (Plumb 1985) as a vector measuring the magnitude and propagation of stationary waves and to trace the factors that generate stationary wave changes since the LGM. WAF vectors diverge from their source areas and converge

into sinks. In Fig. 7a, the deep-summer (July–September) WAF during the PI originates from source areas in the eastern Pacific, with positive WAF divergence, and then WAF vectors propagate across the whole North American continent and decay in the Atlantic. The stationary wave source in the eastern Pacific is likely the altered jet stream impinging on the Rocky Mountains. During the LGM (Fig. 7b), the deep-summer WAF vectors originate from both the eastern Pacific and the Laurentide Ice Sheet, and the teleconnection pattern in WAF crossing North America breaks down due to the presence of stationary waves induced by the Laurentide Ice Sheet. In this section, we use climate model simulations to study the stationary waves from the Rocky Mountains and the Laurentide Ice Sheet, and develop a simple predictive theory to quantify the shifts of the stationary waves.

<figure><img src="image_13.png" /><figcaption>FIG. 6. PMIP3 ensemble mean of LGM-minus-PI precipitation for summertime (colored shading) with NASM domain using the MSE-based index (black contours) and the IPCC index (red contours), during the PI (solid contours) and the LGM (dashed contours).</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>FIG. 7. (a) PI control run PMIP3 wave activity flux (WAF; black arrows) and the WAF divergence (colored shading) at 300 mb in deep summer (JAS). A moving average (20° in the zonal direction and 10° in the meridional direction) is applied for the WAF divergence. (b) As in (a), but for LGM.</figcaption></figure>

a. Evidence that precipitation in the south-central United States during the LGM is enhanced by the eastward shift of stationary waves

PMIP3 ensemble averages of stationary waves at 300 mb during the PI and the LGM summer (Figs. 8a,b, respectively) show southerly winds in the western parts of the United States and northerly winds in the eastern parts of the United States. These stationary waves are generated topographically as jet streams flow across the Rocky Mountains. The southerly winds in the western United States during both the PI and the LGM form crescent-like shapes (red shading in Figs. 8a,b). During the LGM, the crescent-like shape moves slightly eastward compared to the one during the PI, and this eastward displacement of the crescent-like pattern generates southerly wind anomalies in the south-central United States during the LGM (Fig. 8c), which end up bringing more moisture into that region. Meridional winds over the subtropics for the LGM, the PI, and the LGM minus PI in Fig. 8d support that the southerly wind anomalies of the LGM minus PI in the south-central United States are mainly governed by the eastward shift of the stationary waves but not the changing magnitudes or wavelengths of the stationary waves.

We speculate that the eastward shift of the stationary waves forms as a delayed response of the Rocky Mountain-induced stationary waves during the LGM. The phase shift of the topographic stationary waves can be attributed to the surface drag (Charney and Eliassen 1949; Held 1983) with its magnitude proportional to the strength of the background jet stream. Therefore, qualitatively, the enhanced strength of the LGM summer jet stream advects the stationary wave phase downstream to the east, thus generating the southerly wind anomalies that increase precipitation in the south-central United States.

b. Evidence that the eastward shift of stationary wave phase is driven by the thermodynamic effect of the LGM ice sheet

To investigate the primary driver of the eastward shift of the stationary wave phase as shown in Fig. 8, we analyze data from a set of experiments using the fully coupled HadCM3 model (Gordon et al. 2000) in 1) the PI-control climate, PI-control climate with 2) a white plain (WP) case with a flat

LGM ice sheet, 3) a white mountain (WM) case with the full Laurentide ice topography, and 4) the LGM climate. The details of the HadCM3 model setup are described in section A1 and we here present the simulated *V* of four cases in Figs. 9a–d respectively. The *δV* of the WP case minus PI-control scenario is shown in Fig. 9e, and represents the thermodynamic effect of the LGM ice sheet. *δV* of the WM case minus PI-control scenario (Fig. 9f) indicates the full Laurentide ice effect. We also present simulated *δV* of the LGM minus PI scenario in Fig. 9g with full considerations of the LGM climate.

The formation of stationary waves when jets flow across the Rocky Mountains can be found in all four cases of the HadCM3 simulations with southerly winds shown in the western United States (Figs. 9a–d). In the case of the WP simulation (Fig. 9b), the eastward shifting of stationary wave phase can be seen in the southerly wind (red shading in Fig. 9b) that shifts eastward due to the thermodynamic effects of the LGM ice sheet. The northerly–southerly–northerly wind pattern as shown in Fig. 9e is also consistent with PMIP3 ensemble *δV* (Fig. 8c). Figure 9c shows that there is no farther eastward movement of stationary wave phases by adding Laurentide ice topography on the flat LGM ice sheet in the WM case, and the N-S-N wind patterns remain similar in the case of WM minus PI (Fig. 9f). However, the simulation with the Laurentide ice topography (Fig. 9f) shows that southerly wind anomalies extend further toward the southeastern United States and we speculate that the eastward extension of the westerly wind anomalies is a consequence of the deflection of waves by the Laurentide ice topography. The stationary wave phases also have no additional eastward displacement in the case under LGM climate conditions (Fig. 9d), and the N-S-N wind patterns remain the same qualitatively in the case of the LGM minus PI (Fig. 9g). Hence, we conclude that the thermodynamic effects of the LGM ice sheet are qualitatively responsible for the eastward shift of the stationary wave phase, thus generating the southerly wind anomalies in the south-central United States.

We present the annual mean precipitation anomalies ($\delta P$) in HadCM3 experiments for the WP minus PI (Fig. 10a), the WM minus PI (Fig. 10b), and the LGM minus PI (Fig. 10c), and compare with the pollen- and diatom-based reconstruction of $\delta P$. The spatial patterns of the annual mean $\delta P$ from simulations with a WP, showing positive $\delta P$ in the south-central United
<figure><img src="image_15.png" /><figcaption>FIG. 8. (a) PMIP3 ensemble-mean meridional wind (V) at 300 mb during summer PI. (b) As in (a), but for LGM. (c) As in (a), but for LGM minus PI (δV). (d) The meridional average (15°–35°N) of the meridional wind for PI, LGM, and LGM minus PI.</figcaption></figure>

States and negative $\delta P$ in the southeastern United States, are consistent with the proxies. However, the HadCM3 simulations with a WM and the LGM climate show a significant intensification of $\delta P$ in the southeastern United States, which is inconsistent with the proxy data showing negative $\delta P$ trends in the same region. The distributions of $\delta P$ remain similar in the summer-time mean (Figs. 10d–f), and the spatial correlation coefficients of $\delta P$ between the proxy data and each HadCM3 experiment during the summer are 0.35 ± 0.13 for the WP minus PI, -0.50 ± 0.11 for the WM minus PI, and -0.42 ± 0.12 for the LGM minus PI. The fact that $\delta P$ in the WP minus PI compares

more favorably with the proxy data than in other experiments is consistent with the argument that the ice thermodynamic effects play an essential role in controlling the $\delta P$ but implies the possibility that the HadCM3 overestimates the ice topographic effects.

### c. Evidence that stationary wave phase shifts are due to the strengthening of jet streams by the thermodynamic effects of the Laurentide ice sheet

We suspect that the eastward stationary wave phase shifts are induced by the enhancement of the jet stream by the thermodynamic effects of the LGM ice sheet. To verify our conjecture quantitatively, we 1) derive a theory to quantify the magnitude of eastward stationary wave phase shift in response to the jet speed, 2) show the relationship of jet speeds and the meridional temperature gradients, and 3) verify the correlations of meridional temperature gradients and equator-to-pole temperature anomalies, so that we can link to the thermodynamic effects of the LGM ice sheet on temperature in higher latitudes. It is essential to study the dynamics of jet streams and stationary waves since PMIP3 models that compare favorably to the pollen proxies simulate weaker jet streams (Fig. 11a); we discuss PMIP3 model-pollen data constraints in detail in section 5.

To measure the magnitude (in degrees) of the eastward stationary wave phase shifts, we begin with the definition of stationary wave amplitude. Stationary wave amplitudes can be approximated by the average of summer V between 30° and 50°N, the latitude circle where summer jet streams impinge on the Rocky Mountains. We define the phase angle of stationary waves at the location of the V maximum inside the domain of the U.S. continent (75°–135°W), and the amount of phase shifting is the phase angle of stationary waves in the LGM minus PI. We use the maximum of V because the stationary wave has a large amplitude downstream of the Rocky Mountains.

We speculate that the stationary waves are generated as jet streams flow across the Rocky Mountains. Stronger jet streams during the LGM form due to enhanced meridional temperature gradients driven by the thermodynamic effects of the LGM ice sheet, and the phase of the stationary waves shifts when these strong LGM jets flow across the Rocky Mountains. We calculate the relation between $\delta U$, where U is defined as the maximum of the global zonally averaged zonal wind during summertime, and phase shifting (black curve in Fig. 11b) using the Charney–Eliassen model (Charney and Eliassen 1949; Held 1983) that describes midlatitude stationary waves forced by surface topography (e.g., the Rocky Mountains). We derive a new solution to quantify the phase shift. The calculation and its validation are summarized in the appendixes. Figure 11b indicates that the amount of phase shifting positively correlates with the change of jet stream values ($\delta U$) for each PMIP3 model, and their correlations are consistent with the predictions by the Charney–Eliassen model (the black straight line in Fig. 11b). The prediction by the Charney–Eliassen model is consistent with our speculation
<figure><img src="image_16.png" /><figcaption>FIG. 9. (a) HadCM3 simulations of meridional wind (V) under the PI climate, and with (b) a white plain, (c) a white mountain, and (d) under the LGM climate, as well as δV of (e) a white plain minus PI, (f) a white mountain minus PI, and (g) LGM minus PI.</figcaption></figure>

that the stronger jet anomalies cause larger magnitudes of the eastward stationary wave phase shifts.

We note that our analysis is insufficient to rule out other possible processes that could generate a phase shift of the LGM stationary waves. For example, other processes could include the interactions between topography and the thermodynamic effect, the diabatic heating from the Laurentide Ice Sheet, or the responses of waves from tropical forcing (Lee et al. 2015). In addition, changing strengths of the jet stream in the LGM could alter the refraction of stationary waves with northwest-southeast tilting.

To explore δU in the PMIP3 models, we apply thermal wind balance, which describes the linear relation between the vertical zonal-mean zonal wind shear and the meridional temperature gradient ($\partial \bar{T}/\partial y$; $\bar{T}$ represents the vertically averaged zonal mean temperature between the surface and 300 mb). We assume

that U at the surface is small compared with U at 300 mb so that the disagreement of δU in the PMIP3 model at 300 mb can mainly be related to the disagreement of $\delta(\partial \bar{T}/\partial y)$ at the same latitudes of δU based on the correlations as shown in Fig. 11c.

We compare the summertime equator-to-pole temperature anomalies ($\delta T$) by averaging zonal mean surface temperature in the tropics (15°S–15°N) and the Arctic (65°–90°N) respectively for each PMIP3 model (Figs. 11c,d). Figure 11c shows a similar trend in the meridional temperature gradient and the equator-to-pole temperature anomalies. This is consistent with the disagreement of δU between PMIP3 model simulations being primarily driven by the disagreement of the equator-to-pole temperature anomalies. Figure 11d shows that surface temperature anomalies for each PMIP3 model are similar in the tropics but the Arctic is less cold for the models comparing favorably with proxies, and this drives a weaker jet stream response ($\delta U$) to
<figure><img src="image_17.png" /><figcaption>FIG. 10. Annual mean $\delta P$ under (a) a white plain minus PI, (b) a white mountain minus PI, and (c) LGM minus PI in HadCM3 simulations (colored shading), pollen reconstruction of precipitation anomalies (colored circles), and diatom reconstruction of precipitation anomalies (the colored square). (d)–(f) As in (a)–(c), respectively, but during summertime.</figcaption></figure>

the LGM forcing. The averages of $\delta\bar{T}$, summarized in Table 2, are about 5.5°C colder in the LGM Arctic for group 1 than for group 2, but almost the same in the tropics. We conclude that PMIP3 models with weak polar amplification exhibit spatial precipitation anomalies in the United States that compare more favorably to the pollen-based reconstructions of precipitation.

**5. Implications of PMIP3 model-data comparison**

In Fig. 4 we show that group 1 of that the PMIP3 models compares favorably with reconstructions of precipitation, and that these models simulate weaker meridional wind anomalies ($\delta V$) compared to group 2 models. This feature can be attributed to smaller phase shifts of stationary wave anomalies (Fig. 8d), likely due to weaker $\delta U$ (Fig. 11b). Figure 11a shows that models with weaker $\delta U$ are in better agreement with pollen-based reconstructions of precipitation, and the weaker $\delta U$ is a consequence of weaker cooling in the polar regions (Fig. 11d).

Although there is evidence for extreme cooling in Greenland during the LGM (Alley 2000), which might imply strong polar amplification at this time (Braconnot et al.

2012; Shakun and Carlson 2010), we speculate that there may be spatial variability in the magnitude of cooling in the Arctic. The COSMOS-ASO model simulations, which have the strongest agreement with the reconstructions out of all group 1 models, shows weak summertime (and annual) polar amplification and predicts strong localized cooling in Greenland, possibly due to local factors such as freezing of the Labrador Sea. Given that differences between PMIP3 model simulations mostly occur due to their parameterizations (Stocker et al. 2013; Abe-Ouchi et al. 2015), it may be that the parameterizations of Arctic summer sea ice could be an important factor. However, further study is needed to exclude other factors that may be important, such as atmospheric lapse rates, water vapor feedbacks, cloud effects, and ocean circulations. The CERN-CM3 model simulations show relatively warmer surface temperature anomalies in the LGM Arctic (Fig. 11d), deviating from the majority of the PMIP3 models. This anomalous warming mainly occurs in Greenland, the Labrador Sea, and the Arctic Sea near Siberia, but the bias does not affect the spatial patterns of $\delta P$, $\delta V$, and $\delta D$ (not shown).

We note that the PMIP3 ensemble mean in our study has a stronger correlation with pollen reconstruction of precipitation
<figure><img src="image_18.png" /><figcaption>FIG. 11. (a) The change of the summertime global zonal mean jet stream strength, $\delta U$, compared to the correlation between the PMIP3 model and pollen-based precipitation reconstructions. The strength of the jet is determined by the latitude with the strongest jet. The dashed black curve shows the linear regression with a slope of -0.77, an intercept of 0.65, and a *p* value of 0.027. (b) The change of stationary wave phase and $\delta U^2$. The black line is the absolute value of the line determined from the Charney-Eliassen theory. (c) $\delta U$, the zonal mean of the change in vertically averaged meridional temperature gradient $\delta(\partial\bar{T}/\partial y)$ (circles), and the change in equator-to-pole temperature difference (plus signs). The meridional temperature gradient is at the latitude of $\delta U$ at the surface level. (d) The change of surface temperature in the tropics (plus signs) and in the Arctic (circles).</figcaption></figure>

than eight individual models [3(e)]. In general, the ensemble mean compares more favorably than individual models by averaging out unpredictable noise (internal variability) and errors in different models (Murphy 1990; Tracton and Kalnay 1993). Given the fact that the PMIP3 ensemble average of the jet speed anomalies ($\delta U$) and polar surface temperature anomalies ($\delta T$) are medium in the ranking of all PMIP3 models, we speculate that there might be a cancellation effect beyond our discussion, such as the disagreement of meridional displacements of the jet by the Laurentide ice topography. We also note that in this article we only discuss the primary mechanism that enhances summer precipitation during the LGM in the south-central United States, but it is very likely that other mechanisms might also play a minor role and provide a source

of water to the south-central United States during the summer, and we leave those for future study.

## 6. Conclusions and future work

We compared a synthesis of precipitation reconstructions from pollen data for the LGM to PMIP3 model simulations, confirming previous findings of increasing precipitation in the western United States during the LGM. The LGM precipitation increases mostly in the southwestern United States during the winter, while in the south-central United States during the summer.

The mechanisms of the enhancement of precipitation in the south-central United States during the LGM are illustrated in a schematic diagram (Fig. 12). The HadCM3 simulations provide evidence that the thermodynamic effect of the LGM ice sheet causes the eastward shift of stationary wave phase lines in the LGM. A simple model of stationary waves (Charney and Eliassen 1949) implies that stronger jet streams cause stationary waves to propagate farther downstream with greater phase shifts. The positive correlation between the strength of the jet streams and the equator-to-pole surface temperature difference indicates that the stronger jets in the LGM are likely the

<table><caption>TABLE 2. Changes of the zonal-mean surface temperature ($\delta \bar{T}$) in the tropics (15°S–15°N) and in the Arctic (65°–90°N) for groups 1 and 2.</caption><thead><tr><th>δT&#773; (°C)</th><th>Group 1</th><th>Group 2</th></tr></thead><tbody><tr><td>Tropics</td><td>-2.68</td><td>-2.48</td></tr><tr><td>Arctic</td><td>-7.92</td><td>-13.46</td></tr></tbody></table>
<figure><img src="image_19.png" /><figcaption>FIG. 12. Illustration of the mechanism that enhances the LGM precipitation in the south-central United States.</figcaption></figure>

consequence of the cooling effect by the LGM ice sheet, con-
sistent with the thermal wind balance. The eastward shift of
the stationary phase lines generates the southerly wind anoma-
lies in the south-central United States and, in turn, increases
the LGM precipitation locally. The PMIP3 models that simu-
late small phase shifts of stationary waves, weaker jet streams,
and weak polar amplification compare more favorably with
the data.

Our study provides a framework for climate model–paleo
data comparison to better understand regional climate and
improve model predictability. Two areas of future work are of
particular importance: First, this study depends on comparing
pollen-based precipitation reconstructions with annual mean
precipitation from PMIP3 models. Hence, we will incorporate
more proxy reconstructions of precipitation, evaporation, and
P minus E from lakes, vegetation, soils, speleothem, and
groundwater in the future. Also, we point out the influence of
precipitation changes in the United States due to the uncer-
tainty of the polar ice sheet. However, the analysis is only ap-
plicable during the summer, and we propose to perform more
analyses of the paleo-temperature reconstructions in the
Arctic region to revisit the polar climate sensitivity (Tierney
et al. 2020; Seltzer et al. 2021). We will also update our analy-
sis with PMIP4 in the future. It is also likely that the enhance-
ment of summer precipitation is related to biases of climate
models, and in this study the model–data constraint relies on
several pollen datasets showing the drying trend in the south-
eastern United States.

Second, in this study, we investigate the relationship be-
tween precipitation and stationary waves, given their spatial
coincidence, and explore their causality. Chen (2010) pro-
poses a hypothesis that stationary waves during boreal sum-
mer follow Sverdrup balance, and this is consistent with the
phasing of δV, δD, and δP as shown in our analysis. However,
Sverdrup balance implies that ageostrophic processes are non-
negligible, unlike our geostrophic stationary wave mechanism.
In the future, we will perform stationary wave model experiments

to test whether the Charney–Eliassen model applies to different
climates, from the LGM, to the Holocene, and to future warming
scenarios.

*Acknowledgments.* H.-I L. was supported in part by a gov-
ernment scholarship to study abroad (GSSA) from the Ministry
of Education, Taiwan, Republic of China, Furukawa Fellowship,
and Dissertation Year Fellowship from Department of Atmo-
spheric and Oceanic Sciences, UCLA. A.T. acknowledges sup-
port from an NSF CAREER award (EAR-1352212) and from
the Laboratoire d'Excellence LabexMER (ANR-10-LABX-19)
and the French government (Investissements d'Avenir). We
particularly thank Dr. William Roberts at the Department of
Geography and Environmental Sciences, Northumbria Univer-
sity, for providing data from HadCM3 simulations, and further
acknowledge the PMIP3 climate modeling groups. In addition,
we thank Prof. Tiffany Shaw of the Department of Geophysics,
University of Chicago and Prof. Gang Chen of the Department
of Atmospheric and Oceanic Sciences, UCLA, for tightening
the gap between the precipitation dipole and wave dynamics.
We also appreciate careful reviews from Dr. Isla Simpson from
National Center for Atmospheric Research (NCAR) and two
anonymous reviewers.

*Data availability statement.* The analyzed data are avail-
able through Lee (2020) and each PMIP3 model data can be
obtained via https://pmip3.lsce.ipsl.fr/.

APPENDIX A

Data and Model Setup

Pollen-based precipitation anomalies (δP) for 29 sites in the United States with standard errors at each site has been summarized in Table A1, and we label 7 of the 29 sites (marked with a dagger in Table A1) from the south-central United States where this study primarily focuses (see section 2).
<table><caption>TABLE A1. The list of pollen-based precipitation anomalies (mean annual precipitation, $\delta P$) for 29 sites in the United States (Bartlein et al. 2011) with the standard error ($\sigma_M$) of each site. A dagger (†) denotes significant precipitation anomalies ($\delta P$) in the central United States.</caption><thead><tr><th>Lat (°)</th><th>Lon (°)</th><th>δP</th><th>σ<sub>M</sub>(δP)</th></tr></thead><tbody><tr><td>37</td><td>-117</td><td>0.6596</td><td>0.2033</td></tr><tr><td>39</td><td>-115</td><td>0.2616</td><td>0.2875</td></tr><tr><td>37</td><td>-115</td><td>0.0279</td><td>0.1660</td></tr><tr><td>37</td><td>-111</td><td>0.5146</td><td>0.1660</td></tr><tr><td>35</td><td>-111</td><td>-0.6493</td><td>0.4065</td></tr><tr><td>37</td><td>-109</td><td>-0.0795</td><td>0.2875</td></tr><tr><td>35</td><td>-109</td><td>1.9753</td><td>0.2875</td></tr><tr><td>33</td><td>-109</td><td>0.7243</td><td>0.0544</td></tr><tr><td>39</td><td>-99</td><td>-0.7068†</td><td>0.4065†</td></tr><tr><td>31</td><td>-97</td><td>1.6462†</td><td>0.2875†</td></tr><tr><td>39</td><td>-95</td><td>-0.2312†</td><td>0.1936†</td></tr><tr><td>39</td><td>-93</td><td>-0.9309†</td><td>0.1454†</td></tr><tr><td>33</td><td>-93</td><td>-0.6014†</td><td>0.2109†</td></tr><tr><td>37</td><td>-91</td><td>-0.9010†</td><td>0.2347†</td></tr><tr><td>31</td><td>-91</td><td>-1.6258†</td><td>0.2875†</td></tr><tr><td>39</td><td>-89</td><td>-0.4685</td><td>0.4065</td></tr><tr><td>35</td><td>-89</td><td>-0.9476</td><td>0.2875</td></tr><tr><td>37</td><td>-85</td><td>-1.4871</td><td>0.1068</td></tr><tr><td>35</td><td>-85</td><td>-1.7654</td><td>0.1061</td></tr><tr><td>31</td><td>-85</td><td>-1.3890</td><td>0.2037</td></tr><tr><td>35</td><td>-81</td><td>-1.7094</td><td>0.2279</td></tr><tr><td>29</td><td>-81</td><td>-0.2332</td><td>0.2875</td></tr><tr><td>27</td><td>-81</td><td>0.1280</td><td>0.1815</td></tr><tr><td>39</td><td>-79</td><td>-1.3955</td><td>0.2875</td></tr><tr><td>37</td><td>-79</td><td>-0.3616</td><td>0.4065</td></tr><tr><td>35</td><td>-79</td><td>-1.6434</td><td>0.2148</td></tr><tr><td>33</td><td>-79</td><td>-1.9936</td><td>0.1999</td></tr><tr><td>37</td><td>-77</td><td>-1.9843</td><td>0.1469</td></tr><tr><td>39</td><td>-75</td><td>-1.5945</td><td>0.2078</td></tr></tbody></table>

Pollen-based precipitation estimates in the United States during the LGM are mainly derived by modern analog method (Bartlein et al. 2011), categorizing pollen into different species with their characteristic climate conditions. In general, changes of the precipitation ($\delta P$) derived from each pollen site are statistically significant given that the $\delta P$ is greater

than the standard error ($\sigma_M$) of each site. However, it is noted that plants might respond to evapotranspiration or evaporation as well as precipitation, and this fact might lead to the uncertainty of pollen-based proxies in the LGM with low $CO_2$.

The diatom-based mean annual precipitation reconstruction in Lake Chalco, Mexico (19°N, 99°W), is derived based on the transfer function by training 40 sites of modern precipitation nearby the lake [see details in Caballero et al. (2019)] with $\delta P$ equal to 0.2671 mm day$^{-1}$ and its standard error as 0.0343. The LGM precipitation anomalies are evaluated by averaging all the records from 26.5 to 19 ka.

Table A2 summarizes nine climate model simulations during the LGM climate with low $CO_2$ (280 ppm), and the LGM ice sheets for each model simulation are prescribed from the blended ice sheet products, the average of three different ice-sheet reconstructions [see details in the supplement of Braconnot et al. (2012)]. Each individual model in the PMIP3 groups performs spin-up simulations with various periods from 600 to 3500 years, and the PMIP3 data are retrieved by averaging different amounts of years (from the last 100 to 600 years). The products of PMIP3 data (as shown in the data availability statement) provide the climatological mean for each variable. We here clarify that the comparison of PMIP3 model-precipitation reconstruction relies on the hypothesis that the ranking of predictive skills for each model stays the same during the summertime.

We also apply numerical experiments (Roberts and Valdes 2017) from the Hadley Centre Coupled Model, version 3 (HadCM3) (Gordon et al. 2000) with 3.75° × 2° horizontal resolution and 19 pressure levels to investigate the primary driving mechanism of precipitation anomalies ($\delta P$). Each experiment is simulated for 900 years, and we analyze the averaged data for the last 100 years. HadCM3 is usually not considered a “state-of-the-art” GCM model (Roberts and Valdes 2017) given its relatively low resolution, but it is numerically affordable to make numerous experiments in the wide variety of climate forcing. Hence, we here only apply HadCM3 to perform mechanism denial experiments instead of incorporating it into the PMIP3 model ensemble.

<table><caption>TABLE A2. The list of PMIP3 model simulations with spatial resolutions during the LGM and PI climate.</caption><thead><tr><th>Model</th><th>Resolution (lat × lon)</th><th>Reference</th><th>Institute</th></tr></thead><tbody><tr><td>NCAR-CCSM4</td><td>0.94° × 1.25°</td><td>Brady et al. (2013)</td><td>National Center for Atmospheric Research</td></tr><tr><td>CNRM-CM5</td><td>1.4° × 1.4°</td><td>Voldoire et al. (2013)</td><td>Centre National de Recherches Météorologiques/Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique</td></tr><tr><td>COSMOS-ASO</td><td>3.71° × 3.75°</td><td>Raddatz et al. (2007)</td><td>Alfred Wegener Institut</td></tr><tr><td>FGOALS-g2</td><td>2.8° × 2.8°</td><td>Zheng and Yu (2013)</td><td>LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences and CESS, Tsinghua University</td></tr><tr><td>GISS-E2-R</td><td>2.0° × 2.5°</td><td>Ullman et al. (2014)</td><td>NASA Goddard Institute for Space Studies</td></tr><tr><td>IPSL-CM5A-LR</td><td>1.9° × 3.75°</td><td>Dufresne et al. (2013)</td><td>Institut Pierre-Simon Laplace</td></tr><tr><td>MIROC-ESM</td><td>2.8° × 2.8°</td><td>Sueyoshi et al. (2013)</td><td>Japan Agency for Marine-Earth Science and Technology, Atmosphere and Ocean Research Institute (The University of Tokyo), and National Institute for Environmental Studies</td></tr><tr><td>MPI-ESM-P</td><td>1.87° × 1.87°</td><td>Adloff et al. (2018)</td><td>Max Planck Institute for Meteorology</td></tr><tr><td>MRI-CGCM3</td><td>1.12° × 1.12°</td><td>—</td><td>Meteorological Research Institute</td></tr></tbody></table>
<figure><img src="image_20.png" /><figcaption>FIG. B1. (a) The PMIP3 ensemble of summertime (July–September) MSE at 700 mb during the PI (colored shad- ing) and the enclosed monsoon region (black contour). (b) As in (a), but for the LGM. (c) The PMIP3 ensemble of summertime (MJJAS) mean minus wintertime (November–March)-mean precipitation during the PI (colored shad- ing) and the enclosed monsoon region (black contour). (d) As in (c), but for the LGM.</figcaption></figure>

APPENDIX B

The Definitions of the North American Summer Monsoon Indices

We introduce two North American Summer Monsoon (NASM) indices, the precipitation-based IPCC index and the MSE-based index, applied in this study. The precipita- tion-based IPCC index (Wang et al. 2011) is developed based on the phenomenon that precipitation occurs during summer rather than winter in monsoon regions. Hence the monsoon can be defined as where the summertime (May– September) mean precipitation is 2.5 mm day⁻¹ greater than the wintertime mean (November–March).

Monsoons can also be conceived as the seasonal migra-
tions of thermally driven overturning circulations (Bordoni
and Schneider 2008) that occur when low-level moisture
static energy, MSE = C<sub>P</sub>T + gz + Lq, reaches a maximum
meridionally (Privé and Plumb 2007; Bordoni and Schneider
2008) where C<sub>P</sub> is specific heat capacity, T is air tempera-
ture, g is gravitational acceleration, z is geopotential height,
and L is the latent heat of vaporization. Hence, monsoon re-
gions can be determined by an enclosed region with a maxi-
mum MSE.

Figures B1a and B1b show the EASM domains during
the PI and the LGM using the MSE-based index. The crite-
ria of MSE contours are arbitrarily chosen to reflect the
reasonable NASM domains with 314 kJ kg⁻¹ for the PI and
309.5 kJ kg⁻¹ for the LGM. Different MSE criteria might
vary the NASM regions but do not change our conclusion
with a southward displacement of the NASM areas during
the LGM. Figs. B1c and B1d reveal the NASM domains
during the PI and the LGM using the IPCC index. The

NASM domains predicted by the MSE-based index and the
precipitation-based IPCC index are quite different because
air with large amounts of MSE could be warm and humid
but does not guarantee to rain.

APPENDIX C

Predictions of Phase Shift from Application of the
Charney–Eliassen Theory

Charney–Eliassen theory (Charney and Eliassen 1949)
applies to topographic induced atmospheric stationary
waves using the vorticity equation with the additions of
topographic forcing and surface drag. According to this
theory, vorticity decreases due to the shrinking depth of
atmospheric column by surface topography, and decreases
from surface drag. Hence, the vorticity equation can be
expressed as

$$
U \frac{\partial \zeta'}{\partial x} + \beta v' = -r \zeta' - \frac{fU}{H} \frac{\partial h_T}{\partial x}, \quad (C1)
$$

where *U* is the zonally mean zonal wind, *ζ'* is the zonal
anomaly of vorticity, *β* represents a beta-plane approxima-
tion, *v'* is the zonal anomaly of meridional wind, *r* is sur-
face drag, *f* is the Coriolis parameter, *H* is the scale height,
and *h*<sub>*T*</sub> is the surface topography. Assuming the surface to-
pography has the form

$$
h_T = \mathrm{Re}[h_0 \exp(ikx)] \cos ly, \quad (C2)
$$

where $h_0$ is the height of the mountain, $k$ is the zonal wave-
number, and $l$ is the meridional wavenumber, the solution
<figure><img src="image_21.png" /><figcaption>FIG. D1. (a) The PMIP3 ensemble zonal-mean $\delta U$ at 300 mb during summer for groups 1 and 2. (b) As in (a), but for zonal mean $\delta U$ in the Western Hemisphere. (c) As in (b), but for the Eastern Hemisphere. (d) The PMIP3 ensemble zonal-mean $\delta U$ at 45°N during summer for groups 1 and 2. (e) As in (d), but for zonal-mean $\delta U$ in the Western Hemisphere. (f) As in (e), but for the Eastern Hemisphere.</figcaption></figure>

of vorticity, as the Laplacian of streamfunction ($\psi'$, where $\zeta' = \nabla^2\psi'$), hence becomes

$$ \psi' = \frac{\tilde{h}}{(K^2 - K_S^2 - i\epsilon)} \exp(ikx) \cos ly, \quad (C3) $$

where $K^2 = k^2 + \tilde{k}^2$, $K_S^2 = (\beta/U), \tilde{h} = (fU/H)$, and $\epsilon = rK^2/(Uk)$. We take the $x$ derivative of $\psi'$ to obtain $v'$. The phase shift angle $\phi$ can be derived by completing the square of the denominator of Eq. (C3).

$$ \begin{aligned} v' &= \frac{\partial \psi'}{\partial x} \\ &= i\tilde{h}k \frac{(K^2 - K_S^2) + i\epsilon}{(K^2 - K_S^2)^2 + \epsilon^2} \exp(ikx) \cos ly \\ &= \tilde{h} \frac{k}{\sqrt{(K^2 - K_S^2)^2 + \epsilon^2}} \exp[i(kx - \phi)] \cos ly, \end{aligned} \quad (C4) $$

$$ \cos \phi = \frac{-\epsilon}{\sqrt{(K^2 - K_S^2)^2 + \epsilon^2}}, $$

$$ \sin \phi = \frac{K_S^2 - K^2}{\sqrt{(K^2 - K_S^2)^2 + \epsilon^2}}. $$

The maximum amplitude of $\psi$ occurs when $\sigma^2 = (K^2 - K_S^2)^2 + \epsilon^2$ is a minimum. We define $\epsilon = \tilde{\epsilon}K^2$, where $\tilde{\epsilon} = r/(Uk)$, and complete the square of $\sigma$:

$$ \begin{aligned} \sigma^2 &= K^4 - 2K^2K_S^2 + K_S^4 + \tilde{\epsilon}^2K^4 \\ &= (1 + \tilde{\epsilon}^2)(K^2 - \frac{K_S^2}{1 + \tilde{\epsilon}^2})^2 - \frac{K_S^4}{1 + \tilde{\epsilon}^2} + K_S^4. \end{aligned} \quad (C5) $$

Hence, the maximum amplitude of $\psi$ occurs when

$$ K = \frac{K_S}{\sqrt{1 + \tilde{\epsilon}^2}}. \quad (C6) $$

Then, $\phi$ can be obtained by plugging this dispersion relation [Eq. (C6)] into Eq. (C4) with a small angle approximation and set $\tilde{\epsilon}_{lgm} = U_{lgm}k/r = 1$.

$$ \sin \phi = \frac{1}{\sqrt{1 + \tilde{\epsilon}^{-2}}}, $$

$$ \cos \phi = -\frac{1}{\sqrt{1 + \tilde{\epsilon}^2}}, $$

$$ \begin{aligned} \delta\phi &\sim \sin\delta\phi = \sin(\phi_{lgm} - \phi_{pi}) = \sin\phi_{lgm}\cos\phi_{pi} - \cos\phi_{lgm}\sin\phi_{pi} \\ &= -\frac{1}{\sqrt{1 + \tilde{\epsilon}_{lgm}^{-2}\sqrt{1 + \tilde{\epsilon}_{pi}^2}}} + \frac{1}{\sqrt{1 + \tilde{\epsilon}_{pi}^{-2}\sqrt{1 + \tilde{\epsilon}_{lgm}^2}}} \\ &= \frac{\tilde{\epsilon}_{lgm}^{-1} - \tilde{\epsilon}_{pi}^{-1}}{\sqrt{1 + \tilde{\epsilon}_{lgm}^{-2}\sqrt{1 + \tilde{\epsilon}_{pi}^{-2}}} \\ &\approx \frac{k}{2r}\delta U, \end{aligned} \quad (C7) $$
<figure><img src="image_22.png" /><figcaption>FIG. D2. (a) The PMIP3 model ensemble meridional average (15°–35°N) of the meridional wind for PI and LGM. (b) As in (a), but for the HadCM3 simulations in PI climate and PI with WP.</figcaption></figure>

where $\delta U = U_{\text{lgm}} - U_{\text{pi}}$. Then, Eq. (C6) reveals the phase shift angle with regard to zonal wind anomalies. The predicted phase shift in Fig. 11b could be obtained by taking $k/r = 0.03$.

## APPENDIX D

### Assumptions of the Charney–Eliassen Theory

The selection of $\delta U$ is essential for estimating the phase shift using the Charney–Eliassen theory. Figures D1a–c summarize the global zonal-mean zonal wind anomalies ($\delta U$) at 300 mb, the mean over the Western Hemisphere, and the Eastern Hemisphere respectively. The $\delta U$ in mid latitudes are generally stronger in the Western Hemisphere, consistent with the locations of the Laurentide Ice Sheet. The Western Hemispheric meridional wind anomalies at 300 mb for the PMIP3 ensemble mean (Fig. D2a) shows about three full wavelengths in the Western Hemisphere. Since the derivations in appendix C require periodic boundary conditions, the (global) zonal-mean $\delta U$ is used to calculate stationary wave phase shift even though the zonal wind anomaly mainly occurs over the Western Hemisphere during the LGM.

The vertical profiles of global zonal mean $\delta U$ at 45°N (Fig. D1d) and zonal-mean $\delta U$ over the Western Hemisphere (Fig. D2e) and Eastern Hemisphere (Fig. D2f) reveal that $\delta U$ increase with height from the surface and reach maximum at 300 mb. Zonal-mean $\delta U$ over the entire globe and the Western Hemisphere are negative in the lower atmosphere for group 1, and these vertical variations of zonal winds might influence the calculations. Also, the calculations rely on a choice of the surface drag rate $r$. However, selection of the $U$ level and $r$ does not influence the positive correlations between zonal wind anomalies $\delta U$ and phase shift angles $\delta\phi$ according to Eq. (C7).

Figures D2a and D2b show $\delta V$ for the PI and LGM in the Western Hemisphere from the PMIP3 model ensemble mean and the HadCM3 experiments in PI climate with and

without WP. There is no obvious phase shift over the Eastern Hemisphere, given the invariant of the jets (not shown). According to Held (1983), the topographic stationary waves due to a point source should decay exponentially with the characteristic length $c_x/r$, where the $c_x$ is the zonal group speed of Rossby waves, $c_x = 2U(k^2/K^2)$. Given the fact that $K^2 \gtrsim k^2$, the maximum characteristic length with $r = (5 \text{ days})^{-1}$ and $U = 20 \text{ m s}^{-1}$ is less than $7 \times 10^2 \text{ km}$. Hence, we will not worry about the contributions of stationary wave sources traveling along the Eastern Hemisphere with different $\delta U$. Our argument is consistent with the fact that the phase shift of stationary waves occurs at all longitudes in the global warming scenario [see Fig. 4g in Simpson et al. (2016)], and exploring the stationary wave response is left to our future work.

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