1. Introduction
Long-lasting atmospheric circulation patterns, in particular, quasi-stationary Rossby waves (QSWs), are linked to the occurrence of extreme weather events (Röthlisberger et al. 2019; Ali et al. 2021, 2022; Fragkoulidis et al. 2018; Wolf et al. 2018; Coumou et al. 2014). Despite this important link, the mechanism(s) that cause these waves to become quasi stationary and/or high amplitude are still being debated (Horton et al. 2016; Hoskins and Woollings 2015). In the past few decades, the occurrence of an increasing number of extreme weather events has attracted the attention of both the public and the research community (Coumou and Rahmstorf 2012; Wehrli et al. 2019), and there has been increased interest in understanding the mechanisms of QSWs that last for 1–4 weeks. Understanding how such atmospheric circulation patterns may change with anthropogenic climate change is critical to understanding how climate change will impact extreme events (e.g., Petoukhov et al. 2013; Coumou et al. 2014; Woollings et al. 2018); however, the underlying hypotheses and mechanisms are still under discussion (e.g., Woollings et al. 2018; Barnes and Screen 2015; Screen and Simmonds 2013). A better understanding of the underlying mechanisms of QSWs will undoubtedly help in understanding any expected changes under climate change.
It is worth noting that the term “quasi-stationary wave” has no exact definition within the scientific literature. One early use was to refer to the variations of stationary waves on interannual and monthly time scales (Held 1983; Karoly et al. 1989). It has also been used to refer to Rossby waves with a shorter time scale, ranging from 2 weeks to 1 month (e.g., Reinhold and Pierrehumbert 1982), particularly in recent years (Wolf et al. 2018; Kornhuber et al. 2017; Ma and Franzke 2021). Such quasi-stationary behavior on weekly to monthly time scales has also been included under the term “persistent anomalies” (e.g., Mo and Ghil 1987), although this term often refers specifically to atmospheric blocking. Here we use the term “quasi-stationary wave” to refer to wave circulation patterns apparent in a 14-day running mean.
Using an index identifying QSWs using a 14-day running mean and Fourier transform on meridional wind data, Röthlisberger et al. (2019) found that many QSWs have a recurrent signal—multiple consecutive waves with similar phases—rather than one trough or ridge staying at one place persistently with near-constant amplitude. Thus, they named the phenomenon recurrent Rossby waves (RRWs). While the RRW index they define has no requirement about a recurrent signal—it identifies all QSWs—recurrent wave behavior was observed in Hovmöller diagrams in many cases (Röthlisberger et al. 2019). This concept of recurrence, in place of persistence, may help explain the occurrence of QSWs under a linear wave framework (Röthlisberger et al. 2019). In this work, distinguishing between recurrent and persistent waves is not our goal. The index we use identifies all QSWs; however, we use the concept of the potential recurrent behavior of these waves to inform our analysis. For clarity, in this paper we employ the following definitions:
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QSW: A wave that is present in time-filtered data, e.g., using a 14-day running mean
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RRW: A QSW that is formed of multiple consecutive transient waves with similar phases
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Persistent Rossby wave: A QSW with near-constant amplitude during its lifetime
If QSWs are indeed composed of transient waves, they may follow some of the properties of transient waves; however, the underlying phase consistency within individual QSWs suggests some quasi-stationary forcing, and thus the factors shaping stationary waves, such as land–sea contrast, mountains, and sea surface temperatures (SSTs) (e.g., Held et al. 2002) may also contribute to the overall characteristics of the QSW. Previous research shows that tropical convection can act as a source of extratropical Rossby waves, as the vertical motion associated with convection leads to upper-level divergence, and thus the generation of vorticity anomalies (e.g., Sardeshmukh and Hoskins 1988). In idealized flow conditions, Rossby waves generated in the tropics tend to propagate along great circles (Hoskins and Karoly 1981) and can strongly influence the extratropics. Diabatic heating in the extratropics can also act as a source of Rossby waves (e.g., Held et al. 2002; Koster et al. 2016; Teng and Branstator 2019). Recent work has even suggested that constructive interference between transient and stationary waves can occur, acting through diabatic heating anomalies, to produce high-amplitude waves in the extratropics (e.g., Goss et al. 2016; Park and Lee 2019).
Rossby wave propagation from the tropics into the extratropics is facilitated by the extratropical jets (Seo et al. 2016), particularly in midlatitude winter. Strong and narrow jets can act as waveguides, which may retain wave energy in the extratropics, leading to higher-amplitude waves (Hoskins and Ambrizzi 1993; Branstator 2002; Manola et al. 2013; White et al. 2022). The tropical Pacific, in particular, is a known source of Rossby waves (see, e.g., Nie et al. 2019), and the Pacific jet can act to guide these waves toward North America and even farther downstream into Europe. Thus, in addition to Rossby wave sources, background flow conditions may strongly impact quasi-stationary Rossby waves.
This paper focuses on prominent large-amplitude quasi-stationary Rossby wave events (QWEs). Using composite analysis centered on the starting date of discrete events, we analyze the meridional winds to investigate the structure of the QWEs themselves, tropical precipitation and vorticity anomalies to study potential sources of QWEs, and the zonal wind to investigate the background flow. We explore these in both observation-based reanalysis data (ERA5) and in ensemble simulations from the Community Earth System Model version 2 (CESM2) to investigate the features, mechanisms, and GCM biases of QWEs.
2. Data and methods
a. Data
We use ERA5 data (Hersbach et al. 2020) to examine quasi-stationary waves in observation-based data. For consistency with the available historical data from the climate models we use in this study, the analyzed time period is 1979–2014. We also use the Global Precipitation Climatology Project (GPCP) observational dataset to assess the accuracy of the reanalysis precipitation (Huffman et al. 2001). All datasets are interpolated onto a regular 1° × 1° resolution grid using Climate Data Operators (CDO; Schulzweida 2022). The original R metric of Röthlisberger et al. (2019) was defined using data at 250 hPa; however, daily data for the model used in this study are only available in the upper troposphere at 200 hPa—we therefore analyze all data at the 200 hPa level. Sensitivity studies with the ERA5 show little difference in our results when using 200 or 250 hPa.
We compare the observed features and statistics of QWEs to those in the CESM2 in both a coupled atmosphere–ocean configuration (denoted “CESM2 LENS”), and with fixed sea surface temperatures (denoted “CAM”). The fixed SST simulations are experiments from the Community Atmosphere Model version 6 (CAM6) provided as Atmospheric Model Intercomparison Project (AMIP) experiments within the Coupled Model Intercomparison Project phase 6 (CMIP6); these simulations use the same atmosphere model as the CESM2 LENS simulations (Rodgers et al. 2021; Eyring et al. 2016). Simpson et al. (2020) show that, in the Northern Hemisphere, CESM2’s ability to simulate the observed storm tracks, stationary waves, zonal winds, and blocking events is within the top 10% of CMIP6 models. The horizontal resolution is 0.9° × 1.25° before our interpolation to 1° × 1°. To increase the signal-to-noise ratio, and to study the impact of internal variability on QWEs, we use multiple ensemble members data, using 1979–2014 (36 years) from each ensemble. There are 10 ensemble members in the AMIP CESM2 experiments. Due to data availability at the time of research, we selected 14 ensemble members (realizations) from CESM2 LENS: 1001, 1021, 1041, 1061, 1081, 1101, 1121, 1141, 1161, 1181, 1231, 1251, 1281, and 1301; further information on these ensemble members is given by Rodgers et al. (2021).
b. Quasi-stationary Rossby wave index R
To study QWEs that are most likely to have high impacts on human society, we study large-amplitude and long-lasting events centered over land, which occur mainly over Europe and North America. The European longitude range is 0° to 60°E and North American longitude is 120° to 60°W. We define a QWE as a series of n consecutive days during which the regionally averaged R index exceeds a threshold x. We set thresholds for the strength x ≥ 9 m s−1 and duration n ≥ 8 days; sensitivity tests show our conclusions are robust to larger/smaller x or n, e.g., x ≥ 10 m s−1 or x ≥ 8 m s−1, a range that covers the subseasonal variation of R during winter. With our standard thresholds, there is an average of approximately one event every two years in each region in ERA5, confirming that we are studying relatively extreme events. The first day on which R exceeds our threshold x is defined as “day 0.” To investigate the underlying behavior of these QWEs, we create composites for each of the two regions centered on day 0 of each identified event.
c. Statistical significance
In the model simulations our use of multiple ensembles provides us with a large number of QWEs; however, in reanalysis data, we have a relatively small number of QWEs due to the limited time period from 1979 to 2014. To determine whether the signals we find in our composite analysis are statistically significant, we use bootstrapping resampling (Hesterberg 2011). To avoid possible effects from the seasonal cycle within winter, in bootstrapping we retained the date (day-of-year) of the QWEs in the original dataset, and combined each date with a randomly selected start year from 1980 to 2013 (permitting QWEs composites from −15 to +15 days that may span 2 years), obtaining a set of randomized non-QWE-specific dates. We iterate this process 1000 times, obtaining distributions close to a Gaussian distribution. We consider a signal statistically significant if our QWE composite mean is either less than the 5th percentile or greater than the 95th percentile of our bootstrapped distributions.
3. Results
a. The statistics of QWEs in CESM2
We first explore whether the recurrent behavior observed in reanalysis data by Röthlisberger et al. (2019) is reproduced in the CESM2. Figure 1 shows the first QWE occurring in the reanalysis and model datasets using Hovmöller diagrams of meridional wind. In both ERA5 and the GCM simulations, we see persistent wavelike patterns of positive and negative meridional winds composed of seemingly multiple separate wave peaks, all with low, but nonzero, phase speed, suggesting that the recurrent behavior identified by Röthlisberger et al. (2019) can be simulated by the CESM2; individual analysis of other QWEs shows similar recurrent behavior, but an exhaustive study of all QWEs has not been performed. The three examples all show high R values on the right column in a time range of more than two weeks, demonstrating how the R metric represents a phase-independent wave packet envelope. The QWEs identified by the R metric all have persistent phases over time, i.e., they are quasi stationary.
Examples of QWEs from (top) ERA5, (middle) CAM, and (bottom) CESM. (left) The meridional wind averaged between 35° and 65°N (as for the R metric) over the event, with the recurrent signal clearly illustrated. (right) The R metric value at the same time.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0042.1
Figure 2 shows climatological statistics of the R metric and of our identified QWEs. We calculate the mean of these R metrics for each 36-yr ensemble member for the CESM and CAM models, and show the ensemble mean (lines) in addition to the standard deviation across ensemble members (shading) to evaluate the role of internal variability. In Fig. 2a, we see that the climatological R metric distribution in ERA5 is typically located within two standard deviations of that in the CAM and CESM experiments, except over the Pacific and western North America region from 160°E to 100°W. Figures 2b and 2c show the R distribution during QWEs; the ERA5 values are predominantly within two standard deviations of CESM2 LENS and CAM models, indicating that the bias in climatological R seen over North America in Fig. 2a does not have a strong impact on North American QWEs (Fig. 2c). To investigate if model biases of climatological meridional wind cause the lower climatological R value in the CESM, we also calculated the R index based on the meridional wind day-of-year anomaly, of which a climatology with a seasonal cycle is removed. The underestimation of R over the Pacific region in CESM2 LENS remains, with a similar pattern to Fig. 2a. This, together with the reasonable simulation of R during quasi-stationary events (Figs. 2b,c), indicates that the CESM2 LENS bias in climatological R is unlikely due to an underestimation of the intraseasonal variability of meridional winds. There is a substantial spread in QWE R between the model ensembles during European events (Fig. 2b), with relatively less spread during North American events (Fig. 2c). The large variance of the R metric during European events among models and ensembles indicates a stronger role of internal variability on the upstream wave propagation of European events. The influence of SSTs can be seen in the differences between CAM and CESM2 LENS simulations.
(a) R metric averaged over 1979–2014 in winter (December–February). Dark shading shows one standard deviation calculated across the different ensemble members of the CAM and CESM datasets, with the light shading showing two standard deviations. (b),(c) The mean R metric during European and North American QWEs, respectively. (d) The composite R metric before, during, and after QWEs. The lines “day 0” and “R = 9” are shown in black, with gray lines at days −5, 5, 10, and 15 for reference. (e),(f) The events per year as a function of event duration in bins, with a width of 3 days, in (e) Europe and (f) North America. The shading is one standard deviation across the different ensemble members.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0042.1
In Fig. 2d we show the composite temporal progression of R before and after day 0 of the QWEs. The horizontal line shows our threshold, x = 9. The R distribution is similar for European and North American events, so we average across all events. The average progression of R during QWEs is similar between ERA5, CESM LENS, and CAM.
Figures 2e and 2f show the event frequency as a function of event duration in ERA5, CESM LENS, and CAM. The shape of the distributions is relatively similar between the models and reanalysis datasets, with most events lasting between 8 (the minimum duration threshold) and 17 days, and a few events lasting out to a month or longer. Despite a larger climatological R across almost all longitudes in CAM compared to CESM (Fig. 2a) the two models show very similar distributions of QWEs (Figs. 2e,f). The ensemble-mean frequency of QWEs is generally larger in both models than in ERA5, particularly for events of 10–15 day durations. Even though the ERA5 values do lie within ±2 standard deviations (note that the shading in Figs. 2e,f shows only ±1 standard deviation), the large difference in average frequency for events lasting around 10 days between ERA5 and the models suggests that at least some of this difference could be model bias.
b. Composites of high-amplitude R events
Having determined that the CESM and CAM models can generally simulate QWEs, we now explore the common dynamical features of QWEs and how these features are simulated in the models, with the goal of reaching a better understanding of the mechanisms of QWEs. In this section, we look into composites of 200 hPa meridional winds to study the waves themselves (analysis using 500 hPa geopotential height shows similar results, see online supplementary Fig. S1); we analyze potential sources of the events, with a focus on precipitation in the tropics, and vorticity fluxes, prior to QWEs; we also study the zonal wind conditions in which QWEs occur. Analysis of the time evolution of local finite-amplitude wave activity (LWA) index is used to better understand interactions between the waves and the background flow. In all the anomaly composites shown the day-of-year climatological mean has been removed.
1) Meridional winds
Composites of meridional winds during QWEs are investigated to determine whether the QWEs demonstrate any consistent wave patterns. The method used to identify QWEs has no phase requirement for separate events; however, significant anomalous wave patterns are found for both European events (Fig. 3) and North American events (Fig. 4). We analyze composites of 5-day means from −5 to 15 days relative to day 0 of the events (as defined in section 2b) to allow investigation of the circulation patterns during the growth phase (days −5 to 0), during the maximum (days 0 to 10), and during the decay phase (days 10 to 15) of QWEs (see gray dashed vertical lines in Fig. 2d).
Meridional wind anomaly composites associated with winter QWEs in Europe (shading). (left) Composites from ERA5 data, (center) CAM composites, and (right) CESM composites. Rows show (top to bottom) The composites of −5 to 0, 0 to 5, 5 to 10, and 10 to 15 days. The dotted regions show anomalies significant at the 90% level based on bootstrap resampling. The black lines show the climatological meridional wind in winter from 1979 to 2014. The line spacing is 1 m s−1; solid lines represent positive anomalies, while dashed lines represent negative anomalies and the zero contour.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0042.1
As for Fig. 3, but for North American QWEs.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0042.1
For the ERA5 data (first column) in Fig. 3, a strong wave pattern is present during the QWEs, with a negative (northerly winds) meridional wind center over western Europe, which would likely lead to a cold anomaly in this region due to cold-air advection. The CAM and CESM models reproduce this pattern well, with very similar phases of meridional wind anomalies over Europe, although the composite signals in models (second and third columns in Fig. 3) are slightly weaker than in ERA5. Due to the larger number of events in these ensembles, the composites show less noise and a clearer hemispheric pattern, despite the smaller amplitude. The models, particularly CAM, show a more circumglobal pattern than is apparent in the reanalysis. This can at least partially be attributed to the large number of events in model ensembles—the circumglobal pattern is much less apparent in analysis of individual model ensemble members of the same length as the ERA5 dataset, suggesting a role of internal variability on the circumglobal pattern of QWEs seen in the model ensembles.
Similarly, in Fig. 4, focusing on North American QWEs, a wave pattern in the ERA5 data (left column) can be seen, peaking in strength over North America. In ERA5 the wave signal is most prominent during both 0–5 days and 5–10 days for European QWEs (Fig. 3), while for North American events the signal peaks during 5–10 days and is still strong 10–15 days after the events start (Fig. 4), consistent with the longer durations for North American events (see Fig. 2e). Both the CAM and CESM models reproduce the negative meridional wind anomaly over North American land and two positive centers over the ocean, though the shape of centers is different downstream at 10–15 days. As for North American events, the amplitude of QWE meridional wind anomalies in models is much weaker than in reanalysis data.
The analysis in this section has focused on meridional winds as a metric of measuring the waves, as this is the variable used in the R metric. Similar composites for ERA5 using 500 hPa geopotential height anomalies are shown in supplementary Fig. S1, and show similar wave patterns, with the phase preference still apparent.
The strong wave patterns in the QWE composites seen in Figs. 3 and 4 are indicative of a particular “phase preference” for QWEs. The R index itself is phase independent, as it is a measure of the amplitude of the wave envelope in the running mean of meridional winds; thus, the phase preference seen in the composites is not related to the methodology of wave packet detection. Two mechanisms could cause this phase preference in the composites: 1) QWEs have a preference for this phase, and so most QWEs have a similar phase; 2) QWE phases are randomly distributed, but the amplitudes of the QWEs have a phase dependence—those with the particular phase found in the composites in Figs. 3 and 4 tend to have higher amplitude, thus allowing this phase to dominate the composite.
To determine which mechanism is responsible, we regard the positive meridional wind anomalies as positive phases and the same for negative anomalies and negative phases. We calculate the fraction of events that have a similar phase as the composite mean—i.e., how many events show meridional wind anomalies of the same sign as the composite mean in the locations of the composite maximum and minimum values. If the phases of the QWEs were randomly distributed, we would expect only 50% to have the same sign. In fact, in ERA5 approximately 80% of the QWEs have a similar phase for European events, with North American QWEs showing a preference of approximately 70%. This indicates that QWEs indeed display a phase preference. The similar composite wave pattern across ERA5, CAM, and CESM suggest that the phase preference identified in ERA5 is simulated by both the CAM and CESM models; however, the composite anomalies are weaker in the models, especially for North American events. Figure 2d shows that the R amplitude during events is similar between models and ERA5 (particularly for days 0–5). Thus, it is confirmed that the modeled events have similar wave envelope amplitude to those in ERA5. For both CAM and CESM models, about 70% (±0.15 as one standard deviation across ensemble members) of European QWEs have the same wind phase as the composite mean anomaly, but for North American events the modeled phase preference is just over 50%, and its standard deviation across ensemble members is about 0.25. While the ensemble-mean model phase preference is lower than that for ERA5 for both regions, the ERA5 value is within the spread of the individual ensemble members; thus, we cannot rule out that the difference between models and reanalysis data is due to internal variability of the climate system. The weaker phase preference in the model ensemble mean is consistent with the slightly weaker meridional wind composite anomalies in the models for European events and much weaker values for North American events.
Having determined that QWEs demonstrate a phase preference, at least for European events, we investigate potential mechanisms for this phase preference. One possibility is that, because the climatological meridional wind is not excluded from the R index, QSWs with the same phase as stationary waves will constructively interfere with the stationary waves to produce a larger-amplitude R than QSWs with other phases, and are thus more likely to be detected as high-amplitude QWEs. Figures 3 and 4 show that there is some resemblance between the meridional wind anomalies during QWEs and the climatological meridional wind pattern. We calculate the Spearman’s correlation coefficient between the climatological meridional winds and the QWE composite anomalies for a box covering 35°–65°N, 0°–60°Efor European events and a box covering 35°–65°N, 60°–120°W for North American QWEs. The correlation coefficient is around 0.5 for ERA5 for both Europe and North America, with values between 0.5 and 0.7 for CAM and CESM2 LENS, confirming this connection (note that the climatological values are removed prior to forming the event composites). The correlation coefficients between QWE meridional wind composites and the 4 ≤ k ≤ 15 climatological mean wave are even higher—around 0.8 in ERA5 and around 0.6 in CAM and CESM2.
We further explore the relationship between the QWE phase preference and the climatological stationary waves by removing the climatological mean υ prior to calculating R and then identifying QWEs. Although weaker, the same phase preference for both European and North America QWEs is still apparent in these new composites (supplementary Fig. S2). The weaker signal shows that the inclusion of the climatological winds in the R metric does have some impact on the events that are identified as QWEs; however, the phase preference for high-amplitude QWEs events remains, showing that this is an intrinsic feature of high-amplitude long-lasting QWEs. The inclusion of climatological waves in the R metric is discussed by Röthlisberger et al. (2019), and we discuss possible implications for our results and high-impact weather extremes in our discussion section.
One hypothesis for the phase preference exhibited by QWEs is that this phase preference is a feature shared by all extratropical high-amplitude waves, regardless of whether they show quasi-stationary behavior. If waves with this phase are more frequent, or more likely to be amplified (due to zonal asymmetries in SSTs or topography for example), then we might expect to observe this particular phase for any high-amplitude Rossby wave packets (RWPs), without any requirement for quasi stationarity. To investigate this, we make use of a dataset of RWPs from Fragkoulidis and Wirth (2020) based on 6-hourly data. The main difference to the R metric is that there is no time filtering of the data before finding Rossby wave envelopes in this dataset, and thus, the data include both high-frequency and quasi-stationary wave packets. We perform a latitudinal average of the RWP data over the same range as is selected for the R metric. We select large-amplitude RWPs in winter over our two study regions by selecting dates during winter when the wave packet amplitude exceeds a certain threshold (20) over at least 80% of the region. Composites of these large-amplitude RWPs show no significant preference in meridional wind. This result is insensitive to whether we include, or exclude, dates on which there is a QWE within five days; only about 10% of large-amplitude RWPs occur on dates close to a QWE. This result indicates that the phase preference of QWEs is specific to the quasi stationarity of the associated waves.
2) Rossby wave sources
We now investigate tropical precipitation and vorticity fluxes prior to the QWEs, to explore whether any persistent wave sources may help explain the phase preference of the QWEs identified in the section above.
Figure 5 shows maps of composite tropical precipitation anomalies averaged from 5 to 15 days before the start of each QWE. We select 5–15 days as the tropical Pacific is a common source for Rossby waves, and it usually takes about 1 week to 10 days for such waves to propagate to North America or Europe (Seo et al. 2016). In these plots we limit our analysis to 30°S to 30°N; in supplementary Fig. S3 we show diabatic heating anomaly composites from reanalysis data for the whole globe, and find no consistent extratropical signals prior to QWEs. In ERA5 there is a strong positive precipitation anomaly in the South Pacific convergence zone (SPCZ) and east of the Maritime Continent preceding European events, in addition to weak negative precipitation anomalies in the middle and eastern Pacific. These anomalies show some similarity to La Niña anomalies and/or phase 4/5 of the Madden–Julian oscillation (MJO) during the winter season (Yang et al. 2020; Waliser et al. 2009). However, the models fail to reproduce several details of the precipitation pattern observed in ERA5, particularly in the eastern Pacific, although significant anomalies are found (see Figs. 5c,e), with similar extent and pattern to those during El Niño.
Composites of precipitation anomalies in the tropics (30°S–30°N) 5–15 days before the start date of QWEs for (left) European events and (right) North American events. Results are from (top) ERA5, (center) CAM, and (bottom) the CESM LENS are shown. The stippled region shows the 90% significant precipitation based on bootstrap resampling.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0042.1
Prior to North American events, there is also a positive precipitation anomaly in the Maritime Continent and negative anomalies in the middle and eastern Pacific, similar to the precipitation anomaly composite before European events. In contrast to the case of European events, the pattern of precipitation anomalies before North American events shows clear similarities between ERA5 and models over the Pacific Ocean, although the precipitation in models is weaker. Outside of the Pacific, there is little agreement between models and ERA5; this suggests that either the models exhibit biases, or that the statistically significant signals found in ERA5 outside of the Pacific Ocean are spurious, as can happen when performing repeated significance tests (Wilks 2016).
One hypothesis for the difference between ERA5 and the model composites is that the models rely too strongly on an El Niño–like pattern to simulate the positive precipitation pattern west of the date line. To investigate this, we separate the composite precipitation anomalies into seasonal mean and subseasonal anomalies. For simplicity, we show these anomalies averaged between 10°S and 10°N in Fig. 6, with the shaded region showing the 5%–95% confidence range based on bootstrap resampling. In Fig. 6, the composite mean precipitation anomaly (red line) in ERA5 is dominated by subseasonal anomalies (yellow/orange line) with little contribution from the seasonal mean in each year (green dashed lines). This fits the time scale of SPCZ (Brown et al. 2020). The comparison of the precipitation lines before European events in ERA5 and the models depicted in Fig. 6 provides an explanation for the opposite precipitation patterns in Fig. 5. The models tend to simulate the positive precipitation anomaly on the west side of the date line by El Niño on the seasonal time scale. The subseasonal variation, which may be associated with MJO, may be inadequate to drive European QWEs in the models.
Composite precipitation anomalies in the tropics (10°S–10°N) 5–15 days before the start date of QWEs for (left) European and (right) North American events. Results are from (top) ERA5, (middle) CAM, and (bottom) the CESM2 LENS are shown. The shading shows the 90% precipitation range based on bootstrap resampling. The red line shows the QWE composite precipitation anomaly, and the orange line is the composite anomaly with the seasonal mean for each year removed. The green dashed line shows the composite seasonal mean anomaly. Note that the y-axis scale changes between panels, particularly between the ERA5 data in the top row and the model data in the middle and bottom rows.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0042.1
For North American events Fig. 6 suggests that both subseasonal and seasonal variability contribute in ERA5 and CAM. In CESM, however, seasonal mean anomalies dominate the total anomaly. The different ratio of subseasonal and seasonal mean anomalies between ERA5 and CESM suggests that the role of tropical precipitation as a wave source may differ in the model. For example, tropical precipitation can affect models’ stationary wave simulation (e.g., Garfinkel et al. 2022) and therefore may impact QWEs. The weakened precipitation patterns in the models may be related to the weaker phase preference in the meridional wind composites for the model QWEs, as seen in Figs. 3 and 4.
While reanalysis data are the best available data for frequent global and gridded observations of upper troposphere winds, precipitation in reanalysis datasets is known to have biases relative to direct observations (e.g., Hassler and Lauer 2021); we thus repeat this analysis using daily data from the Global Precipitation Climatology Project (GPCP) dataset (Huffman et al. 2001). We find very similar results between ERA5 and the GPCP analysis (see supplementary Fig. S4), though with some regional differences. Using ERA5 provides a longer time series and thus allows us to include more events in this main analysis.
The high-pass filtered vorticity flux divergence shows no clear patterns in the 5-day averages, during or prior to European (or North American) QWEs (not shown); however, this may simply be because these fluxes are dominated by time scales shorter than 5 days, which are averaged out in our 5-day means. In contrast, the composites of low-frequency (10–30 day bandpass) vorticity flux divergence before particularly strong European events shows a clear wave train from North Pacific to the Atlantic, upstream of the location of the QWEs (Figs. 7a,d,g) in ERA5. The CAM and CESM QWE composites, however, do not reproduce this wave train (supplementary Fig. S5). Given that the models reproduce European QWEs and their phase preference (albeit slightly weaker than observed), we would expect to see similar, if weaker, patterns in the eddy vorticity fluxes to that seen in ERA5 if this signal is indeed robust, unless the models exhibit a bias in their mechanisms for producing QWEs. To test the robustness of the signal found in ERA5, we repeat the analysis for QWEs using lower thresholds for R: 9 ≥ R ≥ 8, i.e., R greater than 8 but excluding events in our R ≥ 9 analysis; and 8 ≥ R ≥ 7, i.e., R greater than 7 but excluding events in our R ≥ 9 and 9 ≥ R ≥ 8 analyses. Meridional wind composites for these slightly weaker European QWEs show similar, albeit weaker, wave patterns to that seen for R ≥ 9 (cf. Fig. 3 with supplementary Fig. S6); however, the eddy vorticity flux composites show little resemblance to the signal seen for R ≥ 9 (contrast columns in Fig. 7). Thus, the statistically significant wavelike pattern in vorticity flux anomalies seen in Figs. 7a, 7d, and 7g may be spurious signals due to a relatively small number of events in the ERA5 dataset. The lack of a robust significant signal in the QWE vorticity flux composites does not mean that vorticity fluxes play no role in the development of QWEs, however; vorticity fluxes across individual events may vary in phase, propagation speed, or wavenumber, such that, in the event composites, there is little robust signal.
Composites of the inverse Laplacian of low-frequency (10–30-day bandpass filter) vorticity flux divergence leading up to European events. The stippled regions show anomalies significant at the 90% level based on bootstrap resampling. (left) The composites of QWEs with R ≥ 9 as in our standard definition, (center) QWEs with 9 ≥ R ≥ 8, and (right) QWEs with 8 ≥ R ≥ 7.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0042.1
3) Background flow
In the section 3b(1), we show that many QWEs have similar meridional wind anomalies, and thus, QWEs exhibit a phase preference across different events, particularly for European QWEs. In the section 3b(2), we find significant signals of tropical precipitation anomalies prior to QWEs; there are some similarities between ERA5 and modeled QWEs precipitation anomalies, but with biases in strength and time scale. Vorticity flux divergence composites do not show any robust phase preference across ERA5 and models, suggesting other factors are needed to explain the phase preference seen during European QWEs. In addition to wave sources, Rossby waves are affected by the background flow in which they propagate. In this subsection, we explore the zonal wind anomalies during QWEs to explore the possible impacts of the background flow conditions on the occurrence of QWEs.
Figure 8 shows composites of zonal wind during European QWEs, with the climatological zonal wind shown in black contours. From day −5 to day 0, at the growth stage of the events, a pattern of positive and negative zonal wind anomalies occurs over the Atlantic, similar in both ERA5 and models. This is similar to the northern jet regime identified by Woollings et al. (2010). Over the Pacific, a positive anomaly is seen in the core of the jet, with negative anomalies to the north and south, indicating a narrowed and strengthened Pacific jet. This strengthening and narrowing of the jet is consistent with a stronger jet waveguide over the Pacific (Manola et al. 2013), which may help guide transient waves from tropical sources toward North America and the Atlantic. Both CAM and the CESM2 LENS reproduce the main features of these zonal wind patterns, though the simulated zonal wind anomalies are more persistent in the models than in ERA5. During days 0–10, the zonal wind anomalies in the Atlantic/Europe region become wavier; this is more noticeable in the ERA5 analysis where the phase preference of the QWEs is stronger. During days 5–15, the zonal wind at the same latitude with QWEs largely slows down and the Atlantic jet is therefore forced to follow a more southward track with reduced SW–NE tilt. By days 10–15, the zonal wind at the latitude of the QWEs has been decelerated, with negative anomalies extending over a broad region (almost hemispheric in the models).
Anomalous zonal wind composites associated with winter QWEs in Europe (shading). (left) Composites from ERA5 data, (center) CAM composites, and (right) CESM composites. Rows show composites for (top to bottom) −5 to 0, 0 to 5, 5 to 10, and 10 to 15 days. The black lines show the zonal wind climatology in winter from 1979 to 2014 with labels on the lines, and the dotted regions show anomalies significant at the 90% level based on bootstrap resampling.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0042.1
In Fig. 9 the ERA5 composites of zonal wind for North American events show somewhat similar features over the Pacific as for European events: The Pacific jet is strengthened over the western and central Pacific, with a slight southward shift, and narrowing as the northern edge weakens. This southward shift may make it easier for a tropical divergence to initiate an extratropical Rossby wave train. The zonal wind over northwest North America is also strengthened in days −5 to 0, with negative anomalies to the south, suggesting increased separation between the Pacific jet and the North Atlantic jet; however, the wavelike structures visible in the zonal wind composites suggest some of these features may be due to the waves themselves and/or potential wave–mean flow interaction. The GCMs simulate much weaker anomalies with generally more spatial coherence, but both models show a general weakening and widening of the Pacific jet prior to QWEs, in contrast to that seen in ERA5.
As for Fig. 8, but for composites of North American QWEs.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0042.1
We calculate changes over time in 〈u〉 and
Change of local finite-amplitude wave activity anomalies during ERA5 QWE composites for consecutive 5-day periods (shading) for (left) European and (right) North American QWEs. The black lines show the corresponding changes in zonal wind anomalies with labels on the lines. The density-weighted vertical average of zonal wind and LWA is calculated from 489 hPa, the corresponding pressure of pseudo-height 5000 m in the LWA algorithm, to the top of the atmosphere.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0042.1
4. Further discussion on phase preference of QWEs
One key result from this work is that quasi-stationary Rossby wave events have a preferred phase. This suggests a preferred amplification of quasi-stationary waves with this particular phase. This phase preference can help explain the high correlation between the wavelike pattern of the Rossby wave packets metric and extreme events (Röthlisberger et al. 2019; Ali et al. 2021), despite the fact that extreme events (such as high or low temperatures) are typically associated only with a particular Rossby wave phase (e.g., Pfahl and Wernli 2012). The preferred northerly winds over Europe in Fig. 3 (blue shading) are consistent with the center of a wider European region where a high R metric is associated with a higher duration of cold extremes during winter (see Fig. 10 in Röthlisberger et al. 2019). The preferred high pressure center over the eastern Atlantic (inferred from the meridional winds in Fig. 3, or directly from supplementary Fig. S1), is also collocated with a region where high R leads to longer dry spells over the North Atlantic (Fig. 4 in Ali et al. 2021). The phase preference of QWEs found in this work deepens the understanding of quasi-stationary waves as a mechanism for extreme events, and suggests that this may lead to a spatially heterogeneous distribution of extreme events.
For North America QWEs, the phase preference is slightly weaker than for European events. The phase of the waves in the composites produces northerly winds over central Canada and the United States, likely leading to cold-air advection in these regions. Indeed, the meridional wind pattern in our North America QWE composites (Fig. 4 and Fig. S2) is very similar to the meridional wind anomalies identified as precursors to eastern North America extreme cold weather events by Harnik et al. (2016). These authors also connect this meridional wind pattern to the second EOF of monthly mean seasonal anomalies of meridional winds, offering a potential connection to quasi-stationary waves. Understanding the origins of QWEs with this particular phase may help understand extreme cold events over North America.
The relationship between stationary waves and large-amplitude QWEs supports previous work on stationary waves and large-amplitude waves. Goss et al. (2016) proposed a stationary wave interference theory, arguing that tropical heating can generate large-amplitude waves in midlatitudes through constructive interference between stationary waves and the transients forced by the tropical heating. This theory was extended by Park and Lee (2019) to highlight the role of extratropical diabatic heating in response to tropical forcing. While Goss et al. (2016) focus on waves with wavenumbers 1–3, this mechanism may extend to intermediate-scale stationary waves which tend to be meridionally trapped (e.g., Simpson et al. 2016). A phase locking of high-amplitude QSWs has also been seen in summertime (e.g., Kornhuber et al. 2019, 2020), potentially related to the “quasi-resonant amplification” (QRA) of Rossby waves, in which transient waves resonate with a more stationary wave to produce a high-amplitude wave (Petoukhov et al. 2013). This would be consistent with our results for QWEs—that waves with a similar phase to that seen in the stationary waves (when filtered to similar wavenumbers) are more likely to be high amplitude and quasi stationary. Further research is required to understand whether the phase preference found here relates to that found in summer and/or QRA.
It is interesting that the AMIP (CAM) and CESM2 LENS experiments show more common features than differences despite the prescribed versus modeled SSTs. The insensitivity of the composites, and model biases, to the SSTs (i.e., similarities between the CAM and CESM2 results) suggest that SSTs do not play a substantial role in the mechanisms of QWEs, or the physics related to precipitation weighs much more than SSTs. The SSTs thus are very unlikely to be the explanation for model biases identified in this study. Since the precipitation anomalies in CAM prior to QWEs remain biased toward a stronger seasonal precipitation signal than is seen in ERA5 even with prescribed SSTs, the similarity between precipitation and diabatic heating suggests that precipitation and related variables can be more helpful for subseasonal forecasting than SSTs. The most intriguing outcome of precipitation analysis is that the subseasonal variability can be independent of seasonal precipitation anomaly, and be associated with QSWs in reality, while it cannot be reproduced in the models. Therefore, it is crucial to be cautious when interpreting results about ENSO and QSWs and designing experiments accordingly.
Further research is required to fully investigate the connections of the key QWE signals identified in this study, including the direction of causality. Our results caution about using models, including idealized models, to research and understand QSWs: realistic-seeming QWEs may be found in models, but the underlying mechanisms may be different from those in reality. Our results show a potentially large influence of internal variability, which may also complicate comparisons between observations and model simulations.
5. Summary
To better understand the mechanisms of high-amplitude quasi-stationary Rossby waves, we have investigated the characteristics, precursors and model biases of high-amplitude and long-duration (≥8 days) quasi-stationary Rossby wave events (QWEs) over Europe and North America in ERA5 data and GCM simulations. We analyze both fully coupled GCM (CESM) simulations and simulations using the same atmosphere model with prescribed SSTs (CAM). We select a threshold for the amplitude and duration of these events such that there is approximately one event every two years in each region in ERA5, thus focusing on relatively rare events.
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The recurrent wave signal and distribution of QWE duration can be broadly reproduced by the CESM and CAM simulations. Both models simulate a higher mean frequency of high-amplitude QWE events than ERA5, although the ERA5 frequency does lie within ±2σ calculated from model ensembles; thus, the differences may be due to natural variability.
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We find a preferred phase for high-amplitude QWEs in ERA5, which is robust to excluding the climatological winds when calculating the wave envelope amplitude. Both CESM and CAM simulate this preferred phase of QWEs. The ensemble-mean phase preference (fraction of distinct QWEs with the same phase) is weaker in the model’s ensemble means than in ERA5, particularly for North American events, but the reanalysis value lies within the ensemble members’ spread.
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There is substantial variability between different 36-yr model ensemble members, for both the number of QWEs and the magnitude of phase preference, suggesting that internal variability plays a relatively large role in the statistics of QWEs.
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Zonally localized tropical precipitation signals are found prior to QWEs in reanalysis data and observations. However, this tropical precipitation anomaly is dominated by subseasonal precipitation anomalies in the reanalysis data, but seasonal precipitation anomalies in the CESM model. Furthermore, the magnitude of the precipitation anomalies is approximately 1–2 times smaller in the models compared to the observed anomalies.
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The jet over the Pacific is narrowed and strengthened prior to and during QWEs in ERA5, potentially increasing the waveguidability for Rossby wave propagation.
Our work has shown that high-amplitude long-lasting QSWs over both Europe and North America have a preferred phase. While models can reproduce some of the observed characteristics of QWEs, the underlying mechanisms are not necessarily the same in the models as in observations, even when observed SSTs are imposed. This has important consequences for understanding the impacts of climate change on quasi-stationary waves, and thus extreme weather events.
Acknowledgments.
We thank three anonymous reviewers and the editor for their helpful comments and feedback which greatly improved this manuscript. We thank Olivia Martius, Syed Mubashshir Ali, and Matthias Röthlisberger for helpful discussions and the code of the R metric. We also thank Isla Simpson for valuable suggestions on the intraseasonal variability. Thanks to Georgios Fragkoulidis and Volkmar Wirth for the Rossby wave packets data. RHW would like to acknowledge insightful and useful conversations with Profs. Rodrigo Caballero and Gabriele Messori at the beginning of this project. CF and RHW were funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) (RGPIN-2020-05783).
Data availability statement.
The ERA5 data can be obtained from Copernicus Climate Change Service (C3S) Climate Date Store: https://cds.climate.copernicus.eu (Hersbach et al. 2020). The CESM2 LENS data can be obtained from Climate Data Gateway at NCAR: https://www.earthsystemgrid.org/dataset/ (Eyring et al. 2016). The AMIP trials of CESM2 can be obtained by acccmip6 package: https://acccmip6.readthedocs.io/en/latest (Rodgers et al. 2021). The Rossby wave packets data are obtained directly from Dr. Georgios Fragkoulidis (Fragkoulidis and Wirth 2020). The code of R metric can be obtained from GitHub: https://github.com/avatar101/R-metric.
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