Diagnosing observed extratropical stationary wave changes in boreal winter

Stationary waves are time-averaged zonally asymmetric component of the climatological mean atmospheric circulation, primarily due to the unevenly distributed topography and diabatic heating. Stationary waves are subject to influence from long-term external forcing. In this study, the temporal evolution of the winter (January) Northern Hemisphere stationary waves during 1961–2020 is diagnosed with the fifth generation European Centre for Medium-range Weather Forecasts reanalysis data (ERA5), which shows an overall strengthening in amplitude and an eastward shift in phase. A stationary wave model is used to attribute the stationary wave response to changes in the zonal mean basic state (ΔZM) and the zonally asymmetric diabatic heating forcing ( Δq∗ ). The pattern of stationary wave changes is well captured by the response to ΔZM alone, whereas the contribution of Δq∗ to the amplitude increases in height and becomes dominant in the stratosphere. Δq∗ is also found to be important in driving stationary wave changes in the North Pacific and Western Europe regions. Furthermore, changes in tropospheric stationary waves are probably a result of internal variability, whereas stratospheric changes are more likely to be driven by external forcing.


Introduction
Stationary waves are the zonal deviations of the time-mean atmospheric circulation and manifest as large troughs and ridges in the upper troposphere, and atmospheric high and low pressure near the ground.Due to their important roles in transporting material, energy and momentum (Andrews and McIntyre 1978, Peixóto and Oort 1984, Lee et al 2011) and controlling tracks of synoptic-scale disturbance (Kaspi and Schneider 2013), stationary waves contribute to regional climate anomalies in the Northern Hemisphere (NH) (Broccoli andManabe 1992, Kaspi andSchneider 2011).For example, stationary waves are critical for the propagation of anomalous signals of the general circulation (e.g., the Northern Annular Mode) between the troposphere and the stratosphere (Baldwin and Dunkerton 1999, Hu and Tung 2002, Kushner and Polvani 2004).Chen et al (2005) further pointed out that the changes in upward planetary waves in winter affect the polar vortex as well as the East Asian westerly jet stream, the East Asian trough, the Siberian high, and the Aleutian low.Moreover, large-scale planetary waves are involved in possible mechanisms in explaining Arctic-midlatitude teleconnections (Honda et al 2009, Barnes and Screen 2015, Cohen et al 2020, Zappa et al 2021, Smith et al 2022, Outten et al 2023) from both thermal (Screen et al 2018, He et al 2020) and dynamic perspectives (Zhang et al 2008, Inoue et al 2012), as well as troposphere-stratosphere interactions (Kidston et al 2015).Therefore, it is important to understand the changes in stationary waves and the underlying dynamics.
Deeper research into climate change projections has led to a consensus that atmospheric dynamics are a key source of uncertainty in predicting future climate change (Shepherd 2014).For example, changes in atmospheric circulation are the largest source of uncertainty in European winter precipitation (Fereday et al 2018).The changes in stationary waves are typically investigated by diagnosing the long-term response of future stationary waves in climate models under different scenarios.The eastward shift with global warming (Wills et al 2019) and poleward shift with increasing extreme events (Wolf et al 2018) are found to occur, as a result of the reduction in the zonal wavenumber (Haarsma andSelten 2012, Simpson et al 2016), the weakening of the Walker circulation (Haarsma and Selten 2012), and the poleward shift of the tropospheric jet (Barnes and Polvani 2013).The changes in the stationary wave amplitude under climate scenarios are still under debate.Joseph et al (2004) indicated that the response of stationary waves to greenhouse gases and sulfate aerosols is to reduce amplitude, while Brandefelt and Körnich (2008) found 14 out of 16 climate models show increasing or relatively fixed amplitude in response to increasing CO 2 .The contrast results of increasing amplitude (Wang et al 2020) and invariable amplitude (Barnes 2013, Sellevold et al 2016) are also proposed in the stationary wave response to Arctic amplification.With 39 coupled models in Coupled Model Intercomparison Project 5 (CMIP5), Wills et al (2019) showed that the amplitude of stationary waves in the troposphere (300 hPa and 850 hPa) will decrease by the end of the century, except for an increasing trend between 50 • N and 60 • N at 300 hPa and between 40 • N and 70 • N at 850 hPa.
Since stationary waves arise from uneven distributions of subsurface and thermal forces, both orography and heating are essential forcing factors for stationary waves.Transient waves and nonlinear interactions are relatively minor zonally asymmetric forcing factors (Nigam 1984).In addition, zonal mean zonal winds affect the structure and amplitude of stationary waves (Hoskins and Karoly 1981, Held 1983, Held et al 2002).Based on this, the contribution of each factor was further investigated as follows.With improved models and higher resolution, the contributions of the zonal mean wind field and zonally asymmetric forcings to explain the stationary wave response to climate change are considered to be similar (Joseph et al 2004, Brandefelt andKörnich 2008).As for the specific contribution of zonally asymmetric forcings, the dominant role of diabatic heating forcing has been widely recognized (Stephenson and Held 1993, Wang and Ting 1999, Sellevold et al 2016).Topographic forcing was considered to be less critical in some studies (Stephenson and Held 1993, Joseph et al 2004), but Ting et al (2001) suggested that it is as important as diabatic heating forcing in high latitudes during the NH winter.Ma and Franzke (2021) revealed that the nonlinear interaction between topography and diabatic heating contributes more than the topographic forcing itself.As regards transient forcing, some considered it to be the main forcing term and can offset part of the effect of diabatic heating forcing (Stephenson and Held 1993), but Wang and Ting (1999) showed the opposite.Garfinkel et al (2020) emphasized the large role of nonlinearities on stationary waves, especially in the Pacific and North American sectors.
In addition, decadal climate change is often the combined result of external forcing (e.g., greenhouse gases, aerosols and solar radiation) and internal variability (e.g., ENSO, PDO, AMO) (Dai and Bloecker 2019).Recent studies have shown that internal variability drives climate change on multidecadal time scales and even dominates external forcing on regional to continental scales (Deser et al 2012, Thompson et al 2015).Internal variability plays a leading role in the atmospheric circulation (Deser et al 2014, Shepherd 2014) and induces stationary wave changes through variations within the oceanatmosphere system.Therefore, it is necessary to consider the role of internal variability in the diagnosis of stationary wave changes.
Stationary waves are time averaged over time scales of decades, and studies of stationary waves often lack comparisons and constraints from observational data, instead adopting the wave response obtained from respective calculations or simulations.However, as mentioned above, there is no consensus in the analysis of stationary wave variations arising from different premises of climate change as well as different methods and models adopted.Currently, relatively long and reliable data are available, allowing us to diagnose specific stationary wave variations observed in the past, and thus provide a reference for modelling and related studies.We use ERA5 reanalysis data for the period 1961-2020 to analyze the changes in stationary waves that have taken place in the real atmosphere, and a nonlinear stationary wave model (Wang and Kushner 2011) is employed to investigate the underlying dynamics.

Data
The ERA5 monthly mean horizontal wind data with a horizontal resolution of 0.25 • × 0.25 • are used to calculate the time average of the zonally asymmetric stream function, which is used to represent stationary waves.We also use the 6-hourly ERA5 data including wind field (u, v, w), temperature, and surface pressure to calculate the zonal mean basic state (ZM) and zonally asymmetric diabatic heating (q * ).Their horizontal resolution is 1 • × 1 • .All of the above data are for January from 1961 to 2020 with the standard 37 vertical levels.The past 60 years are divided into three equal periods: early (1961-1980, P1), middle (1981-2000, P2), and late (2001-2020, P3).Here, changes (P3 minus P1) at 250 hPa (where the stationary wave amplitude is greatest in the troposphere) and 70 hPa (where the changes are most significant in the stratosphere) are studied.

Model
A nonlinear stationary wave model (SWM) is employed to investigate the contribution of factors to stationary waves.It is developed from a dry dynamical core from the Geophysical Fluid Dynamics Laboratory and it has been shown that the model can accurately reproduce the climate state of stationary waves with prescribed realistic orography, diabatic heating and ZM (Wang and Kushner 2011).The solution of the stationary waves in the SWM can be considered a function of the ZM and the zonally asymmetric forcings, including the topography forcing, the diabatic heating forcing (q * ), the transient eddy forcing, and the stationary nonlinear forcing.Diabatic heating consists of radiative fluxes (due to atmospheric constituents such as greenhouse gases), latent fluxes (due to phase changes of water substance) and turbulent fluxes of sensible heat from the Earth's surface (Ling and Zhang 2013).Transient forcing describes the effect of eddy momentum flux from transient waves (Wang and Ting 1999).The stationary nonlinear forcing is mainly from the advection term in the equation of atmospheric motion (Wang and Kushner 2010).
The SWM has a horizontal resolution of T42 (2.8 • × 2.8 • ) and 42 unevenly spaced hybrid vertical coordinates (up to 0.4 hPa).The prescribed ZM for the SWM is composed of zonal winds, meridional winds, vertical winds, temperature, and surface pressure.The q * (unit: K/day) is calculated as residuals of the thermodynamic equation (Yanai et al 1973): where T is the temperature, ω the vertical wind, σ = (RT/c P P) − (∂T/∂P) the static stability, V horizontal wind velocity, and ∇ gradient operator.Given the nonlinearity of the model itself and the degraded performance by including transient wave fluxes, only ZM and q * data are prescribed in the model.

Experiments
To attribute the stationary wave changes to ∆ZM and ∆q * , we put different combinations of forcing data in P1 and P3 into the SWM, e.g., experiment LE represents ZM in P3 (L) and q * in P1 (E).Table 2 lists the experiments and the outputs.The difference between the last four experiments and the first reveals the stationary wave responses to different combinations of forcing.

Results
Figure 1 shows the climatological stationary waves in P1 and changes (P3 minus P1) in total waves and wavenumbers 1-3.Higher wavenumbers have small amplitudes (figure S1) and thus are not shown here.
Most of the changes occur before 2000 (P2 minus P1) and changes are relatively small after that (P3 minus P2) (figure S2).Spatially, consistent eastward shifts and strengthening are seen from the upper troposphere to the lower stratosphere, with a barotropic dipole anomaly over the North Pacific and North America (figures 1(a) and (e)) from 700 hPa to lower stratosphere (not shown).Wavenumber 1 contributes most to the structure of stationary wave changes and its enhancement compensates for the decreased amplitude of wavenumber 2 (figures 1(i) and (j)), leading to the strengthening of the total stationary waves.The change in wavenumber 1 is consistent with the change due to sea ice loss: Zhang et al (2016) showed that the wavenumber 1 enhanced by sea ice loss led to the eastward shift of the polar vortex over North America.In addition, the wavenumber 2 anomaly will affect the East Asian winter monsoon (Chen et al 2005).On the other hand, wavenumber 3 is conducive to the eastward shift at 250 hPa.One change in the stratosphere that is different from the troposphere is that the increasing amplitude in central Siberia in P3 minus P2 offset its earlier decreases (figure S2).
The observed changes in stationary waves are further decomposed into the contributions from changes in the zonal mean basic state and zonally asymmetric diabatic heating (i.e., ∆ZM and ∆q * , as shown in figure 2  The stationary wave response to the changes of the zonal mean basic state and zonally asymmetric diabatic heating is diagnosed using the SWM, as shown in figure 4. Table 3 lists the pattern correlation and relative variance (RVar) of stationary waves segregated with different combinations of zonal mean basic state and zonally asymmetric diabatic heating.As shown by the high pattern correlation, using ∆ZM alone can reproduce the structural features of stationary wave changes.The contribution of ∆ZM to the amplitude, as indicated by the relative variance, decreases gradually as the height increases.In contrast, the contribution of ∆q * does not vary monotonically in height, corresponding to the vertical distribution of heating.In mid-high latitudes, the tropospheric heating source is primarily located near the surface, but the radiative heating extends vertically throughout the stratosphere (figure S3).Correspondingly, the contribution of ∆q * decreases in height in the troposphere but increases as entering the lower stratosphere.Moreover, the stationary waves are diagnosed in response to key regions of ∆q * at the North Pacific Ocean and the Greenland Sea to Western Europe (green enclosed area in figure 2(d)).The changes of stationary waves associated with ∆q * at these two regions alone capture most of the main features in the full response (figures 4(d) vs 4(a), 4(h) vs 4(e)).Based on the ERA5 reanalysis, we demonstrate that winter stationary waves have generally shifted eastward and strengthened in the NH over the past 60 years, with the wavenumber 1 component playing a major role.The changes of stationary waves are attributed to changes in the zonal mean basic state and zonally asymmetric diabatic heating.We found that ∆ZM dominates the structure of stationary wave changes, and the amplitude of the changes depends more on ∆ZM in the troposphere and more on ∆q * in the stratosphere, respectively.In particular, ∆q * in the North Pacific Ocean and Greenland to Western Europe exhibits significant effects on stationary waves.
In addition, we found the stationary wave changes in the 51 ensemble-averaged CMIP6 historical simulations do not reproduce the ERA5 results (figure S4).A similar failure of reproducing the stationary wave changes was also discovered in CMIP5 and ascribed to the deviations in the meridional wind (Woods et al 2017, Lee et al 2019) and latent heating (Park and Lee 2021).Besides model bias, failing to replicate can also result from internal variability.As the available 51 r1i1p1f1 coupled simulations from CMIP6 share the same external forcing with different initial conditions, it is reasonable to consider them as a sample set capable of assessing the internal variability.Internal variability is estimated by calculating the multimodal ensemble mean (X) and standard error (SE) of the stationary wave amplitude, which is the standard deviation of the mean fluctuations at 250 hPa (45 • -55 • N) and 70 hPa (60 • -70 • N).It shows that the changes in stationary wave amplitude at 250 hPa did not pass the 95% significance test (X = −0.47,SE = 1.47), while those at 70 hPa did pass the test (X = 6.44,SE = 2.76).Although the observed stationary wave changes over the past 60 years bear some similarity to those forced by longterm greenhouse gas projections (Wang and Kushner 2011), the tropospheric signal may have been masked by internal variability arisen from the climate system itself or deficits in CMIP6 models.In contrast, the observed stratospheric changes are more likely externally driven (e.g., by changes in greenhouse gases and/or ozone depleting substances).More work is needed in the future to better detangle the forced signal from the internal variability.

Figure 1 .
Figure 1.Climatology in P1 (1961-1980, contours) and the changes (P3 minus P1, shaded) of the (a) total and (b)-(d) zonally decomposed stationary waves at 250 hPa.The same as (a)-(d) but for (e) total and (f)-(h) decomposed waves at 70 hPa.The contour intervals are 6 × 10 6 m 2 s −1 , with solid lines representing positive stream function values and dashed lines representing negative values.The color shading intervals are 1 × 10 6 m 2 s −1 .The dotted area is significant at the 95% level according to the Student's t-test.(i), (j) the amplitude of the wavenumbers 1-3 components in P1 (dotted line) and P3 (solid line), with a unit of 10 6 m 2 s −1 .
).The zonal mean zonal wind ([u]) does not change significantly north of 30 • N, albeit showing large variations in the Southern Hemisphere (figure 2(a)).The zonal mean meridional wind ([v]) only shows noticeable changes in the boreal upper stratosphere (figure 2(b)), and zonal

Figure 2 .
Figure 2. Climatology (P1, contours) and changes (P3 minus P1, shaded) of zonal mean (a) zonal wind, (b) meridional wind, (c) temperature, and (d) zonally asymmetric diabatic heating rate (integrated from surface to 1 hPa) in ERA5.The dotted area is significant at the 95% level according to the Student's t-test.The area enclosed by green contours is the selected area for Experiment EL ′ in table 2.

Figure 3 .
Figure 3. Climatology (P1, contours) and changes (P3 minus P1, shaded) of stream function in ERA5 at (a) 250 hPa and (b) 70 hPa in January over 30 • N, and (c), (d) simulation results in SWM.The contour intervals are 6 × 10 6 m 2 s −1 , with solid lines representing positive values and dashed lines negative.The dotted area is significant at the 95% level according to the Student's t-test.

Figure 4 .
Figure 4. Simulated changes to imposed forcing: ((a), (e); experiment LL minus EE) changes of both zonal mean basic state and zonally asymmetric diabatic heating, ((b), (f); experiment LE minus EE) only zonal mean basic state changes, ((c), (g); experiment EL minus EE) only diabatic heating changes, ((d), (h); experiment EL ′ minus EE) only changes of diabatic heating in the specific area.Contour intervals are 6 × 10 6 m 2 s −1 .The dotted area is significant at the 95% level according to the Student's t-test.

Table 1 .
Settings of damping parameters.
* in the area specified in figure2(d)

Table 3 .
Pattern correlation (CorP) and relative variance (RVar) of stationary waves between imposed forcings and full forcings (Experiment LL minus EE).The debates on stationary wave changes in previous studies arise from different analysis methods and various climate assumptions based on (see Sellevold et al 2016 vs Wang et al 2020).Furthermore, diverse metrics of stationary waves (see Francis and Vavrus 2012 vs Barnes 2013) and distinct model parameters (Brandefelt and Körnich 2008) lead to different conclusions as well.Therefore, we show here how the stationary waves have changed over the past 60 years and what contributed to the changes, to provide a reference for further study on the future stationary wave projections under different climate scenarios.
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