We use two simulations from the “Evaluation” experiment in NA-CORDEX , one using the RegCM4 model and the other using the WRF model. Both models are configured on 25 km grids following the NA-CORDEX protocol (Fig. b) . We also analyze another downscaling simulation conducted with the CAM-MPAS model on a global VR grid with a 100 km coarse domain refined smoothly to 25 km grid spacing over North America (Fig. a).

The RegCM model is a widely used regional climate model with a long history . Downscaled data from the fourth-generation RegCM4 are available from NA-CORDEX on both the 50 and 25 km grids . This model version solves the primitive (hydrostatic) equations on a σ coordinate as described in and . Multiple options are available for the cumulus, boundary layer, and land-surface components . The physics parameterizations were selected based on the performance of test simulations over the CONUS region, particularly for warm-season precipitation (Raymond W. Arritt and Melissa S. Bukovsky, personal communication, 2018).

The WRF model is a regional model for weather and climate applications and has been extensively used to study the present-day and future state of North American climate with a wide range of model resolutions and configurations e.g.,. Version 3.5.1 was used for the NA-CORDEX experiment (50 and 25 km). The dynamical core solves the Euler equations without the hydrostatic assumption. The model physics largely follows that of , who focused on the warm-season climate of the western CONUS and the North American monsoon. Spectral nudging is applied to the temperature, winds, and geopotential height fields at the scales larger than approximately 1000 km to retain synoptic-scale variability in the driving GCM or reanalysis data .

CAM-MPAS is an experimental model in which the dynamical core is ported from the MPAS-Atmosphere version 4 to the CAM model within a beta version of the CESM2. The technical description of the model and downscaling experiments are provided in . Briefly, MPAS is a global dynamical core that solves the Euler equations on an unstructured grid . The unstructured grid can be configured as a global quasi-uniform resolution grid or a VR grid, in which one or more regions of interest have finer grid spacing than the rest of the globe. Advantages of the global VR model over LAMs include the absence of LBs and the consistent dynamical and physical schemes in both the high-resolution (downscaling) and coarse-resolution domains, which can avoid artificial shocks or gradients created by LBs in LAMs.

All models use the ERA-Interim reanalysis product for initial and boundary conditions, including six-hourly updates to the LBs and daily updates to SST and sea ice fraction (SIC) at the bottom (surface) boundary. RegCM4 does not have a specific sea ice scheme, so SIC from ERA-Interim is not used. CAM-MPAS uses SST and SIC only since there are no lateral boundaries; therefore, the large-scale circulations are not constrained. Table  lists the model characteristics and configurations. Table  compares the physics parameterizations used by the three models.

The reference data we use is ERA-Interim, which drives the NA-CORDEX simulations for the “Evaluation” experiment. As discussed by , we expect that dynamical downscaling adds value primarily in the small-scale processes while maintaining the large-scale flow provided from the driving data. If this tenet is true, incoming Rossby wave signals are not affected by LBs or other model details (see Appendix ); Rossby wave metrics calculated from the driving data (ERA-Interim) and from downscaling simulations within the LAM domain should be very close to each other. On the other hand, if numerical aspects of the downscaling model affect the circulations, such as artificial sources of divergence over the time scale of quasi-stationary Rossby waves, or the model exhibits mean biases in the general circulations (e.g., jet strength/width/positions), then we would see deviations in the Rossby waves between the driving data and the downscaling simulations. We are aware that this logic ignores a potential upscale effect from the downscaling simulation on the quasi-stationary Rossby waves. We will briefly discuss this assumption in Sect. ; however, such upscaling signals cannot be easily quantified without a priori designed experiments e.g.,, and this is left for future work.

The NH summer season (JJA) in the 30-year period from 1980 to 2010 is analyzed, except for the CAM-MPAS simulation, which starts at 1990. Most analyses are performed using daily-mean variables at 200 hPa (zonal and meridional winds, geopotential height). This particular pressure level is chosen primarily because it is a standard pressure level available in the CORDEX archives . According to the CORDEX protocol's model data requirements, daily mean quantities are calculated from three-hourly data . The CAM-MPAS data follow this requirement. The ERA-Interim data is available only every six hours, from which we calculated daily statistics. We compared the seasonal means and standard deviations of the monthly mean tas calculated from the six-hourly and three-hourly data, and found that the differences in these statistics are significantly smaller (less than 10 %) than the model biases against ERA-Interim (not shown).

Grid boxes adjacent to the LBs, or “sponge/buffer/relaxation zone”, where the external forcing and model-predicted variables are blended (Table , Appendix ), have already been removed in the NA-CORDEX data. This post-processing is designed for the common use case of regional climate assessment within the model domain; for this study, it poses a challenge. This is because we patch the outside of the LAM domain with ERA-Interim data to produce spatially continuous fields, on which Rossby wave propagations are diagnosed. Without the relaxation zone that blends LAMs predictions and ERA-Interim data, our patched diagnostic approach exhibits stronger gradients between the model and ERA-Interim data than with the relaxation zone. We used a brief WRF simulation to evaluate the impact of removing the buffer zone, which was found not to significantly alter the analysis results within the model domain (Fig. ). However, within the blending zone, the strength and spatial pattern of derived quantities (e.g., vorticity, divergence, and WA fluxes) change, and overall, they are notably noisier without the blending zone (Fig. a, b, d and e). The noise and spurious WA fluxes can be reduced to some extent by spatial smoothing applied over the relaxation zone (Fig. c and f). We tested several smoothing methods and present the figures that utilized a Gaussian filter within the buffer zone when the noise is significant. We do not attempt to evaluate Rossby wave sources/sinks along the LBs; those crucial aspects will be assessed in future work.

Prior to the patched analyses, all the data are regridded to a global 0.7° latitude–longitude grid using the patch method available from the Earth System Modeling Framework (ESMF) library . The 0.7° grid is nearly identical to the original ERA-Interim grid and is coarser than the downscaling datasets. We still remap the ERA-Interim data to this grid to more fairly compare variability and extremes with the models, since remapping can smooth the fields and affect those statistics . The patch method first estimates the grid corner values on the source grid using the second-order polynomials, then weight-averages the corner values to obtain the final estimate on a target point value on the destination grid; therefore, the computation is more expensive than the commonly used bilinear method . The patch method estimates the values and their derivatives more accurately than the bilinear method , which is desirable for calculating Rossby wave diagnostics that involve spatial derivatives.

It is critical to rotate the grid-relative u and v winds to the Earth-relative (eastward and northward) winds in the RegCM4 and WRF data before regridding. For the WRF model, the NCAR Command Language (NCL: ) provides a function for wind rotation (wrf_uvmet). For RegCM4, we wrote an NCL function to rotate winds onto the Rotated Mercator projection, which is available in . It is often necessary to spatially smooth the variables, especially for winds at relatively high resolution. In most cases, we used a suite of spherical harmonic functions available in NCL: vhaeC, tri_trunC, and shaeC.

We begin with the evaluation of tas in the dynamical downscaling simulations. Figure b–d shows the JJA-mean tas biases of the three models against ERA-Interim, showing rather distinct spatial patterns across the models. CAM-MPAS and WRF exhibit a warm bias over Canada, whereas RegCM4 tends to have a cold bias there. Over the western CONUS, CAM-MPAS tends to simulate higher tas while RegCM4 and WRF tend to simulate lower tas than ERA-Interim. An exception is central North America, where all three models exhibit warm biases to varying degrees, consistent with previous studies . CAM-MPAS has by far the worst bias centered around the US–Canada border. The notably higher bias of CAM-MPAS implies the importance of LB constraint for simulating tas, assuming that physics parameterizations in each model perform equally well. In the RegCM4 simulation, the largest bias over land occurs in the South Central region. The highest bias of WRF is over Canada and further south in SGP.

To assess the timing and magnitude of seasonal anomalies, we also plot the time series of JJA-mean tas anomalies relative to the all-year JJA climatology in each dataset (Fig. e), averaged over the central North America region (black box in Fig. b–d). The mean bias against ERA-Interim is added to the anomaly time series (also indicated by the horizontal dashed lines). Without the LB constraint, the time evolution of tas anomaly in CAM-MPAS is not expected to precisely follow that of ERA-Interim, except for the years with substantially strong external forcing such as the cold anomaly in 1992 after the Pinatubo eruption in the previous year ; the impact is felt by CAM-MPAS through the anomalously cold SST, but not through the aerosols since the CAM-MPAS model used prescribed aerosol forcing based on the year 2000 condition (the RegCM4 and WRF simulations do not consider aerosol effects either). RegCM4, with the LB constraint, produces a reasonable correlation with ERA-Interim (0.75). In some years, however, tas anomaly in RegCM4 deviates significantly from that in ERA-Interim (e.g., 1995–1999). WRF with spectral nudging achieves the highest correlation of 0.94 with ERA-Interim, and also with the smallest mean bias over the central North America region (2.7, 0.5, and 0.3 °C for CAM-MPAS, RegCM4, and WRF, respectively).

Figure  compares simulated standard deviations (σmodel) of monthly mean tas of each grid box to those in ERA-Interim (σERAI) as the ratio (Rσ=σmodel/σERAI). ERA-Interim shows the strongest variability in the PNW region in the United States (Fig. a). CAM-MPAS is able to capture this variability center as indicated by Rσ being close to one over the region (Fig. b). However, it overestimates the tas variability in western Canada and the central US. RegCM4 overestimates the variability over most of North America, particularly over western Canada, and northern and southern central US, and the east coast (Fig. c). The contrast in RegCM4 skills between the mean and variability indicates that LB forcing can constrain the time mean but not necessarily the temporal variability of tas. The WRF simulation again shows very good agreement with ERA-Interim (Fig. d).

Acknowledging that not only the upper-level dynamics but also the local land–atmosphere interactions and the upscale growth of convective systems play crucial roles in tas bias, we focus on the role of the subseasonal to seasonal scale upper-level circulations through the lens of Rossby wave dynamics. This section reviews some key aspects of the JJA climatology of the upper-level circulations relevant to quasi-stationary Rossby waves. The evaluation of the model-simulated upper-level circulations over North America follows it.

As in other seasons, the JJA-mean zonal winds are characterized by the extratropical and subtropical jets but with lower wind speeds and less zonally uniform structure (Fig. a). The extratropical jet is nearly circum-global except for the discontinuities over the eastern Pacific and Atlantic oceans, where the subtropical jet extends from ≈20° N latitude to merge with the mid-latitude jet. Since jet streams serve as wave guides , we expect that Rossby waves propagate from the Pacific Ocean to North America along the mid-latitude as well as the subtropical jets. When Rossby waves enter the East Pacific and the West Coast of North America, they encounter complex mean wind patterns, where the traditional assumptions for the base state in Rossby wave dynamics – namely, zonally uniform flow with zero meridional winds – are not valid. Indeed, over the Western US, the mean zonal and meridional wind speeds are comparable; the former range from 12 to 20 m s⁻¹, while the latter can be as high as 8 m s⁻¹ (Fig. b). Therefore, the role of the meridional wind in Rossby wave dynamics should not be ignored in this region.

Vorticity and divergence are essential for Rossby wave dynamics and are also shown in the figure (Fig. c and d). One aspect of the jet's waveguide nature stems from the strong horizontal shear at its edges, which enhances the vorticity gradient. Also, the interaction between vorticity and divergence alters the local vorticity balance (via vortex stretching, -η(∇⋅v)), acting as a source of relative vorticity anomalies, often called Rossby Wave Sources (RWS) . In JJA, local maxima and minima of the mean relative vorticity near the jet create the meridionally banded structure over the Pacific and Atlantic oceans. The relative vorticity maxima near the subtropical jets are also strong enough to produce zonal anomalies of the absolute vorticity (Fig. e). As a result, two sharp meridional gradients of absolute vorticity, or the regions of strong restoring force for Rossby waves, exist upstream of North America from the northern and tropical Pacific (Fig. f). In the tropics, strong 200 hPa divergence is co-located with regions of intense deep convective precipitation, most notably in the Asian Monsoon. This massive latent heating drives a downstream dynamical response, maintaining a region of pronounced upper-level convergence over the Mediterranean and the Middle East . Further north, a secondary divergence anomaly is observed along the southern flank of the North Pacific jet, associated with the midlatitude storm track. Those are potential source regions for Rossby waves propagating to North America.

For evaluating the upper-level circulations in the downscaling simulations, we focus on three variables: the mean 200 hPa zonal winds (ua200), meridional winds (va200), and zonal anomalies of geopotential heights (zg200) (Fig. ). As explained in Sect. , the modeled fields from the two LAMs are patched with the same fields of ERA-Interim outside the model domain, a visualization also used by . We use the full fields here instead of the differences between the simulations and ERA-Interim to emphasize the overall spatial patterns and unphysical discontinuities (difference plots are provided in Fig. ). Ideally, for LAMs, the mean circulation across the LBs appears seamless. This is the case with the WRF simulation (Fig. d, h and l), where its spatial patterns are identical to those from the ERA-Interim even inside the model domain; the contour plot for the difference from ERA-Interim confirms negligible bias (Fig. ). The geopotential height is slightly and uniformly higher within the WRF domain than ERA-Interim, but identifying the sources of zg200 bias in WRF is left for future work.

The overall patterns simulated by RegCM4 look reasonable, but discontinuities are apparent along the boundaries (Fig. c, g and k). Inside the model domain, the subtropical jet entering California is weaker than ERA-Interim, and the va200 and zg200 patterns are shifted to the west. These patterns are time-invariant stationary waves that exist in the mean circulation, which we distinguish from quasi-stationary waves defined as the perturbation on the mean. CAM-MPAS captures the general structure of the upper-level circulations without artificial boundary effects (Fig. b, f and j); however, the jet core is weaker and located more northwestward than ERA-Interim, while the meridional wind speeds are overestimated (also see Fig. a and d). Consistent with the overestimated va200 speeds, the zg200 zonal anomaly over North America is too high compared to ERA-Interim (Fig. i and j). In other words, the amplitude of the time-mean stationary waves is too strong. The position of the positive maxima of zg200 anomaly coincides with the spatial structure of the mean warm bias in tas (Fig. b), indicating the contribution of the upper-level mean wind bias to the tas mean bias. On the other hand, in the case of RegCM4, the mean bias in the upper-level winds and warm bias in tas do not spatially overlap.

The mean wind biases shown above imply that the waveguide structure for Rossby waves is also biased in the model simulations. Before evaluating the model-simulated Rossby wave propagations, we first diagnose major waveguides in the ERA-Interim data. A commonly used diagnostic is the stationary wavenumber, Ks , which is derived from the dispersion relationship of Rossby waves under a zonally uniform state with zero meridional winds. The stationary wavenumber is defined as 4Ks2=η‾yu‾M.

It indicates where Rossby waves can propagate and where they are likely to be trapped or reflected; regions with real-valued Ks are conducive to Rossby wave propagation, while those with imaginary Ks are not. Over the regions where Ks is real-valued, it acts as a cut-off filter for stationary waves. When Ks is small (e.g., in strong westerlies), the total wavenumber allowed is low, so only very long waves can exist as stationary waves. Stationary wavenumber also acts like the refractive index for the optical wave solution, such that Rossby wave rays (paths of group velocity vectors) bend toward regions of higher Ks (see , and Appendix ).

In Fig. a, we apply this metric to the JJA climatology of ERA-Interim at each grid point, assuming that the metric Ks is locally applicable e.g.,. The grid boxes with imaginary values are shown in white in the figure. It depicts the two waveguides along the mid-latitude and subtropical jets into North America, consistent with the mean wind patterns. Interpretation of Ks as the refractive index suggests that a Rossby wave excited within the local maximum of Ks, associated with the mid-latitude jet, is trapped within the jet and propagates zonally since the strong vorticity gradients to the north and south refract back the wave. On the other hand, a Rossby wave excited in the subtropical jet would be refracted southward toward the equator with higher Ks (the red arrow in the figure) toward the critical latitude where waves cannot propagate further, rather than traveling into North America. South of the mid-latitude jet over the central Pacific, there is another prohibited region where the zonal winds are near zero, and the meridional gradient of absolute vorticity is slightly negative (see Fig. ). Figure b shows the same diagnostic but calculated from the smoothed background state, which is more appropriate for the WKB approximation underlying the dispersion relationship (Appendix ). The overall waveguide structure is similar to that in Fig. a, but regional maxima (i.e., waveguides) are blurred, and the prohibited region over the central Pacific is replaced by small real values.

The assumptions of zonally uniform flow and zero meridional winds are not valid over the eastern Pacific and western North America, prompting us to perform ray tracing by LZ2015 to confirm the waveguide structure. To do this, we need to specify the locations of the wave sources. Previous studies suggest several remote sources of Rossby waves reaching North America during the summer, including the East Asian and Indian Monsoon regions, the western Pacific, the Tibetan Plateau, and the Mediterranean e.g.,. Most of them found Rossby waves propagating along the mid-latitude jet, but several studies suggested Rossby wave propagation along the subtropical jet from the central and eastern tropical/subtropical Pacific to North America . Those waves can be initiated during the Madden–Julian Oscillation phases 5 and 6, travel across North America, and break over the Atlantic Ocean . A closely related subseasonal variability, the boreal summer intraseasonal oscillation, is also found to enhance convective heating during particular phases, which triggers Rossby wave trains that tend to place a high-pressure ridge over the Pacific Northwest region .

Ensemble ray tracing is performed as described in Sect.  for the source locations suggested by previous studies and by our preparatory analyses (Appendix ). Results from four source locations are shown in Fig.  using the ERA-Interim climatological winds as the base state. Rossby waves excited in the northern North Pacific (NP) (Fig. c) and eastern tropical/subtropical Pacific (EP) (Fig. d) have significantly higher probabilities of propagating over North America than those originating from other areas. Waveguides extending from the eastern subtropical Pacific (20–30° N) to North America are evident for the waves originating from both the NP and EP regions.

Most waves excited in the West Pacific region travel southeast across the equator owing to the tropical easterly zonal wind and the monsoonal northerly meridional winds . The Tibetan Plateau generates Rossby waves that propagate westward; some of these waves arrive in North America from the east, while others turn eastward over North Africa and propagate along the jet stream. Some of those results may appear inconsistent with previous studies, and it is possible that wave activities originated from the other locations to reach North America, particularly in other seasons , through non-linear processes such as Rossby wave breaking and associated wave reflection , or by the interactions of propagating Rossby waves and the background divergent circulation , which are not included in the linear ray theory. Nonetheless, one-point correlation maps for meridional winds are consistent with the ray tracing result, such that statistically significant lead/lag correlations over North America are found only when the base points are specified in the NP and EP regions. Given those results, we consider it reasonable to focus on the upwind source regions of NP and EP to evaluate regional downscaling simulations.

Sub-samples of individual ray trajectories from these two regions are shown in Fig.  to illustrate actual wave rays and their relationship to the initial zonal wavenumber (k0) and background circulations. The figure uses climatological July winds as the base state, but the result is qualitatively similar to those obtained with June or August climatological flows (not shown). Rossby waves excited over the NP region with smaller k0 (i.e., longer wavelengths) tend to travel south/southeast toward the subtropical eastern Pacific, then turn east/northeast to reach North America. The climatological flow patterns immediately south of the NP source region have the northerly meridional winds of ≈-5 m s⁻¹ with comparable or even weaker zonal winds (Fig. a and b). Also, the meridional group velocity is inversely proportional to the second power of the wavenumber; thus, smaller wavenumbers favor larger group velocity (Eq. ). Those two aspects likely facilitate southward propagation from NP. On the other hand, those initiated with larger k0 tend to propagate along the mid-latitude jet and travel across North America near the US–Canada border (Fig. c).

Located more southeastward, the EP region is situated within the northward meridional background winds (Fig. b). Consistently, the waves excited here with smaller k0 first propagate north, and turn around at the northern edge of the mid-latitude jet (Fig. e and f). The initial northward propagation is not obvious in the Ks diagnostics. Similar to the waves from the NP region, waves initiated with larger k0 tend to be trapped within the mid-latitude jet and propagate more zonally. For both source regions, waves with even larger   k0 are not able to propagate across North America (Fig. c, d, g and h) . The result illustrates the sensitivity of Rossby ray propagation to the base state, highlighting the profound impact of mean-circulation bias on modeled Rossby wave propagation.

The stationary wavenumber Ks and LZ15 ray tracing agree on the waveguide formed by the mid-latitude jet, which is more effective for waves initiated with k0≈6 in the LZ15 framework. For waves with smaller k0, LZ15 results diverge from the waveguide depicted by Ks on: (1) southward propagations over the central Pacific, where Ks prohibits wave propagation, (2) northeastward waveguides by the subtropical jet, where Ks implies equatoward propagation toward the critical latitude south of the jet, and (3) northward propagations off the West Coast guided by southerly meridional winds. As shown below, wave activity flux patterns from TN01 are consistent with the LZ15 result, and meridional winds off the West Coast play an important role in understanding model biases in Rossby wave propagation and their downwind impact over North America. More sophisticated constructions of the background state for Ks have been suggested e.g.,, which may produce a waveguide structure that is more consistent with the LZ15 and TN01 diagnostics.

We evaluate the downscaling models by comparing the ray propagation probabilities obtained with the LZ15 framework, using the ratio of model to reanalysis probabilities, Rp=pmodel/pERAI. The ray probabilities in the WRF simulation are almost identical to those in the ERA-Interim (Rp≈1 in Fig. g and h), as expected from the small bias in the upper-level circulations. For the other two models, biases in jet and meridional wind speeds lead to significantly different wave-propagation patterns from those in ERA-Interim. For the waves initiated in the NP region, CAM-MPAS overestimates the probabilities over Canada and northern CONUS, and underestimates them over the southern part of CONUS (Fig. c). This is likely the result of the northward-shifted mid-latitude jet and overestimated southerly winds over the West Coast (Fig. a and d), which promote more zonal propagations at higher latitudes instead of traveling to the south. Such propagations are seen for the waves initiated with relatively small zonal wavenumbers (Fig. a for k0=4). For those waves, the mean circulation patterns in CAM-MPAS support longer-lived, circumglobal propagation that passes over North America twice, thereby increasing propagation probabilities. Such long-lived waves are rare for the same initial zonal wavenumbers with the ERA-Interim base state. For the waves from the EP region, stronger southerly winds over the West Coast region likely allow more waves to travel north, but the slightly weaker and wider jet in CAM-MPAS appears to be a less effective waveguide, spreading the rays more widely over North America, particularly to the south of the jet where CAM-MPAS simulates higher propagation probabilities (Fig. d).

The Rossby wave probabilities in RegCM4 (Fig. e and f) show some similarity with those in CAM-MPAS, likely due to the two models sharing the mean circulation biases over the western part of the NA-CORDEX domain (Fig. d and e). We can see more dense lines of wave rays emanating north from the EP source region (west of 120° W) in the RegCM4 ray tracing than in ERA-Interim (Fig. b vs. Fig. e), where stronger southerly winds are noted in the RegCM4 simulation (Fig. e). At the same time, the overestimated southerly winds appear to limit the southward wave propagation from the NP region, thus shifting the probabilities northward over North America (Fig. e). The mean wind patterns over North America in RegCM4 allow waves from the EP region with larger k0 to travel farther than in the ERA-Interim base state, for example, for the initial zonal wavenumber of eight (Fig. c vs. Fig. h). Those waves also contribute to the higher probabilities from the EP region. LB effects are not apparent in the RegCM4 ray-tracing results. This is due to the smoothing of the base state after the RegCM4 and ERA-Interim data are patched onto the global grid, thereby effectively weakening discontinuities at the lateral boundaries.

Overall, biases in the large-scale circulations in CAM-MPAS and RegCM4 tend to increase wave-propagation probabilities in the northern part of North America, particularly in the RegCM4 base state. Additionally, the probabilities for Rossby waves around 40° N over CONUS from the NP region are underestimated, whereas the waves from the EP region are overestimated by both models. Those two biases would not simply cancel each other out, because Rossby waves propagating from the NP region tend to have higher wavenumbers (shorter wavelengths) over North America than those from the EP region (Fig. ). The smaller waves from the NP region may be more susceptible to breaking. In contrast, those from the EP region in wavenumbers four to six may have higher probabilities of resonating with Rossby waves of similar wavelengths but different frequencies . Those non-linear processes are not part of our diagnostics, though.

The shifted Rossby wave propagations in CAM-MPAS and RegCM4 may disrupt the spatiotemporal correlations between Rossby waves and surface climate, as seen in the 2009 heatwave example in the Introduction. In the two model simulations, the biases in Rossby wave probabilities and tas variability are both large over the Pacific Northwest, suggesting a connection between them. We explore the connection in the following sections.

We begin with a global view of wave activity in ERA-Interim. The area with the most vigorous quasi-stationary Rossby WA in the JJA season is the Pacific Ocean, followed by the Atlantic Ocean, both characterized by large flux and strong divergence of the W-vector (Fig. a). We interpret the areas of divergence as indicating WA sources. Vigorous WA fluxes from the NP and EP wave sources converge on the West Coast of North America, then propagate across the continent to diverge out from the East Coast, in general agreement with the 2009 heatwave case (Introduction). The pathways from the two source regions agree with the ray-theory result, including the initial southward propagation from the NP region and waveguiding by the subtropical jet.

The bottom four panels in Fig.  compare the W-vector patterns over North America in the downscaling models and ERA-Interim. The most notable feature is the bands of strong divergence/convergence pairs along the LBs in the RegCM4 simulation (Fig. d), which strongly suggests inconsistency between ERA-Interim and RegCM4 circulations, even considering the removed relaxation zone (Sect. ). Although some spuriously large W-vectors emanating from LBs should be ignored, those downwind over the Pacific coastal area and the Pacific Northwest region are calculated fully from the model data, thus reliable. There, the W-vector in RegCM4 is oriented more zonally than in ERA-Interim, and some of the WA flux appear to originate at the LB rather than from the NP and EP regions. Not only the coastal region, but also the WA fluxes over the central US differ between RegCM4 and ERAI; RegCM4 simulates a more northerly W-vector, while ERA-Interim suggests a more zonally propagating flux. The W-vectors in the WRF simulation are almost identical to those from ERA-Interim (Fig. e); subtle linear structures in the divergence pattern parallel to the lateral boundaries may be due to the removal of the relaxation zone in the W-vector calculation. The global VR simulation of the CAM-MPAS model does not suffer from such artifacts (Fig. c). However, the zonal propagation of the W-vector is shifted northward from the US to Canada, creating an anticyclonic rotation over the central US, possibly due to the overly strong positive geopotential anomaly (Fig. j). With this northward shift, the W-vector divergence over the East Coast of the US, as seen in ERA-Interim, is replaced with weak convergence in CAM-MPAS. In addition, the W-vector divergence in CAM-MPAS is overly strong near the coastlines and mountain ranges on the West Coast compared to other models and ERA-Interim. We have examined the variance spectra of surface topography, vertical velocity, and horizontal winds, but there is no indication that the topography and wind kinetic energy in CAM-MPAS differ significantly from those in other models (Fig. ). Topography-related processes in the CAM-MPAS downscaling simulations will be investigated in future work, potentially helping to explain the strong WA flux divergence over the mountainous region.

How are the differences in the WA flux patterns reflected in the regional climate? We first look at the lead/lag correlations between the 5 d running mean zg200 anomalies across North America and the divergence of the W-vector averaged over the West Coast region, where we see strong convergence in the climatology of ERA-Interim (Fig. a). Each time series consists of daily data spanning 30 years (20 years for CAM-MPAS) of JJA seasons. The statistical significance is determined following using the two-tailed Student's t-test against the null hypothesis of zero correlation, taking into consideration the autocorrelation of each time series in determining the degrees of freedom (Eq. 1 in ).

In ERA-Interim, statistically significant negative correlations appear upstream over the Gulf of Alaska and positive correlation just off the US West Coast around lag -2 (Fig. a), that is, zg200 anomalies off the US West Coast for a given day is positively correlated with W-vector divergence over the West Coast happening 2 d later (or the higher zg200 anomalies off the US West Coast are, the stronger WA flux divergence will be in 2 d later over the West Coast). CAM-MPAS can reproduce the correlation pattern, albeit weaker than ERA-Interim (Fig. b). RegCM4 misses the negative correlation over the Gulf of Alaska, extending the area with a positive correlation northwest toward the Gulf of Alaska (Fig. c). WRF with the spectral nudging can capture this lag-2 correlation pattern (Fig. d). At lag+4, statistically significant negative correlation appears over the SGP in ERA-Interim, creating a clear wave pattern (Fig. e). This negative correlation means that W-vector convergence (negative values) over the West Coast leads to a positive zg200 anomaly over SGP 4 d later. WRF generally captures this lag +4 correlation pattern, but without statistical significance over SGP (Fig. f). The other two models struggle to reproduce the lag +4 correlation (Fig. f and g).

The tas response to the W-vector divergence closely follows the zg200 response. Again, we calculate lead/lag correlations between 5 d running mean tas anomalies at each grid point and the daily W-vector divergence averaged over the West Coast region (Fig. ). In ERA-Interim, areas of statistically significant negative correlations appear over SGP around a lag of -2, with the maximum extent occurring when the tas anomaly is lagged by 4 d (lag +4). It suggests that the tas over the SGP tends to be higher than normal when the WA flux converges over the West Coast, particularly 4–8 d earlier. We noted that the significant lagged correlation remains when the tas anomaly is band-pass filtered for the periods between 70 and 90 d (not shown). The long timescale may indicate a role for the land surface, particularly soil moisture e.g.,. The actual physical processes underlying the correlation are left for future work.

Focusing on the lag+4 result, WRF is the only model to simulate the significant negative correlation over SGP and the overall structure of the lead-lag correlation (Fig. b and f). CAM-MPAS simulates a weak negative correlation over SGP but misses the statistical significance (Fig. d). Also, the correlation patterns over the Pacific Northwest, western Canada, and Alaska do not agree with those in ERA-Interim. Similarly, RegCM4 misses the negative correlation center over SGP, and also simulates unrealistic negative correlation over the eastern Pacific (Fig. e). The inability to reproduce the W-vector–tas correlation in these two models is likely one reason for the biases of the mean and/or variability of tas over SGP (Figs.  and ).

It is also possible to correlate the W-vector divergence and the errors in the simulated tas. Figure a presents the lead/lag correlations between the W-vector divergence averaged over the West Coast region in the RegCM4 simulation and the difference in the daily mean tas between RegCM4 and ERA-Interim at each grid point. In this case, we observe statistically significant correlations over SGP at lag +4. Those RegCM4 results suggest that accurately receiving Rossby wave signals through LBs and maintaining the large-scale circulation patterns are essential for LAMs to reproduce the cross-scale connections from Rossby waves to tas. Another example is the observation that the W-vector convergence is significantly overestimated by CAM-MPAS over British Columbia, Canada (Fig. ). The lead/lag correlations between the W-vector divergence averaged over British Columbia, and tas errors in CAM-MPAS exhibit significant positive correlation in the same region, which also overlaps the overestimated tas variability by the same model (Fig. b). Part of these tas errors is attributable to out-of-sync temporal evolutions between ERA-Interim and global, free-running CAM-MPAS, which has its own internal variabilities. Nonetheless, this result illustrates another example of how model error can propagate across scales, from the biased mean wind patterns through Rossby wave forcing to tas variability.

In this final subsection, we demonstrate a connection between quasi-stationary Rossby wave forcing and heatwave (HW) events, and assess how the downscaling simulations replicate the connection identified in ERA-Interim. We diagnose HW events using the criteria outlined in (their Appendix A2). Specifically, an HW event is a period of three or more consecutive days with the daily maximum tas exceeding the 95th percentile of the reference period (1981–2010, except for CAM-MPAS, for which we use 1990–2010). All seasons are included in HW identification, and the seasonal cycle is not removed; thus, this criterion favors warm-season occurrences . Figure a shows the spatial distributions of the average fraction of HW days per JJA season (i.e., the number of HW days during one JJA season = 92 d). ERA-Interim indicates two local maxima, one over the southwestern US and the other over the SGP, where ≈15 % of JJA days, or about 14 HW days, are expected each summer.

The same HW definition is applied to the downscaling simulations, also shown in Fig. . CAM-MPAS can simulate the overall spatial patterns with two local maxima over the southwestern and south-central US, but overestimates the number of HW days during the summer across most of North America, except in the eastern part, where it simulates fewer HW days. RegCM4 also simulates too many HWs in the JJA season across North America, except for SGP, where it underestimates the number. The HW distributions simulated by WRF agree best with those in ERA-Interim.

How does the HW distribution change during the days with strong Rossby wave forcing? We examine the days when the WA flux convergence (-∇⋅W) over the West Coast region (the same region used for the lead/lag correlations) exceeds the top 90th percentile of all years. The bar graphs in Fig.  show how those extreme days are distributed across years. Those top 10 percentiles are not uniformly distributed but instead exhibit variability on a 3–4 year timescale, according to the ERA-Interim data. WRF reproduces this temporal distribution reasonably well, whereas RegCM4 does not. Free-running CAM-MPAS does not simulate the 2009 peak or other clusters in sync with ERA-Interim (Fig. b), indicating the significant roles of biased waveguide locations and/or the atmosphere's internal variability. All models agree well with ERA-Interim on the magnitude of the top 10th percentile: the average magnitude of the extreme convergence is -27, -26, -27, and -25 (10⁻⁸ m² s⁻¹) in ERA-Interim, CAM-MPAS, RegCM4, and WRF, respectively.

Calculating the fraction of HW days only on the days with extreme W-Vector convergence over the West Coast area in the ERA-Interim data, we see significantly higher HW fractions over the Midwest and the South Central US and southern Canadian Prairies, and lower fractions in the northern Canadian Prairies, Quebec, and the Southwestern US (Fig. a). That is, extremely rapid accumulations of WA over the West Coast region have a statistically significant impact on HW occurrences across broad regions of North America. Note that the HW fraction is roughly doubled over the Central Plains from ≈0.12–0.15 with all the samples to ≈0.2–0.3 during the extreme WA flux convergence. Examining the composite means of the HW day fractions in the downscaling simulations, the WRF simulation yields the best agreement with ERA-Interim, although it does not accurately capture the reduced HW occurrences over the southwestern US. In CAM-MPAS, the higher HW fractions are seen over British Columbia and Quebec, the Southwest, and some parts of the Great Plains. Those responses differ from what ERA-Interim describes, and are somewhat similar to the areas with overestimated variability of tas by this model (Fig. b). RegCM4 simulates HW surge with the extreme WA flux over the southern part of CONUS, possibly related to the more northerly WA flux over the West Coast and Great Plains in this simulation compared to the westerly WA flux in ERA-Interim (those during the extreme convergence not shown, but similar to Fig. ). More in-depth analysis is required to conclude how the modeled HW response to WA flux is linked to the overall mean and/or variability biases of tas. Nonetheless, this diagnosis reveals a connection between extreme Rossby wave forcing and the occurrence of HWs over North America, which is accurately reproduced only by the WRF model with spectral nudging.