1. Introduction
Large-scale orography such as the Tibetan Plateau, Rocky Mountains, and Andes influence the general circulation of the atmosphere by exciting planetary-scale stationary Rossby waves (Hoskins and Karoly 1981; Held et al. 2002) with consequences for the zonal variation of temperature (Lau 1979; Kaspi and Schneider 2011; Deser et al. 2014; Hoskins and Woollings 2015) and precipitation (Peixóto and Oort 1983, 1992; Broccoli and Manabe 1992; Wills and Schneider 2015, 2016). Orographic Rossby waves are particularly important for the dryness of central North America and Asia in winter (Broccoli and Manabe 1992), the maintenance of summertime subtropical circulation patterns (Rodwell and Hoskins 2001), and the seasonality and strength of precipitation in the East Asian monsoon region (Molnar et al. 2010; Chen and Bordoni 2014). The response of these regional climate features to global warming will thus depend on the response of orographic Rossby waves. Some studies have suggested that the strength of orographic stationary waves should increase with global warming as a result of the decreased lower-tropospheric meridional temperature gradient (Cook and Held 1988; Held 1993), but this is only one influence out of many on the amplitude of stationary waves.
Coupled climate models can give predictions of the stationary wave response to climate change over the next century (Stephenson and Held 1993; Joseph et al. 2004; Vecchi and Soden 2007; Brandefelt and Körnich 2008; Wang and Kushner 2011; Simpson et al. 2014, 2016). However, the stationary wave response in these models is a superposition of the response to several different stationary wave sources, complicating an assessment of the dominant physical mechanisms for these changes. Traditionally, stationary waves have been split into components attributed to different forcings using stationary wave models: general circulation models in which transient eddies are strongly damped (Ting and Yu 1998; Held et al. 2002) or that explicitly solve the linearized primitive equations (Egger 1976; Hoskins and Karoly 1981; Nigam et al. 1986, 1988; Valdes and Hoskins 1989; Ting 1994). The climatology and climate change response of stationary waves are well simulated by such models (Nigam et al. 1988; Stephenson and Held 1993), but they rely on specifying diagnosed diabatic tendencies, which themselves depend on the stationary wave solution. Stephenson and Held (1993) find that changes in latent heating and transient-eddy heat and momentum fluxes, rather than changes in the zonal-mean basic state, dominate the stationary wave response to global warming. This highlights the importance of understanding moist processes and transient eddies as they change with global warming. In the modern climate, much of the zonal asymmetry in latent heating and transient-eddy heating is either directly or indirectly a response to orography (Nigam et al. 1988). Thus, in order to study changes in diabatic tendencies associated with orography and their role in modifying orographic stationary Rossby waves under climate change, we use a moist general circulation model (GCM), where transient eddies are simulated explicitly and latent heating is allowed to feed back on the dynamics.
In Wills and Schneider (2016, hereinafter WS16), we present idealized GCM experiments in which individual topographic and ocean-heating zonal asymmetries are added to an aquaplanet. This allows an analysis of the differing responses to climate change of stationary waves forced by these two types of zonally asymmetric forcings. We simulate a wide range of climates from cold, dry climates (280-K global-mean surface temperature), where the influence of latent heating is negligible, to warm, moist climates (nearly 320-K global-mean surface temperature), where latent heating plays a leading-order role in modifying stationary-eddy circulations. The response of stationary-eddy vertical velocity to warming is characterized by a strong decrease with warming when stationary eddies are forced by equatorial heating, and by a nonmonotonic response to warming when stationary eddies are forced by midlatitude orography (WS16). The vertical velocity response, which is important for changes in precipitation minus evaporation, mirrors changes in the horizontal flow as a result of the linear vorticity balance in the lower troposphere (WS16). In Wills et al. (2017), we study the mechanisms responsible for the zonally asymmetric circulation changes in the equatorial heating experiment. Here, we focus on the mechanisms responsible for changes in the strength of stationary waves in the midlatitude orographic forcing experiment.
There is an expectation from quasi-geostrophic theory that the strength of orographic stationary waves should depend on the strength of the mean low-level winds at the latitude of the mountain (Hoskins and Karoly 1981; Held 1983; Held and Ting 1990). The role of the low-level winds arises from their role in setting the strength of topographic vertical motions, which influence the atmosphere through adiabatic cooling or heating. Nonlinear modification of the topographic vertical motions by the stationary wave response reduces the strength of the adiabatic cooling and heating tendencies (Chen and Trenberth 1988; Valdes and Hoskins 1991; Ringler and Cook 1997), but it does not change the conclusion that these adiabatic tendencies are the primary way in which orography influences the free troposphere. In the midlatitudes, this adiabatic cooling/heating can be balanced by meridional advection along the large meridional temperature gradient. This meridional flow perturbation sets up a stationary Rossby wave with eastward group velocity (Hoskins and Karoly 1981). The meridional wind response can be modified by additional heating tendencies, such as the latent heating associated with orographic vertical motion or the transient-eddy heat fluxes associated with initiation of a storm track downstream. We can analyze the mechanisms governing changes in the strength of the resulting stationary Rossby wave using an analysis of the thermodynamic equation over the orography (cf. Hoskins and Karoly 1981). This is effectively an analysis of the boundary condition for quasi-geostrophic theory, which is the only way in which topography explicitly enters the quasi-geostrophic equations (Hoskins and Karoly 1981; Vallis 2006).
Considering the importance of the low-level wind and the meridional temperature gradient (both of which vary strongly with latitude) for the orographic forcing of stationary waves, we expect the stationary wave response to climate change will depend on the latitude of orographic forcing. We therefore run simulations with topography at different latitudes, described in section 2. The stationary wave response to climate change forced by increased greenhouse gas absorption is described in section 3. In section 4, we analyze the forcing of upslope and downslope orographic winds by surface flow over and around topography. In section 5, we analyze the thermodynamic equation over the orography to understand the mechanisms through which the atmosphere responds to the orographic vertical winds and determine the factors important in the climate change response. In section 6, we discuss how the stationary wave changes lead to a reduction in the midlatitude zonal variance of temperature and precipitation in the warmest climates studied. Section 7 summarizes the paper and presents our conclusions, and section 8 discusses the implications of our results for the real world.
2. Idealized GCM experiments
To isolate the response of orographically forced stationary Rossby waves to global warming, we add a single large-scale mountain ridge to an idealized GCM simulation with otherwise zonally symmetric aquaplanet boundary conditions. These simulations are described in WS16, which focuses on the response of the zonally asymmetric hydrological cycle to global warming. The model used is that of Frierson et al. (2006) and O’Gorman and Schneider (2008b), based on the GFDL Flexible Modeling System and the convection scheme of Frierson (2007). The model is run at T85 spectral resolution, with 38 levels in the vertical. There is a sponge layer in the top eight model levels in order to avoid reflections of planetary waves from the top of the domain, as described in WS16. Our simulations use a 1-m slab ocean mixed layer depth and a specified zonally symmetric ocean heat flux convergence that does not extend into latitudes where topography is present (see WS16).
(a) Barotropic streamfunction and (b) zonally anomalous lower-tropospheric (550–750 hPa) meridional wind forced by orography in the reference simulation (α = 1, global-mean surface temperature Tg = 289 K) and in the 3 times optical depth simulation (α = 3, Tg = 307 K) for both mountain configurations (R45 and R54). The black contours show the 800- and 900-hPa contours of surface pressure.
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
To focus exclusively on the atmospheric response to orography, we keep the surface properties over topography the same as in the surrounding aquaplanet (i.e., the mountains are aquamountains). Note also that the idealized GCM does not consider the liquid–solid phase change, so there is no atmospheric ice condensate or sea ice over the elevated topography. The surface of the mountain differs from a land surface by its higher heat capacity, lower albedo, and lack of water limitation. The philosophy of this decision is that any modification of the surface boundary condition (e.g., albedo) would provide an additional source of stationary waves, with different physics, that should be studied separately.
We have investigated zonal mountain scales σλ ranging from 2° to 50° longitude (Wills 2016). The stationary wave amplitude is maximum at σλ ≈ 15°. We chose σλ = 12.5° to force a large-amplitude Rossby wave while keeping the mountain size realistic. For narrower mountains, the response is confined to the local orographic precipitation influence (cf. Shi and Durran 2014). At T85 resolution, there starts to be substantial grid-scale noise for σλ ≤ 5°. Such resolution issues are likely the cause of grid-scale noise in IPCC models (cf. Wills et al. 2016).
3. Stationary wave response
In each simulation, the time-mean barotropic streamfunction (Fig. 1a) or zonally anomalous meridional winds (Fig. 1b) shows a stationary Rossby wave emanating from the topography, with a wave train extending southeast into the tropics. The zonal wavenumber of the stationary wave is predominantly a mix of wavenumbers 1–4 and does not show a large change with warming. While length scale changes resulting from strengthening of the upper-tropospheric jet stream have been implicated in the stationary wave response to climate change over the next century (Simpson et al. 2016), the length scale of the stationary waves is too large for this to be an important mechanism in our simulations, since it is important primarily for zonal wavenumbers k > 4. Length scale changes may be an important factor in the climate response of stationary waves forced by orography with a smaller zonal scale.
The stationary wave ray paths (i.e., the southeastward trajectory of the wave train into the tropics) can be understood in terms of ray-tracing theory (Hoskins and Karoly 1981) and depend on the structure of the zonal-mean winds. The zonal-mean winds in the idealized GCM (contours in Fig. 2) are not particularly realistic; they lack a distinct upper-tropospheric maximum and show an equatorward shift of the surface westerlies with climate change. Therefore, investigation of the precise ray paths and spatial structure of the stationary Rossby waves in this model would not necessarily lead to insights about stationary wave changes in the real world. Instead, we focus on the mechanisms controlling the strength of stationary waves, which should translate to more realistic settings.
Vertical profile of sEKE [Eq. (3)] superimposed on contours of the zonal-mean zonal wind (contour interval 5 m s−1, thick contour is the 
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
Stationary-eddy kinetic energy is distributed throughout the troposphere and lower stratosphere in the hemisphere of forcing (Fig. 2). The sEKE is high in the lower troposphere at the latitude of forcing, then grows with height following the direction of vertical and meridional propagation to a local maximum in the upper troposphere near the latitude of maximum zonal wind. The upper-tropospheric maximum reflects the refraction of Rossby waves into the jet stream (Hoskins and Karoly 1981). Note that the latitude–pressure profile of zonal-mean zonal wind does not differ significantly between R45 and R54 and can thus be thought of as independent of the latitude of topographic forcing. In most of the simulations shown here, a weaker secondary wave train can be seen propagating poleward from the mountain, consistent with ray-tracing theory (Hoskins and Karoly 1981).
The planetary waves studied here propagate vertically within the troposphere; all but the longest waves are evanescent beyond the tropopause. The vertical structure of sEKE within the troposphere is therefore determined largely by the stationary external mode (Held et al. 1985). The vertical dispersion of stationary Rossby waves is inhibited in warm climates because of the increased upper-tropospheric vertical winds (cf. Charney and Drazin 1961), which results from the stronger warming of the tropical upper troposphere relative to the extratropical upper troposphere–lower stratosphere. This primarily affects the distribution of sEKE in the stratosphere and upper troposphere (compare in particular the decay of sEKE above 500 hPa for α = 1.6 and 4.0 in Fig. 2). This trapping of stationary waves in the lower troposphere may contribute to the reduction of sEKE in the warmest climates. It can equivalently be thought of as a consequence of the contrasting responses of the upper- and lower-tropospheric meridional temperature gradients, since the meridional temperature gradient sets the strength of stationary-eddy meridional wind required to balance a given heat source, as will be discussed in section 5.
To average over any particular differences in stationary wave ray paths between simulations and focus on the global changes in the strength of stationary waves, we compute the global-mean vertically integrated sEKE. This measure still incorporates information about changes in the zonal, meridional, and height extent of stationary wave activity, which can arise because of changes in meridional and vertical dispersion, but it allows easy comparison between simulations with different stationary wave ray paths. There is an increase in global sEKE with warming in the R45 experiment until a global-mean surface temperature of 300 K (α = 1.8) is reached, at which point there is a pronounced reduction in sEKE with further warming (Fig. 3a). In contrast, the R54 experiment shows a monotonic reduction of global sEKE throughout the range of climates (Fig. 3b).
Variation of global-mean vertically integrated sEKE with global warming in the (a) R45 and (b) R54 experiments. The sEKE is vertically integrated over the full column. Filled symbols indicate the reference climate (α = 1).
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
The main goal of this paper is to provide physical mechanisms for the response of the stationary wave amplitude to warming in these simulations. In particular, we hope to explain the reasons for the reduction of sEKE in the warmest climates and the differences in sEKE for mountains at different latitudes. While sEKE is just one possible measure of stationary wave amplitude, we have found that other measures (e.g., variance of barotropic streamfunction and wave activity) agree with the sense of change in stationary wave amplitude as diagnosed from sEKE. In most of what follows, we focus on the sEKE contribution from the meridional wind variance 
4. Mechanical forcing by orography
(a) Map of 
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
Influences on orographic forcing by 
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
Based on this scaling, the topographic vertical velocity reflects the zonal-mean zonal surface wind at the latitude of the mountain (Fig. 5b), which has a different warming response at different latitudes because of the equatorward shift of the surface westerlies with warming in the idealized GCM (Fig. 6). In these simulations, the 54°N mountain sees a monotonic decrease in the zonal-mean westerlies and the 45°N mountain sees a nonmonotonic change in the zonal-mean westerlies, responses that are reflected in the orographic vertical velocity changes. The equatorward shift of the zonal surface winds with warming mirrors the equatorward shift of upper-tropospheric transient-eddy kinetic energy (tEKE) in this idealized GCM (Wills 2016). This is in contrast to lower-tropospheric tEKE, which shifts poleward with warming, as seen also in observations and comprehensive climate models (Fyfe 2003; Yin 2005; Bender et al. 2012; Chang et al. 2012). The opposite direction of response in surface winds compared to observations and comprehensive models stems from the unrealistic structure of the upper-level winds in this gray-radiation idealized GCM and can be corrected by improving the representation of radiation (Z. Tan, 2017 personal communication). This paper aims to understand the mechanisms controlling the amplitude of orographic stationary waves as a function of mean climate state, rather than the climate response for a particular mountain range and mean climate state, so the equatorward shift of the zonal westerlies does not pose a serious problem here.
Variation of zonal-mean zonal surface wind 
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
The orographic vertical winds, which set the initial orographic perturbation to the atmosphere, are sensitive to changes in the zonal-mean zonal surface winds, in the top-of-mountain surface pressure, and in the stationary-eddy modification of surface winds. When the orography lies fully within the surface westerlies, the changes are determined largely by changes in the strength of the zonal-mean zonal surface winds. There is an additional tendency toward weaker forcing in warmer climates as the top-of-mountain surface pressure increases with warming. The orographic vertical velocity variance 
5. Thermodynamic response to orographic forcing
Vertical profile of terms in the stationary-eddy thermodynamic equation [Eq. (11); units of K s−1] averaged over ϕ ∈ [40°, 50°] in the reference climate (α = 1.0, Tg = 289 K) of R45. (a) Meridional, (b) vertical, and (c) zonal advection of potential temperature by stationary eddies. (d) All diabatic tendencies (latent heating, radiation, and parameterized boundary layer mixing). (e) Transient-eddy potential temperature flux convergence. Signs are chosen such that heating is red and cooling is blue. The residual is distributed among the zonal advection and transient-eddy terms as described in appendix B. A black line shows the time-mean surface pressure. (The black box over the mountain shows the averaging region used in Figs. 9 and 10.)
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
As in Fig. 7, but for α = 3.0 (Tg = 307 K) in the R45 experiment.
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
a. Meridional wind response to vertical motions
Particularly in the warmer climate (α = 3.0, Fig. 8), latent heating (which is the dominant contributor to 
We apply the meridional wind variance budget to averages between 550 and 750 hPa, within 5° latitude and 25° longitude of the mountain center (denoted by 
Factors controlling the meridional wind response to vertical motions and zonally anomalous heating. (a) Meridional wind variance 
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
The temperature tendency variance terms on the right-hand side of Eq. (18) make up a diagnostic for 
b. Latent heating
The thermodynamic forcing of orographic stationary waves by zonal anomalies in adiabatic cooling/heating W2 is reduced by zonal anomalies in latent heating. The net result, 
c. Transient-eddy heat fluxes
The transient-eddy heating term is also important, accounting for about half of the stationary-eddy thermodynamic forcing throughout the range of climates (Fig. 9b). This amplification of stationary waves by transient-eddy heat fluxes results from downgradient heat fluxes into the cold region downstream of the mountain (Figs. 7 and 8). Because this heat flux convergence coincides with heating by adiabatic descent, it is in phase with the initial forcing and acts to strengthen the orographic stationary wave. The magnitude of the temperature anomaly and the resulting transient-eddy heat fluxes are proportional to the strength of the stationary-eddy winds, so these transient-eddy heat fluxes can be thought of as a positive feedback on the strength of stationary eddies. Note that we have not considered transient-eddy momentum fluxes, which can modify the climatological winds and overturning and thus influence other thermodynamic terms. For example, transient-eddy momentum fluxes likely play a role in determining 
Zonal variance of transient-eddy heat fluxes (solid line) and scaling based on Eq. (24) (dashed line), where we fix the scaling to be equal to the transient-eddy heat flux variance in the reference climate (filled circle). Averaging is done over the layer immediately above the mountain (550–750 hPa) within 5° latitude and 25° longitude of the mountain.
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
The coupling between stationary and transient eddies also plays an important role in the localization of storm tracks in the Northern Hemisphere (Son et al. 2009; Kaspi and Schneider 2013). The strong heat flux convergence downstream of the mountain results in part from the generation of eddies by baroclinic instability in this region of enhanced meridional temperature gradient. The enhanced baroclinicity helps to generate a localized storm track downstream, which terminates in the region of reduced baroclinicity west of the mountain (cf. Son et al. 2009; Kaspi and Schneider 2013). This interaction between transient eddies and stationary eddies was discussed extensively by Kaspi and Schneider (2013), who demonstrated the role of stationary waves in terminating a storm track forced by a warm ocean region in the midlatitudes. There is currently no simple theory for the coupling between stationary waves and midlatitude transient eddies. Such a theory would be required to make a closure for the influence of transient eddies on the strength of orographic stationary waves or to make a closed theory for the zonal variation of transient-eddy variance, as exists in the Northern Hemisphere storm tracks.
d. Summary of mechanisms
The meridional wind variance budget leaves us with a mechanistic picture of the tropospheric response to orographic forcing, illustrated schematically in Fig. 11. As surface winds impinge upon topography, they force adiabatic cooling (heating) as they ascend (descend) on topographic slopes. Stationary-eddy meridional velocities arise to balance this cooling/heating. Differences in the strength of surface winds and in the isentropic slope account for much of the difference in stationary wave strength with different forcing latitudes and different mean climate states. For the smaller isentropic slopes in warmer climates, larger meridional velocities are required to balance orographically forced ascent or descent, implying a strengthening of stationary waves with warming.
Schematic of the effect of orography on stationary-eddy thermodynamics in the presence of westerly zonal winds (black arrows), overlain on a depiction of a single isotherm over the mountain (black line). Here, a circle with a cross (dot) denotes poleward (equatorward) advection and upward and downward arrows denote vertical motion. The response is asymmetric, with larger meridional winds on the downstream side of the mountain associated with a cold anomaly. Nonlinear terms such as the geostrophic intensification of the vertical wind aloft (big red arrow) and the transient-eddy heat fluxes (squiggly lines) act primarily in this downstream region, while latent heating acts in the upstream region. Processes are color coded red or blue based on whether they have a warming or cooling effect, respectively.
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
This picture is complicated by other types of heating, such as latent heating and transient-eddy heat fluxes. Latent heating on the windward slope is out of phase with the cooling/heating by adiabatic motions and thus acts as a negative feedback on stationary wave amplitude. Latent heating and the eastward group propagation of Rossby waves break the symmetry of the problem, leading to a northerly wind anomaly and cold anomaly downstream of the mountain [described in Hoskins and Karoly (1981) in the absence of latent heating], perturbations that propagate downstream according to Rossby wave dynamics. Through geostrophy, the northerly wind downstream of the mountain is associated with strong subsidence in the upper troposphere, which enhances adiabatic heating and amplifies the meridional velocity and temperature anomaly (cf. Fig. 7). Transient eddies act on the resulting temperature gradients, converging heat into the anomalous cold region. The transient-eddy heat flux convergence is in phase with the cooling/heating by adiabatic motions and thus acts as a positive feedback on stationary wave amplitude. The negative feedback from latent heating becomes stronger with warming because of increased atmospheric moisture content (section 5b; appendix B). The positive feedback from transient-eddy heat fluxes becomes weaker with warming because of reduced horizontal temperature gradients and reduced transient-eddy kinetic energy (section 5c). Both of these effects act to reduce the amplitude of orographic stationary eddies with warming.
6. Influence on midlatitude zonal asymmetry
Stationary-eddy winds play an important role in the zonal asymmetry of temperature and precipitation. In the simple case studied here, where stationary-eddy winds are forced by a single midlatitude mountain, the stationary-eddy winds weaken in warm, moist climates, leading to a reduction in zonal asymmetry across a range of latitudes. The changes in zonal asymmetry of temperature and precipitation result from a combination of the stationary-eddy wind changes and thermodynamic factors.
The implications of the wind changes in these simulations for changes in the zonal asymmetry of precipitation minus evaporation (
(a) Root zonal variance of potential temperature 
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
7. Summary and conclusions
We have diagnosed the mechanisms governing the strength of stationary Rossby waves forced by midlatitude orography across a wide range of climates in an idealized GCM. To summarize, we group the most important influences on stationary wave amplitude by whether they increase or decrease the amplitude with warming.
- Nonmonotonic change with warming:
- Change in zonal surface winds at the latitude of the mountain.
- Decreased stationary wave amplitude with warming:
- Increased damping by latent heating.
- Reduced zonal variation of transient-eddy heat flux convergence as a result of reduced zonal temperature variance (resulting primarily from a reduced mean meridional temperature gradient) and reduced tEKE (in the warmest climates).
- Decreased top-of-mountain surface pressure due to mean warming.
- Increased stationary wave amplitude with warming:
- Decreased isentropic slope resulting from increased extratropical static stability and reduced meridional temperature gradients.
The net result, after the internal cancellation of several of these effects, is that the amplitude of orographic stationary Rossby waves in these simulations scales roughly with the zonal-mean zonal surface winds because of their influence on orographic vertical winds. While the decrease in meridional temperature gradient and increase in extratropical static stability would both lead to an increased amplitude of orographic stationary Rossby waves, the increased damping by latent heating and reduced forcing by transient-eddy heat fluxes win out to give a reduction in strength of orographic stationary Rossby waves with global warming. While the relationship with zonal surface winds no longer applies for simulations where orography is equatorward of the band of surface westerlies, the same balance of physical mechanisms relates topographic vertical winds to the amplitude of orographic stationary Rossby waves (Wills 2016).
In understanding the influence of stationary waves on the zonal asymmetry of temperature and net precipitation, thermodynamic factors become leading order. Reduced meridional temperature gradients are associated with a reduction in the zonal asymmetry of temperature. Increased atmospheric moisture content is associated with an increase in zonal asymmetry of net precipitation. However, the reduced amplitude of orographic stationary Rossby waves is an important factor in reducing the zonal asymmetry of temperature and precipitation in the warmest, wettest climates.
8. Implications
There are numerous important differences between the idealized GCM used here and comprehensive models or the real atmosphere. For example, the idealized GCM simulates an equatorward shift of the surface westerlies with global warming (Fig. 6) while comprehensive models simulate a poleward shift (Yin 2005). However, all of the mechanisms discussed herein emerge from the governing physical equations and should have some relevance to the real climate system. Which of them can we expect to be important and which not? Because of the unusual equatorward shift of the surface westerlies, we do not expect the particular changes in surface winds and orographic vertical velocities in the R45 and R54 experiments to apply to real-world mountain ranges at these latitudes. We do, however, expect that the impinging zonal westerlies at the latitude of the mountain are the relevant surface winds to determine the strength of orographic forcing by midlatitude mountains. This may remain true even in cases where the zonal surface winds change sign, such as in the orographic forcing of Rossby waves by the Zagros Mountains in summer (Simpson et al. 2015).
In our simulations, the strength of the meridional wind response to orographic perturbations scales with the lower-tropospheric static stability and inversely with the meridional temperature gradient. This is a simple consequence of the thermodynamic equation and should equally apply to the real world. As an example of these mechanisms, recent work has shown that Mongolian topography matters more for the wintertime atmospheric circulation over the Pacific than the higher Tibetan Plateau because of the stronger surface winds and larger meridional temperature gradient at their higher latitude (White et al. 2017). The expected increase in static stability and decrease in lower-tropospheric meridional temperature gradients with global warming would both lead to an increase in the strength of orographic stationary waves. However, latent heating and transient-eddy heat fluxes must also be considered. In our simulations, increased damping by latent heating and reduced reinforcement by transient-eddy heat fluxes limits the strength of orographic stationary waves in warm climates. More work is needed to characterize latent heating and transient-eddy feedbacks within stationary waves in the real world.
Based on the mechanisms discussed in this study, the response of orographic stationary Rossby wave amplitude to climate change in the real climate system should depend on the surface wind, meridional temperature gradient, static stability, and latent heating changes in key regions around large-scale orography, such as the Rocky Mountains, Andes, Himalayas, Tibetan Plateau, and Mongolia, as well as on interactions between stationary wave changes and storm track changes. Cloud and water vapor feedbacks, which were not investigated here, could provide an additional influence on the amplitude of orographic stationary Rossby waves, through their role in the zonally anomalous energy budget, and should be investigated in this context.
This work was primarily completed while both authors were at the Department of Earth Sciences, ETH Zürich, Zurich, Switzerland. This research has also been supported by NSF Grant AGS-1019211 while both authors were at the California Institute of Technology, Pasadena, California. The idealized GCM simulations for this study were performed on ETH Zürich’s EULER computing cluster. Code for the idealized GCM is available online (at https://github.com/tapios/fms-idealized). We thank Simona Bordoni, Andy Thompson, Jess Adkins, Rachel White, Momme Hell, Farid Ait Chaalal, Ori Adam, and three anonymous reviewers for useful comments and discussion during the development of this manuscript.
APPENDIX A
Interpolation to Pressure Coordinates
While the idealized GCM uses sigma coordinates, we interpolate to pressure coordinates to avoid large spatial gradients on sigma surfaces as the coordinate surfaces slope over orography. The interpolation is done at every 6-hourly analysis time step, by linearly interpolating the vertical integral of each quantity. This helps ensure that mass is conserved in the interpolation. However, quantities with large vertical gradients, such as potential temperature, can be shifted in the vertical direction, creating large-amplitude grid-scale noise, which integrates to zero. The residual of the thermodynamic equation (11) after interpolation is shown in Fig. A1c. We distribute it to the other terms in a way that minimizes the total variance of all terms (i.e., we put the residual with the term it is most correlated with at each level). Above 600 hPa, this is the zonal advection term (Fig. A1a), which is noisy because of noise in 
Thermodynamic equation terms in the reference climate (α = 1.0, Tg = 289 K) of R45 before the residual has been distributed in a way such that the sum of variances of all terms is minimized. Note that the color bars are rescaled by a factor of 5, as compared to Figs. 7 and 8.
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0700.1
APPENDIX B
Derivation of the Relationship between Zonally Anomalous Latent Heating and Vertical Motions
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We have neglected a contribution to Eq. (6) from the temporal correlations of the surface winds and the surface pressure gradients, 
While the mountain length scale σλ would show up in a scaling for the local 
The transient-eddy potential temperature flux divergence, 
These calculations are based on an additional four years of the R45 reference simulation, because 
