## 1. Introduction

Orography plays a fundamental role in shaping the dynamics of the atmosphere. At small scales (tens to hundreds of kilometers), airflow over mountains generates internal gravity waves, which propagate vertically and horizontally, transporting momentum away from their source and depositing it wherever they break. Alternatively, if the flow is too slow to move up and over the orography, flow splitting may occur, producing long, persistent downstream wakes. Parameterizing these and other impacts of small-scale orography is notoriously difficult, yet is essential for making accurate weather forecasts and for predicting future circulation changes [see Sandu et al. (2019) for a recent review].

At larger scales, orographically forced quasi-stationary planetary waves transport substantial amounts of heat, moisture and momentum through the atmosphere, exerting a strong control on regional climates and also playing a key role in the zonal-mean circulation. Our understanding of stationary waves is based on linear theory, which provides good qualitative agreement with observations in many respects. For example, the Charney–Eliassen model, which approximates the atmosphere as a barotropic, quasigeostrophic fluid in a *β*-plane channel, does a reasonable job of reproducing the observed Northern Hemisphere wintertime stationary wave pattern when forced with observed orography (Charney and Eliassen 1949; Held 1983). However, there are a number of open questions concerning linear theory’s relevance for quantitatively understanding observed large-scale stationary wave patterns. These include the role of nonlinear interactions between stationary waves (Wang and Kushner 2010), and how to account for interactions between stationary waves and transient eddies. Past modeling studies have found that transients damp the stationary wave response to orography (Vallis and Roads 1984) or that orographically forced stationary waves are unaffected by the presence of transients (Nigam et al. 1988), and it is still unclear how to account for transient eddies in linear stationary wave theory. Another issue concerns stationary nonlinearities: when the linear approximations break down (Cook and Held 1992; Lutsko and Held 2016), how to best account for stationary nonlinearities (e.g., Trenberth and Chen 1988; Valdes and Hoskins 1991) and how relevant they are for the observed atmospheric circulation.

One question that has received less attention recently is how orographically forced stationary waves affect the zonal-mean circulation. Early stationary wave studies typically considered channel geometries, in which waves can only propagate zonally and vertically (e.g., Charney and Eliassen 1949; Smagorinsky 1953; Saltzman 1963, 1965; Kasahara 1966; Derome and Wiin-Nielsen 1971; Egger 1978). Resonances appear generically in this setting, which leads to the existence of multiple equilibrium states when coupling to the mean flow is included, with, for instance, a large stationary wave amplitude/weak-mean-flow state coexisting with a small stationary wave amplitude/strong-mean-flow state for a given mountain height and shape (Charney and DeVore 1979).

The realization that stationary waves tend to propagate (approximately) along great circles (Hoskins et al. 1977; Grose and Hoskins 1979; Hoskins and Karoly 1981) shifted the emphasis away from the coupled wave–mean flow problem and toward understanding the propagation of orographic waves, particularly their interactions with subtropical critical layers. Stationary waves may be absorbed, reflected or overreflected by critical layers (Killworth and McIntyre 1985), though the fact that the climatological stationary eddy momentum flux is directed from the tropics toward midlatitudes suggests that on average these waves are absorbed, rather than reflected. In practice, it seems difficult to create a reflecting critical layer for systems resembling the real atmosphere, with transient eddies and a Hadley circulation [though see Walker and Magnusdottir (2003)].

A separate series of papers have investigated the ability of midlatitude jets to act as waveguides for stationary waves (Branstator 1983; Hoskins and Ambrizzi 1993; Branstator 2002; Branstator and Selten 2009; Manola et al. 2013; Petoukhov et al. 2013; Saeed et al. 2014; Lutsko and Held 2016). “Circumglobal” waves that are trapped in these waveguides can propagate over long zonal distances with little meridional motion, and may play an important role in the atmosphere’s response to increased CO_{2} concentrations (Brandefelt and Kornich 2008; Simpson et al. 2016).

All of these studies have taken the mean flow as fixed, and then examined stationary wave propagation under a given mean flow (see also Wills and Schneider 2018). But the momentum transported by stationary waves plays a key role in the dynamics of midlatitude jets and storm tracks (e.g., Kaspi and Schneider 2013), and changes in wave properties, such as a transition from a meridionally trapped wave to a meridionally propagating wave can potentially lead to large changes in stationary eddy momentum fluxes (EMFs), driving jet shifts or changes in jet intensity. As an example, White et al. (2017) found that the combined effects of the Himalayas and the Tibetan Plateau force relatively small amplitude stationary waves that are trapped within a waveguide and have little impact on the jet over eastern Asia and the Pacific. By contrast, the Mongolian mountains, which are farther north, excite stronger waves that propagate meridionally and intensify the Pacific jet. An externally driven shift of the jet could alter these different wave paths, which could in turn amplify or damp the initial jet shift.

Through the nonacceleration theorem, linear theory says that stationary EMFs should balance the torque exerted by the mountain on the atmosphere (see following subsection); however, this balance is not typically seen in simulations.^{1} Manabe and Terpstra (1974) compared simulations of a general circulation model (GCM) with and without topography, and saw an increase in the transient EMF when topography is removed [Park et al. (2013) found a similar compensation between the stationary and transient eddy heat fluxes]; while Cook and Held (1992) found that in an idealized, moist GCM the mountain torque is mostly balanced by a reduction in the surface friction. Past studies have found that orography exerts a strong drag on zonal jets (e.g., Brayshaw et al. 2009) and that orography can accelerate jets (Son et al. 2009; White et al. 2017). Thus, the question of how the atmosphere balances stationary eddy momentum transport, and in particular how the strengths and latitudes of eddy-driven jets respond to the presence of orography, is an open question.

In this study, the response of the angular momentum budget of an idealized, dry GCM to orography is systematically investigated, through a series of simulations in which the height of the orography is increased. These include simulations with the full dynamical GCM, and simulations in which the GCM is converted into a stationary wave model, which does not include transient eddies and so is the most likely setting for the nonacceleration theorem to hold. Together, these simulations are used to investigate how the atmosphere responds to the stationary EMFs induced by the presence of orography, how this response differs from what is expected from the nonacceleration theorem and how the response is affected by the presence of transient eddies. The goals are to provide a better understanding of the practical utility of the nonacceleration theorem, including how transients and stationary nonlinearities affect the nonacceleration balance, and to further investigate how large orography affects the strength and latitude of midlatitude jets.

The paper is structured as follows: After reviewing the nonacceleration theorem for orographically forced stationary waves in the following subsection, the GCM and the stationary wave model are described in section 2. The momentum budget of the unperturbed system is then presented in section 3, and the stationary wave model results are discussed in section 4 and the GCM simulations with small orography in section 5. The GCM experiments with large orography are presented in section 6, and then conclusions are given in section 7.

### The nonacceleration theorem for orographic stationary waves

^{2}Consider the eddy potential vorticity (PV) equation for an inviscid, adiabatic, continuous quasigeostrophic (QG) system, linearized about a zonal-mean flow

*υ** is the eddy meridional wind, and

*ψ** is the eddy streamfunction,

*f*

_{0}is a reference value of the Coriolis parameter,

*ρ*is density, and

*N*

^{2}is the buoyancy frequency. The vertical coordinate

*z*=

*H*ln(

*p*

_{s}/

*p*), where

*H*is the scale height of the atmosphere,

*p*is pressure, and

*p*

_{s}is surface pressure. The surface boundary condition is the QG thermodynamic equation at

*z*= 0,

*h*is the height of the mountain and

*g*is the gravitational constant. Substituting Eq. (3) into Eq. (2) and rearranging then gives

*q**, Eq. (4) by

*s** = ∂

*ψ**/∂

*z*+ (

*N*

^{2}/

*f*

_{0})

*h*and taking zonal means gives

The left-hand sides of these equations are zero for a steady wave and so

*p*

_{s}is surface pressure.

## 2. Models and experiments

### a. GCM description and experiments

The GCM simulations are the same simulations as in Lutsko and Held (2016). These were carried out using the GFDL spectral dynamical core, forced by zonally symmetric Newtonian relaxation to a prescribed equilibrium temperature field and damped by Rayleigh friction near the surface. The parameter settings are the standard Held and Suarez (1994) parameters, with forcing symmetric about the equator. All simulations presented here were run at T42 resolution with 30 evenly spaced sigma levels, and data sampled once per day. A control simulation was integrated for 40 000 days, with the first 2000 days discarded to ensure the model had spun up.

*H*is the maximum height of the mountain (m);

*ϕ*and

*λ*are latitude and longitude, respectively; and

*α*and

*β*are half widths, both set to 15° in the initial-perturbation experiments (see also Cook and Held 1992). The latitude of the orography was chosen to be collocated with the latitude of maximum surface wind speed.

The maximum height of the orography *H* was varied from 250 to 5000 m. For heights less than 250 m the responses are not clearly separable from the noise. Cases with *H* less than 1 km were run for 40 000 days and the responses were obtained by discarding the first 2000 days of each perturbation experiment and averaging over the rest of the integration. These long integration times were required to ensure that the responses had equilibrated, in the sense that the response calculated using half of the data was indistinguishable from the response calculated using all of the data. This ensures that the stationary wave signal is clearly distinguishable from the transients. Cases with larger mountains equilibrated more quickly and so were only run for 20 000 days.

Lutsko and Held (2016) found that these simulations separated into a “linear” regime, in which the model’s response is approximately linear in *H*, and a nonlinear regime, in which the amplitude of the model’s response increases sublinearly, with the transition occurring between *H* = 700 m and *H* = 1 km. Associated with this transition, the stationary wave response transitions from being more zonally oriented (in the linear regime), to propagating more meridionally in the nonlinear regime (cf. Fig. 1a and Fig. 1b).

### b. Stationary wave model and experiments

Following previous studies such as Held et al. (2002) and Chang (2009), the stationary wave model was created by applying strong damping to the same GCM described above and by strongly relaxing the zonal-mean flow to the time- and zonal-mean flow in the control simulation. The results described below come from simulations in which the hyperdiffusion was doubled from 1.157 × 10^{−4} m^{8} s^{−1} (in the original GCM simulations) to 2.31 × 10^{−4} m^{8} s^{−1}, and the Rayleigh friction damping times for both the vorticity and divergence equations were set to 0.3, 0.5, 1.0, and 8.0 days at the lowest four *σ* levels (0.997, 0.979, 0.935, and 0.866), and 15 days throughout the rest of the domain. The Newtonian cooling time scale was decreased from 40 days (the Held–Suarez value) to 15 days at all levels and the relaxation time scale of the zonal-mean winds was set to 1 day. These parameter settings were found to successfully eliminate the transients in the model; however, there is no objective method for choosing the optimal parameter settings for creating a stationary wave model, and the simulations were repeated with several different parameter settings to ensure that the results are robust (e.g., the relaxation time scale of the zonal-mean wind was varied from 0 to 3 days). In all experiments, the stationary wave model was integrated for 200 days and averages were taken over days 50–200. Inspection of the flow indicated that the model equilibrates after 15–20 days.

The normalized eddy streamfunction response of the stationary wave model to *H* = 500 m orography (Fig. 1c) compares well with the *H* = 500 m simulation with the full GCM (Fig. 1a). The patterns of the responses are very similar, though the stationary wave model’s response is roughly 25% weaker than the GCM’s. This difference is discussed further in section 5.

The stationary wave model has no transient eddies or changes to the mean flow, eliminating two factors that are likely to interfere with the nonacceleration balance. However, it is still possible for the friction to alter the momentum balance, or for a stationary nonlinearity to appear. Comparing the stationary wave model results with the full GCM allows the role of transients and mean-flow changes to be made clear.

## 3. Momentum budget of the unperturbed atmosphere

*a*is Earth’s radius,

*F*is friction, square brackets are time means, overbars are zonal means, asterisks are deviations from the zonal mean, and primes are deviations from the time mean. Throughout this study, the transient term is calculated using the daily zonal-mean surface pressure (recall that data are collected once per day):

The profiles of the terms in the angular momentum budget of the control integration are shown in Fig. 2. As expected, the main balance is between the transient EMF convergence and the friction. These are largest at midlatitudes, where the transient eddies accelerate the flow and the friction decelerates the flow, and change sign in the tropics and at high latitudes. Both terms are small in the subtropics, where the momentum flux by the mean flow is the largest term in the budget (Peixoto and Oort 1992).

## 4. Stationary wave model results

The stationary wave model was run with mountains of maximum height *H* = 0.5 m, 10 m, 500 m, and 2 km. In testing, it was found that *H* = 0.5 m is the smallest mountain height that produces a response distinguishable from the noise in this model setup; however, it is worth noting how extreme this case is, representing a mountain with roughly the same horizontal extent as the Tibetan plateau but a maximum height of less than 1 m.

The normalized mountain torque and stationary EMF convergence are similar in the *H* = 0.5 m and *H* = 10 m experiments (Figs. 3a,b), as the torque decelerates the flow between about 30° and 60°, while the stationary EMF convergence accelerates the flow at these latitudes. However, while the torque has a single maximum at 48°N, the stationary EMF convergence has maxima at 42° and 55°. This pattern leads to a slight deceleration of the jet between 40° and 52° (and hence a reduction in the friction at these latitudes; see Fig. 3c^{3}) and an acceleration between 55° and 62°N. The stationary EMF convergence also decelerates the flow in the subtropics, between ~10° and 30°N.

As the height of the mountain is increased, the torque moves equatorward slightly (darker red lines in Fig. 3a), though its normalized magnitude is roughly constant, and the latitude of maximum stationary EMF convergence shifts poleward. This is a result of the poleward stationary EMF convergence maximum growing relative to the equatorward maximum as the height is increased, so that for *H* = 2 km there is a single maximum in the EMF convergence at 52°N. This extends the region over which the mean flow is accelerated to 48°–62°N, and pushes the jet poleward (Fig. 3c).

To understand these response, the left panels of Fig. 4 show the horizontal components of the Plumb flux at 350 hPa (vectors; see appendix for how the Plumb flux is calculated) and the vertical component at 800 hPa (red contours) for the *H* = 0.5 m experiment (Fig. 4a), the *H* = 10 m experiment (Fig. 4c) and the *H* = 500 m experiment (Fig. 4e). In the *H* = 0.5 m experiment the wave source, as measured by the vertical component of the Plumb flux, is centered slightly northeast of the peak of the orography (Fig. 4a). Figure 4b shows the anomalous 800 hPa wind vectors (green arrows) and the zonal anomalies in *θ* (filled contours). The wind vectors indicate that the flow is preferentially deflected north of the orography, so that the anticyclone associated with the orographic high is centered north of the mountain peak. There is also a cyclone immediately downstream of the orography, southeast of the anticyclone, and weak cooling over the mountain, though this is likely an artifact of the algorithm for interpolating from *σ* coordinates to pressure coordinates. The anticyclone–cyclone pair, centered on the northeastern flank of the orography, is responsible for shifting PF_{z} to the northeast of the orography.

The preferential poleward deflection of the flow is caused by the mean isentropic slope, which slants upward with latitude. Hence, air flowing along an isentrope that is deflected equatorward also moves to lower altitudes, and vice versa for air deflected poleward. This makes the orography appear “taller” on its equatorward flank than on its poleward flank, and more of the air flows poleward around the orography (Valdes and Hoskins 1991).

Returning to Fig. 4a, the arrows show that the majority of the wave energy propagates equatorward, and is dissipated by the damping as it propagates into the subtropics, with little evidence of the wave being absorbed near the critical layer, where *u* = 0 (cyan line). Part of the equatorward propagating wave train is also refracted into the waveguide and propagates zonally before being dissipated (the mean flow in this simulation acts as a waveguide for waves with zonal wavenumber *k* = 5; Lutsko and Held 2016). A smaller portion of the wave energy propagates poleward, where it appears to reflect off a turning latitude and propagate equatorward, before dissipating or, possibly, being refracted into the waveguide.

The dissipation of the wave trains as they propagate away from the orography leads to the stationary EMF convergence maximum near 40°N (from the equatorward-moving wave train), the smaller maximum near 55°N (from the poleward-moving wave train) and to the EMF divergence in the subtropics. So in this small-*H* case, the damping is responsible for the lack of exact compensation between the torque and the stationary EMF convergence, by dissipating the stationary wave as it propagates away from the orography.

There are two wave sources in the *H* = 10 m case (Fig. 4c): one to the east and one to the north of the orography. These are associated with negative temperature anomalies over the equatorward flank of the mountain and, more weakly, on the northeast flank of the mountain (Fig. 4d). The flow is similar to the *H* = 0.5 m case, though the axis of the anticyclone–cyclone pair is rotated farther northwest–southeast, rather than the more zonal orientation seen for *H* = 0.5 m. This circulation pattern advects cold air along the eastern and northeastern flanks of the mountain, creating the temperature anomalies. Since the circulation is shifted poleward of the orography, the temperature advection is not balanced by the adiabatic cooling and warming of the air as it rises and sinks over the orography (the flow in the 0.5 m case seems to be too weak to induce substantial temperature anomalies).

Plotting each of the terms in PF_{z} indicates that the temperature anomalies are responsible for the two wave sources, primarily through the ∂[*θ**]/∂*λ* term (not shown). Thus, the preferential poleward deflection of the flow, and the resulting temperature anomalies, are responsible for generating a stationary nonlinearity. The horizontal components of the Plumb flux suggest that the propagation of the wave trains remains similar to the *H* = 0.5 m case, however, as the majority of the wave energy propagates equatorward and is dissipated near the critical line. One difference is that the poleward wave source is close to the turning latitude, and there is less evidence of wave reflection from the turning latitude on the poleward edge of the waveguide. Instead, the poleward wave train propagates roughly parallel to the turning latitude, before dissipating. Despite this difference, the net effect for the *H* = 10 m case is a similar profile of EMF convergence and divergence as in the *H* = 0.5 m case.

The responses to the larger mountains are generally similar to the *H* = 10 m case (Figs. 4e,f), though the wave sources change shape somewhat, causing the poleward shift of the EMF convergence maximum discussed earlier. The stationary waves are also able to propagate farther into the subtropics, and cause the critical layer to be slightly distorted between 100° and 150°E (cyan line in Fig. 4e). Finally, the temperature anomalies induced by the flow in the simulations with larger orography cause *p*_{s} and ∂*h*/∂*λ* to move out of phase and hence shift the mountain torque equatorward (not shown).

## 5. Response to small mountains

The GCM experiments with small (*H* < 1 km) mountain heights are in an approximately linear regime, with a roughly constant normalized torque (blue curves in Fig. 3a) that decelerates the flow over most of the mountain and weakly accelerates the flow between about 55° and 65°N. The normalized stationary EMF convergence is also roughly constant in these simulations (blue curves in Fig. 3b), and is shifted poleward of the torque, accelerating the flow between about 40° and 70° and decelerating the flow at lower latitudes [note that the larger EMF convergence for the 333 m case (orange curve) is a result of sampling error]. A similar poleward displacement of the stationary EMF convergence relative to the torque was seen in the GCM experiments of Cook and Held (1992). Because of this offset, and because the stationary EMF convergence is larger than the torque, the two terms do not cancel (blue curves in Fig. 3c), with the residual decelerating the flow equatorward of ~42°N and accelerating the flow poleward of this latitude. This induces a deceleration and poleward shift of the midlatitude jet (Fig. 5), though the responses of the friction, the transient EMFs and the mean flow are comparable to the sampling error for these small mountain heights,^{4} making it difficult to identify how exactly the momentum budget is balanced in the small-*H* GCM experiments.

The results of the previous section suggest that the poleward displacement of the stationary EMF convergence relative to the torque is partly due to the stationary nonlinearity; however, the EMFs are larger in the GCM experiments than in the stationary wave model. Comparing Figs. 4g and 4h with Figs. 4e and f demonstrates that the patterns of stationary wave propagation and temperature anomalies are similar in the GCM and the stationary wave model, but that the amplitudes of the wave response and of the temperature anomalies are larger in the GCM. Hence, transient eddies appear to amplify the response to orography relative to the stationary wave model, producing larger stationary EMFs and larger potential temperature anomalies over the mountain in the GCM experiments. The propagation of the stationary wave does not differ substantially in the GCM compared to the stationary wave model, implying that mean-flow changes are not responsible for the changes in the stationary EMFs. The deceleration of the zonal-mean winds in the *H* = 500 m GCM experiment (Fig. 5) should weaken the stationary wave source, in contrast to the strengthening seen here.

It is possible that the weaker hyperdiffusion and surface friction in the GCM also contribute to the larger response; however, these should mostly affect the wave propagation and the far-field response, and should have less of an impact in the immediate vicinity of the mountain, where the mountain-induced response is larger. In stationary wave model experiments with other parameter settings, the amplitude of the response is relatively insensitive to the strength of the damping, provided the damping is strong enough to eliminate the transient eddies (not shown).

Finally, in addition to the stationary nonlinearity discussed in the previous section, another factor that may be responsible for the lack of cancellation between the EMF convergence and the torque is diabatic heating over the mountain: in the Held–Suarez setup the Newtonian cooling is applied on constant *σ* levels, so the near-surface air at the top of the mountain is relaxed to the same temperature as air at sea level (at the same latitude), producing a strong radiative heating over the mountain.

To investigate how orographically induced diabatic heating affects the model’s response, the GCM experiments were repeated with new zonally varying equilibrium temperature fields that are functions of pressure, rather than *σ*. So for instance, all grid points at 700 hPa are relaxed to the same temperature. This eliminates the diabatic heating over the orography, though new radiative-equilibrium temperature fields have to be generated for each mountain height. The responses in these experiments have very similar patterns to the responses in the original experiments, but are 25%–50% stronger, depending on the mountain height (e.g., cf. Fig. 1a and Fig. 1d), suggesting that the radiative heating opposes the orographic forcing and, if anything, damps nonlinearities. Whether the diabatic heating that comes from relaxing the zonal-mean temperatures along constant *σ* levels is physically realistic or whether the radiative-equilibrium temperature field should be specified along constant pressure surfaces is an open question [see Hu and Boos (2017) for a discussion of the physics of orographic heating in a radiative-convective equilibrium context].

## 6. Responses to large mountains

The stationary EMF convergence, the transient EMF convergence and the friction all have substantial responses in the experiments with large (*H* ≥ 1 km) mountains, while the changes in the mean momentum flux convergence are small (Fig. 6). The following subsection discusses the responses of the torque and the stationary EMF convergence, the original terms in the nonacceleration balance, in these experiments, and then section 6b describes the responses of the friction and the transient EMF convergence.

### a. Responses of the torque and the stationary EMF convergence

The normalized zonal profiles of the torque and of the stationary EMF convergence are essentially unchanged as *H* is increased (Figs. 6c,e), as the torque decelerates the flow between 30° and 55°N, while the stationary EMFs decelerate the flow between 10° and 35°N and accelerate the flow between 35° and 60°N. As discussed by Lutsko and Held (2016), the torque increases more slowly for these mountains than the *H*^{2} scaling expected from linear theory because of the increased migration of the orographically forced anticyclone away from the center of the mountain, which causes *p*_{s} and ∂*h*/∂*λ* to move more strongly out of phase for larger *H*.

The slower increase of the torque causes the stationary EMF convergence to increase more slowly than expected from linear theory (i.e., more slowly than *H*^{2}); however, some effect of the increased meridional, versus zonal, propagation of the stationary wave at large mountain heights (Fig. 1a vs Fig. 1b) is expected. Intuitively, more meridional propagation should lead to larger normalized stationary EMFs.

Figure 7 confirms this intuition, showing the maximum mountain torque versus the maximum stationary EMF convergence for the GCM experiments.^{5} As the inset shows, for small values of *H* the torque and the stationary EMF convergence nearly follow a one-to-one line, though the maximum stationary EMF convergence is slightly larger, as discussed in the previous section. However, the stationary EMF convergence increases more rapidly than the mountain torque for *H* ≥ 1 km, so that when *H* = 5 km the maximum stationary EMF convergence is ~60% larger than the maximum torque. Hence, the increased meridional propagation of the stationary waves in the nonlinear regime does lead to increased stationary EMF convergence relative to the torque. The extra stationary EMF convergence must be compensated by the responses of the friction and of the transient EMF convergence.

### b. Responses of the friction and the transient EMF convergence

In the *H* = 1 km and *H* = 2 km experiments there is increased transient EMF convergence north of the mountain and decreased convergence to the south, while the friction is enhanced to the north and reduced to the south (Figs. 6b,d). These are associated with poleward shifts of the midlatitude jet (Fig. 5). Conversely, the jet shifts equatorward in the *H* = 4 km and *H* = 5 km experiments, with the transient EMF convergence reduced north and increased south of the mountain, and the friction having the opposite signed response. The *H* = 3 km experiment is intermediate between the 2 and 4 km experiments, though there is a slight equatorward shift of the jet in this case (Fig. 5).

The filled contours in Figs. 8c and 8e show the responses of the 350 hPa transient EMFs in the *H* = 2 km and *H* = 4 km GCM experiments, respectively. In both simulations, the transient EMFs are enhanced upstream and reduced downstream of the orography. While the regions of enhanced transient EMFs are similar, the downstream reduction of the EMFs is much stronger in the *H* = 4 km experiment, such that there is an increase in the zonal-mean transient EMFs in the 2 km experiment and a reduction in the 4 km experiment (Fig. 8d).

**E**vectors (Hoskins et al. 1983), which indicate the direction of eddy propagation, and hence the direction of westerly momentum transport. The horizontal components of the

**E**vector are given by

**E**

_{h}vectors in the control experiment are shown in Fig. 8b. In the absence of orography, the eddies primarily propagate to the northeast, leading to northward momentum transport by transient eddies in the zonal mean (see Fig. 8a).

The responses of the **E**_{h} vectors in the *H* = 2 km and *H* = 4 km experiments are shown in Figs. 8d and 8f. In both cases the vectors downstream of the orography primarily point to the southwest, suggesting that the transient eddies are decelerating and also elongating zonally [see Fig. 4 of Hoskins et al. (1983)]. The region of eddy deceleration and zonal elongation is much larger in the *H* = 4 km experiment than in the *H* = 2 km experiment.

Upstream of the orography, the transient eddies are deflected around the peak of the orography, though unlike the stationary waves, the eddies are deflected equally to the north and to the south of the orography. The jet also widens upstream of the orography in both simulations (black contours in Fig. 8c), suggesting that there is more space for meridional eddy propagation. Confirming this, the anomalous **E** vectors southwest of the orography (e.g., near 30°N and 60°E) point to northeast, representing increased poleward transient momentum transport. In the *H* = 4 km experiment, the vectors to the southeast of the orography (i.e., downstream of the orography) point to the southwest, and the jet also narrows in this region. So there is likely to be less space for meridional eddy propagation downstream of the orography, further damping the downstream transient EMFs. It is difficult to see whether this is the case in the *H* = 2 km experiment, and there is also less narrowing of the jet downstream of the orography in this case.

In the *H* = 2 km case the broadening of the jet upstream of the orography, and the larger space for meridional eddy propagation, wins out and the poleward transient EMFs increase compared to the control (Fig. 9), pushing the jet poleward. In the *H* = 4 km experiment the deceleration of the eddies downstream of the orography wins out, with evidence of the eddies being disrupted more than 90° downstream of the orography, and the poleward transient EMFs decrease compared to the control (dashed line in Fig. 9), favoring an equatorward shift of the jet.

One other potential mechanism by which the orography could cause the jet to shift in latitude is the effect of the locally enhanced baroclinicity on wave breaking. By enhancing downstream temperature gradients, large-scale orography enhances the local downstream baroclinicity (Son et al. 2009; Lutsko et al. 2019). This is primarily a result of the stationary eddy heat flux, which fluxes heat into the region southeast of the orography (not shown), increasing the baroclinicity there, as was also seen in the idealized moist GCM simulations of Kaspi and Schneider (2013). Orlanksi (2003) showed that increased low-level baroclinicity favors cyclonic wave breaking (CWB), which tends to push jets equatorward, rather than anticyclonic wave breaking (AWB), which tends to push jets poleward. The reason for this is that the amplitude of anticyclonic eddies is bounded by −*f*, because if the absolute vorticity *ζ* + *f* goes to zero then the stretching term in the vorticity equation, which drives the eddies, also goes to zero. By contrast, the amplitude of cyclonic eddies is unbounded, so that as eddy amplitudes increase cyclonic eddies tend to become more prominent relative to anticyclonic eddies.

The wave-breaking algorithm of Rivière (2009) was used to estimate changes in wave breaking in the control, 2 and 4 km simulations. This algorithm identifies and classifies (AWB or CWB) local reversals of the absolute vorticity contours, searching along circumglobal contours whose values are multiples of 10^{−5} s^{−1}. Only circumglobal contours are considered in order to avoid detecting isolated patches of high or low vorticity that are unrelated to wave breaking. Applying the wave-breaking algorithm at 250 hPa gives AWB: CWB ratios of 1.78 ± 0.01:1 in the control experiment, 1.78 ± 0.02:1 in the 2 km experiment, and 1.63 ± 0.01:1 in the 4 km experiment.^{6} In the 4 km case then, the locally enhanced baroclinicity downstream of the orography does favor CWB, which may contribute to, or reinforce the equatorward jet shift. In the *H* = 2 km case the enhanced baroclinicity does not appear to be sufficient to cause a major change in wave-breaking characteristics.

### c. Jet speed

The presence of orography causes the midlatitude jet to decelerate in all of the experiments, with the deceleration increasing as *H* is increased (Fig. 5). However, the nature of these decelerations differ substantially in the different experiments. In the *H* = 2 km experiment, both the transient and the stationary EMFs decelerate the equatorward flank of the jet, and weakly accelerate the poleward flank (Figs. 10d,f). This is balanced by reductions of the friction on the equatorward side of the jet and enhancements on the poleward side of the jet, such that the jet shift is accomplished mostly by a deceleration of the jet equatorward of roughly 55°N.

In the *H* = 4 km experiment, the transient and stationary EMFs approximately cancel over most of the orography (Figs. 10e,g). Thus, the friction must balance the mountain torque (Fig. 10i), leading to a strong deceleration in the jet core (where the orography is located). In the subtropics, the response of the transient EMFs is larger than the stationary EMF, causing an acceleration of the jet.

Hence, although the jet decelerates in all of the orographic experiments, the reasons differ considerably in the experiments with smaller (*H* ≤ 2 km) and in the larger (*H* > 2 km) experiments. For smaller mountain heights the jet decelerates in response to the stationary and transient EMFs, whereas for larger mountains the jet decelerates to balance the mountain torque, with the transient EMFs balancing the stationary EMFs. These cases, then, are far from the balance between stationary EMFs and mountain torque expected from linear theory.

## 7. Conclusions

This study has used a stationary wave model and an idealized, dry GCM to investigate the impact of orography on the atmosphere’s momentum budget, with a focus on assessing the nonacceleration theorem, how transients affect the response to orography and how orography affects the strength and latitude of eddy-driven jets. Comparing simulations with a stationary wave model and a GCM, forced with Gaussian mountains of heights ranging from 0.5 m to 5 km has produced the following picture of how the two models respond to the presence of orography:

- For the smallest mountain considered here (
*H*= 0.5 m) the response of the stationary wave model nearly follows what would be expected from the nonacceleration theorem, as the torque exerted by the mountain on the atmosphere is mostly balanced by the stationary EMF convergence. However, the poleward- and equatorward-propagating wave trains excited by the orography are dissipated as they propagate away, leading to stationary EMF convergence maxima north and south of the orography, instead of at the latitude of the orography, as well as to stationary EMF divergence in the subtropics. Friction compensates for the residual of the torque and the stationary EMFs. - A stationary nonlinearity develops for larger mountains (
*H*= 10 m and higher), caused by the preferential deflection of the flow around the poleward flank of the orography. The nonlinearity becomes more prominent as the height of the orography is increased, and causes the primary wave source to shift from being south of the orography to being north of the orography for*H*= 500 m. - The response of the GCM to small mountains is similar to the stationary wave model, but the transient eddies appear to amplify the stationary nonlinearity, and its associated temperature anomalies, leading to a larger response and to larger stationary EMFs. Diabatic heating over the orography, induced by the Newtonian relaxation along constant
*σ*surfaces, damps the model’s response to orography. Whether this heating is physically realistic, or whether studies of orography should instead relax temperatures along constant pressure surfaces is an open question. - For larger mountains (
*H*≥ 1 km), the mountain torque and the stationary EMF increase more slowly than the*H*^{2}scaling suggested by linear theory, though the increasing meridional (as opposed to zonal) propagation of the stationary wave leads to enhanced stationary EMFs relative to the mountain torque. For*H*≤ 2 km the midlatitude jet shifts poleward, as both the stationary and transient EMFs push the jet poleward. For*H*> 2 km the transient EMFs push the jet equatorward, and balance the stationary EMFs, which always push the jet poleward. The cancellation of the stationary and transient EMFs means that the mountain torque is mostly balanced by the friction, causing the jet to decelerate in its core. For large enough orography, changes in wave-breaking characteristics caused by enhanced downstream baroclinicity may reinforce the jet shift. - The transient EMFs changes are caused by a competition between the jet widening upstream of the orography, which provides more space for meridional eddy propagation and hence leads to increased poleward transient EMFs upstream of the orography, and the slowdown of the eddies downstream of the orography, such that the transient EMF weakens downstream of the orography. The former effect wins out in the
*H*= 2 km case, while the latter effect wins out for*H*= 4 km. In the*H*= 4 km case the jet also narrows downstream of the orography, providing less room for meridional eddy propagation and further damping the transient EMFs.

These results have come in the idealized contexts of a stationary wave model and a dry GCM, but provide several general insights into the impact of orography on the atmosphere’s momentum budget, including how the poleward deflection of the flow promotes the development of a stationary nonlinearity, even for a mountain with maximum height as small as 10 m, and the complex changes in the propagation of transient eddies in the presence of large mountains. Future extensions to this work could explore how the models’ responses are affected by moving the orography away from the latitude of maximum wind speeds, the sensitivity of the responses to the shape of the orography (e.g., comparing with meridional and zonal ridges) and how the responses change when the mean flow consists of a double jet (e.g., Son et al. 2009). Finally, a crucial step for connecting these results to the observed atmosphere is adding the effects of moisture (see Wills and Schneider 2018).

From a zonal-mean perspective, the nonacceleration theorem is the starting point for thinking about the atmosphere’s response to orography, but a complete theory requires accounting for a number of other factors, including friction, transients and interactions between eddy-driven jets and stationary waves. Systematically investigating how these factors combine to determine the response to orography across a hierarchy of models of different complexity, for a wide range of mountain heights, is essential for deepening our understanding of large-scale orography’s role in shaping the observed circulation of the atmosphere, and of orography’s role in past and future climates.

## Acknowledgments

I thank Isaac Held for suggestions and comments on early stages of this project, and two reviewers for thorough readings and helpful comments.

## APPENDIX

### Plumb Fluxes

*θ*denoting potential temperature and all other symbols having the same meaning as in the main text.

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^{1}

White (1986) investigated a nonlinear extension of the nonacceleration theorem to a quasigeostrophic system.

^{2}

Note that for transient eddies, the nonacceleration theorem says that the transient eddy momentum flux convergence balances the form drag exerted by one layer on another.

^{3}

Note that in the stationary wave model the friction is the only other term in the momentum budget, so it balances the residual of the torque and the stationary EMF convergence.

^{4}

That is, the changes in the hemisphere with the mountain are comparable to the changes in the hemisphere without the mountain.

^{5}

The maxima are plotted rather than the meridional integrals because the stationary EMFs decelerate the flow at low latitudes, where the torque is zero.

^{6}

Uncertainties were estimated by calculating the AWB-to-CWB ratios for the first and second halves of the simulations. For example, the first half of the control simulation gave a ratio of 1.76:1, the second half gave a ratio of 1.80, and using all the data gave 1.78:1.