1. Introduction
Midlatitude eddies transfer momentum and energy toward the poles, modifying the atmospheric mean flow. Eddy momentum flux convergence determines the location of the jet stream, and eddy energy flux has been used to define the location of storm tracks, which influence midlatitude weather (Chang et al. 2002; Shaw et al. 2016; Barpanda and Shaw 2017). Understanding the response of jets and storm tracks to climate change has been proposed as one of the grand challenges of climate science (Bony et al. 2015).
Climate models generally predict a poleward shift of the annual-mean zonal-mean storm tracks in response to increased greenhouse gas emissions (Yin 2005; Chang et al. 2012; Shaw et al. 2016). The zonal-mean eddy-driven jet stream in CMIP5 models, as indicated by the location of the surface westerlies, also generally exhibits a poleward shift, although there are regional exceptions and a large model spread (Barnes and Polvani 2013; Simpson et al. 2014; Vallis et al. 2015). While several mechanisms have been proposed to explain the poleward jet shift, there is currently no complete theory that predicts the jet response and explains the model spread (see review in Vallis et al. 2015). Furthermore, there does not exist a theory that has been used to connect the eddy-driven jet and storm-track shifts.
Since the greenhouse gas concentration affects the energy budget directly, a natural starting point for explaining the circulation shift in response to increased CO2 is the energy budget. When one defines the storm track energetically—for example, using dry static energy fluxes—then one can attempt to explain the poleward shift using energetic arguments. For example, arguments related to Clausius–Clapeyron scaling of water vapor and energy flux compensation have been put forward to explain the storm-track shift (Held and Soden 2006; Frierson et al. 2007; Shaw and Voigt 2016). A connection between the eddy energy and momentum flux would then allow for a prediction of the eddy-driven jet shift. In the literature, an alternative approach to studying the response of the midlatitude circulation to climate change has been to impose energetic perturbations in the form of diabatic heating perturbations in dry dynamical core models (e.g., Lorenz and DeWeaver 2007; Butler et al. 2010; Tandon et al. 2013; Lu et al. 2014). Representing the thermal response to increased CO2 using an imposed diabatic heating is problematic because it assumes the thermal response depends only on the radiative effect of increased CO2, yet the atmospheric thermal response is influenced by eddy energy and momentum fluxes. Furthermore, this approach cannot be used to connect the storm-track and eddy-driven jet responses.
Here we use a relationship derived by Eliassen and Palm (1961; herein called the EP relation) to infer the eddy momentum flux and eddy-driven jet response to increased CO2 concentration given the eddy energy flux response. The EP relation states that eddy potential energy flux on pressure surfaces is equal to the eddy momentum flux times the Doppler-shifted phase speed (i.e., the eddy phase speed minus the zonal-mean zonal wind) assuming small-amplitude plane waves and neglecting nonconservative effects (see section 2 for the full derivation). According to the EP relation, the response of the eddy momentum flux to climate change is a function of the response of the eddy potential energy flux, the zonal-mean zonal wind, and the eddy phase speed.
Here, we examine the EP relation and assess the extent to which it can be employed to connect the eddy-driven jet and storm-track responses (indicated by the eddy momentum flux convergence and eddy potential energy flux, respectively) to climate change. We review the derivation of the EP relation and examine its validity for two idealized general circulation model (GCM) climate change simulations. The idealized GCMs include two different radiation schemes. The first radiation scheme is the sophisticated Rapid Radiative Transfer Model for GCMs (RRTMG; Iacono et al. 2008), and the other radiation scheme is the idealized gray radiation (GR) scheme (Frierson et al. 2006). We find that the EP relation approximately holds for the climatology and response to climate change in both models. The vertically integrated eddy potential energy flux shifts poleward in response to climate change in both models, whereas the vertically integrated eddy momentum flux convergence shifts poleward in the RRTMG model and equatorward in the GR model. Dwyer and O’Gorman (2017) and Schneider et al. (2010) previously noted the equatorward shift of the eddy-driven jet using GR. The EP relation is used here to explain the opposite shifts of the eddy momentum flux convergence in the two experiments.
In section 2, we derive the EP relation for plane waves, including all nonlinear and nonconservative terms. A derivation of the EP relation on model (sigma) surfaces is given in the appendix. The physical interpretation of the EP relation and the role of the eddy potential energy flux in the total atmospheric energy flux are also discussed in section 2. In section 3, we describe the idealized model experiments and the analysis methods. The results are described in section 4, including 1) the EP relation and its response to climate change in the model simulations; 2) the response of the eddy momentum flux and eddy-driven jet stream to climate change and the extent to which the EP relation succeeds in estimating it; and 3) the role of the Doppler-shifted phase speed in the eddy-driven jet shift in response to climate change. A summary and discussion of the results are presented in section 5.
2. The EP relation
In this section we derive the EP relation, which connects the eddy potential energy flux and the eddy momentum flux. We discuss the physical interpretation of the EP relation and the role of the eddy potential energy flux in the total atmospheric energy flux. We argue that the EP relation can be used to connect the eddy-driven jet stream and storm-track responses to climate change.
a. The EP relation for plane waves


b. Physical interpretation of the EP relation and its connection to the energy budget









c. Using the EP relation to connect the eddy-driven jet and storm track
The meridional component of the EP relation (4) can be used to connect the eddy-driven jet stream (defined by the eddy momentum flux convergence) and storm-track responses to climate change, assuming 1) the eddy potential energy flux response to climate change shifts in the same direction as the storm track defined by the dry static energy flux and 2) the response of the Doppler-shifted phase speed to climate change is negligible. We find that both assumptions are satisfied in the idealized climate change simulations examined here (see section 4).
3. Data and methods
a. Idealized model simulations
We use an idealized aquaplanet GCM [described in Frierson et al. (2006) and O’Gorman and Schneider (2008)]. All the simulations use a spectral dynamical core with T42 resolution and 30 vertical levels, but with different radiative schemes. The solar insolation is in the same equinox configuration as in Frierson et al. (2006). The ocean is represented by a slab ocean with a mixed layer depth of 30 m and no heat transport. The model is run for 18 model years, and the results are averaged over the last 12 years and both hemispheres. The analysis is based on 6-hourly model output.
We made use of two different radiation schemes. The first radiation scheme called GR is a two-stream gray radiation scheme with a specified longwave optical thickness and no water vapor radiative effects (Frierson et al. 2006). The GR experiment uses the model setup and parameters as in O’Gorman and Schneider (2008), except for the ocean mixed layer depth. Climate change is simulated by increasing the optical depth parameter α, which controls the absorption of longwave radiation in the atmosphere; α is set to 1 in the control simulation and 2.5 in the climate change simulation. The mean surface temperature is 289.1 K in the control simulation and 303.9 K in the climate change simulation. Dwyer and O’Gorman (2017) previously noted the eddy-driven jet shifts equatorward in response to increased longwave optical depth in the GR model, which is opposite to the shift of the eddy heat flux.
The second radiation scheme called RRTMG is a sophisticated radiation scheme based on a correlated k-distribution band model (Iacono et al. 2000, 2008). Climate change is simulated by increasing the CO2 mixing ratio from 355 ppmv in the control simulation to 5680 ppmv (16 times the control value) in the climate change simulation. The mixing ratios of other gases are uniform and prescribed as follows: O3 = 30 ppbv, CH4 = 1700 ppbv, and N2O = 320 ppbv. There are no chlorofluorocarbons (CFCs) and no stratospheric ozone layer in the RRTMG simulations. Water vapor is radiatively active and its concentration is determined by the hydrological cycle. The simulations use a clear sky radiation scheme with no clouds. The surface albedo is set to 0.25. The mean surface temperature is 291.1 K in the control simulation and 305.1 K in the climate change simulation. A more detailed discussion of the GR and RRTMG radiative schemes and a comprehensive investigation of the climate change response, including heating rates, will be presented in a separate study (Z. Tan et al. 2018, unpublished manuscript).
Here we use the GR and RRTMG simulations to examine the EP relation and its ability to connect the shift of the eddy-driven jet and eddy potential energy flux. The GR and RRTMG simulations represent qualitatively different climate change experiments, as evident from the different climatologies and responses of the zonal-mean flow to climate change. In the GR experiment there is amplified warming aloft in the tropics and at the surface in high latitudes (Fig. 1a), consistent with coupled climate models (Vallis et al. 2015). However, the temperature response in the stratosphere does not show the expected cooling, except between 30° and 60° latitude. The temperature response to climate change in the RRTMG experiment exhibits the typical tropospheric warming and stratospheric cooling (Fig. 1b). The absence of significant stratospheric cooling in GR shows that increased optical depth in the GR model and increased greenhouse gas concentration in RRTMG have different impacts on the temperature. Further discussion of the temperature response using different radiative schemes will be given in Z. Tan et al. (2018, unpublished manuscript).
(a),(b) Zonal-mean temperature (K) and (c),(d) zonal-mean zonal wind (m s−1) for the (left) GR and (right) RRTMG experiments. Contours show the climatology (control simulation) values and color shading shows the response to climate change. Positive (negative) values are denoted by solid (dashed) contours and the zero contour is omitted. The contour interval is 10 K for the temperature and 5 m s−1 for the zonal wind.
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0792.1
The climatological zonal-mean zonal wind in the GR experiment (Fig. 1c) consists of a strong stratospheric jet with zonal winds increasing toward the top of the atmosphere. The stratospheric jet is connected to a tropospheric subtropical jet, around latitude 25° and 250 hPa (indicated by a maximum of the zonal-mean zonal wind as a function of latitude). A weaker tropospheric eddy-driven jet exists around 50° latitude (indicated by a maximum of the surface westerlies). The GR zonal-mean zonal wind profile is somewhat unrealistic, since the stratospheric jet in the real atmosphere is not connected to the subtropical jet. In response to climate change the stratospheric jet weakens and shifts equatorward and the tropospheric eddy-driven jet shifts equatorward (Fig. 1c). Since the temperature and the zonal-mean zonal wind are connected according to the thermal wind balance, the weakening of the stratospheric jet in the GR experiment is related to the warming in the stratospheric high latitudes, which is opposite to the response to increased greenhouse gas concentration in coupled climate models (Vallis et al. 2015). The RRTMG experiment has a more realistic zonal-mean zonal wind profile (Fig. 1d), with a single jet concentrated around 30° latitude in the upper troposphere. In response to climate change the stratospheric jet strengthens and the tropospheric (eddy driven) jet shifts poleward.
The poleward jet shift in response to increased CO2 concentration using the RRTMG scheme and the equatorward jet shift in response to increased optical thickness using the GR scheme are robust across a wide range of CO2 and optical thickness values (Z. Tan et al. 2018, unpublished manuscript). The relation between the eddy-driven jet shift and the stratospheric jet response in the GR and RRTMG experiments is consistent with previous studies, which showed that an enhanced (weakened) stratospheric jet leads to a poleward (equatorward) shift of the tropospheric eddy-driven jet (see Kidston et al. 2015 and references therein). In section 4, we analyze the EP relation in each experiment and examine its ability to connect the eddy momentum and energy flux responses to climate change for the different responses to increased greenhouse gases simulated by the different radiation schemes.
b. Cospectra
We use the phase speed cospectrum of the eddy momentum flux as in Randel and Held (1991). In particular, we multiply the eddy momentum flux cospectrum by
c. Calculating shifts of the eddy-driven jet and storm track
We use the latitude of maximum mass-weighted vertically integrated eddy momentum flux convergence3 and eddy potential energy flux as indicators of the latitude of the eddy-driven jet and storm track, respectively. The eddy-driven jet latitude is measured by the vertically integrated eddy momentum flux convergence because the zonal-mean surface zonal wind (
4. Results
We start by examining the extent to which the EP relation (4) holds for the simulations described in section 3, using the GR and RRTMG radiation schemes. Next we use the EP relation to estimate the eddy momentum flux response to climate change, given the eddy potential energy flux and the Doppler-shifted phase speed responses. We show that the EP relation can be used to estimate the eddy-driven jet shift in response to climate change, given the response of the eddy potential energy flux and the climatological zonal-mean zonal wind and eddy phase speed, because the response of the Doppler-shifted phase speed is negligible. Finally, we examine the relative roles of the climatological zonal-mean zonal wind and the eddy phase speed in determining the eddy momentum flux and eddy-driven jet responses.
a. The EP relation and its response to climate change
The climatological
The terms in the EP relation (4) for the (top) GR and (bottom) RRTMG experiments: (a),(c)
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0792.1
Next we examine the mass-weighted vertical integral of the terms in the EP relation (see section 2 for the definition of the vertical integral operator). While the magnitudes of minus the vertically integrated eddy potential energy flux and eddy momentum flux multiplied by the Doppler-shifted wind, that is,
The vertically integrated EP relation terms for the (top) GR and (bottom) RRTMG experiments: (a),(c)
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0792.1


In all the simulations the sum of the residual terms explains the discrepancy between
The vertically integrated residual terms in the EP relation [right hand side of (3)] for (a),(c) the control simulations and (b),(d) the climate change simulations of the (top) GR and (bottom) RRTMG experiments. The different terms are denoted in the legend by the components of L′ substituted into (3) [see (9) for the different components of L′, the residual term in the wave zonal momentum equation]. The residual terms were set to zero equatorward of 5° latitude to avoid nonphysical values due to the term
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0792.1
The above results show that the EP relation (4) is approximately valid in our simulations. Next we examine the response of the EP relation to climate change. In both experiments the EP flux terms shift poleward in response to climate change (Table 1). In the GR experiment there is also a reduction of both terms on their poleward flanks around 50° latitude (Figs. 3a,b). In the RRTMG experiment
Latitudes of maximum vertically integrated energy flux terms (in degrees) for the control and climate change GR and RRTMG simulations, and the latitudinal shifts between the control and climate change simulations. Poleward shifts are marked by an upward arrow. The maximum vertically integrated eddy dry static energy flux
We examine the magnitude and direction of the latitudinal shift of the vertically integrated fluxes in order to assess the relation between them (see section 3 for the definition of the latitudinal shift). The shifts of the EP relation terms and the storm track (eddy dry static energy flux) are summarized in Table 1. In both experiments the EP relation terms shift poleward; however, the shift of
b. Using the EP relation to connect the eddy momentum and energy flux response
The results of the previous subsection demonstrate that the EP relation (4), which connects the energy and momentum fluxes, approximately holds for the climatology and climate change simulations in the idealized model. In both GR and RRTMG simulations the climate change response of the eddy potential energy flux is similar to that of the Doppler-shifted phase speed times the eddy momentum flux, that is, a poleward shift. The poleward shift of the eddy potential energy flux is consistent with the poleward shift of the storm track, as indicated by the eddy dry static energy flux.
In the GR experiment the eddy-driven jet shifts equatorward in response to climate change, as indicated by the surface wind stress (Fig. 5a) and the vertically integrated eddy momentum flux convergence (Fig. 5b). The eddy momentum flux convergence and surface wind stress become negative across a larger range of latitudes poleward of the eddy-driven jet in the climate change simulation, which is typical for a transition from a double to a single upper-tropospheric jet state (Son and Lee 2005; Lachmy and Harnik 2016) as indeed occurs in this case. This is consistent with the eddy momentum flux becoming negative at high latitudes in response to climate change (Fig. 5c). Note that the maximum vertically integrated eddy momentum flux does not shift significantly in latitude (Fig. 5c). In the RRTMG experiment there is a consistent poleward shift of the surface wind stress, eddy momentum flux convergence, and eddy momentum flux, without a significant change in structure (Figs. 5d–f).
Eddy momentum flux and its convergence in the (top) GR and (bottom) RRTMG experiments: (a),(d) zonal-mean zonal surface wind stress, (b),(e) vertically integrated eddy momentum flux convergence, and (c),(f) vertically integrated eddy momentum flux. Blue and red curves are for the control and climate change simulations, respectively.
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0792.1

The vertical integral of the right-hand side of (10)
EP relation estimates for (a),(b),(d),(e) the vertically integrated eddy momentum flux convergence
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0792.1
Since the estimate of the eddy momentum flux based on the EP relation captures the structure of the eddy momentum flux well, we can use it to estimate the shift of the eddy-driven jet in response to climate change. Table 2 shows the latitudinal shift of the eddy-driven jet (i.e., the shift of the latitude of maximum vertically integrated eddy momentum flux convergence), surface westerlies, and surface wind stress in response to climate change and compares them with the estimates based on the EP relation. In the GR simulation the surface westerlies, the surface wind stress, and the vertically integrated eddy momentum flux convergence shift equatorward (by 8.4°, 8.0°, and 7.0°, respectively). The estimate of the eddy-driven jet shift based on the EP relation (the shift of
Latitude of maximum surface westerlies, surface wind stress, vertically integrated eddy momentum flux convergence, and estimates of the vertically integrated eddy momentum flux convergence based on the EP relation (in degrees) for the control and climate change GR and RRTMG simulations, and the latitudinal shifts between the control and climate change simulations. Equatorward and poleward shifts are marked by downward and upward arrows, respectively.
In summary, the response to climate change in the RRTMG experiment shows a consistent poleward shift of the storm track, eddy potential energy flux, eddy momentum flux convergence, and eddy-driven jet. In contrast, the storm-track and eddy potential energy flux shift in the opposite direction to the eddy-driven jet and eddy momentum flux convergence in the GR experiment. In both experiments the EP relation captures approximately 77% of the magnitude of the eddy-driven jet shift. Since the eddy potential energy flux shift follows the storm track in the simulations presented here, an analysis of the EP relation can reveal the conditions that allow for the opposite shifts of the storm track and eddy-driven jet in the GR experiment and the role of the Doppler-shifted phase speed, as shown in the next subsection.
c. The role of the Doppler-shifted phase speed for the eddy-driven jet response
According to the EP relation (10) the opposite shift of the storm track and eddy-driven jet in response to climate change using the GR radiation scheme could be due to 1) dividing the eddy potential energy flux response by the climatological Doppler-shifted phase speed or 2) the response of the Doppler-shifted phase speed. In this subsection we examine whether the climatological Doppler-shifted phase speed is sufficient for reproducing the eddy-driven jet shift in response to climate change given the eddy potential energy flux response. In addition, we examine the contributions from the zonal-mean zonal wind and the eddy phase speed to the EP relation response. We focus mostly on the GR experiment in order to explain the opposite shifts of the storm track and eddy-driven jet.
Using the eddy momentum flux convergence estimate based on the EP relation,
The relative roles of the zonal-mean zonal wind and the eddy phase speed in the EP relation response to climate change can be demonstrated by looking at
The vertically integrated components of the right-hand side of the EP relation (4): (a),(c)
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0792.1
The role of the climatological zonal-mean zonal wind in the EP relation response can be demonstrated by looking at the mixed product
The above results demonstrate that the eddy momentum flux convergence response to climate change, and thus the eddy-driven jet response, can be estimated in both the GR and RRTMG experiments using the EP relation given the eddy potential energy flux response because the response of the Doppler-shifted phase speed is negligible. Thus, according to the EP relation, the response of the Doppler-shifted phase speed is not important for the eddy momentum flux response. According to the EP relation, the eddy-driven jet and storm-track shifts in response to climate change have opposite signs in the GR simulation because the storm-track response (eddy potential energy flux response) must be divided by the climatological Doppler-shifted phase speed to infer the eddy-driven jet response.
5. Summary and conclusions
Eddy energy and momentum fluxes dominate the extratropical circulation and can be used to define the location of the storm track and the eddy-driven jet stream. Eddy energy and momentum fluxes are connected according to the EP relation (4), which states that the eddy potential energy flux is equal to the eddy momentum flux times the Doppler-shifted eddy phase speed. The EP relation assumes linear plane waves in the absence of nonconservative processes. According to the EP relation, the response of the eddy-driven jet to climate change can be connected to the storm-track response assuming 1) the storm-track shift is consistent with the eddy potential energy flux and 2) the response of the Doppler-shifted phase speed is negligible. Here we examine the validity of the EP relation and the extent to which it can be used to estimate the eddy-driven jet response to climate change in idealized GCM simulations of the response to increased greenhouse gases.
We use two idealized GCM simulations, which differ in their radiation schemes. The RRTMG simulation uses a sophisticated radiation scheme and the GR simulation uses a gray radiation scheme. The eddy-driven jet shifts poleward in response to climate change in the RRTMG experiment and equatorward in the GR experiment. In contrast, the storm track, as measured by the eddy dry static energy flux, shifts poleward using both radiation schemes. Eddy potential energy flux also shifts poleward using both radiation schemes, consistent with the storm-track shift.
The main conclusions are as follows:
The EP relation (4) approximately holds in the idealized GCM for the climatology and response to climate change. The residual term in the EP relation is dominated by boundary layer drag, nonlinear terms, and vertical advection of momentum. Despite the residual term, the response of the eddy potential energy flux to climate change is well captured by the response of the eddy momentum flux times the Doppler-shifted phase speed, both in terms of its shape and poleward shift.
The eddy momentum flux estimated from the EP relation (10) qualitatively captures the eddy momentum flux response to climate change in the GR and RRTMG experiments. The latitude of maximum vertically integrated eddy momentum flux convergence estimated from the EP relation also captures the shift of the eddy-driven jet in response to climate change. The EP relation captures 77% of the equatorward eddy-driven jet shift in the GR experiment and 76% of the poleward jet shift in the RRTMG experiment.
The EP relation can be used to connect the eddy-driven jet and storm-track shifts because the response of the Doppler-shifted phase speed is negligible. The EP relation estimate of the eddy-driven jet shift using the eddy potential energy flux response to climate change and the climatological zonal-mean zonal wind captures 89% of the shift in the GR experiment and 86% of the shift in the RRTMG experiment.
According to the EP relation, the opposite shift of the storm track and eddy-driven jet in response to climate change in the GR experiment is associated with dividing the storm-track response by the climatological Doppler-shifted phase speed, dominated by the zonal-mean zonal wind, which consists of a strong stratospheric jet with zonal winds increasing toward the top of the atmosphere.
The results presented here show that the eddy-driven jet shift in response to climate change can be connected to the storm-track response using the EP relation because 1) the storm track follows the eddy potential energy flux response and 2) the response of the Doppler-shifted phase speed is negligible.
Explaining the storm track and eddy-driven jet shift in response to climate change is one of the grand challenges of climate science. The general approach in the literature has been to focus on either the eddy-driven jet (e.g., Chen et al. 2008; Rivière 2011; Lorenz 2014; Lu et al. 2014) or storm-track (e.g., Lorenz and DeWeaver 2007; Lu et al. 2008; Butler et al. 2010; Mbengue and Schneider 2013) response to imposed diabatic perturbations. For the eddy-driven jet response, the starting point is typically a zonal-mean zonal wind perturbation, which is then connected to changes in mean potential vorticity gradient, index of refraction, Doppler-shifted phase speed, or eddy length scale. For the storm-track response, the focus is typically on how the imposed diabatic perturbation impacts Eady growth rate or available potential energy. In both cases imposing a diabatic perturbation is problematic because it assumes a connection between the radiative effect of increased CO2 and the thermal response, which is influenced by eddies. Furthermore, both approaches exclusively focus on either the eddy-driven jet or the storm track but not on their connection.
The EP relation provides, to our knowledge, the only relation connecting the storm-track and eddy-driven jet responses to climate perturbations. It can be used to either infer the eddy-driven jet response from the storm-track response or vice versa. Here we take an energetic approach and consider the storm-track (energetic) response as a starting point, because increased CO2 directly perturbs the energy budget, and use it to infer the eddy momentum flux and eddy-driven jet shift. The response of the energy budget to increased greenhouse gas concentration has been used to explain the storm-track shift. For example, Held and Soden (2006) argued that the dry static energy flux response to climate change can be predicted via its compensation for the increased latent energy flux following the Clausius–Clapeyron relation. Dry static energy flux, which peaks poleward of the latent heat flux, is expected to compensate for the increase in latent heat flux by decreasing on the equatorward side of its maximum, leading to a poleward shift of the dry static energy flux. If the eddy potential energy flux shifts in the same direction as the eddy dry static energy flux, this argument could be used to predict the poleward shift of the eddy potential energy flux in response to climate change. The results presented here support the assumed connection between the response of the eddy dry static energy flux and the eddy potential energy flux, but this assumption requires further investigation.
Conversely, if the climate change involves a mechanical forcing (changes in the surface torque), as occurs during the Last Glacial Maximum, then the EP relation could be used to connect the eddy-driven jet response to the storm-track response. Overall, these ideas highlight the usefulness of the EP relation in connecting the midlatitude energy and momentum flux response to climate change.
Acknowledgments
OL and TAS acknowledge support from NSF (AGS-1538944), and TAS is also supported by the David and Lucile Packard Foundation and the Alfred P. Sloan Foundation. The model simulations in this paper were completed with resources provided by the University of Chicago Research Computing Center. The authors thank Zhihong Tan for setting up the RRTMG radiation scheme in the idealized GCM and for helpful discussions, and two anonymous reviewers for useful comments that helped improve the manuscript.
APPENDIX
The EP Relation in Sigma Coordinates
The EP relation includes an eddy potential energy flux term, which depends on the choice of vertical coordinate. We are interested in the EP relation on pressure surfaces [(4)]; however, the idealized model solves the equations in sigma coordinates, defined as σ ≡ p/ps. There are two options for calculating the EP relation from the model output: 1) interpolating all the data to pressure surfaces, which introduces a source of inaccuracy and an additional computational cost, and 2) using the EP relation in sigma coordinates and adding a correction term to the eddy potential energy flux, so that (4) is satisfied, as explained below. In this study we used the second method. We will show below that the two methods give similar results.


To demonstrate the importance of the correction to the eddy potential energy flux on σ surfaces we examine the different terms in (A5) for year 18 of the control RRTMG simulation. The eddy potential energy flux on pressure levels was calculated by interpolating Φ and υ from the full model levels to pressure levels.5 Terms
Vertically integrated potential energy flux terms in pressure and sigma coordinates for year 18 of the RRTMG control simulation. (a) Minus the eddy potential energy flux on pressure levels (blue) and minus the corrected eddy potential energy flux on σ levels (red; see text). (b) Minus the eddy potential energy flux on σ levels (blue), minus the correction term on σ levels (red), and the difference between mean flow potential energy flux on pressure and σ levels (yellow).
Citation: Journal of Climate 31, 18; 10.1175/JCLI-D-17-0792.1





The sensitivity of the eddy-mean flow decomposition to the choice of vertical coordinates was also noted by Trenberth et al. (1993) in the context of time averaging on model surfaces. Another example of this sensitivity is the fact that all the heat flux is included in the mean flow component in isentropic coordinates (Vallis 2006).
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In the original Eliassen–Palm relation [Eq. (10.5) from Eliassen and Palm (1961)], there is an additional heat flux term, which would be equivalent to adding
The vertical integral operator is defined as
We denote the vertically integrated eddy momentum flux convergence by
Unphysical values of
The interpolation from full model levels to pressure levels was done for each longitude, latitude, level, and time grid point, using the NCL code “int2p” with log interpolation and no extrapolation.