## 1. Introduction

The representation of the interactions between large-scale tropical circulations and local convective processes is an issue of fundamental importance to the simulation of tropical circulations. Computational constraints mean that models that explicitly simulate both convection and large-scale motion over domains of appropriate size can rarely be used. Hence, a strategy, often adopted in the development of convection parameterizations for the tropics for GCMs, is to carry out studies using cloud-resolving models (CRMs) that explicitly model convection in conjunction with a parameterization of the large-scale dynamics that is influenced by local conditions. The weak temperature gradient (WTG) approximation (Sobel and Bretherton 2000; Raymond and Zeng 2005) is one such parameterization.

The atmospheric circulation can be analyzed from the perspective of a generalized Carnot heat engine that converts a temperature difference between two reservoirs into mechanical energy [e.g., Emanuel (1986) for the case of tropical cyclones]. It has been shown by Pauluis (2011) that the impact of the hydrological cycle is to reduce the efficiency of such a conversion relative to the generalized Carnot maximum. This paper addresses convective and large-scale circulations as heat engines, unpacking how efficient they are at transforming potential work into kinetic energy dissipation and water lifting, in the face of the thermodynamic penalties associated with irreversible phase transitions in a moist atmosphere. Hitherto, similar analyses have not been applied in a context where the influence of large-scale circulations on local convection is modeled separately. Such an approach has the potential not only to clarify the consequences of different large-scale parameterization approaches and to identify constraints on them but also to shed light on the nature of the interaction between the large-scale and the local dynamics. In particular, we examine whether the mechanical energy budget is well behaved when two convecting regions are coupled by a parameterized large-scale circulation.

The WTG approximation was first suggested by Sobel and Bretherton (2000) and subsequently modified by introducing a short relaxation time (e.g., Raymond and Zeng 2005). Alternative methods such as the damped gravity wave (DGW) approach (Kuang 2008; Romps 2012), a spectral variant of the WTG (Herman and Raymond 2014), and others have been proposed. In this paper, we focus on the weak temperature gradient approximation, but work on the DGW approach is ongoing. Preliminary analysis suggests that the DGW approach would give similar results to those presented in this paper.

A recent project has compared the results produced by different implementations of some of these methods (Daleu et al. 2015b, 2016), suggesting that the WTG approach is more prone to multiple equilibria than the DGW approach and that the latter produces smoother large-scale vertical velocities. One of the advantages of the WTG approach is that its simplicity permits an analytical approach, and for that reason, it is used here.

*τ*a relaxation time scale, and

*z*the height above mean sea level;

*τ*represents the strength of the coupling between regions, and typical values of 2–3 h can be justified as the spatial scale of the system being modeled (~500 km) divided by the speed of the fastest internal gravity waves (~50 m s

^{−1}).

This circulation can be superimposed on the normal convective motions representing an idealized large-scale circulation that provides a linkage between two regions. Most studies have used a reference-column approach, whereby the large-scale circulation is determined by temperature differences between the area modeled and an assumed environmental profile; more recently, Daleu et al. (2012, 2015a) extended this approach to two coupled regions, which enables a more explicit representation of the consequences of the large-scale coupling mechanism. The reference column approach implies an infinite reservoir of energy and entropy, whereas by using coupled regions, it is possible to analyze the mechanical energy budget of a closed system, separately attributing the contribution of the large-scale circulation described above and of the residual convective motions in the two regions.

*ρ*is the density of dry air and

*w*is vertical velocity, is calculated in

*T*, vapor mixing ratio

*S*, and buoyancy

*B*can be obtained. See Pauluis and Mrowiec (2013) for a further discussion of the merits of this approach.

*g*represents the acceleration due to gravity, and

The terms in this equation can be represented by thermodynamic diagrams of streamfunction contours in the appropriate spaces, for example, that for

Thermodynamic diagrams for the main terms in Eq. (3) for a single uncoupled region at RCE over a surface at temperature 302.7 K showing 15 equally spaced contours of the streamfunction. (Numerical values elsewhere in this paper were calculated using 60 contours.)

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Thermodynamic diagrams for the main terms in Eq. (3) for a single uncoupled region at RCE over a surface at temperature 302.7 K showing 15 equally spaced contours of the streamfunction. (Numerical values elsewhere in this paper were calculated using 60 contours.)

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Thermodynamic diagrams for the main terms in Eq. (3) for a single uncoupled region at RCE over a surface at temperature 302.7 K showing 15 equally spaced contours of the streamfunction. (Numerical values elsewhere in this paper were calculated using 60 contours.)

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

This paper applies the diagnostic approach outlined above to unpack the generation of mechanical work by a modeled overturning circulation superimposed on convective processes. In particular, it analyses the relationship between the strength of WTG coupling and the strength of the circulation, partitioned into contributions from the large-scale circulation and convective circulations in relatively cool/dry and warm/moist regions, in order to deepen understanding of how the large-scale circulation can suppress or enhance convection. The impact of varying the SST difference between the regions is also analyzed.

The paper is organized as follows. Section 2 explains how an isentropic analysis can be applied to two coupled regions and also suggests a refinement that improves the accuracy of the decomposition for an anelastic model. Section 3 briefly describes the CRM used and presents numerical results for the mechanical energy budget, partitioned into a large-scale circulation and convective localized flows for varying strengths of coupling as well as further results for varying SST differences between the two regions. Section 4 develops an analytical expression for the generation of kinetic energy by the WTG coupling, which is seen to be consistent with results obtained from the CRM. Scaling arguments are also developed for comparison with other results obtained by the CRM. The implications of this study are discussed and some conclusions are drawn in section 5.

## 2. Isentropic diagnostic framework

### a. General approach

Pauluis (2016) uses a weighted average of equally spaced contours of the streamfunction over the region where

### b. Extension to two-region case

This paper applies the decomposition of work done discussed above to a two-region system, where the regions are coupled by a large-scale circulation consisting of vertical winds specified by the WTG approximation. The cooler region is identified by a subscript 1 and represents a fraction *ε* (all the numerical results shown in this paper are based on

One could, of course, merge the two regions prior to performing any analysis, but this introduces significant error as the dynamics are generally influenced by local rather than merged values of fields (e.g., buoyancy for

*i*= 1, 2 in each region will now be closed, as will that in a notional third “region” that consists of the total of the large-scale circulation within the two regions, weighted by their relative areas, since

(a),(b) Contours of the integrated vertical mass flux for each of the two regions analyzed in isolation. The regions are run to RCE and are coupled with *τ* = 2 h. Note that the contours are not closed and so cannot be represented by a streamfunction. (c) The streamfunction of the total system. (f) Once the large-scale circulation is treated separately, (d),(e) the residual “convective” circulations in the two regions exhibit closed contours. The same values for the contours are used in each of the plots; here, solid lines indicate negative values and dashed lines positive values.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

(a),(b) Contours of the integrated vertical mass flux for each of the two regions analyzed in isolation. The regions are run to RCE and are coupled with *τ* = 2 h. Note that the contours are not closed and so cannot be represented by a streamfunction. (c) The streamfunction of the total system. (f) Once the large-scale circulation is treated separately, (d),(e) the residual “convective” circulations in the two regions exhibit closed contours. The same values for the contours are used in each of the plots; here, solid lines indicate negative values and dashed lines positive values.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

(a),(b) Contours of the integrated vertical mass flux for each of the two regions analyzed in isolation. The regions are run to RCE and are coupled with *τ* = 2 h. Note that the contours are not closed and so cannot be represented by a streamfunction. (c) The streamfunction of the total system. (f) Once the large-scale circulation is treated separately, (d),(e) the residual “convective” circulations in the two regions exhibit closed contours. The same values for the contours are used in each of the plots; here, solid lines indicate negative values and dashed lines positive values.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

As has been mentioned, these integrals are calculated for the entire system using weighted trajectories along contours of the streamfunction in

### c. Anelastic case

The derivation of Eq. (3) uses the hydrostatic approximation for the reference profile, where the total pressure of all gases in the atmosphere (i.e., dry air and water vapor) is related to height. In the case of an anelastic model, such as the Met Office’s large-eddy model (LEM), which is used to derive the numerical results in this paper, it is the partial pressure of dry air that is assumed to be in hydrostatic equilibrium in the reference profile, and as will be seen, the balance of Eq. (3) can be improved by a small correction that is conveniently included in the water vapor component of the Gibbs penalty term and is derived in the appendix.

## 3. Model description and results

### a. Model setup

The numerical simulations in this section have been produced using a configuration of version 2.4 of the Met Office’s LEM in cloud-resolving mode amended to investigate the effect of coupling two regions via the WTG in an idealized context. The underlying LEM is described in Gray et al. (2004), and more complete details of the configuration used here can be found in Daleu et al. (2012). The microphysics are represented by a five-category prognostic scheme, with mixing ratios for cloud water, rain, ice, graupel, and snow and with number concentrations for ice, graupel, and snow. Each region is modeled in a two-dimensional configuration with a width of 128 km (resolution of 500 m) and a height of 20 km (60 vertical levels). To isolate the effect of the coupling strength, a fixed tropospheric cooling profile is used rather than interactive radiation. The cooling rate is fixed at 1.5 K day^{−1} below 220 hPa and decreases linearly with pressure to 0 K day^{−1} at 120 hPa, corresponding to integrated atmospheric cooling close to 150 Wm^{−2}, depending on the precise value of the surface pressure. The subgrid model is parameterized using a modified first-order Smagorinsky–Lilly approach. For consistency with previous studies with this model, a mixing length of 250 m was used, but the impact of this choice on the conclusions of this paper was tested and found to be minor.

The WTG velocity is specified by the region-mean potential temperature difference between the two regions and is used to perform advection of both temperature and moisture between the two regions. The two regions are of equal size, and the temperature difference at the surface between them

The LEM includes a considerable number of approximations, particularly with respect to the thermodynamics of the system. For example, the specific heat capacities of liquid water and water vapor are taken to be equal (and effectively zero), and the specific enthalpy of water vapor

### b. Results for single region

The model was first run for an uncoupled single region over a 302.7-K sea surface temperature. The resulting thermodynamic diagrams (Fig. 2) have already been mentioned. Values for the work terms can be seen in Table 1, in which the effect of the modification described in the appendix is also identified. For comparison, the reference case values calculated in Pauluis (2016; Table 1 herein) are also shown.

Numerical values (W m^{−2}) for the terms of Eq. (3) for the control case of RCE in a single region:

It will be seen that the values obtained here are in general some 30% higher than those in Pauluis (2016). Comparison of the

### c. Dependency of components of work done on strength of coupling

To investigate the impact of the coupling on the generation of mechanical work in the two regions, simulations were performed with values of the WTG time-scale *τ* between 1 and 50 h; results for two uncoupled regions are also shown for comparison

Components of Eq. (3) for (a) the total system, (b) the large-scale circulation, and the convective circulations in the (c) cooler and (d) warmer regions for varying coupling strengths. A high value of the coupling time-scale parameter *τ* indicates weak coupling. The blue line indicates the buoyancy work done

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Components of Eq. (3) for (a) the total system, (b) the large-scale circulation, and the convective circulations in the (c) cooler and (d) warmer regions for varying coupling strengths. A high value of the coupling time-scale parameter *τ* indicates weak coupling. The blue line indicates the buoyancy work done

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Components of Eq. (3) for (a) the total system, (b) the large-scale circulation, and the convective circulations in the (c) cooler and (d) warmer regions for varying coupling strengths. A high value of the coupling time-scale parameter *τ* indicates weak coupling. The blue line indicates the buoyancy work done

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Values for the local circulations in each of the regions (Figs. 4c,d) on the other hand exhibit a significant difference from the overall mean. This difference is already marked at very weak coupling and increases as the value of *τ* decreases, although the values for the cooler region remain almost constant for *τ* < 10 h, possibly reflecting the fact that the region approaches a humidity minimum (Fig. 5). Column-integrated water vapor in that region decreases significantly between *τ* = 50 and 10 h and then shows less variation, indeed a slight increase for *τ* < 5 h. As the cooler region dries, the relative importance of the Gibbs penalty component will increase. Weaker convection is associated with a disproportionate decrease in the buoyancy component, as discussed in Pauluis (2016). Figure 6 shows the vertical mass flux for the two regions for *τ* = 50 h, which confirms that convection in the cooler region is markedly suppressed even for such weak coupling.

Integrated column water vapor in the cool region (blue) and warm region (red) for varying values of WTG coupling relaxation time scale *τ* for an SST contrast of 2 K.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Integrated column water vapor in the cool region (blue) and warm region (red) for varying values of WTG coupling relaxation time scale *τ* for an SST contrast of 2 K.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Integrated column water vapor in the cool region (blue) and warm region (red) for varying values of WTG coupling relaxation time scale *τ* for an SST contrast of 2 K.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Isentropic distribution of the vertical mass flux *τ* = 50 h. The solid curve represents the mean profile of the equivalent potential temperature

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Isentropic distribution of the vertical mass flux *τ* = 50 h. The solid curve represents the mean profile of the equivalent potential temperature

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Isentropic distribution of the vertical mass flux *τ* = 50 h. The solid curve represents the mean profile of the equivalent potential temperature

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

The large-scale circulation (Fig. 4b) shows a different pattern; this includes both regions, which maintain a temperature difference, and hence, *τ* = 5–10 h before decreasing; this can be attributed to increasing coupling first leading to drying of the cooler region and hence to a decrease in the relative humidity at which energy enters the system (see the scaling arguments below for how this impacts *τ* decreases below 5 h, the cooler region marginally moistens, which contributes to the increase in

### d. Components of kinetic energy

The relationship between the strength of the coupling and components of kinetic energy in the total system is shown in Fig. 7. The coupling has an insignificant effect on the kinetic energy in the aggregate system but a strong influence on the vertical kinetic energy associated with the large-scale circulation, which increases with coupling strength. The increase in vertical kinetic energy for smaller *τ* indicates that the effect of reducing *τ* in increasing *w* in Eq. (1) is stronger than the effect that enhanced coupling has on reducing the temperature contrast

Components of kinetic energy. The blue line represents the horizontal component and the green line the vertical component for the aggregate system. The contribution of the large-scale motion to the vertical kinetic energy is shown in black.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Components of kinetic energy. The blue line represents the horizontal component and the green line the vertical component for the aggregate system. The contribution of the large-scale motion to the vertical kinetic energy is shown in black.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Components of kinetic energy. The blue line represents the horizontal component and the green line the vertical component for the aggregate system. The contribution of the large-scale motion to the vertical kinetic energy is shown in black.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

A similar comparison between large-scale and convective horizontal components of kinetic energy would require further assumptions as to the distance between the two regions and their geometry. From a thermodynamic perspective, horizontal motion is of less relevance to the generation of mechanical energy, as it traverses only a small temperature contrast (as forced by the coupling). It is true to say that the horizontal near-surface flow will likely contribute substantially to the total frictional dissipation in the real system, but in our modeled system, this dissipation is constrained to occur in the two regions.

This comparison between large-scale and convective vertical kinetic energies indicates how weak the large-scale circulation is when compared with convective motions. Despite this, the introduction of a large-scale circulation through WTG coupling has very noticeable effects on the strength and nature of the convection within the two regions. In the absence of this large-scale circulation, the mechanical energy budgets in the two regions are very similar. However, even with a very weak large-scale overturning circulation, convection in the warm region is markedly enhanced and that in the cool region suppressed.

### e. Dependence on SST difference between the regions

A further set of numerical experiments was performed for strong coupling (*τ* = 2 h), where the temperature difference between the two regions was varied between 0 and 2 K. In each experiment, the regions were of equal area and the mean SST of the aggregate system was 303.7 K. (Plots of the components of the mechanical budget are shown in Fig. 9.) As in the experiment with varying coupling strength, the values of the four components of the mechanical work budget remain broadly constant for the system in aggregate, independent of the SST difference. The values for the cool and warm regions coincide for

## 4. Analytical expressions and scaling

### a. Energy conversion under the weak temperature gradient approximation

where *N* is the local Brunt–Väisälä frequency *Q* is the external heat input into the system. It is assumed that *N* does not have any dependency on time although it can vary with height.

*A*is equal to

*BQ*/

*N*

^{2}, which is balanced by a conversion term from potential to kinetic energy of

*τ*for those layers—the impact of this adjustment is minor.

Figure 8 compares the conversion term from Eq. (11) and the large-scale components of the mechanical energy budget as in Fig. 4b, which of course includes moist variables. The “mechanical work done” term

As in Fig. 4b, but also including the conversion rate from potential to kinetic energy based on Eq. (11) (violet).

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

As in Fig. 4b, but also including the conversion rate from potential to kinetic energy based on Eq. (11) (violet).

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

As in Fig. 4b, but also including the conversion rate from potential to kinetic energy based on Eq. (11) (violet).

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Components of Eq. (3) for (a) the total system, (b) the large-scale circulation, and the convective circulations in the (c) cooler and (d) warmer regions for varying values of

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Components of Eq. (3) for (a) the total system, (b) the large-scale circulation, and the convective circulations in the (c) cooler and (d) warmer regions for varying values of

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Components of Eq. (3) for (a) the total system, (b) the large-scale circulation, and the convective circulations in the (c) cooler and (d) warmer regions for varying values of

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Components of Eq. (3) for the large-scale circulation (as shown in Fig. 9, but plotted on a log–log scale) for varying values of

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Components of Eq. (3) for the large-scale circulation (as shown in Fig. 9, but plotted on a log–log scale) for varying values of

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Components of Eq. (3) for the large-scale circulation (as shown in Fig. 9, but plotted on a log–log scale) for varying values of

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

### b. Scaling for components of work done

*E*between Eqs. (13) and (16),

where

Combining these various elements as in Eq. (3), one can expect the efficiency with which such a system in aggregate generates mechanical work

Comparison between the Gibbs penalty component of the large-scale circulation (red, left-hand axis) and latent heat transported between the regions (blue, right-hand axis) for (a) the numerical experiment where coupling strength is varied and (b) the numerical experiment where SST difference is varied.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Comparison between the Gibbs penalty component of the large-scale circulation (red, left-hand axis) and latent heat transported between the regions (blue, right-hand axis) for (a) the numerical experiment where coupling strength is varied and (b) the numerical experiment where SST difference is varied.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

Comparison between the Gibbs penalty component of the large-scale circulation (red, left-hand axis) and latent heat transported between the regions (blue, right-hand axis) for (a) the numerical experiment where coupling strength is varied and (b) the numerical experiment where SST difference is varied.

Citation: Journal of the Atmospheric Sciences 75, 8; 10.1175/JAS-D-17-0314.1

## 5. Summary and discussion

This paper shows how the mechanical energy budget for a convecting system proposed in Pauluis (2016) can be applied in the case of two coupled regions, permitting attribution of each of the components to either localized features within one of the two regions or to the large-scale circulation. It also proposes a refinement to the basic approach in the case of an anelastic cloud-resolving model. Our work has not identified any energetic inconsistencies introduced by the use of the WTG approximation as a proxy for large-scale circulation.

Numerical results were obtained using a CRM in which two regions are coupled using the WTG approximation. They demonstrate that the terms in the mechanical energy budget of the system considered as a whole are insensitive to the strength (indeed even the presence) of the coupling, indicating that the coupling does not distort the energy balance of the whole system. However, even very weak coupling has a marked effect on the nature of the convection within each region and its associated energy budget terms. As the strength of the coupling increases, the convection in the cool region is further suppressed and that in the warmer region enhanced. It is possible that some of this insensitivity is a consequence of using a fixed cooling profile rather than an interactive radiative model. However, other broad properties of the system appear to be independent of the coupling strength, which suggests that an interactive radiative scheme would be likely to produce similar results. Further numerical experiments show that the mechanical energy budget for the entire system is equally insensitive to changes in the difference in SSTs between the columns.

The ratios between the component terms contributing to the total energy flows for the entire system and for each of the regions (where the effect of the large-scale velocity is isolated) are relatively constant as a function of coupling strength and are consistent with estimates obtained by theoretical scaling arguments. On the other hand, the values of the component terms for the large-scale circulation exhibit markedly different features in that the buoyancy contribution is more significant than that due to the lifting of precipitation, while the Gibbs penalty reaches a maximum as the cooler region dries out before then decreasing as the differences between the regions are eliminated with stronger coupling.

The buoyancy contribution to the large-scale circulation is consistent with that predicted from an analytical approach to the energetics implied by WTG in a dry setup, which also predicts its quadratic dependency on the SST difference between the columns that is observed. We also present more general scaling arguments for the components of a mechanical energy budget, which could serve to indicate the impact of changing atmospheric conditions on the generation of kinetic energy.

The vertical kinetic energy associated with the large-scale circulation is found to be more than three orders of magnitude smaller than the vertical kinetic energy associated with the convection within each region. Despite this weak magnitude, the coupling has a strong influence on the suppression and enhancement of convection in each of the regions. The imposition of the WTG approximation may be interpreted as a representation of the requirement for low temperature gradients in the tropics due to weak rotational effects. In this sense, the contrast in local convective intensity can be seen as the consequence of a large-scale dynamical constraint and not as a local result of contrast in SSTs.

The techniques introduced in this paper, suitably modified, can be applied to reanalysis products to estimate a mechanical work budget for large-scale circulations such as the Walker or Hadley circulations, which would provide further insight into whether coupling techniques such as the WTG provide a useful approach to reflecting such phenomena in studies of convection. Similarly, they could shed light on the mechanics of convective aggregation, which can be studied using the WTG approximation (Emanuel et al. 2014).

## Acknowledgments

Jan Kamieniecki is supported by a grant from the U.K. Natural Environment Research Council (Grant NE/K004034/1). We thank the Met Office for access to version 2.4 of the LEM, C. L. Daleu for making her model configuration available to us, and David Raymond, Timothy W. Cronin, and one anonymous reviewer for their helpful comments.

## APPENDIX

### Components of Work Done in an Anelastic Model

In this appendix, we explain how the derivation of the components of work [Eq. (3)] in Pauluis (2016) can be refined in the case of an anelastic model, and we derive an approximate expression for a suggested correction term.

#### a. Hydrostatic balance in an anelastic context

The original derivation of Eq. (3) involves the use of the hydrostatic approximation

#### b. Derivation of approximate correction term

*B*in Eq. (12) and where

*z*:

#### c. Numerical values and scalings

The contribution of this new term can be illustrated by comparing numerical values for the components of Eq. (A15) with the modified version produced by the LEM for the case of a single uncoupled convecting region in a state of equilibrium over a surface at a temperature of 302.7 K (as discussed in section 3b). As Table 1 shows, the effect of the correction term is to reduce the discrepancy in the equation from around 5% of the largest term to less than 1%. Similar impacts are observed in other configurations of the model, including in the coupled version.

The additional term expressed in the form in Eq. (A12) will scale as approximately 20% of

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