## 1. Introduction

A major challenge to our understanding of midlatitude storm systems lies in the interplay between the atmospheric circulation and the hydrological cycle. On a global scale, higher temperature and humidity in the tropics relative to the poles drives poleward transport of both sensible and latent heat. On the local scale, ascending parcels undergo adiabatic expansion, condensing excess moisture to release latent heat. This additional energy can induce local hydrodynamical instabilities in conditions that would otherwise be stable. The effect of moisture is not isolated to the scales on which condensation occurs, but rather impacts dynamics across a broad range of scales, including the aggregate behavior of storm tracks (Shaw et al. 2016), the extratropical stratification (Frierson et al. 2006; Schneider and O’Gorman 2008; Wu and Pauluis 2014), and the global atmospheric circulation (Pauluis et al. 2010). Understanding the impacts of moist processes across the full range of geophysical scales is necessary to understand how midlatitude storm dynamics will change in a world becoming more humid as a result of climate change.

Many previous studies have focused on scale changes associated with moisture. Stronger moist effects lead to smaller-scale motions and narrower regions of saturation (Emanuel et al. 1987; Fantini 1990; Lapeyre and Held 2004). This correlation obfuscates the effect of different mechanisms by which moisture induces smaller-scale motion. For instance, does the shift arise as a result of highly localized precipitation associated with the cascade of moisture to small scales? Would a similar result persist even if the precipitation characteristically occurred on larger scales? And how do nonlinearities in precipitation and Clausius–Clapeyron change the dynamics? Many studies also reach opposite conclusions regarding the impact of moisture. For instance, moisture’s impact on eddy kinetic energy has been found to be positive (Emanuel et al. 1987; Lapeyre and Held 2004; Lambaerts et al. 2011), negative (Zurita-Gotor 2005; Bembenek et al. 2020; Lutsko and Hell 2021), or about neutral (Lambaerts et al. 2012).

Questions remain about how to synthesize results from different implementations of moisture in idealized systems. The construction and interpretation of a moist energy (ME) and moist potential vorticity (MPV) are key pieces that can help bridge this gap. The changes moisture introduces to the energetics result in changes to the scale at which energy is injected into the flow, its ability to cascade to different scales, and the scale at which it is dissipated. Furthermore, because moisture introduces new processes, moist systems feature new mechanisms of growth and propagation, the impact of which must be understood both individually and in combination. To this end, the study of moist turbulence benefits from a hierarchy of models with implementations of moisture mechanisms in different combinations and at different levels of complexity, including both linear (Emanuel et al. 1987; Adames and Ming 2018; Adames 2021) and nonlinear (Fantini 1990; Lapeyre and Held 2004) frameworks.

The two-layer quasigeostrophic (QG) model is one of the simplest mathematical models to exhibit the basic features of the turbulent midlatitude atmosphere, from planetary-scale barotropic jets to synoptic-scale baroclinic eddies that organize into storm tracks. Its relative simplicity, coupled with its ability to capture key dynamical features, has made it a good choice for studying the broader statistical and scaling properties of a dry atmosphere (e.g., Vallis 2006). While its utility in assessing the moist case is limited due to significant ageostrophy in precipitation regions (Fantini 1990, 1995; Lambaerts et al. 2011, 2012), moist QG (MQG) models can still provide insight into the dynamics without confounding influences from the tropics. Consequently, MQG models have been used in studies of the fundamental dynamics of baroclinic systems, such as mechanisms of growth (Parker and Thorpe 1995; Moore and Montgomery 2004; de Vries et al. 2010; Adames and Ming 2018) and analysis of turbulent spectra (Edwards et al. 2019). MQG systems are also ideal for developing theories of wave–mean flow interaction in moist systems, which is the portion of theory we seek to advance in this paper.

We use the MQG model of Lapeyre and Held (2004) to bring new intuition to the impacts of moisture on geostrophic turbulence. In section 2, we review the model and derive a conservation law for a moist potential vorticity. We argue that in the limit of high evaporation rate, the model approaches a saturated limit with precipitation active everywhere. We show in appendix A that this saturated limit is mathematically equivalent to the classic two-layer problem after replacing the baroclinic potential vorticity by the MPV, and rescaling both the horizontal and temporal dimensions. In particular, existing theory for dry QG turbulence can be readily tested in the MQG model in the saturated limit.

In section 3, we discuss the numerical implementation of the MQG model and analyze the results of numerical simulations. We show that increasing the amount of moisture leads to three main effects: a systematic intensification of turbulence, a shift of energy injection to smaller scales, and an extension of the inverse cascade to larger scales. In section 4, we analyze the energetics of the MQG model and derive an expression for ME that is converted into available potential energy (APE) through precipitation. We show that the intensification of turbulence with increased moisture is directly tied to the increased generation of ME by the barotropic flow acting on the mean temperature and humidity gradient. In section 5, we argue that the shift of the most unstable baroclinic mode in the linear instability analysis is reflected by the shift in the precipitation injection scale. We derive an expression for the Rhines scale in the saturated limit by accounting for the additional generation of ME and show that this captures the impact of moisture on the energy containing scale in our simulations. The study concludes in section 6.

## 2. Model description

We use the two-layer MQG model of Lapeyre and Held (2004), depicted schematically in Fig. 1. This model consists of two well-stratified layers of equal mean depth *H* in a doubly periodic domain. Rotational dynamics are captured by a *β* plane in which the Coriolis parameter is expressed linearly in the meridional coordinate as *f* = *f*_{0} + *βy*. For guidance, a list of key variables and their definitions can be found in Table 1.

Variables used in the model description.

### a. The dry system

*ψ*

_{BT}= (

*ψ*

_{1}+

*ψ*

_{2})/2, the column-integrated “bulk” movement, and a baroclinic streamfunction

*ψ*

_{bc}= (

*ψ*

_{1}−

*ψ*

_{2})/2, the vertical gradient. The corresponding geostrophic velocities are given by (

*u*,

_{i}*υ*) = (−∂

_{i}*, ∂*

_{y}ψ_{i}*) in mode*

_{x}ψ_{i}*i*= BT, bc, and the corresponding vorticities

*ζ*= ∇

_{i}^{2}

*ψ*. The vorticities evolve as

_{i}*D*

_{BT}/

*Dt*= ∂

*t*+

*J*(

*ψ*

_{BT}, ⋅) indicates the material derivative with respect to the barotropic flow. Both the barotropic and baroclinic vorticities are advected by the barotropic flow and forced by nonlinear interactions between the two modes characterized by the first term of the right-hand side. Baroclinic vorticity is additionally generated when the ageostrophic convergence

*W*/

*H*, explicitly defined in appendix B, transports mass between the two layers. Mass is transported upward (downward) when

*W*is positive (negative), corresponding with a generation of anticyclonic (cyclonic) baroclinic vorticity. Finally, Ekman damping at the bottom surface predominantly dissipates barotropic vorticity at large scales.

*η*between the two layers acts as a proxy for temperature, evolving with both the vertical and horizontal transport of mass. Thermal wind balance relates this interface to the baroclinic mode

*S*indicates the total diabatic forcing, including both radiative cooling and latent heat release.

*q*

_{BT}=

*ζ*

_{BT}+

*βy*; and for the baroclinic mode as

*q*

_{bc}=

*ζ*

_{bc}−

*f*

_{0}

*η*/

*H*. The potential vorticities evolve as

*S*= 0, in which case Eqs. (5) and (6) are a closed set of equations for two quantities which, in absence of dissipation (

*r*= 0), are conserved in the domain average. The inclusion of diabatic forcing terms disrupts this conservation.

### b. Incorporating moisture

*S*. This requires an equation for water content. We assume that the mixing ratio of water is close to a reference value

*m*

_{0}. We introduce a

*thickness equivalent*mixing ratio

*m*—with units of height—such that the total mixing ratio is

*m*

_{0}(1 +

*m*/

*H*). Since the lower atmosphere contains the bulk of the moisture content, this weighted mixing ratio is defined only in the bottom layer of the system. It is continuously replenished by evaporation of water from the surface at rate

*E*, which we will hold constant. The water budget can be written as

*P*, replenished by surface evaporation

*E*, and increased by low-level convergence

*W*.

*m*exceeds a saturation value

*m*set by the Clausius–Clapeyron relation, here represented by a linearization with respect to temperature perturbation

_{s}*η*:

*τ*, such that

*L*is the latent heat of vaporization,

_{q}*m*

_{0}the reference mixing ratio, and

*c*dry stratification, and

_{p}δθ### c. Moist potential vorticity

*η*and baroclinic vorticity

*ζ*

_{bc}exchange energy through vertical motion

*W*with a constant ratio of

*f*

_{0}/

*H*. The baroclinic PV can be thought of as the vorticity after the thickness perturbation is brought back to 0, so that the thickness perturbation to the (dry) PV is −

*f*

_{0}

*η*/

*H*. In this sense, the thickness acts as the “reservoir” available for conversion into baroclinic vorticity. In the moist case, precipitation contributes to the thickness reservoir. This contribution can be characterized by a second reservoir, defined by combining Eqs. (4) and (7) to eliminate the vertical motion

*W*and isolate the precipitation tendency, e.g.,

*m*=

*m*and

_{s}*η*=

_{c}*η*. We propose interpreting

*η*as the condensation level. Indeed, noting that moisture is confined below the height set by the interface value

_{c}*η*, the condensation trigger can be visualized as the condition that the interface rises above the condensation level, as depicted in Fig. 2.

*η*, this reservoir is enhanced through a combination of two effects: first, it includes a contribution from the moisture field in addition to the thickness perturbation; second, the impact of vertical velocity is reduced by a factor

_{c}*total effective reservoir*is brought back to zero by vertical motion. This yields a

*moist baroclinic potential vorticity*of the form

*μ*plays a central role in the MQG system. It characterizes a reduction to the effective static stability of the atmosphere as a result of precipitation (Neelin and Held 1987; Emanuel et al. 1994; Adames 2021): the larger

_{s}*μ*, the lower the effective stratification. Similar parameters have been shown to relate to the efficiency of precipitation as a dehumidification process (Inoue and Back 2015, 2017). A connection between Eq. (13) and the wet-bulb temperature equation of Pauluis et al. (2008) can be achieved by taking

_{s}*W*, increasing

### d. The saturated limit

The nonlinearity of the precipitation trigger (9) introduces a major complication in the study of moist turbulent dynamics. This nonlinearity, however, is absent in two limiting scenarios: a dry atmosphere with no precipitation, and a fully saturated atmosphere with precipitation everywhere. While the first scenario has been well documented, we argue here that the second scenario, which we refer here to as the saturated limit, can offer additional insights on the impacts of moisture on geostrophic turbulence.

The saturated limit can be achieved if one makes the assumption that precipitation acts quickly enough to maintain the system near saturation. Within the MQG system described above, the limit of complete saturation can be nearly achieved by increasing the evaporation parameter *E* and decreasing the precipitation relaxation scale *τ*. The former increases the amount of water vapor added to the system at every time step, ensuring at sufficiently high values that the system is never subsaturated. The latter decreases the amount of time that the system takes to relax to the saturated value, decreasing the value of the moisture surplus *m* = *m _{s}* in a supersaturated system. Applying both of these limits corresponds to the strict quasi-equilibrium approximation of Emanuel et al. (1994).

*m*=

*m*=

_{s}*Cη*, or equivalently that the condensation level is equal to the interface position, i.e.,

*η*=

_{c}*η*. As a result, the MPV can be written as

## 3. Numerical simulations

_{bc}= −

*U*/2

*y*, associated with an externally forced temperature gradient. The total baroclinic streamfunction is

*Q*

_{BT}=

*βy*and

*Q*

_{bc}=

*Uy*/

*λ*

^{2}. In the dry case, instability occurs when the mean baroclinic PV gradient is larger than the gradient of the Coriolis parameter. This can be recast in terms of the criticality

*ξ*as

*Q*=

_{m}*μ*

_{s}Uλ^{−2}

*y*. Last, the implementation of precipitation requires a closure to account for strict nonnegativity. We follow the closure of Lapeyre and Held (2004), described in appendix C.

We perform experiments on a doubly periodic domain in spectral space with a 256 × 256 grid, The domain size *L* is chosen such that 2*πλ* = *L*/9. Simulations were run for a time *T* = 400*λ*/*U*. Time averages are computed over the last half of the run with sampling at intervals of *δt* = 0.1*λ*/*U*. Time stepping uses a third-order Adams–Bashforth method with an integrating factor to remove the stiff portion of the equation and the Jacobian handled pseudospectrally with antialiasing. Time stepping is done for the upper (*q*_{BT} + *q*_{bc}), lower (*q*_{BT} − *q*_{bc}), and moist lower (*q*_{BT} − *q _{m}*) PV, thereby eliminating the need to compute the vertical motion

*W*. The upper and lower streamfunctions are computed diagnostically in Fourier space. Precipitation is computed diagnostically in real space from the moisture surplus,

The simulations used for data in this paper span the parameter space listed in the right column of Table 2. Realistic values are listed in column 3. The estimate for the precipitation relaxation time scale

Tunable parameter space (nondimensionalized), realistic values, and the values used in the simulations here. To enforce the saturated limit, the integrations were done with a very large value of the evaporation

Figure 3 displays snapshots of the barotropic vorticity (first column), baroclinic PV (second column), and MPV (third column) in three of these simulations. The first row shows a dry simulation (*μ _{s}* = 1) at supercriticality

*ξ*= 1.25. As the configuration is only slightly supercritical, the flow is only weakly unstable and is organized in six fairly narrow zonal jets. The second row shows a moist simulation with

*μ*= 4.0 at the same dry criticality

_{s}*ξ*= 1.25. An intensification of the flow is evidenced by the increase in the magnitude of the vorticity anomalies. The range of motions is substantially enhanced both at small scales, with the emergence of closed vortices, and at large scale with the organization of the flow around two zonal jets instead of six. Finally, the third row shows a dry simulation (

*μ*= 1.0), but at criticality

_{s}*ξ*= 5. This supercriticality is chosen as to match the value of

*μ*in the simulation shown in the second row. Qualitatively, the simulations in the second and third rows exhibit similar levels of turbulence, albeit with systematically larger scale of motions in the dry simulation.

_{s}ξ*μ*= 1.0 case to the

_{s}*μ*= 4.0 case in for simulations with

_{s}*ξ*= 1.25 and

*ξ*= 0.8. There is a corresponding shift to larger scales of the Rhines scale

*k*

_{0}, defined by

*V*is the root-mean-square barotropic velocity. This scale provides an estimate for the termination of the inverse cascade.

*μ*, the peak of the baroclinic EKE for both values of the dry criticality

_{s}*ξ*increases by roughly a factor of 2 and shifts to smaller scales. We approximate the location of this peak by a centroid of the baroclinic EKE, given by

*ξ*= 0.8,

*μ*= 1.0) and roughly two orders of magnitudes different in the most energetic simulations. Indeed, the dry system is predicted to be subcritical when

_{s}*ξ*= 0.8, but exhibits supercritical growth at large

*μ*. These scale changes are what we seek to explain in the remainder of this paper.

_{s}## 4. Energetics

The MQG system exhibits features across a broad range of scales. Held and Larichev (1996) argue that the energetics of the (dry) QG system can be understood as an inverse energy cascade associated with barotropic motions, and a direct cascade of APE. The two cascades are coupled in the sense that the APE is mixed to smaller scales by the barotropic flow before being converted into kinetic energy, which in turn sustains the cascade of barotropic kinetic energy to large scales. We revisit how the inclusion of moist processes modifies this picture by quantifying an additional source of energy associated with the poleward transport of moisture.

*E*

_{bc}consists of a kinetic energy component

*τ*→ 0. Since this limit is an assumption of strict quasi equilibrium, the third term will be neglected. The generation of ME can be written as

*E*

_{mb}:

*ε*=

*ε*

_{APE}+

*ε*

_{ME}, which at saturation can be estimated as

*ε*

_{APE}) and ME (

*ε*

_{ME}). In the limit of short adjustment time, the dissipation of ME by precipitation is negligible

*ε*can be used to estimate the barotropic energy injection

*ε*

_{ME}and vice versa. At higher values of

*μ*, less energy is generated from precipitation than predicted at saturation. However, more subsaturated points occur in the simulations with high

_{s}*μ*, so it is possible that this deficit is due to an increased portion of the domain at subsaturation. This also explains the small deficit in barotropic energy generation at large

_{s}*μ*. The ratio of precipitation to sensible heat flux scales as

_{s}*μ*− 1, as predicted. Notably, the sensible heat flux is the dominant contribution to APE for

*μ*< 2, while precipitation dominates for

_{s}*μ*> 2. This is consistent with a shift from a sensible heating to a latent heating dominated regime, which points to a change in growth mechanism such as described in Parker and Thorpe (1995), de Vries et al. (2010), and Adames (2021).

_{s}## 5. Scalings

The previous section argues that geostrophic turbulence is characterized by the generation, conversion and dissipation of three different components of the energy budget. Here, we focus on the scales at which these occur. Scaling arguments for the termination of the inverse cascade (e.g., Held and Larichev 1996) often assume that the system has an inertial range: a sufficient scale separation between the injection scale and dissipation scale. In practice, such scaling arguments still offer useful insights, even in the absence of a clear inertial range.

### a. Linear stability analysis

*mean MPV gradient*is larger than the gradient of the Coriolis parameter, or, expressed in terms of a saturated criticality

*ξ*,

_{s}*ξ*is the dry criticality. As

*μ*≥ 1 (equality holding only in the dry limit), the saturated criticality

_{s}*ξ*≥

_{s}*ξ*. In particular, it is possible for the saturated system to be unstable with

*ξ*> 1, even where the classical dry theory would predict stability,

_{s}*ξ*< 1.

Figure 3 shows that the scalings of moist systems cannot be determined by *ξ* or *ξ _{s}* alone. The top and middle rows depict the potential vorticities of a dry and moist simulation with the same value of

*ξ*, demonstrating the increase in energy at both small and large scales associated with the inclusion of moisture. The middle and bottom rows depict the potential vorticities of a moist and dry simulation with the same value of

*ξ*, demonstrating that the moist system exhibits smaller-scale vortices than the dry with equivalent moist criticality.

_{s}*σ*as function of the wavenumber modulus

*K*and

*λ*. As with previous equations in the saturated linear instability analysis, the linear growth rate matches the expression for the classic two-layer baroclinic instability, with the saturated versions of criticality and a saturated Rossby deformation radius

*k*is the wavenumber corresponding to propagation in the

*x*(zonal) direction. Figure 7 considers the case with

*K*=

*k*.

*K*

_{−}and

*K*

_{+}, defined by

*μ*is twofold: the spectrum tends toward higher wavenumber, and the range of unstable modes increases.

_{s}*ξ*≫ 1), in which case the long- and shortwave cutoff can be written as

_{s}*K*

_{−}exhibits little dependency on

*μ*, the short-wave cutoff

_{s}*K*

_{+}shifts to smaller scales. The latter change was found in Adames (2021) in the limit of instantaneous precipitation relaxation. Consequently, the spectrum of unstable modes broadens with increasing

*μ*. Because Eq. (35) is equivalent to the growth rate of the dry system under rescaling, we can predict that when

_{s}*ξ*is held constant, the unstable modes will shift to smaller scales as

_{s}We return to Fig. 3 to see how well these predictions play out. The simulations depicted in the middle and bottom rows have equal values of *ξ _{s}*. However, the middle row has

*μ*= 4.0, while the bottom row is dry. The results of the linear stability analysis predict instability on a length scale that is a factor

_{s}### b. Baroclinic energy

Baroclinic energy is generated through the downgradient transport of sensible heat *ε*_{APE} and through precipitation *μ _{s}* increases. The energy injection can be decomposed into the sensible heat flux and the precipitation injection. Figure 8b shows the sensible heat flux increasing across all scales, with the peak shifting to larger scales with increasing

*μ*. In contrast, as shown in Fig. 8c, precipitation generates APE at smaller scales as

_{s}*μ*increases. The broadening shown in Fig. 8a arises with a combination of the dry and moist injections, which dominate at large and small scales, respectively.

_{s}For large *μ _{s}*, there is a small but noteworthy removal of APE by precipitation at large scale. As in Bembenek et al. (2020) and Lutsko and Hell (2021), this occurs due to regions where precipitation is anticorrelated with temperature. Surface dissipation of barotropic energy at large scales induces regions of mechanically forced ascent and subsidence through Ekman pumping. Descending motions in regions of large-scale subsidence induce warm and dry anomalies. Conversely, ascending regions are associated with colder but moister conditions.

*μ*. The slope contains information about whether advection is moving energy to (negative slope, dissipation dominates) or from (positive slope, injection dominates) that scale. A well-defined inertial range would exhibit slope zero, indicating that energy is maintained without gain or loss.

_{s}As shown in Fig. 9a, the APE cascade *μ _{s}* increases. The two terms combine in Fig. 9c, which shows a strong convergence of baroclinic energy at scales close to the deformation radius. This convergence spans from roughly the deformation radius scale to half the deformation radius scale in all simulations and is balanced by the baroclinic to barotropic energy conversion term

*μ*≥ 2 exhibit another region of energy injection at scales smaller than

_{s}*K*≈ 2

*λ*

^{−1}. This corresponds to the scales at which precipitation becomes a dominant source of APE. The absence of a similar region in Fig. 9a indicates that APE generated from precipitation is quickly transferred into the baroclinic vorticity at small scales without further advection.

### c. Barotropic energy

*k*indicates that the term is evaluated at the wavenumber

*k*. Figure 10 demonstrates that both of these terms exhibit spectral broadening to both larger and smaller scales, while linear stability analysis only predicted a broadening to smaller scales. In fact, increasing the strength of the bulk moisture stratification shifts the peak of energy generation to larger scales. The smaller linear term (Fig. 10b) peaks at smaller scales than the dominant nonlinear term, but still exhibits growth at larger scales. Additionally, the nonlinear term becomes proportionately larger as the value of

*μ*increases, from roughly a factor of 2 to 10.

_{s}*μ*increases, unstable growth occurs at smaller scales, causing the enstrophy cascade to likewise start at smaller scales. At larger scales, a slight positive slope indicates some injection occurring even close to the large-scale cutoff. However, the portion with the steepest positive slope starts at scales near the Rossby radius and extends to smaller scales, even though Fig. 10 shows the largest injection at scales above the Rossby radius. This is consistent with a significant transport of barotropic energy from small scales to the largest relevant scales observed in Fig. 10. This indicates that the system exhibits an inverse cascade—a transport of energy to larger scales—but not a corresponding inertial range. This muddies the scaling arguments for the slope of the inverse cascade, but the arguments for the

_{s}*termination*of the cascade may still be valid.

### d. Rhines scale and the inverse cascade

*ε*. The first is a dimensional analysis argument which applies when the system has a sufficient inertial range; however, we will relax that assumption here to statistical equilibrium. This approximation is given by

*V*is the RMS barotropic velocity and

*k*

_{0}is the Rhines scale as defined in Eq. (21). This is an assumption that the energy generation can be approximated by an energy scale ∼

*V*

^{2}and a time scale ∼(

*Vk*

_{0})

^{−1}.

*ξ*=

*U*/

*βλ*

^{2}and that Rhines scale is given by Eq. (21),

*k*

_{0}= (

*V*/

*β*)

^{1/2}, we can combine Eqs. (42) and (44) to yield

*λ*by its moist counterpart

*ξ*by the moist supercriticality

*μ*.

_{s}ξThe first Eq. (45) indicates that the ratio between the Rhines scale and the moist deformation radius *μ _{s}ξ*. It indicates that for a constant temperature gradient, the Rhines scale shifts to larger scale as the humidity gradient increases. The second Eq. (46) indicates that the ratio of RMS barotropic velocity

*V*to the vertical wind shear goes as the moist supercriticality

*μ*. Thus, for a constant temperature gradient, the velocity will increase if the humidity gradient increases, e.g., through increasing the global average temperature. Finally, the third Eq. (47) indicates that the energy generation and dissipation

_{s}ξ*ε*varies as

*μ*.

_{s}These scalings are tested in Fig. 12. As in Held and Larichev (1996), the Rhines scale, RMS barotropic velocity, and energy generation increase faster with criticality than predicted. These arguments should apply best in the asymptotic limit *ξ _{s}* → ∞. More data at larger effective criticality would be needed to see if this is the case. A slight shallowing of the slope for values above

*ξ*≈ 4 indicates that convergence might be possible. At subcriticality,

_{s}*ξ*≈ 1, the barotropic and baroclinic EKE are of similar orders of magnitude. As such, the assumption that the baroclinic PV can be treated as a passive tracer no longer holds.

_{s}While the proposed scalings indicate that geostrophic turbulence has a very high sensitivity to moisture content through the parameter *μ _{s}*, it should be noted that these scalings only hold in the saturated limit, i.e., in an atmosphere that is raining everywhere. For partial saturation, Lapeyre and Held (2004) show that moist geostrophic turbulence behaves somewhere between the dry and saturated limit. Further investigations of the impacts of moisture on geostrophic turbulence in a partially saturated atmosphere are left to a future study.

## 6. Conclusions

We have investigated geostrophic turbulence in an idealized moist model analogous to that of Lapeyre and Held (2004). We introduced a framework for the energy centered on the idea that a “condensation level” characterizes a moist energy (ME) that can be transformed into available potential energy (APE) through precipitation. The large-scale gradient of the condensation level provides a reservoir of ME. Eddies extract ME by transporting moisture poleward, which is then converted to APE when precipitation occurs in the warm sector of the eddies. This provides an additional source of baroclinic energy, which can significantly energize moist geostrophic turbulence compared to its dry counterpart under the same temperature gradient. Associated with this new framework is a moist baroclinic potential vorticity which is conserved under latent heat release. This modified PV emphasizes a gradient that takes into account the background meridional configuration of a moist static energy.

By enforcing a high rate of evaporation and a short precipitation relaxation time, we achieve a saturated limit. Under this limit, the MQG system is mathematically equivalent to the dry two-layer QG equations after rescaling both the time and spatial scales. In particular, this saturated limit makes it possible to extend results from geostrophic turbulence to include the effect of moisture on the dynamics of baroclinic eddies.

We analyzed the conditions for baroclinic instability in the saturated limit using a linear stability analysis. We demonstrate that a saturated criticality, the dry criticality rescaled by a parameter *μ _{s}*, better predicts instability, including where dry models would predict stability. We confirm numerically that the fully nonlinear MQG system exhibits instability at smaller scales and an increase in the total energy injection. This conclusion is consistent with the results of Emanuel et al. (1987), Fantini (1995), and Joly and Thorpe (1989), among others.

We examined the impacts of moisture on the injection scale and energy cascade by considering the tendency terms in the full nonlinear equations. The inclusion of moisture results in the energy injection into the barotropic mode broadening to both larger and smaller scales than in the dry case. In particular, as the strength of moist parameters increased, precipitation shifts to smaller scales and becomes the dominant contribution to instability. The increase in eddy kinetic energy generation in from latent heat release enhances the inverse cascade of barotropic kinetic energy, similar to the arguments presented in Held and Larichev (1996). In particular, the Rhines scale associated with the end of the inverse cascade shifts to larger scale in the presence of a moisture gradient. We confirmed this numerically by calculating the Rhines scale from the root-mean-square barotropic velocity.

The moist available potential energy (MAPE) defined here is equivalent to the framework of Lapeyre and Held (2004) with different partitioning. In their framework, a moist static energy and a moisture deficit/surplus term form the basis for the quadratics. Ours keeps the dry APE intact and defines ME as a quadratic of the condensation level, which is conserved in the absence of diabatic forcing. Our ME more closely resembles that of tropical models such as Pauluis et al. (2008) and Frierson et al. (2004). Likewise, the ME of Smith and Stechmann (2017) can be reworked into a quadratic of a quantity proportional to the dry potential temperature of a system brought adiabatically to saturation. This partitioning can be of particular use away from the assumption of strict quasi equilibrium and uniform evaporation rate employed here. These assumptions result in a system that acts as a highly efficient moist heat engine. More realistic moist systems are not fully saturated, and as a result, tend to have reduced mechanical efficiency stemming from diabatic processes that primarily act on the ME. This partitioning emphasizes that moisture contributes to the EKE through the precipitation term, and would allow for a direct study of the factors within the ME tendency that reduce the efficiency in a partially saturated system. More work is needed to determine the analog of the ME described here in more realistic models of moist atmospheric dynamics. Based on our results, it is worth exploring whether the lifting condensation level could be used to define a generalized ME.

The moist criticality parameter suggests that the classic baroclinic adjustment arguments, such as in Stone (1978), might be improved upon by a framework which includes the poleward transport of latent heat and its impact of static stability. Baroclinic adjustment predicts that the dry criticality of Earth’s atmosphere will remain around *ξ* ≈ 1 across different climate forcings. However, if the threshold of instability is instead determined by a moist criticality, this reduces the temperature gradient necessary to trigger efficient heat transport.

For insight, consider a planet warming uniformly at the surface. We can estimate an increase in the moisture availability by 7% K^{−1}, translating to a comparable increase in the gradient. Naively, one could start by simply increasing parameter *μ _{s}*, leaving the temperature gradient fixed (i.e., fixed dry criticality

*ξ*). This will lead to greater instability and more energy transport poleward. The increase in meridional heat transport, however, would demand a change in the energy balance at the top of the atmosphere. If we instead assume that the top of the atmosphere balance remains about the same, then it is the total meridional transport of energy that is fixed. If the energy transport scales with the saturated criticality

*ξ*=

_{s}*μ*, then the dry criticality decreases proportionately to compensate for the increase in the moisture gradient.

_{s}ξFurther research could help clarify the results of this study and its connections to other work. Our spectral analysis partially supports the findings of studies like Bembenek et al. (2020) and Lutsko and Hell (2021), but a more direct comparison can be achieved by applying our energetic framework to partially saturated systems with nonhomogeneous background states. Additionally, warming predicts an increase in both the dry static stability and the moisture content (Frierson et al. 2006). As such, the *absolute* scale changes may differ from the relative changes described here. Indeed, studies that consider the case where moisture compensates for changes in dry static stability (Zurita-Gotor 2005; Juckes 2000; Moore and Montgomery 2004) often reach opposite conclusions to ours.

Another natural follow-up is the case of partial saturation, which introduces additional complications. The decorrelation of moisture from temperature adds an additional degree of freedom, requiring a full consideration of the corresponding terms in the energy and moist PV that were neglected here. This opens new possibilities for the cascade behavior of ME, which could result in energy generation at scales smaller than those predicted in either the dry or saturated case. The energetic framework provided here can be used to analyze these additional terms and determine the changes to scalings associated with them.

## Acknowledgments.

We thank Shafer Smith for guidance on turbulence and implementing QG, and Ángel F. Adames-Corraliza and two anonymous reviewers for helpful comments on a previous version of this manuscript. MLB and EPG acknowledge support from the U.S. National Science Foundation through Award AGS-1852727. MLB and OP acknowledge support from the National Science Foundation under Grant HDR-1940145 and from the New York University in Abu Dhabi Research Institute under Grant G1102.

## Data availability statement.

The code used to generate the data in this study is stored in the repository at https://github.com/margueriti/Moist_QG_public.

## APPENDIX A

### Equivalence of Dry and Saturated Limits

*λ*and

*U*, respectively, the reference length and velocity scales, rescaled as the unit. The resulting perturbation equations can then be written as

*D*

_{BT}/

*Dt*denotes advection by the barotropic flow. In an atmosphere that is everywhere at saturation, the moist baroclinic potential vorticity becomes

*σ*= −Re(

*iω*) and redimensionalizing where relevant, the above can then be used to solve for Eq. (35).

## APPENDIX B

### Omega Equation

*W*in any of our diagnostics here. Nonetheless, it can be diagnosed by the omega equation, given by

## APPENDIX C

### Precipitation Closure

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