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 = f₀ + βy. For guidance, a list of key variables and their definitions can be found in Table 1.

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Structure of the two-layer model. Thick flat lines correspond to surfaces that remain fixed and the wavy curve to the interface η, which varies. Each layer has a streamfunction relating to the barotropic and baroclinic modes as described in the text, an associated potential temperature, and a typical height scale H. The interface η captures variations from this typical thickness, which are corrected by vertical motion W. The moisture m is confined to the lower layer and precipitation conditionally triggers mass transport $\mathcal{L}P$ from the bottom to the top layer. Ekman dissipation $r{\nabla }^2({\psi }_{\text{BT}}-{\psi }_{\text{bc}})$ takes effect at the bottom surface.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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Table 1

Variables used in the model description.

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c. Moist potential vorticity

In the dry QG system, the interface η and baroclinic vorticity ζ_bc exchange energy through vertical motion W with a constant ratio of f₀/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 −f₀η/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.,

${\eta }_c=\eta +\frac{m-m_s}{1+\mathcal{C}},$(12)

which evolves as

$\frac{D_{\text{BT}}}{Dt}{\eta }_c=J({\psi }_{\text{bc}},{\eta }_c)-\frac{\mathcal{L}(P-E)}{{\mu }_s-1}.$(13)

The quantity

${\mu }_s=\frac{1+\mathcal{C}\mathcal{L}}{1-\mathcal{L}}$(14)

characterizes the reduction to the static stability associated with moist effects. A key feature of this reservoir is its conservation in the absence of adiabatic forcings, including precipitation, evaporation, radiative cooling, and dissipative effects. At saturation, m = m_s and η_c = η. We propose interpreting η_c as the condensation level. Indeed, noting that moisture is confined below the height set by the interface value η, the condensation trigger can be visualized as the condition that the interface rises above the condensation level, as depicted in Fig. 2.

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A schematic for the condensation level η_c as a metric for saturation. When the interface η between the top (white) and bottom (blue) layers is below the condensation level η_c, the system is subsaturated. The interface η evolves by dynamical processes and radiative cooling R, while the condensation level η_c evolves with competing effects from evaporation E and radiative cooling R. When η rises above η_c, precipitation P quickly acts to bring the two to parity by removing water vapor and lowering the interface.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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Vertical motions then resolve anomalies in the term

$\eta +({\mu }_s-1){\eta }_c=\frac{\eta +\mathcal{L}m}{1-\mathcal{L}},$(15)

which resembles the moist static energy with a rescaling relating to the moisture stratification. By analogy with the dry model, this quantity can be thought of as a reservoir of baroclinic vorticity that can be converted through vertical motion. When precipitation is the dominant forcing impacting η_c, 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 $1-\mathcal{L}<1$. Moist baroclinic PV (MPV) can be conceptualized as the baroclinic vorticity after the perturbation to the total effective reservoir is brought back to zero by vertical motion. This yields a moist baroclinic potential vorticity of the form

$q_m={\zeta }_{\text{bc}}-\frac{f_0}{H}[\eta +({\mu }_s-1){\eta }_c].$(16)

This is a baroclinic formulation based on the MPV derived in Lapeyre and Held (2004), and also resembles the gross PV of Adames and Ming (2018) and Adames (2021). Its evolution equation is given by

$\frac{D_{\text{BT}}}{Dt}{q}_m=-J\left[{\psi }_{\text{bc}},q_{\text{BT}}+\frac{f_0}{H}({\mu }_s-1){\eta }_c\right]+\frac{r}{2}({\zeta }_{\text{BT}}-{\zeta }_{\text{bc}}).$(17)

The parameter μ_s 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 μ_s, 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 $1-{\mu }_s^{-1}$ to be the moisture stratification of the system. Crucially, their implementation does not include a Clausius–Clapeyron relation, which introduces interactions between temperature and moisture, including in the horizontal gradient and the preferential latent heat release in warm sectors of the atmosphere. In particular, a nonzero $\mathcal{C}$ increases precipitation in regions where low-level convergence shifts the interface upward. Because precipitation has an opposite effect to positive W, increasing $\mathcal{C}$ decreases the efficiency of upward motions in removing temperature anomalies.

3. Numerical simulations

We perform experiments on a doubly periodic domain in spectral space with a 256 × 256 grid, The domain size L is chosen such that 2πλ = L/9. Simulations were run for a time T = 400λ/U. Time averages are computed over the last half of the run with sampling at intervals of δt = 0.1λ/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, $(m-m_s)/(1+\mathcal{C})={\lambda }^2(q_m-q_{\text{bc}})+({\mu }_s-1){\psi }_{\text{bc}}$.

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 ${\tau }^{\ast }$ comes from analyses of the tropics, which suggest time scales from 2 (Betts and Miller 1986) to 12 h (Bretherton et al. 2004). The rationale for all other physical parameters can be found in Lapeyre and Held (2004). The time step was chosen based on the stability of the simulation. The eighth-order hyperdiffusion coefficient ${\nu }^{\ast }$ was chosen to allow for dissipation at small scales without suppressing the smaller-scale instability associated with latent heat release.

Table 2

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 $\mathcal{E}$ and a very small precipitation relaxation time scale ${\tau }^{\ast }$.

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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_s = 4.0 at the same dry criticality 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_s = 1.0), but at criticality ξ = 5. This supercriticality is chosen as to match the value of μ_sξ 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.

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Snapshots of the barotropic, baroclinic, and moist baroclinic potential vorticity perturbation for three cases. (top) A dry case with mild supercriticality (ξ = 1.25, μ_s = 1.0). (middle) A saturated case with the same dry criticality as the first case and moisture (ξ = 1.25, μ_s = 4.0). (bottom) Another dry case with higher criticality (ξ = 5.0, μ_s = 1.0), chosen so that the second and third cases have same total saturated criticality. Both exhibit more energetic flows than the top row; note the change in color scale. The middle row, as the only moist system, is the only row to exhibit a smaller magnitude of (center) dry baroclinic PV compared to (right) the moist. This is associated with the inclusion of moisture in the “reservoir” for conversion into baroclinic vorticity. Additionally, the middle row is dominated by small-scale vorticity, consistent with a shift to smaller scales, in contrast with the bottom row with the same saturated criticality.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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To better quantify the turbulence, Fig. 4 displays the time-averaged eddy kinetic energy (EKE) spectra for the barotropic and baroclinic modes. These energies are defined in section 4. We observe an increase in the barotropic EKE of roughly a factor of 100 from the μ_s = 1.0 case to the μ_s = 4.0 case in for simulations with ξ = 1.25 and ξ = 0.8. There is a corresponding shift to larger scales of the Rhines scale k₀, defined by

$k_0=\sqrt{\frac{\beta }{V}},$(21)

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

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Spectra of the barotropic and baroclinic eddy kinetic energy for a few values of μ_s with (top) ξ = 1.25 and (bottom) ξ = 0.8. The horizontal scale is the wavelength rescaled by the largest wavelength 2π/L, where L is the domain size. The y axis is plotted in symlog scale, such that the figures are linear for values smaller than 10⁻⁴ and log scale for larger values. The vertical lines in the barotropic mode are the Rhines scale, associated with the termination of the inverse cascade. The vertical lines in baroclinic mode mark the centroid of the baroclinic eddy kinetic energy.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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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.

Precipitation acts as a conversion term between ME and baroclinic energy, while total energy—defined as the sum of its barotropic, baroclinic, and moist components—is conserved. To capture the source of this injection into the baroclinic energy, we define ME as

$\text{ME}=\frac{g^{*}}{4H}({\mu }_s-1){{\eta }^{\prime }}_c^2.$(25)

By multiplying Eq. (13) by $({\mu }_s-1){\eta }_c^{\prime }$ and taking a domain average, we obtain that the domain average $\overline{\text{ME}}$ evolves as

${\partial }_t\overline{\text{ME}}={\epsilon }_{\text{ME}}-\mathcal{P}-{\mathcal{D}}_P.$(26)

The first term captures the generation of ME, with further elaboration below. The second term, defined in the context of Eq. (24) captures the transfer from ME to APE by precipitation. The third captures the dissipation of ME due to precipitation ${\mathcal{D}}_P=(g^{*}H/2)[\mathcal{L}/(1+\mathcal{C})]\tau \overline{{P^{\prime }}^2}$, which vanishes in the limit 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

${\epsilon }_{\text{ME}}=-\frac{g^{*}}{2H}({\mu }_s-1){\overline{\eta }}_y\overline{({\upsilon }_{\text{BT}}^{\prime }-{\upsilon }_{\text{bc}}^{\prime }){\eta }_c^{\prime }}.$(27)

This shows that a downgradient transport of the condensation level acts as a source of ME. It results from a combination of a downgradient thickness and humidity transports. In the saturated limit, the two are related and the generation of ME is proportional to the downgradient thickness flux:

${\epsilon }_{\text{ME},s}=-\frac{g^{*}}{2H}({\mu }_s-1){\overline{\eta }}_y\overline{{\upsilon }_{\text{BT}}^{\prime }{\eta }^{\prime }}.$(28)

We can then combine Eqs. (24) and (26) to create a budget for the total moist baroclinic energy E_mb:

${\partial }_t\overline{E_{\text{mb}}}=-\mathcal{B}-{\mathcal{D}}_{\text{bc}}+{\epsilon }_{\text{APE}}+{\epsilon }_{\text{ME}}.$(29)

The total energy generation of the system can then be computed as ε = ε_APE + ε_ME, which at saturation can be estimated as

$\epsilon ={\epsilon }_{\text{APE}}+{\epsilon }_{\text{ME}}\approx -{\mu }_s\frac{g^{*}}{2H}{\overline{\eta }}_y\overline{{\upsilon }_{\text{BT}}^{\prime }{\eta }^{\prime }},$(30)

with corresponding dissipation

$\mathcal{D}=\frac{r}{2}\overline{{\left|\nabla ({\psi }_{\text{BT}}^{\prime }-{\psi }_{\text{bc}}^{\prime })\right|}^2}.$(31)

The energy transfer for moist geostrophic turbulence in the saturated limit is depicted in Fig. 5. The meridional transport of sensible and latent heat acts as energy sources for APE (ε_APE) and ME (ε_ME). In the limit of short adjustment time, the dissipation of ME by precipitation is negligible ${\mathcal{D}}_P\sim 0$, so that ME is fully converted into baroclinic energy by precipitation ($\mathcal{P}$). In addition, as with traditional (dry) geostrophic turbulence, baroclinic energy is converted into baroclinic energy ($\mathcal{B}$), while kinetic energy is lost due to surface friction (${\mathcal{D}}_B$ and ${\mathcal{D}}_b$).

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Energy transfers in the MQG model, with the estimates of the scaling at saturation. At large scales, the background moisture and temperature gradients are redistributed by the barotropic flow, acting as a source for the APE and ME, ε_APE and ε_ME, respectively (purple arrows). The energy is mixed to smaller scales by the barotropic flow until near the Rossby radius. The precipitation $\mathcal{P}$ (blue arrow) transfers ME into APE across, with a peak near the saturated Rossby radius. The baroclinic mode injects energy into the barotropic $\mathcal{B}$ (red arrow). The barotropic flow has an inverse energy cascade to larger scales and a forward enstrophy cascade to smaller scales. Dissipation (orange arrows) occurs primarily through Ekman dissipation of the barotropic mode ${\mathcal{D}}_{\text{BT}}$ at large scales. Additional dissipation occurs with Ekman dissipation of the baroclinic mode ${\mathcal{D}}_{\text{bc}}$ at large scales and precipitation dissipation ${\mathcal{D}}_P$ mostly at small scales. Additional hyperdiffusion occurs in all modes, but can be neglected and is not depicted here.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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If one assumes that the atmosphere is in statistical equilibrium, there must be a balance between the generation and the dissipation of energy over long time scales. In the saturated limit, all of the total injection into the barotropic mode is balanced by the amount of moist available potential energy (MAPE = APE + ME) generated. Hence,

$\begin{array}{ll}\frac{⟨{\epsilon }_{\text{APE}}⟩+⟨{\epsilon }_{\text{ME}}⟩}{⟨\mathcal{D}⟩}\hfill & =1,\hfill \\ \frac{⟨{\epsilon }_{\text{APE}}⟩+⟨{\epsilon }_{\text{ME}}⟩}{⟨B⟩}\hfill & =1,\hfill \end{array}$(32)

where the angle brackets indicate a sufficiently long time average. Figures 6a and 6b demonstrate that both of these assumptions hold up well in our simulations and may be used for scaling arguments. The system is in statistical equilibrium, and ε can be used to estimate the barotropic energy injection $\mathcal{B}$.

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The balance of (a) the total generation of energy vs the Ekman dissipation, (b) the generation of MAPE vs the injection into barotropic energy, (c) the generation of ME and conversion to precipitation, and (d) the ratio of precipitation injection to sensible heat flux, the two contributions to the generation of APE.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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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

We will begin with a linear stability analysis, similar to those done in greater detail by de Vries et al. (2010), Adames and Ming (2018), and Adames (2021), among others. Recall that, in the dry case, instability occurs when there is a sign reversal in the total PV gradient. In the two-layer case, this requires that the mean baroclinic PV gradient be larger than the gradient of the Coriolis parameter. Similarly, in the saturated limit, instability occurs when the mean MPV gradient is larger than the gradient of the Coriolis parameter, or, expressed in terms of a saturated criticality ξ_s,

${\xi }_s\equiv \frac{{\mu }_{s}U}{\beta {\lambda }^2}\equiv {\mu }_s\xi \ge 1,$(34)

where ξ is the dry criticality. As μ_s ≥ 1 (equality holding only in the dry limit), the saturated criticality ξ_s ≥ ξ. In particular, it is possible for the saturated system to be unstable with ξ_s > 1, even where the classical dry theory would predict stability, 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_s, demonstrating that the moist system exhibits smaller-scale vortices than the dry with equivalent moist criticality.

Figure 7 shows the linear growth rate 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 ${\lambda }_s={\mu }_s^{-1/2}\lambda$ replacing their dry counterparts,

$\sigma =\frac{Uk}{{({\lambda }_{s}K)}^4+2{({\lambda }_{s}K)}^2}{\left[\frac{{({\lambda }_{s}K)}^8}{4}-{({\lambda }_{s}K)}^4+{\xi }_s^{-2}\right]}^{1/2},$(35)

where k is the wavenumber corresponding to propagation in the x (zonal) direction. Figure 7 considers the case with K = k.

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Growth rate as a function of scale Kλ = |k|λ and the gross moisture stratification μ_s, with dry criticality ξ fixed as the value indicated in each title. (right) The dashed horizontal line corresponds to the value of μ_s necessary to achieve marginal saturated criticality. The two lines enveloping the contour correspond to asymptotic bounds on the unstable region in the limit ξ_s → ∞.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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The saturated limit also implies changes in the spectrum of unstable modes, which are confined between lower and upper wavenumber moduli K₋ and K₊, defined by

$K_{\pm }^4=2{\mu }_s^2{\lambda }^{-4}\left(1\pm \sqrt{1-{\mu }_s^{-2}{\xi }^{-2}}\right).$(36)

The impact of the moist parameter μ_s is twofold: the spectrum tends toward higher wavenumber, and the range of unstable modes increases.

It is convenient to consider the limit of strong supercriticality (ξ_s ≫ 1), in which case the long- and shortwave cutoff can be written as

$K_{-}\lambda \approx {\xi }^{-1/2},$(37)

$K_{+}\lambda \approx 2^{1/2}{\mu }_s^{1/2}.$(38)

Figure 7 illustrates these asymptotic limits. While the long-wave cutoff K₋ exhibits little dependency on μ_s, the short-wave cutoff 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_s. Because Eq. (35) is equivalent to the growth rate of the dry system under rescaling, we can predict that when ξ_s is held constant, the unstable modes will shift to smaller scales as ${\mu }_s^{1/2}$ and the fastest growth rate will increase as ${\mu }_s^{1/2}$.

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_s = 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 ${\mu }_s^{1/2}=0.5$ smaller in the middle row compared to the bottom row. Indeed, we observe a roughly factor of 2 change in the scale of vortices between the two integrations.

b. Baroclinic energy

Baroclinic energy is generated through the downgradient transport of sensible heat ε_APE and through precipitation $\mathcal{P}$. Figure 8a captures the total injection into the APE. The peak injection increases by nearly a factor of 100 and broadens to both smaller and larger scales as 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_s. In contrast, as shown in Fig. 8c, precipitation generates APE at smaller scales as μ_s 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.

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The spectra of the source terms of the baroclinic energy E_bc. These include a downgradient flux of sensible heat ε_APE and a precipitation injection term $\mathcal{P}$. The y axis is on a symlog scale, such that it is linear for values between ±0.01.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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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.

Turbulence is characterized by the transport of energy across different scales of motion within the same energy mode. Here, we compute the baroclinic energy flux as

$\begin{array}{c}{\mathcal{F}}_{\text{bc}}=-{\displaystyle {\int }_0^K{J({\psi }_{\text{BT}}^{\prime },\text{APE})|}_k-{J\left({\nabla }^2{\psi }_{\text{BT}}^{\prime },\frac{1}{2}{\left|{\psi }_{\text{bc}}^{\prime }\right|}^2\right)|}_{k}dK}\\ ={\mathcal{F}}_{\text{APE}}+{\mathcal{F}}_{\text{bc},\zeta }\end{array}.$(39)

This flux, which can be further decomposed into an APE component ${\mathcal{F}}_{\text{APE}}$ and a vorticity component ${\mathcal{F}}_{\text{bc},\zeta }$, corresponds to the energy transfer across scales. A positive value corresponds to a transfer toward larger wavenumber (and hence smaller scale). As advection conserves the total baroclinic energy, it does not appear in the domain averaged budget Eq. (24). Figure 9 shows these fluxes for increasing values of μ_s. 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.

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The advective flux of (a) the APE, (b) the baroclinic kinetic energy, and (c) the total baroclinic energy. The y axis is on a symlog scale, such that it is linear for values between ±0.1. The advection term transports energy from the scales where the slope is positive to those where the slope is negative. Cascade behavior corresponds to the regions where the slope is near zero, as energy is added and removed at similar rates.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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As shown in Fig. 9a, the APE cascade ${\mathcal{F}}_{\text{APE}}$ corresponds to a transfer of energy from large scale, where APE is generated by downgradient heat fluxes, to smaller scales corresponding to the negative slopes at the largest scales. In contrast, the baroclinic vorticity term ${\mathcal{F}}_{\text{bc},\zeta }$, plotted in Fig. 9b, displays an inverse energy cascade from small to larger scales. The peak of this injection, as with the precipitation, shifts to smaller scales as 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 $\mathcal{B}$. Simulations with values of μ_s ≥ 2 exhibit another region of energy injection at scales smaller than K ≈ 2λ⁻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

The barotropic injection term $\mathcal{B}$ of Eq. (23) can be decomposed into a linear and nonlinear component in spectral space, such that

$\begin{array}{c}{\widehat{\mathcal{B}}}_k={\widehat{\mathcal{B}}}_{k,\text{nonlin}}+{\widehat{\mathcal{B}}}_{k,\text{lin}}\\ ={\widehat{\psi }}_{\text{BT},k}^{*\prime }{J({\psi }_{\text{bc}}^{\prime },{\nabla }^2{\psi }_{\text{bc}}^{\prime })|}_k+\frac{U}{2}{K}^2{\widehat{\psi }}_{\text{BT},k}^{*\prime }{\widehat{\psi }}_{\text{bc},k}^{\prime }\end{array}.$(40)

Here, the subscript 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_s increases, from roughly a factor of 2 to 10.

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The injection into the barotropic energy, decomposed into linear and nonlinear components. The y axis is on a symlog scale, such that it is linear for values less than 0.01.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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The barotropic energy cascade is characterized by

${\mathcal{F}}_{\text{BT}}=-{\displaystyle {\int }_0^K{\psi }_{\text{BT},k}^{*\prime }}{J({\psi }_{\text{BT}}^{\prime },{\nabla }^2{\psi }_{\text{BT}}^{\prime })|}_{k}dK.$(41)

This term, shown in Fig. 11, exhibits a forward enstrophy cascade at small scales and an inverse energy cascade at large scales. As μ_s 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 termination of the cascade may still be valid.

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The advective flux of the barotropic energy. The y axis is on a symlog scale, such that it is linear for values between ±0.1.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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d. Rhines scale and the inverse cascade

Held and Larichev (1996) argued that the termination of the inverse cascade in a dry system can be predicted from the criticality. We here test whether a similar argument holds for the saturated system. Their argument equates two approximations of the energy generation rate 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

$\epsilon \sim V^3{k}_0,$(42)

where V is the RMS barotropic velocity and k₀ is the Rhines scale as defined in Eq. (21). This is an assumption that the energy generation can be approximated by an energy scale ∼V² and a time scale ∼(Vk₀)⁻¹.

The second approximation comes from mixing length theory, which assumes that a tracer anomaly travels a characteristic mixing length before being reassimilated into the large-scale flow. The size of the anomaly can then be approximated by a first-order Taylor expansion, with the gradient of the large-scale background flow dominating the first derivative and the mixing length characterizing the perturbation about the reference position. Held and Larichev (1996) take the Rhines scale to be the mixing length for the baroclinic PV, which at large scales behaves as a passive tracer advected by the dominant barotropic flow. The equivalent argument in the MQG system would predict the same with the MPV, allowing for an estimate of the typical size of its eddies:

$q_m^{\prime }\approx -k_0^{-1}\frac{\partial Q_m}{\partial y}\approx k_0^{-1}{\mu }_{s}U{\lambda }^{-2}.$(43)

The injection into kinetic energy from MAPE can then be approximated as

$\epsilon ={\epsilon }_{\text{APE}}+{\epsilon }_{\text{ME}}=-U\overline{{\upsilon }_{\text{BT}}^{\prime }{q}_m^{\prime }}\approx V{k}_0^{-1}{\mu }_s{U}^2{\lambda }^{-2}.$(44)

Remembering that the criticality is defined as ξ = U/βλ² and that Rhines scale is given by Eq. (21), k₀ = (V/β)^1/2, we can combine Eqs. (42) and (44) to yield

$k_0({\mu }_s^{1/2}\lambda )\approx {({\mu }_s\xi )}^{-1},$(45)

$\frac{V}{U}\approx {\mu }_s\xi ,$(46)

$\frac{\epsilon }{U^3{\lambda }^{-1}}\approx {\mu }_s^{5/2}{\xi }^2.$(47)

These scalings are fully consistent with those of Held and Larichev (1996) after replacing the deformation radius λ by its moist counterpart ${\mu }_s^{1/2}\lambda$ and the dry supercriticality ξ by the moist supercriticality μ_sξ.

The first Eq. (45) indicates that the ratio between the Rhines scale and the moist deformation radius ${\lambda }_s={\mu }_s^{1/2}\lambda$ is equal to the moist supercriticality μ_sξ. 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_sξ. 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 ε varies as${\mu }_s^{5/2}$, and is thus highly sensitive to the moist parameter 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_s ≈ 4 indicates that convergence might be possible. At subcriticality, ξ_s ≈ 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.

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The scaling of (a) the Rhines scale k₀ computed from the RMS barotropic velocity as a function of saturated criticality, (b) the RMS barotropic velocity V as a function of saturated criticality μ_sξ, and (c) the total energy generated by the MQG system. Each marker corresponds to a fixed value of dry criticality ξ, indicated in the legend.

Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0215.1

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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_sξ, then the dry criticality decreases proportionately to compensate for the increase in the moisture gradient.

Further 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.

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.