## 1. Introduction

The main topic of this paper is conserved energies for atmospheric dynamics, in the case including phase changes of water—for example, vapor and liquid phases. As motivation, we will briefly discuss energies for dry dynamics (without water of any phase), followed by motivating discussion of energies for moist dynamics.

*dry*Boussinesq atmosphere. The two energy densities are

**u**is the velocity,

*b*is the buoyancy,

*N*is the constant buoyancy frequency,

*b*

^{tot}=

*b*+

*N*

^{2}

*z*,

*a*is an arbitrary reference height, and the integration in (2) is done while keeping

*b*

^{tot}constant. Each of the energies has its advantageous properties: for example,

*E*

_{2}and

*ρgz*, and

*E*

_{1}has the property of being quadratic and clearly positive definite.

*moist atmosphere with phase changes*, on the other hand, a generalization of the potential energy in (2) has been proposed as

*b*is a function of the equivalent potential temperature

*z*, and again the integral is taken with

*H*

_{u}is a Heaviside function indicating unsaturated regions, and

*H*

_{s}= 1 −

*H*

_{u}is a Heaviside function indicating saturated regions. The search for an energy of this form was motivated by a similar energy for quasigeostrophic equations with precipitation and phase changes (Smith and Stechmann 2017). The energy in (4) has several advantageous properties: it is piecewise quadratic, manifestly positive definite, and the potential energy has been decomposed into buoyant contributions from each phase (proportional to

Another moist energy that has been considered in the past is the moist available potential energy (APE) of Lorenz (1955, 1978; see also Pauluis 2007, and references therein). It can be viewed as a special case of the potential energy, where a specific background state has been selected (e.g., Tailleux 2013; Stansifer et al. 2017). In the present paper, a more general setup is considered where the reference state is not constrained in the sense of Lorenz’s APE but could be chosen in that way if desired. Also, Lorenz’s APE is often considered along with the assumption of hydrostatic balance, whereas the present paper considers nonhydrostatic motions according to the Boussinesq or anelastic equations.

Yet another type of moist energy that is often considered is the moist static energy (e.g., Emanuel 1994). It is a thermodynamic quantity, not including a contribution from kinetic energy, and it is conserved under the assumption of hydrostatic balance (whereas, as mentioned above, the present paper considers nonhydrostatic motions according to the Boussinesq or anelastic equations).

The remainder of the paper is organized as follows. In section 2a we describe in detail the dry Boussinesq equations, as well as discuss *E*_{1} and *E*_{2} in slightly more depth. Thus, section 2b contains a description of the moist, nonprecipitating Boussinesq equations with phase changes. In section 2c we show that it is still possible to obtain a quadratic, positive-definite energy even in the presence of phase changes. While the Boussinesq case provides a natural starting point and simple analytical expressions, the real atmosphere has nonconstant buoyancy frequency and other complications. As steps to moving beyond this simple case, subsequent sections will incorporate increasing amounts of thermodynamic and microphysical complexity. In section 3, we consider the anelastic equations, and while we cannot formulate a quadratic energy, we can still decompose a total energy into buoyant energy and latent energy released at the interface between different phases. In section 4, we consider a Kessler warm-rain parameterization of cloud microphysics, and the Fast Autoconversion and Rain Evaporation (FARE) model of Hernandez-Duenas et al. (2013). In the former case, while we cannot obtain a conserved energy, we can still derive an energy principle that involves the source terms representing cloud microphysical processes. In the latter case, we show that there exists a unique quadratic, positive-definite energy, which is conserved in the absence of rainfall. In section 5, we discuss an application to empirical orthogonal function analysis, and we show that taking phase changes into consideration results in significant differences of the EOF modes. We conclude with a discussion in section 6.

## 2. Boussinesq equations

### a. Energetics for the dry Boussinesq equations

**u**(

**x**,

*t*) is the velocity vector,

*ϕ*=

*p*′/

*ρ*

_{0},

*p*′ is the pressure,

*ρ*

_{0}is a constant background density,

*b*is the buoyancy, and

*N*

^{2}is the squared buoyancy frequency and is taken to be a constant. The buoyancy is

*b*=

*gθ*′/

*θ*

_{0}, where

*θ*′ is the potential temperature, and we have assumed that the total thermodynamic variables of potential temperature

*θ*(

**x**,

*t*) and pressure

*p*(

**x**,

*t*) have been decomposed into background and fluctuating parts;

*E*

_{1}and

*E*

_{2}, given by (1) and (2). These two energies satisfy the equations

*E*can be

*E*

_{1}or

*E*

_{2}. While there are clear differences between these two energies, they are still related. It can be shown that

*b*+

*N*

^{2}(

*z*−

*a*) is a conserved variable, which implies that

*E*

_{1}and

*E*

_{2}differ by a quadratic material invariant. In subsequent sections, we will define total energies analogous to those given in (1) and (2) that include the effects of moisture and phase changes, and show that these energies satisfy an equation similar to (7).

### b. Moist Boussinesq equations with phase changes

**u**is the velocity vector, and

*b*

_{u}is the unsaturated buoyancy, and

*b*

_{s}is the saturated buoyancy. The variables

*b*

_{u}and

*b*

_{s}are defined over the whole domain and can be expressed in terms of equivalent potential temperature

*θ*

_{e}, total water

*q*

_{t}, and prescribed saturation mixing ratio

*q*

_{υs}:

*H*

_{u}and

*H*

_{s}indicate the unsaturated and saturated phases, respectively, and are defined in terms of

*b*

_{u}and

*b*

_{s}:

*H*

_{s}= 1 −

*H*

_{u}. More precisely,

*H*

_{u}(

**x**,

*t*) is a function of

**x**and

*t*and is defined via function composition as

*H*

_{u}(

**x**,

*t*) =

*H*(

*b*

_{u}−

*b*

_{s}), where

*b*

_{u}and

*b*

_{s}are themselves functions of

**x**and

*t*, and where

*H*(

*s*) is the Heaviside function, which takes the value of 1 for

*s*> 0 and 0 for

*s*≤ 0. The constants

*N*

_{u}and

*N*

_{s}are the buoyancy frequencies for the unsaturated and saturated phases, respectively, and are defined as

*θ*

_{e}and total water

*q*

_{t}, is described in the appendix.

### c. Piecewise quadratic energy

In this section, we present one of the main results of the paper: we show that there is a piecewise quadratic, positive-definite conserved energy for the moist, nonprecipitating Boussinesq equations.

*E*

_{1}, is the sum of four components: kinetic energy, unsaturated potential energy, saturated potential energy, and moist energy, which we denote as KE, PE

_{u}, PE

_{s}, and ME, respectively. Explicitly, this means that

*M*is a material invariant.

To motivate the form we have chosen for the moist energy, ME, we make two observations. First, since *M* is a material invariant, ME has no impact on energy transfers within each phase, but it can still impact energy transfers at the phase interface, due to the Heaviside factors in (13d). The coefficients *A*_{u} and *A*_{s} are then determined to provide the appropriate amount of latent energy transferred at the interface. Second, the variable *M* is associated with an additional eigenmode that arises for moist dynamics but is not present for dry dynamics. It essentially emerges from the linearization of (8) in the purely unsaturated or purely saturated case (see, e.g., Hernandez-Duenas et al. 2015; Smith and Stechmann 2017, and references therein). As such, it is natural to suppose that a moist energy should have an additional term related to *M*^{2} and associated with this additional moist eigenmode. Indeed, other moist systems have also been shown to have an *M*^{2} energy component (e.g., Frierson et al. 2004; Stechmann and Majda 2006; Chen and Stechmann 2016), but without the Heaviside factors that arise in ME in (13d) due to the phase changes.

*A*

_{u}and

*A*

_{s}, which are, as of yet, unknown constants. They will be determined by requiring that the material derivative of

*E*

_{1}consist of only divergence terms. Following this idea, we differentiate each component of the total energy. The derivative of the kinetic energy is found by taking the dot product of (8a) with

**u**, and the derivative of PE

_{u}is found by noting that

_{s}and ME are determined similarly. We then have

*wb*

_{s}

*H*

_{s}and −

*wb*

_{u}

*H*

_{u}. In addition, the presence of the terms

*DH*

_{u}/

*Dt*, and

*DH*

_{s}/

*Dt*indicate that they are also involved in exchanges with the moist energy at the phase interface. The role of the moist energy is to exchange energy with PE

_{u}and PE

_{s}, and this exchange occurs only at the phase interface. The material derivative of the total energy is then

*A*

_{u}and

*A*

_{s}, we require the nondivergence terms in (17) to vanish identically:

*DH*

_{u}/

*Dt*and

*DH*

_{s}/

*Dt*are nonzero only at the phase interface between saturated and unsaturated regions, and at this interface,

*b*

_{u}=

*b*

_{s}. Second, the relationship

*H*

_{s}= 1 −

*H*

_{u}allows us to write

*DH*

_{u}/

*Dt*= −(

*DH*

_{s}/

*Dt*). Therefore, the previous equation becomes

*A*

_{u}and

*A*

_{s}:

*A*

_{u}and

*A*

_{s}. In addition, (20) implies that if

*A*

_{s}≥ 0, and if

*A*

_{s}= 0 is an appealing choice, since it allows

*M*to represent latent energy in the unsaturated phase that is transferred to buoyant potential energy upon reaching saturation; this choice results in the particularly simple expression

*E*

_{1}in (21) is therefore the desired energy that is piecewise quadratic.

Several notes about the piecewise quadratic energy in (21) are in order. The first thing to point out is that we could absorb the factor of *M* given in (14), and the resulting moist energy would then have the simpler form of *M*^{2}*H*_{u}/2. Second, notice that, because it is (piecewise) quadratic, the energy in (21) can be used to define an inner product and a norm. Its use as a norm is explored below in section 5 for defining an energy for empirical orthogonal function analysis. Note, though, that (21) defines an inner product or norm only if the Heaviside functions *H*_{u} and *H*_{s} can be treated as given functions, as in some data analysis applications.

Finally, notice that PE_{u}, PE_{s}, and ME are all discontinuous across the phase interface, but the total potential energy, PE_{u} + PE_{s} + ME, is continuous. To see this, note that the potential energy is *b*_{u} and *b*_{s} within the unsaturated and saturated phases, respectively. Thus, to establish continuity of the potential energy over the entire domain, all we need to do is verify that it is continuous at the phase interface. This can be expressed mathematically as *b*_{u} = *b*_{s}, use (20), and simplify. The details are omitted for the sake of brevity.

Recall that the relationship between the coefficients *A*_{u} and *A*_{s} given by (20) was obtained by forcing the material derivative of the total energy to contain only divergence terms. This exact same relationship could have been obtained by instead demanding that the total energy be continuous across the phase interface.

Also note that *M*^{2}*H*_{u} is similar in spirit to the moist latent energy *L*_{υ}*q*_{υ} of a compressible atmosphere (e.g., Emanuel 1994), but also has some differences. It would be interesting to make a thorough comparison in the future, while for the moment we note the following comparison. For instance, they are similar in that they are both latent energies; that is, they are conserved quantities in unsaturated regions, and their energy can be accessed due to phase changes. They differ, however, in the release of their latent energy: the *M*^{2}*H*_{u} energy is transferred to buoyant potential energy only at the phase *interface*, whereas *L*_{υ}*q*_{υ} is a source of heating or cooling throughout the saturated region.

### d. The traditional form of the potential energy

*a*is an arbitrary reference level,

*H*

_{s}= 1 −

*H*

_{u}, and the integration is performed while keeping

*E*

_{2}as

*E*

_{2}satisfies the equation

### e. Relating the two energies

We have thus derived two conserved energies. The piecewise quadratic energy *E*_{1} is shown above in (21), and the more traditional energy *E*_{2} in terms of potential energy Π is shown above in (26).

_{inv}denotes a material invariant term—that is, a term which satisfies

*D*Π

_{inv}/

*Dt*= 0 and does not affect energy transfers. Thus,

*E*

_{1}and

*E*

_{2}differ by a material invariant. So, even after incorporating moisture and phase changes into the Boussinesq equations, we are still able to obtain a relationship between two energies that is similar to that of the dry case in (7).

## 3. Anelastic equations

We now study energetics of the anelastic equations. As above, the energy can be decomposed into four similar components (KE, PE_{u}, PE_{s}, ME) with straightforward physical interpretations, but in this case the energy is not quadratic, since the coefficients

### a. Energetics for the dry anelastic equations

*b*is the buoyancy. In the anelastic case, the squared buoyancy frequency

*N*

^{2}(

*z*) will no longer be constant, which precludes the existence of a conservation equation like that of (6) for a quadratic energy. We can, however, still obtain such an equation for a slightly altered version of (2) that takes into account the varying nature of

*N*

^{2}(

*z*). Thus, we define,

*b*

^{tot}held fixed, and

*Db*

^{tot}/

*Dt*= 0. It can be shown that in the dry anelastic case we have

### b. Energetics for the anelastic equations with moisture and phase changes

_{m}, and Π

_{inv}, are the saturated potential energy, unsaturated potential energy, latent energy, and material invariant terms of Π, respectively. Upon integrating (33), we find that

_{inv}is a material invariant that satisfies

*D*Π

_{inv}/

*Dt*= 0. The calculations leading from (33) to (37) are similar to the ones done in the appendixes and can be found in the online supplemental material. In brief, the terms

*H*

_{u}and

*H*

_{s}, respectively, in this case with phase changes; and the third term Π

_{m}represents an additional moist latent energy.

*z*

_{r}, not

*z*, appears as the upper limit of integration in (37c), where

*z*

_{r}is the solution to the equation

*z*

_{r}is essentially a lifted condensation level (LCL), since (38) defines

*z*

_{r}as the level where the unsaturated and saturated buoyancies,

*b*

_{u}and

*b*

_{s}, are equal, which is the condition for saturation. Also notice that

*z*

_{r}is a material invariant, since it is a function of the material invariants

*z*

_{r}satisfies

*Dz*

_{r}/

*Dt*= 0.

Also notice that Π_{m} in (37c) is analogous to the moist latent energy ∝*M*^{2}*H*_{u} in the Boussinesq case in (21). As such, (37c) could possibly be used as a definition of a variable like *M* in the anelastic case.

_{inv}is a material invariant, and Π

_{m}, which is given by (37c), transfers energy only at the phase interface. Furthermore, by combining (39a)–(39c), one can see that

*z*

_{r}is equal to

*z*at the phase interface. Hence, the material derivative of Π

_{m}is precisely what cancels the interface flux terms that arise in the flux of the buoyant potential energy terms in (39a) and (39b). So from this perspective, Π

_{m}is analogous to the moist energy of (13d) in that both terms ensure that the total potential energy is continuous, by supplying moist latent energy at the phase interface. In addition, it can be shown that

### c. Alternative energy, anelastic M, and connection with Boussinesq case

We end this section by offering a slightly different definition of Π. The alternative Π will serve two purposes: (i) to make a connection with the Boussinesq case, and (ii) to motivate a definition of an *M* variable in the anelastic case, and to offer a physical interpretation of that *M*.

*a*

_{1}and

*a*

_{2}are reference heights. We note that

*a*

_{1}to

*z*into two integrals, one from

*a*

_{1}to

*a*and another from

*a*to

*z*, and similarly for

*a*

_{2}). A convenient and physically relevant choice (as discussed further below) is to define

*a*

_{1}and

*a*

_{2}correspond to the unsaturated and saturated levels of neutral buoyancy, respectively. Then, as was done for Π, we can decompose

*a*

_{1}and

*a*

_{2}as levels of neutral buoyancy as described above. If different values of

*a*

_{1}and

*a*

_{2}are used, or if the earlier Π from (33) is used instead of the alternative

*M*variable be defined in the anelastic case? In particular, note that we did not earlier define an

*M*variable in the anelastic case, whereas in the Boussinesq case we had defined

*M*before describing the energetics. The formula in (43c) now suggests a definition of the

*M*variable in the anelastic case, since

*M*in the special case of constant

*M*variable was proposed by Wetzel et al. (2019). An advantageous property of

*M*

_{aneastic}here is its physical interpretation: it is related to convective available potential energy (CAPE) (e.g., Moncrieff and Miller 1976; Emanuel 1994; Hernandez-Duenas et al. 2019). In particular, notice that (45) involves the difference between an unsaturated CAPE and a saturated CAPE, since

*z*

_{r}is similar to a lifted condensation level [see its definition in (38)] and

*a*

_{1}and

*a*

_{2}are levels of neutral buoyancy. It would be interesting to investigate the variable

*M*

_{aneastic}in more thorough detail in the future.

## 4. Other moist systems

In this section, we will examine several systems featuring other microphysical parameterizations. We will start by looking at energetics of the anelastic equations when Kessler microphysics are used. The Kessler case allows an exploration of the effects of an additional water constituent, rainwater, and associated source terms and their influence on energetics. Then a limiting form of the Kessler scheme will be considered, called the FARE microphysics scheme (Hernandez-Duenas et al. 2013). The FARE case is interesting because it includes rainwater and precipitation, yet, unlike the Kessler case, it has an energy that is conserved (aside from a single dissipation term, due to precipitation at the surface).

### a. Warm-rain microphysics

*q*

_{c}, rainwater

*q*

_{r}, and water vapor

*q*

_{υ}(Kessler 1969; Grabowski and Smolarkiewicz 1996). They are

*A*

_{r}is the autoconversion of cloud water into rainwater,

*C*

_{r}is the collection of cloud water by rain, and

*E*

_{r}is the evaporation of rainwater into water vapor,

*V*

_{T}is the rainfall velocity, and we have assumed that supersaturation cannot occur, and again

*S*

_{r}=

*A*

_{r}+

*C*

_{r}−

*E*

_{r}. The reason for the terms on the right-hand side of (47) is that

*b*

_{u}and

*b*

_{s}are not material invariants, and so their derivatives will appear when we differentiate Π. As a result, an energy principle can be written down for the case of Kessler microphysics, but the energy is not conserved, due to rain falling at velocity

*V*

_{T}and due to microphysical source terms

*S*

_{r}=

*A*

_{r}+

*C*

_{r}−

*E*

_{r}.

### b. FARE microphysics

In this section, we consider energetics for a system that includes precipitation yet still has a piecewise quadratic energy, and the energy is conserved (aside from a single dissipation term, due to precipitation at the surface).

*q*

_{c}variable as well as expressions for

*A*

_{r}and

*C*

_{r}, and constitutes the fast autoconversion assumption. Second, rainwater, which is assumed to fall at a constant rate

*V*

_{T}, is instantly evaporated in unsaturated regions until saturation is reached. This amounts to the fast rain evaporation assumption and completes a basic description of the FARE model, the equations of which are

*V*

_{T}terms are present and represent precipitation.

Recall that in section 2b, we found an infinite family of conserved, quadratic energies of the moist, nonprecipitating Boussinesq equations of (8a)–(8d). We now show that by including the microphysics of the FARE model, that is, by incorporating a nonzero value of *V*_{T}, we can obtain a unique quadratic energy, which under appropriate circumstances, will be positive definite.

*A*

_{u}and

*A*

_{s}that appear in the energy. Differentiating each component of the energy under these new microphysical assumptions now results in

*q*

_{r}/∂

*z*)

*H*

_{u}= 0. Using this, along with the relationship between

*A*

_{u}and

*A*

_{s}given by (20), we can write the material derivative of the total energy as

*A*

_{s}such that (51) contains only divergence terms. This can be accomplished by rewriting that equation in terms of

*M*and

*q*

_{r}. In particular, the goal is to write the

*b*

_{s}∂

*q*

_{r}/∂

*z*term as a sum of a

*q*

_{r}∂

*q*

_{r}/∂

*z*term and a

*M*∂

*q*

_{r}/∂

*z*term, and then to choose

*A*

_{s}to make the

*M*∂

*q*

_{r}/∂

*z*term vanish. To this end, observing that

*A*

_{s}from the constraint that

*A*

_{u}and

*A*

_{s}, the total energy

*E*is given by

Note from (56) and (57) that the coefficients *A*_{u} and *A*_{s} will be positive if *R*_{υd} ≈ 0.61 and *L*_{υ}/(*c*_{p}*θ*_{0}) = *O*(10), so the ratio *O*(10), and so the condition on the ratio of the squared buoyancy frequencies to ensure positivity of *A*_{u} and *A*_{s} allows a reasonably wide range of values.

## 5. Application to EOF analysis

In this section, we use the ideas from the piecewise-quadratic energy and apply them to empirical orthogonal function (EOF) analysis, or principal component analysis (PCA). In the EOF analysis, the goal is to decompose a signal (in our case, a variable of an atmospheric flow) into a set of spatial and temporal patterns, and to identify the patterns with the largest energy or variance. In its most common form, an EOF analysis would use the standard *L*^{2} energy. Here, the goal is to see if we can use the concepts of the piecewise-quadratic energy from (21), in order to include information about phase changes, to potentially better identify the main modes of variability of moist dynamics.

*N*

_{T}time samples of a scalar variable,

*S*(

**x**,

*t*), located at

*N*spatial points. We denote the

*N*values of

*S*at time

*t*

_{i}by the vector

**s**

^{i}:

*modes*are then the eigenvectors of the sample covariance matrix (1/

*N*

_{T})

^{T}.

*H*

_{u}(

**x**,

*t*) and

*H*

_{s}(

**x**,

*t*) can be taken as given values, as in the case of the present data analysis application). While the EOF analysis below will deviate from using the full form of this energy, we explain the deviations further below and momentarily describe the full energy to aid the discussion. Consider the full state vector

**y**, where

*E*

_{1}=

**y**

^{T}

**y**, where

*E*

_{1}=

**y**

^{T}

**y**, we have assumed that

*H*

_{u}

*H*

_{s}= 0. From the standpoint of data analysis, this means that all of our data points are away from the phase interface. In what follows, to examine the effect of phase changes in the simplest possible setting, we will consider the univariate case rather than the case of the full vector

**y**, and we will take the single variable to be either

*B*(as a setup with phase changes) or

*b*

_{u}(as a comparison case with standard

*L*

^{2}energy without phase changes).

We apply EOF analysis to simulations generated by the University of California, Los Angeles, large-eddy simulation (UCLA-LES) model based on data from the Rain in Cumulus Over the Ocean (RICO) study (Stevens et al. 2005; Rauber et al. 2007; Stechmann 2014). The focus of this study is on shallow cumulus clouds, which are clouds that extend only a few kilometers above Earth’s surface. While these clouds do not involve ice, and often do not rain, they nevertheless incorporate both moisture and phase changes. The simulation is for 5 days with *N*_{T} = 7195 time samples. For spatial resolution, we use a two-dimensional setup with *N*_{x} = 128 and *N*_{z} = 100 grid points with grid spacings of 100 and 40 m in the horizontal and vertical directions, respectively. In Fig. 1, snapshots of the vertical velocity and cloud coverage are shown at approximately 2.4 days. (See Fig. 2 for a time series of vertical velocity and cloud fraction.)

*N*

_{e}members, and denote the

*k*th member by

*S*

^{k}(

**x**,

*t*). The value of

*S*

^{k}at time

*t*

_{i}and location

*x*

_{j}is denoted by

*k*is then given by

_{e}is

Second, we use deviations from the horizontal averages of *B* and *b*_{u} as our variables. That is to say, we subtract off the quantities *B* and *b*_{u}, respectively, where *L*_{x} denotes the horizontal domain length. Hence the data are centered to have a horizontal mean of zero at each vertical level. Finally, and perhaps most importantly, ^{−4} s^{−1} at any level where it is originally less than 10^{−4} s^{−1}, as illustrated in Fig. 3. This type of modification is also tacitly in place in other studies whenever a standard *L*^{2} energy is used to analyze data within a well-mixed or unstable layer. For the other buoyancy frequency

The comparison of EOFs for the standard *L*^{2} energy and the piecewise-quadratic energy are shown in Figs. 4–6. In terms of variance or energy, in Fig. 4, the two cases appear to capture roughly similar amounts of energy in each mode. However, in terms of EOF structures, the two cases show substantial differences. Figures 5 and 6 show the first four EOFs of *b*_{u} and *B*, respectively. The most important feature is that the first two *b*_{u} modes are limited to the lower part of the domain, whereas the first two *B* modes extend into the upper part of the domain, which means that the *B* modes capture cloud variability while the *b*_{u} modes capture mainly the subcloud layer.

One way of understanding the significance of these qualitative differences of the modes is to look at time series of the fraction of the domain covered by clouds. As can be seen in Figs. 1 and 2, the domain is covered by few clouds for most of the simulation. It is, however, punctuated by brief periods of relatively high cloud coverage, which suggests that clouds display intermittency in both space and time. In terms of the standard *L*^{2} energy, the cloud variability contributes little to the energy, due to its intermittency. On the other hand, the piecewise-quadratic energy gives different weight,

## 6. Discussion and conclusions

In summary, we have examined energetics for moist atmospheres with phase changes. We first considered a Boussinesq system with moisture and phase changes, and in one of the main results, we found a piecewise-quadratic, positive definite conserved energy. For this same system, we also considered a second definition of an energy, based on a potential energy Π that generalizes a *ρgz*-like potential energy. While the Π-based energy is not manifestly positive definite, it was shown to be equal to the piecewise-quadratic energy, plus some additional material invariant terms.

Two aspects are perhaps most important in generalizing the dry quadratic energy to the moist piecewise-quadratic energy. First, by using Heaviside functions *H*_{u} and *H*_{s} to represent phase changes, it becomes relatively straightforward to account for the distinction between the two phases. For example, the Heaviside functions are useful in carrying out the integration in the Π-based potential energy, which then demonstrates the relationship between the Π-based energy and the piecewise-quadratic energy. Second, the contribution of the variable *M* is related to an additional eigenmode of the moist system that is not present in the dry system, and it is natural that one component of the moist energy would be associated with this additional eigenmode.

Increasing amounts of complexity were then incorporated beyond the moist Boussinesq equations. As a first step, in section 3, the moist anelastic equations were considered. For the anelastic system, the energy is not quadratic, even in the dry case. Nevertheless, it was shown that the potential energy can be decomposed into three components: buoyant potential energy in the unsaturated phase, buoyant potential energy in the saturated phase, and a moist latent energy that is analogous to the *M* contribution from the Boussinesq case. In adding further complexity, we considered Kessler warm-rain microphysics, in which case we cannot in general obtain a conserved energy due to the presence of source terms representing rain and cloud processes. As a final case, we considered FARE microphysics, which is a simplified version of warm-rain microphysics. The case of FARE microphysics is interesting because, like Kessler microphysics, it includes precipitation; however, unlike Kessler microphysics, FARE microphysics was shown to have a piecewise-quadratic energy, and the energy is conserved, aside from a sign-definite dissipation term due to precipitation at the surface.

As an application, we considered an empirical orthogonal analysis that uses a piecewise-quadratic energy as weighted norm, in place of the standard *L*^{2} norm. In comparing the leading EOFs produced by these two cases (i.e., by using piecewise-quadratic norm versus standard *L*^{2} norm), substantially different EOF modes were seen. The piecewise-quadratic norm is a weighted norm that incorporates information about the phase changes, and, as such, it produces EOFs that are representative of cloud variability. It would be interesting in the future to further investigate and refine this type of weighted norm and its applicability to observational and computational datasets.

## Acknowledgments

The authors thank Rupert Klein and two anonymous reviewers for helpful comments. This research is partially supported by NSF Grants AGS-1443325 and DMS-1907667 and by the University of Wisconsin–Madison Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation.

## APPENDIX A

### Reformulation of Moist Boussinesq Equations in Terms of Unsaturated and Saturated Buoyancies and Extension to the Kessler Scheme

*p*′ is the pressure,

*ρ*

_{0}is a constant background density,

*ϕ*=

*p*′/

*ρ*

_{0},

**u**(

**x**,

*t*) is the velocity vector,

*θ*

_{e}is the equivalent potential temperature anomaly, and

*q*

_{t}is the anomalous total water mixing ratio. The buoyancy,

*b*, can be written in terms of

*θ*

_{e},

*q*

_{t}, and

*z*, and will be described below. We assume that all thermodynamic variables have been decomposed into background functions of height and anomalous parts, so that, for example,

*q*

_{t}is the sum of water vapor

*q*

_{υ}and liquid water

*q*

_{l}and can be written as

*q*

_{υ}and

*q*

_{l}, we adopt a prescribed saturation mixing ratio

*T*

^{tot}and

*p*

^{tot}are close to the background states

*θ*is the potential temperature, the latent heat factor is

*L*

_{υ}≈ 2.5 × 10

^{6}J kg

^{−1}, and specific heat is

*c*

_{p}≈ 10

^{3}J kg

^{−1}K

^{−1}; this definition of

*θ*

_{e}is a linearization of usual definitions (e.g., Emanuel 1994; Stevens 2005), and this linearized version is helpful for the exposition here. Using the equations above, we can obtain an expression for the potential temperature:

*b*is defined as

*θ*

_{0}≈ 300 K is a constant background potential temperature,

*g*≈ 9.8 m s

^{−2}is the acceleration due to gravity and

*R*

_{υd}= (

*R*

_{υ}/

*R*

_{d}) − 1 ≈ 0.61, where

*R*

_{d}is the gas constant for dry air and

*R*

_{υ}is the gas constant for water vapor. The buoyancy changes form based on phase, so that we can write

*b*

_{u}and

*b*

_{s}are defined to be the buoyancies in unsaturated and saturated regions, respectively. The functions

*H*

_{u}and

*H*

_{s}are Heaviside functions that indicate unsaturated and saturated phases, respectively, and are defined as

*H*

_{s}= 1 −

*H*

_{u}. Using the equations above, we can show that

*b*

_{u}and

*b*

_{s}are defined as

*b*

_{u}and

*b*

_{s}is that it leads to a particularly simple phase interface condition;

*b*

_{u}=

*b*

_{s}. This can be seen by observing that when

*q*

_{t}=

*q*

_{υs}, (A9a) and (A9b) are equal.

*q*

_{υ}and cloud water

*q*

_{c}are

*q*

_{t}−

*q*

_{r}≥

*q*

_{υs}, and in unsaturated regions,

*q*

_{t}−

*q*

_{r}<

*q*

_{υs}, and it follows that the buoyancy in the unsaturated and saturated regions, are, respectively,

*b*

_{u}and

*b*

_{s}is still conveniently expressed as

*b*

_{u}=

*b*

_{s}.

## APPENDIX B

### Integrating **Π** for the Boussinesq Equations

*H*

_{u}=

*H*(

*b*

_{u}−

*b*

_{s}).

*a*,

*z*], which is

*H*(

*z*′ −

*a*) −

*H*(

*z*′ −

*z*), and integrate over the entire real line. Doing this ensures that the zero of the delta function’s argument,

Notice that, on the right-hand side, the first three terms are components of the piecewise quadratic energy, and the rest of the terms are functions of three material invariants: *M*, *a* = *z*_{r} would result in *E*_{1} = *E*_{2}, where *E*_{1} is given by (21) and *E*_{2} by (26).

## REFERENCES

Andrews, D. G., 1981: A note on potential energy density in a stratified compressible fluid.

,*J. Fluid Mech.***107**, 227–236, https://doi.org/10.1017/S0022112081001754.Bretherton, C. S., 1987: A theory for nonprecipitating moist convection between two parallel plates. Part I: Thermodynamics and “linear” solutions.

,*J. Atmos. Sci.***44**, 1809–1827, https://doi.org/10.1175/1520-0469(1987)044<1809:ATFNMC>2.0.CO;2.Chen, S., and S. N. Stechmann, 2016: Nonlinear traveling waves for the skeleton of the Madden–Julian oscillation.

,*Commun. Math. Sci.***14**, 571–592, https://doi.org/10.4310/CMS.2016.v14.n2.a11.Cuijpers, J. W. M., and P. G. Duynkerke, 1993: Large eddy simulation of trade wind cumulus clouds.

,*J. Atmos. Sci.***50**, 3894–3908, https://doi.org/10.1175/1520-0469(1993)050<3894:LESOTW>2.0.CO;2.Emanuel, K. A., 1994:

. Oxford University Press, 592 pp.*Atmospheric Convection*Frierson, D. M. W., A. J. Majda, and O. M. Pauluis, 2004: Large scale dynamics of precipitation fronts in the tropical atmosphere: a novel relaxation limit.

,*Commun. Math. Sci.***2**, 591–626, https://doi.org/10.4310/CMS.2004.v2.n4.a3.Grabowski, W. W., and T. L. Clark, 1993: Cloud-environment interface instability. Part II: Extension to three spatial dimensions.

,*J. Atmos. Sci.***50**, 555–573, https://doi.org/10.1175/1520-0469(1993)050<0555:CEIIPI>2.0.CO;2.Grabowski, W. W., and P. K. Smolarkiewicz, 1996: Two-time-level semi-Lagrangian modeling of precipitating clouds.

,*Mon. Wea. Rev.***124**, 487–497, https://doi.org/10.1175/1520-0493(1996)124<0487:TTLSLM>2.0.CO;2.Hernandez-Duenas, G., A. J. Majda, L. M. Smith, and S. N. Stechmann, 2013: Minimal models for precipitating turbulent convection.

,*J. Fluid Mech.***717**, 576–611, https://doi.org/10.1017/jfm.2012.597.Hernandez-Duenas, G., L. M. Smith, and S. N. Stechmann, 2015: Stability and instability criteria for idealized precipitating hydrodynamics.

,*J. Atmos. Sci.***72**, 2379–2393, https://doi.org/10.1175/JAS-D-14-0317.1.Hernandez-Duenas, G., L. M. Smith, and S. N. Stechmann, 2019: Weak- and strong-friction limits of parcel models: Comparisons and stochastic convective initiation time.

,*Quart. J. Roy. Meteor. Soc.***145**, 2272–2291, https://doi.org/10.1002/qj.3557.Holliday, D., and M. E. McIntyre, 1981: On potential energy density in an incompressible, stratified fluid.

,*J. Fluid Mech.***107**, 221–225, https://doi.org/10.1017/S0022112081001742.Ingersoll, A. P., 2005: Boussinesq and anelastic approximations revisited: Potential energy release during thermobaric instability.

,*J. Phys. Oceanogr.***35**, 1359–1369, https://doi.org/10.1175/JPO2756.1.Kessler, E., 1969:

*On the Distribution and Continuity of Water Substance in Atmospheric Circulations. Meteor. Monogr.*, No. 32, Amer. Meteor. Soc., 84 pp.Kuo, H. L., 1961: Convection in conditionally unstable atmosphere.

,*Tellus***13**, 441–459, https://doi.org/10.3402/tellusa.v13i4.9516.Lorenz, E. N., 1955: Available potential energy and the maintenance of the general circulation.

,*Tellus***7**, 157–167, https://doi.org/10.3402/tellusa.v7i2.8796.Lorenz, E. N., 1978: Available energy and the maintenance of a moist circulation.

,*Tellus***30**, 15–31, https://doi.org/10.3402/tellusa.v30i1.10308.Moncrieff, M. W., and M. J. Miller, 1976: The dynamics and simulation of tropical cumulonimbus and squall lines.

,*Quart. J. Roy. Meteor. Soc.***102**, 373–394, https://doi.org/10.1002/qj.49710243208.Pauluis, O., 2007: Sources and sinks of available potential energy in a moist atmosphere.

,*J. Atmos. Sci.***64**, 2627–2641, https://doi.org/10.1175/JAS3937.1.Pauluis, O., 2008: Thermodynamic consistency of the anelastic approximation for a moist atmosphere.

,*J. Atmos. Sci.***65**, 2719–2729, https://doi.org/10.1175/2007JAS2475.1.Pauluis, O., and J. Schumacher, 2010: Idealized moist Rayleigh–Bénard convection with piecewise linear equation of state.

,*Commun. Math. Sci.***8**, 295–319, https://doi.org/10.4310/CMS.2010.v8.n1.a15.Pauluis, O., and J. Schumacher, 2011: Self-aggregation of clouds in conditionally unstable moist convection.

,*Proc. Natl. Acad. Sci. USA***108**, 12 623–12 628, https://doi.org/10.1073/pnas.1102339108.Rauber, R. M., and Coauthors, 2007: Rain in shallow cumulus over the ocean: The RICO campaign.

,*Bull. Amer. Meteor. Soc.***88**, 1912–1928, https://doi.org/10.1175/BAMS-88-12-1912.Seifert, A., and K. D. Beheng, 2001: A double-moment parameterization for simulating autoconversion, accretion and self-collection.

,*Atmos. Res.***59–60**, 265–281, https://doi.org/10.1016/S0169-8095(01)00126-0.Seifert, A., and K. D. Beheng, 2006: A two-moment cloud microphysics parameterization for mixed-phase clouds. Part 1: Model description.

,*Meteor. Atmos. Phys.***92**, 45–66, https://doi.org/10.1007/s00703-005-0112-4.Smith, L. M., and S. N. Stechmann, 2017: Precipitating quasigeostrophic equations and potential vorticity inversion with phase changes.

,*J. Atmos. Sci.***74**, 3285–3303, https://doi.org/10.1175/JAS-D-17-0023.1.Stansifer, E. M., P. A. O’Gorman, and J. I. Holt, 2017: Accurate computation of moist available potential energy with the Munkres algorithm.

,*Quart. J. Roy. Meteor. Soc.***143**, 288–292, https://doi.org/10.1002/qj.2921.Stechmann, S. N., 2014: Multiscale eddy simulation for moist atmospheric convection: Preliminary investigation.

,*J. Comput. Phys.***271**, 99–117, https://doi.org/10.1016/j.jcp.2014.02.009.Stechmann, S. N., and A. J. Majda, 2006: The structure of precipitation fronts for finite relaxation time.

,*Theor. Comput. Fluid Dyn.***20**, 377–404, https://doi.org/10.1007/s00162-006-0014-1.Stevens, B., 2005: Atmospheric moist convection.

,*Annu. Rev. Earth Planet. Sci.***33**, 605–643, https://doi.org/10.1146/annurev.earth.33.092203.122658.Stevens, B., 2007: On the growth of layers of nonprecipitating cumulus convection.

,*J. Atmos. Sci.***64**, 2916–2931, https://doi.org/10.1175/JAS3983.1.Stevens, B., and Coauthors, 2005: Evaluation of large-eddy simulations via observations of nocturnal marine stratocumulus.

,*Mon. Wea. Rev.***133**, 1443–1462, https://doi.org/10.1175/MWR2930.1.Tailleux, R., 2013: Available potential energy density for a multicomponent Boussinesq fluid with arbitrary nonlinear equation of state.

,*J. Fluid Mech.***735**, 499–518, https://doi.org/10.1017/jfm.2013.509.Vallis, G., 2006:

*Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation*. Cambridge University Press, 745 pp.Vallis, G. K., D. J. Parker, and S. M. Tobias, 2019: A simple system for moist convection: The Rainy–Bénard model.

,*J. Fluid Mech.***862**, 162–199, https://doi.org/10.1017/jfm.2018.954.Wetzel, A. N., L. M. Smith, S. N. Stechmann, and J. E. Martin, 2019: Balanced and unbalanced components of moist atmospheric flows with phase changes.

, in press.*Chin. Ann. Math. B*Young, W. R., 2010: Dynamic enthalpy, conservative temperature, and the seawater Boussinesq approximation.

,*J. Phys. Oceanogr.***40**, 394–400, https://doi.org/10.1175/2009JPO4294.1.