Non-conservation and conservation for different formulations of moist potential vorticity

Potential vorticity (PV) is one of the most important quantities in atmospheric science. The PV of each fluid parcel is known to be conserved in the case of a dry atmosphere. However, a parcel's PV is not conserved if clouds or phase changes of water occur. Recently, PV conservation laws were derived for a cloudy atmosphere, where each parcel's PV is not conserved but parcel-integrated PV is conserved, for integrals over certain volumes that move with the flow. Hence a variety of different statements are now possible for moist PV conservation and non-conservation, and in comparison to the case of a dry atmosphere, the situation for moist PV is more complex. Here, in light of this complexity, several different definitions of moist PV are compared for a cloudy atmosphere. Numerical simulations are shown for a rising thermal, both before and after the formation of a cloud. These simulations include the first computational illustration of the parcel-integrated, moist PV conservation laws. The comparisons, both theoretical and numerical, serve to clarify and highlight the different statements of conservation and non-conservation that arise for different definitions of moist PV.


| INTRODUCTION
Potential vorticity (PV) is one of the central conserved quantities in geophysical fluid dynamics (Müller, 1995;Salmon, 1998), with its roots traced back to about a century ago in the works of Rossby (1939) and Ertel (1942).The PV conservation law also has a deep connection to the classic Kelvin and Bjerknes' circulation theorems (Bjerknes, 1898;Thomson, 1867;Thorpe et al., 2003).Pointwise conservation enables the use of PV as a tracer of fluid parcels.PV also possesses an inversion principle that allows one to recover the slowly varying component of the wind and temperature fields from the PV distribution with appropriate boundary conditions (Hoskins et al., 1985;Martin, 2013).Owing to these properties, PV has been used extensively to study the dynamics of synoptic and mesoscale weather systems (Hoskins et al., 1985;Thorpe, 1985;Davis and Emanuel, 1991;Lackmann, 2002) and also ocean circulations (Holland et al., 1984;Rhines, 1986;Pollard and Regier, 1990;Marshall and Nurser, 1992;Taylor and Ferrari, 2010;Ruan et al., 2021).
Note, though, that moist PV is not conserved for each fluid parcel, and inversion of moist PV is problematic (e.g., see the sequence of three studies of Cao and Cho, 1995;Schubert et al., 2001;Wetzel et al., 2020).The traditional conservation and inversion properties of dry PV are for idealized single-component flows, not for the more realistic cases of binary or multi-component fluids such as a moist atmosphere with clouds and phase changes, and salty oceans.
For inversion, the moist case is different than the dry case in several ways.For instance, the balanced portion is comprised of not one but two components (PV and M, where M represents a slow moist component), and correspondingly it is not PV inversion but PV-and-M inversion that recovers the balanced portion of the system (Smith and Stechmann, 2017;Wetzel et al., 2019Wetzel et al., , 2020;;Remond-Tiedrez et al., 2023).Also, among many different moist PV quantities that have been used, only certain moist PV quantities are slowly evolving in the presence of phase changes of water and cloud latent heating (Wetzel et al., 2020;Zhang et al., 2021aZhang et al., ,b, 2022)).
For conservation, recently, we generalized the PV conservation laws to cases with phase changes of water, for both a compressible flow (Kooloth et al., 2022) and a Boussinesq flow (Kooloth et al., 2023).We showed that moist PV is not pointwise conserved as in a dry atmosphere; instead it is conserved over certain 'material' volumes that move with flow.Such conservation laws hold for many, but not all, moist PV quantities.
The purpose of this letter is to present a detailed comparison of different statements of PV conservation and non-conservation for various definitions of moist PV.One part of this comparison is the first numerical illustration of the parcel-integrated PV conservation law.From these comparisons, we hope to bring some clarity to the complex landscape of cases including dry versus moist PV, conservation and non-conservation, and parcel-wise versus parcelintegrated conservation.
In what follows, the equations of PV conservation and non-conservation are described for a compressible atmosphere (section 2) and under the Boussinesq approximation (section 3).The setup of the numerical simulation and the simulation results are also presented in section 3. Finally, section 4 includes a concluding discussion and summary of the laws of conservation and non-conservation.

| COMPARISON OF PV CONSERVATION LAWS
In this section, we compare a variety of statements of conservation and non-conservation of moist potential vorticity, including recently discovered conservation laws that apply for binary or multi-component fluids such as an ocean with salinity or an atmosphere with water vapor, and even in the presence of phase changes and clouds (Kooloth et al., 2022(Kooloth et al., , 2023)).
The dry case without water vapor is considered in section 2.1, where PV is conserved for each fluid parcel.Then the moist case with clouds and phase changes is considered, where each fluid parcel's PV is not conserved (section 2.2), but where a local-volume-integrated PV is conserved (section 2.3).
The setting in this section is a compressible atmosphere.See section 3 below for an alternative setting under the Boussinesq approximation.
For the evolution equations and assumptions, for velocity ì u = (u, v , w ) and density ρ, the form of the equations is the same for both the dry and moist cases: where = (∂ x , ∂ y , ∂ z ) is the gradient operator, D /D t = ∂/∂t + ì u • is the material derivative, p is the pressure and φ is the force potential, which could include, for instance, the gravitational potential.In (1) a case without dissipation (friction, viscosity, etc.) is assumed.Rotation could be added with some modifications (Kooloth et al., 2022) but is left out for simplicity here.Also, in (1), the density ρ and pressure p should be interpreted as total density and total pressure, for the dry case and also for the moist case, so that the form of (1) is the same in both cases.To complete the specification of the dynamical evolution, equations are also needed for thermodynamic quantities.
For a dry atmosphere, the thermodynamic evolution equation can be described in terms of potential temperature θ as and an equation of state, θ = θ (p, ρ ).The evolution is assumed to be (dry) adiabatic.
For a moist (and possibly cloudy) atmosphere, the thermodynamic evolution equations can be described in terms of equivalent potential temperature θ e and total water specific humidity q t as along with an equation of state, θ e = θ e (p, ρ, q t ).The evolution is assumed to be (moist) adiabatic, with reversible phase changes between water vapor and liquid water, as in warm, liquid clouds.No rain, ice, nor precipitation are considered.The total water q t can be decomposed as q t = q v + q l , where q v and q l could be recovered from q t by comparison against the saturation specific humidity, q v s = q v s (T , p ), where T is temperature.We refer to states with q t < q v s and q t ≥ q v s as the unsaturated and saturated phases, respectively.
In what follows, we consider various definitions of moist PV that have been proposed in the past.All definitions involve the vorticity, ì ω = × ì u.In analogy with dry PV, ( ì ω • θ )/ρ, the various moist PV definitions are distinguished by which thermodynamic quantity is used in ì ω • ψ, where common choices of ψ include potential temperature θ, virtual potential temperature θ v , equivalent potential temperature θ e , or liquid water potential temperature θ l .

| Dry PV is conserved for each fluid parcel
Dry PV, ( ì ω • θ)/ρ, is a material invariant-i.e., conserved for each fluid parcel.To see this, the starting point is its evolution equation: which can be derived by using ( 1) and (2) (see, e.g., Kooloth et al., 2022).
To see that the "solenoidal" term on the right-hand side of ( 4) is zero for a dry atmosphere, recall a fundamental property of thermodynamics for a dry atmosphere: the potential temperature (or any other thermodynamic property) can be expressed as a function of pressure and density only, so that θ = θ (p, ρ ).Consequently, we have and it follows that the right-hand side of ( 4) is zero, so that Hence, dry PV is conserved for each fluid parcel.

| Moist PV is not conserved for each parcel, due to clouds
We now describe several common choices of moist PV definitions (based on θ, θ v , θ e , and θ l ), and show how each one is not a material invariant, in the presence of clouds and phase changes.
As a way to encapsulate any definition of moist PV, consider a moist PV defined as ( ì ω • ψ )/ρ, for a generic thermodynamic quantity ψ.Assume that the evolution of ψ is given by Dψ/D t = ψ, where ψ represents all sources/sinks of ψ.Then the evolution of the generic moist PV is given by which follows from (1) (see, e.g., Schubert et al., 2001;Kooloth et al., 2022).Two potential sources of non-conservation appear on the right-hand side in (7): the "solenoidal" term involving a cross product and the source term involving ψ.
First choose ψ = θ and consider PV θ based on potential temperature.From ( 7) its evolution equation is In an atmosphere with clouds and phase changes, (8) cannot be further simplified.The θ term arises from cloud latent heating and does not vanish, and the solenoidal term on the right hand side of (8) remains nonzero in (8), since, for a moist atmosphere with phase changes, the potential temperature is no longer completely determined by pressure and density.Consequently, PV θ is not a material invariant if clouds and phase changes are present.
Next consider PV θv based on the virtual potential temperature θ v .Its evolution equation is which follows from choosing ψ = θ v in (7).The solenoidal term has vanished and does not appear in (9)1 , which is one of the desirable properties of PV θv (Schubert et al., 2001).The right hand side of ( 9) still has a source term resulting from non-conservation of θ v , such as cloud latent heating, so that PV θv is not a material invariant in general.As a special case, though, while a parcel remains in the unsaturated phase, we have θ v = 0 due to the absence of latent heating, and consequently PV θv remains materially conserved in the unsaturated phase.
Another commonly used definition of moist potential vorticity is PV θe defined in terms of the equivalent potential temperature θ e .Note that θ e is a materially conserved quantity, i.e., D θ e /D t = 0. Hence, from ( 7), the evolution of PV θe is given by Here, in the case of PV θe , the non-conservation is due to the solenoidal term.
As a final common choice, one may also consider potential vorticity PV θ l defined in terms of liquid water potential temperature θ l .Note that θ l is a materially conserved quantity, i.e., D θ l /D t = 0. Hence, from ( 7), the evolution of and non-conservation of PV θ l is seen to be due to the solenoidal term.
In summary, the four moist PVs here involve four common potential temperature variables: θ, θ v , θ e , and θ l .
The discussion above serves to illustrate the different properties of the four cases.They are all non-conservative in different ways, due to either sources/sinks or the solenoidal term.For instance, θ v has a source term due to cloud latent heating, so PV θv is conserved (a material invariant) in the unsaturated phase but is not conserved in the saturated phase.On the other hand, the variables θ e and θ l are conserved, so non-conservation of PV θe and PV θ l is due to the solenoidal term only.Hence, it appears that there may not be a moist PV quantity that is a material invariant, due to clouds and phase changes.

| Moist PV is conserved over certain local volumes, even with clouds
While moist PV may be non-conservative for each fluid parcel, it has been recently shown by Kooloth et al. (2022Kooloth et al. ( , 2023) ) that moist PV can be conserved when integrated over certain local volumes (for PV θe or PV θ l , but not PV θ nor PV θv ).Here we sketch the key ideas of these conservation principles.
As motivation for integrating over local volumes, start with the PV θe evolution equation in (10), rewritten as 2 Given that a divergence appears on the right-hand side, one might try to integrate in order to remove this divergence term.
Pursuing this direction, we integrate (12) over a material volume 3 V m (i.e., a volume that moves with the fluid flow) and use the divergence theorem to arrive at where S m is the material surface that bounds the material volume.The right-hand side is still non-zero, so conservation has not yet been demonstrated.
To simplify the right-hand side of ( 13), first choose the material volume V m to be a distorted cylinder with base and lid given by surfaces of constant θ e (say θ e = θ e1 and θ e = θ e2 , respectively) and sides given by q t = q t (θ e ).(An illustration of such a cylinder is shown below in section 3.) On the base and lid, d ì S ∥ θ e i.e., the normal to the surface is parallel to θ e and therefore the surface integral over S m is the same as a surface integral over only the sides of the cylinder: Also, to obtain the last line above, a second key observation, from fundamentals of moist thermodynamics, is needed: ρ can be written as ρ = ρ (p, θ e , q t ) as a function of the three moist thermodynamic quantities (p, θ e , q t ).Furthermore, since q t = q t (θ e ) on the sides of the cylinder, we have ρ = ρ (p, θ e , q t (θ e ) ) and ρ is a function of p and θ e alone.It follows that ( 14) and ( 15) are equal for a function g (p, θ e ) that satisfies ∂g /∂θ e = 1/ρ (p, θ e , q t (θ e ) ).
To complete the derivation, by using Stokes theorem, the surface integral in (15) can be converted to two closed line integrals along the edges of the cylinder, C 1 and C 2 , and we have By noting that θ e is a constant on both C 1 and C 2 , the integrands above reduce to exact differentials which integrate to zero on the closed curves.This gives us our final result, of conservation of PV θe when integrated over certain local material volumes. 4 A similar conservation law can be derived for PV θ l or PV based on entropy or even PV qt (Kooloth et al., 2022).
Using vector calculus identities, note that − θe Recall that, for a material volume, we have It is an open question to understand how general the material volumes could be.See Kooloth et al. (2022Kooloth et al. ( , 2023) ) for some other known examples.
The key property shared by θ e , θ l , s (entropy), and q t is that they are all material invariants.On the other hand, θ and θ v are not material invariants in the presence of phase changes and clouds, and hence the derivation above does not hold for PV θ nor PV θv .

| NUMERICAL SIMULATIONS OF PV CONSERVATION AND NONCONSER-VATION
For numerical demonstration of the conservation laws from section 2, we will set aside the compressible setting that includes acoustic/sound waves and use the simpler setting of the Boussinesq approximation.The governing equations under the Boussinesq approximation are described in the Supporting Information, and they are similar to equations of moist Boussinesq dynamics that have been used in other studies (e.g., Kuo, 1961;Bretherton, 1987;Grabowski and Clark, 1993;Pauluis and Schumacher, 2010;Stechmann and Stevens, 2010;Stechmann, 2014;Hernandez-Duenas et al., 2013;Marsico et al., 2019).The Boussinesq case admits statements of PV conservation and non-conservation (Kooloth et al., 2023) that are analogous to the compressible case from section 2. A summary is as follows.
As a particular moist PV quantity for illustration, we use PV u which is based on the total buoyancy b u in the unsaturated phase: The evolution of PV u is then given by The buoyancy b ′ depends on the phase and is nonconservative.Consequently, it appears as a source term in the evolution of PV u , which is then also nonconservative.
As a special case, though, note that the right-hand-side goes to zero in the unsaturated phase since b u • ( × b ′ u H u ẑ ) = 0, and therefore so that PV u is pointwise conserved for any parcels that are not inside a cloud.
For a general scenario involving phase changes, following similar steps as presented for the compressible case in section 2.3, a parcel-integrated PV u conservation principle can be obtained: which follows from integrating (19) over a material volume V m that is a distorted cylinder whose base and lid are given by b u = const .and the sides are given by b s = const ., where b s is the total buoyancy in the saturated phase.This PV u conservation statement can also be shown to be valid for a material volume enclosed by isosurfaces of the more physically relevant quantities, θ e and q t .The main ideas of the derivation are the same as in the compressible case; the interested reader can refer to supporting information for the detailed derivation.
Parcel-integrated conservation principles can also be derived for many other potential vorticity quantities for the Additionally, it can be shown that PV s is materially conserved in the saturated phase using a similar reasoning as for PV u in the unsaturated phase (Kooloth et al., 2023).
F I G U R E 1 Initial conditions for the 3D rising thermal.Plots are shown in the (x, z ) plane with fixed y = 0.5L y = 3500 m.Anomalies of potential temperature θ (left) and total water specific humidity q t (right), defined as anomalies from a horizontally uniform background state, θ (z ) and qt (z ).The units for x and z are meters, and the units for θ and q t are K and kg/kg respectively.
The three-dimensional (3D) numerical simulations in this study are performed using the code of Hernandez-

Duenas et al. (2013)
. The channel domain is periodic in the x and y directions, and assumes a rigid top and bottom.A third-order Runge-Kutta scheme with adaptive time-stepping (CFL = 0.9) is used for time integration.Spatial discretization is based on pseudo-spectral decomposition using Fourier modes in the horizontal directions, and 2ndorder centered differences on a staggered grid in the vertical direction.In our study, the domain size L x × L y × H is 8000 m × 8000 m × 7000 m and the number of grid points is 256 × 256 × 400, corresponding to horizontal and vertical grid spacings of 31.5 m and 17.5 m, respectively.
The case study for illustration is the well-known case of a rising moist thermal, and the basic aspects of the simulation are as follows.A horizontal slice through the initial, unsaturated, spherical perturbation is shown in Figure 1 (at y = 3500 m).Evolution of the perturbation in an (x, z ) plane is shown in Figure 2.For t > 0, the warm bubble rises due to its buoyancy and a non-zero velocity field develops.In the plane at y = 3500 m, the vapor starts to condense near the center of the plane at t ≈ 1.8 min, contributing to the formation of a 3D cloud (Figures 2 and 3).
The size of the cloud grows as more fluid parcels change phase (see Figure 2 at t = 3 min and Figure 3 at t = 1.2 min).

| Local-volume integrated PV u conservation
In order to verify volume-integrated PV u conservation, a material volume is identified and tracked over time in the rising bubble simulation described above.The material volume consists of roughly 5000 grid cells and is specified by certain level surfaces for θ e and q t , such that their intersection encloses a moving material volume.As shown in Figure 3, at the earlier times of t = 0 min and t = 0.6 min, the fluid parcels in the material volume are unsaturated.By F I G U R E 2 Snapshots of the warm bubble in the plane with fixed y = 0.5L y = 3500 m.Rows show θ − θ, q l , u, w and PV u and columns show times t = 0.6, 1.8 and 3 minutes.The units for x and z are meters, and the units for θ, velocities (u, v , w ), q l and PV u are K, m/s, kg/kg and s −3 respectively.

Material curve
Material volume F I G U R E 3 Illustration of a material volume and material curve, at times t = 0, 0.6, and 1.2 min.(Top row) Evolution of a material volume enclosed between θ e > 326 K and 2.0 × 10 −3 < q t < 5.75 × 10 −3 (in kg/kg).The red and the cyan dots represent the unsaturated and saturated parcels respectively.(Bottom row) Evolution of a closed material curve of θ e ≈ 326.48K and q t ≈ 5.66 × 10 −3 kg/kg.
the later time of t = 1.2 min, however, a cloud of liquid water has developed in upper levels of the volume, and 47% of the fluid parcels within the material volume have undergone a change of phase in their water content, from water vapor to liquid water.Concurrently, within the cloud, the local values of PV u are evolving (see Figure 2).
On the other hand, we can numerically check for local volume-integrated PV u by computing where V m is the specified material volume.Noting that the initial velocity field is zero everywhere in the domain, then the I PV should start and stay at the value zero to machine precision.By monitoring ( 22) in time, we found that the IV P within the material volume remains O (10 −16 ) for the entire simulation, verifying the conservation statement for parcel-integrated PV.

| Material PV u conservation and non-conservation
In this section, we investigate the material conservation of PV u .It is shown that PV u is materially conserved prior to the time when fluid parcels undergo phase change from vapor to liquid, but not for later times after the cloud has formed.We identify and track a closed material curve within the material volume, which are both rising along with the bubble, as shown on the second row of Figure 3.The material curve is initially unsaturated, but by t = 1.2 min, all the parcels composing the material curve have undergone a change of phase.
To quantify material conservation and non-conservation, we measured the maximum of the absolute value of PV u F I G U R E 4 Demonstration of parcel-wise non-conservation of PV u (s −1 ), due to phase change between t = 0.6 and 1.2 min.Evolution of PV u on the closed material curve of θ t ot e ≈ 326.48K and q t ot t ≈ 0.00566 kg/kg.The points on the material curve are ordered using the arc length coordinate.
within the material volume (Figure 3).At representative early time t = 0.6, before a significant number of parcels have experienced a change of water phase, the max absolute value of PV u within the entire material volume is 6.8 × 10 −11 s −3 .In comparison, at later time t = 1.2 min, by which time a robust cloud has formed, the max absolute value of PV u within the volume has increased by roughly 3 orders of magnitude, obtaining the value 6.8 × 10 −8 s −3 .As a graphical illustration, Figure 4 shows the PV u values along the selected material curve.The figure shows PV u associated with each grid point along the curve, at three different times t = 0, 0.6, 1.2 minutes.At t = 0 and t = 0.6 min, when the material curve is unsaturated, the PV u values are close to zero.At t = 1.2 min, some PV u values associated with the material curve are of the order of 10 −8 s −1 , demonstrating that PV u does not remain conserved along the material curve once the parcels undergo change of water phase.

| CONCLUSIONS
In this work, a main goal was to compare and contrast the different moist PV conservation statements and nonconservation statements, in light of the rich variety of possibilities in the literature, including recent developments (e.g.Bennetts and Hoskins, 1979;Emanuel, 1979;Schubert et al., 2001;Marquet, 2014;Kooloth et al., 2022Kooloth et al., , 2023)).
As a summary, Table 1 lists the different formulations of PV considered here and their conservation laws.We note that in a moist flow with phase changes, there are no material invariant PVs; the strongest conservation principle realizable in this setting is an integrated PV invariance for certain material volumes.In both the fully compressible and Boussinesq cases, PVs based on θ e and θ l possess a material-volume-integrated conservation principle, even with phase transitions.In the moist Boussinesq setting, PV u and PV s , based on the unsaturated and saturated buoyancy, respectively, also have an integrated PV invariance principle within certain material volumes even with phase changes.
Additionally, as special cases, PV u and PV s remain materially invariant in the unsaturated and saturated phases respectively.
A more complete set of desirable properties is often associated with PV, including conservation, inversion, and TA B L E 1 Summary of various PV formulations and their conservation, inversion, and balance/slow variation properties.V m refers to a material volume.

PV definition Material invariant Material-volume-integrated invariant Inversion
balance/slow variation.Table 1 also summarizes these other properties.As one note about inversion and balance/slow variation, since PV θv is not slowly varying, any inversion with PV θv will not completely remove the fast wave contributions in the presence of phase changes (Wetzel et al., 2020).The moist PV quantities that are defined in terms of conserved thermodynamic variables (e.g., θ e or θ l ) are slowly evolving, and they are associated with a PV-and-M inversion principle (see the discussion of inversion in section 1).
One application of parcel-integrated PV conservation laws is for diagnosing diabatic processes in the atmosphere or ocean (see discussion in section 1).In the past, (parcel-wise) PV non-conservation was often used to indicate cloud latent heating, as a leading diabatic process.In contrast, for the new conservation laws of PV as a parcel-integrated invariant, the conservation law holds even in the presence of phase changes and cloud latent heating; consequently, any non-conservation must be due to other diabatic processes, such as friction/viscosity, radiative cooling/heating, or precipitation.Therefore, the parcel-integrated conservation laws can potentially provide new information about other diabatic processes in diagnostic studies.

acknowledgements
The authors thank Gerardo Hernandez-Duenas for providing the computer code used for the simulations.L.M. Smith and S.N.Stechmann gratefully acknowledge support from US NSF grant DMS-1907667.
Boussinesq system.One such quantity is potential vorticity PV s = ì ω • b s based on the total saturated buoyancy; two others are PV e = ì ω • θ e based on equivalent potential temperature θ e , and PV l = ì ω • θ l based on liquid water potential temperature θ l .
unsaturated V m PV s = ì ω • b s saturated phase certain V m , even with phase changes, PV-and-M Yes and also all saturated V m PV e = ì ω • θ e nowhere certain V m , even with phase changes PV-and-M Yes PV l = ì ω • θ l nowhere certain V m , even with phase changes PV-and-M Yes * to recover the balanced or slowly evolving component(s) of the system * * with phase changes in the moist cases.