# Scaling approaches to quasi-geostrophic theory for moist, precipitating air

###### Abstract

Quasi-geostrophic (QG) theory is of fundamental importance in the study of
large-scale atmospheric flows. In recent years, there has been growing
interest in extending the classical QG plus Ekman friction layer model (QG-
Ekman) to systematically include additional physical processes known to
significantly contribute to real-life weather phenomena. This paper lays the
foundation for combining two of these developments, namely Smith and
Stechmann’s family of _Precipitating Quasi-Geostrophic_ (PQG) models (J.
Atmos. Sci, 74, 3285–3303, 2017) on the one hand, and the extension of QG-
Ekman for dry air by a strongly _Diabatic Layer_ (DL) of intermediate height
(QG-DL-Ekman) in (J. Atmos. Sci, 79, 887–905, 2022) on the other hand. To this
end, Smith and Stechmann’s PQG equations for sound-proof motions are first
corroborated within a general asymptotic modeling framework starting from a
full compressible flow model. The derivations show that the PQG model family
is naturally embedded in the asymptotic hierarchy of scale-dependent
atmospheric flow models introduced by one of the present authors in (Ann. Rev.
Fluid Mech., 42, 249–274). Particular emphasis is then placed on an asymptotic
scaling regime for PQG that accounts for a generic Kessler-type bulk
microphysics closure and is compatible with QG-DL-Ekman theory. The detailed
derivation of a moist QG-DL-Ekman model is deferred to a future publication.

##  1 Introduction

Ever since its inception, the mathematical model provided by quasi-geostrophic
(QG) theory has proven highly successful as a streamlined setting for the
explanation of major features of large-scale atmospheric flow in the
midlatitudes. Its textbook derivation by scale analysis and asymptotic
expansion, as found, e.g., in (Pedlosky, 1987), is a beautiful example of the
interplay between theoretical meteorology and applied mathematics.

This well-established model, however, does not describe the contributions of
moisture to the large-scale flow in an explicit fashion: only balance
equations for dry air are prescribed, and the only way to integrate the
vitally important effect of latent heat on the energy budget without extending
the model itself is its parametrization as a heat source in the temperature
equation. This approach has been pursued, e.g., by De Vries et al. (2010) who
also provide a discussion of different parametrization schemes.

As far as actual extensions of the dry QG model are concerned, they still tend
to treat moisture as a supplement; the model proposed by Lapeyre and Held
(2004) is emblematic of this approach, in which the authors take QG for dry
air as the starting point and then formulate an equation for the mixing ratio
of water vapor based on certain ad hoc assumptions. Steps toward a more
systematic embedding of moisture into QG theory were made by Monteiro and
Sukhatme (2016). These authors _did_ include moisture in their scaling and
asymptotic analysis, but their derivation took place within the confines of a
one-layer shallow-water model.

Seeking a more generally valid extension of QG theory that supplies balance
equations for water vapor and precipitation and explicitly models the impact
of moisture on the energy budget therefore poses an interesting challenge.
Moreover, the mathematical derivation of such a model should proceed along the
same lines as that of the classical theory. To this date, to the best of the
authors’ knowledge, only one model has been proposed that meets these
criteria: the _precipitating quasi-geostrophic_ (PQG) equations of Smith and
Stechmann (2017). To clarify their key ideas, the authors first provided a
detailed derivation starting from a somewhat simplified cloud-resolving model,
the _fast autoconversion and rain evaporation_ (FARE) system of Hernandez-
Duenas et al. (2013). That model adopts the Boussinesq equations for the
dynamics, linearized thermodynamic relations, and the limit of fast conversion
of cloud into rain water. The authors point out, however, that the PQG
equations form an entire model family parameterized by the asymptotic scalings
assumed for the adopted flow models, e.g., Boussinesq or anelastic, the
thermodynamic equations of state, and the bulk microphysics closure (see
sections 6 and 9 of their paper).

Common to the QG and PQG models is the assumption of a strong background
stratification of the (equivalent) potential temperature which, true to the
well-known QG scalings, is equivalent to a small (moist) internal wave Froude
number. In this context, let us quote from Smith and Stechmann (2017): “…
locally in some regions, such as in the vicinity of fronts, assumptions of
strong moist stratification and/or classical QG scaling may not hold.” Flow
regimes of this kind are addressed explicitly by Klein et al. (2022) through
their QG-DL-Ekman triple-deck boundary layer theory. Within its additional
diabatic layer (DL) of intermediate height of $\sim\\!3\,\text{km}$, the
potential temperature is not restricted to small deviations from a given
background stratification but can freely evolve, even towards neutral
stratification, instead. This way, systematically stronger diabatic effects
are allowed for in the DL than those accounted for in the classical (P)QG
models. The QG-DL-Ekman theory has thus far been derived only for dry air
flows, though, and within our current ongoing work we aim to extend it to
include moist processes.

The present paper is our first step in this direction and it provides two main
contributions: In sections 4 and 5 we illuminate the relationship between the
PQG equations in their anelastic form (based on the FARE model for moist
processes) on the one hand, and the system for moist compressible flow with
the generic Kessler-type bulk microphysics closure of Hittmeir and Klein
(2018) on the other hand. To this end, we proceed by systematically deriving
the former from the latter utilizing asymptotic techniques. This will also
demonstrate how the PQG model family can be embedded naturally in the rich
hierarchy of known scale-dependent atmospheric models as discussed in (Klein,
2010).

This paper’s second main contribution is a self-contained derivation of a _a
particular version_ of PQG-type equations that is set up for the subsequent
inclusion in a moist QG-DL-Ekman triple-deck theory. The underlying scalings
combine features of several of the aforementioned versions of PQG, which is
why we place particular emphasis on the justification of and the reasoning
behind our scaling choices. In section 6, this leads indeed to a PQG model
that is tailor-made to be coupled with the diabatic layer of Klein et al.
(2022). We will further explore its implications in future publications. To
provide some first insight into the model’s characteristics, we discuss in
section 7 its related “omega equation”, which often serves to diagnose upward
vertical velocities in weather forecast model output.

Finally, the appendix sketches a derivation of the moist anelastic system of
Hernandez-Duenas et al. (2013). There, we also point to the key differences
between an asymptotic analysis that starts from a fully compressible system
and one that is based on the anelastic approximation. This, in particular,
serves to explain the occurrence of additional background terms in our
derivation of the buoyancy in section 5.

##  2 The governing equations

Our point of departure is a system that not only accurately describes
compressible flow, but also includes fairly detailed moist thermodynamics (see
Cotton et al. (2011) for a broad discussion of possible modelling approaches).
This model formulation goes back to Hittmeir and Klein (2018), and it includes
established bulk microphysics closures as proposed and investigated in
(Kessler, 1995), (Grabowski and Smolarkiewicz, 1996) and (Klein and Majda,
2006):

$\displaystyle
D_{t}\mathbf{u}+2(\mathbf{\Omega}\times\mathbf{v})_{\parallel}+\frac{1}{\rho}\gradient_{\parallel}{p}$
$\displaystyle=0,$ (1a) $\displaystyle
D_{t}w+2(\mathbf{\Omega}\times\mathbf{v})_{\perp}+\frac{1}{\rho}\partial_{z}p$
$\displaystyle=-g,$ (1b) $\displaystyle
D_{t}\rho_{d}+\rho_{d}(\divergence{\mathbf{v}})$ $\displaystyle=0,$ (1c)
$\displaystyle CD_{t}\ln\theta+\Sigma D_{t}\ln p+\frac{L(T)}{T}D_{t}q_{v}$
$\displaystyle=$ $\displaystyle c_{l}V_{r}q_{r}(\partial_{z}\ln\theta$
$\displaystyle+\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln p),$ (1d)
$\displaystyle D_{t}q_{v}$ $\displaystyle=S_{\text{ev}}-S_{\text{cd}},$ (1e)
$\displaystyle D_{t}q_{c}$
$\displaystyle=S_{\text{cd}}-S_{\text{cr}}-S_{\text{ac}},$ (1f) $\displaystyle
D_{t}q_{r}-\frac{1}{\rho_{d}}\partial_{z}(\rho_{d}V_{r}q_{r})$
$\displaystyle=S_{\text{cr}}+S_{\text{ac}}-S_{\text{ev}},$ (1g)

with the additional relations

$\displaystyle\rho$ $\displaystyle=\rho_{d}(1+q_{v}+q_{c}+q_{r}),$ (2a)
$\displaystyle p$
$\displaystyle=R_{d}\rho_{d}T(1+\frac{q_{v}}{\frac{R_{d}}{R_{v}}}),$ (2b)
$\displaystyle\theta$
$\displaystyle=T\left(\frac{p_{\text{ref}}}{p}\right)^{\frac{\gamma-1}{\gamma}},$
(2c) $\displaystyle\mathbf{v}$ $\displaystyle=\mathbf{u}+w\mathbf{k},$ (2d)
$\displaystyle S_{\text{ev}}$
$\displaystyle=C_{\text{ev}}\frac{p}{\rho}(q_{\text{vs}}-q_{v})^{+}q_{r},$
(2e) $\displaystyle S_{\text{cd}}$
$\displaystyle=C_{\text{cn}}(q_{v}-q_{\text{vs}})^{+}q_{\text{cn}}+C_{\text{cd}}(q_{v}-q_{\text{vs}})q_{c},$
(2f) $\displaystyle S_{\text{ac}}$
$\displaystyle=C_{\text{ac}}(q_{c}-q_{\text{ac}})^{+},$ (2g) $\displaystyle
S_{\text{cr}}$ $\displaystyle=C_{\text{cr}}q_{c}q_{r},$ (2h) $\displaystyle C$
$\displaystyle=c_{\text{pd}}+c_{\text{pv}}q_{v}+c_{l}(q_{c}+q_{r}),$ (2i)
$\displaystyle\Sigma$
$\displaystyle=(\frac{c_{\text{pv}}}{c_{\text{pd}}}R_{d}-R_{v})q_{v}+\frac{c_{l}}{c_{\text{pd}}}R_{d}(q_{c}+q_{r}),$
(2j) $\displaystyle L(T)$
$\displaystyle=L_{\text{ref}}+(c_{\text{pv}}-c_{l})(T-T_{\text{ref}})\equiv
L_{\text{ref}}\phi(T).$ (2k)

In the above equations,
$(\mathbf{u}=(u,v,0),w,\rho,\rho_{d},T,\theta,p,q_{v},q_{c},q_{r},q_{\text{vs}})$
denote horizontal and vertical velocity, density, dry air density,
temperature, potential temperature, pressure and the mixing ratios of water
vapor, cloud water and rain, as well as the saturation mixing ratio,
respectively; $g$ is the gravitational acceleration, $\Omega$ the earth
rotation vector, and the subscripts $\parallel$ and $\perp$ indicate
horizontal and vertical components, respectively. We denote the positive part
of a function $f$ by $f^{+}$. As usual, $c_{\text{pd}}$ and $c_{\text{pv}}$
denote the specific heat capacities at constant pressure of dry air and water
vapor, while $c_{l}$ is the heat capacity of liquid water, here assumed
constant for simplicity, and $V_{r}$ is the terminal rainfall velocity;
$R_{d}$ and $R_{v}$ are the gas constants for dry air and water vapor,
$\gamma=c_{\text{pd}}/(c_{\text{pd}}-R_{d})$ is the isentropic exponent of dry
air, $\mathbf{k}$ the vertical unit vector, $p_{\text{ref}}=10^{5}\text{ Pa}$
the reference pressure and the material derivative is given by

$D_{t}=\partial_{t}+\mathbf{v}\cdot\gradient=\partial_{t}+\mathbf{u}\cdot\gradient_{\parallel}+w\partial_{z}.$
(3)

In line with usual assumptions (Cotton et al., 2011), the dry air mass obeys
the continuity equation (1c). Note that the density appearing in the momentum
equations (1a), (1b) is the full density from (2a) and not the dry air
density. Therefore, the effect of moisture on (total) density is properly
accounted for. Individual contributions from the moist constituents are the
following: in the thermodynamic equation (1), $C$ denotes the “total moist
heat capacity”, specified in (2i); $\Sigma$, defined in (2j), collects moist
contributions related to the work done by pressure forces and $L(T)$ is the
latent heat of vaporization, which can be written as a linear function of
temperature under the assumption of constant $c_{l}$. The reference values for
latent heat and temperature are $L_{\text{ref}}=2.5\cdot 10^{6}\text{ J/kg/K}$
and $T_{\text{ref}}=273.15\text{ K}$, respectively. Finally, the right hand
side represents temperature changes caused by precipitation. In the transport
equations for the respective mixing ratios (1e)-(1g), terms of the form
$S_{\star}$ denote the usual Kessler-type closures for the microphysical
processes of condensation (cd), evaporation (ev), autoconversion (ac) and
collection (cr), respectively. In (2e)-(2h), $C_{\star}$ denote rate constants
of the respective processes, while $q_{\text{cn}}$ represents the density of
condensation kernels and $q_{\text{ac}}$ denotes an activation threshold for
the autoconversion of cloud droplets into raindrops.

We do not explicitly consider cold clouds that would necessitate
parametrization of the ice phase. While a comprehensive treatment of cloud
formation and precipitation on synoptic scales in the midlatitudes _should_
include bulk microphysics closures for the ice phase (Houze, 2014), we reserve
this endeavor for future work.

##  3 Overview and comparison of resulting model equations

For reference, we now briefly summarize and compare the results of the
derivations in sections 4-5 (for Smith and Stechmann’s anelastic/FARE PQG
model) and section 6 (for the new model) in dimensional form. For brevity, we
will refer to the former just as PQG and to the latter as
$\text{PQG}_{\text{DL}}$ from here on out.

Our notation here is as follows: for any model variable $f$ with a leading-
order vertical background profile, we denote it by $\bar{f}=\bar{f}(z)$, while
$\tilde{f}=\tilde{f}(t,\mathbf{x},z)$ stands for the corresponding
perturbation. We will always assume $\tilde{f}\ll\bar{f}$. Furthermore, we
write

$D^{g}_{t}=\partial_{t}+\mathbf{u}\cdot\gradient_{\parallel}$ (4)

for the material derivative with respect to the geostrophic horizontal
velocity $\mathbf{u}$.

###  3.1 The PQG model

The following are equivalent to equations (65)-(66) in (Smith and Stechmann,
2017), if a $\beta-$plane approximation is adopted:

_Diagnostic relations:_ These are the usual geostrophic and hydrostatic
balances,

$\displaystyle f\mathbf{k}\times\mathbf{u}=-\gradient_{\parallel}{\phi},$ (5a)
$\displaystyle g\frac{\tilde{\theta}}{\theta_{\text{ref}}}=\partial_{z}\phi,$
(5b)

where $\phi=\frac{\tilde{p}}{\bar{\rho}}$ is the pressure perturbation scaled
by the background density and $f$ the reference value of the Coriolis
parameter.

_Transport equations:_ As in classical dry air theory, we obtain prognostic
equations for the geostrophic vertical vorticity $\zeta$ and for potential
temperature; in PQG, we get an additional prognostic equation for total
moisture $q_{T}$:

$\displaystyle D^{g}_{t}\left[\zeta+\beta y\right]$
$\displaystyle=\frac{f}{\bar{\rho}}\partial_{z}\left(\bar{\rho}\tilde{w}\right),$
(6a) $\displaystyle
D^{g}_{t}\tilde{\theta}_{e}+\tilde{w}\derivative{\bar{\theta}_{e}}{z}$
$\displaystyle=0,$ (6b) $\displaystyle
D^{g}_{t}\tilde{q}_{T}+\tilde{w}\derivative{\bar{q}_{T}}{z}-\frac{1}{\bar{\rho}}\partial_{z}\left(\bar{\rho}V_{r}\tilde{q}_{r}\right)$
$\displaystyle=0,$ (6c)

where $\beta$ denotes the latitudinal variation of the Coriolis parameter,
$\zeta=\partial_{x}v-\partial_{y}u$ the vertical vorticity and $\tilde{w}$ the
small geostrophic vertical velocity; $\theta_{e}$ denotes the (linearized)
equivalent potential temperature, given by

$\theta_{e}=\theta+\frac{L_{\text{ref}}}{c_{\text{pd}}}q_{v}.$ (7)

Owing to the simplified phase changes in the FARE setting (Hernandez-Duenas et
al., 2013), $\tilde{q}_{r}$ and $\tilde{q}_{v}$ can be written in terms of
$q_{T}$, so that only one equation for moisture is needed (see the next
section for details).

As laid out by Smith and Stechmann (2017), the vertical velocity can be
eliminated from this system to yield a potential vorticity formulation. Here
we only state the results of this calculation and refer the reader to the
cited article for more information. Thus, the potential vorticity $Q$ based on
equivalent potential temperature in PQG reads

$Q=\zeta+\beta
y+\frac{f}{\bar{\rho}}\partial_{z}\left(\frac{\bar{\rho}\tilde{\theta}_{e}}{\text{d}\bar{\theta}_{e}/\text{d}z}\right).$
(8)

This quantity can be rewritten in terms of the pressure perturbation $\phi$
and the moisture-related variable

$M=\tilde{q}_{T}+G_{M}\tilde{\theta}_{e},\qquad\text{where}\qquad
G_{M}=-\frac{\text{d}\bar{q}_{T}/\text{d}z}{\text{d}\bar{\theta}_{e}/\text{d}z}.$
(9)

For given $M$ and $Q$, the result is a second-order diagnostic reconstruction
equation for the pressure perturbation, $\phi$,

$\displaystyle\Delta_{\parallel}\phi+f\beta
y+\frac{f^{2}}{\bar{\rho}}\partial_{z}\left[\frac{\bar{\rho}}{\text{d}\bar{\theta}_{e}/\text{d}z}H_{u}\frac{1}{D_{M}}\left(\frac{\theta_{\text{ref}}}{g}\partial_{z}\phi+\frac{L_{\text{ref}}}{c_{\text{pd}}}M\right)\right]$
$\displaystyle+\frac{f^{2}}{\bar{\rho}}\partial_{z}\left[\frac{\bar{\rho}}{\text{d}\bar{\theta}_{e}/\text{d}z}H_{s}\left(\frac{\theta_{\text{ref}}}{g}\partial_{z}\phi+\frac{L_{\text{ref}}}{c_{\text{pd}}}q_{\text{vs}}\right)\right]=fQ\,,$
(10)

where $D_{M}:=1+L_{\text{ref}}G_{M}/c_{\text{pd}}$ and $H_{u}$, $H_{s}$ are
Heaviside switching functions for the unsaturated and saturated state,
respectively. For later comparison with the PQG${}_{\text{Dl}}$ model, and
following Wetzel et al. (2019), we rewrite (3.1) as

$\displaystyle\Delta_{\parallel}\phi+\frac{1}{\bar{\rho}}\partial_{z}\left[\frac{\bar{\rho}f^{2}}{N^{2}}\partial_{z}\phi\right]=$
$\displaystyle f(Q-\beta
y)-\frac{f^{2}}{\bar{\rho}}\partial_{z}\left[\bar{\rho}gL^{*}\left(H_{u}\frac{M}{N_{u}^{2}}+H_{s}\frac{q_{\text{vs}}}{N_{s}^{2}}\right)\right],$
(11)

where

$\displaystyle N^{2}=H_{u}N_{u}^{2}+H_{s}N_{s}^{2}\qquad\text{and}\qquad
L^{*}=\frac{L_{\text{ref}}}{c_{\text{pd}}\theta_{\text{ref}}}\,.$ (12)

Here $N$ is the local buoyancy frequency, while
$N_{u}=\sqrt{(g/\theta_{\text{ref}})\,\text{d}\bar{\theta}/\text{d}z}$ and
$N_{s}=\sqrt{(g/\theta_{\text{ref}})\,\text{d}\bar{\theta}_{e}/\text{d}z}$ are
the buoyancy frequencies in undersaturated and saturated air, respectively.

As Smith and Stechmann point out, the elliptic operator on the left of (3.1)
has non-constant coefficients owing to the switch in the Brunt-Väisälä
frequency defined in (12). Moreover, the switching functions depend on $q_{T}$
and therefore implicitly on $\partial\phi/\partial z$ through the hydrostatic
balance in (5b) together with the definitions of $M$ in (9) and $\theta_{e}$
in (7). As a consequence, the elliptic PDE (3.1) is nonlinear, and its right
hand side involves a surface $\delta$-distribution along the phase boundary
because the vertical derivative is applied to a discontinuous function. As
discussed by Wetzel et al. (2019), the numerical solution of the
$PV$-$M$-inversion problem is, for these reasons, much more involved than it
is for the classical QG model.

The transport equation for the PQG potential vorticity reads

$D_{t}^{g}Q=-\frac{f}{\text{d}\bar{\theta}_{e}/\text{d}z}\partial_{z}\mathbf{u}\cdot\gradient_{\parallel}\tilde{\theta}_{e}.$
(13)

_Remark:_ Smith and Stechmann chose $Q$ and $M$ as given quantities for
potential vorticity inversion, because $q_{T}$ does not constitute a balanced
quantity in the sense that its evolution equation depends on the vertical
velocity (Wetzel et al., 2019).

###  3.2 The $\text{PQG}_{\text{DL}}$ model

Compared to PQG, this model has a fundamentally different aim: as already
stated in the introduction, it is meant to connect to the diabatic layer of
Klein et al. (2022), which necessitates various changes in the scaling of the
moist constituents. In particular, the mixing ratios here do _not_ decompose
into a vertical background and corresponding perturbations. The quantities
$q_{j}$ ($j=v,c,r$) therefore simply denote the respective leading-order
contributions from moist constituents. Moreover, $\text{PQG}_{\text{DL}}$
incorporates more detailed cloud microphysics, keeping the original number of
moisture species in the asymptotic regime.

_Diagnostic relations:_ The geostrophic and hydrostatic balances take the same
form as above, see (5a) and (5b). However, the scaling of the terminal
rainfall velocity $V_{r}$ now produces an additional diagnostic relation for
$q_{r}$, which is determined from

$\partial_{z}q_{r}=-\frac{\bar{\rho}}{V_{r}}\left[S_{\text{ag,m}}+S_{\text{cr}}-S_{\text{ev}}\right].$
(14)

Here, $S_{\text{ag,m}}$ represents an ad hoc parametrization of the generation
of raindrops through aggregation and melting, see section 6 for details.

_Transport equations:_ Since $\text{PQG}_{\text{DL}}$ does not assume fast
autoconversion, the cloud water evolution equation is retained in the leading-
order equations, increasing the number of prognostic equations by one relative
to PQG,

$\displaystyle D^{g}_{t}\left[\zeta+\beta y\right]$
$\displaystyle=\frac{f}{\bar{\rho}}\partial_{z}\left(\bar{\rho}\tilde{w}\right),$
(15a) $\displaystyle
D^{g}_{t}\tilde{\theta}_{e}+\tilde{w}\derivative{\bar{\theta}}{z}$
$\displaystyle=c_{l}V_{r}q_{r}\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln\bar{p},$
(15b) $\displaystyle D^{g}_{t}q_{v}$
$\displaystyle=S_{\text{ev}}-S_{\text{cd}},$ (15c) $\displaystyle
D^{g}_{t}q_{c}$ $\displaystyle=S_{\text{cd}}-S_{\text{ag,m}}-S_{\text{cr}}.$
(15d)

We observe that rain now influences the system’s dynamics in two ways: it
moistens dry air by evaporating, while producing a cooling effect in the
thermodynamic equation even in saturated air. This constitutes the main
departure from PQG, which we will illustrate by means of a diagnostic relation
for the vertical velocity later on. Moreover, due to the fact that the
moisture constituents do not feature a background stratification and that
above heights of $3\,\text{km}$ their mixing ratios are roughly by an order of
magnitude smaller than assumed for PQG, the background potential temperature
stratification and that of the equivalent potential temperature are identical
in the $\text{PQG}_{\text{DL}}$ regime, i.e.,
$\bar{\theta}_{e}\equiv\bar{\theta}$.

Potential vorticity in the $\text{PQG}_{\text{DL}}$ regime can be defined
exactly as in (8). The absence of a background distribution of moisture,
however, removes the vertical velocity from all moist transport equations.
Therefore, instead of (3.1), in this scenario the diagnostic equation for the
pressure perturbation reads:

$\displaystyle\Delta_{\parallel}\phi+\frac{1}{\bar{\rho}}\partial_{z}\left(\frac{\bar{\rho}f^{2}}{N^{2}}\partial_{z}\phi\right)=$
$\displaystyle f(Q-\beta
y)-\frac{f^{2}}{\bar{\rho}}\partial_{z}\left(\frac{\bar{\rho}L^{*}}{N^{2}}q_{v}\right),$
(16)

where now

$N^{2}=\frac{g}{\theta_{\text{ref}}}\frac{\text{d}\bar{\theta}}{\text{d}z},$
(17)

while $Q$ and $q_{v}$ can be chosen as given quantities to initiate potential
vorticity inversion. In addition, $q_{c}$ will be needed to determine $q_{r}$
from (14) and complete the diagnostic determination of all model variables.
Finally, the evolution equation for potential vorticity also takes a different
form, due to the nonzero right hand side in (15b):

$\displaystyle D_{t}^{g}Q=$
$\displaystyle-\frac{f}{\text{d}\bar{\theta}/\text{d}z}\partial_{z}\mathbf{u}\cdot\gradient_{\parallel}\tilde{\theta}_{e}$
$\displaystyle+\frac{f}{\bar{\rho}}\left[\frac{\bar{\rho}c_{l}V_{r}q_{r}\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln\bar{p}}{\text{d}\bar{\theta}/\text{d}z}\right].$
(18)

In contrast to the PV inversion equation for PQG in (3.1), its
PQG${}_{\text{DL}}$ version in (3.2) is linear for given $Q$ and $q_{v}$, and
it features smooth coefficients in the linear operator. Moreover, $q_{v}$ is
generally continuous – though not continuously differentiable – so that the
right hand side in (3.2) involves merely discontinuities at phase boundaries
instead of surface Dirac distributions as in PQG. We conclude that numerical
PV inversion in PQG${}_{\text{DL}}$ is much less of a challenge than it is in
PQG. In the envisioned PQG-DL-Ekman triple deck model, much stronger effects
of moist processes will be included, but confined to the intermediate diabatic
layer (DL).

##  4 Non-dimensionalisation and scaling for PQG

Let us first recap the generic and space-time-scale independent distinguished
limit for the traditional dry air model from Klein (2010): Coupling the usual
dimensionless parameters to one small parameter $\epsilon$ with a physical
magnitude of $\sim\frac{1}{10}$, we choose the following scaling for the Mach,
external Froude and Rossby numbers:

$\displaystyle\text{M}=\frac{u_{\text{ref}}}{\sqrt{\nicefrac{{p_{\text{ref}}}}{{\rho_{\text{ref}}}}}}=\epsilon^{\nicefrac{{3}}{{2}}}$
$\displaystyle=\frac{u_{\text{ref}}}{\sqrt{gh_{\text{sc}}}}=\text{Fr}_{\text{ext}},$
Ro $\displaystyle=\frac{u_{\text{ref}}}{2\Omega l_{\text{ref}}}=O(\epsilon),$
(19)

where the horizontal reference velocity is $u_{\text{ref}}\approx 10\text{ m
s}^{-1}$, the pressure scale height $h_{\text{sc}}\approx 10\text{ km}$ and
$l_{\text{ref}}$ is of the order of a synoptic length scale, such that

$\frac{h_{\text{sc}}}{l_{\text{ref}}}=\epsilon^{2}.$ (20)

Notice that the _internal_ Froude number
$\text{Fr}_{\text{int}}=\frac{u_{\text{ref}}}{Nh_{\text{sc}}}$, with $N$
denoting the buoyancy frequency, is of the same order of magnitude as Ro, in
accordance with classical QG scaling. The buoyancy frequency in this context
will depend on the moisture content, see the appendix of Smith and Stechmann
(2017).

We work with the standard $\beta-$plane approximation, within which the
Coriolis parameter is a linear function of latitudinal distance $y$:

$\mathbf{\Omega}=(f_{0}+\beta y)\mathbf{k}.$ (21)

As in the textbook derivation of classical QG (Pedlosky, 1987), the
$\beta-$effect is of the order $O(\epsilon)$ compared to $f_{0}$.

Regarding moisture, we start by incorporating the simplifications of the FARE
model. As described by Hernandez-Duenas et al. (2013), fast autoconversion and
rain evaporation imply that the cloud phase can be omitted altogether, since
cloud droplets grow to critical size quasi-instantaneously in this regime,
_and_ since rain can be recovered from the total water content
$q_{T}=q_{v}+q_{r}$ by comparison with the saturation mixing ratio
$q_{\text{vs}}$. We can now combine all moisture balances into one by adding
the corresponding equations and obtain

$D_{t}q_{T}-\frac{1}{\rho_{d}}\partial_{z}(\rho_{d}V_{r}q_{r})=0,$ (22)

where

$q_{r}=(q_{T}-q_{\text{vs}})^{+}.$ (23)

The average water vapor content in the mid-latitude troposphere does not
exceed a few percent and we assume

$q_{\text{vs}}=O(\epsilon^{2}),$ (24)

which imposes the same upper bound on the magnitudes of $q_{v}$, $q_{r}$.
Following Smith and Stechmann (2017), we assume
$q_{\text{vs}}=q_{\text{vs}}(z)$ to be a given function of altitude. We take
the terminal velocity $V_{r}$ to be constant and, in accordance with Smith and
Stechmann (2017), comparable to the vertical reference velocity
$w_{\text{ref}}=\epsilon^{2}u_{\text{ref}}$. This scaling is related to the
horizontal reference velocity through the horizontal-to-vertical aspect ratio
(the vertical velocity scaling directly follows the classical derivation of
the QG model, e.g., in (Pedlosky, 1987)). Thus we let

$\frac{V_{r}}{w_{\text{ref}}}=:V_{T}=O(1).$ (25)

These scaling choices were all explicit in Smith and Stechmann’s derivation.
For the remaining nondimensional parameters, which govern the thermodynamics
of moist air, we present a self-contained justification, based on the
following guidelines (which we will more comprehensively refer to in the
derivation of our new model in section 6):

  1. 1.

The “dry air limit” that we obtain by letting moisture $q_{T}\rightarrow 0$ in
our model equations must agree with the dry QG system.

  2. 2.

The leading-order balances resulting from our choice of the distinguished
limit should encapsulate physically meaningful and - at least qualitatively -
accurate relations.

  3. 3.

Finally, we aim for a scaling that is formally consistent and assigns
realistic values to our system parameters.

Following Hittmeir and Klein (2018), we proceed with the following general
scaling ansatz,

$\displaystyle\frac{L_{\text{ref}}}{c_{\text{pd}}T_{\text{ref}}}=\frac{L}{\epsilon^{a}},\quad\frac{R_{d}}{c_{\text{pd}}}$
$\displaystyle=\epsilon^{b}\Gamma,\quad\frac{R_{v}}{c_{\text{pv}}}=\epsilon^{b_{v}}\Gamma_{v},$
$\displaystyle\frac{c_{l}}{c_{\text{pd}}}$
$\displaystyle=\frac{k_{l}}{\epsilon^{b_{l}}},\quad\frac{R_{d}}{R_{v}}=\epsilon^{c}E,$
(26)

where all parameters on the right-hand sides are $O(1)$ as
$\epsilon\rightarrow 0$. Thus, $L$, $\Gamma$ etc. are _defined_ by the above
relations, dependent on nonnegative numbers $a,b,\dots$ that are to be
determined in accordance with the principles stated above.

Staying true to guideline 3 and comparing with the physical reference values

$\frac{L_{\text{ref}}}{c_{\text{pd}}T_{\text{ref}}}\approx
9.1\qquad\text{and}\qquad\frac{R_{d}}{R_{v}}\approx 0.62$ (27)

(cf. Hittmeir and Klein (2018)), we are compelled to choose $a=1$ and $c=0$.
The choice for $b_{l}$ is not as obvious, but with the reference value

$\frac{c_{l}}{c_{\text{pd}}}\approx 4.2$ (28)

and $\epsilon\sim\frac{1}{10}$, $b_{l}=1$ gives the best fit. Moving on to
$b$, we have

$\frac{R_{d}}{c_{\text{pd}}}\approx 0.29,$ (29)

which is quite ambiguous, as far as heuristic matching is concerned, and both
$b=0$ and $b=1$ are reasonable choices. A closer look at the temperature
equation

$\displaystyle CD_{t}\ln\theta+\Sigma D_{t}\ln p+\frac{L(T)}{T}D_{t}q_{v}=$
$\displaystyle
c_{l}V_{r}q_{r}(\partial_{z}\ln\theta+\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln
p),$ (30)

however, reveals that - given our scaling up to this point - $b=0$ would raise
the rainfall term $c_{l}V_{r}q_{r}\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln
p$ on the right-hand side to a leading-order effect. Yet, no term that depends
on $q_{r}$ occurs in the temperature evolution equation of PQG and this is
accomodated here by letting $b=1$. This amounts to the so-called _Newtonian
limit_ for dry air (Parkins et al., 2000) (see also the discussion of
anelastic models and eqn. (2.18) in (Bannon, 1995)). Finally, $b_{v}$ does not
play a part in the leading-order dynamics, and since the derived quantity

$k_{v}:=\frac{c_{\text{pv}}}{c_{\text{pd}}}\approx 1.8\sim O(1),$ (31)

we choose $b_{v}=1$ purely for consistency.

With these scalings in place, the dimensionless equations now read

$\displaystyle
D_{t}\mathbf{u}+\frac{1}{\epsilon}f\mathbf{k}\times\mathbf{u}+\frac{1}{\epsilon^{3}}\frac{1}{\rho}\gradient_{\parallel}{p}$
$\displaystyle=0,$ (32a) $\displaystyle
D_{t}w+\frac{1}{\epsilon^{5}}\frac{1}{\rho}\partial_{z}p$
$\displaystyle=-\frac{1}{\epsilon^{5}},$ (32b)
$\displaystyle\partial_{t}\rho_{d}+\nabla_{\parallel}\cdot(\rho_{d}\mathbf{u})+\partial_{z}(\rho_{d}w)$
$\displaystyle=0,$ (32c) $\displaystyle
C_{\epsilon}D_{t}\ln\theta+\epsilon^{3}\Sigma_{\epsilon}D_{t}\ln
p+\epsilon\frac{L\phi_{\epsilon}}{T}D_{t}q_{v}$ $\displaystyle=$
$\displaystyle\epsilon^{2}k_{l}V_{T}q_{r}(\partial_{z}\ln\theta+\epsilon\Gamma\partial_{z}$
$\displaystyle\ln p),$ (32d) $\displaystyle
D_{t}q_{T}-\frac{1}{\rho_{d}}\partial_{z}(\rho_{d}V_{T}q_{r})$
$\displaystyle=0,$ (32e)

where

$\displaystyle\rho$ $\displaystyle=\rho_{d}(1+\epsilon^{2}(q_{v}+q_{r})),$
(33a) $\displaystyle p$
$\displaystyle=\rho_{d}T\left(1+\epsilon^{2}\frac{q_{v}}{E}\right),$ (33b)
$\displaystyle T$ $\displaystyle=\theta p^{\epsilon\Gamma}\equiv\theta\pi,$
(33c) $\displaystyle C_{\epsilon}$ $\displaystyle=1+\epsilon(\epsilon
k_{v}q_{v}+k_{l}(q_{r})),$ (33d) $\displaystyle\Sigma_{\epsilon}$
$\displaystyle=\Gamma k_{l}(q_{r})+\epsilon\kappa_{v}q_{v},$ (33e)
$\displaystyle\kappa_{v}$
$\displaystyle=\left(\frac{c_{\text{pv}}}{c_{\text{pd}}}\Gamma-\frac{1}{E}\Gamma\right),$
(33f) $\displaystyle\phi_{\epsilon}$ $\displaystyle=1-\frac{k_{l}-\epsilon
k_{v}}{L}(T-1),$ (33g) $\displaystyle q_{r}$
$\displaystyle=(q_{T}-q_{\text{vs}})^{+}.$ (33h)

##  5 Asymptotic expansion and derivation of PQG

In this section, we derive the anelastic version of PQG given by equations
(65)-(66) in (Smith and Stechmann, 2017). As in QG theory for dry air, we
presume an expansion for density, pressure and temperature that admits purely
vertical profiles up to $O(\epsilon)$, and the latter are indicated by
_single_ subscripts. Instead, when the background distribution of some
variable at some given asymptotic order is constant, then this is indicated by
a _double_ subscript. Superscripts in brackets denote perturbation variables
with generic dependence on $(t,x,y,z)$. Thus, the expansions, including the
velocity components, for the PQG regime read

$\displaystyle p$ $\displaystyle=p_{0}+\epsilon
p_{1}+\epsilon^{2}p^{(2)}+o(\epsilon^{2}),$ (34a) $\displaystyle\rho$
$\displaystyle=\rho_{0}+\epsilon\rho_{1}+\epsilon^{2}\rho^{(2)}+o(\epsilon^{2}),$
(34b) $\displaystyle\theta$
$\displaystyle=\theta_{00}+\epsilon\theta_{1}+\epsilon^{2}\theta^{(2)}+o(\epsilon^{2}),$
(34c) $\displaystyle T$ $\displaystyle=T_{00}+\epsilon
T_{1}+\epsilon^{2}T^{(2)}+o(\epsilon^{2}),$ (34d) $\displaystyle\mathbf{u}$
$\displaystyle=\mathbf{u}^{(0)}+\epsilon\mathbf{u}^{(1)}+o(\epsilon),$ (34e)
$\displaystyle w$ $\displaystyle=w^{(0)}+\epsilon w^{(1)}+o(\epsilon).$ (34f)

The constant leading-order terms in the expansions of both $\theta$ and $T$
are a consequence of the Newtonian limit, see (Klein and Majda, 2006; Hittmeir
and Klein, 2018).

One important feature of Smith and Stechmann’s PQG model is that it assumes a
vertical background profile for water vapor while rain is comparatively weak
and of the order of a perturbation of the total water content only. The
expansions of the respective mixing ratios then read

$\displaystyle q_{v}$
$\displaystyle=\epsilon^{2}q_{v0}+\epsilon^{3}q_{v}^{(1)}+o(\epsilon^{3})\,,$
(35a) $\displaystyle q_{r}$
$\displaystyle=\epsilon^{3}q_{r}^{(1)}+o(\epsilon^{3})\,,$ (35b)

implying _small horizontal and temporal variations_ of the total water
content. Informally speaking, this condition will be met as long as the air in
our model atmosphere does not dry out too much. In their PQG model, Smith and
Stechmann consider a regime in which the moisture perturbation variables
determine the local saturation status. To realize this within the asymptotic
framework based on an expansion of the saturation water vapor profile via

$q_{\text{vs}}=\epsilon^{2}q_{\text{vs}0}+\epsilon^{3}q_{\text{vs}1}+\dots,$
(36)

we must assume an atmosphere close to saturation, so that

$q_{v0}=q_{T0}=q_{\text{vs}0}\,.$ (37)

Then, saturation is reached whereever

$q_{T}^{(1)}\geq q_{\text{vs}1}\,.$ (38)

The analysis of the leading-order momentum and mass balances now proceeds
along the same lines as for dry air with the following results:

_Mass balance:_ By the usual reasoning, anticipating (41), we conclude that

$w^{(0)}=0$ (39)

from the balance at leading order, while the _anelastic constraint_

$\rho_{0}\nabla_{\parallel}\cdot\mathbf{u}^{(1)}+\partial_{z}(\rho_{0}w^{(1)})=0$
(40)

holds for the first-order perturbations. This contrasts with derivations using
the Boussinesq approximation, where $(\mathbf{u}^{(1)},w^{(1)})$ would obey an
incompressibility constraint.

_Horizontal momentum balance:_ At leading order, we obtain geostrophic
balance:

$f_{0}\mathbf{k}\times\mathbf{u}^{(0)}+\frac{1}{\rho_{0}}\gradient_{\parallel}{p}=0\,,$
(41)

while the first-order perturbation reads

$\displaystyle D_{t}^{(0)}\mathbf{u}^{(0)}$ $\displaystyle+\beta
y\mathbf{k}\times\mathbf{u}^{(0)}+f_{0}\mathbf{k}\times\mathbf{u}^{(1)}$
$\displaystyle+\frac{1}{\rho_{0}}\gradient_{\parallel}{p^{(3)}}-\frac{\rho_{1}}{\rho_{0}^{2}}\gradient_{\parallel}{p^{(2)}}=0$
(42)

with

$D_{t}^{(0)}=\partial_{t}+\mathbf{u}^{(0)}\cdot\gradient_{\parallel}$ (43)

since $w^{(0)}\equiv 0$. Taking the vertical component of the curl yields

$D_{t}^{(0)}\left[\zeta^{(0)}+\beta
y\right]+f_{0}\nabla_{\parallel}\cdot\mathbf{u}^{(1)}=0\,,$ (44)

which is the equation for transport of vorticity in the quasi-geostrophic
regime.

_Vertical momentum balance:_ The leading-order equation accepts the explicit
solutions

$T_{00}\rho_{0}(z)=p_{0}(z)=p_{0}(0)e^{-\frac{z}{T_{00}}}\,,$ (45)

where we have used the ideal gas law to obtain
$\rho_{0}=\frac{p_{0}}{T_{00}}$, and the leading-order vertical momentum
equation to derive the hydrostatic balance relation,
$\partial_{z}p_{0}=-\rho_{0}=-\frac{1}{T_{00}}p_{0}$. At first order we obtain

$\partial_{z}p_{1}=-\rho_{1}\,,$ (46)

which, by application of the ideal gas law, further implies

$\frac{T_{1}}{T_{00}}=\partial_{z}\left(\frac{p_{1}}{\rho_{0}}\right).$ (47)

For the second-order perturbations, we get the analogous result

$\partial_{z}p^{(2)}=-\rho^{(2)},$ (48)

which we will analyse in more detail shortly.

In summary, we have derived (41) as the nondimensional form of (5a), while the
substitution of (40) into (44) yields (6a).

_Moisture balances:_ Here, we only need to consider one balance equation for
the total water content,

$D_{t}q_{T}-\frac{1}{\rho_{d}}\partial_{z}(\rho_{d}V_{T}q_{r})=0\,.$ (49)

With the expansion scheme summarized in (35)–(38), the leading-order balance
from (49) gives

$D^{(0)}_{t}q^{(1)}_{T}+w^{(1)}\derivative{q_{T0}}{z}-\frac{1}{\rho_{0}}\partial_{z}(\rho_{0}V_{T}q^{(1)}_{r})=0\,.$
(50)

This is the nondimensional equivalent of equation (6c).

_Temperature transport:_ The leading-order balance of (32) reads

$D_{t}^{(0)}\theta^{(2)}+LD_{t}^{(0)}q_{v}^{(1)}+w^{(1)}\derivative{\theta_{1}}{z}+Lw^{(1)}\derivative{q_{v0}}{z}=0\,.$
(51)

Utilizing the (linearised) equivalent potential temperature, as defined in
(7), we can rewrite the above as

$D_{t}^{(0)}\theta_{e}^{(2)}+w^{(1)}\derivative{\theta_{e1}}{z}=0,$ (52)

which is equivalent to equation (6b) in nondimensional form.

_Buoyancy:_ We now take a closer look at equation (48). By the ideal gas law
(33b), we get

$p^{(2)}=\rho_{0}T^{(2)}+\rho_{1}T_{1}+\rho^{(2)}T_{00}+\rho_{0}T_{00}\left(\frac{1}{E}-1\right)q_{v0},$
(53)

which implies

$\displaystyle\partial_{z}\left(\frac{p^{(2)}}{\rho_{0}}\right)=\frac{\partial_{z}p^{(2)}+\frac{1}{T_{00}}p^{(2)}}{\rho_{0}}=\frac{1}{T_{00}}\frac{1}{\rho_{0}}$
$\displaystyle\left[-T_{00}\rho^{(2)}+\rho_{0}T^{(2)}+\rho_{1}T_{1}+\rho^{(2)}T_{00}+\rho_{0}T_{00}\left(\frac{1}{E}-1\right)q_{v0}\right]$
$\displaystyle=\frac{T^{(2)}}{T_{00}}+\frac{\rho_{1}}{\rho_{0}}T_{1}+\left(\frac{1}{E}-1\right)q_{v0}.$
(54)

With the help of (33c), we can further rewrite (5) in the form

$\partial_{z}\left(\frac{p^{(2)}}{\rho_{0}}\right)=\frac{\theta^{(2)}}{\theta_{00}}+\pi_{1}\theta_{1}+\pi_{2}+\frac{\rho_{1}}{\rho_{0}}T_{1}+\left(\frac{1}{E}-1\right)q_{v0}.$
(55)

Now, the _leading-order_ buoyancy, as defined in (Smith and Stechmann, 2017),
reduces to exactly the leading-order (dry) potential temperature perturbation,
that is $\frac{\theta^{(2)}}{\theta_{00}}$. Our result here shows quite a few
extra terms - all background profiles, however, which means that we can
recover the exact form of the anelastic PQG equations at leading order by
shifting them over to the left hand side and then identifying the appropriate
definition of the pressure perturbation: setting

$\displaystyle\phi:=$ $\displaystyle\frac{p^{(2)}}{\rho_{0}}$
$\displaystyle-\int_{0}^{z}\left[\pi_{1}\theta_{1}+\pi_{2}+\frac{\rho_{1}}{\rho_{0}}T_{1}+\left(\frac{1}{E}-1\right)q_{v0}\right]d\zeta,$
(56)

we recover (5b) in nondimensional form. Since all extra terms are functions of
$z$ only, geostrophic balance remains unaffected and our derivation from the
compressible equations with refined moist thermodynamics can be viewed as
complete, at least at leading order.

Still, the question remains: where does the initial discrepancy in the form of
the moist buoyancy stem from? Essentially, this is caused by what we could
call the higher “asymptotic resolution” of our expansion ansatz, where the
background stratification of pressure, density and temperature is comprised of
_two terms of different order_ each instead of one. Appendix A1 explains this
in more detail by describing a sample derivation of the moist anelastic
buoyancy.

##  6 Consequences of a different scaling approach: PQG and the diabatic layer

In section 2, we laid out the guiding principles for our choice of a
distinguished limit and applied them to the thermodynamic parameters of moist
air. Some of our scaling choices, however, are only valid under quite specific
conditions. Motivated by the recent introduction of the novel intermediary
_diabatic layer_ in (Klein et al., 2022), we therefore propose a different
distinguished limit that is better suited to connect with this adjacent
boundary layer. We then sketch the resulting model, which we will develop in
more detail in an upcoming paper.

###  6.1 The framework

First, let us clarify the scope and limitations of our model, which are
summarized in the following points:

  1. 1.

As stated above, our QG model is meant to be incorporated into a triple-deck
boundary layer theory, coupled to the diabatic layer, as well as to the Ekman
layer. In particular, this has the consequence that _our scaling assumptions
have to hold strictly at altitudes $\geq 3\text{ km}$ only, letting certain
model variables attain asymptotically larger or smaller values in the lower
layers._

  2. 2.

On the length - and timescales of QG theory, we cannot capture convective
processes explicitly, since they require much higher vertical velocities and a
drastically different aspect ratio. Therefore, the primary form of
precipitation that we consider here is _stratiform rain,_ which is generated
from nimbostratus, frequently with cold seeder cells aloft - cf. chapter 6 of
(Houze, 2014).

  3. 3.

We are interested in scenarios with a very thick cloud cover. Therefore, we
take the _highest observed liquid water contents_ in stratiform clouds as a
baseline for our scaling; we discuss the consequences for our scaling in
detail further down.

  4. 4.

Since we have not incorporated the ice phase and the associated phase changes
as of yet, we stick with the simplified subdivision of water into vapor, cloud
droplets and rain. However, most stratiform rain actually originates from snow
and other ice particles that, while falling from the upper troposphere through
the cloud, increase in size and eventually melt. As laid out in (Houze, 2014)
it is this two-stage process of aggregation and melting that dominates in
stratiform rain, _not_ accretion, which is the driving force in convective
rain. For the time being, we will therefore - inaccurately - treat liquid
cloud water and cloud ice as one substance, assume that all solid hydrometeors
eventually turn into rain and parametrize aggregation and melting in the same
manner as autoconversion. All this, of course, is subject to future
refinement.

Keeping the above in mind, let us discuss our scaling choices in depth:

###  6.2 Terminal rainfall velocity

The terminal velocity $V_{r}$ of raindrops is strongly dependent on their
diameter. In the context of a bulk paramerization and large-scale dynamics,
however, the subtleties of the size spectrum cannot be captured. Thus, we opt
for the most straightforward approach and assign a constant value to $V_{r}$,
to be understood as an average over the size distribution of raindrops.
Further, it is only the relative magnitude
$\frac{V_{r}}{w_{\text{ref}}}=V_{T}$ that matters for scaling purposes. Next,
we therefore consider typical sedimentation velocities in stratiform rain.
Large raindrops can approach fall speeds up to $10\text{ m s}^{-1}$, while the
average value is considerably lower and also depends on the form of
precipitation under consideration (see Khvorostyanov and Curry (2014), chapter
12). In convective rain, where also hailstones with much higher fall
velocities can occur, the choice $V_{T}\sim\epsilon^{-2}$ would certainly be
appropriate. Stratiform rain, generally speaking, is gentler and does not
produce comparably large hydrometeors, leading us to choose

$V_{T}\sim\epsilon^{-1}$ (57)

for the present regime, which corresponds to velocities comparable to $\sim
1\text{ m s}^{-1}$.

###  6.3 The saturation mixing ratio

Sample calculations, using approximate solutions to the Clausius-Clapeyron
equation as shown in Fig. 1, show that the saturation mixing ratio
$q_{\text{vs}}$ is $\ll 10\text{ g kg}^{-1}$ in the bulk of the mid-latitude
troposphere (see also sample values in (Weischet and Endlicher, 2018)). We
therefore stipulate

$q_{\text{vs}}\sim\epsilon^{3},$ (58)

keeping in mind that by asymptotic matching, this still allows the saturation
mixing ratio to increase up to $O(\epsilon^{2})$ through the diabatic and
Ekman layers below. The scaling then agrees very well with the observed
magnitudes.

Figure 1: Saturation mixing ratio in the mid-latitude troposphere. At 3 km,
the top of the diabatic layer, $q_{\text{vs}}$ is already $<4\text{ g/kg}$,
dropping well below $1\text{ g/kg}$ toward the tropopause.

###  6.4 Moisture components

_Water vapor:_ Supersaturation levels cannot significantly exceed
$q_{\text{vs}}$ in magnitude, so that the following scaling for the water
vapor content,

$q_{v}\sim\epsilon^{3},$ (59)

suggests itself.

_Cloud water:_ Typically, the cloud water mixing ratio $q_{c}$ will be smaller
than $q_{v}$ by an order of magnitude, especially in thinner mid-level clouds,
as evidenced by observational data, e.g., in (Fleishauer et al., 2001). Yet,
datasets, e.g., from Zhao and Lei (2014), p.96, or Zhang et al. (2020) show
that the liquid water content in a nimbostratus cloud frequently approaches
$0.5\text{ g m}^{-3}$ at altitudes $3$-$4\,\text{ km}$, corresponding to
$q_{c}\sim 0.6-0.7\text{ g kg}^{-1}$. With our stated aim to describe an
atmosphere with a consistently thick cloud cover, also keeping in mind that we
implicitly include cloud ice, this justifies the scaling

$q_{c}\sim\epsilon^{3}\,.$ (60)

By letting $q_{c}^{(0)}\rightarrow 0$, we can also discuss “almost cloud-free”
regions later on, once we have derived our model equations.

_Rain:_ Regarding the rain mixing ratio $q_{r}$, we simply make the assumption
that the total amount of precipitation that eventually reaches the ground is
of the same order of magnitude as the total cloud water content. Assuming a
water vapor (saturation) mixing ratio of the order $O(\epsilon^{2})$ near the
ground, this implies $q_{r}$ should be one order of magnitude smaller than
$q_{v}$. Since nimbostratus clouds extend downward to rather low levels and
rain accumulates as it falls through the various cloud layers, we further
expect $q_{r}$ to increase downwards asymptotically in the same manner as
$q_{\text{vs}}$ and $q_{v}$, indicating that it should be as low as

$q_{r}\sim\epsilon^{4}$ (61)

in the free troposphere (3 to 12 km above ground level). Notice that
convective processes are filtered out on QG scales, whence exceptionally heavy
rain is not covered by our model in absence of parametrizations of the related
processes.

###  6.5 Phase changes

Here, we briefly assess which source terms in the Kessler-type parametrization
play a role in the leading-order dynamics. Our guiding principle will be that
all microphysical processes which are considered dominant should be
represented at leading order in our bulk model as well. This principle is also
grounded in the heuristic rule of asymptotic analysis that significant limits
should retain a maximal number of terms, known as the _principle of least
degeneracy_ (Kevorkian and Cole, 1996). In the following, we will argue on
purely qualitative grounds and implicitly assume that the respective rate
constants scale accordingly; for example, the condensation rate needs to
fulfill $C_{\text{cd}}/t_{\text{ref}}=O(\epsilon^{-3})$ for (62) to hold.

_Condensation:_ First, let us recall that nimbostratus is primarily generated
by gentle updrafts, continually lifting air parcels just above their
condensation level. With the QG vertical velocity $w^{(1)}$ being very small,
the leading-order equation for $q_{c}$ will only deal with _horizontal_
transport, which might lead one to conclude that condensation does not
constitute a leading-order effect in our regime. It should be accounted for,
however, that the formation of nimbostratus is significantly fueled by
convective cells on the sides of, above or even embedded in the cloud itself
((Houze, 2014), section 6.4). These processes are external to our model in the
sense that they arise from vertical velocities that exceed $w^{(1)}$ by
several orders of magnitude. Even though we do not intend to model convection
as such, as we have repeatedly emphasized, we argue that the impact of these
local events _on the development of the stratiform cloud_ suffices to raise
condensation to a leading-order effect and therefore opt for the ansatz

$S_{\text{cd}}\sim\epsilon^{3}$ (62)

for the (nondimensionalized) condensation term - which might be supplemented
by a parametrization of the aforementioned convective updrafts.

_Autoconversion, aggregation and melting:_ Autoconversion of cloud water by
coalescence that causes droplets to cross a size threshold is a very slow
process. As already stated above, aggregation and subsequent melting of solid
ice particles is the _dominant microphysical mechanism_ in stratiform
precipitation (Houze (2014), p. 144). Hence, we replace $S_{\text{ac}}$ by a
source term $S_{\text{ag,m}}$ that parameterizes this mechanism in a manner to
be specified later on. Due to our assumption on the maximum cloud water and
rain mixing ratios, the scaling

$S_{\text{ag,m}}\sim\epsilon^{3}$ (63)

is justified.

_Evaporation:_ In undersaturated regions, rain evaporates very quickly on QG
timescales. In a fully developed nimbostratus cloud, though, it is continually
resupplied through melting snow and ice, as described above. The evaporation
term should therefore balance $S_{\text{ag,m}}$ in the equation for $q_{r}$,
leading to the asymptotic rescaling

$S_{\text{ev}}\sim\epsilon^{3}.$ (64)

_Accretion:_ Here, we refer again to Houze (2014), p. 144: while accretion is
considered to be the primary growth mechanism in convective clouds, it is of
minor importance in nimbostratus. Aiming to account for the presence of
convective cells, which might play a significant role in clouds that extend
into the diabatic layer, we tentatively include the corresponding source term
as well and assume

$S_{\text{cr}}\sim\epsilon^{3}.$ (65)

_Remark:_ The scaling choices in (62)-(65) are physically sensible and
necessary to let $\text{PQG}_{\text{DL}}$ stand on its own. We do not,
however, exclude the possibility that connection to the diabatic layer might
permit the consideration of different distinguished limits in future work.

###  6.6 Thermodynamic parameters

As far as the scalings of the latent heat and various heat capacities are
concerned, we stick with the distinguished limit derived in section 3, with
one exception:
Since we do not aim to conserve equivalent potential temperature in our
variant of PQG, thus allowing precipitation to alter its evolution, the ratio

$\frac{R_{d}}{c_{\text{pd}}}=\Gamma$ (66)

does not require the “Newtonian limit” anymore; we thus assume

$\Gamma=O(1)$ (67)

from here on out, which yields the standard leading-order solutions for the
pressure and density in the troposphere.

###  6.7 Asymptotic expansion and derivation

The governing equations, rescaled for the PQG${}_{\text{DL}}$ regime as
described above, now read as:

$\displaystyle
D_{t}\mathbf{u}+\frac{1}{\epsilon}f\mathbf{k}\times\mathbf{u}+\frac{1}{\epsilon^{3}}\frac{1}{\rho}\gradient_{\parallel}{p}$
$\displaystyle=0,$ (68a) $\displaystyle
D_{t}w+\frac{1}{\epsilon^{5}}\frac{1}{\rho}\partial_{z}p$
$\displaystyle=-\frac{1}{\epsilon^{5}},$ (68b)
$\displaystyle\partial_{t}\rho_{d}+\nabla_{\parallel}\cdot(\rho_{d}\mathbf{u})+\partial_{z}(\rho_{d}w)$
$\displaystyle=0,$ (68c) $\displaystyle
C_{\epsilon}D_{t}\ln\theta+\epsilon^{3}\Sigma_{\epsilon}D_{t}\ln
p+\epsilon^{2}\frac{L\phi_{\epsilon}}{T}D_{t}q_{v}$ $\displaystyle=$
$\displaystyle\epsilon^{2}k_{l}V_{T}q_{r}(\partial_{z}\ln\theta+$
$\displaystyle\Gamma\partial_{z}\ln p),$ (68d) $\displaystyle D_{t}q_{v}$
$\displaystyle=$ $\displaystyle S_{\text{ev}}-$ $\displaystyle S_{\text{cd}},$
(68e) $\displaystyle D_{t}q_{c}$ $\displaystyle=$ $\displaystyle
S_{\text{cd}}-S_{\text{ag,m}}-$ $\displaystyle S_{\text{cr}},$ (68f)
$\displaystyle\epsilon
D_{t}q_{r}-\frac{1}{\rho_{d}}\partial_{z}(\rho_{d}V_{T}q_{r})$
$\displaystyle=$ $\displaystyle S_{\text{ag,m}}+S_{\text{cr}}-$ $\displaystyle
S_{\text{ev}},$ (68g)

where

$\displaystyle\rho$
$\displaystyle=\rho_{d}(1+\epsilon^{3}(q_{v}+q_{c}+\epsilon q_{r}))\,,$ (69a)
$\displaystyle p$
$\displaystyle=\rho_{d}T\left(1+\epsilon^{3}\frac{q_{v}}{E}\right)\,,$ (69b)
$\displaystyle T$ $\displaystyle=\theta p^{\Gamma}\equiv\theta\pi\,,$ (69c)
$\displaystyle S_{\text{ev}}$
$\displaystyle=C_{\text{ev}}\frac{p}{\rho}(q_{\text{vs}}-q_{v})^{+}q_{r}\,,$
(69d) $\displaystyle S_{\text{cd}}$
$\displaystyle=C_{\text{cn}}(q_{v}-q_{\text{vs}})^{+}q_{\text{cn}}+C_{\text{cd}}(q_{v}-q_{\text{vs}})q_{c}\,,$
(69e) $\displaystyle S_{\text{cr}}$ $\displaystyle=C_{\text{cr}}q_{c}q_{r}\,,$
(69f) $\displaystyle C_{\epsilon}$
$\displaystyle=1+\epsilon^{3}(k_{v}q_{v}+k_{l}(q_{c}+q_{r}))\,,$ (69g)
$\displaystyle\Sigma_{\epsilon}$ $\displaystyle=\Gamma
k_{l}(q_{c}+q_{r})+\epsilon\kappa_{v}q_{v}\,,$ (69h)
$\displaystyle\phi_{\epsilon}$ $\displaystyle=1-\frac{k_{l}-\epsilon
k_{v}}{L}(T-1)\,.$ (69i)

The newly introduced source term $S_{\text{ag,m}}$ could be parameterized in
analogy with $S_{\text{ac}}$, e.g., as

$S_{\text{ag,m}}=C_{\text{ag,m}}(q_{c}-q_{\text{ag,m}})^{+},$ (70)

where $q_{\text{ag,m}}$ now denotes an activation threshold for the conversion
of melting snow into raindrops. Naturally, this is subject to refinement once
the ice phase is introduced.

The scaling assumptions outlined earlier in this section imply changes in the
asymptotic expansions relative to those for the PQG regime in (34) and (35).
Since we do not invoke the Newtonian limit anymore (see (67)), the leading-
order temperature is no longer constant, so that

$\displaystyle T$ $\displaystyle=T_{0}+\epsilon
T_{1}+\epsilon^{2}T^{(2)}+o(\epsilon^{2})\,.$ (71a)

(Recall that $T_{00}$ denoted a constant in the PQG-expansions, whereas here
$T_{0}$ is a $z$-dependent background distribution.) In contrast, the moisture
variables expand as

$\begin{array}[]{r@{\ }c@{\ }lc@{\ }l}\displaystyle
q_{v}&=&\displaystyle\epsilon^{3}q_{v}^{(0)}+\displaystyle
o(\epsilon^{3})\,,\\\ \displaystyle
q_{c}&=&\displaystyle\epsilon^{3}q_{c}^{(0)}+o(\epsilon^{3})\,,\\\
\displaystyle
q_{r}&=&\displaystyle\epsilon^{4}q_{r}^{(1)}+o(\epsilon^{4})\,.\end{array}$
(72)

That is, their absolute magnitude is reduced by one order in $\epsilon$
relative to the PQG regime (see (35)), and they do not feature time
independent, vertically stratified background states.

With these expansions, the leading-order equations for mass and horizontal
momentum remain unchanged relative to those found in the PQG regime. Note,
however, that from the second-order vertical momentum balance in (48), we may
now infer

$\partial_{z}\left(\frac{p^{(2)}}{p_{0}}\right)=\frac{1}{T_{0}}\left[T^{(2)}+\frac{\rho_{1}}{\rho_{0}}T_{1}\right]$
(73)

which involves the leading-order temperature stratification $T_{0}(z)$. This
equation can be rewritten in terms of the potential temperature perturbation
to yield the hydrostatic balance (5b), with a pressure perturbation $\phi$
defined in the same manner as in section 4.

Let us also recall that by taking the horizontal gradient and utilizing
geostrophic balance, we obtain for the potential temperature perturbation
$\theta^{(2)}$:

$\frac{\partial\mathbf{u}^{(0)}}{\partial
z}\cdot\gradient_{\parallel}{\theta^{(2)}}=0,$ (74)

which can be viewed as a consequence of the thermal wind relation.

The explicit solutions for the leading-order pressure and density now read

$p_{0}=\left(1-\Gamma z\right)^{\frac{1}{\Gamma}}$ (75)

and

$\rho_{0}=\left(1-\Gamma z\right)^{\frac{1-\Gamma}{\Gamma}},$ (76)

respectively, while the leading-order temperature profile drops off linearly
with height:

$T_{0}=1-\Gamma z.$ (77)

_Moisture at leading order:_ The leading-order equation for water vapor is one
for quasi-horizontal transport, since $w^{(0)}=0$:

$D_{t}^{(0)}q_{v}^{(0)}=S_{\text{ev}}^{(0)}-S_{\text{cd}}^{(0)}.$ (78)

In the same fashion, we arrive at the equation

$D_{t}^{(0)}q_{c}^{(0)}=S_{\text{cd}}^{(0)}-S_{\text{ag,m}}^{(0)}-S_{\text{cr}}^{(0)}$
(79)

for the cloud water content - which is again transported only quasi-
horizontally. Cloud formation via condensation is expected to arise through
local updrafts in small-scale convective cells, so that this effect is
included on the present synoptic scales through an effective parameterization
as part of the condensation source term $S_{\text{cd}}^{(0)}$.

For rain, the vertical fallout term dominates:

$\partial_{z}(\rho_{0}q_{r}^{(0)})=-\frac{\rho_{0}}{V_{T}}\left[S_{\text{ag,m}}^{(0)}+S_{\text{cr}}^{(0)}-S_{\text{ev}}^{(0)}\right],$
(80)

while the leading-order potential temperature balance - recalling (7) - reads

$D_{t}^{(0)}\theta_{e}^{(2)}+w^{(1)}\derivative{\theta_{1}}{z}=-k_{l}V_{T}q_{r}^{(0)}\Gamma\left(1-\Gamma
z\right)^{-1}.$ (81)

Here we have used that, owing to the systematically lower moisture content
compared to the PQG regime, the equivalent potential temperature equals the
dry potential temperature to leading and first order. Coupled with the first
order mass balance (40) and the vorticity transport equation (44), this allows
us to derive an equation for the QG potential vorticity based on the
equivalent potential temperature perturbation $Q=\zeta^{(0)}+\beta
y+\frac{f_{0}}{\rho_{0}}\partial_{z}\left(\frac{\rho_{0}\theta_{e}^{(2)}}{\text{d}\theta_{1}/\text{d}z}\right)$:

$\displaystyle D_{t}^{(0)}Q=$
$\displaystyle\frac{f_{0}}{\rho_{0}}\partial_{z}\left[\rho_{0}\frac{-k_{l}V_{T}q_{r}^{(0)}\Gamma\left(1-\Gamma
z\right)^{-1}}{\text{d}\theta_{1}/\text{d}z}\right]$
$\displaystyle-\frac{f_{0}}{\text{d}\theta_{1}/\text{d}z}\partial_{z}\mathbf{u}\cdot\gradient_{\parallel}{q_{v}^{(0)}}.$
(82)

This enables a formulation of our model in terms of potential vorticity
inversion, as mentioned in the introduction: given $Q$ and $q_{v}$, we can
rewrite the definition of $Q$ to yield an elliptic partial differential
equation for the pressure perturbation, which is given in its dimensional form
in (3.2) above. The related PV-inversion problem has a unique solution if we
specify boundary conditions for $p$. Horizontal flow and potential temperature
can then diagnostically be recovered from the geostrophic and hydrostatic
balances, respectively. With given cloud water content $q_{c}$, we can also
use the diagnostic relation (80) to compute the rain water mixing ratio and
thus we obtain the leading-order solutions of _all_ model variables.

###  6.8 Interpretation of the cloud-free limit

As we already mentioned in our scaling of $q_{c}$, we will have
$q_{c}^{(0)}\rightarrow 0$ quite frequently in a realistic setting (this
formal limit simply describes atmospheric environments with a thin cloud
cover, it is not meant to imply assumptions on microphysical processes, such
as fast autoconversion) - what will be the immediate consequences of this
limit for our model? First, we observe that only the evaporation term remains
as a sink in the equation for rain:

$\partial_{z}(\rho_{0}q_{r}^{(0)})=\frac{\rho_{0}}{V_{T}}S_{\text{ev}}^{(0)}.$
(83)

With the constraint

$q_{r}^{(0)}\rightarrow 0$ (84)

as $z\rightarrow\infty$ and $S_{\text{ev}}^{(0)}\geq 0$, this forces us to
conclude

$q_{r}^{(0)}\equiv 0,$ (85)

which in turn means that the water vapor mixing ratio is purely advected by
the geostrophic flow:

$D^{(0)}_{t}q_{v}^{(0)}=0.$ (86)

Thus, _all_ moisture components vanish in the leading-order temperature
equation. The QG dynamics then becomes that of dry air as expected.

##  7 Properties of $\text{PQG}_{\text{DL}}$: the omega equation

As we have emphasized repeatedly, $\text{PQG}_{\text{DL}}$ is set up to
develop a more general model, covering synoptic dynamics through the lower
troposphere and all the way to the planetary boundary layer. Yet, this model
can also stand on its own as a valid extension of etablished QG theory and as
a variant of the PQG family of models. Therefore, we will illustrate some of
its properties in this section, using the _omega equation._

The omega equation is a diagnostic Poisson-type equation for the QG vertical
velocity that constitutes a widely used tool in synoptic meteorology (Hoskins
et al., 1978, 1985, 2003). Since the relationship between moist processes and
up- or downdrafts is of general meteorological interest, we will present the
omega equation in its $\text{PQG}_{\text{DL}}-$form and discuss its
qualitative properties, also in comparison with PQG. Appendix A2 sketches a
derivation of the omega equation in the context of our model.

###  7.1 General form of the equation

As laid out in appendix A2, the omega equation in QG with an arbitrary heat
source $S_{\theta}$ takes the form

$N^{2}\Delta_{\parallel}\tilde{w}+f^{2}\frac{1}{\bar{\rho}}\partial_{z}(\bar{\rho}\partial_{z}\tilde{w})=2\gradient_{\parallel}\cdot\mathbf{Q}+f\beta
v+\Delta_{\parallel}S_{b},$ (87)

where the buoyancy frequency
$N=\sqrt{\frac{g}{\theta_{\text{ref}}}\derivative{\bar{\theta}}{z}}$ is
traditionally assumed constant and we have introduced the _Q-vector_

$\mathbf{Q}=-\left(\gradient_{\parallel}{\tilde{b}}\cdot\partial_{x}\mathbf{u}\right)\mathbf{i}-\left(\gradient_{\parallel}{\tilde{b}}\cdot\partial_{y}\mathbf{u}\right)\mathbf{j},$
(88)

where $\mathbf{i}$, $\mathbf{j}$ are unit vectors in the $x$ and $y$
directions, respectively and
$\tilde{b}=\frac{g}{\theta_{\text{ref}}}\tilde{\theta}$ denotes the buoyancy.
The second term on the right in (87) is due to the $\beta-$effect, which is
frequently neglected.

As demonstrated in (119)-(B), the source term $S_{b}$ in
$\text{PQG}_{\text{DL}}$ reads

$S_{b}=\frac{g}{\theta_{\text{ref}}}\left[c_{l}V_{r}q_{r}\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln\bar{p}-\frac{L_{\text{ref}}}{c_{\text{pd}}}(S_{\text{ev}}-S_{\text{cd}})\right],$
(89)

which leads to

$\displaystyle
N^{2}\Delta_{\parallel}\tilde{w}+f^{2}\frac{1}{\bar{\rho}}\partial_{z}(\bar{\rho}\partial_{z}\tilde{w})=2\gradient_{\parallel}\cdot\mathbf{Q}+f\beta
v$
$\displaystyle+\frac{g}{\theta_{\text{ref}}}\Delta_{\parallel}\left[c_{l}V_{r}q_{r}\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln\bar{p}-\frac{L_{\text{ref}}}{c_{\text{pd}}}(S_{\text{ev}}-S_{\text{cd}})\right]$
(90)

as the omega equation in $\text{PQG}_{\text{DL}}$.

As is apparent from this formulation, the incorporation of cloud microphysics
via parametrization of the various phase conversions allows moisture to
influence vertical motions in multiple ways:

  * •

By condensation in saturated air.

  * •

By “negative condensation” (evaporation of cloud water) as well as evaporation
of rain in undersaturated air.

  * •

By falling rain (regardless of satuation), as indicated by the rightmost term
in (7.1).

In PQG variants that only consider one dynamical moisture variable, such as
the model derived in section 5, no such variety of interactions is possible:
in undersaturated air, moisture cannot influence the dynamics at all.

In the following, we want to make the connection between vertical velocity and
the moist constituents more palpable:

###  7.2 Estimating moist contributions: a sample solution

We can isolate the component of vertical velocity that is _directly_ due to
moist processes by writing any solution of (7.1) as

$\tilde{w}=w_{d}+w_{m},$ (91)

where $w_{d}$ solves the “dry equation” (87) without any heat sources. Then,
$w_{m}$ will be a solution of

$\displaystyle
N^{2}\Delta_{\parallel}w_{m}+f^{2}\partial_{z}\left(\frac{1}{\bar{\rho}}\partial_{z}\left(\bar{\rho}w_{m}\right)\right)=$
$\displaystyle\frac{g}{\theta_{\text{ref}}}\Delta_{\parallel}\left[c_{l}V_{r}q_{r}\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln\bar{p}-\frac{L_{\text{ref}}}{c_{\text{pd}}}(S_{\text{ev}}-S_{\text{cd}})\right].$
(92)

Let us now prescribe a moisture distribution with a particularly convenient
vertical structure that permits an explicit representation of $w_{m}$ in terms
of the sources on the right hand side: in a saturated atmosphere, assume a
state of _incipient condensation_ , such that $S_{\text{ev}}=q_{r}=0$ and

$S_{\text{cd}}=C_{\text{cn}}q_{\text{cn}}(q_{v}-q_{\text{vs}})^{+},$ (93)

i.e., no significant amount of cloud has emerged yet and all condensation is
due to the presence of condensation kernels. We now investigate the case in
which $S_{\text{cd}}$ assumes a vertical structure as follows:

$S_{\text{cd}}(t,\mathbf{x},z)=C_{\text{cn}}T_{0}(z)K(t,\mathbf{x})\sigma(t,\mathbf{x}),$
(94)

where $K$ and $\sigma$ are horizontal distributions of condensation kernels
and supersaturation, respectively. Provided such a distribution, we can then
assume that $w_{m}$ also takes such a vertical structure; we therefore look
for solutions of the form

$w_{m}(t,\mathbf{x},z)=T_{0}(z)W_{m}(t,\mathbf{x}).$ (95)

Substituting this ansatz into (7.2), we get for the left hand side

$\displaystyle
N^{2}\Delta_{\parallel}w_{m}+f^{2}\partial_{z}\left(\frac{1}{\bar{\rho}}\partial_{z}\left(\bar{\rho}w_{m}\right)\right)=$
$\displaystyle
T_{0}\Delta_{\parallel}N^{2}W_{m}+f_{0}^{2}\partial_{z}\left(\frac{1}{\bar{\rho}}\partial_{z}\left(\bar{\rho}T_{0}\right)\right)W_{m}=T_{0}\Delta_{\parallel}N^{2}W_{m},$
(96)

since $\partial_{z}(\bar{\rho}T_{0})=\partial_{z}\bar{p}=-\bar{\rho}$. Thus,
the vertical component drops out of the equation and, using (94), we are left
with

$T_{0}\Delta_{\parallel}N^{2}W_{m}=T_{0}\Delta_{\parallel}\left[\frac{L_{\text{ref}}}{c_{\text{pd}}}C_{\text{cn}}K\sigma\right],$
(97)

which implies

$\Delta_{\parallel}\left[N^{2}W_{m}-\frac{L_{\text{ref}}}{c_{\text{pd}}}C_{\text{cn}}K\sigma\right]=0.$
(98)

Finally, making the assumption that condensation is horizontally restricted to
the interior of the domain under consideration, (98) can be formulated as a
Laplace equation with homogeneous Dirichlet boundary conditions (Evans, 2010),
and its unique solution is simply

$W_{m}=\frac{L_{\text{ref}}C_{\text{cn}}}{c_{\text{pd}}N^{2}}K\sigma.$ (99)

Thus, if (94) holds, updrafts due to condensation intensify _in proportion_
with both supersaturation and the concentration of condensation kernels. We
emphasize that this result only pertains to updrafts directly related to
condensation, not the vertical velocity as a whole.

##  8 Conclusions

In this note, we have demonstrated how to derive the PQG model of Smith and
Stechmann (2017) in its anelastic form systematically from equations for
compressible moist atmospheric flow with bulk microphysics closures. In doing
so, we have not introduced a sound-proof limit from the outset, but relied
only on the distinguished limit for the Rossby, Mach, and Froude numbers
proposed in (Klein, 2010), and on a heuristic correspondence principle for the
asymptotic rescaling of the moist parameters. The derivation was carried out
by systematic asymptotic analysis.

Motivated by the recent introduction of the diabatic layer in the context of
QG theory by Klein et al. (2022), we then put forward a modified scaling
ansatz, designed to yield a system of equations that can connect to this new
intermediate layer (located between the bulk troposphere and the Ekman
friction layer) in a mathematically sound and physically meaningful way. In
contrast to the PQG model, the new PQG${}_{\text{DL}}$ model allows for a
straightforward PV inversion, essentially analogous to that of the classical
QG theory. In turn, PQG${}_{\text{DL}}$ only models the upper 3 to 12 km of
the troposphere, while stronger diabatic effects of moisture at lower levels
are to be addressed by the diabatic layer equations, which we intend to work
out in a future publication. Properties of the PQG${}_{\text{DL}}$ system were
illustrated using the omega equation, an equation that allows to illuminate
the relationship between vertical air motions and moist temperature sources.

We hope to gain more insight into moisture dynamics in the midlatitude
atmosphere on the synoptic scale from future investigations of this system,
regarding for example baroclinic instability, cyclogenesis and the associated
moist processes.

###### Acknowledgements.

D.B. and S.H. acknowledge support by the Austrian Science Fund (FWF) via the
SFB “Taming Complexity in Partial Differential Systems” with project number
F65. S.H. also acknowledges support by the Austrian Science Fund via the
previous Hertha-Firnberg project T-764. R.K. acknowledges support by Deutsche
Forschungsgemeinschaft through Grant CRC 1114 “Scaling Cascades in Complex
Systems”, Project Number 235221301, Project (C06) “Multi-scale structure of
atmospheric vortices”.
The authors are grateful to the Wolfgang Pauli Institute Vienna for continuous
support, e.g. the Pauli fellowship for R.K. and for scientific discussions
with its director, N.J. Mauser.
The authors thank Leslie Smith and Sam Stechmann for helpful comments, in
particular regarding potential vorticity inversion. The paper presents
theoretical work and all derivations should be described in sufficient detail.
No further data are required to reproduce the findings.

##  Appendix A The moist anelastic buoyancy

In this appendix, we do not aim to derive the full set of anelastic equations;
we only want to derive the anelastic buoyancy in the form of Hernandez-Duenas
et al. (2013) under a minimal set of scaling assumptions. In this, we will try
to stay true to the spirit of the traditional derivation, as it can be found
e.g. in (Vallis, 2017).

  1. 1.

The first assumption is that all state variables _as well as the moisture
components_ can be decomposed into a background state $\tilde{f}=\tilde{f}(z)$
and a perturbation $f^{\prime}=f^{\prime}(t,x,y,z)$, where
$f^{\prime}\ll\tilde{f}$. This roughly corresponds to our asymptotic
expansion, with the crucial difference that only two terms are considered;
however, $f^{\prime}$ corresponds to $f^{(2)}$ in our regime. Therefore,
$f_{0}$ and $f_{1}$ here appear “bundled together”, which leads to a less
refined approximation. This is the chief reason for the discrepancy that we
encountered in the main text.

  2. 2.

All thermodynamic relations for the perturbation variables are linearized.
This corresponds to only considering the first dynamically relevant term in
our asymptotic expansion.

  3. 3.

The water content relative to dry air in the atmosphere is always small, that
is $q_{v}\ll 1$. Furthermore, we consider an unsaturated background state,
implying $\tilde{q}_{T}=\tilde{q}_{v}$; as an additional consequence, the
contribution of water vapor $e$ to the total atmospheric pressure is also
small, i.e. $e\ll p$. This does not specify the order of $q_{v}$ or $e$
relative to the perturbations of the state variables, again in contrast to our
regime.

  4. 4.

The depth of the vertical motion is comparable to the density scale height
$-\tilde{\rho}\left(\derivative{\tilde{\rho}}{z}\right)^{(-1)}$; the
(potential) temperature scale height is one order of magnitude larger. The
first part of this assumption is explicitly stated in (Hernandez-Duenas et
al., 2013); the second part makes up for the fact that we do not differentiate
between a leading-order _constant_ temperature and a first-order background
stratification, which again goes back to assumption 1.

  5. 5.

Moisture is present at leading order and does not drop off too quickly at
higher altitudes, that is we require
$\frac{\tilde{q_{v}}}{\tilde{q_{\text{vs}}}}=O(1)$ as well as
$\frac{\derivative{\tilde{q_{v}}}{z}}{\derivative{\tilde{q_{\text{vs}}}}{z}}=O(1)$.

###  A.1 Relations for potential temperature

Since the ideal gas law holds in the form

$p=R_{d}\frac{1+\frac{q_{v}}{E}}{1+q_{T}}\rho T,$ (100)

we can utilize the definition of the potential temperature to derive the
relation

$\ln\theta=\ln C+\frac{1}{\gamma}\ln p-\ln\rho-\ln\bar{R},$ (101)

where $C$ is a constant and

$\bar{R}=R_{d}\frac{1+\frac{q_{v}}{E}}{1+q_{T}}.$ (102)

Now, since (101) has to hold for the background variables alone,
differentiating with respect to $z$ yields

$\displaystyle\frac{1}{\tilde{\theta}}\derivative{\tilde{\theta}}{z}=$
$\displaystyle\frac{1}{\gamma\tilde{p}}\derivative{\tilde{p}}{z}-\frac{1}{\tilde{\rho}}\derivative{\tilde{\rho}}{z}$
$\displaystyle-\frac{1}{E\left(1+\frac{\tilde{q}_{v}}{E}\right)}\derivative{\tilde{q}_{v}}{z}+\frac{1}{1+\tilde{q}_{T}}\derivative{\tilde{q}_{T}}{z}.$
(103)

Since we assume an unsaturated background state, it holds

$\tilde{q}_{T}=\tilde{q}_{v};$ (104)

utilizing hydrostatic balance then yields

$\frac{1}{\tilde{\rho}}\derivative{\tilde{\rho}}{z}=-\frac{g\tilde{\rho}}{\gamma\tilde{p}}-\frac{1}{\tilde{\theta}}\derivative{\tilde{\theta}}{z}-\left(E-1\right)\frac{1}{1+\tilde{q}_{v}}\derivative{\tilde{q}_{v}}{z}.$
(105)

Now, turning to the perturbations, by linearizing (101) we obtain

$\frac{\theta^{\prime}}{\tilde{\theta}}=\frac{1}{\gamma}\frac{p^{\prime}}{\tilde{p}}-\frac{\rho^{\prime}}{\tilde{\rho}}-\frac{q^{\prime}_{v}}{E}+q^{\prime}_{T}.$
(106)

Derivation of the buoyancy term

The assumption of hydrostatic balance for the background profiles yields

$\frac{Dw}{Dt}+\frac{1}{\tilde{\rho}}\frac{\partial\rho^{\prime}}{\partial
z}=-g\frac{\rho^{\prime}}{\tilde{\rho}}$ (107)

in the vertical momentum balance, which can be rewritten as

$\frac{Dw}{Dt}+\frac{\partial}{\partial
z}\left(\frac{p^{\prime}}{\tilde{\rho}}\right)=-\frac{p^{\prime}}{\tilde{\rho}^{2}}\derivative{\tilde{\rho}}{z}-g\frac{\rho^{\prime}}{\tilde{\rho}};$
(108)

substituting (105), we then arrive at

$\displaystyle\frac{Dw}{Dt}+\frac{\partial}{\partial
z}\left(\frac{p^{\prime}}{\tilde{\rho}}\right)=$
$\displaystyle-\frac{p^{\prime}}{\tilde{\rho}}\left[-\frac{g\tilde{\rho}}{\gamma\tilde{p}}-\frac{1}{\tilde{\theta}}\derivative{\tilde{\theta}}{z}-\left(\frac{1}{E}-1\right)\frac{1}{1+\tilde{q}_{v}}\derivative{\tilde{q}_{v}}{z}\right]-g\frac{\rho^{\prime}}{\tilde{\rho}}.$
(109)

With our assumption on the scale heights, the second term in brackets on the
RHS is clearly asymptotically small; the moisture term can be rewritten in the
form

$\frac{1}{\frac{1}{\tilde{q}_{v}}+1}\frac{1}{\tilde{q}_{v}}\derivative{\tilde{q}_{v}}{z}=\frac{1}{\frac{1}{\tilde{q}_{v}}+1}\derivative{\ln\tilde{q}_{v}}{z};$
(110)

due to $\tilde{q}_{v}\ll 1$, it therefore suffices to show that
$\derivative{\ln\tilde{q}_{v}}{z}$ is not asymptotically large in order to
justify the neglect of this term.
To this end, we can use the Clausius-Clapeyron equation for the (background)
saturation vapor pressure $\tilde{e}_{s}$:

$\derivative{\ln\tilde{e}_{s}}{T}=\frac{1}{R_{v}}\frac{L(\tilde{T})}{\tilde{T}^{2}}.$
(111)

The saturation vapor pressure, in turn, is related to the (background)
saturation mixing ratio $\tilde{q}_{\text{vs}}$ by the formula

$\tilde{q}_{\text{vs}}=\frac{E\tilde{e}_{s}}{\tilde{p}-\tilde{e}_{s}}.$ (112)

In our setting, it holds
$\derivative{\ln\tilde{q}_{v}}{z}\sim\derivative{\ln\tilde{q}_{\text{vs}}}{z}$,
due to assumption 5. It therefore suffices to estimate the latter:

$\displaystyle\derivative{\ln\tilde{q}_{v}}{z}=\derivative{\tilde{T}}{z}\derivative{\ln\tilde{e}_{s}}{\tilde{T}}+\frac{g\tilde{\rho}}{\tilde{p}-\tilde{e}_{s}}+\frac{\tilde{e}_{s}}{\tilde{p}-\tilde{e}_{s}}\derivative{\tilde{T}}{z}\derivative{\ln\tilde{e}_{s}}{\tilde{T}}=$
$\displaystyle\frac{g\tilde{\rho}}{\tilde{p}-\tilde{e}_{s}}+\left(\frac{\tilde{q}_{\text{vs}}}{E}+1\right)\derivative{\tilde{T}}{z}\derivative{\ln\tilde{e}_{s}}{\tilde{T}};$
(113)

the first term here is clearly bounded, while we can employ Clausius-Clapeyron
for the second:

$\derivative{\tilde{T}}{z}\derivative{\ln\tilde{e}_{s}}{\tilde{T}}=\derivative{\tilde{T}}{z}\frac{1}{R_{v}}\frac{L(\tilde{T})}{\tilde{T}^{2}}=\frac{1}{R_{v}}\frac{L(\tilde{T})}{\tilde{T}}\frac{1}{\tilde{T}}\derivative{\tilde{T}}{z}.$
(114)

Due to our assumption on the temperature scale height, we can infer that this
term is not just bounded from above, but actually asymptotically small.

Going back to (A.1), the remaining terms yield the expression

$g\left[\frac{1}{\gamma}\frac{p^{\prime}}{\tilde{p}}-\frac{\rho^{\prime}}{\tilde{\rho}}\right],$
(115)

which by (106) can be rewritten as

$g\left[\frac{\theta^{\prime}}{\tilde{\theta}}+\left(\frac{1}{E}-1\right)q^{\prime}_{v}-q^{\prime}_{c}-q^{\prime}_{r}\right].$
(116)

This is the exact form of the buoyancy force in the anelastic system of
Hernandez-Duenas et al. (2013).

##  Appendix B Derivation of the omega equation

In any version of PQG, the hydrostatic and geostrophic balances (5a) and (5b)
hold, as well as the vorticity equation (9a). Since these relations are
equivalent to those in standard QG theory and no vertical velocity appears in
the moist transport equations of $\text{PQG}_{\text{DL}}$, any changes in the
result will stem from the temperature equation. The latter can be reformulated
in terms of potential temperature such that all moist terms appear as sources;
thus, it suffices to sketch a derivation that is analogous to one for _dry_ QG
with a heat source. While the latter can be considered generally known, most
references on the omega equation make use of the Boussinesq approximation and
neglect the $\beta-$effect. Therefore, we show the main steps of a derivation
without these a priori simplifications.

Our starting point are the geostrophic and hydrostatic balances (5a) and (5b),
respectively, and the transport equations for vorticity and the equivalent
potential temperature perturbation (15a) and (15b), respectively, which we
restate here for the reader’s convenience:

$\displaystyle f\mathbf{k}\times\mathbf{u}$
$\displaystyle=-\gradient_{\parallel}{\phi}$ (117a) $\displaystyle
g\frac{\tilde{\theta}}{\theta_{\text{ref}}}$ $\displaystyle=\partial_{z}\phi,$
(117b) $\displaystyle D^{g}_{t}\left[\zeta+\beta y\right]$
$\displaystyle=\frac{f}{\bar{\rho}}\partial_{z}\left(\bar{\rho}\tilde{w}\right)$
(117c) $\displaystyle
D^{g}_{t}\tilde{\theta}_{e}+\tilde{w}\derivative{\bar{\theta}_{e}}{z}$
$\displaystyle=c_{l}V_{r}q_{r}\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln\bar{p}.$
(117d)

As usual, the pressure perturbation $\phi$ assumes the role of a stream
function (up to a constant factor), in particular it holds:

$\zeta=\frac{1}{f}\Delta_{\parallel}\phi.$ (118)

Observing that $\bar{\theta_{e}}=\bar{\theta}$ in $\text{PQG}_{DL}$, since
there is no moist background in this version of PQG and rewriting (117d) in
terms of $\theta$, we get

$D^{g}_{t}\tilde{\theta}+\tilde{w}\derivative{\bar{\theta}}{z}=c_{l}V_{r}q_{r}\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln\bar{p}-\frac{L_{\text{ref}}}{c_{\text{pd}}}D^{g}_{t}q_{v}$
(119)

and, employing (15c),

$D^{g}_{t}\tilde{\theta}+\tilde{w}\derivative{\bar{\theta}}{z}=c_{l}V_{r}q_{r}\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln\bar{p}-\frac{L_{\text{ref}}}{c_{\text{pd}}}(S_{\text{ev}}-S_{\text{cd}}).$
(120)

Finally, assuming a constant Brunt-Väisälä frequency
$N=\sqrt{\frac{g}{\theta_{\text{ref}}}\derivative{\bar{\theta}}{z}}$, we can
write the above in the form

$\displaystyle D^{g}_{t}\tilde{b}+N^{2}\tilde{w}=$
$\displaystyle\frac{g}{\theta_{\text{ref}}}\left[c_{l}V_{r}q_{r}\frac{R_{d}}{c_{\text{pd}}}\partial_{z}\ln\bar{p}-\frac{L_{\text{ref}}}{c_{\text{pd}}}(S_{\text{ev}}-S_{\text{cd}})\right]=:S_{b},$
(121)

where $\tilde{b}:=g\frac{\tilde{\theta}}{\theta_{\text{ref}}}$.
Now, starting with the actual derivation, we can combine (117c) and (118) to
obtain

$D_{t}^{g}(\Delta_{\parallel}\phi)=\frac{f^{2}}{\bar{\rho}}\partial_{z}(\bar{\rho}\tilde{w})-f\beta
v.$ (122)

Taking the vertical derivative of this equation and using hydrostatic balance
(117b), we get

$\displaystyle D_{t}^{g}(\Delta_{\parallel}\tilde{b})$
$\displaystyle=-\partial_{z}\mathbf{u}\cdot\gradient_{\parallel}(\Delta_{\parallel}\phi)+\frac{f^{2}}{\bar{\rho}}\partial_{z}(\bar{\rho}\tilde{w})-f\beta
v$
$\displaystyle=-\mathbf{k}\times\gradient_{\parallel}{\tilde{b}}\cdot\gradient_{\parallel}{\zeta}+\frac{f^{2}}{\bar{\rho}}\partial_{z}(\bar{\rho}\tilde{w})-f\beta
v.$ (123)

In the next step, we take $\Delta_{\parallel}$ of (117d) and get

$\displaystyle
D_{t}^{g}(\Delta_{\parallel}\tilde{b})+\Delta_{\parallel}(\mathbf{u}\cdot\gradient_{\parallel}{\tilde{b}})-\mathbf{u}\cdot\gradient_{\parallel}(\Delta_{\parallel}\tilde{b})=$
$\displaystyle-N^{2}\Delta_{\parallel}\tilde{w}+\Delta_{\parallel}S_{b}.$
(124)

A laborious, but straightforward calculation then shows

$\displaystyle\Delta_{\parallel}(\mathbf{u}\cdot\gradient_{\parallel}{\tilde{b}})=$
$\displaystyle\mathbf{u}\cdot\gradient_{\parallel}(\Delta_{\parallel}\tilde{b})-\gradient_{\parallel}\tilde{b}\cdot\gradient_{\parallel}\mathbf{u}-2\gradient_{\parallel}\cdot\mathbf{Q},$
(125)

where the Q-vector is defined as in (88). Thus, we obtain

$\displaystyle D_{t}^{g}(\Delta_{\parallel}\tilde{b})=$
$\displaystyle\gradient_{\parallel}\tilde{b}\cdot\gradient_{\parallel}\mathbf{u}+2\gradient_{\parallel}\cdot\mathbf{Q}+\Delta_{\parallel}S_{b}-N^{2}\Delta_{\parallel}\tilde{w}.$
(126)

To conclude, we want to combine (B) and (B) in order to arrive at a diagnostic
equation; checking that it holds

$-\mathbf{k}\times\gradient_{\parallel}\tilde{b}\cdot\gradient_{\parallel}\zeta=\gradient_{\parallel}\tilde{b}\cdot\Delta_{\parallel}\mathbf{u},$
(127)

thanks to the incompressibility of $\mathbf{u}$, these two terms cancel when
substracting (B) from (B) and rearranging yields

$N^{2}\Delta_{\parallel}\tilde{w}+f^{2}\frac{1}{\bar{\rho}}\partial_{z}(\bar{\rho}\partial_{z}\tilde{w})=2\gradient_{\parallel}\cdot\mathbf{Q}+f\beta
v+\Delta_{\parallel}S_{b},$ (128)

the $\text{PQG}_{\text{DL}}$ version of the omega equation.

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