General yet compact equations are presented to express the thermodynamic impact of physical parameterizations in a NWP or climate model. By expressing the equations in a flux-conservative formulation, the conservation of mass and energy by the physics parameterizations is a built-in feature of the system. Moreover, the centralization of all thermodynamic calculations guarantees a consistent thermodynamical treatment of the different processes. The generality of this physics–dynamics interface is illustrated by applying it in the AROME NWP model. The physics–dynamics interface of this model currently makes some approximations, which typically consist of neglecting some terms in the total energy budget, such as the transport of heat by falling precipitation, or the effect of diffusive moisture transport. Although these terms are usually quite small, omitting them from the energy budget breaks the constraint of energy conservation. The presented set of equations provides the opportunity to get rid of these approximations, in order to arrive at a consistent and energy-conservative model. A verification in an operational setting shows that the impact on monthly-averaged, domain-wide meteorological scores is quite neutral. However, under specific circumstances, the supposedly small terms may turn out not to be entirely negligible. A detailed study of a case with heavy precipitation shows that the heat transport by precipitation contributes to the formation of a region of relatively cold air near the surface, the so-called cold pool. Given the importance of this cold pool mechanism in the life cycle of convective events, it is advisable not to neglect phenomena that may enhance it.

The conservation of mass and energy are important
characteristics of a numerical atmospheric model. Especially in view of the
application in climate studies, even small violations of the conservation
laws can accumulate over a long integration time, and lead to faulty results

A lot of research has been spent in designing dynamical cores that conserve
mass and energy

A possible explanation is that the thermodynamics of the dynamical core are
less complicated than those of the physical parameterizations. More
specifically, the dynamics are usually considered adiabatic and reversible
(except for numerical diffusion)

There is, however, an increased interest in different aspects of the coupling
of physical parameterizations to the dynamical core. One of the issues is the
organization of the time step. This problem has been studied with academic
toy-models (see, e.g.

The current paper develops the proposal of CGTBT07 further by generalizing it
for a system with an arbitrary number of hydrometeors with arbitrary
interactions between them. It should be emphasized that the scope of this
work is limited to the coupling of the atmospheric physics parameterizations
to the dynamical core. For instance, when energy-conserving equations are
presented, this property does not necessarily hold for the atmospheric model
as a whole, but only regarding the influence of the physical
parameterizations. Other aspects of the model, most notably its dynamical
core, may not be energy conserving. Also the mutual interactions between
different parameterizations are not considered in this paper, as they relate
only indirectly to the time evolution of the prognostic atmospheric
variables. The next section presents the equations of this generalized
system. In Sect. 3, this set of equations is applied in the AROME numerical
weather prediction (NWP) model

Because the behaviour of the atmosphere is too complex to be described exactly, every numerical model needs to make simplifying hypotheses. This is no different for the work described in the current paper. It is not our aim to present a set of equations which is exact in the sense that it is free of approximations. However, a crucial aspect of the work presented in CGTBT07 is that the set of hypotheses that relate to the thermodynamics is defined from the very beginning. This is important for two reasons. First, it ensures that the simplifications act consistently throughout the model. Second, it allows to set some non-negotiable constraints. For instance, the conservation of energy must be satisfied, no matter what other simplifications are made. This approach of setting the simplifying hypotheses from the beginning contrasts with the conventional approach of ignoring supposedly small terms along the way.

The framework of hypotheses is the following:

A fully barycentric view of air parcels is adopted. This means that all hydrometeors (both suspended and precipitating)
are considered as integral parts of the air, and contribute to the parcel's motion, density, and heat capacity. This barycentric
view has been studied and motivated by many researchers

Water condensates are assumed to have zero volume. This is a common approximation in atmospheric modelling.

Gases follow Boyle–Mariotte's and Dalton's laws.

Temperature is homogeneous across all species, even falling hydrometeors. For small hydrometeors, this approximation
is easily justified, given their short relaxation time

The specific heat values of all species are constant with temperature.

The latent heat values of sublimation and evaporation,

It should be mentioned that this same framework of assumptions has been used
by

The system considered in CGTBT07 consists of dry air (specific mass fraction

In these equations,

The pseudoflux

Although a pseudoflux is arguably more difficult to interpret than a tendency, writing conversions between species in terms of pseudofluxes offers the possibility to write the evolution equations in a flux-conservative form. The benefit of this is explained further. Also note that this does not mean that the internals of the physics parameterizations should be formulated in terms of pseudofluxes. Instead, it is only at the moment when the contributions of the physics parameterizations are added to the prognostic variables, that pseudofluxes are beneficial. They can be determined at that point from the more conventional tendencies using the expression above.

The thermodynamic equation for the system with 4 hydrometeors is as follows:

It should be noted that Eq. (

A full discussion of these equations is given in CGTBT07, but we would like
to stress the following characteristics:

All equations are flux-conservative, i.e. every right-hand side is a divergence of a summation of fluxes. The importance of this property cannot be underestimated, because it means that this system intrinsically conserves mass and energy. Put somewhat simplistically, in a flux-conservative system, the only way energy or mass can leave one model layer, is by transporting it to an adjacent layer. Therefore, mass and energy are conserved by design of the system.

The precipitation fluxes

To derive these relations, one starts from the definition of a flux as a product of a density with a velocity. For instance, for rain, one writes

The absolute velocity

The relative velocity of rain is then given by

The latent heat values of sublimation and condensation

Using the before-set assumption that

This shows how the temperature dependence of the latent heat values can be accounted for by considering the tendency of enthalpy.

Although the equations only describe the evolution of water species, similar flux-conservative equations could be formulated for other atmospheric variables like momentum, turbulent kinetic energy, etc. In this paper, only water species and their effect on the thermodynamic equation are studied.

Despite the clear strength of the equations proposed by CGTBT07, their
application is not straightforward because of the fixed number of water
species, and because of the fixed set of interactions between them (six
pseudofluxes). More advanced microphysics schemes often consider more water
species, for instance by including graupel and/or hail

It is, however, possible to generalize the equations from CGTBT07, without
touching the important characteristics. We introduce the following notation:

Variables

Variables

Next, a precipitation flux

These notations make it possible to formulate the specific mass equations and
the thermodynamic equation as follows:

These equations generalize the ones from CGTBT07 in three ways: (i) an
arbitrary number

Some comments should be given on the application area of the physics–dynamics
interface presented in
Eqs. (

The fact that these equations are very general, opens the road for a “plug-compatible” view of physics parameterizations.
Indeed, the only output that is needed from a parameterization are diffusive and precipitative transport fluxes, pseudofluxes for phase
changes, and the radiative and diffusive energy fluxes. The physics–dynamics interface then receives these quantities and determines the
effect on the prognostic variables of the model, thereby ensuring satisfaction of the conservation of mass and energy, as well as consistency in the thermodynamic assumptions.
However, it should be kept in mind that other conditions should be met before parameterizations can really be considered plug-compatible.
A first aspect is that interactions exist between parameterizations. For instance, the parameterization of cloud processes will affect
the radiation scheme. These kinds of interactions should properly be accounted for when plugging a new parameterization into a model.
In this context, it is interesting to see that the technical recommendations that were made in

A common assumption in atmospheric modelling (although it is often made implicitly) is that all vertical mass transport due to the physics
parameterizations is compensated for by a fictitious flux of dry air

The Eqs. (

However, as shown by

The fact that Eq. (

The importance of writing Eq. (

As a side remark, it can be noted that simply adding temperature tendencies from several parameterizations cannot lead to an
energy-conserving atmospheric model, at least not for a process-split coupling strategy

This expression is only valid for a process-split coupling. For a time-split coupling, the total enthalpy change is still equal to the sum
of the enthalpy changes of the separate processes, as indicated in Eq. (

Working out the heat capacity and the temperature at the end of the time step now gives

So with time-split coupling, the total temperature change can be obtained as the summation of the temperature changes from the separate parameterizations. However, it is better to use an enthalpy-based system, as this works both for the process-split and the time-split cases.

The Eqs. (

Operational AROME domain with a resolution of 2.5 km. The markers indicate the temperature stations used
for the monthly scores. The dashed line indicates the area of the case study of Sect.

RMSE (solid line) and bias (dashed line) over the period 1–30 November 2014, for REF (blue circles) and FCI (red triangles).

AROME is a limited area model that was developed at Météo-France and is
now a configuration inside the ALADIN system. It became operational in France
in 2008, and it is currently used in many European countries of the ALADIN
and HIRLAM consortia. AROME uses a nonhydrostatic, fully compressible
dynamical core

A first approximation that is made in the existing AROME physics–dynamics
interface concerns the heat transport by precipitation. From
Eq. (

The combination of these two effects indeed corresponds to the effect of
precipitation on the right-hand side of Eq. (

The approximation made by the existing physics–dynamics interface in AROME is
that it neglects the heat transport effect of precipitation, i.e. the term
given in Eq. (

A second approximation concerns the effect of diffusive moisture transport
(shallow convection and turbulence) in the energy budget. Similar to the
effect of precipitation, diffusive moisture transport modifies the total
specific heat capacity

A third approximation is that the values of specific heat capacity

A final approximation by the existing physics–dynamics interface in AROME is that the total temperature tendency is obtained by summing the temperature tendencies from the individual parameterizations. As indicated in the previous section, such an approach cannot lead to an energy-conserving system in a model with a process-split time step organization.

RMSE (solid line) and bias (dashed line) over the period 6 January–6 February 2015, for REF (blue circles) and FCI (red triangles).

Although it can be expected that the overall effect of these approximations and inconsistencies is quite limited, the generalized physics–dynamics interface as presented in the previous section offers the possibility to get rid of them in order to take a (admittedly small) step towards a more accurate model. A second motivation to equip the AROME model with the generalized flux-conservative physics–dynamics interface is that this opens the route towards importing physics parameterizations from other NWP models, thus allowing a fair comparison of different parameterizations and stimulating scientific progress.

Neighbourhood observation Brier skill score for precipitation between 12:00 and 18:00 UTC over the period 1–30 November 2014,
for REF (blue circles) and FCI (red triangles):

Neighbourhood observation Brier skill score for precipitation between 12:00 and 18:00 UTC over
the period 6 January–6 February 2015, for REF (blue circles) and FCI (red triangles):

The impact of the presented flux-conservative formulation of the
physics–dynamics interface is investigated with the AROME operational
high-resolution LAM model running at Météo-France. Before April 2015,
this model ran on a

At the surface level, precipitation and evapotranspiration imply a net mass
flux across the surface. Since the vertical coordinate of the AROME model is
mass based, correctly accounting for such net mass exchange between
atmosphere and surface has far-reaching implications, especially in the
surface boundary condition of the nonhydrostatic dynamical core. Currently,
this has not been implemented in the dynamical core of the AROME model.
Instead, the above-mentioned approximation is made that all vertical transport due to
the parameterizations is compensated by a fictitious flux of dry air. Taking
full advantage of the barycentric framework of
Eqs. (

All these settings are identical for the operational run (denoted REF) with the temperature tendency-based interface and for the run with the flux-conservative interface (denoted FCI).

Case of heavy precipitation on 19 January 2015. The arrow and the marker in subfigure

The daily forecasts during two periods are considered in this section:
1–30 November 2014 and 6 January–6 February 2015. The first month is
characterized by exceptionally mild weather, with numerous episodes of heavy
precipitation in the southwest of France. The second month was characterized
by strong winds and episodes of heavy snowfall.
Figures

Vertical profiles at 18:00 UTC in the point indicated in Fig.

The scores indicate that the impact of using the flux-conservative set of equations is quite limited when considering time- and space-averaged scores as the ones presented here. It should be stressed that no retuning has been done for the experiments with the flux-conservative equations. As a result, compensating errors can be responsible for masking an improvement of the scores. The fact that the scores do not change substantially, merely indicates that the approximations that are made in the existing temperature tendency-based interface are indeed small on a domain-wide scale. In this context, the limitations of this standard verification against station data should also be mentioned. By taking the average score over a large number of stations, important local differences may be hidden in the scores. In a similar way, the fact that monthly-averaged scores are considered, only allows to detect differences that are systematic in time. Therefore, notwithstanding the neutral impact on the standard scores, some significant differences may be observed under specific circumstances. A case study is presented in the next section to illustrate this.

When precipitation evaporates while falling through unsaturated air, it cools
its environment. As such, a region of relatively cool air, the so-called cold
pool, originates when heavy, localized precipitation occurs, for instance in
precipitating convective systems

Although evaporative cooling is the main cause for a cold pool, a second
mechanism may enhance it. As precipitation falls from colder layers aloft to
hotter layers below, it will be heated by the surrounding air, which in
response will cool down

This is confirmed when looking at the AROME forecasts over the Balearic
islands on 19 January 2015. This case is characterized by convection
developing ahead of an active cold front coming from the south.
Figure

To further illustrate the impact of the heat transport by precipitation on
the cold pool, the vertical profiles in the point as marked in
Fig.

No comparison with observations is done for this case, because the purpose of
this case study is merely to illustrate that even small terms in the energy
budget can have a significant impact under certain conditions. The
conclusions from this case study are in line with the results from

This paper starts from the equations presented in

Notwithstanding these clear advantages, the equation set in the mentioned paper also faces limitations that hinder its application in existing NWP models. This paper presents a generalized set of thermodynamic equations that overcomes these restrictions without touching the sound theoretical foundations. More specifically, the presented equations are valid for an arbitrary number of hydrometeors, and can be applied in a model with an arbitrary number of conversion processes between these water species. This has allowed to use this set of equations in the AROME NWP model, which currently uses a physics–dynamics interface that makes some ad-hoc approximations. By moving to the generalized flux-conservative equations, the effect of these approximations can be studied.

Monthly verification scores show that the overall effect of introducing the flux-conservative equations in AROME is quite limited. There is no significant improvement or degradation of these scores. Given the mentioned theoretical benefits of the presented equations, this means that the presented work is a valuable advancement of the AROME model. Moreover, it appears that substantial differences may exist in specific cases. A detailed study of a heavy-precipitation case gives the example of the formation of a cold pool, which is an essential mechanism in the life cycle of a convective event. As it appears, one mechanism that contributes to the formation of this cold pool is the heat transport by precipitation. This effect is neglected in the existing AROME physics–dynamics interface, while it is correctly accounted for in the presented flux-conservative set of equations. In this specific case, this leads to a different surface temperature and surface pressure within the cold pool. A more systematic study of the effect of heat transport on the life cycle of a cold pool is left for future research. In this paper, this case serves as an illustration of the importance of correctly accounting for supposedly small terms in the energy budget, something that is achieved with the presented set of thermodynamic equations.

Besides offering a direct improvement of the thermodynamic budget of the physics parameterizations of the AROME model, the presented set of equations also paves the way for interesting future research. Especially the impact of the heat from physics parameterizations on the continuity equation, and the effect of accounting for the net mass exchange between the atmosphere and the surface, are topics that deserve to be studied in detail.

The used ALADIN codes, along with all related intellectual property rights, are owned by the members of the ALADIN consortium. Access to the ALADIN system, or elements thereof, can be granted upon request and for research purposes only.

The authors of this paper wish to commemorate Jean-François Geleyn, who in his unique vision and understanding ceaselessly stressed the importance of this topic. The authors thank the reviewers for their remarks that helped improving this manuscript. This research is supported in part by the Belgian Federal Science Policy Office under contract BR/121/A2/STOCHCLIM.Edited by: J. Williams