Introduction
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
. Atmospheric forecast models are usually
constructed by combining a dynamical core with physical parameterizations. In
general, the dynamical core describes the atmospheric behaviour up until the
resolved scales, while the physical parameterizations estimate the effect of
subgrid processes .
A lot of research has been spent in designing dynamical cores that conserve
mass and energy . Common strategies include a careful
selection of the prognostic variables , the formulation of the equations in flux form
, or taking advantage of properties of the Hamiltonian
character of the atmospheric equations . In contrast with these efforts on the dynamical core, the energy
conservation and consistent thermodynamics seem to be less of a priority in
the development of the physical parameterizations, or in the way they are
coupled to the dynamical core.
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) . The physics
parameterizations, on the other hand, include mass and energy exchange with
the surface, as well as radiative fluxes at the top of the atmosphere. They
constitute an open thermodynamic system, for which the conservation laws are
more difficult to enforce. Moreover, it is tempting to consider physics
parameterizations as plug-compatible, i.e. they are considered as a black box
which, given an atmospheric state, returns an effect on the dynamical
prognostic variables. Unfortunately, this plug compatibility seems to go at
the expense of carefully investigating the thermodynamic consistency between
the dynamical core and the physics parameterizations, and inserting a new
parameterization in a model comes with implicit assumptions and ad-hoc
approximations.
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. ), as
well as in 3-D models . The thermodynamic
aspects of the physics–dynamics coupling is another topic that deserves some
attention. Although some attempts have been made to rigorously formulate the
equations for a multicomponent atmosphere , it
remains a fact that many operational models make several ad-hoc
approximations . , hereafter CGTBT07,
presented a set of equations that expresses the effects of physics
parameterizations in a flux-conservative formulation. The advantage of this
approach is that this is an inherently mass- and energy-conservative system.
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 , thus allowing to get rid of
some approximations that are currently made. Section 4 discusses the impact
on the meteorological results, both by means of monthly scores and with an
in-depth case study of a cold pool formation under heavy precipitation.
Section 5 presents the conclusions.
Formulation of the generalized flux-conservative equations
Framework of hypotheses
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 . For larger hydrometeors, it is a cruder approximation,
but it goes together with the barycentric view: since such hydrometeors are considered part of the parcel, they also take the parcel's temperature.
The specific heat values of all species are constant with temperature.
The latent heat values of sublimation and evaporation, Li and Ll, respectively, vary linearly with temperature T:Li|l(T)=Li|l(T0)+(cpv-ci|l)T,where T0=0 K, cpv is the specific heat capacity at constant
pressure of water vapour, and ci and cl are the specific heat capacity values of ice and liquid water, respectively.
It should be mentioned that this same framework of assumptions has been used
by , , and to
cleanly develop moist atmospheric thermodynamic quantities such as moist
entropy, moist potential temperature, and moist Brunt–Väisälä
frequency.
The flux-conservative equations for a system with five water species
The system considered in CGTBT07 consists of dry air (specific mass fraction
qd=ρd/ρtot) plus five prognostic water
species: vapour (specific mass fraction qv), suspended liquid
water droplets (ql), suspended ice crystals (qi),
precipitating rain (qr), and precipitating snow (qs).
For this system, the following equations are derived for the time evolution
of the prognostic species due to physical parameterizations:
∂qv∂t=g∂∂pRr,v+Rs,v-Rv,l-Rv,i+qv(Pr+Ps)1-qr-qs-Dv∂ql∂t=g∂∂pRv,l-Rl,r+ql(Pr+Ps)1-qr-qs-Dl∂qr∂t=g∂∂pRl,r-Rr,v-Pr∂qi∂t=g∂∂pRv,i-Ri,s+qi(Pr+Ps)1-qr-qs-Di∂qs∂t=g∂∂pRi,s-Rs,v-Ps∂qd∂t=g∂∂pqd(Pr+Ps)1-qr-qs-Dd.
In these equations, Pk denotes precipitation fluxes and Dk denotes
diffusive fluxes. Note that it is necessary that ∑k=d,v,i,lDk=0 to ensure that all terms on the right-hand sides cancel out. The terms
Rk1,k2 denote pseudofluxes and represent mass transfer between two
water species. The concept of pseudofluxes is essential to the presented
system and deserves some more explanation. The common and more intuitive way
to express a mass transfer between two species is through a time tendency.
For instance, consider the microphysical process of condensation, which is a
mass transfer from water vapour to liquid cloud water droplets. The effect of
this process on the specific humidities could be expressed as
∂qv∂t=-∂q∂tcond∂ql∂t=∂q∂tcond.
The pseudoflux Rv,l expresses exactly the same effect, only as a
flux instead of as a tendency. This flux is determined by taking the vertical
integral of the tendency:
Rv,l=1g∫0p∂q∂tconddp.
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:
∂∂t(cpT)=-g∂∂p[(cl-cpd)PrT+(ci-cpd)PsT-(c^-cpd)(Pr+Ps)T+Js+Jrad-Ll(T0)(Rv,l-Rr,v)-Li(T0)(Rv,i-Rs,v)],
where c^=cpdqd+cpvqv+ciqi+clql1-qr-qs,
and Js and Jrad are the diffusive and radiative heat
fluxes, respectively. cp is the total heat capacity of the parcel, given
by
cp=cpdqd+cpvqv+cl(ql+qr)+ci(qi+qs).
It should be noted that Eq. () expresses only the
thermodynamic effect of the physical parameterizations. The complete
thermodynamic equation of the atmospheric model would also include terms that
are resolved by the dynamics of the model.
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 Pr and Ps are relative to the (moving) center of mass of the parcel.
They relate to the absolute precipitation fluxes Pr* and Ps* throughPr=(1-qr)Pr*-qrPs*Ps=-qsPr*+(1-qs)Ps*.
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 writesPr*=ρtotqrwr*.
The absolute velocity w* of the center of mass of the parcel is given by the weighted average of the
velocities of the components. In a system where only rain and snow are precipitating, this means thatw*=qrwr*+qsws*.
The relative velocity of rain is then given by wr=wr*-w*, so the relative precipitation flux becomesPr=ρtotqrwr=ρtot(qrwr*-qr2wr*-qrqsws*)=(1-qr)Pr*-qrPs*.
The latent heat values of sublimation and condensation Li and Ll that appear on the right-hand side,
are evaluated at T0=0K. This does not mean that the temperature dependency of these latent heat values is neglected.
Instead, it is accounted for by considering the time derivative of the enthalpy cpT. Considering only the process of
condensation, the traditional way to express its thermodynamic effect would becp∂T∂t=Ll(T)∂ql∂t.
Using the before-set assumption that Ll varies linearly with temperature, and the fact that, still only considering condensation, ∂qv/∂t=-∂ql/∂t, so ∂cp/∂t=(cl-cpv)∂ql/∂t, this expression becomescp∂T∂t=Ll(T0)∂ql∂t-T∂cp∂t,which can be rewritten as∂∂t(cpT)=Ll(T0)∂ql∂t.
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.
The generalized flux-conservative equations
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 ,
or by separating convective and nonconvective fractions of hydrometeors
. Also the fact that only six transfer mechanisms between
the water species are possible is limiting. For instance, snow melting cannot
be represented directly, but it should be written as a combination of snow
sublimation (Rs,v) and rain evaporation (Rr,v). Although
thermodynamically fully correct, it would be better to have a system that
digests all kinds of transfers between water species.
It is, however, possible to generalize the equations from CGTBT07, without
touching the important characteristics. We introduce the following notation:
n is the number of water species, the index k=1,…,n denotes a single
water species, and by convention, k=0 denotes the dry air component. The
specific heat capacity values at constant pressure of the different species
are written generically as ck, the latent heat of evaporation or
sublimation at 0 K is written as Lk0. The index j denotes a
conversion process between a source water species kjs and a
target water species kjt. The effect of this process is
expressed through the pseudoflux Rj. We consider an arbitrary number m
of such conversion processes. We now define variables
λkj=δk,kjs-δk,kjt for
k=1,…,n and for j=1,…,m, where the usual definition of the
Kronecker delta is used. The variable λkj takes a value of 0 if a
species k is not involved in the conversion process j; it takes a value
of -1 if it is the target species of this process; and it takes a value of 1 if it
is the source species of this process. The variable λkj will allow
to write the time tendency of a water species by summing over all conversion
processes, regardless of the role this specific water species plays in each
process. Furthermore, a variable
Λj0=Lkjs0-Lkjt0 is defined. This
variable is the latent heat released at temperature T0 under a conversion
process with source species kjs and target process
kjt. To clarify these notations, consider the original system of
CGTBT07 with five water species and six conversion processes between them. By
convention, we assign k=1,…,5 to water vapour, liquid cloud water,
precipitating rain, cloud ice crystals, and precipitating snow, respectively.
Tables and give the values of λkj and
Λj0 for the different conversion processes.
Variables λkj for the system of CGTBT07.
Process
r→v
v→l
l→r
s→v
v→i
i→s
j
1
2
3
4
5
6
Species
k
v
1
-1
1
0
-1
1
0
l
2
0
-1
1
0
0
0
r
3
1
0
-1
0
0
0
i
4
0
0
0
0
-1
1
s
5
0
0
0
1
0
-1
Variables Λj0 for the system of CGTBT07.
Process
r→v
v→l
l→r
s→v
v→i
i→s
j
1
2
3
4
5
6
Λj0
Ll(T0)
-Ll(T0)
0
Li(T0)
-Li(T0)
0
Next, a precipitation flux Pk is defined for each component, even for the
non-precipitating species (dry air, vapour, liquid cloud water droplets, and
cloud ice crystals). Contradictory as this may sound, it should be stressed
that in our barycentric system, these fluxes express the motion of the
species with respect to the center of mass of the parcel. When precipitating
species are present, the suspended species will move upward with respect to
the mass center. Using a similar calculation as before to describe the motion
with respect to the center of mass of the parcel, the relative precipitation
fluxes Pk are determined from the absolute fluxes Pk* as
Pk=Pk*-qk∑i=0nPi*,
where the absolute precipitation fluxes of suspended species can be taken to
be zero. It should be noted that the strict distinction in CGTBT07 between
suspended and precipitating species is somewhat arbitrary and
scale dependent. Indeed, also the so-called suspended cloud water species can
undergo a slow sedimentation. This arbitrary distinction is no longer
necessary in the generalized set of equations that is presented here.
Similarly to defining (relative) precipitation fluxes for all species, also
diffusive fluxes Dk are defined for all species, where the diffusive
fluxes of precipitating species can be taken equal to zero.
These notations make it possible to formulate the specific mass equations and
the thermodynamic equation as follows:
∂qk∂t=-g∂∂p∑j=1mλkjRj+Pk+Dk,for k=0,…,n∂∂t(cpT)=-g∂∂p∑k=0nckPkT+Js+Jrad-∑j=1mΛj0Rj.
These equations generalize the ones from CGTBT07 in three ways: (i) an
arbitrary number n of water species is considered; (ii) an arbitrary number
m of interspecies conversion processes is considered; and (iii) the strict
distinction between suspended and precipitating species can be abandoned. The
fact that quite compact equations are obtained, which are valid for all
components of the atmosphere, is an additional indication of the strength of
the barycentric approach.
Remarks
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 regarding the design
of parameterizations and their interactions, are still relevant at present. A second aspect is that parameterizations should also obey
the second law of thermodynamics . This condition cannot be enforced at the higher level of the physics–dynamics
interface, and should be taken care of at the level of the parameterization itself.
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 . This assumption ensures the conservation of total mass
in the atmosphere, but makes it impossible to express a net mass exchange with the surface due to, for instance, precipitation. From a barycentric
point of view, this approximation means that the center of mass of an air parcel does not move vertically. The Eqs. ()–()
remain valid under this assumption, if the absolute flux of dry air is defined as Pd*=-∑k=1nPk*.
The Eqs. ()–() are theoretically only valid for a model using the hydrostatic primitive equations.
In a fully compressible system, the diabatic heating from the physics parameterizations does not only affect the temperature equation but also the continuity
equation . CGTBT07 present the extension of their flux-conservative system to the fully compressible case. An entirely equivalent development
can be made for the generalized equations presented in this paper.
However, as shown by , the impact of including the heat from parameterizations as a forcing in the continuity equation is quite limited.
In other words, one can apply the thermodynamic Eq. () also in a non-hydrostatic model.
The fact that Eq. () describes the evolution of enthalpy h=cpT, does not mean that this variable should become the prognostic
thermodynamic variable of the model. A model that uses temperature T as the prognostic thermodynamic variable, can also use the presented interface.
After all, one can easily calculate the total heat capacity tendency as follows:∂cp∂t=∑k=0nck∂qk∂t,which in turn can be used to determine the temperature tendency from the enthalpy
tendency:∂T∂t=1cp∂∂t(cpT)-T∂cp∂t.
The importance of writing Eq. () as a time evolution of enthalpy only becomes clear in the time-discretized
case.cpt+Δt=cpt+Δt∑k=0nck∂qk∂tTt+Δt=1cpt+ΔtcptTt+Δt∂∂t(cpT),where a superscript t denotes variables at the current time step, while a superscript t+Δt denotes variables at
the next time step. Using an enthalpy-based formulation of the interface is reflected in the use of cpt+Δt on
the right-hand side of Eq. (). Although this appears to be a small detail, it is crucial
in ensuring the conservation of energy. The importance of appropriately discretizing a conserved nonlinear variable
such as enthalpy is also indicated by .
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 . For example,
consider a model containing two parameterizations (indicated with a and b), yielding a respective change in temperature of
ΔTa and ΔTb, and a respective change in heat capacity of Δcpa and Δcpb. Suppose that each
of these parameterizations is energy conservative in itself, meaning that the enthalpy changes
are, respectively, Δha=(cpt+Δcpa)(Tt+ΔTa)-cptTt and
Δhb=(cpt+Δcpb)(Tt+ΔTb)-cptTt. Then the joint effect of the
parameterizations cannot be expressed as ΔT=ΔTa+ΔTb, but it should be determined asΔh=Δha+Δhb,from which the total change in temperature is determined asΔT=(cpt+Δcpa)ΔTa+(cpt+Δcpb)ΔTbcpt+Δcpa+Δcpb.
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. (). However, the fact should be
taken into account that process b does not start from cpt and Tt, but rather from the atmospheric state
after accounting for process a, i.e. cp̃=cpt+Δcpa, and T̃=Tt+ΔTa. So for a time-split coupling, the enthalpy change of process b
becomesΔhb=(cp̃+Δcpb)(T̃+ΔTb)-cp̃T̃.
Working out the heat capacity and the temperature at the end of the time step now givescpt+Δt=cpt+Δcpa+ΔcpbTt+Δt=cptTt+Δhcpt+Δt=Tt+ΔTa+ΔTb.
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. ()–() only describe the evolution of the atmospheric prognostic
variables. The prognostic variables of the surface scheme are not part of this system. In this context, the work of
should be mentioned. They present a method to separate the surface scheme from the atmospheric model. The core of this method is to
describe the interaction between atmosphere and surface with fluxes. In this sense, their work matches perfectly with the flux-based 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. .
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RMSE (solid line) and bias (dashed line) over the period 1–30 November 2014, for REF (blue circles) and FCI (red triangles).
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Application of the flux-conservative equations in the AROME model
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 , with the same spectral
semi-implicit, semi-Lagrangian space–time discretization as the ECMWF's IFS
model, and a terrain-following, mass-based vertical coordinate.
. AROME is coupled to the externalized surface scheme
SURFEX with the flux-based interface of .
The physics parameterizations in AROME originate from the Meso-NH research
model . The Meso-NH model has a dynamical core which is
explicit in time, with a staggered spatial grid and a height-based vertical
coordinate, so it is substantially different from the AROME dynamical core.
The plugging of the physics from this model to a different dynamical core was
quite challenging, and several approximations were made during this process.
A first approximation that is made in the existing AROME physics–dynamics
interface concerns the heat transport by precipitation. From
Eq. (), it is clear that precipitation has two
thermodynamic effects. Falling species modify the composition of the
atmosphere, so they also change the specific heat capacity cp=∑k=0nckqk. Secondly, if a vertical temperature gradient exists, falling species
are heated, thus cooling down the surrounding air. The effect on the enthalpy
due to a change in cp is given by
∂∂tcpTprec,cp=T∑k=0nck∂qk∂t=-gT∂∂p∑k=0nckPk,
while the second effect due to a vertical temperature gradient is given by
∂∂tcpTprec,heat=-g∑k=0nckPk∂T∂p.
The combination of these two effects indeed corresponds to the effect of
precipitation on the right-hand side of Eq. ():
∂∂tcpTprec,cp+∂∂tcpTprec,heat=-g∂∂p∑k=0nckPkT
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 cp, and this effect should be accounted for in the
energy budget. However, this effect is neglected in the existing AROME
physics–dynamics interface.
A third approximation is that the values of specific heat capacity cp and
latent heat Li|l(T) are not consistent between the different
parameterizations. For instance, the heat capacity in the radiation scheme
only accounts for water vapour and neglects the other hydrometeors
(cprad=(1-qv)cpd+qvcpv).
This situation stems from the fact that the different physics
parameterizations are developed by different teams, each using their own
conventions.
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).
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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.
Impact on weather forecast
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): (a) threshold 2 mm; (b) threshold 10 mm.
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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): (a) threshold 2 mm; (b) threshold 10 mm.
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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 739×709 grid with a resolution of 2.5 km.
Figure shows the model domain. The time step is 60 s. The
model is provided with lateral boundary conditions by the operational global
model “ARPEGE” from Météo-France. The initial conditions are generated
with a 3DVAR data assimilation .
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. ()–() would require an
adaptation of the dynamical core of AROME, which falls outside the scope of
this work.
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 (b) indicate the location of the profiles of Fig. .
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Monthly scores
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 and show bias and
RMSE for several meteorological variables for the two periods, respectively.
These scores are calculated by comparing the AROME forecasts with
observations throughout the French territory. Figures
and compare the forecasted precipitation over the two
periods. To avoid the problem of the double penalty, the precipitation is
verified with the neighbourhood observation Brier skill
score . This score is determined by calculating the
probability that a precipitation threshold is exceeded in the vicinity of an
observation. By choosing the threshold, one focuses the verification more on
light or on heavy precipitation.
Vertical profiles at 18:00 UTC in the point indicated in Fig. b for the run with the
flux-conservative interface. (a) precipitation fluxes: rain (black solid line), snow (red dashed line), and graupel
(green dash-dotted line); (b) cold-pool-generating phenomena: latent heat effects due to phase changes (black solid line)
and sensible heat advection (red dashed line); (c) same as (b) but focused on near-surface areas.
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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.
Case study of a cold pool originating from heavy precipitation
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 . It has been shown that
the cold pool is in fact a key element in the life cycle of such systems.
On the one hand, new convective cells originate at the border of the cold
pool and its warmer surroundings, but on the other hand, if the cold pool
becomes too strong, it may cut off the supply of warm air to the updraft
. The cold pool is also accompanied by a mesoscale high
pressure area which plays a crucial role in the wind gusts
that go with heavy precipitation. For these reasons, it is no surprise that
an appropriate representation of the cold pool mechanism is essential in a
NWP model .
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 . As explained in
Sect. , this secondary thermodynamic effect (the transport
of sensible heat) of precipitation is neglected in the existing AROME
physics–dynamics interface, while it is correctly accounted for with the
presented set of flux-conservative equations. One can thus expect that the
intensity of a forecasted cold pool depends on which set of equations is
used.
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 a and b show the forecasted 12:00–18:00 UTC
accumulated precipitation with the existing AROME interface (REF) and with
the flux-conservative interface (FCI). It is observed that the overall
structure of the precipitation is quite similar. However, when comparing the
cold pool characteristics of both experiments, important differences appear.
Figure c and d show the differences between both
experiments for the 2 m temperature and the surface pressure. The
temperature is significantly lower with FCI (up to 5 K cooler), and the
surface pressure is higher (up to 1.4 hPa).
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. b are studied for the experiment with the
flux-conservative interface. The vertical profile of the precipitation fluxes
(Fig. a) shows how snow and graupel originate
aloft, they melt to form rain at around 850 hPa, and the rain starts to
evaporate below 930 hPa. Figure b shows the
vertical profile of the two phenomena that are responsible for the
development of the cold pool, averaged between 12:00 and 18:00 UTC: the
latent heat effects from phase changes (solid line), and the falling of cold
hydrometeors into warmer air layers (dashed line). It is clear that the
second effect is orders of magnitude smaller than the first effect, at least
when considering the full vertical extent of the model. However, as shown in
Fig. c, the heat transport by hydrometeors is not
entirely negligible in the range between the surface and 900 hPa, and thus
contributes to the intensity of the cold pool.
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
, where neglecting a supposedly small term in the energy
budget unexpectedly leads to the worst results.
Conclusions
This paper starts from the equations presented in that
describe how the effect of physical parameterizations on the dynamical core
of an NWP model can be expressed in a flux-conservative way. The main
advantage of these equations is that they impose the constraints of energy
and mass conservation at a higher level in the model than at the level of the
individual physical parameterizations. The presented equations only guarantee
conservation of mass and energy regarding the effect of the physics
contributions, not for the dynamical core of the model. A second advantage of
the presented equations is that by gathering the thermodynamic calculations
of all physics parameterizations in a single equation, it is also guaranteed
that a predefined framework of hypotheses is consistently respected.
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.