Part I
General context: Meteorological modelling and the Arp` ege-Aladin
model
Chapter 1
Generalities about atmospheric modelling
Introduction
To ease the understanding by the reader not familiar with weather sciences, we present here a rapid overview of the general frame of meteorological modelling, focusing on aspects that are relevant to the subsequent chap- ters.
The main characteristics of theArp`ege-Aladinlocal area model, which was the target of our developments, are described in chapter 2.
For a deeper discussion of the notions outlined below, refer to the bibliography.
Most of the time in fluid mechanics, the medium can be regarded as a continuum, of which a “point” (air parcelorparticle) corresponds to a volume element that is very small with respect to the volume under study, but still contains a large number of molecules.
The various quantities characterizing the state of the atmosphere (field variables) are assumed to have unique values at each “point” of the continuum, and to be, as well as their derivatives, continuous functions of space and time.
The laws governing the system are written in terms of partial derivatives equations involving the field variables as dependent variables and space and time as independent variables.
1.1 Thermodynamical and aerological considerations
The atmosphere can be regarded as a mixture of a few constituents, of which dry air and water play the main roles (the other elements could be ignored most of the time, except when considering the interaction with radiation).
For dry air, the perfect gas law writes:
p=ρRaT , Ra = 287.0596J/(kg K).
where pis pressure [Pa], T is absolute temperature [K],ρ is density [kg/m3], Ra is the perfect gas constant for dry air.
Moist air results from the mixing of dry air and water vapour. The perfect gas law for the two components writes then:
p−e=ρaRaT , e=ρvRvT , Rv = 461.5249J/(kg K).
whereeis the partial pressure of water vapour,ρv its density.
We define the specific humidityqby q≡ ρv
ρa+ρv
= eRa
Rv(p−e) +Rae ≈e
p , ≡ Ra
Rv
= 0.622.
1.1. THERMODYNAMICAL AND AEROLOGICAL CONSIDERATIONS GENERALITIES
Sometimes one uses rather themixing ratio:
r≡ ρv
ρa
= q
1−q or q= r
1 +r We have then
p= (ρaRa+ρvRv)T =ρRT ≡ρRaTv
with
ρ≡ρa+ρv , R≡(1−q)Ra+qRv
and thevirtual temperature Tv is the temperature of the same mass of dry air that would occupy the same volume at the same pressure.
The virtual temperature is interesting for evaluating to the buoyancy forceFbaffecting a given parcel through Archimedes law, by comparing the virtual temperature of the parcelTvp with its environmentTv:
Fb=−g(ρp−ρ) 1z= g RaTv
Tvp−Tv
Tvp
1z
giving a positive upward force if the virtual temperature of the parcel is warmer than its environment.
The water vapour contents of a parcel may be expressed in several ways. At a given temperature, the partial pressure of water vapoure cannot remain stably bigger than the saturation valueesat(T): the excess water will condensate as soon as a micro-physical process (condensation nuclei) allows it. The variation ofesat(T) is given by the Clausius-Clapeyron law:
desat
esat
≈ LdT
RvT2 or esat≈exp [α−L/(RvT)]
whereαis a constant,L is the condensation latent heat (a slightly more refined equation is used inArp`ege- Aladinmodel, see part II).
Then the moisture contents may be related to the saturation moisture contents at the same temperature, defining therelative humidityRH:
RH≡e/esat(T)
Thedewpoint temperatureis the temperature of the saturated parcel containing the same moisture as the given parcel:
esat(Td) =e(T)
Thewet bulb temperatureis obtained by inducing saturation (by adding or removing water) along an isobaric process. It is the usual way to measure the humidity in meteorological stations, using a thermometer covered by a wet cloth. Evaporation takes place, cooling the parcel, until it is saturated at its new temperature. The thermodynamic path follows
cpdT+L dq= 0
which is amoist pseudo-adiabat. The new state is called theblue point, characterized by wet bulb temperature and specific humidity.
For an adiabatic process with dry air, we can transform the first law of thermodynamics as follows, combining with the state equation:
dE=dQ+dW = cvdT , dQ= 0 =⇒cvdT+p dV = 0 cpdT−1
ρdp = 0 = cpdT−RaT dp
p (1.1)
If the hydrostatic equilibrium is fulfilled,
dp=−ρ g dz≡ −ρ dφ
GENERALITIES 1.1. THERMODYNAMICAL AND AEROLOGICAL CONSIDERATIONS
whereφ=gzis the geopotential,gis the gravity acceleration (average value fixed by WMO at 9.80665m/s2), the heightz is thegeopotential height(it differs from the actual height to compensate the vertical variation of g).
In the atmosphere it is usual to refer either to vertical levels of constant geopotential (drawing maps of isobars on it) or levels of constant pressure (drawing maps ofisohypses, or curves of equal geopotential).
We define thedry static energy
s≡cpT+φ
and it results from (1.1) that it is constant along a (so-calleddry) adiabat:
ds=cpdT+dφ= 0 (1.1) allows to define thepotential temperatureθ:
dT T = Ra
cp
dp
p =⇒ θ≡T
1000hP a p
cpR
so θ is the temperature that the parcel would take if moved adiabatically to the pressure of 1000hP a. So a dry adiabat is characterized by constants andθ.
Figure 1.1:
Example of aerological diagram exploitation
For moist air, as long as there is no saturation, the adiabatic transformation may be considered identical to the transformation of dry air. It is the transformation experienced by a raising air particle.
When saturation is reached and the parcel is raised further, it is realistic to consider a transformation which is no longer adiabatic, but where the condensed water is removed as soon as it is produced: this is themoistor saturated pseudo-adiabatic transformation. As condensation is exothermic, the parcel cools slower than along a dry adiabat.
Theequivalent potential temperatureθeof a parcel is the temperature it takes after having raised it along a dry adiabat up to saturation, then along a moist pseudo-adiabat until all water has condensed, and then getting back to the level 1000 hPa, following the dry adiabat.
Thewet bulb potential temperatureθ0w on the other hand is obtained by moving to the wet bulb temperature along an isobaric path by adding water, then getting to the 1000 hPa level along a moist pseudo adiabat.
1.2. SCALE ANALYSIS IN METEOROLOGICAL MODELLING GENERALITIES
We can see that bothθeandθw0 are constant along a moist pseudo-adiabat.
As the dry static energy was conserved along a dry adiabat, is themoist static energyhconserved in both dry and moist pseudo adiabatic transformations:
h≡s+L q=cpT +φ+L q =⇒ dh=cpdT+dφ+L dq
The thermodynamic evolution of a raising parcel is an important element to assess the atmospherestability.
It is done classically by representing the vertical profile of the atmosphere — as measured by a radio-sounding using a hydrogen balloon with probes launched from the ground — on a skewedT–logpdiagram (figure 1.1).
The horizontal axis bears the temperature, the vertical axis is drawn at 45o (to spare space), and bears the opposite of the logarithm of the pressure. In such a representation, the isobars appear horizontal, while the isotherms are tilted at 45o, parallel to the pressure axis. Dry adiabats (light blue) and saturated pseudo adiabats (dark brown) are also represented, as well as curves of equal water vapour mixing ratio (green).
The environment curves represent the temperature (solid line) and dewpoint temperature (dashed line) in function of pressure. The dewpoint temperature gives access to the moisture contents. On the diagram we can assess the vertical behaviour as follows:
• If we raise an unsaturated particle – characterized by its temperature and moisture (or dewpoint) – from a given pressure level, its temperature follows the dry adiabat, while its dewpoint temperature follows the constant mixing ratio curve.
• At levelLCL– the Lifting Condensation Level – the dewpoint temperature is equal to the temperature of the parcel, and there is saturation.
• Raised above this point the parcel follows the saturated pseudo adiabat (or the dry adiabat if it goes back down).
• If the (virtual) temperature of the raised particle is bigger than the temperature of the local environment, it gets positively buoyant and continues its ascent: there is instability.
• If on the contrary, the raised parcel is colder than the local environment, it gets negatively buoyant, and should go back down: the atmosphere is stable.
The comparison of the slope of the environmental curve with the dry and moist adiabats is a direct indicator of the local stability of the atmosphere.
In the case of figure (1.1), we can see that a parcel raised from level 1000hP a is stable as long as it is not raised above the Level of Free Convection LFC. The energy needed to raise it up to there is the Convec- tive INhibition energy CIN. When raised above the LFC, it continues its ascent up to the Equilibrium Level EL, but as it has accumulated kinetic energy (represented by the red area: Convective Available Potential En- ergyorCAPE), it might raise further to a maximum and reach the equilibrium after some oscillations around it.
Different kinds of instability may be considered; for instance if a layer is lifted “en-bloc” (while flowing over orography) and its top gets saturated while its base is not, the temperature profile of the layer may become unstable.
1.2 Scale analysis in meteorological modelling
Scale analysis provides a systematic method of comparing the magnitudes of the various terms comprising the hydro-dynamical equations governing atmospheric motions. The physical variables are assumed to have characteristic values and scales in space and time as follows [HaltinerandWilliams, 1980]:
L= characteristic horizontal scale (∼1/4 of disturbances wavelengths) T = local time scale (∼1/2π of local period)
V = characteristic horizontal velocity
And it is assumed that the approximate magnitudes of the derivatives follow
∂v
∂x ∼ ∂u
∂x ∼V
L , ∂u
∂t ∼V
T , · · ·
GENERALITIES 1.2. SCALE ANALYSIS IN METEOROLOGICAL MODELLING
Type of motion Horizontal scale (m) Molecular mean free path 10−7
Minute turbulent eddies 10−2−10−1
Small eddies 10−1−1
Dust devils 1−10
Gusts 10−102
Tornadoes 102
Cumulo-nimbus clouds 103 Fronts, squall lines 104−105
Hurricanes 105
Synoptic cyclones 106
Planetary waves 107
Table 1.1:
Scales of atmospheric mo- tions (following Holton [1992])
The separation of different scales is a crucial element in the assessment of the atmospheric system.
The most obvious distinction is between the horizontal and the vertical, as the horizontal dimensions and movements are much larger than the vertical ones, while the vertical gradients are much bigger than the hor- izontal ones (see practical application in following section). Thanks to this, a wide part of the horizontal and vertical computations may be performed separately.
Moreover, there is a nice correlation between the space and time scales of most processes, which may help to simplify the study of specific phenomena.
Scale separation can be grounded by energy spectrum analysis. There are two ways to perform it:
• First, we can use observations from several stations at a given time, and plot the power density of kinetic energy in function of wavelength or wave number.
• Second, using observations at a single station we can plot the power density (ratio of the kinetic energy for a given period to this period) in function of the logarithmic period or frequency. The resulting curve presents two modes, one for the periods between 10sand 1000s, and the other for the periods between 12hand 100 days. The first mode corresponds to the micro-scale, the latter to synoptic and planetary scales; the energy gap at frequencies between 100sand 12his themeso-scale.
Typical scales are presented in table 1.2.
Scale Wavelength (km) Period (h) Synoptic >500 >48
Meso 20−500 1−48
Micro <20 <1
Table 1.2:
Scales of atmospheric mo- tions (followingAtkinson [1981])
Defining the scale of a particular study means that we consider air parcels that have this typical scale, and that all phenomena smaller than this scale are qualified as “subgrid” or“turbulent”.
1.3. GOVERNING EQUATIONS GENERALITIES
1.3 Governing equations
1.3.1 Instantaneous equations
The instantaneous equations of motion are written in a local “meteorological” trihedron, with the z axis oriented upwards,xin W–E orzonaldirection, and y is S–N ormeridiandirection:
du
dt = ∂u∂t +u∂u∂x +v∂u∂y +w∂u∂z = (2Ω +rcosuϕ)(vsinϕ−wcosϕ) −1ρ∂p∂x +Fx dv
dt = ∂v∂t +u∂x∂v+v∂v∂y+w∂v∂z = −(2Ω +rcosuϕ)usinϕ−vwr −1ρ∂p∂y +Fy dw
dt = ∂w∂t +u∂w∂x +v∂w∂y +w∂w∂z = (2Ω +rcosuϕ)ucosϕ+vr2 −g −1ρ∂p∂z +Fz
(1.2)
whereu,v,ware the three velocity components, pis the pressure,ϕis the latitude, Ω is the earth pulsation, gthe acceleration of gravity,ρthe density andFx, Fy, Fz represent the viscous stress.r=a+zis the distance of the parcel to the center of the Earth, and can be considered constant, equal to the mean Earth radius a (thin layer hypothesis). Also terms withr at the denominator can be neglected, except where multiplied by cosϕ→0 (near the poles).
At synoptic scale, all four parts of the acceleration terms in the horizontal equations are of the order 10−4, while in the vertical equation they are of the order 10−7. On the right-hand sides, the Coriolis terms involving uandv are of order 10−3 whereas that involvingw is of order 10−5; The horizontal pressure gradient terms are of order 10−3while the vertical one is of order 103. Viscous stresses are much smaller than all other terms.
So in the vertical equation, the large vertical pressure gradient is nearly balanced by the force of gravity.
Neglecting the vertical acceleration term leads to the hydrostatic approximation, reducing the vertical equation to
∂p
∂z =−ρg (1.3)
The hydrostatic approximation does not assume that the vertical velocity is zero, it simply neglects its deriva- tive in the vertical motion equation. The vertical velocity is then diagnosed from the continuity equation. A big advantage of the approximation is also that rapid waves such as tri-dimensional sound waves are automat- ically excluded when using it.
In the Coriolis effect we may usually neglectwcosϕand writef ≡2Ω sinϕ; note that the Coriolis term in the z equation was also neglected (it is much smaller thang), while the centrifugal acceleration is included ing.
The horizontal equations become then (neglecting a term utana ϕ, if not too close to the poles)
du
dt = f v −ρ1∂p∂x +Fx dv
dt = −f u −ρ1∂p∂y +Fy
The viscous stress terms are of the form Fx = 1
ρ ∂τxx
∂x +∂τxy
∂y +∂τxz
∂z
= 1 ρ
∂
∂x
µ∂u
∂x
+ ∂
∂y
µ∂u
∂y
+ ∂
∂z
µ∂u
∂z
≈ µ ρ
∂2u
∂x2+∂2u
∂y2 +∂2u
∂z2
ifµis constant. ν ≡ µρ is the dynamic viscosity.
To complete the equations set, we have to consider:
• The continuity equation:
∂ρ
∂t +u∂ρ
∂x +v∂ρ
∂y +w∂ρ
∂z =−ρ ∂u
∂x +∂v
∂y+∂w
∂z
(1.4)
• The state equation (see§1.1).
p=ρRT
GENERALITIES 1.3. GOVERNING EQUATIONS
• The thermodynamic equation (see§1.1) dT
dt =∂T
∂t +u∂T
∂x +v∂T
∂y +w∂T
∂z = RT pcp
∂p
∂t +u∂p
∂x+v∂p
∂y +w∂p
∂z
+ sources where the sources include radiation, diffusion and latent heat transfers in phase changes.
• Moisture equations, for instance
∂q
∂t +u∂q
∂x +v∂q
∂y +w∂q
∂z = sources
where the sources include diffusion and condensation–evaporation processes; when writing equations for the condensates, the water substance budget requires also to take into account the precipitation (which leaves the atmosphere at the ground).
1.3.2 Vertical coordinates
As the vertical pressure gradient is much larger than the horizontal ones, the isobar surfaces are nearly horizontal. This justifies to the wide use of pressure as an alternative vertical coordinate. Atmospheric models often use more elaborated vertical coordinates, so it is useful to rewrite the equations with a generalized vertical coordinateζ. Material derivatives are independent of the coordinate systems, so the 3-D velocity may be expressed as
Vz+w1z= (u1x+v1y)z+w1z= (u1x+v1y)ζ+ ˙ζ1ζ =Vζ+w1ζ
and the “horizontal” components of the wind are also affected by the change. Considering a model variable A(x, y, ζ, t)≡A(x, y, z(x, y, ζ, t), t), we have
∂A
∂ζ = ∂A∂z ∂z∂ζ or ∂A∂z = ∂A∂ζ ∂z∂ζ
∂A
∂s
ζ = ∂A∂s
z+∂A∂z ∂z∂s
ζ = ∂A∂s
z+∂A∂ζ ∂ζ∂z ∂z∂s
ζ fors=x, y, ort
dA
dt = ∂A∂t
ζ+ (V· ∇)ζA+ ˙ζ∂A∂ζ
(1.5)
Using the hydrostatic pressure vertical coordinate, we get
∂p
∂z = −ρ g , ∂z
∂p =− 1
ρ g , ∂A
∂z =−ρ g∂A
∂p ∂A
∂x
z
= ∂A
∂x
p
+∂A
∂p ρ g ∂z
∂x
p
, dA
dt = ∂A
∂t
p
+ (V· ∇)pA+ω∂A
∂p whereω≡ dpdt =−ρgwis the vertical velocity in pressure coordinates.
The motion equations become:
∂u
∂t +u∂u∂x+v∂u∂y +ω∂u∂p = 2Ω(vsinϕ−ωcosϕ) −∂φ∂x +Fx
∂v
∂t +u∂v∂x+v∂v∂y +ω∂v∂p = −2Ωusinϕ −∂φ∂y +Fy
∂w
∂t +u∂w∂x +v∂w∂y +ω∂w∂p = 2Ωucosϕ−g −∂φ∂p +Fp
(1.6)
where φ is the geopotential. We may replace w in the last equation by −ρgω. The transformation of the continuity equation yields adiagnosticform
∂u
∂x +∂v
∂y +∂ω
∂p = 0 (1.7)
which is an important advantage of the pressure coordinate.
1.4. TIME INTEGRATION SCHEMES GENERALITIES
1.3.3 Horizontal averaging
Atmospheric modelling needs to choose a scale of modelling — grid mesh length and time step — and treat all smaller scale and faster phenomena assubgrid or “turbulent”. Model equations are written for the mean flow, while subgrid effects areparameterized, i.e. they must be estimated from the large scale information and only their effect on the large scale variables is relevant.
The intensive values of a model variableψis written as the sum of the mean flow value and a perturbation:
ψ=ψ+ψ0 , ψ0≡0
=⇒ψψ= (ψ+ψ0)(ψ+ψ0) =ψ ψ+ψ0ψ0 (1.8) To adapt the motion equations, we multiply the continuity equation by v, add it to the meridian equation, then consider the average:
v∂u∂x +v∂v∂y +v∂ω∂p = 0
∂v
∂t +u∂x∂v +v∂v∂y +ω∂v∂p = −f u−∂φ∂y
∂v
∂t +∂uv∂x +∂vv∂y +∂ωv∂p = −f u−∂φ∂y
Applying (1.8), then subtracting the mean of the continuity equation multiplied byv, and performing a similar treatment to the other two equations, we get finally:
Dv
Dt ≡ ∂v∂t +u∂v∂x+v∂v∂y+ω∂v∂p = −f u−∂φ∂y −n
∂v0u0
∂x +∂v∂y0v0 +∂v∂p0ω0o
Du
Dt ≡ ∂u∂t +u∂u∂x+v∂u∂y +ω∂u∂p = f v−∂φ∂x−n
∂u0u0
∂x +∂u∂y0v0 +∂u∂p0ω0o
Dw
Dt ≡ ∂w∂t +u∂w∂x +v∂w∂y +ω∂w∂p = −g−∂φ∂p −n
∂w0u0
∂x +∂w∂y0v0 +∂w∂p0w0o
(1.9)
The last terms of the RHS represent the turbulent stresses, which are much more effective at transferring energy than the molecular diffusion effects. In our case, they include all subgrid effects and are the object of thephysical parameterisations.
The other equations are transformed as follows:
• State equation:
p=ρ R T , R=Rvq+Ra(1−q)
• Thermodynamical equation:
DT
Dt = R T cp
ω p +∂FT
∂p + Horiz diff
where the vertical turbulent heat fluxFT includes the vertical diffusion and the exchanges between latent and sensible heat.
• Continuity equation:
∂u
∂x +∂v
∂y +∂ω
∂p =−g∂Fm
∂p
where the vertical turbulent mass flux Fm is linked to the surface budget (one may consider that the water exchanged with the surface through evaporation and precipitation runoff is, or not, compensated by an opposite exchange of dry air).
1.4 Time integration schemes
1.4.1 The Eulerian scheme and the semi-implicit computation
When solving the advection equation, an Eulerian time integration scheme discretizes the partial time derivative of the fields (called the “tendencies”, as they represent the evolution of the field at a given location):
∂ψ
∂t
adv
=−u∂ψ
∂x −v∂ψ
∂y −η˙∂ψ
∂η ≡ A(ψ)
GENERALITIES 1.4. TIME INTEGRATION SCHEMES
η is the vertical coordinate ofArp`ege-Aladin(see following chapter). Thetotaltendency of the variableψ may be splitted into four parts:
∂ψ
∂t =A(ψ) +L(ψ) +N(ψ) + Φ(ψ) where
• A(ψ) is the advective part
• L(ψ) is the linear part of the remaining dynamical computations
• N(ψ) is the non-linear part of the remaining dynamical computations
• Φ(ψ) is the contribution from the physical parameterisations (non-linear).
The time scheme has to discretize this equation in an efficient way.
In anexplicitscheme, none of theA,L,N, Φ is evaluated at timet+4t; in animplicitscheme on the contrary all four are estimated att+4t.
In discretization by finite differences there is an additional choice to do, thedifference centering: for instance, we can estimate a finite difference at a point in space or time either by computing the half of the difference between the next and the previous point(centered scheme)or the difference between next and present point (forward decentering)or present and previous point(backward decentering).
Numerical analysis shows that
• Implicit schemes are unconditionally stable, i.e. the length of the time step has only to be chosen for accuracy reasons. But such schemes are very expensive to compute.
• Explicit forward-decentered schemes are unconditionally unstable for the advective part of the equation.
• Explicitleap-frog(i.e. centered in space and time) schemes are conditionally stable for the advective part of the equation. The length of the time step is then limited by the Courant-Friedrich-Levy (CFL) criterion, stating that the scheme becomes unstable when the fastest travelling signal (acoustic wave, gravity wave, passive advective transport,...) can cross a grid box in less than a time step, or reversely:
stable ⇐⇒ c4t 4x ≤1 where cis the speed of the fastest signal.
In any Eulerian scheme, doubling the resolution in all three directions requires to also divide the time step by 2, leading to a factor 16 in computational load.
The criterion also shows the importance of filtering out or slowing down fast travelling signals in the solution of the equation. In the primitive equations, vertically propagating sound waves are excluded when using the hydrostatic approximation, which implies that all tri-dimensional acoustic waves are automatically suppressed.
In the leap-frog scheme all explicit terms are evaluated at time levelt in order to perform a jump from time levelt− 4ttot+4t. So a diffusion-like equation is discretized as
∂ψ
∂t =−α∂ψ
∂x −→ ψj+−ψ−j =−α4t
4x(ψj+10 −ψ0j−1)
where exponent+meanst+4t,0meanstand−meanst−4t, and indexjcorresponds to the space dimension.
As it uses 3 different time levels simultaneously, its characteristic equation is quadratic, allowing two numerical modes: the physical one, and a computational one, representing a 24toscillation polluting the solution. This mode may be eliminated by applying time filtering (see part II).
Despite the elimination of 3D acoustic waves with the hydrostatic approximation, we still have gravity waves, of which the external propagate also at sound speed. Therefore the explicit scheme requires anyway very short time steps.
Noting that the sub-ensemble concerning the propagation of gravity waves is nearly linear, a new idea was to linearise as much as possible this part in order to move it form theN to theLterm, and to solve the equation semi-implicitly, i.e. the advective and remaining non-linear parts are computed explicitly (at time t), while the linear part is estimated as a mean between t+4t andt− 4t. With this method, the CFL criterion is
1.4. TIME INTEGRATION SCHEMES GENERALITIES
dominated by the advective terms, namely the maximum wind velocity in the domain during the integration period. This allows time steps up to 6 times longer than the explicit scheme.
Linearization of the gravity wave equation has to be performed around a basic state, defined in terms of a reference surface pressurep?s and vertical temperature profileT?(η).
Another problem rises from the physical part of the tendency Φ(ψ), as centered schemes are unconditionally unstable for the physical part of the equations dominated byheat equationtype of processes.
This is solved as follows. The centered time derivative writes:
∂ψ
∂t
0
=ψ+−ψ−
24t ,
and the centered space derivatives:
∂ψ
∂x i
= ψi+1−ψi−1
24x , ∂2ψ
∂x2 i
= ψi+1+ψi−1−2ψi
(4x)2 The heat equation, with a leap-frog scheme (centered in both space and time):
∂ψ
∂t = K∂2ψ
∂x2 =⇒ ∂ψ
∂t
0
i
=Kψ0i+1+ψ0i−1−2ψ0i (4x)2 is unstable. The idea of the linearised split-implicitscheme is to write:
∂ψ
∂t
0
i
= ψ∗i −ψi−
24t =Kψ∗i+1+ψ∗i−1−2ψ∗i
(4x)2 (1.10)
and with this, compute aψ∗ values which is similar to aψ+ value but considering only the partial tendency due to the particular process under study, as if it was acting alone in a linear way. This value is then used only to compute the partial tendency
∂ψ
∂t
0
i
= ψ∗i −ψi− 24t to pass to the global leap-frog algorithm.
A typical example of resolution of (1.10), applying the Gauss elimination technique in the tri-diagonal system matrix can be found in section (6.2.3).
This scheme is practically as stable as the semi-implicit ones for the physical part of the equations, dominated byheat equationtype of processes.
The value ψ?is used only for the stabilisation of the algorithm, while the physical tendency is produced for present time. As the physics Φ(ψ) is computed before the dynamics N(ψ), it is associated to the preceding time level.
The final expression of the Eulerian integration scheme is ψ+−ψ−
24t =A(ψ0) +L(ψ++ψ−
2 ) +N(ψ0) + Φ(ψ−)
1.4.2 The Lagrangian and semi-Lagrangian schemes
A Lagrangian approach uses the material derivatives, following the air particles in their motion. The advection equation writes:
dψ dt =∂ψ
∂t +u∂ψ
∂x +v∂ψ
∂y + ˙η∂ψ
∂η = 0
It is not very easy to follow the fluid parcels as their positions may take any continuous value and a uniform grid coverage is no longer guaranteed.
For this reason, the semi-Lagrangian method was introduced. It consists to follow the particles on their way during the time step, but then take back a new set of particles at each time step, in order to keep a uniform coverage of the domain, associated to the grid points.
To compute the values advected to a given grid point, we trace back the origin of the air parcel arriving there
GENERALITIES 1.4. TIME INTEGRATION SCHEMES
from the preceding time step. As the origin position does not fit on the grid, the fields values at preceding time step may be interpolated. For the semi-Lagrangian approach, the limitation on the time step is that the trajectories of two parcels should not cross each other over one time step: it depends on the flow divergence, but it supports practically time steps 3 times as long as the Eulerian scheme, resulting in a twofold increase in efficiency.
In a three time levels scheme (SL3TL), the trajectory is traced using three different time steps: t+4t, t, t− 4t: the advection wind is taken at the medium point of the trajectory (M at timet) and advects the fields from the origin point (O at−24t) to the end point (Gat t+4t). The full semi-implicit algorithm writes:
ψ+G−ψO−
24t =L(ψG++ψ−O
2 ) +N(ψ0M) + Φ(ψ−O/G) where a choice has also to be done about where to compute the physical tendency.
The only things needed at mid-trajectory (time level t, position M) are the residual dynamical non linear terms, corresponding to
• the Coriolis force representation,
• the energy conversion term,
• the non linear residuals of the linearization process leading toL.
The last two are associated with rather small terms that can be easily evaluated by averaging betweenGand O, but at timet. The case of the Coriolis term may to be solved with additional considerations, that we won’t develop here.
This suggests the possibility to multiply the time step by 2, considering the value at t for the origin point, t+4tfor the arrival point, and the extrapolatedvalue att+4t/2 for the medium point.
This is the principle of thetwo-time levels semi-Lagrangian scheme(SL2TL), which is more performing as the extra computations for extrapolation and interpolation are much lighter than a full time step computation.
1.4. TIME INTEGRATION SCHEMES GENERALITIES
Chapter 2
The Arp` ege-Aladin local area model
Introduction
Arp`ege-Aladin is a high resolution local area model, which was constructed through modifying parts of the global model Arp`ege-IFS. It shares with it the interfaces, and all local (grid point) calculations; the only part that had to be distinct was the part depending on horizontal geometry and the spectral computations.
For mesh sizes bigger than around 10km the hydrostatic approximation is completely justified, and it also alleviates greatly the calculations.
When using meshes of a few kilometers, the hydrostatic approximation should be released at least at the places where non hydrostatic processes might occur: in the neighbourhood of fronts, steep orography, and convective systems. The current operational implementation of Aladin-Belgiumruns in hydrostatic mode with a mesh size of 7km.
The non hydrostatic hypothesis implied significative adaptations of the dynamics. It was possible to keep the same form for the hybrid vertical coordinate, under the condition it is based on thehydrostatic pressure instead of thetrue pressure. Two additional prognostic variables are then required, which were chosen as a normalized pressure departure ˆP and a vertical pseudo-divergence ˆd. Those two variables could logically also be used within the physical parameterisations.
The present work refines certain aspects of the hydrostatic physical package, which was also needed to give sense to a future refinement by non hydrostatic variables.
2.1 Spectral treatment
Global models work on the whole sphere, so their horizontal domain is automatically periodic. From this comes the natural approach to treat horizontal linear calculations in spectral space, using spherical harmonics and Legendre polynomials.
In regard to finite difference methods, the spectral method has the advantage of the Galerkin methods, to be of infinite precision in the computation of the derivatives, and also to allow an easy filtering of some undesired waves and the tuning of horizontal diffusion. But,
• it requires to perform direct and reverse spectral transforms, which is also time consuming;
• sharp features (fronts, interfaces, orography...) are less well represented;
• some undesirable effects are to be considered, as the Gibbs effect where spectral fitting of fields may yield locally unphysical values, like a relative humidity lower than zero or bigger than 100%.
If model variables are represented by Fourier expansions, their derivatives are obtained easily by multiplying the (complex) expansion coefficients by the corresponding frequencies.
Linear combinations but also products of variables (or their derivatives) can be handled by arithmetic opera- tions on the expansion coefficients: so the Eulerian advection may be performed this way.
For divisions and more complicated mathematical operators, it would be much more difficult, so thetransform methodis used, which consists to
• choose a regular discretization of the domain (for a spectral truncation ofM, the number of grid points should be>3M+ 1 to keep a good precision of advections computed in spectral space),
2.2. HORIZONTAL DISCRETIZATION Arp`ege-Aladin
• make a direct spectral transform, preferably with Fast Fourier Transform (FFT) code;
• make the complicated calculation at each grid point independently;
• bring back the result in spectral space with a reverse spectral transform.
This method is of course applied only once per time step of the model. So the physical parameterisations (which have anyway a local character), but also certain nonlinear dynamical terms, are computed in grid point.
For a deeper description of the different aspects of the spectral method, refer to the documentation of the model dynamical part, for instance [Geleyn, 1998-2000].
For a local area model (LAM), we do not have the domain periodicity. But it was still decided to use a spectral representation inArp`ege-Aladin– after having rendered the fields periodic by extending them over an imaginary area (the extension zone) with artificial values – as this method has several advantages and allows a larger similarity with the treatment of the global model.
1
NDLON NDLUX
NBZONL NBZONL
1 NDGL
NDGUX
NBZONGNBZONG
central area coupling zone extension zone
Figure 2.1:
Arp`ege-Aladindomain geome- try
Arp`ege-Aladin’s domain geometry appears in figure 2.1. The domain is closed in x and y directions so it could be seen as a tore. The spectral transforms are now Fourier transforms alongx andy.
As the domain is limited, the model needs lateral boundary conditions from a larger scale model (without this, it would never see perturbations born outside its area and propagating to it): so a LAM cannot work stand-alone, and therefore it was also decided to take Arp`ege-Aladin’s initial state from the larger scale model, instead of performing data assimilation from observations within its domain only.
Refreshing of the boundary conditions is performed by modulating the field values within the coupling zone (figure 2.1) with the larger scale model fields, the modulation factor being maximum at the external border of the zone.
Following the localisation and geometry of its domain, Arp`ege-Aladin may take its initial and boundary conditions from the global model Arp`ege-IFS or from another run of Arp`ege-Aladin on a wider area (for instance the coupling of Aladin-Belgiumis done with Aladin-France). As it would be too heavy to output boundary conditions at each time step of the larger model, (and also the time step of the local model is normally shorter), the boundary conditions are provided only at given intervals (for instance 3 hours) and an interpolation is performed by the local model for its intermediate integration steps.
2.2 Horizontal discretization
In the case of the LAM, it appeared preferable to work on a plane area, obtained by applying a conformal projection: polar-stereographic, Mercator, or more generally Lambert.
Arp`ege-Aladin 2.3. VERTICAL DISCRETIZATION AND COORDINATES
This has some implications in the calculation of the horizontal gradients, the occurence of curvature terms and the form of the Coriolis term in the momentum equations.
2.3 Vertical discretization and coordinates
Arp`ege-Aladinuses a hybrid vertical coordinateη, defined implicitly from the (hydrostatic) pressure through the relations
π(x, y, η, t) =A(η)πtop+B(η)πs(x, y, t) (2.1) whereπ is the hydrostatic pressure,πtop is a constant reference pressure,πsis the surface pressure.
Functions A and B are chosen so that in the lower levels the iso-η surfaces follow the terrain while tending higher to isobaric surfaces: soA→0, B→1 at the surface (η→1), andA→1, B→0 for η→0.
This choice of vertical coordinate eased the implementation of the non-hydrostatic version of the dynamical equations, under the condition that it is the hydrostatic pressure which is used in the hybrid coordinate definition [Bubnov´aet al., 1995].
Using (1.5),
m≡∂π
∂η = A0(η)πtop+B0(η)πs(x, y, t)
∂π
∂z = −ρ g
∇z = ∇η−∇π m
∂
∂η
InArp`ege-Aladin, the dynamics is treated inη coordinate, while the physics uses the pressure coordinate.
The variables are anyway stored in arrays, with a discrete vertical index designating model levels, defined as follows:
l+ 1 l+ 1
l l
l−1
Figure 2.2:
full levelsland interface levelsl
Main model variables(pressure, temperature, moisture, horizontal wind, vertical velocity — all variables that are advected) are computed at thefull model levels, indexed fromtop to bottombyl= 1· · ·L.
The verticalfluxesare computed (locally) atinterface levelsor half levels, noted with a bar,l= 0· · ·L.
The levels disposition is illustrated on figure 2.2.
2.4 Time schemes
Eulerian, semi-Lagrangian with three and two time levels are available inArp`ege-Aladin; the semi-Lagrangian scheme can also use or not vertical interpolation, and several tunings are available.
The mostly used scheme is the semi-Lagrangian scheme with two time levels as it is the most efficient. This scheme is nevertheless not yet working properly for the non-hydrostatic version.
2.5 Model interfacing
The model code is organized so that the same executable program is used for pre-processing, assimilation, forecast, post-processing, and several other possibilities. All the control of the tasks performed as well as the tunings are taken fromFORTRAN NAMELISTs given in a single ASCIIfile.
2.5. MODEL INTERFACING Arp`ege-Aladin
Boundary conditions files as well as an initialisation file are required. They must contain fields with the same geometry as the domain of present forecast: the pre-processing extracts and transform fields from another domain to the required operational domain.
When starting a forecast, a model initialisation may be performed by running the model backwards in time (without physics, for instance 2 hours back) then again forward with the physics, applying digital filtering along the whole process, so that the initial state does not induce noise due to discrepancies between fields interpolated from a wider model and the ones that suit present model.
At regular interval, output files can be produced, either historic files containing main model variables and usable for further model initialisation and coupling, or post-processed fields, derived from the latter. Post- processingalso allows to change the geometry, horizontal but also vertical (for instance to get fields on isobar surfaces rather than model levels).
When changing horizontal geometry (either to produce boundary conditions for a nested model, or to extract output fields on a smaller area or with another projection for visualisation), it is better to use additional information from so-calledclimatological files, containing subgrid information for the source and target model grids. This includes static information such as orography, and its standard deviation and main directions, soil composition, or more time dependent as vegetation cover, mean surface and deep temperature and wetness, etc.
Bibliography
Aladin Internal Course. Various notes documenting lessons given in Toulouse, 1998.
B.W.Atkinson. Meso-scale Atmospheric Circulations. Academic Press Inc, 1981.
R. Bubnov´a, G. Hello, P. B´enard, and J.F. Geleyn. Integration of the fully elastic equations in the hydrostatic pressure terrain-following coordinate in the framework of the arpege/aladin nwp system.
Mon.Wea.Rev, 123:515–535, February 1995.
J.-F. Geleyn. Essential training course on dynamics. Notes documenting lessons given in Toulouse and Radostovice, 1998-2000.
J.-F.Geleyn, E.Bazile, P.Bougeault, M.D´equ´e, V.Ivanovici, A.Joly, L.Labb´e, J.-P.Pi´edeli`evre, J.-M.Piriou, and J.-F.Royer. Atmospheric parameterization schemes in M´et´eo-France’s Arp`ege N.W.P.
model. InProceedings of an ECMWF Seminar, 1994.
G. J.Haltiner and R. T.Williams. Numerical prediction and dynamic meteorology. John Wiley & Sons, second edition, 1980.
J.R.Holton. An introduction to dynamic meteorology. Academic Press Inc, third edition, 1992.
A. Quinet. Ondes atmosph´eriques aux latitudes moyennes. Miscellanea Serie C 11, Institut Royal M´et´eorologique de Belgique, 1975.
