Météo-FranceCentre National de Recherches Météorologiques

ARPEGE-Climate Version 6.2
Algorithmic Documentation

JULY 2017

Contents
1 Basic hypotheses and related constants
1

11

Astronomical constants . . . . . . . . . . . . . . . . . . . . . .

12

1.1

Calendar. . . . . . . . . . . . . . . . . . . . . . . . .

12

1.2

Time . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.3

Astronomical elements . . . . . . . . . . . . . . . . . .

13

1.4

Sun trajectory relative to Earth . . . . . . . . . . . . .

13

Geometry, geoid . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.1

Geometry . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.2

Coordinate system . . . . . . . . . . . . . . . . . . . .

15

2.3

Geoid . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3

Fundamental constants . . . . . . . . . . . . . . . . . . . . . .

15

4

Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165Thermodynamics, gas phase . . . . . . . . . . . . . . . . . . .

16

6Thermodynamics, liquid phase . . . . . . . . . . . . . . . . .

17

7Thermodynamics, solid phase . . . . . . . . . . . . . . . . . .

17

8Thermodynamics, phase transition . . . . . . . . . . . . . . .

17

8.1

Vaporization . . . . . . . . . . . . . . . . . . . . . . .

17

8.2

Sublimation . . . . . . . . . . . . . . . . . . . . . . . .

18

8.3

Melting . . . . . . . . . . . . . . . . . . . . . . . . . .

18

9Consequences on saturation . . . . . . . . . . . . . . . . . . .

18

10Thermodynamic functions . . . . . . . . . . . . . . . . . . . .

19

10.1

Absolute Functions. . . . . . . . . . . . . . . . . . .

19

10.2

Functions in the model parametrizations . . . . . . . .

20

2

2

CONTENTS

2 Dynamics equations

23

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2

Primitive equations in Eulerian form . . . . . . . . . . . . . .

24

3

Variable mesh27

. . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

Stretched and tilted grid. . . . . . . . . . . . . . . .

27

3.2

Impact on the equations. . . . . . . . . . . . . . . .

28

3.3

Impact on post-processing . . . . . . . . . . . . . . . .

29

4

Lagrangian form of the primitive equations . . . . . . . . . .

29

5

Vertical discretization . . . . . . . . . . . . . . . . . . . . . .

30

5.1

Model vertical levels . . . . . . . . . . . . . . . . . . .

30

5.2

Vertical discretization of the equations

30. . . . . . . .

3 Spectral transforms

35

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2Spectral representation. . . . . . . . . . . . . . . . . . . . .

35

2.1

Spherical harmonics . . . . . . . . . . . . . . . . . . .

35

2.2

Collocation grid

36

2.3

Spectral transforms

3. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .

37

Horizontal discretization . . . . . . . . . . . . . . . . . . . . .

38

3.1

Spectral truncation . . . . . . . . . . . . . . . . . . . .

38

3.2

Horizontal derivatives . . . . . . . . . . . . . . . . . .

38

3.3

Spectral relationships for wind representation . . . . .

39

3.4

Relationship between dimension in spectral space and in grid point space 40

4 Semi-lagrangian computations
1

2

43

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

1.1

General purpose of this documentation. . . . . . . . .

43

1.2

Distributed memory code. . . . . . . . . . . . . . . . .

44

1.3

Mass corrector. . . . . . . . . . . . . . . . . . . . . . .

45

1.4

Deep layer equations (according to White and Bromley, 1995). 45

Definition of Eulerian and semi-Lagrangian schemes. . . . . .

46

2.1

46

Eulerian scheme. . . . . . . . . . . . . . . . . . . . . .

CONTENTS
2.2
3

4

5

6

7

8

3
Semi-Lagrangian scheme. . . . . . . . . . . . . . . . .

47

The 2D equations. . . . . . . . . . . . . . . . . . . . . . . . .

48

3.1

Notations for the 2D equations. . . . . . . . . . . . . .

48

3.2

The 2D shallow-water system of equations in spherical geometry. 48

The 3D equations in spherical geometry (ARPEGE/IFS). . .

49

4.1

Notations for the 3D equations. . . . . . . . . . . . . .

49

4.2

The thin layer 3D primitive equation model: Lagrangian formulation. 51

4.3

The deep layer 3D primitive equation model according to White and Bromley, 1995: Lagr

Discretisation of the equations: general aspects. . . . . . . . .

54

5.1

Notations. . . . . . . . . . . . . . . . . . . . . . . . . .

54

5.2

Discretisation for a 3D variable in a 3D model: general case where the RHS has non-zero

5.3

Discretisation for a 3D variable in a 3D model: particular case where the RHS has zero lin

5.4

Discretisation for a 2D variable in a 3D model (GMVS variables, for example continuity e

5.5

Discretisation for a 2D variable in a 2D model. . . . .

69

5.6

Additional vertical derivatives. . . . . . . . . . . . . .

72

5.7

Case when some variables are evaluated at half levels.

73

5.8

Remarks for spline cubic vertical interpolations. . . . .

73

Computation of medium and origin points. . . . . . . . . . . .

74

6.1

Medium point M (subroutines LARMES and LARMES2). 74

6.2

Origin point O (subroutines LARMES and LAINOR2). 80

6.3

Refined recomputation of point O. . . . . . . . . . . .

81

6.4

Remarks. . . . . . . . . . . . . . . . . . . . . . . . . .

81

The SL discretisation of the 2D shallow-water system of equations (spherical geometry). 83
7.1

Momentum equation. . . . . . . . . . . . . . . . . . . .

83

7.2

Continuity equation. . . . . . . . . . . . . . . . . . . .

83

7.3

Quantities to be interpolated. . . . . . . . . . . . . . .

84

The SL discretisation of the 3D primitive equation model. . .

85

8.1

Thin layer formulation of the momentum equation. . .

85

8.2

White and Bromley deep layer formulation of the momentum equation. 86

8.3

Thermodynamic equation. . . . . . . . . . . . . . . . .

87

8.4

Thin layer formulation of the continuity equation. . . .

87

4

CONTENTS

98.5

White and Bromley deep layer formulation of the continuity equation. 88

8.6

Moisture equation. . . . . . . . . . . . . . . . . . . . .

88

8.7

Other advectable GFL variables. . . . . . . . . . . . .

89

8.8

Case of lagged physics. . . . . . . . . . . . . . . . . . .

89

8.9

Quantities to be interpolated (computation under subroutine LACDYN). 89

R operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

9.1

No tilting. . . . . . . . . . . . . . . . . . . . . . . . . .

91

9.2

Tilting. . . . . . . . . . . . . . . . . . . . . . . . . . .

91

9.3

Plane geometry (ALADIN). . . . . . . . . . . . . . . .

92

10Computation of longitudes and latitudes on the computational sphere. 93

11Interpolations and weights computations. . . . . . . . . . . .

94

11.1

Interpolation grid and weights (subroutine LASCAW).

94

11.2

Interpolations. . . . . . . . . . . . . . . . . . . . . . . 101

11.3

Code structures to store weights. . . . . . . . . . . . . 104

12

Computation of η̇ at full levels. . . . . . . . . . . . . . . . . . 113

13

Lateral boundary conditions. . . . . . . . . . . . . . . . . . . 114

14

13.1

Extra longitudes. . . . . . . . . . . . . . . . . . . . . . 114

13.2

Extra latitudes. . . . . . . . . . . . . . . . . . . . . . . 114

13.3

Vertical boundary conditions in the 3D model. . . . . 114

Some distributed memory features. . . . . . . . . . . . . . . . 116
14.1

Case LEQ_REGIONS=F. . . . . . . . . . . . . . . . . 116

14.2

Case LEQ_REGIONS=T. . . . . . . . . . . . . . . . . 117

5 Semi-implicit spectral computations and predictor-corrector schemes119
1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
1.1

Interest of semi-implicit and iterative centred-implicit schemes.119

1.2

Distributed memory. . . . . . . . . . . . . . . . . . . . 119

1.3

The different models described. . . . . . . . . . . . . . 120

1.4

Other restrictions of this documentation. . . . . . . . . 120

2

Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3

General considerations. . . . . . . . . . . . . . . . . . . . . . . 123

CONTENTS

5

3.1

Advection schemes. . . . . . . . . . . . . . . . . . . . . 123

3.2

Semi-implicit treatment of linear terms (case where there is no iterative centred-implicit s

3.3

Iterative centred-implicit schemes and combination with semi-implicit schemes.125

3.4

Introduction of uncentering for semi-Lagrangian schemes.126

3.5

Plane geometry (ALADIN). . . . . . . . . . . . . . . . 127

3.6

Finite elements on the vertical. . . . . . . . . . . . . . 127

4

5

6

7

Prognostic variables and quantities involved in the semi-implicit scheme.128
4.1

Prognostic variables. . . . . . . . . . . . . . . . . . . . 128

4.2

Quantities used for vertical discretisations and linear operators.128

4.3

Relationships between some linear operators. . . . . . 131

Semi-implicit scheme, no Coriolis term in the semi-implicit scheme.132
5.1

3D hydrostatic primitive equations model. . . . . . . . 132

5.2

2D shallow-water model. . . . . . . . . . . . . . . . . . 134

5.3

Plane geometry. . . . . . . . . . . . . . . . . . . . . . . 135

5.4

Shortcomings of the formulation of the semi-implicit scheme with “reduced divergence" (LInclusion of Coriolis term in the semi-implicit scheme. . . . . 137
6.1

Semi-implicit scheme including Coriolis term in the 3D hydrostatic model.137

6.2

Semi-implicit scheme including Coriolis term in the 2D shallow-water model.140

6.3

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . 140

Spectral multiplications by polynomial expressions of the mapping factor.141

6 Horizontal diffusion

143

1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

2

Formulation of the horizontal diffusion schemes. . . . . . . . . 144

3

2.1

Main horizontal diffusion scheme. . . . . . . . . . . . . 144

2.2

Rayleigh friction. . . . . . . . . . . . . . . . . . . . . . 146

2.3

Nudging. . . . . . . . . . . . . . . . . . . . . . . . . . 146

Discretisation of the horizontal diffusion schemes. . . . . . . . 146
3.1

Main horizontal diffusion scheme. . . . . . . . . . . . . 146

3.2

Modification of the horizontal diffusion scheme if “SLHD" interpolations are done in the s

3.3

Particular use of a second reference truncation N2 for standard pressure levels ΠST (l) belo

3.4

Application of diffusion to each variable. . . . . . . . . 150

3.5

Other spectral calculations. . . . . . . . . . . . . . . . 151

6

CONTENTS

7 Radiative fluxes
1

153

Shortwave radiation . . . . . . . . . . . . . . . . . . . . . . . 153
1.1

First glance . . . . . . . . . . . . . . . . . . . . . . . . 153

1.2

Spectral integration . . . . . . . . . . . . . . . . . . . 154

1.3

Vertical integration . . . . . . . . . . . . . . . . . . . . 157

1.4

Multiple reflections between layers . . . . . . . . . . . 162

2

Longwave radiation: the RRTM scheme . . . . . . . . . . . . 163

3

Input to the radiation scheme . . . . . . . . . . . . . . . . . . 166

4

3.1

Solar irradiance data . . . . . . . . . . . . . . . . . . . 166

3.2

Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

3.3

Ground albedo and emissivity . . . . . . . . . . . . . . 166

3.4

Aerosols . . . . . . . . . . . . . . . . . . . . . . . . . . 166

3.5

Radiatively active compounds . . . . . . . . . . . . . . 170

3.6

Cloud optical properties . . . . . . . . . . . . . . . . . 171

3.7

Cloud overlap assumption . . . . . . . . . . . . . . . . 173

3.8Interactions with the SURFEX module . . . . . . . . . 173

Simplified radiation scheme . . . . . . . . . . . . . . . . . . . 173

8 The turbulence scheme
1

2

175

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
1.1

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 175

1.2

From CMIP5 to CMIP6 . . . . . . . . . . . . . . . . . 176

The CBR scheme . . . . . . . . . . . . . . . . . . . . . . . . . 177
2.1

The turbulent kinetic energy equation . . . . . . . . . 177

2.2

The constants . . . . . . . . . . . . . . . . . . . . . . . 178

2.3

The vertical mixing and the (dynamic+thermal) productions 179

2.4

A3-d: The sub-grid variability of cloud water . . . . . 180

2.5

The sub-grid variability of cloud water . . . . . . . . . 181

2.6

Min and Max Q1 limitations: LECTQ1=.TRUE. . . . 183

2.7

The mixing and dissipation lengths . . . . . . . . . . . 183

2.8

The turbulent kinetic energy in the surface layer . . . 184

CONTENTS

7

2.9

The “Bulk” formulations in the surface layer. . . . . 185

2.10

The vertical mixing and the dissipation term . . . . . 185

2.11

The Top-PBL vertical entrainment: LPBLE=.TRUE.

2.12

Explicit diffusion of conservative variables: LDIFCEXP=.T.188

2.13

Enhanced turbulence due to convection, LCVTURB=.TRUE.189

186

3

Eddy-Diffusivity Mass Flux parameterization . . . . . . . . . 189

4

Algorithmics

5

Logical keys and main parameters in the current version . . . 191. . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9 Microphysics parameterizations scheme

193

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

2

The microphysics scheme. . . . . . . . . . . . . . . . . . . . 194

2.1

History of the Lopez Scheme . . . . . . . . . . . . . . 194

2.2

Which processes are described in the scheme? . . . . . 195

2.3"Bulk" equations . . . . . . . . . . . . . . . . . . . . . 195

2.4

The condensation/evaporation process . . . . . . . . . 196

2.5

The auto-conversion processes . . . . . . . . . . . . . . 196

2.6

The distribution of particle - Fall-velocities. . . . . . . 199

2.7

The collection processes . . . . . . . . . . . . . . . . . 200

2.8

The evaporation processes . . . . . . . . . . . . . . . . 202

2.9

The melting . . . . . . . . . . . . . . . . . . . . . . . . 203

2.10

The sedimentation process . . . . . . . . . . . . . . . . 203

3

Algorithmics. . . . . . . . . . . . . . . . . . . . . . . . . . . 204

4

Logical keys and main parameters in the current version . . . 204

10 The CMIP6 dry and moist convection scheme: Prognostic Condensates Microphysics and12

Introduction: objectives of convective parametrization schemes 207
1.1

Convective components and developments . . . . . . . 207

1.2

The apparent sources: the Q1, Q2, Q3 terms . . . . . 207

1.3

The mass-flux parametrization schemes . . . . . . . . 209

From CMIP5 to CMIP6 . . . . . . . . . . . . . . . . . . . . . 210
2.1

Limits of the CMIP5 convection scheme (Bougeault, 1985)210

8

CONTENTS
2.2
3

The CMIP6 convection scheme: an unified scheme, separating microphysic and

The PCMT scheme . . . . . . . . . . . . . . . . . . . . . . . . 212
3.1

Prognostic variables and geometry . . . . . . . . . . . 212

3.2

Vertical velocity prognostic equation and triggering conditions216

3.3

Triggering condition . . . . . . . . . . . . . . . . . . . 220

3.4

Closure hypothesis . . . . . . . . . . . . . . . . . . . . 220

3.5

Resolution dependency . . . . . . . . . . . . . . . . . . 221

3.6

Parametrization of cumulus-scale downdraughts . . . . 222

3.7

Convective cloudiness . . . . . . . . . . . . . . . . . . 222

3.8

Vertical transport . . . . . . . . . . . . . . . . . . . . . 223

3.9

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 225

4

Algorithmic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

5

Logical keys and main parameters in the current version . . . 227

11 Gravity wave drag1

231

Parametrization of non-orographic gravity wave drag . . . . . 231
1.1

Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 231

1.2

Launch spectrum . . . . . . . . . . . . . . . . . . . . . 232

1.3

Vertical propagation . . . . . . . . . . . . . . . . . . . 232

1.4

Computation aspects . . . . . . . . . . . . . . . . . . . 232

12 Ozone

235

1

Default ozone . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

2

Parameterization of photochemical ozone sources and sinks . 235

3

The effect of chlorine on ozone . . . . . . . . . . . . . . . . . 236

4Parameterization of “mesospheric drag” . . . . . . . . . . . . . 236

References239

1
Basic hypotheses and related
constants

Arpege-IFS is a complex code designed not only for weather forecast or
climate simulation, but also for data assimilation, forecast pre- and postprocessing. It has been extended, diversified and complexified since 1986
jointly by Météo-France and ECMWF. The present documentation restricts
to the description of the French climate version of Arpege-IFS. Some features are not compatible with the version used by ECMWF. In this case, wewill use the term Arpege. Some features are specific to the French climate
version, and we will use the term Arpege-climat. The core of the model
is cycle 37T1 of Arpege-IFS.
Arpege-climat is now an atmosphere-only model. So the calculations
concerning the surface boundary layer, vegetation, snow cover and soil aredone in Surfex (SURFace EXternalisée), which is another model. Surfex
simulates the exchanges of momentum, heat, water, carbon dioxide concentration or chemical species between the surface and the atmosphere. It uses
the concept of ’tile’ to describe the surface (nature, town, sea, water) and canperform different parametrizations. Each surface grid box is made of the four
adjacent tiles. The coverage of each of these surfaces is known through theglobal ECOCLIMAP database. Surfex receives atmospheric forcing terms,
runs the surface schemes, computes the average surface fluxes over the nature, town, sea and water weighted by their respective fraction and sends
them back to the atmosphere in addition to radiatives terms. All this information is then used as lower boundary conditions for the atmospheric radiation and turbulent schemes. The complete documentation on Surfex (Surfex main scientific documentation v2 and Surfex V8_0 user’s guide) is available on the web site http://www.umr-cnrm.fr/surfex//spip.php?rubrique141.
The model relies upon a geometrical assumption: the thin layer approxima-

10

1. Basic hypotheses and related constantstion, and a certain number of phenomenological assumptions such as the lawof perfect gases or the hydrostatic approximation. With these assumptions
a set of basic constants is presented here.

1 Astronomical constants
This section follows the last recommendations of the International Astronomical Union. It should be noted that the formulas are not valid for dates
too far away from the 1st January 2000 (more than one century).

1.1 Calendar
The calendar used is the Gregorian calendar. The dates are given in theform:
AAAAM M JJ, sssss
with:


AAAA


 MM

JJ




sssss

year,
month,
day,
seconds in the day.

1.2 Time
The length of the day is:
d = 86400 s
Time t is expressed in seconds, the date of reference being 20000101.43200(2451545.0 in Julian calendar). It is negative before this date and positive
afterward. t is deduced from the calendar date by:t = (JD − 2451545)86400 + sssss
with JD, date in the Julian calendar (E indicating the integer part):
JD = 1720994.5 + K + E(365.25a) + E(30.601(m + 1)) + JJ

1. Basic hypotheses and related constants

11

and:
a=

(

AAAA,
if M M > 2
AAAA − 1, if M M ≤ 2(

M M,
if M M > 2
M M + 12, if M M ≤ 2

m=

K = 2 − E(a/100) + E(E(a/100)/4)

1.3 Astronomical elements
In the following we set:
θ = t/(dyj )with:yj = 365.25 days
The constants between square brackets are not used in the model; however,
we provide them because they form a consistent set with those needed by
the model.
half great axis
[ excentricity[ inclination
mean longitude[ longitude of perihelion[ longitude of ascending nodemean anomaly
Sun-Earth distanceea = 149597870000 m ± 5 10−50.016704 ± 10−4
0 ± 2 10−4
el = 1.7535 + 6.283076 θ
1.79661 + 0.0000563 θ
6.1937 if t < 0, 3.0521 if t > 0
eM = 6.240075 + 6.283020 θ
Rs = ea (1.0001 − 0.0163 sin(el )+0.0037 cos(el ))

]
]
]
]

1.4 Sun trajectory relative to Earthmean longitude
true longitude[ true latitude

ls = 4.8951 + 6.283076 θ
Ls = 4.8952 + 6.283320 θ−0.0075 sin(el ) − 0.0326 cos(el )−0.0003 sin(2el ) + 0.0002 cos(2el )
0
]

12

1. Basic hypotheses and related constants

obliquity
declination[ right ascension[[
equation of time(true solar time −mean solar time)ǫs = 0.409093
δs = Asin(sin(ǫs ) sin Ls )0 ≤ αs ≤ 2π]
cos(αs ) cos(δs ) = cos(Ls )]sin(αs ) cos(δs ) = sin(Ls ) cos(ǫs ) ]Et = 591.8 sin(2ls ) − 459.4 sin(eM )+39.5 sin(eM ) cos(2ls )−12.7 sin(4ls ) − 4.8 sin(2eM )Over the period 1980–2020, the relative accuracy on Rs is of 5 10−4 , theaccuracy on the various angles is of 5 10−4 rd, and that on equation of timeis of 10 s. These constants implicitly define the length of the sidereal year:ys =2π d yj
6.283076

and therefore the length of sidereal day:ds =d
1 + d/ysand earth rotation:

Ω=

2π
ds

2 Geometry, geoid
2.1 Geometry
We mentioned in the introduction that the thin layer assumption is the baseof Arpege-IFS equations. To make sense, it requires the choice of one
surface on which the equations are written.
As we write the momentum equation in vorticity-divergence, the Laplacian
operator must have his kernel reduced to constant functions; we suppose
moreover than the surface is of revolution around the axis of rotation of theplanet.

1. Basic hypotheses and related constants

13

2.2 Coordinate system
On the horizontal, we use longitude λ varying from 0 to 2π to parametrize
the circles of revolution. The East is directed towards increasing longitudes.
In the orthogonal direction, we use µ variable from −1 at South pole to +1
at North pole (by definition) to parametrize the surface generator.
On the vertical, we use a coordinate η varying from 0 at the top of the fluid
to 1 at the bottom. This vertical coordinate has no geometrical significancewith the ordinary metrics. The 3d metrics is obtained as the product ofhorizontal metrics by vertical one.

2.3 Geoid
The preceding assumptions imply that the vertical coordinate does not haveany geometrical significance and that gravity is not explicitly used. In place
we need two infinitely close equipotential surfaces between which the equations are written. We make the additional assumption that for the description of the Earth, equipotential surfaces are spheres of radius a (averagevalue of the reference ellipsoid):a = 6371229 m
To transform an elevation value into geopotential in J kg −1 , it should be
multiplied by the conventional value:g = 9.80665 m s−2
If we used an ellipsoid instead of a sphere, gravity would vary with thelatitude according to a formula close to that of Clairault, but it would notappear explicitly in the equations and the preceding remarks would remain
valid.
Because of the sphericity assumption, the notions of “North, East, longitude . . . ” used above should not be taken in their geographical meaning
since, as we will see further, the pole of the coordinate system is not necessarily in the Arctic.

3 Fundamental constants
light speed
Planck’s constantBoltzmann’s constantAvogadro’s number

c
h
k
N

=
=
=
=

299792458 m s−1
6.6260755 10−34 J s
1.380658 10−23 J K −1
6.0221367 1023 mol−114

1. Basic hypotheses and related constants

4 Radiation
Stefan-Boltzmann’s constantσ=

2π 5 k 4
15c2 h3solar constantI0=1370 W m−2

5 Thermodynamics, gas phase
The fluid is a mixture of dry air, of water in gas, liquid and solid phases.
gas constant
dry air molar masswater vapor molar mass
Ra =R
J kg −1 K −1
Ma

Rv =

R
J kg −1 K −1
Mv

R
Ma
Mv

=
=
=

Nk
28.9644 10−3 kg mol−1
18.0153 10−3 kg mol−1

It is supposed here that dry air and water vapor are perfect gases. The
maximum error is 0.1 %.
cpa =

7
Ra
2

c va =

5
Ra
2

These quantities are not constant in the atmosphere. But this assumption iscoherent with the approximation of perfect gases, and the error introduced
is less than 1 %.
c p v = 4 Rv
c v v = 3 Rv
In the case of water vapor, supposing them constant leads to an error lessthan 5 %.
If one activates the optional “prognostic physical parametrizations”, atmosphere contains also four species: cloud water, cloud ice, rain water and rainice.

1. Basic hypotheses and related constants

15

6 Thermodynamics, liquid phasewater molar mass
massic volume

Ml
vl

=
=

Mv
0

cpl = cvl = cl = 4.218 103 J kg −1 K −1 (value at triple point Tt )The identity between cpl and cvl is very well satisfied and is coherent withthe constancy of the massic volume. The fact that cl is constant is satisfied
with less than 1 % error in the temperature range [0◦ C, 30◦ C], but the error
grows for the negative temperatures and reaches 12.5 % at −40◦ C.

7 Thermodynamics, solid phase
Mg = Ml
vg = vl
cpg = cvg = cg = 2.106 103 J kg −1 K −1 (value at Tt )Actually, cg decreasing linearly with the temperature, the error introduced
is 13 % at −40◦ C.

8 Thermodynamics, phase transition
triple pointTt=

273.16 K

8.1 Vaporization
Lv (T ) = Lv (Tt ) + (cpv − cl )(T − Tt )Lv (Tt ) = 2.5008 106 J kg −1
It is supposed that Lv is independent of the pressure, which is accurate at
0.5 %, and is coherent with vl = 0.

16

1. Basic hypotheses and related constants

8.2 Sublimation
Ls (T ) = Ls (Tt ) + (cpv − cg )(T − Tt )Ls (Tt ) = 2.8345 106 J kg −1
cpv −cg is an order of magnitude weaker than cpv −cl ; however, to neglect thevariation of Ls with temperature, it would be necessary to write: cg = cpv .

8.3 Melting
Lf = Ls − Lv

9 Consequences on saturation
With vl = 0, Clapeyron’s equation becomes:
Lvd ln(es )=dT
Rv T 2Using the expression of Lv , and integrating from:(Tt , es (Tt ) = 611.14 P a )yields:ln(es ) = αl −βl
− γl ln T
T

with:

βl
+ γl ln Tt
Tt= Lv (Tt )/Rv + γl Tt
cl − cpv=
Rv

αl = ln(es (Tt )) +βlγl

In the presence of ice, the formula remains valid if the l are replaced by g .

1. Basic hypotheses and related constants

17

10 Thermodynamic functions
The thermodynamic functions are gathered in the block of declarations FCTTRM. They are divided into two groups: the first corresponds to the absolutefunctions and to the second to the approximate functions whose approximations are consistent with the above statements. This module is inserted in all
the subroutines which require thermodynamic calculations. Thus coherence
between the various parts of the code is ensured.
Notations:• Tt temperature of water triple point• γl = (cl − cpv )/Rv
• γi = (ci − cpv )/Rv
• βl = Lv (Tt )/Rv + γl Tt• βi = Ls (Tt )/Rv + γi Tt• αl = ln es (Tt ) + βl /Tt + γl ln Tt• αi = ln es (Tt ) + βi /Tt + γi ln Tt• δ index for liquid/solid calculation: δ =(• δT index of temperature positivity: δT =if liquid whatever T
if solid whatever T

1
0

(

1
0

10.1 Absolute Functions
Latent heat of vaporization:
Lv (T ) = RLV(T ) = Lv (Tt ) + (cpv − cl )(T − Tt )Latent heat of sublimation:
Ls (T ) = RLS(T ) = Ls (Tt ) + (cpv − ci )(T − Tt )Latent heat of fusion:
Lf (T ) = RLF(T ) = Ls (T ) − Lv (T )if T ≥ Tt
otherwise

18

1. Basic hypotheses and related constants

Saturation pressure above liquid water:esl (T ) = ESW(T ) = exp αl −βl
− γl ln T
T

Saturation pressure above solid water:esi (T ) = ESS(T ) = exp αi −βi
− γi ln T
T

Saturation pressure above liquid/solid water:es (T ) = ES(T )βl + (βi − βl )δT− (γl + (γi − γl )δT ) ln T
= exp αl + (αi − αl )δT −T

10.2 Functions in the model parametrizations
Saturation pressure:es (T, δ) = FOEW(T, δ)βl + (βi − βl )δ
= exp αl + (αi − αl )δ −− (γl + (γi − γl )δ) ln TT
Derivative of the logarithm of the saturation pressure:∂ ln esβl + (βi − βl )δ − (γl + (γi − γl )δ)T(T, δ) = FODLEW(T, δ) =∂T
T2
Saturation specific moisture:qs = FOQS(

es
es /p
)=
p
1 + (Rv /Ra − 1) max(0, 1 − es /p)

This formulation makes it possible to have:qs =
es /Rv (p − es )/Ra + es /Rv


es




p

if es (T ) ≤ p
if es (T ) ≥ p1. Basic hypotheses and related constants

19

Derivative saturation of specific moisture:∂qs
es ∂ ln es
qs − qs2 ∂ ln es= FODQS(qs , ,
)=
∂T
p
∂T
1 − es /p ∂T
Latent heat:
L(T, δ) = FOLH(T, δ)= Rv [βl + (βi − βl )δ − (γl + (γi − γl )δ)T ]= Lv (Tt ) + [Ls (Tt ) − Lv (Tt )] δ + [cpv − cl + (cl − ci )δ] (T − Tt )

20

1. Basic hypotheses and related constants

2
Dynamics equations

1 Introduction
Each equation of the model can generally write as:
dX
=A+F
dt
where X is a prognostic variable, the evolution of which one wants to know.
A represents all the effects which can be explicitly represented for the current
resolution (often named “adiabatic effects”). They are:• Coriolis force (momentum equation)• pressure-gradient force term (momentum equation)• conversion term (temperature equation)• divergence term (continuity equation)
F represents all the sub-scale effects (often named “diabatic effects”) which
are calculated by physical parametrization routines. They are:• radiation• clouds and turbulence• large-scale precipitations• vertical diffusion

22

2. Dynamics equations• convection• orographic gravity wave drag• soil, snow and vegetation

The time derivative of X means the total temporal derivative, including
advection, also known as Lagrangian derivative.

2 Primitive equations in Eulerian form
Making the hydrostatic assumption, we use for vertical coordinate a hybrid
coordinate η(p, ps ) derived from the pressure coordinate p and terrain following. It must satisfy:η(0, ps ) = 0
η(ps , ps ) = 1∂η(p, ps ) > 0∂pThis vertical coordinate η is defined by two functions A(η) and B(η), in sucha way that the pressure at a given point is:
p = A(η) + B(η)pswhere ps is surface pressure. We have:A(0) = 0
B(0) = 0

A(1) = 0
B(1) = 1

ensuring for η-surfaces to follow orography at the bottom and to be pressure
surfaces at the top. The model does not need to explicitly know the functional form of A and B, only their values at the interface of the layers arenecessary.
The hydrostatic assumption leads to the equation:
∂Φ
RT ∂p
=−
∂η
p ∂η
which is used as a diagnostic equation to calculate geopotential Φ on level pby an integral starting at the lower boundary condition Φ(ps ) = Φs .

2. Dynamics equations

23

The evolution of the parameters which define the state of the atmosphere,
horizontal wind ~v , temperature T and mass moisture ratio qv is controlled
by the following equations, where the total temporal derivative is written as:
dX
∂X
∂X
=
+ ~v ∇X + η̇
dt
∂t
∂η

(1)

Momentum equation
−
→
→
d−
v
∂η ∂ F~v −
→ −→
+ 2Ω
×
~
v
+ S~v + K~v
+
RT
∇
ln
p
+
∇Φ
=
−g
| {z } |
{z
}
dt
∂p ∂η
Coriolispressure force(2)

To conserve momentum in the vertical discretization, the acceleration term
due to the pressure force is transformed into:∂p ∂Φ∇p
∂η
Φ∇
−
∂p
∂η
∂η



+ ∇Φ

Thermodynamics equation
dT
−
dt

κT

ω
p

=−

| {z }g ∂η ∂Fh+ Sh + Kh
cp ∂p ∂η(3)conversion
Moisture equation∂η ∂Fqv
d qv= −g+ Sqv + Kqv
dt
∂p ∂η

(4)

In the above equations one takes:
R = qa Ra + qv Rv
c p = qa c p a + qv c p v
Rκ =cpThe terms in the right-hand members of Equations (2), (3) and (4) respectively represent vertical fluxes (noted F ), sources (noted S) and horizontal
diffusion (noted K) of momentum, enthalpy, and specific moisture.
The continuity equation is written as:
∂
∂η



∂p
∂t





∂p
+ ∇ · ~v
∂η





∂
∂p
+
η̇
∂η
∂η



= −g

∂Fp
∂η

(5)

24

2. Dynamics equations

Fp is the mass flux, no source term being considered.
By integrating, one obtains the evolution equation of surface pressure:


Z



1
∂p
∂ps
=−
∇ · ~vdη − gFp (1)∂t∂η0vertical velocity in pressure coordinate:
ω=−

Z η
0



∇ · ~v∂pdη + ~v · ∇p − gFp (η)∂η

and vertical velocity:
∂p
∂p
=−
−
η̇
∂η
∂t

Z η





∂p
∇ · ~vdη − gFp (η)∂η
0

The momentum equations are integrated divergence and rotational form:

∂ζ
∂t
∂D
∂t

!
−
→
∂η ∂ F~v −→−→= ∇ × H~v − g+ S~v + Kζ
∂p ∂η
!
−
→
∂η ∂ F~v −→−→= ∇ · H~v − g+ S~v − ∆(Φ + Ec ) + KD
∂p ∂η

with:

∂u ∂η ∂Φ 1 ∂p+∂η∂p ∂η a ∂λ∂v ∂η ∂Φ (1 − µ2 ) ∂p+
= −(ζ + f )u − η̇∂η ∂p ∂η
a
∂µ

1 2
u + v2=2

Hu = (ζ + f )v − η̇
Hv
Ec

The wind is calculated from velocity potential χ and stream function ψ by:~v = ∇χ + ∇ × ψ
Velocity potential and stream function are obtained from divergence andvorticity by solving Poisson equations:2. Dynamics equations25

χ = ∆−1 Dψ = ∆−1 ζIt is in these last three relations, the kernel of the Laplacian ∆ is implicitly
supposed to reduce to constant functions. It is then equivalent to know thewind or divergence and vorticity pair. This property is true on the sphere
as well as on the torus.

3 Variable mesh3.1 Stretched and tilted grid
Arpege makes it possible to increase the horizontal resolution on part ofthe sphere, while preserving locally the isotropy (Courtier and Geleyn, 1988).
For that one uses a new set of coordinates (λ′ , µ′ ). First, North Pole is shiftedat the point of coordinates (λ0 , µ0 ) which becomes the new pole (or tiltedpole). One defines new coordinates (λb , µb ) by:µb= µ0 µ +qp

1 − µ20 1 − µ2 cos(λ − λ0 )1qpcos λb = (1 − µ2b )− 2 (µ 1 − µ20 − µ0 1 − µ2 cos(λ − λ0 ))sin λb1= (1 − µ2b )− 2p

1 − µ2 sin(λ0 − λ)The origin longitude is the one which contains the geographical North Pole.
Reciprocally:µ= µ0 µb +qq

1 − µ20 1 − µ2b cos λb1qqcos(λ − λ0 ) = (1 − µ2 )− 2 (µb 1 − µ20 − µ0 1 − µ2b cos λb )sin(λ0 − λ)1

= (1 − µ2 )− 2q

1 − µ2b sin λb

Then, one carries out a stretching of the latitudes (without modifying longitudes, λ′ = λb ) obtained by homothety of a factor c on stereographic
projection at the pole of stretching. It comes:µ′ =(1 − c2 ) + (1 + c2 )µb(1 + c2 ) + (1 − c2 )µb

and reciprocally:µb =(c2 − 1) + (c2 + 1)µ′(c2 + 1) + (c2 − 1)µ′26

2. Dynamics equations

3.2 Impact on the equations
The conformal transform of η-surfaces described above changes the model
equations. The fields are represented by a base of functions defined on transformed surfaces. The equations are integrated on original surfaces. However
modifications are necessary in the calculation of the horizontal derivative. It
consists of multiplying them by a scale factor m. In the case of the present
transform, this factor is:m=c2 + 1 c2 − 1 ′
+
µ
2c
2c

At the pole of dilation (µ′ = 1) this factor is c. At the pole of contraction(µ′ = −1) it is 1/c. The horizontal wind thus becomes:~v = m ~v ′and the horizontal gradient:∇ = m ∇′In going from the real sphere to the transformed sphere, the velocity potential
χ and the stream function ψ, which are scalars, are invariant. The main
modification consists of solving Poisson equation in a more complicated form:
D
m2
−1 ζ
ψ = ∆′m2
χ = ∆′−1

where ∆′ −1 is the same formal operator as ∆−1 but on the transformed
sphere. As a consequence, the state variable of the model is: ζ ′ = ζ/m2 andD′ = D/m2 (or equivalently ψ and χ).
The equations become then:

∂ζ ′
∂t
∂D′
∂t

"

!#
−
→
1 −→
∂η ∂ F~v −
→
= ∇ ×
+ S~v
H~v − g+ Kζ ′
m
∂p ∂η
"
!#
−
→
1 −→
∂η ∂ F~v −
→
′
= ∇ ·
+ S~v
H~v − g− ∆′ (Φ + Ec ) + KD′
m
∂p ∂η
′

2. Dynamics equations

27

3.3 Impact on post-processing
The files produced by Arpege-climat for restarting as well as for postprocessing contain the model prognostic variables, even though, in the caseof post-processing, ψ and χ are transformed into ~v ′ . This is also true formomentum fluxes. As a consequence, both components of wind velocity or
surface stress have to be multiplied by m before any comparison with observations or other model outputs. This operation can be done, for example,
at the same time as the conversion from Arpege format to another format.

4 Lagrangian form of the primitive equations
The Lagrangian form of the equations of momentum, thermodynamics andmoisture are respectively (2), (3) and (4). The continuity equation (5) has
as a Lagrangian form:
d
dt



∂p
∂η



=−



∂p
∂ η̇
D+
∂η
∂η



−g

∂Fp
∂η

(6)

It can take another form, more adapted to the semi-Lagrangian advection
method :
d
dt



 

∂Fp
∂p
J = −g∂η∂η(7)where J indicates the Jacobian of the transform which associates the positionof a point at time t with its position at a reference time to .
On the stretched and tilted sphere, the continuity equation (6) becomes:
d
dt



∂p
∂η



=−



∂p
∂ η̇
m2 D′ +
∂η
∂η



−g

∂Fp
∂η

and its Lagrangian form (7):
d
dt



∂p
∂η





J′
∂Fp
= −g
m2
∂η

The momentum equation (2) becomes:−→∂η ∂ F~v −d m~v ′
→ −→+ m 2Ω × ~v ′ + RT ∇′ ln p + ∇′ Φ = −g
+ S~v + K~v
dt∂p ∂ηThe equations for thermodynamics (3) and moisture (4) are formally unchanged.

28

2. Dynamics equations

pℓ̃−1

Φℓ̃−1

pℓ

Φℓ , Tℓ

pℓ̃

Φℓ̃

pℓ+1

Φℓ+1 , Tℓ+1

pℓ̃+1

Φℓ̃+1

Figure 1: Position of variables on the vertical.

5 Vertical discretization
5.1 Model vertical levels
The atmosphere is vertically split into L layers, defined by the pressures attheir interfaces, which are calculated by:pℓ̃ = Aℓ̃ + Bℓ̃ psℓ̃ = 0, . . . , L(8)Aℓ̃ and Bℓ̃ are constants which define the vertical coordinate. The vertical
distribution of the variables is presented in Figure 1, indices ℓ relating to
the mid-layers (also named full levels) and indices ℓ̃ to the inter-layers (alsonamed half levels).
The values of Aℓ̃ and Bℓ̃ are imposed to the model. In earlier versions of
Arpege-climat they were calculated from analytical functions. In recent
cycles, one uses the same vertical discretization as in forecast models (MétéoFrance or ECMWF).

5.2 Vertical discretization of the equations
The vertical discretization scheme is defined according to Simmons and Burridge (1981). One introduces the operator δ which represents the variation

2. Dynamics equations

29of a variable between the two ends of a layer:
δpℓ = pℓ̃ − pℓ̃−1Index ℓ will be omitted when there is no ambiguity.
Continuity equation
The continuity equation is written as (Fm being the mass flux due to water
cycle):
∂
∂η



∂p
∂t





+ ∇ · ~v

∂p
∂η



+







+ Fm

∂
∂p
η̇
∂η
∂η

= Fm

For a given layer, one writes it as:


∂(δp)
∂p
= −∇ · (~v δp) − δ η̇∂t∂η

One will note, in the following, the physical term of the discretized equationin the same way as the corresponding term of the continuous equation.
Summing on the vertical, one obtains the evolution equation of surface pressure:LX∂ps[δp D + δB ~v · ∇ps ] − g(P + E)
=−
∂t
ℓ=1

since:∇δp = δB ∇ps
The vertical speed is obtained by summing the continuity equation from the
top to the current level:∂pwℓ̃ = η̇
∂η



ℓ̃

=

ℓ
X

k=1

[−δBk ~vk · ∇ps − δpk Dk + Fmk ] − Bℓ̃

∂ps
∂t

30

2. Dynamics equations

Eulerian advection
The vertical advections in momentum, thermodynamics and moisture equations are calculated using the scheme:


∂X
w
∂p



=

ℓ

i
1 h
wℓ̃ (Xℓ+1 − Xℓ ) + wℓ̃−1 (Xℓ − Xℓ−1 )2δpThis scheme ensures conservation of X and X 2 . It results from the following
form of the vertical advection:
w

∂wX
∂w
∂X
=
−X
∂p
∂p
∂p

with the interpolation:
1
Xℓ̃ = (Xℓ + Xℓ+1 )2
In the case of the semi-Lagrangian scheme, see Chapter ??.
Hydrostatic equation
The equation of hydrostatic balance is integrated by using the centeredscheme:pΦℓ̃−1 = Φℓ̃ − Rℓ Tℓ ln ℓ̃−1pℓ̃Which gives, summing from the surface:Φℓ̃ = Φs +ℓ+1
X

Rk Tk ln

k=L

pk̃
pk̃−1

where Φs indicates surface geopotential.
To calculate the geopotential in the mid-layers one writes:Φℓ = Φℓ̃ + αℓ Rℓ Tℓwhere:

 α1
 αℓ

=1
=1−

pℓ̃−1
p
ln ℓ̃
δpℓ
pℓ̃−1

(9)

The expression of Φℓ is consistent with the discretization of the form:
Φ=

∂pΦ
+ RT
∂p

(10)

2. Dynamics equations

31

Discretization of the pressure force
For conserving angular momentum, one writes the acceleration term due tothe pressure force as:











∂(Φ ∇p)∂p −1−∇Φ + RT ∇ ln p =∂η∂η  −1 ∂(Φ ∇p)∂p∂p−= ∇Φ + Φ∇
∂η
∂η
∂η
∂p
∇ Φ
∂η

The discretization of this term yields:1(RT ∇ ln p)ℓ = Rℓ Tℓ
δpℓ
|

"

p
Cℓ
ln ℓ̃
δBℓ +
δpℓ pℓ̃−1
{z

ZRTGR

#

·∇ps

}

with:
Cℓ = Aℓ̃ Bℓ̃−1 − Aℓ̃−1 Bℓ̃
Term ZRTGR is also used in the calculation of the energy transformation
term. It can be written in the simpler form:"p1
αℓ δBℓ + Bℓ̃−1 ln ℓ̃
δp
pℓ̃−1

#

Energy transformation term
The energy transformation term is written as:
RT ω
cp p
One writes:ω1= ~v · ∇ ln p −
p
p

Z η
0



∂p
~v · ∇
∂η





∂p
+D
+ Fm dη∂η32

2. Dynamics equations

The first term of the right-hand member is evaluated in the same way as thecorresponding term of the momentum equation:
"

#

p
Cℓ
1
ln ℓ̃ ~vℓ · ∇psδBℓ +(~v · ∇ ln p)ℓ =δpℓ
δpℓ pℓ̃−1
|

{z

}

ZRTGR

and the second term:"p1αℓ (∇ · (~vℓ δpℓ ) + Fmℓ ) + ln ℓ̃
−
δpℓ
pℓ̃−1

! ℓ−1
X

(∇ · (~vk δpk ) + Fmk )k=1

where:∇ · (~vℓ δpℓ ) = ~vℓ · ∇δpℓ + δpℓ DℓThis discretization results from writing the last term in the form:
1
p

Z

Xdη =





∂p −1 ∂ ln p
∂η
∂η

Z





Z



∂
Xdη =
ln p Xdη − X ln p∂ηin which one calculates the last logarithm of the pressure by:ln p =d(p ln p) − 1
dp





∂p −1
∂η

#

3
Spectral transforms

1 Introduction
Arpege-IFS is a spectral global model. One part of computations is madein spectral space (semi-implicit scheme, horizontal diffusion scheme), theother part in grid-point space on a grid defined by a Gaussian quadrature.
It is therefore necessary to perform spectral transforms from spectral space
to grid-point space or vice-versa. The present chapter aims at giving a briefsummary of the spectral method. For more algorithmic details on can reportto Rochas and Courtier (1992) or to Temperton (1991). Computation aspects
are described in details in Yessad (2007). Most subroutines are located in
an independent library named TFL.
For a global spectral model, spectral transforms are a combination of a Legendre transform and a Fourier transform. A spectral limited-area model like
Aladin uses a double Fourier representation for spectral fields.

2 Spectral representation
2.1 Spherical harmonics
The spherical Laplacian operator ∆ on a sphere Σ of radius a admits aseigenvalues family −n(n + 1)/a2 with an order of 2n + 1. The eigenvectors
are the surface spherical harmonics. An orthogonal base of an eigenspace isgiven by:Ynm = Pnm (µ)eimλ34

3. Spectral transforms

where µ is the sine of the latitude and λ longitude. The Pnm (µ) are thefirst-type Legendre polynomials. The standardization of the Pnm (µ) is such
as:
ZZ

σ

′
Ynm (λ, µ)Ynm′ (λ, µ)dσ = δn=n′ δm=m′

ZZ

dσ
σ

One has then, for m ≥ 0:
Pnm (µ) =s(2n + 1)n+m(n − m)! 1
2 m/2 d(1−µ)(µ2 − 1)n(n + m)! 2n n!
dµn+m

(1)

And for m ≤ 0:
Pn−m (µ) = Pnm (µ)
As mentioned above, the spherical harmonics satisfy:∆Ynm = −n(n + 1) m
Yn
a2

The Laplacian operator is invariant by rotation, his eigenspaces are thus
also invariant by rotation. We deduce from it that under the effect of arotation i.e. a change of pole, the coefficients of the decomposition of afield in spherical harmonics are exchanged at fixed n. For each n, there is alinear transformation (thus a matrix) which makes it possible to make thebasic change. This property is preserved by the discretization if truncationis triangular. This is why triangular truncation is said to be isotropic.

2.2 Collocation grid
A collocation grid is selected for non-linear calculations which cannot be
carried out directly on the coefficients of the spherical harmonics. At eachtime step, one passes from the spectral coefficients to the grid-point values
and reciprocally. From the expression of the Ynm , the E-W transforms are
Fourier transforms. To use fast Fourier transforms (FFT), one thus needs aregular grid in longitude. In the N-S direction, one uses a Gauss quadrature
for the direct transform, therefore the latitudes are not equidistant.
At high latitudes, one takes less points on a latitude circle than Fouriermodes, in order to maintain the grid almost isotropic. The collocation gridis said to be reduced (Hortal and Simmons, 1991). The grid is said to
be Gaussian quadratic (or simply Gaussian) when the number of latitude

3. Spectral transforms

35

circles is large enough (for a given truncation) so that the Gauss quadrature
is exact for any product of two Legendre polynomials in the truncation.
This is useful in the case of the Eulerian advection (product of velocity bygradient). When the approximate calculation of the integral is exact only
for the Legendre polynomials of the truncation, the grid is said to be linear(Hortal, 1996).

2.3 Spectral transforms
To pass from the spectral coefficients to the grid points values, one uses thetwo formulas of direct evaluation:∞X

Am (µ) =mAmn Pn (µ)

n=|m|

and:
+∞
X

A(λ, µ) =Am (µ)eimλm=−∞where the Am are called the Fourier coefficients.
The horizontal derivatives are calculated exactly by using the derivatives of
the Legendre functions for the N-S direction and by multiplying by im for
the E-W direction.
From the grid point fields, the Fourier coefficients are determined by:1
Am (µ) =2πZ 2πA(λ, µ) e−imλ dλ0a formula which ensures that A0 (µ) is the average of field A along parallelµ. Integration is carried out numerically by using a fast Fourier transform(FFT). The integral in latitude is:
Am
n =

1
2

Z 1

−1

Am (µ)Pnm (µ) dµAs mentioned above, we use a Gauss quadrature, discrete version of thepreceding integral:
Am
n =

K
X

k=1

ω(µk )Am (µk )Pnm (µk )36

3. Spectral transforms

where the µk are the K roots of the Legendre polynomial of degree K and
the Gauss weights ω(µk ) are given by:ω(µk ) =1 − µ2k(N PN −1 )2Here, the Legendre polynomial is the one of the mathematicians, the squared
norm of which is 1/(2n + 1).

3 Horizontal discretization
3.1 Spectral truncation
In practical the expression of A is limited to a finite set of harmonics corresponding to 0 ≤ n ≤ N and −n ≤ m ≤ n. That defines a triangulartruncation N . The truncated expansion of field A reads:
A(λ, µ) =

m=N
X

n=N
X

imλ
m
Am
n Pn (µ)em=−N n=|m|

Due to the properties of Pnm (µ), expression of A becomes for a real scalar
field:A(λ, µ) =

m=N
X
X n=N

m
imλ
Am
n Pn (µ)e

m=0 n=|m|

3.2 Horizontal derivatives
Meridional derivative relative to latitude θ
For a variable A, meridional derivative is discretized in spectral space by thefollowing formula:


∂A
cos θ
∂θ

m
n

m
m
m
= −(n − 1)emn An−1 + (n + 2)en+1 An+1

where e00 = 0 and:
em
n =

s

n2 − m2
4n2 − 1

3. Spectral transforms

37

Zonal derivative relative to longitude λFor a variable A, zonal derivative is discretized in spectral space by thefollowing formula:


∂A
∂λ

m
n

= i mAmnSuch a derivation can be made on Fourier coefficients by a multiplication byi m.

3.3 Spectral relationships for wind representation
The reduced components of the velocity are obtained by dividing the physicalcomponents by the mapping factor M . Divergence and vorticity are divided′
by M 2 . As reduced divergence D is obtained from velocity potential χ by′a laplacian operator, and as reduced vorticity ζ is obtained similarly fromstream function ψ, we have in spectral space:
D′m
n =−

ζ ′m
n =−

n(n + 1) mχna2n(n + 1) mψna2′

Relationship between U , ψ and χ:′(U a cos θ) =∂ψ∂χ
− cos θ∂λ∂θ

the spectral discretization of which is:
′

m
m m
m
m
(U a cos θ)mn = i mχn + (n − 1)en ψn−1 − (n + 2)en+1 ψn+1′

Relationship between V , ψ and χ:′(V a cos θ) =∂χ∂ψ
+ cos θ∂λ∂θ

the spectral discretization of which is:
′

m
m m
m
m
(V a cos θ)mn = i mψn − (n − 1)en χn−1 + (n + 2)en+1 χn+1

38

3. Spectral transforms′′′

Relationship between D , U and V :′′∂U∂(V cos θ)
+
∂λ
∂θ

1
D =
a cos θ′!

which can be rewritten:
1
D = 2
a cos2 θ′′′∂(U a cos θ)∂(V a cos θ)+ cos θ
∂λ
∂θ
′

′

!

′

Relationship between ζ , U and V :1ζ =a cos θ′′′∂V∂(U cos θ)
−
∂λ
∂θ

!

which can be rewritten:1
ζ = 2
a cos2 θ′′′∂(V a cos θ)∂(U a cos θ)− cos θ∂λ∂θ!

Spectral discretizations allow to retrieve easily spectral components of fields′′′
D a2 cos2 θ and ζ a2 cos2 θ, but not directly spectral components of D and′ζ (requiring inversion of a penta-diagonal matrix). In fact, the algorithm′′
involved to retrieve spectral coefficients of D and ζ once known valuesof wind components is slightly different (requiring a division by a cos θ in
Fourier space), and is described in detail in Temperton (1991).

3.4 Relationship between dimension in spectral space and in
grid point space
Quadratic grid, linear grid
Spectral space is defined by a triangular truncation N . Grid point space has
ndgl latitudes and a maximum number of longitudes equal to ndlon. ndlon
and ndgl are always even integers: if ndlon is a multiple of 4, ndgl = ndlon/2;
if ndlon is not a multiple of 4, ndgl = ndlon/2 + 1. For a quadratic Gaussiangrid, there is a relationship between these parameters to avoid aliasing on
quadratic terms.
• If the stretching coefficient c is equal to 1 (no stretching), N is the
maximum integer verifying the relationship 3 ∗ N ≤ (ndlon − 1).

3. Spectral transforms

39• If the stretching coefficient c is greater than 1 (stretching), N is the
maximum integer verifying the relationship 3 ∗ N ≤ min(2 ∗ ndgl −3, ndlon − 1).
In a semi-Lagrangian scheme the advective quadratic terms disappear, so it
is possible to use a smaller grid-point space: a linear grid. It is characterizedby:• If the stretching coefficient c is equal to 1 (no stretching), N is the
maximum integer verifying the relationship 2 ∗ N ≤ (ndlon − 1).
• If the stretching coefficient c is greater than 1 (stretching), N is the
maximum integer verifying the relationship 2 ∗ N ≤ min(2 ∗ ndgl −3, ndlon − 1).
In Arpege-climat with c > 1, some aliasing is allowed, and the same N istaken as in the case c = 1.
Admissible dimensions for longitude
The current algorithm for FFT allows integers ndlon which can factorize as
21+p2 ∗ 3p3 ∗ 5p5 . That limits the possibility of choosing the dimensions in a
discontinuous subset of truncations and dimensions for Gaussian grid. In therange compatible with climate multi-year integrations, the admissible sizes(with even number of latitudes) are:
64 72 80 90 96 100 108 120 128 144 150 160 162 180 192 200 216 240 250
256 270 288 300 320 324 360 384 400 432 450 480 486 500 512 540 576 600
640 648 720
Reduced gridTo save memory and computation time (in particular in the physical parametrizations), the number of longitudes per latitude circle is reduced outside
the tropics, in order to maintain a quasi-isotropic grid (note that the spectral
triangular truncation allows an isotropic representation of the fields, despite
the accumulation of grid points near the poles). This optimization is doneat the expense of an aliasing error (Williamson and Rosinski, 2000). An
algorithm is proposed to compute for a given truncation, the number of
longitudes per latitude circle which is the best compromise between accuracy
in the spectral transform an isotropy in the physical parametrizations.

40

3. Spectral transforms

4Semi-lagrangian computations

1 Introduction.
1.1 General purpose of this documentation.
This chapter describes the set of equations used, and also the way to integrate the dynamics of the model with the semi-Lagrangian method currently
implemented in ARPEGE/IFS. Equations will be written without horizontal
diffusion scheme (which is treated in spectral computations), and without
Rayleigh friction (which is done in grid-point space) in order to give a clearer
presentation of the discretised equations. For additional information about
horizontal diffusion scheme, report to documentation (IDDH). Extensions
to ALADIN (cycle AL37T1) are not described in detail, but differences arebriefly mentioned.
The following sets of equations will be described in this documentation:• The 2D shallow-water equations model (configuration 201).
• The primitive equations hydrostatic model (configuration 1): thin layerand deep layer (according to White and Bromley, 1995) formulations.
In the current version of this documentation, some points are partly described:• The option with finite element vertical discretisations LVERTFE=.T. .
When LVERTFE=.T., the main modifications in the semi-Lagrangianpart of the code are the following ones:

42

4. Semi-lagrangian computations– η̇ ∂Π∂η is directly computed at full levels so the way of computing η̇to find the vertical displacement is modified.
– All the vertical integrals use a matricial multiplication with special coefficients computed in the setup routine SUVERTFE1
or SUVERTFE3; the vertical integration is done by routine
VERINT.
– All the vertical derivatives use a matricial multiplication with special coefficients computed in the setup routine SUVERTFE3D;
the vertical derivation is done by routine VERDER.
• The spline cubic vertical interpolations used when LVSPLIP=.T. (inpractical only for ozone, when YO3_NL%LVSPLIP=.T.). The detail of calculation of the interpolation weights is not currently given.
• The “semi-Lagrangian horizontal diffusion" interpolations (SLHD) used
when LSLHD=.T. .
• Modified interpolations when L3DTURB=.T. .

1.2 Distributed memory code.
Some distributed code has been introduced for the semi-Lagrangian scheme,
for some convenience expressions such “DM-local" or “DM-global" will be
used to describe some distributed memory features.
• Expression “DM-local" for a quantity means “local to the couple ofprocessors (proca,procb)": each processor has its own value for thequantity. Expression “DM-local computations" means that the computations are made independently in each processor on “DM-local"
quantities, leading to results internal to each processor, which can bedifferent from a processor to another one.
• Expression “DM-global" for a quantity means that it has a unic value
available in all the processors. Expression “DM-global computations"means that the computations are either made in one processor, then the
results are dispatched in all the processors, or the same computations
are made in all the processors, leading to the same results in all theprocessors.
• In a routine description the mention “For distributed memory computations are DM-local" means that all calculations made by this routine are DM-local; the mention “For distributed memory computations
are DM-global" means that all calculations made by this routine are
DM-global; when no information is provided it means that a part ofcalculations is DM-local and the other part is DM-global.

4. Semi-lagrangian computations

43

1.3 Mass corrector.
Two different mass correctors are coded (but do not work in all configurations), one used at ECMWF, the other one for METEO-FRANCE climatic
simulations. They are useful especially for climatic simulations, to correct
the lack of conservativity of the semi-Lagrangian scheme. They are not described in this documentation.

1.4 Deep layer equations (according to White and Bromley,
1995).
They have been introduced in the hydrostatic model for the Eulerian scheme
and most options of the semi-Lagrangian scheme.

44

4. Semi-lagrangian computations

2 Definition of Eulerian and semi-Lagrangian schemes.
2.1 Eulerian scheme.
In Eulerian form of equations, the time dependency equation of a variable X writes as:∂X= −U.∇3 X + Ẋ∂t(1)

where U is the 3D wind, ∇3 is the 3D gradient operator, Ẋ is the sum of the dynamicaland physical contributions. X(t + ∆t) is computed knowing X(t − ∆t) at the same gridpoint. Eulerian technique obliges to use a time-step that satisﬁes to the CFL (CourantFriedrich Levy) condition everywhere.
• For the variable-mesh spectral global model ARPEGE, the horizontal CFL condition writes as:
M |V|

Dt
2

r

N (N + 1)
<1
r2

(2)

which can be rewritten:
M| V | Dt p
N (N + 1) < 1
r 2

(3)

where M is the mapping factor, Dt is the time-step at the ﬁrst integration stepand twice the time-step otherwise (leap-frog scheme), | V | is the horizontal windmodulus, N is the truncation, r is the distance between the point and the centreof the Earth.
• For the spectral limited area model ALADIN, the horizontal CFL condition writesas:M| V | Dt
(2π)
r 2

s

1
L2
L2
y
x
2 + a2 N 2
a2 Nm
n

<1

(4)

where M is the mapping factor, Dt is the time-step at the ﬁrst integration stepand twice the time-step otherwise (leap-frog scheme), | V | is the horizontal windmodulus, Nm is the zonal truncation, Nn is the meridian truncation, a is the mean
Earth radius, r is the distance between the point and the centre of the Earth, Lx(resp. Ly ) is the zonal (resp. meridian) length of the ALADIN domain taken on asurface iso r = a.
The vertical CFL condition writes as:| η̇ |

Dt∆η < 1
2(5)

For a TL358L46 (triangular truncation 358, linear Gaussian grid, 46 levels) model withstretching coeﬃcient c=2.4, that gives ∆t ≃ 2 mn.

4. Semi-lagrangian computations

45

2.2 Semi-Lagrangian scheme.
In semi-Lagrangian form of equations, the time dependency equation of a variable X writes
as:
dX
= Ẋ
dt

(6)

In a three-time level semi-Lagrangian scheme X(t + ∆t) is computed at a grid-point F
knowing X(t − ∆t) at the point O (not necessary a grid-point) where the same particle isat the instant t − ∆t. In a two-time level semi-Lagrangian scheme X(t + ∆t) is computed
at a grid-point F knowing X(t) at the point O (not necessary a grid-point) where thesame particle is at the instant t. The semi-Lagrangian technique is more expensive for one
time-step than the Eulerian technique because it is necessary to compute the positionsof the origin point O and the medium point M along the trajectory and to interpolate
some quantities at these points (roughly 1.5 times the cost of the Eulerian scheme in the
TL358L46c2.4 model with full French physics). But it allows to use larger time-steps: thestability condition is now the Lipschitz criterion (trajectories do not cross each other) andis less severe than the CFL condition.
. Lipschitz criterion
D is the divergence of the horizontal wind on the η-coordinates, η̇ = dηdtwrites for a three-time level semi-Lagrangian scheme:|D+∂ η̇ Dt
|
<1
∂η
2

(7)

Lipschitz criterion writes for a two-time level semi-Lagrangian scheme:|D+∂ η̇ ∆t
|
<1
∂η
2

(8)

Expressions “semi-Lagrangian scheme", “three-time level semi-Lagrangian scheme" and“two-time level semi-Lagrangian scheme" will be from now on abbreviated into “SL scheme",“3TL SL scheme" and “2TL SL scheme".

46

4. Semi-lagrangian computations

3 The 2D equations.
3.1 Notations for the 2D equations.
• V is the horizontal wind. Its zonal component (on the Gaussian grid) is denotedby U . Its meridian component (on the Gaussian grid) is denoted by V .
• D is the horizontal wind divergence.
• ζ is the horizontal wind vorticity.

• Φ is the equivalent height. Φs is the surface geopotential height (i.e. the orography).
Φ∗ is a reference equivalent height which is only used in the semi-implicit schemeand the linear model.

• Ω is the Earth rotation angular velocity.

• ∇ is the ﬁrst order horizontal gradient on η-surfaces.

• a is the Earth radius.

• (λbne , θbne ) are the longitude-latitude coordinates on a tilted and not stretchedgeometry, the tilting being the same as the one of the computational sphere.
• k is the unit vertical vector. One can write:
r
r
k=
=
|r|
a

3.2 The 2D shallow-water system of equations in sphericalgeometry.
Momentum equation.
Coriolis force can be treated explicitly (δV~ =0) or implicitly (δV~ =1) in the Lagrangian
equation.
~ + δ ~ (2Ω~ ∧ ~r)
d V
V
dt



~ ∧V
~ )] − ∇Φ= [−2(1 − δV~ )(Ω(9)

Continuity equation.
• Conventional formulation.

d(Φ − (1 − δT R )Φs )~ ∇(Φs )= −(Φ − Φs )D + δT R V
dt• Lagrangian formulation.
d((Φ − Φs )J)=0
dt
J is a “Jacobian" quantity which satisﬁes to:
dJ
= −JD
dt

(10)

(11)

(12)

4. Semi-lagrangian computations

47

4 The 3D equations in spherical geometry (ARPEGE/IFS).
4.1 Notations for the 3D equations.
• V is the horizontal wind. Its zonal component (on the Gaussian grid) is denotedby U . Its meridian component (on the Gaussian grid) is denoted by V .
• D is the horizontal wind divergence.
• ζ is the horizontal wind vorticity.
• T is the temperature.
• q is the humidity.

• qr : is the rain.

• Π is the hydrostatic pressure.

• Πs is the hydrostatic surface pressure.

• Ω is the Earth rotation angular velocity.

• (λbne , θbne ) are the longitude-latitude coordinates on a tilted and not stretchedgeometry, the tilting being the same as the one of the computational sphere.
• (λ, θ) are the geographical longitude-latitude coordinates.

• (Λ, Θ) are the computational sphere longitude-latitude coordinates.

.
• w is the z-coordinate vertical velocity: w = dz
dt

is the total temporal derivative of the hydrostatic pressure.
• ω = dΠdt

• p is the pressure, ps is the surface pressure.

• gz is the geopotential height.

• Φ is the total geopotential. Φ = gz in the thin layer equations, but not in the deep
layer equations formulation of White and Bromley.
• Φs = gzs is the surface geopotential (i.e. the orography).

• r is the vector directed upwards, the length of which is the Earth radius. The lengthof this vector is r. In the deep layer equations according to (White and Bromley,
1995), one uses an approximation of this radius, only depending on the hydrostaticpressure (“pseudo-radius").
• a is the average Earth radius near the surface.

s• W = dris the pseudo-vertical velocity used in some Coriolis and curvature terms
dtin the deep layer equations according to (White and Bromley, 1995). W = 0 in thethin layer equations.

• i (resp. j) is the unit zonal (resp. meridian) vector on the Gaussian grid.

• k is the unit vertical vector. One can write:
r
r
=
k=
rs
r
• g is the gravity acceleration constant.

• In the case where vertical variations of g are taken into account, we denote by G
the reference value of g at rs = a.
• R is the gas constant for air and Rd the gas constant for dry air.

• cp is the speciﬁc heat at constant pressure for air and cp d is the speciﬁc heat atconstant pressure for dry air.
• cv is the speciﬁc heat at constant volume for air and cv d is the speciﬁc heat atconstant volume for dry air.
• ∇ is the ﬁrst order horizontal gradient on η-surfaces.

48

4. Semi-lagrangian computations• αT is a vertical-dependent coeﬃcient used to deﬁne a thermodynamic variablesless sensitive to orography than temperature T . Expression of αT
T + δT R RαdTTΦST
is:
αT = B



Rd
−
g

h

dT
dz

i

ST



TST



ΠST
Πs st

 − Rgd [ dT
−1dz ]ST

(13)

where B is a vertically dependent and horizontally constant quantity which deﬁnes
the vertical hybrid coordinate (see later paragraph “Deﬁnition of the vertical coordinate η", subsection (6.1)): B varies from 1 to 0 from bottom to top. Subscript“st" stands for “standard atmosphere".
• ρ is the mass per volume unit of air.
• M is the mapping factor.
• M is a reference mapping factor for the semi-implicit scheme.
• D3 is the 3D divergence used in the NH model. Its expression is given by equation(2).
• τ , γ, ν, L∗ , ∂ ∗ are linear operators used in the semi-implicit scheme (for moredetails, see documentation (IDSI) about semi-implicit scheme).
• T ∗ is a vertically-constant reference temperature which is used in the semi-implicit
scheme and in some non-hydrostatic equations. IF LSPRT=.T. (use of virtual
temperature in spectral transforms instead of real temperature), T ∗ is used as areference virtual temperature (same default value).
• Ta∗ is a cold vertically-constant reference temperature which is used in the semiimplicit scheme in the NH vertical divergence equation; it is recommended to haveTa∗ lower than the current temperature.

• TST is the reference standard atmosphere surface temperature (288.15 K). dT
dz ST
is the standard atmosphere tropospheric gradient of temperature (-0.0065 K/m).
• Π∗ is a reference hydrostatic pressure and Πs ∗ is a reference hydrostatic surfacepressure, which are used in the semi-implicit scheme and in some non-hydrostatic
equations. These reference quantities are vertically dependent and “horizontally"(i.e. on η surfaces) constant. ∆Π∗ are layer depths corresponding to a surfacehydrostatic pressure equal to Πs ∗ .
• Πs st is a reference hydrostatic pressure equal to the surface pressure of the standard
atmosphere (101325 Pa, variable VP00). ΠST is a reference hydrostatic pressuredeﬁned at full levels and half levels corresponding to the surface reference hydrostatic pressure Πs st (stored in array STPRE).
• ∆Φ∗ is a reference geopotential depth computed on model layers, used in the nonhydrostatic model (more exactly in the non-linear part of the true 3D divergence).
∆Φ∗ is vertically dependent and “horizontally" (i.e. on η surfaces) constant.
• In the Wood and Staniforth deep-layer NH equations, a mass vertically integrated
quantity Π̃ is introduced, in order to hide some metric terms, especially in thecontinuity equation: Π̃ replaces Π in the adiabatic equations (Π becomes, if needed,a diagnostic quantity). The deﬁnition of the hybrid vertical coordinate applies forΠ̃, not for Π. Quantities δ and α (depths of logarithm of hydrostatic pressure) are
replaced by δ̃ and α̃ (depths of logarithm of Π̃).
∂ Π̃rs 2 G= 2∂Πa g

4. Semi-lagrangian computations

49

4.2 The thin layer 3D primitive equation model: Lagrangian
formulation.
Momentum equation.
Vectorial form of momentum equation is used. Coriolis force can be treated explicitly(δV~ =0) or implicitly (δV~ =1).
~ + δ ~ (2Ω~ ∧ ~r)
d V
V
dt



= −2(1 − δV~ )(Ω ∧ V) − ∇Φ − RT ∇(log Π) + FV

(14)

FV is the physical contribution on horizontal wind.

Thermodynamic equation.
sd T + δT R RαdTTΦST

dt



=



s
d δT R RαdTTΦST

dt



+

RT ω
+ FT
cp Π

(15)

FT is the physical contribution on temperature. When δT R = 1 the Eulerian treatment of
orography is applied and the prognostic variable is replaced by one variable less sensitive
to the surface orography. This modiﬁcation has been proposed by Ritchie and TanguayαΦd δT R R TT s

d ST
(1996). See equation (13) for deﬁnition of αT . Term
only contains advection
dtterms linked to horizontal variations of orography and vertical variations of the coeﬃcientαT .

Continuity equation.
The equation which is discretised is the vertically integrated Lagrangian formulation ofcontinuity equation.

R η=1 ∂B
η=0

∂η

h

− Π1s

R η=1
η=0



R η=1 ∂B d log Πs +δT R RdΦTsst
η=0

∂η



dt

h

dη =

i

i

~ ∂Π dη + V~ ∇ log Πs + δT R Φs
∇ V
− Π1s g [Fm ]η=1 dη
(16)∂η
Rd TstDetails leading to this formulation is given in part 6.2.3 of the documentation (IDEUL)
about model equations and Eulerian dynamics.
Variable δT R is 0 or 1; when δT R = 1 the new variable is less sensitive to the orography(new variable proposed by Ritchie and Tanguay (1996) to reduce orographic resonance).
If one assumes that a volume of air occupied by rainfall drops is not replaced by dry
air when drops are falling(case δm = 0, variable NDPSFI is 0 in NAMPHY), Fm is
is equal to zero.
replaced by zero and η̇ ∂Π∂η η=1
If one assumes that a volume of air occupied by rainfall drops is replaced by dry air whendrops are falling (case δm = 1, variable
 NDPSFI is 1 in NAMPHY), Fm (diabatic ﬂux)
is non-zero (more details are given in documenhas to be taken in account and η̇ ∂Π∂η
η=1tation ∂Π  (IDEUL)).
η̇ ∂η
is non-zero only when there is an upper radiative boundary condition (LRUBC=.TRUE.).
η=0

50

4. Semi-lagrangian computations

Advectable GFL variables: moisture, but also for example ozone,
liquid water, ice, cloud fraction, TKE, aerosols and extra GFL
variables equations.
Equation is written for moisture q, and is the same for the other advectable GFL variables.
dq
= Fq
dt

(17)

Fq is the physical contribution on moisture.

Non advectable pseudo-historic GFL variables:
Equation is written for rain qr , and is the same for the other non-advectable GFL variables.
Since there is no advection, the SL equation is identical to the Eulerian equation.
∂qr
= Fqr
∂t

(18)

4.3 The deep layer 3D primitive equation model according toWhite and Bromley, 1995: Lagrangian formulation.
Deep layer equations: basics and new features.
The following modiﬁcations are done, according to (White and Bromley, 1995):• One takes account to the fact, that the distance to the Earth center is no longer
a but a radius varying with the vertical. For conveniency (with the η verticalcoordinate), one approximates the radius by a pseudo-radius rs which depends
only on the hydrostatic pressure Π. Two vertical lines are no longer parallel, so thesection of a vertical column varies with the hydrostatic pressure.
• The vertical velocity is now taken in account in the Coriolis term through a pseudos
vertical velocity W deﬁned by W = dr
. W also appears in some new curvaturedt
terms.
• The total geopotential Φ, which appears in the RHS of the wind equation, is nolonger equal to the geopotential height gz.
Details about deep layer equations (deﬁnition and expression of rs , W , geopotential relationships) is given in the documentation (IDEUL) about model equations and Eulerian
dynamics and is not detailed again here. All calculations giving the below formulation ofprimitive equations is also given in the documentation (IDEUL).

Momentum equation.
Coriolis force can be treated explicitly (δV~ =0) or implicitly (δV~ =1) in the Lagrangian
equation.
~ + δ ~ (2Ω~ ∧ ~r)
d V
V
dt



= (1−δV~ )(−2Ω∧V−2Ω∧W k)−W
V−∇Φ−(RT +µs Rd Tr )∇(log Π)+FV (19)rs

FV is the physical contribution on horizontal wind. See documentation (IDEUL) for
deﬁnition of Φ and µs .

4. Semi-lagrangian computations

51

Thermodynamic equation.
Lagrangian tendency:sd T + δT R RαdTTΦST

dt



=



s
d δT R RαdTTΦST

dt



RT ω
+ FT
cp Π

+

(20)

FT is the physical contribution on temperature. When δT R = 1 the Eulerian treatment of
orography is applied and the prognostic variable is replaced by one variable less sensitive
to the surface orography. This modiﬁcation has been proposed by Ritchie and Tanguay(1996).
This equation is unchanged compared to its expression in the thin layer equations. The
only change is the diagnostic expression of ω (see documentation (IDEUL) for details
about expression and discretisation of ω).

Continuity equation.
The equation which is discretised is the vertically integrated Lagrangian formulation ofcontinuity equation.

R η=1 rs 2 ∂B d log Πs +δT R RdΦTsst
η=0

R η=1 rs 2 ∂B ∂ log(Πs )
a2

η=0

∂η

∂t

a2 ∂η

dη +

dt

R η=1 rs 2 ∂B a
a2

η=0

V
∂η rs

r

∇
a
s

dη =

i

h

log Πs + δT R RdΦTsst dη

(21)

s)
where ∂ log(Πis given by equation:∂t− Π1s η̇ ∂Π∂η∂ log(Πs )
=−
∂t

η=1

+ Π1s

h 2i
a
rs 2

h 2i
a
rs 2

η=1

η=1

h

rs 2
a2

1
Πs

i

R η=1  rs 
η=0

η=0

a

 ∂Π 
η̇ ∂η

∇

η=0

rs
V ∂Π
a
∂η

dη

− Π1s g [Fm ]η=1

Advectable GFL variables.
These equations are identical to the thin layer model ones.

Non advectable pseudo-historic GFL variables.
These equations are identical to the thin layer model ones.



h 2i
a
rs 2

η=1

(22)

52

4. Semi-lagrangian computations

5 Discretisation of the equations: general aspects.
This section does not describe in detail iterative centred-implicit schemes and describes
only non-iterative schemes in detail. For more details about iterative centred-implicit
schemes, see documentation (IDSI).

5.1 Notations.
∗ Upper index:• First integration step: + (resp. m, o, −) for t + ∆t (resp. t + 0.5∆t, t, t) quantity.

• Following integration steps: + (resp. m, o, −) for t + ∆t (resp. t + 0.5∆t, t, t − ∆t)quantity.

∗ Lower index: F (resp. M and O) for ﬁnal (resp. medium and origin) point.
∗ Particular case of the first timestep in a SL3TL scheme: Written

discretisations are valid from the second integration step. ∆t has to be replaced by ∆tfor2
the ﬁrst integration step (in this case the t − ∆t quantities are equal to the t quantities).

∗ The different classes of prognostic variables: Prognostic variables canbe split into diﬀerent classes:• 3D variables, the equation RHS of which has a non-zero adiabatic contribution anda non-zero semi-implicit correction contribution. They are called “GMV" in thecode (“GMV" means “grid-point model variables"). This class of variables includes
wind components, temperature (and the two additional non-hydrostatic variables
in a non-hydrostatic model). The sub-class of thermodynamic variables includes
T , and the two additional non-hydrostatic variables in a non-hydrostatic model.
There are NFTHER thermodynamic variables.
• 3D advectable “conservative" variables. The equation RHS of these variables hasa zero adiabatic contribution, only the diabatic contribution (and the horizontal
diﬀusion contribution) can be non-zero. They are called “GFL" in the code (“GFL"
means “grid-point ﬁelds"). This class of variables includes for example humidity,liquid water, ice, cloud fraction, ozone, TKE, aerosols, and some extra ﬁelds. See
documentation (IDEUL) for a comprehensive list of advectable GFL.
• 3D non advectable pseudo-historic variables. The equation RHS of these variables
looks like the one of the 3D advectable “conservative" variables, but there is no
advection. They are included in the GFL variables. This class of variables includes
for example rain, snow, graupels, hail, convective precipitation ﬂux, stratiform
precipitation ﬂux, SRC (second-order ﬂux for AROME), forcings, easy diagnostics,
greenhouse gases, reactive gases, moisture convergence, total humidity variation,
standard deviation of the saturation depression, convective vertical velocity. See
documentation (IDEUL) for a comprehensive list of non advectable pseudo-historic
GFL.
• 2D variables, the equation RHS of which mixes 3D and 2D terms, has a nonzero adiabatic contribution and a non-zero semi-implicit correction contribution.
They are called “GMVS" in the code (“GMVS" means “grid-point model variables
for surface"). This class of variables includes the logarithm of surface pressure(continuity equation).

4. Semi-lagrangian computations

53

5.2 Discretisation for a 3D variable in a 3D model: generalcase where the RHS has non-zero linear and non-linear
terms (GMV variables).
List of equations.
• Momentum equation.

• Temperature equation.

• Pressure departure variable (non-hydrostatic model only).

• Vertical divergence variable (non-hydrostatic model only).

Generic notations.
Generic notation N(X)LAG stands for:
• NWLAG for momentum equation.

• NTLAG for temperature equation.

• NSPDLAG for pressure departure variable (non-hydrostatic model only).

• NSVDLAG for vertical divergence (non-hydrostatic model only).
Generic notation P(X)L0, P(X)L9, P(X)T1 stands for:• PUL0, PUL9, PUT1 for U-momentum equation.

• PVL0, PVL9, PVT1 for V-momentum equation.
• PTL0, PTL9, PTT1 for temperature equation.
• PSPDL0, PSPDL9, PSPDT1 for Q̂ equation.

• PSVDL0, PSVDL9, PSVDT1 for d equation.
Generic notation P(X)NLT9 stands for:
• PUNLT9 for U-momentum equation.
• PVNLT9 for V-momentum equation.
• PTNLT9 for temperature equation.
• PSPDNLT9 for Q̂ equation.

• PSVDNLT9 for d equation.

Generic notation for total term, linear term, non linear term, physics: A is the total
term (sum of dynamical contributions), B is the linear term (treated in the semi-implicit
scheme), the diﬀerence A − βB is the non-linear term. F is the sum of contributions
computed in the physical parameterizations.
Description stands for the general case where linear and non-linear terms are gathered in
the same buﬀer, and where no additional splitting is required to do diagnostics or to apply
SLHD interpolations to a subset of the terms interpolated by a high-order interpolation.
In some particular cases, additional splitting involving separate buﬀers may be required:• In the NH model for options where linear terms must be separately interpolated(controlled by variables LSLINL, LSLINLC1 and LSLINLC2). Linear terms
are stored in arrays, the name of them has appendix _SI, or in parts of PB1 withpointers, the name of them has appendix _SI.

54

4. Semi-lagrangian computations• Separation between linear and non-linear terms may have to be done for some DDH
diagnostics too (if LRSIDDH=T).
• When some diagnostics (for example DDH) impose that evaluation of the dynamics
and of the physics at the origin point of the SL trajectory must be done separately
in two diﬀerent buﬀers (controlled by variable NSPLTHOI set to -1). Buﬀer
P(X)LF9 is used instead of P(X)L9 to store quantities, the interpolation of which
is a non-SLHD high-order one.
• When SLHD interpolations, if switched on, are applied only to a subset of thequantities interpolated by a high order interpolation (applied to X(t) or X(t − ∆t)but not to the other terms); in particuliar physics do not use a SLHD interpolationin this case (controlled by variable NSPLTHOI set to 1). Buﬀer P(X)LF9 is usedinstead of P(X)L9 to store quantities, the interpolation of which is a non-SLHD
high-order one.

Case N(X)LAG=4 is not described in detail: in this case diabatic terms are interpolatedby trilinear interpolations and they enter the buﬀer containing terms interpolated bytrilinear interpolations. Start from discretisations given for N(X)LAG=3 and move term∆tF from P(X)L9 to P(X)L0.

Other points.
∗ High-order interpolations: In the following discretisations, “high-order interpolations" means 32 points interpolations for 3D terms (vertical interpolations are cubic),
12 points interpolations for 2D terms.

∗ Uncentering: ǫ is a ﬁrst-order “uncentering factor". It allows to remove the noisedue to gravity waves (orographic resonance).
∗ Vectorial variables: The following discretisations are written for scalar variables.

For vectorial variables (for example the horizontal wind) a rotation operator R has to be
applied from interpolation point to ﬁnal point:• expression interpolated at O has to be replaced by ROF {this expression}O .

• expression interpolated at M has to be replaced by RM F {this expression}M .

3TL vertical interpolating SL scheme.
Equation
dX
=A+F
dt

(23)

is discretised as follows:−(X − (1 + ǫ)∆tβB)++ (1 − ǫ)∆t[A − βB]o + [(1 − ǫ)∆tβB + 2∆tF ]− }O
F = {X+{(1 + ǫ)∆t[A − βB]o }FBuﬀers content before interpolations for N(X)LAG=2:(24)

4. Semi-lagrangian computations

55• P(X)L0 is not used.

• P(X)L9: X − + (1 − ǫ)∆t[A − βB]o + [(1 − ǫ)∆tβB + 2∆tF ]− for high-order interpolation at the origin point O.
• P(X)T1: (1+ǫ)∆t[A−βB]o then provisional add of quantity [(1+ǫ)∆tβB]o before
t+dt physics; evaluated at the ﬁnal point F .
Buﬀers content before interpolations for N(X)LAG=3:• P(X)L0: (1 − ǫ)∆t[A − βB]o + [(1 − ǫ)∆tβB]− for trilinear interpolation at theorigin point O.
• P(X)L9: X − + [2∆tF ]− for high-order interpolation at the origin point O.

• P(X)T1: (1+ǫ)∆t[A−βB]o then provisional add of quantity [(1+ǫ)∆tβB]o before
t+dt physics; evaluated at the ﬁnal point F .

2TL vertical interpolating SL scheme: conventional discretisation(LSETTLS=.F.) and first-order uncentering.
The t + ∆t
non-linear term Am − βBm used in the 2TL SL scheme is computed by a linear2
temporal extrapolation using the t and t − ∆t quantities at the same location. At the ﬁrst
are set equal to initial values. This discretisation
time integration step, values at t + ∆t
2of the 2TL SL scheme follows (Mc Donald and Haugen, 1992). Quantity Ao − βBo has tobe saved in a buﬀer P(X)NLT9 to be available as A− − βB− for the following timestep.
Equation (10) is discretised as follows:om∆t(X − (1 + ǫ) ∆tβB)++ [(1 − ǫ) ∆t
βB + ∆tF ]o }O
F = {X + (1 − ǫ) 2 [A − βB]22[A − βB]m }F+{(1 + ǫ) ∆t
2(25)

which can be rewritten, once expanded the extrapolation:ooo∆t(X − (1 + ǫ) ∆tβB)+
F = [X + 2 (1 − ǫ)A + ∆tF ]O
2[(A − βB)o − (A − βB)− ]O + (1 + ǫ) ∆t[1.5(A − βB)o − 0.5(A − βB)− ](F26)+0.5(1 − ǫ) ∆t
2
2
Buﬀers content before interpolations for N(X)LAG=2:• P(X)L0 is not used.

• P(X)L9: X o + ∆t
(1 − ǫ)Ao + 0.5(1 − ǫ) ∆t[(A − βB)o − (A − βB)− ] + ∆tF o for22high-order interpolation at the origin point O.
[1.5(A − βB)o − 0.5(A − βB)− ] then provisional add of quantity• P(X)T1: (1 + ǫ) ∆t
2o∆t[(1 + ǫ) 2 βB] before t+dt physics; evaluated at the ﬁnal point F .
Buﬀers content before interpolations for N(X)LAG=3:• P(X)L0: ∆t(1−ǫ)Ao +0.5(1−ǫ) ∆t[(A−βB)o −(A−βB)− ] for trilinear interpolation22
at the origin point O.
• P(X)L9: X o + [∆tF ]o for high-order interpolation at the origin point O.

[1.5(A − βB)o − 0.5(A − βB)− ] then provisional add of quantity• P(X)T1: (1 + ǫ) ∆t
2o∆t[(1 + ǫ) 2 βB] before t+dt physics; evaluated at the ﬁnal point F .

56

4. Semi-lagrangian computations

2TL vertical interpolating SL scheme: conventional discretisation(LSETTLS=.F.) and pseudo-second order uncentering.
One starts to remove uncentering ǫ from the nonlinear terms and to apply a second-orderuncentering ǫX to linear terms, that yelds a term B− in the discretisation. From property:
B− = 3Bo − 2Bm(27)

one can remove term B− and show that discretisation is equivalent to replace β by (1 +ǫX )β. For more details, see part 5 (equations (37) and (38)) of (Simmons and Temperton,
1996). In equation (25), uncentering ǫ has to be replaced by zero, and β has to be replaced
non-linear term Am − (1 + ǫX )βBm used in the 2TL SL scheme isby (1 + ǫX )β. The t + ∆t
2computed by a linear temporal extrapolation using the t and t − ∆t quantities at the same
location. At the ﬁrst time integration step, values at t + ∆t
are set equal to initial values
2and second-order uncentering is replaced by a ﬁrst order uncentering. This discretisationof the 2TL SL scheme follows (Mc Donald and Haugen, 1992). Quantity Ao − (1 + ǫX )βBo
has to be saved in a buﬀer P(X)NLT9 to be available as A− − (1 + ǫX )βB− for thefollowing timestep.
Equation (10) is discretised as follows:
o
m
∆t
βB)+
+ [(1 + ǫX ) ∆tβB + ∆tF ]o }O(X − (1 + ǫX ) ∆t
F = {X + 2 [A − (1 + ǫX )βB]
2
2

+{ ∆t
[A − (1 + ǫX )βB]m }F2(28)

which can be rewritten, once expanded the extrapolation:
o
o
o
−
∆t o
∆t
βB)+
(X − (1 + ǫX ) ∆t
F = [X + 2 A + ∆tF ]O + 0.5 2 [(A − (1 + ǫX )βB) − (A − (1 + ǫX )βB) ]O
2[1.5(A − (1 + ǫX )βB)o − 0.5(A − (1 + ǫX )βB)− ]F+ ∆t2

Buﬀers content before interpolations for N(X)LAG=2:• P(X)L0 is not used.
• P(X)L9: X o + ∆t
Ao + 0.5 ∆t[(A − (1 + ǫX )βB)o − (A − (1 + ǫX )βB)− ] + ∆tF o for22high-order interpolation at the origin point O.
[1.5(A − (1 + ǫX )βB)o − 0.5(A − (1 + ǫX )βB)− ] then provisional add• P(X)T1: ∆t
2of quantity [(1 + ǫX ) ∆tβB]o before t+dt physics; evaluated at the ﬁnal point F .
2Buﬀers content before interpolations for N(X)LAG=3:• P(X)L0: ∆t
Ao + 0.5 ∆t[(A − (1 + ǫX )βB)o − (A − (1 + ǫX )βB)− ] for trilinear22interpolation at the origin point O.
• P(X)L9: X o + [∆tF ]o for high-order interpolation at the origin point O.
[1.5(A − (1 + ǫX )βB)o − 0.5(A − (1 + ǫX )βB)− ] then provisional add• P(X)T1: ∆t
2of quantity [(1 + ǫX ) ∆tβB]o before t+dt physics; evaluated at the ﬁnal point F .
2(29)

4. Semi-lagrangian computations

57

2TL vertical interpolating SL scheme: stable discretisation (LSETTLS=.T.) and first-order uncentering.
non-linear term Am − βBm used in the 2TL SL scheme if LSETTLS=.F. is
The t + ∆t2
replaced in the case LSETTLS=.T. by a linear spatio-temporal extrapolation comparable
to the one applied to the wind components for the research of trajectory (see formula (81)),
except the fact that there is an additional uncentering. Term Am − βBm is replaced by
0.5(1 + ǫ)[Ao − βBo ]F + 0.5(2 − ǫ)[Ao − βBo ]O − 0.5[A− − βB− ]OThis type of extrapolation is available only for N(X)LAG=3. At the ﬁrst time integration
step, values at t + ∆t
are set equal to initial values. Quantity Ao − βBo has to be saved2
in a buﬀer P(X)NLT9 to be available as A− − βB− for the following timestep.
Equation (10) is discretised as follows:

o
o
−
o
∆t
∆t
∆t
βB)+
(X − (1 + ǫ) ∆t
F = {X + (2 − ǫ) 2 [A − βB] − 2 [A − βB] + [(1 − ǫ) 2 βB + ∆tF ] }O
2

[A − βB]o }F+{(1 + ǫ) ∆t
2(30)

which can be rewritten, once expanded the extrapolation:ooo∆t(X − (1 + ǫ) ∆tβB)+
F = [X + 2 (1 − ǫ)A + ∆tF ]O
2[(A − βB)o − (A − βB)− ]O + [ ∆t
(1 + ǫ)(A − βB)o ]F
+ ∆t
2
2

(31)

Buﬀers content before interpolations for N(X)LAG=3:• P(X)L0: ∆t(1 − ǫ)Ao + ∆t[(A − βB)o − (A − βB)− ] for trilinear interpolation at22
the origin point O.
• P(X)L9: X o + [∆tF ]o for high-order interpolation at the origin point O.

(1+ǫ)(A−βB)o then provisional add of quantity [(1+ǫ) ∆tβB]o before• P(X)T1: ∆t
2
2t+dt physics; evaluated at the ﬁnal point F .

2TL vertical interpolating SL scheme: stable discretisation (LSETTLS=.T.) and pseudo-second order uncentering.
In equation (30), uncentering ǫ has to be replaced by zero, and β has to be replaced by(1 + ǫX )β. At the ﬁrst time integration step, values at t + ∆t
are set equal to initial
2
values and second-order uncentering is replaced by a ﬁrst order uncentering. Quantity
Ao −(1+ǫX )βBo has to be saved in a buﬀer P(X)NLT9 to be available as A− −(1+ǫX )βB−
for the following timestep.
Equation (10) is discretised as follows:(X − (1 + ǫX ) ∆t
βB)+
F
2

[A − (1 + ǫX )βB]− + [(1 + ǫX ) ∆tβB + ∆tF ]o }O= {X o + ∆t[A − (1 + ǫX )βB]o − ∆t22
[A − (1 + ǫX )βB]o }F+{ ∆t2

which can be rewritten, once expanded the extrapolation:oo∆t o(X − (1 + ǫX ) ∆tβB)+
F = [X + 2 A + ∆tF ]O
2
o
−
∆t
∆t
+ 2 [(A − (1 + ǫX )βB) − (A − (1 + ǫX )βB) ]O + [ 2 (A − (1 + ǫX )βB)o ]F (33)Buﬀers content before interpolations for N(X)LAG=3:

(32)

58

4. Semi-lagrangian computations
Ao + ∆t[(A − (1 + ǫX )βB)o − (A − (1 + ǫX )βB)− ] for trilinear inter• P(X)L0: ∆t
2
2polation at the origin point O.
• P(X)L9: X o + [∆tF ]o for high-order interpolation at the origin point O.

(A − (1 + ǫX )βB)o then provisional add of quantity [(1 + ǫX ) ∆tβB]o
• P(X)T1: ∆t
2
2
before t+dt physics; evaluated at the ﬁnal point F .

Specific treatment for some options in the vertical divergence equation.
∗ Option LGWADV=.T. . When this option is activated, the SL scheme treats
the Lagrangian equation of w instead of the one of d (or d4 ). That implies the following
steps in the code:• Change of variable from d or d4 to w.

• SL explicit treatment of the w equation. For ﬁnite diﬀerence vertical discretisation(LVFE_GW=F) w is given at half levels: that means that the SL trajectorymust be computed also for trajectories ending at half levels. For ﬁnite element
vertical discretisation (LVFE_GW=T) w is given at full levels. There is a speciﬁc
treatment for wsurf (diagnostic condition at t + ∆t).
• Calculation of the linear terms of the d equation for the semi-implicit scheme.

• Conversion from w+ into d+ or d4 + . If LVFE_GW=T, this conversion is done
by applying the vertical derivation to the temporal increment of w to obtain thetemporal increment of d.
• Add the linear terms to d+ or d4 + .

It is necessary to keep the prognostic variable d or d4 in the linear model: w in the linear
model would lead to linear instabilities.
This option allows to remove spurious chimneys above slopes, and also spurious noise insome “bubble" tests.

∗ Option NVDVAR=4. In this case we must discretize the term dX, and this isdt
not done by a SL treatment of the equation of X (the RHS of this equation is not easy tocompute).
If there is no predictor-corrector scheme activated, this term is diagnosed as follows:• Three-time level semi-Lagrangian scheme:

0.5XoF + 0.5XoO − X−
dX
O
=
dt
∆t
• Two-time level semi-Lagrangian scheme:(34)

Xm − XoO
dX
= M
(35)
dt
0.5∆t
m
XM is a symbolic denotation for the extrapolated t + 0.5∆t value of X. The wayof extrapolating depends on LSETTLS and is the same as for the other terms
evaluated at t + 0.5∆t (there is no interpolation at M ).

If LPC_FULL=.T. and LPC_NESC=.F. these discretisations also apply to the predictor step; in this case there are two options:• Option 1 (ND4SYS=1): include all the contributions of the evolution of X in thepredictor-corrector scheme iterations.

4. Semi-lagrangian computations

59• Option 2 (ND4SYS=2): include only the advective contributions evolution of X in
the predictor-corrector scheme iterations; after the last step of the corrector step,
include the non advective processes.
In some cases (ECMWF conﬁgurations with lagged physics) option 1 may generate instabilities above slopes.

∗ Assumptions at the surface (velocities). Calculation of dd
for the layer
dt

surf
surf
. There are two options to compute dwdt
, conl = L requires the calculation of dwdt
trolled by the variable LRDBBC of NAMDYN.

• For LRDBBC=.F., one simply discretizes equation (??) with a SL treatment. The
surf
and Vsurf , and the assumptions currently done
RHS of this equation contains dVdt
about these quantities are:. Vsurf = Vl=L.

dVsurf
dt

=

h

~
dV
dt

i

, with an explicit treatment of the Coriolis term, evenl=L,adiabif LADVF=.T. or LIMPF=.T. .
surf• An alternate discretisation (option LRDBBC=.T.) is to evaluate dwdt
by:
SL3TL:−
w + − wO
dwsurf
= F
dt
2∆t

(36)

SL2TL:ow + − wO
dwsurf
= F
dt
∆t

(37)

which requires additional interpolations to compute wO .
This option allows to remove spurious chimneys above slopes.
w+ is computed by:+
w+ = Vl=L,prov∇Φs+where Vl=L,prov
is the provisional t + ∆t value of Vl=L computed just after theinterpolations (this is this provisional value which is used as input for the lagged
physics).
The treatment of dd
for the layer l = L is done according to the following steps:
dtCalculation of

 dw

surf

dt



 dd 

lrdbbc=T

Calculation ofhi(as if LRDBBC were .F.),

dt l=L,lrdbbc=F.

 dd 

dt l=L,lrdbbc=T

h

 dw

surf

dt



lrdbbc=F

andby the following formula:

i

h

dd
dd
=
−
dt l=L,lrdbbc=T
dt l=L,lrdbbc=F

dwsurfdt

i

lrdbbc=T−hdwsurfdt

i

lrdbbc=F

where M h
M is thei decentered middle of O and F (that requires a speciﬁc interpolation ofgp
Rd T ∆Πat O).

l=L

For more details about these calculations and the additional interpolations required,
see documentation (IDVNH3), especially the parts 1 to 3.
The above formulae are valid for thin layer equations. For WS2003 deep-layer
equations change Π into Π̃, g into G, w into (rs 2 /a2 )w in the above formulae.

h

i

o
gp
(38)
Rd T ∆Π l=L,M M

60

4. Semi-lagrangian computations

5.3 Discretisation for a 3D variable in a 3D model: particularcase where the RHS has zero linear and non-linear terms(advectable GFL variables).
List of equations.
Humidity equation, and for example:• Ozone equation.
• Liquid water equation.
• Ice equation.
• Cloudiness equation.
• TKE.
• Aerosols equation.
• Extra GFL variables equations.
See documentation (IDEUL) for a comprehensive list of advectable GFL variables.

Generic notations.
Generic notation P(X)L9, P(X)T1 stands for:
• PGFLL9, PGFLT1 for GFL variables.
Generic notation for total term, linear term, non linear term, physics: A is the total
term (sum of dynamical contributions), B is the linear term (treated in the semi-implicit
scheme), the diﬀerence A − βB is the non-linear term. F is the sum of contributions
computed in the physical parameterizations. In the present case A and B are equal tozero.
Description stands for the general case where diabatic and adiabatic terms are gathered in
the same buﬀer, and where no additional splitting is required to do diagnostics or to apply
SLHD interpolations to a subset of the terms interpolated by a high-order interpolation.
In some particular cases, additional splitting involving separate buﬀers may be required:• When some diagnostics (for example DDH) impose that evaluation of the dynamics
and of the physics at the origin point of the SL trajectory must be done separately
in two diﬀerent buﬀers (controlled by variable NSPLTHOI set to -1). Buﬀer
P(X)LF9 is used instead of P(X)L9 to store quantities, the interpolation of which
is a non-SLHD high-order one.
• When diabatic terms use a diﬀerent interpolation from the adiabatic one. Thatcan be the case for non-zero values of NSPLTHOI, attribute LPHYLIN set
to T. Buﬀer P(X)LF9 is used instead of P(X)L9 to store diabatic terms to beinterpolated.

Other points.
∗ High-order interpolations: In the following discretisations, “high-order interpolations" means: 32 points interpolations for 3D terms (vertical interpolations are
cubic), 12 points interpolations for 2D terms. For ozone, vertical cubic interpolations
can be replaced by vertical Hermite cubic interpolations (switch YO3_NL%LHV in
NAMGFL), or vertical spline cubic interpolations (switch YO3_NL%LVSPLIP in
NAMGFL).

4. Semi-lagrangian computations

61∗ Uncentering: ǫ is a ﬁrst-order “uncentering factor". It allows to remove the noisedue to gravity waves (orographic resonance).

3TL vertical interpolating SL scheme.
Equation (10) is discretised as follows:
XF+ = {X − + [2∆tF ]− }O(39)

Buﬀers content before interpolations:• P(X)L9: X − + [2∆tF ]− for high-order interpolation at the origin point O.
• P(X)T1 contains zero; evaluated at the ﬁnal point F .

2TL vertical interpolating SL scheme.
At the ﬁrst time integration step, values at t + ∆t
are set equal to initial values. This2
discretisation of the 2TL SL scheme follows (Mc Donald and Haugen, 1992).

Equation (10) is discretised as follows:
XF+ = {X o + [∆tF ]o }O(40)

Buﬀers content before interpolations:• P(X)L9: X o + [∆tF ]o for high-order interpolation at the origin point O.
• P(X)T1 contains zero; evaluated at the ﬁnal point F .

∗ Remark for iterative centred-implicit schemes: For options where
this type of scheme involves the momentum equation (this is the case for the option
LPC_FULL=.T.) XF+ has to be recomputed at all iterations of the iterative centredimplicit scheme since the origin point O is recomputed at each iteration.

Non advectable pseudo-historic GFL variables.
For these variables the discretization always writes (3TL SL and 2TL SL):XF+ = {X o + [∆tF ]o }Fand there are never interpolations.

(41)

62

4. Semi-lagrangian computations

5.4 Discretisation for a 2D variable in a 3D model (GMVS
variables, for example continuity equation).
The equation which is now discretised is:[Rinte ](top,surf )Wvei dX∆η dt[Rinte ](top,surf )



Wvei
∆η



= [Rinte ](top,surf )





Wvei
A +[Rinte ](top,surf )
∆η





Wvei
F (42)∆η

where:



(43)

=1

and [Rinte ](top,surf ) is the vertical integral matricial operator (the scalar product [Rinte ](top,surf ) hXi
R η=1
is the discretisation of η=0 Xdη, hXi is the vector containing the layer values of X:(X1 ; X2 ; ...; Xl ; ...; XL )).
In the thin layer equations or in the Wood and Staniforth formulation of deep-layer equations, expression of Wvei at full levels is:(44)[Wvei ]l = ∆Bl

In the deep layer equations (White and Bromley, 1995), expression of Wvei at full levelsis:[Wvei ]l = 

∆Bl

h

rs 2
a2

[Rinte ](top,surf )il∆B rs 2
∆η a2




(45)

When the ﬁnite element vertical discretisation is activated (LVERTFE=.T.), [Rinte ](top,surf )is a vector tricky to compute, it is precomputed in the setup routine SUVERTFE1 or
SUVERTFE3 and stored in the array RINTE of YOMCVER. Vertical integrations
are done in routine VERINT. For more details see the part of the appendix of documentation (IDEUL) which explains the computation of matrix [Rinte ]. When the ﬁnite
diﬀerence vertical discretisation is activated (LVERTFE=.F.), [Rinte ](top,surf ) is simply
the vector of coordinates ([∆η]1 ; [∆η]2 ; ...; [∆η]l ; ...; [∆η]L ) so:• equation (42) rewrites:
L
X[Wvei ]l

l=1



dX
dt

• equation (45) rewrites:





L
X

[Wvei ]l Al +

h

i

=
l

l=1

∆Bl

[Wvei ]l =  Pk=L
k=1

rs 2
a2

[∆Bk ]

 rl 2  
s

a2

k

• routine VERINT is not used in this case.

List of equations.
• Continuity equation.

L
X
l=1

[Wvei ]l Fl

(46)

(47)

4. Semi-lagrangian computations

63

Generic notations.
Generic notation N(X)LAG stands for:• NVLAG for continuity equation.
Generic notations P(X2D)9 (2D term), P(X)T1 (2D term), P(X3D)L9 (3D term),
stand for:• PX9, PSPT1, PCL9 for continuity equation.
Generic notation P(X)NLT9 (3D term) stands for:• PSPNLT9 for continuity equation.
Generic notation for total term, linear term, non linear term, physics:• A is the total term (sum of dynamical contributions): it is assumed to be a 3D
term (sum of 3D and 2D contributions).
• B is the linear term (treated in the semi-implicit scheme): it is assumed to be a 2D
term (vertical integral of a 3D term).
• the diﬀerence A − βB is the non-linear term, considered as a 3D term.

• F is the sum of contributions computed in the physical parameterizations; it isassumed to be a 2D term (vertical integral of a 3D term).

Description stands for the general case where linear and non-linear terms are gathered in
the same buﬀer, and where no additional splitting is required to do diagnostics or to apply
SLHD interpolations to a subset of the terms interpolated by a high-order interpolation.
In some particular cases, additional splitting involving separate buﬀers may be required(see in part 5.2).

Other points.
∗ High-order interpolations: In the following discretisations, “high-order inter-

polations" means: 32 points interpolations for 3D terms (vertical interpolations are cubic),
12 points interpolations for 2D terms.

∗ Uncentering: ǫ is a ﬁrst-order “uncentering factor". It allows to remove the noisedue to gravity waves (orographic resonance).
∗ Horizontal interpolation of 2D terms: Since the horizontal position of
the interpolation point is vertical dependent, horizontal interpolations of 2D quantities
have to be done for each layer. For example,
interpolating a E
2D surface varii
Dh W whenvei[surface quantity]O has no reason
able at the origin point, [Rinte ](top,surf )∆ηFto be equal to Dh[surface iquantity]O(η=1) , theseE quantities are generally diﬀerent:
Dh W ithis is
E
W
vei
vei
[surface quantity]O which has to be computed.
[surface quantity]O[Rinte ](top,surf )∆ηis the vector containing

this is

PL

hF[W

vei

[∆η]l

]l

i

F

∆η

F

[surface quantity]O(l) , l = 1 to L. For LVERTFE=.F.

[[Wvei ]l ]F [surface quantity]O(l) .
l=1

64

4. Semi-lagrangian computations

3TL vertical interpolating SL scheme.
Equation (42) is discretised as follows:= [Rinte ](top,surf )

Dh W

vei

∆η

i

(X − (1 + ǫ)∆tβB)+
FF{X − + (1 − ǫ)∆t[A − βB]o + [(1 − ǫ)∆tβB + 2∆tF ]− }O+{(1 + ǫ)∆t[A − βB]o }F

E

(48)

Buﬀers content before interpolations for N(X)LAG=2:• P(X3D)L9: (1 − ǫ)∆t[A − βB]o + [(1 − ǫ)∆tβB]− for high-order interpolation at
the origin point O(l).
• P(X2D)9: [X − + 2∆tF ]− for horizontal high-order interpolation at the origin
point O(l).
• P(X)T1: (1+ǫ)∆t[A−βB]o then provisional add of quantity [(1+ǫ)∆tβB]o before
t+dt physics; evaluated at the ﬁnal point F .
Buﬀers content before interpolations for N(X)LAG=3:• P(X3D)L9: (1 − ǫ)∆t[A − βB]o + [(1 − ǫ)∆tβB]− for trilinear interpolation at theorigin point O(l).
• P(X2D)9: [X − + 2∆tF ]− for horizontal high-order interpolation at the origin
point O(l).
• P(X)T1: (1+ǫ)∆t[A−βB]o then provisional add of quantity [(1+ǫ)∆tβB]o before
t+dt physics; evaluated at the ﬁnal point F .

2TL vertical interpolating SL scheme: conventional discretisation(LSETTLS=.F.) and first-order uncentering.
The t + ∆t
non-linear term Am − βBm used in the 2TL SL scheme is computed by a linear2
temporal extrapolation using the t and t − ∆t quantities at the same location. At the ﬁrst
time integration step, values at t + ∆t
are set equal to initial values. This discretisation2of the 2TL SL scheme follows (Mc Donald and Haugen, 1992). Quantity Ao − βBo has tobe saved in a buﬀer P(X)NLT9 to be available as A− − βB− for the following timestep.
Equation (42) is discretised as follows:= [Rinte ](top,surf )

Dh W

vei

∆η

i

βB)+
(X − (1 + ǫ) ∆t
F
2F{X o + (1 − ǫ) ∆t[A − βB]m + [(1 − ǫ) ∆t
βB + ∆tF ]o }O22+{(1 + ǫ) ∆t[A − βB]m }F
2

E

(49)

which can be rewritten, once expanded the extrapolation:βB)+(X − (1 + ǫ) ∆t
F = [Rinte ](top,surf )2+0.5[Rinte ](top,surf )

Dh W

vei

i

Dh W

vei

∆η

i

F

{X o + ∆t(1 − ǫ)Ao + ∆tF o }O2{(A − βB)o − (A − βB)− }O(1 − ǫ) ∆t
2

∆η
F
∆t
+ 2 (1 + ǫ)[1.5(A − βB)o − 0.5(A − βB)− ]F

Buﬀers content before interpolations for N(X)LAG=2:

E

E
(50)

4. Semi-lagrangian computations

65[1.5(A−βB)o −0.5(A−βB)− ]+[(1−ǫ) ∆tβB]o for high-order• P(X3D)L9: (1−ǫ) ∆t
2
2interpolation at the origin point O(l).
• P(X2D)9: X o + [∆tF ]o for horizontal high-order interpolation at the origin point
O(l).
• P(X)T1: (1 + ǫ) ∆t[1.5(A − βB)o − 0.5(A − βB)− ] then provisional add of quantity
2o∆t[(1 + ǫ) 2 βB] before t+dt physics; evaluated at the ﬁnal point F .
• remark: another possibility is to add all the content of P(X2D)9 to P(X3D)L9
in order to remove one horizontal interpolation.
Buﬀers content before interpolations for N(X)LAG=3:• P(X3D)L9: (1 − ǫ) ∆t[1.5(A − βB)o − 0.5(A − βB)− ] + [(1 − ǫ) ∆tβB]o for trilinear22interpolation at the origin point O(l).
• P(X2D)9: X o + [∆tF ]o for horizontal high-order interpolation at the origin point
O(l).
• P(X)T1: (1 + ǫ) ∆t[1.5(A − βB)o − 0.5(A − βB)− ] then provisional add of quantity
2o∆t[(1 + ǫ) 2 βB] before t+dt physics; evaluated at the ﬁnal point F .

2TL vertical interpolating SL scheme: conventional discretisation(LSETTLS=.F.) and pseudo-second order uncentering.
One starts to remove uncentering ǫ from the nonlinear terms and to apply a second-orderuncentering ǫX to linear terms, that yelds a term B− in the discretisation. From property
given by formula (27), one can remove term B− and show that discretisation is equivalent
to replace β by (1 + ǫX )β. For more details, see part 5 (equations (37) and (38)) of(Simmons and Temperton, 1996). In equation (49), uncentering ǫ has to be replaced byzero, and β has to be replaced by (1+ǫX )β. The t+ ∆tnon-linear term Am −(1+ǫX )βBm
2
used in the 2TL SL scheme is computed by a linear temporal extrapolation using the t andt − ∆t quantities at the same location. At the ﬁrst time integration step, values at t + ∆t
2
are set equal to initial values and second-order uncentering is replaced by a ﬁrst orderuncentering. This discretisation of the 2TL SL scheme follows (Mc Donald and Haugen,1992). Quantity Ao − (1 + ǫX )βBo has to be saved in a buﬀer P(X)NLT9 to be available
as A− − (1 + ǫX )βB− for the following timestep.
Equation (42) is discretised as follows:= [Rinte ](top,surf )

Dh W

vei

∆η

i

βB)+
(X − (1 + ǫX ) ∆t
F
2F[A − (1 + ǫX )βB]m + [(1 + ǫX ) ∆tβB + ∆tF ]o }O{X o + ∆t22+{ ∆t[A − (1 + ǫX )βB]m }F2(51)

which can be rewritten, once expanded the extrapolation:(X − (1 + ǫX ) ∆tβB)+
F = [Rinte ](top,surf )2+0.5[Rinte ](top,surf )

Dh W

vei

i

Dh W

vei

∆η

i

F

Ao + ∆tF o }O
{X o + ∆t
2E∆t{(A − (1 + ǫX )βB)o − (A − (1 + ǫX )βB)− }O
2

∆η
F
∆t
+ 2 [1.5(A − (1 + ǫX )βB)o − 0.5(A − (1 + ǫX )βB)− ]F

Buﬀers content before interpolations for N(X)LAG=2:

E

E

(52)

66

4. Semi-lagrangian computations[1.5(A − (1 + ǫX )βB)o − 0.5(A − (1 + ǫX )βB)− ] + [(1 + ǫX ) ∆tβB]o
• P(X3D)L9: ∆t2
2
for high-order interpolation at the origin point O(l).
• P(X2D)9: X o + [∆tF ]o for horizontal high-order interpolation at the origin point
O(l).
• P(X)T1: ∆t[1.5(A − (1 + ǫX )βB)o − 0.5(A − (1 + ǫX )βB)− ] then provisional add2βB]o before t+dt physics; evaluated at the ﬁnal point F .
of quantity [(1 + ǫX ) ∆t2
• remark: another possibility is to add all the content of P(X2D)9 to P(X3D)L9
in order to remove one horizontal interpolation.

Buﬀers content before interpolations for N(X)LAG=3:• P(X3D)L9: ∆t[1.5(A − (1 + ǫX )βB)o − 0.5(A − (1 + ǫX )βB)− ] + [(1 + ǫX ) ∆t
βB]o2
2for trilinear interpolation at the origin point O(l).
• P(X2D)9: X o + [∆tF ]o for horizontal high-order interpolation at the origin point
O(l).
• P(X)T1: ∆t[1.5(A − (1 + ǫX )βB)o − 0.5(A − (1 + ǫX )βB)− ] then provisional add2βB]o before t+dt physics; evaluated at the ﬁnal point F .
of quantity [(1 + ǫX ) ∆t
2

2TL vertical interpolating SL scheme: stable discretisation (LSETTLS=.T.) and first-order uncentering.
non-linear term Am − βBm used in the 2TL SL scheme if LSETTLS=.F. is
The t + ∆t2
replaced in the case LSETTLS=.T. by a linear spatio-temporal extrapolation comparable
to the one applied to the wind components for the research of trajectory (see formula (81)),
except the fact that there is an additional uncentering. Term Am − βBm is replaced by
0.5(1 + ǫ)[Ao − βBo ]F + 0.5(2 − ǫ)[Ao − βBo ]O(l) − 0.5[A− − βB− ]O(l)This type of extrapolation is available only for N(X)LAG=3. At the ﬁrst time integration
step, values at t + ∆t
are set equal to initial values. Quantity Ao − βBo has to be saved2
in a buﬀer P(X)NLT9 to be available as A− − βB− for the following timestep.
Equation (42) is discretised as follows:= [Rinte ](top,surf )

Dh W

vei

∆η

i

(X − (1 + ǫ) ∆t
βB)+
F
2
F

{X o + (2 − ǫ) ∆t[A − βB]o − ∆t[A − βB]− + [(1 − ǫ) ∆t
βB + ∆tF ]o }O222[A − βB]o }F+{(1 + ǫ) ∆t
2(53)

which can be rewritten:(X − (1 + ǫ) ∆t
βB)+
F
2

= [Rinte ](top,surf )+[Rinte ](top,surf )

Dh W

vei

∆η

Dh W

vei

∆η

i

iF
F

A + ∆tF ]o }O{X o + (1 − ǫ) ∆t
2∆t{[A − βB]o − [A − βB]− }O2+{(1 + ǫ) ∆t[A − βB]o }F2

Buﬀers content before interpolations for N(X)LAG=3:• P(X3D)L0 is not used.

E

E

E

(54)

4. Semi-lagrangian computations

67[A − βB]o − ∆t[A − βB]− + [(1 − ǫ) ∆tβB]o for trilinear• P(X3D)L9: (2 − ǫ) ∆t2
22interpolation at the origin point O(l) (which can be rewritten: (1−ǫ) ∆t
A+ ∆t
{[A−
2
2
o
−
βB] − [A − βB] }).

• P(X2D)0 is not used.

• P(X2D)9: X o + [∆tF ]o for horizontal high-order interpolation at the origin pointO(l).
[A−βB]o then provisional add of quantity [(1+ǫ) ∆tβB]o before• P(X)T1: (1+ǫ) ∆t
2
2t+dt physics; evaluated at the ﬁnal point F .

2TL vertical interpolating SL scheme: stable discretisation (LSETTLS=.T.) and pseudo-second order uncentering.
In equation (53), uncentering ǫ has to be replaced by zero, and β has to be replaced by
are set equal to initial(1 + ǫX )β. At the ﬁrst time integration step, values at t + ∆t
2values and second-order uncentering is replaced by a ﬁrst order uncentering. Quantity
Ao −(1+ǫX )βBo has to be saved in a buﬀer P(X)NLT9 to be available as A− −(1+ǫX )βB−
for the following timestep.
Equation (42) is discretised as follows:= [Rinte ](top,surf )

Dh W

vei

∆η

i

(X − (1 + ǫX ) ∆t
βB)+
F
2
F

[A − (1 + ǫX )βB]− + [(1 + ǫX ) ∆tβB + ∆tF ]o }O{X o + ∆t[A − (1 + ǫX )βB]o − ∆t22
[A − (1 + ǫX )βB]o }F
+{ ∆t
2

(55)

which can be rewritten:(X − (1 + ǫX ) ∆t
βB)+F2
= [Rinte ](top,surf )+[Rinte ](top,surf )

Dh W

vei

∆η

i

F

Dh W

vei

∆η

i

F

{X o + ∆t
A + ∆tF ]o }O2E∆t{[A − (1 + ǫX )βB]o − [A − (1 + ǫX )βB]− }O2+{ ∆t[A − (1 + ǫX )βB]o }F
2

E

(56)

Buﬀers content before interpolations for N(X)LAG=3:• P(X3D)L0 is not used.

[A − (1 + ǫX )βB]− + [(1 + ǫX ) ∆tβB]o for• P(X3D)L9: ∆t[A − (1 + ǫX )βB]o − ∆t22trilinear interpolation at the origin point O(l) (which can be rewritten: ∆tA+2∆t{[A − (1 + ǫX )βB]o − [A − (1 + ǫX )βB]− }).
2• P(X2D)0 is not used.

• P(X2D)9: X o + [∆tF ]o for horizontal high-order interpolation at the origin point
O(l).

• P(X)T1: ∆t[A − (1 + ǫX )βB]o then provisional add of quantity [(1 + ǫX ) ∆tβB]o22before t+dt physics; evaluated at the ﬁnal point F .

5.5 Discretisation for a 2D variable in a 2D model.
The content of the part (5.2) is generally valid, but there are particular remarks.

E68

4. Semi-lagrangian computations

List of equations.
• Momentum equation.
• Continuity equation.

Generic notations.
Generic notation N(X)LAG stands for:
• NWLAG for momentum equation.

• NVLAG for continuity equation. Negative values of NVLAG are used for Lagrangian formulation of continuity equation (the following discretisations apply tothe absolute value of NVLAG), only NVLAG=-2 is available in this case.
Generic notation P(X)L0, P(X)L9, P(X)T1 stands for:• PUL0, PUL9, PUT1 for U-momentum equation.

• PVL0, PVL9, PVT1 for V-momentum equation.

• PSPL0, PSPL9, PSPT1 for continuity equation.

Generic notation P(X)NLT9 stands for:
• PUNLT9 for U-momentum equation.

• PVNLT9 for V-momentum equation.
• PSPNLT9 for continuity equation.

Generic notation for total term, linear term, non linear term: A is the total term (sum ofdynamical contributions), B is the linear term (treated in the semi-implicit scheme), thediﬀerence A − βB is the non-linear term.

Other points.
∗ High-order interpolations: In the following discretisations, “high-order interpolations" means 12 points interpolations.

∗ Uncentering: ǫ is a ﬁrst-order “uncentering factor". It allows to remove the noisedue to gravity waves (orographic resonance).
∗ Vectorial variables: The following discretisations are written for scalar variables.

For vectorial variables (for example the horizontal wind) a rotation operator R has to be
applied from interpolation point to ﬁnal point:• expression interpolated at O has to be replaced by ROF {this expression}O .

• expression interpolated at M has to be replaced by RM F {this expression}M .

4. Semi-lagrangian computations

69

3TL SL scheme.
Equation:
dX
=A
dt

(57)

is discretised as follows:−(X − (1 + ǫ)∆tβB)++ [(1 − ǫ)∆tA − (1 − ǫ)∆tβB]o + [(1 − ǫ)∆tβB]− }O
F = {X+{[(1 + ǫ)∆tA − (1 + ǫ)∆tβB]o }F(58)

Buﬀers content before interpolations for N(X)LAG=2:• P(X)L0 is not used.
• P(X)L9: X − + [(1 − ǫ)∆tA − (1 − ǫ)∆tβB]o + [(1 − ǫ)∆tβB]− for high-orderinterpolation at the origin point O.
• P(X)T1: [(1 + ǫ)∆tA − (1 + ǫ)∆tβB]o ; evaluated at the ﬁnal point F .
Buﬀers content before interpolations for N(X)LAG=3:• P(X)L0: [(1 − ǫ)∆tA − (1 − ǫ)∆tβB]o + [(1 − ǫ)∆tβB]− for bilinear interpolation
at the origin point O.
• P(X)L9: X − for high-order interpolation at the origin point O.
• P(X)T1: [(1 + ǫ)∆tA − (1 + ǫ)∆tβB]o ; evaluated at the ﬁnal point F .

2TL SL scheme: conventional discretisation (LSETTLS=.F.) andfirst-order uncentering.
non-linear term Am − βBm used in the 2TL SL scheme is computed by a linear
The t + ∆t2
temporal extrapolation using the t and t − ∆t quantities at the same location. At the ﬁrst
are set equal to initial values. This discretisation
time integration step, values at t + ∆t
2of the 2TL SL scheme follows (Mc Donald and Haugen, 1992). Quantity Ao − βBo has tobe saved in a buﬀer P(X)NLT9 to be available as A− − βB− for the following timestep,when non-zero (i.e. only for continuity equation, if β=1).
Equation (57) is discretised as follows:
o
m
∆t
∆t
(X − (1 + ǫ) ∆tβB)++ [(1 − ǫ) ∆tβB]o }OF = {X + [(1 − ǫ) 2 A − (1 − ǫ) 2 βB]
22A − (1 + ǫ) ∆tβB]m }F+{[(1 + ǫ) ∆t
2
2

(59)

Buﬀers content before interpolations for N(X)LAG=2:• P(X)L0 is not used.
• P(X)L9: X o + [(1 − ǫ) ∆t
A − (1 − ǫ) ∆tβB]m + [(1 − ǫ) ∆tβB]o for high-order222interpolation at the origin point O.
• P(X)T1: [(1 + ǫ) ∆t
A − (1 + ǫ) ∆tβB]m ; evaluated at the ﬁnal point F .
2
2
Buﬀers content before interpolations for N(X)LAG=3:
A − (1 − ǫ) ∆tβB]m + [(1 − ǫ) ∆tβB]o for bilinear interpolation• P(X)L0: [(1 − ǫ) ∆t2
2
2
at the origin point O.
• P(X)L9: X o for high-order interpolation at the origin point O.
• P(X)T1: [(1 + ǫ) ∆t
A − (1 + ǫ) ∆tβB]m ; evaluated at the ﬁnal point F .
2
2

70

4. Semi-lagrangian computations∗ Remark for momentum equation: β can only take the value 1 and in
this case the non-linear term A − B is zero; simpliﬁcations are made in the code; arrays
PUNLT9 and PVNLT9 are useless and not allocated.
2TL SL scheme: stable discretisation (LSETTLS=.T.) and firstorder uncentering.
non-linear term Am − βBm used in the 2TL SL scheme if LSETTLS=.F. is
The t + ∆t2
replaced in the case LSETTLS=.T. by a linear spatio-temporal extrapolation comparable
to the one applied to the wind components for the research of trajectory (see formula (81)),
except the fact that there is an additional uncentering. Term Am − βBm is replaced by
0.5(1 + ǫ)[Ao − βBo ]F + 0.5(2 − ǫ)[Ao − βBo ]O − 0.5[A− − βB− ]OThis type of extrapolation is available only for N(X)LAG=3. At the ﬁrst time integration
step, values at t + ∆t
are set equal to initial values. Quantity Ao − βBo has to be saved2
in a buﬀer P(X)NLT9 to be available as A− − βB− for the following timestep, when
non-zero (i.e. only for continuity equation, if β=1).
Equation (57) is discretised as follows:
o
o
−
o
∆t
∆t
∆t
∆t
∆t
βB)+
(X − (1 + ǫ) ∆t
F = {X + [(2 − ǫ) 2 A − (2 − ǫ) 2 βB] − [ 2 A − 2 βB] + [(1 − ǫ) 2 βB] }O
2+{[(1 + ǫ) ∆t
A − (1 + ǫ) ∆tβB]o }F2
2

Buﬀers content before interpolations for N(X)LAG=3:
A−(2−ǫ) ∆tβB]o −[ ∆tA− ∆tβB]− +[(1−ǫ) ∆tβB]o for bilinear• P(X)L0: [(2−ǫ) ∆t
2
2
2
2
2
interpolation at the origin point O.
• P(X)L9: X o for high-order interpolation at the origin point O.

A − (1 + ǫ) ∆tβB]o ; evaluated at the ﬁnal point F .
• P(X)T1: [(1 + ǫ) ∆t
2
2∗ Remark for momentum equation: β can only take the value 1 and in
this case the non-linear term A − B is zero; simpliﬁcations are made in the code; arrays
PUNLT9 and PVNLT9 are useless and not allocated.
2TL SL scheme: pseudo-second order uncentering.
In cycle 37T1 of ARPEGE/IFS the pseudo-second order uncentering is not coded in theshallow-water model.

5.6 Additional vertical derivatives.
If δT R is non-zero, discretisation of temperature equation needs to compute the verticalTadvection η̇ dα(at full levels) of αT . Layers values of αT (array RCORDIF) are useddηs, but half level values of αT (array RCORDIH) are used toto deﬁne T + δT R RαdTTΦST
compute vertical advection.

There is something similar in the Q̂ equation if δP is non-zero (Wood and Staniforthdeep-layer version of the NH-PDVD model, use of an array RCORPDH).
No longer existing option.

(60)

4. Semi-lagrangian computations

71

5.7 Case when some variables are evaluated at half levels.
The prognostic variables and the RHS of equations are generally evaluated at full levels inthe discretisation. The previous general considerations valid to layer variables also apply
to half level variables, but a “half level" trajectory has now to be computed (the origin O
now matches a half level ﬁnal point F ).
• Horizontal displacement at half levels: the coordinates of the half level-trajectoryinterpolation point are computed as the average (with a vertical weight taking
account of η) of the coordinates of the adjacent full level-trajectory interpolation
points; this average is a lon-lat average on the computational sphere in sphericalgeometry, and a x-y average on the projection plane in plane geometry (ALADIN).
• Vertical displacement at half levels: the vertical coordinate of the half level-trajectoryinterpolation point is computed as the average (with a vertical weight taking account of η) of the vertical coordinates of the adjacent full level-trajectory interpolation points; no vertical displacement if there is no vertical displacement for thetwo adjacent full levels.
• No complete iterative recalculation of trajectory is done for half-level trajectories.

5.8 Remarks for spline cubic vertical interpolations.
In this case the vertical interpolation uses all model levels and can be written as the
product of two vertical interpolations: the ﬁrst one uses all model levels and can bedone at F in the unlagged grid-point calculations (the intermediate quantity obtained isstored in the array P(X)SPL9), the second one is a 4 points interpolation, done in thelagged grid-point calculations in the interpolation routine. Interpolation routine uses both
P(X)SPL9 (for interpolations) and P(X)L9 to apply a monotonicity constraint.

72

4. Semi-lagrangian computations

6 Computation of medium and origin points.
Preliminary remark: the subsections (6.1), (6.2) and (6.3) are detailed for a sphericalgeometry and for the trajectory which is used in the advection of full level variables; thesubsection (6.4) gives informations about the other cases (for example plane geometry andhalf level variables).

6.1 Medium point M (subroutines LARMES and LARMES2).
Trajectories are great circles on the geographical sphere. The computation of themedium point M location of the Lagrangian trajectory is performed by an iterative method
described by Robert (1981) and adapted to the sphere by M. Rochas. In a 3TL SL scheme,
the particle is at the point M at the instant t (t + ∆t/2 for the ﬁrst integration step). In a
2TL SL scheme, the particle is at the point M at the instant t + ∆t/2. M is at the middle
position of the origin point O and the ﬁnal point F . Algorithm is described for deep layer
equations; in the thin layer equations, replace simply rs by a in formulae. For conveniency
equations are written with the angular velocity V/rs but actually this is rather (a/rs )V
which is used in the code. Parts (6.1), (6.1) and (6.1) are valid for non implicit iterative
schemes. Part (6.1) is valid for a class of iterative centred-implicit schemes where themomentum equation is treated in an iterative centred-implicit manner (this is the case of
the option LPC_FULL in ALADIN-NH).

∗ Notations:• RM F is the rotation operator from medium point to ﬁnal point (see section 9).
• ROF is the rotation operator from origin point to ﬁnal point (see section 9).
• rF = CF (C Earth centre, F ﬁnal point).

• rM = CM (M medium point).
• r = rs k.

dCF).
• φM F : angle (CM,• θF ,λF : latitude, longitude on the geographical sphere of F .

• θM ,λM : latitude, longitude on the geographical sphere of M .

• VM : interpolated horizontal wind at M (wind at t in 3TL SL scheme, t + 0.5∆t in
2TL SL scheme).

• VO : interpolated horizontal wind at O (wind at t in 3TL SL scheme, t + 0.5∆t in
2TL SL scheme).
• a is the average Earth radius near the surface.
• ∆t: time-step.
• δt:– In a 3TL SL scheme, δt = 0.5∆t at the ﬁrst integration step, δt = ∆t at thefollowing integration steps (leap-frog scheme).
– In a 2TL SL scheme, δt = 0.5∆t.

• L: number of layers of the model.
• A, B deﬁne hydrostatic pressure on the η levels ( Π = A + BΠs , where Πs is thehydrostatic surface pressure).
• Πs st is a reference hydrostatic pressure equal to the surface pressure of the standard
atmosphere (variable VP00).
• ΠST is a reference hydrostatic pressure deﬁned at full levels and half levels corresponding to the surface reference hydrostatic pressure Πs st ( ΠST = A + BΠs st ):stored in array STPRE.

4. Semi-lagrangian computations

73∗ Definition of the vertical coordinate η: Research of medium point needs

an exact deﬁnition of the vertical coordinate η. For the half level number l (l between 0
and L), ηl is deﬁned by:
ηl =

Al
+ Bl
Πs st

(61)

if LREGETA=.F. (in namelist NAMCT0), and:
ηl =

l
L

(62)

if LREGETA=.T. .
For the layer number l (l between 1 and L), ηl is deﬁned by:(63)ηl = 0.5(ηl + ηl−1 )A speciﬁc deﬁnition of η may be required for the VFE operators if LVERTFE=.T.: it iscontroled by the key LVFE_REGETA.

Conventional algorithm (always used for 3TL SL scheme, case
LSETTLST=LELTRA=.F. for 2TL SL scheme).
∗ Extrapolation of the wind for 2TL SL scheme: The quantity [V/rs ]at t + 0.5∆t used in the 2TL SL scheme is computed by a linear temporal extrapolation
using the t and t − ∆t winds at the same location.
∗ Algorithm: The medium point is deﬁned by the following iterative scheme: for theiteration k + 1:

F
[r/rs ]M
k+1 = [r/rs ] cos φk −RM F ([V/rs ]Mk )sin φk| [V/rs ]M
|
k

(64)

where:
φk = δt | [V/rs ]M
k |

(65)

φ is a small angle:
sin φ ≃ φ −

φ3
6

(66)

cos φ ≃ 1 −

φ2
2

(67)

and:

This approximation allows to simplify some calculations and avoids the occurrence of adivision by 0 in the formula deﬁning rM
k+1 . Of course:φ2
sin φ ≃ φ 1 −
6





φ2
≃| [V/rs ] | δt 1 −
6



(68)

74

4. Semi-lagrangian computations

thus:
F[r/rs ]M
k+1 = [r/rs ]



φ2
1− k
2



−R

MF

([V/rs ]M
k )δt



φ2
1− k
6



(69)

On the vertical, for 3D model:
Mηk+1= η F − δtη̇kM

(70)

F
F
∗ First iteration: Let us start with M0 = F , [V/rs ]M
0 = [V/rs ] , φ0 = δt | [V/rs ] |,η̇0M = η̇ F . Horizontal wind V has components (u,v). Thussin θ1M = sin θF cos φ0 −[v/rs ]Fcos θF sin φ0| [V/rs ]F |FF
cos θ1M cos(λM
1 − λ ) = cos θ cos φ0 +Fcos θ1M sin(λM
1 −λ ) = −(71)[v/rs ]Fsin θF sin φ0| [V/rs ]F |[u/rs ]Fsin φ0| [V/rs ]F |

(72)

(73)

η1M = η F − δtη̇0F(74)This deﬁnes the coordinates of M1 . Then [u/rs ], [v/rs ], η̇ are interpolated at this point,Mthat gives [V/rs ]M
1 and η̇1 . Tri-linear interpolations are used in the 3D primitive equation
model, horizontal 12 points interpolations are used in the 2D shallow-water model (seesection 11).
′′′

∗ Following iterations: Let us denote by V (u , v ) = RM F (VkM ).
′M
sin θk+1= sin θF cos φk −[v /rs ]cos θF sin φk| [V′ /rs ] |

(75)

′

M
F
F
cos θk+1
cos(λM
k+1 − λ ) = cos θ cos φk +[v /rs ]sin θF sin φk| [V′ /rs ] |

(76)

′

M
F
cos θk+1
sin(λM
k+1 − λ ) = −Mηk+1= η F − δtη̇kM[u /rs ]sin φk| [V′ /rs ] |

(77)

(78)

This deﬁnes coordinates of Mk+1 . Then, if it is not the last iteration, [u/rs ], [v/rs ], η̇M
are interpolated at this point, that gives [V/rs ]M
k+1 and η̇k+1 . Tri-linear interpolations areused in the 3D primitive equation model, horizontal 12 points interpolations are used in
the 2D shallow-water model (see section 11). This iterative algorithm quickly converges:
3 iterations are generally enough.

4. Semi-lagrangian computations

75

Stable algorithm for 2TL SL scheme (LSETTLST=.T., LELTRA=.F.
in NAMDYN).
∗ Extrapolation of the wind: The previous algorithm with LSETTLST=.F.
can sometimes generate instability (especially when applied to the vertical displacement)so a stable algorithm has been developed by M. Hortal at ECMWF. For more details
about theoretical aspects see (Hortal, 1998), (Hortal, 2002). The basic idea is to replace
the purely temporal extrapolation by a spatio-temporal extrapolation:
RM F [V/rs ]M (t + 0.5∆t) = 1.5RN F [V/rs ]N (t) − 0.5ROF [V/rs ]O (t − ∆t)(79)

where N is the position of the particle at time t for a particle which goes from the origin
point O at time t − ∆t to M at time t + 0.5∆t. Assuming that the wind is constant along
the trajectory one can write:(80)

ON = 2N M = 0.5N F

and evaluate the angular velocity RN F [V/rs ]N (t) by 2/3RM F [V/rs ]M (t)+1/3ROF [V/rs ]O (t)or 1/3[V/rs ]F (t) + 2/3ROF [V/rs ]O (t). Expression of [V/rs ]M (t + 0.5∆t) becomes:
RM F [V/rs ]M (t+0.5∆t) = 0.5[V/rs ]F (t)+0.5ROF (2[V/rs ]O (t)−[V/rs ]O (t−∆t))(81)The same type of extrapolation is done for the η-coordinate vertical velocity. The algorithm of research of trajectory uses directly the RHS of this equation, and for all iterations
the origin point O is computed instead of the medium point M .

∗ Algorithm: The origin point is deﬁned by the following iterative scheme: for theiteration k + 1:

F
[r/rs ]O
k+1 = [r/rs ] cos φk −O
0.5[V/rs ]F (t) + 0.5ROF (2[V/rs ]O
k (t) − [V/rs ]k (t − ∆t))sin φk (82)O
O
F
OF
| 0.5[V/rs ] (t) + 0.5R (2[V/rs ]k (t) − [V/rs ]k (t − ∆t)) |where:
O
φk = 2δt | 0.5[V/rs ]F (t) + 0.5ROF (2[V/rs ]O
k (t) − [V/rs ]k (t − ∆t)) |(83)

Approximations given by equations (66), (67) and (68) are still valid (change δt by 2δt in(68)), thus:



φ


2

[r/rs ]F 1 − 2k[r/rs ]O
k+1 =



φ2



O
k
− (0.5[V/rs ]F (t) + 0.5ROF (2[V/rs ]O
k (t) − [V/rs ]k (t − ∆t)))(2δt) 1 − 6 (84)

On the vertical, for 3D model:
Oηk+1= η F − 2δt(0.5η̇ F (t) + 0.5(2η̇kO (t) − η̇kO (t − ∆t)))(85)∗ First iteration: One starts with M0 = F , [V/rs ]F (t) as a ﬁrst guess for the spatiotemporally extrapolated horizontal angular velocity, φ0 = 2δt | [V/rs ]F (t) |, η̇ F (t) as aﬁrst guess for the spatio-temporally extrapolated η-coordinate vertical wind. Remarkthat quantities at t are taken as a ﬁrst guess and not quantities at (t + 0.5∆t), contrary
to the case LSETTLST=.F. . Use equations (71) to (74) replacing δt by 2δt and thesuperscript M by O. This deﬁnes the coordinates of O1 ; (2[V/rs ](t) − [V/rs ](t − ∆t))and (2η̇(t) − η̇(t − ∆t)) are interpolated at this point, that allows to compute the windcomponents which will be used for the next iteration.

76

4. Semi-lagrangian computations′′′∗ Following iterations: [V /rs ] (of coordinates ([u /rs ], [v /rs ]) ) is a generic noF

OFO(2[V/rs ]O
k (t) − [V/rs ]k (t − ∆t))).

tation for (0.5[V/rs ] (t) + 0.5R
For horizontal displacement use equations (75) to (77) replacing δt by 2δt and the superscript M by O. For
vertical displacement use equation
Oηk+1= η F − 2δt(0.5η̇ F (t) + 0.5(2η̇kO (t) − η̇kO (t − ∆t)))(86)

Alternate stable algorithm for 2TL SL scheme (LELTRA=.T. in
NAMDYN).
∗ Extrapolation of the horizontal angular velocity: The quantity [V/rs ]at t + 0.5∆t used in the 2TL SL scheme is computed using the RHS of the [V/rs ] equation
with explicit formulation of Coriolis term. We denote this RHS by RHSAV in subsubsection (6.1). The extrapolated value of [V/rs ] is given by:[V/rs ]ext = [V/rs ](t) + 0.5∆tRHSAV(87)∗ Extrapolation of the vertical velocity: The RHS is assumed to be zero,so:η̇ext = η̇(t)(88)

There is no extrapolation at all.

∗ Algorithm: The medium point is deﬁned by the following iterative scheme: for theiteration k + 1, a provisional position of the origin point is computed using equations (64)to (70), replacing “M " by “O" and “δt" by “∆t".
F
F∗ First iteration: Let us start with O0 = F , [V/rs ]O
0 = [V/rs ]ext , φ0 = ∆t | [V/rs ]ext |,F
η̇0O = η̇ext. Horizontal angular velocity [V/rs ]ext has components (u/rs ,v/rs ). Use equations (71) to (74), replacing “M " by “O" and “δt" by “∆t" to deﬁne coordinates of O1 .
O
Then u/rs ,v/rs ,η̇ are interpolated at this point, that gives [V/rs ]O
1 and η̇1 (subscript“ext" is omitted). Tri-linear interpolations are used in the 3D primitive equation model,
horizontal 12 points interpolations are used in the 2D shallow-water model (see section
11).

∗ Following iterations: Use equations (75) to (78), replacing “M " by “O" and “δt"by “∆t" to deﬁne coordinates of Ok+1 . Then u/rs ,v/rs ,η̇ are interpolated at this point,
O
that gives [V/rs ]O
k+1 and η̇k+1 (subscript “ext" is omitted). Tri-linear interpolations areused in the 3D primitive equation model, horizontal 12 points interpolations are used in
the 2D shallow-water model (see section 11). This iterative algorithm quickly converges:
3 iterations are generally enough.
Algorithm used with the iterative centred-implicit schemes.
∗ Preliminar remarks: Iterative centred-implicit schemes are used to improve sta-bility and it has been shown that this type of scheme has to be used when non-hydrostatic
equations are advected by a SL2TL scheme. There are several manners to do iterative
centred-implicit schemes; these schemes are not described in detail here; for some of themwhich are not obsolescent, see documentation (IDSI). When the momentum equation istreated by an iterative centred-implicit scheme, the semi-Lagrangian trajectory has to berecomputed at each iteration of the iterative centred-implicit scheme and interpolations
have to be done again. In this case the algorithm of research trajectory is modiﬁed. The
case is currently encountered with a SL2TL scheme and LPC_FULL=.T. . The following algorithm will be described for a SL2TL scheme but it can be extended to a SL3TL
scheme (replacing instant t by instant t − ∆t).

4. Semi-lagrangian computations

77∗ Extrapolation of the wind if non-extrapolating option (LPC_NESCT=.T.):
No extrapolation is done. The ﬁrst iteration of the iterative centred-implicit scheme uses
[V/rs ](t) and η̇(t). The following iterations of the iterative centred-implicit scheme use
the [V/rs ](t+∆t) and η̇(t+∆t) of the previous iteration to start the research of trajectory.
As for the “LSETTLST" option the algorithm computes the origin point O.

∗ Extrapolation of the wind if extrapolating option (LPC_NESCT=.F.):

The only diﬀerence with the previous case is for the ﬁrst iteration of the iterative centredimplicit scheme: the wind which is used is now 1.5[V/rs ](t) − 0.5[V/rs ](t − ∆t) and1.5η̇(t) − 0.5η̇(t − ∆t).

∗ Algorithm: One denotes by index “k" the numbering of the SL-trajectory researchalgorithm iteration, and by index “(i)" the number of the iterative centred-implicit scheme
iteration (in variable NCURRENT_ITER).
The origin point is deﬁned by the following iterative scheme: for the iteration k + 1 of theSL-trajectory research algorithm:







F
0.5ROF [V/rs ]O
k (t) + 0.5 [V/rs ] (t + ∆t) (i−1)
F[sin φk ](i) (89)[r/rs ]O
k+1 (i) = [r/rs ] [cos φk ](i) −F| 0.5ROF [V/rs ]O
k (t) + 0.5 [[V/rs ] (t + ∆t)](i−1) |where:

F[φk ](i) = 2δt | 0.5ROF [V/rs ]O
k (t) + 0.5 [V/rs ] (t + ∆t) (i−1) |(90)

Approximations given by equations (66), (67) and (68) are still valid (change δt by 2δt in(68)), thus:



0.5R

OF

[V/rs ]Ok (t) + 0.5







F
[r/rs ]O
k+1 (i) = [r/rs ]



F

1−



[V/rs ] (t + ∆t) (i−1) (2δt)

 2

φk (i)
2

1−

!

 2

φk (i)
6

−

!

(91)

On the vertical, for 3D model:

 O 







ηk+1 (i) = η F − 2δt 0.5η̇kO (t) + 0.5 η̇ F (t + ∆t) (i−1)



(92)

For i = 0: [[V/rs ](t + ∆t)](i=0) = [V/rs ](t) and [η̇(t + ∆t)](i=0) = η̇(t) if LPC_NESCT=.T. ;[[V/rs ](t + ∆t)](i=0) = 2[V/rs ](t) − [V/rs ](t − ∆t) and [η̇(t + ∆t)](i=0) = 2η̇(t) − η̇(t − ∆t)if LPC_NESCT=.F. .

∗ First  iteration of  the research of SL trajectory: One starts with
M0 = F , [V/rs ]F (t + ∆t) (i−1) as a ﬁrst guess for the spatio-temporally extrapolated








horizontal wind, [φ0 ](i) = 2δt | [V/rs ]F (t + ∆t) (i−1) |, η̇ F (t + ∆t) (i−1) as a ﬁrst guess

for the spatio-temporally extrapolated η-coordinate vertical wind. Use equations (71) to(74) replacing δt by 2δt and the superscript M by O in the SL-trajectory research. Thisdeﬁnes the coordinates of [O1 ](i) ; [V/rs ](t) and η̇(t) are interpolated at this point, thatallows to compute the wind components which will be used for the next iteration of theresearch of SL trajectory.

78

4. Semi-lagrangian computations′∗ Following iterations of the research of SL trajectory:  [V /rs ] (of co- ′′[V/rs ]F (t + ∆t) (i−1) .
ordinates ([u /rs ], [v /rs ]) is a generic notation for 0.5R
For horizontal displacement use equations (75) to (77) replacing δt by 2δt and the superscript M by O otherwise. For vertical displacement use equation

 O 

OF





[V/rs ]Ok (t)+0.5

ηk+1 (i) = η F − 2δt 0.5η̇kO (t) + 0.5 η̇ F (t + ∆t) (i−1)



(93)

∗ Alternate stable algorithm for 2TL SL scheme (LELTRA=.T. in
NAMDYN): For each iteration of the iterative centred-implicit scheme the algorithm

is the same as for explicit schemes; at each iteration the RHS of the momentum equation
is updated with the “provisional" t + ∆t information computed at the previous iteration.

6.2 Origin point O (subroutines LARMES and LAINOR2).
In a 3TL SL scheme, the particle is at the point O at the instant t − ∆t (t for the ﬁrst
integration step). In a 2TL SL scheme, the particle is at the point O at the instant t.
O is on the same great circle arc (on the geographical sphere) as M and F and thedlength of OF is twice the length of M F . If angle ([r/rs ]O, [r/rs ]F ) is small (less than 10o ,what is generally satisﬁed), on can write for horizontal displacement:[r/rs ]O − [r/rs ]F ≃ 2([r/rs ]M − [r/rs ]F )(94)

For vertical displacement on can always write:η O − η F = 2(η M − η F )(95)

One denotes by:d• φ = ([r/rs ]M, [r/rs ]F )′′′• [V /rs ] (of coordinates ([u /rs ], [v /rs ])) the last interpolated horizontal velocity.

Using the following identities:cos 2φ = 2 cos2 φ − 1(96)

sin 2φ = 2 sin φ cos φ(97)

the origin point horizontal coordinates can be computed by:sin θO = sin θF cos 2φ − 2 cos φ′[v /rs ]cos θF sin φ| [V′ /rs ] |cos θO cos(λO − λF ) = cos θF cos 2φ + 2 cos φ
cos θO sin(λO − λF ) = −2 cos φ



′



′



[v /rs ]sin θF sin φ| [V′ /rs ] |[u /rs ]sin φ| [V′ /rs ] |Terms in brackets are already computed to determine M .

(98)



(99)

(100)

4. Semi-lagrangian computations

79

6.3 Refined recomputation of point O.
∗ Option L2TLFF for RW2TLFF=1: Option (switch L2TLFF in YOM-

DYN) controls recomputation of the origin point using the average between the angularvelocity at the origin point and the provisional t + ∆t angular velocity, according to thealgorithm previously described. Only term (2Ω ∧ ak) is computed (always analytically)at this improved position of O (so L2TLFF is active only if LADVF=.T. or LADVFW=.T.). Reﬁned recomputation of point O is available only in a limited set of options.
In the following sections 7, 8 and ?? discretised equations are written with notation O for
all quantities. Equations system is integrated to ﬁnd a ﬁrst guess of VF (t + ∆t) and also
a ﬁrst guess of Πs (t + ∆t) which provides rs (t + ∆t), then 0.5([V/rs ]F + ROF [V/rs ]O )is used to recompute O. A correction (2Ω ∧ ak)(Oimproved ) − (2Ω ∧ k)(O) is analyticallycomputed and added to wind equation to ﬁnd the “improved" value of VF (t + ∆t). In the
deep layer equations and in the cases where a multiplicative factor rs /a is required, thisfactor remains interpolated at O and is never re-interpolated at the reﬁned origin point(this is too tricky to code and too expensive also). Computations are currently made inroutine LAPINEB and LADINE.

∗ Options L2TLFF for RW2TLFF between 0 and 1: In this case theimproved position ON of the origin point O is computed as a linear interpolation between
O and its position for RW2TLFF=1 on a great circle bow on geographical sphere. For
RW2TLFF=0. ON =O. For an idealised straight displacement with a wind of constantacceleration and constant direction, one can show that the “exact" position of the origin
point is given by RW2TLFF=1/3 (Yessad, internal paper in French).

6.4 Remarks.
∗ Shallow water model: Computations remain valid for horizontal coordinates(there is no vertical movement, equations containing η have not to be considered). There
are remaining some old features (compute M in LARMES2 even for LPC_FULL=.T.,
LELTRA=.T. or LSETTLST=.T., then O in LAINOR2).
∗ Trajectory going out of atmosphere for trajectories ending at alayer final point: If the origin point O is found above the top of the atmosphere(resp. under the ground) it is put on the top of the atmosphere (resp. on the ground).
Then the position of the medium point is recomputed (when necessary), that gives necessary a point between the bottom and the top of the atmosphere. At last the origin point isbounded by a vertical position ηO between the top of the atmosphere and the layer l = 1,
according to the namelist variable VETAON in NAMDYN (resp. between the layerl = L and the ground, according to the namelist variable VETAOX in NAMDYN).
Upper bound of O is ηO = ηl=1 + (VETAON − 1)(ηl=1 − ηl=0 ). Lower bound of O is
ηO = ηl=L + (1 − VETAOX)(ηl=L − ηl=L ).

∗ Case of interpolations applied to half level variables: That produces
for example in the non-hydrostatic scheme when the half level variable w is advected
instead of the full level vertical divergence variable (option LGWADV=.T.). In this caseone needs to deﬁne an origin O(l) for a half level trajectory (which ends at a half levelﬁnal point F (l)). The following rules are applied to compute such a kind of trajectory:• The horizontal displacement from F (l) is a weighted average of the horizontal displacements from F (l − 1) and F (l) (see below for vertical displacement).
• At the top (resp. bottom) the horizontal displacement from F (l = 0) (resp. F (l =L)) is equal to the horizontal displacement from F (l = 1) (resp. F (l = L)).

80

4. Semi-lagrangian computations• The rule applied to the horizontal displacement is also applied to the vertical displacement for l between 1 and L − 1. For example, if l is between 1 and L − 1:ηO(l) = + 1 −ηO(l) − ηO(l)ηO(l+1) − ηO(l)

ηO(l) +

ηO(l) − ηO(l)ηO(l+1) − ηO(l)ηO(l+1)• A particle coming from the top or the bottom has no vertical displacement (thatassumes that η̇ = 0 at the top and the bottom, and so that excludes the options
LRUBC=.T., NDPSFI=1 and Πtop > 0 which are not consistent with the constraint (η̇top = 0; η̇surf = 0)).
• If the trajectory goes above the top of the atmosphere, it is bounded at the top of
the atmosphere.
• If the trajectory goes below the surface, it is bounded at the surface.

• Computation of the position of O(l) is done in routine LARCINHA.

∗ Plane geometry (ALADIN): Computation of the SL trajectory is madeon the projected plane geometry. ELARMES and ELARMES2 are called insteadof LARMES and LARMES2.
∗ Treatment of η̇ in the upper stratosphere: For some options of the code
the horizontal interpolations applied on η̇ are replaced by “least-square" interpolations;that allows to remove some instabilities. See documentation (IDSVTSM) for more details.

4. Semi-lagrangian computations

81

7 The SL discretisation of the 2D shallow-water system of equations (spherical geometry).
7.1 Momentum equation.
∗ Definition of X, A and B.
~ + δ ~ (2Ω~ ∧ ~r)
X=V
V

(101)

~ ∧V
~ )] − ∇Φ
A = [−2(1 − δV~ )(Ω(102)

B = −∇Φ(103)∗ Remarks.
• In the cycle 37T1 of ARPEGE/IFS Coriolis term can be treated explicitly (δV~ = 0)
or implicitly (δV~ = 1). Use switch LADVF in namelist NAMDYN. LADVF=.F.
corresponds to (δV~ = 0). LADVF=.T. corresponds to (δV~ = 1). If LADVF=.T.,term (2Ω ∧ r) is analytically computed.

• For a limited set of options, term (2Ω ∧ r) can be recomputed at an improved
position of the origin point (RW2TLFF>0 in NAMDYN).
• Coriolis term can also be put in the semi-implicit scheme by tuning βCo (which
has a sense only if LADVF=.F.). Values βCo = 0 (LIMPF=.F. in namelist
NAMDYN) and βCo = 1 (LIMPF=.T.) are available in the cycle 37T1. Caution:
do not use LIMPF=.T. in variable resolution in cycle 37T1 (formulation of spectral
computations is not correct in this case for the semi-implicit scheme).
~ ∧V~ )] − ∇Φ) − β(−∇Φ) is zero.
• If β=1, the non linear term ([−2(1 − δV~ )(Ω

7.2 Continuity equation.
Conventional formulation (positive value of NVLAG in the namelist
NAMDYN).
∗ Definition of X, A and B.
X = (Φ − Φs )(104)

~ ∇(Φs )A = −(Φ − Φs )D + δT R V

(105)

2

B = −Φ∗ M D′

δT R = 1 plays the same role as the “Tanguay-Ritchie" modiﬁcation for 3D model.

(106)

82

4. Semi-lagrangian computations

Lagrangian formulation (negative value of NVLAG in the namelist
NAMDYN).
∗ Definition of X, A and B.
X = (Φ − Φs )J

(107)

A=0

(108)
2

′

B = −Φ∗ M D J(109)

where J is a “Jacobian" quantity deﬁned by its Lagrangian derivative (see equation (12)).

∗ Calculation of J if 3TL scheme. Jacobian quantities are computed by:
J − = (1 − ∆tDo )/(1 + ∆tDo ), J o = 1/(1 + ∆tDo ) and J + = 1.

∗ Discretisation if 2TL scheme. In cycle 37T1 of ARPEGE/IFS the option
NVLAG=-2 is not coded for the 2TLSL scheme.

7.3 Quantities to be interpolated.
When researching the medium point by an iterative algorithm, the interpolation at themedium point (or the origin point LSETTLST=.T. in NAMDYN) of the two components of the horizontal wind is needed: a 12 points interpolation is used. For otherquantities to be interpolated, see section 5. For more details about interpolations, seesection 11.

4. Semi-lagrangian computations

83

8 The SL discretisation of the 3D primitive equation
model.
Remark: the detailed discretisation of each part of the RHS is given in the documentation(IDEUL). Some notations used in the expression of linear term B (like τ , γ, µ, ν) are givenin the documentation (IDSI).

8.1 Thin layer formulation of the momentum equation.
∗ Definition of X, A, B and F, top and bottom values.
~ + δ ~ (2Ω~ ∧ ~r)
X=V
V

(110)

A = −2(1 − δV~ )(Ω ∧ V) − ∇Φ − RT ∇(log Π)

(111)





Rd T ∗~ ∧V~ )]B = −∇ γT +
Πs + βCo [−2(1 − δV~ )(Ω
Πs ∗

(112)

F = FV(113)

Vη=0 = Vl=1

(114)

Top:

Bottom if δm = 0:Vη=1 = Vl=L

(115)

Bottom if δm = 1:Vη=1 = 0(116)∗ Remarks.
• In the cycle 37T1 of ARPEGE/IFS Coriolis term can be treated explicitly (δV~ = 0)
or implicitly (δV~ = 1). Use switch LADVF in namelist NAMDYN. LADVF=.F.
corresponds to (δV~ = 0). LADVF=.T. corresponds to (δV~ = 1). If LADVF=.T.,term (2Ω ∧ r) is analytically computed.

• For a limited set of options, term (2Ω ∧ r) can be recomputed at an improved
position of the origin point (RW2TLFF>0 in NAMDYN).
• Coriolis term can also be put in the semi-implicit scheme by tuning βCo (which
has a sense only if LADVF=.F.). Values βCo = 0 (LIMPF=.F. in namelist
NAMDYN) and βCo = 1 (LIMPF=.T.) are available in the cycle 37T1. Caution:
do not use LIMPF=.T. in variable resolution in cycle 37T1 (formulation of spectral
computations is not correct in this case for the semi-implicit scheme).

84

4. Semi-lagrangian computations

8.2 White and Bromley deep layer formulation of the momentum equation.
∗ Definition of X, A, B and F, top and bottom values.
~ + δ ~ (2Ω~ ∧ ~r)X=V
V
A = (1 − δV~ )(−2Ω ∧ V − 2Ω ∧ W k) −
′

(117)
W
V − ∇Φ − (RT + µs Rd Tr )∇(log Π)(118)rs

B = −M ∇ [γT + Rd T ∗ log(Πs )] + βCo [−2(1 − δV~ )(Ω ∧ V)](119)

F = FV(120)

Vη=0 = Vl=1

(121)

Top:

Bottom if δm = 0:Vη=1 = Vl=L

(122)

Bottom if δm = 1:Vη=1 = 0(123)∗ Remarks.
• In the cycle 37T1 of ARPEGE/IFS Coriolis term can be treated explicitly (δV~ = 0)
or implicitly (δV~ = 1). Use switches LADVF and LADVFW in namelist NAMDYN. (LADVF;LADVW)=(.F.;.F.) corresponds to (δV~ = 0). (LADVF;LADVW)=(.T.;.T.)
corresponds to (δV~ = 1). If (LADVF;LADVW)=(.T.;.F.) term (2Ω ∧ ak) is
treated implicitly and the remaining Coriolis term is treated explicitly. If (LADVF;LADVW)=(.F.;.T.)
term (2Ω∧(rs −a)k) is treated implicitly and the remaining Coriolis term is treatedexplicitly. If at least LADVF=.T. or LADVFW=.T., term (2Ω ∧ ak) is analytically computed. If LADVFW=.T., rs /a (at t − ∆t if SL3TL or t if SL2TL) has to
be interpolated at O, and rs /a has also to be computed at t+∆t using a provisional
value of log Πs at t + ∆t.
• For a limited set of options, term (2Ω ∧ ak) can be recomputed at an improved
position of the origin point (RW2TLFF>0 in NAMDYN).
• The horizontal part of the Coriolis term can also be put in the semi-implicit
scheme by tuning βCo (which has a sense only if LADVF=.F.). Values βCo = 0(LIMPF=.F. in namelist NAMDYN) and βCo = 1 (LIMPF=.T.) are available
in the cycle 37T1. Caution: do not use LIMPF=.T. in variable resolution incycle 37T1 (formulation of spectral computations is not correct in this case for thesemi-implicit scheme). The combination (LIMPF;LADVW)=(.T.;.T.) is possible
but does not eliminate completely a residual explicit term linked to Coriolis force
in the RHS of the momentum equation.

4. Semi-lagrangian computations

85

8.3 Thermodynamic equation.
∗ Definition of X, A, B and F, top and bottom values.
X = T + δT R

αT Φ sRd TST

(124)

Φs
αT ~
RT ω
A=
V ∇(Φs ) + δT R
+ δT Rcp Π
Rd TST
Rd TST
2



dαT
η̇
dη



(125)

′

B = −τ (M D )(126)

F = FT(127)

Tη=0 = Tl=1

(128)

Top:

Bottom if δm = 0:(129)

Tη=1 = Tl=L
Bottom if δm = 1 (output of physics):(130)

Tη=1 = Ts

8.4 Thin layer formulation of the continuity equation.
∗ Definition of X, A, B, and F.
X = log Πs + δT R

1
A=−
Πs

Z η=1
η=0

Φs
Rd Tst



~ ∂Π
∇ V
∂η

(131)



h

~ ∇ log Πs + δT R Φsdη + V
Rd Tst

i

(132)

2

B=−
F=



M
νD
M2

(133)



(134)

Fm
Πs

∗ Remarks:• A is a sum of 3D terms (the advection term) and 2D terms (the other terms).
• B and F are 2D terms (vertical integrals).

86

4. Semi-lagrangian computations

8.5 White and Bromley deep layer formulation of the continuity equation.
∗ Definition of X, A, B, and F.
X = log Πs + δT R

Φs
Rd Tsth 2i

Z η=1 h

A=−

1

a

rs 2

Πs
η=1

2

η=0

(135)

rs
a

∇

i

rs
a

V

∂Π
∂η



dη+

a
rs

V

h

rs
a

∇

ih

log Πs + δT R′

B = −ν(M D )
F=



Fm
Πs



Φs
Rd Tst

i

−

1
Πs

h

η̇

∂Π
∂η

(137)

(138)

∗ Remarks:• A is a sum of 3D terms (the advection term) and 2D terms (the other terms).
• B and F are 2D terms (vertical integrals).

8.6 Moisture equation.
∗ Definition of X, A, B and F, top and bottom values.
X=q

(139)

A=0

(140)

B=0

(141)

F = Fq(142)

qη=0 = ql=1

(143)

Top:

Bottom if δm = 0:qη=1 = ql=L

(144)

Bottom if δm = 1 (see CPQSOL, relative humidity is the same for η = ηL and η = 1):qη=1 = qsurf

(145)

i

+
η=1

1
Πs

h 2i
a

rs 2

η=1

h

rs 2
a2

4. Semi-lagrangian computations

87

8.7 Other advectable GFL variables.
Equations are discretised as for humidity equation. Vertical boundary conditions: quantities are assumed constant above the middle of the upper layer and below the middle ofthe lower layer in case δm = 0; quantities are assumed constant above the middle of theupper layer in case δm = 1; quantities other than q are assumed to be zero at the surface
in case δm = 1.

8.8 Case of lagged physics.
All the previous discretisations have been written with not lagged physics (interpolated
at the origin point O).
• For lagged physics (LAGPHY=.T., LSLPHY=.F.): the previous discretisations
are done without physics, then the provisional(X + − (1 + ǫX )∆t +∆tβB + (1 + ǫX ) βBo )F
2
2

or

∆t +
∆t
βB + (1 + ǫ) βBo )F2
2
is used as input to the lagged physics.
(X + − (1 + ǫ)

• For split physics used at ECMWF (LEPHYS=.T., LAGPHY=.T., LSLPHY=.T.):
one part of the physics is interpolated at the origin point (t or t−∆t physics according to LTWOTL), the remainder is evaluated at the ﬁnal point (t + ∆t physics).
The physical contribution is put in a separate interpolation buﬀer (name P(X)P9)
and tri-linearly interpolated. The way to compute the non-lagged contribution
is diﬀerent than the way used at METEO-FRANCE: compute it at the previous
timestep as a lagged contribution, then saving it (by the routine GPSAVTEND)
from one timestep to the following one (where it is restored by calling the routine GPGETTEND and added to the interpolation buﬀer by calling the routine
GPADDSLPHY). Partition between the non-lagged and lagged contribution isdone by a linear partition of coeﬃcient RSLWX.

8.9 Quantities to be interpolated (computation under subroutine LACDYN).
Research of trajectory.
When researching the medium point by an iterative algorithm, the interpolation at themedium point (or the origin point LSETTLST=.T., LELTRA=.T. or LPC_FULL=.T.)of ([U/rs ], [V /rs ], η̇) is needed: a tri-linear interpolation is performed. For more details
about interpolations, see section 11.

RHS of equations.
The list of quantities to be interpolated has been described in subsections 5.2, 5.3 and 5.4
for each type of equation.

88

4. Semi-lagrangian computations

Additional quantities to be interpolated at the origin point if RW2TLFF>0.
The two components of the [V/rs ] at time t (if 2TL SL scheme) or t−∆t (if 3TL SL scheme)
when not available after the other interpolations (for example if not lagged physics). These
additional interpolations are useless if lagged physics (or adiabatic run) and NWLAG=3.

4. Semi-lagrangian computations89

9 R operator.
9.1 No tilting.
To transport a vector along a trajectory (part of a great circle) from an origin point O
to a ﬁnal point F the following operator ROF is deﬁned:′

V = ROF (V)

(146)

′

′

′

where V has coordinates (u , v ), V has coordinates (u, v), and the relationship between′′(u, v) and (u , v ) is:



′

u
′
v



=



p
−q

q
p



u
v



(147)

where:
p=cos θF cos θO + (1 + sin θF sin θO ) cos(λF − λO )iF iO + j F j O
=
F
O
1+k k
1 + cos φ

(148)

q=

(sin θF + sin θO ) sin(λF − λO )iF j O + j F iO=
1 + kF kO
1 + cos φ(149)(Notations θO ,θF ,λO ,λF ,φ: see section 6.).
p and q verify the following identity:
p2 + q 2 = 1(150)

Computation of p and q is made in subroutine LARCHE.

9.2 Tilting.
′

The coordinates of V and V are linked by the following relationship:



′

u
′
v



=



GN ORDM−GN ORDL

GN ORDLGN ORDM



p
−q

q
p



cos α
sin α

− sin α
cos α



u
v

where:
cos α =

2c[sin θp cos θO − sin θO cos θp cos(λO − λp )]A cos ΘO(152)

sin α =

2c[cos θp sin(λO − λp )]A cos ΘO

(153)

A = (1 + c2 ) + (1 − c2 )(sin θp sin θO + cos θp cos θO cos(λO − λp ))and where:

(154)



(151)

90

4. Semi-lagrangian computations• c is the stretching coeﬃcient.

• ΘO is the latitude on the computational sphere of the origin point O.

• (θp , λp ) are the latitude and longitude on the geographical sphere of the stretchingpole.
• p and q are computed like in the not tilted case (in subroutine LARCHE).

• cos α and sin α are also computed in subroutine LARCHE).

• (GN ORDL, GN ORDM ) are the coordinates in the computational sphere of theunit vector directed towards the true north, computed in subroutine SUGEM2.

9.3 Plane geometry (ALADIN).
The curvature of the Earth is now taken into account in computing an operator ROF
in the routine ELARCHE instead of computing curvature terms. Expressions of p andq are diﬀerent from the ones of ARPEGE and are not detailed here.

4. Semi-lagrangian computations

91

10 Computation of longitudes and latitudes on thecomputational sphere.
For interpolations it is necessary to compute (ΘO , ΛO ), latitude and longitude of theinterpolation point O in the computational sphere. The iterative algorithm allowing to ﬁnd
O gives (θO , λO ), latitude and longitude in the geographical sphere (more exactly sin θO ,cos θO cos λO − λF and cos θO sin λO − λF where (θF , λF ) are the coordinates of the ﬁnal
point on the geographical sphere). Transform formulae giving (Θ, Λ) on the computational
sphere once knowing (θ, λ) on the geographical sphere are given by equations (155) to(157).

sin Θ =

(1 − c2 ) + (1 + c2 )(sin θp sin θ + cos θp cos θ cos(λ − λp ))
A

(155)

cos Θ cos Λ =2c(cos θp sin θ − sin θp cos θ cos(λ − λp ))
A

(156)

cos Θ sin Λ =2c cos θ sin(λ − λp )
A

(157)

where:• A = (1 + c2 ) + (1 − c2 )(sin θp sin θ + cos θp cos θ cos(λ − λp ))• c is the stretching coeﬃcient.

• (θp , λp ) are the latitude and longitude on the geographical sphere of the stretchingpole.
• Computation of Θ, Λ is made in subroutine LARCHE.

∗ Plane geometry (ALADIN): The SL trajectory is already computed on thecomputational grid, so equivalent transformation formulae from geographical space tocomputational space are useless.

92

4. Semi-lagrangian computations

11 Interpolations and weights computations.
11.1 Interpolation grid and weights (subroutine LASCAW).
Horizontal interpolation grid and weights for bi-linear interpolations.
∗ Definitions: A 16 points horizontal grid is deﬁned as it is shown in ﬁgure 11.1,
but only 4 of these 16 points are used in the interpolations. The interpolation point O(medium or origin point) is between B1 , C1 , B2 and C2 . Λ and Θ are the longitudes andlatitudes on the computational sphere. The following weights are deﬁned as follows:• zonal weight number 1:• zonal weight number 2:• meridian weight:

ZDLO1 =

ΛO − ΛB1
ΛC1 − ΛB1

ZDLO2 =

ΛO − ΛB2
ΛC2 − ΛB2

ZDLAT =

ΘO − ΘB1
ΘB2 − ΘB1

∗ Computations:• The weights ZDLO1 and ZDLO2 are computed then stored in the array PDLO.
• The weight ZDLAT is computed then stored in the array PDLAT.

• The memory address (in SL arrays) of the data concerning the points A1 and A2
are computed then stored in the array KL0 or KLH0. Memory address of the data
concerning the points B1 , B2 , C1 and C2 can be easily computed in interpolations
routines knowing these ones of A1 and A2 .
• Interpolations use data of points B1 , B2 , C1 and C2 .

Vertical interpolation grid and weights for vertical linear interpolations.
∗ Definitions: A 4 points vertical grid is deﬁned as it is shown in ﬁgure 11.2, but
only 2 of these 4 points are used in the interpolations. The interpolation point O (mediumor origin point) is between Tl+1 and Tl+2 . The vertical weight is deﬁned by:
ZDV ER =ηO − ηTl+1
ηTl+2 − ηTl+1∗ Computations:• The weight ZDV ER is computed then stored in the array PDVER.
• The level number l of Tl is stored in the array KLEV.
• Interpolations use data of points Tl+1 and Tl+2 .

∗ Remark: The same formulae and computations are valid for half level data, simply
replace the layer index l by the half level index l.

4. Semi-lagrangian computations

93

Horizontal interpolation grid and weights for 12 points cubic interpolations.
∗ Definitions: A 16 points horizontal grid is deﬁned as it is shown in ﬁgure 11.3,
but only 12 of these 16 points are used in the interpolations. The interpolation point O(medium or origin point) is between B1 , C1 , B2 and C2 . The following weights are deﬁned
as follows:• zonal linear weights for latitudes 0, 1, 2, 3:
ZDLO0 =

ΛO − ΛB0
ΛC0 − ΛB0

ΛO − ΛB1
ΛC1 − ΛB1
ΛO − ΛB2
ZDLO2 =
ΛC2 − ΛB2
ΛO − ΛB3
ZDLO3 =
ΛC3 − ΛB3
• zonal cubic weights for latitude 1:
ZDLO1 =ZCLO11 = f1 (ZDLO1)ZCLO12 = f2 (ZDLO1)ZCLO13 = f3 (ZDLO1)

where:– f1 (α) = (α + 1)(α − 2)(α − 1)/2
– f2 (α) = −(α + 1)(α − 2)α/2– f3 (α) = α(α − 1)(α + 1)/6

• zonal cubic weights for latitude 2:

ZCLO21 = f1 (ZDLO2)
ZCLO22 = f2 (ZDLO2)
ZCLO23 = f3 (ZDLO2)• meridian cubic weights:
ZCLA1 =

(ΘO − ΘB0 )(ΘO − ΘB2 )(ΘO − ΘB3 )(ΘB1 − ΘB0 )(ΘB1 − ΘB2 )(ΘB1 − ΘB3 )(ΘO − ΘB0 )(ΘO − ΘB1 )(ΘO − ΘB3 )(ΘB2 − ΘB0 )(ΘB2 − ΘB1 )(ΘB2 − ΘB3 )(ΘO − ΘB0 )(ΘO − ΘB1 )(ΘO − ΘB2 )ZCLA3 =
(ΘB3 − ΘB0 )(ΘB3 − ΘB1 )(ΘB3 − ΘB2 )ZCLA2 =∗ Computations:• The zonal linear weights ZDLO0, ZDLO1, ZDLO2 and ZDLO3 are computed
then stored in the array PDLO.
• The zonal cubic weights ZCLO11, ZCLO12, ZCLO13, ZCLO21, ZCLO22, ZCLO23
are computed then stored in the array PCLO.
• The meridian cubic weights ZCLA1, ZCLA2 and ZCLA3 are computed thenstored in the array PCLA. One can notice that the denominators of ZCLA1,
ZCLA2 and ZCLA3 do not depend on coordinates of O and can be pre-computedin the subroutine SULEG, in array RIPI.
• The memory address (in SL arrays) of the data concerning the points A0 , A1 , A2
and A3 are stored in the array KL0 or KLH0. Once knowing these addresses one
can easily retrieve the other points addresses in the interpolations routines.
• Interpolations use data of the following 12 points B0 , C0 , A1 , B1 , C1 , D1 , A2 , B2 ,C2 , D2 , B3 and C3 .

94

4. Semi-lagrangian computations∗ Extension to diffusive interpolations (SLHD): The way to take accountof the diﬀusive properties of interpolations has been redesigned, and now it is completely
contained in the calculation of cubic weights: we pass from conventional cubic interpolations to diﬀusive SLHD cubic interpolations simply by changing the way of computing
the cubic weights. The comprehensive way of computing the SLHD cubic weights is notprovided in this documentation (because it leads to rather tricky formulae), but details
about the calculations can be found in documentation (IDSLIF2) and looking in routine
LASCAW (ELASCAW in ALADIN). We just give a sum-up of the way to compute the
cubic weights.
• Meridian diﬀusive cubic weights:
ZCLA1slhd = ZCLA1 + ZIN CRM 1
ZCLA2slhd = ZCLA2 + ZIN CRM 2
ZCLA3slhd = ZCLA3 + ZIN CRM 3
where each of increments ZIN CRM 1, ZIN CRM 2 and ZIN CRM 3 is a linear
combination of cubic Lagrangian weights ZCLA1, ZCLA2 and ZCLA3 and diﬀusive weights (linear for LSLHD_OLD=.T., quadratic otherwise).
Coeﬃcients of these linear combinations depend on:– Constants in time Csldw , describing Laplacian smoother.
– A pre-computed quantity κ (computed in GP_KAPPA) depending on horizontal ﬂow deformation, ranging from 0 to 1; κ is equal to 1 if LSLHD_STATIC=T.
– Lower and upper bounds (κmin and κmax ) respectively stored in variables
SLHDKMIN and SLHDKMAX, used to construct limit interpolators (0- cubic Lagrange, 1 - linear/quadratic; they are not restricted to range 0-1).
• Zonal diﬀusive cubic weights for latitude number 1:
ZCLO11slhd = ZCLO11 + ZIN CRL11
ZCLO12slhd = ZCLO12 + ZIN CRL12ZCLO13slhd = ZCLO13 + ZIN CRL13
where each of increments ZIN CRL11, ZIN CRL12 and ZIN CRL13 is a linear
combination of cubic Lagrangian weights ZCLO11, ZCLO12 and ZCLO13 anddiﬀusive weights (linear for LSLHD_OLD=.T., quadratic otherwise).
Coeﬃcients of these linear combinations depend on:– Constants like Cslhdepsh .
– κ (see above).
– Lower and upper bounds κmin and κmax (see above).
• Zonal diﬀusive cubic weights for latitude number 2:
ZCLO21slhd = ZCLO21 + ZIN CRL21
ZCLO22slhd = ZCLO22 + ZIN CRL22
ZCLO23slhd = ZCLO23 + ZIN CRL23
where each of increments ZIN CRL21, ZIN CRL22 and ZIN CRL23 is a linear
combination of cubic Lagrangian weights ZCLO21, ZCLO22 and ZCLO23 anddiﬀusive weights (linear for LSLHD_OLD=.T., quadratic otherwise).
Coeﬃcients of these linear combinations depend on Cslddepsh , κ, κmin , κmax like thezonal weights for latitude number 1.
Several options of SLHD smoothing are available, according to values of the keys LSLHDQUAD and LSLHD_OLD.
In part 11.2, cubic interpolations are written with conventional cubic weights: replace
conventional cubic weights by their SLHD counterpart for diﬀusive cubic interpolations.

4. Semi-lagrangian computations

95∗ Extension to 3D-turbulence: not described in detail; additional cubic weights
are computed if L3DTURB=T.

Vertical interpolation grid and weights for vertical cubic 4 points
interpolations.
A 4 points vertical grid is deﬁned as it is shown in ﬁgure 11.2. The interpolation point O(medium or origin point) is between Tl+1 and Tl+2 . The vertical weights are deﬁned by:
ZCV E1 =(ηO − ηTl )(ηO − ηTl+2 )(ηO − ηTl+3 )(ηTl+1 − ηTl )(ηTl+1 − ηTl+2 )(ηTl+1 − ηTl+3 )ZCV E2 =(ηO − ηTl )(ηO − ηTl+1 )(ηO − ηTl+3 )(ηTl+2 − ηTl )(ηTl+2 − ηTl+1 )(ηTl+2 − ηTl+3 )ZCV E3 =(ηO − ηTl )(ηO − ηTl+1 )(ηO − ηTl+2 )(ηTl+3 − ηTl )(ηTl+3 − ηTl+1 )(ηTl+3 − ηTl+2 )∗ Computations:• The vertical weights ZCV E1, ZCV E2 and ZCV E3 are computed then stored in
the array PVINTW. One can notice that the denominators of ZCV E1, ZCV E2and ZCV E3 do not depend on coordinates of O and can be pre-computed in thesubroutine SUVERT, in the array VCUICO.
• The level number l of Tl is stored in the array KLEV.
• Interpolations use data of points Tl , Tl+1 , Tl+2 and Tl+3 .

∗ Remark: The same formulae and computations are valid for half level data, simply
replace the layer index l by the half level index l, and use array VCUICOH.
∗ Extension to diffusive interpolations (SLHD): Like we do for horizontal
weights, the semi-Lagrangian diﬀusion can be taken into account by modifying vertical
cubic weights as follow:
ZCV E1slhd = ZCV E1 + ZIN CRV 1
ZCV E2slhd = ZCV E2 + ZIN CRV 2
ZCV E3slhd = ZCV E3 + ZIN CRV 3
Expressions giving vertical increments ZIN CRV 1 to ZIN CRV 3 have a shape similar to
those of horizontal increments, and still depend on quantities like κ, κmin , κmax .
In part 11.2, cubic interpolations are written with conventional cubic weights: replace
conventional cubic weights by their SLHD counterpart for diﬀusive cubic interpolations.

Vertical interpolation grid and weights for vertical cubic Hermite
interpolations.
This part is valid only for interpolations of full level data.
A 4 points vertical grid is deﬁned as it is shown in ﬁgure 11.2. The interpolation point O(medium or origin point) is between Tl+1 and Tl+2 .
First weights to compute vertical derivatives at layers l + 1 and l + 2 are computed.
For a variable X, ∂X/η̇, but with additional
is computed as close as possible as η̇ ∂X
∂η
∂η

.
approximations allowing to avoid horizontal interpolations for term η̇ ∂Π∂η96

4. Semi-lagrangian computations• For layers other than the ﬁrst or the last layer, discretisation follows:



∂X
∂η



=
l+1

0.5(Xl+2 − Xl )
ηl+1 − ηl

(158)



= 0 (valid only if LRUBC=.F.);• For layer l = 1, discretisation assumes that η̇ ∂Π∂η l=0discretisation follows:



∂X
∂η



=
l=1

(Xl=2 − Xl=1 )ηl=1 − ηl=0

(159)

• For layer l = L, discretisation assumes that η̇ ∂Π
∂η



∂X
∂η



=
l=L

(Xl=L − Xl=L−1 )ηl=L − ηl=L−1

l=L= 0; discretisation follows:(160)

The following weights are computed:• For an interpolation point included between layers 2 and L − 1 (l ≥ 1 and l ≤ L − 3):
0.5(ηl+2 − ηl+1 )V DERW 11 =
ηl+1 − ηl
V DERW 21 =0.5(ηl+2 − ηl+1 )ηl+1 − ηl

V DERW 12 =0.5(ηl+2 − ηl+1 )ηl+2 − ηl+1

V DERW 22 =0.5(ηl+2 − ηl+1 )ηl+2 − ηl+1

• For an interpolation point included between layers 1 and 2:
V DERW 11 = 0
V DERW 21 =(ηl=2 − ηl=1 )ηl=1 − ηl=0

V DERW 12 =0.5(ηl=2 − ηl=1 )ηl=2 − ηl=1

V DERW 22 =0.5(ηl=2 − ηl=1 )ηl=2 − ηl=1

such case is extended to the case where the interpolation point is between the top
and the ﬁrst layer; in this case the interpolation becomes an extrapolation.
• For an interpolation point included between layers L − 1 and L:
V DERW 11 =0.5(ηl=L − ηl=L−1 )ηl=L−1 − ηl=L−2

V DERW 21 =0.5(ηl=L − ηl=L−1 )ηl=L−1 − ηl=L−2

V DERW 12 =(ηl=L − ηl=L−1 )ηl=L − ηl=L−1

V DERW 22 = 0such case is extended to the case where the interpolation point is between the lastlayer and the ground; in this case the interpolation becomes an extrapolation.

4. Semi-lagrangian computations

97∗ Computations:• The vertical weights V DERW 11, V DERW 21, V DERW 12 and V DERW 22 arecomputed then stored in the array PVDERW.
• The weight ZDV ER is computed then stored in the array PDVER (see subsubsection 11.1).
• Functions fH1 (ZDV ER) to fH4 (ZDV ER) (involved in any Hermite cubic interpolation), where:– fH1 (α) = (1 − α)2 (1 + 2α)– fH2 (α) = α2 (3 − 2α)– fH3 (α) = α(1 − α)2– fH4 (α) = −α2 (1 − α)are computed and stored in array PHVW.
• The level number l of Tl is stored in the array KLEV.
• Interpolations use data of points Tl , Tl+1 , Tl+2 and Tl+3 .

Vertical interpolation grid and weights for vertical cubic splineinterpolations.
This part is valid only for interpolations of full level data.
A 4 points vertical grid is deﬁned as it is shown in ﬁgure 11.2. The interpolation point O(medium or origin point) is between Tl+1 and Tl+2 . The algorithm of calculation of thevertical weights will be given in a later version of the documentation.

∗ Computations:• Vertical interpolation is the product of two operators. The ﬁrst one uses all thelayers and is done in the unlagged part of the grid-point calculations by a routine
VSPLTRANS and needs to compute top and bottom values and vertical derivatives of the ﬁeld to be interpolated, and also the inversion of a tridiagonal matrix(routine TRIDIA); for this operation it is necessary to use some coeﬃcients stored
in the arrays RVSPTRI and RVSPC and pre-computed in the set-up subroutine
SUVSPLIP; the original ﬁeld is stored in the buﬀer P(X)L9 and the intermediate
result after this ﬁrst part is stored in the buﬀer P(X)SPL9. The second part uses
4 points and is done in the interpolation routine itself.
• The vertical weights ZCV E0, ZCV E1, ZCV E2 and ZCV E3 necessary for thesecond part are computed then stored in the array PVINTWS. Some part of thecalculations can be pre-computed in the set-up subroutine SUVSPLIP (arrays
RFAA, RFBB, RFCC and RFDD).
• The level number l of Tl is stored in the array KLEV.
• Interpolations use data of points Tl , Tl+1 , Tl+2 and Tl+3 , stored in the intermediatearray P(X)SPL9.

Interpolation grid and weights for tri-linear interpolations.
A 64 points grid is deﬁned as it is shown in ﬁgure 11.4, but only 8 of these 64 points areused in the interpolations. The interpolation point O (medium or origin point) is between
B1,l+1 , C1,l+1 , B2,l+1 , C2,l+1 , B1,l+2 , C1,l+2 , B2,l+2 and C2,l+2 . For the two levels l+1 andl +2 see subsubsection 11.1 corresponding to bi-linear horizontal interpolations for weights
computations. For weights needed for vertical interpolations (ZDV ER) see subsubsection
11.1 corresponding to linear vertical interpolations.
• The memory address (in SL arrays) of the data concerning the points A0,l , A1,l ,A2,l and A3,l is computed then stored in the array KL0. Once knowing theseaddresses one can easily retrieve the other points addresses in the interpolations
routines.
• Interpolations use data of points B1,l+1 , C1,l+1 , B2,l+1 , C2,l+1 , B1,l+2 , C1,l+2 ,B2,l+2 and C2,l+2 .

98

4. Semi-lagrangian computations∗ Remark: The same formulae and computations are valid for half level data, simply
replace the layer index l by the half level index l.
Interpolation grid and weights for 32 points interpolations.
A 64 points grid is deﬁned as it is shown in ﬁgure 11.4, but only 32 (48 if vertical spline
cubic interpolations) of these 64 points are used in the interpolations. The interpolation
point O (medium or origin point) is between B1,l+1 , C1,l+1 , B2,l+1 , C2,l+1 , B1,l+2 , C1,l+2 ,B2,l+2 and C2,l+2 . For the two levels l and l + 3 see subsubsection 11.1 corresponding
to bi-linear horizontal interpolations for weights computations. For the two levels l + 1
and l + 2 see subsubsection 11.1 corresponding to 12 points horizontal interpolations forweights computations. For weights needed for vertical interpolations see subsubsection
11.1 for vertical cubic interpolations, see subsubsection 11.1 for vertical Hermite cubic
interpolations, see subsubsection 11.1 for vertical spline cubic interpolations.
• The memory address (in SL arrays) of the data concerning the points A0,l , A1,l ,A2,l and A3,l is computed then stored in the array KL0. Once knowing theseaddresses one can easily retrieve the other points addresses in the interpolations
routines.
• Interpolations use data of the following 32 points B1,l , C1,l , B2,l , C2,l , B0,l+1 ,C0,l+1 , A1,l+1 , B1,l+1 , C1,l+1 , D1,l+1 , A2,l+1 , B2,l+1 , C2,l+1 , D2,l+1 , B3,l+1 , C3,l+1 ,
B0,l+2 , C0,l+2 , A1,l+2 , B1,l+2 , C1,l+2 , D1,l+2 , A2,l+2 , B2,l+2 , C2,l+2 , D2,l+2 , B3,l+2 ,C3,l+2 , B1,l+3 , C1,l+3 , B2,l+3 and C2,l+3 .

∗ Remarks:• The same formulae and computations are valid for half level data, simply replace
the layer index l by the half level index l.
• For vertical spline cubic interpolations a 12 points grid is used on each level l, l + 1,
l + 2, l + 3 (total = 48 points used).

Horizontal interpolation grid and weights for 16 points linear leastsquare fit interpolations.
A 16 points horizontal grid is deﬁned as it is shown in ﬁgure 11.3. The interpolation point
O (medium or origin point) is between B1 , C1 , B2 and C2 . The interpolation is replaced
by a linear least-square ﬁt minimisation of a ﬁrst order polynomial in each direction (ﬁrstzonal interpolations, then meridian interpolations).
The weights used are the same ones as for the 4 points bilinear horizontal interpolation,
but the zonal linear weights are required for the 4 latitudes of the 16 points grid (ZDLO0,
ZDLO1, ZDLO2, ZDLO3, ZDLAT ). The actual weights used in one direction (forexample the meridian direction) are respectively: 0.4 − 0.3ZDLAT , 0.3 − 0.1ZDLAT ,0.2 + 0.1ZDLAT , 0.1 + 0.3ZDLAT .

∗ Computations:• The weights ZDLO0, ZDLO1, ZDLO2 and ZDLO3 are computed then stored in
the array PDLO.
• The meridian weight ZDLAT is computed then stored in the array PDLAT.

• The memory address (in SL arrays) of the data concerning the points A0 , A1 , A2
and A3 are stored in the array KL0 or KLH0. Once knowing these addresses one
can easily retrieve the other points addresses in the interpolations routines.
• Interpolations use data of the following 16 points A0 , B0 , C0 , D0 , A1 , B1 , C1 , D1 ,
A 2 , B 2 , C 2 , D2 , A 3 , B 3 , C 3 , D 3 .

4. Semi-lagrangian computations

99

Horizontal interpolation grid and weights for 32 points linear leastsquare fit interpolations.
A 32 points grid is deﬁned as it is shown in ﬁgure 11.4 where only the two intermediate
layers are retained. The interpolation point O (medium or origin point) is between B1,l+1 ,C1,l+1 , B2,l+1 , C2,l+1 , B1,l+2 , C1,l+2 , B2,l+2 and C2,l+2 . For the two levels l + 1 andl + 2 see subsubsection 11.1 corresponding to 16 points linear least-square ﬁt horizontal
interpolations for weights computations. For weights needed for vertical interpolations,only the linear weight ZDV ER is required; see subsubsection 11.1 for vertical linear
interpolations.
• The memory address (in SL arrays) of the data concerning the points A1,l and A2,lis computed then stored in the array KL0. Once knowing these addresses one caneasily retrieve the other points addresses in the interpolations routines.
• Interpolations use data of the following 32 points A0,l+1 , B0,l+1 , C0,l+1 , D0,l+1 ,A1,l+1 , B1,l+1 , C1,l+1 , D1,l+1 , A2,l+1 , B2,l+1 , C2,l+1 , D2,l+1 , A3,l+1 , B3,l+1 , C3,l+1 ,D3,l+1 , A0,l+2 , B0,l+2 , C0,l+2 , D0,l+2 , A1,l+2 , B1,l+2 , C1,l+2 , D1,l+2 , A2,l+2 , B2,l+2 ,C2,l+2 , D2,l+2 , A3,l+2 , B3,l+2 , C3,l+2 , D3,l+2 .

∗ Remarks:• The same formulae and computations are valid for half level data, simply replace
the layer index l by the half level index l.

Plane geometry (ALADIN).
All previous formulae for weight computation can be used for an irregular latitude
spacing and a diﬀerent number of points on each longitude. The ALADIN grid has ahorizontal regular spacing, so the previous formulae can be simpliﬁed and array RIPI is
no longer necessary. ELASCAW is called instead of LASCAW and is cheaper in CPU
time.

11.2 Interpolations.
Bilinear interpolation (subroutine LAIDLI).
See ﬁgure 11.1 and subsubsection 11.1 for deﬁnition of ZDLO1, ZDLO2, ZDLAT andpoints B1 , C1 , B2 and C2 .
For a quantity X, are computed successively:• a linear interpolation on the longitude number 1:
X1 = XB1 + ZDLO1(XC1 − XB1 ).
• a linear interpolation on the longitude number 2:
X2 = XB2 + ZDLO2(XC2 − XB2 ).
• a meridian linear interpolation:
Xinterpolated = X1 + ZDLAT (X2 − X1 ).

Tri-linear interpolation (subroutine LAITLI).
For layers l + 1 and l + 2 (see ﬁgure 11.4) bilinear horizontal interpolations give twointerpolated values Xl+1 and Xl+2 (see subsubsection 11.2). Then the ﬁnal interpolated
value is given by the following expression:
Xinterpolated = Xl+1 + ZDV ER(Xl+2 − Xl+1 )100

4. Semi-lagrangian computations∗ Remark: The same formulae and computations are valid for half level data, simply
replace the layer index l by the half level index l.
Horizontal 12 points interpolation (subroutine LAIDDI).
See ﬁgure 11.3 and subsubsection 11.1 for deﬁnition of ZDLO0, ZDLO1, ZDLO2,
ZDLO3, ZCLA1, ZCLA2 and ZCLA3 and points B0 , C0 , A1 , B1 , C1 , D1 , A2 , B2 , C2 ,D2 , B3 and C3 .
For a quantity X, are computed successively:• a linear interpolation on the longitude number 0:
X0 = XB0 + ZDLO0(XC0 − XB0 ).

• a cubic 4 points interpolation on the longitude number 1:
X1 = XA1 + ZCLO11(XB1 − XA1 ) + ZCLO12(XC1 − XA1 ) + ZCLO13(XD1 − XA1 ).

• a cubic 4 points interpolation on the longitude number 2:
X2 = XA2 + ZCLO21(XB2 − XA2 ) + ZCLO22(XC2 − XA2 ) + ZCLO23(XD2 − XA2 ).
• a linear interpolation on the longitude number 3:
X3 = XB3 + ZDLO3(XC3 − XB3 ).

• a meridian cubic 4 points interpolation:
Xinterpolated = X0 + ZCLA1(X1 − X0 ) + ZCLA2(X2 − X0 ) + ZCLA3(X3 − X0 ).

There is a shape-preserving option: after cubic 4 points interpolations on longitudes
number 1 and 2, X1 is bounded between XB1 , and XC1 and X2 is bounded between XB2
and XC2 ; after meridian cubic 4 points interpolation Xinterpolated is bounded between X1
and X2 . Use of switches LQMW (momentum equation), LQMT (temperature equation), LQMP (continuity equation), LQMSPD (pressure departure variable equation),
LQMSVD (vertical divergence equation), Y[X]_NL%LQM for GFL variables allow to
use shape-preserving option.

Cubic 4 points vertical interpolation.
See ﬁgure 11.2 and subsubsection 11.1 for deﬁnition of ZCV E1, ZCV E2 and ZCV E3.
The cubic 4 points vertical interpolation gives the ﬁnal interpolated value:
Xinterpolated = Xl + ZCV E1(Xl+1 − Xl ) + ZCV E2(Xl+2 − Xl ) + ZCV E3(Xl+3 − Xl )∗ Remark: The same formulae and computations are valid for half level data, simply
replace the layer index l by the half level index l.
Cubic Hermite vertical interpolation.
This part is valid for interpolations of full level variables. See ﬁgure 11.2 and subsubsection 11.1 for deﬁnition of V DERW 11, V DERW 21, V DERW 12 and V DERW 22.
See subsubsection 11.1 for deﬁnition of ZDV ER. See subsubsection 11.1 for deﬁnition offunctions fH1 to fH4 . The cubic Hermite vertical interpolation gives the ﬁnal interpolated
value:
Xinterpolated = fH1 (ZDV ER)Xl+1 + fH2 (ZDV ER)Xl+2+fH3 (ZDV ER)(V DERW 11(Xl+1 − Xl ) + V DERW 21(Xl+2 − Xl+1 ))+fH4 (ZDV ER)(V DERW 12(Xl+2 − Xl+1 ) + V DERW 22(Xl+3 − Xl+2 ))4. Semi-lagrangian computations

101

Spline cubic 4 points vertical interpolation.
This part is valid for interpolations of full level variables. See ﬁgure 11.2 and subsubsection 11.1 for deﬁnition of ZCV E0, ZCV E1, ZCV E2 and ZCV E3. The spline cubic
4 points vertical interpolation gives the ﬁnal interpolated value:
Xinterpolated = ZCV E0∗XRPl +ZCV E1∗XRPl+1 +ZCV E2∗XRPl+2 +ZCV E3∗XRPl+3
where XRP is the re-proﬁled version of X available in P(X)SPL9. A monotonic constraint can be added, bounding Xinterpolated between Xl+1 and Xl+2 .

32 points 3D interpolation with vertical cubic 4 points interpolation (subroutine LAITRI).
For layers l and l + 3 (see ﬁgure 11.4) bilinear horizontal interpolations give two interpolated values Xl and Xl+3 (see subsubsection 11.2). For layers l + 1 and l + 2 (see ﬁgure11.4) 12 points horizontal interpolations give two interpolated values Xl+1 and Xl+2 (seesubsubsection 11.2). The ﬁnal interpolated value Xinterpolated is a cubic 4 points verticalinterpolation of Xl , Xl+1 , Xl+2 and Xl+3 (see subsubsection 11.2).
There are shape-preserving options for horizontal or both horizontal and vertical interpolations.
Use of switches LQMW (momentum equation), LQMT (temperature equation), LQMP
(continuity equation), LQMSPD (pressure departure variable equation), LQMSVD(vertical divergence equation), Y[X]_NL%LQM (GFL equations), allows to use shapepreserving option for both horizontal and vertical interpolations.
Use of switches LQMHW (momentum equation), LQMHT (temperature equation),
LQMHP (continuity equation), LQMHSPD (pressure departure variable equation),
LQMHSVD (vertical divergence equation), Y[X]_NL%LQMH (GFL equations), allows to use shape-preserving option for horizontal interpolations.

∗ Remark: The same formulae and computations are valid for half level data, simply
replace the layer index l by the half level index l.
32 points 3D interpolation with vertical cubic Hermite interpolation (subroutine LAIHVT).
This part is valid for interpolations of full level variables. For layers l and l + 3 (seeﬁgure 11.4) bilinear horizontal interpolations give two interpolated values Xl and Xl+3
(see subsubsection 11.2). For layers l + 1 and l + 2 (see ﬁgure 11.4) 12 points horizontal
interpolations give two interpolated values Xl+1 and Xl+2 (see subsubsection 11.2). The
ﬁnal interpolated value Xinterpolated is a cubic Hermite vertical interpolation of Xl , Xl+1 ,Xl+2 and Xl+3 (see subsubsection 11.2).
There are shape-preserving options for horizontal or both horizontal and vertical interpolations.
Use of switch Y[X]_NL%LQM (GFL equations), allows to use shape-preserving option
for both horizontal and vertical interpolations.
Use of switch Y[X]_NL%LQMH (GFL equations), allows to use shape-preserving option for horizontal interpolations.

102

4. Semi-lagrangian computations

48 points 3D interpolation with vertical cubic spline interpolation(subroutine LAITVSPCQM).
This part is valid for interpolations of full level variables. This type of interpolation is
activated for GFL variables when Y[X]_NL%LVSPLIP=.T. in NAMGFL (currently
for ozone only). Contrary to what is done for the other 32 points interpolations routines,
the vertical interpolations are performed ﬁrst. 12 vertical interpolations are done on theverticals matching the following points: B0,l+1 , C0,l+1 , A1,l+1 , B1,l+1 , C1,l+1 , D1,l+1 ,A2,l+1 , B2,l+1 , C2,l+1 , D2,l+1 , B3,l+1 , C3,l+1 . A monotonic constraint is added for the
lower levels (currently the 9 lower levels). The projection horizontal plane on the levelof the interpolation point provides a 12-points grid. A 12 points interpolation is done on
this projection (see part 11.2). A ﬁnal monotonic constraint is added: the interpolated
value of X is bounded between the value of X at the points B1,l+1 , C1,l+1 , B2,l+1 , C2,l+1 ,
B1,l+2 , C1,l+2 , B2,l+2 , C2,l+2 , and the overshoots/undershoots found are dispatched on
upper levels to ensure as possible it can be done conservation properties.

Horizontal 16 points linear least-square fit interpolation.
See ﬁgure 11.3 and subsubsection 11.1 for deﬁnition of ZDLO0, ZDLO1, ZDLO2,
ZDLO3, ZDLAT and points A0 , B0 , C0 , D0 , A1 , B1 , C1 , D1 , A2 , B2 , C2 , D2 , A3 , B3 ,C3 , D3 . Let us deﬁne:• f1 (α) = 0.4 − 0.3α
• f2 (α) = 0.3 − 0.1α
• f3 (α) = 0.2 + 0.1α
• f4 (α) = 0.1 + 0.3α
For a quantity X, are computed successively:• a linear least-square ﬁt 4 points interpolation on the longitude number lon for
lon = 0, 1, 2, 3:
Xlon = +f1 (ZDLOlon )XAlon + f2 (ZDLOlon )XBlon + f3 (ZDLOlon )XClon + f4 (ZDLOlon )XDlon .
• a meridian linear least-square ﬁt 4 points interpolation:
Xinterpolated = +f1 (ZDLAT )X0 + f2 (ZDLAT )X1 + f3 (ZDLAT )X2 + f4 (ZDLAT )X3 .

32 points 3D interpolation with linear least-square fit horizontal
interpolations and vertical linear interpolations (subroutine LAISMOO).
For layers l + 1 and l + 2 (see ﬁgure 11.4) 16 points linear least-square ﬁt horizontal
interpolations give two interpolated values Xl+1 and Xl+2 (see subsubsection 11.2). The
ﬁnal interpolated value is then given by the following expression:
Xinterpolated = Xl+1 + ZDV ER(Xl+2 − Xl+1 )∗ Remark: In the current usage of this interpolation (for η̇ in the upper stratosphere)
there is an additional smoothing done in routine LAISMOA before the interpolation by
LAISMOO.
∗ For more information: See the internal paper (IDSVTSM).

11.3 Code structures to store weights.
Two structures have been created in module intdyn_mod.F90:• structure TLSCAW for linear weights.
• structure TRSCAW for non-linear weights.

4. Semi-lagrangian computations
Figure 11.1: Interpolation horizontal grid for bilinear interpolations.

103

104

4. Semi-lagrangian computations

A0

B0

✉

✉

C0

D0

✉

✉

✉

A1

B1

✉

✉

C1

D1

✲

✉

✉

lat 0

x1

x1 = ZDLO1
y=0

✉

lat 1

O ✉y = ZDLA

A2

B2

C2

✉

✉

✉ ✉

✲

x2 = ZDLO2 x2
y=1

A3

B3

✉

✉

✉

y

❄

x1 (B1 ) = x2 (B2 ) = 0; x1 (C1 ) = x2 (C2 ) = 1.

D2

lat 2

✉

C3

D3

✉

✉

lat 3

4. Semi-lagrangian computations

105

Figure 11.2: Interpolation vertical grid for linear and cubic vertical interpolations.

106

4. Semi-lagrangian computations

T

✉ l

T

✉ l+1

z=0

level llevel l+1

O ✉z = ZDV ER

T

✉ l+2

z=1

T

✉ l+3

z

❄

level l+2

level l+3

4. Semi-lagrangian computations
Figure 11.3: Interpolation horizontal grid for 12 points interpolations.

107

108

4. Semi-lagrangian computations

A0

B0

✉

✉

C0

x0

x2 = ZDLO0
y = −ZDA

A1

B1

✉

✉

D0

✲

✉

✉

C1

D1

✲

✉

✉

lat 0

✉

x1

x1 = ZDLO1
y = −ZDB✉

lat 1

O ✉y = 0

A2

B2

C2

✉

✉

✉ ✉

✲

x2 = ZDLO2 x2
y = −ZDC

A3

B3

✉

✉

C3
✉

✉

x2 = ZDLO3
y y = −ZDD

D2

lat 2

✉

✲

x3

D3
✉

lat 3

❄

x0 (B0 ) = x1 (B1 ) = x2 (B2 ) = x3 (B3 ) = 0; x0 (C0 ) = x1 (C1 ) = x2 (C2 ) = x3 (C3 ) = 1.

4. Semi-lagrangian computations
Figure 11.4: Interpolation grid for tri-linear and 32 points interpolations.

109

110

r
❜

4. Semi-lagrangian computations– points used in 32 points interpolations.
– points not used in 32 points interpolations .

A2,l
❜
A3,l
❜

A0,l
❜
A1,l
❜
B2,l
r
B3,l
❜

B0,l
❜
B1,l
r
r C2,l
r
C3,l
❜

C0,l
❜
C1,l
r
D2,l
❜
D3,l
❜

D0,l
❜
D1,l
❜

A2,l+1r
A3,l+1❜A0,l+1B0,l+1
C0,l+1D0,l+1❜rr❜
A1,l+1
B
C
D
1,l+1
1,l+1
1,l+1
r
r
r
r
rC2,l+1
B2,l+1D2,l+1rrr
B3,l+1C3,l+1D3,l+1rr❜level llevel l + 1

O r

A2,l+2r
A3,l+2❜A0,l+2B0,l+2C0,l+2D0,l+2❜rr❜
A1,l+2
B
C
D
1,l+2
1,l+2
1,l+2
r
r
r
r
rC2,l+2
B2,l+2D2,l+2rrr
B3,l+2
C
D
3,l+2
3,l+2
r
r
❜

level l + 2

A2,l+3❜A3,l+3❜A0,l+3B0,l+3C0,l+3D0,l+3❜❜❜❜A1,l+3
B
C
D
1,l+3
1,l+3
1,l+3
❜
r
r
❜
rC2,l+3
B2,l+3
D
2,l+3
r
r
❜
B3,l+3C3,l+3D3,l+3❜❜❜level l + 3

4. Semi-lagrangian computations

111

12 Computation of η̇ at full levels.
∗ General expression: η̇ is needed to ﬁnd the height of the medium and origin

points.

η̇ can be written:

η̇ =



η̇

∂Π
∂η



∂η
∂Π

(161)

∗ Discretisation at full levels for LVERTFE=.F.:is provided atη̇ ∂Π∂ηhalf levels, and ∆η and ∆Π are provided at full levels. Discretisation of (161) is:η̇l = 0.5

"

∂Π
η̇
∂η



+
l



∂Π
η̇
∂η



l−1

#

[∆η]l
[∆Π]l

∗ Discretisation at full levels for LVERTFE=.T.:

(162)



η̇ ∂Π
is provided at∂ηfull levels, and ∆η and ∆Π are provided at full levels. Discretisation of (161) is:

η̇l =



η̇

∂Π
∂η



[∆η]l
[∆Π]
l
l

(163)

112

4. Semi-lagrangian computations

13 Lateral boundary conditions.
13.1 Extra longitudes.
Let us denote by LX the number of longitudes (in the array NLOENG for each latitudein the code). For a quantity X, let us deﬁne:X(longitude number 0)=X(longitude number LX).
X(longitude number LX+1)=X(longitude number 1).
X(longitude number LX+2)=X(longitude number 2).
These extra computations are necessary for all interpolated ﬁelds. For distributed memorycomputations are done when making the halo (routine SLCOMM+SLCOMM2A which
exchange data with other processors).

13.2 Extra latitudes.
Let us denote by lx the number of latitudes (NDGLG in the code): latitudes number-1,0,lx + 1,lx + 2 are respectively the symmetric of latitudes number 2,1,lx,lx − 1. These
extra computations are necessary for all interpolated ﬁelds. For distributed memory computations are done in SLEXTPOL.

13.3 Vertical boundary conditions in the 3D model.
∗ Vertical linear interpolations for layer variables at the medium
point: The medium point has a vertical coordinate always included between ηl=0 andηl=L in case of vertical interpolating scheme. Therefore no extrapolated values are needed.

∗ Vertical cubic 4 points interpolations for layer variables at theorigin point: When the origin point is above the layer number 2 (resp. below thelayer number L − 1), the vertical cubic 4 points interpolations using data of the layers
number 1, 2, 3 (resp. L − 2, L − 1, L) and the extra-layer number 0 (resp. L + 1) aredegenerated into linear interpolations between the layers number 1 and 2 (resp. L − 1
and L). The extrapolated values at the extra-layer number 0 (resp. L + 1) are always
multiplied by a weight equal to 0 and are set to 0 in subroutine LAVABO. This algorithm
extends itself to the case where the origin point is between the top (resp. surface) andthe layer number 1 (resp. L), but in this case the interpolation using data of the layers
number 1 and 2 (resp. L − 1 and L) becomes an extrapolation.

∗ Vertical cubic 4 points interpolations for half level variables atthe origin point: When the origin point is above the half level number 1 (resp.

L − 1), the vertical cubic 4 points interpolations using data of the half levels number -1, 0,
1, 2 (resp. L − 2, L − 1, L and L + 1) are degenerated into linear interpolations between
the half levels numbers 0 and 1 (resp. L − 1 and L).

4. Semi-lagrangian computations

113∗ Vertical cubic Hermite interpolations for layer variables at theorigin point: When the origin point is above the layer number 2 (resp. L − 1),interpolation is still a vertical cubic Hermite one, computation of vertical derivatives is
modiﬁed for layer number 1 (resp. L). This algorithm extends itself to the case where theorigin point is between the top (resp. ground) and the layer number 1 (resp. L), but inthis case the interpolation using data of the layers number 1 and 2 (resp. L − 1 and L)
becomes an extrapolation. For more details see subsubsection 11.1.

∗ Vertical cubic spline interpolations for layer variables at the origin point: Some top and bottom values are computed (the algorithm of computation

will be provided in a later version of this documentation) and the vertical interpolation
always uses 4 points.

114

4. Semi-lagrangian computations

14 Some distributed memory features.
14.1 Case LEQ_REGIONS=F.
Calculations packets.
∗ Grid-point computations: The total number of processors involved in the
A-level parallelisation is NPRGPNS. The total number of processors involved in the
B-level parallelisation is NPRGPEW. One processor treats NGPTOT points (a part of
the Gaussian grid points). The total amount of grid-points for all the processors is NGPTOTG. The maximum value of NGPTOT is NGPTOTMX. In the grid-point space
there is a subdivision of the current processor grid-points into NGPBLKS=int[(NGPTOT+NPROMA1)/NPROMA] packets of length NPROMA (the useful number of values in each packet
is lower or equal than NPROMA). These packets do not contain neither extra-longitudes
nor polar or extra-polar latitudes data. A NPROMA-packet does not always containa set of complete latitudes. This subdivision into NPROMA-packet both concern notlagged and lagged computations. Currently all the not lagged calculations are made forthe int[(NGPTOT+NPROMA-1)/NPROMA] packets of length NPROMA before
calling the lagged computations for the int[(NGPTOT+NPROMA-1)/NPROMA]
packets of length NPROMA. int[(NGPTOT+NPROMA-1)/NPROMA] is stored
in the variable NGPBLKS. More details will be given later for the data transmission
for horizontal interpolations. One 2D ﬁeld has NGPTOTG points divided into NPRGPNS*NPRGPEW sets of NGPTOT points treated by each processor. NGPTOT
does not take account of the extra-longitudes and the extra-polar latitudes. All thesevariables take account of the reduced Gaussian grid. It is assumed and hardcoded that
there are one western extra-longitude and two eastern extra-longitudes. The DM-global
longitude jlon = 1 is always the “Greenwich" meridian of the computational sphere. All
the vertical levels and the variables corresponding to a same grid-point are treated by
the same processor. There are necessary transpositions (reorganisation of data) between
grid point computations and Fourier transforms because Fourier transforms need complete
latitudes.
∗ Additional remarks about the LEQ_REGIONS environment variables. Variables N_REGIONS_NS, N_REGIONS and N_REGIONS_EW
are used even when LEQ_REGIONS=F but in this case:• N_REGIONS_NS=NPRGPNS.

• N_REGIONS=NPRGPEW everywhere.
• N_REGIONS_EW=NPRGPEW.

Transmission of data necessary for semi-Lagrangian horizontal interpolations from the non lagged grid-point computations towards
the lagged grid-point computations: interpolation buffers.
Description is done for the 3D model (the 2D model uses an obsolescent dataﬂow structure and does not work any longer).
First one associates to each interpolation point, which is generally not a model gridpoint, an “associated" model grid-point which currently satisﬁes to the following rule: forthe semi-Lagrangian scheme one associates to the interpolation point the corresponding4. Semi-lagrangian computations

115

arrival point of the SL trajectory. This associated model grid-point is always between
the latitudes 1 and NDGLG (it is never a pole or an extra-polar latitude point). The
processor which treats the interpolation point is the processor which treats this associatedmodel grid-point.
Interpolations use data of points which are not necessary on the same latitude andlongitude as the interpolation point and the arrival grid-point of the semi-Lagrangian
trajectory. Thus interpolation routines need to have access to a limited number of surrounding latitudes and longitudes which are not necessary treated by the current processor. First the not lagged computations are done for the int[(NGPTOT+NPROMA1)/NPROMA] packets of length NPROMA, then the lagged computations are done
for the int[(NGPTOT+NPROMA-1)/NPROMA] packets of length NPROMA interpolation points. The number of surrounding latitudes and longitudes rows necessary
for interpolations but which do not belong to the current processor is precomputed in thesubroutine SUSC2B (variable NSLWIDE). This is a sort of “halo" belonging to someother processors. Due to the “halo" there is still need to split calculations into not lagged
ones and lagged ones. Quantities to be interpolated are computed in the non-lagged partand interpolations are performed in the lagged part. In the non-lagged part, only data of
the current processor (without any extra-longitudinal data nor polar and extra-polar data)
are computed. For all the NPROMA-packages treated by the current processor thesedata are stored in the arrays PB1 in CPG (corresponding to local array ZSLBUF1 inthe routine GP_MODEL). The ﬁrst dimension of PB1 is NASLB1, which is the total
number of points one needs for the interpolations (NASLB1 is greater than NGPTOT).
The second dimension of PB1 is NFLDSLB1: this is the number of 2D quantities tobe interpolated. Inside CPG, an intermediate array ZSLBUF1AU is used and data
are transferred in PB1: this memory transfer uses a precomputed intermediate array
NSLCORE. Then some communication routines are called in CALL_SL to constitutethe halo. SLCOMM does processor communication to ﬁll the halo (receives and sends
data from some other processors). When all the NASLB1 dataset is constituted, thelagged part CALL_SL is called which do horizontal interpolations for all the data to beinterpolated for the current processor.

∗ Particular case of the “on demand" processor communication:

SLCOMM and SLEXTPOL are still called with speciﬁc options, and do the communications only on the part of the buﬀer which contains the information to ﬁnd thesemi-Lagrangian displacement. For the RHS of equations, the interpolations need tocommunicate a subset of points, this subset can be known only when all the interpolations grids have been computed (in LASCAW): communications are currently done by
SLCOMM2A and SLEXTPOL called by CALL_SL just before the interpolations.
SLCOMM2A and SLEXTPOL have less points to exchange so the model integrationis slightly less expensive.
Terms which must be evaluated at F are stored in GFLT1, GMVT1, GMVT1S (forthe RHS of equations) and PB2 (ZSLBUF2 in GP_MODEL) for additional intermediate terms. See documentation (IDEUL) for more details about PB2 and the dataﬂow
in the grid-point calculations.

14.2 Case LEQ_REGIONS=T.
This case is relevant only when NPRGPEW>1 (B-level parallelisation at least in thegrid-point calculations). This is an optimised version of the LEQ_REGIONS=F case

116

4. Semi-lagrangian computations

which is well designed for reduced Gaussian grid and it improves the load balance in thiscase. A comprehensive description can be found in (Mozdzynski, 2006). To sum-up, we
can say that:• the A-level grid-point distribution splits the Earth into N_REGIONS_NS bands.
N_REGIONS_NS can be slightly diﬀerent from NPRGPNS.
• for each band jroca, the B-level grid-point distribution splits the band into N_REGIONS(jroca)zones: the minimum value of N_REGIONS is at the poles of the computational
sphere (equal to 1 in the examples provided by Mozdzynski); the maximum value of
N_REGIONS is at the equator of the computational sphere and this maximum
is equal to N_REGIONS_EW. The meridian variations of N_REGIONS are
highly correlated to those of NLOENG.
• In the examples provided by Mozdzynski, NPRGPNS=NPRGPEW=NPRTRW=NPRTRVand we notice that N_REGIONS_NS is slightly below NPRGPNS, and that
N_REGIONS_EW is slightly below 2∗NPRGPEW.

5
Semi-implicit spectral
computations andpredictor-corrector schemes

1 Introduction.
1.1 Interest of semi-implicit and iterative centred-implicit schemes.
For both hydrostatic and non-hydrostatic models it is necessary to treat implicitly thelinear terms source of (fast moving) gravity waves to ensure a good stability; hence theresolution of equations involve the inversion of a linear system leading to a Helmholtz
equation: inversion of such a system is more convenient to do in spectral space. For nonhydrostatic models the semi-implicit scheme is generally not suﬃcient (especially for a
two-time level semi-Lagrangian scheme) and some non-linear terms have also to be treated
implicitly; for that one uses an iterative centred-implicit (abbreviated into “ICI") scheme.
The iterative centred-implicit schemes are often called “predictor-corrector" schemes, but
in a theoretical point of view one has normally to reserve this appellation for a subset of
iterative centred-implicit schemes with only one iteration. The iterative centred-implicit
scheme can have an incremental formulation or a non-incremental formulation. The one
which is coded in ARPEGE/ALADIN (for both hydrostatic and non-hydrostatic models)
and which will be retained and described is a non-incremental one. All the additional
calculations generated by an iterative centred-implicit scheme are mainly done in thegrid-point calculations and in the spectral transforms.

1.2 Distributed memory.
Some distributed memory features are now introduced in the code and will be brieﬂy
described. For convenience one uses some generic appellations.

118 5. Semi-implicit spectral computations and predictor-corrector schemes• Expression “DM-local" for a quantity means “local to the couple of processors(proca,procb)": each processor has its own value for the quantity. Expression “DMlocal computations" means that the computations are done independently in eachprocessor on “DM-local" quantities, leading to results internal to each processor,
which can be diﬀerent from a processor to another one.
• Expression “DM-global" for a quantity means that it has a unic value available inall the processors. Expression “DM-global computations" means that the computations are either done in one processor, then the results are dispatched in all theprocessors, or the same computations are done in all the processors, leading to thesame results in all the processors.
• In a routine description the mention “For distributed memory computations are
DM-local" means that all calculations done by this routine are DM-local; the mention “For distributed memory computations are DM-global" means that all calculations done by this routine are DM-global; when no information is provided it meansthat a part of calculations are DM-local and the other part is DM-global.
• Expression “main" processor currently refers to the processor number 1: (proca,procb)=(1,1).

1.3 The different models described.
• 2D shallow-water model.

• 3D primitive equations model (denoted as HYD).

1.4 Other restrictions of this documentation.
• Linear systems are written for thin layer equations: in practical, they don’t change
for deep layer equations because the radius r is linearised around a reference value
equal to a.

5. Semi-implicit spectral computations and predictor-corrector schemes 119

2 Notations.
• M is the mapping factor. M is a reference mapping factor for semi-implicit computations. M = c (stretching factor) if semi-implicit scheme with reduced divergence(LSIDG=.F. in YOMDYN). M = M (mapping factor) if semi-implicit schemewith unreduced divergence (LSIDG=.T. in YOMDYN).
• a is the Earth mean radius.

• r is the radius. The reference value of r for the linearisation is the mean radius a.
In the thin layer equations, r = a everywhere. In the LVERCOR=T (White andBromley, 1995) deep-layer equations, r is replaced by a pseudo-radius rs depending
only on the hydrostatic pressure. All the equations involving the radius will be
written with the denotation r.
• V is the horizontal geographical wind. Its zonal component is U . Its meridiancomponent is V .
′• D is the unreduced divergence of horizontal wind, D is the reduced divergence. D′′
and D are linked by the relationship D = (a/r) ∗ M 2 ∗ D .
′• ζ is the unreduced vorticity of horizontal wind, ζ is the reduced vorticity. ζ and′′ζ are linked by the relationship ζ = (a/r) ∗ M 2 ∗ ζ .

• w is the z-coordinate vertical velocity: w = dz.
dt

• T is the temperature. T ∗ is a vertically-constant reference temperature which isused in the semi-implicit scheme and in some non-hydrostatic equations. Default
value is 300 K or 350 K according to conﬁguration (for more details see subsection?? for variable SITR). IF LSPRT=.T. (use of virtual temperature in spectral
transforms instead of real temperature), T ∗ is used as a reference virtual temperature (same default value).
Ta∗ is a cold vertically-constant reference temperature which is used in the semiimplicit scheme in the NH vertical divergence equation; it is recommended to haveTa∗ lower than the current temperature.

• q is the humidity.

• Π is the hydrostatic pressure, Πs is the hydrostatic surface pressure. Π∗ is a reference hydrostatic pressure and Πs ∗ is a reference hydrostatic surface pressure, which
are used in the semi-implicit scheme and in some non-hydrostatic equations. These
reference quantities are vertically dependent and “horizontally" (i.e. on η surfaces)
constant. Default value of Πs ∗ is generally between 800 hPa and 1000 hPa. ∆Π∗are layer depths corresponding to a surface hydrostatic pressure equal to Πs ∗ .
• Πs st is a reference hydrostatic pressure equal to the surface pressure of the standard
atmosphere (variable VP00). Default value is 101325 Pa.
• ω = dΠis the total temporal derivative of the hydrostatic pressure (vertical velocity
dt
in hydrostatic pressure coordinate).
• p is the pressure, ps is the surface pressure.

• Q̂ is the pressure departure variable. Expression of Q̂ is:pQ̂ = log
Π• gz is the geopotential height.

(1)• Φ is the total geopotential (equivalent height in the shallow-water model), Φs is thesurface geopotential (i.e. the orography). In the thin layer equations, Φ = gz. Φsis assumed to be always equal to gzs . Φ∗ is a reference equivalent height which isonly used in the shallow-water model (semi-implicit scheme). Default value of Φ∗
is 100000 J/kg. ∆Φ∗ is a reference geopotential depth computed on model levels.

120 5. Semi-implicit spectral computations and predictor-corrector schemes• Ω is the Earth rotation angular velocity.

• r is the vector directed upwards, the length of which is the Earth radius a.

• g is the gravity acceleration constant, assumed to be vertically constant in the current documentation. For the (Wood and Staniforth, 2003) deep-layer NH equations
with vertical variations of g, only the reference value of g (vertically constant) is
taken into account in the semi-implicit scheme.
• R is the gas constant for air and Rd the gas constant for dry air.

• cp is the speciﬁc heat at constant pressure for air and cp d is the speciﬁc heat atconstant pressure for dry air.
• cv is the speciﬁc heat at constant volume for air and cv d is the speciﬁc heat atconstant volume for dry air.
′

• ∇ is the unreduced ﬁrst order horizontal gradient on η-surfaces. ∇ is the reducedﬁrst order horizontal gradient. These two operators are linked by the relationship′∇ = (a/r) ∗ M ∗ ∇ .

• D3 is the true 3D divergence. In the thin layer equations, expression of D3 is:p~ +D3 = ∇V
∇Φ
∂Π
RT
∂η



∂V
∂η



gp
− ∂Π
RT
∂η



∂w
∂η



(2)

• d is the vertical divergence. In the thin layer equations, the relationship between dand the height-coordinate vertical velocity w is:
gp
d = − ∂Π
Rd T
∂η



∂w
∂η



• Variable d4 = d + ∂ΠpRT ∇Φ
∂η

(3)
∂V
∂η



can be also used as prognostic variable.

• L: number of layers of the model.

• A, B deﬁne hydrostatic pressure on the η levels ( Π = A + BΠs , where Πs is thehydrostatic surface pressure).
• β coeﬃcient for the semi-implicit scheme (between 0 and 1).

• γ, τ , ν, µ, ∂ ∗ , L∗ and T∗ are generic notations for linear operators (see subsection
4.2).
• H, C, N are intermediate constants used in the semi-implicit scheme of the nonhydrostatic model. Deﬁnitions are respectively:H=

Rd T ∗g

C = Rd T ∗
N= p

cp d
cv d

g
cp d T ∗

(4)
(5)
(6)

• For a variable X deﬁned at full levels, hXi is the vector of coordinates (X1 ; ...; Xl ; ...; XL ).
• Rinte is the vertical integration operator used in the case LVERTFE=.T. :
–
–
–

R η=1
η=0

Xdη is discretised by [Rinte ](top,surf ) hXi.

η=0

Xdη is discretised by [Rinte ](top,l) hXi.

η=ηl

Xdη is discretised by [Rinte ](l,surf ) hXi.

R η=ηlR η=1• Rderi is the vertical ﬁrst-order derivative operator used in the case where VFE arealso applied to derivatives.

5. Semi-implicit spectral computations and predictor-corrector schemes 121

3 General considerations.
3.1 Advection schemes.
∗ Explicit Eulerian equations: In Eulerian form of equations, the time dependency equation of a variable X writes as:∂X= −U.∇3 X + A + F∂t(7)

where U is the 3D wind, ∇3 is the 3D gradient operator, A is the dynamical contribution,
and F is the physical contribution. X(t + ∆t) is computed knowing X(t − ∆t) at the samegrid point.

∗ Explicit semi-Lagrangian equations: In semi-Lagrangian form of equations, the time dependency equation of a variable X writes as:
dX
=A+F
dt

(8)

In a three-time level semi-Lagrangian scheme (abbreviated into 3TL SL scheme) X(t+∆t)
is computed at a grid point F knowing X(t − ∆t) at the point O (not necessary a grid
point) where the same particle is at t − ∆t. In a two-time level semi-Lagrangian scheme(abbreviated into 2TL SL scheme) X(t + ∆t) is computed at a grid point F knowing X(t)at the point O (not necessary a grid point) where the same particle is at t.

3.2 Semi-implicit treatment of linear terms (case where there
is no iterative centred-implicit scheme).
∗ Adding of a semi-implicit correction: In all cases the linear terms sourceof gravity waves must be treated implicitly, in order to allow time-steps compatible with
an operational use of the model. Expression of the linear terms is obtained assuming adeﬁnition of a reference state. The reference state is deﬁned by a dry resting isotherm
atmosphere in hydrostatic balance, reference orography is zero. Equations (7) and (8)become respectively (9) and (10):• Eulerian scheme:∂X= −U.∇3 X + A + F + [SIcorrection]∂t(9)• Semi-Lagrangian scheme:dX= A + F + [SIcorrection]
dt

(10)

∗ Discretisation of equations (9) and (10): Equations (9) and (10) givethe following discretized equations, where ∆t is the time step, B is the linear term sourceof gravity waves, β is a tunable parameter (β = 0 corresponds to an explicit formulation,
β = 1 to an implicit formulation):

122 5. Semi-implicit spectral computations and predictor-corrector schemes• Eulerian scheme:[SIcorrection] = −βBt +β t−∆t β t+∆t
B
+ B
2
2

(11)

X t+∆t −β∆tBt+∆t = X t−∆t +2∆t(−U.∇3 X+A+F )−2β∆tBt +β∆tBt−∆t (12)all computations are done at the same grid point.
• Three-time level semi-Lagrangian (3TL SL) scheme (without uncentering factor):[SIcorrection] = −βBt +β t−∆t β t+∆t
B
+ B
2
2

X t+∆t − β∆tBt+∆t = X t−∆t + 2∆t(A + F ) − 2β∆tBt + β∆tBt−∆t

(13)
(14)

where X t+∆t −β∆tBt+∆t is computed at the ﬁnal grid point of the semi-Lagrangian
trajectory, X t−∆t and β∆tBt−∆t are computed at the origin point of the semiLagrangian trajectory, −2β∆tBt is computed as an average between the origin
and ﬁnal points of the trajectory, A is computed either at the medium point or
as an average between the origin and ﬁnal points of the trajectory. If there is auncentering factor ǫ replace ∆t by (1 − ǫ)∆t for terms at the origin point, ∆t by(1 + ǫ)∆t for terms at the ﬁnal point. For more details see documentation (IDSL)
about the semi-Lagrangian scheme.
• Two-time level semi-Lagrangian (2TL SL) scheme (without uncentering factor):[SIcorrection] = −βBt+0.5∆t +β t β t+∆t
B + B
2
2

(15)

X t+∆t − 0.5β∆tBt+∆t = X t + ∆t(A + F ) − β∆tBt+0.5∆t + 0.5β∆tBt (16)where X t+∆t − 0.5β∆tBt+∆t is computed at the ﬁnal grid point of the semiLagrangian trajectory, X t and 0.5β∆tBt are computed at the origin point of thesemi-Lagrangian trajectory, −β∆tBt+0.5∆t and A are computed either at the medium
point or as an average between the origin and ﬁnal points of the trajectory. If there
is a ﬁrst-order uncentering factor ǫ replace ∆t by (1 − ǫ)∆t for terms at the origin
point, ∆t by (1 + ǫ)∆t for terms at the ﬁnal point. For more details see documentation (IDSL) about the semi-Lagrangian scheme.
Bt+0.5∆t , Bt and Bt−∆t are computed in grid point space. The right-hand side membersof equations (12), (14) and (16) are computed in grid point space, then transformed into
spectral space. Entering spectral space a system of equations of the following type must
be solved:
X t+∆t − β∆tBt+∆t = X ∗(17)

for a leap-frog scheme, and:X t+∆t − 0.5β∆tBt+∆t = X ∗(18)

for a two-time level semi-Lagrangian scheme, where X ∗ is known and X t+∆t is unknown.
Now the spectral computations to solve this system of equations are described for a primitive equations 3D model, a 2D shallow water model and several NH 3D models.

5. Semi-implicit spectral computations and predictor-corrector schemes 123

3.3 Iterative centred-implicit schemes and combination withsemi-implicit schemes.
Purpose.
In some cases (especially in the non-hydrostatic models), the model with a semi-implicit
treatment of linear terms may remain unstable, hence a treatment by an iterative centredimplicit scheme may be necessary. In the following description one sticks to non-incremental
formulations.

Iterative centred-implicit scheme.
∗ Algorithm: The total number of iterations is denoted by Nsiter .
t+∆t• The iteration number (i = 0) computes an estimation X(i=0)
of X t+∆t with anormal semi-implicit scheme. Horizontal diﬀusion can be done optionally at thisstage.
t+∆t• Iterations (i > 0): the i-th iteration (i > 0) computes X(i)(after inversion oft+∆tHelmholtz equation) knowing X(i−1). Horizontal diﬀusion is always done at thelast iteration, it can be done optionally at the other iterations. The ﬁnal value oft+∆t
X t+∆t is equal to X(i=N.
siter )• In the cycle 37T1 of ARPEGE/IFS and the cycle AL37T1 of ALADIN, this scheme
is controlled by the key LPC_FULL=.T. . and is coded in the Eulerian schemeand the two-time level semi-Lagrangian scheme only. It is compulsory for the nonhydrostatic model with a SL2TL scheme to ensure stability (but not compulsory
with the hydrostatic model with the two-time level semi-Lagrangian scheme).
• For unlagged physics, the physics has to be computed for the iteration (i = 0) only.
For lagged physics, there are several possible options in test, but in practical the
only one which works is:– adiabatic treatment of iterations 0 to Nsiter − 1.
– diabatic treatment of iteration Nsiter .

∗ Discretisation of algorithm:• First iteration (i = 0): One has to start from the discretisations of equations for amodel with no iterative centred-implicit formulation (see documentations (IDEUL)and (IDSL)). For a leap-frog scheme the calculations are the same ones. For a twotime level semi-Lagrangian scheme, (A − β∆tB)t+0.5∆t
is assumed to be equal to(i=0)
t(A − β∆tB) if no extrapolation is done (case LPC_NESC=.T.), and to 1.5(A −β∆tB)t −0.5(A−β∆tB)t−∆t if extrapolation is done (case LPC_NESC=.F.). For
a SL2TL case with no uncentering factor that yields the following discretisations(physics is assumed to be unlagged):– no extrapolation:
t+∆t
t+∆t
(X(i=0)−0.5∆tβB(i=0))F = [0.5∆tAt −0.5∆tβBt ]F +[X t +0.5∆tAt −0.5∆tβBt +0.5∆tβBt +∆tF t ]O(i=0)(O(i = 0) and F are respectively the origin and ﬁnal points of the semiLagrangian trajectory), which can be rewritten:
t+∆t
t+∆t
(X(i=0)−0.5∆tβB(i=0))F = [0.5∆tAt −0.5∆tβBt ]F +[X t +0.5∆tAt +∆tF t ]O(i=0)

124 5. Semi-implicit spectral computations and predictor-corrector schemes– extrapolation: discretisation is identical to the case with no iterative centredimplicit scheme and conventional extrapolation (type LSETTLS=.F.); theRHS terms other than X t and F t can be replaced by a “spatio-temporal"
average; see documentation (IDSL).
• Following iterations (i > 0): The general iteration writes (no uncentering, unlagged
physics):– Eulerian scheme (ADV stands for advection terms):
t+∆t
t+∆t
X(i)
− ∆tβB(i)t+∆tt+∆t]+[∆tAt−∆t −∆tβBt−∆t ]+∆tβBt−∆t +2∆tF t−∆t−∆tβB(i−1)= X t−∆t +2∆tADV t +[∆tA(i−1)

which can be rewritten:

t+∆t
t+∆t
t+∆t
t+∆t
X(i)
−∆tβB(i)= ∆tA(i−1)−∆tβB(i−1)+[X t−∆t +2∆tADV t +∆tAt−∆t +2∆tF t−∆t ]– Three-time level semi-Lagrangian scheme (without uncentering factor):
t+∆t
t+∆t
[X(i)
− ∆tβB(i)
]F
t−∆t
t+∆t
t+∆t
= XO(i)+[∆tA(i−1)−∆tβB(i−1)]F +[∆tAt−∆t −∆tβBt−∆t ]O(i) +[∆tβBt−∆t +2∆tF t−∆t ]O(i)which can be rewritten:

t+∆t
t+∆t
t−∆t
t+∆t
t+∆t
[X(i)
−∆tβB(i)]F = XO(i)+[∆tA(i−1)−∆tβB(i−1)]F +[∆tAt−∆t +2∆tF t−∆t ]O(i)The iterative centred-implicit algorithm also applies to re-compute the semiLagrangian trajectory (see documentation (IDSL) for more details), the position of the origin point at the i-th (resp i − 1-th) iteration is O(i) (resp.
O(i − 1)).

– Two-time level semi-Lagrangian scheme (without uncentering factor):
t+∆t
t+∆t
]F
− 0.5∆tβB(i)
[X(i)

t
t+∆t
t+∆t
= XO(i)+[0.5∆tA(i−1)−0.5∆tβB(i−1)]F +[0.5∆tAt −0.5∆tβBt ]O(i) +[0.5∆tβBt +∆tF t ]O(i)which can be rewritten:

t+∆t
t+∆t
t
t+∆t
t+∆t
]F +[0.5∆tAt +∆tF t ]O(i)−0.5∆tβB(i−1)+[0.5∆tA(i−1)]F = XO(i)−0.5∆tβB(i)[X(i)

The iterative centred-implicit algorithm also applies to re-compute the semiLagrangian trajectory (see documentation (IDSL) for more details), the position of the origin point at the i-th (resp i − 1-th) iteration is O(i) (resp.
O(i − 1)). Remark: when the iterative centred-implicit scheme is used, thestable extrapolation LSETTLS=.T. is never involved.

∗ Cheap version of this algorithm: In a semi-Lagrangian scheme, it is possible

not to iterate the position of the origin point O (i.e O(i) = O(i = 0)): this cheap version
is activated if LPC_CHEAP=.T. . It is coded only for a non-extrapolating SL2TL
scheme. In this case the quantity to be interpolated (i.e. [0.5∆tAt + ∆tF t ]O ) needs to
be interpolated at the predictor step only. It is then stored in a buﬀer and re-used at thecorrector steps without any interpolation.

3.4 Introduction of uncentering for semi-Lagrangian schemes.
Averages along the semi-Lagrangian trajectory will be weighted by (1 − ǫ) at the origin
point and (1 + ǫ) at the ﬁnal point. If the uncentering coeﬃcient ǫ is horizontally constant
the algorithms remain valid, replacing β by (1 + ǫ)β.

5. Semi-implicit spectral computations and predictor-corrector schemes 125

3.5 Plane geometry (ALADIN).
Particular features of ALADIN when diﬀering from ARPEGE/IFS are not described indetail, only brief comments are mentioned. For more details see ALADIN documentation.
Concerning the semi-implicit and iterative centred-implicit algorithms one can consider
that the major part of this documentation is still valid for ALADIN; the main diﬀerenceswith ARPEGE/IFS are:
• The shallow-water model is not coded in ALADIN.

• Option LESIDG=.T. replaces LSIDG=.T. in ALADIN, and is available onlywith the tilted-rotated Mercator projection. This option is useful only when thehorizontal variations of the mapping factor are signiﬁcant. See (IDESIDG) about
its implementation.
• Option LIMPF=.T. is not coded in ALADIN.

• In the hydrostatic (resp. NH) model, spectral part of the semi-implicit scheme isperformed in routine ESPCSI instead of SPCSI (resp. ESPNHSI instead of
SPNHSI) in ALADIN; the algorithm is the same as in ARPEGE but the truncation of the spectral representation is elliptic and not triangular.

3.6 Finite elements on the vertical.
The option with ﬁnite element vertical discretisations is coded for the hydrostatic modeland partly for the NH models. For VFE, the main modiﬁcations in the semi-implicit
scheme are the following ones:• The discretisation of Π, α and δ at full levels is diﬀerent.

• The model avoids as possible to compute quantities at half levels; all vertical integrals directly provide quantities at full levels.
• The vertical integrals contained in some linear operators (γ, τ and ν) are discretised
diﬀerently, as a matricial multiplication with special coeﬃcients (contained in thematrix Rinte ) computed in the setup routine SUVERTFE1 or SUVERTFE3;
the vertical integration is done by routine VERINT.

126 5. Semi-implicit spectral computations and predictor-corrector schemes

4 Prognostic variables and quantities involved in thesemi-implicit scheme.
4.1 Prognostic variables.
Prognostic variables can be split into diﬀerent classes:• 3D variables, the equation RHS of which has a non-zero adiabatic contribution anda non-zero semi-implicit correction contribution. They are called “GMV" in thecode (“GMV" means “grid-point model variables"). This class of variables includes
the components of the horizontal wind V, temperature T , and the two additional
non-hydrostatic variables in a non-hydrostatic model.
• 3D “conservative" variables. The equation RHS of these variables has a zero adiabatic contribution, only the diabatic contribution (and the horizontal diﬀusion
contribution) can be non-zero. They are called “GFL" in the code (“GFL" means“grid-point ﬁelds"). This class of variables includes for example humidity q, liquidwater, ice, cloud fraction, ozone, and some extra ﬁelds.
• 2D variables, the equation RHS of which mixes 3D and 2D terms, has a nonzero adiabatic contribution and a non-zero semi-implicit correction contribution.
They are called “GMVS" in the code (“GMVS" means “grid-point model variables
for surface"). This class of variables includes the logarithm of surface pressure(continuity equation).
Only the GMV and GMVS variables appear in the semi-implicit scheme. In the shallowwater 2D model, only GMV variables exist, this class of variables includes the componentsof the horizontal wind V, and the equivalent height Φ − Φs (continuity equation).

4.2 Quantities used for vertical discretisations and linear operators.
∗ Operators “alpha" and “delta": these operators are used for discretisationsof some vertical integrals. They have a diﬀerent expression according to the value ofvariables NDLNPR, LVERTFE.
• For LVERTFE=.F., NDLNPR=0 and a layer l between 2 and L (and also l = 1
if the pressure at the top of the model is not zero), α∗ and δ ∗ are discretised asfollows at full levels:αl∗ = 1 −δl∗ = log

∗
Πl−1

∆Π∗l

log

Π∗l
Π∗

l−1

Π∗l
Π∗

l−1

!

!

(19)

(20)

• For LVERTFE=.F., NDLNPR=0 and the layer l = 1 if the pressure at the topof the model is zero:∗– αl=1= 1 at METEO-FRANCE.
∗– αl=1= log(2) at ECMWF.
∗– δl=1
has in theory an inﬁnite value, but in the code it is computed with a top
pressure equal to 0.1 Pa to provide a ﬁnite value.

5. Semi-implicit spectral computations and predictor-corrector schemes 127• For LVERTFE=.F., NDLNPR=1 and a layer l between 2 and L (and also l = 1
if the pressure at the top of the model is not zero), α∗ and δ ∗ are discretised asfollows at full levels:αl∗ = 1 −
δl∗ =

s

Π∗

l−1
Π∗
l

=1−

Π∗l
Π∗

(21)

l

∆Π∗l
∆Π∗l
= q
∗
Πl
Π∗ Π∗
l−1

(22)

l

Π∗ is discretised as follows at full levels:Π∗ l =p(23)

Π∗ l Π∗ l−1∗
is used for quantity 1 −Notation αl−1

Π∗
l−1
Π∗
l

instead of notation βl∗ of (Bubnová etal., 1995).
• For LVERTFE=.F., NDLNPR=1 and the layer l = 1 if the pressure at the topof the model is zero:∗∗= 1.
– αl=1= 1 and αl=0c∗
– δl=1= 1 + Rpdd .

– Π∗l=1 =

∆Π∗
l=1
δ∗
l=1

.

• For LVERTFE=.T., NDLNPR=0 and a layer l between 1 and L, α∗ and δ ∗ arediscretised as follows at full levels:Π∗ − Π∗lαl∗ = l ∗
(24)
Πl
δl∗ =

∆Π∗l
Π∗l

(25)

where Π∗l = Al + Bl Πs ∗ . See documentation (IDEUL) for computation of Al and
Bl in this case. Formulae (24) and (25) provide ﬁnite values of α1∗ and δ1∗ even if
the pressure at the top of the model is zero. αl∗ is not used in the SI scheme inthis case because vertical integrals are directly provided at full levels without theintermediate state of interlayer data.

∗ Linear operator “γ": this operator is applied to temperature and pressure departure variable to compute linear term in momentum equation. Denotation γ has thesame meaning as the denotation Rd ∗ G∗ of (Bubnová et al., 1995) or (IDNH2.1). For avariable Z, (γZ) is a discretisation of vertical integral

Z 1η

Expression of this discretisation is:• Case LVERTFE=.T.:(γZ)l = [Rinte ](l,surf )• Case LVERTFE=.F.:(γZ)l = αl∗ Rd Zl +



L
X

∂Π∗
∂η
Π∗



Rd Zdη

(26)

Rd Zk δk∗(27)

Rd Zδ ∗
∆η

k=l+1

Remark: if LSPRT=.T., γ is applied to virtual temperature instead of real temperature.

128 5. Semi-implicit spectral computations and predictor-corrector schemes∗ Linear operator “τ ": this operator is applied to divergence to compute linear
term∗ in temperature equation. Denotation τ has the same meaning as the denotation
Rd T∗ S∗ of (Bubnová et al., 1995) and (IDNH2.1). For a variable Z, (τ Z) is the discp
d∗cretisation of vertical integral cRpd TΠ∗d• Case LVERTFE=.T.:R η ∂Π∗0∂η

Zdη. Expression of (τ Z) is:

Rd T ∗ δl∗[Rinte ](top,l)cp d ∆Π∗l(τ Z)l =



∆Π∗ Z
∆η



(28)

Remark: according to the expression of δl∗ in this case, this equation can be rewritten:


∆Π∗ Z
Rd T ∗ 1[R](τ Z)l =inte(top,l)cp d Π∗l∆η• Case LVERTFE=.F.:
Rd T ∗(τ Z)l =
cp d

"

δl∗ X
∆Π∗k Zk∆Π∗ll−1αl∗ Zl +k=1

Remark: if LSPRT=.T., Rd T ∗ becomes

#

(29)

2
Rd
T ∗ in expression of (τ Z).
R∗ Linear operator “ν": this operator is applied to divergence to compute linear
term in continuity equation. Denotation ν has the same meaning as the denotation Πs ∗ N∗of (Bubnová et al., 1995) and (IDNH2.1). For a variable Z, deﬁnition of (νZ) is:• Case LVERTFE=.T.:(νZ) =

1[Rinte ](top,surf )
Πs ∗



∆Π∗ Z
∆η



(30)

• Case LVERTFE=.F.:
1 X∆Π∗l Zl
Πs ∗L(νZ) =(31)l=1(νZ) is the discretisation of vertical integral Π1s ∗R 1 ∂Π∗
0

∂η

Zdη.

∗ Linear operator “µ": This operator is applied to log(Πs ) to compute linear termin momentum equation. For a variable Z, Deﬁnition of (µZ) is:(µZ) = Rd T ∗ Z(32)(µZ) is applied to log(Πs ).
Remarks:• if LSPRT=.T., Rd T ∗ becomes2
Rd
T ∗ in expression of (µZ).
R• In some documentations one sometimes ﬁnds the expanded expression Rd T ∗ I instead of µ (where I is the identity matrix).

5. Semi-implicit spectral computations and predictor-corrector schemes 129

4.3 Relationships between some linear operators.
In the NH-PDVD model, operators ν, γ and τ SHOULD be linked by the following
relationship if one wants the complete elimination of variables in order to provide a “onevariable" Helmholtz equation computing the vertical divergence, and all the following partsof this documentation assumes that this condition is fulﬁlled. For more details see section
4 of (IDNH2.1) where this condition is denoted by “C1". The adimensioned formulationof constraint “C1" writes:
ν=

cp
cp d
1
τ − 2 d ∗ γτγ+
Rd
Rd T ∗Rd T(33)This equation can be multiplied by cp d /cv d and rewritten as follows:
COR = 0(34)

where:
COR =

cv
cv d
cv d
cv d
τ + dν
γ−γτ −R d cp d
Rd T ∗
cp d
Rd2 T ∗

(35)

Note that this condition is fulﬁlled at least for LVERTFE=.F. (FD discretisation ofoperators ν, γ and τ ) and NDLNPR=1 (this is the reason of using the NDLNPR=1
discretisation in a NH model), and is SURELY NOT fulﬁlled for LVERTFE=.F. and
NDLNPR=0.
For LVERTFE=.T., this condition is not fulﬁlled. This is the reason why there is apossibility to keep LVERTFE=.F. and NDLNPR=1 in the linear model, even if VFE
are switched on in the explicit model.
Additional studies (see for example Bénard, 2004) show that, when the constraint C1 is
not fulﬁlled, it is not possible to fulﬁll it simply by changing the deﬁnition of one of theoperators ν, γ or τ .
• Changing the deﬁnition of ν to match formula (33) does not ensure any longer thefact that ν is a 2D operator.
• Changing the deﬁnition of γ to match COR = 0 does not work because thatprovides an expression with the inverse of a non-invertible matrix.
• Changing the deﬁnition of τ to match COR = 0 has none of the shortcomingsof the two previous proposals, but provides a linear operator generating spurious
noise.

130 5. Semi-implicit spectral computations and predictor-corrector schemes

5 Semi-implicit scheme, no Coriolis term in the semiimplicit scheme.
Equations are written for a leap-frog scheme (Eulerian scheme of three-time level semiLagrangian scheme). For a two-time level semi-Lagrangian scheme replace ∆t by 0.5∆t.

5.1 3D hydrostatic primitive equations model.
∗ Expression of the linear term B for GMV and GMVS variables:• Continuity equation (X = log(Πs )):
′

2

(36)

B = −ν(M D )′• Divergence equation (X = D ):′B = −∇ 2 (γT + µ log Πs )

(37)

′

• Vorticity equation (X = ζ ):

(38)

B=0

• Temperature equation (X = T ):
′

2

(39)

B = −τ (M D )∗ System to be solved: Equations are written for log(Πs ) as a prognostic variablefor continuity equation.

′2log(Πs )t+∆t + β∆tM νDt+∆t = P ∗

(40)

′

′

′

Dt+∆t + β∆t∇ 2 (γTt+∆t + µlog(Πs )t+∆t ) = D ∗2′

Tt+∆t + β∆tM τ Dt+∆t = T ∗

(41)

(42)

′

P ∗ , D ∗ , T ∗ correspond to X ∗ deﬁned in equation (17) and are available in spectral arrays(SPSP, SPDIV, SPT) at the beginning of the spectral computations. Equations (40)to (42) yield (43) (Helmholtz equation):
′

2

′

′

′

(I − β 2 ∆t2 B ∇ 2 M )Dt+∆t = D ∗ − β∆t∇ 2 (γT ∗ + µP ∗ )(43)where B = γτ + µν is a matricial operator L ∗ L (precomputed in routines SUDYN,SUBMAT and stored in the array SIB).
When M = M (cases LSIDG=.T. or LESIDG=.T.) it is more convenient to rewrite
equation (43) as:
′

′

′

′

(∇ −2 − β 2 ∆t2 B M 2 )Dt+∆t = ∇ −2 D ∗ − β∆t(γT ∗ + µP ∗ )which shows a symmetric matricial operator in the left hand side.

(44)

5. Semi-implicit spectral computations and predictor-corrector schemes 131∗ Spectral computations to solve system of equations (40) to (42).

Algorithm works zonal wave number by zonal wave number m (| m | varies between 0 andthe truncation Ns ) and performed in the routine SPCSI before all horizontal diﬀusion
schemes. For a given zonal wave number m:• After a preliminary memory transfer the right-hand side member of equation (43)is computed for all total wave numbers n between m and Ns .
• Inversion of Helmholtz equation for the case “reduced divergence" (case LSIDG=.F.
and LESIDG=.F.) and method via a diagonalisation in the eigenmodes space.
– First the diagonalisation of B is used: B = Q−1 A Q, where A is a diagonal
L ∗ L matrix, the diagonal coeﬃcients al of which are stored in the array
SIVP. Q is a L ∗ L matrix stored in the array SIMI, Q−1 is stored in thearray SIMO. Note that the vertical operators ν, µ, τ , γ, B , Q commute with′the horizontal operator ∇ 2 .
– Helmholtz equation (43) becomes, for each eigenmode l:
′

2

′

′

′

(I − β 2 ∆t2 al ∇ 2 M )QDt+∆t = Q(D ∗ − β∆t∇ 2 (γT ∗ + µP ∗ ))

(45)
′

2

– For each eigenmode l and each zonal wave number m: (I − β 2 ∆t2 al ∇ 2 M )is a diagonal matricial operator (Ns + 1− | m |) ∗ (Ns + 1− | m |): spectral
coeﬃcients of the right-hand side member of (45) are simply divided by the
diagonal coeﬃcients of this matrix. The result is then multiplied by Q.
• Inversion of Helmholtz equation for the case “unreduced divergence" (case LSIDG=.T.
in ARPEGE, LESIDG=.T. in ALADIN): in this case M = M . Inversion ofHelmholtz equation is more complicated than in the case of semi-implicit schemewith reduced divergence because the left-hand side member of Helmholtz equation
contains values of the divergence for all levels and ﬁve total wave numbers (n − 2
to n + 2). Of course M 2 is a symmetric pentadiagonal matrix, for a given zonal
wave number m. Pay attention to the fact that M 2 does not commute with the′
diagonal operator ∇ 2 .
– First the diagonalisation of B is used: B = Q−1 A Q, where A is a diagonal
L ∗ L matrix, the diagonal coeﬃcients al of which are stored in the array
SIVP. Q is a L ∗ L matrix stored in the array SIMI, Q−1 is stored in thearray SIMO. Note that the vertical operators ν, µ, τ , γ, B , Q commute with′the horizontal operators ∇ 2 and M 2 .
– Helmholtz equation (43) (resp. (44)) becomes, for each eigenmode l:
′

′

′

′

′

′

(I − β 2 ∆t2 al ∇ 2 M 2 )QDt+∆t = Q(D ∗ − β∆t∇ 2 (γT ∗ + µP ∗ ))(46)

resp.:′′(∇ −2 − β 2 ∆t2 al M 2 )QDt+∆t = Q(∇ −2 D ∗ − β∆t(γT ∗ + µP ∗ )) (47)′

– For each eigenmode l and each zonal wave number m: (∇ −2 −β 2 ∆t2 al M 2 ) isa symmetric pentadiagonal matricial operator (Ns +1− | m |)∗(Ns +1− | m |).
The factorisation LU of this matrix is computed, where L is a lower triangulartridiagonal matrix, U is a upper triangular tridiagonal matrix with coeﬃcientsequal to 1 on the main diagonal. All useful coeﬃcients of L, U are computedin the set-up routine SUHEG and stored in the array SIHEG.
– The right-hand side member of (47) is computed, then multiplied by the′
inverse of the symmetric pentadiagonal operator (∇ −2 − β 2 ∆t2 M 2 al ) (resolution of two tridiagonal triangular systems by routine MXTURS). That′′
yields QDt+∆t . Multiplying by Q−1 one obtains Dt+∆t .

132 5. Semi-implicit spectral computations and predictor-corrector schemes– For the zonal wave number m = 0 equation (46) is used rather than (47)′because, for the total wave number n = 0, ∇ 2 is equivalent to a multiplication′−2
by 0 and ∇is equivalent to a division by 0. The only diﬀerence is that the′
pentadiagonal but non-symmetric operator (I − β 2 ∆t2 al ∇ 2 M 2 ) is factorised
and inverted. All useful coeﬃcients of L, U are computed in the set-up routine
SUHEG and stored in the arrays SIHEG and SIHEG2.
′• Once known Dt+∆t equation (40) provides log(Πs )t+∆t and equation (42) provides
Tt+∆t . For the case LSIDG=.T. only (resp. LESIDG=.T. in ALADIN), spectral
multiplications by M 2 are performed by the product of a symmetric pentadiagonal
matrix of dimensions (Ns + 1− | m |) ∗ (Ns + 1− | m |) (useful coeﬃcients computedin routine SUSMAP (resp. SUESMAP in ALADIN) and stored in the array
SCGMAP (resp. ESCGMAP in ALADIN)) by a vector containing spectral
coeﬃcients (m, n) for n varying from | m | to Ns .

• Semi-implicit scheme ends by a ﬁnal memory transfer.

5.2 2D shallow-water model.
∗ Expression of the linear term B:• Continuity equation (X = Φ):2

B = −Φ∗ M D

′

(48)
′

• Divergence equation (X = D ):′B = −∇ 2 (Φ)(49)′• Vorticity equation (X = ζ ):
(50)

B=0

∗ System to be solved:′

2

Φt+∆t + β∆tM Φ∗ Dt+∆t = H∗
′

′

(51)

′

Dt+∆t + β∆t∇ 2 (Φt+∆t ) = D ∗

(52)

′

H∗ , D ∗ correspond to X ∗ deﬁned in equations (17) and are available in spectral arrays(SPSP, SPDIV) at the beginning of the spectral computations. Equations (51) and (52)yield (53) (Helmholtz equation):
′

2

′

′

′

(1 − β 2 ∆t2 Φ∗ ∇ 2 M )Dt+∆t = D ∗ − β∆t∇ 2 (H∗ )(53)

When M = M (case LSIDG=.T.) it is more convenient to rewrite equation (53) as:
′

′

′

′

(∇ −2 − β 2 ∆t2 Φ∗ M 2 )Dt+∆t = ∇ −2 D ∗ − β∆t(H∗ )which shows a symmetric matricial operator in the left hand side.

(54)

5. Semi-implicit spectral computations and predictor-corrector schemes 133∗ Spectral computations to solve system of equations (51) and (52).

Algorithm works zonal wave number by zonal wave number m (| m | varies between 0 andthe truncation Ns ) and performed in the routine SPC2 before all horizontal diﬀusion
schemes. For a given zonal wave number m:• After a preliminary memory transfer the right-hand side member of equation (53)is computed for all total wave numbers n between m and Ns .
• Inversion of Helmholtz equation for the case “reduced divergence" (case LSIDG=.F.):′2
For each zonal wave number m: (1 − β 2 ∆t2 Φ∗ ∇ 2 M ) is a diagonal matricial operator (Ns + 1− | m |) ∗ (Ns + 1− | m |). Spectral coeﬃcients of the right-hand side
member of (53) are simply divided by the diagonal coeﬃcients of this matrix.
• Inversion of Helmholtz equation for the case “unreduced divergence" (case LSIDG=.T.):
Inversion of Helmholtz equation is more complicated than in the case of semiimplicit scheme with reduced divergence because the left-hand side member ofHelmholtz equation contains values of the divergence for all levels and ﬁve total
wave numbers (n − 2 to n + 2).
′– For each zonal wave number m: (∇ −2 − β 2 ∆t2 M 2 Φ∗ ) is a symmetric pentadiagonal matricial operator (Ns +1− | m |)∗(Ns +1− | m |). The factorisation
LU of this matrix is computed, where L is a lower triangular tridiagonal matrix, U is a upper triangular tridiagonal matrix with coeﬃcients equal to 1 on
the main diagonal. All useful coeﬃcients of L, U are computed in the set-up
routine SUHEG and stored in the array SIHEG.
– The right-hand side member of (54) is multiplied by the inverse of the sym′
metric pentadiagonal operator (∇ −2 − β 2 ∆t2 Φ∗ M 2 ) (resolution of two tridi′
agonal triangular systems by routine MXTURS). That yields Dt+∆t .
– For the zonal wave number m = 0 equation (53) is used preferably than (54)′because, for the total wave number n = 0, ∇ 2 is equivalent to a multiplication′−2
by 0 and ∇is equivalent to a division by 0. The only diﬀerence is that the′
pentadiagonal but non-symmetric operator (1−β 2 ∆t2 Φ∗ ∇ 2 M 2 ) is factorised
and inverted. All useful coeﬃcients of L, U are computed in the set-up routine
SUHEG and stored in the arrays SIHEG and SIHEG2.
′• Once known Dt+∆t equation (51) provides Φt+∆t . For the case “unreduced divergence" (case LSIDG=.T.) only, spectral multiplications by M 2 are performed by
the product of a symmetric pentadiagonal matrix of dimensions (Ns + 1− | m |) ∗ (Ns + 1− | m |) (useful coeﬃcients computed in routine SUSMAP and stored
in the array SCGMAP) by a vector containing spectral coeﬃcients (m, n) for nvarying from | m | to Ns .
• Semi-implicit scheme ends by a ﬁnal memory transfer.

5.3 Plane geometry.
For 3D models, semi-implicit calculations are done in ESPCSI, ESPNHSI, ESPNHSI_GEOGW
instead of SPCSI, SPNHSI and SPNHSI_GEOGW.
Option LSIDG=.T. has an equivalent LESIDG=.T. in the cycle AL37T1 of ALADIN,
coded only for a tilted-rotated Mercator projection.

5.4 Shortcomings of the formulation of the semi-implicit schemewith “reduced divergence" (LSIDG=.F.) in case of stretching.
∗ ARPEGE/IFS: In the grid points computations for some equations (for example
temperature and continuity equation in the 3D hydrostatic model), the linear term B

134 5. Semi-implicit spectral computations and predictor-corrector schemes2′

contains the reduced quantity (M D ). This quantity is added to geographical quantities.
That is no problem near the high resolution pole. This reduced quantity becomes very2
= c4 at the low
large near the low resolution pole: if the stretching coeﬃcient is c, M
M2
resolution pole, which is equal to 33.2 if c = 2.4. Thus the order of magnitude of the semiimplicit correction tendency becomes too high and physically absurd in the low resolution
zone (gravity waves are no longer treated implicitly). That leads to instabilities in regionsof the low resolution zone with high orography, in adiabatic eulerien runs, or in semiLagrangian runs with time-steps above the limit imposed by the Courant-Friedrich-Levy
condition. In Eulerian runs with physics, the combination of physics and small timesteps inhibits this instability (at least in the hydrostatic model), but scores are degraded,especially far from the high resolution pole. In order to avoid this instability, we have
implemented a new formulation of the semi-implicit scheme which allows to avoid mixingof reduced and geographical quantities in the grid-point computations, and which gives animplicit treatment of the gravity waves everywhere on the sphere and not only near thehigh resolution pole. This formulation is a formulation with unreduced divergence (simply
by replacing the quantity M by the mapping factor M ).
Remark: in the deep layer equations, the implicit treatment involves in this case the′′′quantity M 2 D and not D. The small residual term D − M 2 D = (a/r − 1)M 2 D has an
explicit treatment.

∗ ALADIN: The previous point is not an issue in most applications of ALADIN

where the mapping factor M has small variations in the forecast domain (which is not toolarge). It can become an issue for large domains (use of large domains with a tilted-rotatedMercator projection), and option LESIDG becomes useful in this case.

5. Semi-implicit spectral computations and predictor-corrector schemes 135

6 Inclusion of Coriolis term in the semi-implicit scheme.
Although this term is non linear, it is added to linear terms in the semi-implicit scheme.
Equations are written for a leap-frog scheme (Eulerian scheme of three-time level semiLagrangian scheme). For a two-time level semi-Lagrangian scheme replace ∆t by 0.5∆t. In
ARPEGE/IFS this option is coded for both Eulerian and semi-Lagrangian schemes (3Dmodel and shallow-water model). This option is available only in unstretched untilted
spherical geometry, in setting LIMPF=.T. in namelist NAMDYN. The reason to use
such option is accuracy for longe range forecasts (but not stability issues: LIMPF=.F. is
as stable as LIMPF=.T. but may lead to sligntly degraded scores for long-range forecasts).

6.1 Semi-implicit scheme including Coriolis term in the 3Dhydrostatic model.
Thin layer equations.
∗ Expression of the linear term B: Equations (37) and (38) become respec-tively:′• Divergence equation (X = D ):′′B = −∇ 2 (γT + µ log Πs ) − 2∇ (Ω ∧ V)

(55)

′

• Vorticity equation (X = ζ ):′(56)

B = −2k.[∇ ∧ (Ω ∧ V)]∗ System to be solved: Equations (40) and (42) are unchanged. Equation (41) is
replaced by the two following equations, for divergence and vorticity.
′

′

′

′

Dt+∆t + β∆t∇ 2 (γTt+∆t + µlog(Πs )t+∆t ) + β∆t(2∇ (Ω ∧ V)) = D ∗′′′ζt+∆t + β∆t(2k.[∇ ∧ (Ω ∧ V)]) = ζ ∗

(57)
(58)

′

D ∗ , ζ ∗ correspond to X ∗ deﬁned in equation (17) and are available in spectral arrays(SPDIV, SPVOR) at the beginning of the spectral computations.

∗ Restriction to not stretched and not tilted model: Inclusion of Coriolis

term im the semi-implicit scheme will be treated only in not stretched and not tilted
geometry for diﬀerent reasons, including the following considerations:• In not stretched and not tilted geometry, Coriolis parameter f = 2Ω sin θ writes as aﬁrst degree polynomial of the sinus of computational sphere latitude. This property
leads to invert pentadiagonal matrices in the algorithm which will be described.
• In stretched and not tilted geometry, Coriolis parameter f = 2Ω sin θ writes an homographical function of the sinus of computational sphere latitude. This property
leads to invert full matrices in the algorithm which will be described (which in ourcase is very expensive in memory).
• In stretched and tilted geometry, Coriolis parameter f = 2Ω sin θ depends on thecomputational sphere longitude and latitude. This property leads to a coupling
between all spectral coeﬃcients and obliges to solve the semi-implicit system in the
spectral space of geographical sphere, which is very expensive in memory and cost.

136 5. Semi-implicit spectral computations and predictor-corrector schemes• There is another way to treat implicitly Coriolis term in the semi-Lagrangian
scheme which works as well in the two-time level semi-Lagrangian scheme, replacing
the prognostic variable V by V + 2Ω ∧ r, where r is the vertical vector from Earthcentre to computational point (r = ak), and keeping the semi-implicit scheme unchanged. Contrary to inclusion of Coriolis term in the semi-implicit scheme, thismethod does not increase diﬃculty in stretched or tilted geometry.
In not stretched and not tilted geometry, equations (57) and (58) become:Dt+∆t + β∆t∇2 (γTt+∆t + µlog(Πs )t+∆t ) + β∆t(2∇(Ω ∧ V)) = D∗(59)

ζt+∆t + β∆t(2k.[∇ ∧ (Ω ∧ V)]) = ζ ∗(60)∗ Divergence and vorticity in spherical geometry: For a vector Y ofhorizontal components Yx and Yy , the divergence and vertical component of vorticity
write as:1
∇Y =a cos θ∂(Yy cos θ)
∂Yx
+
∂λ
∂θ

1
k.(∇ ∧ Y) =a cos θ





(61)

∂(Yx cos θ)
∂Yy
−
∂λ
∂θ



(62)

∗ Helmholtz equation: Using relations (61) and (62) in equations (59) and (60)lead to the following equations:Dt+∆t +β∆t∇2 (γTt+∆t +µlog(Πs )t+∆t )+β∆t(−2Ω sin θ)ζt+∆t +β∆tζt+∆t + β∆t(2Ω sin θ)Dt+∆t + β∆t







2Ω cos θ
Ut+∆t = D∗ (63)a2Ω cos θ
Vt+∆t = ζ ∗a(64)

The four following expressions are used, for each complex spectral coeﬃcient:• Relationship between divergence and velocity potential χ:D = ∇2 χ(65)• Relationship between vorticity and stream function ψ:ζ = ∇2 ψ(66)• (65) and (66) can be rewritten, for each complex spectral component:n(n + 1)
χ(m,n)a2(67)

n(n + 1)
ψ(m,n)a2(68)

D(m,n) = −ζ(m,n) = −• Relationship between U , ψ and χ:(U a cos θ)(m,n) = imχ(m,n) +(n−1)e(m,n) ψ(m,n−1) −(n+2)e(m,n+1) ψ(m,n+1) (69)

5. Semi-implicit spectral computations and predictor-corrector schemes 137• Relationship between V , ψ and χ:(V a cos θ)(m,n) = imψ(m,n) −(n−1)e(m,n) χ(m,n−1) +(n+2)e(m,n+1) χ(m,n+1) (70)where e(0,0) = 0 and:e(m,n) =rn 2 − m2
4n2 − 1(71)

Equations (67) to (70) allow to eliminate Ut+∆t and Vt+∆t in equations (63) and (64).

h

1 − i 2Ωβ∆tm
D(m,n),t+∆t + β∆t∇2 (γT(m,n),t+∆t + µlog(Πs )(m,n),t+∆t ) − β∆t(2Ω(sin θ)ζ)(m,n),t+∆tn(n+1)hζ(m,n),t+∆t + β∆t(2Ω(sin θ)D)(m,n),t+∆t + β∆t
1 − i 2Ωβ∆tm
n(n+1)i−β∆ti

2Ωe(m,n)2Ωe(m,n+1)∗
ζ(m,n−1),t+∆t + β∆tζ(m,n+1),t+∆t = D(m,n)
n
n+1

−β∆t

(72)

2Ωe(m,n)D(m,n−1),t+∆tn2Ωe(m,n+1)∗
D(m,n+1),t+∆t = ζ(m,n)
n+1

(73)

Multiplication by sin θ is eliminated by using the following relationship valid for any
variable X, in not stretched and not tilted geometry:(74)[(sin θ)X](m,n) = e(m,n) X(m,n−1) + e(m,n+1) X(m,n+1)
Equations (72) and (73) become:h

1 − i 2Ωβ∆tm
D(m,n),t+∆t + β∆t∇2 (γT(m,n),t+∆t + µlog(Πs )(m,n),t+∆t ) − β∆tn(n+1)

h1 − i 2Ωβ∆tm
ζ(m,n),t+∆t + β∆tn(n+1)i−β∆ti

2Ωe(m,n) (n+1)ζ(m,n−1),t+∆tn2Ωe(m,n+1) n∗
ζ(m,n+1),t+∆t = D(m,n)
n+1

(75)

2Ωe(m,n) (n+1)
2Ωe(m,n+1) n
D(m,n−1),t+∆t + β∆tD(m,n+1),t+∆tnn+1∗= ζ(m,n)(76)

ζ is eliminated in equation (75) by using (76) in replacing n by n − 1 then by n + 1.
Equation (75) becomes:22(n−1)(n+1)
2+ (2β∆tΩ)+ (2β∆tΩ)
I − i 2Ωβ∆tm
2Ωβ∆tm e(m,n)
2Ωβ∆tmn(n+1)n21−i1−i(n−1)n2(n+1)(n+2)(n+1)+ (2β∆tΩ)2Ωβ∆tm e(m,n) e(m,n−1) (n−1) D(m,n−2),t+∆t +1−i (n−1)n(2β∆tΩ)22Ωβ∆tm1−i (n+1)(n+2)

e2(m,n+1) (n)(n+2)(n+1)2

D(m,n),t+∆tne(m,n+1) e(m,n+2) (n+2)
D(m,n+2),t+∆t+β∆t∇2 (γT(m,n),t+∆t + µlog(Πs )(m,n),t+∆t )∗+= D(m,n)(2β∆tΩ)
2Ωβ∆tm1−i (n−1)n∗+ζ(m,n−1)e(m,n) (n+1)
n(2β∆tΩ)
2Ωβ∆tm1−i (n+1)(n+2)∗
nζ(m,n+1)e(m,n+1) (n+1)(77)

138 5. Semi-implicit spectral computations and predictor-corrector schemesTt+∆t and log(Πs )t+∆t are eliminated by using equations (40) and (42). That leads to
Helmholtz equation (78):22(n−1)(n+1)
2
I − i 2Ωβ∆tm+ (2β∆tΩ)+ (2β∆tΩ)2Ωβ∆tm e(m,n)
2Ωβ∆tmn(n+1)n21−i1−i(n−1)n(n+1)(n+2)(2β∆tΩ)22(n+1)+ (2β∆tΩ)2Ωβ∆tm e(m,n) e(m,n−1) (n−1) D(m,n−2),t+∆t +1−i (n−1)n+(2β∆tΩ)
2Ωβ∆tm1−i (n−1)ne2(m,n+1) (n)(n+2)− β 2 ∆t2 B ∇2(n+1)22Ωβ∆tm1−i (n+1)(n+2)(2β∆tΩ)

2Ωβ∆tm1−i (n+1)(n+2)

D(m,n),t+∆ne(m,n+1) e(m,n+2) (n+2)
D(m,n+2),t+∆t∗∗∗= D(m,n)− β∆t∇2 (γT(m,n)+ µP(m,n))∗e(m,n) (n+1)ζ(m,n−1)
+
n



∗
n
e(m,n+1) (n+1)ζ(m,n+1)

Equation (78) is solved in eigenmodes space. The right-hand side member needs a multiplication by a tridiagonal complex matrix, for each zonal wave number m. Inversion
of Helmholtz equation is equivalent to invert a pentadiagonal complex matrix (done inroutine SIMPLICO), for each zonal wave number m. All computations are currently (incycle 37T1) done in SPCSI. For m = 0 complex operators become real operators.

∗ Determination of other quantities at t + ∆t: Equation (77) is used tocompute ζt+∆t (for each zonal wave number m, multiplication by a complex tridiagonal
matrix). Then equation (40) is used to compute log(Πs )t+∆t and equation (42) is used tocompute Tt+∆t .

Deep layer equations (according to White and Bromley, 1995).
Equations (55) to (58) remain valid. Only the horizontal part of the Coriolis term (−2Ω ∧V) can be included in the semi-implicit scheme. The term (−2Ω ∧ W k) remains explicit.
Equations (59) to (78) remain valid, replacing D by (rs /a)D, ζ by (rs /a)ζ, ∇ by (rs /a)∇.
The code of spectral computations is unchanged.

6.2 Semi-implicit scheme including Coriolis term in the 2D
shallow-water model.
Such an algorithm is also coded in the 2D shallow-water model in not stretched and nottilted geometry.
• Replace (γT + µlog(Πs )) by Φ in equation (55).

• Replace (γTt+∆t + µlog(Πs )t+∆t ) by Φt+∆t in equations (57), (59) and (63).

• Replace (γT(m,n),t+∆t + µlog(Πs )(m,n),t+∆t ) by Φ(m,n),t+∆t in equations (72), (75)and (77).
∗∗∗• Replace (γT(m,n)+ µP(m,n)) by H(m,n)
in equation (78).
′• Once computed Dt+∆t equation (51) provides Φt+∆t .

6.3 Conclusion.
The simplest conﬁguration to treat the case LIMPF=.T. is the case where the unknown
is the horizontal divergence in the Helmoltz equation: this is the case in the hydrostatic
model and the shallow-water model, and treatment of implicit Coriolis term is similar in
all these cases, with only one call to SIMPLICO.

(

5. Semi-implicit spectral computations and predictor-corrector schemes 139

7 Spectral multiplications by polynomial expressionsof the mapping factor.
∗ Expression of mapping factor M in spectral space in ARPEGE.
Let us denote by:• ac = 0.5(c + 1c )• bc = 0.5(c − 1c )• e(0,0) = 0• e(m,n) =qn2 −m24n2 −1

Expression of M is:(79)

M = ac + b c ξ
where ξ is the sinus of computational sphere latitude. Expression of [M X](m,n) is:[M X](m,n) = bc e(m,n) X(m,n−1) + ac X(m,n) + bc e(m,n+1) X(m,n+1)(80)

It is easy from (80) to retrieve the coeﬃcients of spectral multiplication by any ﬁrst degree
polynomial of M . This is equivalent to a multiplication by a tridiagonal symmetric matrix
in spectral space.

∗ Expression of M 2 in spectral space in ARPEGE.
[M 2 X](m,n) = b2c e(m,n) e(m,n−1) X(m,n−2) + 2ac bc e(m,n) X(m,n−1) + (a2c + b2c (e2(m,n) + e2(m,n+1) ))X(m,n)+2ac bc e(m,n+1) X(m,n+1) + b2c e(m,n+1) e(m,n+2) X(m,n+2)
This is equivalent to a multiplication by a pentadiagonal symmetric matrix in spectral
space.

∗ Expression of M and M 2 in spectral space in ALADIN. This formulais only valid on a tilted-rotated Mercator projection, and it assumes that the reference
latitude of the projection is at the middle of sub-domain C+I.

If we assume that the ALADIN plane coordinates will be not slanted relatively to thelongitudes and latitudes of the Mercator projection, the mapping factor M always depends
only on the y coordinate and never vary along the x coordinate.
M=11= cosh(y/a)
= p
cos θ
1 − µ2where µ = sin θ, the y coordinate (this is an distance measured on the plane projection)
assumes that y = 0 at the apparent equator, and a is the mean Earth radius. M is not a
low order polynomial function of y so even in this case it needs to be approximated. The
approximation used is, for a Fourier decomposition of M 2 , to have only two harmonics.

(81)

140 5. Semi-implicit spectral computations and predictor-corrector schemes

6
Horizontal diffusion

1 Introduction.
∗ General considerations for ARPEGE/IFS: In the code of ARPEGE/IFS
horizontal diﬀusion computations are generally spectral computations. There is a main
horizontal diﬀusion scheme in spectral space. In case of constant resolution, the main
horizontal diﬀusion scheme is discretised by a purely diagonal operator in spectral space.
The main horizontal diﬀusion scheme is implicit in order to remain stable even with highdiﬀusion coeﬃcients and are called after the semi-implicit scheme. Horizontal diﬀusion iscalled after inversion of the semi-implicit scheme and before nudging.

∗ Horizontal diffusion in ALADIN: Horizontal diﬀusion computations arestill spectral computations. There is a main horizontal diﬀusion scheme only. For themain horizontal diﬀusion scheme, the formulation looks like the unstretched ARPEGE
one. There are speciﬁc routines for spectral calculations.
∗ Rayleigh friction: there is a Rayleigh friction in the grid-point space computations which acts only in upper atmosphere for the U -component of the horizontal wind(properly coded only for untilted spherical geometry or untilted plane geometry).
∗ Two-time level semi-Lagrangian scheme: equations are written for aleap-frog scheme. The algorithm remains the same for a two-time level semi-Lagrangian
scheme.

∗ Deep layer equations: For conveniency, equations are written with the geographical horizontal gradient operator ∇, which is used in the horizontal diﬀusion scheme 
for thin layer equations. For deep layer equations, the operator actually used is ar ∇ ,where r is the actual radius, to simulate a “pseudo-geographical" diﬀusion, instead of ∇:this approximation is reasonable and allows to avoid any tricky modiﬁcation in the horizontal diﬀusion spectral computations; use of the true geographical gradient ∇ sticks to
the grid-point calculations.

142

6. Horizontal diffusion∗ Distributed memory: some distributed memory features are now introduced in
the code and will be brieﬂy described. For convenience one uses some generic appellations.
• Expression “DM-local" for a quantity means “local to the couple of processors(proca,procb)": each processor has its own value for the quantity. Expression “DMlocal computations" means that the computations are done independently in eachprocessor on “DM-local" quantities, leading to results internal to each processor,
which can be diﬀerent from a processor to another one.
• Expression “DM-global" for a quantity means that it has a unique value available
in all the processors. Expression “DM-global computations" means that the computations are either done in one processor, then the results are dispatched in all
the processors, or the same computations are done in all the processors, leading to
the same results in all the processors.
• In a routine description the mention “For distributed memory computations are
DM-local" means that all calculations done by this routine are DM-local; the mention “For distributed memory computations are DM-global" means that all calculations done by this routine are DM-global; when no information is provided it meansthat a part of calculations are DM-local and the other part is DM-global.
• Expression “main" processor currently refers to the processor number 1: (proca,procb)=(1,1).

2 Formulation of the horizontal diffusion schemes.
2.1 Main horizontal diffusion scheme.
∗ General formulation: The main horizontal diﬀusion formulation is close to:
′′
′
∂X
~ rX
= −KX M ∇
∂t

(1)

′′

′′

where KX is a vertically dependent and horizontally constant coeﬃcient. KX is generally
complex: exp(0.5πir) multiplied by a real positive coeﬃcient. r is the power of thehorizontal diﬀusion scheme. M is the mapping factor.
In ARPEGE:′′

KX = exp(−0.5πir)"r

Ns (Ns + 1)a2#−r

Ω hX g(2)(a is the mean Earth radius, Ns is the truncation). g is horizontally constant, verticallydependent. Ω is the angular velocity of the Earth rotation (0.00007292115 s−1 ). hX
is a constant coeﬃcient for each prognostic variable. There are seven constants, one
for vorticity (hζ ), one for divergence (hD ), one for temperature (hT ), one for humidity(hq ), one for ozone (hO3 ), one for the extra GFL variables (hEXT ), one for the surfacehydrostatic pressure (hSP ) which is also used for the equivalent height in the 2D model.
There are additional constants for non-hydrostatic variables in the non-hydrostatic model.
For divergence expression of hD matches:
12πa (1 + 0.5rnlginc )2.5=
rdampdivΩhD
Ndlon
rdxtau(3)

6. Horizontal diffusion

143

rnlginc is the increment from linear to quadratic grid (rnlginc = 0 for a linear grid, 1 fora quadratic grid); rdxtau is a tuning parameter; Ndlon is the number of grid-points on alatitude situated near the equator of the computational sphere.
In ALADIN:′′

KX = exp(−0.5πir) (2π)

s

1
L2
L2
x
+ Ny2
2
Nms
s

−r


(4)

Ω hX g(a is the mean Earth radius, Lx and Ly are respectively the zonal and the meridian lengthsof the ALADIN domain, these lengths being taken on a surface situated at a distance aof the Earth center, Nms is the zonal truncation, Ns is the meridian truncation).
(1 + 0.5rnlginc )2.5
1
= [∆X]gp
rdampdivΩhD
rdxtauwhere:[∆X]gp =q(5)

0.5 [∆X]2gp,zonal + [∆X]2gp,merid[∆X]gp,zonal and [∆X]gp,merid are respectively the zonal and meridian grid-point meshsizes taken at a location where the mapping factor is equal to 1.
In both ARPEGE and ALADIN, for the other 3D upper air ﬁelds:
rdampX 1
1=ΩhX
rdampdiv ΩhD(6)which can be rewritten:(ΩhX ) =rdampdiv(ΩhD )
rdampX(7)∗ Modifications brought to this scheme if the SLHD (semi-Lagrangian
horizontal diffusion) interpolations are done in the semi-Lagrangian
scheme: When the advection scheme is a semi-Lagrangian one, it is possible to activate more diﬀusive interpolations in the semi-Lagrangian scheme (“SLHD" diﬀusion), andin this case the horizontal diﬀusion scheme must be modiﬁed:• Less diﬀusion in the above formulation (generally by reducing the value of functiong(l)).
• Adding of a second horizontal diﬀusion scheme for divergence and vorticity (andalso vertical divergence in the NH model): this scheme has the same formulation
as the previous one, but the following variables are changed:
1– hX becomes hhdsX (currently Ωhhds=
X
rdamphds (ΩhX )).

1
1,rdamphds ΩhXi.e.

(ΩhhdsX ) =′′– KX becomes KhdsX .

– the power r becomes s (s is generally above r).

The “SLHD" diﬀusion can now be activated on a subset of prognostic variables and theabove modiﬁcations are applied only to this subset of variables.

144

6. Horizontal diffusion∗ Additional remarks: In the unstretched version of ARPEGE, M = 1. In AL-

ADIN, the horizontal variations of M are generally neglected and M is replaced by a′′constant hidden in the coeﬃcients KX and KhdsX . So the ALADIN diﬀusion looks likethe unstretched ARPEGE one. This approximation may become not very appropriate forlarge domains (used particularly with the rotated-tilted Mercator projection).

2.2 Rayleigh friction.
The following scheme is applied on the U -component of the momentum equation (i.e. thezonal component on the Gaussian grid).





∂U
= −Kfric U∂t fric(8)−Kfric U is a Rayleigh friction which is activated if LRFRIC=.T. in NAMCT0. Kfric
is a vertical dependent coeﬃcient which is non-zero only for a standard pressure lower
than a threshold stored in variable RRFPLM.

2.3 Nudging.
This scheme called after horizontal diﬀusion is a linear relaxation of prognostic variables
towards pre-deﬁned ﬁelds and is not properly saying a diﬀusion scheme, so it will not be
described in detail in this documentation. Pre-deﬁned ﬁelds (for example climatologicalﬁelds) can be read on a ﬁle. Nudging coeﬃcients can be diﬀerent for each variable. Nudging
coeﬃcients are not vertically dependent, except for the upper atmosphere where they canbe set to zero. Nudging is applied to all variables present in spectral space. There is also
nudging for grid-point variables in grid-point space, such surface temperature, surface
moisture, deep moisture and snow depth. Nudging is controlled by LNUDG in namelist
NAMNUD.

3 Discretisation of the horizontal diffusion schemes.
3.1 Main horizontal diffusion scheme.
Main horizontal diffusion scheme in unstretched ARPEGE/IFS.
∗ Discretisation: Calculations are done in spectral space. The discretised equation
which is coded is:+−+X(m,n)− X(m,n)= −Ω hX g(l)f (n, N, n0 (X), x0 , r)DtX(m,n)(9)

where the superscripts + and − indicate respectively variables after horizontal diﬀusionand variables before horizontal diﬀusion, Dt is the time step in the ﬁrst integration stepand twice the time step later (leap frog scheme), n is the total wave number (between 0
and the truncation Ns ), m is the zonal wave number (between −n and n in a triangulartruncation), r is the order of the horizontal diﬀusion operator, N is a reference wave
number, n0 (X) is a threshold depending on variable X (generally zero except for vorticity′
where it is 2 ), x0 is a threshold between 0 and 1. ∇ is the ﬁrst order horizontal reduced′
derivative operator (i.e in the grid point space, ∇ = M ∇ , where M is the mapping
factor). This horizontal diﬀusion scheme is an implicit horizontal diﬀusion scheme (in the6. Horizontal diffusion

145

+
−
right-hand side member of the equation there is X(m,n)
and not X(m,n)). Equation (9)becomes (10):1X−
1 + KX (n)Dt (m,n)(10)

KX (n) = Ω hX g(l)f (n, Ns , n0 (X), x0 , r)

(11)+
X(m,n)=

where:

Expression of f (n, N, n0 (X), x0 , r) is currently:











f (n, N, n0 (X), x0 , r) = max 0., min 1., max(0,n(n+1)−n0 )max(0,N (N +1)−n0 )1 − x0

 12

− x0

r 

 
 (12)

An exact discretisation of equation (1) would give:f (n, N, n0 (X), x0 , r) = f (n, Ns , 0, 0, r)(13)

One can notice that this discretisation is equivalent to a purely diagonal operator inspectral space.

∗ Vertical dependency of g: g(l) only depends on the altitude. For the layer
number l, expression of g(l) is:g(l) = min y0

Πref
1
,
ΠST (l) y3



(14)

where ΠST (l) is the standard atmosphere pressure for the layer number l, Πref is a reference pressure (sea level pressure for the standard atmosphere: 1013.25 hPa). y0 is between
0 and 1: g(l) = 1 if ΠST (l)/Πref is above y0 and g(l) is above 1 if ΠST (l)/Πref is below
y0 . y3 is between 0 and y0 , in practical it is signiﬁcantly lower than y0 and it is used
to avoid too much diﬀusion in the high stratosphere and in the mesosphere. In the 2Dmodel, g(l) can only take one value equal to 1. Deﬁnition of g(l) is diﬀerent in ECMWF
conﬁguration (LECMWF=.T.).

Main horizontal diffusion scheme in ALADIN.
Compared to ARPEGE, the main diﬀerences are:
• KX also depends on the zonal wavenumber m.
• Function f also depends on m and on the zonal truncation Nms , and has a diﬀerent
expression.
• n0 is replaced by zero in the code (for conveniency one will let n0 in the following
formulae).
• Equation (9) becomes:+−+X(m,n)− X(m,n)= −Ω hX g(l)f (m, n, Nm , N, n0 (X), x0 , r)DtX(m,n)

where Nm is a reference zonal wavenumber.

(15)

146

6. Horizontal diffusion• Equation (10) becomes:+X(m,n)=

1X−
1 + KX (m, n)Dt (m,n)(16)• Equation (11) becomes:(17)

KX (m, n) = Ω hX g(l)f (m, n, Nms , Ns , n0 (X), x0 , r)
• Expression of f (m, n, Nm , N, n0 (X), x0 , r) is currently:











f (m, n, Nm , N, n0 (X), x0 , r) = max 0., min 1., 

2
n2
+ Nm2
Ns2
ms

 21

− x0

1 − x0• An exact discretisation of equation (1) would give:f (m, n, Nm , N, n0 (X), x0 , r) = f (m, n, Nms , Ns , 0, 0, r)

r 

 
 (18)(19)

Main horizontal diffusion scheme in stretched ARPEGE/IFS.
∗ Expression of mapping factor in spectral space. Let us denote by:• ac = 0.5 c + 1c
• bc = 0.5 c − 1c

• e(0,0) = 0• e(m,n) =

q





n2 −m24n2 −1

Expression of M is:(20)

M = ac + b c µwhere µ is the sinus of computational sphere latitude. Expression of [M X](m,n) is:[M X](m,n) = bc e(m,n) X(m,n−1) + ac X(m,n) + bc e(m,n+1) X(m,n+1)(21)

A multiplication by M in spectral space is equivalent to a multiplication by a symmetrictridiagonal matrix.

∗ Discretisation: Calculations are done in spectral space. The discretised equation
which is coded is:−X(m,n)=++bc e(m,n) Ω hX g(l)Dtf (n − 1, N, n0 (X), x0 , r)X(m,n−1)++(1 + ac Ω hX g(l)Dtf (n, N, n0 (X), x0 , r))X(m,n)++bc e(m,n+1) Ω hX g(l)Dtf (n + 1, N, n0 (X), x0 , r)X(m,n+1)(22)Equation (22) is equivalent to invert a tridiagonal matrix for each zonal wave number
m. The computation of this matrix and a decomposition into a product of two triangularbidiagonal matrixes are performed in the set-up routine SUHDU. At each time step these+
two triangular bidiagonal matrixes are inverted in order to compute the X(m,n).

6. Horizontal diffusion

147

3.2 Modification of the horizontal diffusion scheme if “SLHD"interpolations are done in the semi-Lagrangian scheme.
∗ Modified expression of g(l): Expression of g(l) is now modiﬁed; new expressionof g(l) is:



Πref
1
,
g(l) = min y0ΠST (l) y3



(23)

− sdred∗ Unstretched ARPEGE/IFS: A double diﬀusion is applied on vorticity anddivergence. It writes:+
X(m,n)=



1
1 + KhdsX (n)Dt

1
1 + KX (n)Dt

−X(m,n)(24)

where:(25)

KhdsX (n) = Ω hhdsX ghds (l)f (n, Ns , n0 (X), x0 , s)

∗ ALADIN: A double diﬀusion is applied on vorticity and divergence. It writes:






1
−
X(m,n)
1 + KX (m, n)Dt

(26)

KhdsX (m, n) = Ω hhdsX ghds (l)f (m, n, Nms , Ns , n0 (X), x0 , s)(27)+
X(m,n)=

1
1 + KhdsX (m, n)Dt

where:∗ Stretched ARPEGE/IFS: A double diﬀusion is applied on vorticity and di-vergence. It writes:−
X(m,n)=+++bc e(m,n) Ω hX g(l)Dtf (n − 1, N, n0 (X), x0 , r)X(m,n−1)+++(1 + ac Ω hX g(l)Dtf (n, N, n0 (X), x0 , r))X(m,n)+++bc e(m,n+1) Ω hX g(l)Dtf (n + 1, N, n0 (X), x0 , r)X(m,n+1)

(28)

then:
++
X(m,n)=++bc e(m,n) Ω hhdsX ghds (l)Dtf (n − 1, N, n0 (X), x0 , s)X(m,n−1)++(1 + ac Ω hhdsX ghds (l)Dtf (n, N, n0 (X), x0 , s))X(m,n)++bc e(m,n+1) Ω hhdsX ghds (l)Dtf (n + 1, N, n0 (X), x0 , s)X(m,n+1)(29)This discretisation requires to invert two tridiagonal symmetric matrices in spectral space.

148

6. Horizontal diffusion∗ Expression of ghds (l): Expression of ghds (l) is:


Πref
1
,
ghds (l) = min yS0
ΠST (l) y3



(30)

3.3 Particular use of a second reference truncation N2 forstandard pressure levels ΠST (l) below y0 ∗ Πref .
In ARPEGE, it is possible in the code to modify g(l) in order to simulate a replacingof f (n, Ns , n0 (X), x0 , r) by f (n, N2 , n0 (X), x0 , r), where N2 < Ns , for standard pressuresΠST (l) < y1 ∗ Πref , where y1 < y0 . In this case g(l) also depends on n for standard
pressures below y0 ∗ Πref .
If ΠST (l)/Πref is below y1 (y3 is assumed to be lower than y1 ):



Πref
1
g(l, n) = min y0,ΠST (l) y3



f (n, N2 , n0 (X), x0 , r)f (n, Ns , n0 (X), x0 , r)(31)

If ΠST (l)/Πref is over y1 and below y0 :g(l, n) =y0 ∗ Πref − b(n)ΠST (l) − b(n)(32)

where:
b(n) =(y0 ∗ Πref )(y1 ∗ Πref )(f (n, N2 , n0 (X), x0 , r) − f (n, Ns , n0 (X), x0 , r))(y0 ∗ Πref )f (n, N2 , n0 (X), x0 , r) − (y1 ∗ Πref )f (n, Ns , n0 (X), x0 , r)(33)

One can check that (32) and (33) yield g(l, n) = 1 for ΠST = y0 ∗ Πref and that (31) and(32) are continuous for ΠST = y1 ∗Πref . When taking N2 = Ns , one retrieves formula (14).
Choice of N2 < Ns is useful in climate modelisation.
The same type of modiﬁcation can be extended to ghds .
This option is not coded in ALADIN.

3.4 Application of diffusion to each variable.
Application to GMV variables and to GFL variables in 3D models.
For GMV variables, horizontal diﬀusion is applied to:′• “reduced" vorticity ζ with n0 (ζ) = 2 at METEO-FRANCE, 0 at ECMWF.
′• “reduced" divergence D with n0 (D) = 0.
• for temperature, a variable which looks like T − α log(Πs ) (see part 3.4), withn0 (T ) = 0.
• log(Πs ) should not be diﬀused (even if some diﬀusion code is implemented for thisvariable).
For GFL variables, horizontal diﬀusion is applied to:• speciﬁc humidity q if spectral one, with n0 (q) = 0.
• ozone O3 if spectral one, with n0 (O3) = 0.
• extra GFL variables if spectral ones, with n0 = 0.
• For the other advectable GFL variables and for the non-advectable GFL variables,
no diﬀusion is currently coded.

6. Horizontal diffusion

149

Application to temperature.
Diﬀusion has to be done on a variable less sensitive to the orography than T , so one chooses
a variable looking like T −α log(Πs ) with a “good" formulation of α. T −α log(Πs ) is diﬀused
instead of T , where α is a coeﬃcient depending on the altitude and using parameters
related to standard atmosphere. Let us denote by TsST the surface temperature in astandard
atmosphere (288.15 K), R the air constant, g the acceleration due to gravity,
 dT 
thetropospheric temperature vertical gradient in a standard atmosphere (-0.0065
dz STK/m), TtST the tropopause temperature in a standard atmosphere (217.15 K), and Πrefa reference temperature equal to 1013.25 hPa. If:(TsST )(Π/Πref )(R/g)[ dz ]ST > (TtST )dT

the expression of αl for the layer number l is:αl = −Bl (R/g)

h

dT
dz

i

dT
(TsST )(Π/Πref )−((R/g)[ dz ]ST +1)

(34)

ST

where Bl is used in the deﬁnition of pressure on layers and interlayers (hybrid verticalcoordinate). In the other cases:(35)

αl = 0Temperature horizontal diﬀusion writes as:+(T(m,n)− α log(Πs )−(m,n) ) =1
(T −− α log(Πs )−(m,n) )
1 + KT (n)Dt (m,n)(36)

Equation (36) yields (37):+−T(m,n)= (T(m,n)+ KT (n)Dtα log(Πs )−(m,n) )/(1 + KT (n)Dt)(37)

For temperature n0 (T ) = 0.

Application to GMV variables in 2D models.
Horizontal diﬀusion is performed for vorticity, divergence and equivalent height.

3.5 Other spectral calculations.
∗ Numerical horizontal diffusion for post-processed fields by FULLPOS. See documentation (IDFPOS) about FULL-POS. This diﬀusion is done in routine SPOS.

∗ Rayleigh friction (ARPEGE/IFS only). The following scheme is discre-

tised in grid-point space for the U -component of the momentum equation (i.e. the zonal
component on the Gaussian grid). In equation (8) −Kfric U is a Rayleigh friction whichis activated if LRFRIC=.T. in NAMCT0. Kfric is a vertical dependent coeﬃcientwhich is non-zero only for a standard pressure lower than Πrfric . Kfric is computed in
SURAYFRIC according the following formula for a layer l of standard pressure ΠST (l)lower than Πrfric :Kfric (l) =1tanh (Rfz1 − 7 log(100000/ΠST (l)))
1 − 7.7
3 ∗ 86400(38)

Rfz1 is stored in variable RRFZ1 in YOMDYN. Kfric is stored in variable RKRF
in YOMDYN. This scheme has never been used at METEO-FRANCE, but only at
ECMWF. It is suited for models having layers in the high stratosphere, with an untilted
geometry (i.e. NSTTYP=1).

150

6. Horizontal diffusion

7
Radiative fluxes
This chapter describes the salient features concerning the radiation scheme, used in NWP
as well as in ARPEGE-CLIMAT. The present chapter is based on the IFS DocumentationCy37r2. The detailed original ECMWF scientiﬁc documentation is available online at
http://www.ecmwf.int/sites/default/files/elibrary/2012/9239-part-iv-physical-processes.pdf.
The package calculates the radiative ﬂuxes taking into account absorption-emission of longwave radiation and reﬂection, scattering and absorption of solar radiation by the earth’s
atmosphere and surfaces. The longwave radiation scheme is based on that of the ECMWF
model, the Rapid Radiation Transfer Model (RRTM). The shortwave part of the scheme,
originally developed by Fouquart and Bonnel (1980), solves the radiation transfer equationand integrates the ﬂuxes over the whole shortwave spectrum between 0.2 and 4 mm. Note
that, since 2007, a new approach to the inclusion of the cloud eﬀects on radiation ﬁelds(the Monte-Carlo Independent Column Approximation, McICA) has been introduced in
the ECMWF radiation schemes , but not used yet in the ARPEGE NWP (where tests
have been done) and Climate models. The radiative heating rate is computed as thedivergence of net radiation ﬂuxes F so thatg ∂F∂T)rad = −
∂t
cp ∂p

(1)

where F is a net ﬂux: i.e. F = F↑ + F↓ sum of the upward F↑ and downward F↓ ﬂuxes,
and a total ﬂux: i.e. F = FLW + FSW sum of the solar or shortwave FSW and atmosphericor longwave FLW ﬂuxes and cp is the speciﬁc heat at constant pressure of moist air.
Sections 1 and 2 describe the computation of the shortwave and longwave radiative ﬂuxes
respectively. A description of the inputs, in particular the climatologically deﬁned quantities of radiative importance is given in Section 3. In the CMIP6 version, the full radiation
scheme is called every hour (corresponding to NRADFR=-1). There is no modiﬁcationof the ECMWF code done in the ARPEGE code, except the choices made for opticalparameters and overlap hypothesis. Note also that, since 2007, a new approach to theinclusion of the cloud eﬀects on radiation ﬁelds (the Monte-Carlo Independent Column
Approximation, McICA approximation) has been introduced in the ECMWF radiation
schemes, but is not used in the ARPEGE NWP and Climate current versions.

1 Shortwave radiation
1.1 First glance
The rate of atmospheric warming by absorption and scattering of shortwave radiation is:

152

7. Radiative fluxesg ∂FSW
∂T
=
∂t
cp ∂p

(2)

where FSW is the net total shortwave ﬂux, expressed in W m−2 and positive when downward.

F =

Z ∞

dν

0

Z 2π

dφ

0

Z +1

µ Lν (δ, µ, φ) dµ dφ

(3)

−1

Lν is the diﬀuse radiance at wavenumber ν, in a direction given by φ, the azimuth angleand ϑ the zenith angle such as µ = cos ϑ. In (3), we assume a plane parallel atmospherewith the optical depth δ, as a convenient vertical coordinate when the energy source is
outside the medium

δ(p) =Z 0(4)

βν (p) dppwhere βνext (p) is the extinction coeﬃcient equal to the sum of the scattering coeﬃcient βνscaof the aerosol and cloud particle absorption coeﬃcient βνabs and of the purely molecularabsorption coeﬃcient kν . The diﬀuse radiance Lν is governed by the radiation transfer
equationµdLν (δ, µ, φ)dδ=Lν (δ, µ, φ) −−ω ν (δ)4ω ν (δ)− δPν (δ, µ, φ, µ0 , φ0 ) Eν0 e µ04Z 2πZ +10Pν (δ, µ, φ, µ′ , φ′ ) Lν (δ, µ′ , φ′ ) dµ′ dφ′ .

(5)
(6)

−1

Eν0 is the incident solar irradiance in the direction µ0 = cosϑ0 , ω ν is the single scattering
albedo (= βνsca /kν ) and Pν (δ, µ, φ, µ′ , φ′ ) is the scattering phase function which deﬁnes theprobability that radiation coming from direction (µ′ , φ′ ) is scattered in direction (µ, φ).
The shortwave part of the scheme, originally developed by Fouquart and Bonnel (1980 ),solves the radiation transfer equation and integrates the ﬂuxes over the whole shortwave
spectrum between 0.2 and 4 µm. Upward and downward ﬂuxes are obtained from thereﬂectances and transmittances of the layers, and the photon-path-distribution method
allows to separate the parametrization of the scattering processes from that of the molecular absorption.

1.2 Spectral integration
Solar radiation is attenuated by absorbing gases, mainly water vapor, uniformly mixed
gases (oxygen, carbon dioxide, methane, nitrous oxide) and ozone, and scattered bymolecules (Rayleigh scattering), aerosols and cloud particles. Since scattering and molecular absorption occur simultaneously, the exact amount of absorber along the photon
path length is unknown, and band models of the transmission function cannot be used directly as in the longwave radiation transfer. The approach of the photon path distribution
method is to calculate the probability p(U ) dU that a photon contributing to the ﬂux Fc
in the conservative case (i.e. no absorption, ω ν = 1, kν = 0) has encountered an absorber7. Radiative fluxes

153

amount between U and U + dU . With this distribution, the radiative ﬂux at wavenumber
ν is related to Fc by

Fν = Fc

Z ∞(7)p(U ) exp(−kν U ) dU

0

and the ﬂux averaged over the spectral interval ∆ν can then be calculated with the helpof any band model of the transmission function t∆ν

F =

1
∆ν

Z

Fν dν = Fc

Z ∞

(8)

p(U ) t∆ν (U ) dν.

0∆ν

To ﬁnd the distribution function p(U ), the scattering problem is solved ﬁrst, by any
method, for a set of arbitrarily ﬁxed absorption coeﬃcients kl , thus giving a set of simulated ﬂuxes Fkl . An inverse Laplace transform is then performed on (7) to get p(U )(Fouquart 1974 ). The main advantage of the method is that the actual distribution p(U )is smooth enough that (7) gives accurate results even if p(U ) itself is not known accurately.
In fact, p(U ) needs not be calculated explicitly as the spectrally integrated ﬂuxes are, inthe two limiting cases of weak and strong absorption:

F = Fc t∆ν (< U >)where

< U >=Z ∞p(U ) U dU

(9)

0

1

F = Fc t∆ν (< U 2 >)

where

1

< U 2 >=Z ∞1p(U ) U 2 dU

(10)

0

respectively. The atmospheric absorption in the water vapor bands is generally strong1
and the scheme determines an eﬀective absorber amount Ue between < U > and < U 2 >derived from

Ue =

Fk
1
ln( e )
ke
Fc

(11)

where ke is an absorption coeﬃcient chosen to approximate the spectrally averaged transmission of the clear-sky atmosphere:

ke = (

Utot
Utot −1) ln(t∆ν
)
µ0
µ0

(12)

with Utot the total amount of absorber in a vertical column and µ0 = cos ϑ0 . Once theeﬀective absorber amounts of H2 O and uniformly mixed gases are found, the transmissionfunctions are computed using Padé approximants:

154

7. Radiative fluxest∆ν (U ) =NPai U i−1

i=0
N

P

.

(13)

bj U j−1

j=0

Absorption by ozone is also taken into account, but since ozone is located at low pressurelevels for which molecular scattering is small and Mie scattering is negligible, interactions
between scattering processes and ozone absorption are neglected. Transmission throughozone is computed using (13) where the amount of ozone UO3 is:

d
UO
=M
3

Z 0

dUO3

p

for the downward transmission of the direct solar beam, and:

u
UO
=r
3

Z p

d
(ps )
dUO3 + UO
3

psfor the upward transmission of the diﬀuse radiation with r = 1.66 the diﬀusivity factor
and M , the magniﬁcation factor (Rodgers 1967 ) used instead of µ0 to account for the
sphericity of the atmosphere at very small solar elevations:

M= p

35
µ20 + 1.

(14)

To perform the spectral integration, it is convenient to discretize the solar spectrum into
subintervals in which the surface reﬂectance, molecular absorption characteristics, andcloud optical properties can be considered as constants. One of the main causes forsuch a spectral variation is the sharp increase in the reﬂectivity of the vegetation in thenear-infrared. Also, water vapour does not absorb below 0.69 µm nor do liquid water
clouds. Till June 2000, the ECMWF shortwave scheme considered only two spectralintervals, one for the visible (0.2/0.69 µm), one for the near-infrared (0.69/4.00 µm) partsof the solar spectrum. From June 2000 to April 2002, the near-infrared interval was subdivided into three intervals (0.69/1.19/2.38/4.00 µm) to account better for the spectral
variations of the cloud optical properties. Till April 2002, all the molecular absorption
coeﬃcients (for O3, H2O, uniformly mixed gases) were derived from statistical models of
the transmission function using spectroscopic parameters derived from various versionsof the HITRAN database (Rothman et al.,1987,1992 ). In April 2002, following therecomputation of all the molecular absorption coeﬃcients from an updated version of theshortwave lineby- line model of Dubuisson et al. (1996) using spectroscopic data fromHAWKS (2000, http://www.hitran.com) the ultraviolet and visible part of the spectrum
are now considered in three spectral intervals (0.20/0.25/0.69 µm) making the scheme
having a total of six spectral intervals over which the aerosol and cloud optical properties
are also deﬁned. The cut-oﬀ at 0.69 µm allows the scheme to be more computationaleﬃcient, in as much as the interactions between gaseous absorption (by water vapour anduniformly mixed gases) and scattering processes are accounted for only in the near-infraredinterval(s).

7. Radiative fluxes

155

1.3 Vertical integration
Considering an atmosphere where a fraction Ctot (as seen from the surface or the top of
the atmosphere) is covered by clouds (the fraction Ctot depends on which cloud overlap
assumption is assumed for the calculations), the ﬁnal ﬂuxes are given as↓↓F↓ = Ctot Fcloudy+ (1 − Ctot ) Fclear(15)

with a similar expression holding for the upward ﬂux. Contrarily to the scheme of Geleyn
and Hollingsworth (1979 ), the ﬂuxes are not obtained through the solution of a systemof linear equations in a matrix form. Rather, assuming an atmosphere divided into N
homogeneous layers, the upward and downward ﬂuxes at a given interface j are given by:F↓ (j)=F0N
Y

Tb (k),(16)

k=jF↑ (j)=F↓ (j) Rt (j − 1),(17)

where Rt (j) and Tb (j) are the reﬂectance at the top and the transmittance at the bottomof the j th layer. Computations of Rt ’s start at the surface and work upward, whereas
those of Tb ’s start at the top of the atmosphere and work downward. Rt and Tb accountfor the presence of cloud in the layer:

Rt
Tb

=
=

Cj Rcdy + (1 − Cj ) Rclr ,Cj Tcdy + (1 − Cj ) Tclr .

(18)
(19)

The subscripts clr and cdy respectively refer to the clear-sky and cloudy fractions of thelayer with Cj the cloud fraction of the layer j.

Cloudy fraction of the layers
Rtcdy and Tbcdy are the reﬂectance at the top and transmittance at the bottom of thecloudy fraction of the layer calculated with the Delta-Eddington approximation. Givenδc , δa and δg , the optical thicknesses for the cloud and the aerosol, and the molecularabsorption of the gases, gc and ga the cloud and aerosol asymmetry factors, Rtcdy andTbcdy are calculated as functions of:• the total optical thickness of the layer δ ∗ :δ ∗ = δc + δa + δg

• the total single scattering albedo:ω∗ =δc + δaδc + δa + δg

(20)

156

7. Radiative fluxes• the total asymmetry factor:
g∗ =

δa
δc
gc +
ga
δc + δaδc + δa(21)of the reﬂectance R_ of the underlying medium (surface or layers below the j th interface)and of the eﬀective solar zenith angle µe (j) which accounts for the decrease of the direct
solar beam and the corresponding increase of the diﬀuse part of the downward radiationby the upper scattering layers.
The eﬀective solar zenith angle µe (j) is equal to:µe (j) =



ef f
(j))
(1 − Ccld
ef f+ r Ccld
(j)
µ

−1

,

(22)

ef f
with Ccld(j) the eﬀective total cloudiness over level j and r the diﬀusivity factor.

ef f
Ccld
(j) = 1 −

N
Y

i=j+1

(1 − Ccld (i)E(i))

and(1 − ωc (i)gc (i)2 )δc (i)E(i) = 1 − exp −
µ



(23)

where δc (i), ωc (i) and gc (i) are the optical thickness, single scattering albedo and asymmetry factor of the cloud in the ith layer.
The scheme follows the Eddington approximation, ﬁrst proposed by Shettle and Weiman(1970), then modiﬁed by Joseph et al. (1976) to account more accurately for the large
fraction of radiation directly transmitted in the forward scattering peak in case of highly
asymmetric phase functions. Eddington’s approximation assumes that, in a scattering
medium of optical thickness δ ∗ , of single scattering albedo ω, and of asymmetry factor g,
the radiance L entering (5) can be written as:

L(δ, µ) = L0 (δ) + µ L1 (δ).

(24)

In that case, when the phase function is expanded as a series of associated Legendre
functions, all terms of order greater than one vanish when (5) is integrated over µ and φ.
The phase function is therefore given by

P (θ) = 1 + β1 (θ) cos θ,
where θ is the angle between incident and scattered radiances. The integral in (5) thus
becomes

Z 2πZ +10

−1

P (µ, φ, µ′ , φ′ ) L(µ′ , φ′ ) dµ′ dφ′ = 4π (L0 + πL1 )(25)

7. Radiative fluxes

157

where g, the asymmetry factor identiﬁes as

1
β1
=
g=
3
2

Z +1

P (θ) cos θ d(cos θ).

−1

Using (25) in (5) after integrating over µ and dividing by 2π, we getµd(L0 + µL1 )1−δ) (1 + 3gµ0 µ).(26)= −(L0 + µL1 ) + ω (L0 + gµL1 ) + ω F0 exp(dδ
4µ0

where ω is the single scattering albedo.
We obtain a pair of equations for L0 and L1 by integrating (26) over µ:d(L0 )3−δ),= −3(1 − ω) L0 + ω F0 exp(
dδ
4
µ0

(27)

d(L1 )3−δ).
= −(1 − ωg) L1 + ω g µ0 F0 exp(
dδ
4
µ0

(28)

For the cloudy layer assumed non-conservative (ω < 1), the solutions to (27) and (28) are,in the range 0 ≤ δ ≤ δ ∗ :L0 (δ)=

L1 (δ)

=

−δ
),
µ0
−δ
),
p(C1 exp(−kδ) − C2 exp(+kδ)) − β exp(
µ0
C1 exp(−kδ) + C2 exp(+kδ) − α exp(

(29)
(30)

where

k

=

p

=

α

=

β

=





1

3 (1 − ω) (1 − ωg) 2

1

2
3 (1 − ω)/(1 − ωg)
1 + 3g (1 − ω)
3 ω F 0 µ02 2
 4 (1 − k µ0 ) 2 
1 + 3 g (1 − ω) µ0.
3 ω F 0 µ0
4 (1 − k2 µ20 )The two boundary conditions allow to solve the system for C1 and C2 . First, the downward
directed diﬀuse ﬂux at the top of the layer is zero

158

7. Radiative fluxesF↓ (0) = L0 (0) +which translates into
(1 +


2
L1 (0) = 0,
3

2p
2β
2p
) C1 + (1 −) C2 = α +
.
3
3
3

(31)

For the second condition, one assumes that the upward directed ﬂux at the bottom of thelayer is equal to the product of the downward directed diﬀuse and direct ﬂuxes by thecorresponding diﬀuse and direct reﬂectances (Rd and R_, respectively) of the underlying
mediumF↑ (δ ∗ ) = L0 (δ ∗ ) −




2
2
−δ ∗
),
L1 (δ ∗ ) = R_ L0 (δ ∗ ) + L1 (δ ∗ ) + Rd µ0 F0 exp(33µ0which translates into

(1 − R_ −2p2p(1 + R_)) C1 exp(−k δ ∗ ) + (1 − R_ +(1 + R_)) C2
3
3
2= ((1 − R_)α − (1 + R_)β + Rd µ0 F0 )3

exp(+k δ ∗()32)∗

exp( −δ)(33).
µ0

In the Delta-Eddington approximation, the phase function is approximated by a Diracdelta function (forward scatter peak) and a two-term expansion of the phase function
P (θ) = 2f (1 − cos θ) + (1 − f ) (1 + 3g ′ cos θ),
where f is the fractional scattering into the forward peak and g ′ the asymmetry factor of
the truncated phase function. As shown by Joseph et al. (1976), these parameters are:

f
g

′

=

g2

(34)

=

g
.
1+g

(35)

The solution of the Eddington’s equations remains the same provided that the total optical
thickness, single scattering albedo and asymmetry factor entering (26)-(32) take theirtransformed values:

δ
ω

′

′

=
=

(1 + ωf ) δ ∗ ,(1 − f ) ω.
1 − ωf(36)(37)

Practically, the optical thickness, single scattering albedo, asymmetry factor, and solar
zenith angle entering (26)-(31) are δ ∗ , ω ∗ , g ∗ and ue deﬁned in (21) and (22).

7. Radiative fluxes

159

Clear-sky fraction of the layers
In the clear-sky fraction of the layers, the shortwave scheme accounts for scattering andabsorption by molecules and aerosols. The following calculations are practically done
tot
twice, once for the clear-sky fraction (1 − Ccld
) of the atmospheric column µ with equal toµ0 , simply modiﬁed for the eﬀect of Rayleigh and aerosol scattering, the second time fortot
the clear-sky fraction of each individual layer within the fraction Ccldof the atmosphericcolumn containing clouds, with µ equal to µe .
As the optical thickness for both Rayleigh and aerosol scattering is small, Rclr (j − 1)
and Tclr (j) the reﬂectance at the top and transmittance at the bottom of the j th layer
can be calculated using respectively a ﬁrst and a second-order expansion of the analyticalsolutions of the two-stream equations similar to that of Coakley and Chylek (1975). For
Rayleigh scattering, the optical thickness, single scattering albedo and asymmetry factor
are respectively δR , ωR = 1 and gR = 0, so that

RR

=

TR

=

δR
,
2 µ + δR2µ.
2 µ + δR

(38)
(39)

The optical thickness δR of an atmospheric layer is simply:∗
δR = δR(p(j) − p(j − 1))
,
psurf

∗
where δR
is the Rayleigh optical thickness of the whole atmosphere parameterized as afunction of solar zenith angle (Deschamps et al. 1983):

∗
δR
=

5
X

ai µi−10 .

i=0For aerosol scattering and absorption, the optical thickness, single scattering albedo andasymmetry factor are respectively δa , ωa (with 1 − ωa ≪ 1) and ga so that:den = 1 + (1 − ωa + back(µe ) ωa )R(µe ) =back(µe ) ωa δa /µeden

T (µe ) =

1
den

δa 2
δa
+ (1 − ωa ) (1 − ωa + 2 back(µe ) ωa ) 2µeµe

where back(µe ) = (2 − 3µe ga )/4 is the backscattering factor.

(40)

160

7. Radiative fluxes

Practically, Rclr and Tclr are computed using (40) and the combined eﬀect of aerosol andRayleigh scattering comes from using modiﬁed parameters corresponding to the additionof the two scatters with provision for the highly asymmetric aerosol phase function througha Delta-Eddington approximation of the forward scattering peak (as in (34)-(36)):

δ + = δR + δa (1 − ωa ga2 )
g+ =

δa
ga
1 + ga (δR + δa )ω+ =

ωa (1 − ga2 )
δR
δa
ωR +
δR + δaδR + δa 1 − ωa ga2

As for their cloudy counterparts, Rclr and Tclr must account for the multiple reﬂections
due to the layers underneath:

Rclr

=

Tclr

=

T (µe )
R_,
1 − R∗ R_
T (µe ),
(1 − R∗ R_)

R(µe ) +

(41)
(42)

with R∗ = R(1/r), T ∗ = T (1/r), R_ = Rt (j − 1) is the reﬂectance of the underlying
medium and r is the diﬀusivity factor.
Since interactions between molecular absorption and Rayleigh and aerosol scattering are
negligible, the radiative ﬂuxes in a clear-sky atmosphere are simply those calculated from(16) and (41) attenuated by the gaseous transmissions (13).

1.4 Multiple reflections between layers
To deal properly with the multiple reﬂections between the surface and the cloud layers,
it should be necessary to separate the contribution of each individual reﬂecting surface tothe layer reﬂectances and transmittances in as much as each such surface gives rise to aparticular distribution of absorber amount. In case of an atmosphere including N cloudlayers, the reﬂected light above the highest cloud consists of photons directly reﬂectedby the highest cloud without interaction with the underlying atmosphere and of photons
that have passed through this cloud layer and undergone at least one reﬂection on theunderlying atmosphere. In fact, (8) should be written

F =

N
X
l=0

Fcl

Z ∞

pl (U ) t∆ν (U ) dν,

(43)

0

where Fcl and pl (U ) are the conservative ﬂuxes and the distributions of absorber amountcorresponding to the diﬀerent reﬂecting surfaces.

7. Radiative fluxes

161

Fouquart and Bonnel (1980) have shown that a very good approximation to this problem is obtained by evaluating the reﬂectance and transmittance of each layer (using (32)and (41)), assuming successively a non-reﬂecting underlying medium (R_ = 0), then areﬂecting underlying medium (R_ 6= 0). First calculations provide the contribution to
reﬂectance and transmittance of those photons interacting only with the layer into consideration, whereas the second ones give the contribution of the photons with interactions
also outside the layer itself.
From these two sets of layer reﬂectances and transmittances (Rt0 , Tt0 ) and (Rt6= , Tt6= ) respectively, eﬀective absorber amounts to be applied to computing the transmission functions for upward and downward ﬂuxes are then derived using (11) and starting from thesurface and working the formulas upward:

U↓e0 =Tb1
ln( 0 ),keTb cU↓e6= =

Tb6=
1
),
ln(
ke
Tb c

U↑e0 =

1
Rt
ln( 0 ),
keRtc

U↑e6= =

Rt6=
1
),
ln(
ke
Rtc

where Rtc and Tbc are the layer reﬂectance and transmittance corresponding to a conservative scattering medium. Finally the upward and downward ﬂuxes are obtained as:↑

F (j) = F0↓

F (j) = F0

Rt0 t∆ν (U↑e0 ) + (Rt6= − Rt0 ) t∆ν U↑e6= )

Tb0 t∆ν (U↓e0 ) + (Tb6= − Tb0 ) t∆ν U↓e6= )



(44)



(45)

2 Longwave radiation: the RRTM scheme
The main characteristics of RRTM are:
• Solution of radiative transfer equation: Two-stream method• Number of spectral intervals: 16• Absorbers : H2 O, CO2 , O3, CH4 , N2 O, CF C11, CF C12, aerosols• Spectroscopic data base: HITRAN 1996
• Absorption coeﬃcient: From LBLRTM line-by-line model• Cloud handling: True cloud fraction

162

7. Radiative fluxes• Cloud optical properties: 16-band spectral emissivity• Cloud overlap assumption: Maximum random• References : Mlawer et al. (1997)

As stated in Mlawer et al. (1997), the objective in the development of RRTM has been
to obtain an accuracy in the calculation of ﬂuxes and heating rates consistent with thebest line-by-line models. It utilizes the correlated-k methode and shows its ﬁliation to
the Atmospheric and Environmental Research, Inc. (AER) line-by-line model (LBLRTM;
Clough et al. 1989, 1992; Clough and Iacono 1995) through its use of absorption coeﬃcients for the relevant k-distributions derived from LBLRTM. Therefore the k-coeﬃcients
in RRTM include the eﬀect of the CKD2.2 water vapour continuum (Clough et al. 1989).
The main point in the correlated-k method (Lacis and Oinas 1991; Fu and Liou 1992 )is the mapping of the absorption coeﬃcient k(ν) from the spectral space (where it varies
irregularly with wavenumber ν) to the g-space (where g(k) is the probability distributionfunction, i.e. the fraction of the absorption coeﬃcients in the set smaller than k). The
eﬀect of this reordering is a rearrangement of the sequence of terms in the integral overwavenumber in the radiative transfer equation (RTE), which makes it equivalent to what
would be done for monochromatic radiation.
In the ECMWF (hence, ARPEGE-CLIMAT) model, no provision is presently taken for
scattering in the longwave. Therefore, in order to get the downward radiance, the integration over the vertical dimension is simply done starting from the top of the atmosphere,
going downward layer by layer. At the surface, the boundary condition (in terms of spectral emissivity, and potential reﬂection of downward radiance) is computed, then, in orderto get the upward radiance, the integration over the vertical dimension is repeated, thistime from the surface upward.
The spectrally averaged radiance (between ν1 and ν2 ) emerging from an atmospheric layeris1R=(ν1 − ν2 )

Z ν1
ν2

dν



R0 (ν) +

Z 1
tν



B(ν, T (t′ν )) − R0 (ν)



dt

′



(46)

where R0 is the incoming radiance to the layer, B(ν, T ) is the Planck function at wavenumber ν and temperature T , tν is the transmittance for the layer optical path, and t′ν is thetransmittance at a point along the optical path in the layer. Under the mapping ν → g,this becomes

R=

Z 1

dg

0





Beff (g, Tg ) + [R0 (g) − Beff (g, Tg )] exp −k(g, P, T )

ρδz
cos φ



(47)

where Beff (g, T ) is an eﬀective Planck function for the layer that varies with the layer’stransmittance such as to ensure continuity of ﬂux across layer boundaries for opaqueconditions. The dependence of the transmittance is now written in terms of the absorption
coeﬃcient k(g, P, T ) at layer pressure P and temperature T , the absorber density ρ, thevertical thickness of the layer δz, and the angle φ of the optical path.
For a given spectral interval, the domain of the variable g is partitioned into subintervals(see Table 7.1, number of g-points), each corresponding to a limited range of k(g) values
and for which a characteristic value κj of the absorption coeﬃcient is chosen. These κj
are then used to compute the outgoing radianceR = Σj Wj





ρδz
Beff j + (R0 (g) − Beff j ) exp −κjcos φ

where Wj is the size of the sub-intervals (ΣWj = 1).



(48)

7. Radiative fluxes

163

The accuracy of these absorption coeﬃcients has been established by numerous and continuing high-resolution validations of LBLRTM with spectroscopic measurements, in particular those from the Atmospheric Radiation Measurement program (ARM). Compared tothe original RRTM (Mlawer et al. 1997), the version used at ECMWF (hence ARPEGECLIMAT) has been slightly modiﬁed to account for cloud optical properties and surface
emissivity deﬁned for each of the 16 bands over which spectral ﬂuxes are computed. Foreﬃciency reason, the original number of g-points (256 = 16 × 16) has been reduced to 140(see Table 7.1). Other changes are the use of a diﬀusivity approximation (instead of thethree-angle integration over the zenith angle used in the original scheme) to derive upwardand downward ﬂuxes from the radiances, and the modiﬁcation of the original cloud random overlapping assumption to include (to the same degree of approximation as used in
the operational SW scheme) a maximum-random overlapping of cloud layers. Given the
monochromatic form of the RTE, the vertical integration is simply carried out one layer
at a time from the top-of-the-atmosphere to the surface to get the downward ﬂuxes. The
downward ﬂuxes at the surface are then used with the spectral surface emissivities andthe surface temperature to get the upward longwave ﬂuxes in each of the 140 subintervals.
Then the upward ﬂuxes are obtained in a similar fashion from the surface to the top of
the atmosphere.

For the relevant spectral intervals of the RRTM schemes, ice cloud optical propertiesare derived from Ebert-Curry (1992), and water cloud optical properties from Smith andShi (1992) . Whereas in the previous operational scheme the cloud emissivity used tocompute the eﬀective cloud cover is deﬁned over the whole LW spectrum from spectrally
averaged mass absorption coeﬃcients and the relevant cloud water and/or ice paths, in
RRTM, the cloud optical thickness is deﬁned as a function of spectrally varying mass
absorption coeﬃcients and relevant cloud water and ice paths, and is used within the true
cloudy fraction of the layer. Alternate sets of cloud optical properties are also available
for RRTM (Section 3).

Table 7.1: Spectral distribution of the absorption by atmospheric gases in
RRTM
Spectral intervals Number
Gases included−1
cm
g-points Troposphere
Stratosphere10-2508
H2O
H2O
250-500
14
H2O
H2O
500-630
16
H2O, CO2
H2O, CO2
630-700
14
H2O, CO2
O3, CO2
700-820
16
H2O, CO2, CCl4
O3, CO2, CCl4
820-980
8
H2O, CFC11, CFC12 CFC11, CFC12980-1080
12
H2O, O3
O3
1080-1180
8
H2O, CFC12, CFC22 O3, CFC12, CFC221180-1390
12
H2O, CH4CH41390-1480
6
H2O
H2O
1480-1800
8
H2O
H2O
1800-2080
8
H2O
2080-2250
4
H2O, N2O
2250-2380
2CO2CO22380-2600
2
N2O, CO2,
2600-3000
2
H2O , CH4

164

7. Radiative fluxes

3 Input to the radiation scheme
3.1 Solar irradiance data
Depending on the experiment, the solar constant can be speciﬁed on a yearly basis in
the model. The recommendations for CMIP6 solar forcing data are available there:
http://solarisheppa.geomar.de/cmip6

3.2 Clouds
Cloud fraction, and liquid/ice water content are provided in all layers by the cloud scheme.

3.3 Ground albedo and emissivity
Ground albedo and emissivities are calculated in the SURFEX module (see chapter “Surfex
processes schemes”). The main features are the following ones:
1. one emissivity by type of vegetation (coming from the Ecoclimap 1 km resolution
data base) and for ocean and sea-ice.
2. Albedo:• over continental areas. It is deduced from the Ecoclimap data base, one by typeof vegetation, the total mean albedo on a mesh is weighted with the vegetation
fraction. This version is updated by the annual mean MODIS cycle (Carrer etal.,RSE, 2014), introducing annual cycles of vegetation, and also soil albedo.
• over ocean: a new parametrization of the ocean surface albedo has been developedin collaboration with LMD. It includes dependencies on wavelength, surface windspeed, chlorophyll content as well as the distribution of solar zenith angle. The
albedo is decomposed into a direct and a diﬀuse contribution, each of them being
divided according to the reﬂection process, i.e a Fresnel surface albedo and an ocean
volume albedo. The computation is done in the routine albedors14.F90.
• over snow: albedo varies with the age of snow following Douville et al. (1995)

3.4 Aerosols
Aerosols are included in ARPEGE-Climat using either monthly 2D aerosol optical depth(AOD) ﬁles or an interactive aerosol scheme providing 3D concentrations. In both cases,
ﬁve tropospheric aerosol types are taken into account: desert dust, sea-salt, sulfate, black
carbon and organic matter.
In the ﬁrst case, these AOD data come from ARPEGE-Climat simulations using theinteractive aerosol scheme (TACTIC), a running average on 11 years is calculated tosmooth interannual variability. The 2D optical thicknesses are vertically distributed in
the RADAER subroutine according to the aerosol type.
In the second case, the aerosol vertical proﬁle of extinction is directly computed in theaerosol scheme. In both cases, these 5 tropospheric aerosol types are redistributed in the
4 following types, to be taken into consideration in the radiation scheme :• Continental : Organic + Sulfate• Maritime (sea-salt aerosols)• Desert (soil dust type aerosols)• Urban (black-carbon type aerosols)

7. Radiative fluxes

165

Two additional types are also included in ARPEGE-Climat to represent volcanic andstratospheric aerosols:• Volcanic class (background) and Stratospheric (background) + volcanic eruptions

Direct forcing of aerosols
Through their optical properties, aerosols can scatter and/or absorb solar and thermalradiation. The following optical properties, namely normalized optical depth (RTAUA),
single scattering albedo (RPIZA) and asymmetry factor (RCGA), all deﬁned in SUAERSN
and used in SWCLR, are used in the SW radiation scheme (6 spectral bands), dependingon the value of NCOEFAERO (noted NCA in the tables below). The values of RPIZA
and RCGA have been updated according to Mie calculations (see Nabat et al., (2013) for
more details). Note that the spectral dependance of all these optical properties is for the
moment not taken into consideration inside the visible spectrum (i.e. the ﬁrst three bands
have the same coeﬃcients), but it should be the case in a future version of the model.
• NCOEFAERO=0 : updated values for continental, sea-salt, desert dust and black
carbon• NCOEFAERO=1 : updated values for desert dust only, old coeﬃcients (with nomore than three signiﬁcant numbers)• NCOEFAERO=2 : old coeﬃcients
Normalized optical depth (RTAUA) :Figure 1: Average global profile for each aerosol type in simulations using either 2D AOD data (full lines) or the interactive aerosol scheme (dotted lines).
Note that the full lines for black carbon, sea-salt and continental aerosols
are superimposed. Altitudes have been calculated for standard atmosphericconditions.

166

7. Radiative fluxes

Aerosols

NCA

Continental
Sea-salt
Desert dust
Black carbon
Volcanic
Stratospheric

0/1/2
0/1/2
0/1/2
0/1/2
0/1/2
0/1/2

0.185
0.25
µm
1.69446
1.11855
1.092121.721451.03858
1.12044

0.25
0.44
µm
1.69446
1.11855
1.092121.721451.03858
1.12044

0.44
0.69
µm
1.69446
1.11855
1.092121.721451.03858
1.12044

0.69
1.19
µm
0.52838
0.932850.934490.53078
0.671480.46608

1.19
2.38
µm
0.205430.846420.84958
0.21673
0.28270
0.10915

2.38
4.0
µm
0.10849
0.66699
0.65255
0.11600
0.065290.04468

Single scattering albedo (RPIZA) :Aerosols
NCA
0.185
0.25
µm
Continental
0
0.99
1
0.91
2
0.9148907
Sea-salt
0
0.996
1
0.996
2
0.9956173
Desert dust
0/1
0.90
2
0.7504584
Black carbon
0
0.32
1
0.81
2
0.8131335
Volcanic
0/1
0.94
20.9401905
Stratospheric
0/1
0.999
2
0.9999999

0.25
0.44
µm
0.99
0.91
0.9148907
0.996
0.996
0.9956173
0.90
0.7504584
0.32
0.81
0.8131335
0.94
0.9401905
0.999
0.9999999

0.44
0.69
µm
0.99
0.91
0.9148907
0.996
0.996
0.9956173
0.90
0.7504584
0.32
0.81
0.8131335
0.94
0.9401905
0.999
0.9999999

0.69
1.19
µm
0.99
0.897
0.8970131
0.995
0.998
0.9984940
0.93
0.9245594
0.27
0.777
0.7768385
0.953
0.9532763
0.999
0.9999999

1.19
2.38
µm
0.99
0.828
0.8287144
0.994
0.994
0.9949396
0.94
0.9279543
0.24
0.677
0.6765051
0.946
0.9467578
0.995
0.9955938

2.38
4.0
µm
0.99
0.52
0.5230504
0.99
0.79
0.7868518
0.95
0.8531531
0.20
0.405
0.4048149
0.875
0.8748231
0.235
0.2355667

Asymmetry factor (RCGA) :Aerosols
NCA
0.185
0.25
µm
Continental
0
0.69
1
0.73
2
0.729019
Sea-salt
0
0.72
1
0.80
2
0.803129
Desert dust
0/1
0.785
2
0.784592
Black carbon
0
0.43
1
0.71
2
0.712208
Volcanic
0/1
0.70
20.7008249
Stratospheric
0/1
0.727
2
0.7270548

0.25
0.44
µm
0.69
0.73
0.729019
0.72
0.80
0.803129
0.785
0.784592
0.43
0.71
0.7122080.70
0.7008249
0.727
0.7270548

0.44
0.69
µm
0.69
0.73
0.729019
0.72
0.80
0.803129
0.785
0.784592
0.43
0.71
0.7122080.70
0.7008249
0.727
0.7270548

0.69
1.19
µm
0.64
0.67
0.668431
0.73
0.79
0.788530
0.75
0.698682
0.31
0.66
0.657422
0.673
0.6735182
0.652
0.6519706

1.19
2.38
µm
0.62
0.64
0.636342
0.73
0.80
0.802467
0.75
0.691305
0.30
0.63
0.627497
0.611
0.6105750
0.476
0.4760794

2.38
4.0
µm
0.60
0.70
0.700610
0.73
0.82
0.818871
0.75
0.702399
0.30
0.69
0.689886
0.463
0.4629866
0.191
0.1907639

The dependance of these optical properties on relative humidity can be activated using
the key LHUMDEP : it concerns only sea-salt (for RCGA) and continental (for RPIZA
and RCGA) aerosols. In the case LHUMDEP=False, values in the tables above are used

7. Radiative fluxes

167(corresponding to an average relative humidity of 50%). In the case LHUMDEP=True,
RCGA (for sea-salt and continental aerosols) and RPIZA (for continental aerosols) are
replaced by the following values:• Single scattering albedo (RPIZA) for continental aerosols:
RH
0.185
0.25
0.44
0.69
1.19
2.38
0.25
0.44
0.69
1.19
2.38
4.0
µm
µm
µm
µm
µm
µm
0-15%
0.988 0.988 0.988 0.988 0.988 0.988
15-25%
0.989 0.989 0.989 0.989 0.989 0.989
25-35%
0.989 0.989 0.989 0.989 0.989 0.989
35-45%
0.990 0.990 0.990 0.990 0.990 0.990
45-55%
0.991 0.991 0.991 0.991 0.991 0.991
55-65%
0.993 0.993 0.993 0.993 0.993 0.993
65-75%
0.994 0.994 0.994 0.994 0.994 0.994
75-85%
0.995 0.995 0.995 0.995 0.995 0.995
85-92.5% 0.997 0.997 0.997 0.997 0.997 0.997
92.5-97% 0.998 0.998 0.998 0.998 0.998 0.998
97-100%0.999 0.999 0.999 0.999 0.999 0.999• Asymmetry factor (RCGA) for sea-salt aerosols:
RH
0.185 0.25 0.44 0.69 1.19 2.38
0.25
0.44 0.69 1.19 2.38
4.0
µm
µm
µm
µm
µm
µm
0-15%
0.69
0.69 0.69 0.70 0.70 0.70
15-25%
0.70
0.70 0.70 0.71 0.71 0.71
25-35%
0.71
0.71 0.71 0.72 0.72 0.72
35-45%
0.72
0.72 0.72 0.73 0.73 0.73
45-55%
0.73
0.73 0.73 0.74 0.74 0.74
55-65%
0.74
0.74 0.74 0.75 0.75 0.75
65-75%
0.75
0.75 0.75 0.76 0.76 0.76
75-85%
0.76
0.76 0.76 0.77 0.77 0.77
85-92.5%
0.78
0.78 0.78 0.78 0.78 0.78
92.5-97%0.79
0.79 0.79 0.79 0.79 0.79
97-100%0.82
0.82 0.82 0.79 0.79 0.79
• Asymmetry factor (RCGA) for continental aerosols:
RH
0.185 0.25 0.44 0.69 1.19 2.38
0.25
0.44 0.69 1.19 2.38
4.0
µm
µm
µm
µm
µm
µm
0-15%
0.66
0.66 0.66 0.38 0.38 0.38
15-25%
0.67
0.67 0.67 0.39 0.39 0.39
25-35%
0.68
0.68 0.68 0.40 0.40 0.40
35-45%
0.68
0.68 0.68 0.41 0.41 0.41
45-55%
0.69
0.69 0.69 0.43 0.43 0.43
55-65%
0.70
0.70 0.70 0.45 0.45 0.45
65-75%
0.71
0.71 0.71 0.47 0.47 0.47
75-85%
0.72
0.72 0.72 0.51 0.51 0.51
85-92.5%0.74
0.74 0.74 0.58 0.58 0.58
92.5-97%0.76
0.76 0.76 0.64 0.64 0.64
97-100%0.80
0.80 0.80 0.74 0.74 0.74

The following optical properties (absorption coeﬃcients) are used in the LW radiation
scheme (16 spectral bands, the aerosol coeﬃcients, deﬁned in SUAERL, are gathered in 5

168

7. Radiative fluxes

classes and used in RRTM_ECRT_140GP)
Aerosols
NCA Spectral bands
1/2
3/4/5
Continental
0/1/2
0.036271
0.030153
Sea-salt
0/1/2
0.026561
0.032657
Desert dust
0/1
0.1
0.1
2
0.014897
0.016359
Black carbon 0/1/2
0.001863
0.002816
Volcanic
0/1/2
0.011890
0.016142
Stratospheric 0/1/2
0.013792
0.026810

6/8/9
0.017343
0.017977
0.1
0.019789
0.002355
0.021105
0.05220370.015002
0.0142100.1
0.030777
0.002557
0.028908
0.066338

10-16
0.008806
0.016775
0.1
0.013341
0.001774
0.011890
0.013792

Indirect forcing of the sulfate aerosols
The aerosols can act as cloud condensation nuclei. At constant cloud liquid water, increasing aerosol concentration leads to a larger concentration of cloud droplets of small
radius and increases cloud reﬂectivity. This eﬀect, referred as the ﬁrst aerosol indirect
eﬀect (Twomey, 1974), can be taken into account depending on the value of NRADLP
used in RADLSW to calculate the eﬀective radius of cloud droplets:• NRADLP=2 : no aerosol indirect eﬀect• NRADLP=3 : A simple parametrization from Quaas and Boucher (2005) simulates
this eﬀect in the case of sulfate aerosols. The cloud droplet concentration CDN (incm−3 ) is given by:
lnCDN = 1.7 +0.2 ln mSO2−
4

with mSO2− expressed in µgm−3 (calculated from the sulfate aerosol optical depth).
4
The mean cloud droplet radius is then calculated from the cloud liquid water content
ql and liquid water density ρl by:ql
rv = 3 4πρ3q
l CDN

and the eﬀective radius re =1.1 rv .
The sulfate aerosols are available as output of the routine RADAER in the 3D
array ZAERINDS.
• NRADLP=4 : the same as 3, but relative humidity is taken into account in thecalculation of mSO2− .
4• NRADLP=5 : organic matter and sea-salt aerosols are also taken into account in
the calculation of the cloud droplet concentration according to Menon et al. (2002).
lnCDN = 2.41 + 0.5 ln mSO2− + 0.05 ln mSS + 0.13 ln mOM
4

Only the hydrophilic part of organic matter and the ﬁne mode of sea-salt aerosols
are considered for this calculation (climatological fractions, respectively 77.64% and1.37%).
Note that the second aerosol indirect eﬀect (modiﬁcation of cloud life time and precipitation by aerosols) is not included in ARPEGE-Climat.

3.5 Radiatively active compounds
Concentrations of carbon dioxide, methane, nitrous oxide, CFC-11, CFC-12 and chlorine
are given in namelist NAERAD. During the simulation, values are evolving yearly following
the IPCC data. The CFC-12 compound also includes the CFC-12 equivalent of halocarbon
species. Outputs from the CNRM REPROBUS chemical model, simulating coeﬃcients
which describe 3D ozone distributions, are used as input of a linear interactive scheme(Cariolle and Teyssèdre, 2007). The value NVCLIS = 7 (in NAMDPHY) means thatseven of the climatological coeﬃcients of the REPROBUS outputs are used. Evolution ofozone chemistry is due to chlorine and meteorological parameters (temperature, humidity)
evolutions.

0.006865
0.022123
0.1
0.014321
0.001780
0.011890
0.013792

7. Radiative fluxes

169

3.6 Cloud optical properties
For the SW radiation, the cloud radiative properties depend on three diﬀerent parameters:
the optical thickness, the asymmetry factor and the single scattering albedo of particles.
For LW the cloud properties are linked with emissivity and spectral optical thickness.
All these properties are deﬁned for water and ice particles in the RADLSW routine.
They depend on liquid (or ice) water path or on the eﬀective radius of the particles.
These characteristic parameters can be calculated by diﬀerent ways according to NRADLP,NRADIP, NLIQOPT, NICEOPT parameters. During the tuning of the CMIP6 model,
corrections of some options have been done. New combinations of optical properties havebeen coded: see comments on NICEOPT/NLIQOPT and NRADLP/NRADIP below.
Cloud liquid particles• NRADLP = 0; cloud water eﬀective radius is calculated as a function of pressure(old parameterisation).
• NRADLP = 1; the cloud eﬀective radius is equal to 10 µm over land and equal to13 µm over the ocean.
• NRADLP = 2; parameterisation based on Martin et al. (1994).
• From NRADLP=3, a parametrization of the ﬁrst indirect eﬀect of aerosols is proposed: Effective radius of cloudy droplets
The eﬀective radius of cloudy droplets writes:re = 100 ∗ RT U N HRM ∗ (3 ∗ ZLW C)1/3
4 ∗ πZCCDN C(49)

where re is the eﬀective radius in µm, RTUNHRM is a parameter (RTUNHRM=1.1),
ZLW C= 1000 ∗ ρair ∗ P QLW P (unit: g/m3 ) with P QLW P input of radlsw.F90(unit: kg/kg) and ZCCDN C the number of CCN (cm−3 ). If NRADLP=5, themin value of re is 2 µm and the max one is 24 µm.
• NRADLP = 3: the eﬀective radius of cloud particle is calculated taking into accountthe indirect eﬀects of sulfate aerosols only. The number of cloud condensation nuclei
is computed by Quaas, pers. comm.
• NRADLP = 4: as NRADLP =3 but with tabulated extinction coeﬃcients vs theaerosol types and relative humidity. CNN number of Quaas.
• NRADLP = 5: eﬀect of sea salt and organic aerosols. Tabulated extinction coeﬃcient as for NRADLP=4, Menon’s (2002) parameter for the computation of CCN
number.
Cloud ice particles• NRADIP = 0; the ice particle eﬀective radius is ﬁxed at 40 µm.
• NRADIP = 1; ice particle eﬀective radius calculated as f(T) from Ou and Liou
(1995).
• NRADIP = 2; ice particle eﬀective radius calculated as f(T) from Ou and Liou
(1995) and ﬁxed between 30 and 60 microns .
• NRADIP = 3; Ice eﬀective radius calculated as a function of temperature and ice
water content from Sun and Rikus (1999) and revised by Fu and Sun (2001).
• NRADIP = 4 is equivalent to NRADIP = 2 for LW computations and NRADIP =3 for SW computations.
SW radiation
Cloud water optical properties depend on NLIQOPT:• NLIQOPT diﬀerent from zero, refers to Slingo (1989).

170

7. Radiative fluxes• NLIQOPT = 0, refers to Parol et al. (1991).

Ice water optical properties depend on NICEOPT:• NICEOPT lower than or equal to 1 , refers to Ebert and Cury (1992).
• NICEOPT = 2, refers to Fu and Liou (1993).
• NICEOPT = 3 or NICEOPT =4, refers to Fu (1996).
LW radiation
Cloud water emissivity depends on NLIQOPT:• NLIQOPT = 0 or greater than or equal to 3, refers to Smith and Shi (1992).
• NLIQOPT = 1 , refers to Savijarvi and Raisanen (1997).
• NLIQOPT = 2 , refers to Lindner and Li (2000).
• NLIQOPT > ou = 3 , refers to Smith and Shi(1992). NLIQOPT = 4 is equivalent
to NLIQOPT= 0 in LW computations.
Ice cloud emissivity depends on NICEOPT:• NICEOPT = 0, refers to Smith and Shi (1992).
• NICEOPT = 1, refers to Ebert-Curry (1992).
• NICEOPT = 2, refers to Fu and Liou (1993).
• NICEOPT = 3, refers to Fu et al. (1998) including parametrisation for LW scattering eﬀect .
• NICEOPT = 4, equivalent to NICEOPT = 1 in LW computations.
Inhomogeneity factors
The cloudy inhomogeneity factors are RSWINHF for SW and RLWINHF for LW. During
the tuning of the model, a diﬀerence between the liquid and ice clouds (RSWINHFLIQ
and RSWINHFICE for the SW and RLWINHFLIQ /RLWINHFICE for the LW) has been
introduced for computation of optical depth and spectral emissivity. According to J.J
Morcrette (pers. comm.) the SW coeﬃcients can be chosen in the range (0.6-0.8) and the
LW one in the range (0.8-1), depending also on the water condensate vertical, horizontal
distribution and eﬀective radius. These parameters have been used to obtain closed TOA
and surface budgets.
The current values of the parameters are:
• NRADLP = 5,
• NRADIP = 4,
• NLIQOPT = 4• NICEOPT = 4, Note that the couple (NLIQOPT=4, NICEOPT=4) is equivalent
to the couple (2,3) in SW computations and (0,1) in the LW computations.
• RTUNHRM=1.1,• NCOEFAERO =0,• LHUMDEP=.T.,• RSWINHF=1.,• RLWHINF=1.,• RSWINHFLIQ =0.74,• RSWINHFICE =0.74,• RLWINHFLIQ =0.95,• RLWINHFICE =0.76.

7. Radiative fluxes

171

3.7 Cloud overlap assumption
The cloud overlap assumption is used for the determination of Ctot (see Section 1). It isdetermined by the NOVLP key: NOVLP = 1 means "maximum-random overlap", NOVLP= 2 means "maximum overlap " and NOVLP = 3 means "random overlap". An option
NOVLP=6 has been added in the code for CMIP6 following the work of Thouron (Thouron
et al 2017 ) which proposes to use diﬀerent overlap assumptions for the computation ofclear-sky fraction and for total eﬃciency of diﬀusion. This option corresponds to a maxrandom overlap for the computation of the clear-sky fraction and a random one for thecomputation of the solar zenithal angle. This formulation is particularly adapted to multilayers case.

3.8 Interactions with the SURFEX module
To obtain consistent radiative ﬂuxes in ARPEGE and in SURFEX, downward ﬂuxes
calculated by ARPEGE and provided to the SURFEX scheme have to be cut into spectral
bands. Moreover, for the long-wave radiative ﬂuxes, additional corrections have to bemade in the APLPAR subroutine to ensure equal radiative net budgets in ARPEGE as
well as in SURFEX.

4 Simplified radiation scheme
Given the computation cost of the radiation scheme, it seems interesting to call it only
at some time steps, named radiative time steps. Nevertheless, it is necessary to produce
radiative ﬂows at each time step. One thus needs a way to calculate these ﬂuxes at low
cost. This is carried out in Arpege-climat by calling subroutine RADHEAT.
The idea of this routine is to save between two radiative time steps the solar transmissivity
and long-wave emissivity. Let FSW (PFRSO in APLPAR) and FLW (PFRTH) be total
net ﬂuxes of solar and infrared radiation calculated at the last radiative time step. The
approximation consists in maintaining constant the values of transmissivities t0 (PTRSOL)
and emissivities (PEMTD) ǫ0 calculated at this time:
t0 =

FSW 0
µ0 E 0

(50)

ǫ0 =

FLW 0
σT04

(51)

Then, in the next time steps, one recomposes solar and infrared ﬂuxes by modifying only
the solar incidence µ and temperature T :FSW = t0 µ0 E 0(52)

FLW = ǫ0 σT 4

(53)

Note that with the logical key LABSDIR=.TRUE., a modiﬁcation to diagnose layers’
radiative properties as function of infrared ﬂuxes of the last radiative step is introduced(in APLPAR). The surface "Delta" (diﬀerence between the surface infrared ﬂux with Ts
from SURFEX and the surface infrared ﬂux computed in the last call of the full radiative
scheme) infrared ﬂux is computed as function of Delta Ts. The modiﬁcation consists in a
progressive absorption of the "Delta IR" from the surface to the top of the atmosphere. It
starts with a diagnostic of the IR absorption rate in each atmospheric layer and modiﬁes
the corresponding heating rate. The radiative budget is modiﬁed taking into account thedelta IR upward ﬂux reaching TOA.

172

7. Radiative fluxes

8
The turbulence scheme

1 Introduction
1.1 Objectives
Among all the diﬀerent processes which need to be represented in the numerical prediction
model for the atmosphere, valid from the short term range forecasting to the simulationof the general climate, one of the most important one is the decreasing of the verticalgradients or heterogeneous feature in potential temperature, wind or humidity. These
processes correspond to “vertical transport” or “dissipation”, represented either by what is
usually called “turbulence” or “convection”.
The splitting of the mixing processes between “turbulent” or “convective” ones are made interms of asymmetric buoyancy eﬀects for the convection, whereas the turbulence is thinkin terms of symmetric and horizontal rolls, mainly located within the dry PBL, bellow thecloudy part of the atmosphere.
This separation was clear until the 80’s and 90’s. Since then, more and more moist
eﬀects are taken into account in the turbulent schemes, and more and more dry thermal
convective processes are associated to the shallow convection (inside the cloud) and in thedry PBL (bellow the cloud base).
Nowadays, several uniﬁed “turbulent + mass-ﬂux” schemes exist. They are often called“EDMF” (or “EDKF”), for “Eddy Diﬀusivity Mass-Flux”. The aim of the scheme describedin this documentation is to described only the vertical mixing of atmospheric variables
due to turbulent processes, with the vapor and condensed waters taken into account. The
dry and moist convection processes are not explicitly managed, though they impact insome way on the turbulent scheme via the two “turbulent” and “convective” tendencies
which are fully combined for computing the next time-step, with possible covering of thetwo schemes on the vertical.
The aim the the parameterization is to compute the exchange coeﬃcients Km (valid forthe momentum, i.e. the 2 wind components u and v) and Kh (valid for the potential
temperature and the speciﬁc humidity). These exchange coeﬃcients are used to computethe turbulent ﬂuxes, used in the implicit solver for the vertical mixing and obtained by
the inversion of a tri-diagonal matrix.
The choice of the present parameterization, implemented both in the NWP and the GCM
versions of ARPEGE, has been mainly motivated by the hope to better represent themarine Strato-cumulus in the model. Indeed, there is a lack of strato-cumulus in most of
the CMIP5 GCM and NWP models in the Eastern part of the tropical oceans, close to
the coast. These low-level clouds are important features for determining even the sign ofthe local impacts associated with the Climate Change.

174

8. The turbulence scheme

1.2 From CMIP5 to CMIP6
For CMIP5, a turbulent scheme, based on the stationary equation for the turbulent kinetic
energy, using the 2.0 order scheme of Mellor and Yamada (1977, 1982), has been used in the
Climate version of ARPEGE. The numerical computations of the exchange coeﬃcients Km
and Kh are described more precisely in Ricard and Royer (1993). The moist processes aretaken into account in the Mellor-Yamada set of equations, by using the subgrid varianceof atmospheric humidity, following the ideas of Sommeria, Deardorﬀ, Bougeault (1976,
1977, 1982) where a certain statistical law is prescribed (an asymmetric and exponential
ﬁxed PDF).
The turbulent kinetic energy variable is deﬁned on the full-levels of ARPEGE (samelocation as for the wind or the temperature). As a consequence the advective processes
could be activated, but they are switched-oﬀ since it is a stationary (diagnostic) turbulent
kinetic energy equation. A lot of half-summations are made to go from the full-levels tothe half-levels, and vice versa. With the crude spacing of the vertical levels, such halfsummations processes create an artiﬁcial and strong mixing at the top of the PBL andat the inversion. It is probably the reason why the shallow convection processes are - insome way - represented by this moist diagnostic turbulent scheme, via a synergy between
the moist subgrid representation and the (too) strong vertical mixing.
The present prognostic scheme for CMIP6 corresponds to a continuous series of developments, starting in 1998-99.
The diﬀerent processes represented in this new set of parameterizations are• the turbulent kinetic energy (e) is computed with a prognostic equation where
the horizontal and vertical advections are switched-oﬀ, with the vertical mixing ofe, the dissipation, the shear (dynamical) production and the buoyancy (thermal)
production occur ;• the turbulent kinetic energy (e) is computed on the “half-levels”, the levels where theexchange coeﬃcients Km and KT , also the vertical velocity, are computed (in between the “half-levels” where are computed the wind components, the temperatureand the speciﬁc humidity) ;• the turbulent kinetic energy (e) has a minimum value (typically of 10−6 m2 s−2 );• the formulations for the turbulent ﬂuxes are given in Redelsperger and Sommeria(1981) and Cuxart, Bougeault, Redelsperger (2000) ;• the “moist” versions of the ﬂuxes are computed by using the Lilly (1968) potential
temperature (i.e. θvl ) and the Betts (1973) variables (i.e. for θl and qt ), for thevertical mixing of the conservative variables ;• the computations of the sub-grid variance of cloud liquid water are made withthe hypotheses of Bougeault (1982) and Bechtold (1995), by using some mixed
symmetric (Gaussian) and asymmetric (Exponential), in order to represent the
Cumulus and the Strato-Cumulus, respectively ;• the mixing and dissipation lengthes are given by the non-local formulation of
Bougeault and LacarrÃ¨re (1989) ;• the surface layer value for the turbulent kinetic energy is given by AndrÃ c etal.(1978) ;• the surface layer values for the exchange coeﬃcients are given by Louis (1979, 1981);• the vertical mixing of the turbulent kinetic energy is made in an implicit way, by
taking into account a linearized version of the dissipation term, i.e. the non-linear
term (e)3/2 , leading to the weighting factors αexp = −0.5 and αimp = 1.5 ;• the top-PBL vertical entrainment is parameterized following the ideas of Grenier
and Bretherton (2000) and Grenier (2002).

8. The turbulence scheme

175

2 The CBR scheme
2.1 The turbulent kinetic energy equation
The turbulent kinetic energy equation gives the change in time of the grid-cell mean value
(e).
∂e
∂t

=

[ Advect. ] + Diﬀvert + Pdyn. + Pther. − Diss ,(1)

Diﬀvert

=

−

(2)

Pdyn.

=

Pther.

=

Diss

=


1 ∂
ρ e′ w ′ ,
ρ ∂z
h
i
∂u
∂v
− u′ w ′,+ v′ w′∂z∂z′
β w′ θvl
,
√
e e
.
Cǫ
Lǫ

(3)
(4)
(5)

Except the (neglected) advective part (Advect.), it is the sum of 4 terms. The ﬁrst
term (Diﬀvert ) is the vertical mixing (or diﬀusion) of e. It represents the change ofe by the turbulent processes plus, in some way, the impact of the (unknown) pressocorrelation term. There are two production terms: the dynamical production Pdyn. isalways positive and represents the impact of the shear of the wind components, the thermal
production Pther. can be positive or negative, depending on the vertical ﬂuxes of the Lilly(1968) potential temperature θvl , computed in a complex way by using the ﬂuxes of theconservative Betts (1973) variables θl and qt . The last dissipation term (Diss) depends
on a constant Cǫ and on a dissipation length Lǫ .
For the thermal production, the formulation for β and θvl writes:
β

=

θvl

=

g
,
θ
θ (1 + 0.608 q v − q c ) .

(6)
(7)

The potential temperature of Lilly depends on the grid-cell average of both the water
vapor q v and the condensed water q c (either liquid or solid). The main diﬃculty is tocompute the ﬂuxes of these quantities, whereas the moist ﬂuxes are only known for theBetts variables w′ θl′ and w′ qt′ .
The conservative variables of Betts write in terms of the mean water vapor (q v ), the meanliquid cloud water (q l ), the mean solid cloud water (q i ) and the mean total condensed
cloud water (q c = q l + q i ) :

1−

Lv q l + Lf q i
cp T

θl

=

θ

qt

=

qv + qc .



(8)

,

(9)

The two terms Lv and Lf are the latent heats of vaporization and fusion, respectively.
The ﬁrst order turbulent ﬂuxes are written in terms of the vertical gradients of the meanvariables, following the formulations of Redelsperger and Sommeria (1981 ; or RS81), alsoof Cuxart, Bougeault and Redelsperger (2000 ; or CBR00).
u′ w ′

=

v′ w′

=

√ ∂u
e
∂z
√ ∂v
− Cm Lm e∂z− Cm Lm; Km = C m L m√; Km = C m L m

√

e,

(10)

e,

(11)

176

8. The turbulence schemee′ w ′=

w′ θl′=

w′ qt′

=

√ ∂e
e
∂z
√ ∂ θl− Cθ Lm e
φ3∂z√ ∂ qt− Cq Lm eψ3∂z− Ce Lm; Ke = C e L m√; KT = C θ L m√; Kq = C q L m

√

e,

(12)

e φ3 ,(13)e ψ3 .

(14)

Note that the unit of turbulent exchange coeﬃcients computed in acturb.F90 is kg.m−2 .s−1 .
Indeed, in the relation:∂Ψ∂(−ρ ∗ (−K ∂z ))∂(−ρw′ Ψ′ )
∂Ψ
= −g
= −g
∂t
∂p
∂p

(15)

unit of K is m2 s−1 . The computed turbulent exchange coeﬃcients (PKTROV, PKQROV,.
PKQLROV,PKUROV) are gρK∆Ψ
The ﬁrst order turbulent ﬂuxes and the associated exchange coeﬃcients (10) to (14) depend
on four unknown constants (Cm , Ce , Cθ , Cq ), with the mixing length denoted by Lm andthe dissipation length by Lǫ .
This turbulent scheme includes two stability functions (φ3 and ψ3 ), which are found to beequal in that case of an “1D-vertical” turbulence, leading to
φ3 = ψ 3 =1.
1 + C (Rθ + Rq )(16)

The term C = Cθ Cǫθ is another unknown constant. The stability functions φ3 and ψ3
describe the enhancement or inhibition of turbulent transfers by stability eﬀects.
Hypothesis 1 : In the Meso-NH and AROME LAM, the true computations of Rθ + Rq
need to keep some pseudo-prognostic variables (kept in memory from one time step for the
next one). In the CBR00 scheme implemented in ARPEGE, the computation of (16) is
made with a more straightforward method, without pseudo-prognostic variables but witha direct computation of the vertical gradient of θvl instead, with the assumption
Rθ + Rq ≈ β

Lm Lǫ ∂θvl
.
e
∂z

(17)

The vertical gradient is computed from (7), with no hypothesis concerning the sub-gridvariability of the humidity.
The stability function φ3 = ψ3 varies from 0.78 (for the stables cases) to possible large oreven inﬁnite values for unstable cases, for instance if C(Rθ +Rq ) = −1. As a practical rule,values for φ3 = ψ3 are limited to the maximum value of ACBRPHIM = 2.2 (or possiblyless), available in the NAMELIST of ARPEGE.

2.2 The constants
The set of constants used in ARPEGE are diﬀerent from the one used in Meso-NH and
AROME. Three of the independent constants deﬁned in Meso-NH and AROME are Cpv ,Cpθ and Cǫθ , leading to the four ARPEGE constants Cm , Cθ , Cq and C, according to
Cm

=

4
,
15 Cpv

Cθ

=

Cq =

(19)

C

=

Cθ
Cǫθ

(20)

2
,
3 Cpθ
2
=
.
3 Cpθ Cǫθ(18)

8. The turbulence scheme

177

In addition to Cpv , Cpθ and Cǫθ , the other independent constants are Cǫ in (5) and Ce in(12).
There exist diﬀerent sets of values for these ﬁve independent constants, depending on
diﬀerent papers or diﬀerent tuning of the models. The set of constants proposed byCheng, Canuto and Howard (2002)(below CCH02) is used here.
The choice of the Cheng et al. constants provides an enhancement of the momentum ﬂuxes(a decrease in Cpv ), an increase of the dissipation (a decrease in Cǫ ) and with almost thesame thermal and moisture ﬂuxes (not so diﬀerent values for Ce , Cpθ and Cǫθ ), comparedwith other tested sets of constants (e.g those of Cuxart, Bougeault, Redelsperger (2000)
ou Redelsperger, Sommeria (1981)). The constant values are: Cǫ = 0.845, Ce = 0.34,
Cpv = 2.11, Cpθ = 4.65 and Cǫθ = 1.01.
The corresponding values for C, Cm and Cθ write: C = 2/(3 Cpθ Cǫθ ) =0.143, Cm =4/(15 Cpv )= 0.126, Cθ = Cq = 2/(3 Cpθ )= 0.143.
In ARPEGE, there are ﬁve constants which are all set in the NAMELIST of the model.
In addition to the same constant C used in (16) for the deﬁnition of φ3 = ψ3 , the fourother constants are AKN, αT , αe and ALD, corresponding to√44; Cpv =⇒ AKN = Cm =,(21)
Km = AKN Lm e
15 Cpv
15 AKNCθ
5 Cpv
2
KT = Kq = α T Km φ3 ⇒ α T ==(22),; Cpθ =Cm
2 Cpθ
3 AKN αT15
Ce=
Cpv Ce ; Ce = αe AKN (23),Ke = α e Km
⇒ αe =
Cm
4
√
1
1
e e
Diss =⇒ ALD =; Cǫ =.(24)
ALD LǫCǫ
ALD
The values used in this version are AKN = Cm = 4/(15 Cpv )= 0.126;ALPHAT = αT =(5 Cpv ) /(2 Cpθ )= 1.13 and ALD = 1/Cǫ = 1.18 and ALPHAE = αe = Ce /Cm = 2.7.

2.3 The vertical mixing and the (dynamic+thermal) productions
The dissipation term is given directly by (5). The vertical mixing term and the dynamical
production term are computed by putting (10) and (12) into (2) and (3), with dp =− ρ g dz = − ρ dφ, leading toDiﬀvert

=

−g

Pdyn.

=

Km

∂
∂p





ρ g Ke

∂u
∂z

2

+

∂e
∂φ



∂v
∂z



= −g

2 

∂
∂p



= Cm Lmρ g Ce Lm
√

e



∂u
∂z

√ ∂e
e
∂φ

2

+





∂v
∂z

,

(25)

2 

. (26)The thermal production term (4) is computed diﬀerently. It is not computed as the
product of an exchange coeﬃcient with the associated gradient, i.e.
′w′ θvl6= − Cθ Lm√ ∂ θvl
e
φ3 .
∂z

(27)

The usual method is rather to express Pther. in terms of the conservative Betts variables,
given by (8) and (9), leading to a formulation′= β Eθ w′ θl′ + β Eq w′ qt′ ,Pther. ≡ β w′ θvl

(28)

178

8. The turbulence scheme

where the two terms Eθ and Eq are two non-trivial coeﬃcients to be determined, which
depends on the normalized saturation deﬁcit and to the the sub-grid variance of thehumidity (cloud water).
The two following additional hypotheses are made in (28)w′ θl′=

w′ qt′=√ ∂ θle
φ3 ,∂z√ ∂ qt− Cq Lm e
ψ3 ,
∂z

(29)

− Cθ Lm(30)

with φ3 = ψ3 given by (16) and (17).

2.4 A3-d: The sub-grid variability of cloud water
The two coeﬃcients Eθ and Eq are to be determined in order to allow the computationof the moist thermal production (28). To do so, some additional hypotheses must be
done, concerning the sub-grid variability of the humidity. It is the reason why the moist
turbulent scheme is so dependent on the way to represent the moist processes.
We will describe the same method that has been already used from the old version 3 to theactual version 6 of the ARPEGE-GCM (see Ricard and Royer, 1993). Let us deﬁne theparameter “s” which depends on the sub-grid departure terms θl′ and qt′ of the conservative
Betts variables :
s
a
α1

=
=
=

a
2



1+

T
θ



qt′ − α1 θl′ ,



Lv/f
cp
∂ qsat
∂T





(31)

∂ qsat∂T(T =Tl )

−1

(32)

,

(33)

.
(T =Tl )Both a and α1 depend on the derivative with respect to T of the saturating water vapor
qsat , where the derivative is computed at the value Tl .
Let us denote by Q1 the normalized saturation deﬁcit, which depends both on a and on
the standard deviation of s, denoted by σs , leading to
Q1

=

a



qt − qsat (Tl )
2 σs



(34)

.

The standard deviation of s, i.e. σs , writes in terms of a, α1 and the second order ﬂuxes,
givingσs=1/2
a ′ 2(qt ) − 2 α1 (qt′ θl′ ) + (α1 )2 (θl′ )2
.
2

(35)

From Cuxart et al. (2000), the second order ﬂuxes write(qt′ )2(θl′ )2(qt′ θl′ )===C (Lm )2 ψ3
C (Lm )2 φ3
C (Lm )2





∂qt
∂z
∂θl
∂z

2

2

(φ3 + ψ3 )
2

(36)

,



(37)

,
∂qt
∂z



∂θl
∂z



.

(38)

8. The turbulence scheme179

Let us replace the second order ﬂuxes (36) to (38) into (35), with the property φ3 = ψ3
valid for the present "1D-column" version of the scheme. The result can be factorized into
the square of a quantity expressed in terms of the vertical gradients of θl and qt , leading
to
σs =

a p
C (Lm )2 φ3
2

∂qt
∂θl
− α1
∂z
∂z

(39)

.

p

It is possible to write diﬀerently the term C (Lm )2 φ3 , in order to express the result interms of the exchange coeﬃcient KT given by (13), the constant Cǫθ given by (20), themixing length Lm and the turbulent kinetic energy e, leading top1
C (Lm )2 φ3 = p
CǫθThe coeﬃcient 1/r

L m KT
√
.
e

(40)

p

Cǫθ (ARSB2 in the code) equals 0.833.

2.5 The sub-grid variability of cloud water
The method, described in Mellor (1977), starts with the deﬁnition of (7), (8) and (9) for
the temperature of Lilly and the conservative variables of Betts.
In order to get an equation like (28) for the ﬂux of θvl , let us write θvl in terms of theBetts variables θl and qt :
θvl
D1

=
=

D2

=

(41)
(42)

θl + D1 qt + D2 qc ,0.608 θ ,

Lv/f
− 1.608
cp T



(43)

θ.

From (41), the ﬂux of the Lilly temperature writes′
w′ θvl=(44)

w′ θl′ + D1 w′ qt′ + D2 w′ qc′ .

The last terms of (41) or (44), i.e. D2 qc or D2 w′ qc′ , cannot be computed without making
additional hypotheses. There are 3 such hypotheses, all involving the ﬂuxes, not directly
the variables.
Hypothesis 2 : established by Mellor (1977) for the Gaussian distributions, it has been
extended for non-Gaussian cases by Bougeault (1982). From Bougeault (1982), Cuijpers
and Bechtold (1995), Bechtold et al. (1995), it is assumed that the residual term w′ qc′ in(44) can be expressed in terms of ‘s’ deﬁned by (31), leading tow′ qc′
where w′ s′=

w ′ s′≡a 2

λ3 (Q1 , AS )′s′ qt′ − α1 s′ θls′ qc′(σs )2
,



,

(45)
(46)

where λ3 is a function of the two variables (Q1 , AS ), function deﬁned hereafter, dependingof the normalized saturation deﬁcit Q1 and the asymmetry factor AS .
Hypothesis 3 : from Bougeault (1982), it is assumed that the last term of (45) can be
written ass′ qc′(σs )2=

2 F2 (Q1 , AS ) ,

(47)

180

8. The turbulence scheme

Table 8.1: Value of AS and λ3 in terms of Q1 .
Exp / Asym.
(Cu)
Q1 < −2
AS = 2λ3 = 3Mixed regime( Cu / Sc)−2 < Q1 < 0
AS = −Q1
λ3 = 1 − Q 1

Gaussian(Sc)
Q1 > 0AS = 0λ3 = 1

where F2 , like λ3 , is a function of the two variables (Q1 , AS ).
Finally, from (44) to (47), the ﬂux of θvl can be rewritten as′
w′ θvl=

w′ θl′ + D1 w′ qt′







− {a D2 F2 λ3 } α1 w′ θl′ − w′ qt′ .

(48)

It is the sum of two terms. The ﬁrst term into brackets doesn’t depend on F2 or the variability in cloud water and moisture. The second term directly depends on the variability
in cloud water and moisture, via the product of the four terms a D2 F2 λ3 .
Hypothesis 4 : from Bougeault (1982), it is assumed that the probability density function G(s) is a possibly mixed symetric (Gaussian) and asymmetric (exponential) function.
This PDF gives the average values over the grid-cell for the cloud cover N s , for the cloudwater content qc s and the normalized second order ﬂux s′ qc′ / [ 2 (σs )2 ]. It results
Ns

=

qc s
2 σs

=

s′ qc′2 (σs )2=

F0 (Q1 , AS ) =F1 (Q1 , AS ) =F2 (Q1 , AS ) =

Z +∞

Z

Z

G(t) dt ,(49)(Q1 + t) G(t) dt ,(50)

t (Q1 + t) G(t) dt .

(51)

−Q1
+∞

−Q1
+∞
−Q1

For the given PDF, i.e. G(s), all the functions F0 , F1 , F2 and λ3 can be analyticallydetermined and numerically computed. They depend on the two variables (Q1 , AS ) andthey determine the ﬂux (48), and thus the thermal production term (28).
The Table (8.1) gives the deﬁnitions for AS and λ3 in terms of the normalized saturationdeﬁcit Q1 . The symmetric Gaussian distributions regime (Q1 > 0) is supposed to represent
the Strato-Cumulus. The asymmetric Exponential regime (Q1 < −2) is supposed torepresent the Cumulus regime. The regime in between, a mix of the Gaussian and theExponential PDFs, has been pre-computed and tabulated in the Meso-NH model.
The “large-scale” (or “stratiform”) cloud cover N s is computed at each time step and foreach grid-point with (49). Similarly, the “large-scale” cloud water content qc s is computedwith (50), with 2 σs given by (39) and (40).
These “large-scale” cloud cover and cloud water content are merged with the “convective”
quantities, with some overlapping assumptions, in order to transmit them to the radiation
code.
Also, the “large-scale” cloud water content is transmitted to the Bulk prognostic schemeof condensation and precipitation (Lopez’s scheme).
The identiﬁcation of (28) with (48) leads to the following formulas for Eθ et Eq :
Eθ
Eq

=
=

1 − α 1 { a D2 F 2 λ3 } ,D1 + { a D2 F 2 λ 3 } ,

(52)
(53)

8. The turbulence scheme

181

2.6 Min and Max Q1 limitations: LECTQ1=.TRUE.
During the development of the high-resolution version of the model (T359) for CMIP6,
it appears some instabilities, for which we have decided to reactivate the limitation of
Q1, already used in CMIP5 conﬁguration. This limitation is run with the logical key
LECTQ1= .TRUE. in the acturb.F90 subroutine. It corresponds both to a minimum andmaximum value of Q1. As in CMIP5 Ricard-Royer scheme, it appears essential to limit
the occurence of negative moistures (otherwise unrealistic values of the mixing length canappear) through a condition on the standard deviation of qw . The following inequality is
met:|Q1| >

w|
A|∆q|| ∂q∂zwlqw | ∂q− α1 ∂θ
|
∂z
∂z

(54)

= Q1minIn the code, Q1min is ZQ1MIN, A is STTBMIN (STTBMIN=1.73), |∆q| is ZDELTQH,

w
w
l
| ∂q
| is ABS(ZDQW), qw is ZQWF, and | ∂q
−α1 ∂θ
| is ZDIFFC. This minimum Q1 value∂z∂z∂z
gives a maximum value for the mixing length. For Q1 max value, ﬁrstly the minimum
value for Φ3 (neutral cases, see Cuxart et al.), is computed, in a consistent manner withthe computation done in ACBL89.F90, by:

φ3min =

1
1 + 2.ARSC1(55)

Then the maximum of Q1 (ZQ1MAX in the code) is computed as:
qt − qsat (Tl ) ∂qt∂θl− α1
Q1max == p2∂z∂z
C (Lm ) φ3(56)

2.7 The mixing and dissipation lengths
The mixing length Lm and the dissipation length Lǫ are computed in a non-local way,following Bougeault et Lacarrère (1989). Starting from a level at the altitude z where theenergy is set to the local turbulent kinetic energy e (z), the non-local lengths Lup andLdown are computed as the possible upward and downward displacements, respectively,
until the energy e (z) is equal to the integral of the work of the buoyancy force, expressedin terms of θvl give by (7).
It results
e (z)

=

Z z+Lup
z

e (z)

=

Z z



β θvl (z ′ ) − θvl (z)z−Ldown

β θvl (z) − θvl (z ′ )dz ′ ,

(57)

dz ′ .

(58)

i−3/2

(59)

The formulation for Lm in ARPEGE-GCM is:
Lm =

h

1
{ (Lup )−2/3 + (Ldown )−2/3 }2.

Close to the ground, the Karman law Lm ≈ 0.4 z is not set. Instead, following thetheoretical arguments of Redelsperger, Mahé and Carlotti (2001), it could be normal toapproach Lm ≈ 2.8 z in true convective cases (values of Lm ≈ 2 z are simulated in 1D
cases as RICO or ARM-Cu).

182

8. The turbulence scheme

For the stable regimes, the numerical scheme for the computation of Lup and Ldown are
made in a very accurate way, with a second order scheme, in order to be as close as possible
to the Deardoﬀ’s length:
LD =

s

√
2e
2√
 =
e,
N
β ∂θvl /∂z(60)

where N 2 = β ∂θvl /∂z is the square of the Brunt Väisälä frequency.
The Deardoﬀ’s length can be very small for very stable cases (Lm ≈ LD ≪ 1 m), because
in that cases e is small, reaching the minimum value of 10−4 to 10−6 m2 s−2 , also because∂ θ/∂z ≫ 0 is large.

As a consequence, the small values for both Lm and e in the stable regions lead to smallvalues√ for the exchange coeﬃcients Km to Kq given by (10) to (14), which all vary as
Lm e.
The large scale ARPEGE-GCM must be able to manage all the cases, from the surface
layer to the mesosphere, including the planetary boundary layer, the troposphere and thestratosphere. In order to maintain a minimum vertical mixing in all stable regions, somemodiﬁcations have been included after the computation of Lm in (87).
Lm = Max [ Lm ; Min (λE ; 0.4 z) ] .

(61)

In the code g ∗ Lm is ZGLMIX and 0.4gz is ZGLKARMN.

As a consequence, the asymptotic value λE replaces the Deardorﬀ’s length in the stableregions. Close to the ground, the formulae (61) is a security, leading to Lm ≥ 0.4 z forz < λE /0.4 and Lm ≥ λE for z > λE /0.4. Typically, λE = 10 m (ALMAVE in the code)and λE /0.4 = 25 m. The mixing length Lm is PLMECT in the code and the dissipation
length Lǫ (1/PUSLE in the code) is computed as ALD*PLMECT, with ALD=1.18.

2.8 The turbulent kinetic energy in the surface layer
The values of e are given by the prognostic equation (1) for each upper-air half-levels,
i.e. on the half-levels where the exchange coeﬃcients Km to Kq are computed, at the
middle points between the full-levels where all the thermodynamics variables are available(temperature, wind, speciﬁc humidity, ...)
But the vertical mixing processes represented by (2) needs the knowledge of e at the
groung level or in the surface layer, denoted by eS .
In the current version of the code (for .LECTREP=.TRUE.), es =eklev , value at the lastvertical level. This hypothesis avoids strong and unrealistic values of es over mountains.
For the key LECTREP=.FALSE., eS ≡ [(u′S )2 + (vS′ )2 + (w′ )2 ]/2 is computed following
André et al. (1978), see their Eq.(29), p.1866, with the term (−ζ)2/3 (u∗ )2 dropped in the
instable case, leading to :eS = 3.75 (u∗ )2 + 0.3 (w∗ )2 (1 − δstab ) .

(62)

The ﬁrst part, valid for both stable and instable conditions, depends on the friction velocity u∗ . The values of u∗ are given by the “Bulk” scheme of Louis (1979, 1982), previously
computed in ACHMT in ARPEGE. The second part, only valid for the instable (convective) cases, i.e. for 1 − δstab = 1, and including the term 0.3 (w∗ )2 , depends on theconvective velocity w∗ (Deardorﬀ, 1980), deﬁned by :w∗
where Q0=

( β HP BL Q0 )1/3 ,

(63)

=

(w′ θ′ )

(64)

surf

= − Ch |uN | (∆θ)N .

8. The turbulence scheme

183

In stable regime, (62) reduces to eS = 3.75 (u∗ )2 .
The term HP BL in (63) is a diagnostic PBL height. It is computed starting from theground as the height of the ﬁrst level where an important decrease in e occurs, corresponding to the top of the PBL and the beginning of the stable layers located above the
PBL.
The term Ch in (64) is the thermal surface Drag coeﬃcient given by (67). The term |uN |is the norm of the wind speed at the ﬁrst upper-air level above the ground. The term(∆θ)N is the diﬀerence in θ between the ground level and the ﬁrst upper-air level above
the ground.

2.9 The “Bulk” formulations in the surface layer
The neutral, dynamical and thermal versions of the surface Drag are denoted by Cdn ,Cd
and Ch , respectively. They are computed following the “Bulk” scheme of Louis (1979,
1982) :Cdn (z, z0 )=Cd (Ri , z, z0 )
Ch (Ri , z, z0 )

=
=



0.4
log(1 + z/z0 )

2

(65)

,

(66)
(67)

fd (Ri ) Cdn (z, z0 ) ,fh (Ri ) Cdn (z, z0 ) .

The functions fd (Ri ) and fh (Ri ) only depend on the Richardson’s number Ri . The coefﬁcient Ch given by (67) is the one used to deﬁne Q0 in (64), in order to compute w∗ by(63).

2.10 The vertical mixing and the dissipation term√
The dissipation term (5) writes Cǫ e e/Lǫ . It is numerically computed by using either
explicit or implicit methods, depending of the value of two coeﬃcients αexp (ADISE in
the code) and αimp (ADISI). In the current version, ADISE=-0.5 and ADISI=1.5.

p√√
The term e in the numerator of Cǫ e e/Lǫ always appears in the explicit forme(t) .
The two coeﬃcients αexp and αimp determine how the other term e is computed, leading
to the following discretization for the term (e)3/2 , in terms of the time steps t and t + dt
:
p

(68)

e(t) [ 1.5 e(t + dt) − 0.5 e(t) ] .

The last formulation is obtained when the term X 3/2 = (e)3/2 is written as the following
Taylor expansion (personal communication of V. Masson)

(e)

3/2

X 3/2

=

(X0 )3/2 + 1.5 (X0 )1/2 (X − X0 ) ,(t + dt)

=

e(t)

p

e(t)

+ 1.5

p

e(t) [ e(t + dt) − e(t) ] ,where 1.5 (X0 )1/2 is the derivative of X 3/2 at X0 , leading indeed to (68).

(69)
(70)

184

8. The turbulence scheme

2.11 The Top-PBL vertical entrainment: LPBLE=.TRUE.
The properties of stratocumulus clouds are greatly aﬀected by cloud-top entrainment
process corresponding to the mixing of free-tropospheric air above the inversion withthe turbulent cloudy air. Stratocumulus cloud layers are maintained by an energeticbalance between radiative cooling and surface and cloud top entrainment warming, and amoisture balance between moisture ﬂuxes and drying from cloud top entrainment. Cloud
top entrainment of dry, warm air from the free troposphere in stratocumuls is strongly
aided by the evaporation of cloud water into a thin layer just above cloud, which cools
the air parcels in the mixing region, making them suﬃciently negatively buoyant to sink
into the cloud layer. Strength of cloud top entrainment increases with stratocumulus
cloud thickness, as thicker stratocumuli have more cloud-top condensate than thinner
ones. Cloud top entrainment mixes in warm, dry air into the boundary layer, raising thecloud base and thereby thinning the cloud layer. A review on stratocumulus can be foundthere:
http : //www.atmos.washington.edu/ robwood/teaching/535/StratusStratocumulusW oodJ uly22.pdf
A historical review up to 1986 can be found in Reuter (1986).
Although the importance of entrainment in aﬀecting cloud properties was realized early(Lilly 1968), the parameterization of entrainment for stratocumulus cloud is yet to be fully
resolved (Wood 2012; Bretherton and Blossey 2014). The ﬁrst eﬀort to understand cloudtop entrainment can be found in the classic work by Lilly (1968), in which a budget for
turbulence kinetic energy (TKE) was applied to the whole BL. Subsequent treatments ofentrainment rate closures considered the TKE budget of the inversion layer (e.g., Tennekes
1973; Tennekes and Driedonks 1981). More recently, substantial theoretical, modeling, andobservational work has focused on the application of a simpliﬁed TKE budget to relateentrainment rates to vertical velocity variance in the cloud-topped BL and the convective
velocity scale w* (e.g., Nicholls 1984; Nicholls and Turton 1986; Bretherton et al. 1999;
Lock et al. 2000). The vertical resolution of the large scale models in GCM mode, remains
rather coarse. Above an altitude of 1000 to 1500 m height, the layer depths are typicallyas high as ∆z > 200 to 300 m.
With such coarse vertical resolutions (above the critical value of (∆z)crit ≈ 20 m to50 m) it is not possible to represent in a realistic way the vertical entrainment in theStratocumulus. This vertical entrainment occurs at the very top of the PBL, where thedry air above the Top-PBL is mixed into the underlying cloudy air, located below the
Top-PBL height.
In order to parameterize this sub-grid process (sub-grid on the vertical), the old ideas ofTenekes (1973), revisited by Nicholls and Turton (1986) or Grenier et Bretherton (2001),
and summarized in the note of Grenier (2002), has been implemented in the CBR00 versionof ARPEGE.
The activation of top-PBL entrainment is also speciﬁc of the climate version (LPBLE=.TRUE)
vs the NWP one (LPBLE=.FALSE.) for positive impact in terms of stratocumulus coverin the eastern part of the oceans.
The ﬁrst step is to compute the inversion height zinv . It could be deﬁned as a jump in,either ∂θ/∂z or θ, or when the Richardson number Ri becomes greater than a thresholdvalue (Ri )crit . It is rather computed in the code starting from the ground as the heightof the ﬁrst level where an important decrease in e occurs, corresponding to the top of the
PBL and the beginning of the stable layers located above the PBL.
The second step is to replace the exchange coeﬃcients previously computed by (10) to(14) by another coeﬃcient, computed at the inversion, denoted by Kinv and verifying(w′ θvl ′ )(zinv )==− went ∆inv (θvl ) ,∆inv (θvl )− Kinv∆inv (z)(71)⇒

Kinv = went ∆inv (z) ,(72)

where went is called “entrainment velocity” (at the inversion) and where ∆inv (θvl ) is thebuoyancy jump in θvl and across the layer surrounding this inversion ∆inv (z).

8. The turbulence scheme

185

Following the results of Grenier and Bretherton (2001) and some unpublished ideas of Grenier (Workshop EUROCS at Utrecht, in April 2002), the Top-PBL entrainment exchangecoeﬃcient Kinv is deﬁned at the inversion level by
Kinv=

went ∆inv (z) = Ainv

< e >3/2.
2
Linv Ninv(73)

Presently, there is no Prandtl number and all the exchange coeﬃcients (10) to (14) are
replaced by the same value (73).
Grenier (2002) has introduced the average value over the whole PBL of the turbulent
kinetic energy, denoted by
<e>

1

=

zinv

Z zinv(74)

e(z) dz .

0

2
The unknown parameters in (73) are the Richardson number at the inversion Ninv
, themixing length at the inversion Linv and an adimensional coeﬃcient Ainv . These parameters are deﬁned by:

2
Ninv

=

Linv

β

=

Ainv



∆inv (θvl )∆inv (z)

0.085 zinv ,

=

A1



(75)

,
(z=zinv )Lv/f < qc >1 + A2cp ∆inv (θlt )



(76)
(77)

.

for positive values of AGREF. And for negative values of AGREF, as proposed by GuÃ c rÃ c my
(2011):Ainv=−A1 ∗ AGREFh

1 − (1 +1) ∗ sin2 (f (inversion))
AGREF

i

.

(78)

For AGREF <0 (the current choice is AGREF=-0.036), the sinus is zero for strong inversions (∆inv (θlt ) > 1.5K corresponding to stratocumulus case, whereas the sinus equals 1
for smaller inversion (convective case). If the sinus = 0, Ainv = - AGREF*AGRE1 ( 0.2
with the current values, and AGRE1 (=5.5) for the convective cases. Then the ratio for
top PBL entrainment between convective clouds and stratocumulus is around 27. In the2code, Ninv
is ZBI (acturb.F90) There is a security for the computation of Kinv in (73)2where, from (75), the division by Ninv
leads to a division by the term ∆inv (θvl ) which
can be tiny or even equal to 0. A minimum value of 1.5 K or so is managed by setting
AJBUMIN= 0.005 in the NAMELIST, with ∆inv (θvl ) > AJBUMIN ⋆ θ.
The part of the entrainment coeﬃcient Ainv which is controlled by A2 is called the “evaporative enhancement of entrainment” in Grenier and Bretherton (2001). It is not used in
the current version. The jump in potential temperature ∆inv (θlt ) corresponds to anothermoist potential temperature, diﬀerent from the Lilly one θvl given in (7). This conservative
potential temperature θlt is a mix of the Betts variables, deﬁned byθlt=θl (1 + 0.608 q t ) .

(79)

The average value over the whole PBL (liquid + solid) < qc > is deﬁned by (ZQCBLK inthe code):< qc >=

1zinv

Z zinv0qc (z) dz .

(80)

186

8. The turbulence scheme

During CMIP6 development, the choice for the numerical values of A1 and A2 has been
made with analysis of 1D cases and checking the stratocumulus cover on the eastern sideof oceans.
It is possible to rewrite Kinv given by (73) in a form similar to (13) or (14), in order todeﬁne a kind of “stability function” (the term into brackets).
Kinv

=

Ainv
Linv < e >1/2
2



2 <e>
2
(Linv )2 Ninv



.

(81)

The term into brackets is a kind of bulk stability function [ (LD )2inv / (Linv )2 ], similar to2
the functions φ3 and ψ3 deﬁned in (13) and (14), where (LD )2inv ≡ 2 < e > /Ninv
is thesquare of a kind of bulk Deardorﬀ length, similar to (60).

2.12 Explicit diffusion of conservative variables: LDIFCEXP=.T.
Note that this option is speciﬁc of climate simulations (GuÃ c rÃ c my, 2011) and corresponds to the logical keys LDIFCEXP=.T. and LDIFCONS=.F.. In the NWP version,
LDIFCEXP=.F. and LDIFCONS=.T.
The conservative variables are:sl = s −LLql −qi = cp Tl + gz
Cp
Cp

qt = qv + ql + qi

(82)

, with s = Cp T + gz, the dry static energy, sl the moist static energy, qt the total water
speciﬁc content, and Tl = T − CLp ql (Bougeault, 1982 for example). Using the subgrid
distribution of ql , the deﬁnition of Tl and the s variable, it can be shown that the equation
for turbulent ﬂuxes of ql and of conservative variables is of the type:w′ ql′ = aλ3 F 2(Q1) ∗ (w′ qt′ − α1 w′ Tl′ )(83)

In the code: a corresponds to ZAH variable, λ3 F 2(Q1) to PL3F2 and α1 to ZQSLTLH.
For computing the corresponding turbulent exchange coeﬃcients,Kql (ZGKQLH in thecode), Ks (ZGKTAH in the code), and Kqv (ZGKQH in the code), the following relaK ∆ql
Kqt
ltions are used: ql) because the turbulent exchange= aλ3 F 2(Q1) ∆Φ∗ (∆qt − α1 ∆s∆Φcp
coeﬃcient for moist static energy Ks equals that for the total water content Kqt .
For computing Kqv , as qt = qv + ql + qi , we have:
Kqv ∆qv = Kqt ∆qt − Kql ∆ql(84)

and, for Ks ,Ks ∆s = Ksl ∆sl + Lv Kql ∆ql(85)

The subroutine ACTURB computes the turbulent coeﬃcients of conservative variables (sl ,qt ) and the transformation of model variables (s,qv and ql ) into conservative variables. As
output of the routine, when LDIFCEXP =.TRUE., the turbulent coeﬃcients of the modelis inverted, starting with qlvariables are computed: the generic relation Fψ = −Kψ ∗ ∂ψ∂z(equation 83) and then s and qv with equation 82.
The subroutines ACDIFV1 and ACDIFV2 then compute the turbulent ﬂuxes (implicit resolution) with as input model variables and their turbulent coeﬃcient (if LDIFCEXP=.TRUE.)
or the conservative variables and their exchange coeﬃcients for LDIFCONS=.TRUE..

8. The turbulence scheme

187

2.13 Enhanced turbulence due to convection, LCVTURB=.TRUE.
At high levels, in presence of convection and when the grid is saturated, the turbulenceis generally insuﬃcient. As a consequence, due to the formulation of cloudiness, the gridis too often overcast. There is a need to add an extra source of turbulence in order todecrease the cloudiness. The source of turbulence due to convection could be introduced
by induced gravity wave or by a convective Ri (from convective transport of dynamics
and thermodynamics). For sake of simplicity a ﬁrst attempt has been designed using
a minimum value of σs (= 5.10−5 kg/kg) in case of presence of convective condensatesand grid saturation above 600 hPa (mean level of beginning of detrainment for deep
convection).

3 Eddy-Diffusivity Mass Flux parameterization
The algorithmic treatment of transport is based on the "EDMF" concept and uses an implicit combined resolution. The "EDMF" concept (Hourdin et al 2002, SOares et al 2004,
Siebesma 2007) is used in all NWP and climates models in MF (Arome, ARPEGE/ALADIN):w′ φ′ = −K∂φMu+
(φu − φ)
∂z
ρ

(86)

√
with K = cLBL89 T KE and

LBL89] = Lǫ =h1
{ (Lup )−2/3 + (Ldown )−2/3 }
2

i−3/2

.

(87)

where Lup and Ldown are computed using dry buoyancy following Bougeault and LacarrÃ¨re(1989). Due to numerical stability problems encountered with EDKF ( shallow convection scheme of EDMF type; Pergaud et al., 2009) in ARPEGE with "large time-steps",a common implicit solver for Eddy-Diﬀusivity and Mass Flux part has been developed,
enabling stability for large time-steps.
1.Implicit treatment of the mass ﬂux equations:



∂ψ
∂t

 Fψ
MF

=
=

ρw′ ψ ′ = M (ψu − ψ)1 ∂Fρ ∂z ψ

(88)

The second equation is solved implicitely for Fψ = (1 − zi )Fψ− + zi Fψ+ ,Fψ+ = Fψ− + δFψ = Fψ− +∂Fψδψ = Fψ− − M (ψ˜+ − ψ˜− )
∂ψ

(89)

then:



∂ψ
∂t



=
MF

1 ∂
(F − − zi M (ψ˜+ − ψ˜− ))ρ ∂z ψ
|
{z
}

(90)

with upward increasing j levels and Fψ , M deﬁned at levels where ψ̃(j) = 0.5(ψ(j) + ψ(j −1)) The second rhs term corresponds to the implicit correction.

188

8. The turbulence scheme

We obtain: +ψ − (j) = ψ∆t(j) −−(Fψ (j + 1) − Fψ− (j)+(j + 1) + 0.5ψ + (j) − 0.5ψ − (j + 1) − 0.5ψ − (j))
 −zi M (j + 1)(0.5ψ++zi M (j)(0.5ψ (j) + 0.5ψ + (j − 1) − 0.5ψ − (j) − 0.5ψ − (j − 1)))
ρ∆z

(91)

Grouping ′ +′ terms in the lhs of the equation the following tridiagonal system is obtained: +∆tψ (j + 1)(0.5 ρ∆zzi M (j + 1))



+
∆t
∆t
 +ψ (j)(1 + 0.5 ρ∆zzi M (j + 1) − 0.5 ρ∆zzi M (j))
−ψ + (j − 1)(0.5 ∆t zi M (j)) = ψ − (j) + ∆t (Fψ− (j + 1) − Fψ− (j))

ρ∆z
ρ∆z

∆t

+0.5 ρ∆zzi M (j + 1)(ψ − (j + 1) + ψ − (j))

 −0.5 ∆t z M (j)(ψ − (j) + ψ − (j − 1))

(92)

ρ∆z i

2.Implicit treatment of the Eddy-Diﬀusivity equation. The Eddy diﬀusivity equation:



∂ψ
∂t



eddy

=−

1 ∂
∂ψ
(k
)
ρ ∂z ∂z(93)

is discretized as follows:ψ + (j)−ψ − (j) = −k(j + 1)k(j)∆t((ψ + (j+1)−ψ + (j))−(ψ + (j)−ψ + (j−1)))(94)ρ∆z(j) ∆z(j + 1)∆z(j)This yields to the simple tridiagonal system:
k(j+1)+∆t ψ (j + 1)( ρ∆z(j) ∆z(j+1) )+ψ + (j)(1 −∆t( k(j+1) + k(j) ))ρ∆z(j) ∆z(j+1)∆z(j)
 +ψ + (j − 1)( ∆t k(j) ) = ψ − (j)(95)ρ∆z(j) ∆z(j)3. Common implicit resolution of the EDMF equation The discretization of the full EDMF
equation:



∂ψ
∂t



=
edmf

1 ∂
∂ψ
(−k
+ M (ψu − ψ))
ρ ∂z
∂z

(96)

yields to the following tridiagonal system: +k(j+1)∆t( ∆z(j+1)+ 0.5M (j + 1)))+ψ (j + 1)( ρ∆z(j)k(j+1)
k(j)
+
∆t

(
+ ∆z(j)+ 0.5M (j + 1) − 0.5M (j)))+
ψ
(j)(1
−

ρ∆z(j) ∆z(j+1)ψ + (j − 1)((∆t

( k(j) + 0.5M (j)))ρ∆z(j) ∆z(j)

∆t
∆t


(ψ − (j + 1) − (ψ − (j)) + 0.5 ρ∆z(j)M (j + 1)(ψ − (j + 1) + ψ − (j))= ψ − (j) + ρ∆z(j)


−
−
∆t

−0.5 ρ∆z(j) M (j)(ψ (j) + ψ (j − 1))

The value of zi in the code is 0, meaning that a combined (convection, turbulence) CHECK
treatment is used but the coeﬃcient for implicitation of convection equals 0. Sensitivity
tests around this choice have been run.
This work is done in subroutines acdifv1/acdifv2.

(97)

8. The turbulence scheme

189

4 Algorithmics
The monitor of the ARPEGE physics is APLPAR. The prognostic TKE scheme is calledby the main subroutine ACTKE, with the TKE variable “PTKE” as input array and theﬂux of the tendencies “PFTKE” as output array.
The list of subroutines called if LVDIF.AND.LECT=.TRUE. is:• APLPAR : monitor of the ARPEGE physics• ACHMTLS : general surface layer computations, PCPS (chaleur massique Ã pression constante de l’air au sol), surface latent heat...
• ACTKE : monitor of the TKE computations ; possible half<->full<->half averages(if LECTFL)• ACBL89 : computes the upward and downward buoyancy lengths Lup and Ldown ,then the mixing length Lm and the dissipation length Lǫ by (87) and (61) ; computes
the stability functions φ3 = ψ3 given by (16) and (17). It is assumed that the gridcell mean liquid and solid cloud water contents are available as input of ACBL89
and ACTURB.
• ACTURB : compute eS by (62) ; computes the exchange coeﬃcients Km and KT ;computes the moist thermal production term (28) with the help of (48) and (51)computes the grid-cell mean cloud cover N s , the grid-cell mean cloud water content
qc s and the subgrid normalized second order ﬂux F2 (Q1 , AS ) with the use of (49)to (51) ; compute the Top-PBL height and the exchange coeﬃcient at the Top-PBL
inversion.
• ACEVOLET : numerical computation of the TKE equation (1), with an implicit
solver for the vertical diﬀusion (25), with αimp = 1.5 and αexp = −0.5 givenby (68). The dissipation term (5) is computed with the help of the same (1.5
; −0.5) implicit scheme. The moist thermal production term is an input of thesubroutine (computed in ACTURB). The dynamical production term is computedin this subroutine.
Note on the interaction with the radiation scheme.
The “large scale” variables N s and qc s are computed in (49) and (50), in the turbulent
scheme. They correspond to the grid-cell averaged values for the Cloud Cover and theCondensed Cloud Water. They are added to the same informations coming from the
PCMT convections scheme, with some suitable assumptions for the overlapping of thecloud layers on the vertical (Maximum overlap for LGPCMT + LRNUMX).

5 Logical keys and main parameters in the current
versionLogical keys
&NAMPHY
LDIFCEXP= .TRUE. : explicit diﬀusion of conservative variables
LDIFCONS= .FALSE. : use the Betts conservative variables (LDIFCONS and LDIFCEXP work together: if one is TRUE, the other is False)
LDISTUR= .TRUE.
: alternative discretization in ACTURB (cf JEF)
LECT= .TRUE.
: the main switch for the turbulent scheme
LECTFL= .FALSE.
: no half<->full<->half averages in ACTKELECTQ1= .TRUE.
: for limitation of Q1 in acturb
LNEBECT= .TRUE. : use of the Bougeault-Bechtod values (F0 , F1 , F2 )LNSMLIS= .TRUE. : key for activating smoothing of Smith’s cloudiness
LPBLE= .TRUE.
: use of the Top-PBL vertical entrainment

190

8. The turbulence scheme

Parameters &NAMPHY0ACBRPHIM= 2.2: a maximum value for (φ3 ). ZPHI3MAX (in acbl89) is(1-ACBRPHIM)/ACBRPHIM.
ADISE= −0.5
: Dissipative parameter for TKE (explicit part)
ADISI= 1.5: Dissipative parameter for TKE (implicit part)
AGREF= −0.036 Parameter for Top-PBL entrainment; Ainv = AGRE1∗AGREF
AGRE1= 5.5: ﬁrst Sc value A1 – Top-PBL entrainment ; see the Table (??)
AGRE2= 0.
: second Sc value A2 – Top-PBL entrainment ; see the Table (??)
AJBUMIN= 0.00525 : a threshold value to avoid divisionby zero√AKN= 0.126: CCH02 value for (Km /Lm / e) – see the Table (??)
ALD= 1.18: CCH02 value for (1/Cǫ ) – see the Table (??)
ALMAV= 20.
: momentum exchange asymptotic mixing length
ALMAVE= 10.
: an asymptotic and minimum value for Lm
ALPHAE= 2.7: CCH02 value for (Ke /Km ) – see the Table (??)
ALPHAT= 1.13
: CCH02 value
p for (KT /Km /φ3 ) – see the Table (??)
ARSB2= 0.833 : the coeﬃcient 1/ Cǫθ in (40)
ARSC1= 0.1389 : parameter for φ3min
ECTMIN= 1.0E − 6 : a minimum value for (e)ECTMAX= 100 : a maximum value for (e)Top-PBL entrainment ; see the section (2.11)
STTBMIN= 1.73 : parameter for Q1 limitation (min)9
Microphysics
parameterizations scheme

1 Introduction
There exists two main types of microphysics scheme, Bulk and Bin schemes:• spectral (Bin) microphysics aim to compute microphysics as accurately and generally as possible: it divides microphysical particles into bins for diﬀerent sizes andcompute evolution of each bin separately; in that case the particle size distribution (PSD, see below) is an output not an input, but these schemes are much too
expensive for operational use.
• Bulk schemes are used to calculate microphysics with a semi-empirical PSD.
• There exist also hybrid schemes which aim the accuracy of bin schemes with efﬁciency of bulk schemes ( for example, Onishi and Takahashi (2012) who usedbin for warm processes and bulk for ice) but they remain too expensive for NWP
and Climate models. There exist also bin-emulating schemes which calculate rates
oﬄine with complex bin scheme and develop lookup tables (Verlinde, 1990).
Microphysics processes are complex, but can be modelled very eﬀectively with a bin approach, which is only a research tool, much too expensive for use in NWP or Climates.
Note also the deﬁnition of microphysical moments: for PSDs f (m), m being the particle
mass, the kth moment is:
M (k) =Z ∞mk f (m)dm(1)

0

It is possible to distinguish three types of schemes according to the "k" value:• One-moment schemes (1M) correspond to k=1 (mass)• Two-moments schemes (2M) correspond to k=0 and 1 (number concentration andmass)• Three-moments schemes (3M) correspond to k=0 and 1 and 2 (number concentration, mass and radar reﬂectivity)
A partial history of schemes with main steps and papers is proposed below. Note theincrease in complexity and reducing gap between bulk and bin methods:

192

9. Microphysics parameterizations scheme• Kessler (1969) : ﬁrst warm rain bulk parameterization• Lin et al. (1983) : 1M, includes hail
• Cotton et al. (1986) : ﬁrst bin parameterization (RAMS)• Murakami (1990) : 1 M, snow includes crystals and aggregates• Verlinde et al.(1990) : development of lookup tables• Ferrier (1993) : 2M for ice and precipitating species• Cohard and Pinty (2000) : 2M for warm microphysics• Saleeby and Cotton (2004) : 2M bin-emulating bulk scheme. Fully interactive withprognostic CCN and IN aerosol schemes• Morrison et al. (2005) : 2M scheme for droplets, cloud ice, rain and snow
• Milbrandt and Yau (2005) : 3M scheme for hail
• Lim and Hong (2010) : WRF 2M 6 classes; prognostic treatment of cloud condensation nuclei

Processes to account for are:
• Droplet nucleation (condensation)• Droplet growth by vapour diﬀusion• Collisions between droplets and between hydrometeors• Sedimentation (diﬀerential motion)• Freezing/Melting• Ice multiplication• Raindrop breakup• Eﬀects of aerosol on all these
Much "large-scale" rain originates from melting ice. Much convective rain and some largescale rain originates from collision and coalescence of cloud droplets.
In bulk schemes diﬀerent collisions can be described:• self-collection (X+X–>X): droplet + droplet –> droplet• self-collection: rain + rain —> rain
• autoconversion: droplet + droplet —> rain
• accretion: droplet + rain —> rain
Self-collection does occur due to fall speed diﬀerences between particles of diﬀerent sizes
but of same type.

2 The microphysics scheme
2.1 History of the Lopez Scheme
Mainly developed for variational assimilation of cloud and precipitation observations, theLopez’ scheme (2002) describes both the large-scale cloud water content and the largescale precipitations (liquid and solid), by using prognostic equations and describing several
physical processes acting between the diﬀerent species. Since then, the Lopez has beentested, modiﬁed and improved, for both Climate and NWP mode (see Bouyssel, 2005),
also in the ALADIN-ALARO team (see Gerard, 2005 ). This is also the microphysics
scheme used in the current NWP ARPEGE model. This scheme has been deeply modiﬁed
by Bouteloup et al. (2010) introducing a new statistical sedimentation scheme. Since
2009, this scheme has been used and tested in diﬀerent research versions of the climate
model, with a convection diagnostic scheme (Bougeault, 1985) or a prognostic one (PCMT,
Piriou, 2007, GuÃ c rÃ c my, 2011). In the second case, the Lopez scheme is called twice,
one time by the APLPAR routine for the large-scale part, and one time by the ACPCMT
routine for the convective part.

9. Microphysics parameterizations scheme

193

2.2 Which processes are described in the scheme?
The physical processes described by the scheme of Lopez (2002) are based on a “bulk”
assumption. They are close to the one described in Fowler et al. (1996).
• The cloud water content and the precipitations ( 4 prognostic variables, either
for large-scale or for convective part = 8 prognostic variables) are supposed to bepronostic variables, available as input of the scheme.
• The cloud water content (liquid and ice) and the precipitations (rain and snow)
form two separate distributions of dropplets. The smaller ones are the suspended
cloud condensate, whereas the larger ones are precipitating species.
• The condensation/deposition process acts as a source for the liquid/ice condensed
cloud water.
• The evaporation/sublimation process acts as sinks for the rain/snow precipitation
content• The condensed cloud water (liquid and ice) can transform into the precipitation
content species (rain and snow) via the processes of auto-conversion and collections(accretion + aggregation + riming).
• The precipitation content species (rain and snow) fall with diﬀerent fall speeds.
The sedimentation processes is that of Bouteloup et al. (2010), described in chapter
PCMT.
• The Freezing-Melting-Bergeron processes between cloud liquid water and cloud ice
are not explicitly described. Only the snow melting is computed (between snow andrain). The detrainment from the convection schemes (neither shallow nor deep) are
not taken into account.

2.3 "Bulk" equations
The either grid-averaged or convective condensed water species, input of the scheme, will
be denoted below by
ql
qi
qr
qs

liquid cloud water content
ice cloud water content
rain precipitation content
snow precipitation content

The system of four prognostic equations corresponding to the physical processes describedin the section (??) can be written as∂(ql )/∂t =





 ∂(qi )/∂t =∂(qr )/∂t =





∂(qs )/∂t=+ Cl − Al+ Ci− COL(l/r)− Ai− Er + Al− Es− COL(l/s)− COL(i/s)+ COL(l/r)
+ Ai

(2)
− Fr

+ COL(i/s) + COL(l/s)

The condensation processes Cl and Ci act as source terms of the cloud water contents(liquid and solid). The evaporation and the falling processes, i.e. Er , Es , Fr and Fs , act
as sink terms for the precipitation contents (rain and snow). The auto-conversion terms
Al and Ai and the collection terms COL(l/r) , COL(i/s) and COL(l/r) act as conversion
terms, transforming one species into another.
The three collection processes parameterized in the scheme of Lopez are caused by thediﬀerential (constant) fall speed between the cloud species and the precipitating ones.
They represent the capture of the cloud species by the precipitations.

− Fs

194

9. Microphysics parameterizations scheme
COL(l/r)
COL(i/s)
COL(l/s)accretion
aggregation
rimingcollection of cloud liquid water by rain
collection of cloud ice by snow
collection of cloud liquid water by snow

The fourth collection (of cloud ice by rain) is not parameterized. Indeed, the cloud ice
regions are usually located above the rain regions.

2.4 The condensation/evaporation process
The large-scale cloud-cover and condensed cloud water are computed in the prognostic
TKE scheme, using the Bougeault functions (LNEBECT=TRUE). If the statistical cloudwater content is denoted by (qc )stat , the partitioning of the liquid part (ql )stat and thesolid part (qi )stat is obtained with

δice =(

1 − exp (T − Tf )2 /(2 ∆T )20



for T < Tf ,otherwise .

(3)

The value of δice is given by the function FONICE(T ), contained in the header fctdoi.h,with ∆T = RDT∗RDTFAC, RDT= 11.82 K and RDTFAC= 0.5, leading to ∆T = 5.91 K.
The ﬂuxes of condensation/evaporation are denoted by PFCSQL for the liquid cloud waterand by PFCSQN for the solid cloud water. They are computed in ACPLUIZ, with (qc )stat
as input, following∆ (PFCSQL)∆p∆ (PFCSQN)
g
∆p
g

=
=

(1 − δice ) ∗ (qc )stat − ql,∆t
δice ∗ (qc )stat − qi
.
∆t

(4)
(5)

2.5 The auto-conversion processes
Rain is initiated in liquid water clouds by collision and coalescence of cloud droplets
wherein larger droplets with higher settling velocities collect smaller droplets and become
embryonic raindrops. This so-called autoconversion process is usually the dominant process that leads to the formation of drizzle in stratiform clouds.
Kessler (1969) proposed a simple parametrization that linearly relates the autoconversion
rate to the cloud liquid water content, and this parametrization is widely used in cloudrelated modeling studies because of its simplicity. This simple parametrization supposesthat the autoconversion rate is a function of not only of the liquid water content but also
the cloud droplet number concentration and the spectral shape of the cloud droplet size
distribution (Berry, 1967; Berry and Reinhardt 1974).
Without loss of generality, all of the Kessler-type parametrizations can be written as( Liuet al., 2003):
P = cLH(y − yc )(6)where P is the autoconversion rate (in gcm−3 s−1 , c is an empirial coeﬃcient (in s−1 ) and
L is the cloud liquid water content (in gcm−3 ). The Heaviside step function H(y − yc ) is
introduced to describe a threshold yc below which the autoconversion is negligibly small.
The meaning of y is diﬀerent in diﬀerent parametrizations. In our scheme, it represents
the cloud liquid water content, as in the original Kessler parametrization. Liu et al.
(2003) propose a reexamination of typical Kessler-type parametrizations and show thatapplication of the generalized mean value theorem for integrals to the general equation for
the autoconversion rate provides a rigourous theoretical basis for Kessler’s formulation.

9. Microphysics parameterizations scheme

195

As shown in eq. 6, the auto-conversion processes start to occur when the values of thesuspended condensed water content are larger than some threshold value qcrit . It is important to think in terms of the in-cloud values (close to the saturating state), and notwith the grid-cell averages (often far from the saturating state).
Let us denote by qc the grid-cell average of the condensed water content, and by qc∗ thecorresponding in-cloud value, obtained from the cloud cover Ns = Max[ εN ; Ns ] as
qc∗

≡

h

Max 0 ;

qc
Ns

i

(7)

.

The auto-conversion process can be represented as the in-cloud conversion rate A∗c fromone of the (liquid or solid) condensed water in-cloud value qc∗ to the corresponding (liquid
or solid) precipitating in-cloud value qp∗ , with the Kessler (1969) assumption, leading to





∂qc∗
∂t
∂qc∗
∂t




auto

auto

≡

− A∗c ,≡−

(8)qc∗ − qcrit
.
τc

(9)

The threshold value qcrit and the time scale τc are supposed to be two constant terms.
They correspond to the physical assumption that the dropplets smaller than a given radiusremain suspended cloud condensate, whereas the dropplets larger than a given radius are
becoming precipitating species, with a time scale τc . The threshold used in this “bulk”
scheme is not a critical radius, but a critical speciﬁc content qcrit for the suspended cloud
condensate.
The threshold values for qcrit are set from the NAMELIST of ARPEGE. They are not thesame for the liquid and the solid species.
(ql )crit(qi )crit (T )RQLCR ,[ RQICRMIN ; RQICRMAX ] .

=
∈

(10)
(11)

The threshold for the liquid water content is a constant, whereas for the solid water(ice), the formulation is more complex, with a variation between the two extremum values
max(qi )mincrit = RQICRMIN and (qi )crit = RQICRMAX. Indeed, in order to allow the autoconversion of ice even at very low temperature, (qi )crit (T ) is assumed to decrease withtemperature, according to the formulae (A.1) of Lopez (2002):(qi )crit (T )





max
min
(qi )max{ 1 + tanh [ α (T − Tf ) + β (12)]} .
crit − 0.5 (qi )crit − (qi )crit=

The coeﬃcients α and β are two constant terms in Lopez (2002), see his Appendix-D
where α = −0.1572 K−1 and β = −4.9632. In the ARPEGE code, due to additionaltunings made in the NWP team, α and β depend on the temperatures T1 = RQICRT1
and T2 = RQICRT2, both available in the NAMELIST:(qi )max
crit (1 − 0.999),
min(qi )max
crit − (qi )crit

γ1=−1 + 2γ2=−1 + 2min(qi )max
crit − 1.5 (qi )crit
,
min
max
(qi )crit − (qi )crit

0.5 log

abs

γ3
γ4

=
=

0.5 log





abs





1 + γ11 − γ1
1 + γ2
1 − γ2




(13)
(14)

,

(15)

.

(16)

196

9. Microphysics parameterizations scheme

From (13) to (16), α and β are deﬁned by
α

=

β

=

γ3 − γ4,T2 − T1
γ 3 − T2 α .

(17)
(18)

Over the ice-shell sea regions or over the snow-covered land areas, there are fewer Cloud
Condensation Nuclei (CNC) than over the open-sea or free land areas, where more aerosol
are available. A ﬁrst attempt to take into account these diﬀerential icy/warm properties ismade in the ACMICRO subroutine, by computing the term ZFACICE and by multiplying(qi )crit (T ) by 1−ZFACICE∗(1−RQICRSN).
The term ZFACICE is equal to 0 over the open-sea and free land areas, equal to 1 overthe icy regions. For RQICRSN = 0.5 (available in the NAMELIST), (qi )crit (T ) is not
modiﬁed over the open-sea and free land areas, whereas it is divided by a factor 2 overthe icy regions. For RQICRSN = 1, (qi )crit (T ) is not modiﬁed over the icy regions.
The in-cloud auto-conversion rate A∗c is computed with both the ﬁnite diﬀerence (19) plus
the analytic method (20). These schemes are obtained by integrating (8) and (9) from thetime t with the value qc ∗(−) to the time t + ∆t with the value qc ∗(+) , considering qcritand τc as two constant terms.
It results the two following formulationsA∗c≈qc ∗(+) − qcritqc ∗(−) − qcrit≈qc ∗(+) − qc ∗(−)
,
∆t


∆t
.
exp −
τc

(19)

−

(20)

They respectively correspond to discrete solutions of (8) and (9).
From (19) and (20), the auto-conversion rates for the liquid and the solid (ice) watercontent write
A∗l
A∗i

≈
≈

h

h



∆t
1 − exp −τl

1 − exp −∆tτi

 i  q ∗(−) − (q )l crit

c

i 

∆t



qc ∗(−) − (qi )crit (T )
∆t

(21)

,



.

(22)

The terms A∗l and A∗i are denoted by ZDUM in ACMICRO.
The threshold values (ql )crit and (qi )crit (T ) are given by (10) and (12). The two time
constant terms τl and τi are denoted by ZCAUT in ACMICRO, they write
1/τl
1/τi

=
=

RAUTEFR ,RAUTEFS ∗ exp[ − (β)ice (T − Tf ) ] ,

(23)
(24)

with RAUTEFR and RAUTEFS available in the NAMELIST.
The exponential term in (24) is the temperature dependant ice conversion eﬃciency of
Pruppacher and Klett (1998), with the coeﬃcient (β)ice = RAUTSBET = 0.025 available
in the NAMELIST.
The ﬁnal grid-cell average or convective auto-conversion rate for the liquid and solid species
are computed from (21) and (22) by inverting (7) and by multiplying by the large-scale
cloud cover Ns , for the large-scale contribution (ZNEBS in the code), and convective
cloudiness (ZN EBCM ICRO in acpcmt.F90).
Al
Ai

=
=

Max { 0 ; A∗l ∗ Ns } ,Max { 0 ; A∗i ∗ Ns } .

The terms Al = PAUTOL and Ai = PAUTOI are available as output of ACMICRO.

(25)
(26)

9. Microphysics parameterizations scheme

197

2.6 The distribution of particle - Fall-velocities.
The classical equations for the collection of suspended cloud species by the precipitating
ones are derived in Fowler et al. (1996) by using the Marshall and Palmer (1948) distribution for the rain and the Gunn and Marhall (1958) distribution for the snow.
In
the paper of Lopez (2002), the same distribution of Marshall and Palmer (1948) is used
for the rain, with the fall speed given by Sachidananda and Zrnić (1986) and Foote and
Du Toit (1969). For the ice particles, the ideas of Houze et al. (1979) and Cox (1988)
have been taken into account.
For the rain, the assumed spectra of particle number Nr (D), mass Mr (D) and fall velocity
Vr (D) express in terms of the particle diameter D as follows:
Nr (D)Mr (D)Vr (D)

=
=
=

(27)

N0r exp(− λr D) ,
ρw
ν1

π D3
,
6



ρ0
ρ

0.4

(28)
D ν2 .

(29)

with N0r the intercept parameter in raindrop size distribution ( N0r = 8.106 m−4 ) , λr theslope parameter of the raindrop size distribution, ρw the liquid water density (ρw = 1000
kgm−3 ) and ν1 a coeﬃcient in rain fall-speed distribution (ν1 = 377.8 m1/3 s−1 ). The
slope λr of the Marshall and Palmer distribution is related to the grid-cell average rain
content qr by

λr



=

π ρw N0r
ρ qr

1/4

(30)

.

For the snow, the equivalent spectra of particle number Ns (D), mass Ms (D) and fallvelocity Vs (D) express as follows (with D the maximum dimension of ice particles):Ns (D)Ms (D)==

N0s (T ) exp(− λs D) ,σ1 D σ 2 ,Vs (D)

=

τ1



ρ0
ρ

0.4

(31)
(32)

D τ2 .

(33)

with N0s(T ) the intercept parameter in ice particle size distribution (N0s(T ) = 2.106 exp(−0.1222(T −Tf ))m−4 , Houze et al. (1979)) , λs the slope parameter of the ice particle size distribution,
σ1 a coeﬃcient in ice-mass distribution (σ1 = 0.069 kgm−2 ), τ1 a coeﬃcient in ice fallspeed distribution (τ1 = 21 m1/2 s−1 ) and ν2 the exponent in snow fall-speed distribution(ν2 = 2/3). The mass-diameter relationship Ms (D) is given by Cox (1988).
The slope λs of the distribution is related to the snow content qs by:

λs



=

Γ(1 + σ2 ) N0s (T ) σ1
ρ qs

1/(1+σ2 )

.

(34)

where σ2 = 2, and the Gamma function Γ(z) is deﬁned for any real z by

Γ(z)

=

Z +∞
0

tz−1 exp(−t) dt,

(35)

198

9. Microphysics parameterizations schemewith Γ(1) = 1 and the usual properties Γ(z + 1) = z Γ(z), leading to Γ(n + 1) = n ! valid
for any positive integer n.
The mass-weighted average fall speed of particles can be computed as:R∞
N (D)M (D)V (D)dD
V = 0 R∞
0(36)

N (D)M (D)dD

and the derived associated fall speed of particles Vr for rain and Vs for snow should begiven byVr
Vs=

17.116 ∗

=



ρ0
ρ

0.4

(ρ qr )1/6 ,4.323 ∗ exp [ 0.0204 (T − Tf ) ]

(37)



ρ0
ρ

0.4

(ρ qs )1/6 .

(38)

These expressions are very similar to the ones derived by Heymsﬁeld (1977).
During the development of CMIP6 ARPEGE model, SCM and GCM tests of these formulations have been done with rather positive impacts in 1D, but detrimental ones in 3D.
So constant values are used as in CMIP5: Vr = TFVR and Vs = TFVS are set equal tosome “bulk” constant values, with both TFVR ≈ 5 m s−1 and TFVS ≈ 1 m s−1 available
in the NAMELIST.

2.7 The collection processes
The distributions (27) to (29) for the rain spectra and (31) to (33) for the snow spectra
lead the three collection rates (39) to (41) for COL(l/r) , COL(i/s) and COL(l/s) .
COL(l/r)
COL(i/s)
COL(l/s)≡accretion

(l/r)

(l/r)

(39)

(i/s)

(40)

Eacc ql qr ,= Kacc(i/s)≡

aggregation = Kagg Eagg (T ) qi qs ,≡riming

(l/s)

(l/s)

(41)

= Krim Erim ql qs .

They express in terms of the product of the 3 collection coeﬃcients (K), the 3 collection
eﬃciencies (E) and appropriate couples of grid-cell mean water species content, i.e. ql qr
or qi qs or ql qs .
The collection eﬃciencies write
(l/r)

Eacc
(i/s)
Eagg (T )=RACCEF ,(42)=

RAGGEF ∗ exp [ − 0.025 (T − Tf ) ] ,

(43)

(l/s)

=

RRIMEF ,

(44)

Erim

with RACCEF, RAGGEF and RRIMEF three constants, all available in the NAMELIST.
The collection coeﬃcients write
(l/r)

Kacc

(i/s)

Kagg

=
=

12.695 ν1 Γ(3 + ν2 )
4 ρw
(l/s)

Krim =



ρ0
ρ

0.4

≈ 4.81080.0485 τ1 π Γ(3 + τ2 )3+τ2

4 σ1 [ Γ(1 + σ2 ) ] 1+σ2



ρ0
ρ



0.4

ρ0
ρ

0.4

(45)

,

≈ 17.1623



ρ0
ρ

0.4

(46)
.

9. Microphysics parameterizations scheme

199

The numerical values 4.8108 and 17.1623 are the same as the values given in Lopez (2002,
Appendix-D), providing that ν1 = 377.8 m1/3 s−1 , ν2 = 2/3, τ1 = 21 m1/2 s−1 , τ2 = 0.5,σ1 = 0.069 kg m−2 , σ2 = 2, ρw = 1000 kg m−3 and ρ0 = 1.2 kg m−3 .
In the subroutine ADVPRCS of the ARPEGE code, the coeﬃcients ZCOEFF1, ZCOEFF2
and ZCOEFF2b correspond to the products of the collection coeﬃcients by the collection
eﬃciencies for accretion, riming and aggregation resp.
The three collection tendencies are computed with an analytic scheme applied to the terms
ql and qi . As an exemple, let us derive the result for the accretion term. The equivalentof the auto-conversion results (8), (9), (20) and (21) write





∂ql
∂t acc ∂ql∂t acc∂ ln( ql )⇒∂tacc
ql (+)
ql (−)and so :≈ql (+) − ql (−)∆t≡− Kacc=− Kacc≈exp − Kacc

= −

(l/r)

Eacc ql qr ,(l/r)

Eacc qr ,h[ ∆qr ]acc
,
∆t

(47)

(l/r)

(48)

(l/r)

(49)

(l/r)

(l/r)

Eacc qr (−) ∆t

i

(50)

.

Since the collection processes correspond to conversion terms from the suspended cloud
condensate ql or qi into the precipitating species qr or qs , the following properties holds









qr (+) − qr (−) acc = − [ ∆ql ]acc = − ql (+) − ql (−) acc
(51)
,


 (+)
 (+)
(−)
(−)
− qs
− qi
(52)
,
[ ∆qs ]agg = qs
agg = − [ ∆qi ]agg = − qiagg[ ∆qr ]acc =[ ∆qs ]rim =

qs (+) − qs (−) rim = − [ ∆ql ]rim = −

ql (+) − ql (−) rim(53).

From (47) and (50), the ﬁnal formulae for the decrease in ql due to the accretion process,i.e. −[ ∆ql ]acc , is exacty equal to the corresponding opposite increase in qr , i.e. [ ∆qr ]acc .
The impact of the accretion process is ﬁnally computed as a change in qr , given by (54)[ ∆qr ]acc ≡[ ∆qs ]agg ≡[ ∆qs ]rim ≡







n

1 − exp − Kacc Eacc qr (−) ∆t (54),n.
1 − exp − Krim Erim qs (−) ∆t (56)

qs (+) − qs (−) agg = qi (−)
qs (+) − qs (−) rim = ql (−)

n

h

io



qr (+) − qr (−) acc = ql (−)

h

(l/r)

(l/r)

(i/s)

(i/s)

(l/s)

(l/s)

io

1 − exp − Kagg Eagg qs (−) ∆t (55),hio

The equivalent formulas for the aggregation and the riming processes are given in (55)and (56), with conversion of ql or qi into qs , generating positive values of [ ∆qs ].
From the general set of equations (2), and from (51) to (53), the accretion and riming
processes cannot transform into the time step ∆t more cloud liquid water than the available
existing amount at time t, i.e. ql . Similarly, the aggregation process cannot transform
more cloud ice water than qi .
It results the two following important limitations[ ∆ql ]acc + [ ∆ql ]rim

<

ql ,

(57)

[ ∆qi ]agg

<

qi .

(58)

200

9. Microphysics parameterizations scheme

2.8 The evaporation processes
The tendencies due to the evaporation of rain Er and the sublimation of snow Es are
given by Eq.(9) in Lopez (2002). The distributions (27) and (29) for the rain spectra and(31) and (33) for the snow spectra are used, together with some linearized versions of(C.7) and (C.8) made in Lopez (2002).
Let us deﬁne the four constant terms c3 to c6 by
c3

=

c4

=

c5

=

c6

=

2 ∗ 0.78 ∗2 ∗ (π)√π ≈ 2.765 ,(3−ν2 )/8

∗ 0.31 ∗√ν1 ∗ (ρref )0.2



5 + ν2
∗ Γ
2

(59)



≈ 30.1 ,4 ∗ 0.65≈ 9.736 ,(2 ∗ σ1 )2/3

√
5.784 ∗ 4 ∗ 0.44
5 + τ2≈ 478.1 .
∗ τ1 ∗ (ρref )0.2 ∗ Γ(5+τ2 )/62
(2 ∗ σ1 )

(60)
(61)
(62)

The coeﬃcients c3 to c6 are computed in the subroutine ADVPRCS in the local variables
ZCOEFF3 to ZCOEFF6.
Let us deﬁne the terms f3 (p) and f4 (T, p), two functions of the temperature and thepressure, by
f3 (p)=

f4 (T, p)=h√1.669 10−5

∗ 2 10−5∗ ( pref /p)

i1/3

Rv T
≈ 230.8 105 ∗ T ∗2 10−5 ∗ ( pref /p)≈ 4.34 10



pref
p

−1

−3

.

∗



1/3

pref
(63),
p
(64)

The coeﬃcients f3 and f4 are computed in the subroutine ADVPRCS in the local variables
ZFACT3 and ZFACT4.
Let us deﬁne the evaporation and sublimation terms cevap and csubl bycevap (T, p)=csubl (T, p)=[ 1 − ql /(qsat )l (T ) ] [ 1 − Ns ]∗ RNINTR ,(65)
ρ f5 (T, p)[ 1 − qi /(qsat )i (T ) ] [ 1 − Ns ]∗ RNINTS ∗ exp [ − 0.122 (T − Tf ) (66)],ρ f6 (T, p)

where RNINTR and RNINTS are both available in the NAMELIST, with f5 and f6 deﬁnedbyf5 (T, p)=

f6 (T, p)=12.31 10−2 ∗ Rv
1
2.31 10−2 ∗ Rv

∗
∗




Lv (T )T
Lf (T )
T

2

2

+

f4 (T, p),(esat )l (T )

(67)

+

f4 (T, p).
(esat )i (T )(68)

The coeﬃcient f5 (resp. f6 ) is computed in the subroutine ADVPRCS in the local variable
ZCONDT+ZDIFFV, just before the computations of ZCEV= cevap (resp. ZCSU= csubl ).
Let us deﬁne
cE1 (T, p)
cE2 (T, p)
cS1 (T, p)
cS2 (T, p)

=
=
=
=

cevap (T, p) ∗ c3 ,cevap (T, p) ∗ c4 / f3 (p) ,csubl (T, p) ∗ c5 ,csubl (T, p) ∗ c6 / f3 (p) .

(69)
(70)
(71)
(72)

9. Microphysics parameterizations scheme

201

The coeﬃcients cE1 , cE1 , cS1 and cS2 are computed in the subroutine ADVPRCS in thelocal variables ZCEV1, ZCEV2, ZCSU1 and ZCSU2, respectively.
For given amount of precipitations, either for rain qr or for snow qs , the parts of them tobe evaporated express in terms of the variables< qr >< qs >

qr / { RNINTR ∗ ρw } ,qs / { RNINTS ∗ exp [ − 0.122 (T − Tf ) ] } .

=
=

(73)
(74)

In the subroutine ADVPRCS, < qr > and < qs > are denoted by ZQRNR and ZQRNS,
respectively.
The ﬁnal formulas (75) and (76), analogous of Eqs. (9), (C.7) and (C.8) in Lopez (2002),
writes
EVA
SUB

=
=

cE1 ∗ < qr >1/2 + cE2 ∗ < qr >17/24 ,2/3

cS1 ∗ < qs >+ cS2 ∗ < qs > .

(75)
(76)

The ﬂux terms EVA and SUB are denoted in ADVPRCS by ZEVAPPL and ZEVAPPN,
respectively. The corresponding decrease in liquid or solid precipitations are denoted by∆ < qr >=ZTQEVAPPL and ∆ < qs >=ZTQEVAPPN, respectively. They are computedby choosing the minimum values among three terms, leading to∆ < qr > = Min∆ < qs > = Min







EVA
∆p
∆p
< qr > ; ∆t,EVA ;[ (qsat )l (T ) − qv ] (77)g
g EVA + SUB



SUB
∆p
∆p
< qs > ; ∆t.
SUB ;[ (qsat )i (T ) − qv ] (78)gg EVA + SUB

These limitations for the evaporations and sublimation processes are equivalent to thelimitations (57) and (58), valid for the collection processes.

2.9 The melting
The snow melting is computed in ADVPRCS (under the logical key LLMELTS = .TRUE.
at the beginning of the routine) in the term ZQMLTX,ZQMLTX=Max∆p cp (T − Tf )0 ;g
Lf − Lv



,

(79)

where Lf − Lv is the latent heat of fusion of snow into rain.

2.10 The sedimentation process
In the original Lopez scheme, a kind of semi-lagrangian method was applied for the sedimentation of precipitating species. It permits to use NWP and GCM longest time-step.
A new statistical scheme for the sedimentation of precipitation has been introduced later
by Bouteloup et al. (2010), (subroutine ADVPRCS) and enables, for example, to use thecomplete formulas for the fall speed of the liquid and solid particles instead of the "bulk"
constant values TFVR and TFVS or to describe sedimentation of cloudy condensates(TFVI/TFVL larger than zero). This scheme is described in part ?. Note that the fall
speed for liquid and solid precipitation can be either computed as a combined constantvalue for NSEDIM=0 (same value for liquid and solid), or as two diﬀerent constant fall
speeds, one for liquid precipitation, one for solid one (NSEDIM=1).

202

9. Microphysics parameterizations scheme

3 Algorithmics
As the same microphysics scheme is used for convective and large-scale condensates, theLopez scheme is called twice:• in aplpar.F90 subroutine, for the large-scale part, under the logical keys
(LPROCLD.AND.LCONDWT)=.TRUE.
• in acpcmt.F90 subroutine, for the convective part.
The following sequence of subroutines is used (the example is given here for large-scalecondensates; the sequence under ACPLUIZ is the same for the convective part). Only thesequence used in our options is described:
APLPAR : monitor of the ARPEGE physics> ACPLUIZ : general call to the Lopez subroutine> in ACPLUIZ : compute the condensation/evaporation of cloud water,
with the statistical cloud water coming as output fromthe prognostic TKE scheme (LNEBECT=TRUE) for
the stratiform part, and the convective cloud water coming as output of the convection
scheme for the convective part
> ACMICRO : compute the auto-conversion ﬂuxes
with PAUTOL and PAUTOI as output ;> ADVPRCS : compute the vertical sedimentation of the precipitation,plus the collection, the evaporation of precipitationand the melting (snow<->rain) processes.

4 Logical keys and main parameters in the current
versionLogical keys (in NAMPHY)• LCOLLEC=.TRUE. to switch-on the collection processes• LCONDWT=.TRUE. use of prog. condensed water in the Physics• LEVAPP=.TRUE. to switch-on the evaporation of precipitations• LNEBECT=.TRUE. use of Bougeault-Bechtold PDFs (<-> TKE-CBR)• LPROCLD=.TRUE. to switch-on the Lopez scheme
Main parameters used in the current climate version (NAMPHY0). During the development of the atmospheric model for CMIP6, tunings of some not-well constrained parameters have been done to balance TOA and surface budgets. Following Mauritsen et al.
(2012) and Hourdin et al. (2012), these tuned parameters are either microphysics ones (seebelow), or parameters describing clouds-radiation interaction (see chapter on radiation).
• NSEDIM= 1, sedimentation options in advprcs• RACCEF=1.0, collection eﬃciencies (accretion)• RAGGEF=0.3, collection eﬃciencies (aggregation)• RAUTEFR = 10 E-04 eﬃciency for auto-conversion water –> rain: value for 1/τl(unit in s−1 )• RAUTEFS = 52 E-04 auto-conversion: value for 1/τl (unit in s−1 )• RAUTSBET=0.025 ; auto-conversion: the eﬃciency coeﬃcient (β)ice for autoconversion ice –> snow
• RDTFAC=0.5, auto-conv.: tunes the width of δice in term of RDT= 11.82 K• REVASX=1 E-7., to limit (if > 0) the evaporation of precipitations (maximum
evaporation rate)9. Microphysics parameterizations scheme203• RNINTR=0.8 E+7, collection: a global tuning for cevap• RNINTS=0.2 E+7, collection: a global tuning for csubl• RQICRMAX=0.21 E-4, maxi auto-conv. threshold (solid) (kg/kg)• RQICRMIN=0.14 E-6 ; mini auto-conv. threshold (solid) (kg/kg)• RQICRSN=1., auto-conv.: tunes the impact of CNC concentration on (qi )crit• RQICRT1=-80., auto-conversion: ﬁrst temperature for (qi )crit (T1 , in K)• RQICRT2=0., auto-conversion: second temperature for (qi )crit (T2 , in K)• RQICVMAX=0.3 E-4, maximum critical ice content for autoconversion of convective cloud ice (kg/kg)• RQICVMIN=0.2 E-6, minimum critical ice content for autoconversion of convective
cloud ice (kg/kg)• RQLCR=200. E-6, auto-conversion: threshold of liquid water content (liquid) (kg/kg)• RQLCV=0.2 E-3, critical liquid water content for autoconversion of convective
cloud water (kg/kg)• RRAUTCS=1, ratio (convective/stratiform) of eﬃciency for autoconversion of water —> rain. In the convective case, the eﬃciency of autoconversion water —>rain is RAUTEFR*RRAUTCS, RAUTEFR in the stratiform case.
• RREVASXCS = 1, ratio of the maximum evaporation rate (convective/stratiform).
In the convective case, the evaporation rate is REVASX*RREVASXCS, REVASX
in the stratiform case,• RRIMEF=1.3, collection eﬃciencies (riming)• RSPCRR=1, supercooled rain switch (1= yes, 0= no)• TFVR=5.0, Fall speed for rain (m/s)• TFVI=0.08, Sedimentation speed of cloud ice water (m/s)• TFVL=0.02, Sedimentation speed for cloud liquid water (m/s)• TFVS=1., Fall speed for snow (m/s)

204

9. Microphysics parameterizations scheme

10
The CMIP6 dry and moist
convection scheme: Prognostic
Condensates Microphysics andTransport

1 Introduction: objectives of convective parametrization schemes
Following Arakawa (2004), the deep cumulus parameterization problem may be deﬁned
as the problem of formulating the statistical eﬀects of moist convection to obtain a closed
system for predicting weather and climate, trying to predict the grid-averaged eﬀect of an
ensemble of subgrid-scale drafts. Since cumulus parametrization is an attempt to formulate
the statistical eﬀects of cumulus convection without predicting individual clouds, it is a
closure problem in which a limited number of equations that govern the statistics of asystem with huge dimensions is looked for. The closure consists in a hypothesis that links
the occurrence and overall intensity of cumulus activity (e.g., cloud-base mass ﬂux) tolarge-scale processes.

1.1 Convective components and developments
Three main aspects for the causes of convection in the atmosphere can be highlighted: thebuoyancy force (responsible for free convection), convergence (due to orography or frontal
systems and that causes forced convection), and vertical wind shear (potential additionalrole in the formation of organized thunderstorms). In the convective development the ﬁrst
phase is the thermal, driven by the buoyancy, followed by the second phase with cumulus,
when the water vapour condensation starts (cumulus humilis in the sky), then towering
cumulus (cumulus congestus), and cumulonimbus at a late stage.

1.2 The apparent sources: the Q1, Q2, Q3 terms
The core of parameterization consists in evaluating the apparent source terms, as deﬁned
in Yanai (1973), looking for a closed expression for the apparent sources in terms of the10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
206
Microphysics and Transport
known resolved-scale variables (grid-averaged quantities).
Following Yanai et al (1973), the total eﬀect of convection on large-scale may be deduced
from the basic equations of thermodynamic and dynamics: averaging these equations
on a grid (.), using the Reynolds axioms and assuming that the horizontal transport ofsubgrid scale quantities is small relative to the vertical transport, the large-scale tendencies
due to convection for s = CpT + gz dry static energy (unit: m2 .s−2 ), q speciﬁc humidity(unit:kg.kg −1 ), u, v zonal and meridional component of wind (unit: m.s−1 ) can be written
as:











∂s
∂t
∂q
∂t
∂u
∂t
∂v
∂t


c
c
c

=
=
=
=

c

′ ′

Lv (c − e) − ∂ω∂ps = Q1 − Qr′ ′2−(c − e) − ∂ω∂pq = −Q
Lv

(1)

′ ′

− ∂ω∂pu = Q3 u′ ′− ∂ω∂pv = Q3 v

with ω (P a.s−1 ) vertical velocity in pressure coordinate, Qr (W.kg −1 ) radiative heating rate, Q1 (W.kg −1 ) apparent source of heat, Q2 (W.kg −1 ) apparent sink of moisture,Q3 u, Q3 v (m.s−2 ) apparent sources of momentum, Lv (J.kg −1 ) water vaporization latentheat, c and e (kg.kg −1 .s−1 ) condensation and evaporation rate.
Then, following Yanai et al (1973) :






















Q1

=

Qr

|{z}

radiative ef f ect

Q2



Q3 u












Q3 v






=

| {z }

condensation/evaporation+

Lv (c − e)

| {z }

condensation/evaporation=∂ω ′ s′∂p| {z }

vertical transport by unresolved eddies∂ω ′ q ′Lv∂p| {z }

vertical transport by unresolved eddies∂ω ′ u′−∂p=−

Lv (c − e)+

| {z }(2)vertical transport by unresolved eddies∂ω ′ v ′−∂p| {z }

vertical transport by unresolved eddies

These terms represent the mean eﬀect of convective processes on large-scale, as a functionof the grid-averaged variables. The apparent heat source, Q1 , depends on condensation/evaporation rate, radiative eﬀects and dry static energy vertical transport by unresolved (i.e subgrid) eddies. Q1 >0 for atmospheric heating, Q1<0 for atmospheric cooling. Condensation/(evaporation) is a source/(sink) of heat (Q1>0/(<0)) and of drying(Q2>0/(<0)).
Q2 depends on condensation/evaporation rate and of vertical transport of water vapor byunresolved eddies. Q3 depends only on the vertical transport of momentum by unresolved
eddies.
Some basic constraints for the apparent sources can be derived (Yanai et al 1973): ﬁrst,
the sum of the two ﬁrst equations of 1 leads to a budget equation for the moist static
energy h = s + Lv q :Q1 − Q2 − Qr = −1 ∂ρw′ h′ρ ∂z(3)Furthermore, a vertical integral of the source terms can be expressed to a good approximation by:
C p Q 1 = Q r + L v P + Fs

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
Microphysics and Transport
207
Cp
Q 1 = P − FL
Lv
where . designates the vertical integral from the surface to the top of the atmosphere, P
the precipitation rate, Fs the surface sensible heat ﬂux and FL is the surface evaporation
rate. Any given parameterization must be consistent with those constraints but there are
not strong enough to guide the construction of a sub-grid scale parameterization. Then,
determining the vertical distribution of the terms Q1 − Qr , Q2 and Q3 is one of the objectives of the parameterization schemes.
Generally speaking, a convection parameterization scheme follows three main steps:- deﬁnition of a triggering condition to describe when and where convection appears- determine the vertical distribution of Q1 , Q2 , Q3 u and Q3 v
- determine the grid-averaged convective intensity, through the closure of the scheme
As CMIP5 one, CMIP6 deep convection parameterization scheme uses a mass-ﬂux formulation, as the vast majority of NWP and Climate models. This formulation is quickly
examined below.

1.3 The mass-flux parametrization schemes
The hypothesis used in the mass-ﬂux formulation relies on a self-consistency argument forthe mean heat-budget in the tropical atmosphere: the tropical mean heat budget can not
be closed without assuming the existence of tall, isolated, convective vertical heat transportthat happens without substantial interaction with the environment (named "hot tower"by (Riehl and Marcue (1958)). To represent such an entity as a subgrid-scale process, an"isolated" vertical transport process is required, which may be described by introducing
a mass-ﬂux, a measure of vertical transport rate (Yano and Plant (2016)). Once a massﬂux is deﬁned, it is relatively straightforward to deﬁne the vertical transport of heatand moisture. Thus, the key issue of mass-ﬂux parameterization is to deﬁne the mass-ﬂux
associated with convection. The mass-ﬂux schemes allow to write the convective transportof the variables s, q, u and v from the mass ﬂux and the Cloud/Environment diﬀerencesof these variables. Each model grid is divided in two sub-domains: the convective one(cloudy) and the environmental one (clear sky). Then, for any variable χ:χ = σχc + (1 − σ)χe(4)

and for the vertical velocity ω:ω = σωc + (1 − σ)ωe(5)

where χ is the grid-averaged variable, ω the averaged vertical velocity, σ the convective
fraction in the grid, indices c and e stand for convective and environmental variables.
The subgrid vertical transport of χ is :
ω ′ χ′ = ωχ−ω χ = σωc χc +(1−σ)ωe χe −(σωc +(1−σ)ωe )χ = σ(ωc −ωe )(χc −χ)(6)Assuming ωc >> ωe , ω ∗ mass ﬂux of the updraughts relative to their environment in thegrid is :ω ∗ = σ(ωc − ωe ) = σωc, and the convective transport ﬂux is: ω ′ χ′ = ω ∗ (χc − χ).

(7)

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
208
Microphysics and Transport
Assuming that the area with updraught are small (σ << 1), and steadiness of the cloudbudget for mass, heat and water vapor (Yanai et al 1973), the apparent source terms canbe written (as in Yanai et al., 1973):






Q1
Q2
Q

 3u

Q3 v

=
=
=
=

∂s
Qr + ω ∗ ∂p
+ D(sc − s)∗ ∂q
−Lv ω ∂p − Lv D(qc − q)
+ D(uc − u)ω ∗ ∂u∂p+
D(vc − v)ω ∗ ∂v∂p(8)

with D (unit : s−1 ) the cloudy air detrainment rate.
The convective tendencies on the large-scale are given as functions of convective mass ﬂux
ω ∗ , cloudy detrainment rate D and diﬀerences between cloudy and environmental variables. Then the impact of convection on the environment in terms of heating (Q1 − Qr )is due to heating associated with compensating subsidence between cumulus clouds (term∂sω ∗ ∂p) and detrainment of cloudy air in the environment (term D(sc − s)). One of the
drawback of mass ﬂux formulation is the need of specifying not well-known parameters as
entrainment/detrainment rates.

2 From CMIP5 to CMIP6
2.1 Limits of the CMIP5 convection scheme (Bougeault, 1985)
The objective here is not to describe the CMIP5 scheme, rather to give some ideas about
the limits of the CMIP5 scheme and the way followed from CMIP5 to CMIP6 convection
schemes. Some limits of the Bougeault’s scheme are known: misrepresentation of diurnal
cycle of convection on continents (too early) linked with the triggering condition, too easy
triggering with large areas of simulated low precipitation. More precisely, during the EUROCS1 project (from 2000 to 2003), with the purpose to improve the clouds simulation in
RCM and GCM, it has been shown, in SCM conﬁgurations of deep convection triggeringin a oceanic stratocumulus case (Duynkerke et al 2004), that the scheme includes a moisture convergence closure not well adapted to extreme moisture conditions (Derbyshire etal 2004), a simulated diurnal cycle of clouds depth and precipitation too linked with the
maximum of solar heating and too small downdrafts (Guichard et al 2004).
In parallel of the development of the PCMT scheme, many tests have been done in a
climate conﬁguration with the prognostic physics used in NWP, i.e similar to that described in this documentation, except the deep and shallow convection schemes that arethat of Bougeault for the deep convection and Kain-Fritsh-Bechtold (Kain and Fritsch
1993, Bechtold et al 2001) for the shallow convection (see description in table 10.1, column "NWP"). Even if many sensitivity tests have been run, this physics has never beenchosen for CMIP exercise due to a very strong double ITCZ (even in forced simulations).
Moreover, in CMIP5 conﬁgurations (left column), no speciﬁc scheme exists for shallow
convection and thermals are not represented.

2.2 The CMIP6 convection scheme: an unified scheme, separating microphysic and transport in grid-scale equations
In a parametrization context, it is common to talk about "dry" convection as that produced by thermals which are not associated with cloud and which are conﬁned within theboundary layer; to talk about "shallow" convection as relatively weak, non-precipitating
cumuli; and to talk about "deep" convection as cumulonimbus clouds. These aspects are
1

EUROpean Cloud Systems

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
Microphysics and Transport
209

Table 10.1: Table with main characteristics of CMIP5, NWP (2015 version)and CMIP6 atmospheric physics.
Parametrization
CMIP5
NWP
Shallow convection
No specific scheme, Mass-flux scheme Bechpartly through moist told et al 2001
PDF
Deep convection
Bougeault 1985
Bougeault 1985 withmodifications
Turbulence
Diagnostic
TKE Prognostic TKE Cuxart
Ricard-Royer 1993
et al 2000
Mixing length
Quadratic
profile Mixing
lengthLenderink-Holtslag
Bougeault-LacarrÃ¨re
2004
1989
CloudsBougeault 1981 PDF
Bougeault 1981 PDF
Microphysics
Diagnostic Smith 1990
Pronostic scheme Lopez
2002
Radiation
ECMWF
scheme ECMWF
scheme(Fouquart-Bonnel 1980 (Fouquart-Bonnel 1980
and Mlawer et al 1997) and Mlawer et al 1997)

often treated separately in numerical models, within the boundary-layer parameterization, the shallow convection parameterization and the deep-convection parameterization
respectively. In the PCMT scheme an uniﬁed approach of the three types of convection
is proposed. For example, a continous formulation of turbulent entrainment (see eq. 24)
is used, allowing treatment of both shallow and deep convection. For the vertical proﬁle,both dry (i.e starting from the absolute temperature) and moist (starting from the wet
bulb temperature) adiabats are considered, the choice being done through the comparisonof the temperatures at the top level.
The idea behind separating microphysics and transport is given in Piriou et al. 2007.
The proposed separation consists in deﬁning the role of convective parameterizations as"estimating the subgrid-scale rate of convective transport (subgrid-scale motions) andmicrophysical processes (condensation, evaporation, downdrafts...) as a feedback to resolved forcing". In this paper the separation is used both as a way to introduce into theparametrization a more explicit causal link between all involved processes and as a vehicle
for an easier representation of convective cells. Doing that the equations of parameterization become closer to those of CRM and LES.
Actually, the "Microphysics-Transport" spirit consists in externalizing microphysical computations from the subgrid convection code, in order to share microphysical routines between subgrid and resolved convection, and use "up-to-date" microphysics (see chapter:"The CMIP6 microphysics scheme"). The grid-averaged convective tendencies at resolved
scale are expressed in terms of change of phase (Microphysics) and Transport, enabling amore direct comparison with CRM or LES data. The generic equation for q, speciﬁc huc −q))= ∂(M (q− C, separating transport and microphysics, instead
midity, is written as ∂q
∂t
∂p
∂q
∂q
of ∂t = −M ∂p + D(qc − q), with M the mass ﬂux, C the condensation rate and D thedetrainment rate.

CMIP6
PCMT (Gueremy 2011 ,Piriou et al 2007)
PCMT (Gueremy 2011,
Piriou et al 2007)
Prognostic
TKE(Cuxart et al 2000)
MixinglengthBougeault-LacarrÃ¨re1989
Bougeault 1981 PDF
Prognostic
schemeLopez 2002
ECMWF
scheme(Fouquart-Bonnel 1980
and Mlawer et al 1997)

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
210
Microphysics and Transport

3 The PCMT scheme
The PCMT scheme (Piriou 2007, Gueremy 2011) is a parametrization scheme in which
the grid-scale budget equations of parametrization use separate microphysics and transport terms. The role of convective parameterizations is then to estimate the subgrid-scalerate of convective transport (subgrid-scale motions) and microphysical processes (condensation, evaporation, downdrafts...) as a feedback to resolved forcing. The prognostic
microphysics scheme is that of LOpez (2002), largely modiﬁed by Bouteloup et al (2010).
Some adjustments of microphysics parameters have been done (see Part Microphysics) forthe convective part. In this documentation only the options for closure and entrainement
used in the Climate version are described for simplicity. The closure hypothesis is basedon CAPE relaxation. A prognostic equation is used for convective vertical velocity and thetriggering condition is based on upward vertical velocity. A continous treatment of convection, from thermals to deep precipitating convection, is obtained through a continous
formulation of entrainment/detrainment rates depending on convective vertical velocity.

3.1 Prognostic variables and geometry
Geometry
The variables ql , qi , qr , qs are prognostic, enabling to use the same microphysics forconvective and resolved part. Vertical velocity is also a prognostic variable.
The gridpoint is divided in three parts: updraft (of area αu ), downdraft (of area αd ),environment (of area 1 − αu − αd ).

Table 10.2: Geometry of the three subgrid components of PCMT.
Component
updraft downdraft environment
Area fractionαuαd
1 − αu − αd−αu wu −αd wd
Vertical velocitywuwd
1−αu −αdLiquid water
qlc
qlc
qlr
Precipitation
qrc
qrc
qrr
Be careful that vertical velocities deﬁned here are those for convection, then relative tothe large-scale vertical velocity: the absolute velocity in the updraft is w = w + wu . At
the moment the downdrafts are only diagnostic ones, not prognostic.

Tendency equations for qv and s
For qv , the tendency due to subgrid convection is (see table 10.3) :
(

1 ∂
1 ∂
∂
q v )c = −ραu wu (qvu −qv )−ραd wd (qvd −qv )−Condens+Evapqrc +Evapqsc (9)
∂t
ρ ∂z
ρ ∂z

The enthalpy (s = cp T + φ) tendency due to the convection is (see also Table 10.4):( ∂ s)c∂t=− ρ1 ∂ ραu wu (su − s)∂z
− ρ1 ∂ ραd wd (sd − s)∂z
+CondensEvapqlc + CondensEvapqic + Evapqrc + Evapqsc+MeltingIcingqlc(10)10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
Microphysics and Transport
211

Table 10.3: Name of fortran variables in the code∂ ρα w (q − q ) − 1 ∂ ρα w (q − q )Convective flux of specific humidity − ρ1 ∂z
u u vuveve
d d vdρ ∂z(not rain/snow)
Convective condensation flux
−Condens
for liquid/ice water
precipitation flux due to+Evapqrc
convective evaporation
precipitation flux due to+Evapqsc
convective sublimation

PDIFCQPFCCQL + PFCCQNPFPEVPCLPFPEVPCNPFPEVPCN

Table 10.4: Name of fortran variables in the code
Convective flux of enthalpy∂ ρα w (s − s)(not rain/snow)− ρ1 ∂zu u u
1 ∂− ρ ∂z ραd wd (sd − s)
PDIFCS
Convective condensation flux
−CondensEvapqlc + CondensEvapqic
PFCCQL/PFCCQN times L
for liquid water/ice
in CPTEND_NEW
Precipitation flux due to+Evapqrc + Evapqsc
PFPEVPCL/PFPEVPCN times L
convective evaporation/sublimationin CPTEND_NEW
q-flux due to icing-melting following
convective transport of convective liquid+MeltingIcingqlc
PFIMCC times Lf usion
and ice waterin CPTEND_NEW10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
212
Microphysics and Transport

Table 10.5: Name of fortran variables in the code
Convective flux
∂ ρ [−α w − α w ] q
of liquid water (not rain/snow)− ρ1 ∂zu u
d d lr
Entrainment/detrainmentqlr (liquid convective and resolved)−(Eu + Ed ) qlr + (Du + Dd ) qlc
Stratiform condensation flux for liquid water
CondensEvapqlr + MeltingIcingqlr
Flux of resolved precipitation: the generation term
−AutoconvCollqlr
Using p instead of z as vertical coordinate, equations include only a transport term
and a condensation one, (microphysics, downdraughts and other phase changes (melting/freezing) are taken into account as induced processes), and with M the mass-ﬂux
deﬁned as M = −ασωu :(M (qvu − qv )qvu∂q v )c =
−M
∂t
∂p
∂p

(11)

vu
The last rhs term corresponds to the condensation term C= M q∂pbeing positive. Andfor enthalpy the tendency equation in p-coordinate writes:(

M (su − s)
qvu
∂
s)c =
+ Lv M
∂t
∂p
∂p

(12)

Tendency equations for cloudy variables
The environment includes liquid water qlr , (the same for ice qir ), and precipitating condensates qrr for rain and qsr for snow, with grid-averaged value: qlr = (1 − αu − αd ) qlr .
These condensates correspond to the large-scale condensation tendency, and to a contribution of entrainment/detrainment of updraft/downdraft, as can be seen in equation 13,forcloudy liquid water from resolved condensation (see part 2.5 CBR documentation) :
∂q
∂t lr

=

Advec(qlr )− ρ1 ∂ ρ [−αu wu − αd wd ] qlr∂z−(Eu + Ed ) qlr + (Du + Dd ) qlc+CondensEvapqlr − AutoconvCollqlr + MeltingIcingqlr(13)

In this equation, the ﬁrst four rhs terms correspond to transport (the third and fourth tohorizontal one) and the last rhs terms to microphysics. Table 10.5 gives the name of thediﬀerent terms in the code.
The updraft and downdraft contributions appear in the budget of subgrid convective
liquid water qlc ( the same for ice qic , and precipitating condensates qrc and qsc ), withgrid-averaged value qlc = (αu + αd ) qlc . These condensates are due to condensation in thesubgrid convective updraft. Equation 14 gives the tendency of cloudy convective subgrid
liquid water:
∂q
∂t lc

=

Advec(qlc )− ρ1 ∂ ρ [αu wu + αd wd ] qlc∂z+(Eu + Ed ) qlr − (Du + Dd ) qlc+CondensEvapqlc − AutoconvCollqlc + MeltingIcingqlc

Table 3.1 gives the names of the diﬀerent terms in the code.

(14)

PDIFCQL-PFEDQLC-PFCSQLPFPFPSL10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
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213

Table 10.6: Name of fortran variables in the code∂ ρ [α w + α w ] q
Convective transport flux of qlc (liquid convective)− ρ1 ∂zu u
d d lcEntrainment/Detrainment of qlc and qlr+(Eu + Ed ) qlr − (Du + Dd ) qlc
Convective condensation flux for liquid water
CondensEvapqlc
Flux of convective precipitation: generation term
−AutoconvCollqlc
q flux due to icing-melting following
convective transport of convective liquid and ice water+MeltingIcingqlc

PDIFCQLCPFEDQLC-PFCCQLPFPFPCL

Table 10.7: Name of fortran variables in the code∂ ρ [α (w + w ) + α (w + w )] q
Convective precipitation as rain
− ρ1 ∂z
u
u
s
s
rc
d
d
Entrainment/detrainment of qrc and qrr(convective and resolved rain)+(Eu + Ed ) qrr − (Du + Dd ) qrc
Flux of convective precipitation: generation term
+AutoconvCollqlr
precipitation flux due to convective evaporation−Evapqrc + MeltingIcingqrc

PFIMCCPFPLCLPFEDQRC-PFPFPCLPFPEVPCL

Tendency equations for prognostic precipitation
The equation for convective precipitation is:
∂q
∂t rc

=

Advec(qrc )− ρ1 ∂ ρ [αu (wu + ws ) + αd (wd + ws )] qrc∂z+(Eu + Ed ) qrr − (Du + Dd ) qrc+AutoconvCollqlr − Evapqrc + MeltingIcingqrc(15)

and Table 3.1 gives the names of the diﬀerent variables. The equation for large-scaleprecipitation from resolved + subgrid condensation Smith type (table 3.1) is:
∂q
∂t rr

=

Advec(qrr )− ρ1 ∂ ρ [−αu wu − αd wd + (1 − αu − αd )ws ] qrr∂z−(Eu + Ed ) qrr + (Du + Dd ) qrc+AutoconvCollqlr − Evapqrr + MeltingIcingqrr(16)

Table 10.8: Name of fortran variables in the code∂ ρ [−α w − α w + (1 − α − α )w ] q
Stratiform precipitation as rain
− ρ1 ∂z
u u
u
s rr
d d
d
Entrainment/Detrainment of qrc and qrr(convective and resolved rain)−(Eu + Ed ) qrr + (Du + Dd ) qrc
Flux of resolved precipitation: generation term
+AutoconvCollqlr
precipitation flux due to resolved evaporation−Evapqrr + MeltingIcingqrr

PFPLSL-PFEDQRC-PFPFPSLPFPEVPSL

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
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Microphysics and Transport

3.2 Vertical velocity prognostic equation and triggering conditions
Vertical velocity prognostic equation
The cloud vertical velocity is that of Gueremy(2011 ), derived from Simpson and Wiggert(1969) and Chen and Bougeault (1992) (corresponding to the logical key LCVNHD=F,
see parameters below):

1 ∂ωu 2∂ωu
ρ g 2 (Tvu − Tv )=−−+ (ǫt + ǫ0 + Kd )ωu 2
∂t
2 ∂p
1+γ
|
{z
}
Tv

| {z }

advection|{z

buoyancy}(17)

entrainment

where:• Tvu = Tu (1 + 0.608qu − qlu ) (K) is the cloud virtual temperature, and Tv = T (1 +0.608q) (K) the grid averaged virtual temperature• the total friction coeﬃcient is the sum of turbulent (ǫt ), organized (ǫ0 ) entrainments and aerodynamic drag (Kd ), unit of all three being P a−1 , following SimpsonWiggert (1969), Simpson et al (1965) and Simpson (1971).
• γ describes the buoyancy reduction due to non-hydrostatic eﬀects (γ = 0.5 as inSimpson, 1971), analog to the impact of the motion of a solid in a ﬂuid, this"virtual mass coeﬃcient" has been introduced by Simpson and Wiggert (1969) tocompensate for a too strong buoyancy acceleration in their model. Essentially thisis a factor through which the eﬀective mass of a bubble (an air parcel) is enhanced.
The evolution of ωu depends on three terms: advection, buoyancy (source term) and sinkterm (entrainment).
An implicit time discretization is used following Chen and Bougeault (1992):2+
2+0.5(ω(j−1)+ +ω(j)+ )−0.5(ω(j−1)− +ω(j)− )= Buoyancy − 12 ω(j) −ω(j−1)
+
∆t
∆p
2
(ǫt + ǫ0 + Kd )( 41 (ω(j − 1)+ + ω(j)+ ) )(18)

where ω(j) is the vertical velocity at level j of the model (j decreasing upward) andthe superscript stands for the temporal discretization. Then the cloud vertical velocity
ω(j − 1)+ is obtained as the negative root of a second order polynomial of ω(j − 1)+ .
Buoyancy (second rhs term of equation 17) is a forcing term of the prognostic verticalvelocity equation. This equation is then used to deﬁne triggering conditions, mass ﬂux
and entrainment/detrainment rates. Buoyancy is also used for the closure assumption,
through its vertically integrated form: the CAPE. Unlike what has been done in part 1.3,
where ω ∗ and D are given from steady cloud budget equations, no more steady hypothesis
is assumed. The mass-ﬂux M is written as :M = −ασωu(19)with ωu (P a.s−1 ) the convective vertical velocity, σ the convective fraction and α theconvective fraction at the cloud bottom, both unitless.

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
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Entrainment/detrainment
Dilution of a cumulus cloud by entrainment of environmental air was described for the ﬁrst
time by Stommel (1947). Numerous observational studies of cumulus clouds using aircraft
followed (e.g Warner, 1955), then more precise quantitative descriptions of entrainment
originated from laboratory water tank experiments of thermal plumes (Morton et al. ,1956;
Turner, 1962) describing an increasing upward mass ﬂux M (with M = ρωu σu (unitkgm−2 s−1 ), ωu the plume updraught velocity and the associated plume fractional area
σc ), with height:
1 ∂M
=ǫ
M ∂z
, where ǫ denotes the fractional entrainment. An important reﬁnement of the entrainment formulation of this equation was ﬁrst pointed out by Houghton and Cramer (1951),
who made a distinction between dynamical -organized- entrainment due to larger scaleorganized inﬂow and turbulent entrainement caused by turbulent mixing at the cloud
edge:
1 ∂M
= ǫo + ǫt
M ∂z
The idea behind dynamical (or organized) entrainment is simple: there must be an inﬂow
to a convective plume (or tower) proportional to a large-scale convergence. The same idea
is termed by Tiedtke (1989) the "entrainment by organized ﬂow" and Nordeng (1994)
invoked an approximate relationship between the fractional entrainment and the fractionalvertical divergence:
ǫo =

1 ∂ρ
1 ∂wu
+
wu ∂z
ρ ∂z

Assuming that the density stratiﬁcation (second rhs term) is negligible, this relation reduces to:
ǫo =

1 ∂wu
wu ∂z

Whereas the organized entrainment has the characteristics of advective transport acrossthe interface, turbulent entrainment is of a diﬀusive character. Since both fractional
entrainment rates are by deﬁnition positive, they cause the mass ﬂux to increase withheight. For a cumulus cloud, due to the turbulent mixing at the cloud edge, a mixtureof in-cloud and environmental air is made; this mixture can become negatively buoyant
by evaporative cooling and will in this case detrain from the cloud, represented by aturbulent detrainment rate. The cloud or thermal itself can become negatively buoyant,stops rising and is usually dissolved into the environment. This process is called organizedor dynamical detrainment, and is represented by an organized detrainment rate.
Qualitatively the following behaviours have to be represented by entrainment/detrainment:• Two spatial scales of entrainment/detrainment: organized and turbulent. The organized entrainment is linked with the cloud-scale global velocity ﬁeld (bottom
entrainment, top detrainment, role of the vertical divergence of the mass ﬂux),
whereas the turbulent entrainment/detrainment is of smaller scale, located at thecloud edges and due to vertical shear of horizontal wind and horizontal shear ofvertical wind.
• When in-cloud vertical velocity is large, the eﬀect of turbulent entrainment/detrainmenton the updraft is smaller as the entrainment/detrainment processes have less time
to impact on a parcel.

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
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Microphysics and Transport• Wind shear is conducive to turbulent detrainment, as dry environmental air (saturation deﬁcit qsat − qv ), because liquid and ice water evaporation at the edge ofthe clouds is a source of cooling and then of turbulence.
Following Tiedtke (1989), the total entrainment/detrainment in PCMT is partitioned inthese two parts, corresponding to two diﬀerent scales at which the entrainment/detrainment
processes occur : ǫ = ǫ0 + ǫt where ǫ0 is the meso-scale entrainment, and ǫt the turbulent
one.
As the organized entrainment corresponds to the mixing due to the ﬂow associated withlarge-scale divergence/convergence, the organized inﬂow is assumed to occur in the lower
part of the cloud layer where there is large-scale moisture convergence, whereas the organized outﬂow is assumed to exist only in the top layer of the deepest clouds. The meso-scale
entrainment is derived from the mass conservation in the convective ascent, modulated
by a buoyancy-sorting process following Bretherton et al (2004). The buoyancy-sorting
approach assumes that the lateral mixing of the updraught and its environment generates
a spectrum of mixtures, and is used to determine which mixtures are incorporated into the
updraught and which are rejected (negatively buoyant mixtures are assumed to detrain,
whereas positively buoyant mixtures entrain). First, a maximum value of the fractional
entrainment rate is deﬁned as:ǫox = |

1 ∂ωu
|
ωu ∂p

(20)

, derived from the classical equation (from Yanai 1973) for a cumulus ensemble budget ofmass as
∂M
=D−E
∂p

(21)

,M being the mass-ﬂux, D and E the rates of mass detrainment/entrainment per unitpressure interval (unit: s−1 ). To obtain eq. 20, convective fractional area coverage isassumed to be constant and no detrainment is assumed. Then, following Bretherton(2004), if µ is the mass-mixing fraction of environmental air in the mixtures, ranging from
0 for undiluted updraught to 1 for pure environmental air, µ0 the critical mixing fraction
partioning positive versus negative buoyancy mixtures, the fractional entrainment anddetrainment rates are ﬁnally:ǫo = ǫox µ20(22)δo = ǫox (1 − µ0 )2(23)uThis buoyancy-sorting scheme is not applied as long as ω1u ∂ω> Kǫ ǫt , with Kǫ a constant∂P(parameter FENTRT in the code), to take into account the fact that some negatively
buoyant parcels are not immediately rejected from the updraught and that the buoyancysorting scheme is not applied as long as the maximum fractional organized entrainment
is signiﬁcantly larger than its turbulent counterpart. The organized entrainment is thena function of ωu but also of the thermodynamical state of the convective fraction and ofenvironment (parameter µ).

The turbulent entrainment is also a function of ωu (unit: P as−1 ) deﬁned by ǫt = ǫtn +(ǫtx − ǫtn ) × fǫ (ωu , ωu x, ωu n) with ǫtx the maximum value of ǫt and ǫtn the minimum one(unit: m−2 s2 ) or.

ǫt (ωu ) =

(

ǫtx
sin2
ǫtn

π ωx −ωu2 ωx −ωn

if |ωu | > |ωx |if |ωn | ≤ |ωu | ≤ |ωx |if |ωu | < |ωn |(24)

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
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with :
ǫtx = 0, 47.10−4 P a−1 the maximum value of entrainment for small ωu (TENTRX parameter in the code)
ǫtn = 0, 5.10−5 P a−1 the minimum value for large ωu (TENTR in the code),ωx = −27, 5 P a.s−1 the maximum threshold of ωu to get minimum entrainment (VVX in
the code).
ωn = −2 P a.s−1 the minimum threshold of ωu to get maximum entrainment (VVN in thecode).
This formulation is consistent with: more intense ascent, more isolated updraft fromthe environment and smaller interaction. This formulation allows the treatment of bothshallow and deep convection, giving a large value of entrainment for shallow clouds (small
absolute value of ω) and a small value of entrainment for deep clouds (large absolute valueof ωc ), using a continous transition.
The aerodynamic drag coeﬃcient Kd follows similar equations as ǫt :Kd (ωu ) =

(

Kdx
sin2
Kdn

π ωx −ωu2 ωx −ωn

if |ωu | > |ωx |if |ωn | ≤ |ωu | ≤ |ωx |if |ωu | < |ωn |(25)

with :
Kdx = 3, 57.10−4 P a−1
Kdn = 3, 810−5 P a−1
The lateral entrainment for a variable χ is given by:∂χu= ρ(ǫ0 + ǫt )(χ − χu )
∂φ

(26)

For a variable χ with an updraft value χu , a downdraft one χd , and an environmental
value χr , the tendency of χu in the updraft associated with entrainment/detrainment is:∂χu= (Eu + Ed )χr − (Du + Dd )χu
∂t

(27)

with, for a vertical ﬂow of perimeter L, of section S, uE the entrainment velocity and α
the surface fraction,


Eu




 Ed
Du




 Dd

=

u
α
uE u LSu u=d
αuE d L
Sd d

=

u
uDu L
α
Su u

=

d
α
uDd L
Sd d

(28)

with unit of Eu , Ed , Du , Dd : kg.m−3 s−1 .
The generic equation 27 is that used to give the entrainment/detrainment process in the
PCMT resolved prognostic equations (as eq. 14). A protection against large time-step
instability has been introduced: for any time-step the mixing process for the updraft
variable ψu can neither give negative value of ψu nor exceeding ψu + ψr . So we have:0 ≤ ψu +∂ψu∆t ≤ ψu + ψr∂t

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and the condition
−ψu ≤

∂ψu
∆t ≤ ψr∂t(29)

An equal rate for both turbulent entrainment and detrainment is assumed as in Tiedtke(1989). The ACMTENTR subroutine computes equation 27 and then limits the tendency
in the range given by 29. ψr is the variable PPSIR whereas ψu is the variable PPSIC.

σ computation
Using equation 21, we can write:
1 ∂σωu= ǫo − δoσωu ∂p(30)Note that the contribution of turbulent entrainment/detrainment is zero as the entrainment and detrainment rates are equal. The normalized convective fraction σ is computed
thanks to this mass-conservation equations and decreases from value 1 at the surface. The
computation is done in local variable ZSIGMA in acmtud.F90 subroutine. In the formula,for maximum organized entrainment rate, there is no variation of σ in the vertical.

3.3 Triggering condition
At this point of the algorithm, the triggering condition of the considered layer is determinedby the sign of the convective vertical velocity at both ﬂux levels surrounding the top
variable level. If the sign of ωu is negative, at least at one of the two ﬂux levels, thelayer is considered as convectively unstable. Then, in this scheme, it is possible to trigger
convection even if CIN (Convective INhibition, corresponding to the amount of energy
required to overcome the negatively buoyant energy the environment exerts on an air
parcel, from the ground to the level of free convection) is not equal to zero. It is possible,
using parameter GCVCINC (see below), to choose to add a threshold of CIN to trigger
convection. Then only α needs to be known to close the system.

3.4 Closure hypothesis
A CAPE relaxation closure is used for that (corresponding to the choice N CV CLOS = 1,
see below):

CAP E =Z PN ET
PLF C

Ra (Tvc − Tv )

dP
P

(31)

with PLF C the pressure at the pressure level of free convection, PN ET the level of thermal
equilibrium of the lifted parcel and Ra the gas constant of dry air. The closure hypothesis assumes an equilibrium between the CAP E production by large-scale and sink byconvection reached on a typical time-scale τ (see also part 3.5). We write:

∂CAP E
∂t



c

=−

CAP E
τ

(32)

The relaxation time τ can be related to the adjustment time deﬁned by Arakawa andSchubert (1974) as the "time required for convective processes to produce a neutral state,10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
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reducing the CAPE to zero, in the absence of large-scale forcing". Neglecting(following Nordeng 1994)

∂CAP E
∂t



c

= −Ra

Z PN ET (1 + 0.608q)(

PN CL

∂T
∂q
)c + 0.608T ( )c
∂t
∂t



∂Tvc
∂t

dP
P



c

(33)

Then the equation 32 is linear with α (see equations 11 and 12) and CAPE is consumed
by convection on a characteristic time proportional to the ratio of convective depth on themean convective vertical velocity. The proportionality is deﬁned with a linear functionof the inverse of the resolution (i.e. linear function of the grid size) in order to keep thesame magnitude between the mass ﬂux and the resolved vertical velocity, which is itself
proportional to the resolution (through the continuity equation).

R
PN ET
PN CL

τ = fτ (resolution) R P

N ET

PN CL

dp

2

(34)

|ωc |dp

The resolution dependency is described below in more details below.

3.5 Resolution dependency
At higher horizontal resolution, convection becomes more and more explicitely resolved,
and subgrid convection must decrease: mean subgrid mass ﬂux on a given domain decreases
to zero for inﬁnite higher resolution.
In this approach, the convective response time τ (ZTAU in acmtud.F90 subroutine) writes:
τ =

2
1 I12
fτ I11

where : f1τ (ZFDXTAUX variable in acmtud.F90 subroutine) is a function of T EQC,
1
, tuned for climate resolutions ( from T127 to T359):
P GM (scale factor) and ∆x
1
P GM 2P GM= T EQC.103 (2.23.10−4 () − 0.425+ 252.5)fτ∆x∆xwith :• T EQC (in m−1 ) is linked with REF LCAP E parameter through the mesh size(computed in sugem1b.F90 subroutine): TEQC = REFLCAPE
∆x
•

2
I12
I11

is a response time, (unit s), I12 is the convective depth (variable ZS12 in theacmtud.F90 subroutine, in P a) and I11 is the mean vertical velocity (in P as−1 ).
Then the CAPE relaxation time τ is proportional to the convective depth andinversely proportional to the vertical velocity.

τ is used to compute the active fraction at the convective cloud bottom:
α = min(ALF X,

I15)
I16 .τ

where I15 is the CAPE and I16 is such that the equation 32 linear versus α isαI16 .

1 ∂CAP E
τ
∂t



c

=

Then, for a constant horizontal resolution, for higher REFLCAPE, higher TEQC, smallerτ and higher α.

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3.6 Parametrization of cumulus-scale downdraughts
A downdraught is a commonly observed feature of the circulation pattern of many convective systems: from an individual cumulus which may be associated with an adjacent areaof rapidly descending air which may in turn be associated with strong surface winds andthe formation of a pool of cold, low-lying air , to organized convective systems associatedwith coherent mesoscale streams of descending air. Downdraughts can have contrasting effects on the life cycle and organization of convective systems: on the one hand, it modiﬁes
the structure of the underlying boundary layer, which may dampen or inhibit convection.
On the other, particularly when embedded within a sheared environment, the associatednear-surface outﬂow can produce near-surface convergence and uplift that induces newconvective elements. Convective cells generate precipitation that intensiﬁes downdraughts
by evaporation and by inertia. The downdraughts from each cell create a cold pool near
the Earth’s surface, which spreads out at the surface as a density current , triggering
new convective cells and limiting the lifetime of the parent cells. Mesoscale subsidence
is driven by evaporation of precipitation from the stratiform part (the anvil shield) andthis reinforces the density current, which in turn helps to initiate a rear-to-front mid-levelﬂow and the convective ascent with a jump updraught driven by a propagating positive
pressure jump near the edge of the cold pool. The jump updraught is sensitive to the
low-level vertical shear of the environment. A mass-ﬂux scheme with an explicit cold-pool
representation was created by Grandpeix and Lafore (2010). Circular cold pools (thewakes) are treated parametrically with cooling by the precipitating downdraughts while
the outside area is warmed by the subsidence induced by the saturated draughts.
The current version of PCMT scheme includes only a diagnostic downdrafts parameterization described in (Gueremy 2011 ). The deﬁnition of the downdraught proﬁle is very
similar to that of the updraught: the bulk proﬁle is deﬁned starting from the top of the
updraught with a saturated parcel having the thermodynamical properties of the environmental air. If this starting level is located below the level of minimum equivalent potential
temperature, no downdraught is considered. In most cases, the downdraught starting levelis actually the level of minimum equivalent potential temperature. Entrainment is taken
into account and uses a turbulent entrainment rate equal to ǫtx and an organized entrainment rate deﬁned with a value µ0 equal to 1 if ωdowndraf t is increasing with the pressure, 0
otherwise. Moist adiabatic process is considered until the updraught lifting condensationlevel where dry adiabat takes over. The vertical downdraught velocity is computed using
the steady version of eq. 17. The downdraught triggering condition is the same as thatof the updraught, with a change of sign. The same ασ value obtained for the updraught
is used to get the downdraught mass ﬂux. As the downdraught is taking its energy from
evaporation of convective rain or condensate over a fractional area ασ/4 (only one-fourthof the convective area is assumed to be aﬀected by the evaporation process, according
to SCM simulations, see parameter FEVAPC below), the downdraught mass ﬂux is multiplied again by this value, and the value of the downdraught mass ﬂux is equal to the
opposite of ωdowndraf t .(ασ)2 /4. This work is done in subroutine acmtddd.F90

3.7 Convective cloudinessConvective cloudiness( Gueremy 2011) nc is computed by nc = α σ m, where α is theconvective fraction at the cloud bottom (obtained from the scheme closure), σ is thevertical variation of this convective buoyancy (linked with the buoyancy-sorting). The m
constant describes the fact that, in convective clouds, cloudiness nc is larger than the area
occupied by the strongest ascents (α σ).
Algorithmics• In ACMTUD, convective cloudiness PNEBC is computed by PNEBC = α σ FNEBC(FNEBC = 10.). The cloudy condensate (liquid + ice) is PQLIC=ZLN * α * σ *FQLIC, where ZLN is the condensate issued from the diagnostic computation ofascent (FQLIC = 1),• In ACPCMT, PNEBC and PQLIC are multiplied by PAIPCMT, a binary indicator: 1 if PCMT computes the shallow convection, 0 if an other shallow-convection
scheme is used.

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221• In APLPAR: cloudiness named PNEBC in ACPCMT becomes ZNEBC0 in APLPAR.
ZNEBC0 is used by ACNEBN. The condensate PQLIC in ACPCMT becomes
ZQLIC_CVP in APLPAR.
• In ACNEBN: APLPAR/ZNEBC0 becomes ACNEBN/PNEBC. Two possible cases:
1. where PAIPCMT equals 1, convective cloudiness PNEBC issued from PCMTis combined with resolved cloudiness, (logical key LGPCMT) according to
max or random hypothesis (driven by logical key LRNUMX).
2. where PAIPCMT equals 0., PNEB is computed as large-scale cloudiness +that of shallow convection.

• In APLPAR, total cloudiness PNEB may be multiplied by a constant value (GAEPS).
• Then PNEB is send to the radiation scheme to compute radiative eﬀects.

3.8 Vertical transport
Vertical transport of prognostic subgrid and large-scale variables ql and qi by the subgrid
updrafts, downdrafts, and their overturn circulation uses the scheme, developed by Yves
Bouteloup et al. (2010) and coded in subroutine acadvec.F90. This semi-lagrangian
scheme of probabilistic type enables to transport subgrid variables with CFL conditions
greater than 1 in a stable and conservative manner, and constitutes a simpliﬁed approachof Geleyn (2008).

Description of the sedimentation scheme: equations of the scheme
As in Bouteloup et al. (2010), the reasoning is applied here to rain but could be extended
to any precipitating species. The scheme appears as a generalization of the approach of
Rotstayn (1997). The budget equation for rain water mass can be written as:
1 ∂
∂qr
−
→−
→
= − V . ∇qr +Fr + s0r − sir
∂t
ρ ∂z
−
→
with qr the mixing ratio of rain, Fr the precipitation ﬂux, ρ the density of air, V the wind0iand sr -sr the sources and sinks of rain, respectively. For rain, sources are auto-conversion,collection and melting of snow (see chapter microphysics); sinks are evaporation and freezing. The advection term is computed by the dynamical core of the model. The equation
to be solved by the microphysics scheme is:
∂qr
1 ∂
=
Fr + s0r − sir
∂t
ρ ∂z

(35)

If levels are supposed numbered from top to bottom (as in ARPEGE), the vertical andtemporal discretization of eq. 35 gives:
qr+ (j) = qr− (j) +∆t(Fr (j − 1) − Fr (j)) + Sr0 − Sriρ∆zwhere ∆z is the thickness of the current layer j, ∆t the time-step, Sr = ∆t.sr , Fr (j − 1)
the incoming precipitation ﬂux at the top of the layer, Fr (j) the output ﬂux, qr− the valueof the rain mixing ratio before microphysics (including sedimentation) and qr+ the value
after microphysics. There are two unknown quantities in eq. 35, qr and Fr ; to solve theproblem, Fr is generally related to qr by:
Fr = ρqr Vrt

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
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where Vrt is the mass-weighted bulk terminal fall velocity of rain. Then eq. 35 writes:
∂qr
1 ∂
=
(ρqr Vrt ) + s0r − sir∂tρ ∂zEquation 3.8 is not easy to solve because s0r , sir and Vrt are non-linear functions of qr .
Lopez (2002) gives a Lagragian solution. In many other schemes, eq. 3.8 is split into two
equations, sequentially solved. The ﬁrst one takes microphysical processes into account(without sedimentation):∂qr= s0r − sir∂t microwhile the second takes the sedimentation process alone into account:
∂qr
1 ∂Fr
=
∂t sedρ ∂z
All microphysical processes are computed before the sedimentation. This sequential approach requires small time-steps. Following Rotstayn (1997), it is possible to deﬁne avertical discretized form of eq. 3.8 where the output ﬂux is written as a function of qr andVrt :Fr (j − 1)
V t (j)∂qr(j) =− qr (j) r
∂t
ρ∆z
∆z
Assuming that the ﬂux coming from the upper-level Fr (j − 1) is constant during thetime-step, the integration of eq. 3.8 yields:
qr+ (j) = qr− (j)exp(−C) +Fr (j − 1)∆t
(1 − exp(−C))/Cρ∆z

where C = (Vrt ∆t/∆z) is the sedimentation Courant number. It can be written as:
qr+ (j) = (1 − P1 )qr− (j) + (1 − P2 )Fr (j − 1)∆tρ∆zwith P1 = 1 − exp(−C) and P2 = 1 − P1 /C. In the Bouteloup ’s formulation, the verticaland temporal discretization of eq. 3.8 is written:
qr+ (j) = qr− (j) +∆t(Fr (j − 1) − Fr (j))ρ∆zCombining eqs. 3.8 et 3.8, a diagnostic equation for Fr can be obtained:
Fr (j) = P1ρ∆z −qr (j) + P2 Fr (j − 1)∆tThen the solution of the microphysics parameterization (microphysics + sedimentation)
can be solved ﬁrst by diagnostically computing Fr from the top to the bottom of theatmosphere without taking microphysical processes into account, using eq. 3.8, then bycomputing the temporal evolution of qr through eq.3.8. So, for any precipitating species,
the vertical transport ﬂux is computed by vertical integration of:gψ(j)F (j − 1) − F (j)g= P1− (1 − P2 )
F (j)
∆p
∆t
∆p

(36)

where P1 is the probability for the top layer to receive, in a time-step, the total of thelocal quantity (ψl ∆p/g, local source), and P2 the probability for total transmission of thebottom ﬂux to the top (non-local source).

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
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Description of the sedimentation scheme: computation of the basic
proportions.(Routine acadvec.F90)
If a set of raindrops with a terminal velocity Vrt is considered, the proportion P0 (z, t) ofV tt

drops able to pass at least a vertical distance z during a time t is 0 if zr less than 1,
is 1 otherwise. For a layer of the model, containing rain and assuming an homogeneousvertical distribution,
R ∆z the proportion P1 of rain which leaves the layer during the time-step
1∆t is P1 = ∆z
P0 (z, ∆t)dz. Secondly, if we consider drops of rain which are not in0
the layer under consideration at the beginning of the time-step but which come from thelevel above, the proportion P2 of those which also leave the layer by the bottom during
the time-step (assuming a constant incoming ﬂux of rain mass from the level above) is:R ∆t1
P0 (∆z, t)dt. Using the expression for P0 , it is found that:
P2 = ∆t0
P1 = min(1, C)orP1 = min(1,Vrt ∆t)∆z

with C the sedimentation Courant number. For P2 ,P2 = max(0, 1 −

1
)
C

or
∆z log ǫt

P2 = max(0, min(e Vr ∆t , 1 −∆z))
Vrt ∆t∆z log ǫt

If .LADVLIM=.T., P2 is limited to e Vr ∆t (with ǫ = 0.01), to ensure that it is not possible
to transport during one time-step on a vertical distance greater than Vrt ∆t. Then it is
not possible to transport during one time-step on a vertical distance greater than Vrt ∆t(LADVLIM=.FALSE. in our code).

3.9 Conclusion
Compared with Bougeault’s scheme, the PCMT scheme provides an improvement of the
distribution of precipitation regime and of convection diurnal cycle (Couvreux et al 2015),
intensity and propagation of precipitation. Main defaults are a lack of triggering overcontinents with convective inhibition and an insuﬃcient propagation, maybe linked withtoo large cumulated precipitation for intense convective events.
Table 3.9 gives the main characteristics of the diﬀerent convection schemes used in MF
models.

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
224
Microphysics and Transport

Table 10.9: Main characteristics of the deep and shallow convective schemesof MF
Convection
Bougeault CMIP5 Bougeault NWP
PCMT
schemeClosure of deep Large-scale
As
Bougeault CAPE relaxation
convection
advection + mois- CMIP5
ture convergence=
convective
precipitation +detrainment
Entrainment and Exponential
As
Bougeault Organized
enDetrainment
decreasewith CMIP5
trainment+height of entrainturbulent entrainment;detrainment
ment
deducedfromconservationofcloud moist static
energyTriggering condi- Moisture conver- As
Bougeault Prognostic equations
gence + buoyancy with a minimum tion for conveccloud depth of 3 tive vertical velockm
ity. Triggering if
w>0
Downdraught
No
No
Yes-Diagnostic
Closure of shallow No
shallow CAPE relaxation CAPE relaxation
convection
scheme(Kain-FritshBetchold)10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
Microphysics and Transport
225

4 Algorithmic• Input of ACPCMT: prognostic convective variables of the preceeding time-step:wu (variable PUDOM), wd (variable PDDOM), αu (variable PUDAL), αd (variable PDDAL), qlc (variable PQLCONV) , qic (PQICONV), qrc (PQRCONV), qsc(PQSCONV).
• From wu , ACMTUD computes entrainment E and detrainment D, and their fractional analog ǫ and δ (P EN T RU , P DET RU , P EN T RD ,P DET RD , entrainment/detrainment
in updraft/downdraft in s−1 ).
• ACMTUD uses ǫ and δ to compute the proﬁle of a diagnostic ascent, the associated buoyancy, the new value of wu through a prognostic equation, αu through aprognostic equation, and gives the condensation rate of equation 14• ACMTENTR computes the tendency of prognostic variables with the entrainment/detrainment process.
• ACPCMT updates the intra time-step prognostic variables with condensation andentrainment/detrainment• ACPLUIZ computes microphysics (autoconversion, collection, precipitation evaporation).
• ACADVEC computes non-precipitating condensate transport: qlc and qic subgrid
and resolved.
• ACPCMT updates intra time-step prognostic variables with transport.
• ACPCMT computes freezing/melting of qlc , qic after transport.
• ACPCMT updates all the ﬂuxes of the prognostic equations for qlc , qic , qrc , qsc ,if a shallow convection scheme, diﬀerent from PCMT, is activated (like that ofcurrent NWP ARPEGE -Kain-Fritsch-Bechtold- or PMMC09, Arome’s one) and if
the convective depth of the PCMT updraft is smaller than a threshold (THPCMT).
• As for the other parametrization schemes, all the processes are described as ﬂuxes as
output of the subroutines. These ﬂuxes are read in the subroutin CPTEND_NEW
which computes tendencies of prognostic variables.

5 Logical keys and main parameters in the current
versionLogical keys
• NCVCLOS: type of convective closure (α computation: NCVCLOS=1, CAPE relaxation)• NCVENTR: type of entrainment (NCVENTR=1, GuÃ c rÃ c my, Tellus, 2011)• NCVSIG: type of σ computation (NCVSIG=1, equation 30)• LADVLIM: in subroutine acadvec.F90, minimises p2 to the value such as the transport scheme can not transport beyond a distance of w.∆t in a time-step (LADVLIM=.F.)• LGDDD: activates diagnostic downdrafts (LGDDD=.TRUE.)• LGMT: activates separate microphysics and transport resolved-scale tendencies(LGMT=.TRUE.)• LGPCMT: for using PCMT scheme (LGPCMT=.T.)• LGPSMI: activates Smith-Type convective precipitation (LGPSMI=.F.)• LCVFEVV: feedback of evaporation on vertical velocity (LCVFEVV=.F.)• LCVCONTAU: control response time of convection min/max values (LCVCONTAU=.F.)• LCVEOD: activates large entrainment/detrainment rates below or over the cloud(LCVEOD=.F.)10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
226
Microphysics and Transport• LCVIMPT: implicit transport (LCVIMPT=.F.)• LCVNAUV: new algorithmics to compute U and V convective eﬀects (LCVNAUV=.F.)• LCVNHD: activates non-hydrostatic drafts dynamics (LCVNHD=.F.)• LCVUVM: estimates the U and V updraft values as uniform, equal to the mean of
PU (LCVUVM=.F.)• LEDKFI: key for EDMF concept (LEDKFI=.TRUE.)• LRNUMX: key for maximum overlap of adjacent radiative clouds (LRNUMX=.T.)• LSBUO: smooth buoyancy (LSBUO=.F.)
Main parameters used in the current climate version• ALFX: maximum convective updraft fraction (ALFX= 0.04)• ECMNP: minimum critical thickness for precipitating clouds (ECMNP=8000 m)• FENTRT: factor of turbulent entrainment at detrainment level (FENTRT=1.75)• FEVAPC: factor for precipitation evaporation (FEVAPC=4)• FNEBC: factor for convective cloudiness (FNEBC=10)• FQLIC: factor for convective liquid water (FQLIC=1)• GAEPS: constant used in APLPAR to multiply total cloudiness issued from ACNEBN (GAEPS=1).
• GAMAP1: virtual mass parameter +1: parameter γ+1 of equation 17 (GAMAP1=
1.5)• GCLOEB: link between cloud depth and eﬀective transport (GCLOEB=0)• GCVCINC: CIN threshold for triggering condition (GCVINC=0 Jkg −1 , a thresholdof 5 Jkg −1 between LCL and LFC corresponds to 0.5 K of diﬀerence between cloudand environment)• GCVADET: fraction of zero buoyancy variable in the detrainment (GCVADET=1)• GCVADS: Coeﬃcient to switch from an equipressure adiabat computation to an
equipotential one (GCVADS=0)• GCVRE: Reduce convective entrainment (GCVRE=1)• GFRIC: aerodynamic of wu and wd (GFRIC=4.2 ∗ 10−3 , used only for diagnostic)• GFSURF: fraction of surface vs lowest model in the computation of diagnostic
updraft ascent initial state (GFSURF=0, meaning that the proﬁle starts at lowest
model level)• GREDDRS: reduction of detrainment for convective rain and snow into the environment (parameter without unit between 0 and 1, GREDDRS=1 for no reduction).
• GREMAX: maximum value of the "reducing entrainment factor" (GREMAX=1.0,
not used if GCLOEB=0)• GREMIN:minimum value of the "reducing entrainment factor" (GREMIN=0,01,
not used if GCLOEB =0)• GSDMAX: maximum value of saturation deﬁcit, to compute entrainment reduction(GSDMAX=0.13E-02 kg.m−2 ), not used if GCLOEB=0• GSDMIN: minimum value of saturation deﬁcit, to compute entrainment reduction(GSDMIN=0.1E-02 kg.m−2 ), not used if GCLOEB=0• PAIPCMT: Binary indicator for treatment of shallow-convection (1 if PCMT)• RDTFAC: Modulation factor of "RDT" in "FONICE" function (RDTFAC=0.5)• REFLCAPE: Parameter for convective response time (REFLCAPE =1)• RKDN: minimum aerodynamic friction (RKDN = 3.8 ∗ 10−5 P a−1 )• SCO: threshold under which convective precipitation are not taken into account(SCO=-1 kgm−2 )• TENTR: convective updraft entrainment rate (0.5 ∗ 10−5 P a−1 )10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
Microphysics and Transport
227• TENTRX: maximum convective updraft’s entrainment rate (0.47 ∗ 10−4 P a−1 )• THPCMT: threshold value, that deﬁnes the activation of PCMT versus shallow
convection schemes (THPCMT=-1 as PCMT is used for both convections in thisconﬁguration)• TUDGP: Updraught horizontal pressure gradient eﬀect coeﬃcient (Kershaw and
Gregory) (TUDGP= 0)• TVFC: parameter for convective precipitation (TVFC= 1, not used if LCVNAUV=.F.)• VVN: minimum vertical velocity for turbulent entrainment (VVN=-2 Pa/s)• VVX: maximum vertical velocity for turbulent entrainment (VVX= -27.5 Pa/s)

10. The CMIP6 dry and moist convection scheme: Prognostic Condensates
228
Microphysics and Transport

11
Gravity wave drag1 Parametrization of non-orographic gravity wave
dragThis parametrization of non-orographic gravity wave drag is mainly described in Lott etal. (2012) and is treated by subroutine ACNORGWD.

1.1 Formalism
A broadband spectrum of gravity waves (GWs) is represented via the superposition of alarge ensemble of statistically independent monochromatic ones. To produce this large
ensemble at a reasonable numerical cost, one launchs at each model time step δt a ﬁnite
number M (typically 8) of waves with characteristics chosen randomly and compute thetendencies due to them. As they are independent realizations the averaged tendency they
produce (the gravity wave drag (∂t u)gw ) is the average of these M tendencies. One takes
into account that the lifetime of these waves (typically ∆t ∼ 1 day) exceeds the modeltime step (typically δt < 1h).
We then redistribute this averaged tendency over a longer time scale ∆t by ﬁrst rescaling
it by δt/∆t and second by using a lag-one auto-regressive (AR-1) relation between the
GW tendencies at two successive time steps:
δt 1 X 1 ∂Fn∆t − δt(∂t u)tgw +
∆t
∆t M
ρ ∂z
M

(∂t u)t+δt
gw =

(1)

n=1

In other words, and at each time step, M new waves are emitted, and ones reduces theprobabilities of all the others by the multiplicative factor (∆t − δt)/∆t. A few hundred
waves are then active at each model step, which gives an excellent spectral resolution ata reasonable computational cost.
At each time step, the GWs horizontal wave number is choosed randomly within theinterval kmin ≤ k ≤ kmax , with kmax and kmin are related to the grid dimensions. The
intrinsic phase speed is also randomly chosen from a Gaussian distribution of mean 0 andstandard deviation cmax .

230

11. Gravity wave drag1.2 Launch spectrum
The Eliassen-Palm (EP) momentum ﬂux carried by each wave is speciﬁed at a given
launching altitude zl by

F(zl ) = ρr Gp

RLw
ρr Hcp22∆zf
k2 e−(m∆zp ) 2 k
P +Gf
N Ω3
k
4f

Z ztopρ0 (z)N (z)ζ 2 (z)e−π

√

J(z)

0

The ﬁrst term in the right-hand side of equation (2) represents the contribution fromconvective sources. The second term in the right-hand side represents the contribution
from the background ﬂux and accounts for GWs from nonconvective (and nonorographic)
sources.
To relate the gravity waves to the convective forcing, the surface precipitation P is used.
And to translate precipitation into diabatic heating, the latent heat ﬂux produced by
precipitation is distributed into the vertical over a Gaussian distribution with standard
deviation ∆zp , that characterizes the heating depth (see Lott et al. (2013) for mathematical derivation). In (2), ρr = 1 kg.m−3 is the reference density, Lw is the latent heat ofcondensation, cp is the heat capacity at constant pressure, H is the characteristic height of
the atmosphere, N is the Brunt-Vaisala frequency, k = ||k|| is the horizontal wavenumber,Ω is the intrinsic frequency Ω = ω − k.u and m is the vertical wavenumer m = N k/Ω. Gp
is a tuning parameter of order 1.
To relate the gravity waves to frontal sources, the potential vorticity ζ is used (see De la
Camara et al. (2015) for details). In (2), f is the Coriolis parameter, ztop is the model top
altitude, ρ0 = ρr e−z/H is the background density, J = N 2 /Λ2 is the Richardson number(Λ is the vertical shear of the horizontal wind), ∆zf is the tunable depth of the frontal
source, and Gf is also a tunable parameter of order 1 that controls the amplitude of theEP ﬂux.

1.3 Vertical propagation
First, the momentum ﬂux carried by gravity waves is set to zero where the waves encountera critical level. In the absence of critical levels, the momentum ﬂux remains nearly constant
except for a small dissipation µ/ρ0 in order to guarantee that the waves are ﬁnally erased
over the last few model layers. The GW momentum ﬂux is also limited by that produced
by a saturated monochromatic wave. The passage of EP ﬂux from one level z to the nextz + δz can be written as



2
µm3
|Ω|3 kmink−2δzF(z + δz) = Θ(Ω(z)Ω(z + δz)) min |F(z)|e ρ0 Ω , ρr Sc
k
N k4



(3)

where the Heaviside function Θ handles critical levels and Sc is a tunable parameter
controlling the saturated momentum ﬂux.

1.4 Computation aspects
Subroutine ACNORGWD is active under LNORGWD logical key and calculates the gravity wave drag (PTEND_U, PTEND_V) = ((∂t u)gw , (∂t v)gw ) from the vertical proﬁlesof temperature PTT, zonal wind PUU, meridional wind PVV and potential vorticity
PVOVO. The total (stratiform plus convective) precipitation PPREC is also used as a
input of the subroutine.
The parameters of the scheme are deﬁned in the module YOMNORGWD.
• NORGWD_RDISS is the dissipation coeﬃcient (µ).

kdz (2)k11. Gravity wave drag231

• NORGWD_SAT is the saturation parameter (Sc ).
• NORGWD_KMIN is the mininal horizontal wavenumber (kmin ).
• NORGWD_KMAX is the maximal horizontal wavenumber (kmax ).
• NORGWD_CMAX is the ’maximal’ intrinsic phase velocity (cmax ).
• NORGWD_DELTAT is the time scale (∆t) of the life cycle of the waves parameterized• NORGWD_DZ is the characteristic depth of the convective sources (∆zp ).
• NORGWD_RUWMAX is the tunable parameter that controls the amplitude of
the EP ﬂux from convective GWs (Gp ).
• NORGWD_DZFRON : is the characteristic depth of the frontal sources (∆zf ).
• NORGWD_GFRON is the tunable parameter that controls the amplitude of theEP ﬂux emitted by frontal GWs (Gf ).

232

11. Gravity wave drag

12
Ozone

1 Default ozone
When ozone is not a historical variable advected by dynamics and modiﬁed by photochemistry, it is simply speciﬁed by a climatological monthly ﬁle interpolated onto modellevels from a climatology coming from University of Reading calculated by Li and Shine(1995). See:
http://badc.nerc.ac.uk/data/ugamp-o3-climatology/ugamp_help.html

2 Parameterization of photochemical ozone sources
and sinks
The explicit representation of stratospheric photochemistry is too complex to be able to
be introduced into a general circulation model. This is why one uses a linearization of
the terms of ozone sources and sinks starting from a 2d latitude-pressure model (Cariolle
and Déqué, 1986). The 2d model MOBIDYC (64 latitudes and 40 levels pressure) utilized
a zonal circulation of stratosphere and 168 chemical reactions concerning 59 components(with in addition 51 reactions of photo-dissociation). The 2d model reaches an equilibrium
at the end of 30 years of integration (certain reactions have characteristic times of several
years). Stating at the equilibrium situation, one advances one time step ahead in eachgrid point of the photochemical model, after having disturbed by ±10% the ozone mixing
ratio rO3 . One thus obtains the derivative of the term of ozone photochemical production
or destruction P −L versus rO3 . This derivative can be regarded as a relaxation coeﬃcientof the ozone ﬁeld. Time characteristic of this relaxation (calculated for each latitude andeach level of the model 2d and each month of the year) varies from 0.1 day towards 1 hP a
to 1 year at the tropopause. One calculates in same manner the derivative of P − L versus
temperature T . Indeed the rates of the chemical reactions (Chapman cycle) depend on
temperature. One calculates ﬁnally the derivative with respect to ozone thickness above
the grid point, ΣO3 . When this quantity decreases, ultraviolet ﬂux reaching the point
increases, and the production of ozone due to photo-dissociation increases.
The following linear model is thus considered:
∂rO3
∂t

=

∂P − L[rO3 − rO3 ]∂rO3
 ∂P − L ∂P − L +
T −T +
ΣO3 − ΣO3
∂T
∂ΣO3

P −L+

(1)

234

12. Ozone

where the over-lined quantities were obtained using the 2d model MOBIDYC. This linear
model was substituted for the photochemical model in the 2d model, and the results of thetwo versions were compared. The relative error remains lower than 10%; the maximum
error is in low stratosphere where chemistry is strongly non-linear (just as in troposphere,
but the concentrations are weak). The linearized model is thus validated compared to thefull model from which it results.
Linear parameterization thus utilizes the 7 coeﬃcients of Equation (1) which are regarded
as two-dimensional ﬁelds as long as the poles are not tilted, three-dimensional otherwise.
Subroutine ACOZONE calculates initially the ozone quantity above each point:ΣO3 (l) =l−1
Xp(ei) − p(ei − 1)
i=0

grO3 (i) +p(l) − p(˜l − 1)
rO3 (l)g

The same algorithm is used for term ΣO3 starting from rO3 to ensure the coherenceof the vertical discretization of the model. Then, one calculates the ozone tendency of
Equation (1) which is written as:∂rO3= P K2 + P K3 (rO3 − P K1 ) + P K5 (T − P K4 ) + P K7 (ΣO3 − ΣO3 )
∂t
with:

ΣO3 (l) =l−1
Xp(ei) − p(ei − 1)
i=0g

P K1 (i) +p(l) − p(˜l − 1)P K1 (l)g

In fact, the way it is written is less simple as one uses an implicit temporal discretization
for the term in rO3 (but not for ΣO3 because it would be too complicated) to ensure
numerical stability with large time steps.
Finally one calculates the ozone ﬂow per vertical integration of the temporal tendencies
which one has just calculated and with the assumption of null ﬂow at the top.

3 The effect of chlorine on ozone
Parameterization above does not make it possible to calculate the eﬀect of the application of the Montreal protocol on the reduction of Clx in stratosphere. Heterogeneous
chemistry responsible for “ozone hole” is very complex and here a particularly simpliﬁed
formulation is used. When the solar zenith angle is positive (i.e. during daylight) andwhen temperature is lower than TPSCLIM, namely 195 K, one adds to the P K3 term
the quantity P K6 RCLX2 , where P K6 is a new 2d ﬁeld and RCLX is a parameter whose
temporal evolution is controlled by namelist. There is no compatibility with the former
versions of the scheme for which ﬁelds P K1 to P K7 corresponded to other coeﬃcients ofEquation (1) (caution: do not mix boundary conditions ﬁles!).

4 Parameterization of “mesospheric drag”
The goal of this parameterization is to mitigate the absence of physics in the highest
levels of the model (i.e. in the mesosphere) when the stratospheric vertical discretization
is high. This parameterization consists simply of a linear relaxation of the wind towards
0, of speciﬁc moisture towards qmin = 3.725 10−6 (RFMESOQ) to avoid excessive drying

12. Ozone

235

and represent chemical sources of H2 O and of the temperature towards the temperatureof standard atmosphere Tsta (Fels, 1986) deﬁned in subroutine SUSTA by:

 z=0

 0 < z < 11.0 km
 11.0 km < z < 20.0 km

 20.0 km < z < 32.0 km
 32.0 km < z < 47.4 km

 47.4 km < z < 51.4 km
 51.4 km < z < 71.7 km

 71.7 km < z < 85.7 km

85.7 km < z

Tsta = 288.15 K∂Tsta= −6.5 K km−1
∂z
∂Tsta
=0
∂z
∂Tsta
= 1.00 K km−1∂z
∂Tsta
= 2.75 K km−1
∂z
∂Tsta
=0
∂z
∂Tsta
= −2.75 K km−1∂z
∂Tsta
= −1.97 K km−1
∂z
∂Tsta
=0
∂z

The proﬁle of the relaxation coeﬃcients K(l) is deﬁned in subroutine SUTOPH. One
chooses for the wind a reference level pressure pref and a coeﬃcient α. In the same way,for moisture and temperature, one takes another set of coeﬃcients. Proﬁle K(l) is then:K(l) = α maxpref − psta (l),0psta (l)

where psta (l) is the standard air pressure at level l. In practice, these two levels are
selected above 1 hP a and the constants are adjusted in order to obtain time-constants ofa few hours in the highest level.
The writing of linear parameterization is a little complicated by the fact that on the one
hand an implicit discretization is used, and that on the other hand one must calculate a
ﬂux instead of a tendency. For example, the enthalpy ﬂux at inter-level l is given by:

PFRMH = −

KLEV
K(i)
1 X
δp(i)
cp (i)(T (i) − Tsta (i))
g
1 + ∆tK(i)i=l+1

where cp is the speciﬁc heat of the air, g gravity, KLEV the number of levels, δp(i)the thickness of the layer, ∆t the time step of physics and T (i) the temperature. Contrary to ﬂuxes produced by the other parameterizations, this ﬂux is zero at surface. The
momentum and energy exchange is done with space and not with earth.

236

12. OzoneReferences

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