Météo-FranceCentre National de Recherches Météorologiques

ARPEGE-Climate Version 5.2
Algorithmic Documentation

September 2011

Contents
1 Basic hypotheses and related constants
1

3

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

4

1.1

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

4

1.2Time . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

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

5

1.4

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

5

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

6

2.1

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

6

2.2

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

7

2.3

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

7

3

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

7

4

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

85Thermodynamics, gas phase . . . . . . . . . . . . . . . . . . .

86Thermodynamics, liquid phase . . . . . . . . . . . . . . . . .

97Thermodynamics, solid phase . . . . . . . . . . . . . . . . . .

98Thermodynamics, phase transition . . . . . . . . . . . . . . .

9

8.1

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

9

8.2

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

10

8.3

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

109Consequences on saturation . . . . . . . . . . . . . . . . . . .

10

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

11

10.1

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

11

10.2

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

12

2

2

CONTENTS

2 Dynamics equations

15

1

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

15

2

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

16

3

Variable mesh19

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

3.1

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

19

3.2

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

20

3.3

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

21

4

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

21

5

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

22

5.1

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

22

5.2

Vertical discretization of the equations

22. . . . . . . .

3 Spectral transforms

27

1

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

27

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

27

2.1

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

27

2.2

Collocation grid

28

2.3

Spectral transforms

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

29

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

30

3.1

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

30

3.2

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

30

3.3

Spectral relationships for wind representation . . . . .

31

3.4

Relationship between dimension in spectral space andin grid point space . . . . . . . . . . . . . . . . . . . .

32

4 Semi-lagrangian discretization
1

2

35

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

35

1.1

Purpose . . . . . . . . . . . . . . . . . . . . . . . . . .

35

1.2

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

35

1.3

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

36Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.1

36

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

CONTENTS
2.2
3

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

38

Generic discretization of the equations . . . . . . . . . . . . .

39

3.1

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

39

3.2

Discretization for a 3D variable: general case where the
RHS has non-zero linear and non-linear terms (GMV
variables). . . . . . . . . . . . . . . . . . . . . . . . . .

40

Discretization for a 3D variable: particular case where
the RHS has zero linear and non-linear terms (advectable GFL variables) . . . . . . . . . . . . . . . . .

42

3.4

Non advectable pseudo-historic GFL variables . . . . .

44

3.5

Discretization for a 2D variable in a 3D model (GMVS
variables, for example continuity equation) . . . . . . .

44

3.6

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

46

3.7

Remarks for spline cubic vertical interpolations . . . .

46

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

47

3.3

4

3

4.1

Medium point M. . . . . . . . . . . . . . . . . . . .

47

4.2

Origin point O . . . . . . . . . . . . . . . . . . . . . .

50

4.3

Refined re-computation of point O . . . . . . . . . . .

51

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

51

5.1

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

51

5.2

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

52

5.3

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

52

5.4

GFL variables . . . . . . . . . . . . . . . . . . . . . . .

53

5.5

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

53

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

54

6.1

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

54

6.2

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

54

7

Longitudes and latitudes on the computational sphere . . . .

55

8Interpolations and weights computations . . . . . . . . . . . .

56

8.1

Interpolation grid and weights. . . . . . . . . . . . .

56

8.2

Interpolations . . . . . . . . . . . . . . . . . . . . . . .

61

Computation of η̇ on layers . . . . . . . . . . . . . . . . . . .

65

5

6

9

4

CONTENTS
10

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

65

10.1

Extra longitudes . . . . . . . . . . . . . . . . . . . . .

65

10.2

Extra latitudes . . . . . . . . . . . . . . . . . . . . . .

65

10.3

Vertical boundary conditions . . . . . . . . . . . . . .

66

5 Semi-Implicit spectral computations
1

2

3
4

General considerations . . . . . . . . . . . . . . . . . . . . . .

67

1.1

Purpose . . . . . . . . . . . . . . . . . . . . . . . . . .

67

1.2

Advection scheme . . . . . . . . . . . . . . . . . . . . .

67

1.3

Semi-implicit treatment of linear terms . . . . . . . . .

68

Prognostic variables and quantities involved in the semi-implicit
scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.1

Prognostic variables . . . . . . . . . . . . . . . . . . .

69

2.2

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

69

2.3

Quantities used for vertical discretizations and linear
operators . . . . . . . . . . . . . . . . . . . . . . . . .

70

Semi-implicit scheme with reduced divergence . . . . . . . . .

71

3.1

71

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

Semi-implicit scheme with unreduced divergence. . . . . . .

73

Shortcomings of this formulation of the semi-implicit
scheme in case of stretching . . . . . . . . . . . . . . .

73

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

74

Spectral multiplications by polynomial expressions of the mapping factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4.1
4.2
5

67

6 Horizontal diffusion and nudging

77

1

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

77

2Formulation of the horizontal diffusion . . . . . . . . . . . . .

77

2.1

General considerations . . . . . . . . . . . . . . . . . .

77

Discretization of the horizontal diffusion . . . . . . . . . . . .

78

3.1

Unstretched model . . . . . . . . . . . . . . . . . . . .

78

3.2

Stretched model . . . . . . . . . . . . . . . . . . . . .

79

3.3

Second reference truncation N2 in the stratosphere . .

80

3.4

Discretized equations . . . . . . . . . . . . . . . . . . .

81

Nudging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

3

4

CONTENTS

5

7 Boundary conditions

85

1

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

85

2

Surface boundary conditions . . . . . . . . . . . . . . . . . . .

86

3Additional boundary forcings . . . . . . . . . . . . . . . . . .

88

3.1

Ozone . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

3.2

Aerosols

88

4. . . . . . . . . . . . . . . . . . . . . . . . .

Coupling with an ocean model. . . . . . . . . . . . . . . . .

89

4.1

Ocean model . . . . . . . . . . . . . . . . . . . . . . .

90

4.2

Coupler . . . . . . . . . . . . . . . . . . . . . . . . . .

91

8 Radiation

93

1

Radiative fluxes. . . . . . . . . . . . . . . . . . . . . . . . .

93

2

Shortwave radiation . . . . . . . . . . . . . . . . . . . . . . .

94

2.1

First glance . . . . . . . . . . . . . . . . . . . . . . . .

94

2.2

Spectral integration . . . . . . . . . . . . . . . . . . .

95

2.3

Vertical integration . . . . . . . . . . . . . . . . . . . .

98

2.4

Multiple reflections between layers . . . . . . . . . . . 105

3

Longwave radiation: the RRTM scheme . . . . . . . . . . . . 106

4

Input to the radiation scheme . . . . . . . . . . . . . . . . . . 109

5

4.1

Solar irradiance data . . . . . . . . . . . . . . . . . . . 109

4.2

Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3

Ground albedo and emissivity . . . . . . . . . . . . . . 109

4.4

Aerosols . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.5

Radiatively active compounds . . . . . . . . . . . . . . 113

4.6

Cloud optical properties . . . . . . . . . . . . . . . . . 113

4.7

Cloud overlap assumption . . . . . . . . . . . . . . . . 115

4.8Interactions with the SURFEX module . . . . . . . . . 115

Simplified radiation scheme . . . . . . . . . . . . . . . . . . . 115

6

CONTENTS

9 Clouds and turbulence
1

2

117

General description of the scheme. . . . . . . . . . . . . . . 117

1.1

Condensate assumptions . . . . . . . . . . . . . . . . . 117

1.2

Turbulent kinetic energy assumptions . . . . . . . . . 117

1.3

Sub-grid condensation scheme . . . . . . . . . . . . . . 118

1.4

Sub-grid turbulence scheme . . . . . . . . . . . . . . . 122

1.5

Solving an implicit equation

1.6

Summing up . . . . . . . . . . . . . . . . . . . . . . . 135

1.7

Tuning the scheme . . . . . . . . . . . . . . . . . . . . 139. . . . . . . . . . . . . . 129

Turbulent coefficients in stratosphere . . . . . . . . . . . . . . 141
2.1

Stability. . . . . . . . . . . . . . . . . . . . . . . . . 142

2.2

Calculations for the wind components . . . . . . . . . 143

2.3

Calculations fore temperature and moisture . . . . . . 143

2.4

Brunt-Vaïsala frequency . . . . . . . . . . . . . . . . . 143

3

Convective cloudiness. . . . . . . . . . . . . . . . . . . . . . 144

4

Total cloudiness . . . . . . . . . . . . . . . . . . . . . . . . . . 144

10 Large-scale precipitation

147

1

Description of the scheme . . . . . . . . . . . . . . . . . . . . 147

2

Precipitation in a cloud layer (Smith) . . . . . . . . . . . . . . 148
2.1

Liquid phase . . . . . . . . . . . . . . . . . . . . . . . 149

2.2

Solid phase . . . . . . . . . . . . . . . . . . . . . . . . 149

2.3

Precipitation flux at the base of the saturated part ofthe mesh . . . . . . . . . . . . . . . . . . . . . . . . . 149

2.4

Recombination of the total condensation flux . . . . . 151

3

Evaporation in the unsaturated parts (Kessler) . . . . . . . . 151

4

Melting or freezing (Kessler). . . . . . . . . . . . . . . . . . 152

4.1

Saturated part of the mesh . . . . . . . . . . . . . . . 153

4.2

Unsaturated part of the mesh. . . . . . . . . . . . . 153

CONTENTS

7

11 Vertical diffusion

155

1

General comment the scheme . . . . . . . . . . . . . . . . . . 155

2Equation of the vertical diffusion for a conservative quantity ψ 155

3

Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . 156

4

3.1

Parametrization in the free atmosphere . . . . . . . . . 156

3.2

Parametrization at ground level . . . . . . . . . . . . . 156

Algorithm of flux calculation . . . . . . . . . . . . . . . . . . 156
4.1

Flux of momentum:Fu and Fv

4.2

Flux of heat Fqv and Fs . . . . . . . . . . . . . . . . . 158. . . . . . . . . . . . . 157

5

Parametrization of planetary boundary layer depth . . . . . . 161

6Additional surface parameters. . . . . . . . . . . . . . . . . 163

12 Convection165

1

Principle of the convection scheme . . . . . . . . . . . . . . . 165

2Preliminary calculations . . . . . . . . . . . . . . . . . . . . . 166

3

2.1

Initialization of the convergence of moisture

2.2

Wet-bulb thermometer calculation . . . . . . . . . . . 167

Elements of the convection scheme . . . . . . . . . . . . . . . 169
3.1

Determination of the profile of (sn , qn )3.2

Algorithm of the reversible saturated adiabatic . . . . 170

3.3

Advection of compensatory subsidence

3.4

Determination of coefficient K

3.5

Effect on horizontal wind . . . . . . . . . . . . . . . . 176

3.6

Energetics in the scheme . . . . . . . . . . . . . . . . . 179

13 Gravity wave drag1

2

. . . . . 166. . . . . . . . 169. . . . . . . . 174. . . . . . . . . . . . . 175

183

Parametrization of orographic gravity wave drag . . . . . . . 183
1.1

Calculation at the surface . . . . . . . . . . . . . . . . 183

1.2

Calculation at a given level . . . . . . . . . . . . . . . 183

Refinements for orographic gravity wave drag . . . . . . . . . 184
2.1

Effective surface wind . . . . . . . . . . . . . . . . . . 184

2.2

Mountain anisotropy. . . . . . . . . . . . . . . . . . 185

8

CONTENTS

3

4

5

2.3

Resonance. . . . . . . . . . . . . . . . . . . . . . . . 185

2.4Trapping . . . . . . . . . . . . . . . . . . . . . . . . . 186

2.5

Sub-grid orography . . . . . . . . . . . . . . . . . . . . 187

2.6

The lift effect. . . . . . . . . . . . . . . . . . . . . . 187

Numerical securities in orography waves . . . . . . . . . . . . 189
3.1

Implicit formulation versus surface wind . . . . . . . . 189

3.2

Implicit advective formulation. . . . . . . . . . . . . 189

Parametrization of convective gravity wave drag. . . . . . . 191

4.1

Calculations at the top of the cloud. . . . . . . . . . 191

4.2

Calculations above the cloud . . . . . . . . . . . . . . 191

4.3

Calculations inside the cloud . . . . . . . . . . . . . . 192

Numerical securities for convective waves . . . . . . . . . . . . 192

14 Surface processes scheme

195

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

2

Soil, snow and vegetation : ISBA surface scheme . . . . . . . 196

3

2.1

Force restore approach . . . . . . . . . . . . . . . . . . 197

2.2

Treatment of the canopy water . . . . . . . . . . . . . 204

2.3

Spatial variability of precipitation intensities . . . . . . 205

2.4

Treatment of the snow . . . . . . . . . . . . . . . . . . 207

2.5

The surface fluxes . . . . . . . . . . . . . . . . . . . . 208

2.6

Summary of Useful Parameters . . . . . . . . . . . . . 212

Water surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 213
3.1

4

Simple parameterization . . . . . . . . . . . . . . . . . 213

Parametrization of the surface boundary layer . . . . . . . . . 220

15 Ozone

225

1

Default ozone . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

2

Parameterization of photochemical ozone sources and sinks . 225

3

The effect of chlorine on ozone . . . . . . . . . . . . . . . . . 227

4

Parameterization of “mesospheric drag” . . . . . . . . . . . . . 227

CONTENTS

916 Diabatic terms

229

1

Physical parametrizations . . . . . . . . . . . . . . . . . . . . 229

2Calculation of altitude tendencies . . . . . . . . . . . . . . . . 229

3

4

2.1

Equation of wind

. . . . . . . . . . . . . . . . . . . . 229

2.2

Equation of water vapor

2.3

Equation of temperature and enthalpy . . . . . . . . . 230. . . . . . . . . . . . . . . . 230

Flux of enthalpy due to wet processes . . . . . . . . . . . . . 231
3.1

Sensible heat flux . . . . . . . . . . . . . . . . . . . . . 231

3.2

Enthalpy flux related to condensation . . . . . . . . . 232

3.3

Total precipitation flux

3.4

Total enthalpy flux related to precipitation and condensation . . . . . . . . . . . . . . . . . . . . . . . . . 232

Time advance in altitude. . . . . . . . . . . . . . . . . 232. . . . . . . . . . . . . . . . . . . . 232

4.1

Evolution of wind and of vapor, liquid and solid water 232

4.2

Evolution of temperature . . . . . . . . . . . . . . . . 233

4.3

Dissipation of the kinetic energy . . . . . . . . . . . . 233

17 Diagnostics

235

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

2

Post-processable fields . . . . . . . . . . . . . . . . . . . . . . 235

3

4

2.1

3D dynamical fields (DYN3D) . . . . . . . . . . . . . . 236

2.2

2D dynamical fields (DYN2D)

2.3

Physical fields (PHYSOL)

2.4

Physical accumulated fluxes . . . . . . . . . . . . . . . 241

2.5

Physical instantaneous fluxes . . . . . . . . . . . . . . 243. . . . . . . . . . . . . 237. . . . . . . . . . . . . . . 239

Horizontal interpolations . . . . . . . . . . . . . . . . . . . . . 245
3.1

Bilinear horizontal interpolations . . . . . . . . . . . . 245

3.2

12-point horizontal interpolations . . . . . . . . . . . . 246

Vertical interpolations and extrapolations . . . . . . . . . . . 248
4.1

General considerations . . . . . . . . . . . . . . . . . . 248

4.2

Notations . . . . . . . . . . . . . . . . . . . . . . . . . 248

4.3

More details for 3D dynamical variables . . . . . . . . 248

10

CONTENTS
4.4
5

2D dynamical variables which need extrapolations . . 252

Filtering in spectral space . . . . . . . . . . . . . . . . . . . . 253
5.1

THX filtering on derivatives . . . . . . . . . . . . . . . 253

5.2

Bell-shaped filtering on non-derivative fields . . . . . . 254

References257

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 32T0 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) andcan perform different parametrizations. Each surface grid box is made of
the four adjacent tiles. The coverage of each of these surfaces is known
through the global ECOCLIMAP database. Surfex receives atmosphericforcing 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 present document gives some
information about Surfex and how Surfex interacts with the atmosphere.
The complete documentation on Surfex (algorithmic and user’s guide) isavailable on the site.

12

1. Basic hypotheses and related constants

The model relies upon a geometrical assumption: the thin layer approximation, 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

13

and:
(

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

M M,
if M M > 2
M M + 12, if M M ≤ 2

a=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
]

14

1. Basic hypotheses and related constants

obliquity
declination[ right ascension[[
equation of time(true solar time −mean solar time)s = 0.409093
δs = Asin(sin(s ) sin Ls )0 ≤ αs ≤ 2π]
cos(αs ) cos(δs ) = cos(Ls )]sin(αs ) cos(δs ) = sin(Ls ) cos(s ) ]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

15

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16

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

cva =

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 %.
cpv = 4 Rv
cvv = 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

17

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
M g = 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.

18

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:d ln(es )Lv=
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

19

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 =10(• δT index of temperature positivity: δT =if liquid whatever T
if solid whatever T1
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

20

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:βi
esi (T ) = ESS(T ) = exp αi −− γi ln TT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 /p
es
)=
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

if es (T ) ≤ p



es




if es (T ) ≥ p

p





1. Basic hypotheses and related constants

21

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 )

22

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

24

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, p ) > 0s∂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

25

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{z× ~v} + RT ∇ ln p + ∇Φ = −g
+ S~v + K~v
{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:∂η∂p ∂Φ∇p
Φ∇
−
∂p
∂η
∂η




+ ∇Φ

Thermodynamics equation
dT
−
dt

κT

ω
p

=−

g ∂η ∂Fh+ Sh + Kh
cp ∂p ∂η

(3)| {z }

conversion
Moisture equationd qv∂η ∂Fqv= −g+ Sqv + Kqv
dt
∂p ∂η

(4)

In the above equations one takes:
R = qa Ra + qv Rv
cp = qa cpa + qv cpv
Rκ =cp
The 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)

26

2. Dynamics equations

Fp is the mass flux, no source term being considered.
By integrating, one obtains the evolution equation of surface pressure:1∂ps∂p∇ · ~vdη − gFp (1)
=−
∂t
∂η
0



Z



vertical velocity in pressure coordinate:
ω=−

Z η



∂p
dη + ~v · ∇p − gFp (η)∂η
∇ · ~v

0and 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 + v22

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 equations27

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 (λ0 , µ0 ). 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 )q1cos λb = (1 − µ2b )− 2 (µ 1 − µ20 − µ0 1 − µ2 cos(λ − λ0 ))sin λb1= (1 − µ2b )− 2pp

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 λbq

1qcos(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, λ0 = λb ) obtained by homothety of a factor c on stereographic
projection at the pole of stretching. It comes:µ0 =(1 − c2 ) + (1 + c2 )µb(1 + c2 ) + (1 − c2 )µb

and reciprocally:µb =(c2 − 1) + (c2 + 1)µ0
(c2 + 1) + (c2 − 1)µ028

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 0
+
µ
2c
2c

At the pole of dilation (µ0 = 1) this factor is c. At the pole of contraction(µ0 = −1) it is 1/c. The horizontal wind thus becomes:~v = m ~v 0
and the horizontal gradient:∇ = m ∇0
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 = ∆0m2
χ = ∆0−1

where ∆0 −1 is the same formal operator as ∆−1 but on the transformed
sphere. As a consequence, the state variable of the model is: ζ 0 = ζ/m2 andD0 = D/m2 (or equivalently ψ and χ).
The equations become then:

∂ζ 0
∂t
∂D0
∂t

!#
−
→
1 −→
∂η ∂ F~v −→= ∇ ×
H~v − g
+ S~v
+ Kζ 0
m
∂p ∂η
"
!#
−
→
∂
F
1
∂η
−
→
−
→
~
v
= ∇0 ·
H~v − g+ S~v
− ∆0 (Φ + Ec ) + KD0
m
∂p ∂η
"

0

2. Dynamics equations

29

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 0 . 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



∂p
∂Fp
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 D0 +
∂η
∂η




−g

∂Fp
∂η

and its Lagrangian form (7):
d
dt



∂p
∂η



J0
∂Fp
= −g
m2
∂η


The momentum equation (2) becomes:−→d m~v 0∂η ∂ F~v −→ −→+ m 2Ω × ~v 0 + RT ∇0 ln p + ∇0 Φ = −g
+ S~v + K~v
dt∂p ∂ηThe equations for thermodynamics (3) and moisture (4) are formally unchanged.

30

2. Dynamics equations

p`−1
˜

Φ`−1
˜

p`

Φ` , T`

p`˜

Φ`˜

p`+1

Φ`+1 , T`+1

p`+1
˜

Φ`+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:p`˜ = A`˜ + B`˜ ps`˜ = 0, . . . , L(8)A`˜ and B`˜ are constants which define the vertical coordinate. The vertical
distribution of the variables is presented in Figure 1, indices ` relating to
the mid-layers (also named full levels) and indices `˜ to the inter-layers (alsonamed half levels).
The values of A`˜ and B`˜ 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

31

of a variable between the two ends of a layer:δp` = p`˜ − p`−1
˜
Index ` 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



∂p
+ ∇ · ~v
∂η




∂
∂p
+
η̇
∂η
∂η




= Fm

For a given layer, one writes it as:∂p∂(δp)= −∇ · (~v δp) − δ η̇
∂t
∂η




+ Fm

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:
L
X
∂ps
=−
[δp D + δB ~v · ∇ps ] − g(P + E)∂t`=1

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

∂p
∂η



=
`˜

`
X
k=1

[−δBk ~vk · ∇ps − δpk Dk + Fmk ] − B`˜

∂ps
∂t

32

2. Dynamics equations

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

∂X
w
∂p



=
`

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

∂X
∂wX
∂w
=
−X
∂p
∂p
∂p

with the interpolation:
1X`˜ = (X` + X`+1 )2
In the case of the semi-Lagrangian scheme, see Chapter 4.
Hydrostatic equation
The equation of hydrostatic balance is integrated by using the centeredscheme:p˜Φ`−1= Φ`˜ − R` T` ln `−1˜p`˜Which gives, summing from the surface:Φ`˜ = Φs +`+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:Φ` = Φ`˜ + α` R` T`
where:

 α1

=1

 α`

=1−

p`−1
p˜
˜
ln `
δp`
p`−1
˜

(9)

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

∂pΦ
+ RT
∂p

(10)

2. Dynamics equations

33

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∂p∂(Φ ∇p)∂p= ∇Φ + Φ∇
−
∂η
∂η
∂η
∂p
∇ Φ
∂η











The discretization of this term yields:1(RT ∇ ln p)` = R` T`
δp`

"

p˜
C`
δB` +
ln `
δp` p`−1
˜

|

{z

ZRTGR

#

·∇ps
}

with:
C` = A`˜B`−1− A`−1˜˜ B`˜Term ZRTGR is also used in the calculation of the energy transformation
term. It can be written in the simpler form:"p`˜1α` δB` + B`−1
˜ ln
δp
p`−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34

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˜
1
C`
(~v · ∇ ln p)` =δB` +ln ` ~v` · ∇ps
δp`
δp` p`−1
˜
{z

|

}

ZRTGR

and the second term:
"

p˜
1
−
α` (∇ · (~v` δp` ) + Fm` ) + ln `
δp`
p`−1
˜

! `−1
X

#

(∇ · (~vk δpk ) + Fmk )k=1

where:∇ · (~v` δp` ) = ~v` · ∇δp` + δp` D`
This discretization results from writing the last term in the form:
1
p



Z

Xdη =

∂p −1 ∂ ln p
∂η
∂η


Z

∂
ln p Xdη − X ln p
Xdη =
∂η




Z



in 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 (2007a). 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36

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

0
Ynm (λ, µ)Ynm0 (λ, µ)dσ = δn=n0 δm=m0

ZZ

dσ
σ

One has then, for m ≥ 0:s

Pnm (µ) =(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

37

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 )38

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

m
imλ
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:
s

em
n =

n2 − m 2
4n2 − 1

3. Spectral transforms

39

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
0
by M 2 . As reduced divergence D is obtained from velocity potential χ by0
a laplacian operator, and as reduced vorticity ζ is obtained similarly fromstream function ψ, we have in spectral space:
D0m
n =−

ζ 0m
n =−

n(n + 1) mχna2n(n + 1) mψna20

Relationship between U , ψ and χ:0(U a cos θ) =∂χ∂ψ
− cos θ∂λ∂θ

the spectral discretization of which is:
0

m
m m
m
m
(U a cos θ)mn = i mχn + (n − 1)en ψn−1 − (n + 2)en+1 ψn+1
0

Relationship between V , ψ and χ:0(V a cos θ) =∂ψ∂χ
+ cos θ∂λ∂θ

the spectral discretization of which is:
0

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

40

3. Spectral transforms
000

Relationship between D , U and V :01
D =a cos θ0∂U∂(V cos θ)
+
∂λ
∂θ

0

!

which can be rewritten:
1
D = 2
a cos2 θ
0

00∂(U a cos θ)∂(V a cos θ)+ cos θ
∂λ
∂θ
0

0

!

0

Relationship between ζ , U and V :1ζ =a cos θ000∂V∂(U cos θ)
−
∂λ
∂θ

!

which can be rewritten:1
ζ = 2
a cos2 θ000∂(V a cos θ)∂(U a cos θ)− cos θ∂λ∂θ!

Spectral discretizations allow to retrieve easily spectral components of fields
000
D a2 cos2 θ and ζ a2 cos2 θ, but not directly spectral components of D and0ζ (requiring inversion of a penta-diagonal matrix). In fact, the algorithm
00involved 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

41• 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.

42

3. Spectral transforms

4Semi-lagrangian discretization

1 Introduction
1.1 Purpose
This chapter describes the semi-Lagrangian scheme used in Arpege-climat.
The Arpege-IFS code contains many other options. See Yessad (2007b) for
a comprehensive description of all available features. Equations will be written without horizontal diffusion scheme (which is treated in spectral computations, see Chapter ??), in order to give a clearer presentation of thediscretized equations.

1.2 Eulerian scheme
In Eulerian form of equations, the time dependency equation of a variable
X writes as:
∂X
→
~ .−
= −U
∇3 X + Ẋ
∂t

(1)

→
~ is the 3D wind, −where U
∇3 is the 3D gradient operator, Ẋ is the sum of
the dynamical and physical contributions. X(t + ∆t) is computed knowing
X(t − ∆t) at the same grid-point. Eulerian technique obliges to use a timestep that satisfies to the CFL (Courant Friedrich Levy) condition everywhere.
For the variable-mesh spectral global model Arpege, the horizontal CFL
condition writes as:sm | U | ∆t

N (N + 1)
<1
a2

(2)

44

4. Semi-lagrangian discretization

where m is the mapping factor, | U | is the 3D wind module, N is thetruncation, a is the Earth radius. The vertical CFL condition writes as:| η̇ | ∆t∆η < 1(3)

For a T359L31 (triangular truncation 359, 31 levels) model with stretchingcoefficient c=2.5 (maximum horizontal resolution about 20 km), that gives∆t ' 2 min.

1.3 Semi-Lagrangian scheme
In semi-Lagrangian form of equations, the time dependency equation of avariable X writes as:
dX
= Ẋ
dt

(4)

In a three-time level semi-Lagrangian scheme X(t + ∆t) is computed at agrid-point F knowing X(t − ∆t) at the point O (not necessary a grid-point)where the same particle is at the instant t − ∆t. In a two-time level semiLagrangian 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 the
instant t. The semi-Lagrangian technique is more expensive for one time-stepthan the Eulerian technique because it is necessary to compute the positionsof the origin point O and the medium point M along the trajectory and tointerpolate some quantities at these points (1.5 times the cost of the Eulerian
scheme in the T199L31 model with physics). But it allows to use larger timesteps: the stability condition is now the Lipschitz criterion (trajectories do
not cross each other) and is less severe than the CFL condition.
dη
D is the divergence of the horizontal wind on the η-coordinates, η̇ =.
dt
Lipschitz criterion writes for a two-time level semi-Lagrangian scheme:|D+∂ η̇ ∆t
|
<1
∂η 2

(5)

Arpege-climat using a two-time level scheme (2TL), the three-time level
scheme (3TL) is not described in this chapter.

2 Equations
2.1 Notations~ is the horizontal wind. Its zonal component (on the Gaussian grid)• V
is denoted by U

4. Semi-lagrangian discretization

45• D is the horizontal wind divergence• w is the z-coordinate vertical velocity: w =dzdt• T is the temperature. T ∗ is a vertically-constant reference temperature
which is used in the semi-implicit scheme. Default value is 300 K or
350 K according to configuration (for more details see documentation
about semi-implicit scheme). TsST is the reference standard atmospheredTsurface temperature (288.15 K).
is the standard atmospheredz ST
tropospheric gradient of temperature (−0.0065 K m−1 )• q is the humidity• p is pressure, ps is surface pressure. p∗ is a reference pressure and p∗s is areference surface pressure, which are used in the semi-implicit scheme.
These reference quantities are vertically dependent and “horizontally"(i.e. on η-surfaces) constant. Default value of p∗s is 1000 hP a for a 2TL
SL scheme. ∆p∗ are layer depths corresponding to a surface pressure
is a reference pressure equal to the surface pressure
equal to p∗s . pSTsof the standard atmosphere (101325 P a, variable VP00). pST is areference pressure defined on layers and inter-layers corresponding to
the surface reference pressure pSTs (stored in array STPRE).
• ω=

dp
is the total temporal derivative of pressuredt• Φ is the geopotential, Φs is the surface geopotential (i.e. the orography)~ is the Earth rotation angular velocity• Ω• ~r is the vector directed upwards, the length of which is the Earth radius• ~ı is the unit zonal vector on the Gaussian grid• g is the gravity acceleration constant• R is the gas constant for air, Rd the gas constant for dry air and Rv
the gas constant for water vapor• cp is the specific heat at constant pressure for air and cpa is the specificheat at constant pressure for dry air• cv is the specific heat at constant volume for air and cva is the specificheat at constant volume for dry air→−• ∇ is the first order horizontal gradient on η-surfaces

46

4. Semi-lagrangian discretization• B is the quantity which defines the vertical hybrid coordinate (seechapter 2): B varies from 1 to 0 from bottom to top. Layer value of B
is defined as the average of the adjacent inter-layer value of B.
• αT is a vertical-dependent coefficient used to define a thermodynamic
αT Φsvariable T + δT R
less sensitive to orography than temperature
Rd T STT . Expression of αT is:
Rd dT
αT = B −
g dz








T
ST

ST

pST
pST
s




! − Rd dT

g

dz



−1

ST

(6)

δT R being the switch to come back to temperature.

2.2 3D primitive equations hydrostatic model
Momentum equation:
Vector form of momentum equation is used. Coriolis force can be treatedexplicitly (δV~ = 0) or implicitly (δV~ = 1).
~ + δ ~ (2Ω~ ∧ ~r)
d V
V
dt

h

i

~ ∧V
~ ) − ∇Φ~ − RT ∇~ ln p + F~ ~= −2(1 − δV~ )(Ω
V(7)

FV~ is the physical contribution on horizontal wind.
Temperature equation:
dT
RT ω
=
+ FT
dt
cp p

(8)

FT is the physical contribution on temperature. This equation can be modiαT Φsfied by replacing temperature T by T + δT R
which is less sensitive to
Rd T ST
orography, as it is made for continuity equation (see next paragraph relative
to continuity equation). This modification has been proposed by Ritchie andTanguay (1996).
αT Φsd
T + δT R
dt
Rd T ST




d
αT Φs
=
δT R
dt
Rd T ST




+

RT ω
+ FT
cp p

(9)

d
αT Φs
See Equation (6) for definition of αT . Term
δT R
only contains
dt
Rd T STadvection terms linked to horizontal variations of orography and verticalvariations of coefficient αT .


4. Semi-lagrangian discretization

47

Humidity equation:
dq
= Fq
dt

(10)

Fq is the physical contribution on specific humidity, This equation is also
valid for other advective variables (GFL).
For non-advectable GFL variables (for example rain in the optional physics
qr ), the equation is identical to the Eulerian equation:
∂qr
= Fqr
∂t

(11)

Fq is the liquid precipitation flux
Continuity equation:
The impact of water phase changes on atmosphere mass balance is no longer
taken into account in Arpege-climat because of the increasing complexityof the code and the unsafety of an option which is not supported by ECMWF.
The conservation of dry air mass, also known as continuity equation, is
written as:
Φs
Z 1
d ln ps + δT R
∂B
Rd T ST
dη
dt
0 ∂η


Z 1
0

∂B
1
−
∂η
ps




Z 1

=

~ V
~ ∂p dη + V~∇~ ln p + δT R Φs∇∂η
Rd T ST
0








dη

(12)

Variable δT R is 0 or 1; when δT R = 1 the new variable is less sensitive to theorography (see temperature equation about orographic resonance).

3 Generic discretization of the equations
3.1 Notations
Upper index:• δt is half time-step• First integration step: + (resp. m, o, −) for t + ∆t (resp. t + δt, t, t)
quantity.

48

4. Semi-lagrangian discretization• Following integration steps: + (resp. m, o, −) for t + ∆t (resp. t + δt,t, t − ∆t) quantity.

Lower index: F (resp. M and O) for final (resp. medium and origin)
point.
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 and a non-zero semi-implicit correction contribution. They
are called “GMV” in the code (“GMV” means “grid-point model variables”). This class of variables includes wind components and temperature.
• 3D advectable “conservative” variables. The equation RHS of thesevariables 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 may include liquid/ice cloud water, ozone and TKE.
• 3D non advectable pseudo-historic variables. The equation RHS ofthese 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 may include rain, snow, graupels,convective precipitation flux, stratiform precipitation flux, moisture
convergence, total humidity variation or convective vertical velocity.
• 2D variables, the equation RHS of which mixes 3D and 2D terms, hasa non-zero 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).

3.2 Discretization for a 3D variable: general case where the
RHS has non-zero linear and non-linear terms (GMV variables).
List of equations:• Momentum equation• Temperature equation
Generic notations: Generic notation N(X)LAG stands for:

4. Semi-lagrangian discretization

49• NWLAG for momentum equation.
• NTLAG for temperature equation.
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.
Generic notation P(X)NLT9 stands for:
• PUNLT9 for U-momentum equation.
• PVNLT9 for V-momentum equation.
• PTNLT9 for temperature 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)• A − βB is the non-linear term
• F is the sum of contributions computed in the physical parametrizations
Other points:• High-order interpolations: in the following discretizations, “high-orderinterpolations” means 32-point interpolation for 3D terms (vertical interpolations are cubic), 12-points interpolations for 2D terms• Uncentering:  is a first-order “uncentering factor”; it allows to remove
the noise due to gravity waves (orographic resonance)• Vectors: for vectors like horizontal wind, a rotation operator R has tobe 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

50

4. Semi-lagrangian discretization

2TL vertical interpolating SL scheme: stable discretization (LSETTLS=.T.) and first-order uncentering The basic equation can be written as:
dX
=A+F
dt

(13)

When N(X)LAG=3, Equation (13) is discretized by averaging the non-linear
term along the trajectory in the RHS.
The t + δt non-linear term Am − βB m is calculated by a linear space andtime extrapolation.
1
1
1(1 + )[Ao − βB o ]F + (2 − )[Ao − βB o ]O − [A− − βB − ]O222This type of extrapolation is available only for N(X)LAG=3. At the firsttime integration step, values at t+δt are set equal to initial values. Quantity
Ao − βB o has to be saved in a buffer P(X)NLT9 to be available as A− − βB −for the following time-step.
Equation (13) is discretized as follows:oo(X − (1 + )δt βB)+F = {X + [(2 − )δt A − (2 − )δt βB]−δt[A − βB]− + [(1 − )δt βB + ∆tF]o }O+{[(1 + )δt A − (1 + )δt βB]o }FBuffers content before interpolations for N(X)LAG=3:• P(X)L0: [(2 − )δtA − (2 − )δt βB]o − δt[A − βB]− + [(1 − )δt βB]ofor tri-linear interpolation at the origin point O.
• P(X)L9: X o + [∆tF]o for high-order interpolation at the origin pointO.
• P(X)T1: [(1 + )δtA − (1 + )δt βB]o then provisional add of quantity[(1 + )δt βB]o before t + ∆t physics; evaluated at the final point F .

3.3 Discretization for a 3D variable: particular case where
the RHS has zero linear and non-linear terms (advectable
GFL variables)
List of equations:
Humidity equation, and for example:

4. Semi-lagrangian discretization

51• Specific humidity equation• Ozone equation• Liquid water equation• Ice equation• TKE equation• Extra GFL variables equations
Generic notations
Generic notation P(X)L9, P(X)T1 stands for:
• PGFLL9, PGFLT1 for GFL variables
In the present case A and B are equal to zero.
Other points:• High-order interpolations: in the following discretizations, “high-orderinterpolations” means 32-point interpolations for 3D terms (verticalinterpolations are cubic), 12-point 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).
• Uncentering:  is a first-order “uncentering factor”; It allows to remove
the noise due to gravity waves (orographic resonance)
2TL vertical interpolating SL scheme
At the first time integration step, values at t + δt are set equal to initialvalues. This discretization of the 2TL SL scheme follows (Mc Donald andHaugen, 1992 ).
Equation (13) is discretized as follows:
XF+ = {X o + [∆tF]o }O(14)

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

52

4. Semi-lagrangian discretization

3.4 Non advectable pseudo-historic GFL variables
For these variables the discretization always writes:
XF+ = {X o + [∆tF]o }F(15)

and there are never interpolations.

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



Wvei
A
∆η


Wvei
+[Rinte ](top,surf )
F
∆η




= [Rinte ](top,surf )where:
[Rinte ](top,surf )

Wvei
∆η



=1

(16)

and [Rinte ](top,surf ) is the vertical integral matrix operator (the scalar prodR
uct [Rinte ](top,surf ) hXi is the discretization of 01 Xdη, hXi is the vector
containing the layer values of X: (X1 ; X2 ; ...; Xl ; ...; XL )).
In the thin layer equations, expression of Wvei at full levels is:[Wvei ]l = ∆Bl(17)

List of equations:• Continuity equation
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 (3Dterm), stand for4. Semi-lagrangian discretization

53• 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.
Other points
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, when interpolating
a 2D


W
vei
surface variable SV at the origin point, [Rinte ][SV ]O(top,surf )∆ηF

has no reason to be equal to [SV ]O(η=1) , these quantities are generally dif

ferent: this is [Rinte ](top,surf )


L.

W
vei
∆η





[SV ]O

W
vei
∆η





[SV ]OF

is the vector containingFwhich has to be computed.

[Wvei ]l[∆η]l[SV ]O(l) , for l = 1 toF

2TL vertical interpolating SL scheme: stable discretization (LSETTLS=.T.) and first-order uncentering
The t + δt non-linear term Am − βB m is calculated by a linear space andtime extrapolation:
1
1
1(1 + )[Ao − βB o ]F + (2 − )[Ao − βB o ]O(l) − [A− − βB − ]O(l)222This type of extrapolation is available only for N(X)LAG=3. At the firsttime integration step, values at t+δt are set equal to initial values. Quantity
Ao − βB o has to be saved in a buffer P(X)NLT9 to be available as A− − βB −for the following time-step.
Equation (13) is discretized as follows:

54

4. Semi-lagrangian discretization(X − (1 + )δtβB)+
F

Wvei= [Rinte ](top,surf )
{X o
∆η F
+[(2 − )δtA − (2 − )δtβB]o − δt[A − βB]−+ [(1 − )δtβB + ∆tF]o }O+{[(1 + )δtA − (1 + )δtβB]o }FBuffers content before interpolations for N(X)LAG=3:• P(X3D)L0 is not used.
• P(X3D)L9: [(2 − )δtA − (2 − )δtβB]o − [δtA − δtβB]− + [(1 − )δtβB]ofor tri-linear interpolation at the origin point O(l)• P(X2D)0 is not used
• P(X2D)9: X o + [∆tF]o for horizontal high-order interpolation at theorigin point O(l)• P(X)T1: [(1 + )δtA − (1 + )δtβB]o then provisional add of quantity[(1 + )δtβB]o before lagged physics; evaluated at the final point F

3.6 Additional vertical derivatives
If δT R is non-zero, discretization of temperature equation needs to computedαT(at full levels) of αT . Layers values of αT
the vertical advection η̇dη
αT Φs(array RCORDIF) are used to define T + δT R
, but inter-layer values
Rd T ST
of αT (array RCORDIH) are used to compute vertical advection.

3.7 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 modellevels and can be done at F in the unlagged grid-point calculations (theintermediate quantity obtained is stored in the array P(X)SPL9), the second
one is a 4 points interpolation, done in the lagged grid-point calculations in
the interpolation routine. Interpolation routine uses both P(X)SPL9 (forinterpolations) and P(X)L9 to apply a monotonicity constraint.

4. Semi-lagrangian discretization

55

4 Computation of medium and origin points
4.1 Medium point M
The medium point is calculated in subroutines LARMES and LARMES2.
Trajectories are great circles on the geographical sphere. The computation of
the medium point M location of the Lagrangian trajectory is performed byan iterative method described by Robert (1981) and adapted to the sphereby Rochas. In a 3TL SL scheme, the particle is at the point M at time t(t + δt for the first integration step). In a 2TL SL scheme, the particle is at
the point M at t + δt. M is at the middle position of the origin point O andthe final point F .
Notations:• RM F is the rotation operator from medium point to final point (seesection 6)• ROF is the rotation operator from origin point to final point (see section 6)~ (C Earth center, F final point)• ~rF = CF
~ (M medium point)• ~rM = CM~d
~ )• φM F : angle (CM, CF• θF ,λF : latitude, longitude on the geographical sphere of F
• θM ,λM : latitude, longitude on the geographical sphere of M
~ M : interpolated horizontal wind at M (at t + δt )• V
~ O : interpolated horizontal wind at O (at t + δt )• V
• a is the average Earth radius near the surface• ~r = a~k• ∆t: time-step• δt: half time-step• L: number of layers of the model

56

4. Semi-lagrangian discretization

Definition of the vertical coordinate η:
Research of medium point needs an exact definition of the vertical coordinate
η. For the inter-layer number l (l between 0 and L), ηl is defined by:
ηl =

l
L

(18)

For the layer number l (l between 1 and L), ηl is defined by:1ηl = (ηl + ηl−1 )2(19)Stable algorithm for 2TL SL scheme (LSETTLS=.T.)
Extrapolation of the wind:This algorithm has been developed by Hortal (2002). The basic idea is toreplace the purely temporal extrapolation by a space and time extrapolation:~ ]M (t + δt) = 3 RN F [V~ ]N (t) − 1 ROF [V~ ]O (t − ∆t)
RM F [V22(20)

where N is the position of the particle at time t for a particle which goesfrom the origin point O at time t − ∆t to M at time t + δt. Assuming thatthe wind is constant along the trajectory one can write:
ON = 2N M = 0.5N F(21)~ ]N (t) by
and evaluate the angular velocity RN F [V
M
F
M
OF
O
~ ] (t) + 1/3R [V~ ] (t) or 1/3[V~ ]F (t) + 2/3ROF [V~ ]O (t).
2/3R [V~ ]M (t + δt) becomes:
Expression of [V~ ]M (t + δt) = 1 [V~ ]F (t) + 1 ROF (2[V~ ]O (t) − [V~ ]O (t − ∆t)) (22)
RM F [V22
The same type of extrapolation is done for the η-coordinate vertical velocity. The algorithm of research of trajectory uses directly the RHS of thisequation, and for all iterations the origin point O is computed instead of themedium point M .
Algorithm
The origin point is defined by the following iterative scheme: for the iterationk + 1:[~r]Or]F cos φk −k+1 = [~~ ]F (t) + ROF (2[V~ ]O (t) − [V~ ]O (t − ∆t))[Vkksin φk (23)~ ]F (t) + ROF (2[V~ ]O (t) − [V~ ]O (t − ∆t)) |
| [V
k

k

4. Semi-lagrangian discretization57

where:~ ]F (t) + ROF (2[V~ ]O~ O
φk = δt | [Vk (t) − [V ]k (t − ∆t)) |(24)

Some geometrical approximations (small angles on the sphere) yield:
!

[~r]O
r]F
k+1 = [~2φ2~ ]F (t) + ROF (2[V~ ]O (t) − [V~ ]O (t − ∆t)) δt 1 − φk
1 − k − [Vkk2
6On the vertical:
Oηk+1= η F − 2δt(0.5η̇ F (t) + 0.5(2η̇kO (t) − η̇kO (t − ∆t)))(26)

First iteration~ ]F (t) as a first guess for the space and time
One starts with M0 = F , [V~ ]F (t) |, η̇ F (t) as a firstextrapolated horizontal angular velocity, φ0 = ∆t | [Vguess for the space and time extrapolated η-coordinate vertical wind. Quantities at t are taken as a first guess and not quantities at (t + δt), contrary to~ ](t) − [V~ ](t − ∆t))the case LSETTLS=.F. . The coordinates of O1 are (2[V. (2η̇(t) − η̇(t − ∆t)) is interpolated at this point, which allows to computethe wind components which will be used for the next iteration.
Following iterations[V~ 0 ] (of coordinates ([u ], [v ]) ) is a generic notation for~ ]F (t) + ROF (2[V~ ]O (t) − [V~ ]O (t − ∆t))).
1/2([V00kk

For horizontal displacement use equations:0O
sin θk+1= sin θF cos φk −[v ]cos θF sin φk
0~| [V ] |

(27)

0

O
F
F
cos θk+1
cos(λO
k+1 − λ ) = cos θ cos φk +[v ]sin θF sin φk~ 0] |
| [V

(28)

0

O
F
cos θk+1
sin(λO
k+1 − λ ) = −[u ]sin φk~ 0] || [V(29)

For vertical displacement use equation:Oηk+1= η F − δt(η̇ F (t) + (2η̇kO (t) − η̇kO (t − ∆t)))

(30)

!

(25)

58

4. Semi-lagrangian discretization

4.2 Origin point O
The origin point is calculated in subroutines LARMES and LAINOR2.
In a 2TL SL scheme, the particle is at the point O at time t.
O is on the same great circle arc (on the geographical sphere) as M and
F and the length of OF is twice the length of M F . If angle ([~r]Od, [~r]F ) isosmall (less than 10 , what is generally satisfied), on can write for horizontal
displacement:[~r]O − [~r]F ' 2([~r]M − [~r]F )(31)

For vertical displacement on can always write:η O − η F = 2(η M − η F )(32)

One denotes by:• φ = ([~r]Md, [~r]F )000~ ] (of coordinates ([u ], [v ])) the last interpolated horizontal velocity.
• [VUsing the following identities:cos 2φ = 2 cos2 φ − 1(33)

sin 2φ = 2 sin φ cos φ(34)

the origin point horizontal coordinates can be computed by:
"

0

[v ]
sin θO = sin θF cos 2φ − 2 cos φ
cos θF sin φ~ 0] |
| [V
"

#

0

(35)

#

[v ]
cos θ cos(λ − λ ) = cos θ cos 2φ + 2 cos φsin θF sin φ (36)0~| [V ] |
O

O

F

F

"

0

[u ]
cos θO sin(λO − λF ) = −2 cos φsin φ~ 0] || [V#Terms in brackets are already computed to determine M .

(37)

4. Semi-lagrangian discretization

59

4.3 Refined re-computation of point O
Option L2TLFF controls re-computation of the origin point using the average
between the angular velocity at the origin point and the provisional t + ∆tangular velocity, according to the algorithm previously described. Only term
~ ∧ a~k) is computed (always analytically) at this improved position of O(2Ω
(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. Equations~ F (t + ∆t) and also a first guesssystem is integrated to find a first guess of VFOF~~ ]O ) is used to recompute O.
of ps (t + ∆t) which provides 1/2([V ] + R [V~ ∧ a~k)(O improved) − (2Ω~ ∧ ~k)(O) is analytically computed
A correction (2Ω~ F (t + ∆t).
and added to wind equation to find the “improved” value of V
Computations are currently made in routine LAPINEB and LADINE.

5 The SL discretization of the 3D primitive equation
model
5.1 Momentum equation
Definition of X, A, B and F, top and bottom values~ + δ ~ (2Ω~ ∧ ~r)
X=V
V

(38)

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

(39)

∗

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

(40)

F = F~V~(41)~η=0 = V
~l=1
V

(42)





Top:

Bottom:~η=1 = V
~l=L
V

(43)

Remarks• Coriolis term is treated implicitly (δV~ = 1, LADVF=.T.) This means~ ∧ ~r) is analytically computedthat (2Ω~ ∧ ~r) is recomputed at an improved• With option L2TLFF, term (2Ω
position of the origin point

60

4. Semi-lagrangian discretization

5.2 Thermodynamic equation
Definition of X, A, B and F, top and bottom values

X = T + δT R

αT Φs
Rd TST(44)

RT ω
αT ~ ~ΦsA=+ δT RV ∇(Φs ) + δT R
cp p
Rd TST
Rd TST2

B = −M τ D



dαT
η̇
dη



(45)

0

(46)

F = FT(47)

Tη=0 = Tl=1

(48)

Top:

Bottom:Tη=1 = Tl=L(49)

5.3 Continuity equation
Definition of X, A, B, and F

X = log ps + δT R

1
A=−
ps

Z η=1

Φs
Rd Tst(50)~ V~ ∂p dη + V~∇~ log ps + δT R Φs∇∂η
Rd Tst
η=0








(51)

2

B=−

M
νD
M2

(52)

B0 = 0

(53)

F =0

(54)

4. Semi-lagrangian discretization

61

5.4 GFL variables
We detail here the case of specific moisture.
Definition of X, A, B and F, top and bottom values
X=q

(55)

A=0

(56)

B=0

(57)

F = Fq(58)

qη=0 = ql=1

(59)

Top:

Bottom:qη=1 = ql=L(60)

The other GFL are treated similarly. Quantities are assumed constant above
the middle of the upper layer and below the middle of the lower layer.

5.5 Quantities to be interpolated
The computation is performed in subroutine LACDYN.
Research of trajectory
When researching the medium point by an iterative algorithm, the interpolation at the origin point of ([U ], [V ], η̇) is needed: a tri-linear interpolationis performed. For more details about interpolations, see section 8.
RHS of equations
The list of quantities to be interpolated has been described in subsections
3.2, 3.3 and 3.5 for each type of equation.
Additional quantities to be interpolated at the origin point if L2TLFF~ ] at time t
The two components of the [V62

4. Semi-lagrangian discretization

6 R operator
6.1 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:~ 0 = ROF (V
~)
V

(61)

~ 0 has coordinates (u0 , v 0 ), V
~ has coordinates (u, v), and the relationwhere V00ship between (u, v) and (u , v ) is:
0

u
0
v

!

=

p q
−q p

!

u
v

!

(62)

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

(63)

q=

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

(64)

(Notations θO ,θF ,λO ,λF ,φ: see section 4).
p and q verify the following identity:
p2 + q 2 = 1(65)Computation of p and q is made in subroutine LARCHE.

6.2 Tilting~ 0 and V~ are linked by the following relationship:
The coordinates of V

0

u
0
v

!

=

GN ORDM GN ORDL−GN ORDL GN ORDM
cos α − sin α
sin α cos α

!

u
v

!

p q
−q p

!

×

!

(66)

4. Semi-lagrangian discretization

63

where:
cos α =

2c[sin θp cos θO − sin θO cos θp cos(λO − λp )]A cos ΘO(67)

sin α =

2c[cos θp sin(λO − λp )]A cos ΘO

(68)

A = (1 + c2 ) + (1 − c2 )(sin θp sin θO + cos θp cos θO cos(λO − λp )) (69)and where:• 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 stretching pole
• 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 the unit vector directed towards the true north, computed insubroutine SUGEM2

7 Longitudes and latitudes on the computational sphere
For interpolations it is necessary to compute (ΘO , ΛO ), latitude and longitude of the interpolation 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 andcos θO sin λO − λF where (θF , λF ) are the coordinates of the final point on
the geographical sphere). Transform formulas giving (Θ, Λ) on the computational sphere once knowing (θ, λ) on the geographical sphere are givenby:sin Θ =

(1 − c2 ) + (1 + c2 )(sin θp sin θ + cos θp cos θ cos(λ − λp ))(70)Acos Θ cos Λ =2c(cos θp sin θ − sin θp cos θ cos(λ − λp ))A(71)cos Θ sin Λ =2c cos θ sin(λ − λp )A(72)

where:

64

4. Semi-lagrangian discretization• 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 stretching pole
• Computation of Θ, Λ is made in subroutine LARCHE

8 Interpolations and weights computations
8.1 Interpolation grid and weights
Computation is done in subroutine LASCAW.
Horizontal interpolation grid and weights for bi-linear interpolations
A 16 points horizontal grid is defined as it is shown in figure ??. The interpolation point O (medium or origin point) is between B1 , C1 , B2 and C2 .
Λ and Θ are the longitudes and latitudes on the computational sphere. The
weights are defined as follows:• zonal weight number 1:
ZDLO1 =

ΛO − ΛB1
ΛC1 − ΛB1

• zonal weight number 2:
ZDLO2 =

ΛO − ΛB2
ΛC2 − ΛB2

• meridional weight:
ZDLAT =

ΘO − ΘB1
ΘB2 − ΘB1

Vertical interpolation grid and weights for vertical linear interpolations
A 4 points vertical grid is defined as it is shown in figure ??. The interpolation point O (medium or origin point) is between Tl+1 and Tl+2 . The
vertical weight is defined by:
ZDV ER =ηO − ηTl+1
ηTl+2 − ηTl+1

4. Semi-lagrangian discretization

65

Horizontal interpolation grid and weights for 12-point cubic interpolations
A 16 points horizontal grid is defined as it is shown in figure ??. The interpolation point O (medium or origin point) is between B1 , C1 , B2 and C2 .
The weights are defined as follows:• zonal weight number 0:
ZDLO0 =

ΛO − ΛB0
ΛC0 − ΛB0

• zonal weight number 1:
ZDLO1 =

ΛO − ΛB1
ΛC1 − ΛB1

• zonal weight number 2:
ZDLO2 =

ΛO − ΛB2
ΛC2 − ΛB2

• zonal weight number 3:
ZDLO3 =

ΛO − ΛB3
ΛC3 − ΛB3

• meridional weights:
ZCLA2 =

(ΘO − ΘB0 )(ΘO − ΘB2 )(ΘO − ΘB3 )(ΘB1 − ΘB0 )(ΘB1 − ΘB2 )(ΘB1 − ΘB3 )ZCLA3 =

(ΘO − ΘB0 )(ΘO − ΘB1 )(ΘO − ΘB3 )(ΘB2 − ΘB0 )(ΘB2 − ΘB1 )(ΘB2 − ΘB3 )

ZCLA4 =

(ΘO − ΘB0 )(ΘO − ΘB1 )(ΘO − ΘB2 )(ΘB3 − ΘB0 )(ΘB3 − ΘB1 )(ΘB3 − ΘB2 )Vertical interpolation grid and weights for vertical cubic 4-pointinterpolations
A 4-point vertical grid is defined as it is shown in figure ??. The interpolation
point O (medium or origin point) is between Tl+1 and Tl+2 . The verticalweights are defined by:
ZCV E2 =(ηO − ηTl )(ηO − ηTl+2 )(ηO − ηTl+3 )(ηTl+1 − ηTl )(ηTl+1 − ηTl+2 )(ηTl+1 − ηTl+3 )66

4. Semi-lagrangian discretizationZCV E3 =(ηO − ηTl )(ηO − ηTl+1 )(ηO − ηTl+3 )(ηTl+2 − ηTl )(ηTl+2 − ηTl+1 )(ηTl+2 − ηTl+3 )ZCV E4 =(ηO − ηTl )(ηO − ηTl+1 )(ηO − ηTl+2 )(ηTl+3 − ηTl )(ηTl+3 − ηTl+1 )(ηTl+3 − ηTl+2 )Vertical interpolation grid and weights for vertical cubic Hermite
interpolations
A 4-point vertical grid is defined as it is shown in figure ??. 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 +2com are ∂X∂X
puted. For a variable X,
is computed as close as possible as η̇
/η̇,
∂η
∂η
but with additional
approximations allowing to avoid horizontal interpola∂ptions for term η̇.
∂η• For layers other than the first or the last layer, discretization follows:


∂X
∂η



=
l+1

1 Xl+2 − Xl
2 ηl̄+1 − ηl̄(73)• For layer l = 1, discretization assumes that η̇

∂p
∂η



= 0; discretizal̄=0tion follows:


∂X
∂η



=
l=1

Xl=2 − Xl=1
ηl̄=1 − ηl̄=0

(74)

∂p
• For layer l = L, discretization assumes that η̇∂ηtion follows:




∂X
∂η



=
l=L

Xl=L − Xl=L−1
ηl̄=L − ηl̄=L−1



= 0; discretizal̄=L(75)

The following weights are computed:• For an interpolation point included between layers 2 and L − 1 (l ≥ 1
and l ≤ L − 3) :V DERW 11 =1 ηl+2 − ηl+1
2 ηl̄+1 − ηl̄4. Semi-lagrangian discretization

V DERW 21 =1 ηl+2 − ηl+1
2 ηl̄+1 − ηl̄

V DERW 12 =1 ηl+2 − ηl+1
2 ηl̄+2 − ηl̄+1

V DERW 22 =1 ηl+2 − ηl+1
2 ηl̄+2 − ηl̄+1

67• For an interpolation point included between layers 1 and 2:
V DERW 11 = 0
V DERW 21 =

ηl=2 − ηl=1
ηl̄=1 − ηl̄=0

V DERW 12 =1 ηl=2 − ηl=12 ηl̄=2 − ηl̄=1

V DERW 22 =1 ηl=2 − ηl=12 ηl̄=2 − ηl̄=1

such a 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 =1 ηl=L − ηl=L−1
2 ηl̄=L−1 − ηl̄=L−2

V DERW 21 =1 ηl=L − ηl=L−1
2 ηl̄=L−1 − ηl̄=L−2

V DERW 12 =

ηl=L − ηl=L−1
ηl̄=L − ηl̄=L−1

V DERW 22 = 0such a case is extended to the case where the interpolation point is
between the last layer and the ground; in this case the interpolation
becomes an extrapolation.
Functions fH1 (ZDV ER) to fH4 (ZDV ER) (involved in any Hermite cubic
interpolation) are:68

4. Semi-lagrangian discretization• fH1 (α) = (1 − α)2 (1 + 2α)• fH2 (α) = α2 (3 − 2α)• fH3 (α) = α(1 − α)2• fH4 (α) = −α2 (1 − α)Interpolation grid and weights for tri-linear interpolations
A 64-point grid is defined as it is shown in figure ??. 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 and l + 2 seesection 8.1 corresponding to bi-linear horizontal interpolations for weights
computations. For weights needed for vertical interpolations (ZDV ER) see
section 8.1 corresponding to linear vertical interpolations.

Interpolation grid and weights for 32-point interpolations
A 64-point grid is defined as it is shown in figure ??. 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 section 8.1 corresponding to bi-linear horizontal interpolations for weights computations.
For the two levels l + 1 and l + 2 see section 8.1 corresponding to 12-point
horizontal interpolations for weights computations. For weights needed forvertical interpolations (ZDV ER) see section 8.1 corresponding to linear vertical interpolations.

Other grids and weights
The following interpolation systems are available but not described here:• vertical cubic splines• horizontal 16-point linear least-square fit• horizontal 32-point linear least-square fit• horizontal 12-point spline cubic• 32-point spline cubic

4. Semi-lagrangian discretization

69

8.2 Interpolations
Bilinear interpolation
It is done in subroutine LAIDLI. See figure ?? and section 8.1 for definition
of ZDLO1, ZDLO2, ZDLAT and points 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 meridional linear interpolation:
Xinterpo = X1 + ZDLAT (X2 − X1 )Tri-linear interpolation
It is done in subroutine LAITLI. For layers l + 1 and l + 2 (see figure ??)bilinear horizontal interpolations give two interpolated values Xl+1 and Xl+2
(see section 8.2). Then the final interpolated value is given by the following
expression:
Xinterpo = Xl+1 + ZDV ER(Xl+2 − Xl+1 )Horizontal 12-point interpolation
It is done in subroutine LAIDDI or its shape-preserving version LAIDQM.
See figure ?? and section 8.1 for definition of ZDLO0, ZDLO1, ZDLO2,
ZDLO3, ZCLA2, ZCLA3 and ZCLA4 and points B0 , C0 , A1 , B1 , C1 , D1 ,
A2 , B2 , C2 , D2 , B3 and C3 . Let us define:• f2 (α) = (α + 1)(α − 2)(α − 1)/2
• f3 (α) = −(α + 1)(α − 2)α/2• f4 (α) = α(α − 1)(α + 1)/6
For a quantity X, are computed successively:• a linear interpolation on the longitude number 0:
X0 = XB0 + ZDLO0(XC0 − XB0 )70

4. Semi-lagrangian discretization• a cubic 4-point interpolation on the longitude number 1:
X1 = XA1 + f2 (ZDLO1)(XB1 − XA1 ) + f3 (ZDLO1)(XC1 − XA1 )+f4 (ZDLO1)(XD1 − XA1 )• a cubic 4-point interpolation on the longitude number 2:
X2 = XA2 + f2 (ZDLO2)(XB2 − XA2 ) + f3 (ZDLO2)(XC2 − XA2 )+f4 (ZDLO2)(XD2 − XA2 )• a linear interpolation on the longitude number 3:
X3 = XB3 + ZDLO3(XC3 − XB3 )• a meridional cubic 4-point interpolation:
Xinterpo = X0 + ZCLA2(X1 − X0 ) + ZCLA3(X2 − X0 )+ZCLA4(X3 − X0 )There is a shape-preserving version LAIDQM of routine LAIDDI: after cubic 4-point 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-point interpolation Xinterpo is bounded between X1 andX2 . Use of switches LQMW (momentum equation), LQMT (temperatureequation), LQMQ (humidity equation), LQMV (passive scalar equations),
LQMP (continuity equation), allow to use shape-preserving interpolation
routine LAIDQM instead of LAIDDI.
Cubic 4-point vertical interpolation
See figure ?? and section 8.1 for definition of ZCV E2, ZCV E3 and ZCV E4.
The cubic 4-point vertical interpolation gives the final interpolated value:

Xinterpo = Xl + ZCV E2(Xl+1 − Xl ) + ZCV E3(Xl+2 − Xl ) +ZCV E4(Xl+3 − Xl )Cubic Hermite vertical interpolation
See figure ?? and section 8.1 for definition of V DERW 11, V DERW 21,
V DERW 12 and V DERW 22. See section 8.1 for definition of ZDV ER.
See section 8.1 for definition of functions fH1 to fH4 . The cubic Hermite
vertical interpolation gives the final interpolated value:
Xinterpo = 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 discretization

71

32-point 3D interpolation with vertical cubic 4-point interpolation
It is done in subroutine LAITRI or its shape-preserving version LAITQM.
For layers l and l + 3 (see figure ??) bilinear horizontal interpolations give
two interpolated values Xl and Xl+3 (see section 8.2). For layers l + 1 andl + 2 (see figure ??) 12-point horizontal interpolations give two interpolated
values Xl+1 and Xl+2 (see section 8.2). The final interpolated value Xinterpo
is a cubic 4 points vertical interpolation of Xl , Xl+1 , Xl+2 and Xl+3 (seesection 8.2).
There is a shape-preserving version LAITQM of routine LAITRI. In LAITQM
12 points horizontal interpolations for layers l + 1 and l + 2 are shapepreserving interpolations (see description of routine LAIDQM in section 8.2)
and vertical cubic 4-point interpolation is shape-preserving: Xinterpo is boundedby Xl+1 and Xl+2 . Use of switches LQMW (momentum equation), LQMT
(temperature equation), LQMQ (humidity equation), LQMV (passive scalar
equations), LQMP (continuity equation), allows to use shape-preserving interpolation routine LAITQM instead of LAITRI.
32-point 3D interpolation with vertical cubic Hermite interpolation
It is done in subroutine LAIHVT or its shape-preserving version LAIHVTQM.
For layers l and l + 3 (see figure ??) bilinear horizontal interpolations give
two interpolated values Xl and Xl+3 (see section 8.2). For layers l + 1 andl + 2 (see figure ??) 12-point horizontal interpolations give two interpolated
values Xl+1 and Xl+2 (see section 8.2). The final interpolated value Xinterpo
is a cubic Hermite vertical interpolation of Xl , Xl+1 , Xl+2 and Xl+3 (seesection 8.2).
There is a shape-preserving version LAIHVTQM of routine LAIHVT. In
LAIHVTQM 12-point horizontal interpolations for layers l + 1 and l + 2
are shape-preserving interpolations (see description of routine LAIDQM in
section 8.2) and vertical cubic Hermite interpolation is shape-preserving:
Xinterpo is bounded by Xl+1 and Xl+2 . Use of switch LQMV (passive scalar
equations), allows to use shape-preserving interpolation routine LAIHVTQM
instead of LAIHVT.
Other algorithms
The following interpolation methods are also available but are not describedhere:• diffusive bilinear interpolation (subroutine LAIDLIHD)• diffusive tri-linear interpolation (subroutine LAITLIHD)

72

4. Semi-lagrangian discretization• 48-point 3D interpolation with vertical cubic spline interpolation (subroutine LAITVSPCQM)• semi-lagrangian horizontal diffusion (subroutine LAITSLD)• horizontal 16-point linear least-square fit interpolation• 32-point with linear least-square fit horizontal and linear vertical interpolation (subroutine LAISMOO)• horizontal 12-point spline interpolation (subroutine LAIDSP)• 32-point 3D cubic spline interpolation• semi-lagrangian horizontal diffusion with cubic splines (subroutine LAITSLDSP)

inputfigslag

4. Semi-lagrangian discretization

73

9 Computation of η̇ on layers
This quantity is needed to find the height of the medium and origin points.
η̇ (resp. η̇l ) is the η-coordinate vertical velocity η̇ at the inter-layer ¯l (resp.
l̄

layer l), ∆pl is the pressure depth of layer l.
η̇ can be written:η̇ = η̇∂pwhere η̇
∂η


∂p
∂η



∂η
∂p

(76)



is computed at inter-layers (and stored in the array PEVEL)using a vertical integration of continuity equation, and

∂η
∂p



is computed

at layers.
Discretization of Equation (76) is:
1
η̇l =
2

"

∂p
η̇
∂η



∂p
+ η̇
∂η
l̄


#


l̄−1

ηl̄ − ηl̄−1
∆pl

(77)

10 Lateral boundary conditions
10.1 Extra longitudes
Let us denote by LX the number of longitudes (in the array NLOENG foreach latitude in 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 memory computations are done when making the halo (routine
SLCOMM or SLCOMM1+SLCOMM2A which exchange data with otherprocessors).

10.2 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 or
SLEXTPOL1A+SLEXTPOL2.

74

4. Semi-lagrangian discretization

10.3 Vertical boundary conditions
Vertical linear interpolations for layer variables at the medium
point: The medium point has a vertical coordinate always included between ηl=0 and ηl=L in case of vertical interpolating scheme. Therefore no
extrapolated values are needed.
Vertical cubic 4-point interpolations for layer variables at the origin
point: When the origin point is above the layer number 2 (resp. below thelayer number L − 1), the vertical cubic 4-point 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) are degenerated into linear interpolations between the layers
number 1 and 2 (resp. L − 1 and L). The extrapolated values at the extralayer 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 tothe case where the origin point is between the top (resp. surface) and thelayer number 1 (resp. L), but in this case the interpolation using data of thelayers number 1 and 2 (resp. L − 1 and L) becomes an extrapolation.
Vertical cubic 4-point interpolations for inter-layer variables at theorigin point: When the origin point is above the inter-layer number 1(resp. L−1), the vertical cubic 4-point interpolations using data of the interlayers number -1, 0, 1, 2 (resp. L − 2, L − 1, L and L + 1) are degenerated
into linear interpolations between the inter-layers numbers 0 and 1 (resp.
L − 1 and L).
Vertical cubic Hermite interpolations for layer variables at the origin point: When the origin point is above the layer number 2 (resp. L−1),
interpolation is still a vertical cubic Hermite one, computation of verticalderivatives is modified for layer number 1 (resp. L). This algorithm extends
to the case where the origin point is between the top (resp. ground) and thelayer number 1 (resp. L), but in this case the interpolation using data of thelayers number 1 and 2 (resp. L − 1 and L) becomes an extrapolation.
Vertical cubic spline interpolations for layer variables at the origin point: Some top and bottom values are computed and the verticalinterpolation always uses 4 points.

5
Semi-Implicit spectral
computations

1 General considerations
1.1 Purpose
This chapter describes the semi-implicit scheme used in Arpege-climat.
The Arpege-IFS code contains many other options. See Yessad (2007c) for
a comprehensive description of all available features.
It is necessary to treat implicitly the linear terms source of (fast moving)gravity waves to ensure a good numerical stability at reasonable time steps;
hence the solution of the equations involves the inversion of a linear system leading to a Helmholtz equation: inversion of such a system is moreconvenient to do in spectral space.

1.2 Advection scheme
In semi-Lagrangian form of equations, the time dependency equation of avariable X writes as:
dX
=A+F
dt

(1)

where A is the dynamical contribution and F the contribution of the physicalparametrizations. In a two-time level semi-Lagrangian scheme (abbreviatedinto 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.

76

5. Semi-Implicit spectral computations

1.3 Semi-implicit treatment of linear terms
Adding of a semi-implicit correction: The linear terms source of gravity waves must be treated implicitly, in order to allow time-steps compatible
with an efficient use of the model. Expression of the semi-implicit linear
terms is obtained assuming a definition of a reference state. Reference stateis defined by a dry resting isotherm atmosphere in hydrostatic balance, reference orography is zero. Equation (1) becomes, in 2TL SL scheme without
uncentering factor:dXββ= A + F + −βB t+0.5∆t + B t + B t+∆t
dt
2
2




(2)

where B is the linear term source of gravity waves, β is a tunable parameter. β = 0 corresponds to an explicit formulation, β = 1 to an implicit
formulation.
Discretization of Equation (2):Equation (2) gives the following discretized equation, where ∆t is the time
step (without uncentering factor):
X t+∆t − 0.5β∆tB t+∆t = X t + ∆t(A + F) − β∆tB t+0.5∆t + 0.5β∆tB t (3)where X t+∆t − 0.5β∆tB t+∆t is computed at the final grid point of the semiLagrangian trajectory, X t and 0.5β∆tB t are computed at the origin point of
the semi-Lagrangian trajectory, −β∆tB t+0.5∆t and Ẋ are computed either
at the medium point or as an average between the origin and final points ofthe trajectory.
If there is a first-order uncentering factor , averages along the semi-Lagrangian
trajectory will be weighted by (1 − ) at the origin point and (1 + ) at the
final point. In this case, ∆t is replaced by (1 − )∆t for terms at the origin
point, and ∆t is replaced by (1 + )∆t for terms at the final point. For moredetails see Chapter 4.
B t+0.5∆t , B t and B t−∆t are computed in grid point space. The right-hand side
member of Equation (3) is computed in grid point space, then transformed
into spectral space. Once in spectral space, a system of equations of thefollowing type must be solved:
X t+∆t − 0.5β∆tB t+∆t = X ∗(4)

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 3Dmodel.

5. Semi-Implicit spectral computations77

2 Prognostic variables and quantities involved in thesemi-implicit scheme
2.1 Prognostic variables~ , temperature
Prognostic variables are components of the horizontal wind V
T , humidity q and logarithm of surface pressure ln ps (for continuity equation).

2.2 Notations• M is the mapping factor. M is a reference mapping factor for semiimplicit computations. M = c (stretching factor) if semi-implicit
scheme with reduced divergence (LSIDG=.F.). M = M (mapping factor) if semi-implicit scheme with unreduced divergence (LSIDG=.T.).
The latter option is used when horizontal resolution is not uniform• a is the Earth mean radius~ is the horizontal geographical wind. Its zonal component is U . Its• V
meridional component is V0

• D is the unreduced divergence of horizontal wind, D is the reduced00divergence. D and D are linked by the relationship D = M 2 ∗ D
0• ζ is the unreduced vorticity of horizontal wind, ζ is the reduced vor00ticity. ζ and ζ are linked by the relationship ζ = M 2 ∗ ζ• T is the temperature. T ∗ is a vertically-constant reference temperature
which is used in the semi-implicit scheme. Default value is 300 K or
350 K according to configuration• q is the humidity• p is pressure, ps is surface pressure. p∗ is a reference pressure and p∗s is
a reference surface pressure, which are used in the semi-implicit scheme
and in some non-hydrostatic equations. These reference quantities arevertically dependent and horizontally (i.e. on η surfaces) constant.
Default value of p∗s is 1000 hP a for a 2TL SL scheme. ∆p∗ are layer
depths corresponding to a surface pressure equal to p∗s• pSTs is a reference pressure equal to the surface pressure of the standard
atmosphere (variable VP00). Default value is 101325 P a
dp
is the total temporal derivative of (vertical velocity in pressure
dt
coordinate)

• ω=

78

5. Semi-Implicit spectral computations• Φ is geopotential, Φs is the surface geopotential (i.e. the orography)~ is the Earth rotation angular velocity• Ω• ~r is the vector directed upwards, the length of which is the Earth radiusa• g is the gravity acceleration constant• R is the gas constant for air, Rd the gas constant for dry air and Rv
the gas constant for water vapor• cp is the specific heat at constant pressure for air and cpa is the specificheat at constant pressure for dry air• cv is the specific heat at constant volume for air and cva is the specificheat at constant volume for dry air→−→−0• ∇ is the unreduced first order horizontal gradient on η-surfaces. ∇is the reduced first order horizontal gradient. These two operators are→−→−0
linked by the relationship ∇ = M ∗ ∇• L: number of layers of the model• A, B define pressure on the η levels ( p = A + Bps , where ps is thehydrostatic surface pressure)• β coefficient for the semi-implicit scheme (between 0 and 1)• γ, τ , ν, µ are generic notations for semi-implicit linear operators (seesection 2.3)

2.3 Quantities used for vertical discretizations and linear operators
Operators α∗ and δ ∗ : these operators are used for discretizations of somevertical integrals.
αl∗ = 1 −δl∗ = ln

p∗l−1
∆p∗l
p∗l

ln

p∗l

!

∗
pl−1

!

p∗l−1

∗
∗
αl=1
= 1 , δl=1
= ln

p∗1
0.1

5. Semi-Implicit spectral computations

79

Linear operator γ: this operator is applied to temperature for semi-implicit
term in momentum equation.
LX(γZ)l = αl∗ Rd Zl +Rd Zk δk∗

k=l+1

Linear operator τ : this operator is applied to divergence for semi-implicit
term in temperature equation.
l−1
Rd T ∗ ∗δ∗ X(τ Z)l =αl Zl + l ∗∆p∗k Zk
cp d∆pl k=1"#

Linear operator ν: this operator is applied to divergence for semi-implicit
term in continuity equation.

(νZ) =

L
1 X∆p∗l Zlps ∗ l=1

Linear operator µ: this operator is applied to ln ps for semi-implicit termin momentum equation. If ln ps is the prognostic variable in continuity equation, definition of (µZ) is:(µZ) = Rd T ∗ Z(5)(µZ) is applied to ln ps .

3 Semi-implicit scheme with reduced divergence3.1 3D hydrostatic primitive equations model
Expression of the semi-implicit term B:• Continuity equation (X = ln ps ):2

B = −M νD

0

(6)
0

• Divergence equation (X = D ) :~ 0 2 (γT + µ ln ps )B = −∇(7)

80

5. Semi-Implicit spectral computations0• Vorticity equation (X = ζ ):
B=0

(8)

• Temperature equation (X = T ):2

B = −M τ D

0(9)• Humidity equation (X = q) and more generally GFL variables:
B=0

(10)

System to be solved:
ln ps t+∆t + β
0(11)∆t ~ 0 2
0∇ (γTt+∆t + µ ln ps t+∆t ) = D ∗2(12)∆t 2 0M τ Dt+∆t = T ∗2(13)

Dt+∆t + βTt+∆t + β∆t 2 0M νDt+∆t = P ∗20P ∗ , D ∗ , T ∗ correspond to X ∗ defined in Equation (4) and are available
in spectral arrays (SPSP, SPDIV, SPT) at the beginning of the spectral
computations. Equations (11) to (13) yield (Helmholtz equation):(1 − β 2∆t 2 2 ~ 0 2 0∆t ~ 0 2
0(γT ∗ + µP ∗ )M B ∇ )Dt+∆t = D ∗ − β ∇
4
2

(14)

where B = γτ + µν is a matrix operator L ∗ L (precomputed in routines
SUDYN, SUBMAT and stored in the array SIB).
Spectral computations to solve system of equations (11) to (13)Algorithm works zonal wave number by zonal wave number m (| m | varies
between 0 and the 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 ofEquation (14) is computed for all total wave numbers n between m
and Ns .
• Inversion of Helmholtz equation: method via a diagonalization in the
eigenmodes space (LSITRIC=.F. in NAMCT0).

5. Semi-Implicit spectral computations

81– First the diagonalization of B is used: B = Q−1 AQ, 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 thearray SIMI, Q−1 is stored in the array SIMO. Note that verticaloperators ν, µ, τ , γ, B, Q commute with the horizontal operator~ 02.
∇– Helmholtz equation (14) becomes, for each eigenmode l:∆t 2∆t ~ 0 200
2 ~ 02
al M ∇)QDt+∆t = Q(D ∗ − β ∇(γT ∗ + µP ∗ ))(15)
4
2
– For each eigenmode l and each zonal wave number m: (1 −∆t 2
2 ~ 02
al M ∇
β2) is a diagonal matrix operator (Ns + 1− | m |) ×
4
(Ns + 1− | m |): spectral coefficients of the right-hand side member of (15) are simply divided by the diagonal coefficients of thismatrix. Then the result is multiplied by Q−1 .

(1 − β 20• Once known Dt+∆t Equation (11) provides ps t+∆t and Equation (13)provides Tt+∆t .
• Semi-implicit scheme ends by a final memory transfer.

4 Semi-implicit scheme with unreduced divergence
4.1 Shortcomings of this formulation of the semi-implicit schemein case of stretching
In the grid points computations for some equations (for example temperature and continuity equation), the semi-implicit term B contains the reduced
2 0quantity (M D ). This quantity is added to geographical quantities. That
is no problem near the high resolution pole. This reduced quantity becomesvery large near the low resolution pole: if the stretching coefficient is c,(M /M )2 = c4 at the low resolution pole, which is equal to 81 if c = 3. Thus
the order of magnitude of the semi-implicit correction tendency becomes toohigh and physically absurd in the low resolution zone (gravity waves are
no longer treated implicitly). That leads to instabilities in regions of the
low resolution zone with high orography, in adiabatic Eulerian runs, or insemi-lagrangian runs with time-steps over the limit imposed by the CourantFriedrich-Levy condition. In Eulerian runs with physics, the combination ofphysics and small time-steps inhibits this instability. In order to avoid thisinstability, another formulation of the semi-implicit scheme can be used (controlled by LSIDG) which provides an implicit treatment of the gravity waves
everywhere on the sphere and not only near the high resolution pole. This

82

5. Semi-Implicit spectral computations

formulation is a formulation with unreduced divergence (simply by replacing
the quantity M by the mapping factor M ).

4.2 3D hydrostatic primitive equations model
Expression of the semi-implicit term B:
As in section 3. Just replace M by M in Equations (6) to (10).
System to be solved:
As in section 3. Just replace M by M in Equations (11) to (14).
Spectral computations to solve system of equations (11) to (13).
Algorithm works zonal wave number by zonal wave number m (| m | varies
between 0 and the truncation Ns ). Algorithm is performed in the routine
SPCSI. For a given zonal wave number m:• After a preliminary memory transfer the right-hand side member of
equation (14) is computed for all total wave numbers n between m
and Ns .
• Inversion of Helmholtz equation is more complicated than in the caseof semi-implicit scheme with reduced divergence because the left-handside member of Helmholtz equation contain values of the divergence forall levels and five total wave numbers (n − 2 to n + 2). Of course M 2 isa symmetric penta-diagonal matrix, for a given zonal wave number m.
Pay attention to the fact that M 2 does not commute with the diagonal~ 02.
operator ∇– First the diagonalization of B is used: B = Q−1 AQ, 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 thearray SIMI, Q−1 is stored in the array SIMO. Note that verticaloperators ν, µ, τ , γ, B, Q commute with the horizontal operators~ 0 2 and M 2 .
∇– Helmholtz Equation (14) becomes, for each eigenmode l:(1 − β 2∆t 2 ~ 0 2 2∆t ~ 0 2
00al ∇ M )QDt+∆t = Q(D ∗ − β ∇(γT ∗ + µP ∗ ))
4
2

(16)

~ 0 −2 −– For each eigenmode l and each zonal wave number m: (∇∆t 2
β2
al M 2 ) is a symmetric penta-diagonal matrix operator of4
size:(Ns + 1− | m |) × (Ns + 1− | m |).
The factorization LU of this matrix is computed, where L is a

5. Semi-Implicit spectral computations

83

lower triangular tri-diagonal 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.
– For the zonal wave number m = 0 Equation (16) is not multi~ 0 −2 because, for the total wave number n = 0, ∇~ 0 2 isplied by ∇~ 0 −2 is equivalent to aequivalent to a multiplication by 0 and ∇division by 0. The only difference is that the penta-diagonal but∆t 2 ~ 0 2 2
non-symmetric operator (1−β 2al ∇ M ) is factorized and in4
verted. All useful coefficients of L, U are computed in the set-up
routine SUHEG and stored in the arrays SIHEG and SIHEG2.
0• Once known Dt+∆t equation (11) provides ln pst+∆t and Equation (13)provides Tt+∆t . Spectral multiplications by M 2 are performed by the
product of a symmetric penta-diagonal matrix of size:(Ns + 1− | m |) × (Ns + 1− | m |)(useful coefficients computed in routine SUSMAP and stored in thearray SCGMAP) by a vector containing spectral coefficients (m, n) forn varying from | m | to Ns• Semi-implicit scheme ends by a final memory transfer.

5 Spectral multiplications by polynomial expressionsof the mapping factor
Expression of mapping factor M in spectral space.
Let us denote by:1
1
ac = (c + )2
c1
1
bc = (c − )2ce(0,0) = 0se(m,n) =n2 − m2
4n2 − 1

Expression of M is:
M = ac + bc ξ

(17)

84

5. Semi-Implicit spectral computations

where ξ is the sine 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)(18)

It is easy from (18) to retrieve the coefficients of spectral multiplication by
any first degree polynomial of M . This is equivalent to a multiplication bya tri-diagonal symmetric matrix in spectral space.
Expression of M 2 in spectral space[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)(19)This is equivalent to a multiplication by a penta-diagonal symmetric matrix
in spectral space.

6
Horizontal diffusion andnudging

1 Introduction
In the code of Arpege-IFS horizontal diffusion computations are spectral
computations. The main horizontal diffusion scheme acts in spectral space.
It is implicit in order to remain stable even with high diffusion coefficientsand is called after the semi-implicit scheme.
Details on options of this scheme not used in Arpege-climat, on the otherhorizontal schemes (enhanced horizontal diffusion scheme, semi-lagrangian
horizontal diffusion) may be found in Yessad (2007d).

2 Formulation of the horizontal diffusion
2.1 General considerations
In the spectral space the horizontal diffusion formulation is close to:∂X~ 0r X= −KX M ∇∂t(1)where KX is a vertically dependent and horizontally constant coefficient.
π
KX is generally complex: exp(i r) multiplied by a real positive coefficient.
2
The power of the scheme is r. M is the mapping factor.
sπ
KX = exp(−i r) 2

−r

Ns (Ns + 1) 
a2

ΩhX g

(2)

86

6. Horizontal diffusion and nudging

a is the mean Earth radius, Ns is the truncation (maximum total wavenumber), g is a vertical profile, Ω is the angular velocity of the Earth rotation(7.29 10−5 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 (hSV ), and one for surface pressure (hSP ). For
divergence, expression of hD matches:
12πa (1 + 0.5RLG)2.5= rdampD
ΩhD
N LO
DIF

(3)

RLG is the increment from linear to quadratic grid (RNLGINC), RLG = 0
for a linear grid, 1 for a quadratic grid); DIF is a tuning parameter; N LO is
the maximum number of longitudes of the computational sphere (NDLON),
2πa/N LO is the grid size before stretching.
For the other 3D upper air fields:hX =rdampDhD
rdampX(4)

The basic control of horizontal diffusion in the namelist is done by RRDXTAU (DIF ), the different RDAMPx (rdampX ), and REXPDH (r).

3 Discretization of the horizontal diffusion
3.1 Unstretched model
The discretized equation which is coded is:+−+= −ΩhX g(l)f (n, N, n0 (X), x0 , r)∆t X(m,n)
X(m,n)− X(m,n)(5)

where the superscripts + and − indicate respectively variables after horizontal diffusion and variables before horizontal diffusion, ∆t is the time step, n
is the total wave number (between 0 and the truncation Ns ), m is the zonal
wave number (between −n and n in a triangular truncation), r is the order ofthe horizontal diffusion operator, N is a reference wave number, n0 (X) is a
threshold depending on variable X (generally zero except for vorticity where~ 0 is the first order horizontal
it is 2), x0 is a threshold between 0 and 1. ∇~ = M∇~ 0 , where
reduced derivative operator (i.e in the grid point space, ∇M is the mapping factor). hX is a coefficient depending on the variable X.
The vertical dependency is expressed by g(l).

6. Horizontal diffusion and nudging

87

This horizontal diffusion scheme is an implicit horizontal diffusion scheme (in+−the right-hand side member of the equation there is X(m,n)
and not X(m,n)).
Equation (5) becomes (6):+X(m,n)=

1X−
1 + ΩhX g(l)f (n, Ns , n0 (X), x0 , r)∆t (m,n)(6)

Expression of f (n, N, n0 (X), x0 , r) is currently:
x − x0f (n, N, n0 (X), x0 , r) = max 0, min 1,
1 − x0






r 

(7)

with:
max(0, n(n + 1) − n0 )max(0, N (N + 1) − n0 )



x=

1

2

An exact discretization of equation (1) would give:f (n, N, n0 (X), x0 , r) = f (n, Ns , 0, 0, r)(8)

For layer number l, expression of g(l) is:g(l) =y0STmin(y0 , max(y3 , pSTl /ps ))

(9)

where pST
is the standard atmosphere pressure for the layer number l,lST
ps is a reference pressure (sea level pressure for the standard atmosphere:
1013.25 hP a). The dimensionless parameter y0 is between 0 and 1: g(l) = 1
STST
ST
if pSTl /ps is above y0 and g(l) > 1 if pl /ps is below y0 . The dimensionless parameter y3 is between 0 and y0 . In practical it is much less than y0
and is used to avoid too high diffusion in the upper stratosphere.

3.2 Stretched model
Approximation of the mapping factor
Expression of mapping factor in spectral space
Let us denote by:
1
1
• a=
c+
2
c




88

6. Horizontal diffusion and nudging
1
1
c−
• b=
2
c




• (0,0) = 0s• (m,n) =n2 − m2
4n2 − 1

Expression of M is:
M = a + bµ(10)where µ is the sine of computational sphere latitude. Expression of [M X](m,n)
is:[M X](m,n) = b(m,n) X(m,n−1) + aX(m,n) + b(m,n+1) X(m,n+1)(11)

A multiplication by M in spectral space is equivalent to a multiplication bya symmetrical tri-diagonal matrix.
Discretization+−+ ΩhX g(l)∆t= X(m,n)X(m,n)h+b(m,n) f (n − 1, N, n0 (X), x0 , r)X(m,n−1)++af (n, N, n0 (X), x0 , u)X(m,n)++b(m,n+1) f (n + 1, N, n0 (X), x0 , r)X(m,n+1)iSolving this equation is equivalent to invert a tri-diagonal matrix for eachzonal wave number m. The computation of this matrix and a decomposition
into a product of two triangular bi-diagonal matrices are performed in the
set-up routine SUHDU. At each time step these two triangular bi-diagonal+
matrices are inverted in order to compute X(m,n).

3.3 Second reference truncation N2 in the stratosphere
It is possible in the code to modify g(l) in order to simulate a replacement off (n, Ns , n0 (X), x0 , r) by f (n, N2 , n0 (X), x0 , r), where N2 < Ns , for standard0
pressure pST< y1 ∗ pST
s , where y1 < y0 . In this case g(l) (and KX ) alsolSTST
depends on n for standard pressures below y0 ∗ pST
s . If pl /ps is below y1 :g(l, n) =y0f (n, N2 , n0 (X), x0 , r)STST
min(y0 , (max(y3 , pl /ps )) f (n, Ns , n0 (X), x0 , r)

(12)

6. Horizontal diffusion and nudging

89

ST
If pSTl /ps is over y1 and below y0 :g(l, n) =y0 ∗ pSTs − b(n)pST− b(n)
l(13)

where:
b(n) =

ST(y0 ∗ pSTs )(y1 ∗ ps )(f (n, N2 , n0 (X), x0 , r) − f (n, Ns , n0 (X), x0 , r))ST(y0 ∗ pSTs )f (n, N2 , n0 (X), x0 , r) − (y1 ∗ ps )f (n, Ns , n0 (X), x0 , r)

One can check that (13) yields g(l, n) = 1 for pST= y0 ∗ pST
and that (12)
s
l
ST
ST
and (13) are continuous for pl = y1 ∗ ps . When taking N2 = Ns , one
retrieves formula (9). Choice of N2 < Ns is useful in simulations with highvertical resolution in the stratosphere.

3.4 Discretized equations
GMV and GFL variables0• reduced vorticity ζ (n0 (ζ) = 2)0

• reduced divergence D (n0 (D) = 0)
• specific humidity q (n0 (q) = 0)
• other spectral GFL variables (e.g. ozone) (n0 = 0)
For grid point GFL variables (e.g. liquid water) and non-advectable GFL
variables, no horizontal diffusion is applied

Temperature
The role of horizontal diffusion for temperature is not to reduce the horizontal
gradient between mountains and valleys. With a terrain-following verticalcoordinate like η, we cannot apply the scheme directly to T . For this reason,
T − α ln ps is diffused instead of T , where α is a coefficient depending on thealtitude and using parameters related to standard atmosphere. Let us denoteby TsST the surface temperature in a standard atmosphere (288.15 K), R thedT
the tropospheric
air constant, g the acceleration due to gravity,dz ST
temperature vertical gradient in a standard atmosphere (−0.0065 K m−1 ),ST the tropopause temperature in a standard atmosphere (217.15 K).
Ttro

90

6. Horizontal diffusion and nudging

If:

TsST

pST
pST
s


 
! R dT

g

dz

ST



>

ST
Ttro

the expression of αl for layer number l is:

R dT
g dz


αl = −Bl


ST

TsST

pST
pST
s


 
!− R dT

g

dz



+1

ST

(14)

where Bl is used in the definition of pressure on layers and inter-layers (hybrid
vertical coordinate).
In the other cases:
αl = 0(15)

Thus temperature horizontal diffusion reads:+− α (ln ps )−T(m,n)(m,n) =−− α (ln ps )−(T(m,n)(m,n) )

1 + ΩhT g(l)f (n, N, n0 (T ), x0 , r)∆t

(16)which yields:+=
T(m,n)−+ αΩhT g(l)f (n, N, n0 (T ), x0 , r)∆t (ln ps )−(T(m,n)(m,n) )

1 + ΩhT g(l)f (n, N, n0 (T ), x0 , r)∆t

(17)This scheme is fully implicit only when hSP = 0, which is the default. For
temperature n0 (T ) = 0.

4 NudgingThis scheme called just after horizontal diffusion (for spectral variables) is alinear relaxation of prognostic variables towards pre-defined fields and is notproperly saying a diffusion scheme. The generic equation is:∂X1= − (X − Xref )∂t
τXPre-defined fields Xref (for example reanalyzed fields) can be read on a file
every NFRHIS time steps (typically at 6 h frequency) and interpolated at

6. Horizontal diffusion and nudging

91

each time step linearly. A cubic spline for time interpolation is also possible,
but it requires more IO and more memory for a tiny impact. Nudging coefficients 1/τX can be different for each variable. A vertical profile (common
to each variable) can be applied to the coefficients.
Nudging can be applied to most prognostic variables in spectral space or in
grid point space. When applied to spectral variables, it can be restricted towave numbers less than a critical value. This allows to drive the large-scalefeatures and let the model adjust the small scales freely. When applied togrid-point variables (the same as spectral ones, except that velocity potential and stream function are replaced by u- and v- components) there is apossibility of applying a 2D mask which modulates horizontally the strengthof the relaxation. One can thus mimic the behavior of a limited area modelor impose the atmosphere variables in a particular region of the globe. As
for the vertical or the spectral modulation, this horizontal modulation is thesame for all variables.
There is also nudging for some grid-point variables in grid-point space, suchas surface temperature, surface moisture, deep moisture and snow depth. In
the case of surface temperature, when the nudging coefficient is not zero,
it is considered as infinite over sea (i.e. model values are prescribed by thedriving field). This allows to replace an obsolete feature of Arpege-climat
which updated some fields at daily frequency from two monthly boundary
condition files (subroutine UPDCLIDM, switch LMCC01). See Chapter 7
for information about boundary conditions. The grid-point nudging is notcalled in the spectral subroutine SPCHOR, but in the grid-point subroutine
MFPHYS.
Another interesting feature of nudging (spectral or grid-point variables) isthe behavior in case of negative nudging coefficient. In this case, the model
is no longer relaxed toward the driving field, but the driving field is justadded to the time derivative:∂X= Xref + · · ·∂tThis allows to introduce flux correction for some variables. Nudging andcorrection can be easily combined:1∂X= − (X − Xref ) + C∂tτX
can be rewritten∂X1= − (X − (Xref − CτX ))
∂t
τX

92

6. Horizontal diffusion and nudging

Each variable can be treated in nudging or correction mode separately, sinceeach variable has its own relaxation coefficient (one for spectral and one forgrid point in the case of atmospheric fields).
In any case, the time discretization is implicit for numerical stability. The
control is made in NAMNUD; each coefficient is expressed as an inverse of
the time step, e.g. a value of one means that the time relaxation is one
time step and a value of zero (the default value) means that no relaxation
is applied to the variable. The driving data are found in Arpege files withname RX$DATE with DATE expressed as YYYYMMDDHH.

7
Boundary conditions

1 Introduction
To carry out an integration of the atmosphere equations, the model needs
to know the values of a certain number of variables (named historical or
prognostic) at time t = 0 (initial conditions), but also of a certain numberof variables for which the evolution is not calculated (boundary conditions).
At the end of an integration, the model provides the values of the historicalvariables at time t = T , which are used as initial conditions for the next
integration.
In the current version of Arpege-climat, the initial conditions are now
split into 2 files : one file for the atmosphere variables and one file for thesurface variables. The boundary conditions are read on the same file as theinitial conditions. For the atmosphere variables, this work is carried out
by subroutines SUSPECA (fields in spherical harmonics like orography) andSUGRIDADM (fields on the collocation grid) at the beginning of any integration. These variables come from monthly climatological files. For thesurface variables, the subroutine AROINI_SURF (fields on the collocationgrid) is called at the beginning of any integration. These variables (physiography and orography fields) come from the ECOCLIMAP, orography, sandand clay fractions files.
Those climatological files as well as the physiographic fields need to be calculated for any geometry (truncation, rotation and stretching, vertical discretization).
The files of boundary conditions are obtained first by external scripts, followed by run of specific binaries in charge of aerosols and ozone. In themodel script, the binaries UPDCLI and UPDOZO updates the atmospheric94

7. Boundary conditions

restart file with monthly (and possibly yearly varying) fields taken from theclimatological files.
The fields of sea surface temperature, sea-ice extent and sea-ice albedo areread in the course of integration (in principle once per day) from the data
produced by the ocean model and prepared by the coupler Oasis in subroutine UPDCPL .

2 Surface boundary conditions
The surface physiographic fields The data come from :- the ECOCLIMAP-II data base ( see Surfex - Scientific Documentation II LAND USE: ECOCLIMAP ) which is a global database of land surface
parameters (land covers) at 1-km resolution. It is intented to be used toinitialize the soil-vegetation-atmosphere transfer schemes (SVATs) in meteorological and climate models.
- the orography file GTOPO30 (source : U.S Geological Survey) : resolutionof 30 ” XXX A REVOIRXXXX- the HSWD database for the clay fraction and the sand fraction of thenear-surface soil : resolution of 5’
The physiographic fields are averaged or interpolated on the specified grid by
the program PGD provided in the Arpege-climatsource code. The fields
are stored in a file, called PGD file, but only with the physiographic 2D
fields, the geographic and grid data written in it. Therefore, the PGD filecontains the spatial characteristics of the surface and all the physiographic
data necessary to run the interactive surface schemes for vegetation andtown.
Output data
1. land cover
2. orography parameters (averaged, silhouette, maximum, mininum)1
3. subgrid-scale orography parameters (slope , anisotropy, standard deviation, direction of sso ...)
4. sub-grid surface runoff slope
5. continuous drainage
6. topographic index statistics
1

During Arpege-climatrun, the model is forced to use the orography coming fromthe atmospheric boundary conditions and not from the surface boundary conditions.

7. Boundary conditions

95

7. bathymetry
8. clay fraction
9. sand fraction
10. percentage of sand
11. percentage of clay
The surface prognostic fields. The surface init file will contain the prognostic fields. The PREP binary (source code provided within Arpegeclimat) performs the initialization of the surface scheme prognostic variables, as temperature profiles, water and ice soil contents, interception reservoirs, snow reservoirs.
Output data
1. surface temperature
2. deep soil temperature
3. third layer temperature
4. sea surface temperature
5. water temperature
6. surface liquid volumetric water content
7. root liquid volumetric water content
8. deep liquid volumetric water content
9. surface frozen volumetric water content
10. root frozen volumetric water content
11. third layer frozen volumetric water content
12. liquid water retained by the foliage
13. 1st layer snow water equivalent
14. 1st layer snow water density
15. glacier ice storage reservoir
16. snow albedo
17. aerodynamical resistance96

7. Boundary conditions

18. leaf area index
19. vegetation fraction
20. surface roughness length witout snow
21. fraction of each patch (12 ISBA vegetation types/patches)
22. soil layer thicknesses
23. sea/nature/water/town fractions (the 4 Surfex tiles)
24. orography roughness length
25. roughness length over the ocean
26. water surface roughness length
27. rainfall rate
28. snowfall rate

3 Additional boundary forcings
3.1 Ozone
In former versions, Arpege-climat could use the variable ozone as a passive
tracer. To parametrize the photochemical sources and sinks, Cariolle andDéqué (1986) introduced a linear parametrization of the form:∂rO3
2= A2 + (A3 + A6 RCl)(rO3 − A1 ) + A5 (T − A4 ) + A7 (ΣrO3 − ΣA1 )∂twhere rO3 is the ozone mass mixing ratio, ΣrO3 the vertically integrated
ozone between the current level and the top of the atmosphere, T the temperature, RCl a chlorine index, and Ai coefficients which depend on latitude,pressure and month.

3.2 Aerosols
There are 6 types of aerosols: continental (organic), maritime (salt), urban(soot), desert (dust), sulfate and stratospheric. The default horizontal distribution of the first 5 is proposed by Tegen et al. (1997). The horizontal
resolution is very coarse, but the data are used on a grid of 72 longitude by
46 latitudes. The horizontal interpolation is performed by basic barycentricinterpolation (no masking) in prepaero.sh script. The stratospheric aerosols

7. Boundary conditions

97

are uniformly distributed on the sphere and does not need interpolation. The
global effect of volcanic eruption can be taken into account by changing eachyear the solar constant.
The horizontal distribution of the anthropogenic sulfate aerosols may also
be obtained from Boucher and Pham (2002) on a grid of 96 longitudes by 73
latitudes with decadal variation and anthropogenic scenarios. A fraction of
the sulfate of each decade divided by the sulfate of decade 1990 is interpolatedon the target grid. This ratio is multiplied by the Tegen sulfate distribution
already interpolated on the target grid.
For future use, an additional field with volcano aerosols containing uniformly
10.5 10−3 is written.

4 Coupling with an ocean model
Arpege-climat can be used in coupled mode with an ocean model. In thiscase sea surface temperature, sea-ice mask and the attached variables (surface albedo . . . ) are no longer provided externally at the beginning of eachmonth, but at a proper frequency (NFRCPL) during the course of a monthly
integration. Because of the coupling, there is a feedback between the surface fluxes calculated by Arpege-climat and the sea surface temperature
imposed to it.
In the present version Arpege-climat is coupled with NemoV2-3 (from
CNRS/IPSL) as the ocean model through OasisV3 (from Cerfacs) as thecoupler with the switch LMCC03.
At each coupling time step, the atmosphere and ocean models exchange fields
with the help of the coupler in charge of the communication between themodels and the interpolation between the two different grids. The exchanged
fields are averaged over the coupling time period preceding each couplingtime step. The atmospheric model sends wind stress fluxes, surface totalenergy flux, surface net solar flux, surface net hydrological flux and run-off
flux; while the ocean model sends the sea surface temperature, the ice cover,the sea surface albedo and the surface currents (to be used in the wind stresscomputation). In the present version, no sea-ice model is taken into accountand a monthly climatology is used by Oasis to mimic the presence of a seaice model. Through the switch LMCC02, Arpege-climat treats sea-ice asland and balances the surface temperature with the surface heat flux (seeChapter 14)

98

7. Boundary conditions

4.1 Ocean model
The Nemo ocean model web site is
http://www.locean-ipsl.upmc.fr/NEMO/
and its reference publication is Madec (2008).
The ocean model Nemo is a primitive equation model adapted to regionaland global ocean circulation problems. Prognostic variables are the threedimensional velocity field, a linear or nonlinear sea surface height, the temperature and the salinity. In the horizontal direction, the model uses acurvilinear orthogonal grid and in the vertical direction, a full or partial
step z-coordinate, or s-coordinate, or a mixture of the two. The distributionof variables is a three-dimensional Arakawa C-type grid. Various physicalchoices are available to describe ocean physics, including TKE and KPP
vertical physics.
In more details, Nemo presents the following characteristics:
The model is discretized on a staggered grid (Arakawa C grid) with maskingof land areas and uses a leap-frog environment for time-stepping. Vertical
discretization used depends on both how the bottom topography is represented and whether the free surface is linear or not. Full step or partial
step z-coordinate or s- (terrain-following) coordinate is used with linear freesurface (level position are then fixed in time). In nonlinear free surface, thecorresponding rescaled height coordinate formulation (z* or s*) is used (thelevel position then vary in time as a function of the sea surface height). Explicit, split-explicit and implicit free surface formulations are implemented
as well as rigid-lid case. A number of numerical schemes are available formomentum advection, for the computation of the pressure gradients, as well
as for the advection of tracers (second or higher order advection schemes,
including positive ones).
The model allows penetration of solar radiation There is an optional geothermal heating at the ocean bottom.
Global configurations of the model make use of the ORCA tripolar grid,with special north fold boundary condition. Free-slip or no-slip boundary
conditions are allowed at land boundaries.
The model includes an implicit treatment of vertical viscosity and diffusivity.
The lateral laplacian and bi-harmonic viscosity and diffusion can be rotatedfollowing a geopotential or neutral direction. There is an optional eddy induced velocity (Gent and Mc Williams 1990) with a space and time variable
coefficient Tréguier et al. (1997). The model has vertical harmonic viscosityand diffusion with a space and time variable coefficient, with options to compute the coefficients with Blanke and Delecluse (1993), Large et al. (1994),
or Pacanowski and Philander (1981) mixing schemes7. Boundary conditions

99

Specific on-line diagnostics are available in the model : output of all the tendencies of the momentum and tracers equations, output of tracers tendencies
averaged over the time evolving mixed layer.
The model is implemented in Fortran 90, with preprocessing (C preprocessor). It is optimized for vector computers and parallelized by domain
decomposition with MPI. All input and output is done in NetCDF with a
optional direct access format for output.

4.2 Coupler
The Oasis coupler web site is
http://www.cerfacs.fr/globc/software/oasis/and reference publication is Valcke (2006).
Oasis 3 is the direct evolution of the Oasis coupler developed since morethan 10 years at Cerfacs. Oasis 3 is a portable set of Fortran 77, Fortran 90
and C routines. At run-time, Oasis 3 acts as a separate mono process executable, which main function is to interpolate the coupling fields exchanged
between the component models, and as a library linked to the component
models, the Oasis 3 PRISM Model Interface Library (Oasis 3 PSMILe).
Oasis 3 supports 2D coupling fields only. To communicate with Oasis 3,
directly with another model, or to perform I/O actions, a component modelneeds to include few specific PSMILe calls. Oasis 3 PSMILe supports inparticular parallel communication between a parallel component model andOasis 3 main process based on Message Passing Interface (MPI) and fileI/O using the mpp-io library from GFDL.
Different transformations and 2D interpolations are available in Oasis 3 toadapt the coupling fields from a source model grid to a target model grid.
They are divided into five general classes that have precedence one over
the other in the following order: time transformation (with PSMILe only),
preprocessing, interpolation, adjustments, and postprocessing. This order ofprecedence is conceptually logical, but is also constrained by the Oasis 3
software internal structure.
Oasis 3 notably includes the interpolation techniques offered by Los Alamos
National Laboratory SCRIP 1.4 library. These interpolation techniques
are the following: distance weighted nearest neighbor interpolation, nearest
neighbor interpolation weighted by their distance and a Gaussian function,bilinear and bi-cubic interpolation, 1st or 2nd order conservative remapping,which means that the weight of a source cell is proportional to area intersected by target cell.

100

7. Boundary conditions

8
Radiation

1 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 basedon the IFS Documentation-Cy31r1 and on the Meso-NH documentation:
http://mesonh.aero.obs-mip.fr/mesonh/
The detailed original ECMWF scientific documentation is available onlineat
http://www.ecmwf.int/research/ifsdoc/CY31r1/
The package calculates the radiative fluxes taking into account absorptionemission of longwave radiation and reflection, scattering and absorption ofsolar 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.
The radiative heating rate is computed as the divergence of net radiation
fluxes F so that
(

∂T
g ∂F
)rad = −
∂t
cp ∂p

(1)

where F is a net flux: i.e. F = F↑ + F↓ sum of the upward F↑ and downwardF↓ fluxes, and a total flux: i.e. F = FLW +FSW sum of the solar or shortwave

102

8. Radiation

FSW and atmospheric or longwave FLW fluxes and cp is the specific heat atconstant pressure of moist air.
Sections 2 and 3 describe the computation of the shortwave and longwave
radiative fluxes respectively. The solution of the radiative transfer equation
to obtain the fluxes is unfortunately very expensive, and we cannot afford to
do it more than every 3 hours. A description of the inputs, in particular the
climatologically defined quantities of radiative importance is given in Section
4.

2 Shortwave radiation
2.1 First glance
The rate of atmospheric warming by absorption and scattering of shortwave
radiation is:
∂T
g ∂F
=
∂t
cp ∂p

(2)

where F = FSW is the net total shortwave flux, expressed in W m−2 andpositive when downward.
Z ∞

Z +1

Z 2π

dφ

dν

F =

0

0

−1

µ Lν (δ, µ, φ) dµ dφ(3)Lν is the diffuse radiance at wavenumber ν, in a direction given by φ, theazimuth angle and ϑ the zenith angle such as µ = cos ϑ. In (3), we assume aplane parallel atmosphere with the optical depth δ, as a convenient verticalcoordinate when the energy source is outside the medium
Z 0δ(p) =βν (p) dp(4)p

where βνext (p) is the extinction coefficient equal to the sum of the scattering
coefficient βνsca of the aerosol and cloud particle absorption coefficient βνabs
and of the purely molecular absorption coefficient kν . The diffuse radianceLν is governed by the radiation transfer equation

8. Radiationµ103dLν (δ, µ, φ)dδ= Lν (δ, µ, φ) −−ω ν (δ)4ω ν (δ)− δPν (δ, µ, φ, µ0 , φ0 ) Eν0 e µ04Z 2πZ +1
0

−1

(5)

0
Pν (δ, µ, φ, µ0 , φ0 ) Lν (δ, µ0 , φ0 ) dµ0 dφ(6).

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, µ0 , φ0 ) is the scattering
phase function which defines the probability that radiation coming fromdirection (µ0 , φ0 ) is scattered in direction (µ, φ). The shortwave part of thescheme, originally developed by Fouquart and Bonnel (1980) , solves theradiation transfer equation and integrates the fluxes over the whole shortwave
spectrum between 0.2 and 4 µm. Upward and downward fluxes are obtained
from the reflectances and transmittances of the layers, and the photon-pathdistribution method allows to separate the parametrization of the scattering
processes from that of the molecular absorption.

2.2 Spectral integration
Solar radiation is attenuated by absorbing gases, mainly water vapor, uniformly mixed gases (oxygen, carbon dioxide, methane, nitrous oxide) andozone, and scattered by molecules (Rayleigh scattering), aerosols and cloudparticles. Since scattering and molecular absorption occur simultaneously,
the exact amount of absorber along the photon path length is unknown, andband 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 amount between U and U + dU . With this
distribution, the radiative flux at wavenumber ν is related to Fc by
Z ∞Fν = Fc

p(U ) exp(−kν U ) dU

(7)0

and the flux averaged over the spectral interval ∆ν can then be calculatedwith the help of any band model of the transmission function t∆ν
1
F =
∆ν

Z ∞

Z

Fν dν = Fc
∆νp(U ) t∆ν (U ) dν.

(8)0

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

104

8. Radiation

giving a set of simulated fluxes Fkl . An inverse Laplace transform is thenperformed on (7) to get p(U ) (Fouquart 1974 ). The main advantage of themethod 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:

Z ∞F = Fc t∆ν (< U >)wherep(U ) U dU (9)< U >=
0

1

F = Fc t∆ν (< U 2 >)

where

1

Z ∞

1

p(U ) U 2 dU
(10)

< U 2 >=
0respectively. The atmospheric absorption in the water vapor bands is generally strong and the scheme determines an effective absorber amount Ue
1between < 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 −1
Utot) ln(t∆ν
)
µ0
µ0

(12)

with Utot the total amount of absorber in a vertical column and µ0 = cos ϑ0 .
Once the effective absorber amounts of H2 O and uniformly mixed gases arefound, the transmission functions are computed using Padé approximants:NPt∆ν (U ) = i=0NPai U i−1.

(13)

bj U j−1

j=0

Absorption by ozone is also taken into account, but since ozone is located at
low pressure levels for which molecular scattering is small and Mie scattering
is negligible, interactions between scattering processes and ozone absorption

8. Radiation

105

are neglected. Transmission through ozone is computed using (13) where theamount of ozone UO3 is:

UOd 3 = M

Z 0
p

dUO3

for the downward transmission of the direct solar beam, and:UOu 3 = rZ p
ps

dUO3 + UOd 3 (ps )for the upward transmission of the diffuse radiation with r = 1.66 the diffusivity factor and M , the magnification factor (Rodgers 1967 ) used insteadof µ0 to account for the sphericity of the atmosphere at very small solar
elevations:

M=q

35

.

(14)

µ20 + 1

To perform the spectral integration, it is convenient to discretize the solar
spectrum into subintervals in which the surface reflectance, molecular absorption characteristics, and cloud optical properties can be considered as
constants. One of the main causes for such a spectral variation is the sharp
increase in the reflectivity of the vegetation in the near-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 spectral intervals, one for the visible (0.2/0.69 µm), one for the near-infrared (0.69/4.00µm) parts of the solar spectrum. From June 2000 to April 2002, the nearinfrared interval was sub-divided into three intervals (0.69/1.19/2.38/4.00µm) 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 the recomputation of all the molecular absorption coefficients froman updated version of the shortwave lineby- line model of Dubuisson et al.
(1996) using spectroscopic data from HAWKS (2000):
http://www.hitran.com

106

8. Radiation

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 ofsix spectral intervals over which the aerosol and cloud optical properties arealso defined. The cut-off at 0.69 µm allows the scheme to be more computational efficient, in as much as the interactions between gaseous absorption(by water vapour and uniformly mixed gases) and scattering processes areaccounted for only in the near-infrared interval(s).

2.3 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 thescheme of Geleyn and Hollingsworth (1979) , the fluxes are not obtained
through the solution of a system of linear equations in a matrix form. Rather,assuming an atmosphere divided into N homogeneous layers, the upward anddownward fluxes at a given interface j are given by:F↓ (j) = F0

NY

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 transmittanceat the bottom of the j th layer. Computations of Rt ’s start at the surface andwork upward, whereas those of Tb ’s start at the top of the atmosphere andwork downward. Rt and Tb account for the presence of cloud in the layer:

Rt = Cj Rcdy + (1 − Cj ) Rclr ,(18)

Tb = Cj Tcdy + (1 − Cj ) Tclr .

(19)

8. Radiation

107

The subscripts clr and cdy respectively refer to the clear-sky and cloudy fractions of the layer 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 the cloudy fraction of the layer calculated with the Delta-Eddington
approximation. Given δc , δa and δg , the optical thicknesses for the cloud andthe aerosol, and the molecular absorption of the gases, gc and ga the cloud
and aerosol asymmetry factors, Rtcdy and Tbcdy are calculated as functionsof:• the total optical thickness of the layer δ ∗ :δ ∗ = δc + δa + δg
• the total single scattering albedo:ω∗ =δc + δaδc + δa + δg(20)

• the total asymmetry factor:
g∗ =

δc
δa
gc +
ga
δc + δaδc + δa(21)of the reflectance R_ of the underlying medium (surface or layers below thej th interface) and of the effective solar zenith angle µe (j) which accounts forthe decrease of the direct solar beam and the corresponding increase of the
diffuse part of the downward radiation by the upper scattering layers.
The effective solar zenith angle µe (j) is equal to:ef f−1(1 − Ccld(j))
ef fµe (j) =+ r Ccld
(j)
,
µ





(22)

ef f
with Ccld(j) the effective total cloudiness over level j and r the diffusivity
factor.

108

8. Radiation
N
Y

ef f
Ccld
(j) = 1 −(1 − Ccld (i)E(i))

i=j+1

and


E(i) = 1 − exp −(1 − ωc (i)gc (i)2 )δc (i)
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 accountmore 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, theradiance 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 associatedLegendre 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 integralin (5) thus becomes
Z 2πZ +10

−1

P (µ, φ, µ0 , φ0 ) L(µ0 , φ0 ) dµ0 dφ0 = 4π (L0 + πL1 )where g, the asymmetry factor identifies as

g=

β1
1
=
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 get(25)

8. Radiation

µ

109

1
−δ
d(L0 + µL1 )= −(L0 +µL1 )+ω (L0 +gµL1 )+ ω F0 exp( ) (1+3gµ0 µ).(26)dδ
4µ0where 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 ∗ :

−δ
),
µ0
−δ
L1 (δ) = p(C1 exp(−kδ) − C2 exp(+kδ)) − β exp( ),µ0

L0 (δ) = C1 exp(−kδ) + C2 exp(+kδ) − α exp(

(29)
(30)

where
1

k = [3 (1 − ω) (1 − ωg)] 2
1

p = [3 (1 − ω)/(1 − ωg)] 2[1 + 3g (1 − ω)]
α = 3 ω F0 µ0
4 (1 − k 2 µ20 )[1 + 3 g (1 − ω) µ20 ]β = 3 ω F0 µ0.
4 (1 − k 2 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 zero2F↓ (0) = [L0 (0) + L1 (0)] = 0,3which translates into
(1 +

2p
2p
2β
) C1 + (1 − ) C2 = α +
.
3
3
3

(31)

110

8. Radiation

For the second condition, one assumes that the upward directed flux at thebottom of the layer is equal to the product of the downward directed diffuse
and direct fluxes by the corresponding diffuse and direct reflectances (Rdand 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

In the Delta-Eddington approximation, the phase function is approximatedby a Dirac delta function (forward scatter peak) and a two-term expansionof the phase function
P (θ) = 2f (1 − cos θ) + (1 − f ) (1 + 3g 0 cos θ),
where f is the fractional scattering into the forward peak and g 0 the asymmetry factor of the truncated phase function. As shown by Joseph et al.
(1976), these parameters are:

f

= g2

g0 =

g
.
1+g

(34)
(35)

The solution of the Eddington’s equations remains the same provided thatthe total optical thickness, single scattering albedo and asymmetry factor
entering (26)-(32) take their transformed values:

δ
ω

0

0

= (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).

exp(+k δ ∗(32))∗

exp( −δµ0 )(33).

8. Radiation

111

Clear-sky fraction of the layers
In the clear-sky fraction of the layers, the shortwave scheme accounts for
scattering and absorption by molecules and aerosols. The following calculatot ) of
tions are practically done twice, once for the clear-sky fraction (1 − Ccld
the atmospheric column µ with equal to µ0 , simply modified for the effectof Rayleigh and aerosol scattering, the second time for the clear-sky fraction
tot of the atmospheric columnof each individual layer within the fraction Ccld
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 thebottom of the j th layer can be calculated using respectively a first and asecond-order expansion of the analytical solutions 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 thatδR,
2 µ + δR2µ.
2 µ + δR

RR =
TR =

(38)
(39)

The optical thickness δR of an atmospheric layer is simply:∗
δR = δR(p(j) − p(j − 1)),
psurf∗ is the Rayleigh optical thickness of the whole atmosphere paramwhere δR
eterized as a function 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 and asymmetry factor are respectively δa , ωa (with 1 − ωa  1) andga so that:

112

8. Radiation

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 ) 2
µe
µe
(40)

where back(µe ) = (2 − 3µe ga )/4 is the backscattering factor.
Practically, Rclr and Tclr are computed using (40) and the combined effect ofaerosol and Rayleigh scattering comes from using modified parameters corresponding to the addition of the two scatters with provision for the highly
asymmetric aerosol phase function through a Delta-Eddington approximation of the forward scattering peak (as in (34)-(36)):
δ + = δR + δa (1 − ωa ga2 )
g+ =

ga
δa
1 + ga (δR + δa )

ω+ =

δR
δa ωa (1 − ga2 )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:

T (µe )
R_,
1 − R∗ R_
T (µe ),
(1 − R∗ R_)

Rclr = R(µe ) +(41)

Tclr =(42)

with R∗ = R(1/r), T ∗ = T (1/r), R_ = Rt (j − 1) is the reflectance of theunderlying 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 aresimply those calculated from (16) and (41) attenuated by the gaseous transmissions (13).

8. Radiation

113

2.4 Multiple reflections between layers
To deal properly with the multiple reflections between the surface and thecloud layers, it should be necessary to separate the contribution of eachindividual reflecting surface to the layer reflectances and transmittances inas much as each such surface gives rise to a particular distribution of absorberamount. In case of an atmosphere including N cloud layers, the reflected
light above the highest cloud consists of photons directly reflected by thehighest cloud without interaction with the underlying atmosphere and ofphotons that have passed through this cloud layer and undergone at least
one reflection on the underlying atmosphere. In fact, (8) should be written

F =

N
X
l=0

Z ∞

Fcl

0

pl (U ) t∆ν (U ) dν,(43)where Fcl and pl (U ) are the conservative fluxes and the distributions ofabsorber amount corresponding to the different reflecting surfaces.
Fouquart and Bonnel (1980) have shown that a very good approximation tothis problem is obtained by evaluating the reflectance and transmittance ofeach layer (using (32) and (41)), assuming successively a non-reflecting underlying medium (R_ = 0), then a reflecting underlying medium (R_ 6= 0).
First calculations provide the contribution to reflectance and transmittanceof 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 thenderived using (11) and starting from the surface and working the formulas
upward:U↓e0 =1Tb
ln( 0 ),
keTbc

U↓e6= =Tb1ln( 6= ),ke
Tbc

U↑e0 =

1
Rt
ln( 0 ),
ke
Rtc

114

8. RadiationU↑e6= =Rt1
ln( 6= ),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 Rt0 t∆ν (U↑e0 ) + (Rt6= − Rt0 ) t∆ν U↑e6= )F (j) = F0 Tb0 t∆ν (U↓e0 ) + (Tb6= − Tb0 ) t∆ν U↓e6= )
↓

(44)

(45)

3 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• 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 the best 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; Cloughand Iacono 1995) through its use of absorption coefficients for the relevant kdistributions derived from LBLRTM. Therefore the k-coefficients in RRTM

8. Radiation

115

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 andLiou 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 distribution function, i.e. the fraction of theabsorption coefficients in the set smaller than k). The effect of this reordering
is a rearrangement of the sequence of terms in the integral over wavenumberin 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 order to get
the upward radiance, the integration over the vertical dimension is repeated,
this time from the surface upward.
The spectrally averaged radiance (between ν1 and ν2 ) emerging from an
atmospheric layer is1R=(ν1 − ν2 )

Z ν1



Z 1

dν R0 (ν) +
ν2

tν


B(ν, T (t0ν )) − R0 (ν) dt0





(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 t0ν is the transmittance at a point along the optical pathin the layer. Under the mapping ν → g, this becomesZ 1

ρδz
dg Beff (g, Tg ) + [R0 (g) − Beff (g, Tg )] exp −k(g, P, T )
R=
cos φ
0






where Beff (g, T ) is an effective Planck function for the layer that varies withthe layer’s transmittance such as to ensure continuity of flux across layerboundaries for opaque conditions. 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, the vertical thickness of thelayer 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 8.1, number of g-points), each corresponding to alimited range of k(g) values and for which a characteristic value κj of the

(47)

116

8. Radiation

absorption coefficient is chosen. These κj are then used to compute theoutgoing radianceR = Σj Wj Beff j + (R0 (g) − Beff j ) exp −κj

ρδz
cos φ



(48)

where Wj is the size of the sub-intervals (ΣWj = 1).
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 to the original RRTM (Mlawer etal. 1997), the version used at ECMWF (hence ARPEGE-CLIMAT) has beenslightly modified to account for cloud optical properties and surface emissivity defined for each of the 16 bands over which spectral fluxes are computed.
For efficiency reason, the original number of g-points (256 = 16 × 16) has
been reduced to 140 (see Table 8.1). Other changes are the use of a diffusivity approximation (instead of the three-angle integration over the zenith
angle used in the original scheme) to derive upward and downward fluxes
from the radiances, and the modification of the original cloud random overlapping assumption to include (to the same degree of approximation as usedin 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 thesurface to get the downward fluxes. The downward fluxes at the surface arethen used with the spectral surface emissivities and the 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 topof the atmosphere.
For the relevant spectral intervals of the RRTM schemes, ice cloud optical properties are derived from Ebert-Curry (1992), and water cloud opticalproperties from Smith and Shi (1992) . Whereas in the previous operational
scheme the cloud emissivity used to compute 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, thecloud optical thickness is defined as a function of spectrally varying mass
absorption coefficients and relevant cloud water and ice paths, and is usedwithin the true cloudy fraction of the layer. Alternate sets of cloud opticalproperties are also available for RRTM (Section 4).

8. Radiation

117

Table 8.1: Spectral distribution of the absorption by atmospheric gases in
RRTM
Spectral intervals Number of g-points
Gases included
cm−1
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

4 Input to the radiation scheme
4.1 Solar irradiance data
Depending on the experiment, the solar constant can be specified on a yearlybasis in the model. For instance, for XXth century simulations, the historicreconstruction recommended by CMIP5 is used:

http://www.geo.fu-berlin.de/en/met/ag/strat/forschung/SOLARIS/Input_data/CMIP5_solar_irra

4.2 Clouds
Cloud fraction, and liquid/ice water content are provided in all layers by thecloud scheme.

4.3 Ground albedo and emissivity
Ground albedo and emissivities are calculated in the SURFEX module (seechapter “Surfex processes schemes”). The main features are the following
ones:

118

8. Radiation

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 type of vegetation.
• over ocean: the diffuse albedo is taken as a constant (0.061); the direct
albedo is computed as a function of the zenithal angle, the function ischosen to obtain similar results than with the value 0.061 on a wholeannual cycle.
• over snow: albedo varies with the age of snow following Douville et al.
(1995)

4.4 Aerosols
The aerosol distributions are given by outputs of simulations run with the
LMDZ-INCA chemical model (Balkanski, personnal communication; Schulz(2007)), forced with observed emissions. The vertically integrated monthly
averaged optical thicknesses are then available from 1850 to 2000, and from2000 to 2100 for each RCP scenario. To smooth interannual variability,
a running average on 11 years is calculated. Five aerosol types are used:maritime, sulfate, continental, soot and desert. An option for volcanic classis available. The 2D optical thicknesses are vertically distributed in the
RADAER subroutine according to the aerosol type. In the same subroutine
aerosols are also redistributed in 6 types:• Continental (Organic) + Sulfate + Background tropospheric.
• Maritime (seasalt aerosols)• Desert (soil dust type aerosols)• Urban (mainly black-carbon type aerosols)• Volcanic class (background)• Stratospheric (background) + volcanic eruptions
Diffusion

Asymmetry

Normalized

RPIZA

factor RCGA

optical depth

8. Radiation

119

0.25-0.68 µm

0.68-4.0 µm

0.25-0.68 µm

0.68-4.0 µm

0.25-0.68 µm

0.68-4.0 µm

0.9148907

0.8814597

0.729019

0.663224

1.694460.40174

Maritime

0.9956173

0.9920407

0.803129

0.793746

1.11855

0.89383

Desert0.7504584

0.9239428

0.784592

0.696315

1.092120.89546

Carbonic

0.8131335

0.7546879

0.712208

0.652612

1.721450.40741

Volcanic

0.9401905

0.9515548

0.7008249

0.6608509

1.03858

0.51143

Stratospheric

0.9999999

0.9938563

0.7270548

0.6318786

1.12044

0.32646

Spectral bands
Organic + Sulfate+ Tropospheric

Coefficients for the solar radiation and for the two spectral intervals case.
For the other cases, see routine SUAERSN for short-wave coefficients and
SUAERL for long-wave coefficients.
Spectral bands>2.85 µm

1.25-2.0 µm

1.25-2.0 µm

0.9-1.0 µm

2.0-2.85 µm

Organic + sulfate + tropospheric

0.036271

0.030153

0.017343

0.015002

0.008806

0.006865

Maritime

0.026561

0.032657

0.017977

0.0142100.016775

0.022123

Desert0.014897

0.016359

0.019789

0.030777

0.013341

0.014321

Carbonic

0.001863

0.002816

0.002355

0.002557

0.001774

0.001780

Volcanic

0.011890

0.016142

0.021105

0.028908

0.011890

0.011890

Stratospheric

0.013792

0.026810

0.052203

0.066338

0.013792

0.013792

Absorption coefficients for long-wave radiation (SUAERL).
Indirect forcing of the sulfate aerosols
The aerosols can act as cloud condensation nuclei. At constant cloud liquidwater, increasing aerosol concentration leads to a larger concentration ofcloud droplets of small radius and increases cloud reflectivity. A simple
parametrization from Quaas and Boucher (2005) simulates this effect in thecase of sulfate aerosols. The cloud droplet concentration CDN (in cm−3 ) is
given by:lnCDN = 1.7 +0.2 ln mSO2−
4

with mSO2− expressed in µgm−3 . The mean cloud droplet radius is then4
calculated from the cloud liquid water content ql 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.

120

8. Radiation

Figure 1: Mean global profiles of types of aerosols as input of the RRTM
scheme: optical thickness by level (sum of all the levels gives the verticallyintegrated optical thickness, normalized to 1 for each aerosol type).

8. Radiation

121

4.5 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) meansthat seven of the climatological coefficients of the REPROBUS outputs areused. Evolution of ozone chemistry is due to chlorine and meteorologicalparameters (temperature, humidity) evolutions.

4.6 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 withemissivity and spectral optical thickness. All these properties are defined forwater and ice particles in the RADLSW routine. They depend on liquid (orice) 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:
Cloud liquid particles• NRADLP = 0; cloud water effective radius is calculated as a functionof pressure (old parameterisation).
• NRADLP = 1; the cloud effective radius is equal to 10 µm over landand equal to 13 µm over the ocean.
• NRADLP = 2; parameterisation based on Martin et al. (1994).
• NRADLP =3: the effective radius of cloud particle is calculated taking
into account the indirect effects of sulfate aerosols.
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).

122

8. Radiation• 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).

SW radiation
Cloud water optical properties depend on NLIQOPT:• NLIQOPT different from zero, refers to Slingo (1989).
• 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, 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).
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 .
The current values are NRADLP = 3, NRADIP = 2, NLIQOPT = 0 and
NICEOPT = 1.

8. Radiation

123

4.7 Cloud overlap assumption
The cloud overlap assumption is used for the determination of Ctot (seeSection 2). It is determined by the NOVLP variable: NOVLP = 1 means"maximum-random overlap", NOVLP = 2 means "maximum overlap " andNOVLP = 3 means "random overlap" (NOVLP = 1 in the present code).

4.8 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 be made in the APLPAR subroutine
to ensure equal radiative net budgets in ARPEGE as well as in SURFEX.

5 Simplified radiation scheme
Given the computation cost of the radiation scheme, it seems interesting tocall it only at some time steps, named radiative time steps. Nevertheless, itis 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) andFLW (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) 0 calculated at this time:
t0 =

FSW 0
µ0 E 0

(49)

0 =

FLW 0
σT04

(50)

Then, in the next time steps, one recomposes solar and infrared fluxes bymodifying only the solar incidence µ and temperature T :FSW = t0 µ0 E 0(51)

FLW = 0 σT 4

(52)

124

8. Radiation

9
Clouds and turbulence

1 General description of the scheme
This scheme was developed by Ricard and Royer (1993) and has been usedin the former versions of Arpege-climat. It calculates the stratiform cloud
fraction, the stratiform liquid water content and the coefficients of turbulent
vertical mixing as well as the Brunt-Vaïsala frequency.

1.1 Condensate assumptions
The scheme of sub-grid condensation is based on Deardorff (1977) and Mellor (1977), resumed and adapted by Bougeault (1981, 1982). These studies
use the conservative variables of Betts (1973), namely the potential temperature of liquid water θl and the specific moisture of total water qw (vapor +condensed).
Arpege-climat does not have a prognostic equation for condensate, where,by definition, all condensate precipitates. Ricard and Royer considered thatthe original variables of the model T and qv were to play the role of theconservative variables in the equations. Calculations will thus be made withthe variables of Betts (θl , qw ), by replacing them by (θ, qv ) in the code.

1.2 Turbulent kinetic energy assumptions(B) For the turbulence part , the problem of closure is solved by the sub-grid
scheme of order 2 described in Mellor and Yamada (1974, 1975, 1982) and
Galperin et al. (1988), in which the effects of liquid water and the water
vapor are taken into account.

126

9. Clouds and turbulence

These order 2 schemes use a diagnostic turbulent kinetic energy eT = (u02 +v 02 + w02 )/2 which comes from a stationary equation d(eT )/dt = 0. The
moments of order 2 such as the correlations and the variances (for example
0 2 ) are calculated using a parametrization according to
w0 u0 , w0 θl0 , θl0 2 or qw
eT and of the wet conservative variables (θl , qw ).
√The exchange coefficients are in the form KX = l 2 eT SX , for X = (u, v, T, q).
Even if the mixing length l has a known and imposed analytical formulation,
the functions of stability SX do not have a fully determined analytical formulation. One difficulty of the scheme comes from that functions SX depend on
the Richardson number Ri and the condensate ql which must be determinedaccording to the deficit to standardized saturation Q1 = a (q − qsat )/(2σs )and of the functions introduced by Bougeault (1981, 1982) F0 , F1 and F2 .
One obtains R = F0 (Q1 ), ql = (2σs ) F1 (Q1 ) and s0 ql0 = (2σs2 ) F2 (Q1 ).
It is necessary to solve implicit equations of type Q1 [ql (Q1 )], admitting one
or more solutions. After the study of the various cases which can arise, amethod of resolution by successive iterations was developed by Ricard (1992)
or Ricard and Royer (1993).

1.3 Sub-grid condensation scheme
This scheme was developed by Sommeria and Deardorff (1977) and Mellor(1977). It accounts for sub-grid condensation in the case of non-precipitating
clouds. It makes it possible to evaluate the cloudy fraction and the liquid
quantity of water from the small scale turbulence. In this chapter, the term“stratiform” is understated when the terms “cloudy fraction” and “liquid water” are mentioned.
In order to describe the thermodynamical properties of the planetary boundary layer, one uses the “conservative variables” θl (potential temperature ofliquid water) and qw (total water content), in the place of potential temperature θ, of specific moisture of vapor q and of liquid water content ql . Thesevariables are defined as:




θl


 q

w

L θ
ql
cp T
= q + ql=θ−(1)Resumed and adapted by Bougeault (1981, 1982), this scheme relies on threeassumptions:• a systematic condensation.
• fluctuations of θl and qw are weak.
• a knowledge of the joint distribution function of θl and qw .

9. Clouds and turbulence

127

Systematic condensation
At the scale of the particle, one supposes that there is immediate condensation as soon as the specific moisture of the water vapor, q, reaches the
specific moisture at saturation for the ambient temperature, qs . Liquid total
water content formed locally is expressed then using the equation:
ql = (qw − qs ) H(qw − qs )(2)(

where H is Heavyside function: H : x −→1 si x > 0
0 si x ≤ 0

Fluctuations of θl and qw are weak
The formulation of ql being defined, cloud fraction R and liquid water content
ql are expressed using these two double integrals:



R







 ql

ZZ

=
IR2

e l , qw ) dθl dqw
H(qw − qs ) G(θ

(3)
ZZ

=
IR2

e l , qw ) dθl dqw(qw − qs ) H(qw − qs ) G(θe indicates the joint distribution of θl and qw .
where GThe difficulty in resolving these two integrals comes from the fact that qs
depends on T, which depends in turn on qw and θl . The way of solving thisproblem is to find an expression of qs as a function of Tl , temperature ofliquid water.
In a volume around a given grid point, it is supposed that fluctuations of
the potential temperature of liquid water (θl ) and of the total water content(qw ) remain weak. In this way, one can carry out a Taylor development of qs
around Tl to the first order, with the help of the approximation which consists
in neglecting the influence of the fluctuations in pressure on qs compared tothose of temperature on it:

qs (T ) = qs (Tl ) + (T − Tl )

∂qs
(Tl )
∂T

(4)

This approximation is justified by an analysis in order of magnitude which
relies on these two formulas:

128

9. Clouds and turbulence• the formula giving saturated specific moisture as a function of pressurep and temperature T :qs (T, p) = 0.622

es (T )∂qs
qs (T, p), hence=−p − 0.378 es (T )∂pp − 0.378 es (T )• Clapeyron equation:
qs L
∂qs
=
∂T
Rv T 2∂qs∂qs∆T ∼ 104∆p, which justifies the last approximation
∂T
∂p
T
Tl
T
Assuming that
∼ , one can rewrite Equation (4) in the following
∼
θ
θl
θ
way:

It is shown that

qs (T ) = qs (Tl ) +T 0 L 0
θ l + qlcpθ!

L qs (Tl )
Rv Tl

(5)

2

When qw > qs , ql can then be put in the form:0− α1 θl0 )ql = a∆q + a(qw(6)

in which:
"

L ∂qs
a= 1+
(Tl )
cp ∂T

#−1

α1 =

T ∂qs
(Tl ) ∆q = qw − qs (Tl )
θ ∂T

(7)

Introduction of variable s
0 and θ 0 which allows
At this stage, one introduces a linear combination of qw
lan integration with respect to single variable s, defined as:s=

a 0(q − α1 θl0 )
2 w

(8)

In a diagram (θl , qw ), curves ql = cste are parallel to the curve of saturation
qs (θl ); s represents the local algebraic variation to saturation. The liquidwater ql depends only on variable s which is measured on a perpendicularaxis. The algebraic deviation to saturation averaged on the mesh is given bya∆q.
Under these conditions, ql is written:(

ql = 0ql = a∆q + 2s

if s < smin
if s > sminwith smin = −

a∆q
2

(9)

9. Clouds and turbulence

129

0
0
Using variables s and r, r being defined by r = a
2 (qw + α1 θl ), one carries0 out
0 )a transformation which aims at replacing the conservative variables (θl , qw
Z +∞e r − s , r + s ) dr, where J is the Jacobianby (r, s). Posing G(s) = |J|G(
a α1 a α1−∞of the transformation (J = − 22 ), one obtains:
a α1

Z +∞




R






G(s) ds

=
Zsmin
+∞

q

=

l






 s0 ql0

Zsmin
+∞

=
smin

ql (s) G(s) ds(10)s ql (s) G(s) ds

Introduction of variable t
One carries out a last transformation by introducing the standardized variable defined by t = s/σs where σs is the standard deviation of s. One
obtains:
q

σs =

s02 =

a
2

rh

i

0 2 − (2 α ) θ 0 q 0 + (α )2 θ 0 2 .
qw
1 l w
1
l

(11)

Let Q1 be an algebraic measurement of the distance to the curve of saturation:
Q1 =

a ∆q
smin
=−
.
2 σs
σs

(12)

Under these conditions, the local value of ql is given by the following relation:ql= (Q1 + t) H(Q1 + t) .
2 σs

(13)

Posing G = σs G(σs t), System (10) becomes:



R





 ql

2 σs





s0 ql0



2
2 σs

Z +∞

= F0 (Q1 ) == F1 (Q1 ) == F2 (Q1 ) =G(t) dt
1
Z−Q
+∞

−Q1
Z +∞
−Q1

(Q1 + t) G(t) dt(14)t (Q1 + t) G(t) dt

where F0 , F1 , F2 , functions of Q1 , depend only on the of distribution function
G.
From System (14), in order to calculate cloud fraction R and mean liquidwater content q l on the mesh, it is sufficient to know the distribution functionof t and the standard deviation of s.

130

9. Clouds and turbulence

Distribution function G(t)Sommeria and Deardorff (1977) and Mellor (1977) supposed that the joint
distribution of θl and qw is Gaussian or bi-normal.
Three functions F0 , F1 , F2 were proposed by Bougeault (1981, 1982) who
stated that the form of the distribution of s (or t) was not very significant
for calculations of cloud fraction R and ql /(2σs ). However, he showed thatan asymmetric distribution seemed more effective than a Gaussian distribution to parametrize the cumuliform trade wind layer: in such a layer, strong
ascents in the clouds balance weak subsidences out of the clouds. This phenomenon cannot be represented by a Gaussian distribution.
For this scheme, an asymmetric distribution was thus selected:
G(t) = e−(t+1) H(t + 1)

(15)

System (14) can then be written:

Q1 < 1
F0 (Q1 ) = eQ1 −1







F1 (Q1 ) = eQ1 −1








Q1 ≥ 1F2 (Q1 ) = (2 − Q1 ) eQ1 −1F0 (Q1 ) = 1







F1 (Q1 ) = Q1








(16)

F2 (Q1 ) = 1

0 and variances
The evaluation of σs requires an estimate of correlation θl0 qw
0 2 , which justifies the use of the sub-gridof temperature θl0 2 and moisture qw
turbulence scheme which is described hereafter.

1.4 Sub-grid turbulence scheme
For reasons of simplicity of setting up, reliability and robustness, the model
used is a level 2 Mellor and Yamada (1974, 1975, 1982) model.
This model relies on the following assumptions:• Boussinesq hypotheses.
• horizontal homogeneity for turbulent motion: cloudy regions are produced by turbulent fluctuations only. This hypothesis is valid outside
regions with strong advection.

9. Clouds and turbulence131• in the prognostic equation for turbulent kinetic energy eT = (u02 +v 02 + w02 )/2, the divergence of mean turbulent kinetic energy flux, i.e.
the diffusion term:
−

∂
p0 w0
∂eT
(w0 e0T +
−ν
)
∂z
ρ
∂z

is neglected• the turbulent kinetic energy equation is stationary.
Under these assumptions, the equations are:


 ∂u



 ∂t








∂v





∂t







∂θ



∂t







∂p





∂z








∂eT




∂t

=−

∂w0 u0 1 ∂p
∂2u−
+ ν 2 + fv
∂z
ρ ∂x
∂z

=−

∂w0 v 0 1 ∂p
∂2v−
+ ν 2 − fu
∂z
ρ ∂x
∂z

=−

∂w0 θ0∂2θ+ νθ 2
∂z
∂z

(17)

= −ρg


= 0 = − w0 u0∂u∂v+ w0 v 0∂z∂z+ β w0 θv0 − CeT 3/2l

where β = g/T and where θv is the virtual temperature defined by Lilly(1968), Deardorff (1976), Sommeria and Deardorff (1977), see Equation (26).
N.B: a level 2.5 Mellor and Yamada model (with a prognostic equation for
the turbulent kinetic energy) could have been used but it has the disadvantage of being less robust. Indeed, Helfand and Labraga (1988) showedthat the model can explode when the two following conditions are met: theRichardson number is negative and the turbulent kinetic energy is lower thanthe equilibrium turbulent kinetic energy given by the level 2 model.
In order that the sub-grid condensation scheme is compatible with the subgrid turbulence scheme, it is necessary to introduce the effects of the water
vapor and liquid water into it. To go from the dry air system (in θ and q)
to the moist air system (in θl and qw ), the steps are:• replace θ by θl ,• add prognostic equations for qw and correlations between qw and (u,v, w, θl ),• modify the expression of virtual temperature, taking θv defined by
Equation (26).

132

9. Clouds and turbulence

Mean motion equations
The realization of the first two points makes it possible to lead to the following system:

∂u





∂t








∂v





∂t






∂θl




 ∂t







 ∂qw




∂t






ET

=−

∂w0 u0 1 ∂p
∂2u−
+ ν 2 + fv
∂z
ρ ∂x
∂z

=−

∂w0 v 0 1 ∂p
∂2v−
+ ν 2 − fu
∂z
ρ ∂x
∂z

=−

∂w0 θl0∂ 2 θl+ νθ 2∂z∂z=−

0∂w0 qw∂ 2 qw
+ νq
∂z
∂z 2

(18)

= Pd + Pθ

where

Pθ

= β w0 θv0Pd= −(w0 u0

ET

= C

∂u
∂v
+ w0 v 0)∂z∂z

eT 3/2eT 3/2=lΛ1Pθ is the term of production by buoyancy of the turbulent kinetic energy,
Pd is the dynamic production term and ET is the dissipation of turbulent
kinetic energy term. As previously, β = g/T .

Second order moments
It is necessary to introduce the effects of liquid water and of water vapor into
the second order moments of the level 2 Mellor and Yamada (1974) scheme,as redefined in Yamada and Mellor (1975) and in Mellor and Yamada (1982).
One √should pay attention that, in these papers, the quadratic speed averageq = 2 eT is used, with l1 = A1 l, l2 = A2 l, Λ1 = B1 l = l /C and Λ2 = B2 l.
But in this documentation, one will use directly kinetic turbulent energyeT = q 2 /2, with the mixing length l and the 5 coefficients (A1 , A2 , B1 , B2 ,C1 ).

9. Clouds and turbulence

133

Under these conditions, one has:




 u02










v 02












 w02




 u0 θl0












v0 θ0

l












0

 w 0 θl
 u0 v 0










2 eT
∂u
∂v
A1 l
=
−4w0 u0+√+ 2w0 v 0
− 2Pθ
3
∂z
∂z
2 eT


∂u
∂v
2 eT
A1 l
2w0 u0+√− 4w0 v 0
− 2Pθ
3
∂z
∂z
2 eT



2 eT
A1 l
∂u
∂v
=
+√
+ 2w0 v 0
+ 4Pθ
2w0 u0
3
∂z
∂z
2 eT





=



"

#

"

#

3 A2 l
∂θl
∂u
= −√
w0 u0
+ w0 θl0
∂z
∂z
2 eT
∂θl
∂v
3 A2 l+ w0 θl0
w0 v 0
= −√
∂z
∂z
2 eT
"

∂θl
3 A2 lw02= −√− β θl0 θv0
∂z
2 eT



0

u0 qw










0
v 0 qw











0

 w 0 qw#

3 A1 l
∂v
∂u
= −√
w0 u0
+ w0 v 0
∂z
∂z
2 eT






3 A1 l∂v= −√(w02 − 2 eT C1 )− βv 0 θv0
∂z
2 eT


3 A2 l
∂qw
∂u
0
= −√
w0 u0
+ w 0 qw
∂z
∂z
2 eT



3 A2 l
∂qw
∂v
0
= −√
w0 v 0+ w 0 qw
∂z
∂z
2 eT







3 A2 l
∂qw
0 θ0
= −√
w02
− β qw
v
∂z
2 eT


(19)

(20)

3 A1 l∂uu0 w0 = − √(w02 − 2 eT C1 )− βu0 θv0

∂z
2
e
T










 v 0 w0







(21)



(22)

134

9. Clouds and turbulence




θl0 2











0 2
qw











0 0


 θ l qw

B2 l
∂θl
= −√
w0 θl0
∂z
2 eT
B2 l
∂qw
0
= −√
w 0 qw
∂z
2 eT

(23)

"

∂θl
B2 l
∂qw
0
w 0 qw=− √+ w0 θl0∂z∂z
2 2 eT#

The exchange coefficients are defined by:



w0 u0







 w0 v 0


0 0


 w θl



 0 0


w qw

∂u
∂z
∂v
= −Km
∂z
∂θl
= −Kh
∂z
∂qw
= −Kc
∂z
= −Km

"



Km











Gm











l2
=
2 eT

 Gh

=−

l2 ∂θlβ2 eT ∂z=−l2 ∂qwβ
2 eT ∂z












 Gc

∂u
∂z

2

Kh

=l







 K

=l



+

=l

∂v
∂z

c

√
√
√

g
2 eT Smfh
2 eT S(24)

fc
2 eT S

2 #(25)

Expression of the virtual potential temperature
The third point is dealt by introducing the expression of the virtual potential
temperature defined in Lilly (1968), Deardorff (1977) and Sommeria andDeardorff (1977):
θv = θ [1 + (RET V ) q − ql ](26)

where, in the code, RET V = Rv /Rd − 1 ≈ 0.601.
L
Using conservative variables θl = θ −cppreceding equation can be rewritten as:
θv = θl + CT0 qw + D(z) ql
where:



p0
p

κ

ql and qw = q + ql , the(27)

9. Clouds and turbulence135• CT0 = (RET V ) θ• D(z) =

p0
p

κ

L
− (1 + RET V ) θ =
cp

L
Rv
−
cp T
Rd

!

θ

In system (18), the order two moments using virtual temperature are ex0 and of q 0 :pressed as a function of variables θl0 , qw
l
0 + D(z) m0 q 0m0 θv0 = m0 θl0 + CT0 m0 qw
l(28)

m ∈ {u, v, w, θl , qw }The correlations involving liquid water potential temperature (m0 θl0 ) and0 ) do not generate calculation problems. However,total water content (m0 qw
correlations involving liquid water content (m0 ql0 ) are difficult to estimate.
To solve this problem, Bougeault (1982) proposed the following assumption:










 wheres0 ql0σs 2!

m0 ql0= m0 s0m0 s0a0 − α m0 θ 0 )= (m0 qw
1
l
2

(29)

The calculations carried out by Mellor (1977) show that this assumption isvalid if the joint distribution for the pair of variables (s, m) is Gaussian.
However, Bougeault (1982) assumes that System (29) can be applied to the
non-Gaussian casess0 ql0, the order two moments using liquid water content are
2 σs2
written in the form:
Like F2 (Q1 ) =0 − α m0 θ 0 )m0 ql0 = a F2 (Q1 ) (m0 qw
1
l

(30)

Reformulation of the Richardson number
The flux Richardson number, ratio of the production by buoyancy Pθ andthe dynamic production Pd of turbulent kinetic energy, is defined as follows,with β = g/T :Rf = −Pθ=Pd

βw0 θv0∂u∂v
w0 u0+ w0 v 0
∂z
∂z

(31)

136

9. Clouds and turbulence

In Equation (28), replacing w0 ql0 by the second member of Equation (30)(m = w) yields a reformulation of the flux Richardson number:




0

 w0 θl0 + CT0 w0 qw
 + F2 (Q1 ) β a D(z) 

∂u
∂v 

0 − α w0 θ0

w 0 qw
1
l

∂u
∂v 
0
0
0
0
wu
+wv
∂z
∂z



Rf = β 




w0 u0

∂z

+ w0 v 0∂zor:Rf = (Rf )h + F2 (Q1 ) (Rf )c(32)

Rf is therefore the sum of two terms using:• a flux Richardson number (Rf )h taking into account water vapor effect:(Rf )h = β 0w0 θl0 + CT0 w0 qw

w0 u0



∂v 

∂u
+ w0 v 0∂z∂z• a complementary flux Richardson number (Rf )c taking account of theeffects of liquid water:(Rf )c = β a D(z) 
0 − α w0 θ0

w 0 qw
1
l

∂u
∂v 
0
0
0
0
wu
+wv
∂z
∂z

In a similar way, the gradient Richardson number is split into two terms:
Ri = (Ri)h + F2 (Q1 ) (Ri)cwhere (Ri)h =gSm(Rf )h and (Ri)c =

(33)
g
S
m

g
f
(Rf )c , S
m and Sh being terms
fh
fh
S
S
involved in fluxes and second order moments: see System (24) and consequences on Systems (19) to (22).

A property arises from this last equation: as function F2 is a bijection of IR
onto ]0,1[, the Richardson number function, defined as follows, is a bijection:IR −→ ] (Ri)min , (Ri)max [ R : Q ,→ (Ri) + F (Q ) (Ri)
i
1
2
1
c
h

(34)

9. Clouds and turbulence

where

137

(Ri)min = min[ (Ri)h , (Ri)h + (Ri)c ](Ri)max = max[ (Ri)h , (Ri)h + (Ri)c ]This last property shows that the knowledge of the Richardson number Ri
in interval ] (Ri)min , (Ri)max [ leads to the knowledge of the position of Q1
a ∆q
in IR, therefore leads to evaluation of σs since σs =.
2 Q1

1.5 Solving an implicit equation
Mixing length
In old versions of Arpege-climat, the mixing length was the same as inroutine ACCOEFK, with a profile which depended on ALMAV, BEDIFV
and UHDIFV.
Further work allowed, as an option, a boundary layer depth variable in timeand space. If the profile of mixing length l(z) was supposed to be constant
above the planetary boundary layer, the boundary layer depth H (ZHCPLV)
either was imposed and equal to AHCLPV, or was calculated in subroutine ACPBLH at previous time step (see Chapter 11) when AHCLPV= 0.
The value of this mixing length in the free atmosphere is λ =ALMAV, theasymptotic length used in ACCOEFK. The need to limit the number of adjustable parameters in the scheme was dictated by the constraint to have aparametrization which is robust in simulations of climate change. To thisgoal, a cubic profile according to z + z0 was used for the mixing length
between the ground and the top of the boundary layer.
In order to better mimic the quadratic feature of the mixing length l(z)usually observed within the boundary layer (i.e. the PBL located betweenz = 0 and z = Zi = ZHCLPV), the cubic profile decribed above is replaced
in by a new formulation, adapted from the general formulas from Lenderinkand Holtslag (2004).
This new quadratic profile is computed for l(z) if LMLH is set to TRUE.
in the namelist. If LMLH is set to .FALSE., l(z) is equal to the old cubic
profile.
From the study of Lenderink and Holtslag (2004), the mixing length is deduced from a general function F (Ri ), a prescribed function of the local
Richarson number Ri (z). For the simple case of a constant value F (Ri ) =RLMLH1 (with RLMLH1 available in the namelist), the master “integral”
mixing length at the height z > 0 is defined in terms of “(z + z0 )” (if z0 is
the roughtness length), by

Lint (z) ≡ Max(z + z0 ) [Zi − (z + z0 )]0 ; RLMLH1
Zi



.

(35)

138

9. Clouds and turbulence

Zi is the top PBL height and it is denoted by ZHCLPV in the code, depending on the choice for AHCLPV. As for g Lint , it is denoted by ZGLINT in
the code.
The method described in Lenderink and Holtslag (2004) for computing this“integral” mixing length Lint (z) - and then the corresponding mixing lengthl(z) - is considered as a “poor man’s parcel method”, obtaining rather similar
results to those of the non-local method of Bougeault and Lacarrere (1989)
for convective situations, though at a much lower computational cost.
For a constant value F (Ri ) = RLMLH1, the maximum of Lint (z) is reached
for the height Zi / 2 − z0 , located very close to the middle of the PBL, since
z0 is much smaller than Zi . This maximum is equal to RLMLH1 ∗ Zi /4 andit depends linearly of the local top PBL height. For RLMLH1 = 0.4 andZi = 1500 m, this maximum is equal to 150 m. The quadratic profile ofLint (z) is ensured from the second order formulation (35), with Lint (0) ≈ 0
and Lint (Zi ) ≈ 0, assuming small values for z0 .
Close to the surface Lint (z) ≈ RLMLH1 ∗ (z + z0 ). Thus, RLMLH1 can bechosen close to the Von Karman constant, equal to κ = 0.4. But possible
higher or lower values for RLMLH1 can be set in the namelist (for instance
between 0.2 and 0.8).
Still close to the surface, the length scale is also limited to half the neutral
length scale ln /2 = 0.5cn κ(z +z0 ), with cn = (c0 )−1/2 and c0 = 3.75 (leading
to cn ≈ 0.5164). In the code 0.5 cn is denoted by ZLMCS.
Above the Planetary Boundary Layer, the too small value of Lint (z), forz > 0.9 Zi or so, are avoided by a relaxation toward the asymptotic value
λ = ALMAV, with ALMAV available in the namelist. Above the top PBL
height Zi , Lint (z) is presently set to the value λ = ALMAV.
The two limitations l(z) > 0.5cn κ(z +z0 ) close to the surface and l(z) ≈ λ =ALMAV close to the top of the PBL are realized via a common minimum
value Lmin (z), defined by the Blackadar formulae (B.3) of the Appendix B
in Lenderink and Holtslag (2004)
1
Lmin (z)

≡

1
1
+
.
λ
0.5 cn κ (z + z0 )(36)

In the code g Lmin is denoted by ZGLMIN.
Another similar limiting value L∗min is defined byL∗min (z) ≡ Min ( λ ; 0.5 cn κ (z + z0 ) ) .
In the code g L∗min is denoted by ZGLMIN2.

(37)

9. Clouds and turbulence

139

The final formulation for the quadratic mixing length l(z) is given by Eq.(B.4)
in the Appendix B of Lenderink and Holtslag (2004), where the stable mixing length ls is dropped and with the special choice p = 1 and q = 2 for thetwo free exponentsl(z) ≡ Max

q(Lint )2 + (Lmin )2 ; L∗min (z)



.

(38)

In the code g l(z) is denoted by ZGLUZ.
Presentation of the implicit equation
A new formulation of σs is seeked, avoiding calculations of correlation between the fluctuations of temperature and those of moisture, and calculationsof variances of fluctuations of temperature and moisture. For that, one uses
the three equations of System (22), namely:




θ0 2


 l








0 2
qw











0 0


 θ l qw

B2 l∂θl
w0 θl0
= −√
∂z
2 eT
B2 l
∂qw
0
= −√
w 0 qw∂z
2 eT
1 B2 l=− √2 2 eT∂qw∂θl0
w0 θl0+ w 0 qw
∂z
∂z

(39)
!

Here, B2 is one of the constants specified in the Mellor and Yamada (1982)
scheme and defined as follows:(A1 , A2 , B1 , B2 , C1 ) = (0.92, 0.74, 16.6, 10.1, 0.08)
In addition, l is the mixing length introduced above. One should pay attention that another set of values was defined and used in Mellor and Yamada(1974) and Yamada and Mellor (1975). It is necessary to take that of the1982 paper.
One rewrites Equation (11) by carrying out substitutions of the variables
using System (39). The coefficients of vertical exchange Kh and Kc are
introduced into the equation obtained:



0

 w 0 θl0
 w 0 qw√∂θlfh
and Kh = l 2 eT S
∂z
√
∂qw
fc
= −Kc
and Kc = l 2 eT S
∂z
= −Kh

(40)

140

9. Clouds and turbulence

fh = Sfc , we have Kh = Kc and we obtain for σs defined by Equation (11):
As S

al
2

σs =

q
fh
B2 S

∂qw
∂θl
.
− α1
∂z
∂z

(41)

Replacing in this last equation σs by a∆q
2Q , the following equation is obtained:1q

fh (Q1 ) =
Q1 S

∆q
∂qw
∂θl
l B2
− α1
∂z
∂z
√

= TTB .

(42)

The second member of Equation (42) – T T B – being known, Q1 is thesolution to the implicit equation:
Q1qfh (Q1 ) = T T B
S(43)

fh depends on two variables (Gh , Gm ), theWhereas in the level 2.5 model, S
fh as a function of a single
level 2 model has the advantage of expressing S
variable, namely the flux Richardson number. Indeed:fh = 3 A2 γ1 − γ2SRf
1 − Rf

where γ1 and γ2 are calculated starting from Mellor and Yamada constants
fh is a function of(see hereafter for definitions of γ1 and γ2 ). Ultimately, S
single variable Q1 , which facilitates the solution of the implicit Equation (43).
Studying the curvesqfh (Q1 ) according to the sign
Ricard (1992) studied of curves Q1 ,→ Q1 S
of (Ri)c and the relative position of (Ri)min , (Ri)moy = (Ri)h + (Ri)c /2,(Ri)max and (Ri)crit . His study thus considered eight cases reported inTable 9.1 from which one canqdraw some information about monotony of thecurve representing Q1 ,→ Q1

fh (Q1 ) .
S

We need to know how many solutions Equation Q1qfh (Q1 ) = T T B has,Sqfh (Q1 ) is not monotonous. The solution of suchwhen function Q1 ,→ Q1 S
a problemqinvolves the localization of the pairs [(Ri)h , (Ri)c ] for which curves

fh (Q1 ) are not monotonous. Refer to Ricard (1992) for a graphiQ1 ,→ Q1 S
cal discussion of the three zones (no extremum, one extremum, two extrema).

9. Clouds and turbulence

141(Ri)c < 0(Ri)crit < (Ri)min(Ri)min < (Ri)crit < (Ri)moy(Ri)moy < (Ri)crit < (Ri)max(Ri)max < (Ri)crit(Ri)c > 0(Ri)crit < (Ri)min(Ri)min < (Ri)crit < (Ri)moy(Ri)moy < (Ri)crit < (Ri)max(Ri)max < (Ri)crit

Table 9.1: Positions of minimum, mean, maximum and critical Richardson
numbers according to the sign of (Ri)c .
Successive iterations method and convergence accelerator
The method of successive substitutions is adopted to solve Equation (42).
To improve the speed of convergence of this method, one uses Wegsteinconvergence accelerator.
Sufficient convergence criterion
This method consists in rewriting Equation (42) in the following way:
TTB
Q1 = F (Q1 ) , where F (Q1 ) = q.
fhSA sufficient criterion so that successive substitutions method converges is:| F 0 (Q1 ) |< 1 ∀ Q1 ∈ IR
In other words, if in a given point Q1 ,


0 < Q1q

0fh (Q1 )Sq

fh , then | F 0 (Q1 ) |< 1 .
<2 SThus, the method of resolution by successive substitutions converges locally.
On the other hand, if in Q1 the curve meetsqfh (Q1 )Q1 S

0< 0, then| F 0 (Q1 ) |> 1 and this method diverges locally.

Application of this criterionqfh (Q1 ) admitting two extrema. For certain
One considers a curve Q1 ,→ Q1 S
q

fh (Q1 ) = T T B admits three solutions, notedvalues of T T B, Equation Q1 S
S1 , S2 and S3 .

142

9. Clouds and turbulence

Between the two extrema, the part of curve has a negative slope –local
divergence of the method–, while the other two parts have a positive slope –local convergence of the method–. Consequently, the successive substitutions
method discards solution S2 .
From a physical point of view, one rejects solution S2 located on the decreasing part of the curve for the following reason: in the vicinity of S2 , theresponse to an increase in ∆q is at the same time an increase in T T B (cf
Equations (42) and (43)) and a reduction in Q1 ; this is not compatible with
Equation (12) which shows that Q1 and ∆q vary in the same direction (aand σs are positive).
The two physically acceptable solutions S1 and S3 correspond to the samelarge-scale environment, but the cloud cover and sub-grid turbulence differ.

Wegstein convergence accelerator
The Wegstein convergence accelerator method consists in modifying the successive substitutions method in order to increase its convergence speed andsystematically impose it. Details on this method are given in Gourdin and
Boumahrat (1983). In order to maintain the divergent solutions generated
by the successive substitutions method, it is necessary to apply Wegsteinmethod only in the case of convergence:• One calculates α, coefficient which optimizes convergence of the iterative process, in the following way:

xn








∆






 α

= F (xn−1 )F (xn ) − xn
xn − xn−1
1
=
1−∆

=

(44)

• If α < 21 , one applies the method of successive substitutions.
• If α > 12 , one calculates xn+1 by the formula of Wegstein method :xn+1 = xn + α [F (xn ) − xn ]Number of iterations
A study by Ricard (1992) highlighted two significant remarks:• the transition from the stable to the unstable state is perfectly reachedafter two steps

9. Clouds and turbulence143• the transition from the unstable to the stable state is slower: after 10
iterations a perfect accuracy is not yet reached
From this study, it proved to be necessary to choose an iteration number
allowing the best compromise between precision of the results and speed of
calculation. This number was fixed to 2.

Initialization of Q1
The best initialization of Q1 would consist of starting from the value of Q1
at the previous time step for each grid point. Indeed, this method would beof two interests:• a better accuracy with respect to an arbitrary initialization of Q1• a tendency to remain in the same state as at the previous time step,
namely stable or turbulent: this introduces a hysteresis effect
In Arpege, saving the value of Q1 at the previous time step is heavy to set
up because it requires the introduction of a 3-d array (latitude, longitude,level). This kind of process is usually done for 2-d variables like Halsteadcoefficient, planetary boundary layer depth or roughness length. Recently,
convective precipitation of the previous time step, necessary to the calculation of convective cloud cover, was fully saved in memory (pseudo-historicalvariables).
In the current code, the initial value of Q1 is zero, and no pseudo-historicalvariable is used.

1.6 Summing up
The use of both a sub-grid scale condensation scheme and sub-grid scale level
2 turbulence scheme gives all the ingredients for the calculation of the cloud
fraction and the liquid water content.
As far as the coefficients of turbulent vertical exchange are concerned, theircalculation involves 2.5 level formulas (Galperin et al., 1988) in order to avoid
the feedback, in further calculations, of the inaccuracy from the solution of
implicit Equation (42). Indeed, it is not unlikely that numerical solution Q∗1fh (Q∗ ) does not satisfy
is somewhat q
far from real solution. In some cases, S
1fh (Q1 ) = T T B.
Equation Q1 S

144

9. Clouds and turbulence

In the current code, some constants derived from Mellor and Yamada (1974,
1975, 1982) and Galperin et al. (1988) are defined in modules (for GALP,TURB, TYM[1-5] ) or locally at the beginning of the subroutine:
ZGAMMA1 = γ1 =1(B2 + 6 A1 )2 A1
; ZGAMMA2 = γ2 =
−
3
B1
B1

ZBETA0 = β0 = γ1 + γ2 − 3


A2
2 A1



A1
A2

ZBETA1 = β1 =ZBETA2 = β2 =





β0
γ1 − C1 + (6 A1 + 3 A2 /B1 )γ 1 − C1
β0

ZGAX1 = g1 = A2

A1
B1

1−6

A1
B1



; ZBETA3 = β3 =



γ1
β0



ZGAX2 = g2 = (−3A2 ) (6A1 + B2 ) ; ZGAX3 = g3 = A1 (1 − 3C1 − 6A1 /B1 )ZGAX4 = g4 = (−3A1 A2 ) [(B2 − 3A1 )(1 − 6A1 /B1 ) − 3C1 (B2 + 6A1 )]ZGAX5 = g5 = −9A1 A2 ; ZGALP2 = (GALP)2 /TURB
These constants being available, one deduces from them the limit values for
the Richardson numbers and Mellor and Yamada functions:
γ1
γ1 + γ2ZRFCRIT = (Rf )crit ="(Rf )crit
ZRICRIT = (Ri )crit =β3 − (Rf )critfh )crit =ZSHCRIT = (S# β2 − (Rf )crit
2 β1



g1
fh )max = 3A2 (γ1 + γ2 ); ZSHMAX = (S1 − g2 ZGALP2g
ZSMCRIT = (Sm )crit =g3 − (ZGALP2) g4[1 − (ZGALP2) g2 ] [1 − (ZGALP2) g5 ]Here is a summary of the steps which must be followed:

9. Clouds and turbulence145

1. Calculation of vertical gradients between layer (JLEV − 1) and(JLEV + 1), for the variables ZDPHI = ∆(qv ), ZDTL = ∆(T + φ/cp )and
ZDVENT = ∆(u2 + v 2 ).
2. Calculation of gradient Richardson number ZRIH = (Ri)h .
3. Calculation of complementary gradient Richardson number ZRIC =(Ri)c .
4. Calculation of mixing length ZLE.
5. Calculation of ZTTB = T T B for implicit Equation (43)"

T T B = ∆q l

p

B2

∂qw
∂θl
− α1
∂z
∂z

#−1

6. Calculation of ZQ11 = Q1 from (Ri)h , (Ri)c and T T B at mesh scale:solution of implicit Equation (43) using the successive
substitutionsqmethod and Wegstein convergence accelerator for Q1
One iterates ITER = 2 times the following actions:fh (Q1 ) = T T B.
S(a) calculation of ZRI = Ri = (Ri)h + F2 (ZQ11) (Ri)c(b) calculation of ZRF = F ORF(ZRI),
hipwith F ORF (ZRI) = β1 ZRI + β2 − F ODD(ZRI)
and F ODD(ZRI) = (ZRI)2 + 2 ZRI (β2 − β3 )/β1 + (β2 )2p(c) calculation of ZQ12 = T T B/ F OSH(ZRF),
with F OSH(ZRF) = A2 [γ1 − γ2 F OGAM M A(ZRF)],
where F OGAM M A(ZRF) = ZRF/(1 − ZRF)(d) calculation of ZRI = Ri = (Ri)h + F2 (ZQ12) (Ri)c ,then of ZRF = F ORF (ZRI)p(e) calculation of the new ZQ13 = T T B/ F OSH(ZRF)(f) calculation of the new ZQ11 = ZQ12 + ZALPHA(ZQ13 − ZQ12),
where
ZALPHA = α is the parameter for convergence acceleration usedand defined in Equation (44). It depends on
ZDELTA = ∆ = (ZQ13 − ZQ12) / (ZQ12 − ZQ11).
7. Use of the last Q1 = ZQ11 to calculate ZIGMAS = σs by solving
Equation (12), i.e.: σs = (a ∆q)/(2 Q1 )8. Calculation of F2 (Q1 ), assuming that the distribution function of t isasymmetrical (System (16))

146

9. Clouds and turbulence

9. Calculation of the gradient Richardson number
ZRI = Ri = (Ri)h + F2 (Q1 ) (Ri)cfh by solving implicit Equation (42). One ob10. Calculation of ZSH = S
tains:
fh = (T T B/Q1 )2
S

11. Calculation of ZGH = Gh using the level 2.5 formula of Galperin et al.
(1988):
fh =S

g1
1 + g2 Gh; then : Gh =fh
g1 − Sfh
g2 S

12. Calculation of ZGM = Gm defined by: Gm = − Gh /Rig
13. Calculation of S
m by the level 2.5 formula of Galperin et al. (1988):

ZSM = Sm =( g3 + g4 Gh )( 1 + g2 Gh ) ( 1 + g5 Gh )14. The calculation of the turbulent kinetic energy (eT ) can be made bysolving Equation (24) for Gh , in order to deduce the “mean quadratic
speed”:
ZKE =

√

2 eT

l
= √
Gm

s



∂u
∂z

2



+

∂v
∂z

2

15. One deduces turbulent vertical exchange coefficients:√g
ZKM = Km = l 2 eT S
m








ZKH = Kh = l√fh
2 eT S

coefficient of vertical exchange for u and v

coefficient of vertical exchange for T and q

16. Calculation of F0 (Q1 ) and F1 (Q1 ), assuming that the distribution function of t is asymmetrical (System (16))
17. Calculation of stratiform cloud fraction R and stratiform liquid water
ql by System (14). One obtains PNEBS = R = F0 (Q1 ) and PQLIS =ql = (2 σs ) F1 (Q1 )18. Calculation of the Brunt-Vaïsala frequency used in ACDRAG:PNBVNO = N 2 /(ρ g)2 , where N 2 = β (∂θv /∂z) is a symbolic squared
value which can be negative. Concretely, in the code of ACNEBR,
with ρ = p/(R T ), one obtains:
PNBVNO = N 2 /(ρ g)2 =R2 T∆(T + φ/cp )p2 ∆φ

9. Clouds and turbulence147

1.7 Tuning the scheme
At the time of the set up of the parametrization, numerical problems appeared:• mixing length reaches too high values in stable zones• cloud liquid water becomes unrealistic when mixing length is too large
These two problems are solved by limitating Gh , l and T T B.

Lower limitation of Gh in stable zones
It is necessary to limit l in the stable layers, i.e. where the Richardson
number is higher than the critical Richardson number. The restriction usedby André et al. (1978) and Galperin et al. (1988) is defined as:

l≤

0.53

√

2 eT

N

(45)

v 1/2 is Brunt-Vaïsala frequency.
where N = (β ∂θ∂z )22As for Gh = − l2 N
eT , the limitation on l is equivalent to the following limitation on Gh :Gh ≥ −(0.53)2 = −0.28 = (Gh )crit(46)

Ricard (1992) noted that taking this critical value produced unrealistic downward heat fluxes in the polar areas (up to −22.6 W/m2 in zonal mean). This
remark resulted in introducing coefficient T U RB such as:

Ghcrit = −(0.53)2
0.28=−
T U RBT U RB(47)

During the first adjustments of this coefficient, a value 0.2 had been retained,
for which the downward heat fluxes showed more realistic values in theseareas, namely lower than −10.6 W/m2 . But currently the neutral value
T U RB = 1 is used.

148

9. Clouds and turbulence

Negative moisture
A study of the mixing length showed that unrealistic results can appear
when the probability of negative moistures is not negligible. Indeed, it isshown thereafter that for very high mixing lengths the cloud liquid waterobtained takes too large values. This phenomenon can easily be highlighted.
qalfh ∂qw − α1 ∂θl shows that σs ∼ ( K l ) when l −→Relation σs =
B2 S
2
∂z
∂z
+∞ (K being a positive constant), therefore that lim l→+∞ (Q1 ) = 0 since
a ∆q
Q1 =
. Thus, ql = 2 σs F1 (Q1 ) drives to:2 σslim (ql ) = 2σs F1 (0) i.e. ql ∼ Cl as l −→ +∞

(48)

l→+∞

It was thus shown that when the mixing length tends towards infinity, thecloud liquid water grows towards infinity so that it reaches values higher
than qw .
To solve this problem, it appears essential to limit the occurrence of negative
0 ≤q )
moistures. However, a direct limitation on qw (to impose qw ≥ 0 i.e. qw
w
is not easy to implement due to the use of variable s, linear combination of
the fluctuations of moisture and temperature.
Hence, a condition is imposed not on qw but on its standard deviation σqw .
It is supposed that specific moisture follows a Gaussian law of the type:"1(qw − qw )2
G(qw ) = √exp −2σqw 2
2πσqw

#

(49)

Thus the limitation of the occurrence of negative specific moistures is carriedout by the following inequality:
Z 0−∞

G(qw ) dqw <  where  is an arbitrary constant(50)

This inequality is equivalent to:qw
1> A , where A is such that √
σqw
2π

Z A
0

2

e−t /2 dt =

1
−
2

(51)

It appears that  = 0 and  = 1/2 correspond to A = +∞ and A = 0,
respectively. To impose the preceding condition on qw comes to eliminating9. Clouds and turbulence149

the values from Q1 in the vicinity of 0. Indeed, it is sufficient that thefollowing inequality is met:∂qw∂z
|Q1 | >
∂qw
∂θl
qw
− α1
∂z
∂z
A |∆q|≡ (Q1 )min

(52)

q

fh (Q1 ) = T T B, one can obtain a value (T T B)min from
Using equation Q1 S
T T B satisfying inequality | T T B |> (T T B)min . Consequently, the equation:

TTB =

1
∆q
∆q
√
=⇒ lmax =(T T B)min B2 ∂qw
√
∂qw
∂θl
∂θl
l B2
− α1
− α1
∂z
∂z
∂z
∂z

allows to determine the maximum value for mixing length, that is to say
lmax , by taking for T T B its minimal value (T T B)min .
The above mentioned algorithm, which aims at limiting the probability ofnegative moistures, resulted in introducing into the scheme a new adjustableparameter ( or A). A study by Ricard showed that any value of A ranging between √1 and 2 appeared reasonable. In practice, this value is set to
STTBMIN ( 3).

2 Turbulent coefficients in stratosphere
On the model levels where cloudiness is calculated, one uses the above described turbulence scheme to calculate the exchange coefficients for the vertical diffusion (starting from level KTDIAN). At the higher levels, one uses
Louis scheme (ACCOEFK). For more details on this parametrization of atmospheric turbulence, see Louis (1979), Louis et al. (1982) and Geleyn(1986).
We have:
∂ψ
1 ∂
∂ψ
=
ρK
∂t
ρ ∂z
∂z




where K is the exchange coefficient (m2 /s). One seeks flux Fψ such as
∂Fψ
∂ψ
= −g
∂t
∂p

150

9. Clouds and turbulence

but as:Fψ =ρ K g ∆ψ∆φone stores:− KU ROV = −

ρK g
∆φ

and one expresses:
K = g l2 |∂~u
| f (Ri )∂φ

Thepmixing length l takes the value λU (ALMAV) for momentum and λT =λU 3d/2 for energy.

2.1 Stability
Let Ri be the Richardson number:
Ri =

g
cp T

∂s
∂z
∂~v 2
∂z

This number is calculated by taking into account, under the approximation
to the first order (with a 15% accuracy):
cp a
Ra
≈
cp v
Rv
of the influence of moisture on buoyancy. Thus one “moves” the particles, tocompare their level of buoyancy, in the middle (in geopotential) of the layer(see ACHMT in chapter 11). One thus replaces ∆s/Cp T by:
Rj Tj − Rj+1 Tj+1 + ∆φRa /cp a(Rj Tj + Rj+1 Tj+1 )/2
As in ACHMT, one can impose an upper limit (Ri)cr on Ri by taking a
non-zero value for USURIC (= 1/(Ri)cr ):if Ri > 0 then Ri ←Ri
1 + Ri/(Ri)cr

One represents atmospheric stability by value ZIS. The atmosphere is stable
if Ri > 0 and unstable if Ri < 0 . One calculates the sign of Ri startingfrom ZSTA=Ri · k∆~v k2 .

9. Clouds and turbulence

151

2.2 Calculations for the wind components
One calculates the coefficient of vertical diffusion for wind (PKUROV).
In the stable case:f (Ri ) =1√
1 + 2bRi / 1 + dRi

and in the unstable case:f (Ri ) = 1 −2bRi
1
1 + 3bc √
27




λT
z + z02p| Ri |

2.3 Calculations fore temperature and moisture
One calculates the coefficient of vertical diffusion for energy (PKTROV).
In the stable case:f (Ri ) =1√

1 + 3bRi 1 + dRi

and in the unstable case:f (Ri ) = 1 −3bRi
1
1 + 3bc √
27




λT
z + z02p| Ri |

2.4 Brunt-Vaïsala frequency
One calculates here the squared Brunt-Vaïsala frequency N divided by g andby density:


N
ρg

2

=

ZSTA
ZSTA(RT )2=(ρ∆φ)2(p0 ∆φ)2

p0 is the pressure at half-levels. RT is calculated there by simple averaging.
The symbolic square of the above formula can be negative in the case of
unstable atmosphere.

152

9. Clouds and turbulence

3 Convective cloudiness
The convective clouds are poorly represented by the above scheme as theycorrespond to a small fraction of the mesh, but to a large quantity of liquidwater. The cloud condensate and corresponding cloud cover are describedby diagnostic relations. These relations involve convective fluxes which must
thus be saved at the previous time step, since calculations of the convection
are carried out after the vertical diffusion, which uses the results of theradiation scheme to perform an implicit temporal discretization. However
the calculation of cloudiness must precede the radiation scheme. If LNEBCO
is activated, the following calculations are carried out.
Let qcc be the specific cloud moisture (in liquid and solid form); it is written
as:









qcc = qcx  1 − exp −

αc g∂Fpc  ∆t  
∂p


qcx

with
• Fpc convective precipitation flux
• ∆t time step
• αc tunable coefficient (QSSUSC).
• qcx maximum specific humidity of the condensate in a model mesh
(QSUSX).
One deduces convective cloudiness:
nc = αnc qcc
where nc convective cloudiness and αnc adjustable coefficient (QSNEBC).

4 Total cloudiness
If LRNUMX is activated (option LRAY), or if NOVLP< 3 (option LRAYFM),
or if NOVLP15< 3 (option LRAYFM15), the two types of condensate andcloudiness (stratiform and convective) are combined by maximum overlap:
qc = max(qcs , qcc )n = max(ns , nc )

9. Clouds and turbulence

153

Otherwise, random overlap is used:
qc = qcs + qcc
n = ns + (1 − ns ) nc
Then one divides the condensate by cloudiness (limited to XNBMAX), to
obtain condensate in the cloud. One obtains a proportion of condensate perunit of cloud mesh (qc /ns ). Then one makes the partition of qc /ns in aliquid part (PQLI) and a solid part (PQICE). One introduces a function oftemperature giving the proportion of ice into a cloud cell. The experimental
measurements carried out by Matveev (1984) and resumed by Rockel et al.
(1991) recommend a curve having a complement-Gaussian shape, null for
T = Tt and having for width 15 K at e−1 .
This function is modeled by postulating that the proportion of ice, at temperature T < Tt , is related to the integral of the difference between thesaturation functions of liquid and solid water. One thus writes:f (T ) =

 Z
Tt



ew (t) − ei (t) dt



 ZTT

t

if T < Tt

ew (t) − ei (t) dt










0

0

otherwise

The usable form of this expression is obtained by a Gaussian adjustment:f (T ) = δTt 1 − exp

−1(T − Tt )2
2∆T 2



where Tt is the temperature the triple point and δTt the Heavyside function,
being 1 if T < Tt , 0 otherwise
One uses for ∆T the the difference between the temperature of triple point
and the abscissa of the maximum of ew (T ) − ei (T ), that is to say 11.82 K,
which provides a good approximation of the integral. The variable used in
the code is RDT (in YOMCST).
The value of f (T ) being known, one calculates the liquid/ice partition in thecloud by:• ql = [1 − f (T )] qc• qi = f (T ) qc

154

9. Clouds and turbulence

10Large-scale precipitation

1 Description of the scheme
This scheme results from the statistical precipitation scheme of Smith (1990).
It must be associated with the statistical cloud scheme (ACNEBR, see Chapter 9). This scheme has been used in the former versions of Arpege-climat.
Ricard (1992) explained why the first tests of the Kessler-type precipitation scheme (1969), did not give satisfaction when the old cloud scheme
ACNEBT were replaced by ACNEBR. The cloud amount was too weak,reaching 30% in global average. The Kessler-type scheme type eliminatedany supersaturation at the mesh scale. Coarse tests to allow supersaturation
up to 110% showed positive impacts and it was decided to connect the rateof precipitation to the liquid water amount which is calculated by statisticalscheme ACNEBR, thanks to Smith (1990) scheme, which itself results froma simplification of the Sundqvist (1978) scheme.
Thus, in ACPLUIS, one takes into account the quantity of condensate (PQLIS=liquid+solid) and of stratiform cloudiness (PNEBS) in each mesh, bothalready calculated by statistical scheme ACNEBR. ACPLUIS thus fits insame approach as the statistical cloud generation scheme, in the sense thatit includes the concept of sub-grid variance.
The atmosphere is scanned from top to bottom, in the direction of precipitation. These calculations led in a layer cannot be independent of calculationsof the directly upper layer, because there is a mass transport. Therefore,
there is a necessarily transmission of the information from layer to layer(precipitation flux and proportion of snow).
The random overlap assumption is made here, which supposes that the precipitation flux coming from the upper layer is the same in term of unit of

156

10. Large-scale precipitation

mesh area as in term of unit saturated (or unsaturated) mesh area. This assumption makes it possible to treat independently the saturated part and theunsaturated part of the mesh, since each one receives the same precipitation
flux at the top of the mesh.
It should be noted that in Arpege-climat, the maximum-random overlap assumption is used in ACNEBR to calculate “radiative cloudiness”, (fortransmission to the radiation scheme).
The parametrization used is mainly that of Smith (1990), however the Kessler(1969) formulation concerning the evaporation of precipitations has beenused.
Three processes are involved: precipitation, evaporation and melting or freezing. They are dealt in the next three sections.

2 Precipitation in a cloud layer (Smith)
The Smith (1990) model requires a parametrization of the reduction in cloudwater due to precipitations. The conversion rate of cloud water into precipitation depends on the phase of water. One supposes here an abrupt transition between the liquid state and the solid state at the temperature of triple
point Tt .
One uses a partition of condensate qc (PQLIS) in liquid water ql and solid
water qif (T ) = δTt
where Tt is the temperature of triple point and δTt Heavyside function ,yielding 1 if T < Tt , 0 if not.
It should be noted that in the other parts of the code of Arpege-climat
one uses a Gaussian and continuous function in the transition from the liquid
form to the solid form (with a characteristic width ∆T = 11.82 K (RDT)
and with a coexistence of the two phases between Tt − ∆T and Tt ).
The value of f (T ) being known, one calculates the partition liquid/ice qc by:• ql = [1 − f (T )] qc• qi = f (T ) qc

10. Large-scale precipitation

157

2.1 Liquid phase
The parametrization of the tendency of liquid cloud water due to precipitation is defined as (Smith, 1990):
"

(

"

∂ql
ql /ns= − Ct 1 − exp −
∂t
Cw


2 #)

#

+ Ca FPh ql = − [ A ] ql

(1)

where
• Ca , Ct and Cw are constants, respectively TCA, TCT and TCW in
namelist• ns is stratiform cloud fraction (PQLIS) and ql stratiform liquid water• FPh is precipitation flux per unit area coming from the upper layer
The exponential factor inhibits the liquid water conversion per unit of cloud
mesh ql /ns into precipitations, if this one is weak compared to Cw .

2.2 Solid phase
The tendency of cloud solid water (per unit of mesh) due to precipitation isexpressed in the following way (Smith, 1990):
∂qi
=
∂t



FPh
ρ ∆z





−

vf
∆z



qi = B − D q i

(2)

where
• ∆z is layer depth• vf = TVF is falling speed for ice/snow. One supposes here that TVF =1 m/s.

2.3 Precipitation flux at the base of the saturated part of themesh
In the cloud part, the principle consists in calculating the tendencies ofmoisture due to the process of condensation in the sub-grid cloud, then to158

10. Large-scale precipitation

convert these tendencies into precipitation flux. One can write the formulas(1) and (2) in the form:

∂ql



 ∂t

= −A ql




 ∂qi

= B − D qi solid case∂tliquid case(3)

where A, B and D are defined in Equations (1) and (2).
In the current version of ACPLUIS, the total value qc = ql + qi intervenes
everywhere in Equation (3). One obtains tendencies (∂qc /∂t)(l) = −A qc and(∂qc /∂t)(i) = B − D qc which undergo the later recombination:∂qc /∂t = [1 − f (T )] (∂qc /∂t)(l) + [f (T )] (∂qc /∂t)(i)(see the following sub-section). One should use separately ql and then qi in
System (3), to obtain directly∂ql /∂t = −A ql and ∂qi /∂t = B − D qi
as envisaged in Smith (1990). There would be more simply∂qc /∂t = ∂ql /∂t + ∂qi /∂t.
The scheme remains consistent as long as it does not make coexist the solidand liquid phases, as it is currently the case in this part of the code whichuses Heavyside function f (T ) = δTt .
For numerical stability reasons, one uses an implicit scheme where the second
member of each one of these two equations is taken at time t + ∆t. For
example, the first equation of System (3) is written as:


∂ql
∂t

(t)

"

∂ql
= −A (ql )t+1 = −A (ql )t + ∆t∂t
(t) #(4)

One then obtains classically the Smith (1990) equations of moisture tendency
in their implicit form:

∂ql





 ∂t

−A qlt≡
1 + A ∆t



=




∂qi




B − D qit=≡
1 + D ∆t



∂t

∂qc
∂t



∂qc
∂t



liquid case(l)(5)
solid case(i)

10. Large-scale precipitation

159

2.4 Recombination of the total condensation flux
With the above values for function f (T ), the total tendencies (solid andliquid) due to the condensation of the Smith (1990) scheme are obtained
starting from the combination of the partial tendencies given by System (5):∂qc
∂qc
= [1 − f (T )]
∂t
∂t






+ [f (T )]
(l)

∂qc
∂t


(i)

The conversion of the moisture tendencies into precipitation flux is given by:


∂qn
g
=
(FPs − FPh )∂t precip ∆p

(6)where ∆p is the layer thickness in pressure (positive).
In the code, the moisture tendencies are positively counted if they contributeto precipitations.
One keeps the total precipitations flux of Smith for the values of qv which
are greater than QSMIN = 10−4 . For values less than QSMIN, a Kessler(1969) scheme is coded and replaces that of Smith (1990). Consistently, thesame threshold QSMIN is used in ACNEBR to put cloudiness at a residual
value (10−12 ) if qv < QSMIN. The motivation of this threshold is to avoid
having too strong cloudiness and precipitation in higher troposphere and polar stratosphere. Elsewhere, for high altitudes, safety relies on a limitation
along the vertical which is ensured through maximum levels given in NAMTOPH (where ETNEBU=100 hP a is the highest level for both ACNEBR
and ACPLUIS).

3 Evaporation in the unsaturated parts (Kessler)
A precipitation flux falling into an unsaturated zone in the lower layer isreduced at the base of this sub-mesh by evaporation phenomena for which
Kessler (1969) proposes the following parametrization:√∂ FP= Evap (qw − q)
∂ (1/p)where(7)

Evap = CEvap · [1 − rme (1 − RV )]Here, CEvap = EVAP = 0.48 107 is an empirical coefficient, RV = REVGSL =80 is the ratio of evaporation speed between snow and liquid water, and rme
are the fictitious proportion of snow in precipitations.

160

10. Large-scale precipitation

By integrating Equation (7) between the top and the base of each layer, one
obtains FPc , precipitation flux, at the base of the layer per unit of area of
unsaturated mesh:
FPc =

q

∆p 2
p2


FPh − Evap (qw − q)(8)

Applying these two processes, namely condensation and evaporation, leads
to the knowledge of the precipitation flux at the base of the layer per unitof area of unsaturated mesh (FPc ) and of saturated mesh (FPs ). Obtaining
the precipitation flux at the base of the layer per unit of area of mesh (FPb )is done by:
FPb = ns FPs + (1 − ns )FPc

(9)

4 Melting or freezing (Kessler)
In the configuration which consists in taking into account the cryoscopic
cycle, the equation governing the phenomenon of snow melt or freezing of
liquid water is:∂rf
T − Tt= Fmelt √,∂ (1/p)FP(10)

where Fmelt represents the melting of precipitation:
Fmelt = CF onte · [1 − rme (1 − RV )] .

(11)

Here, CF onte = FONT = 0.24 105 , with the same definitions as previously
for RV = REVGSL = 80 and for rme .
The component of the snow proportion (rf ) due to melting or freezing of
new precipitation results from the integration of Equation (10) between the
top and the base of the layer, whose result is:
rf = Fmelt

Tt − T∆pp
0.5( FPh + FPb ) p2
p

(12)

To obtain the actual snow proportion (rn ), one adds to rf the snow proportion in the precipitation flux (rg ).
The term rg is calculated from the snow proportion rn of the upper layer
and precipitation flux. It depends on the part of the mesh in which it iscalculated: saturated part or unsaturated part.

10. Large-scale precipitation

161

4.1 Saturated part of the meshTwo cases arise according to whether one is above or below the triple point:• If T > Tt
There cannot be condensation in solid phase. If vapor condensates, liquidwater is formed. Noting rn the snow fraction in the upper layer and FPbc
the precipitation flux at layer base, due only to condensation in the cloudy
sub-mesh, the snow fraction produced by precipitation (rg ) is:
rg = rn

FPh
FPbc

(13)

• If T < Tt
If there is condensation, snow is formed; rg is expressed using theequation:rg = 1 − (1 − rn )

FPh
FPbc

(14)

4.2 Unsaturated part of the mesh
The snow proportion due to precipitation does not change in the non-cloudy
zone: the evaporation and the sublimation of precipitation are done in thesame proportions.
The process of melting or freezing intervenes in the determination of thenature of the precipitation. One obtains stratiform precipitation fluxes inliquid form and solid form by multiplying the precipitation flux at the baseof each layer by (1-rn ) and rn respectively.
One uses here a partition of condensate in rain and/or snow according to a
Gaussian and continuous formulation:
n

h

io

f (T ) = δTt 1 − exp −(T − Tt )2 /(2 ∆T 2 )One uses for ∆T the difference between the temperature of triple point andthe abscissa of the maximum of ew (T ) − ei (T ), i.e. ∆T = 11.82 K, which
provides a good approximation of the integral. The variable used in the codefor ∆T is noted RDT in YOMCST.

162

10. Large-scale precipitation

11
Vertical diffusion

1 General comment the scheme
This scheme is associated with the cloud scheme ACNEBR and has beenused in the former versions of Arpege-climat. The coupling with Surfex implies that surface fluxes, radiative properties and surface diagnostics
variables are computed inside the Surfex routines. The parametrization of
turbulent fluxes in the surface boundary layer, i.e. between the surface andthe last level of the model is described in details by Louis (1979), Louis etal. (1982), and Geleyn (1988).

2 Equation of the vertical diffusion for a conservative
quantity ψ
Variable ψ is used here in the place of the horizontal components of wind (uand v), of specific moisture vapor (qv ) or of dry static energy (s).
(with :

s = cp T + φ (φ : geopotential)
cp = cpa + (cpv − cpa ) qv

One has:
∂ψ
1 ∂Fψ
=
∂t
ρ ∂z

(

with

Fψ = −ρψ 0 w0 : flux oriented downwards
w : vertical velocity

Taking into account the hydrostatic relation (∂w/∂z = 0), the vertical balance of the atmosphere is written as:
dp = −ρgdz = −ρdφ164

11. Vertical diffusion

One thus obtains:
∂Fψ
∂ψ
= −g
∂t
∂p

3 Parametrization
3.1 Parametrization in the free atmosphereLet:Fψ = ρgK∆ψ
= Ḱ∆ψ
∆φ(with

K : turbulent exchange coefficient (in m2 s−1 )Ḱ : computed coefficient (see ACNEBR)

3.2 Parametrization at ground level
All details are given in the Surfex Scientific Documentation.

4 Algorithm of flux calculation
The implicit coupling of the atmosphere model with Surfex implies thecomputation, by the atmosphere model, of the coefficients for vertical turbulent diffusion from the top to the last but one layer for : momentum, dry
static enrgy, potential temperature, specific humidity (including a negative
humidity correction), liquid and solid water (ACDIFV1).
The atmosphere calls the surface model (ARO_GROUND_PARAM). Surfex will use atmospheric variables from the lowest atmospheric layer :• air temperature• specfic humidity• wind components• pressure• surface pressure• CO2 concentration

11. Vertical diffusion165• dry air density• height of model interlayers• orography of atmospheric model• cosine of zenithal angle at t
• cosine of zenithal angle at t+1• liquid precipitation surface flux
• snow precipitation surface flux
• graupel precipitation surface flux
to compute the surface fluxes and variables.
The surface model send back to the atmosphere :• surface flux of potential temperature• surface flux of water vapor• surface flux of scalar or flux of chemical variables• surface flux of CO2• surface fluxes of horizontal momentum• direct albedo for each spectral band• diffuse albedo for each spectral band• surface emissivity• surface radiative temperature
Then, the atmosphere model procedes with the computation of the vertical turbulent diffusion by back substitution, vertical turbulent fluxes, andthermal radiation fluxes correction (ACDIFV2).

4.1 Flux of momentum:Fu and Fv
Let ψi be ui or vi . One has then:ψi+ − ψi− = ψit+∆t − ψit−∆t = −g∆t[Fψi − Fψi−1 ]
δpi

166

11. Vertical diffusion⇐⇒ ψi+ − ψi− = −g∆t++[Ḱi (ψi+ − ψi+1
) − Ḱi−1 (ψi−1− ψi+ )]
δpi

This equation is written in matrix form (wi,j = g Ḱi ∆t/δpj ) and (xN =g Ćd ∆t/δpN ):
ψ1−
ψ1+

 ψ+  
ψ2−


 
2


 
.
.


=

 +
 
−

 ψN −1  
ψN
−1
−
+
ψN
+ ψs xs
ψN


M







where M is the matrix:


1 + w1,1
−w1,1
 −w
1
+
w
1,2
1,2 + w2,2 −w2,2


.
.
.



.
.
(0)
−wN −1,N(0)−wN −1,N −1
1 + wN −1,N + xN









The surface conditions impose that ~vs = (us , vs ) = ~0 thus ψs xs = 0 One uses
the method of Gauss to solve this system. The variables then are considered:
ZELIM to indicate the element sub-diagonal by which it is necessary tomultiply the preceding line in order to eliminate it by subtraction, ZMUL
to indicate the multiplicative factor intended to put the diagonal at 1. A
system is then obtained of the type:


1








a1,2
1 a2,3
.

(0)

(0)
.
1








aN −1,N 

1+
By going up the values of ψNwith ψ1+ , one deduces from this system atmo++
spheric fluxes in the form: wk,l ∆ψi,i+1
and surface flux: ρCd k~v k ~vN.

4.2 Flux of heat Fqv and Fs
Let ψi be si or qvi . One has in the same way:ψi+ − ψi− = −g∆t++[Ḱi (ψi+ − ψi+1
) − Ḱi−1 (ψi−1− ψi+ )]
δpi

11. Vertical diffusion

167

But the equation of evolution of Ts is written as:
X∂Ts
2π(Tp − Ts )= Cs ( Fenergy ) +∂t
τ1

It couples through the latent heat flux the evolutions of sN , Ts and qvN .
One thus will simultaneously solve the 2 implicit systems by adding to thema linearized equation for Ts :
∂Ts
∂t

2π
(Tp − Ts+ ) + Cs {ε(F T − σTs4 ) + F S (1 − α)τ1− ρCh k~v k {[cpa + (cpv − cpa )(HU qsat (Ts ) + HQqvN )]Ts=

+ φs − (cp T + φ)+N++ [LT =0 + (cpa − Cw )Ts ][HU qsat (Ts ) + (HQ − 1)qvN]}}

By considering the following relations of linearization:

+4
−4
−3
+
−

 σTs = σTs + 4σTs (Ts − Ts )∂q qsat (Ts+ ) = qsat (Ts− ) + sat (Ts+ − Ts− )∂Ts−

and by removing terms (Ts+ − Ts− ) of the second order, one obtains:
2π∆t3+ Cs ∆t{4εσTs− + ρCh k~v k (cps −
τ1

∂qsat
−
−
+Ls − HU
+
(c
−
C
)(q
−
q
))}
pa
w
vs
vN
∂Ts −
h
2π∆t
Ts− +
Tp + Cs ∆t εF T + F S (1 − α) + 3εσ(Ts− )4
τ
(1
)
+
−
− ∂qsat−+ρ Ch k~v k (φ + cp T )N − φs + Ls HU (Ts− qsat Ts )∂Ts−

Ts+ 1 +n−−−+Ts− (cpa − Cw )(qvs− qvN) + (cpv − cpa )qvs

o

+
−
+ L−
qvN
s (1 − HQ) − (cpv − cpa )Ts



=

i

(1)

Remark: in Equation (1), in contradiction with the principle of linearization,
one considers first term HQ qvN only in index − , in order to obtain a perfectly
tri-diagonal matrix (see term 0∗ in matrix A below).

168

11. Vertical diffusion

These equations are written in matrix form:
s−
s+
1
1
 s+   s−  2   2   .   .  +  
 sN   +  
 T = B  s  
 q+   vN    .   . 
 +   − 
 qv2   qv2 
−
+
qv1
qv1


N







where N is the matrix:


1 + w1,1
−w1,1
 −w
1
+
w
1,2
1,2 + w2,2 −w2,2


.
.
.



d










(0)

e
A

b

(0)

c
. .
.

.
.
−w1,2
−w1,1 1 + w1,1


















where central matrix A is in the form:−wN −1,N

0

0

A21
−CSV T
0−wN,N CT V S
0∗
1 + CT V T−CQV T−wN,N CT V Q
A4300−wN −1,Nwith A21 = 1 + wN −1,N + wN,N and A43 = 1 + wN −1,N + wN,N CQV Q , andcentral vector B in the form:s− + wN,N RHSS N −
Ts + RHST−qvN+ wN,N RHSQ

Note: one uses the elimination method of the preceding paragraph, but with
2N +1 lines. Terms CT V S with RHSQ are obtained by formal identification
with the complex expression of Equation (1) and the formation of surface
fluxes computed by Surfex.

11. Vertical diffusion

169

5 Parametrization of planetary boundary layer depth
Subroutine ACPBLH calculates the height of the planetary mixing layer
according to surface fluxes and of the vertical stability of the air column. This
pseudo-historical variable is used in the formulation of the vertical profile ofmixing length (see Chapter 9). For more details on this parametrization, see
Troen and Mahrt (1986).
This parametrization is activated if LNEBCO and LNEBR are .T., and if
AHCLPV is equal to zero.
If AHCLPV> 0, the parametrization is not taken into account and the mixing layer depth takes exactly the value of AHCLPV. There is thus no variation in time and space of this parameter.
If AHCLPV=0, the the mixing layer depth becomes a pseudo-historical variable of the model. This variable is given according to the variation of the
Richardson number for mass Ri (bulk number ) to a critical value Rc .
Ri (j) =with(θv (j) − θs )g Zj
θvs |~v (j)|2



 θv

 |~
v|

Zj




s

: potential virtual temperature: wind modulus: height of level j: index for surface value

Ri is diagnosed between each level of the model, (sub-scripted j in the equation) and surface.
Let j and j − 1 be the two levels such as:
Ri (j) < Rc
Ri (j − 1) > Rch, the top of the mixing layer satisfies:
Ri (h) = Rc
By supposing that Ri varies linearly between these two levels, one deduces
h as the linear interpolation between j and j − 1.
The recommended value for Rc generally lies between 0.25 and 0.5. This
makes it possible to take into account the inertia and the particle entrainment.

170

11. Vertical diffusion

The value of θvN is preferred to that of To used in the simplified theories, inagreement with Holtslag and Boville (1993).
θs represents the temperature of the air near surface.
In the stable cases, i.e. if the average value of surface turbulent flux (w0 θv0 )s <0, one gets:θs = θvsIn the unstable cases, if (w0 θv0 )s > 0, the kinematic heat flux of the low
layers is taken into account, θs is corrected by θT , heat flux resulting fromconvective thermals.
θs = θvs + θT
θT = Cwith(w0 θv0 )s
wm



 C

: empirical dimensionless constant (8.5)(w0 θv0 )s : surface virtual heat flux
 w
: scale of turbulent speedm(w0 θv0 )s = (w0 θ0 )s + 0.608 · θvs (w0 q 0 )s
(w0 θv0 )s = −(1
1
Flat )
Fsens + RETV · θvsρ s Cpρs L

Fsens and Flat are respectively the surface sensible and latent heat fluxes(calculated by Surfex and sent back to the atmosphere). wm is a functionof friction speed and of convective speed scale:1

wm = (u∗3 + c w∗3 ) 31w∗ = ( θg (w0 θv0 )s h∗) 3



vs

u∗



 u∗

2

2 1

= [(w0 u0 )s + (w0 v 0 )s ] 4
1= [( Fρus s )2 + ( Fρvss )2 ] 4

c is an empirical dimensionless constant of 0.6 (Dyer, 1974, Troen and
Mahrt, 1986). F u and F v are turbulent fluxes of momentum calculatedby ARO_GROUND_PARAM (interface to call Surfex) . Instead of an
implicit system for the calculation of h in the equation of w∗, one uses h∗
as an approximation to h. h∗ is the value of h calculated at previous timestep.

11. Vertical diffusion

171

6 Additional surface parameters
Subroutine ACHMTLS, calculated the surface density (PGWDCS) and BruntVaïsala frequency Ns for use in the gravity wave drag parameterization .
ps
P GW DCS = RT

One calculates, as in ACNEBR but for the surface, the Brunt-Vaïsala frequency Ns divided by g and the density:


Ns
ρg

2

=

ZSTA
ZSTA(RT )2=(ρ∆φ)2(ps ∆φ)2ZST A = approximation of ∆φ ∗ ∆ ln θ
ps being pressure of surface (uncentered calculation). The symbolic squareof the above formula can be negative in case of unstable atmosphere.

172

11. Vertical diffusion

12
Convection

1 Principle of the convection scheme
The scheme used has been described by Bougeault (1985) and further developed in successive cycles of Arpege. The version described here corresponds
to the parametrization used in the former versions of Arpege-climat.
The deep convection occurs under two conditions: a convergence of humidity
at low layers is required and the vertical temperature profile must be unstable. The convection adjusts the unstable profile to a cloudy profile, which isassumed to be moist adiabatic.
The scheme uses the mass-flux concept where the vertical ascent in the cloud(ω ∗ = −Mc , where Mc is the mass-flux) is compensated by a large-scale
subsidence.
A Kuo-type closure is assumed where the available moisture is either precipitated or recycled into the environment by the detrainment term.
The scheme equations are written as : ∂u conv∂u= ω∗+ K (uc − u)



∂t
∂p






 conv


∂v
∂v


= ω∗
+ K (vc − v)



∂t
∂p

 conv

dif

∂s

∗ ∂s + K (s − s) + g ∂Fs


=
ω
c


∂t
∂p
∂p







 conv∂Fqdif
∂q
∂q


= ω∗
+ K (qc − q) + g


∂t

∂p

∂p

174

12. Convection

where K is a relaxation coefficient (to be determined). The index c represents
the cloud profile.
As the convection includes vertical diffusion effects of temperature and moisture, the diffusive fluxes Fsdif and Fqdif are subtracted from the equations
for dry static energy and moisture to avoid double counting.
The mass-flux −ω ∗ is due to buoyancy which causes a conversion between
moist static and kinetic energy: ω ∗ is thus proportional to the square root of
the difference between the moist static energy inside and outside the cloud.
The proportionality factor α is obtained by the Kuo-type closure assumption: the available humidity is the sum of the large-scale advection and thehumidity tendency produced by the vertical diffusion; it must be equal to
the sum of convective precipitation and detrainment:
Z pb
pt

Z pb
pt

∂Fqdif∂q−ω− u∇q − g
∂p
∂p
∂q
−
∂t


conv

!

dp
=
g

∂Fqdif+ K (qc − q) + g
∂p

!

dp
g

where pb is the pressure at the bottom of the cloud, pt at the top. Thus, one
obtains :Z pb 

α=

pt

∂q
dp
− u∇q
+ Fqdif (pt ) − Fqdif (pb )∂pg
Z pb rp∂q dp−(hc − h)
T
∂p g
pt


−ω

The factor K is then deduced from the moist static energy conservation
between the top and the bottom of the cloud. The wind components uc andvc in the cloud are calculated considering entrainment effect of surrounding
air, horizontal pressure gradient between cloud and surrounding air, andmomentum conservation in the convective column.

2 Preliminary calculations
2.1 Initialization of the convergence of moisture
Moisture available for theis calculated by subroutine APLPAR
 convection∂qin the form of tendency ∂t(PCVGQ):
HDC


∂q
∂t

~ + ω ∂q
= −r (~v ·∇)q
∂p
HDC





dif _tur



−g

∂Fq

∂p

12. Convection
dif _tur

−g

Term



Term

∂Fq

175
!

is provided by the vertical diffusion scheme.

∂p~ + ω ∂q(~v ·∇)q∂p

is provided to APLPAR by the dynamics of themodel; taking into account the fact that the convection scheme provides
only the part of precipitation corresponding to the ascents unresolved by
the model dynamics, the convective effects should decrease as resolution
increases. One tries to represent this effect by a reduction of moisture convergence. Coefficient r reducing convergence has as an expression:r=

1
1 + Tle /Tc

with Tle equivalent local spectral truncation of the model and critical Tc
truncation: Tc corresponds to the truncation beyond which a part of theconvective phenomena becomes resolved by dynamics. In APLPAR, theabove expression involves the geometry of Arpege: r = 1/(1 + χfe ) with
χ (TEQK) ratio of a critical mesh size REFLKUO by the unstretched meshsize and fe (PGM) scale factor.

2.2 Wet-bulb thermometer calculation
Calculation the temperature of the wet-bulb thermometer and corresponding
moisture is carried out in subroutine ACTQSAT. One starts from an origin
state defined by (T0 , qv 0 ) and one seeks a final state defined by (Tf , qv f )with qv f = qsat (Tf , p) such as the transformation is done at constant energy
and pressure. In the presence of snow (index ZDELTA=1), calculations are
modified: it is then necessary to replace the terms in cpv − Cw by terms in
cpv − Ci which intervene inter alia in the calculation of the latent heat, qs
and ∂qs /∂t.
(

cp dT + Ldqv = 0

with

cp = cp 0 + (cpv − Cw )(qv − qv 0 )L = L0 + (cpv − Cw )(T − T0 )hence:
cp

dL
dcp
+L
=0
cpv − Cw
cpv − Cw⇒

Lcp = constant = L0 cp 0

which can take the two following forms:
cp 0 (Tf − T0 ) + Lf (qv f − qv 0 ) = 0
or cp f (Tf − T0 ) + L0 (qv f − qv 0 ) = 0

indifferently

176

12. Convection

qs not being linear in T , one uses a Newton algorithm with linearization in
the vicinity of the result of the preceding iteration:
qv i+1 = qsat (Ti ) +∂qsat(Ti+1 − Ti )∂Tiwhich combines with:
cp 0 (Ti+1 − T0 ) + Li+1 (qv i+1 − qv 0 ) = 0
to give:
cp 0 [(Ti+1 − Ti ) + (Ti − T0 )] + [L0 + (cpv − Cw )((Ti+1 − Ti ) + (Ti − T0 ))]


∂qsat
× (qsat (Ti ) − qv 0 ) +(Ti+1 − Ti ) = 0∂Ti
One neglects the square terms in Ti+1 − Ti to have the linearized solution:[cp 0 (Ti − T0 ) + Li (qsat (Ti ) − qv 0 )]
∂qsat+(Ti+1 − Ti ) cp0 + (cpv − Cw )(qsat (Ti ) − qv 0 ) + Li
=0
∂Ti


(1)

One defines qv i by:
cp (Ti − T0 ) + Li (qv i − qv 0 ) = 0(2)

and thus qv i+1 by:
cp (Ti+1 − Ti ) + Li+1 (qv i+1 − qv i ) = 0(3)

cp i+1 (Ti+1 − Ti ) + Li (qv i+1 − qv i ) = 0

(4)

or:

The combination of Equations (1), (2) and (3) gives:
Li (qsat (Ti ) − qv i ) +(Ti+1 − Ti ) cp i + (cpv − Cw )(qsat (Ti ) − qv i ) + Li

∂qsat
=0
∂Ti

⇒

∂qsat
]=0
∂Ti



Li+1 (qsat (Ti ) − qv i ) + (Ti+1 − Ti )[cp i + Li

12. Convection⇒

177(qsat (Ti ) − qv i ) = (qv i+1 − qvi )[1 +Li ∂qsat]
cp i ∂Ti(5)

One uses Equations (4) and (5) to build the algorithm:
qv i+1 − qv i = (qsat (Ti ) − qv i )/(1 +Li ∂qsat)
cp i ∂Ti

cp i+1 = cp i + (cpv − Cw )(qv i+1 − qv i )Ti+1 − Ti = −Li
cp i+1(qv i+1 − qvi )Li+1 = Li + (cpv − Cw )(Ti+1 − Ti )(L/cp ) = (L0 /cp i+1 )with as first guess (T0 , qv 0 ). The algorithm converges towards (Tw , qw ) after
NBITER iterations (a number fixed at the beginning). At the time of thefirst iteration one stores in PQSAT value qsat (T0 ).
Relative humidity Hr is calculated by:
Hr =q(1 + (Rv /Ra − 1)qs )qs (1 + (Rv /Ra − 1)q)

3 Elements of the convection scheme
3.1 Determination of the profile of (sn , qn )(sn , qn ) is determined: the cloudy ascent is defined as the pseudo-adiabatic
resulting from “blue” point Tw of a level, involving environmental air; thecalculation of values (Tn , qn ) is carried out by the process:1. Calculation of “blue” point Tw : full calculation is described below.
2. Construction of the wet pseudo-adiabatic up to the next level, whileconsidering entrainment of environmental air; the taking into accountof entrainment of environmental air is done through a relaxation of thecloudy variables towards the environmental ones at a rate λ:∂µn= λ(µ − µn )
∂φ

178

12. Convection
with µ = T , q or `; the omission of entrainment would over-estimate
the altitude of the top of the clouds. Rate λ varies with altitude, withtwo possible formulations: λ is λx at the base of the cloud; it tends
towards λn to the top; the rate of decrease of λ is taken arbitrarily
3/4 1/4
equal to λx λn which represents an average value of λ:3/4 1/4

λ = λn + (λx − λn ) e−λx λn (φ−φb )
or:
λ=

Cλ
min(z, zclp )with Cλ a constant and zclp the planetary boundary layer depth.
3. Liquid lift in the cloud: in order to take into account the reduction ofcloudy instability due to the presence of liquid water, one diagnoses a
specific quantity of liquid water `n intervening later on in the calculation of instability below, then in calculation of cloud mass flux.
`n is diagnosed by:∂(qn + `n )
`n
=−
∂φ
φ0
This equation yields weak `n in clouds with low geopotential thickness.
φ0 is a constant corresponding to critical thickness of precipitating
clouds.
4. Test of convective activity of the level: a level is declared convective if
cloudy buoyancy and total moisture available are both positive.
5. Comparison of the temperature of the cloud to that of the environment;
if it is lower than that of the environment one takes the blue point of thelevel to continue the ascent. If not, one continues with the propertiesof the current cloud.

3.2 Algorithm of the reversible saturated adiabatic
One seeks here to determine the final temperature of a particle rising froma bottom level defined by (Tb , qb , pb ) to reach a level defined by (Ts , qh , ph ),with conservation of wet static energy (pb and ph are constants of calculationand qh = qs (Ts , ph )).

12. Convection

179

Differential equation
cp dT + Ldq + dφ = 0

with:

cp = cpb + (cpv − cw/i )(q − qb )L = Lb + (cpv − cw/i )(T − Tb )This differential equation is also written:
cp

dL
dcp
+L
+ dφ = 0
cpv − cw/i
cpv − cw/i

posing φb = 0 arbitrarily (and thus φh = ∆φ):Lcp + (cpv − cw/i )φ = Cte = Lb cpb
One thus has:Lb cpb
Lh cph+ φb =+ φhcpv − cw/i
cpv − cw/iand the exact integral of b with h take indifferently one of the two following
forms:
cpb (Ts − Tb ) + Lh (qh − qb ) + ∆φ = 0(6)cph (Ts − Tb ) + Lb (qh − qb ) + ∆φ = 0(7)Geopotential calculation
For ∆φ, if one knew the (∆ ln p)b and (∆ ln p)h having been used for calculation of the environment:pb(∆ ln p)b = lnpi
∆ ln p)h = lnpiphone would have then:∆φ = Rb Tb (∆ ln p)b + Rh Ts ∆ ln p)h= Rb Tb (∆ ln p)b + [Rb + Rv (qh − qb )] Ts (∆ ln p)hthat one writes:= R̃b− Tb + R̃b+ Ts + R̃v+ Ts (qh − qb )R̃b− , R̃b+ and R̃v+ being variables independent of later calculation of qh and
Ts .

180

12. Convection

Linearized equation h = i + 1
qs being nonlinear in T , one will calculate the solution by a Newton algorithmwith linearization about the previous iteration:
qi+1 = qs (Ti ) +∂qs(Ti+1 − Ti )∂Tiwhich one combines with:cpb (Ts − Tb ) + Lh (qh − qb ) + R̃b− Tb + R̃b+ Ts + R̃v+ Ts (qh − qb ) = 0
where subscript h is replaced by i + 1 for temperature, to obtain:R̃b− Tb + R̃b+ Ti + R̃b+ (Ti+1 − Ti ) + cpb [(Ti+1 − Ti ) − (Ti − Tb )]hi+ Lb + R̃v+ Tb + (cpv − cw/i + R̃v+ ) [(Ti+1 − Ti ) − (Ti − Tb )]∂qs
qs (Ti ) − qb +(Ti+1 − Ti )
∂Ti





=

0

One neglects the square terms in (Ti+1 −Ti ) to obtain the linearized equation:R̃b− Tb + R̃b+ Ti + cpb (Ti − Tb ) + (Li + R̃v+ Ti )(qs (Ti ) − qb )+(Ti+1 − Ti ) cpb + (cpv − cw/i + R̃v+ )(qs (Ti ) − qb )∂qs+(Li + R̃v+ Ti )+ R̃b+
∂Ti



=

0

Equation h = i
Starting from:cpb (Ti − Tb ) + Li (qi − qb ) + φi − φb = 0
one gets:
cpb (Ti − Tb ) + Li (qi − qb ) + R̃b− Tb + R̃b+ Ti + R̃v+ Ti (qi − qb ) = 0(8)12. Convection

181

and:
h

i

(R̃b+ + cpb )(Ti − Tb ) + R̃v+ Tb + Li + R̃v+ (Ti − Tb ) (qi − qb )hi

+ R̃v+ Tb + Li + R̃v+ (Ti − Tb ) (qi − qb ) + (R̃v+ + R̃b− )Tb=0

One poses:c̃p = R̃b+ + cp + R̃v+ (q − qb )(9)L̃ = R̃v+ Tb + L + R̃v+ (T − Tb )(10)

Note: we have still:
∂ L̃
∂c̃p
=
∂q
∂q

(= cpv − cw/i + R̃v+ )(11)

It comes:c̃pb (Ti − Tb ) + L̃i (qi − qb ) + (R̃b+ + R̃b− )Tb = 0(12)c̃pb (Ti+1 − Tb ) + L̃i+1 (qi+1 − qb ) + (R̃b+ + R̃b− )Tb = 0

(13)

and:

Subtracting (13) − (12) gives:c̃pi (Ti+1 − Ti ) + Li+1 (qi+1 − qi ) = 0(14)which, according to (6) and (7), can be also written as:c̃p(i+1) (Ti+1 − Ti ) + Li (qi+1 − qi ) = 0(15)

Equation (12) with b = i leads to:R̃b+ + R̃b− = 0(16)Equation (8) with b = i and Equation (16) leads to:L̃i (qs (Ti ) − qi ) + (Ti+1 − Ti ) c̃pi

182

12. Convection∂qs+(cpv − cw/i + R̃v+ )(qs (Ti ) − qi ) + L̃i
∂Ti



=

0

using Equation (10) with b = i on level i + 1:


∂qs
=0
∂Ti


L̃i+1 (qs (Ti ) − qi ) + (Ti+1 − Ti ) c̃pi + L̃iand with Equation (14):"L̃i ∂qs
qs (Ti ) − qi = (qi+1 − qi ) 1 +c̃pi ∂Ti#(17)

Newton algorithm
One uses Equations (17), (15) and (11) to build the algorithm:
qi+1 − qi =qs (Ti ) − qiL̃i ∂qs1+c̃pi ∂Tic̃p(i+1) − c̃pi = (cpv − cw/i + R̃v+ )(qi+1 − qi )Ti+1 − Ti = −L̃ic̃p(i+1)(qi+1 − qi )L̃i+1 − L̃i = (cpv − cw/i + R̃v+ )(Ti+1 − Ti )(18)

the starting point being:


 T0

 L̃

=
=
0

L̃
0 (q0 − qb ) =



c̃p0
=

Tb + ∆T (e.g. ∆T = Ts − Tb )L̃b + R̃v+ Tb = (cpv − cw/i + R̃v+ )(T0 − Tb )−(R̃b+ + R̃b− )Tb − c̃ph (T0 − Tb )cpb + R̃b+ + (cpv − cw/i + R̃v+ )(q0 − qb )3.3 Advection of compensatory subsidence
It is obtained by resolving the equation:
∂ψ impl∂= − [ω ∗ ψ]∂t conv_ tur∂pwith Mc = −ω ∗ and ψ = q, h, u and v, the discretization of this equation iscarried out implicitly for stability reason:
ψl+ − ψl = −cl

+
+
ψl+1
+ ψl+
ψ + + ψl−1
+ cl−1 l
2
2

12. Convection

183

where cl = ωl∗ ∆t/δpl is a coefficient of vertical CFL and ψ + indicates thevalue of ψ at time t + ∆t. But the linear tri-diagonal system obtained forψ + may be ill-conditioned (non-dominant diagonal) for non negligible valuesof cl . The formulation is thus:+
ψl + 2ψl−1− ψl−1
ψl+1 + 2ψl+ − ψl+
+ cl−1ψl − ψl = −cl22which leads to a triangular system which can be written as:+(1 + cl )ψl+ − cl−1 ψl−1
= ψl + cl−12ψl+ − ψl−1 − ψl+1
ψl − ψl+1+ (cl − cl−1 )2
2

a nonlinear instability can appear if|cl − cl−1 |> 1. One replaces cl by c0l by|1 + cl |calculating:c0l = c0l−1 +(cl − c0l−1 )(1 + c0l−1 )1 + |cl − c0l−1 ||c0l − c0l−1 |<|1 + c0l |1. For more details on this algorithm, to see Geleyn et al. (1982). One thus
obtains final value ψ + of ψ after effect of compensatory subsidence, so that:
One thus modifies to second order the mass flux by ensuring that∂ψ explψ+ − ψ
conv _ tur= Tψ=∂t conv_ tur∆t

3.4 Determination of coefficient K
It is carried out by ensuring the integral conservation of wet static energy;
indeed, convection involves reorganizations of mass and condensate losses in
an atmospheric column, and thus the vertical integral of h = cp T + gz + Lq
must be conserved.
One writes:
Z p2
p1

∂h dp
=0
∂t g

Z p2 hconv _ tot

Tsp1

Z p2 

ω∗

p1

conv _ tot

+ LTq

i dp

g

=0

∂h
dif _ tur dp+ K(hn − h) − Th
=0
∂p
g


184

12. Convection

which leads, by posing −g

−

Z p2

ω∗

p1

K=

∂ dif _ tur
dif _ tur
Fs
= Ts
, to:
∂p

∂h dp
dif _ tur
dif _ tur+ Fh(p1 ) − Fh
(p2 )
∂p g
Z p2
dp
(hn − h)gp1

3.5 Effect on horizontal windWind profile
One can first write vertical balance, according to the assumption that convective processes just produce a reorganization on the vertical of heat, moistureand the momentum. This can be written, for the ascending current:
Z pb 
pt

∂~v
∂t

ud

dp = 0conv

To build the profile, one must consider the effect of entrainment on thesurrounding air in the ascending current, and the effect of pressure gradient
between the cloud and his environment. One can write:
ω ˆ∗

∂~vu
λu ˆ∗
= − (∇φ)u +ω (~vu − ~v )
∂p
ρ

ω ˇ∗

∂~vd
λd
= − (∇φ)d + ω ˇ∗ (~vd − ~v )∂pρThe entrainment coefficients λu and λd are the same as for thermodynamic
variables s and q, since the air is driven with all its properties.
Kershaw and Gregory (1997) showed that the horizontal pressure gradient
between the air and its environment is proportional to the mass flux andthe vertical shearing of large-scale. In their experiments, the proportionality
factor was practically the same for the upward and downward currents: G d 'G u ' 0.7.
∂~vu
λu
∂~v
=
(~vu − ~v ) + G u
∂p
ρ
∂p

or

∂~vu
∂~v
= −λu (~vu − ~v ) + G u
∂φ
∂φ

∂~vd
λd
∂~v
=
(~vd − ~v ) + G d
∂p
ρ
∂p

or

∂~vd
∂~v
= −λd (~vd − ~v ) + G d
∂φ
∂φ

(19)

12. Convection

185

Calculation for the upward currents is developed here. One will obtain thesame thing, except for some changes of sign, for the downward currents. In
Arpege-climat the downward currents are not coded.
To discretize Equation (19), one must keep in mind the way in which one
discretized the entrainment equation for q and s:∂ψc= −λ(ψc − ψ)
∂φ






ψcl = ψcl+1 + φl − φl+1 λl+1 ψ l+1 − ψcl+1



instead of using the values at the intermediate levels:ψcl = ψcl+1 + φl − φl+1 λl̄ ψ l̄ − ψcl̄

To avoid this approximation, it would be necessary to initially calculate avalue for ψ l which would depend on the value of ψcl in the loop of Newton,
which would imply a double iterative calculation. Thus, we will make thesame approximation for Equation (19):

λl+1 > 0









~vcl − ~vcl+1 − G u ~v l − ~v l+1 = − φl − φl+1

λl+1



~vcl+1 − ~v l+1

That is to sayξ l ≡ φl − φl+1=⇒ ~vcl = ~vcl+1 1 − ξ l + G u~v l − G u − ξ l ~v l+1
We impose the wind at the cloud base: ~vcb = ~vc0 .
By taking an expression of the form:~vcl = βl~vc0 + (1 − βl ) ~v

lor ~vcl+1 = βl+1~vc0 + (1 − βl+1 ) ~v

l+1

we obtain~vcl = 1 − ξ l
=⇒

n



βl+1~vc0 + (1 − βl+1 ) ~v
βl = 1 − ξ l βl+1

l+1

o





+ G u~v l − G u − ξ l ~v l+1

186

12. Convection
βb = 1l(1 − βl ) ~v = 1 − ξ l (1 − βl+1 ) ~vl~v = 1 − ξ l

=⇒



= ~v

l+1

 (1 − βl+1 ) l+1~v

(1 − βl )+G u~v l − (G u − ξl ) ~v l+1(1 − βl )G u~v l − G u − ξ l ~v l+1
+



= ~v





+ G u~v l − G u − ξ l ~v l+1

1 − ξ l − βl(1 − βl )

l+1

l+1

ξ l ~v l+1 − ~v

(1 − βl )

l+1

+





+ G u ~v l − ~v l+1(1 − βl )At the cloud base:βb = 1βb−1 = 1 − ξ b

=⇒



~v

b−1

= ~v b +G u ~v b−1 − ~v b

(1 − βb−1 ) = ξ b−1

⇐⇒



ξ b−1

If G u = 0 we obtain:


~v

b−1

l

= ~v band ~v = ~v

l+1

ξ l ~v l+1 − ~v
+

l+1



(1 − βl )which is exactly the parametrization used before (without the effects of pressure gradient ).
Calculation of the wind at the base
One writes the conservation of momentum on the vertical:
Z pb 
pt

∂~v
∂t

u

Z pb 
pt

conv

Z pb 

=
pt

∂~v
∂t
−

=⇒

ω ˆ∗

dp = 0 =~vc0 =

u

dp + K u

pt

Z p



b

~vu dp −pt

cs

Z pb 

∂~v+ K u (~vu − ~v ) dp
∂p

∂~v
∂t

u

Z pb



~v dp
pt

dp + K u

Z pb~v dp − K u

pt

cs

Ku

pt

Z pb

βdp
pt

Z pb

(1 − β)~v dp12. Convection

187

Calculation of flux and tendency


∂~v
∂t

u

∂F
= −g ~vu =
∂p
conv



∂~v
∂t



+ K u (~vu − ~v )cs=~vcs − ~v− K u ~v + K u (1 − β)~v + β K u ~vc0∆t=~vcs − ~v (1 + K 0u ) + K 0u (1 − β)~v + β K 0u ~vc0∆tposing, for stability reasons :K 0u ≡ ZALF S =K∆t
1 + K∆t

The parametrization must return fluxes:
F~vu l = F~vu l−1 − δstab= F~vu l−1 − δstabi∆pl h l(~vcs − ~v l ) + K u (~vul − ~v l )
g∆t

i
∆pl h
~vcs − ~v (1 + K 0u ) + K 0u (1 − β)~v + βK 0u ~vc0g∆t

3.6 Energetics in the scheme
Latent heat of flux
One investigates here the relation between the precipitation flux Fp and the
dry static energy flux (Fs )conv_ prec , these two fluxes being defined by:• dry static energy flux:


rη

∂s
∂t



=
conv _ prec∂∂s(Fs )conv_ prec + δm Fp
∂η
∂η

1 ∂p
pseudo-density in η coordinate. This form of the teng ∂ηdency of s due to precipitations separates the pseudo-advection partfrom the remainder by the vacuum left by precipitation which corresponds to a mass flux rη η̇ = δm Fp .
with rη = −188

12. Convection• precipitation flux:


rη

∂q
∂t



=
conv _ prec∂Fp∂ηand one seeks Lf lux such as:(Fs )conv_ prec = −Lf lux Fp
On the basis of:


rη cp

∂T
∂t



= −L(T )conv _ prec

∂Fp
∂φ
∂T
+ δm Fp+ Fp(c − cpa (1 − δm ))∂η∂η∂η w/iand of:


∂s
∂t




conv _ prec

= cp

∂T
∂t





+
conv _ prec

∂cp
∂t



T
conv _ prec

one expresses cp in q to obtain:


rη

∂s
∂t



=−
conv _ prec∂∂s(Fp [L(T ) − (1 − δm qv )(cpv − cpa )T ]) + δm Fp∂η∂η

One thus has:Lf lux = L(T ) − (1 − δm qv )(cpv − cpa )T(20)

Effective local latent heat
One tries here to answer the following question: “which energy contribution
to the whole column in sensible heat form does produce a local variation inwater vapor due to precipitation/evaporation ?” the answer will be provided
by Lef f such as:∂Fp− Lef fdη = d
∂η

"Z

η2



cp
η1

∂T
∂t

#


conv _ prec

rη dη

(21)

Taking into account the assumptions carried out on the water cycle, Lef fwill consist of three terms:
1. Conversion of latent heat into sensible heat: L(T ) (true mass latent
heat of the water vapor at this temperature).

12. Convection

189

2. Transformation of potential energy of water vapor going in a time stepfrom the ground to the level considered (or opposite way according to
the sign of ∂Fp /∂η): δm (φ − φs )3. Redistribution of sensible heat; indeed, precipitation is adjusted oneach level at the local temperature: (cw/i − cpa (1 − δm ))(T − Ts )Effective local latent heat:
Lef f = L(T ) + δm (φ − φs ) + (cw/i − cpa (1 − δm ))(T − Ts )which is also written in the form:
Lef f = L(Ts ) + δm (φ − φs ) + (cpv − cpa (1 − δm ))(T − Ts )satisfies Equation (21), and thus:Z η2 "
η1

∂Fp
∂T
Lef f
+ cp
∂η
∂t


#


conv _ precrη dη = 0

Latent heat of budget
One takes again the problem of the preceding paragraph, while being interested this time not in sensible heat but in dry static energy s = cp T + φ:“which energy contribution to the whole column in dry static form does produce a local water vapor variation due to precipitation/evaporation?” Theanswer will be provided by Lbil such as:
∂Fp
−Lbil
dη = d
∂η

"Z
η2



cp
η1

∂s
∂t

#


conv _ precrη dηOne has:


∂s
∂t





conv _ prec

= cp

∂T
∂t





+
conv _ prec

∂cp
∂t





T+
conv _ prec

∂φ
∂t


conv _ prec∂φ
One neglects. The first term of right-hand side is the same∂t conv_ prec
as that of the preceding paragraph (related to the effective local latent heat).
Let us detail the second:




∂cp
∂t



∂q
T = T (cpv − cpa )∂tconv _ prec
conv _ prec

190

12. Convection= T (cpv − cpa )(1 − δm q)∂Fp∂η

One thus has:
Lbil = Lef f + (cpv − cpa )(1 − δm q)T
and finally:
Lbil = L(Ts ) − (cpv − cpa (1 − δm ))Ts + δm (cp T + φ − φs )

(22)

13
Gravity wave drag1 Parametrization of orographic gravity wave drag1.1 Calculation at the surface
The momentum flux due to a wave excited by the displacement of a largescale flow, vertically, on a small scale mountain, is given by:~τs = ρs Ns~vs Kg hs(1)

This vector corresponds in the code to PSTRDU and PSTRDV at level
KLEV. It represents the drag exerted by the lowest atmospheric layer on
surface. In the right-hand side:• ρs is the density at surface (PGWDCS)• ~vs is the effective wind at surface (see section 2.1)
• Ns is the effective Brunt-Vaïsala frequency at surface (see section 2.1)
• hs is the standard deviation of unresolved orography• Kg is a dimensionless coefficient for the adjustment of the parametrization (GWDSE)

1.2 Calculation at a given level
The momentum flux of a wave of amplitude a and wavelength λ is given by:τ = ρN U

a2
λ

192

13. Gravity wave dragwhere U is the projection of wind ~v on the effective wind at surface ~vs :U=~v · ~vsk~vs kAs long as the wave remains linear (low amplitude compared to the wavelength), there is no interaction with the large-scale flow (surge or saturation)and τ remains constant on the vertical. It is admitted moreover that:• the wave breaks at the surface.
• the condition of saturation is: λ = CU/N (Lindzen criterion) where C
is a constant on the vertical.
Thus, posing:
Γ=

ρ
ρs



U
Us

3

Ns
N

(ZRAPP)(2)

there will be saturation if τ = Γτs . One thus has τl = min(Γτs , τl+1 ). In fact,
one defines a factor of proportionality to τs . One takes at surface ΓL = 1,
and one goes up while posing Γl = min(Γ, Γl+1 ). As soon as Γl ≤ 0, one
poses Γl = 0 and one sets it to zero above (there is no more flux because thewave has been completely absorbed). That occurs as soon as wind ~vl forms
an angle higher than 90◦ with the surface wind, or, at last, at the top of theatmosphere (ρ = 0).

2 Refinements for orographic gravity wave drag2.1 Effective surface wind
The formulas above utilize the surface wind ~vs . In the model, the surface
wind is zero, and using the lowest model level wind would introduce a dependence of the scheme with the vertical discretization. It is supposed that thewave generating orography induces a boundary layer thickness H = Kh hs
where Kh is an empirical dimensionless coefficient (HOBST). One calculates
the average parameters in this mountainous boundary layer:
1~vs =< ~v >=
H

Z H

~v dz

(ZSUMU,ZSUMV)

0

in the same way for < Ns2 /(ρs g)2 > (ZSUMF). To maintain the continuityof Γl , one modifies the vertical profile of wind in the mountainous boundary
layer:~vl = ~vs (1 − ∆pl /∆pobst ) + ~vl ∆pl /∆pobst (ZU,ZV)

13. Gravity wave drag193

where ∆pl is the thickness in pressure between the surface and level l, and∆pobst the thickness between the surface and the top of the mountainous
boundary layer. The same way is applied for the standardized Brunt-Vaïsala
frequency (ZNFNO).

2.2 Mountain anisotropy
The theory supporting this parametrization makes the assumption that theflow is two-dimensional. In fact the sub-grid scale orography can be represented by the anisotropic coefficient γ 2 (PVRLAN) which is the ratio of thetwo eigenvalues of the tensor of variance of unresolved orography, and angle ψ (PVRLDI) which are the direction (null in the direction of increasing
pseudo-longitudes) of the first eigenvector of this tensor. The surface stressis not co-linear to the surface wind. One calculates a fictitious surface wind
(uf s , vf s ), in the code (ZUSUR, ZVSUR), which would give in the isotropic
case same flux as the one obtained with (us , vs ):uf s = Aus + D(us cos 2ψ + vs sin 2ψ)vf s = Avs + D(us sin 2ψ − vs cos 2ψ)A = γ2 +D=

1(4(1 − γ)(1 + α1 γ) − γ(1 + α2 ) ln γ)2π

1(4(1 − γ)(1 + δ1 γ) + 3γ(1 + δ2 ) ln γ)
2π

Coefficients α1 , α2 , δ1 , and δ2 are calculated from elliptic integrals and areabout 1.44, 0.22, 0.67 and 0.44 respectively (more accurate values are usedin the code, based on asymptotic approximations). One thus uses ~vf s to calculate the surface flux and the projection of the wind in the free atmospherein the place of ~vs in Equations (1) and (2).

2.3 ResonanceThis theory assumes the wave does not deposit all its momentum on thesaturation level, but is reflected downwards, and can, after a certain numberof reflections, produce a resonance. The theory has been developed by Clark
and Peltier (1977). They showed from very simplified considerations, andalso with results from simulations with a mesoscale model, that resonant
amplification takes place when altitude z of the critical level checks:
N
3π
z≡
U
2

[2π]

194

13. Gravity wave dragOne defines a function of resonance f , periodical and of average value 1, asa reference for the parametrization:1f (θ) = |1 − iKa eiθ − Ka2 e2iθ + · · · | = (1 + 2Ka sin θ + Ka2 )− 2
where Ka is an empirical dimensionless parameter (GWDAMP). This function reaches its maximum 1/(1 − Ka ) for θ = 3π/2 (amplification) and its
minimum 1/(1 + Ka ) for θ = π/2 (destruction).
In the parametrization of the friction of the orographic gravity waves, one
includes the assumption of resonance as formulated above. One locates pcrit
the first critical level reached (Γl lower than 1, for the first time), where thewave is likely to be reflected in phase with itself; one evaluates amplification
by resonance by calculating:
Z zcrit

θ=
0

N
dz
U

One deduces f (θ) from it. One defines Γcrit as min(1, f (θ)), and one modifies
factor Γl :• between the surface and the critical level, one takes a linear profile
with pressure, with f (θ) at the surface and Γcrit at the critical level.
• above the critical level, one takes min(Γl , Γcrit ).
One does not take into account the phenomena of reflection on the levels
above the first critical level.

2.4 Trapping
In the case of a neutral or unstable stratification of the atmosphere, theBrunt-Vaïsala frequency goes to zero as well as the drag due to the orographic
gravity waves. One seeks the first level starting from bottom where N iszero or changes sign. If this level is below the critical level of resonance,
one does not apply the modification of the profile by resonance (by imposing
f (θ) = 1). Profile Γl is modified:• above level N = 0 by setting it to zero.
• below level N = 0, by cutting off a linear function of pressure in order
not to modify the surface stress and to set it to zero at level N = 0.

13. Gravity wave drag195

2.5 Sub-grid orography
The effect of blocking of flow by the succession of the mountains and thevalleys was taken into account in certain models by envelope orography,
method which consisted in replacing the average orography by an increaseof a fraction of the standard deviation of the unresolved fluctuations. Lottand Miller (1995) using the results of the PYREX experiment, proposed tobetter define the local impact of the unresolved mountains by means of anincrease in friction by gravity waves only in the model levels occupied by
the actual orography. The adaptation to Arpege consists in multiplying
coefficient Γl (and consequently tension ~τ ) by:
v
u
3
u
u 1− zubd H
1 + ad u
u 
z
t

1+

H

on the levels whose altitude z is lower than bd H, where H is the thickness ofboundary layer calculated above. From projection Us of the effective surface
wind ~vs onto the fictitious wind (which takes into account the anisotropy oforography) ~vf s , one introduces a standardized thickness HN :HN = HNs k~vf s k
Us2

Two empirical parameters are introduced: a factor of friction at the surface Kd (GWDCD) and a critical value for standardized thickness HN C
(1/GWDBC). When standardized thickness HN is higher than the criticalvalue, coefficients ad (ZAA) and bd (ZBB) are given by:
ad = Kdk~vf s k2 HN − HN C
2
Us2
HN

and

bd =

HN − HN C
HNIn the opposite case, ad and bd are zero and the stress is not modified.

2.6 The lift effect
Here one takes into account the transverse force exerted on the wind attacking an unresolved mountain due to the effect of terrestrial rotation combinedwith the deviation of the flow which circumvents the obstacle (in an asymmetrical way). In theory it would be necessary that this force is exerted
transversely to a hypothetical flow undisturbed by the obstacle (geostrophic
wind for example). But this calculation would be difficult to carry out in

196

13. Gravity wave dragArpege: the force defined by this way would produce a work, a property
contrary with the principle of the effect of rising (hereafter called lift ), andwhose conjugation with the effect of form-drag of the sub-grid orography(see section 2.5) would be unexpected. One could also have constrained
this force to be exactly orthogonal with the forces due to orographic friction
previously defined in this chapter, but this would have answered only thesecond of the two criticisms formulated above. As a simplification here, we
just have defined a (positive) complement to Coriolis force, with an intensity
proportional to the volume of the sub-grid mountain.
This force of lift is exerted on the volume of atmosphere of the model between
the ground and H and varies according to (1 − z/H)/(1 + z/H) to remain inagreement with the choice of the shape of the sub-grid mountain implicitly
made in section 2.5. There is a multiplying coefficient Lt whose intensity
corresponds to the vertical integral of the effects layer by layer (and thus
with the total stress applied): GWDLT. For a value 1 of this coefficient, one
obtains on average a doubling of the “Coriolis” effect on depth H. Fromthe phenomenological point of view, this amounts (when GWDLT = 1) toincreasing the rotational effects of the mountain solved by what would givean effect of envelope orography with coefficient Kh , without having now theeffects of vertical displacement and blocking associated with the modification
with the large-scale orography seen by the model.
The effect of lift is calculated with gravity waves by convenience and because the effects of the sub-grid orography intervene there in a way close towhat is made in the section 2.5. But as this additional force physically does
not have anything common with the remainder of calculations previously
described, it will be added in an absolutely independent way. In particular
it will not undergo to the numerical securities described hereafter. By applying it without caution to a wind which is perhaps not the average of thevector wind over the duration of the time step, this force would consume or
produce energy, even marginally, and this could cause problem for very longintegrations. A trapezoidal split-implicit calculation is thus carried out sothat the wind vector conserves its module, when this effect is the only one
taken into account during the time step.
All this leads to the following formulations. For z < H :1 − z/H
1 + z/Hf ∗ = Lt f
Kh (2 ln(2) − 1)




∂u
∂t



∂v
∂t



= +f ∗

v − u ∆t f ∗ /2
1 + (∆t f ∗ /2)2

= −f ∗

u + v ∆t f ∗ /2
1 + (∆t f ∗ /2)2

lif t

lif t

13. Gravity wave drag197

For z > H there is no lift force.

3 Numerical securities in orography waves
3.1 Implicit formulation versus surface wind
All fluxes are proportional to projection Us of wind ~vs on ~vf s , therefore thetendency of Us also:∂Us= −KUs∂t
However, for large time steps the system can become unstable. Consequently
the following relations are used (Us indicating the value with t and Us+ witht + ∆t):Us+ − Us
= −KUs+∆t
Us+ − Us= −ḰUs∆t
One replaces K by Ḱ = 1 +K
K∆t (split-implicit scheme). This is equivalent
to multiplying all relations of proportionality between flux at a given leveland Us by:1∆t ∂Us1−
Us ∂t∂Us
is calculated by integrating the vertical derivative of flux inside the∂tmountainous boundary layer defined in the section 2.1.

3.2 Implicit advective formulation
One has just seen how to implicitly take into account the dependence of thetendency of ~v on ~vs ; in fact, one has treated only component of the windin the direction of ~vf s , but the tendency of wind is zero in the direction
orthogonal to ~vf s . However one does not take into account the dependence

198

13. Gravity wave dragon ~v itself, since it does not appear in a linear way in the equations. One
wants to discretize:
∂U
∂τ
= −g
∂t
∂p
where U is the projection of ~v on ~vf s . Here τ = Γl τs and Γl depends on U
(Equation (2) and corrections). One poses:
τl =1
U l Mlg

If one regards the phenomenon as a vertical advection of moment, Ml plays
the role a vertical velocity (or of a mass flux, by analogy with the convection). One introduces the dimensionless quantity, similar to a vertical CFL
criterion:
Al =

Ml ∆t
δpl

One discretizes the pseudo-advection with an upstream implicit scheme:+
Ul+ − Ul = −(Al Ul+ − Al−1 Ul−1)

which one can write as:+(1 + Al )(Ul+ − Ul ) = −Al Ul + Al−1 Ul−1 + Al−1 (Ul−1− Ul−1 )+If one knows the implicit tendency on level l −1, Ul−1
−Ul−1 , and the explicit
tendency calculated by parametrization on level l, −Al Ul + Al−1 Ul−1 , one
deduces from it the implicit tendency on level l. One starts at level l = 1
since the tendency is null at the top and one carries out a downward loop.
One will replace stress τ by a stress τ́ such as:

U+ − U
∂ τ́
= −g
∆t
∂p
It comes:τ́0 = 0τ́l = τ́l−1 −In the code,1(τ́l−1 − τl )1 + Al1
is called ZALPHA.
(1 + Al )13. Gravity wave drag199

4 Parametrization of convective gravity wave dragThis parametrization of impact of tropospheric convection on the stratospheric jets was introduced by Bossuet et al. (1998).

4.1 Calculations at the top of the cloud
In a way similar to the flow over a mountain, the convective zones produce
vertical small-scale velocities which generate gravity waves in the stable atmosphere layers located above. The parametrization described here supposesthat the intensity of these waves is a function of the precipitation flux at thebase of the cloud (as an index of the intensity of the convection). It is supposed that these waves move vertically in the reference frame related to thebase of the cloud, and that, similarly with the waves of orographic origin,
the interaction with the large-scale flow occurs only in the direction of theabsolute wind at the top of the cloud, level of the source of the gravity waves.
The momentum flux due to a wave excited by convective motions can be
empirically represented by:~τs = Kc Pcon~vsk~vs k(3)This vector corresponds in the code to ZSTUS and ZSTVS. It also corresponds to vector PSTRCU and PSTRCV at level KIUPC (stress at the topof the cloud). This vector represents the stress exerted by a layer on thelayer below. In the right-hand side:• Kc is a tuning parameter• Pcon is convective precipitation (rain+snow)• ~vs is wind at the top of the cloud

4.2 Calculations above the cloud
As long as the wave remains linear (low amplitude compared with the wavelength), there is no interaction with the large-scale flow (surge or saturation)and τ remains constant on the vertical. By using Lindzen criterion for thesurge, one introduces, in the same way as for the orographic waves, profile
Γ:
ρ
Γ=
ρs



U
Us

3

Ns
N

(ZRAPP)(4)

200

13. Gravity wave dragwhere U is projection on the wind at the top of the cloud of the relative
wind in the reference frame related to the base of the cloud:U=(~v − ~vb ) · ~vsk~vs kThere will be saturation if τ = Γτs . One thus has τl = min(Γτs , τl+1 ). In
fact, one defines a factor of proportionality to τs . One takes at the top ofcloud Γs = 1, and one goes up while posing Γl = min(Γ, Γl+1 ). As soon asΓl ≤ 0, one poses Γl = 0 and one sets it to zero above (flux is zero because
the wave has been completely absorbed). That occurs as soon as the windat level l forms an angle greater than 90◦ with the wind at the top of thecloud (in the mobile reference frame), or, at last, at the top of atmosphere(ρ = 0).

4.3 Calculations inside the cloudIn the case of the waves of orographic origin, the momentum borrowed fromthe atmosphere is yielded to the earth, according to the principle of conservation. In the case here, there is no exchange of momentum with the ground,and the quantity must be yielded to the lower atmospheric layers. It is reasonable to think that the stable layers located under the base of the cloudare not affected either. One thus takes a linear variation of flow between
value τ́s at the top and zero at the base.
In the above calculation, one considers that if there are several convective
layers disjointed, one takes the bottom of the lowest layer and the top of thehighest layer.

5 Numerical securities for convective waves
The temporal evolution of the wind at level l depends, via Ul on the winditself. As a consequence, an explicit formulation of the temporal advance canlead to numerical instabilities. It would be too complicated to solve the fullsystem with an implicit scheme because it is non-linear (with cubic terms
and thresholds) and multidimensional. In fact one wants to discretize:
∂U
∂τ
= −g
∂t
∂p
where U is the projection of the relative wind. We have τ = Γl τs , and Γldepends on U (Equation (4)). One poses:
τl =1
U l Mlg

13. Gravity wave drag201

If one considers the phenomenon as a vertical advection of momentum, Ml
plays the role a vertical velocity (or of a mass flux, by analogy with theconvection). One introduces the dimensionless quantity:
Al =

Ml ∆t
δpl

One discretizes the pseudo-advection with an upstream implicit scheme:+
Ul+ − Ul = −(Al Ul+ − Al−1 Ul−1)

which can be written as:+(1 + Al )(Ul+ − Ul ) = −Al Ul + Al−1 Ul−1 + Al−1 (Ul−1− Ul−1 )+If one has calculated the implicit tendency on level l −1, Ul−1
−Ul−1 , and the
explicit tendency calculated by parametrization on level l, −Al Ul +Al−1 Ul−1 ,one deduces from it the implicit tendency at level l. One starts at level l = 1
since the tendency is null at the top and one carries out a downward loop.
One will replace stress τ by a stress τ́ such as:∂ τ́
U+ − U
= −g
∆t
∂p
Then:
τ́0 = 0τ́l = τ́l−1 −1(τ́l−1 − τl )1 + AlThis is the same procedure as in section 3.2.

202

13. Gravity wave drag14
Surface processes scheme

1 Introduction
The ISBA land surface scheme has been used in the former versions of
Arpege-climat. A more recent version is now available in Surfex. Surfex (stands for SURFace EXternalisée) gathers all developments and improvements made in surface schemes. Not only physical parameterizations
have been externalized, but also the preparation of specific surface parameters needed by physical schemes and the initialization of all state variablesof the different models. More recently, the modelling of urban areas hasbegan to be of great interest in the research community. TEB (Town Energy Balance) model, specially designed to represent the exchanges between
a town and the atmosphere is available in Surfex . Moreover, the surface
representation has been improved and thus Surfex system has been enhanced with the specific treatment for water surfaces. Indeed, up to now,the exchanges of energy between water surfaces and the atmosphere weretreated in a very simple way, while now a physically based model have been
introduced to build a more complex but accurate surface model, available
for all atmospheric models.
In Surfex, the exchanges between the surface and the atmosphere are realized by mean of a standardized interface (Polcher et al., 1998; Best et al.,2004 ) that proposes a generalized coupling between the atmosphere andsurface. During a model time step, each surface grid box receives the upper
air temperature, specific humidity, horizontal wind components, pressure,
total precipitation, long-wave radiation, short-wave direct and diffuse radiations and possibly concentrations of chemical species and dust. In return,Surfex computes averaged fluxes for momentum, sensible and latent heatand possibly chemical species and dust fluxes and then sends these quanti-

204

14. Surface processes scheme

Figure 1: Description of the exchanges between an atmospheric model sending meteorological and radiative fields to the surface and Surfex composedof a set of physical models that compute tiled variables F∗ covering a fractionf∗ of a unitary grid box and an interface where the averaged variables F aresent back to the atmosphereties back to the atmosphere with the addition of radiative terms like surface
temperature, surface direct and diffuse albedo and also surface emissivity.
All this information is then used as lower boundary conditions for the atmospheric radiation and turbulent schemes. In Surfex, each grid box ismade of four adjacent surfaces: one for nature, one for urban areas, one
for sea or ocean and one for lake. The coverage of each of these surfaces
is known through the gobal ECOCLIMAP database (Masson et al., 2003),
which combines land cover maps and satellite information. The Surfex
fluxes are the average of the fluxes computed over nature, town, sea/oceanor lake, weighted by their respective fraction.
The complete description of Surfex is in it’s proper scientific documentation.

2 Soil, snow and vegetation : ISBA surface scheme
The ISBA scheme (Noilhan and Planton, 1989; Noilhan and Mahfouf, 1996
) computes the exchanges of energy and water between the continuum soilvegetation-snow and the atmosphere above. In its genuine version, the evapotranspiration of the vegetation is controlled by a resistance like proposed by(Jarvis, 1976). A more recent version of the model named Isba-A-gs accounts

14. Surface processes scheme

205

for a simplified photosynthesis model where the evaporation is controlled by
the aperture of the stomates, the component of the leaves that regulates thebalance between the transpiration and the assimilation of CO2. The current
version Arpege-climat will use the ISBA scheme with the Force restore
method (Noilhan and Planton, 1989) to treat the transfer of water and heatin the soil ( option : 3 layers for tempertaure, liquid water and ice) the snow
scheme from Douville (1995a, 1995b ,).

2.1 Force restore approach
Treatment of the soil heat content
The prognostic equations for the surface temperature Ts and its mean value
T2 over one day τ , are obtained from the force-restore method proposed by
Bumralkar (1975) and Blackadar (1976):

∂Ts
∂t
∂T2
∂t

= CT (Rn − H − LE) −=2π(Ts − T2 ),
τ

(1)

1
(Ts − T2 ),τ(2)where H and LE are the sensible and latent heat fluxes, and Rn is the net
radiation at the surface. The surface temperature Ts evolves due to both
the diurnal forcing by the heat flux G = Rn − H − LE and a restoring term
towards its mean value T2 . In contrast, the mean temperature T2 only varies
according to a slower relaxation towards Ts .
The coefficient CT is expressed by(1 − veg)(1 − psng ) veg(1 − psnv ) psn
CT = 1/
+
+
Cg
Cv
Cs

!

(3)

where veg is the fraction of vegetation, Cg is the ground heat capacity, Cs
is the snow heat capacity, Cv is the vegetation heat capacity, andWs
Ws + Wcrn
hspsnv =
hs + 5000z0
psn = (1 − veg)psng + vegpsnv
psng =

(4)
(5)
(6)

are respectively the fractions of the bare soil and vegetation covered by snow,
and the fraction of the grid covered by snow. Here, Wcrn = 10 mm, and206

14. Surface processes scheme

Figure 2: Partitioning of the gridhs = Ws /ρs is the thickness of the snow pack (ρs is the snow density). The
partitioning of the grid into bare soil, vegetation, and snow areas, is indicatedin Fig.(2) .

Treatment of the soil water
Equations for wg and w2 are derived from the force-restore method applied
by Deardorff (1977) to the ground soil moisture:
∂wg
∂t
∂w2
∂t

C1
C2
(wg − wgeq );(Pg − Eg ) −ρw d1
τ0 ≤ wg ≤ wsat
1
C3=
(Pg − Eg − Etr ) −max [0., (w2 − wf c )] ;
ρw d2
d2 τ
0 ≤ w2 ≤ wsat
=

(7)
(8)
(9)
(10)

where Pg is the flux of liquid water reaching the soil surface (including themelting), Eg is the evaporation at the soil surface, Etr is the transpiration
rate, ρw is the density of liquid water, and d1 is an arbitrary normalization
depth of 1 centimeter. In the present formulation, all the liquid water fromthe flux Pg goes into the reservoirs wg and w2 , even when snow covers fractions of the ground and vegetation. The first term on the right hand side ofEq. (12) represents the influence of surface atmospheric fluxes when the contribution of the water extraction by the roots is neglected. The coefficients
C1 and C2 , and the equilibrium surface volumetric moisture wgeq , have beencalibrated for different soil textures and moistures.
The expression for C1 differs depending on the moisture content of the soil.

14. Surface processes scheme

207

For wet soils (i.e., wg ≥ wwilt ), this coefficient is expressed as

C1 = C1sat

wsat
wg

!b/2+1(11)

For very dry soils (i.e., wg < wwilt ), the vapor phase transfer needs to beconsidered in order to reproduce the physics of water exchange. These transfers are parameterized as a function of the wilting point wwilt , the soil watercontent wg , and the surface temperature Ts , using the Gaussian expression(Braud et al., 1993; Giordani, 1993).
"(wg − wmax )2
C1 = C1max exp −
2σ 2

#

(12)

where wmax , C1max , and σ are respectively the abscissa of the maximum,
the mode, and the standard deviation of the Gaussian functions (see 4). The
other coefficient, C2 , and the equilibrium water content, wgeq , are given by
w2
wsat − w2 + 0.01

 "

 #
w2 p
w2 8p
= w2 − awsat
1−
wsat
wsat


C2 = C2ref
wgeq



(13)
(14)

For the w2 evolution, Eq. (13) represents the water budget over the soil
layer of depth d2 . The drainage, which is proportional to the water amountexceeding the field capacity (i.e., w2 − wf c ), is considered in the second termof the equation (see Mahfouf et al., 1994). The coefficient C3 does not depend
on w2 but simply on the soil texture (see 4). Similarly, run-off occurs whenwg or w2 exceeds the saturation value wsat or when a sub-grid runoff scheme
is used. Coefficients C1sat , C1max , C2ref and p are made dependant on thesoil texture (Noilhan and Mahfouf, 1996)
Root zone soil layer option
In the standard two-soil layer version of ISBA, it is not possible to distinguishthe root zone and the total soil water reservoirs. With the three-layer version,
the deepest soil layer may provide water to the root zone through capillaryrises only, and the available water content for transpiration is defined as(wsat − wsat ) × d2 .
The bulk soil layer (referred to as w2 in the previous sections) is divided into
a root-zone layer (with a depth d2 ) and base-flow layer (with a thickness

208

14. Surface processes scheme

defined as d3 − d2 ). The governing equations for the time evolution of soil
moisture for the two sub-surface soil layers are written following Boone etal. (1999) as
∂w2
∂t

∂w3
∂t

1
C3
(Pg − Eg − Etr ) −max [0, (w2 − wf c )]
ρw d2
d2 τ
C4
−
(w2 − w3 )
τ
(
)
d2
C3
C4
=
max [0, (w2 − wf c )] +(w2 − w3 )(d3 − d2 ) d2 ττC3−max [0, (w3 − wf c )] ;0 ≤ w3 ≤ wsat
(d3 − d2 ) τ
=

(15)
(16)
(17)
(18)

where C4 represents the vertical diffusion coefficient. It is defined as
C4 = C4 ref w2,3 C4b(19)

where w2,3 represents the interpolated volumetric water content representative of the values at the layer interface (d2 ). The C4 ref and C4b coefficientsare defined using the soil sand and clay contents, consistent with the othermodel parameters (see the section on model coefficients). In addition, theC4 ref coefficient is scaled as a function of grid geometry. The equations areintegrated in time using a fully implicit method.
Exponential profile of ksat
In this version, the soil column assumes an exponential profile of the saturated hydraulic conductivity, ksat , with soil depth (Decharme et al., 2006).
This parameterization depends only on two parameters, which represent therate of decline of the ksat profile and the depth where ksat reaches its so-called"compacted" value.
ksat (z) = ksat,c e−f (z−dc )(20)where z(m) is the depth of the soil profile, f (m−1 ) is the exponential profile
decay factor and dc (m) the compacted depth where ksat reaches its compacted value, ksat,c given by Clapp and Hornberger (1978). In the standard
approach, f varies with soil properties (texture and/or rooting depth) but
can not exceed 2m−1 and dc assumes to be equal to rooting depth d2 . Sensitivity tests to these parameters and a detailed discussion about this parameterization can be found in Decharme et al. (2006). The main hypothesisis that roots and organic matter favor the development of macropores andenhance the water movement near the soil surface, and that soil compaction

14. Surface processes scheme

209

is an obstacle for vertical water transfer in the deeper soil. This exponential
soil profile increases the saturated hydraulic conductivity at the surface byapproximately a factor 10, and its mean value increases in the root zone
and decreases in the deep layer in comparison with the values given by
Clapp and Hornberger (1978). In ISBA, all hydraulic force-restore coefficients (C1, C2, C3 and C4) are re-formulated to take into account this ksat
profile.
Treatment of runoff in the Isba initial version
Run-off occurs when w2 exceeds the saturation value wsat . In its standard
version, ISBA simulates surface runoff through the saturation excess mechanism (also known as Dune mechanism), therefore, runoff is only produced
when the soil is saturated (i.e. w2 exceeds the saturation value wsat ). Notethat if w3 exceeds the saturation, the excess water is added to the drainage
term.
When the scale of variability of runoff production is smaller than the typicalscale of the grid scale (which is common in most applications), the soil almost
never saturates and the runoff production is very low, even though, in reality,
a fraction of the cell is saturated and does produce surface runoff.
In order to account for subgrid scale runoff, three parametrisations are available- the variable Infiltration Capacity (VIC) scheme (Dümenil and Todini, 1992)
- the TOPography based MODEL (TOPMODEL) approach- the Horton runoff approach
Arpege-climat is using the TOPMODEL parametrisation for the subgrid
runoff and the Horton runoff. These two parametrisations are describedhereafter.
TOPMODEL approach
TOPMODEL (TOPography based MODEL) attempted to combine the important distributed effects of channel network topology and dynamic contributing areas for runoff generation (Beven et al., 1979; Silvapalan et al.,1987). This formalism takes into account topographic heterogeneities explicitly by using the spatial distribution of the topographic indices, λi (m), ineach grid-cell defined as follows:
λi = ln (ai / tan βi )

(21)

210

14. Surface processes scheme

where ai (m) is the drainage area per unit of contour of a local pixel, i,and tan βi approximates the local hydraulic gradient where βi is the local
surface slope. If the pixel has a large drainage area and a low local slope, its
topographic index will be large and thus, its ability to be saturated will behigh. Then, this topographic index can be related to a local water deficit,
and using the spatial distribution of the topographic indices over the gridcell, a saturated fraction, fsat, inversely proportional to the grid cell mean
deficit, Dt (m), can be defined. The "coupling" between TOPMODEL and
ISBA was proposed by Habets et al. (2001) and generalized by Decharme
et al. (2006). The active layer used for the ISBA-TOPMODEL couplingis the rooting layer, and not the total soil column. TOPMODEL describes
generally the evolution of a water storage deficit near the soil surface thatreacts quasi-instantaneously following rainy events (Beven et al., 1979). In
that case, the root zone appears to be a reasonable compromise in ISBA.
So, the relation between the grid cell mean deficit and the soil moisturecomputed by ISBA is simply expressed as:
0 ≤ Dt = (wsat − w2 ) × d2 ≤ d0

(22)

where d2 (m) is the rooting depth and d0 (m) the maximum deficit computed
as the difference between the saturation, ws at, and the wilting point, ww ilt:
d0 = (wsat − wwilt ) × d2

(23)

So for a given rooting soil moisture, w2 , a mean deficit, Dt , is calculated
and it is therefore possible to determine the saturated fraction of the gridcell. The runoff, Qtop , is thus simply given by: Qtop = Pg × fsat where Pg
is the throughfall rain rate. For w2 lower than the wilting point, the mean
deficit is a maximum, Dt = d0 , fsat = 0 and no surface runoff occurs. Note
that, the spatial distribution of the topographic index in each grid-cell canbe computed with the three- parameter gamma distribution introduced by
Silvapalan et al. (1987). The three parameters are derived from the mean,
standard deviation, and skewness of the actual distribution that can be doneby the HYDRO1K dataset at a 1 km resolution or another database. This
TOPMODEL approach has been intensively validated both at the regionaland global scale Decharme et al. (2006 , 2007 ).
Horton runoff approach.
The Horton runoff occurs for a rainfall intensity that exceeds the effective
maximum infiltration capacity. This infiltration excess mechanism tends to14. Surface processes scheme

211

dominate the overland flow production in most desert or semiarid regions
where short rainfall events can be very intense, but also where the absenceof vegetation and other organic matter prevents the development of a porous
soil structure through which water can move easily. The development of athin crust at the soil surface can also inhibit the infiltration (arid or frozensoil). So the Horton runoff, Qhort , is calculated using two infiltration functions following Decharme et al. (2006):

Qhort = (1 − δf ) × max (0, Sm + Pg − Iunf )(24)+δf max (0, Sm + Pg − If )(25)

where Sm is snowmelt, Pg the throughfall rain rate, Iunf and If the infiltration functions over unfrozen and frozen soil, and δf the fraction of the frozen
soil. These functions depend on root zone soil moisture conditions as well as
on soil hydraulic properties. When the Horton runoff (being estimated only
on the non-saturated fraction of the grid) is activated with the VIC or the
TOPMODEL runoff, the surface runoff is given by :Qs = Qtop_or_vic + (1 − fsat ) Qhort(26)

Treatment of drainage
The gravitational drainage when w > wf c is given by the following equations(Mahfouf and Noilhan, 1996 ; Boone et al ,1999) :
K2 =
K3 =

C3 d3
τ d2 max[0, (w2 − wf c )]C3 d3
τ d3 −d2 max[0, (w3 − wf c )]

(27)
(28)

where τ is a characteristic time (one day).
C3 is the force-restore parameter which account for the velocity at which thehumidity profile is restored to the field capacity. This parameter depends
on the hydraulic properties of the soil. In ISBA, it can be described by anempirical equation and depends on the proportion of clay in the grid cell.
−1.043C3 = 5.327 · Xclay

(29)

212

14. Surface processes scheme

Subgrid drainage
In the original formulation, the drainage stops below the field capacity wf c .
Within the framework of the Safran-Isba-Modcou model (Habets et al., 2008)a subgrid drainage was introduced in order to account for unresolved aquifers
in the model. A residual drainage was introduced in ISBA. The equations
above are slightly modified :
K2 =
K3 =

C3 d3
τ d2 max[wd2 , (w2 − wf c )]C3 d3
τ d3 −d2 max[wd3 , (w3 − wf c )]

(30)
(31)

In this formulation, wdi (for each layer i) is expressed as :min(wf c , wi ) − wg min
wdi = wdrain max 0,
wf c − wg min

!

(32)

where wdrain is a parameter to be calibrated, and wg min a small parameterto avoid numerical problems.
wdrain must be calibrated using discharge measurements during dry periods.
See Caballero et al. (2007) and Habets et al. (2008) for calibration with
discharge for the Safran-Isba-Modcou model.
Treatment of soil ice
The inclusion of soil freezing necessitates the addition of so-called phasechange to the thermal and hydrologic transfer equations. In addition, a freezing/drying wetting/thawing analogy is used to model changes in the forcerestore coefficients so that they must also be modified accordingly. Terms
which have been added to the baseline ISBA scheme are underlined in thissection, while terms which are modified are denoted using an ∗ superscript.
Additional details related to soil freezing scheme can be found in Boone etal. (2000, 2001).

2.2 Treatment of the canopy water
Rainfall and dew intercepted by the foliage feed a reservoir of water content
Wr . This amount of water evaporates in the air at a potential rate from thefraction δ of the foliage covered with a film of water, as the remaining part
(1 − δ) of the leaves transpires.


δ=

Wr
Wrmax

2/3

(33)

14. Surface processes scheme

213

Following Deardorff (1978), we set∂Wr= vegP − (Ev − Etr ) − Rr ; 0 ≤ Wr ≤ Wrmax
∂t

(34)

where P is the precipitation rate at the top of the vegetation, Ev is the evaporation from the vegetation including the transpiration Etr and the direct
evaporation Er when positive, and the dew flux when negative (in this case
Etr = 0), and Rr is the runoff of the interception reservoir. This runoff occurs when Wr exceeds a maximum value Wrmax depending upon the densityof the canopy, i.e., roughly proportional to vegLAI. According to Dickinson(1984), we use the simple equation:
Wrmax = 0.2vegLAI

[mm]

(35)

2.3 Spatial variability of precipitation intensities
Surfex offers 2 options for spatial distribution of rainfall intensity : toactivate an homogeneous distribution or an exponential distribution which
depends on the fraction of the mesh where it rains. Arpege-climat is set
with the second option.
With this option, the main assumption is that, generally, the rainfall intensity is not distributed homogeneously over an entire grid cell. As a first-orderapproximation, the sub- grid variability in liquid precipitation, Pi , can begiven by an exponential probability density distribution, f (Pi ):f (Pi ) =µ −µ Pi
e P
P

(36)

where P represent the mean rainfall rate over the grid cell and µ a fractionof the grid cell affected by rainfall. µ is calculated using the results of Fan etal. (1996), who showed an exponential relationship between the fractional
coverage of precipitation and rainfall rate, based on their analyses of over 2
years radar observations and rain gauge measurements over the ArkansasRed river basin in the southern plains of the United States. This relationshipis:µ = 1 − e−βP(37)where β is a parameter which depends on grid resolution, dx :β = 0.2 + 0.5e−0.001dx

(38)

214

14. Surface processes schemedx represents represents lengths of square grid cells ranging from 40km to
500km. In consequence, the µ parameter is fixed to 1 at high resolution(≤ 10km). This Spatial variability of precipitation intensities induces a new
expression for the runoff from the interception reservoir, Wr :Wr = P × eµ(Wr −Wrmax )
P ∆t

(39)

The second consequence is that the Horton runoff, Qhort , is calculated byintegrating the difference between the local rainfall and the local maximum
infiltration capacity, Ii , as follows:
Z ∞Qhort = µ(Pi − Ii ) f (Pi )dPi

(40)

Ii

Another assumption is made on the spatial heterogeneity of the local maximum infiltration capacity. Its spatial distribution can also be approximatedby an exponential probability density distribution:g(Ii ) =1 − Iie I
I(41)

where I is the mean maximum infiltration rate over the grid cell. As previously said, I is calculated for unfrozen and frozen soil conditions. So Eq.40, without snowmelt, can be noted as :Qhort = µ(1 − δf )Z ∞Z ∞
0

Z ∞Z ∞

+ µδf

0

If,i

Iunf,i(Pi − Iunf,i )f (Pi )g(Iunf,i )dPi dIunf,i(Pi − If,i )f (Pi )g(If,i )dPi dIf,i(42)

After some mathematical developments, the Horton runoff in presence ofrainfall and snowmelt, Sm , is given following Decharme et al. (2006):

!

Qhort

P
= (1 − δf )+ max(0, Sm − Iunf )1 + Iunf Pµ
!

+δf

P
+ max(0, Sm − If ) dPi dIf,i
1 + If Pµ(43)14. Surface processes scheme

215

2.4 Treatment of the snow
One-layer snow scheme option and Multi-layer snow scheme option are available with Surfex. This section will describe the one-layer snow scheme
option used by Arpege-climat.
The evolution of the equivalent water content of the snow reservoir is givenby∂Ws= Ps − Es − melt
∂t

(44)

where Ps is the precipitation of snow, and Es is the sublimation from thesnow surface.
The presence of snow covering the ground and vegetation can greatly influence the energy and mass transfers between the land surface and the atmosphere. Notably, a snow layer modifies the radiative balance at the surface
by increasing the albedo. To consider this effect, the albedo of snow αs istreated as a new prognostic variable. Depending if the snow is melting or
not, αs decreases exponentially or linearly with time.
If there is no melting (i.e., melt = 0):αs (t) = αs (t − ∆t) − τa∆t Ps ∆t+(αsmax − αsmin )τWcrn
αsmin ≤ αs ≤ αsmax

(45)
(46)

where τa = 0.008 is the linear rate of decrease per day, αsmin = 0.50 andαsmax = 0.85 are the minimum and maximum values of the snow albedo.
If there is melting (i.e., melt > 0):αs (t) = [αs (t − ∆t) − αsmin ] exp −τf

∆t
Ps ∆t
+ αsmin +(αsmax − αsmin )(47)τWcrn
αsmin ≤ αs ≤ αsmax (48)

where τf = 0.24 is the exponential decrease rate per day. Of course, thesnow albedo increases as snowfalls occur, as shown by the second terms of
Eqs. (21) and (23).
The average albedo of a model grid-area is expressed as
αt = (1 − psn )α + psn αs(49)

Similarly, the average emissivity t is also influenced by the snow coverage:
t = (1 − psn ) + psn s

(50)

216

14. Surface processes scheme

where s = 1.0 is the emissivity of the snow. Thus, the overall albedo andemissivity of the ground for infrared radiation is enhanced by snow.
Because of the significant variability of thermal properties related with thesnow compactness, the relative density of snow ρs is also considered as aprognostic variable. Based on Verseghy (1991), ρs decreases exponentiallyat a rate of τf per day:∆t
Ps ∆t
ρs (t) = [ρs (t − ∆t) − ρsmax ] exp −τf+ ρsmax +ρsmin (51)τWs
ρsmin ≤ ρs ≤ ρsmax (52)where ρsmin = 0.1 and ρsmax = 0.3 are the minimum and maximum relative
density of snow.
Finally, the average roughness length z0t is
z0t = (1 − psnz0 )z0 + psnz0 z0s

(53)

where
psnz0 =Ws
Ws + Wcrn + βs gz0(54)Here, βs = 0.408 s2 m−1 and g = 9.80665 ms−2 are physical constants,
whereas z0s is the roughness length of the snow.

2.5 The surface fluxes
Only one energy balance is considered for the whole system ground-vegetationsnow (when the 3-layer snow scheme option is not in use). As a result, heatand mass transfers between the surface and the atmosphere are related tothe mean values Ts and wg .
The net radiation at the surface is the sum of the absorbed fractions of theincoming solar radiation RG and of the atmospheric infrared radiation RA ,reduced by the emitted infrared radiation:Rn = RG (1 − αt ) + t RA − σSB Ts 4



(55)

where σSB is the Stefan-Boltzmann constant.
The turbulent fluxes are calculated by means of the classical aerodynamic
formulas. For the sensible heat flux:
H = ρa cp CH Va (Ts − Ta )(56)

14. Surface processes scheme

217

where cp is the specific heat; ρa , Va , and Ta are respectively the air density,
the wind speed, and the temperature at the lowest atmospheric level; andCH , as discussed below, is the drag coefficient depending upon the thermal
stability of the atmosphere. The explicit snow scheme sensible heat flux is
calculated using the same formulation (but with Tsn ). The water vapor flux
E is the sum of the evaporation of liquid water from the soil surface (i.e.,
Eg l ), from the vegetation (i.e., Ev ), and sublimation from the snow and soilice (i.e, Es and Eg f ):LE = LEg l + LEv + Li (Es + Eg f )(57)

Eg l = (1 − veg)(1 − psng ) (1 − δi ) ρa CH Va (hu qsat (Ts ) − qa )(58)

Ev = veg(1 − psnv )ρa CH Va hv (qsat (Ts ) − qa )(59)

Es = psn ρa CH Va (qsat (Ts ) − qa )

(60)

Eg f

=

(1 − veg) (1 − psng ) δi ρa CH Va (hui qsat (Ts ) − qa )(61)

where L and Li are the specific heat of evaporation and sublimation, qsat (Ts )is the saturated specific humidity at the temperature Ts , and qa is the atmospheric specific humidity at the lowest atmospheric level. The water vapor
flux E from the explicit snow surface is expressed as
LE (Tsn 1 ) = LEsl + Li Es(62)

Esl = δsn ρa CHs Va (qsat (Tsn 1 ) − qa )(63)

Es = (1 − δsn ) ρa CHs Va (qsat (Tsn 1 ) − qa )(64)
0 ≤ δsn (65)≤1

δsn = wsl 1 /wsl max 1 ;where evaporation of liquid water is zero when Tsn 1 < T0 . The transfer
coefficient (CHs ) is calculated over the snow covered surface using the same
formulation as CH .
The surface ice fraction is is used to partition the bare soil latent heat flux
between evaporation and sublimation, and it is defined as
0 ≤ δi < 1 .

δi = wg f / (wg f + wg ) ;(66)

The relative humidity hu at the ground surface is related to the superficial
soil moisture wg following
"

hu =

1
wg
1 − cosπ2
wf c ∗

hu = 1, if wg ≥ wf c∗!#

, if wg < wf c ∗ (67)
(68)
(69)

218

14. Surface processes scheme

where the field capacity with respect to the liquid water is defined using the∗ /w
modified soil porosity so that wf c ∗ = wf c wsat
sat . The humidity for theice covered portion of the grid box is calculated in a similar fashion as
"

hui =

wg f
1
1 − cosπ2
wf c ∗∗

hui = 1
, if wg f ≥ wf c!#∗∗
, if wg f < wf(70)
c

(71)

∗∗

(72)

where wf c ∗∗ = wf c (wsat − wg )/wsat . In case of dew flux when qsat (Ts ) < qa ,hu is also set to 1 (see Mahfouf and Noilhan, 1991 for details). When theflux Ev is positive, the Halstead coefficient hv takes into account the direct
evaporation Er from the fraction δ of the foliage covered by interceptedwater, as well as the transpiration Etr of the remaining part of the leaves:
hv = (1 − δ)Ra /(Ra + Rs ) + δδEr = veg(1 − psnv )(qsat (Ts ) − qa )Ra1−δ
(qsat (Ts ) − qa )Etr = veg(1 − psnv )Ra + Rs

(73)
(74)
(75)

When Ev is negative, the dew flux is supposed to occur at the potential rate,and hv is taken equal to 1.
Following Deardorff (1978), δ is a power function of the moisture content of
the interception reservoir:
δ = (Wr /Wrmax )2/3

(76)

The aerodynamic resistance is Ra = (CH Va )−1 . The surface resistance, Rs ,depends upon both atmospheric factors and available water in the soil; it isgiven by:
Rs =

Rsmin
F1 F2 F3 F4 LAI(77)

with the limiting factors F1 , F2 , F3 , and F4 :f + Rsmin /Rsmax
1+f
w2 − wwilt=
and 0 ≤ F2 ≤ 1
wf c − wwilt= 1 − γ (qsat (Ts ) − qa )

F1 =

(78)

F2

(79)

F3

−3

F4 = 1 − 1.6 × 10

2

(Ta − 298.15)

(80)
(81)

14. Surface processes scheme

219

where the dimensionless term f represents the incoming photosynthetically
active radiation on the foliage, normalized by a species-dependent thresholdvalue:
f = 0.55

RG 2
RGl LAI(82)

Moreover, γ is a species-dependent parameter (see Jacquemin and Noilhan,1990) and Rsmax is arbitrarily set to 5000 sm−1 .
The surface fluxes of heat, moisture, and momentum can be expressed asHρa cp Ta /θaE=ρa (1 − qa )= CD |Va |2 = u2∗(w0 θ0 )s =(83)

(w0 rv0 )s(84)|w0 V 0 |s(85)

where rv is the water vapor mixing ratio, w is the vertical motion, θa is
the potential temperature at the lowest atmospheric level. The primes andoverbars denote perturbation and average quantities.
For the drag coefficients CH and CD , the formulation of Louis (1979) was
modified in order to consider different roughness length values for heat z0
and momentum z0h (Mascart et al., 1995):
CD = CDN Fm ; CH = CDN Fh

(86)

with
CDN =

k2
[ln(z/z0 )]2(87)

where k is the Von Karmann constant. Also
Fm = 1 −
Fm =

10Ri
p
1 + Cm |Ri|
1

1 + √10Ri
1+5Ri

if Ri > 0

if Ri ≤ 0

(88)
(89)
(90)

and
"

#

ln(z/z0 )15RipFh = 1 −×ln(z/z
1 + Ch |Ri|
0h )if Ri ≤ 01
ln(z/z0 )√Fh =×ln(z/z0h )1 + 15Ri 1 + 5Ri
if Ri > 0




(91)
(92)
(93)
(94)

220

14. Surface processes scheme

where Ri is the gradient Richardson number. The coefficients Cm and Ch of
the unstable case are given by
Cm = 10Cm ∗ CDN (z/z0 )pm(95)

ln(z/z0 )= 15Ch ∗ CDN (z/z0h )ph ×
ln(z/z0h )


Ch



(96)

∗ , C ∗ , p , and p are functions of the ratio µ = ln(z /z ) only:
where Cm
0 0h
m
h
h

Ch∗ = 3.2165 + 4.3431 × µ + 0.5360 × µ2 − 0.0781 × µ3

(97)

∗
Cm

2

3

(98)

2

3

(99)

2

3

(100)

= 6.8741 + 2.6933 × µ − 0.3601 × µ + 0.0154 × µ

ph = 0.5802 − 0.1571 × µ + 0.0327 × µ − 0.0026 × µ
pm = 0.5233 − 0.0815 × µ + 0.0135 × µ − 0.0010 × µ

2.6 Summary of Useful Parameters
The parameters have been chosen in order to characterize the main physicalprocesses, while attempting to reduce the number of independant variables.
They can be divided into two categories: primary parameters needing to be
specified by spatial distribution, and secondary parameters which values canbe associated with those of the primary parameters.
In the present state of the method, the primary parameters describe the nature of the land surface and its vegetation coverage by means of only four
numerical indices: the percentage of sand and clay in the soil, the dominant
vegetation type, and the land-sea mask.
The secondary parameters associated with the soil type are evaluated fromthe sand and clay composition of the soil, according to the continuous formulation discussed in Giordani (1993) and Noilhan et al. (1995) (see 4). These
parameters are:• the saturated volumetric moisture content wsat ;• the wilting point volumetric water content wwilt ; NoilhanL1995• the field capacity volumetric water content wf c ;• the slope b of the retention curve;• the soil thermal coefficient at saturation CGsat ;• the value of C1 at saturation (i.e., C1sat );14. Surface processes scheme221• the reference value of C2 for w2 = 0.5wsat (i.e., C2ref );• the drainage coefficient C3 ;• the diffusion coefficients C4 ref and C4b ;• and the coefficients a, p for the wgeq formulation.
On the other hand, the parameters associated with the vegetation can either
be derived from the dominant vegetation type, or be specified from existingclassification or observations. They are• the fraction of vegetation veg;
• the depth of the soil column d2 (or the root zone depth);• the depth of the soil column d3 (if third soil layer option in use);• the minimum surface resistance Rsmin ;• the leaf area index LAI;
• the heat capacity Cv of the vegetation;• the RGl and γ coefficients found in the formulation of the surface resistance Rs ;• and the roughness length for momentum z0 and for heat z0h .
Other necessary parameters are• the albedo α• the emissivity .
• and characteristic time scale for phase changes (currently constant) τi .

3 Water surfaces
3.1 Simple parameterization
Free water surfaces
For ocean surfaces and over inland waters, all the prognostic variables are
kept constant.

222

14. Surface processes scheme

The surface fluxes are calculated using Eqs. 55, 56, 58 and Eqs. 83, 84, 85 ofIsba, taking the relative humidity of the ocean hu = 1, and veg = psn = 0.
The roughness length is given by Charnock’s relation:u2
z0sea = 0.015 ∗
g

(101)

Sea ice
Sea ice is detected in the model when sea surface temperature (SST) is
two degrees below 0◦ C (i.e. 271.15 K). In this case, in order to avoid anoverestimation of the evaporation flux, the calculations are performed withthe roughness length of flat snow surfaces:
z0ice = 10−3 m(102)

In the same manner, the sea ice albedo is set equal to the fresh snow albedo
instead of the free water albedo. This leads to a much brighter surface. This
has no effect on the sea ice cover (since there is no evolution of the sea surface
parameters), but modifies the lower boundary shortwave flux input for theatmospheric radiative scheme.
Sea surface turbulent fluxes
Various sea surface fluxes parameterizations available in the Surfex surface
scheme. In addition to the direct parameterization from Louis (1979), twoiterative parameterizations are avalaible : the COARE3.0 (Fairall et al.,2003) and ECUME (Belamari, 2005) parameterizations. Arpege-climat
uses the ECUME parametrization.
Bulk equations
Bulk parameterizations estimate the surface fluxes from mean meteorological gradients in the atmospheric boundary layer. This method ’s aim is todetermine the transfer coefficients that directly link the surface flux with themeteorological gradients between the surface and a “measurement’s height”(Liu et al, 1979).
The surface turbulent fluxes, i.e. the stress or the momentum flux τsea , thesensible heat flux Hsea and the latent heat flux LEsea are expressed by:
τ |sea = ρa w0 u0 = −ρa u2∗
 |~H= ρ c w0 θ0 = −ρ c u θ

sea
a pa
a pa ∗ ∗ LE
0 0
sea = ρa Lv w q = −ρa Lv u∗ q∗(103)

14. Surface processes scheme

223

where u0 , θ0 and q 0 are the vertical perturbations of wind, temperature potential and specific humidity, u∗ , θ∗ and q∗ are the characteristic scale parameters from Monin-Obukhov.
Considering the bulk parameterizations using transfer coefficients:τ |sea = −ρa CD U 2
 |~= ρ c C U (θ − θ )

H

s
a
sea
a pa H LE
sea = ρa Lv CE U (qs − qa )(104)

s indicates sea surface variables whereas a indicates atmospheric variables
at first level. U is the mean value of the relative wind. Here, we choose theatmospheric convention, i. e. fluxes are defined positive in case of energy
benefit for the atmosphere.
From equations (103) and (104), we can write:

u∗ 2

 CD = Uθ∗CH = U (θua∗−θs)q∗
CE = U (qua∗−q
s)

(105)

In a general way, the transfer coefficient for the X variable is:CX =w 0 x0U ∆X(106)with X equal D for drag, H for heat and E for evaporation and ∆X is thegradient of x (= u, θ or q) between the ocean surface and the atmospheric
low level.
Each coefficient is divided in two components:
1

1

CX = cx2 cd2(107)

that could be expressed following the Monin-Obukhov’s similitude theory as
a function of the first atmospheric level height z, of atmospheric stratification
with a parameter ζ, of roughness lengths (z0 , z0t and z0q ) and of the Von
Karman’s constant κ:
1

1

1

2
2
Fx (ζ, κ, Cx10n)
cx2 (ζ) = Cx10n

1
2
Cx10n
=

κ
z
ln( z0x
)

(108)

(109)

224

14. Surface processes scheme

Roughness lengths are generally computed thanks to the relationship:z0 =αu2∗ βν
+
g
u∗

(110)

where α (also called the Charnock’s constant) and β are numerical constantsand ν is the dynamical viscosity.
Each of the following parameterizations uses its own closure hypothesis witha theoretical method or resulting from experimentation to determine theexchange coefficients from neutral transfer coefficients at 10m CD10n , CH10n
and CE10n (i.e. for ζ = 0) and from a stability function Fx and roughness
lengths (Lebeaupin et al., 2007 ).
Iterative parameterizations
Bulk equations could be resolve with iterative methods on the stability parameter and the characteristic scale parameters from Monin-Obukhov. Convergence criteria vary according to the parameterizations. They also differ inthe representation of various processes as waves effects, sea spray, seawatersalinity effect on evaporation, wind gusts and especially in the calculation of
the roughness lengths or of the transfer coefficients.
.
The Liu et al (1979) algorithm is the most used iterative algorithm for the
turbulent air-sea fluxes computation and was also a base for new parameterizations developments (for example, COARE or the ECUME parameterization).
The ECUME parameterization
The unified parameterization or ECUME (Exchange Coefficients from Unified Multi-campaigns Estimates) is a bulk iterative parameterization developed in order to obtain an optimized parameterization covering a wide rangeof atmospheric and oceanic conditions.
Based on the LKB algorithm, ECUME includes an estimation of neutraltransfer coefficients at 10m from a multi-campaign calibration derived fromthe ALBATROS database that collects data from five flux measurement
campaigns:• POMME “Programme Océanique Multidisciplinaire à Moyenne Echelle”,• FETCH “Flux, Etat de la mer et Télédetection en Condition de Fetch”,• SEMAPHORE “Structure des Echanges Mer-Atmosphère, Propriétésdes Hétérogénéités Océaniques : Recherche Expérimentale”,

14. Surface processes scheme

225

Figure 3: Multi-campaign calibration of the neutral drag coefficient at 10
meters CD10n .
• CATCH “Couplage avec l’ATmosphère en Conditions Hivernales”,• EQUALANT99.
A more detailed description of each campaign could be found in Weill et al(2003 ) and Belamari ().
A similar post-processing was applied to the five campaigns data to derive
the drag coefficient CD10n , the heat coefficient CH10n and the evaporation
coefficient CE10n as neutral 10m-wind functions (Figures 3, 4 and 5).
The ECUME parameterization main characteristic are:
1. An important effort was done on the ECUME algorithm in order toassure the convergence in maximum 20 iterations for every kind of conditions. The iterative sequence is stopped when the difference between
the scale parameters between two iterations is inferior to prescribed
threshold that are 2.10−4 m s−1 for u∗ , 2.10−4 K for θ∗ and 2.10−7
kg/kg for q∗ .
The closure relationship is the multi-campaign calibration of the neutral transfer coefficients at 10 meters.

226

14. Surface processes scheme

Figure 4: Multi-campaign calibration of the neutral heat coefficient at 10
meters CH10n .

Figure 5: Multi-campaign calibration of the neutral evaporation coefficient
at 10 meters CE10n .

14. Surface processes scheme

227

2. The stability functions are Businger’s functions with different coefficients than COARE3.0: ψm and ψh depend on the Monin-Obukhov’s
length ζ = Lz which is computed as described in the following equations:• For wind:ZL =gκz(T∗ (1 + r0 q) + r0 T q∗ )T (1 + r0 q) × [M AX(u∗ , 1.10−9 )]2

with r0 = Rv /Ra − 1.
si ZL ≥ 0z/L = M IN (ZL, 0.25)(111)

si ZL < 0 z/L = M AX(ZL, −200)(112)• For temperature and humidity:zqzt;(z/L)q = z/L ×(z/L)t = z/L ×zzFinally:ζ = Lzstable(ζ ≥ 0)
unstable:(ζ < 0)KansasConvective

ψm (ζ) =−Γζψh (ζ) =−Γζ

Γ=7(1 − f )ψmK + f ψmC(1 − f )ψhK + f ψhC
ζ2
f = (1.0+ζ 2 )21+xψmK = 2ln( 1+x
2 ) + ln( 2 )−2arctan(x) + π2
1with x = (1 − 16ζ) 4
2
ψmC = 32 ln( y +y+1)3√π√ )+ √− 3arctan( 2y+1
3
3

with x = (1 − 16ζ) 22
ψhC = 32 ln( y +y+1)3√π√ )+ √− 3arctan( 2y+1
3
3

with y = (1 − 12.87ζ) 3

with y = (1 − 12.87ζ) 3

1

ψhK = 2ln( 1+x
2 )
1

1

3. The roughness length is given by (Eq. 110) with α = 0.011 and β =0.11.
4. The reduction of 2% of the specific humidity at saturation due to seawater salinity is applied (eq. 113, Kraus (1972).

qs = 0.98 × qsat (θs )

(113)

228

14. Surface processes scheme

5. The gustiness correction could be applied (Eq. 114).
1

wg = βgust (bf.zbl ) 3(114)

6. The corrections due to precipitation τp and Hp according to Fairall etal. (1996) and Gosnell et al. (1995) could also be computed in the
ECUME parameterization ( Eq. 115 and 116).

τp =

RU
3600

(115)


Hp = R̃cpr (Ts − Ta ) 1 +

1
B



(116)

7. The Webb’s correction (LEwebb ) is a correction applied to the latent
heat flux. It is due to air density variations when the humidity vary
under the evaporation action. If w̄ is the mean value of the vertical
perturbations,w = 1.61w0 q 0 + (1 + 1.61q)

w0 T 0T(117)

the Webb’s correction expression is:
LEW ebb = ρa Lwq(118)

where L is the latent heat of vaporization for water.
8. No waves effects are taking into account in the ECUME parameterization.

4 Parametrization of the surface boundary layer
Diagnostics computed by Surfex within the routine atmosphere ’ARO_GROUND_DIAG’:• surface humidity• roughness lenght for momentum• roughness length for heat
• temperature at 2m• specific humidity at 2m

14. Surface processes scheme229• relative humidity at 2m• zonal wind at 10 m
• meridional wind at 10 m
• longwave budget between Arpege and Surfex• shortwave budget between Arpege and Surfex• surface latent heat flux
For more details you can refer to the Scientific Documentation at the http://www.cnrm.meteo.
fr/surfex/.

Appendix A: Continuous formulation of the soil secondary parameters
The sand and clay composition (i.e., SAN D and CLAY ) are expressed inpercentage.
The saturated volumetric water content (m3 m−3 ):wsat = (−1.08SAN D + 494.305) × 10−3

(119)

The wilting point volumetric water content (m3 m−3 ):wwilt = 37.1342 × 10−3 (CLAY )0.5

(120)

The field capacity volumetric water content (m3 m−3 ):wf c = 89.0467 × 10−3 (CLAY )0.3496(121)

The slope of the retention curve:b = 0.137CLAY + 3.501(122)

The soil thermal coefficient at saturation (Km2 J −1 ):CGsat = −1.557 × 10−2 SAN D − 1.441 × 10−2 CLAY + 4.7021 (123)230

14. Surface processes scheme

The value of C1 at saturation:
C1sat = (5.58CLAY + 84.88) × 10−2

(124)

The value of C2 for w2 = 0.5wsat :C2ref = 13.815CLAY −0.954(125)

The coefficient C3 :C3 = 5.327CLAY −1.043(126)

The coefficient C4b :C4b = 5.14 + 0.115 CLAY(127)

The coefficient C4 ref :3 X2(d3 − d2 )
−1
j
j
log
β
+
β
SAN
D
+
α
CLAY
(128)
0
j
j
10
(d2 d3 2 )j=1"

C4 ref=#where the βj (j = 0, 3) coefficients are 4.42 × 10−0 , 4.88 × 10−3 , 5.93 × 10−4
and −6.09 × 10−6 . The αj (j = 1, 3) coefficients are defined as −2.57 ×
10−1 , 8.86 × 10−3 and −8.13 × 10−5 .
The coefficients for the wgeq formulation:a = 732.42 × 10−3 CLAY −0.539(129)

p = 0.134CLAY + 3.4(130)

Appendix B: Gaussian formulation for the C1 coefficient
For dry soils (i.e., wg < Wwilt ), the C1 coefficient in Eq. (13) is approximatedby the Gaussian distribution:"(wg − wmax )2C1 (w) = C1max exp −
2σ 2

#

(131)

14. Surface processes scheme

231

In this expression,
C1max = (1.19wwilt − 5.09) × 10−2 Ts + (−1.464wwilt + 17.86) (132)
wmax = ηwwilt(133)

η = (−1.815 × 10−2 Ts + 6.41)wwilt + (6.5 × 10−3 Ts − 1.4)

(134)

with

and
σ2 = −

2
Wmax

2ln



0.01
C1max



(135)

232

14. Surface processes scheme

15
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 fileinterpolated onto model levels from a climatology coming from University ofReading 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 froma 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 havecharacteristic times of several years). Stating at the equilibrium situation,
one advances one time step ahead in each grid 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 coefficient of the ozone field. Time characteristic of this relaxation

234

15. Ozone(calculated for each latitude and each level of the model 2d and each monthof 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 tophoto-dissociation increases.
The following linear model is thus considered:∂rO3∂t∂P − L[rO3 − rO3 ]∂rO3ii ∂P − L h∂P − L h+
T −T +
ΣO3 − ΣO3
∂T∂ΣO3

= P −L+(1)

where the over-lined quantities were obtained using the 2d model MOBIDYC.
This linear model was substituted for the photochemical model in the 2dmodel, and the results of the two 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 the fullmodel 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 eachpoint:
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 thecoherence of 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15. Ozone

235

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 toocomplicated) 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) and when 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 of Equation (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 thehighest levels of the model (i.e. in the mesosphere) when the stratosphericvertical discretization is high. This parameterization consists simply of alinear relaxation of the wind towards 0, of specific moisture towards qmin =3.725 10−6 (RFMESOQ) to avoid excessive drying and represent chemicalsources of H2 O and of the temperature towards the temperature of standard

236

15. Ozone

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 anotherset of coefficients. Profile K(l) is then:pref − psta (l)K(l) = α max,0
psta (l)where psta (l) is the standard air pressure at level l. In practice, these twolevels are selected above 1 hP a and the constants are adjusted in order toobtain time-constants of a few hours in the highest level.
The writing of linear parameterization is a little complicated by the fact thaton 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:X
1 KLEVK(i)PFRMH = −δp(i)cp (i)(T (i) − Tsta (i))
g i=l+1
1 + ∆tK(i)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) thetemperature. Contrary to fluxes produced by the other parameterizations,this flux is zero at surface. The momentum and energy exchange is donewith space and not with earth.

16
Diabatic terms

1 Physical parametrizations
In Arpege-climat, grid point computations at each time step (except postprocessing) are carried out in subroutine CPG which calls MF_PHYS. If one
excludes horizontal diffusion and nudging terms if any, the diabatic termsof the model equations come from physical parametrizations. This chapter describes what occurs in the model at MF_PHYS level. Subroutine
APLPAR is the manager of the physical parametrizations. It takes as input
the model prognostic variables at time t − ∆t and provides vertical fluxes
for these variables. Before making calculations of advection, i.e. at the origin points of the trajectories, these fluxes are used to evolve the prognostic
variables. The next chapters will describe for each parametrization or groupof parametrizations how the fluxes are calculated for altitude variables. The
surface variables are now computed by Surfex .

2 Calculation of altitude tendencies
The physical tendencies of the atmospheric variables are calculated fromphysical fluxes in subroutine CPTEND. We do not take into account here asin the rest of the documentation the option of variable atmospheric mass. Seealgorithmic documentation of Arpege-climat version 4 for the complete
formulas.

2.1 Equation of wind
∂U
∂Fu
= −g
∂t
∂p

238

16. Diabatic terms
∂Fv
∂V
= −g
∂t
∂p

with
Fu = Futurb + Fugwd + Fuconv
Fv = Fvturb + Fvgwd + Fvconv

2.2 Equation of water vapor
turb_conv

 h∂qv= −g 
∂t∂ Fqturb
+ Fqv
v
∂p

i



+

∂Fcond ∂pwithcond_conv

Fcond = Fqi

cond_conv+ Fql

cond_stra+ Fqi

cond_stra+ Fql

Here the condensation fluxes are simply the precipitation fluxes.

2.3 Equation of temperature and enthalpy
In Arpege parametrizations, the variable is not temperature, but moist
static energy. The equation of change of the temperature is given by:

cp

∂T
∂t

∂Fqv
∂Fql
∂Fqi
+ (cl − cpa )+ (ci − cpa )
∂p
∂p
∂p
cond
cond
∂Fql
∂Fqi
∂Fh
+
− Ll (T )− Li (T )
∂p
∂p
∂p
∂
∂
+Fqcond((cl − cpa )T ) + Fqcond((ci − cpa )T )
i
l
∂p
∂p
1 ∂(u2 + v 2 )
−
2
∂t




= −g −T (cpv − cpa )

with:
∂Φ
=0
∂t
turb_conv

Fqv = Fqturb
+ Fqv
v



16. Diabatic terms

239
turb_conv

Fqi = Fqturb+ Fqii

turb_conv+ Fql
Fql = Fqturblturb_conv

Fh = Fhturb + Fh

+ Fhrayt + Fhrays

Since we do not take into account here the time evolution of liquid and solid.
water, we have Fpi = Fqcond
and Fpl = FqcondilOne names here enthalpy the quantity:
h = cp T + Φ +u2 + v 22

with:
cp = cpa + (cpv − cpa )qv + (cl − cpa )ql + (ci − cpa )qi
After some algebraic calculations, one can arrive at an equation of evolution
for enthalpy:
∂
∂h
= −g
[Fh + Fhp ]
∂t
∂p
with:
Fhp = −Ll (0)Fq cond
− Li (0)Fq cond
+ Fpl (cl − cpa ) T + Fpi (ci − cpa ) Tilwhere Li (0) is the latent heat of sublimation to 0 K and Ll (0) the latent
heat of vaporization to 0 K.

3 Flux of enthalpy due to wet processes
Subroutine CPHPFS calculates fluxes of enthalpy and sensible heat generated by precipitation and condensation.

3.1 Sensible heat flux
Fs conv= Fp conv(cl − cpa )Tll
Fs conv= Fp conv(ci − cpa )Tii
Fs stra= Fp stra(cl − cpa )Tll
Fs stra= Fp stra(ci − cpa )T
i
i

240

16. Diabatic terms

3.2 Enthalpy flux related to condensation
cond_conv

Fh conv= −Ll (0)Fqllcond_conv

Fh conv= −Li (0)Fqi
i

cond_stra

Fh stra= −Ll (0)Fqllcond_stra

Fh stra= −Li (0)Fqi
i

3.3 Total precipitation flux
Fp = Fp conv
+ Fp conv+ Fp stra+ Fp stralili

3.4 Total enthalpy flux related to precipitation and condensation

Fhp = Fs conv
+ Fs conv+ Fs stra+ Fs stralili+Fh conv+ Fh conv+ Fh stra+ Fh stralili

4 Time advance in altitude
One modifies the prognostic variables in the atmosphere at time t + ∆t
(index 1): u1 , v1 , T1 , qv1 , ql1 and qi1 by adding to it the physical tendencies
in CPUTQY.

4.1 Evolution of wind and of vapor, liquid and solid water
One has:u1 = u1 + ∆t PTENDU
v1 = v1 + ∆t PTENDV
qv1 = qv1 + ∆t PTENDQ16. Diabatic terms

241

4.2 Evolution of temperatureFirst of all, enthalpy at t − ∆t (index 9) is calculated:
h9 = cp9 T9 +u29 + v92
2

Then, specific moisture at t + ∆t, due only to physics:
qv9+ϕ = qv9 + ∆t PTENDQ
ql9+ϕ = ql9 + ∆t PTENDQL
qi9+ϕ = qi9 + ∆t PTENDQI
cp1 = cpa (1 − qv9+ϕ − ql9+ϕ − qi9+ϕ ) + cl ql9+ϕ + ci qi9+ϕ
as well as kinetic energy at t + ∆t, due to physics:
K1 =(u9 + ∆t PTENDU)2 + (v9 + ∆t PTENDV)22

One can thus write the new temperature:
T1 = T1 +h9 − K1 + ∆t PTENDH
− T9
cp1

4.3 Dissipation of the kinetic energy
By definition flux at the top is equal to zero.
PFDIS(l) = PFDIS(l − 1) +δp (K1 − K9 )
g ∆t

242

16. Diabatic terms

17
Diagnostics

1 Introduction
Arpege-climat calculates a large number of diagnostics during the courseof the integration. This software, controlled by the model namelist, is known
as “Full Pos”. It includes pressure level, model level, and surface instantaneous as well as time accumulated fields. One Arpege file and one Surfex
file are created at each postprocessing time step (NFRPOS/NFRSFXHIS),
prefixed by PF for the atmosphere and SFX+ for the surface diagnostics.
A more detailed description of the “Full Pos” features is available in Yessad(2007e). A comprehensive “Full Pos” User’s guide is maintained by Ryad El
Khatib who is also in charge of the code development.
In the standard script of Arpege-climat, an additional postprocessing is
performed at run time, but it does not use Arpege-IFS executable. Fromthe many PF and SFX files generated by the model, a monthly mean fileis built (in Arpege format) and various ieee-binary files of time series areproduced. See the Arpege-climat User’s guide for details. In all files, theprinciples are that IS units are used (P a for pressure, K for temperature),
vectors like wind have their components on the computational sphere andupward fluxes are counted negatively.

2 Post-processable fields
Post-processing can be made on pressure levels, height levels, potential vorticity levels, potential temperature levels or η-levels.

244

17. Diagnostics

2.1 3D dynamical fields (DYN3D)
Note that availability of some fields depends on the physical parametrizations
which have been activated. The standard Arpege-climat does not offer
all features.
Non derived:• Aerosols AERO
• Cloud fraction qa• Downdraft mesh fraction qdal
• Downdraft vertical velocity qdom
• Equivalent potential temperature Θe• Extra GFL fields• Geopotential height gz• Graupel qg• Greenhouse gases GHG• Horizontal wind components U and V• Ice content qi• Liquid water content ql
0• Moist (irreversible) pseudo-adiabatic potential temperature Θw• Montgomery geopotential Φmg• Ozone O3• Potential temperature Θ• Pressure p• Prognostic pseudo-historic entrainment quen
• Pseudo-historic convective cloudiness qunebh• Rain qr• Reactive gases GRG• Relative humidity HU

17. Diagnostics

245• Simulated reflectivity• Snow qs• Specific humidity (moisture) q• Stream function ψ (has to be fitted)• Temperature T• Tracers T RAC• Turbulent kinetic energy T KE
• Updraft mesh fraction qual
• Updraft vertical velocity quom
• Velocity potential χ (has to be fitted)• Virtual potential temperature Θv• Wind velocity k~uk
Derived:• Absolute vorticity ζ + f• Divergence D• Potential vorticity P V• Pressure coordinate vertical velocity ω• Relative vorticity ζ• Shearing deformation SHD• Stretching deformation ST D

2.2 2D dynamical fields (DYN2D)
Fields marked with ♣ are in the Surfex file:• Altitude of isotherm 0• Altitude of isotherm -100• Altitude of iso-Θw =0

246

17. Diagnostics• Convective available potential energy• Convective inhibition energy• ICAO jet meridian component of wind
• ICAO jet zonal component of wind
• ICAO tropopause pressure• ICAO tropopause temperature• Interpolated (spectral) model orography• Logarithm of surface pressure ln ps• Mapping factor M• Maximum relative humidity at 2 m Rx
• Maximum temperature at 2 m Tx ♣• Mean sea level pressure pM SL• Minimum relative humidity at 2 m Rn ♣• Minimum temperature at 2 m Tn ♣• Moisture convergence• Planetary boundary layer depth• Pressure jet• Pressure of isotherm 0• Pressure of isotherm -10• Relative humidity at 2 m Rcls ♣• Specific humidity at 2 m qcls ♣• Surface pressure ps• Temperature at 2 m Tcls
• Total water vapor content in a vertical column• Tropopause folding indicator of the iso-2 PVU surface• U-component of wind at 10 m Ucls ♣• V-component of wind at 10 m Vcls ♣• Wind velocity at 10 m ♣• Max Wind velocity at 10 m ♣17. Diagnostics

247

2.3 Physical fields (PHYSOL)These fields are Surfex fields except for the aerosols and ozone which arearchived in the Arpege file.
• Aerosols horizontal distribution (continental)• Aerosols horizontal distribution (desert)• Aerosols horizontal distribution (marine)• Aerosols horizontal distribution (soot)• Aerosols horizontal distribution (sulfate)• Aerosols horizontal distribution (volcano)• Anisotropy coefficient of orography• Anisotropy vector U-momentum• Anisotropy vector V-momentum• Budget values• Climatological relative deep soil wetness• Climatological relative surface soil wetness• Deep soil temperature• Deep soil wetness• Direction of main axis of orography• Emissivity• Free surface fields• Frozen deep soil wetness• Frozen surface soil wetness• Index of vegetation• Instantaneous surface heat flux
• Instantaneous U-component of surface stress• Instantaneous V-component of surface stress• Interception content

248

17. Diagnostics• Interpolated dynamic surface g ∗ z0
• Interpolated surface temperature• Interpolated thermal surface g ∗ z0
• Land/sea mask
• Leaf area index• Maximum soil depth• Minimum stomatal resistance• Output grid-point orography times g• Ozone horizontal distribution (O3A)• Ozone horizontal distribution (O3B)• Ozone horizontal distribution (O3C)• Percentage of clay within soil
• Percentage of land
• Percentage of sand within soil
• Percentage of urbanization• Percentage of vegetation• Relaxation deep soil wetness• Resistance to evapotranspiration• Roughness length for heat times g• Roughness length of bare surface times g• Snow depth• Soil depth• Standard deviation of orography times g• Surface albedo• Surface albedo for bare soil
• Surface albedo for snow-free areas• Surface albedo for soil with vegetation

17. Diagnostics• Surface relative moisture• Surface roughness length times g• Surface snow albedo• Surface snow density• Surface soil wetness• Surface temperature• Vegetation roughness length times g

2.4 Physical accumulated fluxes• Contribution of convection to cp T flux
• Contribution of convection to q flux
• Contribution of convection to U flux
• Contribution of convection to V flux
• Contribution of gravity wave drag U flux
• Contribution of gravity wave drag V flux
• Contribution of turbulence to cp T flux
• Contribution of turbulence to T flux
• Contribution of turbulence to U flux
• Contribution of turbulence to V flux
• Convective cloud cover• Convective rainfall• Convective snow fall
• Deep soil runoff flux
• Evapotranspiration flux
• Filtered duration of total precipitation• Heat flux in soil
• High cloud cover

249

250

17. Diagnostics• Interception layer runoff flux
• Large scale rainfall (stratiform)• Large scale snow fall (stratiform)• Latent heat flux
• Liquid evaporation flux
• Liquid latent heat flux
• Liquid specific moisture• Low cloud cover• Medium cloud cover• Sensible heat flux
• Snow evaporation flux
• Snow mass
• Snow melt flux
• Soil moisture• Solid latent heat flux
• Solid specific moisture• Surface clear sky solar radiation flux
• Surface clear sky thermal radiation flux
• Surface downward moon radiation flux
• Surface downward solar flux
• Surface downward thermal flux
• Surface enthalpy flux (due to the dissipation of kinetic energy)• Surface parallel solar flux
• Surface soil runoff flux
• Surface solar radiation flux
• Surface thermal radiation flux
• Tendency of surface pressure17. Diagnostics• Top clear sky solar radiation flux
• Top clear sky thermal radiation flux
• Top mesospheric enthalpy flux (+ dissipation)• Top parallel solar flux
• Top solar radiation flux
• Top thermal radiation flux
• Total cloud cover• Total ozone• Total precipitable water• Transpiration flux
• Water flux in soil

2.5 Physical instantaneous fluxes• CAPE• Cloudiness• Contribution of convection to cp T
• Contribution of convection to q
• Contribution of convection to U
• Contribution of convection to V
• Contribution of gravity wave drag to U
• Contribution of gravity wave drag to V
• Contribution of turbulence to cp T
• Contribution of turbulence to q
• Contribution of turbulence to U
• Contribution of turbulence to V
• Convective cloud cover• Convective precipitation

251

252

17. Diagnostics• Convective snow fall
• Height of the PBL out of the model• High cloud cover• IR clear sky radiance• IR cloudy sky radiance• Large scale precipitation• Large scale snow fall
• Low cloud cover• Maximum relative humidity at 2 m• Maximum temperature at 2 m• Medium cloud cover• Minimum relative humidity at 2 m• Minimum temperature at 2 m• Moisture convergence• Relative humidity at 2 m• Specific humidity at 2 m• Surface solar radiation• Surface thermal radiation• Temperature at 2 m• Top solar radiation• Top thermal radiation• Total cloud cover• U-component of wind at 10 m
• V-component of wind at 10 m
• Wind modulus at 10 m
• WV clear sky radiance• WV cloudy sky radiance17. Diagnostics

253

3 Horizontal interpolations
Horizontal interpolations can be bilinear interpolations or 12-point cubic
interpolations.

3.1 Bilinear horizontal interpolations
Horizontal interpolation grid and weights for bi-linear interpolations
A 16-point horizontal grid is defined as it is shown in Figure ?? of Chapter 4.
The interpolation point O is between B1 , C1 , B2 and C2 . Λ and Θ are thelongitudes and latitudes on the computational sphere (departure geometry).
The following weights are defined as follows:
zonal weight number 1:
ZDLO1 =

ΛO − ΛB1
ΛC1 − ΛB1

zonal weight number 2:
ZDLO2 =

ΛO − ΛB2
ΛC2 − ΛB2

meridional weight:
ZDLAT =

ΘO − ΘB1
ΘB2 − ΘB1

Bilinear interpolation
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 meridional linear interpolation:
Xinterpo = X1 + ZDLAT (X2 − X1 )In the FULL-POS code the weights are pre-computed in routines SUHOW2and SUHOWLSM, so the separation of zonal and meridian interpolations is
not visible in the interpolation routines.

254

17. Diagnostics

3.2 12-point horizontal interpolations
Horizontal interpolation grid and weights for 12-point cubic interpolations
A 16-point horizontal grid is defined as it is shown in Figure ?? of Chapter 4.
The interpolation point O is between B1 , C1 , B2 and C2 . The following
weights are defined as follows:
zonal weight number 0:

ZDLO0 =

ΛO − ΛB0
ΛC0 − ΛB0

zonal weight number 1:

ZDLO1 =

ΛO − ΛB1
ΛC1 − ΛB1

zonal weight number 2:

ZDLO2 =

ΛO − ΛB2
ΛC2 − ΛB2

zonal weight number 3:

ZDLO3 =

ΛO − ΛB3
ΛC3 − ΛB3

meridional weights:

ZCLA2 =

(ΘO − ΘB0 )(ΘO − ΘB2 )(ΘO − ΘB3 )(ΘB1 − ΘB0 )(ΘB1 − ΘB2 )(ΘB1 − ΘB3 )ZCLA3 =

(ΘO − ΘB0 )(ΘO − ΘB1 )(ΘO − ΘB3 )(ΘB2 − ΘB0 )(ΘB2 − ΘB1 )(ΘB2 − ΘB3 )

ZCLA4 =

(ΘO − ΘB0 )(ΘO − ΘB1 )(ΘO − ΘB2 )(ΘB3 − ΘB0 )(ΘB3 − ΘB1 )(ΘB3 − ΘB2 )17. Diagnostics

255

Horizontal 12-point interpolation
Let us define:f2 (α) = (α + 1)(α − 2)(α − 1)/2
f3 (α) = −(α + 1)(α − 2)α/2
f4 (α) = α(α − 1)(α + 1)/6

For a quantity X, are computed successively:a linear interpolation on longitude number 0:
X0 = XB0 + ZDLO0(XC0 − XB0 )a cubic 4-point interpolation on longitude number 1:
X1 = XA1 + f2 (ZDLO1)(XB1 − XA1 ) + f3 (ZDLO1)(XC1 − XA1 )+f4 (ZDLO1)(XD1 − XA1 )a cubic 4-point interpolation on longitude number 2:
X2 = XA2 + f2 (ZDLO2)(XB2 − XA2 ) + f3 (ZDLO2)(XC2 − XA2 )+f4 (ZDLO2)(XD2 − XA2 )a linear interpolation on longitude number 3:
X3 = XB3 + ZDLO3(XC3 − XB3 )a meridional cubic 4-point interpolation:
Xinterpo = X0 + ZCLA2(X1 − X0 ) + ZCLA3(X2 − X0 ) + ZCLA4(X3 − X0 )In the FULL-POS code the weights are pre-computed in routines SUHOW2and SUHOWLSM, so the separation of zonal and meridian interpolations is
not visible in the interpolation routines.

256

17. Diagnostics

4 Vertical interpolations and extrapolations
4.1 General considerations
For 3D variables to be vertically interpolated, vertical interpolations are
generally linear interpolations between the layers where are defined model
variables. The treatment of the extrapolations above the highest layer, theextrapolations below the lowest layer or the surface depend on the variableconsidered. In particular cases some variables can be diagnosed using thevertically interpolated value of some other variables.

4.2 Notations
dT: standard atmosphere vertical gradient of the temperature in
dz ST
the troposphere (0.0065 K m−1 )•• Rd : dry air constant• g: gravity acceleration• L: number of layers of the model

4.3 More details for 3D dynamical variables
Wind components, wind velocity
Way of interpolating (subroutine PPUV):• Linear interpolation between layer 2 and the lowest layer.
• The coordinate used for linear interpolation is the logarithm of thepressure.
• Linear interpolation between layer 1 and layer 2 using the values oflayers 1, 2 and 3.
• Linear interpolation between the top and layer 1 using the values of
the top, layers 1 and 2; the value of the top is obtained by a linear
extrapolation from the values of layers 1 and 2.
• Extrapolation below the middle of the lowest layer and below the surface assumes that the quantity is constant.

17. Diagnostics

257

Temperature
Applies to temperature if the vertical coordinate of post-processing is not
the potential vorticity, otherwise see routine PP2DINT.
Way of interpolating (subroutine PPT):• Quadratic interpolation between the middles of the upper and lowest
layers.
• Quadratic interpolation between the top and the middle of the highest
layer: the top value of the temperature is assumed to be equal to thevalue of the middle of the highest layer; due to the fact that the interpolation is a quadratic one, that does not mean that the temperature
is constant in this atmosphere depth.
• The coordinate used for quadratic interpolation is the logarithm ofpressure.
• A surface temperature TSU RF is computed by Surfex on land surface
and the ocean model on sea surface.
Geopotential height
Applies to geopotential height if the vertical coordinate of post-processing is
not the potential vorticity, otherwise see routine PP2DINT.
Way of interpolating (subroutine PPGEOP):• The variable interpolated is a geopotential height departure from a reference defined by a standard atmosphere without any orography. Afterthe interpolation an increment is added, sum of the surface orography
and the “standard” geopotential height depth between the pressurelevel of interpolation and the actual surface. This method avoids tointroduce interpolations for the standard component of the geopotential height which can be computed analytically (in routine PPSTA).
• Quadratic interpolation between the middles of the upper and lowest
layers.
• Quadratic interpolation between the top and the middle of the highest
layer.
• The coordinate used for quadratic interpolation is the logarithm ofpressure. The interpolation is a quadratic analytic expression of thelogarithm of pressure of the same type as the one used to post-process
the temperature.

258

17. Diagnostics• Linear interpolation between the lowest layer and the surface• Extrapolation below surface uses the surface temperature TSU RF of
Equation (??).
psgzextrapo = Φs − Rd TSU RF ln
pL




y y2
1+ +
2
6

dT
where y is defined by formula (??) with Γ =
dz


!

(1)



in all cases.
ST

Variables interpolated using routine PP2DINT
List of variables:• Geopotential height gz if vertical coordinate is potential vorticity• Temperature T if vertical coordinate is potential vorticity• Relative vorticity ζ• Divergence D• Potential temperature Θ if vertical coordinate is not potential temperature• Velocity potential χ• Stream function ψ• Equivalent potential temperature Θe• Absolute vorticity ζ + f• Stretching deformation ST D
• Shearing deformation SHD• Potential vorticity P V
Way of interpolating:• Linear interpolation (between the upper and the lowest layer for quantities defined on the middle of layers, between the layer 1 and thesurface for quantities defined on inter-layers).
• The coordinate used for linear interpolation is the pressure.
• Extrapolation above the middle of the highest layer assumes that thequantity is constant.
• Extrapolation below the middle of the lowest layer and below the surface assumes that the quantity is constant.

17. Diagnostics259

GFL variables (moisture, . . . )These variables use subroutine PPQ.
• Linear interpolation (between the upper and the lowest layer).
• The coordinate used for linear interpolation is the pressure.
• Extrapolation above the middle of the highest layer assumes that thequantity is constant.
• Extrapolation below the middle of the lowest layer and below the surface assumes that the quantity is constant.

Relative humidity
This variable uses subroutine PPRH.
• Linear interpolation (between the upper and the lowest layer).
• The coordinate used for linear interpolation is the pressure.
• Extrapolation above the middle of the highest layer assumes that thequantity is constant.
• Extrapolation below the middle of the lowest layer and below the surface assumes that the quantity is constant.

Pressure coordinate vertical velocity ωThis variable uses subroutine PPVVEL.
• Linear interpolation (between the upper and the lowest layer).
• The coordinate used for linear interpolation is the pressure.
• Extrapolation above the middle of the highest layer is a linear interpolation between a zero value at the top and the value of the highest
layer.
• Extrapolation between the middle of the lowest layer and the surface
assumes that the quantity is constant.
• Extrapolation below the surface assumes that the quantity is zero.

260

17. Diagnostics
0

Moist (irreversible) pseudo-adiabatic potential temperature Θw
Subroutine PPTHPW provides this diagnostic. It takes as input the vertically post-processed pressure, temperature, moisture, liquid water and ice0
and computes Θw at the post-processing levels using a diagnostic (and rather
complicated) algorithm.

4.4 2D dynamical variables which need extrapolations
Mean sea level pressure pM SL
Subroutine PPPMER provides this diagnostic. It takes as input the vertically If | Φs | is lower than 0.001 J kg −1 the mean sea level pressure is set tothe surface pressure. In the other cases one uses the following algorithm:• One computes the surface temperature TSU RF of Equation(??) anddT
Φsthe mean sea level temperature T0 = TSU RF +.
dz ST g• To avoid extrapolation of too low pressures over high and warm surfaces
the following modifications are done:– if T0 > 290.5 K and TSU RF ≤ 290.5 K, Γ is defined by:gΓ = (290.5 − TSU RF )(2)Φs
– if T0 > 290.5 K and TSU RF > 290.5 K, Γ is set to 0, TSU RF is
modified and set to 0.5 ∗ (290.5 K + old value of TSU RF ).
• To avoid extrapolation of too high pressures over cold surfaces thefollowing
modifications are done when TSU RF < 255 K: Γ is set todT
and TSU RF is modified and set to 0.5 ∗ (255 K + old value ofdz ST
TSU RF ).


• In the other cases Γ is set to

dT
dz



.
ST

• Mean sea level pressure is computed as follows:"ΦspM SL = ps exp
Rd TSU RF

x x2
1− +
2
3

!#

(3)

where:x=ΓΦsgTSU RF(4)

17. Diagnostics

261

5 Filtering in spectral space
Filtering is done in routine SPOS and processed differently on derivative andnon-derivative fields.

5.1 THX filtering on derivatives
Cases where this filtering is used: This filtering applies in the spectral
space to absolute vorticity, relative vorticity, divergence, vertical velocity,
stretching and shearing deformations, and potential vorticity (and extends
to all variables if the vertical coordinate of post-processing is the potential
vorticity). This filter is active if:• CFPFMT is not ’MODEL’ in namelist NAMFPC• LFPFIL=.TRUE. in namelist NAMFPF• Variable NFMAX is smaller than the equivalent unstretched spheretruncation Nc (in practical Nc is between 1.1 c Ns and 1.2 c Ns , wherec is the stretching factor)
If one wishes to keep these fields unfiltered, then just set LFPFIL=.FALSE.
(namelist NAMFPF). On the other hand, you can keep the filter active but
you can tune the filtering function.
Function of filtering: This function looks like a smoothed step function;
for a given total wave-number n in the unstretched spectral space (i.e. thespectral space of the equivalent unstretched sphere of truncation Nc ), theformula is:fT HX (n) =1 − tanh(e−k (n − n0 ))2(5)(The use of the function hyperbolic tangent is the reason of the nickname
THX for this filter). It means that this function equals roughly 1 if n is lessthan n0 , and 0 if it is greater than n0 .
Tunable parameters in the previous function: k and n0 are tunable
parameters:• n0 is in the variable NFMAX. If CFPFMT=’GAUSS’ or ’MODEL’ in
namelist NAMFPC, the default value is NFPMAX*FPSTRET; else, itis the truncation of the Gaussian grid which would have been definedby NLAT latitudes and NLON longitudes with default so that 3 ∗N F M AX + 1 = min(N LAT , N LON )262

17. Diagnostics• k is stored in variable RFPBED

Operations done in SPOS for this filtering: One assumes that CFPFMT is not ’MODEL’.
• LFPFIL=.T. and Ns < Nc : the previous function fT HX (n) is computed in the equivalent unstretched sphere of truncation Nc , so reading dilatation and contraction matrices respectively denoted by D andC (computed by the configuration 911) is necessary. The operator
applied to spectral fields in the computational sphere is a matrix operator C fT HX (n) D pre-computed in the routine FPFILTER (called by
SU3FPOS) and stored in the array RFPMAT. In SPOS the initially
unfiltered fields are in SPDFP and the filtered fields are put in SPBFP.
Filtering is done only if NFMAX < Nc , elsewhere there is a simple copyof SPDFP in SPBFP without filtering.
• LFPFIL=.T. and Ns = Nc : identity Ns = Nc is satisfied if the model
resolution has no stretching. fT HX (n) is stored in the array RFPFIL.
This function is directly applied in SPOS to SPDFP and the filteredfields are put in SPBFP. Filtering is done only if NFMAX < Ns , elsewhere there is a simple copy of SPDFP in SPBFP without filtering.
• LFPFIL=.F.: no filtering, simple copy of SPDFP in SPBFP.

5.2 Bell-shaped filtering on non-derivative fields
Way of filtering: It is also possible to filter the non-derivative postprocessed fields through bell-shaped filters. Separate bell-shaped filters areavailable for geopotential height, temperature, mean sea level pressure, relative humidity and all other non-derivatives. By default, these filters areactive for geopotential height, temperature, mean sea level pressure and relative humidity. For a given wave-number n in the stretched spectral space,
the formula is:
 !

k n 2
fbs (n) = exp −
2 Ns


(6)

where Ns is the model truncation (NSMAX) and k a tunable variable. In
SPOS, the bell-shaped filtering is done by multiplying the array SPAFP byfbs (n), the result is still in SPAFP.
Variables controlling bell-shaped filtering:17. Diagnostics

263• Switches LFPBEG for geopotential height, LFPBET for temperature,
LFPBEH for relative humidity, LFPBEP for mean sea level pressure,
LFPBEL for other non-derivatives,: .TRUE. if filtering, .FALSE. if nofiltering.
• Variables RFPBEG for geopotential height, RFPBET for temperature,
RFPBEH for relative humidity, RFPBEP for mean sea level pressure,
RFPBEL for other non-derivatives, : to control the intensity of thefiltering (variable k of formula (6) ).

264

17. Diagnostics

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