operational model Arp` ege-Aladin

Part II

Physical parameterizations in

operational model Arp` ege-Aladin

Chapter 3

Introduction

Confronted to the lack of a comprehensive up-to-date documentation of the Arp` ege-Aladin operational model physics, we decided to re-make a complete English documentation, starting from various texts available mainly in French, inquiring for undocumented features, and completing the picture with theoretical bases supporting the model implementation.

The physical parameterisations are shared between the global model Arp` ege and the local area model Arp` ege-Aladin; the global model is also run in climate mode, or other specialised applications.

Our goal was to document the operational model and especially the LAM, so we focused on the parameterisa- tions relevant to the operational part, which besides are common to the LAM and to the global model.

As the documentation is also intended to the model implementors, there are sometimes detailed descriptions of the precise implementation, with names of variables, parameters, etc. Also, we did not discuss all implications and variants of the theory, and rather sticked to the way things are coded in the model.

Seen the interest of such a documentation for the Aladin community, we had the chance to receive feedback and corrections from various developers of the model.

We also gave lectures in April 2000 in Radostovice (Czech Republic) about some parts of this work to Aladin project participants.

This documentation is now available on-line on the Aladin web site, and should receive upgrades on a regular basis.

We are very grateful to the persons who brought corrections and remarks, in particular Jean-Fran¸cois Geleyn ¹ , Alfred Quinet ³ , Jean-Marcel Piriou ¹ , Doina Banciu ² , Eric Bazile ¹ .

For the logical structuration of present document, we reported the presentation of the convection scheme, on which we based most of our developments, to part III. The other parameterizations are presented here. As we give them with no specific adaptation for this document, there could appear a few redundancies with the general presentation given in part I, but they are scarce.

First in Chapter 4 we review the interface between the physical package and the rest of the model.

General auxiliary functions called by various packages, as well as particular energy budget-related concepts are presented in chapter 5.

Chapter 6 presents the planetary boundary layer and turbulent fluxes treatment, which is up to now done diagnostically.

In Chapter 7 we present different subgrid orographic effects, grouped in the model under the label “gravity wave drag”, but covering actually a wider range of phenomena.

Chapter 8 concerns the soil-vegetation scheme, providing the lower boundary interface for heat and moisture fluxes.

1

M´et´eo France, CNRM/GMAP, Toulouse

2

INMH, Bucharest, Romania

3

RMIB, Brussels

INTRODUCTION

Chapter 9 presents the large scale cloudiness and precipitation schemes, and Chapter 10 the radiation scheme.

Notes:

• A table of essential notations has been given at the beginning of this document.

• Model Fortran names (parameters, variables, subprograms) are often mentioned, and typed in sanserif font.

Many of these technical indications might be skipped by the general reader.

• General naming conventions apply in the entire model code:

– Real-type local variables (in a FORTRAN sense) begin with Z, integer local variables begin with I;

– Real-type subroutine arguments begin with P, integer arguments with K, Logical arguments with LD and character arguments with CD

– All other names are global variables (gathered in various FORTRAN-90 Modules).

Chapter 4

Physics-Dynamics interface

4.1 Equations

4.1.1 Coordinate systems

In the physical grid point routines, the equations are generally written in hydrostatic pressure or geopotential φ = gz coordinates. The conversion rules between different vertical coordinates are given by (1.5). The relevant coordinates yield:

• Hydrostatic pressure coordinate:

∂π

∂z = − ρg ∂z

∂π = − 1 ρg

∂π

∂φ = − ρ ∂φ

∂π = − 1

ρ π ˙ ≡ ω z ˙ ≡ w yields:

∇ ^z = ∇ ^π + ρg( ∇ ^π z) ∂

∂π

∂

∂t

z

= ∂

∂t

π

+ ρg( ∇ ^π z) ∂

∂π

• η coordinate:

π(x, y, η, t) = A(η)π top + B(η)π s (x, y, t) ∂π

∂η ≡ m = A ⁰ (η)π top + B( ⁰ η)π s (x, y, t)

= ⇒ ∂η

∂φ = − ρ m yields:

∇ ^z = ∇ ^η + ρg

m ( ∇ ^η z) ∂

∂η ∇ ^π = ∇ ^η − 1

m ( ∇ ^η π) ∂

∂η

4.1.2 Variable Mass option

Parameter NDPSFI ≡ δ m controls mass conservation:

• δ m = 0: Total mass is conserved, water rained out is replaced by dry air, while the vapour produced by evaporating precipitation is compensated by a removal of dry air.

• δ m = 1: Total mass is not conserved, mass changes are controlled by the precipitation-evaporation budget at the surface. Condensation produces a local mass deficit, while precipitation evaporation produces a local mass increase.

The budget for both situations is illustrated on figures 4.1 and 4.2.

Presently, the operational model is run with δ m = 0, while the more refined case δ m = 1 is used in longer range climatological runs.

As we do not want to model explicitly the surface viscous layer ( ∼ 1mm thickness) where the mixing between

evaporating water (q = 1) and atmospheric air (q 1) takes place, we will write the different mass fluxes at

4.1. EQUATIONS Physics-dynamics interface

E

qE (1 − q)E (1 − q)E − (1 − q)E

advection diffusion

L l

J v J a P _L

P _l

q = q sat

q = 1 q

Figure 4.1: Surface budget with δ

m

= 1. Blue arrows represent water, beige arrows represent dry air

E

E − E

diffusion

q = q sat

q = 1 q L

l

J v J a P

L

P l

−P _L

−P l

Figure 4.2: Surface budget with δ

m

= 0.

the top of the viscous layer and use them as boundary condition at the ground.

Within the surface layer, both mass and water are conserved.

In the atmosphere, mass is conserved if δ m = 0, but water is never conserved, as there are exchanges with the ground reservoirs.

Considering the case δ m = 1 (fig. 4.1):

• the net evaporation flux E from the soil produces an equal mass vertical advection of moist air charac- terized by the local specific humidity q. This implies to advect qE of water and (1 − q)E of dry air from the surface layer.

• Moisture steady state in the viscous layer requires the diffusion of J v = (1 − qE) of water, replaced by the diffusion of J a = − (1 − qE) of dry air.

• the precipitation flux P L leaving at the lower interface implies a loss of mass of the atmosphere.

while in the case total mass is conserved (δ m = 0, fig. 4.2), the evaporation flux E implies that the same mass of dry air be removed from the atmosphere. Similarly, the precipitation flux P is compensated by an opposite dry air flux, keeping the mass constant.

Hence the different fluxes at the top of the viscous layer write

• Water vapour advection: ˙ η ^∂p _∂η q = δ m Eq

• Water vapour diffusion: gJ v = gE(1 − δ m q)

Physics-dynamics interface 4.1. EQUATIONS

• Dry air advection: ˙ η ^∂p _∂η q a = gEδ m (1 − q)

• Dry air diffusion: gJ a = − gE(1 − δ m q)

So the total water flux from ground evaporation entering the atmosphere is always 1

g

˙ η ∂p

∂η q + J v

= E (4.1)

while we have:

• J v + J a = 0: no net mass transfer by diffusion. This remains valid along the whole vertical, i.e we assume that the vertical flux linked to turbulence is mass-conserving in the atmosphere.

• gJ a + ˙ η ^∂p _∂η q a =

0 if δ m = 1 gJ a = − gE if δ m = 0

so a dry air flux exiting the atmosphere ensures mass conservation in the case δ m = 0.

The surface budget has also an impact on the budget of the other layers, higher in the atmosphere:

• With δ m = 0, the local vertical divergence of the precipitation flux in layer l is compensated by an equivalent mass of dry air, added or removed from the atmosphere.

• On the contrary, with δ m = 1, the vertical divergence of the precipitation flux makes the mass of layer l change.

The vertical mass flux (counted positively downwards) writes, in η vertical coordinate:

1 g η ˙ ∂p

∂η

The surface budget affects primarily the surface pressure π s . The mass flux at interface level l 6 = L is affected by the mass change due to δ m = 1 as following:

˙ η ∂p

∂η δ

m

=1

l

=

˙ η ∂p

∂η δ

m

=0

l

+ δ m g ∂p _l

∂π s

∂π s

∂t

s.b. − P l

=

˙ η ∂p

∂η δ

m

=0

l

+ δ m g

B _l ( P L + E) − P l

where ^∂π _∂t

^s

s.b. is the partial tendency of the surface pressure due to the local vertical budget, and P is the precipitation flux.

4.1.3 Continuity equation

In η coordinates, where ∇ ^η and V η remind that the “horizontal” derivatives are computed along a constant η surface,

∂

∂t ∂p

∂η

= −∇ ^η ·

V η

∂p

∂η

− ∂

∂η

˙ η ∂p

∂η

− g ∂F m

∂η (4.2)

with the turbulent vertical mass flux

F m ≡ F _P + J v + J a = F _P

as J v + J a ≡ 0, and noting F _P the vertical mass flux due to the precipitation flux.

At the top of the atmosphere, we have F m = 0 =

˙ η ^∂p _∂η

. Using

∂

∂t ∂p

∂η

= B ⁰ (η) ∂π s

∂t and integrating vertically (4.2) from the top (η = 0) to level η:

B(η) ∂π s

∂t = − Z η

0

∇ ^η ·

V η

∂p

∂η

dη − η ˙ ∂p

∂η − gF _P (η)

4.1. EQUATIONS Physics-dynamics interface

If we integrate down to the surface we have η ˙ ^∂p _∂η

= δ m gE gF _P = δ m g P L

)

at η = 1

and as B(1) = 1 we obtain the evolution equation for the surface pressure:

∂π s

∂t = − Z 1

0

∇ ^η ·

V η

∂p

∂η

dη − gδ m (E + P L ) (4.3)

For levels above, the vertical mass advection term is zero, and the generalised vertical velocity writes

˙ η ∂p

∂η = − B(η) ∂π s

∂t − Z η

0

∇ ^η ·

V η

∂p

∂η

dη − gF _P (4.4)

The pressure vertical velocity:

ω ≡ p ˙ = ∂p

∂t + (V · ∇ ) η p + ˙ η ∂p

∂η = (V · ∇ ) η p + Z η

0

∂

∂η ∂p

∂t + ˙ η ∂p

∂η

dη

= (V · ∇ ) η p − Z η

0

∇ ^η ·

V η

∂p

∂η

+ g ∂F _P

∂η

dη (4.5)

4.1.4 Moisture equation

Conservation of moisture has a form similar to (4.2) (tendency=advection + turbulent source), where the turbulent fluxes are the diffusion flux J v and the precipitation flux P .

∂

∂t

q ∂p

∂η

= −∇ ^η ·

V η ∂p

∂η q

− ∂

∂η

˙ η ∂p

∂η q

− g ∂

∂η (J v + P ) subtracting q ∗ (4.2) and multiplying by ^∂η _∂p yields

dq

dt ≡ ∂q

∂t + (V · ∇ ) _η q + ˙ η ∂q

∂η = − g ∂η

∂p

∂

∂η (J v + P − qF _P )

= − g

(1 − δ m q) ∂ P

∂p + ∂J v

∂p

≡ ∂q

∂t

Φ

(4.6) labelling () Φ the partial tendencies provided by the physics.

With the boundary condition at the surface: P = P L and J v = E(1 − δ m q), while the total vertical water flux produced by evaporation is given by (4.1):

˙

η ^∂p _∂η q + J v

surf = gE

4.1.5 Momentum equations

Labelling the 3-D vectors with a ~, d~ V

dt = − f~k × V ~ − ∇ ~ φ − RT ~ ∇ ln p − gδ m P ∂V

∂p + Vertic diff

| {z } (

^∂∂t^V

)

_Φ

+Horiz diff (4.7)

The term ^∂ _∂t ^V

Φ being the partial tendency of the horizontal momentum provided by physics. The boundary condition at the ground is

˙ η ^∂p _∂η V

surf = 0

Physics-dynamics interface 4.1. EQUATIONS

4.1.6 Thermodynamic equation

We have:

dE = dQ + dW = c v dT dW = − pdV = ⇒ c v dT + pdV = dQ + dE cin + dE pot

p = ρRT = ⇒ c p dT = RT dp

p + dQ + dE cin + dE pot

dT dt = RT

c p

ω ρ + 1

c p { F vertic + F horiz + dE cin + dE pot }

where F vertic represents all subgrid heat production and vertical transfers, F horiz represents horizontal heat transfers - for instance horizontal diffusion. Practically physics provides subgrid contributions not only of heat but also of potential and kinetic energy conversions. Hence the material large scale heating writes:

dT dt = RT

c p

ω p + 1

c p

L v g ∂ P

∂p − g P c `

∂T

∂p − δ m g P ∂φ

∂p + Q + Vertic diff

| {z }

(

^∂T∂t

)

_Φ

+Horiz diff (4.8)

so the physical tendency ^∂T _∂t

Φ includes

• heat source terms

– L v g ^∂ _∂p ^P comes from the evaporation of the precipitation;

– − g P c ` ∂T

∂p is the heating of the liquid precipitation;

– Q is the radiative heating

• energy conversion terms:

– − δ m g P ^∂φ ∂p is the conversion of the potential energy of the precipitation;

• vertical transfer terms:

– ”Vertic diff”: includes all vertical exchanges: turbulent vertical diffusion, convection, kinetic energy variations...

4.1.7 Communication between physics and dynamics

Actual communication between physics and dynamics proceeds in a quite different manner from what was outlined above.

Computational constraints lead to a time step algorithm using a semi-implicit integration scheme.

4.1.7.1 Principles

We could represent model evolution as

∂X

∂t = M (X ) explicit CF L c ₄ ⁴ _x ^t < 1 implicit 4 t → precision

where X is the model state, 4 x and 4 t the space mesh and time step, c the velocity of the fastest waves (Lamb’s waves, i.e. sound velocity). An implicit discretization remains stable for much longer time steps, but requires solving nonlinear equations which would burden the computation.

M describes a variety of processes such as for example energy conversions, advection by the wind or wave propagation. Some of these processes are well approximated by a linear operator L .

The semi-implicit resolution scheme computes a first guess by solving the model equation explicitly in real

4.1. EQUATIONS Physics-dynamics interface

grid point space, and then applies a semi-implicit correction stabilizing the system. The stability criterion is then linked to the maximum wind speed instead of the fastest wave speed. It writes:

X ⁺ − X ⁻ = 2 4 t M (X) + 2 4 tβ L ( X ⁺ + X ⁻

2 − X)

X ⁺ (1 − 4 tβ L ) = 2 4 t M (X ) + X ⁻

| {z } X _expl ⁺ first guess

+ 4 tβ L (X ⁻ − 2X )

| {z }

semi − implicit correction

where parameter β sets the degree of implicitness. The linear implicit part, implying the inversion of a Lapla- cian, takes place in spectral space, while the nonlinear explicit part is computed in real grid point space, under the control of routine CPG. Physics contribution to this computation appear in CPG as tendencies of the model variables.

4.1.7.2 Eulerian time scheme

Noting ψ a model state variable, the Eulerian time scheme computes the advective tendency as A (ψ) ≡

∂ψ

∂t

adv

= − V · ∇ ψ − η ˙ ∂ψ

∂η Total tendency of ψ is given by

∂ψ

∂t = A (ψ)

| {z }

advection

+ L (ψ)

| {z }

linear part

+ N (ψ)

| {z }

nonlinear part

| {z }

remaining dynamics

+ Φ(ψ)

| {z }

physic tendencies

In the leap-frog scheme, all explicit terms are evaluated at time t in order to perform a jump from t − 4 t to t + 4 t. As it is taking into account three time levels simultaneously, its characteristic equation is quadratic and there are two numerical modes, the physical mode corresponding to the meteorological signal, while the second computational mode represents a 2 4 t oscillation polluting the solution. To alleviate this problem, a temporal filter has to be applied (GPTF1, GPTF2).

A decentered temporal filter is applied in two steps:

ψ e ⁰ = ψ ⁰ + 1 (ψ ⁻ − ψ ⁰ ) − 2 ψ ⁰ ψ ⁰ = ψ e ⁰ + 2 ψ ⁺

The semi-implicit discretization used in Arp` ege-Aladin writes:

ψ ⁺ − ψ ⁻

2 4 t = A (ψ ⁰ ) + L ( ψ ⁺ + ψ ⁻

2 ) + N (ψ ⁰ ) + Φ(ψ ⁻ )

Operator L linearizes the model around a reference state, with an isothermal atmosphere at 300K and a surface pressure of 1000hP a.

4.1.7.3 Semi-Lagrangian time schemes

The semi-Lagrangian scheme treats the advective part by solving the equation:

dψ dt = ∂ψ

∂t + V · ∇ ψ + ˙ η ∂ψ

∂η = 0

and computing the values at grid point G as resulting from advection of values at point O, by the (3D) wind at medium point M. As neither O nor M have to be on the model grid, 3D interpolations are needed.

The semi-implicit algorithm writes now:

ψ _G ⁺ − ψ _O ⁻

2 4 t = L ( ψ ⁺ _G + ψ _O ⁻

2 ) + N (ψ ⁰ _M ) + Φ(ψ _O/G ⁻ )

The CFL criterion no longer determines the length of the time step, but it is replaced by the fact that trajectories (O − M − G) should not cross each other.

Other specific dynamic considerations lead to decenter the leapfrog scheme.

A choice must be made about where to compute the physic tendencies: in Arp` ege-Aladin it is computed at

t − 4 t at the origin point O, and thus interpolated.

Physics-dynamics interface 4.2. GP CALCULATIONS

4.2 Grid Point calculations flowchart: CPG

CPG essentially computes the physical and dynamical tendencies.

Grid point calculations are performed on packets of horizontal rows by separate calls of the subroutine chain below SCAN2H, which eases the explicit multitasking. This is perfect for the Eulerian version of the time step where all computations are purely local.

But the semi-Lagrangian version also implies values at the origin and middle point of the trajectory, which need to be interpolated from values at neighbouring grid points. To do this, semi-Lagrangian buffers (SLB) were introduced, and the grid point computations splitted as next:

• the non-lagged part (CPG):

– evaluation of all grid point tendencies from physics (APLPAR, CPTEND...);

– evaluation of grid point quantities (LACDYN).

– extension to the C+I domain

– storage in SLB1 of all inputs for trajectory construction and interpolation:

– storage in SLB2 of quantities used for evaluation at the arrival points of the trajectories

• the lagged part starts in CPGLAG, using one SLB2 packet (arrival point) and several SLB1 packets for interpolation. It calls LAPINE which is the main interface to the semi-lagrangian computations.

CPG proceeds as follows.

With the Eulerian time scheme:

• SC2RDT9: read work buffers of time t − 4 t;

• GPTF2: apply time filter part 2;

• various initialisations for auxiliary variables;

• Call physic parameterisations: APLPAR

• Compute tendencies: CPFHPFS, CPTEND

• Compute T , q, wind at time t + 4 t: CPUTQY

• Compute surface tendencies: CPTENDS, CPWTS

• Compute surface T , q, wind ad time t + 4 t: CPUTQYS

• CPDYN: compute dynamic tendencies;

• Call Diagnostics

• Store cumulated and instantaneous fluxes

• GPTF1: apply time filter part 1;

• SC2WRT9: write to work files.

With a Semi-Lagrangian time scheme:

• SC2RDT9: read work buffers of time t − 4 t;

• GPTF2: apply time filter part 2;

• various initialisations for auxiliary variables;

• get and initialize semi-Lagrangian buffers;

• Call physical parameterisations: APLPAR

• Compute tendencies: CPFHPFS, CPTEND

• Compute T , q, wind at time t + 4 t: CPUTQY

• Compute surface tendencies: CPTENDS, CPWTS

• Compute surface T , q, wind ad time t + 4 t: CPUTQYS

• LACDYN: evaluate grid point quantities at t and t − 4 t for subsequent SL computations

• Call Diagnostics

• Store cumulated and instantaneous fluxes

• GPINISLB9: fill semi-Lagrangian buffers for lagged physics;

• GPTF1: apply time filter part 1;

• SC2WRT9: write to work files.

4.3. TIME STEP Physics-dynamics interface

4.3 Physical tendencies and time step calculations

The local physical tendencies include the vertical divergence of vertical physical fluxes, and a vertical pseudo- advection term linked to the precipitation flux in the case mass is not conserved.

This pseudo-advection is applied to large scale momentum and large scale enthalpy (or dry static energy and kinetic energy). Concerning moisture, it would be dubious to consider advection of large scale moisture by the precipitation flux, as the moisture in the neighbourhood of the precipitation flux has little relation with the large scale moisture, and is closer to the wet bulb moisture associated to the large scale temperature. So the pseudo-advection term does not appear in the expression of the moisture tendency.

4.3.1 Model levels tendencies and time step CPTEND, CPUTQY

4.3.1.1 Momentum ∂V

∂t

Φ

= − g ∂

∂p

n J V ^turb + J V ^gwd + J V ^med + J V ^conv

o − δ m g P ∂V

∂p (4.9)

with

• PSTRTU, PSTRTV ≡ J V ^turb : turbulent momentum flux

• PSTRCU, PSTRCV ≡ J V ^conv : convective momentum flux

• PSTRDU, PSTRDV ≡ J V ^gwd : gravity wave drag

• PSTRMU, PSTRMV ≡ J V ^med : mesospheric drag

• P ≡ PFPLCL + PFPLCN + PFPLSL + PFPLSN: total precipitation flux Tendency is stored into PTENDU, PTENDV.

The physical time step is performed by CPUTQY by modifying the values at t + 4 t through the addition of the physical tendencies.

V 1 = V 1 + 4 t phys · ∂V

∂t

Φ

(4.10) and stored into PUT1, PVT1.

4.3.1.2 moisture

∂q

∂t

Φ

= − g ∂

∂p

J _q ^turb + J _q ^conv − g(1 − δ m q) ∂ P

∂p (4.11)

with

• PDIFTQ ≡ J _q ^turb : turbulent moisture flux

• PDIFCQ ≡ J _q ^conv : convective moisture flux

we consider here the turbulent and convective moisture fluxes, and the effect of precipitation evaporation and condensation. We cannot consider a pseudo-advection of the large scale moisture. The precipitation flux is a moisture flux, and the air is always close to saturation in its neighbourhood: it would be meaningless to introduce an advection of the large scale moisture by the precipitation flux. On the other hand, whatever the mass conservation parameter, variation of the precipitation flux has always an impact on the specific moisture.

The last term of the RHS represents the partial tendency of specific moisture due to precipitation evapora- tion/recondensation. This expression was already derived in equation (4.6) and is shown more intuitively in the chapter about energetic aspects.

Tendency is stored into PTENDQ.

The physical time step is performed by CPUTQY by modifying the value at t + 4 t:

q v1 = q v1 + 4 t phys · ∂q

∂t

Φ

(4.12)

and stored into PQT1.

Physics-dynamics interface 4.3. TIME STEP

4.3.1.3 moisture equations with prognostic condensed phases

The operational model does not yet consider prognostic (large scale) condensed phases, but the structure is already present to handle them in the dynamics and within the budgets.

Model key LCONDWT activates the condensed phases: in this case, we have to consider the fluxes representing the different phase changes. we split equation (4.11) between vapour, solid and liquid phases:

• Vapour:

∂q

∂t

Φ

= − g ∂

∂p

J _q ^turb + J _q ^conv − g ∂F cond

∂p + gδ m q ∂ P

∂p (4.13)

• Solid water:

∂q i

∂t

Φ

= − g ∂

∂p

J _q ^turb

_i

+ J _q ^conv

_i

− g ∂ P ⁱ − F cond i

∂p + gδ m q i ∂ P

∂p (4.14)

• Liquid water: ∂q w

∂t

Φ

= − g ∂

∂p

J _q ^turb

_w

+ J _q ^conv

_w

− g ∂ P ^w − F cond w

∂p + gδ m q w

∂ P

∂p (4.15)

where

• P i | w represent the total (convective + large scale) solid and liquid precipitation fluxes,

• F cond the total (convective + large scale, ice and water) condensation flux linked to the precipitation flux

— if LCONDWT = .FALSE., this flux is equal to the total precipitation flux, and we find back equation 4.11;

• F cond w | i are the liquid and solid parts of the same flux.

The “condensation flux” corresponds actually to the total tendency of the condensate present above a given level, while the precipitation flux is the part of it travelling downwards (and it is only this part that implies a net advection under δ m = 1).

4.3.1.4 enthalpy and temperature

Enthalpy is not a model variable, but we need it to compute the temperature evolution. We define the pseudo-enthalpy as

¯

h ≡ c p T + φ + u ² + v ²

2 = c p T + φ + E kin

with c p = c pa + (c pv − c pa )q + (c w − c pa )q w + (c i − c pa )q i

The physical tendency PTENDH is ∂¯ h

∂t

Φ

= − g ∂

∂p

F + F _† + J _h ^meso + J _s ^turb + J _s ^conv + F s P − δ m g P ∂¯ h

∂p (4.16)

where

• PFRSO ≡ F and PFRTH ≡ F _† are the enthalpy fluxes from the solar and thermal radiation

• PFRMH ≡ J _h ^meso is the enthalpy flux from the mesosphere (this is used only in specific climatological runs of the Arp` ege model)

• PDIFTS ≡ J _s ^turb is the turbulent diffusion dry static energy flux

• PDIFCS ≡ J _s ^conv is the convective dry static energy flux

• PFHP ≡ F s P represents the dry static energy flux associated to the total precipitation flux at the given level (convective and large scale, liquid and solid) or associated to total precipitation and condensation fluxes under LCONDWT: this flux is computed by routine CPFHPFS.

With no large scale condensate we have:

F s P ≡ PFHP = PFHPCL + PFHPCN + PFHPSL + PFHPSN = − L flux P

Under LCONDWT we must distinguish

4.3. TIME STEP Physics-dynamics interface

– the part linked to condensation: − L v − w | i (0K) · F cond w | i (with the latent heats of the different phase changes, taken at 0K)

– the sensible heat fluxes linked to the precipitation flux: P w | i · T · (c w | i + c pa (δ m − 1)), – the pseudo enthalpy flux born by the precipitation under δ m = 1: − δ m · c p · T · P The physical time step is performed by CPUTQY as follows. Enthalpy at t 9 ≡ t − 4 t is

¯

h 9 = c p9 T 9 + u ² ₉ + v ² ₉ 2

First we evaluate the evolution of moisture, specific heat , kinetic energy, enthalpy, due to physics only (labelled as t 9+Φ ):

q v9+Φ = q v9 + 4 t · PTENDQ q l9+Φ = q l9 + 4 t · PTENDQL q i9+Φ = q i9 + 4 t · PTENDQI c p9+Φ =

c pa − (c pa − c pv )q v9+Φ

c pa · (1 − q v9+Φ − q l9+Φ − q i9+Φ ) + c pv q v9+Φ + c w · q l9+Φ + c i · q i9+Φ if LCONDWT

¯

h 9+Φ = ¯ h 9 + 4 t · PTENDH E kin9+Φ = u ² _9+Φ + v ² _9+Φ

2 We have then:

T 1 = T 1 + ¯ h 9+Φ − E _kin9+Φ c p9+Φ − T 9

A Kinetic energy dissipation flux PFDIS, zero at the top, is obtained by:

J _diss ^l = J _diss ^{l − 1} + δp(E kin9+Φ − E kin9 )

g 4 t (4.17)

4.3.2 Variable mass effect on surface pressure and vertical velocity: CPMVVPS

4.3.2.1 surface pressure

Computed according to equation(4.3). The integrated divergence PSDIV is input, and the δ m term is added to it to output the surface pressure tendency as PSDIV:

− PSDIV(KLEV) = ∂π s

∂t = − Z 1

0

∇ ·

V ∂p

∂η

dη − gδ m (F evn + F evl + P ^s ) (4.18) where

• PFEVL = F evl is the evaporation over water or moist soil,

• PFEVN = F evn is the evaporation flux over snow or ice.

4.3.2.2 vertical velocity

Equation(4.4) gives the generalised vertical velocity in η coordinates, PEVEL:

˙ η ∂p

∂η = − B(η) ∂π s

∂t − Z η

0

∇ ·

V ∂p

∂η

dη − gδ m P and (4.5) gives the vertical velocity in pressure coordinates, PVVEL:

ω

p = = V · ∇ ^η ln(p) − 1 p Z η

0

∇ ^η ·

V ∂p

∂η

dη − 1 p Z η

0

gδ m

∂ P

∂η dη The discretization uses

1 X

Z

Xdη = ∂p

∂η

− 1 ∂

∂η

ln(p) Z

Xdη

− X · ln(p)

and also p l = e ^{− α}

^l

· p _l if LAPRXPK=.FALSE..

Physics-dynamics interface 4.4. APLPAR

4.3.3 Surface tendencies and physical time step: CPTENDS , CPWTS

Definitions, fluxes, tendencies, and evolution of ground variables are covered in the chapter about the soil- vegetation scheme.

4.4 Physical Parameterisations Call: APLPAR

The different physical parameterisations called by routine APLPAR are described in the following chapters.

They are organized as follows:

5. Auxiliary tools and initialisations

• Energetics

• Thermodynamical functions

• Wet bulb computation

• Initialisations

6. PBL and turbulent processes

• ACHMT: Turbulent fluxes between surface and lowest model level

• ACCOEFK: Turbulent fluxes between model levels

• ACDIFUS:Vertical turbulent diffusion

• Anti-Fibrillation scheme

7. Subgrid dynamical and other processes

• ACDRAG: Orographic gravity wave drag

• ACDRAC: Convective gravity wave drag

• ACDRME: Mesospheric drag 8. Soil processes

• ISBA soil-vegetation scheme

• Practical implementation

– ACSOL, ACSOLW: Surface characteristics – ACVEG: Vegetation characteristics

– ACDROV: Water transfers inside the soil-vegetation scheme – ACHMT, ACDIFUS

– APLPAR

– CPTENDS, CPWTS

9. Cloudiness and Large scale precipitation

• ACNEBN: Combined cloudiness scheme

• ACPLUIE: Large scale Precipitation 10. Radiation and chemicals

• ACRANEB: Economic radiation scheme

• ACOZONE: Ozone and chemicals

11. ACCVIMP, ACCVIMPD: Deep convection – this chapter is detailed in part III.

4.5. DIAGNOSTICS Physics-dynamics interface

4.5 Diagnostics

Namelist NAMZDI allows to order 4 different kinds of zonal diagnostics (i.e. performing an average over all longitudes, for a range of latitudes), each one with its own frequency. They are:

• LZDIDY (routine ZODIDY): global diagnostics on dynamic fields:

– maximal values on the sphere (and their location): horizontal wind, vertical velocity, horizontal and vertical CFL numbers, critical semi-Lagrangian divergence.

• LZDIFI (routine ZODIFI): zonal mean diagnostics on fields: wind, vorticity, divergence, ∆p, C p ∗ T , specific and relative humidity, cloudiness, solid and liquid moisture.

• LZDIGI (routine ZODIGI): zonal mean diagnostics on invariants: mass, potential and kinetic energy, kinetic momentum (3 components), entropy, potential temperature, potential vorticity.

• LZDIFF (routine ZODIFF): zonal mean diagnostics on physical fluxes:

– Convective and turbulent fluxes of moisture, enthalpy, horizontal momentum;

– gravity wave drag flux of horizontal momentum;

– convective and large scale rain and snow;

– enthalpy flux from precipitation and from kinetic energy dissipation;

– solar and thermal radiation fluxes;

– fictive flux of negative specific humidity;

– mesospheric fluxes of horizontal momentum and enthalpy;

– ozone flux.

Beside this system, the DDH package provides a more flexible way to generate various diagnostics over arbitrary

horizontal domains, through namelist NAMDDH. Refer to [Piriou, 1999].

Chapter 5

Initialisations and auxiliary tools

5.1 Energetics and conservation aspects

5.1.1 Basic hypotheses

• The atmosphere is composed of a mixture of two perfect gases, dry air and water vapour.

• The system is in local thermodynamic equilibrium (so we may use the state equation).

• Condensed phases of water may appear in the atmosphere:

– their volume is zero;

– they leave the system immediately (in one time step);

– while falling, they exchange mass, momentum and energy with the layers they cross. The precipi- tation temperature is the same as the layer they cross, as well as their horizontal velocities. Their potential energy is converted into kinetic energy, dissipated into heat;

– when leaving the system, they bring away mass, momentum, and energy from the atmosphere.

The fact that the condensed phases are evacuated in one time step means that the environmental mixing ratio of condensed waters is kept to zero; but it does not prevent that there may be some condensed water present locally, as the condensed water diagnosed inside the updraught (subscript u ), through the equation:

∂(q u + ` u )

∂φ = − ` u

ECMNP (5.1)

where ECMNP is the critical minimum thickness of precipitating clouds, and ` represents the condensed water specific contents.

The volume of the condensate is neglected everywhere, so we write

` = ρ i + ρ l

ρ

where ρ i and ρ l are the mass of ice and liquid water per unit volume, and the specific humidity is defined as

ρ v = qρ or q ≡ ρ v

ρ a + ρ v

(5.2) The precipitation flux (kg m ^{− 2} s ^{− 1} ) crossing a layer of thickness 4 p in time 4 t can be seen as follows:

P = ^m _S

^prec

_{4 t} = ρ prec · w w = _ρg ⁴ ₄ ^p _t

)

= ⇒ P = ρ prec

ρ 4 p

g 4 t (5.3)

This represents condensed water present in the layer crossed by a precipitation flux, but this water is evacuated

in one time step. It does not take part to the definition of local q. It is supposed constant over the duration

of the time step, even if there is precipitation evaporation into the layer, because the precipitation flux is fed

continuously.

5.1. ENERGETICS AND CONSERVATION ASPECTS Initialisations and auxiliary tools

5.1.2 Variable mass option

Parameter NDPSFI ≡ δ m controls mass conservation:

• δ m = 0: Total mass is conserved, water rained out is replaced by dry air, while the vapour produced by evaporating precipitation is compensated by a removal of dry air.

• δ m = 1: Total mass is not conserved, mass changes are controlled by the precipitation-evaporation budget at the surface. Condensation produces a local mass deficit, while precipitation evaporation produces a local mass increase.

5.1.3 Thermo-dynamical properties

The model considers the variation of the specific heat c p and gas constant R with the specific moisture; and also the variation of the vaporisation and sublimation latent heats with temperature.

We have:

c p = c pa + (c pv − c pa )q + (c w | i − c pa )q w | i

= c pa (1 − q − q w | i ) + c pv q + c w | i q w | i

(5.4) If we compare two states such that water substance is conserved:

q + q _{w | i} = q 0 + q _{w | i0} (5.5)

we have:

c p = c p0 + c pv (q − q 0 ) + c w | i (q w | i − q w | i0 )

= c p0 + c pv (q − q 0 ) + c w | i (q 0 − q)

= c p0 + (c pv − c w | i )(q − q 0 ) (5.6)

This expression will be used in the updraught, where we consider intra-cloud moisture and condensed water.

Since the model doesn’t contain large scale condensed water, equation (5.6) is of less interest at large scale.

For the latent heat, we consider the following:

liq(T ) L(T - ) vap(T )

? liq(T 0 )

− c w (T − T 0 ) -

L 0 vap(T 0 ) 6

c pv (T − T 0 )

And the dependency writes

FOLH(T, δ) ≡ L(T ) = L 0 + (c pv − c w | i )(T − T 0 ) (5.7) with δ = 0 for liquid water, δ = 1 for ice. For T 0 we can take the triple point temperature T t

This temperature dependence of the latent heat has little impact on short/medium range NWP integrations, but it provides the framework to have an exact description of all energy cycles in the atmosphere — including the Clausius-Clapeyron cycle ( § 5.2.1)— and it allows to construct a simpler Newton algorithm computing the saturation specific humidity function q sat (T, p) for wet bulb ( § 5.3) and moist adiabat (see chapter on deep convection) calculations.

5.1.3.1 Local effective latent heat

State changes must take into account the hypotheses made around the condensed phases.

Initialisations and auxiliary tools 5.1. ENERGETICS AND CONSERVATION ASPECTS

First let’s consider local intra-cloud phase changes, i.e. the condensation of some cloud water vapour or evaporation of locally present liquid water. The heat produced is computed using the local effective latent heat, defined by:

p

b

Z

p

t

L eff

− g ∂ P

∂p

+ c p

∂T

∂t

P

dp = 0 (5.8)

where the boundaries p t and p b are the pressures at the top and the bottom of the modelled atmospheric system.

Condensation −4 m v at a model level l implies the following:

• Latent heat for state change at T : L v − w | i (T ) 4 m v

• Move the condensed water to the surface: 4 m v c w | i (T − T surf);

• If δ m = 0, move the same mass of dry air from the surface to level l: 4 m v c pa (T − T surf);

• If δ m = 1, we must convert into heat the potential energy of the mass we removed: 4 m v (φ − φ surf ) The resulting local effective latent heat is then

L eff = L(T ) +

(c w | i − c pa ) + δ m c pa (T − T surf ) + δ m (φ − φ surf ) (5.9) or using (5.7),

L eff = L(T surf ) + { (c pv − c pa ) + δ m c pa } (T − T surf ) + δ m (φ − φ surf ) (5.10) Actual code writes

L eff = L(T ^l ) + n

(c w − c pa ) + (c i − c w )ZSNP ^l + δ m c pa

o (T ^l − T surf ) + δ m (φ ^l − φ surf ) (5.11)

where ZSNP is the solid fraction of the precipitation.

5.1.3.2 Precipitation flux latent heat

We want to express here the total amount of dry static energy born by the precipitation flux, i.e. the flux of dry static energy corresponding to the precipitation flux:

F s P ≡ − L flux P (5.12)

We do not consider suspended condensed phases (LCONDWT=.FALSE.). Local specific moisture is linked to precipitation flux P (kg m ^{− 2} s ^{− 1} ) through (cfr. equation (4.6))

∂q

∂t

P

= − g(1 − δ m q) ∂ P

∂p (5.13)

Intuitively we can feel this as follows.

q = m

v

m

v

+ m

a

g 4 t

4 p P = m

p

m

v

+ m

a

and we consider a variation µ (evaporation) of the condensed phase.

4 m

v

= µ = −4 m

p

With δ

m

= 0 this yields:

4 q = µ

m

v

+ m

a

= − g 4 t 4 p 4P while with δ

m

= 1 we get:

4 q = m

v

+ µ

m

v

+ m

a

+ µ − m

v

m

v

+ m

a

= µ

m

v

+ m

a

+ µ (1 − q) = − g 4 t

4 p 4P (1 − q)

The local tendency of dry static energy, due to P : ∂s

∂t

P

= − g ∂F s P

∂p − g P [pseudo − advection] (5.14)

includes

5.1. ENERGETICS AND CONSERVATION ASPECTS Initialisations and auxiliary tools

• the variation brought by the variation of P (evaporation/re-condensation), which is the s-flux we want to express,

• if the mass is not conserved, the part advected by P (pseudo-advection resulting of the void created by the precipitation, i.e. the mass loss due to the precipitation flux, as the condensate is not replaced by air).

Now, definition of s implies

∂s

∂t

P

= c p

∂T

∂t

P

+ T ∂c p

∂t

P

+ ∂φ

∂t

| {z } P

=0

(5.15)

The local tendency of φ due to the precipitation flux is actually zero.

We may express the temperature tendency as:

c p

∂T

∂t

P

= gL(T ) ∂ P

∂p − gδ m P ∂φ

∂p − g P ∂T

∂p (c w | i − c pa + δ m c pa ) (5.16) where

• the first term of the RHS represents the evaporation,

• the second term is the conversion of potential energy into heat when δ m = 1,

• the third term is a pseudo-advection term by the precipitation flux:

– the liquid or solid precipitation exchanges heat with the layer to keep all the time the temperature of the environment;

– if mass is conserved, an equivalent flux of dry air has to be moved upwards and get the layer temperature.

Note that from (5.8) we should verify:

pb

Z

pt

c

p

_∂T

∂t

P

dp =

pb

Z

pt

gL

eff

∂ P

∂p dp

or

p

Z

_surf

pt

gL(T ) ∂ P

∂p − gδ

m

P ∂φ

∂p − g P ∂T

∂p (c

_w|i

− c

pa

+ δ

m

c

pa

)dp

=

? p

Z

surf

pt

gL(T ) + g

(c

_w|i

− c

pa

) + δ

m

c

pa

(T − T

surf

) + gδ

m

(φ − φ

surf

) ∂ P

∂p dp

=

p

Z

_surf

pt

gL(T ) ∂ P

∂p dp + g

(c

w|i

− c

pa

) + δ

m

c

pa

 

 ^{[(T − T}

^surf

^{) P ]}

psurf pt

| {z }

=0

−

p

Z

_surf

pt

P ∂T

∂p dp

 



+gδ

m

 

 ^{[(φ − φ}

^surf

^{) P ]}

p_surf pt

| {z }

=0

−

p

Z

_surf

pt

P ∂φ

∂p dp

 



= QED.

As we have no suspended condensate,

∂c p

∂t = (c pv − c pa ) ∂q

∂t = ⇒ ∂c p

∂t

P

= − g(c pv − c pa )(1 − δ m q) ∂ P

∂p and (5.7) implies

∂L(T )

∂p = ∂L(T )

∂T

∂T

∂p = (c pv − c w | i ) ∂T

∂p

Initialisations and auxiliary tools 5.1. ENERGETICS AND CONSERVATION ASPECTS

(5.15) becomes ∂s

∂t

P

= gL(T ) ∂ P

∂p − g P ∂T

∂p (c w | i − c pa + δ m c pa ) − g(c pv − c pa )(1 − δ m q)T ∂ P

∂p − gδ m P ∂φ

∂p

= g ∂L(T ) P

∂p − g P ∂T

∂p (c pv − c w | i + c w

i

− c pa + δ m c pa )

− gT ∂ P

∂p (c pv − c pa )(1 − δ m q) − gδ m P ∂φ

∂p

= g ∂L(T ) P

∂p − g(c pv − c pa )(1 − δ m q) ∂ P T

∂p +g P (c pv − c pa )(1 − δ m q) ∂T

∂p − g P (c pv − c pa + δ m c pa ) ∂T

∂p − gδ m P ∂φ

∂p

= g ∂

∂p [ P (L(T ) − (c pv − c pa )T (1 − δ m q))]

− g P ∂T

∂p ((c pv − c pa )δ m q + δ m c pa ) − gδ m P ∂φ

∂p

= g ∂

∂p [ P (L(T ) − (c pv − c pa )T (1 − δ m q))] − g P δ m

c p

∂T

∂p + ∂φ

∂p

Considering

∂c p

∂p = ∂c p

∂q

∂q

∂p = (c pv − c pa ) ∂q

∂p 1 because q 1 and p 1, we obtain

∂s

∂t

P

= g ∂

∂p [ P (L(T ) − (c pv − c pa )T (1 − δ m q))]

| {z }

≡ F

sP

− gδ m P ∂s

| {z } ∂p

pseudo advection

yielding the flux latent heat:

L flux = L(T ) − (c pv − c pa )T (1 − δ m q) (5.17) This value is computed by routine ACCLFP:

− L flux ≡ PLH L

N = − FOLH(T ^l , 0

1 ) + (c pv − c pa )T ^l (1 − δ m q ^l ) 5.1.3.3 Budget latent heat

L bud represents the static energy brought to the whole vertical column by a local unit variation of q due to precipitation/evaporation effects or any external moisture input.

Physically, since the model doesn’t have condensed water storage (LCONDWT=.FALSE.), every water drop appearing at a model level has either to be evaporated or to be moved to the ground.

When making a global energy budget, it cannot make a difference that such a drop condensed near the tropopause or in the PBL: it could occur that the water has done several courses around the cloud with phase changes in both directions, if we consider the downdraughts.

The energy lost by the atmosphere is equal to the dry static energy brought by the precipitation flux when it reaches the surface, i.e.

L flux (surf) · P ^s

If we assess the energy brought by the large scale moisture convergence for instance, the large scale brings

water vapour at a given level into the vertical column. If this vapour condensates, somewhere into the column,

the corresponding net precipitation reaching the surface is the only part of it that has released its latent heat

into sensible heat in the energy budget of this column. Can we then conclude that the budget latent heat is

equal to the surface precipitation flux latent heat ? Actually the part of the fed moisture that doesn’t finish

into precipitation, it is still available as latent heat, and the budget is the same...

5.2. THERMODYNAMICAL FUNCTIONS Initialisations and auxiliary tools

L bud = L(T s ) − (c pv − c pa )(1 − δ m q s )T s = L flux (surf) (5.18) L bud will multiply a moisture flux (at any particular level) resulting either from advection or a local variation of the precipitation flux.

Remember that the values T

_s

and q

_s

intervening inside the latent heat expression correspond to the environment at the place where we decide to compute the state change, and it does not contradict the fact that the same latent heat has to be multiplied by a moisture flux occurring at a given level to obtain the corresponding heat (sensible and latent) brought by the large scale to the column.

The budget latent heat is relevant whenever we compute budgets between the whole vertical column and the en- vironment, while the local effective latent heat must be used when we consider intra-cloud water phase changes.

Actually, the budget latent heat may be quite different from the local value of the latent heat corresponding to the wet bulb characteristics (T w , q w ), which are used implicitly when computing the saturated pseudo-adiabat:

if we decided to use the budget latent heat to multiply the large scale moisture convergence in the convective cloud calculation, we would be at risk to change the cloud limits: for this reason, in this particular computation L eff is used.

On the contrary, in the final budgets of the convection routine, the convective dry static energy flux for output is computed with the budget latent heat.

5.2 Thermodynamical functions

Include module FCTTRM contains 2 separate sets of thermo-dynamical function, as described below.

5.2.1 Absolute thermodynamical functions

Latent heats are derived from the triple point values: T t ≡ RTT, L v − w (T t ) ≡ LVTT, L v

i

(T t ) ≡ LSTT.

Vaporisation, sublimation and melting latent heats are then:

RLV ≡ L v − w (T ) = L v − w (T t ) + (c pv − c w )(T − T t ) RLS ≡ L v − i (T ) = L v − i (T t ) + (c pv − c i )(T − T t ) RLF ≡ L w − i (T ) = L v − w (T ) − L v − i (T )

The saturating vapour tension is given by Clausius-Clapeyron equation:

de sat

e sat

= LdT

R v T ² (5.19)

Writing

L R v

= β − γT we get readily:

ln(e sat (T )) = α − β

T − γ ln(T )

with α corresponding to e sat (T t ), and α, β, γ being different for liquid and ice phases.

The saturating vapour tensions with respect to liquid water and ice, are computed as:

ESW(T ) = exp h

α w − ^β T

^w

− γ w ln T i

ESS(T ) = exp h

α i − ^β T

ⁱ

− γ i ln T i γ w = (c w − c pv )/R v γ i = (c i − c pv )/R v

β w = L v − w (T t )/R v + γ w T t β i = L v − w (T t )/R v + γ i T t

α w = ln(e sat (T )) + β w /T + γ i T t α i = ln(e sat (T )) + β i /T + γ i T t

Checking temperature positivity through index δ T ≡

0 if T ≥ T t

1 if T < T t

we can express the saturating vapour tension ES(T).

Initialisations and auxiliary tools 5.3. WET BULB COMPUTATION

5.2.2 Model physics functions

Using a phase indicator

δ ≡

0 referencing to water 1 referencing to ice we get the saturating vapour function

FOEW(T, δ) ≡ e sat (T, δ) = exp

α w + (α i − α w )δ − β w + (β i − β w )δ

T − (γ w + (γ i − γ w )δ) ln T

FODLEW(T, δ) ≡ ∂ ln e sat

∂T = β w + (β i − β w )δ − (γ w + (γ i − γ w )δ)T T ²

The saturating specific humidity is then FOQS( e sat

p ) = e sat /p

1 + (R v /R a − 1) max(0, 1 − e sat /p) FODQS(q sat , e sat

p , ∂ ln e sat

∂T ) ≡ ∂q sat

∂T = q sat − q _sat ² 1 − e sat /p

∂ ln e sat

∂T so that

q sat =

 

 

 



e

sat

/R

v

(p − e

sat

)/R

a

+e

sat

/R

v

if e sat (T ) ≤ p

e

sat

p if e sat (T ) ≥ p

And finally the latent heat function

FOLH(T, δ) ≡ L(T, δ) = L v − w (T t ) + [L v − i (T t ) − L v − w (T − t)] δ + [c pv − c w + (c w − c i )δ] (T − T t )

5.3 Wet bulb computation

Module ACTQSAT computes the wet bulb temperature and specific humidity.

dh = 0 , dp = 0 = ⇒ c p dT + Ldq = 0 with c p (q) and L(T ) given by (5.6) and (5.7):

c p = c p0 + (c pv − c w )(q − q 0 ) = ⇒ dc p = (c pv − c w )dq L(T ) = L 0 + (c pv − c w )(T − T 0 ) = ⇒ dL = (c pv − c w )dT

= ⇒ c p

dL c pv − c w

+ L dc p

c pv − c w

= 0 = ⇒ Lc p = L 0 c p0 = cste

hence

c p0 (T f − T 0 ) + L f (q f − q 0 ) = 0

c pf (T f − T 0 ) + L 0 (q f − q 0 ) = 0 (5.20) The wet bulb final state is saturated: q f = q sat (T f ), but as q sat (T ) is not linear, we use a Newton algorithm linearizing around the previous iteration k:

q k+1 = q sat (T k ) + ∂q sat (T k )

∂T k

(T k+1 − T k ) (5.21)

with (5.20)

c p0 [(T k+1 − T k ) + (T k − T 0 )]

+ [L 0 + (c pv − c w ) ((T k+1 − T k ) + (T k − T 0 ))]

(q sat (T k ) − q 0 ) + ∂q sat (T k )

∂T k (T k+1 − T k )

= 0

5.4. INITIALISATIONS Initialisations and auxiliary tools

Dropping second order terms in (T k+1 − T k ):

[c p0 (T k − T 0 ) + L k (q sat (T k ) − q 0 )]

+(T k+1 − T k )

c p0 + (c pv − c w )(q sat (T k ) − q 0 ) + L k ∂q sat (T k )

∂T k

= 0 (5.22)

Defining q k and q k+1 with

c p0 (T k − T 0 ) + L k (q k − q 0 ) = 0 , c pk (T k+1 − T k ) + L k+1 (q k+1 − q k ) = 0 (5.23) yields

L k (q sat (T k ) − q k ) + (T k+1 − T k )

c pk + (c pv − c w )(q sat (T k ) − q k ) + L k

∂q sat (T k )

∂T k

= 0 L k+1 (q sat (T k ) − q k ) + (T k+1 − T k )

c pk + L k

∂q sat (T k )

∂T k

= 0

(q sat (T k ) − q k ) = (q k+1 − q k )

1 + L k

c pk

∂q sat

∂T k

(5.24) Starting from (T 0 , q 0 ) the algorithm follows:

(q k+1 − q k ) = q sat (T k ) − q k

1 + _c ^L

^k

pk

∂q

sat

∂T

_k

c pk+1 = c pk + (c pv − c w )(q k+1 − q k ) T k+1 − T k = − L k

c pk+1

(q k+1 − q k )

L k+1 = L k + (c pv − c w )(T k+1 − T k ) converging to (T w , q w ) after NBITER iterations.

Use of c pk+1 in the third statement is the only difference with the “intuitive” algorithm, but it is essential to ensure the scheme coherence.

Routine ACTQSAT receives as input the 3D fields of T , q, c p and p, and returns the 3D fields of T w , q w , L, L/c p , q sat , RH, using:

L/c p = L(T 0 )/c p (q 0 ) q ≡ ρ v

ρ , q sat ≡ ρ vsat

ρ sat

= ⇒ RH ≡ ρ v

ρ vsat

= qρ

q sat ρ sat

= qR sat

q sat R = q(1 + (R v /R a − 1)q sat ) q sat (1 + (R v /R a − 1)q)

5.4 Initialisations

Initialisations are required for

• Chemistry:

– Ozone: only in the CLIMATE version, is ozone a historic model variable; in other cases (LOZONE = .FALSE.) a climatologic profile is used:

ZQO3 ^l = 2a 1 + r

b/p ^l 3 , a = 6.012 · 10 ^{− 2} , b = 3166Pa – CO 2 : a constant profile is taken (namelist parameter QCO2).

– Aerosols: an empiric profile is taken, using namelist parameters AERCS1, AERCS3, AERCS5.

ZDAR ^l = ^{l − 1} − ^{l l} = AERCS1 p ^l

π ref

!

+ AERCS3 p ^l π ref

! 3

+ AERCS5 p ^l π ref

! 5

p ^l π ref

!

= A(η) + B(η)

where p is the horizontally averaged pressure, as π ref corresponds to average surface pressure.

Initialisations and auxiliary tools 5.4. INITIALISATIONS

• Critical moisture for ACNEBN is computed in CPG and APLPAR: see chapter VII.

• Moisture convergence for convection is computed in APLPAR: see chapter VI.

• Meteorological geopotential heights for wind, temperature and moisture PBL diagnostics.

• Flux latent heats are computed in ACCLFP (see § 5.1.3.2).

5.4. INITIALISATIONS Initialisations and auxiliary tools

Chapter 6

Turbulent fluxes and PBL processes

6.1 Theoretical context

Interaction with earth surface influences strongly the flow up to a certain height above the surface, depending itself of the specific situation.

The no-slip boundary condition imposes that the velocity must vanish at the surface, causing an important vertical shear near the surface. Viscosity itself acts directly only in a thin layer of a few millimeters, while above it it becomes much smaller than the other terms of the momentum equation.

The relay is taken by turbulent eddies, first with very small spatial and temporal scales, but becoming wider with increasing height. Those shear-induced eddies, together with convective eddies caused by surface heating, are very effective at transferring momentum down to the surface and heat away from the surface - much faster than could be done by molecular diffusion alone (Kolmogorov’s cascade).

The turbulent layer height can range from 30m to more than 3 km in highly convective conditions. Remember that a boundary layer of 1 km contains around 10% of the mass of the atmosphere. Above, in the free atmo- sphere and not too close to fronts, jet streams, and convective clouds, turbulence can generally be ignored, while the flow rejoins a geostrophic behaviour.

Turbulent eddies in the Planetary Boundary Layer (PBL) tend to have similar scales in the horizontal and the vertical, ranging thus between 10 ^{− 3} m and 10 ³ m: those scales are not resolved in operational mod- els, while these eddies are of critical importance for momentum, energy and moisture budgets at the surface.

Momentum transport to the surface also affects dramatically the large scale flow, so the geostrophic approxi- mation is no longer adequate.

Given the importance of buoyancy forces, density fluctuations cannot be ignored. Boussinesq’s approximation replaces density everywhere by a constant value ρ 0 except in the buoyancy term of the vertical momentum equation.

We define then (within the grid box we are studying):

ρ ≡ ρ 0 (z) + ρ(x, y, z, t) e , θ ≡ θ 0 (z) + θ(x, y, z, t) e , p ≡ p(z) + p(x, y, z, t) e i.e. ρ, e e θ, p e represent departures from the basic state values ρ 0 , θ 0 , p 0 .

As we have

θ θ 0 = T

T 0

p 0

p

_cp^R

= ρ 0

ρ p 0

p

^cv_cp

≈ ρ 0

ρ and the hydrostatic equation for the basic state:

1 ρ 0

∂p 0

∂z + g = 0 we can write the vertical equation

dw

dt = − g − 1 ρ

∂p

∂z + F rz = − g 1 ρ ( ∂p 0

∂z + ∂ e p

∂z ) + F rz

= − 1 ρ

∂ p e

∂z + g( ρ 0

ρ − 1) + F rz ' − 1 ρ 0

∂ e p

∂z + g( θ

θ 0 − 1) + F rz

= ⇒ dw

dt = − 1 ρ 0

∂ p e

∂z + g θ e θ 0

+ F rz = − 1 ρ 0

∂p

∂z + g θ θ 0

+ F rz (6.1)