Net Exchange Reformulation of Radiative Transfer in the CO2 15Um Band on Mars Jean-Louis Dufresne, Richard Fournier, Christophe Hourdin, Frérédric Hourdin

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Net exchange reformulation of radiative transfer in the CO2 15um band on Mars Jean-Louis Dufresne, Richard Fournier, Christophe Hourdin, Frérédric Hourdin

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Jean-Louis Dufresne, Richard Fournier, Christophe Hourdin, Frérédric Hourdin. Net exchange reformulation of radiative transfer in the CO2 15um band on Mars. Journal of the Atmospheric Sciences, American Meteorological Society, 2005, 62 (9), pp.3303-3319. 10.1175/JAS3537.1. hal-00113218

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Net exchange reformulation of radiative transfer in the CO2 15µm band on Mars.

Jean-Louis Dufresne 1∗, Richard Fournier2 Christophe Hourdin 1†, Fred´ eric´ Hourdin1

1 Laboratoire de Météorogogie Dynamique, IPSL; CNRS and Université Pierre et Marie Curie; Paris; France 2 Laboratoire d’Energétique; Université Paul Sabatier; Toulouse; France To be pulished in J. Atm. Sci.

Abstract vertical layers and no more proportionally to its square N 2. We also investigate some specific The Net Exchange Formulation (NEF) is an al- points such as numerical instabilities that may ternative to the usual radiative transfer formu- appear in the high atmosphere and errors that lation. It was proposed by two authors in 1967, may be introduced if inappropriate treatments but until now, this formulation has been used are performed when reflection at the surface oc- only in a very few cases for atmospheric stud- curs. ies. The aim of this paper is to present the NEF and its main advantages, and to illustrate them in the case of planet Mars. 1 Introduction In the NEF, the radiative fluxes are no more considered. The basic variables are the net ex- In the past decades, numerical modeling of the change rates between each pair of atmospheric atmospheric circulation of Mars has been tak- layers i, j. NEF offers a meaningful matrix ing an increasing importance, in particular in representation of radiative exchanges, allows to the frame of the spatial exploration of the red quantify the dominant contributions to the local planet (Leovy and Mintz, 1969; Pollack et al., heating rates and provides a general framework 1981; Hourdin et al., 1993; Forget et al., 1998). to develop approximations satisfying reciprocity With the increased number of missions to Mars of radiative transfer as well as first and second and especially with the use of aero-assistance for principle of thermodynamic. This may be very orbit injection, there is an increasing demand useful to develop fast radiative codes for GCMs. for improvements of our knowledge of Martian We present a radiative code developed along physics, in particular of Martian upper atmo- those lines for a GCM of Mars. We show that sphere. computing the most important optical exchange Computation of radiative transfer is a key factors at each time step and the others ex- element in the modeling of atmospheric circu- change factors only a few times a day strongly lation. Absorption and emission of visible and reduces the CPU time without any significant infrared radiation are the original forcing of at- precision lost. With this solution, the CPU time mospheric circulation. With typical horizontal increases proportionally to the number N of the grids of a few thousands to 10.000 points, and since we want to cover years with explicit repre- ∗ Corresponding author: Laboratoire de Météorologie sentation of diurnal cycle, operational radiative Dynamique, Université P. et M. Curie - Boite 99, F-75252 Paris Cedex 05 - France. E-mail: codes must be extremely fast. Representation of [email protected] radiative transfer must therefore be drastically †Now at Laboratoire d’Océanographie Physique, Mu- simplified and parameterized. seum National d’Histoire Naturelle, Paris For Mars, the main contributors to atmo-

1 spheric radiation are by far carbon dioxide In standard flux formulations, it is difficult (which represents about 95% of the atmospheric to quantify the relative importance of the var- mass) and airborne dust particles (even outside ious contributions to the local heating rates large planetary scale dust-storms, extinction of because the individual contributions are not solar light by dust is of several tens of percent). identified as such in the formalism. Green Carbon dioxide is dominant at infrared frequen- (1967) suggested that a reformulation of radia- cies with a vibration-rotation line spectrum that tive transfers in terms of net exchanges allows must be properly accounted for. quantifying the relative importance of physically In the development phase of the Laboratoire distinct contributions to the local heating rates de Météorologie Dynamique (LMD) Martian at- and could help design more efficient models. In mospheric circulation model, a major step was Green’s approach, called here the Net Exchange the derivation of a radiative transfer code for the Formulation (NEF), the quantity under consid- CO2 15 µm band (Hourdin, 1992). This model eration is directly the net energy exchanged be- was based upon the Wide Band Model approach tween two atmospheric layers (or more generally developed by Morcrette et al. (1986), and used two surfaces or gas volumes). Joseph and Bursz- in the operational model of the European Cen- tyn (1976) attempted to use the net exchange ter for Medium-Range Weather Forecasts (Mor- approach to compute radiative exchanges in the crette, 1990). This model is based on a two terrestrial atmosphere. Despite some numeri- stream flux formulation. Wide band transmi- cal difficulties, they showed that radiative net tivities are fitted as Padé approximants (ratio exchanges between an atmospheric layer and of two polynomials) as functions of integrated boundaries (space and ground) are dominant al- absorber amounts, including simple representa- though the net exchanges with the rest of the at- tions of temperature and pressure dependencies. mosphere are not negligible as they contribute For application to Martian atmosphere, the fit to approximatively 15% of the total energy bud- was somewhat adapted in order to account for get. With NEF, Bresser et al. (1995) did elabo- Doppler line broadening which becomes signifi- rate analytical developments for particular cases cant above 50 km. in order to compute the radiative damping of Altogether, in standard configurations of the gravity waves. The well known Curtis matrix LMD Martian model with a 25 layer vertical ((Curtis, 1956), see for instance,[]Goody.Yung- discretization, infrared computations represent 1989) may be related to NEF, but in the Curtis a significant part (up to one half) of the total matrix approach one way exchanges are consid- computational cost. Similar reports are made ered instead of net exchanges, which means that concerning terrestrial models. The transmis- useful properties of NEF, such as the strict si- sion functions being not multiplicative for band multaneous satisfaction of energy conservation models, the determination of radiative fluxes at and reciprocity principle, are abandoned. Fels each level requires independent calculations of and Schwarzkopf (1975) and Schwarzkopf and the contributions of all atmospheric layers. The Fels (1991) take advantage of the importance of corresponding computation cost increases as the the cooling to space to develop an accurate and square of the number of vertical layers. This rapid longwave radiative code. They don’t use quadratic dependency undoubtedly represents a the NEF but their work may be easily under- severe limitation when thinking of further model stand in the net exchange framework. refinements, in particular as far as near sur- Similar developments were also motivated face and high atmosphere processes are con- by various engineering applications. Hottel’s cerned, both requiring significant vertical dis- method (Hottel and Sarofim, 1967), also named cretization increases. However, it is commonly the zone method, is originally based on NEF. recognized that, despite of this formal difficulty, However, difficulties were encountered consid- infrared radiative transfers are dominated by a ering multiple reflection configurations and the few terms such as cooling to space and short NEF symmetry was practically abandoned. distance exchanges (e.g. Rodgers and Walshaw, Cherkaoui et al. (1996, 1998), Dufresne et al. 1966; Fels and Schwarzkopf, 1975). In prac- (1998) and De Lataillade et al. (2002) showed tice, the quadratic dependency of absorbtivity- that NEF can be used to derive efficient Monte emissivity methods is widely over costly, a sig- Carlo algorithms. Dufresne et al. (1999) used nificant part of the computations resulting in NEF to identify and analyze dominating spec- fully negligible contributions. tral ranges, emphasizing the contrasted behav-

2 ior of gas-gas and gas-surfaces exchanges. Fi- In this expression, ν is the frequence, A repre- nally, this formulation was recently used to ana- sents the entire system (for an atmosphere, the lyze longwave radiative exchanges on Earth with entire atmosphere plus ground and space bound- a Monte-Carlo method (Eymet et al., 2004). aries), ΓM,P is the space of all optical paths join- In the present paper, we show how NEF can ing locations M and P . For each optical path ν help derive efficient operational radiative codes γ, τγ is the spectral transmission function along ν ν for circulation models. This code is now op- the path. BP and BM are the spectral black- erational in the general circulation model de- body intensities at the local temperatures of P ν veloped jointly by Laboratoire de Météorologie and M, and KM is the absorption coefficient in Dynamique and the University of Oxford (For- M. The differential dVP around location P is get et al., 1999). As an example, this circu- either an elementary volume or an elementary ν lation model has been used to produce a cli- surface and KP,γ is either the absorption coeffi- mate database for Mars for the European Space cient in P (if P is within the atmosphere) or the Agency 1 (Lewis et al., 1999). In Sec. 2 the NEF directional emissivity (if P is at the boundaries). is presented in the specific case of stratified at- Expressed this way, the radiative budget of mospheres and analysis are performed for typi- elementary volume dVM can be seen as the dif- cal Martian conditions. Section 3 discusses the ference of two terms: the radiative power ab- questions related to operational radiative code sorbed by dVM coming from the whole atmo- derivations, in particular those related to verti- sphere plus surface and space (BP part of Eq. 2) cal integration procedures and reflections at the minus the power emitted by dVM toward all surface. The time integration scheme is consid- other locations (BM term). When separated ered in Sec. 4, first by investigating the numer- this way, the equation can be simplified fur- ical instabilities that may occur in the high at- ther noticing that the BM part (total emission mosphere, then by finding how computer time +∞ ν ν of volume dVM ) reduces to 4π 0 KM BM dν. may be saved without loosing accuracy. Sum- This approach is the current basis for engineer- mary and conclusions are in section 5. ing zone method and Curtis matrixR (see for instance, Goody and Yung, 1989). In NEF, equation 2 is rather interpreted 2 Net exchange formulation keeping the formal symmetry as the sum of the individual net exchanges between volume dVM 2.1 General approach and all other elementary volumes or surfaces (in- Longwave atmospheric radiative codes are gen- cluding space in the case of an atmosphere). An erally based on flux formulations. Angular in- individual spectral net exchange rate between tegration of all intensities at each location leads dVM and dVP to the radiative flux field, q~R, the divergence of ν which gives the radiative budget of an elemen- ψ (dVM , dVP ) = dVM dVP dγ (3) Γ tary volume dVM around point M as Z M,P ν ν ν ν ν τγ KM KP,γ (BP − BM ) dQ = −div (q~R) dVM (1) is just the power emitted by dVP and absorbed In an exchange formulation, the volumic ra- by dVM minus that emitted by dVM and ab- diative budgets are addressed directly without sorbed by dVP . For a discretized atmosphere, explicit formulation of the radiative intensity the spectral net exchange rate between two and radiative flux fields. Corresponding for- meshes i and j reads: mulations include complex spectral and optico- geometric integrals that may come down to ν ν ψ = ψ (dVM , dVP ) (4) (Dufresne et al., 1998) i,j ZAi ZAj +∞ where Ai and Aj are the volumes or surfaces of dQ = dVM dν dVP dγ (2) meshes i and j. The spectral radiative budget 0 A ΓM,P Z Z Z ψν of mesh i is the sum of the net exchange rates τ ν Kν Kν (Bν − Bν ) i γ M P,γ P M between i and all other meshes j: 1The database is accessible both with a Fortran ν ν interface for engineering and through the WEB at ψi = ψi,j (5) http://www.lmd.jussieu.fr/mars.html j X

3 A very specific feature of this formulation Under these assumptions, the space of relevant lies in the fact that both the reciprocity prin- optical paths reduces to straight lines between ciple, the energy conservation principle and the exchange positions. Dividing the atmosphere ν second thermodynamic principle may be strictly into N layers, spectral net exchange ψi,j be- satisfied whatever the level of approximation is tween layer i and layer j can be derived from retained to solve Eq. 3 and 4. The reciprocity Eq. 3 and 4 as principle states that the light path does not de- π Zi+ 1 Zj+ 1 2 pend on the direction in which light propagates, ψν = 2 2 (7) which means that the integrals of the optical i,j Z − 1 Z − 1 0 ν Z i 2 Z j 2 Z transmission τγ over both the optical path space ν ν ν ν 2(Bz − Bz )k (zi)ρ(zi)k (zj)ρ(zj ) ΓM,P and over the reciprocal space ΓP,M are the j i same: zj kν (z)ρ(z) exp − dz tan θ ν ν cos θ dγτγ = dγτγ (6) µ ¯Zzi ¯¶ ΓM,P ΓP,M ¯ ¯ Z Z dθdzidzj¯ ¯ Using Eq. 3, the reciprocity principle reduces ¯ ¯ ν ν where Z 1 and Z 1 are the altitudes at the to ψ (dVM , dVP ) = −ψ (dVP , dVM ). This con- i− 2 i+ 2 dition may be satisfied provided that the same lower and higher boundaries of atmospheric ν ν computation is used for both ψ (dVM , dVP ) and layer i, ρ is the gas density, k is the spectral ν −ψ (dVP , dVM ). In other words, when pho- absorption coefficient and θ is the zenith angle. tons emitted by dVM and absorbed by dVP are The previous equation may be rewritten as: counted as an energy loss for volume dVM , the Zi+ 1 Zj+ 1 same approximate energy amount is counted as ν 2 2 ψi,j = (8) an energy gain for dVP . As a direct conse- Z − 1 Z − 1 Z i 2 Z j 2 quence, the energy conservation principle is also ∂2Υν (z , z ) satisfied. Finally, provided that the difference (Bν − Bν ) i j dz dz zj zi ∂z ∂z i j (Bν − Bν ) that appears as such inside the op- ¯ i j ¯ P M ¯ ¯ tical path integral is preserved, Eq. 3 ensures ν ¯ ¯ where Υ is the spectral¯ integrated transmission¯ that warmer regions heat colder regions in ac- function defined as cordance with the second thermodynamic prin- π 2 ciple. ν 0 Altogether, the NEF allows the derivation of Υ (z, z ) = 2 (9) 0 approximate numerical schemes strictly satisfy- 0 Z z kν (x)ρ(x) ing both the reciprocity principle, the energy exp − dx sin θ cos θ dθ conservation principle and the second thermo- Ã ¯ z cos θ ¯! ¯Z ¯ dynamic principle. Any approximation may be ¯ ¯ ¯ ¯ retained for the integration over the optical path With the same¯ assumptions, the¯ spectral net ex- ν domain without any risk of inducing artificial change rate ψi,b between layer i and ground or global energy sources, or non physical energy space can be derived from Eq. 3 as redistributions. π Zi+ 1 2 ν 2 ψi,b = (10) Z − 1 0 2.2 Application to the CO2 band Z i 2 Z ν ν ν 15µm in the Martian context 2(B (Tb) − Bzi )k (zi)ρ(zi) zb kν (z)ρ(z) After these general considerations, we illustrate exp − dz sin θ dθdz cos θ i the net exchange approach in the case of the µ ¯Zzi ¯¶ CO 15µm band on Mars. At this first stage, ¯ ¯ 2 with T = T for¯ exchanges with¯ the planetary we make the following simplifying assumptions: b s ¯ ¯ surface (at temperature Ts) and Tb = 0 K for • the atmosphere is perfectly stratified along cooling to space. This equation may be rewrit- the horizontal (plan parallel assumption). ten as:

Zi+ 1 ν • the surface is treated as a black-body ν 2 ν ν ∂Υ (zi, zb) ψi,b = (B (Tb) − Bzi ) dzi (emissivity ²=1). ∂zi Z − 1 Z i 2 ¯ ¯ ¯ ¯ • the atmosphere is assumed dust free. ¯ ¯ (11) ¯ ¯

4 This last equation is well known as it is com- with m m ∂τ¯ (z , z ) monly used to compute the cooling to space. ξ = i b (15) zi,zb ∂z In practice, net exchange computations re- ¯ i ¯ quire therefore angular, spectral and vertical in- ¯ ¯ Finally the net exc¯hange ψs,e ¯between the tegrations. In the present study, the following ground surface s and¯space e becomes:¯ choices are made: m ψ = −B¯m(T )ξ ∆ν (16) 1. As in most GCM radiative codes, the an- s,e s zs,ze m m gular integration is computed by applying X the diﬀusive approximation which consists with in the use of a mean angle θ (1/ cos θ = m m ξz ,z = τ¯ (zs, ze) (17) 1.66, see Elsasser, 1942). s e 2. As in the original martian model, the spectral integration is replaced by a band model approach in which the Planck functions and wide band transmitivities are separated (Morcrette et al., 1986; Hour- din, 1992). 2 3. The vertical integration is what we con- centrate on in Section 3 with various levels of approximation. With the diﬀusive approximation and the use of wide band transmitivities, the net exchange ψi,j becomes after spectral integration of Eq. 8 over wide bands m:

Z + 1 Z + 1 i 2 j 2 ψi,j = (12) Zi− 1 Zj− 1 m Z 2 Z 2 X ¯m ¯m m (Bzj − Bzi )ξzi,zj ∆νmdzidzj Figure 1: Temperature vertical proﬁle (K, bot- with 2 m tom axis, continuous line) with 500 layers (thin m ∂ τ¯ (z , z ) ξ = i j (13) line) and 25 layers (bullet, thick line). Each zi,zj ∂z ∂z ¯ i j ¯ temperature of the 25 layers grid is the mean of m ¯ ¯ where τ¯ (zi, zj ) is the¯ wide band ¯transmitivity 20 layers of the high resolution grid (ﬁnite vol- ¯ ¯ between zi and zj with a mean angle θ and ∆νm ume representation). The reference heating rate is the band width. (K/day, top axis, dotted line) is also displayed The net exchange ψi,b between layer i and for the two grids. boundary b (ground or space) becomes after spectral integration of Eq. 11:

Z + 1 i 2 2.3 Reference case ψi,b = (14) Zi− 1 m We first present a computation of net exchanges Z 2 X m m m on a typical 25 layers GCM grid (Table 1), with B¯ (Tb) − B¯ ξ ∆νmdzi zi zi,zb refined discretization near the surface. To avoid 2This approac¡ h is exact for ¢a spectral interval nar- problems with the vertical integration for this row enough to use a constant value of the Planck func- reference computation, exchanges are first com- tion. For larger intervals, temperature variations affect puted on an over-discretized grid of 500 layers the correlation between the gaz absorbtion spectrum and the Planck function. Following Morcrette et al. (1986), (Fig. 1), each layer of the coarse GCM grid cor- this effect was accounted for in the original model by responding exactly to 20 layers of the 500 lay- using different sets of fitting parameters for the trans- ers grid. An exchange between two atmospheric mission function depending upon the temperature of the layers of the coarse grid is simply obtained as emitting layers. Here, we only use one set of parameters and control tests indicated that this simplification has a the sum of the 20x20 exchanges from the finer negligible effect on the estimated radiative heat sources. grid.

5 We use a reference temperature profile known property has been widely used to de- (Fig. 1) derived from the measurement taken by rive approximate solutions in atmospheric con- two Viking probes during their entry in the Mar- text (e.g. Rodgers and Walshaw, 1966; Fels and tian atmosphere (Seiff, 1982) and already used Schwarzkopf, 1975; Schwarzkopf and Fels, 1991). by Hourdin (1992). In the upper atmosphere, Internal exchanges within the atmosphere are by there is no systematic temperature increase as far dominated by the exchanges with adjacent there is no significant solar radiation absorption layers (note that there is a factor of ≈ 3 between equivalent to that the ozone layer on Earth. In two consecutive colors). As a consequence, the the middle atmosphere, gravity waves and ther- net exchange matrix is very sparse. A very few mal tides disrupt the temperature profile. Near terms dominate all the others. These important the surface, the quasi isothermal part of this av- terms are the exchanges with boundaries (space eraged profile hides a strong diurnal cycle. The and surface) and the exchanges with adjacent surface pressure is fixed to 700 Pa. layers. Thanks to the NEF, the relative magnitude of these terms can be quantified. 2.4 Net exchange matrix 2.4.2 Thermal aspects NEF offers a meaningful matrix representation m of radiative exchanges. A graphical example of In each spectral band, each contribution χi,j to such a matrix is shown in Fig. 2. Each element the total net exchange rate χi,j is the integral of displays the net exchange rate χi,j for a given the product of two terms: the blackbody inten- pair i, j of meshes converted in terms of heating sity difference between zi and zj and the optical m rate: exchange factor ξzi,zj (e.g. Eq. 12). The sign g 1 ψi,j of the net exchange rate χm only depends on χi,j = (18) i,j Cp δt δpi the temperature difference between i and j as m where g is the gravity, Cp the gas mass heat the optical exchange factor ξzi,zj is always pos- capacity, and δt the length of the Martian day itive. Layer i heats layer j only if its tempera- (δt = 88775s). For the ground , the heating rate ture Ti is greater than Tj . The direct influence is arbitrary computed using a thermal capaci- of the temperature profile on the exchange ma- tance of 1 J.K−1.m−2.day−1. The total heating trix can be seen on Fig. 2. For instance layer 10 rate of a layer i is: is heated by the warmer underlying atmosphere and surface and looses energy toward the colder

χi = χi,j . (19) layers above and toward space. The picture is j more complex in the upper atmosphere where X strong temperature variations are generated by χi,j and χj,i are of opposite sign but the ex- atmospheric waves. In this region, a layer can change matrices expressed in K/day are not an- be heated by both adjacent warmer layers (e.g. tisymmetric as they would be if expressed in layer 20). In particular these radiative exchange 2 W/m (ψi,j = −ψj,i but χi,j 6= −χj,i). between adjacent layers are known to damp the possible temperature oscillations due to atmo- 2.4.1 Matrix characteristics spheric waves (Bresser et al., 1995). Finally, the gas radiative properties depend As an example of reading Fig. 2, consider layer much less on the temperature than the black i = 10 (marked in the figure). The temperature body intensity. Therefore the metric of optical profile and total heating rate χ are also plot- i exchange factors ξ may be assumed as constant ted on both sides. The horizontal line of the for qualitative exchange analysis, and even to matrix shows the decomposition of the heating some extent for practical computations as dis- rate in terms of net exchange contributions (see cussed later on. Eq. 19). This partitioning of the heating rates first emphasizes some well established physical pictures. The cooling to space is the dominant 2.4.3 Spectral aspects part of the heating rate: it essentially defines The exchange factors ξ between two meshes are the general form and the order of magnitude proportional to the gas transmission τ¯ for the of the heating rate vertical profile. This well exchange between surface and space (Eq. 17), proportional to the first derivative of τ¯ for the

6 Layer σ Approx. Layer σ Approx. Layer σ Approx. # height (m) # height (km) # height (km) 1 0.99991 3.6 10 0.9251 3.030 19 0.3256 43.69 2 0.99958 16.4 11 0.8787 5.037 20 0.2783 49.80 3 0.99898 39.8 12 0.8157 7.934 21 0.2359 56.24 4 0.99789 82.1 13 0.7403 11.70 22 0.1975 63.15 5 0.99592 159.0 14 0.6597 16.19 23 0.1613 71.04 6 0.99238 297.9 15 0.5803 21.19 24 0.1275 80.21 7 0.98605 547.0 16 0.5061 26.52 25 0.0842 96.35 8 0.97494 988.4 17 0.4388 32.07 9 0.95598 1753 18 0.3788 37.80

Table 1: Low resolution (i.e. 25 layers) vertical grid characteristics: layer number, σ levels (σ = P/Ps, with Ps the surface pressure) and approximate corresponding heights.

Heating from ground Cooling to space Exchanges with layer

Below Above Atmospheric Layer

Net Exchange Matrix (K/day) Temperature Profile (K) Heating Rate (K/day)

Figure 2: Graphical representation of radiative net exchange rates in the Martian atmosphere. Left: temperature proﬁle, Middle: net exchange matrix, Right: heating rate. The vertical axis is the layer number. Same conditions as in Fig. 1.

7 three normalized functions, τ¯, 1/²¯zi .∂τ¯/∂X and 2 2 1/(²¯zi .²¯zj ).∂ τ¯/∂X , that may also be seen as normalized exchange factors, are displayed in Fig. 3 for the two bands. The three normalized exchange factors go to 1 when X goes to 0. Indeed, when the two extremities are adjacent, the exchange factor is the product of the emissivity at both extremities3. When X increases, the normalized exchange factor slowly decreases (in particular for band #2) if the two extremities are black surfaces (here space and ground); it decreases faster if one extremity is a gas layer and decreases even faster if the two extremities are gas layers Figure 3: Gas transmission τ (solid) and its (Fig. 3). This is an illustration of the so-called two first derivatives normalized by the emissiv- “spectral correlation effect” (e.g. Zhang et al., ity ²¯ of the gas layer, 1/²¯.|∂τ¯/∂X| (dash) and 1988; Modest, 1992; Dufresne et al., 1999). For 1/²¯2.|∂2τ¯/∂X2| (dot), as a function of the nor- the exchange between two gas layers, both ab- malized integrated mass of atmosphere X = sorption and emission are maximum in spectral (P − Ps)/Ps, for the central part (band #1, up- regions near the center of the absorbing lines. per plot) and for the wings part (band #2, lower But exactly at the same frequencies gas absorp- plot) of the CO2 15µm band. The atmospheric tion creates a strong decrease of the transmis- temperature is assumed uniform (T = 200K) sion when the distance between extremities in- and the surface pressure is Ps = 700P a. creases. Thus the exchange strongly decreases with distance. On the contrary, the exchange between ground and space is most important in exchanges between a layer and surface or space spectral regions where the spectral transmission (Eq. 15) and proportional to the second deriva- is high, that is where the gas absorption is low. tive of τ¯ for the exchanges between two atmo- Thus the exchange factor between ground and spheric layers (Eq. 13). The behavior of optical space is much less sensitive to the integrated air exchange factors can be understood by analyz- mass between them. Exchange between a layer ing these three functions. To allow comparison, and ground or space is an intermediate case. we normalize them by the product of the emis- When X increases, the decrease of the three sivity at both extremities. If the extremity i normalized exchange factors is faster for the cen- is a gas layer of differential thickness dzi, the tral part of the CO2band (band #1) than for the emissivity is: wings (band #2) (Fig. 3). As a consequence, the decrease with distance of the exchange be- m ν tween two atmospheric layers is more important ²¯zi = dzi k (zi)ρ(zi)dν (20) Zν in band #1 (left panel of Fig. 4) than in band m #2 (right panel of Fig. 4). The exchanges be- If the extremity i is ground or space, ²¯i = 1. We also use a normalized integrated mass X of tween adjacent layers are much greater for band atmosphere #1 than for band #2, whereas distant exchanges have the same magnitude for the two bands. For z g 0 the cooling to space, the competition between X(z) = ρdz (21) the decrease of |∂τ¯/∂X| and the increase of the Ps s Z local black body intensity yields noticeably dif- where Ps is the ground pressure. ferent vertical profiles: The absolute value of The wide band model used in this study has the cooling to space decreases when the layer two spectral bands chosen empirically (Hourdin, is closer to the surface for band #1 whereas it 1992). The first one (band #1), ranging from increases for band #2. 635 to 705 cm−1, corresponds to the central part 3This is only true in the limit where the gas layer(s) of the CO2 15 µm band. The second one (band −1 is(are) optically thin. In the example presented here, the #2), ranging from 500 to 635 cm and from emissivities are computed for layers with a normalized 705 to 865 cm−1 corresponds to the wings. The thickness ∆X = 0.01.

8 Net Exchange MatrixBand 1 (K/day) Net Exchange MatrixBand 2 (K/day) Heating Rate (K/day)

Figure 4: Graphical representation of the net exchange matrix for spectral bands # 1 (left) and # 2 (middle). Same conditions as in Fig. 2. On the right side, the total heating rate is shown for band # 1 (black squares), for band # 2 (crosses) and over the whole spectrum (open circles).

3 Vertical integration with

m m m m m ξi,j = τ¯ 1 1 − τ¯ 1 1 − τ¯ 1 1 + τ¯ 1 1 In the above reference computation, net ex- i+ 2 ,j− 2 i− 2 ,j− 2 i+ 2 ,j+ 2 i− 2 ,j+ 2 changes have been computed using given sub- ¯ (23) ¯ ¯ ¯ grid scale temperature profiles(Fig. 1). In prac- In the¯equivalent flux formulation, the individ- ¯ tice, with circulation models, only mean tem- ual contributions of the radiation emitted by peratures are known for each layer4 and assump- 1 layer i to the flux at interface j + 2 tions are required concerning sub-grid scale pro- ¯m m m files. First we present the very simple assump- Fi→j+ 1 = Bi τ¯ 1 1 − τ¯ 1 1 2 i+ 2 ,j+ 2 i− 2 ,j+ 2 tion of a uniform temperature within each at- m X ³ ´ mospheric layer. This allows us to highlight the (24) link between NEF and flux formulations. Then are first summed over i to compute the radiative 1 flux F 1 at each interface j + . The radiative we address the more general case of non isother- j+ 2 2 mal layers and finally we highlight the modifica- budget of each layer j then reads: tions that are required in the case of a reflective ψj= Fj− 1 − Fj+ 1 (25) surface. 2 2

= F 1 − F 1 i→j− 2 i→j+ 2 i i 3.1 Isothermal layers X X For isothermal layers, the individual exchanges An exact equivalence between the net exchange contributing to the radiative budget of layer i and ﬂux formulations is obtained by noting that

(ψi = j ψi,j ) take a simple form (Eq. 12 and ψi,j = Fj→i+ 1 − Fj→i− 1 − Fi→j+ 1 − Fi→j− 1 13) reducing to 2 2 2 2 P ¯ ¯ ¯ (26)¯ ¯ ¯ ¯ ¯ m ¯m ¯m and using the property τ¯ = τ¯ . ψi,j = ξi,j (Bj − Bi ) (22) ¯ i,j¯ ¯ j,i ¯ m For developing a radiative code, NEF how- X ever presents advantages. With flux formula- 4We assume that circulation models make use of finite volume representations and that GCM outputs are rep- tion, fluxes are first integrated over altitude z resentative of mean temperatures rather than mid-layer and then differentiated. When temperature con- temperatures trasts are weak (here near the surface for instance), net exchange rates can be by orders of

9 magnitude smaller than fluxes. Computing first the fluxes and then the differences may lead to strong accuracy loss which we observed could in- duce reciprocity principle violations (colder layers heating warmer layers for instance)5. In Fig. 5, we show the error on net exchanges if the isothermal approach is retained for all exchanges with respect to the reference 500-layer simulation. The error is very large around the diagonal and often larger than the exchange itself. Indeed, at frequencies where significant CO2 emission occurs (close to absorption lines centers) the atmosphere is extremely opaque, which means that most emitted photons have very short path lengths compared to layer thicknesses. Consequently, exchanges between adjacent layers are mainly due to photon exchanged Figure 5: Error matrix with the isothermal layer in the immediate vicinity of the layer interface. assumption. Line i column j gives the error in In this thin region, temperature contrasts are K/day for the heating rate of layer i due to much weaker than the differences between mean its exchange with layer j. The error is com- layer temperatures. The isothermal approxima- puted with respect to the reference computation tion thus results in a strong overestimation of performed with 20 sub-layers inside each layer. the net exchanges. Other conditions are the same as in Fig. 2. The relative error due to the isothermal hypothesis strongly decreases with distance be- Note that in this approach, the assumed temper- tween layers. This feature is further commented ature profile inside a layer is different when com- in appendix. puting the exchange with the layer just above or just below. 3.2 Net exchanges between adja- With this assumption, exact integration pro- cent layers cedures could be designed, for instance using the analytical solution available for the best fitted The specific difficulty of exchange estimations Malkmus transmission function (Dufresne et al., in the case of adjacent layers is commonly iden- 1999), or integrating by parts and tabulating in- tified and solutions have been implemented in tegrated transmission function from line by line flux computation algorithms (e.g. Morcrette computations. Here we test a more basic solu- et al., 1986). In most GCMs, only the average tion by dividing the linear profile into isother- layer temperatures and compositions are avail- mal sub-layers, with thiner sub-layers closer to able. Here a linear approximation is retained the interface (Fig. 6). The sub-discretization for B to describe the atmosphere close to the scheme was tested against reference simulations. mesh interface. Because of the symmetry of For the present application, a satisfactory ac- Eq. 8 in zi and zj, the linear approximation is curacy is reached with a sub-discretization into strictly equivalent to a quadratic approximation three isothermal sub-layers of increasing thick- in the limit case of two layers of identical thicknesses (∆z/7, 2∆z/7 and 4∆z/7) away from the nesses (see Appendix). When computing ψi,i+1 interface. we therefore assume that B(z) is linear between Whatever the integration procedure, a direct zi−1/2 and zi+3/2, satisfying the following con- consequence of the previous linear black body straints for j = i § 1 (Fig. 6): intensity assumption is that the net exchange between two adjacent layers may be still written Zj+1/2 B(z)dz = Bj.∆zj (27) formally like the net exchange between isother- Zj−1/2 mal layers (Eq. 22). Only the expression of the Z m exchange coefficient ξ depends on the tem- 5This problem may be partially overcome in the flux i,i§1 formulation by introducing the blackbody differential perature profile hypothesis. fluxes F˜ = πB − F (e.g. Ritter and Geleyn (1992)) We finally adopt the following solution

10 exchange i, i+1 exchange i−1, i

i+1

i−1 Figure 7: Vertical profile of the heating rate error in K/day due to the vertical integration B scheme (left), and part of this error due to the computation of the net exchange with ad- Figure 6: A linear black body intensity pro- jacent layers (middle, square), with distant lay- file approximation is used for computation of ers (middle, circle), with ground (right, square) the net exchanges between adjacent layers. The and with space (right, circle). The error is com- black body intensity profile inside a layer is dif- puted with respect to the reference computation ferent when computing the exchange with the performed with 20 sub-layers inside each layer. layer just above (black line) or just below (grey Other conditions are the same as in Fig. 2. The line). The sub-grid discretization is also shown vertical axis is the layer number. (thin lines). emissivity can differ from 1. The mean emissiv- for the vertical integration: the above sub- ity of the Martian surface is believed to be of discretization into three isothermal sub-layers the order of 0.95 (Santee and Crisp, 1993), and is used to compute the radiative exchanges emissivity is believed to be lower in some regions between adjacent layers whereas the simple (Forget et al., 1995). isothermal layer assumption is used to compute When reflection at the surface is present, two the exchange between distant layers. For the atmospheric layers can exchange photons, either exchange between ground and first layer, we as- directly, or through reflection at the surface. For sume the temperature of gas just above the sur- instance, the net exchange between two atmo- face to be T0 = (T1 +Ts)/2 and a linear B profile spheric layers i and j (Eq. 12) becomes: between T1 and T0. An isothermal description is Z + 1 Z + 1 i 2 j 2 retained for the exchange between the optically ψi,j = (28) thin upper layer and space. The global error due Zi− 1 Zj− 1 m Z 2 Z 2 X to the vertical integration scheme as well as the (B¯m − B¯m) origin of the error are displayed Fig. 7. The an- zj zi m m alytical expression of these errors are presented ξd (zi, zj ) + ξs (zi, zj ) dzidzj in appendix for some cases. One should have h i in mind that results are compared with a high m where ξd is the optical exchange factor for di- resolution vertical grid where the temperature rect exchanges profile has a more precise description than in 2 the low resolution grid. ∂ τ(zi, zj ) ξ (z , z ) = (29) d i j ∂z ∂z ¯ i j ¯ ¯ ¯ 3.3 Exchanges with reflection at m ¯ ¯ and ξs the optical exchange¯ factor¯through re- the surface flection at the surface: The above presentation assumes that the surface 2 ∂ Γs(zi, zj ) behaves as a black-body. In practice, surface ξ (zi, zj ) = (1 − ²s) (30) s ∂z ∂z ¯ i j ¯ ¯ ¯ ¯ ¯ 11 ¯ ¯ where ²¯s is the surface emissivity and Γs(zi, zj ) is the transmission function from zi to zj via the surface for a spectral interval. Assuming the dif- fusive approximation this transmission writes:

ν ν Γs(zi, zj ) = (τ (zi, 0)τ (0, zj )) dν (31) Zν In the original flux formulation, as well as in other radiative codes based on the so-called absorbtivity/emissivity method, the downward flux is first integrated from the top of the atmosphere to the surface. The reflected part of this downward flux is then added to the flux emitted Figure 8: Left: Vertical profile of the heating by the grey surface. This flux is then used as a rate when the surface is perfectly black (cross, limit condition to integrate the upward flux up continuous line) and when the surface has an to the atmospheric top. This assumption corre- emissivity ² = 0.9, the computation being ei- sponds to the following approximation : s ther exact (circle, dash line) or neglecting spectral correlation when reflection at the surface Γs(zi, zj ) ≈ τ(zi, 0)τ(0, zj ) (32) occurs (square, dotted line). Same atmospheric which is wrong for wide and narrow band mod- conditions as in Fig. 1. The vertical axis is the els because the spectral information is forgotten layers number. Right: Same, but with a per- at the surface. The error on the heating rate is fectly reflecting surface (²s = 0). particularly strong for the layers near the surface. For a surface emissivity of 0.9, as expected, the exact solution displays small changes in the physical processes (radiation, turbulent vertical heating rate compare to the case where the sur- mixing...) are active, we present hereafter re- face is black (plus signs and circles in the left sults obtained with a 1D model that corresponds on Fig. 8). On the other side, the computa- to a single vertical column of the 3D GCM. tion which neglects the spectral correlation at When the temperature profile is prescribed, the surface (squares) displays a very large and a decrease of the surface emissivity reduces the unrealistic change of the heating rate near the cooling of the surface but increases the cooling of surface. the atmosphere above (Fig. 8). With the full 1D A useful property can be used to check the model, a decrease of the surface emissivity re- results with a reflective surface. Let us consider duces the cooling of the surface which increases an atmosphere with a thin layer near the sur- its temperature (Fig. 9). The temperature of face having the same temperature as the sur- the atmosphere above also increases, but less, face itself. This layer will never exchange en- as the decrease of emissivity increases the cool- ergy with the surface because both are at the ing of the atmosphere. same temperature. If the surface emissivity dif- When the temperature profile is prescribed, fers from 1, the exchange of this layer with the we previously noticed that neglecting the spec- atmosphere above and with the space will be tral correlation at the surface leads to strongly increased through reflection at the surface. For overestimate the atmospheric cooling just above an optically thin layer and for a perfect mirror the reflective surface (Fig. 8). With the 1D (² = 0) all those exchanges, and hence the radia- model, neglecting the spectral correlation leads tive cooling, will be exactly twice that without to underestimate by a factor of 0.5 to 0.7 the reflection (² = 1) (Cherkaoui et al., 1998). The temperature increase in the boundary layer due net exchange computation fulfills this property to the emissivity decrease (Fig. 9). This un- (plus signs and circles in the right hand of Fig. 8) derestimate is even more important if the tur- but the original flux model does not (square). bulent vertical mixing is neglected (not shown). The above computations were performed Neglecting the spectral correlation also slightly with a prescribed vertical temperature profile. increases the diurnal cycle of the atmospheric In order to evaluate the error associated to temperature near the surface (not shown). incorrect treatment of reflection when all the

12 space. The optical exchange factors ξ present in Eq. 34 are comparable to the so-called weighting functions used to invert satellite radiative measurements. Therefore NEF should be a useful framework to assimilate those measurements in GCMs. In the atmosphere, the net flux at level i + 1 2 is equal to the net exchange between all the 1 meshes below i + 2 and all the meshes above 1 i + 2 : N+1 i n F 1 = ψj,k (35) i+ 2 j=i+1 X kX=0 If one realy wants the values of the upward and downward fluxes, they may be approximated assuming each atmospheric layer is isothermal. If the optical exchange factors ξ have been computed with this assumption, the upward flux at 1 level i + 2 is equal to the fluxe emitted by all 1 the meshes below i + 2 and absorbed by all the 1 meshes above i + 2 :

Figure 9: Vertical profile of the daily mean i N+1 + ¯ temperature difference due to a change in the F 1 = ξj,kBj (36) i+ 2 surface reflectivity with an exact computation j=0 X k=Xi+1 (open circle) and neglecting spectral correlation e at the surface (closed circle). The tempera- Note that the errors on fluxes arising from the tures of a run with a slightly reflective surface isothermal hypothesis are much smaller than the (²s = 0.9) are compared to a run with a non error on net exchanges between adjacent layers reflective surface (²s = 1). The runs are 10 days due to the same hypothesis. The same way, the 1 long, have the same initial state and are per- downward flux at lever i + 2 is equal to the flux 1 formed with the single column version of the emitted by all the meshes above i + 2 and ab- 1 Martian GCM. Diurnally averaged temperature sorbed by all the meshes below i + 2 : differences are plotted for the last day. The ver- i N+1 tical axis is the atmospheric layer number. − ¯ F 1 = ξj,kBk (37) i+ 2 j=0 X k=Xi+1 3.4 Computing vertical fluxes e In a general way there is no direct relationship 4 Time integration between upward and downward fluxes and net exchanges. Only the net radiative fluxes may 4.1 Numerical instabilities in the be directly expressed as a function of net ex- high atmosphere changes. For instance, the net fluxes at the top n When the atmospheric vertical resolution in- of atmosphere FN+ 1 reads: 2 creases, numerical instabilities appear in the N Martian GCM in the high atmosphere and n they may increase dramatically. This prob- FN+ 1 = ψN+1,k (33) 2 lem has also been encountered in some GCM kX=0 N of the Earth atmosphere and specific stabiliza- m ¯m tion techniques are commonly used to bypass = ξN+1,kBk (34) k=0 m this difficulty. Here we analyze the reasons of X X this difficulty and we propose a solution that where N is the number of vertical layers, k = 0 takes advantage of the NEF. stands for ground and k = N + 1 stands for

13 In the original Martian model, the radi- 4.2 Saving computer time tative transfer is integrated with an explicit The computation cost of the LW radiative code time scheme, the evolution between times t and is known to be very important in most GCMs. t + δt of the temperature of layer i beeing com- Solutions have been proposed and implemented puted from a computation of the heating rate to reduce this computation time. Generally the ψt = ξt (Bt − Bt) at time t as i j ij j i full radiative code is computed only one out of P ψtδt N time steps, and approximations are used to T t+δt = T t + i (38) interpolate the LW cooling rates between those i i m C i p N time steps. The simplest time interpolation scheme is to maintain constant the cooling rates In the upper atmosphere, the mass mi of the atmospheric layers becomes very low, reducing during this period. This is the case in the origi- they thermal capacitance. As a consequence, nal Martian GCM where the radiative code is strong numerical oscillations appear for large computed one out of two time steps (each 1 hour). time steps if the variations of ψi with temperature, within the time-step, are not taken into On Mars, the surface temperature diurnal account. cycle is as high as 100K and the time interpola- A well-known solution to this problem con- tion method has to reproduce the effects of this t (α) diurnal cycle. The NEF provides an easy answer sists in replacing ψi in Eq 38 by ψi = (1 − t t+δt to this problem. Since the Planck function dom- α)ψi + αψi . With those notations, the tem- poral scheme covers the cases of explicit (α = 0), inates the variations of the radiative exchanges, implicit (α = 1) and semi-implicit (α = 1/2) the Planck function will be computed at each schemes. For α 6= 0, the scheme is no more ex- time step while computing the optical factors plicit and requires an inversion procedure. The ξi,j only one out of N time steps. A second level net exchange formalism offers a simple practical of optimisation consist in computing the optical solution to this problem. Based on the analysis exchange factors corresponding to the most im- above, it can be assumed that only the black- portant exchange rates (see Sec. 2.4) more frequently than the others. Once again, the NEF body emissions Bi vary during the time-step while the optical coefficients do not. Also it ensures that the above approximation will not can be assumed that only the exchanges with alter the energy conservation and the reciprocity adjacent layers (i § 1) and boundaries (b) vary principle (Sec. 2.1). Practically all the optical while the exchanges with distant layers are un- exchange factors are scattered in three groups : modified. With these approximations, and after the exchange factors between each atmospheric linearization of the Planck function, layer i and (1) its adjacent atmospheric layers, (2) the distant atmospheric layers (i.e. the other (α) t t atmospheric layer) and (3) the boundaries (i.e. ψi = ψi + α ξij (39) j=i§1,b surface and space). X We present numerical tests performed using dB dB | (T t+δt − T t) − | (T t+δt − T t) the single column version of our GCM. The runs dT Tj j j dT |Ti i i · ¸ last 50 days and the comparison between runs If in addition we do not consider the variation of is performed using the results of the two last days. For these two days, we computed the at- Tb within the time-step (which is exact for space and not a problem for the surface when comput- mospheric temperature difference between each ing the heating rates in the upper atmosphere), run and the reference run. The mean and the the temperature at time t + δt is obtained from RMS of this difference allows a quick compar- that at time t through the inversion of a tridi- ison between them (Table 2). In the reference agonal matrix, for a low CPU cost. run (case # 1), the full LW code is called at each This approach, implemented in the Martian time step of the physics, i.e. every 30 Martian GCM with α = 1/2, is very efficient and su- minutes). Case # 2 corresponds to what was im- presses all the numerical oscillations in the up- plemented in the original version of our GCM: per atmosphere. the full LW code is called one out of two time steps of the physic, i.e. every Martian hour, and the LW cooling rates are constant during this period. Computing all the optical exchange fac-

14 tors only once a day (case # 3) leads to an error The graphical representation of the net ex- only slightly greater than computing all the LW change matrix appears to be a meaningful tool radiative code one out of two time steps (case # to analyze the radiative exchanges and the ra- 2), but requires a much smaller CPU time. This diative budgets in the atmosphere. In the case result illustrates that the diurnal variations of of Mars, the exchange between a layer and space the optical exchange factors are not very impor- (the cooling to space) and the exchanges be- tant compared to the diurnal variations of the tween a layer and its two adjacent layers are by Planck function. far the dominant contributions to the radiative As mentioned above, one may compute budgets. The exchange with space explains the the most important exchange factors more fre- general trend of the radiative budget with alti- quency than the other. Computing the opti- tude. The exchanges with adjacent layers play cal exchange factors with boundaries at each a key role as they dump the temperature oscil- time step highly reduces the error while it lations due to the various atmospheric waves. only slightly increases the CPU time (case #4). A key point of the NEF is that it ensures Computing the exchange factors with adjacent both the energy conservation and the reciprocity layers at each time step slightly reduces the er- principle whatever the errors or approximations ror while strongly increasing the CPU time (case are made when computing the optical exchange #5). factors. Computing exchange factors only once a day The net exchange between meshes is equal may introduce a significant bias for long term to the product of an optical exchange factors simulations as the diurnal cycle is very badly and the Planck function difference between the sampled. We choose to compute all the ex- two meshes. This allows one to analyze sep- change factors at least four times a day (case arately the role of the optical properties of the # 7-10). Computing the exchange factors with atmosphere and the role of the temperature pro- boundaries at each time step (30’) and the other file. The optical exchange factors are very ex- exchange factors every 6 hours (case # 8) pro- pensive to compute and they vary slowly with duces much smaller errors than the original so- time. On the contrary, the Planck function lution (case #2) while being two times less con- strongly depends on temperature, that strongly suming. Another important advantage of this varies during a day, but is very fast to com- solution is that the number of exchange factors pute. Computing the optical exchange factors with boundaries increases linearly with the num- and the Planck function at different time steps ber N of vertical layers. The number of the is therefore of immediate interest. Moreover, be- other exchange factors are still proportional to cause the NEF ensures both the energy conser- N 2 but they are computed much less frequently vation and the reciprocity principle, some of the and the required CPU time is therefore negligi- exchange factors (the most important) may be ble: the CPU time will increase almost linearly computed more frequently than others. These with N and no more as a function of N 2 as for possibilities give various opportunities to reduce all the absorbtivity-emissivity methods. the CPU time without loosing accuracy. Some If higher accuracy levels are required, a more possibilities have been explored in this paper. frequent computation of the exchanges factors In particular we have shown that the most im- with both the boundaries and the adjacent lay- portant terms are the exchanges with bound- ers is a good solution (case # 10). The errors aries, number of which is proportional to the are negligible and the CPU time is divided by a number N of vertical layers. Computing those factor of two compared to the reference solution terms more frequently than the others leads the (case #1). CPU time to increase proportionally to N and not proportionally to N 2 as in all the absorbtivity/emissivity methods. 5 Summary and conclusion Another consequence of the splitting of the net exchange rates into optical exchange factors In the present paper, a radiative code based on and Planck function differences is the possibil- a flux formulation has been reformulated into a ity to linearize the Planck function for all or radiative code based on the NEF. This formu- parts of the net exchanges. This allows us to lation has been proposed by Green (1967) but implement implicit or semi-implicit algorithms has not be often used since this time. at a low numerical cost (inversion of a tridiag-

15 Computation periode of Atm. temperature Normalized case the net exchanges the exchange factors with diﬀerence (K) CPU time of # and the adjacent boundaries distant mean RMS the LW radiative budgets layers layers radiative code 1 30´ 30´ 30´ 30´ 0.00 0.00 1.00 2 1hr 1hr 1hr 1hr -0.10 0.38 0.50 3 30´ 1dy 1dy 1dy 0.32 0.38 0.15 4 30´ 1dy 30´ 1dy -0.08 0.16 0.21 5 30´ 30´ 1dy 1dy 0.27 0.33 0.46 6 30´ 30´ 30´ 1dy -0.10 0.13 0.54 7 30´ 6hr 6hr 6hr 0.21 0.20 0.16 8 30´ 6hr 30´ 6hr 0.01 0.05 0.25 9 30´ 30´ 6hr 6hr 0.18 0.20 0.47 10 30´ 30´ 30´ 6hr -0.01 0.02 0.52

Table 2: Comparison between the various time interpolation schemes. onal matrix associated with exchanges between only absorption by the aerosols is considered, adjacent layers). scattering being neglected. This is consistent In our original radiative code (as well as in with previous studies that show that scattering other codes), reflections at the surface are con- by dust aerosols has the highest impact in win- sidered in a crude way: the reflected part of the dow regions of the atmosphere (e.g. Dufresne downward radiation and the radiation emitted et al., 2002). Currently a radiative code based by the surface are supposed to have the same on the NEF, that uses a ck-method for the spec- spectrum. We have shown that this approxi- tral integration and that also considers scatter- mation leads to highly overestimate the cooling ing is under progress for the Venus planet. of the atmosphere above the surface. The rea- son is that the spectrum of the downward radiation strongly depends on the gas absorption 6 Acknowledgments spectrum and is therefore very different from This work was supported by the European the spectrum of the radiation emitted by the Space Agency through ESTEC TRP con- surface. An exact computation is possible but tract 11369/95/NL/JG. The graphics have double the CPU time. been made with the user friendly and pub- A drawback of the NEF is that only the net lic domain graphical package GrADS origi- flux in the atmosphere can be directly deduced nally developed by Brian Dotty (COLA, sup- from the net exchanges, not the upward and [email protected]). We thanks the two anony- downward fluxes (although they are of experi- mous reviewers for their very constructive com- mental interests). Nevertheless we have shown ments. that they can be estimated with a few more assumptions. On the other hand, the optical exchange factor between each gas layer and space References does correspond to the so-called weighting function used to invert satellite flux measurements. Bresser, G., A. Manning, S. Pawson, and Therefore a radiative code based on the NEF C. Rodgers, 1995, A new parameterization of might be well suited for assimilation of satellite scale-dependent radiative rates in the strato- radiances. sphere, J. Atmos. Sci., 52, 4429–4447. The radiative code presented here is used in the last version of the LMD GCM of Mars (For- Cherkaoui, M., J.-L. Dufresne, R. Fournier, J.- get et al., 1999). In addition to the absorption Y. Grandpeix, and A. Lahellec, 1996, Monte- by gases presented in this paper, the effects of Carlo simulation of radiation in gases with a aerosols is also considered. Outside the two CO2 narrow-band model and a net-exchange for- wide bands, both absorption and scattering ef- mulation, ASME J. of Heat Transfer, 118, fect are computed using the algorithm of Toon 401–407. et al. (1989). Inside the two CO2 wide bands,

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18 7 Appendix 7.1 Sub-grid temperature quadratic profile We consider here an atmosphere with a temperature profile that is compatible with a second order blackbody intensity profile within the considered spectral band

B(z) = az2 + bz + c. (40)

We also assume that absorption coeﬃcients are uniform. Under these assumptions, the net exchanges between two layers i and j (Eq. 12) of thickness e separated by a layer of thickness l lead to the following double integral:

−l/2 +l/2+e 2 ψi,j = dν dx dy kν exp[−kν (y − x)][B(y) − B(x)] (41) Z∆ν Z−l/2−e Z+l/2 e e 2 = dν dx dy kν exp[−kν (x + y)] exp(−kν l)[B(y + l/2) − B(−x − l/2)] (42) Z∆ν Z0 Z0 = b dν exp (−kν l) [1 − exp (−kν e)] {(l + 2/kν ) [1 − exp (−kν e)] − 2e exp (−kν e)} Z∆ν (43)

This is obtained by expanding the expression of B in Eq. 41. The terms with c directly disapear. The double integral of the terms with a is equal to zero. Only the terms with b remain. Under the same assumptions, Eq. 40 leads to

B − B b = j i (44) l + e with Bi and Bj the average blackbody intensities of layers i and j. Therefore, Eq. 43 depends only on the layer average of B, which means that all quadratic profile that meet the layer averages have the same first order terms and therefore lead to identical net exchange rates. In particular, the linear subgrid profile approximation used in Sec. 3.2 for adjacent layer net exchange computations is therefore equivalent to a second order approximation. Note that this demonstration is only valid for conditions in which layer thicknesses are comparable and that the reasoning is made with blackbody intensity layer averages, whereas GCM outputs are temperature averages. Practical use therefore requires that blackbody intensities may be confidently linearized as fonction of temperature, which implies limited temperatures gradients.

7.2 Errors due to the isothermal layer approximation The same analysis may also be used to justify the use of the isothermal layer assumption for non adjacent layer net exchange computations. We consider an atmosphere with the same previous temperature proﬁle that is compatible with a second order blackbody intensity proﬁle within the considered spectral band. Under this assumption, the net exchange between two layers i and j (Eq. 12) of thickness e separated by a layer of thickness l may be approximated as

2 ψ˜i,j = dν exp(−kν l) [1 − exp(−kν e)] (Bj − Bi) (45) Z∆ν 2 = dν exp(−kν l) [1 − exp(−kν e)] b(l + e) (46) Z∆ν ψ−ψ˜ Using the exact expression of ψi,j (Eq. 43), the corresponding relative error E = ψ in the optically thick limit writes: 2 − kν e if kν e À 1 then E ≈ . (47) 2 − kν l

19 If the optical thickness between the two layers distant of l is also high (kν l À 1), the relative error e on the net exchange is E ≈ l , which means that the relative error on the net exchange between two layers due to the isothermal layer approximation decreases when the distance between the two layer increases. One can be more precise when considering three contiguous layers, numbered 1, 2 and 3, of thickness e. We estimate the error made when computing the net exchanges between layer 1 and the two other layers, ψ = ψ1,2 + ψ1,3, accounting for the exact sublayer proﬁle for ψ1,2 (i.e. for the adjacent layer) and using the isothermal layer assumption for ψ1,3 (i.e. for the distant layer). With the same notation as here above with l = e, the corresponding relative error becomes:

(kν e − 2)[1 − exp(−kν e)] + 2kν e exp(−kν e) Eν = (k e + 2)[1 − exp(−k e)] − 2k e exp(−k e) + 1−ae/b {2[1 − exp(−k e)] − 2k e exp(−k e)} ν ν ν ν exp(−kν e) ν ν ν

This relative error is 0 for small values of the optical thickness kν e, reaches 5% for an optical thickness of 2, then decreases toward 0 when the optical thickness increases.