Net Exchange Reformulation of Radiative Transfer in the CO m Band on Mars

Transcription

1 SEPTEMBER 2005 D U F R E S N E E T A L Net Exchange Reforulation of Radiative Transfer in the CO Band on Mars JEAN-LOUIS DUFRESNE Laboratoire de Météorologie Dynaique, Institut Pierre Sion Laplace, CNRS, and Université Pierre et Marie Curie, Paris, France RICHARD FOURNIER Laboratoire d Energétique, Université Paul Sabatier, Toulouse, France CHRISTOPHE HOURDIN* AND FRÉDÉRIC HOURDIN Laboratoire de Météorologie Dynaique, Institut Pierre Sion Laplace, CNRS, and Université Pierre et Marie Curie, Paris, France (Manuscript received 6 October 2004, in final for 16 February 2005) ABSTRACT The net exchange forulation (NEF) is an alternative to the usual radiative transfer forulation. It was proposed by two authors in 1967, but until now, this forulation has been used only in a very few cases for atospheric studies. The ai of this paper is to present the NEF and its ain advantages and to illustrate the in the case of planet Mars. In the NEF, the radiative fluxes are no longer considered. The basic variables are the net exchange rates between each pair of atospheric layers i, j. NEF offers a eaningful atrix representation of radiative exchanges, allows qualification of the doinant contributions to the local heating rates, and provides a general fraework to develop approxiations satisfying reciprocity of radiative transfer as well as the first and second principles of therodynaics. This ay be very useful to develop fast radiative codes for GCMs. A radiative code developed along those lines is presented for a GCM of Mars. It is shown that coputing the ost iportant optical exchange factors at each tie step and the other exchange factors only a few ties a day strongly reduces the coputation tie without any significant precision lost. With this solution, the coputation tie increases proportionally to the nuber N of the vertical layers and no longer proportionally to its square N 2. Soe specific points, such as nuerical instabilities that ay appear in the high atosphere and errors that ay be introduced if inappropriate treatents are perfored when reflection at the surface occurs, are also investigated. 1. Introduction In the past decades, nuerical odeling of the atospheric circulation of Mars has been taking on an increased iportance, in particular in the fraework of the spatial exploration of the red planet (Leovy and * Current affiliation: Laboratoire d Océanographie Physique, Museu National d Histoire Naturelle, Paris, France. Corresponding author address: Jean-Louis Dufresne, Laboratoire de Météorologie Dynaique, Institut Pierre Sion Laplace, Université Pierre et Marie Curie, Boite 99, F Paris Cedex 05, France. E-ail: dufresne@ld.jussieu.fr Mintz 1969; Pollack et al. 1981; Hourdin et al. 1993; Forget et al. 1998). With increased nubers of issions to Mars and especially with the use of aero assistance for orbit injection, there is an increasing deand for iproveents of our knowledge of Martian physics, in particular of the Martian upper atosphere. Coputation of radiative transfer is a key eleent in the odeling of atospheric circulation. Absorption and eission of visible and infrared radiation are the original forcing of atospheric circulation. With typical horizontal grids of a few thousands to ten thousand points, and since we want to cover years with explicit representation of diurnal cycle, operational radiative codes ust be extreely fast. Representation of radiative transfer ust therefore be drastically siplified and paraeterized Aerican Meteorological Society

2 3304 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 For Mars, the ain contributors to atospheric radiation are by far carbon dioxide (which represents about 95% of the atospheric ass) and airborne dust particles (even outside large planetary-scale duststors, extinction of solar light by dust is several tens of percent). Carbon dioxide is doinant at infrared frequencies with a vibration rotation line spectru, which ust be properly accounted for. In the developent phase of the Laboratoire de Météorologie Dynaique (LMD) Martian atospheric circulation odel, a ajor step was the derivation of a radiative transfer code for the CO band (Hourdin 1992). This odel was based upon the wide-band odel approach developed by Morcrette et al. (1986) and used in the operational odel of the European Centre for Mediu-Range Weather Forecasts (Morcrette 1990). This odel is based on a two-strea flux forulation. Wide-band transissivities are fitted as Padé approxiants (ratio of two polynoials) as functions of integrated absorber aounts, including siple representations of teperature and pressure dependencies. For application to Martian atosphere, the fit was soewhat adapted in order to account for Doppler line broadening, which becoes significant above 50 k. Altogether, in standard configurations of the LMD Martian odel with a 25-layer vertical discretization, infrared coputations represent a significant part (up to one-half) of the total coputational cost. Siilar reports are ade concerning terrestrial odels. The transission functions being not ultiplicative for band odels, the deterination of radiative fluxes at each level requires independent calculations of the contributions of all atospheric layers. The corresponding coputation cost increases as the square of the nuber of vertical layers. This quadratic dependency undoubtedly represents a severe liitation when thinking of further odel refineents, in particular as far as near-surface and high-atosphere processes are concerned, both requiring significant vertical discretization increases. However, it is coonly recognized that, despite of this foral difficulty, infrared radiative transfers are doinated by a few ters such as cooling to space and short distance exchanges (e.g., Rodgers and Walshaw 1966; Fels and Schwarzkopf 1975). In practice, the quadratic dependency of absorptivity eissivity ethods is too costly with respect to the contributions of the various ters. In standard flux forulations, it is difficult to quantify the relative iportance of the various contributions to the local heating rates because the individual contributions are not identified as such in the foralis. Green (1967) suggested that a reforulation of radiative transfers in ters of net exchanges allows quantifying the relative iportance of physically distinct contributions to the local heating rates and could help design ore efficient odels. In Green s approach, called here the net exchange forulation (NEF), the quantity under consideration is directly the net energy exchanged between two atospheric layers (or ore generally two surfaces or gas volues). Joseph and Bursztyn (1976) attepted to use the net exchange approach to copute radiative exchanges in the terrestrial atosphere. Despite soe nuerical difficulties, they showed that radiative net exchanges between an atospheric layer and boundaries (space and ground) are doinant although the net exchanges with the rest of the atosphere are not negligible as they contribute to approxiately 15% of the total energy budget. With NEF, Bresser et al. (1995) did elaborate analytical developents for particular cases in order to copute the radiative daping of gravity waves. The well-known Curtis atrix (Curtis 1956; see, e.g., Goody and Yung 1989) ay be related to NEF, but in the Curtis atrix approach, one-way exchanges are considered instead of net exchanges, which eans that useful properties of NEF, such as the strict siultaneous satisfaction of energy conservation and reciprocity principle are abandoned. Fels and Schwarzkopf (1975) and Schwarzkopf and Fels (1991) take advantage of the iportance of the cooling to space to develop an accurate and rapid longwave radiative code. They do not use the NEF but their work ay be easily understood in the net exchange fraework. Siilar developents were also otivated by various engineering applications. Hottel s ethod (Hottel and Sarofi 1967), also naed the zone ethod, is originally based on NEF. However, difficulties were encountered considering ultiple reflection configurations and the NEF syetry was practically abandoned. Cherkaoui et al. (1996, 1998), Dufresne et al. (1998), and De Lataillade et al. (2002) showed that NEF can be used to derive efficient Monte Carlo algoriths. Dufresne et al. (1999) used NEF to identify and analyze doinating spectral ranges, ephasizing the contrasted behavior of gas gas and gas surface exchanges. Finally, this forulation was recently used to analyze longwave radiative exchanges on Earth with a Monte Carlo ethod (Eyet et al. 2004). In the present paper, we show how NEF can help derive efficient operational radiative codes for circulation odels. This code is now operational in the general circulation odel developed jointly by Laboratoire de Météorologie Dynaique and the University of Oxford (Forget et al. 1999). As an exaple, this circulation odel has been used to produce a cliate database for

3 SEPTEMBER 2005 D U F R E S N E E T A L Mars for the European Space Agency (the database is accessible both with a FORTRAN interface for engineering and online at ars.htl; Lewis et al. 1999). In section 2, the NEF is presented in the specific case of stratified atospheres and analyses are perfored for typical Martian conditions. Section 3 discusses the questions related to operational radiative code derivations, in particular those related to vertical integration procedures and reflections at the surface. The tie-integration schee is considered in section 4, first by investigating the nuerical instabilities that ay occur in the high atosphere, then by finding how coputer tie ay be saved without loosing accuracy. Suary and conclusions are in section Net exchange forulation a. General approach Longwave atospheric radiative codes are generally based on flux forulations. Angular integration of all intensities at each location leads to the radiative flux field, q R the divergence of which gives the radiative budget of an eleentary volue dv M around point M as dq div q R dv M. In an exchange forulation, the voluic radiative budgets are addressed directly without explicit forulation of the radiative intensity and radiative flux fields. Corresponding forulations include coplex spectral and optico-geoetric integrals that ay coe down to (Dufresne et al. 1998) dq dv M 0 1 dv d d A K M K P, B P B M. P M,P In this expression, is the frequency, A represents the entire syste (for an atosphere, the entire atosphere plus ground and space boundaries), and M,P is the space of all optical paths joining locations M and P. For each optical path, is the spectral transission function along the path; B P and B M are the spectral blackbody intensities at the local teperatures of P and M, and K M is the absorption coefficient in M. The differential dv P around location P is either an eleentary volue or an eleentary surface and K P, is either the absorption coefficient in P (if P is within the atosphere) or the directional eissivity (if P is at the boundaries). Expressed this way, the radiative budget of eleentary volue dv M can be seen as the difference of two 2 ters: the radiative power absorbed by dv M coing fro the whole atosphere plus surface and space [B P part of Eq. (2)] inus the power eitted by dv M toward all other locations (B M ter). When separated this way, the equation can be siplified further noticing that the B M part (total eission of volue dv M ) reduces to 4 0 KMB M d. This approach is the current basis for engineering zone ethod and Curtis atrix (see, e.g., Goody and Yung 1989). In NEF, Eq. (2) is rather interpreted keeping the foral syetry as the su of the individual net exchanges between volue dv M and all other eleentary volues or surfaces (including space in the case of an atosphere). An individual spectral net exchange rate between dv M and dv P, dv M, dv P dv M dv d K M K P, B P B M, P M,P is just the power eitted by dv P and absorbed by dv M inus that eitted by dv M and absorbed by dv P. For a discretized atosphere, the spectral net exchange rate between two eshes i and j reads i, j dv M, dv P, 4 Ai Aj where A i and A j are the volues or surfaces of eshes i and j. The spectral radiative budget i of esh i is the su of the net exchange rates between i and all other eshes j: i j i, j. A very specific feature of this forulation lies in the fact that both the reciprocity principle, the energy conservation principle and the second therodynaic principle ay be strictly satisfied whatever the level of approxiation is retained to solve Eqs. (3) and (4). The reciprocity principle states that the light path does not depend on the direction in which light propagates, which eans that the integrals of the optical transission over both the optical path space M,P and over the reciprocal space P,M are the sae: M,P d P,M d. Using Eq. (3), the reciprocity principle reduces to (dv M, dv P ) (dv P, dv M ). This condition ay be satisfied provided that the sae coputation is used for both (dv M, dv P ) and (dv P, dv M ). In other words, when photons eitted by dv M and absorbed by dv P are counted as an energy loss for volue dv M, the sae approxiate energy aount is counted as an en

4 3306 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 ergy gain for dv P. As a direct consequence, the energy conservation principle is also satisfied. Finally, provided that the difference (B P B M) that appears as such inside the optical path integral is preserved, Eq. (3) ensures that warer regions heat colder regions in accordance with the second therodynaic principle. Altogether, the NEF allows the derivation of approxiate nuerical schees strictly satisfy the reciprocity principle, the energy conservation principle, and the second therodynaic principle. Any approxiation ay be retained for the integration over the optical path doain without any risk of inducing artificial global energy sources or nonphysical energy redistributions. b. Application to the CO band in the Martian context After these general considerations, we illustrate the net exchange approach in the case of the CO band on Mars. At this first stage, we ake the following siplifying assuptions: the atosphere is perfectly stratified along the horizontal (plan parallel assuption), the surface is treated as a blackbody (eissivity 1), and the atosphere is assued dust free. Under these assuptions, the space of relevant optical paths reduces to straight lines between exchange positions. Dividing the atosphere into N layers, spectral net exchange i,j between layer i and layer j can be derived fro Eqs. (3) and (4) as Z i 1 2 i,j Zi 1 2 Z j 1 2 Z j B zj B zi k z i z i k z j z j z j k z z exp z i cos dz tan d dz i dz j, 7 where Z i (1/2) and Z i (1/2) are the altitudes at the lower and higher boundaries of atospheric layer i, is the gas density, k is the spectral absorption coefficient, and is the zenith angle. The previous equation ay be rewritten as Z i 1 2 Z j 1 2 i,j B Zi 1 2 Z zj B zi j ϒ z i, z j z i z j dz i dz j, 8 where ϒ is the spectral integrated transission function defined as 2 ϒ z, z 2 0 z k x x exp z cos dx sin cos d. With the sae assuptions, the spectral net exchange rate i,b between layer i and ground or space can be derived fro Eq. (3) as Z i i,b 2 B Zi T b B zi k z i z i z b k z z exp z i cos dz sin d dz i, Z i 1 2 i,b B T b B zi Zi with T b T s for exchanges with the planetary surface (at teperature T s ) and T b 0 K for cooling to space. This equation ay be rewritten as ϒ z i, z b z i dz i. 11 This last equation is well known as it is coonly used to copute the cooling to space. In practice, net exchange coputations require angular, spectral, and vertical integrations. In the present study, the following choices are ade. 1) As in ost GCM radiative codes, the angular integration is coputed by applying the diffusive approxiation, which consists in the use of a ean angle (1/cos 1.66, see Elsasser 1942). 2) As in the original Martian odel, the spectral integration is replaced by a band odel approach in which the Planck functions and wide-band transissivities are separated (Morcrette et al. 1986; Hourdin 1992). 1 3) The vertical integration is what we concentrate on in section 3 with various levels of approxiation. 1 This approach is exact for a spectral interval narrow enough to use a constant value of the Planck function. For larger intervals, teperature variations affect the correlation between the gas absorption spectru and the Planck function. Following Morcrette et al. (1986), this effect was accounted for in the original odel by using different sets of fitting paraeters for the transission function depending upon the teperature of the eitting layers. Here, we only use one set of paraeters and control tests indicated that this siplification has a negligible effect on the estiated radiative heat sources.

5 SEPTEMBER 2005 D U F R E S N E E T A L With the diffusive approxiation and the use of wideband transitivities, the net exchange i,j becoes, after spectral integration of Eq. (8) over wide bands, Z i 1 2 i,j Z i 1 2 with Z j 1 2 Z j 1 2 B zj B zi zi, z j dz i dz j, 12 zi, z j 2 z i, z j, 13 z i z j where (z i, z j ) is the wide-band transissivity between z i and z j with a ean angle and is the bandwidth. The net exchange i,b between layer i and boundary b (ground or space) becoes, after spectral integration of Eq. (11), with z i 1 2 i,b z i 1 2 B T b B zi zi, z b dz i, 14 zi, z b z i, z b. 15 z i Finally the net exchange s,e between the ground surface s and space e becoes with s,e c. Reference case B T s zs, z e, 16 zs, z e z s, z e. 17 We first present a coputation of net exchanges on a typical 25-layer GCM grid (Table 1), with refined discretization near the surface. To avoid probles with the vertical integration for this reference coputation, exchanges are first coputed on an overdiscretized grid of 500 layers (Fig. 1), each layer of the coarse GCM grid corresponding exactly to 20 layers of the 500-layer grid. An exchange between two atospheric layers of the coarse grid is siply obtained as the su of the exchanges fro the finer grid. We use a reference teperature profile (Fig. 1) derived fro the easureent taken by two Viking probes during their entry in the Martian atosphere (Seiff 1982) and already used by Hourdin (1992). In the upper atosphere, there is no systeatic teperature increase, as there is no significant solar radiation absorption equivalent to that the ozone layer on Earth. In the iddle atosphere, gravity waves and theral tides disrupt the teperature profile. Near the surface, the quasi-isotheral part of this averaged profile hides a strong diurnal cycle. The surface pressure is fixed to 700 Pa. d. Net exchange atrix NEF offers a eaningful atrix representation of radiative exchanges. A graphical exaple of such a atrix is shown in Fig. 2. Each eleent displays the net exchange rate i, j for a given pair i, j of eshes converted in ters of heating rate: i,j i,j g 1, Cp t p i 18 where g is the gravity, Cp the gas ass heat capacity, and t the length of the Martian day ( t s). For the ground, the heating rate is arbitrary coputed using a theral capacitance of 1JK 1 2 day 1. The total heating rate of a layer i is i j i,j, 19 where i, j and j,i are of opposite sign but the exchange atrices expressed in K day 1 are not antisyetric as TABLE 1. Low-resolution (i.e., 25 layers) vertical grid characteristics: layer nuber, levels ( P/P s, with P s the surface pressure), and approxiate corresponding heights. Layer No. Approx height () Layer No. Approx height (k) Layer No. Approx height (k)

6 3308 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 As an exaple of reading Fig. 2, consider layer i 10 (arked in the figure). The teperature profile and the total heating rate i are also plotted on both sides. The horizontal line of the atrix shows the decoposition of the heating rate in ters of net exchange contributions [see Eq. (19)]. This partitioning of the heating rates first ephasizes soe well-established physical pictures. The cooling to space is the doinant part of the heating rate: it essentially defines the general for and the order of agnitude of the heating rate vertical profile. This well-known property has been widely used to derive approxiate solutions in atospheric context (e.g., Rodgers and Walshaw 1966; Fels and Schwarzkopf 1975; Schwarzkopf and Fels 1991). Internal exchanges within the atosphere are by far doinated by the exchanges with adjacent layers (note that there is a factor of 3 between two consecutive colors). As a consequence, the net exchange atrix is very sparse. A very few ters doinate all the others. These iportant ters are the exchanges with boundaries (space and surface) and the exchanges with adjacent layers. Thanks to the NEF, the relative agnitude of these ters can be quantified. 2) THERMAL ASPECTS In each spectral band, each contribution i, j to the total net exchange rate i, j is the integral of the product of two ters: the blackbody intensity difference between z i and z j and the optical exchange factor zi,z j FIG. 1. Teperature vertical profile (K, botto axis, continuous line) with 500 layers (thin line) and 25 layers (bullet, thick line). Each teperature of the 25-layer grid is the ean of 20 layers of the high-resolution grid (finite volue representation). The reference heating rate (K day 1, top axis, dotted line) is also displayed for the two grids. they would be if expressed in W 2 ( i,j j,i but i,j j,i ). 1) MATRIX CHARACTERISTICS [e.g., Eq. (12)]. The sign of the net exchange rate i,j only depends on the teperature difference between i and j as the optical exchange factor zi,z j is always positive. Layer i heats layer j only if its teperature T i is greater than T j. The direct influence of the teperature profile on the exchange atrix can be seen on Fig. 2. For instance layer 10 is heated by the warer underlying atosphere and surface and looses energy toward the colder layers above and toward space. The picture is ore coplex in the upper atosphere where strong teperature variations are generated by atospheric waves. In this region, a layer can be heated by both adjacent warer layers (e.g., layer 20). In particular these radiative exchange between adjacent layers are known to dap the possible teperature oscillations due to atospheric waves (Bresser et al. 1995). Finally, the gas radiative properties depend uch less on the teperature than the blackbody intensity. Therefore the etric of optical exchange factors ay be assued as constant for qualitative exchange analysis, and even to soe extent for practical coputations as discussed later on. 3) SPECTRAL ASPECTS The exchange factors between two eshes are proportional to the gas transission for the exchange between surface and space [Eq. (17)], proportional to the first derivative of for the exchanges between a layer and surface or space [Eq. (15)] and proportional to the second derivative of for the exchanges between two atospheric layers [Eq. (13)]. The behavior of optical exchange factors can be understood by analyzing these three functions. To allow coparison, we noralize the by the product of the eissivity at both extreities. If the extreity i is a gas layer of differential thickness dz i, the eissivity is

7 SEPTEMBER 2005 D U F R E S N E E T A L FIG. 2. Graphical representation of radiative net exchange rates in the Martian atosphere: (left) teperature profile, (iddle) net exchange atrix, and (right) heating rate. The vertical axis is the layer nuber. Sae conditions as in Fig. 1. zi dz k v z i z i d. i v 20 If the extreity i is ground or space, i 1. We also use a noralized integrated ass X of atosphere X z g P s s z z dz, 21 where P s is the ground pressure. The wide-band odel used in this study has two spectral bands chosen epirically (Hourdin 1992). The first one (band 1), ranging fro 635 to 705 c 1, corresponds to the central part of the CO band. The second one (band 2), ranging fro 500 to 635 c 1 and fro 705 to 865 c 1 corresponds to the wings. The three noralized functions,, 1/ zi / X and 1/( zi zj ) 2 / X 2, that ay also be seen as noralized exchange factors, are displayed in Fig. 3 for the two bands. The three noralized exchange factors go to 1 when X goes to 0. Indeed, when the two extreities are adjacent, the exchange factor is the product of the eissivity at both extreities. 2 When X increases, the noralized exchange factor slowly decreases (in particular for band 2) if the two 2 This is only true in the liit where the gas layer(s) is (are) optically thin. In the exaple presented here, the eissivities are coputed for layers with a noralized thickness X extreities are black surfaces (here space and ground); it decreases faster if one extreity is a gas layer and decreases even faster if the two extreities are gas layers (Fig. 3). This is an illustration of the so-called spectral correlation effect (e.g., Zhang et al. 1988; Modest 1992; Dufresne et al. 1999). For the exchange between FIG. 3. Gas transission (solid) and its two first derivatives noralized by the eissivity of the gas layer, (1 / ) / X (dash) and (1/ 2 ) 2 / X 2 (dot), as a function of the noralized integrated ass of atosphere X (P P s )/P s, for (upper) the central part (band 1) and (lower) the wings part (band 2) of the CO band. The atospheric teperature is assued unifor (T 200 K) and the surface pressure is P s 700 Pa.

8 3310 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 FIG. 4. Graphical representation of the net exchange atrix for spectral bands (left) 1 and (iddle) 2. Sae 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 spectru (open circles). two gas layers, both absorption and eission are at a axiu in spectral regions near the center of the absorbing lines. But exactly at the sae frequencies gas absorption creates a strong decrease of the transission when the distance between extreities increases. Thus the exchange strongly decreases with distance. On the contrary, the exchange between ground and space is ost iportant in spectral regions where the spectral transission is high, that is where the gas absorption is low. Thus the exchange factor between ground and space is uch less sensitive to the integrated air ass between the. Exchange between a layer and ground or space is an interediate case. When X increases, the decrease of the three noralized exchange factors is faster for the central part of the CO 2 band (band 1) than for the wings (band 2) (Fig. 3). As a consequence, the decrease with distance of the exchange between two atospheric layers is ore iportant in band 1 (left panel of Fig. 4) than in band 2 (right panel of Fig. 4). The exchanges between adjacent layers are uch greater for band 1 than for band 2, whereas distant exchanges have the sae agnitude for the two bands. For the cooling to space, the copetition between the decrease of / X and the increase of the local blackbody intensity yields noticeably different vertical profiles: The absolute value of the cooling to space decreases when the layer is closer to the surface for band 1 whereas it increases for band Vertical integration In the above reference coputation, net exchanges have been coputed using given subgrid scale teperature profiles (Fig. 1). In practice, with circulation odels, only ean teperatures are known for each layer 3 and assuptions are required concerning subgrid scale profiles. First, we present the very siple assuption of a unifor teperature within each atospheric layer. This allows us to highlight the link between NEF and flux forulations. Then we address the ore general case of nonisotheral layers and finally we highlight the odifications that are required in the case of a reflective surface. a. Isotheral layers For isotheral layers, the individual exchanges contributing to the radiative budget of layer i ( i j i,j ) take a siple for [Eqs. (12) and (13)], reducing to with i,j i,j i 1 2, j 1 2 i 1 2, j 1 2. i,j B j B i, i 1 2, j 1 2 i 1 2, j In the equivalent flux forulation, the individual contributions of the radiation eitted by layer i to the flux at interface j We assue that circulation odels ake use of finite volue representations and that GCM outputs are representative of ean teperatures rather than idlayer teperatures.

9 SEPTEMBER 2005 D U F R E S N E E T A L F i j 1 2 B i i 1 2,j 1 2 i 1 2,j are first sued over i to copute the radiative flux 1 F j at each interface j 1. The radiative budget of each 2 2 layer j then reads j F j 1 2 F j 1 2 i F i j 1 2 i F i j An exact equivalence between the net exchange and flux forulations is obtained by noting that i,j F j i 1 2 F j i 1 2 F i j 1 2 F i j and using the property i, j j,i. For developing a radiative code, NEF however presents advantages. With flux forulation, fluxes are first integrated over altitude z and then differentiated. When teperature contrasts are weak (here near the surface for instance), net exchange rates can be by orders of agnitude saller than fluxes. Coputing first the fluxes and then the differences ay lead to strong accuracy loss, which we observed could induce reciprocity principle violations (colder layers heating warer layers for instance). 4 In Fig. 5, we show the error on net exchanges if the isotheral approach is retained for all exchanges with respect to the reference 500-layer siulation. The error is very large around the diagonal and often larger than the exchange itself. Indeed, at frequencies where significant CO 2 eission occurs (close to absorption lines centers) the atosphere is extreely opaque, which eans that ost eitted photons have very short path lengths copared to layer thicknesses. Consequently, exchanges between adjacent layers are ainly due to photon exchanged in the iediate vicinity of the layer interface. In this thin region, teperature contrasts are uch weaker than the differences between ean layer teperatures. The isotheral approxiation thus results in a strong overestiation of the net exchanges. The relative error due to the isotheral hypothesis strongly decreases with distance between layers. This feature is further coented in appendix. 4 This proble ay be partially overcoe in the flux forulation by introducing the blackbody differential fluxes F B F (e.g., Ritter and Geleyn 1992). FIG. 5. Error atrix with the isotheral layer assuption. Line i colun j gives the error in K day 1 for the heating rate of layer i due to its exchange with layer j. The error is coputed with respect to the reference coputation perfored with 20 sublayers inside each layer. Other conditions are the sae as in Fig. 2. b. Net exchanges between adjacent layers The specific difficulty of exchange estiations in the case of adjacent layers is coonly identified and solutions have been ipleented in flux coputation algoriths (e.g., Morcrette et al. 1986). In ost GCMs, only the average layer teperatures and copositions are available. Here a linear approxiation is retained for B to describe the atosphere close to the esh interface. Because of the syetry of Eq. (8) in z i and z j, the linear approxiation is strictly equivalent to a quadratic approxiation in the liit case of two layers of identical thicknesses (see the appendix). When coputing i,i 1 we therefore assue that B(z) is linear between z i 1/2 and z i 3/2, satisfying the following constraints for j i 1 (Fig. 6): z j 1 2 B z dz B j z j. z j Note that in this approach, the assued teperature profile inside a layer is different when coputing the exchange with the layer just above or just below. With this assuption, exact integration procedures could be designed, for instance, using the analytical solution available for the best fitted Malkus transission function (Dufresne et al. 1999), or integrating by parts and tabulating integrated transission function fro line by line coputations. Here we test a ore basic solution by dividing the linear profile into isotheral sublayers, with thinner sublayers closer to the in-

10 3312 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 FIG. 6. A linear blackbody intensity profile approxiation is used for coputation of the net exchanges between adjacent layers. The blackbody intensity profile inside a layer is different when coputing the exchange with the layer just above (black line) or just below (gray line). The subgrid discretization is also shown (thin lines). FIG. 7. Vertical profile of the heating rate error in K day 1 due to the (left) vertical integration schee, and part of this error due to the coputation of the net exchange with (iddle) adjacent layers (black circles) and distant layers (open circles), and (right) ground (black circles) and space (open circles). The error is coputed with respect to the reference coputation perfored with 20 sublayers inside each layer. Other conditions are the sae as in Fig. 2. The vertical axis is the layer nuber. terface (Fig. 6). The subdiscretization schee was tested against reference siulations. For the present application, a satisfactory accuracy is reached with a subdiscretization into three isotheral sublayers of increasing thicknesses ( z/7, 2 z/7, and 4 z/7) away fro the interface. Whatever the integration procedure, a direct consequence of the previous linear blackbody intensity assuption is that the net exchange between two adjacent layers ay be still written forally like the net exchange between isotheral layers [Eq. (22)]. Only the expression of the exchange coefficient i,i 1 depends on the teperature profile hypothesis. We finally adopt the following solution for the vertical integration: the above subdiscretization into three isotheral sublayers is used to copute the radiative exchanges between adjacent layers whereas the siple isotheral layer assuption is used to copute the exchange between distant layers. For the exchange between ground and first layer, we assue the teperature of gas just above the surface to be T 0 (T 1 T s )/2 and a linear B profile between T 1 and T 0. An isotheral description is retained for the exchange between the optically thin upper layer and space. The global error due to the vertical integration schee, as well as the origin of the error, are displayed Fig. 7. The analytical expression of these errors is presented in an appendix for soe cases. One should have in ind that results are copared with a high-resolution vertical grid where the teperature profile has a ore precise description than in the low resolution grid. c. Exchanges with reflection at the surface The above presentation assues that the surface behaves as a blackbody. In practice, surface eissivity can differ fro 1. The ean eissivity of the Martian surface is believed to be of the order of 0.95 (Santee and Crisp 1993), and eissivity is believed to be lower in soe regions (Forget et al. 1995). When reflection at the surface is present, two atospheric layers can exchange photons, either directly, or through reflection at the surface. For instance, the net exchange between two atospheric layers i and j [Eq. (12)] becoes z i 1 2 i,j z i 1 2 z j 1 2 z j 1 2 s z i, z j dz i dz j, B zj B zi d z i, z j 28 where d is the optical exchange factor for direct exchanges d z i, z j 2 z i, z j z i z j 29

11 SEPTEMBER 2005 D U F R E S N E E T A L FIG. 8. (left) Vertical profile of the heating rate when the surface is perfectly black (cross, continuous line) and when the surface has an eissivity s 0.9, the coputation being either exact (circle, dash line) or neglecting spectral correlation when reflection at the surface occurs (square, dotted line). Sae atospheric conditions as in Fig. 1. The vertical axis is the layer nuber. (right) Sae as left, but with a perfectly reflecting surface ( s 0). and s the optical exchange factor through reflection at the surface: s z i, z j 1 s 2 s z i, z j, 30 z i z j where s is the surface eissivity and s (z i, z j )isthe transission function fro z i to z j via the surface for a spectral interval. Assuing the diffusive approxiation, this transission writes s z i, z j z i,0 0, z j d. 31 In the original flux forulation, as well as in other radiative codes based on the so-called absorbtivity/ eissivity ethod, the downward flux is first integrated fro the top of the atosphere to the surface. The reflected part of this downward flux is then added to the flux eitted by the gray surface. This flux is then used as a liit condition to integrate the upward flux up to the atospheric top. This assuption corresponds to the following approxiation: s z i, z j z i,0 0, z j, 32 which is wrong for wide and narrow band odels because the spectral inforation is forgotten at the surface. The error on the heating rate is particularly strong for the layers near the surface. For a surface eissivity of 0.9, as expected, the exact solution displays sall changes in the heating rate copare to the case where the surface is black (plus signs and circles in the left on Fig. 8). On the other side, the coputation that neglects the spectral correlation at the surface (squares) displays a very large and unrealistic change of the heating rate near the surface. A useful property can be used to check the results with a reflective surface. Let us consider an atosphere with a thin layer near the surface having the sae teperature as the surface itself. This layer will never exchange energy with the surface because both are at the sae teperature. If the surface eissivity differs fro 1, the exchange of this layer with the atosphere above and with the space will be increased through reflection at the surface. For an optically thin layer

12 3314 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 and for a perfect irror ( 0) all those exchanges, and hence the radiative cooling, will be exactly twice that without reflection ( 1) (Cherkaoui et al. 1998). The net exchange coputation fulfills this property (plus signs and circles on the right-hand side of Fig. 8) but the original flux odel does not (square). The above coputations were perfored with a prescribed vertical teperature profile. To evaluate the error associated to incorrect treatent of reflection when all the physical processes (radiation, turbulent vertical ixing...) are active, we present hereafter results obtained with a 1D odel that corresponds to a single vertical colun of the 3D GCM. When the teperature profile is prescribed, a decrease of the surface eissivity reduces the cooling of the surface but increases the cooling of the atosphere above (Fig. 8). With the full 1D odel, a decrease of the surface eissivity reduces the cooling of the surface, which increases its teperature (Fig. 9). The teperature of the atosphere above also increases, but less, as the decrease of eissivity increases the cooling of the atosphere. When the teperature profile is prescribed, we previously noticed that neglecting the spectral correlation at the surface leads to strongly overestiate the atospheric cooling just above the reflective surface (Fig. 8). With the 1D odel, neglecting the spectral correlation leads to underestiate by a factor of the teperature increase in the boundary layer due to the eissivity decrease (Fig. 9). This underestiate is even ore iportant if the turbulent vertical ixing is neglected (not shown). Neglecting the spectral correlation also slightly increases the diurnal cycle of the atospheric teperature near the surface (not shown). d. Coputing vertical fluxes In a general way there is no direct relationship between upward and downward fluxes and net exchanges. Only the net radiative fluxes ay be directly expressed as a function of net exchanges. For instance, the net fluxes at the top of atosphere F n N 1/2 reads FIG. 9. Vertical profile of the daily ean teperature difference due to a change in the surface reflectivity with an exact coputation (open circle) and neglecting spectral correlation at the surface (closed circle). The teperatures of a run with a slightly reflective surface ( s 0.9) are copared to a run with a nonreflective surface ( s 1). The runs are 10 days long, have the sae initial state, and are perfored with the single-colun version of the Martian GCM. Diurnally averaged teperature differences are plotted for the last day. The vertical axis is the atospheric layer nuber. The optical exchange factors present in Eq. (34) are coparable to the so-called weighting functions used to invert satellite radiative easureents. Therefore NEF should be a useful fraework to assiilate those easureents in GCMs. In the atosphere, the net flux at level i 1 is equal 2 to the net exchange between all the eshes below i 1 and all the eshes above i 1 : 2 2 n F N 1 2 N k 0 N 1,k 33 n F i 1 2 N 1 j i 1 i k 0 j,k. 35 N k 0 N 1,k B k, 34 where N is the nuber of vertical layers, k 0 stands for ground, and k N 1 stands for space. If one really wants the values of the upward and downward fluxes, one ay be approxiated assuing each atospheric layer is isotheral. If the optical exchange factors have been coputed with this assuption, the upward flux at level i 1 is equal to the flux eitted by 2

13 SEPTEMBER 2005 D U F R E S N E E T A L all the eshes below i 1 and absorbed by all the 2 eshes above i 1 : 2 i N 1 F i 1 2 j 0 j,kb j. k i 1 36 Note that the errors on fluxes arising fro the isotheral hypothesis are uch saller than the error on net exchanges between adjacent layers due to the sae hypothesis. The sae way, the downward flux at level i 1 is equal to the flux eitted by all the eshes above 2 i 1 and absorbed by all the eshes below i 1 : Tie integration i N 1 F i 1 2 j 0 j,kb k. k i 1 37 a. Nuerical instabilities in the high atosphere When the atospheric vertical resolution increases, nuerical instabilities appear in the Martian GCM in the high atosphere and they ay increase draatically. This proble has also been encountered in soe GCM of the Earth atosphere and specific stabilization techniques are coonly used to bypass this difficulty. Here we analyze the reasons of this difficulty and we propose a solution that takes advantage of the NEF. In the original Martian odel, the radiative transfer is integrated with an explicit tie schee, the evolution between ties t and t t of the teperature of layer i being coputed fro a coputation of the heating rate t i j t ij(b t j B t i) at tie t as T t t i T t i t i t. i C p 38 In the upper atosphere, the ass i of the atospheric layers becoes very low, reducing they theral capacitance. As a consequence, strong nuerical oscillations appear for large tie steps if the variations of i with teperature, within the tie step, are not taken into account. A well-known solution to this proble consists in replacing t i in Eq. (38) by ( ) i (1 ) t i t t i. With those notations, the teporal schee covers the cases of explicit ( 0), iplicit ( 1) and seiiplicit ( 1/2) schees. For 0, the schee is no ore explicit and requires an inversion procedure. The net exchange foralis offers a siple practical solution to this proble. Based on the analysis above, it can be assued that only the blackbody eissions B i vary during the tie step while the optical coefficients do not. Also it can be assued that only the exchanges with adjacent layers (i 1) and boundaries (b) vary while the exchanges with distant layers are unodified. With these approxiations, and after linearization of the Planck function, i t i t db dt T j T t t j T t j j i 1, b ij db dt T i T i t t T i t. 39 If in addition we do not consider the variation of T b within the tie step (which is exact for space and not a proble for the surface when coputing the heating rates in the upper atosphere), the teperature at tie t t is obtained fro that at tie t through the inversion of a tridiagonal atrix, for a low coputational cost. This approach, ipleented in the Martian GCM with 1/2, is very efficient and suppresses all the nuerical oscillations in the upper atosphere. b. Saving coputer tie The coputation cost of the LW radiative code is known to be very iportant in ost GCMs. Solutions have been proposed and ipleented to reduce this coputation tie. Generally the full radiative code is coputed only one out of N tie steps, and approxiations are used to interpolate the LW cooling rates between those N tie steps. The siplest tie interpolation schee is to aintain constant the cooling rates during this period. This is the case in the original Martian GCM where the radiative code is coputed one out of two tie steps (each 1 h). On Mars, the surface teperature diurnal cycle is as high as 100 K and the tie interpolation ethod has to reproduce the effects of this diurnal cycle. The NEF provides an easy answer to this proble. Since the Planck function doinates the variations of the radiative exchanges, the Planck function will be coputed at each tie step while coputing the optical factors i,j only one out of N tie steps. A second level of optiization consists in coputing the optical exchange factors corresponding to the ost iportant exchange rates (see section 2d) ore frequently than the others. Once again, the NEF ensures that the above approxiation will not alter the energy conservation and the reciprocity principle (section 2a). Practically all the optical exchange factors are scattered in three groups: the exchange factors between each atospheric layer i and 1) its adjacent atospheric layers, 2)

14 3316 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 the distant atospheric layers (i.e., the other atospheric layer), and 3) the boundaries (i.e., surface and space). We present nuerical tests perfored using the single colun version of our GCM. The runs last 50 days and the coparison between runs is perfored using the results of the two last days. For these two days, we coputed the atospheric teperature difference between each run and the reference run. The ean and the rs of this difference allows a quick coparison between the (Table 2). In the reference run (case 1), the full LW code is called at each tie step of the physics, that is, every 30 Martian inutes). Case 2 corresponds to what was ipleented in the original version of our GCM: the full LW code is called one out of two tie steps of the physic that is, every Martian hour and the LW cooling rates are constant during this period. Coputing all the optical exchange factors only once a day (case 3) leads to an error only slightly greater than coputing all the LW radiative code one out of two tie steps (case 2), but requires a uch saller CPU tie. This result illustrates that the diurnal variations of the optical exchange factors are not very iportant copared to the diurnal variations of the Planck function. As entioned above, one ay copute the ost iportant exchange factors ore frequency than the other. Coputing the optical exchange factors with boundaries at each tie step highly reduces the error while it only slightly increases the coputation tie (case 4). Coputing the exchange factors with adjacent layers at each tie step slightly reduces the error while strongly increasing the coputation tie (case 5). Coputing exchange factors only once a day ay introduce a significant bias for long-ter siulations as the diurnal cycle is very badly sapled. We choose to copute all the exchange factors at least four ties a day (cases 7 10). Coputing the exchange factors with boundaries at each tie step (30 ) and the other exchange factors every 6 h (case 8) produces uch saller errors than the original solution (case 2) while being two ties less consuing. Another iportant advantage of this solution is that the nuber of exchange factors with boundaries increases linearly with the nuber N of vertical layers. The nuber of the other exchange factors are still proportional to N 2 but they are coputed uch less frequently and the required coputation tie is therefore negligible: the CPU tie will increase alost linearly with N and no ore as a function of N 2 as for all the absorptivity eissivity ethods. If higher accuracy levels are required, a ore frequent coputation of the exchanges factors with both the boundaries and the adjacent layers is a good solution (case 10). The errors are negligible and the coputation tie is divided by a factor of 2 copared to the reference solution (case 1). 5. Suary and conclusions In the present paper, a radiative code based on a flux forulation has been reforulated into a radiative code based on the NEF. This forulation has been proposed by Green (1967) but has not been often used since this tie. The graphical representation of the net exchange atrix appears to be a eaningful tool to analyze the radiative exchanges and the radiative budgets in the atosphere. In the case of Mars, the exchange between a layer and space (the cooling to space) and the exchanges between a layer and its two adjacent layers are by far the doinant contributions to the radiative bud- Case No. Net exchanges and radiative budgets TABLE 2. Coparison between the various tie interpolation schees. Coputation period Exchange factors Adjacent Distant At teperature difference (K) layers Boundaries layers Mean Rs Noralized coputation tie of the LW radiative code h 1h 1h 1h day 1 day 1 day day 30 1 day day 1 day day h 6 h 6 h h 30 6 h h 6 h h