Anja Scherwass
email: anja.scherwass@uni-koeln.de

Tanja Bergfeld

Andreas Schoel

Markus Weitere

Hartmut Arndt

A Schöl

& Arndt

De Ruyter Van Steveninck

Admiraal E D De

W Zanten

B Van

Changes in the plankton community along the length of the River Rhine: Lagrangian sampling during spring

Keywords: protozoa, phytoplankton, metazooplankton, bacteria, river, Lagrangian sampling protozoa, phytoplankton, metazooplankton, bacteria, river, 25 Lagrangian sampling

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

21 Despite these studies, the dynamics of the microbial plankton community 22 and their regulating factors during downstream transport over longer river 23 stretches remain poorly understood (e.g. Sorokin, 1990). 24 The present study was conducted to investigate the composition and changes 25 within the different groups of the plankton community, in particular on the 26 components of the microbial food web, during downstream transport in a 27 large river. Due to the canalisation of its bed, the River Rhine is particularly The development of plankton was investigated on a stretch of the River 13 Rhine in Germany between km-173.1 and km-844.9 (Fig. 1; the Rhine km 14 numbers refer to the distance from the outflow at Lake Constance). The 11.8 (± 1.6) µg L -1 measured at Rhine-km-844.9 (Fig. 2a). Seston content 19 and turbidity increased markedly during downstream transport as well (Fig. 20 2b).

21

Phytoplankton abundance showed an increase (up to eightfold, Fig. 3a) from 22 the Upper to the Lower Rhine (positively correlated with the increase of the 23 chlorophyll a). Most tributaries and the "Rest-Rhine" showed higher 24 abundances of phytoplankton than the Rhine itself. The largest difference 25 between the main stream and the tributaries was found in the middle stretch 26 of the Rhine: in the rivers Moselle and Lahn, phytoplankton abundances 27 were about nine times higher than in the Rhine at this location (Fig. 3a). in the River Rhine (where they averaged 2.2 x 10 6 (± 6.9 x 10 5 ) bacteria 1 mL -1 ).

2

The abundances of both heterotrophic nanoflagellates and ciliates declined 3 during downstream transport (Fig. 4 a,b). Moreover, abundances of both 4 protozoan groups were significantly negatively correlated with the flow time 5 (b = -0.385, p < 0.01 for HNF; b = -0.424, p < 0.01 for ciliates). The 6 average HNF abundance dropped from 407 (± 50) ind. mL -1 at km-173.1 to 7 328 ind. mL -1 (± 10) at km-844.9, and did not exceed a value of 400 ind. The pattern of downstream development of ciliate abundance was partly 13 similar to that of the HNF (Fig. 4 b). A significant reduction from a mean of 14 263 ciliates L -1 (± 21) at Rhine-km-173.1 to 64 ind. L -1 (± 32) at Rhine-km-15 844.9 was observed. Ciliates showed high numbers in the Lahn, Moselle and 16 "Rest-Rhine" (up to a mean of 838 ind. L -1 in the latter), whereas low 17 abundances were recorded for the small tributaries. The biovolume of total zooplankton was about one order of magnitude 5 lower than that of phytoplankton (ranging between 0.03 mm³ L -1 and 6 0.16 mm³ L -1 in the Rhine), being dominated by heterotrophic 7 nanoflagellates (Fig. 5 b), which contributed between 50 and 88% to the 8 total zooplankton biovolume. The absolute biovolume of HNF (between 9 0.02 and 0.09 mm³ L -1 ) was much higher than that of metazooplankton. In The share of metazooplankton decreased from 43 % at km-173.1 to 13 % at 16 km-844.9 (0.002 -0.03 mm³ L -1 ). In the tributaries metazooplankton 17 reached between 9 and 91 % ("Rest-Rhine" at km-291).

(

  two decades research on the microbial food webs of different 4 aquatic habitats has increased, giving insight into the characteristics and 5 interactions of this thus far poorly understood part of the aquatic food web 6 Weisse et al., 1990; Sanders et al., 1992; Gilbert et al., 1998 etc.). Our 7 understanding of the functions and dynamics of the microbial components in 8 riverine systems is still at an early stage (see Servais et al., 2000; Weitere et 9 al., 2005; Joaquim-Justo et al., 2006; Kiss et al., 2009). Moreover, the 10 longitudinal development of planktonic organisms has only been studied for 11 selected groups, i.e. phytoplankton (Reynolds and Glaister, 1993; Reynolds 12 and Descy, 1996; Ietswaart et al., 1999), metazooplankton (de Ruyter van 13 Steveninck et al., 1992; Welker and Walz, 1998; Viroux, 2002) and 14 heterotrophic nanoflagellates ( Weitere and Arndt, 2002 a, b). 15 However, studies on the components of the microbial loop in rivers have 16 indicated the importance of protozoans, which often make up a high 17 proportion of the planktonic biomass and have a high potential grazing 18 impact on the bacterial biomass (Carlough and Meyer, 1989; Arndt and 19 Mathes, 1991; Sleigh et al., 1992; Lair et al., 1999; Servais et al., 2000; 20 Weitere et al., 2005; Joaquim-Justo et al., 2006; Bergfeld et al., 2009).

28 suited

 28 for following a water parcel during downstream passage: the 29 influence of back waters is negligible, which facilitates the investigation of 30 the influence of the major tributaries. The abundances of all plankton 31 components were studied along a 660 km stretch of the Rhine.

(

  internal production resulting in a characteristic river 1 plankton community in the potamal region (as has been proposed 2 previously; e.g. Vannote et al., 1980; Allan, 2007). Such a downstream 3 increase in abundance has been shown in the River Rhine on a 660 km 4 stretch in an earlier study of phytoplankton and rotifers (de Ruyter van 5 Steveninck et al., 1992). However, more recent studies at different stations 6 along a river stretch have shown that there can be a significant reduction of 7 some plankton components, probably due to losses to benthic filter feeders 8 Weitere and Arndt, 2002 a, b; Scherwass and Arndt, 2005).

  15investigation took place from May 22 nd to 28 th 2000 following a water 16 parcel moving downstream. The study was carried out towards the end of 17 the phytoplankton spring bloom, so that we assumed a high productivity of 18 all components of the microbial food web. Twenty-four sampling stations 19 including eight tributaries were investigated (TableI). Between km-173.1 20 and km-334, most of the river flows in an artificial river bed regulated by 21 ten dams; only below 10% of the total discharge (usually between 20 and 22 150 m³ sec -1 ) flows in the former river bed. The part of the river which flows 23 in the original bed is hereafter referred to as the "Rest-Rhine". The Rhine 24 flows in its natural bed with no further dams from km-334 to the end of the 25 investigated area (km-844.9).26The samples between km-173.1 and km-483.2 were taken at 1 m depth from side and one in the middle of the river. If this was not possible, 1 three consecutive samples were taken from the middle of the river within 2 30 minutes.3Immediately after sampling, temperature as well as chlorophyll a and 4 oxygen content were measured using a multi-probe (Yellow Spring 5 Instrument Inc.). In order to assess the seston content, 500 mL of river water 6 were filtered through preweighted GF/F glass fibre filters (Whatman). The 7 filters were then dried for 24 h at 100 °C and weighed. As a preparation for 8 determining the bacterial density, water samples were fixed with formalin 9 (2 % final concentration). 1 mL of sample was filtered onto 0.2 µm black 10 polycarbonate filters within 24 hours after sampling, stained with 0.2 mL of 11 DAPI (working solution 0.01 mg mL -1 ) and then stored at -20 °C until 12 further processing. Bacteria were counted under an epifluorescence 13 microscope (Zeiss Axioskop) under UV-excitation (365 nm; Filter: BP 395-14 440 BP, beam splitter 460, LP 470). At least 400 bacteria were counted per 15 filter, grouped into size classes according to measurements with an eye-16 piece micrometer. Because the delicate heterotrophic and autotrophic 17 nanoplankton are often destroyed during fixation, they were live-counted on 18 board. The live-count was conducted directly after sampling under a phase 19 contrast microscope (Zeiss Axiostar) with magnifications of 100 -400x to 20 determine their abundance and taxonomic composition. At least five 21 subsamples of 5 -10 µL each were counted until sufficient numbers of 22 protists for statistical analysis were obtained. The lengths and widths of 23 individual nanoplanktonic organisms were measured using video 24 microscopy to calculate biovolumes (based on simple geometrical forms). 25 Unfiltered samples of microplankton (ciliates and larger algae) were fixed 26 with Bouin's fluid (5% final concentration), concentrated in the laboratory 27 and processed as described in Scherwass et al. (2002) using a method 28 combining density gradient centrifugation (to exclude inorganic seston 29 particles) and quantitative protargol staining (Skibbe, 1994). The resulting 30 permanent slides were examined under a microscope (Axiophot 2, Zeiss) 31 with a magnification of up to 1000x. Biovolumes were estimated based on 32 simple geometric forms assuming a shrinkage factor of 30%. This shrinkage 33 factor refers to observations for several ciliate groups (Pfister et al., 1999). In addition to the analysis of protozooplankton, 2 or 4 L of water were 1 sieved through a plankton net (10 µm-mesh size) and the metazooplankton 2 was counted at 200x magnification. Metazooplankton lengths, abundance 3 and composition were determined, and published length-weight ratios were 4 used to calculate the biovolume (e.g. Bottrell et al., 1976). These data were 5 collected to estimate the abundance and biomass of metazooplankton 6 organisms for the present study; a more detailed investigation of 7 metazooplankton was conducted at one station (Cologne Rhine-km 684.5) in 8 a parallel study (Weitere et al., 2005). 9 Statistical analyses were conducted using SPSS for Windows (version 15.0). 10 Spearman rank coefficients were used to analyse correlations between 11 plankton abundances and flow time.

  abundances of plankton organisms 15 Water temperature and oxygen concentration varied only slightly along the 16 length of the river stretch, whereas the chlorophyll a concentration increased 17 significantly during downstream transport, resulting in a maximum of

  18

28

  Particularly the Moselle contributed a significant plankton load to the Rhine 29 (Tab. II). 30 Bacterial abundances remained constant over the stretch of the Rhine 31 averaging 3.7 x 10 6 (± 5.5 x 10 5 ) bacteria mL -1 (Fig. 3 b) with a slight 32 tendency towards higher values in the upper part of the river. Generally, 33 bacterial abundances were lower in the tributaries and the 'Rest-Rhine' than 34

  8 mL -1 in the Middle and Lower Rhine. Tributaries of the middle and lower 9 reaches of the Rhine showed abundances which were signficantly higher 10 than in the main stream reaching mean values between 703 and 1,683 ind. 11 mL -1 in the Main, Lahn and Moselle.

  12

18

  Though only roughly estimated in the present study, the metazooplankton 19 abundance (mainly rotifers) seem to show the same trend as HNF and 20 ciliates along the river stretch with decreasing abundances during 21 downstream transport from about 70 to about 20 ind. L -1 . 22 The plankton load contributed by the largest tributaries in the reach of the 23 Rhine investigated was calculated based on the abundances of organisms 24 (Tab. II). The Moselle contributed a large part to the total plankton 25 community in the Rhine: up to 42% of phytoplankton, 10.1% of bacteria, contributed by the River Main (57 %).

28 29

 28 Share of different planktonic components in terms of biovolume30 Phytoplankton made up the largest proportion of the total plankton 31 biovolume in spring 2000. Its biovolume ranged from 0.36 mm³ L -1 (Rhine-32 km-173.1) to 1.70 mm³ L -1 (Rhine-km-844.9); up to 3.4 mm³ L -1 (in the 33 Moselle) were observed in the tributaries. The bacterial biovolume ranged 34 21 to 0.34 mm³ L -1 . Phytoplankton exceeded bacterial biovolume at 1 all stations (Fig. 5 a). Bacteria contributed the most to the total heterotrophic 2 biovolume (comprising bacterial and total zooplankton biovolume, data not 3 shown).

  4

  10 the tributaries, HNF accounted for up to 86 % ("Rest-Rhine" km-225.5) of 11 total zooplankton biovolume. Ciliates contributed the least to the total 12 zooplankton in the River Rhine as well as in all tributaries; their biovolume 13 reached a maximum of 8 % (km-469.3) in the Rhine and 21 % (River 14 Kinzig) in the tributaries (Fig. 5 b, 0.1 and 4 x 10³ mm³ L -1 , respectively).

  15

710(and 30 Hydrobiol78 28 Ges7

 3028 of phytoplankton on the basis of biovolume 21 (Fig. 6) showed a distinct change during downstream transport. 22 Cryptophytes dominated the plankton community in the Upper Rhine and 23 diminished in importance during downstream transport. The proportion of 24 diatoms increased significantly in parallel to the reduction of the 25 cryptophytes in the River Rhine, whereas the tributaries showed a 26 considerable number of chlorophytes.27In contrast to phytoplankton, protozooplankton did not show a clear change 28 in taxonomic composition during downstream transport (Fig.6). Among 29 HNF, chrysomonads were the dominant group during the investigation. This 30 was true for the River Rhine itself as well as for all tributaries. Their relative 31 contribution remained comparatively constant throughout downstream 32 transport. For ciliates, oligotrichs (typical planktonic forms) formed the 33 largest component of the biomass, followed by peritrichs (mainly benthic 34 Their relative taxonomic composition did not change significantly 1 during downstream transport. Metazooplankton was always dominated by 2 rotifers, other metazooplankton groups were only found in a few cases such 3 as nauplii larvae, mussel larvae and, less frequently, cladocerans.Changes of the planktonic community during downstream transport8 The abundance and biovolume of plankton components in the Rhine is 9 generally low throughout the year as has been shown in several long-term 10 studies (ICPR, 2002; Scherwass and Arndt, 2005; Weitere et al., 2005; 11 Bergfeld et al., 2009), especially when compared to other large rivers, e.g. 12 River Loire (e.g. Lair et al., 1999). The season with the highest plankton 13 abundance in the River Rhine is spring (ICPR, 2002; Weitere et al., 2005). 14 As a consequence, the spring situation in May 2000 was chosen for the 15 present Lagrangian investigation in order to be able to observe changes in 16 the plankton community composition along the river stretch. 17 Within the plankton community of the River Rhine, we found a high 18 proportion of protozooplankton (on average 75% of the total zooplankton 19 biovolume) in the present study. This observation agrees with long-term 20 studies of the River Rhine at Cologne (Rhine-km 684.5, Weitere et al., 21 2005) and Koblenz (Rhine-km 590, Bergfeld et al., 2009) and thus seems to 22 be a general phenomenon in the middle and lower reaches of the River 23 Rhine. The present study indicates that this is characteristic along almost the 24 total river stretch, at least in spring/early summer. The proportion of 25 protozoans to the total zooplankton in the River Rhine is high in comparison 26 to other large rivers (e.g. in the Loire, Meuse and Elbe; see Lair et al., 1999; 27 Servais et al., 2000; Wörner and Zimmermann-Timm, 2005), despite the 28 fact that their proportion is generally higher in rivers compared to lakes .. A 29 similarly high contribution of protozoans to the total zooplankton biovolume 30 as shown for the River Rhine was only reported for the River Yenisei in 31 Siberia by Sorokin (1990) and for two large rivers belonging to the Rhine 32 River system, the Moselle and its tributary, the Saar (Bergfeld et al., 2009). Lagrangian sampling allowed further insights into changes of the plankton 1 community during downstream transport within the River Rhine in spring. 2 The development of the abundance of the microbial plankton differed 3 between the groups of organisms. Bacterioplankton remained at a nearly 4 constant low level during downstream transport despite their high potential 5 growth rates. This is in contrast to the results of earlier studies in the River 6 Rhine, where an increase of bacterioplankton over the course of the Rhine 7 was observed during the same season (April and May; de Ruyter van 8 Steveninck et al., 1992; Ietswaart et al., 1999). Also the increase of 9 phytoplankton along the river stretch was much lower during the present 10 study than during earlier studies (de Ruyter van Steveninck et al., 1992; 11 Ietswaart et al., 1999; Friedrich and Pohlmann, 2009). One possible 12 explanation for these changes in plankton dynamics might be the drastic 13 changes of the community structure of invertebrates due to massive growth 14 of invasive species immigrated into the River Rhine within the last two 15 decades (ICPR 1996, 2002, see below). 16 17 Potential factors causing losses of plankton during downstream transport in 18 spring 19 Sedimentation of plankton due to decreasing flow velocities in lower river 20 reaches may be one reason for their decline during downstream transport, as 21 has been shown e.g. in the lowland River Spree (Koehler et al., 2002) and 22 the River Meuse (Marneffe et al., 1996). However, due to the strong 23 canalisation of the River Rhine combined with a high discharge, flow 24 velocity remained high up to the end of the investigated reach (km-844.9) 25 and sedimentation was assumed to play a significant role only in the very 26 low Dutch reaches of the Rhine (de Ruyer van Steveninck et al., 1990). 27 Additionally, turbidity increases with increasing river length and may affect 28 autotrophic as well as heterotrophic plankton (Reynolds and Descy, 1996; 29 Pourriot et al., 1997). However, the light-dependent phytoplankton seems 30 not to be affected by increased turbidity in spring, as its abundance increased 31 along the river stretch. For metazooplankton, a disturbance of feeding due to 32 high concentrations of inorganic particles in combination with high 33 turbulence might negatively affect metazooplankton (Miquelis et al., 1998). This could be the reason for the reduced metazooplankton abundance in the 1 lower reaches of the River Rhine. The abundance of heterotrophic protists 2 also decreased along the river stretch, but protists can be positively affected 3 by inorganic sediment particles as they offer a habitat for colonization 4 Wörner et al., 2002).This indicates that the dynamics of the different 5 plankton groups cannot be explained by turbidity alone. Regarding 6 mechanical forces potentially affecting plankton organisms it has to be 7 considered that protists and bacteria have much lower Reynolds numbers 8 than metazoans and are unlikely to be physically damaged in their relatively 9 laminar surroundings (cf. Vogel, 1993). Thus turbulence does not seem to 10 directly influence the unicellular organisms, whereas a negative influence on 11 metazoans is possible (Picard and Lair, 2003; Sluss et al., 2008). 12 Another reason for decreasing plankton numbers in the course of the river 13 stretch may be dilution by the tributaries. However, during our 14 investigations tributaries generally had higher abundances of plankton than 15 the River Rhine (see Tab. II). By taking these high plankton loads of the 16 tributaries into account, our present data show that within-stream processes 17 rather than effects of the tributaries might be the reason for the plankton 18 dynamics observed in the Rhine. 19 In an earlier study, we demonstrated with a combination of size fractionation 20 experiments and Lagrangian sampling, that planktonic heterotrophic 21 flagellates showed high losses during the downriver passage in the Lower 22 Rhine (km 685 to 845, Weitere et al., 2002a). The losses strongly depended 23 on temperature (high losses in summer, low losses in winter), indicating 24 biotic interactions. Furthermore, the losses could not be explained by 25 planktonic grazers as derived from size-fractionation experiments. This gave 26 rise to the conclusion that benthic grazers contribute significantly to losses 27 of plankton in the Rhine. This conclusion is in accordance with the 28 application of a water quality model in the Rhine for the year 1990 which 29 revealed that metazooplankton grazing on the algal production was 30 negligible, but that algae were partly reduced by grazing of the benthic 31 mussel Dreissena polymorpha (Schöl et al., 2002). Studies from other rivers 32 pointed to the importance of benthic predation on the components of the 33 planktonic microbial food web of rivers (e.g. Basu and Pick, 1997; Welker 34 Walz 1998; Descy et al., 2003). Some efficient benthic filter-feeding 1 macroinvertebrates like Corbicula spp., Dreissena polymorpha and 2 Corophium curvispinum are abundant in the Lower Rhine (e.g. Rajagopal et 3 al., 1995, 1998) and have been shown to capture particles of a wide size 4 range (see e.g. Sprung and Rose, 1988; Scherwass et al., 2001; Viergutz et 5 al., 2007; Weitere et al., 2008). In their study in the River Rhine, Ietswaart 6 et al. (1999) calculated a significant loss rate of algae due to predation by 7 the benthic macrofauna community of 0.38 d -1 for the warm season for the 8 year 1995. Losses to the benthic grazers may probably be even larger as only 9 a few 'focus' grazers were considered in these estimations. The role of other 10 potentially important benthic filter feeders including sponges, bryozoans and 11 ciliates is as yet unknown (Weitere et al., 2003; Kathol et al., 2009; 12 Vohmann et al., 2009). 13 A possible control of the plankton by benthic filter feeders could explain 14 group-specific patterns observed in the present study. The slow growing 15 groups (metazoans, ciliates) decrease much more than fast growing groups 16 such as heterotrophic flagellates, possibly due to different abilities to 17 compensate for grazing losses (Weitere and Arndt, 2002a; Scherwass and 18 Arndt, 2005). Bacteria probably experience a low grazing pressure by 19 benthic filter feeders due to their small size (Sprung and Rose, 1988), but a 20 strong grazing pressure may affect heterotrophic flagellates within the water 21 column (Weitere et al., 2002b). The loss rates of phytoplankton seem to be 22 governed by factors different to those of protozooplankton. Further detailed 23 longitudinal studies are required to explain these discrepancies in future. 24 25 The present case study for a spring scenario is the first that considered the 26 longitudinal dynamics not only of classical components but also of 27 organisms of the microbial food web in a large river. Additional studies at 28 particular locations along the river stretch of the Rhine showed that general 29 patterns such as stable or even decreasing plankton abundance occurs 30 throughout the year despite high potential productivity (Weitere and Arndt, 31 2002a,b; Weitere et al., 2005; Bergfeld et al., 2009). Despite clear 32 indications of a significant impact of the benthos on the plankton dynamics 33 as found in different studies in the Rhine (see above), the interplay between 34 in controlling plankton in the Rhine (e.g. grazing, 1 residence time, turbidity, turbulence, sediment infiltration) is as yet poorly 2 understood. The importance of biotic versus non-biotic factors varies with 3 the river stretch and each plankton group (Strayer et al., 2008). Future 4 empirical studies as well as model approaches need to focus on the interplay 5 of the multiple factors in controlling plankton in rivers. In this context, it has 6 to be pointed out that dynamics of plankton in the River Rhine shown here 7 were observed only for the last ten years. There was still a considerable 8 downstream increase of all plankton components until 1995 (de Ruyter van 9 Steveninck et al., 1992; ICPR, 1996; Friedrich and Pohlmann, 2009). 10 Efforts to reduce the nutrient load (ICPR, 2002) and the massive increase of 11 populations of invasive species (especially the effective plankton filterers 12 such as Corbicula spp., ICPR, 2002) are probably important reasons for the 13 recent shift. Understanding the regulation of the plankton dynamics in 14 changing riverine environments is a major challenge for future studies. A. Scherwass and from the Deutsche 19 Bundesstiftung Umwelt to M. Weitere. We thank Mr. Vobis and his 20 colleagues from the Landesanstalt für Umweltschutz Baden-Württemberg 21 (LUBW) as well as the crew of the R/V 'Max Honsell' for the support 22 during sampling. We are indebted to the scientific and technical assistance 23 by Annick Eimer, Claudia Günster and Walter Krings. We would also like 24 to thank Frederic Bartlett for linguistic comments. Gilbert, D., Amblard, C., Bourdier, G. and Francez, A. J. (1998) The 1 microbial loop at the surface of a peatland: structure, function and impact of 2 nutrient input. Microb. Ecol. 35, 83--93. ., Breebaart, L., van Zanten, B. and Bijkerk, R. (1999) Plankton 11 dynamics in the river Rhine during downstream transport as influenced by 12 biotic interactions and hydrological conditions. Hydrobiologia, 410, 1-20. 13 14 Joaquim-Justo, C., Pirlot, S., Viroux, L., Servais, P., Thomé, J. P. and 15 Descy, J. P. (2006) Trophic links in the lowland River Meuse (Belgium): 16 assessing the role of bacteria and protozoans in planktonic food webs. J.17Plankt. Res., 9, 857 --870. 18 19 Kathol, M., Norf, H., Arndt, H. and Weitere, M. (2009) Effects of 20 temperature increase on the grazing of planktonic bacteria by biofilm-21 associated consumers. Aquat. Microb. Ecol., 55, 65--79. 22 23 Kiss, A. K., Acs, E., Kiss, K. T. and Torok, J. K. (2009) Structure and 24 seasonal dynamics of the protozoan community (heterotrophic flagellates, 25 ciliates, amoeboid protozoa) in the plankton of a large river (River Danube, 26 Hungary). Europ. J. Protist., 45, 121--138. 27 28 Koehler, J., Bahnwart, M. and Ockenfeld, K. (2000) Growth and loss 29 processes of riverine phytoplankton in relation to water depth. Int. Rev. ., Jacquet, V. and Reyes-Marchant, P. (1999) Factors related to 33 autotrophic potamoplankton, heterotrophic protists and micrometazoan 34 two sites in a lowland temperate river. Hydrobiologia, 394, ., Descy, J. P. and Thomé, J. P. (1996) The zooplankton of the 4 lower river Meuse, Belgium: seasonal changes and impact of industrical and 5 municipal discharges. Hydrobiologia, 319, 1-13. 6 Miquelis, A., Rougier, C. and Pourriot, R. (1998) Impact of turbulence and 8 turbidity on the grazing rate of the rotifer Brachionus calyciflorus (Pallas). ., Sonntag, B. and Posch, T. (1999) Comparison of a direct live 12 count and an improved quantitative protargol stain (QPS) in determining 13 abundance and cell volumes of pelagic freshwater protozoa. Aquat. Microb. . and Lair, N. (2003) Laboratory approach of the growth of rotifers 17 sampled in the Middle Loire (France) under turbulence. Journal de 18 Recherche Océanographique, 28, 196 -199. 19 20 Pourriot, R., Rougier, C. and Miquelis, A. (1997) Origin and development 21 of river zooplankton: Example of the Marne. Hydrobiol. 345, 143-148. Reynolds, C. S. and Descy, J.-P. (1996) The production, biomass and 1 structure of phytoplankton in large rivers. Arch. Hydrobiol. Suppl., 113, . S. and Glaister, M. S. (1993) Spatial and temporal changes in 5 phytoplankton abundance in the upper and middle reaches of the River 6 Servan. Arch. Hydrobiol. Suppl., 101, 1--22. 7 Sanders, R. W., Caron, D. A. and Berninger, U. G. (1992) Relationships 9 between bacteria and heterotrophic nanoplankton in marine and fresh waters 10 -an inter-ecosystem comparison. Mar. Ecol. Progr. Ser., 86, 1-14. 11 12 Scherwass, A. and Arndt, H. (2005) Structure, dynamics and control of the 13 ciliate fauna in the potamoplankton of the river Rhine in Cologne. Arch. ., Wickham, S. A. and Arndt, H. (2002) Determination of the 17 abundance of ciliates in highly turbid waters -an improved method tested 18 for the River Rhine. Arch. Hydrobiol., 156, 135--143. 19 20 Scherwass, A., Eimer, A. and Arndt, H. (2001) Selective influence of filter-21 feeding benthic bivalves (Corbicula spec., Mytilus spec.) on planktonic 22 ciliates. Verh. Int. Verein. Limnol., 17, 3315 --3318. 23 24 Schöl, A. , Kirchesch, V., Bergfeld, T., Schöll, F., Borcherding, J. and 25 Müller, D. (2002) Modelling the chlorophyll a content of the River Rhine-26 interrelation between riverine algal production and population biomass of 27 grazers, rotifers and the zebra mussel Dreissena polymorpha. Internat. Rev. ., Gosselain, V., Joaquim-Justo, C., Becquevort, S., Thomé, J. P. 31 and Descy, J. P. (2000) Trophic relationships between planktonic 32 microorganisms in the river Meuse (Belgium): a carbon budget. Arch. 33 Hydrobiol., 149, 625--653. . (1994) An improved quantitative protargol stain for ciliates and 2 other planktonic protists. Arch. Hydrobiol., 130 ,339--347. . A., Baldock, B. M. and Baker, J. H. (1992) Protozoan 5 communities in chalk streams. Hydrobiologia, 248, 53--64. 6 Sluss, T. D., Cobbs, G. A. and Thorp, J. H. (2008) Impact of turbulence on 8 riverine zooplankton: a mesocosm experiment. Freshw. Biol., 53, . I. (1990) Heterotrophic microplankton in plankton successions 12 and self purification processes along the Yenisei river. Arch . Hydrobiol.

5Figure 1 :Figure 2 : 18 Figure 3 : 25 Figure 4 :

 12183254 Figure 1: Map of the river stretch investigated from km 173.1 to km 844.9

29 30 Figure 5 :Figure 6 :

 3056 Figure 5: Biovolume partitioning of (a) bacteria and phytoplankton and (b)

Table I :

 I Kommission für die Hydrologie des Rheins, List of sampling stations on the Rhine and in the tributaries at the

	1 1 1
	2 2 2	Viroux , L. (2002) Seasonal and longitudinal aspects of microcrustacean Weitere, M., Dahlmann, J., Viergutz, C. and Arndt, H. (2008) Differential
	3 3 3	(Cladocera, Copepoda) dynamics in a lowland river. J. Plankt. Res., 24, grazer-mediated effects of high summer temperatures on pico-and confluence
	4 4 4	281-292. nanoplankton communities. Limnol. Oceanogr., 53, 477 -486.
	5 5
	6 6	Vogel, S. (1993) Life in Moving Fluids: The Physical Biology of Flow. Welker, M. and Walz, N. (1998) Can mussels control the plankton in rivers?
	7 7	Princeton University Press: New Jersey. -a planktological approach applying a Lagrangian sampling strategy.
	8 8	Limnol. Oceanogr. 43, 753--762.
	9 9	Vohmann, A., Mutz, M., Arndt, H. and Weitere, M. (2009) Grazing impact
	10 10	and phenology of the freshwater sponge Ephydatia muelleri and the Woerner, U., Zimmermann-Timm, H. and Kausch, H. (2002) Aggreate-
	11 12 13 14 14 11 12 13	F o r bryozoans Plumatella emarginata and Fredericella sultana under experimental warming. Freshw. Biol., 54, 1078 -1092. Weisse, T., Müller, H., Pintocoelho, R.M., Schweizer, A., Springmann, D. F o r associated bacteria and heterotrophic flagellates in the River Elbe -Their relative significance along the longitudinal profile from km 46 to km 583. Int. Rev. Hydrobiol., 87, 255 -266. F o r
	15 15	P and Baldringer, G. (1990) Response of the microbial loop to the P Wörner, U. and Zimmermann -Timm, H. (2006) Plankton und benthische P
	16 16	e phytoplankton spring bloom in a large prealpine lake. Limnol. Oceanogr., e Besiedlung in Buhnenfeldern. Aggregate in Buhnenfeldern. In: Stoffdynamik e
	18 17 18 17 18	e r e r und Habitatstruktur in der Elbe. Ed.: Pusch, M, Fischer, H. Weißensee 35, 781--794. Verlag: Berlin. pp. 118-146. e r
	19 20 21 22 23 19 20 19 20 21 22 23	Lelystad. Sprung, M. and Rose, U. (1988) Influence of food size and food quantity on the feeding of the mussel Dreissena polymorpha. Oecologia, 77, 526 --532. R e v i e Weitere, M. and Arndt, H. (2002 a) Top-down effects on pelagic heterotrophic nanoflagellates (HNF) in a large river (River Rhine): Do losses to the benthic environment play a role? Freshw. Biol. , 46, 1437--R e R e 1450. v i e v i e
	24 25 24 25	Strayer , D. L., Pace, M. L., Caraco, N. F., Cole, J. J., Findlay, S. E. G. and Stuart, E. G. (2008) Hydrology and grazing jointly control a large-river food w Weitere, M. and Arndt, H. (2002 b) Water discharge-regulated bacteria/heterotrophic nanoflagellates (HNF) interactions in the water w w
	26	web. Ecology, 89, 12 -18. column of the River Rhine. Microb. Ecol. 44, 19--29.
	27
	28	Weitere, M., Schmidt-Denter, K., Auer, B. and Arndt, H. (2003) Laboratory
	29	experiments on the impact of biofilms on the plankton of a large river.
	30	Freshw. Biol., 48, 1983--1992.
	31
	32	Weitere, M., Scherwass, A., Sieben, K. T. and Arndt, H. (2005) Planktonic
	33	food web structure and potential carbon flow in the lower River Rhine with
	34	a focus on the role of protozoans. Riv. Res. and Appl., 21, 535--549.
		20 21

13

Beih. Ergebn. Limnol., 34, 267--273. 14 15 Spreafico, M. and van Mazijk, A. (Red.) (1993): Alarmmodell Rhein. Ein 16 Modell für die operationelle Vorhersage des Transportes von Schadstoffen 17 im Rhein. Bericht Nr. 1-12.

Table II :

 II Plankton load contributed by the main tributaries Neckar (at Rhine-

manuscriptcentral.com/jplankt Journal of Plankton Research

		100%
											Others
		100% 40% 60% Phytoplankton 80%		Neckar	Main	Mosel				Bacteria Chlorophytes Diatomes Cyanobacteria
		80% 20%									Chrysophytes Algae
	Biovolum e [%] a) Biovolum e [%] Taxonomic composition of	20% 40% 60% 80% 100% 40% 60% Ciliates 80% 0% 20% 40% 60% 0% 20% 40% 60% 80% 100% Flagellates 0% 100%		F o r P e e r R e v i e w v i e w F o r P e e r R e	HNF Ciliates Metazoopl. Suctorians Nassophoreans Peritrichs Hymenostomatids Cryptophytes Others Protista incertae sedis Choanoflagellates Bicosoecids Chrysomonads Euglenids Kinetoplastids Others
											Prostomatids
	b)	0% 0% km 173.1 20%	km 190.5 km 173.1	km 214.6 km 190.5 km 214.6 km 267.3 km 267.3 km 279.4 km 279.4 km 296.4 River Rhine km 342. 5 km 359.2 km 395 km 457.9 km 296.4 km 342.5 km 359.2 km 395 km 457.9 km 4693 km 469.3 km 483.2 km 483.2 km 571 km 571 km 590.9	km 590.9 km 684.5	km 684.5 km 844.9	km 844.9 Restrh.km 225.5	Restrh. km 291	Restrhein 225.5 Kinzig km 298 Tributaries Restrhein 291 Kinzig km 298 Ill km 312 Main km 496 Ill km 312 Neckar km 429 Main km 496 Lahn km 585 Mosel km 592 Neckar km 429	Lahn km 585 Heterotrichs Mosel km 592 Oligotrichs Hypotrichs Stichotrichs Haptorids
	Scherwass, A. et al., Fig. 5 River Rhine					Tributaries

http://mc.manuscriptcentral.com/jplankt Journal of Plankton Research

20% of HNF, and 43.0 % of ciliates. The largest metazooplankton load was

Acknowledgements:


Christian Obrecht
email: christian.obrecht@insa-lyon.fr

Frédéric Kuznik
email: frederic.kuznik@insa-lyon.fr

Bernard Tourancheau
email: bernard.tourancheau@ens-lyon.fr

Jean-Jacques Roux
email: jean-jacques.roux@insa-lyon.fr

Global Memory Access Modelling for Efficient Implementation of the Lattice Boltzmann Method on Graphics Processing Units

Keywords: GPU computing, CUDA, lattice Boltzmann method, CFD

In this work, we investigate the global memory access mechanism on recent GPUs. For the purpose of this study, we created specific benchmark programs, which allowed us to explore the scheduling of global memory transactions. Thus, we formulate a model capable of estimating the execution time for a large class of applications. Our main goal is to facilitate optimisation of regular data-parallel applications on GPUs. As an example, we finally describe our CUDA implementations of LBM flow solvers on which our model was able to estimate performance with less than 5% relative error.

Introduction

State-of-the-art graphics processing units (GPU) have proven to be extremely efficient on regular data-parallel algorithms [START_REF] Dongarra | Exploring new architectures in accelerating CFD for Air Force applications[END_REF]. For many of these applications, like lattice Boltzmann method (LBM) fluid flow solvers, the computational cost is entirely hidden by global memory access. The present study intends to give some insight on the global memory access mechanism of the nVidia's GT200 GPU. The obtained results led us to optimisation elements which we used for our implementations of the LBM.

The structure of this paper is as follows. First, we briefly review nVidia's compute unified device architecture (CUDA) technology and the algorithmic aspects of the LBM. Then, we describe our measurement methodology and results. To conclude, we present our CUDA implementations of the LBM.

1 Compute Unified Device Architecture CUDA capable GPUs, i.e. the G8x, G9x, and GT200 processors consist in a variable amount of texture processor clusters (TPC) containing two (G8x, G9x) or three (GT200) streaming multiprocessors (SM), texture units and caches [START_REF]nVidia: Compute Unified Device Architecture Programming Guide version 2[END_REF]. Each SM contains eight scalar processors (SP), two special functions units (SFU), a register file, and shared memory. Registers and shared memory are fast but in rather limited amount, e.g. 64 KB and 16 KB per SM for the GT200. On the other hand, the off-chip global memory is large but suffers from high latency and low throughput compared to registers or shared memory.

The CUDA programming language is an extension to C/C++. Functions intended for GPU execution are named kernels, which are invoked on an execution grid specified at runtime. The execution grid is formed of blocks of threads. The blocks may have up to three dimensions, the grid two. During execution, blocks are dispatched to the SMs and split into warps of 32 threads.

CUDA implementations of data intensive applications are usually bound by global memory throughput. Hence, to achieve optimal efficiency, the number of global memory transactions should be minimal. Global memory transactions within a half-warp are coalesced into a single memory access whenever all the requested addresses lie in the same aligned segment of size 32, 64, or 128 bytes. Thus, improving the data access pattern of a CUDA application may dramatically increase performance.

Lattice Boltzmann Method

The Lattice Boltzmann Method is a rather innovative approach in computational fluid dynamics [START_REF] Mcnamara | Use of the Boltzmann Equation to Simulate Lattice-Gas Automata[END_REF][START_REF] Qian | Lattice BGK models for Navier-Stokes equation[END_REF][START_REF] Humières | Multiplerelaxation-time lattice Boltzmann models in three dimensions[END_REF]. It is proven to be a valid alternative to the numerical integration of the Navier-Stockes equations. With the LBM, space is usually represented by a regular lattice. The physical behaviour of the simulated fluid is determined by a finite set of mass fractions associated to each node. From an algorithmic standpoint, the LBM may be summarised as: The propagation phase follows some specific stencil. Figure 1 illustrates D3Q19, the most commonly used three-dimensional stencil, in which each node is linked to 18 of its 27 immediate neighbours. CUDA implementations of the LBM may take advantage of its inherent data parallelism by assigning a thread to each node, the data being stored in global memory. Since there is no efficient global synchronisation barrier, a kernel has to be invoked for each time step [START_REF] Ryoo | Optimization principles and application performance evaluation of a multithreaded GPU using CUDA[END_REF]. CPU implementations of the LBM usually adopt an array of structures (AoS) data layout, which improves locality of mass fractions belonging to a same node [START_REF] Pohl | Optimization and Profiling of the Cache Performance of Parallel Lattice Boltzmann Codes[END_REF]. On the other hand, CUDA implementations benefit from structure of arrays (SoA) data layouts, which allows coalesced global memory accesses [START_REF] Kuznik | LBM Based Flow Simulation Using GPU Computing Processor[END_REF]. However, this approach is not sufficient to ensure optimal memory transactions, since propagation corresponds to one unit shifts of global memory addresses for the minor spatial dimension. In other words, for most mass fractions, the propagation phase yields misalignments. A way to solve this issue consists in performing propagation partially in shared memory [START_REF] Tölke | TeraFLOP computing on a desktop PC with GPUs for 3D CFD[END_REF]. Yet, as shown in [START_REF] Obrecht | A new approach to the lattice Boltzmann method for graphics processing units[END_REF], this approach is less efficient than using carefully chosen propagation schemes in global memory.

for

Methodology

To study transactions between global memory and registers, we used kernels performing the following operations :

1. Store time t 0 in a register. 2. Read N words from global memory, with possibly L misalignments.

3. Store time t 1 in a register. 4. Write N words to global memory, with possibly M misalignments. 5. Store time t 2 in a register. 6. Write t 2 to global memory. Time is accurately determined using the CUDA clock() function which gives access to counters that are incremented at each clock cycle. Our observations enabled us to confirm that these counters are per TPC, as described in [START_REF] Papadopoulou | Micro-benchmarking the GT200 GPU 12[END_REF], and not per SM as stated in [START_REF]nVidia: Compute Unified Device Architecture Programming Guide version 2[END_REF]. Step 6 may influence the timings, but we shall see that it can be neglected under certain circumstances.

The parameters of our measurements are N , L, M , and k, the number of warps concurrently assigned to each SM. Number k is proportional to the occupancy rate α, which is the ratio of active warps to the maximum number of warps supported on one SM. With the GT200, this maximum number being 32, we have: k = 32α.

We used a one-dimensional grid and one-dimensional blocks containing one single warp. Since the maximum number of blocks supported on one SM is 8, the occupancy rate is limited to 25%. Nonetheless, this rate is equivalent to the one obtained with actual CUDA applications.

We chose to create a script generating the kernels rather than using runtime parameters and loops, since the layout of the obtained code is closer to the one of actual computation kernels. We processed the CUDA binaries using decuda [12] to check whether the compiler had reliably translated our code. We carried out our measurements on a GeForce GTX 295 graphics board, featuring two GT200 processors. 4

Modelling

At kernel launch, blocks are dispatched to the TPCs one by one up to k blocks per SM [START_REF] Collange | Power Consumption of GPUs from a Software Perspective[END_REF]. Since the GT200 contains ten TPCs, blocks assigned to the same TPC have identical blockIdx.x unit digit. This enables to extract information about the scheduling of global memory access at TPC level. In order to compare the measurements, as the clock registers are peculiar to each TPC [START_REF] Papadopoulou | Micro-benchmarking the GT200 GPU 12[END_REF], we shifted the origin of the time scale to the minimal t 0 . We noticed that the obtained timings are coherent on each of the TPCs.

For a number of words read and written N ≤ 20, we observed that:

-Reads and writes are performed in one stage, hence storing of t 2 has no noticeable influence. -Warps 0 to 8 are launched at once (in a determined but apparently incoherent order). -Subsequent warps are launched one after the other every ∼ 63 clock cycles.

For N > 20, reads and writes are performed in two stages. One can infer the following behaviour: if the first n warps in a SM read at least 4,096 words, where n ∈ {4, 5, 6}, then the processing of the subsequent warps is postponed. The number of words read by the first n warps being n × 32N , this occurs whenever n × N ≥ 128. Hence, n = 4 yields N ≥ 32, n = 5 yields N ≥ 26, and n = 6 yields N ≥ 21.

Time t 0 for the first 3n warps of a TPC follow the same pattern as in the first case. We also noticed a slight overlapping of the two stages, all the more as storing t 2 should here be taken into account. Nonetheless, the read time for the first warp in the second stage is noticeably larger than for the next ones. Therefore, we may consider, as a first approximation, that the two stages are performed sequentially.

In the targeted applications, the global amount of threads is very large. Moreover, when a set of blocks is assigned to the SMs, the scheduler waits until all blocks are completed before providing new ones. Hence, knowing the average processing time T of k warps per SM allows to estimate the global execution time.

For N ≤ 20, we have T = +T R +T W , where is time t 0 for the last launched warp, T R is read time, and T W is write time. Time only depends on k. For N > 20, we have T = T 0 + + T R + T W , where T 0 is the processing time of the first stage, (i) = (i -3n + 9) with i = 3k -1, T R and T W are read and write times for the second stage.

Fig. 2. Launch delay in respect of warp rank

To estimate , we averaged t 0 over a large number of warps. Figure 2 shows, in increasing order, the obtained times in cycles. Numerically, we have (i) ≈ 0 for i ≤ 9 and (i) ≈ 63(i -10) + 13 otherwise.

Throughput

N ≤ 20

Figures 3 and4 show the distribution of read and write times for 96,000 warps with N = 19. The bimodal shape of the read time distribution is due to translation look-aside buffer (TLB) misses [START_REF] Volkov | Benchmarking GPUs to tune dense linear algebra[END_REF]. This aspect is reduced when adding misalignments, since the number of transactions increases while the number of misses remains constant. Using the average read time to approximate T is acceptable provided no special care is taken to avoid TLB misses. We observed that average read and write times depend linearly of N . Numerically, with k = 8, we obtained:

T R ≈ 317(N -4) + 440 T W ≈ 562(N -4) + 1,178 T R ≈ 575(N -4) + 291 T W ≈ 983(N -4) + 2,030
where T R and T W are read and write times with L = N and M = N misalignments. Hence, we see that writes are more expensive than reads. Likewise, misalignments in writes are more expensive than misalignments in reads.

21 ≤ N ≤ 39

As shown in figures 5 and 6, T 0 , T R , and T W depend linearly of N in the three intervals {21, . . . 25}, {26, . . . 32}, and {33, . . . 39}. As an example, for the third interval, we obtain:

Complementary studies

We also investigated the impact of misalignments and occupancy rate on average read and write times. Figures 7 and8 show obtained results for N = 19.

Fig. 7. Misaligned reads

For misaligned reads, we observe that the average write time remains approximatively constant. Read time increases linearly with the number of misalignments until some threshold is reached. From then on, the average read time is maximal. Similar conclusion can be drawn for misaligned writes.

Implementations

We implemented several LBM fluid flow solvers: a D3Q19 LBGK [START_REF] Qian | Lattice BGK models for Navier-Stokes equation[END_REF], a D3Q19 MRT [START_REF] Humières | Multiplerelaxation-time lattice Boltzmann models in three dimensions[END_REF], and a double population thermal model requiring 39 words per node [START_REF] Peng | A 3D incompressible thermal lattice Boltzmann model and its application to simulate natural convection in a cubic cavity[END_REF]. Our global memory access study lead us to multiple optimisations. For each implementation, we used a SoA like data layout, and a two-dimensional grid of one-dimensional blocks. Since misaligned writes are more expensive than misaligned reads, we experimented several propagation schemes in which misalignments are deferred to the read phase of the next time step. The most efficient appears to be the reversed scheme where propagation is entirely performed at reading, as outlined in figure 9. For the sake of simplicity, the diagram shows a two-dimensional version. Fig. 9. Reversed propagation scheme Performance of a LBM based application is usually given in million lattice node updates per second (MLUPS). Our global memory access model enables us to give an estimate of the time T (in clock cycles) required to process k warps per SM. On the GT200, where the number of SMs is 30 and the warp size is 32, k warps per SM amounts to K = 30 × k × 32 = 960k threads. Since one thread takes care of one single node, T is therefore the number of clock cycles needed to perform K lattice node updates. Hence, using the global memory frequency F in MHz, the expected performance in MLUPS is: P = (K/T ) × F .

With our D3Q19 implementations, for instance, we have N = 19 reads and writes, L = 10 misaligned reads, no misaligned writes, and 25% occupancy (thus k = 8). Using the estimation provided by our measurements, we obtain: T = + T R + T W = 15,594. Since K = 7,680 and F = 999 MHz, we have P = 492 MLUPS.

To summarize, table 1 gives both the actual and estimated performances for our implementations on a 128 3 lattice. Our estimations appear to be rather accurate, thus validating our model.

Summary and discussion

In this work, we present an extensive study of the global memory access mechanism between global memory and GPU for the GT200. A description of the scheduling of global memory accesses at hardware level is given. We express a model which allows to estimate the global execution time of a regular dataparallel application on GPU. The cost of individual memory transactions and the impact of misalignments is investigated as well. We believe our model is applicable to other GPU applications provided certain conditions are met:

-The application should be data-parallel and use a regular data layout in order to ensure steady data throughput. -The computational cost should be negligible as compared with the cost of global memory reads and writes. -The kernel should make moderate use of branching in order to avoid branch divergence, which can dramatically impact performance. This would probably not be the case with an application dealing, for instance, with complex boundaries.

On the other hand, our model does not take possible TLB optimisation into account. Hence, some finely tuned applications may slightly outvalue our performance estimation.

The insight provided by our study, turned out to be useful in our attempts to optimize CUDA implementations of the LBM. It may contribute to efficient implementations of other applications on GPU.

3

 3

Fig. 1 .

 1 Fig. 1. The D3Q19 stencil

Fig. 3 .

 3 Fig. 3. Read time for N = 19

Fig. 4 .

 4 Fig. 4. Write time for N = 19

T 0 ≈Fig. 5 .Fig. 6 .

 056 Fig. 5. First stage duration

Fig. 8 .

 8 Fig. 8. Occupancy impact

Table 1 .

 1 Performance of LBM implementations (in MLUPS)

	Model	Occupancy	Actual	Estimated	Relative error
	D3Q19 LBGK	25%	481	492	2.3%
	D3Q19 MRT	25%	516	492	4.6%
	Thermal LBM	12.5%	195	196	1.0%

Taking the stationary mass fraction into account, the number of mass fractions per node amounts to 19, hence D3Q19.

In the CUDA environment, the GPUs of the GTX 295 are considered as two distinct devices. It should be noted that our benchmark programs involve only one of those devices.

Acknowledgement

We thank the reviewers for their helpful comments improving the clarity of our paper.


Laurent Véron

Maximal solutions of nonlinear parabolic equations with absorption

Keywords: 1991 Mathematics Subject Classification. 35K60, 34. Key words. Parabolic equations, singular solutions, self-similarity, removable singularities

We study the existence and the uniqueness of the solution of the problem (P): ∂tu -∆u + f (u) = 0 in Q := Ω × (0, ∞), u = ∞ on the parabolic boundary ∂pQ when Ω is a domain in R N with a compact boundary and f a continuous increasing function satisfying super linear growth condition. We prove that in most cases, the existence and uniqueness is reduced to the same property for the associated stationary equation in Ω.

Introduction

Let Ω be a bounded domain in R N with boundary ∂Ω := Γ, Q Ω T := Ω × (0, T ) (0 < T ≤ ∞) and ∂ p Q = Ω × 0 ∪ ∂Ω × (0, T ]. We denote by ρ ∂Ω (x) the distance from x to ∂Ω and by d P (x, t) = min{ρ ∂Ω (x), t} the product distance from (x, t)

∈ Q Ω ∞ to ∂ p Q Ω ∞ . If f ∈ C(R), we say that a function u ∈ C 2,1 (Q Ω ∞ ) solution of u t -∆u + f (u) = 0, (1.1) in Q Ω ∞ is a large solution of (1.1 ) in Q Ω ∞ if it satisfies lim d P (x,t)→0 u(x, t) = ∞. (1.2)
The existence of such a u is associated to the existence of large solutions to the stationary equation -∆w + f (w) = 0, (1.3) in Ω, i.e. solutions which satisfy lim

ρ ∂Ω (x)→0 w(x) = ∞, (1.4)
and solutions of the ODE φ ′ + f (φ) = 0 in (0, ∞). A natural assumption on f is to assume that it is nondecreasing with f (0) ≥ 0. If f (a) > 0, a necessary and sufficient condition for the existence of a maximal solution w Ω to (1.3 ) is the Keller-Osserman condition,

∞ a ds F (s) < ∞, (1.7)
where F (s) = s 0 f (τ )dτ . A necessary and sufficient condition for the existence of a solution φ of (1.6 ) with initial blow-up is

∞ a ds f (s) < ∞. (1.8)
Furthermore the unique maximal solution φ is obtained by inversion from the formula ∞ φ(t) ds f (s) = t ∀t > 0.

(1.9)

It is known that, if f is convex, (1.7 ) implies (1.8 ). If (1.7 ) holds and there exists a maximal solution to (1.3 ), it is not always true that this maximal solution is a large solution. In the case of a general nonlinearity, only sufficient conditions are known, independent of the regularity of ∂Ω. We recall some of them.

If N ≥ 3 and f satisfies the weak singularity assumption ∞ a s -2(N -1)/(N -2) f (s)ds < ∞ ∀a > 0.

(1.10)

If N = 2 and the exponential order of growth of f defined by

a + f = inf a ≥ 0 : ∞ 0 f (s)e -as ds < ∞ (1.11)
is finite.

When f (u) = u q with q > 1, (1.10 ) means that q < N/(N -2). When q ≥ N/(N -2) the regularity of ∂Ω plays a crucial role in the existence of large solutions. A necessary and sufficient condition involving a Wiener type test which uses the C R N 2,q ′ -Bessel capacity has been obtained by probabilistic methods by Dhersin and Le Gall [START_REF] Dhersin | Wiener's test for super-Brownian motion and the Brownian snake[END_REF] in the case q = 2 and extended to the general case by Labutin [START_REF] Labutin | Wiener regularity for large solutions of nonlinear equations[END_REF].

Uniqueness of the large solution of (1.3 ) has been obtained under three types of assumptions (see [START_REF] Marcus | Uniqueness and asymptotic behaviour of solutions with boundary blow-up for a class of nonlinear elliptic equations[END_REF], [START_REF] Marcus | Existence and uniqueness results for large solutions of general nonlinear elliptic equations[END_REF] and [START_REF] Marcus | Maximal solutions for -∆u + u q = 0 in open and finely open sets[END_REF]):

If ∂Ω = ∂Ω c and f (u) = u q with 1 < q < N/(N -2) or if N = 2 and f (u) = e au .
If ∂Ω is locally a continuous graph and f (u) = u q with q > 1 or f (u) = e au .

If f (u) = u q with q ≥ N/(N -2) and C R N 2,q ′ (∂Ω \ Ωc ) = 0, where Ẽ denotes the closure of a set in the fine topology associated to the Bessel capacity C R N 2,q ′ . In this article we extend most of the above mentioned results to the parabolic equation (1.1 ). We first prove that, if f is super-additive, i. e.

f (x + y) ≥ f (x) + f (y) ∀(x, y) ∈ R × R, (1.12)
and satisfies (1.7 ) and (1.8 ), there exists a maximal solution u

Q Ω to (1.1 ) in Q Ω , and it satisfies u Q Ω (x, t) ≤ w Ω (x) + φ(t) ∀(x, t) ∈ Q Ω . (1.13)
If we assume also that ∂Ω = ∂Ω c , there holds

max{w Ω (x), φ(t)} ≤ u Q Ω (x, t) ∀(x, t) ∈ Q Ω . (1.14)
Under the assumption ∂Ω = ∂Ω c , it is possible to consider a decreasing sequence of smooth bounded domains Ω n such that Ω n ⊂ Ω n-1 , Ω = ∩Ω n , and prove that the increasing sequence of large solutions u

Q Ω n of (1.1 ) in Q Ω n := Ω n × (0, ∞), converges to the exterior maximal solution u Q Ω of (1.1 ) in Q Ω .
If we proceed similarly with the large solutions w Ω n of (1.3 ) in Ω n and denote by w Ω their limit, then we prove that

max{w Ω (x), φ(t)} ≤ u Q Ω (x, t) ∀(x, t) ∈ Q Ω . (1.15)
The main result of this article is the following

Theorem 1. Assume Ω is a bounded domain such that ∂Ω = ∂Ω c , f ∈ C(R) is nondecreas-
ing and satisfies (1.7 ), (1.8 ) and (1.12 ). Then, if w Ω = w Ω , there holds u

Q Ω = u Q Ω .
Consequently, if (1.3 ) admits a unique large solution in Ω, the same holds for (1.1 ) in

Q Ω ∞ .

The maximal solution

In this section Ω is a bounded domain in R N and f ∈ C(R) is nondecreasing and satisfies (1.7 ) and (1.8 ). We set k 0 = inf{ℓ ≥ 0 : f (ℓ) > 0} and assume also that, for any m ∈ R there exists

L = L(m) ∈ R + such that ∀(x, y) ∈ R 2 , x ≥ m, y ≥ m =⇒ f (x + y) ≥ f (x) + f (y) -L.
(2.1)

Theorem 2.1 Under the previous assumptions there exists a maximal solution

u Q Ω in Q Ω ∞ . Proof.
Step 1-Approximation and estimates. Let Ω n be an increasing sequence of smooth domains such that Ω n ⊂ Ω n+1 and ∪Ω n = Ω. For each of these domains and (n, k) ∈ N 2 * we denote by w = w n,k the solutions of

-∆w + f (w) = 0 in Ω n w = k in ∂Ω n . (2.2)
where

∂ p Q Ωn ∞ := ∂Ω n × (0, ∞) ∪ Ω n × {0}
. By [START_REF] Keller | On solutions of ∆u = f (u)[END_REF] there exists a decreasing function g from R + to R, with limit ∞ at zero, such that

w n,k (x) ≤ g ρ ∂Ωn (x) ∀x ∈ Ω n . (2.
3)

The mapping k → w n,k is increasing, while n → w n,k is decreasing. If we set

w Ω = lim n→∞ lim k→∞ w n,k , (2.4)
it is classical that w Ω is the maximal solution of (1.3 ) in Ω, and it satisfies

w(x) ≤ g (ρ ∂Ω (x)) ∀x ∈ Ω. (2.5)
We denote also by u = u n,k the solution of

u t -∆u + f (u) = 0 in Q Ωn ∞ u = k in ∂ p Q Ωn ∞ .
(2.6)

By the maximum principle k → u n,k is increasing and n → u n,k decreasing. If we denote by φ the maximal solution of the ODE (1.5 ), then φ(t) is expressed by inversion by (1.9 ). If

t k = φ-1 (k), there holds, since φ is decreasing, φ(t + t k ) ≤ u n,k (x, t) in Q Ωn ∞ . (2.7) Furthermore, if f (k) ≥ 0 (which holds if k ≥ k 0 ), w n,k ≤ k. Therefore w n,k (x) ≤ u n,k (x, t) in Q Ωn ∞ . (2.8)
Combining (2.7 ) and (2.8 ), we derive

max{w n,k (x), φ(t + t k )} ≤ u n,k (x, t) ∀(x, t) ∈ Q Ωn ∞ .
(2.9)

Next we obtain an upper estimate. Let T > 0 and m ∈ R such that min{w Ω (x) : x ∈ Ω} > m ≥ φ(T ).

For n ≥ n 1 and k ≥ k 1 there holds min{w n,k (x) :

x ∈ Ω} ≥ m. Let L = L(m) ≥ 0 be the corresponding damping term from (2.1 ). If v n,k = w n,k (x) + φ(t + t k ), then it satisfies v t -∆v + f (v) = f (v) -f ( φ(. + t k )) -f (w n,k ) ≥ -L if (x, t) ∈ Ω n × [0, T -t k ]. (2.10) Since L ≥ 0, the function ṽn,k := v n,k + Lt is a supersolution for (1.1 ) in Q Ωn T -t k := Ω n ×(0, T -t k ) which dominates u n,k on ∂ p Q Ωn T -t k , thus in Q Ωn T -t k by the maximum principle. Therefore u n,k (x, t) ≤ w n,k (x) + φ(t + t k ) + Lt ∀(x, t) ∈ Q Ωn T -t k . (2.11)
Step 2-Final estimates and maximality. Using the different monotonicity properties of the mapping (k, n) → w n,k and the estimates (2.9 ) and (2.11 ), it follows that the function defined by

u Q Ω := lim n→∞ lim k→∞ u n,k (2.12) is a solution of (1.1 ) in Q Ω ∞ . Furthermore max{w Ω (x), φ(t)} ≤ u Q Ω (x, t) ∀(x, t) ∈ Q Ω ∞ , (2.13)
and

u Q Ω (x, t) ≤ w Ω (x) + φ(t) + tL(φ(T )) ∀(x, t) ∈ Q Ω T . (2.14) since φ(T ) ≤ min{w Ω (x) : x ∈ Ω}. Next, we consider u ∈ C 2,1 (Q Ω ∞ ), solution of (1.1 ) in Q Ω ∞ . Then, for ǫ > 0 and n ∈ N, there exists k * > 0 such that for k ≥ k * , u n,k (x, t -ǫ) ≥ u(x, t) ∀(x, t) ∈ Ω n × (ǫ, ∞). Letting successively k → ∞, n → ∞ and ǫ → 0, yields to u Q Ω ≥ u in Q Ω ∞ .
Since w Ω be a large solution in Ω implies the same boundary blow-up for u Q Ω on ∂Ω × (0, ∞), we give below some conditions which implies that u Q Ω is a large solution.

Corollary 2.2 Assume the assumptions of Theorem 2.1 are fulfilled. Then u Q Ω is a large solution if one of the following additional conditions is satisfied: (i) N ≥ 3 and f satisfies the weak singularity condition (1.10 ).

(ii) N = 2 and the exponential order of growth of f defined by (1.11 ) is positive. (iii) N ≥ 3 and ∂Ω satisfies the Wiener regularity criterion. Proof. Under condition (i) or (ii), for any x 0 ∈ ∂Ω, there exists a solution w c,x0 of

-∆w + f (w) = cδ x0 in B R (x 0 ) w = 0 in ∂B R (x 0 ), (2.15)
where R > 0 is chosen such that Ω ⊂ B R (x 0 ) and c > 0 is arbitrary under condition (i) and smaller that 2/a + f in case (ii). The function w c,x0 is radial with respect to x 0 and

lim x→x0 w c,x0 (x) = ∞.
If x ∈ Ω, we denote by x 0 a projection of x on ∂Ω. Since

w n (x) ≥ w c,x0 (x) =⇒ w Ω (x) ≥ w c,x0 (x),
we derive from (2.13 ), lim

ρ ∂Ω (x)→0 u Q Ω (x, t) = ∞,
uniformly with respect to t > 0. In case (iii) we see that, for any k > 0

w Ω (x) ≥ w k,∞ (x) ∀x ∈ Ω, (2.16)
where w k,∞ is the solution of (2.2 ), with Ω n replaced by Ω. This again implies (2.13 ).

Using estimate (2.13 ) leads to the asymptotic behavior of u Q Ω (x, t) when t → ∞.

Corollary 2.3 Assume the assumptions of Theorem 2.1 are fulfilled. Then u Q Ω (x, t) → w Ω (x) locally uniformly on Ω when t → ∞.

Proof. For any k > k 0 and n ∈ N * and any s > 0, there holds by the maximum principle,

u n,k (x, s) ≤ k = u n,k (x, 0) ∀x ∈ Ω n .
Using the monotonicty of f , we derive u n,k (x, t+s) ≤ u n,k (x, t) for any (x,

t) ∈ Q Ωn ∞ . Letting k → ∞ and then n → ∞ yields to u Q Ω (x, t + s) ≤ u Q Ω (x, t) ∀(x, t) ∈ Q Ω ∞ .
(2.17)

It follows that u Q Ω (x, t) converges to some W (x) as t → ∞ and w Ω ≤ W from (2.13 ). Using the parabolic equation regularity theory, we derive that the trajectory T := t≥0 {u Q Ω (., t)} is compact in the C 1 loc (Ω)-topology. Therefore W is a solution of (1.3 ) in Ω. It coincides with w Ω because of the maximality.

Large solutions

In this section we construct a minimal-maximal solution of (1.1 ) which is the minimal large solution whenever it exists. If ∂Ω is regular enough, the construction of the minimal large solution is easy.

Q Ω to (1.1 ) in Q Ω ∞ . Furthermore max{w Ω (x), φ(t)} ≤ u Q Ω (x, t) ∀(x, t) ∈ Q Ω ∞ , (3.1)
and, for any T > 0,

u Q Ω (x, t) ≤ w Ω (x) + φ(t) + tL(φ(T )) ∀(x, t) ∈ Q Ω T , (3.2)
where L(φ(T )) is as in (2.16 ), and w Ω denotes the minimal large solution of (1.3 ) in Ω.

Proof. For k ≥ k 0 (see Section 2), we denote by u k the solution of

u t -∆u + f (u) = 0 in Q Ω ∞ u = k in ∂ p Q Ω ∞ . (3.3)
When k increases, u k increases and converges to some large solution u

Q Ω of (1.1 ) in Q Ω ∞ . If u is any large solution of (1.1 ) in Q Ω
∞ , then the maximum principle and (1.2 ) implies u ≥ u k . Therefore u ≥ u Q Ω . The same assumption allows to construct the solution w k of

-∆w + f (w) = 0 in Ω w = k in ∂Ω, (3.4)
and, by letting k → ∞, to obtain the minimal large solution w Ω of (1.3 ) in Ω. Next we first observe, that, as in the proof of Theorem 2.1, (2.10 ) applies under the form

φ(t + t k ) ≤ u k (x, t) in Q Ω ∞ , (3.5)
where, we recall it, t k = φ -1 (k). In the same way, for k ≥ k 0 (with f (k) ≥ 0), (2.11 ) holds under the form

w k (x) ≤ u k (x, t) in Q Ω ∞ . (3.6) Letting k → ∞ yields to max{w Ω (x), φ(t)} ≤ u Q Ω (x, t) ∀(x, t) ∈ Q Ω ∞ . (3.7)
In order to prove the upper estimate we consider the same m as it the proof of Theorem 2.1 such that min{min{w k (x) : x ∈ Ω}, φ(t)} ≥ m, and for k ′ > k, there holds

w k ′ + φ ≥ k = w k ∂pQ Ω T . Since w k ′ (x) + φ(t) + tL is a supersolution for (1.1 ) in Q Ω T it follows w k ′ + φ + tL ≥ w k in Q Ω T . Letting successively k ′ → ∞ and k ′ → ∞, we derive (3.2 ).
From this result we can deduce uniqueness results for solution of Corollary 3.2 Under the assumptions of Theorem 3.1, if we assume moreover that f is convex and, for any θ ∈ (0, 1), there exists r θ such that

r ≥ r θ =⇒ f (θr) ≤ θf (r). (3.8) Then w Ω = w Ω =⇒ u Q Ω = u Q Ω . (3.9)
Proof. We fix T ∈ (0, 1] such that tL(φ(1)) ≤ φ(t) ∀t ∈ (0, T ],

(remember that L is always positive) and

2w Ω (x) + φ(t) ≥ 0 ∀(x, t) ∈ Q Ω T .
Then w Ω (x) + φ(t) ≥ 0 and

w Ω (x) + φ(t) + tL(φ(1)) ≤ w Ω (x) + 2φ(t) ≤ w Ω (x) + 2φ(t) ≤ 3 w Ω (x) + φ(t) ,
from which inequality follows

2 -1 w Ω (x) + φ(t) ≤ u Q Ω (x, t) ≤ 3 w Ω (x) + φ(t) ∀(x, t) ∈ Q Ω T .
Therefore, if w Ω = w Ω , it follows

u Q Ω ≤ u Q Ω ≤ 6u Q Ω in Q Ω T . (3.10) Next we assume u Q Ω < u Q Ω and set u * = u Q Ω - 1 6 u Q Ω -u Q Ω .
Since f is convex, u * is a supersolution of (1.1 ) in Q Ω T (see [START_REF] Marcus | The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case[END_REF], [START_REF] Marcus | Existence and uniqueness results for large solutions of general nonlinear elliptic equations[END_REF]) and u * < u Q Ω . Up to take a smaller T , we can also assume from (3.8 )

that min{u Q Ω (x, t) : (x, t) ∈ Q Ω T } ≥ r 1/12 , thus f (u Q Ω /12) ≤ 1 12 f (u Q Ω ) in Q Ω T . Therefore u Q Ω /12 is a subsolution for (1.1 ) in Q Ω T and 12 -1 u Q Ω < u * .
Using a standard result of sub and super solutions and the fact that f is locally Lipschitz continuous, we see that there exists some u # solution of (1.1 ) in Q Ω T such that

1 12 u Q Ω ≤ u # ≤ u * < u Q Ω in Q Ω T . (3.11)
Then u # is a large solution, which contradicts the minimality of

u Q Ω on Q Ω T . Finally u Q Ω = u Q Ω in Q Ω ∞ . Lemma 3.3 Let Ω be a bounded domain in R N and, for ǫ > 0, Ω ǫ := {x ∈ R N : dist (x, Ω) < ǫ}.
The four following assertions are equivalent:

(i) ∂Ω = ∂Ω c .
(ii) For any x ∈ ∂Ω, there exists a sequence {x n } ⊂ Ω c such that x n → x.

(iii) For any x ∈ ∂Ω and any ǫ > 0,

B ǫ (x) ∩ Ω c = ∅. (iv) For any x ∈ ∂Ω, lim ǫ→0 dist (x, Ω c ǫ ) = 0. (v) Ω = o Ω.
Proof. There always holds ∂Ω

c = Ω c ∩ Ω ⊂ Ω c ∩ Ω = ∂Ω.
(i)=⇒ (iii). Assume (iii) does not hold, there exist x 0 ∈ ∂Ω and ǫ 0 > 0 such that B ǫ0 (x 0 ) ∩ Ω c = ∅. Thus x 0 / ∈ Ω c , and x 0 / ∈ ∂Ω c . Therfore (i) does not hold.

(iii)=⇒ (i). Let x 0 ∈ ∂Ω. If, for any ǫ > 0, B ǫ (x)∩Ω c = ∅, then x ∈ Ω c . Because x ∈ Ω c ∩Ω, it implies that x ∈ Ω ∩ Ω c = ∂Ω c .
The equivalence between (iii) and (ii) is obvious.

(ii))=⇒ (iv). We assume (iv) does not hold. There exist x 0 ∈ ∂Ω, α > 0 and a sequence of positive real numbers {ǫ n } converging to 0 such that dist (x 0 , Ω c ǫn ) ≥ α. Since for ǫ ≥ ǫ n , Ω c ǫ ⊂ Ω c ǫn , there holds dist (x 0 , Ω c ǫ ) ≥ α. Furthermore, this inequality holds for any ǫ > 0. If there exist a sequence {x n } ⊂ Ω c such that x n → x 0 , then dist (x n , Ω) = δ n > 0, thus

x n ∈ Ω c δn . Consequently |x n -x 0 | ≥ α, which is impossible. Therefore (ii) does not hold. (iv)=⇒ (iii). Let x ∈ ∂Ω and x n ∈ Ω c 1/n such that |x -x n | = dist (x, Ω c 1/n ) → 0. Since Ω c 1/n ⊂ Ω, x n ∈ Ω c and x n → x.
(iii)=⇒ (v). We first notice that Ω = ∩ ǫ>0 Ω ǫ = ∩ ǫ>0 Ω ǫ and Ω ⊂ o Ω. If there exists some

x ∈ o Ω \ Ω, then for some ǫ > 0, B ǫ (x) ⊂ Ω which implies B ǫ (x) ∩ Ω c = ∅. But x /
∈ Ω implies x ∈ ∂Ω. Thus (iii) does not hold.

(v)=⇒ (iii). If (iii) does not hold, there exists x ∈ ∂Ω and ǫ > 0 such that

B ǫ (x) ∩ Ω c = ∅ ⇐⇒ B ǫ (x) ⊂ Ω. Therefore x ∈ o Ω \ Ω. Definition 3.4 A solution U (resp. W to problem (1.1 ) in Q Ω ∞ (resp. (1.3 ) in Ω) is called an exterior maximal solution if it is larger than the restriction to Q Ω ∞ (resp. Ω) of any solution of (1.1 ) (resp. (1.3 ) ) defined in an open neighborhood of Q Ω ∞ (resp. Ω)). Proposition 3.5 Assume Ω is a bounded domain in R N such that ∂Ω = ∂Ω c and f ∈ C(R)
is nondecreasing and satisfies (1.7 ). Then there exists an exterior maximal solution w * Ω to problem (1.3 )

j from H j ∩ O j into R + such that ∂Ω ∩ O j = x = x ′ + h j (x ′ )ν j : ∀x ′ ∈ H j ∩ O j
where ν j is a fixed unit vector orthogonal to H j and ∂Ω ⊂ ∪ j O j . We can assume that H j ∩ O j = B j is a (N-1) dimensional closed ball and,

G j := {x = x ′ + tν j : x ′ ∈ B j , 0 ≤ t < h j (x ′ )} ⊂ Ω, G # j := {x = x ′ + tν j : x ′ ∈ B j , h j (x ′ ) < t ≤ a} ⊂ Ω c .,
for some a > 0 such that a/4 < h j (x ′ ) < 3a/4 for any x ′ ∈ B j . Finally, we can assume that

O j = {x = x ′ + tν j : x ′ ∈ B j , 0 ≤ t ≤ a}.
Let ǫ ∈ (0, a/8) and

G j,ǫ := {x = x ′ + tν j : x ′ ∈ B j , ǫ ≤ t < h j (x ′ ) + ǫ}.
There exists a smooth bounded domain Ω ′ such that Ω ⊂ Ω ′ and

∂Ω ′ ∩ O j = {x = x ′ + ℓ(x ′ )ν j : x ′ ∈ B j , h(x ′ ) + ǫ/2 ≤ ℓ(x ′ ) ≤ h(x ′ ) + 3ǫ/2}, where ℓ ∈ C ∞ (B j ). We denote G j := G j,0 , ∂ p G j,ǫ := {x = x ′ + tν j : x ′ ∈ ∂B j , ǫ ≤ t ≤ h j (x ′ ) + ǫ} ∪ {x = x ′ + ǫν j : x ′ ∈ B j }, and
∂ u G j,ǫ := {x = x ′ + (h j (x ′ ) + ǫ)ν j : x ′ ∈ B j }.
Let w ′ be the minimal large solution of (1.3 ) in Ω ′ , α ′ = min{w ′ (x) : x ∈ Ω ′ } and W ǫ the minimal solution of

     -∆W + f (W ) = 0 in G j,ǫ W = α ′ in ∂ p G j,ǫ lim t→h(x ′ )+ǫ W (x ′ + tν j ) = ∞ ∀x ′ ∈ B j . (3.12)
Then w ′ ≥ W ǫ in G j,ǫ ∩ Ω ′ . Furthermore W ǫ (x) = W ǫ (x ′ + tν j ) = W 0 (x ′ + (tǫ)ν j ) for any x ′ ∈ B j and ǫ < t < h(x ′ ) + ǫ. Therefore, given k > 0, there exists δ k > 0 such that for any

x ′ ∈ B j and h j (x ′ )δ k ≤ t < h j (x ′ ) =⇒ W 0 (x ′ + tν j ) ≥ k.

As a consequence, lim inf t→hj (x ′ ) w * Ω (x ′ + tν j ) ≥ k, uniformly with respect to x ′ ∈ B j . This implies that w * Ω is a large solution. Remark. We conjecture that the equality w * Ω = w Ω holds under the mere assumption that the Wiener criterion is satisfied. Theorem 3.7 Assume Ω is a bounded domain in R N such that ∂Ω = ∂Ω c and f ∈ C(R)

satisfies (1.7 ), (1.8 ) and (2.1 ). Then there exists a exterior maximal solution u * Q Ω to problem (1.1 ). Furthermore estimates (3.1 ) and (3.2 ) hold with w Ω replaced by the exterior maximal solution w * Ω to problem (1.3 ) in Ω.

Proof. The construction of u * Q Ω is similar to the one of w Ω , since we can restrict to consider open neighborhoods Q 1/n = Ω 1/n ×(-1/n, ∞). Then u * Q Ω is the increasing limit of the minimal large solutions u n of (1.1 ) in

Q 1/n , since Q Ω ∞ = ∩ n Q 1/n and, by Lemma 3.3-(v), Q Ω ∞ = o Q Ω ∞ .
We recall that the minimal large solution w n of (1.3 ) in Ω 1/n is the increasing limit, when k → ∞, of the sequence of solution {w k n } of

-∆w + f (w) = 0 in Ω 1/n w = k on ∂Ω 1/n , (3.13)

Theorem 3 . 1

 31 Let Ω be a bounded domain in R N the boundary of which satisfies the Wiener regularity condition. If f ∈ C(R) is nondecreasing and satisfies (1.7 ),(1.8 ) and (2.1 ), then there exists a minimal large solution u

  in Ω. Proof. Since ∂Ω = ∂Ω c we can consider the decreasing sequence of the Ω 1/n defined in Lemma 3.3 with ǫ = 1/n and, for each n, the minimal large solutions w n of (1.3 ) in Ω 1/n : this possible since ∂Ω 1/n is Lipschitz. The sequence {w n } is increasing. Its restriction to Ω is bounded from above by the maximal solution w Ω . It converges to some function w * Ω . By Lemma 3.3-(v), w * Ω is a solution of (1.3 ) in the interior of ∩ n Ω 1/n which is Ω. If w is any solution of (1.3 ) defined in an open neighborhood of Ω, it is defined in Ω 1/n for n large enough and therefore smaller than w n . Thus w Ω ≤ w * Ω . Consequently, w * Ω coincides with the supremum of the restrictions to Ω of solutions of (1.3 ) defined in an open neighborhood of Ω. Consequently, if w * Ω is a large solution, it coincides with the minimal large solution w Ω . Because ∂Ω is compact, there exists a finite number of bounded open subset O j , hyperplanes H j and continuous functions h

	Proposition 3.6 Let f ∈ C(R) be a nondecreasing function for which (1.7 ) holds and Ω a bounded domain in R N such that ∂Ω = ∂Ω c . Then w * Ω is smaller than any large solution.
	Furthermore, if ∂Ω satisfies the Wiener regularity criterion and is locally the graph of a
	continuous function, then w Ω = w * Ω .
	Proof. We first notice that Wiener criterion implies statement (iii) in Lemma 3.3, hence
	∂Ω = ∂Ω

c

. If w Ω is a large solution, it dominates on ∂Ω, and therefore in Ω by the maximum principle, the restriction to Ω of any function w solution of

(1.3 )

in an open neighborhood of Ω. Then w * Ω ≤ w Ω .

while the minimal large solution u n of (1.1 ) in Q 1/n is the (always increasing) limit of the solutions u k n of

The upper estimate is proved in the following way.

T , thus it dominates the minimal large solution of (1.1 ) in Q is the minimal large solution in Q Ω ∞ for the same reasons as w * Ω . Therefore the proof of Corollary 3.2 applies and it implies the result.

Remark. We conjecture that (3.17 ) holds, even if w * Ω is not a large solution.


A W Liu

S Kassi

A Campargue
email: alain.campargue@ujf-grenoble.fr

L R Brown

Michel Herman
email: mherman@ulb.ac.bepgj.

High Sensitivity CW-Cavity Ring Down Spectroscopy of CH 4 in the 1.55 m transparency window

Keywords: methane, CH 4, near infrared, Cavity ring down spectroscopy, weak absorption

INTRODUCTION

The quality of the analysis of the near infrared spectrum of the giant outer planets and Titan is strongly dependant on the knowledge of the methane absorption spectrum in the experimental conditions existing on these planets (see for instance Ref. [1]). Despite considerable progress achieved on the theoretical modelling of the CH 4 absorption spectrum [2], individual line parameters cannot presently be satisfactorily modelled or predicted below 1.7 m. As a consequence, the modelling of the spectrum of the above planetary atmospheres relies on band models which are extrapolation from laboratory measurements performed at higher temperatures and/or shorter path lengths. In particular, the actual CH 4 absorption in the low opacity spectral regions is of first importance to investigate the deeper atmosphere and determine the surface reflectivity (with implications on its composition). However, as recently discussed in Ref. [3], the methane band models have a limited accuracy in these "transparency windows" corresponding to very weak absorption of methane. Empirical line by line spectroscopic parameters is the alternative to the band models.

But the present situation of databases such as HITRAN [4] or GEISA [5] are far from fulfilling the needs in the case of methane. For instance, above 5000 cm -1 , HITRAN is limited to empirical line parameters which were obtained at room temperature by high resolution Fourier Transform Spectroscopy (FTS) with path lengths up to 433 meters [6]. In absence of rovibrational assignments or at least of the determination of the lower energy of the transitions, the line intensity cannot be extrapolated at low temperature and are then of limited use in planetology. Another drawback of these data in particular, in the transparency windows, is their sensitivity which is too limited to account for the methane spectrum with kilometric atmospheric scale heights. To our knowledge, apart from the FTS results of Ref.

[6], the only previous report devoted to the high resolution and high sensitivity spectrum of methane near 1.55 m, is the very recent contribution of Deng et al [7]. These authors used a Distributed Feed-Back (DFB) diode near 1.51 m and a 1159 meters path length (obtained with a 7.7 meters base cell) to record a 17 cm -1 wide spectral section around 6620 cm -1 by direct absorption spectroscopy.

As a contribution to the characterization of the methane absorption spectrum in spectral regions of very weak absorption, we have investigated the whole 1.60-1.50 m transparency window by high sensitivity CW-Cavity Ring Down Spectroscopy (CW-CRDS).

After a brief description of the spectrometer used in the next section, the obtained spectra will be presented. The high sensitivity of our experimental set up has allowed for the observation of a considerable number of new transitions and revealed an impressive complexity and congestion of the spectrum. The impact of the newly observed transitions to the total absorption in the considered region will be discussed.

EXPERIMENT

We have developed a fibered CW-CRDS spectrometer using DFB diode lasers dedicated to the characterization of the important 1.5 m atmospheric window of transparency. The typical achieved sensitivity of 2-5×10 -10 cm -1 and the 4 to 5 decades dynamic range on the intensity scale, have allowed for a significant improvement of the knowledge of the absorption spectrum of species of major atmospheric interest: H 2 O [8, 9],

12 CO 2 [10], 13 CO 2 [11,12], N 2 O [13] and ozone [14,15]. The set up has been described in Ref. [16] and its application for routine spectroscopic measurements has been described in Ref. [11]. Briefly, a single-mode fibre delivers laser radiation to one end of a vacuum-tight ringdown cell, made of a stainless steel tube. The fibered laser diode (NEL 20 mW butterfly mounted) was slowly temperature tuned from -10 to 60°C in order to cover a 30 cm -1 spectral region within about 70 minutes. The cavity mirrors (R= 1 m) are mounted on tilt stages, one of which includes a piezoelectric tube that modulates the cavity length by slightly more than /2. This makes the cavity modes oscillate by more than one Free Spectral Range thus producing a passage through resonance with the laser line, twice per modulation cycle. The cavity output was focused on an amplified InGaAs photodiode.The cavity losses at each laser wavelength were obtained by averaging the results of exponential fits to 70 ring-down events, thus giving one data point in the spectrum. The absorption coefficient(cm -1 ), was then directly calculated from the decrease of the cavity ring-down time  (in s) induced by the molecular absorption:

    c      



where c is the light velocity and  0 is the ring-down time of the empty cell (on the order of 60 ms) depending of the mirror losses

The ringdown cell temperature (typical value 298 K) and the gas pressure, measured by a capacitance gauge (MKS Baratron, 10 Torr range) are monitored during the spectrum acquisition. Two series of recordings were performed at 2.5 and 10 Torr. In these pressure conditions, the collisional broadening (9.0×10 -4 cm -1 HWHM at 10 Torr [4]) is significantly weaker than the Doppler width (HWHM 9.8×10 -3 cm -1 ). As the DFB linewidth is about one thousandth of the Doppler broadening, the observed line profile is mostly Gaussian.

Each 30 cm -1 wide spectrum was calibrated independently on the basis of the wavelength values provided by the Michelson-type wavemeter (Burleigh WA-1650, 60 MHz resolution and 100 MHz accuracy at this wavelength). The calibration was further refined by stretching the whole spectrum in order to match accurate positions of reference lines. For this purpose, we used CH 4 line positions measured by L. Brown [6] and adopted in the HITRAN database. The standard deviation error of the differences between CRDS and HITRAN line positions was minimized (typical rms values are less than 10 -3 cm -1 ). A maximum deviation of 2×10 -3 cm -1 was observed for a few reference lines which is then claimed here to be the maximum position error.

RESULTS AND DISCUSSION

The whole investigated region which extends from 6250 cm -1 (1.60 m) to 6680 cm -1

(1. arises when estimating the minimum measurable intensity value: as illustrated on the different plots of Fig. 2 where the intensity scale has been expanded from 10 -5 to 4 ×10 -8 cm -1 , it seems impossible to localize a spectral section free of absorption lines and then difficult to evaluate the zero absorption level. We have estimated the density of lines detected in the 6600 -6630 cm -1 section and obtained a value of about 20 lines/cm -1 which leads to an extrapolated value of about 9000 transitions for the whole 6250 -6680 cm -1 region …

The considered region of weak opacity lies between two strongly absorbing regions corresponding to the tetradecad ( 14) and the icosad (20) of vibrational states. Interactions connecting the different upper rovibrational levels are expected to be so strong and numerous that an accurate theoretical treatment is probably intractable. The observed spectral congestion is then the result of the superposition of (i) transitions from highly excited rotational levels reaching these strongly interacting rovibrational levels, (ii) hot bands arising from the first dyad near 1500 cm -1 and (iii) transitions due to minor isotopologues present in natural abundance in our sample. It is worth noting that, as a consequence of the isotopic shift of the band centers, the minor isotopologues may have a relative contribution in the transparency windows which largely exceeds their respective abundance. Interestingly, the appearance of the observed spectrum show similarities with the high resolution spectrum of methane recorded at much higher energies in the visible range [17,18].

As spectra of planetary atmospheres are generally obtained at low resolution (several cm -1 or more), it is interesting to estimate the importance of the newly observed transitions on the integrated absorption in the region. For this purpose, a "low resolution CW-CRDS spectrum" was simulated by convolution with a 10 cm -1 wide (FWHM) Gaussian function.

The resulting spectrum is displayed on Fig. 3 and compared to a "low resolution HITRAN spectrum" obtained by affecting the same normalized Gaussian profile function to each transition provided by HITRAN for methane. The obtained curve multiplied by the molecular density (at 10.0 Torr) gives the absorption coefficient allowing for a quantitative comparison with the low resolution CRDS spectrum. The spectra displayed in Fig. 3 show that an additional absorbance ranging between 1.4 10 -8 and 10 -7 cm -1 is induced by the newly detected lines. ((Note that we estimate to less than 10 -9 cm -1 uncertainty on the base line of the CRDS spectrum). This absorption depends smoothly on the wavelength and could be interpreted as a "continuum" in absence of our high resolution spectrum.

As expected, the relative impact of the newly detected transitions is much higher in the regions of lower opacities. The additional absorbance represents 100 % of the CRDS absorption near 6325 cm -1 and about 25 % in the low and high energy regions. Fig. 4 shows the region of highest transparency: While the HITRAN absorption is nonexistent between 6310 and 6350 cm -1 , the CRDS absorption is always larger than 1.4×10 -8 cm -1 On the other hand, the low and high energy regions are dominated by stronger transitions which have a dominant contribution to the absorption, leading to a better agreement between CRDS and HITRAN.

We have calculated the sum of the line intensities reported in Ref.

[7] from high sensitivity direct absorption spectroscopy between 6607.9 and 6624.6 cm -1 . The obtained value is 9.23×10 -24 cm/molecule which exceeds by 15 % the integrated intensity value calculated from the HITRAN database (7.14 ×10 -24 cm/molecule). For comparison, the CRDS absorption exceeds by about 35 % HITRAN absorption in this particular region.

CONCLUSION

The highly sensitive absorption spectrum of methane has been recorded at Doppler limited resolution between 6250 and 6680 cm -1 corresponding to the full coverage of the 1.55 m transparency window. The achieved experimental sensitivity has evidenced a highly complex and congested structure with a density of transitions on the order of 20 per cm -1 . The high dynamics on the line intensities allowed by the CW-CRDS technique reveals a kind of "fractal" character with no region free of absorption lines. The total number of observed transitions is estimated to be about 9000 while 684 lines are included in the HITRAN database. From simulations at low resolution (10 cm -1 FWHM), it has been found that the impact of the newly detected transitions is considerable in the centre of the region (near 1.57 m) corresponding to the highest transparency.

Unfortunately the present results will be of little benefit to interpret the methane opacities in the low temperature conditions existing in the outer planets or Titan. In absence of determination of the lower energy level of the observed transitions, the room temperature spectrum cannot be extrapolated at temperatures different from 295 K. This is why we are planning to combine our CW-CRDS spectrometer with a cell cooled at liquid nitrogen. Such combination of high sensitive laser absorption techniques with low temperature cells has been applied to methane for photoacoustic spectroscopy [17] and Intracavity Laser Absorption spectroscopy (ICLAS) [19]. The combination with CW-CRDS is more challenging as the use of optical windows is excluded. Even if the spectral congestion and line width will be drastically reduced at 77 K, we anticipate a considerable task for the line by line retrieval of the intensities.

  50 m), was recorded by the help of 22 DFB diodes. An overview of the central part of the CW-CRDS spectrum is presented in Fig. 1. The high sensitivity and high dynamics achieved with our CW-CRDS spectrometer is illustrated in Fig. 2: five successive zooms reveal the considerable complexity of the room temperature spectrum of methane in the considered region. The CH 4 transitions present in the HITRAN database has been marked with an open circle on Fig. 2. The HITRAN database provides line parameters of 684 transitions in the 6250-6680 cm -1 region, with minimum line intensity of 3.7×10 -26 cm/molecule. A gain of sensitivity of about 3 orders of magnitude is obtained by CW-CRDS allowing for the measurements of intensity values smaller than 10 -28 cm/molecule. However a major difficulty

Figure 1 .

 1 Figure 1.Overview of the room temperature spectrum of methane recorded by CW-CRDS between 6250 and 6680 cm -1 . The whole spectral region was covered by the help of 22 DFB diode lasers which correspond to the alternated colors. The gas pressure was 10.0 Torr.

Figure 2 .

 2 Figure 2. Five successive enlargements of the CW-CRDS spectrum of methane revealing a "fractal" structure of the highly congested spectrum of methane in the 1.55 m transparency window. The pressure was 10.0 Torr. The open circles mark the lines included in the HITRAN database corresponding to the right hand intensity scale. Note the high dynamics of the CW-CRDS spectrometer allowing for the measurement of absorption coefficient differing by four orders of magnitude.

Figure 3 .

 3 Figure 3. Comparison of the "low resolution" absorption spectrum of methane (p = 10.0 Torr) simulated from the CW-CRDS spectrum (solid line) and from the HITRAN database (dashed line). The spectra were obtained by convolution with a normalized Gaussian function (FWHM= 10.0 cm -1 ) modeling the apparatus function of a low resolution spectrometer. The intensity given on the right hand scale corresponds to the recorded CW-CRDS spectrum displayed on the background plot.

Figure 4 .

 4 Figure 4. Same as Fig. 3 for the 6300-6400 cm -1 high transparency region. The open circles mark the lines included in the HITRAN database. Their vertical position corresponds to the HITRAN intensity values which have been scaled to match the CW-CRDS spectrum.

Figure

  Figure

We thank the referee for the minor corrections he noted. They have been taken into account in the revised version Regards, Alain Campargue Acknowledgements This work is supported by the Programme National de Planétologie (CNRS, INSU) and a collaborative project between CNRS and CAS-China (PICS grant No 3359). A. Liu thanks the European Space Agency for financial support.

Abstract

The high resolution absorption spectrum of methane has been recorded at room temperature by high sensitivity CW-Cavity Ring Down Spectroscopy between 1.50 and 1.60 m (6250-6680 cm -1 ). This spectral region has a low opacity and is then of particular interest for the spectral analysis of the outer planets and Titan.

The recordings were performed with a sensitivity corresponding to a noise equivalent absorption of  min ~510 -10 cm -1 and allowed for the detection of lines with an intensity smaller than 10 -28 cm/molecule far beyond the weakest lines provided by the HITRAN database. The spectrum exhibits a highly complex and congested structure with a kind of "fractal" character and a density of transitions of a few tens per cm -1 . The impact of the weak newly observed transitions on the total absorption in the region is discussed.