Ronald Enrique Morales-Vargas
email: ronald.mor@mahidol.ac.th

Dorothée Missé
email: dorothee.misse@ird.fr

Irwin F Chavez
email: irwin.cha@mahidol.edu

Pattamaporn Kittayapong
email: pattamaporn.kit@mahidol.ac.th

Vector Competence for Dengue-2 Viruses Isolated from Patients with Different Disease Severity

Keywords: dengue virus, disease severity, vector competence Conceptualization, methodology, formal analysis, investigation, data curation, and writing -original draft preparation,

Dynamics of dengue serotype 2 virus isolated from patients with different disease severity, namely flu-like classic dengue fever (DF) and dengue shock syndrome (DSS) were studied in its mosquito vector Aedes aegypti. We compared isolate infectivity and vector competence (VC) among thirty two A. aegypti-viral isolate pairs. Mosquito populations from high dengue incidence area exhibited overall greater VC than those from low dengue incidence area at 58.1% and 52.5%, respectively. On the other hand, the overall infection rates for the isolates ThNR2/772 (DF, 62.3%) and ThNR2/391 (DSS, 60.9%), were significantly higher than those for isolates ThNR2/406 (DF, 55.2%) and ThNR2/479 (DSS, 54.8%). These results suggest that the efficacy of dengue virus circulation was likely to vary according to the combination between the virus strains and origin of the mosquito strains, and this may have epidemiologic implications toward the incidence of flu-like classic dengue fever (DF) and dengue shock syndrome (DSS).

Introduction

Flulike classic dengue fever (DF) and dengue hemorrhagic fever (DHF) are increasingly important public health problems issues in the tropical region. Annually, an estimated 390 million dengue infections per year and 96 million of symptomatic dengue infections occur [START_REF] Bhatt | The global distribution and burden of dengue[END_REF]. It has been predicted that population at risk of dengue will rise by 2.25 (1.27-2.80) billion by the year 2080 compared to 2015, which will put over 6.1 (4.7-6.9) billion at risk-nearly 60% of the world's projected 2080 population [START_REF] Messina | The current and future global distribution and population at risk of dengue[END_REF]. DF/DHF and DSS has re-emerged in tropical and subtropical zones, with DHF as the prominent cause of pediatric mortality and hospitalization during the past 20 years in Southeast Asia [START_REF] Gubler | Dengue/dengue haemorrhagic fever, the emergence of a global health problem[END_REF][START_REF] Monath | Yellow fever and dengue: The interactions of virus, vector and host in the reemergence of epidemic disease[END_REF]. All dengue viruses (DENV) can result to DHF cases, but the severe form is mostly associated with DENV-2 and DENV-3 [START_REF] Nisalak | Serotype-specific dengue virus circulation and dengue disease in Bangkok, Thailand from 1973 to 1999[END_REF]. It has been suggested that DHF epidemics occur as a consequence of a very complex mechanism comprising the intersection of three groups of factors (viral, host, and epidemiological) [START_REF] Kouri | Why dengue haemorrhagic fever in Cuba? 2: An integral analysis[END_REF]. In addition, some amino acid changes on the premembrane (prM) and envelope (E) proteins of DEN-2 strains have been related to DHF epidemics [START_REF] Leitmeyer | Dengue virus structural differences that correlate with pathogenesis[END_REF]. Furthermore, Pandy and Igarashi classified Thailand DENV-2, according to the non-synonymous amino acid replacement, in three subtypes (I, II, and III) and proposed that viruses' molecular structures and patients' serological responses influence clinical severity [START_REF] Pandey | Severity-related molecular differences among nineteen strains of dengue of dengue type 2[END_REF].

Pathogens 2020, 9, 859 2 of [START_REF] Apostol | Population genetics with RAPD-PCR markers: The breeding structure of Aedes aegypti in Puerto Rico[END_REF] Aedes aegypti, the primary vector of dengue, is well-adapted to cohabitate with humans by breeding in man-made containers. Several population genetic and vector competence (VC) studies have been done on A. aegypti because of its ability to transmit dengue and yellow fever flaviviruses [START_REF] Aitken | Aedes aegypti strain fitness for yellow fever virus transmission[END_REF][START_REF] Rosen | Comparative susceptibility of mosquito species and strain to oral and parenteral infection with dengue and Japanese encephalitis viruses[END_REF][START_REF] Gubler | Variation in susceptibility to oral infection with dengue viruses among geographic strains of Aedes aegypti[END_REF][START_REF] Tabacnick | A world-wide survey of genetic variation in the yellow fever mosquito, Aedes aegypti[END_REF][START_REF] Apostol | Population genetics with RAPD-PCR markers: The breeding structure of Aedes aegypti in Puerto Rico[END_REF][START_REF] Bosio | Quantitative genetics of vector competence for dengue-2 virus in Aedes aegypti[END_REF][START_REF] Gorrochotegui-Escalante | Genetic isolation by distance among Aedes aegypti populations along the northeastern coast of Mexico[END_REF][START_REF] Bosio | Quantitative trait loci that control vector competence for dengue-2 virus in the mosquito Aedes aegypti[END_REF]. Genetic markers have been used to identify biotypes, biting behavior, and other epidemiologically relevant traits [START_REF] Tabanick | Evolutionary genetics and arthropod-borne disease. The yellow fever mosquito[END_REF]. Vector movement among disease foci or from a site of initial colonization can be investigated through genetic similarity between geographically heterogeneous populations [START_REF] Ballinger-Crabtree | Use of genetic polymorphisms detected by RAPD-PCR for differentiation and identification of Aedes aegypti subspecies and populations[END_REF]. The genetically intrinsic leniency of a vector to infection, replication, and virus transmission, vector competence [START_REF] Hardy | Susceptibility and resistance of vector mosquitoes[END_REF], present significant variability among A. aegypti populations particularly for DENV-2es [START_REF] Rosen | Comparative susceptibility of mosquito species and strain to oral and parenteral infection with dengue and Japanese encephalitis viruses[END_REF][START_REF] Gubler | Variation in susceptibility to oral infection with dengue viruses among geographic strains of Aedes aegypti[END_REF][START_REF] Tardieux | Variation among strains of Aedes aegypti in susceptibility to oral infection with dengue virus type 2[END_REF]. Genes or sets of genes controlling the midgut infection and escape barriers have been found to be associated with the VC of A. aegypti for flaviviruses [START_REF] Bosio | Quantitative genetics of vector competence for dengue-2 virus in Aedes aegypti[END_REF][START_REF] Gorrochotegui-Escalante | Genetic isolation by distance among Aedes aegypti populations along the northeastern coast of Mexico[END_REF][START_REF] Bosio | Quantitative trait loci that control vector competence for dengue-2 virus in the mosquito Aedes aegypti[END_REF].

There are several possible mechanisms that can affect the transmission potential of different dengue virus strains. Some strains may infect and replicate target cells more efficiently, sustain higher viremia in the human host, and infect more mosquitoes [START_REF] Vaughn | Dengue viremia titer, antibody response pattern, and virus serotype correlate with disease severity[END_REF][START_REF] Wu | Human skin Langerhans cells are targets of dengue virus infection[END_REF][START_REF] Morens | Growth of dengue type 2 virus isolates in human peripheral blood leukocytes correlates with severe and mild dengue disease[END_REF]. Studies have shown that dengue virus serotypes and strains within a serotype, between and within a genotype may vary in their ability to infect and disseminate in mosquitoes [START_REF] Rosen | Comparative susceptibility of mosquito species and strain to oral and parenteral infection with dengue and Japanese encephalitis viruses[END_REF][START_REF] Gubler | Variation in susceptibility to oral infection with dengue viruses among geographic strains of Aedes aegypti[END_REF][START_REF] Armstrong | Differential susceptibility of Aedes aegypti to infection by the American and Southeast Asian genotypes of Dengue type 2 virus[END_REF][START_REF] Morales | Infection and dissemination of two dengue type-2 viruses isolated from patients exhibiting different disease severity in orally infected Aedes aegypti from different geographic origin[END_REF][START_REF] Armstrong | Efficiency of dengue serotype 2 virus strains to infect and disseminate in Aedes aegypti[END_REF].

Several studies have reported variation in the oral receptivity of A. aegypti for DENV-2 at a regional geographic scale [START_REF] Tran | Aedes aegypti in Ho Chi Minh city (Viet Nam): Susceptibility to dengue type 2 virus and genetic differentiation[END_REF][START_REF] Vazeille-Falcoz | Variation in oral susceptibility to dengue type 2 virus of population of Aedes aegypti from the islands of Tahiti and Moorea, French Polynesia[END_REF][START_REF] Fouque | Aedes aegypti in French Guiana: Susceptibility to a dengue virus[END_REF], as well as VC [START_REF] Bennet | Variation in vector competence for dengue 2 virus among 24 collections of Aedes aegypti from Mexico and the United States[END_REF]. Furthermore, it has also been reported that DENV-2 vary according to the genotype in their efficiency to infect and disseminate in A. aegypti [START_REF] Armstrong | Differential susceptibility of Aedes aegypti to infection by the American and Southeast Asian genotypes of Dengue type 2 virus[END_REF][START_REF] Morales | Infection and dissemination of two dengue type-2 viruses isolated from patients exhibiting different disease severity in orally infected Aedes aegypti from different geographic origin[END_REF][START_REF] Armstrong | Efficiency of dengue serotype 2 virus strains to infect and disseminate in Aedes aegypti[END_REF]. However, these studies were done using only a few mosquito collections or virus strains resulting in the evaluation of patterns of differential infection in a small number of virus-vector pairs.

In the present study, we address the potential variation in VC and patterns of oral infection of dengue virus isolates representing distinct disease severity in sympatric DENV-2-A. aegypti pairs on a regional geographic scale in Thailand. In particular, eight geographically heterogeneous mosquito strains collected from areas with high and low dengue incidence were orally challenged with four virus isolates recently isolated from patients exhibiting flu-like classic dengue fever (DF) and dengue shock syndrome (DSS).

Results

Disseminated infection rates: Variations in the disseminated infection rates of the virus isolates in Aedes aegypti were observed according to the mosquito origin (dengue incidence at the area and location), mosquito strain, disease severity of the patients from whom the virus was isolated, and the virus isolate (Table 1 and Figure 1). Overall, the disseminated infection rates of dengue viruses in mosquito strains ranged from 66.7% (TAK, 136 of 204 mosquitoes) to 39.2% (RBK, 98 of 250 mosquitoes). Mosquito strains collected from high dengue incidence areas showed a higher infection rate (58.1% of 854 mosquitoes) compared to mosquitoes from low dengue incidence area (52.5% of 953 mosquitoes). However, no significant statistical difference was observed (Figure 1A). Significant differences in infection rates were observed among geographic origin. Mosquitoes collected from Bang Pakong district exhibited the highest infection rate (62.1% of 422 mosquitoes) while mosquitoes collected from Ratchasan district exhibited the lowest disseminated infection rate (46.9% of 484 mosquitoes). Mosquito strains collected from the Muang district, despite being the most urbanized and with the highest reported dengue incidence, showed lower disseminated infection rate (54.9% of 432 mosquitoes) compared to mosquito strains from the rural Krong Kruang district (59.5% of 459 mosquitoes) that has lower reported dengue incidence (Table 2 and Figure 1B). Among mosquito strains, significantly different disseminated infection rates were observed, except for the BPA-GKE, BPA-TAL, BPA-KAO, GKE-RBA, and GKE-TAL pairs of mosquito strains (Figure 1C). For virus isolates grouped according to the disease severity of the patients from whom they were isolated, no differences in disseminated infection rate was observed between DF and DSS virus isolates (Figure 1D). However, disseminated infection rates of the isolates ThNR2/772 (from DF patient) and ThNR2/391 (from DSS patient), were overall significantly different from isolates ThNR2/406 (DF) and ThNR2/479 (DSS) with p < 0.005 (Figure 1E). Vector competence variation: As a proxy of vector competence (VC), the VC was calculated as the number of mosquitoes positive for the viral RNA in head tissues divided by the number of mosquitoes exposed to the respective viruses. To better understand the geographical variation in the A. aegypti vector competence for dengue viruses isolated from patients exhibiting different disease severity (DF and DSS), the mosquito strains were grouped by dengue incidence at the area of mosquito origin (sub-district level). High dengue incidence sub-districts comprises TAL, KAO, BPA, and TAK strains, while low incidence sub-districts were KKR, GKE, RBK, and RBA strains. Mosquito strains were further classified location at the district administrative level Muang (TAL and KAO), Bang Pakong (BPA and TAK), Krong Kruang (KKR and GKE) and Ratchasan (RBK and RBA).

For the first grouping, the VC among mosquitoes from high dengue incidence for DSS isolates ranged from 19.6% (KAO mosquito strain for the ThNR2/479 virus isolate) to 85.4% (TAK for the ThNR2/479 virus isolate). DF isolates ranged from 37.0% (TAK mosquito strain for the ThNR2/406 virus isolate) to 73.1% (TAK mosquito strain for the ThNR2/772 virus isolate). Although no statistically significant differences were observed between the overall VC for DF and DSS, mosquito strains collected from high dengue incidence areas showed an overall higher competence for viruses isolated from DSS patients than from DF patients. Whereas mosquito strains collected from low dengue incidence areas exhibited a similar overall VC for either DF or DSS virus isolates, with VC ranging from 40.6% (GKE mosquito strain for the ThNR2/391 virus isolate) to 67.3% (KKR mosquito strain for the ThNR2/391) for DSS isolates, and 18.8% (RBK mosquito strain for the ThNR2/406) to 81.8% (KKR mosquito strain for the ThNR2/406) for DF isolates (Table 1). For the second grouping, mosquito strains from Bang Pakong and Ratchasan districts showed a significantly higher VC for viruses isolated from patients with DSS compared to patients with DF symptoms (Figure 2B). The VC for mosquito strains from Bang Pakong ranged from 53.7% (for ThNR2/391 isolate virus) to 85.4% (ThNR2/479 isolate virus) for DSS isolates, whereas DF isolates ranged from 37.0% (for ThNR2/406 isolate virus) to 73.1% (ThNR2/772). The VC for mosquito strains from Ratchasan ranged from 47.2% (for ThNR2/479) to 65.5% (for ThNR2/391) for DSS isolates, and from 18.8% (for ThNR2/406) to 61.7% (ThNR2/772) for DF isolates (Table 1). In contrast, mosquito strains from Muang and Krong Kruang districts showed a significantly higher VC for viruses isolated from patients with DF than from those with DSS, (Figure 2B). The VC for mosquito strains from Muang ranged from 52.3% (for ThNR2/772) to 70.4% (ThNR2/406) for DF isolates and from 19.6% (for ThNR2/479) to 61.4% (for ThNR2/391) DSS. The VC for mosquito strains from Krong Kruang ranged from 56.2% (for ThNR2/772) to 81.8% (for ThNR2/406 isolate virus) for DF, and from 40.6% (for ThNR2/391) to 67.3% (for ThNR2/391) for DSS isolates (Table 1).

The VC of individual mosquito strains significantly differed between virus isolated from patients with DSS and DF, p < 0.05. Females of the TAL, KAO, GKE, and KKR strains exhibited a significantly higher VC for DF virus isolates than for DSS, while BPA, TAK, and RBK showed a significantly higher VC for DSS than for DF virus isolates, (p < 0.05) as shown on Figure 2C. As summarized on Table 1, the VC among mosquito strains for the four isolates differed according to the A. aegypti strain-virus isolate combinations which ranged from 18.8% (RBK-ThNR2/406) to 85.4% (TAK-ThNR2/479).

Viral isolates infectivity variations: The infectivity of a virus isolate for A. aegypti was calculated as the number mosquitoes positive for viral RNA in the abdomen divided by the number exposed to the virus isolate. Overall, ThNR2/772 (DF) and ThNR2/391 (DSS) were the most infectious virus isolates, where 62.3% of 467 and 60.9% of 422 mosquitoes were positive, respectively; while ThNR2/406 (DF) and ThNR2/479 (DSS) were significantly less infectious where 55.2% of 458 and 54.8% of 449 mosquitoes were positive, respectively, (p < 0.05; Figure 1E).

Different patterns of infection dynamics among individual virus isolates in mosquitoes were observed. Infection rates for ThNR2/772 (DF) and ThNR2/391 (DSS) isolates were more uniform. ThNR2/772 ranged from 40% (RBK mosquito strain) to 73.1% (KAO mosquito strain), while ThNR2/391 ranged from 40% (GKE mosquito strain) to 80% (TAK mosquito strain). In contrast, virus isolates ThNR2/406 (DF) and ThNR2/479 (DSS), exhibited more phenotypic variation where infection rates ranged from 18.8% (RBK mosquito strain) to 88.6% (KKR mosquito strain), and from 21.6 (KAO mosquito strain) to 85.4% (TAK mosquito strain), respectively, (Table 1). Furthermore, according to the mosquito strains, significant differences in the number of infected mosquitoes were observed among individual viral isolates. The proportion of mosquitoes infected by the ThNR2/406 (DF) virus isolate was not significantly different only between KAO-GKE and TAK-RBA mosquito strain pairs, and by the ThNR2/772 (DF) virus isolate was not significantly different between GKE-KKR-RBA mosquito strains, and by the ThNR2/391 (DSS) virus isolate was not significantly different between TAL-KAO-BPA-RBK mosquito strains, and by the ThNR2/479 (DSS) viral isolate was not significantly different only between KKR-RBA mosquito strains.

patients with DSS and DF, p < 0.05. Females of the TAL, KAO, GKE, and KKR strains exhibited a significantly higher VC for DF virus isolates than for DSS, while BPA, TAK, and RBK showed a significantly higher VC for DSS than for DF virus isolates, (p < 0.05) as shown on Figure 2C. As summarized on Table 1, the VC among mosquito strains for the four isolates differed according to the A. aegypti strain-virus isolate combinations which ranged from 18.8% (RBK-ThNR2/406) to 85.4% (TAK-ThNR2/479). Viral isolates infectivity variations: The infectivity of a virus isolate for A. aegypti was calculated as the number mosquitoes positive for viral RNA in the abdomen divided by the number exposed to the virus isolate. Overall, ThNR2/772 (DF) and ThNR2/391 (DSS) were the most infectious virus isolates, where 62.3% of 467 and 60.9% of 422 mosquitoes were positive, respectively; while ThNR2/406 (DF) and ThNR2/479 (DSS) were significantly less infectious where 55.2% of 458 and 54.8% of 449 mosquitoes were positive, respectively, (p < 0.05; Figure 1E).

Different patterns of infection dynamics among individual virus isolates in mosquitoes were observed. Infection rates for ThNR2/772 (DF) and ThNR2/391 (DSS) isolates were more uniform. ThNR2/772 ranged from 40% (RBK mosquito strain) to 73.1% (KAO mosquito strain), while ThNR2/391 ranged from 40% (GKE mosquito strain) to 80% (TAK mosquito strain). In contrast, virus isolates ThNR2/406 (DF) and ThNR2/479 (DSS), exhibited more phenotypic variation where infection rates ranged from 18.8% (RBK mosquito strain) to 88.6% (KKR mosquito strain), and from 21.6 (KAO mosquito strain) to 85.4% (TAK mosquito strain), respectively, (Table 1). Furthermore, according to the mosquito strains, significant differences in the number of infected mosquitoes were observed among individual viral isolates. The proportion of mosquitoes infected by the ThNR2/406 (DF) virus isolate was not significantly different only between KAO-GKE and TAK-RBA mosquito strain pairs,

Discussion

Vector, viral, and environmental factors determine the capacity of dengue virus to infect and disseminate in its mosquito vector. In the present study, we challenged each mosquito strain with all four virus isolates at the same time and mosquitoes were held together during the same environmental fluctuations of the incubation period (14 days), to minimize the effect of environmental variation. Even though all mosquito strains may not have experienced the same environmental conditions no statistical significant differences were observed when mean temperature and humidity that each mosquito strain experienced during the incubation period were compared among strains (data not shown). Nonetheless, some of the observed mosquito VC and viral isolate infectivity variations may have been influenced by environmental variation among experimental groups.

Variations in infection rates of A. aegypti experimentally infected with DEN virus has been reported [START_REF] Rosen | Comparative susceptibility of mosquito species and strain to oral and parenteral infection with dengue and Japanese encephalitis viruses[END_REF][START_REF] Gubler | Variation in susceptibility to oral infection with dengue viruses among geographic strains of Aedes aegypti[END_REF][START_REF] Tardieux | Variation among strains of Aedes aegypti in susceptibility to oral infection with dengue virus type 2[END_REF][START_REF] Tran | Aedes aegypti in Ho Chi Minh city (Viet Nam): Susceptibility to dengue type 2 virus and genetic differentiation[END_REF][START_REF] Vazeille-Falcoz | Variation in oral susceptibility to dengue type 2 virus of population of Aedes aegypti from the islands of Tahiti and Moorea, French Polynesia[END_REF][START_REF] Fouque | Aedes aegypti in French Guiana: Susceptibility to a dengue virus[END_REF][START_REF] Bennet | Variation in vector competence for dengue 2 virus among 24 collections of Aedes aegypti from Mexico and the United States[END_REF]. Differences on the proportion of infected mosquitoes between DENV-2 with SEA and American genotype have also been reported in geographically separated A. aegypti populations [START_REF] Armstrong | Differential susceptibility of Aedes aegypti to infection by the American and Southeast Asian genotypes of Dengue type 2 virus[END_REF][START_REF] Morales | Infection and dissemination of two dengue type-2 viruses isolated from patients exhibiting different disease severity in orally infected Aedes aegypti from different geographic origin[END_REF][START_REF] Armstrong | Efficiency of dengue serotype 2 virus strains to infect and disseminate in Aedes aegypti[END_REF]. In contrast to previous studies, the present study used low-passage virus isolates that were recently isolated from patients exhibiting different disease severity (i.e., DF and DSS), which were of sympatric geographic origin with the used A. aegypti strains. Although the A. aegypti were collected in the same province, considerable variability in overall infection rates was observed according to the dengue incidence in the collection area, geographic origin (districts), mosquito strain, individual virus isolate, and disease severity of the patients from whom the virus was isolated (Figure 1A-E).

From our observations, although neither high or low dengue incidence mosquito strains showed significantly different VC for DF or DSS viral isolates (Figure 2A), the mosquito strains collected from high dengue incidence sites exhibited higher overall VC for virus isolates either isolated from patients exhibiting mild (DF) or severe (DSS) disease severity than mosquito strains from sites with low dengue incidence site (Table 3). However, significant differences in VC for DF and DSS virus isolates were observed among mosquito strains from different geographic origins (districts) (Figure 2B). Furthermore, most individual mosquito strains showed significantly different VC either for DF or DSS virus isolates. Only the RBA mosquito strain did not show significantly different VC between DF and DSS viral isolates (Figure 2C). Therefore, we speculate that factors such as the origins of mosquitoes, the viral strains (DF or DSS) circulating in a given locale, and mosquito factors, per se play an important role in determining A. aegypti VC. A detailed genetic analysis is needed to determine if these mosquito strains are actually isolated and constitute independent populations that differ in susceptibility to the infection with dengue virus. In previous studies, it has been shown that mosquitoes from populated and urbanized areas are genetically highly differentiated and exhibit high and heterogeneous infection rates, and this genetic differentiation has been related to the intensity of insecticide control and human population density [START_REF] Bosio | Quantitative trait loci that control vector competence for dengue-2 virus in the mosquito Aedes aegypti[END_REF][START_REF] Tran | Aedes aegypti in Ho Chi Minh city (Viet Nam): Susceptibility to dengue type 2 virus and genetic differentiation[END_REF][START_REF] Vazeille-Falcoz | Variation in oral susceptibility to dengue type 2 virus of population of Aedes aegypti from the islands of Tahiti and Moorea, French Polynesia[END_REF][START_REF] Fouque | Aedes aegypti in French Guiana: Susceptibility to a dengue virus[END_REF]. As for mosquito factors, the VC for arboviruses is associated with anatomic barriers to productive vector infection, including a midgut infection barrier (MIB), a midgut escape barrier (MEB), and a salivary gland barrier [START_REF] Bennet | Variation in vector competence for dengue 2 virus among 24 collections of Aedes aegypti from Mexico and the United States[END_REF][START_REF] Franz | Tissue Barriers to Arbovirus Infection in Mosquitoes[END_REF]. A. aegypti VC for DENV-2 is considered as being universally proportional to the level of MIB and MEB found in mosquito collections, where strong MIBs and MEBs decrease transmission potential [START_REF] Bennet | Variation in vector competence for dengue 2 virus among 24 collections of Aedes aegypti from Mexico and the United States[END_REF]. It is also important to remember that there is potential for different loci to affect VC in different populations [START_REF] Bosio | Quantitative genetics of vector competence for dengue-2 virus in Aedes aegypti[END_REF][START_REF] Bosio | Quantitative trait loci that control vector competence for dengue-2 virus in the mosquito Aedes aegypti[END_REF]. Thus, the need to look for genetic variation is evident. This study also raised the possibility that VC for a virus isolate is not randomly distributed across the disease severity of the patient from whom the virus was isolated. We found that mosquito strains from the same district exhibited the same VC pattern, namely TAL, KAO, and GKE, KKR mosquito strains collected from Muang and Krong Kruang districts, respectively, were significantly more competent for DF virus isolates than for DSS viral isolates. In addition, the BPA and TAK mosquito strains collected from Bang Pakong districts and RBK and RBA from Rachasan district, though RBA did not show significant different VC, exhibited the same VC pattern, indeed, were more competent for DSS than for DF viral isolate (Figure 2C).

Dengue viruses' efficiency to infect and disseminate in their vector mosquitoes can vary greatly. In the present study, virus infectivity was evaluated by measuring the proportion of mosquitoes infected by individual virus isolates. Overall, we found variation in the proportion of mosquitoes infected with individual virus isolates. Specifically, the ThNR2/772 and ThNR2/391 isolates were more infective than the ThNR2/406 and ThNR2/479 viral isolates (Figure 1E). However, the virus isolate infectivity seems likely to be differentiated according to mosquito strain-viral isolate pairs, Figure 3. Therefore, we speculate that differences in the proportions of infected mosquitoes among individual virus isolate are probably due to certain genetic elements of the isolated virus. Few genetic differences among dengue virus isolates may have a significant effect on their infection, dissemination and transmission by vector mosquitoes. Previous studies on DENV-2 revealed differences in the proportions of mosquitoes infected by individual virus isolates and have been associated with molecular differences among virus isolates [START_REF] Armstrong | Differential susceptibility of Aedes aegypti to infection by the American and Southeast Asian genotypes of Dengue type 2 virus[END_REF][START_REF] Morales | Infection and dissemination of two dengue type-2 viruses isolated from patients exhibiting different disease severity in orally infected Aedes aegypti from different geographic origin[END_REF][START_REF] Armstrong | Efficiency of dengue serotype 2 virus strains to infect and disseminate in Aedes aegypti[END_REF][START_REF] Pompon | Dengue Subgenomic Flaviviral RNA Disrupts Immunity in Mosquito Salivary Glands to Increase Virus Transmission[END_REF]. differences among dengue virus isolates may have a significant effect on their infection, dissemination and transmission by vector mosquitoes. Previous studies on DENV-2 revealed differences in the proportions of mosquitoes infected by individual virus isolates and have been associated with molecular differences among virus isolates [START_REF] Armstrong | Differential susceptibility of Aedes aegypti to infection by the American and Southeast Asian genotypes of Dengue type 2 virus[END_REF][START_REF] Morales | Infection and dissemination of two dengue type-2 viruses isolated from patients exhibiting different disease severity in orally infected Aedes aegypti from different geographic origin[END_REF][START_REF] Armstrong | Efficiency of dengue serotype 2 virus strains to infect and disseminate in Aedes aegypti[END_REF][START_REF] Pompon | Dengue Subgenomic Flaviviral RNA Disrupts Immunity in Mosquito Salivary Glands to Increase Virus Transmission[END_REF].

Materials and Methods

Collection sites: Four districts in Chachoengsao province situated in eastern Thailand, with the highest (Muang and Bang Pakong) and lowest (Krong Kruang and Ratchasan) dengue (DF/DHF/DSS) incidence (per 100,000 population) reported during the past 20 years were selected. Within these districts the sub-districts (tambon) with the highest (Na Muang, Bang Pakong, and Tha Kam) and lowest (Krong Kruang, Gon Kheo, and Bang Kla) dengue incidence from 1999 to 2002 were selected as mosquito collection sites (Figure 4). Sub-districts were classified as high and low dengue risk areas based on annual incidence of DHF/DSS (cases/100,000 habitants), and the number of houses and inhabitants per square kilometer. The selected locations covered a wide geographical distribution of A. aegypti, including urban and rural environments. The shortest distance between study areas (districts) was ca. 20 km, and between sub-districts was ca. more than 30 km.

Materials and Methods

Collection sites: Four districts in Chachoengsao province situated in eastern Thailand, with the highest (Muang and Bang Pakong) and lowest (Krong Kruang and Ratchasan) dengue (DF/DHF/DSS) incidence (per 100,000 population) reported during the past 20 years were selected. Within these districts the sub-districts (tambon) with the highest (Na Muang, Bang Pakong, and Tha Kam) and lowest (Krong Kruang, Gon Kheo, and Bang Kla) dengue incidence from 1999 to 2002 were selected as mosquito collection sites (Figure 4). Sub-districts were classified as high and low dengue risk areas based on annual incidence of DHF/DSS (cases/100,000 habitants), and the number of houses and inhabitants per square kilometer. The selected locations covered a wide geographical distribution of A. aegypti, including urban and rural environments. The shortest distance between study areas (districts) was ca. 20 km, and between sub-districts was ca. more than 30 km. Mosquito collections: A. aegypti were collected from two sites in each sub-district, (Table 2). Samples from high dengue incidence areas were collected from houses with recent report of dengue cases and from low incidence areas in house without dengue cases for the last two years. The mosquitoes were collected as fourth stage larvae and/or pupae, at least 200 per collection site, from water containers found outside and inside the house within a radius of ca. 2 km, at collection sites separated by at least ca.5 km. Field collected mosquitoes (F0 generation) were reared to adults under room conditions (26 °C ± 2 °C, RH 75% ± 10%). After morphological identification, based on the illustrated key to the mosquitoes of Thailand [START_REF] Rattanarithikul | Illustrated keys to the mosquitoes of Thailand. II. Genera Culex and Lutzia[END_REF], adults were kept in cages and females were bloodengorged on mice to produce eggs. The onward generations of mosquitoes were reared in plastic pans (33 cm × 25 cm × 11 cm) containing 3 L of aged tap water with a density of 180-210 larvae per container. The larvae were fed on a diet of mouse food powder and maintained at 26 °C ± 2 °C, RH 75% ± 10%. Adults were fed on 3% sucrose soaked cotton and females were allowed to mate and then blood-feed on mice to enable egg production. The F0 mosquitoes were stored at -80 °C for future genetic analysis. Oral infection experiments were performed with females from the first (F1) or second (F2) generations generated from the F0 proved to be free of dengue virus infection. The mosquito strains used for oral infection experiments and the dengue background of the district where the mosquitoes were collected are given in Table 2 and Figure 4. Mosquito collections: A. aegypti were collected from two sites in each sub-district, (Table 2). Samples from high dengue incidence areas were collected from houses with recent report of dengue cases and from low incidence areas in house without dengue cases for the last two years. The mosquitoes were collected as fourth stage larvae and/or pupae, at least 200 per collection site, from water containers found outside and inside the house within a radius of ca. 2 km, at collection sites separated by at least ca.5 km. Field collected mosquitoes (F0 generation) were reared to adults under room conditions (26 • C ± 2 • C, RH 75% ± 10%). After morphological identification, based on the illustrated key to the mosquitoes of Thailand [START_REF] Rattanarithikul | Illustrated keys to the mosquitoes of Thailand. II. Genera Culex and Lutzia[END_REF], adults were kept in cages and females were blood-engorged on mice to produce eggs. The onward generations of mosquitoes were reared in plastic pans (33 cm × 25 cm × 11 cm) containing 3 L of aged tap water with a density of 180-210 larvae per container. The larvae were fed on a diet of mouse food powder and maintained at 26 • C ± 2 • C, RH 75% ± 10%. Adults were fed on 3% sucrose soaked cotton and females were allowed to mate and then blood-feed on mice to enable egg production. The F0 mosquitoes were stored at -80 • C for future genetic analysis. Oral infection experiments were performed with females from the first (F1) or second (F2) generations generated from the F0 proved to be free of dengue virus infection. The mosquito strains used for oral infection experiments and the dengue background of the district where the mosquitoes were collected are given in Table 2 and Figure 4.

Virus strains: The four virus isolates were obtained from the sera of patients from Nakorn Ratchasima Provincial Hospital, Northeastern Thailand, diagnosed with DF and DSS in 2002. Clinical diagnosis confirmation and serological assays (ELISA IgG and IgM) were performed at the hospital, while virus isolation and serotype determination were performed by the staff of the Arbovirus Section, National Institute of Health, Medical Science Center, Nonthaburi. The clinical diagnosis and clinical severity gradings of each isolate was classified using the World Health Organization criteria (WHO, 1986) [START_REF]Dengue Guidelines for Diagnosis, Treatment, Prevention and Control[END_REF]. The serotype was determined as dengue virus type 2 by reverse transcription-polymerase chain reaction (RT-PCR). Relevant patient information is summarized in Table 3.

All isolates were passaged four times in Aedes albopictus clone C6/36 cell line from the patient serum before using for experiments. To ensure that the virus specimens were not altered significantly from their wild type character as found in the host patient, none of the isolate has been purified by any methodology before inoculation into mosquito cells.

Plaque assay: A seed virus was prepared by inoculation into a monolayer culture of Ae. albopictus clone C6/36 cell line and incubated at 28 • C for 8 days in Eagle's medium supplemented with 2% heat-inactivated fetal bovine serum (FBS) and 0.2 mM each of nonessential amino acids plus antibiotics (20 µg/mL Gentamicin, 5 µg/mL Amphotericin B, 200 U/mL). The infected culture fluid was harvested 8 days after inoculation, aliquoted, and stored at -80 • C until used. The virus titer of the isolates was measured by the focus formation test by using BHK-21 cells on 96-well plates [START_REF] Mangada | Molecular and in vitro analysis of eight dengue type 2 viruses isolated from patients exhibiting different disease severities[END_REF]. The virus isolates were then used for preparation of the infectious meal, which was a blood virus sucrose solution (BVS).

Oral Infection: Only 4 to 5 day-old female mosquitoes were used to minimize age factors effects. About 60 females were placed in cylindrical pint cardboard cages covered at one end with fine non-wettable nylon mesh. These females were deprived of food for 36-40 h prior to the infectious meal, then were allowed to feed on the infectious meal (BVS) consisting of equal volumes of isolate virus suspension, washed rabbit erythrocytes, and 10% sucrose solution (as a source of energy). Drops of the infectious meal were placed on the mesh covering the cardboard cage containing mosquitoes as previously described [START_REF] Gubler | Variation in susceptibility to oral infection with dengue viruses among geographic strains of Aedes aegypti[END_REF][START_REF] Morales | Infection and dissemination of two dengue type-2 viruses isolated from patients exhibiting different disease severity in orally infected Aedes aegypti from different geographic origin[END_REF]. Feeding time was limited to 1 h, mosquitoes were cold anesthetized and fully engorged mosquitoes were collected with an aspirator at 30 min intervals, transferred into clean carton cups and maintained for up to 14 days at 30 ± 1 • C as a virus extrinsic incubation period (EIP). After the EIP completed survived mosquitoes were frozen at -80 • C, then transfer individually into 1.5 mL tube and kept at -80 • C until further use for viral RNA detection. Mosquito strains were orally challenged on different days; however, each strain was challenged simultaneously with the four virus isolates on the same day. All feeding suspensions contained the same virus titer (1 × 10 2 PFU/mL) with an aliquot of the same virus pool. The titers of the post feeding virus suspensions did not change significantly.

Detection of virus infection: After has completed the EIP (14 days) mosquito head and abdomen were severed and homogenized in Trizol LS reagent for total RNA extraction, and then assayed for dengue virus genome by using RT-PCR method. The detection of viral RNA in the homogenized abdomen was interpreted as indicating that the mosquito midgut had become infected. Detection of RNA viral in the homogenized head indicated disseminated from the infected midgut to a secondary target organs.

RNA extraction and RT-PCR: Total RNA was extracted from individual mosquito head and abdomen severed using Trizol LS (Invitrogen) according to the manufacturer's recommendations. Briefly, the head and abdomen of mosquito were separately homogenized in 300 µL of Trizol reagent then 99.9 µL of chloroform was added. The resulting mixed suspension was centrifuged at 14,000 rpm/min for 10 min at 4 • C. RNA was precipitated from the aqueous phase (ca. 145 µL) by mixing with 133.2 µL of isopropyl alcohol. Isopropyl alcohol-RNA precipitate was recovered by centrifugation and the RNA pellet was washed once with 90 µL of 75% ethanol followed by a second wash with absolute ethanol and then air dried. The RNA pellet was resuspended in 10 µL of RNase-free water and used as a template RNA in reverse transcriptase PCR (RT-PCR). Synthetic oligonucleotide primer pairs were designed based on published sequence data for dengue virus serotype specific primer, sense (D2-S) and complementary (D2-C) genome nucleotide (nt) regions 1203 to 1222 and 1432 to 1413 respectively [START_REF] Deubel | Nucleotide sequence and deduced aminoacid sequence of the structural proteins of Dengue type 2 virus: Jamaica genotype[END_REF][START_REF] Woodring | Natural Cycles of Vector-Borne Pathogens[END_REF].

Dengue virus RNA was assayed by using one step RT-PCR kit (Quiagen) following the manufacturer's recommendations with slight modifications. Briefly, 2 µL of RNA, 8 µL of reaction mix, and RNase free water was reverse transcribed and amplified following the thermal cycle protocol recommended in the kit instruction manual, with some modifications, as follows: one cycle at 53 • C for 30 min, one cycle at 94 • C for 2 min, followed by 45 cycles at 94 • C for minute, 53 • C for 1 min, and 68 • C for 2 min, followed by one cycle at 68 • C for 7 min. The reaction mixture solution contained a mixture of 5 µL of 5× buffer (contains 12.5 mM MgCl 2 ), 1 µL of dNTP mix (containing 10 mM of each dNTP), 0.5 µL (25 pmol) of corresponding primers, D2-S (5 -GTTCGTCTGCAAACACTCCA-3 ) and D2-C (5 -GTGTTATTTTGATTTCCTTG-3 ), and 1 µL of enzyme mix (an optimized combination of Omnscript Reverse Transciptase, Sensiscript Reverse Transcriptase, and HotStarttTap DNA Polymerase). Nine microliters of PCR product was subjected to agarose gel electrophoresis, and amplified DNA fragments were visualized with ethidium bromide staining.

Statistical analysis: One-way analysis of variance was used to test the variation among disseminated infection rates, vector competence measures, and virus infectivity rates. Tukey's HSD was used for pairwise comparisons. Analogous non-parametric test were used when departures from normality or equality of variance were encountered. Statistical Package for the Social Sciences (SPSS) 11.5 software package was used to generate all the results and graphs in this study.

Conclusions

In this study, we analyzed 32 recently collected vector-virus pairs (eight geographically different mosquito strains versus four isolate virus strains). Among vector-virus pairs infection rates were different; indeed, many of those were significantly different. This leads us to conclude that the observed variations in susceptibility among mosquito strains is probably due to the combination of certain genetic elements of the vector and the virus as has been previously observed in similar researches. Furthermore, these results suggest that the efficacy of dengue virus circulation in a given locale may vary according to the interaction of virus strains with the origin of the vector mosquitoes. This study contributes to the knowledge of the role of the association between mosquito genetic determinants of susceptibility and genetic variation of dengue virus in the occurrence of different disease severity.

Figure 1 .

 1 Figure 1. Disseminated infection rate variations of dengue virus serotype 2 isolated from patients with different disease severity in Aedes aegypti according to dengue incidence at the mosquito origin area (A), mosquito geographic origin and districts (B), mosquito strains (C), patient disease severity (D), and virus isolate (E). Mosquitoes were scored positive if RNA dengue virus was detected in head tissue by RT-PCR. Bars and error bars show the mean percentage of positive mosquitoes and the standard error, respectively. Bars with the same letter are not significantly different (p < 0.05), while bars without or different letter are significantly different (p > 0.05), based on Tukey's test for means comparison.

Figure 1 .

 1 Figure 1. Disseminated infection rate variations of dengue virus serotype 2 isolated from patients with different disease severity in Aedes aegypti according to dengue incidence at the mosquito origin area (A), mosquito geographic origin and districts (B), mosquito strains (C), patient disease severity (D), and virus isolate (E). Mosquitoes were scored positive if RNA dengue virus was detected in head tissue by RT-PCR. Bars and error bars show the mean percentage of positive mosquitoes and the standard error, respectively. Bars with the same letter are not significantly different (p < 0.05), while bars without or different letter are significantly different (p > 0.05), based on Tukey's test for means comparison.

Figure 2 .

 2 Figure 2. Vector competence of Aedes aegypti for dengue serotype 2 virus isolated from patients exhibiting different disease severity according to dengue incidence at the mosquito strain origin area (A), mosquito geographic origin, districts (B), and mosquito strains (C). Vector competence is expressed as the number of mosquitoes positive for viral RNA in head tissues/number tested. Bars and error bars show the mean percentage of positive mosquitoes and the standard error, respectively. Bars with the same letter within the same group are not significantly different (p < 0.05) based on Tukey's test for means comparison.

Figure 2 .

 2 Figure 2. Vector competence of Aedes aegypti for dengue serotype 2 virus isolated from patients exhibiting different disease severity according to dengue incidence at the mosquito strain origin area (A), mosquito geographic origin, districts (B), and mosquito strains (C). Vector competence is expressed as the number of mosquitoes positive for viral RNA in head tissues/number tested. Bars and error bars show the mean percentage of positive mosquitoes and the standard error, respectively. Bars with the same letter within the same group are not significantly different (p < 0.05) based on Tukey's test for means comparison.

Figure 3 .

 3 Figure 3. Infectivity of four dengue 2 viruses isolated from patients exhibiting different disease severity for geographically different Aedes aegypti strains. Virus infectivity is expressed as the number of mosquitoes positive for viral RNA in abdomen/number tested. Bars and error bars show the mean percentage of positive mosquitoes and the standard error, respectively. Bars with the same letter within the same virus isolate are not significantly different (p < 0.05), while bars without or different letter are significantly different (p > 0.05), based on Tukey's test for means comparison.

Figure 3 .

 3 Figure 3. Infectivity of four dengue 2 viruses isolated from patients exhibiting different disease severity for geographically different Aedes aegypti strains. Virus infectivity is expressed as the number of mosquitoes positive for viral RNA in abdomen/number tested. Bars and error bars show the mean percentage of positive mosquitoes and the standard error, respectively. Bars with the same letter within the same virus isolate are not significantly different (p < 0.05), while bars without or different letter are significantly different (p > 0.05), based on Tukey's test for means comparison.

Figure 4 .

 4 Figure 4. Geographic localization of the collection sites of Aedes aegypti used depicting the habitat type corresponding to each district.

Figure 4 .

 4 Figure 4. Geographic localization of the collection sites of Aedes aegypti used depicting the habitat type corresponding to each district.

Table 1 .

 1 Infection and dissemination rates (%) A of DENV-2es isolated from patients with different disease severity in orally challenged Aedes aegypti strains.

	Mosquito	ThNR2/406 (DF) B	ThNR2/772 (DF)	ThNR2/391 (DSS)	ThNR2/479 (DSS)
	Strains	Infec	Disse	Infec	Disse	Infec	Disse	Infec	Disse
	TAL	66.7	58.3 (60) C	57.1	52.4 (63)	56.4	56.4 (55)	55.5	52.4 (63)
	KAO	75.0	70.4 (44)	73.1	67.3 (52)	61.3	61.4 (44)	21.6	19.6 (51)
	BPA	48.1	42.6 (54)	66.7	63.6 (66)	55.5	53.7 (54)	72.7	72.7 (44)
	TAK	40.7	37.0 (54)	73.1	73.1 (52)	80.0	78.0 (50)	85.4	85.4 (48)
	KKR	88.6	81.8 (44)	64.0	60.0 (50)	70.1	67.3 (48)	48.1	46.3 (54)
	GKE	70.7	69.3 (75)	59.4	56.3 (64)	40.6	40.6 (64)	62.7	59.3 (59)
	RBK	18.8	18.8 (69)	40.0	40.0 (60)	61.2	55.1 (49)	51.4	47.2 (72)
	RBA	44.8	41.4 (58)	68.3	61.7 (60)	67.2	65.5 (58)	51.7	51.7 (58)

A Infection and dissemination: number of mosquitoes for viral RNA in abdomen and head tissue, respectively; B DF: dengue fever; DSS: dengue shock syndrome; C Total number of mosquitoes tested for each mosquito/virus combination.

Table 2 .

 2 Demographic information and dengue (DEN) background of the origin of Aedes aegypti strains from Chachoengsao province, central eastern Thailand, used in this study.

	Strain	Subdistrict	At Collection Site Demographic A	Incidence B	District	At Collection Area Demographic A	Incidence B
	TAL KAO	Na Muang	4480.9/1640.4	153.2	Muang	369.2/106.8	141.3
	BPA TAK	Bang Paakong Tha Kam	207.4/89.5 542.5/345.2	284.9 204.2	Bang Pakong	306.1/89.6	105.8
	KKR GKE	Krong Kruang Gon Kheo	114.2/30.9 99.7/21.2	117.3 88.45	Krong Kruang	106.5/25.9	91.7
	RBK RBA	Bang Kla	70.4/18.3	40.3	Ratchasan	92.4/23.4	72.1
	A Numbers of people/houses per square Km. Based on census of 2002, Public Health Office, Chachoengsao; B Dengue
	(DHF + DSS) annual average incidence, cases/100,000 habitants. Public Health Office, Chachoengsao, 1999-2002.

Table 3 .

 3 Medical data and in vitro infectivity results of four DEN-2 isolates virus isolated patients of Nakorn Ratchasima provincial hospital from Nakorn Ratchasima province, northeast Thailand. Due to the lack of a second serum sample; B Second serum sample was obtained 2 weeks after the first sample.

	Pathogens 2020, 9, x FOR PEER REVIEW					8 of 13
	Isolate Name	Sex Age	Clinical Diagnosis	Medical Data Antibody Response	ELISA Assay (Unit) Den-IgG Den-IgM	Infectivity LLC-MK2 (PFU/mL)
	ThNR02/406 M	16	DF	NA A	7	0	1.2 × 10 5
	ThNR02/772 M	14	DF	NA	14	9	8.5 × 10 3
	ThNR02/391 M	12	DSS	NA	29	2	1.38 × 10 5
	ThNR02/479 M	12	DSS	Secondary	3	0	1.0 × 10 2
						149 B	56

A

Table 3 .

 3 Medical data and in vitro infectivity results of four DEN-2 isolates virus isolated patients of Nakorn Ratchasima provincial hospital from Nakorn Ratchasima province, northeast Thailand.

	Isolate Name	Sex Age	Clinical Diagnosis	Medical Data Antibody Response	ELISA Assay (Unit) Den-IgG Den-IgM	Infectivity LLC-MK2 (PFU/mL)
	ThNR02/406	M	16	DF	NA A	7	0	1.2 × 10 5
	ThNR02/772	M	14	DF	NA	14	9	8.5 × 10 3
	ThNR02/391	M	12	DSS	NA	29	2	1.38 × 10 5
	ThNR02/479	M	12	DSS	Secondary	3	0	1.0 × 10 2
						149 B	56

A Due to the lack of a second serum sample; B Second serum sample was obtained 2 weeks after the first sample.

Acknowledgments:

The authors are grateful to Pathon Swanpanyalert and Surapee Anantapreecha for providing viral isolates; Sutee Yoksan for his assistance in the virus seed preparation and titration; Noppawan Phumala Morales and Somboon Srimarat for their help during the mosquito collection. We also thank the Chachoengsao provincial and local (districts and subdistricts) public health officials for field assistance and the many homeowners who generously let us into their homes.

Funding: This research was funded by the TRF/BIOTEC Special Programme for Biodiversity Research and Training (BRT R245004), the UNDP/World Bank/World Health Organization/Special Programme for Tropical Diseases Research and Training (TDR/RCS/A00786), and the Mahidol University Research Grant (SCBI-47-T-217).

Conflicts of Interest:

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.


L J O L Y

J F O N T A N E 1 A N

D P C H A S S A I N G

The Rayleigh-Taylor instability of two-dimensional high-density vortices

We investigate the stability of variable-density two-dimensional isolated vortices in the frame of incompressible mixing under negligible gravity. The focus on a single vortex flow stands as a first step towards vortex interactions and turbulent mixing. From heuristic arguments developed on a perturbed barotropic vortex, we find that high-density vortices are subject to a Rayleigh-Taylor instability. The basic mechanism relies on baroclinic vorticity generation when the density gradient is misaligned with the centripetal acceleration field. For Gaussian radial distributions of vorticity and density, the intensity of the baroclinic torque due to isopycnic deformation is shown to increase with the ratio δ/δ ρ of the vorticity radius to the density radius. Concentration of mass near the vortex core is confirmed to promote the instability by the use of an inviscid linear stability analysis. We measure the amplification rate for the favoured azimuthal wavenumbers m = 2, 3 on the whole range of positive density contrasts between the core and the surroundings. The separate influence of the density-contrast and the radius ratio is detailed for modes up to m = 6. For growing azimuthal wavenumbers the two-dimensional structure of the eigen mode concentrates on a ring of narrowing radial extent centered on the radius of maximum density gradient. The instability of the isolated high-density vortex is then explored beyond the linear stage based on high Reynolds number numerical simulations for modes m = 2, 3 and a moderate density contrast C ρ = 0.5. Secondary roll-ups are seen to emerge from the non-linear evolution of the vorticity and density fields. The transition towards m smaller vortices involves vorticity exchange between initially-rotating dense fluid particles and the irrotational less-dense medium. It is shown that baroclinic enstrophy production is associated with the centrifugal mass ejection away from the vortex center.

Introduction

Baroclinic vorticity production has been identified as the counterpart of buoyancy forces in stratified geophysical flows where misalignment between the gravity acceleration field and the density gradient produces and destroys vorticity, see [START_REF] Turner | Buoyancy Effects in Fluids[END_REF]. The baroclinic vorticity production has been much less commented in high Froude number mixing flows. The transition stage of such flows have deserved some interest since the experimental evidence that low-density jets develop specific instability modes as in [START_REF] Monkewitz | Absolute instabilities in hot jets[END_REF]. The development of the Kelvin-Helmholtz instability in high-Froude number incompressible variable-density mixing layers was later subjected to numerical simulations by [START_REF] Soteriou | Effects of the free-stream density ratio on free and forced spatially developing shear layers[END_REF], revealing the crucial role of baroclinic vorticity sources. [START_REF] Reinaud | The baroclinic secondary instability of the two-dimensional shear layer[END_REF] found a secondary instability in the variabledensity mixing layer providing a two-dimensional spatial cascade towards smaller scales.

It was demonstrated to be related to the instability of the strained layer of vorticity and density-gradient. [START_REF] Turner | Buoyant vortex rings[END_REF] early mentioned the stabilizing effect of the centrifugal force due to rotation within vortex rings lighter than the surrounding medium. He compared it to the action of gravity on stably stratified shearing flows. Much later, [START_REF] Chomiak | Modeling variable density effects inturbulent flames -some basic considerations[END_REF] referred briefly to the instability of denser vortices into their variable-density turbulence model. The two-dimensional direct numerical simulations by [START_REF] Chassaing | Variable density mixing in kinematically homogeneous turbulence[END_REF] also confirmed the robustness of low-density vortices and the short life-time of high-density ones. Additional simulations of binary-mixing two-dimensional turbulence, summarized in [START_REF] Joly | Inertia effects in variable-density flows[END_REF], described the underlying process of mass-segregation by vorticity measuring negative correlation ξ 2 ρ , with ξ the vorticity and ρ the density fluctuation. Also displayed in [START_REF] Joly | Inertia effects in variable-density flows[END_REF] were accompanying non-linear simulations of isolated vortices showing axisymmetrization of a low-density vortex and the blow-up of a high-density one into fine-grained smaller structures. More recently, [START_REF] Miller | Visualizations of the dynamics of a vortical flow[END_REF] have observed such a two-dimensional mechanism in the case of an impulsively started density interface at high Schmidt number. After a first roll-up into a primary structure embedding folded density spirals, this mechanism was responsible for a secondary vorticity baroclinic generation. This secondary instability, resulting from the streamlines curvature, is of the same type as the one occurring in the case of an isolated, high-density vortex.

Besides, the stability of swirling flows has deserved much attention due to the interest in the breakdown of trailing vortices. Both axisymmetric and non-axisymmetric perturbations of swirling flows were analysed in the light of both asymptotic and normal-mode analysis. Though a sufficient condition for centrifugal instability has been extended to the compressible case by [START_REF] Eckhoff | A note on the instability of columnar vortices[END_REF] in the limit of large axial wavenumbers, it is demonstrated in a complementary paper by [START_REF] Sipp | Stability of a vortex with a heavy core[END_REF] that such a condition cannot be formulated for two-dimensional non-axisymmetric perturbations. Only an asymptotic analysis in the limit of vanishing density contrast and radius ratio, yields that vortices are unstable to non-axisymmetric perturbations if the density decreases radially somewhere. In contrast with these studies relying on the formulation of classical Sturm-Liouville eigenvalue problems, the present paper favours a heuristic approach and explores the mechanism of the two-dimensional instability of heavy vortices in the light of baroclinic vorticity generation associated with the centripetal acceleration field. This mechanism, originally discussed in [START_REF] Joly | Inertia effects in variable-density flows[END_REF], is demonstrated to be of the same nature as the Rayleigh-Taylor instability in the unstable atmosphere. It is explained in the preliminary part of §2 for a perturbed density field providing a tractable expression of the baroclinic torque associated with departure from barotropy. It succeeds in stressing the destabilizing role of density concentration near the vortex center. The second part of §2 is then devoted to the derivation of a inviscid linear stability analysis of the heavy Lamb-Oseen vortex. The growth rate and spatial structure of the eigen modes, corresponding to the more amplified first azimuthal wavenumbers, are detailed in §3. The separate influence of the radius ratio and density contrast are commented for modes up to m = 6. In §4, high Reynolds number numerical simulations are carried out beyond the linear stage. Based on these simulations, we describe the main features of the breakdown of a heavy vortex submitted to optimal perturbations with m = 2 and m = 3.

The statement of the problem

Equilibrium solutions to the variable-density Euler equations

In the description of inviscid incompressible flows under negligible gravity, the Navier-Stokes equations reduce to the Euler momentum equations supplemented with a purely convective continuity equation for the density :

∇• u = 0,
(2.1a)

d t u = - 1 ρ ∇P, (2.1b) d t ρ = 0, (2.1c)
where d t = ∂ t + (u • ∇) stands for the material derivative. Let χ = ∇ × u be the vorticity that reduces to (0, 0, ξ) in the two-dimensional case. Taking the curl of (2.1b) yields the transport equation for the vorticity scalar field

d t ξ = a × ∇ρ ρ = a × ∇(ln ρ/ρ 0 ). (2.2)
where a = d t u is the acceleration of the particle along a streamline and hereinafter the specific density gradient is quoted g = ∇(ln ρ/ρ 0 ) where ρ 0 can be any density reference.

The right-hand side of (2.2) exhibits one source term in contrast with the conservative nature of vorticity in homogeneous two-dimensional inviscid flows. The baroclinic torque b = a × g is felt when an inhomogeneous mass field is subjected to an acceleration not aligned with the local density gradient. A two-dimensional vortex with circulation Γ and characteristic radius δ is unsensitised to the gravity field if the the inertia force due to the vortex flow itself, ρu 2 θ /δ with u θ ∼ Γ/δ a characteristic azimuthal velocity, is large compared to the projection g ∆ρ of the buoyancy force on the vortex plane with ∆ρ a characteristic density difference between the vortex core and its surroundings. Defining the density contrast as C ρ ∼ ∆ρ/ρ, we form the Froude number Fr = Γ 2 /(δ 3 C ρ g ) and we consider Fr 1. A steady solution to the constant-density Euler equation will stand as a solution to the variable-density Euler equations if the density gradient is everywhere aligned with the pressure gradient and isopycnic or iso-density lines are globally invariant by advection. This is verified for any axisymmetric vorticity field associated with any centered axisymmetric density distribution where the baroclinic torque remains inactive as rings of fluids are moving around the vortex center on circular streamlines. We explore the effect of the density distribution on the stability characteristics of such equilibrium solutions that we term barotropic vortices. In the unperturbed diffusive situation, an initially barotropic vortex evolves as a barotropic vortex with the Schmidt number setting the relative spreading rate of density and vorticity fields.

The baroclinic torque distribution on a perturbed barotropic vortex

The purpose of this section is to give a heuristic explanation of the instability mechanism occurring on a high-density barotropic vortex. We give the expression of the baroclinic vorticity production resulting from an harmonic deformation of the otherwise circular density contours. The instability mechanism is demonstrated by considering vorticity sources and sinks as secondary vortices due to the loss of barotropy. We consider a Gaussian axisymmetric vortex of circulation Γ and core radius δ, defined by its vorticity in polar coordinates, Ω(r, θ) = Γ πδ 2 exp(-r 2 /δ 2 ).

(2.3) This vortex induces a purely azimuthal velocity u θ and a purely radial acceleration a r ,

u θ (r, θ) = [1 -exp(- r 2 δ 2 )] Γ 2π r
, and a r (r, θ) = -u 2 θ /r.

(2.4)

In order to obtain a tractable expression of the baroclinic torque in a particular case, we apply a harmonic deformation with amplitude e < 1 and azimuthal wavenumber m > 0 on a Gaussian axisymmetric density field with characteristic radius

δ ρ = δ/ε, ρ(r, θ) = ρ b + (ρ c -ρ b ) exp(- r 2 ρ (θ) δ 2 ρ ), and r ρ (θ) = r[1 + e cos(mθ)] (2.5)
where ρ b is the density of the background fluid and ρ c the density at the vortex center. Hence, the azimuthal component of the specific density gradient reads

g θ (r, θ) = 1 rρ ∂ρ ∂θ = (1 - ρ b ρ ) 2mer δ 2 ρ
[1 + e cos(mθ)] sin(mθ).

(2.6)

The radial profiles of these flow variables are illustrated on figure 1 for m = 2 and along the ray θ = π/4 where g θ is maximum and for two radius ratio ε = 1 and ε = 4. Increasing the radius ratio concentrates mass towards the vortex core and increases the peak value of g θ while the radius of maximum gradient decreases. The deformation of isopycnic lines yields the misalignment between density and pressure gradients resulting in the distribution, b, of baroclinic vorticity sources and sinks

b(r, θ) = a r g θ = 2 m e ε 2 u 2 θ δ 2 [1 + e cos(mθ)] ρ b -ρ ρ sin(mθ). (2.7)
The baroclinic torque is of the sign of (ρ b -ρ c ) sin(mθ) and its amplitude increases with the amplitude e of the isopycnic deformation, the azimuthal wavenumber m, the density contrast (ρ b /ρ c -1) and the circulation Γ. Both denser and less-dense vortices are considered on figure 2, where we illustrate the sign of vorticity sources and sinks in the four quadrants of the vortex perturbed on mode m = 2. On the interval θ ∈ [0, 2π[, the sign of sin(mθ) changes on (2m -1) rays. Therefore, a system of 2m alternate sign contributions corresponding to m counter-rotative dipoles are superimposed on the base vorticity field. Consequently, less-dense fluid particles are engulfed in and denser fluid particles are ejected off the vortex core between consecutive dipoles. Axisymmetrization of the density distribution is expected if the vortex core is less dense than the surrounding fluid, whereas vortices with a denser core are unstable to perturbations. The arbitrary harmonic deformation of the density field serves here as an illustration of the generic distribution of vorticity sources and sinks on a perturbed barotropic vortex. The instability of the massive vortex is the result of the departure from barotropy or, as developed in §2.1, proceeds from the loss of axisymmetry. The sensitivity of the baroclinic torque intensity to the mass-concentration ε towards the vortex axis is described on figure 3 in the case of a dense vortex. The harmonic deformation of the density iso-contours indicates that the Rayleigh-Taylor instability of the denser vortex is favoured by a characteristic density radius smaller than the vorticity radius.

This mechanism is similar to the one occurring in Rayleigh-Taylor instability where a quiescent, stratified medium is submitted to the external acceleration of the gravity field. There the acceleration is of constant direction and intensity, and the stability of the flow is governed by the misalignment between a fixed acceleration field and a perturbed density gradient. Here the acceleration field is due to the kinematics of the vortex flow which initially yields a purely radial acceleration. Figure 4 illustrates the analogy between the unstable atmosphere and the equivalent unrolled heavy vortex. This is the basic mechanism mentioned by [START_REF] Turner | Buoyant vortex rings[END_REF] and observed in another context by [START_REF] Miller | Visualizations of the dynamics of a vortical flow[END_REF]. Now we explore the parameter space of the isolated dense vortex with a normal mode linear stability analysis.

The equations for the inviscid stability analysis

The base flow is the Gaussian vortex with azimuthal velocity U (r) as given by (2.4). We superimpose the following density distribution R(r) also resulting from similarity solutions of the diffusive problem,

R = ρ b + (ρ c -ρ b ) exp(- r 2 δ 2 ρ
).

(2.8)

We choose the density contrast C ρ = (ρ c -ρ b )/(ρ c + ρ b ) as the relevant parameter, lying in [-1, 1], to explore the whole range of density ratios. This parameter, elsewhere invoked as an Atwood number, gives a straightforward logarithmic scale for the normalisation of relative density gradients in the equations. Denoting s = ρ c /ρ b the density ratio, both measures of the fluid inhomogeneity are of course equivalent by C ρ = (s -1)/(s + 1). Besides, the radius of the density Gaussian distribution provides a second length scale δ ρ that will be related to the vorticity radius by the ratio ε = δ/δ ρ . We will often refer to vortices larger then their associated density field for which ε > 1.

The variable-density Euler equations are linearised around the base flow for the velocity, pressure and density perturbations [û r , ûθ , p, ρ](r, θ, t). The perturbations are assumed to be of the form

[û r , ûθ , p, ρ] = [iu r (r), u θ (r), p(r), ρ(r)] exp[i(mθ -ωt)],
where m is the positive integer azimuthal wavenumber and ω is the complex disturbance phase speed. With γ = mU/rω, () * = () + ()/r and prime denoting r-differentiation, we write the set of equations for the disturbance amplitudes :

γρ + R u r = 0,
(2.9)

u * r + mu θ /r = 0, (2.10) R(γu r + 2U u θ /r) + U 2 ρ/r + p = 0, (2.11) R(γu θ + U * u r ) + mp/r = 0.
(2.12)

The loss of barotropy being associated with the loss of axisymmetry, there is no unstable modes for axisymmetric perturbation and hereafter m > 0. The eigenfunctions must vanish at r = ∞ and considering leading-order terms in the Taylor series we get the limit behaviour at r = 0:

u r , u θ ∼ O(r m-1 ), p, ρ ∼ O(r m ).
(2.13) We reduce the system (2.9)-(2.12) to a minimalist set of two coupled equations for the amplitudes of the radial velocity and the density perturbations:

γρ + R u r = 0, (2.14) (r 2 Rγ/m)u r + [2RU + r(rRγ) /m + 2rRγ/m -rRU * ] u r + [2RU/r -mRγ -(rRU * ) + (rRγ) /m] u r + (mU 2 /r) ρ = 0. (2.15)
Included in the function γ, the phase speed ω = ω r + i ω i appears linearly in the above system which forms an eigenvalue problem. The sign of the imaginary part of ω discriminates between stable vortices for negative ω i and unstable ones for positive ω i . We use a spectral collocation method on a basis of Chebyshev polynomials mapped algebraically on the interval [0, ∞]. This discretization produces a generalized eigenvalue problem under the matrix form Ax = ωBx that is solved by the QZ algorithm under Matlab. The number of Chebyshev modes for each variable is N which sets the dimension of matrices A and B to (2N ) 2 . Due to the presence of critical layers where γ = 0 the search of the frontier of neutral stability requires contouring the singularities in the complex plane, see [START_REF] Boyd | Complex coordinate methods for hydrodynamic instabilities and sturm-liouville eigenproblems with an interor singularity[END_REF]. Increasing the number of collocation points shrinks the discretization spectrum towards the real axis but does not succeed in unveiling the neutral limit. We use contour deformation according to the proposal of [START_REF] Leibovich | A sufficient condition for the instability of columnar vortices[END_REF] which deforms the discretization spectrum below the real axis. The real coordinate r is mapped to r = r[1 -iλγ (r)], where λ ∈ [0, 1] sets the intensity of the contour deformation. All real coefficients of the system (2.14),(2.15) are complexified by substitution of r to r. For the detection of the most amplified mode, we increase simultaneously the number of collocation points N and the displacement factor λ until one fixed point is detected in the (ω r , ω i ) plane. The invariance to the couple (N, λ) is judged to a relative accuracy of 10 -4 . This results in rapid convergence and N ∼ 60 for largely amplified modes remote to the neutral limit whereas marginally stable modes require a higher resolution raising up to N ∼ 300.

Topography of the instabilities

Figure 5 displays the contours of the amplification rate in the (C ρ , ε)-plane for modes m = 2 and m = 3 and positive values of the density contrast. As expected from above considerations no unstable modes are found for vortices less dense then the background fluid, i.e. for negative C ρ . We choose to apply a logarithmic scale to the radius ratio to stretch the range around unity. For m = 2 the neutral curve stands above ε = 1 for density contrasts below 0.5. The sharp transition around this value is due to weakly amplified modes detected for extremely dense vortices. For this azimuthal wavenumber the amplification rate peaks up to 0.48 in the upright corner of the (C ρ , ε)-plane. The map of the amplification rate for m = 3 is quite similar with a maximum amplification rate of 0.62 for unrealistic extreme density ratios (C ρ ∼ 1) and very small density cores (ε 1). Given a density contrast, the vortex is stable to small perturbations below a critical radius ratio. For moderate density ratios encountered in binary mixing and thermal mixing, instability will occur only if the denser core is smaller than the vortex core. This has to be related with the preliminary results displayed on figure 3, stressing the increase of the perturbation baroclinic torque with decreasing density radii. For a given density contrast, the density gradient increases when reducing the density radius and so does the cause of the Rayleigh-Taylor instability. For small enough dense cores the growth rate of the instability in the linear regime increases with the density contrast. Consequently the larger growth rates are found in the up-right corner of the (C ρ , ε)-plane.

These main conclusions are further illustrated on figure 6 where we separate the influences of the radius ratio and the density contrast for the first six modes. For a mean density contrast C ρ = 0.5, the critical radius ratio is seen to increase with the azimuthal wavenumber. The increase of the growth rate with increasing radius ratio saturates before ε = 10 when only one length scale, δ, becomes relevant to the base flow. The asymptotic value at saturation is seen to increase with the azimuthal wave number. As noticed for modes 2 and 3, the amplification rate increases with the density contrast for a given radius ratio. It is seen from figure 6 that for a radius ratio ε = 2 mode m = 2 is the most Table 1. Eigenvalues for the unstable mode of the vortex defined by Cρ = 0.5 and ε = 2.

unstable for moderate density contrasts below 0.35 and is superseded by mode m = 3 above that level.

Mode m = 1 exhibits much lower amplification rates than modes m = 2, 3 almost everywhere except in a small region of the parameter space lying around (C ρ , ε) = (0.2, 1.4). As also displayed in a complementary paper by [START_REF] Sipp | Stability of a vortex with a heavy core[END_REF], we found that when mode m = 1 is the most amplified mode, its amplification rate remains below 0.01. From [START_REF] Joly | The merger of two-dimensionnal radially-stratified high-froude number vortices[END_REF], it is seen that vortex interactions occurring in a typical twodimensional vortex merger are producing large scale perturbations similar to the m = 2 case. We focus here on these largely amplified modes likely to occur in viscous flows with vortex interactions.

From these results we select the vortex with a density core twice smaller than the vortex core, ε = 2, and C ρ = 0.5, corresponding to a density ratio of s = 3. This vortex is unstable to mode m = 1 to m = 7 with eigenvalues reported in table 1. The absolute value of the eigenfunctions of velocity and density perturbations is described on figure 7 for modes m = 2, 3, 4, 6. The asymptotic behaviour at the origin as prescribed by (2.13) is verified. In particular, the velocity eigenfunctions of mode m = 2 exhibit a linear growth from the origin. As m is increased, the velocity and density eigenfunctions spread over a region of narrowing radial extent. This region is centered on the characteristic radius r = δ ρ of the density distribution. The spatial structure of these modes is compared on figure 8 displaying the two-dimensional vorticity perturbation fields. The vorticity perturbation and density perturbation (not displayed) are organized as 2m spiral arms of alternate sign. In the next section, we investigate the development of eigen-modes m = 2, 3 beyond the linear stage.

-1 0 1 -1 0 1 y/δ m=2 -1 0 1 -1 0 1 m=3 -1 0 1 -1 0 1 x/δ y/δ m=4 -1 0 1 -1 0 1 x/δ m=6

Non-linear simulations of the breakdown of a dense vortex

The numerical procedure

A viscous numerical simulation is carried out on the previously selected base field perturbed by the most unstable inviscid mode. In the limit of zero Mach number the divergence of the velocity field is due to the molecular diffusion of species only, and the continuity equation comes out as an advection-diffusion one. This is relevant to the mixing of two incompressible fluids with Fickian diffusion allowing the density to vary between the densities of the pure species, ρ c and ρ b . For the high Reynolds number flows consid-ered here, the divergent contribution of the viscous diffusion is discarded. We consider also constant diffusivities in order to retain first order density effects of inertial nature in a minimalist code avoiding contributions due to variations of the fluid properties, see [START_REF] Chassaing | Variable Density Fluid Turbulence[END_REF] for details. The characteristic scales are determined by the vortex circulation Γ, the vortex radius, the density contrast and the radius ratio. In particular, the time scale τ is the inverse of the maximum vorticity πδ 2 /Γ. Given a unitary Schmidt number and Re = Γ/ν = 20000 the Reynolds number based on the vortex circulation Γ, we solve the following normalized transport equations for the primitive variables ( , u), with = ln(ρ/ρ b ),

d t = 1 ReSc ∆ , (4.1a) d t u = - 1 ρ ∇p + 1 Re ∆u. (4.1b)
The numerical procedure is a two-third dealiased pseudo-spectral code based on a variable-density transpose of the projection method ensuring the diffusive nature of the velocity divergence. The non-solenoidal part of the velocity field is prescribed when solving the Poisson equation for the pressure by ensuring : ReSc∇ • u = -C ρ ∆ . The variables are time-advanced using a low-storage third-order Runge-Kutta scheme with semi-implicit treatment of right-hand-side terms. The time-step is variable and adjusted according to the current maximum of velocity ensuring the Courant-Friedrichs-Lewy condition is respected with CFL= ∆t/∆x × max(u) kept under 0.7. As will be described later, the spatial structure of the flow evolves towards much smaller length scales. We opted for a time-dependent spatial resolution to minimize the early-time simulation cost. Starting with a 2048 2 grid, a spectral sensor placed near the maximum wavenumber triggers the increase of the spectral radius raising the final resolution up to 5760 2 in the m = 3 case. At the initial time, the amplitude of the perturbation is normalized to represent 2% of the enstrophy of the base flow.

The boundary conditions are periodic in both directions resulting in a bias compared to the case considered in the stability analysis with an infinite spatial extent. As discussed by [START_REF] Pradeep | Effects of boundary conditions in numerical simulations of vortex dynamics[END_REF], periodicity of the boundary conditions yields zero normal and tangential velocities on the square computational boundaries and a zero circulation over the computational domain. Due to its slow radial decrease the azimuthal velocity of the base flow is the variable most affected by this constraint. The vorticity field is also added a background negative value ξ b such that ξ b × L x × L y = -Γ. The bias amplitude depends on the relative size of the computational domain compared to the vortex core. In the simulations commented here, the vortex circulation is π/4 while the domain is (2π) 2 with a vortex radius δ = 0.5. This corresponds to a distance of the vortex center to the domain boundary of a little more than 6 radius and a peak vorticity ξ p = 1 + ξ b . In this case ξ b = (16π) -1 which amounts to less than 2% of ξ p . This allows for a good compromise between representing the isolated vortex flow and the required spatial resolution of the very thin layers arising during the non-linear stage of the instability. But clearly the eigen-modes deduced from the stability analysis only approach the eigen-modes of the effective base flow as seen from the numerical simulation tool. The simulation of the linear stage thus suffers from the lack of representativity of the boundary conditions of the isolated vortex. However, in the non-linear stage, the instability mechanism produces baroclinically enhanced vorticity up to 10 times the initial peak value ξ p , or 1000 times the artificial background vorticity ξ b . Simulations were performed in worse conditions with a vortex radius twice the present one and exhibited vorticity and density field very similar to the ones that follow. We conclude that the non-linear stage is properly represented despite the periodization of the flow. Besides, the purpose of this numerical approach is to give a first illustration of this original instability mechanism beyond the linear stage. the vorticity and density perturbations by,

Description of the flow

ξ = (ξ -Ω)dxdy, ρ = (ρ -R)dxdy. (4.2)
The radius of the unperturbed Gaussian vortex is known to evolve as δ = δ 0 1 + 4πt/Re in normalized time units. Considering the very high Reynolds number of the simulation, the radius of the unperturbed field would have gained less than 1% of the initial value at t = 20. Figure 13 displays the time evolution of the perturbations amplitudes, normalized by their initial value. The exponential growth rates of these perturbations as predicted by the linear stability analysis are σ m=2 = 0.192 and σ m=3 = 0.204. We compare the effective growth rates to an average growth rate σ = 0.2. The vorticity and density perturbations are seen to grow after a few time units because of the non-optimal nature of the eigen-modes within the numerical frame. Then exponential growth is observed that compares well with the predicted rate despite the misrepresentation of the boundary conditions. On figure 13(b) the density perturbation of mode m = 3 is seen to grow faster than the density perturbation of mode m = 2 confirming the slight precedence of mode m = 3. From the evolution of the amplitude of vorticity and density perturbations, the saturation is seen to occur near t = 30 for both modes. The departure from the linear exponential growth is effective by t = 20. On figure 14(a), enstrophy production by the baroclinic torque is seen to peak at saturation time. The peak value is higher for mode m = 2 and approximately six times the initial value. In the development of mode m = 3, a secondary instability of the vortex sheets results in a small burst of enstrophy at t = 35. Then for both modes the enstrophy monotonously decreases due to viscous dissipation at high wavenumbers. The baroclinic vorticity generation has been proposed to be the basic mechanism of this instability yielding the centrifugal ejection of the denser fluid particle away from the vortex core. In order to prove this point, we measure the correlation between the radial velocity and the density difference relative to the mean density ρ, where the overbar denotes an average over the whole simulation domain. The correlation is normalized by their respective variances to form a correlation coefficient,

C ρur = (ρ -ρ)u r (ρ -ρ) 2 u 2 r 1/2 . (4.3)
From the time evolution of this correlation coefficient in figure 14(b), the ejection of high density fluid particles and convergence of low-density ones towards the vortex center is seen to be significant after several time units. It occurs sooner for mode m = 2 and reaches a value of 0.4 well before saturation time indicating a quite coherent mass ejection as seen from density iso-contours on figure 10 at t = 20, 25. After the two smaller vortices have formed far from the center of the domain, the radial velocity is not relative to a vortex center and C ρur is no more relevant. Mass ejection during the non-linear development of mode m = 3 is less violent and occurs later in two phases associated with the two peaks in enstrophy generation. After the first stage between t = 20 and t = 25, three small vortices are seen to rotate near the initial vortex center on figure 11 at t = 30. At this point the radial velocity from the domain center is still relevant to the global vortex system. Then another centrifugal mass ejection is observed by t = 30 signaled by a second increase of enstrophy and C ρur . The measure of the density at the center of the m smaller structures at t = 30 reveals a density ratio of s = 1.44 against the background medium, corresponding to a density contrast C ρ = 0.18. Due to the unitary Schmidt number the resulting roll-ups have similar density and vorticity radii, hence ε ∼ 0(1). We refer to figure 5 to conclude that these new vortices, slightly denser than the surrounding fluid, are stable pertaining to the Rayleigh-Taylor instability. From inspection of the contours of figures 10 and 12, we estimate that the characteristic length scale of the m generated vortices compared to the initial vorticity radius has been divided by 6 for m = 2 and by 10 for m = 3. Moreover, very high levels of vorticity have been generated and the pick values at the new vortex centers are ten times higher than the initial maximum vorticity for both cases. A clear picture of the situation after the completion of the Rayleigh-Taylor instability then emerges from these measurements. The baroclinic production of vorticity has produced smaller stable rollups of much higher vorticity. During the process, high-density samples of fluid have been shed forming strained lumps much more sensitive to molecular diffusion than the initial barotropic distribution.

Concluding remarks

The analysis of the linear inviscid stability of the variable-density isolated vortex has been derived. The heavy vortex is subjected to a Rayleigh-Taylor instability promoted by the concentration of dense fluid on a smaller radius than the vortex core. Low azimuthal wavenumbers, m = 2, 3 are seen to be the more amplified for moderate radius ratios and realistic density ratios. The maps of their growth rates have been established and the neutral curve delineated in the density-contrast versus radius-ratio parameter space. Non-linear numerical simulations of the breakdown of a vortex three times denser than the surrounding fluid have been carried out for eigen-perturbations derived for m = 2 and m = 3. The baroclinic torque is seen to produce thinning vorticity sheets partially nucleating into m robust smaller vortices of lower density and higher vorticity than the initial one. This relaxation towards a more stable flow proceeds from vorticity exchange between dense fluid particles from the vortex core and less-dense ones from the background irrotational field.

This mechanism shall be examined further as being central to variable-density mixing at high Froude-number. The instability of high-density vortices and the robustness of lowdensity ones are expected to result in mass-segregation by vorticity in two-dimensional flows resulting in low-density fluid particles near vortex centers and high-density fluid particles occupying the interlacing medium. Measuring the anti-mixing effect of the masssegregation mechanism should be addressed in turbulent flows where vortex interactions are crucial and lower the relevance of the isolated vortex configuration. However, the spectral reconditioning of the vorticity field towards higher wavenumbers associated to the breakdown of high-density vortices increases dissipation and mixing at small scales. The resulting strained high-density fluid lumps arising during the development of the Rayleigh-Taylor instability are more efficiently smeared out by molecular mass-diffusion. The resolution of this contradictory effects of the baroclinic torque on the mixing efficiency is the subject of ongoing efforts. The influence of the Schmidt number may well be of first importance in the discussion of such a question. In the perspective of variable-density turbulent situations, the two-dimensional and three-dimensional vortex interactions are also under consideration.

We acknowledge fruitful discussions with J.N. Reinaud and D.G. Dritschel. Numerical simulations were performed on the scalar resources of IDRIS center under project number 41552.

Figure 1 .

 1 Figure 1. Radial profiles along ray θ = π/4 of two perturbed dense barotropic vortices with Γ = π, δ = 1, ρc/ρ b = 3, m = 2 and e = 0.3: (a) azimuthal velocity u θ (solid) and density profile ρ-ρ b for ε = 1 (dashed) and ε = 4 (dot-dashed); (b) radial acceleration ar (solid) and azimuthal component of the specific density gradient g θ for ε = 1 (dashed) and ε = 4 (dot-dashed).

Figure 2 .Figure 3 .

 23 Figure 2. Sketch of the baroclinic torque contributions on Gaussian vortices bearing denser and less-dense elliptic density distributions.

Figure 4 .

 4 Figure 4. Analogy between the dense vortex and the unstable density stratification submitted to a downward gravity field (ρ1 > ρ2).

Figure 5 .Figure 6 .

 56 Figure 5. Contours of constant amplification rate ωi in the (Cρ, ln10(ε))-plane for azimuthal wavenumbers m = 2 (left), m = 3 (right). Increment between contours is 0.02 and the thick line is the neutral curve.

Figure 7 .

 7 Figure 7. Absolute value of the eigenfunctions of the denser vortex flow with Cρ = 0.5 and ε = 2: radial velocity ur (solid), azimuthal velocity u θ (dashed) and density ρ (dot-dashed). Normalisation is performed to ensure a unitary kinetic energy e = 1 2 R ∞ 0 (u 2 r + u 2 θ )dr.

Figure 8 .

 8 Figure 8. Two-dimensional structure of the vorticity perturbation corresponding to eigen-modes m = 2, 3, 4, 6 of the vortex with Cρ = 0.5 and ε = 2. Contour spacing : max( ξ)/6.

Figure 9 .

 9 Figure 9. Sequence of iso-vorticity contours for m = 2. Length scale δ is represented by the line segment at t = 0. Contour spacing 0.1/τ from t = 0, 5, 10, 15, 0.4/τ for t = 20, 25 and 0.8/τ for t = 30, 35. Contours at negative levels are dashed. N is the number of grid point along one direction of the squared simulation domain.

Figure 10 .

 10 Figure 10. Sequence of iso-density contours for m = 2. Length scale δρ is represented by the line segment at t = 0. Fixed contour spacing (ρcρe)/9.

Figures 9 -

 9 Figures 9-12 illustrate the time evolution of the vorticity and density fields of non-linear realizations of the Rayleigh-Taylor instability for modes m = 2, 3 of the vortex three times denser than the surrounding fluid. As the instability develops, the baroclinic torque

Figure 11 .

 11 Figure 11. Sequence of iso-vorticity contours for m = 3. Same legend as for m = 2 on figure 9.

Figure 12 .Figure 13 .

 1213 Figure 12. Sequence of iso-density contours for m = 3. Same legend as for m = 2 on figure 10.

Figure 14 .

 14 Figure 14. (a) Time evolution of the total enstrophy on the simulation domain z = R ξ 2 dxdy. (b) Time evolution of the correlation coefficient between the density fluctuation and the radial velocity.