Anne Delextrat

Véronique Tricot

Thierry Bernard

Fabrice Vercruyssen

Christophe Hausswirth

Jeanick Brisswalter
email: brisswalter@univ-tln.fr.

Pr Jeanick Brisswalter

Drafting during Swimming Improves Efficiency during Subsequent Cycling

Keywords: TRIATHLETES, HYDRODYNAMIC DRAG, OXYGEN KINETICS, HEMODYNAMICS, CADENCE

triathlon determinants highlighted that the metabolic demand induced by swimming could have detrimental effects on subsequent cycling or running adaptations (e.g., 3).

Experimental studies on the effect of prior swimming on subsequent cycling performance have led to contradictory results. Kreider et al. [START_REF] Kreider | Cardiovascular and thermal responses of triathlon performance[END_REF] have found that an 800-m swimming bout resulted in a significant decrease in power output (17%) during a subsequent 75-min cycling exercise. More recently, Delextrat et al. [START_REF] Delextrat | Effect of wet suit use on energy expenditure during a swim-to-cycle transition[END_REF] have observed a significant decrease in cycling efficiency (17.5%) after a 750-m swim conducted at a sprint triathlon competition pace when compared with an isolated cycling bout. In contrast, Laursen et al. [START_REF] Laursen | The effects of 3000-m swimming on subsequent 3-h cycling performance: implications for ultraendurance triathletes[END_REF] indicated no significant effect of a 3000-m swim performed at a long-distance triathlon competition pace on physiological parameters measured during a subsequent cycling bout. It is therefore suggested that the swimming section could negatively affect the subsequent cycling, especially during sprint triathlon, where the intensity of the swim is higher than during longdistance events.

Within this framework, we showed in a recent study [START_REF] Delextrat | Effect of wet suit use on energy expenditure during a swim-to-cycle transition[END_REF] that decreasing the metabolic load during a 750-m swim by using a wet suit resulted in a 11% decrease in swimming heart rate (HR) values and led to a 12% improvement in efficiency during a subsequent 10-min cycling exercise, when compared with swimming without a wet suit. The lower relative intensity when swimming with a wet suit is classically explained by a decrease in hydrodynamic drag. This decrease in hydrodynamic drag results from an increased buoyancy that allows the subjects to adopt a more horizontal position, thus reducing their frontal area [START_REF] Chatard | Effects of wetsuit use in swimming events[END_REF].

During swimming, hydrodynamic drag could also be reduced when swimming in a drafting position (i.e., swimming directly behind another competitor). The effects of drafting during short swimming bouts have been widely studied in the recent literature [START_REF] Bassett | Metabolic responses to drafting during front crawl swimming[END_REF][START_REF] Chatard | Performance and drag during drafting swimming in highly trained triathletes[END_REF][START_REF] Chollet | The effects of drafting on stroking variations during swimming in elite male triathletes[END_REF][START_REF] Millet | Effects of drafting behind a two-or a six-beat kick swimmer in elite female triathletes[END_REF]. The main factor of decreased body drag with drafting seems to be the depression made in the water by the lead swimmer [START_REF] Bentley | Specific aspects of contemporary triathlon[END_REF]. This low pressure behind the lead swimmer decreases the pressure gradient from the front to the back of the following swimmer, hence facilitating his displacement through the water [START_REF] Chatard | Performance and drag during drafting swimming in highly trained triathletes[END_REF]. Within this framework, significant decreases in passive drag (i.e., drag forces exerted on subjects passively towed through the water in prone position [START_REF] Chatard | Performance and drag during drafting swimming in highly trained triathletes[END_REF]) from 10% to 26% have been reported in a drafting position compared with isolated conditions (for review, (3)). Moreover, swimming in drafting position is associated with significant reductions in oxygen uptake (10%), HR (6.2%), and blood lactate concentration (11-31%) [START_REF] Bassett | Metabolic responses to drafting during front crawl swimming[END_REF][START_REF] Chatard | Performance and drag during drafting swimming in highly trained triathletes[END_REF].

During a multidisciplinary event, such as triathlon, the effect of drafting on subsequent performance has been studied only during the cycling leg. Hausswirth et al. [START_REF] Hausswirth | Effects of cycling alone or in a sheltered position on subsequent running performance during a triathlon[END_REF] showed that drafting during the cycle portion of a sprint triathlon led to a significant decrease in cycling energy expenditure (14%) compared with an individual effort, leading to a 4.1% improvement in performance during the subsequent 5-km run. To the best of our knowledge, no similar study has been conducted during a swim-bike trial, in order to evaluate the effects of drafting during swimming on subsequent cycling performance

The objective of the present study was therefore to investigate the effects of drafting during swimming on energy expenditure in the context of a swim-bike trial. We hypothesized that swimming in drafting position would be associated with a lower metabolic load during swimming and would reduce energy expenditure during subsequent cycling.

MATERIALS AND METHODS Subjects

Eight male triathletes competing at interregional or national level (age: 26 ± 6 yr, height: 183 ± 7 cm, weight: 74 ± 7 kg, body fat: 13 3%) participated to this study. They were all familiarized with laboratory testing. Average training distances per week were 6.6 km in swimming, 59 km in cycling, and 34 km in running, which represented 150 min, 135 min, and 169 min for these three disciplines, respectively. This training program included only one cross-training session (cycle-to-run) per week. The low distance covered by the triathletes during training, especially in cycling could be partly explained because the experiment was undertaken in winter, when triathletes usually decrease their training load in the three disciplines. Written consent was given by all the subjects before ail testing and the ethics committee for the protection of individuals gave their approval of the project before its initiation (Saint-Germain-en-Laye, France).

Protocol

Maximal oxygen uptake (V0 2 ".) and maximal aerobic power (MAP) determinations. The first test was a laboratory incremental test on a cycle ergometer to determine V0 2 . and MAP. After a 6-min warm-up at 150 W, the power output was increased by 25 W every 2 min until volitional exhaustion. The criteria used for the determination of V0 2 . were: a plateau in VO 2 despite the increase in power output, a HR over 90% of the predicted maximal HR, and a respiratory exchange ratio (RER) over 1.15 [START_REF] Howley | Criteria for maximal oxygen uptake: review and commentary[END_REF]. Because V0 2 max was used as a simple descriptive characteristic of the population for the present study and was sot a primary dependent variable, the attainment of two out of three criteria was considered sufficient [START_REF] Howley | Criteria for maximal oxygen uptake: review and commentary[END_REF]. The ventilatory threshold (VT) was calculated using the criteria of an increase in VE/VCO 2 with no concomitant increase of VE/ VCO 2 [START_REF] Wasserman | Anaerobic threshold and respiratory gas exchange during exercise[END_REF].

Submaximal sessions. After this first test, each triathlete underwent three submaximal sessions separated by at least 48 h. The experimental protocol is described in Figure 1. All swim tests took place in the outdoor Olympic swimming pool of Hyères (Var, France) and were performed with a neoprene wet suit (integral wet suit Aquaman ® , Pulsar 2000, thickness: shoulders: 1.5 mm, trunk: 4.5 mm, legs: 1.5 mm, arms: 1.5 mm) The cycling tests were conducted adjacent to the swimming pool in order to standardize the duration of the swim-to-cycle transition (3 min). The first test was always a 750-m swim performed alone at a sprint triathlon competition pace (SA trial). It was used to determine the swimming intensity for each subject. The two other tests, presented in a counterbalanced order, comprised one swim-to-cycle transition performed alone (SAC trial) and one swim-to-cycle transition with a swimming bout performed in drafting position (SDC trial).

The SAC trial consisted of a 750-m swim at the pace adopted during SA, followed by a 15-min ride on the bicycle ergometer at 75% of MAP and at a freely chosen cadence (FCC). This intensity was chosen to be comparable with the cycling competition pace during a sprint triathlon reported in subjects of the same level by previous studies (e.g., [START_REF] Hausswirth | Effects of cycling alone or in a sheltered position on subsequent running performance during a triathlon[END_REF]). Moreover, it was similar to those used in recent works studying the cycle-to-run transition in trained triathletes (e.g., 27). During the SDC trial, the subjects swam 750 m in drafting position (i.e., swimming directly behind a competitive swimmer in the same lane) at the pace adopted during SA. They then performed the 15-min ride at the same intensity as during SAC. The lead swimmer, who was used for ail the triathletes, was a highly trained swimmer competing at international level. To reproduce the swimming pace adopted during SA, the lead swimmer was informed of his performance every 50 m via visual feedback.

Measured Parameters

Swimming trials. During each swimming trial, the time to cover each 50 m and overall time were recorded. Subjects were instructed to keep the velocity as constant as possible. Stroke frequency (SF), expressed as the number of complete arm cycles per minute, was measured for each 50 m on a 20-m zone situated in the middle of the pool. The stroke length (SL) was calculated by dividing the mean velocity of each 20-m swim by the mean SF of each 20-m swim.

Immediately after each trial, the triathletes were asked to report their perceived exertion (RPE) using the 15-graded Borg scale (from 6 to 20 [START_REF] Borg | Perceived exertion as an indicator of somatic stress[END_REF]).

Blood sampling. Capillary blood samples were collected from subjects' earlobes at the following times: 1 and 3 min after swimming (L1, L2), and at the third and 15th minutes of cycling (L3, L4). Blood lactate concentration (LA, mmol.L -1 ) was then measured by the lactate Pro TM LT-1710 portable lactate analyzer (Arkray, KDK, Japan). The highest of the two postswim (L1, L2) concentrations was considered as the postswim lactate value, because the time delay for lactate to diffuse from the muscles to the blood during the recovery from a swimming exercise has not been precisely established [START_REF] Lepers | Swimming-cycling transition modelisation of a triathlon in laboratory. Influence on lactate kinetics[END_REF].

Measurement of respiratory gas exchange. During the cycling trials, oxygen uptake (VO 2 ), HR, and respiratory parameters (expiratory flow: VE; respiratory frequency: RF) were monitored breath-bybreath and recorded by the Cosmed K4b 2 telemetric system (Rome, Italy).

HR was continuously monitored during swimming and cycling using a cardiofrequency meter (Polar Vantage, Kempele, Finland). Physiological solicitation of cycling was assessed using oxygen kinetics analysis (e.g., 29); energy expenditure was analyzed by gross efficiency calculation [START_REF] Chavarren | Cycling efficiency and pedalling frequency in road cyclists[END_REF].

Curve fitting. Oxygen kinetics were modeled according to the method used by Barstow et al. [START_REF] Barstow | Influence of muscle fiber type and pedal frequency on oxygen uptake kinetics of heavy exercise[END_REF]. Breath-by-breath VO 2 data were smoothed in order to eliminate the outlying breaths (defined as those that were lying outside two standard deviations of the local mean). For each trial (SDC and SAC), the time course of the VO 2 response after the onset of the cycling exercise was described by two different exponential models that were fit to the data with the use of nonlinear regression techniques in which minimizing the sum of squared error was the criterion for convergence.

The first mathematical model was a mono-component exponential model:

The second mathematical model was a two-component exponential model:

The use of one of these models depends on the relative exercise intensity [START_REF] Xu | Oxygen uptake kinetics during exercise[END_REF]. The mono-component exponential model characterizes the VO 2 response during an exercise of moderate intensity (i.e., below the lactate threshold). After a time delay corresponding to the transit time of blood flow from the exercising muscle to the lung (TD), VO 2 increases exponentially toward a steady state level. The VO 2 response is characterized by an asymptotic amplitude (A) and a time constant (T) defined as the time to reach 63% of the difference from final plateau value and baseline (V0 2 (b), corresponding to the value recorded on the bicycle before the onset of cycling). At higher intensities, the VO 2 response is modeled by a two-component exponential function. The first exponential term describes the rapid rise in VO 2 previously observed (the parameters TD 1 , A 1 , and τ 1 are identical to TD, A, and T of the monocomponent exponential model), whereas the second exponential term characterizes the slower rise in VO 2 termed "VO 2 slow component" that is superimposed on the rapid phase of oxygen uptake kinetics.

The parameters TD 2 , A2, and τ 2 represent, respectively, the time delay, asymptotic amplitude, and time constant for this exponential term. The computation of best-fit parameters was chosen by a computer program (SigmaPlot 7.0) so as to minimize the sum of the squared differences between the fitted function and the observed response.

Determination of cycling gross efficiency. Cycling gross efficiency (GE, %) was calculated as the ratio of work accomplished per minute (kJ•min-1 ) to metabolic energy expended per minute (kJ•min- 1) . Because relative intensity of the cycling bouts could be superior to VT, the aerobic contribution to metabolic energy was calculated from the energy equivalents for oxygen (according to respiratory exchange ratio value) and a possible anaerobic contribution was estimated using blood lactate increase with time (à lactate: 63 J-kg -1 .mM -1 ; 13). For this calculation, VO 2 and lactate increase was estimated from the difference between the 15th and the third minutes.

Pedal rate. All the cycling tests were performed on an electromagnetically braked cycle ergometer (SRM Jülich, Welldorf, Germany) The cycle ergometer was equipped with the triathletes' own pedals, and the handlebars and racing seat were fully adjustable both vertically and horizontally to reproduce conditions known from their own bicycles. The SRM system can maintain a constant power output independent of the pedal rate spontaneously adopted by the subjects.

Statistical Analysis

All the results were expressed as mean and standard deviation (mean SD). Differences between the two conditions (swimming alone or in drafting position) in physiological and biomechanical parameters were analyzed using a Student t-test for paired samples. The level of confidence was set at P < 0.05.

RESULTS

Maximal Test

The subjects' physiological characteristics recorded during the incremental cycling test are presented in Table 1. VO 2m a x values were close to those previously obtained for triathletes of the same level [START_REF] Brisswalter | Energetically optimal cadence vs. freely-chosen cadence during cycling: effect of exercise duration[END_REF][START_REF] Vercruyssen | Influence of cycling cadence on subsequent running performance in triathletes[END_REF]. From VT values, it could be observed that the cycling bouts were performed at an intensity close to VT + 2%.

Swimming Trials

Performance. No significant difference in performance was observed between the two swimming trials (respectively for SAC and SDC: 638 ± 38 s and 637 ± 39 s, P > 0.05). The two 750-m swims were therefore performed at a mean velocity of 1.18 m-s -1 . In addition, the stroke characteristics (SR and SL) were not significantly different between SAC and SDC trials (mean SR: 33.2 ± 4.5 cycles•min - 1 vs 33.1 ± 5.1 cycles.min -1 , respectively, for SAC and SDC, P > 0.05; mean SL: 2.13 ± 0.29 rn•cycle -1 vs 2.15 ± 0.30 m•cycle -1 , respectively, for SAC and SDC, P > 0.05). During the SDC trial, the mean distance between the subjects (draftees) and the lead swimmer did not exceed 1 m.

Physiological parameters and RPE.

The HR values recorded during the last 5 min of swimming are presented in Figure 2. The main result shows that swimming in drafting position resulted in a significant mean decrease of 7% in HR values during the last 4 min of swimming in comparison with the isolated swimming bout (160 ± 15 beats.min -1 vs 172 ± 18 beats•min -1 , respectively, for SDC and SAC trials, Fig. 2, P < 0.05). Furthermore, postswim lactate values were significantly lower (29.3%) after the SDC session when compared with the SAC session (5.3 ± 2.1 mmol-L -1 vs 7.5 ± 2.4 mmol•L - 1 , respectively, for SDC and SAC trials, P < 0.05).

Finally, RPE values recorded immediately after swimming indicated that the subjects' perception of effort was significantly lower in the SDC trial than in the SAC trial (13 ± 2 vs 15 ± 1, corresponding to "rather laborious" versus "laborious" respectively for SDC and SAC trials, P < 0.05).

Cycling trials

VO 2 kinetics. All VO 2 responses were best fitted by a mono-component exponential model, except the 'O 2 responses of one subject during the SAC trial that were best described by a two-component exponential function. The occurrence of a slow component in this latter case is representative of a heavy-intensity exercise whereas the other subjects have exercised in a moderate-intensity domain [START_REF] Xu | Oxygen uptake kinetics during exercise[END_REF]. Therefore, the parameters of the model for this subject are different (two-component exponential model vs mono-component exponential model) and could not be included in the same analysis. Figure 3 shows the breath-by-breath VO 2 responses during SAC and SDC trials for a representative subject (responses best fitted by a mono-component exponential model, Fig. 3A) as well as the breath-bybreath \ . /0 2 responses for the subject eliminated (responses best fitted by a two-component exponential model, Fig. 3B). Statistical analysis shows that baseline VO 2 values were sot significantly different between SAC and SDC trials (P > 0.05). However, we have observed that during the SAC trial, higher VO 2 values at the steady state level were attained more quickly than during the SDC trial (time constant values for SAC and SDC trials were, respectively, 17.1 ± 7.8 s vs 23.6 ± 10.1 s for VO 2 kinetics, P < 0.05).

Mean physiological parameters and RPE.

The influence of drafting, during prior swimming, on the mean physiological values measured during subsequent cycling is presented in Table 2. The statistical analysis shows that cycling efficiency was significantly higher in the SDC trial (4.8%) in comparison with the SAC trial (P < 0.05). The V02 . HR and lactate values measured during cycling were significant higher when previous swimming was performed alone compared with the drafting condition (Table 2, P < 0.05). However, no significant increase in blood lactate concentration with time was observed, indicating the main contribution of aerobic metabolism [START_REF] Di Prampero | Energetics of muscular exercise[END_REF]. Therefore, the decrease in gross efficiency during the SAC trial is related to higher VO 2 values. Furthermore, the subjects' RPE was significantly lower in the SDC trial compared to the SAC trial (15 ± 2 vs 17 ± 2, corresponding to "laborious" vs "very laborious," P < 0.05).

Pedal rate. The statistical analysis indicated a significant difference in pedal rate measured during cycling between the two conditions. A significantly lower pedal rate (5.6%) was observed in the SDC trial in comparison with the SAC trial (Table 2, P < 0.05).

DISCUSSION

The main result of the present study indicated a significant effect of swimming metabolic load on oxygen kinetics and efficiency during subsequent cycling at competition pace. Within this framework, a prior 750-m swim performed clone resulted in faster oxygen kinetics and a significantly higher global energy expenditure during subsequent cycling, in comparison with an identical swimming bout performed in a drafting position (P < 0.05).

Drafting during swimming and swimming metabolic load. The effects of drafting on energy expenditure during short-or long-distance events have been investigated over a variety of physical activities. Drafting has been shown to significantly reduce the metabolic load during swimming (2), cycling [START_REF] Hausswirth | Effects of cycling alone or in a sheltered position on subsequent running performance during a triathlon[END_REF], cross-country skiing [START_REF] Spring | Drag area of a cross-country skier[END_REF], and speed skating [START_REF] Van Ingen Schenau | The influence of air friction in speed skating[END_REF]. The lower energy cost observed in a drafting position is classically attributed to a decrease in aerodynamic or hydrodynamic drag [START_REF] Bassett | Metabolic responses to drafting during front crawl swimming[END_REF][START_REF] Hausswirth | Effects of cycling alone or in a sheltered position on subsequent running performance during a triathlon[END_REF]. In this context, Bassett et al. [START_REF] Bassett | Metabolic responses to drafting during front crawl swimming[END_REF] have suggested that the decrease in drag associated with drafting was lower in swimming in comparison with terrestrial activities. This is because of the characteristics of swimming such as the relatively low velocity, the prone position, and the turbulence owing to the kicks of the lead swimmer. Decreases in passive hydrodynamic drag in drafting position from 10% to 26% have been reported in the literature [START_REF] Chatard | Performance and drag during drafting swimming in highly trained triathletes[END_REF][START_REF] Millet | Effects of drafting behind a two-or a six-beat kick swimmer in elite female triathletes[END_REF]. It should be noted that the active drag experienced by a subject while swimming is approximately 1.5-2 times greater than passive drag [START_REF] Di Prampero | Energetics of swimming in man[END_REF].

In this study, the HR values recorded during the two swimming bouts (Fig. 2) shows mean HR values corresponding respectively for SDC and SAC trials to 84.2% and 90.5% of HR max measured during cycling. Consequently, drafting involved a significant 7% decrease in HR during a 750-m swim (P < 0.05). Furthermore, the SDC trial was characterized by significant reductions in postswim lactate values (29.3%) and RPE values (20%), in comparison with the SAC trial.

The main factor classically evoked in the literature to explain the lower swimming energy cost in drafting position is the reduction of hydrodynamic drag owing to the body displacement of the leading swimmer. The extent to which hydrodynamic drag could be reduced in a drafting position depends on several factors, such as swimming velocity and the distance separating the draftee and the lead swimmer. Concerning the distance between the swimmers, there seems to be a compromise between the positive effect of the hydrodynamic wake created by the lead swimmer and the negative effect of the turbulence generated by his kicks [START_REF] Bassett | Metabolic responses to drafting during front crawl swimming[END_REF][START_REF] Chatard | Performance and drag during drafting swimming in highly trained triathletes[END_REF][START_REF] Millet | Effects of drafting behind a two-or a six-beat kick swimmer in elite female triathletes[END_REF]. However, during triathlon, the draftee could follow the lead swimmer quite closely because triathletes usually adopt a two-beat kick that does not generate excessive turbulence.

The effects of drafting during short-distance swimming bouts have been well documented in the literature [START_REF] Bassett | Metabolic responses to drafting during front crawl swimming[END_REF][START_REF] Chatard | Performance and drag during drafting swimming in highly trained triathletes[END_REF][START_REF] Chollet | The effects of drafting on stroking variations during swimming in elite male triathletes[END_REF]. However, during these experiments, the race conducted in drafting position was performed either at the same relative velocity as the isolated condition (2), or the subjects were asked to swim as fast as possible during the second half of the race [START_REF] Chatard | Performance and drag during drafting swimming in highly trained triathletes[END_REF][START_REF] Chollet | The effects of drafting on stroking variations during swimming in elite male triathletes[END_REF]. Using a protocol comparable to the present study, Bassett et al. [START_REF] Bassett | Metabolic responses to drafting during front crawl swimming[END_REF] have observed during a 549-m swim (600 yards) performed at 1.20 m•s-1 (1.18 m•s -1 in the present study) significantly lower HR (6.2%), lactate (31%), and RPE values (21%) when the swimming bout was performed in a drafting position (P < 0.05), compared with an isolated effort. These results are in agreement with this previous study. One interesting result of this study is that the significant effect of drafting previously reported in the literature was observed even though our subjects were wearing a wet suit. It has been reported that the use of wet suit induced significant decreases in energy cost (from 7% to 22%) and active drag (from 12% to 16%) among different speeds (for review, [START_REF] Chatard | Effects of wetsuit use in swimming events[END_REF]). It could be concluded that during triathlon events, where subjects are wearing wet suits, drafting could further increase the reduction in metabolic load during swimming.

Drafting during swimming and cycling exercise. In the present study, the decrease in metabolic load associated with swimming in a drafting position involved two main modifications in physiological parameters during subsequent cycling. First, VO 2 kinetics, at the onset of cycling, were significantly slowed when the prior swimming bout was performed in a drafting position (slower time constant, τV02) compared with swimming alone (P < 0.05). Second, a significantly higher cycling efficiency, measured at steady state level, was observed in the SDC trial versus the SAC trial (+4.8%, P < 0.05).

The modification in VO 2 kinetics observed in the present study is in accordance with previous results reported in the literature. During the last decade, several investigations have analyzed the influence of previous exercise metabolic load on the rate of VO 2 increase at the onset of subsequent exercise. Gerbino et al. [START_REF] Gerbino | Effects of prior exercise on pulmonary gasexchange kinetics during high-intensity exercise in humans[END_REF] have found that VO 2 kinetics during a high-intensity cycling exercise (i.e., greater than the lactate threshold) was significantly increased by a prior high-intensity cycling bout, whereas no effect was reported after a prior low-intensity exercise (i.e., lower than the lactate threshold). In addition, Bohnert et al. ( 4) have observed an acceleration of VO 2 kinetics when a cycling trial was preceded by a high-intensity arm-cranking exercise.

Many studies have been conducted in order to identify the mechanisms underlying the rate of VO 2 increase at the onset of exercise (e.g., [START_REF] Xu | Oxygen uptake kinetics during exercise[END_REF]. Although these mechanisms are not clearly established, two major hypotheses are reported in the literature. Some authors suggest that VO 2 kinetics are limited by the rate of oxygen supply to the active muscle mass, whereas others report that the capacity of muscle utilization is the most important determinant of VO 2 responses at the onset of exercise [START_REF] Xu | Oxygen uptake kinetics during exercise[END_REF]. Concerning the hypothesis of oxygen transport limitation, Hughson et al. [START_REF] Hughson | Kinetics of ventilation and gas exchange during supin and upright cycle exercise[END_REF] investigated the influence of an improved perfusion of active muscle mass during cycling on the rate of VO 2 increases at the onset of exercise. These authors found that VO 2 kinetics at the onset of exercise were significantly faster when the perfusion of active muscle mass was augmented. In our study, several factors could be evoked to increase perfusion in the muscles of the lower limbs during cycling, such as previous metabolic load and pedal rate.

Gerbino et al. [START_REF] Gerbino | Effects of prior exercise on pulmonary gasexchange kinetics during high-intensity exercise in humans[END_REF] suggested that the faster VO 2 kinetics observed during the second bout of two repeated high-intensity cycling exercises could be accounted for by the residual metabolic acidemia from previous high-intensity exercise, involving a vasodilatation and thus an enhancing blood flow to the active muscle mass at the start of subsequent cycling bout. In favor of this hypothesis, a higher metabolic acidemia was observed in the present study immediately after the swimming stage of the SAC trial in comparison with the SDC trial (postswim lactate values: 7.5 ± 2.4 mmol.L -1 vs 5.3 ± 2.1 mmol•L -1 for SAC and SDC trials, respectively, P < 0.05). Therefore, we suggest that the higher contribution of anaerobic metabolism to energy expenditure when swimming alone has involved a better perfusion of active muscular mass at the start of subsequent cycling exercise.

However, in this study, subjects adopted a higher pedal rate after the swimming bout performed alone. There is little information on the effects of pedal rate manipulation on cardiovascular adjustments during cycling. However, Gotshall et al. [START_REF] Gotshall | Cycling cadence alters exercise hemodynamics[END_REF] have indicated an enhanced muscle blood flow with increasing cadences from 70 to 110 rpm. Indeed, the frequency of contraction and relaxation of the muscles of the lower limbs increases throughout high cadences, improving venous return and therefore heart filling. As a consequence, the skeletal muscle pump is progressively more effective, resulting in an over perfusion of the active muscle mass [START_REF] Gotshall | Cycling cadence alters exercise hemodynamics[END_REF]. According to this hypothesis, the significantly higher pedal rate reported in the present study in the SAC trial in comparison with the SDC trial (Table 2, P < 0.05) could have involved an increased blood flow to the muscles of the lower limbs. Therefore, both the higher contribution of anaerobic metabolism to energy expenditure during prior swimming and the higher pedal rates adopted during subsequent cycling in the SAC trial could account for the faster VO 2 kinetics observed at the onset of cycling in this trial in comparison with the SDC trial.

The second principal result of the present study indicated a significantly higher cycling efficiency during the SDC trial in comparison with the SAC trial (Table 2, P < 0.05). To the best of our knowledge, the effects of drafting during swimming on subsequent cycling adaptation have never been investigated. However, these results were similar to another study from our laboratory showing that wearing a neoprene wet suit reduced the metabolic load at the end of swimming and led to a 12% increase in subsequent cycling efficiency [START_REF] Delextrat | Effect of wet suit use on energy expenditure during a swim-to-cycle transition[END_REF]. In our study, subjects were wearing a wet suit, and our results indicated that drafting could lead to a further improvement of cycling efficiency.

In the context of multidisciplinary events, the effect of drafting on subsequent performance has been mainly studied during the cycle-to-run portion of a simulated sprint triathlon [START_REF] Hausswirth | Effects of cycling alone or in a sheltered position on subsequent running performance during a triathlon[END_REF][START_REF] Hausswirth | Effect of two drafting modalities in cycling on running performance[END_REF]. For example, Hausswirth et al. [START_REF] Hausswirth | Effects of cycling alone or in a sheltered position on subsequent running performance during a triathlon[END_REF] reported that the significant reductions in VO 2 VE, HR, and blood lactate concentration during the cycle stage of a simulated sprint triathlon (0.75-km swim, 20-km cycle, 5-km run), observed when cycling was performed in drafting position in comparison with an isolated cycling stage, were related to significant increases in subsequent running velocity (4.1%). More recently, Hausswirth et al. [START_REF] Hausswirth | Effect of two drafting modalities in cycling on running performance[END_REF] observed that drafting continuously behind a leader during the 20-km stage of a sprint triathlon resulted in a significantly lower cycling metabolic cost, in comparison with alternating drafting and leading every 500 m at the same pace. This lower metabolic cost led to a 4.2% improvement in velocity during a subsequent 5-km run [START_REF] Hausswirth | Effect of two drafting modalities in cycling on running performance[END_REF]. These authors suggested that during the drafting conditions (drafting position vs isolated cycling, or continuous vs alternate drafting), the decrease in energy cost of cycling is the main factor of running performance improvement. In the present study, the cycling bouts were conducted at constant speed. Therefore, no improvement in performance (i.e., velocity) could be observed. However, we recorded a 4.8% increase in cycling efficiency after a swimming bout performed in drafting position compared with an isolated swimming bout. This improvement in cycling efficiency could be mainly accounted for by the lower swimming relative intensity involving a lower state of fatigue in the muscles of the lower limbs at the beginning of subsequent cycling. Consequently, in long-distance events such as triathlon, where performance depends on the capacity to spend the lowest amount of metabolic energy during the whole race [START_REF] Hausswirth | Effects of cycling alone or in a sheltered position on subsequent running performance during a triathlon[END_REF][START_REF] Hausswirth | Effect of two drafting modalities in cycling on running performance[END_REF], we suggest that the increase in cycling efficiency could lead to an improvement in performance. However, further studies are needed to investigate the effects of this improved cycling efficiency on running and total triathlon performance.

However, it should be noted that the possibility for athletes and coaches to put the results of the present study into practice could be limited by the Jack of cycling training of our subjects and by the difference between the intensity and duration of the cycling trials in this study and the metabolic load encountered during a sprint triathlon [START_REF] Hausswirth | Effects of cycling alone or in a sheltered position on subsequent running performance during a triathlon[END_REF]. Because cycling experience could lead to a lower variability in energy cost of locomotion, more training in cycling would be associated with a lower benefit of drafting. Furthermore, even if a measure of actual cycling performance improvements after drafting (such as time or power output) would have been more applicable to competition, the constant power output set in this study allowed the quantification of the modifications in energy expenditure during cycling, which is a main determinant of triathlon performance [START_REF] Hausswirth | Effects of cycling alone or in a sheltered position on subsequent running performance during a triathlon[END_REF][START_REF] Hausswirth | Effect of two drafting modalities in cycling on running performance[END_REF]. Further studies are necessary to validate the effects observed in this study during a real triathlon event.

In conclusion, the results of the present study show that the metabolic load during swimming could have a significant effect on subsequent cycling performance during a sprint triathlon. In particular, a decrease in swimming relative intensity could lead to a significantly higher efficiency during subsequent cycling. These findings highlight that swimming behind another athlete is beneficial during triathlon events. Within this framework, further studies could include a running session to investigate more precisely the effects of drafting during the swimming bout of a sprint triathlon on total triathlon performance.

FIGURES and TABLES

FIGURE 1 -

 1 FIGURE 1-Experimental protocol. L: blood sampling, K4 b 2 : installation of the Cosmed K4 b 2 analyzer.

FIGURE 2 -FIGURE 3 -

 23 FIGURE 2-Changes in HR values during the fast 5 min of the two swimming trials (SAC and SDC). *Significant difference between SDC and SAC trials, P < 0.05.

TABLE 1 . Subjects' physiological characteristics during the cycling incremental test. VO2max (mL•min-1 .k -1 )

 1 V0 2max , maximal oxygen uptake; MAP, maximal aerobic power; HR max, maximal heart rate; RER max , maximal respiratory exchange ratio; power output at VT, power output corresponding to the ventilatory threshold.

		MAP (W)	75% of MAP (W)	HRmax (beats•min-1)	RERmax.	Power Output at VT (W)
	66.2 ± 6.8	343 ± 39	262 ± 29	190 ± 9	1.06 ± 0.05	258 ± 42

Table 2 . Effect of drafting during prior swimming on mean values of physiological parameters and pedal rate recorded during subsequent cycling exercice.

 2 V0 2 , oxygen uptake; LA, blood lactate concentration; GE, gross efficiency; HR, heart rate; VE, expiratory flow; RF, respiratory frequency. * Significant difference between SAC and SDC trials; P < 0.05.

	SDC	SA

The authors acknowledge all the triathletes who took part to the experiment for their high cooperation and motivation. We are also grateful to Rob Suriano for his assistance with the language.


Francisco J Blanco-Rodríguez

Stéphane Le Dizès

Curvature instability of a curved Batchelor vortex

In this paper, we analyse the curvature instability of a curved Batchelor vortex. We consider this short-wavelength instability when the radius of curvature of the vortex centerline is large compared to the vortex core size. In this limit, the curvature instability can be interpreted as a resonant phenomenon. It results from the resonant coupling of two Kelvin modes of the underlying Batchelor vortex with the dipolar correction induced by curvature. The condition of resonance of the two modes is analysed in detail as a function of the axial jet strength of the Batchelor vortex. Contrarily to the Rankine vortex, only a few configurations involving m = 0 and m = 1 modes are found to become the most unstable. The growth rate of the resonant configurations is systematically computed and used to determine the characteristics of the most unstable mode as a function of the curvature ratio, the Reynolds number, and the axial flow parameter. The competition of the curvature instability with another short-wavelength instability, which was considered in a companion paper [Blanco-Rodríguez & Le Dizès, Elliptic instability of a curved Batchelor vortex, J. Fluid Mech. 804, 224-247 (2016)], is analysed for a vortex ring. A numerical error found in this paper which affects the relative strength of the elliptic instability is also corrected. We show that the curvature instability becomes the dominant instability in large rings as soon as axial flow is present (vortex ring with swirl).

Introduction

Vortices are ubiquitous in nature. They are subject to various instabilities induced by the interaction with their surroundings. In this work, we analyse the so-called curvature instability which is a short-wavelength instability induced by the local curvature of the vortex. We provide theoretical predictions for a curved vortex when the underlying vortex structure is a Batchelor vortex (Gaussian axial velocity and axial vorticity). This work is the follow-up of Blanco-Rodríguez & Le [START_REF] Blanco-Rodríguez | Elliptic instability of a curved Batchelor vortex[END_REF], hereafter BRLD16, where another short-wavelength instability, the elliptic instability, was analysed using the same theoretical framework.

These two instabilities are different from the long-wavelength instabilities which occur in vortex pairs [START_REF] Crow | Stability theory for a pair of trailing vortices[END_REF]) and helical vortices [START_REF] Widnall | The stability of a helical vortex filament[END_REF][START_REF] Quaranta | Long-wave instability of a helical vortex[END_REF]. Their characteristics strongly depend on the internal vortex structure and their wavelength is of the order of the vortex core size. When the vortex is weakly deformed, both instabilities can be understood as a phenomenon of resonance between two (Kelvin) modes of the underlying vortex and a vortex correction. For the elliptic instability, the resonance occurs with a quadripolar correction generated by the background strain field [START_REF] Moore | The instability of a straight vortex filament in a strain field[END_REF], while for the curvature instability, it is associated with a dipolar correction created by the vortex curvature [START_REF] Fukumoto | Curvature instability of a vortex ring[END_REF]. Numerous works have concerned the elliptic instability in the context of straight vortices [START_REF] Tsai | The stability of short waves on a straight vortex filament in a weak externally imposed strain field[END_REF][START_REF] Eloy | Three-dimensional instability of Burgers and Lamb-Oseen vortices in a strain field[END_REF]Fabre & Jacquin 2004a;[START_REF] Lacaze | Elliptic instability in a strained Batchelor vortex[END_REF]. The specific case of the curved Batchelor vortex has been analysed in BRLD16. Contrarily to the elliptic instability, the curvature instability has only been considered for vortices with uniform vorticity [START_REF] Fukumoto | Curvature instability of a vortex ring[END_REF][START_REF] Hattori | Modal stability analysis of a helical vortex tube with axial flow[END_REF].

Both elliptic and curvature instabilities have also been analysed using the local Lagrangian method popularized by [START_REF] Lifschitz | Local stability conditions in fluid dynamics[END_REF] [see [START_REF] Bayly | Three-dimensional instability of elliptical flow[END_REF]; [START_REF] Waleffe | On the three-dimensional instability of strained vortices[END_REF] for the elliptic instability, [START_REF] Hattori | Short-wavelength stability analysis of thin vortex rings[END_REF][START_REF] Hattori | Short-wavelength stability analysis of a helical vortex tube[END_REF][START_REF] Hattori | Effects of axial flow on the stability of a helical vortex tube[END_REF] for the curvature instability]. This method can be used to treat strongly deformed vortices but it provides a local information on a given streamline only. When the vortex is uniform, as the Rankine vortex, the local instability growth rate is also uniform. In that case, a connection can be made between the local results and the global results obtained by analyzing the mode resonances [START_REF] Waleffe | On the three-dimensional instability of strained vortices[END_REF][START_REF] Eloy | Stability of the Rankine vortex in a multipolar strain field[END_REF][START_REF] Fukumoto | The three-dimensional instability of a strained vortex tube revisited[END_REF][START_REF] Hattori | Short-wave stability of a helical vortex tube: the effect of torsion on the curvature instability[END_REF][START_REF] Hattori | Modal stability analysis of a helical vortex tube with axial flow[END_REF]. Le [START_REF] Dizès | Theoretical predictions for the elliptic instability in a twovortex flow[END_REF] used the local prediction at the vortex centre to estimate the global growth rate of the elliptic instability in a non-uniform vortex. Although a good agreement was demonstrated for the Lamb-Oseen vortex, no such link is expected in general.

The goal of the present work is to obtain global estimates for the curvature instability using the framework of [START_REF] Moore | The instability of a straight vortex filament in a strain field[END_REF] for the Batchelor vortex. Such an analysis was performed by [START_REF] Hattori | Modal stability analysis of a helical vortex tube with axial flow[END_REF] for a Rankine vortex. The passage from the Rankine vortex to the Batchelor vortex will turn out not to be trivial. The main reason comes from the different properties of the Kelvin modes in both vortices. In smooth vortices, Kelvin modes are affected by the presence of critical layers (Le Dizès 2004) which introduce singularities and damping [START_REF] Sipp | Widnall instabilities in vortex pairs[END_REF][START_REF] Fabre | The Kelvin waves and the singular modes of the Lamb-Oseen vortex[END_REF]. These singularities have to be monitored and avoided in the complex plane to be able to obtain the properties of the Kelvin modes from the inviscid equations as shown in [START_REF] Lacaze | Elliptic instability in a strained Batchelor vortex[END_REF]. In the present work, we shall also use the asymptotic theory of Le [START_REF] Dizès | An asymptotic description of vortex Kelvin modes[END_REF] to obtain an approximation of the Kelvin mode dispersion relation and analyse the condition of resonance.

The dipolar correction responsible of the curvature instability is also obtained by an asymptotic theory in the limit of small vortex core size [START_REF] Callegari | Motion of a curved vortex filament with decaying vortical core and axial velocity[END_REF]. This correction appears as a first order correction to the Batchelor vortex. The detail of the derivation can be found in [START_REF] Blanco-Rodríguez | Internal structure of vortex rings and helical vortices[END_REF]. As for the elliptic instability, the coupling terms, as well as weak detuning and viscous effects are computed using an orthogonality condition. The final result is an expression for the growth rate of a given resonant configuration close to the condition of resonance. Each resonant configuration provides a growth rate expression. We shall consider up to 50 resonant configurations to extract the most unstable one. This will allows us to obtain the curvature instability diagram as a function of the curvature ratio and the Reynolds number.

The paper is organized as follows. In §2, the base flow and perturbation equations are provided. In §3, the analysis leading the growth rate expression of a resonant configuration is presented. The results for the Batchelor vortex are obtained in §4. We first provide the characteristics of the resonant modes, then the stability diagrams for the Batchelor vortex for a few values of the axial flow parameter. Section §5 provides an application of the results to a vortex ring with and without swirl (axial flow). In that section, we analyse the competition of the curvature with the elliptic instability using the results of BRLD16. A numerical error affecting the strength of the elliptic instability has been found in this paper. It is corrected in a corrigendum which is presented in appendix D. The last section §6 gives a brief summary of the main results of the paper.

Problem formulation

Base flow

The first description of the base flow was provided by [START_REF] Callegari | Motion of a curved vortex filament with decaying vortical core and axial velocity[END_REF]. Here, as in BRLD16, we mainly follow the presentation given in [START_REF] Blanco-Rodríguez | Internal structure of vortex rings and helical vortices[END_REF]. The vortex is considered in the local Frenet frame (t, n, b) attached to the vortex centerline and moving with the structure. We assume that the vortex is concentrated (i.e. thin), which means that its core size a is small compared to the local curvature radius R c of the vortex centerline and the shortest separation distance δ to other vortex structures. For simplicity, we consider a single small parameter ε = a/R c , and assume that δ = O(R c ).

The internal vortex dynamics is described using the "cylindrical" coordinate system (r, ϕ, s) constructed from the Frenet frame (see Fig. 1).

The velocity-pressure field of the base flow is expanded in power of ε as U = U 0 + εU 1 + • • •. The leading order contribution is the prescribed Batchelor vortex of velocity field U 0 = (0, V (0) (r), W (0) (r), P (0) (r)) with

V (0) = 1 -e -r 2 r , W (0) = W 0 e -r 2 . (2.1)
As in BRLD16, spatial and time scales have been non-dimensionalized using the core size a and the maximum angular velocity of the vortex Ω (0) max = Γ/(2πa 2 ), Γ being the vortex circulation. The axial flow parameter W 0 is defined as the ratio

W 0 = W (0) max Ω (0) max a . (2.2)
We assume that W 0 0.5 such that the vortex remains unaffected by the inviscid swirling jet instability [START_REF] Mayer | Viscous and inviscid instabilities of a trailing vortex[END_REF]. We also implicitly assume that the weak viscous instabilities occurring for small values of W 0 (Fabre & Jacquin 2004b;[START_REF] Dizès | Large-Reynolds-number asymptotic analysis of viscous centre modes in vortices[END_REF] remain negligible in the parameter regime that is considered. In the following, we shall also use the expression of the angular velocity Ω (0) (r) and vorticity ζ (0) (r):

Ω (0) (r) = 1 -e -r 2 r 2 , ζ (0) (r) = 2e -r 2 . (2.3)
As explained by [START_REF] Blanco-Rodríguez | Internal structure of vortex rings and helical vortices[END_REF], the first order correction is a dipolar field which can be written as

U 1 ∼ ε Re U (1) e iϕ = ε 2     iU (1) (r) V (1) (r) W (1) (r) P (1) (r)     e iϕ + c.c. , (2.4)
where expressions for U (1) , V (1) , W (1) and P (1) are provided in appendix A. It is worth emphasizing that these expressions only depend on the local characteristics of the vortex at leading order. In particular, they do not depend on the local torsion. For helices, torsion as well as the Coriolis effects associated with the change of frame appear at second order [START_REF] Hattori | Short-wavelength stability analysis of a helical vortex tube[END_REF]. The above expression then describes the internal structure of both helices and rings up to the order ε. This contrasts with the quadripolar correction responsible of the elliptic instability which appears at second order. This quadripolar correction varies according to the global vortex geometry and is different for rings and helices even if they have the same local curvature.

Perturbation equations

The perturbations equations are obtained by linearizing the governing equations around the base flow

U = U 0 + εU 1 + • • •.
As shown in BRLD16, if the perturbation velocitypressure field is written as u = (-iu, v, w, p) we obtain up to o(ε) terms a system of the form :

(i∂ t I + i∂ s P + M) u = ε e iϕ N (1)
+ + e -iϕ N

(1) -

u + i Re Vu (2.5)
where the operators I, P, M = M(-i∂ ϕ ), N

± = N (1) ± (-i∂ ϕ , -i∂ s ), V = V(-i∂ ϕ , -i∂ s ) are defined in Appendix B. (1)
The left-hand side corresponds to the inviscid perturbation equations of the undeformed Batchelor vortex. The first term on the right-hand side is responsible of the curvature instability, while the second term accounts for the viscous effects on the perturbations. By introducing viscous effects in this equation, we implicitly assume that the Reynolds number

Re = Ω (0) max a 2 ν = Γ 2πν ,
with ν the kinematic viscosity, is of order 1/ε.

Instability description

Curvature instability mechanism

The mechanism of the curvature instability is similar to that of the elliptic instability.

The instability results from a resonant coupling of two Kelvin modes of the undeformed axisymmetric vortex with non-axisymmetric corrections. Two Kelvin modes of characteristics (ω A , k A , m A ) and (ω B , k B , m B ) are resonantly coupled via the dipolar correction if they satisfy the condition of resonance (assuming m A < m B )

ω A = ω B , k A = k B , m A = m B -1. (3.1)
Fukumoto ( 2003) further demonstrated that the coupling is destabilizing only if the energy of the modes is opposite or if the frequency vanishes. It leads to a growth of the Kelvin mode combination with a maximum growth rate scaling as ε.

Formal derivation of the growth rate formula

For each resonant configuration, a growth rate expression can be obtained from an orthogonality condition as we did for the elliptic instability (see BRLD16). We consider a combination of two Kelvin modes of azimuthal wavenumber m A and m B = m A + 1 close to the their condition of resonance (3.1):

u = Aũ A (r) e im A ϕ + Bũ B (r) e im B ϕ e iks -iωt , (3.2)
where

k is close to k A = k B = k c , and ω close to ω A = ω c and ω B = ω c + i Im(ω B ).
We assume that the resonance is not perfect. The mode B will exhibit a weak critical layer damping given by Im(ω B ) (imaginary part of ω B ). The functions ũA (r) and ũB (r) are the eigenfunctions of the Kelvin modes which satisfy

(ω A I -k A P + M(m A )) ũA = 0, (3.3) (ω B I -k B P + M(m B )) ũB = 0, (3.4)
with a prescribed normalisation:

pA ∼ r→0 r |m A | , pB ∼ r→0 r |m B | . (3.5)
If we plug (3.2) in (2.5), we obtain for the components proportional to e im A ϕ and e im B ϕ :

A ωI -kP + M(m A ) - i Re V(m A , k) ũA = BεN (1) -(m B , k)ũ B , (3.6)
B ωI -kP + M(m B ) - i Re V(m B , k) ũB = AεN (1) + (m A , k)ũ A . (3.7)
Relations between the amplitudes A and B are obtained by projecting these equations on the subspace of the adjoint Kelvin modes. We define the adjoint eigenfunctions ũ † A and ũ † B of the Kelvin modes as the solutions to the adjoint equations of (3.3)-(3.4) with respect to the scalar product

< u 1 , u 2 >= ∞ 0 u 1 • u 2 r dr = ∞ 0 (u 1 u 2 + v 1 v 2 + w 1 w 2 + p 1 p 2 ) r dr.
(3.8)

We then obtain

ω -ω c -Q A (k -k c ) -i V A Re A = εC AB B, (3.9) ω -ω c -i Im(ω B ) -Q B (k -k c ) -i V B Re B = εC BA A, (3.10)
where the coefficients of these equations are given by

Q A = < ũ † A , P ũA > < ũ † A , I ũA > , Q B = < ũ † B , P ũB > < ũ † B , I ũB > , (3.11) V A = < ũ † A , V ũA > < ũ † A , I ũA > , V B = < ũ † B , V ũB > < ũ † B , I ũB > , (3.12) C AB = < ũ † A , N (1)
+ (m B , k B ) ũB > < ũ † A , I ũA > , C BA = < ũ † B , N (1)
-(m A , k A ) ũA > < ũ † B , I ũB > . (3.13)
The formula for the complex frequency ω is then finally given by

ω -ω c -i Im(ω B ) -Q B (k -k c ) -i V B Re ω -ω c -Q A (k -k c ) -i V A Re = -ε 2 N 2 , (3.14) with N = -C AB C BA .
(3.15)

The right-hand side of (3.14) represents the coupling terms responsible of the curvature instability. The left-hand side of (3.14) gives the dispersion relation of each Kelvin mode close to the resonant point. It is important to mention that none of the coefficients

Q A , Q B , V A , V B
and N depends on the normalization chosen for the Kelvin modes.

Instability results for the Batchelor vortex profile

4.1. Resonant Kelvin modes The main difficulty of the analysis is to determine the Kelvin modes that satisfy the condition of resonance (3.1). A similar problem was already addressed in [START_REF] Lacaze | Elliptic instability in a strained Batchelor vortex[END_REF]. The Kelvin modes are here defined from the inviscid equations. Two kinds of Kelvin modes are found to exist: the regular and neutral Kelvin modes which can easily be obtained by integrating the inviscid perturbation equations in the physical domain and the singular and damped Kelvin modes which require a particular monitoring of the singularities of the perturbation equations in the complex plane. We shall see below that the condition of resonance always involves a singular mode.

The singularities of the inviscid perturbation equations are the critical points r c where ω -kW (0) (r c ) -mΩ (0) (r c ) = 0. When Im(ω) > 0, these singularities are in the complex plane, and do not affect the solution in the physical domain (real r). However, one of such critical points may cross the real axis when Im(ω) becomes negative. As explained in Le [START_REF] Dizès | Viscous critical-layer analysis of vortex normal modes[END_REF], the inviscid equations must in that case be integrated on a contour in the complex r plane that avoids the critical point from below (resp. above) if the critical point has moved in the lower (resp. upper) part of the complex plane. On such a contour, the solution remains regular and fully prescribed by the inviscid equations. On the real axis, the inviscid solution is however not regular anymore. As illustrated in [START_REF] Fabre | The Kelvin waves and the singular modes of the Lamb-Oseen vortex[END_REF], it no longer represents the vanishing viscosity limit of a viscous solution in a large interval of the physical domain. The Kelvin mode forms by the contour deformation technique is damped and singular. The inviscid frequency of the mode then possesses a negative imaginary part, which corresponds to what we call the critical layer damping rate. By definition, the critical layer damping rate is independent of viscosity.

A mode cannot be involved in a resonance if it is too much damped. In the asymptotic framework, the growth rate associated with the resonance is expected to be O(ε), so the damping rate of the modes should a priori be asymptotically small of order ε. However, in practice, we shall consider values of ε up to 0.2, and the maximum growth rate will turn out to be around 0.05 ε. We shall then discard all the modes with a damping rate whose absolute value exceeds 0.01.

Predictions from the WKBJ analysis

Le [START_REF] Dizès | An asymptotic description of vortex Kelvin modes[END_REF] showed that information on the spectrum of the Kelvin modes can be obtained using a large k asymptotic analysis. They applied their theory to the Batchelor vortex and were able to categorize the neutral Kelvin modes in four different types: regular core modes, singular core modes, regular ring modes, singular ring modes. For each m, they provided the region of existence of each type of mode in a (kW 0 , ω)

-2.5 -2 -1.5 -1 -0.5 0 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 k W0 ω (1,2) (0,1)
(2,3)

Figure 2. Prediction from the WKBJ analysis of the domains of parameters in the (kW0, ω) plane where resonance between two Kelvin modes (mA, mA + 1) is possible. Only positive frequencies are considered. A symmetrical plot is obtained for negative frequencies.

plane. The energy of the waves can also be deduced from the asymptotic expression of the dispersion relation as shown in Le [START_REF] Dizès | Inviscid waves on a Lamb-Oseen vortex in a rotating stratified fluid: consequences on the elliptic instability[END_REF]. It is immediately found that regular core modes and regular ring modes are always of negative energy, while singular modes have positive energy. The condition of resonance can then easily be analysed. One just needs to superimpose the domains of existence of each pair of modes of azimuthal wavenumbers m and m+1, to find the regions of possible resonance. The final result is summarized in Fig. 2. For positive frequencies, only three different regions are obtained corresponding to (m A , m B ) = (0, 1), (1, 2) and (2, 3) (Negative frequencies are obtained by symmetry changing m → -m and k → -k). No intersection of the domains of existence of the modes m and m + 1 are obtained for m larger than 2. In each region of Fig. 2, we always find that the mode A is a regular core mode of negative energy, while the mode B is a singular core mode of positive energy. Each branch crossing is therefore expected to provide an instability.

As shown in Le [START_REF] Dizès | An asymptotic description of vortex Kelvin modes[END_REF], both types of core modes have an asymptotic dispersion relation of the form

k rt 0 ∆(r) Φ(r) dr = (|m| + 2l) π 2 l = 0, 1, 2, . . . (4.1)
where

∆(r) = 2Ω (0) (r)ζ (0) (r) -Φ 2 (r), (4.2) Φ(r) = ω -mΩ (0) (r) -kW (0) (r), (4.3)
and r t is a turning point defined by ∆(r t ) = 0. The integer l is a branch label which measures the number of oscillations of the mode in the vortex core. The larger l, the more oscillating is the mode. Singular modes differ from regular modes by the presence of a critical point r c > r t where Φ(r c ) = 0 in their radial structure. In the WKBJ description, this critical point does not create any damping at leading order. However, it makes the eigenfunction singular. It will justify the use of a complex integration path in the numerical resolution of the mode.

Numerical determination of the Kelvin modes

The characteristics of the resonant modes are obtained by integrating numerically Eqs. (3.3)-(3.4). The numerical code is based on a Chebyshev spectral collocation method, essentially identical to that used in Fabre & Jacquin (2004b). The eigenvalue problem is solved in a Chebyshev domain (-1,1) on 2(N + 1) nodes which is mapped on a line in the complex-r plane using the mapping

r(x; A c , θ c ) = A c tanh(x) e i θc , (4.4)
where A c is a parameter close to 1 that controls the spreading of the collocation points, and θ c is the small inclination angle of the path in the complex r plane. We typically take θ c ≈ π/10 such that the critical point of the singular mode is avoided. As in Fabre & Jacquin (2004b), we take advantages of the parity properties of the eigenfunctions by expressing for odd m (resp. even m), w and p on odd polynomials (resp. even) and ũ and ṽ on even polynomials (resp. odd). It leads to a discretized eigenvalue problem of order 4N , which is solved using a global eigenvalue method. We also use an Arnoldi algorithm in order to follow specific eigenvalues and easily find the condition of resonance. In most computations, the value N = 200 was found to be adequate. This collocation method was also used to determine the adjoint modes and compute the integrals that define the coefficients in the growth rate equation (3.14).

Typical results for the eigenvalues are shown in Fig. 3. In this figure, we compare the numerical results with the theoretical formula (4.1). The good agreement demonstrates the usefulness of the asymptotic approach to obtain valuable estimates for the condition of resonance.

Stability diagram

Lamb-Oseen vortex

In this section, we assume that there is no axial flow. The underlying vortex is then a Lamb-Oseen vortex. For this vortex, Kelvin mode properties have been documented in Le [START_REF] Dizès | An asymptotic description of vortex Kelvin modes[END_REF] and [START_REF] Fabre | The Kelvin waves and the singular modes of the Lamb-Oseen vortex[END_REF] for m = 0, 1, 2, 3. It was shown that the singular core modes become strongly damped as soon as the critical layer singularity moves in the vortex core. This gives a constraint on the frequency of the mode B which has to be small. As a consequence, we immediately see that the only modes (m A , m B ) that can possibly resonate are the modes (m A , m B ) = (0, 1). Moreover, the constraint on the frequency implies that only large branch labels of the mode m A = 0 will be able to resonate with a weakly damped mode m B = 1. In figure 4, we show the crossing of the first m A = 0 and m B = 1 branches in the (k, ω) plane. Only the modes with a damping rate smaller (in absolute value) than 0.01 are in solid lines. We observe that the branch label of the m A = 0 modes must be 6 or larger to cross the first m B = 1 branch in the part where it is only weakly damped. The characteristics of these first resonance points are given in table 1. We also give in this table, the value of the coefficients of Eq. (3.14) at each resonant point. For each resonant configuration, we can then plot the growth rate Im(ω) of the curvature instability as a function of the wavenumber k for any Re and ε. An example of such a plot is provided in Fig. 5. In this figure, we have plotted only the first four resonant configurations. Other configurations have been computed corresponding to labels [2, 6], [2, 7], etc but their growth rates were found to be much weaker for Re 10 5 . The spatial structure of the most unstable resonant configurations are also been given in Fig. 5. We have plotted the vorticity contours for a particular phase which maximizes the relative amplitude of the Kelvin mode m A = 0. This mode is then clearly visible in each case. If we had chosen a phase such that e iks-iωt = i, we would have seen the mode m B = 1 only.

l B = 2 l A = 2 l A = 1 l B = 1 l A = 9
We have systematically computed the maximum growth rate and obtained the most unstable mode characteristics for all ε 0.22 and Re 10 5 . The result is displayed in Fig. 6 where the maximum growth rate is shown in the (ε, Re) plane. The labels of the most unstable configurations are also shown in this plot. We can note that only 3 resonant configurations can become the most unstable corresponding to the crossing of the first branch of the Kelvin mode m B = 1 with the 7th to 9th branch of the Kelvin mode m A = 0. In particular, the resonant configuration [6, 1] observed in Fig. 5 never becomes the most unstable configuration although this configuration possesses the largest coupling coefficient N (see table 1). This is directly related to the property mentioned above: the critical layer damping rate Im(ω B ) of the mode m B = 1 is too large.

Figure 6 provides the stability diagram of the Lamb-Oseen vortex with respect to the curvature instability. It is important to emphasize the large value of the Reynolds number needed for instability. Even for a value as large as ε = 0.2, the critical Reynolds number for instability is Re c ≈ 6000. We shall see in the next section that axial flow will strongly decrease this value.

-1.6 -1.5 -1.4 -1.3 -1.2 -1.1 0 1 2 3 4 x 10 -3 k Im(ω) [6,1] [7,1] [8,1] [9,1] [7, 1] [8, 1] [9, 1]

Effects of the axial flow

The characteristics of the Kelvin modes strongly vary with the parameter W 0 . Additional branch crossings involving smaller branch labels are obtained as W 0 increases. As explained in the previous section, resonances between m A = 1 and m B = 2, as well as between m A = 2 and m B = 3 become a priori possible (see Fig. 2). However, they involve very high branch labels which implies that they never become the most unstable modes for moderate Reynolds numbers (Re 10 5 ) .

W0 = 0.2 W0 = 0.4 -3 -2.5 -2 -1.5 -1 0 0.002 0.004 0.006 0.008 0.01 0.012 k Im(ω) [4,2] [3,1] [4,4] [4,1] [5, 1] [3,2] [2,1] [4,5] [5, 3] [4,3] -2 -1.8 -1.6 -1.4 -1.2 -1 0 0.005 0.01 0.015 k Im(ω) [2,2] [2,4] [2,5] [3,2] [2,3] [2,1] [3,1] [5, 1] [3, 1] [4, 1] [4, 2] [3, 1] [2, 2] -1 0 1 -1 0 1 -5 0 5 -1 0 1 -1 0 1 -4 -2 0 2 4 -1 0 1 -1 0 1 -5 0 5 -1 0 1 -1 0 1 -4 -2 0 2 4 -1 0 1 -1 0 1 -4 -2 0
For the parameters W 0 = 0.1, 0.2, 0.3, 0.4 and 0.5, we have considered the crossing points of the seven first branches of the Kelvin modes m A = 0 and m B = 1. Each crossing point corresponds to a mode resonance. At each crossing point, we have computed the coefficients of the growth rate expression. In Fig. 7, we have plotted the growth rate curves obtained from Eq. (3.14) for W 0 = 0.2 and 0.4, and for ε = 0.2 and Re = 5000. Contrarily to the Lamb-Oseen vortex, more resonant configurations can now become unstable. Moreover, they involve smaller branch labels. The spatial structure of the main resonant configurations have also been provided in Fig. 7 for this set of parameters. As in Fig. 5, we have plotted the vorticity contour for a particular phase which maximizes the relative amplitude of the Kelvin mode m A = 0 . Note that the spatial structure of the resonant mode [3, 1] is different for W 0 = 0.2 and W 0 = 0.4: this difference is not only associated with the different values of the coefficient B/A obtained from (3.6), but also with an effect of W 0 on the Kelvin modes.

If we take the maximum value of the growth rate over all possible k for each ε and Re, we obtain the plots shown in Fig. 8. The same colormap and contour levels have been used as in Fig. 6 for comparison. We clearly see that the growth rates are larger in the presence of axial flow. The region of instability is also much larger. In these plots, we have indicated the labels of the most unstable modes. As for the Lamb-Oseen vortex, the most unstable configuration changes as ε or Re varies. However, the branch labels of the Kelvin modes are now smaller in the presence of axial flow. This property explains in part the larger growth rates of the configurations with jet. Indeed, the viscous damping of the modes with the smallest labels is the weakest. The impact of viscosity is therefore weaker on these modes. Yet, the resonant configuration with the smallest labels are not necessarily the most unstable because they may also exhibit a larger critical layer damping, or a smaller coupling coefficient N [see equation (3.14)].

In tables 2 and 3 of appendix C, we have provided the characteristics of the main resonant configurations for W 0 = 0.2 and W 0 = 0.4. The data for the other resonant configurations and for other values of W 0 are available as supplementary material.

Competition with the elliptic instability in a vortex ring

The results obtained in §4 can readily be applied to the vortex ring by using ε = a/R where R is the radius of the ring and a the core radius.

As first shown by [START_REF] Widnall | The instability of short waves on a vortex ring[END_REF], the vortex ring is also subject to the elliptic instability. This instability appears at the order ε 2 , so it is a priori smaller. Yet, the short wavelength instability observed experimentally in a vortex ring without swirl has always been attributed to the elliptic instability (see the review by [START_REF] Shariff | Vortex rings[END_REF]. It is therefore natural to provide a more precise comparison of the growth rates of both instabilities.

In BRLD16, we have obtained theoretical predictions for the elliptic instability in a vortex ring with a Batchelor profile. As for the curvature instability, growth rate contour plots can be obtained for the elliptic instability in a (ε, Re) plane using the data of this paper. It should be noted that an error of a factor 2 was found in some of the coefficients of the elliptic instability growth rate formula. This error, which is corrected in appendix D, does not affect the main conclusion of this paper but modifies the relative importance of the elliptic instability with respect to the curvature instability.

The comparison of the elliptic instability with the curvature instability is shown in Fig. 9 for three values of the axial flow parameters (W 0 = 0, 0.2, 0.4). In this figure, we have plotted the largest value of both instability growth rates in the (ε, Re) plane. We have also indicated where each instability appears and becomes dominant over the other one. Interestingly, we observe that depending on the value of W 0 the region of dominance of the curvature instability changes. For the case without axial flow [Fig. 9(a)], the elliptic instability domain is larger than the curvature instability domain and the elliptic instability is always the dominant instability. For the other two cases W 0 = 0.2 and W 0 = 0.4, the situation is different: there is a balance between both instabilities. For both cases, curvature instability is dominant over the elliptic instability for small ε while it is the opposite for large ε. Yet, there are some differences between both cases. For W 0 = 0.2, we observe that the curvature instability is the first instability to appear as Re is increased for all ε < 0.2. For W 0 = 0.4, the elliptic instability domain is larger and extends to smaller values of the Reynolds numbers than for the other two cases. It is also the dominant instability for all Reynolds numbers as soon as ε is larger than 0.1.

These plots have interesting implications. First, it explains why the curvature instability has never been observed in vortex ring without swirl. For such a vortex ring, the elliptic instability is always stronger than the curvature instability. Second, it implies that the curvature instability should be visible in a vortex ring with swirl if ε is smaller than 0.1 and the Reynolds number larger than 10000.

It should also be noted that due to the different inviscid scalings, which are in ε for the curvature instability growth rate and in ε 2 for the elliptic instability growth rate, the curvature instability should always become dominant over the elliptic instability whatever W 0 if ε is sufficiently small and the Reynolds number sufficiently large. This tendency is clearly seen in figures 9(b,c) for W 0 = 0.2 and W 0 = 0.4. For W 0 = 0 (fig. 9(a)), the change of dominance of both instabilities occurs for a much larger Reynolds number.

Conclusion

In this work, we have provided the characteristics of the curvature instability for a Batchelor vortex for several axial flow parameters. We have shown that although a same resonant coupling is active as in a Rankine vortex, the characteristics of the resonant configurations are very different owing to the critical layer damping of many Kelvin modes. We have shown that this effect precludes the resonance of Kelvin modes with azimuthal wavenumbers larger than m = 3. Moreover, when it occurs, the resonance of modes (m A , m B ) = (1, 2) or (2, 3), involves a Kelvin mode with a very high complexity (large branch label) which is strongly sensitive to viscous effects. For moderate Reynolds numbers (Re 10 5 ), we have then found that the most unstable configuration always involves Kelvin modes of azimuthal wavenumbers m A = 0 and m B = 1. We have analysed the condition of resonance of the 7 first branches (9 for the Lamb-Oseen vortex) for several axial flow parameters to identify the most unstable configuration.

For the case without axial flow (Lamb-Oseen vortex), we have shown that the most unstable configuration involves the first branch of the Kelvin mode of azimuthal wavenumber m B = 1 and the seventh to nineth branch of the Kelvin mode of azimuthal wavenumber m A = 0, depending on the Reynolds number and ε (for Re 10 5 ). The high value of the branch label implies a larger viscous damping and therefore a weaker growth rate of the curvature instability for this case. In the presence of axial flow, resonant configura-tions with smaller branch labels were shown to become possible. The instability growth rate was then found to be larger than without axial flow. We have presented the characteristics of the most unstable configurations for two axial flow parameters W 0 = 0.2, W 0 = 0.4. The data provided as supplementary material can be used to obtain the instability characteristics for other values of W 0 (W 0 = 0, 0.1, 0.2, 0.3, 0.4, 0.5).

We have applied our results to the vortex ring and analysed the competition of the curvature instability with the elliptic instability. We have shown that the elliptic instability is always dominant without axial flow. However, the situation changes in the presence of axial flow which provides hope in observing this instability in vortex rings with swirl.

The present results can also be applied to helical vortices as they only depend on the local vortex curvature. By contrast, the elliptic instability characteristics in helices depend on the helix pitch and on the number of helices (Blanco-Rodríguez & Le Dizès 2016). Whether the curvature instability dominates the elliptic instability must then be analysed on a case by case basis. All the elements to perform such an analysis are now available.

Our analysis has been limited to a particular model of vortices. In the very large Reynolds number context of aeronautics, other models have been introduced to describe the vortices generated by wing tips [START_REF] Moore | Axial flow in laminar trailing vortices[END_REF][START_REF] Spalart | Airplane trailing vortices[END_REF]. It would be interesting to analyse the occurrence of the curvature instability in these models as well as the competition with the elliptic instability (Fabre & Jacquin 2004a;[START_REF] Feys | Elliptical instability of the Moore-Saffman model for a trailing wingtip vortex[END_REF]. 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0

      , P =       W (0) 0 0 0 0 W (0) 0 0 0 0 W (0) 1 0 0 -1 0       , ( B 1)
M(-i∂ϕ) =       Ω (0) i∂ ϕ -2 Ω (0) 0 ∂ r -ζ (0) Ω (0) i∂ ϕ 0 i r ∂ ϕ -W (0) r 0 Ω (0) i∂ ϕ 0 1 r + ∂ r -i r ∂ ϕ 0 0       , (B 2) V(-i∂ϕ, -i∂ s ) =           ∆ - 1 r 2 2i r 2 ∂ ϕ 0 0 2i r 2 ∂ ϕ ∆ - 1 r 2 0 0 0 0 ∆ 0 0 0 0 0           , ( B

N

(1)

± (-i∂ϕ, -i∂ s ) = 1 2           D (1) ± ± U (1) r U (1) r + 2 V (1) r -2W (0) 0 V (1) r + V (1)
r D

(1)

± ± V (1) r ± U (1) r ±2W (0) 0 W (1) r -W (0) ± W (1) r ∓ W (0) D (1) ± ∓ V (0) -ri∂ s 1 ±1 ri∂ s 0           , ( B 4)
where

D
(1)

± = ±U (1) ∂ r - V (1)
r i∂ ϕ -T w i∂ s , T w = W (1) + rW (0) , (B 5) 6)

T v = V (1) + rV (0) , ∆ = ∂ 2 r + 1 r ∂ r + 1 r 2 ∂ 2 ϕ + ∂ 2 s . ( B

Figure 1 .

 1 Figure 1. Sketch of the vortex structure and definition of the local Frenet frame (adapted from BRLD16).

Figure 3 .

 3 Figure 3. Analysis of the branch crossing for the Batchelor vortex at W0 = 0.2. Plot of Re(ω) versus kW0 of the first branches of the Kelvin modes of azimuthal wavenumber mA (in blue) and mB = mA + 1 (in green). The branch labels are also indicated. Solid lines: numerical results. Dashed lines: WKBJ predictions. The domains shown in figure 2 where branch crossings are expected have also been indicated. (a): (mA, mB) = (0, 1); (b): (mA, mB) = (1, 2).

Figure 4 .

 4 Figure 4. Frequency versus wavenumber of the Kelvin modes of the Lamb-Oseen vortex for mA = 0 (blue) and mB = 1 (red) in the frequency-wavenumber domain where resonance exists. The real part of the frequency is plotted in solid lines when |Im(ω)| < 0.01 (neutral or weakly damped modes) and in dotted lines when |Im(ω)| > 0.01 (strongly damped modes).

Figure 5 .Figure 6 .

 56 Figure 5. Top: Temporal growth rate of the curvature instability as a function of the axial wavenumber for the Lamb-Oseen vortex (W0 = 0) at ε = 0.1, Re = ∞ (dashed line) and Re = 10 5 . The label [lA, lB] corresponds to the branch indices of the resonant configuration. It means that the resonant configuration is formed of the lAth branch of the Kelvin mode mA = 0 and the lBth branch of the Kelvin mode mB = 1. Bottom: Vorticity contours in a (x, y) cross section of modes [7, 1], [8, 1], and [9, 1] for the parameters indicated by a star on the top graph (that is at k = kc). The vorticity is defined by (3.2) at a time t and location s such that e iks-iωt = 1 with A = 1. The black circle indicates the vortex core radius.

2 4Figure 7 .

 27 Figure 7. Top: Temporal growth rate of the curvature instability as a function of the axial wavenumber for the Batchelor vortex at ε = 0.2 and Re = 5000 for W0 = 0.2 (left) and W0 = 0.4 (right). Bottom: Vorticity contours in a (x, y) cross section of modes [3, 1], [4, 1], and [4, 2] for W0 = 0.2 (left) and modes [3,1] and [2.2] for W0 = 0.4 (right). See caption of Fig. 5 for more information.

Figure 8 .Figure 9 .

 89 Figure 8. Maximum growth rate contours of the curvature instability in the (ε,Re) plane for the Batchelor vortex. (a): W0 = 0.2; (b): W0 = 0.4. See caption of Fig. 6.

Table 1 .

 1 Characteristics of the first resonant configurations (mA, mB) = (0, 1) of label[lA, lB] for the Lamb-Oseen vortex (W0 = 0).

	3)

Table 2 .

 2 Same as table 1 for the Batchelor vortex with W0 = 0.2.

† Email address for correspondence: ledizes@irphe.univ-mrs.fr

Acknowledgments

This work received support from the French Agence Nationale de la Recherche under the A*MIDEX grant ANR-11-IDEX-0001-02, the LABEX MEC project ANR-11-LABX-0092 and the ANR HELIX project ANR-12-BS09-0023-01.

Appendix A. Dipolar correction

The first order correction is given by

where

with

We have used the index r to denote derivative with respect to r (for example

Appendix B. Operators

The operators appearing in equation (2.5) are given by

Appendix C. Tables

In this section, we provide the coefficients of the growth rate formula (3.14) for the dominant instability modes for the three cases W 0 = 0, W 0 = 0.2 and W 0 = 0.4.

Blanco-Rodríguez and Le Dizès

Appendix D. Elliptic instability of a curved Batchelor vortex -Corrigendum

Due to a normalisation mistake, a systematic error has been made in the values of the coefficients R AB and R BA in the dispersion relation (4.7) of Blanco-Rodríguez & Le [START_REF] Blanco-Rodríguez | Elliptic instability of a curved Batchelor vortex[END_REF]. The correct values are twice those indicated in this paper for all the modes. This modifies the values given in table 2 and formulas (C2n-q), (C3n-q), (C4n-q). For instance, in table 2 the correct value of R AB for the mode (-2, 0, 1) at

This error affects the y-scale of the plots (c) and (d) of figure 5 which has to be multiplied by two, and those of figure 6, which has to be divided by two. It also changes all the figures obtained in section 8. The correct figures (available on request) are nevertheless qualitatively similar if we multiply the y-scale of all the plots by a factor 2.

The comparison with [START_REF] Widnall | The instability of the thin vortex ring of constant vorticity[END_REF] done in section 8.1 for a vortex ring is also slightly modified. With the correct normalisation, the inviscid result of [START_REF] Widnall | The instability of the thin vortex ring of constant vorticity[END_REF] for the Rankine vortex is σ max /ε 2 = [(0.428 log(8/ε) -0.455) 2 -0.113] 1/2 while we obtain for the Lamb-Oseen vortex σ max /ε 2 = 0.5171 log(8/ε) -0.9285. The Lamb-Oseen vortex ring is thus less unstable than the Rankine vortex ring as soon as ε > 0.039 for the same reason as previously indicated.