Julien Herrou
email: jherrou@imm.cnrs.fr

Tam Mignot
email: tmignot@imm.cnrs.fr

Tâm Mignot

tunable protein oscillator in bacteria

Keywords: Myxococcus xanthus, MglA, MglB, RomRX, FrzX, FrzZ, oscillator, polarity, motility

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

Introduction

In all three kingdoms of life, cell polarization plays an essential role in many developmental and cellular processes including molecule transport, cell shape and differentiation, cell growth and division, motility, and organelle development and localization. Cell polarity is driven by the asymmetrical distribution of proteins within a cellular compartment; this asymmetric distribution of proteins enables polarized functions by establishing a gradient of activity across a cell or the spatial confinement of an activity to a specific location. Thus, the polarization of a cell is a dynamic phenomenon, involving the active diffusion and accumulation of polarized proteins to a precise location [1][2][3][4][5][6]. Like many spatially organized organisms, bacterial cells present asymmetrically distributed polar proteins that vary widely in function. This asymmetric distribution can be dynamic over time and regulate a number of important cellular processes including cell division, DNA segregation, cell differentiation and motility etc. Specific features of the cell poles and the cell envelope facilitate protein relocalization and accumulation to these regions. Indeed, at the cell extremities, proteins can be recruited through specific interactions with polar proteins already present at the cell poles. Protein polarization can also be favored by the low chromosomal DNA density present in these regions and the curved geometry and lipid composition of the cell membrane at the poles [2,3,6]. Cell polarity can be fixed and, for example, dictate the assembly and activity of specific cellular organelles, such as flagella, pili, stalks etc. [7]. In other instances, cell polarity is a dynamic process and is intimately associated with protein movement between poles. These movements can originate from the activity of biochemical oscillators. The MinCDE system, used by Escherichia coli to define the position of the division septum at mid-cell, is a good example of an oscillating protein system (Figure 1A). At one pole, MinC forms a complex with MinD, an ATPase, which associates with the membrane when bound to ATP. MinC is only active when bound to MinD; its function is to prevent FtsZ ring polymerization everywhere but the mid-cell and, thus, its polar localization prevents the formation of aberrant mini-cells. When MinE ring is recruited to the membrane by MinD, this interaction activates hydrolysis of ATP, resulting in the dissociation of the MinD cluster from the pole and the release of MinC. After binding ATP, free MinD rapidly relocates at the opposite pole, and reassociates with the MinC division inhibitor. Because of the continuous pole-to-pole oscillation of MinC and MinD, over time, the lowest concentration of MinC is at mid-cell, allowing division at the cell midpoint only, so that daughter cells are equivalent in size and shape [3,8,9]. In this system the oscillatory period is dictated by the slow recruitment of MinE which defines a limiting relaxation step. The Cyanobacterial McdAB protein system is another example in which protein oscillations dictate the localization of protein complexes (Figure 1B). In Cyanobacteria, specialized compartments called carboxysomes contain essential enzymes for photosynthesis. In Synechococcus elongatus, McdA is an ATPase that interacts with the nucleoid in its ATP-bound form. McdB is a protein that localizes to carboxysomes and has the ability to directly stimulate McdA ATPase activity. In its ADP-bound form, McdA is released from the DNA, leading to the formation of McdA-free regions on the nucleoid. Because McdB is on the carboxysome and has the propensity to localize at regions rich in McdA, it promotes the carboxysome relocalization to those regions on the nucleoid. Thus, the carboxysomes become evenly distributed along the length of a cell. In this review, we describe a novel type of oscillator that, similar to the two examples above, uses a nucleotide switch to regulate the oscillatory regime, but that also contains a "gate" modulating the oscillatory regime. In Myxococcus xanthus, this system controls the direction of motile cells in response to environmental cues and allows the formation of complex multicellular patterns. M. xanthus polarity switch during reversals M. xanthus is a Gram negative rod-shaped Deltaproteobacteria commonly found in soil and in marine sediments [14,15]. This organism has the ability to prey on other microorganisms and to form spores embedded in fruiting bodies when nutrients are scarce in its environment [16,17]. M. xanthus has been extensively studied for its social behavior, its complex life cycle and its motility strategies [18]. M. xanthus cells can indeed adopt a "social" motility (S-motility) or an "adventurous" motility (A-motility). During S-motility, large groups of cells move in a coordinated manner, using a form of bacterial "twitching" motility involving the so-called Type-IVa pili (T4aP) that assembles at the bacterial leading pole (Figure 2). In this process, the pili are polymerized by a multiprotein apparatus and bind like "grappling hooks" to a self-secreted exopolysaccharide. After adhesion, the pili retract by depolymerization, pulling the cell forward [19,20]. During A-motility (also known as gliding motility), single cells move at the colony periphery, exploring their environment for food. Unlike S-motility, A-motility is not T4aP dependent but, instead, involves a motility machinery named Agl-Glt. This protein complex assembles at the leading pole of the cell and traffics directionally toward the lagging cell pole, attaching to the substratum thus powering the forward movement of the cell. Aglt-Glt disassembles when it reaches the lagging pole (Figure 2) [20,21]. Therefore, M. xanthus presents a front-rear polarity, with the leading pole corresponding to the pole where the T4aP and gliding motility apparatus assemble. A striking feature of M. xanthus motility is the presence of periodic directional reversals, where cells switch direction by 180˚ due to the inversion of cell polarity and thus redirection of pili and Agl-Glt assembly to the opposite cell pole. Regulated reversals are essential for the formation of multicellular patterns, the formation of so-called rippling waves and fruiting bodies [17, [START_REF] Stevens | Making waves: pattern formation by a cell-surface-associated signal[END_REF][START_REF] Zhang | The mechanistic basis of Myxococcus xanthus rippling behavior and its physiological role during predation[END_REF][START_REF] Berleman | Rippling is a predatory behavior in Myxococcus xanthus[END_REF][START_REF] Thutupalli | Directional reversals enable Myxococcus xanthus cells to produce collective one-dimensional streams during fruiting-body formation[END_REF].

MglA, MglB and RomR form a biochemical oscillator

A reversal provokes the activation of the two M. xanthus motility machineries at the new leading pole which is orchestrated by the small Ras-like GTPase protein MglA. MglA binds to the leading pole in its active GTP-bound form and presumably recruits key proteins of each motility systems to be assembled / activated (the exact activation mechanisms are only partially characterized and not the topic of this review) [START_REF] Leonardy | Regulation of dynamic polarity switching in bacteria by a Ras-like G-protein and its cognate GAP[END_REF][START_REF] Mauriello | Bacterial motility complexes require the actin-like protein, MreB and the Ras homologue, MglA[END_REF][START_REF] Treuner-Lange | The small G-protein MglA connects to the MreB actin cytoskeleton at bacterial focal adhesions[END_REF][START_REF] Zhang | A bacterial Ras-like small GTPbinding protein and its cognate GAP establish a dynamic spatial polarity axis to control directed motility[END_REF]. The polarity of MglA is controlled by two protein complexes, the newly identified RomRX system (formed by two proteins, RomR and RomX) and MglB [START_REF] Leonardy | Regulation of dynamic polarity switching in bacteria by a Ras-like G-protein and its cognate GAP[END_REF][START_REF] Zhang | A bacterial Ras-like small GTPbinding protein and its cognate GAP establish a dynamic spatial polarity axis to control directed motility[END_REF][START_REF] Szadkowski | Spatial control of the GTPase MglA by localized RomR-RomX GEF and MglB GAP activities enables Myxococcus xanthus motility[END_REF][START_REF] Keilberg | A response regulator interfaces between the Frz chemosensory system and the MglA/MglB GTPase/GAP module to regulate polarity in Myxococcus xanthus[END_REF]. During reversals, the RomRX complex recruits MglA to the new leading pole, apparently acting as a Guanine nucleotide Exchange Factor (GEF) and thus allowing MglA-GTP to bind polar effectors of the motility complexes [START_REF] Szadkowski | Spatial control of the GTPase MglA by localized RomR-RomX GEF and MglB GAP activities enables Myxococcus xanthus motility[END_REF]. On the other hand, binding to the lagging pole is prevented by MglB, a GTPase activating protein present at the opposite pole that converts MglA-GTP to the inactive GDP-bound state. MglA-GDP is diffuse in the cytoplasm and cannot interact with the poles [20]. Thus, MglA, RomRX and MglB define a polarity axis that controls the direction of movement. During reversals, MglA relocalizes to the opposite pole, switching the polarity axis, and allowing the cells to move in the opposite direction. As further discussed below, this switch operates due to the combined action of oscillating RomRX and signal transduction (Figure 3A andB). Following a reversal and the targeting of MglA-GTP to the new leading pole (and hence activation of motility from this pole), RomRX slowly dissociates from the pole and accumulates at the lagging pole as RomR also appears to directly interact with MglB [START_REF] Szadkowski | Spatial control of the GTPase MglA by localized RomR-RomX GEF and MglB GAP activities enables Myxococcus xanthus motility[END_REF][START_REF] Keilberg | A response regulator interfaces between the Frz chemosensory system and the MglA/MglB GTPase/GAP module to regulate polarity in Myxococcus xanthus[END_REF][START_REF] Zhang | A dynamic response regulator protein modulates G-protein-dependent polarity in the bacterium Myxococcus xanthus[END_REF]. Remarkably, this gradual accumulation is driven by the slow dissociation of RomR, which acts as a slow pendulum for the oscillation and thus defines a typical relaxation step for the system (Figure 3A andB) [START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF]. The dynamics of RomR does not appear to be regulated by signal transduction and operates at the same rate, independent of the genetic background or the environmental conditions. The system reaches steady state when RomRX molecules are fully relocalized to the lagging pole, ready to recruit MglA-GTP at this pole. However and most importantly, MglA-GTP cannot readily relocalize to this pole, likely because the GAP activity of MglB predominates and must be inhibited for MglA to be recruited effectively by RomRX. In this situation, the GATE is closed and its opening requires a signal, which is provided by the socalled Frz system.

The Frz system activates the polarity switch

The Frz chemosensory pathway is essential to trigger the polarity switch. Frz mutants are perfectly motile but are unable to reverse and consequently are blocked in rippling and fruiting body formation. The Frz system is constituted of a chemosensory-like apparatus, centrally formed by a receptor-type methyl accepting protein (FrzCD) and a cognate CheA-type histidine kinase (FrzE) [20,[START_REF] Zhang | A dynamic response regulator protein modulates G-protein-dependent polarity in the bacterium Myxococcus xanthus[END_REF][START_REF] Eckhert | Dual biochemical oscillators may control cellular reversals in Myxococcus xanthus[END_REF][START_REF] Guzzo | Evolution and design governing signal precision and amplification in a bacterial chemosensory pathway[END_REF][START_REF] Igoshin | A biochemical oscillator explains several aspects of Myxococcus xanthus behavior during development[END_REF][START_REF] Kaimer | Chemosensory signaling controls motility and subcellular polarity in Myxococcus xanthus[END_REF][START_REF] Kaiser | Myxococcus xanthus swarms are driven by growth and regulated by a pacemaker[END_REF][START_REF] Kaimer | Regulation of cell reversal frequency in Myxococcus xanthus requires the balanced activity of CheY-like domains in FrzE and FrzZ[END_REF]. The connection between Frz and MglA has long remained unclear but recently, two direct FrzE-substrate response regulators (named FrzX and FrzZ) have been shown to interact with the MglAB polarity complex [START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF]. FrzX acts as a phosphorylation-dependent trigger: when phosphorylated by FrzE, it binds at the lagging pole, where it has been proposed to antagonize the action of MglB [START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF]. Thus, the action of FrzX opens the GATE, allowing RomRX to recruit MglA to the new leading pole and provoke a reversal (Figure 3A andB). Reversals thus require that two threshold concentrations are reached at the lagging cell pole: [RomRX] allows efficient recruitment of MglA-GTP when MglB is efficiently antagonized by [FrzX~P]. Controlling reversals this way combines the advantages of a switch and an oscillator: the RomR relaxation step causes the polarity apparatus to naturally reverse poles, whereas a gating mechanism uncouples the dynamics of RomR and the reversal switch. It follows, that at low Frz signaling levels (ie when environmental signals are not present), the cell is therefore in a poised state, fully primed for reversal and the system can rapidly switch as soon as FrzX~P levels increase due to signal activation. Remarkably, at high Frz signaling levels (ie when environmental signals are persistent), FrzX~P is in excess and the dynamics of RomR become limiting. Given that the dynamics of RomR are highly regular, the system oscillates as a typical relaxation oscillator in this regime (Figure 3A andB). Thus, the genetic architecture of the Frz-Mgl system allows highly adjustable responses, poised and excitable or oscillatory, depending on environmental stimulations [START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF].

FrzZ modulates the relaxation period

The relaxation property of RomR also implies that FrzX~P-dependent stimulation is only possible if sufficient amount of RomR has accumulated at the lagging pole, which times a so-called refractory period during which no new reversal can be activated [START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF]. However, the relocalization of RomR is a slow process (minutes), greatly limiting the maximum reversal frequency. The FrzZ response regulator acts to limit the length of this refractory period by lowering the amount of RomR necessary at the lagging cell pole. How precisely FrzZ performs this function is unknown, but it can bind to the leading cell pole when phosphorylated and mathematical simulations suggest that FrzZ could accelerate MglA dissociation from the leading cell pole [START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF][START_REF] Kaimer | Phosphorylation-dependent localization of the response regulator FrzZ signals cell reversals in Myxococcus xanthus[END_REF]. Thus, FrzZ acts as a rheostat, tuning the refractory period in a phosphorylation-dependent manner and allowing fast reversal frequencies at high signal concentrations despite the slow dynamics of RomR (Figure 3A andB).

Molecular mechanism of the polarity switch

The exact molecular sequence of events that lead to Frz activation of the polarity switch remains to be determined. At the lagging cell pole, the accumulation of both RomRX and MglB lead to antagonizing GAP and GEF activities. However, no accumulation of MglA is observed until FrzX accumulates at the lagging cell pole in an MglB-dependent manner. Thus, FrzX~P could directly shift the balance between GAP and GEF activities in favor of the GEF and thus allow MglA to relocalize to the lagging cell pole (Figure 4). Consistent with this, MglA and MglB co-localize at the lagging cell pole for up to 30 s when the switch is provoked, suggesting that during this time window, the GAP activity of MglB is no-longer efficient [START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF]. The mechanism by which MglB is then relocalized to the opposite cell pole is not yet clear. Guzzo et al. [START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF] postulated that MglA induces the detachment of MglB, which then interacts cooperatively with itself and the membrane at the opposite pole. While this scenario is plausible, there is currently no evidence to support it and alternative mechanisms are possible. Other proteins could be involved as well, for example the MglB-like protein MglC and the PilZ-like protein PlpA [20, [START_REF] Mcloon | MglC, a paralog of Myxococcus xanthus GTPaseactivating protein MglB, plays a divergent role in motility regulation[END_REF][START_REF] Pogue | PlpA, a PilZ-like protein, regulates directed motility of the bacterium Myxococcus xanthus[END_REF]. The exact function of these proteins in the switch mechanism remains, however, mysterious. In particular, similar to MglB, PlpA is essential for MglA polarity and cells bearing a plpA deletion reverse like the mglB deletion mutant [START_REF] Pogue | PlpA, a PilZ-like protein, regulates directed motility of the bacterium Myxococcus xanthus[END_REF]. Thus, PlpA and MglB might function in the same molecular pathway, which will require further investigation in the future. (Figure 4) The mechanism by which MglA detaches from the leading cell pole is also intriguing. While it has been proposed that RomR acts as a localization factor for MglA, MglA remains stably anchored at the leading cell pole even when the most of the RomR pool has relocalized to the lagging cell pole [START_REF] Zhang | A dynamic response regulator protein modulates G-protein-dependent polarity in the bacterium Myxococcus xanthus[END_REF]. This apparent conundrum could be explained if following its activation, MglA-GTP interacts with other polar proteins, ie A-and S-motility effector proteins. However, it remains to be established how MglA-GTP detaches from the leading pole at the time of reversals; FrzZ~P likely participates in this mechanism, but it cannot be the sole mechanism given that cells still reverse (albeit at lower frequencies) in a frzZ mutant [START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF][START_REF] Bustamante | Analysis of the Frz signal transduction system of Myxococcus xanthus shows the importance of the conserved C-terminal region of the cytoplasmic chemoreceptor FrzCD in sensing signals[END_REF].

Conclusion remarks

In this review, we describe how a complex biochemical oscillator regulates M. xanthus polarity switch and its direction of movement. Because the Myxococcus polarity system incorporates a checkpoint into an oscillator, it allows excitable or oscillatory behaviors, depending on the stimulation intensity [START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF]. This design allows unique developmental transitions as mutants that cannot enter fast oscillations (ie the frzZ mutant) are unable to form fruiting bodies, and mutants that cannot escape from oscillations are incapable to form motility swarms [START_REF] Bustamante | Analysis of the Frz signal transduction system of Myxococcus xanthus shows the importance of the conserved C-terminal region of the cytoplasmic chemoreceptor FrzCD in sensing signals[END_REF][START_REF] Zusman | Frizzy" mutants: a new class of aggregation-defective developmental mutants of Myxococcus xanthus[END_REF]. In the future, it will be important to determine where and when motility oscillations are exactly required during the predatory lifecycle. The molecular signals that activate the Frz pathway remain unidentified which largely complicates this analysis. Remarkably, the Frz receptor-kinase complex is not assembled in the bacterial inner membrane, as most receptors do, but it localizes to the cytoplasm, interacting directly with the nucleoid [START_REF] Moine | The nucleoid as a scaffold for the assembly of bacterial signaling complexes[END_REF]. The cognate response regulators (FrzX and FrzZ) thus act as diffusible messengers between the bacterial chromosome and the cell poles. Thus, it is possible that rather than sensing extracellular cues, the Frz complex senses drastic intracellular transitions (ie metabolic) and changes as a function of global physiology. Direct coupling with the bacterial chromosome could also couple Frz (and thus cell polarity) directly with the cell cycle, potentially linking cell growth to pattern formation. In the future, it will be essential to link molecular studies in single cells to large scale pattern formation to elucidate how these regulations lead to remarkable self-organization properties. From a broader perspective, the Frz-Mgl network likely evolved from the cooption of a bacterial chemosensory-type system (Frz) to a Ras-like polarity complex (MglA) [START_REF] Guzzo | Evolution and design governing signal precision and amplification in a bacterial chemosensory pathway[END_REF]. As a result, a biochemical oscillator became tuned by a signal transduction. Given that these functional modules are broadly conserved, it is possible that similar regulations might occur in other rhythmic biological systems, converting linear regulations into oscillations (and vice-versa) as a function of stimulation intensity. Hence, the layout of the Myxococcus regulatory network could be used as a framework to facilitate the elucidation of the properties and evolution of tunable biological oscillators. MglA activates two motility systems at the bacterial cell pole. The S-motility system (otherwise known as twitching motility) is involved in the movement of cells in groups and involves retractile Type-IV pili that pull cells forward like grappling hooks. The A-motility system (otherwise known as gliding motility) requires the Agl-Glt complex that assembles at the leading cell pole and moves directionally toward the lagging cell pole. Propulsion is produced when moving complexes adhere to the underlying surface. Active Aglt-Glt complexes are disassembled by MglB. Figure 3: A gated relaxation oscillator controls cell polarity switch in M. xanthus. A) Before a reversal, MglA-GTP is localized at the leading pole. MglB and RomRX localize to the lagging cell pole. The cell does not reverse because the GATE is closed. When FrzE becomes active, FrzX~P accumulates at the lagging cell pole and opens the GATE. The RomRX GEF complex then recruit MglA to the new leading pole and provoke a reversal. Following the reversal, RomRX slowly dissociates from the pole and accumulates at the lagging pole interacting directly with MglB. This slow process defines the relaxation step for the system and introduces a refractory period during which no new reversal can be activated. FrzZ~P, which also accumulates when FrzE is active, acts to limit the length of this refractory period set by RomR. B) Simulated profiles of MglA, MglB, RomR and FrzX~P during the reversal cycle (adapted from Guzzo et al. [START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF]). Note that the simulation show that reversals only occur when requirements for both RomR and FrzX~P are fulfilled at the lagging cell pole. Therefore, if the [FrzX~P] is limiting the cell is primed and in an excitable state. On the contrary, if [FrzX~P] is high the slow dynamics of RomR set the reversal period and the cell oscillates. The dashed lines indicate the time of parity of the MglB levels. an example of self-organized cellular protein waves. Philos Trans R Soc Lond B Biol Sci 373. This paper gives an overwiev of the Min-oscillating system and reviews the experimental and theoretical works that unveiled the spatio-temporal pattern emerging from interactions among the Min proteins and with the cytoplasmic membrane. Figure 4

Figure legends Figure 1 :

 1 Figure legends Figure 1: Spatial oscillators control the positioning of cellular structures in bacteria. A) The E. coli Min system positions the septal FtsZ ring at mid-cell. Pole-to-pole oscillations of MinCD ensures that the concentration of the MinC FtsZ inhibitor is minimal at mid-cell, allowing FtsZ tubulin polymerization at this site only. B) McdB-driven intracellular oscillations of McdA position bacterial carboxysomes along the nucleoid, which here functions as a subcellular scaffold for organelle assembly.Figure 2: Motility systems in Myxococcus xanthus.MglA activates two motility systems at the bacterial cell pole. The S-motility system (otherwise known as twitching motility) is involved in the movement of cells in groups and involves retractile Type-IV pili that pull cells forward like grappling hooks. The A-motility system (otherwise known as gliding motility) requires the Agl-Glt complex that assembles at the leading cell pole and moves directionally toward the lagging cell pole. Propulsion is produced when moving complexes adhere to the underlying surface. Active Aglt-Glt complexes are disassembled by MglB. Figure3: A gated relaxation oscillator controls cell polarity switch in M. xanthus. A) Before a reversal, MglA-GTP is localized at the leading pole. MglB and RomRX localize to the lagging cell pole. The cell does not reverse because the GATE is closed. When FrzE becomes active, FrzX~P accumulates at the lagging cell pole and opens the GATE. The RomRX GEF complex then recruit MglA to the new leading pole and provoke a reversal. Following the reversal, RomRX slowly dissociates from the pole and accumulates at the lagging pole interacting directly with MglB. This slow process defines the relaxation step for the system and introduces a refractory period during which no new reversal can be activated. FrzZ~P, which also accumulates when FrzE is active, acts to limit the length of this refractory period set by RomR. B) Simulated profiles of MglA, MglB, RomR and FrzX~P during the reversal cycle (adapted from Guzzo et al.[START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF]). Note that the simulation show that reversals only occur when requirements for both RomR and FrzX~P are fulfilled at the lagging cell pole. Therefore, if the [FrzX~P] is limiting the cell is primed and in an excitable state. On the contrary, if [FrzX~P] is high the slow dynamics of RomR set the reversal period and the cell oscillates. The dashed lines indicate the time of parity of the MglB levels. Figure 4: Molecular model of the polarity switch. Upon activation, nucleoid-bound Frz receptor-kinase complexes phosphorylate the two diffusible response regulators FrzX and FrzZ that localize to opposite

Figure 2 :

 2 Figure legends Figure 1: Spatial oscillators control the positioning of cellular structures in bacteria. A) The E. coli Min system positions the septal FtsZ ring at mid-cell. Pole-to-pole oscillations of MinCD ensures that the concentration of the MinC FtsZ inhibitor is minimal at mid-cell, allowing FtsZ tubulin polymerization at this site only. B) McdB-driven intracellular oscillations of McdA position bacterial carboxysomes along the nucleoid, which here functions as a subcellular scaffold for organelle assembly.Figure 2: Motility systems in Myxococcus xanthus.MglA activates two motility systems at the bacterial cell pole. The S-motility system (otherwise known as twitching motility) is involved in the movement of cells in groups and involves retractile Type-IV pili that pull cells forward like grappling hooks. The A-motility system (otherwise known as gliding motility) requires the Agl-Glt complex that assembles at the leading cell pole and moves directionally toward the lagging cell pole. Propulsion is produced when moving complexes adhere to the underlying surface. Active Aglt-Glt complexes are disassembled by MglB. Figure3: A gated relaxation oscillator controls cell polarity switch in M. xanthus. A) Before a reversal, MglA-GTP is localized at the leading pole. MglB and RomRX localize to the lagging cell pole. The cell does not reverse because the GATE is closed. When FrzE becomes active, FrzX~P accumulates at the lagging cell pole and opens the GATE. The RomRX GEF complex then recruit MglA to the new leading pole and provoke a reversal. Following the reversal, RomRX slowly dissociates from the pole and accumulates at the lagging pole interacting directly with MglB. This slow process defines the relaxation step for the system and introduces a refractory period during which no new reversal can be activated. FrzZ~P, which also accumulates when FrzE is active, acts to limit the length of this refractory period set by RomR. B) Simulated profiles of MglA, MglB, RomR and FrzX~P during the reversal cycle (adapted from Guzzo et al.[START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF]). Note that the simulation show that reversals only occur when requirements for both RomR and FrzX~P are fulfilled at the lagging cell pole. Therefore, if the [FrzX~P] is limiting the cell is primed and in an excitable state. On the contrary, if [FrzX~P] is high the slow dynamics of RomR set the reversal period and the cell oscillates. The dashed lines indicate the time of parity of the MglB levels. Figure 4: Molecular model of the polarity switch. Upon activation, nucleoid-bound Frz receptor-kinase complexes phosphorylate the two diffusible response regulators FrzX and FrzZ that localize to opposite

Figure 4 :

 4 Figure legends Figure 1: Spatial oscillators control the positioning of cellular structures in bacteria. A) The E. coli Min system positions the septal FtsZ ring at mid-cell. Pole-to-pole oscillations of MinCD ensures that the concentration of the MinC FtsZ inhibitor is minimal at mid-cell, allowing FtsZ tubulin polymerization at this site only. B) McdB-driven intracellular oscillations of McdA position bacterial carboxysomes along the nucleoid, which here functions as a subcellular scaffold for organelle assembly.Figure 2: Motility systems in Myxococcus xanthus.MglA activates two motility systems at the bacterial cell pole. The S-motility system (otherwise known as twitching motility) is involved in the movement of cells in groups and involves retractile Type-IV pili that pull cells forward like grappling hooks. The A-motility system (otherwise known as gliding motility) requires the Agl-Glt complex that assembles at the leading cell pole and moves directionally toward the lagging cell pole. Propulsion is produced when moving complexes adhere to the underlying surface. Active Aglt-Glt complexes are disassembled by MglB. Figure3: A gated relaxation oscillator controls cell polarity switch in M. xanthus. A) Before a reversal, MglA-GTP is localized at the leading pole. MglB and RomRX localize to the lagging cell pole. The cell does not reverse because the GATE is closed. When FrzE becomes active, FrzX~P accumulates at the lagging cell pole and opens the GATE. The RomRX GEF complex then recruit MglA to the new leading pole and provoke a reversal. Following the reversal, RomRX slowly dissociates from the pole and accumulates at the lagging pole interacting directly with MglB. This slow process defines the relaxation step for the system and introduces a refractory period during which no new reversal can be activated. FrzZ~P, which also accumulates when FrzE is active, acts to limit the length of this refractory period set by RomR. B) Simulated profiles of MglA, MglB, RomR and FrzX~P during the reversal cycle (adapted from Guzzo et al.[START_REF] Guzzo | A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus[END_REF]). Note that the simulation show that reversals only occur when requirements for both RomR and FrzX~P are fulfilled at the lagging cell pole. Therefore, if the [FrzX~P] is limiting the cell is primed and in an excitable state. On the contrary, if [FrzX~P] is high the slow dynamics of RomR set the reversal period and the cell oscillates. The dashed lines indicate the time of parity of the MglB levels. Figure 4: Molecular model of the polarity switch. Upon activation, nucleoid-bound Frz receptor-kinase complexes phosphorylate the two diffusible response regulators FrzX and FrzZ that localize to opposite

  The McdA oscillations thus arise from the presence of multiple McdB-containing carboxysomes, causing McdA to repetitively dissociate from and then reassociate with the nucleoid. Hence, McdB drives emergent pole-to pole oscillatory patterning of McdA [10,11]. A common feature shared by the Min and the Mcd systems is the presence of a hydrolyzable nucleotide that regulates the interaction and localization of the oscillating proteins. Thus, MinD and McdA can be considered as molecular switches that exist in an ATP-bound "ON" state and an ADP-bound "OFF" state. When bound to ATP, these proteins can form heteromers that are biologically active [8,10,12,13]. In both examples, gradual activation of ATPase activity controls the oscillatory period and provokes abrupt transitions in the oscillation regime.


H Imine1
email: imine@robot.uvsq.fr

Y N K M'sirdi1

Delanne2

Sliding-mode observers for systems with unknown inputs: application to estimating the road profile

Keywords: road profile, tyre-road friction, vehicle modelling, sliding-mode observers

In this paper, a sliding-mode observer for systems with unknown inputs is presented. The system considered is a vehicle model with unknown inputs that represent the road profile variations. Coefficients of road adhesion are considered as unknown parameters. The tyre-road friction depends essentially on these parameters. The developed observer permits these longitudinal forces acting on the wheels to be estimated. Then another observer is developed to estimate the unknown inputs. In the first part of this work, some results are presented which are related to the validation of a full-car modelization, by means of comparisons between simulations results and experimental measurements (from a Peugeot 406 as a test car).

INTRODUCTION

with the inertial profiling method, as currently used, is the impossibility of building up a three-dimensional Road profile unevenness through road-vehicle profile from the elementary measurements needed dynamic interaction and vehicle vibration affects for a road-vehicle interaction simulation package. It safety (tyre contact forces), ride comfort, energy conis worthwhile to mention that these methods do not sumption, and wear. The road profile unevenness is take into consideration the dynamic behaviour of the consequently basic information for road maintenvehicle. However, it has been shown that modifiance management systems [1]. In order to obtain this cations of the dynamic behaviour may lead to biased road profile, several methods have been developed. results. Measurement of road roughness has been the subject Finding a way to obtain a three-dimensional proof numerous research studies for more than 70 years file from the dynamic response of an instrumented [2-5]. Methods developed can be classified in two car driven on a chosen road section is the general types: the response type and the profiling method.

purpose of research engaged in at the Laboratoire Nowadays the profiling method, which gives a road Central des Ponts et Chausse ´es (LCPC) in cooperation profile along a measuring line, is generally preferred.

with the Laboratoire de Robotique de Versailles. These methods pertain to two basic techniques: the The method proposed estimates the unknown rolling-beam or the inertial profiling method. The inputs of the system corresponding to the height of latter method, which was first proposed in 1964 [6],

the road by the use of sliding-mode observers [8][9][10][11][START_REF] Xia | Observing systems with unmeasurable inputs[END_REF]. is now used worldwide. Inertial profiling methods

The design of such observers requires a dynamic consist in analysing the signal coming from displacemodel. In the first step, a model is built up for a ment sensors and accelerometers [5,7] Section 2 of this paper deals with the vehicle and damping respectively. M is the inertia matrix, B is related to the damping effects, and K is the spring description and modelling. Section 3 is devoted to some comparison results to evaluate the accuracy of stiffness matrix (Fig. 1).

A dynamic model of the vehicle can be defined as the full-car model. Then the observer design is presented in section 4. The main results are presented in section 5 to show the accuracy of the estimated road profile coming from the observer-based method.

m C v ˙x v ˙y v ˙z D =F
(3) Finally, section 6 concludes on the effectiveness of the method.

where v=[v x ,v y ,v z ]T is the vector of the vehicle velocities (along the x, y and z axes respectively)

VEHICLE DYNAMIC MODEL

and F is the vector of the tyre-road frictional forces. By assuming that the longitudinal forces are pro-When considering the vertical displacement along portional to the transverse forces, these forces can the z axis, the dynamic model of the system can be be expressed as written as (2) front and rear wheels respectively. Figure 2 represents the variations in the road where (q ˙, q ¨) represent the vectors of the velocities and accelerations respectively. G is related to the adhesion m with respect to the longitudinal slip l.

F xf =mF zf (

Many researchers have proposed different methods gravity effects, U=(u 1

,u 2 ,u 3 ,u 1 )T is the vector of unknown inputs which characterize the road profile, to measure these two coefficients. Bakker et al. [START_REF] Bakker | tire model with an application in vehicle dynamics studies[END_REF] proposed 'a magic formula'. In the linear area of The wheel angular motion is given by

J fi v ˙fi =T ei -rF xfi J ri v ˙ri =-rF xri (7)
where T ei , i=1, 2, is the engine torque, r is the wheel radius, and J fi and J ri are the wheel inertias.

Remark 1

The engine torque is deduced using the vehicle speed and the throttle position, without an explicit model for the engine behaviour. The steering and braking angles, the braking torque, and rolling resistance are measured.

l= K v r -v x max(v r , v x )K (6) x ˙=f (x)+CU+DU ẏ=h(x)
where v r is the wheel velocity. Figure 3 represents the longitudinal slip during the (8) test reported here at a speed of 20 m/s.

where the state vector x=(x 11 , x 12 )T=(q, q ˙)T, and It should be noted that this longitudinal slip is y=q(yµR8) is the vector of measured outputs of located in the linear area (longitudinal slip is between the system. 0 and 0.1; see Fig. 2). Therefore, this justifies the use Thus, of the Burkhardt model in this work.

x ˙11 =x 12 x ˙12 =M-1(-Bx 12 -Kx 11 -G)+M-1(Cx 3 +Dx 4 ) x ˙3=x 4 x ˙4=0 (9)
where x 3 =U. Before developing the sliding-mode observer, the following assumptions must be considered.

1. The state is bounded (dx(t)d<2, Y t0). 2. The system is inputs bounded (Z a constant mµR4 such that: U ˙<m). 3. The vehicle rolls at constant speed on a defect road of the order of millimetres, without bumps. In the state form, the wheel angular motion

Convergence analysis becomes

The dynamics estimation errors can be written as

j ˙1=j 2 =J-1(C-RY ) x ˜11 =x ˜12 -H 1 sgn(x ˜11 ) y 1 =j 1 x ˜12 =-M-1(Bx ˜12 +Kx ˜11 ) (10) +M-1Cx ˜3+M-1Dx ˜4-H 2 sgn(x ˜11 ) x ˜3=x ˜4-H 3 sgn(x ˜11 ) j=(j 1 , j 2 )T where j 1 =y 1 =[v r1 , v r2 , v f1 , v f2
]T represents the measured angular velocity vector and j 2 is x ˜4=-H 4 sgn(x ˜11 ) the vector of angular accelerations.

(13) Because of small variations in the longitudinal forces, is assumed that Y ˙=0,

In order to study the observer stability and to find the gain matrices H i , i=1, …, 4, firstly, the convergence of x ˜11 to the sliding surface x ˜11 =0, in finite time t 1 , must be proved. Then, some conditions

J= C J r1 000 0 J r2 00 00 J f1 0 000 J f2 D about x ˜12
must be deduced to ensure its convergence towards 0. Finally, it must be proved that the input estimation errors (namely x ˜3 and x ˜4) converge towards 0. C = [0, 0, T e1 ,T e2 ]T, R = r 1 l (with lµR4×4 is the Consider the Lyapunov function identity matrix) and

Y=[F xr1 ,F xr2 ,F xf1 ,F xf2 ]T repre- V 1 = 1 2 x ˜T 11 x ˜11
(14) sents the vector of the longitudinal forces to be

The time derivative of this function is given by estimated.

V ˙1=x

˜T 11

[x ˜12 -H 1 sgn(x ˜11 )] (15)

Observer design

By considering the gains matrix

H 1 =diag(h i1 ) with h i1 >|x ˜i2 |, i=1, …, 8, then V ˙1<0.
Therefore, from In order to estimate the state vector x and to deduce sliding-mode theory [START_REF] Utkin | Sliding mode observer[END_REF], the surface defined by both the unknown inputs vector U and its derivative x ˜11 =0 is attractive, leading x ˆ11 to converge towards U ˙, the sliding mode observer:

x 11 in finite time t 0 . Moreover, x ˜=0 Y tt 0 . Con-

x ˆ11 =x ˆ12 +H 1 sgn(x ˜11 )
sequently and according to equation (13), for tt 0 ,

x ˆ12 =-M-1(Bx ˆ12 +Kx ˆ11 +G) sgn eq (x ˜11 )=H-1 1 x ˜12 ( 16
)
where sgn eq represents an equivalent form of the sgn +M-1(Cx ˆ3+Dx ˆ4)+H 2 sgn(x ˜11 ) function on the sliding surface. Then the equation x ˆ3=x ˆ4+H 3 sgn(x ˜11 ) system (13) can be written as

x ˆ4=H 4 sgn(x ˜11 ) x ˜11 =x ˜12 -H 1 sgn eq (x ˜11 ) 0 (11) x ˜=-M-1Bx ˜12 +M-1Cx ˜3+M-1Dx ˜4-H 2 H-1 1 x ˜12 x ˜3=x ˜4-H 3 H-1 1 x ˜12 is proposed,
V ˙2=-x ˜T 12 (B+MH 2 H-1 1 )x ˜12 ( 22
)
Recalling that M and H 1 are positive definite matrices, and by choosing H 2 of the form

H 2 =M-1(Q-B)H 1 ( 23
)
such that Q=B+MH 2 H-1 1 is a positive definite diagonal matrix (with QµR8×8), then V ˙2<0. Therefore, the surface x ˜12 =0 is attractive, leading x ˜12 to converge towards x 12 . According to equations (18), it can then be deduced that the estimation error x ˜3 of the road profile and its time derivative x ˜4 also converge towards Fig. 4 LPA 0. The dynamic estimation error of j 1 is given by

j ˜1=-J-1RY ˜-L 1 sgn(j ˜1) ( 24
)
The force estimation error Y ˜is defined by

Y ˜=-x-L 2 sgn(j ˜1) ( 25
)
Consider the Lyapunov function

V∞ 1 = 1 2 jT 1 j 1 + 1 2 Y ˜TPY ˜(26)
where PµR4×4 is a diagonal positive matrix.

According to equations ( 24) and (25), the time derivative of this function gives

V ˙∞ 1 =j ˜T 1 j ˜1+Y ˜TPY =-j ˜T 1 L 1 sgn(j ˜1)-j ˜T 1 J-1RY ˜Fig. 5 Vehicle speed -Y ˜TPx-Y ˜TPL 2 sgn(j ˜1) (27)
known. To mitigate this disadvantage, observers are vertical displacement z and the yaw angle y of the chassis are presented. It is shown that these displaceused to estimate the longitudinal forces which are ments can be estimated rapidly and fairly accurately. related to these parameters. The system outputs

The lower two plots of Fig. 7 represent the velocities. are the displacements of the wheels and the chassis, It can be seen that the estimated vertical velocity z ẇhich correspond to signals given by sensors.

is accurate compared with the true signal. Different measurements are made with the vehicle However, an error occurs concerning the estimoving at several speeds. mation of y ˙. This error is mainly due to sensor Figure 5 shows a vehicle speed average of 70 km/h calibration (the sensor that was used in the measure-(20 m/s) with an error which does not exceed ments had an error of calibration that could not be 1.2 m/s (Fig. 6). corrected). Figure 7 shows the measured and the estimated In Fig. 8, it can be seen that the estimated angular displacements. In the upper two plots of Fig. 7 the velocity of the wheel converges well towards the actual values in finite time. Indeed, there is a convergence time of only 1 s. The convergence of the states is very fast and the estimation is of good quality. The good reconstruction of these states allows the unknown inputs to be estimated.

In Fig. 9 the behaviour of the road profile estimator is presented. This figure presents both the measured road profile (measured using the LPA) and the estimated road profile. Thus it can be observed that the estimated values are quite close to the true values. These profiles have the same shape and the differences are not important. Figure 10 shows the power spectral density of the estimated road profile and the It should be noted that low and average waves of 5 CONCLUSION the road (i.e. mean high and average frequencies) respectively are well reconstructed. However, there

In this paper, sliding-mode observers have been developed to estimate the longitudinal tyre-road are limitations of this method in estimating high waves of the road. forces of the system and the unknown inputs which Automotive Engineers, New York). 9 Barbot, J. P., Boukhobza, T., and Djemai, M.

  Fig. 1 Vehicle model

Fig. 2

 2 Fig. 2 Variations in the road adhesion with longitudinal slip

Fig. 3

 3 Fig. 3 Longitudinal slip during the testIt can then be assumed that U ¨=0.

Fig. 6

 6 Fig. 6 Vehicle speed error profile measured using the LPA.

Fig. 7

 7 Fig. 7 Estimated and measured states

Fig. 8

 8 Fig. 8 Estimated and measured wheel velocities

Fig. 9

 9 Fig. 9 Comparison of the profile measured using the LPA estimated profile

Fig. 10

 10 Fig. 10 Power spectral density (PO, low wave; MO, average wave; GO, high wave)

4) Mq ¨+Bq ˙+Kq+G=CU+DU ˙(1)

	qµR8 is the coordinates vector defined by q=[z 1 , z 2 , z 3 , z 4 , z, h, w, y]T	where F zfi of the front and rear wheels respectively. F xfi and F zri , i=1, …, 2, are the vertical forces and F , xri i=1, …, 2, represent the longitudinal forces of the

  Triangular input observer form and sliding mode observer. In Proceedings of the 35th IEEE Con-REFERENCES ference on Decision and Control, Kobe, Japan, 1996, pp. 1489-1491 (IEEE, New York). 10 Bestle and Zeitz Canonical form observer design for 1 Van Der Jagt, P. and Parson, A. Road surface nonlinear time varying systems. Int. J. Control, 1988, correction of tire test data. In Proceedings of the 47, 1823-1836. First International Colloqium on Tyre Models for 11 Drakunov, S. V. Sliding-mode observers based on Vehicles Dynamics Analysis, Delft, The Netherlands, equivalent control method. In Proceedings of the 21-22 October 1991. 31st IEEE Conference on Decision and Control, 2 Misum Simulation of the interaction between Tucson, Arizona, USA, 1992, pp. 2368-2369 (IEEE, vehicle wheel and the unevenness of the road surface. Veh. System Dynamics, 1990, 19, 237-253. New York).


F Y Moulin

J.-B Flór
email: flor@hmg.inpg.fr

Experimental study on wave breaking and mixing properties in the periphery of an intense vortex

Keywords: Rotating stratified flows, Wave-vortex interaction, Mixing

We experimentally investigate the effects of the breaking of inertia-gravity waves on the dynamics and transport properties of a vortex in a rotating stratified fluid. A tall cyclonic vortex with vertical vorticity axes, and Gaussian velocity profile is perturbed by planar waves, emitted by a horizontally oscillating cylinder. We show that the trapping and breaking of the waves inside the vortex lead to a deposit of negative momentum in the periphery of the vortex. Periodic waves destabilize the streamlines and chaotically mix the outer part of the vortex while the core remains coherent. In case waves are trapped, and break, mushroom-like structures form that may transport fluid in the radial direction. The two different mixing processes are discussed. Further, we consider the effects of vortex strength and wave-energy on the mixing.

Introduction

In large-scale geophysical flows, the dominant presence of background rotation and stratification not only prompts the inverse energy cascade towards large scales leading to the formation of large-scale vortices, but also support waves, e.g. generated during geostrophic adjustment. The interaction of waves with shear layers in rotating stratified fluids have been studied in detail experimentally, numerically and analytically. In studies on the interaction of waves with a jet, reflection, absorption or breaking of the waves occurred as shown experimentally (e.g. [START_REF] Koop | A preliminary investigation of internal gravity waves with a steady shearing motion[END_REF], theoretically [START_REF] Booker | The critical layer for internal gravity waves in a shear flow[END_REF][START_REF] Ivanov | Deformation of internal gravity waves by a stream with horizontal shear[END_REF][START_REF] Olbers | The propagation of internal waves in a geostrophic current[END_REF][START_REF] Badulin | The trapping and vertical focusing of internal waves in a pycnocline due to the horizontal inhomogeneities of density and currents[END_REF][START_REF] Badulin | On the irreversibility of internal-wave dynamics due to wave trapping by mean flow inhomogeneities. Part 1. Local analysis[END_REF] and in numerical studies by [START_REF] Dornbrack | Turbulent mixing by breaking gravity waves[END_REF] and [START_REF] Staquet | On transport across a barotropic shear flow by breaking inertia-gravity waves[END_REF]. Wave-vortex interaction is a very rich and relatively unexplored subject of study. In the context of the polar vortex and the transport of ozone depleting radicals, the effect of wave breaking and the transport of material across the edge of the polar vortex, have been discussed by [START_REF] Mcintyre | The stratospheric polar vortex and sub-vortex: fluid dynamics and mid-latitude ozone loss[END_REF]. Recently, [START_REF] Bühler | Remote recoil: a new wave-mean flow interaction effect[END_REF] have investigated wave-vortex interactions in the shallow-water approximation and showed that the vortex core of a potential vortex is advected by horizontally refracted surface waves. Here, we focus on the wave-induced mixing within the vortex and the related vortex dynamics. The breaking of waves inside a jet or a vortex results from an accumulation of energy of the waves trapped in the background flow. Such a trapping process is well predicted by the WKB theory (see [START_REF] Bretherton | The propagation of groups of internal gravity waves in a shear flow[END_REF], which, to summarize, assumes that the phase of the wave evolves on time and space scales shorter than the scales associated with the jet or vortex. The time dependence of the wave field is given by its absolute frequency, ω abs , corresponding to the wave frequency before the beginning of the interaction. During the interaction, this time dependency is conserved, and the intrinsic frequency ω 0 , defined as its frequency in a referential moving with the local mean flow velocity U, is inferred from the Doppler shifting relationship that reads

ω abs = ω 0 + k • U, ( 1
)
where k is the wave vector. In the WKB theory, ω 0 satisfies the dispersion relationship for inertia-gravity waves, ω 2 0 = (k 2 h N 2 + k 2 z f 2 )/(k 2 h + k 2 z ), where N is the buoyancy frequency of the stratified fluid, f the Coriolis parameter, and k h and k z the horizontal and vertical components of the wave vector, respectively.

In the case of interaction of a planar monochromatic wave with a jet, ω 0 is a function of the spatial coordinates only. Space is then divided into regions where waves can propagate or are arrested. For a barotropic jet (without vertical shear), waves approaching the layer ω 0 = N are trapped, the component of the wave number perpendicular to the layer increasing towards infinity. The accumulation of energy combined with the decrease in wavelength eventually makes the wave overturning and breaking. Here we consider interactions of inertia-gravity waves with a tall baroclinic vortex in a rotating linearly stratified fluid.

For the interaction of a planar monochromatic wave with a vortex, the equations of the WKB theory have to be solved numerically in cylindrical coordinates (see [START_REF] Moulin | Vortex-wave interactions in a rotating stratified fluid: WKB simulations[END_REF]. In the case of the interaction with tall vortices, the incident wave field is partially trapped in a broad region. This is in contrast to the trapping in a unique layer predicted by the WKB theory for a barotropic jet. In addition, the wave breaking is observed only in experiments where the time scale associated with the trapping is far shorter than the time scale of the viscous damping of waves (see [START_REF] Moulin | Vortex-wave interactions in a rotating stratified fluid: WKB simulations[END_REF]. For this condition to be fulfilled, we need stable vortices with a high level of shear. In experiments with isolated vortices, having net zero circulation at a finite distance, the shear values were far too low for wave breaking to occur, while highly sheared isolated vortices are unstable (see e.g. [START_REF] Carton | The emergence of tripole from unstable axisymmetric vortex structures[END_REF]. Therefore, vortices were generated by siphoning off fluid through a perforated tube. These vortices are stable and confined to a small radius so that the shear is relatively high.

In the present paper, we investigate experimentally the breaking of waves and its effects on the dynamics of such a vortex and on the radial transport of material. The results presented here and in [START_REF] Moulin | Vortex-wave interactions in a rotating stratified fluid: WKB simulations[END_REF] as well as related subjects on vortex-wave interactions are also presented in [START_REF] Moulin | Interactions onde-vortex en milieu stratifié tournant et transport à travers une barrière dynamique[END_REF]. In Section 2, we describe the experimental methods and the modelling of the vortex, followed in Section 3 by experimental observations of the wave trapping and of a deposit of momentum inside the vortex due to the wave breaking. Section 4 reports on the transport of material driven by the wave-vortex interaction and the breaking of waves. Conclusions are drawn in Section 5.

Experimental procedure

Experiments were carried out in a 1 m 2 tank of 45 cm depth, mounted on a rotating table .  A side view of the experimental set-up is sketched in Fig. 1. The tank was filled with a linearly salt-stratified fluid. The Coriolis parameter f = 2Ω varied between 0.4 and 0.5 rad s -1 , and the buoyancy frequency N of the water solution, defined as N 2 = -(g/ρ 0 )∂ρ/∂z, varied between 1.15 and 1.5 s -1 . Experimental values of N were calculated measuring the density at different depths in the tank with a refractometer. To measure the horizontal velocity field, the motion of neutrally buoyant particles of 70 m diameter, illuminated by a horizontal laser sheet, was recorded by a B&W camera from the top and processed with the CIV method developed by [START_REF] Fincham | Advanced optimization of correlation imaging velocimetry algorithms[END_REF]. Values of the vertical vorticity and the horizontal divergence were then calculated using their algorithms. To observe the mixing induced by wave-vortex interaction, we introduced fluorescein dye inside the vortex, and illuminated the flow either by a horizontal or a vertical laser sheet.

To generate an elongated vortex, a 1.5 cm diameter tube which was perforated over a vertical extent of 10 cm, was introduced vertically at the centre of the tank. A radial inflow was generated by siphoning off fluid with a pump, leading to a cyclonic vortex by conservation of angular momentum. In order to siphon off an equal amount of fluid at each depth, a 15-mm diameter tube was filled with seven small tubes of about 2 mm in diameter that ended at different levels in the tube. The circulation Γ = U θ (r)dr, with U θ the azimuthal velocity, depends on the total volume, V, pumped through the tube. After removing the tube, this forcing results in a vortex with a Gaussian vorticity profile at middepth, which slowly diffuses radially outwards.

In the inviscid approximation, an inwards radial flow that is uniform over a vertical extent of H = 10 cm and of a total volume V, would yield a vortex with circulation Γ = fV/H. Measurements of Γ in the mid-plane just after the removal of the tube were found to be lower than this theoretical value, most likely, because we underestimated the vertical extent of the pumped fluid which is larger than the 10 cm distribution of holes.

In order to reproduce the evolution of the vortex structure, the diffusion equation for momentum was integrated numerically (see [START_REF] Beckers | Dynamics of pancake-like vortices in a stratified fluid: experiments, model and numerical simulations[END_REF] for the initial condition of a tube of constant vorticity of height 10 cm and radius 1 cm. Once the value of the circulation is prescribed by the experimental measurements just after the removal of the tube, an excellent agreement was found between the experimental data and the prediction of this model. We also found a good agreement between measurements and predictions for a diffusing Lamb-Oseen vortex, which is an exact solution of the diffusion equation. Its azimuthal velocity, U LO , evolves as

U LO = Γ 2πr (1 -e -(r/R(t)) 2 ), (2)
with

R(t) = 2 √ ν(t -t 0 )
, where ν is the kinematic viscosity and t 0 a reference time. The vortex radius, R max , defined as the radial position of the maximal velocity in the mid-plane, satisfies R max (t) = 1.12R(t) for the velocity field (2). The experimental value of R max just after removal of the tube was around 1.6 cm. To reproduce the experimental data, the reference time t 0 was chosen to be equal to 50 s after the tube removal.

Inertia-gravity waves were generated by vertical oscillations of a horizontal cylinder, of length L = 30 cm and diameter D = 5 cm, placed near the bottom of the tank. The amplitude of the oscillations, A cyl , ranged from 0.5 to 3 cm. The vertical distance between the vortex centre and the cylinder was around 15 cm. The cylinder was placed such that the mid of the wave front propagated towards the vortex centre during the interaction. Thus, at one side, waves propagated in opposite directions to the vortex flow, while at the other side, they propagated in the same direction.

Since the cylinder was very long compared to its radius, the wave pattern was almost planar, with wave fronts parallel to the cylinder axis. The wave energy propagates along trajectories whose angle with the horizontal plane, θ, depends only on the oscillation frequency, and is perpendicular to the propagation direction of the front. The wave fronts propagate downwards while the energy propagates upwards towards the perforated tube.

Only one wavelength appears in the velocity field, associated with the bimodal structure of the wave field generated by an oscillating cylinder. Though the spatial spectrum of such a wave field is large, it is observed experimentally that the dominant wavelength which appears in the velocity field, is approximately equal to twice the diameter of the cylinder.

Impact of the wave breaking on the dynamics of the vortex

To investigate the behaviour of waves propagating towards the vortex flow, the WKB theory is used. The planar wave generated by the oscillating cylinder is well described by a group of parallel rays originating from the cylinder, and the behaviour of these rays in the framework of the WKB theory agrees with the experimental observations.

Waves propagating along the vortex flow experience a Doppler shift (1) leading to a decrease of the intrinsic frequency ω 0 . As ω 0 decreases, the intrinsic group velocity, v g , becomes increasingly horizontal. The wave energy is then advected by the horizontal flow of the vortex, reflects on the vortex flow and escapes without being trapped. This behaviour is similar to that of waves propagating along a barotropic jet. Wave rays rays associated with the wave field propagating against the vortex flow, experience a Doppler shift (1) leading to an increase of the intrinsic frequency ω 0 towards N. As ω 0 increases, the intrinsic group velocity, v g , becomes increasingly vertical, and the wave vector increases gradually in the direction of the vortex axis because of the strong radial shear. Most of these rays are then propagating locally in a flow with strong radial shear and weak vertical shear, and exhibit a similar behaviour as rays propagating against a barotropic jet: they are trapped along a critical layer and their wave vector increases towards infinity in the framework of the WKB theory. In the experiments, trapped waves are either damped by viscosity, or eventually break (see [START_REF] Moulin | Vortex-wave interactions in a rotating stratified fluid: WKB simulations[END_REF]). In the scope of the present paper, only results on experimental waves-breaking will be discussed.

Fig. 2 shows typical vertical vorticity and horizontal divergence fields of an experiment with wave breaking. Waves propagating along the flow are advected horizontally and scattered away from the field of view, and are therefore not visible in Fig. 2. Waves propagating against the flow are trapped, as shown in the divergence field Fig. 2(b). They are hardly visible in the vertical vorticity field (Fig. 2(a)) because their intrinsic frequency is close to N and their polarisation is very close to pure gravity waves. The energy of these trapped waves is transported along the flow and wrapped around the vortex centre, while the horizontal wavelength of the wave pattern decreases when approaching and turning along the vortex core. It is hard to detect wave-breaking events by PIV measurements, because of intrinsic limitations of this method, observations by other methods had to be used to ensure the occurrence of wave breaking.

To measure the impact of the wave breaking on the vortex dynamics, we compare a vortex in the absence of waves in experiment A, with a vortex in the presence of wave-breaking

f = 0.4 s -1 , N = 1.5 s -1 , V = 3.2 L, D = 5.0 cm, A cyl = 2
.0 cm and ω abs = 0.53 rad s -1 . The measurement is taken at t = 45 s after forcing, and R max ≈ 2.5 cm and U max ≈ 2.5 cm s -1 , yielding Γ/2π ≈ 8.3 cm 2 s -1 . during a finite time interval in experiment B. By using exactly the same experimental conditions for the vortex generation, we are able to create identical vortices. By generating waves during a finite time interval, we are able to measure accurately the vortex velocity profiles before and after the interaction, and compare them quantitatively. Fig. 3 shows the results of these experiments. The azimuthal velocity profiles of the vortex just after the removal of the tube are plotted on the left of Fig. 3. The experimental data agree well with a Lamb-Oseen vortex with the same circulation Γ and the same radius R max for experiments A (top) and B (bottom), showing that the vortex generation by aspiration is well reproducible. For experiment B, Fig. 3(B,2) shows the radial profiles of the azimuthal velocity at the onset of trapping; the scatter in the tail is due to the non-axisymmetric wave field. Fig. 3(B,3) shows the radial profiles of the azimuthal velocity field after wave breaking.

The continuous lines in Fig. 3 correspond to the predictions of the Lamb-Oseen vortex, and agree very well with the measurements in experiment A without waves. In contrast, a clear discrepancy is found between the Lamb-Oseen profile and the measurements of experiment B after wave breaking. In the outer part of the vortex, the azimuthal velocity has decreased, revealing a deposit of negative momentum in the vortex periphery. This is in agreement with the trapping of waves which transport linear momentum oriented against the flow. Another interesting point is that this deposit of negative momentum leads to a steepening of the velocity profile, corresponding to negative vorticity.

In order to estimate the vortex momentum and energy variations due to the wave interactions, we compare our vortex with a Lamb-Oseen vortex. In Fig. 4(a), the radial profiles of the azimuthal velocity field before the interaction with the waves Fig. 3(B,1) and after in Fig. 3(B,3), have been smoothed and fitted with the profile of a Lamb-Oseen vortex, while the deviation in angular momentum of the two profiles from the Lamb-Oseen vortex is shown in Fig. 4(b). The difference in angular momentum between the initial and final profile is estimated by the surface of the grey region in Fig. 4(b) and, for 5 < r < 14 cm, amounts M dep = -92 cm 3 s -1 . This is equal to 12% of the total angular momentum of the vortex calculated for r < 15 cm. With the values equally derived for the kinetic energy, we deduced a decrease of 255 cm 4 s -1 due to the wave breaking from the discrepancy between the evolution of energy of the Lamb-Oseen vortex and of the experimental vortex. This decrease is around 20% of the amount of kinetic energy the vortex at the end of the experiment.

Let us compare these values with the energy density of the incident wave field, e wave . This energy was measured from the amplitude of the vertical displacement of dye in a similar experiment with the same experimental conditions and amounted, for an amplitude of 0.5 cm, e wave ≈ 0.82 cm 2 s -2 . If this energy would be completely converted into a barotropic horizontal flow, it would produce a velocity ũ = √ 2e wave = 1.28 cm s -1 . The waves are generated during t = 66 s and propagate away from the cylinder at the group velocity v g = 1.57 cm s -1 along a beam of width roughly equal to Λ = 2D and of half a cylinder length, L/2, since only waves propagating against the flow are trapped and may provide energy and momentum to the vortex field. We therefore estimate the kinetic energy and momentum of the incident wave field by LDv g ( t)e wave = 12745 cm 5 s -2 and Fig. 4. (a) Smoothed profiles of the azimuthal velocity U θ deduced from experimental data of Fig. 3(B,1) and (B,3), respectively, before wave generation, and after wave breaking. The dashed lines represent predictions of the Lamb-Oseen vortex, whose azimuthal velocity U LO is given by ( 2). (b) Difference in negative angular momentum, rv θ , between the smoothed experimental profiles and the predictions from the Lamb-Oseen vortex plotted in (a). The difference in azimuthal velocity, v θ , is defined as v θ (r) = U LO (r) -U θ (r). (• • •) Experiment of Fig. 3(B,1), before the generation of waves. (-) Experiment of Fig. 3(B,3), after the wave breaking. The grey-shaded area indicates the amount of momentum transferred by wave breaking from the incident wave field towards the vortex field.

LDv g ( t)ũ = 19895 cm 4 s -1 , respectively. For a uniform transfer in the vertical direction, along a vortex of height H = 10 cm, the kinetic energy and momentum at each level, would roughly read Ec av = 1275 cm 4 s -1 and M av = 1990 cm 3 s -1 , respectively.

Apparently, only a very small part of the wave energy and momentum are converted to the horizontal flow. As can be inferred from Figs. 5, 6 and 11, this loss in wave energy and momentum is most likely due to the (vertical) diapycnal mixing, horizontal mixing and viscous dissipation on small scales. Such a weak deposit of momentum in the background field was also observed numerically for waves interacting with a uniform shear flow by [START_REF] Staquet | On transport across a barotropic shear flow by breaking inertia-gravity waves[END_REF].

Mixing induced by the wave interaction and breaking

Method and observations

To study the mixing induced by the interaction of waves with the vortex and their breaking inside the vortex, we used fluorescein dye illuminated by either a horizontal or a vertical laser sheet. To generate a dye sheet, we employed the method of [START_REF] Hopfinger | Internal waves generated by a moving sphere and its wake in a stratified fluid[END_REF] and introduced a 1 mm thin cotton wire, soaked in concentrated fluorescein, vertically in the fluid just before vortex generation. The dye sheet was made by translating the vertical wire horizontally. During the vortex generation, the radial inflow and the increasing azimuthal flow wraps the vertical plane of dye around the vortex centre. Top view of the dye illuminated by a horizontal plane takes the form of a spiral, like in Fig. 5(a), while a typical side view is given by Fig. 6(a). If the radial inflow had been uniform in the vertical direction, we would

First stage: chaotic mixing

The zigzag patterns that initially appear in the top and side views of the dye distribution shown, respectively, in Figs. 5(b) and 6(b), are horizontal and vertical cross-sections of an inclined folding of the dye sheet transported by the mean flow. To understand how the propagation without breaking of waves inside the vortex could produce such patterns, we performed numerical simulations of the transport of passive tracers in a time-dependent flow. For simplicity, we modeled the vortex by a time-independent azimuthal velocity field u θ deduced from the expression (2). Let R f be the radius of the vortex at time t, R f = R(t). We freeze the velocity field and scale the radial coordinate r → R f r, the velocity U θ → (Γ/2πR f )U θ . The non-dimensional azimuthal velocity of the vortex is then given by the expression (2) with R(t) = 1, Γ = 2π and U θ ≈ 1/r away from the core. The time coordinate t, is then scaled as t → (2πR 2 /Γ )t. To simulate numerically the perturbation by the waves, we superposed a non-divergent time-dependent velocity field on the vortex whose non-dimensional form reads

u θ r = - 2 m e -r 2 sin(Φ) - k r mr (1 -e -r 2 ) cos(Φ) f (t) u r = 1 r (1 -e -r 2 ) cos(Φ)f (t), (3)
where Φ = mθ + k r r -ωt is the wave phase, m the azimuthal wave number, k r the radial wave number, the non-dimensional wave amplitude, and f (t) is a time function aimed to describe a gradual formation of the wave field from zero. To generate a wave field on a time roughly equal to its period we chose f (t) = 1exp(-ωt). The description of the wave field by (3) reproduces the basic relevant features of the wave field: it is time periodic and the constant-phase surfaces for the trapped waves form a spiral-like pattern. Fig. 7 shows results of numerical simulations of the transport of passive tracers for m = 2. In accordance with the experiment, we generated a spiral of passive tracers by advecting a straight segment of tracers with the velocity field of a Lamb-Oseen vortex (2). We added the wave velocity field (3), once the spiral pattern was well developed. Zigzag patterns clearly appear around a trajectory that satisfies n = T t /T w = 2, where T t is the time of revolution of the considered trajectory, T t = 2πr/u θ , and T w is the period of the perturbation, T w = 2π/ω. From the Lagrangian point of view (see [START_REF] Wiggins | Chaotic Transport in Dynamical Systems[END_REF], trajectories with n rational can be destabilized by the superposition of a periodic disturbance; the destabilized trajectories split in a succession of elliptic and hyperbolic points. In the neighbourhood of elliptic points, the distribution of tracers rotates gradually at every period while near the hyperbolic points, chaotic regions form which drive a rapid separation of initially close tracers. To reveal these elliptic and hyperbolic points, results of the same numerical simulation with tracers distributed initially along the destabilized trajectory n = 2 are plotted in Fig. 7(b). The elliptic points are located at the centre of the zigzag patterns, while the hyperbolic points are found in the centre of regions with a low tracer density. For other values of m zigzag patterns formed near trajectories satisfying n = m.

In the experiments, the wave field is more complicated than the description of Eq. (3). However, the time dependence is well defined, so that we may expect a destabilization of trajectories corresponding to integer values of n. Careful analysis of the dye distribution in the mid-plane shows that the zigzag patterns appear on trajectories satisfying n = 2 and n = 3, as illustrated in Fig. 8 for n = 2. Some zigzag patterns appear also near the trajectory n = 1 but are more difficult to detect since they are close to the well-mixed region of the vortex core. The correspondence with the experimental observations strongly suggests that the fold and stretch patterns, associated to the chaotic mixing, are due to the wave-induced perturbations of the vortex flow.

To explain the appearance of zigzag patterns in both vertical and horizontal crosssections, we should take into account of the vertical dependence of the wave field. Therefore, we consider a wave with phase Φ = mθ + k r r -ωt + k z z + Φ 0 in Eq. ( 3), with k z the vertical component of the wave vector and Φ 0 a phase reference. Then we may expect that the elliptic points will be distributed along constant phase lines at a fixed time, and that small-scale structures (see rectangular selections in Fig. 9(d-g)). These structures have a vertical extent which does not correspond to the vertical wave length of the trapped waves, and are therefore likely to be associated to the non-linear destabilization of incident waves by parametric resonance (see [START_REF] Benielli | Excitation and breaking of internal gravity waves by parametric instability[END_REF]. Eventually, mushroom-like patterns begin to appear in the dye field (see circular selections in Fig. 9(h) and (i)), indicating the occurrence of the Rayleigh-Taylor instability, associated with a local inversion of the density field due to the high amplitude wave motions. This instability is studied in the case of an unstable linear density-stratification in a thin gap by [START_REF] Voropayev | Experiments on the evolution of gravitational instability of an overturned, initially stably stratified fluid[END_REF], revealing the formation of two-dimensional mushroom-shaped structures, that move vertically by gravity. The transport by the mean flow of the present convective structures across the laser sheet explains the brief appearance (see Fig. 10), and reveals their almost spherical shape, not dissimilar to the wave breaking of orographic gravity-wave structures observed by [START_REF] Gheusi | A numerical study of three-dimensional orographic gravity-wave breaking observed in a hydraulic tank[END_REF] and [START_REF] Eiff | Lee-wave breaking over obstacles in stratified flow[END_REF]. They show that instabilities take the form of toroidal or ring-like vortex structures, inclined with respect to the horizontal. The preferential horizontal orientation of these rings may provoke a transport of fluid through the dynamical barrier at the edge of the vortex.

Radial transport of fluid

The chaotic mixing (see Fig. 8) is confined to a narrow ring and is, therefore, not very efficient to transport fluid radially. The non-linear generation of small-scale patterns by parametric resonance does not affect the transport of fluid particles in the neighbourhood of the vortex core; only the mushroom-like patterns associated with the overturning instability are able to transport fluid efficiently in both vertical and radial directions. The location and strength of these convective patterns should be controlled by the properties of the trapped incident waves, described to some extent by the WKB theory. In an attempt to check this, we performed two experiments with different vortex strengths, and different amounts of energy of the incident wave field prescribed by the amplitude of the cylinder oscillations. For comparison, the dye distributions are shown in Fig. 11 after an equal number of wave periods. For the interaction of an intense vortex (V = 5 L) with a wave with moderate energy (A cyl = 2 cm), experiment A, the mixing occurs in the region r ∈ [5, 8], far away from the vortex core for which the radius R max is predicted to be equal to 2.7 cm by the Lamb-Oseen vortex model. For the interaction of a relatively weak vortex (V = 4 L) with more energetic incident waves (A cyl = 3 cm), experiment B, we observe a displacement of the mixed region to the region r ∈ [2.5, 7], closer to the vortex core. Here, the vortex radius R max is predicted to be equal to 3.0 cm. The broadening of the mixed region is due to the larger amplitude of the incident wave field, which provides a larger amount of energy to the mushroom-like structures associated with the wave breaking (the energy of the trapped waves in experiment B amounts about twice of that in experiment A). Besides, since the dynamical barrier of the vortex in experiment B is weaker than in experiment A (the vortex has a smaller circulation), the convective structures penetrate more deeply into the vortex structure.

Conclusions

We have investigated the impact of breaking waves on the dynamics and transport properties of a tall vortex. A planar wave propagating against the flow deforms and wraps around the vortex core, while the energy accumulates in the vortex periphery until its amplitude is high enough to trigger overturning and breaking. We have shown that this wave breaking may lead to a deposit of negative momentum in the periphery of the vortex. This deposit of momentum modifies the velocity profile of the vortex from a Lamb-Oseen vortex to an isolated vortex with a positive core embedded in a negative vorticity ring, and experimentally demonstrates an energy transfer from the wave field to the vortex field.

We characterize two distinct mixing processes. The periodic wave field destabilizes resonant trajectories of the vortex, leading to local chaotic mixing; the radial transport of dye is limited to a narrow region of the vortex. The breaking of waves leads to convective mushroom-like patterns; the radial penetration of these structures into the vortex core depends on the amount of energy provided by the trapped waves.

Fig. 1 .

 1 Fig. 1. Sketch of the experimental set-up.

Fig. 2 .

 2 Fig. 2. Vertical vorticity (a) and horizontal divergence field (b) of waves propagating into the vortex flow, obtained from PIV measurements at mid-level in the fluid. The waves are generated at the right near the bottom of the tank and propagate from below to the left, towards the vortex. Experimental parameters: f = 0.4 s -1, N = 1.5 s -1 , V = 3.2 L, D = 5.0 cm, A cyl = 2.0 cm and ω abs = 0.53 rad s -1 . The measurement is taken at t = 45 s after forcing, and R max ≈ 2.5 cm and U max ≈ 2.5 cm s -1 , yielding Γ/2π ≈ 8.3 cm 2 s -1 .

Fig. 3 .

 3 Fig. 3. Azimuthal velocity profiles ,U θ , measured at mid-depth for (A) an experiment without waves and (B) an experiment with wave breaking. Experimental parameters: f = 0.5 s -1 , N = 1.5 s -1 , V = 3.5 L, D = 5.0 cm, A cyl = 2.0 cm and ω abs = 0.8 rad s -1 . The times of measurement are indicated below the figures, t = 0 s corresponding to the beginning of the PIV acquisition, 15 s after removal of the perforated tube. Waves are generated between t = 12 and 78 s, and break around t = 120 s in experiment B. The lines represent the Lamb-Oseen vortex, given by (2).

Fig. 5 .

 5 Fig. 5. Top-view dye-visualisation of the trapping and breaking of the wave in the vortex for t = (a) 15 s, (b) 65 s, (c) 103 s and (d) 124 s after removal of the tube. Experimental parameters: f = 0.5 s -1 , N = 1.3 s -1 , V = 3.0 L, D = 5.0 cm, A cyl = 3.0 cm, ω abs = 0.85 s -1 . Waves are generated at t = 34 s near the bottom of the tank and propagate from right to left.

Fig. 6 .

 6 Fig. 6. Side-view dye-visualisation of the trapping and breaking of the wave for (a) 18 s, (b) 59 s, (c) 70 s and (d) 120 s after the removal of the tube. Waves are generated at t = 24 s from the right near the bottom of the tank and propagate upwards to the left, towards the vortex. Experimental parameters: f = 0.5 s -1 , N = 1.21 s -1 , V = 5.0 L, D = 5.0 cm, A cyl = 2.0 cm, ω abs = 0.9 s -1 .

Fig. 7 .

 7 Fig. 7. Numerical simulation of the transport of passive tracers in a periodically perturbed vortex with (a) results for an initial distribution of passive tracers along a vertical line and (b) results for an initial distribution along the resonant trajectory that satisfies n = T t /T w = 2 (circle near r = 2). The background flow is given by expression (2) with fixed R(t) = 1 and Γ = 2π. The periodic perturbation is given by expression (3) with m = 2, ω = 0.5, k = 2.0π and = 0.01. The period of the perturbation reads T w = 2π/ω and the period along a trajectory, T t , is given by T t = 2πr/U θ . their angular position, θ e , satisfies θ e = θ 0 + k z z/m. Since the zigzag patterns form around elliptic points, they are located along the same trajectory and generate the inclined folding of the dye sheet observed in the experiments.

Fig. 9 .

 9 Fig. 9. Time sequence of the wave trapping, with wave frequency, T w = 7 s for the experiment shown in Fig. 6. The time sequence begins at t = 30 s.

Fig. 10 .

 10 Fig.10. Evolution of a mushroom-shaped structure generated during the wave breaking crossing the vertical laser plane at t = 96, 96.6, 97.3 and 98 s after the removal of the tube. The spatial scale is given by the thick vertical line that corresponds to a length of 2.3 cm. Experimental parameters: see Fig.6.

Fig. 11 .

 11 Fig. 11. Side view dye-visualization of the mixing induced by the wave breaking for experiment A with moderate waves breaking into an intense vortex, and for experiment B with strong waves breaking into a relatively moderate vortex. The experimental parameters are (A) f = 0.5 s -1 , N = 1.21 s -1 , V = 5.0 L, D = 5.0 cm, A cyl = 2.0 cm, ω abs = 0.8 s -1 and (B) f = 0.5 s -1 , N = 1.16 s -1 , V = 4.0 L, D = 5.0 cm, A cyl = 3.0 cm, ω abs = 0.9 s -1 . The waves were generated at t = 24 s in experiment A and t = 60 s in experiment B after the removal of the tube, and the dye distribution observed at, respectively, t = 124 s and t = 160 s, 100 s after the onset of wave generation. The thick crosses of height and width 4 cm indicate the spatial scale.

Acknowledgements

The authors greatly acknowledge the help of Pierre Carecchio for his technical assistance with the experiments. This research was financed by PATOM and INSU contract ACI-PCN2002.