István Szalai

Patrick De Kepper

Pattern formation in the ferrocyanide-iodate-sulfite reaction: The control of space scale separation

We revisit the conditions for the development of reaction-diffusion patterns in the ferrocyanideiodate-sulfite bistable and oscillatory reaction. This hydrogen ion autoactivated reaction is the only example known to produce sustained stationary lamellar patterns and a wealth of other spatiotemporal phenomena including self-replication and localized oscillatory domain of spots, due to repulsive front interactions and to a parity-breaking front bifurcation ͑nonequilibrium Ising-Bloch bifurcation͒. We show experimentally that the space scale separation necessary for the observation of stationary patterns is mediated by the presence of low mobility weak acid functional groups. The presence of such groups was overlooked in the original observations made with hydrolyzable polyacrylamide gels. This missing information made the original observations difficult to reproduce and frustrated further experimental exploitation of the fantastic potentialities of this system. Using one-side-fed spatial reactors filled with agarose gel, we can reproduce all the previous pattern observations, in particular the stationary labyrinthine patterns, by introducing, above a critical concentration, well controlled amounts of polyacrylate chains in the gel network. We use two different geometries of spatial reactors ͑annular and disk shapes͒ to provide complementary information on the actual three-dimensional character of spatial patterns. We also reinvestigate the role of other feed parameters and show that the system exhibits both a domain of spatial bistability and of large-amplitude pH oscillations associated in a typical cross-shape diagram. The experimental method presented here can be adapted to produce patterns in the large number of oscillatory and bistable reactions, since the iodate-sulfite-ferrocynide reaction is a prototype of these systems.

In a large variety of natural and artificial systems, the development of spatial structures results from the interplay of nonlinear processes operating on different time and length scales. Chemical processes commonly proceed through competing sequences of elementary steps that operate at very different rates and may involve species with very different molecular weights, thus with eventual different diffusion coefficients in solution. Accordingly, nonlinear chemical reaction-diffusion systems naturally serve as paradigms for nonequilibrium self-organization phenomena. In particular, they are often thought to offer simplified versions of time and space patterning mechanisms at work in biological systems. [1][START_REF] Meinhardt | Models of Biological Pattern Formation ͑Academic[END_REF][START_REF] Murray | 4 Oscillations and Traveling Waves in Chemical Systems[END_REF][4][START_REF] Kapral | Chemical Patterns and Waves[END_REF][START_REF] Epstein | An Introduction to Nonlinear Chemical Dynamics ͑Oxford[END_REF] Reactions leading to such patterning phenomena rely on competing chemical activation and inhibition kinetic pathways. [4][START_REF] Kapral | Chemical Patterns and Waves[END_REF][START_REF] Epstein | An Introduction to Nonlinear Chemical Dynamics ͑Oxford[END_REF] During the past three decades, more than a dozen families of aqueous solution reactions, with hundreds of variants, were discovered to have the above required kinetic properties. [START_REF] Epstein | An Introduction to Nonlinear Chemical Dynamics ͑Oxford[END_REF] In well stirred solutions, these reactions can lead to steady-state multiplicity, periodic, or chaotic oscillations when they operate far from their thermodynamic equilibrium. 4, [START_REF] Epstein | An Introduction to Nonlinear Chemical Dynamics ͑Oxford[END_REF] These phenomena and the underlying instabilities at their origin are currently well documented. [START_REF] Murray | 4 Oscillations and Traveling Waves in Chemical Systems[END_REF] In parallel, there has also been progress in the associated experimental problem of the formation of chemical spatial patterns through reaction-diffusion instabilities, but in a more limited way. Following the advancement of open spatial reactors, in which the reaction processes could be maintained at a controlled distance from thermodynamic equilibrium, sustained traveling-wave patterns have been observed in an increasing number of different reactions. [START_REF] Noszticzius | [END_REF][8][9][10][11][12][13] Beside waves, the spontaneous development of stationary reaction-diffusion patterns was predicted by the British mathematician Alan Turing 1 more than 50 years ago. Such stationary concentration patterns typically develop in systems in which the competing activatory and inhibitory kinetic processes act on different space scales due to appropriate differences in the diffusion coefficients of species. In two variable kinetic systems, the onset of Turing patterns requires that the effective diffusion coefficient of the activator be sufficiently smaller than that of the inhibitory species, i.e., "short-range activation." 1,[START_REF] Murray | 4 Oscillations and Traveling Waves in Chemical Systems[END_REF][START_REF] Epstein | An Introduction to Nonlinear Chemical Dynamics ͑Oxford[END_REF] In more recent years, it has been shown that stationary patterns could also be generated by interactions between traveling fronts in bistable [START_REF] Hagberg | Chaos 4, 477[END_REF]15 and excitable 16 systems. As in the previous case, the space scale separation of the kinetic processes plays a central role in the onset of those stationary patterns. Though not sufficient, the control of the relative mobility of species is a key factor in the quest for stabilized spatial patterns in a reacting solution. Two main approaches have been developed to selectively control the effective diffusivity of species: (i) The addition of uniformly distributed immobile (or reduced mobility) functional sites able to reversibly bind a targeted species. 17,18 (ii) The use of finely dispersed multiphase systems where some species stay trapped in droplets of the minority phase, while others can diffuse in the continuum phase. 19 Whether they result from a Turing instability or from front pairing interactions, up to now stationary patterns have been observed only in three different chemical systems: the chlorite-iodide-malonic acid (CIMA) reaction family, 8 the ferrocyanide-iodate-sulfite (FIS) reaction, 9,10 and in the Belousov-Zhabotinsky reaction dispersed in a water-in-oil reverse microemulsion. 20 However, this multiphase technique seems to be restricted to those systems that can exhibit long-lived transient oscillatory behavior in a batch reactor, a kinetic property only fulfilled by less than a handful of reactions among the hundreds of well documented oscillating and bistable reactions. 4,[START_REF] Epstein | An Introduction to Nonlinear Chemical Dynamics ͑Oxford[END_REF] Based on the first approach for the control of the relative diffusion of species and on the systematic examination of front interactions in spatial bistable systems, we have proposed an orderly method to discover stationary patterns in chemically different systems. 21 This led us to revisit the mechanisms at the origin of pattern formation in the FIS reaction, a chemical system prototypic of a large class of chemical oscillating reactions but which was left aside in recent experimental developments.

INTRODUCTION

The first clear experimental observation 8 of Turing patterns was made by using the chlorite-iodide-malonic acid 22 ͑CIMA͒ family of reactions. The chemical 23 and physical mechanisms 17,18 at play in the patterning processes of this system are well understood and controlled. 17,[24][25][26] Soon after the initial discovery, several teams could reproduce and expand the original observations. Many aspects of patterns were studied: one-, 24 two-, 25 and three-dimensional [27][28][29] stationary periodic patterns, interaction between the Turing instability and the oscillatory Hopf instability, 30 localized patterns, 31 pattern growth dynamics, 26 and the effect of temporary or permanent spatiotemporal forcing on the pattern mode selection and on the wavelength. 32 In this system, it was theoretically proposed 17 and experimentally demonstrated 24,26 that the required short-range activation is provided by complexing agents, such as starch or poly͑vinyl͒-alcohol, that reversibly bind the iodide ions, the activatory species. Much more recently, the Belousov-Zhabotinsky reaction dispersed in a water-in-oil reverse microemulsion was shown to also produce Turing and other types of patterns, including packet waves, dash waves, and segmented antispirals. 20 Though quite complex, the transport and kinetic mechanisms at work in this system are reasonably well understood. However, this multiphase approach to the development of chemical patterns seems to be restrained to reactions that can exhibit long transient oscillatory dynamics in a batch reactor, a peculiar kinetic property very seldom observed in nonbiological systems and thus not exportable to the large majority of other documented oscillatory reactions. The majority of the chemical reactions that produce oscilla-tions work only in continuous ͑flow͒ stirred tank reactors ͑CSTR͒.

Soon after the discovery of Turing patterns in the CIMA systems, new types of stationary and dynamic patterns were discovered in the acid autoactivated ferrocyanide-iodatesulfite ͑FIS͒ reaction. 9,10 The FIS reaction is a typical CSTR oscillator. The gross kinetic mechanism of this reaction is known. [33][34][35][36] It can be described by two composite processes, where a hydrogen ion driven autocatalytic oxidation of hydrogen sulfite ͓͑1͒ positive feedback͔ is coupled with a hydrogen ion consuming oxidation of ferrocyanide ͓͑2͒ negative feedback͔,

IO 3 -+ 3HSO 3 -→ 3SO 4 2-+ I -+ 3H + , ͑1͒ IO 3 -+ 6H + + 6͓Fe͑CN͒ 6 ͔ 4-→ I -+ 6͓Fe͑CN͒ 6 ͔ 3-+ 3H 2 O.

͑2͒

Although, reaction ͑1͒ is autocatalytic both for protons and iodide ions, Rabai and co-workers assert that the kinetic role of the latter ones is negligible in unbuffered systems. 36 The main activation is provided by the free protons.

A large variety of spatial patterns 9,10,37,38 arising from planar front multiplicity, front morphological instabilities, and front interactions were found in the FIS system. In particular, stationary labyrinthine patterns, self-replicating spots, and bouncing waves were observed when this reaction is operated in an open one-side-fed reactor ͑OSFR͒. Numerical studies based on a two-10 and later on a four-variable kinetic model revealed that most of the observed spatial patterns could be qualitatively accounted for by assuming that the diffusivity of hydrogen ions is less than that of the hydrosulfite ions, which conversely act as the main dynamic negative feedback species. 37 However, there was no tested explanation as to how this assumption could be fulfilled. A priori, protons are the fastest diffusing ions in aqueous solutions. In a recent short report, 39 we showed that in an agarose gel reactor, a number of original observations, in particular the development of stationary patterns, could not be reproduced if a supercritical amount of poly͑acrylate͒, a charged polymer, was not introduced in the gel network. It is well established that due to the fast protonation equilibrium, weak acid functions with very low mobility can considerably decrease the effective diffusivity of protons. 40,41 Though the presence of such low mobility acid functions was never considered, we argue that the same effect was provided by the hydrolysis of the poly͑acrylamide͒ gel used in the original experiments but in an uncontrolled way. The more or less rapid degradation of amide functions to carboxylate functions, especially in alkaline aqueous solutions, is well known. 42,[START_REF] Sandler | Polymer Synthesis ͑Academic[END_REF] In our approach, the low mobility carboxylate functions are provided by long acid polymer chains that impregnate the gel network, the concentration of which is controlled by the feed composition of the CSTR. This makes the complex pattern dynamics observed earlier easier to reproduce and their development more comprehensive. Here, we present a more detailed description of this renewed approach to the spatiotemporal patterns observed in this reaction and their dependence on different control parameters.

EXPERIMENTAL TECHNIQUES

Two geometries of OSFR were used in the present experiments ͑Fig. 1͒: The first consists of a transparent thin disk ͑22 mm diam and 0.75 mm thick͒ made of 4% agarose gel ͑Fluka, BioChemika 05070͒. One face of the disk is in direct contact with the contents of a CSTR through a circular hole ͑18 mm diam͒ in a mask that holds the gel tightly immobilized against a flat observation window. Beyond the contact surface, the disk extends ͑2 mm͒ under the mask ͑for details, see Ref. 25͒. Patterns are observed by transparency and are monitored by a CCD video camera, connected to a video tape recorder and to a black and white frame grabber board. To enhance the contrast of the transmitted light patterns, a stabilized soft orange light illumination was shone opposite to the camera. The observations bring information on the integrated light absorption patterns across the thickness of the disk of gel but not on how these changes organize within the thickness. This complementary information is provided by a second OSFR geometry. It consists in a flat annular gel reactor ͑w = 1 mm deep and 25 mm outer diameter͒ ͑for details, see Ref. 13͒. All the faces of the annulus are tightly pressed against impermeable walls except the outer edge, which is in direct contact with the contents of a CSTR. The observations, in reflected light, are made in a direction orthogonal to the feed surface and thus allow solving the color changes in the feed direction. This geometry of OFSR is self-sustained and offers periodic boundary conditions. It is free of boundary limit perturbations. In this respect, it mimics an infinite boundary system. However, in a series of experiments, an impermeable mask was glued over a small part of the outer rim of the annulus to mirror the edge effect that is present in the disk OSFR. Note that both reactors are confined in the direction of the feed, but while the disk reactor is extended in the two other directions ͑apparent twodimensional system͒, the annular reactor is extended only in one other direction ͑apparent one-dimensional system͒. All reported experiments were made at 30 °C with a residence time of the CSTRs of = 220 s. The chemical solutions were stored in three separated reservoirs and injected by precision pumps ͑Pharmacia P500͒ with equal flow rates at the base of the CSTRs. They enter through a single port in the close vicinity of a turbine rotating at 1000 rpm. An overflow pipe maintains the reaction volume constant and at room pressure, with no air interface. The respective reservoirs contain the following: ͑1͒ potassium iodate ͑Fluka͒; ͑2͒ sodium sulfite ͑Fluka͒, potassium ferrocyanide ͑Acros Organics͒, and sodium poly͑acrylate͒ ͑Aldrich, MW:15 000 Dalton͒; ͑3͒ sulfuric acid ͑Roth͒. In the disk OSFR experiments, bromothymol blue ͑Acros Organics͒, a pH color indicator that switches from blue to yellow in the 7.6-6.2 pH range, was added ͑0.3 g / L͒ in reservoirs 1 and 2, while bromocresol green was used at the same weight concentration in the annular OSFR experiments. This pH indicator switches from blue to light yellow in the 5.4-3.8 pH range. All the chemicals are analytical grade and solutions are prepared in deionized water. The feed concentration of potassium iodate ͓KIO 3 ͔ 0 and sodium sulfite ͓Na 2 SO 3 ͔ 0 was fixed at 75 and 89 mM, respectively. Here ͓ ͔ 0 denotes the concentration that the input species would have after mixing and prior to any reaction. To explore the dynamics of the system, we varied the feed concentration of ferrocyanide ͓K 4 Fe͑CN͒ 6 ͔ 0 , sulfuric acid ͓H 2 SO 4 ͔ 0 , and poly͑acrylate͒ ͓PA͔ 0 in the ranges 0 -20, 0 -4.17, and 0 -4 mM, respectively. The value of ͓PA͔ 0 corresponds to the concentration of carboxylic groups in the feed solution introduced by the poly͑acrylate͒ polymer chains. These polyacid chains slowly diffuse in the agarose gel where their asymptotic concentration is expected to be close to that of the CSTR solution. For each new PA feed concentration, the gel parts of the reactors were left to impregnate in a solution with the targeted value, for at least 24 h.

EXPERIMENTAL RESULTS

Let us first describe the CSTR dynamics. In the range of chemical feed parameter used in this paper, the CSTR content exhibits bistability between a high pH ͑8-6͒ and a low pH ͑3.5-4.0͒ steady-state branch. They correspond, respectively, to the so-called "flow" and "thermodynamic" branches. 33 On the flow branch, the extent of the reaction is small and the concentration of the input species generally does not differ much from the composition in the input flow, while on the thermodynamic branch the extent of the reaction is large. The switch from one branch to the other depends sensitively on the sulfuric acid feed concentration ͓͑H 2 SO 4 ͔ 0 ͒, which is conveniently used as the tunable control parameter throughout the experiments. Introducing ferrocyanide, the antagonist feedback species, decreases the range of steady-state bistability in the CSTR. Above a critical ferrocyanide feed concentration, the CSTR content may In this paper, the content of the CSTR was always maintained on the stationary state branch with the lowest extent of reaction, i.e., the high pH flow branch.

Spatial bistability, oscillations, and front interaction in an annular OSFR

In Fig. 2, we present a section of phase diagram in the ͓͑K 4 Fe͑CN͒ 6 ͔ 0 , ͓H 2 SO 4 ͔ 0 ͒ plane for the annular OSFR. The phase diagram was established by gradually changing ͓H 2 SO 4 ͔ 0 for different fixed values of the ͓K 4 Fe͑CN͒ 6 ͔ 0 . Different spatial states can be observed in the gel depending on ͓H 2 SO 4 ͔ 0 and on the initial state of the gel contents. In the absence of ferrocyanide at ͓H 2 SO 4 ͔ 0 = 0 mM, the gel is in a quasiuniform dark color state ͓Fig. 3͑A͔͒, the F-state of the OSFR. In the pictures, the dark and light gray colors correspond, respectively, to high and low pH compositions in gel. The F-state of the gel is stable up to ͓H 2 SO 4 ͔ 0 = 2.38 mM. A further increase of ͓H 2 SO 4 ͔ 0 to 2.68 mM makes the innermost part of the annulus suddenly turn clear. A stable pHindicator color switch settles parallel to the rim of the annulus, as illustrated in Fig. 3͑B͒; this is the mixed state M of the OSFR. Now, decreasing ͓H 2 SO 4 ͔ 0 , the M-state remains stable down to 1.68 mM. Lowering ͓H 2 SO 4 ͔ 0 to 1.04 mM makes the contents of the annulus return to the uniformly dark F-state. The stability range of the F and the M states of the gel overlaps between ͓H 2 SO 4 ͔ 0 = 1.68 and 2.38 mM; this is spatial bistability. [START_REF] Blanchedeau | [END_REF] In the range of spatial bistability, an appropriate perturbation can create interfaces between the M and the F states. The direction of the propagation of the interface depends on the actual value of ͓H 2 SO 4 ͔ 0 . Clear differences can be observed between the acid producing ͑+͒front ͑transition from F-to M-state͒ and the acid consuming ͑-͒front ͑transition from M-to F-state͒. A ͑+͒front exhibits a small overshoot of the radial extension of the acid core while the acidic part is narrower in the vicinity of a ͑-͒front ͑Fig. 4͒.

The extent of the domain of spatial bistability decreases as ͓K 4 Fe͑CN͒ 6 ͔ 0 is increased and vanishes at a critical value ͓͑K 4 Fe͑CN͒ 6 ͔ 0 =10 mM͒. Let us describe the changes of states above this limit, e.g., at ͓K 4 Fe͑CN͒ 6 ͔ 0 = 15 mM. Starting from ͓H 2 SO 4 ͔ 0 = 0, the content of the gel is in the F-state until ͓H 2 SO 4 ͔ 0 = 2.92 mM. A further increase to ͓H 2 SO 4 ͔ 0 = 3.09 mM makes the innermost part of the gel suddenly become acidic. As above, a mixed state appears but now the switching position of the color indicator is no longer stable but moves back and forth across the width w of the annulus, in a way somewhat similar to what was observed with the iodate-sulfite reaction. 13 The domain of existence of this oscillatory state interchanges with the domain of spatial bistability through a typical cross-shaped phase diagram topology 45 ͑Fig. 2͒.

We show later that in this system, pattern developments are mainly governed by front interactions. One special feature of the front dynamics in the disk OSFR is the "bouncing" of ͑+͒fronts as they approach the rim of the circular mask. To understand this bouncing phenomenon, we have glued a mask over a small part of the annular OSFR. In the parts of the gel under the mask, the extent of the reaction is naturally greater than in the parts directly in contact with the contents of the CSTR. The composition under the mask turns acidic before the rest of the gel. However, as illustrated in Fig. 4, the M-state does not propagate from under the mask but develops at a distance. The observations are summarized in Fig. 4. A ͑+͒front approaches the mask, slows down, stops, and then reverses direction. At the same time, the acid part under the mask extends slightly and then recedes. The propagation of the ͑+͒front ͑ϳ1.6 mm/ min͒ is faster than that of the ͑-͒front ͑ϳ0.5 mm/ min͒.

Introducing PA, as a low mobility proton binding agent, changes significantly the dynamics of the system. Figure 5 shows the interaction of a front with the composition under the masked area of the annular reactor, when ͓PA͔ 0 = 4 mM. Starting with the annular reactor uniformly in the F-state at ͓H 2 SO 4 ͔ 0 = 2.92 mM, an increase to ͓H 2 SO 4 ͔ 0 = 3.33 mM leads to the formation of a ͑+͒front, which approaches the masked area and bounces, in a somewhat similar way to that shown in Fig. 4. However, the initial rate of approach and withdrawal of the ͑+͒front and ͑-͒front are a factor 3 slower than in the absence of PA. Then, the F/M-state interface reverses once more and approaches again the masked area, whereas at the same time the acidic zone under the mask outstretches again. The F/M-state interface finally stops at a finite distance from the edge of the mask. A stable F-state pulse ͑ϳ0.9 mm wide͒ forms in between two M-state regions ͑Fig. 5͒. These observations show that by introducing PA, a nontrivial stationary structure may develop spontaneously. Under these feed conditions, we did not observe the formation of other pulses far from the edge of the mask.

Fronts and labyrinthine patterns in a disk OSFR

The disk OSFR is characterized by a higher dimensionality where the curvature of fronts in the plane parallel to the feed can play an important role in the dynamics of the system.

Let us start again with observations made in the absence of PA. Consistently with the phase diagram in Fig. 2, the dynamic and patterning capacities of the system depend critically on the ͓K 4 Fe͑CN͒ 6 ͔ 0 value. For ͓K 4 Fe͑CN͒ 6 ͔ 0 Յ 10 mM, only the uniform steady states F or M are asymptotically observed as a function of ͓H 2 SO 4 ͔ 0 , whereas for ͓K 4 Fe͑CN͒ 6 ͔ 0 = 20 mM, a value leading to M-state oscillations in Fig. 2, more or less complex traveling domain patterns delimited by ͑+͒fronts and acid ͑-͒fronts, can be observed. In Fig. 6, the ͑+͒ and ͑-͒fronts are distinguished, respectively, by a sharp change to a bright M-state braid and to a darker F-state zone. While distinctive colors can be seen with the naked eye in the state of the disk ͑blue for the F-state and brownish blue for the M-state͒, these two states cannot always be discriminated by their gray level but their interfaces always exhibit a sharp switch in the gray level, as explained above. Starting with the gel reactor uniformly in the F-state at ͓H 2 SO 4 ͔ 0 = 2.80 mM, a supercritical increase of ͓H 2 SO 4 ͔ 0 to 2.92 mM leads to the spontaneous growth of an M-state domain at pacemaker regions close to the rim of the mask but never in direct contact with the contents of the gel under the mask, as was the case in the annular OSFR experiments. The M-state domain starts to expand in all directions, but this expansion stops at a finite distance from the mask. The sign of the front reverses, and leads to a recovery of the F-state. After some time, a new patch of M-state grows. As a result, except for a narrow ring area ͑ϳ1.3 mm wide͒ in the vicinity of the mask, every space point of the disk undergoes large-amplitude pH oscillations with a period of about 16 min. This behavior can be related to the periodic "shrinking ring" phenomenon reported in the original publications. 38 However, in our case the domain oscillations become more complex due to different possible front interactions. Fronts of the same sign annihilate on head-on collision. A head-to-back interaction between fronts of opposite sign can cause the ͑+͒front to change sign, while the connection of fronts of opposite sign along the same M-state domain usually leads to the development of domain states rotating around this singularity, i.e., spiraling domain growths ͑Fig. 7͒. When the connection occurs in the close vicinity of the M-state excluded area next to the rim of the mask, the rotation is impeded and the singularity drifts along the above quiescent area. Remarkably, at high enough ͓H 2 SO 4 ͔ 0 and still in the absence of PA, a repulsion develops between ͑-͒fronts and leads to the formation of more or less long-lived "filaments" of the M-state which eventually swell again as the ͑-͒fronts revert to ͑+͒fronts. This is seen when ͓H 2 SO 4 ͔ 0 is increased to 3.09 mM in the two lower snap- shots in Fig. 8. Another general observation, in the domain of parameter explored, is the fact that contrary to the M-state, which can spontaneously emerge inside an F-state domain, F-state "bubbles" never spontaneously form in the middle of an M-state region. Patchy F-state patterns such as that in Fig. 8 are the result of F-state trapping between growing domains of M-state, as seen in the first two snapshots in Fig. 8. In this trapping mechanism, an F-state bubble survives only if surrounded by a ͑-͒front and if it is large enough. Let us continue the description of pattern development at ͓K 4 Fe͑CN͒ 6 ͔ 0 = 20 mM, a feed value at which oscillations are observed in the absence of PA. When ͓PA͔ 0 is increased to 0.5 mM, dynamical patterns similar to those described above are observed. However, when ͓PA͔ 0 = 1 mM, stationary lamellar patterns can be observed in addition ͑see Fig. 2E in Ref. 39͒. This is exemplified in Fig. 9, which shows the formation of a stable M-state network pattern at ͓PA͔ 0 = 2 mM: After changing ͓H 2 SO 4 ͔ 0 from 2.67 mM, a value for which the disk is in the uniform F-state, to ͓H 2 SO 4 ͔ 0 = 2.98 mM, a stationary pattern develops with the initial spontaneous appearance of an M-state domain, which grows, bounces at the rim, and starts to form persistent M-state filaments. Once they are formed, these filaments contribute to their further invasion by different oscillatory strategies. A free tip may shrink and swell while other parts of the filaments develop a breathing dynamics that makes them move and bud when no other front is at a close critical distance. This mechanism is continuous until the mutual "repulsive" interaction between the filaments freezes the space-filling dynamic. A stationary lamellae pattern with a relatively narrow wavelength distribution settles 2 h after the supercritical feed change.

Starting with the above stationary pattern, if ͓H 2 SO 4 ͔ 0 is increased further to 3.09 mM, the relative stability of the M-state increases at the expense of the F-state and the M-state becomes the majority state, as shown in Fig. 10. After this ͓H 2 SO 4 ͔ 0 jump, many F-state bubbles are trapped and the dynamic of the pattern is then ruled by that of the F-state bubbles. These may divide, pulsate, and eventually die by overgrowth with the spontaneous development of the M-state in their middle or a local asymmetric commutation of a ͑-͒front to a ͑+͒front, which deforms the circular symmetry of the bubble. The system no longer settles into a stationary pattern mode. Note that the spot pattern is inserted in a large discoid M-state domain disconnected from the mask ͑except at one point, at 12 o'clock in the snapshots-Fig. 10, probably due to some local "boundary imperfection"͒. The diameter of this discoid M-state domain undergoes small-amplitude oscillations with a period of about 24 min, similar to that of the oscillatory F-state spots. The sequence of snapshots in Fig. 11 details the breathing dynamics of a small F-state spot of Fig. 10. Note the color changes inside the spot when it grows or shrinks, indicating that the pH decreases when the spot shrinks and vice versa.

We have also explored the effect of ferrocyanide feed concentration on pattern formation in the disk OSFR. For simplicity reasons, let us only consider the case of stationary patterns at ͓PA͔ 0 = 4 mM and ͓H 2 SO 4 ͔ 0 = 2.86 mM. In these conditions, the gel is in the stationary F-state when ͓K 4 Fe͑CN͒ 6 ͔ 0 Ͻ 5 mM. At ͓K 4 Fe͑CN͒ 6 ͔ 0 = 5 mM, the gel switches to the stationary uniform M-state; no nontrivial patterns are observed. Above a critical ͓K 4 Fe͑CN͒ 6 ͔ 0 around 10 mM, the M-state become unstable and a stationary M-state labyrinthine pattern forms, as illustrated in Fig. 12͑a͒ for ͓K 4 Fe͑CN͒ 6 ͔ 0 = 15 mM. The area covered by the M-state and the interconnections in the network increase with increasing ͓K 4 Fe͑CN͒ 6 ͔ 0 ͓Fig. 12͑b͔͒. At ͓K 4 Fe͑CN͒ 6 ͔ 0 = 30 mM, similarly to observations made at high ͓H 2 SO 4 ͔ 0 ͑Fig. 9͒, the stationary network pattern loses stability and the system is dominated by the birth and death dynamics of F-state spots ͓Fig. 12͑c͔͒. Test experiments were also made as a function of temperature. At ͓PA͔ 0 = 2 mM and ͓K 4 Fe͑CN͒ 6 ͔ 0 = 20 mM, stationary lamellae patterns could be observed at 22 and 33 °C by adjusting the value of ͓H 2 SO 4 ͔ 0 . The pattern parameter domain shifts to higher acid feed with decreasing temperature. Noteworthy, at 22 °C, no CSTR oscillation is observed but only bistability. 33,34

DISCUSSION

In this paper, we revisit the conditions for the observations of reaction-diffusion patterns when operating the ferrocyanide-iodate-sulfite reaction in a one-side-fed spatial reactor. Despite the fantastic patterning capacity of this system ͑due to morphological front instabilities and an Ising-Bloch front bifurcation͒, it was left dormant for about 15 years, contrary to the case of the CIMA reaction. Due to some unnoticed and thus uncontrolled charge contents of the gels, the original experimental observations were difficult to reproduce. Lets us point out the similarities and differences between the original and our present experimental method. Both sets of experiments were made in the same range of chemical feed parameters and temperature. The CSTR residence time in the original experiments ranged from 36 to 240 s, which is shorter by a factor 6 or equivalent to ours. Conversely, the volume of our stirred tank reactor is significantly greater to minimize feedback effects of the contents of the gel on the actual composition of the CSTR-i.e., to minimize global coupling effects. In this type of reactor, global coupling cannot be totally avoided but in the present experiments it does not drive the overall pattern dynamics. 46 The main differences come from the polymer gels used in the two sets of experiences and the way they are set in contact with the CSTR contents. The original experiments used 16% weight polyacrylamide gels films ͑0.2-0.6 mm thick͒ in contact with the CSTR contents through an aluminum oxide and a nitrocellulose membrane to provide, respectively, structural rigidity and a white backing for visualization. We use no such composite of porous membrane with unknown chemical adsorption properties. In our experiments, a single porous material is used-agarose ͑4% weight͒-a better characterized gel that should make experimental reproducibility easier. Our gels are thicker ͑w = 0.75 mm for the disk geometry OSFR and 1.0 mm in the annular one͒. Pattern development can be very sensitive to the thickness w as shown in the original set of publications 38 and in recent studies with other bistable systems. [47][48][49] Li et al. 38 report observation of no patterns for gels thicker than 0.4 mm, which is not our case. However, because the acrylamide and agarose gels have very different structures and because of the presence of additional membranes in the original experiments, it is difficult to compare the effective thickness-i.e., characteristic exchange time between the feed surface and the opposite face of the gel reactors-in the old and new sets of experiments. Further studies on size effect would be required to understand some of the differences between the two sets of experiments. There are different theoretical approaches in the literature to describe the formation of labyrinthine patterns and spot division, [START_REF] Hagberg | Chaos 4, 477[END_REF]50 but the presence of a long-range inhibition and correlatively a short-range activation is always considered to be an important condition. We argue that in original experiments, as in the ones introduced here, an effective sufficiently short activation range was mediated by supercritical concentrations of low mobility "weak" acid functions. Here, the low mobility protonable functions are mostly provided by poly͑acrylate͒ chains directly fed at controlled concentration in the CSTR, while in the original experiments they were provided by the uncontrolled hydrolysis of the polyacrylamide gel itself. It is known that the amide functions can be readily hydrolyzed into carboxylate functions in alkaline solution. 42,[START_REF] Sandler | Polymer Synthesis ͑Academic[END_REF] This hydrolysis is favored by the increase of alkalinity of the solutions but also by the presence of different ions. In standard polyacrylamide gels, the amide units are in molar concentration and only a few tenths of a percent need to be hydrolyzed to generate supercritical amounts of carboxylate functions. It was also reported 9 that similar patterns could be observed using agar gels but that these gels would rapidly degrade. Agar gels are heavily charged gels; they are a mixture of about 70% agarose ͑uncharged polysaccharide͒ and 30% agaropectin ͑charged sulfated polysaccharides including D-glucuronic and pyruvic acids͒. The effective diffusivity of protons would naturally be quite low in such gels. Our agarose gels do not degrade over periods of time of the order of the week and are only very slightly charged. 50 We think that it is not necessary to invoke more complex indirect mechanisms such as the immobilization of other active species such as iodine. 52 However, this qualitatively seductive hypothesis was not experimentally tested and would not account for previous observations in gels as different as polyacrylamide and agar.

We observe most of the qualitative behaviors reported in the original publications. While in the original publications stationary patterns are found in both the spatial bistability parameter region 37 and in the oscillatory region, 38 here we have observed such patterns only in the oscillatory region of the parameter. Our patterns develop for ͓K 4 Fe͑CN͒ 6 ͔ 0 Ͼ 10 mM. At this value, spatial bistability has vanished and no external perturbation is needed to trigger an interfaces between the F-and M-states. We have made no spatial light or feed perturbation by, i.e., stopping the stirring of the CSTR as in Ref. 38. The patterns develop spontaneously beyond a critical ͓H 2 SO 4 ͔ 0 value through the periodic growth of M-state domains at random pacemaker positions. Note that boundaries and pacemakers that break the uniformity of the system can be thought of as permanent extrinsic perturbations of the system. We should also mention that within our experimental acid feed steps of 0.06 mM, we observed no hysteresis in the onset of stationary or traveling patterns.

In the absence of poly͑acrylate͒, special attention should be paid to the observation of long-lived M-state filaments at high ͓H 2 SO 4 ͔ 0 . These transient filamentous structures result from the flattening of colliding ͑-͒fronts associated with the growth of circular domains of F-states. This sort of "repulsive" interaction could suggest that a long-range inhibitory and short-range activatory mechanism is already at play, inferring that the proton is already significantly slowed down. Let us examine this possibility. Besides the oxygenated iodine and the iron cyanide compounds which have diffusion coefficients of the same order of magnitude as that of the hydrosulfite ions, and thus would not make the associated protons diffuse slower than this ion, there are two possibilities: the pH color indicators that have a molecular mass around 700 D or the residual impurities in the agarose gel. 51 To provide suitable color contrasts for the detection of patterns, relatively high concentrations ͑ϳ0.3 mM͒ of pH color indicators were used, concentrations that are in the same range of magnitude as those of protons in the acid part of the M-state. However, recently the same ͑-͒front flattening was observed in the absence of a pH indicator, using polyvinylalcohol, which acts as a polyiodide ion color indicator, 53 a species that forms in the fronts. In this vein, we are left with the possible contributions of the residual charges in the agarose gels. These residual gel charges are likely provided by agaropectin compounds such as D-glucuronic and pyruvic acids, which have a pK a , respectively, of 2.9 and 2.4.

Another possibility is that on increasing ͓H 2 SO 4 ͔ 0 , the system comes very close to the nonequilibrium Ising-Bloch front bifurcation limit. In this critical parameter region, the front flattening mechanism would not necessarily require short-range activation. It is now well established [START_REF] Hagberg | Chaos 4, 477[END_REF] that in the vicinity of a nonequilibrium Ising-Bloch front bifurcation, the velocity of the fronts becomes very sensitive to intrinsic parameters such as curvature or front pair interactions. In this case, the velocity and direction of propagation of two fronts could depend critically on their distance. This latter front interaction mechanism would be supported by the observation that after some time, the filamentous structures swell again. A clear discrimination between the two possible mechanisms is probably not an easy experimental task since the two mechanisms can combine. In all cases, experiments with purer agarose gels would be necessary.

Finally, we would like to emphasize that this revival of observation of labyrinthine patterns and other involved spot dynamics in the FIS reaction, in a quite different gel-filled open spatial reactor, stems from the systematic design method that we have recently proposed to discover standing reaction-diffusion patterns in chemically different reactions. 21 The method is based on the finding of the spatial bistability phenomenon in OSFRs and the study of front interactions in these and neighboring regions of parameter space, and then selectively reducing the effective diffusivity of the main activator species by introducing controlled amounts of low mobility complexing agents. Our method was first successfully applied to the chlorine-dioxide-iodidemalonic acid ͑CDIIMA͒ reaction, 21 a spatially bistable reaction that is an extended version of the reaction that first produced 8 Turing patterns. Though the patterns observed in the CDIIMA system are not as rich as those observed in the FIS reaction, we find strong similar requirements for the development of stationary patterns. In particular, besides a critical concentration of an appropriate complexing agent, it is necessary also to introduce a critical concentration of an independent species that could act as a source or sink of the activator and the diffusivity of which is not reduced by the low mobility complexing agent. This is the case of malonic acid in the CDIIMA reaction and of ferrocyanide in the FIS reaction. In this case, it is expected that the latter species could be replaced by other proton consuming substrates. The lessons we have learned from the study of these two systems should lead the way to discovering stationary and nontrivial front dynamic patterns other than excitability waves in chemically different systems in the near future.

FIG. 1 .

 1 FIG. 1. Sketches of the reactors used in the experiments: disk-shaped OSFR ͑a͒ and annular OSFR ͑b͒, where w is the thickness of the disk and width of the annulus.

FIG. 2 .

 2 FIG. 2. Experimental nonequilibrium phase diagram in the ͓͑K 4 Fe͑CN͒ 6 ͔ 0 , ͓H 2 SO 4 ͔ 0 ͒ plane. Symbols attributed to states of the gel: ᭢ stable M state; ᭡ stable F state; b oscillatory M state. The lines indicate the limit of the spatial states.

FIG. 4 .

 4 FIG. 4. Front interaction and oscillations observed in the annular OSFR at ͓H 2 SO 4 ͔ 0 = 3.09 mM and ͓K 4 Fe͑CN͒ 6 ͔ 0 = 20 mM. The left part of the annular gel is masked, no direct contact with the CSTR. The location of the mask is indicated by a white stripe ͑a͒. The white rectangle indicates the area used in the spatio-temporal representation ͑b͒, where the sampling time is 1 min.

FIG. 5 .

 5 FIG. 5. Formation of a stable pulse in the presence of poly͑acrylate͒ ͓͑PA͔ 0 =4 mM͒ at ͓H 2 SO 4 ͔ 0 = 3.33 mM and ͓K 4 Fe͑CN͒ 6 ͔ 0 = 20 mM. The spatio-temporal representation shows the same part of the gel as in Fig. 4. FIG. 6. Traveling M-state domains in the disk OSFR at ͓H 2 SO 4 ͔ 0 = 2.86 mM; ͓K 4 Fe͑CN͒ 6 ͔ 0 = 20 mM; ͓PA͔ 0 = 0. Sampling time of snapshots 6 min.

FIG. 7 .

 7 FIG. 7. Spiraling M-state domains in the disk OSFR at ͓H 2 SO 4 ͔ 0 = 2.92 mM; ͓K 4 Fe͑CN͒ 6 ͔ 0 = 20 mM; ͓PA͔ 0 = 0. Sampling time of snapshots from top left to right and down 4 min.

FIG. 11 .

 11 FIG. 11. Breathing small F-state domain. Period of oscillations 24 min. Conditions as in Fig. 10. Sampling time of snapshots from left to right and down 4 min. Size of the frames 3.5ϫ 3.5 mm.

1054-1500/2008/18͑2͒/026105/9/$23.00 © 2008 American Institute of Physics 18, 026105-1 Downloaded 26 Aug 2011 to 147.210.83.224. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions

026105-2 I. Szalai and P. De Kepper Chaos 18, 026105 ͑2008͒ Downloaded 26 Aug 2011 to 147.210.83.224. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions

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ACKNOWLEDGMENTS

We thank J. Boissonade and P. Borckmans for stimulating discussions. I.S. acknowledges the support of OTKA ͑F049666, K67701͒ and the MTA Bolyai Fellowship. P.D.K. is supported by CNRS and the French Agence Nationale de la Recherche.


MD Francesco Barone-Adesi
email: fbaroneadesi@yahoo.it

Loredana Vizzini

Franco Merletti

Lorenzo Richiardi

It is just a game: lack of association between watching football matches and the risk of acute cardiovascular events

Keywords: Acute Myocardial Infarction, football, cardiovascular triggers Football, Acute Myocardial Infarction, Cardiovascular Triggers

It is just a game: lack of association between watching football matches and the risk of acute cardiovascular events.

Introduction

The role of trigger factors in acute cardiovascular events has been much studied in the last few years. Studies have investigated the effects of triggers acting on both the individual level, such as episodes of anger, heavy physical exertion, sexual activity and drug consumption [START_REF] Tofler | Triggering of acute cardiovascular disease and potential preventive strategies[END_REF], and on a large number of people during a limited time span, from earthquakes and armed conflicts to political and sporting events [START_REF] Bergovec | Acute myocardial infarction among civilians in Zagreb city area[END_REF][START_REF] Leor | The Northridge earthquake as a trigger for acute myocardial infarction[END_REF][START_REF] Witte | Cardiovascular mortality in Dutch men during 1996 European football championship: longitudinal population study[END_REF]. Watching sporting events could contribute to cardiovascular risk by increases in sympathetic nervous system stimulation and circulating catecholamines induced by the emotional involvements in the game, associated with other behaviours such as smoking, binge drinking and overeating [START_REF] Witte | Cardiovascular mortality in Dutch men during 1996 European football championship: longitudinal population study[END_REF].

A recent study by Wilbert-Lampen and collaborators [START_REF] Wilbert-Lampen | Cardiovascular events during World Cup soccer[END_REF] analysed changes in the rates of cardiac emergencies in Bavaria (Germany) during the last football world cup, held in Germany in 2006. The population incidence of cardiac emergencies increased by 2.7-fold during the 12 hours before and after matches involving the German team [START_REF] Wilbert-Lampen | Cardiovascular events during World Cup soccer[END_REF]. The authors claimed that this finding has relevant public health implications, which sparked a debate on the necessity of the introduction of ad-hoc preventive measures [START_REF] Wilbert-Lampen | Cardiovascular events during World Cup soccer[END_REF][START_REF]Watching football can kill you[END_REF][START_REF] Spiegel Online | Watching Football Is Hazardous to Your Health[END_REF].

We have investigated the association between football matches involving the Italian team in the last three international competitions and the rates of hospital admissions for acute myocardial infarction (AMI) among the Italian population (60 million inhabitants). Since the Italian team participated in the final phase of all of the last competitions, our study is based on a very large number of events and takes advantage of several repeated episodes of exposure. Furthermore, we evaluated the results of our study and that of Wilbert-Lampen and collaborators in the context of published studies on the association between football matches and the risk of cardiovascular events.

Methods

We studied daily hospital admissions for AMI in Italy during three international football The exposure period included the days on which the Italian team played, whereas the control period included all other days of the three competitions. We used Poisson regression to estimate the relative risk of hospital admission for AMI comparing the exposure days to the control days, for each single competition and for the three competitions combined.

All models were adjusted for day of the week. The pooled effect estimate was obtained fitting a model including tournament as a random-effect. As the comparison was made between exposed and control days within the same competition, possible confounding by long term trends and seasonal variations in incidence of AMI were inherently taken into account. As the the control period consisted of days before and after the matches, we considered unlikely that temperature and air pollution could change between control days and exposed days at such an extent to introduce bias. We checked this assumption collecting data on temperature and concentration of nitrogen Death, Sudden. Furthermore, we scrutinised all cited references from the retrieved articles. When studies reported both single-match and pooled estimates, we considered the summary estimate in our review. In addition, if only gender-specific estimates were reported, we calculated a pooled estimate for men and women combined.

Results

In total, 25 159 hospital admissions for acute myocardial infarction (AMI) occurred in the Italian population during the three football competitions. Out of these, 4,395 events occurred on 1 of the 14 days of matches involving the Italian team ( Finally, when we included in the analysis only matches disputed in the final stage of each competition, no increase in the rates of admissions for AMI was observed (RR: 0.99, 95% CI 0.94-1.06).

(Table 2 here)

Our review identified ten published articles on the association between football matches and cardiovascular events (Table 3). There were important methodological heterogeneities among these studies that hampered a quantitative summary measure of effect. Regarding the definition of the outcome, five studies analysed mortality rates for AMI (9, 10) or AMI and stroke combined [START_REF] Witte | Cardiovascular mortality in Dutch men during 1996 European football championship: longitudinal population study[END_REF][START_REF] Brunekreef | No association between major football games and cardiovascular mortality[END_REF][START_REF] Kirkup | A matter of life and death: population mortality and football results[END_REF], two studies evaluated hospital admissions with a discharge diagnosis of either AMI or stroke [START_REF] Carroll | Admissions for myocardial infarction and World Cup football: database survey[END_REF][START_REF] Bauman | The hazards of watching football--are Australians at risk?[END_REF], two studies investigated the occurrence of out-of-hospital cardiac arrests and sudden cardiac death [START_REF] Katz | Increase of out-of-hospital cardiac arrests in the male population of the French speaking provinces of Switzerland during the 1998 FIFA World Cup[END_REF][START_REF] Katz | Increase of sudden cardiac deaths in Switzerland during the 2002 FIFA World Cup[END_REF] and one study investigated hospital admissions with a preclinical diagnosis of acute cardiovascular events (AMI, unstable angina, symptomatic cardiac arrhythmia, cardiac arrest leading to cardiopulmonary resuscitation or therapeutic discharge of an implantable cardioverterdefibrillator) [START_REF] Wilbert-Lampen | Cardiovascular events during World Cup soccer[END_REF]. Regarding the definition of the exposure period, three studies evaluated the effect of a single football match (4, 9, 10), while the studies by Bauman and colleagues [START_REF] Bauman | The hazards of watching football--are Australians at risk?[END_REF], Carroll and colleagues [START_REF] Carroll | Admissions for myocardial infarction and World Cup football: database survey[END_REF], Brunekreef and colleagues [START_REF] Brunekreef | No association between major football games and cardiovascular mortality[END_REF] and Wilbert-Lampen and colleagues (5) were based on two, four, five and seven matches, respectively. (Table 3 here)

Figure 1 shows the results of the different studies plotted against their precision. In the absence of publication bias, the effect estimates are expected to be distributed in a triangular ("funnel") pattern, with less precise studies (on the right side of the graph) randomly dispersed around the mean. The two small studies by Katz and colleagues [START_REF] Katz | Increase of out-of-hospital cardiac arrests in the male population of the French speaking provinces of Switzerland during the 1998 FIFA World Cup[END_REF][START_REF] Katz | Increase of sudden cardiac deaths in Switzerland during the 2002 FIFA World Cup[END_REF] had the same study design and were carried out on the same population during two different competitions. For these reasons they have been pooled together. With the exception of the study by Wilbert-Lampen and colleagues

(5) (RR for both genders: 2.66) and the two small studies by Katz et colleagues (15, 16) (pooled RR for both genders: 1.8), all RR estimates were between 0.70 (10) and 1.3 [START_REF] Witte | Cardiovascular mortality in Dutch men during 1996 European football championship: longitudinal population study[END_REF]. The effect estimate found by Wilbert-Lampen and colleagues is a clear outlier, as there is no overlap between its confidence interval and most of the confidence intervals reported by the other authors.

(Figure 1 Nevertheless, we found no evidence of increased rates of hospital admissions for acute myocardial infarction (AMI) associated with football matches played by the Italian national team in three international competitions held between 2002 and 2006. Our estimates are based on large numbers of events and football matches distributed over a period of five years. The control period consisted of the days before and after the matches. For these reasons it is unlikely that our results are due to the effects of unmeasured potential confounders such as temperature and pollution.

Although a fraction of patients with AMI die before they reach hospital, it is unlikely that the proportion of deaths outside hospital increased selectively during days when the Italian team played biasing our observed incidence rates downwards.

Our findings are consistent with most of previous studies (4, 9-14) that report a weak, if any, increase in cardiovascular events during the days of football matches. A remarkable exception is the recently published study by Wilbert-Lampen and colleagues, which reported a 2.7-fold increase in hospital admissions for cardiac emergencies on the days of matches involving the German team [START_REF] Wilbert-Lampen | Cardiovascular events during World Cup soccer[END_REF].

The main difference in study design between the German study and the other studies is that the former used prospectively collected information, whereas the latter studies were based on routinely collected administrative data that lack information on the exact time of the onset of the symptoms.

Moreover, while Wilbert-Lampen and colleagues (5) analysed an interval of 24 hours beginning at noon, the other studies used a daily interval beginning at midnight. The latter approach could introduce a certain degree of misclassification as some of the matches were watched by European viewers at dinner time; hence, some of the cardiovascular events in excess could have been admitted to the hospital after midnight and thus been counted on a control day. However, we consider unlikely that this could explain the differences in the results. First, watching a football

F o r R e v i e w O n l y
match is considered to be a trigger for cardiovascular events, implying that its effects are likely to act during a short time period, typically 1-2 hours (1, 5, 17). Therefore, the proportion of cases admitted to the hospital the day after the matches would probably be small even when matches were disputed at evening. Second, we did not find an increased incidence of AMI in either the World Cup 2002, in which all matches were played in Korea and Japan and therefore watched by Italians during the morning, nor in single matches disputed during afternoon in the World Cup 2006 and the

European Championship 2004

One should also consider the magnitude of the estimated relative risks, which were averaged over a 24-hour period in all studies, including that of Wilbert-Lampen and colleagues (5), while triggers typically increase cardiovascular risk over a short time period. Assuming that the risk is increased within the first 2 hours from the start of a match, the RR of 2.66 over a 24-hour period found by the German study would correspond to an acute relative risk of 21 (i.e., a RR of 2.66 is the 24-hour average between a RR of 1.00 during 22 hours and a RR of 21 during 2 hours). However, only a proportion of the population actually watches the matches (and is thus exposed). Assuming a proportion of viewers of 50% (higher than that reported in Italy during the final match of the World Cup 2006 (http://it.wikipedia.org/wiki/Audience_(media)), the corresponding acute RR among viewers would be of 41 (set to 1.0 the acute RR for non-viewers). This estimate is considerably larger than that of any other cardiovascular trigger studied in case-crossover studies (1). Indeed, with the exception of cocaine consumption which entails an acute RR of 24 [START_REF] Mittleman | Triggering of myocardial infarction by cocaine[END_REF], sexual activity, an episode of anger, strenuous physical exercise, smoking marijuana, drinking coffee and having a heavy meal are all associated with acute RRs ranging between 2 and 7 (1). Even assuming a conservative scenario, in which the effect of watching a football match lasts for 8 hours, the corresponding acute RR would still be 10. The inconsistency between the effect estimates reported by Wilbert-Lampen and the magnitude of the effects expected on the basis of previous evidence on other recognised cardiovascular triggers further suggests that the results by Wilbert-Lampen et al.

should be regarded cautiously. The publication of the German study stimulated intense media coverage and debate (6, 7, 19-21), but neither in the scientific literature nor in the media was it mentioned that its relative risk estimate was much higher than those previously reported. This resulted in public concern over the possible harmful effects of watching sports events and in proposals to introduce ad-hoc preventive measures, including changes in the pharmacological treatments for individuals with pre-existing coronary artery disease (5-7). Our study and review contribute to the debate and suggest that the population cardiovascular effects of watching a football match are likely to be, if anything, very small. This is reassuring in terms of public health. A recent study carried out in Bavaria (Germany) during the last Football World Cup reported a 2.7fold increase in the incidence of cardiac emergencies in the 12 hours before and after football matches involving the German team.

We investigated this issue in the entire Italian population. We found no evidence of increased rates of hospital admissions for acute myocardial infarction associated with football matches played by the Italian national team in three international competitions held between 2002 and 2006.

We also reviewed the ten published studies on the association between football matches and the risk of cardiovascular events. With the exception of the German study and two small Swiss studies, all relative risk estimates are spread around the null value, ranging between 0.7 and 1.3.

Our study and review suggest that the population cardiovascular effects of watching a football match are likely to be, if anything, very small. 3. A summary of the published studies on the association between football matches and cardiovascular events. Relative risks refer to the numbers of cardiovascular events on days in which major football matches were played compared with events during control periods.

F

F o r R e v i e w

O n l y

  It is just a game: lack of association between watching football matches and the risk of acute cardiovascular events.

7 The

 7 English study byKirkup and Merrick (12) evaluated all competitive matches played over five years by four local football teams to investigate if the result of the matches (local team lost at home vs. other results) was associated with cardiovascular mortality. Finally, in the two studies by Katz and colleagues[START_REF] Katz | Increase of out-of-hospital cardiac arrests in the male population of the French speaking provinces of Switzerland during the 1998 FIFA World Cup[END_REF][START_REF] Katz | Increase of sudden cardiac deaths in Switzerland during the 2002 FIFA World Cup[END_REF], the entire duration of the studied competitions was considered as the exposure period, regardless of whether a match involving the national team was played or not.

F o r R e v i e w O n l y 15 Table

 15

F o r R e v i e w O n l y 17 Figure 1 .

 171 Figure 1. Funnel plot of the relative risks (RR) vs. their standard errors [SE Ln(RR): standard error of natural logarithm of the RR] for studies on football matches and the risk of cardiovascular events. Bars represent 95% confidence intervals.

  , the European Championship 2004 (held in Portugal between June 12 and July 4) and the 2006 FIFA World Cup (held in Germany between June 9 and July 9). Records of hospital admissions with a primary diagnosis of acute myocardial infarction (ICD-9 code 410) were obtained from the National Hospital Discharge Database in the form of individual anonymous records, which included age, date of admission and gender. The National Hospital Discharge Database, created in 1994, collects information on all hospitalisations recorded in Italy. The data collection has gradually improved in quality and completeness during the first five years and is currently regarded as highly complete (8).

	30)
	F o r
	R
	e v
	i e
	w
	O n l
	y
	4

competitions: the 2002 FIFA World Cup (held in South Korea and Japan between May 31 and June

Table 2

 2 the final match of the World Cup 2006 (data not reported in tables). Restriction of the analyses to men aged less than 65 years (pooled RR= 1.03, 95% CI: 0.97-1.09) did not substantially affect the results. Further secondary analyses were carried out stratifying matches by time of the day when they were broadcasted in Italy. No effect was observed in any of three groups

	penalty shootout in -matches of World cup 2002 (viewed in Italy in the morning), matches disputed in the afternoon
	(RR 0.97; 95 %CI: 0.92-1.03), matches disputed in the evening (RR 1.02; 95% CI 0.96-1.08).
	F o r
	R
	e v
	i e
	w
	O n l
	y
	6

). During the 2002 FIFA World Cup, matches involving Italy were viewed in Italy between 9am and 11am. In the European Championship 2004, they started between 4pm and 7pm. Finally, in the 2006 FIFA World Cup matches involving Italy started between 4pm and 9pm. The days of the football matches were not associated with an increase in admissions for AMI in the three competitions combined (pooled relative risk [RR] 1.01, 95% confidence interval 0.98-1.05) or in each single competition. The incidence did not increase (RR=1.03, 95% CI: 0.90-1.17) when Italy and France played a dramatic

  DiscussionFootball is the most popular sport in Italy. The final match of the World Cup 2006, when Italy beat France, was watched by more than 24 million television viewers (approximately 40% of the population), corresponding to a share of 84% (http://it.wikipedia.org/wiki/Audience_(media)).

	F o r
	R
	e v
	i e
	w
	O n l
	y
	here)

Table 1 .

 1 Descriptive statistics of temperature and concentrations of nitrogen dioxide in different Italian cities during three international football competitions (2002 FIFA World Cup, the European Championship 2004, and 2006 FIFA World Cup). Mean values and standard deviations (SD) are stratified by days when the Italian team played and days when the Italian team did not play. : figures are based on measurements obtained during 2002 FIFA World Cup and the European Championship 2004.

		Temperature (C°)	Nitrogen Dioxide (µg/m 3 )
		When Italy	When Italy	When Italy	When Italy
		played	did not play	played	did not play
		Mean (SD)	Mean (SD)	Mean (SD)	Mean (SD)
	Milan (Northern Italy)	24.8 (3.1) a	25.1 (2.9) a	47.8 (13.6)#	39.8 (11.8)#
	Rome (Central Italy)	23.9 (3.0) a	24.6 (3.4) a	57.3 (14.8) a	57.8 (15.5) a
	Palermo (Southern Italy )	r o 25.5 (3.6) a	26.1 (3.3) a	50.3 (14.1)#	48.2 (12.2)#
		R
		e v
			i e
			w
			O n l
				y

a : figures are based on measurements obtained during all the three competitions. #

Table 2 .

 2 Hospital admissions for acute myocardial infarction (ICD9:410) in the Italian population during three international football competitions. The relative risk of acute myocardial infarction on match days involving the Italian team is compared with the other days of the competitions.

	Days

of the competitions When Italy played When Italy did not play N. Events (N. Days) RR a (95% CI) a N. Events (N. Days) RR a

	World Cup 2006	2236	1.00	7377	1
		(7)	(0.95-1.05)	(24)	(ref)
	EURO 2004	929	0.98	5800	1
		(3)	(0.91-1.06)	(20)	(ref)
	World Cup 2002	1230	1.05	7587	1
		(4)	(0.99-1.12)	(27)	(ref)
	Overall	4395	1.01	20764	1
		(14)	(0.98-1.05)	(71)	(ref)

a RR, relative risk adjusted for day of the week; CI, confidence interval

16 *

 16 RR adjusted for year, month, day of the week, temperature, air pollution, influenza epidemics ** RR adjusted for gender, year, month, day of the week, temperature. § RR adjusted for month and day of the week § § RR adjusted for competition and day of the week #: RR adjusted for year, day of the week, temperature, air pollution, barometric pressure °: Control days were May 1 to June 8 and July 10 to July 31 in 2006; May 1 to July 31 in 2003 and 2005 List of abbreviations: N: number of events during the exposure period; AMI: Acute myocardial infarction E.C.: European Championship; W.C: World Cup; E.F.T: English Football Tournament; A.F.L.: Australian Football League; W.C.Q: World Cup Qualifications; M: men; W: women; M+W: both genders; Penalty: match lost by the team at a penalty shootout; Win: matches won by the team; Loss: match lost by the team not at penalty shootout; NA: not available

Acknowledgements

The authors are grateful to Daniela Galeone, Lucia Lispi and Lorenzo Spizzichino (Italian Ministry of Health) for providing data from the National Hospital Discharge Database.

Funding

The work was carried out within the framework of projects that were partially supported by the Special Project Oncologia, San Paolo Foundation/FIRMS, the Italian Association for Cancer Research (AIRC) and Regione Piemonte.

Study

Events within 2 days after a the AFL final match and after the regional World Cup qualifying match vs. events during the preceding [START_REF] Bauman | The hazards of watching football--are Australians at risk?[END_REF]


E Tassi
email: tassi@cpt.univ-mrs.fr

F L Waelbroeck

D Grasso

Gyrofluid simulations of collisionless reconnection in the presence of diamagnetic effects

The effects of the ion Larmor radius on magnetic reconnection are investigated by means of numerical simulations, with a Hamiltonian gyrofluid model. In the linear regime, it is found that ion diamagnetic effects decrease the growth rate of the dominant mode. Increasing ion temperature tends to make the magnetic islands propagate in the ion diamagnetic drift direction. In the nonlinear regime, diamagnetic effects reduce the final width of the island. Unlike the electron density, the guiding center density does not tend to distribute along separatrices and at high ion temperature, the electrostatic potential exhibits the superposition of a small scale structure, related to the electron density, and a large scale structure, related to the ion guidingcenter density.

Introduction

Magnetic reconnection is responsible for many phenomena occurring in laboratory and astrophysical plasmas. [START_REF] Priest | Magnetic Reconnection[END_REF][START_REF] Biskamp | Magnetic Reconnection in Plasmas[END_REF] In this paper, we restrict consideration to reconnection in lowbeta configurations (i.e. m e /M i < β ≪ 1, with m e and M i indicating electron and ion mass, respectively ) characterized by the presence of a magnetic "guide" field perpendicular to the reconnection plane. The guide field ensures that the ions undergo rapid Larmor gyration as they cross the reconnection region. Guide field reconnection is thought to be responsible, in particular, for the sawtooth crash in tokamaks. Kinetic simulations of guide-field reconnection are generally more costly and difficult than in the alternative case of "component reconnection," due to the separation of scales imposed by the guide field. [START_REF] Wesson | [END_REF]4,5] In the linear [6] as well as in the nonlinear regime, [7,8,9] magnetic reconnection is characterized by processes occurring in very narrow resonant layers. For high-temperature plasmas, such as those existing in the present generation of tokamaks, the resonant layer is narrower than the ion Larmor radius, so that the ions may traverse the entire layer during a cyclotron period. [10] Their response is then a nonlocal functional of the fields they sample along their orbit. [11,12] Conventional fluid models treat finite Larmor radius (FLR) effects by expanding the ion response to first order in (k ⊥ ρ i ) 2 . [13,14,15,16] In the linear regime, the predictions of fluid models that include FLR effects [17] are qualitatively similar to the results of kinetic calculations [11,12] that account for the nonlocal response of the ions. In particular, both fluid and kinetic models predict the stabilization of ideal instabilities when the growth rate γ 0 computed in the absence of diamagnetic drifts is less than half of ω * i , the ion diamagnetic drift angular frequency. [11,17,18] It is unclear a priori, however, whether fluid FLR calculations such as the investigations of the nonlinear saturation of reconnecting modes in Refs. [13,15,19], offer an adequate description of ion dynamics in the nonlinear regime. In particular, it is unclear whether they are adequate for the task of extrapolating the experimental observations to different values of ρ i /L, where L describes the plasma size. Some indications are provided by nonlinear simulations using gyrokinetic codes, [20] but numerical limitations make scaling studies with such codes impractical.

The need for a simpler nonlinear model for the nonlocal ion response has motivated the derivation and investigation of gyrofluid models that account for these physical effects in a fluid context. Grasso et al. [21] introduced the first electromagnetic gyrofluid model and applied it to the study of magnetic reconnection. Their gyrofluid model, although unable to describe kinetic phenomena such as wave particle resonance (Landau damping), [12] nevertheless illuminated the qualitative properties of the reconnection dynamics. More elaborate models that do account for Landau damping were subsequently developed by Snyder and Hammett, but the applications of their models have been limited to the study of electromagnetic effects on turbulence driven by pressure gradients. [22] In the present contribution, we investigate a recently derived 3-field gyrofluid model [23] by means of numerical simulations. This gyrofluid model is part of a family of Hamiltonian fluid models [24,21,25,26] that have been investigated in past years, but is distinguished from these other Hamiltonian models by its nonlocal treatment of the ion density evolution. In the linear regime, the nonlocal description of the ion density evolution improves on FLR models by describing the evanescence of drift waves excited at frequencies in the ion drift direction but below the diamagnetic frequency. [23] The propagation and evanescence properties of drift waves can be important for determining the role of the polarization current in island evolution. [27] In recent work, [28] we have used the 3-field model of Ref. [23] to revisit earlier investigations [21] of gyrofluid magnetic reconnection in a homogeneous plasma, where ion diamagnetic effects vanish. The present article extends these studies to the case of inhomogeneous plasma, where ion diamagnetic drifts affect the reconnection. Such ion drift effects can be important for sawtooth dynamics in tokamaks [9,13,15], where it accounts for the saturation of the resistive kink instability which can prevent complete reconnection during sawtooth crashes. [29,30] It is also important for reconnection at the dayside magnetopause [31,32] where the interface between the incoming solar wind and the Earth's magnetosphere is characterized by a strong density gradient. Unfortunately, our assumption of a strong guide field makes our model inapplicable to the magnetopause.

The paper is organized as follows. In Sec. 2 we review the model equations. Sec. 3 is devoted to the simulation results in the linear phase and to the comparison with analytical predictions. In Sec. 4 we consider the evolution of the system in the nonlinear phase. We conclude in Sec. 5.

The model

We consider the two-dimensional version of the Hamiltonian gyrofluid model described in Ref. [23]. The model equations are

∂n i ∂t + [Φ, n i ] = 0, (1)
∂n e ∂t + [φ, n e ] -[ψ, ∇ 2 ψ] = 0, (2)
∂ ∂t (ψ -d 2 e ∇ 2 ψ) + [φ, ψ -d 2 e ∇ 2 ψ] + ρ 2 s [ψ, n e ] = 0, (3)
n e = Γ 1/2 0 n i + (Γ 0 -1)φ/ρ 2 i . (4)
These represent the continuity equation for the ion guiding centers, the electron continuity equation, the Ohm's law and the Poisson's equation, respectively. In Eqs.( 1)-( 4) n i is the ion guiding center density, n e the electron density, φ the electrostatic potential, ψ the poloidal magnetic flux function, d e the electron skin depth, ρ s the sonic Larmor radius and Φ = Γ 1/2 0 φ is the gyro-averaged electrostatic potential. For our analysis the Padé approximant version Γ

1/2 0 = (1 -ρ 2 i ∇ 2 /2) -1
of the gyro-average operator will be adopted, where ρ i is the ion Larmor radius. Given a Cartesian coordinate system (x, y, z), we assume all the fields be translationally invariant along z and we define the canonical bracket between two generic fields f and g by [f, g] = ẑ • ∇f × ∇g. We also recall that we normalize the time with respect to the Alfvèn time and distances with respect to a magnetic equilibrium scale length L. Dependent variables are normalized in the following way

n i = L di ni n 0 , n e = L di ne n 0 , ψ = Âz BL , φ = ρ2 s L 2 L di e φ T e ,
where carets denote dimensional variables, d i is the ion skin depth, n 0 a background density amplitude, A z the magnetic potential, e the unit charge, B a characteristic magnetic field amplitude and T e the electron temperature, which is assumed to be constant.

Numerical simulations in the linear regime

We devote this section to the analysis of the numerical solutions of the model ( 1)-( 4) in the phase in which nonlinear terms are not dominant. We solve the model equations over the domain {(x, y) : -π ≤ x < π, -aπ ≤ y < aπ}, where we prescribe the value of the number a at each simulation. The grid is composed by 1024×128 points and double periodic boundary conditions are imposed on the field perturbations, which are the quantities that the code advances in time. Unlike Ref. [28] we consider now inhomogeneous density equilibria. The simulations are then started by perturbing the equilibrium

n ieq (x) = n ′ 0 x, n eeq (x) = n ′ 0 x, ψ eq (x) = 11 n=-11 a n exp(inx), (5)
where n ′ 0 is a constant and the a n are the Fourier coefficients of the function f (x) = 1/ cosh 2 x. With respect to the simulations of Ref. [28] the presence of equilibrium density gradients makes the evolution of n i no longer negligible and introduces the diamagnetic effects. A global dispersion relation for the system in the presence of diamagnetic effects was derived in Ref. [23]. This showed also how the Padé approximant version of the gyrofluid model was able to reproduce adequately dispersive properties of the system, such as a spectral gap in the frequencies, which are not caught by Finite Larmor Radius (FLR) models. The equilibrium (5) is perturbed in n i with a four-cell pattern disturbance of the form ñi ∝ cos(x + y/a) -cos(x -y/a). The field φ is also perturbed according to (4), in such a way that the initial perturbation on n e is zero.

If we indicate with χ(x, y, t) any of the three dynamical variables of our system, i.e. n i , n e and ψ, then we can decompose the fields as χ = χ eq (x) + χ(x, y, t). We write then the perturbation χ as a Fourier series truncated at N modes, so that χ = N -N χk exp(ik • x + λt), where the hat indicates the Fourier transform, k is the wave vector and λ = γ -iω is a complex number having the growth rate γ as real part and the rotation frequency -ω as imaginary part. Simulations with a relatively small amplitude of the initial perturbation, corresponding to 10 -6 , have been carried out, in order to obtain a long linear phase (∼ 25 Alfvèn times for Table 1. Table displaying values of linear growth rate γ and rotation frequency ω, for the dominant mode. A comparison is made between values obtained from numerical simulations (ω num ,γ num ) and values (ω theor ,γ theor ) predicted by the asymptotic theory of Ref. [11]. The agreement between numerical and analytical results is better for ∆ ′ = 59.9, which is a value closer to the asymptotic regime of validity of the theory. For all cases d e = 0. most of the simulations) that can be effectively investigated. In particular, we are interested in extracting from the simulations the values of γ and ω, which can permit us to observe the effects of equilibrium density gradients and finite gyro-radius, on the linear mode stabilization and island rotation frequency. We find it also of interest to compare the obtained numerical results with those predicted by the theory developed in Ref. [11]. The linear analysis by Porcelli refers namely to the collisionless case in the presence of finite ion temperature. Here we limit ourselves to recall the results of the theory that are relevant for our analysis. By adopting a matching asymptotic technique, in Ref. [11], the following dispersion relation, for the main mode, is obtained

λ 2 -ω * e ω * i + iλ(ω * e + ω * i ) ≈ γ 2 0 . (6)
In ( 6) the electron diamagnetic frequency ω * e is defined by ω * e = -k y cT e n ′ 0 /(eBn 0 v A ), where v A is the Alfvèn speed based on B. The ion diamagnetic frequency is ω * i = -τ ω * e , with τ = ρ 2 i /ρ 2 s , and γ 0 , for our case, reads γ 0 = 2k y (2d e (ρ 2 s + ρ 2 i )/π) 1/3 . Splitting the dispersion relation into its real and imaginary part yields

ω ≈ ω * e 2 (1 -τ ), (7)
γ 2 ≈ γ 2 0 - ω * e 2 (1 + τ ) 2 , (8)
which give us a prediction for the growth rate and the rotation frequency. Such results, however, are valid only in the limit ρ i ≫ d e and ∆ ′ ≫ ρ

1/3 i d 2/3
e , where ∆ ′ is the standard stability parameter for tearing modes. Examples of theoretical and numerical values are provided in Table 1. Two main cases are considered, corresponding to two different values of ∆ ′ (which correspond to the values a = 1.5 and a = 2 of the parameter scaling the length of the domain along y). For each of these two cases the same sets of parameters ρ i , ρ s and v * = ω * e /k y are considered. From these representative examples we can see that the numerical results follow the trends predicted by the theory, for the dominant mode (the latter is k y = 0.66 for ∆ ′ = 34.2 and k y = 0.5 for ∆ ′ = 59.9). In particular, for fixed ρ i and ρ s , the cases with v * = -0.4 exhibit smaller growth rate than those with v * = -0.16. This indicates the stabilizing effect operated by diamagnetic flows and corresponding to the term proportional to ω 2 * e in (8). We report also that another simulation, again with ρ i = ρ s = 0.4, but with v * = -0.64, showed no growth of the magnetic island, indicating the complete linear stabilization due to diamagnetic effects. For these two sets of parameters, the numerical solutions show almost no rotation frequency. On the other hand, for the case ρ i = 0.35, ρ s = 0.2 and v * = -0.04, the numerical solutions show a positive value of ω, which corresponds to a propagation along the direction of the ion diamagnetic drift (which in our case corresponds to the positive y direction). The propagation along the ion diamagnetic drift direction is indeed also what is predicted by the theory, according to (7), when ions are hotter than electrons. With regard to the quantitative agreement with the theory, Table 1 shows that this improves when increasing ∆ ′ . In the case with larger ∆ ′ the distances between the numerical and predicted values get shorter and we can quantify that the relative error drops from an average of 28% to 18.7%. The reduction of the error is of course a consequence of the fact that, by increasing ∆ ′ , we get closer to the asymptotic regime in which the theory is valid.

Numerical simulations in the nonlinear regime

In this section we focus on the evolution of the system in the nonlinear phase. Fig. 1 shows a comparison between contour plots of ψ for v * = -0.16 and v * = -0.4, at times at which the two simulations reached the same phase of the dynamical evolution. From the figure we can see that the presence of stronger diamagnetic effects, apart from reducing the linear growth rate, also leads to a nonlinear island of smaller width. We recall that a nonlinear stabilizing effect of n ′ 0 was observed in a cold-ion, three-field model in cylindrical geometry [15]. In that context diamagnetic flows were indeed suggested to be responsible for the saturation of islands at a small width. We also note that, in spite of the fact that, in the linear phase, the islands exhibited very little rotation for these parameters, in the nonlinear regime, upward shifts of the O-points are present, in particular in the case v * = -0.4. Island propagation in the ion diamagnetic drift direction, caused either by viscosity [33,16] or by nonlinear zonal flows, [34,35] has been reported in FLR simulations of visco-resistive reconnection. An effective way to look at the nonlinear dynamics of our system is to express the model in terms of its "normal fields", suggested by the Hamiltonian structure. We recall that the model equations ( 1)-( 4) can be rewritten as [23]

∂n i ∂t + [Φ, n i ] = 0, (9)
∂G + ∂t + [φ + , G + ] = 0, (10)
∂G - ∂t + [φ -, G -] = 0, (11)
where

G ± = ψ -d 2 e ∇ 2 ψ ± d e ρ
s n e and φ ± = φ ± (ρ s /d e )ψ. The fields n i and G ± , which we denote as normal fields, represent natural variables for the system because, in terms of them, the model equations take the form of simple advection equations. As pointed out in a number of previous papers on reconnection based on Hamiltonian fluid models [36,37,38,39,26,40], the stream functions φ ± correspond to velocity fields that rotate in opposite directions. The fields G ± , advected by such velocity fields, get then stirred in opposite directions. The stirring process causes G ± to form finer and finer structures, leading to an energy cascade toward small scales. Because n e is proportional to the difference between G + and G -, such cascade inevitably reflects on the electron density structures.

In order to investigate the nonlinear structures, we double the number of grid points in the y direction, which enables us to better resolve small scale structures that form nonlinearly. We also bring up to 10 -3 the amplitude of the initial perturbation. From Fig. 2 indeed we see that electron density, as is typical of Hamiltonian reconnection, develops lobes around the separatrices. These emerge as the difference between the spiral arms present in G + and G -(the latter not shown here but such that G -(x, y, t) = G + (-x, y, t)). Unlike the above mentioned studies of Hamiltonian reconnection, however, the structures of Fig. 2 are of course not symmetric with respect to the y = 0 axis, because of diamagnetic effects. The pattern of φ + (and, analogously of φ -(x, y, t) = -φ + (-x, y, t)) are such to create thinner and shorter layers in the lower part of the magnetic island whereas wider and longer lobes are present in the upper part of the magnetic island. Inside the island, stirring of G ± takes place and the initial linear density profile gets flattened. The ion guiding center density gradient also gets partly flattened in the center of the island, as a consequence of the advection operated by Φ. Such stream function, however, has a circulation pattern quite different from that obtained by the combination of φ + and φ -. Therefore n i does not tend to accumulate around the separatrices, unlike n e . The four convective cells of the gyroaveraged electrostatic potential, on the other hand, tend to lie along the magnetic separatrices. Such behavior is also evident in the electrostatic potential φ. By virtue of the relation φ = (1 -ρ 2 i ∇ 2 /2)Φ, we can also infer that φ exhibits, in general, finer structures, compared to Φ. We can indeed observe regions with steep gradients of electrostatic potential, around the separatrices, similarly to what happens to the electron density. Nevertheless, in this respect, it is important to recall that, for n ′ 0 = 0, the growth of n i was negligible, and one had φk ∼ -ρ 2 i ñek , as ρ i → +∞. This led to conclude [28] that the filamentation in n e implied an analogous phenomenon in φ, with consequent formation of strong electric fields. In the presence of diamagnetic effects, on the other hand, Eq. ( 4), in the large

ρ i limit, yields φk ∼ -ρ 2 i ñek + 2 k 2 ñik . (12)
Therefore, φ is no longer proportional to ñe , but is given by the superposition of two contributions: one, related to ñe , which is responsible for the formation of steep gradients, and a second one, due to -2∇ -2 ñi which, on the contrary, is relevant at large scales.

Conclusions

Numerical simulations of a Hamiltonian electromagnetic gyrofluid model have been carried out and investigated. The present analysis represents a step forward, with respect to a recent related work [28], because it accounts for inhomogeneous density equilibria, which lead to diamagnetic effects. In the linear phase, diamagnetic effects are shown to stabilize the growth rate of the dominant mode and induce a propagation of the magnetic island. Simulations also indicate that the direction of propagation tends to move from that of the electron to that of the ion diamagnetic drift, as the ion temperature is increased. Such features are in agreement with what is predicted by the linear theory of Ref. [11] and the quantitative agreement improves as the values of parameters approach the asymptotic regime of validity of the theory. In the nonlinear phase, stronger equilibrium gradients tend to yield smaller "saturated" islands, as already observed in Ref. [15]. We note also a different distribution of the ion guiding center density with respect to the electron density, with the former retaining a four-cell pattern, whereas the latter concentrates around separatrices, although in an asymmetric way, also with respect to the y = 0 axis, as a consequence of the diamagnetic terms. Compared to the homogeneous equilibrium case [28], the potential possesses now a smoother contribution, due to the finite n i fluctuations. We finally indicate the derivation of a refined version of the linear theory, and the extension of the model to include parallel dynamics, as subjects of work in progress.

Figure 1 .

 1 Figure 1. Contour plots of ψ for v * = -0.16, at t = 39 (left) and v * = -0.4 at t = 50 (right). The values of the other parameters are: ρ s = ρ i = 0.4, d e = 0.2. The plots indicate that stronger diamagnetic effects lead to a smaller saturated island.

Figure 2 .

 2 Figure 2. Contour plots of n e , φ, n i , Φ, G + and φ + at t = 19, for v * = -0.16, ρ i = ρ s = 0.4 and d e = 0.2. The magnetic island at t = 19 is superimposed onto each plot.

Acknowledgments

This work was supported by the European Community under the contracts of Association between EURATOM and ENEA and between EURATOM, CEA, and the French Research Federation for fusion studies. FW was supported in part by a fellowship from the Katholieke Universiteit Leuven and by the US Department of Energy under contract DE-FG02-04ER-54742. The views and opinions expressed herein do not necessarily reflect those of the European Commission. Financial support was also received from the Agence Nationale de la Recherche (ANR EGYPT).