M Lingam

⇤ P J Morrison

E Tassi

Inertial magnetohydrodynamics

A version of extended magnetohydrodynamics (MHD) that incorporates electron inertia is obtained by constructing an action principle. Unlike MHD which freezes in magnetic flux, the present theory freezes in an alternative flux related to the electron canonical momentum. The associated Hamiltonian formulation is derived and reduced models that have previously been used to describe collisionless reconnection are obtained.

I. INTRODUCTION

Ideal magnetohydrodynamics (MHD), which arose in the early 20th century, ranks amongst the great breakthroughs in plasma physics. The simplicity of MHD has led to its extensive usage in the arenas of fusion, space and astrophysical plasmas; see e.g. [START_REF] Krall | Principles of plasma physics[END_REF][START_REF] Goedbloed | Principles of Magnetohydrodynamics[END_REF][START_REF] Kulsrud | Plasma physics for astrophysics[END_REF] and references therein. In addition, MHD is an attractive theory as it is endowed with several geometric properties such as flux freezing and the conservation of magnetic and cross helicities, amongst others. The latter duo, in particular, have attracted much attention owing to their topological properties [START_REF] Mo↵att | [END_REF]5] and intimate connections with selforganization and relaxation [START_REF] Woltjer | Proc. Natl. Acad. Sci[END_REF][START_REF] Taylor | [END_REF][8]. As the advantages of ideal MHD are far too numerous to elaborate, we refer the reader to the aforementioned references.

However, in the latter half of the 20th century, an increasing awareness arose amongst the plasma community that MHD could serve as an e↵ective theory only in certain regimes. To counteract these limitations, several fluid and kinetic models were developed, of which we shall only focus on the former. Amongst them, the most famous are the two-fluid model [START_REF] Krall | Principles of plasma physics[END_REF][START_REF] Goedbloed | Principles of Magnetohydrodynamics[END_REF], Hall MHD [9], electron MHD [10] and extended MHD [START_REF] Spitzer | Physics of Fully Ionized Gases[END_REF][START_REF] Lüst | [END_REF]. Extended versions of ideal MHD often incorporate additional terms into Ohm's law, but they (mostly) su↵er from a common deficiency -a failure to conserve the energy, in the absence of dissipative e↵ects. This fact was first pointed out in [13], who also presented an analysis of the di↵erent terms in the extended Ohm's law and their role in energy conservation; also see [14] for a related analysis. Even amongst MHD models, most versions tend to neglect the electron inertia, which can be of considerable interest when their characteristic velocity is much faster than their ionic counterparts. In order to retain electron inertia while simplifying other features, several reduced MHD (RMHD) models have been proposed [15,16]. Such models are of considerable interest, as they represent alternative methods of driving reconnection, which in turn ⇤ Electronic address: manasvi@physics.utexas.edu † Electronic address: morrison@physics.utexas.edu ‡ Electronic address: tassi@cpt.univ-mrs.fr is expected to play a crucial role in terrestrial and astrophysical plasmas [17].

The energy-conserving property of MHD and its invariants are closely linked to its Hamiltonian structure, i.e. it can be shown that such theories can be described via a noncanonical bracket and an appropriate Hamiltonian. The existence of such a Hamiltonian structure for MHD was first recognized in the seminal paper of Morrison & Greene [18], and was consequently employed in a wide range of contexts in the 1980s [19][20][21][22]. We refer the reader to [START_REF] Morrison | Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems[END_REF][START_REF] Salmon | [END_REF][25][26], and references therein, for comprehensive reviews of the same. The Hamiltonian formalism has advantages that extend far beyond the mere ability to reproduce the equations of motion; it can be used to obtain invariants, equilibria and conduct stability analyses [25,[27][28][29].

Associated with Hamiltonian structure is the action principle formalism, and this can serve as a starting point for obtaining the noncanonical Poisson bracket. Such an action principle dates back to Lagrange's pioneering work in the 18th century [START_REF] Lagrange | Mécanique Analytique[END_REF], but only relatively recently was such employed in the context of MHD [START_REF] Newcomb | [END_REF] and, subsequently, for incompressible gyroscopic systems [32,33].

Here, we construct an action by employing a method for building actions that was described in [START_REF] Morrison | New Developments in Nonlinear Plasma Physics: Proceedings for the 2009 ICTP College on Plasma Physics[END_REF]. An important feature of this method is the Eulerian closure principle (ECP) that insures the resulting theory can be written in terms of an Eulerian set of variables. The method has proven to be successful for deriving compressible gyroviscous MHD [START_REF] Morrison | [END_REF], extended MHD [36] and generalized gyroviscous fluids [37]. The method begins with an action in terms of a set of Lagrangian variables, treating the particle trajectory as the sole variable, together with the ECP. The ECP provides a generalization of, and indeed a justification of, constrained variations that were called Euler-Poincaré variations in [38][39][40], because of its antecedents in the work of Poincaré [41], although the idea was previously elucidated for ideal fluids and MHD plasmas in, e.g., [START_REF] Newcomb | [END_REF]42]. More recently, Euler-Poincaré variations were employed for gyrokinetic theory in [43] and gyro fluids in [44].

The action formalism also has several other advantages in addition to its ability to reproduce energy conserving models. A second advantage of this approach is that one can obtain reduced or simplified fluid models by performing suitable orderings directly within the action. These modifications still preserve energy conservation, which is not guaranteed when an ordering is undertaken at the level of the equations of motion. Thirdly, one can recover the noncanonical bracket and the Hamiltonian described previously; the former via a systematic reduction of the canonical bracket, and the latter via a Legendre transform. As the action principle formulation is intimately linked with the Hamiltonian approach, we refer to them collectively as the Hamiltonian and Action Principle (HAP) formalism, and illustrate its use in the current paper. Lest we lose sight of our goal, we summarize it before proceeding further. In this paper, we shall use the HAP formalism to construct a MHD model with electron inertia.

The organization of the paper is as follows. In Section II, we introduce the requisite mathematical preliminaries. In Section III, we introduce a new dynamical variable, construct the action and obtain the equations of motion. In Section IV, we obtain the noncanonical Hamiltonian formalism via reduction, and demonstrate how it constitutes a generalization of the widely employed Ottaviani-Porcelli model [15]. We also briefly comment on possible extensions of this model, and finally conclude in Section V.

II. THE ACTION PRINCIPLE AND THE LAGRANGIAN COORDINATES

We commence with a very brief synopsis of the Lagrangian approach and the Lagrangian coordinate. For finite dimensional systems, the action is

S[q] = Z t1 t0 dt L (q, q, t) , (1)
where L = T V represents the Lagrangian, with the kinetic and potential energies denoted by T and V respectively. Here, q represents the generalized coordinate(s).

One can obtain the equations of motion by extremizing the action via S[q]/ q k = 0. The functional derivative is defined via

S[q; q] = dS[q + ✏ q] d✏ ✏=0 =: ⌧ S[q] q i , q i . (2)
The continuum version of the action principle is equally straightforward. We introduce a label a that tracks a given fluid element; q is now a function of a and t. We introduce the deformation matrix, @q i /@a j =: q i , j and the Jacobian, J := det(q i , j ). The volume element for the fluid obeys

d 3 q = J d 3 a , (3)
and an area element is regulated via

(d 2 q) i = J a j , i (d 2 a) j , (4)
where J a j , i is defined to be the transpose of the cofactor matrix of q j , i . There are many identities that can be constructed involving q i , j and J , but we desist from doing so; a detailed discussion is present in [25].

A. Attributes, observables and the Lagrange-Euler maps

Our discussion has solely revolved around q, but a fluid element can also possess a certain density, entropy, etc. We refer to these quantities as attributes, since they are inherent to the fluid element. These quantities are dependent on a alone and are Lagrangian constants of motion. The subscript 0 is used to denote the attributes, to distinguish them from their Eulerian versions.

The Eulerian version is commonly employed since it describes fields in terms of r := (x 1 , x 2 , x 3 ) and t, each of which can be tracked through experiments. Hence, we refer these fields as observables. However, a connection between the attributes and the observables, between the Lagrangian and Eulerian pictures is unclear. In order to transition back and forth, we must introduce maps that permit such an activity. We refer to these as the Euler-Lagrange or Lagrange-Euler maps, depending on the context.

Before proceeding further, we remark that the model developed in this paper is 2D in nature, i.e. it has one ignorable coordinate. Hence, we shall treat our system as 3D in nature, but we note the existence of an implicit constraint z = q z (a, t) = a z , indicating that the z-component of the trajectory remains fixed.

A natural starting point is the velocity field v(r, t). In the Eulerian picture, we detect this velocity at a given instance in space and time. Intuitively, we expect the same result to hold true in the Lagrangian picture, i.e. the relation q(a, t) = v(r, t) must hold true. The LHS is the Lagrangian velocity, and the overdot indicates the derivative at fixed a. By the same logic, we expect the Eulerian and Lagrangian positions to coincide at this moment, ensuring that r = q(a, t). We assume that this map is invertible, permitting us to obtain a = q 1 (r, t) =: a(r, t). From the two conditions, we find that v(r, t) = q(a, t)| a=a(r,t) .

(

) 5
The above map constitutes a Lagrange to Euler map, as it Eulerianizes the Lagrangian version. Amongst the attributes transported by a fluid element, one of them is the entropy s 0 (a). The entropy is advected in an ideal fluid, implying that it is constant along a fluid trajectory. In other words, the Eulerian entropy s(r, t) must equal the Lagranian entropy s 0 (a). Thus, we conclude that s behaves as a zero form, or as a scalar. Next, we consider the density whose attribute-observable pair are denoted by ⇢ 0 (a) and ⇢(r, t). Our fluids must (typically) obey mass conservation, which is expressible as ⇢(r, t)d 3 r = ⇢ 0 d 3 a in an infinitesimal volume. One can geometrically interpret this as ⇢ behaving as a three form or as a scalar density. By using (3) we obtain ⇢ 0 = ⇢J . Hitherto, our attributes (and observables) have been purely thermodynamic. Let us consider the magnetic field and denote the attribute-observable pair by B 0 (a) and B(r, t). At this point, we make an important observation regarding ideal MHD: it has the very special property that the flux is frozen. The frozen flux constraint is expressible as B • d 2 r = B 0 • d 2 a, and from (4) we obtain J B i = q i , j B j 0 . In other words, the magnetic field B can be interpreted as a two-form or as a vector density, but only in the case of ideal MHD. In general, one cannot introduce two forms into extended MHD theories since the frozen flux condition is (apparently) not obeyed. In the next section, we shall tackle the issue of frozen flux, and present a new dynamical variable that, by construction, satisfies the frozen flux constraint.

In the preceding expressions, we evaluate the attributes at a = q 1 (r, t) =: a(r, t), thereby completing the Lagrange-Euler maps. We can also express the above relations in an integral form. This is done by demanding that the Eulerian and Lagrangian observation points coincide, which is accomplished via the judicious use of the delta function (r q(a, t)). Before proceeding further, we remind the reader that our model is actually 2D in nature.

As an example, the relation for the density is shown

⇢(r, t) = Z D d 2 a ⇢ 0 (a) (r q (a, t)) = ⇢ 0 J a=a(r,t) . (6)
Instead of the velocity, we introduce the canonical momentum

M c = (M c 1 , M c
2 ), which is related to the Lagrangian canonical momentum as follows

M c (r, t) = Z D d 2 a ⇧(a, t) (r q(a, t)) = ⇧(a, t) J a=a(r,t) . (7)
In the case of ideal MHD, ⇧(a, t) = (⇧ 1 , ⇧ 2 ) = ⇢ 0 q. One can always determine ⇧(a, t) via ⇧(a, t) = L/ q, and use it in the above expression. In the case of gyroviscous fluids, one finds that ⇧(a, t) 6 = ⇢ 0 q as shown in [START_REF] Morrison | [END_REF]37].

B. The Eulerian Closure Principle and action-building

Thus far, much of our analysis has been predicated on the notion that the Eulerian and Lagrangian pictures are related to one another, i.e., one can find a set of maps that allows us to go back and forth between the two descriptions. A natural consequence of this postulate is that our action must also obey such a property. This property is the ECP, referred to in the Introduction. The ECP amounts to the following: given an action expressed in terms of the Lagrangian variables, it must be equally expressible fully in terms of the Eulerian variables. For instance, consider the kinetic energy functional

Z Z 1 2 ⇢ 0 q2 d 2 a dt, (8)
and let us invoke the Lagrange-Euler maps introduced in the previous subsection. Through suitable use of ( 6) and [START_REF] Taylor | [END_REF], one can show that (8) reduces to

Z Z 1 2 ⇢v 2 d 2 r dt. (9)
As a result, we conclude that (8) satisfies the ECP since the kinetic energy functional in Lagrangian variables has been expressed in terms of Eulerian variables. The ECP dictates that all the other terms in the action also exhibit identical behavior as the kinetic energy functional. In summary, we follow a two step procedure to construct the action. The first involves the choice of the domain and the observables. The second involves the construction of each term in the action from first principles (when possible) while ensuring that they obey the ECP.

III. THE INERTIAL MHD ACTION

In this section, we shall present a new dynamical variable, one that determines a frozen flux for our model. An action principle in terms of this new variable is developed, and the equations of motion are obtained and analyzed.

A. The inertial magnetic field: a new dynamical variable

In Section II A, we discussed the implications of magnetic flux freezing in ideal MHD. Extended MHD lacks this feature, which implies that the magnetic field can no longer be interpreted as a two form. From a purely geometric point-of-view, it would be logical to look for a new dynamical variable, not B, which could play a similar role.

Hence, we introduce the variable B e and its corresponding attribute B e0 . The relation between the two is akin to that obeyed by the magnetic field in ideal MHD, viz. J B i e = q i , j B j e0 . Since we claim that our new theory is still a magnetofluid model, it is necessary for B e to be a function of the MHD variables v, B, n and s. We make the choice

B e = B + m e e 2 r ⇥ ✓ J n ◆ = B + m e µ 0 e 2 r ⇥ ✓ (r ⇥ B) n ◆ . (10)
In other words, this is also equivalent to stating that we replaced the vector potential A by A e , the latter of which is given by

A e = A + m e e 2 ✓ J n ◆ = A + m e µ 0 e 2 ✓ r ⇥ B n ◆ . (11)
Although the expressions ( 10) and ( 11) may appear ad hoc, there are good reasons that justify these choices.

The first stems from the inclusion of electron inertia, which is exemplified by the presence of an additional factor involving m e and it also satisfies the consistency requirement, i.e. in the limit m e /m i ! 0, we recover the usual magnetic field and vector potential. Secondly, we note that B e serves as a natural dynamical variable in extended MHD theories; for instance, if one takes the curl of equation ( 20) in [13] and uses Faraday's law, we recover a dynamical equation for B e . It is possible to carry out a similar procedure for the extended MHD models presented in [START_REF] Goedbloed | Principles of Magnetohydrodynamics[END_REF][START_REF] Spitzer | Physics of Fully Ionized Gases[END_REF] and arrive at the same conclusion. Lastly, the statement of flux freezing in ideal MHD is equivalent to stating that H A • dl is an invariant, which is altered in our model. To understand the alteration consider the canonical momentum for the electrons, which is proportional to A (m e v e /e). Assuming v e v i permits the approximation J ⇡ env e ; consequently, the canonical momentum is (approximately) equal to A e . If we let m e /m i ! 0, the canonical momentum reduces to A. As this is a 2D theory, with z serving as the ignorable coordinate, the corresponding canonical momentum in the z-direction is conserved. This yields H (A e ) z dz, which is akin to H A • dl being conserved in ideal MHD. Later, we shall show that even better reasons can be advanced, albeit a posteriori, that further justify the choice of B e .

Before we proceed to the next section, we introduce the nomenclature 'inertial magnetic field' to refer to B e . The choice is natural since B e plays the role of a magnetic field, and incorporates the e↵ects of electron inertia. Hence, we refer to this theory as inertial MHD (IMHD).

B. The IMHD action

We introduce the action for IMHD below, and then comment on its significance and interpretation. Our variables are chosen to be the density (3-form) ⇢, the inertial magnetic field (2-form) B e , the entropy (0-form) s and the velocity v.

S = Z Z  ⇢v 2 2 ⇢U (⇢, s) B e • B 2µ 0 d 2 r dt. (12)
The first term in ( 12) is the kinetic energy, which was already shown to obey the ECP in Section II B. The second term in [START_REF] Lüst | [END_REF] is the internal energy density, and is the product of density and the specific internal energy (per unit mass). This term generates the temperature and the pressure, given by @U/@s and ⇢ 2 @U/@⇢ respectively. The third term in the above expression is the unusual part, since it deviates from the usual ideal MHD action. In the limit where m e /m i ! 0 we have noted that B e ! B, which in turn reduces the last term of ( 12) to the conventional magnetic energy density. Although [START_REF] Lüst | [END_REF] is expressed in terms of the Eulerian variables, the ECP and the Euler-Lagrange maps allow us to express [START_REF] Lüst | [END_REF] purely in terms of the Lagrangian coordinate q and the attributes. In order to do so, we express the magnetic field B in terms of the inertial magnetic field B e as follows

B (r, t) = Z K (r|r 0 ) B e (r 0 , t) d 2 r 0 , ( 13
)
where K is a complicated kernel. Using the kernel is quite complex, but we note that the self-adjoint property is preserved. Alternatively, one can use the Euler-Poincaré approach, described in Section I, to obtain the same result. A short summary of this method is presented in Appendix A. Before proceeding to the next section, a couple of remarks regarding [START_REF] Lüst | [END_REF] are in order. Firstly, the only term involving q is the kinetic energy term. Hence, one can perform a Legendre transformation, and recover the Hamiltonian (in Lagrangian variables). Upon Eulerianizing the Hamiltonian, we obtain

H = Z  ⇢v 2 2 + ⇢U (⇢, s) + B e • B 2µ 0 d 2 r. (14)
We can use the definition of B e , as given in (10), and simplify the resultant expression. The result is

H = Z  ⇢v 2 2 + ⇢U (⇢, s) + B 2 2µ 0 + m e ne 2 J 2 2 d 2 r. ( 15
)
The above expression is identical to equation ( 23) of [13]. Furthermore, we see that ( 15) is identical to MHD Hamiltonian, except for the last term. As a consistency check, we verify that the last term does vanish in the limit m e /m i ! 0. These facts lend further credence to our choice of B e and the action [START_REF] Lüst | [END_REF].

C. The IMHD equations of motion

The Lagrange-Euler maps outlined in Section II A permit us to obtain the corresponding dynamical equations for the observables by applying @/@t on both sides of the map. We obtain

@s @t + v • rs = 0, (16)
@⇢ @t + r • (⇢v) = 0, (17)
@B e @t + B e (r • v) (B e • r) v + (v • r) B e = 0. ( 18
)
The equations ( 16), ( 17) and ( 18) correspond to the Liedragging of zero, three and two forms respectively. From the definition of (10), we see that r • B e = 0, and this implies that one can rewrite (18) as follows

@B e @t = r ⇥ (v ⇥ B e ) = r ⇥ (v ⇥ B) + m e e 2 r ⇥  v ⇥ ✓ r ⇥ ✓ r ⇥ B n ◆◆ . (19)
The equation of motion is obtained by extremizing the action in Lagrangian variables, or by extremizing the Eulerian action via the Euler-Poincaré approach. It is found to be

⇢ ✓ @v @t + (v • r) v ◆ = rp + J ⇥ B m e e 2 (J • r) ✓ J n ◆ . ( 20
)
Equations ( 19) and ( 20) constitute the heart of our model. Let us first consider the latter. We see that it is nearly identical to the usual ideal MHD momentum equation, except for the presence of the last term, which can be neglected in the limit m e /m i ! 0. However, this term represents more than a correction -in the extended MHD models with electron inertia, this term is absolutely crucial for energy conservation, as pointed out in [13]. Secondly, we note that our equation of motion is exactly identical to equations ( 2) and ( 19) of [13], thereby lending further credence to our choice of the inertial magnetic field and action.

We turn our attention to (19), which represents the extended Ohm's law. It is instructive to compare this against the inertial Ohm's law of [13], represented by their equation (20). We find that our expression is exactly identical to equation ( 20) of [13], when the 2D limit of their model is considered and B z ! const (constant guide field) is assumed. Under these assumptions, the two results are exactly identical, irrespective of whether the fluid is compressible or incompressible. A few comments on the 3D generalization of this model are presented in Section IV C.

To summarize thus far, we find that the momentum equations of our model and that of [13] are identical. The Ohm's laws are also in perfect agreement with one another in the 2D, constant guide field limit. In addition, both of them yield the same (conserved) energies and momenta. Collectively, it is self-evident that these represent ample grounds for justifying the form of the inertial magnetic field B e and the IMHD action.

IV. THE HAMILTONIAN FORMULATION OF INERTIAL MHD

In this section, we describe the methodology employed in recovering the (Eulerian) noncanonical Hamiltonian picture from the (Lagrangian) canonical action. After obtaining the bracket-Hamiltonian pair, we comment on potential extensions of this framework.

A. Derivation of the inertial MHD bracket

Our first step is the determination of the Hamiltonian, which is done via a Legendre transformation and Eulerianizing the resultant expression. The exercise was already performed in Section III B, and the Hamiltonian is given by ( 14). An alternative route is to invoke Noether's theorem, which also leads to the same result.

Next, we need to obtain the noncanonical bracket. A detailed description of this procedure can be found in [START_REF] Morrison | New Developments in Nonlinear Plasma Physics: Proceedings for the 2009 ICTP College on Plasma Physics[END_REF]; here, we shall merely present the salient details. Before proceeding on to the derivation, we reformulate our observables. We replace the velocity v by the momentum M c , and the entropy s by the entropy density = ⇢s. The new set of observables result in a simpler and compact noncanonical Poisson bracket. Let us recall from Section II A that the Lagrange-Euler maps can be represented in an integral form. We present them below

⇢ = Z d 2 a (r q(a, t)) ⇢ 0 (a), (21)
= Z d 2 a (r q(a, t)) 0 (a), (22)
B j e = Z d 2 a (r q(a, t)) q j ,k B k e0 (a), (23)
M c = Z d 2 a (r q(a, t)) ⇧(a, t). (24)
The last expression is also equivalent to M c = ⇢v, which can be found by computing ⇧ from the Lagrangian, and then obtaining the Eulerian equivalent. We drop the subscript c henceforth, since the canonical momentum M c is the same as the kinetic momentum M = ⇢v. Next, we note that a given functional can be expressed either in terms of the canonical momenta and coordinates, ⇧ and q, or in terms of the observables. Hence, we can denote the former by F and the latter by F , and note that F ⌘ F . As a result, we find that

Z d 2 a F ⇧ • ⇧ + F q • q (25) = Z d 2 r F M • M + F B • B + F ⇢ ⇢ + F .
From (21), we can take the variation on the LHS and RHS, thereby obtanining

⇢ = Z d 2 a ⇢ 0 r (r q(a, t)) • q. (26)
A similar procedure can also be undertaken for ( 22), ( 23) and ( 24) as well. We substitute (26) into the second line of ( 25) and carry out an integration by parts, and a subsequent interchange of the order of integration. This process is repeated for the rest of the variables. By doing so, we can determine the functional derivatives F / q and F / ⇧ in terms of the functional derivatives of the observables. Next, we note that the canonical bracket is given by

{ F , Ḡ} = Z d 2 a ✓ F q • Ḡ ⇧ Ḡ q • F ⇧ ◆ . (27)
We can now substitute the expressions for F / q and F / ⇧, obtained as per the procedure outlined above, into (27) and derive the noncanonical bracket. It is found to be

{F, G} = Z d 2 r " M i ✓ F M j @ j G M i G M j @ j F M i ◆ + ⇢ ✓ F M j @ j G ⇢ G M j @ j F ⇢ ◆ + ✓ F M j @ j G G M j @ j F ◆ + B i e ✓ F M j @ j G B i e G M j @ j F B i e ◆ + B i e ✓ G B j e @ i F M j F B j e @ i G M j ◆ # . (28)
The inertial MHD bracket, derived above, possesses a couple of remarkable features. Firstly, we note that the bracket is precisely identical to the ideal MHD bracket of [18], if we replace B e in (28) with B instead. Secondly, if one replaces M by M c in the above expression, one can obtain an expression for the gyroviscous inertial MHD bracket, yielding results identical to those of [START_REF] Morrison | New Developments in Nonlinear Plasma Physics: Proceedings for the 2009 ICTP College on Plasma Physics[END_REF][START_REF] Morrison | [END_REF]. We must reiterate the importance of the bracket, because it further highlights the merits of B e as a dynamical variable. Our simple postulate in Section III A, that B e behaves as a two form, ensures that inertial MHD and ideal MHD are identical to one another under the exchange B e $ B. Not only does B e yield equations of motion that are highly similar to those of extended MHD, but it also maintains a close connection with ideal MHD via its notion of flux freezing. Owing to the nearidentical nature of the inertial and ideal MHD brackets, an independent analysis of inertial MHD is not required; instead, one can simply migrate the results pertaining to the Casimirs, equilibria and stability of ideal MHD models, by replacing B by B e in the suitable places. In particular, we note that the Casimir

C 1 = Z d 3 r ⇢f (s), (29)
still remains an invariant in inertial MHD. On the other hand, the counterpart of the magnetic helicity of ideal MHD is

C 2 = Z d 3 r A e • B e = Z d 3 r " A • (r ⇥ A) (30)
+ 2m e µ 0 ne 2 B • (r ⇥ B) + m 2 e e 4 ✓ J n ◆ • ✓ r ⇥ ✓ J n ◆◆ # ,
and it is seen that each of the three terms is of the form W • (r ⇥ W ). Note that the terms in the second line vanish when m e /m i ! 0, thereby yielding the ideal MHD magnetic helicity. The cross helicity of ideal MHD morphs into

C 3 = Z d 3 r v • B e = Z d 3 r v • (r ⇥ A e ) (31)
=

Z d 3 r  v • B + m e e 2 v • ✓ r ⇥ ✓ J n ◆◆ ,
and we see that it reduces to the ideal MHD cross helicity if we assume m e /m i ! 0. It is easily seen that the ideal and inertial MHD cross helicities are both expressible as v • (r ⇥ W ).

B. The six-field model and its subcases Hitherto, we have not fully exploited the 2D nature of our model. The choice was deliberate since the bracket and the equations of motion could be expressed in a relatively compact form. However, it comes at the cost of obtaining a narrower class of Casimirs, and an inability to clearly demarcate the behavior of the di↵erent fields. We shall now exploit this 2D symmetry.

First, let us consider B e , defined via (10). We see that it is divergence free. As z serves as our ignorable coordinate, we can immediately express it as

B e = B ez ẑ + r e ⇥ ẑ. (32)
Next, we consider the momentum, recognizing that it involves two components. Hence, the most general possible representation is

M = r + r' ⇥ ẑ. ( 33
)
We note that a similar analysis, albeit in terms of B instead of B e , was carried out in [45,46]. The advantage of inertial MHD is that the bracket is identical in structure to that of ideal MHD under the interchange B e $ B. Upon substituting (32) and ( 33) into (28) and using the functional derivative chain rule, we obtain a bracket identical to that of equation (98) in [45], except for two di↵erences. The bracket obtained involves an integration over d 2 r, as opposed to d 3 r in [45]. Secondly, one must set M z = 0 in equation (98) in [45] as our model lacks the z-component of the velocity.

In summary, we have a 6-field model with our observables given by ( , ', B ez , e , ⇢, ). Each of these six fields are now scalar and possess clear physical interpretations. We can whittle the model down to a 5-field model by assuming it to be isentropic, which eliminates . If we assume incompressibility, we eliminate ⇢ and -the last of which follows from the condition r • v = 0. Lastly, we can eliminate the guide field B ez by making it constant, and our resultant model now involves just ' and e . We introduce the notation ! = ', implying that the two functional derivatives are related via F ! = F ' , and the final bracket is given by

{F, G} = Z d 2 r h ![F ! , G ! ] + e ⇣ [F ! , G e ] [G ! , F e ] ⌘i , (34)
and the corresponding Hamiltonian takes on the form

H = Z d 2 r 1 2  d 2 e (r 2 ) 2 µ 0 + |r | 2 µ 0 + |r'| 2 ⇢ , (35)
where B = r ⇥ ẑ, and the relation between e and is determined via (10). We note that d e represents the electron skin depth. The bracket and Hamiltonian, given by ( 34) and ( 35) are of considerable importance, as they give rise to the well known Ottaviani-Porcelli model [15], used in modelling collisionless magnetic reconnection.

C. Extensions of the inertial MHD bracket

In the preceding subsection, we have obtained the inertial MHD noncanonical bracket, with the corresponding expression given by (28). A crucial feature of inertial MHD was also identified, namely, the close a nity with the ideal MHD bracket, as one can be transformed into the other via B e $ B.

The analogy between B and B e also makes it possible to import the results of 2D gyroviscous MHD, and recast them in an inertial MHD framework. As noted in the previous subsection, the noncanonical brackets derived in [START_REF] Morrison | [END_REF] can be adapted for such a purpose. They are easily distinguishable from the non-gyroviscous brackets owing to the presence of the canonical momentum M c in place of the kinetic momentum M . If the same methodology is employed herein, we can obtain a model for 2D gyroviscous inertial MHD. It must be cautioned, however, that these methods are only applicable to the inertial and ideal brackets, as they are equivalent under B e $ B. Modifying the Hamiltonians is a trickier task, as it requires us to explicitly use the relation (10).

As we have stated thus far, our model of inertial MHD possesses an ignorable coordinate, thereby rendering it 2D. A natural generalization of the procedure is to undertake the same work in a 3D framework. Our central results thus far were the equation of motion (20) and the Ohm's law (19). We find that the former is unmodified, and is identical to that of [13]. However, in the 3D limit, we find that the Ohm's law of [13] and our model are not in agreement, although most of the terms are identical to one another. Next, we consider the incompressible 3D limit, and we find that there is a near-exact match; in fact, we find that the Ohm's law of [13] reduces to our Ohm's law (19) when the flow is irrotational.

V. CONCLUSION

In this paper, we have approached the issue of electron inertia in an unusual manner -via the inclusion of geometric constraints. We generalized the flux freezing condition of ideal MHD, by replacing the vector potential A with an extended vector potential A e , and motivated it via the conservation of canonical momentum. Our model, dubbed 2D inertial MHD, comes with an intrinsic advantage -it is endowed with flux conservation, albeit not for the magnetic field.

2D inertial MHD was shown to possess a couple of pleasing properties. Firstly, it yielded an equation of motion and an Ohm's law that were identical to the ones derived in [13,36], when the 2D case of the latter, in the constant guide field limit, was taken. Secondly, we demonstrated that 2D inertial MHD could be expressed as a six (scalar) field model. A limiting subcase of 2D inertial MHD was shown to reproduce the Ottaviani-Porcelli model [15] of magnetic reconnection. Lastly, we demonstrated the inertial MHD bracket was identical to that of ideal MHD under the interchange B e $ B, thereby cementing the close connection between the two models.

There are several avenues that open up for investigation. It is possible, akin to ideal MHD, to derive expressions for compressible waves and nonlinear Alfvén-like solutions for inertial MHD. A second possibility is to move to the weak 3D limit, and obtain a suitable extension of the Ottaviani-Porcelli model. We expect to tackle such issues in our subsequent work.

the Euler-Poincaré approach is @M c i @t + @T j i @x j = 0, (A1)

where M c is the canonical momentum introduced in (7), and is given by M c i = @L/@v i . The stress tensor in the above equation is given by

T j i = M c i v j + @L @B i e B j e + j i ✓ L ⇢ @L @⇢ B k e @L @B k e ◆ . ( A2
)
Note that the above expression is independent of the variable s. Upon taking the functional derivatives of S with respect to the given variables, and substituting them into (A2) and (A1), we obtain the equation of motion (20).

Acknowledgment

The work of ML and PJM was supported by U.S. Dept. of Energy Contract # DE-FG05-80ET-53088. ET acknowledges financial support from the CNRS through the PEPS project GEOPLASMA 2, and from the Agence Nationale de la Recherche (ANR GYPSI) n. 2010 BLAN 941 03.

Appendix A: The Euler-Poincaré approach to magnetofluids Descriptions of the Euler-Poincaré formalism can be found in [37,38,42]. Let us represent the Lagrangian density given in [START_REF] Lüst | [END_REF] by L. The equation of motion, via


Jeremy J Kiszka
email: jeremy.kiszka@gmail.com

Aurore Aubail

Nigel E Hussey

Michael R Heithaus

Florence Caurant

Paco Bustamante

Plasticity of trophic interactions among sharks from the oceanic south-western Indian Ocean revealed by stable isotope and mercury analyses

Keywords: elasmobranchs, Indian Ocean, δ 15 N, δ 13 C, trace elements, intra-species variations

Introduction

Sharks are a major component of the top predator guild in open-ocean ecosystems along with tunas, billfish, and cetaceans. These "pelagic sharks" occur from cold temperate to tropical waters, from the surface to 1,000 m depth or more [START_REF] Compagno | Pelagic elasmobranch diversity[END_REF]) and include some of the most wide-ranging marine predator species (Pikitch et al. 2008). Their extensive movements and ocean-basin scale migrations are most likely related to oligotrophy and the patchy nature of food resources in open-ocean ecosystems as well as directed movements for social and reproductive purposes. Many pelagic sharks use coastal/continental shelf waters in addition to open oceanic waters throughout ontogeny although certain biological functions, for example, gestation may be restricted to the open ocean [START_REF] Compagno | FAO species catalogue. Vol. 4. Sharks of the world. An annotated and illustrated catalogue of shark specimen known to date. Part 2. Carcharhiniformes[END_REF][START_REF] Bonfil | The biology and ecology of the silky shark, Carcharhinus falciformis[END_REF][START_REF] Nakano | The biology and ecology of the blue shark, Prionace glauca[END_REF]. Even though pelagic sharks commonly occur in an environment far from human populations, they commonly overlap and interact with offshore fisheries. Currently, three-quarters of pelagic elasmobranchs are classified as Threatened or near Threatened (IUCN Red List Status), and 11 species are globally threatened with a high risk of extinction [START_REF] Dulvy | You can swim but you can't hide: the global status of and conservation of pelagic sharks and rays[END_REF]. Despite clear evidence for shark population declines, including oceanic ecosystems [START_REF] Baum | Collapse and conservation of shark populations in the Northwest Atlantic[END_REF], Myers and Worm 2003, Estes et al. 2007[START_REF] Heithaus | Predicting ecological consequences of marine top predator declines[END_REF], relatively little is known on the feeding ecology of many species and the ecological importance of this guild is poorly understood (Ferreti et al. 2010, Heithaus et al. 2010[START_REF] Kitchell | The role of sharks and longline fisheries in a pelagic ecosystem of the central Pacific[END_REF].

Sharks can play important roles in marine ecosystems through diverse mechanisms, but their relative importance may vary significantly among ecosystems, species and contexts [START_REF] Heithaus | Predicting ecological consequences of marine top predator declines[END_REF](Heithaus et al. , 2010)). While the decline of large predatory sharks in some coastal ecosystems may initiate trophic cascades and affect overall community structure (see [START_REF] Heithaus | Predicting ecological consequences of marine top predator declines[END_REF][START_REF] Ferretti | Patterns and ecosystem consequences of shark declines in the ocean[END_REF] for reviews), it is still unclear whether the removal of pelagic sharks affects community structure in open-ocean ecosystems [START_REF] Stillwell | Food, feeding habits, and estimates of daily ration in the shortfin mako (Isurus oxyrinchus) in the Northwest Atlantic[END_REF]Kohler 1982, Ward andMyers 2005). Evidence from ecosystem models from the central Pacific Ocean suggest pelagic shark declines may be mitigated by compensatory increases in populations of large teleosts [START_REF] Kitchell | The role of sharks and longline fisheries in a pelagic ecosystem of the central Pacific[END_REF]. However, these models generally categorize pelagic sharks into a single homogenous trophic group when in fact there may be interspecific variation in habitat use and trophic interactions resulting in greater trophic complexity and less trophic redundancy than assumed for pelagic ecosystems. More ecological data on the relative trophic position and trophic interactions of oceanic sharks is required to test such model assumptions. If species are functionally redundant (i.e. they occupy similar trophic niches, hence consume the same prey base and perform the same functional role; [START_REF] Walker | Biodiversity and ecological redundancy[END_REF], the loss of one or several may not result in large scale community rearrangements because of compensation by other species (e.g. [START_REF] Kitchell | The role of sharks and longline fisheries in a pelagic ecosystem of the central Pacific[END_REF], but rather density compensation effects among the remaining species [START_REF] Walker | Biodiversity and ecological redundancy[END_REF].

Recent studies suggest that sympatric shark species show considerable variation in trophic interactions [START_REF] Hussey | Stable isotope profiles of large marine predators: viable indicators of trophic position, diet and movement in sharks?[END_REF][START_REF] Kinney | Communal or competitive? Stable isotope analysis provides evidence of resource partitioning within a communal shark nursery[END_REF][START_REF] Speed | Trophic ecology of reef sharks determined using stable isotopes and telemetry[END_REF]. However, patterns of trophic redundancy within a given community are likely to be biologically and environmentally context-dependent (e.g. ocean productivity, intra-guild species richness), and are expected to vary accordingly.

In the tropical Indian Ocean (between 10°N and 10°S), the most commonly caught pelagic sharks are blue (Prionace glauca), silky (Carcharhinus falciformis), pelagic thresher (Alopias pelagicus) and short-fin mako sharks (Isurus oxyrinchus). Between 10°S and 25°S, sharks are less abundant and the community is dominated by P. glauca and I. oxyrinchus [START_REF] Huang | Bycatch and discards by Taiwanese large-scale tuna longline fleets in the Indian Ocean[END_REF]. Despite evidence for high bycatch levels of pelagic sharks in the Indian Ocean (Romanov 2002, Huang and[START_REF] Huang | Bycatch and discards by Taiwanese large-scale tuna longline fleets in the Indian Ocean[END_REF], there is a paucity of reliable data to facilitate assessing historical changes in shark catch rate trends [START_REF] Smale | Pelagic shark fisheries in the Indian Ocean[END_REF]. Similarly, knowledge on the trophic ecology of pelagic sharks is limited in the Indian Ocean (and globally), particularly at the community level [START_REF] Rabehagasoa | Isotopic niches of the blue shark Prionace glauca and the silky shark Carcharhinus falciformis in the southwestern Indian Ocean[END_REF].

The use of naturally occurring carbon and nitrogen stable isotopes provide chemical tracers to examine the ecology of organisms in a given ecosystem. Carbon and nitrogen stable isotopic ratios (δ 13 C and δ 15 N) can depict the food webs in which consumers are foraging and their relative trophic position, respectively [START_REF] Hobson | Tracing origins and migration of wildlife using stable isotopes: a review[END_REF]. Stable isotope analyses may provide complementary or, in some cases, an alternative method to traditional diet (e.g. based on analyses of stomach contents) and costly habitat and movement methodologies (e.g. telemetry). They have already been successfully used to understand elasmobranch ecology, including community trophic interactions and ontogenetic variation in trophic position and foraging habitats (Hussey et al. 2011[START_REF] Kinney | Communal or competitive? Stable isotope analysis provides evidence of resource partitioning within a communal shark nursery[END_REF][START_REF] Vaudo | Dietary niche overlap in a nearshore elasmobranch mesopredator community[END_REF].

Mercury (Hg) is a non-essential metal that is released from natural (e.g. volcanism) and anthropogenic sources (e.g. discharge by hydroelectric industries, mining), reaching the open ocean through runoff, riverine input and atmospheric deposition [START_REF] Fitzgerald | Marine biochemical cycling of mercury[END_REF]). Mercury provides an indicator of foraging habitats and trophic position of large marine predators because body burden concentrations are highly correlated to size/age, environmental parameters and geographic location [START_REF] Rivers | Total and organic mercury in marine fish[END_REF][START_REF] Power | Mercury accumulation in the fish community of a sub-Arctic lake in relation to trophic position and carbon sources[END_REF][START_REF] Colaço | Bioaccumulation of Hg, Cu, and Zn in the Azores triple junction hydrothermal vent fields food web[END_REF][START_REF] Cai | Bioaccumulation of mercury in pelagic fishes from the northern Gulf of Mexico[END_REF].

Total Hg levels in pelagic fishes increase with median depth of occurrence in the water column and mesopelagic habitats are probably major entry points of mercury into marine food webs as a result of increased methylation at these depths [START_REF] Monteiro | Mercury concentrations in prey fish indicate enhanced bioaccumulation in mesopelagic environments[END_REF][START_REF] Choy | The influence of depth on mercury levels in pelagic fishes and their prey[END_REF][START_REF] Chouvelon | Enhanced bioaccumulation of mercury in deep-sea fauna from the Bay of Biscay (north-east Atlantic) in relation to trophic positions identified by analysis of carbon and nitrogen isotopes[END_REF].

The aim of this work was to investigate the trophic interactions and potential overlap in foraging habitats of pelagic sharks from the south-western Indian Ocean using three complementary chemical tracers: δ 15 N and δ 13 C and Hg. Because tropical oceanic ecosystems are characterized by low productivity, we hypothesized that trophic plasticity would drive isotopic niche overlap among pelagic shark species, similarly to tropical seabirds and epipelagic tunas from the same region [START_REF] Catry | The role of stable isotopes and mercury concentrations to describe seabird foraging ecology in tropical environments[END_REF][START_REF] Cherel | Resource partitioning within a tropical seabird community: new information from stable isotopes[END_REF], Kojadinovic et al. 2008). As a consequence of diet overlap, we hypothesized that Hg levels would be comparable among species reflecting a similar rate of Hg bioaccumulation. In addition, we also hypothesized that that size and δ 15 N values would be positively correlated with Hg concentrations due to its bioaccumulation with age and its bioamplification within marine food webs.

Materials and methods

Sampling

Shark muscle samples were collected in the south-western Indian Ocean, from southern Madagascar to the Mascarene Islands (Fig. 1). Sharks were sampled during scientific longline fishing cruises or by observers on commercial longline fishing vessels from January 2009 to November 2010. The ecological characteristics of the pelagic shark species sampled are summarized in Table 1. For each individual shark caught, white muscle tissue was sampled from the base of the dorsal fin and immediately stored frozen. Once at the laboratory, the samples were freeze-dried, ground into a homogenous powder using a porcelain mortar and pestle, and then analyzed for stable isotopes (δ 15 N and δ 13 C) and total Hg. A total of 92 samples from 7 species were analyzed, including 31 Prionace glauca, 29 Isurus oxyrinchus, 13 Carcharhinus longimanus, 6 Sphyrna lewini, 5 Alopias pelagicus, 5 Pseudocarcharias kamoharai and 3

Carcharhinus falciformis (Table 2).

Stable isotope analyses

Because lipids are highly depleted in 13 C relative to other tissue components [START_REF] De Niro | Mechanism of carbon fractionation associated with lipid synthesis[END_REF] and urea and trimethylamine oxide (TMAO) present in shark muscle tissue can alter  15 N values, samples were lipid and urea/TMAO-extracted by two successive extractions prior to stable isotope analysis (1 h shaking in 4ml cyclohexane at room temperature and subsequent centrifugation). This standardized data among individuals and across species within the food web (Hussey et al. 2012a(Hussey et al. , 2012b)). After drying, lipid-free sub-samples (0.35 to 0.45 mg ±0.01 mg) were weighed in tin cups for stable isotope analysis. Stable isotope measurements were performed with a continuous-flow isotope-ratio mass spectrometer (Delta V Advantage, Thermo Scientific, Germany) coupled to an elemental analyzer (Flash EA1112 Thermo Scientific, Italy). Reference gas was calibrated against International Reference Materials (IAEA-N1, IAEA-N2 and IAEA-N3 for nitrogen; NBS-21, USGS-24 and IAEA-C6 for carbon). Results are expressed in the  notation relative to PeeDee Belemnite and atmospheric N 2 for δ 13 C and δ 15 N, respectively, according to the equation:

δX = [(R sample / R standard ) -1] × 10 3
Where X is 13 C or 15 N and R is the isotope ratio 13 C/ 12 C or 15 N/ 14 N, respectively. Replicate measurements of a laboratory standard (acetanilide) indicated that analytical errors were <0.1‰ for  13 C and  15 N. Percent C and N elemental composition of tissues were obtained using the elemental analyzer and the calculated C:N ratios were always below 3.5, indicating good lipid removal efficiency (Table 2).

Hg analyses

Total Hg measurements were performed using a solid sample atomic absorption spectrometer AMA-254 (Advanced Mercury Analyser-254 from Altec © ). At least two aliquots of 5 to 15 mg of homogenized dry muscle tissue subsamples for each individual were analyzed. The analytical quality (i.e. accuracy and reproducibility) of the Hg measurements by the AMA-254 was assessed by the analyses of blanks and TORT-2 Certified Reference Material (Lobster Hepatopancreas Reference Material from the National Research Council of Canada) at the beginning and at the end of the analytical cycle, and by running it every 10 samples. Results of quality controls showed a satisfactory precision with a relative standard deviation of 6.0%. The accuracy was 93% of the assigned concentration (n = 14). The detection limit was 0.005 µg g -1 dry weight (dwt). All Hg concentrations in tissues reported below are expressed in µg g -1 dwt.

Trophic position (TP 15N ) calculation

Trophic position (TP) was calculated using a scaled  15 N framework approach based on a dietary δ 15 N value-dependent  15 N model (Hussey et al. 2014a and2014b). In brief, with knowledge of the δ 15 N value of a known baseline consumer (δ 15 N base ), the δ 15 N value of the consumer of interest (δ 15 N TP ), the dietary δ 15 N value at which 15 N incorporation and 15 N elimination are equal (δ 15 N lim ) and the rate at which the ratio between 15 N incorporation and 15 N elimination changes relative to dietary δ 15 N averaged across the food-web (k), TP is calculated as follows:

Mobula japonica, a zooplanktivore [START_REF] Last | Sharks and rays of Australia[END_REF], was used as the baseline consumer (TP=3). Stable isotope values in white muscle samples from four individuals caught in offshore waters off southern Madagascar, during the sampling period, were (mean δ 15 N ± SD) 9.7 ± 0.1, similar to those reported for Mobula spp. off the east coast of South Africa (9.9 ± 0.5; Hussey et al. 2011). A value of k = 0.14 and δ 15 N lim = 21.9 were used following the meta-analysis of experimental isotopic studies conducted by Hussey et al. (2014a and2014b).

Data analysis

We used Shapiro-Wilk tests to assess the assumptions of parametric tests. Hg data were logtransformed to reduce skewness and fit parametric requirements. ANOVA were used to test the influence of size (fork length), latitude, longitude, sex and species on δ 13 C and δ 15 N values and on log-transformed Hg concentrations. Post-hoc Tukey HSD (Honesty Significance Difference) tests were used to investigate paired differences for significant factors. To investigate isotopic niche space and isotopic niche overlap among species we employed a Bayesian approach based on multivariate ellipse metrics (SIBER -Stable Isotope Bayesian Ellipses within the R-package siar; Jackson et al. 2011). Standard ellipses are bivariate data calculated from the variance and covariance of the data matrix (x, y parameters) and represent the core isotopic niche for a species. This analysis generates standard ellipse areas corrected for small sample sizes (SEAc) which can

TP = log(d 15 N lim -d 15 N base ) -log(d 15 N lim -d 15 N TP ) k + TP base
be compared among species to determine differences in core isotopic niche space and niche overlap (Jackson et al. 2011). The calculation of standard ellipses (SEAc) was undertaken for δ 15 N vs. δ 13 C data, but also for combinations of δ 13 C vs. Hg and δ 15 N vs. Hg data to provide complementary tracer information. The degree of niche overlap among species for each combination of the above data parameters, i.e. the overlap in species' standard ellipse (SEAc)

areas was then calculated using the OVERLAP command. Pearson's correlations were used to examine  13 C and  15 N values vs. log Hg concentrations, as well as the relationship between  13 C and  15 N values. Statistical analyses were performed using R 2.12.0 (R Development Core Team, 2010).

Results

Inter-species variations

Values of δ 15 N and δ 13 C and Hg concentrations varied significantly among species (one-way ANOVA, F 6, 85 = 5.94; p < 0.001; F 6,85 = 11.23, p < 0.001; F 6, 85 = 5.41, p < 0.001, respectively;

Figs. 2 and 3, Table 2). There were, however, relatively few significant pairwise comparisons (Table 3). For  15 N, I. oxyrinchus had significantly higher values than C. falciformis (p < 0.01),

C. longimanus (p < 0.01) and P. glauca (p < 0.001). For  13 C, the only significant pairwise comparison was between P. glauca and all species except C. falciformis. Concerning log Hg concentrations, pairwise comparisons between P. kamoharai and the other species were all significant (Table 3). Moreover, log Hg concentrations were significantly higher in I. oxyrinchus than in C. falciformis (p < 0.01, Table 3).

Standard  15 N - 13 C ellipses (SEAc) identified considerable niche overlap among species, with few exceptions (Fig. 4, Table 4). For P. glauca, no ellipse overlap occurred with C. longimanus, P. kamoharai and S. lewini and there was very limited overlap with A. pelagicus (Fig. 4). S.

lewini showed a high degree of niche overlap with C. longimanus and P. kamoharai. Niche space represented by ellipse size was the largest for S. lewini (2.7), similar for I. oxyrinchus, P. glauca, C. longimanus and A. pelagicus (1.0, 1.2, 1.5 and 1.2, respectively) and smallest for P. kamoharai (0.3, Fig. 4; Table 4). When considering the  13 C vs. Hg and  15 N vs. Hg standard ellipses (SEAc; Figs. 5 and6), P. kamoharai showed significant niche space separation from all other species with the exception of C. longimanus (Tables 6 and7). Most species showed considerable niche overlap in terms of Hg vs.  13 C and Hg vs.  15 N space (Figs. 5 and 6; Tables 5 and6). For both  13 C vs. Hg and  15 N vs. Hg, C. longimanus had the largest ellipse area (12.3 and 27.4, respectively), while P. glauca had the smallest (2.3 and 5.3, respectively; Figs. 5 and6;

Table 5 and6).

Estimated TPs of all pelagic shark species based on δ 15 N values (TP 15N ) spanned 1.8 trophic positions ranging from TP 3.5 for a 99 cm FL C. longimanus to TP 5.3 for a 173 cm FL I.

oxyrinchus (Table 2). Overall I. oxyrinchus fed at the highest mean TP (4.7 SD ± 0.3) and C.

falciformis at the lowest mean TP (3.8 ± 0.1), with all other species feeding at a similar level (mean TP ~4.3). High intraspecific variation in TP 15N was observed with most species estimated to be feeding over one trophic level.

Geographic, size and sex-related variation

Values of  13 C, δ 15 N and Hg, increased with increasing longitude for I. oxyrinchus (F 1, 90 = 6.41, p < 0.05). For P. glauca, there were significant increasing log Hg concentrations when latitude increased (F 1, 90 = 13.93, p < 0.001) and δ 13 C values decreased with increasing latitude (F 1, 90 = 10.31, p < 0.01). A significant positive effect of longitude on δ 15 N values was also detected in P.

glauca (F 1, 90 = 8.12, p < 0.01). When considering all species combined, there was a significant positive effect of size on log Hg concentrations (one-way ANOVA, F 1, 90 = 7.03, p < 0.01) and on δ 13 C values (F 1, 90 = 6.48, p < 0.05), but not for δ 15 N (F 1, 90 = 3.70, p > 0.05).

At the species level, the effect of size on log Hg concentrations was significant for I. oxyrinchus (F 1, 27 = 16.1, p < 0.001, Fig. 7a), P. glauca (F 1, 29 = 23.7, p < 0.001, Fig. 7b) and C. longimanus (F 1, 11 = 15.9, p < 0.01, Fig. 7c). There was a positive relationship between size and both δ 15 N and δ 13 C in P. glauca (F 1, 11 = 11.3, p < 0.01 and F 1, 11 = 25.1, p < 0.001, respectively, Fig. 7b), but not for any other species examined. There was no detectable variation in δ 15 N and δ 13 C values and Hg concentrations between sexes (one-way ANOVA for log Hg, δ 13 C and δ 15 N; all p > 0.05).

Relationships between  15 N,  13 C and Hg concentrations

For all data combined, there was a significant correlation between log Hg concentrations and  15 N values (Pearson: r = 0.38, df = 90, p < 0.001) but not for  13 C (r = 0.12, df = 90, p > 0.05, Table 7). At the species level, positive correlations between log Hg and  15 N, and log Hg and  13 C were only significant for P. glauca (r = 0.64, df = 29, p < 0.001 and r = 0.71, p < 0.001, respectively). There was a significant positive correlation between  13 C and  15 N for C.

13 falciformis (r = -0.99, p < 0.05), P. glauca (r = 0.71, p < 0.001) and I. oxyrinchus (r = 0.67, df = 27, p < 0.001; Table 7).

Discussion

Understanding interspecific variation in trophic interactions and foraging base of pelagic sharks based on size, sex and geographic location is important for elucidating the role of this predator guild in pelagic ecosystems. Traditionally considered to be a trophically redundant group, our data suggest the potential for high levels of intraspecific variation in trophic interactions or foraging locations that obscures differences among species within the guild as a whole. These data identify species trophic plasticity as would be predicted in the oligotrophic pelagic realm of the western Indian Ocean. In addition, this study extends our understanding of ontogenetic shifts (studied for P. glauca in the Mozambique Channel and off eastern Madagascar; [START_REF] Rabehagasoa | Isotopic niches of the blue shark Prionace glauca and the silky shark Carcharhinus falciformis in the southwestern Indian Ocean[END_REF], and provides the first Hg concentrations for this data poor region. However, considering the limited sample size for a number of species, our results should be interpreted with caution.

Patterns of isotopic niche partitioning

Intraspecific variation in estimated trophic positions was considerable, with most species feeding over more than a single trophic level. However, as hypothesized and assumed by models of pelagic ecosystems, interspecific variation in average trophic positions was limited among species in our study area. The notable exception to this pattern was I. oxyrinchus, which exhibited significantly higher δ 15 N values, likely because of their propensity to forage on large predatory fish, including small elasmobranchs and marine mammals [START_REF] Cliff | Sharks caught in the protective gill nets off Natal, South Africa. 3. The shortfin mako shark Isurus oxyrinchus (Rafinesque)[END_REF]). Our mean species TP results provided similar estimates to those based on stomach contents by Cortes (1999), but overall the TP range was slightly higher, a result of using the more accurate scaled ∆ 15 N framework (Hussey et al. 2014a(Hussey et al. , 2014b)). Nonetheless, our results suggest that pelagic sharks from the study region generally occupy similar trophic positions, and that intraspecific variation is much greater than interspecific variation. Surprisingly, calculated TP 15N for P. kamoharai was relatively high (mean value=4.5) given the small body size of this species (<100 cm). Stomach contents of P. kamoharai suggest a diet of small mesopelagic teleosts, cephalopods and shrimps [START_REF] Compagno | FAO species catalogue. Vol. 4. Sharks of the world. An annotated and illustrated catalogue of shark specimen known to date. Part 2. Carcharhiniformes[END_REF]. The high δ 15 N values and estimated trophic position for this species may be a result of a higher nitrogen baseline value in the deep-sea food web where this species forages.

This would bias the comparison of δ 15 N values and estimated TP among species, given the other species feed predominantly in the epipelagic zone. Alternatively, this could suggest the existence of a longer food chain at smaller body sizes in these deep-water environments than previously anticipated. Further analyses are required in the future to test these hypotheses.

Most species had similar δ 13 C values suggesting a high degree of overlap in habitat use. The principal exception was P. glauca which was more depleted in 13 C compared to the other species.

There is a limited diversity of carbon sources in open-ocean systems, particularly in oligotrophic environments, and detrital carbon sources are likely to be the most available. δ 13 C values commonly decrease from productive continental shelf waters to pelagic systems with coastal species having amore enriched 13 C values [START_REF] Cherel | Resource partitioning within a tropical seabird community: new information from stable isotopes[END_REF][START_REF] Kiszka | Ecological niche segregation within a community of sympatric dolphins around a tropical island[END_REF][START_REF] Chouvelon | Enhanced bioaccumulation of mercury in deep-sea fauna from the Bay of Biscay (north-east Atlantic) in relation to trophic positions identified by analysis of carbon and nitrogen isotopes[END_REF]). The δ 13 C values of most shark species were intermediate to these environments, indicating the use of both pelagic and coastal waters [START_REF] Hussey | Stable isotope profiles of large marine predators: viable indicators of trophic position, diet and movement in sharks?[END_REF]. Lower δ 13 C values of P. glauca, indicate this species spends longer periods of time in pelagic vs. coastal waters and/or forages in more 13 C-depleted deeper waters. This is supported by satellite tracking data of P. glauca in the North Atlantic (Campana et al. 2011).

Overall, our results confirm high core isotopic niche overlap among pelagic sharks in the western Indian Ocean. A similar pattern was previously reported for other pelagic sharks in the temperate waters of the North Atlantic, specifically P. glauca and I. oxyrinchus [START_REF] Estrada | Predicting trophic position of sharks in the north-west Atlantic Ocean using stable isotope analysis[END_REF]) and other oceanic top predator guilds, including tunas and tropical seabirds in the western Indian

Ocean [START_REF] Catry | The role of stable isotopes and mercury concentrations to describe seabird foraging ecology in tropical environments[END_REF][START_REF] Cherel | Resource partitioning within a tropical seabird community: new information from stable isotopes[END_REF], Kojadinovic et al. 2008). This degree of overlap in core isotopic niches of pelagic sharks and other pelagic predator guilds is likely driven largely by the homogeneity of carbon sources in oligotrophic open-ocean ecosystems, characterized by low productivity and little seasonality [START_REF] Cherel | Resource partitioning within a tropical seabird community: new information from stable isotopes[END_REF].

Hg as a complementary dietary tracer

Hg levels which provide insights into the trophic position of organisms (e.g. [START_REF] Mcmeans | Diet and resource use among Greenland sharks (Somniosus microcephalus) and teleosts sampled in Icelandic waters using δ 13 C, δ 15 N, and mercury[END_REF], were relatively similar among species. A positive relationship between Hg concentrations and δ 15 N has been previously documented for many marine taxa, including pelagic sharks from other regions in the Atlantic and Pacific Oceans (e.g. [START_REF] Branco | Total and organic mercury concentrations in muscle tissues of the blue shark (Prionace glauca L. 1758) from the northeast Atlantic[END_REF][START_REF] Maz-Courrau | Bioaccumulation and biomagnification of total mercury in four exploited shark species in the Baja California peninsula, Mexico[END_REF]. Our results for P. glauca, I. oxyrinchus and C. longimanus (species with satisfactory sample size) are consistent with this pattern. Overall, Hg concentrations in muscle tissues were very high in pelagic sharks sampled in the south-western Indian Ocean (mean = 6.34 µg.g -1 dw), compared to other pelagic fish from the same region, including Xiphias gladius (mean = 3.97 µg.g -1 dw), Thunnus albacares (1.15 µg.g -1 dw), Thunnus obesus (0.67 µg.g -1 dw) and

Coryphaena hippurus (0.21 µg.g -1 dw) [START_REF] Kojadinovic | Mercury content in commercial pelagic fish and its risk assessment in the Western Indian Ocean[END_REF]. The total Hg levels in blue shark muscle samples of a similar size were also much higher in the south-western Indian Ocean compared to those reported off Baja California (central Pacific Ocean) (1.96±1.48 µg.g -1 dw vs.

5.3 ±2.17 in our study, [START_REF] Maz-Courrau | Bioaccumulation and biomagnification of total mercury in four exploited shark species in the Baja California peninsula, Mexico[END_REF]). These concentrations highlight not only the high trophic position of these species but the potential risks of shark meat consumption for human health that need to be further investigated.

The high Hg concentrations observed in P. kamoharai (mean = 17.25 µg.g -1 dw) were unexpected, since this species appears to feed at a similar trophic position to most other pelagic species, based on δ 15 N despite being much smaller in size (<100 cm). In addition the mean δ 13 C value of P. kamoharai was also similar to other species indicating it is unlikely feeding from a different carbon pool. P. kamoharai is a poorly known shark species that occurs in oceanic circumtropical waters around the world [START_REF] Compagno | FAO species catalogue. Vol. 4. Sharks of the world. An annotated and illustrated catalogue of shark specimen known to date. Part 2. Carcharhiniformes[END_REF]. The morphology of the P. kamoharai, particularly large eye size, suggests a deep-water habitat for this species, but with movements toward the surface at night [START_REF] Compagno | FAO species catalogue. Vol. 4. Sharks of the world. An annotated and illustrated catalogue of shark specimen known to date. Part 2. Carcharhiniformes[END_REF]. Methyl-Hg is produced by the methylation of inorganic Hg in deep oceanic waters [START_REF] Blum | Methylmercury production below the mixed layer in the North Pacific Ocean[END_REF], resulting in its enhanced bioaccumulation in mesopelagic organisms [START_REF] Monteiro | Mercury concentrations in prey fish indicate enhanced bioaccumulation in mesopelagic environments[END_REF][START_REF] Chouvelon | Enhanced bioaccumulation of mercury in deep-sea fauna from the Bay of Biscay (north-east Atlantic) in relation to trophic positions identified by analysis of carbon and nitrogen isotopes[END_REF]. Therefore, top marine predators feeding on mesopelagic prey, such as seabirds and large predatory fishes, exhibit significantly higher concentrations than epipelagic predators [START_REF] Thompson | Seabirds as biomonitors of mercury inputs to epipelagic and mesopelagic marine food chains[END_REF][START_REF] Kojadinovic | Mercury content in commercial pelagic fish and its risk assessment in the Western Indian Ocean[END_REF][START_REF] Choy | The influence of depth on mercury levels in pelagic fishes and their prey[END_REF]. In this respect, Hg can be considered as an indicator of the feeding depth [START_REF] Carravieri | Penguins as bioindicators of mercury contamination in the Southern Ocean: birds from the Kerguelen Islands as a case study[END_REF]). Higher Hg concentrations in P. kamoharai, are most likely due to its occupying a significantly deeper feeding habitat than the other, more surface-oriented, pelagic species sampled.

Spatial and ontogenetic variation

Spatial variation in δ 13 C and δ 15 N values and Hg concentrations of individual sharks sampled were not consistent, especially for P. glauca and I. oxyrinchus. Drawing conclusions on these inconsistent patterns is difficult because there are currently no complementary independent datasets from this region to provide insights into isoscapes or geographical patterns of Hg concentrations. Information on size and sex-based differences in trophic interactions of marine top predators, including sharks, is critical to better understand ecosystem function and to assess the ecological role of these organisms. Ontogenetic shifts in diet are common in elasmobranchs, for large predatory species such as tiger (Galeocerdo cuvier, [START_REF] Lowe | Ontogenetic dietary shifts and feeding behavior of tiger sharks, Galeocerdo cuvier, in Hawaiian waters[END_REF] and great white sharks (Carcharodon carcharias, [START_REF] Estrada | Use of isotopic analysis from vertebrae in reconstructing ontogenetic feeding ecology in white sharks[END_REF] to zooplanktivorous whale sharks (Rhincodon typus, [START_REF] Borrell | Stable isotope profiles in whale shark (Rhincodon typus) suggest segregation in the diet depending on sex and size[END_REF]. Ontogenetic variation in the foraging ecology and habitat use of most pelagic sharks investigated in this study are poorly known, except for P. glauca, including in the western Indian Ocean. Previous studies suggest a significant increase in δ 13 C and δ 15 N values with body length in P. glauca in the surrounding oceanic waters of Madagascar, but not in C. falciformis [START_REF] Rabehagasoa | Isotopic niches of the blue shark Prionace glauca and the silky shark Carcharhinus falciformis in the southwestern Indian Ocean[END_REF]. Our data suggest an identical pattern for P.

glauca (not tested for C. falciformis due to small sample size), which is also consistent with a significant dietary shift with increasing size based on stomach content analyses in Baja California [START_REF] Markaida | Food and feeding habits of the blue shark Prionace glauca caught off Ensenada, Baja California, Mexico, with a review on its feeding[END_REF].

Conclusion

In this study, we found a high degree of isotopic niche overlap among pelagic sharks in the oligotrophic waters of the south-western Indian Ocean that is consistent with previous studies on other epipelagic marine top predators. Nevertheless, our data also found large intra-species variation in δ 15 N values identifying dietary and foraging habitat plasticity that masks variation among species. This large intra-species variation in δ 15 N values would suggest pelagic species are either opportunistic feeders or have evolved variable and complex feeding behaviors, potentially related to variable regional prey bases and the migratory nature of these animals. Nevertheless a high degree of dietary plasticity would be expected in oligotrophic waters and intra-species variation should be investigated further in other pelagic guilds. Although, at the species level, significant core isotopic and Hg niche overlap among species is consistent with the hypothesis of trophic redundancy, these results must be viewed with caution due to limited differentiation in carbon pools in this ecosystem coupled with high intra-species variation.

Alopias pelagicus

Epipelagic, oceanic 0 -152 Small-to medium sized fish and squids [START_REF] Compagno | FAO species catalogue. Vol. 4. Sharks of the world. An annotated and illustrated catalogue of shark specimen known to date. Part 2. Carcharhiniformes[END_REF], [START_REF] Smith | The biology and ecology of thresher sharks (Alopiidae)[END_REF] Carcharhinus falciformis Epipelagic, oceanic 0 -500 Large squids, pelagic fishes, pelagic crabs [START_REF] Compagno | FAO species catalogue. Vol. 4. Sharks of the world. An annotated and illustrated catalogue of shark specimen known to date. Part 2. Carcharhiniformes[END_REF], [START_REF] Bonfil | The biology and ecology of the silky shark, Carcharhinus falciformis[END_REF], Cabrera-Chavez et al. ( 2010)

Carcharhinus longimanus

Epipelagic, oceanic 0 -152 Large epipelagic teleosts and squids [START_REF] Compagno | FAO species catalogue. Vol. 4. Sharks of the world. An annotated and illustrated catalogue of shark specimen known to date. Part 2. Carcharhiniformes[END_REF], [START_REF] Bonfil | The biology and ecology of the oceanic whitetip shark, Carcharhinus longimanus[END_REF] Prionace glauca Epi-to mesopelagic, oceanic 0 -600 Small pelagic fishes, squids and elasmobranchs [START_REF] Compagno | FAO species catalogue. Vol. 4. Sharks of the world. An annotated and illustrated catalogue of shark specimen known to date. Part 2. Carcharhiniformes[END_REF], [START_REF] Nakano | The biology and ecology of the blue shark, Prionace glauca[END_REF] Isurus oxyrinchus Epi-to mesopelagic, oceanic 0 -600 Pelagic fishes, squids and elasmobranchs [START_REF] Stevens | The biology and ecology of the shortfin mako shark, Isurus oxyrinchus. In: Sharks of the Open Ocean[END_REF] Pseudocarcharias kamoharai Mesopelagic 0 -600 Deep-water fishes, squids and shrimps [START_REF] Compagno | FAO species catalogue. Vol. 4. Sharks of the world. An annotated and illustrated catalogue of shark specimen known to date. Part 2. Carcharhiniformes[END_REF], [START_REF] Stevens | Epipelagic oceanic elasmobranchs[END_REF] Sphyrna lewini Coastal and semioceanic 0 -450 Pelagic to demersal fishes, squids and elasmobranchs [START_REF] Compagno | FAO species catalogue. Vol. 4. Sharks of the world. An annotated and illustrated catalogue of shark specimen known to date. Part 2. Carcharhiniformes[END_REF], [START_REF] Kimley | Highly directional swimming by scalloped hammerhead sharks, Sphyrna lewini, and subsurface irradiance, temperature, bathymetry and geomagnetic field[END_REF], [START_REF] Hussey | Stable isotope profiles of large marine predators: viable indicators of trophic position, diet and movement in sharks?[END_REF]

Fig. 1 .

 1 Fig. 1. Distribution of sampled oceanic sharks in the south-western Indian Ocean (n = 92).

Fig. 2 .

 2 Fig. 2. Relationship (means ± SD) between δ 15 N and δ 13 C values (‰) of muscle.

Fig. 3 .

 3 Fig. 3. Comparisons of Hg concentrations in muscle (in μg.g -1 dw) of C. falciformis (Cf), A. pelagicus (Ap), C. longimanus (Cl), P. glauca (Pg), S. lewini (Sl), I. oxyrinchus (Io) and P. kamoharai (Pk). The upper and lower hinges represent the quartiles, the vertical lines, the maximum and minimum data values and the bold line represents the median value while the points are outliers. Horizontal lines represent homogeneous groups determined by Tukey HSD (Honestly Significant Difference) test.

Fig. 4 .

 4 Fig. 4. Standard ellipses for corrected sample sizes (SEAc, in ‰, δ 15 N and δ 13 C) of A. pelagicus, C. longimanus, P. glauca, S. lewini, I. oxyrinchus and P. kamoharai.

Fig. 5 .

 5 Fig. 5. Standard ellipses for corrected sample sizes (SEAc, in ‰, δ 13 C) and Hg concentrations (in μg.g -1 dw) of A. pelagicus, C. longimanus, P. glauca, S. lewini, I. oxyrinchus and P. kamoharai.

Fig. 6 .

 6 Fig. 6. Standard ellipses for corrected sample sizes (SEAc, in ‰, δ 15 N) and Hg concentrations (in μg.g -1 dw) of A. pelagicus, C. longimanus, P. glauca, S. lewini, I. oxyrinchus and P. kamoharai.

Fig. 7 .

 7 Fig. 7. Influence of size (FL: fork length) on log Hg concentrations (top), δ 15 N (middle) and δ 13 C values (bottom) in a-I. oxyrinchus, b-P. glauca and c-C. longimanus.

Table 1 :

 1 Ecological characteristics ( * minimal depth range, ** major prey types) of pelagic shark species investigated in this study.

	Species	Habitat	Depth range	Diet**	References
			(m)*

Table 2 :

 2 Characteristics of pelagic sharks sampled in the south-western Indian Ocean. Values are mean (±SD). TP 15N = trophic position based onHussey et al. (2014a and 2014b).

	Species	n	Fork Length (cm)	Sex ratio	δ 15 N (‰)	δ 13 C (‰)	C:N	Hg (µg.g -1 dw)	TP15N
			Mean	Range	(m:f)	Mean ± SD	Mean ± SD	Mean ± SD	Mean ± SD	Mean ± SD
	Alopias pelagicus	5	200	147 -268	1:0	13.91 ± 0.69	-16.71 ± 0.43	2.9 ± 0.02	4.97 ± 3.60	4.4 ± 0.3
										(3.9 -4.7)
	Carcharhinus falciformis	3	114	96 -148	2:1	12.35 ± 0.36	-16.53 ± 0.35	2.9 ± 0.1	2.43 ± 2.15	3.8 ± 0.1
										(3.7 -3.9)
	Carcharhinus longimanus	13	161	99 -229	0.2:1	13.39 ± 1.03	-16.46 ± 0.44	2.9 ± 0.1	7.14 ± 7.44	4.2 ± 0.4
										(3.5 -4.7)
	Prionace glauca	31	202	160 -269	1:0.8	13.57 ± 1.10	-17.47 ± 0.49	2.8 ± 0.2	5.30 ± 2.17	4.2 ± 0.4
										(3.6 -5.0)
	Isurus oxyrinchus	29	185	122 -304	0.8:1	14.62 ± 0.68	-16.74 ± 0.58	2.9 ± 0.1	5.96 ± 2.78	4.7 ± 0.3
										(3.9 -5.3)
	Pseudocarcharias kamoharai	5	87	82 -95	1:0.7	14.35 ± 0.57	-16.58 ± 0.57	2.9 ± 0.1	17.25 ± 6.45	4.5 ± 0.2
										(4.3 -4.8)
	Sphyrna lewini	6	152	100 -190	0.5:1	13.88 ± 1.01	-16.12 ± 0.69	3.0 ± 0.1	5.86 ± 2.73	4.4 ± 0.4
										(3.9 -5.0)

Table 3 :

 3 Mean differences and p-values based on post hoc tests Tukey's tests for paired comparisons of Hg concentrations (top), δ 15 N (middle), and δ 13 C (bottom). Significant differences are in bold.

		A. pelagicus	C. falciformis	C. longimanus	I. oxyrinchus	P. glauca	P. kamoharai
	Log Hg
	C. falciformis	-0.701, 0.612
	C. longimanus	0.246, 0.981	0.947, 0.129
	I. oxyrinchus	0.368, 0.824	1.069, <0.05	0.122, 0.995
	P. glauca	0.274, 0.970	0.946, 0.090	-0.0004, 0.999	-0.122, 0.995
	P. kamoharai	1.443, <0.01	2.143, <0.001	1.197, <0.01	1.075, <0.01	1.197, <0.001
	S. lewini	0.342, 0.952	1.042, 0.131	0.096, 0.999	-0.026, 0.999	0.096, 0.999	-1.101, <0.05
	δ 15 N
	C. falciformis	-1.561, 0.243
	C. longimanus	-0.523, 0.931	1.037, 0.576
	I. oxyrinchus	0.710, 0.684	2.271, <0.01	1.233, <0.01
	P. glauca	-0.340, 0.987	1.221, 0.307	0.183, 0.997	-1.050, <0.001
	P. kamoharai	0.434, 0.989	1.995, 0.056	0.957, 0.434	-0.276, 0.996	0.774, 0.585
	S. lewini	-0.036, 1.000	1.525, 0.234	0.488, 0.933	-0.746, 0.544	0.304, 0.989	-0.470, 0.980
	δ 13 C
	C. falciformis	0.175, 0.999
	C. longimanus	0.244, 0.973	0.068, 0.999
	I. oxyrinchus	-0.035, 0.999	-0.210, 0.994	-0.278, 0.676
	P. glauca	-0.765, <0.05	-0.940, 0.052	-1.009, <0.001	-0.730, <0.001
	P. kamoharai	0.126, 0.999	-0.049, 0.999	-0.118, 0.999	0.161, 0.995	0.891, <0.05
	S. lewini	0.585, 0.508	0.410, 0.920	0.342, 0.833	0.620, 0.119	1.350, <0.001	0.459, 0.765

Table 4 :

 4 Percentage of δ 13 C vs. δ 15 N overlap of corrected standard ellipses (SEAc) among the six species investigated; note two overlap values per species comparison are given, e.g. I. oxyrinchus vs. C. longimanus and vice versa.

	Species	C. longimanus	I. oxyrinchus	P. glauca	P. kamoharai	S. lewini	A. pelagicus
	C. longimanus	*	23.9	1.8	62.5	38.6	65.7
	I. oxyrinchus	15.4	*	5.7	83.4	16.9	32.0
	P. glauca	2.9	1.4	*	0	1.5	8.7
	P. kamoharai	11.3	23.5	0	*	8.5	17.2
	S. lewini	71.1	48.4	1.7	87.0	*	64.7
	A. pelagicus	54.1	41	4.5	78.2	28.9	*

Table 5 :

 5 Percentage of Hg vs. δ 13 C overlap of corrected standard ellipses (SEAc) among the six species investigated; note two overlap values per species comparison are given, e.g. I. oxyrinchus vs. C. longimanus and vice versa.

	Species	C. longimanus	I. oxyrinchus	P. glauca	P. kamoharai	S. lewini	A. pelagicus
	C. longimanus	*	68.3	8.1	34.7	59.8	75.9
	I. oxyrinchus	29.3	*	25.8	2	39.3	69.6
	P. glauca	1.8	13.1	*	0	1.4	1.5
	P. kamoharai	10.0	1.4	0	*	1.1	2.7
	S. lewini	29.0	44.4	3.1	1.8	*	20.5
	A. pelagicus	26.0	55.7	2.4	3.2	14.5	*

Table 6 :

 6 Percentage of Hg vs. δ 15 N overlap of corrected standard ellipses (SEAc) among the six species investigated; note two overlap values per species comparison are given, e.g. I. oxyrinchus vs. C. longimanus and vice versa.

	Species	C. longimanus	I. oxyrinchus	P. glauca	P. kamoharai	S. lewini	A. pelagicus
	C. longimanus	*	36.4	63.6	6.3	82.4	91.1
	I. oxyrinchus	8.1	*	33.2	6.6	38.2	28
	P. glauca	21.7	50.6	*	1.2	56.7	44.9
	P. kamoharai	3.3	1.6	1.9	*	7.6	6.4
	S. lewini	32.4	67.2	65.4	5.7	*	77.9
	A. pelagicus	34.1	46.9	49.3	4.5	74.2	*

Table 7 :

 7 Correlations between δ 13 C, δ 15 N values and log Hg concentrations of pelagic sharks based on Pearson's correlations. Significant values are bold.

	Pearson	Log Hg/δ 15 N		Log Hg/δ 13 C		δ 15 N/δ 13 C
	Species	t, p-value	r	t, p-value	r	t, p-value	r
	A. pelagicus	-0.79, 0.49	-0.41	2.67, 0.08	0.84	-0.35, 0.75	-0.20
	C. falciformis	1.61, 0.35	0.85	-1.77, 0.33	-0.87	-25.06, <0.05	0.99
	C. longimanus	1.86, 0.09	0.49	-0.47, 0.65	-0.14	1.71, 0.11	0.46
	I. oxyrinchus	0.32, 0.75	0.06	0.28, 0.78	0.05	4.67, >0.001	0.67
	P. glauca	4.49, <0.001	0.64	5.37, <0.001	0.71	5.43, <0.001	0.71
	P. kamoharai	0.97, 0.40	0.49	-1.13, 0.34	-0.55	-1.50, 0.23	-0.65
	S. lewini	0.15, 0.89	0.07	-1.39, 0.24	-0.57	0.19, 0.86	0.10
	All species	3.84, <0.001	0.38	1.11, 0.27	0.12	4.27, <0.001	0.41

Acknowledgements

Our particular thanks are addressed to sample collectors during longline fishing cruises and IFREMER (IOSSS-Espadon project) for providing shark samples. Special thanks are addressed to J. Bourjea (IFREMER, La Réunion) for coordinating the collation of shark samples. We are also very grateful to G. Guillou and P. Richard (University of La Rochelle, lab. LIENSs) for running the mass spectrometer analyses. Sample analysis was supported by the CPER (Contrat de Projet Etat-Région). Ward, P., Myers, R.A. 2005. Shifts in open-ocean fish communities coinciding with the commencement of commercial fishing. Ecology 86, 835-847. Zar, J.H. 2009. Biostatistical analysis: International edition. Pearson Education. Zeitzschel, B. 1969. Primary productivity in the Gulf of California. Marine Biology 3, 201-207.