2.1 Evolution equations

The barotropic XMHD equations, presented in a series of recent articles (Kimura & Morrison Reference Kimura and Morrison2014; Abdelhamid et al. Reference Abdelhamid, Kawazura and Yoshida2015; Lingam et al. Reference Lingam, Morrison and Miloshevich2015b ; Grasso et al. Reference Grasso, Tassi, Abdelhamid and Morrison2017; Kaltsas et al. Reference Kaltsas, Throumoulopoulos and Morrison2017), in Alfvén units, are given by:

where

The parameters $d_{i}$ and $d_{e}$ are the normalized ion and electron skin depths respectively, $p=p(\unicode[STIX]{x1D70C})$ is the total pressure and $\unicode[STIX]{x1D70C}$ , $\boldsymbol{v}$ , $\boldsymbol{B}$ and $\boldsymbol{J}$ represent the mass density, the velocity, the magnetic field and the current density, respectively.

2.2 Hamiltonian formulation

It has been recognized that (2.1)–(2.3) possess a non-canonical Hamiltonian structure, i.e. the dynamics can be described by a set of generalized Hamiltonian equations (Morrison Reference Morrison1982, Reference Morrison1998)

where $\unicode[STIX]{x1D702}=(\unicode[STIX]{x1D70C},\boldsymbol{v},\boldsymbol{B}^{\ast })$ are non-canonical dynamical variables (not consisting of canonically conjugate pairs), ${\mathcal{H}}[\unicode[STIX]{x1D70C},\boldsymbol{v},\boldsymbol{B}^{\ast }]$ is a real valued Hamiltonian functional and $\{F,G\}$ is a Poisson bracket acting on functionals of the variables $\unicode[STIX]{x1D702}$ , which is bilinear, antisymmetric, and satisfies the Jacobi identity. The appropriate Hamiltonian for our system is the following:

where $D\subseteq \mathbb{R}^{3}$ and $U$ is the internal energy function ( $p=\unicode[STIX]{x1D70C}^{2}\,\text{d}U/\text{d}\unicode[STIX]{x1D70C}$ ), while the corresponding non-canonical Poisson bracket is

where $F_{\boldsymbol{z}}:=\unicode[STIX]{x1D6FF}F/\unicode[STIX]{x1D6FF}\boldsymbol{z}$ denotes the functional derivative of $F$ with respect to the dynamical variable $\boldsymbol{z}$ , defined by $\unicode[STIX]{x1D6FF}F[\boldsymbol{z},\unicode[STIX]{x1D6FF}\boldsymbol{z}]=\int _{D}\,\text{d}^{3}x\,\unicode[STIX]{x1D6FF}\boldsymbol{z}\boldsymbol{\cdot }(\unicode[STIX]{x1D6FF}F/\unicode[STIX]{x1D6FF}\boldsymbol{z})$ . For the computation of the functional derivatives of the field variables we make use of $\unicode[STIX]{x1D6FF}z_{i}(\boldsymbol{x}^{\prime })/\unicode[STIX]{x1D6FF}z_{j}(\boldsymbol{x})=\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}(\boldsymbol{x}^{\prime }-\boldsymbol{x})$ .

For non-canonical (degenerate) Poisson brackets, such as the bracket (2.7), there exist functionals ${\mathcal{C}}[\unicode[STIX]{x1D702}]$ that commute with every arbitrary functional $F[\unicode[STIX]{x1D702}]$

These functionals ${\mathcal{C}}$ are called Casimir invariants and obviously they do not change the dynamics if ${\mathcal{H}}\rightarrow \mathfrak{F}={\mathcal{H}}-\sum _{i}{\mathcal{C}}_{i}$ , that is

describes the same dynamics as (2.5).

Equilibrium solutions satisfy $\{\unicode[STIX]{x1D702},\mathfrak{F}\}=0$ , which is true if the first variation of the generalized Hamiltonian functional $\mathfrak{F}$ vanishes at the equilibrium point, i.e.

is a sufficient but not necessary condition for equilibria (Morrison Reference Morrison1998; Yoshida, Morrison & Dobarro Reference Yoshida, Morrison and Dobarro2014). To obtain stability criteria one may take the second variation of the EC functional. It is known that if the second variation $\unicode[STIX]{x1D6FF}^{2}\mathfrak{F}$ at the equilibrium point is positive definite, then it provides a norm which is conserved by the linear dynamics, so the equilibrium is linearly stable (Kruskal & Oberman Reference Kruskal and Oberman1958; Holm et al. Reference Holm, Marsden, Ratiu and Weinstein1985; Morrison Reference Morrison1998).

The aim of the following sections is to derive the Casimir invariants of the helically symmetric XMHD and then to find the corresponding equilibrium equations via the condition (2.10). For the general 3-D version of the model described by means of (2.6) and (2.7), the Casimir invariants are

with $\boldsymbol{B}^{\ast }=\unicode[STIX]{x1D735}\times \boldsymbol{A}^{\ast }$ and $\unicode[STIX]{x1D6FE}_{\pm }$ the two roots of the quadratic equation $\unicode[STIX]{x1D6FE}^{2}-d_{i}\unicode[STIX]{x1D6FE}-d_{e}^{2}=0$ , i.e. $\unicode[STIX]{x1D6FE}_{\pm }=(d_{i}\pm \sqrt{d_{i}^{2}+4d_{e}^{2}})/2$ .

3.1 Helical symmetry and Poisson bracket reduction

The helical symmetry can be imposed by assuming that in a cylindrical coordinate system $(r,\unicode[STIX]{x1D719},z)$ all equations of motion depend spatially on $r$ and the helical coordinate $u=\ell \unicode[STIX]{x1D719}+nz$ , where $\ell =\sin (a)$ and $n=-\text{cos}(a)$ with $a$ being the helical angle. For $a=0$ we obtain the axisymmetric case and for $a=\unicode[STIX]{x03C0}/2$ the translationally symmetric case. The contravariant unit vector in the direction of the $u$ coordinate is $\boldsymbol{e}_{u}=\unicode[STIX]{x1D735}u/|\unicode[STIX]{x1D735}u|=\ell k\boldsymbol{e}_{\unicode[STIX]{x1D719}}+nkr\boldsymbol{e}_{z}$ , where $k$ is

The tangent to the direction of the helix $r=\text{const.}\;u=\text{const.}$ is given by $\boldsymbol{e}_{h}=\boldsymbol{e}_{r}\times \boldsymbol{e}_{u}$ and one can prove that the following relations hold:

where $\boldsymbol{h}=k\boldsymbol{e}_{h}$ , hence $\boldsymbol{h}\boldsymbol{\cdot }\boldsymbol{h}=k^{2}$ . Helical symmetry means that $\boldsymbol{h}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f=0$ where $f$ is arbitrary scalar function. The relations (3.2) give us the opportunity to introduce the so-called poloidal representation for the divergence-free magnetic field and also a poloidal representation for the velocity field, including additionally a potential field contribution accounting for the compressibility of the flow, i.e.

For incompressible flows $\unicode[STIX]{x1D6F6}$ is harmonic or constant. In view of (3.2), the divergence and the curl of (3.3) and (3.4) are given by

where $\unicode[STIX]{x1D6E5}:=\unicode[STIX]{x1D735}^{2}$ and ${\mathcal{L}}:=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }(k^{2}\unicode[STIX]{x1D735}())$ is a linear, self-adjoint differential operator. For convenience we define the following quantities: $w:=\unicode[STIX]{x0394}\unicode[STIX]{x1D6F6}$ or $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D6E5}^{-1}w$ and $\unicode[STIX]{x1D6FA}={\mathcal{L}}\unicode[STIX]{x1D712}$ or $\unicode[STIX]{x1D712}={\mathcal{L}}^{-1}\unicode[STIX]{x1D6FA}$ .

Having introduced the representation of (3.3)–(3.4) for the helically symmetric fields, in order to derive the helically symmetric Hamiltonian formulation we need to express the Hamiltonian (2.6) and the Poisson bracket (2.7) in terms of the scalar field variables $\unicode[STIX]{x1D702}_{HS}=(\unicode[STIX]{x1D70C},v_{h},\unicode[STIX]{x1D712},\unicode[STIX]{x1D6F6},B_{h}^{\ast },\unicode[STIX]{x1D713}^{\ast })$ . This is achieved not only by expressing the fields $\unicode[STIX]{x1D702}_{3D}=(\unicode[STIX]{x1D70C},\boldsymbol{v},\boldsymbol{B}^{\ast })$ in terms of the scalar field variables but it requires also the transformation of the functional derivatives from derivatives with respect to $\unicode[STIX]{x1D702}_{_{3D}}$ to functional derivatives with respect to the scalar fields $\unicode[STIX]{x1D702}_{HS}$ . As in Andreussi, Morrison & Pegoraro (Reference Andreussi, Morrison and Pegoraro2010), Andreussi et al. (Reference Andreussi, Morrison and Pegoraro2012) and Kaltsas et al. (Reference Kaltsas, Throumoulopoulos and Morrison2017), we accomplish this transformation by employing a chain rule reduction,

where

which follow from

upon introducing the relations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FA}={\mathcal{L}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D712}$ , $\unicode[STIX]{x1D6FF}w=\unicode[STIX]{x0394}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F6}$ and exploiting the self-adjointness of the operators $\unicode[STIX]{x1D6E5}$ and ${\mathcal{L}}$ . Also we observe that in (2.7) there exist bracket blocks which contain the curl of $F_{\boldsymbol{B}^{\ast }}$ , which is

The helically symmetric Poisson bracket occurs by substituting (3.3), (3.6), (3.8) and (3.13) into (2.7) and assuming that any surface-boundary terms which emerge due to integrations by parts, vanish due to appropriate boundary conditions, for example periodic conditions or vanishing field variables $\unicode[STIX]{x1D702}_{HS}$ on $\unicode[STIX]{x2202}D$ , except for the mass density $\unicode[STIX]{x1D70C}$ because various terms diverge as $\unicode[STIX]{x1D70C}$ approaches zero, as is evident even from (2.7). However, in view of the actual physical situation one can assume that the mass density on the boundary is sufficiently small. The Poisson bracket takes the form

where $[f,g]:=(\unicode[STIX]{x1D735}f\times \unicode[STIX]{x1D735}g)\boldsymbol{\cdot }\boldsymbol{h}$ is the helical Jacobi–Poisson bracket. One may prove that with appropriate boundary conditions, e.g. such as those mentioned above, the identity

holds for arbitrary functions $f,g,h$ . These conditions are necessary to derive the bracket (3.14) and also for finding the Casimir determining equations.

It is not difficult to show that if we set $a=\unicode[STIX]{x03C0}/2$ the bracket (3.14) reduces to the translationally symmetric XMHD bracket derived in Kaltsas et al. (Reference Kaltsas, Throumoulopoulos and Morrison2017). The corresponding axisymmetric bracket can be obtained by setting $a=0$ . In this case the purely helical terms which contain a coefficient $2n\ell$ vanish and the scale factor $k$ becomes $1/r$ .

To complete the Hamiltonian description of helically symmetric XMHD dynamics we need to express the Hamiltonian (2.6) in terms of the scalar fields $\unicode[STIX]{x1D702}_{HS}$ , leading to

Also from the definition of the generalized magnetic field $\boldsymbol{B}^{\ast }$ (2.4) and the helical representation (3.3) one may derive the following relations for the generalized variables $B_{h}^{\ast }$ and $\unicode[STIX]{x1D713}^{\ast }$ :

where $B_{h}$ is the helical component and $\unicode[STIX]{x1D713}$ the poloidal flux function of the magnetic field $\boldsymbol{B}$ . Note that terms containing the parameters $n$ and $\ell$ are purely helical, i.e. they vanish in the cases of axial and translational symmetry. Also the last term of (3.17) is purely compressible, i.e. it vanishes if we consider incompressible plasmas. Another interesting observation is that due to the non-orthogonality of the helical coordinates, there is a poloidal magnetic field contribution in the helical component of the generalized magnetic field $B_{h}^{\ast }$ and helical magnetic contribution $B_{h}$ in the poloidal flux function $\unicode[STIX]{x1D713}^{\ast }$ . This mixing makes the subsequent dynamical and equilibrium analyses appear much more involved than in our previous study, however it can be simplified upon observing that

where $J_{h}=k^{-1}{\mathcal{L}}\unicode[STIX]{x1D713}-2n\ell k^{2}B_{h}$ is the helical component of the current density. Therefore the variation of the magnetic part of the Hamiltonian can be written as

leading to the following relations for the functional derivatives of the Hamiltonian:

In addition, the functional derivatives with respect to the velocity related variables are given by

3.2 Helically symmetric dynamics

The helically symmetric dynamics is described by means of the Hamiltonian (3.16) and the Poisson bracket (3.14) as $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D702}_{HS}=\{\unicode[STIX]{x1D702}_{HS},{\mathcal{H}}\}_{HS}^{XMHD}$ . Due to the helical symmetry and the compressibility, the equations of motion appear much more involved than the corresponding equations of motion in Grasso et al. (Reference Grasso, Tassi, Abdelhamid and Morrison2017). For this reason we present here the dynamical equations for incompressible plasmas ( $\unicode[STIX]{x1D70C}=1$ ). Incompressible equations are obtained from the Hamiltonian and the Poisson bracket that correspond to $\unicode[STIX]{x1D70C}=1$ and $w=0$ , or equivalently by the compressible equations of motion by neglecting the dynamical equations for $\unicode[STIX]{x1D70C}$ and $w$ and substituting ${\mathcal{H}}_{w}=0$ and ${\mathcal{H}}_{\unicode[STIX]{x1D6FA}}=\unicode[STIX]{x1D712}$ in the rest (see the appendix A),

Equations (3.27)–(3.30) differ from the corresponding dynamical equations of reference (Grasso et al. Reference Grasso, Tassi, Abdelhamid and Morrison2017) owing to the presence of the scale factor $k$ and the purely helical terms with the coefficients $n\ell$ . Setting $n=0$ we recover the equations of motion for incompressible, translationally symmetric plasmas, whereas for $\ell =0$ we restrict the motion to respect axial symmetry.

3.3 Bracket transformation

In Lingam et al. (Reference Lingam, Morrison and Miloshevich2015b ) the authors proved that the XMHD bracket (2.7) can be simplified to a form identical to the HMHD bracket by introducing a generalized vorticity variable

This transformation was utilized in Grasso et al. (Reference Grasso, Tassi, Abdelhamid and Morrison2017) and Kaltsas et al. (Reference Kaltsas, Throumoulopoulos and Morrison2017) in order to simplify the bracket and the derivation of the symmetric families of Casimir invariants. For this reason we perform the transformation also for the helically symmetric bracket (3.14), rendering the subsequent analysis more tractable. One can see that the corresponding scalar field variables, necessary for the poloidal representation of $\boldsymbol{B}^{\pm }$ , are connected to the variables $\unicode[STIX]{x1D702}_{HS}$ as follows:

Transformation of the bracket requires expressing the functional derivatives in the new representation $(v_{h},\unicode[STIX]{x1D712},\unicode[STIX]{x1D6F6},B_{h}^{\pm },\unicode[STIX]{x1D713}^{\pm })$ . Following an analogous procedure to that employed in Lingam et al. (Reference Lingam, Morrison and Miloshevich2015b ), Grasso et al. (Reference Grasso, Tassi, Abdelhamid and Morrison2017) and Kaltsas et al. (Reference Kaltsas, Throumoulopoulos and Morrison2017) we find

where $\bar{F}$ denotes the functionals expressed in the original variable representation. Upon inserting the transformation of the functional derivatives of (3.34)–(3.36) into (3.14) and expressing $B_{h}^{\ast }$ and $\unicode[STIX]{x1D713}^{\ast }$ in terms of $B_{h}^{\pm }$ and $\unicode[STIX]{x1D713}^{\pm }$ we obtain the following bracket:

where $\unicode[STIX]{x1D708}_{\pm }:=d_{i}-2\unicode[STIX]{x1D6FE}_{\pm }$ . Note that the helically symmetric XMHD dynamics is described correctly by either using the parameter $\unicode[STIX]{x1D708}_{+}$ or the parameter $\unicode[STIX]{x1D708}_{-}$ .

4.1 MHD limit

In Kaltsas et al. (Reference Kaltsas, Throumoulopoulos and Morrison2017) various limits of the symmetric XMHD Casimirs to the Casimirs of the simpler models of Hall MHD, ordinary MHD and inertial MHD were obtained. Here, to corroborate that the computed invariants are correct, we take the MHD limit, anticipating the recovery of the invariants found in Andreussi et al. (Reference Andreussi, Morrison and Pegoraro2012). For the MHD limit we set $d_{e}=0$ (Hall MHD) and $d_{i}=0$ . If we set only $d_{e}=0$ we exclude electron inertial contributions and we obtain the Hall MHD Casimirs

For the corresponding MHD families of invariants we additionally require $d_{i}\rightarrow 0$ in (4.15)–(4.18). From the resulting set of Casimirs the cross-helicity and the helical momentum are absent. This is a characteristic peculiarity, encountered when the MHD limit of models with Hall physics contributions is considered (e.g. see Hazeltine, Hsu & Morrison Reference Hazeltine, Hsu and Morrison1987; Yoshida & Hameiri Reference Yoshida and Hameiri2013; Abdelhamid et al. Reference Abdelhamid, Kawazura and Yoshida2015; Kaltsas et al. Reference Kaltsas, Throumoulopoulos and Morrison2017). In Kaltsas et al. (Reference Kaltsas, Throumoulopoulos and Morrison2017) we resolved this peculiarity by expanding the invariants ${\mathcal{C}}_{1}^{HMHD}$ , ${\mathcal{C}}_{3}^{HMHD}$ about $\unicode[STIX]{x1D713}$ , then by rescaling the arbitrary functions we managed to show that, since the terms that diverge when $d_{i}\rightarrow 0$ are already Casimirs, the rest of the terms translate into the MHD Casimirs. Doing so for the helically symmetric Casimirs we arrive at

The functionals (4.19)–(4.22) are indeed the correct helically symmetric MHD Casimir invariants (Andreussi et al. Reference Andreussi, Morrison and Pegoraro2012).

4.2 Equilibrium variational principle with helical symmetry

With the helically symmetric Casimirs at hand, we can build the EC variational principle to obtain equilibrium conditions. For analogous utilizations of this methodology for symmetric or 2-D plasmas the reader is referred to Holm et al. (Reference Holm, Marsden, Ratiu and Weinstein1985), Almaguer et al. (Reference Almaguer, Hameiri, Herrera and Holm1988), Andreussi & Pegoraro (Reference Andreussi and Pegoraro2008), Tassi et al. (Reference Tassi, Morrison, Waelbroeck and Grasso2008), Andreussi et al. (Reference Andreussi, Morrison and Pegoraro2010, Reference Andreussi, Morrison and Pegoraro2012), Moawad (Reference Moawad2013), Morrison, Lingam & Acevedo (Reference Morrison, Lingam and Acevedo2014) and Kaltsas et al. (Reference Kaltsas, Throumoulopoulos and Morrison2017). As mentioned in § 2, the EC principle states that the phase space points that nullify the first variation EC functional $\mathfrak{F}$ are equilibrium points. In our case requiring the vanishing of $\unicode[STIX]{x1D6FF}\mathfrak{F}$ amounts to

where $\unicode[STIX]{x1D711}:=\unicode[STIX]{x1D713}^{\ast }+\unicode[STIX]{x1D6FE}k^{-1}v_{h}$ , $\unicode[STIX]{x1D709}:=\unicode[STIX]{x1D713}^{\ast }+\unicode[STIX]{x1D707}k^{-1}v_{h}$ and

Since the variations of the field variables are independent, equation (4.23) is satisfied if the coefficients of the variations of the field variables vanish. This requirement, upon exploiting the relations (3.21)–(3.26), leads to the following equilibrium conditions:

Note that the left-hand side of (4.25)–(4.30) are the coefficients of the variations $(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F6},\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D712},\unicode[STIX]{x1D6FF}v_{h},\unicode[STIX]{x1D6FF}B_{h}^{\ast },\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}^{\ast })$ in $\unicode[STIX]{x1D6FF}\mathfrak{F}$ . In addition to these terms, some surface-boundary terms emerged in $\unicode[STIX]{x1D6FF}\mathfrak{F}$ , due to integration by parts. We assumed that those terms vanish, which is true if the variations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F6},\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D712},\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}^{\ast }$ vanish on the boundary $\unicode[STIX]{x2202}D$ . The first equation (4.25) represents a Bernoulli law

where $\tilde{p}:=\unicode[STIX]{x1D70C}[\unicode[STIX]{x1D70C}U(\unicode[STIX]{x1D70C})]_{\unicode[STIX]{x1D70C}}=\unicode[STIX]{x1D70C}h(\unicode[STIX]{x1D70C})$ where $h(\unicode[STIX]{x1D70C})$ is the total enthalpy ( $\tilde{p}=\unicode[STIX]{x1D6E4}p/(\unicode[STIX]{x1D6E4}-1)$ if we adopt the equation of state $p\propto \unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D6E4}}$ with $\unicode[STIX]{x1D6E4}$ being the adiabatic constant). It describes the effect of macroscopic equilibrium flow including the electron inertial effect, expressed via the magnetic terms, in the total plasma pressure. The rest of the equations can be cast into a Grad–Shafranov or a JFKO system as in the case with translational symmetry.

4.3 The JFKO–Bernoulli system

The system (4.25)–(4.30) can be cast in a JFKO–Bernoulli PDE form that describes completely helically symmetric XMHD equilibria. This can be done by exploiting (4.26), (4.27), (4.29) and (3.17) in order to turn (4.28) and (4.30) into a coupled system for the flux functions $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D709}$ . These equations, except of their coupling to the Bernoulli equation, are additionally coupled to the definition (3.18) given in terms of $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D709}$ expressing essentially the Ampere’s law. The derivation of the system requires tedious algebraic manipulations that we omit here; however, the steps are analogous to those used in Kaltsas et al. (Reference Kaltsas, Throumoulopoulos and Morrison2017) for the derivation of the corresponding system. The JFKO equations for barotropic XMHD are

where $s:=4n^{2}\ell ^{2}d_{e}^{2}\unicode[STIX]{x1D70C}^{-1}k^{4}$ . The equations above coupled to the Bernoulli equation (4.31) describe completely the equilibria in terms of the flux functions $\unicode[STIX]{x1D713}$ , $\unicode[STIX]{x1D711}$ , $\unicode[STIX]{x1D709}$ and of the density $\unicode[STIX]{x1D70C}$ , for given free functions ${\mathcal{F}}(\unicode[STIX]{x1D711})$ , ${\mathcal{G}}(\unicode[STIX]{x1D709})$ , ${\mathcal{M}}(\unicode[STIX]{x1D711})$ , ${\mathcal{N}}(\unicode[STIX]{x1D709})$ and a thermodynamic closure $p=p(\unicode[STIX]{x1D70C})$ , since all physical quantities of interest can be expressed in terms of $\unicode[STIX]{x1D713}$ , $\unicode[STIX]{x1D711}$ , $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D70C}$ . Namely, the helical component of the flow is

the helical magnetic field is given by (4.29), the poloidal field is simply $\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\times \boldsymbol{h}$ , while for the poloidal velocity we need to observe that (4.26) and (4.27) can be written as

with

Therefore the mutual solution of (4.26) and (4.27), should satisfy

Upon taking the cross-product of (4.38) with $\boldsymbol{h}$ , we obtain the poloidal velocity

Note that (4.38) was necessary in the derivation of (4.32)–(4.33). Due to the number of the PDEs that have to be solved simultaneously and the consequence of the symmetry that inserts additional terms and the strong coupling among the equations, the solution of this system is not trivial even in the context of numerical computing. For this reason we present below special cases of equilibria, including axisymmetric XMHD, incompressible XMHD, barotropic and incompressible Hall MHD equilibria with helical symmetry. To our knowledge all these reduced kinds of equilibria have not been studied so far. In the next section we present the corresponding system of Grad–Shafranov or JFKO equations for each of the aforementioned equilibria.

5.1 Axisymmetric barotropic XMHD

The axisymmetric equilibrium equations are obtained by setting the helical angle $a$ to zero, i.e. $\ell =0$ and $n=-1$ , so the parameter $s$ is zero and the scale factor $k=1/r$ and $\boldsymbol{h}=r^{-1}\hat{e}_{\unicode[STIX]{x1D719}}$ . With these parameters, equations (4.32)–(4.34) reduce to the following Grad–Shafranov system:

where $\unicode[STIX]{x1D6E5}^{\ast }:=r^{2}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\unicode[STIX]{x1D735}/r^{2}\right)$ is the so-called Shafranov operator. The Bernoulli equation (4.31) assumes the form:

where $J_{\unicode[STIX]{x1D719}}=-r^{-1}\unicode[STIX]{x1D6E5}^{\ast }\unicode[STIX]{x1D713}$ is the toroidal current density. For $d_{e}=0$ we obtain the axisymmetric Hall MHD Grad–Shafranov–Bernoulli system (Throumoulopoulos & Tasso Reference Throumoulopoulos and Tasso2006).

5.2 Incompressible equilibria

To obtain the equilibrium system for incompressible plasmas with uniform mass density, we set $\unicode[STIX]{x1D70C}=1$ . Note that incompressibility may refer also to the kind of the flows, i.e. flows with divergence-free velocity fields, that renders the mass density a Lagrangian invariant, that is, $\unicode[STIX]{x1D70C}$ is advected by the flow. Here we address the simpler case where the mass density is constant. One should be careful when adopting this assumption because it has to be imposed a priori, i.e. before varying the EC functional. This is because, if we use the barotropic version of the EC functional to derive the equilibrium equations and then impose the uniformness of the mass density, this will result in a restricted class of equilibria because the Bernoulli equation (4.31) will act as an additional constraint on the permissible equilibria. However for uniform mass density, no Bernoulli equation occurs via the variational principle and the computation of the pressure decouples from the PDE problem. Ultimately the resulting equilibrium equations will be given by (4.27)–(4.30) with $\unicode[STIX]{x1D70C}=1$ . This system leads to the equilibrium system of (4.32)–(4.34) with $\unicode[STIX]{x1D70C}=1$ , that is, the differential operators on the left-hand side of (4.32) and (4.33) reduce to the operator $-{\mathcal{L}}$ acting on ${\mathcal{F}}$ and ${\mathcal{G}}$ , respectively. Those equations can alternatively be derived directly by taking projections of the starting stationary XMHD equations. We have verified that this method leads to the same JFKO system. The pressure can be computed from (2.2) by setting $\unicode[STIX]{x2202}_{t}\boldsymbol{v}=0$ , taking the divergence of the resulting equation and acting with the inverse of the Laplacian operator in order to solve for the pressure, leading to the following equation:

If we employ the helically symmetric representation (3.3), (3.4) for the fields $\boldsymbol{B}^{\ast }$ , $\boldsymbol{v}$ and $\boldsymbol{B}$ and use the equilibrium equations (4.27)–(4.30) with $\unicode[STIX]{x1D70C}=1$ , then we can prove that

so from (5.5) and (5.6), we deduce that the incompressible pressure is given by

5.3 Hall MHD equilibria

The Hall MHD limit is effected by setting $d_{e}=0$ and thereby neglecting electron inertial effects. Thus, $\unicode[STIX]{x1D6FE}=d_{i}$ , $\unicode[STIX]{x1D707}=0$ , and the flux functions become $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}+d_{i}k^{-1}v_{h}$ and $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D713}$ . In this model, only ion drift effects are considered and the electron surfaces coincide with the magnetic field surfaces. The JFKO system for computing the poloidal ion and magnetic fluxes is

These equilibria are completely determined through the coupling of the equations above with a Bernoulli law, which can be deduced from (4.31) for $d_{e}=0$ , allowing the computation of the mass density $\unicode[STIX]{x1D70C}$ self-consistently given an equation of state $P(\unicode[STIX]{x1D70C})$ . So the HMHD Bernoulli equation is simply

Also from (4.35) and (4.39) we have

For $\ell =0$ , equations (5.8)–(5.10) reduce to the axisymmetric Grad–Shafranov–Bernoulli system of Throumoulopoulos & Tasso (Reference Throumoulopoulos and Tasso2006). For the baroclinic version of the axisymmetric HMHD equilibrium equations the reader is referred to Hameiri (Reference Hameiri2013), Guazzotto & Betti (Reference Guazzotto and Betti2015).

5.4 Incompressible HMHD equilibria and the double-Beltrami solutions

As in the case of XMHD, to obtain an equilibrium system that is not constrained by an equation arising from the $\unicode[STIX]{x1D70C}$ functional derivative of the EC functional, we take $\unicode[STIX]{x1D70C}=1$ before varying the EC functional. If we do so for the HMHD model then the equilibrium system reduces to (5.8)–(5.9) with $\unicode[STIX]{x1D70C}=1$ , i.e. we have

The pressure can be computed using (5.7) with $d_{e}=0$ . To obtain solutions for the fluxes $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D713}$ , we need to specify the free functions ${\mathcal{F}}$ , ${\mathcal{G}}$ , ${\mathcal{M}}$ and ${\mathcal{N}}$ . There exists a particular ansatz for the free functions, for which the system (5.12)–(5.13) assumes an analytic solution. In this case the magnetic and velocity fields are superpositions of two Beltrami fields and the functions $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D713}$ are expressed as linear combinations of the corresponding poloidal flux functions of the Beltrami fields. The generic linear ansatz, for the system (5.12)–(5.13) is

where $f_{0},f_{1},g_{0},g_{1},m_{0},n_{0}$ are constant parameters, leads to the following equations for helically symmetric HMHD equilibria:

where

For $n,\ell \neq 0$ we can find a solution to this system assuming $m_{1}=n_{1}=f_{0}=g_{0}=0$

where $\unicode[STIX]{x1D713}_{\pm }$ are solutions of the equation

and parameters $\unicode[STIX]{x1D706}_{\pm }$ are given by

Either solving (5.18) directly or following the construction of Barberio-Corsetti (Reference Barberio-Corsetti1973) (see also Chandrasekhar & Kendall (Reference Chandrasekhar and Kendall1957)) we obtain the following analytic solutions $\unicode[STIX]{x1D713}_{\pm }$ :

where $\unicode[STIX]{x1D70E}_{\pm }:=\sqrt{m^{2}n^{2}-\unicode[STIX]{x1D706}_{\pm }^{2}}$ and $\text{I}_{\ell m}$ denotes the modified Bessel function of the first kind with order $\ell m$ . The parameters $c_{\pm }$ and $a_{m}^{\pm }$ can be specified in connection with the desirable boundary conditions. The functions $\unicode[STIX]{x1D713}_{\pm }$ are poloidal flux functions of helically symmetric Beltrami fields with Beltrami parameters $\unicode[STIX]{x1D706}_{\pm }$ . Their combination (5.17) is a homogeneous solution of the system (5.15). Since the solution is a linear combination of two Beltrami fields, the resulting solution is called double Beltrami (DB). Another reason for adopting this terminology is that the resulting velocity and magnetic fields satisfy conditions that involve the double curl operator. Such states, are not only natural solutions of the incompressible Hall MHD equilibrium equations (see Mahajan & Yoshida (Reference Mahajan and Yoshida1998)) but they occur also as relaxed states via minimization principles (Yoshida & Mahajan Reference Yoshida and Mahajan2002). They have been used to construct high-beta equilibria with flows for 1-D (Mahajan & Yoshida Reference Mahajan and Yoshida1998; Iqbal et al. Reference Iqbal, Mirza, Murtaza and Yoshida2001) and axisymmetric systems (Yoshida et al. Reference Yoshida, Mahajan, Ohsaki, Iqbal and Shatashvili2001) but not for helically symmetric ones. Here we compute a helical DB equilibrium in view of (5.17) and (5.20). The computed configuration is depicted in figure 1. The flux function $\unicode[STIX]{x1D713}$ labels the magnetic surfaces while the function $\unicode[STIX]{x1D719}$ labels the ion flow surfaces. We obtained the configuration of figure 1, possessing closed surfaces, for normalized ion skin depth $d_{i}=0.09$ , $f_{1}=4.2$ , $g_{1}=2.01$ and imposing the vanishing of $\unicode[STIX]{x1D713}$ on some predetermined boundary points, yielding the values of the free parameters in the truncated expansions (5.20). We observe that the ion surfaces depart significantly from the electron–magnetic surfaces, although in a manner consistent with other computations for axisymmetric (Guazzotto & Betti Reference Guazzotto and Betti2015) and translationally symmetric (Kaltsas et al. Reference Kaltsas, Throumoulopoulos and Morrison2017) HMHD equilibria, resulting in a configuration with distinct helical structures for the ions and the electrons.

Figure 1.

The magnetic (solid red) and the ion (dashed blue) surfaces of the analytic DB equilibria with helical symmetry in connection with (5.17) and (5.20) in three different sections, namely $z=0$ , $z=\unicode[STIX]{x03C0}/12$ , $z=\unicode[STIX]{x03C0}/3$ . The parameters $\ell$ and $n$ are $\ell =1$ and $n=5$ corresponding to five helical windings for distance $2\unicode[STIX]{x03C0}$ covered in the $z$ -direction. The contours have been plotted on the $(x,y)$ plane (perpendicular to the $z$ -direction).