Helically symmetric extended MHD: Hamiltonian formulation and equilibrium variational principles
Hamiltonian extended MHD (XMHD) dynamics is restricted to respect helical symmetry by reducing the Poisson bracket for 3D dynamics to a helically symmetric one, as an extension of the previous study for translationally symmetric XMHD dak_gnt_pjm (). Four families of Casimir invariants are obtained directly from the symmetric Poisson bracket and they are used to construct Energy-Casimir variational principles for deriving generalized XMHD equilibrium equations with arbitrary macroscopic flows. The system is then cast into the form of Grad-Shafranov-Bernoulli equilibrium equations. The axisymmetric and the translationally symmetric formulations can be retrieved as geometric reductions of the helical symmetric one. As special cases, the derivation of the corresponding equilibrium equations for incompressible plasmas is discussed and the helically symmetric equilibrium equations for the Hall MHD system are obtained upon neglecting electron inertia. An example of an incompressible double-Beltrami equilibrium is presented in connection with a straight-stellarator configuration.
pacs:Valid PACS appear here
Extended MHD (XMHD) is perhaps the simplest consistent, in terms of energy conservation kimu_morr (), fluid plasma model containing both Hall drift and electron inertial effects. It is obtained by reduction of the standard two-fluid plasma model, when the quasineutrality assumption is imposed and expansion in the smallness of the electron-ion mass ratio is performed lust (); kimu_morr (), although the latter expansion need not be done (see Sec. VI of pjmKM17 ()). The Hamiltonian structure of this model, first identified in abde_kawa_yosh () for its barotropic version and corroborated in ling_morr_milo (), where transformations to the Hamiltonian structures of Hall MHD (HMHD) (e.g. lighthill ()), Inertial MHD (IMHD) kimu_morr (); ling_morr_tass () and the ordinary ideal MHD model were identified. The Hamiltonian structure of XMHD served as the starting points for two subsequent papers that dealt with applications of its translationally symmetric counterpart to magnetic reconnection gras_tass_abde_morr () and equilibria dak_gnt_pjm (). In the former publication the incompressible case with homogeneous mass density was considered, while in the latter the analysis concerned the compressible barotropic version of the model.
Here we present the Hamiltonian formulation of the barotropic XMHD model in the presence of continuous helical symmetry, an extension of our previous work dak_gnt_pjm () that was concerned with translationally symmetric plasmas. Helical symmetry is a general case of continuous spatial symmetry that includes both the cases of axial and translation symmetry. Therefore the results obtained within the context of a helically symmetric formulation can be directly applied to the sub-cases of axial symmetry and translation symmetry. This provides a unified framework for the study of equilibrium and stability of symmetric configurations, which is important because purely or nearly helical structures are very common in plasma systems. For example, bifurcated 3D equilibrium states with internal helical structures with toroidicity, e.g., helical cores, have been observed experimentally weller (); lorenz (); berge_prl () and simulated cooper_1 (); cooper_2 () in Tokamaks and RFPs (e.g. rfp_exp (); rfp_sim ()). Another example of helical structures that emerge from plasma instabilities, such as the resistive or collisionless tearing modes, or as a result of externally imposed symmetry-breaking perturbations are magnetic islands wael_1 (). In addition the helix may serve as a rough approximation of the Stellarator spitzer (); hela (), the second major class of magnetic confinement devices alongside the Tokamak, in the large aspect-ratio limit. Also helical magnetic structures are common in astrophysics, e.g., in astrophysical jets pino (); pudri (). Therefore it is of interest to derive a joint tool for two-fluid equilibrium and stability studies of systems with helical symmetry, with the understanding that for most cases of laboratory application helical symmetry is an idealized approximation.
As in our previous work dak_gnt_pjm (), we use the Energy-Casimir (EC) variational principle to obtain equilibrium conditions. However it is known that the EC principle can be extended for the study of linear, and nonlinear stability holm_mars_rati (); morr_rev () by investigating the positiveness of the second variation of the EC functional, an idea that dates to the early plasma literature KO (). Many works that employ such principles for the derivation of equilibrium conditions and sufficient MHD stability criteria, arising as consequences of the noncanonical Hamiltonian structure of ideal MHD morr_gree (), have been published over the last decades for several geometric configurations holm_mars_rati (); holm_hall (); alma (); morr_tass_tron (); moawad (); amp2a (); amp2b (); moawad_ibra (). In amp1 (); amp2a (); amp2b () Energy-Casimir equilibrium and stability principles were used in the case of helically symmetric formulation. Also similar equilibrium variational principles were applied in the case of XMHD dak_gnt_pjm () for plasmas with translation symmetry. Therefore the use of such principles in the case of helically symmetric XMHD seems a natural generalization of the previous studies. To accomplish this task we first derive the Poisson bracket of the helically symmetric barotropic XMHD and its corresponding families of Casimir invariants. Those invariants, along with the symmetric version of the Hamiltonian function are used in an Energy-Casimir variational principle in order to obtain the equilibrium equations for helical plasmas described by XMHD. To our knowledge this is the first time that equilibrium equations containing two-fluid physics are derived for helical configurations, especially exploiting Hamiltonian techniques.
The present study is organized as follows: in Sec. II we present briefly the Hamiltonian field theory of barotropic XMHD. In Sec. III we introduce the requisite description of the helical coordinate and representations of the helically symmetric magnetic and velocity fields. Then, the XMHD Poisson bracket is reduced to its helically symmetric counterpart. In Sec. IV the Casimir invariants of the symmetric bracket are obtained and their MHD limit is considered. Also we establish the symmetric EC variational principle, from which we derive generalized equilibrium equations for helical systems. Special cases of equilibria such as the Hall MHD equilibria, are discussed in detail in Sec. V. We conclude with Sec. VI, where we discuss the results of our study.
Ii Barotropic XMHD
ii.1 Evolution equations
The parameters and are the normalized ion and electron skin depths, respectively, is the total pressure and , and represent the mass density, the velocity and the magnetic field, respectively.
ii.2 Hamiltonian formulation
It has been recognized that the equations (1)-(3) possess a noncanonical Hamiltonian structure, i.e. the dynamics can be described by a set of generalized Hamiltonian equations morr_aip (); morr_rev ()
where are noncanonical dynamical variables (not consisting of canonically conjugate pairs), is a real valued Hamiltonian functional, and is a Poisson bracket acting on functionals of the variables , which is bilinear, antisymmetric, and satisfies the Jacobi identity. The appropriate Hamiltonian for our system is the following:
where and is the internal energy function (), while the corresponding noncanonical Poisson bracket is
where denotes the functional derivative of with respect to the dynamical variable .
For noncanonical (degenerate) Poisson brackets, such as the bracket (7), there exist functionals that commute with every arbitrary functional
These functionals are called Casimir invariants and obviously they do not change the dynamics if , that is
describes the same dynamics as Eq. (5).
Equilibrium solutions satisfy , which is true if the first variation of the generalized Hamiltonian functional vanishes at the equilibrium point, i.e.,
is a sufficient but not necessary condition for equilibria morr_rev (); yos_morr (). To obtain stability criteria one may take the second variation of the EC functional. It is known that if the second variation 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 KO (); holm_mars_rati (); morr_rev ().
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 (10). For the general 3D version of the model described by means of (6) and (7), the Casimir invariants are
with and being the two roots of the quadratic equation , i.e. .
Iii Helically symmetric formulation
As mentioned above, the helically symmetric formulation includes both the translational and axial symmetric cases, while being the most generic case for which a globally exact “poloidal” representation of the vector fields is possible. In a series of papers this symmetry was employed for deriving equilibrium equations of the Grad-Shafranov-type, i.e. PDEs with solutions being a poloidal magnetic flux, jfko (); thro_tass_heli (); amp1 (); evan_kuir_thro () in the context of standard MHD theory. Particularly in amp1 () the equilibrium Grad-Shafranov or JFKO (Johnson-Frieman-Kulsrud-Oberman jfko ()) equation was derived using a Hamiltonian variational principle. The same approach is adopted also for our derivation, however, for the more complicated XMHD theory.
iii.1 Helical symmetry and Poisson bracket reduction
The helical symmetry can be imposed by assuming that in a cylindrical coordinate system all equations of motion depend spatially on and the helical coordinate , where and with being the helical angle. For we obtain the axisymmetric case and for the translationally symmetric case dak_gnt_pjm (). The contravariant unit vector in the direction of the coordinate is , where is
The tangent to the direction of the helix is given by and one can prove that the following relations hold:
where , hence . Helical symmetry means that where is arbitrary scalar function. The relations (14) 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, adding though a potential field contribution accounting for the compressibility of the velocity field, i.e.,
where and is a linear, self-adjoint differential operator. For convenience we define the following quantities: or and or .
Having introduced the representation of (15)–(16) for the helically symmetric fields, in order to derive the helically symmetric Hamiltonian formulation we need to express the Hamiltonian (6) and the Poisson bracket (7) in terms of the scalar field variables . This is accomplished not only by expressing the vector fields in terms of the scalar field variables but it requires also the transformation of the functional derivatives from derivatives with respect to vector fields to functional derivatives with respect to the scalar fields . As in dak_gnt_pjm (); andr_morr_pego (); amp1 (), we express the functional derivatives with respect to in terms of the functional derivatives with respect to the fields by employing a chain rule reduction,
which follow from
upon introducing the relations , and exploiting the self-adjointness of the operators and . Also we observe that in (7) there exist bracket blocks which contain the curl of , which is
The helically symmetric Poisson bracket occurs by substituting Eqs. (15), (18), (20) and (25) into (7) and assuming that any surface-boundary terms which emerge due to integrations by parts, vanish due to appropriate boundary conditions (e.g. periodic):
where is the helical Jacobi-Poisson bracket. One may prove that with appropriate boundary conditions the identity
holds for arbitrary functionals . These conditions are necessary to derive the bracket (26) and also for finding the Casimir determining equations.
It’s not difficult to show that if we set the bracket (26) reduces to the translationally symmetric XMHD bracket derived in dak_gnt_pjm (). The corresponding axisymmetric bracket can be obtained by setting . In this case the purely helical terms which contain a coefficient vanish and the scale factor becomes .
To complete the Hamiltonian description of helically symmetric XMHD dynamics we need to express the Hamiltonian (6) in terms of the scalar fields , leading to
Note that terms containing the parameters and are purely helical, i.e., they vanish in the cases of axial and translational symmetry. Also the last term of Eq. (29) 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 and helical magnetic contribution in the poloidal flux function . 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 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
iii.2 Helically symmetric dynamics
The helically symmetric dynamics is described by means of the Hamiltonian (28) and the Poisson bracket (26) as . Due to the helical symmetry and the compressibility, the equations of motion appear much more involved than the corresponding equations of motion in gras_tass_abde_morr (). For this reason we present here the dynamical equations for incompressible plasmas (). Incompressible equations are obtained from the Hamiltonian and the Poisson bracket that correspond to the case and , or equivalently by the compressible equations of motion by neglecting the dynamical equations for and and substituting and in the rest (see Appendix A),
Equations (39)–(42) differ from the corresponding dynamical equations of reference gras_tass_abde_morr () owing to the presence of the scale factor and the purely helical terms with the coefficients . Setting we recover the equations of motion for incompressible translationally symmetric plasmas, while for we restrict the motion to respect axial symmetry.
iii.3 Bracket transformation
This transformation was utilized in dak_gnt_pjm (); gras_tass_abde_morr () in order to simplify the bracket and the derivation of the symmetric families of Casimir invariants. For this reason we should be able to perform this transformation also for the helically symmetric bracket (26), rendering the subsequent analysis more tractable. For helically symmetric plasmas one can see that the corresponding scalar field variables, necessary for the poloidal representation of , are connected to the variables as follows:
Transformation of the bracket requires expressing the functional derivatives in the new representation . Following an analogous procedure to that employed in dak_gnt_pjm (); ling_morr_milo (); gras_tass_abde_morr () we find
where denotes the functionals expressed in the new field variable representation. Upon inserting the transformation of the functional derivatives of (46)–(48) into (26) and expressing and in terms of and we obtain the following bracket:
where . Note that the helically symmetric XMHD dynamics is described correctly by either using the parameter or the parameter .
Iv Casimir invariants and equilibrium variational principle with helical symmetry
where are expressions obtained by manipulating the bracket so as to extract as common factors the functional derivatives of the arbitrary functional . Requiring (50) to be satisfied for arbitrary variations is equivalent to the independent vanishing of the expressions , i.e.,
The expressions for the , read