This paper studies the nonlinear evolution of magnetic field turbulence in proximity of steady ideal Magnetohydrodynamics (MHD) configurations characterized by a small electric current, a small plasma flow, and approximate flux surfaces, a physical setting that is relevant for plasma confinement in stellarators. The aim is to gather insight on magnetic field dynamics, to elucidate accessibility and stability of three-dimensional MHD equilibria, as well as to formulate practical methods to compute them. Starting from the ideal MHD equations, a reduced dynamical system of two coupled nonlinear partial differential equations for the flux function and the angle variable associated with the Clebsch representation of the magnetic field is obtained. It is shown that under suitable boundary and gauge conditions such reduced system preserves magnetic energy, magnetic helicity, and total magnetic flux. The noncanonical Hamiltonian structure of the reduced system is identified, and used to show the nonlinear stability of steady solutions against perturbations involving only one Clebsch potential. The Hamiltonian structure is also applied to construct a dissipative dynamical system through the method of double brackets. This dissipative system enables the computation of MHD equilibria by minimizing energy until a critical point of the Hamiltonian is reached. Finally, an iterative scheme based on the alternate solution of the two steady equations in the reduced system is proposed as a further method to compute MHD equilibria. A theorem is proven which states that the iterative scheme converges to a nontrivial MHD equilbrium as long as solutions exist at each step of the iteration.
I. INTRODUCTION
Despite its relevance for the confinement of magnetized plasmas and the development of nuclear fusion reactors known as stellarators, a general theory concerning the existence of regular solutions of the MHD equilibrium equations (1) is not available at present.2 This is because the characteristic surfaces associated with the first-order system of partial differential equations (PDEs) (1) depend on the unknown B.3,4 The existence of regular solutions is known for the special cases in which the pressure is constant or the fields are invariant under some combination of continuous Euclidean isometries of (rotations and translations). In the first case the magnetic field is a Beltrami field.5 In the second case, Eq. (1) reduces to the Grad–Shafranov equation, a nonlinear second order elliptic PDE for the flux function.6–8 Both cases are however not relevant for stellarators,9 which consist of toroidal vessels without trivial symmetries surrounded by coils with complex shapes whose purpose is to generate the field line twist required to minimize particle losses caused by cross-field drifts. In principle, stellarators achieve steady plasma confinement mostly through an externally produced vacuum magnetic field, and are therefore more suitable for continued operation compared to an axially symmetric tokamak where the field line twist is obtained by driving an electric current within the plasma. However, the lack of axial symmetry results in the breaking of conservation of vertical angular momentum, a fact that deteriorates the quality of plasma confinement. For this reason, in addition to (1) the equilibrium magnetic field within a stellarator must satisfy additional conditions, such as quasisymmetry, a property that ensures particle confinement by constraining particle orbits close to a given flux surface.10–12
In this context, the aim of the present paper is (i) to obtain a closed set of reduced equations preserving the Hamiltonian structure13,14 of ideal MHD and describing the nonlinear evolution of the magnetic field in proximity of MHD equilibria (1) and in a physical regime relevant for stellarator plasmas, (ii) to use the derived equations to elucidate the stability properties of such equilibria, and (iii) to apply the derived equations to formulate dissipative and iterative schemes to construct nontrivial MHD equilibria (1) with nested flux surfaces and a non-vanishing pressure gradient in toroidal domains of arbitrary shape. In addition to providing a toy model of magnetic field turbulence in a physically relevant setting, we conjecture that the derived results may serve as a starting point for a mathematical proof of existence of nontrivial MHD equilibria (1) in toroidal domains without Euclidean symmetries.
The present paper is organized as follows. In Sec. II the ideal MHD equations (2) are reduced according to an ordering in which the plasma flow u, the electric current ∇ × B, and time derivatives ∂/∂t are small, and the pressure field P is related to the mass density ρ and the velocity field u by a generalized Bernoulli principle. This ordering also implies the existence of approximate flux surfaces Ψ for the magnetic field. This fact is used to enforce at leading order a Clebsch representation15,16 of the magnetic field B = ∇Ψ × ∇Θ with Ψ the flux function and Θ a multi-valued (angle) variable. In the reduced system, the dynamics of the magnetic field is thus described by a pair of coupled equations for the two Clebsch potentials Ψ and Θ.
In Sec. III it is shown that under suitable boundary conditions and gauge conditions for the magnetic vector potential the reduced dynamics preserves magnetic energy, magnetic helicity, and total magnetic flux. These conservation laws are then applied to describe steady states in terms of critical points of a functional of Ψ and Θ given by a linear combination of magnetic energy and total magnetic flux. This variational formulation can be physically interpreted in analogy with Taylor relaxation18,19 in which magnetic energy is minimized under the constraint of magnetic helicity.17 Here, functionals involving higher order derivatives of the dynamical variables are dissipated at a faster rate by non-ideal (dissipative) mechanisms.20
In Sec. IV the noncanonical Hamiltonian structure of the reduced equations is identified in terms of a Poisson bracket and a Hamiltonian functional.21,22 In particular, it is shown that the Poisson bracket satisfies all the Poisson bracket axioms, including the Jacobi identity.23
In Sec. V the Hamiltonian structure obtained in Sec. IV is used to prove that steady solutions of the reduced dynamics are nonlinearly stable24–27 against turbulent fluctuations involving only one of the two Clebsch potentials. Here, we recall that a positive second variation of the Hamiltonian is not sufficient to guarantee nonlinear stability. In general, a norm on the space of solutions must be found such that the deviation of the perturbed solution at a given time is bound by the discrepancy of initial conditions in the prescribed topology. The type of nonlinear stability shown here effectively constrains the deviation of the perturbed Clebsch potential from initial conditions on the level sets of the other unperturbed Clebsch potential.
In Sec. VI the method of double brackets28,29 is used to formulate a dissipative dynamical system for the Clebsch potentials Ψ and Θ with the property that, instead of being constant, the Hamiltonian is progressively dissipated. This pair of equations has the structure of coupled diffusion equations. Double bracket dynamics is obtained by applying twice the Poisson bracket and represents an effective tool to compute energy minima while preserving the Casimir invariants that span the kernel of the Poisson bracket.30,31 The derived dissipative dynamical system may therefore be applied to compute MHD equilibria (1) corresponding to critical points of the Hamiltonian in a dynamical fashion.
In Sec. VII a second iterative scheme based on the alternate solution of the two steady equations of the reduced dynamical system obtained in Sec. II is discussed. Here, a theorem is proven which states that the iterative scheme converges to a nontrivial MHD equilbrium (with a non-vanishing pressure gradient) as long as solutions exist at each step of the iteration. The dissipative and iterative schemes obtained in Secs. VI and VII may be regarded as alternative methods to the constrained minimization of the plasma energy , where γ ≥ 0 is the adiabatic index, often used to numerically compute MHD equilibria.32,33
Concluding remarks are given in Sec. VIII.
II. A REDUCED IDEAL MHD SYSTEM FOR MAGNETIC FIELD TURBULENCE IN PLASMAS WITH SMALL CURRENTS AND APPROXIMATE FLUX SURFACES
The aim of this section is to derive a reduction of the ideal MHD equations that determines the nonlinear evolution of magnetic field turbulence within a plasma characterized by approximate flux surfaces. This system should be appropriate to describe the confinement regime of a tokamak or a stellarator provided that flux surfaces exist to some degree throughout time evolution. This statement will be made quantitatively precise later.
III. CONSERVATION LAWS AND RELAXED STATES
In this section we first discuss the invariants of systems (11), (30), and (32). Then, these invariants are used to construct a variational principle describing steady configurations of system (32). These steady states correspond to MHD equilibria (1) and can be understood as the result of a constrained relaxation process in which the weakest invariant is dissipated while the others are kept constant.
A. Conservation of magnetic energy, helicity, and flux
Invariant . | Expression . | Field conditions . | Boundary conditions . |
---|---|---|---|
Magnetic energy MΩ | B · n = 0, ∇ × B · n = 0 | ||
Magnetic helicity KΩ | B · n = 0, ∇ × B · n = 0 |
Invariant . | Expression . | Field conditions . | Boundary conditions . |
---|---|---|---|
Magnetic energy MΩ | B · n = 0, ∇ × B · n = 0 | ||
Magnetic helicity KΩ | B · n = 0, ∇ × B · n = 0 |
Invariant . | Expression . | Field conditions . | Boundary conditions . |
---|---|---|---|
Magnetic energy MΩ | B = ∇Ψ × ∇Θ | Ψ = constant | |
Magnetic helicity KΩ | B = ∇Ψ × ∇Θ, A = q0 + Ψ∇Θ | Ψ = constant, | |
∂Ω not connected | |||
Magnetic flux FΩ | B = ∇Ψ × ∇Θ | Ψ = constant |
Invariant . | Expression . | Field conditions . | Boundary conditions . |
---|---|---|---|
Magnetic energy MΩ | B = ∇Ψ × ∇Θ | Ψ = constant | |
Magnetic helicity KΩ | B = ∇Ψ × ∇Θ, A = q0 + Ψ∇Θ | Ψ = constant, | |
∂Ω not connected | |||
Magnetic flux FΩ | B = ∇Ψ × ∇Θ | Ψ = constant |
B. Steady states and relaxation
IV. HAMILTONIAN STRUCTURE
The aim of this section is to show that the model system (32) is endowed with a Hamiltonian structure. This property will later be used to discuss the nonlinear stability of steady solutions.
Let δF/δΨ denote the functional derivative of the functional on the state space with respect to . We have
V. REMARKS ON THE NONLINEAR STABILITY OF STEADY SOLUTIONS
The Proof of Proposition 2 follows by evaluating the difference (66) for δΘ = 0. Similarly,
Again, the Proof of Proposition 3 follows by evaluation of the difference (66) for δΨ = 0.
We conclude this section by observing that the distance functions and behave as seminorms due to the degeneracy brought by the cross product. For example, the distance function is degenerate since for any function . Such degeneracy can be removed by restricting the state space to include only those functions Ψ such that where is a level set of Θ with surface element dS. Then, if and only if . On the other hand, if and only if g = 0. Alternatively, one may simply interpret Propositions 2 and 3 as constraints on the size of and across ∇Θ0 and ∇Ψ0 respectively (the gradients of the variations are constrained on the submanifolds defined by level sets of Θ0 and Ψ0).
VI. CONSTRUCTION OF MHD EQUILIBRIA BY DOUBLE BRACKET DISSIPATION
Hence, if solutions exist in the limit t → +∞, they are nontrivial critical points of HΩ. We therefore suggest that the dissipative system (69) can be applied to numerically compute MHD equilibria with nested flux surfaces.
VII. CONSTRUCTION OF MHD EQUILIBRIA BY ITERATION
Finally, we observe that if a solution of system (82) is found after a finite number of iterations n, successive iterations will simply return the same solution, e.g., . Hence, in this case .□
If Ω is a hollow toroidal volume with boundary ∂Ω corresponding to two distinct level sets of a smooth function , with ∇Ψ0 ≠ 0 in Ω, and if level sets of Ψ0 foliate Ω with nested toroidal surfaces, Theorem 1 of Ref. 42 ensures that Eq. (85) always has a nontrivial solution Θ0 such that ∇Ψ0 × ∇Θ0 ≠ 0. Furthermore, the angle variable Θ0 is not unique, but solutions exist in the form Θ0 = Mμ + Nν + χ0, where μ, ν are toroidal and poloidal angle variables, the integers M, N determine the rotational transform of the vector field ∇Ψ0 × ∇Θ0, and the function is single-valued. The same result applies when solving for Θi at any step (84b) of the iteration provided that Ψi satisfies the same properties listed above for Ψ0 in Ω.
An argument analogous to that used in the Proof of Theorem 1 in Ref. 42 shows that for a given angle variable Θi−1 in (84a), a solution Ψi can be obtained by reducing Eq. (84a) to a two-dimensional elliptic equation on each level set of Θi−1 and by joining solutions corresponding to adjacent level sets.
In light of Remarks 1 and 2 above, if one could show that at each step of the iteration the solutions Θi−1 and Ψi, i ≥ 1 preserve their properties (in particular, Θi remains an angle variable and Ψi foliates the domain with nested toroidal surfaces) then, combining this result with Theorem 1 proved in this section, one would have obtained a proof of the existence of MHD equilibria (70) in hollow toroidal volumes of arbitrary shape. In such construction, although no control is available on the form of the flux surfaces Ψ∞ within Ω, one can conjecture that, if they exist, solutions B = ∇Ψ∞ × ∇Θ∞ with different rotational transforms can be obtained by appropriate choice of the integers M, N mentioned in Remark 1.
VIII. CONCLUDING REMARKS
In this study, starting from the ideal MHD equations (2) and with the aid of Clebsch potentials, we derived a reduced set of Eqs. (11) and (30), as well as (32) describing the nonlinear evolution of magnetic field turbulence in proximity of MHD equilibria (1). The ordering (8) used to arrive at these equations is appropriate for a plasma with small flow, small electric current, approximate flux surfaces, and slow time variation. This setting is expected to be relevant for stellarator plasmas. The same governing equations can be obtained under the more general ordering (34) in which both the plasma flow and the electric current are not small. We showed that the reduced equations possess invariants. In particular, the closed system (32) preserves magnetic energy, magnetic helicity, and total magnetic flux under suitable boundary and gauge conditions. Furthermore, it exhibits a noncanonical Hamiltonian structure with Poisson bracket (56) and Hamiltonian (57) (Proposition 1 of Sec. IV). Such Hamiltonian structure can be used to examine the stability properties of steady solutions: we found that MHD equilibria (70) are nonlinearly stable against perturbations involving a single Clebsch potential in the sense of Propositions 2 and 3 of Sec. V. The Hamiltonian structure can also be applied to obtain a dissipative dynamical system (69) with the property that the Hamiltonian of the system is progressively dissipated as described by Proposition 4 of Sec. VI. System (69), which comprises two coupled diffusion equations for the Clebsch potentials, thus provides a dynamical method to compute nontrivial MHD equilibria (70) by minimizing the Hamiltonian (57). We further proposed a second scheme to compute MHD equilibria (70) based on the iterative solution of the two coupled Eq. (82). Here, Theorem 1 shows that, if solutions exists at each step of the iteration, the process must converge toward a solution of system (82) and thus to a nontrivial MHD equilibrium of the type (70).
The reduced equations derived in the present paper can be regarded as a toy model of turbulence that can be useful to assess dynamical accessibility and stability of MHD equilibria in physically relevant regimes. Furthermore, they provide two practical approaches (a dissipative one and an iterative one) to numerically compute MHD equilibria. Finally, as outlined in the remarks at the end of Sec. VII, we conjecture that the iterative scheme of Sec. VII may represent the basis for a mathematical proof of the existence of MHD equilibria with a non-vanishing pressure gradient in hollow tori of arbitrary shape (that is, configurations in which the boundary ∂Ω is not invariant under some combination of Euclidean isometries).
ACKNOWLEDGMENTS
N.S. would like to thank Z. Yoshida for useful discussion.
The research of N.S. was partially supported by JSPS KAKENHI Grant Nos. 21K13851. and 22H04936.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Naoki Sato: Conceptualization (lead); Formal analysis (lead); Funding acquisition (lead); Methodology (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead). Michio Yamada: Conceptualization (supporting); Formal analysis (supporting); Methodology (supporting); Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (supporting).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.