A linear dual-field finite element method for incompressible MHD, part I: cross-helicity and Alfv\'enic alignment in periodic domains

1. Introduction

• Cross helicity measures global velocity–magnetic correlation, which energy balance alone does not determine; local directional alignment can weaken nonlinear interactions.The introduction links alignment to MHD turbulence and solar-wind fluctuations.Source: §1 · p. 1
• In the unforced ideal case, energy and cross-helicity conservation preserve both Elsässer energies; if either is initially zero, its exact Alfvénic relation persists.The nonnegative sum-of-squares representation connects these conserved quantities to alignment.Source: §1 · pp. 1–2
• The paper situates its method among existing cross-helicity-preserving discretizations and builds specifically on the dual-field Mao–Xi formulation, which does not generally preserve the discrete cross helicity considered here.The introduction also outlines the paper's formulation, scheme, conservation analysis, and numerical tests.Source: §1 · pp. 2–3
• The method modifies the magnetic-field time staggering and Lorentz and induction couplings to preserve cross helicity while retaining sequential linear updates and energy balances.It also retains divergence-free properties and magnetic- and fluid-helicity balances.Source: §1 · p. 2
• In the ideal aligned test, both methods conserve energy, but only the proposed method also conserves cross helicity and preserves alignment; under weak dissipation, it exhibits increasing alignment.The Mao–Xi method departs from the aligned initial state in the ideal test and shows declining normalized cross helicity for the weakly dissipative data considered.Source: §1 · p. 3

2.1. MHD systems and periodic Sobolev spaces; 2.2. The dual-field mixed weak formulation

• The periodic formulation uses zero-mean spaces and two sets of vector fields, with pressure variables, to express the MHD equations in dual-field weak form.The momentum equation uses velocity, vorticity, pressure, viscosity, and Lorentz-force terms; the induction equation evolves the magnetic field directly rather than a magnetic potential.Source: §2.1 · p. 3, §2.2 · p. 4

2.3. Divergence constraints and balance laws

• With the stated initial magnetic constraints, the formulation preserves strong and weak divergence-free properties for velocity, magnetic field, and current density.The magnetic divergence-free initial conditions are preserved in time.Source: §2.3 · pp. 4–5
• The two energy balances and the cross-helicity balance include their corresponding diffusion and forcing terms, and energy, E1, E2, and cross helicity are conserved when forcing is zero and Re = Rm = ∞.The energy balances follow from cancellations between the Lorentz and induction terms.Source: §2.3 · p. 5
• In the unforced ideal case, each Elsässer energy is conserved; if either starts at zero, the corresponding exact relation ui = ±√sBi holds for both field pairs at all times.The result follows from a nonnegative squared-norm representation of the Elsässer energies.Source: §2.3 · p. 6
• The two magnetic helicities coincide, and their common value is conserved when Rm = ∞.The section defines magnetic helicity using potentials associated with the magnetic fields.Source: §2.3 · p. 6
• The two fluid helicities coincide; fluid helicity is conserved in the unforced inviscid hydrodynamic case with both magnetic fields zero.Its MHD balance includes viscous terms.Source: §2.3 · p. 6

3.1. Periodic finite element de Rham complex

• The method uses zero-mean periodic Lagrange, Nédélec, Raviart–Thomas, and discontinuous polynomial spaces that form an exact reduced complex.Periodic degrees of freedom are identified, and the scalar and vector means are removed.Source: §3.1 · p. 7
• The weak curl from the Raviart–Thomas space to the Nédélec space is defined as the L2 adjoint of the strong discrete curl.The framework uses finite element de Rham complexes and commuting projections.Source: §3.1 · p. 7
• When a nonlinear product lies outside the Nédélec space, its L2 projection may be included or omitted in pairings with the weak curl without changing the functional.The spatial formulation replaces the continuous Sobolev spaces with these finite element spaces.Source: §3.1 · p. 7

3.2. Staggered integer-step and half-integer-step updates

• Magnetic potentials are reconstructed from the magnetic fields for helicity analysis and are not unknowns in the time-stepping systems.The algorithm implements the induction term using either the assembled weak-curl matrix or a mass-matrix projection followed by discrete curl.Source: §3.2 · pp. 7–8
• The first half-integer fields are computed by a startup solve using a half time increment; subsequent steps alternate integer- and half-integer-grid updates.Each update uses the most recently available state on the other grid.Source: §3.2 · p. 7, Figure 1 · p. 8
• The proposed method changes the velocity–magnetic unknown pairs at integer and half-integer times relative to Mao–Xi, and pairs the Lorentz and induction couplings accordingly.Table 1 compares the staggered fields and couplings.Source: §3.2 · p. 8
• Each substep remains linear because its coupling coefficients are fields already computed on the other staggered grid.The chosen pairings and couplings provide the nonlinear cancellations needed for an additional cross-helicity balance while retaining energy balances.Source: §3.2 · p. 8

3.3.1. Divergence-free constraints

• The scheme preserves its velocity, magnetic-field, and current-density divergence constraints in strong or weak form at the stated time levels.The constraints follow from the updates and curl identities, with initialization and startup supplying them at n = 0.Source: §3.3.1 · p. 9

3.3.2. Energy and cross-helicity balances

• In the unforced ideal case, both discrete energy sequences and cross helicity are conserved on their respective index ranges.The theorem gives energy and cross-helicity balances with diffusion and forcing contributions; these vanish when f = 0 and Re = Rm = ∞.Source: §3.3.2 · p. 9

3.3.3. Els¨asser energies and Alfv´enic alignment

• The Elsässer energies have a nonnegative sum-of-squares representation, so either initially zero Elsässer energy preserves the corresponding exact alignment relation in the ideal unforced case.The discrete corollary applies for either sign when Re = Rm = ∞ and f = 0.Source: §3.3.3 · p. 11

3.3.4. Magnetic- and fluid-helicity balances

• The common discrete magnetic and fluid helicities satisfy one-step balance laws, with conservation in the stated inviscid, unforced cases.Magnetic helicity is conserved when Rm = ∞; fluid helicity is conserved in the unforced inviscid hydrodynamic case with B1 = B2 = 0.Source: §3.3.4 · pp. 12–13

3.3.5. Energy lower bounds from magnetic helicity

• With Rm = ∞, nonzero conserved magnetic helicity and no forcing give positive lower bounds for both staggered energies, so neither energy can decay to zero.The bounds are a discrete analogue of the Arnold inequality.Source: §3.3.5 · p. 13
• The energy bounds involving |H_M|^2 have constants determined by the initial data, s, and C1, and independent of n.The passage states that these bounds prove (3.43).Source: §3.3.5 · p. 14

• The experiments examine spatial accuracy, the role of cross helicity, conservation properties, and planar vortex evolution.Source: §4 · p. 14
• The section tests conservation laws identified in Table 2.Source: §4 · p. 14
• Planar vortex evolution is included as a numerical illustration.Source: §4 · p. 14

4.1. Convergence tests

• The tests set s = 1, Re = Rm = 1, T = 1, and ∆t = 0.01 on meshes with h = 2π/m for m = 4, 6, 8, and 10.Each cube is divided into six tetrahedra.Source: §4.1 · p. 14
• Errors in the integer- and half-integer-grid fields are evaluated at tN = 1 and tN+1/2 = 1.005, respectively.Here N = 100.Source: §4.1 · p. 14
• Reference triangles indicate O(h^k) rates for vector fields and p3, and O(h^(k+1)) for p0.The tests use k = 1 and k = 2.Source: §4.1 · p. 14

4.2.1. Ideal aligned test

• In the ideal aligned test, both methods conserve energy, but only the proposed method conserves cross helicity and retains its initial discrete state.Mao–Xi departs from its initial state despite energy conservation.Source: §4.2.1 · p. 15, Figure 4 · p. 16, Figure 5 · p. 16

4.2.2. Weakly dissipative non-aligned test

• In the weakly dissipative test, the proposed method's cumulative cross-helicity-balance residual remains below 10^-10.The Mao–Xi method has no plotted residual because the balance law established here does not apply to it.Source: §4.2.2 · p. 16, Figure 6 · p. 17
• Under weak dissipation, the proposed method's global alignment diagnostic increases towards one, while the Mao–Xi diagnostic decreases.The tests use non-aligned initial data.Source: §4.2 · p. 14, §4.2.2 · p. 17
• Locally, the proposed method's alignment distribution concentrates near cos θ = 1, whereas the Mao–Xi distribution remains broader.The local distribution agrees with the distinct global alignment trends.Source: §4.2.2 · p. 17, Figure 8 · p. 18

4.3.1. Energy and cross-helicity conservation and divergence-free constraints

• In the unforced ideal test, energy and cross-helicity increments remain at roundoff level, and the strong divergence-free constraints are verified by near-zero divergence norms.The plotted norms are for u2, B2, and J2.Source: §4.3.1 · p. 18, Figure 9 · p. 18

4.3.2. Magnetic-helicity conservation and energy lower bound

• With Re = 1 and Rm = ∞, magnetic-helicity increments are near zero while both energy sequences decrease due to viscosity.This short-time test uses h = 1/8, ∆t = 0.01, and T = 2.Source: §4.3.2 · p. 19, Figure 10 · p. 19
• In the long-time test, both energy sequences decrease but remain well above zero, consistently with the theoretical lower bounds.The test runs to T = 100 with h = 1/6 and ∆t = 0.05; magnetic-helicity increments again remain near zero.Source: §4.3.2 · p. 19, Figure 10 · p. 19

4.3.3. Fluid-helicity conservation

• In the unforced inviscid hydrodynamic limit, the two fluid-helicity diagnostics have near-zero increments, confirming fluid-helicity conservation.The magnetic fields remain zero in this test.Source: §4.3.3 · p. 19

4.4. Orszag–Tang vortex

• The planar reduction sets fields independent of z and their z-components to zero, giving two-dimensional incompressible MHD on a periodic square.The usual three-dimensional magnetic helicity is trivial for these zero-mean planar magnetic fields, so the tests assess field evolution rather than a nontrivial magnetic-helicity invariant.Source: §4.4 · pp. 19–20

4.4.1. Periodic Cartesian spaces

• The periodic Cartesian spaces use cellwise tensor-product polynomials, identify tangential and normal degrees of freedom periodically, and form exact complexes after constant modes are removed.The section also defines the rotated gradient and scalar curl used in the spaces.Source: §4.4.1 · p. 20

4.4.2. Viscous and resistive Orszag–Tang vortex

• At t = 10, the velocity and magnetic fields vary more gradually in space than at t = 2.The two velocity approximations have similar spatial patterns, as do the two magnetic approximations; Figures 12 and 13 show the snapshots.Source: §4.4.2 · p. 20, Figure 12 · p. 21, Figure 13 · p. 22

4.4.3. Nearly ideal Orszag–Tang vortex

• From t = 0.3 to T = 0.7, both current-density approximations show progressively thinning sheets at similar locations.The evolution is qualitatively consistent with results reported in [19] and.Source: §4.4.3 · p. 21, Figure 14 · p. 23

5. Conclusions

• The periodic dual-field finite element method uses sequential linear staggered updates to reproduce energy and cross-helicity balances while retaining strong and weak divergence-free constraints and magnetic- and fluid-helicity balances.Source: §5 · p. 24
• In the unforced ideal case, simultaneous conservation of energy and cross helicity preserves both discrete Elsässer energies and normalized cross helicity.If either Elsässer energy is initially zero, its nonnegative sum-of-squares representation guarantees persistence of the corresponding exact Alfvénic relation.Source: §5 · p. 24
• For the weakly dissipative initial data considered, the method satisfies cumulative cross-helicity balance and develops increasing Alfvénic alignment, while Mao–Xi shows declining normalized cross helicity.In the ideal aligned comparison, energy conservation alone does not ensure preservation of the aligned state; extending the analysis to nonperiodic boundaries remains future work.Source: §5 · p. 24