Exact force-free plasma equilibria with axial and with translational symmetries

1 Introduction

Equations of the ideal plasma equilibria have the form [1], [2], [3]:

Magnetic field B(x) for a force-free plasma equilibrium with pressure P(x)=const satisfies the Beltrami equation

where ∇ × B(x) = J(x) is the electric current and α(x) is a differentiable function of the coordinate vector x. Equation (1.2) with α(x)≠const is nonlinear because function α(x) depends on the vector field B(x). Indeed, the well-known equation ∇α(x)⋅B(x)=0 follows from Equation (1.2) and means that function α(x) is constant along the magnetic field B(x) lines, see Chapter 1 of monograph [3]. Hence for a general case function α(x) is constant on magnetic surfaces ψ(x) = const (ψ(x) is the magnetic function).

Another relation connecting α(x) and B(x) is

Equation (1.3) means that vector field ∇α(x) × B(x) (that is also tangent to the magnetic surfaces α(x) = C = const) is divergence free. Equation (1.3) follows from Equations (1.2), ∇⋅B(x) = 0 and identity

where X and Y are arbitrary smooth vector fields.

There are well-known exact solutions to Equation (1.2) with α(x)=α=const, for example the spheromak magnetic field [1], [2]. The general solutions to the linear Beltrami equation with α(x)=const were presented in terms of Bessel and Legendre functions in papers [1], [2], [4] and later analysed in detail in Refs. [3], [5], [6], [7], [8]. We proved in Refs. [9], [10], [11] that any solution to the linear Beltrami Equation (1.2) with α(x) = α = const has the form

where T(k) is an arbitrary smooth vector field tangent to the unit sphere S²: k⋅k = 1 and dσ is an arbitrary measure on S². We presented in papers [9], [10], [11] the unsteady generalizations of exact solutions Equation (1.4) as exact solutions to the Navier–Stokes equations and to the viscous magneto-hydrodynamics equations.

The term “spheromak” was first introduced in Ref. [12], see review [13]. Moduli spaces of vortex knots for the spheromak Beltrami field in different invariant domains were presented in Ref. [14], and for another Beltrami field in Ref. [15]. The vortex knots for non-Beltrami fluid flows were studied in Refs. [16], [17], [18].

In section 2 we introduce a transformations T_β of the axially symmetric plasma equilibria and show that transformations T_β satisfy equation T_γ⋅T_β = T_β+γ and hence for β≥0 form a Lie semi-group. In section 3 we construct using transformations T_β an abundance of exact force-free plasma equilibria satisfying the nonlinear Beltrami Equation (1.2) with α(x)≠const.

For all previously known force-free plasma equilibria (Equation 1.2) with α(x) = α = const the relation J(x) = αB(x) holds. Therefore the magnetic field B(x) and electric current J(x) vanish at the same isolated points. For the equilibria constructed in this paper magnetic field B(x) also vanishes at some isolated points x but the electric field J(x) vanishes additionally on the entire magnetic surfaces defined by equation ψ‾(x)=0 where ψ‾(x) is the magnetic function. This follows from the fact that the new force-free plasma equilibria satisfy equation

where α and β>0 are arbitrary constants. Equation (1.5) yields that for the constructed equilibria electric current J(x) changes its direction to the opposite when point x crosses the magnetric surface ψ‾(x)=0. We present in section 3 two examples of axially symmetric force-free plasma equilibria where magnetic surface ψ‾(x)=0 has infinitely many components in ℝ³ and therefore the switching of direction of electric current J(x) occurs on infinitely many surfaces.

Translationally invariant force-free plasma equilibria are constructed in section 4. We present equilibria based on exact solutions to the nonlinear equation

and on exact solutions to the Helmholtz equation ∇²ψ(x, y) = −α²ψ(x, y) .

Exact force-free plasma equilibria connected with the elliptic Sine–Gordon equation are constructed in section 5. The solutions are smooth and bounded in the whole space ℝ³.

As known, equilibria of an ideal incompressible fluid with a constant density ρ obey the equations equivalent to Equations (1.1). Therefore the presented constructions are equally applicable to the exact axisymmetric ideal fluid equilibria and to the translationally invariant ones.

2 A transformation of the axisymmetric plasma equilibria

I. A steady z-axisymmetric magnetic field B(r, z) satisfying equation ∇⋅B = 0 has the form

where ψ‾(r, z) is the magnetic (or flux) function and e^r, e^z, e^φ are unit vector fields in the directions of the cylindrical coordinates r, z, φ. The corresponding electric current J(r, z) = ∇×B(r, z) is

The magnetic field B(r, z) Equation (2.1) and electric current J(r, z) Equation (2.2) are tangent to the magnetic surfaces ψ‾(r, z)=const.

The plasma equilibrium Equations (1.1) for the z-axisymmetric magnetic field B(r, z) Equation (2.1) were reduced in 1958 to the Grad–Shafranov equation [19], [20]:

where P(ψ‾) is the plasma pressure.

II. Substituting into Equation (2.2) the expression of ψ‾rr−1rψ‾r+ψ‾zz from the Grad–Shafranov Equation (2.3) we get:

Using here Equation (2.1) we arrive at equation

Equation (2.5) yields that for P(ψ‾)=const the magnetic field B(r, z) satisfies the Beltrami equation

that has the form of Equation (1.2) with α(x)=dG(ψ‾)/dψ‾.

III. The last term in Equation (2.3) equals to −12dG2(ψ‾)/dψ‾. It is evidently unchanged after the simple nonlinear transformation

Therefore the same magnetic function ψ‾(r,z) satisfies also Equation (2.3) with Gβ(ψ‾) instead of G(ψ‾). Substituting Gβ(ψ‾)=±β+G2(ψ‾) into Equations (2.1) we get a new magnetic field

for that Equation (2.5) takes the form

Equation (2.9) for P(ψ‾)=const becomes the Beltrami equation

where

Hence transformation T_β (Equation (27)) produces from any axisymmetric solution to the Beltrami Equation (2.6) a new solution B_β (r, z) (Equation 2.8) to the Beltrami Equation (2.10) with another function α_β(x) Equation (2.11), β > 0. Equation (2.11) yields

Therefore function α_β(x) is changing in the range −|α(x)|<α_β(x)<|α(x)| .

Indeed, Equation (2.7) yields

Hence Equation (2.12) holds. For β < 0 transforms (Equation 2.7) are defined only in the domain G²(ψ‾(x))≥|β|. For 0 ≤ β < ∞ transforms (Equation 2.7) are defined everywhere in ℝ³. Evidently T₀(G) is the identity transformation. Therefore Equation (2.12) yields that transformations T_β (Equation (2.7)) with sign + and 0 ≤ β< ∞ form a one-dimensional Lie semi-group.

3 Exact axisymmetric force-free plasma equilibria

I. Consider the Grad–-Shafranov Equation (2.3) with P(ψ‾)=const and G(ψ‾)=Gζ(ψ‾)=±ζ+α2ψ‾2 with an arbitrary constant α and ζ ≥ 0. Evidently we have dGζ(ψ‾)/dψ‾=α2ψ‾/Gζ(ψ‾). Therefore Gζ(ψ‾)dGζ(ψ‾)/dψ‾=α2ψ‾ and hence Equation (2.3) takes the linear form

Let B_ζ (r,z) be the corresponding magnetic field (Equation 2.1):

Substituting the Grad–Shafranov Equation (2.3) with P(ψ‾)=const and Gζ(ψ‾)=±ζ+α2ψ‾2 into Equation (2.2) we get the electric current

Inserting Equation (3.2) into Equation (3.3) we find

Equation (3.4) is the Beltrami Equation (1.2) with the non-constant function

Therefore for any solution ψ‾(r, z) to the linear Equation (3.1) we constructed the force-free magnetic fields B_ζ (r, z) (3.2) and electric currents J_ζ (r, z) (3.3) satisfying the Beltrami Equation (1.2) with non-constant function α(x) (Equation 3.5). Only for ζ=0 function dG0(ψ‾)/dψ‾=±α becomes constant.

For function α_ζ (r, z) (Equation 3.5) we have (for ζ > 0)

Hence function α_ζ (r,z) (Equation 3.5) satisfies inequalities −|α|<α_ζ(r,z)<|α| and α_ζ (r,z)=0 at the points (r,z) where ψ‾(r,z)=0.

Here R = r2+z2 is the spherical radius in ℝ³. The corresponding to the solution (Equation 3.6) magnetic fields (Equation 3.2)

with any 0 ≤ ζ < ∞ satisfy the nonlinear Beltrami Equation (1.2) with the non-constant functions α(r, z):

Equation (3.8) yields that for the new force-free plasma equilibria (Equation 3.7) the electric current

vanishes on the magnetic surface ψ₂ (r,z) = 0. Equation (3.6) implies that the latter has infinitely many components that all are spheres Sk2:R=Rk where R_k are the roots of equation

The first four numerical solutions to Equation (3.10) are

At k→∞ the roots R_k have asymptotics |α|Rk≈(k+12)π. The electric current J_ζ.2 (r, z) (Equation 3.9) switches its direction to the opposite at the infinitely many spheres Sk2. Equation (3.9) yields that electric current J_ζ.2 (r, z) is smooth everywhere in ℝ³ and has zero current density on the axis z (r = 0).

Function G₂(u) in Equation (3.6), G₂(u) = u⁻²(cos u−u⁻¹sin u) where u = αR, is connected with the Bessel function J_3/2(u) [21] of order 3/2 by the relation

The functions G_k(u) are analytic for all u and have the following values at u = 0: G₂(0) = −1/3, G₃(0) = 1/15, G₄(0) = −1/105 [15].

Magnetic field B_ζ.3(r, z) has the form

and satisfies the nonlinear Beltrami equation

The electric current

vanishes on the magnetic surface ψ₃(r, z) = 0 that according to Equation (3.11) contains the plane z = 0 and infinitely many spheres Sm2: R = Rm where R_m satisfy equation G₃(αR) = 0:

The electric current J_ζ.3 (r, z) (Equation 3.15) switches its direction to the opposite at the plane z = 0 and at infinitely many spheres Sm2. Equation (3.15) demonstrates that electric current J_ζ.3 (r, z) is smooth everywhere in ℝ³ and has zero current density on the axis z (r = 0).

The first four numerical solutions to Equation (3.16) are

At m→∞ the roots R_m have asymptotics |α|R_k≈(m+1)π.

II. Analogous construction exists for the magnetic function ψ₄ (r, z) (Equation 3.12). The corresponding electric current J_ζ.4 (r, z) = ∇ × B_ζ.4(r, z) vanishes on the magnetic surface ψ₄ (r, z) = 0 that according to Equations (3.12) and (3.14) satisfies equation

Equation (3.17) yields that the surface intersects the plane z = 0 at infinitely many circles 𝕊m1:z=0, R=Rm where R_m are roots of equation G₃(αR) = 0 (Equation 3.16). Therefore the magnetic surface ψ₄ (r, z) = 0 (Equation 3.17) has infinitely many components that are not spheres but are z-axially symmetric.

The linearity of Equation (3.1) yields that any linear combination

obeys Equation (3.1). Let us consider the corresponding magnetic fields (Equation 3.2):

with 0 ≤ ζ < ∞. Equation (3.4) yields that the magnetic fields B_ζ.N (r, z) (Equation 3.18) satisfy the nonlinear Beltrami Equation (1.2):

The electric currents

vanish on magnetic surfaces ψ_N (r, z) = 0 that have infinitely many non-spherical axisymmetric components. The current density vanishes on the axis z (r = 0).

4 Exact translationally invariant force-free plasma equilibria

I. In the Cartesian coordinates x, y, z, the z-independent magnetic fields B(x, y) satisfying the equilibrium Equations (1.1) have the form

where ψ = ψ(x, y) is the magnetic function, G(ψ) is an arbitrary differentiable function of ψ and e^x, e^y, e^z are unit vectors in directions of coordinates x, y, z.

The electric current J = ∇ × B takes the form

where ∇²ψ = ψ_xx + ψ_yy . Hence we get

Therefore the translationally invariant plasma equilibrium Equations (1.1) take the form

where pressure P = P(ψ) is an arbitrary differentiable function of ψ.

Substituting Equation (4.4) into Equation (4.2) we find

From Equations (4.1) and (4.5) we derive

Hence for the z-independent force-free plasma equilibria with P(ψ) = const magnetic field B satisfies Beltrami equation: ∇ × B = α(x)B with function α(x) = −dG(ψ)/dψ.

II. Consider magnetic field

with exponential function G(ψ) = 2me^ψ/2. Applying transformation T_β(Equation 2.7) we get magnetic fields

with function G_β(ψ) = ±β+2m2eψ. Beltrami Equation (4.6) with P(ψ) = const for the field (Equation 4.8) becomes

For the both magnetic fields Equations (4.7) and (4.8)Equation (4.4) with P(ψ) = const has the form

III. Exact solutions to the nonlinear Equation (4.10) were first derived by Vekua [22]. Vekua’s method consists of the following. Let x + iy be a complex variable and f(x + iy) = u(x, y) + iv(x, y) be any analytic function of x + iy. Then the Cauchy–Riemann equations u_x = v_y , u_y = −v_x hold. Let function ψ(x, y) has the form

Since f′ = df(x + iy)/d(x + iy) = u_x + iv_x = u_x−iu_y we get log|f′|² = log(u_x² + u_y²). Since f′ also is an analytic function we have ∇²log|f′|² = 2∇²log|f′| = 0. Hence ∇²log(u_x²+u_y²) = 0. Therefore we get

Using here equations ∇²u = 0, ∇²v = 0 and the Cauchy–Riemann equations we find

Equation (4.11) yields

Equations (4.12) and (4.13) imply that for arbitrary analytic functions f(x + iy) functions ψ(x,y) (Equation 4.11) satisfy Equation (4.10).

IV. Consider analytic function f(x + iy) = a(x + iy)^k where a = const and k ≥ 1 is an integer. Then f′ = ka(x + iy)^k−1 and function (Equation 4.11) becomes

For k ≥ 2 function ψ(x, y) (Equation 4.14) has singularity (tends to −∞) at x = 0, y = 0. For k = 1 we get

Function ψ₁(x, y) is smooth everywhere. The corresponding force-free magnetic field B_β.1(x, y) (Equation 4.8) is a generalization of the Bennett pinch solution [23], [24]. It satisfies Equation (4.9) that takes the form

V. Consider analytic function f(x+iy) = e^α(x+iy) where α is real. Hence |f(x+iy)|=eαx and function ψ(x, y) (Equation 4.11) becomes

Hence Equation (4.9) for the corresponding force-free magnetic field B_β.α(x, y) (Equation 4.8) takes the form

VI. Let us consider functions

where α and ζ > 0 are arbitrary parameters. We have G_ζ(ψ)dG_ζ(ψ)/dψ = α²ψ. Hence Equation (4.4) with P(ψ) = const takes the linear form

The two-dimensional Helmholtz Equation (4.16) evidently has exact solutions

where 0 ≤ θ < 2π and f(θ) is any piece-wise continuous function of angle θ.

Any finite sum of functions (Equation 4.17) is an exact solution to the linear Equation (4.16):

where C_k, θ_k are arbitrary constants. For the corresponding magnetic fields Equation (4.1), Equation (4.15) with ζ > 0

Equation (4.18) is the Beltrami Equation (1.2) with the non-constant function

VII. Integrating functions ψ(x,y,θ) (Equation 4.17) with respect to the angle θ and using the linearity of Equation (4.16) we derive the general exact solution to the Helmholtz Equation (4.16):

The corresponding magnetic fields Equation (4.1), Equation (4.15) with ζ > 0 have the form

Equation (4.6) with P(ψ^)=const and G(ψ)=Gζ(ψ^)=±ζ+α2ψ^2 with ζ > 0 takes the form

Equation (4.21) is the Beltrami Equation (1.2) with non-constant function

Thus Equations (4.19), (4.20) provide an abundance of exact solutions to the nonlinear Beltrami Equation (1.2), (4.21). The exact solutions (Equations 4.19, 4.20) for ζ > 0 are bounded for all x, y. It is evident that −|α|<α^ζ(x, y)<|α|.

5 Plasma equilibria connected with the Sine–Gordon equation

Let us consider a trigonometric function G(ψ) = A sin [α(ψ + γ)] where A, α and γ are arbitrary constants. We get

Therefore Equation (4.4) with P(ψ) = const takes the form

that coincides with the elliptic Sine-Gordon equation. The Beltrami Equation (4.6) (with P(ψ) = const) is

Hence function α(x) in the corresponding Beltrami Equation (1.2) is α(x) = −αAcos[α(ψ+γ)].

To construct exact solutions to the nonlinear Equation (5.1) we consider equation of first order

Differentiating Equation (5.3) with respect to x we get ψ_xx=−αA² sin[α(ψ+γ)] cos[α(ψ+γ)]=−(αA²/2) sin[2α(ψ+γ)]. Hence any solution to Equation (5.3) satisfies Equation (5.1). Integrating Equation (5.3) we find its exact solutions

that satisfy also Equation (5.1). Solutions (Equation 5.4) lead (after rotation of variables x, y for an angle θ) to the more general solutions to Equation (5.1):

where λ = cosθ, 1−λ2 = sinθ. For functions ψ_λ(x, y) (Equation 5.5)Equation (5.2) with function G(ψ_λ) = A sin [α(ψ_λ+γ)] takes the form

Equation (5.6) shows that electric current J = ∇ × B always has the same direction as vector field −αAB and no switching of its direction occurs.

Applying transformations T_β(Equation 2.7) with β > 0: G(ψ_λ)→G_β(ψ_λ) = ±β+G2(ψλ) we get the magnetic fields (Equation 4.1):

Magnetic fields (Equation 5.7) with the exact magnetic functions ψ_λ (x, y) (Equation 5.5) satisfy Beltrami equation

Using here exact solution (Equation 5.5) we obtain