Invariants of the Axisymmetric Plasma Flows

1 Introduction

As is known [1], [2], the magnetohydrodynamic (MHD) equations with isotropic viscosity and zero electrical resistance have the form

For incompressible plasma flows with velocity V(x, t), (2) implies that the magnetic field B(x, t) is transported in time with the plasma flow (or “is frozen in the flow”). Here, ρ(x, t) is the plasma density, which according to the last equation of (3) is preserved along the plasma streaklines, μ is the magnetic permeability, p(x, t) is the pressure, ν is the kinematic viscosity, ▽ is the nabla operator, and Δ is the Laplace operator. We assume that during some finite time t, the vector functions V(x, t), B(x, t), and the scalar functions ρ(x, t), p(x, t) are of class C². The special regimes of plasma relaxation are derived in the explicit form in [3]. Symmetry transforms for the ideal MHD equilibria are derived in [4]. Exact axisymmetric plasma equilibria are presented in [5] and the helically symmetric ones in [6].

The concrete form νΔV of the dissipative term in (1) can be altered for a more sophisticated formula, for example, for the one recently introduced in [7]. However, this would not have changed our results because they are based entirely on the MHD equations (2) and (3). Therefore, we use in (1) the classical viscosity term νΔV as in the Navier-Stokes equations.

As is known [1], [2], the frozenness of magnetic field B into the plasma flow leads to such invariants as the integer-valued Gauss linking number for any two closed magnetic field lines [8] and the discrete topological invariants of magnetic field knots [9].

Woltjer’s integral H_D [10] of the magnetic helicity has the form

where A is the vector potential of the magnetic field B=∇×A.

Woltjer’s magnetic helicity H_D=∫_DA·[∇×A]d³x [10], [11] has an analogue, which is the hydrodynamic helicity ℋ_D=∫_DV·[∇×V]d³x. The well-known distinction between the two helicities is that the magnetic helicity H_D (4) is not uniquely defined by the magnetic field B because H_D depends on the gauge transformations of the vector potential A. In contradistinction to this, the hydrodynamic helicity ℋ_D is uniquely defined by the velocity field V because there are no gauge transformations of the velocity V.

The hydrodynamic helicity ℋ_D satisfies the equation

where the surface S=∂D is the boundary of the domain D⊂ℝ³, vector field n(x) is the outward unit vector field orthogonal to S, and ds is the area element of surface S. Equation (5) was first derived by Moffatt in [8], where it was assumed that the pressure p is a function of the density ρ only (the barotropic condition p=p(ρ)) and ∇F(ρ)=ρ⁻¹∇p. In ([12], p. 108) Serre has shown that, in general (if [∇×V]·n≠0), the helicity ℋ_D inside an invariant domain D (with the non-penetration condition V·n=0 on S) is not constant in time.

Helicity was studied by Kudryavtseva [13], [14] and by Enciso et al. [15]. The authors stated that for divergence-free vector fields the integral invariants (i.e. the integrals of certain densities “over compact three-dimensional manifold M without boundary” ([15], p. 2035) are often related to helicities.

We derive in this paper new integral invariants of the axisymmetric plasma flows with zero electrical resistance, which depend on the magnetic field B and in some cases on the plasma density ρ but do not depend on any gauge transformations of the vector potential A. Our key idea is to study the safety factor q [2], [16], [17] for the time-dependent axisymmetric magnetic fields B and to prove that the emerging function q(x, t) is conserved along the plasma streaklines.

In Section 2 we define the magnetic rings and magnetic blobs for the axisymmetric dynamics of plasma with zero electrical resistance and prove that the former are frozen into the plasma flows. In Section 3 we calculate the Woltjer’s helicity H_D (4) for the axisymmetric magnetic rings and blobs and demonstrate how it is altered by the axisymmetric gauge transformations of the vector potential A. In Section 5 we prove that the safety factor q(x, t) (which is uniquely defined by the magnetic field B in the domains where all magnetic surfaces are tori or magnetic axes) is conserved along the plasma streaklines. In Sections 6 and 7 we prove that the axisymmetric plasma flows preserve infinitely many new invariants, which are either functional invariants or the integrals of the arbitrary C²-functions of the safety factor q(x, t) over the magnetic rings and magnetic blobs, for example, qⁿ(x, t), n=1, 2, …. The functional invariants are the values of certain axisymmetric functions on the magnetic axes of the axisymmetric field B(x, t). In Section 7 we demonstrate that infinitely many new integral invariants with possibly one exception are functionally independent of the magnetic helicity.

2 Magnetic Rings and Blobs

Let A(r, z, t) be a z-axisymmetric vector potential

where ê_r, ê_z, ê_ϕ are vectors of unit length along the axes of the cylindrical coordinates r≥0, z, φ. The z-axisymmetric gauge transformation of the vector potential A is

where u, v, f, ψ are C²-functions of r, z, and t, and χ(t) is a C¹-function. The gauge transformation (7) is well defined in any (non-simply-connected) toroidal domain T³=D²×𝕊¹ that lies in the space r>0. Here, D² is a compact domain in the plane (r>0, z) and 𝕊¹ is the circle corresponding to the anglular variable φ. The gauge transformation of function ψ(r, z, t) is

The magnetic field B=∇×A has the form

and is evidently gauge-invariant:

The gauge transformation (8) implies that the flux function ψ(r, z, t) is not uniquely defined by the magnetic field B(r, z, t) (9).

Equation (9) yields that the derivative of the flux function ψ(r, z, t) along the plasma streaklines is

Here, the initial poloidal components B_r, B_z of the magnetic field B(r, z, t) and V_r, V_z for the plasma velocity V(r, z, t) can be choosen independently. For unsteady solutions, (11) yields that the generic flux function ψ˜(r, z, t) is not conserved along the plasma streaklines. Indeed, assuming that a special flux function ψ(r, z, t) is conserved (Dψ/Dt=0), we get that all functions ψ˜(r, z, t) (8) with dχ(t)/dt≠0 are not conserved because

For any constant time t, the magnetic field B(r, z, t) (9) is tangent to the z-axisymmetric magnetic surfaces ψ(r, z, t)=C(t). The family of magnetic surfaces for the arbitrary function C(t) is invariant under the gauge transformations (8).

Dynamics along magnetic field lines for t=const is defined by the system

dxdτ=dxdτe^x+dydτe^y+dzdτe^z=drdτe^r+rdφdτe^φ+dzdτe^z=B(r, z, t),

which by virtue of (9) has the form

The system (13), (14) is evidently invariant under the gauge transformations (8), (10). After the time change dτ₁/dτ=r⁻¹, the system (13) for t=const becomes a τ₁-autonomous Hamiltonian system.

Hamiltonian systems were applied to the “plasma-confining magnetic fields” by Kerst [18], see also pages 1062–1063 of review [19]. We derive the Hamiltonian system (13) in the poloidal coordinates (r, z) and not in the Cartesian ones (x, y) as in [18], [19]. The system (13) is derived above straight from the definition B=∇×A and without any assumption of “a uniform B_z” as in ([18], p. 253).

Boozer introduced in [20] for the Hamiltonian treatment of the plasma equilibria “generalised magnetic coordinates”, see also the review ([21], p. 1076). We do not use them because the standard cylindrical ones (r, z, φ) are sufficient for our study.

We will use the following classical properties of any autonomous Hamiltonian system in ℝ² [22]:

• (α)Its Hamiltonian function is a first integral (for system (13) it is the flux function ψ(r, z, t) with a fixed time t);

• (β)All its non-degenerate equilibria are either saddles s_ℓ or centers c_k;

• (γ)It preserves the area.

These properties imply that every closed trajectory C_ψ(t) (ψ(r, z, t)=const) of system (13) for a fixed time t belongs to one of the two-dimensional sets D_k(t) in the poloidal plane (r, z) satisfying the following conditions:

• D_k(t) is invariant with respect to the system (13);

• the set D_k(t) is connected and closed;

• all trajectories of system (13) in the set D_k(t) have finite Euclidean length and are either closed curves or separatrices of the equilibrium points in D_k(t);

• a dense open subset of D_k(t) is filled with closed trajectories of system (13);

• the set D_k(t) is the largest among the sets satisfying conditions 1–4 and having a non-empty intersection with D_k(t).

We will call each set D_k(t) satisfying conditions 1–5 a maximal set (the term “maximal” refers to the condition 5). Figure 1 shows the phase portrait of a concrete system (13) that has eight maximal sets D₁, …, D₈.

Figure 1:

Poloidal sections of magnetic rings D₃, D₆, D₇, D₈ (pink) and magnetic blobs D₁, D₂, D₄, D₅ (blue) for the exact plasma equilibrium with the flux function ψ(r, z)=r²[0.005−zR⁻⁴((3−R²)R⁻¹sinR−3cosR)], where R=r2+z2. Rotation of the sets D₃, D₆, D₇, D₈ around the axis z defines four magnetic rings and rotation of the sets D₁, D₂, D₄, D₅ around the axis z defines four magnetic blobs.

The boundary ∂D_k(t) is a one-dimensional set 𝒞_k(t) that is invariant with respect to the dynamical system (13). Therefore, 𝒞_k(t) satisfies the equation ψ(r, z, t)=b_k(t). System (13) preserves the poloidal area dS=2πr dr dz.

The closed trajectories C_ψ(t) of system (13) (for t=const) after rotation around the axis of symmetry z define the two-dimensional tori Tψ(t)2=Cψ(t)×S1⊂ℝ3, which are invariant with respect to the system (13), (14) and therefore are called B(r, z, t)-invariant. Here, the circle 𝕊¹ corresponds to the angular variable φ. The rotation of a maximal set D_k(t) around the z-axis defines a three-dimensional set Dk3(t)=Dk(t)×S1⊂ℝ3, which is closed and B(r, z, t)-invariant. Its 3D volume is

We call Ar_mD_k(t) the poloidal area of D_k(t); it is equal to the 3D volume VolDk3(t). Rotation of the boundary curve 𝒞_k(t)=∂D_k(t) around the axis z defines the boundary ∂Dk3(t)=Ck(t)×S1.

3 Helicity in the Magnetic Rings and Blobs

Using formulas (6) and (9) and dv=rdrdzdφ, we find the following for Woltjer’s helicity (4) in the magnetic ring or blob Dk3(t):

Using Green’s theorem and the equation ψ(r, z, t)=b_k(t) on the boundary curve 𝒞_k(t)=∂D_k(t), we get

∫Dk(t)[(ψv)r−(ψu)z]drdz=∮Ck(t)ψ(vnr−unz)ds =bk(t)∮Ck(t)(vnr−unz)ds,

where ds is the element of the arc length of the curve 𝒞_k(t), and the vector (n_r, n_z) is the outward unit normal vector to it. Applying Green’s theorem again, we find

bk(t)∮Ck(t)(vnr−unz)ds=bk(t)∫Dk(t)(vr−uz)drdz =−bk(t)∫Dk(t)wdrdz.

Hence we get

Substituting formula (17) into (16), we derive for the magnetic helicity

As shown above, the gauge transformations (7), (8) are well defined in the magnetic rings ℛk3(t). Formulas (8) and (10) imply b˜k(t)=bk(t)+χ(t), w˜(r, z, t)=w(r, z, t). Therefore, (18) yields for the magnetic helicity H˜k(t) after the gauge transformation (8), (10)

Formula (19) demonstrates that the magnetic helicity H_k(t) (16) is not invariant under the gauge transformations (8), (10) if ∫Dk(t)w(r, z, t)drdz≠0. Indeed, the helicity H_k(t) (16) becomes an arbitrary function Fk(t)=H˜k(t) after the gauge transformation (8) with χ(t)=[Fk(t)−Hk(t)](2π∫Dk(t)w(r, z, t)drdz)−1. The helicity H˜k(t) becomes a constant C_k after the gauge transformation (8) with the gauge function

4 Dynamics in the Poloidal Coordinates (r, z)

Suppose a z-axisymmetric magnetic field B(r, z, t₁) at t=t₁ has a closed magnetic field line L. Its dynamics in the poloidal coordinates (r, z) is described by system (13), which defines the corresponding closed trajectory P(L) that satisfies the equation ψ(r, z, t₁)=C. The closed trajectory P(L) in the plane (r, z) belongs to a maximal set D(t₁)⊂ℝ², which is densely filled with closed trajectories Cψ(t1) of system (13), see Figure 1.

For the z-axisymmetric plasma flows, system (13) defines the so-called induced diffeomorphisms of the poloidal plane (r, z). The plasma flow by virtue of the equation divV=0 preserves the 3D volume dΩ=rdrdzdφ. For the z-axisymmetric flows, we get that the induced diffeomorphisms of the poloidal plane (r, z) preserve the poloidal area dS=2πrdrdz, the axis of symmetry z (r=0), and the invariant domain r>0. The frozenness of the magnetic field lines into the plasma flow [1] implies for the z-axisymmetric case that the trajectories of system (13) are frozen into the induced dynamics of plasma in the poloidal coordinates (r, z). Therefore, the phase portrait of dynamical system (13) is changed in time t by the induced diffeomorphisms of the poloidal plane (r, z). Let Mt1t be the induced diffeomorphism defined by the plasma dynamics from time t₁ to time t>t₁. Any closed trajectory Cψ(t1) of system (13) in the invariant domain r>0 is transformed by the diffeomorphism Mt1t into another closed trajectory C_ψ(t) of system (13) in the same domain r>0. Since any maximal set D_k(t₁) is densely filled with the closed trajectories Cψ(t1), it is transformed by the diffeomorphism Mt1t into the maximal set D_k(t). Using the frozenness of the magnetic field B(r, z, t) into the plasma flow, it is easy to prove that the image Mt1t(Dk(t1)) coincides with the maximal set D_k(t).

Because of the conservation of the area dS=2πrdrdz by system (13), we get the equality of the poloidal areas ArmDk(t1)=ArmDk(t)=2π∫Dk(t)rdrdz. Since the induced diffeomorphisms Mt1t preserve the axis of symmetry z (r=0) and the domain r>0, we get that the total number N_r of magnetic rings ℛk3(t)=Dk(t)×S1 and their 3D volumes (15) are constant in time. The same is true for the total number N_s of the magnetic blobs ℬk3(t) and for their 3D volumes. The frozenness of the magnetic rings and blobs into the plasma flows with zero electrical resistance implies that no collapses or touchings between magnetic rings or blobs can occur during the axisymmetric dynamics of plasma.

5 Safety Factor for the Time-Dependent Axisymmetric Magnetic Fields

In this section, we prove the conservation along the plasma streaklines of the safety factor q for the general helical magnetic field lines on invariant tori 𝕋² of a non-stationary axisymmetric magnetic field B(x, t). The safety factor q was studied previously in [2], [8], [16], [17], [23].

Let for a z-axisymmetric field B(x, t) a magnetic field line lie on a torus 𝕋²=C×𝕊¹ and is closed and goes (say) m times a long way (along the circle 𝕊¹) and n times a short way (along the closed curve C). In [17], the safety factor q for such a closed curve is defined as q=m/n. Other definitions of the pitch p=2πq for the vortex lines in the fluid equilibria are given in [8], [24].

We propose the following definition of the safety factor of the non-closed helical magnetic field lines on the time-dependent tori Tψ(t)2=Cψ(t)×S1. Let τ(C_ψ(t)) be the period of the corresponding closed trajectory C_ψ(t) of the system (13). The generalised safety factor q(r, z, t) of the helices is equal to the increment of the angle φ during one period τ(ψ(t)), divided by 2π:

Formula (21) defines the safety factor for both the non-closed, infinite helical trajectories of system (13), (14) on the torus Tψ(t)2 and for the closed ones. The safety factor q(r, z, t) (21) has the same value for all magnetic field lines on a given torus Tψ(t)2 because the circle integral in (21) does not depend on the starting point. If the number q(r, z, t) is rational (=m/n), then during n periods τ(C_ψ(t)) the angle φ increases for n·2πm/n=2πm. Therefore, the corresponding curve on the torus Tψ(t)2=Cψ(t)×S1 is closed and makes m complete turns a long way and n turns a short way. Hence our definition (21) being applied to the closed magnetic field lines reduces to the definition given in [17] only for the closed ones. Such closed curves are called “torus knots K_m,n” [9]. As is known, the rational number m/n is a topological invariant of the torus knot K_m,n [9].

6 Functional Invariants of Plasma Flows

The boundary ∂D_k(t) is a one-dimensional set 𝒞_k(t) that is invariant with respect to the dynamical system (13) and hence satisfies the equation ψ(r, z, t)=C(t). Therefore, for any flux function ψ(r, z, t) (t=const) of class C³, there exists in the interior of each maximal set D_k(t) at least one point c_k(t)=(r_k(t), z_k(t)) of local maximum or minimum of ψ(r, z, t). The point c_k(t) is an equilibrium point of system (13) and is the limit of the neigbouring invariant, small, closed curves C_ψ(t), which are transported by the induced diffeomorphisms of the poloidal plane (r, z). Therefore, the points c_k(t) also are trasported with the plasma flow. The same is true for the corresponding magnetic axes 𝕊_k(t)=c_k(t)×𝕊¹. Let A_k(t) and B_k(t) be the limits of the safety factor q(r, z, t) at the point c_k(t) and at the boundary 𝒞_k(t)=∂D_k(t), respectively. The limits A_k(t) are finite if the local maximum or minimum c_k(t) of function ψ(r, z, t) is non-degenerate. The limits B_k(t) can be infinite, see [25]. Since the safety factor q(r, z, t) is preserved along the plasma streaklines, we get that its limits A_k(t) and B_k(t) also are preserved. Therefore, the limits do not depend on time t and hence are the new invariants A_k and B_k of the viscous plasma flows. We call them the functional invariants.

Suppose that, for some t, the function ψ(r, z, t) has a non-degenerate local maximum or minimum at a point c_k(t)=(r_k(t), z_k(t)). This means that ∂ψ(c_k(t))/∂r=0, ∂ψ(c_k(t))∂z=0, and the Hessian

is positive, ℋ(c_k(t))>0. Then the neighbouring curves ψ(r, z, t)=ψ=const are closed curves C_ψ(t) encircling the point c_k(t). The curves C_ψ(t) are closed trajectories of system (13). Let τ(C_ψ(t)) be their periods. In the limit ψ(r, z, t)→ψ(c_k(t)), the closed curves C_ψ(t) (ψ(r, z, t)=const) tend to the equilibrium point c_k(t): r(τ)→r_k(t) and z(τ)→z_k(t) for all τ. Applying formula (21), we find

Ak(t)=lim(r,z)→(rk(t),zk(t))q(r, z, t)=τ(ψ(ck(t)))w[rk(t), zk(t), t]2πrk(t),

where τ(ψ(c_k(t)))=limτ(C_ψ(t)) when ψ(t)→ψ(c_k(t)).

As is shown above, the value A_k(t) does not depend on t: A_k(t)=A_k=const. Using the formula derived in [25] τ(ψ(ck))=2πrk/ℋ(ck) for the limit of the periods τ(ψ) at ψ→ψ(c_k(t)), we get

Formula (23) defines the explicit form of the new functional invariants A_k of the MHD equations (1)–(3). The Hessians ℋ(c_k(t)) (22) and invariants A_k (23) are independent of the gauge transformations (8).

Remark 1: Formula (23) proves that for any time-dependent axisymmetric magnetic field B(x, t), the safety factor q(x, t) has a finite limit A_k at a magnetic axis 𝕊_k(t)=c_k(t)×𝕊¹, which means at ψ(t)→ψ(c_k(t)), provided that the non-degeneracy condition ℋ(c_k(t))>0 is met. Moffatt mistakenly stated in [16] that for the plasma equilibria the limits A_k are always infinite and the limits B_k in case of blobs ℬk3 are always zero. These two statements have led Moffatt to the erroneous conclusions about the structure of the magnetic field knots. ^[2]

7 Infinite Families of Integral Invariants

Let Dk3(t1) be a maximal ring or blob, and let dv=rdrdzdφ be the 3D volume element. For any n, m=0, 1, 2, 3, … the formulae

define the infinite family of invariants of the axisymmetric MHD equations (1)–(3). Indeed, the plasma flow preserves all functions qⁿ(x, t)ρ^m(x, t)); it transforms each set Dk3(t1) into Dk3(t) for t>t₁ and preserves the 3D volume. Hence the integrals (24) are preserved.

Let us consider also the functionals

which are defined for any integer n≥0 and form an infinite matrix of invariants of (1)–(3). The matrix I_kn (25) has N=N_r+N_s rows because k=1, 2, …, N_r+N_s and infinitely many colums because n=0, 1, 2, 3, ….

Formulas (24) for n=m=0 and (25) for n=0 imply that the volume and the total mass M_k of each magnetic ring ℛk3(t) and each magnetic blob ℬk3(t) are invariants of plasma dynamics.

Each invariant |I_kn| (25) is bounded by the total mass M_k of plasma inside the set Dk3(t). Indeed, because |cos(nq)|≤1, formulas (25) yield |Ikn|≤∫Dk3(t)|cos(nq(x, t))|ρ(x, t)dv≤∫Dk3(t)ρ(x, t)dv=Mk.

The mutual functional independence of invariants I_kn follows from that for the functions cosnq for n=1, 2, 3, ….

The conservation of functions qⁿ(x, t)ρ^m(x, t)) along the plasma streaklines is equivalent to the equation

∂[qn(x, t)ρm(x, t))]/∂t +∑i=13Vi(x, t)∂[qn(x, t)ρm(x, t))]/∂xi=0,

where V_i(x, t) are components of the plasma velocity. This equation, together with equation divV=0, defines an infinite family of conservation laws

∂[qn(x, t)ρm(x, t))]∂t+∑i=13∂[Vi(x, t)qn(x, t)ρm(x, t))]∂xi=0,

where n, m=0, 1, 2, 3, ….

8 Conclusion

The following main results were obtained in this article:

1. For the axisymmetric dynamics of viscous plasmas with zero electrical resistance, we have proved the existence of magnetic rings ℛ_k(t) and magnetic blobs ℬ_m(t) which are frozen into the plasma flows and therefore cannot intersect each other during the plasma dynamics.

2. We constructed the family of functional invariants A_k which are presented by the explicit formula (23).

3. We presented infinitely many integral invariants J_knm (24) and I_kn (25) and proved that they are functionally independent of the magnetic helicity (4), (27).

4. The axisymmetric dynamics of plasmas with zero electrical resistance between two given states is possible only if the corresponding total numbers of magnetic rings N_r are equal (the same for the total numbers of magnetic blobs N_s) and all new invariants A_k, J_kmn, I_kn for them coincide.