Exact flows of an incompressible fluid satisfying the Beltrami equation inside a spherical shell are constructed in the Cartesian coordinates in terms of elementary functions. Two scale-invariant equations defining two infinite series of eigenvalues λ n and of the operator curl in the shell with the nonpenetration boundary conditions on the boundary spheres are derived. The corresponding eigenfields are presented in explicit form and their symmetries are investigated. Asymptotics of the eigenvalues λ n and at n, m → ∞ are obtained.
Here, V(x, t) is fluid velocity, B(x, t) is magnetic field, p(x, t) is the pressure, Φ(x) is the gravitational potential, ρ is constant fluid density, ν(t) is viscosity coefficient, η(t) is magnetic viscosity, and Δ is the Laplace operator.
Equation (1.1) for V(x, t) = 0 reduce to
The force-free plasma equilibria are defined by condition p(x, t) + Φ(x) = const and are equivalent to the Beltrami equation curl B = λ B, where λ = const in view of equation div B = 0.
If magnetic field B = 0 and viscosity ν(t) = 0, Eq. (1.1) becomes Euler equations for ideal incompressible fluid:
For the steady case and with the additional condition p/ρ + |V|2/2 + Φ = const, the Euler equations reduce to the Beltrami equation
where c ≠ 0 is an arbitrary parameter.
where V 1(x) is any solution to the Beltrami Eq. (1.4) and pressure p is defined from equation p/ρ + |V|2/2 + Φ = const.
If B(x, t) = 0 then Eq. (1.1) reduce to the Navier–Stokes equations
In this paper, we construct two infinite series of exact Beltrami flows satisfying Eq. (1.4) inside the spherical shells
We assume that on the two boundary spheres : R = R 1 and : R = R 2 the nonpenetration boundary condition n(x) ⋅ V(x) = 0 is satisfied. Here, n(x) is vector field of unit normals to the boundary. Since for the spheres we have n(x) = x/|x| the boundary conditions have the equivalent form
for , k = 1, 2. The same solutions with magnetic field B(x) instead of velocity V(x) describe the force-free magnetic fields inside a spherical shell (1.9). All constructed exact solutions define by formulas (1.6) and (1.7) exact flows of viscous magnetic fluid inside the spherical shell (1.9) with the nonpenetration conditions (1.10).
The main problem consists of finding the eigenvalues λ n of the operator curl and its eigenfields V n (x) that satisfy Eq. (1.4) inside the spherical shell and the nonpenetration condition (1.10) at its two boundaries. Cantarella, DeTurck, Gluck, and Teytel studied in paper  vector fields V that “satisfy the equation ∇ × V = λ V , where λ is the eigenvalue of curl having smallest nonzero absolute value among such fields. It is shown that on the ball the energy minimizers are axially symmetric spheromak fields found by Woltjer and Chandrasekhar-Kendall, and on spherical shells they are spheromak-like fields” , p. 2766. The authors used the spherical coordinate r, θ, φ in and the theory of Bessel functions, and the Legendre and Gegenbauer functions applied earlier by Woltjer  and by Chandrasekhar and Kendall [7, 8]. The methods of papers [6–8] were used also in the works [1, 9], , ,  where the eigenvalue problem (1.4), (1.10) was studied in spherical coordinates in the ball using the Bessell functions.
Cantarella, DeTurck, Gluck, and Teytel presented in Table 1 of paper  results of their calculations (using Bessel’s functions) of the minimal eigenvalues for different values of R 1 and R 2.
In this paper, we calculate for the first time in the literature the products of the correspondent numbers and R 1 − R 2 from Table 1 of . The results of our calculations are presented in the right column of Figure 1.
Reading the right column of Figure 1 down from the top to the bottom we see that numbers monotonously decrease and tend to the number of π = 3.1415926535…. Therefore our calculations prove that the dimensionless eigenvalues converge to the number of π when (R 1 − R 2) → 0 (hence the dimensionless parameter μ = R 1 R 2/(R 1 − R 2) → ∞). As a consequence of this we get the asymptotics
Note that for R 1 = 10 and R 2 = 9.996665555, the corresponding value of parameter μ is μ = 8996999.5. For R 1 = 100 and R 2 = 99.99996667 (in the bottom line of Figure 1), the value of μ is enormous: μ = 9001797 × 106.
where c > 0 is an arbitrary parameter. We define the dimensionless parameters
that are invariant under the group of scaling transformations (1.12). We will call parameter x the dimensionless eigenvalue.
We construct in this paper two infinite series of eigenvalues λ n and and the corresponding eigenfields V n (x, A) and which depend on arbitrary constant vectors . Each eigenvalue λ n has multiplicity 3 and each eigenvalue has multiplicity 5. We study the eigenvalue problem (1.4), (1.10) in an arbitrary shell (1.9) and use only Cartesian coordinates x 1, x 2, x 3 and elementary functions of them. In Section 3, we derive the first series of eigenvalues λ n = x n /(R 1 − R 2) where x n are roots of equation
In Section 4, we derive the second series of eigenvalues where are the roots of equation
We prove that any root of Eq. (1.17) defines the eigenvalue :
Cantarella, DeTurck, Gluck, and Teytel wrote on p. 2767 of :
“Theorem B. For the spherical shell B 3(a, b) of inner radius a and outer radius b, the eigenvalue of curl having least absolute value is , where is the smallest of the infinite sequence of positive numbers x k that satisfy
Let us prove that for any given radii R 1 and R 2 no one eigenvalue λ n coincides with an eigenvalue . Indeed, let λ be the common value of . Then the dimensionless eigenvalue x = λ(R 1 − R 2) satisfies both Eqs (1.14) and (1.17). Subtracting Eqs (1.14) and (1.17) and reducing to the common denominator we arrive at equation
In Section 3, we prove that all eigenvalues λ k for R 2 → R 1 have the following asymptotics
This asymptotics for k = 1 reduces to the special asymptotics (1.11). Equation (1.14) with arbitrary values of parameter μ > 0 (that means for arbitrary R 1 and R 2) yields the following asymptotics for eigenvalues λ n :
In Section 4, we prove the following asymptotics for the eigenvalues :
In Section 3, we present the three-dimensional (3D) space of eigenfields V n (x, A) in explicit form that yields that any linear combination of the axisymmetric eigenfields V n (x, A) with arbitrary axes of symmetry is again axisymmetric.
We present in Section 5 in an explicit form the linear five-dimensional (5D) space of eigenfields and prove in Section 6 that the eigenfields with noncollinear constant vectors A, B are not axisymmetric. The corresponding dynamical systems do not have first integrals. A recent numerical investigation  proves that some of these systems possess chaotic streamlines and therefore are not integrable.
The special eigenfields with B = κ A are axisymmetric. The general linear combinations of the axisymmetric eigenfields are not axisymmetric. In Section 7, we demonstrate that topologies of the axisymmetric fluid flows and V n (x, A) are completely different. All eigenfields are presented in explicit form in terms of elementary functions of the Cartesian coordinates x 1, x 2, x 3.
2 Functions H n (λR) and their properties
Using the method of Chandrasekhar and Kendall , we study vector fields
where U(x) satisfies the vector Helmholtz equation
In view of the identity curl curl U(x) = grad(div U(x)) − ΔU(x) and Eq. (2.1), we get
We consider vector field U(x) (2.1) of the form U(x) = H 1(λR)A, where A is an arbitrary vector and function H 1(λR) satisfies the spherically symmetric Helmholtz equation
variable R is the spherical radius . All solutions to the Helmholtz Eq. (2.3) have the form
where v = λR, and a, b are arbitrary constants. Here, function H 0(v) is
In this paper, we will use elementary functions
Let us define by induction the functions
Functions H 0(v), H 1(v), …, H n (v), … with arbitrary constants a and b satisfy an infinite series of identities
3 The first series of eigenvalues λ n and axisymmetric eigenfields V n (x, A)
Substituting formula U(x) = H 1(v)A into Eq. (2.2) we derive
Vector field (3.1) evidently is invariant with respect to rotations around the axis having direction A. Vector fields (3.1) for any vector A and for λ = 1 and H 1(v) = (sin R)/R (that corresponds to b = 0) are equivalent to the well-known in plasma physics spheromak exact solution [1, 7, 13, 14].
Hence we get
where we put λ 2(x ⋅ x) = λ 2 R 2 = v 2. Identity (2.9) for n = 1 has the form
Multiplying Eq. (2.6) with v 2/(a cos v) we find
Hence equation H 2(v) = 0 takes the form
The second equality implies
Collecting the similar terms we get
Hence we find
Substituting v 1 = λR 1 and v 2 = λR 2 into Eq. (3.9) and using the trigonometric identity
we get from Eq. (3.9) the equation for the eigenvalues λ:
Substituting v 1 = λR 1 and v 2 = λR 2 into Eq. (3.8) we get
Hence function H 1(λR) (2.4) has the form
where a is an arbitrary parameter.
Let us express the dimensionless quantities λR 1 and λR 2 in terms of variables (1.13). We have
Evidently α 1 − α 2 = 1 and α 1 α 2 = μ. Hence α 1 satisfies equation . Therefore by Vieta formula we find
Since function tan(x) is π-periodic and tan(x) → ±∞ as x → (n ± 1/2)π we see that Eq. (3.19) for any constant μ > 0 has infinitely many roots x n that satisfy the inequalities . The maximal value of function f(x) = x/(1 + μx 2) is that is attained at . Therefore at μ ≫ 1 function for all x. Hence solutions x k of Eq. (3.19) are close to the roots of function tan(x) that means to kπ. Hence we get the asymptotics at μ ≫ 1 or at R 2 → R 1:
Comparing these sequences with the sequence of numbers z k = kπ:
corresponding to the eigenvalue λ n = x n /(R 1 − R 2). Vector field V n (x, A) (3.1) and (3.2) corresponding to function H 1.n (λ n R) (3.27) satisfies the Beltrami Eq. (1.4) with λ = λ n and the two nonpenetration boundary conditions (1.10). Therefore it is one of the corresponding eigenfields.
Any linear combination of eigenfields (3.2) depending on arbitrary vectors A k has the form
due to the linearity of formula (3.2) with respect to vector A. Hence an arbitrary linear combination (3.28) is another Beltrami vector field (3.2). Therefore for any eigenvalue λ n the linear space of the corresponding eigenfields (3.2) has dimension 3. Hence each eigenvalue λ n = x n /(R 1 − R 2) where x n is a root of Eq. (3.19) has multiplicity 3.
In view of formula (3.2), the eigenfield (3.28) is invariant under rotations around vector . Formula (3.28) provides an explanation why any linear combination of the axisymmetric eigenfields is also axisymmetric. For the second series of eigenfields in Section 4, this is not so.
4 The second series of eigenvalues and eigenfields
For functions H k (v) we find using the equality v = λR and definition (2.8):
Permuting here vectors A and B we get vector field
Formula (4.5) and λ 2(x ⋅ x) = λ 2 R 2 = v 2 yield
Identity (2.9) for n = 2 takes the form
Equation (4.8) implies that on each sphere satisfying equation R = R *, where H 3(v *) = H 3(λR *) = 0 we have . That means vector field is tangent to the spheres : R = R *. Hence the spheres are invariant submanifolds for the flows (4.5) with arbitrary vectors A and B. Therefore the two nonpenetration boundary conditions (1.10) are satisfied for the fluid flows (4.5) if and only if
Multiplying function H 3(v) (2.7) with v 4/(a cos v) we get
Hence equation H 3(v) = 0 is equivalent to
where v 1 = λR 1 and v 2 = λR 2. Equation (4.11) leads to
Collecting here similar terms we find
This equation evidently implies
Applying here the trigonometric identity (3.10) and substituting v 1 = λR 1 and v 2 = λR 2 we get the equation for the eigenvalues λ:
Equation (4.12) evidently is invariant with respect to the scale transformations (1.12). Using formulas (3.17) it is easy to verify that in the dimensionless variables x, μ (1.13) Eq. (4.12) takes the form
Since tan(x) is periodic with period π, tan(kπ) = 0 and tan(x) → ±∞ as x → (k ± 1/2)π we get that Eq. (4.13) has infinitely many roots. The function f 2(x) in the right hand side of Eq. (4.13) tends to zero when either μ → ∞ (that means R 2 → R 1) or when x → ∞. Hence we get the asymptotics
Therefore the corresponding function H 1(λR) (2.4) takes the form
where a is an arbitrary constant. Vector fields (4.5) with and constant b/a (4.17) and corresponding to the function H 1.m (v) (4.18) satisfy the Beltrami Eq. (1.4) with in the spherical shell (1.9) and the two nonpenetration boundary conditions (1.10) on the spheres R = R 1 and R = R 2.
Vector fields (4.5) are bilinear symmetric functions of vectors A and B because of equality . Expanding vectors A and B in an orthonormal basis , , and we find that any eigenfield is a linear combination of six eigenfields . However, in view of identity (5.1) proved in Section 5 below only five of these eigenfields are linearly independent. Therefore for any eigenvalue the space of eigenfields has dimension 5. Hence the multiplicity of each eigenvalue is equal to 5.
5 An identity for axisymmetric eigenfields
Vector field (4.5) with B = A has the form
Vector field (5.1) evidently is invariant under rotations around the axis generated by vector A. The corresponding fluid flow is integrable and its streamlines belong to certain axisymmetric tori , see Section 7.
Let vectors A 1, A 2, and A 3 are mutually orthogonal: (A i ⋅ A j ) = 0 and i ≠ j. Then the identity holds:
Vector fields have unit lengths and are mutually orthogonal. Evidently we have . Vector field (4.5) has the form
The following identities evidently hold in the basis , , and :
Hence using Eq. (5.3) and equation λ 2 R 2 = v 2 we find
This expression vanishes due to the identity x × x = 0 and identity (4.7). Hence we get
that proves the identity (5.2). □
An arbitrary linear combination of two axisymmetric eigenfields and is not axisymmetric. However, if vectors A 1 and A 2 are orthogonal and have equal lengths A = |A 1| = |A 2| then using identity (5.2) of Lemma 2 we get
where A 3 = (A 1 × A 2)/A, |A 3| = A. Equations (5.4), (5.1) imply that for this case the eigenfield is axisymmetric with respect to the axis generated by vector A 3 that is orthogonal to both vectors A 1 and A 2.
6 Eigenfields with any noncollinear vectors A, B are not axisymmetric
Fluid streamlines defined by an eigenfield are trajectories of the dynamical system
For any noncollinear vectors A and B vector field and dynamical system (6.1) are not axisymmetric.
On the boundaries of the spherical shell (1.9) : R = R k and v = v k = λR k we have H 3(v k ) = 0. Hence from identity (4.7) we get . Note that H 2(v k ) ≠ 0 because if both H 3(v k ) = 0 and H 2(v k ) = 0 then from identities (2.9) for n = 1 and n = 0 we get H 1(v k ) = 0 and H 0(v k ) = 0 that implies a 2 + b 2 = 0. Therefore in view of Eq. (4.5) dynamical system (6.1) at H 3(v k ) = 0 takes the form
System (6.2) evidently has critical points
Note that A × B ≠ 0 because vectors A and B are noncollinear. The critical points (6.3) are intersections of the sphere with the axis
Substituting x = r(A × B) into formula (4.5) and using ((A × B) ⋅ A) = 0 and ((A × B) ⋅ B) = 0 we get
Equation (6.5) shows that if (A ⋅ B) = 0 then the axis L consists of critical points of the system (6.1). Equation (6.5) defines nontrivial dynamics on the line L if (A ⋅ B) ≠ 0. In both cases, if system (6.1) is axisymmetric then its axis of symmetry can be only the axis L and then system (6.1) has no isolated critical points outside of the axis L.
However, let us show that system (6.2) has another critical points which belong to the intersection of the sphere with the plane P orthogonal to the axis L and have the form
Equation (6.2) at a critical point yields
Since vectors A and B are linearly independent, we get from Eq. (6.7) two equations
Using expression (6.6) we find
Hence we get β = ±α|A|/|B|. Substituting this into Eq. (6.10) we find
The bracket here is nonzero due to the Cauchy inequality and the noncollinearity of vectors A and B. Therefore we derive
Hence we find the critical points (6.6):
If system (6.1) were axisymmetric then the orbit of any critical point under the rotations around the axis of symmetry L would be a continuous set of critical points. Each of the four critical points (6.11) belong to the plane P orthogonal to the axis of symmetry L. Hence their one-dimensional orbits consisting of critical points also would belong to the plane P. However, we have demonstrated above that the critical set of system (6.1) in the plane P is not continuous but consists only of the four isolated critical points (6.11). The obtained contradiction proves that the dynamical system (6.1) and eigenfield are not axisymmetric if vectors A and B are noncollinear. □
For any collinear vectors A and B, the axis L (6.4) degenerates into one point x = 0 and therefore the above proof is not applicable. A recent numerical investigation  proves that some systems (6.1) with noncollinear vectors A and B possess chaotic streamlines and hence are not integrable.
7 Topology of axisymmetric eigenfields
The axisymmetry and incompressibility of eigenfields (5.1) lead to the existence of a first integral of the corresponding dynamical systems
Dynamical system (7.1) possesses a first integral
Indeed, differentiating function F 2(x(t)) along the flow (7.1) we find
The existence of the first integral F 2(x) = H 3(v)(x ⋅ A)Z(x) implies that all surfaces of its constant levels F 2(x) = const are invariant submanifolds of dynamical system (7.1). Inside the spherical shell all surfaces of nonzero levels of first integral F 2(x) = c are compact axisymmetric tori . Dynamics of the fluid streamlines (7.1) on the tori is quasi-periodic. The invariant surface of zero level F 2(x) = 0 consists of the plane (x ⋅ A) = 0, the straight line x = c A, and infinitely many spheres : R = R m where R m are roots of equation H 3(λR m ) = 0.
The axisymmetry and incompressibility of the flow with velocity field V(x, A) (3.1) lead to the existence of a first integral for the fluid streamlines defined by equation
Dynamical system (7.4) has first integral
Indeed, differentiating function F 1(x(t)) along the flow (7.4) we get equation
The existence of the first integral F 1(x) (7.5) yields that all surfaces of its constant level F 1(x) = const are invariant submanifolds of the system (7.4). Inside the spherical shell all nonzero levels of first integral F 1(x) = c are axisymmetric tori . Dynamics of the fluid streamlines (7.4) on the tori is quasi-periodic. The surface of zero level F 1(x) = H 2(λR)Z(x) = 0 consists of the straight line x = c A and infinitely many spheres : R = R k satisfying equation H 2(λR k ) = 0.
Topology of the axisymmetric fluid flows and that of V n (x, A) are completely different. Indeed, invariant tori for belong either to the half space (x ⋅ A) > 0 or to the half space (x ⋅ A) < 0 and there is invariant plane (x ⋅ A) = 0. For fluid flow V n (x, A) each invariant torus belongs to the both sides of any plane (x ⋅ C) = 0 where C is an arbitrary constant vector.
We have studied the boundary eigenvalue problem for vector fields V(x) satisfying the Beltrami equation curl V(x) = λ V(x) inside a spherical shell : with nonpenetration boundary conditions x ⋅ V(x) = 0 on the spheres R = R 2 and R = R 1. We have introduced the scale invariance (1.12) of the eigenvalue problem and have defined the dimensionless parameters x = λ(R 1 − R 2) and . We have shown that eigenvalues λ are connected with parameters x, μ by equation
where radius R 1 is arbitrary due to the scale invariance. Radius . We have derived equations
for the dimensionless parameters x, μ. For any μ > 0 the roots x n of Eq. (8.2) define by formula (8.1) eigenvalues λ n . Any root of Eq. (8.3) specifies eigenvalue by formula (8.1). Using Eqs. (8.2) and (8.3) we have obtained asymptotics of eigenvalues λ n , at R 2 → R 1 (that is equivalent to μ → ∞) and at n, m → ∞ and have proved that the eigenvalues λ n and are not equal for any n and m.
We have derived in elementary functions the corresponding to λ n 3D linear space of eigenfields V n (x, A) and the corresponding to 5D linear space of eigenfields where A and B are arbitrary constant vectors in . We have proved that eigenfields with noncollinear vectors A and B are not axisymmetric and eigenfields are axisymmetric. The topologies of fluid flows V n (x, A), , and with A × B ≠ 0 are completely different.
Author contribution: The author has accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The author declares no conflicts of interest regarding this article.
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