New axisymmetric equilibria with flow from an expansion about the generalized Solov'ev solution

Kuiroukidis, A. I.; Kaltsas, D. A.; Throumoulopoulos, G. N.

doi:10.1063/5.0249174

We construct analytic solutions to the generalized Grad–Shafranov equation, which incorporates both toroidal and poloidal flows. This is achieved by adopting a general linearizing ansatz for the free-function terms of the equation and expanding the generalized Solov'ev solution [Ch. Simintzis, G. N. Throumoulopoulos, G. Pantis and H. Tasso, Phys. Plasmas 8, 2641 (2001)]. On the basis of these solutions, we examine how the genaralized Solov'ev configuration is modified as the values of the free parameters associated with the additional pressure, poloidal current, and electric field terms changed. Thus, a variety of equilibria of tokamak, spherical tokamak, and spheromak pertinence are constructed, including D-shaped configurations with positive and negative triangularity and diverted configurations with either a couple of X-points or a single X-point.

I. INTRODUCTION

The magnetohydrodynamics axisymmetric equilibrium states are governed by the well-known Grad–Shafranov (GS) equation, a quasi-linear elliptic partial differential equation for the poloidal magnetic flux-function, containing a couple of arbitrary functions of that flux (surface functions), namely, the pressure and the poloidal-current functions. This equation is generally solved numerically under appropriate boundary conditions, and to this end, several codes have been developed, e.g., the HELENA code.¹ Also, to easier gain physical intuition and for benchmarking equilibrium codes, analytic solutions have been constructed to linearized forms of the GS equation. In connection with the present study, we first mention the widely employed in plasma confinement studies Solov'ev solution.² This solution, corresponding to constant surface-function terms in the GS equation, describes an up-down symmetric equilibrium with D-shaped magnetic surfaces surrounded by a spontaneously formed separatrix with a couple of X-points (cf. Fig. 1). Also, known is the Maschke–Hernegger solution,^3,4 expressed in terms of the Whittaker functions, corresponding to linear choices of the surface-function terms and describing more realistic equilibria with current–density profiles vanishing on the plasma boundary. Several additional analytic solutions to the GS equation are available, e.g., extending the Solov'ev solution by polynomial contributions to the homogeneous counterpart of the GS equation in order to construct diverted tokamak equilibria;⁵ for constant pressure and linear current–density choices of the free terms in the GS equation;⁶ for linear choices of both the pressure and current–density terms by imposing D-shaped boundaries or boundaries with a lower X-point;^7–9 by differentiation of known separated solutions with respect to the separation parameter in order to get simpler solutions;¹⁰ and in elliptic prolate geometry.¹¹ Also, for the case of the generic linearized form of the GS equation an analytic solution involving infinite series was derived in Ref. 12 and was employed to construct equilibrium configurations pertinent to the tokamak ASDEX-Upgrade.

FIG. 1.

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The equilibrium configuration of the GS solution (12) on the poloidal plane $x - y$ ⁠, with x corresponding to the horizontal axis and y to the vertical axis. The red dashed curve represents the separatrix with $x_{s 1} = \sqrt{ϵ}$ ⁠, $x_{s 2} = {(- ϵ λ - 3 δ^{2} + \sqrt{3} \sqrt{- ϵ^{2} λ^{2} - 2 δ^{2} ϵ λ + 3 δ^{4} + δ^{2} λ + 4 λ^{2}} / (2 λ))}^{1 / 2}$ and $y_{s} = {[{(1 - ϵ)}^{2} (2 δ^{2} + λ (ϵ + 1))]}^{1 / 2}$ ⁠.

Extended equilibrium investigations have been conducted to include plasma flows and associated electric fields, which play an important role in the creation of advanced confinement regimes in tokamaks, e.g., the transition of the low to high confinement mode. Specifically, to include axisymmetric toroidal flows, inherently incompressible because of the symmetry, elliptic PDEs were derived in Ref. 13 for either isentropic or isothermal magnetic surfaces and analytic solutions were constructed therein. For compressible flows of arbitrary direction, the problem reduces to a set of a PDE coupled with a Bernoulli equation involving the pressure.^14–16 Depending on the value of the poloidal velocity, this PDE can be either elliptic or hyperbolic, that is, there are three critical transition poloidal velocities, thus forming two elliptic and two hyperbolic regions. In the frame of two-fluid model, the electron and ion surfaces depart from the magnetic surfaces and the problem reduces to a set of three PDEs involving respective flux-function labels.^17,18 The problem was also investigated within the framework of simplified two-fluid models, such as Hall-MHD that neglects the electron inertia^19,20 and extended MHD.^21–23 The impact of pressure anisotropy was examined in Refs. 24–28. In addition, to include fast particle populations the problem was addressed in the frame of hybrid fluid-kinetic modes.^29–33

For incompressible MHD flows having a poloidal component the generalized GS equation becomes elliptic [Eq. (3)]^34,35 and decouples from the Bernoulli equation, which can be employed as a formula for the pressure [Eq. (8)]. Incompressibility, implying that the density becomes uniform on the magnetic surfaces, is an acceptable approximation for fusion laboratory plasmas for two reasons: First, the poloidal velocities of those plasmas lie well within the first elliptic region. Second, for typical flows in laboratory fusion plasmas, i.e., for Alfvén Mach numbers on the order of 0.01, density variations on the magnetic surfaces are very low.

Equation (3), which contains five free surface quantities, has an additional electric-field term through the electrostatic potential $Φ$ ⁠, associated with the component of plasma velocity non-parallel to the magnetic field. For parallel velocity and arbitrary assignment of the free surface-function terms, an extension of the HELENA code was developed in,³⁶ while the generalized Solov'ev solution to Eq. (3) [given by Eq. (12)], which constitutes a basic ingredient of the present study, was obtained in Ref. 35. Owing to the electric field, the respective equilibrium in addition to a separatrix, similar to that of the usual static Solov'ev equilibrium, possess a third X-point outside the separatrix (cf. Fig. 11 of Ref. 35). Other analytic solutions to the generalized GS equation of tokamak or space plasma pertinence were obtained by adopting a Solov'ev-like linearizing ansatz and imposing a diverted boundary;³⁷ alternative linearizing ansatzes;^38–40 nonlinear forms of the generalized GS equation;⁴¹ employing Lie-point symmetries for linear and nonlinear choices of the free-function terms;^42–44 and by the method of similarity reduction.^45,46

Aim of the present work is to construct analytically solutions to the generalized GS equation in its generic linearized form by employing an alternative method. This consists in pursuing solutions as expanded generalized Solov'ev ones, i.e., as superposition of the generalized Solov'ev solution and a function to be determined. In addition to the novelty of this method we are particularly interested in examining how the Solov'ev configuration is modified when the influence of the expanding function becomes stronger by changing the pertinent new free parameters.

In Sec. II, the generalized GS equation and the generalized Solov'ev solution are briefly reviewed and the method of solving the generalized GS equation adopting the generic linearized ansatz is presented. Then, analytic solutions to the generic linearized generalized GS equation are constructed in Sec. III. In Sec. IV, we examine how the Solov'ev configuration is modified by varying the additional free parameters associated with the pressure, the poloidal current in conjunction with the parallel component of the velocity, and electric field, thus obtaining a variety of new configurations. The conclusions are summarized in Sec. V.

II. GENERIC LINEAR AXISYMMETRIC EQUILIBRIA WITH NON-PARALLEL FLOW

The generalized GS equation is given in SI units by (cf. Refs. 34 and 35)

(1 - M_{p}^{2}) Δ^{⋆} ψ - \frac{1}{2} \frac{d M_{p}^{2}}{d ψ} | \nabla ψ |^{2} + \frac{1}{2} \frac{d}{d ψ} (\frac{X^{2}}{1 - M_{p}^{2}}) + μ_{0} R^{2} \frac{d P_{s}}{d ψ} + μ_{0} \frac{R^{4}}{2} \frac{d}{d ψ} [\frac{ρ}{1 - M_{p}^{2}} {(\frac{d Φ}{d ψ})}^{2}] = 0 .

(1)

Here,

(z, R, ϕ)

are cylindrical coordinates with z corresponding to the axis of symmetry;

ψ (R, z)

is the poloidal magnetic flux function which labels the magnetic surfaces;

M_{p} (ψ)

is the Mach function of the poloidal velocity with respect to the poloidal-magnetic-field Alfvén velocity;

ρ (ψ)

and

Φ (ψ)

are the density and the electrostatic potential;

X (ψ)

relates to the toroidal magnetic field and to the poloidal current [cf. relations (4) and (7)]; for vanishing flow the surface function

P_{s} (ψ)

coincides with the pressure [cf. Eq. ];

Δ^{⋆} = R^{2} \nabla \cdot (\nabla / R^{2})

and

μ_{0}

are the permeability of free space.

Equation can be simplified by the transformation

u (ψ) = \int_{0}^{ψ} {[1 - M^{2} (g)]}^{1 / 2} d g,

(2)

which reduces Eq. to

Δ^{*} u + \frac{1}{2} \frac{d}{d u} [\frac{X^{2}}{1 - M_{p}^{2}}] + μ_{0} R^{2} \frac{d P_{s}}{d u} + μ_{0} \frac{R^{4}}{2} \frac{d}{d u} [ρ {(\frac{d Φ}{d u})}^{2}] = 0 .

(3)

Note that no quadratic term as

| \nabla u |^{2}

appears anymore in Eq. . The transformation [Eq. ] does not affect the shape of magnetic surfaces; it just relabels them in terms of the function u. It is noted here that there is a misprint of the respective

R^{4}

-term in Eq. (17) of Ref. 35, i.e., the density factor should be linear instead of quadratic. To solve Eq. , the surface-quantity terms (⁠

(1 / 2) (d / d u) [X^{2} / (1 - M_{p}^{2})]

⁠,

μ_{0} R^{2} d P_{s} / d u

and

μ_{0} (R^{4} / 2) (d / d u) [ρ {(d Φ / d u)}^{2}]

⁠) should be assigned as functions of u. Also, the equilibrium can be completely determined by calculations in the u-space; specifically, the magnetic field, velocity, current density, pressure, and electric field are given by the relations

B = I \nabla ϕ + {(1 - M_{p}^{2})}^{- 1 / 2} \nabla ϕ \times \nabla u,

(4)

with

I = {(1 - M_{p}^{2})}^{- 1 / 2} [\frac{X}{{(1 - M_{p}^{2})}^{1 / 2}} - μ_{0} R^{2} \sqrt{\frac{ρ}{μ_{0}}} M_{p} \frac{d Φ}{d u}],

(5)

v = \frac{M_{p}}{\sqrt{μ_{0} ρ}} B - R^{2} {(1 - M_{p}^{2})}^{- 1 / 2} \frac{d Φ}{d u} \nabla ϕ,,

(6)

j = \frac{1}{μ_{0}} {{(1 - M_{p})}^{- 1 / 2} [Δ^{⋆} u - \frac{1}{2 (1 - M_{p}^{2})} \frac{d (1 - M_{p}^{2})}{d u} | \nabla u |^{2}] \nabla ϕ - \nabla ϕ \times \nabla I},

(7)

P = P_{s} - ρ [\frac{v^{2}}{2} - R^{2} {(\frac{d Φ}{d u})}^{2}],

(8)

E = - \frac{d Φ}{d u} \nabla u

(9)

with

v^{2} = v \cdot v

⁠.

Following Ref. 35, we first make the following choice of the free surface-function terms in Eq.

\begin{matrix} \frac{d (μ_{0} P_{s})}{d u} = - \frac{P_{s 0}}{U_{0}}, \\ \frac{1}{2} \frac{d}{d u} [\frac{X^{2}}{1 - M_{p}^{2}}] = ϵ \frac{P_{s 0} R_{0}^{2}}{U_{0} (1 + δ^{2})}, \\ \frac{1}{2} \frac{d}{d u} [μ_{0} ρ {(\frac{d Φ}{d u})}^{2}] = - λ \frac{P_{s 0}}{U_{0} (1 + δ^{2}) R_{0}^{2}} . \end{matrix}

(10)

It is noted that a couple of the five surface quantities involved in the generalized GS equation remain free. Normalizing as

x : = R / R_{0}

⁠,

y : = z / R_{0}

⁠,

U : = u / u_{0}

with

u_{0} : = P_{s 0} R_{0}^{4} / (2 U_{0} (1 + δ^{2}))

⁠, Eq. assumes the form

U_{x x} - \frac{1}{x} U_{x} + U_{y y} - 2 λ x^{4} - 2 (1 + δ^{2}) x^{2} + 2 ϵ = 0,

(11)

and the generalized Solov'ev solution is given by

U_{g s} = [y^{2} (x^{2} - ϵ) + \frac{δ^{2} + λ}{4} {(x^{2} - 1)}^{2} + \frac{λ}{12} {(x^{2} - 1)}^{3}] .

(12)

The equilibrium configuration for

λ > 0

is shown in Fig. 1.

As already mentioned in Sec. I, the outermost closed surface (red dashed line) is a separatrix consisting of an elliptic outer part and a straight inner part parallel to the axis of symmetry; thus, a couple of X-points are created. Expressions of the X-point coordinates and of the inner and outer points of the configuration on the mid-plane $y = 0$ in terms of the parameters $δ$ ⁠, $ϵ$ and $λ$ are given in the caption of Fig. 1. The magnetic axis is located at (⁠ $x = 1, y = 0$ ⁠). The parameter $ϵ$ determines the compactness of the configuration, i.e., for $ϵ > 0$ the equilibrium is diamagnetic and describes either a tokamak or a spherical tokamak, while for $ϵ = 0$ the inner part of the separatrix touches the axis of symmetry forming a spheromak. The parameter $δ$ relates to the elongation of the magnetic surfaces in the vicinity of the magnetic axis, i.e., the higher the value of $δ$ ⁠, the larger the elongation parallel to the axis of symmetry. For $λ = 0$ the velocity becomes parallel to the magnetic field, while for $λ < 0$ a third X-point appears outside the separatrix. When the velocity vanishes (⁠ $M_{p} = λ = 0$ ⁠), the usual static Solov'ev solution is recovered. Here, we will consider equilibria with $λ > 0$ ⁠.

We now try the following new generic linearizing ansatz to Eq.

\begin{matrix} \frac{d (μ_{0} P_{s})}{d u} = - \frac{P_{s 0}}{U_{0}} + \bar{α} \frac{P_{s 0}}{U_{0}^{2}} u, \\ \frac{1}{2} \frac{d}{d u} [\frac{μ_{0} X^{2}}{1 - M_{p}^{2}}] = ϵ \frac{P_{s 0} R_{0}^{2}}{U_{0} (1 + δ^{2})} - \bar{β} ϵ \frac{P_{s 0} R_{0}^{2}}{U_{0}^{2} (1 + δ^{2})} u, \\ \frac{1}{2} \frac{d}{d u} [μ_{0} ρ {(\frac{d Φ}{d u})}^{2}] = - λ \frac{P_{s 0}}{U_{0} (1 + δ^{2}) R_{0}^{2}} + \bar{γ} \frac{λ P_{s 0}}{U_{0}^{2} (1 + δ^{2}) R_{0}^{2}} u, \end{matrix}

(13)

where

\bar{α}, \bar{β}, \bar{γ}

are dimensionless parameters. The linear terms in u in Eq. result in more realistic equilibrium characteristics, e.g., the constant

ϵ [P_{s 0} R_{0}^{2} / (U_{0} (1 + δ^{2}))]

in Eq. results in monotonically increasing toroidal–current–density profiles while the additional linear in u term,

- \bar{β} ϵ [P_{s 0} R_{0}^{2} / (U_{0}^{2} (1 + δ^{2})] u

⁠, in Eq. results in peaked toroidal–current–density profiles. Also, the linear term

\bar{γ} [λ P_{s 0} / (U_{0}^{2} (1 + δ^{2} R_{0}^{2}))] u

in (13) produces larger velocity shear. Setting

G : = 2 ϵ

⁠,

F : = - 2 (1 + δ^{2})

⁠,

E : = - 2 λ

⁠,

α : = \bar{α} (u_{0} / U_{0})

⁠,

β : = \bar{β} (u_{0} / U_{0})

⁠, and

γ : = \bar{γ} (u_{0} / U_{0})

⁠, as well as substituting Eq. into Eq. we straightforwardly obtain the following equation:

U_{x x} - \frac{1}{x} U_{x} + U_{y y} + E x^{4} + F x^{2} + G - E x^{4} γ U - F x^{2} α U - G β U = 0 .

(14)

Note that the solution to Eq. for

α = β = γ = 0

is given by Eq. . We now will pursue a solution to Eq. in the form of an expansion about the GS solution (12), i.e.,

U = U_{g s} + W (x, y),

(15)

with the function

W (x, y)

to be determined so that Eq. to satisfy Eq. . Substituting Eq. into Eq. yields

\begin{matrix} W_{x x} - \frac{1}{x} W_{x} + W_{y y} - γ E x^{4} W - α F x^{2} W - β G W \\ = (γ E x^{4} + α F x^{2} + β G) U_{g s} . \end{matrix}

(16)

This is an inhomogeneous linear second-order PDE for the unknown function $W (x, y)$ ⁠. The general solution consists of the sum of the general solution to the counterpart homogeneous equation plus a particular solution of the complete inhomogeneous equation.

III. SOLUTIONS

We first consider the homogeneous counterpart of Eq.

W_{x x} - \frac{1}{x} W_{x} + W_{y y} - (γ E x^{4} + α F x^{2} + β G) W = 0 .

(17)

The general solution of Eq. can be found by separation of variables, i.e., in the form

W (x, y) = X (x) Y (y) .

(18)

Inserting Eq. into Eq. we find

Y^{″} + k^{2} Y = 0,

(19)

and

X^{″} - \frac{1}{x} X^{'} - (γ E x^{4} + α F x^{2} + β G + k^{2}) X = 0,

(20)

where k is the parameter of separation. The general solution of Eq. is

Y (y) = C cos (k y) + D sin (k y) .

(21)

Setting

ξ = x^{2}

⁠, Eq. takes the form

4 ξ X^{″} (ξ) - (γ E ξ^{2} + α F ξ + β G + k^{2}) X = 0 .

(22)

Since

ξ = 0

is a regular singular point of Eq. its solution can be derived by the method of Frobenius in terms of infinite converging series of the form

X (ξ) = \sum_{n = 0}^{\infty} a_{n} ξ^{n + r} .

(23)

Substituting Eq. into Eq. results in the following recursion relations

\begin{matrix} 4 r (r - 1) a_{0} = 0, \\ 4 r (r + 1) a_{1} - (β G + k^{2}) a_{0} = 0, \\ 4 (r + 1) (r + 2) a_{2} - α F a_{0} - (β G + k^{2}) a_{1} = 0, \\ (n \geq 2) 4 (n + r) (n + r + 1) a_{n + 1} - γ E a_{n - 2} - α F a_{n - 1} - (β G + k^{2}) a_{n} = 0, \end{matrix}

(24)

where

a_{0}

is a freely specified parameter. The roots of the indicial polynomial,

r (r - 1)

⁠, are

r_{1} = 1

and

r_{2} = 0

⁠. Therefore, for

r_{1} = 1

⁠, one solution of Eq. is

X_{1} (ξ) = ξ \sum_{n = 0}^{\infty} a_{n} ξ^{n} .

(25)

Since

r_{1} - r_{2}

is a non-zero integer, the second linearly independent solution is of the form

X_{2} (ξ) = c X_{1} (ξ) ln ξ + \sum_{n = 0}^{\infty} {\bar{a}}_{n} ξ^{n},

(26)

where the coefficients c and

{\bar{a}}_{n}

can be determined by substituting Eq. into Eq. . Here, we will employ the solution

X_{1} (ξ)

⁠. By means of Eqs. , , and and introducing a new parameter s by

k = s l

with

l = 0, 1, 2, \dots

⁠, the solution of the homogeneous PDE (17) is written as

\begin{matrix} W (x, y) = \sum_{l}^{\infty} X_{1} (ξ) [C_{l} cos (sly) + D_{l} sin (sly)] \\ = \sum_{l = 0}^{\infty} \sum_{n = 0}^{\infty} a_{n, s l} ξ^{n} [C_{l} cos (sly) + D_{l} sin (sly)] . \end{matrix}

(27)

The values of the free coefficients

C_{l}

and

D_{l}

may be properly fixed in order to impose a boundary. However, here since in the Solov'ev equilibrium the boundary (separatrix) is spontaneously formed we will not impose any boundary condition; accordingly, in Eq. we will keep only the first two l-terms (for

l = 0

and 1).

Furthermore, we will pursue a particular solution of Eq. in the form

W (x, y) = W_{1} (x) y^{2} + W_{2} (x) .

(28)

Inserting Eq. into Eq. yields

W_{1}^{″} - \frac{1}{x} W_{1}^{'} - (γ E x^{4} + α F x^{2} + β G) W_{1} = (x^{2} - \frac{G}{2}) (γ E x^{4} + α F x^{2} + β G),

(29)

and

\begin{matrix} W_{2}^{″} - \frac{1}{x} W_{2}^{'} + 2 W_{1} - (γ E x^{4} + α F x^{2} + β G) W_{2} \\ = - [\frac{(F + E + 2)}{8} {(x^{2} - 1)}^{2} + \frac{E}{24} {(x^{2} - 1)}^{3}] (γ E x^{4} + α F x^{2} + β G) . \end{matrix}

(30)

Equation involves only the function

W_{1} (x, y)

⁠. Thus, once Eq. is solved, the solution contributes to the inhomogeneous part of Eq. .

To solve Eq. , we again change variable,

ξ = x^{2}

⁠, to get

4 ξ W_{1}^{″} (ξ) - (γ E ξ^{2} + α F ξ + β G) W_{1} = (ξ - \frac{G}{2}) (γ E ξ^{2} + α F ξ + β G) .

(31)

Setting

W_{1} + (ξ - \frac{G}{2}) : = {\bar{W}}_{1},

(32)

Equation is put in the form

4 ξ {\bar{W}}_{1}^{″} - (γ E ξ^{2} + α F ξ + β G) {\bar{W}}_{1} = 0 .

(33)

For

β = 0

⁠, the solution of Eq. is

{\bar{W}}_{1} = c_{1} A i (\frac{α F + γ E ξ}{{(2 γ E)}^{2 / 3}}) + c_{2} B i (\frac{α F + γ E ξ}{{(2 γ E)}^{2 / 3}}),

(34)

where Ai and Bi are the Airy functions of the first and second kind. For

β \neq 0

apart from the

k^{2}

-term, Eq. is identical to Eq. . Therefore, the solution of Eq. is already included in Eq. for

l = 0

⁠.

Finally, we will obtain a particular solution of the inhomogeneous equation . Introducing once more the variable

ξ = x^{2}

⁠, Eq. is written as

\begin{matrix} 4 ξ W_{2}^{″} (ξ) - (γ E ξ^{2} + α F ξ + β G) W_{2} \\ = - 2 W_{1} (ξ) - [C_{5} ξ^{5} + C_{4} ξ^{4} + C_{3} ξ^{3} + C_{2} ξ^{2} + C_{1} ξ + C_{0}], \end{matrix}

(35)

where

W_{1} = {\bar{W}}_{1} - (ξ - G / 2)

by Eq. and

\begin{matrix} C_{5} = γ \frac{E^{2}}{24}, \\ C_{4} = \frac{γ E (F + 2)}{8} + α F \frac{E}{24}, \\ C_{3} = β G \frac{E}{24} + α F \frac{(F + 2)}{8} - γ E \frac{(2 F + E + 4)}{8}, \\ C_{2} = β G \frac{(F + 2)}{8} - α F \frac{(2 F + E + 4)}{8} + γ E \frac{(3 F + 2 E + 6)}{24}, \\ C_{1} = - β G \frac{(2 F + E + 4)}{8} + α F \frac{(3 F + 2 E + 6)}{24}, \\ C_{0} = β G \frac{(3 F + 2 E + 6)}{24} . \end{matrix}

(36)

In view of the converging series solutions (25) to the homogeneous equations and , we pursue a similarly structured solution to Eq. .

W_{2} = ξ \sum_{n = 0}^{\infty} b_{n} ξ^{n} .

(37)

Inserting Eqs. into Eq. , we obtain after some tedious but straightforward algebra that

b_{0}

is freely specified and

\begin{matrix} C_{0} + G = 0, \\ 8 b_{1} - β G b_{0} + 2 d_{0} - 2 + C_{1} = 0, \\ 24 b_{2} - α F b_{0} - β G b_{1} + 2 d_{1} + C_{2} = 0, \\ 48 b_{3} - γ E b_{0} - α F b_{1} - β G b_{2} + 2 d_{2} + C_{3} = 0, \\ 80 b_{4} - γ E b_{1} - α F b_{2} - β G b_{3} + 2 d_{3} + C_{4} = 0, \\ 120 b_{5} - γ E b_{2} - α F b_{3} - β G b_{4} + 2 d_{4} + C_{5} = 0, \\ (n \geq 6) 4 n (n + 1) b_{n} - γ E b_{n - 3} - α F b_{n - 2} - β G b_{n - 1} + 2 d_{n - 1} = 0 . \end{matrix}

(38)

Inspection of the last of Eqs. and

C_{0} + G = 0

[first of Eqs. ] together with the definitions

G : = 2 ϵ

⁠,

F : = - 2 (1 + δ^{2})

and

E : = - 2 λ

implies the following:

When $β = 0$ ⁠, then $ϵ = 0$ and therefore the second of Eq. (13) becomes $(1 / 2) d / d u (X^{2} / (1 - M_{p}^{2})) = 0$ ⁠. In the absence of flow then Eq. (5) implies that the plasma is confined by a vacuum toroidal magnetic field (⁠ $β_{p} = 1$ equilibrium); when the non-parallel flow is there the electric-field term in Eq. (5) additionally contributes to the poloidal current.
When $ϵ \neq 0$ and $β \neq 0$ ⁠, the last of Eq. (36) becomes $β (3 F + 2 E + 6) + 24 = 0$ ⁠. In this case, all the linear terms in the ansatz (13) can remain finite. This relation just acts as a constraint on the parameters $β$ ⁠, $δ$ ⁠, and $λ$ ⁠. In Sec. IV, we intend to examine the impact of the linear in u electric-field term in Eq. (13) on the equilibrium by varying the parameter $β$ ⁠; accordingly, leaving $β$ free we will employ the constraint to express $δ$ in terms of $β$ and $λ$ ⁠, i.e.

$δ = {[\frac{(2 / 3) (6 - β λ)}{β}]}^{1 / 2} .$

(39)
When $ϵ = 0$ ⁠, $β$ does not necessarily vanish and both $δ$ and $λ$ remain free.

Additional free parameters are $α$ ⁠, $β$ ⁠, $γ$ ⁠, $a_{0}$ ⁠, $b_{0}$ ⁠, $C_{0}$ ⁠, $C_{1}, D_{1}$ ⁠, $R_{0}$ ⁠, $P_{s 0}$ ⁠, $U_{0}$ ⁠, and s. It is recalled here that $R_{0}$ is a reference length determining the R-position of the magnetic axis; the parameters $P_{s 0}$ and $U_{0}$ are reference parameters associated with the pressure and poloidal magnetic flux; $δ$ and $ϵ$ are geometrical (shaping) parameters of the starting generalized Solov'ev equilibrium (12) shown in Fig. 1; $λ$ relates to the electric field term in Eq. (12) through the ansatz (10); $α$ ⁠, $β$ ⁠, and $γ$ are physical parameters of the current density, pressure, and electric field, respectively, in connection to the additional linear terms in the ansatz (13); and $a_{0}$ ⁠, $b_{0}$ ⁠, $C_{0}$ ⁠, $C_{1}$ ⁠, $D_{1}$ ⁠, and s are shaping parameters associated with the new constructed solution. A list of the free parameters is given in Table I. Also, it is noted that the series expansion (37) was shown numerically to converge and we kept terms up to the order of $ξ^{20}$ in our calculations.

TABLE I.

List of the free parameters.

Parameters	Description
$R_{0}$	Reference R-position of the magnetic axis
$P_{s 0}$ ⁠, $U_{0}$	Reference pressure and poloidal magnetic flux
$δ$ ⁠, $ϵ$	Shaping of the generalized Solov'ev equilibrium of Fig. 1
$λ$	Parameter related to the electric field term in the
	generalized Solov'ev solution (12)
$α$ ⁠, $β$ ⁠, $γ$	Current density, pressure and electric field parameters
	in the ansatz (13)
$a_{0}$ ⁠, $b_{0}$ ⁠, $C_{0}, C_{1}$ ⁠, $D_{1}$ ⁠, s	Shaping parameters associated with the new constructed solution
$ρ_{0}$ ⁠, $l_{ρ}$ ⁠, $M_{p 0}$ ⁠, $l_{M_{p}}$	Density and poloidal Mach function parameters [Eq. (40)]

Parameters	Description
$R_{0}$	Reference R-position of the magnetic axis
$P_{s 0}$ ⁠, $U_{0}$	Reference pressure and poloidal magnetic flux
$δ$ ⁠, $ϵ$	Shaping of the generalized Solov'ev equilibrium of Fig. 1
$λ$	Parameter related to the electric field term in the
	generalized Solov'ev solution (12)
$α$ ⁠, $β$ ⁠, $γ$	Current density, pressure and electric field parameters
	in the ansatz (13)
$a_{0}$ ⁠, $b_{0}$ ⁠, $C_{0}, C_{1}$ ⁠, $D_{1}$ ⁠, s	Shaping parameters associated with the new constructed solution
$ρ_{0}$ ⁠, $l_{ρ}$ ⁠, $M_{p 0}$ ⁠, $l_{M_{p}}$	Density and poloidal Mach function parameters [Eq. (40)]

In summary, a solution of the general linearized form of the generalized GS equation (16) consists of a separated solution of its homogeneous counterpart in terms of Eqs. (18), (21), and (25) plus a particular solution of the form (28), expressed in terms of the series (27) for $l = 0$ and Eq. (37).

IV. EQUILIBRIUM CONFIGURATIONS

By employing the solution to Eq. derived in the Sec. III, we have constructed several new equilibria. Particularly, we examined how the generalized Solov'ev equilibrium is modified by varying the additional parameters

α

⁠,

β

⁠, and

γ

⁠, associated with the respective linear-in-u terms in Eq. of pressure, poloidal current in conjunction with the parallel velocity [Eqs. and ], and electric field. To impose a stagnation point,

(x_{0}, y_{0})

⁠, in addition to the conditions

d U / d x |_{x_{0}, y_{0}} = 0

and

d U / d y |_{x_{0}, y_{0}} = 0

we considered the sign of the quantity

{\frac{d^{2} U}{d x^{2}} \frac{d^{2} U}{d y^{2}} - {(\frac{d^{2} U}{dxdy})}^{2} |}_{(x_{0}, y_{0})},

which should be positive for a magnetic axis and negative for an X-point. For the up-down symmetric configurations considered, the magnetic axis is located at the same position,

(x = 1, y = 0)

as the generalized Solov'ev configuration and

U = 0

thereon. Note that in this case the condition

d U / d y |_{(1, 0)} = 0

is identically satisfied because of symmetry. If not otherwise specified, we assigned the following parametric values:

s = 0.1

⁠,

δ = 0.1

⁠,

λ = 0.1

⁠, and

a_{0} = 1

⁠. It is recalled that when

β \neq 0

and

ϵ \neq 0

⁠,

δ

should be calculated from Eq. . The parameters

C_{0}

⁠,

C_{1}

⁠, and

D_{1}

were determined as solutions to the set of the algebraic equations derived by the conditions associated with the imposition of a stagnation point. Since we employed dimensionless quantities, the reference dimensional parameters

R_{0}

⁠,

P_{s 0}

(in units of 4π×10⁻⁷ H/mPA) and

U_{0}

remained free. Depending on the value of

ϵ

we constructed tokamak, spherical tokamak and spheromak equilibria, the latter corresponding to

ϵ = 0

⁠.

First, we examined the impact of the $α$ -, $β$ -, and $γ$ -terms individually on up-down symmetric equilibria by imposing the same position of the magnetic axis, $(x = 1, y = 0)$ ⁠, as that of the respective generalized Solov'ev equilibrium and the same value $U = 0$ thereon. In the presence of $α$ -term, a D-shaped configuration is formed with negative triangularity, the boundary of which is shown in Fig. 2-left by the red dashed curve. As $α$ gradually takes larger values, the configuration with increasing triangularity extends toward the axis of symmetry, eventually touching it to form a separatrix with a single X-point located at the origin of the coordinate system (blue, dot dashed line) and then becomes a peculiar configuration having a separatrix with a couple of X-points (green dotted line).

FIG. 2.

View large Download slide

Left: different equilibrium boundaries are given on the poloidal plane $x - y$ as the additional linear-in-U pressure term in the first of Eq. (13) varies in connection with different values of $α$ ⁠. The black continuous curve represents the separatrix of the respective generalized Solov'ev spheromak configuration. Right: a set of magnetic surfaces for one of the configuration represented by blue, dot dashed U-curves are shown together with the respective curves of the generalized Solov'ev equilibrium. $U_{b}$ is the value of the bounding u-curve (separatrix).

The $β$ -term creates configurations with small triangularity elongated along the x axis as shown in Fig. 3. The red dashed curves represent the magnetic surfaces of a configuration with the same poloidal magnetic flux as the respective generalized Solov'ev tokamak configuration represented by the black continuous curve. As $β$ takes larger values, the configuration extends further and eventually reaches the axis of symmetry, transforming into a spheromak. It is noted that for $ϵ = β = γ = 0$ ⁠, the equilibrium becomes independent of $β$ ⁠. The impact of the $γ$ -term is similar to that of the $α$ -term as shown in Fig. 4.

FIG. 3.

View large Download slide

The impact of the additional linear-in-U, poloidal-current term in the second of Eq. (13) in connection with different values of $β$ ⁠. The black continuous curves represent the magnetic surfaces of the respective generalized Solov'ev tokamak configuration. $U_{b}$ is the value of the bounding u-curve (separatrix).

FIG. 4.

View large Download slide

Left: different equilibrium boundaries are given as the additional linear-in-U, electric-field term in the third of Eq. (13) varies in connection with different values of $γ$ ⁠. The black continuous curve represents the separatrix of the respective generalized Solov'ev spheromak configuration. Right: A set of magnetic surfaces for one of the configuration represented by green dotted U-curves is shown together with the respective curves of the Solov'ev equilibrium. $U_{b}$ is the value of the bounding u-curve (separatrix).

Subsequently, the impact of combinations of the $α$ -, $β$ -, and $γ$ -terms was examined by imposing an X-point at several positions. Thus, a variety of configurations were created. Three examples are given in Figs. 5–7. In Fig. 5, the blue, dot dashed curves represent an up-down symmetric spherical tokamak configuration (⁠ $ϵ = 0.1$ ⁠) the X-points of which are common with those of the respective generalized Solov'ev equilibrium, but the poloidal magnetic flux of this former equilibrium is larger than that of the latter. The red dashed curve represents the boundary of a D-shaped configuration with poloidal magnetic flux equal to that of the generalized Solov'ev solution. In Fig. 6 is shown an up-down symmetric diverted configuration with the imposed X-points located at (⁠ $x = 0.84, y = \pm 14.5$ ⁠), while Fig. 7 shows an up-down asymmetric configuration with the magnetic axis located at (⁠ $x = 1.02, y = 0.83$ ⁠) and the single lower X-point at $(x = 0.94, y = - 22.6)$ ⁠.

FIG. 5.

View large Download slide

A spherical tokamak diverted configuration (⁠ $ϵ = 0.1$ ⁠, $δ = 20$ ⁠) with a couple of X-points common with the respective Solov'ev equilibrium. The red dashed curve represents the boundary of a D-shaped tokamak equilibrium with poloidal magnetic flux equal to that of the Solov'ev one. $U_{b}$ is the value of the bounding u-curve (separatrix).

FIG. 6.

An up-down symmetric, diverted tokamak configuration ( ϵ=0.1, δ=20) with a couple of X-points located at ( x=0.84,y=±14.5). The black continuous curves represent the magnetic surfaces of the respective Solov'ev equilibrium. Ub is the value of the bounding u-curve (separatrix).

View large Download slide

An up-down symmetric, diverted tokamak configuration (⁠ $ϵ = 0.1, δ = 20$ ⁠) with a couple of X-points located at (⁠ $x = 0.84, y = \pm 14.5$ ⁠). The black continuous curves represent the magnetic surfaces of the respective Solov'ev equilibrium. $U_{b}$ is the value of the bounding u-curve (separatrix).

FIG. 7.

View large Download slide

An up-down asymmetric, diverted tokamak configuration for $k = 0.2$ ⁠, $ϵ = 0.2$ ⁠, $δ = 31.6$ ⁠, $D_{1} = 0.5$ ⁠, and $D_{2} = 0.01$ ⁠. The magnetic axis is located at $(x = 1.02, y = 0.83)$ and the single lower X-point at $(x = 0.94, y = - 22.6)$ ⁠. The black continuous curves represent the magnetic surfaces of the respective Solov'ev equilibrium. $U_{b}$ is the value of the bounding U-curve (separatrix).

Finally, to completely determine the equilibrium two from the five free functions involved should be specified. For example, considering as free functions the density and the poloidal Mach function we make the choice

ρ = ρ_{0} {(1 - \frac{U}{U_{b}})}^{l_{ρ}}, M_{p} = M_{p 0} {(1 - \frac{U}{U_{b}})}^{l_{M_{p}}},

(40)

where

ρ_{0}

⁠,

M_{p 0}

⁠,

l_{ρ}

⁠, and

l_{M_{p}}

are free parameters. Introducing the dimensionless quantities

\begin{matrix} \tilde{ρ} = \frac{ρ}{ρ_{0}}, \tilde{X} = \frac{X}{u_{0} / R_{0}}, \tilde{B} = \frac{B}{u_{0} / R_{0}^{2}}, \tilde{j} = \frac{j}{u_{0} / (R_{0}^{3} μ_{0})}, \tilde{v} = \frac{v}{u_{0} / (R_{0}^{2} \sqrt{μ_{0} ρ_{0}})}, \\ \tilde{Φ} = \frac{Φ}{u_{0}^{2} / (R_{0}^{3} \sqrt{μ_{0} ρ_{0}})}, \tilde{P} = \frac{P}{P_{s 0} / (u_{0}^{2} / R_{0}^{4})}, \end{matrix}

integration of the relations (13) with respect to u yields (in the dimensionless form and omitting the tildes)

\begin{matrix} P_{s} = A - 2 (1 + δ^{2}) U + α (1 + δ^{2}) U^{2} \\ \frac{X^{2}}{1 - M_{p}^{2}} = B + 4 ϵ U - 2 β U^{2} \\ ρ {(\frac{d Φ}{d U})}^{2} = C - 4 λ U + 2 γ U^{2}, \end{matrix}

(41)

where A, B, and C are free parameters. The reference density

ρ_{0}

remains free. Using Eq. in Eq, (8) we obtain

P = A - 2 (1 + δ^{2}) U + α (1 + δ^{2}) U^{2} - \frac{1}{2} ρ v^{2} + x^{2} (C - 4 λ U + 2 γ U^{2}) .

(42)

The values of A and C are fixed so that the pressure vanish on the boundary. Isobaric contours for the D-shaped tokamak configuration of Fig. 5, the boundary of which is indicated by the red dotted curve therein, are given in Fig. 8 together with pressure profiles on the poloidal plane at $y = 0$ and at the vertical line passing from the position of the pressure maximum. It is noted that although the values of the dimensionless pressure are quite high, realistic pressure values can be obtained by appropriate values of the free parameters $P_{s 0}$ and $U_{0}$ ⁠. For flows of experimental fusion pertinence, the deviation of the magnetic surfaces from the isobaric surfaces is very small.

FIG. 8.

View large Download slide

Upper: Isobaric (blue-continuous) curves on the poloidal plane for the D-shaped tokamak configuration of Fig. 5 for $B = 0.2$ ⁠, $M_{p 0} = 0.1$ ⁠, $l_{ρ} = 2$ and $l_{M_{p}} = 2$ ⁠. The red dashed curves represent magnetic surfaces. Lower left: Respective pressure profile on the mid-plane $y = 0$ ⁠. Lower right: Respective vertical pressure profile at the position of the pressure maximum.

V. CONCLUSIONS

Making a generic linearizing choice of the free function-terms involved in the generalized GS equation (3) we solved the resulting linearized equation (16) analytically. The requested solution was expressed as an expansion of the generalized Solov'ev solution (12) in the form of Eq. (15), involving the determinable function W. Then, the solution was obtained as a superposition of the counterpart homogeneous equation and a particular solution of the complete inhomogeneous equation in the form of Eq. (28). The solutions of the pertinent radial ODEs were expressed in terms of infinite converging series.

Employing these solutions, we constructed several configurations by changing the values of the free parameters, either individually or in combination, associated with the new terms of pressure, poloidal current in conjunction with the parallel velocity, and electric field. The equilbria constructed are pertinent to tokamaks, spherical tokamaks and spheromaks; they include D-shaped configurations with either positive or negative triangularity and diverted configurations with either a couple of X-points or a single X-point.

The present study can be extended to a more generic generalized GS equation accounting for CGL pressure anisotropy [Ref. 27, Eq. (34) therein]. Another extension could be pursued in the framework of Hall-MHD, a simplified two-fluid model with inertialess electron-fluid elements keeping on magnetic surfaces and ion-fluid elements departing from them.

ACKNOWLEDGMENTS

This work was conducted in the framework of participation of the University of Ioannina in the National Programme for the Controlled Thermonuclear Fusion, Hellenic Republic. The authors would like to thank the anonymous reviewers for constructive comments, which have resulted in a remarkably improved version of the manuscript.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Apostolos Kuiroukidis: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal). Dimitrios A. Kaltsas: Formal analysis (equal); Investigation (equal); Software (equal). George N. Throumoulopoulos: Formal analysis (equal); Investigation (equal); Writing – original draft (equal).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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New axisymmetric equilibria with flow from an expansion about the generalized Solov'ev solution

I. INTRODUCTION

II. GENERIC LINEAR AXISYMMETRIC EQUILIBRIA WITH NON-PARALLEL FLOW

III. SOLUTIONS

IV. EQUILIBRIUM CONFIGURATIONS

V. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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New axisymmetric equilibria with flow from an expansion about the generalized Solov'ev solution Open Access

I. INTRODUCTION

II. GENERIC LINEAR AXISYMMETRIC EQUILIBRIA WITH NON-PARALLEL FLOW

III. SOLUTIONS

IV. EQUILIBRIUM CONFIGURATIONS

V. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

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pubs.aip.org

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New axisymmetric equilibria with flow from an expansion about the generalized Solov'ev solution