I Introduction

Axisymmetric equilibria with pressure anisotropy

and plasma flow

A. Evangelias, G. N. Throumoulopoulos

University of Ioannina, Physics Department,

Section of Astrogeophysics, GR 451 10 Ioannina, Greece

A generalised Grad-Shafranov equation that governs the equilibrium of an axisymmetric toroidal plasma with anisotropic pressure and incompressible flow of arbitrary direction is derived. This equation includes six free surface functions and recovers known Grad-Shafranov-like equations in the literature as well as the usual static, isotropic one. The form of the generalised equation indicates that pressure anisotropy and flow act additively on equilibrium. In addition, two sets of analytical solutions, an extended Solovev one with a plasma reaching the separatrix and an extended Hernegger-Maschke one for a plasma surrounded by a fixed boundary possessing an X-point, are constructed, particularly in relevance to the ITER and NSTX tokamaks. Furthermore, the impacts both of pressure anisotropy and plasma flow on these equilibria are examined. It turns out that depending on the maximum value and the shape of an anisotropy function, the anisotropy can act either paramagnetically or diamagnetically. Also, in most of the cases considered both the anisotropy and the flow have stronger effects on NSTX equilibria than on ITER ones.

## I Introduction

It has been established in a number of fusion devices that sheared flow both zonal and mean (equilibrium) play a role in the transitions to improved confinement regimes as the L-H transition and the Internal Transport Barriers zonal (), flow (). These flows can be driven externally in connection with electromagnetic power and neutral beam injection for plasma heating and current drive or can be created spontaneously (zonal flow). An additional effect of external heating, depending on the direction of the injected momentum, is pressure anisotropy which also may play a role in several magnetic fusion related problems.

In many important plasmas as the high temperature ones the collision time is so long that collisions can be ignored. It would appear that for such collisionless plasmas a fluid theory should not be appropriate. However, for perpendicular motions because of gyromotion the magnetic field plays the role of collisions, thus making a fluid description appropriate. Macroscopic equations for a collisionless plasma with pressure anisotropy have been derived by Chew, Goldberger, and Low CGL () on the basis of a diagonal pressure tensor consisting of one element parallel to the magnetic field and a couple of identical perpendicular elements associated with two degrees of freedom.

The MHD equilibria of axisymmetric plasmas, which can be starting points of stability and transport studies, is governed by the well known Grad-Shafranov (GS) equation. The most widely employed analytic solutions of this equation is the Solovev solution so () and the Hernegger-Maschke solution hema (), the former corresponding to toroidal current density non vanishing on the plasma boundary and the latter to toroidal current density vanishing thereon. In the presence of flow the equilibrium satisfies a generalised Grad-Shafranov (GGS) equation together with a Bernoulli equation involving the pressure (see for example moso (); ha (); T-Th ()). For compressible flow the GGS equation can be either elliptic or hyperbolic depending on the value of a Mach function associated with the poloidal velocity. Note that the toroidal velocity is inherently incompressible because of axisymmetry. In the presence of compressibility the GGS equation is coupled with the Bernoulli equation through the density which is not uniform on magnetic surfaces. For incompressible flow the density becomes a surface quantity and the GGS equation becomes elliptic and decouples from the Bernoulli equation (see section II). Consequently one has to solve an easier and well posed elliptic boundary value problem. In particular for fixed boundaries, convergence to the solution is guaranteed under mild requirements of monotonicity for the free functions involved in the GGS equation couhi (). For plasmas with anisotropic pressure the equilibrium equations involve a function associated with this anisotropy [Eq. (8) below]. To get a closed set of reduced equilibrium equations an assumption on the functional dependence of this function is required (cf. Cotsaftis ()-Kuznetsov () for static equilibria and iabo ()-ivma () for stationary ones).

In this work we derive a new GGS equation by including both anisotropic pressure and incompressible flow of arbitrary direction. This equation consists of six arbitrary surface quantities and recovers known equations as particular cases, as well as the usual GS equation for a static isotropic plasma. Together we obtain a Bernoulli equation for the quantity [Eq. (9)], which may be interpreted as an effective isotropic pressure. For the derivation we assume that the function of pressure anisotropy is uniform on magnetic surfaces. In fact, as it will be shown, for static equilibria as well as for stationary equilibria either with toroidal flow or incompressible flow parallel to the magnetic field, this property of the anisotropy function follows if the current density shares the same surfaces with the magnetic field. Then for appropriate choices of the free functions involved we obtain an extended Solovev solution describing configurations with a non-predefined boundary, and an extended Hernegger-Maschke solution with a fixed boundary possessing an X-point imposed by Dirichlet boundary conditions. On the basis of these solutions we construct ITER-like, as well as NSTX and NSTX-Upgrade-like equilibria for arbitrary flow, both diamagnetic and paramagnetic, to examine the impact both of pressure anisotropy and plasma flow on the equilibrium characteristics. The main conclusions are that the pressure anisotropy and the flow act on equilibrium in an additive way, with the anisotropy having a stronger impact than that of the flow. Also the effects of flow and anisotropy are in general more noticeable in spherical tokamaks than in conventional ones.

The GGS equation for plasmas with pressure anisotropy and flow is derived in section II. In section III the generalised Solovev and Hernegger-Maschke solutions are obtained and employed to construct ITER and NSTX pertinent configurations. Then the impact of anisotropy and flow on equilibrium quantities, as the pressure and current density, are examined in section IV. Section V summarizes the conclusions.

## Ii The Generalised Grad-Shafranov Equation

The ideal MHD equilibrium states of an axially symmetric magnetically confined plasma with incompressible flow and anisotropic pressure are governed by the following set of equations:

 →∇⋅(ρ→v)=0 (1)
 ρ(→v⋅→∇)→v=→J×→B−→∇⋅\lx@stackrel↔P (2)
 →∇×→B=μ0→J (3)
 →∇×→E=0 (4)
 →∇⋅→B=0 (5)
 →E+→v×→B=0 (6)

The diagonal pressure tensor , introduced in CGL (), consists of one element parallel to the magnetic field, , and two equal perpendicular ones, , and is expressed as

 \lx@stackrel↔P=p⊥\lx@stackrel↔I+σdμ0→B→B (7)

where the dimensionless function

 σd=μ0p∥−p⊥|→B|2 (8)

is a measure of the pressure anisotropy. Particle collisions in equilibrating parallel and perpendicular energies will reduce and therefore a collision-dominated plasma can be described accurately by a scalar pressure. However, because of the low collision frequency a high-temperature confined plasma remains for long anisotropic, once anisotropy is induced by external heating sources.

At this point we define the quantity

 ¯¯¯p=p∥+p⊥2 (9)

which may be interpreted as an effective isotropic pressure, and which should not be confused with the average plasma pressure



=13Tr(\lx@stackrel↔P)=p∥+2p⊥3=¯¯¯p−σdB26μ0 (10)

On the basis of Eqs. (8) and (9), the following instructive relations arise for the two scalar pressures:

 p⊥=¯¯¯p−σdB22μ0 (11)

and

 p∥=¯¯¯p+σdB22μ0 (12)

Owing to axisymmetry, the divergence-free fields, i.e., the magnetic field, the current density, , and the momentum density of the fluid element, , can be expressed in terms of the stream functions , , and as

 →B=I→∇ϕ+→∇ϕ×→∇ψ (13)
 →J=1μ0(Δ∗ψ→∇ϕ−→∇ϕ×→∇I) (14)

and

 ρ→v=Θ→∇ϕ+→∇ϕ×→∇F (15)

Here, (, , ) denote the usual right-handed cylindrical coordinate system; constant surfaces are the magnetic surfaces; is related to the poloidal flux of the momentum density field, ; the quantity is related to the net poloidal current flowing in the plasma and the toroidal field coils; ; is the elliptic operator defined by ; and .

Equations (1)-(6) can be reduced by means of certain integrals of the system, which are shown to be surface quantities. To identify two of these quantities, the time independent electric field is expressed by and the Ohm’s law, (6), is projected along and , respectively, yielding

 →∇ϕ⋅(→∇F×→∇ψ)=0 (16)

and

 →B⋅→∇Φ=0 (17)

Equations (16) and (17) imply that and . An additional surface quantity is found from the component of Eq. (6) perpendicular to a magnetic surface:

 Φ′=1ρR2(IF′−Θ) (18)

where the prime denotes differentiation with respect to . On the basis of Eq. (18) the velocity [Eq. (15)] can be written in the form

 →v=F′ρ→B−R2Φ′→∇ψ (19)

Thus, is decomposed into a component parallel to and a non parallel one associated with the electric field in consistence with the Ohm’s law (6). Subsequently, by projecting Eq. (2) along we find a fourth surface quantity of the system:

 X(ψ)≡(1−σd−M2p)I+μ0R2F′Φ′ (20)

Here we have introduced the poloidal Mach function as:

 M2p≡μ0(F′)2ρ=v2polB2pol/μ0ρ=v2polv2Apol (21)

where is the Alfvén velocity associated with the poloidal magnetic field. From Eqs. (18) and (20) it follows that, neither is a surface quantity, unlike the case of static, isotropic equilibria, nor .

With the aid of Eqs. (16)-(19) and (20), the components of Eq. (2) along and perpendicular to a magnetic surface are put in the respective forms

 →B⋅{→∇[v22+ΘΦ′ρ]+1ρ→∇¯¯¯p}=0 (22)

and

 {→∇⋅[(1−σd−M2p)→∇ψR2]+[μ0F′F′′ρ−(1−σd)′]|→∇ψ|2R2−μ0σ′dB22μ0}|→∇ψ|2 (23) + {μ0ρ→∇(v22)−μ0ρ2R2→∇(Θρ)2+(1−σd)2R2→∇I2+μ0→∇¯¯¯p}⋅→∇ψ=0

Therefore, irrespective of compressibility the equilibrium is governed by the equations (22) and (23) coupled through the density, , and the pressure anisotropy function, . Equation (23) has a singularity when , and so we must assume that .

In order to reduce the equilibrium equations further, we employ the incompressibility condition

 →∇⋅→v=0 (24)

Then Eq. (1) implies that the density is a surface quantity,

 ρ=ρ(ψ) (25)

and so is the Mach function

 M2p=M2p(ψ) (26)

In addition to obtain a closed set of equations following Cotsaftis (); Clement (); Kuznetsov (); clst () we assume that is uniform on magnetic surfaces

 σd=σd(ψ) (27)

For static equilibria this follows from Eq. (20), which becomes , if in the presence of anisotropy the current density remains on the magnetic surfaces (). Since , the same implication for holds for parallel incompressible flow as well as for toroidal flow. Also, the hypothesis , according to Cotsaftis (), may be the only suitable for satisfying the boundary conditions on a rigid, perfectly conducting wall.

From Eqs. (18) and (20) it follows that axisymmetric equilibria with purely poloidal flow cannot exist because of the following contradiction: from Eq. (20) it follows that is a surface function, but also, from Eq. (18), implying that has an explicit dependence on ; so it cannot be a surface function. On the other hand, there can exist an equilibrium with purely toroidal flow, either “compressible”, in the sense that the density varies on the magnetic surfaces, or an incompressible one with uniform density thereon. For isotropic plasmas both kinds of these equilibria were examined in pothta ().

With the aid of Eq. (25), Eq. (22) can be integrated to yield an expression for the effective pressure, i.e.,

 ¯¯¯p=¯¯¯ps(ψ)−ρ[v22−(1−σd)R2(Φ′)21−σd−M2p] (28)

Therefore, in the presence of flow the magnetic surfaces in general do not coincide with the surfaces on which is uniform. In this respect, the term containing is the static part of the effective pressure which does not vanish when .

Finally, by inserting Eq. (28) into Eq. (23) after some algebraic manipulations, the latter reduces to the following elliptic differential equation,

 (1−σd−M2p)Δ∗ψ+12(1−σd−M2p)′|→∇ψ|2+12(X21−σd−M2p)′ (29) + μ0R2¯¯¯p′s+μ0R42[(1−σd)ρ(Φ′)21−σd−M2p]′=0

This is the GGS equation that governs the equilibrium for an axisymmetric plasma with pressure anisotropy and incompressible flow. For flow parallel to the magnetic field the -term vanishes. For vanishing flow Eq. (29) reduces to the one derived in Clement (), when the pressure is isotropic it reduces to the one obtained in T-Th (), and when both anisotropy and flow are absent it reduces to the well known GS equation. Equation (29) contains six arbitrary surface quantities, namely: , , , , and , which can be assigned as functions of to obtain analytically solvable linear forms of the equation or from other physical considerations.

### A Isodynamicity

There is a special class of static equilibria called isodynamic for which the magnetic field magnitude is a surface quantity () Palumbo (). This feature can have beneficial effects on confinement because the grad- drift vanishes and consequently plasma transport perpendicular to the magnetic surfaces is reduced. Also, it was proved that the only possible isodynamic equilibrium is axisymmetric PaBo (). For fusion plasmas the thermal conduction along is fast compared to the heat transport perpendicular to a magnetic surface, so a good assumption is that the parallel temperature is a surface function, . Then, assuming that the plasma obeys the ideal gas law, it follows that the parallel pressure becomes also a surface function, .

With the aid of these assumptions, and on the basis of Eqs. (19) and (22) it follows that the magnitude of the magnetic field is related with the perpendicular pressure as

 |→B|2=2G(ψ)M2p(ψ)−(p⊥−ρR2(Φ′)2)1M2p(ψ) (30)

where . We note that becomes a surface function when the perpendicular pressure satisfies the relation . This implies that

 σd=σd(ψ,R)=μ0p∥(ψ)|→B|2(ψ)−R2μ0ρ(ψ)(Φ′)2(ψ)|→B|2(ψ) (31)

which is in contradiction with the hypothesis that the function is a surface quantity. Consequently, the only possibility for isodynamic magnetic surfaces to exist is that for field aligned flow, , because then Eq. (30) reduces to

 |→B|2=2G(ψ)M2p(ψ)−p⊥M2p(ψ) (32)

Eqs. (8) and (32) imply that both and .

Thus, the conclusions for the isotropic case T-Th () are generalised for anisotropic pressure, i.e. all three , and become surface quantities. We note here that the more physically pertinent case that and remain arbitrary functions would require either compressibility or eliminating the assumption . However, in this case tractability is lost and the problem requires numerical treatment.

### B Generalised Transformation

Using the transformation

 u(ψ)=∫ψ0√1−σd(g)−M2p(g)dg,σd+M2p<1 (33)

Eq. (29) reduces to

 Δ∗u+12ddu(X21−σd−M2p)+μ0R2d¯¯¯psdu+μ0R42ddu[(1−σd)ρ(dΦdu)2]=0 (34)

Transformation (33) does not affect the magnetic surfaces, it just relabels them by the flux function , and is a generalisation of that introduced in Simintzis () for isotropic equilibria with incompressible flow () and that introduced in Clement () for static anisotropic equilibria (). Note that no quadratic term as appears anymore in (34). Once a solution of this equation is found, the equilibrium can be completely constructed with calculations in the -space by using (33) and the inverse transformation

 ψ(u)=∫u0(1−σd(g)−M2p(g))−1/2dg (35)

Before continuing to the construction of analytical solutions, we find it convenient to make a normalization by introducing the dimensionless quantities: . The index can be either or 0, where denotes the magnetic axis, and 0 the geometric center of a configuration. Thus, the normalization constants are defined as follows: is the radial coordinate of the configuration’s magnetic axis/geometric center, and , , are the magnitude of the magnetic field, the plasma density, and the Afvén velocity thereon. Consequently, with the use of the generalised transformation (33), Eqs. (13)-(15), (20), (28), and (34), are put in the following normalized forms in -space:

 ˜→B=˜I˜→∇ϕ+(1−σd−M2p)−1/2˜→∇ϕ×˜→∇˜u (36)
 ˜→v=Mp√˜ρ˜→B−ξ2(1−σd−M2p)1/2(d˜Φd˜u)˜→∇ϕ (37)
 ˜→J = [(1−σd−M2p)−1/2˜Δ∗˜u−12(1−σd−M2p)−3/2dd˜u(1−σd−M2p)|˜→∇˜u|2]˜→∇ϕ (38) − ˜→∇ϕ×˜→∇˜I
 ˜X=(1−σd−M2p)[˜I+ξ2(d˜Fd˜u)(d˜Φd˜u)] (39)
 ˜¯¯¯p=˜¯¯¯ps(˜u)−˜ρ⎡⎣˜v22−(1−σd)ξ2(d˜Φd˜u)2⎤⎦ (40)

and

 ˜Δ∗˜u+12dd˜u(˜X21−σd−M2p)+ξ2d˜¯¯¯psd˜u+ξ42dd˜u⎡⎣(1−σd)˜ρ(d˜Φd˜u)2⎤⎦=0 (41)

where . In section III for appropriate choices of the surface functions, Eq. (41) will be linearised and solved analytically.

### C Plasma beta and safety factor

The safety factor, measuring the rate of change of toroidal flux with respect to poloidal flux through an infinitesimal annulus between two neighboring flux surfaces, is given by the following expression

 q≡dψtordψpol=12π∮IdlR|→∇ψ| (42)

Expressing the length element in Shafranov coordinates Fr () the above formula becomes

 q=12π∫2π0I√r2+(ψθψr)2R|→∇ψ|dθ (43)

where, , and . On the basis of the generalised transformation (33) and the adopted normalization, Eq. (43) is put in the following form

 (44)

from which we can calculate numerically the safety factor profile.

For the local value of the safety factor on the magnetic axis there exists the simpler analytic expression

 qa=(1−σd−M2p)1/2˜Iξ{∂2˜u∂ξ2∂2˜u∂ζ2}−1/2ξ=ξa,ζ=ζa (45)

obtained by expansions of the flux function close to the magnetic axis. From equations (39) and (45), one observes that when the flow is parallel to the magnetic field, , then the value of has no dependence on the anisotropy. Indeed becomes independent on , where is the local value of the anisotropy function on the magnetic axis, .

In the case of anisotropic pressure we represent the plasma pressure by the effective pressure, so that the plasma beta can be defined as

 β≡¯¯¯pB2/2μ0 (46)

It also useful to define separate toroidal and poloidal quantities measuring confinement efficiency of each component of the magnetic field. The toroidal beta to be used here is

 βt=¯¯¯pB20/2μ0 (47)

Recent experiments on the National Spherical Torus Experiment (NSTX) have made significant progress in reaching high toroidal beta Menard (), while on ITER the beta parameter is expected to take low values, ITERbeta ().

## Iii Analytic Equilibrium Solutions

### A Solovev-like solution

According to the Solovev ansatz, the free function terms in the GGS equation are chosen to be linear in as

 ˜¯¯¯ps=˜¯¯¯psa(1−˜u˜ub),˜u≥0 ˜X21−σd−M2p=2ϵ˜¯¯¯psa(1+δ2)˜u˜ub+1 ˜ρ(1−σd)(d˜Φd˜u)2=2λ˜¯¯¯psa(1+δ2)(1−˜u˜ub) (48)

Here, denotes the magnetic axis and the plasma boundary; determines the elongation of the magnetic surfaces near the magnetic axis; for the plasma is diamagnetic (paramagnetic); and is a non-negative parameter related with the non-parallel component of the flow. In addition, we impose that the solution vanishes on the magnetic axis, .

With this linearising ansatz the GGS equation (41) reduces to

 ˜Δ∗˜u+˜¯¯¯psa˜ub[ϵ(1+δ2)−ξ2−ξ4λ(1+δ2)]=0 (49)

which admits the following generalised Solovev solution valid for arbitrary , and :

 ˜u(ξ,ζ)=˜¯¯¯psa2(1+δ2)˜ub[ζ2(ξ2−ϵ)+δ2+λ4(ξ2−1)2+λ12(ξ2−1)3] (50)

This solution does not include enough free parameters to impose desirable boundary conditions, but has the property that a separatrix is spontaneously formed. Thus, we can predefine the position of the magnetic axis, , chosen as normalization point and the plasma extends from the magnetic axis up to a closed magnetic surface which we will choose to coincide with the separatrix.

For an up-down symmetric (about the midplane ) magnetic surface, its shape can be characterized by four parameters, namely, the coordinates of the innermost and outermost points on the midplane, and , and the coordinates of the highest (upper) point of the plasma boundary, (see Fig. 14). In terms of these four parameters we can define the normalized major radius

 ξ0=ξin+ξout2 (51)

which is the radial coordinate of the geometric center, the minor radius

 ˜α=ξout−ξin2 (52)

the triangularity of a magnetic surface

 t=ξ0−ξup˜α (53)

defined as the horizontal distance between the geometric center and the highest point of the magnetic surface normalized with respect to minor radius, and the elongation of a magnetic surface

 κ=ζup˜α (54)

Usually, we specify the values of , , , and , instead of (, , , ) to characterize the shape of the outermost magnetic surface. On the basis of solution (50) the latter quantities can be expressed in terms of , , . Subsequently, in order to make an estimate of realistic values for the free parameters , and the radial coordinate of the magnetic axis in connection with the tokamaks under consideration, we employ the relations (51)-(54) to find (, and ) in terms of the known parameters (, , and ). (For ITER: , , , / for NSTX: , , , ). In the static limit () this estimation procedure can be performed analytically and when the plasma is diamagnetic we find

 ϵ=(R0−α)2R20+α2 δ=κ√αR0 Ra=√R20+α2 (55)

Also, for a diamagnetic equilibrium it holds (cf. Fig. 1). The respective relations for a paramagnetic equilibrium Arapoglou () for which are

 ϵ=(t−1)44(t2−2t−1 δ=κ√2√−t2+2t+1 Ra=√2R0 (56)

Relations (A)-(A) will also be employed to assign values of the free parameters , and for non parallel flows ( because in this case the above estimation procedure becomes complicated. Afterwards, since the vacuum magnetic field at the geometric center of a configuration is known (ITER: / NSTX: ), we can also estimate its value on the magnetic axis by using the relation , and therefore the value of from the relation , once the maximum pressure for each device is known (ITER: / NSTX: ).

Thus, we can fully determine the solution from Eq. (50), as well as the position of the characteristic points of the boundary and obtain the ITER-like and NSTX-like, diamagnetic and paramagnetic configurations, whose poloidal cross-section with a set of magnetic surfaces are shown in Figs. (1)-(3).

We note that by expansions around the magnetic axis it turns out that the magnetic surfaces in the vicinity of the magnetic axis have elliptical cross-sections (see also Lao ()-Bizarro ()). In the diamagnetic configurations presented in Figs. (1) and (2) the inner part of the separatrix is defined by the vertical line , and, for the NSTX it is located very close to the tokamak axis of symmetry in accordance with the small hole of spherical tokamaks. It may be noted that such D-shaped configurations are advantageous for improving stability with respect to the interchange modes because of the smaller curvature on the high field side. On the other hand, in a paramagnetic configuration the plasma reaches through a corner the axis of symmetry implying values for the minor radius different from the actual ones, and thus, such a configuration is not typical for conventional tokamaks. However, a configuration with a similar corner was observed recently in the QUEST spherical tokamak as a self organized state mizu () (Fig. 5 therein).

### B Hernegger-Maschke-like solution

Since the charged particles move parallel to the magnetic field free of magnetic force, parallel flows is a plausible approximation. In particular for tokamaks this is compatible with the fact that the toroidal magnetic field is an order of magnitude larger than the poloidal one and the same scaling is valid for the toroidal and poloidal components of the fluid velocity. Also, for parallel flows the problem remains analytically tractable and leads to a generalised Hernegger-Maschke solution to be constructed below.

In the absence of the electric field term ( -term) the GGS equation (41) becomes

 ˜Δ∗˜u+12dd˜u(˜X21−σd−M2p)+ξ2d˜¯¯¯psd˜u=0 (57)

where all quantities have now been normalized with respect to the geometric center. Choosing the free function terms of Eq. (57) to be quadratic in as

 ˜¯¯¯ps(˜u)=˜p2˜u2 ˜X2(˜u)1−σd(˜u)−M2p(˜u)=1+˜X1˜u2 (58)

it reduces to the following linear differential equation

 ∂2˜u∂ξ2+∂2˜u∂ζ2−1ξ∂˜u∂ξ+˜X1˜u+2˜p2ξ2˜u=0 (59)

The values of the parameters and , will be chosen in connection with realistic shaping and values of the equilibrium figures of merit, i.e. the local toroidal beta and the safety factor on the magnetic axis. The solution to Eq. (59) is found by separation of variables,

 ˜u(ξ,ζ)=G(ξ)T(ζ) (60)

on the basis of which, it further reduces to the following form

 1T(ζ)d2T(ζ)dζ2=−1G(ξ)d2G(ξ)dξ2+1ξG(ξ)dG(ξ)dξ−˜X1−2˜p2ξ2=−η2 (61)

where is the separation constant.

Therefore, the problem reduces to a couple of ODEs. The one for the function is

 d2T(ζ)dζ2+η2T(ζ)=0 (62)

having the general solution

 T(ζ)=a1cos(ηζ)+a2sin(ηζ) (63)

with the coefficients and to be determined later. The second equation satisfied by the function is

 d2G(ξ)dξ2−1ξdG(ξ)dξ+(˜X1−η2)G(ξ)+2˜p2ξ2G(ξ)=0 (64)

Introducing the parameters , , and , so that and , Eq. (64) becomes

 d2G(ϱ)dϱ2+[iη2−γ4√δ1ϱ−14]G(ϱ)=0 (65)

Furthermore, if we set then Eq. (65) is put in the form

 d2G(ϱ)dϱ2+[νϱ−14]G(ϱ)=0 (66)

which is a special case of the Whittaker’s equation for , and thus, it admits the general solution

 G(ϱ)=b1Mν,12(ϱ)+b2Wν,12(ϱ) (67)

Here, and are the Whittaker functions, which are independent solutions of the homonemous differential equation. Consequently, a typical solution of the original equation (59) is written in the form

 ˜u(ϱ,ζ)=[b1Mν,12(ϱ)+b2Wν,12(ϱ)][a1cos(ηζ)+a2sin(ηζ)] (68)

For further treatment it is convenient to restrict the separation constant to positive integer values . Therefore, by superposition the solution can be expressed as

 ˜u(ϱ,ζ)=∞∑j=1[ajMνj,12(ϱ)cos(jζ)+bjMνj,12(ϱ)sin(jζ)+cjWνj,12(ϱ)cos(jζ)+djWνj,12(ϱ)sin(jζ)] (69)

Following the analysis given in Appendix A we fully specify the solution (69) and construct the diverted equilibrium with ITER-like characteristics shown in Fig. (4). Note that the magnetic axis is located outside of the midplane at .

In a similar way we constructed NSTX-Upgrade-like equilibria (, , , , and ) on the basis of the extended Hernegger-Maschke solution, shown in Fig. (5). The magnetic axis of the respective configuration is located at the position , while the flux function on axis takes the value .

## Iv Effects of Anisotropy and Flow on Equilibrium

To completely determine the equilibrium we choose the plasma density, the Mach function and the anisotropy function profiles to be peaked on the magnetic axis and vanishing on the plasma boundary. Specifically, for the Solovev solution we choose: , and with and constant quantities, while for the Hernegger-Maschke solution we choose: , , , with and constant quantities, respectively. It is noted here that the above chosen density function, peaked on the magnetic axis and vanishing on the boundary is typical for tokamaks. Also, the Mach function adopted having a similar shape is reasonable at least in connection with experiments with on axis focused external momentum sources. The functions , and chosen depend on two free parameters; their maximum on axis and an exponent associated with the shape of the profile; the exponent of the function , connected with flow shear, is held fixed at .
The value of depends on the kind of tokamak (conventional or spherical). On account of experimental evidence Brau (); Pantis (), the toroidal rotation velocity in tokamaks is approximately which for large conventional ones implies , while the flow is stronger for spherical tokamaks () Menard (). In addition, from the requirement of positiveness for all pressures within the whole plasma region, we find that the pressure anisotropy parameter takes higher values on spherical tokamaks than in the conventional ones, as shown on Table I; also it must be .

An argument why the flow and pressure anisotropy are stronger in spherical tokamaks is that in this case the magnetic field is strongly inhomogeneous, as the aspect ratio is too small. In contrast, for the generalised Hernegger-Maschke equilibrium the pressure anisotropy takes a little higher values on ITER rather than on the NSTX-U tokamak, and this may be attributed to a peculiarity of this solution.

When the plasma is diamagnetic the toroidal magnetic field inside the plasma decreases from its vacuum value, and consequently the profile of the function is expected to be hollow (). As shown in Fig. (6), as becomes larger the field increases, and for sufficient high it becomes peaked on the magnetic axis. This means that increasing pressure anisotropy acts paramagnetically in terms of its maximum value on axis, .