Some axisymmetric equilibria for certain ideal and resistive magnetohydrodynamics with incompressible flows

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Results in Physics 7 (2017) 3163–3175

Contents lists available at ScienceDirect

Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics

Some axisymmetric equilibria for certain ideal and resistive magnetohydrodynamics with incompressible flows S.M. Moawad a,⇑, O.H. El-Kalaawy a, H.M. Shaker b a b

Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt Future High Institute of Engineering in Fyoum, Ministry of High Education, Fyoum, Egypt

a r t i c l e

i n f o

Article history: Received 28 March 2017 Received in revised form 16 August 2017 Accepted 17 August 2017 Available online 21 August 2017 Keywords: Magnetohydrodynamics Axisymmetric plasma Resistivity Incompressible flows Exact equilibria Magnetic confinement devices

a b s t r a c t In this paper, the equilibrium properties of some ideal and resistive magnetohydrodynamics (MHD) are investigated. The governing equations are taken in the steady state for parallel and non-parallel flow to magnetic filed. The governing equations are reduced to Bernoulli-Grad-Shafranov system. The problem of finding exact equilibria to the governing equations in the presence of incompressible mass flows is studied. Several nonlinear equilibria of the governing equations are obtained with aid of constructed constraints. The obtained results cover several previously configurations and include new considerations about the nonlinearity of magnetic flux stream variables. The possibility of applying the obtained results to magnetic confinement devices are discussed. Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction MHD equilibrium is one of the basic objectives in astrophysical and geophysical problems, plasma confinement systems and fusion research. The mathematical complexity of the MHD equations has stood in the way of understanding hydromagnetic phenomena, e.g. stellar winds [1–4], solar prominence magnetic fields [5–8], and plasma confinement devices [9–11]. They are still waiting general solutions for which few representative classes of MHD steady state flows has been achieved. MHD equilibria with incompressible flows in cylindrical and axisymmetric domains have been carried out by many authors [12–32]. The equilibrium of a toroidal plasma was found that it is governed by a second-order elliptic differential equation for the poloidal magnetic flux function include five surface quantities along with an algebraic relation for the pressure [12–15]. The poloidal and toroidal plasma rotation can be obtained in experiments like the electric tokamak [33], the Joint European Torus (JET) [34], and the National Spherical Tokamak Experiment [35]. A general theory of axisymmetric MHD equilibria with relativistic flows was developed and applied to disks associated with rotating magnetized stars and black holes by Lovelace et al. [36]. Bogoyavlenskij [37] demonstrated the existence of exact axisymmetric ⇑ Corresponding author. E-mail address: [email protected] (S.M. Moawad).

equilibria in the magnetostatic case. Astrophysical jets were modelled by an exact axisymmetric equilibrium of plasma in the gravitational field of an attracting center [38]. Intrinsic symmetries of the ideal MHD equilibrium equations were introduced for the divergence-free plasma flows [39]. Magnetic dipole equilibria with flow effects was studied by Catto and Krasheninnikov [40]. Axisymmetric stellar wind equilibria with both open and closed magnetic field regions were examined by Keppens and Goedbloed [41]. The derivation of the equilibrium equations of MHD plasmas has been extensively discussed in the literature by many authors [13,19,42–44]. A similar analysis was investigated for a system with translational and axial symmetry by Goedbloed and Lifschitz [45]. Many attempts has been made over the past years to investigate different cases of the problem of finding general solutions to MHD equations by means of essential mathematical simplifications. Magnetostatic case, force-free field condition, parallel flow and field condition and other similar conditions on the symmetry and the physics of the problem were imposed to make the mathematics tractable. The problem of finding exact solutions of MHD equations in the presence of flow has been carried out by many authors [15–18,30,31,46–48]. Numerical solutions of MHD boundary layer flow of two dimensional tangent hyperbolic fluid towards a stretching sheet were discussed in [49]. The nonexistence of static axisymmetric equilibria with constant resistivity was suggested in [44,50].

http://dx.doi.org/10.1016/j.rinp.2017.08.033 2211-3797/Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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In the present work we consider the problem of finding nonlinear equilibria to resistive MHD of axisymmetric incompressible flows. The paper is organized as follows: In Section ‘‘Non-parallel ideal MHD flows”, we explain how to obtain the equilibrium physical variables of non-parallel ideal MHD flows besides finding exact solutions to the equations governing these flows. In Section ‘‘Parallel resistive MHD flows” we consider the basic equations and investigate the problem formulation of the equilibrium equations of resistive incompressible MHD flows in the steady state. In Sect ion ‘‘Nonlinear axisymmetric equilibria for parallel resistive MHD flows”, we obtain nonlinear equilibria to the equilibrium equations of the MHD flows presented in Section ‘‘Parallel resistive MHD flows”. Results and discussion are investigated in Section ‘‘Results and discussion”. Finally, the conclusion of the paper is presented in Section ‘‘Conclusion”.

with the following expression for the pressure:

2 v U0 H : P ¼ ps ðwÞ q þ 2 q

The dash denotes differentiation with respect to w. The functions X and ps are arbitrary functions of w (surface quantities) and the symbol M denotes the poloidal Alfvénic Mach number which is defined by

M2 ¼

1

The ideal MHD incompressible flows are governed by the following set of equations, written in the steady-state and quite simple units: the mass conservation equation:

I 1

the momentum equation:

qðv rÞv ¼ rP þ j ^ B;

ð2Þ

ðG0 Þ

r ^ E ¼ 0;

2

q

Z

! þ r2 G0 U0 ¼ XðwÞ:

w

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 M 2 ðuÞdu;

the divergence-free Gauss law:

r B ¼ 0;

ð5Þ

Ohm’s law for MHD:

E þ v ^ B ¼ 0;

ð6Þ

where q; v ; P; B and j stand as usual for the mass density, flow velocity field, gas pressure, magnetic field and current density. We use the cylindrical coordinates ðr; /; zÞ so with the aid of Eq. (4) the fields B, j and v are expressed as

B ¼ Ir/ þ r/ ^ rw;

ð7Þ

j ¼ M wr/ r/ ^ rI;

ð8Þ

1

q

ðHr/ þ r/ ^ rGÞ;

q ¼ qðwÞ;

ð9Þ

with the stream functions wðr; zÞ, Iðr; zÞ, Gðr; zÞ and Hðr; zÞ. Here, D is the elliptic operator defined by

r D r r 2 : r

ð15Þ

1 d X2 D Uþ 2 dU 1 M2

!

" # 2 dps r4 d dU q þ þr ¼ 0: dU 2 dU dU 2

ð16Þ

Several classes of exact solutions to Eq. (16) can be derived for some choices of M, as we show later. Moreover, we show how to obtain the associated physical quantities of the full MHD system (1)–(6). Construction for the equilibrium physical variables In this subsection we explain how to obtain the equilibrium physical variables, B; j; E; r; P and q for the MHD system (1)–(6). From Eq. (15) we have

Q

dw 1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : dU 1 M2

ð17Þ

Hence,

and

v¼

M 2 < 1:

Eq. (10) is reduced to

Ampére’s law:

ð4Þ

ð14Þ

0

ð3Þ

r ^ B ¼ j;

ð13Þ

For the above MHD flows, there are six surface quantities GðwÞ; UðwÞ; XðwÞ; qðwÞ; ps and MðwÞ five of them are arbitrary. The solution w can be determined from Eq. (10), that can be solved analytically. Under the transformation [14,15]:

UðwÞ ¼

Faraday’s law:

ð12Þ

ðIG0 HÞ ¼ U0 ;

qr2

ð1Þ

v 2p ðG0 Þ2 ¼ ; q v 2Ap

where v p and v Ap are the poloidal flow velocity and Alfvén velocity, respectively. In [8] it is shown the functions G, U, X, q, I and H are provided the following two relations:

Non-parallel ideal MHD flows

qr v ¼ v rq ¼ 0;

ð11Þ

2

The electric field is expressed by E ¼ rU. Using Eqs. (7)–(9), the MHD system (1)–(6) is reduced to the following generalized Grad-Shafranov equation [13]: 0 1 1 X2 ð1 M ÞD w ðM2 Þ jrwj2 þ 2 2 1 M2 ! 0 2 r 4 qðU0 Þ þ ¼ 0; 2 1 M2 2

!0

0 XG0 U0 þ r ps 1 M2

jrwj2 ¼ Q 2 jrUj2 ;

D w ¼ Q D U þ

dQ jrUj2 : dU

ð18Þ ð19Þ

To express the physical quantities in terms of the new variable U we introduce the following new vector:

¼ r/ ^ rU: B

ð20Þ

From Eqs. (9) and (20), the velocity field reads

v¼

1

q

Hr/ þ

dG B ; dU

ð21Þ

2

ð10Þ

Using Eqs. (13) and (17)–(19), the equilibrium physical variables v ; B; j; P can be expressed as

þ Ir/; B ¼ QB

ð22Þ

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S.M. Moawad et al. / Results in Physics 7 (2017) 3163–3175

v

G0 1 B dG dU r2 ¼ B r 2 U0 r/ ¼ r/ ; Q q dU q dU

ð23Þ

dQ dI Q D U þ jrUj2 r/ B; dU dU

j¼

ð24Þ

Table 2 The magnetic field and velocity field corresponding to the choices of Mach number dG r 2 ddUU r/ is used. described in Table 1. A new vector field, v H ¼ qB dU B

1

B pﬃﬃﬃﬃﬃﬃﬃﬃ 1e2 p3ﬃﬃ 2

2

dU E¼ rU; dU

ð25Þ

" # " 2 2 # 1 1 B2 dG dU dU 2 þr q : P ¼ ps ðUÞ þH Q 2Q q dU dU dU

ð26Þ

Exact solution classes In this subsection we consider some nonlinear cases of Eq. (16). We introduce a new variable depends on r and z to obtain several classes of exact solutions to Eq. (16). Consequently, we obtain the associated physical quantities to the full MHD system (1)–(6). Using the representations:

!

¼ f ðUÞ;

dps ¼ gðUÞ; dU

" 2 # 1 d dU ¼ hðUÞ; q 2 dU dU

ð27Þ

1

@ 2 U 1 @U @ 2 U þ 2 ¼ f ðUÞ r 2 gðUÞ r4 hðUÞ: @r 2 r @r @z

ð28Þ

Consider now the following new variable:

n ¼ ar2 þ vðzÞ;

ð29Þ

3 4

1 þ I r/ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B 2

5 6

þ Ir/ secðUÞB eU þ I r/ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B

1

ð23UÞ3 p3 ﬃﬃ 2

þ Ir/

cosðUÞv H pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2U e

e2U 1

v3 ¼

vH

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 U 2 Þv H qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 ð1 UÞ2 v H

eU

1

vH

rﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ k coshðc3 azÞ; a

ð33Þ

where c1 , c2 , and c3 are integration constants. Inserting Eqs. (29) and (30) into Eq. (28) yields

d2 U dU 1 þ f ðUÞ þ ðn vÞgðUÞ 4aðn vÞ þ av2 þ k þ av dn a dn2 1 þ 2 ðn vÞ2 hðUÞ a ¼ 0:

ð34Þ

Equating the coefficients of like powers of v in both sides of Eq. (34) we get

v0 : ð4an þ kÞ v1 : 4a v2 :

dn

2

dn2

2

d U dn2

2

d U

2

d U

¼

þa

1 1 þ f ðUÞ þ ngðUÞ þ 2 n2 hðUÞ ¼ 0; a a

ð35Þ

dU 1 2 gðUÞ 2 nhðUÞ ¼ 0; dn a a

ð36Þ

1 hðUÞ: a3

ð37Þ

Substitution of Eq. (37) into Eq. (36) yields

with

2 dv ¼ av2 þ k; dz

ð30Þ

where a and k are real constant, that has the following general solutions: if k > 0 and a > 0,

rﬃﬃﬃ pﬃﬃﬃ k sinhðc1 azÞ; ¼ a

ð31Þ

if k > 0 and a < 0,

v2

þ I r/ B

ð23UÞ3 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃB ð1U 2 Þ

in Eq. (16), it is re-written as:

v1

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 e2 v H

þ Ir/

1ð1UÞ

With aid of Eq. (15) we can obtain exact solutions to the original Eq. (10). Table 1 shows some solutions uðr; zÞ corresponding to solutions wðr; zÞ in accordance with choices for the Mach number. The physical quantities v ; B; j and P can be obtained using Tables 2 and 3.

1 d X2 2 dU 1 M2

v

No.

rﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ k ¼ sinðc2 azÞ; a

ð32Þ

gðUÞ ¼

1 dU ð4 2nÞhðUÞ þ a2 : a dn

ð38Þ

Substitution of Eqs. (38) and (37) into Eq. (35) yields

f ðUÞ ¼

1 2 4a2 4 k dU hðUÞ an n þ n þ : a2 a2 a dn

ð39Þ

To seek solutions satisfy Eqs. (35)–(37), first we solve Eq. (37) by putting the solution as UðnÞ ¼ a þ bðFðnÞÞc where a and b are real constants to be determined later. The function FðnÞ expresses a trigonometric or hyperbolic function, and c is a real constant. The constant c can be determined according to the choice of the function FðnÞ. We consider

hðUÞ ¼ A þ B UðnÞ þ C ðUðnÞ aÞn ;

if k < 0 and a > 0,

ð40Þ

Table 1 Some choices for the Mach number and corresponding quantities w; Q and dQ =dU as functions of U. No.

M

wðUÞ

Q ðUÞ

dQ =dU

1

e ¼ const

1 pﬃﬃﬃﬃﬃﬃﬃﬃ U 1e2

1 pﬃﬃﬃﬃﬃﬃﬃﬃ 1e2

0 p ﬃﬃﬃ 3 2ð2 3UÞ4=3

2

pﬃﬃﬃﬃ w

3

sinðwÞ

sin1 ðUÞ

cosðwÞ

1

4

1 ð1 32 UÞ cos

ð1 UÞ

1

5

tanhðwÞ

sinh

6

sechðwÞ

cosh

2=3

½tanðUÞ

1

ðeU Þ

p ﬃﬃﬃ 3 2ð2 3UÞ1=3

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2

3=2

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1ð1UÞ2

Uð1 U 2 Þ 3=2 ðU 1Þ 2U U 2

secðUÞ

secðUÞ tanðUÞ

e pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e2U 1

ð1U Þ

U

eU 3=2 ð1þe2U Þ

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S.M. Moawad et al. / Results in Physics 7 (2017) 3163–3175 Table 3 The gas pressure and axial electric current corresponding to the choices of Mach number described in Table 1. No.

P

rj/ q

1

h

dG 2

U 2 r 2 ddU

i

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ U H 1 e2 ddU

2 3 4 5 6

B2

where A , B , C and n are real constants, and FðnÞ ¼ sin n; 2 . Hence we obtain four classes cos n; sinh n; cosh n. In this case c ¼ 1n of exact solutions to Eq. (37) as shown in Tables 4 and 5. For these solutions, the constants A , B and C are determined in Table 5. Using Eq. (39), the function gðUÞ is determined for each solution U i , i ¼ 1; . . . ; 4 as:

" !# 1n 2 1 1 U 1 a hðU 1 Þ 4 2 sin gðU 1 Þ ¼ a b qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þn 18a2 þ ðU 1 aÞ 2 b1n ðU 1 aÞ1n ; 1n

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ D U þ 2 ð1U Þ

UjrUj2

3=2

ð1U 2 Þ

2

Uj 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D U þ ðU1Þjr 2 2 3=2 ð2UU Þ

1ð1UÞ

secðUÞD U þ secðUÞ tanðUÞjrUj2 e pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D U e2U 1 U

eU jrUj2 3=2 ðe2U 1Þ

Table 5 The constants A , B and C appeared in Eq. (40) for each solution U i , i ¼ 1; . . . ; 4 in Table 4. U

FðnÞ

U1

sin n

U2

cos n

U3

sinh n

U4

cosh n

A ; B ; C 3 A ¼ 4aa 2 , B ¼

ð1nÞ

3 A ¼ 4aa 2 , B ¼

ð1nÞ

ð41Þ

" !# 1n 2 1 1 U 2 a hðU 2 Þ 4 2 cos gðU 2 Þ ¼ a b qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þn 18a2 ðU 1 aÞ 2 b1n ðU 1 aÞ1n ; 1n

A ¼

4aa3 , ð1nÞ2

B ¼

A ¼

4aa3 , ð1nÞ2

B ¼

4a3 , ð1nÞ2 4a3 , ð1nÞ2

3 4a 2 , ð1nÞ

4a3 , ð1nÞ2

C ¼ 2a C ¼

ð1þnÞb1n ð1nÞ2

2a3 ð1þnÞb1n ð1nÞ2

C ¼ 2a C ¼ 2a

3

3

3 ð1þnÞb1n

ð1nÞ2

ð1þnÞb1n ð1nÞ2

2

3 !2 1n 1n 2 2 1 4a2 4 k5 1 U 2 a 1 U 2 a 4 f ðU 2 Þ ¼ 2 cos þ cos þ hðU 2 Þ a a2 a b b

ð42Þ

" !# 1n 2 1 1 U 3 a hðU 3 Þ 4 2 sinh gðU 3 Þ ¼ a b qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þn 18a2 ðU 3 aÞ 2 b1n þ ðU 3 aÞ1n ; 1n

þ

1n qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þn 2a U2 a 2 1n cos1 ðU 2 aÞ 2 b1n ðU 2 aÞ ; 1n b ð46Þ

2

ð43Þ

" !# 1n 2 1 1 U 4 a hðU 4 Þ gðU 4 Þ ¼ 4 2 cosh a b qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þn 18a2 ðU 4 aÞ 2 ðU 4 aÞ1n b1n : 1n

3

!2 1n 1n 2 2 1 4a2 4 k 1 U 3 a 1 U 3 a f ðU 3 Þ ¼ 4 2 sinh þ sinh þ 5hðU 3 Þ b b a a2 a þ

1n qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1þn 2a 1 U 3 a ðU 3 aÞ 2 b1n ðU 3 aÞ1n ; sinh 1n b ð47Þ

ð44Þ

The function f ðUÞ is determined from Eq. (39) for each solution U i , i ¼ 1; . . . ; 4 of Table 4, respectively, as: 2 3 !2 1n 1n 2 2 1 4a2 4 k5 1 U 1 a 1 U 1 a 4 f ðU 1 Þ ¼ 2 sin þ sin þ hðU 1 Þ a a2 a b b

1 pﬃﬃﬃﬃﬃﬃﬃﬃ D U 1e2 p3ﬃﬃ p3ﬃﬃ 2 2D U þ 2jrUj4=3 ð23UÞ1=3 ð23UÞ

ps 2 ð1 e Þ q2 dU þ h 2 2 2=3 1=3 i B dG 2 dU 2 dU p3ﬃﬃ p3ﬃﬃ ps q2 ð23UÞ H ð23UÞ dU q2 dU þ r dU 4 2 h 2 2 i p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ

dG U 2 H 1 U 2 dU þ r 2 ddU ps q2 ð1 U 2 Þ qB2 dU dU h 2 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ U 2 i dG U ps q2 ð2U U 2 Þ qB2 dU þ r 2 ddU H 2U U 2 ddU h 2 2 i

2 q dG U U ps 2 cos2 ðUÞ qB2 dU þ r 2 ddU H cosðUÞ ddU h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i q e2U 1 B2 dG 2 2 dU 2 e2U 1 dU ps 2 e2U H eU dU q2 dU þ r dU 2

1n qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1þn 2a 1 U 1 a 1n sin ðU 1 aÞ 2 b1n ðU 1 aÞ ; 1n b

2 f ðU 4 Þ ¼ 4

3

!2 1n 1n 2 2 1 4a2 4 k 1 U 4 a 1 U 4 a cosh þ cosh þ 5hðU 4 Þ a2 a2 a b b

þ

1n qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1þn 2a 1 U 4 a 1n ðU 4 aÞ 2 ðU 4 aÞ b1n ; cosh 1n b ð48Þ

One can use solutions Uðr; zÞ in Tables 4 and 5 to find the physical variables of the original MHD equations. ð45Þ

Parallel resistive MHD flows The problem formulation of equilibrium equations

Table 4 Solutions for Eq. (37) where hðUÞ ¼ A þ B UðnÞ þ C ðUðnÞ aÞn . The vector field pﬃﬃ pﬃﬃ pﬃﬃﬃ is used when n ¼ ar 2 þ v1 ðzÞ, BH ¼ rk BH ¼ rk coshðc1 azÞer 2aez , pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ cosðc2 azÞer 2aez is used when n ¼ ar 2 þ v2 ðzÞ and BH ¼ k sinh r pﬃﬃﬃ ðc3 azÞer 2aez is used when n ¼ ar2 þ v3 ðzÞ. B

FðnÞ

UðnÞ

sin n

U 1 ¼ a þ bðsin nÞ1n

cos n

1þn 2b 1n 1n cos n sin

U 2 ¼ a þ bðcos nÞ

2b 1n sin n cos1n nBH

sinh n

U 3 ¼ a þ bðsinh nÞ1n

cosh n

2 1n

2

2 1n 2

U 4 ¼ a þ bðcosh nÞ

nBH

1þn

1þn 2b 1n nBH 1n cosh n sinh 1þn H 2b 1n sinh n cosh nB 1n

ps R gðU 1 ÞdU 1 R gðU 2 ÞdU 2 R gðU 3 ÞdU 3 R gðU 4 ÞdU 4

The MHD equilibrium states of a plasma with scalar conductivity are governed by Eqs. (1)–(5) and Ohm’s law for resistive MHD:

Eþv^B¼

j

r

;

ð49Þ

where r denotes conductivity. We consider, here, MHD flows under the following field-aligned condition on the fields v and B:

v¼

K

q

B;

ð50Þ

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S.M. Moawad et al. / Results in Physics 7 (2017) 3163–3175

where K is a function of r and z. Also, the flow density q is a variable function depends upon r and z. Using Eqs. (1) and (5), the divergence of Eq. (50) implies K ¼ KðwÞ and q ¼ qðwÞ. The toroidal component, the component along B and the component perpendicular to magnetic surface for both of the momentum Eq. (1) and Ohm’s law (49) yield the following equations [48]: 0 1 1 X2 ð1 M ÞD w ðM 2 Þ jrwj2 þ 2 2 1 M2

2

P ¼ ps ðwÞ

qv 2 2

¼ ps

!0 þ r2 p0s ¼ 0;

K 2 B2 ; 2q

D w ; Vc !

rðr; wÞ ¼ 1

K

2

q

ð51Þ

ð52Þ ð53Þ

I ¼ XðwÞ;

ð54Þ

To obtain the solution wðr; zÞ of the original Eq. (51), the Mach number should be chosen as a function of w using Table 1. The corresponding magnetic field, conductivity, axial electric current density, gas pressure and electric field are described in Tables 6 and 7. Nonlinear axisymmetric equilibria for parallel resistive MHD flows In this section we consider some nonlinear cases of the equilibrium equations given in Section ‘‘Parallel resistive MHD flows”. We obtain the magnetic flux and the other physical quantities of the full MHD system (1)–(5) and Eq. (49). Case 1: Put the choices:

1 d X2 2 dU 1 M2

!

¼ k1 Uðr; zÞn ;

dps ¼ k2 Uðr; zÞm ; dU

ð61Þ

in Eq. (56), we get the nonlinear partial differential equation

@ 2 U 1 @U @ 2 U þ 2 ¼ k1 U n r 2 k2 U m ; @r 2 r @r @z

where V c is the constant toroidal loop voltage divided by 2p and the electric field is expressed as E ¼ rU þ V c r/. XðwÞ is a function defined later (see Eq. (55) below). Previously, special cases of ideal incompressible MHD equilibria were investigated by Avinash et al. [51] and Andruschenko et al. [52]. The Mach number M can be represented by the functions K and q as follows:

where k1 , k2 , n and m are real constants. We seek solutions to Eq. (62) by considering the following constraint assumption:

K M ¼ pﬃﬃﬃﬃ :

where a is real constant and g is a function of z. Using expression (63) into Eq. (62), we get

ð55Þ

q

From the analysis above, we have four surface quantities (K; q; X and ps ) are provided for the above MHD flows. The surface quantity XðwÞ is identified from the toroidal component of the momentum conservation Eq. (2). From Eq. (54) we find that the function I is a surface quantity, I ¼ IðwÞ. Also, from Eqs. (54) and (55) we find that the Mach number is a surface quantity, M ¼ MðwÞ. Therefor, we have six surface quantities, KðwÞ; qðwÞ; XðwÞ; ps ðwÞ; IðwÞ and MðwÞ, for the MHD flows considered here. Biased on the definition pﬃﬃﬃﬃ of the Mach number ðM ¼ K= qÞ and Eq. (54), four of these six quantities remain arbitrary in w. The solution w can be determined from Eq. (51), that can be solved analytically. Using the Eq. (15), Eq. (51) is reduced to

1 d X2 D Uþ 2 dU 1 M 2

!

þ r2

dps ¼ 0: dU

ð56Þ

Eq. (51) with the relation of pressure in Eq. (52) are BernoulliGrad-Shafranov system. Several classes of exact solutions to Eq. (51) can be derived for some choices of M as we show later. Moreover, we show how to obtain the associated physical quantities to the full MHD system (1)–(5) and Eq. (49).

ð62Þ

n ¼ 9ar 2 þ gðzÞ;

ð63Þ

"

2 # 2 2 dg d U d g dU 36aðn gÞ þ þ dz dn2 dz2 dn ¼ k1 U n

k2 ðn gÞU m : 9a

ð64Þ

Now, we seek solutions to Eq. (64) such that the function gðzÞ satisfies the following equation:

2 dg ¼ 4k1 g þ b; dz

ð65Þ

that has the general solution

gðzÞ ¼

i 1 h 2 4k1 ðz þ cÞ2 b ; 4k1

ð66Þ

where k1 and b are real constants, and c is the integration constant. Inserting Eq. (65) into Eq. (64) yields

½36aðn gÞ þ 4k1 g þ b

2

d U 2

dn

þ 2k1

dU k2 ¼ k1 U n ðn gÞU m : dn 9a ð67Þ

The equilibrium physical variables Using Eqs. (17)–(20), the conductivity, electric current density field, gas pressure and electric field can be expressed as

rj 1 dQ r¼ /¼ Q D U þ jrUj2 ; dU Vc Vc j ¼ V c rr/

dI B; dU

K 2 ðUÞB2 K2 P ¼ ps ðUÞ ¼ ps 2 Q 2 jrUj2 þ I2 ; qðUÞ r q E¼

dI B þ V c r/: r dU

ð57Þ

ð58Þ

Table 6 The magnetic field, conductivity and axial electric current density corresponding to the choices of Mach number described in Table 1. No.

B

V c r ¼ rj/

1

1 þ I r/ pﬃﬃﬃﬃﬃﬃﬃﬃ B 1e2 p3ﬃﬃ 2 þ Ir/ B ð23UÞ1=3

1 pﬃﬃﬃﬃﬃﬃﬃﬃ D U 1e2 p3ﬃﬃ p3ﬃﬃ 2 2D U þ 2jrUj4=3 ð23UÞ ð23UÞ1=3

2 3

ð59Þ

ð60Þ

1 þ Ir/ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ B 1U 2

4

1 þ Ir/ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃB

5

secðUÞ þ Ir/ B eU þ I r/ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B

6

2UU 2

e2U 1

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ D U þ 2 ð1U Þ

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2UU 2

D Uþ

UjrUj2 ð1U 2 Þ

3=2

ðU1ÞjrUj2 3=2

ð2UU 2 Þ

secðUÞD U þ secðUÞ tanðUÞjrUj2 e pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D U e2U 1 U

eU jrUj2 3=2 ðe2U 1Þ

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Table 7 The gas pressure and electric field corresponding to the choices of Mach number described in Table 1. No.

P

1

ps

2

ps

3

h

2

h

ps

i

2 k ðUÞ B I2 2qðUÞ 1e2 þ r 2 h i p ﬃﬃ 2 3 2 k ðUÞ 4B I2 2qðUÞ ð23UÞ2=3 þ r2

ps 2kqðUÞ ðUÞ

4

2

2

þ rI 2

B 1U 2

h

2 2 k ðUÞ B 2qðUÞ 2UU 2

r/ þ

2

6

pﬃﬃﬃﬃﬃﬃﬃﬃ 1e2 BdI=dU D U 4=3

ð23UÞ BdI=dU pﬃﬃ r/ þ p3ﬃﬃ2ð23UÞ D Uþ 32jrUj2

i

2 3=2

Bð1U Þ dI=dU r/ þ ð1U 2 ÞD UþUjrUj2

i

þ rI 2 h i 2 2 k ðUÞ 2 ps 2qðUÞ B sec2 ðUÞ þ rI 2 h 2U 2 i 2 e B I2 ps 2kqðUÞ ðUÞ e2U 1 þ r2

5

ps ¼ ps1 ¼ A1 þ

VcE 2

2 3=2

ð2UU Þ BdI=dU r/ þ ð2UU 2 ÞD UþðU1ÞjrUj2

cosðUÞ r/ þ DBðdI=dUÞ UþtanðUÞjrUj2

ðe2U 1Þ

3=2

BdI=dU

r/ þ eU ðe2U 1ÞD UeU jrUj2

Equating the coefficients of like powers of g in both sides of Eq. (67) we get

g0 : ð36an þ bÞ

2

d U

þ 2k1

dn2

dU k2 ¼ k1 U n nU m ; dn 9a

ð68Þ

2

d U

k g : ð36a þ 4k1 Þ 2 ¼ 2 U m : 9a dn 1

ð69Þ

To find solution satisfies both of Eqs. (68) and (69), we substitute Eq. (69) into Eq. (68) which yields 2

ðb þ 4k1 nÞ

d U dn2

þ 2k1

dU þ k1 U n ¼ 0: dn

Using the transformation

Integrating the second part of Eq. (61) with respect to u, the static pressure is obtained as

ð70Þ

2

" # 1 32 2 1n 3 3 ð1 nÞ 2 2 5: w1 ðr; zÞ ¼ 1 41 9ar þ k1 ðz þ cÞ 2 2ð1 þ nÞ

KðU 1 Þ B qðU 1 Þ " p ﬃﬃﬃ # 3 KðU 1 Þ k1 2ðn 1ÞU n1 ; 9ae ¼ I r/ þ ðc þ zÞe r z 1 qðU 1 Þ ðn þ 1Þð2 3U 1 Þ3 r

v¼

Substituting this solution into Eq. (71) we find that the constants N and a must be related to the constants n and k1 by the following relations:

1 ; 1n

a¼

9að1 nÞ2 ðk1 9aÞ

2ð1 þ nÞ2 " pﬃﬃﬃ 3 2 K 2 ðU 1 Þ 4B

2

ð1 nÞ 8k1 ð1 þ nÞ

1 #1n

:

ð73Þ

# I2 ; þ 2qðU 1 Þ ð2 3U 1 Þ23 r 2

ð81Þ

" pﬃﬃﬃ # p ﬃﬃﬃ 3 3 2D U 1 2jrU 1 j2 1 ; r¼ þ V c ð2 3U 1 Þ1=3 ð2 3U 1 Þ4=3

ð82Þ

j/ ¼

" pﬃﬃﬃ # p ﬃﬃﬃ 3 3 1 2 D U 1 2jrU 1 j2 ; þ r ð2 3U 1 Þ1=3 ð2 3U 1 Þ4=3

D U 1 ¼ k1 U n1

m ¼ 2n 1;

jrU 1 j2 ¼

9anð1 nÞ2 ðk1 9aÞ ð1 þ nÞ2

:

ð75Þ

Using the transformation v ¼ b þ 4k1 n, Eq. (63) and Eq. (66) into Eq. (72), we obtain the following exact solution class for Eq. (72):

" U1 ¼

9anð1 nÞ2 ðk1 9aÞr2 ð1 þ nÞ2

U 2n1 ; 1

2 1n 2 2 2 2 U 2n 1 ½81a r þ k1 ðc þ zÞ : nþ1

ð84Þ

ð85Þ

Case 2: Consider the following choices

and

k2 ¼

ð83Þ

where

To satisfy both of Eqs. (68) and (69), we substitute the solution (72) into Eq. (69). Therefor we have the following relations between the constants n; m; a; k1 and k2 :

ð74Þ

ð80Þ

U 2n 1

1

ð72Þ

ð79Þ

pﬃﬃﬃﬃ Using Eq. (50) and the choice M ¼ w, the velocity and magnetic fields, gas pressure, conductivity and axial current density are obtained as

ð71Þ

U ¼ U 1 ¼ avN :

N¼

ð78Þ

dom of the functional dependence of M 2 on w to obtain exact solutions wðr; zÞ to Eq. (51). This is shown in Table 1. If we choose pﬃﬃﬃﬃ M ¼ w as shown in Table 1, then w ¼ 1 ½1 ð3=2ÞU2=3 . Hence, we obtain exact solution to Eq. (51) as follows:

Eq. (71) has a solution of the form

"

U 2n 1 ;

one takes n ¼ 2; 3 and m ¼ 3; 5 for [41], X 2 =ð1 M 2 Þ ¼ const and m ¼ 1 for [48], n ¼ 3 and m ¼ 7 for [53,54]. We can use the free-

P ¼ A1 þ

d U dU 1 n 2v 2 þ þ U ¼ 0: dv dv 8k1

2ð1 þ nÞ2

where A1 is the integration constant. Some solutions of the equilibrium equations of resistive MHD flows were obtained in [18,48,53,54]. Such solutions can be obtained again as special cases of the solution class in Eq. (76) if

v ¼ b þ 4k1 n, Eq. (70) becomes

2

9að1 nÞ2 ðk1 9aÞ

1 #1n ð1 nÞ2 : ½9ar 2 þ k1 ðz þ cÞ2 2ð1 þ nÞ

ð76Þ

!

¼ k1 f ðUÞ;

dps ¼ k2 f ðUÞ: dU

ð77Þ

ð86Þ

For these choices, we seek exact solutions to Eq. (56) using two transformations as we explain in the following two subcases. Case 2.1: Insert Eq. (86) into Eq. (56), we get

@ 2 U 1 @U @ 2 U þ 2 ¼ k1 f ðUÞ k2 r 2 f ðUÞ; @r 2 r @r @z

is given by For this solution, the quantity B

1 ¼ n 1 U n k1 ðc þ zÞ er 9aez ; B nþ1 1 r

1 d X2 2 dU 1 M 2

ð87Þ

where f ðUÞ is a function of U. Substituting Eqs. (63) and (65) into Eq. (87), we obtain

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S.M. Moawad et al. / Results in Physics 7 (2017) 3163–3175

½36aðn gÞ þ 4k1 g þ b

2

d U dn2

þ 2k1

dU k2 þ k1 f ðUÞ þ ðn gÞf ðUÞ ¼ 0: dn 9a ð88Þ

Equating the coefficients of like powers of g in both sides of Eq. (88), we get 2

d U

g0 : ð36an þ bÞ

dn2

þ 2k1

dU k2 þ k1 f ðUÞ þ nf ðUÞ ¼ 0; dn 9a

ð89Þ

" pﬃﬃﬃ # p ﬃﬃﬃ 3 3 1 2D U 2 2jrU 2 j2 ; r¼ þ V c ð2 3U 2 Þ1=3 ð2 3U 2 Þ4=3

ð101Þ

" pﬃﬃﬃ # p ﬃﬃﬃ 3 3 2 D U 2 2jrU 2 j2 1 j/ ¼ ; þ r ð2 3U 2 Þ1=3 ð2 3U 2 Þ4=3

ð102Þ

where

D U 2 ¼

2

d U

k g : ð4k1 36aÞ 2 2 f ðUÞ ¼ 0: 9a dn 1

ð90Þ

2

9að4k1 36aÞ d U : k2 dn2

ð91Þ

Substituting Eq. (91) into Eq. (89), we get 2

ð4k1 n þ kÞ

d U dn2

þ 2k1

dU ¼ 0; dn

ð92Þ

where k ¼ b þ 9ak1 ð4k1 36aÞ=k2 . Integration of Eq. (92) yields

ð93Þ

where c1 and c2 are the constants of integration. Therefore after using Eq. (63), we obtain the following exact solution class for Eq. (87).

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ U 2 ðr; zÞ ¼ c1 4k1 ½9ar2 þ k1 ðz þ cÞ2 þ k b þ c2 :

ð94Þ

From Eq. (91) we get,

k2 ðU 2 c2 Þ

3

:

ð95Þ

ð96Þ

Integrating the second part of Eq. (86) with respect to U, the static pressure is obtained as 2

ðU 2 c2 Þ2

ð104Þ

ð105Þ

@2U @2U k2 þ 4 a x þ k þ x f ðUÞ ¼ 0: 1 @z2 @x2 a

;

ð97Þ

where A2 is the integration constant. As mentioned in the text after Eq. (78) we can use the freedom of the functional dependence of M 2 on w to obtain exact solutions pﬃﬃﬃﬃ wðr; zÞ to Eq. (51). If the Mach number M is chosen as M ¼ w then from Table 1 the solution wðUÞ is given by w ¼ 1 ½1 ð3=2ÞU2=3 . Hence we obtain an exact solution class to Eq. (51) as:

2 1 3 3 w2 ðr; zÞ ¼ 1 1 c1 ½4k1 ð9ar 2 þ k1 ðz þ cÞ2 Þ þ k b2 : 2

ð98Þ

Therefore the velocity field, gas pressure, conductivity and axial current density are obtained, respectively, as

KðU 2 Þ B qðU 2 Þ " !# pﬃﬃﬃ 2 2 KðU 2 Þ 16 3 2c41 k1 k1 ðc þ zÞ2 2 ; 81a þ ¼ Ir/ þ 1 qðU 2 Þ r2 ðU 2 c2 Þ2 ð2 3U 2 Þ3

v¼

ð99Þ " pﬃﬃﬃ # 2 2 3 2 72að9a k1 Þc41 k1 k ðU 2 Þ I2 4B 2 ; þ P ¼ A2 2qðU 2 Þ ð2 3U 2 Þ23 r 2 k2 ðU 2 c2 Þ2

ð106Þ

Equating the coefficients of like powers of x in both sides of Eq. (106), we get

x0 :

@2U þ k1 f ðUÞ ¼ 0; @z2

ð107Þ

x1 :

@2U k2 þ f ðUÞ ¼ 0: @x2 4a2

ð108Þ

From Eqs. (107) and (108), we have

2 k1 ðc þ zÞ 2 ¼ 4k1 c1 B er 9aez : r ðU 2 c2 Þ

ps ¼ ps2 ¼ A2

2

½81a2 r 2 þ k1 ðc þ zÞ2 :

x ¼ ar 2 ;

is given by For solution (95), the quantity B

72að9a k1 Þk1 c41

ðU 2 c2 Þ2

Case 2.2: In this subcase we consider the same choices shown in Eq. (86) but we use a different method to find exact solutions to Eq. (94). We present a new independent variable x as a quadratic function of r only where the stream function n is considered as a function of x and z. For this consideration we put

2

144ak1 c41 ð9a k1 Þ

16k1 c41

where a is a real number. Inserting expression (105) into Eq. (86), we obtain

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ U ¼ U 2 ¼ c1 4k1 n þ k þ c2 ;

f ðUÞ ¼ f ðU 2 Þ ¼

ð103Þ

2

jrU 2 j2 ¼

From Eq. (89), we have

f ðUÞ ¼

2

144að9a k1 Þk1 c41 ðk1 þ k2 r 2 ÞðU 2 c2 Þ3 ; k2

ð100Þ

2

@ U @2U ¼ l2 2 ; 2 @x @z where

ð109Þ

l2 ¼ k2 =ð4a2 k1 Þ. The general solution of Eq. (109) is

zÞ ¼ u ðz þ lxÞ þ u ðz lxÞ; Uðx; zÞ ¼ Uðx; 1 2

ð110Þ

where u1 and u2 are arbitrary functions. If f ðUÞ is a linear function of U, several exact solutions can be derived from the general solution in Eq. (110) when the functions u1 and u2 are chosen. In Table 8 are considered when several nonlinear cases for the function f ðUÞ u2 ¼ 0. The function f ðUÞ in Table 8 is determined using Eqs. (107) and (110) which imply 2 2 2 ¼ 1 @ U ¼ 1 @ U ¼ 1 @ u1 f ðUÞ ¼ f ðUÞ k1 @z2 k1 @z2 l @x2

¼

1 @ 2 u1 : l @x2

ð111Þ

and static pressure for solutions U 1 U 14 , which The quantity B are shown in Table 8, are obtained in Table 9. One can use [Tables zÞ in 1, 6, 7] to obtain the solution wðr; zÞ for each solution Uðr; Table 8 and the corresponding other physical variables of the original MHD Eqs. (1)–(5). A solution presented in [55,56] can be 4 of 3 and U obtained as a special case of the solution classes U Table 8 if one takes n ¼ 1. Case 3: Insert the following choices:

1 d X2 2 dU 1 M2

!

¼ f ðUÞ;

dps ¼ gðUÞ; dU

ð112Þ

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S.M. Moawad et al. / Results in Physics 7 (2017) 3163–3175

in Eq. (56), we get 2

2

@ U 1 @U @ U þ 2 ¼ f ðUÞ r 2 gðUÞ: @r 2 r @r @z Using Eq. (18) and

2 dg dz

ð113Þ

¼ 4ag þ b, Eq. (112) becomes

2

d U

dU 1 ¼ f ðUÞ ðn gÞgðUÞ: ½36aðn gÞ þ 4ag þ b 2 þ 2a dn 9a dn ð114Þ Equating the coefficients of like powers of g in both sides of Eq. (114) we get

g0 : ð36an þ bÞ

2

d U dn2

þ 2a

dU 1 ¼ f ðUÞ ngðUÞ; dn 9a

1 g : 2 ¼ gðUÞ: 288a2 dn

ð118Þ

where a and b are real constants to be determined later. The function FðnÞ expresses a trigonometric or hyperbolic function, and c is a real constant. The constant c can be determined according to the choice of the function FðnÞ. It can be any real number according to the form of the function gðuÞ and the choice of the function FðnÞ as shown in Table 10. For applying the solution method presented in Eq. (118) and Table 10, we consider the following examples.

gðUÞ ¼

n¼8 X

eighth

degree

an U n1 :

and

ð119Þ

n¼1

Table 8 Exact solutions of Eq. (106) for several nonlinear forms of the function f ðUÞ. The parameters used are l2 ¼ k2 =ð4a2 k1 Þ and n is any real number. No.

zÞ ¼ u ðx; zÞ Uðr; 1

k1 f ðUÞ

1

1 ¼ ðlar 2 þ zÞn U

n nðn 1ÞU 1

2

2 jÞ nðn 1Þðln jU

3

2 ¼ eðlar þzÞ U 3 ¼ sinn ðlar 2 þ zÞ U

4

4 ¼ cosn ðlar 2 þ zÞ U

5

5 ¼ tann ðlar 2 þ zÞ U

6

6 ¼ secn ðlar 2 þ zÞ U

2

n

7

7 ¼ cscn ðlar 2 þ zÞ U

8

8 ¼ cotn ðlar 2 þ zÞ U

9

9 ¼ sinh ðlar 2 þ zÞ U

10

10 ¼ coshn ðlar 2 þ zÞ U

11

11 ¼ tanhn ðlar 2 þ zÞ U

12

12 ¼ sechn ðlar 2 þ zÞ U

13

13 cschn ðlar 2 þ zÞ U

14

14 ¼ cothn ðlar 2 þ zÞ U

n

n2

n2 n

2 jÞ þ n2 ðln jU

2ðn1Þ n

2 U

3 n2 U nðn 1ÞU 3 n2 n

4 n n2 U nðn 1ÞU 4 n2

nþ2

5 n þ 2n2 U n þ nðn 1ÞU nðn þ 1ÞU 5 5 n2

6 n2 U nðn þ 1ÞU 6

6

;

a4 ¼

6

;

a6 ¼

b 4480a2 a3 b 128a2

32a2 ð28a6 b6 Þ b6

4480a2 a4 b6 2688a2 a2 6

b

;

; ;

a7 ¼

896a2 a b6

;

8 þ nðn 1ÞU n n þ 2n2 U nðn þ 1ÞU 8 8

n2

9 þ n2 U nðn 1ÞU 9

32a2 ð4a7 þ ab6 Þ 6

b

11 þ nðn 1ÞU n n 2n2 U nðn þ 1ÞU 11 11 nþ2

14 þ nðn 1ÞU n n 2n2 U nðn þ 1ÞU 14 14

n2

:

ð121Þ

ð122Þ

ð123Þ

ð124Þ

ð125Þ

1152aa2 ð1 nÞ

2

;

B¼

1152a2 2

ð1 nÞ

;

C¼

;

C¼

;

C¼

576a2 ð1 þ nÞb1n ð1 nÞ2

:

ð126Þ

:

ð127Þ

For FðnÞ ¼ sinh n:

1152aa2 ð1 nÞ

2

;

B¼

1152a2 2

ð1 nÞ

576a2 ð1 þ nÞb1n ð1 nÞ2

For FðnÞ ¼ cosh n:

nþ2

13 n þ n2 U nðn þ 1ÞU 13

b6

Example 2. gðUÞ ¼ A þ BUðnÞ þ CðUðnÞ aÞn where A; B; C, are real constants, n is not positive integer and FðnÞ ¼ sin n; cos n; sinh n; cosh n. 2 . Hence we obtain four In this situation, from Table 10, c ¼ 1n classes of exact solutions to Eq. (116) as shown in Table 12. For these solutions, the constants A, B and C are determined as follows. For FðnÞ ¼ sin n; cos n:

n2

12 n þ n2 U nðn þ 1ÞU 12

32a2 ð28a6 þ b6 Þ

; a2 ¼

" # 3 n¼8 X 1 1 U 4 a f ðU 4 Þ ¼ 4a cosh þb an U n1 4 288a2 b n¼1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2a þ 3 ðU 4 aÞ b6 ðU 4 aÞ6 : 3b

A¼

n2

ð120Þ

" # 3 n¼8 X 1 1 U 3 a 4a sinh þ b an U n1 f ðU 3 Þ ¼ 3 288a2 b n¼1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2a þ 3 ðU 3 aÞ ðU 3 aÞ6 þ b6 ; 3b

n2 n

10 n þ n2 U nðn 1ÞU 10

:

b6

" # 3 n¼8 X 1 1 U 1 a f ðU 1 Þ ¼ 4a sin þb an U n1 1 2 288a b n¼1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2a þ 3 ðU 1 aÞ ðU 1 aÞ6 b6 ; 3b " # 3 n¼8 X 1 1 U 2 a 4a cos þ b an U n1 f ðU 2 Þ ¼ 2 2 288a b n¼1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2a 3 ðU 2 aÞ ðU 2 aÞ6 b6 ; 3b

nþ2

nþ2

a2 ¼

The function f ðUÞ is determined from Eq. (117) for each solution U i ; i ¼ 1; . . . ; 4 of Table 11, respectively, as

7 n n2 U nðn þ 1ÞU 7

nþ2

2688a2 a5

A¼

nþ2 n

nþ2

b

;

ð116Þ

To seek solutions satisfy both of Eqs. (115) and (116), we apply a solution method that constructed in [31]. For such method we put

UðnÞ ¼ a þ bðFðnÞÞc ;

6

For FðnÞ ¼ sinh n; cosh n, the constants ai ; i ¼ 3; . . . ; 8 are the same as described in Eq. (120) while a1 and a2 are determined as:

ð117Þ

of

a5 ¼

a1 ¼

1 dU ð4an þ bÞgðUÞ 2a : 288a2 dn

Example 1. gðUÞ is a polynomial FðnÞ ¼ sin n; cos n; sinh n; cosh n. Consider the polynomial

a3 ¼

32a2 ð4a7 ab6 Þ

a8 ¼

Substitution of Eq. (116) into Eq. (115) yields

f ðUÞ ¼

a1 ¼

ð115Þ

2

d U

1

In this situation from Table 10 we get c ¼ 1=3. Using Eqs. (118) and (119) into Eq. (116) we obtain four classes of exact solutions to Eq. (116) (see Table 11). For these solutions the coefficients ai ; i ¼ 1; . . . ; 8 appeared in Eq. (119) are determined as follows. For FðnÞ ¼ sin n; cos n:

A¼

1152aa2 ð1 nÞ

2

;

B¼

1152a2 2

ð1 nÞ

576a2 ð1 þ nÞb1n ð1 nÞ2

:

ð128Þ

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S.M. Moawad et al. / Results in Physics 7 (2017) 3163–3175 Table 9 14 shown in Table 8. The symbols v , and static pressure for solutions U 1 U The quantity B i i ¼ 1; . . . ; 14 denote integration constants and the vector quantity B is defined as B ¼ er 2larez . No.

B

1

n r

2

n r

3

ps

lar þ z 2

n1

B

n1 ðlar2 þzÞn e B

n1 n 2 sin l a r þ z cos l ar2 þ z B r

lar2 þ z

4

nr cosn1

5

n1 n r tan

6

n nþ1 r sec

7

nr cscnþ1

8

nr cotn1

9

n r

sinh

10

n r

cosh

11

n r

tanh

12

nr sech

13

nr csch

14

nr coth

lar2 þ z sec2 lar2 þ z B

lar2 þ z sin lar 2 þ z B

lar2 þ z cos lar2 þ z B

n1

lar2 þ z csc2 lar2 þ z B

lar2 þ z cosh lar 2 þ z B

n1

lar2 þ z sinh lar 2 þ z B

n1

lar2 þ z sech2 lar2 þ z B

nþ1

lar 2 þ z sinh lar2 þ z B

nþ1

FðnÞ ¼ tan n; cot n; tanh n; coth n

gðUÞ is a polynomial of degree n

gðUÞ is a polynomial of second or third degree c2 2 c ¼ 2 for n ¼ 2, c ¼ n ) c ¼ 1n, for an integer n > 1,

c ¼ 2, for n ¼ 1. c¼

2 1n,

n is not

cþ2 c ¼m

c

3 1 2

1 3 4

1 1

1 2 2 3

3 2 4 3

4

.. .

.. .

2 3

13

U 2 ¼ a þ bðcos nÞ

sinh n

f ðU 4 Þ ¼

6 .. .

Using Eq. (117), the function f ðUÞ is determined for each solution U i , i ¼ 1; . . . ; 4 of Table 12, respectively, as

" # 1n 2 1 1 U 1 a f ðU 1 Þ ¼ 4a sin þ b gðU 1 Þ 288a2 b 1n qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4ab 2 ðU 1 aÞ ðU 1 aÞn1 bn1 ; 1n " # 1n 2 1 1 U 2 a 4a cos þ b gðU 2 Þ f ðU 2 Þ ¼ 288a2 b 1n qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4ab 2 þ ðU 2 aÞ ðU 2 aÞn1 bn1 ; 1n " # 1n 2 1 1 U 3 a f ðU 3 Þ ¼ 4a sinh þ b gðU 3 Þ 288a2 b 1n qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4ab 2 ðU 3 aÞ bn1 þ ðU 3 aÞn1 ; 1n

U 1 ¼ a þ bðsin nÞ

cos n

13

U 3 ¼ a þ bðsinh nÞ

ð129Þ

b3 ¼

13

4 3

U n1 þ h1

n¼1 n

U n2 þ h2

cosh n csch nB

n¼1 n

U n3 þ h3

n¼1 n

U n4 þ h4

Pn¼8 an

Pn¼8 an

sinh nsechnB

" # 1n 2 1 1 U 4 a 4a cosh þ b gðU 4 Þ 288a2 b 1n qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4ab 2 ðU 4 aÞ bn1 ðU 4 aÞn1 : 1n

ð132Þ

576a2 ða3 þ ab2 Þ 2

b

1728a2 a 2

b

;

;

b4 ¼

b2 ¼ 576a2 b2

576a2 ð3a2 þ b2 Þ b2

:

; ð133Þ

For FðnÞ ¼ tanh n; coth n:

b1 ¼ ð131Þ

3b 3b

n¼1 n

Pn¼8 an

P n1 Example 3. gðUÞ ¼ n¼4 where bi , i ¼ 1; . . . ; 4 are real n¼1 bn U constants and FðnÞ ¼ tan n; cot n; tanh n; coth n. In this situation, from Table 10, c ¼ 1. Hence we obtain four classes of exact solutions to Eq. (116) as shown in Table 13. For these solutions, the constants bi , i ¼ 1; . . . ; 4 are determined as follows. For FðnÞ ¼ tan n; cot n:

b1 ¼

ð130Þ

U 4 ¼ a þ bðcosh nÞ

ps Pn¼8 an

4 b 3 3 cos ncsc nB 4 b 3 3 sin n sec nB

13

cosh n

a positive integer

B

UðnÞ

sin n

gðUÞ ¼ A þ BU þ CðU aÞn þ DðU aÞm c2 c ¼n

Table 11 Solutions for Eq. (116) where g(U) is a polynomial of eight degree and and the static FðnÞ ¼ sin n; cos n; sinh n; cosh n. The corresponding vector field B pressure ps are determined. A new vector field, B ¼ 2a ðz þ cÞer 18aez , is used. The r parameters hi , i ¼ 1; . . . ; 4 are integration constants and n ¼ 9ar 2 þ aðz þ cÞ2 . FðnÞ

c ¼ 1 for n ¼ 3.

gðUÞ ¼ A þ BU þ CðU aÞn

c2 c ¼n)

lar2 þ z csch2 lar2 þ z B

Table 10 Determination of c for solution (118). The parameters A, B and C are real constants. FðnÞ ¼ sin n; cos n; sinh n; cosh n

lar2 þ z cosh lar2 þ z B

n1

2ðn1Þ

2

þ v1

n2kk12 U 22 ðlnðU 2 ÞÞ n þ v2 2ðn1Þ 2 n2kk12 U 3 n U 23 þ v3 2ðn1Þ 2 n2kk12 U 4 n U 24 þ v4 2ðnþ1Þ 2ðn1Þ 2 n2kk12 U 5 n þ U 5 n þ 2U 25 þ v5 2ðnþ1Þ 2 n2kk12 U 6 n U 26 þ v6 2ðnþ1Þ 2 n2kk12 U 7 n U 27 þ v7 2ðnþ1Þ 2ðn1Þ 2 n2kk12 U 8 n þ U 8 n þ 2U 28 þ v8 2ðn1Þ 2 n2kk12 U 9 n þ U 29 þ v9 2ðn1Þ 2 n2kk12 U 10n þ U 210 þ v10 2ðnþ1Þ 2ðn1Þ 2 n2kk12 U 11n þ U 11n 2U 211 þ v11 2ðnþ1Þ 2 n2kk12 U 12n þ U 212 þ v12 2ðnþ1Þ 2 n2kk12 U 13n þ U 213 þ v13 2ðnþ1Þ 2ðn1Þ 2 n2kk12 U 14n þ U 14n 2U 214 þ v14

lar2 þ z sin lar 2 þ z B

2ðn1Þ

2

n2kk12 U 1 n

b3 ¼

576a2 ða3 ab2 Þ 2

b

1728a2 a b2

; b2 ¼

; b4 ¼

576a2 b2

:

576a2 ð3a2 b2 Þ b2

; ð134Þ

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S.M. Moawad et al. / Results in Physics 7 (2017) 3163–3175

Table 12 Solutions for Eq. (116) where gðUÞ ¼ A þ BUðnÞ þ CðUðnÞ aÞn . A; B; C are real constants, n is not a positive integer and FðnÞ ¼ sin n; cos n; sinh n; cosh n. The parameters qi , i ¼ 1; . . . ; 4 are integration constants and n ¼ 9ar2 þ aðz þ cÞ2 . FðnÞ

B

UðnÞ

sin n

U 1 ¼ a þ bðsin nÞ

cos n

U 2 ¼ a þ bðcos nÞ1n

2 1n 2

sinh n

U 3 ¼ a þ bðsinh nÞ

cosh n

2 1n

2b 1n 2b 1n

2 1n

U 4 ¼ a þ bðcosh nÞ

Using Eq. (117), the function f ðUÞ is determined for each solution U i , i ¼ 1; . . . ; 4 of Table 12, respectively, as

f ðU 1 Þ ¼

1 2a 1 U 1 a þ b gðU 1 Þ ½b2 þ ðU 1 aÞ2 ; 4a tan 288a2 b b

ð135Þ

1 4a cot1 f ðU 2 Þ ¼ 288a2

U2 a 2a þ b gðU 2 Þ þ ½b2 þ ðU 2 aÞ2 ; b b ð136Þ

f ðU 3 Þ ¼

1 2a 1 U 3 a 4a tanh þ b gðU 3 Þ ½b2 ðU 3 aÞ2 ; 288a2 b b ð137Þ

f ðU 4 Þ ¼

ps C AU 1 þ B2 U 21 þ 1þn ðU 1 aÞ1þn þ q1

1þn 2b 1n nB 1n cos n sin 1þn 2b 1n sin n cos1n nB

1 2a 1 U 4 a þ b gðU 4 Þ ½b2 ðU 4 aÞ2 : 4a coth 288a2 b b ð138Þ

One can use Tables 1, 6 and 7 to obtain the solution wðr; zÞ for each solution Uðr; zÞ in Tables 11–13 and the corresponding other physical variables of the original MHD Eqs. (1)–(5) and (49). Results and discussion It should be pointed out that Eqs. (29) and (63) constrain the solution class, in which Eqs. (29) and (63) actually define the shape of the flux surfaces, since the L.H.S of Eq. (29) or Eq. (63) is eventually a constant along the flux surface, by construction. Eq. (51) is identical in form to the corresponding ideal equilibrium equation. Also, the resistive effect (Eq. (53)) decouples from the rest of the equations which govern the ideal MHD equilibrium. Therefore, the equilibrium solutions found here are effectively the ideal MHD solutions. The only difference is that the plasma conductivity, or the toroidal loop voltage if the latter is also allowed to be a two-dimensional function of ðr; zÞ, has been specifically chosen to satisfy Eq. (53). The requirement of a constant conductivity can be include the existence of the resistive MHD equilibrium solution here by taking M ¼ const, ps ¼ const and X 2 / w. These choices imply, form Eq. (51), D w ¼ const which is equivalent to the conductivity being constant from Eq. (53). Previously, in [48] it is shown that the axisymmetric MHD equilibrium states with finite resistivity and flows parallel to the magnetic field are governed by a second-order partial differential equation for the poloidal magnetic flux function coupled with a Bernoulli type equation for the plasma density. For incompressible flows that partial differential equation becomes elliptic and decouples from the Bernoulli equation [13]. The equilibrium presented in [48] was a solution for linear type of that equation where the 2 d X s choices 12 dU ¼ const and dp ¼ const were considered. Here, dU 1M 2 we have considered several nonlinearities in the equilibrium equations for finding exact equilibria to the governing equations. These nonlinearities are polynomials, fractional functions and algebraic functions of the magnetic flux. Also, solutions were constructed

1þn 1n

cosh n sinh sinh n cosh

1þn 1n

C AU 2 þ B2 U 22 þ 1þn ðU 2 aÞ1þn þ q2

nB

C AU 3 þ B2 U 23 þ 1þn ðU 3 aÞ1þn þ q3

nB

C AU 4 þ B2 U 24 þ 1þn ðU 4 aÞ1þn þ q4

Table 13 Solutions for Eq. (116) where g(U) is a polynomial of fourth degree and FðnÞ ¼ tan n; cot n; tanh n; coth n. The parameters li , i ¼ 1; . . . ; 4 are integration constants and n ¼ 9ar 2 þ aðz þ cÞ2 . FðnÞ

Uðr; zÞ

B

tan n

U 1 ¼ a þ b tan n

b sec2 nB

cot n

U 2 ¼ a þ b cot n

bcsc nB

tanh n

U 3 ¼ a þ b tanh n

coth n

U 4 ¼ a þ b coth n

ps

2

Pn¼4 bn

bsech nB 2

b csch nB

U n1 þ l1

n¼1 n

U n2 þ l2

n¼1 n

U n3 þ l3

n¼1 n

U n4 þ l4

Pn¼4 bn

2

n¼1 n

Pn¼4 bn

Pn¼4 bn

by many authors [11,13,15,35,52] can be recovered as special cases of the solutions we presented. From the physical point of view we just describe the characteristics of one the obtained equilibria and its relevance to axisymmetric confinement systems. The toroidal equilibria depends crucial on the existence of magnetic surfaces. It is known that such surfaces in general do not exist in three dimensional toroidal geometry without any symmetry. As a matter of fact to construct tokamak or reversed-field pinch equilibria, closed toroidal magnetic surfaces can be well defined for these confinement systems because of axial symmetry. Confinement of plasma requires magnetic field with poloidal and toroidal components. Taking also into account the fact that the poloidal flow in the edge region of these systems plays a role in the transition from the low-confinement mode to the high-confinement mode, in the present paper we have obtained MHD equilibria with flows having non-vanishing poloidal components in addition to toroidal ones. The solutions obtained here have well defined magnetic surfaces which would be described by the relation wðx; yÞ ¼ const. Fig. 1(a)–(f) with the values of parameters listed in their captions show magnetic flux surfaces for the first solution class presented in Table 4 with Mach number given in Table 1. We have used the function v3 in Eq. (33). They describe O-shaped field lines which are formed providing a strong toroidal magnetic field in the plasma. Magnetic field structures for the poloidal field are shown in Fig. 2(a)–(f). These structures may help in the formation and stability of plasma on the hot magneto-fluid inside a device. Finally, the solutions obtained here may be of some relevance to axisymmetric confinement systems. Conclusion The equilibrium properties of some non-parallel ideal and some parallel resistive MHD plasmas with incompressible flows are investigated. In our study, we addressed many nonlinear issues of equations of motion and dealt with more than nonlinear term in the same equation. Transformation properties of a generalized Grad-Shafranov equation are analyzed for many cases with spatial symmetry in a framework that makes it possible to recover both the magnetized static and dynamic limits. The analysis of the symmetry properties of the Grad-Shafranov equation that describes axisymmetric MHD plasma equilibria without flows was developed in previous papers [57–61]. In partic-

S.M. Moawad et al. / Results in Physics 7 (2017) 3163–3175

3173

Fig. 1. Contour plots of magnetic surfaces describe plasma in axisymmetric confinement systems. The magnetic surfaces are plotted for the first solution class presented pﬃﬃﬃﬃ in Table 4 with Mach number given in Table 1. The values of parameters used are: a ¼ 0; b ¼ 1; n ¼ c3 ¼ 0 with n ¼ r 2 þ cosh z and v3 ¼ cosh z. (a) M ¼ const, (b) M ¼ w, (c) M ¼ sinðwÞ, (d) M ¼ cosðwÞ, (e) M ¼ tanhðwÞ and (f) M ¼ sechðwÞ.

ular, in [57], conditional symmetries were introduced for the GradShafranov equation. The presence of such symmetries made it possible to identify some additional classes of solutions which describe D-shaped toroidal plasma equilibria. In [58], an analysis of the symmetry properties of some partial differential equations of relevance for plasma physics, including the Grad-Shafranov equation as a special case, has been performed. The present paper is devoted to the investigation of generalized forms of the Grad-

Shafranov equation that describes axisymmetric plasma equilibria in the presence of poloidal and toroidal incompressible flows. We have proposed constraint assumptions to deal with many cases of the equilibrium equations in the presence of mass flows. So, the results obtained in the above references can be recovered in our treatment. Also, linear magnetostatic and MHD equilibria were investigated in [18,46,48,53–56] can be recovered by the solutions obtained here.

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S.M. Moawad et al. / Results in Physics 7 (2017) 3163–3175

Fig. 2. Ellipsoidal contour plots of the poloidal magnetic field for the fist solution in Table 4 with Mach number given in Table 1. The same parameters of Fig. 1 are used.

Examples of the obtained solutions to magnetic confinement fusion devices are discussed in which they are relevant to the study of stability of plasmas in laboratory. Acknowledgments The authors would like to thank the referee for his constructive and useful comments, which helped in putting the manuscript into its final form.

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Some axisymmetric equilibria for certain ideal and resistive magnetohydrodynamics with incompressible flows

Some axisymmetric equilibria for certain ideal and resistive magnetohydrodynamics with incompressible flows

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