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Parker problem in Hall magnetohydrodynamics Parker problem in Hall magnetohydrodynamics

Bhimsen K. Shivamoggi

University of Central Florida

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Phys. Plasmas 16, 052111 (2009); https://doi.org/10.1063/1.3140055 16, 052111

© 2009 American Institute of Physics.

Parker problem in Hall

magnetohydrodynamics

Cite as: Phys. Plasmas 16, 052111 (2009); https://doi.org/10.1063/1.3140055

Submitted: 02 September 2008 . Accepted: 28 April 2009 . Published Online: 28 May 2009

Bhimsen K. Shivamoggi

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Physics of Plasmas 18, 052304 (2011); https://doi.org/10.1063/1.3581092

Parker problem in Hall magnetohydrodynamics

Bhimsen K. Shivamoggi

a兲

Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

共Received 2 September 2008; accepted 28 April 2009; published online 28 May 2009兲

The Parker problem in Hall magnetohydrodynamics 共MHD兲 is considered. Poloidal shear

superposed on the toroidal ion ﬂow associated with the Hall effect is incorporated. This is found to

lead to a triple deck structure for the Parker problem in Hall MHD, with the magnetic ﬁeld falling

off in the intermediate Hall-resistive region more steeply 共like 1/ x

3

兲 than that 共like 1/ x兲 in the outer

ideal MHD region. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3140055兴

I. INTRODUCTION

Numerical simulations

1

suggested that magnetic recon-

nection in resistive magnetohydrodynamics 共MHD兲 can be

driven by a magnetic ﬂux pile-up. In this process, magnetic

ﬁeld builds up upstream of a Sweet

2

–Parker

3

current sheet

共this ﬁeld buildup strengthens as the resistivity is decreased兲,

which leads to an increase in the outﬂow downstream of the

current sheet. This provides for a remedy for the Sweet–

Parker “bottleneck”

4

limiting the outﬂow and enabling re-

connection to proceed at the externally imposed rate. How-

ever, as the resistivity is decreased further, the development

of large magnetic pressure gradients upstream of the current

sheet opposes the ion inﬂow,

5

which is the only means in

resistive MHD to transport magnetic ﬂux into the reconnec-

tion layer. The magnetic ﬂux transport into the reconnection

layer 共and hence the reconnection rate兲 is reduced—the so-

called pressure problem.

6

The Hall effect

7

can overcome the

pressure problem

8,9

thanks to the decoupling of electrons

from ions on length scales below the ion skin depth d

i

. So, if

the reconnection-layer width is less than d

i

, the electron in-

ﬂow can keep on going, which transports the magnetic ﬂux

into the reconnection layer and hence reduces the ﬂux pile-

up. Dorelli

10

considered the role of Hall effect in magnetic

ﬂux pile-up driven antiparallel magnetic ﬁeld merging and

gave analytical solutions of the resistive Hall MHD equa-

tions describing stagnation-point ion ﬂows in a thin current

sheet—Parker problem.

11

However, Dorelli’s solution for the

Hall regime turned out to be basically Parker’s solution for

the ideal MHD regime. Indeed, the Hall contribution can be

transformed away from Dorelli’s solution by suitably rede-

ﬁning the velocity gradient associated with the stagnation-

point ion ﬂow. A more complete formulation of the Parker

problem in Hall MHD is therefore in order—this is the ob-

jective of this paper.

II. GOVERNING EQUATION FOR HALL MHD

Consider an incompressible, two-ﬂuid, quasineutral

plasma. The governing equations for this plasma dynamics

are 共in usual notation兲

nm

e

冋

v

e

t

+ 共v

e

· ⵜ兲v

e

册

=−ⵜp

e

− ne

冉

E +

1

c

v

e

⫻ B

冊

+ ne

J, 共1兲

nm

i

冋

v

i

t

+ 共v

i

· ⵜ兲v

i

册

=−ⵜp

i

+ ne

冉

E +

1

c

v

i

⫻ B

冊

− ne

J, 共2兲

ⵜ · v

e

=0, 共3兲

ⵜ · v

i

=0, 共4兲

ⵜ · B =0, 共5兲

ⵜ ⫻ B =

1

c

J, 共6兲

ⵜ ⫻ E =−

1

c

B

t

, 共7兲

where

J ⬅ ne共v

i

− v

e

兲. 共8兲

Neglecting electron inertia 共m

e

→ 0兲, Eqs. 共1兲 and 共2兲 can

be combined to give a modiﬁed ion equation of motion,

nm

i

冋

v

i

t

+ 共v

i

· ⵜ兲v

i

册

=−ⵜ共p

i

+ p

e

兲 +

1

c

J ⫻ B, 共9兲

and a generalized Ohm’s law,

E +

1

c

v

i

⫻ B =

J +

1

nec

J ⫻ B. 共10兲

Nondimensionalize distance with respect to a typical

length scale a, magnetic ﬁeld with respect to a typical mag-

netic ﬁeld strength B

0

, time with respect to the reference

Alfvén time

A

⬅a / V

A

0

, where V

A

0

⬅B

0

/

冑

and

⬅m

i

n, and

introduce the magnetic and velocity stream functions accord-

ing to

a兲

Permanent address: University of Central Florida, Orlando, FL 32816-

1364.

PHYSICS OF PLASMAS 16, 052111 共2009兲

1070-664X/2009/16共5兲/052111/4/$25.00 © 2009 American Institute of Physics16, 052111-1

B = ⵜ

⫻ i

ˆ

z

+ bi

ˆ

z

共11兲

v

i

= ⵜ

⫻ i

ˆ

z

+ wi

ˆ

z

and assume that the physical quantities of interest have no

variation along the z-direction. The Hall magnetic ﬁeld b is

believed to be produced by the dragging of the in-plane mag-

netic ﬁeld in the out-of-plane direction by the magnetized

electrons near the magnetic neutral surface.

12,13

Equations

共9兲 and 共10兲 then yield

t

+ 关

,

兴 +

关b,

兴 =

ˆ

ⵜ

2

, 共12兲

b

t

+ 关b,

兴 +

关

,ⵜ

2

兴 + 关

,w兴 =

ˆ

ⵜ

2

b, 共13兲

w

t

+ 关w,

兴 = 关b,

兴, 共14兲

where

关A,B兴⬅ⵜA ⫻ ⵜB · i

ˆ

z

,

⬅ d

i

/a,

ˆ

⬅

c

2

A

/a

2

.

III. PARKER PROBLEM IN HALL MHD

Consider a stagnation-point ion ﬂow at a current sheet

separating plasmas of opposite magnetizations

11

in Hall

MHD, governed by Eqs. 共12兲–共14兲. Let us assume that the

magnetic ﬁeld lines are straight and parallel to the current

sheet. Here, pure resistive annihilation without reconnection

of antiparallel magnetic ﬁelds 共in the x , y-plane兲 occurs. Spe-

ciﬁcally, consider a unidirectional magnetic ﬁeld

B = B

y

共x兲i

ˆ

y

, 共15兲

with the boundary condition

B

y

共0兲 =0, 共16兲

which is carried toward a neutral sheet at x=0 by a

stagnation-point ion ﬂow,

v

i

=−axi

ˆ

x

+ ayi

ˆ

y

+ wi

ˆ

z

. 共17兲

Noting that the process in question is steady and that the

magnetic ﬁeld is prescribed as in Eq. 共15兲, Eqs. 共12兲–共14兲

become

E +

x

y

−

b

y

x

=

ˆ

2

x

2

, 共18兲

b

x

y

−

b

y

x

+

x

w

y

=

ˆ

ⵜ

2

b, 共19兲

w

x

y

−

w

y

x

+

x

b

y

=0, 共20兲

where

E ⬅

t

.

Dorelli

10

looked for a solution of Eqs. 共18兲–共20兲 of the

form

b = yf共x兲, 共21a兲

w =

␣

g

1

共x兲, 共21b兲

where

␣

is a constant. However, this solution restricts the

role played by the Hall effect in the process in question.

Indeed, the Hall contribution can be transformed away from

Dorelli’s solution by suitably redeﬁning the velocity gradient

a associated with the stagnation-point ion ﬂow 共see below兲.

One way to remedy this situation is to recognize the presence

of the poloidal shear in the toroidal ion ﬂow, which is intrin-

sic to the Hall effect. This aspect is also recognized in the

Hall resistive tearing mode formulation.

14

The presence of

poloidal shear in the toroidal ion ﬂow basically signiﬁes the

generic variation in the out-of-plane ion ﬂow velocity along

the outﬂow direction. On the other hand, the physical mecha-

nism proposed for the generation of the Hall magnetic ﬁeld b

共Refs. 12 and 13兲 involves a poloidal shear in the toroidal

electron ﬂow, some of which appears to be coupled also to

the toroidal ion ﬂow. Therefore, we incorporate a poloidal

shear into the toroidal ion ﬂow according to

w =

␣

g

1

共x兲 +

2

y

2

g

2

共x兲, 共22兲

where

is a constant characterizing this poloidal shear. The

particular form of poloidal shear superposed on the toroidal

ion ﬂow used in Eq. 共22兲 is motivated by the symmetry prop-

erties of Eqs. 共12兲–共14兲 in the ideal limit, which indicate

and w to be even functions of both x and y, and

and b to be

odd function of both x and y;

15

these properties are sustained

by the solutions in the following. Using Eqs. 共21a兲 and 共22兲,

Eq. 共20兲 gives

冉

␣

g

1

⬘

+

2

y

2

g

2

⬘

冊

共− ax兲 − 共

yg

2

兲共− ay兲 +

x

f

⬘

=0, 共23兲

where primes denote differentiation with respect to x.We

obtain from Eq. 共23兲

共− ax兲共

␣

g

1

⬘

兲 =−

x

f

⬘

, 共24兲

g

2

共x兲 = x

2

. 共25兲

Using Eqs. 共21a兲, 共22兲, 共24兲, and 共25兲, Eq. 共19兲 then

gives

yf

⬘

共− ax兲 − f共− ay兲 +

x

共

yx

2

兲 =

ˆ

yf

⬙

, 共26兲

or

f

⬙

+

a

ˆ

共xf

⬘

− f兲 =

ˆ

x

2

x

. 共27兲

Recognizing that the Hall effect becomes important

away from the current sheet at x =0 共in what was called the

“intermediate” layer by Terasawa

16

in his investigation of the

Hall resistive tearing mode兲, a reasonable approximate solu-

tion of Eq. 共27兲 is

052111-2 Bhimsen K. Shivamoggi Phys. Plasmas 16, 052111 共2009兲

f共x兲⬇Ax −

a

x

2

x

, 共28兲

where A is a constant.

Using Eqs. 共21a兲 and 共28兲, Eq. 共18兲 gives

− E +

x

共− ax兲 −

冉

Ax −

a

x

2

x

冊

x

=

ˆ

2

x

2

,

or

E =

ˆ

B

y

⬘

+ 共a +

A兲xB

y

+

a

x

2

B

y

2

, 共29兲

which is a generalization of Parker ’s equation to Hall MHD.

Observe that, for Dorelli’s solution 共which corresponds to

=0兲, the Hall contribution 共represented by the term corre-

sponding to

兲 can be simply transformed away by redeﬁn-

ing the velocity gradient a associated with the stagnation-

point ion ﬂow.

It is important to note that Eq. 共29兲 shows that the triple-

deck structure 共borrowing the terminology from boundary

layer theory in ﬂuid dynamics

17

兲 in Hall resistive MHD

共originally pointed out by Terasawa

16

兲 is operational for the

Parker problem in Hall MHD, as to be expected. Thus, we

have

共i兲 a resistive region near the current sheet at x =0,

共ii兲 an ideal MHD region away from the current sheet at

x= 0, and

共iii兲 a Hall resistive region in between 共i兲 and 共ii兲—called

the intermediate region by Terasawa.

16

In the resistive region, Eq. 共29兲 may be approximated by

E ⬇

ˆ

B

y

⬘

, 共30兲

which gives Parker’s solution,

B

y

⬇

E

ˆ

x. 共31兲

In the ideal MHD region, Eq. 共29兲 may be approximated

by

E ⬇共a +

A兲xB

y

+

a

x

2

B

y

2

, 共32兲

which gives modiﬁed Dorelli’s solution,

B

y

⬇

冋

2a共a +

A兲

册

冋

−1+

冑

1+

4E共a +

A兲a

2

2

2

册

1

x

.

共33兲

In the Hall resistive region, Eq. 共29兲 may be approximated

by

E ⬇

ˆ

B

y

⬘

+

a

x

2

B

y

2

, 共34兲

which gives

B

y

⬇

冉

3

ˆ

a

冊

1

x

3

. 共35兲

The triple-deck structure given by Eqs. 共31兲, 共33兲, and

共35兲 is shown in Fig. 1. Noting from Eqs. 共21a兲 and 共28兲 that

b = y

冉

Ax +

a

x

2

B

y

冊

, 共36兲

one observes that both in the

• Hall resistive region, B

y

⬃1 / x

3

, and

• ideal MHD region, B

y

⬃1 / x,

b has a quadrupolar structure. So, Hall effects materialize

only via their signature—the quadrupolar out-of-plane mag-

netic ﬁeld pattern.

18

IV. DISCUSSION

In recognition of the fact that a signature of the Hall

effect is the generation of out-of-plane “separator” compo-

nents of the magnetic and ion-ﬂow velocity ﬁelds, a more

accurate representation of the latter appears to be in order for

a more complete formulation of the Parker problem. In this

paper, this is accomplished by incorporating poloidal shear

into the toroidal ion ﬂow associated with the Hall effect. This

is found to lead to a triple-deck structure for the Parker prob-

lem in Hall MHD, in accordance with the idea originally put

forward by Terasawa.

16

The magnetic ﬁeld is found to fall off

in the intermediate Hall-resistive region more steeply 共like

1/ x

3

兲 than that 共like 1/ x兲 in the outer ideal MHD region.

ACKNOWLEDGMENTS

I acknowledge with gratitude the stimulating interactions

with Dr. Luis Chacon that led to this work. I am thankful to

the referee for his helpful remarks. My thanks are also due to

Dr. Michael Shay and Dr. Michael Johnson for helpful com-

munications and discussions.

1

2

3

0

x

B

y

FIG. 1. Magnetic ﬁeld proﬁle for the Parker problem in Hall MHD. 共1兲

B

y

⬃x, 共2兲 B

y

⬃1 / x

3

, and 共3兲 B

y

⬃1 / x

052111-3 Parker problem in Hall magnetohydrodynamics Phys. Plasmas 16, 052111 共2009兲