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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