Lear Theory of Tearing Instability with Viscosity, Hyper-Resistivity, and improved WKB-approximation

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齋胄代壼警
（
愛媛大学宇宙進化研究センター）
Tohru Shimizu, (RCSCE, Ehime University, Japan)
shimizu@cosmos.ehime-u.ac.jp
◊✲┠ⓗ
脅胄三的 (Research Objective):
  In past years, the linear perturbation equations of tearing instability derived

by Loureiro (Loureiro,PoP2007 named as LSC theory)) has been deeply and
widely explored by numerically solving as initial value problem (IVP) (Shimizu,
AAPPS-DPP2018, KDK Research Report2018, Shimizu&Kondoh, arXiv4472111).
The Loureiro’s equations are based on Non-Viscous case. In this paper, Viscosity
and Hyper resistivity are introduced to the equations (Shimizu, KDK Research
Report2020,21&22, Shimizu&Fujimoto, AOGS2021&22). Then, Non-Uniform
viscosity and improved WKB-approximation are also studied (Shimizu,
AAPPS-DPP2021&22 and ICNSP2022). This paper summaries those variations
of the perturbation equations with some highlighted numerical results.
1.
l
J。 Introduction:
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四o
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:
This paper starts from the linear perturbation equations of tearing instability
shown next, which were derived by Loureiro,et.al. (PoP2007).

￠”K
,
2
E
2q
J=-f(
＜）
（
ゅ”―代％％）
／
入＋f
”（
く
）ゆ
／ 入，

ゅ”-K%
％＝ 位屈ー 吋く）
（ の．

(1-1)
(1-2)

g

! (~) =f
oe予／ 2
1c
,dzez2;
2
'

(1-3)

゜

Every notation in this paper is based on the Loureiro’s definitions, where ȭand
ȯare respectively perturbed potential functions of flow and magnetic fields. The
prime is the derivative for the direction normal to the current sheet, where f(ȥ)
is the equilibrium function of magnetic field Bxo, as shown below.

Bxo(~) =VA f （~) ,

四o

=-I
'
o
y,

(1-4, 1-5)

Eq.(1-4) is based on the equilibrium linear flow field of Eq.(1-5), where y is
translated toȥwith y=1.307ȥ. In Eqs.(1-1&2), Ȣis the linear growth rate andȡ
is the wave number of ȭand ȯ along the current sheet. In the original LSC
theory (PoP2007) and most of my previous studies, Eqs.(1-4&5) are applied only
－ 53 －

for ȥ<1.307, i.e., inside of the current sheet. However, in this paper, Eqs.(1-4&5)
are applied also for ȥ>1.307 to rigorously keep the equilibrium even in the
introduction of viscosity. ResistivityȜand Lundquist number S are defined below,
with the sheet thicknessțcs and sheet length Lcs of Sweet-Parker model.
E

= 2妬 /Le
s

2
/
v
i
s

(1-6)

Loureiro analytically solved Eqs.(1-1)~(1-6) under the upstream condition of ȭ
=ȯ=0 at ȥ=+҄ (PoP2007). To do so, the traditional approximation introduced
by FKR theory (Fruth,PhysFluids1963) was employed, where the outer region of
the current sheet was assumed to be ideal-MHD, and hence, the inner region is
only solved in resistive-MHD. In contrast, Shimizu solved Eqs.(1-1)~(1-6) without
the assumption of the ideal-MHD, i.e., every region was seamlessly solved in
resistive-MHD. To do so, Shimizu did not consider the upstream condition of ȭ=
ȯ=0 at ȥ=+҄. Instead, ȭ=ȯ=0 at ȥc<+҄, i.e., a finite point ȥc, was studied.
The condition can be considered to be open boundary condition at finite upstream
pointȥc. The concept of the open boundary may be close to what is widely
employed in MHD simulations. Then, Eqs.(1-1)~(1-6) were numerically solved as
initial value problem (IVP) fromȥ=0 toȥc (Shimizu, AAPPS-DPP2018, KDK
Research Report2018, Shimizu&Kondoh, arXiv4472111). The LSC theory
modified by Shimizu was named as modified-LSC theory. The modified-LSC can
explore the case of nj=0, which gives the critical (marginal) unstable condition of
tearing instability, as shown in theȤ=0 line of Figs.1 and 2.
2. U
Uniform
Viscosity:
2
n
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f
o
:
r
m巧
s
c
o
s
i
ば

When viscosity is added to Eqs(1-1&2), the equations to be solved are as below,
whereȤis the viscosity coefficient in isotropic viscosity.

v
>
/
<
"
"＝氏召（（入＋ 2氏V
)q>"-

（入＋代レ）氏2 召 cp + f (＜）（ゅ”—氏2E2 ゆ ）一 f”(（） ゆ ）

ゅ”-K2E2
μ
1 =K入ゆー吋(()￠.

(2-1)
(2-2)

These equations can be also numerically solved as IVP fromȥ=0 with initial
values of ȭ’(0) and ȭ’’’(0), whereȭ(0)=ȭ’’(0)=ȯ’(0)=0 and ȯ(0)=1 are fixed for
the symmetric current sheet. In addition toȭ’(0) andȭ’’’(0), changing Ȣ, ȡ, Ȝ
andȤ,ȭandȯ can be solved as IVP so that ȭ=ȭ’=ȯ=0 are satisfied atȥc. Let
us call it Zero-Contact solution, where “Zero-Contact” means ȭ=ȭ’=0. Hence,
once a set of Ȣ, ȡ, Ȝand Ȥ is specified, ȥc is determined. It means that the
linear growth rateȢdepends on the locationȥc of upstream boundary. Then,
Zero-Converging solution which satisfies ȭ(+҄)=ȯ(+҄)=0 may be deduced by

－ 54 －

examining larger ȥ c (see Fig.1). Then, if Ȣ =0 is set, the critical unstable
condition is found, as shown inȤ>0 in Fig.2.
3.
Viscosity:
3。 Non-Uniform
Non{
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立o
:
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m巧
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:
In traditional studies such as FKR(1963) and LSC(2007) theory, the outer
region (ȥ>1.307) is assumed to be ideal-MHD which means Ȝ=Ȥ=0. It seems
that they expect that the assumption is also applicable for when the outer region
is solved in resistive-MHD. In other words, they expect that whether the outer
region is ideal-MHD or resistive-MHD is not essential to study the linear growth
of tearing instability. However, such an expectation fails at some points.
In this section, Eqs.(2-1&2) are solved only in the inner region of the current
sheet (ȥ<1.307). Meanwhile,Ȥ=0 is assumed in the outer region (1.307<ȥ),
where Eqs.(1-1&2) is solved. Hence, viscosity works only in the current sheet.
Note that resistivityȜ(>0) is uniformly kept even in the outer region. In this case,
to compensate the discontinuity ofȤat ȥ=1.307, there are two strategies for the
combination of the differential continuity of ȭ at ȥ=1.307 and the upstream
boundary condition atȥc.

Strategy 1:
To keep the continuity of ȭ’’ at Ǐ=1.307, the next equation must be satisfied
at ȥ=1.307-0, where is close toȥ=1.307 from ȥ<1.307.

￠"
"=2足召 ¢"―代4 €%

(3-1)

Then, find ȭandȯ which satisfies ȭ=ȯ=0 at ȥc. In this strategy, ȭ’=0 is not
required at Ǐc. Hence, that is not Zero-Contact solution. Rather, that is called as
Zero-Crossing solution. This will be the most rigorous solution. Mathematically,
this may be a kind of “strong” solution. To study the case ofȢ=0, Eq.(1-1&2) for
ȥ>1.307 is replaced by below. Fig.1 includes this numerical result, and Fig.3
shows the highlighted summary of Ȣ=0.

0=り
（e
)
2>
1
<+J
"
(
e
)
，
心

(3-2)

ゅ”
ー
氏2げゅ＝
一r
;
,
f(~) ¢ =”
f（
く
） ゆ／
f（
C
)．

(3-3)

Strategy 2:
In this strategy, Eq.(3-1) is ignored. Hence, the continuity of ȭ’’ at Ǐ=1.307 is

not satisfied but ȭ’ is still continuous. Then, find ȭandȯ which satisfies ȭ=
ȭ’=ȯ=0 atȥc. Hence, this is Zero-Contact solution. Mathematically, this may be
a kind of “weak” solution.

－ 55 －

Physically, Non-Uniform viscosity may be considered to be “anomalous”
viscosity. That is physics. On the other hand, most numerical simulations show
not only strong solutions but often also weak solutions, depending on the
employed numerical scheme. That is not physics but numerical error. In actual,
when Non-Uniform viscosity is steadily included in numerical dissipation to
numerically stabilize MHD simulations. For example, in the shock-capturing
schemes such as TVD and HLLD, higher-order differential continuity of the
solutions may not necessarily be kept around the extremely thin current sheet.
Strategy 2 may be able to examine how tearing instability is disturbed by such
numerical Non-Uniform viscosity. Fig.1 includes this numerical result.
4.
Hyper Resistivity
viscosity):
4。旦エ早墨
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紐たもy(electron
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:
Hyper resistivity means the fourth-order differential magnetic diffusion, while
usual resistivity is the second-order magnetic diffusion. Some kinematic
full-particle simulations of the magnetic reconnection process predict that such
higher-order magnetic diffusion is dominant rather than the second-order. That
is physics. In another viewpoint, every finite-differential MHD simulations
have numerical dissipations of such higher-order diffusion to stably simulate
extremely thin current sheets. That is not physics. Such numerical diffusive error
must be examined for how tearing instability is affected or not. For these reasons,
hyper resistivity is worth to be studied in comparison with usual resistivity.
For simplicity, the viscosityȤexamined in the preceding section is ignored in
this section. First, equilibriums f(ȥ) are studied in the mixture of hyper and
usual resistivities. Second, the perturbed solutions are studied on the basis of the
equilibrium.
Equilibrium 1:
Note that Eq.(1-3) is applicable only for usual resistivity. When hyper
resistivity effect is added to the usual resistivity effect, f(ȥ) must satisfy the
following equation.
( l / SH』 !"'(~)

=(
1
/
Si)!'(~)+~! (~) +c

(4-1)

where 1/Si and 1/SHi are respectively the intensity of usual resistivity and hyper
resistivity. Si and SHi are “each” Lundquist number for inflow speed to neutral
sheet (Shimizu&Fujimoto, AOGS2022). At this point, Ȝ defined in Eq.(1-6)
remains as “total” Lundquist number on the basis of Sweet-Parker model. In
other words, Lundquist number referred in this section consists ofȜ(1st step) and
either of Si and SHi (2nd step).

－ 56 －

To appropriately normalize Eq.(4-1), let us fix the inflow speed uyo=-1.307 at
ȥ=1.307 in Eq.(1-5). In addition, let us fix f(1.307)=1.0 and f ’(1.307)=0. With
these setups, the second term of the rhs of Eq.(4-1), i.e., the convection electric
field (V™B) is fixed at -1.307f(1.307)=-1.307, where ȥ=1.307 is the boundary
point of the inner and outer region of the current sheet. This normalization is an
extension of the concept employed in the original LSC (PoP2007).
Eq.(4-1) can be numerically solved as IVP from ȥ =0 with initial values
f(0)=f ’’’(0)=0 and f ’(0). Note that f(0)=f ’’’(0)=0 is fixed for the symmetric current
sheet. ...