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Mamatsashvili, G, Stefani, F, Hollerbach, R et al. (1 more author) (2019) Two types of
axisymmetric helical magnetorotational instability in rotating flows with positive shear.
Physical Review Fluids, 4 (10). 103905. ISSN 2469-990X
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arXiv:1810.13433v2 [physics.flu-dyn] 19 Sep 2019
New type of axisymmetric helical magnetorotational instability in rotating flows with
positive shear
George Mamatsashvili
∗
Niels Bohr International Academy, Niels Bohr Institute,
Blegdamsvej 17, 2100 Copenhagen, Denmark and
Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400, D-01328 Dresden, Germany
Frank Stefani
Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400, D-01328 Dresden, Germany
Rainer Hollerbach
Department of Applied M athematics, University of Leeds, Leeds LS2 9JT, U.K.
G¨unther R¨udiger
Leibniz-Institut f¨ur Astrophysik Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany
(Dated: September 20, 2019)
We reveal and investigate a new type of linear axisymmetric helical magnetorotational instability
which is capable of destabilizing viscous and resistive rotational flows with radially increasing angular
velocity, or positive shear. This instability is double-diffusive by nature and is different from the
more familiar helical magnetorotational instability, operating at positive shear above the Liu limit, in
that it works instead for a wide range of the positive shear when (i) a combination of axial/poloidal
and azimuthal/toroidal magnetic fields is applied and (ii) the magnetic Prandtl number is not
too close to unity. We study this instability first with radially local WKB analysis, deriving the
scaling properties of its growth rate with respect to Hartmann, Reynolds and magnetic Prandtl
numbers. Then we confirm its existence using a global stability analysis of the magnetized flow
confined between two rotating coaxial cylinders with purely conducting or insulating boundaries
and compare the results with those of the local analysis. From an experimental point of view, we
also demonstrate the presence of the new instability in a magnetized viscous and resistive Taylor-
Couette flow with positive shear for such values of the flow parameters, which can be realized in
upcoming experiments at the DRESDYN facility. Finally, this instability might have implications
for the dynamics of th e equatorial parts of the solar tachocline and dynamo action there, since
the above two necessary conditions for the instability to take place are satisfied in this region.
Our global stability calculations for the tachocline-like configuration, representing a thin rotating
cylindrical layer with the appropriate boundary conditions – conducting inner and insulating outer
cylinders – and the values of the flow parameters, indicate that it can indeed arise in this case with
a characteristic growth time comparable to the solar cycle period.
I. INTRODUCTION
According to Rayleigh’s criterion [1], rotating flows of
ideal fluids with radially increasing specific angular mo-
mentum are linearly stable. This result has severe astro-
physical consequences, implying hydrodynamic stability
of Keplerian rotation in accretion disks. Nowadays, the
magnetorotational instability (MRI) [2–4] is considered
to be the most likely destabilizing mechanism for these
disks, driving radially outward transport of angular mo-
mentum and inward accretion of mass.
The standard MRI (SMRI, with a purely ax-
ial/poloidal magnetic field, [2–4]), as well as the non-
axisymmetric azimuthal MRI (AMRI, with a purely az-
imuthal/toroidal magnetic field, [5]) and the axisymmet-
ric helical MRI (HMRI, with combined axial and az-
imuthal magnetic fields, [6]) have all been extensively
∗
Electronic address: george.mamatsashvili@nbi.ku.dk
studied theoretically (see a recent review [7] and refer-
ences therein). The inductionless forms of AMRI and
HMRI have also been obtained in liquid metal experi-
ments [8–10], while unambiguous experimental evidence
for inductive SMRI remains elusive, despite promising
first results [11, 12].
In contrast to Keplerian-like rotation with increas-
ing angular momentum but decreasing angular velocity,
much less attention is usually devoted to flows with in-
creasing angular velocity. Until recently, such flows have
been believed to be strongly stable, even under magnetic
fields. However, for very high enough Reynolds num-
bers Re ∼ 10
7
, they can yield non-axisymmetric linear
instability [13]. Apart from this hydrodynamic instabil-
ity, there is also a special type of AMRI operating in
flows with much lower Reynolds number but sufficiently
strong positive shear [14–16]. This restriction to strong
shear makes, however, this so-called Super-AMRI astro-
physically less significant. One of few positive shear re-
gions is a portion of the solar tachocline extending ±30
◦
about the Sun’s equator. Even there, the shear measured
2
in terms of Rossby number Ro = r(2Ω)
−1
dΩ/dr is only
around 0.7 [17, 18], much less than the so-called upper
Liu limit (ULL) Ro
ULL
= 2(1 +
√
2) ≈ 4.83 [19] required
for Super-AMRI. Another astrophysical system in which
positive shear is expected is the boundary layer between
an accretion disk and its host star [20, 21].
Given a general similarity between AMRI and HMRI
and the universal nature of the Liu limits [22, 23], one
might expect a similar result to hold also for Super-
HMRI. However, as we report in this paper, there exists
a new type of axisymmetric HMRI, which we refer to
as type 2 Super-HMRI, that operates in positive shear
flows with arbitrary steepness, whereas the more famil-
iar HMRI operating only at high enough positive shear
above the Liu limit, Ro > Ro
ULL
, is labelled type 1
Super-HMRI. The only requirements are (i) the presence
of both axial and azimuthal magnetic field components
and (ii) magnetic Prandtl number is neither zero (the
inductionless limit) nor too close to unity. These condi-
tions are indeed satisfied in the solar tachocline, where
this new instability can possibly play an important role
in its dynamics and magnetic activity. Although this re-
quires a detailed separate study and is out of scope of
the present paper, we have also done calculations at the
end of this paper, showing the possibility of occurrence of
this instability for the tachocline-like configuration and
parameters, but still remaining in the framework of cylin-
drical flow. The resulting growth time (inverse of the ex-
ponential growth rate) of the most unstable mode in fact
turns out to be comparable to the solar cycle period.
In this paper, we carry out a linear stability analy-
sis of a magnetic rotational flow in cylindrical geome-
try mainly using the Wentzel-Kramers-Brillouin (WKB)
short-wavelength formulation of the underlying magneto-
hydrodynamics (MHD) problem [23, 24], which is espe-
cially useful for understanding the basic features and scal-
ing properties of the new instability. This local analysis
is then complemented by global, radially one-dimensional
(1D) calculations of the corresponding unstable eigen-
modes with the primary aim to demonstrate the exis-
tence of this new version of Super-HMRI beyond the lo-
cal WKB approximation as well as to draw a compari-
son with the results obtained using this approximation.
A more comprehensive global linear analysis exploring
parameter space, and subsequently nonlinear analysis of
this double-diffusive type 2 Super-HMRI at positive shear
will be presented elsewhere.
The paper is organized as follows. Main equations and
the formulation of a problem are given in Sec. II. The
local WKB analysis of the instability is presented in Sec.
III. The global stability analysis of a differentialy rotating
flow between two coaxial cylinders at positive shear both
in the narrow and wide gap cases as well as a comparison
with the results of the local analysis are presented in Sec.
IV. A summary and discussion on the relevance of this
new version of Super-HMRI to the solar tachocline are
given in Sec. V.
II. MAIN EQUATIONS
The motion of an incompressible conducting medium
with constant viscosity ν and ohmic resistivity η is gov-
erned by the equations of non-ideal MHD
∂U
∂t
+(U·∇)U = −
1
ρ
∇
P +
B
2
2µ
0
+
(B · ∇)B
µ
0
ρ
+ν∇
2
U,
(1)
∂B
∂t
= ∇ × (U × B) + η∇
2
B, (2)
∇ · U = 0, ∇ · B = 0. (3)
where ρ is the constant density, U is the velocity, P is
the thermal pressure, B is the magnetic field and µ
0
is
the magnetic permeability of vacuum.
Consider a flow between two coaxial cylinders at in-
ner, r
i
, and outer, r
o
, radii, rotating, respectively, with
angular velocities Ω
i
and Ω
o
in the cylindrical coordi-
nates (r, φ, z). Since we are primarily interested in the
flow stability in the case of positive shear, or so-called
“super-rotation” [15, 16], the inner cylinder is assumed
to rotate slower than the outer one, Ω
i
< Ω
o
, induc-
ing an azimuthal nonuniform flow U
0
= (0, rΩ(r), 0) be-
tween the cylinders with radially increasing angular ve-
locity, dΩ/dr > 0, and hence positive Rossby number,
Ro > 0. The pressure associated with this base flow and
maintaining its rotation is denoted as P
0
. The imposed
background helical magnetic field B
0
= (0, B
0φ
(r), B
0z
)
consists of a radially varying, current-free azimuthal com-
ponent, B
0φ
(r) = βB
0z
r
o
/r, and a constant axial compo-
nent, B
0z
, where the constant parameter β characterizes
field’s helicity.
We investigate the linear stability of this equilibrium
against small axisymmetric (∂/∂φ = 0) perturbations,
u = U − U
0
, p = P − P
0
, b = B − B
0
, which are all
functions of r and depend on time t and axial/vertical z-
coordinate via ∝ exp(γt+ik
z
z), where γ is the (complex)
eigenvalue and k
z
is the axial wavenumber. There is in-
stability in the flow, if the real part of any eigenvalue, or
growth rate is positive, Re(γ) > 0, for any of the eigen-
nmodes. In such cases, for a given set of parameters, we
always select out the mode with the largest growth rate
from a corresponding eigenvalue spectrum.
III. WKB ANALYSIS
In this section, we use a radially local WKB approx-
imation, where the radial dependence of the perturba-
tions is assumed to be of the form ∝ exp(ik
r
r) with k
r
being the radial wavenumber. The resulting dispersion
relation, which follows from Eqs. (1)-(3) after linearizing
and substituting the above exponential form of the per-
turbations, is represented by the fourth-order polynomial
[23, 24]:
γ
4
+ a
1
γ
3
+ a
2
γ
2
+ (a
3
+ ib
3
)γ + a
4
+ ib
4
= 0, (4)
3
with the real coefficients
a
1
= 2
k
2
Re
1 +
1
P m
,
a
2
= 4α
2
(1 + Ro) + 2(k
2
z
+ 2α
2
β
2
)
Ha
2
Re
2
P m
+
k
4
Re
2
1 +
4
P m
+
1
P m
2
,
a
3
= 8(1 + Ro)α
2
k
2
ReP m
+ 2[k
4
+ (k
2
z
+ 2α
2
β
2
)Ha
2
]
k
2
Re
3
P m
1 +
1
P m
b
3
= −8α
2
βk
z
Ha
2
Re
2
P m
,
a
4
= 4α
2
k
4
P m
2
(1 + Ro)
1
Re
2
+ β
2
Ha
2
Re
4
+ 4α
2
k
2
z
Ro
Ha
2
Re
2
P m
+
k
2
z
Ha
2
+ k
4
2
1
Re
4
P m
2
,
b
4
= 4βk
3
z
Ro
1 −
1
P m
−
2
P m
Ha
2
Re
3
P m
.
Henceforth γ is normalized by the outer cylinder’s angu-
lar velocity Ω
o
, and the wavenumbers by its inverse ra-
dius, r
−1
o
. Other nondimensional parameters are: α =
k
z
/k, where k = (k
2
r
+ k
2
z
)
1/2
is the total wavenum-
ber; the Reynolds number Re = Ω
o
r
2
o
/ν, the magnetic
Reynolds number Rm = Ω
o
r
2
o
/η, and their ratio, the
magnetic Prandtl number P m = ν/η = Rm/Re; the
Hartmann number Ha = B
0z
r
o
/(µ
0
ρνη)
1/2
that mea-
sures the strength of the imposed axial magnetic field.
Another quantity characterizing the field is Lundquist
number S = Ha · P m
1/2
, which, like Rm, does not in-
volve viscosity. Since we focus on positive Rossby num-
bers, Ro > 0, or positive shear, the flow is generally
stable both hydrodynamically, according to Rayleigh’s
criterion (but see Ref. [13]), as well as against SMRI
with a purely axial field (β = 0) [24–26].
In the inductionless limit, P m → 0, the roots of Eq.
(4) can be found analytically [19, 23, 24, 27, 28]. For
positive and relatively large Ro > Ro
ULL
, one of the
roots always has a positive real part, implying instability
with the growth rate
Re(γ) =
q
2X + 2
p
X
2
+ Y
2
− (k
2
z
+ 2α
2
β
2
)
Ha
2
k
2
Re
−
Re
k
2
, (5)
where
X = α
2
β
2
(α
2
β
2
+ k
2
z
)
Ha
4
Re
2
k
4
− α
2
(1 + Ro),
Y = βα
2
k
z
(2 + Ro)
Ha
2
k
2
Re
,
which we call type 1 Super-HMRI. Our main goal though
is to reveal that apart from this type 1 Super-HMRI at
large positive shear, Eq. (4) also yields a completely new
type of dissipation-induced double-diffusive instability at
finite P m, which we call type 2 Super-HMRI.
Regarding the dependence on β parameter in Eq. (4),
it is readily seen that, as long as β 6= 0, it enters the coef-
ficients of these dispersion relations through the re-scaled
wavenumbers, Hartmann, Lundquist and Reynolds num-
bers, k
∗
z
≡ k
z
/β, k
∗
≡ k/β, Ha
∗
≡ Ha/β, S
∗
≡ S /β,
Re
∗
≡ Re/β
2
, Rm
∗
≡ Rm/β
2
, in terms of which we
carry out the following WKB analysis. It is easy to check
that β disappears in the polynomial Eq. (4) after sub-
stituting these re-scaled parameters (denoted with aster-
isks) in its coefficients.
Figure 1(a) shows the growth rate, Re(γ), as a func-
tion of the re-scaled axial wavenumber, as determined
from a numerical solution of Eq. (4) at finite but very
small P m = 10
−6
, together with solution (5) in the in-
ductionless limit, for fixed Ha
∗
and Re
∗
. For the Rossby
number we take the values lower, Ro = 1.5, 2, and higher,
Ro = 6, than Ro
ULL
. Two distinct instability regimes
are clearly seen in this figure. Type 2 Super-HMRI is
concentrated at small k
∗
z
and exists at finite P m both for
Ro < Ro
ULL
and Ro > Ro
ULL
, i.e., it is insensitive to
the upper Liu limit, but disappears for P m → 0 at fixed
Hartmann and Reynolds numbers. By contrast, type 1
Super-HMRI, concentrated at larger k
∗
z
, exists only for
Ro > R o
ULL
, and approaches the inductionless solution
as P m → 0. This latter branch is basically an extension
of the more familiar HMRI operating at negative shear,
which in the inductionless limit also satisfies Eq. (5), but
at Ro < Ro
LLL
, where Ro
LLL
= 2(1 −
√
2) ≈ −0.83 is
the lower Liu limit [19, 24, 27].
At large P m ≫ 1, type 1 Super-HMRI disappears and
there remains only type 2 Super-HMRI, as shown in Fig.
2(a). The corresponding dispersion curves as a function
of axial wavenumber have a shape similar to those at
small P m in Fig. 1(a), but now the instability occurs
at order of magnitude larger k
∗
z
and several orders of
magnitude smaller Ha
∗
and Re
∗
at the same values of
Ro adopted in these figures.
Thus, type 2 Super-HMRI represents a new,
dissipation-induced instability mode at positive shear,
which appears to require the presence of both finite vis-
cosity and resistivity. As we will see below though, it
does not operate in the immediate vicinity of P m = 1,
that is, it is double-diffusive in nature, operating for both
small and large P m, but not for P m = O(1). Just as
all previous MRI variants, this one also derives energy
4
solely from the shear, since the imposed magnetic field
is current-free, thereby eliminating current-driven insta-
bilities, such as the Tayler instability. Energy is drawn
from the background flow rΩ(r) to the growing perturba-
tions due to the coupling between meridional circulation
and azimuthal field perturbations brought about by the
imposed azimuthal field, a mechanism also underlying
HMRI at negative shear [6, 29].
Our main goal is to describe the properties of this new
type 2 Super-HMRI. Type 1 Super-HMRI, existing only
for Ro > Ro
ULL
and persisting even in the inductionless
limit P m → 0 [19, 23, 24, 28], is also relatively new and
interesting in its own right, but will not be considered
here further.
Like normal HMRI at negative shear, type 2 Super-
HMRI is an overstability, that is, its growth rate comes
with an associated non-zero imaginary part, ω = Im(γ),
which is the frequency of temporal oscillations of the
solution at a given coordinate and, together with ax-
ial wavenumber, defines its propagation speed. Figure 3
shows these frequencies as a function of k
∗
z
, correspond-
ing to the growth rates plotted in Figs. 1(a) and 2(a).
They monotonically increase with k
∗
z
by absolute value,
but are positive at small P m and negative at large P m,
implying opposite propagation directions of the wave pat-
terns at these magnetic Prandtl numbers. Also, ω re-
mains smaller than the frequency of inertial oscillations,
ω
io
= 2α(1 + Ro)
1/2
, and tend to the latter only at small
P m as the solution changes from type 2 to type 1 Super-
HMRI with increasing k
∗
z
and do not change afterwards.
This reflects the fact that type 1 Super-HMRI represents
weakly destabilized inertial oscillations, like the normal
HMRI at negative shear [19].
To explore the behavior of type 2 Super-HMRI further,
we first vary α as well as the re-scaled Hartmann and
Reynolds numbers. The growth rate, maximized over
the last two numbers and k
∗
z
, increases linearly with α
and scales as ∝ Ro
1.75
at small P m = 10
−6
(Fig. 1(b))
and as ∝ Ro
1.18
at large P m = 100 (Fig. 2(b)), while its
dependence on Ha
∗
and Re
∗
, when maximized over k
∗
z
and α, is shown in Fig. 1(c) at P m = 10
−6
and in Fig.
2(c) at P m = 100 with Ro = 1.5 < Ro
ULL
(when type 1
Super-HMRI is absent) in both cases. The most unstable
region is quite localized, with the growth rate decreasing
for both small and large Ha
∗
and Re
∗
, implying that this
instability relies on finite viscosity and resistivity, i.e., it
is indeed of double-diffusive type. The overall shape of
the unstable area in (Ha
∗
, Re
∗
)-plane does not change
qualitatively at other P m and Ro; the unstable region
always remains localized and shifts to larger Ha
∗
and
Re
∗
with decreasing P m. In particular, the maximum
growth rate, γ
m
, occurs for (Ha
∗
m
, Re
∗
m
) ≈ (700, 9 · 10
4
)
when Pm is small (Fig. 1(c)), but for orders of magnitude
smaller (Ha
∗
m
, Re
∗
m
) = (2.54, 0.23) when P m is large
(Fig. 2(c)) The actual values of the characteristic vertical
wavenumber, Hartmann and Reynolds numbers for type
2 Super-HMRI at different β are obtained by simply mul-
tiplying the values of re-scaled quantities k
∗
z
, Ha
∗
, Re
∗
10
-2
10
-1
10
0
k
z
10
-3
10
-2
10
-1
Re( )
Ro=1.5
Ro=2
Ro=6
Ro=6, Pm=0
0 0.2 0.4 0.6 0.8 1
0
0.01
0.02
m
/Ro
1.75
10
1
10
2
10
3
10
4
10
5
Ha
10
3
10
4
10
5
10
6
10
7
Re
0.01
0.02
0.03
0.04
Type 2 Super-HMRI
Type 1 Super-HMRI
(b)
(a)
(c)
FIG. 1: Panel (a) shows the growth rate Re(γ) v s. k
∗
z
at
fixed Ha
∗
= 90, Re
∗
= 8 · 10
3
, α = 0.71 (i.e., k
∗
r
= k
∗
z
) and
P m = 10
−6
for different Ro = 1.5 (blue ), 2 (green), 6 (red).
New type 2 Super-HMRI branch exists at smaller k
∗
z
and finite
P m, for all three Ro values. By contrast, type 1 Super-HMRI
branch at larger k
∗
z
appears only for Ro = 6 > Ro
ULL
from
these th ere values of the Rossby number, but persists also
in the inductionless limit (Eq. 5, dashed-black line). For the
same P m, panel (b ) shows the growth rate of type 2 Super-
HMRI, maximized over a set of the parameters (k
∗
z
, Ha
∗
,
Re
∗
) and normalized by Ro
1.75
, vs. α, while panel (c) shows
the growth rate, maximized over k
∗
z
and α, as a function of
Ha
∗
and Re
∗
at Ro = 1.5 and the same P m = 10
−6
.
10
-2
10
-1
10
0
k
z
10
-3
10
-2
10
-1
10
0
Re( )
Ro=1.5
Ro=2
Ro=6
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
m
/Ro
1.18
10
-4
10
-2
10
0
10
2
10
4
Ha
10
-4
10
-2
10
0
10
2
Re
0.05
0.1
0.15
0.2
(a)
(c)
Type 2 Super-HMRI
(b)
FIG. 2: Same as in Fig. 1, but at Ha
∗
= 5, Re
∗
= 0.1,
α = 0.71 in panel (a) and P m = 100 in all panels. New
type 2 Super-HMRI branch exists at h igher k
∗
z
than those at
small P m, while type 1 Super-HMRI b ranch is absent. In
panel (b), the maximum growth rate now exhibits the scaling
with Rossby number, Ro
1.18
, different from that at small P m.
In panel (c), the maximum growth occurs now at orders of
magnitude smaller Ha
∗
m
and Re
∗
m
than those at small P m in
Fig. 1(c) at the same Ro = 1.5.