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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.