Dissipation-induced instabilities
in the
Taylor-Couette flow
of a
liquid metal
Oleg N. Kirillov
Oleg N. Kirillov, http://www.onkirillov.com
Preliminaries. A magnetized Taylor-Couette flow
Seite 2
Oleg N. Kirillov, http://www.onkirillov.com
1823 Navier, 1845 Stokes
formulated equations of motion for viscous fluid
How to measure viscosity?
 Stokes, 1848:
“Consider the motion of a mass of uniform inelastic
fluid comprised between two cylinders having a
common axis, the cylinders revolving uniformly about
their axis.
If the inner one were made to revolve too fast, the
fluid near it would have a tendency to fly outwards in
consequence of the centrifugal force, and eddies
would be produced.”
Seite 3
Oleg N. Kirillov, http://www.onkirillov.com
1888: Arnulph Mallock and Maurice Couette
 Measurements of the kinematic viscosity of water
 Liquid in a gap between rotating concentric cylinders
Seite 4
Oleg N. Kirillov, http://www.onkirillov.com
Mallock & Couette confirmed anticipations of
Stokes
 Couette rotated the outer cylinder
(radius R2, velocity W2)
 Observed laminar flow at low speeds,
turbulent at high
 Mallock rotated the inner cylinder
(radius R1, velocity W1)
 Observed instability at all speeds that
he used
Seite 5
Oleg N. Kirillov, http://www.onkirillov.com
1917: Rayleigh’s criterion of centrifugal stability1
 Inviscid fluid, planar flow
 Axisymmetric perturbation
Stability: if and only if
the angular momentum increases radially
1 d
2 2
(ÐR
) >0
3
R dR
Corollary for the rotating Couette flow:
Ð2R22 > Ð1R12
1Maxwell
Seite 6
proposed this as a problem for the Cambridge Mathematical Tripos as early as 1866
Oleg N. Kirillov, http://www.onkirillov.com
1923: Geoffrey Taylor’s theory and experiment
Motivated by the limitations
in the design of Mallock and
Couette setups:
 Small length-to-diameter
ratio caused deviation from
the planar flow
 Only one of the cylinders
could rotate
 Impossible to check the
Rayleigh criterion in full
Seite 7
Oleg N. Kirillov, http://www.onkirillov.com
1923: Taylor’s stability diagram for the viscous
rotating Couette flow
 Perfect agreement of the linear stability analysis
with the experiment for moderate Reynolds numbers
 Rayleigh’s line is an asymptote in case of co-rotation
Seite 8
Oleg N. Kirillov, http://www.onkirillov.com
2000s: New generation of Couette-Taylor
experiments
 Unprecedentedly high
Reynolds numbers Re~106
Controversial results:
 Princeton reported laminar
motion (2006, 2011)
Princeton (J. Goodman)
Maryland (D. Lathrop)
Seite 9
 Maryland detected
turbulence (2011)
Oleg N. Kirillov, http://www.onkirillov.com
The modern Couette-Taylor race
 Ji, H., Burin, M. J., Schartman, E., Goodman, J. Hydrodynamic
turbulence cannot transport angular momentum effectively in
astrophysical disks, Nature 444, 343 (2006)
 Paoletti, M. S. & Lathrop, D. P. Angular momentum transport
in turbulent flow between independently rotating cylinders,
Phys. Rev. Lett. 106, 024501 (2011)
 Schartman, E., Ji, H., Burin, M. J., Goodman, J. Stability of
quasi-Keplerian shear flow in a laboratory experiment,
Astronomy & Astrophysics 543, A94 (2012)
 Balbus, S. A turbulent matter, Nature 470, 475-476 (2011)
 ...
Seite 10
Oleg N. Kirillov, http://www.onkirillov.com
The modern Couette-Taylor race
 Ji, H., Burin, M. J., Schartman, E., Goodman, J. Hydrodynamic
turbulence cannot transport angular momentum effectively in
astrophysical disks, Nature 444, 343 (2006)
 Paoletti, M. S. & Lathrop, D. P. Angular momentum transport
in turbulent flow between independently rotating cylinders,
Phys. Rev. Lett. 106, 024501 (2011)
 Schartman, E., Ji, H., Burin, M. J., Goodman, J. Stability of
quasi-Keplerian shear flow in a laboratory experiment,
Astronomy & Astrophysics 543, A94 (2012)
 Balbus, S. A turbulent matter, Nature 470, 475-476 (2011)
 ...
WHY?
Seite 11
Oleg N. Kirillov, http://www.onkirillov.com
The modern Couette-Taylor race
 Ji, H., Burin, M. J., Schartman, E., Goodman, J. Hydrodynamic
turbulence cannot transport angular momentum effectively in
astrophysical disks, Nature 444, 343 (2006)
 Paoletti, M. S. & Lathrop, D. P. Angular momentum transport
in turbulent flow between independently rotating cylinders,
Phys. Rev. Lett. 106, 024501 (2011)
 Schartman, E., Ji, H., Burin, M. J., Goodman, J. Stability of
quasi-Keplerian shear flow in a laboratory experiment,
Astronomy & Astrophysics 543, A94 (2012)
 Balbus, S. A turbulent matter, Nature 470, 475-476 (2011)
 ...
Answer: to detect
Magnetorotational instability
in the laboratory
Seite 12
Oleg N. Kirillov, http://www.onkirillov.com
1959: Evgeny Velikhov discovers
new instability of the magnetized CT-flow
Magnetorotational instability (MRI)




CT-flow of inviscid fluid
Perfect electrical conductor
Uniform axial magnetic field
Axisymmetric perturbation
Sufficient condition of stability
dÐ2
> 0 or Ð2 > Ð1
dR
E. P. Velikhov, Sov. Phys. JETP 36 995–998 (1959)
Seite 13
Oleg N. Kirillov, http://www.onkirillov.com
Paradox of Velikhov (1959) and Chandrasekhar (1960)
Couette-Taylor flow
Ð1
When magnetic field
tends to zero, the MRI
domain does not shrink
to the domain of
centrifugal instability
Stability
Ð2
MRI
Centrifugal instability
Velikhov-Chandrasekhar
Rayleigh
R22Ð2 < R12Ð1
Ð2 < Ð1
S. Chandrasekhar, Proc. Natl. Acad. Sci. 46, 253–257 (1960)
Seite 14
Oleg N. Kirillov, http://www.onkirillov.com
1991: Steven Balbus and John Hawley established
the astrophysical relevance of MRI
Observations: flow in accretion discs is turbulent
Disc flow follows Keplerian orbits
Ð(r) = (GM)1=2=r3=2;
dÐ=dr < 0
Disc flow is centrifugally stable
d(r2Ð)=dr > 0;
Re > 1012
S.A. Balbus and J. F. Hawley, Astrophys. J. 376, 214–222 (1991).
Seite 15
Oleg N. Kirillov, http://www.onkirillov.com
HH30 By HST
Accretion
Disk + Black Hole in the Core of Galaxy NGC 4261
Accretion disc is gas, dust, and plasma accumulated by massive stars
and black holes. Accretion is responsible for many important
astrophysical processes:
 Star and planet formation
 Mass transfer in binary systems
 Huge amounts of radiation from quasars and active galactic nuclei
S.A. Balbus and J.F. Hawley, Rev. Mod. Phys. 70, 1–53 (1998)
Seite 16
Oleg N. Kirillov, http://www.onkirillov.com
Mechanism of MRI in accretion discs
Fluid is a perfect conductor
Magnetic field lines frozen into the fluid
A line ‘tethers’ a couple of fluid particles
Inner mass retarded, goes to lower orbit
Outer mass accelerated, goes to higher orbit
Computations
Balbus and Hawley
1991
S.A. Balbus and J. F. Hawley, Astrophys. J. 376, 214–222 (1991).
Seite 17
Oleg N. Kirillov, http://www.onkirillov.com
Leading order WKB equations for the onset of MRI
Ã
!
¯
dÐ ¯
2
x
Ä ¡ 2Ð0y_ + R0
+
!
A x=0
¯
dR R=R0
(Hill equation
2¯
for two tethered
2
yÄ + 2Ð0x_ + !A
y =0
– Rossby number
Solid body:
Keplerian disc:
Seite 18
W(R) = const,
W(R) ~ R-3/2,
¯
1 R dÐ ¯¯
Ro :=
2 Ð dR ¯R=R0
RoSolid = 0
RoKepler = -0.75
Rayleigh line:
RoRayleigh = -1
Onset of the divergence instability:
2
!A
Ro = ¡ 2
4Ð0
Velikhov-Chandrasekhar
paradox:
Ro = 0 6= ¡1
wA = 0,
satellites)
Oleg N. Kirillov, http://www.onkirillov.com
Experiments on MRI with the Couette-Taylor cells
Despite the evident successes of the numerical modeling of MRI, the
range of parameters that are typical in the astrophysical applications
had still not been reached in the computer simulations. By this reason,
increasing efforts are being taken during the last decade in order to
reproduce the MRI in the laboratory.
PROMISE
Helical MRI
Princeton
Seite 19
Maryland
Oleg N. Kirillov, http://www.onkirillov.com
Dresden
Standard MRI: Not observed yet
Helical MRI: Successful PROMISE experiment since 2006
Stefani, F. et al., Phys. Rev. Lett. 97(2006), 184502
Rüdiger, G. et al., Astrophys.J. Lett. 649 (2006), L145
Stefani, F. et al., New J. Phys. 9 (2007), 295
Stefani, F. et al., Astron. Nachr. 329 (2008), 652-658
Priede, J., Gerbeth, G., Phys. Rev. E 79 (2009), 046310
Stefani et al., Phys. Rev. E 80 (2009), 066303
Seite 20
Oleg N. Kirillov, http://www.onkirillov.com
Standard, Helical, and Azimuthal MRI
Standard MRI (SMRI) – axial magnetic field perpendicular to
the disc
Helical MRI (HMRI) – helical magnetic field, i.e. axial plus
azimuthal (parallel to the disc) magnetic field
Standard MRI – theoretically exists for Keplerian discs,
however it is difficult for observation in the laboratory (Re~106)
Helical MRI – detected in the experiment (2006) owing to
substantially more moderate requirements for rotating speeds
than in case of SMRI, role in accretion discs is not clear
What laws of differential rotation are susceptible to the
instabilities?
Seite 21
Oleg N. Kirillov, http://www.onkirillov.com
A. Rotational flow in an azimuthal magnetic field
Seite 22
Oleg N. Kirillov, http://www.onkirillov.com
Observation of inductionless azimuthal MRI with m=1,-1
Recently the PROMISE
experiment has shown
expected wave structure
and confirmed the
critical current of 10 kA
M. Seilmeier et al. Phys. Rev. Lett. 113(2), 024505 (2014)
Seite 23
Oleg N. Kirillov, http://www.onkirillov.com
Start with the incompressible MHD equations
Navier-Stokes equation
for the fluid velocity u
+
Induction equation
for the magnetic field B
@u
1
1
+ u ¢ ru ¡
B ¢ rB + rP ¡ ºr2u = 0
@t
¹0 ½
½
@B
+ u ¢ rB ¡ B ¢ ru ¡ ´r2 B = 0
@t
Incompressibility and solenoidality
r ¢ u = 0;
r¢B = 0
p : pressure, r = const : density, n = const : kinematic viscosity
h = (m0s)-1 : magnetic diffusivity, s : conductivity of the fluid
m0 : magnetic permeability of free space
O.N.K., F. Stefani Phys. Rev. Lett. 111(6), 061103 (2013)
Seite 24
Oleg N. Kirillov, http://www.onkirillov.com
Assume the magnetized Taylor-Couette flow as a
steady state
u0 (R) = R Ð(R) eÁ ;
General perturbation:
p = p0 (R);
u = u0 + u0 ;
B0 (R) = BÁ0 (R)eÁ
p = p0 + p0 ;
B = B0 + B0
Hydrodynamic (Ro) and magnetic (Rb) Rossby numbers:
Ro =
R @Ð
2Ð @R
Rb =
R
@ (BÁ =R)
2 (BÁ =R)
@R
For the current-free azimuthal field
BÁ(R) / R¡1 ) Rb = ¡1
For the Keplerian flow profle
Ð(R) / R¡3=2 ) Ro = ¡3=4
O.N.K., F. Stefani Phys. Rev. Lett. 111(6), 061103 (2013)
Seite 25
Oleg N. Kirillov, http://www.onkirillov.com
Assume the magnetized Taylor-Couette flow as a
steady state
Rotating flow
u0 (R) = R Ð(R) eÁ ;
General perturbation:
p = p0 (R);
u = u0 + u0 ;
B0 (R) = BÁ0 (R)eÁ
p = p0 + p0 ;
B = B0 + B0
Hydrodynamic (Ro) and magnetic (Rb) Rossby numbers:
Ro =
R @Ð
2Ð @R
Rb =
R
@ (BÁ =R)
2 (BÁ =R)
@R
For the current-free azimuthal field
BÁ(R) / R¡1 ) Rb = ¡1
For the Keplerian flow profle
Ð(R) / R¡3=2 ) Ro = ¡3=4
O.N.K., F. Stefani Phys. Rev. Lett. 111(6), 061103 (2013)
Seite 26
Oleg N. Kirillov, http://www.onkirillov.com
Assume the magnetized Taylor-Couette flow as a
steady state
Rotating flow
Azimuthal field
u0 (R) = R Ð(R) eÁ ;
General perturbation:
p = p0 (R);
u = u0 + u0 ;
B0 (R) = BÁ0 (R)eÁ
p = p0 + p0 ;
B = B0 + B0
Hydrodynamic (Ro) and magnetic (Rb) Rossby numbers:
Ro =
R @Ð
2Ð @R
Rb =
R
@ (BÁ =R)
2 (BÁ =R)
@R
For the current-free azimuthal field
BÁ(R) / R¡1 ) Rb = ¡1
For the Keplerian flow profle
Ð(R) / R¡3=2 ) Ro = ¡3=4
O.N.K., F. Stefani Phys. Rev. Lett. 111(6), 061103 (2013)
Seite 27
Oleg N. Kirillov, http://www.onkirillov.com
Assume the magnetized Taylor-Couette flow as a
steady state
Rotating flow
Azimuthal field
u0 (R) = R Ð(R) eÁ ;
General perturbation:
p = p0 (R);
u = u0 + u0 ;
B0 (R) = BÁ0 (R)eÁ
p = p0 + p0 ;
B = B0 + B0
Hydrodynamic (Ro) and magnetic (Rb) Rossby numbers:
Ro =
R @Ð
2Ð @R
Rb =
R
@ (BÁ =R)
2 (BÁ =R)
@R
For the current-free azimuthal field
BÁ(R) / R¡1 ) Rb = ¡1
For the Keplerian flow profle
Ð(R) / R¡3=2 ) Ro = ¡3=4
O.N.K., F. Stefani Phys. Rev. Lett. 111(6), 061103 (2013)
Seite 28
Oleg N. Kirillov, http://www.onkirillov.com
Assume the magnetized Taylor-Couette flow as a
steady state
Rotating flow
Azimuthal field
u0 (R) = R Ð(R) eÁ ;
General perturbation:
p = p0 (R);
u = u0 + u0 ;
B0 (R) = BÁ0 (R)eÁ
p = p0 + p0 ;
B = B0 + B0
Hydrodynamic (Ro) and magnetic (Rb) Rossby numbers:
Ro =
R @Ð
2Ð @R
Rb =
R
@ (BÁ =R)
2 (BÁ =R)
@R
For the current-free azimuthal field
BÁ(R) / R¡1 ) Rb = ¡1
For the Keplerian flow profle
Ð(R) / R¡3=2 ) Ro = ¡3=4
O.N.K., F. Stefani Phys. Rev. Lett. 111(6), 061103 (2013)
Seite 29
Oleg N. Kirillov, http://www.onkirillov.com
Assume the magnetized Taylor-Couette flow as a
steady state
Rotating flow
Azimuthal field
u0 (R) = R Ð(R) eÁ ;
General perturbation:
p = p0 (R);
u = u0 + u0 ;
B0 (R) = BÁ0 (R)eÁ
p = p0 + p0 ;
B = B0 + B0
Hydrodynamic (Ro) and magnetic (Rb) Rossby numbers:
Ro =
R @Ð
2Ð @R
Rb =
R
@ (BÁ =R)
2 (BÁ =R)
@R
For the current-free azimuthal field
BÁ(R) / R¡1 ) Rb = ¡1
For the Keplerian flow profle
Ð(R) / R¡3=2 ) Ro = ¡3=4
O.N.K., F. Stefani Phys. Rev. Lett. 111(6), 061103 (2013)
Seite 30
Oleg N. Kirillov, http://www.onkirillov.com
Assume the magnetized Taylor-Couette flow as a
steady state
Rotating flow
Azimuthal field
u0 (R) = R Ð(R) eÁ ;
General perturbation:
p = p0 (R);
u = u0 + u0 ;
B0 (R) = BÁ0 (R)eÁ
p = p0 + p0 ;
B = B0 + B0
Hydrodynamic (Ro) and magnetic (Rb) Rossby numbers:
Ro =
R @Ð
2Ð @R
Rb =
R
@ (BÁ =R)
2 (BÁ =R)
@R
For the current-free azimuthal field
BÁ(R) / R¡1 ) Rb = ¡1
For the Keplerian flow profle
Ð(R) / R¡3=2 ) Ro = ¡3=4
O.N.K., F. Stefani Phys. Rev. Lett. 111(6), 061103 (2013)
Seite 31
Oleg N. Kirillov, http://www.onkirillov.com
Linearize the MHD equations about the steady state
µ
2
@t +U+u0 ¢ r¡ºr
B ¡ B0 ¢ r
0 ¢r
¡ B+B
½¹0
¶µ
@t ¡U+u0 ¢ r¡´r2
r ¢ u0 = 0;
0
u
B0
Ã
¶
+
r
½
0
p
(B0 ¢B0 )
+ ¹0
!
0
r ¢ B0 = 0
Gradients of the background fields
0
1
0
¡1 0
U(R) = ru0 = Ð @ 1 + 2Ro 0 0 A ;
0
0 0
0
1
0
¡1 0
@ 1 + 2Rb 0 0 A
B(R) = rB 0 =
R
0
0 0
BÁ0
O.N.K., F. Stefani Phys. Rev. Lett. 111(6), 061103 (2013)
Seite 32
Oleg N. Kirillov, http://www.onkirillov.com
=0
Introduce a ‘fast’ phase and a ‘slow’ amplitude
Geometrical optics asymptotic series
0
i©(x;t)=²
u (x; t; ²) = e
(0 < ² ¿ 1)
³
´
(0)
(1)
u (x; t) + ²u (x; t) + ²ur (x; t)
º = ²2 ºe
³
´
0
(0)
(1)
i©(x;t)=²
B (x; t; ²) = e
B (x; t) + ²B (x; t) + ²Br (x; t)
´ = ²2 é
³
´
0
i©(x;t)=²
(0)
(1)
p (x; t; ²) = e
p (x; t) + ²p (x; t) + ²pr (x; t)
Ã
²¡1 :
0 ¢r©)
¡ (B½¹
0
@t © + (u0 ¢ r©)
¡(B0 ¢ r©)
@t © + (u0 ¢ r©)
!µ
(0)
u
B(0)
Ã
¶
+
r©
½
S. Friedlander, M. M. Vishik Chaos 5, 416–423 (1995)
Seite 33
Oleg N. Kirillov, http://www.onkirillov.com
(0)
p
+
(B0 ¢B(0) )
¹0
0
!
=0
Introduce a ‘fast’ phase and a ‘slow’ amplitude
Geometrical optics asymptotic series
0
i©(x;t)=²
u (x; t; ²) = e
(0 < ² ¿ 1)
³
´
(0)
(1)
u (x; t) + ²u (x; t) + ²ur (x; t)
º = ²2 ºe
³
´
0
(0)
(1)
i©(x;t)=²
B (x; t; ²) = e
B (x; t) + ²B (x; t) + ²Br (x; t)
´ = ²2 é
³
´
0
i©(x;t)=²
(0)
(1)
p (x; t; ²) = e
p (x; t) + ²p (x; t) + ²pr (x; t)
Ã
²¡1 :
0 ¢r©)
¡ (B½¹
0
@t © + (u0 ¢ r©)
¡(B0 ¢ r©)
@t © + (u0 ¢ r©)
!µ
(0)
u
B(0)
Ã
¶
+
r©
½
(0)
p
+
(B0 ¢B(0) )
¹0
0
Particular solution
@t © + (u0 ¢ r©) = 0;
(B 0 ¢ r©) = 0;
(0)
p
(B 0 ¢ B (0) )
+
=0
¹0
S. Friedlander, M. M. Vishik Chaos 5, 416–423 (1995)
Seite 34
Oleg N. Kirillov, http://www.onkirillov.com
!
=0
Introduce a ‘fast’ phase and a ‘slow’ amplitude
Geometrical optics asymptotic series
0
i©(x;t)=²
u (x; t; ²) = e
(0 < ² ¿ 1)
³
´
(0)
(1)
u (x; t) + ²u (x; t) + ²ur (x; t)
³
´
0
(0)
(1)
i©(x;t)=²
B (x; t; ²) = e
B (x; t) + ²B (x; t) + ²Br (x; t)
³
´
0
i©(x;t)=²
(0)
(1)
p (x; t; ²) = e
p (x; t) + ²p (x; t) + ²pr (x; t)
Ã
²0 : i
r©
½
(1)
p
+
(B0 ¢B(1) )
¹0
0
´ = ²2 é
! µ
¶ µ (0) ¶
0 ¢r
@t + U + u0 ¢ r + º~(r©)2
¡ B+B
u
½¹0
+
=0
B ¡ B0 ¢ r
@t ¡ U + u0 ¢ r + ´~(r©)2
B(0)
S. Friedlander, M. M. Vishik Chaos 5, 416–423 (1995)
Seite 35
º = ²2 ºe
Oleg N. Kirillov, http://www.onkirillov.com
Introduce a ‘fast’ phase and a ‘slow’ amplitude
Geometrical optics asymptotic series
0
i©(x;t)=²
u (x; t; ²) = e
(0 < ² ¿ 1)
³
´
(0)
(1)
u (x; t) + ²u (x; t) + ²ur (x; t)
³
´
0
(0)
(1)
i©(x;t)=²
B (x; t; ²) = e
B (x; t) + ²B (x; t) + ²Br (x; t)
³
´
0
i©(x;t)=²
(0)
(1)
p (x; t; ²) = e
p (x; t) + ²p (x; t) + ²pr (x; t)
Ã
²0 : i
r©
½
(1)
p
+
(B0 ¢B(1) )
¹0
0
º = ²2 ºe
´ = ²2 é
! µ
¶ µ (0) ¶
0 ¢r
@t + U + u0 ¢ r + º~(r©)2
¡ B+B
u
½¹0
+
=0
B ¡ B0 ¢ r
@t ¡ U + u0 ¢ r + ´~(r©)2
B(0)
Isolate pressure, substitute it back
Ã
ir©
(1)
p
(B0 ¢ B
+
½
½¹0
(1)
!
)
·
¸
r©(r©)T
B
+
B
¢
r
0
=¡
2Uu(0) ¡
B(0)
2
jr©j
½¹0
S. Friedlander, M. M. Vishik Chaos 5, 416–423 (1995)
Seite 36
Oleg N. Kirillov, http://www.onkirillov.com
Introduce a ‘fast’ phase and a ‘slow’ amplitude
Geometrical optics asymptotic series
0
i©(x;t)=²
u (x; t; ²) = e
(0 < ² ¿ 1)
³
´
(0)
(1)
u (x; t) + ²u (x; t) + ²ur (x; t)
³
´
0
(0)
(1)
i©(x;t)=²
B (x; t; ²) = e
B (x; t) + ²B (x; t) + ²Br (x; t)
³
´
0
i©(x;t)=²
(0)
(1)
p (x; t; ²) = e
p (x; t) + ²p (x; t) + ²pr (x; t)
Ã
²0 : i
r©
½
(1)
p
+
(B0 ¢B(1) )
¹0
0
º = ²2 ºe
´ = ²2 é
! µ
¶ µ (0) ¶
0 ¢r
@t + U + u0 ¢ r + º~(r©)2
¡ B+B
u
½¹0
+
=0
B ¡ B0 ¢ r
@t ¡ U + u0 ¢ r + ´~(r©)2
B(0)
Isolate pressure, substitute it back
Ã
ir©
(1)
p
(B0 ¢ B
+
½
½¹0
(1)
!
)
·
¸
r©(r©)T
B
+
B
¢
r
0
=¡
2Uu(0) ¡
B(0)
2
jr©j
½¹0
S. Friedlander, M. M. Vishik Chaos 5, 416–423 (1995)
Seite 37
Oleg N. Kirillov, http://www.onkirillov.com
Introduce a ‘fast’ phase and a ‘slow’ amplitude
Phase equation:
k = r©
r@t © + r(u0 ¢ r©) = 0
D
= @t + u0 ¢ r
Dt
(B0 ¢ k) = 0
Transport equations for amplitudes:
(0)
Ã
T
Du
kk
=¡ I ¡2 2
Dt
jkj
!
Ã
Uu(0) ¡ º~jkj2u(0) +
1
½¹0
I¡
T
!
kk
jkj2
(B + B0 ¢ r) B(0)
DB(0)
= UB(0) ¡ ´~jkj2B(0) ¡ (B ¡ B0 ¢ r)u(0)
Dt
S. Friedlander, M. M. Vishik Chaos 5, 416–423 (1995)
Seite 38
Oleg N. Kirillov, http://www.onkirillov.com
Introduce a ‘fast’ phase and a ‘slow’ amplitude
Phase equation:
k = r©
r@t © + r(u0 ¢ r©) = 0
¡
¢
@t r© + u0 ¢ r + U T r© = 0
D
= @t + u0 ¢ r
Dt
(B0 ¢ k) = 0
Transport equations for amplitudes:
(0)
Ã
T
Du
kk
=¡ I ¡2 2
Dt
jkj
!
Ã
Uu(0) ¡ º~jkj2u(0) +
1
½¹0
I¡
T
!
kk
jkj2
(B + B0 ¢ r) B(0)
DB(0)
= UB(0) ¡ ´~jkj2B(0) ¡ (B ¡ B0 ¢ r)u(0)
Dt
S. Friedlander, M. M. Vishik Chaos 5, 416–423 (1995)
Seite 39
Oleg N. Kirillov, http://www.onkirillov.com
Introduce a ‘fast’ phase and a ‘slow’ amplitude
Phase equation:
k = r©
r@t © + r(u0 ¢ r©) = 0
¡
¢
@t r© + u0 ¢ r + U T r© = 0
D
= @t + u0 ¢ r
Dt
Dk
= ¡U T k
Dt
(B0 ¢ k) = 0
Transport equations for amplitudes:
(0)
Ã
T
Du
kk
=¡ I ¡2 2
Dt
jkj
!
Ã
Uu(0) ¡ º~jkj2u(0) +
1
½¹0
I¡
T
!
kk
jkj2
(B + B0 ¢ r) B(0)
DB(0)
= UB(0) ¡ ´~jkj2B(0) ¡ (B ¡ B0 ¢ r)u(0)
Dt
S. Friedlander, M. M. Vishik Chaos 5, 416–423 (1995)
Seite 40
Oleg N. Kirillov, http://www.onkirillov.com
Introduce a ‘fast’ phase and a ‘slow’ amplitude
Phase equation:
Dk
= ¡U T k
Dt
k = r©
D
= @t + u0 ¢ r
Dt
(B0 ¢ k) = 0
Transport equations for amplitudes:
(0)
Ã
T
Du
kk
=¡ I ¡2 2
Dt
jkj
!
Ã
Uu(0) ¡ º~jkj2u(0) +
1
½¹0
I¡
T
!
kk
jkj2
(B + B0 ¢ r) B(0)
DB(0)
= UB(0) ¡ ´~jkj2B(0) ¡ (B ¡ B0 ¢ r)u(0)
Dt
S. Friedlander, M. M. Vishik Chaos 5, 416–423 (1995)
Seite 41
Oleg N. Kirillov, http://www.onkirillov.com
Introduce a ‘fast’ phase and a ‘slow’ amplitude
Phase equation:
Dk
= ¡U T k
Dt
k = r©
k_ R = ¡R@R ÐkÁ ;
k_ Á = 0;
D
= @t + u0 ¢ r
Dt
k_ z = 0
(B0 ¢ k) = 0
Transport equations for amplitudes:
(0)
Ã
T
Du
kk
=¡ I ¡2 2
Dt
jkj
!
Ã
Uu(0) ¡ º~jkj2u(0) +
1
½¹0
I¡
T
!
kk
jkj2
(B + B0 ¢ r) B(0)
DB(0)
= UB(0) ¡ ´~jkj2B(0) ¡ (B ¡ B0 ¢ r)u(0)
Dt
S. Friedlander, M. M. Vishik Chaos 5, 416–423 (1995)
Seite 42
Oleg N. Kirillov, http://www.onkirillov.com
Introduce a ‘fast’ phase and a ‘slow’ amplitude
Phase equation:
Dk
= ¡U T k
Dt
k = r©
k_ R = ¡R@R ÐkÁ ;
k_ Á = 0;
kR = const;
D
= @t + u0 ¢ r
Dt
k_ z = 0
kÁ ´ 0;
kz = const
(B0 ¢ k) = 0
Transport equations for amplitudes:
(0)
Ã
T
Du
kk
=¡ I ¡2 2
Dt
jkj
!
Ã
Uu(0) ¡ º~jkj2u(0) +
1
½¹0
I¡
T
!
kk
jkj2
(B + B0 ¢ r) B(0)
DB(0)
= UB(0) ¡ ´~jkj2B(0) ¡ (B ¡ B0 ¢ r)u(0)
Dt
S. Friedlander, M. M. Vishik Chaos 5, 416–423 (1995)
Seite 43
Oleg N. Kirillov, http://www.onkirillov.com
Write the transport equations in cylindrical coordinates
®=
kz
;
jkj
2
jkj2 = kR
+ kz2
Bounded and asymptotically non-decaying solution
to the phase equation: kÁ ´ 0; kR; kz = const
(0)
Solenoidality: kR BR + kz Bz(0) = 0
(0)
(0)
¡
¢ (0)
B0 B
BÁ0 @Á BR
(0)
2
2
2 Á Á
@t + Ð@Á + ºejkj uR ¡ 2® ÐuÁ + 2®
¡
=0
R ½¹0
R ½¹0
(0)
(0)
¡
¢ (0)
2BÁ0
BÁ0 @Á BÁ
BR
(0)
2
@t + Ð@Á + ºejkj uÁ + 2Ð(1 + Ro)uR ¡
(1 + Rb)
¡
R
½¹0
R ½¹0
=0
¡
¢ (0) BÁ0
(0)
2
@t + Ð@Á + éjkj BR ¡
@ÁuR = 0
R
¡
¢ (0)
BÁ0 (0) BÁ0
(0)
(0)
2
@t + Ð@Á + éjkj BÁ ¡ 2ÐRoBR + 2Rb uR ¡
@ÁuÁ = 0
R
R
B. Eckhardt, D. Yao Chaos, Solitons & Fractals 5, 2073–2088 (1995)
Seite 44
Oleg N. Kirillov, http://www.onkirillov.com
Solve the amplitude equations in the modal form
be°t+imÁ;
u(0) = u
b °t+imÁ
B(0) = Be
Viscous & resistive frequencies and Alfven angular velocity
!º = ºejkj2;
Hz = °z;
!´ = éjkj2 ;
where
!AÁ
BÁ0
= p
;
R ½¹0
Rb =
R
@R!AÁ
2!AÁ
bR ; B
bÁ )T
z = (b
uR ; u
bÁ ; B
0
B ¡imÐ ¡ !º
B
¡2Ð(1 + Ro)
H=B
B
@ im!AÁ p½¹0
p
¡2!AÁ Rb ½¹0
m!A
i p½¹0Á
2®2 Ð
¡imÐ ¡ !º
0
p
im!AÁ ½¹0
2!AÁ
p
½¹0 (1
+ Rb)
¡imÐ ¡ !´
0
2ÐRo
¡imÐ ¡ !´
O.N.K., F. Stefani, Y. Fukumoto Fluid Dyn. Res. 46, 031403 (2014)
Seite 45
2!AÁ ®2
¡ p½¹0
m!A
i p½¹0Á
Oleg N. Kirillov, http://www.onkirillov.com
1
C
C
C
C
A
Solve the amplitude equations in the modal form
be°t+imÁ;
u(0) = u
b °t+imÁ
B(0) = Be
Viscous & resistive frequencies and Alfven angular velocity
!º = ºejkj2;
Hz = °z;
!´ = éjkj2 ;
where
!AÁ
BÁ0
= p
;
R ½¹0
Rb =
R
@R!AÁ
2!AÁ
bR ; B
bÁ )T
z = (b
uR ; u
bÁ ; B
0
B ¡imÐ ¡ !º
B
¡2Ð(1 + Ro)
H=B
B
@ im!AÁ p½¹0
p
¡2!AÁ Rb ½¹0
m!A
i p½¹0Á
2®2 Ð
¡imÐ ¡ !º
0
p
im!AÁ ½¹0
2!AÁ
p
½¹0 (1
+ Rb)
¡imÐ ¡ !´
0
2ÐRo
¡imÐ ¡ !´
O.N.K., F. Stefani, Y. Fukumoto Fluid Dyn. Res. 46, 031403 (2014)
Seite 46
2!AÁ ®2
¡ p½¹0
m!A
i p½¹0Á
Oleg N. Kirillov, http://www.onkirillov.com
1
C
C
C
C
A
Solve the amplitude equations in the modal form
be°t+imÁ;
u(0) = u
b °t+imÁ
B(0) = Be
Viscous & resistive frequencies and Alfven angular velocity
!º = ºejkj2;
Hz = °z;
!´ = éjkj2 ;
where
!AÁ
BÁ0
= p
;
R ½¹0
Rb =
R
@R!AÁ
2!AÁ
bR ; B
bÁ )T
z = (b
uR ; u
bÁ ; B
0
B ¡imÐ ¡ !º
B
¡2Ð(1 + Ro)
H=B
B
@ im!AÁ p½¹0
p
¡2!AÁ Rb ½¹0
m!A
i p½¹0Á
2®2 Ð
¡imÐ ¡ !º
0
p
im!AÁ ½¹0
2!AÁ
p
½¹0 (1
+ Rb)
¡imÐ ¡ !´
0
2ÐRo
¡imÐ ¡ !´
O.N.K., F. Stefani, Y. Fukumoto Fluid Dyn. Res. 46, 031403 (2014)
Seite 47
2!AÁ ®2
¡ p½¹0
m!A
i p½¹0Á
Oleg N. Kirillov, http://www.onkirillov.com
1
C
C
C
C
A
Solve the amplitude equations in the modal form
be°t+imÁ;
u(0) = u
b °t+imÁ
B(0) = Be
Viscous & resistive frequencies and Alfven angular velocity
!º = ºejkj2;
Hz = °z;
!´ = éjkj2 ;
where
!AÁ
BÁ0
= p
;
R ½¹0
Rb =
R
@R!AÁ
2!AÁ
bR ; B
bÁ )T
z = (b
uR ; u
bÁ ; B
0
B ¡imÐ ¡ !º
B
¡2Ð(1 + Ro)
H=B
B
@ im!AÁ p½¹0
p
¡2!AÁ Rb ½¹0
m!A
i p½¹0Á
2®2 Ð
¡imÐ ¡ !º
0
p
im!AÁ ½¹0
2!AÁ
p
½¹0 (1
+ Rb)
¡imÐ ¡ !´
0
2ÐRo
¡imÐ ¡ !´
O.N.K., F. Stefani, Y. Fukumoto Fluid Dyn. Res. 46, 031403 (2014)
Seite 48
2!AÁ ®2
¡ p½¹0
m!A
i p½¹0Á
Oleg N. Kirillov, http://www.onkirillov.com
1
C
C
C
C
A
Get the dispersion relation for the non-ideal MHD
(Kirillov, Stefani & Fukumoto ‘14)
p(°) := det(H ¡ °I) = 0
0
B ¡imÐ ¡ !º
B
¡2Ð(1 + Ro)
H=B
B
@ im!AÁ p½¹0
p
¡2!AÁ Rb ½¹0
m!A
i p½¹0Á
2®2 Ð
¡imÐ ¡ !º
0
p
im!AÁ ½¹0
2!AÁ
p
½¹0 (1
2!AÁ ®2
¡ p½¹0
m!A
i p½¹0Á
+ Rb)
¡imÐ ¡ !´
0
2ÐRo
¡imÐ ¡ !´
1
C
C
C
C
A
Ideal MHD (Friedlander & Vishik ‘95, Ogilvie & Pringle ‘96)
!º = 0;
2
p(°) = 4®
2
2
(!A
Rb
¡
Ð
Ro)
Á
!´ = 0
³
´
2
2 2
(i° ¡ mÐ) ¡ m !AÁ
³
´2 ³
´2
2
2
2 2
¡4® Ð(i° ¡ mÐ) + m!AÁ + (i° ¡ mÐ) ¡ m !AÁ = 0
2
G. I. Ogilvie, J. E. Pringle Mon. Not. R. Astron. Soc. 279, 152–164 (1996)
Seite 49
Oleg N. Kirillov, http://www.onkirillov.com
Get the dispersion relation for the non-ideal MHD
(Kirillov, Stefani & Fukumoto ‘14)
p(°) := det(H ¡ °I) = 0
0
B ¡imÐ ¡ !º
B
¡2Ð(1 + Ro)
H=B
B
@ im!AÁ p½¹0
p
¡2!AÁ Rb ½¹0
m!A
i p½¹0Á
2®2 Ð
¡imÐ ¡ !º
0
p
im!AÁ ½¹0
2!AÁ
p
½¹0 (1
2!AÁ ®2
¡ p½¹0
m!A
i p½¹0Á
+ Rb)
¡imÐ ¡ !´
0
2ÐRo
¡imÐ ¡ !´
1
C
C
C
C
A
Ideal MHD (Friedlander & Vishik ‘95, Ogilvie & Pringle ‘96)
!º = 0;
2
p(°) = 4®
2
2
(!A
Rb
¡
Ð
Ro)
Á
!´ = 0
³
´
2
2 2
(i° ¡ mÐ) ¡ m !AÁ
³
´2 ³
´2
2
2
2 2
¡4® Ð(i° ¡ mÐ) + m!AÁ + (i° ¡ mÐ) ¡ m !AÁ = 0
2
G. I. Ogilvie, J. E. Pringle Mon. Not. R. Astron. Soc. 279, 152–164 (1996)
Seite 50
Oleg N. Kirillov, http://www.onkirillov.com
B. Chandrasekhar’s equipartition solution
Seite 51
Oleg N. Kirillov, http://www.onkirillov.com
Look closer at the remarkable relation
2
2
!A
Rb
=
Ð
Ro
Á
Under the above constraint the roots of the ideal MHD
dispersion relation are always imaginary (marginal stability)
r
°1;2 = ¡i(m + ®)Ð § i
(m +
+
2
®)2 !A
Á
³
´
2
+ ®2 Ð2 ¡ !AÁ
r
°3;4 = ¡i(m ¡ ®)Ð § i
Chandrasekhar’s
equipartition solution:
(m ¡
³
´
2
2
Ð ¡ !AÁ
2
®)2 !A
Á
®2
!AÁ = Ð;
°1;4 = 0; °2;3 = ¡2i(m § ®)Ð
S. Chandrasekhar, PNAS, (1956) 42, 273-276
Seite 52
Oleg N. Kirillov, http://www.onkirillov.com
Rb = Ro = ¡1
Recall the Chandrasekhar’s theorem ‘56
“A special stationary solution of MHD equations in the case of
ideal incompressible infinitely conducting fluid with
(a) total constant pressure,
for which
(b) the fluid velocity is parallel to the direction of the magnetic field
and
(c) magnetic and kinetic energies are finite and equal,
is marginally stable”
What is the influence of viscosity and magnetic diffusivity
on the stability of Chandrasekhar’s equipartition solution?
S. Chandrasekhar, PNAS, (1956) 42, 273-276
Seite 53
Oleg N. Kirillov, http://www.onkirillov.com
C. AMRI as a dissipation-induced instability
Seite 54
Oleg N. Kirillov, http://www.onkirillov.com
Use the Bilharz criterion to derive the instability threshold
Modified azimuthal wavenumber and magnetic Reynolds number:
m
n= ;
®
Ð
Rm = ®
!´
Assume:
!AÁ = Ð;
Instability threshold:
Ro = Rb;
!º = 0
16(n2 ¡ Rb2 )(n2 ¡ Rb ¡ 2)2Rm4
+(n6 ¡ 12n2 Rb2 + 32n2 (Rb + 1) ¡ 16Rb2 (Rb + 2))Rm2
¡4Rb2 + 4n2(Rb + 1) = 0
O.N. Kirillov, Nonconservative stability problems of modern physics, De Gruyter, Berlin, Boston 2013
Seite 55
Oleg N. Kirillov, http://www.onkirillov.com
Magnetic diffusivity transforms marginal stability to AMRI
AMRI domain is the widest in the inductionless limit: Rm ! 0
AMRI domain vanishes in the ideal MHD limit: Rm ! 1
Rb = Ro = ¡
O.N.K., F. Stefani, Y. Fukumoto Journal of Fluid Mechanics, (2014) in press. arXiv:1401.8276
Seite 56
Oleg N. Kirillov, http://www.onkirillov.com
3
4
Inductionless AMRI of a Keplerian flow on the line Ro=Rb
Inductionless AMRI of a Keplerian flow
1 (Ro + 2)2
Rb > ¡
:
8 Ro + 1
3
25
Ro = ¡ ) Rb > ¡
4
32
O.N.K., F. Stefani Phys. Rev. Lett. 111(6), 061103 (2013)
Seite 57
Oleg N. Kirillov, http://www.onkirillov.com
Transition from the inductionless AMRI to the Tayler instability
Ð
Re = ® ;
!º
!AÁ
Hb = ® p
;
!º !´
Ro = ¡1;
1
Onset of TI: Hb = q
p
2Rb + 2 Rb2 + n2 ¡ n2
n ¼ 1:27
n2
At Rb =
¡ 1; Hb ! 1
4
O.N.K., F. Stefani, Y. Fukumoto Journal of Fluid Mechanics, (2014) in press. arXiv:1401.8276
Seite 58
Oleg N. Kirillov, http://www.onkirillov.com
Monograph on dissipation-induced instabilities (2013)
Seite 59
Oleg N. Kirillov, http://www.onkirillov.com
References
1.
2.
3.
4.
5.
Seite 60
Kirillov, O.N., Stefani, F. “Extending the range of the inductionless
magnetorotational instability”, Physical Review Letters, 111: 061103 (2013)
Kirillov, O.N. “Nonconservative stability problems of modern physics”
De Gruyter, Berlin, Boston, (2013) 429 pp
Kirillov, O.N., Stefani, F., Fukumoto, Y. “Instabilities of rotational flows in
azimuthal magnetic fields of arbitrary radial dependence“.
Fluid Dynamics Research, 46: 031403 (2014)
Kirillov, O.N., Stefani, F., Fukumoto, Y. “Inductionless magnetorotational
instability beyond the Liu limit“ in “Fundamental and Applied MHD,
Thermo Acoustic and Space Technologies” (eds. A. Alemany and
J. Freibergs), Proceedings of the 9th International PAMIR conference,
June 16-20, 2014, Riga, Latvia, 273-277
Kirillov, O.N., Stefani, F., Fukumoto, Y. Instabilities in magnetized rotational
flows: A comprehensive short-wavelength approach. Journal of Fluid Mechanics,
subm. (arXiv:1401.8276v1) (2014)
Oleg N. Kirillov, http://www.onkirillov.com