Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 1/29
Magnetic phase transition and
confinement regimes
Emilia R. Solano1,2, Richard D. Hazeltine3
1 Laboratorio Nacional de Fusión, CIEMAT, Madrid, Spain2JET EFDA CSU
3 Institute for Fusion Studies, Univ. of Texas at Austin, Austin, USA
15th European Fusion Theory Conference
Oxford, United Kingdom
23rd-26th September 2013
Acknowledgements: E. Fable (IPP Garching), P. McCarthy (U.C. Cork, Ireland)
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 2/29
Inspiration and motivation
• Since the discovery of the H-mode, many models of the L-H transition have
been proposed
• The now conventional model is based on electrostatic turbulent vortices
shredded by rotation shear. It is highly developed, can include sophisticated
predator-prey model, it is now beginning to move towards electromagnetic
consequences of electrostatic fluctuations=
• It is a very attractive model to many, but then I saw this movie of an L-H
transition in MAST
http://www.ccfe.ac.uk/videos.aspx?currVideo=24&currCateg=0
(L-mode 10-18 s, H-mode later)
and I started thinking about phase transitions.
One of the better studied phase transitions in physics
is the magnetic phase transition.
So that got me thinking some more#
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 3/29
Microturbulence
γ > 0
electrostatic
drift velocity
Sophisticated,
non-linear models
λ~ (1-10) ρs
MHD
γ > 0
electro-magnetic
Vthermal or VAlfvén
Linear models
Energy principle
λ~ p/p’ or λ~ a
Usual set-ups
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 4/29
Microturbulence
γ > 0
electrostatic
drift velocity
Sophisticated,
non-linear models
λ ~ (1-10) ρρρρs
MHD
γ > 0
electro-magnetic
Vthermal or VAlfvén
Linear models
Energy principle
λ~ p/p’ or λ~ a
Usual set-ups
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 5/29
Index
• Plasma overall magnetisation (cylindrical tokamak approximation)
• Magnetisation of “tubes” (field aligned pressure perturbations)
• Motion of magnetised tubes in magnetised plasma (Bz gradient)
• Effect on profiles
• Connections to interchange stability theory
• Experimentally testable criterion
• Data!
References:
E. R. Solano, Plasma Phys. Control. Fusion 46 L7 (2004)
E. R. Solano & R. D. Hazeltine Nucl. Fusion 52 114017 (2012)
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 6/29
Plasma Equilibrium
Plasma force balance:
p ζ θ θ ζ∇ = × = × + ×j B j B j B
( )( )0
0
dpp '
d
j R p ' FF'/ R
j F 'B , F( ) RB
ζ
θ θ ζ
≡Ψ
=− + µ
µ =− Ψ =
In cylindrical approximation :
2 2 2
z
0 0
B B Bdp
dr 2 r
θ θ + + =− µ µ
( )( )z 0 0 0
z0
j R p ' FF'/ R
dBj
drθ
=− + µ
µ =−
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 7/29
Plasma magnetization of a “cylindrical” tokamak
( )a
2 2
za z za za z0
2 2 2
a 0 a a
pdS B B 2B B B1
B / 2 B Bθ
θ θ θ
− −β ≡ = +
µ∫
≃
Integrating cylindrical force balance:
(βθ -1) is related to normalised average plasma magnetisation
Bz
r
Bz0
Bz
r
Bz0
βθ <1 Bz increased by jθparamagnetism,
low pressure
βθ> 1 Bz reduced by jθdiamagnetism,
high pressure
jp Bζ θ<∇ ×� �
jp Bζ θ>∇ ×� �
z0
dBj
drθ=−µ
Bz
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 8/29
MAGNETS
The tokamak plasma is a magnet.
the difference between poloidal magnetic and kinetic pressure
determines if it is a para-magnet or a dia-magnet
so far I have just reviewed well known facts
avac 2 vac
z z 0 a 0 z0
B B B / 2 pdS Bθ − ≅ µ µ − ∫
Paramagnets
increase the background magnetic field
move towards high field regions
Diamagnets
decrease the background magnetic field
move towards low field regions
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 9/29
Diamagnetic levitation
A frog (diamagnetic) dropped in a
strong magnetic field levitates because
it tries to get away from the high field.
It moves towards the lower field,
arranged to be upwards.
M V Berry and A K Geimz
Eur. J. Phys. 18 (1997) 307–313.
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 10/29
Magnetised plasma element
The poloidal current density around a
tokamak plasma is responsible for
plasma magnetisation: the difference
between the externally applied toroidal
field and the local toroidal field inside
the plasma
Next consider a field-
aligned plasma element
with a pressure
perturbation relative to
the background plasma
pressure.
What is its
magnetisation?
Not conventional current
“filaments”, with
parallel current density
X
Y
Kinetic
Pressure
Contours
para
dia
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 11/29
Magnetism in cylindrical tube with pressure hill/hole
B
ɶM
ρ
pɶ
( )dmn p
dt
ρρ ρ ρ= =−∇ + ×
vF j Bɶɶ
p
B⊥
×∇=
ɶɶ bj
Diamagnetic current:
if inside the tube there is a pressure hill (more pressure
than in the background plasma), the associated
perpendicular current reduces Bz: diamagnetism
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 12/29
p
B⊥
×∇=
ɶɶ bj
Magnetism in cylindrical tube with pressure hill/hole
B
ɶM
ɶM
ρ
pɶ
pɶ
Paramagnetic current:
if inside the tube there is a pressure hole (less pressure
than in the background plasma), the associated
perpendicular current increases Bz: paramagnetism
Diamagnetic current:
if inside the tube there is a pressure hill, the associated
perpendicular current reduces Bz: diamagnetism
( )dmn p
dt
ρρ ρ ρ= =−∇ + ×
vF j Bɶɶ
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 13/29
Magnetism in cylindrical tube with pressure hill/hole
0
||0
||
b p dMˆ
B dr
1 p( ') pd '
B ' B
ρ
×∇∇× =µ =−
∂ ρ= λ ρ ≈−λ ∂ρ∫
M r
bΜ b
ɶɶɶ
ɶ ɶɶ
Magnetization of the blob:
< 0, dia
> 0, para
B
ɶM
ɶM
Paramagnetic current:
if inside the tube there is a pressure hole, the associated
perpendicular current increases Bz: paramagnetism
ρ
pɶ
pɶ
p
B⊥
×∇=
ɶɶ bj
Diamagnetic current:
if inside the tube there is a pressure hill, the associated
perpendicular current reduces Bz: diamagnetism
( )dmn p
dt
ρρ ρ ρ= =−∇ + ×
vF j Bɶɶ
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 14/29
Movement of magnetized object in field gradient (see Jackson)
Bz0
( )
( )( )
V
V
0z
dm ( . ) dV
dt
dV B dV
= ∇
× ∇
∫
∫ ∫
vMB
r j
ɶ ≃
ɶ≃
the cold tube (paramagnetic) seeks high field
the hot tube (diamagnetic) seeks low field
Bz0
blob magnetization
0 0B B r. B ...= + ∇ +� � ��
averaged dBz/dr controls motion of magnetised plasma tubes:
Anti-potential leads to magnetic phase separation
rr
V z z0 0 z
dmn M B M j
dtθ∇ =−µ
vɶ ɶ≃
zMɶ
zMɶ
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 15/29
paramagnetic cold blobs move inward,
diamagnetic hot blobs move outward
Paramagnetic plasma: L-mode
Motion of pressure blobs depends on dBz/dr
0rV
dBdvmn M
dt dr
ζζɶ≃
p(r)
r
outward thermal energy convection
at the expense of
inward magnetic energy convection
r
p(r)
Bz(r)
Paramagnetism
blobs “grow”, “instability”
r
L-mode
pɶ
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 16/29
Diamagnetic plasma: H-mode
p(r)
r
rBz(r)
p(r)
diamagnetism
diamagnetic hot blobs move inward,
paramagnetic cold blobs move outward
Motion of pressure blobs depends on dBz/dr
0rV
dBdvmn M
dt dr
ζζɶ≃
inward thermal energy convection
at the expense of
outward magnetic energy convection
H-mode
blobs “decrease”, “saturation”pɶ
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 17/29
Magnetic Boundary: phase transition
p(r)
r
Bz(r)
r
increases somewhere,
creating diamagnetic region
at plasma edge.
p∇
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 18/29
At a magnetic phase boundary blobs
of the same type accumulate
diamagnetic blobs (heat) seek wells
paramagnetic blobs seek hilltops
With multiple blobs moving,
p and Bz profiles evolve
Magnetic Boundary: phase transition
p(r)
rBz(r)
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 19/29
Magnetic Boundary: phase transition
p(r)
r
Bz(r)
r
Magnetization,
of both signs, increases.
Phase transition is self-reinforcing.
Pressure pedestal forms, grows.
Pressure gradient
increases in diamagnetic region
Decreases in paramagnetic region
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 20/29
Evolution equations
Ideal 1 fluid MHD evolution magnetization force terms
( )
i
ni
dnm
dt
3 p. H
2 t
( )t
=
∂=−∇ +
∂∂
=∇× × −∇×η −∂
vF
Q
Bv B j j ( )
rV i 0z
M
r
M
zr z
M
dv dn m M B
dt dr
3 p d(p v )
2 t dr
v Bt
ζ=
∂=−
∂
∂=∇×
∂B
θ
ɶ
ɶ
For now, consider what the magnetisation force does,
disregard other transport mechanisms
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 21/29
Pedestal formation at magnetisation boundary
Assume dashed Bz(r), p(r) initial profiles
Ideal MHD with magnetization force
rV i 0z
M
r
M
zr 0z
M
dv dn m M B
dt dr
3 p d(p v )
2 t dr
B d(v B )
t dr
ζ=
∂=−
∂
∂=
∂
ɶ
ɶ
ɶ
Magnetic phase separation drives pedestal formation
Integrating one temporal step t∆
pressure steepens in diamagnetic regions,
increases diamagnetism
flattens in paramagnetic regions,
increases paramagnetism
p(r)
Bζ-Bζ0
Initial profiles - - -
Final profiles
~ζdB
drvr~ Mζ~
dia para dia
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 22/29
Recap and applicability conditions
• plasma tubes with an excess or defect of pressure are convected
radially, depending on the plasma magnetisation.
• phase separation occurs at the flux surface in which jq changes sign.
Under what conditions?
• The seed pressure perturbation is strong enough to protrude above or
below the background pressure profile.
• plasma elements must be long enough to average out the 1/R variation
of the vacuum field: λ||> qR. Otherwise conventional, ballooning-like
transport would drive short diamagnets towards low R.
• Edge pressure must be high enough to allow negative perturbations
as well as positive.
• Collisionality/resistivity: the temperature must be high enough for
particles to sample LFS and HFS before being scattered out of the
tube.
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 23/29
Caveats
• So far we have treated the plasma tubes as “test particles” with a
magnetic moment
• ignored geometrical magnetisation from j||
• No evolution equation for tube magnetisation.
• the magnetic interaction between plasma elements and bulk plasma is
quite complex, and our model very simple (too simple?)
• We hope that a more detailed calculation, up to second order on the
spatial variation of F=RBtor, can be carried out. Kind of neoclassical
magnetisation, instead of classical
• Much harder to do …
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 24/29
1M.N. Rosenbluth and C.L. Longmire, Annals of Physics, Volume 1, Issue 2, May 1957,120
light fluid
g
• present when a radial force acts equally on electrons and ions
• equivalent to the Rayleigh-Taylor instability in a fluid.
• magnetization gradient acting on magnetized plasma blobs replace
“gravitational field” or “curvature”.
Interchange instability1
heavy fluid
Magnetization interchange growth faster for
high magnetisation, strong seed, low field & mass
Magnetization interchange
g / ⊥γ = λ
m,blob
dB1 p 1
B dr
ɶ ζ
ρ
γ = −ρ λ
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 25/29
Suydam criterion for interchange instability
22 2
r
2
s 0
BRq q ''r 4q
κ β > µ
22 2
0zrz 2
s 0
dBBRq q '' Mr dr 4q
κ β + > µ ɶ
magnetic shear opposes interchange of tubes
driven by cylindrical curvature and
Generalization:
add magnetization force to cylindrical curvature
In magnetically mixed states
magnetisation force adds to curvature, instability,
until the magnetic shear q’ or the variation of dBz/dr changes.
0zz
dBM 0
dr<ɶ
B. R. Suydam, Proc. 2nd UN Conf. on Peaceful Uses of Atomic Energy, Geneva, 1958.
∇β
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 26/29
Evolution towards magnetic phase transition
As heating is applied, low pressure paramagnetic plasmas have
degraded confinement, driven by low ∇p
When sufficient heating is applied, ∇p grows until zero magnetization is
obtained somewhere inside the plasma:
p 0ζ θ θ ζ∇ = × + × =j B j B
Estimate critical pressure gradient as
loop Spitzer
dpj B E B
drζ θ θ= = η
Need database of typical ∇p, loop voltage, resistivity and Bθ
to test predictionsor measurements of jθ
Explaining Te threshold for L-H transition via ηSpitzer ?
and associated pressure gradient (βθ) threshold
j 0θ =
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 27/29
Experimental evidence? AUG
(10×) jθ zero crossing at L→H transition
P. J. McCarthy, P4.115, 40th EPS Conference
on Plasma Physics, Espoo, 2013
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 28/29
In a slow L-H transition followed by “type III” ELMs
jθ remains diamagnetic after transition
But at least one counter example has been found in a slow transition (still
unpublished). More analysis needed, as well as more refined model.
Experimental evidence? AUG II
Emilia R. Solano “Magnetic Phase Transitions & Confinement Regimes” 15th Eur. Fus. Th. Conf., Oxford, 24/09/2013 29/29
• First-principles model of plasma magnetization and magnetic phase transition
as the basis for triggering confinement transitions
• The magnetic state of the plasma determines convective motion of high and
low pressure tubes.
• Paramagnetic plasma regions attract cold tubes, become more paramagnetic.
• Diamagnetic plasma regions attract hot tubes, becoming more diamagnetic.
• A pedestal structure builds up in the magnetic phase boundary.
• Magnetic boundary defines critical magnetization:
• Magnetization force drives the magnetic interchange mechanism in closed
field line region, similar to curvature interchange in SOL.
Summary and comments