Transcript Slide 1

1/19
Axisymmetric two-fluid plasma equilibria
with momentum sources and sinks
K G McClements1 & A Thyagaraja2
1EURATOM/CCFE
Association, Culham Science Centre, Abingdon,
Oxfordshire, OX14 3DB, United Kingdom
2University
of Bristol, H. H. Wills Physics Laboratory, Bristol, BS8 1TL,
United Kingdom
Plasma physics seminar, Australian National University,
Canberra, April 4 2011
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2/19
Introduction (1)
 Basic tool used to describe axisymmetric plasma equilibria (e.g.
tokamaks, accretion discs) is Grad-Shafranov equation1, obtained from
MHD force balance in absence of equilibrium flows & viscosity (p  j B):
  1 Ψ   2Ψ
df
2 dp
R
f

  2   μ0R
R  R R  Z
dΨ
dΨ
(R,,Z) – right-handed cylindrical coordinates; (R,Z) - poloidal magnetic
flux, defined such that
B  Ψ  φ  RBφφ
f = f() = RB - stream function for poloidal current; pressure p = p()
 Can be generalised to include toroidal rotation – necessary when flow
approaches or exceeds local sound speed cs, e.g. in Joint European Torus
(JET)2 or Mega Ampère Spherical Tokamak (MAST)3 at Culham
1 Shafranov
3
Sov. Phys.-JETP 6, 545 (1958)
2
de Vries et al. Nucl. Fusion 48, 065006 (2008)
Akers et al. Proc. 20th IAEA Fusion Energy Conf., paper EX/4-4 (2005)
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3/19
Introduction (2)
 Flows ~350 km s-1 occurred in MAST
during counter-current beam injection1
 driven by jB torque associated with
radial bulk ion current balancing current
due to beam ion losses2
 Midplane density profile ne(R) shifted
outboard with respect to temperature
(assumed to be flux function due to
rapid parallel heat transport)
 If flow is purely toroidal, Te & Ti are flux functions, flux surfaces rotate as
rigid bodies at rate , & momentum sources/sinks are neglected, then3
ne  n0 Ψexp[mi Ω2ζ R2 / 2(Te  Ti )]
1 Akers
et al. Proc. 20th IAEA Fusion Energy Conf., paper EX/4-4 (2005)
2 McClements
3
& Thyagaraja Phys. Plasmas 13, 042503 (2006)
Maschke & Perrin Plasma Phys. 22, 579 (1980)
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4/19
Introduction (3)
 Rigid body rotation implied by ideal MHD Ohm’s law + axisymmetry:
Φ
Ψ
Φ
Ψ
 Ωζ
,
 Ωζ

Ω ζ  Ω ζ Ψ 
R
R
Z
Z




 - electrostatic potential
What happens when all possible relevant terms in force balance
equation(s) are taken into account?
Due to dissipation (in particular neoclassical & turbulent viscosity) flows in
tokamaks must be continuously driven
Poloidal flows, when measurable, usually found to be very small (~ few
km s-1), in accordance with neoclassical predictions, but occasionally
observed to be significant fraction of cs, e.g. close to internal transport
barriers in JET1 – such flows could affect equilibrium2
In this talk I will consider purely toroidal flows
1 Crombé
2
et al. PRL 95, 155003 (2005)
McClements & Hole Phys. Plasmas 17, 082509 (2010)
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5/19
Plasma coordinates (1)
 Often convenient to use right-handed plasma-based coordinates (,,)
where toroidal angle  = - & poloidal angle  is defined such that
Jacobian of transformation from laboratory coordinates does not
generally vanish in domain of interest:
1 Ψ, θ  Ψ θ
J  Ψ  θ   ζ 

R R, Z 
R 
 - arc length along flux surface in (R,Z) plane
 We set J = J() – generalisation of Hamada coordinates (J = constant)1;
facilitates evaluation of flux-surface averages, since volume element is
RddζdΨ / Ψ  dθdζdΨ / J Ψ
Such coordinate systems are quasi-orthogonal –  ≠ 0 in general
 B  ζ  Ψ  RBζ ζ where B = -B; we denote RB by F = -f
1
Hamada Nucl. Fusion 2, 23 (1962)
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Plasma coordinates (2)
 In steady-state & in absence of poloidal flows, momentum sources &
dissipation, F = F() in both ideal MHD1 & 2-fluid theory (in limit me → 0)2
 Here we assume only that F is axisymmetric, i.e. F = F(R,Z) or F(,)
1 F
1 F
 Ampère’s law 
j  Rjζ ζ 
ζ  Ψ 
ζ  θ
μ0 Ψ
μ0 θ
where
1    1 Ψ   2Ψ 
1 
jζ 
R


ΔΨ



2
μ0R  R  R R  Z  μ0R
1  
F 
F F
J F
Hence j  B  
Δ
Ψ

F

Ψ


θ

ζ

2 
2
μ0R 
Ψ 
μ0R θ
μ0 θ
Axisymmetric ideal MHD with purely toroidal flow & no sources/sinks
requires that toroidal component of jB vanishes  F = F()
1
McClements & Thyagaraja Mon. Not. R. Astron. Soc. 323, 733 (2001)
2
Thyagaraja & McClements Phys. Plasmas 13, 062502 (2006)
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Ion momentum balance
 Quasi-neutral plasma with singly-charged ions & electrons, each with
scalar pressure; ions have toroidal flow vi = R2
 ion momentum balance equation can be written as

1
mi nΩ 2ζ R 2   pi  neΦ  neΩ ζ Ψ  Fext  Fdrag  nR ie
2
applied torque
momentum loss rate momentum
(e.g. due to viscosity) exchange with
electrons
For inductive tokamak operation we can write
R ie 
eVL
 ζ  e ηj
2π
VL – loop voltage (we assume uniform toroidal voltage across plasma);
 - resistivity (assumed to be isotropic)
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Electron momentum balance
 In limit me → 0 electron momentum balance equation (≡ generalised
Ohm’s law) can be written as
0  pe  neΦ  neΩζ Ψ  j  B  nRei
 Momentum conservation  R ei  R ie  
momentum
exchange with ions
eVL
ζ  eηj
2π
 Momentum sources & sinks neglected in electron momentum balance –
any momentum acquired by electron via interaction with e.g. beam ions
ext
drag
very rapidly transferred to bulk ions, hence F  F
for electrons (if
this were not the case, beam injection would produce large numbers of
highly superthermal electrons, which are not observed)
 We neglect external current sources (driven e.g. by beams or radiofrequency waves) & bootstrap current (diamagnetic current associated
with drift orbits of trapped particles)
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Single-fluid momentum balance (1)
 Adding ion & electron equations yields
1
(1)
mi nΩ 2ζ R 2   p  j  B  Fext  Fdrag
2
where p = pi + pe; to ensure compatibility with neglect of poloidal flows we
consider only toroidal components of Fext & Fdrag, & assume that drag
term can be characterised by phenomenological relaxation time  :
2
m
n
Ω
R
Fdrag   i ζ
ζ
τζ

(for momentum losses arising from neoclassical or turbulent viscosity,
more exact expression would involve spatial derivatives of vi)
 Substituting our expression for jB into (1) yields
1
μ0R 2
F 
F F
J F
1
 
2
2
Δ
Ψ

F

Ψ



θ


ζ

m
n
Ω

R


i
ζ
Ψ 
μ0R 2 θ
μ0 θ
2

 ext
mi nΩ ζ R 2 
 p  Fζ R 
ζ
τζ


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Single-fluid momentum balance (2)
 Regarding R2 & p as functions of  & , & equating components, we obtain
2
1  
F 
p 1
2 R
ΔΨF
 mi nΩ ζ

2 
μ0R 
Ψ 
Ψ 2
Ψ
2
p 1
F F
2 R
 mi n

 2

0R 2 
2
J F mi nΩ ζ R
F R

μ0 θ
τζ
ext
ζ
(1)
(2)
(3)
 In limit Ωζ  0 , Fζext  0 (3) implies F = F() & (2) implies p = p();
(1) then reduces to standard form of Grad-Shafranov equation
 When  ≠ 0 & F = F() (1) is equivalent to Grad-Shafranov equation for
purely toroidal flow obtained by previous authors1
 F = F() to leading order in v ζ2 / cA2 /Ωζ τ ζ 
1
Maschke & Perrin Plasma Phys. 22, 579 (1980)
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11/19
Variation of density on flux surfaces (1)
 Relation between momentum drive & dissipation depends on beam
deposition & momentum transport; we consider simple cases to illustrate
influence of torque & relaxation time on density distribution
 1st case: F = F() & Fζext τ ζ  KR where K = K(); from (2) & (3) on
previous slide we obtain min = K &
n2  n2


2
n
K2

R2  R2 
2miT
1  mi Ω 2ζ ( R 2  R 2 ) / 2T
… - flux surface average
- differs from result for rigidly-rotating flux surfaces when momentum
sources & sinks are neglected:
n  n0 Ψexp[mi Ω2ζ R2 / 4T ]
 In both cases n increases with R on flux surface – arises from inertial term
in ion momentum balance equation, & is qualitatively consistent with
measurements in spherical tokamak plasmas with transonic toroidal flows
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Variation of density on flux surfaces (2)
 2nd case: F & Fζext τ ζ are flux functions → M ≡ minR (toroidal linear
momentum per unit volume) is a flux function &
2
2
2


M
R
n
2
min
  nmin
Ψ 
n2 
ln 2
2
2miT  Rmin Ψ  
1  (mi Ω 2ζ R 2 / 2T ) ln(R 2 / Rmin
)
 3rd case: F & Fζext τ ζ R are flux functions → L ≡ minR2 (toroidal angular
momentum per unit volume) is a flux function &
n2  N 2 (Ψ)  L2 / 2miTR2  N 2 /(1 mi Ω2ζ R2 / 2T )
 In all cases we find that at magnetic axis
2 2
d ln n mi Ω ζ R
R

dR
2T
- agrees well with measurements in National Spherical Torus Experiment
(NSTX) at Princeton1; but measurements at magnetic axis alone cannot be
used to determine density variation on flux surface
1
Menard et al. Nucl. Fusion 43, 330 (2003)
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Temperature – density relation: theory
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 Eliminating n we obtain  = (T ); e.g. in 1st case
KT 1/ 2
Ωζ 
mi [IT  K 2R 2 / 2mi ]1/ 2
where I, K are flux functions
 Expressions for  in terms of Ti & Te can be obtained from ion & electron
momentum balance equations in limit Fζext  0 , τζ   , VL  0 ;1 result
for rigidly-rotating flux surfaces is
1/ 2
T T 
   0  i e 
 2T0 
  T ' d 
exp  e

 0 2Ti  Te 
0, T0, 0 – constants; when Ti =Te =T this reduces to
Ωζ  Ωζ 0 (T / T0 )1/ 4
→ rotation profile predicted to be much flatter than temperature profile
1
Thyagaraja & McClements Phys. Plasmas 13, 062502 (2006)
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14/19
Temperature – density relation: experiment
 Spectroscopic measurements in JET
plasmas with Ti Te show that  & Ti
have similar profiles, contradicting
prediction based on (i) assumption of
rigid body rotation & (ii) neglect of
momentum sources/sinks
 Similar relation between profiles
observed in MAST
Rotation, temperature & momentum/ion
heat diffusivity profiles in JET #57865; solid
black curves show experimental profiles1
1
 In order to account for measured
rotation & temperature profiles it is
necessary to invoke either
momentum sources/sinks or nonrigid rotation of flux surfaces
Tala et al. Nucl. Fusion 47, 1012 (2007)
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Ohm’s law for rotating tokamak plasma
15/19
When F = F() poloidal component of electron momentum balance yields
2
 p 1
J pe J Φ
VL
F 'B 2 
2 R



 η
 mi nΩ ζ

2
neF θ F θ 2πR
Ψ
F 
 Ψ 2
Flux surface average of this yields
2
μ
F
FF '   0 2
B
2
 VL

1
1

R
'
2
 p  mi nΩ ζ


2
2
πη
R
2

Ψ


Substituting into Grad-Shafranov equation & integrating over (R,Z) → plasma
current in terms of loop voltage & resistivity:
0
I p  2π 
Ψ0
0 B 2 1/ R 2
2
Bθ2  1


R
d
Ψ
dΨ
ζ
'
 mi nΩ2ζ


p

V
L 
 J
Ψ
ηJ
B 2  2
B2
Ψ0

- could be used e.g. in fluid turbulence codes to calculate loop voltage
required to maintain specified current, given profiles of , , J, B2 etc.
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16/19
Radial electric field
 Radial component of ion momentum balance equation yields
mi Ω2ζ R 2
Φ
1 pi
θ  ζ

 Ωζ 
 ηj 
Ψ
en Ψ
2e Ψ
J
 Last term on right hand side negligible under typical tokamak conditions;
averaging remaining terms with respect to  we obtain simple expression
for  / - measure of flux surface-averaged radial electric field; shear
of this is thought to play important role in transport barrier physics
 Radial electric fields fields can be determined from motional Stark effect
measurements;1 - equilibrium electric field given by above expression could
thus be compared directly with experiment
1
Rice et al. Nucl. Fusion 37 517 (1997)
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Conclusions
17/19
 Previous analyses of axisymmetric plasma equilibria with toroidal flows extended
to take into account continuous drive of such flows when dissipative effects are
present (invariably the case in tokamaks)
 Ion & electron fluid momentum equations yield three coupled PDEs which indicate
that flux surfaces do not rotate rigidly, as frequently assumed & required by ideal
MHD in absence of momentum sources/sinks & poloidal flows
 For specific assumed relations between momentum drive & damping, expressions
can be obtained for variation of density on flux surfaces - in principle can be tested
experimentally
 Either momentum sources/sinks or non-rigid rotation of flux surfaces must be
invoked to account for measured rotation & temperature profiles
 Relation derived between loop voltage & plasma current in tokamak plasma with
toroidal flow - could be used to determine voltage required to maintain particular
current in slowly-evolving discharge
 Simple expression obtained for equilibrium radial electric field - can be compared
directly with experiment
 For further details see McClements & Thyagaraja Plasma Phys. Control. Fusion
53, 045009 (2011)
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Postscript: ripple transport in MAST (1)
18/19
 MAST has R  0.85m, minor radius
a  0.65m, B  0.5T, Ip  1MA & is heated
by 70keV deuterium neutral beams
– plot shows trapped beam ion orbit in
(R,Z) plane
 Due to presence of N = 12 toroidal field
coils at Rcoil = 2m, toroidal & radial field
components are no longer axisymmetric:
~
B 
B0R0  R

R  Rcoil
N

 cosN 

~
BR 
B0R0  R

R  Rcoil
N

 sinN 

 even in absence of prompt losses, collisions & turbulence, beam ions are no
longer perfectly confined in plasma because

d
H
P   mRv  Ze   
0
dt

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Postscript: ripple transport in MAST (2)
19/19
 Cyclotron resonance between particle motion & ripple field can produce
additional transport & loss;1 resonance condition for zero frequency field
perturbation is kIIvII = i where i is beam ion cyclotron frequency
 approximately satisfied by beam ions in MAST
 Significant anomalous beam ion transport has been reported in MAST2
- ripple effects may be contributing to this
 Transport of fast ions due to ripple, or other non-axisymmetric perturbations,
can be modelled using CUEBIT full orbit test-particle code
 Principal aim of my visit is to pursue this project in collaboration with MJH
1
Putvinskii JETP Lett. 36 397 (1982)
2 Turnyanskiy
et al. Nucl. Fusion 49 065002 (2009)
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