EDA H-mode: a theoretical model*
A.L. Rogister
Institut für Plasmaphysik, Forschungszentrum
Jülich GmbH, EURATOM Association,
Trilateral Euregio Cluster
* 9th EU-US TTF workshop, Cordoba, Spain,
September 9-12, 2002
1
Content
1. Introduction
2. Toroidal momentum equation, revisited;
simplification for edge plasmas
3. Parallel velocity shear Kelvin-Helmholtz
Instability & EDA H-mode model
4. Trial function approach
5. Discussion
6. Future works
2
1. Introduction
At the L to H-mode transition, the
toroidal velocity in the core experiences a
jump from a few kms-1 in the counter
direction to some tenths kms-1 in the co
direction [1].
The H-mode is
ELM-free if q95 < 3.5
and
EDA
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if q95 > 4
At the ELM-free to EDA transition [2, 3]
(i) a quasi-coherent oscillation with
f ~ 120 kHz
and
k ~ 400 m-1
sets in, and
(ii) the particle confinement time decreases
dramatically:
p /E   2 - 3
Neoclassical theory [4] explains quite
well the large co-rotation in an ELMfree H-mode discharge with q95 = 3.4 [5].
Large jump in the toroidal velocity
occurs across the pedestal.
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My purpose here is to show that
(i)
the parallel velocity gradient
predicted by neoclassical theory for
the q95  3.4 discharge is also close
to the threshold value for the
parallel velocity shear KelvinHelmholtz (PVS K-H) instability;
(ii) the frequency and poloidal mode
number of the threshold PVS K-H
mode correspond to those of the
coherent oscillation;
(iii) a non-linear model based on the
assumption that anomalous
transport keeps the plasma near
marginal (in)stability leads to the
observed p /E in EDA.
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2. Toroidal momentum equation
(neoclassical theory), revisited [5];
simplification for edge plasmas
U  ,i
0.107q 2  ln Ti B
 r  2 ,i (

U  ,i )
2
2
r
r B
1 Q / S

 mi N iU r ,i U  ,i  mi (  t   iz   cx )N iU  ,i
In edge relevant temperature range:
 v  cx  2  v  iz
Albedo factor  0.5   cx   iz  0 
(  r  D 1U r ,i )U  ,i
0.107q 2  ln Ti B

U  ,i
2
2
r B
1 Q / S
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which can again be integrated! Here,
D  1.2 i ai2
is the classical diffusion coefficient.
In the following, I define
L1  0.107q 2 ( 1  Q 2 / S 2 )1 ( B / B ) r ln Ti

(  r  D1U r ,i )U ,i  L1U ,i
Note: U,i is provided by neoclassical
theory [4] & [6]; the equation has two
unknowns: U,i and Ur,i
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3. Parallel velocity shear KelvinHelmholtz instability & EDA H-mode
model
PVS K-H oscillations are unstable if (in
absolute values)
 rU ,i  2ci r ln N i
where  1 in the fluid limit [7];   1 if
Landau damping [8] or other dissipation
mechanisms or electromagnetic effects are
taken into account.
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3.1. Below instability threshold,
the radial particle flow is small (e.g.,
neoclassical) and can be neglected (ELMfree H-mode). Thus, the toroidal velocity
gradient and the radial flux are given by
 rU  ,i  L1U ,i
and
N iU r ,i  0
if
L1U  ,i  2ci  r ln N i
Those equations have been applied
successfully to explain the large core corotation in an ELM-free q95 = 3.4 Hdischarge [5].
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3.2. At instability threshold,
I assume that anomalous transport clamps
the density profile at marginal (in)stability.
The EDA H-mode model is therefore
represented by the equations:
 rU  ,i  2ci r ln N i
( N iU r ,i )1  Di N i
2ci r ln N i  L1U ,i
r

2ci r ln N i dr' U ( rs )
rs
if
L1U  ,i  2ci  r ln N i
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.
Those are completed by the continuity
equations for charged particles and
neutrals:
 r ( N iU r ,i )   iz N i
r
( NiU r ,i )2   N 0V0  ( N 0V0 )rs exp  01dr'
rs
where the neutrals velocity V0 is negative
(directed inwards) and
0  V0 /  v  iz N i
is their mean-free path.
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4. Trial functions approach
Identifying the two expressions of the
flux yields a non-linear integrodifferential equation for Ni (alternatively
a 2nd order differential equation).
Rather than solving the above, as well as
the energy equation for the temperature
profile, I introduce the trial functions
Ti  ( Ti )inf [1  tanh( r  rinf ) / N ] i
and
N i  ( N i )inf [ 1  tanh( r  rinf ) /  N ]
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Then, (from continuity equation)
( NiU r ,i )2  ( N0V0 )inf [ 2 y /( 1  y )]
 N /( 0 )inf
and [from toroidal momentum equation; I
assume here i  2 for convenience and
U,i(rs) = 0 (as no momentum input)]:
( Di N i )inf 1  ys y ( yys  1)
( N iU r ,i )1  4
N
ys (1  y ) 4
where y  e2x and x  r-rinf.
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The latter expression vanishes at y = 1/ys ,
showing that instability threshold is not
reached inside that point: indeed, the
destabilising term is  (Ti’)2/Ti ; the
stabilising one to Ni’/Ni.
I request:
1 / ys
 [( N U
i
)  ( N iU r ,i )2 ] dy / y  0
r ,i 1
ys
which yields the relation
(  N 0V0 )inf  ( N iU r ,i )inf  ys ( Di N i )inf / 3N ln ys
between the particle flux and the density
[I have assumed ( 0 )inf   N ].
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A last relation is obtained by identifying
V0 with the mean velocity of half a
Maxwellian distribution at T0 = (Ti)s 
( N0 / Ni )inf  0.5( ys2 / ln ys )( i ai )inf / i N
which yields the neutral density, the ion
density being known.
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5. Discussion
(i)
Marginal stability condition
The marginal stability condition
L1U ,i  2ci r ln Ni
at rrs can be rewritten as
  0.28q 2i2 ( 2 / ys )0.5 ( 1  Q2 / S 2 )r1( ai ,p )inf / N
i
s
The ELM-free shot discussed earlier [5] is
close to the transition, since q95  3.4; the
profiles should thus be close to instability
threshold. Introducing the corresponding
pedestal parameters indeed yields   0.91,
in agreement with fluid theory.
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(ii) The characteristic mode number of
the most unstable PVS K-H mode is
kai  0.5  k  1.400 m-1
which may be reconciled with the
measured 400 m-1 if Landau damping or
electromagnetic effects are taken into
account.
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(iii) Particle confinement time in EDA
Experimental results suggest that ion
energy transport is little affected by the
transition from ELM-free to EDA
operation. Comparing therefore the subneoclassical heat flux [9] to the particle
flux in EDA yields
(qi ,r )inf
 ln ys
Ti
2 
 4q i 
2
2
( NiU r ,i ) rs
1

Q
/
S

 inf ys
As observed, the ratio of the particle and
energy confinement times is thus
 p /  E  2( qr ,i )inf /( Ti )Max( N iU r ,i )r  3  4
s
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(i)
Fluctuation level
In those theories, the marginal stability
condition provides one profile (e.g.,
density); the quasi-linear anomalous
transport equation
NiU r ,i   Dan r Ni
in which
Danomalous  n~ / N
2
then provides locally the turbulence level.
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(ii) Stabilisation mechanism
Stabilisation will most likely occur through
profile relaxation owing to extremely high
transport rates of toroidal momentum once
the instability is excited.
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(iii) Other collisionality regimes
The model can easily be extended to other
collisionality regimes, provided that the
structure of the neoclassical toroidal
momentum equation remains valid. In the
banana regime, all signs are reversed
(poloidal and toroidal velocities, ).
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6. Future works
(i)
Numerical (analytical?) solution of
the non-linear integro-differential
equation for Ni;
(ii) Use of Eirene code for neutrals;
(iii) Reformulation of the PVS K-H
instability theory in the H-mode
context: stiff profiles  non-local
theory; role of ExB velocity shear,...
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References
[1] J.E. Rice et al., Phys. Plasmas 7 (2000)
1825.
[2] M. Greenwald, et al., Phys. Plasmas 6
(1999) 1943;
[3] I.H. Hutchinson, Nucl. Fusion 41
(2001) 1391.
[4] H.A. Claassen et al., Phys. Plasmas 7
(2000) 3699.
[5] A.L. Rogister et al, Nucl. Fusion 42
(2002) 1144.
[6] R.D. Hazeltine, Phys. Fluids 17 (1974)
961.
[7] N. D’Angelo, Phys. Fluids 8 (1965)
1748.
[8] C.G. Smith & S. von Goeler, Phys.
Fluids 11 (1968) 2665.
[9] A. Rogister, Phys. Rev. Lett. 81 (1998)
3663
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