Warm-Ion Drift Alfven Turbulence and the L-H Transition
B. Scott
Max-Planck-Institut fur Plasmaphysik, D-85748 Garching, Germany
1. Introduction and Summary
Recent years have seen the emergence of three dimensional computational models of
turbulence and transport in tokamak plasmas near the plasma edge [1{4]. The edge region
is important since not only is it a signicant if not the dominant part of the plasma deciding
the L-H transition [5,6], the parameters of the edge plasma before and during the transition
are deeply within the regime in which the electrons dynamics is controlled by the balance
between magnetic induction and parallel forces [3]. These initial eorts have tended to leave
one or more of the main background gradients and their corresponding uctuations out
of the computations. Due to the rich nature of plasma edge prole phenomena in general
[5,7], increasingly realistic computational models will be needed before they become useful
to the state of the art in edge modelling [8].
This work involves an initial eort to model both plasma uids computationally on
an equal footing, electromagnetically and with all three main gradients. The electrons are
important since they are the route to electromagnetic uctuation dynamics [3], and particle
transport with a strong eect on energy transport is known to be signicant [9]. The ions
are important since they can carry much or most of the energy transport in L-mode [10],
and provide the route to the rTi driven turbulence receiving the most attention in core
transport studies [11{13]. The electron dynamics by itself is still well described as drift
Alfven turbulence, but the Ti dynamics is the almost purely two-dimensional electrostatic
toroidal rTi mode, which then combines with the electron physics to produce a hybrid
with some of the properties of both. This can be thought of as an \ion mixing" mode [14].
The most important result of these computations is the discovery of a complicated synergy
between the prole gradients, leading not only to a self-consistent particle pinch but also to
a sharp bifurcation in the energy transport when the prole scale lengths become unequal.
This may provide an important route to the L-H transition.
2. Model and Equations
The basic model is of low frequency two-uid drift dynamics. The coordinate system
(x; y; s) is aligned to the unperturbed magnetic eld, with x the distance down the pressure
gradient, s the distance along the magnetic eld, and ry in the electron drift direction.
Normalisation is in terms of drift scales, s and cs=L?, where L? is the reference prole
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scale (see [3]). The system of equations for the uctuating quantities desribes Alfven
dynamics (1,2), and the electron (3-5) and ion (6-8) responses:
(1)
r? dtd r?( + pi ) = rkJk ? K (pe + pi )
@ A + ^ d J = r (p ? ) ? (! + ! ) @Ak ? ^ J + 0:71 ?q + 0:71J (2)
^ @t
n
t @y
k
k
k e
k 1:6 e k
dt k
d n = ?! @ + r ?J ? u ? K (p ? )
(3)
e
n @y
k k k
dt
3 d T = ? 3 ! @ + r ?J ? u ? q ? K p ? + 5 T
(4)
e
e
k k k ek
2 dt e
2 t @y
2
?
@Ak 5
d
(5)
^ dt qek = ? 2 rkTe ? !t @y ? ^aLe qe k ? ^ 51=:62 qe k + 0:71Jk
3 d T = ? 3 ! @ + r ?J ? u ? r q ? K p ? ? 5 T
(6)
e
k ik i
2 dt i
2 i i @y i k k k
2 i
@A
(7)
^ dtd uk = ?rk (pe + pi) + [(1 + i)!n + !t + i!i ] @yk
@A
^dtd qi k = ? 52 i rk Ti ? i !i @yk ? ^aLi qi k
(8)
Here, pe = n + Te and pi = in + Ti , where i is the background ion/electron temperature ratio, and Jk = ?r2?Ak . The dierential operators are a property of the geometry;
here we use the rst order inverse aspect ratio form for the curvature operator,
@
@
(9)
K = !B (cos s + Ss sin s) @y + sin s @x ;
where S gives the shear, and the zeroth order forms for everything else [2,16,17]:
2
2
@A
@A
@
@
@
@
@
@
k
k
r? = @x + Ss @y + @y2
rk = @s ? ^ @x @y ? @y @x
(10)
d = @ + v r = @ + @ @ ? @ @ (11)
dt @t E
@t
@x @y @y @x
The main parameters are ^ = (4nTe=B2 )(qR=L?)2 and ^ = (me =Mi )(qR=L?)2 ,
determining the relative transit Alfven and electron thermal frequencies, respectively. The
normalised gradients are !n, !t, and !i , for the density and electron and ion temperatures.
Other parameters are the collisionality, = eL?=cs, ion mass normalised by the scale
ratio, ^ = (qR=L?)2 , and the curvature parameter, !B = 2L?=R. The constants are
= 0:71 and = 1:6. Landau damping parameters are aLe = Ve =qR and aLi = cs=qR,
where Ve is the electron thermal frequency.
2
2
3. Computational Results Involving the Ion Temperature
The computational domain was 64 256s in the (x; y) drift plane and one connection length of 2 in s. The periodicity constraint was strictly observed as the boundary
condition in the s-direction, which is important in correctly representing the parallel electron coupling among the state variables. Further details are as in [3]. The baseline set of
parameters was ^ = ^ = 10, !B = 0:03, = 0:2, and !n = !t = !i = i = 1.
Dependence of electron and ion
heat transport with beta.
Bifurcation due to synergy in the
prole gradients.
One of the important results of [3] was that the electron turbulence is equally well
driven by either gradient, !n or !t . This result extends to the warm-ion model. All
three gradients, not just !i through the rTi mode, were found to be eective when their
scale lengths are equal. At this , almost all of the parallel dissipation is electron Landau
damping. At this ^, the electrons are deeply electromagnetic. The total heat transport
is nevertheless evenly divided among the electrons and ions (see the gure, left), showing
that these two branches of the dynamics are self-consistently interacting. The importance
of the electrons is shown by the fact that the scaling of the heat transport with ^ is similar
to that in the cold-ion model.
When !n was set to zero with !t = !i = i = 1, a pinch eect, ? = hnvxi < 0
with !n = 0, was found. This is the rst self-consistent computation of such an eect not
involving trapped electrons [11], suggesting that parallel electron dynamics, as in the ion
mixing model [14], can deliver the pinch eect in the edge [9].
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Finally, it was found that the transport was very much weaker in these !n = 0 cases
than otherwise (gure, right). It was also somewhat weaker for !t = !i < !n . This is
another indication of synergy in the prole gradients, with obvious bifurcation implications.
Although further study is needed to understand this phenomenon, it would be of interest
for experiments to investigate whether such changes in d log T=d log n indeed happen before
and during the L-H transition.
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