wall potential and φf ≈3Te [31] is the floating potential. This can be rewritten in terms of the difference of the plasma potential to a reference potential φ∗:
jk,sheath=−necs 1−e−e(φp−φ∗)/Te
≈ ne2cs(φp−φ∗) Te
, (3.2)
with the ion sound speed cs =p
Te/mi. In the absence of temperature fluctuations, however, φ∗ is only a constant reference potential and it will be seen below that its numerical value has no influence on the blob dynamics.
d
w/λ
D≈10 ϕ
pϕ
SEϕ
fn
e<n
in
en
in
e=n
iplasma pre-sheath
sheath ϕ
Figure 3.1: The potential in the presence of a wall as a function of the distance dw in units of the Debye length λD. Towards the wall, the potential drops from the plasma potential Φp to the sheath entrance potential Φse in the approximately quasi-neutral pre-sheath and drops further until it reaches the floating potential Φf at the wall.
3.2 Radial blob propagation
The basic mechanism of this radial motion is as follows: Due to charge-separating drifts, the blob is polarized. The resulting electrical field ˜E then leads to ˜E×B drifts in the background magnetic field B with a radial velocity component of up to ten percent of the ion sound velocity cs [16].
To estimate the radial blob velocity vr,b the first two features stated above are used (long lifetime and kk = 0). Since the blobs seem to be stable on turbulence
time scales it is assumed that the polarization has reached an equilibrium state and no further charges are accumulated. The current perpendicular to the magnetic field j⊥ (the currents arising from charge separating drifts that are responsible for the blob polarization and currents counteracting this polarization) are balanced by a current parallel to the magnetic fieldjk leading to the vanishing divergence of the total current:
−∇⊥·j⊥=∇kjk. (3.3)
It is shown below how the radial blob velocity can be derived using this model.
It is clear that the solution depends on the choice of the dominant contributions to j⊥ and jk. Different drifts have been identified to cause charge separation in blobs [16] and, hence, radial propagation. However, in magnetized toroidal plasmas the dominant drift is almost always caused by the magnetic field geometry [16, 32, 33]. In the literature often the single particle drifts are discussed for the blob polarization namely the curvature and gradient drift. The same results are obtained in the fluid picture, where the divergence of the diamagnetic drift is used. For the balancing of the polarization caused by these drifts, often the ion-polarization current and a parallel current flowing along the filament to the sheath in front of a limiter (more general a plasma facing component) are considered. In Ref. [34] it is shown that in this case Eq. (3.3) can be written as
mi
B2∇ ·
n d
dt∇⊥φ+nνin∇⊥φ
=∇kjk+ 2
Bˆb·κ× ∇p , (3.4) with d/dt = (∂/∂t+vE×B · ∇), the plasma potential φ, the unit vector in the direction of the magnetic field ˆb = B/B, the curvature vector κ = (ˆb· ∇)b, theˆ neutral-ion collision frequencyνinand the plasma pressurep=pe+pi =neTe+niTi. It should be noted that in contrast to Ref. [34] Eq. (3.4) is given in SI units and the term accounting for ion-neutral collisions from Ref. [35] has been added. The terms describe from left to right the divergence of the ion-polarization current, the divergence of perpendicular currents due to collisions, the divergence of the parallel sheath currents (which will be treated in more detail in Sec. 3.5) and the divergence of the diamagnetic current. The last term is responsible for the polarization of the filament and is, therefore, sometimes called forcing term orblob drive. To solve this equation analytically a number of approximations are employed in Ref. [34]:
1. The Boussinesque approximation:
∇ ·
n d dt∇⊥φ
=n d
dt∇2⊥φ . (3.5)
This approximation assumes low fluctuation amplitudes ˜n/n ≪ 1, the ab-sence of a background electrical field and structure sizes much smaller than the background profile scale length. The discussion in Ref. [34] shows that these requirements are not strictly fulfilled for blobs in the SOL, but that the approximation is adequate to describe the blob danymics.
3.2. Radial blob propagation 35 2. The assumption of a simple magnetic field geometry with a local coordinate
system where B =Beˆz and κ=−eˆx/R (R is the major radius).
3. No pressure variations in the blob parallel to the magnetic field (∂zp= 0).
With these approximations Eq. (3.4) can be written as 2
RB
∂p
∂y = nmi
B2 ∂
∂t+vE×B· ∇
∇2φ− ∇kjk+nνin∇2⊥φ . (3.6) Figure 3.2 illustrates the meaning of the different terms in this equation.
L I M
I T E R
η
∥η
shj
∥j
pol+
-E v
ExBFigure 3.2: Illustration of a blob extended along a curved magnetic field line. Cur-vature induced drifts polarize the filament. The polarization is balanced by parallel cur-rents jk, which are damped due to sheath resistivity ηsh, neutral-ion collisions ηk and ion-polarization currents perpendicular to the field line. The resulting poloidal (upwards in the figure) electrical field gives rise to radial E×B drifts. Reproduced after [34, 36].
Blobs are observed in plasmas with very different conditions. Hence, usually not all of the terms in Eq. (3.6) are of equal importance and further approximations can be used to find simple solutions. Two such solutions are briefly presented in the following, one for plasmas with cold ions as in TJ-K and one for warm ions as in ASDEX Upgrade.
3.2.1 Cold plasmas
Equation (3.6) can be simplified when the ions are cold (Ti ≪ Te) and electron temperature fluctuations can be neglected ( ˜Te= 0). Then ˜p= ˜pe+ ˜pi ≈p˜e ≈Ten˜e. To derive the radial blob velocity vr,b a model for jk is required. In Refs. [32, 33]
it is assumed that the parallel current flowing along the blob filament matches the sheath current at the sheath of the wall/limiter (see Sec. 3.1). Since it is assumed that kk = 0 for the density and potential perturbation associated with the blob
and that the filament extends between two plasma facing components (i. e. sheath contact at both ends of the filament), the parallel derivative of jk is [34]
∇kjk = 2
lkjk,sheath. (3.7)
In Refs. [35, 37, 38] a number of approximations are applied to simplify Eq. (3.6):
1. The blob drive is explained by the interchange mechanism. Hence, the growth rate of the ideal interchange instability is used to replace the total time deriva-tive in the first term of Eq. (3.6): d/dt≈√
2cs/√
Rδb with the blob size δb. 2. It is assumed that at the position of the positive potential pole ∂p/∂y =
−Teδn/δb, where δn = nmax − n0 is the difference between maximum and background density.
3. Linearization of the derivatives [35, 38]: ∇⊥φ ≈0, ∇2φ ≈ −φ∗/δb2, and φ∗ ≈ Bvr,bδb (approximating the E×B velocity).
With these approximations Eq. (3.6) can be solved for vr,b:
vr,b=
q2δb
R cs
1 + ρ21 slk
qR
2δ5/2b + νin√√2cRδb
s
δn n0
, (3.8)
withlk the parallel length of the blob (e. g. the distance between two limiter plates) and ρs the drift scale:
ρs =
√miTe
eB . (3.9)
An agreement with this predicted velocity has been found in experiments in the magnetized torus TORPEX [35]. It should be noted that this model assumes a density perturbation like for the interchange instability, described in Sec. 2.2, to obtain the velocity calculated above. This implies a pure interchange instability with a cross phase between density and potential of αφ,n=π/2.
The three terms in the denominator of Eq.(3.8) describe from left to right the influence of the ion-polarization current, the parallel currents through the sheath and neutral-ion collisions. If one of the three terms dominates the denominator, three different branches or regimes of Eq.(3.8) can be identified that were derived independently before (references given below). Assuming δn/n0 = 1, these are the sheath limited regime [33, 34]
vr,sheath≈2cs
ρs
δb 2
lk
R, (3.10)
3.2. Radial blob propagation 37 the so-called inertial regime [37, 39]
vr,inertial= r2δb
R cs, (3.11)
and a regime dominated by neutral friction [40]
vr,fric.= 2c2s
νinR . (3.12)
All three regimes predict a different dependence of vr,b on the blob size δb.
3.2.2 Warm plasmas
The approximations justified for cold plasmas are no longer valid in fusion plasmas, where in the SOL it is usually the case that Ti > Te together with a fluctuating part ˜Te,i 6= 0 [41]. While there are numerical simulations taking this into account, not much effort has been spent yet to find scaling laws for the blob velocity as those shown above for the cold ion case. Only very recently this problem has been treated analytically in Ref. [42]. In analogy to Eq. (3.6) the evolution of the charge separation is described by
d∇2⊥(φ+τip˜e)
dt =∇kjk−(1 +τi)2L⊥ R
∂p˜e
∂y . (3.13)
Here, τi =Ti/Te, ˜peis the pressure amplitude normalized to the background pressure pe,0 and L⊥ is the mean profile scale length. Additional equations to describe the electron pressure evolution dpe/dtand to predict the parallel current jk are needed, which depend on the dominant reduction mechanism for the blob polarization and are given in Ref. [42]. For the sheath dissipation regime, where parallel currents to the walls are important, the following velocity scaling is obtained:
vr,sheath≈(1 +τi)cs ρs
δb
2
lk
R, (3.14)
which agrees with the cold ion scaling according to Eq. (3.14) beside an additional factor (1 + τi)/2 describing the pressure contribution by the ions. If, however, collisions get important, the following scaling is obtained
vr,b= Λ(1 +τi) R
lk LSOL
2 ρs δb
2
, (3.15)
with the SOL connection length LSOL and the collision parameter [43]
Λ = νieLSOL
Ωceρs ≈1.7·10−22ne[m−3]LSOL[m]
Te[eV]2 . (3.16)
Here, νie is the electron-ion collision frequency and Ωce the cyclotron frequency of the electrons. Eq. (3.15) has the same size dependence as Eq. (3.14), but is much more sensitive to the background densityneand electron temperatureTe. According to Refs. [15, 42], the collisional scaling is valid for Λ>1.