Plasma Physics Reports, Vol. 29, No. 10, 2003, pp. 803–815. Translated from Fizika Plazmy, Vol. 29, No. 10, 2003, pp. 867–879. Original Russian Text Copyright © 2003 by Vlasov, Trubnikov. TOKAMAKS A Possible Mechanism for the Generation and Motion of Blobs in Tokamaks V. P. Vlasov and B. A. Trubnikov Nuclear Fusion Institute, Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia Received January 15, 2003; in final form, March 7, 2003 Abstract—A possible mechanism for the generation and motion of so-called blobs—peculiar perturbations that are observed in a tokamak edge plasma—is proposed. It is suggested that blobs are self-contracting plasma filaments generated either by the thermal-radiative instability of a plasma with impurities or by the nonradiative resonant charge-exchange instability resulting from the presence of neutral hydrogen atoms near the tokamak wall. Instability occurs in a narrow temperature range in which pressure is a decreasing function of density. Under these conditions, the most typical perturbations are the local ones that originate spontaneously in the form of separate growing hills and wells in the density. The temperature at the centers of the hills is lower than that in the surrounding plasma, but they are denser and, consequently, brighter than the background. The (denser) hills should move (“sink”) toward the separatrix, while the (less dense) wells should “rise” in the oppo- site direction, as is observed in experiments. It may even be said that they behave in accordance with a peculiar Archimedes’ principle. © 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION some papers, it was pointed out that the temperature at Observations in some tokamaks have revealed fila- the center of a blob is close to 20 eV, which seems to be ments—commonly referred to as “blobs”—that more than mere coincidence. The fact that a blob appeared sporadically in the edge plasma, were divides into two parts in crossing the separatrix likely stretched along the main magnetic field, and moved provides evidence for its double structure. toward the chamber wall. Figure 1, borrowed from the The mechanism by which blobs move was consid- paper by Marmar [1], presents six successive film ered by Krasheninnikov [2], who supposed that a blob frames of an individual blob that moved at a speed of is a magnetic tube that becomes polarized in a nonuni- 2 km/s toward the edge of the plasma column in the form peripheral tokamak magnetic field by the differ- Alcator C-Mod tokamak (the frames were taken with µ ence between the gradient drifts of the electrons and an exposure time of 4 s). Blobs have also been ions and then continues to move with the velocity V = observed in the DIII-D tokamak, whose cross section is cE/B toward the wall, while at the same time being shown in Fig. 2, in which the arrow indicates the obser- spread out by diffusion. However, in [2], nothing was vation region. said about the mechanism by which blobs originate. From Fig. 1, we can see that, as a blob crosses the separatrix magnetic surface, it breaks into two parts, In the present paper, we consider possible causes of one of which stays inside the separatrix and the other the generation of blobs and the mechanism for their passes through the separatrix and continues to move motion. Although the mechanism of motion to be con- toward the wall. Blobs originate in the edge plasma, in sidered here differs somewhat from that proposed in which neutral impurities play an important role. In [2], it yields the same motion picture of the blobs. 2 cm ∆t = 4 µs Fig. 1. Six successive frames of a blob in the Alcator C-Mod tokamak. The dashed curve shows the separatrix. 1063-780X/03/2910-0803 $24.00 © 2003 MAIK “Nauka/Interperiodica” 804 VLASOV, TRUBNIKOV z, m 2. THERMAL-RADIATIVE INSTABILITY We think that one of the possible mechanisms for 1 the generation of blobs is thermal-radiative instability, during which the plasma heating power W is balanced by the power Wrad of radiative losses from impurities arriving from the chamber wall. In this case, the pres- sure p, as a rule, becomes a decreasing function of den- sity, thereby giving rise to the instability. It is important to note that, in such specific unstable 0 media (in [3, 4], they were called “quasi-Chaplygin media”), there are no running waves that causally con- nect the neighboring regions. Consequently, more typi- cal disturbances in these media are individual local per- turbations that are spontaneously generated and grow until they reach their saturation level. These perturba- tions qualitatively resemble blobs. In addition, the Ð1 blobs observed in experiments originate in a rather nar- row temperature range around a value of ~20 eV. It is noteworthy that the model based on the thermal-radia- tive instability gives the same temperature value. Note that thermal-radiative instability is also referred to as radiative condensational instability and, in the particu- lar case of Joule heating, is called thermal-ionizational instability [5]. Let us discuss this hypothesis in more detail. The inductive voltage around the circumference of the torus is approximately equal to V [V] ≈ 3.6/B|| [T] [6], so that 021 the longitudinal electric field can be assumed to be con- R, m stant during the discharge, E|| = const. The Joule heating 2η power WJ = E is deposited in the central plasma and Fig. 2. Cross section of the DIII-D tokamak. The arrow indi- heats it. However, blobs originate in the scrape-off cates the region where the blobs are observed. layer (SOL) plasma, in which an important role is played by impurities arriving from the tokamak wall. This is why, in considering the first illustrative Q, W cm3 example, we assume that the electrical conductivity in the peripheral plasma is created by collisions of elec- trons with neutral atoms and is described by the formu- W las 10Ð25 η 2 ν ν σ ==e ne/me ea, ea na eav Te, Mo (2.1) Ð26 σ π 2 10 Fe ea ==a , v Te T e/me, σ where ea is the cross section for collisions and vTe is 10Ð27 the electron thermal velocity. In the SOL plasma, impurity atoms are ionized and O become ions with a certain density nZ. The electrons 10Ð28 C recombine with impurity ions, emitting radiation with the specific power Wrad = nenZQ. The dependence of the factor Q on the electron temperature Te for typical 10Ð29 10Ð3 10Ð2 10Ð1 110 impurities (such as C, O, Fe, Mo, and W) is shown in Fig. 3, borrowed from [7]. Te, keV We can see that, for heavy impurities, this factor is nearly constant and is approximately equal to Q ≈ − Fig. 3. Factor Q(T) for C, O, Fe, Mo, and W impurities. 10 17 [cm3 erg/s]. In the simplest model at hand, the PLASMA PHYSICS REPORTS Vol. 29 No. 10 2003 A POSSIBLE MECHANISM FOR THE GENERATION AND MOTION 805 ρ * equality WJ = Wrad yields the dependence nZ T e = C1 = 2 2 (‡) E|| e /n σ Qm = const. a ea e 2 If we introduce the ratio of the density of impurity 3 α ions to the electron density, nZ/ne = , and take into 1 1 account the electron pressure p = n T , then we arrive at e e 2 the following inverse dependence of the pressure of the electrons on their density: Ð3π/2 3π/2 x ρ 2 * C2 C1 (b) p ===------, C2 -----α- const. (2.2) ne 3 It is this inverse dependence that leads to a peculiar 2 kind of quasi-Chaplygin instability, which will be ana- lyzed below. 1 1 Ð3π/8 3π/8 x 3. THEORY OF QUASI-CHAPLYGIN INSTABILITIES Fig. 4. Density profiles in an individual well and an individ- Here, the theory of quasi-Chaplygin instabilities, ual hill that develop spontaneously against the background of an initially homogeneous medium (borrowed from [3, which was presented in detail in [3, 4], is described 4]). Curves 1 refer to t = Ð∞. The evolution proceeds from only briefly. We begin with the conventional hydrody- curves 2 to curves 3. namic equations ∂ -----ρ +0,— ⋅ ()ρv = The simplest solutions to Eqs. (3.3) are those describ- ∂t ing spatially periodic standing perturbations that grow (3.1) ∂ exponentially in time: Ψ(t, r) ~ exp(γt + ikr). However, ()⋅ 1 ∂-----vv+ — v = – ρ---—p, these perturbations can only originate from spatially t periodic seeds, which, in turn, cannot appear for no rea- but supplement them with an unusual equation of state son whatever because, in such media, there are no run- in which the pressure decreases according to a power ning waves that causally connect the neighboring ρ ρ Ð|s| regions. That is why the most interesting solutions are law as the density increases, p = p0( / 0) , where p0 and ρ are the unperturbed pressure and density of a those corresponding to spontaneous local perturba- 0 tions. We require that these solutions vanish at an infi- homogeneous background medium, respectively. Intro- ∞ ducing the dimensionless density ρ = ρ/ρ , we find it nitely distant past (t Ð), thereby modeling the ∗ 0 property of perturbations to originate spontaneously expedient to rewrite the equations as and grow progressively. ∂ Thus, in three-dimensional spherical geometry, in -----ρ +0,— ⋅ ()ρ v = ∂t * * which Ψ(t, x, y, z) = ψ(R)exp(γt) (where R = (3.2) 2 2 2 ∂ 2 1/µ x ++y z ), we obtain the following equation and -----vv+ ()⋅ — v = µc —ρ , ∂t 0 * local solution for ψ(R): where c = sp/ρ is the growth rate of the perturba- ∆ ψ 1 ()ψ 2ψ ψ sinkR 0 0 0 3 ==--- R RR'' –k , =A-------------- , (3.4) tions and the parameter µ = Ð1/(1 + |s|) will be referred R kR to as the azimuthal number.