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Impurity drift modes at the plasma edge
PHYSICS LETTERS A
Physics LettersA 175 (1993) 41—44 North-Holland
Impurity drift modes at the plasma edge Stefano Migliuolo Research Laboratory ofElectronics, Massachusetts Institute ofTechnology, Cambridge, MA 02139-4307, USA Received 23 December 1992; revised manuscript received 2 February 1993; accepted for publication 3 February 1993
Communicated by M. Porkolab
The linear stability of modes, due to the presence of impurities and driven unstable by the combination of the free energy contained in the ion temperature gradient and the dissipation due to collisions, is investigated forparameters appropriateto the edge of the plasma in toroidal magnetic confinement experiments. These modes are low frequency (compared to the drift wave frequency), propagate in the electron diamagnetic direction (even though ~ = din T,/d ln n>O), and are sensitive to the overall charge concentration, Z 1n1/n0, the collisionality of the plasma, and the relative peaking of the density gradients of the different ion species. They are not stabilized by magnetic shear.
Theoretical understanding of transport phenomena in high temperature plasmas has long been a goal of the magnetic fusion program, since the early observation that particle and energy losses from the plasma are much faster than predicted by classical and neo-classical theories (cf. the review articles by Tang [1] and Liewer [2] as well as the transport analyses and simulations carried out by the Texas [3] and GA [4,5] groups). With respect to this “anomalous” energy loss, theoretical modelling was given impetus by experiments [6—8],which indicated that peaked density profiles were beneficial to the containment of plasma energy, and the observation [9,10] of fluctuations propagating in the ion diamagnetic direction, under conditions appropriate for the excitation of the ion temperature gradient instability [11—13].Since these early papers, a great deal of analysis has been published on the theory ofthese ion temperature gradient (ITG) modes (see, e.g., ref. [14] for an exhaustive list of references). The drawback of models that depend on ITG modes (or, for that matter, any drift wave type of instability) for the anomalous transport of energy is that predicted 2 and diffusivities with vTipI/LP cx T~’ thus decreasealways sharplyscale at the plasma edge, while experimentally inferred diffusivities peak at the edge (here p~,P~are the ion thermal velocity and gyroradius, while L~is a typical pressure gradient scale
length). Though this “defect” has been partially overcome by recent models that take into account the toroidal coupling oflinear harmonics centered at different mode rational surfaces (thereby creating a radially extended linear eigenmode) [15,161, this can only account for the region r~O.75a, leaving the edge “uncovered”. One possibility is that modes that are particular to the plasma edge can take up the slack and “carry” the energy over the last interval to the outside. One such candidate is the impurity drift mode, that is driven unstable by the temperature gradient of the plasma fuel (the primary ion population, which we assume to be hydrogenic, Z, =1), but which appears only in conjunction with the presence of a second ion species (the impurity). This mode has been studied previously, primarily in the local approximation [17—21] (Tang, White and Guzdar [22] analysed this mode in collisionless plasmas and sheared magnetic fields, for reversed profiles ~ <0). In the present work, we consider these impurity drift modes anew, in a sheared slab magnetic geometry,toB=B0(e~ +I~x/L,), for parameters appropriate the plasma edge. In particular, the effect of finite ion parallel thermal conductivity is examined, as it represents the “prototypical” dissipation required for the linear destabilization of these modes. Since the parallel phase velocity of the perturbations
0375-9601/93/S 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
41
Volume 175, number 1
PHYSICS LETTERS A
is much smaller than the electron thermal speed, the common approximation of an adiabatic electron response, ~e = en~ Ø~/T~,is adequate (note: ~ is the perturbed electrostatic potential; perturbations may be taken to be electrostatic at the plasma edge). Fluid theory is used to describe the ions. For reasonably small impurity density concentrations, the equations for conservation of parallel momentum and energy of the impurity ion species are dominated by the impurity—primary collisional friction term (cf. the appendix of ref. [18]). As a consequence, we set = ü~,and ?~ =?, (where i, I refer to the primary and impurity ion species, respectively). The remaining equations (cf.I):ref. [23]) are the two continuity equations (j=i,
29 March 1993
drift wave, we consider the limit in which ji= 0, and Z1 tIll fle ~ 1. Simple a!-
w << wI,, v~ w << k~
gebra leads to ~2
~,+1(-~-) (1 ~
~?5~ p~V~
—
(~ ?7j)
~
]
(1)
Larmorradius (relative to species mass), tasTe/Ti, and co~as k~cT~/eB0L~ is the drift frequency, defined with the electron temperature and the density scale length, L~, of the primary ion, and k1 sek~x/L5, = d~/dx~k~), the momentum conservation equation (primary ions) —
(~+i ~
=
t V~~J
k
— —
+
fl~
—
Te
+ ~
)
(2)
and the energy conservation equation (primary ions) I 2xk~c~)i~ 1 (co+i——— ‘,, 3r £‘~~ ~ = — 11iW*~1+
(4)
—
2] ,
(5)
‘0HN(~~’P~1) exp[—~cx(x/p,1) where
In1
(6) This dispersion relation (6) yields the eigenfrequency
z (0= —
1 n1
—
(a1—! )w~ 2
2—1
x[!+x+~il1i___~r2N+~)]
(7)
which shows a mode propagating in the electron diamagnetic direction, driven unstable by the primary ion temperature gradient, with the usual ,~,~ ~ fluid threshold. Note that, like standard ITG modes [131, dius, possibly providing the dominant contribution higher harmonics (N> 1) are more extended in rato energy transport (cf. ref. [241). Clearly, optimal conditions for instability are those where the three characteristic frequencies of the system, a)K (denoting dissipation), W*T = ~ (denoting the ITG fre-
(3)
quency), (fljlfle)0iW* (the impurity drift wave frequency, weighted by the impurity fraction) are commensurate. When one of the three frequencies is much higher than the rest, the three “modes” de-
where ~t and x are coefficients of order unity that characterise the strength of the parallel transport coefficients (perpendicular transport coefficients are neglected; we shall return to them at the end of this work), and C~= m1 is the sound speed. In order to illustrate the instability of the impurity
coupleand the instability is turned off (this criterion is “elastic” with respect to W*T, this frequency can be an order ofmagnitude greater than the other two and still cause an instability; this is because ~, represents the free energy for the mode). Also when dissipation (thermal conductivity) is either absent or infinite (isothermal primary ions), the instability disappears.
+
42
—
3r 1(
(0—
1 ~
k~ UHi
(i)J
2/v where w,,, ~(k~p51c5) 1~L~. This Weber equation is easily solved, leading to a solution in terms of Hermite functions:
(~v= e~/Te denotes the normalised perturbed potential, 0= d ln n1/d ln n, measures the relative peaking of the density profiles, Ps~ ,JTfl~TeIZ3C is the sound
(a1 —1)
~,
16i =w—~w*(l+~i))P~V~ 11 o,
/
Volume 175, number 1
PHYSICS LE’flERS A
In order to study trends, eqs. (1)— (3) are solved numerically with the help of a shooting code. For purposes of illustration, we adopt parameters appropriate to the edge region of, e.g. JET [25], choose oxygen (A1 =16, Z1 = 6; we consider temperatures in the 100—200 eV range) as the impurity ion, take a~= 2 as an example and vary the rest of the parameters (note that we assume equal temperatures for all species, thus ‘h = ~/a~). As expected, charge concentration p1 as Zifli/fle, rather than density concentration (fli/’Ze), plays the important role (note that, as p’ tends toward unity, our assumption of strong impurity—primary collisional friction becomes progressively invalid) in determining the growth rate. The charge of the impurity tends to mostly influence the real part of the frequency (as Z1 decreases, the mode frequency also decreases in absolute value). Parallel ion viscosity (au) has a small stabilizing effect, notable only near marginal stability. The growth rate follows a nonlinear trend with any one parameter (L5/L~,,j,, p1, w,~).Typically, it increases from zero as the parameter exceeds a threshold, reaches a maximum (where the three “modes” couple most strongly) and then decays asymptotically as the parameter is taken to very large values. Note that there never is re-stabilization (i.e., y remains positive) in the absence of parallel viscosity (p = 0). Fixing k~p~=0.0l, r= 1 in all that follows, we show the trend against magnetic shear (L5/L~)in fig. 1. Clearly
0.1
strong shear, L5=L~,is unable to stabilize the mode. Weak shear (L5 >>L~) makes for low thermal conductivity and poor coupling to the temperature perturbation. Figure 2 shows the parameters for maximum growth (in terms of impurity charge concentration and ion collisionality, for L5 = 6L~) indicating that the mode has appreciable growth (a finite fraction of an inverse transit time for the acoustic wave) over a large region of parameter space. A typical eigenmode will extend over several ion gyral radii: its wavelength of oscillation will often be (2—3 )p51, while the envelope, I ~‘I,will span (10— 15 )p51. Perpendicular transport coefficients can also be considered, in the limit of small parallel thermal conductivity and parallel viscosity, leading to a second order differential equation in conjugate (i.e., Fourier) space (cf. ref. [23]). An exhaustive search has failed to find any unstable impurity drift eigenmodes for parameters that can reasonably be applied to the edge of the plasma (the standard ITG root,
\
2
N~—~ 6 8 10 L S I/ L N
4
\~472
\
\0406
0.4
\
-
29 March 1993
0.2
0:3
04
0.5
K 2
4
Fig. 2. Loci of maximum growth rate, in £,~-p~ space (p1~Z1n,/
Fig. 1. Eigenfrequency, Q+ i1~co/k1,c~,where k1,~k7p.,/L,,, versus the magnetic shear parameter L5/L~.The (normalised) frequency of thermal conduction is varied along with the magnetic shear, k,p~c,/v~L, =0.Oi5L5/L5. Other parameters are: /c~,p~=0.Ol, r=l,u=0, aj=2, Z1=6 (oxygen), n1/n~=0.05,,~=2,
n~ is the impurity charge concentration, while (1,, k,,p,~c. = lk~p~,c5/v~L,). Values of the normalised growth rate, y,/k~,p,,c.,are indicated at sample points (the real part of the frequency, normaiised in the same manner, is of order —0.4, —0.7, —0.8 for ,~= 2, 4, 6 on these curves). Other parameters are L5/L5=6, k,p,=0.01, r=l, ~t=O,oi=2, Z1=6 (oxygen). The three curves (in ascending order, top to bottom) con~espondto
4.
~
43
Volume 175, number 1
PHYSICS LETFERS A
mildly stabilized by p’, was easily identified). In conclusion, the instability of the “impurity drift wave”, destabilized by the ion temperature gradient, has been shown to survive in the presence of magnetic shear and parallel thermal conduction, and to be a good candidate for the anomalous transport of energy over the outer portion of the plasma column. A nonlinear analysis, in the same spirit as that carried out for the ITG mode [26], will begin shortly and will, hopefully, elucidate the properties of the nonlinear convective cells that are ultimately responsible for transport.
29 March 1993
[9] D.L. Brower, W.A. Peebles, S.K. Kim, N.C. Luhmann Jr., W.M. Tang and P.E. Phillips, Phys. Rev. Lett. 59 (1987)
48. [10] D.L. M.H. Redi, Tang, R.V. Bravenec, Durst,Brower, S.P. Fan, Y.X. He, W.M. S.K. Kim, N.C. Luhmann Jr.,R.D. S.C. McCooi, A.G. Meigs, M. Nagatsu, A. Ouroua, W.A. Peebles, P.E. Phillips, T.L. Rhodes, B. Richards, C.P. Ritz, W.L. Rowan and A.J. Wooton, NucI. Fusion 29 (1989)1247. [11] L.I. Rudakov and R.Z. Sagdeev, Soy. Phys. Dokl. 6 (1961) 415. [12] M. Porkolab, Phys. Lett. 22 (1966) 427; Nucl. Fusion 8
(1968) 29.M.N. Rosenbluth and R.Z. Sagdeev, Phys. Fluids [13] B. Coppi, 10 (1967) 582. [141S. Migliuolo, Nucl. Fusion 33 (1993) 3.
This work was sponsored in part by the U.S. Department of Energy, under contract DE-FGO2-91 ER54109. The author wishes to thank B. Coppi for encouraging this research.
References [1] W.M. Tang, Nucl. Fusion 18 (1978)1089. [2]P.C. Liewer,Nucl. Fusion 25 (1985) 543. [3] W. Horton and R.D. Estes, NucI. Fusion 19 (1979) 203. [4] RJ. Groebner, W. Pfeiffer, F.P. Blau, K.H. Burrell, E.S. Fairbanks, R.P. Seraydarian, H.St. John and R.E. Stockdale, Nucl. Fusion 26 (1986) 543. [5]R.R. Dominguez and R.E. Waltz, Nuci. Fusion 27 (1987) 65. [6] S.M. Wolfe, M. Greenwaid, R. Gandy, R. Granetz, C. Gomez, D. Gwinn, B. Lipschultz, S. McCool, E. Marmar, J. Parker, R.R. Parker and J. Rice, Nuci. Fusion 26 (1986) 329. [7]0. Gehre, 0. Gruber, H.D. Murmann, D.E. Roberts, F. Wagner, B. Bomba, A. Eberhagen, H.U. Fahrbach, G. Fussmann, J. Gernhardt, K. Hubner, G. Janeschitz, K. Lacker, E.R. Muller, H. Niedermeyer, H. Rohr, G. Staudenmeier, K.H. Steuer and 0. Voilmer, Phys. Rev. Lett. 60(1988) 1502. [8] F.X. Soidner, E.R. Muller, F. Wagner, H.S. Bosch, A.
Eberhagen, H.U. Fahrbach, G. Fussmann, 0. Gehre, K. Gentle, J. Gernhardt, 0. Gruber, W. Herrmann, G. Janeschitz, M. Kornherr, H.M. Mayer, K. McCormick, H.D. Murmann, J. Neuhauser, R. Nolte, W. Poschenrieder, H. Rohr, K.H. Steuer, U. Stroth, N. Tsois and H. Verbech, Phys. Rev. Lett. 61 (1988) 1105.
44
[15] M.G. Gray, M.J. LeBrun, T. Tajima, G. Furnish and W. Horton, Bull. Am. Phys. Soc. 37 (1992)1433. [16] W.M. Tang, Bull. Am. Phys. Soc. 37 (1992)1527. [17] B. Coppi, H.P. Furth, M.N. Rosenbluth and R.Z. Sagdeev, Phys. Rev. Lett. 17(1966)377. 118] B. Coppi, G. Rewoldt and T. Schep, Phys. Fluids 19 (1976)
1144. [19] R. Paccagnella, F. Romanelli and S. Briguglio, Nucl. Fusion 30(1990) 545.
[20] M. Frojdh, M. Liljeström and H. Nordman, Nucl. Fusion 32(1992)419. [21] S. Migliuolo, Nucl. Fusion 32 (1992) 1331. [22] W.M. Tang, R.B. White and P.N. Guzdar, Phys. Fluids 23
(1980) 167. [23] A.B. Hassam, T.M. Antonsen Jr., J.F. Drake and P.N. Guzdar, Phys. Fluids B 2 (1990) 1822. [24] P.W. Terrl, J.N. LeBoeuf, P.H. Diamond, D.R. Thayer,i.E. Sedlak and G.S. Lee, Phys. Fluids 31(1988) 2920. [25] L. DeKock, P.E. Stott, S.K. Erents, P.J. Harbour, M. Laux, G.M. McCracken, CS. Pitcher, M.F. Stamp, P.C. Strangeby, D.R.R. Summers and AJ. Tagle, in: Plasma physics and controlled nuclear fusion research 1988, Vol. 1 (IAEA, Vienna, 1989) p. 467. [26] J.F. Drake, P.N. Guzdar and A. Dimits, Phys. Fluids B 3 (1991)1937.
Physics LettersA 175 (1993) 41—44 North-Holland
Impurity drift modes at the plasma edge Stefano Migliuolo Research Laboratory ofElectronics, Massachusetts Institute ofTechnology, Cambridge, MA 02139-4307, USA Received 23 December 1992; revised manuscript received 2 February 1993; accepted for publication 3 February 1993
Communicated by M. Porkolab
The linear stability of modes, due to the presence of impurities and driven unstable by the combination of the free energy contained in the ion temperature gradient and the dissipation due to collisions, is investigated forparameters appropriateto the edge of the plasma in toroidal magnetic confinement experiments. These modes are low frequency (compared to the drift wave frequency), propagate in the electron diamagnetic direction (even though ~ = din T,/d ln n>O), and are sensitive to the overall charge concentration, Z 1n1/n0, the collisionality of the plasma, and the relative peaking of the density gradients of the different ion species. They are not stabilized by magnetic shear.
Theoretical understanding of transport phenomena in high temperature plasmas has long been a goal of the magnetic fusion program, since the early observation that particle and energy losses from the plasma are much faster than predicted by classical and neo-classical theories (cf. the review articles by Tang [1] and Liewer [2] as well as the transport analyses and simulations carried out by the Texas [3] and GA [4,5] groups). With respect to this “anomalous” energy loss, theoretical modelling was given impetus by experiments [6—8],which indicated that peaked density profiles were beneficial to the containment of plasma energy, and the observation [9,10] of fluctuations propagating in the ion diamagnetic direction, under conditions appropriate for the excitation of the ion temperature gradient instability [11—13].Since these early papers, a great deal of analysis has been published on the theory ofthese ion temperature gradient (ITG) modes (see, e.g., ref. [14] for an exhaustive list of references). The drawback of models that depend on ITG modes (or, for that matter, any drift wave type of instability) for the anomalous transport of energy is that predicted 2 and diffusivities with vTipI/LP cx T~’ thus decreasealways sharplyscale at the plasma edge, while experimentally inferred diffusivities peak at the edge (here p~,P~are the ion thermal velocity and gyroradius, while L~is a typical pressure gradient scale
length). Though this “defect” has been partially overcome by recent models that take into account the toroidal coupling oflinear harmonics centered at different mode rational surfaces (thereby creating a radially extended linear eigenmode) [15,161, this can only account for the region r~O.75a, leaving the edge “uncovered”. One possibility is that modes that are particular to the plasma edge can take up the slack and “carry” the energy over the last interval to the outside. One such candidate is the impurity drift mode, that is driven unstable by the temperature gradient of the plasma fuel (the primary ion population, which we assume to be hydrogenic, Z, =1), but which appears only in conjunction with the presence of a second ion species (the impurity). This mode has been studied previously, primarily in the local approximation [17—21] (Tang, White and Guzdar [22] analysed this mode in collisionless plasmas and sheared magnetic fields, for reversed profiles ~ <0). In the present work, we consider these impurity drift modes anew, in a sheared slab magnetic geometry,toB=B0(e~ +I~x/L,), for parameters appropriate the plasma edge. In particular, the effect of finite ion parallel thermal conductivity is examined, as it represents the “prototypical” dissipation required for the linear destabilization of these modes. Since the parallel phase velocity of the perturbations
0375-9601/93/S 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
41
Volume 175, number 1
PHYSICS LETTERS A
is much smaller than the electron thermal speed, the common approximation of an adiabatic electron response, ~e = en~ Ø~/T~,is adequate (note: ~ is the perturbed electrostatic potential; perturbations may be taken to be electrostatic at the plasma edge). Fluid theory is used to describe the ions. For reasonably small impurity density concentrations, the equations for conservation of parallel momentum and energy of the impurity ion species are dominated by the impurity—primary collisional friction term (cf. the appendix of ref. [18]). As a consequence, we set = ü~,and ?~ =?, (where i, I refer to the primary and impurity ion species, respectively). The remaining equations (cf.I):ref. [23]) are the two continuity equations (j=i,
29 March 1993
drift wave, we consider the limit in which ji= 0, and Z1 tIll fle ~ 1. Simple a!-
w << wI,, v~ w << k~
gebra leads to ~2
~,+1(-~-) (1 ~
~?5~ p~V~
—
(~ ?7j)
~
]
(1)
Larmorradius (relative to species mass), tasTe/Ti, and co~as k~cT~/eB0L~ is the drift frequency, defined with the electron temperature and the density scale length, L~, of the primary ion, and k1 sek~x/L5, = d~/dx~k~), the momentum conservation equation (primary ions) —
(~+i ~
=
t V~~J
k
— —
+
fl~
—
Te
+ ~
)
(2)
and the energy conservation equation (primary ions) I 2xk~c~)i~ 1 (co+i——— ‘,, 3r £‘~~ ~ = — 11iW*~1+
(4)
—
2] ,
(5)
‘0HN(~~’P~1) exp[—~cx(x/p,1) where
In1
(6) This dispersion relation (6) yields the eigenfrequency
z (0= —
1 n1
—
(a1—! )w~ 2
2—1
x[!+x+~il1i___~r2N+~)]
(7)
which shows a mode propagating in the electron diamagnetic direction, driven unstable by the primary ion temperature gradient, with the usual ,~,~ ~ fluid threshold. Note that, like standard ITG modes [131, dius, possibly providing the dominant contribution higher harmonics (N> 1) are more extended in rato energy transport (cf. ref. [241). Clearly, optimal conditions for instability are those where the three characteristic frequencies of the system, a)K (denoting dissipation), W*T = ~ (denoting the ITG fre-
(3)
quency), (fljlfle)0iW* (the impurity drift wave frequency, weighted by the impurity fraction) are commensurate. When one of the three frequencies is much higher than the rest, the three “modes” de-
where ~t and x are coefficients of order unity that characterise the strength of the parallel transport coefficients (perpendicular transport coefficients are neglected; we shall return to them at the end of this work), and C~= m1 is the sound speed. In order to illustrate the instability of the impurity
coupleand the instability is turned off (this criterion is “elastic” with respect to W*T, this frequency can be an order ofmagnitude greater than the other two and still cause an instability; this is because ~, represents the free energy for the mode). Also when dissipation (thermal conductivity) is either absent or infinite (isothermal primary ions), the instability disappears.
+
42
—
3r 1(
(0—
1 ~
k~ UHi
(i)J
2/v where w,,, ~(k~p51c5) 1~L~. This Weber equation is easily solved, leading to a solution in terms of Hermite functions:
(~v= e~/Te denotes the normalised perturbed potential, 0= d ln n1/d ln n, measures the relative peaking of the density profiles, Ps~ ,JTfl~TeIZ3C is the sound
(a1 —1)
~,
16i =w—~w*(l+~i))P~V~ 11 o,
/
Volume 175, number 1
PHYSICS LE’flERS A
In order to study trends, eqs. (1)— (3) are solved numerically with the help of a shooting code. For purposes of illustration, we adopt parameters appropriate to the edge region of, e.g. JET [25], choose oxygen (A1 =16, Z1 = 6; we consider temperatures in the 100—200 eV range) as the impurity ion, take a~= 2 as an example and vary the rest of the parameters (note that we assume equal temperatures for all species, thus ‘h = ~/a~). As expected, charge concentration p1 as Zifli/fle, rather than density concentration (fli/’Ze), plays the important role (note that, as p’ tends toward unity, our assumption of strong impurity—primary collisional friction becomes progressively invalid) in determining the growth rate. The charge of the impurity tends to mostly influence the real part of the frequency (as Z1 decreases, the mode frequency also decreases in absolute value). Parallel ion viscosity (au) has a small stabilizing effect, notable only near marginal stability. The growth rate follows a nonlinear trend with any one parameter (L5/L~,,j,, p1, w,~).Typically, it increases from zero as the parameter exceeds a threshold, reaches a maximum (where the three “modes” couple most strongly) and then decays asymptotically as the parameter is taken to very large values. Note that there never is re-stabilization (i.e., y remains positive) in the absence of parallel viscosity (p = 0). Fixing k~p~=0.0l, r= 1 in all that follows, we show the trend against magnetic shear (L5/L~)in fig. 1. Clearly
0.1
strong shear, L5=L~,is unable to stabilize the mode. Weak shear (L5 >>L~) makes for low thermal conductivity and poor coupling to the temperature perturbation. Figure 2 shows the parameters for maximum growth (in terms of impurity charge concentration and ion collisionality, for L5 = 6L~) indicating that the mode has appreciable growth (a finite fraction of an inverse transit time for the acoustic wave) over a large region of parameter space. A typical eigenmode will extend over several ion gyral radii: its wavelength of oscillation will often be (2—3 )p51, while the envelope, I ~‘I,will span (10— 15 )p51. Perpendicular transport coefficients can also be considered, in the limit of small parallel thermal conductivity and parallel viscosity, leading to a second order differential equation in conjugate (i.e., Fourier) space (cf. ref. [23]). An exhaustive search has failed to find any unstable impurity drift eigenmodes for parameters that can reasonably be applied to the edge of the plasma (the standard ITG root,
\
2
N~—~ 6 8 10 L S I/ L N
4
\~472
\
\0406
0.4
\
-
29 March 1993
0.2
0:3
04
0.5
K 2
4
Fig. 2. Loci of maximum growth rate, in £,~-p~ space (p1~Z1n,/
Fig. 1. Eigenfrequency, Q+ i1~co/k1,c~,where k1,~k7p.,/L,,, versus the magnetic shear parameter L5/L~.The (normalised) frequency of thermal conduction is varied along with the magnetic shear, k,p~c,/v~L, =0.Oi5L5/L5. Other parameters are: /c~,p~=0.Ol, r=l,u=0, aj=2, Z1=6 (oxygen), n1/n~=0.05,,~=2,
n~ is the impurity charge concentration, while (1,, k,,p,~c. = lk~p~,c5/v~L,). Values of the normalised growth rate, y,/k~,p,,c.,are indicated at sample points (the real part of the frequency, normaiised in the same manner, is of order —0.4, —0.7, —0.8 for ,~= 2, 4, 6 on these curves). Other parameters are L5/L5=6, k,p,=0.01, r=l, ~t=O,oi=2, Z1=6 (oxygen). The three curves (in ascending order, top to bottom) con~espondto
4.
~
43
Volume 175, number 1
PHYSICS LETFERS A
mildly stabilized by p’, was easily identified). In conclusion, the instability of the “impurity drift wave”, destabilized by the ion temperature gradient, has been shown to survive in the presence of magnetic shear and parallel thermal conduction, and to be a good candidate for the anomalous transport of energy over the outer portion of the plasma column. A nonlinear analysis, in the same spirit as that carried out for the ITG mode [26], will begin shortly and will, hopefully, elucidate the properties of the nonlinear convective cells that are ultimately responsible for transport.
29 March 1993
[9] D.L. Brower, W.A. Peebles, S.K. Kim, N.C. Luhmann Jr., W.M. Tang and P.E. Phillips, Phys. Rev. Lett. 59 (1987)
48. [10] D.L. M.H. Redi, Tang, R.V. Bravenec, Durst,Brower, S.P. Fan, Y.X. He, W.M. S.K. Kim, N.C. Luhmann Jr.,R.D. S.C. McCooi, A.G. Meigs, M. Nagatsu, A. Ouroua, W.A. Peebles, P.E. Phillips, T.L. Rhodes, B. Richards, C.P. Ritz, W.L. Rowan and A.J. Wooton, NucI. Fusion 29 (1989)1247. [11] L.I. Rudakov and R.Z. Sagdeev, Soy. Phys. Dokl. 6 (1961) 415. [12] M. Porkolab, Phys. Lett. 22 (1966) 427; Nucl. Fusion 8
(1968) 29.M.N. Rosenbluth and R.Z. Sagdeev, Phys. Fluids [13] B. Coppi, 10 (1967) 582. [141S. Migliuolo, Nucl. Fusion 33 (1993) 3.
This work was sponsored in part by the U.S. Department of Energy, under contract DE-FGO2-91 ER54109. The author wishes to thank B. Coppi for encouraging this research.
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