It is shown that besides the ion orbit loss mechanism, which occurs in a region a {minus} {var_epsilon}{sub t}{rho}{sub p} < r < a, the collisionless drift-orbit transport flux can also drive the poloidal {rvec E} {times} {rvec B} velocity in a region r < a {minus} {var_epsilon}{sub t}{rho}{sub p} in stellarators. Here, r is the minor radius, a is the plasma radius, {var_epsilon}{sub t} is the toroidal amplitude of the magnetic field spectrum, {rvec E} is the electric field, {rvec B} is the magnetic field, and {rho}{sub p} is the poloidal ion gyroradius. The transport-fluxdriven {rvec E} {times} {rvec B} velocity can be triggered most efficiently by an increase of the ion temperature gradient. The theory is applied to the high-mode (H-mode) phenomenon observed in stellarators

Shaing, K. C.

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A Theory of the High-Mode Phenomenon for S tellarators" K. C. Shaing Oak Ridge National Laboratory, Oak Ridge, Tennessee 3783 -8Gli ABSTRACT It is shown that besides the ion orbit loss mechanism, which occurs in a region a - qpp < r < a, the collisionless drift-orbit transport flux can also drive the poloidal ,?? x g velocity in a region r <  a - q p ,  in stellarators. Here, r is the minor radius, a is the plasma radius, E, is the toroidal amplitude of the magnetic field spectrum, ,?? is the electric field, is the magnetic field, and p ,  is the poloidal ion gyroradius. The transport-flux- driven Ex$ velocity can be triggered most efficiently by an increase of the ion temperature gradient. The theory is applied to the high-mode (H-mode) phenomenon observed in stellarators. The high-mode (H-mode) phenomenon has been observed in both tokamaks and stellarators.1-6 The general behavior is similar in both kinds of devices. It can be understood by a logical sequence originally proposed in Refs. 7 and 8. First, the poloidal ,?? x flow velocity driven by the ion orbit loss bifurcates over the local maximum of the plasma viscosity located at M, = 1. Here, E is the electric field, is the magnetic field, and M,  is the poloidal EX$ Mach number. Second, the turbulence fluctuation is suppressed by the radial gradients of the EX g angular velocity and the diamagnetic angular velocity, and the plasma confinement is thus improved. The results of this simple theory are in agreement with many H-mode phenomena observed in tokamaks. However, the theory may not adequately explain some of experimental results because realistic plasmas usually have neutral particles, magnetic islands, error fields, and other "non-ideal" effects in the edge region. Incorporating these realistic plasma conditions in the theory produces better agreement between the theory and the experimental results, as demonstrated in Refs. 9 and 10. The absence of unquantifiable parameters from the transition theory is what makes possible the comparisons with experimental results. This letter is directed toward pursuing this quantitative theory to explore the consequences of the differences between tqkamak and stellarator transport properties. It is shown that in stellarators the poloidal x B velocity can be driven to bifurcation not only by the ion orbit loss mechanism but also by the collisionless drift-orbit transport flux. The collisionless drift-orbit transport flux is the radial particle flux driven by the collisionless helically trapped particles. The ion orbit loss mechanism is important only in the region where a - ~ , p ,  < r < a ,  where a is the plasma minor radius, r is the local minor radius, E, is the toroidal amplitude of the model magnetic field spectrum B = Bo[' - E, cos$ - E, cos(m0 - ng)], Bo is the magnetic field strength on the axis, E, is the helical amplitude, (m,n) is the poloidal and toroidal mode numbers of the helical magnetic field, and p, is the ion poloidal gyroradius. In this region, the bifurcation of the ,?? x 13 velocity in stellarators is similar to that in tokamaks and is not discussed in-detail. * Research sponsored by the Office of Fusion Energy, U.S. Department of Energy, under contract DE- ACO5-84OR21400 with Lockheed Martin Energy Systems, Inc. 1 DISC LA1 M E R This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. However, in the region r < a - q p p ,  the collisionless ion drift-orbit transport flux can also drive the poloidal E x i velocity to bifurcation. This bifurcation mechanism does not exist in axisymmetric tokamaks. It can be triggered most efficiently by an increase in the ion temperature gradient. A similar mechanism was employed in Ref. 11 to show that the sign of the radial electric field E, can be positive12 in H-mode in a rippled tokamak owing to the electron drift-orbit transport flux. The poloidal and toroidal components of the momentum equation in Hamada coordinates13 in a toroidal plasma are NM a( Ep . F) Jat = -(ZP . v . x) - veff N M ( ~ ~  . F) ---( y?' J . VV) , where the angular brackets denote the flux surface average, N is the plasma density, M is the ion mass, n is the viscous tensor, J' is the current density, c is the speed of light, E, = y ' V V x V 8 ,  sp =-x 'VVxVs,  y ' = 8 . V 5 ,  x ' = 8 . V 8 ,  V is the volume enclosed in the flux surface, and 8 and 5 are poloidal and toroidal angle, res ectively. The radial current density (7 - VV) in Eqs. 1) and (2) is related to d( E .  V V ) b  through Ampkre's law ( J ' .  VV) = -(1/ 4z) d( E .  VV\/dt. The effective collision frequency of the charge- exchange momentum loss is defined as veff = N,(OV)~,  where N, is the neutral density and ( O V ) ~ ~  is the charge-exchange reaction cross section.I4 The plasma viscous forces f p  V F )  and ( 8, V .  z) contains information about the ion orbit loss, the collisionless ift-orbit transport flux, and the nonlinear collisional viscosities. The nonlinear collisional viscosities are from particles in the plateau-fluid- Pfirsch-Schluter regime.15 The particles in the collisionless regime contribute to either the ion orbit loss or the drift-orbit transport flux, depending on how far they are from the plasma boundary. The relevant collisionality parameter here is v," for the helically trapped particles, defined as v," = vRq/(~~'~v~lrn - nql), where v is the ion-ion collision frequency, R is the major radius, q is the safety factor, v, is the ion thermal speed, and m and n are the poloidal and toroidal mode numbers of the helical amplitude of the magnetic field spectrum. When v," < 1, the helically trapped particles become collisionless. We assume that nq >> rn, which leads us to conclude that the conventional ion collisionality of the toroidally trapped particles v, is larger than v,h if E, = E , ,  where E, and E, are the amplitudes of the toroidal and helical components of the magnetic field spectrum. This indicates that the most relevant particle orbit topology for the stellarator H-mode is that of the helically trapped particles. The ra_dial- drift-orbit size Ar of the helically trapped particles in the presence of the poloidal E X B drift is of the order of Ar = v,At , (3) where v,is the bounce-averaged radial drift velocity and At is the period of the drift orbit. Because the poloidal EX B velocity is larger than the V 8  and curvature drifts if of the order of r /  V, with V, = cE, / B .  Here, Q, is the electrostatic , e is the charge of the ions, and T is the ion temperature. Note that v the particle speed, and Q the ion gyrofrequency, one concludes Ar=EtPp/Mp , (4) that where M p  is the poloidal Ex 8 Mach number defined as VEB/vtBp, Because at the H-mode bifurcation M p  is of the order of unity, the helically trapped drift ion orbits can intersect the plasma boundary if they are within a distance &,pp away from the boundary; 2 that is, if they are in the region a - E,P, < r < a .  If they are in the region Y < a - E,P, , they cannot intersect the plasma boundary when M p  = 1. In that case the physical consequence of their contribution to plasma viscosities is the well-known collisionless transport flux discussed extensively in Refs. 16 and 17. The relationship between the drift-orbit transport flux and the plasma viscosities is shown to beis ( m v )  =-- C (ip.v.Z)="-(Et.v..) , non ex'w' ex'w (5) where F. VV) is the ridial component ok the drift-orbit transport flux. In general, (F-VV! con%sts of three regimes as shown in Ref. 16. Here, for simplicity, we employ 8Aly the l/v regime flux to demonstrate the bifurcation of the poloidal Ex velocity for H-mode application. In the cylindrical coordinates (r,@,5), Eq. (5) can be expressed as ( Ep . V . Z) = -(Bt . V .  E )  = -- e BpBr, , c and r, is the radial component of the particle flux. The particle flux in the 1 / v regime is where P = N T  is the ion pressure, P'=dPldr,  T '=dTldr ,  @'=d@/dr,  and the integrals I-* and Z-2 are defined as f At the steady state, Eqs. (1) and ( 2 )  in cylindrical coordinates become 32 E," 1 - 9 ( 2 ~ ) ~ ' ~  lm - nql 7[ v, 1-1 (M, - vp,p) -r26.4 "e# 4 v, Rq vt +-- , where 4 is the parallel (to E) flow speed; M p  =-cE,/Bpvt, =-cP'/Nev,B,, Vp,T = -cT'/evtBp, and E,, is the Fourier amplitude of the magnetic field spectrum B= Bo[l+~,,~,cos(rn6-ni;)]. The integrals I, and Lmn in Eqs. ( 8 )  and (9) are defined as 3 R,? V ,  (mu, -nu, i ’ + V i l ;  W, = (VI, + I.;,)x’/B + V, * V0,  4, L J  w h e r e  u), = (v,, + y,) y ’ /B  + V, - yI is the parallel article speed, v, = 3v, + v, + ve8, v, is the deflection frequency,lg and v, is the energy exchange frequency. l9 Note that we have employed the poloidal and toroidal viscosities calculated in Hamada coordinates in Ref. 15 from the solution of the drift kinetic equation and the definitions given in Ref. 20 and converted the results to the cylindrical coordinates based on the conversion formula derived for a large-aspect-ratio tokamak in Ref. 2 1. Note that the energy integrals Zmn and L,,, are truncated. This indicates that we are assuming that the particle distribution function is a Maxwellian. The hot particles contribute to drift-orbit transport flux, and the cold particles contribute to the collisional viscosities. For simplicity, we use the simple model field B = Bo[ 1 - E, cos@ - E, cos( rn@ - 4 1 .  We also assume that the toroidal flow speed =: V;, is heavily damped by toroidal viscous force and charge-exchange momentum loss so that V;/v, =x/vt  =O.  However, this assumption is not necessary. The bifurcation solutions for the coupled nonlinear Equations (8) and (9) will be presented separately. With this simplification Eq. (8) becomes a nonlinear equation of M p  for the set of plasma parameters employed. This equation is solved by plotting the left and the right sides of Eq. (8) as functions of M p .  The solution is the intersection of these two curves. For the plasmas parameters, similar to those of Wendelstein 7-AS, E, = 0.053, E, = 0.025, q = 1.92, rn = 2 ,  n = 5 ,  and vef = 0.01, the results are shown in Figs. 1-3. In Fig. 1, the collisionality is high, v, = 15, and Vp,p = Q,T = 0.25. The Ex E Mach number is subsonic for these parameters. This is the Low-mode (L-mode) solution. When the collisionality decreases from the Pfirsch-Schluter regime to the plateau regime, the neoclassical ion energy confinement improves which leads to higher values of ion temperature and ion temperature gradient dT/dr . In that case, the value of M p  increases. If the ion collisionality keeps decreasing, there can be three equilibrium solutions for M p  , as shown in Fig. 2, where %,p = Vp,T = 0.50 and V ,  = 7.5. The one with the smallest value is the continuation of the L-mode solution. The one with the largest value is the new H-mode solution. Both of these solutions are stable. The one in the middle is unstable and is not relevant. If the ion collisionality decreases further to v, = 7.0, only the H-mode solution exists, as shown in Fig. 3, where VP+ = = 0.55. Note that the H-mode solution has M p  = 2 .  One can easily conclude that increasing the ion temperature gradient is more efficient than increasing the density gradient in driving M p  to the H-mode value. The reason is that the magnitude of Z-2 is about a factor of 2.5 larger than I - l .  Thus, the increase in dT/dr needed to push the zero of the left side of Eq. (8) to a higher value, to facilitate bifurcation, is smaller than the necessary increase in dN/dr . Furthermore, the ion energy confinement in the edge region could be close to be neoclassical. In that case, lower ion collisionality can improve the ion energy confinement and increase dT/dr . However, particle confinement in the edge region is likely to be anomalous. Lower ion collisionality may not increase dN/dr . After H-mode bifurcation, particle confinement should improve, and dN/dr should increase. We emphasize that the process described here is applicable in the region r < a - ~ , p ~ . .  Because etpp is much less than 1 cm for typical edge parameters, this is the region that IS most likely to be measured with limited spatial resolution. Only in the region 4 a - c tpp  < r < a can the ion orbit loss process be observed. As noted earlier, the drift- orbit-transport-driven E x flow only exists in non-axisymmetric toroidal plasmas such as stellarators and not in axisymmetric tokamaks. We summarize the H-mode transition sequence in the region r<a-c,p,  in stellarators as follows: 1. The ion collisionality decreases due to plasma heating, which leads to higher values of T and dT/dr . 2 .  Lower collisionality and larger dT/dr drive the poloidal ,!? x flow to bifurcation. 3 .  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Then an h e  equilibrium solutions of M,. The one in the middle is unstable. 4 -2 -6 -4 -2 0 2 4 6 MP Fig. 3. J, and i, versus M, for the samc panmeters as in Fig. 1 ex- v. = 7.0 and Vp,, = V, = 0.55. Then IS only one equilibrium solution of M, . 

A theory of the high-mode phenomenon for stellarators

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A theory of the high-mode phenomenon for stellarators

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