理想的托克馬克，其磁場在大軸方向是對稱的。但因往往無法創造出完美的對稱，在有微小的磁場誤差(error fields)下，會破壞托克馬克裡大軸方向的對稱性，其結果會增加大軸方向電漿的阻尼並且增加流體阻尼速率(flow damping rate)。於是大家反其道而行，創造出有微小誤差的磁場，並用此誤差的磁場來控制大軸方向的電漿轉速。因此在可以控制誤差磁場狀況下，計算出阻尼的大小就變的很重要。
　　現今的托克馬克，都是在有效碰撞頻率(effective collision frequency)小於鎖住粒子(trapped particles)反彈頻率(bounce frequency)的情況下做實驗，在這樣的情況下，流體的阻尼會正比於電漿通量(transport flux)。我們利用有效碰撞頻率小於反彈頻率的條件，將漂移運動方程式(drift kinetic equation)做反彈平均(bounce average)。然後用漸近分析(asymptotic analysis)的方式，分別探討出不同範圍下的粒子分布函數，並利用此粒子分布函數來計算出各個不同範圍的電漿通量。
　　在知道不同範圍的電漿通量之後，我們利用粒子馬克斯威爾能量的分布，將三個不同範圍的通量superbanana plateau regime， 1/nu regime與 sqrt(nu) regime的結果連接在一起，其中 是碰撞頻率。再藉由電腦模擬出數值。這個方法的優點是不用去計算有效頻率再去找其範圍。
　　這篇論文將報告仔細的計算結果；包括平衡狀態下，有多於一個電場解的可能性。

In ideal tokamaks, the magnetic field is toroidally symmetric. Because it is difficult to create perfect symmetry, the toroidal symmetry in tokamaks is often destroyed by the error fields. This results in increased neoclassical toroidal plasma viscosity and the toroidal flow damping rate. Thus, the error fields are created to control toroidal plasma rotation.
Current tokamak experiments are all operated under the condition that the effective collision frequency is less than the bounce frequency of the trapped particles. In this case, the toroidal plasma viscosity is approximately proportional to the transport fluxes. Also under this condition, we can bounce average the drift kinetic equation to solve for the particle distribution function in different collisionality regimes by using asymptotic analysis.
After knowing the transport fluxes in all the asymptotic limits, we can connect the results according to the energy distribution in a Maxwellian distribution function. We then evaluate the transport fluxes numerically.
Because the ambipolarity condition Gamma_i=Gamma_e, where Gamma_i is the transport flux of ions and Gamma_e is the transport flux of electrons, is a nonlinear function of the radial electric field, there can be multiple equilibrium solutions for the ambipolar radial electric field.

引用文獻
[1] K. C. Shaing, Phys. Rev. Lett. 87, 245003 (2001).
[2] K. C. Shaing, Phys. Plasmas 10, 1443 (2003).
[3] W. Zhu, S. A. Sabbagh, R. E. Bell, J. M. Bialek, M. G. Bell, B. P. LeBlanc, S. M. Kaye, F. M. Levinton, J. E. Menard, K. C. Shaing, A. C. Sontag, and H. Yuh, Phys. Rev. Lett. 96, 225002 (2006).
[4] E. Lazzaro, R. J. Buttery, T. C. Hender, P. Zanca, R. Fitzpatrick, M. Bigi, T. Bolzonella, R. Coelho, M. DeBenedetti, S. Nowak and O. Sauter, Phys.plasma 9, 3906 (2002).
[5] Y. Sun, Y. Liang, H. R. Koslowski, S. Jachmich, A. Alfier, O. Asunta, G. Corrigan5, C. Giroud5, M. P. Gryaznevich5, D. Harting, T. Hender, E. Nardon, V. Naulin, V. Parail, T. Tala, C. Wiegmann, S. Wiesen1 and JET-EFDA contributors, 36th EPS Conference on Plasma Phys. Sofia 33E O-2.019 (2009).
[6] M. Coronado and H. Wobig, Phys. Fluid 29, 527 (1986).
[7] K. C. Shaing and J. D. Callen, Phys. Fluids 26, 3315 (1983).
[8] K. C. Shaing, Phys. Plasmas 3, 4276 (1996).
[9] K. C. Shaing, S. A. Sabbagh and M. S. Chu. Plasma Phys. Control. Fusion 51, 035009 (2009).
[10] A. A. Galeev and R. Z. Sagdeev: Review of Plasma Physics, ed. M. A. Leontovich (Consultants Bureau, New York, 1979) Vol. 7, p. 307.
[11] K. C. Shaing, S. Sabbagh, M. S. Chu, M. Becoulet and Cahyna, Phys. Plasmas 15, 082505 (2008).
[12] K. C. Shaing and S. A. Hokin, Phys. Fluids 26, 2136 (1983).
[13] K. C. Shaing, P. Cahyna, M. Becoulet, J.-K. Park, S. A. Sabbagh, and M. S. Chu Phys. Plasmas 15, 082506 (2008).
[14] http://www-fusion-magnetique.cea.fr/gb/index.html
[15] http://en.wikipedia.org/wiki/Tokamak
[16] G. Bateman, MHD Instability, MIT Press, Cambridge (1979).
[17] F. F. Chen, Introduction to plasma physics and controlled fusion, Plenum Press New York (1984).
[18] F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48, 239 (1976).
[19] S. P. Hirshman and D.J. Sigman, Nucl. Fusion 21 1079 (1981).
[20] P. Helander, D. J. Sigmar, Collisional transport in magnetized plasmas, Cambridge University Press New York (2002).
[21] K. C. Shaing, Physics of Fluids 27, 1567 (1984).