Catto, P.J.

Simakov, A.N.

DSpace@MIT

                      *Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545    Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge  MA  02139  USA               This work was supported by the U.S. Department of Energy, Grant No.  DE-FG02-91ER-54109 . Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.  Submitted for publication in the Physics of Plasma (March 2009) PSFC/JA-09-8    Pfirsch-Schlüter Electric Field in a Tokamak  P.J. Catto and A.N. Simakov*       1Pfirsch-Schlüter Electric Field in a TokamakPeter J. Catto1 and Andrei N. Simakov21 MIT Plasma Science and Fusion Center, Cambridge, MA 021392 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545AbstractA concise and complete differential equation determining the Pfirsch-Schlüter radialelectric field in an up-down symmetric tokamak is presented in the limit of weak poloidalmagnetic field.The differential equation for the Pfirsch-Schlüter electric field in a tokamak hasbeen evaluated for large aspect ratio circular flux surfaces [1], for general cross sections[2,3], and for tokamaks having the poloidal magnetic field Bp small compared to thetoroidal magnetic field Bt with a weak ion diamagnetic drift [4]. Recent measurements inAlcator C-Mod [5] and DIII-D [6] for subsonic ion flows indicate that the iondiamagnetic flow must be retained since it tends to cancel with the      E × B drift velocity,where      B = I(ψ)∇ζ + ∇ζ×∇ψ is the axisymmetric tokamak magnetic field with ψ the fluxfunction, ζ  is the toroidal angle variable, I = RBt, R is the major radius, and     E = −(∂Φ/∂ψ)∇ψ is the lowest order electric field with the electrostatic potential Φ a fluxfunction to lowest order. Consequently, it is convenient to use the results of [2-4] togeneralize the Bp << Bt expression of [4] so that it retains the ion diamagnetic drift. To doso is very straightforward since the result of [4] is consistent with the Bp << Bt limit of[2,3] within the ion diamagnetic drift term that now has to be retained. Defining the iondensity, temperature and pressure as n, T and p, and taking Z = 1 for the ion chargenumber, the lowest order ion flow velocity      V on a flux surface is the usual Pfirsch-Schlüter result of Hazeltine [7]:     V = ωR2∇ζ + u(ψ) B (1)with  ω = −c[(∂Φ/∂ψ) + (en)−1(∂p/∂ψ)], (2)and2    u = − cIe1.78〈B2〉 +0.057〈(∇||nB)2〉〈(∇||B)2〉⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ∂T∂ψ , (3)where     B = | B |,     ∇|| = B−1 B ⋅ ∇ , and 〈...〉 denotes a flux surface average. The departure of  ∂Φ/∂ψ  from a generalized radial Maxwell-Boltzmann relation (ω = constant) for an up-down symmetric tokamak is then found from the radial flux of toroidal angularmomentum by employing    − 56pν 〈R2∇ζ ⋅ π ⋅ ∇ψ〉 =R4Bp2Ω2∂ω∂ψ+ 0.19 cI2Te(∂T /∂ψ)2eT(T + Te)〈Ω2〉R2 1− B2〈B2〉⎛ ⎝ ⎜ ⎞ ⎠ ⎟   −0.53cT1/ 2en2∂∂ψI2n2〈B2〉T1/ 2〈Ω2〉∂T∂ψ〈R2〉 1B −B〈B2〉⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2− R2B2 1−〈R2〉R2⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪  , (4)where Te is the electron temperature,     ν = (4π1/ 2e4nnΛ) /3M1/ 2T3/ 2  is the ion-ioncollision frequency, and      π is the ion stress tensor. Setting    〈R2∇ζ ⋅  π ⋅ ∇ψ〉  = 0 (5)gives the differential equation determining the steady state Pfirsch-Schlüter radial electricfield once a boundary condition is specified. Except for the retention of the ∂p/∂ψ term inthe ω appearing in (4), Eqs. (2) and (4) are the same as Eqs. (68) and (75) of Ref. [4] foran up-down symmetric tokamak. Moreover, Eqs. (2) and (4) correspond precisely to theBp << Bt limit of Eq. (60) of Ref. [2], and are consistent with the Te = T large aspect ratioresult given in Ref. [1] for which the terms in the second line of Eq. (4) may be neglectedas smaller than those in the first line by an inverse aspect ratio squared. The up-downasymmetric term of Refs. [2,4] may be added into (4) as necessary.The correct isothermal limit is now recovered [8] by (5), while for the moretypical non-isothermal cases the ∂p/∂ψ term in ω is comparable to the small numericalcoefficient second term on the right when T∂[n-1∂p/∂ψ] ~ 0.2q2(∂T/∂ψ)2, with q the safetyfactor. In fact, the ion pressure gradient term is required to understand the Alcator C-Modmeasurements in collisional to semi-collisional pedstals [5] because it nearly cancels the  ∂Φ/∂ψ  term in ω. Moreover, based on the entropy production argument of [9], in abanana regime pedestal of poloidal ion gyroradius width, T∂p/∂ψ >> p∂T/∂ψ is required,as has been observed in DIII-D [6].3The temporal evolution of the electrostatic potential is found from the flux surfaceaverage of conservation of toroidal angular momentum:    ∂∂t Mn(ω〈R2〉 + Iu)[ ] + 1′ V ∂∂ψ′ V 〈R2∇ζ ⋅  π ⋅ ∇ψ〉[ ] = 0  , (6)where V' = 〈...〉.In summary, we have presented a concise and complete expression for thedifferential equation determining the Pfirsch-Schlüter radial electric field in an up-downsymmetric tokamak in the limit of weak poloidal magnetic field.ACKNOWLEDGEMENTSThe authors are indebted to Kai Wong and Vincent Chan for a number of importantconversations regarding the neoclassical electric field in tokamaks and appreciate theirinsightful comments during the preparation of this brief communication. Researchsupported by U.S. Department of Energy grants DE-FG02-91ER-54109 at MIT,  and DE-AC52-06NA-25396 at LANL.REFERENCES[1] H. A. Claassen and H. Gerhauser, Czech. J. Phys. 49, 69 (1999) and H. A. Claassen,H. Gerhauser, A. Rogister and C. Yarim, Phys. Plasmas 7, 3699 (2000).[2] P. J. Catto and A. N. Simakov, Phys. Plasmas 12, 012501 (2005).[3] P. J. Catto and A. N. Simakov, Phys. Plasmas 12, 114503 (2005) and Phys. Plasmas13, 129901 (2006).[4] S. K. Wong and V. S. Chan, Phys. Plasmas 14, 112505 (2007).[5] R. M. McDermott, B. Lipschultz, J. W. Hughes, P. J. Catto, A. E. Hubbard, I. H.Hutchinson, M. Greenwald, B. LaBombard, K. Marr, M. L. Reinke, J. E. Rice, D.Whyte and the Alcator C-Mod Team, to appear in Phys. Plasmas 16, (2009).[6] J. S. deGrassie, private communication (September 18, 2008).[7] R. D. Hazeltine, Phys. Fluids 17, 961 (1974).[8] P. J. Catto and R. D. Hazeltine, Phys. Plasmas 13, 122508 (2006).[9] G. Kagan and P. J. Catto, Plasma Phys. Control. Fusion 50, 085010 (2008).

Pfirsch-Schlüter Electric Field in a Tokamak

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