7 0 0 2 On Vlasov approach to tokamaks near n a magnetic axis J 3 2 H. Tasso ] h Max-Planck-Institut fu¨r Plasmaphysik, Euratom Association, p - D-85748 Garching, Germany m s a l G.N. Throumoulopoulos p . University of Ioannina, Association Euratom - Hellenic Republic, s c i Section of Theoretical Physics, GR 451 10 Ioannina, Greece s y h February 2, 2008 p [ 2 v Abstract 7 2 Apreviousproofofnonexistenceoftokamakequilibriawithpurely 0 poloidalflowwithinmacroscopictheory[Throumoulopoulos,Weitzner, 1 0 Tasso, Physics of Plasmas 13, 122501 (2006)] motivated this micro- 7 scopicanalysisnearmagnetic axisfortoroidal and”straight” tokamak 0 / plasmas. Despite the new exact solutions of Vlasov’s equation found s c here, the structure of macroscopic flows remains elusive. i s y h p : v i X r a 1 1 Introduction Some time ago (see [1, 2]), it was possible to prove non existence of tokamak equilibria with purely poloidal incompressible flow. Recently, anextension to compressible plasmas appeared in Ref.[3] including Hall term and pressure anisotropy. The proof for the incompressible case given in Refs.[1, 2] was globalwhiletherecent proof[3]islimitedtotheneighbouringofthemagnetic axis through a kind of Mercier expansion. This last result motivated the idea to extend the analysis to Vlasov- Maxwell equations examined near axis. An important ingredient is to write the Vlasov equation in cylindrical coordinates in a tokamak geometry, which simplifies the subsequent analysis. We use for that purpose the calculation done in an old ICTP report [4] where the Vlasov equation is written in arbitrary orthogonal coordinates. In Section 2 the expression of the Vlasov equation is obtained in toroidal geometry. In Section 3 the ODEs of the characteristics are derived while Section 4 is devoted to ”straight tokamaks” and Section 5 to discussion and conclusions. 2 Vlasov equation in orthogonal coordinates As explained in Ref.[4] we consider a general system of orthogonal coordi- nates x1, x2, x3 with the metric ds2 = g (dx1)2 +g (dx2)2 +g (dx3)2 and 11 22 33 unit vectors e = ∇xi where i goes from 1 to 3. The velocity vector of a i |∇xi| ”microscopic” fluid element is then projected on the unit vectors e as i v = vie , (1) i where the components vi are independent upon space variables. The total derivative of v is ∂v +v·∇v = E+v×B, (2) ∂t where E and B are the electric and magnetic fields consistent with Maxwell equations and the charge to mass ratio e is set to one. Projecting Eq.(2) on m the unit vectors we obtain dvi = e ·(E+v×B)+e ·v×∇×v. (3) i i dt 2 Finally, the Vlasov equation in orthogonal coordinates is given by ∂f ∂f ∂f +v·∇f +e ·(E+v×B) +(e ·v×∇×v) = 0, (4) ∂t i ∂vi i ∂vi where f is a function of the xi, vi and time while v is given by Eq.(1). For more details see Ref. [4]. f stays here for the ion distribution while the distribution function for the electrons is governed by an equation similar to Eq.(4). Let us now specialize on cylindrical coordinates x1 = r, x2 = φ, x3 = z. Then ∇×e = 0 for i = 1 and 3 and ∇×e = e ×∇φ. If we replace the i 2 1 indices 1, 2, 3 by r,φ,z we have ∇×v = vφe ×∇φ and r vrvφe (vφ)2e φ r v×∇×v = − . (5) r r So the last term of Eq.(4) becomes [(vφ)2 ∂f −vrvφ ∂f ]. Setting B = e I near r ∂vr r ∂vφ φr axis and ∂f = 0 for steady state, Eq.(4) reads ∂t ∂f [vzI −(vφ)2] ∂f vrI ∂f vrvφ ∂f v·∇f +(e ·∇Φ) − + − = 0. (6) i ∂vi r ∂vr r ∂vz r ∂vφ Assuming ∇f = ∇Φ = 0 on axis the final equation to solve is ∂f ∂f ∂f −[vzI −(vφ)2] −vrvφ +vrI = 0. (7) ∂vr ∂vφ ∂vz 3 ODEs for characteristics Let us start with the simpler case I = 0, then the characteristics of Eq.(7) are given by the solution of dvr dvφ − = , (8) (vφ)2 vrvφ whosesolutionis(vr)2+(vφ)2 = C. Sincef = f(C,vz) = f[((vr)2+(vφ)2),vz] on axis we obtain for the toroidal flow vφfd3v = 0, (9) Z 3 which means zero toroidal flow on axis. For I 6= 0 the charasteristics are given by dvr dvφ dvz − = − = . (10) vzI −(vφ)2 vrvφ vrI The last equality delivers C = vz +Iln|vφ|, the second characteristic being 1 the particle energy C = (vr)2 +(vφ)2 +(vz)2. C is ”antisymmetric” in vz 2 1 but symmetric in vφ, which leads to vφf(C ,C )d3v = 0, vzf(C ,C )d3v 6= 0. (11) Z 1 2 Z 1 2 It means that the φ-flow is zero while the unphysical z-flow is finite. This is obviously not acceptable. 4 ”Straight” tokamaks The straight tokamaks do have magnetohydrodynamic solutions with purely poloidal flow as known from previous work [5]. For the purpose of a micro- scopic theory the appropriate coordinate system is the cartesian one x1 = x, x2 = y, x3 = z so that the toroidal angular coordinate is replaced by y and the toroidal field I by By. Since ∇×e vanishes for all i, the term v×∇×v i in Eq.(4) disappears. For the steady state with finite By, Eq.(7) is replaced by ∂f ∂f −vz +vx = 0, (12) ∂vx ∂vz whose characteristic is given by dvx dvz − = . (13) vz vx The solution of Eq.(13) is C = (vx)2 + (vz)2, which leads to f = f((vx)2 + (vz)2,vy). Purely poloidal flows are possible, which is consistent with Ref.[5]. 4 5 Discussion and Conclusions The result of section 3 obliges us to change the assumptions leading from Eq.(6) to Eq.(7) i.e. ∇f 6= 0 instead of zero on the magnetic axis. The special canonical φ-momentum solution is of that kind, and leads naturally to toroidal flows but no poloidal flows. However, a comprehensive discussion of the problem cannot be done since the complete set of characteristics of Eq.(6) is not known. Finally, though we know from section 3 that f must be a function of C 1 and C , we could, in addition, choose f to have different values for different 2 signs of, for instance, vφ. A known example of that kind of solutions is the case of BGK waves [6], in which the ”free particles” have different distribu- tions for different signs of their velocities. See also Ref.[7] for a quasi-neutral treatment. Though toroidal flows can then be constructed, physical con- straints like isotropy of the pressure tensor or constraints on other moments or geometrical symmetries and, ultimately, collisions could exclude such so- lutions. Again we are led to look for the general solution of Eq.(6) with ∇f 6= 0 on axis in order to discuss the structure of the macroscopic flows. Unfortunately, as mentioned before, the answer to this problem is quite un- certain. Acknowledgements The authors would like to thank Prof. Harold Weitzner for useful discus- sions. Part of this work was conducted during a visit of the author G.N.T. to theMax-Planck-Institut fu¨rPlasmaphysik, Garching. Thehospitalityofthat Institute is greatly appreciated. The present work was performed under the Contract of Association ERB 5005 CT 99 0100 between the European Atomic Energy Community and the Hellenic Republic. The views and opinions expressed herein do not necessar- ily reflect those of the European Commission. 5 References [1] H. Tasso, Phys. Fluids 13, 1874 (1970). [2] H. Tasso, G.N. Throumoulopoulos, Phys. Plasmas 8, 2378 (1998). [3] G.N. Throumoulopoulos, H. Weitzner, H. Tasso, Phys. Plasmas 13, 122501 (2006). [4] F. Santini, H. Tasso, Internal Report IC/70/49, (1970). (See URL: streaming.ictp.trieste.it/preprints/P/70/049.pdf). [5] G.N.Throumoulopoulos, G.Pantis, Plasma Phys. Controlled Fusion38, 1817 (1996). [6] I.B. Bernstein, J.M. Greene, M.D. Kruskal, Phys. Rev. 108, 546 (1957). [7] H. Tasso, Plasma physics 11, 663 (1969). 6