Cheng Zhang, Deng Zhou, Sizheng Zhu, J. E. Menard* Institute of Plasma Physics, Chinese Academy of Science, P.O. Box 1126, Hefei , P. R. China * Princeton Plasma Physics Lab, P.O.Box 451, Princeton, NJ 08543, USA An Analysis of the Plasma Relaxed States of the Plasma Relaxed States for Tokamaks for Tokamaks An analysis of plasma relaxed state from the minimum energy dissipation principle subject to the helicity and energy balance is presented for Ohmic tokamak with an arbitrary aspect ratio. We get the solution on toroidal current distribution from the resulting Euler-lagrangian equations. Different typical forms of current profile are found. Our results predict the possible structures of minimum dissipation state for tokamak. A B S T R A C T Minimum dissipation principle and Euler_Lagrange equations for ohmically driven tokamaks Analytical results for plasma current density and toroidal field self-consistent solutions for Euler-Lagrange equations Main theoretical results and comparison with experiment on NSTX Summary and discussion O U T L I N E Euler_Lagrange Equation for ohmic tokamak (1) Euler_Lagrange Equation for ohmic tokamak (1) From the total energy dissipation : The magnetic helicity balance condition : and the energy balance condition : We have the variational functional : where 0 and 0 are Lagrangian multipliers Euler_Lagrange Equation (2) The cylindrical coordinates are used for Tokamak The minor cross section is assumed a rectangle The plasma resistivity is assumed uniform through the whole plasma Fig. 1. Coordinate system For stationary plasma, it is reasonable to assume that the applied electric field are inversely proportional to r (the distance from the symmetric axis ): E = E 1 r 1 / r = E 0 r 0 /r Toroidal magnetic field on the boundary is determined by TF coils, it also can be expressed as inversely proportional function of rb B b = B 1 r 1 / r b = B 0 r 0 /r b Euler_Lagrange Equation (3) Taking the first variation, introducing = 0 /(1+ 0 ) = 0 /2(1+ 0 ), we finally obtain the Euler-Lagrange equations 1.1 1.2 (1.3) And natural boundary condition Equation for j and B are non- homogenous Helmhotz equation (1.4) We found a particular solution and write the solution j as the sum of this particular solution and term Y(r,z) Where Y satisfies the related homogenous equation as: with following boundary condition Solving equation of Y under boundary condition (4), We finally obtain the solution of Y as the sum of two parts Y=Y 1 +Y 2 (5) 33 44 of the Euler_Lagrange Equation Analytical Solution of the Euler_Lagrange Equation for Plasma Current Density (1) (2) Analytical Solution of the Euler_Lagrange Equation for Plasma Current Density (2) Y 1 and Y 2 are presented as following (7) and (8) respectively: For with u m 2 = 2 -(k 1 m ) 2 for 1 m n, u m 2 = (k 1 m ) for m>n. In which k 1 m = x 1 m / r 00, is the mth zero point of Bessel function of order 1. Using the boundary condition of Y 1, we have (7) Analytical Solution of the Euler_Lagrange Equation for Plasma Current Density (3) For, ( ) where J 1, N 1, I 1 and K 1 are respectively Bessel and modified Bessel functions. Coefficients c n and d n are obtained by applying the boundary conditions. (8) The analytical solution for plasma current density is obtained: The analytical solution for plasma current density is obtained: j (r,z) = Y 1 (r,z,, ( , )) + Y 2 (r,z,, ( , ))+ E 0 r 0 /2 r The global analysis for plasma current distribution can be obtained analytically Subjecting (2) into (1.2) equation of B is From the analysis for Y, we have Y= (, ) F(, R, z) B solution is presented as following (9): The second term is a particular solution of E-L equation for B it is related to the current profile. The first term is the solution of the corresponding homogeneous equation -- it is a decreasing function of R Analytical solution for B from E-L equations (9) Equations are solved numerically employing Buneman method. Poloidal flux on the boundaries is considered to be a constant, without lose of generality, set to be 0. Powell optimization method is employed to search for the point satisfying both energy and helicity balance conditions in the (, ) space. Numerical Self-Consistent Solutions of Whole Euler-Lagrange Equations Two balance conditions are needed to determine and self- Two balance conditions are needed to determine and self- consistently. However, the process can not be accomplished using consistently. However, the process can not be accomplished using analytical method. analytical method. The self-consistent solutions of whole Euler-Lagrange equations The self-consistent solutions of whole Euler-Lagrange equations as well as both helicity and energy balance equations are obtained as well as both helicity and energy balance equations are obtained numerically for a set of given parameters and boundary conditions. numerically for a set of given parameters and boundary conditions. A global analysis for current profile For a given dimensions of device j (r,z) = Y(r,z,, ( , )) + E 0 r 0 / 2 r Y is the solution of the homogenous equation related to the E-L equation and related to Lagrange multiplier and ( , ) determines the form of Y (as analyzed in Zhang et.al, Nuclear Fusion, 2001), ( , ) determines the magnitude of Y. The final current profile is determined by and the relative magnitude of two terms of Y and E 0 R 0 / 2 R. Main Results (p1) The form of Y is determined by There exists some critical values c Different forms of Y are obtained in different ranges The first form transfers smoothly to the second as increases up to > c1 When increases up to c2 the distribution changes violently, like a phase transition, Y is reversed in the central part. Main Results (p2) for Y(r,z) Some conclusion for Y(r,z) FIG.2 Some forms of Y on mid- plane for NSTX- like. Y Y Fig. 3. The dependence of c 1 and c 2 on aspect ratio a) with fixed a = 0.67m, h = 2.7m. b) with fixed R0=0.85 m, h/a=4. Main Results (p3) Analysis about the difference of typical minimum dissipation state between low and general aspect tokamaks It is found that the region between c1 and c2 is getting smaller as tokamak aspect decreases, and becomes a very narrow region for a low aspect tokamak. Main Results (p4) from analytical and numerical Main Results (p4) from analytical and numerical For a low aspect tokamak always dominant in this region Region between c1 and c2 is very narrow, meanwhile the second part of j is always dominant in this region, so the total current on mid-plane for < c2 is always a decreasing function of r It is the typical minimum dissipation state on low aspect ratio tokamak and similar with the typical experimental result, where the current peaks in the edge region of the high field side. For a large aspect tokamak, Region lower than c1 is very narrow, and the second part of j is almost uniform, therefore we can obtain a typical current profile with a peak in the central region for < c2, which corresponds to the typical experimental form for a large aspect general tokamak. Difference of typical minimum dissipation state between low and general aspect tokamaks Main Results (p5) Analysis for low aspect tokamak Can be achieved by adjusting controllable parameters such as plasma resistivity, boundary toroidal magnetic field or electric field. Three forms of current profile are presented under different experimental conditions for a low aspect ratio tokamak The first type similar with the typical experimental form peaks in the edge region of the high field side as shown in Fig.4. could be transformed violently from the first when increases to a value higher than c2 ( c2 = 2.86 for NSTX-like). (Fig.5 and Fig.6) Two other possible types Each current profile mode FIG. 4. Toroidal current on equatorial plane (a) and in minor cross section (b) with the Parameters B 0 =0.29 T, E 0 / =0.38MA/m2, = 0.1MA/m2, =2.0m-1 The typical form with < c 2 for NSTX-like. The current peaks in the edge region on the high field side. Main Results (p6) the first form FIG. 5. Toroidal current on equatorial plane (a) and in minor cross section (b) with the Parameters B 0 =0.266 T, E 0 / =1.694, = , =3.8 The second form with a negative value and c 2 < < c 3 for NSTX-like. The current peaks in the central region. Main Results (p7) the second form The third form with a positive value and c 2 < < c 3 for NSTX-like. The current may have a hole or reverse in the central part FIG.6. Toroidal current on equatorial plane (a) and in minor cross section (b). Parameters: B 0 = T, E 0 / = 1.8, ( = 0.212, =4.85) Main Results (p8) Main Results (p8) Current hole or reversed in the central region is expected as a possible relaxed state Main Results (p9) Main Results (p9) Current hole or reversed in the central region is expected as a possible relaxed state Calculated current profile with a hole or reversed in the central region for general tokamaks JT-60U. Fig.7. The current profile reversed in the central region for JT-60U dimensions (R 0 =3.4 m, a =1.2m, h = 4.6m), with parameters B 0 =3.51 T, E 0 / =11.61 ( = 2.3 m -1, = 1.71) Main Results (p10) key parameter for mode change Numerical results show that only when E 0 /( B 0 ) is larger than a critical value, E 0 /( B 0 )~5.8m -1 for NSTX-like, can we obtain solutions with larger than critical value c2. Both the second and the third types could be obtained violently by increasing key parameter E 0 /( B 0 ) to be above its critical value. The rapid transformation from the typical current profile to a central peak form has been observed in the experiment with a high loop voltage on NSTX, which seems to agree with our results. We found there exits a key parameter in determining the final relaxed state. It is the boundary parameter (E/ B) b, or E 0 /( B 0 ) for our model results on Comparison with results on NSTX (p1) There exits two typical current profile modes: One peaks close to edge region of high field side and the other peaks in central region on mid- plane. There exits rapid transformation from the typical current profile to a central peak form ( J. Menard PPPL, APS_ DPP, 1999) Rapid transformation of plasma current profile observed in NSTX experiment shot Jonathan E. Menard (PPPL) results on Comparison with results on NSTX (p2) Main Results (p11) calculated Comparison of calculated B profile with NSTX (p3) The first term of analytical solution is a decreasing function of R, the second term will be related to the current profile, so B (R) is a deceasing function for the first type and it is raised in the center region for the second type experiment on NSTX (shot Menard) Experiment support theoretical predictions qualitatively calculation First type Second type q-profiles calculated from the solutions of Euler-Lagrange equations are in good agreement with the experiment. calculation experiment (shot J.E. Menard) Main Results (p12) Main Results (p12) Calculated Calculated q profile and Comparison with NSTX (p4) First type Second type (1) For low and general aspect ratio tokamaks, there exist different typical minimum dissipation states, corresponding respectively to different typical current profiles observed in experiments (2) For a selected device geometry, there exist different types of relaxed states in the different regions of the parameter space. (3) Each current profile mode can be achieved by adjusting controllable parameters such as plasma resistivity, boundary toroidal magnetic field or boundary electric field. (4) It is found that there exists a key boundary parameter E b / B b in determining the final relaxed states. The typical minimum dissipation state may evolve to other forms abruptly by increasing key parameter to be above a critical value. Summary and discussion (5) Three forms of current profile are presented for low aspect ratio tokamaks NSTX. The first peaks in the edge of the high field side. The second peaks in the central region. The third may have a hole or reverse in the central region. Both the second and the third states could be obtained by increasing E 0 / B 0 above the critical value. It is expected that the typical state could transform to other states. Especially when key parameter E 0 /( B 0 ) is close to the critical value, the current profile may be modified rapidly by MHD perturbation event, similar to that observed in experiment. Summary and discussion Comparison with NSTX experiment 1.The first type of current profile peaks in the edge of the high field side agree with the typical experimental form on NSTX. 2.The second type expected from theory has been observed in the experiment with a high loop voltage on NSTX 3.The rapid transformation from the typical current profile to a central peak form has been observed on NSTX. Meanwhile, the inverse process has been observed. The plasma current broadens transiently during I P ramp- down. 4.The key parameter of transformation expected from theory agree with experimental condition on NSTX. 5.The features of toroidal field and q-profile before and after the transformation agree with experiment Our theoretical results get the support from the experimental results on NSTX The analysis of plasma relaxation state can help us to understand the GLOBAL STRUCTURE of a system Acknowledgement This work is supported by the National Science Foundation of China (Project No ; ). THANK YOU Euler_Lagrange Equation (2) and are Lagrangian multiplies. Taking the first variation, we have: E-L equation and natural boundary conditions are obtained if both the volume integral and surface integral are zero. This is resulting E-L equation : Euler_Lagrange Equation (3) This is resulting Natural boundary condition Redefining Lagrangian multiplies and as and we obtain the equation and boundary condition as following: Equation Boundary condition It is found that tokamak plasmas tend to evolve to a self- consistent natural profile. This means a relaxation mechanism in tokamak plasmas. To avoid dealing with the complex time- dependent nonlinear relaxation processes, variation principles are employed to predict the features of plasma relaxation. I N T R O D U C T I O N There are three variational principles used in plasmas. ----The Minimum Magnetic Energy Principle (Taylor, 1974), It successfully predicted the features of RFP experiments. ----The Principle of Minimum Entropy Production Hameiri and Bhattacharjee, 1987), employed in Tokamak plasmas. ----The Principle of Minimum rate of Energy Dissipation (Montegomery, et al, 1988 ), employed in description of RFP (Wang, et al, 1991) helicity injection current drive (Farengo, et al, 1994 ), helicity injection current drive tokamak (Zhang, et al, 1998) and Ohmically driven tokamak (Farengo, 1994). I N T R O D U C T I O N In this paper, An analysis of plasma relaxed state from the minimum energy dissipation principle is presented for Ohmic tokamak with an arbitrary aspect ratio. We get the solution on toroidal current distribution from the resulting Euler-lagrangian equations. Different typical forms of current profile are found. Our results predict the possible structures of minimum dissipation state for tokamak. I N T R O D U C T I O N (1)For < HIT experiment mode (2) For 7.1 < < The typical form on general tokamak. The much larger driven current values than the first case are expected. (3) For > There exists the reversion of both j and B in the central part of plasmas. Their reversion points are quite close to each other when is near c. ANALYSIS for HICD TOKAMAK HIT Minimum dissipation state modes are mainly decided by Lagrange multiplier. Some critical value c is found. quite different current profiles are in different ranges. c is mainly dependent on device geometry The mode changes violently like a phase transition. Typical Current Profiles from E-L Equation Fig.2 Typical mid-plane current profile for low (