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Cheng Zhang, Deng Zhou, Sizheng Zhu, J. E. Menard*

An Analysis of the Plasma Relaxed States for Tokamaks. Cheng Zhang, Deng Zhou, Sizheng Zhu, J. E. Menard* Institute of Plasma Physics, Chinese Academy of Science, P.O. Box 1126, Hefei 230031, P. R. China * Princeton Plasma Physics Lab, P.O.Box 451, Princeton, NJ  08543, USA.

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Cheng Zhang, Deng Zhou, Sizheng Zhu, J. E. Menard*

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  1. An Analysis of the Plasma Relaxed States for Tokamaks Cheng Zhang, Deng Zhou, Sizheng Zhu, J. E. Menard* Institute of Plasma Physics, Chinese Academy of Science, P.O. Box 1126, Hefei 230031, P. R. China *Princeton Plasma Physics Lab, P.O.Box 451, Princeton, NJ  08543, USA

  2. A B S T R A C T 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.

  3. O U T L I N E • 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

  4. Euler_Lagrange Equation for ohmic tokamak (1) • From the totalenergy dissipation : • Themagnetic helicity balance condition : • and theenergy balance condition : • We have thevariational functional : where0 and 0 are Lagrangian multipliers

  5. Euler_Lagrange Equation (2) • Thecylindrical coordinatesare used for Tokamak • The minor cross section is assumedarectangle • The plasma resistivity is assumed uniform through the whole plasma • For stationary plasma, it is reasonable to assume that theapplied electric fieldare inversely proportional to r (the distance from the symmetric axis ): • E = E1r1 / r = E0 r0 /r • Toroidal magnetic field on the • boundary is determined by TF coils, it also can be expressed asinversely proportional function of rb • Bb = B1r1 / rb = B0 r0 /rb Fig. 1. Coordinate system

  6. 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 (1.4) Equation for j and B are non- homogenous Helmhotz equation

  7. Analytical Solution of the Euler_LagrangeEquation for Plasma Current Density (1) • 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=Y1+Y2 (5) (2) （3） （4）

  8. Analytical Solution of the Euler_LagrangeEquation for Plasma Current Density (2) • Y1 and Y2 are presented as following (7) and (8) respectively: • For ， • with um2 = 2-(k1m)2 for 1m  n, um2 = (k1m)2 -2 for m>n. • In which k1m = x1m/ r00 , is the mth zero point of Bessel function of order 1. • Using the boundary condition of Y1 , we have (7)

  9. Analytical Solution of the Euler_LagrangeEquation for Plasma Current Density (3) • For , ( ) • where J1, N1, I1 and K1 are respectively Bessel and modified Bessel functions. Coefficients cn and dn are obtained by applying the boundary conditions. (8) • The analytical solution for plasma current density is obtained: • j(r,z) = Y1(r,z, , (,)) + Y2(r,z, , (,))+ E0 r0/2r • The global analysis for plasma current distribution can be obtained analytically

  10. Analytical solution for B from E-L equations 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): (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

  11. Numerical Self-Consistent Solutions of Whole Euler-Lagrange Equations • Two balance conditions are needed to determine  and  self- • consistently. However, the process can not be accomplished using • analytical method. • The self-consistent solutions of whole Euler-Lagrange equations • as well as both helicity and energy balance equations are obtained • numerically for a set of given parameters and boundary conditions. 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.

  12. Main Results (p1) A global analysis for current profile • For a given dimensions of device • j (r,z) = Y(r,z, , (,)) + E0 r0 / 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 inZhang 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 E0R0 / 2R.

  13. Main Results (p2) Some conclusion for Y(r,z) • 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. Y Y FIG.2 Some forms of Y on mid-plane for NSTX- like.

  14. 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. Fig. 3. The dependence of c1 and c2 on aspect ratio a) with fixed a = 0.67m, h = 2.7m. b) with fixed R0=0.85 m, h/a=4.

  15. Main Results (p4) from analytical and numerical Difference of typical minimum dissipation state between low and general aspect tokamaks • For a low aspect tokamak • Region between c1and c2is 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 thecurrent peaks in the edge region of the high field side. • For alarge aspect tokamak, • Region lower than c1is 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 alarge aspect general tokamak.

  16. Main Results (p5) Analysis for low aspect tokamak 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 Can be achieved by adjusting controllable parameters such as plasma resistivity, boundary toroidal magnetic field or electric field.

  17. Main Results (p6) the first form The typical form with < c2 for NSTX-like. The current peaks in the edge region on the high field side. FIG. 4. Toroidal current on equatorial plane (a) and in minor cross section (b) with the Parameters B0=0.29 T, E0 / =0.38MA/m2,  = 0.1MA/m2, =2.0m-1

  18. Main Results (p7) the second form The second form with a negative  value and c2 << c3 for NSTX-like. The current peaks in the central region. FIG. 5. Toroidal current on equatorial plane (a) and in minor cross section (b) with the Parameters B0=0.266 T, E0 / =1.694,  = - 0.371, =3.8

  19. Main Results (p8) Current hole or reversed in the centralregionis expected as a possible relaxed state The third form with a positive  value and c2 << c3 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: B0= 0.266 T, E0 / = 1.8, ( = 0.212, =4.85)

  20. Main Results (p9) Current hole or reversed in the centralregionis 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 regionfor JT-60U dimensions (R0=3.4 m, a =1.2m , h = 4.6m), with parameters B0=3.51 T, E0/ =11.61 ( = 2.3 m-1,  = 1.71)

  21. Main Results (p10) key parameter for mode change We found there exits a key parameter in determining the final relaxed state. It is the boundary parameter (E/B)b, or E0/(B0) for our model • Numerical results show that only when E0/(B0) is larger than a • critical value, E0/(B0)~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 E0/(B0) 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.

  22. Comparison with results on NSTX (p1) ( J. Menard PPPL, APS_ DPP, 1999) • 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

  23. Comparison with results on NSTX (p2) Rapid transformation of plasma current profile observed in NSTX experiment shot100857 Jonathan E. Menard (PPPL)

  24. Main Results (p11) 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, soB(R) is a deceasing function for the first type and it is raised in thecenter region for the second type Second type First type calculation experiment on NSTX (shot100857 Menard) Experiment support theoretical predictions qualitatively

  25. Main Results (p12) Calculated qprofile and Comparison with NSTX (p4) q-profiles calculated from the solutions of Euler-Lagrange equations are in good agreement with the experiment. First type Secondtype calculation experiment (shot100857 J.E. Menard)

  26. Summary and discussion (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 Eb/Bb 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.

  27. 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 obtainedby increasing E0/B0 above the critical value. It is expected that the typical state could transform to other states. Especially when key parameter E0/(B0) is close to the critical value, the current profile may be modified rapidly by MHD perturbation event, similar to that observed in experiment.

  28. 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 IP 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

  29. The analysis of plasma relaxation state can help us to understand the GLOBAL STRUCTURE of a system

  30. THANK YOU Acknowledgement This work is supported by the National Science Foundation of China (Project No. 10135020; 10475077).

  31. 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 :

  32. Euler_Lagrange Equation (3) • This is resulting Natural boundary condition • RedefiningLagrangianmultiplies and • as and， • we obtain the equation and boundary condition as following: • Equation • Boundary condition

  33. I N T R O D U C T I O N 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.

  34. 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).

  35. 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.

  36. 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. • For  <7.1 -- HIT experiment mode (2)For 7.1 < < 9.65 -- The typical form on general tokamak. The much larger driven current values than the first case are expected. (3)For > 9.65 -- There exists the reversionof both j and B in the central part of plasmas. Their reversion points are quite close to each other when  is near c.

  37. Typical Current Profiles from E-L Equation calculation experiment Fig.2 Typical mid-plane current profile for low ( <7.1) agree well with HIT experiment when  =2.91

  38. Poloidal Flux Contour of State Poloidal flux contour of state in Fig.2. The magnetic surface construction with R=0.3m, a=0.178m, k=1.82,A=1.68,(=0.41) are in good agreement with HIT experiment.

  39. Compare of Calculated Parameters with Experiment on HIT CalculatedExp. 0.300m 0.32m R a 0.178m 0.19m A 1.68 1.68 k 1.82 1.85  0.41 0.62(up)0.91(low) Itclosed 169.21 kA 171 kA Itotal 238.51 kA 222kA

  40. Three Typical Current Profiles(2) Fig.3. Typical mid-plane j profile profile for  =7.1- 9.65

  41. Three Typical Current Profiles(3) Fig.4 The profiles of toroidal current density(solid line) and magnetic field (dash line) on mid-plane for  =10.5. There exists the reversion of both j and B in central part of plasmas

  42. Dependence of  value and corresponding current profiles on experimental parameters • Numerical analysis shows that the different  and corresponding profiles, driven current value can be achieved by adjusting parameters like Vinj, , ed and vacuum Bt. • There exist critical values of plasma temperature, bias voltage and vacuum toroidal magnetic field to induce the transformation of current profile mode.

  43. ASIPP Different State () Achieved by Adjusting Plasma Temperature Reversed field state Fig.5 Different state () achieved by adjusting plasma temperature. There is the critical temperature value for mode transition to RFS.

  44. ASIPP Different State () Achieved by Adjusting Bias Voltage Vinj RF-STATE Fig.6 Different state () achieved by adjusting bias voltage Vinj. There is the critical Vinj value for mode transition to RFS.

  45. ASIPP Different State () Achieved by Adjusting vacuum toroidal magnetic field BV RF-state Fig.6 Different state () achieved by adjusting BV on r= R0. There is the critical Bv value for mode transition to RFS.

  46. Summary • For HICD tokamak(1) • Driven current profile modes are mainly determined by • Lagrange multiplier . Some critical valuec are found. Within one parameter  space the solution structure is relatively robust, a set of compatible parameters can change the quantity of solutions but not the structure. Different current profiles are obtained in different  ranges. • Three typical current profiles are predicted for HIT. The key features agree well with present experiments on HIT as the first case of <7.1. We predict other different states: Larger driven plasma current and the typical current profiles of normal tokamaks can be obtained in the region of  from 7.1 to 9.65. There exist reversions of both j and B in the central part when  becomes higher than c9.65.

  47. For HICD tokamak(2) • For a selected geometry, the values of c weakly depend on other • parameters. This means c determined by machine geometry. • The different  values and corresponding current profiles can be • achieved by adjusting some parameters. • There exist critical values of plasma temperature, bias voltage and • vacuum toroidal magnetic field to induce the transformation of • current profile mode. • Accounting to the natural boundary condition from variation • jb =Bb/2, it is found that there exits an important • boundary parameter (j/B)b for HICD tokamak, • which will decide current profile modes of relaxed state.

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