Control of linear modes in cylindrical resistive MHD with a resistive wall, plasma rotation, and complex gain

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1 Control of linear modes in cylindrical resistive MHD with a resistive wall, plasma rotation, and complex gain Dylan Brennan 1 and John Finn 2 contributions from Andrew Cole 3 1 Princeton University / PPPL 2 Los Alamos National Laboratory 3 Columbia University October 3, 214

2 Outline Final step before full toroidal, resistive MHD control study: Linear cylindrical tokamak model with finite β for control of resistive-plasma resitive-wall modes, using a combination of normal and tangential magnetic field measurements Comparison of Full Resistive MHD with diffuse profiles compared with Analytic Reduced MHD in step function profiles, facilitates understanding of the origin physics results Marginal stability values β rp,rw < β rp,iw < β ip,rw < β ip,iw (resistive or ideal, plasma or wall) indicate transition points Imaginary gain plasma rotation stabilizes below β rp,iw because rotation supresses the diffusion of flux from the plasma out through the wall More surprisingly, imaginary gain or rotation destabilizes above β rp,iw because it prevents the feedback flux from entering the plasma through the resistive wall. Method of using complex gain to optimize in the presence of rotation for β > β rp,iw.

3 Full MHD Computation compared with Reduced MHD Analytic model for Intuition Model 1: Reduced MHD stepfunction profiles j z (r)=2θ(a 1 r) and p (r)=p ()Θ(a 2 r). Ideal outer region, Tearing layers either RI regime (γ d τ t ) /4 1 or VR regime γ d τ 1 Model 2: Full MHD smooth profiles 2 j z (r)= from (1+(r/a 1 ) 8 ) /4 Furth, Rutherford, Selberg (Flattened) and p (r)= p (1+(r/a 2 ) 16 ) 9/8 Plasma resistivity, viscosity constant: S = 1 1 8, P =.1 a 1 < r t a 2 Pressure drive outside r t enhances wall interaction; (somewhat mimics toroidal external kink) Resistive wall by thin wall approximation. γτ w ψ(r w )=[ ψ ] rw Feedback by control equation. ψ(r c )= ψ(r w )+ ψ (r w ) Rotation by Doppler shift. γ d γ + iω.

4 Equilibrium Models are comparable a) 4 b) B z 4 J z P q 3 2 J z P q r B θ r Model 1: Reduced MHD a 1 =., a 2 =.8, r w = 1, r c = 1., q()=.9 q(r)=q() for r < a 1 and q(r)=q() r 2 for r > a a1 2 1 where q()=b /R and R is the major radius Model 2: Full MHD a 1 =., a 2 =.7, r w = 1, r c = 1., q()=.8. B z (r) from radial force balance j θ B z j z B θ = p (r) by Bz 2 2 = B2 2 + p p (r) r j z(r )B θ (r )dr. B = B z () specifies q()

5 Model 1: Analysis simplified to a matching problem between regions Outer region 2 ψ = mj z (r) rf (r) ψ + 2m2 Bθ 2 (r)p (r) B 2r ψ 3 F (r) 2 = Aδ(r a 1 ) ψ Bδ(r a 2 ) ψ Solution can be expressed by a basis set ψ(r)=α 1 ψ 1 (r)+α 2 ψ 2 (r)+α 3 ψ 3 (r) With matching conditions at γ d τ t ψ(r t )= ψ r t γτ w ψ(r w )= ψ r w Tearing layer Resistive wall ψ(r c )= ψ(r w )+ ψ (r w ) Feedback

6 Model 1: Basis functions allow for analytic solution construction ψ 1 ψ 2 1 ψ 3 ψ φ φ 1 t φ 2 φ w a r t a 1 r 2 r w r c Basis function method (from early Culham years?) of separating the solution into zones with superimposed solutions shielded from neighboring resonant surfaces or conducting walls. Further separating the solution into the plasma response (ψ 1 ) and the resistive wall / control coil external solution ( ψ 2 ) simplifies the analysis into a 2x2 matrix structure for coefficients of ψ 1 and ψ 2.(ψ 3, the control flux then determined)

7 Model 1 : Simple 2x2 Matrix Structure offers Intuitive Understanding For VR regime 1 γ d τ t l 21 l 12 l 32 l 12 2 γτ w l 32 + l 32 l ( ) 22 α1 α 2 = VR: τ t S 2/3 for Pr = 1. 1 = is at β rp,iw 2 = is at β ip,rw. Equivalence of wall rotation and i (Finn-Chacon 24) For RI regime τ t S 3/ and γ d τ t (γ d τ t ) /4 For VR or RI can in principle stabilize up to β ip,iw ( 1 = 2 = ) using both and.

8 Model 2: Full MHD model includes finite β, compressibility, parallel dynamics, resistivity, viscosity γ~v = ~B B + j B p + ν 2 ~v γ~b = ~v B η ~B γ p = ṽ p Γp ṽ Finite difference discretization (with variable grid density) leads to the standard matrix form: ṽ ṽ γ B p = M B p

9 Model 2: Boundary Conditions in Full MHD: Resistive Wall and Control Boundary conditions at resistive wall include effect of control coil and complex gain, equivalent to reduced model γ d B r (r w )=ik B ṽ r imṽ r /r + r r (ṽ θ /r)= ikṽ r + r ṽ z = r (r B θ ) im B r = γ d p = ṽ r r p (r w ) Γp (r w )( v) rw γτ w B r =[ B r ] rw B r (r c )=[ (r w ) B r (r w )+r w B r (rw )]/r c

10 Results: γ vs. β showing β rp,rw < β rp,iw < β ip,rw β ip,iw ; analytic and numerical.2 a) Reduced MHD 4 x 1 3 b) Full MHD S=1x1 Pr=.1 S=1x1 6 Pr=.1 3 γτ A.1 γτ A Δ 1 Δ Δ 1, Δ 2 γτ A rp,rw rp,iw γτ A S=1 S=1 6 rp,rw rp,iw ip,rw ip,iw.2 ip,iw ~ ip,rw S=1x β β β rowth rate γ for the analytic model (a), with β rp,rw =.4, β rp,iw =.11, β ip,rw =.383, β ip,iw =.44. At β rp,iw, 1 equals zero and at β ip,rw, 2 equals zero. Numerical results in (b), showing β rp,rw =.4 β rp,iw =.11, and β ip,rw β ip,iw. Large ideal limits are due to diffuse profiles (computaitonally advantageous), while the focus is on the lower limits β rp,rw and β rp,iw.

11 In toroidal (DIII-D) configurations the upper limits at far lower β, limits in same order Resistive / ideal plasma limits with / without a (perfectly) conducting wall indicates (without rotation or control) four β limits a) b) PEST-III DCON β N n=1 Unstable NIMROD γ τ A Ideal stability boundaries ~~NIMROD n=1 boundary~~.6 Q.4 Q.2 Q PEST-III n=1 resistive Q γ q 6 a) x b) c) ψ (Wb) DIII D PEST-III resistive plasma β limits q min =1.8, flat top wall conformal at r w =1.1a no wall wall at r w =1.1a 1 P (pa) Z (m) 1 1 q=1. q=2 1 2 R (m) S=τ R /τ A =1 8 non-viscous δw= ideal plasma Z (m) R (m) β rp,rw β rp,iw β ip,rw β ip,iw q min β N Next steps: resistive wall and control in toroidal. For now: cylindrical.

12 - Maps with Ω= i = i = Show similar structure between reduced and full MHD models (a) (b) (c) Reduced MHD 1 S=1x1 6 β =.7,.9,.11,.13 1 S=1x1 β =.7,.9,.11,.13 β =.2,.7,.1, (a) Analytic model with β =.2,.7,.1, and.22. The left boundary is vertical at β =.11 = β rp,iw. (b) Full MHD with S = 1 6 with β =.7,.9,.11 = β rp,iw, and β =.13, and the left boundary is indeed vertical at β rp,iw. (c) Full MHD with S = 1 with the same β values as (b) and the left boundary is also vertical as β rp,iw =.12 is crossed. Qualtivative structure of the maps is captured by reduced MHD (a). Both results have vertical line at β rp,iw. Effects of Ω, i and i change above β rp,iw,where Ω becomes destabilizing.

13 Results qualitatively similar between Analytic and Full MHD models for β < β rp,iw : Increasing Ω Stabilizing (a) (b) Ω = 1 1 Ω = Ω = Ω = Ω = Ω = Ω = Ω = (a) Analytic with β =.68 < β rp,iw =.11. (b) Full MHD with β =.9 < β rp,iw =.12. The results show that increasing Ω increases the stable area for β < β rp,iw except for small Ω.

14 Results: β > β rp,iw where Ω is Destabilizing; analytic and numerical (a) 2 2 (b) Ω =,.1,.3,. Ω =,.1,.3, S = 1, i = i = and varying rotation Ω. (a) simplified model with β =.12 (b) full MHD model with β =.13. The plasma Doppler shift frequencies in (a) and (b) are Ω=,.1,.3,.. These results show that for β > β rp,iw the stable region shrinks as Ω increases.

15 iven a particular Ω, i can optimally counter and have the largest stable window equivalent to Ω=; analytic and numerical (a) (b) i =1 Ω =. 1 i =3 i = Ω =. i = i =4 1 i =6 i =4 i = In (a) we have reduced MHD same as above except for the wall time, which is made equal to the numerical case, τ w = 2 1 4, and i = 1,3,4,6. In (b) we have full MHD with parameters same as above, with i =, 2, 4, 8. There is an optimal value of i ; for this value the effective wall rotation rate Ω w is equal to Ω and the stable region is maximized. i equivalent to Ω w.

16 Analytic Results: With Ω=, finite i is destabilizing for β < β rp,iw and β > β rp,iw (a) (b) i =, ±2, ±4 i =, ±1, ±2 1 1 Stability diagrams for the reduced MHD model only (a) β =.68 < β rp,iw with i =, ±2, ±4 (b) β =.1 > β rp,iw with i =, ±1, ±2. In both regimes of β, i decreases the size of the stable region.

17 With i =, Ω=., i is destabilizing for β < β rp,iw ; analytic and numerical (a) i = i = 3 2 (b) i = i = i = i = i = i = (a) Reduced MHD model with β =.68 (b) the full MHD model with β =.9. Optimal value of i is small and larger values destabilize in the β < β rp,iw regime.

18 With i =, Ω=., i has an optimal value for β > β rp,iw ; analytic and numerical (a) (b) i = 4 1 i = i = i = i = 1 i =1 2 1 i = 2 1 i = (a) Reduced MHD model with β =.1 (b) Full MHD model with β =.13 In (a) the optimal value of i is -1. In (b) the stability regions are more complex, but optimal for i small.

19 Summary Feedback with complex gain multiplying normal component of B and complex gain multiplying tangential component. i and i represent simple phase shift of coils. Full resistive MHD model agrees with reduced resistive MHD model using stepfunction profiles For β < β rp,iw rotation Ω and i Ω stabilize, as expected. i stabilizes in different way For β > β rp,iw rotation Ω and i destabilize. i destabilizes too In β > β rp,iw regime with Ω: can optimize the feedback stable region by applying i such that Ω w =Ω. There is an optimal i too, but no obvious equivalence

20 Example Eigenfunction shows layer response v r x 1 4 rowth Rate: B r x v x B x v z r B z r

21 Example Eigenfunction shows layer response.1. p v div(b)/(kb) x