Summary of FRC Theory Status and Planning Meeting at 1998 APS
Thursday Nov 19, 1998
Fairmont Hotel (University Room)
5:30 - 8:00 pm
agenda
Dan Barnes (LANL)
, Outstanding Problems in FRC Stability TheoryElena Belova (PPPL)
, Application of M3D-B to FRCsY. Omelchenko (GA),
Application of an Ion Ring Code to the FRCLoren Steinhauer (UW),
Relaxed Equilibria of FRC PlasmasC.Z. Cheng (PPPL),
Rotating Equilibrium for the FRCJ. Breslau (PPPL),
Simulation of Magnetic ReconnectionA. Tarditi (SAIC)
Formation and stability studies at SAICA. Hassam (UM)
Flow stabilization studies at UMS. Cohen(PPPL)
Stabilization of FRCs by rotating magnetic fieldsH. Ji (PPPL)
Rigid body analysis of global stability of FRCsM. Yamada (PPPL)
Experimental Needs and Plans
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Dan Barnes (LANL), Outstanding Problems in FRC Stability Theory
In the MHD approximation, all prolate FRC configurations are unstable to an internal kink. Nonlinear effects are found not to be stabilizing unless there is a rotation velocity of 0.5 the sound velocity, or higher. It has been concluded that a kinetic ion model is needed to predict macroscopic FRC stability. [Barnes, et al Phys. Fluids 29, 2626 (1986)] analyzed a elliptical FRC with elongation in the range 5-7 for n=1 tilt stability, using a particular trial function approach but retaining ion kinetic effects. They found stability for s values below about 2-4. Experimentally, FRX-C/LSM finds plasmas with s > 2 "disrupt", which may be a tilt. In LSX, plasmas with s values up to 4-5 have been obtained with no tilt signature. The conjecture is that tilt stability of FRCs is profile and elongation dependent. Tilt stability is expected to occur when s < C Z/R_s, where C is an equilibrium dependent constant, Z is the half-height of the FRC, and R_s is the mid-plane radius of the separatrix. A suggested action is to repeat the previous kinetic stability calculations for more realistic profiles. The numerical methods are well documented in the Phys. Fluids paper, and Barnes is willing to support reviving this kinetic stability method as a consultant or, if support is available, as a collaborator.
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Elena Belova (PPPL), Application of M3D-B to FRCs
Initial results were presented of a nonlinear 3D code in cylindrical geometry for the stability studies of FRC, M3D-B. The code utilizes a MHD/particle representation in which background plasma is described by MHD equations and energetic ions are treated using particle simulations. The MHD equations are advanced on a finite difference mesh in a cylindrical coordinate system, while particle pushing is done on a 3D Cartesian grid. Full ion dynamics are retained in order to include large-orbit effects (with s~1), which are important for the tilt mode stabilization in FRC. In contrast to the previous work, the delta-f method is utilized to reduce numerical noise in the particle part of the simulation. A 2D self-consistent equilibrium is found by solving Grad-Shafranov equation including equilibrium flows, and it is used as an initial condition for 3D stability calculations. The code was benchmarked against previous MHD simulation results for the tilting instability in FRC [Milroy, et al. (1989)]. It was found that rigid rotation reduces the growth rate, but does not stabilize the mode even for rotation rates equal to the Alfven time. Sheared rotation is found to be destabilizing for the two velocity profiles considered. Results were presented for 1% energetic ions (up to 25% of bulk pressure) showing very little difference from the MHD results. Plans are to study the stability as a function of profiles, shape, and s as the energetic ion fraction increases, and for a fully kinetic ion description. Bulk rotation could also be added to the kinetic ion description.-----------------
Y. Omelchenko (GA), Application of an Ion Ring Code to the FRC
Preliminary FRC results were discussed. This code uses full orbit particles, and fluid electrons. Methods for finding an initial equilibrium were developed. Very preliminary runs were performed with both s=4 and s=16 configurations, with somewhat ambiguous results. The present runs utilized 700,000 particles on a 100 x 60 x 1 mode grid. The toroidal field has been zeroed out for numerical reasons.-----------------
Loren Steinhauer (UW), Relaxed Equilibria of FRC Plasmas
A formalism for describing two-fluid flowing equilibria has been presented. This generalizes the Grad-Shafranov (GS) system by allowing flow of both ion and electron species and taking into account more electron physics (inertia, pressure, Hall effect). The characteristic surfaces of the two-fluid equilibria are the guiding-center surfaces for each species. The system of equations governing these equilibria reduce to a pair of second order equations for the magnetic flux and ion stream functions plus a Bernoulli equation for the density. The system involves six arbitrary functions rather than the two appearing in GS equilibria. For a relaxed state these functions take special forms. This system is general to all two-fluid equilibria and is not restricted to FRCs. The next step is to calculate 2D (r,z) flowing equilibria of FRCs in a realistic geometry. -----------------------------------------------------------------------C.Z. Cheng (PPPL), Rotating Equilibrium for the FRC
A PPPL fixed boundary flux-coordinate equilibrium code has been modified to compute FRC equilibrium with flow. The equilibrium code has 3 free functions, corresponding to normalized 1D functions for the pressure p(psi), the density rho(psi), and the toroidal rotation frequency omega(psi). These equilibria will eventually be interfaced with the NOVA-K stability code. Several equilibrium runs for the proposed SPIRIT project have been performed. The results showed that at high rotating velocity (but still less than the ion sound speed) hollow and sheet current can be formed and the weak magnetic field region near the magnetic axis enlarges. The code results will be used to help design the SPIRIT device as well as input for SPIRIT stability analysis.
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J. Breslau (PPPL), Simulation of Magnetic Reconnection
A new 2D resistive MHD code was shown that was designed to operate efficiently on parallel computers, especially for reconnection problems with boundary layers. This is done by variable mesh spacing, and using implicit differencing only in the direction across the boundary layer. The storage is laid out so that those operations which require a lot of memory references are contained within the same processor memory. Initial reconnection studies show that antisymmetric spheromak merging occurs about 4 times the rate as symmetric merging. Reconnection rates scale approximately as nu**(-.4) x eta**.55 , where nu is the [fluid] viscosity and eta is the resistivity. Pressure, two-fluid, and possibly 3D effects will be added.-----------------
A. Tarditi (SAIC) Formation and stability studies at SAIC
The TRIM code is a nonlinear resistive MHD code with a triangular adaptive mesh. It is 3D with pseudo-spectral applied to the 3rd dimension. This has been used to simulate the nonlinear growth of the MHD tilt mode in the absence of equilibrium plasma flow. The instability remained in the linear growth regime for a long time, and finally completely tilted and destroyed flux surfaces. These results were in substantial agreement with earlier published results by Milroy, et al. (1989) NIMROD is a nonlinear two-fluid resistive MHD code using a triangular/ rectangular mesh. It is planned to repeat the TRIM calculations with NIMROD as a benchmarking exercise, and then to turn on the 2-fluid terms and study the effect of different s parameters and of rotation on the tilt mode.
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A. Hassam (UM) Flow stabilization studies at UM
Two numerical studies were reported. In the first, the thermoelectric force term was added to Ohms law to demonstrate that the FRC magnetic configuration can be maintained by heating at the plasma center. In the second, a infinite-aspect-ratio rectangular pinch was nonlinearly stabilized with a sheared flow with Mach number 1.6. It was speculated that this effect may also work to stabilize the tilt mode in the FRC configuration.
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S. Cohen(PPPL) Stabilization of FRCs by rotating magnetic fields
A high frequency rotating magnetic field applied to the FRC may provide dynamic stabilization in analogy with the inverted pendulum. The frequency must high enough to provide stabilization but low enough to penetrate. If B_omega is the RMF field strength, B_a is the axial field strength, V_A is the Alfven velocity, R_s is the separatrix mid-plane radius, and L is the axial length, then the approximate criterion for stability isomega x B_omega / B_a > 1.8 V_A / L
The second criteria comes from the condition that the rotating magnetic field fully penetrate the plasma. This condition can be expressed as an inequalityomega < 10**9 x (B_omega / B_a)**2 x (R_s B_a )**-2 x T_e **3.5
Here, T_e is the electron temperature in eV, B_a is in gauss, and R_s is in cm. Someone should try and simulate this in a kinetic-ion MHD code. -----------------H. Ji (PPPL) Rigid body analysis of global stability of FRCs
A rigid-body analysis of the FRC tilting, which keeps the viscous stress tensor terms in Bragenski, shows that the s value one needs for stabilization increases with elongation. The effective plasma rotation due to ion diamagnetic drift (as a two-fluid effect) also provides tilt stabilization when s/E >2. Additional plasma rotation increases stabilization in the large E regime. Future plans are to include this self-consistently in the equilibrium calculations.
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M. Yamada (PPPL) Experimental Needs and Plans
The proposed SPIRIT experiment needs stronger supporting theory analysis for fundamental plasma stability property of FRC. We would like to positively identify stable operating regimes against interchange modes (with intermediate-n mode numbers) as well as tilt/shift modes. These calculation should include the effects of finite gyroradius and shear flows. We expect that SPIRIT will use stabilizing shells. If available, some theoretical support is also desirable in the more design specific area of formation calculations.
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It was agreed that we will create a FRC theory web page, that S. Jardin will maintain. We will list parameters for "standard" test cases on this page. The next get-together will probably be in conjunction with the Sherwood meeting .