cth sawtoothing convergence and scalingsย ยท extended mhd sawtooth relaxation in cth non-linear...
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Extended MHD Sawtooth Relaxation in CTHLinear Evolution
โข After ๐0 is driven below 1, a tearing mode becomes unstable and is excited with a small amount of energyโข In simulations of CTH operating in tokamak mode, the mode is ๐ = 1โข When stellarator fields are added, ๐ is no longer a good quantum number and the unstable mode is represented
with Fourier numbers 1,4,6,9,11,14,16, โฆ
Extended MHD Sawtooth Relaxation in CTHNon-linear Evolution
โข The nonlinear evolution is the growth of an island at ๐ = 1 The island drives the reconnection of the plasma core, and the center of the island becomes the magnetic axis after complete reconnection of the core
โข This is the basic picture of Kadomptsev reconnectionโข When a 3D stellarator field is added, the core (red) and island (black) are helically deformed.
Spatial Convergence
โข Sawtooth Simulations of CTH operating in tokamak mode are spatially resolved with only 11 Fourier numbers (or less)
โข When 3D stellarator field is added, as many as 86 modes are required to resolve the reconnection current layer
๐ = ๐ยฐ ๐ = ๐๐ยฐ
๐ต = ๐๐๐ต = ๐๐๐ต = ๐๐
Toroidal current plotted on the ๐ = 0 midplane.
Cu
rren
t D
ensi
ty ๐๐ด
๐2
Time Step Convergenceโข The semi-implicit operator NIMROD uses for axisymmetric cases (left) is extremely efficient, allowing for large
time steps without loss of accuracy during the linear phase of internal kink growth. โข The isotropic operator used when 3D fields (right) are added is less efficient, requiring very small timesteps
for convergence during the linear phase of evolution.
Linear phase of internal kink mode in tokamak operation (left) and with 3D fields added (right). When 3D fields are added, a time step of 2๐ธ โ 8 or smaller
is needed for convergence.
Algebraic System Convergence
โข Strongly anisotropic thermal diffusion is seen to make the temperature advance matrix badly conditioned for cases with 3D equilibrium fields.
โข Many GMRES steps are required in these cases and the required CPU time is significantly increased.
โข Computations having the same resolution of axisymmetric systems or 3D systems without strongly anisotropic thermal diffusion proceed much faster.
๐๐ ๐๐ค decreases as 3D fields are added
โข The strength of the 3D stellarator field is defined ๐๐ฃ๐๐, the rotational transform at the limiter when there is no plasma.โข ๐๐ฃ๐๐ โก 0 for tokamak operation
โข As ๐๐ฃ๐๐ is increased, we see reduced sawtooth period ๐๐ ๐๐ค
This scaling is observed experimentally
๐พ๐๐๐ ๐๐๐๐ (๐๐)
0 0.56
0.044 0.38
0.12 0.3
Confinement is Reduced as ๐๐ฃ๐๐ is Increased
โข ๐๐ decreases as ๐๐ฃ๐๐ is increasedโข At the same time ๐ผ๐๐๐๐ ๐๐ is decreased and total Ohmic heating power ๐ is
increased
โข The energy confinement time is reduced as 3D fields are addedโข Defining ๐๐ธ as
๐๐ธ =๐
32๐๐๐ต๐ ๐๐
๐
โข V is the volume inside ๐๐ = 50 ๐๐
โข ๐๐ธ is evaluated immediately after sawtooth relaxations
Why is Energy Confinement Reduced?
โข Increasing ๐๐ฃ๐๐ leads toโข Smaller generalized minor radius
โข Energy confinement scales as ๐2 given perpendicular diffusion and nested flux surfaces
โข Chains of small islands in the equilibrium fieldsโข Rapid parallel temperature diffusion means heat flows efficiently across chains of
islands.
๐พ๐๐๐ ๐๐๐๐ (๐๐) ๐ป๐ (๐๐ฝ) S๐ โก
๐๐ฝ
๐บ(๐)
๐E (๐๐)
0 0.56 200 1.7E5 0.256 0.29
0.044 0.38 165 1.3E5 0.205 0.18
0.12 0.3 145 1.1E5 0.186 0.14
Sometimes Activity Follows Immediately After Relaxationsโข Some relaxations are followed by a flux rearrangement
โข May be due to a strong reconnection return flow that is not efficiently dissipated after reconnection of the core is complete
โข Can be suppressed by โข Increasing viscocity
โข Reducing ๐โฅ while holding ๐ constantโข This causes faster reheating of the core
โข Core does not stay in low shear configuration as long