CTH Sawtoothing Convergence and Scalings

Auburn UniversityNicholas Roberds

November 14, 2015

Extended MHD Sawtooth Relaxation in CTHLinear Evolution

• After 𝑞0 is driven below 1, a tearing mode becomes unstable and is excited with a small amount of energy• In simulations of CTH operating in tokamak mode, the mode is 𝑛 = 1• When stellarator fields are added, 𝑛 is no longer a good quantum number and the unstable mode is represented

with Fourier numbers 1,4,6,9,11,14,16, …

Extended MHD Sawtooth Relaxation in CTHNon-linear Evolution

• The nonlinear evolution is the growth of an island at 𝑞 = 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

• This is the basic picture of Kadomptsev reconnection• When a 3D stellarator field is added, the core (red) and island (black) are helically deformed.

Spatial Convergence

• Sawtooth Simulations of CTH operating in tokamak mode are spatially resolved with only 11 Fourier numbers (or less)

• When 3D stellarator field is added, as many as 86 modes are required to resolve the reconnection current layer

𝝓 = 𝟎° 𝝓 = 𝟑𝟎°

𝑵 = 𝟖𝟔𝑵 = 𝟒𝟑𝑵 = 𝟐𝟐

Toroidal current plotted on the 𝑍 = 0 midplane.

Cu

rren

t D

ensi

ty 𝑀𝐴

𝑚2

Spatial Convergence

N=22 N=43 N=86

Time Step Convergence• 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. • 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 2𝐸 − 8 or smaller

is needed for convergence.

Algebraic System Convergence

• Strongly anisotropic thermal diffusion is seen to make the temperature advance matrix badly conditioned for cases with 3D equilibrium fields.

• Many GMRES steps are required in these cases and the required CPU time is significantly increased.

• Computations having the same resolution of axisymmetric systems or 3D systems without strongly anisotropic thermal diffusion proceed much faster.

𝜏𝑠𝑎𝑤 decreases as 3D fields are added

• The strength of the 3D stellarator field is defined 𝜄𝑣𝑎𝑐, the rotational transform at the limiter when there is no plasma.• 𝜄𝑣𝑎𝑐 ≡ 0 for tokamak operation

• As 𝜄𝑣𝑎𝑐 is increased, we see reduced sawtooth period 𝜏𝑠𝑎𝑤

This scaling is observed experimentally

𝜾𝒗𝒂𝒄 𝝉𝒔𝒂𝒘 (𝒎𝒔)

0 0.56

0.044 0.38

0.12 0.3

Confinement is Reduced as 𝜄𝑣𝑎𝑐 is Increased

• 𝑇𝑒 decreases as 𝜄𝑣𝑎𝑐 is increased• At the same time 𝐼𝑝𝑙𝑎𝑠𝑚𝑎 is decreased and total Ohmic heating power 𝑄 is

increased

• The energy confinement time is reduced as 3D fields are added• Defining 𝜏𝐸 as

𝜏𝐸 =𝑉

32𝑛𝑘𝐵𝑇 𝑑𝑉

𝑄

• V is the volume inside 𝑇𝑒 = 50 𝑒𝑉

• 𝜏𝐸 is evaluated immediately after sawtooth relaxations

Why is Energy Confinement Reduced?

• Increasing 𝜄𝑣𝑎𝑐 leads to• Smaller generalized minor radius

• Energy confinement scales as 𝑎2 given perpendicular diffusion and nested flux surfaces

• Chains of small islands in the equilibrium fields• Rapid parallel temperature diffusion means heat flows efficiently across chains of

islands.

𝜾𝒗𝒂𝒄 𝝉𝒔𝒂𝒘 (𝒎𝒔) 𝑻𝒆 (𝒆𝑽) S𝒂 ≡

𝟐𝑽

𝑺(𝒎)

𝝉E (𝒎𝒔)

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 Relaxations• Some relaxations are followed by a flux rearrangement

• May be due to a strong reconnection return flow that is not efficiently dissipated after reconnection of the core is complete

• Can be suppressed by • Increasing viscocity

• Reducing 𝑘⊥ while holding 𝑆 constant• This causes faster reheating of the core

• Core does not stay in low shear configuration as long