D. D. Schnack

S. E. Kruger

Center for Energy and Space Science

Science Applications International Corp.

San Diego, CA 92121

and

The NIMROD Team

**NIMROD is designed for high Lundquist number calculations:**

**Grid aligned with flux surfaces****Grid packing at selected rational surfaces**

**Push the limits of the NIMROD code for linear and nonlinear calculations at large***S*

**Modern tokamaks operate at**

**Present***nonlinear*calculations limited to

*Linear calculations are "easier", but accuracy and efficiency at high S is difficult to achieve*

**Non-circular poloidal cross-sections are difficult**

APPROACH

*Remain within the resistive MHD model*

**"Easy" problems => difficult problems**

**"Easy" problems (linear):**

**Linear resistive tearing mode***Geometry:*

**Circular cross-section, doubly periodic cylinder****Circular cross-section torus****Shaped cross-section torus**

**Difficult problems (nonlinear):**

**Single resistive tearing mode in circular cylindrical geometry****Multiple resistive tearing modes in circular cylindrical geometry****Tearing modes in a circular cross-section torus****Tearing modes in a shaped cross-section torus**

**Graduate school**

**Neo-classical tearing modes and two-fluid physics**-

THE NIMROD CODE

**Physics model**

**Contains full***resistive*,*two-fluid*, and*neo-classical*physics, including:

**Anisotropic non-linear heat flux****2 choices for neo-classical closure**

Use only resistive MHD model for this study

**Can be run in linear or nonlinear mode**

**Geometry**

*Toroidal, with arbitrarily shaped poloidal cross-section, including R*= 0**Linear, doubly periodic circular cylinder****Slab**

**Numerical methods**

*Finite elements for poloidal plane, FFTs for toroidal (or axial) direction**Semi-implicit time advance*

**Parallel implementation**

*MPI**Nearly ideal scaling demonstrated**Machine independent:*

**Serial Linux PCs to the T3E!**

THE PRESENT STUDY

*Linear resistive tearing mode in doubly periodic circular cylinder**Well-known test cases:*

Case I:

**Linearly unstable to 2/1 and 3/2 tearing modes****Flat q-profile makes interior Suydam unstable to many modes with***m*/*n*slightly greater than 1, eg., 8/7, 9/8, 10/9, 11/10, …..**Extremely difficult nonlinear case**

**Linearly unstable to 2/1 tearing mode***Advertised*to be relatively benign nonlinear case

**Concentrate on***Case I linear 2/1 tearing mode***Obtain***g*as a function of*S*for*S*> 10^{5}

Saftey Factor vs. Radius

**CASE III**

**Safety Factor vs. Radius**

**Essential for efficient computation at high S**

**Algorithm gives D***r**/*D*r*=_{0}*f*(*r*), with D*r**small near rational surface:*

2/1 Tearing Mode

S=10^{7}

2/1 Tearing Mode S=10^{7}

**(Zoomed)**

2/1 Tearing Mode S = 10^{7}

(Zoomed) with Grid Points

**Obtained converged***linear*growth rates for Case I 2/1 tearing mode for

**NIMROD scaling:****Compare with analytic scaling:**

**Grid packing essential for efficient calculations,***but*:

**Linear calculations have required up to 128 processors****Computational requirement will only increase as S becomes larger**

**Large time steps (D***t**~ 10t*are possible,_{A})*but*:

**Accuracy (as measured by linear growth rate) requires to decrease with***S*, not

Time step seems tied to the Alfvén time, not the tearing
mode growth time

**Nonlinear calculations have been less successful**

**Case I calculations are dominated by***m*/*n*~ 1 Suydam modes in the interior. Spectrum always fills and code crashes**Case III calculations with 2/1 mode at***S*= 10^{5}have begun

**Continue to push Case I linear calculations to higher***S*

**S = 10**^{8}calculations are underway

**Pursue nonlinear Case III calculations**

Are there more "interesting" nonlinear cylindrical cases?

**Repeat in more interesting geometry**

**Circular cross-section toroidal****Shaped cross-section toroidal**

**Document the parameter space available to the NIMROD code**