14th IEA-RFP Workshop, Padova 26th-28th April 2010

14th IEA-RFP Workshop, Padova 26th-28th April 2010

The SHEq code: an equilibrium calculation tool for SHAx states

Emilio Martines, Barbara Momo

Consorzio RFX, Associazione Euratom-ENEA sulla fusione, Padova, Italy


SHAx states

SHAx

new helical axis

DAx magnetic axis

X-point

O-point

Experimentally found in RFX-mod [R. Lorenzini et al., PRL 101, 025005 (2008)]

DoubleAxis

SingleHelicalAxis


The Single Helicity Equilibrium (SHEq) code

The SHEq code was an important ingredient of the Nature Physics paper

[R. Lorenzini et al., Nature Phys. 5, 570 (2009)].

The code computes for SHAx states in toroidal geometry:• Shape of flux surfaces (also for DAx);• Average over flux surfaces of any quantity;• Safety factor profile;• Metric coefficients to be used by ASTRA for transport calculations.

Limitations:• Force-free; • First order in dominant mode amplitude;

• Fixed model for parallel current density profile (-0).


toroidal flux

The approach

Canonical magnetic field representation:

In general, F=F(r,,) and = (r, ,).

In Single Helicity, F=F(r,u) and = (r,u), where u = m-n. In this case, it can be shown that B·=0, where the helical flux is defined as

Thus, the contours of the helical flux give the shape of the flux surfaces.

The SHEq code uses the helical flux obtained as superposition of an axysymmetric equilibrium and of the dominant mode (1,7) eigenfunction given by Newcomb’s equation, as in:

P. Zanca and D. Terranova, Plasma Phys. Control. Fusion 46, 1115 (2004)

FBpoloidal flux


Step 1: Axysymmetric equilibrium

circular cross section

Origin of cylindrical coordinates (torus axis)

Vacuum vessel center

Origin of “geometric” and “flux” coordinates

GEOM. FLUX

Shafranov shift

Assuming circular flux surfaces, one defines “geometric” coordinates (r,,) describing the shifted surfaces, and then redefines the poloidal angle, obtaining “flux” coordinates (r,f,), i.e. straight field lines coordinates, through f = + (r, ).

)r()r(F 0f00B

Equations are derived to compute F0(r), 0(r) and (r), assuming a (r) profile given by the -0 model.


Step 2: Newcomb’s equation solution

Calculate amplitudes and phases of perturbed fluxes m,n and fm,n solving the first-order force balance equation J1B0 + J0B1 = 0.

For each n, coupled equations for m = -1, 0, 1, 2 are obtained.

The solution involves an unknown derivative discontinuity on resonant surfaces. Thus, it is required to impose both the Br and B harmonics at plasma edge, obtained from measurements.

Notice that the perturbed fluxes, F(r,f,) and (r,f,), are not flux functions any

more. However, for the Single Helicity case, = m -nF is a flux function.

Include perturbations:

n,m

fn,m0f

n,m

fn,m0f

)]nm(iexp[)r(f)r(F),,r(F

)]nm(iexp[)r()r(),,r(


Example of flux surfaces computed by SHEq


Example of mapping of Te and SXR over flux surfaces

reproduced from: R. Lorenzini et al., Nature Phys. 5, 570 (2009)

Te

SXRemissivity


Helical coordinates for flux surface averaging

P

R

Z

RAR

Z

ZA

helical axis

Helical coordinates:

Toroidal angle

Poloidal-like angle

Helical flux (flux surface label)

Coordinate origin on helical axis

label flux surface

angular variables

Geometric relationship linking helical coordinates to cylindrical coordinates.

NB: 3D Equilibrium

We use as “radius” and define a new poloidal angle, , with respect to the new axis.


Relating helical coordinates to cartesian ones

• Relate the Jacobian to that of the flux coordinates defined by Zanca and Terranova for the axisymmetric equilibrium

>0

Result:

Positive defined Jacobian

In order to compute flux surface averages, we need to write the metric tensor elements of the new coordinate system, and in particular the Jacobian.

Helical coordinate Jacobian

Function remapped on the helical coordinates

Average of a function F(x):


Examples of flux surface averages

We use as “radial” coordinate the square root of the normalized helical flux:

(now in progress, change to poloidal flux, for better comparison with VMEC)

Btor

Bpol

Jtor

Jpol


Example: power balance

g11 element of metric tensor

V’

n

1

g11 and V’ are computed by SHEq and fed into ASTRA, which calculates the thermal conductivity.

courtesy of Rita Lorenzini


Safety factor

The safety factor q is computed using the formal equivalence to Hamiltonian dynamics (method suggested by D. F. Escande).

fFB

uFB

By substitution:

We have now a “time-independent Hamiltonian” (F,u).Flux coordinates (straight field lines) in Hamiltonian language are action-angle coordinates. Compute action by averaging over constant- orbit:

'du)'u,(F2

1)(Fh

(new equivalence: H, F p, u q, t)

The motion frequency in action-angle coordinates is:h

h dF

d

Taking into account the n-fold twisting of the helical axis, the actual rotational transform can be computed as:

nh

(equivalence: H, F p, f q, t)


Example of safety factor in SHAx states

The safety factor takes an almost constant value around 1/8 inside the bean-shaped region, where the electron temperature is also flat.


Ohmic constraint

BjB 2/Vt

(many thanks to A. Boozer for useful discussions)

In stationary conditions, the parallel Ohm’s law, E·B = j·B, gives

where is the electrostatic potential and Vt is the toroidal loop voltage.Flux-surface averaging removes the electrostatic term, so that

BjBB 2/V)( t

The SHEq equilibria do not satisfy the Ohmic constraint.

the (,0) model is not adequate.Bj

2/Vt B

uniform Zeff profile


Outlook

The SHEq code is operational for RFX-mod. It also provides input for VMEC calculations (at the moment essential to ensure VMEC convergence).

Possible improvements include:

Write output in format which can be read by other codes (DKES, ....).

Adapt profile, so as to reduce the discrepancy in Ohmic constraint.

Better treatment of DAx cases (presently only flux surface plotting).

More ambitiously, iteratively compute an ohmic equilibrium, which simultaneously satisfies force balance and parallel Ohm’s law.

The use of SHEq on other RFP devices is encouraged (requires some adaptation, but we are eager to collaborate).

A closer interaction with the stellarator community would also be important.


An off-topic slide

Results from RFX-mod point to the need of providing the RFP configuration with a divertor.

We have recently proposed to use the intrinsic m=0 islands to build, for a RFP operating in SHAx state, the equivalent of the “island divertor” used in stellarators.

[E. Martines et al., Nucl. Fusion 50, 035014 (2010)]

This is an issue to be considered when designing new experiments.

Limiter-like condition Divertor-like condition




14th IEA-RFP Workshop, Padova 26th-28th April 2010

Technology

flux coordinates r

poloidal flux

flux function

normalized helical flux

example of flux surfaces

circular flux surfaces

shape of flux surfaces

fluxsurface averaging