Minimally constrained model of self-organised

helical states in RFXGraham Dennis1, Stuart Hudson2, David Terranova3, Paolo Franz3, Robert Dewar1, and Matthew Hole1

1Plasma Research Laboratory, Australian National University2Princeton Plasma Physics Laboratory, Princeton University3Consorzio RFX, Associazione Euratom-ENEA sulla Fusione

Fisica e ingegneria della fusione:la ricerca verso una nuova fonte di energia

Consorzio RFXAssociazione Euratom-ENEA sulla Fusione

Soci: CNR, ENEA, Università di Padova, INFN, Acciaierie Venete S.p.A.Corso Stati Uniti, 4 – 35127 Padova, Italy

Tel +39 049 8295000-1 - Fax +39 049 8700718 - Email direzione@igi.cnr.it - www.igi.cnr.it

A self-organized helical state has been observed in RFP experiments

�

�

✓

Limited confinement observed in “traditional” axisymmetric RFP states

Better confinement now observed when helical state forms in RFX-mod

A self-organized helical state has been observed in RFP experiments

�

�

✓

Limited confinement observed in “traditional” axisymmetric RFP states

Better confinement now observed when helical state forms in RFX-mod

This structure occurs even for an axisymmetric plasma boundary, i.e. it is self-organized.

Ideal MHD can model the Single-Helical Axis state

ReconstructedPoincaré plot1

P. Martin et al., Nuclear Fusion 49, 104019 (2009).D. Terranova et al., PPCF 52, 124023 (2010).

[1][2]

Plasma Phys. Control. Fusion 52 (2010) 124023 D Terranova et al

Figure 1. Safety factor profiles for an axisymmetric (dashed) and helical (continuous) RFP.Resonance radii are shown with dots. Note that the ! = 7 resonance is lost in the helical case.(Colour online.)

associated with the dominant mode (this single helical equilibrium—SHEq—model isdescribed in [LOR09] and in [SHEq]). Even though the model is simple in its ingredientsand neglects pressure effects, the results are good in most cases and highlight the need for afull 3D approach not only for equilibrium but also for stability and transport. Furthermorethis approach allowed the determination of the safety factor profile with respect to the helicalaxis that is different from the monotonic profile of the axisymmetric RFP (figure 1): helicalstates are characterized by a null/reverse magnetic shear in the region corresponding to thetemperature barrier region [ME-EPS09] and the dominant helicity is no longer resonant.

As the magnetic configuration shares similarities with the stellarator, the VMEC spectralcode [HW83] was considered as a complementary equilibrium study of the helical RFP. Tothis end the code had to be modified in order to correctly deal with the RFP toroidal fieldreversal: the toroidal flux is non-monotonic and cannot be used as a flux surface label as iscustomary when dealing with the tokamak and stellarator configurations. The code is runproviding as input data global plasma quantities such as plasma current, total toroidal flux,q and pressure profiles, as well as a helical axis guess [PB09] (inferred from experimentalmeasurements such as Te profile or SXR tomographic reconstruction). The solution is thendetermined in a fixed-boundary mode, i.e. defining the shape of the last closed flux surface,LCFS. To reduce as much as possible the harmonic content, VMEC can be run with a definedtoroidal machine periodicity (Nfp) so that only this mode and its harmonics are considered.The value of Nfp in a stellarator is defined by the structure of the device, while in the helicalRFP it corresponds to the periodicity selected by the plasma itself as dominant mode of thespectrum and is the most internally resonant mode (in RFX-mod the m = 1, n = 7, so thatNfp = 7). While the issue of the LCFS will be considered in the next section, here we wouldlike to underline the key role of the safety factor profile that is an input constraint to VMECand has to be determined independently. This is done either from the SHEq equilibrium in asemi-analytical way or numerically estimated with the FLiT [INN07] or ORBIT [ORBIT] fieldline tracing codes following field lines as they wind around the helical axis. Both solutionsprovide the same result though the second one requires much more computational time.

The present RFP version of VMEC was benchmarked against a typical RFP equilibrium[ISHW09] both in the axisymmetric [ZT04] and in the helical (from SHEq) cases showing agood match between the codes when run with the same constraints and boundary conditions.Currently VMEC is being run to provide equilibria as close as possible to experimental

4

Theoretical reconstruction using VMEC2

Theoretical safety factor profile2

Ideal MHD can model the Single-Helical Axis state

ReconstructedPoincaré plot1

P. Martin et al., Nuclear Fusion 49, 104019 (2009).D. Terranova et al., PPCF 52, 124023 (2010).

[1][2]

Plasma Phys. Control. Fusion 52 (2010) 124023 D Terranova et al

Figure 1. Safety factor profiles for an axisymmetric (dashed) and helical (continuous) RFP.Resonance radii are shown with dots. Note that the ! = 7 resonance is lost in the helical case.(Colour online.)

associated with the dominant mode (this single helical equilibrium—SHEq—model isdescribed in [LOR09] and in [SHEq]). Even though the model is simple in its ingredientsand neglects pressure effects, the results are good in most cases and highlight the need for afull 3D approach not only for equilibrium but also for stability and transport. Furthermorethis approach allowed the determination of the safety factor profile with respect to the helicalaxis that is different from the monotonic profile of the axisymmetric RFP (figure 1): helicalstates are characterized by a null/reverse magnetic shear in the region corresponding to thetemperature barrier region [ME-EPS09] and the dominant helicity is no longer resonant.

As the magnetic configuration shares similarities with the stellarator, the VMEC spectralcode [HW83] was considered as a complementary equilibrium study of the helical RFP. Tothis end the code had to be modified in order to correctly deal with the RFP toroidal fieldreversal: the toroidal flux is non-monotonic and cannot be used as a flux surface label as iscustomary when dealing with the tokamak and stellarator configurations. The code is runproviding as input data global plasma quantities such as plasma current, total toroidal flux,q and pressure profiles, as well as a helical axis guess [PB09] (inferred from experimentalmeasurements such as Te profile or SXR tomographic reconstruction). The solution is thendetermined in a fixed-boundary mode, i.e. defining the shape of the last closed flux surface,LCFS. To reduce as much as possible the harmonic content, VMEC can be run with a definedtoroidal machine periodicity (Nfp) so that only this mode and its harmonics are considered.The value of Nfp in a stellarator is defined by the structure of the device, while in the helicalRFP it corresponds to the periodicity selected by the plasma itself as dominant mode of thespectrum and is the most internally resonant mode (in RFX-mod the m = 1, n = 7, so thatNfp = 7). While the issue of the LCFS will be considered in the next section, here we wouldlike to underline the key role of the safety factor profile that is an input constraint to VMECand has to be determined independently. This is done either from the SHEq equilibrium in asemi-analytical way or numerically estimated with the FLiT [INN07] or ORBIT [ORBIT] fieldline tracing codes following field lines as they wind around the helical axis. Both solutionsprovide the same result though the second one requires much more computational time.

The present RFP version of VMEC was benchmarked against a typical RFP equilibrium[ISHW09] both in the axisymmetric [ZT04] and in the helical (from SHEq) cases showing agood match between the codes when run with the same constraints and boundary conditions.Currently VMEC is being run to provide equilibria as close as possible to experimental

4

Theoretical reconstruction using VMEC2

Theoretical safety factor profile2

…but the safety factor profile must be carefully chosen

Helical states with non-trivial topology are also observed

Double-Helical Axis state

Single Helical Axis state

Axisymmetric multiple-helicity

state

Tomographic inversions of soft x-ray imaging

Helical states with non-trivial topology are also observed

Double-Helical Axis state

Single Helical Axis state

Axisymmetric multiple-helicity

state

Tomographic inversions of soft x-ray imaging

Ideal MHD (with assumed nested flux surfaces) cannot model the Double-Helical Axis state.

Helical states with non-trivial topology are also observed

Double-Helical Axis state

Single Helical Axis state

Axisymmetric multiple-helicity

state

Tomographic inversions of soft x-ray imaging

We seek a minimally constrained model for all RFX helical states

Taylor’s theory is a good description of axisymmetric Reversed Field Pinches

Taylor’s theory:

Ideal MHD:

Plasma quantities are only conserved globally

Plasma quantities conserved on every flux surface

Taylor’s theory is a good description of axisymmetric Reversed Field Pinches

Taylor’s theory:

Ideal MHD:

Plasma quantities are only conserved globally

Plasma quantities conserved on every flux surface

Goal: minimal description of helical states in RFP

Taylor’s theory is a good description of axisymmetric Reversed Field Pinches

Taylor’s theory:

Ideal MHD:

Plasma quantities are only conserved globally

Plasma quantities conserved on every flux surface

Fewer constraints

More constraints

Ideal MHDTaylor’s theory

Goal: minimal description of helical states in RFP

Taylor’s theory is a good description of axisymmetric Reversed Field Pinches

Taylor’s theory:

Ideal MHD:

Plasma quantities are only conserved globally

Plasma quantities conserved on every flux surface

Fewer constraints

More constraints

Ideal MHDTaylor’s theory

✓✗

Goal: minimal description of helical states in RFP

Taylor’s theory is a good description of axisymmetric Reversed Field Pinches

Taylor’s theory:

Ideal MHD:

Plasma quantities are only conserved globally

Plasma quantities conserved on every flux surface

Fewer constraints

More constraints

Ideal MHDTaylor’s theory

✓✗

Multiregion

Relaxed MHD

Goal: minimal description of helical states in RFP

Taylor’s theory is a good description of axisymmetric Reversed Field Pinches

Taylor’s theory:

Ideal MHD:

Plasma quantities are only conserved globally

Plasma quantities conserved on every flux surface

Fewer constraints

More constraints

Ideal MHDTaylor’s theory

✓✗

Multiregion

Relaxed MHD?

Goal: minimal description of helical states in RFP

Taylor’s theory is that a turbulent plasma will minimize its energy…

E =

Z ✓p

� � 1+

1

2B2

◆dV

Taylor’s theory is that a turbulent plasma will minimize its energy…

E =

Z ✓p

� � 1+

1

2B2

◆dV

…with conserved magnetic helicity

H =

ZA ·B dV (+ gauge terms)

H = �1�2

50 Taylor's Theory of Plasma Relaxation

f (Vl/>xA).ndS = I V.(Vl/>xA) dV (3.8a)sz Vz

= r B· V¢dV = O. (3.8b)Jvz

Thus each of the integrals Kl is invariant in a perfectly conducting plasma. This result, first obtained by Woltjer [1958], depends only on the JOundary condition n . B =0, and on the relationship E =-v x B + Vl/>.

fhe Topological Properties of the Woltjer Constraints

We are thus motivated to seek states that minimize W subject to the nfinity of constraints Kl = constant, for o:s; I :s; 00. Before proceeding, lowever, we remark on the physical significance of these constraints, as )riginally given by Moffatt [1969, 1978], and later by Berger and Field [1984], md by Taylor [1986]. Consider two infinitesimal flux tubes that follow two :losed space curves Cz and C2, with magnetic fluxes <PI and l1>2, and volumes 11 and V2, as shown in Figure 3-1. These flux tubes link each other once.

Figure 3-1. Sketch of linked flux tubes.

3.2 Energy Minimization with the Constraints of Ideal MHO 51

Then, for the first flux tube, VI, we can write BdV = B'ndS dl = <Pldl, and hence

Kl = f A·BdV = <Pt" A·d1 <Pt<1>:2, (3.9)VI 7CI

while, for the second flux tube, V2, we have

K2 = r A·BdV = <1>:2" A·d1 = <Pt<1>:2 (3.10)JV2 7C2

Thus, K1 and K2 measure the linkage of the two tubes of flux. If the tubes were not interlinked, the line integrals would vanish, as would Kl and K2; if they had been linked N times, we would find K1 = K2 = ± N <Pll1>2, with the sign determined by the "handedness" of the linkage. A single knotted flux tube with flux <P, and a flux tube with two twists, have also the same helicity. The integrals Kl are thus seen to measure a topological property of the field. The invariance of the Kl with respect to ideal MHO motions (for which E = - v x B + Vl/» is another expression of the well known property of integrity of flux tubes, as derived in Chapter 2: they cannot be broken or reconnected. The topological complexity of the field, once initially established, is retained for all time.

3.2 Energy Minimization with the Constraints of Ideal MHO

We now continue with the minimization of W. We have just shown that ideal MHO, in which E = -v x B + V l/>, implies that the Kl are constant. We may thus seek to directly minimize W with respect to this infinity of constraints. This requires an extension of the usual Lagrange multiplier technique [Taylor, 1986]. However, it can be shown [Freidberg, 1987] that this minimization procedure is identically equivalent to the unconstrained minimization of W under the assumption of ideal MHO. (Thus, the statement that W is minimized while the Kl remained fixed is equivalent to the assumption of ideal MHO, and vice versa: each implies the other.) Since the latter is mathematically simpler, we adopt this approach. Furthermore, we assume that the internal energy density is much less than the magnetic energy density, or f3 = pi B2« 1. Then the minimization of W reduces to the minimization of the magnetic energy. We discuss this point further later in this chapter.

To perform the minimization, we imagine an infinitesimal displacement of the plasma away from some static background state, and then calculate the resulting change in energy. Setting this energy change to zero determines a differential equation that must be satisfied by the background magnetic field. The imagined displacement is arbitrary; its effect on the

Taylor’s theory is that a turbulent plasma will minimize its energy…

E =

Z ✓p

� � 1+

1

2B2

◆dV

…with conserved magnetic helicity

H =

ZA ·B dV (+ gauge terms)

H = �1�2

50 Taylor's Theory of Plasma Relaxation

f (Vl/>xA).ndS = I V.(Vl/>xA) dV (3.8a)sz Vz

= r B· V¢dV = O. (3.8b)Jvz

Thus each of the integrals Kl is invariant in a perfectly conducting plasma. This result, first obtained by Woltjer [1958], depends only on the JOundary condition n . B =0, and on the relationship E =-v x B + Vl/>.

fhe Topological Properties of the Woltjer Constraints

We are thus motivated to seek states that minimize W subject to the nfinity of constraints Kl = constant, for o:s; I :s; 00. Before proceeding, lowever, we remark on the physical significance of these constraints, as )riginally given by Moffatt [1969, 1978], and later by Berger and Field [1984], md by Taylor [1986]. Consider two infinitesimal flux tubes that follow two :losed space curves Cz and C2, with magnetic fluxes <PI and l1>2, and volumes 11 and V2, as shown in Figure 3-1. These flux tubes link each other once.

Figure 3-1. Sketch of linked flux tubes.

3.2 Energy Minimization with the Constraints of Ideal MHO 51

Then, for the first flux tube, VI, we can write BdV = B'ndS dl = <Pldl, and hence

Kl = f A·BdV = <Pt" A·d1 <Pt<1>:2, (3.9)VI 7CI

while, for the second flux tube, V2, we have

K2 = r A·BdV = <1>:2" A·d1 = <Pt<1>:2 (3.10)JV2 7C2

Thus, K1 and K2 measure the linkage of the two tubes of flux. If the tubes were not interlinked, the line integrals would vanish, as would Kl and K2; if they had been linked N times, we would find K1 = K2 = ± N <Pll1>2, with the sign determined by the "handedness" of the linkage. A single knotted flux tube with flux <P, and a flux tube with two twists, have also the same helicity. The integrals Kl are thus seen to measure a topological property of the field. The invariance of the Kl with respect to ideal MHO motions (for which E = - v x B + Vl/» is another expression of the well known property of integrity of flux tubes, as derived in Chapter 2: they cannot be broken or reconnected. The topological complexity of the field, once initially established, is retained for all time.

3.2 Energy Minimization with the Constraints of Ideal MHO

We now continue with the minimization of W. We have just shown that ideal MHO, in which E = -v x B + V l/>, implies that the Kl are constant. We may thus seek to directly minimize W with respect to this infinity of constraints. This requires an extension of the usual Lagrange multiplier technique [Taylor, 1986]. However, it can be shown [Freidberg, 1987] that this minimization procedure is identically equivalent to the unconstrained minimization of W under the assumption of ideal MHO. (Thus, the statement that W is minimized while the Kl remained fixed is equivalent to the assumption of ideal MHO, and vice versa: each implies the other.) Since the latter is mathematically simpler, we adopt this approach. Furthermore, we assume that the internal energy density is much less than the magnetic energy density, or f3 = pi B2« 1. Then the minimization of W reduces to the minimization of the magnetic energy. We discuss this point further later in this chapter.

To perform the minimization, we imagine an infinitesimal displacement of the plasma away from some static background state, and then calculate the resulting change in energy. Setting this energy change to zero determines a differential equation that must be satisfied by the background magnetic field. The imagined displacement is arbitrary; its effect on the

Taylor’s theory is that a turbulent plasma will minimize its energy…

E =

Z ✓p

� � 1+

1

2B2

◆dV

…with conserved magnetic helicity

H =

ZA ·B dV (+ gauge terms)

…and conserved enclosed fluxes

H = �1�2

50 Taylor's Theory of Plasma Relaxation

f (Vl/>xA).ndS = I V.(Vl/>xA) dV (3.8a)sz Vz

= r B· V¢dV = O. (3.8b)Jvz

Thus each of the integrals Kl is invariant in a perfectly conducting plasma. This result, first obtained by Woltjer [1958], depends only on the JOundary condition n . B =0, and on the relationship E =-v x B + Vl/>.

fhe Topological Properties of the Woltjer Constraints

We are thus motivated to seek states that minimize W subject to the nfinity of constraints Kl = constant, for o:s; I :s; 00. Before proceeding, lowever, we remark on the physical significance of these constraints, as )riginally given by Moffatt [1969, 1978], and later by Berger and Field [1984], md by Taylor [1986]. Consider two infinitesimal flux tubes that follow two :losed space curves Cz and C2, with magnetic fluxes <PI and l1>2, and volumes 11 and V2, as shown in Figure 3-1. These flux tubes link each other once.

Figure 3-1. Sketch of linked flux tubes.

3.2 Energy Minimization with the Constraints of Ideal MHO 51

Then, for the first flux tube, VI, we can write BdV = B'ndS dl = <Pldl, and hence

Kl = f A·BdV = <Pt" A·d1 <Pt<1>:2, (3.9)VI 7CI

while, for the second flux tube, V2, we have

K2 = r A·BdV = <1>:2" A·d1 = <Pt<1>:2 (3.10)JV2 7C2

Thus, K1 and K2 measure the linkage of the two tubes of flux. If the tubes were not interlinked, the line integrals would vanish, as would Kl and K2; if they had been linked N times, we would find K1 = K2 = ± N <Pll1>2, with the sign determined by the "handedness" of the linkage. A single knotted flux tube with flux <P, and a flux tube with two twists, have also the same helicity. The integrals Kl are thus seen to measure a topological property of the field. The invariance of the Kl with respect to ideal MHO motions (for which E = - v x B + Vl/» is another expression of the well known property of integrity of flux tubes, as derived in Chapter 2: they cannot be broken or reconnected. The topological complexity of the field, once initially established, is retained for all time.

3.2 Energy Minimization with the Constraints of Ideal MHO

We now continue with the minimization of W. We have just shown that ideal MHO, in which E = -v x B + V l/>, implies that the Kl are constant. We may thus seek to directly minimize W with respect to this infinity of constraints. This requires an extension of the usual Lagrange multiplier technique [Taylor, 1986]. However, it can be shown [Freidberg, 1987] that this minimization procedure is identically equivalent to the unconstrained minimization of W under the assumption of ideal MHO. (Thus, the statement that W is minimized while the Kl remained fixed is equivalent to the assumption of ideal MHO, and vice versa: each implies the other.) Since the latter is mathematically simpler, we adopt this approach. Furthermore, we assume that the internal energy density is much less than the magnetic energy density, or f3 = pi B2« 1. Then the minimization of W reduces to the minimization of the magnetic energy. We discuss this point further later in this chapter.

To perform the minimization, we imagine an infinitesimal displacement of the plasma away from some static background state, and then calculate the resulting change in energy. Setting this energy change to zero determines a differential equation that must be satisfied by the background magnetic field. The imagined displacement is arbitrary; its effect on the

Taylor’s theory is that a turbulent plasma will minimize its energy…

E =

Z ✓p

� � 1+

1

2B2

◆dV

…with conserved magnetic helicity

H =

ZA ·B dV (+ gauge terms)

H = ⌘

ZJ ·B dV ⇠ ⌘

X

k

k1B2k

Motivation: with small resistivity, both energy and helicity will decay

E = ⌘

ZJ · J dV ⇠ ⌘

X

k

k2B2k

... but energy more quickly (for short length-scale turbulence)

…and conserved enclosed fluxes

• Relaxed regions , separated by• nested, ideal, toroidal barrier

interfaces , which• independently undergo Taylor

relaxation.• Magnetic islands and chaos are

allowed between the toroidal current sheets• Each plasma region has constant

pressure, creating a piecewise constant pressure profile

Ri

Ii

Multiregion Relaxed MHD (MRxMHD) extends Taylor Relaxation

RN

R1

R2 I1

I2

ININ�1

Multiregion Relaxed MHD (MRxMHD) approaches ideal MHD as N→∞

s

p

N

!

G. Dennis et al., Phys. Plasmas 20, 032509 (2013).[1]

RN

R1

R2 I1

I2

ININ�1

Fewer constraints

More constraints

Ideal MHDTaylor’s theory

✓Multiregion

Relaxed MHD

Goal: minimal description of helical states in RFP

✗ ?

Fewer constraints

More constraints

Ideal MHDTaylor’s theory

✓Multiregion

Relaxed MHD

Goal: minimal description of helical states in RFP

N = 1

N = 2

N = 1

is Taylor’s theory

is Ideal MHD

✗ ?

Fewer constraints

More constraints

Ideal MHDTaylor’s theory

✓Multiregion

Relaxed MHD

Goal: minimal description of helical states in RFP

N = 1

N = 2

N = 1

is Taylor’s theory

is Ideal MHD

✗

✗

✓

?

Fewer constraints

More constraints

Ideal MHDTaylor’s theory

✓Multiregion

Relaxed MHD

Goal: minimal description of helical states in RFP

N = 1

N = 2

N = 1

is Taylor’s theory

is Ideal MHD

✗

✗

✓?

?

A two-volume MRxMHD model (without pressure) is defined by 5 parameters

{ t,1; H1}

{ t,2; p,2; H2}

H = 2

Z t d p (+ gauge terms)

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

s = normalized enclosed poloidal flux

q −

safe

ty fa

ctor

A two-volume MRxMHD model (without pressure) is defined by 5 parameters

We take the ideal MHD constraints and compute the 5 parameters; i.e. examine 1D line in 5D parameter space

{ t,1; H1}

{ t,2; p,2; H2}

H = 2

Z t d p (+ gauge terms)

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

s = normalized enclosed poloidal flux

q −

safe

ty fa

ctor

A two-volume MRxMHD model (without pressure) is defined by 5 parameters

We take the ideal MHD constraints and compute the 5 parameters; i.e. examine 1D line in 5D parameter space

{ t,1; H1}

{ t,2; p,2; H2}

H = 2

Z t d p (+ gauge terms)

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

s = normalized enclosed poloidal flux

q −

safe

ty fa

ctor

Transport barrier

Outer volumeInner

volume

t,2; p,2; H2H1

t,1

A two-volume MRxMHD model (without pressure) is defined by 5 parameters

We take the ideal MHD constraints and compute the 5 parameters; i.e. examine 1D line in 5D parameter space

{ t,1; H1}

{ t,2; p,2; H2}

H = 2

Z t d p (+ gauge terms)

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

s = normalized enclosed poloidal flux

q −

safe

ty fa

ctor

Transport barrier

Outer volumeInner

volume t,1 t,2; p,2; H2H1

A two-volume MRxMHD model (without pressure) is defined by 5 parameters

We take the ideal MHD constraints and compute the 5 parameters; i.e. examine 1D line in 5D parameter space

{ t,1; H1}

{ t,2; p,2; H2}

H = 2

Z t d p (+ gauge terms)

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

s = normalized enclosed poloidal flux

q −

safe

ty fa

ctor

Transport barrier

Outer volumeInner

volume

t,2; p,2; H2 t,1; H1

0 0.2 0.4 0.6 0.84.028

4.03

4.032

4.034

4.036

4.038

4.04

4.042

4.044

x 105

s

Pla

sma

En

erg

y (

J)

MRXMHD (Full 3D)

MRXMHD (Axisymmetric)

Ideal MHD (Full 3D)

Ideal MHD (Axisymmetric)

Single!volume solution

The plasma equilibrium is a minimum energy state

Ideal MHD flux surface chosen as ideal barrier

0 0.2 0.4 0.6 0.84.028

4.03

4.032

4.034

4.036

4.038

4.04

4.042

4.044

x 105

s

Pla

sma

En

erg

y (

J)

MRXMHD (Full 3D)

MRXMHD (Axisymmetric)

Ideal MHD (Full 3D)

Ideal MHD (Axisymmetric)

Single!volume solution

The plasma equilibrium is a minimum energy state

Taylor (single-volume) solution

Ideal MHD flux surface chosen as ideal barrier

0 0.2 0.4 0.6 0.84.028

4.03

4.032

4.034

4.036

4.038

4.04

4.042

4.044

x 105

s

Pla

sma

En

erg

y (

J)

MRXMHD (Full 3D)

MRXMHD (Axisymmetric)

Ideal MHD (Full 3D)

Ideal MHD (Axisymmetric)

Single!volume solution

The plasma equilibrium is a minimum energy state

}Energy difference between axisymmetric and helical ideal MHD solutions (same q profile)

Taylor (single-volume) solution

Ideal MHD flux surface chosen as ideal barrier

0 0.2 0.4 0.6 0.84.028

4.03

4.032

4.034

4.036

4.038

4.04

4.042

4.044

x 105

s

Pla

sma

En

erg

y (

J)

MRXMHD (Full 3D)

MRXMHD (Axisymmetric)

Ideal MHD (Full 3D)

Ideal MHD (Axisymmetric)

Single!volume solution

The plasma equilibrium is a minimum energy state

}Energy difference between axisymmetric and helical ideal MHD solutions (same q profile)

Taylor (single-volume) solution

Ideal MHD flux surface chosen as ideal barrier

(e) (f) (g) (h)

� = 0(2⇡/7) � =1

4(2⇡/7) � =

2

4(2⇡/7) � =

3

4(2⇡/7)

(a) (b) (c) (d)

Comparison of VMEC and SPEC RFX-mod equilibria

Reconstructed Poincaré plots

Quasi-single helicity

Single Helical Axis

Top figure source: P. Martin et al., Nuclear Fusion 49, 104019 (2009).

1.5 2 2.5−0.05

0

0.05

0.1

0.15

0.2

R

Safe

ty fa

ctor

(q)

1.5 2 2.5−0.4

−0.2

0

0.2

0.4

Radius (R)

Elev

atio

n (Z

)

Flux surfaces at q =0

Reconstructed Poincaré plots

Poincaré section Safety factor profile (q)

1.5 2 2.5−0.05

0

0.05

0.1

0.15

0.2

R

Safe

ty fa

ctor

(q)

1.5 2 2.5−0.05

0

0.05

0.1

0.15

0.2

R

Safe

ty fa

ctor

(q)

Reconstructed Poincaré plots

Poincaré section Safety factor profile (q)

1.5 2 2.5−0.05

0

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(q)

Reconstructed Poincaré plots

Poincaré section Safety factor profile (q)

1.5 2 2.5−0.05

0

0.05

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0.15

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R

Safe

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(q)

1.5 2 2.5−0.05

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0.15

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0

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Reconstructed Poincaré plots

Poincaré section Safety factor profile (q)

1.5 2 2.5−0.05

0

0.05

0.1

0.15

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R

Safe

ty fa

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1.5 2 2.5−0.05

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1.5 2 2.5−0.05

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1.5 2 2.5−0.05

0

0.05

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ty fa

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Reconstructed Poincaré plots

Poincaré section

1.5 2 2.5−0.05

0

0.05

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Safe

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(q)

Safety factor profile (q)

MRxMHD gives a good qualitative explanation of the high-confinement state in Reversed Field Pinches

With a minimal model we reproduced the helical pitch and structure of the Quasi-Single Helicity state in RFP

MRxMHD is a well-formulated model that interpolates between Taylor’s theory and ideal MHD

With MRxMHD we reproduced the second magnetic axis. This is the first equilibrium model to be able to reproduce

the Double-Axis state.

Conclusions

G. Dennis et al., PRL 111, 055003 (2013).[1]

Considering RFX helical states with pressure

More detailed experimental comparisons with RFX

Future Work

Generalize MRxMHD to include flow

Apply the same methodology to 3D structures in tokamaks, in particular, the sawtooth crash