minimally constrained model of self-organised helical ... · 9/17/2013 · customary when dealing...
TRANSCRIPT
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 [email protected] - 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
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)
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
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)
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
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)
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
1.5 2 2.5−0.05
0
0.05
0.1
0.15
0.2
R
Safe
ty fa
ctor
(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