FLOW: an equilibrium code for rotating anisotropic axysimmetric plasmas

L. Guazzotto, T. Gardiner and R. Betti

Laboratory for Laser Energetics

University of Rochester

NSTX Theory Meeting

Princeton Plasma Physics Lab.

September 9-13, 2002,Princeton NJ

• The code Flow:►) The system of equations►) The numerical solution

• NSTX-like equilibria with toroidal flow

• NSTX-like equilibria with poloidal flow

• Conclusions

Outline

The relevant equations

• Continuity:

0 v

PBJvv"

0 Bv

0 B

BBIpP""

2

|| / Bpp

• Momentum:

•Maxwell equations:

• Faraday’s law yields the plasma flow:

The previous system of equations can be reduced to a “Bernoulli”

and a “Grad-Shafranov” equation: 

2

21A

A

I R MB R

M

HWRBM A 2

2

2

1

2

1

• The φ-component of the momentum equation gives an equation for the toroidal component of the magnetic field:

• The B-component of the momentum equation reduces to a “Bernoulli-like” equation for the total energy along the field lines:

E. Hameiri, Phys. Fluids (1983)

ˆ RvMv AA

/AM

• Finally, the Ψ-component of the momentum equation gives a “GS-like” equation :

The modified Grad-Shafranov equation 

W

d

dH

d

dvR

d

dBv

d

dI

R

Bp

RM A

||

221

• W(ρ,B,Ψ) is the enthalpy of the plasma, and its definition depends on the description of the plasma (MHD, CGL, or kinetic)

• I(Ψ), Φ(Ψ), Ω(Ψ), H(Ψ), (p||/ Ψ), (W/ Ψ) are free functions of Ψ

R. Iacono et al., Phys. Fluids B (1990)

• The input is a set of free functions representing quasi-physical variables

The code input requires a “user friendly” set of free functions.

D(Ψ)P||(Ψ)P(Ψ)B0(Ψ)Mθ(Ψ)Mφ(Ψ)

Quasi-DensityQuasi-Parallel pressureQuasi-Perpendicular pressureQuasi-Toroidal magnetic fieldQuasi-Poloidal sonic Mach numberQuasi-Toroidal sonic Mach number

The numerical algorithm: the multi-grid solver

• The Bernoulli equation is solved for

• The Grad-Shafranov equation is solved for Ψ using a red-black algorithm

• If the system is anisotropic, the equation for Bφ is also solved

• The procedure is repeated until convergence, then the solution is interpolated onto the next grid

NSTX-like equilibria with toroidal flow

2|| ||

CP P

CM M

||P P

0M

CD D

vacuumBBB 300

M. Ono et al., Nucl. Fusion (2001)

DC = 3.4*1019 [m-3] B0C = 0.29 [T] 0 M

C 1

T = 9% I = 0.9 [MA]

R0 = 0.86 [m] a = 0.69 [m] k = 1.9

•The following set of free functions is used as input to compute anisotropic equilibria with toroidal flow

The centrifugal force causes an outward shift of the plasma

R [m]

Den

sit

y [m

-3]

0.5

4E+19

3E+19

2E+19

1E+19

1 1.5

MC = 0

MC = 0.8

MC = 0.4

MC = 1

MC

MC

0 1 2R [m]

-1

0

1

XYZ

MC = 1

The parallel anisotropy (p|| > p) causes an inward shift

Θ = 0

Θ = 0.2

Θ = 0.6

Θ = 1

R [m]

Den

sit

y [m

-3]

1

5E+19

0.5 1.5

4E+19

3E+19

2E+19

1E+19

P

PP||

Flow and parallel anisotropyhave opposite effect

R [m]

De

ns

ity

[m

-3]

0.5

5E+19

1 1.5

4E+19

3E+19

2E+19

1E+19

MC=1, Θ=0

MC=1, Θ=0.6

MC=1, Θ=0.2

MC=0, Θ=1

MC=1, Θ=1

FLOW can also be applied to equilibria with poloidal flow

• The free function determining the poloidal flow is M(Ψ) representing (approximately) the sonic poloidal Mach number (poloidal velocity/ poloidal sound speed). Poloidal sound speed = CsB/B.

• Radial discontinuities in the equilibrium profiles develop when the poloidal flow becomes transonic (M(Ψ) ~ 1).

• As the poloidal sound speed is small at the plasma edge, transonic flows may develop near the edge.

• FLOW can describe MHD or CGL equilibria with poloidal flow.

R. Betti, J. P. Freidberg, Phys. Plasmas (2000)

NSTX-like equilibria with poloidal flow can exhibit a pedestal structure at the

edge

R [m]

[%

]

0.5

10 Subsonic

Transonic9

8

7

6

5

4

3

2

1

01 1.5

Discontinuity

V

[km

/s]

130

120

110

100

90

80

70

60

50

40

30

20

10

0

R [m]0.5 1 1.5

Subsonic

Transonic

Poloidal sound speed

Conclusions

• The code FLOW can compute axysimmetric anisotropic equilibria with arbitrary flow.

• MHD, CGL, and kinetic closures are implemented. At the moment, poloidal flow is described only by the MHD and CGL closures.

• Fast-rotating anisotropic NSTX-like equilibria have been computed with FLOW.

• Future work: develop a stability code for NSTX rotating equilibria computed with the code FLOW.