M. Onofri, F. Malara, P. Veltri

Compressible magnetohydrodynamics Compressible magnetohydrodynamics simulations of the RFP with anisotropic simulations of the RFP with anisotropic

thermal conductivitythermal conductivity

Dipartimento di Fisica, Università della Calabria

Compressible MHD equationsCompressible MHD equations

BJFV

t

EB

t

0)(

Vt

ijijjiij μσPδ+VρV=F

ijσ is the stress tensor

BVJE η=

Cylindrical coordinates

10 <r< π<θ< 20 Rφ=z π<φ< 20

V

3

2

2

12ijijp σσμ+ηJ=H

|11|||| pHPTTPt

P

VV

Compressible MHD equations include heat production and heat transport, which are not present in the incompressible case.

Case 1: Adiabatic

Case 2: Heat production but

Case 3: Isotropic thermal conductivity

Case 4: Anisotropic thermal conductivity

const=pρ γ

0=κ

Boundary conditionsBoundary conditions

z

0rB

•Boundary conditions at r=a for the other variables are calculated using a characteristic wave decomposition

•Periodic conditions in and directions

•In the r direction the conducting wall gives:

0V

θ

const=Bθ0,0

0=Jθ 0=J nm,z

(Poinsot et al., J Comp. Phys.,1992)

Initial conditionsInitial conditions

Poloidal field Toroidal field

Equilibrium field (force-free magnetic field):

Perturbations:

Rnz+θrε=v=nr /cos11

4

2

Rnz+θrε=v=nθ /sin11

4

2

42 0.832321.874810 r+rq=rq (Robinson, Nuclear Fusion 1978)

To test the numerical code, we verified that it reproduces previous results obtained when the density and pressure dynamics are neglected. (Cappello et al, Phys. Plasmas, 1996)

In this case, the results are similar to those obtained in previous works, for example the reversal parameter F becomes positive in the beginning of the simulation, but, due to the action of the unstable modes the reversal is recovered ad later times.

Test: comparison with previous resultsTest: comparison with previous results

0 0P

Field reversal parameter:

410=μ 310=η R=2

N r=100 32=Nφ 128=N z

z

z

B

aBF

)(

Time evolution of the modes m=1,n=-4 (solid line), , m=1,n=-3 (dashed line) and m=1,n=-5 (dotted line)

The initial perturbations destabilize the system and the unstable modes begin to grow exponentially in the linear phase. After , the modes m=1,n=-4 and m=1,n=-3 have comparable amplitudes, indicating the formation of a MH state.

800t

Adiabatic caseAdiabatic case4102 =μ

4102 =η R=2

Spectral spread

2

21,

21,1

n

ns B

B=N

Reversal parameter

z

z

B

aB=F

Isosurface of the density in a SH state at t=400 (left) and ina MH state at t=1000 (right)

Non-adiabatic caseNon-adiabatic case

Time evolution of the modes m=1,n=-4 (solid line), , m=1,n=-3 (dashed line) and m=1,n=-5 (dotted line)

In the linear phase, the unstable modes grow exponentially and the m=1,n=-4 mode becomes the most energetic. After t=180 most of the energy is contained in the modes m=1,n=-3 and m=1,n=-5. The evolution is faster than in the adiabatic case.

2

21,

21,1

n

ns B

B=N

Spectral spread Reversal parameter

z

z

B

aB=F

Isosurface of the density in a SH state at t=100 (left) and in a MHstate at t=200 (right).

Contours of temperature in a SH state (left) and in a MH state (right) and Poincaré maps of magnetic field lines.

In the SH state, a magnetic island characterized by high temperature is present. In the MH state the magnetic field is chaotic

Simulation with thermal conductivity

pt HPTkPt

P)1(])1()([

VV

4102 =

Boundary conditions at r=a

T=cost

const=Bz0,0

0rB 0V

const=Bθ0,0 0=J nm,

θ 0=J nm,z

•Boundary conditions at r=a for the density is derived using a characteristic wave decomposition

R=4

Spectral spread

2

21,

21,1

n

ns B

B=N

Reversal parameter

z

z

B

aB=F

P

Radial profile of density, pressure

0,0ρ 0,0P

t=50

t=1000

Contours of temperature in a SH state and in a MH state and Poincaré plots of magnetic field lines.

Anisotropic thermal conductivity

The thermal conductivity in a magnetized plasma is anisotropicwith respect to the direction of the magnetic field and for a fusionplasma the ratio may exceed (Fitzpatrick, Phys. Plasmas ,1995)κκ /||

1010

Thermal conduction occurs on different time scales in the paralleland perpendicular direction, so that magnetic field lines tend to become isothermal.

In a simulation it is not possible to use a realistic value of because the time step would become too small

||κ

|11|||| pHPTTPt

P

VV

Multiple-time-scale analysis(Frieman J. Math. Phys., 1963)

We separate the evolution on fast time scales from the evolution onslow time scales

Tκ=τ

P||||

0

pHγ+Pγ+TκP=τ

P11

1

VV

(1)

(2)

At each time step we look for an asymptotic solution of (1) and use itin (2)

Extend the number of time variables

1,0 =dt

dτ ε,=dt

dτ1,ε=

dt

dτ 22

.... 1ε

t=50

t=1000

Contours of temperature in a SH state and in a MH state and Poincaré plots of magnetic field lines.

Reversal parameter

z

z

B

aB=F

Spectral spread

2

21,

21,1

n

ns B

B=N

Radial profile of density and pressure

0,0ρ 0,0P

PP

P

. .ET

EB

BT EE +

dSJBBJη+dSr

T

γ

κ=dτ

γ

P+ρV+

B

dt

dθzθz

112

1

22

2

BT E+E=dτγ

P+ρV+

B

dt

d

12

1

22

2

Energy flux

ConclusionsConclusions

The density and pressure dynamics is important for the evolution of the system.

Using an anisotropic thermal conductivity we can producehot structures corresponding to closed magnetic surfaces andalmost flat temperature when the magnetic field is chaotic.