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Three-dimensional numerical simulations of
magnetohydrodynamics equations
M. Onofri
Dipartimento di Fisica, Università della Calabria
In collaboration with
L. Primavera, P. Veltri, F. Malara, P. Londrillo
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• Incompressible cartesian code

Solves the incompressible magnetohydrodynamics
equations in slab geometry

• Compressible cylindrical code

Solves the compressible magnetohydrodynamics
equations in cylindrical geometry.
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Incompressible cartesian code
We study the magnetic reconnection in an
incompressible plasma bounded by two conducting
walls in three-dimensional slab geometry.
Different resonant surfaces are simultaneously
present in different positions of the simulation
domain and nonlinear interactions are possible
not only on a single resonant surface, but also
between adjoining resonant surfaces.
The nonlinear evolution of the system is
different from what has been observed in
configurations with an antiparallel magnetic
field.
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Previous results

• In two-dimensional situations it has been found
  that a
• chain of magnetic islands formed as a result of
  a tearing
• instability is subject to coalescence driven by
  the stretching
• of the most intense X point (Malara et al.).

Dahlburg and Einaudi have studied
three-dimensional configurations in
an antiparallel magnetic field

only one resonant surface is present at the
centre of the current sheet
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1st case the equilibrium is perturbed with a
large dose of the most unstable 2D
eigenfunction and a small dose of
2D velocity noise The evolution is dominated by
coalescence.
2nd case the equilibrium is perturbed with the
same 2D velocity noise and with a
3D eigenfunction to allow 3D
instability
Coalescence is observed, but it is overcome by
the growth of the (0,1) mode, the longest
wavelength in the third direction.
The final state is dominated by the longest
wavelength in the direction perpendicular to the
plane where coalescence is observed in 2D
simulations
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Description of the simulations equations and
geometry
Incompressible, viscous, dimensionless MHD
equations
B is the magnetic field, V the plasma velocity
and P the kinetic pressure.
and
are the magnetic and kinetic Reynolds numbers.
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Geometry
The MHD incompressible equations are solved to
study magnetic reconnection in a current layer
in slab geometry with the plasma confined between
two conducting walls
Periodic boundary conditions along y and z
directions
Dimensions of the domain -lx lt x lt lx, 0 lt y lt
2ply, 0 lt z lt 2plz
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Description of the simulations the initial
conditions
Equilibrium field plane current sheet (a c.s.
width)
Incompressible perturbations superposed
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Description of the simulations the numerical code

• Boundary conditions
• periodic boundaries along y and z directions
• in the x direction, conducting walls give

• Numerical method
• FFT algorithms for the periodic directions (y and
  z)
• fourth-order compact difference scheme in the
  inhomogeneous direction (x)
• third order Runge-Kutta time scheme
• code parallelized using MPI directives to run on
  a 16-processor Compaq a -server

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Numerical results characteristics of the runs
Magnetic reconnection takes place on resonant
surfaces defined by the condition
where
is the wave vector of the perturbation.
The periodicity in y and z directions imposes the
following conditions
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Testing the code growth rates
The numerical code has been tested by comparing
the growth rates calculated in the linear stage
of the simulation with the growth rates predicted
by the linear theory.
Two-dimensional modes
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Numerical results instability growth rates
Parameters of the run
Perturbed wavenumbers -4 ? m ? 4, 0 ? n ? 12
Resonant surfaces on both sides of the current
sheet
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Numerical results spectrum along z for m0
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Numerical results B field lines and current at
y0
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Numerical results time evolution of the spectra
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Three-dimensional structure of the current
Isosurfaces of
at t500
J
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Summary

• The two-dimensional modes (m0) are not the most
  unstable ones, contrary to what is
  predicted from the linear theory
• Initially, the modes with n3 (m0,1) grow
  faster
• At later times an inverse cascade transports the
  energy towards longer wavelengths
• This corresponds, in the physical space, to a
  coalescence of the magnetic islands

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Summary

• The spectrum of the fluctuations is anisotropic,
  it develops mainly in one specific direction,
  which changes in time
• Coalescence occurs in the centre of the current
  sheet and formation of small scale structures is
  observed in the lateral regions.
• The final state is different from what has been
  found by
• Dahlburg and Einaudi, this must be due to the
  presence of
• a guide field in the y direction and more than
  one resonant surface

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Reversed field pinch
Toroidal direction
Poloidal direction
The plasma is confined by a magnetic field with
toroidal and poloidal components
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Compressible MHD equations
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The numerical method

• FFT algorithms for the periodic directions (
  and )
• fourth-order compact difference scheme along the
  inhomogeneous direction (r)
• third order Runge-Kutta time scheme
• code parallelized using MPI directives to run on
  a 16-processor Compaq a -server

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Boundary conditions
At r0 the following

regularity conditions on the
-tranformed variables must be respected
Is an even function for even m and an
odd function for odd m if f is a scalar or an
axial vector component.
Is an odd function for even m and an
even function for odd m if f is a poloidal vector
component.

• Periodic conditions in and directions
• In the r direction the conducting wall gives

at ra,

• Boundary conditions at ra for the other
  variables are
• calculated according to the characteristic
  decomposition method

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Characteristic boundary conditions
The solution is written as a superposition of
waves propagating in one direction and
appropriate conditions are imposed on the
amplitudes of the waves
In cartesian geometry
The eigenvalues of the matrix A are the
velocities of the waves propagating in the x
direction
In cylindrical geometry
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• Write the radial derivatives as a linear
  combination of the
• eigenvectors
• Substitute in the one-dimensional ideal
  equations with the
• conditions

and
are the amplitudes of the outcoming and incoming
fast waves
and
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gives
The condition
is the amplitude of the outcoming fast wave
and can be expressed in terms of the radial
derivatives of the fields calculated from
internal points
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Initial conditions
Equilibrium field
Poloidal field
Toroidal field
Perturbation
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Energy conservation
Physical dissipation
Numerical dissipation
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Parameters of the run
Perturbed wavenumbers
m1, n-2
(-1,2)
(-2,4)
(-3,6)
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Magnetic energy spectrum
t50
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Conclusions

• We have developped a numerical code for the
  compressible
• MHD equations in cylindrical geometry

• Particular attention has been paid to boundary
  conditions, which
• have been calculated according to the
  characteristic method

• Some runs have been performed as a test for
  the
• numerical code.

• The present code can be used to investigate the
  nonlinear evolution
• of a compressible plasma in cylindrical
  geometry to simulate RFP
• dynamics.