Advection of a magnetic field loop. More...

#include "pluto.h"

Include dependency graph for init.c:

Functions
void	Init (double *v, double x1, double x2, double x3)

void	Analysis (const Data d, Grid grid)

void	UserDefBoundary (const Data d, RBox box, int side, Grid *grid)

Detailed Description

Advection of a magnetic field loop.

This problem consists of a weak magnetic field loop being advected in a uniform velocity field. Since the total pressure is dominated by the thermal contribution, the magnetic field is essentially transported as a passive scalar. The preservation of the initial circular shape tests the scheme dissipative properties and the correct discretization balance of multidimensional terms.

Following [GS05][MT10][MTB10] (see also references therein), the computational box is defined by $x\in[-1,1],\, y\in[-0.5,0.5]$ discretized on $2N_y\times N_y$ grid cells (Ny=64). Density and pressure are initially constant and equal to 1. The velocity of the flow is given by

$\vec{v} = V_0(\cos\alpha, \sin\alpha)$

with $V_0 = \sqrt{5},\,\sin \alpha = 1/\sqrt{5},\, \cos \alpha = 2/\sqrt{5}$ . The magnetic field is defined through its magnetic vector potential as

$A_z = \left\{ \begin{array}{ll} A_0(R-r) & \textrm{if} \quad R_1 < r \leq R \,, \\ \noalign{\medskip} 0 & \textrm{if} \quad r > R \,, \end{array} \right.$

with $A_0 = 10^{-3},\, R = 0.3,\, r = \sqrt{x^2+y^2}$ . A slightly different variant is used for the finite difference schemes as explained in [MTB10]:

$A_z = \left\{ \begin{array}{ll} a_0 + a_2r^2 & \textrm{if} \quad 0 \leq r \leq R_1 \,, \\ \noalign{\medskip} A_0(R-r) & \textrm{if} \quad R_1 < r \leq R \,, \\ \noalign{\medskip} 0 & \textrm{if} \quad r > R \,, \end{array} \right.$

where $R_1=0.2 R,\, a_2 = -0.5A_0/R_1,\, a_0 = A_0(R-R_1) - a_2R_1^2$ .

Double periodic boundary conditions are imposed.

A snapshot of the solution on a 128x64 grid at t=0.2 is shown below.

Magetic pressure at t=0.2 (configuration #03).

Author: A. Mignone (migno.nosp@m.ne@p.nosp@m.h.uni.nosp@m.to.i.nosp@m.t)

Date: Aug 6, 2015

References

[GS05] "An unsplit Godunov method for ideal MHD via constrained transport", Gardiner & Stone JCP (2005) 205, 509
[MT10] "A second-order unsplit Godunov scheme for cell-centered MHD: The CTU-GLM scheme", Mignone & Tzeferacos, JCP (2010) 229, 2117.
[MTB10] "High-order conservative finite difference GLM-MHD schemes for cell-centered MHD", Mignone, Tzeferacos & Bodo, JCP (2010) 229, 5896.

Definition in file init.c.

Function Documentation

void Analysis	(	const Data *	d,
		Grid *	grid
	)

Perform runtime data analysis.

Parameters

[in]	d	the PLUTO Data structure
[in]	grid	pointer to array of Grid structures

Definition at line 117 of file init.c.

 {
 
 }

void Init	(	double *	v,
		double	x1,
		double	x2,
		double	x3
	)

The Init() function can be used to assign initial conditions as as a function of spatial position.

Parameters

[out]	v	a pointer to a vector of primitive variables
[in]	x1	coordinate point in the 1st dimension
[in]	x2	coordinate point in the 2nd dimension
[in]	x3	coordinate point in the 3rdt dimension

The meaning of x1, x2 and x3 depends on the geometry:

$\begin{array}{cccl} x_1 & x_2 & x_3 & \mathrm{Geometry} \\ \noalign{\medskip} \hline x & y & z & \mathrm{Cartesian} \\ \noalign{\medskip} R & z & - & \mathrm{cylindrical} \\ \noalign{\medskip} R & \phi & z & \mathrm{polar} \\ \noalign{\medskip} r & \theta & \phi & \mathrm{spherical} \end{array}$

Variable names are accessed by means of an index v[nv], where nv = RHO is density, nv = PRS is pressure, nv = (VX1, VX2, VX3) are the three components of velocity, and so forth.

Definition at line 64 of file init.c.

 {
   double c, s, v0=sqrt(5.0);
   double A=1.e-3, R=0.3, R1, r, Bphi;
 
   r = sqrt(x1*x1 + x2*x2);
   c = 2.0/sqrt(5.0);
   s = 1.0/sqrt(5.0);
 /*
   c = 1.0/sqrt(2.0);
   s = 1.0/sqrt(2.0);
 */
   v[RHO] = 1.0;
   v[VX1] = v0*c;
   v[VX2] = v0*s;
   v[VX3] = 0.0;
   v[PRS] = 1.0;
   v[TRC] = 0.0;
 
 #if PHYSICS == MHD || PHYSICS == RMHD
 
   v[AX1] = 0.0;
   v[AX2] = 0.0;
   v[AX3] = A*(R - r)*(r <= R);
   
   #ifdef FINITE_DIFFERENCE
   R1 = 0.2*R;  
   if (r < R1) {
     double a0, a2; 
     a2 = -A/(2.0*R1);
     a0 = A*(R-R1) - a2*R1*R1;
     v[AX3] = a0 + a2*r*r;
     A      = -2.0*a2*r;
   } else if (r <= R){
     v[AX3] = A*(R - r);
   } else {
     v[AX3] = 0.0;
     A      = 0.0;
   }
   #endif
 
   v[BX1] = -A*x2/r*(r <= R);  /*  =   dAz/dy  */
   v[BX2] =  A*x1/r*(r <= R);  /*  = - dAz/dx  */
   v[BX3] = 0.0;
   
 #endif
 }

void UserDefBoundary	(	const Data *	d,
		RBox *	box,
		int	side,
		Grid *	grid
	)

Assign user-defined boundary conditions.

Parameters

[in,out]	d	pointer to the PLUTO data structure containing cell-centered primitive quantities (d->Vc) and staggered magnetic fields (d->Vs, when used) to be filled.
[in]	box	pointer to a RBox structure containing the lower and upper indices of the ghost zone-centers/nodes or edges at which data values should be assigned.
[in]	side	specifies the boundary side where ghost zones need to be filled. It can assume the following pre-definite values: X1_BEG, X1_END, X2_BEG, X2_END, X3_BEG, X3_END. The special value side == 0 is used to control a region inside the computational domain.
[in]	grid	pointer to an array of Grid structures.

Assign user-defined boundary conditions in the lower boundary ghost zones. The profile is top-hat:

$V_{ij} = \left\{\begin{array}{ll} V_{\rm jet} & \quad\mathrm{for}\quad r_i < 1 \\ \noalign{\medskip} \mathrm{Reflect}(V) & \quad\mathrm{otherwise} \end{array}\right.$

where $V_{\rm jet} = (\rho,v,p)_{\rm jet} = (1,M,1/\Gamma)$ and M is the flow Mach number (the unit velocity is the jet sound speed, so $ v = M$ ).

Assign user-defined boundary conditions:

left side (x beg): constant shocked values
bottom side (y beg): constant shocked values for x < 1/6 and reflective boundary otherwise.
top side (y end): time-dependent boundary: for $x < x_s(t) = 10 t/\sin\alpha + 1/6 + 1.0/\tan\alpha$ we use fixed (post-shock) values. Unperturbed values otherwise.

Assign user-defined boundary conditions at inner and outer radial boundaries. Reflective conditions are applied except for the azimuthal velocity which is fixed.

Assign user-defined boundary conditions.

Parameters

	[in/out]	d pointer to the PLUTO data structure containing cell-centered primitive quantities (d->Vc) and staggered magnetic fields (d->Vs, when used) to be filled.
[in]	box	pointer to a RBox structure containing the lower and upper indices of the ghost zone-centers/nodes or edges at which data values should be assigned.
[in]	side	specifies on which side boundary conditions need to be assigned. side can assume the following pre-definite values: X1_BEG, X1_END, X2_BEG, X2_END, X3_BEG, X3_END. The special value side == 0 is used to control a region inside the computational domain.
[in]	grid	pointer to an array of Grid structures.

Definition at line 127 of file init.c.

131 { }

Functions

Detailed Description

Function Documentation