Types of boundary conditions

  • Dirichlet: The variable's value is kept constant over time (e.g. $\rho=cte$ or $\delta\rho = 0$)

  • Natural: A condition on the variable's normal derivative through a boundary integral term (e.g. $\nabla T\cdot\mathbf{n} = c T$). The implementation is explained in the JOREK overview article section 2.3.2. See also an example in "sheath boundary conditions for the heat-flux"

  • Special conditions: Special conditions can be applied to some variables. For example we typically set the parallel velocity to be the sound speed ($v_\parallel=c_s$) in the divertor region (the Mach=1 condition).

  • No explicit conditions: We can choose not to apply any boundary condition for the variable. For some variables this can cause the code to crash since the system is not determined (e.g. for $\psi$). However there are variables in which not applying explicitly a condition determines implicitly another one. For example for $\rho$ and $T$ not applying any condition directly implies that ($\nabla T\cdot\mathbf{n} = 0$ and $\nabla \rho\cdot\mathbf{n} = 0$), since is equivalent to implement a boundary term that is 0.

Types of boundaries (boundary type indices)

The JOREK boundary can be separated into different parts, for example the divertor targets (with finite $\mathbf{B}\cdot\mathbf{n}$) and the boundary which is aligned with the magnetic field ($\mathbf{B}\cdot\mathbf{n}=0$). To apply boundary conditions in different boundary regions, such regions are labeled with a "boundary type". This labeling depends on the grid that you choose, see below some examples (not complete)

  • Polar grid (grid_polar_bezier): "2" for all the boundary
  • Polar-aligned grid (grid_flux_surface): "2" for all the boundary
  • X-point grid (grid_xpoint): "1" for the divertor targets, "2" for the flux-aligned boundary, "3" for the grid corners

Boundary types > 3 exist for more complex grids. You can check the boundary types that you use with jorek2_postproc and the command grid_diagnostics.

xpt_grid_types

The plot was generated by using gnuplot and the following command:

p 'postproc/nodes_by_boundary_type_00000001_s00000.dat' t 'type 1', 'postproc/nodes_by_boundary_type_00000002_s00000.dat' t 'type 2', 'postproc/nodes_by_boundary_type_00000003_s00000.dat' t 'type 3'

Boundary types with patches

!!! RECAP !!! DEFINED BY GUIDO

  • 1: TARGET, side 2
  • 2: TANGENT, side 3
  • 3: CORNER, between type-1 and type-2
  • 4: TARGET, side 2 (Same as type-1)
  • 5: TARGET, side 3
  • 9: CORNER, between type-4 and type-5

!!! RECAP !!! DEFINED BY STAN TO INCLUDE FIELD DIRECTION, COMPATIBLE WITH OLD GRIDS

  • 1: TARGET, side 2 (inward field)
  • 11: TARGET, side 2 (outward field)
  • 5: TARGET, side 3 (inward field)
  • 15: TARGET, side 3 (outward field)
  • 9: CORNER, free (inward field)
  • 19: CORNER, free (outward field)
  • 2: TANGENT, side 3 (tangent field)
  • 12: TANGENT, side 2 (tangent field)
  • 20: CORNER, fixed (tangent field)
  • 21: CORNER, inverted (3 elements) (special case)

! NOT USED IN NEW GRID

  • 3: Corner between type-1 and type-2 (only for old grid)
  • 4: Same as type-1 (only for old grid)

!!! MAPPING !!! TO GO FROM STAN'S DEFINITION TO GUIDO'S

  • 1 -> 1
  • 11 -> 1
  • 5 -> 5
  • 15 -> 5
  • 9 -> 9
  • 19 -> 9
  • 2 -> 2
  • 12 -> not defined!
  • 20 -> 9 (because not defined!)
  • 21 -> not defined!
  • 3 -> 3

Specifying the boundary conditions in the input file

The "bcs" structure allows you to add the boundary conditions in the input file. For example, to apply a Dirichlet BC for variable "rho" and boundary type "j", add the following to the input file:

bcs(j)%dirichlet%rho = .true.

IMPORTANT: Trying to specify a boundary condition for all boundary types with the following statement will not work: bcs(:)%dirichlet%rho. You have to retype each line for each boundary type index separately: bcs(1)%dirichlet%rho=.t., bcs(2)%dirichlet%rho=.t., etc.

IMPORTANT: If you want to apply any kind of natural boundary conditions, you must set bc_natural_open=.t. in the input file.

Available boundary conditions

To see all available boundary conditions, see the table below for boundary type "i":

Variable Dirichlet Natural (bc_natural_open=.t.) Special None applied
$\psi$ bcs(i)%dirichlet%psi None free-boundary ?
$u$ bcs(i)%dirichlet%u None None ?
$j$ bcs(i)%dirichlet%zj None free-boundary Determined by induction equation
$w$ bcs(i)%dirichlet%w None None ?
$\rho$ bcs(i)%dirichlet%rho bcs(i)%natural%rho

Condition: $D\nabla\rho\cdot\mathbf{n} = -r_\rho\rho\mathbf{v}\parallel \cdot\mathbf{n}$
with $r\rho$ = density_reflection		 None $\nabla\rho\cdot\mathbf{n}=0$
$T$ bcs(i)%dirichlet%T bcs(i)%natural%T

see here		 None $\nabla T \cdot\mathbf{n}=0$
$T_i$ bcs(i)%dirichlet%Ti bcs(i)%natural%Ti

see here		 None $\nabla T_i\cdot\mathbf{n}=0$
$T_e$ bcs(i)%dirichlet%Te bcs(i)%natural%Te

see here		 None $\nabla T_e\cdot\mathbf{n}=0$
$v_\parallel$ bcs(i)%dirichlet%vpar bcs(i)%natural%vpar
mach_one_bnd_integral=.t.

Condition: $v_\parallel = c_s$
via penalization method		 bcs(i)%mach1

Condition: $v_\parallel = c_s$		 ?
$\rho_n$ bcs(i)%dirichlet%rhon bcs(i)%natural%rhon

Condition: $D_n\nabla\rho_n\cdot\mathbf{n} = -r_{\rho_n}\rho\mathbf{v}\parallel \cdot\mathbf{n}$
with $r{\rho_n}$ = neutral_reflection		 None $\nabla\rho_n\cdot\mathbf{n}=0$
$n_{RE}$ bcs(i)%dirichlet%nre None None ?

Default values

  • $\psi, u, j, w, n_{RE}$: Dirichlet for all boundary types

  • $\rho, T, T_i, T_e, \rho_n$:
    • None applied for boundary types "1, 4, 5, 9, 11, 15, 19"
    • Natural for boundary types "1, 4, 5, 9, 11, 15, 19" (if bc_natural_open=.t.)
    • Dirichlet for all other boundary types
  • $v_\parallel$:
    • Special Mach=1 for boundary types "1, 3, 4, 5, 9, 11, 15, 19"
    • Dirichlet for all other boundary types

Conflicts between boundary conditions

Conflicts can occur when specifying multiple conditions. For example, what happens if you specify both bcs(i)%dirichlet%vpar=.t. and bcs(i)%mach1=.t.? In such cases, the following priority order applies (1 = highest priority):

  1. Special conditions (e.g., free-boundary and Mach=1)
  2. Dirichlet conditions
  3. Natural conditions
  4. No explicit conditions

Example 1: Apply Mach=1 to $v_\parallel$ 

bcs(i)%mach1 = .t.

Example 2: Apply Dirichlet to $v_\parallel$ 

bcs(i)%mach1          = .f.
bcs(i)%dirichlet%vpar = .t.

Example 3: Apply natural conditions to $T$ 

bc_natural_open    = .t.    ! Switches on boundary integral calls
bcs(i)%dirichlet%T = .f.
bcs(i)%natural%T   = .t.

Example 4: Set no conditions to $\rho$ 

bc_natural_open      = .f.
bcs(i)%dirichlet%rho = .f.
bcs(i)%natural%rho   = .f.