x_ops Module Reference

Operations considering perturbation quantities. More...

Interfaces and Types
interface	calc_x
	Calculates either vectorial or tensorial perturbation variables. More...
interface	redistribute_output_x
	Redistribute the perturbation variables. More...
interface	print_output_x
	Print either vectorial or tensorial perturbation quantities of a certain order to an output file. More...

Functions/Subroutines
integer function, public	init_modes (grid_eq, eq)
	Initializes some variables concerning the mode numbers.
integer function, public	setup_modes (mds, grid_eq, grid, plot_name)
	Sets up some variables concerning the mode numbers.
integer function, public	check_x_modes (grid_eq, eq)
	Checks whether the high-n approximation is valid:
integer function, public	calc_res_surf (mds, grid_eq, eq, res_surf, info, jq)
	Calculates resonating flux surfaces for the perturbation modes.
integer function, public	resonance_plot (mds, grid_eq, eq)
	plot \(q\)-profile or \(\iota\)-profile in flux coordinates with \(nq-m = 0\) or \(n-\iota m = 0\) indicated if requested.
integer function, public	calc_magn_ints (grid_eq, grid_x, eq, x, x_int, prev_style, lim_sec_x)
	Calculate the magnetic integrals from \(\widetilde{PV}_{k,m}^i\) and \(\widetilde{KV}_{k,m}^i\) in an equidistant grid where the step size can vary depending on the normal coordinate.
integer function, public	divide_x_jobs (arr_size)
	Divides the perturbation jobs.
integer function, public	print_debug_x_1 (mds, grid_x, x_1)
	Prints debug information for X_1 driver.
integer function, public	print_debug_x_2 (mds, grid_x, x_2_int)
	Prints debug information for X_2 driver.

Variables
logical, public	debug_check_x_modes_2 = .false.
	plot debug information for check_x_modes_2()

Detailed Description

Operations considering perturbation quantities.

Function/Subroutine Documentation

calc_magn_ints()

integer function, public x_ops::calc_magn_ints	(	type(grid_type), intent(in)	grid_eq,
		type(grid_type), intent(in)	grid_x,
		type(eq_2_type), intent(in), target	eq,
		type(x_2_type), intent(in)	x,
		type(x_2_type), intent(inout)	x_int,
		integer, intent(in), optional	prev_style,
		integer, dimension(2,2), intent(in), optional	lim_sec_x )

Calculate the magnetic integrals from \(\widetilde{PV}_{k,m}^i\) and \(\widetilde{KV}_{k,m}^i\) in an equidistant grid where the step size can vary depending on the normal coordinate.

All the variables should thus be field-line oriented. The result is saved in the first index of the X variables, the other can be ignored.

Using prev_style, the results of this calculation on X can be combined with the ones already present in X_int.

• prev_style=0: Overwrite [def].
• prev_style=1: Add to current integral.
• prev_style=2: Divide by 2 and add to current integral. Also modify the indices of the current integral:
  • for magn_int_style = 1: 1 2 2 2 2 2 1 -> 2 2 2 2 2 2,
  • for magn_int_style = 2: 1 3 3 2 3 3 1 -> 3 2 3 3 2 3.
• prev_style=3: Add to current integral. Also modify the indices of the current integral as in prev_style = 2.
• else: ignore previous magnetic integrals.

Therefore, it is to be used in order. For example:

1. \(\phantom{\rightarrow}\) (R=1,E=1): style 0,
2. \(\rightarrow\) (R=1,E=2): style 1,
3. \(\rightarrow\) (R=2,E=1): style 2,
4. \(\rightarrow\) (R=2,E=2): style 3.

Afterwards, it would cycle through 2 and 3, as style 0 and 1 are only for the first Richardson level.

Note

1. The variable type for the integrated tensorial perturbation variables is also X_2, but it is assumed that the first index has dimension 1, the rest is ignored.
2. Simpson's 3/8 rule converges faster than the trapezoidal rule, but normally needs a better starting point (i.e. higher min_n_par_X)
3. For alpha style 1, Weyl's lemma is used to convert the surface integral into a line integral along the magnetic field. This means that the resulting line integral still needs to be multiplied by \(\frac{2\pi}{M}\), where \(M\) is the number of times the poloidal or toroidal angle completes a full \(2\pi\) tour, depending on use_pol_flux_F [6].
4. For alpha style 2, this factor is just the step size in alpha.

See also

See x_vars.x_2_type.

Returns

ierr

Parameters

[in]	grid_eq	equilibrium grid
[in]	grid_x	perturbation grid
[in]	eq	metric equilibrium
[in]	x	tensorial perturbation variables
[in,out]	x_int	interpolated tensorial perturbation variables, containing all mode combinations
[in]	prev_style	style to treat X_prev
[in]	lim_sec_x	limits of m_X (pol flux) or n_X (tor flux) for both dimensions

Definition at line 3079 of file X_ops.f90.

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calc_res_surf()

integer function, public x_ops::calc_res_surf	(	type(modes_type), intent(in)	mds,
		type(grid_type), intent(in)	grid_eq,
		type(eq_1_type), intent(in)	eq,
		real(dp), dimension(:,:), intent(inout), allocatable	res_surf,
		logical, intent(in), optional	info,
		real(dp), dimension(:), intent(inout), optional, allocatable	jq )

Calculates resonating flux surfaces for the perturbation modes.

The output consists of mode number, resonating normal position and the fraction \(\frac{m}{n}\) or \(\frac{n}{m}\). for those modes for which a solution is found that is within the plasma range.

It contains three pieces of information:

• (:,1): the mode index
• (:,2): the radial position in Flux coordinates
• (:,3): the fraction \(\frac{m}{n}\) or \(\frac{n}{m}\)

for every single mode in sec of mds, which can be tabulated in an arbitrary grid, not necessarily the equilibrium one.

Optionally, the total safety factor or rotational transform can be returned to the master.

Also, information can be displayed with info.

Returns

ierr

Parameters

[in]	mds	general modes variables
[in]	grid_eq	equilibrium grid_eq
[in]	eq	flux equilibrium
[in,out]	res_surf	resonant surface
[in]	info	if info is displayed
[in,out]	jq	either safety factor or rotational transform

Definition at line 1637 of file X_ops.f90.

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check_x_modes()

integer function, public x_ops::check_x_modes	(	type(grid_type), intent(in)	grid_eq,
		type(eq_1_type), intent(in)	eq )

Checks whether the high-n approximation is valid:

This depends on the X_style:

For X_style 1 (prescribed): every mode should resonate at least somewhere in the whole normal range:

• \(\frac{\left|nq-m\right|}{\left|n\right|} < T\) and \(\frac{\left|nq-m\right|}{\left|m\right|} < T\) for poloidal flux
• \(\frac{\left|q-\iota m\right|}{\left|m\right|} < T\) and \(\frac{\left|n-\iota m\right|}{\left|n\right|} < T\) for toroidal flux

where \(T\) = tol << 1.

This condition is determined by the sign of \(q\) (or \(\iota\)) and given by:

• \(\max\left(q_\text{min}-T,\frac{q_\text{min}}{1+T}\right) < \frac{m}{n} < \min\left(q_\text{max}+T,\frac{q_\text{max}}{1-T}\right)\), \(q>0\)
• \(\max\left(q_\text{min}-T,\frac{q_\text{min}}{1-T}\right) < \frac{m}{n} < \min\left(q_\text{max}+T,\frac{q_\text{max}}{1+T}\right)\), \(q<0\)

for poloidal flux.

For toroidal flux, \(q\) should be replaced by \(\iota\) and \(\frac{m}{n}\) by \(\frac{n}{m}\).

For X_style 2 (fast): the resonance has been taken care of, but it remains to be checked whether the number of modes is efficient.

Returns

ierr

Parameters

[in]	grid_eq	equilibrium grid
[in]	eq	flux equilibrium

Definition at line 1349 of file X_ops.f90.

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divide_x_jobs()

integer function, public x_ops::divide_x_jobs

(

integer, intent(in)

arr_size

)

Divides the perturbation jobs.

This concerns the calculation of the magnetic integrals of blocks of tensorial perturbation variables. These are set up using the equilibrium and vectorial perturbation variables that are stored in memory, but only the integrated tensorial result is stored in memory, with negligible variables size.

The size of the (k,m) pairs to be calculated is determined by looking at what fits in memory when the equilibrium and vectorial perturbation variables are stored in memory first.

Returns

ierr

Parameters

[in]

arr_size

array size (using loc_n_r)

Definition at line 3363 of file X_ops.f90.

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init_modes()

integer function, public x_ops::init_modes	(	type(grid_type), intent(in)	grid_eq,
		type(eq_1_type), intent(in)	eq )

Initializes some variables concerning the mode numbers.

Setup minimum and maximum of mode numbers at every flux surface in equilibrium coordinates

• min_n_X, max_n_X
• min_m_X, max_m_X,

It functions depending on the X_style used: 1 (prescribed) or 2 (fast). For the fast style, at every flux surface the range of modes is sought that resonates most:

• \(m = nq \pm \frac{N}{2}\) for poloidal flux
• \(n = \iota m \pm \frac{N}{2}\) for toroidal flux

where \(N\) = n_mod_X.

However, at the same time, both m and n have to be larger, in absolute value, than min_sec_X. Therefore, the range of width n_mod_X can be shifted upwards.

Returns

ierr

Parameters

[in]	grid_eq	equilibrium grid
[in]	eq	flux equilibrium

Definition at line 918 of file X_ops.f90.

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print_debug_x_1()

integer function, public x_ops::print_debug_x_1	(	type(modes_type), intent(in)	mds,
		type(grid_type), intent(in)	grid_x,
		type(x_1_type), intent(in)	x_1 )

Prints debug information for X_1 driver.

Parameters

[in]	mds	general modes variables
[in]	grid_x	perturbation grid
[in]	x_1	vectorial X variables

Definition at line 3489 of file X_ops.f90.

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print_debug_x_2()

integer function, public x_ops::print_debug_x_2	(	type(modes_type), intent(in)	mds,
		type(grid_type), intent(in)	grid_x,
		type(x_2_type), intent(in)	x_2_int )

Prints debug information for X_2 driver.

Parameters

[in]	mds	general modes variables
[in]	grid_x	perturbation grid
[in]	x_2_int	tensorial X variables

Definition at line 3632 of file X_ops.f90.

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resonance_plot()

integer function, public x_ops::resonance_plot	(	type(modes_type), intent(in)	mds,
		type(grid_type), intent(in)	grid_eq,
		type(eq_1_type), intent(in)	eq )

plot \(q\)-profile or \(\iota\)-profile in flux coordinates with \(nq-m = 0\) or \(n-\iota m = 0\) indicated if requested.

The plot will be done in the grid in which mds is tabulated, which is not necessarily the equilibrium one.

Returns

ierr

Parameters

[in]	mds	general modes variables
[in]	grid_eq	equilibrium grid
[in]	eq	flux equilibrium

Definition at line 1813 of file X_ops.f90.

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setup_modes()

integer function, public x_ops::setup_modes	(	type(modes_type), intent(inout), target	mds,
		type(grid_type), intent(in)	grid_eq,
		type(grid_type), intent(in)	grid,
		character(len=*), intent(in), optional	plot_name )

Sets up some variables concerning the mode numbers.

Apart from the mode numbers

• n, m,

also the variables of the secondary modes

• sec,

are set up in the coordinates of the grid passed. The normal values of this grid are saved as well, in

• r_F.

All variables are tabulated in the full normal grid. The mode indices, however, are chosen so that there is maximum overlap between different normal positions. This is trivial for X_style 1 (prescribed), but for X_style 2 (fast), this is best explained by an example:

kd m1 m2 m3

1 10 11 12 <- start 2 10 11 12 <- no change in limits 2 13 11 12 <- limits shift up by one 3 13 11 12 <- no change in limits 4 13 14 12 <- limits shift up by one 5 16 14 15 <- limits shift up by two 6 13 14 15 <- limits shift down by one

This procedure makes use of the global variables

• min_m_X, max_m_X, min_n_X, max_m_X

that have to be set up using init_nm_x().

Optionally, n and m can be plot.

Returns

ierr

Parameters

[in,out]	mds	modes variables
[in]	grid_eq	equilibrium grid
[in]	grid	grid at which to calculate modes
[in]	plot_name	name to be used when plotting n and m

Definition at line 1059 of file X_ops.f90.

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Variable Documentation

debug_check_x_modes_2

logical, public x_ops::debug_check_x_modes_2 = .false.

plot debug information for check_x_modes_2()

Note

Debug version only

Definition at line 26 of file X_ops.f90.