Operations considering perturbation quantities. More...
Interfaces and Types | |
| interface | calc_x |
| Calculates either vectorial or tensorial perturbation variables. More... | |
| interface | print_output_x |
| Print either vectorial or tensorial perturbation quantities of a certain order to an output file. More... | |
| interface | redistribute_output_x |
| Redistribute the perturbation variables. More... | |
Functions/Subroutines | |
| integer function, public | init_modes (grid_eq, eq) |
| Initializes some variables concerning the mode numbers. More... | |
| integer function, public | setup_modes (mds, grid_eq, grid, plot_name) |
| Sets up some variables concerning the mode numbers. More... | |
| integer function, public | check_x_modes (grid_eq, eq) |
| Checks whether the high-n approximation is valid: More... | |
| integer function, public | calc_res_surf (mds, grid_eq, eq, res_surf, info, jq) |
| Calculates resonating flux surfaces for the perturbation modes. More... | |
| 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. More... | |
| integer function | calc_u (grid_eq, grid_X, eq_1, eq_2, X) |
| Calculate \(U_m^0\), \(U_m^1\) or \(U_n^0\), \(U_n^1\). More... | |
| integer function | calc_pv (grid_eq, grid_X, eq_1, eq_2, X_a, X_b, X, lim_sec_X) |
calculate \(\widetilde{PV}_{k,m}^i\) (pol. flux) or \(\widetilde{PV}_{l,n}^i\) (tor. flux) at all equilibrium loc_n_r values. More... | |
| integer function | calc_kv (grid_eq, grid_X, eq_1, eq_2, X_a, X_b, X, lim_sec_X) |
calculate \(\widetilde{KV}_{k,m}^i\) (pol. flux) or \(\widetilde{KV}_{l,n}^i\) (tor. flux) at all equilibrium loc_n_r values. More... | |
| 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. More... | |
| integer function, public | divide_x_jobs (arr_size) |
| Divides the perturbation jobs. More... | |
| integer function, public | print_debug_x_1 (mds, grid_X, X_1) |
| Prints debug information for X_1 driver. More... | |
| integer function, public | print_debug_x_2 (mds, grid_X, X_2_int) |
| Prints debug information for X_2 driver. More... | |
Variables | |
| logical, public | debug_check_x_modes_2 = .false. |
| plot debug information for check_x_modes_2() More... | |
Detailed Description
Operations considering perturbation quantities.
Function/Subroutine Documentation
◆ calc_kv()
| integer function x_ops::calc_kv | ( | type(grid_type), intent(in) | grid_eq, |
| type(grid_type), intent(in) | grid_X, | ||
| type(eq_1_type), intent(in) | eq_1, | ||
| type(eq_2_type), intent(in), target | eq_2, | ||
| type(x_1_type), intent(in) | X_a, | ||
| type(x_1_type), intent(in) | X_b, | ||
| type(x_2_type), intent(inout) | X, | ||
| integer, dimension(2,2), intent(in), optional | lim_sec_X | ||
| ) |
calculate \(\widetilde{KV}_{k,m}^i\) (pol. flux) or \(\widetilde{KV}_{l,n}^i\) (tor. flux) at all equilibrium loc_n_r values.
Like in calc_U(), use is made of variables Ti that are set up in the equilibrium grid and are afterwards converted to Ti_X in the perturbation grid, which needs to have the same angular coordinates as the equilibrium grid:
\[\begin{aligned} T_1 &= \rho \mathcal{J}^2 \frac{h^{22}}{\mu_0} g_{33} \\ T_2 &= \rho \frac{1}{h^{22}} \ , \end{aligned}\]
with \(T_i\) = Ti.
The interpolated Ti_X are then used to calculate \(\widetilde{KV}_{k,m}^i\):
\[\begin{aligned} \widetilde{KV}_{k,m}^0 &= T_1 U_k^{0*} U_m^0 + T_2 \\ \widetilde{KV}_{k,m}^1 &= T_1 U_k^{0*} U_m^1 \\ \widetilde{KV}_{k,m}^2 &= T_1 U_k^{1*} U_m^1 \ , \end{aligned}\]
where
\[DU_k^i = i (nq-m) U_k^i + \frac{\partial U_k^i}{\partial\theta} . \]
This is valid for poloidal Flux coordinates and where \(n\) is to be replaced by \(m\) and \((nq-m)\) by \((n-\iota m)\) for toroidal Flux coordinates.
- See also
- See [15] .
- Returns
- ierr
- Parameters
-
[in] grid_eq equilibrium grid [in] grid_x perturbation grid [in] eq_1 flux equilibrium [in] eq_2 metric equilibrium [in] x_a vectorial perturbation variables of dimension 2 [in] x_b vectorial perturbation variables of dimension 2 [in,out] x tensorial perturbation variables [in] lim_sec_x limits of m_X(pol flux) orn_X(tor flux) for both dimensions
Definition at line 2879 of file X_ops.f90.
Here is the call graph for this function: Here is the caller graph for this function:◆ 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.
- for
prev_style=3: Add to current integral. Also modify the indices of the current integral as inprev_style= 2.- else: ignore previous magnetic integrals.
Therefore, it is to be used in order. For example:
- \(\phantom{\rightarrow}\) (R=1,E=1): style 0,
- \(\rightarrow\) (R=1,E=2): style 1,
- \(\rightarrow\) (R=2,E=1): style 2,
- \(\rightarrow\) (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
- 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. - 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) - 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]. - For alpha style 2, this factor is just the step size in alpha.
- The variable type for the integrated tensorial perturbation variables is also
- 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) orn_X(tor flux) for both dimensions
Definition at line 3081 of file X_ops.f90.
Here is the call graph for this function: Here is the caller graph for this function:◆ calc_pv()
| integer function x_ops::calc_pv | ( | type(grid_type), intent(in) | grid_eq, |
| type(grid_type), intent(in) | grid_X, | ||
| type(eq_1_type), intent(in), target | eq_1, | ||
| type(eq_2_type), intent(in), target | eq_2, | ||
| type(x_1_type), intent(in) | X_a, | ||
| type(x_1_type), intent(in) | X_b, | ||
| type(x_2_type), intent(inout) | X, | ||
| integer, dimension(2,2), intent(in), optional | lim_sec_X | ||
| ) |
calculate \(\widetilde{PV}_{k,m}^i\) (pol. flux) or \(\widetilde{PV}_{l,n}^i\) (tor. flux) at all equilibrium loc_n_r values.
Like in calc_U(), use is made of variables Ti that are set up in the equilibrium grid and are afterwards converted to Ti_X in the perturbation grid, which needs to have the same angular coordinates as the equilibrium grid:
\[\begin{aligned} T_1 &= \frac{h^{22}}{\mu_0} g_{33} \\ T_2 &= \mathcal{J}S + \mu_0 \sigma \frac{g_{33}}{\mathcal{J}} h^{22} \\ T_3 &= \frac{\sigma}{\mathcal{J}} T_2 \\ T_4 &= \frac{1}{\mu_0} \mathcal{J}^2 h^{22} \\ T_5 &= 2 p' \kappa_n \ , \end{aligned}\]
with \(T_i\) = Ti.
The interpolated Ti_X are then used to calculate \(\widetilde{PV}_{k,m}^i\):
\[\begin{aligned} \widetilde{PV}_{k,m}^0 &= T_1(DU_k^{0*} - T_2)(DU_m^0 - T_2) - T_3 + (nq-m)(nq-k)T_4 - T_5 \\ \widetilde{PV}_{k,m}^1 &= T_1(DU_k^{0*} - T_2) DU_m^1 \\ \widetilde{PV}_{k,m}^2 &= T_1 DU_k^{1*} DU_m^1 \ , \end{aligned}\]
where
\[DU_k^i = i (nq-m) U_k^i + \frac{\partial U_k^i}{\partial\theta} . \]
This is valid for poloidal Flux coordinates and where \(n\) is to be replaced by \(m\) and \((nq-m)\) by \((n-\iota m)\) for toroidal Flux coordinates.
- See also
- See [15] .
- Returns
- ierr
- Parameters
-
[in] grid_eq equilibrium grid [in] grid_x perturbation grid [in] eq_1 flux equilibrium [in] eq_2 metric equilibrium [in] x_a vectorial perturbation variables of dimension 2 [in] x_b vectorial perturbation variables of dimension 2 [in,out] x tensorial perturbation variables [in] lim_sec_x limits of m_X(pol flux) orn_X(tor flux) for both dimensions
Definition at line 2646 of file X_ops.f90.
Here is the call graph for this function: Here is the caller graph for this function:◆ 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 1638 of file X_ops.f90.
Here is the call graph for this function: Here is the caller graph for this function:◆ calc_u()
| integer function x_ops::calc_u | ( | type(grid_type), intent(in) | grid_eq, |
| type(grid_type), intent(in) | grid_X, | ||
| type(eq_1_type), intent(in), target | eq_1, | ||
| type(eq_2_type), intent(in), target | eq_2, | ||
| type(x_1_type), intent(inout) | X | ||
| ) |
Calculate \(U_m^0\), \(U_m^1\) or \(U_n^0\), \(U_n^1\).
This is done at eq loc_n_r values of the normal coordinate, n_par values of the parallel coordinate and size_X values of the poloidal mode number, or of the toroidal mode number, as well as the poloidal derivatives
Use is made of variables Ti that are set up in the equilibrium grid and are afterwards converted to Ti_X in the perturbation grid, which needs to have the same angular coordinates as the equilibrium grid:
\[\begin{aligned} T_1 &= \frac{B_\alpha}{B_\theta} \\ T_2 &= \Theta^\alpha + q' \theta \\ T_3 &= \frac{B_\alpha}{B_\theta} q' + \frac{\mathcal{J}^2}{B_\theta} \mu_0 p' \\ T_4 &= \frac{B_\alpha}{B_\theta} q' \theta - \frac{B_\psi}{B_\theta} \\ T_5 &= \frac{B_\alpha}{B_\theta} \frac{\partial \Theta^\theta}{\partial \theta} - \frac{\partial\Theta^\theta}{\partial \alpha} \\ T_6 &= \frac{B_\alpha}{B_\theta} \Theta^\theta \ , \end{aligned}\]
with \(T_i\) = Ti.
which is valid for poloidal Flux coordinates.
For toroidal Flux coordinates, \(q'\) has to be replaced by \(-\iota'\) and \(\theta\) by \(\zeta\).
The interpolated Ti_X are then used to calculate \(U\):
\[\begin{aligned} U_0 =& -T_2 &\text{(order 1)}\\ &+ \frac{i}{n}\left(T_3 + i(nq-m) T_4\right) &\text{(order 2)}\\ &+ \left(\frac{i}{n}\right)^2 i(nq-m) \left(-T_5 - (nq-m) T_6\right) &\text{(order 3)}\\ U_1 =& \frac{i}{n} &\text{(order 1)}\\ &+ \left(\frac{i}{n}\right)^2 i(nq-m)(-T_1) &\text{(order 2)} . \end{aligned}\]
This is valid for poloidal Flux coordinates and where \(n\) is to be replaced by \(m\) and \((nq-m)\) by \((n-\iota m)\) for toroidal Flux coordinates.
For VMEC, these factors are also derived in the parallel coordinate.
- See also
- See [15].
- Returns
- ierr
- Parameters
-
[in] grid_eq equilibrium grid variables [in] grid_x perturbation grid variables [in] eq_1 flux equilibrium [in] eq_2 metric equilibrium [in,out] x vectorial perturbation variables
Definition at line 2097 of file X_ops.f90.
Here is the call graph for this function: Here is the caller graph for this function:◆ 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 1350 of file X_ops.f90.
Here is the call graph for this function: Here is the caller graph for this function:◆ 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 3364 of file X_ops.f90.
Here is the call graph for this function: Here is the caller graph for this function:◆ 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_Xmin_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 919 of file X_ops.f90.
Here is the call graph for this function: Here is the caller graph for this function:◆ 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 | ||
| ) |
◆ 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 | ||
| ) |
◆ 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 1814 of file X_ops.f90.
Here is the call graph for this function: Here is the caller graph for this function:◆ 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 nandm
Definition at line 1060 of file X_ops.f90.
Here is the call graph for this function: Here is the caller graph for this function: