In situations when multiple beams are traced, either when allowing
multiple plasma crossings (raytracing.ipass > 0) or the initial polarisation
is mixed (raytracing.ipol == .true.), the couplings of all but the first
beam (with least index_rt) were invalid.
The bug is due to the re-use of the psipol,chipol variables as the beams
are traced sequentially over the beam_loop. For the first beam being
traced the psipol,chipol are correctly initialised to the user-defined
value and the resulting coupling is correct.
However, in each subsequent beam the values were not set to those of the
parent beam (or to the user-defined value in the case of the first X
mode beam), but to those of the previous beams (current index_rt - 1).
This change repurposes the psipv,chipv arrays to store the polarisation
of the parent beams, including the initial user-defined value and makes
plasma_in always use these to compute the coupling.
In addition, in the case the polarisation is not immediately known (i.e.
if raytracing.ipol == .false.), this change postpones the computation of
the Jones vector (ext, eyt) from the launch point, if the magnetic
equilibrium is available, to when the ray actually crosses the
plasma boundary.
The original code, besides being strictly incorrect, can lead to
non-negligible alterations to the coupling. This change also mean:
1. most of the functionality of `set_pol` has been merged with
`plasma_in`
2. the polarisation is undefined and the Jones vector is set to the
placeholder value [1, 0] till `plasma_im` is called
Finally, `polarcold` is removed because it's unused.
While technically accepted by GRAY, these indices do not carry a special
meaning, as wrongly implied by the documentation: they are equivalent
to 8, 18 and specifically don't change the meaning of sgnbi,sgni.
Since 24edfdc4 print_prof tabulates the input profiles up to ψ_bnd,
however the torr_curr_psi may fail (producing FPEs) outside ψ=1.
Moreover, torr_curr_psi gives the current on the LFS on the specific
line z=z_maxis, while the flux surface average is usually more
interesting, say when comparing J_φ with Jcd.
This change fixes both issues.
If frhopol is called with a very small ρ_p, less than then step size of
ε_machine^⅓, the two sided finite step differentiation evaluates frhotor
at ρ_p < 0, producing a floating point exception.
In this case we use the single- sided definition.
This adds a proper procedure to rescale and shift a 2D B-spline by
manipulating the coefficients.
Note: the previous code in set_equil_spline did work, but by
transforming the whole partial(i,j) triggered warnings about operations
on uninitialised memory.
When a call to regrid results in ier=-1 (data needs to be refitted with
interpolating spline) the computation of the sum of the residuals can
overflow. In this case we can exit immediately (avoiding raising a FPE)
ignoring the results. The latter need to be recomputed anyway.
The minimisation algorithm may try to evaluate a function outside of its
domain triggering a floating point exception. However, this is not a
concern because it won't affect the final result, so they can be ignored
1. Introduces enumerations (and some booleans) intended to replace all
the magic numbers used throughout the code to represent multiple
choices.
2. Replace the gray_params.sh script a new one that automatically
generates code for all the GRAY parameters by parsing
gray_params.f90.
3. Also generate extra code to accept the enum identifiers as valid
values in the configuration files and command line arguments.
4. Set sensible default values for parameters that are rarely changes.
This change limits the evaluation of the poloidal flux spline ψ(R,z) to
a particular domain where ψ(R,z) is strictly monotonic. This choice
ensures that ψ_n < 1 only inside the plasma boundary; so plasma
parameters like q(ψ), Te(ψ), etc. can be mapped to the physical space
without ambiguities.
Before this change, anywhere ψ_n happened to decrease below 1 (either as
a result of a physical current or simply from the spline extrapolation),
Gray would effectively create a spurious plasma region.
This behavior is seriously problematic because it completely invalidates
the simulation: it can alter the ray directions, the power disribution
and total power if the beam enters one such region.
The domain is computed by applying a transformation to the contour of
the plasma boundary: for each contour point we cast a ray from the
magnetic axis to that point and extend the ray until the restriction of
ψ(R,z) on the ray starts decreasing or reach a maximum scaling factor.
The endpoint of the ray is then taken as the new point.
If ψ(R,z) is globally monotonic, the transformation is a homotety wrt
the magnetic axis, so the domain will be an enlarged boundary; otherwise
the shape will be more irregular (an intersection of the enlarged
boundary and several level curves of ψ).
Finally, each `pol_flux(r, z)` call is now guarded behind a check
`inside(psi_domain, r, z)`. For points outside the domain the subroutine
returns, as usual, -1 for ψ and 0 for derivatives.
Previously the ray initial positions were set to the wavefront
S_R(x,y,z) = 0, with (x,y) chosen such that S_I(x,y,0) = const.
The wavefront itself, however, was determined using the value of the
beam parameters (k_ξ, k_η, w_ξ, w_η, etc.) fixed at z=0, which is valid
only when the initial wavefront is approximately flat.
Moreover, since the ray are distributed according to S_I(z=0), this
choice creates an inconsistency between the phase (from S_R at z≠0) and
the power (from S_I at z=0) assigned to the rays.
Satisfying both conditions on S_R and S_I exactly is really hard;
however, given we do not really care about the phase and we want to
precisely track the power, so it's more sensible to simply set z=0.
This means that when integrating in the phase (idst=2), gray will no
longer construct wavefronts, but merely transport the initial phase
on the z=0 plane.
Note that k₀⋅s will still give the correct phase. so if necessary, a
wavefront could be reconstructed by interpolating the (s, x̅, y̅, z̅)
points.
ar does not support multiple builder writing to the same file.
This disable parallel building for .a targets.
See also "Archive Pitfalls" in the GNU Make manual.
When building the static library the Nixpkgs stdenv `ar` defaults to not
writing timestamps for the members file. This imply make will always try
to add all members to the archive, which defeats the purpose of the
file-level granularity.
To improve the analytical model correctness this changes the formula
of the toroidal flux to lift the large aspect ratio approximation (a << R₀).
In fact, Φ(r) = B₀πr² is technically inconsistent with the field varying
as B₀R₀/R. The exact expression is:
Φ(r) = B₀πr² 2/[1 + √(1 - r²/R₀²)],
which is approximately equal to the former for r << R₀.
Note that this change introduces a divergence in the poloidal field at
r=R₀ (since ∂Φ/∂r → +∞), so the domain of the equilibrium has been
restricted to r<R₀, as expected.
This should not be a concern because the field outside the plasma
boundary is never directly, particularly not by the integrator.
Directly mapping r(R,z) to ρ_p greatly simplifies the implementation of
pol_flux, but also produces a weird current profile and resonance curve.
This keeps the previous changes to the model but reverts the mapping to
the previous one: r → ρ_t.
This changes all the print_* subroutines to not depend on q_spline,
either for the ψ data points or just their number.
In fact, in the new analytical model q_spline doesn't exist anymore.
Pro: This allows to mark q_spline as private.
This change combines `equian` and `equinum_psi` into a new `pol_flux`
subroutine that computes ψ(R,z) and derivatives for either numerical or
analytical equilibria.
Similarly, `equian` (the fpol and dfpol outputs) and `equinum_fpol` are
combined intro `pol_curr` that computes F(ψ) and its derivative.
Callers of these subroutines do not select a specific version based on
the value of `iequil` anymore.
- Hide the implementation of the re-normalisation of the ψ(R,z) spline
by adjusting the spline coefficients, instead of shifting and
rescaling after each evaluation.
- Correct the value of `psia` = ψ(X point) - ψ(O point) after the ψ(R,z)
spline has been re-normalised.
This fixes another instance of decoupling between the values of X and
∇X that introduce a systematic error in the numerical integration of
the raytracing equations.
Here the issue is caused by the different normalisations used,
specifically X=X(ψ_n') and ∇X=∇X(ψ_n), where ψ_n' is the re-normalised
spline of the normalised flux and ψ_n the spline of the normalised
flux.
See d0a5a9f for more details on problems resulting from this error.
- Change `equian` and `equinum_psi` to return the normalised values
for both the flux and its derivatives, to avoid confusions.
Callers that needed the unnormalised derivatives now multiply
explicitly by `psia`.
This change modifies the analytical equilibrium in order to simplify the
computation of the poloidal flux normalization and the derivatives.
In the power law parametrisation of the safety factor, ρ_t is replaced
with ρ_p and, similarly, the normalised poloidal radius is now
identified with ρ_p, instead of ρ_t.
With the same parameters (q₀,q₁,α...), this choice slightly changes the
plasma current distribution, but enables us to obtain a closed form for
ψ_a = ψ(r=a) and the relation ρ_t(ρ_p). In fact, both expressions are
now obtained by integrating the q(ρ_p), instead of 1/q(ρ_t), which has
no elementary antiderivative.
As the normalisation is now computed exactly, the values of the
normalised flux ψ_n = ψ/ψ_a and the gradient ∇ψ (entering the raytracing
equations in X and ∇X, respectively) are computed to the same precision.
Previously, ψ_n was computed to a lower precision due to the use of a
simple trapezoid integration of 1/q(ρ_p) for ψ_a, while ∇ψ was computed
up to machine precision using an exact formula.
This error effectively caused a very slight decoupling between X=ω_p²/ω²
and ∇X that introduced a systematic error in the numerical solution of
the raytracing equations.
The error manifests itself as a bias with a weak dependency on X in the
values taken by the dispersion function Λ(r̅, n̅) on the phase-space
points generated by the integrator. More specifically,
lim h→0 Λ(r̅_i, n̅_i) = -kX(r̅_i)
where h is the integrator step size;
r̅_i is the position at the i-th step;
k ≈ -3.258⋅10⁻⁵ and depends only on the number of points used to
perform the trapedoid integral for ψ_a (as ~ 1/n²).
After this change Λ behaves consistently with being a conserved quantity
(zero) up to the cumulative integration error of the 4° order
Runge-Kutta method. In fact we now have that:
Λ(r̅_i, n̅_i) ∝ - h⁴ ‖∂⁴X(r̅_i)/∂r̅⁴‖
It must be said that within this model the relation ρ_p(ρ_t) can't be
computed analytically (inverting ρ_t(ρ_p) produces a trascendental
equation of the form b = x + c x^α). However, this relation is not
necessary for raytracing and is easily solved, up to machine
precision, using minpack.
In addition, this change also makes the model consistetly use the
cocos=3 and fully implements the ability to force the signs of I_p, B_φ
(via equilibrium.sgni,sgnb) and rescaling the field (via
equilibrium.factb).
In the case of analytic equilibrium without a limiter contour, the
simple limiter was built incorrectly due to an unnecessary conversion
from cm (the equilibrium data are already in metre).
