Toroidal and Poloidal Magnetic Currents

This is a work in progress and does not give correct results at the moment!

Definition

Toroidal current is defined as

\[-\Delta_2 \psi = \frac{\partial J_y}{\partial x} - \frac{\partial J_x}{\partial y} J_{toro} = ∇×ψ\]

Poloidal current is defined as

\[-\Delta_2 \phi = J_z J_{polo} = ∇×∇×ϕ\]

Where

\[\Delta_2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\]

Green's function that solves the system is 2D Poisson kernel

\[G(x_1, y_1; x_0, y_0) = \frac{1}{4\pi} \ln (\Delta x^2 + \Delta y^2)\]

Submodule

There is a submodule for toroidal-poloidal decomposition of magnetic currents. To access it just

using TensorOperations
using HKQM
using HKQM.ToroidalCurrent

SYSMOIC files

There are two options to read Sysmoic files. A low level one

data = read_sysmoic(file_name)

and a high level one

J, dJ = read_current(file_name)

You can control the precision of integration grid

ne = 4  # Number of elements per dimension
ng = 32 # Number of Gauss points per element per dimension
J, dJ = read_current(file_name, ne, ng)

This affects memory usage. So, if you have problems with memory, try to lower precision.

Toroidal and Poloidal Currents

To calculate toroidal current type

J_toro = toroidal_current(dJ)

To calculate poloidal current type

J_polo = poloidal_current(J)

To check that everything is correct type

# This should be zero current
ΔJ = J[:,3] - ( J_toro + J_polo )

# If everything is correct this returns zeros
bracket.(ΔJ, ΔJ)

Change Magnetic Field direction

The default direction of magnetic field is z-axis. You can change the direction by giving different direction

# Magnetic field in x-direction
Bx = 1.u"T"
By = 0.u"T"
Bz = 0.u"T"

B = [Bx, By, Bz]

J_toro = toroidal_current(dJ; B=B)
J_polo = poloidal_current(J; B=B)

# Magnetic field between x- and y-directions
Bx = cos(π/4) * u"T"
By = sin(π/4) * u"T"
Bz = 0.u"T"

B = [Bx, By, Bz]

J_toro = toroidal_current(dJ; B=B)
J_polo = poloidal_current(J; B=B)

Adjust integration parameters

Poisson kernel includes a calculation of logarithm of zero distance, which is minus infinity. To counter this, there is a small additional number added to the logarithm

\[\ln (\Delta x^2 + \Delta y^2) \approx \ln (\Delta x^2 + \Delta y^2 + \epsilon)\]

This will make sure no logarithm is taken from zero.

You can change this variable by calling

# 1.0 * 10^-7 Å change in distance
J_toro = toroidal_current(dJ; eps=1E-7u"Å")

# 1.0 * 10^-5 bohr change in distance
J_polo = poloidal_current(J; eps=1E-5u"bohr")

Writing results

You can write toroidal and poloidal current with

data = write_currents("output_file.csv", J_toro, J_polo)

# change number of points per dimension
write_currents("output_file.csv", J_toro, J_polo; n_points=25)

The output is a CSV file.