# ElectricField¶

The ElectricField class holds settings for the parallel electric field solved for by DREAM, which is described by the unknown E_field. This quantity is defined as

$\mathrm{E\_field} = \frac{\langle \boldsymbol{E}\cdot\boldsymbol{B} \rangle}{\sqrt{\langle B^2 \rangle}},$

where angle brackets $$\langle\cdot\rangle$$ denote a flux-surface average, $$\boldsymbol{E}$$ denotes the electric field, $$\boldsymbol{B}$$ the magnetic field and $$\varphi$$ the toroidal angle.

The electric field can be solved for in two different ways:

1. By prescribing the electric field profile in time $$E_\parallel = \tilde{E}(t,r)$$ (TYPE_PRESCRIBED)

2. By solving the induction equation (TYPE_SELFCONSISTENT):

(1)$\begin{split} \begin{cases} \frac{\partial\psi_{\rm p}}{\partial t} &= V_{\rm loop},\\ \mu_0\frac{j_\parallel}{B}\left\langle \boldsymbol{B}\cdot\nabla\varphi \right\rangle &= \frac{1}{V'}\frac{\partial}{\partial r}\left[ V'\left\langle \frac{\left|\nabla r\right|^2}{R^2} \right\rangle \frac{\partial\psi_{\rm p}}{\partial r} \right] \end{cases}\end{split}$

where $$\psi_{\rm p}$$ is the poloidal flux, $$\mu_0$$ is the vacuum permeability, $$V'$$ the spatial Jacobian and $$r$$ the minor radius in the outer midplane. The loop voltage $$V_{\rm loop}$$ is defined by

$V_{\rm loop} = 2\pi \frac{\langle \boldsymbol{E}\cdot\boldsymbol{B} \rangle}{\langle \nabla \varphi \cdot \boldsymbol{B} \rangle}.$

## Prescribed evolution¶

The electric field can be prescribed in both space and time. The full syntax for prescribing an electric field evolution is

where E_field is an array of shape (nt, nr) specifying the parallel electric field strength in space and time, E_field_r is a vector with nr elements representing the radial grid on which the electric field is prescribed, and E_field_t is a vector with nt elements representing the time grid on which the electric field is prescribed.

For convenience, it is possible to prescribe a constant and radially uniform electric field by only specifying it as a scalar:

ds.eqsys.E_field.setPrescribedData(0.3)

This will cause the electric field to be $$E_\parallel = 0.3\,\mathrm{V/m}$$ at every radius, in every time step, during the simulation.

## Self-consistent evolution¶

In this mode the electric field is evolved by solving the system (1). The evolution of the electric field is thus coupled to the poloidal flux $$\psi_{\rm p}$$ and, by extension, to the total plasma current density $$j_{\rm tot}$$. To evolve the electric field self-consistently, the first thing one must do is to set the type of the electric field to TYPE_SELFCONSISTENT via a call to setType():

import DREAM.Settings.Equations.ElectricField as ElectricField

...

ds.eqsys.E_field.setType(ElectricField.TYPE_SELFCONSISTENT)

By default, the initial electric field profile will be identically zero. This is automatically overridden if the settings are loaded from an old output file (see restart), but the initial profile can also be set explicitly via a call to setInitialProfile():

where E_field is a vector of size nr giving the initial electric field profile, and E_field_r is a vector representing the radial grid on which the initial electrc field profile is defined.

### Boundary condition¶

The second equation in (1) requires a boundary condition at $$r=r_{\rm wall}$$ to be given. In DREAM, three different boundary conditions can be applied at the tokamak wall.

The first boundary condition, BC_TYPE_PRESCRIBED, prescribes the time evolution of the loop voltage $$V_{\rm loop,wall}$$ on the tokamak wall (normalized to the tokamak major radius $$R_0$$). This is particularly useful for simulating experimental scenarios where the parameter $$V_{\rm loop,wall}$$ has been measured.

import DREAM.Settings.Equations.ElectricField as ElectricField

ds = DREAMSettings()
...
R0        = 1.65
# Define evolution of V_loop/R0
Vmax      = 1
V_loop_t  = np.linspace(0, 1, 100)
V_loop_R0 = (Vmax/R0)*(1 - (1-t)**2)

ds.eqsys.E_field.setBoundaryCondition(bctype=ElectricField.BC_TYPE_PRESCRIBED,
V_loop_wall_R0=V_loop_R0, times=V_loop_t, R0=R0)

The second boundary condition, BC_TYPE_SELFCONSISTENT, instead lets the user specify the (inverse) tokamak wall time, $$1/\tau_{\rm wall}$$, which directly corresponds to the wall resistivity. The perfectly conducting limit $$\tau_{\rm wall} = \infty$$ is supported, and is obtained by setting inverse_wall_time = 0. In case a cylindrical geometry is used, the major radius R0 can be explicitly set, independently of the geometry used, since the external inductance otherwise diverges for infinite major radius.

import DREAM.Settings.Equations.ElectricField as ElectricField

ds = DREAMSettings()
...
R0       = 1.65
# Wall time
tau_wall = .01   # (s)

ds.eqsys.E_field.setBoundaryCondition(bctype=ElectricField.BC_TYPE_SELFCONSISTENT,
inverse_wall_time=1/tau_wall, R0=R0)

The third boundary condition, BC_TYPE_TRANSFORMER, can be seen as a combination of the two other boundary conditions, with a resistive wall and a prescribed loop voltage (although at the transformer, which passes through the axis of symmetry at $$R=0$$).

import DREAM.Settings.Equations.ElectricField as ElectricField

ds = DREAMSettings()
...
R0        = 1.65
# Define evolution of V_loop/R0
Vmax      = 1
V_loop_t  = np.linspace(0, 1, 100)
V_loop_R0 = (Vmax/R0)*(1 - (1-t)**2)
# Wall time
tau_wal   = .01  # (s)

ds.eqsys.E_field.setBoundaryCondition(bctype=ElectricField.BC_TYPE_PRESCRIBED,
inverse_wall_time=1/tau_wall, R0=R0,
V_loop_wall_R0=V_loop_R0, times=V_loop_t)

Note

With all boundary conditions, the tokamak wall location wall_radius must be specified. This parameter denotes the minor radial coordinate of the tokamak wall (i.e. the distance of the wall from the center of the plasma). This is done via the call ds.radialgrid.setWallRadius(b), where b is the desired wall radius.

## Class documentation¶

class DREAM.Settings.Equations.ElectricField.ElectricField(settings, ttype=1, efield=None, radius=0, times=0)

Bases: DREAM.Settings.Equations.PrescribedParameter.PrescribedParameter, DREAM.Settings.Equations.PrescribedInitialParameter.PrescribedInitialParameter, DREAM.Settings.Equations.PrescribedScalarParameter.PrescribedScalarParameter, DREAM.Settings.Equations.UnknownQuantity.UnknownQuantity

__getitem__(index)

Returns the value of the prescribed electric field at the given indices.

Constructor.

Parameters
• settings (DREAM.DREAMSettings) – Parent settings object.

• ttype (int) – Method to use for evolving electric field.

• efield – Prescribed or initial electric field (profile).

• radius – Radial grid on which the prescribed or initial electric field (profile) is defined.

• times – Time grid on which the prescribed electric field is defined.

fromdict(data)

Sets this paramater from settings provided in a dictionary.

setBoundaryCondition(bctype=2, V_loop_wall_R0=None, times=0, inverse_wall_time=None, R0=0)

Specifies the boundary condition to use when solving for the electric field self-consistently, i.e. with TYPE_SELFCONSISTENT. Possible boundary condition types are:

Name

Description

BC_TYPE_PRESCRIBED

Set $$V_{\rm loop}$$ on the tokamak wall.

BC_TYPE_SELFCONSISTENT

Specify the tokamak wall time and solve self-consistently for $$V_{\rm loop,wall}$$.

BC_TYPE_TRANSFORMER

Same as BC_TYPE_SELFCONSISTENT, but with prescribed loop voltage via transformer.

Parameters
• bctype (int) – Type of boundary condition to use (see table above for available options).

• V_loop_wall_R0 – Prescribed value of $$V_{\rm loop}/R_0$$ on the tokamak wall (or at transformer in case of type=BC_TYPE_TRANSFORMER), normalized to the tokamak major radius $$R_0$$.

• times – Time grid on which V_loop_wall_R0 is given.

• inverse_wall_time – Inverse wall time for the tokamak, used when solving for $$V_{\rm loop,wall}$$ self-consistently.

• R0 – Major radius for the tokamak, only used when solving for $$V_{\rm loop,wall}$$ self-consistently (independent of radial-grid major radius).

When TYPE_SELFCONSISTENT, sets the initial electric field profile. The parameter efield may be either a scalar (in which case the profile is taken to be uniform) or an array. The associated radial grid radius must be of the same type and dimension.

Parameters
• efield – Initial electric field profile.

• radius – Radial grid on which the initial profile is defined.

When TYPE_PRESCRIBED, sets the spatiotemporal evolution of the electric field during the simulation. The parameter efield may be either a scalar (in which case the electric field is taken to be constant and uniform in time and radius) or a 2D array of shape (nt, nr). The associated time grid times must be of size nt and the radial grid must be of size nr.

Parameters
• efield – Prescribed electric field.

• radius – Radial grid on which the electric field is prescribed.

• times – Time grid on which the electric field is prescribed.

setType(ttype)

Set the type of equation to use for evolving the electric field. The available types are

Name

Description

TYPE_PRESCRIBED

Prescribe spatiotemporal evolution of electric field.

TYPE_SELFCONSISTENT

Evolve electric field consistent with the evolution of the poloidal flux and plasma current.

Parameters

ttype (int) – Type of electric field evolution to use.

todict()

Returns a Python dictionary containing all settings of this ColdElectrons object.

verifySettings()

Verify that the settings of this unknown are correctly set.