Setting Up the Resistivity

The input parameter eta sets the resistivity at the equilibrium plasma centre ($\psi_n=0$), in JOREK units. The resistivity used elsewhere in the plasma can be constant or can follow a Spitzer-like temperature dependence. Variations of the Coulomb logarithm and, in impurity models, $Z_{eff}$ can also be included.

Simple Spitzer-like setup

A typical input setup is

eta                  = 1.d-8 ! Central resistivity in JOREK units
eta_T_dependent      = .true.
eta_coul_log_dep     = .true.
T_max_eta            = 1.d99

Here, eta is the value at the equilibrium central electron temperature $T_{e0}$. When eta_T_dependent = .true., JOREK evaluates

\[\eta_T = \eta\left(\frac{T_{e,\mathrm{corr}}}{T_{e0}}\right)^{-3/2}.\]

For a model with separate temperatures, the evolved $T_e$ is used. For a one-temperature model, JOREK uses $T_e=T/2$; the factor of two cancels in the ratio above.

If eta_coul_log_dep = .true., the result is additionally multiplied by the local Coulomb logarithm normalized to its central value:

\[\eta_T \longrightarrow \eta_T\frac{\ln\Lambda}{\ln\Lambda_0}.\]

Thus, eta remains the central value even when the Coulomb-logarithm dependence is enabled. This option is currently available in models that call the common resistivity function, including models 600 and 750.

Choosing eta

The physical Spitzer resistivity used for reference is

\[\eta_{\mathrm{Spitzer}}^{SI} = 1.65\times10^{-9}\,\ln\Lambda_0 \left(T_{e0}[\mathrm{keV}]\right)^{-3/2}\;\Omega\,\mathrm{m}.\]

you will need to convert it to JOREK units using the normalization. For convenience, in the logfile you will find

eta_Spitzer (not input parameter; printed for reference in JOREK units)

eta_Spitzer is a diagnostic reference, not an input parameter: JOREK does not automatically assign it to eta. To use the nominal Spitzer value, read the printed value from a short initialization run and set eta to that value in the input file.

You can deliberately choose a different eta to change the resistive timescale, provided that the consequences for the Lundquist number and Ohmic heating are understood.

Available options

Constant resistivity

For a resistivity that is constant in both temperature and density, disable both dependencies:

eta                  = 1.d-6
eta_T_dependent      = .false.
eta_coul_log_dep     = .false.

Disabling only eta_T_dependent is not sufficient for a strictly constant profile when eta_coul_log_dep remains enabled, because local variations of $\ln\Lambda$ still modify eta.

Temperature-dependent resistivity

Enable the Spitzer-like $T_e^{-3/2}$ scaling with

eta_T_dependent = .true.

The function uses the positive corrected temperature $T_{e,\mathrm{corr}}$. The correction prevents a non-integer power from being evaluated with a negative temperature; see Correct Negative Densities / Temperatures.

When the raw temperature falls below T_min, the resistivity is frozen at

\[\eta_T = \eta\left(\frac{T_{min}}{T_{e0}}\right)^{-3/2}.\]

This avoids an unbounded resistivity in very cold or numerically negative-temperature regions.

Upper temperature cutoff

T_max_eta limits the temperature used in the Spitzer scaling:

T_max_eta = 1.d99 ! Effectively no upper cutoff (default)

For $T_{e,\mathrm{corr}}>T_{max,\eta}$, JOREK uses

\[\eta_T = \eta\left(\frac{T_{max,\eta}}{T_{e0}}\right)^{-3/2}.\]

This imposes a minimum resistivity in the hottest regions. T_max_eta is expressed in the same normalized temperature units as the evolved temperature and should be changed only for a specific numerical reason.

Coulomb-logarithm dependence

Enable local density and temperature effects in $\ln\Lambda$ with

eta_coul_log_dep = .true.

The common plasma-function implementation calculates $n_e$ in $\mathrm{cm}^{-3}$ and the corrected $T_e$ in eV. It uses

\[\ln\Lambda = \begin{cases} 23.0 - \frac{1}{2}\ln n_e + \frac{3}{2}\ln T_e, & T_e < 10\;\mathrm{eV}, \\ 24.1513 - \frac{1}{2}\ln n_e + \ln T_e, & T_e \ge 10\;\mathrm{eV}. \end{cases}\]

The density passed to this expression is bounded from below at $10^{10}\;\mathrm{cm}^{-3}$ to prevent nonphysical values of the logarithm.

Impurity correction

When the selected model evolves impurities, the temperature-dependent resistivity is automatically multiplied by a $Z_{eff}$ correction. The factor implemented in the common plasma function is

\[f(Z_{eff}) = \frac{ Z_{eff}\left(1+1.198Z_{eff}+0.222Z_{eff}^2\right) }{ 1+2.966Z_{eff}+0.753Z_{eff}^2 } \left[ \frac{1+1.198+0.222}{1+2.966+0.753} \right]^{-1},\]

which is normalized so that $f(1)=1$. This factor is applied by the current implementation when eta_T_dependent = .true..

Input parameters at a glance

Parameter	Purpose	Default
eta	Central resistivity in JOREK units	1.d-5
eta_T_dependent	Apply the $T_e^{-3/2}$ dependence	.true.
eta_coul_log_dep	Multiply by $\ln\Lambda/\ln\Lambda_0$ where supported	.true.
T_max_eta	Upper temperature used to evaluate eta	1.d99

Other considerations

• In RMHD there is formally no poloidal resistivity
• Neoclassical effects may increase the resisitivity via trapped particles

Trapped electrons do not carry a current, thus neoclassical reduction leads to an additional radial dependency: \(\eta_{||,neo}=\eta_{||}\;\left(1-\sqrt{r/R_0}\right)^{-2}\) For $r/R=0.3$ the correction factor is $5$, but this also depends on the collisionality and de-trapping time scale.