Setting Up Parallel Heat Conduction

The ZKpar parameters control the parallel heat-conduction coefficient in JOREK units. You can use a constant coefficient or the Spitzer–Härm-like temperature dependence $T^{5/2}$. In models with separate ion and electron temperatures, the two coefficients are configured independently.

This page describes the common conductivity_parallel implementation used by models 600 and 750.

One-temperature setup

For a model that evolves a single MHD temperature $T=T_i+T_e$, use ZK_par:

ZK_par             = 1.d0  ! Central parallel heat-conduction coefficient
ZKpar_T_dependent  = .true.
ZK_par_max         = 1.d20

Here, ZK_par is the coefficient at the equilibrium central temperature T_0. When ZKpar_T_dependent = .true., JOREK evaluates

\[\kappa_{\parallel}(T) = \kappa_{\parallel,0} \left(\frac{T_{\mathrm{corr}}}{T_0}\right)^{5/2}.\]

Separate ion and electron temperatures

When $T_i$ and $T_e$ are evolved separately, use ZK_i_par and ZK_e_par:

ZK_i_par            = 1.d0 ! Central ion coefficient
ZK_e_par            = 1.d0 ! Central electron coefficient
ZKpar_T_dependent   = .true.
ZK_par_max          = 1.d20

The common routine is applied independently to each species:

\[\begin{aligned} \kappa_{i,\parallel}(T_i) &= \kappa_{i,\parallel,0} \left(\frac{T_{i,\mathrm{corr}}}{T_{i0}}\right)^{5/2}, \\ \kappa_{e,\parallel}(T_e) &= \kappa_{e,\parallel,0} \left(\frac{T_{e,\mathrm{corr}}}{T_{e0}}\right)^{5/2}. \end{aligned}\]

ZK_par_max is shared by the ion and electron coefficients.

Choosing the central coefficient

JOREK calculates nominal Spitzer–Härm coefficients from the central plasma parameters and prints them in the log for reference. For separate-temperature models, look for

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

For a one-temperature model, look for

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

These are diagnostic reference values, not input parameters. JOREK does not automatically assign them to ZK_par, ZK_i_par, or ZK_e_par. To use the nominal Spitzer–Härm value, read it from a short initialization run and copy it to the corresponding input parameter.

Available options

Constant parallel conduction

Disable the temperature scaling with

ZKpar_T_dependent = .false.

The coefficient is then equal to ZK_par, or to ZK_i_par and ZK_e_par in a separate-temperature model. The negative-temperature override described below is evaluated independently and can still replace this value.

Maximum coefficient

When the $T^{5/2}$ scaling produces a value larger than ZK_par_max, the coefficient is clipped:

\[\kappa_{\parallel}=\kappa_{\parallel,max}.\]

Configure the limit with, for example,

ZK_par_max = 1.d20 ! Default

This limit is mainly a numerical safeguard against excessively large parallel conduction in hot regions. It is applied to the single-temperature, ion, and electron coefficients.

Minimum temperature used in the scaling

The $T^{5/2}$ expression uses the positive corrected temperature. See Correct Negative Densities / Temperatures for the temperature correction.

When ZKpar_T_dependent = .true., the decision to freeze the coefficient is based on the raw, uncorrected temperature. If $T_{raw}<T_{min,ZK\parallel}$, JOREK uses

\[\kappa_{\parallel} = \kappa_{\parallel,0} \left(\frac{T_{min,ZK\parallel}}{T_0}\right)^{5/2}.\]

The corresponding inputs are

T_min_ZKpar  = 1.d-6 ! Single-temperature model
Ti_min_ZKpar = 1.d-6 ! Ion temperature
Te_min_ZKpar = 1.d-6 ! Electron temperature

All values are in JOREK temperature units. Their preset sentinel value is -1.d12; during initialization, any value below -1.d10 is replaced by the general T_min setting. Therefore, if these parameters are not specified, their effective value is T_min.

Input parameters at a glance

Parameter	Purpose	Preset value
ZK_par	Central coefficient for a single temperature	1.d0
ZK_i_par	Central ion coefficient	1.d0
ZK_e_par	Central electron coefficient	1.d0
ZKpar_T_dependent	Apply the $T^{5/2}$ dependence	.true.
ZK_par_max	Maximum temperature-dependent coefficient	1.d20
T_min_ZKpar	Minimum single temperature used in the scaling	-1.d12, then replaced by T_min
Ti_min_ZKpar	Minimum ion temperature used in the scaling	-1.d12, then replaced by T_min
Te_min_ZKpar	Minimum electron temperature used in the scaling	-1.d12, then replaced by T_min