Bohm term

The mixed model is derived using the dimensional analysis approach, whereby the diffusivity in a Tokamak plasma can be written as:

$$\chi = \chi_0 F(x_1, x_2, x_3, ...)$$

where $\chi_0$ is some basic transport coefficient and F is a function of the plasma dimensionless parameters $(x_1,\ x_2,\ x_3,\ ...)$. We choose for $\chi_0$ the Bohm diffusivity:

$$\chi_0 = \frac{cT_e}{eB}$$

The expression of the dimensionless function F is chosen according to the following criteria:

• The diffusivity must be bowl-shaped, increasing towards the plasma boundary
• The functional dependencies of F must be in agreement with scaling relationships of the global confinement time, reflecting trends such as power degradation and linear dependence on plasma current
• The diffusivity must provide the right degree of resilience of the temperature profile

It easily shown that a very simple expression of F that satisfies the above requirements is:

$$F = q^2/\vert L_{pe}^*\vert$$

where q is the safety factor and $L_{pe}^*=p_e(dp_e/dr)^{-1}/a$, being a the plasma minor radius. The resulting expression of the diffusivity can be written as:

$$\chi \propto \vert v_d\vert \Delta G$$

where $v_d$ is the plasma diamagnetic velocity, $\Delta=a$ and $G=q^2$, so that it is clear that this model represents transport due to long-wavelength turbulence.
The evidence coming up from the simulation of non-stationary JET experiments [4](such as ELMs, cold pulses, sawteeth, etc.) suggested that the above Bohm term should depend non-locally on the plasma edge conditions through the temperature gradient averaged over a region near the edge:

$$<L_{T_e}^*>_{\Delta V}^{-1} = \frac{T_e(x=0.8) - T_e(x=1)}{T_e(x=1)}$$

where x is the normalized toroidal flux coordinate. The final expression of the Bohm-like model is:

$$\chi_{e,i}^B = \alpha_{Be,i} \frac{cT_e}{eB} L_{pe}^{*-1} q^2 <L_{T_e}^*>_{\Delta V}^{-1}$$

where $\alpha_{e,i}^B$ is a parameter to be determined empirically, both for ions and electrons.