View source for Heat flux width

The [[Scrape-Off Layer]] (SOL) heat flux width <math>\lambda_q</math> is the length scale of the decaying exponential heat flux profile on the open flux surfaces.
Here, there is a competition between heat transport parallel to the field, which conducts heat to the divertors, and perpendicular diffusion.
Since parallel conduction is much faster than perpendicular diffusion, heat flux widths (<math>\lambda_q</math>) are fairly narrow&mdash;usually a few mm to a cm.
<math>\lambda_q</math> is assumed to be set at or near the outboard midplane,<ref name="eich_2013">[[doi:10.1016/j.jnucmat.2013.01.011|T. Eich, et al., J. Nucl. Mater. '''438''' (2013) S72-S77]]</ref> which is where the dominant heat source from the core into the SOL is located.
So when <math>\lambda_q</math> is quoted, it should be understood that the value at the outboard midplane is given unless otherwise stated.

As heat flows through the SOL to the divertor, the profile is broadened by [[magnetic flux expansion]].
After passing the [[Divertor|X-point]], perpendicular diffusion can go in both directions; outboard and deeper into the SOL as before, and inward to the [[Divertor|private flux region]] (PFR).<ref name=eich_2011>[[doi:10.1103/PhysRevLett.107.215001|T. Eich, et al., Phys. Rev. Lett. '''107''' (2011) 215001]]</ref>
This has the effect of smoothing the profile.

The heat flux <math>q</math> profile at the divertor (neglecting dissipation in the SOL, such as in the case of detachment) is then<ref name="eich_2013" />
:<math>q(\bar{s})=\frac{q_0}{2} \exp\left(\left(\frac{S}{2\lambda_q}\right)^2-\frac{\bar{s}}{\lambda_q f_x}\right) \cdot \mathrm{erfc}\left(\frac{S}{2\lambda_q}-\frac{\bar{s}}{S f_x}\right) +q_{BG}</math>
where <math>\bar{s}=s-s_0</math>, <math>s</math> is distance along the divertor plate, <math>s_0</math> is the [[magnetic strike point]] position, <math>S</math> is the width of the Gaussian blur effect that is convoluted with the exponential profile, <math>f_x</math> is the flux expansion (distance between flux surfaces at the divertor / distance between the same surfaces at the midplane), and <math>q_{BG}</math> is a background heat flux (which could come from radiated heat, for example).
An equation for the heat flux at the divertor is useful because the divertor heat flux profile can be measured by infrared thermography, Langmuir probes, or surface thermocouples.

This functional form was fit to heat flux profiles from several devices, and the resulting <math>\lambda_q</math> values were regressed versus several important parameters, such as field, power, safety factor, and device size.<ref name=eich_2013_nf>[[doi:10.1088/0029-5515/53/9/093031|T. Eich, et al., Nucl. Fusion '''53''' (2013) 093031]]</ref>
The regression analysis had several variants with different subsets of available devices and different parameters.
An example is
:<math>\lambda_q = C B_T^{\alpha_B} q_{cyl}^{\alpha_q} P_{SOL}^{\alpha_P} R_{geo}^{\alpha_R}</math>
with a fit to [[:Wikipedia:Joint European Torus|JET]], [[:Wikipedia:DIII-D (fusion reactor)|DIII-D]], and [[:Wikipedia:ASDEX Upgrade|ASDEX Upgrade]] resulting in 

<math>C=0.86\pm 0.25</math> mm, <math>\alpha_B=-0.80\pm 0.21</math>, <math>\alpha_q=1.11\pm 0.15</math>, <math>\alpha_P=0.11 \pm 0.09</math>, <math>\alpha_R=-0.13 \pm 0.16</math>,

where <math>B_T</math> is the toroidal magnetic field, <math>q_cyl</math> is the cylindrical safety factor, <math>P_{SOL}</math> is the power flowing into the SOL, and <math>R_{geo}</math> is the geometric major radius of the plasma.

There have been other regression fits by different researchers using different subsets of devices and different parameters.<ref>[[doi:10.1016/j.jnucmat.2013.01.028|M.A. Makowski, et al., J. Nucl. Mater. '''438''' (2013) S208-S211]]</ref><ref>[[doi:10.1088/1741-4326/ab472c|H. Niemann et al 2020 Nucl. Fusion '''60''' (2020) 016014]]</ref>

== References ==
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