A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
Figure 11. Inclusion of Ettingshausen and Nernst effects (simulations show that the self-similar
solution acts as an attractor) in a 1D time-dependent simulation [113, figure 4].
2.6. Nernst and Ettingshausen effects
In the previous section we ignored two cross-phenomena that arise from the velocity
dependence of the collision frequency (ν ∝ v −3 ). The radial component of the electron
heat flux q er and the axial component of Ohm’s law in a 1D model are
q er =−κ ⊥e
∂T e
∂r − T e
e β ∧J z , (2.59)
E z + v r B θ = η ⊥ J z + β ∧
e
∂T e
∂r , (2.60)
where in Braginskii’s notation [98] η ⊥ is α ⊥ /(ne) 2 . The Ettingshausen effect in equation (2.59)
is an additional heat flux in the direction of J × B while the Nernst term in equation (2.60)
describes the convection of the magnetic field by the hotter electrons associated with the heat
flux, having a common coefficient β ∧ through Onsager’s relations. In the limit of e τ ei ≫ 1
the ratio of the Ettingshausen term to the thermal conduction term is 0.32J z B θ /n e ∂T /∂r while
the ratio of the Nernst term to η ⊥ J is half the inverse of this. Therefore at least one of these
two effects will always be important. From [101] we note that equations (2.59) and (2.60) can
be combined to give
E z + v r B θ + q (
erB θ
5
2 n ≈ η ⊥ 1 − β2 ∧ T )
J
eT e e 2 z ≈ η ‖ J z , (2.61)
κ ⊥e η ⊥
where for Z = 1, the parallel resistivity is 0.506 times the perpendicular resistivity. This
clearly shows the advection of magnetic field by the heat flow with a velocity approximately
q er /(5/2n e T e ). The physical meaning of the Ettingshausen effect can best be envisaged as an
inward E z /B θ radial guiding-centre drift of the hotter, relatively collisionless electrons in the
applied axial electric field.
Chittenden and Haines [113] included these two terms in a 1D time-dependent two
temperature simulation. The main effect, shown in figures 11(a) and (b), is to remove the
sharp electron temperature rise and the associated skin current from the plasma edge, and to
cause a slight peaking of the electron temperature at r = 0, i.e. an inward heat-flow occurs.
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