Page 15 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-15 Egedal et al.
DE 2p Ð v2Pl0 ð@f0s=@tÞJ v;ndv dn
Phys. Plasmas 25, 072510 (2018)
FIG. 17. Schematic illustration of a MP drive circuit. Oppositely oriented coils should be explored to generate plasma perturbations with optimal drive efficiency.
x=ð2pÞ 1⁄4 1=ð10sb0Þ ’ 14 kHz. The volume of the plasma is approximately V ’ 7 m3. Furthermore, the assumed mag- netic field of B0 1⁄4 1 T is consistent with the confinement of a plasma at n’1020 m􏰂3 and Ti ’30 keV (so vth ’5:5). The total ion kinetic energy in the configuration is then about nVTi 1⁄4 3:5 MJ. Again, for the scenario of heating through 1% parallel compressions, from Fig. 16 with vth ’ 5:5 we obtain the estimate ðDE=EÞ ’ 2:5 􏰃 10􏰂5, such that the total heating power will be of the order Wp 1⁄4 ðDE=EÞðx= 2pÞnVTi ’ 1:2 MW. Thus, during about 3.5 MJ/1.2 MW or 3 s, the entire ion energy will be replaced, comparable to the expected FRC reactor energy confinement time.
We believe that the above example for modest 1% com- pression ratio illustrates the good potential for heating FRC plasmas using MP. In our model, the optimal pumping fre- quency decreases slowly as a function of the plasma temper- ature. This is opposite to transit-time heating where the optimal frequency is given by fTP 1⁄4 0:17vth=a, where a is the size of the applied localized magnetic perturbation.7 For example, at the above configuration with Ti ’ 30 keV and using a 1⁄4 zs =2, we obtain for transit-time pumping fTP ’ 209 kHz, much larger than the 14 kHz of our model.
Compared to transit-time heating, a main advantage of the lower optimal pump frequency is the ability for antenna systems to launch large scale magnetic perturbations that penetrate the plasma. Practical implementations will benefit from simultaneous radial and axial compression with a coil arrangement sketched in Fig. 17. By running the coils p out of phase in a resonant LC circuit, we can have the FRC change shape nearly adiabatically and at near constant inter- nal energy. However, the pitch angle mixing inside the FRC, as described by our model in Sec. III C, is non-adiabatic and leads to energy absorption from the circuit. Using such opti- mized magnetic perturbations will significantly reduce the excess circulating power compared to that required for the separate radial and axial compressions considered above.
V. CONCLUSIONS
FRC plasmas include non-standard fast ion orbits whose behavior is not capÞtured by standard guiding center theory. Applying the Jq 1⁄4 vqdq action variable, we here develop a generalized drift kinetic approach applicable for describing the ion behavior during periodic and adiabatic modification of the FRC equilibrium. The FRC includes distinct orbit topologies that can be characterized as regular cyclotron motion and figure-eight type orbits in area of weak magnetic fields. A class of trajectories includes transitions between
E 1⁄4 x Ð v2f0sJv;ndvdn ; (37)
where it should be noted that Pl0ðnÞ 1⁄4 0 for n < nc and
Pl0 ðnÞ 1⁄4 1 for n > nc. From Eq. (37), it is clear that the heat-
ing rate depends on the instantaneous temperature of f0s,
which we here characterize through the thermal speed,
pffiffiffiffiffiffiffiffiffiffiffi
vth 1⁄4 2T=m for ions with temperature T and mass m. In
Fig. 16, we use f0s / exp ð􏰂v2 =v2th Þ and show DE=E as a
function of vth. Consistent with Fig. 15, the highest efficiency
is observed for x=xb0 ’ 0:1 yielding a relative energy
increase, DE=E ’ 10􏰂4 for vth ’ 1 and declining to DE=E
’ 2:5 􏰃 10􏰂5 for vth ’ 5. For convenience, we also cast the
results for DE=E as a function of the FRC s-parameter.14
Using the definition s 􏰄 Ð qs qdq=ðq q Þ, where q is the q0 si 0
radius of the magnetic null, we find for the considered Solov􏰌ev equilibrium s 1⁄4 qB0qs=ð8mvthÞ 1⁄4 87=ð8vthÞ.
For fixed pumping conditions, it may appear counterin-
tuitive that the relative heating efficiency decreases with
increasing vth. For example, because more energetic ions
have shorter bounce times, it could be expected that pitch-
angle-mixing is stronger and that the relative heating effi-
ciency will increase at larger energies. However, because the
orbit changes with velocity, the average bounce time only
pffiffi
decreases as hsbin 1⁄4 T0;0 / 1= v. In addition, the pitch-
angle-diffusion per orbit bounce scales as D/B / 1=v0:8
yielding an effective mixing frequency of 􏰎eff / 1=v0:3. We
also note that constant DE=E would require the velocity dif-
fusion coefficient in Eq. (36) to scales as Dv / v2, while we
observe Dv / v0:7 for x2 􏰅 􏰎2 . Meanwhile, because Dv eff
does increase with v, the absolute heating by pumping E 􏰀 ðDE=EÞ is, in fact, increasing (nearly linearly) as a function of vth. This is in contrast to collisional pumping, where the pitch-angle scattering is provided by ion-ion collisions, 􏰎ii / 1=v3, and the absolute heating rates decline rapidly with increasing vth.
We can apply the above heating rates for collisionless FRC pumping to estimate the overall heating power that can be expected for the configuration considered in the present study. With the normalization introduced in Sec. IIA, we find that the optimal pumping frequency is
FIG. 16. Relative energy increase DE=E per pump cycle as a function of vth, calculated based on Eq. (36) using eight separate values of x=xb0. Here, vth is the thermal speed in the applied Maxwellian form of the background dis- tribution, f0s / e􏰂v2 =v2th . More generally, the results can be cast in terms of the s-parameter (for the considered equilibrium, s 1⁄4 87=ð8vthÞ.