Intense non-neutral beam propagation in a periodic solenoidal field using a macroscopic fluid model with zero thermal emittance

A macroscopic fluid model is developed to describe the nonlinear dynamics and collective processes in an intense high-current beam propagating in the z-direction through a periodic focusing solenoidal field $B_z{\mathrm{(z+S)=B}}_z\mathrm{(z)}$ , where $S$ is the axial periodicity length. The analysis assumes that space-charge effects dominate the effects of thermal beam emittance, ${\mathrm{Kr}}_b^2{\mathrm{\gg \epsilon }}_{\mathrm{th}}^2$ , and is based on the macroscopic moment-Maxwell equations, truncated by neglecting the pressure tensor and higher-order moments. Here, ${\text{K=2N}}_b{Z}_i^2{e}^2{\mathrm{/\gamma ̂}}_b^3{\mathrm{m\beta }}_b^2{c}^2$ is the self-field perveance, $N_b$ is the number of particles per unit axial length, and $r_b$ is the characteristic beam radius. Assuming a thin beam with $r_b\mathrm{\ll S}$ , azimuthally symmetric beam equilibria with $\text{∂/∂t=0=∂/∂θ}$ are investigated, allowing for an axial modulation of the beam density $n_b\mathrm{(r,z)}$ and macroscopic flow velocity $V_{\mathrm{rb}}\mathrm{(r,z)}{{ê}}_r{\mathrm{+V}}_{\mathrm{\theta b}}\mathrm{(r,z)}{{ê}}_{\theta }{\mathrm{+V}}_{\mathrm{zb}}\mathrm{(r,z)}{{ê}}_z$ by the periodic focusing field. To illustrate the considerable flexibility of the macroscopic formalism, assuming (nearly) uniform axial flow velocity $V_b$ over the beam cross section, beam equilibrium properties are calculated for two examples: (a) uniform radial density profile over the interval ${\text{0⩽r<r}}_b\mathrm{(z)}$ , and (b) an infinitesimally thin annular beam centered at ${\mathrm{r=r}}_b\mathrm{(z)}$ . The analysis generally allows for the azimuthal flow velocity $V_{\mathrm{\theta b}}\mathrm{(r,z)}$ to differ from the Larmor frequency, and the model is used to calculate the (leading-order) correction ${\mathrm{\delta V}}_{\mathrm{zb}}\mathrm{(r,z)}$ to the axial flow velocity for the step-function density profile in case (a) above.