An adiabatic equilibrium theory is presented for an intense, axisymmetric charged-particle beam propagating through a periodic solenoidal focusing field. The thermal beam distribution function is constructed based on the approximate and exact invariants of motion, i.e., a scaled transverse Hamiltonian and the angular momentum. The approximation of the scaled transverse Hamiltonian as an invariant of motion is validated analytically for highly emittance-dominated beams and highly space-charge-dominated beams, and numerically tested to be valid for cases in between with moderate vacuum phase advances ͑ v Ͻ 90°͒. The beam root-mean-square ͑rms͒ envelope equation is derived, and the self-consistent nonuniform density profile is determined. Other statistical properties such as flow velocity, temperature, total emittance and rms thermal emittance, equation of state, and Debye length are discussed. Numerical examples are presented, illustrating the effects of the beam perveance, emittance, and rotation on the beam envelope and density distribution. Good agreement is found between theory and a recent high-intensity beam experiment performed at the University of Maryland Electron Ring ͓S. Bernal, B. Quinn, M. Reiser, and P. G. O'Shea, Phys. Rev. ST Accel. Beams 5, 064202 ͑2002͔͒.

A warm-fluid equilibrium theory is presented which describes a new thermal equilibrium of a rotating charged-particle beam in a periodic solenoidal focusing field. Warm-fluid equations are solved in the paraxial approximation. The rms beam envelope, the density and flow velocity profiles, and the self-consistent Poisson equations are derived. Density profiles are calculated numerically for high-intensity and low-intensity beams. Temperature effects in such beams are investigated. Radial confinement of the beam is discussed.