Computer simulations are employed to examine the transfer of transvers
e thermal energy to the longitudinal direction via a collective electr
ostatic instability which is driven by an initial temperature anisotro
py. The nature of the instability is illustrated using electrostatic p
article-in-cell simulations in an axisymmetric geometry, but with thre
e components of the velocity. Results are presented for a long coastin
g space-charge-dominated beam, propagating in a uniform focused channe
l to illustrate the development of a coherent mode structure with an a
xial wavelength comparable with the beam diameter, and growth times co
mparable with the betatron period.