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By decomposing velocity dispersion into non-spin and spin-induced, mean flowand dispersion are analytically solved for axisymmetric rotating and growinghalos. The polar flow can be neglected and azimuthal flow is directly relatedto dispersion. The fictitious ("Reynolds") stress acts on mean flow to enableenergy transfer from mean flow to random motion and maximize system entropy.For large halos (high peak height $\nu$ at early stage of halo life) withconstant concentration, there exists a self-similar radial flow (outward incore and inward in outer region). Halo mass, size and specific angular momentumincrease linearly with time via fast mass accretion. Halo core spins fasterthan outer region. Large halos rotate with an angular velocity proportional toHubble parameter and spin-induced dispersion is dominant. All specific energies(radial/rotational/kinetic/potential) are time-invariant. Both halo spin($\sim$0.031) and anisotropic parameters can be analytically derived. For"small" halos with stable core and slow mass accretion (low peak height $\nu$at late stage of halo life), radial flow vanishes. Small halos rotate withconstant angular velocity and non-spin axial dispersion is dominant. Smallhalos are spherical in shape, incompressible, and isotropic. Radial andazimuthal dispersion are comparable and greater than polar dispersion. Due tofinite spin, kinetic energy is not equipartitioned with the greatest energyalong azimuthal direction. Different from normal matter, small halos are hotterwith faster spin. Halo relaxation from early to late stage involves variationof shape, density, mean flow, momentum, and energy. During relaxation, haloisotopically "stretches" with conserved specific rotational kinetic energy,increasing concentration and momentum of inertial. Halo "stretching" leads todecreasing angular velocity, increasing angular momentum and spin parameter.

The mean flow, velocity dispersion, energy transfer and evolution of rotating and growing dark matter halos