  In this paper we perform a numerical study of the tranverse expansion of
hadronic scattering amplitudes in the dipole picture of high energy QCD. We go
beyond the mean field approximation by including fluctuations and also wave
function saturation effects, and the evolution with both a fixed and a running
coupling is investigated. We also study the nonperturbative aspects, and as has
been predicted earlier, our results indicate that the Froissart-Martin bound is
saturated once confinement effects are included in the evolution. Thus the
total cross section increases proportional to the square of the logarithm of
the cms energy. Furthermore it is seen that we obtain a reasonable value for
the proportionality coefficient. The impact of saturation and non-leading
effects on this coefficient is also studied.
