  We study the transfer of angular momentum in high energy nuclear collisions
from the colliding nuclei to the region around midrapidity, using the classical
approximation of the Color Glass Condensate (CGC) picture. We find that the
angular momentum shortly after the collision (up to times ~ 1/Q_s, where Q_s is
the saturation scale) is carried by the "beta-type" flow of the initial
classical gluon field, introduced by some of us earlier. beta^i ~ mu_1 nabla^i
mu_2 - mu_2 nabla^i mu_1 (i=1,2) describes the rapidity-odd transverse energy
flow and emerges from Gauss' Law for gluon fields. Here mu_1 and mu_2 are the
averaged color charge fluctuation densities in the two nuclei, respectively.
Interestingly, strong coupling calculations using AdS/CFT techniques also find
an energy flow term featuring this particular combination of nuclear densities.
In classical CGC the order of magnitude of the initial angular momentum per
rapidity in the reaction plane, at a time 1/Q_s, is |dL_2/d eta| ~ R_A/Q_s^3
epsilon_0/2 at midrapidity, where R_A is the nuclear radius, and epsilon_0 is
the average initial energy density. This result emerges as a cancellation
between a vortex of energy flow in the reaction plane aligned with the total
angular momentum, and energy shear flow opposed to it. We discuss in detail the
process of matching classical Yang-Mills results to fluid dynamics. We will
argue that dissipative corrections should not be discarded to ensure that
macroscopic conservation laws, e.g. for angular momentum, hold. Viscous fluid
dynamics tends to dissipate the shear flow contribution that carries angular
momentum in boost-invariant fluid systems. This leads to small residual angular
momentum around midrapidity at late times for collisions at high energies.
