Inverse and direct cascade of kinetic and potential energy for
  self-gravitating collisionless dark matter flow and effects of halo shape on
  energy cascade
  Halo-mediated mass and energy cascade are key to understand dark matter flow.
Both cascades origin from mass exchange between halo and out-of-halo
sub-systems. Kinetic energy can be from motion of halo and particle motion in
halo, while potential energy can be from inter- and intra-halo interactions.
Intra-halo equilibrium is established much faster than inter-halo. All energy
change of entire system comes from virilization in halos. At statistically
steady state, continuous mass exchange is required to sustain growth of total
halo mass $M_h$~$a^{1/2}$ and energy $E$~$a^{3/2}$, where $a$ is scale factor.
Inverse cascade is identified for kinetic energy that is injected at the
smallest scale and transferred to larger scales until consumed to grow halos.
This is sustained by direct cascade of potential energy from large to small
scale. Both energies have scale- and time-independent flux in propagation range
that is proportional to mass flux. Energy cascade is mostly facilitated by mass
cascade. It can be quantitatively described by typical halos growing with a
constant waiting time. Halo radial, rotational moment and angular velocity are
also modelled and inverse cascade is identified for coherent motion in halos.
In hydrodynamic turbulence, vortex stretching (shape changing) along its axis
of spin enables energy cascade from large to small length scale. However, halo
shape change does not contribute much to energy cascade with halo moment of
inertial gained from shape changing less than 2 times. Large halos exhibit
preference for prolateness over oblateness and most halos have spin axis
perpendicular to major axis. Since mass cascade is local in mass space, halo
shape also evolves continuously in mass space with large halo formed by
incrementally inheriting structure from progenitor halo. A unique evolution of
halo structure is found that gradually approaches sphere with increasing size.
