  We study the fundamental physics of cascades and spectra in 2D
Cahn-Hilliard-Navier-Stokes (CHNS) turbulence, and compare and contrast this
system with 2D MagnetoHydroDynamic (MHD) turbulence. The important similarities
include basic equations, ideal quadratic invariants, cascades and the role of
linear elastic waves. Surface tension induces elasticity, and the balance
between surface tension energy and turbulent kinetic energy determines a length
scale (Hinze scale) of the system. The Hinze scale may be thought of as the
scale of emergent critical balance between fluid straining and elastic
restoring forces. The scales between the Hinze scale and dissipation scale
constitute the elastic range of the 2D CHNS system. By direct numerical
simulation, we find that in the elastic range, the mean square concentration
spectrum $H^\psi_k$ of the 2D CHNS system exhibits the same power law ($-7/3$)
as the mean square magnetic potential spectrum $H^A_k$ in the inverse cascade
regime of 2D MHD. This power law is consistent with an inverse cascade of
$H^\psi$, which is observed. The kinetic energy spectrum of the 2D CHNS system
is $E^K_k\sim k^{-3}$ if forced at large scale, suggestive of the direct
enstrophy cascade power law of 2D Navier-Stokes (NS) turbulence. The difference
from the energy spectra of 2D MHD turbulence implies that the back reaction of
the concentration field to fluid motion is limited. We suggest this is because
the surface tension back reaction is significant only in the interfacial
regions. The interfacial regions fill only a small portion of the 2D CHNS
system, and their interface packing fraction is much smaller than that for 2D
MHD.
