Discrete numerical approximations with additional conserved quantities
are developed here both for barotropic geophysical flows generalizing
the 2D incompressible fluid equations and suitable discretizations of
the Burgers-Hopf equation. Mathematical, numerical, and statistical
properties of these approximations are studied below in various different
contexts through the symbiotic interaction of mathematical theory and
scientific computing. The new contributions include an explicit concrete
discussion of the sine-bracket spectral truncation with many conserved
quantities for 2D incompressible flow, a theoretical and numerical
comparison with the standard spectral truncation, and a rigorous proof
of convergence to suitable weak solutions in the limit as the number of
Fourier modes increases. Systematic discretizations of the Burgers-Hopf
equation are developed, which conserve linear momentum and a non-linear
energy; careful numerical experiments regarding the statistical behavior
of these models indicate that they are ergodic and strongly mixing but
do not have equipartition of energy in Fourier space. Furthermore, the
probability distribution of the values at a single grid point can be
highly non-Gaussian with two strong isolated peaks in this distribution.
This contrasts with earlier results for statistical behavior of difference
schemes which conserve a quadratic energy. The issues of statistically
relevant conserved quantities are introduced through a new case study
for the Galerkin-truncated Burgers-Hopf model.