An active scalar system refers to a system with a scalar field that is coupled to the fluid dynamics and gives feedback to the velocity field through local forces. Active scalar turbulence systems are ubiquitous, and the study of these systems is a central focus of research in theoretical plasma physics. As examples, the 2D Cahn-Hilliard Navier-Stokes (CHNS) system and 2D Magnetohydrodynamics (MHD) system are studied in this dissertation.

The similarities and differences between 2D CHNS and 2D MHD are discussed. These are both elastic (i.e., self-restoring) systems, and display a memory, governed by freezing-in laws. The CHNS system supports an elastic wave, which is analogous to Alfven wave in MHD. Cascades and spectra in 2D CHNS are investigated, with focus on the interaction between inverse and forward cascades. The inverse cascade of mean square concentration $\langle\psi^2\rangle$, which is closely related to the real space dynamics of blob formation and merger, is found to be the dominant nonlinear transfer process. The spectrum of $\langle\psi^2\rangle_k$ exhibits a scaling law of $\sim k^{-7/3}$, and this exponent is the same as the corresponding one in 2D MHD. On the other hand, the kinetic energy spectrum follows $E_k\sim k^{-3}$. This exponent is closer to that for 2D Navier-Stokes, instead of that for 2D MHD. We suggest this is because the restoring force is significant only in the interfacial regions. The packing fraction of interfacial regions is small because of the formation and merger of blobs.

This suggests that the inverse cascade of $\langle \psi^2 \rangle$ - related to blob coalescence - modifies the forward cascade in 2D CHNS.

The evolution of the concentration field of the Cahn-Hilliard system in the background of a single eddy is studied. This is analogous to the flux expulsion phenomenon in 2D MHD. Though the system is simple, complex evolution is observed. 3 stages are observed: the ``jelly roll'' pattern stage, the stage of topological evolution, and the ``target'' pattern stage. The target pattern is metastable, as the bands gradually merge with time.

We also study turbulent transport in active scalar systems. We intended to first explore the classic problem of the suppression of turbulent transport in 2D MHD as an exercise in code verification, and then move to 2D CHNS. However, novel blob-and-barrier real space structures were observed with higher magnetic Reynolds number $\mathrm{Rm}$ in 2D MHD. We argue that the conventional approach of mean field theory is not applicable for the case without an external large scale magnetic field. The magnetic energy is observed to be concentrated in the intermittent, thin transport barrier regions, which located in the interstices between blobs of magnetic potential. The turbulent transport is quenched primarily because of these barriers. Barrier formation is linked to the inverse cascade of mean square magnetic potential $\langle A^2\rangle$ and negative turbulent resistivity. For small scale forcing, spontaneous formation of layering occurs.

More generally, we demonstrate that synergistic studies of related but different systems -- 2D CHNS and 2D MHD -- can lead to improved understanding. These studies can provide insights for all active scalar turbulence systems, since these systems share important common properties such as memory, elastic waves, and conservation laws.