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.

The interaction of turbulent microscales in a plasma can conspire to produce transport on mesoscopic or macroscopic scales, often with extremely important consequences for the system. In this dissertation, we present research on mesoscopic turbulent transport in plasmas relevant to fusion energy and astrophysics.

The research is divided into three projects. First, we study drift-wave turbulence, a paradigm for gradient-driven turbulence in the tokamak, using a novel data-driven method. Deep learning is used to infer, from direct numerical simulation of the 2-D Hasegawa-Wakatani system, the dependencies of the turbulent fluxes which control the nonlinear dynamics on mesoscales, thus closing a simplified 1-D mean-field model. Using this approach, we show that the gradient of vorticity drives a nondiffusive particle flux. This nondiffusive flux, which we also recover analytically, modulates the profile in the presence of a quasiperiodic zonal flow. We also show that zonal flow formation is described by a Cahn-Hilliard-type model, both concurring with and expanding upon previous theoretical work.

The self-propagation of turbulence, called ``turbulence spreading,'' can lead to turbulence invading linearly stable regions of a tokamak. In the second project, we introduce a new model for turbulence spreading based on turbulence bistability. This model takes the form a reaction-diffusion equation with cubic nonlinearity and nonlinear diffusion. We find that the bistable model remedies certain deficiencies of the popular, unistable Fisher model, for instance predicting far more robust penetration of turbulence into stable regions. We also find that the model exhibits a threshold for propagation of an initial seed of turbulence, which we liken to an avalanche. We analytically estimate this threshold using a simple physical argument.

Finally, we study momentum transport in magnetohydrodynamic (MHD) turbulence on a $\beta$-plane, which serves as a simple model for the solar tachocline. We show that the cross-helicity, which is conserved in pure MHD turbulence, builds up to a predictable level in this system. Using weak turbulence theory, we also show that the cross helicity spectrum is equivalent to the Els\"asser alignment spectrum, which determines momentum transport. We supplement and verify our results with direct numerical simulations.

We develop a simple model for the generation and amplification of intrinsic axial flow in a linear device, Controlled Shear Decorrelation Experiment (CSDX). This model develops a novel dynamical symmetry breaking mechanism in drift wave turbulence, which does not require complex magnetic field structure, such as shear. Thus, the model is applicable to both tokamaks and linear devices. This mechanism is, essentially, a form of negative viscosity phenomenon.

Negative compressibility ITG turbulence can also induce a negative viscosity increment.

However, we show that no intrinsic axial flow can be generated by pure ITG turbulence in a straight magnetic field. When the flow gradient is steepened by any drive mechanism, the flow profile saturates at a level close to the value above which parallel shear flow instability (PSFI) becomes dominant over the ITG instability. This saturated flow gradient exceeds the PSFI linear threshold, and grows with $\nabla T_{i0}$ as $|\nabla V_\parallel| / |k_\parallel c_s| \sim|\nabla T_{i0}|^{2/3} / (k_\parallel T_{i0})^{2/3}$.

The coupling of azimuthal and axial flows in CSDX--in absence of magnetic shear--is investigated.

In particular, we focus on the apportionment of turbulence energy between azimuthal and axial flows, and how the azimuthal flow shear affects axial flow generation and saturation by drift wave turbulence.

Detailed measurements of intrinsic axial flow parallel to the magnetic field are performed on CSDX, with no axial momentum input.

The results present a direct demonstration that the broken spectral symmetry of drift wave turbulence causes the development of axial mean flows in cylindrical magnetized plasmas.

Measurements suggest the axial flow is parasitic to the drift wave--zonal flow system.

Besides, we show that consideration of wave--flow resonance resolves the long-standing problem of how zonal flows (ZFs) saturate in the limit of weak or zero frictional drag and also determines the ZF scale directly from analysis.

We show that resonant vorticity mixing, which conserves potential enstrophy, enables ZF saturation in the absence of drag, and so is effective at regulating the Dimits up-shift regime.

Vorticity mixing is incorporated as a nonlinear, self-regulation effect in an extended 0D predator--prey model of drift--ZF turbulence.

Tangled magnetic fields, often coexisting with an ordered mean field, have a major impact on turbulence and momentum transport in many plasmas, including those found in the solar tachocline and magnetic confinement devices. In this dissertation, we present research on the turbulent transport in presence of a stochastic magnetic field, and the discuss implications on the formation of astronomical objects and on the turbulence in edge plasma in fusion devices.

The research is divided into three projects. First, we present a novel mean field theory of potential vorticity mixing in β -plane magnetohydrodynamic (MHD). Our results show that mean square stochastic fields strongly reduce Reynolds stress coherence. This decoherence of potential vorticity flux due to stochastic field scattering leads to suppression of momentum transport and zonal flow formation. We discuss a model of stochastic fields as a resisto-elastic network.

In second project, we shows the breaking of the shear-eddy tilting feedback loop by stochastic fields is the key underlying physics mechanism. A simple calculation suggests that the breaking of the shear-eddy tilting feedback loop by stochastic fields is the key underlying physics mechanism. A dimensionless parameter that quantifies the increment in power threshold is identified and used to assess the impact of stochastic field on the L-H transition in fusion devices.

Finally, we study the turbulent transport of parallel momentum and ion heat by the interaction of stochastic magnetic fields and turbulence in third project. Attention is focused on determining the kinetic stress and the compressive energy flux. A critical parameter is identified as the ratio of the turbulent scattering rate to the rate of parallel acoustic dispersion. For the parameter large, the kinetic stress takes the form of a viscous stress. For the parameter small, the quasilinear residual stress is recovered. In practice, the viscous stress is the relevant form, and the quasilinear limit is not observable. This is the principal prediction of this project.