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Why we study stochastic Magnetic Fields?
Why we study stochastic Magnetic Fields?
Momentum transport and the formation of zonal flows are central research areas in the study of fluid and plasma dynamics, especially in two-dimensional (2D) and compressible three-dimensional (3D) systems. A fundamental concept in these systems is the freezing law for fluid vorticity. Consequently, it is natural to describe such systems in terms of potential vorticity (PV). In this framework, the transport of momentum and the development of flow patterns are governed by the inhomogeneous mixing of PV, which is intricately linked to the cross-phase of vorticity flux.
Recently, there has been a growing interest in understanding the physics of PV transport in the presence of stochastic magnetic fields. This interest arises from the fact that even a modest mean magnetic field can lead to significant magnetic perturbations, as predicted by Zel'dovich's theorem and its high magnetic Reynolds number implications. Examples of such scenarios can be observed in the interstellar and intergalactic medium, the solar tachocline, and fusion devices' edge regions.
In practical applications, stochastic magnetic fields manifest in various ways. For instance, in the solar tachocline, they arise due to the 'pumping' of the convective zone. In fusion devices, resonant magnetic perturbations (RMPs) are imposed to control edge-localized modes, which are responsible for high transient heat loads and damage to the device's wall components. In both cases, a central question revolves around the phase dynamics: how do stochastic magnetic fields influence the coherence of fluctuating velocities, which in turn impacts the cross-phase of PV flux.
My research has delved into several interconnected topics, including the dynamics of momentum and heat transport, the emergence of zonal flows, and the impact of small-scale stochastic fields on larger-scale waves and flows. Within this context, several pivotal questions have arisen:
What role does turbulence play in the redistribution of momentum and heat in the context of β-plane magnetohydrodynamics (MHD) turbulence?
What factors influence the cross-phase during the transport of mean potential vorticity (PV), thereby affecting the growth of zonal flows? Which key dimensionless parameters dictate the quenching effect?
How is the symmetry disrupted by zonal shear and its consequences for the cross-phase of stress, especially in the presence of stochastic magnetic fields?
I have addressed these inquiries in a series of research papers, which primarily focus on understanding how stochastic magnetic fields impact the turbulent transport of momentum and heat in both stratified, incompressible 2D MHD (Chen & Dimond, 2020) and compressible 3D MHD turbulence (Chen et al., 2021).
Random Fields in beta-plane Solar tachocline
The turbulent transport of momentum plays a pivotal role in the formation of the Solar tachocline, yet its underlying nature remains elusive. Proposed mechanisms for suppressing the 'burrowing' of the tachocline include latitudinal viscous diffusion and potential vorticity (PV) mixing. These proposals are both problematic, however, for the viscous or mixing mechanism due to meridional cells driven by baroclinic torque seems to be ineffective.
In this study, I investigated the effects of stochastic magnetic fields on momentum transport in β-plane turbulence at high magnetic Reynolds numbers and low Prandtl numbers. The simulation results revealed that Reynolds stress and momentum transport are inhibited by stochastic fields when the mean field intensity is lower than that required for complete Alfvénization of the system. By formulating an effective 'mean-field theory' applicable to this regime of high magnetic perturbations, I derived an explicit expression for the suppression of PV flux due to stochastic magnetic fields, consequently explaining the reduction in Reynolds stress as per the Taylor Identity.
This result explained that the Reynolds stress (or flow generation) suppression is due to disordered magnetic fields for a relative weak mean field intensity. I also found that the dispersion relation can be rewritten with a form that contains the conventional dissipation and a drag from stochastic fields. These 'dissipation' and 'drag' effects imply that stochastic fields create an effective resisto-elastic network in which Alfvén wave dynamics evolve. This enhances the memory of stochastic-field-induced elasticity and drag, ultimately leading to the suppression of turbulent momentum transport in the tachocline. A critical dimensionless parameter pinpointing at which intensity of mean field the growth of zonal flow ceases is also obtained.
Our work: Chang-Chun Chen & Patrick Diamond (2020), "Potential Vorticity Mixing in a Tangled Magnetic Field", The Astrophysical Journal, 892(1), 24. DOI:10.3847/1538-4357/ab774f
L-H transition at the Edge of Tokamaks with Resonant Magnetic Perturbations
Further, we asked how stochastic fields affect the 3D turbulence? Experiments showed that the Reynolds stress burst drops when RMPs are applied at the edge of DIII-D in pre-L-H transition stage, and that a increment in L-H power thresholds in fusion devices. In my second paper, we shedded light on these two phenomena and addressed the more general question of Reynolds stress decoherence in stochastic fields. This decoherence requires stochastic broadening larger than the natural turbulence linewidth. By considering the effect of a stochastic magnetic field on the “shear-eddy tilting feedback loop” in presence of stochastic field, I found that stochastic fields can break this feedback loop, since the phase correlation in Reynolds stress is no longer set by flow shear merely. This explains the strong suppression of poloidal Reynolds stress at the edge of tokamak experiments with RMPs in DIII-D . Finally, the increment in L-H power thresholds was calculated—the stochastic magnetic fields raises the power thresholds linearly in proportional to the mean-square of normalized stochastic field intensity and one over the normalized gyro radius.
Our work: Chang-Chun Chen, Patrick H. Diamond, Rameswar Singh, and Steven M. Tobias (2021), "Potential vorticity transport in weakly and strongly magnetized plasmas" , Physics of Plasmas 28, 042301 . DOI: 10.1063/5.0041072.
Ion Heat and Parallel Momentum Transport by Stochastic Magnetic Fields and Turbulence
We explore relatively untouched issues—the interaction of stochastic magnetic field and turbulence, and how they together drive transport in compressible MHD turbulence. I studied how the transport of ion heat and parallel momentum via the pressure and parallel flow responses to stochastic fields in strong/weak turbulence regimes. I found in strong turbulent regime (i.e. turbulent viscosity dissipation rate larger than other rates), the turbulent viscosity will dissipate the parallel flow perturbation, in response to the pressure excess. I explicitly derived that the transport mechanism is dominated by the hybrid turbulent diffusivity, which contains a stochastic magnetic scattering term and a fluid turbulent scattering term. While in weak turbulent regime, pressure gradient builds up along the mean field line in response of the pressure slug. The momentum and energy transport occur only through magnetic fields in this regime, with familiar transport coefficient that contains the sound speed and the magnetic diffusivity.
Our work: "Ion Heat and Parallel Momentum Transport by Stochastic Magnetic Fields and Turbulence", Plasma Physics and Controlled Fusion 64, 015006. DOI: 10.1088/1361-6587/ac38b2
Ambipolar Diffusion on β-plane Partially Ionized MHD
A project currently in preparation is to study the stochastic field effect on a β-plane partially ionized MHD (PIMHD) system. Jupiter-like gas giant planet atmospheres are characterized by two regions—the outer neutral envelop and the fully ionized interior. This transition from neutral to the fully ionized regime is happen continuously as a function of radius. In this transition region, physics is depicted by PIMHD which contains neutral and charged flows interaction. Compared with studies in β-plane MHD, this two-fluid interaction yields a nonlinear ambipolar diffusion in the induction equation and a Lorentz drag in the equation of motion for neutral particles. I will work on the theory of momentum transport in presence of modest stochastic magnetic field.
Shear layer and staircase formation in a stochastic magnetic field
Another project in preparation is the shear layer and staircase formation in a stochastic magnetic field. This staircase is analogous to the “salinity staircases” observed in oceanography. Staircase-like structure of particle density and pressure (or density/pressure corrugation) will forms a “barrier” that prevents momentum transport and hence quenches the turbulence at the edge of fusion devices. I will study in the spatial scale of the layer and how resilient is this barrier with a prescribed stochastic field. This can be start from a turbulence mixing length that involves two scales—the forcing scale and Rhines scale. Preliminary calculation suggests a large magnetic Kubo number is required for significant change in mean turbulence coupling. For cases where magnetic Kubo number is small, this results indicates the staircase is resilient.
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