7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Elucidating Mesoscopic Dynamics Through Simple Systems
P.H. Diamond1, F.R. Ramirez1 and Runlai Xu2
1
University of California San Diego, 2 Oxford University
email: pdiamond@ucsd.edu

We report on studies of mesoscopic dynamics in
simple systems. The aim is to avoid the geometricalcomplexity and hidden assumptions of conventionallarge-scale ‘simulations’, and instead to examine
dynamics in depth. We focus on two related, classicproblems — namely staircase structure and evolution,and turbulence spreading.
a)Staircase Resiliency
Layered profile structures, or staircases, have been
observed in many systems, including drift wave–zonal
flow turbulence. Here, we explore the resiliency andevolution of staircases. The simplest type of staircase
occurs in passive scalar convection in a vortex array, andis a consequence of the existence of two disparate timescales, characteristic of cellular over-turning anddiffusion. Since transport thru cell boundaries is purely
diffusive, steep concentration gradients form there,
yielding a global concentration profile staircase. The
mean field scalar transport coefficient for the system isthe geometric mean of cellular diffusion and collisional
diffusion [1]. This result has important implications forconfinement near criticality, and also is related to thewell-known Chalker–Coddington model [2] of theQuantum Hall Effect. Interestingly, no assumptions
concerning “feedback loops”, “shear suppression”, etc.
are a priori necessary. Of course, this model is quite
idealized, so we are motivated to explore staircase
resiliency to disorder in the presence of fluctuations,mixing, etc. Thus, the fixed cellular array is replaced by
a fluctuating vortex array, the model for which wasoriginally developed for a finite-temperature vortex
crystal [3]. A series of numerical experiments at modest
Reynolds number were performed. Results indicate:i) Scalar concentration mixing is inhomogeneous,and staircase profiles form.
ii) Staircases persist and are resilient over a broad
range of Reynolds number, though coarsening
due to step condensations occurs.
iii) Scalar concentration travels along streamlines
in regions of strong shear. Hence, inter-cellbarriers form first, while scalar concentration
homogenizes in vortices later. An image of thescalar concentration field thus resembles a web,which gradually fills in.
iv) Scattering by vortices reduces the speed of
scalar concentration front propagation. Thesepaths are those of least time.
v) If flow velocity and collisional diffusion areheld fixed, only cell geometric properties
ultimately determine the mean field scalardiffusivity. The latter does not deviatesignificantly from the value for the fixed
cellular array. This is a consequence of the factthat array fluctuations do not induce vortex
boundary crossings.
Extensions of this study to the case of an active
scalar (2D MHD) are under study and will be reported.
The implications of this basic study for confinement
physics will be discussed.
b) Turbulence Spreading
The spreading and propagation of turbulence arewidely discussed [4] in the context of fast transients,
non-local transport, etc. Amazingly, little or no detailed
basic studies of spreading and entrainment dynamics
have been pursued in the context of relevant models. We
examine the spreading of a turbulent patch resulting froma locally forced excitation in a 2D fluid with viscosityand drag. The forcing is a linear array of stirring, withvariable bandwidth. Results indicate that:i) The turbulent patch expands at roughly constantvelocity, so width ~ time.
ii) Finite drag limits expansion, thus defining afinite stopping length. This length is thesought-after “depth of penetration into thestable region”.
iii) Vortex dipoles form and contribute to spreading.
Recall that a vortex dipole moves at constantvelocity, and so is relevant to result (i).
iv) Spreading dynamics are sensitive to theproperties of the forcing — especially thebandwidth.
Ongoing work is concerned with extension tosystems with waves and zonal flows (i.e. Hasegawa–Mima) and to 2D MHD. 2D MHD is of particularinterest, so as to determine the spreading dynamics of(kinetic and magnetic) energy, mean square magnetic
potential, etc. The effects of a weak magnetic field
threading the system are examined.

References[1] M.N. Rosenbluth et al., Phys. Fluids 30, 2636 (1987).
[2] J.T. Chalker and P.D. Coddington, J. Physics C: Solid
State Physics 21, 2665 (1988).
[3] P. Perlekar and R. Pandit, New J. Phys. 12, 023033(2010).
[4] T.S. Hahm and P.H. Diamond, J. Kor. Phys. Soc. 73,747 (2018).