Skipp, Jonathan Matthew (2021) Turbulent cascades and condensation in nonlinear wave systems. PhD thesis, University of Warwick.

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Abstract

This thesis explores different aspects of large-scale structure formation by turbulent processes. We work in the context of wave turbulence (WT): the statistical description of large ensembles of weakly nonlinear waves. There are two relevant aspects of WT theory that we focus on, and each different aspect is exemplified by a different equation of motion that we treat in the WT description.

The first aspect we explore is the WT description of absolute thermodynamic equilibria, including the phenomenon of condensation. Condensates of dynamical invariants into the fundamental mode of the system constitute the large-scale structures we seek to characterise. We study the equilibria and condensates of the Charney-Hasegawa-Mima (CHM) equation. The CHM equation is unique in WT theory as it possesses three adiabatic invariants, in contrast the two invariants in related models such as the 2D Euler equation, and the Gross-Pitaevskii equation. We explore how the third, anisotropic, invariant of the CHM system enriches the description of condensation, and show how both zonal ows and isotropic large-scale (and indeed small-scale) vortex condensates arise out of the equilibrium WT description of the CHM equation. We also discuss the role of negative thermodynamic potentials (viz. negative temperatures) in describing these states.

The second aspect of large-scale structure formation by WT processes is the dual cascade (in systems with two invariants) of particles, or waveaction, to large scale, and energy to small scale. We study the dual cascade in the Schr�odinger- Helmholtz Equation (SHE) which is a new object of inquiry in WT theory. We examine the nature of the SHE dual cascade in two and three dimensions, and interpret the inverse cascade of particles as the nonequilibrium process of structure formation at large scales. We show that in both the fully local and fully nonlocal limits of the SHE, the inverse particle and direct energy fluxes are carried by small deviations from thermodynamic distributions, rather than the Kolmogorov- Zakharov cascades that are familiar in WT. We develop a differential approximation model to characterise such \warm cascade" states.

Finally we report initial results on direct numerical simulations of the SHE in two dimensions. These results are a numerical proof-of-concept demonstration that the general phenomenology of the dual cascade that we have characterised theoretically is qualitatively correct. We indicate the directions that a future research programme might take in fully characterising the SHE dual cascade numerically.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics Q Science > QC Physics T Technology > TA Engineering (General). Civil engineering (General)
Library of Congress Subject Headings (LCSH):	Turbulence, Nonlinear waves, Condensation, Cascades (Fluid dynamics)
Official Date:	September 2021
Dates:	Date Event September 2021 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Centre for Complexity Science
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Nazarenko, Sergey
Format of File:	pdf
Extent:	xv, 130 leaves : illustrations
Language:	eng

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