Interactions of disparate scales in drift-wave turbulence

Abstract

Renormalized statistical theory is used to calculate the interactions between short scales (wave vector k) and long scales (wave vector q<<k) in the Hasegawa-Mima model of drift-wave turbulence (generalized to include proper nonadiabatic response for k( ||)=0 fluctuations). The calculations include the zonal-flow growth rate as a special case, but also describe long-wavelength fluctuations with q oriented at an arbitrary angle to the background gradient. The results are fully renormalized. They are subtly different from those of previous authors, in both mathematical form and physical interpretation. A term arising in previous treatments that is related to the propagation of short-scale wave packets is shown to be a higher-order effect that must consistently be neglected to lowest order in a systematic expansion in q/k. Rigorous functional methods are used to show that the long-wavelength growth rate gamma(q) is related to second-order functional variations of the short-wavelength energy and to derive a heuristic algorithm. The principal results are recovered from simple estimates involving the first-order wave-number distension rate gamma;((1))(k)equals, single dot abovek small middle dotnablaOmega;(k)/k(2), where Omega;(k) is a nonlinear random advection frequency. Fokker-Planck analysis involving gamma;((1))(k) is used to heuristically recover the evolution equation for the small scales, and a random-walk flux argument that relates gamma;((1))(k) to an effective autocorrelation time is used to give an independent calculation of gamma(q). Both the rigorous and heuristic derivations demonstrate that the results do not depend on, and cannot be derived from, properties of linear normal modes; they are intrinsically nonlinear. The importance of random-Galilean-invariant renormalization is stressed.

Fundamental statistical descriptions of plasma turbulence in magnetic fields

“…However, those authors considered the unrealistic situation of frozen short-wavelength statistics and obtained results that did not properly reduce to Kraichnan's 2D NS formula. Krommes and Kim (2000) reconsidered the problem allowing for self-consistent interactions between long and short scales, and I follow that work here (the work of Chechkin et al is discussed in the appendix of that paper). For simplicity I shall assume that the spectrum is isotropic although this is not strictly correct in the presence of a diamagnetic frequency.…”

“…However, recently employed the formalism in his discussion of higher-order statistics (Sec. 10.2, p. 221), and Krommes and Kim (2000) found it to be useful in their discussion of the theory of longwavelength growth rates nonlinearly driven by short scales (Sec. 7.3.3,p.…”

“…Nevertheless, it provides a compelling unification of various earlier cumbersome techniques, highlights strong and beautiful links between a variety of fields, and permits an economy of description that can be extremely useful in practice (Krommes and Kim, 2000). I therefore give a relatively thorough discussion.…”

“…Smagorinsky's result is the only dimensionally correct diffusion coefficient that can be constructed from ν eddy and d. 233 I have changed the wave-number notation slightly to conform to current practice in plasma physics. Namely, following the work of Krommes and Kim (2000) and references therein, I use q for the small or resolved wave numbers and k for the large or unresolved wave numbers.…”

On Imaging of Plasma Turbulence

“…For recent development of statistical theory in turbulent plasmas, readers are referred to Refs. [1,6,[36][37][38][39][40] The fluctuating force from background broadband turbulence can induce stochastic transition in global dynamics. This is one of the processes that determine the lifetime of structures.…”