Turbulence Spreading into Linearly Stable Zone and Transport Scaling Page: 4 of 18
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tokamak fusion reactor. In recent years, progress in experiment, theory and computation has been dramatic,
yet the 'Holy Grail' of predictive capacity by other than brute-force, case-by-case direct numerical simula-
tion, remains elusive. Serious challenges remain due to the fact that virtually all models of fluctuation levels
and turbulent transport are built on an assumption of local balance of linear growth with linear damping and
nonlinear coupling to dissipation. Here, 'local balance' refers to balance at a point or in a region compara-
ble in extent to the modal width. Such models thus necessarily exclude mesoscale dynamics which refers
to dynamics on scales larger than a mode or integral scale eddy size, but smaller than the system size or
profile scale length. In particular, transport barriers, avalanches, heat and particle pulses all are mesoscale
phenomena[1, 2, 3].
In this paper, we identify and study in depth the simplest, most minimal problem in the mesoscale
dynamics category before proceeding to consider more complicated examples. In this case, the 'minimal
problem' is that of the spatio-temporal propagation of a patch of turbulence from a region where it is locally
excited to a region of weaker excitation, or even local damping. This process can be described by a single
model equation for the local turbulence intensity I(x, t), which includes the effects of local linear growth
and damping, spatially local nonlinear coupling to dissipation and spatial scattering of turbulence energy
induced by nonlinear coupling. These effects combine to give an energy equation loosely of the form
ca(I)M = 'Y(z)I - caI -+,(1)
the terms of which correspond to nonlinear spatial scattering (i.e. typically x(I) - xoI/ where # = 1
for weak turbulence, and 0 = 1/2 for strong turbulence), linear growth and damping, and local nonlinear
decay, respectively. Here a is a nonlinear coupling coefficient. Note that a and xo could be functions of
radius. This energy equation is the irreducible minimum of the model, to which additional equations for
other fields, and contributions to dynamics which feedback on I, may be added. Note that the above energy
equation manifests the crucial effect of spatial coupling in the nonlinear diffusion term. This implies that
the integrated fluctuation intensity in a region of extent A about a point x (i.e. f o I(x')dx') can grow,
even for negative- (x), so long as x(I)OI/OxIi is sufficiently large. Alternatively, I can decrease, even
for positive- (x), should x(I)OI/Ox Io be sufficiently negative. Thus, the profile of fluctuation intensity
is crucial to its spatio-temporal evolution. These simple observations nicely illustrate the failure of the
conventional local saturation paradigm[4], and strongly support the argument that propagation of turbulence
energy is a crucial, fundamental problem in understanding confinement scalings for fusion devices in which
growth and damping rate profiles vary rapidly in space.
While the radial spreading of turbulence has been widely observed in previous global nonlinear gyroki-2
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Hahm, T.S.; Diamond, P.H.; Lin, Z.; Itoh, K. & Itoh, S.-I. Turbulence Spreading into Linearly Stable Zone and Transport Scaling, report, October 20, 2003; Princeton, New Jersey. (https://digital.library.unt.edu/ark:/67531/metadc740443/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.