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It | In the context of fusion plasmas, the Reynolds stress is a mechanism for generation of sheared flow from turbulence. | ||
<ref>[http://dx.doi.org/10.1088/0741-3335/43/10/308 S.B. Korsholm et al, ''Reynolds stress and shear flow generation'', Plasma Phys. Control. Fusion '''43''' (2001) 1377]</ref> | |||
Starting from the incompressible momentum balance equation, neglecting the dissipative pressure tensor, in slab coordinates (think of ''x'' as radial, ''y'' as poloidal, and ''z'' as toroidal): | |||
<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref> | |||
:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = -\nabla_y P + \frac{1}{\rho} \left ( \vec{j} \times \vec{B} \right )_y</math> | |||
Averaging over a [[Flux surface|magnetic surface]] (i.e., over ''y''), the right-hand side cancels: | |||
:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = 0</math> | |||
It may seem as if one has lost all information concerning the background field. | |||
However, this is not true, as the choice of the ''x,y,z'' coordinate system depends, precisely, on the background magnetic field (and, in particular, on the cited flux surfaces). | |||
The corresponding anisotropy is in fact essential to the effectiveness of the Reynolds Stress mechanism. | |||
Now, writing the flow as the sum of a mean and a fluctuating part | |||
:<math>u = \bar{u} + \tilde{u}</math> | |||
one obtains | |||
:<math>\frac{\partial \bar{u}_y}{\partial t} + \nabla_x \left \langle \tilde{u}_x \tilde{u}_y \right \rangle = 0</math> | |||
Here, the Reynolds stress tensor appears: | |||
:<math>R_{xy} = \left \langle \tilde{u}_x \tilde{u}_y \right \rangle</math> | |||
Thus, a non-zero value of the gradient of the Reynolds stress (of fluctuating flow components) can drive a laminar flow. Obviously, <math>\tilde{u}_x</math> and <math>\tilde{u}_y</math> must be ''correlated'' for this to work. | |||
This correlation occurs naturally in the presence of a background (mean) gradient driving turbulent transport. | |||
Once the laminar flow shear develops, it may suppress small-scale turbulence, leading to a reduction of transport. | |||
<ref>[http://link.aps.org/doi/10.1103/RevModPhys.72.109 P. W. Terry, ''Suppression of turbulence and transport by sheared flow'', Rev. Mod. Phys. '''72''' (2000) 109–165]</ref> | |||
== See also == | |||
* [[H-mode]] | |||
* [[Internal Transport Barrier]] | |||
== References == | |||
<references /> | |||