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| In the context of fusion plasmas, the Reynolds stress is a mechanism for generation of sheared flow from turbulence.
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| <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>
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| 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):
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| <ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>
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| :<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>
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| Averaging over a [[Flux surface|magnetic surface]] (i.e., over ''y''), the right-hand side cancels:
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| :<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = 0</math>
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| 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).
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| The corresponding anisotropy is in fact essential to the effectiveness of the Reynolds Stress mechanism.
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| Now, writing the flow as the sum of a mean and a fluctuating part
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| :<math>u = \bar{u} + \tilde{u}</math>
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| one obtains
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| :<math>\frac{\partial \bar{u}_y}{\partial t} + \nabla_x \left \langle \tilde{u}_x \tilde{u}_y \right \rangle = 0</math>
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| Here, the Reynolds stress tensor appears:
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| :<math>R_{xy} = \left \langle \tilde{u}_x \tilde{u}_y \right \rangle</math>
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| 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.
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| This correlation occurs naturally in the presence of a background (mean) gradient driving turbulent transport.
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| Once the laminar flow shear develops, it may suppress small-scale turbulence, leading to a reduction of transport.
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| <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>
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| == See also ==
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| * [[H-mode]]
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| * [[Internal Transport Barrier]]
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| == References ==
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| <references />
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