# Reynolds stress

In the context of fusion plasmas, the Reynolds stress is a mechanism for generation of sheared flow from turbulence. 

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): 

$\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = -\frac{1}{\rho}\nabla_y P + \frac{1}{\rho} \left ( \vec{j} \times \vec{B} \right )_y$

Averaging over a magnetic surface (i.e., over y), the right-hand side cancels:

$\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = 0$

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

$u = \bar{u} + \tilde{u}$

one obtains

$\frac{\partial \bar{u}_y}{\partial t} + \nabla_x \left \langle \tilde{u}_x \tilde{u}_y \right \rangle = 0$

Here, the Reynolds stress tensor appears:

$R_{xy} = \left \langle \tilde{u}_x \tilde{u}_y \right \rangle$

Thus, a non-zero value of the gradient of the Reynolds stress (of fluctuating flow components) can drive a laminar flow. Obviously, $\tilde{u}_x$ and $\tilde{u}_y$ 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.