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:

${\frac {\partial u_{x}}{\partial t}}+\nabla _{y}\left(u_{y}u_{x}\right)=-\nabla _{x}P-{\frac {1}{\rho }}\left({\vec {j}}\times {\vec {B}}\right)_{x}$

Averaging over a magnetic surface (assuming it exists), the right-hand side cancels (MHD equilibrium):

${\frac {\partial u_{x}}{\partial t}}+\nabla _{y}\left(u_{y}u_{x}\right)=0$

Now, writing the flow as the sum of a mean and a fluctuating part

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

one obtains

${\frac {\partial {\bar {u}}_{x}}{\partial t}}+\nabla _{y}\left\langle {\tilde {u}}_{y}{\tilde {u}}_{x}\right\rangle =0$

Here, the Reynolds stress tensor appears:

$R_{xy}=\left\langle {\tilde {u}}_{y}{\tilde {u}}_{x}\right\rangle$

and it is clear that a non-zero value of the gradient of the Reynolds stress (of fluctuating flow components) can drive a laminar flow.