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:

$\frac{\partial u_x}{\partial t}+{\mathrm{\nabla }}_y\left(u_y{u}_x\right)=-{\mathrm{\nabla }}_{x}P-\frac{1}{\rho }{(\overrightarrow{j}\times \overrightarrow{B})}_x$

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

$\frac{\partial u_x}{\partial t}+{\mathrm{\nabla }}_y\left(u_y{u}_x\right)=0$

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

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

one obtains

$\frac{\partial {\overline{u}}_x}{\partial t}+{\mathrm{\nabla }}_y⟨{\tilde{u}}_y{\tilde{u}}_x⟩=0$

Here, the Reynolds stress tensor appears:

$R_{xy}=⟨{\tilde{u}}_y{\tilde{u}}_x⟩$

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