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:

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

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

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

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

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

one obtains

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

Here, the Reynolds stress tensor appears:

$R_{xy}=⟨{\tilde{u}}_x{\tilde{u}}_y⟩$

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, which will depend on the details of the (equations describing the) turbulence.

See also

• H-mode
• Internal Transport Barrier