67 Turbulent Viscosity

The derivation for the turbulent viscosity (aka the turbulent momentum diffusivity) follows similar steps as for the turbulent diffusivity for a tracer. Here we will consider flow in a two-dimensional (xz) plane, but similar expressions hold for each pairwise correlation of velocity components.

(u=\overline u+u') and (w=\overline w+w'). The advective flux of x-momentum in the z-direction

    \[wu = \overline w \overline u + \overline w u' + \overline u w' + u'w'\]

Now we will take the time average of each of these terms

    \[wu = \overline {\overline w \overline u } + \overline {\overline w u'} + \overline {\overline u w'} + \overline {u'w'} \]

(1) For the first term, the time average of the product of the two time averages is redundant since these are just constants in time.

(2) The second term is identically zero.  The time-average concentration is just a number and the fluctuating part has zero time average, so their product is zero.

(3) Similarly, the third term is identically zero.

(4) The fourth term represents a time averaged correlation between the horizontal and vertical velocity component fluctuations.  Physically, this term represents the turbulent transport of x-momentum in the z-direction.


    \[wu = \overline w \overline u + \overline {u'w'} =\overline w \overline u + F_{Turb}^x\]

As before, we often model turbulent transport using Newton’s Law of viscosity, but with a viscosity that is much (several orders of magnitude in the ocean) larger than the molecular viscosity.

    \[F_{Turb}^x = \overline {u'w'} =-\frac{\tau^x}{\rho}= - {K_{Turb}}\frac{{\partial \overline u }}{{\partial z}}\]

We usually drop the overbars from the time-mean quantities.