Calculation of eddy viscosity and eddy diffusivity

According to Garcia et al. (2013), vertical eddy diffusivity is

eddy_diffusivity = beta * eddy_viscosity

Eddy viscosity in a parabolic profile is calculated as

eddy_viscosity = von_karman_constant * shear_velocity * vertical_location * (1 - vertical_location/depth).

However, the code in calculateKz.m calculates the eddy diffusivity (I'm assuming Kz) as

eddy_diffusivity = beta * von_karman_constant * shear_velocity * Zprime * (1 - Zprime/depth)

where Zprime is vertical_location + (0.5 * Kprime * Dt), and Kprime is calculated as

Kprime = beta * von_karman_constant * shear_velocity * (1 - 2*vertical_location/depth).

The vertical gradient of eddy viscosity is von_karman_constant * shear_velocity * (1 - 2*vertical_location/depth), so this make sense.

A few questions:

• Is my interpretation of the code correct?
• Can we verify that FluEgg calculates eddy diffusivity differently than the paper describes?
• If so, what is the reference?
• Why is Zprime used instead of vertical_location to calculate eddy viscosity?

Most relevant to the current issue:

From the code, it doesn't seem that Zprime is a property that's independent of the eggs because the equation for beta contains the settling velocity of the eggs (which in turn is dependent on the diameter and density of the eggs), which makes the eddy diffusivity and viscosity dependent on properties of the eggs.

• Is this correct, or can we make eddy viscosity a property independent of the eggs?
• What is Zprime and the term 0.5 * Kprime * Dt?

Also, how is the calculation for ZR working?

If H is the cell depth, and Z is the vertical location of the egg from the water surface

• When the egg is at the top and Z is 0, ZR would be H
• When the egg is at the bottom and Z is equal to H, ZR will be 2H

Is this accurate?

Edit: Added link to Garcia et al. (2013)

Edit: Added questions about calculation of ZR

Edit: Changed question from eddy diffusivity to viscosity