Both roughness length $z_0$ and zero plane displacement $d$ seemed to be defined as the height above the ground at which wind speed theoretically becomes zero. But wind speed is also supposed to go to zero at $d+z_0$. What is the difference between the two, and how should they actually be defined?

  • 1
    $\begingroup$ It would be useful if you included a reference for the equivalent definition of $z_0$ and $d$. $\endgroup$ Apr 8, 2015 at 17:57

2 Answers 2


First some definitions:

$z_0$: Roughness length is defined as the height at which the mean velocity is zero due to substrate roughness. Real walls/ground are not smooth and often have varying degrees of roughness, this parameter (which is determined empirically) accounts for that effect.

$d$: Zero Plane displacement is defined as the height at which the mean velocity is zero due to large obstacles such as buildings/canopy.

The two parameters are not the same because they describe the effects of two fundamentally different processes. $d$ can be anywhere from $6$ to $20$ times larger than $z_0$.

The basis for most turbulence modeling is the eddy viscosity model:

$$-\overline{u'w'} = \nu_t \frac{\partial U}{\partial z}$$

where $\nu_t$ is the eddy viscosity. Employing some scaling arguments with basis on the Prandtl mixing length model and integrating one arrives at the logarithmic law of the wall:

$${U} = \frac{u^*}{\kappa} \ln\, \frac{z}{z_0}$$

Your equation is

$${U} = \frac{u^*}{\kappa} \ln\, \frac{z-d}{z_0}$$ which is the law of the wall with $d = 0$ because it applies to flat plates. It is easy to see then, that by subtracting $d$ from $z$ the effect is to reduce $U$ at that height, which makes sense because large obstacles remove energy from the mean flow and slow it down.

Note that if there are no large obstacles then $d \approx 0$, but $z_0$ is still larger than zero.

  • $\begingroup$ Did you intend one of your equations to include d? $\endgroup$ Apr 8, 2015 at 22:51
  • $\begingroup$ Added for clarity. $\endgroup$ Apr 8, 2015 at 23:02

Let me add to the previous answer: In mixing length theory the mixing length is often defined as the friction velocity ${u_*}$ divided by the vertical shear $\frac{dU}{dz}$. Both of these quantities can be measured more or less directly at different heights. In the absence of buildings/canopy the mixing length is to a first approximation proportional to ${z}$.

The displacement height ${d}$ is given by the height at which the mixing length approaches zero. The roughness length ${z_0}$ is, however, still needed to match the wind speed profile.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.