<p>First some definitions:</p>

<p>$z_0$: <em>Roughness length</em> 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.</p>

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

<p>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 $z_0$.</p>

<p>The basis for most turbulence modeling is the eddy viscosity model:</p>

<p>$$-\overline{u'w'} = \nu_t \frac{\partial U}{\partial z}$$</p>

<p>where $\nu_t$ is the eddy viscosity. Employing some scaling arguments with basis on the <a href="https://en.wikipedia.org/wiki/Mixing_length_model" rel="noreferrer">Prandtl mixing length model</a> and integrating one arrives at the <a href="https://en.wikipedia.org/wiki/Law_of_the_wall" rel="noreferrer">logarithmic law of the wall</a>:</p>

<p>$${U} = \frac{u^*}{\kappa} \ln\, \frac{z}{z_0}$$</p>

<p>Your equation is </p>

<p>$${U} = \frac{u^*}{\kappa} \ln\, \frac{z-d}{z_0}$$
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.</p>

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

CC BY-SA 3.0
2015-04-08T22:51:17.553Z
Did you intend one of your equations to include d?
Semidiurnal Simon
CC BY-SA 3.0
2015-04-08T23:02:59.443Z
Added for clarity.
Isopycnal Oscillation
CC BY-SA 3.0
2015-04-08T18:30:46.990Z
2015-04-08T23:02:48.160Z
2015-04-08T23:02:48.160Z
Isopycnal Oscillation
<p>Let me add to the previous answer:
In mixing length theory the mixing length is often defined as the friction velocity <span class="math-container">${u_*}$</span> divided by the vertical shear <span class="math-container">$\frac{dU}{dz}$</span>. 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 <span class="math-container">${z}$</span>.</p>
<p>The displacement height <span class="math-container">${d}$</span> is given by the height at which the mixing length approaches zero. The roughness length <span class="math-container">${z_0}$</span> is, however, still needed to match the wind speed profile.</p>

CC BY-SA 4.0
2021-09-17T09:09:00.860Z
2021-09-22T12:04:20.037Z
2021-09-22T12:04:20.037Z
Martin
<p>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?</p>

CC BY-SA 3.0
2015-04-08T17:57:52.500Z
It would be useful if you included a reference for the equivalent definition of $z_0$ and $d$.
milancurcic
<atmosphere><wind><planetary-boundary-layer>
What is the difference between roughness length and zero plane displacement?
environmentstudent
# What is the difference between roughness length and zero plane displacement?

(Asked by: username_0 on 2015-04-08)

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?

**Tags:** atmosphere, wind, planetary-boundary-layer

> **Comments:**
> **username_1**: It would be useful if you included a reference for the equivalent definition of $z_0$ and $d$.  

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## 2 answers

### Answer by username_2 on 2015-04-08. Score: 12

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](https://en.wikipedia.org/wiki/Mixing_length_model) and integrating one arrives at the [logarithmic law of the wall](https://en.wikipedia.org/wiki/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.

> **Comments:**
> **username_3**: Did you intend one of your equations to include d?  
> **username_2**: Added for clarity.  

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### Answer by username_4 on 2021-09-17. Score: 1

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.


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