<p>$z_0$ is a theoretical construct that, while useful in its intended uses, cannot be thought of in too much detail as a physical reality. When using a log law to describe wind speed, it represents the distance above the surface at which that log curve decreases to zero. However, if a measurement of speed were made at this height, it would be unlikely to be zero - more detailed, finer-scaled processes dominate here.
(I don't know wind modelling, but by analogy from water I'm guessing a that thin linear boundary layer exists between the ground and the point at which the log curve becomes dominant)</p>

<p>In practice the log-law approach for wind speed is used when dealing with speeds large distances (tens of metres) above the surface, and is not applicable when $z$ approaches $z_0$. Far more detailed techniques would be needed in this realm.</p>

CC BY-SA 3.0
2015-08-13T08:16:55.530Z
2015-08-13T08:16:55.530Z
Semidiurnal Simon
<p>There are additional mathematical models for the profile of the wind speed above the ground. 
For instance the power law: $u$ $=$ $bz^b$ (where $u$ is the speed of the ground at an height $z$ ;  $a$ and $b$ are numerical coefficients (usually it is assumed that $b$ $=$ $1/7$)</p>

<p>Another expression for the wind speed profile is the exponential formula:
$u$ $=$ $a e^{-bz}$ where $u$, $z.a$ and $b$ are as previously defined (but here $b$ is not $1/7$)</p>

CC BY-SA 3.0
2015-08-18T11:11:21.770Z
2015-08-18T11:47:20.897Z
2015-08-18T11:47:20.897Z
Jailcu45
<p>The key text here is "for $z&gt;z_0$".  It's telling you that, while you can evaluate the equation for other values of $z$, outside of that range the equation is not a valid description of the physical system.  The equation could be written piece-wise to be complete:</p>

<p>$u(z) = \begin{cases} (u_*/k) \ln(z/z_0)&amp; z&gt;z_0 \\ 0 &amp; z\le z_0\end{cases}$</p>

<p>But this doesn't really add anything useful.  In practice, the "log-law" is used to describe the wind profile over 10s of metres and values of $z_0$ range from 1 mm to 2 m, so values of $z$ are likely to be in the valid region.  If you do need to make calculations that close to the surface (in the interfacial sublayer) then you'll need a different equation anyway.</p>

CC BY-SA 3.0
2017-07-17T21:23:47.697Z
I was under the impression that beneath $z_0$, molecular diffusion was dominant, and the profile would break down, or return back to $\frac{\partial u}{\partial z}=\nu \frac{\partial^2 u}{\partial z^2}$ with boundary conditions $u(z_0)=0$ and $u(0)=0$, except for over the ocean.
BarocliniCplusplus
CC BY-SA 3.0
2015-08-12T00:47:10.873Z
Deditos, out of curiosity, what kind of equations would be needed to model the interfacial sublayer that you mentioned?
user4624937
CC BY-SA 3.0
2015-08-13T08:58:32.783Z
@user4624937 It's not really my area, so I'll just quote Brutsaert (1982): "In this sublayer... there are as many different types of flow as there are types of surface."  It's very dependent on the nature of the surface obstacles and their arrangement.
Deditos
CC BY-SA 3.0
2015-08-14T21:30:03.277Z
@user4624937 You would need the Navier-Stokes equations which are notoriously difficult to solve in general. So one would typically employ a DNS/LES simulation or use a reduction of the NS equation that has the terms relevant to the BL. There are a lot more interesting details that would make this a great question to ask in this site.
Isopycnal Oscillation
CC BY-SA 3.0
2015-08-11T13:13:22.503Z
2015-08-11T13:13:22.503Z
Deditos
<p>The vertical profile of horizontal windspeed is often given by the so-called "log-law":  </p>

<p>$$u(z)=(u^*/k)\ln(z/z_0),\ \mathrm{for}\  z&gt;z_0,$$   </p>

<p>which can be found <a href="http://mstudioblackboard.tudelft.nl/duwind/Wind%20energy%20online%20reader/Static_pages/wind_shear.htm">here</a>. </p>

<p>As can be seen from the equation, when $z=z_0$, $u(z)=0$. But what happens below $z_0$? If we continue to use the same equation, for $z&lt;z_0$, $u(z)$ becomes negative and changes direction, which seems absurd. </p>

CC BY-SA 3.0
2015-08-11T10:14:00.850Z
Isn't $z_0$ the surface of the Earth?
gerrit
CC BY-SA 3.0
2015-08-11T11:15:49.730Z
Related: http://earthscience.stackexchange.com/questions/4665
Deditos
CC BY-SA 3.0
2015-08-11T16:28:31.963Z
A good start is: Garratt, J. R. (1978). Transfer characteristics for a heterogeneous surface of large aerodynamic roughness. Quarterly Journal of the Royal Meteorological Society, 104(440), 491-502. http://twister.ou.edu/QJ/CD1-1971-1980/1978/v104n440/s18.pdf
arkaia
CC BY-SA 3.0
2015-08-11T16:30:13.173Z
@gerrit, Zo is the roughness length. It is related to the typical height of closely spaced surface obstacles, often called roughness elements.
arkaia
<meteorology><atmosphere><wind><turbulence><planetary-boundary-layer>
Wind profile below the aerodynamic roughness length
user4624937
# Wind profile below the aerodynamic roughness length

(Asked by: username_0 on 2015-08-11)

The vertical profile of horizontal windspeed is often given by the so-called "log-law":

$$u(z)=(u^\*/k)\ln(z/z\_0),\ \mathrm{for}\ z>z\_0,$$

which can be found [here](http://mstudioblackboard.tudelft.nl/duwind/Wind%20energy%20online%20reader/Static_pages/wind_shear.htm).

As can be seen from the equation, when $z=z\_0$, $u(z)=0$. But what happens below $z\_0$? If we continue to use the same equation, for $z<z\_0$, $u(z)$ becomes negative and changes direction, which seems absurd.

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

> **Comments:**
> **username_1**: Isn't $z_0$ the surface of the Earth?  
> **username_2**: Related: http://earthscience.stackexchange.com/questions/4665  
> **username_3**: A good start is: Garratt, J. R. (1978). Transfer characteristics for a heterogeneous surface of large aerodynamic roughness. Quarterly Journal of the Royal Meteorological Society, 104(440), 491-502. http://twister.ou.edu/QJ/CD1-1971-1980/1978/v104n440/s18.pdf  
> **username_3**: @username_1, Zo is the roughness length. It is related to the typical height of closely spaced surface obstacles, often called roughness elements.  

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## 3 answers

### Answer by username_2 on 2015-08-11. Score: 10 | Accepted answer

The key text here is "for $z>z\_0$". It's telling you that, while you can evaluate the equation for other values of $z$, outside of that range the equation is not a valid description of the physical system. The equation could be written piece-wise to be complete:

$u(z) = \begin{cases} (u\_\*/k) \ln(z/z\_0)& z>z\_0 \\ 0 & z\le z\_0\end{cases}$

But this doesn't really add anything useful. In practice, the "log-law" is used to describe the wind profile over 10s of metres and values of $z\_0$ range from 1 mm to 2 m, so values of $z$ are likely to be in the valid region. If you do need to make calculations that close to the surface (in the interfacial sublayer) then you'll need a different equation anyway.

> **Comments:**
> **username_0**: username_2, out of curiosity, what kind of equations would be needed to model the interfacial sublayer that you mentioned?  
> **username_2**: @username_0 It's not really my area, so I'll just quote Brutsaert (1982): "In this sublayer... there are as many different types of flow as there are types of surface."  It's very dependent on the nature of the surface obstacles and their arrangement.  
> **username_7**: @username_0 You would need the Navier-Stokes equations which are notoriously difficult to solve in general. So one would typically employ a DNS/LES simulation or use a reduction of the NS equation that has the terms relevant to the BL. There are a lot more interesting details that would make this a great question to ask in this site.  
> **username_6**: I was under the impression that beneath $z_0$, molecular diffusion was dominant, and the profile would break down, or return back to $\frac{\partial u}{\partial z}=\nu \frac{\partial^2 u}{\partial z^2}$ with boundary conditions $u(z_0)=0$ and $u(0)=0$, except for over the ocean.  

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### Answer by username_4 on 2015-08-13. Score: 7

$z\_0$ is a theoretical construct that, while useful in its intended uses, cannot be thought of in too much detail as a physical reality. When using a log law to describe wind speed, it represents the distance above the surface at which that log curve decreases to zero. However, if a measurement of speed were made at this height, it would be unlikely to be zero - more detailed, finer-scaled processes dominate here.
(I don't know wind modelling, but by analogy from water I'm guessing a that thin linear boundary layer exists between the ground and the point at which the log curve becomes dominant)

In practice the log-law approach for wind speed is used when dealing with speeds large distances (tens of metres) above the surface, and is not applicable when $z$ approaches $z\_0$. Far more detailed techniques would be needed in this realm.


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### Answer by username_5 on 2015-08-18. Score: 1

There are additional mathematical models for the profile of the wind speed above the ground.
For instance the power law: $u$ $=$ $bz^b$ (where $u$ is the speed of the ground at an height $z$ ; $a$ and $b$ are numerical coefficients (usually it is assumed that $b$ $=$ $1/7$)

Another expression for the wind speed profile is the exponential formula:
$u$ $=$ $a e^{-bz}$ where $u$, $z.a$ and $b$ are as previously defined (but here $b$ is not $1/7$)


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