# Wind profile below the aerodynamic roughness length

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.

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.

• Isn't $z_0$ the surface of the Earth?
– gerrit
Aug 11 '15 at 10:14
• Aug 11 '15 at 11:15
• 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. twister.ou.edu/QJ/CD1-1971-1980/1978/v104n440/s18.pdf Aug 11 '15 at 16:28
• @gerrit, Zo is the roughness length. It is related to the typical height of closely spaced surface obstacles, often called roughness elements. Aug 11 '15 at 16:30

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.

• Deditos, out of curiosity, what kind of equations would be needed to model the interfacial sublayer that you mentioned? Aug 12 '15 at 0:47
• @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. Aug 13 '15 at 8:58
• @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. Aug 14 '15 at 21:30
• 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. Jul 17 '17 at 21:23

$z_0$ 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.

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 $1/7$)