# 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
• – Deditos 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 – arkaia 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. – arkaia 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? – user4624937 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. – Deditos 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. – Isopycnal Oscillation 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. – BarocliniCplusplus 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 $1/7$)