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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.
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.
$z_0$ 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 $1/7$)
