Logarithmic wind profile

The logarithmic wind profile is used as an approximation to describe speed profiles that arise from the roughness of the ground or the development in wind currents . The logarithmic wind profile applies to the Prandtl layer close to the ground (up to approx. 60 m above the ground). It is important for wind energy .

The logarithmic vertical wind profile only applies to neutral stratification . In the case of a non-neutral stratification (i.e. stable and unstable ) the relationship does not apply .

definition

dynamic roughness lengths z ₀ for different substrates

Surface texture	z 0 [m]
offshore (v 10 = 5 m / s)	0.0001
offshore const. (Class 0)	0.0002
smooth snow	0.001
offshore (v 10 = 25 m / s)	0.003
smooth earth	0.005
Airport / short-cut meadow	0.01
Cultural landscape with very few buildings, trees, etc. (grade 1)	0.03
Cultivated landscape with a closed appearance (grade 2)	0.1
Class 3	0.4
Suburbs	0.5
Forest	0.8
city	1

In the Prandtl layer, the wind speed increases (approximately) logarithmically with height , i.e. first very quickly and then increasingly slowly: ${\displaystyle u}$ ${\displaystyle z}$

${\displaystyle u(z)={\frac {u^{*}}{\kappa }}\ln \left({\frac {z}{z_{0}}}\right)}$

With

• the shear stress velocity (i.e. the root of the specific vertical momentum flux) ${\displaystyle u^{*}}$
• the Von Karman constant ${\displaystyle \kappa \approx 0{,}4}$
• the dynamic roughness length (height above ground at which the vertical logarithmic wind profile assumes the value 0 or the speed no longer changes; see adjacent table). ${\displaystyle z_{0}}$

Often the logarithmic profile is related to the wind speed at a reference height , e.g. B. for the data of the German weather service DWD at a height above ground. The following formula can be used to convert to any other amount: ${\displaystyle u_{r}}$ ${\displaystyle z_{r}}$ ${\displaystyle z_{r}=10\,{\text{m}}}$

${\displaystyle u(z)=u_{r}{\frac {\ln(z/z_{0})}{\ln(z_{r}/z_{0})}}}$

because ${\displaystyle {\frac {u_{r}}{\ln(z_{r}/z_{0})}}={\frac {u(z)}{\ln(z/z_{0})}}={\frac {u^{*}}{\kappa }}}$

The strong, logarithmic wind increase with height (vertical wind shear ) in the Prandtl layer as well as the wind rotation in the Ekman layer above the Prandtl layer is z. B. to be considered when building wind turbines : due to the uneven wind pressure , high tensions can act on the individual rotor blades .

Derivation

The atmospheric boundary layer is dominated by turbulence and the vertical turbulent flow of horizontal momentum results from the Reynolds equations as the covariance between vertical and horizontal speed. Assuming that mean wind direction and mean shear stress coincide, one can write: ${\displaystyle \tau }$

${\displaystyle \tau =\rho {\overline {u'w'}}=\rho u_{*}^{2}}$

with the shear stress speed , the air density, u and w the horizontal and vertical wind speed and z. B. the deviations or fluctuations from the mean . ${\displaystyle u_{*}}$ ${\displaystyle \rho }$ ${\displaystyle u'=u-{\overline {u}}}$ ${\displaystyle {\overline {u}}}$

Mixing path length approach

The original derivation according to Ludwig Prandtl is very clear and is based on the assumption that the turbulence transports certain flow properties from a different height to a certain location:

If one assumes that the fluctuations u 'and w' result from vertical, turbulent transport over a mixing path length or from the horizontal wind, so that one ${\displaystyle l_{u}'}$ ${\displaystyle l_{w}'}$

${\displaystyle u'=l_{u}'\cdot {\frac {d{\overline {u}}}{dz}}}$ and ${\displaystyle w'=l_{w}'\cdot {\frac {d{\overline {u}}}{dz}}}$

can approach, this leads to the equation by substitution:

${\displaystyle u_{*}^{2}={\overline {l_{u}'l_{w}'}}\cdot \left({\frac {d{\overline {u}}}{dz}}\right)^{2}}$

The mixing path lengths should increase with the height z (larger turbulence elements) and be positively correlated (the same upwash or downwind transports the u 'and w' disturbance). The simplest approach to this is:

${\displaystyle {\sqrt {\overline {l_{u}'l_{w}'}}}=\kappa \cdot z}$

with a constant of proportionality. Insertion leads to ${\displaystyle \kappa }$

${\displaystyle {\frac {d{\overline {u}}}{dz}}={\frac {u_{*}}{\kappa \cdot z}}}$

If is constant with altitude, then this equation can be integrated. The lower integration limit is defined in such a way that the horizontal wind disappears here. The logarithmic wind profile is obtained: ${\displaystyle u_{*}}$ ${\displaystyle z_{0}}$

${\displaystyle {\overline {u}}(z)={\frac {u_{*}}{\kappa }}\cdot \ln \left({\frac {z}{z_{0}}}\right)}$

In the atmospheric boundary layer, the lower 100 m only decreases by a few percent, so that the assumption of a constant shear stress velocity is justified. In honor of Prandtl, who was the first to derive the logarithmic wind profile, the lowest hundred meters of the atmospheric boundary layer, in which the turbulent flows are approximately constant, are called the Prandtl layer . ${\displaystyle u_{*}}$

The logarithmic wind profile is found in nature with neutral stratification, that is, when the potential temperature does not change with altitude. In other stratifications there are deviations from this, which are described in the Monin-Obuchow theory . The above equation, which links the gradient and the shear stress velocity, then contains a correction function that can be interpreted as a correction of the mixing path length: ${\displaystyle \phi _{u}}$

${\displaystyle {\frac {d{\overline {u}}}{dz}}=u_{*}\ {\frac {\phi _{u}(z/L_{*})}{\kappa \cdot z}}}$

The parameter is called Obuchow length and describes the thermal stratification of the atmosphere. It is positive when it is stable (temperature increases upwards) and negative when the stratification is unstable (temperature decreases upwards, the air near the ground is warmer). With stable stratification, vertical movements are suppressed, the mixing path length must be smaller than with neutral stratification and consequently must be greater than one. In the case of unstable stratification, vertical movements extend further than in the case of neutral stratification, the mixing path length must be greater and consequently less than one. ${\displaystyle L_{*}}$ ${\displaystyle \phi _{u}}$ ${\displaystyle \phi _{u}}$

swell

• Helmut Pichler: Dynamics of the atmosphere
• John A. Dutton: Dynamics of Atmospheric Motion