14.5. Wind Profile¶

Wind profile in neutral conditions

The variation of mean wind speed \(\bar{u}\) with height z (in surface layer, above the RSL) above an aerodynamically rough surface can be represented by a logarithmic relation:

(14.12)¶\[\bar{u}(z) = \frac{u_{*}}{k}\ln\left\lbrack \frac{(z - d)}{z_{0}} \right\rbrack\]

where \(u_{*}\) is the friction velocity (\(u_{*}^{2} = - \overline{u'w'}\)) rate of vertical transfer by turbulence of horizontal momentum per unit mass of air), \(z_{0}\) is the roughness length of the surface for momentum, d is the zero-plane displacement and k is von Karman’s constant (0.4). The logarithmic law is strictly valid only in neutral conditions, i.e. when the effect of buoyancy on turbulence is small compared to the effect of wind shear. In such conditions, the temperature profile in the surface layer will be close to adiabatic (i.e. \(dT/dz=–9.8 \textrm{ K km}^{-1}\) or potential temperature is constant with height). When the sensible heat flux is significantly different from zero, Monin-Obukhov theory (or other) must be used.

Wind and temperature profile in non-neutral conditions

Modifications to the logarithmic profile are required in conditions of non-neutral stability. Here we use Monin-Obukhov theory of the surface layer that derives relations between the vertical variation of wind speed u(z) and potential temperature \(\theta(z)\) (which approximates the measured temperature T close to the surface), the scaling factors for momentum and temperature, \(u^*\) and \(T^*\), and the Monin‑Obukhov stability parameter

(14.13)¶\[\frac{z'}{L} = - \frac{k\left( z - d \right)\frac{g}{\theta_{0}}\frac{H}{\rho c_{p}}}{u_{*}^{3}}\]

where \(L\) is the Obukhov length and \(z’= z - d\). NB: the surface temperature \(\theta_0\) is an absolute temperature (units: K). The logarithmic profile relation can be rewritten for wind speed to include the stability corrections

(14.14)¶\[\bar{u}(z) = \frac{u_{*}}{k}\left\lbrack \ln\left( \frac{z - d}{z_{0}} \right) - \Psi_{m}\left( \frac{z - d}{L} \right) + \Psi_{m}\left( \frac{z_{0}}{L} \right) \right\rbrack\]

and similarly, for potential temperature:

(14.15)¶\[\bar{\theta}(z) = \theta_{0} + \frac{T_{*}}{k}\left\lbrack \ln\left( \frac{z - d}{z_{h}} \right) - \Psi_{h}\left( \frac{z - d}{L} \right) + \Psi_{h}\left( \frac{z_{h}}{L} \right) \right\rbrack\]

where the turbulent temperature scale is given by \(T_{*} = - \overline{w^{'}T^{'}}/u_{*} = - Q_{H}/(\rho c_{p}u_{*})\), \(\Psi_{m}\) is the integral stability correction function for momentum and \(\Psi_{h}\) is the integral stability correction function for heat. Note that both \(T_*\) and \(z’/L\) have the opposite sign to \(Q_H\) (which is positive in unstable conditions and negative in stable conditions). If \(z'/z \gg 1\) then the third term can assumed to be negligible (Garratt 1994)

Profiles and fluxes of moisture

Just as surface layer profiles of momentum and temperature follow a logarithmic form in neutral conditions, humidity in the surface layer has the same form, being transported by the same eddies. Thus, the profile is given by

(14.16)¶\[\overline{q}\left( z \right) - {\overline{q}}_{0} = \frac{q_{*}}{k}\ln\left( \frac{z - d}{z_{q}} \right)\]

where q is specific humidity, subscript 0 denotes a surface measurement, zq is the equivalent “roughness length” for humidity, and d is the zero-plane displacement height of a plant canopy over which measurements are being made). q* is a scaling variable, defined as

(14.17)¶\[q_{*} = \frac{- \overline{w'q'}}{u_{*}}\]

and thus the dimensionless moisture profile is defined as

(14.18)¶\[\phi_{q} = \frac{k\left( z - d \right)}{q_{*}}\frac{\partial\overline{q}}{\partial z}.\]

The moisture flux can be written in various equivalent forms

(14.19)¶\[E = \rho\overline{w'q'} = u_{*}q_{*} = - \rho K_{q}\frac{\partial\overline{q}}{\partial z}\]

where \(K_{q}\) is the eddy diffusivity for moisture. In neutral conditions it is assumed \(K_m =K_h =K_q=k(z-d)u_*\). Moisture follows Monin-Obukhov similarity just as other scalar variables do. This was not established at the Kansas experiments due to limitations in the accuracy of the measurements.

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