14.4. Stability

Modifications to the logarithmic profile are required in conditions of non-neutral stability, using the results of Monin-Obukhov theory. This theory of the surface layer 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

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

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

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

and similarly, for potential temperature:

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

where the turbulent temperature scale \(T_*\) 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.

There are a number of forms of \(\Psi_{m}\) and \(\Psi_{h}\); one set of forms from Foken (2008) are as follows:

under unstable conditions:

(14.9)\[\begin{split}\begin{array}{c} {\psi_{m}(\zeta)=\ln \left[\left(\frac{1+x^{2}}{2}\right)\left(\frac{1+x}{2}\right)^{2}\right]-2 \tan ^{-1} x+\frac{\pi}{2} \text { for } \frac{z}{L}<0} \\ {\psi_{h}(\zeta)=2 \ln \left(\frac{1+y}{2}\right) \text { for } \frac{z}{L}<0} \end{array}\end{split}\]

with \(x=(1-19.3 \zeta)^{1 / 4} \quad y=0.95(1-11.6 \zeta)^{1 / 2}\).

under stable conditions:

(14.10)\[\begin{split}\begin{array}{l} {\psi_{m}(\zeta)=-6 \frac{z}{L} \quad \text { for } \quad \frac{z}{L} \geq 0} \\ {\psi_{h}(\zeta)=-7.8 \frac{z}{L} \quad \text { for } \quad \frac{z}{L} \geq 0}\end{array}\end{split}\]

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 \(z’/z_0 \gg 1\) then the third term can assumed to be negligible (Garratt 1992).

Other stability metrics include the Richardson number:

  • Gradient

  • Bulk

  • Flux

Bulk Richardson number is the ratio of thermally produced turbulence and turbulence generated by vertical shear or the ratio of free or forced convection (thermal: mechanical)

(14.11)\[R_{B}=\frac{\left(g / T_{v}\right) \Delta \theta_{v} \Delta z}{(\Delta U)^{2}+(\Delta V)^{2}}\]

where \(g\) acceleration due to gravity, \(T_V\) virtual temperature, \(\Delta \theta_{v}\) change (difference) in potential temperature, \(\Delta z\) change in height \(\Delta U\) change in \(U\) wind-speed, and \(\Delta V\) change in \(V\) wind-speed.

Tip

  1. Stuck? the help and FAQ pages are useful places to start.

  2. Please report workshop manual issues at GitHub Issues. Go from the page with the problem - an automatical link will be inserted. Thanks.