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
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
and similarly, for potential temperature:
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:
with \(x=(1-19.3 \zeta)^{1 / 4} \quad y=0.95(1-11.6 \zeta)^{1 / 2}\).
under stable conditions:
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)
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
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