Monin Obuchow theory

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The Monin-Obuhov theory describes the relationship between the gradients of wind and temperature and the turbulent flows in the near-ground boundary layer of the atmosphere. It is based on Buckingham's Π-theorem d. H. the assumption that dimensionless equations can be derived in a complex non-linear system that describe the system universally. For this purpose, state variables must be found that adequately describe the system.

The turbulence in the boundary layer close to the ground is essentially determined by the turbulent flows of momentum and sensible heat and influences the course of wind speed and potential temperature with altitude. Scaling variables for wind speed and temperature can be constructed from and : ${\displaystyle \tau }$ ${\displaystyle H}$ ${\displaystyle u}$ ${\displaystyle \theta }$ ${\displaystyle \tau }$ ${\displaystyle H}$

${\displaystyle {\begin{aligned}u_{*}&={\sqrt {-\tau /\rho }}\\\theta _{*}&={\frac {-H}{\rho c_{P}u_{*}}}\end{aligned}}}$

Here is the speed of shear stress , the density of the air, and the heat capacity of the air at constant pressure. With these scaling variables, the height dependence of the vertical gradients can be represented via dimensionless functions , which should be universally valid according to the Π-theorem: ${\displaystyle u_{*}}$ ${\displaystyle \rho }$ ${\displaystyle c_{P}}$ ${\displaystyle \phi _{M},\phi _{H}}$

${\displaystyle {\begin{aligned}{\frac {du}{dz}}\cdot {\frac {\kappa z}{u_{*}}}&=\phi _{M}(z/L_{*})\\{\frac {d\theta }{dz}}\cdot {\frac {\kappa z}{\theta _{*}}}&=\phi _{H}(z/L_{*})\end{aligned}}}$

with the vertical coordinate (height above the ground), the von Karman constant from the logarithmic wind profile , and the Monin-Obukhov longitude. This in turn can be omitted and calculated: ${\displaystyle z}$ ${\displaystyle \kappa }$ ${\displaystyle L_{*}}$ ${\displaystyle u_{*}}$ ${\displaystyle \theta _{*}}$

${\displaystyle L_{*}={\frac {\overline {\theta }}{\kappa g}}\cdot {\frac {u_{*}^{2}}{\theta _{*}}}}$

with the gravitational acceleration and the mean potential temperature in the layer close to the ground. ${\displaystyle g}$ ${\displaystyle {\overline {\theta }}}$

The theory was derived in 1947 by AS Monin and AM Obukhov on a theoretical basis alone. When it became possible in the early 1970s to measure turbulent flows with sufficient accuracy, first attempts were made to determine the functions and (Dyer and Bradley 1970, Businger et al. 1971). The theory is used today in almost all meteorological forecast models to describe the exchange of momentum, heat and water vapor between the surface and the atmosphere. ${\displaystyle \phi _{M}}$ ${\displaystyle \phi _{H}}$

See also

• Logarithmic wind profile
• Prandtl's mixing path hypothesis