# Intuitive definition of the Monin-Obukhov length?

The Monin-Obukhov length (L) is defined as:

$$L = - \frac{\rho C_p T_v u^3_*}{kgH}$$

where $\rho$=moist air density; $C_p$=air specific heat capacity; $T_v$=virtual temperature; $u^3_*$=wind friction velocity; $k$=0.41 (von Karman constant); $g$=gravity; and $H$=sensible heat. Note that there are lots of different formulations - this one is taken from this video on eddy covariance.

It is used when describing how turbulence is affected by bounancy, e.g. from rising thermals from the earth's surface. I would like to get some intuition for what the length actually physically represents. The wikipedia page states that:

A physical interpretation of L is given by the Monin–Obukhov similarity theory. During the day -L it is the height at which the buoyant production of turbulence kinetic energy (TKE) is equal to that produced by the shearing action of the wind (shear production of TKE).

However, that is still pretty opaque to me. To me, that sounds something like a parameter of a distribution that describes at what height wind shear starts to break up up-welling packets of air, but I really don't have any idea. Is there a good intuitive description of this quantity?

The way I understand it is the height above the surface at which buoyant (heat) production of turbulence first equals the mechanical (shear) production of turbulence.

A more useful definition is given by the American Meteorological Society:

A parameter with dimension of length that gives a relation between parameters characterizing dynamic, thermal, and buoyant processes.

At altitudes below this length scale, shear production of turbulence kinetic energy dominates over buoyant production of turbulence. It is defined by where k is von Kármán's constant, u* is the friction velocity (a measure of turbulent surface stress), g is gravitational acceleration, Tv is virtual temperature, and Qv0 is a kinematic virtual temperature flux at the surface. The parameter was first described by Obukhov in 1946, and therefore should not be called the Monin–Obukhov length, even though there is a Monin–Obukhov similarity theory that uses it. The Obukhov length, of order one to tens of meters, is the characteristic height scale of the dynamic sublayer. The Obukhov length is zero for neutral stratification, and positive (negative) for stable (unstable) stratifications. The dimensionless Obukhov length z/ L (where z is height above the surface) is used as a stability parameter, with z/L = 0 for statically neutral stability, and is positive (negative) in a typical range of 1 to 5 (-5 to -1) for stable (unstable) stratification.

The Obukhov length ($L$) measures the

relative importance of mechanical shear-generated turbulence and density-driven (buoyancy) fluxes (1)

$L$ can be used to determine the eddy structure of the flow and it provides a measure for the hydrodynamic stability of the boundary layer - it effectively imposes an upper limit on the vertical excursion of fluid particles due to (negative) buoyancy forces.

Using the example of a marine layer, if $Z_l$ is the boundary layer height, the dominant eddy structure (roll vortex motion, free convection or both) can be estimated with the parameter $-Z_l/L$ (its absolute value measures the stability of the boundary layer). Some common values categorizing the dominant eddy structure and morphology are given in (2). A summary is found in the table below: It is not clear that the Obukhov length has an exact physical interpretation. The length L is certainly a length that dimensional argument shows to follow from the set of basis parameters that Monin and Obukhov proposed was sufficient to describe turbulence in the bottom 10% or so of atmospheric boundary layers.

We may ask several questions: is the MO basis set complete and unique; is the dimensional argument sound in the face of possible mixed scales (i.e.scales of the form $$\ell_1^{1/2}\ell_2^{1/2}$$), many of which are known in fluid mechanics; is dynamical interpretation possible when the variables in the Reynolds-averaged equations, on which the standard interpretation is based, are defined as ensemble averages (i.e. constructed using flow realizations that have no connection in time)?

The answers are no, no and no.

• Please excuse my ignorance, but what does \ell_ mean in \ell_1^{1/2}? – Fred Oct 12 '19 at 4:17
• @Fred: presumably for variables where the l1 norm should be used in some dimensions, and the L2 norm in others? – naught101 Oct 12 '19 at 4:28
• \ell is a relevant length that has influence on the flow system, such as the depth of the atmospheric boundary layer or the height of an observation, or even the Obukhov length itself. For reasons unknown mixed scales seem always to involve half powers. – KG McNaughton Nov 3 '19 at 5:44