# Can Monin-Obukhov Similarity theory be generalized to tracers?

According to Monin-Obukhov similarity theory, Reynolds-averaged surface layer winds can be expressed by $$\frac{\partial \bar{U}}{\partial z}=\frac{u_*\psi_M(\frac{z}{L})}{kz}$$ The potential temperature can be expressed as$$\frac{\partial \bar{\theta}}{\partial z}=\frac{\theta_*\psi_H(\frac{z}{L})}{kz}$$ and I found one for moisture $$\frac{\partial \bar{q}}{\partial z}=\frac{q_*\psi_q(\frac{z}{L})}{kz}$$My question is whether this could be generalized to a tracer $\chi$ such that $$\frac{\partial \bar{\chi}}{\partial z}=\frac{\chi_*\psi_\chi(\frac{z}{L})}{kz}$$

If so, would this question the validity of the Gaussian Plume model?

Note: $u_*, \theta_*, q_*$ and $\chi_*$ are the scaled quantities. $\psi_M, \psi_H, \psi_q,$ and $\psi_\chi$ are the universal functions for each quantity. $k$ is the Von Karman constant.

Yes, the equation can be generalized as you mentioned. The method used to drive it is pretty close to humidity and temperature formulas.

I will start with neutral case and apply law of the wall, it is pretty simple to drive the following equation with Fc as diffusive flux and K as an empirical constant (NOT von-Karman) and also you can get C* from here by setting z = z0:

Remind yourself that Monin-Obukhov length scale (L) is simply the ratio of shear production to buoyancy in TKE budget and is NOT tracer dependent.

Now back to dimensional analysis. The group of variables are unique to our problem are :

1) z/z0 , 2) C/C* and 3) **z/L

where C is mean concentration (C bar in previous equation). Similar to what we do for temperature and humidity. A ψc can be calculated by choosing the second and the third group as independent variables and we would arrive at your equation.

Regarding the Gaussian plume, Remember that the equations you posted are only applicable near a boundary and only used to drive eddy diffusivity close to the wall layer e.g. wind tunnel, ocean, and earth. If I recall correctly Gausian plume model use different diffusivities compared to these and are more appropriate to interior of the flow.