All,
Let $S(x,y,t)$ be a variable function in $x$, $y$, and $t$. After
performing Reynold averaging over area $\frac{1}{A}\int S(x,y,t) dA$, could $S$ still be a function in $x$, and $y$?


[Equations (1-18)](https://www.ecmwf.int/sites/default/files/elibrary/2017/Atmospheric_moist_convection.pdf) of dry statics energy ($s$) and specific humidity ($q$) find the gradient of area-averaged variables ($s$ and $q$) indicating that that Reynolds-averaged variables might be still a function in space. Any idea how that might be the case?
Thanks.



Could variable be still function in x and y after performing Reynolds averaging over area
Based on the comment by @CarloBeenakker, Coarse-grain averaging could be performed on a subset of the domain (spatial or temporal) so that the averaged value still varies with the independent dimension. I just wanted to document some resources (below) that might be helpful for others.


The classical Reynold average in which the average is performed on the whole data is based on the assumption that the data is stationary, yet if the data is not statistically stationary then the coarse-grain average might do the job by calculating the average at each subset of the data (recall running-average). I am not sure if the coarse-grain averaging is a subtype of the Reynold average.


The coarse-grain technique is discussed in the following textbooks


* Storm and cloud dynamics by Cotton and Anthes [Chapter 3](https://www.elsevier.com/books/storm-and-cloud-dynamics/cotton/978-0-12-088542-8)
* Mesoscale Meteorological modeling by Pielke Sr [Chapter 4](https://www.elsevier.com/books/mesoscale-meteorological-modeling/pielke-sr/978-0-12-385237-3)


