Kubunutu 12.10 motivation screenshot

Drawing in Latex

Here are some pictures I have created using LaTex and TikZ.

Figure 1

Figure 2

The code TEX used for generation of the images is shown below:

Gradient of scalar product or divergence of a tensor.

When you read a scientific work with equations there could be surprises and confusion even in the notation. Here I want to speak about the one I encountered recently, when looking at the Navier-Stokes equation.

First I thought, the result of the operation should be a vector since it appears as a term in a vectorial equation so it was natural for me to consider it as a gradient of a scalar product of two vectors. Then I started remembering the formula for the gradient of a scalar product.

Here the star represents the vector on which the nabla operator acts. Those could be expressed as follows( (*,*) - scalar product, [*,*] - vector product ):

and swapping a and b we also get

Then combining the last 3 equations we get:

Which differs from the first equation. Then It came to me that the term

is a flux of a momentum, that is a flux of a vector quantity. This means that each coordinate of the quantity should be multiplied by each coordinate of the velocity, which gives us the following tensor:

Then each component of the divergence D of the flux will have the following form

which is equivalent to

Which, I suppose, is exactly what the author had in mind.

Summation along a dimension

Suppose you have a netcdf file with temporal data of say evaporation (in meters). And you want to know how much water evaporates during a year. And also you do not want to loose time writing a program for this. Fortunately we have NCO. I did the procedure for the ERA-Interim reanalysis file downloaded from the ECMWF site. I picked the year 1985. The header of the file looks like this:


Here is the description of the procedure that I used:

1. Calculate the mean of the variable along the specified dimension (time in my case).

   ~# ncra -d time,0,729 -v e evap-1985.nc mean_evap_1985.nc
   

   Here I was not sure whether I should take care of the scale_factor and add_offset. The answer is no, because nco takes care of it for you.
2. Then, since sum = mean * ntimes, I use the mean to get the sum. Minus is because of the model convention, they consider the flux negative if it is upwards.

   ~# ncap -O -s "e=-730*e" mean_evap_1985.nc sum_evap_1985.nc
   

3. So we have our resulting field in the file sum_evap_1985.nc, which can be viewed using ncview. The field looks as shown below. I do not know how to save the legend in ncview, so the figure is not very informative. The result corresponds to the means given by ECMWF: E-P and P.
   
   
   

Data source: "ECMWF ERA-Interim data used here have been obtained from the ECMWF data server."

ECMWF variables description

The table of variable names and units in the ECMWF files: http://www.ecmwf.int/publications/manuals/d/gribapi/param/