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Based on an answer from ECMWF support which OP can confirm to my best understanding relative vorticity in the ECMWF model is not calculated using grid points and finite differences (centered and forward and backward). Instead it is calculated using spectral approaches in meteorology. There is a package in Fortran called spherepack and a python wrapper as ...

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When I look at problems like these I first check to see if there is a well tested and well documented implementation already rather than reinventing the wheel. In this case MetPy temperature advection is a well tested software that does many of the things meteorologists want including calculating finite differences with the right map scale factors. Since ...

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So i don't know for how many timesteps you've integrated. And I don't really understand what you have done to obtain the temperature $\rm T[t,s,y,x]$ in your second box, but this seems a bit like a redundant operation: the end result is $T:=T$, so of course your right side looks like the left side, in terms of the temperature colourmap. I don't understand ...

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