Here is the thing that I am looking for. I want to plot the cumulative sum of the flux let's say starting from location of Melbourne covering all around Australia and ending at Melbourne.
So it will give me line plot which show where (lat,lon) the flux is increasing or decreasing.
If I understand correctly, that's exactly what I did in my paper:
The basic principle is "easy": Let our "coast line" exactly follow the outline of the gridboxes that form Australia. In this case, the "cross-shore direction" is either zonal or meridional exactly. Therefore, the cumulative sum you want is the running sum of u Δx or v Δy, depending on the orientation of the local "coastline".
Unfortunately, implementing this is complicated in my experience.
If all you want is the total flux, instead of a running sum, then you can use the divergence theorem as Billy said. . . . Come to think of it, you could set up an artificial domain (box) that gradually "grows". At the artificial boundaries through which you don't want the flux to go, you set the flux to be zero. Then you apply Gauss's theorem to get the flux across the portion of the coast the domain currently occupies. As the domain "grows", the coastline extends and you get the cumulative flux. . . .
Cheers,
Ryo