If you had limits from 1 to 100, you would expect about 90% of the colors to be in the 10 to 100 range and about 10% to be in the 1 to 10 range, right? (Actually 90/99 and 9/99, respectively.) That's the same situation as you have here, except that the limits are scaled up by 1e10.
What you are really after is some sort of log-scale colorbar, which as far as I know, there is no built-in support for (maybe on the file exchange you can find something). What you can try is to use log10(C) for your color values and use log10() of your color limits:
pcolor(X,Y,log10(C));
caxis([10 12]);
colorbar();
then manually set the YTickLabels of the colorbar axes to {'10^{10}', '10^{11}', '10^{12}'}. It may be a little tricky to maintain the correct YTickLabels on the colorbar as the data change, but it is feasible.
