Is there some specific function? No. But what does a high density of contour lines mean? It says the gradient is large in that vicinity. For example, I have no idea what this surface looks like in advance. Just some waves, some bumps.
fun = @(X,Y) sin(X+Y).*cos(X-2*Y);
[x,y] = meshgrid(linspace(0,pi));
We can clearly see where the contour lines pile up in the plot. But we can also get a measure of that from this plot:
pcolor(x,y,sqrt(gx.^2 + gy.^2))
You can see the flat spots, generally the tops or bottoms of the bumps are represented in blue. Of course, we can also have a local flat spot at a saddle point. The regions of high contour density are yellow. Regions of intermediate contour density are seen as green in the pcolor plot.
Do you see that what I am computing here is the local norm of the gradient vector? Where that norm is a large number, then the function is changing rapidly. Where the norm of the gradient is small, we are in a locally flat region, and so there are few contour lines piling up.
Try it on another function. This time I'll use peaks. While I know the basic shape of the peaks function, I don't know where to expect the regions of maximum contour density will be in advance. I can make a guess though.
[x,y] = meshgrid(linspace(-3,3));
pcolor(x,y,sqrt(gx.^2 + gy.^2))
So the steepest region, with the highest contour density is the bright yellow blob near the bottom middle, near the point (x,y) = (0,-1). There is another region of high contour density near (0,1). And of course on the perimeter of the plot, the function is almost flat, so there are very few contour lines seen at all. That entire region will be dark blue.
If the function is known, so we have a functional form for it, then we could use an optimizer to search for the point where the norm of the gradient is maximized. On the peaks function, there would clearly be at least 3 local maxima of the norm f the gradient. Actually, I'd guess we could find at least 5 local maxima. (Not difficult to do, but please don't make me write code to find all of the local maxima of the norm of the gradient function. I'm just too lazy.)
Anyway, you should get the general idea from these examples. Why does the norm of the grient work for this? If a contour line represents the locus of points where the function takes on a specific fixed value, then if you have many contour lines in one region, it means the function is changing rapidly in that vicinity. Therefore the gradient would have a large norm. And while there is no explicit function that computes the contour density directly, it is almost trivial to do the computation.
