You can see the “edge” of a cloud from far away so it should be obvious what ∂cloud means. But up close (from an airplane) you can see there is no edge. The mist fades gradually into blue sky.
Here’s another job for schwartz functions: to define a “fuzzy boundary” ∂ that looks sharp from far away but blurred up close. In other words, to map each Cartesian 3-point to a fuzzy inclusion %∈ in the set {this cloud}.
Jan Koenderink, in his masterpiece Solid Shape, notes that a typical European cumulus cloud has density 𝓞(100 droplets) per cm³ (times 16 in inch⁻³). Droplets are 3–30 μicrons in diameter. (3–30 hair widths across) Typical clouds have a density of .4g/m³ or 674 pounds of water per cubic football field of cloud.
To lift directly from page 508:
What is actually meant by “density” here? Clearly the answer depends on the inner scale or resolution.
At a resolution of 1 μm the density is either that of liquid water or that of air, depending critically on the position within the cloud. At a resolution of ten miles the density is near zero because the sample in the window is diluted.
Both results are essentially useless. The right scale is about a meter, with maybe an order of magnitude play on both sides.
Rather than having just one sharp boundary, ∂cloud is a sequence of level surfaces that enclose a given density at a given resolution. To avoid having to choose an arbitrary resolution parameter, we can define the fuzzy inclusion with a schwartz function. We get a definite beginning and end (compact support) without going too into particulars (like rate of the %∈ dropoff) and this is true at any sensible resolution.
We can’t say exactly where the boundary is, but we can point to a spot in the sky that’s not cloud and we can point to a spot in the sky that is cloud.
