Or what is the surface area of any fractal solid?
This is strongly related to the Coastline Paradox:
Just with the added dimension of a solid.
Clouds are ‘self similar’ in many cases, yet also can have very different forms too. Watching clouds form, they grow, shrink, shift, stretch and connect, or dissipate a connection. So there is a dynamic aspect too. By the time you figure out what the surface area might be, it is different.
Now further consider that the “surface” does not, in fact, exist. Clouds form from water VAPOR transparently dissolved in air. As that vapor finds a “condensation nuclei” it starts to collect and make a droplet. The droplets grow and continue to coalesce. So what we see from the ground as a cloud is, in fact, “Billions and Billions” of water droplets. Each with a surface area and an edge; but each distinct from the next (or surface tension would form them into one bigger drop). So we can see (in a fuzzy way) the boundary where water droplets are big enough to interact with light and look like a cloud; but that is NOT the surface of a cloud. The cloud has no surface, only the droplets doo.
As the droplets coalesce into larger drops, there are more of them but with less surface area. Held up by Brownian Motion of air molecules. Eventually the drops get big enough that random air motion can’t hold them up, and we get rain. After enough rain, what is left of the cloud dissipates back to water vapor again. Along the way the cloud moved from zero mass / surface area, to some clear mass (measured by total precipitation if nothing else) and large surface area visable, then back to zero mass and surface area again. So if density is mass / volume, and we don’t (can’t?) know the mass or the surface area (and from surface area the volume); if we can’t know those basics, how can we compute the density of the cloud?
I started this ponder in the middle of the Arizona Desert, watching the sun poke though bits of cloud in the rain. How much sun gets to the ground? So what is the density and optical opacity / reflectivity of the clouds? How to compute it? One spot was bright sun. Next to that a patch of solid dark. Some gray in between and around edges. What is the albedo of that? Even the average albedo requires some idea how much is clear and how much “solid” cloud. No surface area. No volume. No density. No mass. How to model that? How to calculate it? How to even measure it? What cell size could even start to capture that? A meter? Maybe 100 meters? A cm?
In the end, it all comes down to parameters. Picking “plug numbers” that you think or guess are close enough to what you think you observed. It is not possible to calculate the surface area of a fractal from first principles. It is not possible to measure it. You must “make a good guess”. Use some kind of “rule of thumb” to shrink the problem (like sailing 200 m off the coastline).
But if all your basic model paraneters for core properties like albedo, solar absorption in clouds, transmission to ground, precipitation, and thermal mass are all essentially “plug numbers” then your model is disconnected from reality. It is a dream (or nightmare) embodied in software guesses.
All this leads me to conclude that any hope of a representative Climate Model is folly. Simply because you can not accurately compute the basics of the water cycle and clouds.
Even the ocean is a fractal surface. What is the surface area of the ocean? First compute all the coastlines of all the land masses and remove that area from the computed surface of the earth… oh, yeah, the coastline problem… Then figure out the ‘roughness’ of the ocean. It changes with the wind. Waves from a few inches to 100 meters high. With swells and ripples on them. With wind blown spray and crashing crests. What is the surface area of the ocean? A “plug number”. Guess well… So what is the heat and water vapor transmission through that surface? W/m^2 with unknown m? Kg/m^2 with unknown m?
It all comes down to the interaction of a bunch of surfaces and volumes with fractal geometry. Good luck with that…
So I’m now trying to figure out some way past this conundrum. How do you weigh a cloud? Without mass, volume, and surface area; how do you compute thermo properties and effects?
I think I see why they say “modeling clouds is hard”. It may be impossible.