[Space] Weather on planets & moons

[Agemegos]

I'm trying to work out how to do a "near enough is good enough" estimate of three aspects of the weather on worlds generated using the world generation sequence in GURPS Space 4th edition. I want to be able to spot weather effects severe enough to challenge habitability.

1. Daily temperature variation
   My idea here is that it ought to be proportional to the day length and the average temperature of the planet, and inversely related to the Greenhouse factor (which depends on the thickness and composition of the atmosphere). But it ought to be capped for planets that rotate slowly: no greater than the difference between dayside and nightside temperature for such a planet in such an orbit if it were tide-locked (and without atmospheric collapse).
2. Latitudinal temperature variation
   How much warmer are the tropics than the average? How much cooler are the poles than the average? I reckon that the effectiveness with which the atmosphere transports heat from equator to poles is proportional to the temperature difference driving that transport, and that therefore the temperature difference ought to be proportional to the square root of (average temperature * diameter / thickness of atmosphere). But things will be different on a tide-locked planet or one rotating so slowly that heat is drawn to the terminator rather than the poles, i.e. distributed by atmospheric circulation faster than by planetary rotation.
3. Windiness
   I reckon that this is proportional to the square root of (daily temperature variation squared plus twice the difference between equator and poles) times the relative strength of the Coriolis Effect (which is the diameter of the world divided by its rotational period).

Comments?

[Trachmyr]

Not a helpful comment, but I would be very interested in using what-ever you come up with. I list weather and temp variations for the worlds I write up, but have to guestimate (more guessing than estimating), so I would love some sort of formulas to use as a guidline.

One thing though... wouldn't Hydrodynamic coverage play a role into Heat dispersal. I currently assume the greater the Hydrodynamics, the lower the lattitude and nighttime variations.

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I would also love a formula to determine nightside and dayside temps on rotating bodies w/o atmosphere. Consider our own moon for example, and "Average Surface Temp" hardly tells the story.

And while your at it with windiness... perhaps use that same coriolis effect (with mass/density) to give a rough estimate of the planet's magnetosphere strength (i.e., None, Weak, Average, etc.), something which has a major impact on the development of life.

[RyanW]

Obviously, the day/night difference will approach zero as the rotational period approaches zero, and approach the tide-lock numbers as the rotational period approaches the year length.

If it helps, the data for the moon:

Equatorial: 100 K minimum, 220 K mean, 390 K maximum
Near poles: 70 K minimum, 130 K mean, 230 K maximum
Permanent shadow: as low as 26K

[martinl]

Originally Posted by Brett

1. Daily temperature variation
   My idea here is that it ought to be proportional to the day length and the average temperature of the planet, and inversely related to the Greenhouse factor (which depends on the thickness and composition of the atmosphere). But it ought to be capped for planets that rotate slowly: no greater than the difference between dayside and nightside temperature for such a planet in such an orbit if it were tide-locked (and without atmospheric collapse).
2. Latitudinal temperature variation
   How much warmer are the tropics than the average? How much cooler are the poles than the average? I reckon that the effectiveness with which the atmosphere transports heat from equator to poles is proportional to the temperature difference driving that transport, and that therefore the temperature difference ought to be proportional to the square root of (average temperature * diameter / thickness of atmosphere). But things will be different on a tide-locked planet or one rotating so slowly that heat is drawn to the terminator rather than the poles, i.e. distributed by atmospheric circulation faster than by planetary rotation.
3. Windiness
   I reckon that this is proportional to the square root of (daily temperature variation squared plus twice the difference between equator and poles) times the relative strength of the Coriolis Effect (which is the diameter of the world divided by its rotational period).

Comments?

Hydrographic coverage and distribution will have a significant effect on all of these, as will major ocean currents.

[Agemegos]

Originally Posted by RyanW

Obviously, the day/night difference will approach zero as the rotational period approaches zero, and approach the tide-lock numbers as the rotational period approaches the year length.

I think it is more the case of the daily temperature variation approaching the tide-lock numbers as apparent day length approaches some large value that depends on the transportation of heat by atmospheric and oceanic circulation. Basically, heat is transported from the midday point to the midnight point by two mechanisms (three if you count mantle conduction, but I'm pretty sure that is negligible).

First, heat is carried to the nightside by the rotation of the planet: air, water, and rock/soil get warmed in the sunlight and carry heat with them as they rotate to the night side. Air, water, and rock near the subsolar point warm most and carry their heat strictly eastward. When this mode dominates you get heat carried across the terminator mostly near the equator, and so the tropics are warm and the poles are cold.

Second, heat is carried away from the subsolar point by the circulation of the air and oceans. In the first approximation that is uniform around the terminator, and the poles ought to be no colder than the points where the terminator intersects the equator. But the Coriolis Effect interferes with that. Warm winds off the equator tend to be heading polewards, and get turned to the East (ie. they become westerlies). Cold winds off the equator tend to be heading equatorwards, and get deflected towards the West (they become easterlies). This gets complicated (I have seen simulations, I didn't really comprehend the results), but the belts of easterlies and westerlies limit the latitudinal transportation of heat. So the poles are somewhat cooler than other points on the Terminator, but not as cold at the midnight point.

When the planet is rotating at a speed large compared with the typical speed of the winds, then more heat is carried to the nightside by rotation than circulation, and you get a big difference between poles and tropics and a modest difference between day an night (such as we are used to: Earth's equatorial speed of rotation is about 465 m/s, and winds are typically much slower than that, ocean currents even slower). But as a planet rotates slower and slower the transportation of heat by circulation comes to dominate the transportation of heat by rotation, and the pattern comes to approach the pattern with tide lock.

In the case of a planet in spin:orbit resonance the day may be over 5,000 hours, in which case the equatorial speed of rotation is only about 2 metres per second, which is slower than typical winds and some ocean currents. Circulation will dominate rotation as a mechanism for transporting heat, and something like the tide-locked pattern will emerge. (one important difference will result from the advection of latent heat: no great ice-cap will form over the midnight point (as it does not a tide-locked world if the night-side temperature is below freezing) because the time-scale for such masses of ice to accumulate is centuries or millennia, and the equatorial ice gets carried into sunlight by the rotation of the planet on a scale of months.

Originally Posted by martinl

Hydrographic coverage and distribution will have a significant effect on all of these, as will major ocean currents.

Yes, though the effect on daily temperature variation can be ignored if I report a figure standardised to, say, a continental west coast at 45° latitude rather than an average over the whole planet. That might be a reasonable indicator the quality of experience on the planet, a basis on which it is easy to apply modifiers for a more continental or more maritime set of circumstances.

I might include a factor for the extent of the oceans, but I think that that is as far as I can go towards estimating the effects of things like the Gulf Stream. I'm not even generating maps!

[Fish]

What about polar ice caps?

[Agemegos]

Originally Posted by Fish

What about polar ice caps?

They'll form if the polar regions are consistently colder than freezing point, which depends on the average temperature and the latitudinal temperature variation.

For example, Earth's average surface temperature is 14.7 C, and the latitudinal temperature variation is approximately 50 K. The thermal equator is about 30 C (annual average around the world: warmer over Africa, cooler over the Pacific) and the poles are very roughly -20 C (annual average reduced to sea level: the North Pole is not quite so cold because of the sea water under it, the South Pole is a bit colder because of altitude and distance from the sea. We have icecaps where there are extensive bodies of land with average annual temperatures below freezing, and pack ice when the air temperature is below about -9 C.

Supposing that a planet 80% of Earth's diameter with 1.25 bar of atmospheric pressure had 80% Earth's latitudinal range of temperatures, then its equator might be 12 K warmer and its poles 28 K cooler than its average. So if its average surface temperature were 28 C or more it would be ice-free except for winter snow and sea ice, and a few localised glaciers at high latitude and altitude, whereas if its average surface temperature were -12 C it would probably be frozen from pole to pole except for a few isolated warm spots near the equator and volcanoes.

[martinl]

Originally Posted by Brett

Yes, though the effect on daily temperature variation can be ignored if I report a figure standardised to, say, a continental west coast at 45° latitude rather than an average over the whole planet. That might be a reasonable indicator the quality of experience on the planet, a basis on which it is easy to apply modifiers for a more continental or more maritime set of circumstances.

This should work as a rule of thumb in many cases, but at very high (90%+) or moderately low (40%-) hydrographics, you might not have a "continental west coast at 45° latitude." Admittedly, these sorts of places are probably less habitable.

You are going to be pretty limited is defining climate without maps.

[Agemegos]

Originally Posted by martinl

You are going to be pretty limited is defining climate without maps.

Yes. But mapping every planet and doing a full circulation model is a task for a climatologist with a supercomputer. It is not a minor extension to the star system generation sequence in GURPS Space. You are suggesting that I graft a thousand-ton tail onto a fifteen-pound dog.

[su_liam]

As a disclaimer, I should say that I use a planet design sequence based on 2nd, 3rd and 4th ed GSpace, GTFirstIn, 2300AD, bits of Space Opera and MT World Builders Handbook as well as about 300 gigs of scholarly planetary science, climatology, and astrophysics pdfs. I'm up to nine three-ring-binders of dense, crabbed manuscript so far. I figure this subject would be worth about 128 pages of charts. So YMMV, and that would probably be a good thing. Also as a fresh victim of a physical geography degree, I'd have to agree with Martini about maps!

But what can be done for folks who are perhaps less psychotically obsessive about the subject. Let's try to simplify. The first problem is temperature variation is going to be dependent on the rate of atmospheric heat transfer. The rate of atmospheric heat transfer is dependent on wind. And, ooh we're getting close, DOH! Wind depends on temperature variation. How do you solve for that? If you know a bit of differential calculus, Newton's Method is a pretty girl that wants your body(assuming you're a boy). Barring that, the Bisection Method is uglier and only likes you for your money, but it might find a solution. The Newton's Method will get you there. Eventually. Here, we find ourselves back in the realm of computer modeling. It is a lot simpler than even a baby global circulation models that Creationist Global Warming Deniers would point at and laugh, but a little much for an RPG that I ain't GMing. On top of that I think we do need to take the hydrosphere rating into account. For verisimilitude if nothing else.

For GMs between, "It rained on Mongo that day,"(Who, really don't care about this kind of detail), and, "Oh my Gaawd, I just got a supercomputer cluster and a team of PhD Climatologists and Programmers for my birthday!!!"(NSFW, and that should be ALLCAPS!!!!), a set of charts would be a good idea. The more I think about this the more complicated it gets. I've spent four hours pacing, thinking and writing. The more time I spend the more insane this gets. Okay, so I'll throw out a formula for mean latitudinal temperature I found in a book on paleoclimatology(Brooks, 1949)*:

T(phi) = T0 + a*cos(phi) - b*cos(2*phi) + c*n(phi)*cos(2*phi),

where phi is the latitude band we're interested in, T0, a, b, and c are a set of constants we need to figure out for a planet, n is the fraction of the given latitude band covered by land, and Tavg is the average temperature of the planet surface which we may need later. So far the simplification isn't quite simple. For Earth in the present interglacial, T0 = 27.6ºF, a = 32 Fº, b = -13 Fº, c = 35 Fº and Tavg = 59ºF.

The equation for Tavg, which we should know from the planet design sequence, I calculated to be:

Tavg = T0 + pi/4*a - b/3 + c*[definite integral from phi = 0 to phi = 90º](n(phi)*cos(2*phi)*cos(phi)*d[phi]).

Taking into account the variation of surface area at different latitudes. That looks fun... :\ ... Let us assume that land is evenly distributed across all latitudes, such that n(phi) = 1 - ([%hydrosphere] / 100)

Tavg = T0 + pi/4*a - b/3 + c*n/3

To solve this we also need maximum and minimum temperatures.

Tmax = T(phi = 0º) = Tequator = T0 + a - b + c*n(equator)

Tmin = T(phi = 90º) = Tpolar = T0 + b - c*n(pole).

Ah, dang! All this work and right now it looks like we are right back where we started from. I still think this could be useful for weather modeling down the line. Keep in mind that increased moderating effects such as axial tilt, greenhouse effect and hydrosphere as well as high wind speeds will decrease b and c, with a commensurate increase in T0. Larger land masses and a larger planetary diameter will make variations more extreme, largely due to retarded heat exchange, so higher a,b, and c, and lower T0. I think higher gravity would result in a larger proportion of atmospheric mass near the surface, causing more wind resistance, so higher a, much higher b and c and lower T0. A more mountainous surface would result in a higher c value and probably b, with no real effect on a and reduced T0 in consequence.

I need to think on this some more.

*Brooks, C.E.P., 1949, Climate through the Ages, New York, McGraw-Hill