What would the weather be like on a infinite gentle slope?

Question

Imagine a Yorkshire hill. This place is like that Yorkshire hill in all aspects except for those I describe. Its size is 100km by 100km, with a slope of 0.5%.

When going up the slope and crossing the edge one would pop up seamlessly on the other side but 200m lower down with regards to the way gravity is pointing. The opposite would happen when crossing the border when going down (being popped to the top of the hill 200m higher in elevation and 100km over). The sides do the same to each other but without the change in elevation.

If visibility was good and nothing was blocking your view then you would see yourself infinitely many times all 100km apart. There will be some forest, some meadows, plenty of water, air, and light. Just an idyllic place to live. Were it not for being trapped within.

It would be like the Truman Show but contained by something like invisible portals. Working in a similar way as those in the portal games.

A diagram for clarity

You can see meadows(light green), forest(dark green) and rivers(blue)

The two red borders are the sides and just plop you (anything) out of the other the same distance away from the dark blue border. The dark blue and purple borders do the same with respect to the right red border. Only there is the added change in elevation. I have made sure that the blue river and streams are continuously flowing downhill.

What would the weather be like?

Would there be a constant wind? In which direction? Could it be quite regular due to its small size? Would you often see clouds? What kinds?

Answer 1

With is what I can say from my physics studies :

The need for a homogeneous pressure

The pressure is continuous in space, otherwise a sudden air movement occurs to get back to continuity (we usually call it an explosion). However, in a real atmosphere (take Earth's as a representative exemple), the pressure changes with altitude. Here, this would imply that their is a pressure discontinuity between the sides of different altitude.

As we've seen, this cannot be stable, so the pressure must not change with altitude. We end up having a toroidal atmosphere with homogeneous pressure.

As a scenario, you can imagine the beginning of this world like EdwinW in his answer : a single normal cell with real atmosphere suddenly wrapped around itself in a torus. Then there is an initial pressure discontinuity, but the initial bang (one could actually call it the "big bang", lol) would neutralize this discontinuity. As there would be no pressure-related acceleration mechanism, the initial wind created by this bang would eventually stop as the atmosphere would evolve towards a homogeneous pressure, which is essentially the case I will develop further on.

This homogeneous pressure makes wind

The pressure being the same everywhere, the only non-neutral force is weight, pulling air down the bottom. However, usually the ground counters this effort vertically (that is, perpendicular to its surface), but here is always counters it by a sideways force because of the slope.

This phenomenon would be the genesis of wind : at the beginning of times, this resultant effort pushes the lower layer of air downslope. As the lower "layer" flows down, it pulls with it "layers" from higher in the air, building a boundary layer which would grow up without end.

As the winds gets faster and faster as a result of "falling" down an infinite slope, it is more and more slowed down by the ground itself (mostly by trees I would say), and it will eventually reach a maximal velocity. Note that it can be quite high, it's up to you to imagine how much if you don't want to run the computations. It's mostly here that you can trick with the physics of your universe to make the wind speed compatible with life. But since the slope in infinite, there is nothing that can stop the wind : you get a homogeneous downslope wind.

From then on, at a given altitude (perpendicular to the surface) the wind speed would be constant or periodic, since it must be the same at each boundary. I guess obstacles could create a small periodic variability of windspeed, but not so much that it could turn around for instance.

This effect of the ground slowing down air would become less visible in altitude, so the higher (perpendicular to the surface), the faster the wind is.

If the wind reaches supersonic velocity, each obstacle on the ground would create a shock, and complexities would arise in the overall picture...

Other weather features

Having a very fast homogeneous downslope wind is already quite an interesting weather feature. However we can imagine other perturbations on top of this, as long as they are periodic with regard to the boundaries.

For instance, there could be a constant or periodic wind sideway in either direction.

I wonder if any rotating speed variation of the wind (like a hurricane) could exist with a diameter larger than 100 km... it's an interesting question.

How loud sounds behave

There are interesting phenomenons that happen when a sound crosses a whole cell (100 km x 100 km). I agree that a sound must be quite loud to cross 100 km but let's imagine.

You can hear an echo of it from the other direction, but it decreases in strength at it usually does in air.

If you have a constant loud sound (say you shout with a tremendous voice), the sound will interfere on the borders. You would have both constructive and destructive interferences, and thus a whole interference pattern. It may be feasible to have a constant self-powering noise if the original source is loud enough.

We must take the wind into account and it adds a doppler effect. So if you sing really loud, you may here yourself on different frequencies (depending on the wind speed). Depending on the musical faithfulness of the frequencies, you may be able to sing a polyphony despite being alone. You definitely created a very interesting universe !

Answer 2

Perpetual Downhill Cyclone

Imagine a bowling ball placed at the top of the slope. It rolls downhill for 100km and then teleports back to the top. Now it is where it started but moving faster than it was before. It rolls downhill another 100km and then teleports a second time, moving faster still. This proceeds until finally it is moving fast enough that the air resistance on the next downwards trip cancels with the increase in speed.

Now imagine a million billion tiny bowling balls (air patricles). They are pulled downhill under gravity and teleport to the other side and are pulled down again. This results in a perpetual wind blowing downhill around the world. It might be a gentle breeze or it might be a hurricane. I don't know the relevant equations to model this.

This of course is a perpetual motion machine. Every teleport adds energy to the system. Air friction causes the world to heat up over time.

Note: I am assuming gravity is straight downwards. The direction does not change from one point of the slope to another.

Answer 3

Edit: I expanded my answer in response to comments by @Daron.

There would be a steady downhill wind, possibly quite a strong one. 0.5% of 100 km is is half a kilometre, 500 meters.

Suppose upper side would not be connected to the lower side and we take the lower side is assumed to be at sea level. The pressure at 500 m elevation is about 60 millibar lower than at sea level, about 950 millibar. This would be a equilibrium state.

If we now connect the two sides, air will start to flow from the high-pressure zone to the low-pressure zone until the pressure equalises everywhere.

In the real world, pressures as low as 950 millibar are normally strong enough to produce moderate hurricanes. The mechanisms are not necessarily the same, but it's the only clue I find on how much wind you get when equalising the pressure. If I'm to guess I'd think you get stronger winds in your world than would be suggested by the hurricanes, but that's conjecture based on the fact your system would be permanent.

What you want to find is the equilibrium state where the air is moving fast enough to stabilise the pressure.

Without some mechanism significantly slowing the wind down, not just on the ground but at high altitudes as well, your world would experience a perpetual storm.

Edit: I should give some references...

According to the diagram at this site the pressure at 500 m is about 955 millibar, and according to this site 955 millibar corresponds to a category 3 hurricane.

Answer 4

Fun things

This is a fun question, although other answers have covered the main points already. I'll look a bit at how not to doom your world.

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Infinite tiled plane (doomed if left as-is)

First, I'll look at it from a different perspective, slightly counterintuitive. The ground is flat, but gravity is crooked (and height-invariant). We're standing on an infinite plane that happens to start as a precisely-equal repeating pattern. (That's what it'll look like from inside, and it's equivalent.)

Air will fall sideways, and so will water. This will cause erosion, and there's nothing but the inertia of the ground to stop the erosion. Your atmosphere will tumble down this slope, never pooling at the nonexistent bottom. Soon voracious winds tear the ground up. If gravity fades out with altitude, it's disturbingly easy to achieve escape velocity, and most of your atmosphere probably will.

Preservation

Suppose that the portals cease to function below ground level. This means the ground has something to push against, and will be held up as if by terraces on the infinite plane. The wind still shreds everything, but the debris doesn't necessarily lift into orbit in a plasma cloud.

Suppose that the ground rests on a frictionless solid plane. Then the whole world accelerates at the same pace, and settles into balance. Survival, but the world is constantly accelerating, so the wind and water will be moving pretty fast. However, you can fiddle with your slope to moderate that, or erect large sails or specialised trees. This only controls for the slope, the portals/tiling can still doom your world.

Because you're accelerating, you'll get a Doppler effect for light as well as sound, although it's unlikely to be redshift-worthy outside specialist studies.

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Sunshine

Another answer mentioned this could destroy your world, purely because of the tiling. The theory goes that the walls will also make the sky look tiled into 100km squares, and that the Sun will appear in every square at once, giving your world a few moments of "touch the Sun" heat at high noon and complete night the rest of the time. In theory, this is balanced because the Earth has the entire night to radiate that heat outward, and it might be a nice plot point to have your world's temperature spike to lethal levels once per day. But it'll need some exotic vegetation to survive in it.

This is not unsolvable. To save some time with altitude-shenanigans, we'll assume that light shines "down" matching the direction of gravity exactly. We need an average light-level over thewhole sky to match what the Earth sees (more or less). That's about one part in 100000, and will look like a little over-bright speck overhead. Shadows are crisp and always downward.

To make things more interesting, let's consider a diffuse light source. Now the entire sky glows faintly, constantly, and evenly; and you're living in the Twilight Forest. (The Edge? Minecraft mod? There are a few of them about.) Shadows are very blurry.

(NB: Your portals are still injecting energy into the atmosphere. You might want to turn down the Sun a bit to counter that heating, though most of the energy is accelerating the ground along its infinite plane.

What if we liked day and night?

To get day and night, we only have to shine down from some areas more and others less. For the fun of it, we'll say that day and night look like great stripes moving downhill (and across a bit), 100km wide. By some trickery with angles (we're running lighting for The Truman Show here), we can have morning and evening. The fun part is when the atmosphere distorts the "sun", and makes it seem to twinkle: it will blink out for one moment, then stretch along the day-line the next. It will also seem to wobble back and forth in the direction that day moves, but remain fairly consistent in brightness.

At 100km/24h, that means that day and night move at a comfortable walking pace.

(Also, the remains of past morning/evening light could in theory form a tiny rainbow-aurora near the horizon, doppler-shifts affecting the angle. But atmospheric diffusion would blur this into oblivion unless you went into space. "Orbit" isn't a thing in this world, but you can stay up with minimal push ... in one direction.)

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Sound

The idea of singing to yourself is fun, but your sound will disperse just as it would in any other medium. A large shock (explosion?) will die out much like a ripple in a tank of water. If you have a very loud sound, you might be able to exploit resonance to create a standing wave - but this is mostly a concern for the weather, since it will take 50 minutes for a sound to do a lap of your world (ignoring wind).

That very low drone might build up based on landforms, but I can't say whether it would effect or even much affect erosion. It would probably result in fun cloud-patterns, though.

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Weather

OK, so to give a proper answer to the original question rather than just playing with your world:

The weather will still be extremely windy. The entire world is accelerating constantly, and air cycles through the portals more often than the ground. This means that the atmosphere is dragging the ground faster and faster, and the ground is slowing the atmosphere ... a bit. Since the amount of energy you need to impart to the ground is a function of mass, I suggest the thinnest ground you can get away with, and a lot of forest to act as sails. A wet climate (more water in the air) will help your vegetation not to lose too much moisture to transpiration. Also avoid rivers, they will become boiling torrents very easily. If you do have one, let it be a small stream and human-made. This argues for a more arid climate.

For a good idea of what the weather might do, look at an oil-and-water-between-moving-plates demonstration. There should be quite a few online.

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Thanks for asking this, and for reading an largely tangential answer. I hope it's helped you to avoid accidentally sandblasting/baking/sending-to-orbit your world.

To quote the Discworld: "You might not get what you asked for, but you'd get what you wanted."

I'm inclined to think you wanted solutions rather than problems.

Answer 5

infinite downhill wind

Imagine having 2 portals. One above the other. Drop a stone in the bottom portal and it'll fall out the top one, only to fall into the bottom one again. The stone will accelerate to terminal velocity, where it can't accelerate anymore because of the air resistance equalising to the pull of gravity.

Your situation is much the same. You have a gentle slope, but one end is slightly higher than the other. With physics what they are your air column should be slightly heavier on one side. This is because the air column us higher, has more air and slightly more air pressure. Air behaves much like water. It moves from high to low places. So this is the first situation I would imagine.

Only this isn't a stable situation. Right next to the downhill higher pressure and heavier air is a lower area with a lower air pressure. It'll move there to equalise the air pressure as well as to continue moving down the slope due to gravity. It'll appear at the top of the hill, meaning it can go further down or at least put pressure on the lower laying air pressures. That means the lower air column will be pressurised, making it a higher pressure than the higher air column. The cycle repeats, much like the stone falling in the first example.

There are some extra factors here at play, like that the high pressure air column will heat up faster and generally be hotter than the low pressure air column. This will in general increase this effect.

Will this lead to high winds? Incredible devastation of hurricane proportions that scrape the land clean of anything, playing with the dust in an infinite loop of speeding up wind?

Probably not. Just like the stone there is a terminal velocity. Each air particle does bump into others when moving from one side to the other. Though other people can still conclude that in effect the whole column will start moving quicker and quicker, as the last air particle has moved a bit faster so there is less air resistance overall, I think it'll not be such a case. The column is small for air sizes. It'll probably interfere with itself too much. In addition, the gravity isn't very powerful and the air pressure differences are tiny on such a small area. It'll probably sort of equalise at a small breeze.

I mean if you had an infinite water stream you would also not conclude it would be a torrent of high speed water after some time. It reaches a 'terminal velocity' rather sooner than later.

Vegetation

For growing vegetation and life it'll be more simple than you might think. Though large complex life will be relatively little due to limited resources, they can survive. The rest is very possible to live. It is much like those jars you can put some plants in and close off to the world. As long as there is sunlight it'll become a tiny ecosystem. Water evaporates from the plants but can't escape, which condensates and comes back to the plants. All nutrients are absorbed by the life and released again upon death. Your world is much the same only without walls. The water and nutrients stay within the 100×100km area. Only the sun is needed for the cycle to continue (near) indefinitely.

That means if you start it during a time where there's a lot of water in the air and ground, you'll have more frequent rainfall. If you start it with less there is less to evaporate and reuse. It'll be a dryer climate.

As the internal area does not change in the composition at all, the only factor of influence is the sun with the seasons. Assuming the sun is regular the climate can hardly change, save for incredible ecological destabilising things that release a lot of carbon for example.

Answer 6

It would quickly turn into a hellscape. As some comments and answers already explained, each teleportation would add energy to the system.

As already said, if you had a bowling bowl falling in the air, it would accelerate up to a point where the acceleration cancels out with air friction, reaching terminal velocity.

But this is much worse here, since it is the air itself that is moving, it has no "air friction" working against it, so the only thing that slows it down is not air friction, but friction against anything that isn't moving in the same direction. As things like dust start to move with the wind, they contribute less to friction, and help erode everything else faster, which reduces the amount of stuff that slows the air down.

Unless there is no "ceiling" to the world, then I guess the atmosphere would spread out and "climb" as it accelerates, spreading thinner and thinner. If a particle has enough horizontal speed it could "do a lap" around the world fast enough to counter its own vertical fall, then, the upper layers of the atmosphere would be in low enough gravity, and encounter no resistance against the void, so that it could keep climbing this way indefinitely, leaving room for other particles below to go through the same process.

But if there is a ceiling high enough that there is no "chokepoint effect" with stuff on the ground (which would still produce heat, it would go the same way but it would happen much slower as the wind isn't as free to accelerate), then the only thing slowing down the wind would be stuff on the ground, dust in the air would only slow it down a bit by hitting against grounded stuff, but the ceiling would probably be high enough (probably doesn't need to be very high for that) that the upper parts of the atmosphere can keep accelerating, only getting "second-hand drag" from the air below. But as things keep eroding, more particles enable more erosion, until a point is reached where enough stuff has eroded that the whole world can cascade into an infinitely accelerating cyclone, definitely turning into, as a comment said, some plasma soup.

Answer 7

Easy-peasy, just plant lotsa windmills and extract the extra energy. Let's see if it's feasible (spoiler: it is. Yes, it will spoil the vista, but it makes the place livable and... it's renewable :grin:)

So, lets see how much energy the teleportation injects into the system at each cycle. The most defavourable case is when the entire landscape is isolated from the external environment, so that the air can't expand outside and carry an amount of energy in the process. Something like in the diagram below, representing a cross-section of the skewed prism (with the note that, at 0.5% slope, the $sin(\alpha)$ and $tan(\alpha)$ start to differ at $10^{-7}-10^{-8}$, so don't bother).

So, over the duration of an entire cycle of atmospheric recycle, the prism on top descends and replaces the equal volume of the bottom prism. With the note that the mass center of the triangle is 1/3 from its height (and thus $\Delta h = 2/3\cdot{h}$), lets compute the variation in the potential energy. $\Delta E_p$

A triangular prism with a height of 100km (either top or bottom) will have a volume of $0.005\cdot10^5\cdot10^5\cdot10^5 = 5\cdot10^{12}m^3$ and a mass of $m = 5\cdot10^{12}\cdot1.22 kg = 6.1\cdot 10^{12} kg$.
The potential energy difference is $$\Delta E_p = m\cdot g\cdot\frac{2}{3}h = 6.1\cdot10^{12}\cdot9.8\cdot\frac{2}{3}\cdot(0.005\cdot10^{5})J = 2\cdot10^{16}J$$

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Assume we want the wind to be limited to a $v_{wind} = 10\frac{m}{s} (= 36\frac{km}{h})$. How much power we would need to dissipate in windmills?

The entire prism moving downhill at $10\frac{m}{s}$ will have a kinetic energy of $2\frac{6.1\cdot 10^{12}\cdot{10}^2}{2} J = 6.1\cdot10^{14}J$, which is negligible with $\Delta E_p$, so we'll ignore it in the balance of power. However, $10\frac{m}{s}$ comes into play in the time required for the entire prism of air to be recycled through the upper outlet port $T_{cycle} = \frac{100\cdot10^3m}{10\frac{m}{s}} = 10,000s$.

Which means the entire power that need to be captured by windmills is $P = \frac{\Delta E_p}{T} = 2^{11}W$.

One of the current most common size and power of the wind turbines is

the GE 1.5-megawatt model, has 116-foot blades [=35m] on a 212-foot [=65m] tower

To dissipate $2^{11}W$ one would need to use $2^{11}/1.5^{6}=133333$ such turbines.
At 35m of "wind turbine space personal" (lets make if 50), one will need 67 rows of such turbines, each row 100 km in length.
This will lead to a lost of a real estate strip about 3.5km in width.

Answer 8

Lets work out how fast the wind is!

First pass, lets try a perfectly flat world. In that case, if the atmosphere is in steady state, then each square meter of land is producing ~100,000 Newtons of air pressure vertically, which is decomposed into a 100,000 newton air pressure normal to the surface, + 500 Newtons of skin drag parallel to the surface. For this square meter of surface,

$$ 500\:\mathrm{N} = F_D = \frac12 C_D \cdot \rho \cdot v^2 \cdot A $$ $$ C_D \approx 0.003 $$ $$ \rho = 1.2\:\mathrm{\tfrac{kg}{m^3}} $$ $$ A = 1\:\mathrm{m^2} $$ $$\begin{align} V_\text{surface} \approx \sqrt{\frac{500\:\mathrm{N}}{.003 \times 1.2\:\mathrm{\frac{kg}{m^3}}\cdot1\:\mathrm{m^2}}} \approx 1200\:\mathrm{mph} \end{align}$$

This provides an upper bound

In general, every square meter of your world is responsible for pushing uphill on the atmosphere with $50\:\mathrm{kg}$ of force. So, if your world is covered by a crowd of unmovable people standing 1 meter apart in a grid, the wind will exert $50\:\mathrm{kg}$ on each of them, or roughly their bodyweight.

So, as a lower bound, the wind whistling through this crowd will be around $120\:\mathrm{mph}$, the typical terminal velocity of a person. Of course, the winds above their heads will be significantly higher.

We now have an upper bound of $1200\:\mathrm{mph}$ and a lower bound of $120\:\mathrm{mph}$.

We can also go to the literature: this study

https://journals.ametsoc.org/view/journals/atsc/72/12/jas-d-14-0383.1.xml

suggests groundcover has a $C_D$ of around 0.01, which would mean 450 mph winds. They only find $C_D$ for wind speeds up to around 20 mph, but the trend is downwards. While this calculation is still pretty dubious, I think it's honestly a pretty good estimate for how fast the wind would be.

How hot is the atmosphere?

A lower bound to the heat input to the atmosphere is $120\:\mathrm{mph}\times 500\:\mathrm{N} = 26\:\mathrm{kW}$ per square meter. Assuming that all this energy is being radiated to space, then the stefan boltzmann law says that the world is 557 degrees celcius

An upper bound to the heat input to the atmosphere is $1200\:\mathrm{mph}\times 500\:\mathrm{N} = 260\:\mathrm{kW}$ per square meter. This requires a world at 1204 degrees celcius to stay steady state by radiating to space.

Final answer: Yorkshire is somewhere between a hurricane-force pizza oven and a supersonic blast furnace.

Answer 9

Quite apart from perpetual wind and a perpetual mudslide that would mash everything to a fine sludge and wear the surface and any unevenness away, there would be no way and nowhere for the sun to set or alternatively rise - how would it cross an infinite plane? Either there would be perpetual night (too cold) or there would be perpetual day (too hot). Let us assume that the sun is always in the sky. In that case, there will effectively be an infinite number of suns each above one square section of the plane. The problem with this is that each sun will not only shine on the square directly beneath it; it will also shed its light and heat on neighbouring squares. Thus there will be an overlap whereby the 'land' will receive heat from all directions. The effect will be catastrophic. This is true whether there are infinite suns and infinite squares or there is just one sun and one square - the effect will be identical.

Finally, how will a sun of diameter 1.39 million kilometres, fit in a sky that is only 100 kilometres across? Answer: It won't. Even if we see only a square portion of the sun, each square of sky above us will be filled with sun - in other words, the sky will be permanently ablaze with infinite sun.

Answer 10

Exponentially growing disaster

At first, air would relatively gently "roll" down the infinite slope. As the wind blows, the whole system would have more and more energy pumped in it by the portals. The faster the wind blows, more energy would be pumped in the system and the more energy is in the system, the faster would the wind blow. It turns out that the rate of change of kinetic energy would be roughly proportional to the kinetic energy itself (plus probably some constant rate of pumping the energy $P_0$, present at the beginning): $$ \frac{dE_k}{dt} \approx \omega E_k + P_0, $$ where $\omega$ is just some constant which would be greater for greater slopes. Perhaps $E_k$ on the RHS should be raised to some power, but this will not change the solution much, which would be roughly $$ E_k(t) \approx \frac{P_0}\omega \left(e^{\omega t} - 1 \right) $$

The problem is that this situation grows exponentially from the initial $E(0)=0$. Even for the gentlest slopes, the exponential growth will still happen, although it will take more time to reach the disastrous levels, but it will still be inevitable.

At first, only some wind will start to blow. The wind will blow faster and faster, with nothing to stop it because the energy is only pumped into the system. Sure, some of it is lost due to friction, but that will only cause a constantly growing rise in temperature, which might be insignificant at first, but it will also grow exponentially. As the wind speeds continue to grow, the wind will tear everything on the landscape and eventually the landscape itself. Strong wind will turn into combination of wind and violent ever accelerating landslides and sooner or later everything this initially pristine landscape was made out of will be rolling down the slope ridiculously fast. The temperature of that disastrous chaos will also grow, so at some point it would turn into plasma, later perhaps even into something more exotic like quark-gluon plasma and eventually even the structure of the spacetime itself might get affected and everything may turn into some sort of singularity.

tl;dr: Everything would be completely destroyed.