Sunday, 25 April 2021

Acceleration ≈ gravity

The effects of acceleration and gravity aren't exactly the same of course, but we can generate 'artificial gravity' by spinning things round, like swinging a bucket of water in a horizontal plane - the water stays in, the same as it would if you just put the bucket down. As a reality check, I put some marbles in the centre of a smooth, flat plate on a record player turntable. When I lifted the needle to get it turning, all the marbles hit the edge of the plate within a second or so.

For all the 'Physics Deniers' out there, let's work out what would happen if you had a large, well-insulated tube of air (the larger and longer the better) being spun around an axis at one end (the faster the better).

1. While standing still, the density, pressure and temperature are constant along the whole length (I haven't bothered with units or numbers on the axes, this is about basic principles):
2. When it is spinning around the axis, the gas will be pushed towards the outer end (the same as the marbles on a plate). So density goes up at the outer end and down at the axis end. I've assumed that the increase is linear for now:
3. The pressure at any point along the tube is the mass of all the gas to the left of it pressing 'down' on it, which is the area of the 'density' shape to the left of that point (multiplied by the strength of the artificial gravity). The length and height of the upper triangle both increase by the same multiple with each step to the right, so the area increases by that multiple to the power of 2. So pressure increases geometrically from left to right: 4. Temperature is proportional to pressure divided by density. There are no physical barriers along the tube, it's all in equilibrium, so you can use the Gas Laws to calculate T if you know D and P (or indeed, calculate any variable if you know the other two). The ratio of P/D increases from left to right. So temperature goes up from left to right in a straight line (crudely speaking, T ∝ P/D and P ∝ D^2, so T ∝ D^2/D => T ∝ D): This is the bit the Physics Deniers can't or won't understand. The gas can't become isothermal again, because the density and pressure at each point in the tube are dictated by the amount of gas, the length of the tube, how fast it is spinning etc, and temperature is the result of all that, temperature can't just make up its own mind what to do. So the inner surface at the outer end becomes warmer than when it was standing still, and the inner surface at the axis end becomes cooler. No thermal energy has been added to the gas - it is just distributed differently.

A real life application of a similar effect is the vortex tube, which uses centrifgual forces to split air at a certain starting temperature into a stream of much colder air and a stream of much warmer air.

5. What relevance does this have to the gravito-thermal effect? Simple, you just rotate Diagram 4 clockwise by 90 degrees and relabel the axes. This now resembles the measured profile of the troposphere. You can multiply T and D at any altitude, divide that by P at that altitude and you get the same answer.
The other and even easier to understand explanation for the temperature profile in the atmosphere is that when air rises, it converts thermal energy to potential energy (cools); when air falls (or presses down on air beneath it), it converts potential energy to thermal energy (warms). Actually it is gravity and the Gas Laws that says there must be a lapse rate and the formula g/Cp just tells us how much the lapse rate is.

So the other way of modelling the whole thing is by just knowing the effective/average temperature and the lapse rate. We know that Earth receives enough sunshine to warm the atmosphere to an average of ~255 degrees (effective temperature), which is (by definition) the temperature half-way up the troposphere (the same as the temperature half-way along the spinning tube = the starting temperature). So you subtract ~5.5 x lapse rate ~6.5 degrees to find temperature at tropopause and add the same amount to find the temperature at sea level. You can work backwards to find pressure and density at different altitudes (for example by using the Barometric Formula or trial and error).

The equilibrium and actual observed profile is that for every 100m fall, density increases by ~1.115%; pressure increases by ~1.373%; and the ratio pressure-to-density increases to give a lapse rate of ~6.5 degrees/km. These three variables are inter-dependent, one does not 'cause' the others, they are in balance, like two playing cards balanced in an inverted 'V'. Each card is holding the other one up.

And if we have two different approaches which both give you the correct answer, you can be fairly confident that the approaches themselves are correct.

But whatever the explanation, it reminds us that the effects of gravity and artificial gravity/acceleration are very similar; and acceleration due to gravity is a lot, it is slightly more than the acceleration of a car that can do 0 to 60 mph in three seconds*. The laws of physics that apply to a bucket of water also apply to a spinning tube also apply to the troposphere.
6. "But what about radiation and greenhouse gases?" shout the Physics Deniers. Well, what about them? You can explain what happens in the spinning tube or the vortex tube without mentioning them and they are irrelevant when you rotate the diagram and have real gravity instead of artificial gravity. In the spinning tube, there was a fixed amount of thermal energy to start off with, which was recycled towards the outer end. Earth gets enough sunshine to keep the whole system at a steady average ~255K (fixed - see 7.) and the laws of physics ensure that the thermal energy is recycled downwards to cool the air at higher altitudes to ~220K and warm the surface to ~288K.

7. For sure, land and the ocean surface are emitting more radiation than actually gets to space (which must be approx. equal to the amount received from the Sun each day) but there's no point subtracting one from the other, it's not comparing like with like. We calculate the effective temperature ~255K by treating clouds as part of the 'surface', so the 288K at the hard surface/ocean surface is a red herring; the effective temperature should be compared with the overall average temperature of the top surface of clouds (two-thirds) and land and ocean surface (one-third), which is a lot closer to ~255K than ~288K.

The surface is as warm as it is, and emits the corresponding amount of radiation, it simply doesn't care what happens higher up. Somehow or other, the whole system will ensure that incoming and outgoing radiation balance, whether we point the finger at greenhouse gases trapping radiation or the low emissivity of clouds. Nature finds a way**.

* People picture the atmosphere (or anything else) resting on the Earth's surface as a static situation. In some ways, you can imagine the surface accelerating upwards at 9.8 m/s (a car doing 0 to 60 mph in three seconds) and pushing everything up accordingly. Imagine a ship moored in a river flowing at 10 knots, there's a bow wave in front of it. Or the same ship travelling up a canal at 10 knots, there's the same bow wave. If you lean over the bow and look straight down, you can't tell whether the ship is moored or moving. So it's like the Earth's surface is a giant piston pushing and compressing (hence warming) a ten-ton 'bow wave' of air in front of it.

** The top 1m of the ocean surface only radiates away about 1.4% of its total (thermal) energy overnight, sufficient to cool it by ~4K, which matches up with the typical diurnal temperature range for the surface of a fairly still ocean and the air immediately above it. Losing energy by radiation is a slow and inefficient process - blacksmiths quench red hot steel by plunging it into water; how much longer would it take if they held the red hot steel just above the surface of the cold water and allowed radiation to do its thing?

Here endeth today's lesson. I'd rather be a Climate Denier than a Physics Denier.


Doonhamer said...

Rendezmous With Rama by AC Clarke.
A book I enjoyed as a teenager.

Mark Wadsworth said...

D, yes, that's the general idea. Or the satellite in "Space Odyssey" for the less well-read.

MrMC said...

Not many studying physics nowadays, those making the most noise seem to be recipients of meeja degrees and the like

Bayard said...

'Tis a pity she's a commentwhore.

Mark Wadsworth said...

MC, yup.

B, 'tis a pity I deleted her comment.

James Higham said...

Vortices are also critical to the science of my form of forward motion, also delamination and twist.

MrMC said...

I am a big fan of delamination I am sick of hearing him bang on about racism

Dinero said...

The conclusion of Point 4 is wrong . As the the ratio is constant therefore the the temperature is actually constant.

Mark Wadsworth said...

Din, spoken like a true Physics Denier! "you are wrong but I'm not going to point out what I think the explanation is"

MrMC said...

Indiana Once Tried to Change Pi to 3.2

Any high school geometry student worth his or her protractor knows that pi is an irrational number, but if you’ve got to approximate the famed ratio, 3.14 will work in a pinch. That wasn’t so much the case in late-19th-century Indiana, though. That’s when the state’s legislators tried to pass a bill that legally defined the value of pi as 3.2.

The very notion of legislatively changing a mathematical constant sounds so crazy that it just has to be an urban legend, right? Nope. As unbelievable as it sounds, a bill that would have effectively redefined pi as 3.2 came up before the Indiana legislature in 1897.

Mark Wadsworth said...

MC, they should have changed it to 3, that's even easier than 3.2

Doonhamer said...

I remember 22/7, or 22 divided by 7. Easy when using a primitive calculator.

Mark Wadsworth said...

D, I prefer 355/113.

Dinero said...

now point 4 is edited. " The ratio of P/D increases from left to right. "

That is not correct. Pressure goes up and Density goes up left to right.

now the gas equation T=PV/nR .

V/n is the reciprocal of Density .
And so as n/V goes up V/n goes down.
And so P *(n/V) is constant and so T is constant.

The temperature profile cannot be arrived at by the ideal gas law alone.

Mark Wadsworth said...

Din, this is basic maths.

For a small step to the right, D increases from (say) 100 to 110, by a factor of 1.1

For the same small step, P increases by a factor of 1.1^2 = 1.21. So it increases from (say) 100 to 121.

121/110 is clearly greater than 100/100.

Density goes up A BIT and pressure goes up A LOT from left to right.
So P/D goes up from left to right.

Understand basic maths first, then I will explain the gas law stuff.

But your takeaway for now is this:
Pressure is NOT the same as density.
Units of pressure are N/m2 or lbs/sq foot. It is a FORCE.
Density is simply "how much MASS is there in a certain volume?" so the units are kg per cubic metre.

Dinero said...
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Dinero said...

As P increases as a square of a step to the right the density will also increase by a square of a step to the right.

because (n/V) = P/TR

and so the temperature is constant as pressure increases and density increases.

Mark Wadsworth said...

Din, nope.

You can see that the top bit of the "density" graph is a triangle.

Density = height at any point.
Pressure = area to the left of any vertical line drawn from that point.

Draw a triangle.
Choose a point from left to right.
Measure the height at that point = density.
Calculate area to the left = height x base x 0.5 = pressure

Now choose a point further right.
Repeat the calculations.

You will see that area increases by 'increase in density to the power of two'.

Try to understand 'basic maths' before you dip your toe in the 'gas laws' paddling pool.

Dinero said...

The density graph is drawn incorrectly. As density is a function of pressure, it would follow the curve of pressure if it was drawn correctly.

Mark Wadsworth said...

Din, nope.

"density is a function of pressure" Nope.

You are making the Physics Denier mistake of assuming constant temperature. Why assume something for which there is no real life back up and which clashes with all logic?

If in doubt, understand basic maths and look up the density, pressure and temperature gradients in our atmosphere. They match my diagrams.

Density is a function of gravity
Pressure is proportional to change in density to the power of two
Temperature is P/D.

Funny how basic maths and real life agree so nicely. Greater minds that our came up with the Gas Laws and clever weather people took all the readings. Clever scientists worked out specific heat capacity and measured the force of gravity.

And they all tie in very nicely.

Dinero said...

(n/V) = P/TR


P = (n/V)TR

They are directly proportional relationships and so the curves would match if they were drawn correctly.

Kevin the Chimp said...

Why stop at the surface?
Check out this paper on air pressure in mines:

Mark Wadsworth said...

Din, are you actually trolling me? You just used the more scientific expression for T ∝ P/D, expressed as P ∝ D x T, which is the formula I used in step 3.

KTC, thanks, clearly, exactly the rules apply to deep holes (but there's extra temperature from the rocks deep down).

Kevin the Chimp said...

The most interesting part to me was when that when they cooled the 'Western Deep' air to 28°C, the air pressure went up to twice that at the surface - seems counter intuitive that increasing pressure would decrease temperature

Mark Wadsworth said...

KTC, that does seem a bit mysterious.

Bayard said...

KTC, if you cool air, it becomes denser, which is why hot air rises and cool air falls. If you have a column of air in a mine shaft, the pressure at the bottom is the pressure at the surface, plus the weight of all that air between the bottom and the surface. If you make the air weigh more by cooling it, then the pressure at the bottom of the mine shaft will increase. Increasing the pressure of a gas will cause it to become hotter, as energy is being added to it. However, in this case energy is being extracted from the system and no work is being done to gas, so the temperature decreases.

Dinero said...
This comment has been removed by the author.
Dinero said...
This comment has been removed by the author.
Dinero said...

The density and pressure both increase exponentially as the density of the gas at the lower levels is caused by the pressure. ie D is proportional to P. The temperature is constant. Atmospheric density and pressure vs altitude are both curves.

Do a google image search for air density and pressure for altitude. They are all curves for both.

also see here

Increasing the pressure of a gas by reducing the volume it occupies does not cause it to be hotter because V has reduced. T=PV/nR. I do however concur that kinetic deceleration of air molecules by gravity has something to do with the Lapse rate.

Mark Wadsworth said...

Din: "The temperature is constant"

Classic Physics Denier stuff!

It is true that density changes in a geometrically with altitude and not linearly, but that does not change the principle one iota. Pressure must change by more than density for any change in altitude, that is basic addition.