(I didn't actually read the whole paper, it's possible this is addressed.) …Is this not just a case of "oops it's the IVT"? The equator is non-arbitrary, so albedo symmetry across it is notable;...
(I didn't actually read the whole paper, it's possible this is addressed.)
…Is this not just a case of "oops it's the IVT"? The equator is non-arbitrary, so albedo symmetry across it is notable; but if we presume that the Earth's albedo is continuous (in the mathematical sense, a reasonable enough assumption given the granularity of possible measurements), I'm pretty sure the IVT guarantees the existence of a plane of longitude which divides it into equal hemispheres.
CC @Barney That was my initial thought too, and I was going to post this here to see if it got ripped to shreds for being obvious because obviously any shape that loops back onto itself can be...
That was my initial thought too, and I was going to post this here to see if it got ripped to shreds for being obvious because obviously any shape that loops back onto itself can be divided into two equal parts in a real, three-dimensional space.
But no, this is not the case here.
the EH–WH symmetry encapsulates a distinctive ‘triple symmetry’ in which clear-sky albedo, cloud radiative effect and open-ocean fraction all exhibit hemispheric symmetry around this meridian.
In other words, all three are equal on each side of this meridian. It seems entirely reasonable that they could be linked—at the very least, the Pacific Ocean seems to have a rotational symmetry around the intersection of 163° W and some line in the tropics, possibly the equator—but it isn't self-evident.
I guess the more interesting thing about it is that all of the world's major high cloud decks are west of 27° E and all (most?) of the major low cloud decks are east of it, so they cancel out in some way. I don't know enough about atmospheric science to know why one wouldn't expect all clouds to reflect the same, but that's what I've read.
Ah, when I skimmed that portion, I read it as explanation—"here's why we think the symmetric meridian is here in particular". If they are in fact three notionally-independent values whose symmetry...
Ah, when I skimmed that portion, I read it as explanation—"here's why we think the symmetric meridian is here in particular". If they are in fact three notionally-independent values whose symmetry happens to align, that's definitely an interesting finding.
The intermediate value theorem only guarantees that you can divide it into 2 parts of equal overall albedo, not that those parts are equal in size, which the 27° E meridian does. That is the...
The intermediate value theorem only guarantees that you can divide it into 2 parts of equal overall albedo, not that those parts are equal in size, which the 27° E meridian does. That is the surprising part of this finding.
No, I think the IVT gets you a guaranteed equal division for any adequately-large set of hemispherical divisions. Consider, for the case of longitude, the function from meridian angle to...
No, I think the IVT gets you a guaranteed equal division for any adequately-large set of hemispherical divisions. Consider, for the case of longitude, the function from meridian angle to difference in albedo between the hemispheres. This function is a distorted sinusoid (it returns to its original value after a full revolution), and it takes on the negative of its original value at a half-revolution. Therefore it crosses zero, therefore by the IVT there is guaranteed to be an angle of longitude which divides the Earth into hemispheres of equal albedo.
Note this argument doesn't depend on the choice of poles, so it holds for any such revolution of dividing planes, though obviously the longitudinal divisions are more interesting to humans for non-mathematical reasons.
The IVT does tend to be like that. It's a remarkably powerful theorem for how obvious it seems on its face.
Your argument makes sense to me. I don't know much about geophysics so I won't wade too deep into the paper, but they say the longitude where there's East-West symmetry in albido stayed within the...
Your argument makes sense to me. I don't know much about geophysics so I won't wade too deep into the paper, but they say the longitude where there's East-West symmetry in albido stayed within the range of 20 and 30 degrees with it oscillating around 27. That such a longitude exists at a fixed point in time is almost trivial as you point out, but that it remains relatively stable over time is probably more interesting.
Can we assume that the function is sinusoidal? Its derivative need not be symmetric between hemispheres either. The Earth's surface is very heterogeneous, so I don't think we can assume that it...
Can we assume that the function is sinusoidal? Its derivative need not be symmetric between hemispheres either. The Earth's surface is very heterogeneous, so I don't think we can assume that it is.
Imagine a hypothetical planet where one half of it is covered in ice while the other half is made up of forests. The IVT still guarantees that you can choose a part of the planet that's albedo equals the rest, but it need not be two equal parts. And in this hypothetical planet's case, it trivially isn't. IVT only results in two equal hemispheres if the planet itself is homogenous.
I'll leave the above example here for posterity but it's obviously incorrect and in no way refutes the claim lmao.
Edit:
After thinking about it a bit more, the only requirement for this to hold is that the albedo of the Earth, when measuring across a longitude, or indeed any dividing plane that passes through the centre, is continuous, since the choice of meridian is free. It doesn't even need to be sinusoidal. It truly is just the IVT after all.
Thank you for the great thought exercise @whbboyd, it would never have occured to me that this could be due to such a fundamental theorem.
As @updawg mentioned, the paper's conclusion is actually that three separate albedos have coincident symmetries, something which is not guaranteed by the IVT. The function here isn't literally a...
As @updawgmentioned, the paper's conclusion is actually that three separate albedos have coincident symmetries, something which is not guaranteed by the IVT.
The function here isn't literally a sine, but it is periodic (we're just revolving the plane around the Earth's axis) and symmetric (for half the revolution, the values are inverse in sign from the other half; the plane is in the same place but "the other way around"). As you note, we do need the albedo field to be continuous to guarantee this function is continuous and the IVT applies, but… I guess it's a question of philosophy, but I'm not sure I'd accept that mathematically discontinuous objects can physically exist. Given all that, the detailed shape of the function is irrelevant; we know enough to apply the IVT and conclude that a plane of symmetry exists. Finding it is left as an exercise. =)
I like the IVT because, again, it seems really obvious, but has some very surprising consequences.
It isn't (necessarily). It would be like finding one line with equal numbers of people, equal numbers of fish, and equal numbers of clouds on each side.
It truly is just the IVT after all.
It isn't (necessarily). It would be like finding one line with equal numbers of people, equal numbers of fish, and equal numbers of clouds on each side.
If you consider clear-sky albedo, cloud radiative effect and open-ocean fraction combined, yes. However, purely for Earth's albedo as a whole, it's guaranteed to exist somewhere. Still, the fact...
If you consider clear-sky albedo, cloud radiative effect and open-ocean fraction combined, yes. However, purely for Earth's albedo as a whole, it's guaranteed to exist somewhere.
Still, the fact all three are equal is fascinating. There's still so much we don't understand about global weather phenomena and geophysics. Very interesting times.
Peer review file (provided because I think it's interesting, not because it necessarily contributes)
Abstract
Earth’s albedo is fundamental to the planetary energy budget1. The Northern Hemisphere (NH) and Southern Hemisphere (SH) contribute essentially equally to the planetary albedo—a remarkable yet puzzling phenomenon known as hemispheric albedo symmetry1,2,3,4,5,6. Although such symmetry is rare, it is not unique7. Nevertheless, other symmetry pairs have remained unexplored, despite their potential to illuminate possible causes of albedo symmetries and implications for the planetary energy budget. Using a 25-year satellite record, here we show that Earth also exhibits a unique and persistent east–west (E–W) albedo symmetry: the 27° E meridian divides the planet into an Eastern Hemisphere (EH) and a Western Hemisphere (WH) that reflect nearly identical amounts of sunlight. In contrast to the NH–SH symmetry, the EH–WH symmetry encapsulates a distinctive ‘triple symmetry’ in which clear-sky albedo, cloud radiative effect and open-ocean fraction all exhibit hemispheric symmetry around this meridian. This EH–WH symmetry arises from greater high-cloud reflection in the EH balancing greater low-cloud reflection in the WH. Furthermore, interannual variability in the EH–WH symmetry tracks the phase of the El Niño–Southern Oscillation (ENSO), indicating a potential connection to general circulation. This discovery of the EH–WH albedo symmetry and its emergence as a triple symmetry provides a reduced degree-of-freedom constraint for Earth system models (ESMs) and stresses the critical nature of continued Earth radiation budget observations under a rapidly changing climate.
Peer review file (provided because I think it's interesting, not because it necessarily contributes)
(I didn't actually read the whole paper, it's possible this is addressed.)
…Is this not just a case of "oops it's the IVT"? The equator is non-arbitrary, so albedo symmetry across it is notable; but if we presume that the Earth's albedo is continuous (in the mathematical sense, a reasonable enough assumption given the granularity of possible measurements), I'm pretty sure the IVT guarantees the existence of a plane of longitude which divides it into equal hemispheres.
CC @Barney
That was my initial thought too, and I was going to post this here to see if it got ripped to shreds for being obvious because obviously any shape that loops back onto itself can be divided into two equal parts in a real, three-dimensional space.
But no, this is not the case here.
In other words, all three are equal on each side of this meridian. It seems entirely reasonable that they could be linked—at the very least, the Pacific Ocean seems to have a rotational symmetry around the intersection of 163° W and some line in the tropics, possibly the equator—but it isn't self-evident.
I guess the more interesting thing about it is that all of the world's major high cloud decks are west of 27° E and all (most?) of the major low cloud decks are east of it, so they cancel out in some way. I don't know enough about atmospheric science to know why one wouldn't expect all clouds to reflect the same, but that's what I've read.
Ah, when I skimmed that portion, I read it as explanation—"here's why we think the symmetric meridian is here in particular". If they are in fact three notionally-independent values whose symmetry happens to align, that's definitely an interesting finding.
The intermediate value theorem only guarantees that you can divide it into 2 parts of equal overall albedo, not that those parts are equal in size, which the 27° E meridian does. That is the surprising part of this finding.
No, I think the IVT gets you a guaranteed equal division for any adequately-large set of hemispherical divisions. Consider, for the case of longitude, the function from meridian angle to difference in albedo between the hemispheres. This function is a distorted sinusoid (it returns to its original value after a full revolution), and it takes on the negative of its original value at a half-revolution. Therefore it crosses zero, therefore by the IVT there is guaranteed to be an angle of longitude which divides the Earth into hemispheres of equal albedo.
Note this argument doesn't depend on the choice of poles, so it holds for any such revolution of dividing planes, though obviously the longitudinal divisions are more interesting to humans for non-mathematical reasons.
The IVT does tend to be like that. It's a remarkably powerful theorem for how obvious it seems on its face.
Your argument makes sense to me. I don't know much about geophysics so I won't wade too deep into the paper, but they say the longitude where there's East-West symmetry in albido stayed within the range of 20 and 30 degrees with it oscillating around 27. That such a longitude exists at a fixed point in time is almost trivial as you point out, but that it remains relatively stable over time is probably more interesting.
Can we assume that the function is sinusoidal? Its derivative need not be symmetric between hemispheres either. The Earth's surface is very heterogeneous, so I don't think we can assume that it is.
Imagine a hypothetical planet where one half of it is covered in ice while the other half is made up of forests. The IVT still guarantees that you can choose a part of the planet that's albedo equals the rest, but it need not be two equal parts. And in this hypothetical planet's case, it trivially isn't. IVT only results in two equal hemispheres if the planet itself is homogenous.I'll leave the above example here for posterity but it's obviously incorrect and in no way refutes the claim lmao.
Edit:
After thinking about it a bit more, the only requirement for this to hold is that the albedo of the Earth, when measuring across a longitude, or indeed any dividing plane that passes through the centre, is continuous, since the choice of meridian is free. It doesn't even need to be sinusoidal. It truly is just the IVT after all.
Thank you for the great thought exercise @whbboyd, it would never have occured to me that this could be due to such a fundamental theorem.
As @updawg mentioned, the paper's conclusion is actually that three separate albedos have coincident symmetries, something which is not guaranteed by the IVT.
The function here isn't literally a sine, but it is periodic (we're just revolving the plane around the Earth's axis) and symmetric (for half the revolution, the values are inverse in sign from the other half; the plane is in the same place but "the other way around"). As you note, we do need the albedo field to be continuous to guarantee this function is continuous and the IVT applies, but… I guess it's a question of philosophy, but I'm not sure I'd accept that mathematically discontinuous objects can physically exist. Given all that, the detailed shape of the function is irrelevant; we know enough to apply the IVT and conclude that a plane of symmetry exists. Finding it is left as an exercise. =)
I like the IVT because, again, it seems really obvious, but has some very surprising consequences.
It isn't (necessarily). It would be like finding one line with equal numbers of people, equal numbers of fish, and equal numbers of clouds on each side.
If you consider clear-sky albedo, cloud radiative effect and open-ocean fraction combined, yes. However, purely for Earth's albedo as a whole, it's guaranteed to exist somewhere.
Still, the fact all three are equal is fascinating. There's still so much we don't understand about global weather phenomena and geophysics. Very interesting times.
Peer review file (provided because I think it's interesting, not because it necessarily contributes)