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If there's no information on where Earth's geomagnetic quadrupoles are located, I'll settle for Uranus or Neptune, which have weak enough dipole moments that their N-S magnetic pole axis passes far from the planet's center (true for Earth too to a much smaller degree, which is why I asked this question initially). I'm hoping to visualize such a magnetic field -- imagining where the 4 poles of quadrupole moment are would be a big help. If I figure this right, any magnetic field can be disassembled into monopole, dipole, quadrupole, octopole etc. moments, analogous to a Taylor series for example. The equations for the tensor don't really help me visualize this. 95.168.121.104 (talk) 20:55, 12 October 2020 (UTC)[reply]
You can be right, but I cannot find any source about 1) describing every dipole as the sum of monopoles, dipoles, quadrupoles and so on and 2) the Earth possessing a quadrupole. As far as I know the Earth possesses a main dipole field, our classical nord and south poles, plus several secondary poles. But they are indipendent dipoles, caused by the fact that the fields are generated by flowing layers of molten metal in the outer core, flows which consist of rings moving at different velocity and producing eddies and vortexes, each of them the origin of separated and independent, temporary dipoles, some of which can be detected at the surface of the planet. See the picture in Earth's_magnetic_field2003:F5:6F11:9700:C4E2:290F:31BC:51B2 (talk) 14:11, 13 October 2020 (UTC) Marco PB[reply]
(OP) Yeah if they use that then presumably adding secondary dipoles converges quicker than multipole expansion for Earth's field. But from a mathematicsl perspective they're both equally valid functions which converge to the real magnetic field, which is impossible to describe exactly for us (being made of quintillions of interacting particles). I'd in any case settle for a real life example where multipole expansion becomes useful (other than CMB which unfortunately isn't very picturable). 95.168.121.89 (talk) 19:07, 13 October 2020 (UTC)[reply]
I think we have previously tried to answer this exact question, but let me try again using different words.
In a multipole expansion, each "pole" is not a "place": it is a coefficient in a mathematical expression that formalizes the numerical expansion that approximates the function. That sounds like a lot of technical jargon - but if you want to use it, it's really really important that you understand it, conceptually.
We can construct any method we like to project the numerical representation of any particular pole onto a geographical location. But, in exactly the same way that we use map projection, we have a free choice of how we want to do the projection. Each choice provides some advantages and disadvantages. All choices are a simplification of the complicated nature of the original function, and are subject to all sorts of distortions. We can express our multipole expansion in spherical coordinates, and that will lend itself to certain obvious projective representations at specific locations - but those locations aren't the poles - they're projections - they're mathematical fictions.
If what you hope to find is the four points of latitude-and-longitude that correspond to the "quadrupole" representation of Earth's magnetic field (or, really, any quadrupole representation of anything), then you have a a core misunderstanding of what a quadrupole exactly is. It's not a list of four locations. If you want to understand a multipole expansion, you will have a better shot if you start looking at pole decompositions of polynomial functions: a pole isn't a place: it is a feature of an equation. We can point to a place, and say "well, at this specific place, the function that approximates the magnetic field strength is chiefly influenced by the second or third pole of the multipole expansion..."; we could even use mathematical formalism or exhaustive search to point to the place where such-and-such effect is strongest; but that doesn't mean that the pole is there. The pole is a term in a mathematical expression - the pole is not a place.
Perhaps most importantly, all of the above is equally true for a simple dipole expansion, too - but because of certain nice properties of symmetry, the weirdness is a lot less obvious; and we can point out the "north pole" and "south pole" with the sort of mathematical imprecision that is inherent in any simplified explanation.
The article says "Most commonly, the [multipole expansion] is written as a sum of spherical harmonics." I'm presuming that, since we can take the best-fitting dipole and calculate the positions of geomagnetic poles (which are of course not the actual magnetic poles), then there'll be a process to do that for eg. a quadrupole magnetic field. Applying that, if we take only the Earth's quadrupole moment's spherical harmonics, the resulting function may well correspond to some kind of a magnetic field which has four poles. If so, these poles will correspond to some locations on Earth (forgetting for the moment however useless these coordinates would be obviously). Is this correct? 95.168.118.134 (talk) 15:59, 15 October 2020 (UTC) (OP)[reply]
Sure. If you were interested, and mathematically-savvy, you could start down that proverbial rabbit-hole of linear algebra and complex analysis, and you could construct some way - a "correspondence," if you will indulge me in some mathematical jargon; so that each pole "corresponds" to some set of spherical-coordinates, expressible as a latitude-longitude projected on to the surface of the Earth.
But the hard part of your question is not addressed: what method will you use to construct this correspondence? The reason I do not try to address it is because you can use any method you like. No particular method is obviously "better" or "worse". You're asking for the relationship between two things that do not have a relationship: the pole is a numerical coefficient, and you want to relate it to a specific place; but there is no obviously-useful relationship between this number and any particular place. The only useful relationship that would exist is the one that you invent to satisfy your own preferences.
If you really love calculus and linear algebra, you might enjoy looking for ways to define an optimal pole decomposition for an arbitrary function; and if you can do that, you might find some fun geometric way to project the eigenvalues onto the surface of the sphere, ... but I mean, what kind of person enjoys doing that?
Some further reading, for the physicists-in-training:
Principal axes, from a course on advanced dynamics at the University of Manchester, with references to several very good textbooks if you need help following the math. Of course, the usual caveats about physicists also apply here: the author rushes through with some sloppy shortcuts that are quickly recognizable as solid math - but only after you spend several years of full-time study memorizing these mathematical methods and techniques by working them across thousands of variations in homework problems. This ain't easy stuff.
Please don't talk down to me. I'm not a physicist but I know full well what eigenvalues are. I took some linear algebra in college but not enough to get to tensors. That's why I was hoping that someone who has done this math (or even better, someone whose field is geophysics) could think a little and tell me, if nothing else, some basic information about what that what I'm looking for should look like.
Let me try one last time with a simpler question. Imagine the Earth has a magnetic field of a perfect quadrupole magnet. It has four poles (as in points where the field lines are vertical to the surface). Suppose we take this field and find a function describing it and do the multipole expansion. 1. Do you mean that this function wouldn't be unique (to the point of scale, etc.) and/or the corresponding multipole expansion will not be unique? 2. Will the expansion be invariant so that e.g. you always get a positive quadrupole moment and zero dipole, octopole etc. moments, or not? 93.136.27.169 (talk) 10:26, 18 October 2020 (UTC)[reply]
I did not mean to offend.
You are correct: the decomposition of the magnetic field is non-unique. There is no preferred orientation of the basis functions, so you are free to select their orientation. You could use some kind of method of optimization, or a method for spectral decomposition, to find the orientation of the basis functions that yields, say, minimized cost-function for the coefficients - in other words, principal component analysis. As that article introduces, this is a method that "defines coordinate systems that optimally describe..." a complex system. You could then use those coordinates as to define some kind of projection into latitude-longitude.
The challenging part of your example is that the Earth doesn't have a magnetic field that resembles a "perfect quadrupole magnet." Even the picture in that article isn't really suitable - first of all, it assumes bar-magnets are perfect dipoles - they are not. If you use a perfect mathematical idealization to construct a new perfect mathematical idealization, it's no surprise that you get perfect mathematically-ideal results! We can take this right back to the simplest and most theoretical core of the problem: in the case of the single dipole "bar magnet" - where is the pole?
The answer is, no-where! A pole is not a place: there is no position in the perfect ideal bar magnet where you can precisely and correctly say "this is the place where the pole is." The pole is not a point inside the material bar magnet; the pole is not located along the top edge of the magnet; the pole is an element of the mathematical expression that defines the field at all positions. To the extent that the real bar-magnet is perfectly modeled by a mathematical equation, we can map the pole to a single point with well-defined coordinates: and that is the exact point where the perfect model ceases to work. The pole is the exact place where the real magnet is most dissimilar from its mathematical model.
A function f of a complex variablez is meromorphic in the neighbourhood of a point z0 if either f or its reciprocal function 1/f is holomorphic in some neighbourhood of z0 (that is, if f or 1/f is differentiable in a neighbourhood of z0). If z0 is a zero of 1/f, then it is a pole of f.
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So the poles of a quadrupole are set of the four complex numbers zi such that they yield singularity in the field strength.
Now, which function f are you using to define the Earth's magnetic field?
If you use mathematical idealizations and write a quadrupole equation f, then you have already defined all four zi. In which-ever coordinate system you wrote f, you already know the arguments for the poles.
If you instead are trying to fit an ideal equationf(r,θ,φ) so that it is a best-fit for some set of empirical measurements of the magnetic field around Earth at various positions (here, using some kind of spherical coordinates - let's call them radius, latitude, longitude, ... well, now you have to solve for those poles. If you sample Earth's field at more than four points, you have an overconstrained system, and you are free to resolve that using your favorite numerical method, and you can solve for the poles. But as you see, those aren't the poles of the field: those are the poles of your approximate solution, fitted via your favorite method of numerical optimization.
No amount of finagling can ever resolve this core theoretical concern: idealized equations define a pole as the exact function-argument for which the idealized equation stops working. (This is couched in a bunch of jargon about singularity and function-differentiability, so I'm trying to translate to plain English without insulting anybody's intelligence).
There is no place where the real, actual, geomagnetic field is actually equal to infinity. The pole is not a place: the pole is a term in the equation that is used to approximate real measurements.
If you want to approximate using only two terms, you get two poles. If you want to use four terms, you get four poles. If you want to approximate using NOAA's very high-precision World Magnetic Model 2020, you can fit a 168-term vector model, and if anybody feels like it, they can solve for its 168 poles - but at a certain point, this stops being useful for any purpose. I would propose - (being, as I am, slightly trained in some areas of geoscience,), that the "certain point" where it stopped being useful was the very beginning - there is no utility in solving for any number of poles: not two, not four, not 168, not a large but finite countable number, not an infinite but still countably-large number of terms. Or perhaps we're playing with the most horrible mathematically-behaved magnetic field in this or any other universe: well, maybe its approximation requires a number of terms that defies countability because the pathological field cannot be decomposed in series-representation. (Exercise for the reader, can such a field even satisfy the constraints of electromagnetism as we know it?)
And that is why I insinuated that nobody would pursue this fruitless pursuit, except for those eccentric individuals who enjoy playing fruitless games with mathematics. Such people do exist - perhaps you are one of them - and if you are, I hope you will enjoy reading some of the other items I linked: although it may superficially seem like I'm being dismissive, in actual fact I am hoping to provide some useful mathematical background that may give you more context to at least one other way to think about the problem you have posed.
When you asked initially, you said that you had a hard time visualizing these equations: well, the truth of the matter is that adding more terms will not make visualizations any easier. It will be more useful and productive for you to play with the visualization of a very simple equation, and completely understand how that equation works, because when you hit four or eight or 168 dimensions, the human visual perception system simply isn't well-suited for those high orders. In fact, most people who work with high-order fields (like the NOAA magnetic model project them into only two dimensions, like a map of declination (only); and some people reduce that dimensionality to a single parameter: azimuth. Like so many other incarnations of mathematics in engineering and science, it's simply not useful or practical for a human to work directly with the higher-order terms.
