Why is modelling the Earth as having geomagnetic poles useful? I'm reading about geomagnetic poles and wondering what their signifcance is. It seems one (and perhaps the main) purpose of using this type of model is for understanding the aggregation of magnetic particles from outside of earth. I feel as though I'm missing a step as to why this model is used, surely the magnetic particles from space still experience the irregular magnetic field of earth?
My guesses from the reading I've done so far are that:

*

*At a great distance, the magnetic particles do actually experience equivalent attraction as though the Earth had a bar magnet that gave it its geomagnetic poles.


*Over some time period, and over a number of particles the force experienced by all particles averages to what would be experienced if the Earth had a bar magnet that gave it its geomagnetic poles.
Are one, both, neither of the above close to correct?
And in general, realting to (1), at great distances can magnets be suitably modelled as equivalent bar magnets (even if they're complex, or a horseshoe magnet)?
 A: The geomagnetic poles are the two points where the magnetic field is vertical if the Earth's field is replaced by the best-fitting dipole field. This dipole is part of a spherical harmonic fit to the Earth's magnetic field at the surface - like a polynomial fit, but for a sphere. Such fits are useful for many reasons. Each component has a different dependence on $R$, the distance from the center of the Earth. A dipole field falls of as $1/R^3$, while more complex components like quadrupole and octupole fall off more rapidly ($1/R^4$ and $1/R^5$ for those two). That allows us to not only predict the field at a particular point on the surface, but also extrapolate above and below the surface. And because of the dependence on R, a magnetic field looks more and more like a dipole field as you move away from the center. So InkTide is dead wrong about the behavior of the magnetic field at a distance. The magnetic field of every magnet asymptotically approaches a dipole field as you go further away. That is a rigorous theorem (see, for example, Jackson, Classical Dynamics chapter 5). Going the other direction, we can predict that the magnetic field looks much less dipolar at the surface of the Earth's core, and that is useful information when we try to understand the origin of the geomagnetic field.
The geomagnetic poles themselves are useful for at least two reasons. First, the dipole approximation accounts for a large part of the Earth's field. Second, they are a lot easier to think about than the full complexity of the field. We can point to their location on the Earth's surface, interpret paleomagnetic results using the Geocentric Axial Dipole Hypothesis (InkTide is correct about that) and talk about how they move with time (most dramatically, when the dipole flips).
A: Possibly 2 is true over geological timescales, but not 1 because it's a first order approximation - the simplifying assumptions are what yield the geomagnetic poles from the real irregular magnetic field data.
For 1, at large distances, the overall field strength declines at the same rate as the irregularities - the ratio of "effects from irregularities" to "effects from an approximated geomagnetic pole bar magnet" shouldn't change. This means the error remains constant over distance as long as the field remains constant. EDIT: see A. Newell's answer, this is not the case.
However, for timescales and distances relevant to particles interacting with the magnetic field of Earth, the importance of the higher order approximations increases as the distance of the particle to the center of the field's source (ostensibly the Earth's core) decreases.
For 2, I believe this would only be true if the average magnetic field encountered by magnetic particles over that time period was symmetric about the average axis from the direction of the particle in question - in other words, radially symmetric about the axis of the geomagnetic poles. This assumption of an average dipole is core to current paleomagnetism, and is called the Geocentric Axial Dipole hypothesis. As far as I'm aware, we have some measured evidence of this, but not a huge amount that isn't circumstantial (and partly reliant on the GAD hypothesis being true, like the interpretation of locked-in magnetism in ocean basin basalts). Needless to say we need more and better quality paleomagnetism data.
EDIT: the following is only correct for multipoles of a higher order than dipoles, see A. Newell's answer.
The general suggestion is incorrect, for reasons that you probably intuit already: as distance increases, the magnetic field tends towards 0 at the same rate the difference from an actual ideal bar magnet's field also tends towards 0, meaning again the ratio remains the same and the relative error remains the same.
