[1.1] I first note that 'geomagnetic activity' is a very complex phenomenon. In order to cope with the complexity, researchers have long ago come up with a device called a 'geomagnetic index' which is a 'short-hand' encapsulation of the complex phenomenon. A very early one was the C-index, where one would simply look at a magnetogram [a recording of the wiggles in the geomagnetic field] at a given location for a day and classify that day as 0 [quiet], 2 [disturbed], or 1 [moderate, in between]. By averaging such C-values from many locations ['stations'] one would get a 'global' average Ci-index ['i' for 'international']. A lot of good science was done with the Ci-index.
[1.2] It was, of course, obvious that only part of a day could have disturbances while the rest remained quiet, so for a better result, a finer time resolution was needed. It turns out that a time resolution of three hours is a useful choice. This is also the time it takes the solar wind to pass the Earth's magnetosphere, so there is even a good physical reason for that choice. Julius Bartels in the late 1930s introduced the so-called K-index with 3-hour resolution. Like so many indices [Richter scale, Stellar Magnitudes, ...] the index [which is a single digit 0 through 9] is logarithmic: each step up corresponds to a doubling of the amplitude of the wiggles over the three hours. Actually, for the higher index values, slightly less than a doubling is used, as otherwise the highest index values would never be reached [but we'll sacrifice such arcane details on the altar of understanding from time to time]. To be able to compute average values we need a linear index, as averages of logarithms have dubious physical meaning. So, to each K-index value, one assigns an 'equivalent amplitude' as the size of the wiggle [in physical units such as nanoTeslas] lying at the midpoint of the interval for that K-index value. Say that K=2 is assigned to wiggles between 20 and 30 nT, then the amplitude would be 25 nT.
[1.3] There are several issues that need to be addressed concerning the derivation of the K-index [and its equivalent 'amplitudes', a-indices]. We shall get to these in due time. At this point we note that just as with the C-index, one can [carefully] average a-indices from a worldwide network of stations and obtain a 'global' a-index. Several such exist [depending on the network] and one is the aa-index [one Northern and one Southern (antipodal - that is the second 'a' of 'aa') station] going back to 1868. Another is the am-index [23 stations worldwide - that is the 'm' in 'am' (from the French word 'mondial' for 'worldwide')] going back to 1959. Both of these indices were conceived and initially provided by the late P.-N. Mayaud.
[1.4] The importance of these indices lies in the fact that they are very good proxies of properties of the solar wind and therefore provide a means to probe the solar wind and solar conditions in the past, long before spacecraft measurements of the solar wind. It was realized long ago [Bartels, 1932] that geomagnetic indices "yield supplementary independent information about solar conditions". In these notes we shall explore how well this goal is realized.
[1.5] In a sense, the Earth is a large sensing device, and we have learned how to 'calibrate' the 'response function' of this instrument in terms of solar conditions. To whet your appetite, the following Figure [1.6] shows the observed am-index [black curve] versus the am-index calculated from solar wind data measured in Space for several solar rotations. The scale is logarithmic to show how well we can account for the observed variation over the whole range of the index. The agreement is less good for very small values of the am-index [say, below 5 nT]. We shall see later that for such low values, it is very difficult to measure geomagnetic activity, so perhaps it is no surprise that the fit is poorer there. (But see also [2.x])
[2.1] The Solar Wind stems from the outer solar atmosphere being heated [by basically unknown processes] to such a high temperature that its random thermal velocity exceeds the gravitational escape velocity of the Sun, thus in a sense 'evaporating' or 'boiling off' and expanding into free Space. The wind consists mostly of protons [and electrons, of course] with a small [and varying] admixture of [mostly] Helium and heavier ions. At the Earth, the wind is so tenuous [a few protons per cubic centimeter] that it is collision-less and would behave as just a bunch of unconnected particles, except that it is pervaded by a magnetic field. This field gives the solar wind 'fluid like' properties meaning that a lot of its behavior can be 'understood' in terms of the same concepts that apply to fluids, like 'pressure', 'waves', 'shocks' and such.
[2.2] Let us examine what properties of the solar wind might be important for the generation of geomagnetic activity. If you stand in front of a firehose you'll certainly feel the effect of the 'directed' pressure of the fast-moving water. This 'dynamic pressure' P is the flux of momentum P = (nmV) V, where n is the number density of particles with mass m coming at you with velocity V. As the solar wind approaches the Earth, the wind will press up against the magnetic field of the Earth. Where the pressure of the solar wind balances the pressure of the magnetic field is the boundary between the solar wind and the Earth's magnetosphere. Electric currents flow along the boundary. Because the density and velocity of the solar wind varies continuously [at times abruptly, e.g. at shock waves] the boundary is very unsteady and 'flaps' around all the time so the magnetic effects [measured at the ground] of the currents are constantly changing. Furthermore, whatever configurations of magnetic fields, plasma regimes, and electric currents that were established to maintain the pressure balance are constantly 'buffeted' and changing, often explosively. All of these processes involve electric currents having magnetic effects felt on the ground as geomagnetic 'activity'. So, the dynamic pressure P is one of the inputs that controls the activity.
[2.3] The magnetic field of the Earth is to first order a dipole with magnetic poles about 11 degrees away from the rotational poles. As the Earth rotates, the solar wind will be up against a wobbling magnetic 'obstacle' during the [e.g. Universal Time] day. In addition, the rotation axis is inclined 23.5 degrees towards the Earth's orbital plane, giving rise to an annual wobble. The combined wobble about both axes results in the solar wind seeing a complicated, varying terrestrial magnetic field with which to maintain pressure balance. By properties of a dipole, the magnetic field at the 'nose' of the Earth's 'magnetosphere' is given by B = Bo sqrt(1 + 3 (cos(psi))^2), where psi is the angle that the Earth's magnetic axis makes towards the solar wind direction [coming from the Sun, but see later posts on this] and Bo is the magnetic field strength at the equator [at the same distance from the center of the Earth]. So, the 'attack' angle psi is another variable that controls [i.e. modulates] the activity. Note, that this variation does not arise from a solar wind property, but from properties of the Earth.
[2.4] The influence of the psi-angle gives rise to a semiannual variation with minima near the solstices when the solar wind sees the strongest geomagnetic field. Figure [2.5] shows the variation of the am-index [left] and then of the so-called Svalgaard-function S(psi) = [1 + 3 (cos(psi))^2]^(-2/3) as functions of Universal Time [UT] and time of year [the month]. Note the deep minima in June at 16:30 UT and in December at 4:30. Ed Cliver has described the result as 'valley digging'. High 'ridges' separate the minima. This is a Universal Time effect [not a day/night effect] that happens because the magnetic poles are tipped towards the Sun at specific times (16:30 Greenwich Mean Time=UT for the Northern pole and 4:30 for the Southern Pole).
When we try to extract solar conditions from geomagnetic variations, we need to remove the effect of this purely terrestrial effect. In addition to the Svalgaard-function modulation there are other [generally smaller] effects [to be discussed later] that give rise to further semiannual variations. The semiannual variations were discovered ~150 years ago, and we are still [heatedly] debating what causes them, and not everybody agrees with the above.