GITM is a three-dimensional Navier–Stokes equation solver that models the Earth's thermosphere and ionosphere system using a spherical coordinate grid. The GITM code solves the coupled continuity, momentum and energy equations of neutrals and ions in a time-marching fashion. For the ions, the velocity variation as a function of time is ignored, so the steady-state ion flow velocity is solved. Moreover, for this model the ion temperature consists of a mixture of the electron and neutral temperatures, and the neutrals are solved for using the Navier–Stokes equations. GITM explicitly solves for the neutral densities of O, O2, N(2D), N(2P), N(4S), N2 and NO, as well as ion species O+(4S), O+(2D), O+(2P), O2+, N+, N2+ and NO+. The simulations are initiated using the neutral/ion densities and temperatures from the MSIS-90 and IRI semi-empirical models. In general, the results of the simulations become independent of the initial conditions after 72 h of simulated time. In contrast to other general circulation models of the thermosphere, GITM solves the vertical and horizontal advection in separate equations without assuming any hydrostatic equilibrium. This allows the code to take into account the non-constant gravity in the vertical direction and the Coriolis forces .

Because GITM 3-D computations are based on a structured spherical grid with adjustable spatial discretisation in longitude and latitude, the accuracy of the results needs to be verified as a function of the grid refinement. A reference case simulation on 19 April 2004 has been run for 4 days (3 days of “spin-up” + 1 day of simulation). From the grid analysis results shown in Figs. –, it is observed how the grid refinement increases the accuracy of the simulations. This is illustrated, for example, in Fig. , where the neutral density values of the atmosphere at an altitude of 200 km at 0∘ latitude are displayed as a function of the longitudinal direction. Even if the density values are confined in a very narrow range between 2×10-10 and 3×10-10 kg m-3, the different longitudinal profiles are scattered as a function of the grid refinement. By increasing the number of cells in the mesh, the density profiles tends to converge to similar results. In fact, the relative error between the density profiles with a refinement of 144×72 cells (2.5∘ long × 2.5∘ lat) and 104×7252 cells (3.46∘ long × 3.46∘ lat) is less than 0.5 %, whereas the relative error between the coarser (36×18 cells or 10∘ long × 10∘ lat) and the finer (144×72 cells or 2.5∘ long × 2.5∘ lat) grid is 7 %. In both cases, the density profiles seem to have peaks and valleys at similar longitudinal locations. With the same grid analysis repeated at an altitude of 400 km (Fig. ), the relative error between the coarser and finer grid is increased to almost 19 %. However, the relative error between the two most refined grids is as low as 1.2 %. The analysis suggests that the quality of the results obtained from GITM simulations are in general grid-dependent below a certain refinement threshold. Because GITM uses a spherical grid for its computations, the cell dimensions change as a function of the altitude for a constant grid refinement. This implies that, for a grid with 144×72 cells (5∘ long × 5∘ lat), the side length of a squared cell at 200 km is about 286 km, whereas this value linearly increases to about 295 km length for an altitude of 400 km.

Figure  again shows a grid sensitivity analysis of the density profiles over a meridian at 0∘ longitude and 400 km altitude. It can be seen from also looking at the meridian profiles that the relative difference between the two finer grids (144×72 and 104×7252 cells) is less than 1.5 %. Because of the low relative difference observed between the two finer grids in Figs. , and , it can be assumed that the accuracy of the results will not benefit from a further refinement of the mesh in both longitude and latitude. In order to find a trade-off between the accuracy of the simulations and the computation time, it was decided to select a grid with 72×36 cells in longitude and latitude respectively. This grid, used hereafter, allows for results to be obtained with only 3 % difference from the most refined grid in 400 CPU hours instead of 1500 CPU hours (for 4 simulated days). The GITM results, discussed in Sect. , have been obtained from 3-D runs with a total of 15 simulated days each (3 days of “spin-up” + 12 days of simulation) in order to be fully representative of the model performances.

CHAMP is a German small-satellite mission for geoscientific and atmospheric research and applications, managed by the GeoForschungsZentrum (GFZ). Between 2000 and 2010, CHAMP simultaneously generated, for the first time, high-precision gravity and magnetic field measurements. These measurements allowed for the detection of not only the spatial variations in both fields but also their variability with time . The accelerations on CHAMP are measured by the STAR (Space Three-axis Accelerometer for Research) instrument developed by ONERA . Its objective is to measure all non-gravitational accelerations of the satellite (drag, solar and Earth radiation pressure) in order to determine the Earth's gravity field from purely gravitational orbit perturbations. The accelerometer measurement principle is based on electrostatic suspension of a proof mass in a cage. The instantaneous position of the proof mass is measured by three capacitive sensors, which permit a determination of the acceleration vector and resulting drag and density. The instrument has a dynamic range of ±1×10-4 m s-2, a resolution of better than ±3×10-9 m s-2, and a frequency range of 10-1 to 10-4 Hz. The STAR instrument is positioned at the centre of gravity of CHAMP to minimise the effect of rotational accelerations and gravity gradients on the measurements.

Data derived from STAR can be used to calculate the neutral density with an uncertainty level of 10–15 % (1σ) for moderate geomagnetic activity . shows an overestimation of the CHAMP density values of about 20 % when compared to the HASDM semi-empirical model. observed a similar overestimation when comparing the same semi-empirical model data with CHAMP densities processed with a different retrieval algorithm. Both and conclude that the bias in the density values is caused by the approximations in the modelling of the external satellite geometry and the aerodynamic interaction of the gas with the spacecraft walls. The design of the CHAMP satellite, with its complex elongated shape and protruding instruments, antennae and baffles, is not ideal for the purpose of density measurements.

CHAMP has also a planar Langmuir probe (PLP) in order to measure the atmosphere electron density with an uncertainty within 10 % (1σ) . The PLP provides measurements of the electron density and temperature every 15 s for 1 s in the voltage sweep mode. Thus, CHAMP data from the database v2.2 are available every 15 s, which corresponds to a spatial resolution of about 115 km considering a satellite velocity of 7.6 km s-1. Further details of the PLP design and performance are summarised by and . In the present paper, novel comparisons have been performed between numerical models and electron density data from CHAMP.

On top of CHAMP data, GITM simulations have also been compared with GRACE flight data. GRACE was launched in 2002 and consists of two almost identical satellites which follow each other on the same orbit at a distance of about 220 km . The orbits are circular polar, with an initial altitude of 485 km at launch (near solar maximum 23) and an inclination of 89∘. The design of the GRACE satellites is similar to the CHAMP satellite. Indeed, these twin satellites contain a GPS receiver, an extremely precise accelerometer and star tracker cameras. In addition, each of the GRACE satellites carries a K-band horn directed at the other satellite for inter-communication. Like CHAMP, these satellites have the main objective of investigating Earth's gravity field and its small variations.

The solar flux F10.7 is a good proxy to describe the daily solar activity; Fig.  shows the cyclic behaviour of this index between the years 1980 and 2015. The grey line in Fig.  shows the daily variation in the F10.7 proxy, whereas the black line represents the 81-day-averaged variation. The grey area indicates the corresponding solar activity period during the descending phase of the 23rd solar cycle, where the F10.7 values are ranging from 66.7 to 233.1 W m-2 Hz-1. Table  summarises the boundary conditions of the cases under investigation in this paper. These specific simulation windows have been selected in periods of low geomagnetic activity (Ap < 30) in order to avoid any geomagnetic storm that could interfere with the study. Figure  shows the impact of a solar storm on CHAMP flight data and GITM simulations. Three distinct storms (Ap >= 150) impacted the thermosphere between 22 and 27 July 2004. Both the data and simulations show an increase of the neutral density by almost 1 order of magnitude.

The neutral density of the GITM simulations from Table  has been averaged over a period of 12 days and presented as a function of altitude, latitude and local time in order to extract the long term effect of the solar flux input. Figures  and  show the effect of the F10.7 values on the simulated density in the thermosphere at 0∘ and 80∘ N latitude as a function of altitude at 02:00 and 14:00 local time (LT) respectively. As can be observed from Fig. , the sensitivity of the atmosphere density to the solar flux is increasing with the increasing altitude. At about 150 km altitude and below there is a small effect of the Sun's activity, and the air density can be considered almost independent of the energy radiated by the Sun. However, at an altitude of 400 km, the air density value can increase by 2 orders of magnitudes between low and high solar activity (i.e. 66.7 to 233.1 W m-2 Hz-1 respectively). The comparison between Figs.  and  shows that the increment of density between low and high solar activity is dependent on both local time and latitude. In particular, the density increase is slightly bigger at 80∘ N with respect to 0∘ (at constant local time) and on the nightside with respect to the dayside (at constant latitude). Because of this large variation in density in the mid-lower thermosphere, satellites in low Earth orbit must increase their orbital altitude to take into account the increased aerodynamic drag during periods of high solar activity. An example is the International Space Station .

The results of the GITM simulations are compared with neutral densities and electron densities measured by CHAMP and GRACE. As an example, the trend of the simulations and the flight data is presented in Figs.  and  as a function of time (only 6 h) for the neutral and the electron density respectively. As can be observed in Fig. , the density shows a sinusoidal-like behaviour typical of the day–night sequence. The higher peaks correspond to a high density over the Equator on the dayside, whereas the valleys correspond to the passage over the South Pole on the nightside. The agreement between the flight data and the semi-empirical models is very good, especially with the NRLMSISE-00 model. However, a marked overestimation of the order of 50 % can be observed from the comparison between the GITM simulations and the flight data.

In Fig.  the electron densities obtained from CHAMP, the GITM simulations and the semi-empirical model are compared. The agreement with the flight data and the IRI-2012 model is good, and again less good agreement with the GITM simulations is observed. In particular, on the nightside, the simulations around the Equator show a highly over-predicted electron density with respect to flight data. The reason for such an overestimation can be attributed to the missing divergence term in the electron continuity equation in GITM. Such a missing term together with the missing electrodynamic model does not allow GITM to account for the proper electron downwelling which causes accumulations of electrons on the nightside in the equatorial region.

Figures – show a comprehensive comparison between GITM simulations and CHAMP neutral densities as a function of latitude and time for all cases presented in Table . All figures represent the ratio between GITM-simulated density and CHAMP flight data. A density ratio value of 1 corresponds to a good validation between the simulations and the experiments, whereas values larger or smaller than 1 indicate that the GITM density is over-predicted or under-predicted with respect to CHAMP data. The ratio values are shown as a function of time and latitude for the ascending/descending branch of the orbit. Local times of ascending/descending nodes and orbit altitude are given for each case under investigation. In general, Figs. – show that GITM simulations are consistent in time and they are representative of the behaviour of the model under the different solar activity periods.

For all cases (a–f), GITM simulations present a different prediction behaviour over the poles than over the Equator. Over the poles the computed neutral density tends always to under-predict the flight data with respect to the predictions at the Equator. This is true both for the ascending/descending branch of the orbit. Figures , , and show that, around ±30∘ latitude, there are two distinct zones of high relative ratio values with respect to the rest of the nightside orbit branch. These high relative ratio values correspond to the under-prediction or the over-prediction of the equatorial mass anomaly (EMA) by GITM. This phenomenon, generated by the equatorial electrojet (EEJ) and recently described by using CHAMP data, creates an accumulation of density between 15 and 30∘ latitude poleward with respect to the magnetic equator. The interaction between the Earth magnetic field and the ions transported by the EEJ causes a Lorentz lift-off mechanism that pushes ions and electrons to higher altitudes over the magnetic equator. When the particles reach the maximum altitude, they fall down and generate an accumulation of density with a fountain-like shape. Because of the experimental nature of the electrodynamics model in GITM (i.e. dynamo modelling), it was decided to disable such a feature during the investigation so that the EMA phenomenon cannot be correctly modelled in the current simulations. As can be seen from Figs.  and , when the orbit crosses the ascending/descending nodes at dawn and dusk respectively, the ratio values of the ascending/descending branch of the orbit are quite similar to each other. Moreover, for both cases the under/over-prediction of the EMA is visible during the ascending branch as well as the descending branch of the orbit.

Figures – represent the comparison between GITM simulations and CHAMP electron number densities as a function of latitude and time for all cases presented in Table . The values shown in the figures are again the ratios between the simulations and the flight data. As discussed above for Fig. , in the comprehensive analysis shown in Figs. – it can be observed that GITM simulations are unable to correctly predict the electron density on the nightside at ±30∘ over the Equator. All cases indicate that the ratio between the simulations and the flight data is equal to 5 (or higher) on the nightside. Again, in Figs. – it can be observed that the results are consistent over time and therefore representative of the behaviour of the model. The comparison between the simulated and measured electron densities shows similar features to those in the analysis carried out for the neutral densities. In particular, on the dayside the EMA phenomenon is still visible at ±30∘ poleward over the magnetic equator, where the typical double peak is either over- or underestimated by the simulations because of the missing electrodynamic model. The analysis over the poles and over the Equator shows again a different behaviour of the model in the two regions. In general, it can be observed that the ratios over the poles are closer to a value of 1 for all cases under investigation, whereas the ratio values over the Equator are more scattered.

Figure  summarises the behaviour on GITM for a wide spectrum of solar flux values. The ratios between the measured and simulated neutral densities are given as a function of the F10.7 index and for CHAMP and GRACE satellites. Every point shown in Fig.  is the result of an orbital average of the ratios between the simulations and the flight data. Figure  indicates that there is a linear dependency between the accuracy of the GITM simulations and the solar activity. In particular, the simulated neutral densities are underestimating the flight data by a factor of 0.4 during low solar activity (85 W m-2 Hz-1), whereas they are overestimating the flight data by a factor of 1.7 during high solar activity (260 W m-2 Hz-1). The density simulated by GITM agrees well (1 : 1 ratio) with the flight measurement only for F10.7 values between 150 and 170 W m-2 Hz-1 (i.e. during medium solar activity).

Figure  shows the results obtained from the comparison between the simulated and measured electron densities as a function of the solar flux. The results shown are affected by a high degree of scattering, in particular on the nightside. As explained in Sect. , a wrong implementation of the electron continuity equation in GITM is the reason for a high density of electrons on the nightside. Because of such an implementation, electrons are not correctly downwelling to lower altitudes on the nightside, which is the reason for their accumulation and the extremely high ratio on the nightside. However, the results from the comparison on the dayside, even if they are still affected by a marked scattering, again show a linear dependency with the solar flux. In conclusion, the sensitivity of GITM to the solar flux is higher for the neutral density (solid line) than for the electron density (dashed line).

An investigation of the reasons for such sensitivity is currently ongoing and some preliminary causes are presented here.

The transport properties, namely the dynamic viscosity and the thermal conductivity, are not correctly implemented in GITM. For example, the viscosity model presents a wrong trend as a function of temperature if compared to a simple Sutherland formulation. Nevertheless, a preliminary comparison by between neutral densities computed with GITM using the original viscosity model and a Sutherland-based model shows a change of less than 5 % in density. The latter comparison suggests that the effect of the viscosity model in GITM is negligible; nevertheless, the effect of a different thermal conductivity model is yet to be investigated.

In general, showed that, using the broad published values of thermal conductivity, NO cooling rates and other parameters, there is a significant variation in the temperatures and densities derived from the model. This study shows that the default parameters in GITM cause the model to be too sensitive to F10.7 when compared to the mass density. The results presented by using an improved chemistry module (which includes both the updated reaction rates and the corrected excess energy allocation) show that the neutral densities computed with the new module are lower with respect to the same densities simulated with the original implementation of the chemistry module. also shows the performances of the new chemistry module over a limited range of solar activities and observes that the sensitivity of GITM to the F10.7 solar flux is not yet fixed.