During geomagnetic storms, Joule heating is known to be the dominant solar wind energy dissipation mechanism. Joule heating maximizes in the lower thermosphere-ionosphere (LTI) region, within the 100–200 km altitude range, where current density and conductivity (Pedersen and Hall) maximize . The quantification of Joule heating is a subject of intense research, as it is critical in determining the structure and evolution of the Lower Thermosphere-Ionosphere, and is responsible for a number of effects of societal importance, such as for determining atmospheric drag and predicting the resulting deorbiting times of satellites and space debris within this region . For example, the loss of 40 Space-X's Starlink satellites in February 2022 is thought to have been caused by an underestimate of the enhancement of thermospheric neutral density that resulted from enhanced Joule heating during a moderate geomagnetic storm . It is for this reason that quantifying the heating rates is critical in order to accurately determine satellite drag and orbital lifetime estimations.

Whereas the physics of the collisional processes leading to Joule heating is well understood and is captured in Global Circulation Models (GCMs) of the ionosphere-thermosphere (IT) system, the quantification of Joule heating is still largely unknown, and large discrepancies appear between different models and estimation methodologies . This is in part because the exact quantification of Joule heating requires the simultaneous and co-located measurement of all relevant parameters that are involved in the calculations of conductivity, electrical currents and fields, and in part because an unknown amount of Joule heating is related to small-scale or sub-grid variability that can not be captured by current models. Also contributing to the above uncertainty, the lower thermosphere-ionosphere (LTI) region, where Joule heating maximizes, is the least sampled of all atmospheric regions (see, e.g. , and references therein): the altitude range from ∼100–200 km is too high for balloon experiments and too low for Low-Earth Orbit (LEO) satellites, due to the large air drag. Thus, the majority of available measurements for this region comes from ground based observatories, such as Incoherent Scatter Radars, and very few in-situ space missions, such as the Atmosphere Explorers of the early 1980s. Measurements from the above are used in formulating empirical models of the upper atmosphere, such as the International Reference Ionosphere (IRI) , NRLMSISE-00 and the Horizontal Wind Model (HWM) . Furthermore, physics-based global circulation Models (GCM), such as the Global Ionosphere/Thermosphere Model (GITM) or the National Center for Atmospheric Research (NCAR) Thermosphere-Ionosphere-Electrodynamics General Circulation Model (TIE-GCM) simulate the energetics, dynamics and chemistry of this region. However, there are great discrepancies in describing the basic state of the LTI between empirical models and physics-based models, such as neutral temperature and density, which is largely due to the uncertainty in estimating the amount of Joule heating in the LTI.

Among physics-based models, GITM and TIE-GCM are widely used by the upper atmosphere scientific community. Both are 3D gridded numerical models that are used to simulate the state of the thermosphere and ionosphere in response to external driving by solar wind conditions. GITM and TIE-GCM are both based on a set of equations that describe the physical processes that occur within the thermosphere and ionosphere, such as radiation, convection, and dynamical forcing. From the outputs of these models, which include all essential variables or geophysical observables of the thermosphere and ionosphere, Joule heating can be directly computed at each model grid point.

Together with the above physics-based models, a number of empirical formulations have been derived as proxies of Joule heating, driven by solar and geomagnetic conditions. For example, Joule heating has been found to be closely related to the AE and AL indices see, e.g.. Seasonal and hemispherical differences have been examined as well to establish a more accurate relation between Joule heating and the geomagnetic indices . Further to the above, estimated Joule heating with a quadratic fit to the Polar Cap (PC) index, whereas expanded on the work of by proposing a formula that is based on both the PC and the Disturbance Storm Time (Dst) indices; and calculated Joule heating empirically, based on a model of Poynting flux that is derived using measurements of the Dynamics Explorer 2 satellite. It is noted that most of the above empirical formulations do not take into account the effects of neutral winds, which are known to impact Joule heating significantly see, e.g..

Empirical models are often designed to describe large-scale climatology and can thus underestimate the real-time magnetospheric energy input. Such differences between empirical models and observations have been discussed extensively in the literature see, e.g.. Various data assimilation methods have been developed which can be used to mitigate the discrepancy, e.g. the standard AMIE procedure and methods like SECS , Local Divergence-Free Fitting (LDFF) of SuperDARN observations and Lattice Kriging . Data assimilation methods are used to replace the empirical high-latitude drivers in GCMs, resulting in higher levels of variability, and enhanced Joule heating. These usually show a better agreement with observations. Some examples of data-driven modeling include the studies by , who performed extensive comparisons between simulation results from TIE-GCM using AMIE as a driver, and , who also used TIE-GCM using the Lattice Kriging method to derive their high-latitude auroral and convection patterns.

In the following, Joule heating estimates are calculated and presented based on simulation results of the solar storm of 17 March 2015, the largest geomagnetic storm of solar cycle 24, which is also known as St. Patrick's Day 2015 storm. As part of this study, globally integrated Joule heating rates are calculated in both GITM and TIE-GCM, and are compared against estimates obtained from various empirical formulations. GITM and TIE-GCM simulations are performed using both the Weimer 2005 empirical high-latitude electric field model and the AMIE data assimilation method as drivers. It is noted that the integration of other available electric field data assimilation models such as those listed above (i.e. the models by , and ) with TIE-GCM and GITM was not available to the authors at the time of this study, and is not considered herein. The integration of the above data assimilation methods as drivers of commonly used GCMs is a topic of future research. Joule heating estimates are presented as time series over the course of St. Patrick's Day 2015 storm. Together with the time series of the evolution of Joule heating during the storm, the cumulative globally integrated Joule heating is compared as calculated by each model. Furthermore, hemispherically-integrated Joule heating rate estimates are compared between GITM and TIE-GCM.

This paper is organized as follows: Sect.  presents details of the GITM and TIE-GCM, including their external drivers, and describes the derivation of Joule heating in both models. Section  presents the results of the implementation of the simulations for St. Patrick's Day 2015 storm as well as the resulting Joule heating as obtained from various empirical formulations. Section  discusses the results, highlighting potential causes of the observed discrepancies. Finally, Sect.  summarizes the conclusions of this work.

Based on the simulation results shown in Figs.  and , a significant disagreement is observed in the values of the globally-integrated Joule heating rates as obtained through TIE-GCM and GITM during the storm main phase, and in particular between 17 March, 06:00 UT, which marks the first southward turning of Bz and is noted with line A in the above figures, and 17 March, 22:50 UT, which is the time of minimum Dst, and is marked with line D. This disagreement is noted both in terms of amplitude as well as in terms of the overall shape and evolution of Joule heating, and is more prominent when the Weimer 2005 model is used for the specification of the high-latitude electric field model. A better agreement between the two GCMs is found when the AMIE model is used, even though at times there are significant differences, such as around 17 March, 22:00 UT. As detailed in Sect. , the two GCMs use similar formulations for the calculation of Joule heating; also the electric field and the particle precipitation models are the same. Thus, a possible reason for these differences is related to the different spatial resolutions that are employed in the two models. Another possible reason is the different way that the two models treat vertical momentum and eddy diffusion: As discussed in Sect. , in GITM the complete vertical momentum equation is solved see, e.g., which could lead to significant differences, in particular in the lower regions of the model, where also the eddy diffusion is expected to be larger. A more detailed parametric study could shed more light onto the causes of these discrepancies. A general order-of-magnitude agreement is observed in the first part of the storm, from 17 March, 00:00–12:00 UT, as well as after the time of minimum Dst and in the recovery phase of the storm.

Further to the models that have been used in this study, used the OpenGGCM to calculate the global ionospheric Joule heating during St. Patrick's Day 2015 geomagnetic storm. The values of global Joule heating rates that are presented in Fig. 7 of are considerably higher than the values reported herein. Such disagreements in the comparisons of simulated Joule heating have been discussed extensively by, e.g. , who investigated the underestimation of Joule heating caused by high-latitude electric field variability in electric field models and addressed the notion of “electric field variability” as a potential source of this underestimation. Similarly, used TIE-GCM driven with AMIE to gain insights into the 2015 St. Patrick's Day storm's impact on the ionosphere-thermosphere (IT) system by utilizing observations of high-latitude forcings, specifically aurora and electric fields, along with the TIE-GCM. , utilized GITM along with empirical models and proxies derived from in situ measurements to estimate the energy distribution within the IT system during the solar storms events of 16–19 March of 2013 and 2015, while , calculated Joule heating rates through GITM simulations and revealed that the globally averaged thermospheric temperature (Tn) was underestimated under quiet geomagnetic conditions.

It is noted that the comparative analyses presented above in Figs. –, and also the analyses in the works by, e.g. , , , , and other similar studies can not, on their own, bring closure as to which model provides the most accurate estimates of storm-time Joule heating. Instead, the analyses herein can provide the range of variability of Joule heating according to the different models and drivers, from which the uncertainty in estimating and predicting the state of the LTI can be approximated. Furthermore, the large discrepancies that are demonstrated by these model runs indicate that the exact quantification of Joule heating is a critically missing parameter in the LTI energetics. This is due to the lack of direct, comprehensive measurements in this region of the Earth geospace environment, as the altitude range where Joule heating maximizes is too high to be sampled with probes on aerial vehicles and balloons and too low for typical spacecraft, making direct measurements therein challenging (see, e.g. ). Thus, the results of this study reinforce the current motion in the scientific community that efforts must be taken to close this gap, by dedicated missions that can provide all missing parameters required to estimate Joule heating, as outlined in Eqs. (), (), and (). Such mission concepts have been proposed and are being actively investigated (see, e.g. ).

In order to identify the key driving parameters for the discrepancies in Joule heating between the different model runs, a Spearman's Rank Correlation analysis has been performed between the four Joule heating rate time series and each of the input parameter time series that are shown in panels (a) through (e) of Fig. . The results are shown comprehensively in Table . Through this correlation analysis, it is found that Joule heating is strongly related to the SME electrojet strength in both the GITMAMIE, TIE-GCMAMIE, and GITMWeimer, with correlation coefficients of 0.86, 0.85, and 0.80 respectively; in TIE-GCMWeimer, this correlation drops to 0.53. The correlation coefficients of the four runs with By are similar, but slightly smaller in TIE-GCMWeimer, whereas the correlation coefficients of the four runs with Bz are similar but negative, indicating anti-correlations, with, again, a smaller correlation (in absolute value) in TIE-GCMWeimer. The anti-correlation between Joule heating and Bz is attributed to the enhanced Joule heating during southward turnings of the IMF; the dependence of Joule heating on Bz has been examined in more detail in various studies, such as, e.g. by . The correlation with solar wind plasma density is lower for TIE-GCMWeimer and similar between the other three runs. The correlation with the absolute value of Dst is similar among all four model runs, and is the second most significant correlation, after SME. Finally, the correlation with SYM-H is similar for all four model runs, and is comparable to the correlation with AE and Bz.

In Figs.  and a notable difference can be seen in Joule heating when it is calculated in TIE-GCMWeimer compared to GITMWeimer. This is particularly evident in the period from ∼12:00UT to ∼19:30UT on 17 March 2015, which is shown in the gray-shaded region that is bounded by lines B and C. During this time, it can be seen that GITMWeimer is well correlated with the SME electrojet strength, with an increase and subsequent decrease in SME being accompanied by a corresponding increase followed by a gradual decrease in Joule heating, whereas, in contrast, Joule heating in TIE-GCMWeimer shows an initial drop followed by a gradual increase. This increase appears to be correlated with the prolonged southward turning of IMF Bz during this time, which does not appear to affect in the same way the calculations of Joule heating in GITMWeimer. Furthermore, Joule heating as computed by GITMWeimer exhibits a higher magnitude compared to the results from TIE-GCMWeimer, as well as the results from GITMAMIE and TIE-GCMAMIE. This is attributed to the differences in the auroral precipitation model that is used for the two runs that utilize the Weimer 2005 model as input, GITMWeimer and TIE-GCMWeimer, as is also indicated by the differences in the HP power presented in Fig. f. As discussed above, the GITMWeimer run uses the FTA model , while TIE-GCMWeimer run uses the analytical auroral model of and . On the other hand, GITMAMIE and TIE-GCMAMIE use the same auroral precipitation model, described in , and show better agreement in terms of shape and magnitude, as shown in Fig. f, especially between lines B and C. These results highlight the significance of particle precipitation in the overall electrodynamic coupling within the LTI (e.g. ), since precipitation leads to increased ionospheric conductivity , which is an essential contributor to Joule heating. A further parametric study based on model runs under different parameterizations of particle precipitation would enable a thorough evaluation on the inter-relationship between particle precipitation, conductivity and Joule heating, and a quantitative assessment of the extent to which the differences in the model run results are indeed correlated to differences in particle precipitation and conductivity. It is also noted that in the TIE-GCMWeimer run a much lower variability is observed in Joule heating compared to the other three runs, which have a larger peak-to-peak fluctuation in the amplitudes of Joule heating. It is speculated that this is due to the lower level of correlation between the TIE-GCMWeimer and the SME index, compared to the other three runs, as discussed above and as shown in Table . Finally, it is noted that all empirical formulations tend to underestimate the overall Joule heating when compared to all four GCM runs, as shown in Fig. f. A possible reason is that empirical models in general rely on historical data and statistical models that may not adequately capture the complex, non-linear dynamics and feedbacks that drive extreme events, such as solar storms. GCMs on the other hand, while still having limitations, attempt to simulate these complex processes, potentially offering a more realistic representation of extreme events.

Comparing the time series of the hemispherically-integrated Joule heating from the two GCMs with the corresponding values from the empirical models, as is plotted in Fig. a, it can be seen that there is a closer agreement between the empirical models and TIE-GCMWeimer rather than with GITMWeimer. Furthermore, comparing the percentage differences between the hemispherically-integrated Joule heating in the northern and Southern Hemispheres as calculated with GITMWeimer and TIE-GCMWeimer, which are plotted in Fig. c, it can be seen that the two model runs show significantly different inter-hemispheric asymmetries, especially during times of enhanced Joule heating, between lines B and D. During this time, GITMWeimer calculations show higher Joule heating in the Southern Hemisphere by up to ∼55%, while TIE-GCMWeimer shows initially higher Joule heating in the Northern Hemisphere by up to ∼35% and subsequently higher Joule heating in the Southern Hemisphere by up to ∼40%, with a time-lag of approximately 5 h. The percentage differences between the hemispherically-integrated Joule heating from GITMAMIE and TIE-GCMAMIE are presented in Fig. d and show almost the same inter-hemispheric asymmetry during the simulation period. Both GITMAMIE and TIE-GCMAMIE show higher Joule heating in the Northern Hemisphere by up to ∼116% and ∼100% respectively during the storm main phase, while during the recovery face of the storm Joule heating deposition in the Southern Hemisphere becomes higher by up to ∼60% for the GITMAMIE and up to ∼74% for the TIE-GCMAMIE.

By comparing the time-integrated (cumulative) hemispherically-integrated Joule heating, as shown in Fig. , it can be seen that, whereas higher Joule heating is observed in the Southern Hemisphere (SH; dashed lines) than the Northern Hemisphere (NH; solid lines) for the GITMWeimer (blue) and TIE-GCMWeimer (brown) simulations, the opposite is observed in the case of GITMAMIE (purple) and TIE-GCMAMIE (green), with the Northern Hemisphere receiving larger amounts of heating over the course of the storm.

Such asymmetries in the hemispherically-integrated Joule heating have been identified by several studies, and have been associated with the Earth's asymmetric magnetic field configuration: as discussed in, e.g. and references therein, the dipole tilt and eccentricity shift in the Earth's magnetic field leads to a displacement between the geographic and geomagnetic poles, which is larger in the Southern Hemisphere, and to a difference in the magnetic field strength between north-south conjugated latitudes. used GITM to study the impacts of different causes on the inter-hemispheric asymmetry of the ionosphere-thermosphere system, including inter-hemispheric differences associated with the solar irradiance, the geomagnetic field, and the magnetospheric forcing under moderate geomagnetic conditions. derived an index of inter-hemispheric asymmetry for Joule heating, which, for solar equinox conditions, was found to be as large as ∼43% due to the asymmetric geomagnetic field, ∼28% due to asymmetric particle precipitation and ∼35% due to asymmetric ion convection pattern. , using Swarm satellite observations, demonstrated that the Northern Hemisphere generally receives a higher electromagnetic energy input across all seasons. This preference has also been observed using DMSP satellites . also investigated the appearance of such asymmetries, and revealed a considerable asymmetry in the hemispherically integrated Poynting flux between the Northern Hemisphere (NH), which generally showed higher flux, and the Southern Hemisphere (SH). More recently, , also investigated inter-hemispheric asymmetries, looking into the role of solar wind driving conditions and the accuracy of Joule heating estimates using GITM (Weimer and AMIE) runs during the 2013 St. Patrick's Day geomagnetic storm (note the different year compared to the 2015 St. Patrick's Day geomagnetic storm that was simulated herein). They showed that AMIE driven simulations lead to stronger inter-hemispheric asymmetries in Joule heating compared to Weimer 2005 driven runs, and found higher Joule heating deposition in the Southern Hemisphere for the first phase of the 2013 St. Patrick's Day geomagnetic storm for both Weimer and AMIE simulation, which reversed in later phases of that storm.

The results presented herein indicate that the appearance of inter-hemispheric asymmetries are largely dependent on the external drivers that are used (primarily the electric field and auroral precipitation specifications). The results show that both the GITMAMIE and TIE-GCMAMIE runs have almost the same inter-hemispheric asymmetry and Joule heating magnitude, as observed in Fig. d, denoting that the implementation of the two GCMs delivers almost the same results under the same driving conditions (electric field and auroral precipitation). This is not confirmed in the case of the GITMWEIMER and TIE-GCMWEIMER runs in their default configuration, as observed in Fig. c, where the Joule heating deposition is largely variable, as discussed above. Taking into account that both of these runs use the Weimer 2005 model as a high latitude electric field driver, these results show that Joule heating is highly dependent on the auroral precipitation model.

It is noted that, as discussed above, both GITM and TIE-GCM use the International Geomagnetic Reference Field (IGRF) magnetic field model, and hence the asymmetries in the magnetic field are the same; thus the differences in the observed behavior are more likely attributed to the asymmetric particle precipitation and the asymmetric ion convection pattern. However the exact causes of the different behavior of TIE-GCM and GITM when run using different external drivers with respect to the inter-hemispheric differences is a subject that requires further research through parametric studies.

Based on GITM and TIE-GCM driven by the Weimer 2005 and AMIE models, as well as on various empirical formulations, globally- and hemispherically-integrated Joule heating rates were calculated during St. Patrick's Day storm of 2015. It is found that Joule heating rate estimates in the global circulation models, GITM and TIE-GCM, for all external drivers, are generally higher in magnitude than any of the empirical models. Comparing GITMAMIE and TIE-GCMAMIE, it is found that they are in better agreement compared to GITMWeimer and TIE-GCMWeimer in terms of amplitudes, peak-to-peak variability and inter-hemispheric asymmetry. In comparing the latter two models, it is found that Joule heating derived using TIE-GCMWeimer has lower amplitudes and also a lower peak-to-peak variability than GITMWeimer. Significant variations that are observed are most likely attributed to the different precipitation models that are employed. This comparison is essential, as it reflects the default parameterizations of the two GCMs. On the other hand, Joule heating results derived using TIE-GCM and GITM driven with AMIE data can be directly compared. Through a correlation analysis, and also by comparing the heating rates for a period of clear anti-correlation in the heating rate trend between the two models, it is found that both the TIE-GCMAMIE and GITMAMIE runs are strongly driven by the SME index; this is not the case for the two Weimer runs, where GITMWeimer is strongly driven by the SME index, which is not present in TIE-GCMWeimer. It is also found that Joule heating in TIE-GCMWeimer is affected by the southward turnings of Bz to a larger extent than GITMWeimer. Joule heating calculated by the two GCMs using AMIE inputs shows similar values and peak-to-peak variability, compared to Joule heating derived using GCMs Weimer runs.

By integrating the Joule heating estimates separately in each hemisphere, it is found that GITMWeimer shows a larger degree of asymmetry during the main phase of the storm than TIE-GCMWeimer. Furthermore, by integrating Joule heating in time it is found that the cumulative Joule heating input to the thermosphere is larger as calculated in GITMWeimer in the SH and NH, followed by the various GCM runs, and then by the various empirical models. A factor of 2 difference is observed between the largest and smallest cumulative Joule heating when comparing different GCM runs, whereas a factor of 25 is observed between the largest and smallest cumulative Joule heating, when all models (GCMs and empirical) are inter-compared. The localization (latitudinal and longitudinal distribution) of Joule heating in the two models exhibits slight variations, as depicted in characteristic time steps, as presented in Fig. .

In conclusion, as also demonstrated by the discrepancies in the above cross-comparisons between physics-based and empirical models, Joule heating remains to this date a quantity with many discrepancies in its estimation, showing large gaps in its understanding and parameterization. At the same time, it is a quantity of great significance in LTI processes, as it determines to a great extent the overall energy budget, in particular during active solar and geomagnetic conditions. Thus, characterizing its magnitude, time evolution and variability within the latitude and altitude region where it maximizes and accurately parameterizing Joule heating by solar and geomagnetic conditions are critical missing pieces in accurately understanding and modeling LTI processes. This demonstrates the currently limited knowledge about Joule heating and emphasizes the need for comprehensive measurements, such as outlined in , to accurately quantify Joule heating.