Earth's overlapping ionosphere-thermosphere (IT) region is the site of mechanical and electrodynamic coupling between the neutral atmosphere and plasmas of both terrestrial (magnetospheric and ionospheric) and extraterrestrial (solar wind) origin. Much of the electrodynamics within this region can be described in terms of a three-fluid model consisting of neutral, ion, and electron fluids in the presence of a strong background magnetic field see Sect. 9.5 in.

A common point of reference for a vast number of experimental investigations of high-latitude IT electrodynamics is the component of the ionospheric Ohm's law that is perpendicular to the main magnetic field, assuming steady-state stress balance between Lorentz and collisional drag forces and neglecting all other forces in the ion momentum equation (Sect. 5 in , ; Sect. 3.2.1 in , ): 1j⟂=σPE⟂+u×B+σHb×E⟂+u×B, with j⟂ the perpendicular current density, σP and σH the Pedersen and Hall conductivities, and E⟂, B, and u the perpendicular ionospheric electric field, the total (background plus perturbation) magnetic field, and the thermospheric wind. The unit vector b points in the direction of B. The corresponding Joule heating (heating rate per volume) is given by 2wJ=j⟂⋅E⟂+u×B=σP|E⟂+u×B|2. Both j⟂ and wJ are independent of reference frame in (magnetic) Galilean relativity by virtue of the fact that they are defined in terms of the sum of the electric field and the “neutral wind dynamo” u×B.

Some notable experimental investigations of Eqs. () and () include that of and those based on the Joule II suborbital sounding rocket campaign , while others have focused on experiment-model comparisons, modeling, or theoretical aspects e.g.. used incoherent scatter radar measurements at a temporal resolution of ∼17 min to present the first examination of how the neutral wind u modifies height profiles of Joule heating. The Joule II-based study of used in situ measurements of electric field, bulk ion drift, neutral wind, and electron density in the vicinity of a relatively quiescent auroral breakup in northern Alaska to estimate the errors in E-region ion-neutral momentum transfer collision frequency and Joule heating altitude profiles that would arise if the neutral winds were assumed to be zero. In their separate analysis of these and additional rocket and ground-based observations from the same campaign, found evidence for horizontal or vertical structuring in ion-neutral collision frequencies on scales of 1–10 km.

Comprehensive measurements of altitude profiles such as those presented by and are rare. Much more frequently one encounters a height-integrated form of the ionospheric Ohm's law: 3J⟂=ΣPE⟂+U×B+ΣHb×E⟂+U×B, where ΣP and ΣH are the Pedersen and Hall conductances (conductivities integrated over altitude, explicitly Σc=∫σcdh with c either H or P). This form is obtained by integrating Eq. () over altitude assuming that B is radial, that B and E⟂ are independent of altitude over ionospheric E- and F-region altitudes (∼100–250 km), and that U is a height-averaged neutral wind, or a representative or effective neutral wind. Most often the wind is “neglected”, which is generally equivalent to assuming it is zero in Earth's rotating frame of reference e.g.. (Appendix  provides a brief summary of the implications of assuming the E-field is constant with altitude.) One also encounters an expression for the height-integrated Joule heating rate, WJ=∫wJdh=J⟂⋅E⟂+U×B4=ΣPE⟂+U×B2, that is also defined in terms of a typically unspecified or unused effective neutral wind U. Last, one encounters estimates of ionospheric conductances via the expressions likewise defined in terms of U, 5ΣH=-b^⋅J⟂×E⟂+U×B/|E⟂+U×B|2;6ΣP=J⟂⋅E⟂+U×B/|E⟂+U×B|2.

The approach represented by Eq. () for estimating height-integrated Joule heating based on height-integrated quantities dates at least back to the work of and continues to be used widely seeand references therein. The study of stands out as the only observational study of global distributions of ionospheric Joule heating that has included information, via an empirical model, about an effective wind pattern. On the other hand, the approach represented by Eqs. () and () for estimating conductances based on height-integrated quantities originates with . This approach has been used by a number of studies over the past two decades, and has elsewhere been referred to as the “electrodynamic method” e.g.. To our knowledge, all published studies in which conductance distributions have been estimated experimentally via Eqs. () and () have assumed U=0 .

The height-integrated approach to ionospheric electrodynamics represented by Eqs. ()–() is heavily simplified: Earth's magnetic field lines are not radial, ionospheric electric fields do not necessarily map along field lines , and the horizontal components of the neutral wind exhibit the nearly permanent presence of vertical shears over altitudes of 80–140 km . More advanced treatments of ionospheric electrodynamics that account for such complications, such as that presented by , nevertheless remain unused in a large number of experimental studies and data assimilation techniques e.g., primarily for lack of a body of robust 3D measurements of the IT system needed to make use of them . For similar reasons, in the majority of existing global MHD models the coupling between the magnetosphere and the IT system is founded on what term a “key magnetosphere-ionosphere coupling equation” derived from current continuity their Eq. 11; see alsoand references therein in which the neutral wind is assumed to be zero or constant with altitude.

In this study, we explore some of the implications of the commonly employed assumption that the neutral wind is independent of altitude that is necessary to arrive at Eqs. ()–() as well as the current continuity equation used in magnetosphere-ionosphere coupling. In Sect.  we summarize central quantities and equations in a height-integrated description of IT electrodynamics when the neutral wind is not assumed to be independent of altitude. In Sect.  we present three basic questions that are raised by this alternative formalism, and answer them using neutral wind profiles derived from rocket-borne trimethylaluminum (TMA) chemical release experiments launched from the Poker Flat Research Range (PFRR) between 2007–2018 together with conductivity profiles calculated from measurements made by the Poker Flat Incoherent Scatter Radar (PFISR). In Sect.  we summarize our findings, and discuss how they can be used to understand the role of the neutral winds in 2D descriptions of IT electrodynamics.

If one assumes that u does vary with height, integration of Ohm's law (Eq. ) over ionospheric altitudes yields 7J⟂=ΣPFP+ΣHb^×FH where 8Fc=E⟂+Uc×B;9Uc=1Σc∫σcudh; and c is either H or P. The integration is performed over all ionospheric altitudes. We refer to the conductivity-weighted neutral wind terms UH and UP as the Hall-weighted or Pedersen-weighted neutral winds. The latter has been termed the “effective neutral wind” by and plays a role in estimates of height-integrated Joule heating and the Pedersen conductance, which we discuss later in this section. The distinction between the Hall-weighted and Pedersen-weighted neutral winds disappears when the two conductivity profiles differ by no more than a constant factor, or when the neutral wind u does not vary with altitude. As noted in the Introduction, these idealized conditions are rarely manifest in measured altitude profiles of the conductivities and neutral winds e.g..

Integrating the right-hand side of the expression for Joule heating density (Eq. ), we obtain the height-integrated Joule heating rate WJ=∫wJdh=∫σP|E⟂+u×B|2dh10=ΣPE⟂2+2E⟂⋅UP×B+∫σP|u×B|2dh. Because of the integral in the last term involving |u×B|2, this expression cannot directly be brought into the more familiar form ΣPE⟂+U×B2 for some appropriately chosen U (cf. Eq. ). However, one may show explicitly via the Cauchy–Bunyakovsky–Schwarz inequality (see Appendix ) that ∫σP‖u×B‖2dh≥ΣPUP×B2. Then the following inequalities hold: WJ≥ΣPE⟂+UP×B2=ΣPFP2;11ΣP≤WJ/FP2. Thus when the “effective” neutral wind U is specified to be the Pedersen-weighted neutral wind UP in Eqs. () and (), these equations are in fact lower and upper bounds, respectively, on the true height-integrated Joule heating and Pedersen conductance. When the neutral wind does not vary with altitude we have UP=UH=U in Inequality (), such that the height-integrated Joule heating WJ=ΣP(E+U×B)2 and the Pedersen conductance ΣP=WJ/(E+U×B)2 as given by Eqs. () and ().

From the foregoing we see that for estimation of height-integrated Joule heating, the Pedersen-weighted neutral wind UP constitutes the most natural definition of the “effective” neutral wind, as indirectly suggested by .

Equations ()–() illustrate that there is in general not one, but two, neutral wind terms in height-integrated treatments of IT electrodynamics. The neutral wind appears as two separate terms weighted separately by the Hall and Pedersen conductivity profiles, here defined by Eq. () and respectively denoted by UH and UP. This situation raises the following questions:

How large is the observed difference between UH and UP? (Sect. )

To answer these questions we use horizontal neutral wind profiles derived from TMA chemical release experiments carried by sounding rockets during five campaigns launched from PFRR and references therein, as well as vertical conductivity profiles derived from PFISR measurements and empirical models of ionospheric and atmospheric composition, atmospheric temperature, and Earth's magnetic field.

This study aims to clear up misconceptions about how to incorporate thermospheric winds, which are sparsely sampled, and highly variable and structured in altitude, into IT data assimilation techniques that rely on 2D uniform slabs to describe IT electrodynamics. The assumption of a steady-state force balance between Lorentz and collisional drag forces is central to the expressions for the ionospheric Ohm's law and Joule heating we have used. The accuracy of assumed steady-state stress balance can be assessed by comparing the ion inertial term to the collisional drag term in the ion momentum equation, i.e. by the ratio τrelax/τvar, where τrelax∼νin-1 is the ion–neutral relaxation time and τvar is the characteristic timescale of variability. In the 110–130 km altitude range νin is large, making τrelax of order ∼0.01–0.1 s; here inertial effects become significant only for variability on comparable sub-second timescales. Even under highly active conditions, typical electrodynamic variability occurs on timescales of seconds to minutes, implying that the inertial term is smaller than the drag term by at least an order of magnitude.

Even for fluctuations at several Hz (representative of extreme Alfvénic forcing), the inertial contribution would remain ≲10 % of the drag term, and substantially smaller for more typical variability. Other neglected terms, such as pressure gradients and gravity, are smaller still (∼10-4 of the drag term). Therefore, in the altitude range where the conductivities maximize, the steady-state balance between Lorentz and collisional drag forces is expected to be accurate to within a few tens of percent under extreme conditions, and considerably more accurate under typical conditions. Thus while transient departures from steady state may occur during highly dynamic events or at very small spatial scales, the quasi-steady approximation we have used is generally robust for describing high-latitude electrodynamics in this altitude range e.g..

The analysis in Sect.  shows how height integration of the ionospheric Ohm's law yields two separate neutral wind vectors (defined by Eq. ), and that height-integrated variables can only produce a lower-bound estimate of the isolated contribution of neutral winds to Joule heating (third term on the right-hand side of Eq. ). This analysis has some bearing on the “key MI coupling equation” derived from current continuity that is mentioned in the introduction, 12∇⋅J⟂=-j∥, where the integrated perpendicular current J⟂ is given by a height- or field line-integrated form of Ohm's law (i.e. J⟂=Σ¯¯⋅E⟂+U×B) in terms of an ionospheric conductance tensor Σ¯¯, the potential electric field E⟂=-∇⟂Φ, and an effective neutral wind U. The development in Sect.  shows that it is strictly speaking not possible to formulate the height-integrated ionospheric Ohm's law as given by Eq. () in terms of the conductance tensor Σ¯¯, because FH and FP are generally not identical. However, it is unclear from the results presented in Fig.  how much the differences between UH and UP might affect the description of IT electrodynamics in models that employ some form of this equation. This topic may be the subject of a future investigation.

Regarding representation of MI coupling in global models, Eq. () is not the state of the art: Ionosphere-thermosphere models such as the Whole Atmosphere Community Climate Model With Thermosphere and Ionosphere Extension WACCM-X, and the Thermosphere-Ionosphere-Electrodynamics General Circulation Model use the 2D continuity equation given in the more advanced treatment of . The equation generalizes Eq. () and does, in fact, take stock of the three-dimensional nature of the neutral wind field. Approaches similar to that of have also been presented .

The results in Sect.  constitute the first experimental comparison of the two neutral wind vectors, showing that with increasing geomagnetic activity the difference in magnitude of these vectors increases (Fig. b) while the vectors themselves become more aligned (Fig. c). It is nevertheless difficult to draw general conclusions about UH and UP solely on the basis of the estimates shown in Fig. a, as these estimates are representative of only one location (PFRR) for a sparsely sampled range of universal times (∼06:00–14:00 UT) and one season (January to March), as indicated in Table . It nevertheless seems reasonable that the range of magnitudes of UH and UP shown in Fig. a is typical at high latitudes during low to moderate geomagnetic activity.

Section  addresses the commonly employed assumption in experimental studies that the winds are zero in Earth's corotating frame of reference, as well as the concept of a two-dimensional “effective neutral wind” pattern introduced by and employed by both and . These studies take the effective neutral wind pattern to be the neutral winds at 160 km altitude, u160, although analysis in Sect.  indicates that a more suitable definition for the effective neutral wind is the Pedersen-weighted neutral wind UP, which appears in both the height-integrated Ohm's law (Eq. ) and the lower-bound estimate of Joule heating (Inequality ).

Given the scarcity of simultaneous measurements of the quantities needed to calculate UP and UH (altitude profiles of conductivities and neutral wind), another motivation for this study is to investigate potential proxies for UP and UH. Figure a explores the accuracy of four proxies for UP. Statistically speaking, the best proxy for UP is uσPpeak, the neutral wind at the altitude of the peak Pedersen conductivity (located between 115 and 128 km altitude for the 15 rocket flights presented in Table ). In contrast, the proxy for UH with the lowest median error is the zero vector (pink).

Regardless, all proxies perform similarly in the sense that all are associated with errors of order at least several tens of ms-1. While this result is subject to the same caveat given above regarding limited sampling, it does suggest that one should be skeptical of “quick fixes” for the neutral wind problem, including ignoring the winds or using wind estimates from higher altitudes such as those estimated via Fabry–Perot interferometers attuned to the 630-nm (red) line e.g.and references therein, which tend to cover the F-region ionosphere. On the other hand Fabry–Perot interferometers attuned to 557.7 nm (green-line) emissions, which tend to peak between 100 and 130 km altitude, could provide neutral wind estimates relevant for analysis of high-latitude IT electrodynamics. The primary caveat with these measurements is that the peak height of green-line emissions can range over ∼80 to 180 km altitude via the influence of auroral precipitation .

Section  provides an experimental demonstration of the analytic result that the isolated contribution of neutral winds to Joule heating (∫σP|un×B|2dh, third term on the right-hand side of Eq. ) is underestimated when calculated from height-integrated quantities as ΣP|UP×B|2. Figure c shows that the contribution estimated from height-integrated quantities tends to approach the true contribution with increasing Kp.

To further illustrate how the neutral wind affects height-integrated Joule heating, each panel in Fig.  plots the three terms in Eq. () as a function of a hypothetical electric field, assuming the electric field is either perpendicular or parallel to UP×B (left and right columns, respectively). The values for the neutral wind and Pedersen conductance are taken from the Super Soaker and Auroral Jets campaigns, corresponding respectively to quiet (top row) and active (bottom row) geomagnetic conditions. The total Joule heating (thick gray line) is also plotted in each panel, along with the approximate total Joule heating (dotted gray line) given by replacing the third term ∫σP|un×B|2dh with the approximation ΣP|UP×B|2, as in Inequality . Electric field values in the top row vary from -10 to 10 mVm-1, and from -50 to 50 mVm-1 in the lower row, corresponding to typical ranges during geomagnetically quiet and active periods e.g..

When the electric field is perpendicular to UP×B (Fig. a and c), the second term (orange line) is zero by definition. In this case, the total Joule heating WJ=ΣP|E|2+∫σP|u×B|2dh. The contribution of the neutral winds is to shift the parabola representing Joule heating upward. That is, the neutral winds make a constant and non-negative contribution to the overall Joule heating. The difference between the true total Joule heating and the approximate total Joule heating, dictated by Inequality , can be seen by comparing the thick gray line and dotted gray line that respectively represent them.

When the electric field is parallel to UP×B (Fig. b and d) the situation is more complicated, since the second term offsets total Joule heating via a linear dependence on the electric field. The two cases shown in these panels illustrate that the vertex, or minimum, is shifted horizontally such that it is located at E=-UP×B. The minimum Joule heating value would be zero if the third term ∫σP|un×B|2dh were equal to the approximation ΣP|UP×B|2. It is precisely the difference between the true term and its approximation that causes true Joule heating (thick gray line) to be everywhere greater than zero, in agreement with Inequality . When the third term is replaced by the approximation ΣP|UP×B|2, the minimum of the approximate total Joule heating (dotted gray lines) is zero as expected.

Two additional general conclusions can be drawn from Fig. : (1) During quiet geomagnetic conditions (top row), the neutral wind influences total Joule heating via both its orientation with respect to the convection electric field (second term, horizontal shifting of the parabola) and its magnitude (third term, vertical shifting). (2) During active conditions (bottom row), the magnitude of the neutral wind is likely less important, with its primary influence being to reduce or enhance the the total Joule heating. This can most easily be seen in Fig.  as the difference between the parabola that defines the total Joule heating (thick gray line) and the parabola that defines the first term (blue line).

In summary, in this study we have pointed out a fundamental limitation of 2D descriptions of IT electrodynamics: In a 2D representation, the governing equation (ionospheric Ohm's law) contains two neutral wind terms that are separately weighted by the altitude profiles of Hall and Pedersen conductivities. Furthermore, even when an appropriately defined neutral wind term is used (the Pedersen-weighted neutral wind), any estimate of height-integrated Joule heating on the basis of height-integrated and averaged quantities is mathematically guaranteed to be a lower bound of the actual height-integrated Joule heating, with some tendency for the pure neutral wind term to be less strongly underestimated with increasing Kp.

At a basic level this study is one more voice in the chorus of recent literature calling for additional neutral wind measurements . However, results in Fig.  show that estimates of height-integrated electrodynamic quantities might be improved simply by having an estimate of the neutral wind at a single altitude near the altitude at which the Pedersen conductivity peaks. This points to the utility of techniques and measurements that enable estimation of the neutral wind profiles over limited ranges of altitudes between 120–200 km where the Pedersen conductivity profile may peak, including Doppler spectroscopy based on emissions from atomic oxygen at 558 nm and references there in as well as next-generation incoherent scatter radar systems such as PFISR and the upcoming EISCAT_3D facility and references therein.

It remains to be seen how important the distinction between the Hall- and Pedersen-weighted neutral winds is for calculations of the height-integrated current density as given by Eq. (). It likewise remains to be seen whether estimates of the Hall and Pedersen conductances on the basis of estimates of the height-integrated perpendicular current and electric field, as is done by and , can be improved by inclusion of limited information about the neutral winds. We suggest that it would be valuable to compare height-integrated Joule heating estimated directly from altitude profiles (Eq. 10) and from height-integrated variables (Eq. 4), and to carry out similar comparisons for height-integrated current density (Eq.  vs. Eq. ) and conductance (∫σH,Pdh vs. Eq. 3). Such a study would be experimentally very demanding, as it would require comprehensive information about the state of the coupled ionosphere-thermosphere system.