Space Weather Center, Center for Atmospheric and Space Sciences, Utah State University, Logan, Utah, USA
Corresponding author: H. C. Carlson, Space Weather Center, Center for Atmospheric and Space Sciences, Utah State University, 4405 Old Main Hill, Logan, UT 84322-4405, USA. (firstname.lastname@example.org)
Corresponding author: H. C. Carlson, Space Weather Center, Center for Atmospheric and Space Sciences, Utah State University, 4405 Old Main Hill, Logan, UT 84322-4405, USA. (email@example.com)
 This first-principles examination of physics driving the cusp/polar upper thermosphere response to significant input energy impulses discloses previously unappreciated factors essential to thermospheric input-response relationships. The physics essential to coupling of cusp input-response processes is detailed, to make previously unexplained up-to-doubling of air density and drag near 400 km not only understandable but expected, if not inevitable. Presented as a common natural consequence of magnetic reconnection near the magnetopause, this energy-coupling from sun to upper atmosphere is through familiar processes, but by inadequately appreciated linkages. The underlying physics applies more broadly than this. We trace a logic path that should clarify the input-response, and lay out a path which if followed should enable most existing time-dependent 3-D global thermospheric models to significantly improve the realism of their representation and prediction of cusp/polar thermosphere disturbances to transient energy sources. We illustrate the concept with a sample model-run incorporating representative data.
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 Magnetopause reconnection generates energy that converges along the Earth's magnetic field, and as Poynting Flux [Kivelson and Russell, 1995] enters the top of the cusp ionosphere-thermosphere region.Carlson provided the basic physical principles to why the ionospheric cusp is to be dramatically more responsive to Joule heating than the night-side aurora [Hays et al., 1973]. The first observations to direct attention of the community to the then generally unexpected upper atmospheric cusp neutral density enhancements (up to doubling), was Lühr et al. , expanded upon by Schlegel et al. . The challenge of understanding this cusp thermosphere behavior has gone unmet for eight years, and received prioritized attention by the CEDAR/GEM (Coupling, Energetics, and Dynamics of Atmospheric Regions/Geospace Environment Modeling) research community over the past couple of years. It has practical interest for satellite drag. The first global 3-D time dependent model calculation [Demars and Schunk, 2007] and a simultaneous first principle 1-D calculation [Carlson, 2007], though reproducing a doubling of the density in the cusp upper-thermospheric ∼400 km, were exploratory. Modelling has advanced [e.g.,Crowley et al., 2010; Knipp et al., 2011] but without resolution of this challenge. This letter is devoted to detailing the four principal features that distinguish cusp and polar-cap heating [Carlson, 1998] as dramatically different from thermospheric heating elsewhere, and the underlying physics essential to achieving more realistic models for the polar cap. In this paper, first we present the physics of the four key elements of cusp upwelling. Then we present data key to the physics, and next a model run to illustrate this new methodology, followed by discussion and conclusion.
2. The Four Key Elements of Cusp Upwelling
2.1. Strong Velocity Shears and Flow Channels Are Ubiquitous Near the Cusp
Burns et al. treating the thermospheric response to night-side auroral heating, highlights that the perturbation expansion of the heated gas cell is upwards, because the horizontal width of the night-side aurora heated band is many vertical neutral scale heights (H) wide. For auroral energetic-particle heating most of the energy flux is at altitudes where this clearly applies. Electrons of a few keV deposit most of their energy near ∼100–120 km, where H is ∼10 km. On the dayside, cusp electrons deposit their energy closer to ∼150–200 km altitude, where H is ∼30 km. The lower thermospheric response to reconnection-event driven flow channels, typically ∼150–200 km wide, is thus initially mostly upwards over a representative flow channel heated by frictional, Joule hard and soft particle heating.
 Upwards expansion increases the atmospheric density above the heated flow channel by a factor increasing with altitude, since H increases as the mean particle mass decreases and T increases. Density increases only above the isopycnic crossing point, below which a small mass reduction feeds mass transport upwards to enhance density at all heights above. The isopycnic is a fulcrum altitude: density decreases below, increases above. To illustrate how much H compounds the increase with altitude, comparing the Jacchia  model for 900 K vs. 800 K yields an increased neutral density of ∼12% at 180 km, yet ∼80% at 400 km.
 Sideways expansion becomes important at heights where H is no longer ≪ the channel width. By ∼400 km for representative Tex ∼ 800–1000 K, H ∼ 50–60 km, so sideways expansion must become significant, limiting the overhead density increase from upwelling, and by mass continuity yielding down-welling to the sides of the heated flow channel.
 Frictional or Joule heating can be expressed as a function of temperatures (Ti − Tn), velocities (Vi − Vn)2, or currents, electric fields, and conductivities (J, E, σ). Thayer and Semeter [2004, Appendix A] define the conditions for the equivalence of these formulations (equations (1), (2), and (3) below) with clarity that should remove past confusion. They lucidly show equivalence of the MHD and individual species approach to redistribution of energy among particles in a gas for altitudes >90 km, and conditions (as here) where <500 km heat flow through the ion gas has stabilized within a fraction of a minute.
 It is crucial to use height-resolved heat inputs. For brevity we here quote the equations derived byThayer and Semeter , using the formulation and terminology in their Table 1.
 Their height-resolved Joule heating rate, withj ionospheric current density, E′ electric field in the neutral gas rest frame, σp Pedersen conductivity, is
This heating rate of the neutral gas can equivalently be expressed in terms of frictional heating, using the energy equation [∂En/∂t] for the neutral species n,
where En is the internal energy of the neutral gas, Σ is summed over the ion species i, ni mi are number density over species i, νinis the ion-neutral collision frequency, and V is the vector velocity of the ion and neutral species gas. Subscript i denotes an ion species summed over, except in Ti and Vi where i refers to bulk ion gas.
 Replacing the frictional heating term in this equation by the heat energy exchange term (below ∼500 km) gives a third equivalent equation in term of ion and neutral species temperatures Ti, Tn:
The data to be used determines which of these three formulations may be most usefully applied. The equality that nn mnνni = ni miνin is used in these derivations. Realistic altitude dependencies of ni(h), νin (h), and σp are critical.
2.4. Critical Dependence of Upper Thermospheric Density Response Upon Ne(h)
 For a given topside input energy flux, Ne(h) defines the height of energy injection. Assuming hydrostatic diffusive equilibrium yields the altitude distribution of atmospheric density given by the balance of the partial pressure gradient against gravity [Rishbeth and Garriott, 1969, pp. 5–9]
for pressure and gas concentration of the atmosphere as a whole, and its constituents. Using the proper scale height, z is calculated in units of the neutral pressure scale height H, (dz = dh/H).
 The pressure p at any height h, is what holds up the sum of all the particles above in a column of unit cross-section. For a single gas H = kT/mg, with k Boltzman's constant, T neutral gas temperature, m neutral particle mass, g acceleration of gravity. The total number of particles in a column of unit cross section above height ho, from the gas law p = n k T is simply:
The static models by Jacchia [1970, pp. 7–10] apply this same physics.
 This lets us see that to raise the geo-potential height of a mass of air by increasing its temperature (and H), if one increases T only near and above 190 km vs. above 110 km, the increase of thermospheric density near 400 km is ∼70% as large as if T increased from110 km (fromequation (5) or Jacchia  referenced to 190 vs. 110 km). So even if the gas is heated only above ∼190 km, little density increase is lost near ∼400 km, and the isopycnic point moves up. However, the atmosphere's weight above 110 km is ∼300 times that above 190 km (for T here). This reduction in weight and heat capacity relieves the energy deposition energy burden ∼102 and offsets past objections to the exaggerated total input energy required in cusp modeling by Demars and Schunk .
 Thus Ne(h) is critical to the derivation of height resolved energy deposition rates, and these rates <200 km are essential to the thermospheric response >300 km. Since Ne(h) is rarely found in the literature at night 100–200 km, we present new such data here in Figure 1. Figure 1a shows Ne(h) for 100–600 km, while Figure 1b focuses in on the region 100–200 km. The time resolution is one minute and altitude resolution well less than a scale height. The data are during three representative magnetic reconnection events overhead the EISCAT Svalbard Radar (ESR), allowing profiles up the magnetic field. The 5 signatures characterizing magnetic reconnection as detailed in Carlson et al.  are all present. Figure 1e shows the three optical flashes, the first signal as fast electrons precipitate out the newly opened magnetic flux tubes. In addition to impact excitation of optical emissions, the particle precipitation simultaneously produces the ionization seen in Figure 1b, which has such a fast recombination rate (seconds) that it is seen only during active production. Its depth of penetration is a measure of the particle characteristic energy, while the density is a measure of its production rate and the particle number flux. As detailed by Carlson et al. , within tens of seconds heating by this fast electron flux raises the electron gas temperature Te. Order a minute later plasma flow jets >km/s flow east/west under the IMF magnetic tension force, and drift poleward with time as seen in panel E, recognized by part of the community as poleward moving auroral forms (PMAFs). Frictional drag in these flow jets heats the ion temperature Ti rapidly as seen in Figure 1c. The thermosphere responds more slowly and combined Te and Ti heating drives upwards plasma transport shown in Figure 1d. Similar effects are detailed further in Skjæveland et al.  for other reconnection events: their Figure 6 motivates including ion upflow in our future modeling.
Figure 1f(right) further details repeatable Ne(h) enhancements observed during the reconnection events (07:07 and 07:30 UT), exceeding lower background Ne profiles (e.g., 07:13 UT). The left hand side to the same altitude scale shows calculated ion production rates for a set of characteristic energies (0.1–10 keV) of a fixed precipitation flux (Gaussian 1.0 erg cm-2 s-1) to illustrate altitude dependence. The enhanced Ne(h) 07:07 UT profile during the reconnection event seen clearly inFigure 1f(right) (with associated clear signatures as described above for all panels) was used then to calculate directly the height-resolved energy deposition rate, using the exospheric temperature for this time derived from the ESR data [Skjæveland et al., 2011], fully consistent in this case with an MSIS derivation. The calculation was done using the equations from Schunk and Nagy [2009, sections 4.1 and 3.3] as described by Thayer and Semeter , appendix A. For our conditions and ∼3 km/s flow speed, our frictional drag energy deposition rate exceeded the rates shown in their Figure 2 for solar VUV, particle, and ion frictional heating they showed, while we reproduced their Figure 2 values when using their Sondrestrom input data.
4. Model Run
 We next input the altitude-resolved energy deposition rate, derived from the ESR data, to the UCL CMAT2 global time-dependent 3-D model [Harris et al., 2002], which solves the continuity, momentum, energy equations using mean molecular mass and MSIS composition. It solves for mass density, neutral temperature Tn, the 3 components of vector neutral wind, and we show results vs. altitude in km. We use rotating earth coordinates, mass flux transport, and have, within defined flow channels, replaced model values with input-observed electric fields (flow channels) and Ne(h). The input altitude-resolved energy deposition rate, based on ion frictional-drag heating, led to calculation of energy and momentum balance, upwelling, outward flow, down-welling, heating/cooling rates, and mapped thermospheric density, neutral temperature enhancements and wind circulation field changes all vs. altitude/latitude/time.
 The model input energy deposition rate profile derived directly from the data was fit by a Chapman function for computational convenience but fit the data from 145–300 km quite well, within ∼15%. Observations show typical plasma flow-shear channel widths are about 100–200 km wide, but sharply square shouldered; onset times are typically 1–2 minutes. To deal with difficulties models have with “sharp edged” functions in space and time, we smoothed these input shapes. As input, for model stability, we have approximated the latitude width of the energy deposition by a Gaussian with a standard deviation of 4 degrees in magnetic latitude. We have turned on the energy deposition linearly from zero to maximum in 10 minutes, then relaxed it linearly back to zero in 10 minutes. Hence, the total heating is 20 minutes, or half-width 10 minutes. Typical flow jets last ∼10–20 minutes. The Ne(h) enhancements seen inFigure 1bare seen only while the precipitation is directly up the magnetic field line within the field of view of the narrow beam radar. Thus one needs to go to panel E for the duration of the precipitation event; onset is seen before it convects into the ESR beam and continues after it exits to the north of the radar beam. The first, second, and third event have peak production durations of ∼5, 9, and 4 minutes long; typical bottom-side Ne(h) enhancement durations are ∼5–10 minutes. Our heating duration and channel width serve to test proof of concept here.
 The other critical input is magnitude of plasma velocity, ranging typically from ∼1–3 km/s, with a square law heating dependence (i.e., ∼1–9). To explore density doubling, at the high end of the Lühr et al. observations, we use the high end of typical peak flow-shear, ∼3 km/s as model input. Note thatFigure 2 in Thayer and Semeter compares energy deposition rates for hard and soft particles (representative of night-side and noon aurora), and Joule heating for three sunlit plasma velocities <∼1 km/s representative of average velocities. Here we must represent transient reconnection event flow shears, which from ESR radar and DMSP data are observed up through 2–3 km/s.
 Here, our Figure 2a shows the model output thermospheric temperature enhancement, with calculated wind vectors superposed on the same altitude/latitude plot. Transport is upwards near the heat input in the central column; divergence increasing outwards at higher heights significantly contributes to cooling in the overall thermal balance. Our Figure 2b shows the peak model output density ratio of 2.1 for the thermospheric density at 450 km, over the flow channel vs. background density. Other like models should get similar results simply by using realistic input data as illustrated here.
 Magnetic reconnection morphology well fits that of cusp density disturbances including fine structure currents [Oksavik et al., 2005].
Energy Input/Thermospheric Response. Downward Poynting flux into the top of the ionosphere/thermosphere determines the amount of electromagnetic energy input available to drive the thermospheric response. The response to this input is driven by Ne(h) via its control of the height discriminated energy deposition rate.
 In sunlight or darkness, this energy input scales as the square of plasma flow shear. Observed 3 km/s flow jets vs. model 1 km/s averages yield nine times the energy deposition rates. Where Ne(h) is dominated by particle production, hard vs. soft particle fluxes can determine whether the upper thermosphere is susceptible to strong or weak disturbance. Hard (>keV) electron precipitation enhances E region Ne(h) near 100–120 km, leading to non-dissipative Hall currents, and particles deposit energy into a large heat-capacity lower thermosphere that is sluggish and massive to heat. Soft (∼hundreds eV) electron precipitation enhances lower F region Ne(h) around 160–200 km, leading to Pedersen dissipative currents or frictional drag heating, which with particle energy deposition heats a much more tenuous rapidly responding thermosphere. Typical ¼ hour flow jets have duration adequate for important temperature increases above ∼170 km, but not much below ∼140 km. Frictional heating scales linearly with Ne(h), so realistic Ne(h) matter. Our model run here was for a dark cusp, for which the data-starved IRI underestimated observed 150–200 km Ne(h) tenfold. IRI values of Ne(h) are usually good for sunlit data-rich areas. Also key was the transient enhancement of Ne(h) during the heating event, vs. average values. We have referenced magnetic reconnection not only because it is commonly seen in the cusp, but its morphology well fits that of cusp density disturbances including location, range of magnitudes of flow-shear velocities (and consequent density enhancements), and even strong colocation with fine structure currents [Lühr et al., 2004; Oksavik et al., 2005].
 Temporal/spatial structure in plasma flow and Ne(h) require more realistic representation in modeling cusp and polar regions. The only other paper successfully to model observed doubling of air density, Crowley et al.  using matched observed data, spoke well for the community view when stating “the … density enhancement … resulted from unexpectedly large amounts of energy…”. We hope to update community expectations.
 We have defined a logic path connecting principles of aeronomy, plasma physics, and atmospheric dynamics, to offer a blueprint of input-response relationships controlling the upper thermospheric disturbance-response to injection of electromagnetic and particle transient input energy. Realistic transient shear velocities and Ne(h) profiles are essential to accurate modeling and prediction of upper atmospheric density and drag perturbations near the cusp or elsewhere in polar regions with plasma shears and soft-particle fluxes. Use of average plasma flows and densities, can each lead and have led to ten-fold underestimates of such perturbations.
 Following the first-principles logic and realistic data input guidelines given here leads naturally to prediction of cusp and polar cap density and drag disturbances near 400 km, of from ∼10% to doubling for common plasma flow shears ∼1 to 3 km/s. The largest disturbances should be for large-magnitude Bycomponents of the IMF, which favor large-magnitude flow shears. The principles presented here lead to agreement with the morphology found byLühr et al.  and Rentz and Lühr , and their invoking joule heating and upwelling. Use of the above principles/guidelines should in the future remove significant obstacles to realistic 3-D time-dependent thermosphere/satellite drag modeling at high latitudes.
 Work was sponsored in part by the USTAR Program, SWC, USU and NSF add AGS-1011921, and Air Force Office of Scientific Research, AFMC, USAF, under grant FA8655-10-1-3003. EISCAT is an international assoc. supported by Finland (SA), France (CNRS), the Fed. Rep. Germany (MPG), Japan (NIPR), Norway (NFR), Sweden (NFR), United Kingdom (PPARC), the Research Council of Norway.
 The Editor thanks William Denig and Hermann Lühr for assisting in the evaluation of this paper.