Separation and Quantification of Ionospheric Convection Sources: 1. A New Technique
J. P. Reistad, K. M. Laundal, N. Østgaard, A. Ohma, S. Haaland, K. Oksavik, S. E. Milan
SSeparation and Quantification of Ionospheric ConvectionSources: 1. A New Technique
J. P. Reistad , K. M. Laundal , N. Østgaard , A. Ohma , S. Haaland ,K. Oksavik , and S. E. Milan Birkeland Centre for Space Science, University of Bergen, Bergen, Norway, Max Planck Institute for Solar SystemResearch, Göttingen, Germany, Arctic Geophysics, University Centre in Svalbard, Longyearbyen, Norway, Department of Physics and Astronomy, University of Leicester, Leicester, UK
Abstract
This paper describes a novel technique that allows separation and quantification of differentsources of convection in the high-latitude ionosphere. To represent the ionospheric convection electricfield, we use the Spherical Elementary Convection Systems representation. We demonstrate how thistechnique can separate and quantify the contributions from different magnetospheric source regions tothe overall ionospheric convection pattern. The technique is in particular useful for distinguishing thecontributions of high-latitude reconnection associated with lobe cells from the low-latitude reconnectionassociated with Dungey two-cell circulation. The results from the current paper are utilized in acompanion paper (Reistad et al., 2019, https://doi.org/10.1029/2019JA026641) to quantify how the dipoletilt angle influences lobe convection cells. We also describe a relation bridging other representations of theionospheric convection electric field or potential to the Spherical Elementary Convection Systemsdescription, enabling a similar separation of convection sources from existing models.
1. Introduction
For decades, patterns of high-latitude plasma circulation have been inferred from measurements of the iono-spheric convection velocity ⃗ v . In addition to in situ measurements from the ionosphere and magnetosphere,ground-based high-frequency radars, for example, the Super Dual Auroral Radar Network (SuperDARN;Chisham et al., 2007; Greenwald et al., 1995), have been an important tool in these investigations. In additionto describe climatological patterns of the high-latitude ionospheric convection (e.g., Cousins & Shepherd,2010; Haaland et al., 2007; Heppner & Maynard, 1987; Greenwald et al., 1995; Thomas & Shepherd, 2018;Ruohoniemi & Greenwald, 2005; Weimer, 1995), the instantaneous ionospheric convection is routinelyderived from SuperDARN by including statistical “fill-in-data” from an empirical model, known as the “mappotential technique” (Ruohoniemi & Baker, 1998), representing a likely snapshot of the present large-scaleionospheric convection pattern.In the reference frame of the radar, the F region convection corresponds to a convection electric field ⃗ E = − ⃗ v × ⃗ B , where ⃗ v is the ionospheric convection velocity vector relative to the radar and ⃗ B is the mag-netic field at the measurement location. On time scales longer than a few 10 s, the convection electric fieldcan be considered curl free (Milan, 2013). Then, the electric field can be written as the gradient of a scalarfield, referred to as the electric potential Φ . As Φ can be considered constant along the magnetic field lineswhen no parallel electric fields are present, it is convenient to represent Φ on a spherical shell as an expan-sion of spherical harmonic functions. This approach is by far the most common way to represent Φ , whereobservations of the plasma drift are used to estimate the coefficients of the spherical harmonic functionsdescribing Φ .Despite providing an efficient and powerful framework to reproduce a scalar field on a sphere, there are fun-damental limitations of the spherical harmonic description. One inherent property is the repetitive natureof the spherical harmonic functions. A good data fit in one part of the sphere can affect the solution at a dif-ferent location. Furthermore, the shape of the analysis area is either global or restricted to a spherical cap byapplying boundary conditions (Haines, 1985), and the level of detail (degree and order) of the reconstructionis the same over the entire sphere or spherical cap. TECHNICALREPORTS: METHODS
This article is a companion to Reistadet al. (2019), https://doi.org/10.1029/2019JA026641.
Key Points: • Spherical Elementary ConvectionSystems (SECS) are used to representthe high-latitude convection electricfield• A novel technique is presented thatallows separation and quantificationof different sources of ionosphericconvection• The convection pattern can beseparated to quantify the magneticflux transport associated withdifferent reconnection sites
Correspondence to:
J. P. Reistad,[email protected]
Citation:
Reistad, J. P., Laundal, K. M.,Østgaard, N., Ohma, A., Haaland, S.,Oksavik, K., & Milan, S. E. (2019).Separation and quantification ofionospheric convection sources: 1.A new technique.
Journal ofGeophysical Research: Space Physics , , 6343–6357. https://doi.org/10.1029/2019JA026634Received 20 FEB 2019Accepted 8 JUN 2019Accepted article online 4 JUL 2019Published online 22 JUL 2019©2019. The Authors.This is an open access article under theterms of the Creative CommonsAttribution-NonCommercial-NoDerivsLicense, which permits use anddistribution in any medium, providedthe original work is properly cited, theuse is non-commercial and nomodifications or adaptations are made. REISTAD ET AL. 6343 ournal of Geophysical Research: Space Physics
This paper describes a different approach to represent the ionospheric convection. We outline a method thatallows the ionospheric convection electric field ⃗ E to be represented as the sum of the electric fields from alarge number of nodes , each having their own electric field ⃗ E 𝑗 associated with them. This is a specific appli-cation of the Spherical Elementary Current Systems (SECS) technique developed by Amm (1997) and Ammand Viljanen (1999). Hence, we also refer to the method as SECS, but for our application C refers to convec-tion rather than current. In this paper we will show that one of the benefits of representing the convectionelectric field in this way is the ability to segment the ionospheric convection field into regions correspondingthe magnetospheric source of the ionospheric convection. A particular application is the ability to isolateand quantify the contributions to the ionospheric convection from dayside and lobe reconnection. In ourcompanion paper Separation and quantification of ionospheric convection sources: 2. The dipole tilt angleinfluence on reverse convection cells during northward IMF , referred to as Paper II, this method enables us toquantify the influence of the dipole tilt angle on the lobe cell circulation in the ionosphere.Although segmenting the convection electric field into source regions is a new ability, the SECS representa-tion of ⃗ E has also other advantages compared to the spherical harmonic representation of Φ . This techniqueallows ⃗ E to be represented locally in limited regions of any shape on a spherical surface. Furthermore, thedensity of nodes can change across the analysis domain to compensate for data coverage or varying degreeof structure of ⃗ E at different locations.Section 2 introduces the SECS representation of a vector field on a sphere, as developed by Amm (1997)and Amm and Viljanen (1999), and how this can be used to represent the convection electric field at highlatitudes, highlighting the ability to separate and quantify the contributions from lobe and Dungey-typeconvection as driven from the solar wind-magnetosphere interactions. Although Amm et al. (2010) alreadypresented how the SECS technique can be used to describe the ionospheric convection velocity field, we hererepeat that description with one important difference, which is to represent the convection electric field. Aswill be shown in the next section, this difference is what enables us to do the segmenting of the ionosphericconvection. Section 3 presents the physical interpretations that can be made from the SECS description ofthe convection electric field, as well as a comparison of the results from the SECS description with a recentspherical harmonic model of Φ . Section 4 concludes the paper.
2. Using SECS to Represent the Ionospheric Convection Electric Field
By convection electric field we refer to the electric field due to the ionospheric plasma motion, as describedby the Lorentz transformation ⃗ E = − ⃗ v × ⃗ B , since we assume no electric field in the frame of the plasmamotion. Hence, ⃗ v is the bulk plasma velocity in the F region or above, relative to an observer, typically aground station, and ⃗ B is the magnetic field where ⃗ v is measured. According to the Helmholtz decomposition, any three-dimensional vector field, ⃗ u , can be represented by asuperposition of a curl-free, ⃗ u c 𝑓 , and a divergence-free, ⃗ u d 𝑓 , vector field: ⃗ u = ⃗ u c 𝑓 + ⃗ u d 𝑓 . Based on this, Amm(1997) developed the functional form of a set of curl-free and divergence-free elementary vector fields thathe showed would reconstruct any smooth vector field on a spherical surface by placing n elementary fieldsources on the sphere, which we will refer to as nodes using the subscript j . In this description, the sum ofthe elementary fields from nodes with a curl-free field would represent ⃗ u c 𝑓 , and the sum of the elementaryfields from nodes with a divergence-free field would represent ⃗ u d 𝑓 : ⃗ u c 𝑓 = n ∑ 𝑗 = ⃗ u c 𝑓,𝑗 ⃗ u d 𝑓 = n ∑ 𝑗 = ⃗ u d 𝑓,𝑗 (1)From a set of requirements that we will return to later, Amm (1997) derived the functional form of thesebasis functions to be ⃗ u c 𝑓,𝑗 ( ⃗ r i ) = A 𝑗 𝜋 R cot ( 𝜃 i 𝑗 ∕ ) ̂𝜃 i 𝑗 ⃗ u d 𝑓,𝑗 ( ⃗ r i ) = A 𝑗 𝜋 R cot ( 𝜃 i 𝑗 ∕ ) ̂𝜙 i 𝑗 (2)REISTAD ET AL. 6344 ournal of Geophysical Research: Space Physics Figure 1. (a) Illustration of the elementary current function derived by Amm (1997) and the geometry involved in theSpherical Elementary Convection Systems (SECS) description. The global magnetic local time/magnetic latitudecoordinate system is shown in black in a spherical shell representing the ionosphere at F region heights (300 km). Themagnitude of the curl-free field from node j , located at ⃗ r 𝑗 is shown in color. The unit vector of this field at position ⃗ r i isindicated as ̂𝜃 i 𝑗 ( ⃗ r i ) . 𝜃 ij is the angle between ⃗ r i and ⃗ r 𝑗 . (b) Three nodes, each representing a curl-free field, where the signand magnitude of its associated amplitude A j is reflected by its color and size. At two arbitrary locations (black dots)the curl-free field from each node is shown as black vector pins, as well as its vector sum (green) representing thecurl-free electric field described by the three nodes at that location. Here, A 𝑗 𝜋 R is scaling the strength of the elementary field, where A j is an amplitude for node j , placed on thespherical surface at radial distance R . A sketch illustrating the curl-free field from a single node and therelevant vectors involved is shown in Figure 1a. In Figure 1a we use the double subscript ij on quantities thatdepend both on the node location ⃗ r 𝑗 and where the field is evaluated ⃗ r i . Hence, the 𝜃 ij argument is the anglefrom the position of the node, ⃗ r 𝑗 , to the position where the field is being evaluated, ⃗ r i . In this description,equation (2) refers to a local coordinate system for each node. The strength of the field from node j is onlydependent on this local polar angle 𝜃 ij . The unit vectors ̂𝜃 i 𝑗 and ̂𝜙 i 𝑗 also refer to this local frame, ̂𝜃 i 𝑗 pointingaway from the node, and ̂𝜙 i 𝑗 in the perpendicular direction, along contours of constant 𝜃 ij . As mentioned, ⃗ u c 𝑓 and ⃗ u d 𝑓 are the superposition of the fields from n nodes. To compute the superposition field from everynode at a specific location ( ⃗ r i ), the node fields need to be converted into a common coordinate system beforethe sum is calculated. Hence, ⃗ u c 𝑓 ( ⃗ r i ) = n ∑ 𝑗 = ( A 𝑗 𝜋 R cot ( 𝜃 i 𝑗 ∕ ) ̂𝜃 i 𝑗 ) t ⃗ u d 𝑓 ( ⃗ r i ) = n ∑ 𝑗 = ( A 𝑗 𝜋 R cot ( 𝜃 i 𝑗 ∕ ) ̂𝜙 i 𝑗 ) t (3)where the subscript t highlights this coordinate conversion. Figure 1b illustrates the superposition field inan idealized case from three nodes each having a curl-free field, as expressed by equation (3a). The threenodes are shown as blue and red dots, where the color and its size indicate the sign and magnitude of itsassociated amplitude A j . The curl-free field described by these three nodes is evaluated at two arbitrarylocations (black dots) where the contributions from each node is shown (black vector pins) as well as thesum of all contributions at that location (green).Amm and Viljanen (1999) used this approach to represent the equivalent currents in the ionosphere frommeasurements of their associated magnetic field perturbations detected on the ground. This approach theytermed SECS, where the nodes could be interpreted as sources of divergence-free and curl-free currents,which they attributed to equivalent and field-aligned currents, respectively. In their application the nodesrepresent the source of a continuous divergence-free or curl-free current field (sheet current on a sphere)centered at the nodes. In the case of a curl-free node, this source is a field-aligned current. The amplitude A j in the elementary function (equation (2)), having units of amperes in their application, has the role ofREISTAD ET AL. 6345 ournal of Geophysical Research: Space Physics scaling the strength of the vector field from each node, normalized to the sphere radius R . Hence, the vectorfield itself represents sheet current density (A/m) in their application.The SECS technique is a tool to represent any vector field on a sphere, where its curl-free and divergence-freecomponents are represented as the superposition of curl-free or divergence-free fields from nodes placed onthe surface of the same sphere. Given a distribution of n such SECS nodes, the task is to estimate all thecorresponding n amplitudes that optimally fit observations of the same vector field. The SECS techniquetherefore has a wide range of applications. Amm et al. (2010) used the same technique and basis functions todescribe the ionospheric convection velocity field. In their description, the ionospheric convection velocityfield is purely divergence free (assume ⃗ B is constant and the flow is incompressible). From benchmarkingwith a synthetic data set, they demonstrated that even when the line-of-sight (LOS) velocity data coveragewas limited (25% of the analysis area), the SECS technique could reconstruct the velocity field with relativeerrors of less than ∼
5% when the velocity field had a scale size of ∼
100 km. Furthermore, in contrast to otherways of reconstructing the ionospheric convection velocity field, this technique requires in principle no apriori knowledge of the convection or boundary conditions but is only constrained by measurements andthe assumption that the velocity field is incompressible.
The present paper uses a slightly different approach to describe the ionospheric convection compared toAmm et al. (2010). Instead of representing the convection velocity field as a divergence-free field, we expressthe convection electric field as a curl-free field. This has a few advantages, as become evident when expand-ing the divergence-free condition using ⃗ E = − ⃗ v × ⃗ B . If assuming ∇ × ⃗ E = in the steady state situation,and considering a magnetic field only due to sources inside the Earth, for example, as represented by theInternational Geomagnetic Reference Model, ∇ × ⃗ B = . Then ∇ · ⃗ v = ⃗ E · ∇ ( B ) × ⃗ B . (4)Hence, inhomogeneities in the magnetic field along the convection path will contribute to a divergence of ⃗ v .Since we will combine measurements from different longitudes to make averages at specific magnetic localtimes (MLTs) and magnetic latitudes (MLATs), the representation in terms of a convection electric field istherefore beneficial when we want to relate the observed convection velocities to magnetic flux transportrates at different locations.Since the convection electric field can be considered as curl free, we only need to use one of the elementaryfunctions in equation (2) for the present application. The SECS electric field at location ⃗ r i in the globalcoordinate system (e.g., an MLT/MLAT system) then becomes ⃗ E ( ⃗ r i ) = ⃗ E c 𝑓 ( ⃗ r i ) = n ∑ 𝑗 = ( A 𝑗 𝜋 R I cot ( 𝜃 i 𝑗 ∕ ) ̂𝜃 i 𝑗 ) t . (5)Here, 𝜃 ij is the angle between ⃗ r 𝑗 and ⃗ r i , as illustrated for one specific node in Figure 1a. ̂𝜃 i 𝑗 is the correspond-ing unit vector of the curl-free elementary field of node j evaluated at ⃗ r i , and the subscript t is to indicate thatthe contribution from each node need to be converted to a common coordinate system before the superpo-sition field can be calculated. For this particular application of the SECS description, the SECS nodes mustbe placed at a minimum radial distance to ensure that ⃗ E = − ⃗ v × ⃗ B . We use the Earth radius plus the height ofthe F region ionosphere, set to 300 km, referred to as R I . Since the electric field has units of volts per meter, A j has units of volts in this description, or equivalently weber per second, highlighting its interpretations interms of magnetic flux transport. We here express the relationship between the SECS node amplitudes A j and Φ , allowing a detailed compar-ison with the more standard representation of the ionospheric convection. The potential at a given pointin the analysis domain is given by the sum of the potential from all electric field sources. As will be shownin the next subsection, a physical interpretation of the sources of the node electric field is indeed a chargedistribution. Since ⃗ E = −∇Φ , the potential is found by integrating equation (5). The potential at location ⃗ r i then becomes Φ( ⃗ r i ) = n ∑ 𝑗 = − A 𝑗 𝜋 ln ( sin ( 𝜃 i 𝑗 ∕ ) ) . (6)REISTAD ET AL. 6346 ournal of Geophysical Research: Space Physics Note that we have multiplied by R I as the integration of the elementary function is done along ̂𝜃 i 𝑗 in adistance R I from origo, resulting in Φ having units of volts. As the potential is a scalar quantity, no coordinatetransformation is needed to sum the contributions from each node. A j Earlier we described how the set of amplitudes A j describes the electric field at an arbitrary location in theanalysis domain, ⃗ r i . Now we use the relationship from equation (5) to describe how A j can be estimated fromobservation of the electric field. Hence, when describing the inversion for A j in the following, ⃗ r i refer to thelocation of observation i .Equation (5) shows that the observations of the electric field are linearly dependent on the node amplitudes A j we seek to infer since 𝜃 ij is independent of A j ( 𝜃 ij can solely be calculated from the location of observation i , ⃗ r i , and the SECS node location, ⃗ r 𝑗 ). With m observations of the convection electric field distributed acrossthe analysis domain, the problem of finding the SECS node amplitudes that describe the observed convectionelectric field can be formulated as a set of m linear equations (one per observation, index i ) where we solvefor the n unknown amplitudes (index j ).Following Amm et al. (2010), this system of linear equations can be written in matrix form as d = GA . (7)Here, d is of size ( m , ) and contains the observational data of the convection electric field in the ̂ k i direction.If the full horizontal ⃗ E i is observed, we have information from two independent directions and it is treated astwo observations in this description. If LOS observations of the plasma velocity is used, ̂ k i = − ̂ k los , i × ̂ B where ̂ k los , i is the LOS direction of plasma velocity observation v los, i and ̂ B is a unit vector along ⃗ B , assumed to bevertical at high latitudes. Hence, d i = v los, i B i in the ̂ k i direction when using LOS measurements, and B i is themagnetic field strength at the location of observation i , ⃗ r i . A is a column vector of size ( n , ) containing thenode amplitudes. G = G ij is a geometry matrix of size ( m , n ) relating the effect of a unity amplitude curl-freeSECS node with pole at ⃗ r 𝑗 at the location of observation i , ⃗ r i , in direction of ̂ k i . The elements of G become G i 𝑗 = ( 𝜋 R I cot ( 𝜃 i 𝑗 ∕ ) ̂𝜃 i 𝑗 ) t · ̂ k i (8)As illustrated in Figure 1a, 𝜃 ij is the angle between node j and the point of observation i . ̂𝜃 i 𝑗 is the unitvector along the great circle connecting ⃗ r 𝑗 and ⃗ r i , pointing away from ⃗ r 𝑗 . As emphasized earlier, a coordinateconversion is needed to bring the vectors into the same coordinate system before the dot product in equation(8) can be computed, indicated by the subscript t . Note that the geometry matrix G is solely determined by thelocation and LOS direction of the observations and the locations of the SECS nodes, making it independentof the observed magnitudes d , hence the linearity of the equations.For many applications, and from physical considerations, we would like the solution to behave in a specificmanner in specific regions. This could, for example, be a boundary condition where the plasma velocityshould approach zero at some specific latitude, or that the convection should or should not cross a specificboundary, for example, into or out from the polar cap. Such constraints can be imposed by adding syntheticobservations at the desired locations and in the desired direction before the inversion for A in equation (7).The exact strategy for obtaining the node amplitudes A j from the set of equations given in equation (7)depends on the particular application of the technique. As G is an ( m , n ) matrix ( m is number of measure-ments, n is number of nodes), regularization will in many applications be necessary. In section 2.6 we willpresent details on the specific steps of the inversion used to make the plots in Figures 4 and 5, which is alsoidentical to how the inversion is done in Paper II.When the SECS node amplitudes A have been found, the estimated curl-free electric field, ⃗ E SECS , can inprinciple be evaluated at any location (except for close to the SECS nodes) by computing G for the desiredlocation(s) in the desired direction(s) and use equation (7) to compute d . The observation vector d nowrepresents components of E SECS at the desired location(s) in the desired directions, typically east and northto represent the convection electric field vector.When applying the SECS analysis on the convection electric field across the entire high-latitude regionusing LOS velocity data from SuperDARN, we have found the grid displayed in Figure 2 appropriate whenREISTAD ET AL. 6347 ournal of Geophysical Research: Space Physics
Figure 2.
An example of a regular Spherical Elementary ConvectionSystems (SECS) grid (blue cells where the SECS nodes are located at thecenter as blue dots), here an equal area grid defined along parallels ofmagnetic latitude with a latitudinal width of 2 ◦ . The innermost ring of bins,[87 ◦ ,89 ◦ ] are separated into four magnetic local time sectors. This densityof nodes are found sufficient to represent the large-scale features of theionospheric convection electric field. Red dots illustrate locations thatalways have similar distance to the neighboring SECS nodes (shifted half ofthe magnetic local time separation in each ring of bins). These locations arefound as suitable evaluation points of the electric field described by theSECS nodes. describing ionospheric convection features with scale sizes ≳
200 km(node separation distance). This is an equal area grid that follows theMLATs, and the spatial resolution is 2 ◦ in the latitudinal direction, andthe innermost ring of bins ∈ [87 ◦ ,89 ◦ ] MLAT is divided into four MLTbins, resulting in 480 grid cells, starting at [59 ◦ ,61 ◦ ] MLAT. The SECSnodes are placed at the center of the grid cells, indicated as blue dotsinside the blue grid in Figure 2. As the elementary functions go to infinityat the node, one has to avoid evaluating the field too close to the node. Itis beneficial to evaluate the field where the distances to the neighboringnodes are similar across the entire analysis domain. For this particulargrid, this is achieved by evaluating ⃗ E SECS on locations that are shiftedfrom the nodes in MLT by half of the MLT separation between nodes inthe same ring. These locations are shown as the red dots in Figure 2. Inthis way, the grid that we evaluate for ⃗ E SECS reflects the level of structurethat can be resolved, limited by the density of the nodes. G Matrix
For the specific application of the SECS technique in Paper II, where theconvection electric field is represented globally above 60 ◦ MLAT, an addi-tional step in the preprocessing of the LOS convection data is introducedto overcome challenges with the inversion of equation (7), involving alarge number of LOS observations, typically ∼ . This strategy is toreduce the large number of LOS observations into a smaller number ofaverage electric field vectors before inverting for A . We found it beneficialto again use the same grid as for the SECS nodes for this reduction, onlyshifted in MLT by half the MLT separation between nodes in the sameMLT ring. Hence, this new grid where we compute the binned averageelectric field from LOS observations within the grid cell has the same center as the red dots in Figure 2 andtherefore also consist of 480 cells. Due to this intermediate step, the geometry matrix G in equation (7) canbe reduced to size ( m , m ) . In addition, it will lead to a uniform spatial weighting (when observations areprovided in every bin). The nonuniformity of LOS data coverage (observations tend to cluster) is one of themain challenges with a direct inversion for A from equation (7) (excluding the intermediate step of comput-ing binned averages). Furthermore, using the binned average ⃗ E as input to the inversion at the locations ofthe red dots in Figure 2 removes the problem of having observations very close to the SECS nodes. If usinga direct inversion for A , a minimum allowed distance to the SECS nodes could be used, which will dependon the density of the SECS grid.The binned average ⃗ E in each grid cell is found using the same approach as described by Reistad et al. (2018),following a method outlined by Förster et al. (2008), but here applied to the convection electric field ratherthan the convection velocity. The average electric field vector in each grid cell, ⃗ E = ( E east , E north ) is found bysolving the set of m linear equations, one per LOS observation ( i ) within the grid cell, by the method of leastsquares: v los , i B i = ( ̂ k los , i × − ̂ B i ) · ⃗ E . (9)Here, v los , i , ̂ k los , i , and ⃗ B i are the LOS ionospheric plasma velocity, its associated unit vector along the LOSdirection, and the magnetic field vector at the observation location, respectively. The cross product arisesfrom the fact that we constrain the electric field in the direction perpendicular to ̂ k los , i since ⃗ E = − ⃗ v × ⃗ B .The LOS observations of F region plasma convection used in Paper II is from SuperDARN. The SuperDARNdata specify ̂ k los , i as degrees from local Altitude Adjusted Corrected GeoMagnetic (AACGM) coordinatesystem (Baker & Wing, 1989) poleward direction, positive in the eastward direction. Hence, each observationis represented in its own local coordinate system. Therefore, before equation (9) can be solved for ⃗ E , theobservations are converted into a common frame, where ̂ k los , i is now specified as degrees from the 12 MLTmeridian, positive when pointing duskward. This conversion into a global coordinate system is especiallyimportant for the highest-latitude grid cells, where the equal area grid cells span several hours in MLT (6 hrin the [87 ◦ , 89 ◦ ] cells). If not correcting for this, observations toward the eastern and western edges of theREISTAD ET AL. 6348 ournal of Geophysical Research: Space Physics grid cell would lead to erroneously large squared distances when estimating ⃗ E , as equation (9) would thendetermine the distance according to the center of the grid cell. Details of the SuperDARN data processingprior to the SECS analysis described here can be found in Paper II. As emphasized in this paper, there are a number of choices needed to be made to arrive at a SECS descrip-tion of the convection electric field. An optimal combination of the different ingredients can be very difficultto find and depends much on the problem at hand and the available data. In our work with representing theaverage ionospheric convection electric field above 60 ◦ MLAT, we have found that in particular three differ-ent quantities need to be tuned together to produce a reliable result. These quantities are the density of thegrid, the degree of forcing of the solution at the low-latitude boundary, and the singular value decomposi-tion (SVD) cutoff value used in the inversion of equation (7). For all plots shown in this and the companionpaper, we use the following values, which we have found highly suitable for this particular analysis:•
Grid : Equal area grid with 480 cells above 60 ◦ MLAT, Δ MLAT = ◦ . Furthermore, we evaluate for E SECS onan equally dense grid, but at locations that have similar distance to the neighboring nodes, see Figure 2.•
Boundary : 600 synthetic observations of E east = and E north = at 59 ◦ MLAT, evenly separated in MLT.These synthetic data are displaced 1 ◦ from the 60 ◦ MLAT analysis boundary to avoid getting very close tothe SECS nodes at that latitude. This weakly imposed boundary condition make the solution behave asexpected toward the boundary, where observations are sparse, while having a minor influence at the polarcap latitudes.•
SVD cutoff : Singular values of less than 6% of the largest singular value are set to 0. In choosing a cutoffvalue, one has to compromise by level of spatial detail the solution can reconstruct and the amount ofnoise in the solution.When these parameters are fixed, a more detailed comparison of the result of the inversion during differentdriving conditions can be made. We note that the exact choice of the values mentioned above does not affectthe conclusions of the data analysis presented in Paper II. For the sake of reproducibility we explicitly statethe values used here.Although we use a large data set of F region LOS plasma velocities from SuperDARN, observations are sparsetoward the low-latitude part of the analysis domain, especially on the dayside. Hence, some of the binnedaverage ⃗ E vectors in these regions are based on very few observations and are poorly defined. To reducethe importance of these grid cells in the inversion of the node amplitudes, we have introduced a weightingof the binned average ⃗ E vectors found using equation (9) based on how well this region is sampled. Fromexperimenting with data selection, we have found that the number of unique hours of observations withina grid cell is a good indicator of the data coverage. This parameter is therefore used to identify grid cells thathave a potentially poorly determined ⃗ E . To reduce the impact of poorly determined binned average ⃗ E onthe solution of A , a weighting of the different cells based on their number of unique observational hours isapplied. For grid cells having observations from more than 50 unique hours, an equal weight of 1 is appliedas this is considered as good coverage. The rest of the cells (mainly below 64 ◦ MLAT on the dayside) aredownweighted (in both northward and eastward direction) according to w j = h j ∕ where h j is the number ofunique hours cell j has observations from. The synthetic boundary observations are given a weight of 1. Theweighting is implemented by constructing a weight matrix w with weights on the diagonal correspondingto how d in equation (7) was constructed. Then, w is multiplied on both sides of equation (7), before this setof equations are solved for A with the method of SVD.
3. Interpretation and Validation of the SECS Representation
In this section we discuss the physical interpretations that can be made of the inferred node amplitudes in theSECS representation of the convection electric field. We also show a comparison of the SECS representationwith the corresponding convection patterns derived using spherical harmonic cap analysis on the same dataset (Thomas & Shepherd, 2018) for validation purposes.
To get a better understanding of the assumptions made by representing ⃗ E as a finite number of SECS nodes,and the physical interpretations that can be made from the node amplitudes A j , we need to realize theREISTAD ET AL. 6349 ournal of Geophysical Research: Space Physics underlying assumptions made by Amm (1997) in deriving the elementary function of the curl-free node fieldand its scaling to the radius and amplitude. We emphasize that the unit of A j depends on the field that isdescribed. In our application it has units of volts, while for the original application (Amm, 1997) it has unitsof amperes. Since Amm (1997) described a height-integrated current field ⃗ J cf (A/m), the following criterianeeded to be met by the elementary current field from node j , ⃗ J cf ,𝑗 : ) ∇ × ⃗ J cf ,𝑗 ≡ ) Birkeland current entering at node 𝑗 [ A ] = lim 𝜃 i 𝑗 → ∫ 𝜋 J c 𝑓,𝑗 R I d 𝜙 ≡ A 𝑗 ) ∇ ⟂ · ⃗ J cf ,𝑗 ≡ const. for 𝜃 ≠ ) ⃗ J c 𝑓,𝑗 ( 𝜃 = ◦ ) ≡ (10)1) The requirement of the current field to be curl free. 2) The Birkeland current entering at the pole, definedto be the value of A j , is distributed horizontally from the pole. 3) The requirement of current continuityensures that there is no pileup of current on the sphere. Away from the node location ⃗ r 𝑗 , current is thenleaving the spherical surface at a constant density where const. = − A 𝑗 𝜋 R I . 4) Ensures that there is no currentat the point opposite to the node. These requirements uniquely define the elementary function as expressedin equation (2). From this definition, we will have the following constraints on A j from point (2) when wedescribe ⃗ E cf through equation (5): lim 𝜃 i 𝑗 → ∫ 𝜋 E 𝑗 R I d 𝜙 = A 𝑗 . (11)We can use this to relate A j to a charge density by using Gauss law and the divergence theorem around thenode location: Q 𝑗 𝜖 = lim V → ∫ V ∇ · ⃗ E 𝑗 d V = lim S → 𝑗 ∮ S ⃗ E 𝑗 · d ⃗ a = lim 𝜃 i 𝑗 → ∫ 𝜋 E 𝑗 R I d 𝜙 · h = A 𝑗 h , (12)where h is the height interval where ⃗ E 𝑗 is given by the elementary function, needed to get a finite electricflux into/out of S . The closed surface S we choose to be a cylindrical box of height h , centered at node j ,whose radial extent is R I 𝜃 ij → . This surface encloses the electric charge Q j . Assuming vertical magneticfield lines representing equipotentials ( E || = ), the magnetic field line threading node j has an associated1-D charge density, 𝜆 j (C/m): 𝜆 𝑗 = Q 𝑗 h = 𝜖 A 𝑗 . (13)Hence, at high latitudes and above the base of the ionosphere, the nodes can be seen as infinite conductingvertical field lines with charge density 𝜆 j = A j 𝜖 .If the distance between the nodes is constant across the analysis region, as is the case in Figure 2, it isstraightforward to relate the 1-D vertical charge density 𝜆 j to a volume charge density 𝜌 j . This can be ofinterest as Gauss law relates 𝜌 to ∇ · ⃗ E . Such a relation will therefore bridge any other representation of ⃗ E to corresponding node amplitudes in a SECS representation. This means that by computing ∇ · ⃗ E from anempirical model of Φ at the node locations in Figure 2, one can calculate the corresponding amplitudesof the curl-free nodes in a SECS representation of ⃗ E . Going the other way, calculating the correspondingspherical harmonic coefficients based on the SECS description is not straightforward. When approximating 𝜌 j from 𝜆 j , or vice versa, we assume that the 1-D charge density from the node is uniformly distributed withinthe area 𝜎 defined by the uniform SECS grid cells (blue grid in Figure 2). For the grid shown in Figure 2, 𝜎 = · m . The corresponding volume charge density then becomes 𝜌 j (C/m ) = 𝜆 j ∕ 𝜎 . By also dividingby the electron charge e , this can be related to the corresponding perturbation in electron density within the j th grid cell, Δ n e , j ( ∕ m ) associated with the charged field lines, or equivalently, Δ n e , j due to ∇ · ⃗ E . Δ n e , 𝑗 = 𝜌 𝑗 e = 𝜖 A 𝑗 e 𝜎 = 𝜖 ∇ · ⃗ Ee (14)REISTAD ET AL. 6350 ournal of Geophysical Research: Space Physics Figure 3.
A synthetic distribution of Spherical Elementary Convection Systems (SECS) nodes (left column), its resulting ⃗ E above 60 ◦ magnetic latitude (middlecolumn), and Φ (right column). (upper row) SECS nodes placed in two circles with amplitudes chosen to reflect typical Region 1/Region 2 current morphologyresult in the common two-cell ionospheric convection pattern. (bottom row) An additional region of negative amplitude SECS nodes in the dayside polar cap isadded. The same amount of positive amplitude is added to the Region 1 band of nodes, leaving the total charge 0. This is to mimic the influence frominterplanetary magnetic field B y , and correspond to adding a pattern similar to the F A panel in Figure 2 in Milan et al. (2015; describing the current systemrelated to interplanetary magnetic field B y ) to the SECS node pattern in the upper left panel. The white circle in the rightmost panels indicate the latitude of theinnermost band of Region 1 current and can be interpreted as the open/closed field line boundary. As shown in Paper II, these values are very small ( ∼ m − ) compared to the F region density ( ∼ m − ),hence not violating the assumption of quasi neutrality. The interpretation of 𝜌 = 𝜖 ∇ · ⃗ E , corresponding tocharged vertical field lines, was also briefly mentioned by Untiedt and Baumjohann (1993, p. 292). However,we are not aware of any studies taking advantage of this representation in investigations of the ionosphericconvection. Note that when expressing the charge density as in equation (14) we have not taken into accountthe constant divergence of ⃗ E 𝑗 over the entire sphere as described in point (3) above. When applying the SECSdescription locally, only the spatial structures in the charge density can be resolved as the constant offsetlevel will need to be determined from nodes all over the sphere. However, this constant contribution to thecharge density is expected to be very small compared to the spatial variations across the high latitudes andis found to be ∼ − of the perturbations seen in Figure 4c when taking the sum of A j above 60 ◦ MLAT asrepresentative of the entire globe.
In this subsection we show the electric field and potential from a synthetic, idealized distribution of SECSnodes. This is to get an impression of how the node amplitudes relate to the more familiar electric field andpotential across the high latitudes. In Paper II we mainly focus on the interpretation of the distribution of Δ n e . Hence, this subsection as well as subsection 3.4 is of particular relevance for the interpretation of themaps in the companion paper.The left column in Figure 3 shows two synthetic SECS node amplitude distributions above 60 ◦ MLAT inan orthogonal MLAT/MLT coordinate system. The top row has two rings, one placed at 75 ◦ MLAT and theother at 61 ◦ MLAT, with opposite sign at dawn and dusk. Each half circle band consists of 100 nodes evenlyspaced in MLT. The corresponding electric field, found using equation (5) and evaluated on the same equalarea grid as shown in Figure 2, is shown in the middle column as black vector pins. Here, also the locationsof the SECS nodes are shown for reference (blue dots). It is evident that ⃗ E SECS points toward and away fromthe locations where SECS nodes have nonzero amplitudes. Using equation (6), we calculate the value of theelectric potential Φ at the same locations as the ⃗ E vectors shown in the middle panel. A contour plot of theREISTAD ET AL. 6351 ournal of Geophysical Research: Space Physics potential values are shown in the rightmost column. The example shown in the top row of Figure 3 aimsto illustrate that information about the source of the ionospheric convection can be revealed from lookingat the corresponding distribution of node amplitudes that represent the convection electric field. We heredemonstrate that a configuration of SECS nodes oriented in bands similar to the Region 1 and Region 2Birkeland current pattern (Iijima & Potemra, 1978) reproduce the familiar two-cell convection pattern. Wenote that this convection pattern is almost identical to the idealized model of ionospheric convection aspresented by Milan (2013), where the convection was modeled in response to similar bands of Region 1 andRegion 2 currents as shown here.Another example of the interpretation of the SECS node amplitude distribution is shown in the bottom rowof Figure 3. As was pointed out for the symmetric two-cell pattern in the top row, the SECS node amplitudesare closely related to the Birkeland currents. This will be discussed in more detail in section 3.4. Milan et al.(2015) showed that the most common Birkeland current pattern except for the Region 1 and Region 2 bandsis the one associated with interplanetary magnetic field (IMF) B y . We adopt their IMF B y pattern as seenin their Figure 2, panel F A , and add SECS nodes with a corresponding amplitude distribution on top ofthe SECS nodes in the top row of Figure 3. Specifically, we add 100 nodes within the dayside polar capbetween 80 ◦ and 85 ◦ MLAT with negative amplitudes, corresponding to NBZ (northward B z ) currents duringIMF B y -dominated periods (Iijima et al., 1984). In addition, an amount of positive amplitude (charge) isadded to every node in the innermost ring of SECS nodes at 75 ◦ , so that the sum of added IMF B y SECSnode amplitudes become 0. The physical interpretation of this is that there can be no pileup of charge inthe ionosphere. Hence, the upward and downward currents must on average cancel, as is the case in theelementary patterns derived by Milan et al. (2015). The sum of this “two-cell + IMF B y ” pattern of SECSnodes is seen in the bottom row left panel of Figure 3. The corresponding electric field and potential revealthe main large-scale features of the convection pattern during southward and B y -dominated IMF, namely,a “banana-” and “orange-” shaped convection cell. Furthermore, the dusk convection cell has its minimumvalue inside the inner ring of SECS nodes, which can be interpreted as the open/closed field line boundary,indicated by the white circle. This is similar to how lobe reconnection during positive IMF B y affects theconvection pattern in the Northern Hemisphere, leading to plasma circulation inside the polar cap in thedirection shown here (Crooker & Rich, 1993). However, from looking at the electric potential only, it is notstraightforward to separate and quantify the two source regions (the rings and the high-latitude region) of theconvection electric field. By utilizing the technique described here, this can now be achieved by examiningthe estimated amplitudes of the SECS nodes in various regions. In order to validate the above described methodology and demonstrate one of its applications, we investigatethe ability to reproduce known variability of the electric potential with the IMF clock angle. Figure 4a showsthe electric potential calculated with the above described technique for eight different orientations of theIMF clock angle based on SuperDARN LOS measurements from the same conditions as used by Thomasand Shepherd (2018) in making their Figure 5, also shown for reference in Figure 4b. These conditions are − ◦ < tilt < ◦ and . < E sw < . mV/m and based on data from Northern Hemisphere only during theyears 2010–2016. Furthermore, only echoes from ranges between 800 and 2,000 km are considered to reducethe likelihood of geolocation inaccuracies with multihop propagation and low-velocity E region echoes. Fordetails on the preprocessing of the SuperDARN data and selection based on IMF, see Paper II. As can beseen from Figures 4a and 4b, the potential patterns derived using these two completely different techniquesagree to a high level of detail. However, we note that our values of the maximum potential difference areslightly lower than those reported by Thomas and Shepherd (2018), typically by ∼ ⃗ E before the inversion, as describedin section 2.5. For a detailed comparison of the magnitudes of the binned average ⃗ E versus ⃗ E SECS , see Figure 4in Paper II.This comparison shows the ability of the described technique to reproduce known features of the iono-spheric convection. Also shown in Figure 4c is the derived values of the SECS node amplitudes correspond-ing to the potential in Figure 4a. For comparison, we also show maps of the Birkeland currents correspondingto the same conditions from the Average Magnetic field and Polar current System (AMPS) model (Laundalet al., 2018). This is an empirical model of the high-latitude current system, parameterized by solar windspeed, IMF B y , IMF B z , dipole tilt angle, and the F . index. When making Figure 4d we have used thevalues v sw = km/s, IMF B T = . nT, dipole tilt = 0 ◦ , and F . = sfu. The striking similaritiesREISTAD ET AL. 6352 ournal of Geophysical Research: Space Physics Figure 4. (a) Electrostatic potential calculated using the technique described in this paper using the same data selection criteria and data set as used by Thomasand Shepherd (2018) when making their Figure 5, shown here for reference in (b). The same contour spacing of 6 kV is used in (a) and (b) as well as a similarcolormap for direct comparison. (c) The SECS node amplitudes that was used to make (a). (d) Statistical average pattern of Birkeland currents during the sameconditions from the AMPS model (Laundal et al., 2018). AMPS = Average Magnetic field and Polar current System; SECS = Spherical Elementary ConvectionSystems. between the patterns of the SECS node amplitudes and Birkeland currents will be discussed in the followingsubsection.
One of the main benefits of the SECS technique applied to ionospheric convection is the ability to separatethe sources of convection. One particular application is to distinguish and quantify the contributions to theionospheric convection from lobe and Dungey-type reconnection. In this subsection we demonstrate thisability and describe how this can be used to quantify the amount of lobe cell convection during northwardIMF, which is the topic of Paper II.REISTAD ET AL. 6353 ournal of Geophysical Research: Space Physics
Figure 5.
Process of identifying the sources of ionospheric convection from the maps of SECS node amplitudes. (a) The estimated SECS amplitudes convertedto units of excess electrons per cubic meter using equation (14). SECS nodes within 80 ◦ magnetic latitude and between 06 and 18 magnetic local time areindicated as lobe cells and highlighted with black dots. (b) The electric potential corresoponding to (a). (c) The potential from only the nodes identified as thelobe cells in (a) (black dots). (d) The potential based only on SECS nodes outside the dayside polar cap. (e) Same as (a) but the SECS node amplitudes areestimated only within 75 ◦ magnetic latitude and based on data only from the same region. (f) Same as (c) but based on the node amplitudes from (e). IMF =interplanetary magnetic field; SECS = Spherical Elementary Convection Systems. Figure 5 shows an example of how we identify the different sources for the ionospheric convection andthe magnetic flux transport rate (potential) associated with these sources. These plots are made fromSuperDARN LOS velocity measurements during northward IMF and equinox conditions, identical to howFigure 4c, middle top panel, was made.Figure 5a shows the estimated SECS node amplitudes converted to units of excess electrons per cubic meterusing equation (14), as color-filled grid cells. A pattern very similar to the average Birkeland currents duringthe same conditions (Figure 4d, northward IMF panel) is seen in Figure 5a. A close relationship to the fieldaligned currents is indeed expected. Following Milan (2013, equations (2)–(4)), current continuity, and thecommon decomposition of the ionospheric currents along and perpendicular to the convection electric field,one arrives at the following expression for the Birkeland current: 𝑗 || = Σ P ∇ · ⃗ E + ∇Σ P · ⃗ E + ∇Σ H · ( ̂ B × ⃗ E ) . (15)Here, 𝛴 H and 𝛴 P are the height-integrated ionospheric Hall and Pedersen conductivities, respectively, and ̂ B is a unit vector along ⃗ B . If we assume that gradients in the Hall and Pedersen conductances are of minorimportance in this regard, we can relate j || to our estimated node amplitudes A j or Δ n e through equation(14): 𝑗 || ≈ Σ P ∇ · ⃗ E = Σ P A 𝑗 𝜎 = Σ P e Δ n e 𝜖 . (16)Hence, the part of the Birkeland currents not associated with conductivity gradients are directly propor-tional to Δ n e , or A j . The close relationship seen between the panels in Figure 4c and 4d is therefore expected.We again emphasize that our patterns of Δ n e , or equivalently 𝜖 ∇ · ⃗ E ∕ e , or 𝜖 A 𝑗 e 𝜎 , is to the first order onlydetermined by the plasma motion and not by the ionospheric conductivity. This is in contrast to the Birke-land currents that strongly depend on the ionospheric conductivity, as shown in equation (16), and Figures1 and 2 in Paper II. When the magnetospheric configuration can be considered stationary and there are noparallel electric fields, the large-scale ionospheric convection electric field are often considered as a “mirrorimage” of the coupled magnetosphere. Although some deviations from this ideal treatment may also existin the steady state as pointed out by Hesse et al. (1997), this strong coupling allows us to relate and quantifythe steady state large-scale ionospheric convection to its magnetospheric counterpart.REISTAD ET AL. 6354 ournal of Geophysical Research: Space Physics From the comparison of Figure 4d northward IMF panel and Figure 5a, it is easy to point out the SECSnodes that correspond to the NBZ currents seen in Figure 4d. These nodes are highlighted with a black dotin Figure 5a. In this identification we have used a threshold value of | Δ n e , j | > m − and be located at ⩾ ◦ MLAT and MLT ∈ [ , ] . In addition, we require that the sign of Δ n e , j should match the expected NBZcurrent direction at the 80 ◦ MLAT bin to avoid selecting nodes related to the poleward edge of the Region 1current.The electric potential Φ associated with the SECS node amplitudes using all nodes are shown in Figure 5b.Here, two small lobe cells are seen at the same locations as where we identified the lobe cells in panel(a). The maximum and minimum locations of the lobe cells are indicated with two “+” symbols, and theirpotential difference is 7 kV. In addition, a two-cell pattern at slightly lower latitudes is seen. This potentialdifference of 7 kV can be a measure of the flux circulation in the lobe cells, but we will here argue thatthere is a different way to estimate this lobe potential that is more beneficial, reflecting their source region.In Figure 5c we show the potential due to the lobe SECS nodes only (black dots). This separation of thesources of the ionospheric convection is one of the benefits of representing the convection electric fieldas a sum of nodes, as this representation also allows for a quantification of the associated magnetic fluxtransport. The potential from the SECS nodes outside this dayside polar cap region is for reference shownin Figure 5d. One can see that this decomposition efficiently isolates the four-cell pattern seen in Figure 5binto the contribution from the lobe cells and the Dungey-type convection. We suggest that the potentialdifference related to the lobe nodes only, as shown in Figure 5c to be 15 kV, is a more realistic value of theinfluence from lobe reconnection, as Figure 5a demonstrates that the local value of Δ n e appear to be closelyrelated to j || and hence a magnetospheric source, and not as much influenced by the surrounding SECSnodes. This property of Δ n e , which is mainly reflecting the influence of the local sources of convection, isinvestigated in Figures 5e and 5f. We here show plots in the same format as the upper row, but now Δ n e and the potential from the identified lobe nodes are calculated from observations above 75 ◦ MLAT only. Inaddition, SECS nodes are only placed above 75 ◦ MLAT. Figures 5e and 5f are therefore a local variant of thesame analysis that should give a similar result if the influence from distant processes is not important forthe locally determined values of Δ n e . We see similar patterns and amplitudes in the dayside polar cap, andthe associated potential is also similar as when including the entire high-latitude region in the inversion for A j . This suggests that the value of Δ n e , j reflects the local sources of influence on the ionospheric convectionelectric field. We therefore suggest to use the values obtained by the analysis in Figure 5c when quantifyingthe influence from lobe reconnection on the ionospheric convection during northward IMF, which is whatwe do in Paper II. If assuming a strong coupling between the ionosphere and magnetosphere ( E || = insidethe polar cap), this potential difference inferred from the lobe nodes, representing the ionospheric magneticflux transport rate around these nodes, can be interpreted as the lobe reconnection rate.
4. Discussion and Conclusion
This paper describes a novel technique that makes it possible to separate and quantify the different sourcesof ionospheric convection. The separation of the sources of lobe cell convection from the typical two-cellpattern is an obvious application of this technique, which is applied in Paper II to quantify how much thedipole tilt angle can alter the lobe cell convection.Although very similar patterns as derived from the SECS node amplitudes (Figure 4c) can be obtained fromtaking the divergence of ⃗ E from any representation of ⃗ E (due to the relation expressed in equation (14)),the SECS representation is the most direct approach to perform this task, as the amplitudes, or equivalentlythe charges (see equation (13)), is what is solved for, and not a derived quantity. However, if expressing asnapshot of the global ionospheric convection, data coverage is always an issue, and additional assumptionsare needed to describe the global convection. Then, including information from existing empirical modelsin regions without data using, for example, the map potential technique (Ruohoniemi & Baker, 1998) torepresent Φ , and subsequently use the relation in equation (14), we expect that a pattern reflecting themagnetospheric source of the ionospheric convection will be obtained.The vorticity of the ionospheric convection (∇× ⃗ v ) has strong similarities to the divergence of the ionosphericconvection electric field discussed in this paper and has been much used as a proxy for the Birkeland current(Sofko et al., 1995; Chisham et al., 2009; Chisham, 2017), as expressed in equation (16). We would like topoint out that although there is a strikingly good correspondence between the two quantities on averageREISTAD ET AL. 6355 ournal of Geophysical Research: Space Physics (see Figure 4c vs. Figure 4d), the ionospheric convection is, at least to the first order, independent of theionospheric conductivity. Hence, the two quantities (Birkeland current and convection) provide differentinformation about the ionospheric electrodynamics. Taken together, they can provide information aboutthe conductivity in the ionosphere without assuming a constant conductance. In some regions, a constantconductance might be an acceptable approximation. In Paper II we make this assumption and use equation(16) to make estimates of 𝛴 P in the dayside polar cap. However, there exist more sophisticated methods likethe “method of characteristics” (Amm, 2002) where the Hall and Pedersen conductances can be solved foronly by assuming their ratio. References
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