Forecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational Framework
Martin A. Reiss, Peter J. MacNeice, Leila M. Mays, Charles N. Arge, Christian Möstl, Ljubomir Nikolic, Tanja Amerstorfer
FF ORECASTING THE A MBIENT S OLAR W IND WITH N UMERICAL M ODELS : I. O
N THE I MPLEMENTATION OF AN O PERATIONAL F RAMEWORK
A P
REPRINT
Martin A. Reiss
Heliophysics Science DivisionNASA GoddardGreenbelt, MD 20771, USA [email protected]
Peter J. MacNeice
Heliophysics Science DivisionNASA GoddardGreenbelt, MD 20771, USA
Leila M. Mays
Heliophysics Science DivisionNASA GoddardGreenbelt, MD 20771, USA
Charles N. Arge
Heliophysics Science DivisionNASA GoddardGreenbelt, MD 20771, USA
Christian Möstl
Space Research InstituteAustrian Academy of Sciences8042 Graz, Austria
Ljubomir Nikolic
Canadian Hazards Information ServiceNatural Resources CanadaOttawa, Canada
Tanja Amerstorfer
Space Research InstituteAustrian Academy of Sciences8042 Graz, Austria2018 December 12 A BSTRACT
The ambient solar wind conditions in interplanetary space and in the near-Earth environment aredetermined by activity on the Sun. Steady solar wind streams modulate the propagation behavior ofinterplanetary coronal mass ejections and are themselves an important driver of recurrent geomagneticstorm activity. The knowledge of the ambient solar wind flows and fields is thus an essentialcomponent of successful space weather forecasting. Here, we present an implementation of anoperational framework for operating, validating, and optimizing models of the ambient solar windflow on the example of Carrington Rotation 2077. We reconstruct the global topology of the coronalmagnetic field using the potential field source surface model (PFSS) and the Schatten current sheetmodel (SCS), and discuss three empirical relationships for specifying the solar wind conditions nearthe Sun, namely the Wang-Sheeley (WS) model, the distance from the coronal hole boundary (DCHB)model, and the Wang-Sheeley-Arge (WSA) model. By adding uncertainty in the latitude about thesub-Earth point, we select an ensemble of initial conditions and map the solutions to Earth by theHeliospheric Upwind eXtrapolation (HUX) model. We assess the forecasting performance from acontinuous variable validation and find that the WSA model most accurately predicts the solar windspeed time series (RMSE ≈ km/s). We note that the process of ensemble forecasting slightlyimproves the forecasting performance of all solar wind models investigated. We conclude that theimplemented framework is well suited for studying the relationship between coronal magnetic fieldsand the properties of the ambient solar wind flow in the near-Earth environment. K eywords Solar wind · Solar-terrestrial relations · Sun: heliosphere · Sun: magnetic fields a r X i v : . [ phy s i c s . s p ace - ph ] M a y orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational Framework GONG
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OMNIMagnetic Maps Solar WindoMeasurementsSolaroImagesoVEUV) C o r on a I np u t H e li o s ph e r e Figure 1: Overview of coronal and heliospheric model combinations for forecasting the ambient solar wind conditionsin the near-Earth environment. The orange colored lines highlight models hosted at NASA’s Community CoordinatedModeling Center online platform (see, https://ccmc.gsfc.nasa.gov/models/ ). The knowledge of the evolving ambient solar wind is an essential component of successful space weather forecasting [26].The ambient solar wind flows and fields play an important role in understanding the propagation of coronal massejections (CMEs) and are themselves an important driver of recurrent geomagnetic activity. More specifically, high-speed solar wind streams contribute about
70 % of geomagnetic activity outside of solar maximum and about
30 % atsolar maximum [34]. Even when the occurrence rate of CMEs increases from . per day during solar minimum toabout – per day during solar maximum [50], key properties in interplanetary space, such as the solar wind bulk speed,magnetic field strength, and orientation, are determined by the ambient solar wind flow [17].The topology of open magnetic field lines along which ambient solar wind flows accelerate to supersonic speedsplays a fundamental role in understanding phenomena that drive our evolving space weather. To date, there are noroutine measurements of the coronal magnetic field. Magnetic models for the solar corona, therefore, have to relyon extrapolations calculated from the observed line-of-sight component of the photospheric field. The most widelyapplied extrapolation technique to reconstruct global solutions for the coronal magnetic field is the Potential FieldSource Surface model [PFSS; 1, 47]. Using the current-free (or potential field) assumption for regions above thephotosphere, solutions for the magnetic field can be expressed as the gradient of a magnetic scalar potential. Sincepotential fields give closed magnetic fields, a spherical source surface, where the magnetic field is assumed to be onlyradial is added as an outer boundary condition. The radius of the spherical source surface is an adjustable parametertypically set to a reference height of . R to best match observations [8]. To account for Ulysses measurementsshowing latitudinal invariance of the radial magnetic field component [55], the so-called Schatten current sheet (SCS)model [46] is typically added beyond the PFSS model to create a more uniform radial field strength solution.The effect of the solar wind is to drag and distort magnetic field lines, and thus to distort the coronal field from theassumed current-free configuration. Ideally, a model should account for the complex dynamics of the solar windflow by solving a set of nonlinear partial differential equations of magnetohydrodynamics (MHD) [e.g., 1]. As such,the Magnetohydrodynamics Algorithm outside a Sphere model [MAS; 16, 21] and the Space Weather ModelingFramework [SWMF; 52] are three-dimensional MHD models. The PFSS solutions are usually used as an initialcondition, and the MHD equations are integrated in time until the plasma and magnetic fields relax into equilibrium.The final steady-state solution is characterized by closed magnetic fields that confine the solar wind plasma and openmagnetic field lines along which solar wind flows accelerate to supersonic speeds.The state-of-the-art framework for forecasting the ambient solar wind couples models of the corona with those of theinner heliosphere [37, 13]. The coronal part spans the range from 1 solar radii ( R ) to . R (PFSS), or R to R (MHD). The inner boundary condition of the heliospheric part either results directly from the coronal modelor is computed from the topology of the coronal magnetic field. The outer boundary condition computed from thecoronal model is used as inner boundary condition for the heliospheric model ( – R to 1 au). Typically, the2orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational Framework GONGpMagnetograms
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Figure 2: Breakdown structure of the framework implementation. The sections explaining the corresponding frameworkcomponent are indicated.heliospheric model is based on the MAS model [16, 21], Enlil [25], or the recently developed European HeliosphericForecasting Information Asset [EUHFORIA; 32], each of which are three-dimensional numerical MHD models thatderive stationary solutions for the ambient solar wind in interplanetary space. Figure 1 shows some different coronaland heliospheric model combinations, together with other approaches for forecasting the ambient solar wind based onempirical relationships [e.g., 43, 53, 49, 33], and statistics or machine learning techniques [e.g., 29, 42]. The orangecolored lines highlight models accessible online at NASA’s Community Coordinated Modeling Center (CCMC) onlineplatform.Many recent studies [e.g., 28, 9, 6, 33] have assessed the performance of operational frameworks for forecastingthe ambient solar wind and reported typical uncertainties of about 1 day in the arrival time of high-speed streams.Furthermore, it is now well established that the performance of models of the ambient solar wind is, if at all, onlyslightly better than a 27 day persistence model [e.g., 29, 12], assuming that the near-Earth solar wind conditions willreplicate after each Carrington Rotation (CR). Overall, these results highlight the fact that forecasting the conditions inthe ambient solar wind in interplanetary space and in the near-Earth environment is challenging, even during times oflow solar activity when the large-scale interplanetary magnetic field configuration evolves less rapidly and disturbancesdue to CMEs are less frequent [26]. As outlined in the space weather roadmap for the years 2015–2025 [see, 48],continued efforts are needed to improve our capabilities for forecasting the conditions in the ambient solar wind.In this paper, we present an implementation of a numerical framework for operating, validating, and optimizing modelsof the evolving ambient solar wind. To study a large number of initial conditions, we rely on the coupling between well-3orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational FrameworkFigure 3: Comparison of the observed line-of-sight component of the photospheric field and the PFSS model solutionas a function of the heliographic latitude and Carrington longitude for CR2077 (i.e., 2008 November 20–December 17).The magnetic field strength is saturated at ± G. (a) Remeshed GONG synoptic magnetogram with N θ = 181 ; (b)–(d)Magnetic field solutions B r , B θ , and B φ computed from the PFSS model using the spherical harmonics expansion upto the order of 80 spherical harmonics. 4orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational Frameworkestablished coronal models [54, 37, 5] and the Heliospheric Upwind eXtrapolation (HUX) model [35] that bridges thegap of ballistic mapping and MHD modeling. While this paper outlines the breakdown structure and the mathematicalfoundation of the operational framework, a subsequent paper will be devoted to the validation and optimization of thenumerical models by examining the relationship between the coronal magnetic fields and properties of the ambient solarwind. This paper is organized into three sections. In Section 2, we discuss the numerical framework for forecasting theambient solar wind, including the remeshing of magnetograms (Section 2.1), the magnetic model of the solar corona(Section 2.2), the features of the coronal field solution (Section 2.3), the empirical relationships for specifying thesolar wind conditions near the Sun (Section 2.4), the mapping of the solar wind solutions to Earth (Section 2.5), theapplication of a sensitivity analysis (Section 2.6), and the quantification of the forecasting performance (Section 2.7);and in Section 3 we conclude with a summary of the results and outline future applications of the framework. We present an implementation of a numerical framework for operating, validating, and optimizing models of the ambientsolar wind. Figure 2 illustrates the breakdown structure of the current framework implementation. The coronal domainspans the range from R to R , where the outer boundary condition computed from the coronal part is used as aninner boundary condition for the heliospheric domain. The model in the heliospheric domain where the solar wind flowis supersonic then uses the boundary condition as an input for propagating the solar wind solutions near the Sun to 1 au.The subsequent sections are concerned with explaining the components of the framework in detail. We use full-disk photospheric magnetograms from the Global Oscillation Network Group (GONG) from the NationalSolar Observatory (NSO) as an inner boundary condition for the coronal model. The global maps of the solar magneticfield measured in Gauss (G) are given on the (sin θ, φ ) grid with × grid points, where θ ∈ [0 , π ] and φ ∈ [0 , π ] are the latitude and longitude coordinates, respectively. They are available as near-real-time magnetic maps or fullCR maps at the GONG online platform . Throughout this paper, we illustrate our model solutions on the example ofCR2077 (2008 November 20–December 17), that is, during solar minimum when the global magnetic field is dominatedby its dipolar component and pronounced polar coronal holes are observable.We use the full-disk magnetic maps as an inner boundary condition for the coronal model. The coronal model is basedon a spherical harmonic decomposition of the input magnetogram. When the raw magnetogram on the (sin θ, φ ) grid isused as an input and the order of spherical harmonics expansion is large compared to the resolution of the magnetic map,inaccurate results, especially at the crucial polar regions, are expected. Tóth et al. [51] concludes that the informationfrom magnetograms is used more efficiently when the input magnetograms (i) are remeshed to a grid that is uniform inthe latitude θ , (ii) contain both poles at θ = 0 , and θ = π , and (iii) have an odd number of grid points.We remesh the magnetograms according to the linear interpolation procedure discussed in Tóth et al. [51]. To do so, webegin with adding two additional grid cells at the north and south poles of the input magnetic map M i,j with N θ = 180 and N φ = 360 grid points in the latitude and longitude, respectively. The values for the extra grid cells at the poles M and M N θ +1 are given by M = 1 N φ N φ (cid:88) j =1 M ,j , (1)and M N θ +1 = 1 N φ N φ (cid:88) j =1 M N θ ,j . (2)The latitude of the uniform θ grid is θ (cid:48) i (cid:48) = π i (cid:48) − N (cid:48) θ − , (3) http://gong.nso.edu N (cid:48) θ is the number of grid points at the uniform θ grid, and the index i (cid:48) = 1 . . . N (cid:48) θ . Finally, we interpolate thegrid points from the raw magnetogram mesh to the uniform θ mesh using a linear interpolation relation (no magneticflux conservation) of the form M (cid:48) i (cid:48) ,j = αM i,j + (1 − α ) M i +1 ,j , (4)where the index i is selected so that θ ≤ θ (cid:48) i (cid:48) ≤ θ i +1 , (5)and α = θ i +1 − θ (cid:48) i (cid:48) θ i +1 − θ i . (6)Notice that the maximum degree for the expansion of spherical harmonics is now limited only by the anti-alias limit [see,51] given by min (cid:18) N (cid:48) θ , N φ (cid:19) . (7)By utilizing remeshed magnetograms as an inner boundary condition for the coronal model, we can, therefore, expectaccurate solutions up to the order of 120 spherical harmonics [51]. Figure 3(a) depicts the computed remeshedmagnetogram as a function of the heliographic latitude and Carrington longitude for CR2077. The magnetic model of the solar corona couples the PFSS model and the SCS model to reconstruct the coronal magneticfield. The PFSS model is based on the assumptions that the region above the photosphere is current free, and that themagnetic field at an imaginary reference sphere, called the source surface with radius R is radial only [1, 47]. Asdetailed in Appendix A, Eq.(19) can be solved to derive the magnetic field components at any point in the coronaldomain ( R ≤ r ≤ R ). Figure 3 (b)–(d) present the magnetic field components B r , B θ , andB φ derived from the PFSSmodel at the solar surface as a function of the latitude and Carrington longitude for CR2077. We note that we used acolor scheme for all illustrations that is color-blind friendly [see, 15]. From the comparison in Figure 3, it is apparentthat the PFSS solution using the spherical harmonics expansion up to n = 80 , is in reasonable agreement with theobserved line-of-sight component of the photospheric magnetic field.To extend the magnetic field solution from the outer boundary of the PFSS model at . R to a distance of R = 5 R ,we employ the SCS model [46]. The SCS model uses the PFSS solutions at the source surface as an inner boundarycondition. The SCS model is similar to the PFSS model but involves solving an additional Laplace equation withdifferent boundary conditions. In the first step, the magnetic field on the source surface is oriented to point away fromthe Sun. In other words, the signs of the magnetic field components B r , B θ , and B φ are reversed if B r < at the sourcesurface. While the magnetic field solution is defined to vanish at infinity, the non-negative magnetic field values asmodel input ensure that a thin current sheet is retained in the field solution. The underlying idea of the SCS model is toextend the magnetic field solution in a nearly radial way. To retain the resolution, we match the grid of the SCS modelwith the PFSS model with equal steps of ◦ in latitude and longitude. As discussed in Appendix B, we use estimatesof the coefficients g mn and h mn from the least mean square fit to compute the magnetic field above the source surface.Finally, the polarity needs to be corrected to match the polarities in the region r ≤ R .The imposed boundary conditions of the PFSS model and the SCS model are not compatible, causing discontinuitiesin the tangential component of the magnetic field across the source surface. When coupling the PFSS model andSCS model solutions, discontinuities in the form of kinks in the magnetic field topology are observable. In order toaccount for this effect, [20] proposed a more flexible coupling between the two models. The authors concluded thattheir approach leads to more accurate forecast results. Following the procedure of [20], we set the radius of the sourcesurface to . R but use the PFSS solution at . R as an inner boundary condition for the SCS model. By doing so,we couple the PFSS and SCS model solutions and reconstruct the global topology of the coronal magnetic field. Models for specifying the solar wind conditions near the Sun rely on the areal expansion factor f p and the great-circleangular distance from the nearest open-closed boundary d , both of which are derived from the coronal magnetic field6orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational FrameworkFigure 4: Illustration of the topology of the magnetic field between the photosphere and the source surface at . R computed from the described magnetic model of the solar corona for CR2077. Red and blue colors indicate positiveand negative magnetic field lines, respectively. While the lower plane shows the great-circle angular distance from thenearest coronal hole boundary at the photosphere d , the upper plane shows the areal expansion factor f p indicating howmuch the corresponding flux tube expands between the photosphere and the source surface at . R .solution. It is noteworthy that both features are linked to different fundamental theories on the origin of slow solarwind [see, 36, 41]. While the expansion factor assumes that the solar wind flow along open field lines that diverge themost leads to the slow solar wind [54], the distance to the coronal hole boundary is more related to the boundary layeridea of interchange reconnection for the origin of the slow solar wind [3].To trace magnetic field lines and reconstruct the magnetic field topology in the coronal model domain R ≤ r ≤ R ,we use a fourth-order Runge-Kutta method (RK4) and solve the following set of equations, expressed as drds = B r B ,dθds = 1 r B θ B ,dφds = 1 r sin θ B φ B , (8)where ds is a segment along the magnetic field line.More specifically, our approach for constructing the large-scale topology of the coronal field is two-fold. First, we startfrom the base of the model at the solar surface ( r = R ) and trace magnetic field lines upwards. When the field linereturns to the solar surface, we label the corresponding footpoint at the solar surface as a “closed field”. In contrast,when the field line reaches the upper boundary of the coronal model ( r = R ) we label the footpoint as a “open field”.In this way, we compute coronal hole regions, i.e., magnetically open regions expected to guide high-speed solar windstreams out into the heliosphere. Using the location of coronal holes at the solar surface, we run a perimeter tracingalgorithm to compute coronal hole boundaries. Then, we compute the great-circle angular distance d between footpointsof open field lines and their nearest coronal hole boundary. Second, we start from the outer boundary of the coronalmodel ( r = R ) and trace the field lines down to the solar surface. By doing so, we compute the areal expansion factor f p as f p = (cid:18) R R (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) B r ( R , θ , φ ) B r ( R , θ , φ ) (cid:12)(cid:12)(cid:12)(cid:12) , (9)7orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational Frameworkwhere B r is the radial magnetic field component at a given reference height. The areal expansion factor is the amount bywhich a flux tube expands from the solar surface to another reference height in the corona. Figure 4 shows the topologyof the coronal magnetic field and illustrates the features d and f p at the photosphere and the source surface, respectively. The structure in the solar wind flow is governed by the dynamic pressure term in the momentum equation ( ∝ ρ v ). Thisindicates that the near-Sun solar wind solution used as an inner boundary condition for heliospheric models is highlysensitive to errors in the computed solar wind bulk speed [41]. Recently, Riley et al. [41] highlighted three empiricalrelationships for specifying the solar wind speed ( v = v ( d, f p )) at a reference sphere of R (or R for the MASmodel). In the literature, these empirical relationships are known as the Wang-Sheeley (WS) model [54], the Distancefrom the Coronal Hole Boundary (DCHB) model [37], and the Wang-Sheeley-Arge (WSA) model [5].The WS model is based on the inverse relationship between the solar wind speed and the magnetic field expansionfactor [54], namely v ws ( f p ) = v + ( v − v ) f p α . (10)Previous research has established that low magnetic field expansion between the photosphere and some reference heightin the corona is correlated with a fast solar wind speed [e.g., 14]. For the coefficients in Eq.(10), we use v = 250 , v = 660 , and α = 2 / [see, 4].The DCHB model relates the speed at the photosphere with the distance of an open field footpoint from the coronalhole boundary and maps the calculated solar wind speed solution out along the magnetic field lines to a given referencesphere [37]. The DCHB relation is defined as v dchb ( d ) = v + 12 ( v − v ) (cid:20) (cid:18) d − (cid:15)w (cid:19)(cid:21) , (11)where (cid:15) is a measure for the thickness of the slow flow band, and w denotes the width over which the solar wind reachescoronal hole values. For an open field footpoint at the coronal hole boundary, the solar wind speed is equal to v . For afootpoint located deep inside a coronal hole, the solar wind speed is equal to v . Hence, the farther away the footpointis from a coronal hole boundary, the faster the expected solar wind speed. In this study, we use v = 350 , v = 750 , (cid:15) = 0 . , and w = 0 . [see, 37].Finally, the WSA model is a hybrid of the WS model and the DCHB model, combining aspects of the expansion factorcomputed from the topology of the magnetic field and the distance to the coronal hole boundary [5]. The WSA modelfor specifying solar wind speed is given by v wsa ( f p , d ) = v + ( v − v )(1 + f p ) α (cid:110) β − γ exp (cid:104) − ( d/w ) δ (cid:105)(cid:111) , (12)where α , β , γ , and δ are additional model parameters. For the coefficients in Eq.(12) we use v = 285 , v = 910 , α = 2 / , β = 1 , γ = 0 . , w = 2 , and δ = 3 . Figure 5 shows a comparison between the different empirical relationshipsfor specifying a solar wind speed of v ( f p , d ) near the Sun. Numerous models dealing with the mapping of solar wind solutions near the Sun to Earth have been developed. Thespectrum extends from a simple ballistic approximation where each parcel of plasma is assumed to travel with aconstant speed through the heliosphere to global heliospheric MHD models that aim to cover all relevant dynamicalprocesses [e.g., 38, 25]. In an attempt to bridge the gap, Riley and Lionello [35] developed the Heliospheric UpwindeXtrapolation (HUX) model. Riley and Lionello [35] have simulated the kinematic of the solar wind flow numericallyby simplifying the MHD equations as much as possible. By neglecting the pressure gradient and the gravity terms inthe fluid momentum equation [e.g., 31] the solar wind solution on a discretized grid may be written as v r +1 ,φ = v r,φ + ∆ r Ω v r,φ (cid:18) v r,φ +1 − v r,φ ∆ φ (cid:19) , (13)8orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational FrameworkFigure 5: Comparison of empirical relationships for specifying the solar wind speed near the Sun on the example ofCR2077. The magnetic field lines illustrate the magnetic connectivity between the computed coronal hole regions andthe sub-Earth orbit at the outer boundary of the coronal model ( R ). The gray colored pixels indicate closed magneticfield topology with negative (dark gray) and positive (light gray) polarity. The full black and blue lines indicate positiveand negative magnetic field lines, respectively. (a) Wang-Sheeley (WS) Model; (b) Distance from the Coronal HoleBoundary (DCHB) Model; (c) Wang-Sheeley-Arge (WSA) Model.9orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational FrameworkFigure 6: The Heliospheric Upwind eXtrapolation (HUX) model for mapping the WSA solar wind speed solution to1 au (or R ). (a)–(b) Solar wind bulk speed as a function of the heliocentric distance. (c) The input solar windspeed at R (blue), and the resulting solar wind speed at R with (yellow) and without (red) the residual speedcontribution v acc .where the subscripts r and φ refer to radial and longitudinal grid cells with cell spacings ∆ r and ∆ φ , and Ω denotesthe equatorial angular rotation rate of the Sun (neglecting differential rotation) [see, 35]. In order to match the spatialresolution of the coronal model, we use ∆ φ = 2 ◦ and ∆ r = 1 R , respectively. As the step size in ∆ r is increasedby two orders of magnitude, the solutions are not significantly different, indicating that the Courant-Friedrichs-Lewycondition for numerical convergence is well met.Furthermore, Riley and Lionello [35] concluded that it is convenient to account for the residual wind accelerationbeyond the coronal model. According to previous model results [e.g., 37, 35], the residual acceleration v acc expectedbeyond the coronal model is v acc ( r ) = α v r (cid:20) − exp (cid:18) − rr h (cid:19)(cid:21) , (14)where v r is the initial speed at the outer boundary of the coronal model, r is the heliocentric distance, α is a factor bywhich v r is enhanced, and r h is the scale length for the acceleration (not crucial when mapping the solar wind to 1 au).For the coefficients in Eq.(14) we use α = 0 . and v r = 50 R .Figure 6 illustrates the HUX model for mapping the solar wind solution near the Sun to Earth. While the top panelshows the propagation of the solar wind solutions in steps of ∆ r from R to R (or 1 au), the bottom panel showsthe propagation in spherical coordinates (left) and the calculated speed time series at Earth with and without the residualspeed contribution (right). It is also noteworthy that computing the solar wind solution for other heliospheric distancesis straightforward. The benefit of the HUX model is that it can match the dynamical evolution explored by global10orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational FrameworkFigure 7: Illustration of the ensemble of initial conditions and the process of mapping the solar wind solutions toEarth by the HUX model. (a) Solar wind solution for CR2077 from the WSA model at R overlaid with 3 out of576 individual trajectories together with the median value (dashed red line) and the ± σ quantiles (red line); (b) Acomparison of observed and WSA model solar wind speed time series at R . The ensemble median of solar windsolutions at Earth is indicated by the dashed blue line and the ± σ quantiles are shown in blue color.heliospheric MHD models that demand only low computational requirements [35]. This is particularly useful to study alarge number of initial conditions in the context of ensemble modeling. The knowledge on the sensitivity of the model performance to initial conditions and model parameter settings is crucialfor models of the ambient solar wind. In recent years, the concept of ensemble forecasting has been successfully appliedto a number of applications in the space weather forecasting regime. As an example, the sensitivity of PFSS and MHDcoronal models on magnetic maps from different observatories has been studied by Riley et al. [39, 40]. More detailson recent activities can be found in the reviews by Knipp [11] and Murray [22]. Ensemble forecasting is a techniquebased on the use of a sample of possible future states to forecast a future state. Intuitively, one would expect that theforecast of the ambient solar wind conditions is very uncertain when the ensemble members (e.g., different solar windmodels) are very different from one another. The strength of ensemble forecasting is its capability to deduce confidencebounds by quantifying the uncertainty in the ensemble of possible future states.Owens et al. [30], for instance, has pointed out that the forecast uncertainty in solar wind models can be studied byadding uncertainty in the sub-Earth orbit at which the coronal solutions near the Sun are sampled. Using the approach11orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational FrameworkTable 1: The skill of model predictions of solar wind speed for CR2077 in terms of Arithmetic Mean (AM), StandardDeviation (SD), Mean Error (ME), Mean Absolute Error (MAE), and Root-Mean-Square Error (RMSE). A 4-day and27-day persistence model of near-Earth solar wind conditions provides a baseline against which solar wind models canbe compared.Model AM [km/s] SD [km/s] ME [km/s] MAE [km/s] RMSE [km/s]WS 428.79 62.63 -27.90 74.09 85.27DCHB 379.47 30.93 21.42 83.83 103.43WSA 437.14 69.14 -36.25 68.54 82.62Ensemble Median (WS) 435.55 51.84 -34.66 71.52 83.36Ensemble Median (DCHB) 367.47 4.62 33.43 78.27 100.04Ensemble Median (WSA) 437.48 63.64 -36.56 62.24 74.86Persistence (4-days) 394.76 99.23 6.13 130.48 161.99Persistence (27-days) 409.94 108.91 -9.05 66.54 78.86Observation 400.89 96.12 - - -of Owens et al. [30], we perform a sensitivity analysis that considers ensemble members at perturbed latitudes θ p around the sub-Earth orbit given by θ p ( φ ) = θ E + θ sin ( n φ + φ ) , (15)where φ is the corresponding Carrington longitude; θ E is the sub-Earth latitude; θ is the amplitude of the deviation; n is the wave number, indicating how many oscillations the perturbed trajectory completes; and φ is the phase offset,indicating how two perturbed trajectories can be out of synchronicity with each other. For the coefficients in Eq.(14),we select θ ∈ [0 ◦ , ◦ ] in ◦ steps, n ∈ [0 , in . steps, and θ ∈ [0 ◦ , ◦ ] in ◦ steps, which ensures that alllatitudes are well represented at each longitude [see, 30].Figure 7(a) shows three individual trajectories together with the median value and the ± σ quantiles of the ensemblemembers. In this way, we obtain information from a sampling of 576 initial speed solutions, each of which are mappedto Earth using the HUX model as outlined in Section 2.5. Figure 7(b) compares the resulting ensemble median at R and the observed solar wind speed in the near-Earth environment. In this study, the ensemble median is thepreferred average measure as the ensembles of solar wind solutions around the sub-Earth latitude are often highlyskewed, such that the ensemble mean can yield very biased measures of the ensemble average. Further, we notethat ensemble averaging of possible future states is not necessarily expected to provide a systematic improvement ofdeterministic forecasts [see, 10]. As discussed in Owens et al. [30] and Henley and Pope [7], it is the assessment offorecast uncertainty that is the key advantage of the ensemble approach rather than the ensemble average providing animproved deterministic forecast. The information from the implemented models is used most efficiently when the uncertainty of their results is constantlyvalidated. To enable that, we measure the performance of the framework by the Operational Solar Wind EvaluationAlgorithm [OSEA; 33]. OSEA runs various validation procedures to compare the forecasts and the observationsto which they pertain. Traditionally, the relationship between forecast and observation can be studied in terms ofcontinuous variables and binary variables. While the former can take on any real values, the latter is restricted to twopossible values such as event/non-event. In the context of solar wind forecasting, the solar wind speed time series canbe interpreted in terms of both aspects. The forecasting performance can either be evaluated in terms of an average erroror in terms of the capability of forecasting events of an enhanced solar wind speed [27, 18, 19]. OSEA is capable ofquantifying both aspects, i.e., a continuous variable evaluation that uses simple point-to-point comparison metrics andan event-based validation analysis that assesses uncertainty of the arrival time of high-speed solar wind streams at Earth.As an example, Table 1 lists the results obtained from the continuous variable validation of different solar windmodels for CR2077 in terms of Arithmetic Mean (AM), Standard Deviation (SD), Mean Error (ME), Mean AbsoluteError (MAE), and Root-Mean-Square Error (RMSE). A 4 day and 27 day persistence model of near-Earth solar windconditions provides a baseline against which the performance can be compared. We find that the RMSE for the WSmodel, DCHB model, and WSA model is 85.27 km/s, 103.43 km/s, and 82.62 km/s, respectively. The 27 day persistencemodel has the same statistics as the measurements and greatly benefits from the quasi-steady and recurrent nature ofthe evolving ambient solar wind in the solar minimum phase and thus is expected to score very high in all measures https://bitbucket.org/reissmar/solar-wind-forecast-verification = 78 . km/s). We conclude that the WSA model gives the best forecast results in terms of a continuousvariable validation and that the process of ensemble forecasting slightly improves the performance of all solar windmodels in this study. It is important to note that the present analysis is indented to illustrate the application of theimplemented framework components, which means that our results apply only to CR2077 and that our conclusions arenot reliable as a general guide. We present a numerical framework for forecasting the evolving ambient solar wind that uses magnetic maps as an inputfor the PFSS and SCS model to reconstruct the global topology of the coronal magnetic field, specifies the solar windspeed using different established empirical relationships (WS, DCHB, and WSA models) based on the areal expansionfactor and the distance to the nearest coronal hole boundary, maps the near-Sun solar wind solution outward to Earth bythe HUX model, creates an ensemble of initial conditions by adding uncertainty in the latitude about the sub-Earth point,and uses an automated forecast validation module to quantitatively assess the forecasting skill. The framework relies onestablished models of the ambient solar wind and is conceptually very similar to already existing numerical frameworks.We carefully compared our model solutions to existing frameworks [e.g., 24, 5], and found that our modular frameworkimplementation in the C ++ and Matlab programming language using tools from the Armadillo library [45] is bothrobust and fast.The coronal part of the framework relies on empirical relationships between the magnetic field topology and thenear-Sun solar wind conditions [54, 37, 5]. Ideally, one would prefer the application of a physics-based MHD modelfor the corona to capture the complex dynamics at solar wind stream interaction regions, which are not included inthe described model approach. Nevertheless, recent studies have shown that the forecast skill of empirical and fullphysics-based coupled corona-heliosphere models in terms of established metrics is very similar. As an example,[28] studied the performance of different forecast models (WSA, WSA-Enlil, and MAS-Enlil) over an 8 yr periodand concluded that the coupled empirical approach currently gives the best forecast results in terms of the meansquare error. Considering the trade-off between accuracy and computational requirements of a full MHD code, it thusseems reasonable to follow the described methodology in the context of solar wind forecasting. We conclude that theefficient implementation of the framework using the heliospheric HUX model is well suited for studying the long-termrelationship between coronal magnetic fields and the properties of the ambient solar wind.In the future, we shall work on several topics to try to improve the forecasting performances. While this study presentsthe implementation of the numerical framework, a subsequent paper will be devoted to the validation and optimizationof the present solar wind models. By studying the coupling between magnetic models of the corona and those ofthe inner heliosphere, we aim toward optimizing the empirical relationships for specifying solar wind properties toadvance further their predictive capabilities. In context, a reliable forecast of the ambient solar wind might be beneficialfor forecasting the arrival of CMEs. Here, we plan to combine the ambient solar wind framework with the ELlipseEvolution model based on Heliospheric Imager observations [ELEvoHI; 44, 2]. The kinematics of the elliptical-shapedCME front in the ELEvoHI model is governed by the ambient solar wind flow. In the current version of ELEvoHI, thesolar wind speed is assumed to be constant during the propagation of the CME in the inner heliosphere. Therefore,we would also like to include information on the complex dynamics of the evolving ambient solar wind to simulatethe dynamic deformation of the CME front during the propagation phase. To make the model runs accessible to thespace weather community, we also plan to install a later version of the solar wind framework at NASA’s CCMC onlineplatform. The release of this online resource is scheduled for 2019 July. Acknowledgments
The work utilizes data obtained by the Global Oscillation Network Group (GONG) Program, managed by the NationalSolar Observatory, which is operated by AURA, Inc. under a cooperative agreement with the National ScienceFoundation. The data were acquired by instruments operated by the Big Bear Solar Observatory, High AltitudeObservatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de Astrofísica de Canarias, and CerroTololo Interamerican Observatory. L.N. performed this work as part of Natural Resources Canada’s Public SafetyGeoscience program. M.A.R., C.M. and T.A. acknowledge the Austrian Science Fund (FWF): J4160-N27, P26174-N27and P31265-N27. 13orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational Framework
A PFSS Model
The PFSS model is based on the assumption that the magnetic field B above the photosphere is current free ( ∇× B = 0 ).With this approximation, the magnetic field can be expressed as the gradient of a scalar potential Ψ , B = −∇ Ψ . (16)Using ∇ · B = 0 , which expresses the fact that there are no magnetic monopoles, we can write the Laplace equation forthe potential Ψ as ∇ Ψ = 0 . (17)The solution of the Laplace equation in spherical coordinates with θ ∈ [0 , π ] and φ ∈ [0 , π ] in the region R ≤ r ≤ R can be expressed as a infinite series of spherical harmonics expressed as Ψ( r, θ, φ ) = (cid:34) R (cid:18) R r (cid:19) n +1 − R (cid:18) R R (cid:19) n +2 (cid:18) rR (cid:19) n (cid:35) × ∞ (cid:88) n =1 n (cid:88) m =0 ( g mn cos mφ + h mn sin mφ ) P mn (cos θ ) , (18)where R = 2 . R is the source surface radius, P mn (cos θ ) are the associated Legendre polynomials, and g mn and h mn are coefficients that are computed from the input magnetograms. Using Eq.(A1), the solution for the magnetic fieldcomponents is given by B = ( B r , B θ , B φ ) = (cid:18) − ∂ψ∂r , − r ∂ψ∂φ , − r sin θ ∂ψ∂φ (cid:19) . (19)The magnetic field components are computed as follows: B r = (cid:34) ( n + 1) (cid:18) R r (cid:19) n +2 + (cid:18) R R (cid:19) n +2 (cid:18) rR (cid:19) n − (cid:35) × ∞ (cid:88) n =1 n (cid:88) m =0 ( g mn cos mφ + h mn sin mφ ) P mn (cos θ ) , (20) B θ = − (cid:34)(cid:18) R r (cid:19) n +2 − (cid:18) R R (cid:19) n +2 (cid:18) rR (cid:19) n − (cid:35) × ∞ (cid:88) n =1 n (cid:88) m =0 ( g mn cos mφ + h mn sin mφ ) P mn (cos θ ) , (21) B φ = (cid:34)(cid:18) R r (cid:19) n +2 − (cid:18) R R (cid:19) n +2 (cid:18) rR (cid:19) n − (cid:35) × ∞ (cid:88) n =1 n (cid:88) m =0 m sin θ ( g mn sin mφ − h mn cos mφ ) P mn (cos θ ) . (22)To compute the coefficients g mn and h mn , we multiply Eq.(A3) for r = R with P m (cid:48) n (cid:48) cos m (cid:48) φ and P m (cid:48) n (cid:48) sin m (cid:48) φ ,respectively. Using the orthogonality of the Schmidt normalized Legendre polynomials and integrating over thespherical surface, we find π (cid:90) π (cid:90) π P mn ( θ ) (cid:26) cos mφ sin mφ (cid:27) P m (cid:48) n (cid:48) ( θ ) (cid:26) cos m (cid:48) φ sin m (cid:48) φ (cid:27) sin θ dθ dφ = 12 n + 1 δ n (cid:48) n δ m (cid:48) m . (23)14orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational FrameworkThis yields the solution of the coefficients g mn and h mn in the form (cid:26) g mn h mn (cid:27) = 2 n + 14 π (cid:18) n + 1 + n (cid:16) R R (cid:17) n +1 (cid:19) (cid:90) π dθ sin θ P mn ( θ ) (cid:90) π dφ B r ( R , θ, φ ) (cid:26) cos mφ sin mφ (cid:27) . (24)For the use of remeshed magnetograms as described in Section 2.1, we modify this relation to a discrete representation.To do so, we use the Clenshaw-Curtis quadrature rule given by (cid:90) π dθ sin θF ( θ ) ≈ N θ (cid:88) i =1 (cid:15) i w i F ( θ i ) , (25)where (cid:15) i is / for i = 0 or i = H , and elsewhere. The weights w i are given by w i = − H H (cid:88) k =0 (cid:15) (cid:48) k k − (cid:18) πk ( i − H (cid:19) (26)where H = ( H θ − / , (cid:15) (cid:48) k is / for i = 0 or i = H , and 1 elsewhere. Using this equation, we write Eq.(A9) as (cid:26) g mn h mn (cid:27) = 2 n + 14 π (cid:18) n + 1 + n (cid:16) R R (cid:17) n +1 (cid:19) πN φ N θ (cid:88) i =1 N φ (cid:88) j =1 (cid:15) i w i P mn ( θ i ) B r ( R , θ i , φ j ) (cid:26) cos mφ j sin mφ j (cid:27) . (27)Using Eq.(A4) and Eq.(A10), we compute the magnetic field components at any point in the region between the solarsurface and the source surface [e.g., 1, 23]. B SCS Model
The solution for the Laplace equation not bounded by the spherical source surface is given by
Ψ = R ∞ (cid:88) n =0 n (cid:88) m =0 (cid:34)(cid:18) R r (cid:19) n +1 ( g mn cos mφ + h mn sin mφ ) P mn ( φ ) (cid:35) . (28)In the SCS model, solutions for the PFSS model for the magnetic field on the source surface are first oriented to pointaway from the Sun. This means that if B r < on the source surface, the sign of all components B r , B θ , and B φ arereversed. In this study, we match the grid of the PFSS model to the SCS model with equal steps of ◦ in latitude andlongitude. The components of the magnetic field beyond the source surface are B r = − d Ψ dr = ∞ (cid:88) n =1 ( n + 1) (cid:18) R r (cid:19) n +2 n (cid:88) m =0 ( g mn cos mφ + h mn sin mφ ) P mn (cos θ ) ,B θ = − r d Ψ dθ = − ∞ (cid:88) n =1 (cid:18) R r (cid:19) n +2 n (cid:88) m =0 ( g mn cos mφ + h mn sin mφ ) dP mn (cos θ ) dθ ,B φ = − r sin θ d Ψ dφ = ∞ (cid:88) n =1 (cid:18) R r (cid:19) n +2 n (cid:88) m =0 m sin θ ( g mn sin mφ − h mn cos mφ ) P mn (cos θ ) . (29)In this way, we ensure that no closed magnetic field exists beyond the source surface. We use the least-squares approachto best fit the vector field on the source surface. In order to minimize the sum of squared residuals, we write F = N i (cid:88) i =1 N j (cid:88) j =1 3 (cid:88) k =1 (cid:34) B k ( θ i , φ j ) − N s (cid:88) n =0 n (cid:88) m =0 ( g mn α nmk ( θ i , φ j ) + h mn β nmk ( θ i , φ j )) (cid:35) , (30)15orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational Frameworkwhere B k ( θ i , φ j ) is the reoriented field of the source surface, and the index k = 1 , , and 3 refers to the radial,latitudinal, and azimuthal fields component of the grid point ( θ i , φ j ) of the reoriented magnetic field. Further, α nmk and β nmk are α nm = ( n + 1) cos mφ P mn ( θ ) α nm = − cos mφ dP mn ( θ ) dθα nm = m sin θ sin mφ P mn ( θ ) β nm = ( n + 1) sin mφ P mn ( θ ) β nm = − sin mφ dP mn ( θ ) dθβ nm = m sin θ cos mφ P mn ( θ ) (31)We compute the derivations to minimize F given by ∂F/∂g mn , and ∂F/∂h mn . For each ( n, m ) we can write (cid:88) i (cid:88) j (cid:88) k α nmk ( θ i , φ j ) (cid:32) B k ( θ i , φ j ) − N s (cid:88) t =0 t (cid:88) s =0 g st α tsk ( θ i , φ j ) + h st β tsk (cid:33) = 0 , (cid:88) i (cid:88) j (cid:88) k β nmk ( θ i , φ j ) (cid:32) B k ( θ i , φ j ) − N s (cid:88) t =0 t (cid:88) s =0 g st α tsk ( θ i , φ j ) + h st β tsk (cid:33) = 0 . (32)The same expression in matrix form is (cid:99) αβ · (cid:98) B = (cid:99) αβ · (cid:99) αβ (cid:62) · (cid:100) GH, (33)with (cid:98) B = B ( θ , φ ) B ( θ , φ ) ... B ( θ I , φ J ) B ( θ , φ ) ... B ( θ I , φ J ) , (cid:100) GH = g g ... g NN h ... h NN , (cid:99) αβ = α . . . α ... ... α N s N s . . . α N s N s β . . . β ... ... β N s N s . . . β N s N s . (34)Here, the dimension of (cid:98) B is IJ × , the dimension of (cid:100) GH is ( N s +1) × , and the dimension of (cid:99) αβ is ( N s +1) × IJ .By choosing (cid:100) AB = (cid:99) αβ · (cid:99) αβ (cid:62) we find that the solution for GH is of the form GH = AB − · αβ · B (35)This means that the solution of GH requires the inversion of the square matrix (cid:100) AB . Finally, we use the estimates of g mn and h mn from the least mean square fit to compute the magnetic field solution above the source surface. We note that thepolarity needs to be refined to match the polarities for r ≤ R .16orecasting the Ambient Solar Wind with Numerical Models: I. On the Implementation of an Operational Framework References [1] M. D. Altschuler and G. Newkirk. Magnetic Fields and the Structure of the Solar Corona. I: Methods of CalculatingCoronal Fields.
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