Adventures in the magnetosheath: two decades of modeling and planetary applications of the Spreiter magnetosheath model

Planetary and Space Science 50 (2002) 421 – 442

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Adventures in the magnetosheath: two decades of modeling and planetary applications of the Spreiter magnetosheath model Stephen S. Stahara RMA Aerospace, Inc., Cupertino, CA 95014, USA

Abstract A review is given of the author’s two decades of magnetosheath modeling work with John Spreiter Beginning in the early 1970s, a sequence of computational magnetosheath submodels that extended and re-ned John’s basic model were developed and successfully applied to virtually every planet in the solar system including the terrestrial planets Mercury, Venus, Earth, and Mars and the giant planets Jupiter, Saturn and Neptune. A real-time version was also developed as one segment of an integrated global model for terrestrial space weather forecasting. The scope and depth of these successful applications taken together have demonstrated the remarkable accuracy and versatility of Spreiter’s basic modeling concepts and have con-rmed the validity of his legacy magnetosheath modeling ideas. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Magnetosheath; Solar wind=magnetosphere interactions; Numerical modeling

1. Introduction It is my pleasure to recall almost two decades of modeling and planetary applications with the Spreiter magnetosheath model. John and I had already been collaborating for almost a decade on aerodynamic applications before we began our joint planetary magnetosheath work. Those prior aerodynamic studies were to stand us in good stead as we initiated our magnetosheath program. They all involved extensions of numerical 8uid dynamic modeling techniques then under development at NASA=Ames and were to form an essential part of our computational realization of the Spreiter magnetosheath model. The paper will begin by providing a brief review of the status of the Spreiter magnetosheath model in the late 1970s when we began our joint work. It will then describe the sequential developments and applications that we undertook involving a series of computational magnetosheath submodels that extended and re-ned the basic model. These developments were all driven by planetary applications, and proceeded from Earth to Venus, Jupiter, Saturn, Neptune, and -nally back to Earth to focus on terrestrial space weather forecasting. 2. Status of the Spreiter model ca. late 1970s Beginning in the early 1960s, John had formulated the theoretical foundation and framework for a complete theory of E-mail address: [email protected] (S.S. Stahara).

continuum high-Alfven Mach number solar wind 8ow about the magnetosphere (Spreiter and Briggs, 1962; Spreiter et al., 1966, 1968; Spreiter and Alksne, 1969). The fundamental assumption of the model is that the bulk properties of solar wind 8ow past planetary magnetopause or ionopause (magnetoionopause) obstacles can be adequately described by solutions of the limiting forms at high Alfven and sonic Mach numbers of the continuum equations of magnetohydrodynamics for a single-component perfect gas having in-nite electrical conductivity and zero viscosity and thermal conductivity. The governing equations express the conservation of mass, momentum, energy, and magnetic 8ux, augmented by jump conditions appropriate for a fast shock wave and a tangential discontinuity at, respectively, the bow shock wave and magnetoionopause surface. The model then determines the magnetoionopause and bow shock locations and the magnetosheath plasma and -eld properties. Now known as the gasdynamic-convected magnetic -eld (GDCF) magnetosheath model, John had developed a capability through the 1960s and early 1970s at NASA=Ames to carry out initial magnetosheath calculations based on that model for solar wind 8ow past Earth (Spreiter and Alksne, 1969; Spreiter and Rizzi, 1974), nonmagnetic planets (Spreiter et al., 1970a) and the moon (Spreiter et al., 1970b). The model calculations require a predetermined magnetoionopause obstacle. For Earth, John and Ben Briggs (Spreiter and Briggs, 1962) developed a model based on Chapman–Ferraro concepts for the 3-D magnetospheric boundary separating a uniformly 8owing solar wind and the

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Fig. 1. Early validation of the GDCF magnetosheath model. Comparisons of model predictions with observations of Pioneer VI.

magnetic dipole -eld of Earth. For nonmagnetic planets, John developed a corresponding model for the ionopause obstacle separating planetary ionospheres from the solar wind (Spreiter et al., 1970a). A number of examples determined from the model were published with results based on nominal solar wind conditions. These were used widely in many aspects of space research and, in fact, continue to be used even today. Fig. 1 from Spreiter et al. (1968) illustrates one of those results and compares Pioneer VI plasma and magnetic -eld observations throughout the magnetosheath with results from the GDCF model. The model results are based on observed solar wind conditions when the spacecraft was just outside the bow shock. The comparisons show good agreement with bow shock and magnetopause locations as well as with the variation of plasma and -eld through the magnetosheath. These original magnetosheath calculations were extremely tedious to carry out, requiring much human intervention to obtain the -nal result, and always pushed the limits of the largest computer then available at NASA=Ames. After John accepted his full professorship at Stanford University in 1969, the computational capability for the model was gradually lost in the early 1970s with the dispersal of the group and the replacement of the computers on which the work had been done. 3. Program of magnetosheath model development It was in the late 1970s then that John and I began our long collaboration on magnetosheath model development. The initial goals were two-fold: (1) to restore the lost capability of producing GDCF magnetosheath solutions, and (2) to signi-cantly improve and extend the basic model. The overall technical approach was to use initially the same physics as the basic model was based upon, to employ the latest advances in computational methods that were cur-

rently being developed at NASA, and to develop an eEcient, user-oriented model that would allow routine application to space research problems. The overall development strategy that we adopted for our magnetosheath modeling development was to begin with the basic GDCF model and then develop a sequence of related computational submodels that would ultimately lead to the full 3-D MHD level. Through the years that we worked together, we were able to accomplish just that and our list of completed computational magnetosheath submodels presently includes: • • • • • • •

Basic GDCF interaction model. Mass loading interaction model. Aligned 8ow MHD level interaction model. 3-D GDCF interaction model. Interplanetary shock impingement model. Terrestrial space weather interaction model. Full 3-D MHD interaction model.

For steady applications, all of the above submodels contain: (1) a coupled two-8ow solver nose=tail combination which we will discuss in detail below, and (2) exact -tted-discontinuity representations for the bow shock and magnetoionopause boundary. Both of these computational enhancements were unique to our computational magnetosheath model development and signi-cantly enhanced both the utility and accuracy of the models. 3.1. Basic GDCF magnetosheath model The basic GDCF model solves a subset of the general 3-D MHD conservation equations based on: • steady 8ow. • predetermined magnetoionopause obstacle. • high Alfvenic Mach number.

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Fig. 2. Illustration of the coupled nose=tail plasma 8ow solver combination incorporated in the GDCF magnetosheath model.

Since for the GDCF model and related submodels development, we were primarily interested in steady 8ows, we chose to enhance all of these models by incorporating two separate plasma 8ow -eld solvers (Spreiter and Stahara, 1980a) as shown in Fig. 2. These were then linked together to determine the total 8ow -eld. In the subsolar nose region, where mixed subsonic=supersonic 8ow occurs, we employed an unsteady, time-marching solver that proceeded to the -nal steady state to determine the solution from the nose region to a selected downstream location, usually the terminator plane, where the downstream axial component of the Mach number was entirely supersonic at that plane from magnetopause to bow shock. From that location, a steady, spatially marching solver was then employed to march the 8ow -eld solution to any arbitrary downstream distance. Because the tail region solver is computationally more than an order of magnitude faster than that for the nose region, we then had the capability to routinely perform distant magnetotail studies. A number of space science investigations have taken advantage of this capability (see for example Slavin et al., 1984). The last assumption of high-Alfvenic Mach number, typical of solar wind 8ows past the planets, results in the magnetic terms in the momentum and energy equations being signi-cantly smaller than the gasdynamic terms. This implies that the solution for the 8uid motion can be decoupled from the magnetic -eld. The simpli-ed equations governing the 8ow -eld then reduce to the familiar gasdynamic Euler equations. The magnetic -eld can then be subsequently determined based upon the previously determined 8ow -eld. It has been shown (Spreiter et al., 1966) that the magnetic -eld determined in this manner obeys the conservation principle that the magnetic 8ux passing through an arbitrary surface moving with the 8uid is conserved or, in more picturesque terms, that the magnetic -eld is convected with the 8uid.

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The decoupling of the B -eld calculation from the 8ow -eld determination is a particularly convenient and valuable feature in the basic GDCF interaction model. The 8ow -eld determination is the most time consuming portion (∼ 90%) of the total computational process for a single case. However, once the 8ow -eld is determined and archived, many diHerent magnetosheath -elds due to diHerent IMFs can be quickly determined as a separate process. Because the B calculation is very rapid, this feature now imbedded in the model has been exploited in a number of investigations. These studies, as well as many others at Earth and other planets as will be described below, have con-rmed the early results that the magnetic -eld calculated in this way i.e., when employing the approximate gasdynamic 8ow -eld, can accurately predict not only the general features but also many details of magnetosheath observations. These magnetic -eld comparisons provide both an extreme test as well as con-rmation of the general accuracy of the entire GDCF modeling concept since the magnetic -eld determined in this manner is the last quantity determined by the model. Consequently, the cumulative inaccuracies inherent in the approximations of the magnetoionopause obstacle and the gasdynamic 8ow -eld are subsequently embedded in the predicted convected magnetic -eld. Nevertheless, we and others continue to be impressed by the agreement that is generally obtained in such comparisons with observations not only at Earth but at all the other planets as well. Fig. 3 from Luhmann et al. (1986) illustrates this point. In the plots on the left of Fig. 3, results are displayed for the vector projections of the three components of the magnetic -eld as observed by Pioneer Venus OMAG on Orbit 438 compared to predictions by the GDCF model. For this orbit, the IMF was exceptionally steady and the agreement between the GDCF model results and observations is remarkably good. The plots on the right of Fig. 3 display corresponding results for Orbit 190. In this case, the IMF underwent a sudden reversal when the PVO spacecraft was near periapsis and inside the ionopause. For the GDCF results presented here, those for the inbound portion of the orbit are based on IMF conditions prior to inbound bow shock crossing while those for the outbound portion of the orbit are based on IMF conditions just after outbound bow shock crossing. Again, comparisons of the GDCF model results with observations for a situation where the IMF undergoes a sudden change are again seen to be quite good. These comparative magnetosheath studies with the GDCF model, and many similar studies, have served to illustrate two key points. First, the magnetosheath plasma and magnetic -eld respond very rapidly to changes in solar wind plasma and IMF conditions with small lag time. We will examine the underlying basis of this further in reviewing results from our interplanetary shock impingement model. This fact implies that use of steady-state magnetosheath model results for comparative observational studies can be quite accurate as long as the appropriate oncoming solar wind conditions are used for driving the model. The second

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Fig. 3. Comparison of the magnetic -eld predictions from the GDCF model with PVO observations at Venus: (a) Orbit 438 when the IMF is exceptionally steady, and (b) Orbit 190 when the IMF changes abruptly while the spacecraft is near periapsis.

point is that the need for a continuous solar wind monitor for these comparative studies is so clearly demonstrated. At Venus, because of the small spatial scale of the interaction region compared, for example, to Earth, comparisons such as those in Fig. 3 can be meaningful since the time lag between the inbound and outbound bow shock crossing is relatively short. However, these studies clearly illustrate the fundamental problem of trying to carry out comparative magnetosheath studies of observations and model predictions using a single spacecraft—that is, solar wind input data needed for the model predictions are unavailable while the spacecraft is in the magnetosheath. The extent of improvement in predicted magnetosheath properties when simultaneous solar wind data are available is discussed and illustrated in detail in our discussion of the terrestrial space weather forecast magnetosheath model below. 3.2. Early applications with the basic GDCF magnetosheath model Throughout the 1980s, a series of collaborative investigations were carried out with the newly enhanced basic GDCF model. We provide a very brief review of some of these since they serve to illustrate the breath and depth of utility provided by results from the GDCF interaction model. In Spreiter and Stahara (1980b) and Mihalov et al. (1982), the model was employed to make predictions of the detailed

plasma and magnetic -eld in the Venusian ionosheath. These were then successfully compared with quiet-time observations of the ionosheath plasma and magnetic -eld obtained from the Pioneer Venus Orbiter plasma analyzer (OPA) and magnetometer (OMAG) instruments. The results suggested that additional phenomena not included in these initial model predictions are occurring throughout the Venusian ionosheath. These included mass loading from photoionization and charge exchange and the possible presence of plasma boundary layer eHects near the ionopause. In Slavin and Holzer (1981) and Slavin et al. (1983), the model was employed to provide the theoretical basis of the most comprehensive study undertaken up to that date of the mean bow shock shape and position of all the terrestrial planets (Mercury, Venus, Earth, Mars). A number of signi-cant -ndings and conclusions, based on results from the model, emerged from this study. These included: (1) the average bow shock position of the solar wind 8ow about Earth is predicted within 2% of its spatial location by the present model, and (2) no East–West asymmetry of Earth’s bow shock as previously suggested by others is discernable within present observational error when appropriate account is taken of the average aberrated direction of the oncoming solar wind. In Slavin et al. (1984), advantage was taken of the distant downstream predictive capability of the model to investigate the far--eld behavior of planetary Mach cones for

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Venus, Earth and Mars. The results veri-ed that the GDCF model, already known to predict good results in the magnetosheath nose regions ahead of and up to the terminator of these planets, could also provide good results to certain downstream distances of these planets; that is, −4 Rob at Venus, −6 Rob at Earth, and −10 Rob at Mars, where Rob denotes the particular obstacle nose radius. The results further suggested that the high-Alfven Mach number approximation that leads to the gasdynamic theory on which the GDCF model is based will most likely be very accurate for describing 8ows about the large bodies (Jupiter, Saturn, Uranus, Neptune) in the outer solar system. This has indeed turned out to be the case as will be shown below in our subsequent studies at these giant planets. Another series of studies focused primarily on the rapid magnetic -eld computational capability of the GDCF interaction model to perform a variety of diHerent investigations about Earth, Venus and Mars. In Luhmann et al. (1984a), the model was employed to investigate patterns of magnetic -eld merging sites on the magnetopause. Predictions of the magnetospheric -eld based on the Hedgecock and Thomas model and predictions of the magnetosheath -eld based on the current model were used to determine the relative orientations between the two -elds at locations in the vicinity of the dayside magnetopause. Areas on the magnetopause with various degrees of antiparallelness between the two -elds for various orientations of the IMF were obtained for the purpose of locating potential -eld merging sites and displayed as contour diagrams. The results suggest that large fractions of the magnetopause surface are suitable for merging for IMFs that are primarily southward or radial in direction. Fig. 4 summarizes these results which are still widely used today in merging site studies. In Luhmann et al. (1984b), a study was made of the magnetospheric source of energetic particles observed upstream of Earth’s bow shock. Calculations were performed in which those magnetosheath -eld lines predicted by the model to drape over the magnetopause were traced from the magnetopause to the bow shock in order to locate regions at the shock that should be populated with magnetospheric particles. Subsets of those -eld lines that connect to potential sites of magnetic merging on the magnetopause were also traced in the event that leakage occurs preferentially where normal components of the -eld are present across the boundary. In Russell et al. (1984), the model was used to investigate the continuing question of whether the Mars-3 spacecraft observation of January 21, 1972 was of a Martian magnetosphere or of a compressed IMF in the magnetosheath. In this study, the gasdynamic model was employed to generate the global 8ow -eld and then the magnetic -eld computational module was repeatedly employed to investigate whether an appropriate IMF could be determined which would produce the time history variation of magnetic -eld that was observed. The model results indicated that a good simulation of the observed magnetic variation could be made purely on the basis of the magnetosheath -eld without invoking an

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entry into a Martian magnetosphere to explain the observations. In Crooker et al. (1985a), the model was employed to study merging site asymmetries on Earth’s dayside magnetopause. Model calculations and comparisons with observations were performed as a function of IMF orientation. Results indicated that asymmetries occur due to the x component of the IMF, and that the asymmetry favors the dawn region when the IMF has a southward component and the dusk region for a northward component in agreement with observations. In Crooker et al. (1985b), magnetic -eld draping against Earth’s dayside magnetopause was studied. IMFs observed upstream of Earth’s magnetosphere by ISEE 3 were used as input to the theoretical model. Predicted results for the magnetic -eld near the magnetopause were then compared with appropriately lagged observations at ISEE 1. In 16 of 24 cases studied, the angle between the transverse component of ◦ the model and observed -elds is less than 20 , a remarkably good result in view of the uncertainty introduced by the large distances between ISEE 1 and ISEE 3. In Slavin et al. (1985), the model was employed to study the bow shock shape and position of the Jupiter and Saturn bow shocks. Based on comparisons with observations from the Voyager 1, 2 and Pioneer 10, 11 spacecraft, the GDCF modeling results indicated that signi-cant nonaxisymmetric distortion of both the Jupiter and Saturn magnetopauses was present. The quantitative resolution of this discrepancy was studied and resolved in Section 3.5. In Luhmann et al. (1985), several possible mechanisms of mass loading of the Venusian magnetosheath were examined. A simple correction was added to the nonmass-loaded model results to simulate the mass loading from planetary ions. The model results provided qualitative agreement with observation. In Luhmann et al. (1986), the model was applied in a quasi-unsteady manner to simulate a temporally varying IMF orientation associated with the passage of rotational discontinuities and MHD waves through planetary magnetosheaths. Model results for several examples of discrete rotational IMF changes at Venus displayed good agreement with observation. Models results of 3-D -eld line topology changes were provided for several diHerent IMF rotations. 3.3. Mass loading magnetosheath interaction model Initial applications of the GDCF interaction model to study the detailed plasma and -eld properties in the Venusian ionosheath (Spreiter and Stahara, 1980a, b; Slavin and Holzer, 1981; Mihalov et al., 1982) consistently predicted bow shock locations closer to the planet than were indicated by observations. Moreover, observational particle data at Venus had established the presence of atmospheric ions, primarily O+, in the Venusian ionosheath. The presence of these additional ions in the ionosheath, not included in the

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Fig. 4. Contours on the magnetopause of equal values of the cosine of the angle between the magnetospheric and the GDCF magnetosheath magnetic -eld for various values (Bx ; By ; Bz ) of the IMF as viewed from the sun.

initial theoretical predictions, globally modi-es the position of the bow shock, with the net eHect being the displacement of the bow shock further outward from the planet as more ions are introduced into the ionosheath. In order to investigate this phenomena in a quantitative manner, we extended the basic GDCF interaction model to include eHects of mass loading of the solar wind plasma 8ow past Venus or a comet (Stahara et al., 1987). This required development of new computational algorithms to handle the high gradient regions known to occur in the vicinity of the planetary or cometary obstacle surfaces as a result of the mass loading. Fig. 5 displays results from the mass loaded model for the location of the Venusian bow shock when mass loading eHects due to photoionization of hot oxygen, believed to be the major contributor, are included in the

solar wind interaction. Shown in that plot are a comparison of results from our model and that of Belotserkovskii et al. (1987) for the bow shock displacement due to hot oxygen mass loading when using the values of hot oxygen number density and scale height suggested by Nagy et al. (1986). The result from our model suggests almost no displacement of bow shock location in contrast to that of Belotserkovskii et al. (1987) which shows substantial movement. Other modeling results using a hybrid model (Moore et al., 1991) and a two-dimensional full MHD model (McGary and Pontius Jr, 1994) have found agreement with our modeling results when using the same hot oxygen parameters. We investigated this discrepancy between the two modeling results further, using numerical information for both mass loaded and zero-mass loaded cases provided by the authors of Belotserkovskii

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Fig. 5. Comparison of results for Venusian bow location from the mass-loaded GDCF model with those from Belotserkovskii et al. (1987) for photoionization of hot oxygen.

et al. (1987). We employed three completely diHerent computational codes to study this case and in fact incorporated the identical implementation of the no-8ow boundary condition at the ionopause boundary used by Belotserkovskii et al. (1987). That is, they incorporated phantom points inside the ionopause in their model and employed a re8ective condition at those points in order to zero out 8uxes at the ionopause boundary. The results of all of these calculations, using both the diHerent numerical codes and the two diHerent boundary condition implementations, showed no change and were identical to our original mass loaded calculation. We conclude that there is an error somewhere in their mass loaded code. We further conclude that the outward displacement of the Venusian bow shock observed at solar maximum due to planetary mass loading eHects cannot be accounted for by the simple hot oxygen photoionization model alone with constants suggested by Nagy et al. (1986). Additional eHects need to be included. The numerical simulations indicate that the mass loading dramatically alters the region where ion pickup is large, i.e. near the ionopause boundary. In the present simulations, representing the boundary between the mangetosheath and ionosphere as a hard impenetrable surface may be too great a simpli-cation for the real nature of this transition region. More recent results by Bauske et al. (1998) employing a global MHD model that simultaneously encompasses both the magnetosheath and

ionosphere have displayed encouraging agreement with observations for the Venusian bow shock movement between solar min and max. 3.4. Aligned 1ow MHD level magnetosheath interaction model One of the -rst computational magnetosheath submodels that we developed to provide an extension beyond the basic GDCF model was the aligned 8ow full MHD magnetosheath model. This submodel was an important step for two primary reasons: -rst, it extended the accuracy level of the magnetosheath simulation beyond the gasdynamic-convected magnetic -eld level that was employed in the other interaction submodels developed up to that point; second, the model was numerically implemented on an extremely accurate, re-ned and robust computational basis. In fact, the aligned 8ow computational submodel, applied at the high-Alfven Mach number limit, has been our computational model of choice to determine the gasdynamic 8ow -elds required for GDCF magnetosheath model applications. As the name implies, this submodel is capable of predicting the interaction of the solar wind with planetary obstacles at the full MHD accuracy level for situations when the IMF is aligned with the oncoming solar wind 8ow

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Fig. 6. Bow shock locations predicted by the aligned 8ow MHD magnetosheath model for a series of aligned 8ows past a Venusian ionopause obstacle for a sonic Mach number of 4.5, ratio of speci-c heats 5=3, and Alfven Mach numbers from 2.5 to 20.

direction. The submodel is appropriate for determining supersonic, super-Alfvenic solar wind 8ows past general axisymmetric magnetoionosphere obstacle shapes. As with the basic GDCF model, the aligned 8ow model also employs two separate but coupled 8ow -eld solvers to determine the complete steady 8ow -eld: an asymptotic time-marching procedure for the subsolar to terminator region, and a spatially marching procedure for the postterminator downstream region. Both of the computational procedures implemented in the aligned 8ow model are based on fully implicit numerical algorithms, both employ -tted discontinuity representations for the bow shock and magnetoionospheric obstacle surfaces, and both implement the boundary conditions at bow shock and obstacle surface in a fully implicit manner. Applications of the model have been successfully made to both Earth and Venus. At Earth, the new model predicts the identical variation of the bow shock shape as a function of Alfven Mach number in the oncoming solar wind that was previously obtained by Spreiter and Rizzi (1974) using a completely diHerent numerical method. In Fig. 6 we show results for application of the model to predict bow shock locations for a series of aligned MHD 8ows past a Venusian ionopause obstacle. Results are presented for a sonic Mach number of 4.5, ratio of speci-c heats of 5=3, and

various Alfven Mach numbers ranging from 2.5 to 20.0. These results display the identical variation of bow shock shape with Alfven Mach number observed by Spreiter and Rizzi (1974) for their application to Earth: that is, as the oncoming Alfven Mach number decreases, the bow shock simultaneously moves inward toward the magnetopause in the nose region and 8ares outward away from the magnetopause along its 8anks. This parametric behavior of the shock bow has also been veri-ed through application of our full 3-D MHD model described in Section 3.9. Moreover, the results show that MHD eHects are signi-cant for Alfven Mach numbers less than about 5 when M∞ =4:5 and =5=3. They also show that the MHD solutions approach the GDCF solution smoothly as Alfven Mach number increases. More recent studies (Spreiter and Stahara, 1994) with the new model have also demonstrated the parametric importance of the magnetosonic Mach number for revealing MHD 8ow similitude and characterizing MHD eHects in planetary bow shocks. These results show that MHD solutions organize better when expressed in terms of magnetosonic Mach number and either sonic or Alfven Mach number rather than when expressed in terms of sonic and Alfven Mach number. They also suggest that for the case of a low-Alfven Mach number that use of the magnetosonic Mach number may be a preferable alternative than simply using the sonic number in analyzing these 8ows with the GDCF model, a fact previously noted by us and others (Russell, 1985). In fact, as shown in the -nal case study presented in the space weather forecast model described below, we have employed this suggestion in a variety of low-Alfven Mach number applications with good success. However, use of this approximation is a matter that deserves further examination to understand the underlying implications and limitations before general application and acceptance. 3.5. 3-D GDCF magnetosheath model As a result of studies undertaken with the new GDCF model, Slavin et al. (1985) had showed that the low-latitude bow shocks of Jupiter and Saturn are observed to be significantly closer to their respective magnetopauses than predicted by GDCF modeling results, as shown in Fig. 7. Those modeling results were based on axisymmetric magnetopause obstacles for Jupiter and Saturn obtained by rotating the observed magnetopause pro-les determined from spacecraft observations of the low-latitude magnetopause crossings at these planets. The cause of the discrepancy between prediction and observation was suspected to be due to signi-cant equatorial 8attening of the magnetopause cross sections at these planets due to the eHects of rapid spin, large size, and substantial ring currents. These eHects were believed to cause the planetary magnetospheres of Jupiter and Saturn to be signi-cantly broader near the equatorial plane than near the noon–midnight polar meridian plane, as illustrated in the bottom cartoon of Fig. 7.

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Fig. 7. Discrepancy in bow shock locations at Jupiter and Saturn between observations and GDCF model results indicative of polar 8attening of the magnetopause cross sectional shapes at these planets.

In order to try to quantitatively account for this 8attening of the Jupiter and Saturn magnetopauses, we were led to develop in the late 1980s a signi-cant extension of the basic GDCF model. As described in Stahara et al. (1989), this was the development of a 3-D gasdynamic model capable of accounting for general three-dimensional magnetoionopause obstacle shapes. That model was then applied to determine

the degree of 8attening of the Jupiter and Saturn magnetopauses in the following manner. First, the magnetopause equatorial trace of the planet was -xed from observations. The magnetopause cross section was then assumed to be elliptical with major=minor axis ratio a=b. Next, the 3-D GD model was successively employed to predict the bow shock location for a sequence of a=b ratios. From these results, the

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ratio a=b was then determined that best -t the observational bow shock location in the equatorial plane. Fig. 8 displays the results of these calculations for Jupiter and Saturn. The plot on the left provides comparison of the 3-D GD model predictions with observation for the Jovian bow shock locations in the equatorial plane. The corresponding result for Saturn is provided in the plot on the right. For Jupiter, the predicted results for a=b = 1:75 are in almost perfect agreement with the observational result all the way from the nose region to almost the terminator, and are in agreement with the independent result of Engle and Beard (1980) in which the three-dimensional shape of the Jovian magnetosphere was calculated for a planetary dipole -eld supplemented by an equatorial current sheet chosen to reproduce the observed Jovian magnetic -eld near the equatorial plane. The corresponding 3-D GD model results for Saturn for a=b = 1:25 also display good agreement with observations in the nose region. However, near the Saturnian terminator the least-squares -tted conic curve (Slavin et al., 1985) used to represent the observational shape of the bow shock 8ares out considerably more than indicated by any of the model predictions, including that for an axisymmetric magnetopause. The inescapable conclusion is that for Saturn the available observational data for bow shock crossings are too few in number and too scattered to provide a well-de-ned representation of the shape of the bow shock in the terminator region. 3.6. Application of the GDCF model to Neptune One of the planetary applications of the GDCF model that John was most impressed with was the application to Neptune. Notwithstanding all the successful modeling applications made to Earth and Venus, of which he was rightfully proud, and the conjecture that the basic assumptions underlying the GDCF model of high-Alfven and sonic Mach numbers would be even more accurate at the giant planets, it seemed to us that the various planetary applications that we were studying were like children from a family—the one constant about them that always seemed to hold true and surprise was that they are all diHerent! This has been so true of each of the planetary magnetosheath interactions that we have applied modeling predictions to throughout the solar ◦ system. Neptune was no exception. The 29 tilt of Neptune’s ◦ spin axis with respect to its orbital plane and the 47 oHset of the magnetic -elds and spin axis (Ness et al., 1989) combine to cause a shift from a nearly pole-on (magnetic pole facing the solar wind) magnetosphere to an Earth-like con-guration (dipole axis perpendicular to the solar wind velocity) every half rotation—a magnetosphere con-guration that is unique in the solar system (Richardson et al., 1994). Application of the GDCF model was undertaken to compare with the plasma and magnetic -eld data obtained by the Voyager 2 spacecraft. Fig. 9 displays a comparison of the results from the GDCF model and observations for the BT

magnetic -eld component through the magnetosheath. The cartoon on the upper part of Fig. 9 displays the trajectory of the Voyager 2 spacecraft in a meridional plane together with models of the bow shock and magnetopause positions. The inbound magnetosheath crossing from bow shock to magnetopause took approximately 4 h, while the outbound crossing from magnetopause to bow shock required a little more than 42 h. Input data for the inbound model predictions are solar wind observations just prior to inbound bow shock crossing, while for the comparisons on the outbound leg, solar wind data were used subsequent to the -rst outbound bow shock crossing. The plot on the left of Fig. 9 displays the comparisons of observations and model predictions for the BT magnetic -eld component on the inbound portion of the spacecraft trajectory, while the corresponding comparisons for the outbound segment of the trajectory are shown in the plot on the right. For the inbound segment, the GDCF model results displayed for the BT magnetic -eld component, as well as the (BR ; Bn ) components not shown here, are quite reasonable from bow shock crossing up until about 1 h prior to magnetopause crossing where some discrepancies are observed to occur. As pointed out by Richardson et al. (1994), the Neptunian inbound magnetopause was analyzed in depth by Szabo et al. (1991) and Lepping et al. (1992). They both found that the magnetopause is a rotational or rotational-like discontinuity at the high magnetic altitude of the Voyager 2 crossing, implying the possibility of plasma 8ow from the sheath to the upper cusp or mantle region. The discrepancy between model results and observations near the magnetopause are consistent with this scenario. For the outbound segment, results for the BT component displayed here, as well as the BR component not shown, are reasonably well predicted across the entire magnetosheath. However, the radial component BR was noted to exhibit a discrepancy near the magnetopause. This discrepancy was studied by Richardson et al. (1994) in detail and their conclusion was that the solar wind IMF most likely changed while the spacecraft was in the magnetosheath. A brief study to investigate this premise was carried out in which the normal component of the IMF was reversed in sign, and then the model results compared with observations. The result was that all three components of the predicted magnetic -eld in the magnetosheath displayed good agreement with observations within 30 min after crossing the magnetopause, but then began to deviate from the observations. As our results from the terrestrial space weather magnetosheath model will show, the occurrence of such discrepancies in comparative magnetosheath studies when simultaneous solar wind are unavailable are not only typical but unavoidable. Finally, I would like to note a remarkable fact of these comparisons. The solar wind proton number density outside the Neptunian bow shock when these comparisons were made was approximately 5 protons=l3 ! John was amazed at this since it was, and will undoubtedly remain for some time, the smallest solar wind density for which the GDCF model predictions have been compared with data.

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Fig. 8. Comparison of results from the 3-D GD magnetosheath model with observations for the bow shock locations at Jupiter and Saturn.

Fig. 9. Application of the GDCF magnetosheath model to Neptune: (a) Voyager 2 spacecraft trajectory through the Neptunian magnetosheath, and (b) comparison of the GDCF model results for the BT magnetic -eld component compared with Voyager 2 observations.

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Fig. 10. Results from the interplanetary shock impingement magnetosheath model for the bow shock location and plasma pressure contours in the terrestrial magnetosheath during passage of an interplanetary shock.

3.7. Interplanetary shock impingement model All of the magnetosheath interaction models discussed previously provide steady-state solutions for time-invariant solar wind 8ow past a planet as viewed from a coordinate system -xed to the planet. However, their use need not be con-ned to absolutely steady conditions. In many instances, conditions in the incident solar wind change suEciently slowly compared with the relatively short response time of conditions in the magnetosheath that the 8ow may be considered to be quasi-steady and approximated satisfactorily as a succession of steady-state solutions for the instantaneous or suitably time-lagged conditions in the solar wind. In other instances, such as shown previously in Fig. 3 from Luhmann et al. (1986) where the change in solar wind conditions occurs more rapidly, useful results can still be obtained by simply joining portions of two separate steady-state solutions at the appropriate location. At certain times, however, conditions in the solar wind change too rapidly to allow use of a quasi-steady approximation. The most extreme example of such an event for magnetosheath studies is provided by the passage of an interplanetary shock wave past an established bow shock magnetopause con-guration. In order to gain insight into the nature of such an interaction, and the rapidity with which the location of the bow shock and the conditions in the magne-

tosheath change from the preshock to postshock states, we have extended the basic GDCF model (Spreiter and Stahara, 1994) to provide time-accurate solutions for such an interaction. The model has been developed to apply initially to a rigid axisymmetric magnetoionopause obstacle about which a steady-state bow shock con-guration has been previously established and that is subsequently impacted by a planar interplanetary shock wave traveling normal to the solar wind velocity vector. As such, the results should be considered more indicative of the general nature of such an interaction and, in particular, of the rapidity of the adjustment of the changes produced by the impact of the interplanetary shock wave rather than as a sample output of a more general model capable of simultaneously accounting for both the deformation of the obstacle surface as well as for the range of possible oblique shock impingement events that can actually occur. Fig. 10 displays results from the interplanetary shock impingement model. Snapshot views are presented of the calculated results for the interacting shocks together with plasma pressure contours at four sequential times as a planar interplanetary shock passes a terrestrial magnetopause obstacle. The results show that the trace of the interplanetary shock in the magnetopause remains nearly planar as it moves through the magnetosheath and that the extent of the transition 8ow region is relatively short. That is, the 8ow in the magne-

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Fig. 11. Illustration of the temporal solar wind dynamics embodied in the terrestrial space weather magnetosheath interaction model.

tosheath adjusts very quickly, i.e. in about 1–2 min if the interplanetary shock is moving at 500 km=s, to the change in conditions following passage of the interplanetary shock. These results are illuminating in the sense that passage of an interplanetary shock through the magnetosheath represents the most rapid and severe change in oncoming solar wind conditions; and even for this event the lag time for the magnetosheath to adjust to the new conditions is quite rapid and relatively short. This result implies that, for the relatively smooth and continuous changes that are constantly ongoing in solar wind plasma and IMF properties, the magnetosheath can be expected to adjust even more rapidly. This conclusion supports use of steady-state magnetosheath model solutions to provide meaningful results when used in a quasi-unsteady mode. Observational evidence supporting this conclusion is shown in detail from comparative testing of our terrestrial space weather forecasting magnetosheath model discussed in the section below. 3.8. Terrestrial space weather magnetosheath interaction model Development of the terrestrial space weather magnetosheath interaction model was perhaps the most in-depth and revealing application of the GDCF magnetosheath model. The space weather model was developed to be one segment of a layered sequence of coupled forecasting models planned for operational use by the Space Forecast Center. These models would ultimately provide a forecasting capability of the plasma and -eld environment from the sun through interplanetary space and down to low Earth orbit. The total space weather forecasting system would be comprised of both satellite and ground-based observational elements linked with the integrated forecasting model to provide real-time forecasts of space weather throughout the terrestrial geospace.

Within its de-ned operational domain, the space weather forecast magnetosheath model was speci-cally developed for routine real-time forecasting of the plasma and -eld environment associated with the transport of the solar wind from the L1 Lagrangian location of an upstream monitor up to and through the terrestrial bow shock, throughout the 3-D volume of the magnetosheath region, down to and across the magnetopause boundary, and into the outer region of the magnetosphere. To be eHective, the entire forecasting process must proceed substantially faster than natural events occur. Our goal was to develop an operational forecasting capability approximately 100 times faster than the actual events within the operational domain of the model. This was the fundamental constraint placed on the development of our model, and required operational times for predictions to be made in less than a minute. This would then provide overall warning times from approximately 20 to 50 min, depending upon oncoming solar wind conditions. By clever use of archived libraries of precomputed solutions together with eEcient search and interpolation procedures, we were able to achieve this in the model. A very brief overview is given here of the theoretical and computational aspects of the forecast model, which is described in detail in Stahara et al. (1993). Fig. 11 illustrates how the temporal dynamics of the solar wind, bow shock and magnetopause are embodied in the model. The temporal dynamics representing the constantly changing solar wind plasma and -eld properties are approximated in the forecast model by a sequence of closely spaced steady states. As shown in the cartoon of Fig. 11, the temporal changes in the solar wind are anticipated to be observed by a spacecraft in the vicinity of L1 at time intervals of approximately 1–2 min. By using the measured solar wind velocity vector, the plane is determined that contains the spacecraft and is normal to the measured solar wind direction. The observed solar wind properties are assumed to be constant in that normal plane and are then propagated along the solar wind direction to

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Fig. 12. Comparison of space weather model forecasts and ISEE 2 observations for Orbit 178 inbound for velocity and magnetic -eld components.

Earth. The time delay for propagation from the monitor to Earth is established from the measured solar wind velocity and the computed distance to the normal plane. Hence, the constantly changing solar wind properties are approximated by 1–2 min thick slabs of plasma propagating along ever-changing solar wind directions and speeds as they move toward Earth. For a particular plasma slab at the propagation time when it reaches Earth, those measured solar wind properties are then used to establish the complete GDCF solution throughout the magnetosheath, including the updated positions of the magnetopause and bow shock. It is interesting to note what this solution methodology actually means in terms of the constantly changing magnetopause and bow shock lo-

cations. Because of dynamic pressure changes between each of the solar wind plasma slabs, the magnetopause and associated bow shocks will be seen as dynamically moving in and out in response to these changes. Similarly, changes in direction of the solar wind velocity vector will cause the entire magnetopause=bow shock con-guration to respond in a manner similar to a wind sock as it moves to constantly realign itself with the new solar wind direction. How important these changes can be, and they are in fact -rst order, are illustrated below in the initial comparative case studies that we carried out with the space weather forecast model. In Fig. 12, we provide results of one of the -rst comparisons of the space weather model forecasts and ISEE 2

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Fig. 13. Detail of the space weather model prediction for the bow shock crossing at 14:0 UT: comparison of the By magnetic -eld component with ISEE 2 orbit 178 observations.

observations for orbit 178 inbound. This data set was selected as one of the candidates for the initial validation study since while this ISEE 2 data set contains solar wind conditions that were considered mostly favorable for good model forecasts, it also contains some solar wind conditions believed unfavorable that would subsequently serve as a further test of the forecast model. In addition, for this case the solar wind input data for the model were provided by the IMP 8 spacecraft which was upstream only minutes away in solar wind propagation time from Earth. This is in contrast to all of the other cases considered in this initial study which employ ISEE 3 observations taken near L1, and which subsequently involve almost an hour of solar wind propagation time to reach Earth. For this particular case, the solar wind conditions are: high magnetosonic Mach number (∼ 8:0), Bz northward, moderately unsteady solar wind plasma conditions, moderately unsteady IMF, with multiple bow shock crossings observed by ISEE 2. The plot on the upper left of Fig. 12 provides a view of the ISEE 2 spacecraft orbital trajectory as seen in the solar wind coordinate system -xed to the magnetopause that is employed in the magnetosheath forecast model computations. Included in the plot is the nominal location of the bow shock during the interval from 10.0 to 18:0 UT. Such a view of the spacecraft trajectory is quite informative as it immediately displays the magnitude of the temporal eHects caused by unsteadiness in the solar wind dynamic pressure and solar wind direction. These are revealed, respectively, in this view of the spacecraft trajectory by: (1) abrupt radial jumps in the spacecraft trajectory corresponding to the sudden in8ation or de8ation of the magnetopause obstacle size in response to solar wind dynamic pressure changes, and (2) abrupt lateral excursions of the spacecraft trajec-

tory corresponding to sudden changes in the solar wind direction. These changes will be displayed to a much greater degree in the subsequent case study shown here. For this case, however, we observe from the ISEE 2 trajectory plot an interesting phenomenon near the bow shock. Because of in8ation of the magnetopause due to a solar wind dynamic pressure reduction at the time when the ISEE 2 spacecraft was nearing the bow shock on its inbound orbital segment, the spacecraft is seen to appear to essentially loiter about the bow shock. This results in the multiple bow shock crossings that are subsequently observed, for example, in the axial Vx plasma velocity component comparison shown in the plot on the upper right of Fig. 12. The IMP 8 data gap, extending from about 10.0 to 12:0 UT during which period the ISEE 2 spacecraft initially crossed the bow shock, undoubtedly causes the discrepancy for the -rst model forecasted bow shock crossing. However, the prediction of the later multiple bow shock crossings by the forecast model and the magnitude of the in-plane velocity components throughout the magnetosheath are in excellent agreement with the ISEE 2 fast plasma analyzer observations. The plots in the bottom of Fig. 12 which display results for the (By ; Bz ) components con-rm this high level of agreement for the magnetic -eld as well. In addition, they show that even a diEcult detail, such as the multiple bow shock crossing at about 14:0 UT, is remarkably well captured by the magnetosheath model magnetic -eld forecasts. Just how well that detail is captured by the forecast model is shown in Fig. 13, which displays an expanded plot of the By magnetic -eld component centered about 14:0 UT. The ability of the model to predict the precise occurrence and the exact magnitude of the rise and fall of the shock-induced variation is remarkable. For this case study recall that the solar wind input data for the model

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Fig. 14. Comparison of space weather model forecasts and ISEE 2 observations for Orbit 331 inbound for velocity and magnetic -eld components.

is provided just a few minutes upstream of the bow shock. With this input information, the forecast model is able to predict plasma velocity components and the three components of the magnetic -eld with exceptional accuracy, even though there exists moderate unsteadiness in both the oncoming solar wind plasma 8ow as well as the IMF. Results for the -nal case study presented here are for ISEE 2 orbit 331 inbound and are displayed in Fig. 14. For this case, the magnetosheath observations are for solar wind conditions involving very low magnetosonic Mach number (∼ 2:9). This dataset was among several that were purposely selected to provide a further challenge to the forecast model, since the fundamental assumptions under which

the basic GDCF computational model used to determine the global magnetosheath plasma and magnetic -eld properties employed in the forecast model become highly strained at low magnetosonic Mach numbers. With the use of the solar wind magnetosonic rather than sonic Mach number in the global gasdynamic plasma calculation, however, we have found that the forecasted results for both plasma velocity and magnetic -eld properties for all of these low magnetosonic Mach number case studies were found to be in quite good agreement with the ISEE 2 observations. This result was not anticipated, and warrants further study. We further note that at these low magnetosonic Mach numbers, the bow shock is substantially displaced outward from its usual

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Fig. 15. Results from the 3-D MHD model for the variation of bow shock location with IMF cone angle for M∞ = 6, = 5=3, MA∞ = 10 in the magnetic plane of symmetry and in the plane perpendicular to the magnetic plane of symmetry.

location and is quite sensitive to small changes in the solar wind magnetosonic Mach number. In view of this sensitivity, in the ISEE 2 trajectory plot displayed in the upper left of Fig. 14 we have displayed the range of predicted bow shock locations for solar wind conditions occurring from 4:20 to 8:00 UT. As with the previous case study shown in Fig. 12, the ISEE 2 spacecraft is forecasted to encounter multiple bow shock crossings. The ISEE 2 observations display some, but not all, of these predicted crossings. This result is not unexpected, since the ISEE 3 solar wind input data employed in these comparisons have a propagation time of almost an hour, in contrast to the IMP 8 solar wind data taken just minutes away that were used for the results in Fig. 12. For the comparisons of the axial Vx plasma velocity component shown in the plot in the upper right of Fig. 14, we note that the forecasts display good agreement with the observations, with the only signi-cant exception being the model forecast of a bow shock crossing at about 7:30 UT which does not appear in the observations. The corresponding results for the magnetic -eld components shown in the plots on the bottom of Fig. 14 also display very good agreement, essentially capturing all of the variations exhibited by the data, in par-

ticular, for the Bz component which displays a signi-cant variation across the magnetosheath. The space weather magnetosheath forecast model has been able to achieve in a quantitative way the validation of a number of important aspects of John Spreiter’s ideas of magnetosheath modeling. It clearly demonstrates the modeling accuracy that can be achieved when simultaneous upstream solar wind data are available for input to drive the magnetosheath model predictions. As we noted in the comparisons above, not only are the overall aspects of the magnetosheath interaction well predicted, but so many of the details are captured as well. As a result of the comparative case studies it is fair to say: (1) that the forecast model is able to provide a remarkable quality of forecasts for both steady and moderately unsteady solar wind conditions, (2) that a major portion of observed variations in the magnetosheath are in direct response to incident solar wind changes with very small relaxation time, and (3) that previous unsteady wave and turbulence interpretations of magnetosheath observations may require reexamination in light of the strong correlation with temporal solar wind variations. Finally, we note that the space weather magne-

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Fig. 16. Results from the 3-D MHD model for the variation of bow shock location with IMF cone angle for M∞ = 6, = 5=3, MA∞ = 5 in the magnetic plane of symmetry and in the plane perpendicular to the magnetic plane of symmetry.

Fig. 17. Prediction of plasma deplection layer eHects: comparison of GD and 3-D MHD results for plasma density along the stagnation streamline for various Alfven Mach numbers for perpendicular IMF cone angle.

tosheath forecast model has also been successfully used in collaborative studies with others. Song et al. (1999) have employed the model in an important detailed study of wave

processes occurring in the terrestrial magnetosheath, and have carefully tested the forecast model in a series of case studies.

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3.9. Full 3-D MHD magnetosheath interaction model Successful development of the full 3-D MHD magnetosheath interaction model represented the achievement of a major goal in our long-term magnetosheath model development plan. One of the features incorporated in the 3-D MHD numerical model is use of the Powell correction (see for example, Bauske et al. (1998)) for maintaining a divergence-free magnetic -eld, and which we have found to be very eHective. We will not address the numerical aspects of the 3-D MHD model here, but instead will provide some preliminary summary computational results from the model and comparative testing with the corresponding GDCF results. All of the 3-D MHD results shown below are for a model axisymmetric terrestrial magnetopause as used in our previous GDCF work (see for example, Spreiter and Stahara, 1985). As with our previous models, -tted-discontinuity surfaces are used at the magnetopause obstacle, where zero normal 8ow and zero normal B -eld are enforced, and at the bow shock where the unsteady 3-D MHD shock conditions are imposed. Fig. 15 displays results from the 3-D MHD model for the variation of bow shock location with IMF cone angle for a sonic Mach number of 6.0, ratio of speci-c heats of 5=3, and Alfven Mach number of 10.0 for three diHerent IMF cone ◦ ◦ angles, i.e. aligned 8ow (0 ), Parker spiral angle (45 ), and ◦ perpendicular (90 ). The bow shocks for these three cases are shown in the magnetic plane of symmetry in the plot of the left, and in the plane perpendicular to the magnetic plane of symmetry in the plot of the left. We note for this high-Alfven Mach number situation, that there is a small but not signi-cant diHerence between the three results as we would expect. Also, we note that the gasdynamic result not shown on these plots is indistinguishable from the aligned 8ow result. However, when the Alfven Mach number is lowered to 5.0, the corresponding results displayed in Fig. 16 exhibit a very notable diHerence. For reference, we note that there is little diHerence between the aligned 8ow result in this case and that shown in Fig. 15. However, for both the Parker spiral angle and perpendicular IMF case, the bow shock displacements are remarkably large. For example, at the nose the diHerence in bow shock displacement between the aligned 8ow and perpendicular result is almost equal to the standoH distance of the aligned 8ow bow shock from the magnetopause nose. This displacement expands enormously when the Alfven Mach number is lowered to 3.0. In that result not shown here, the displacement diHerence between the aligned 8ow and perpendicular bow shock at the nose is twice the standoH distance of the aligned 8ow bow shock from the magnetopause. These diHerences are huge, and clearly demonstrate that at low-Alfven Mach numbers the bow shock structure is dominated by the IMF cone angle. Another key feature of strong MHD eHects in the magnetosheath that the 3-D MHD model can examine in detail is the plasma depletion layer expected near the magnetopause

Fig. 18. Prediction from the 3-D MHD model of the separate contributions of plasma and magnetic pressure to total pressure along the magnetopause surface for various Alfven Mach numbers for perpendicular IMF cone angle.

surface. Fig. 17 displays a comparison of gasdynamic and 3-D MHD results for the density variation along the stagnation streamline from the bow shock to the magnetopause nose for the case of perpendicular solar wind magnetic -eld. Results are shown from the 3-D MHD model for Alfven Mach numbers of {10:0; 5:0; 3:0} together with the gasdynamic result (in-nite Alfven Mach number). The plots have been normalized with the shock standoH distance in each case. The gasdynamic result displays a smooth monotonically increasing variation from bow shock to magnetopause nose. The corresponding 3-D MHD results for the three different Alfven Mach numbers closely track the gasdynamic result for about a third of the standoH distance from the bow shock, but depart signi-cantly from the gasdynamic result and show a strong plasma depletion layer eHect as the

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Fig. 19. Comparison of the variation of Bz magnetic -eld component along the stagnation streamline from the GDCF model and the 3-D MHD model for various Alfven Mach numbers for perpendicular IMF cone angle.

magnetopause nose is approached. Interestingly enough, all three of the 3-D MHD results go to about the same value at the magnetopause nose, i.e. about half of the postshock value. In results not shown here, analogous predictions when the oncoming IMF is at the Parker spiral angle display plasma depletion layer results remarkably similar to the perpendicular case shown in Fig. 17. However, for the aligned 8ow case, the corresponding results display essentially no plasma depletion layer eHects, with all three 3-D MHD results virtually in correspondence with the gasdynamic result. Such results clearly illustrate how exceptionally diHerent from neighboring IMF cone angle solutions that the aligned 8ow solution topology is at low-Alfven Mach number. In Fig. 18, we display a prediction from the 3-D MHD model of the separate contributions of plasma and magnetic pressure to total pressure along the magnetopause surface from the nose to the terminator. Results are shown for the ◦ IMF cone angle of 90 for various Alfven Mach numbers. The upper plot in Fig. 18 is for an Alfven Mach number of 10.0, and shows that the magnetic pressure dominates the plasma pressure at the magnetopause surface by a factor of 3. The corresponding results for Alfven Mach numbers of 5.0 and 3.0, shown in the two lower plots, display an even greater dominance of the magnetic pressure. The -nal 3-D MHD result shown in Fig. 19 addresses a well-known characteristic behavior of the convected magnetic -eld in the GDCF model, namely for the two transverse components of the convected magnetic -eld, the magnetopause surface is a singularity. This behavior of the convected -eld has been recognized from the earliest development of the GDCF model (Spreiter et al., 1966). Numerically, the singularity is weak, varying approximately ∼ 1=n where n is the local normal distance from a point near the magnetopause surface to the surface. Fig. 19 provides results displaying this behavior for perpendicular IMF and compares results for the variation of the Bz magnetic -eld

component along the stagnation streamline from the bow shock to the MP nose as predicted from the GDCF model with those from the 3-D MHD model for three diHerent Alfven Mach numbers. The results from the 3-D MHD model agree well with the GDCF result for about one third of the total standoH distance from the bow shock but, as expected, the GDCF result departs signi-cantly starting at about two-tenths of the standoH distance from the nose. In implementing the GDCF model predictions for the magnetic -eld, rather than permitting the magnetic -eld to grow without bound as the magnetopause is approached, we have used a simple but eHective limiting criteria to bound the magnitude of the transverse convected -eld components. As shown in the formula on the top of Fig. 20, we have applied a cutoH limit on the magnetic -eld predicted by the GDCF model, whereby as the magnetopause surface is approached, the magnitude of the local convected -eld is limited to the local stagnation pressure on the magnetopause surface. The eHect that this has on the convected -eld result near the MP surface is illustrated in the three plots shown in Fig. 20. These plots display a comparison of the GDCF result with cutoH compared to the 3-D MHD result for the three Alfven Mach numbers of {10:0; 5:0; 3:0}. The cutoH criteria depends on Alfven Mach number, and is diHerent for each case. As can be seen, the resulting GDCF result is now in reasonable agreement with the exact 3-D MHD result. This cutoH criteria has been implemented in the space weather forecast model, and is undoubtedly responsible to a signi-cant degree for the good agreement of magnetic -eld forecast predictions in the vicinity of the magnetopause surface that are observed in the comparative case studies displayed in Figs. 12–14. We note that detailed testing and veri-cation of the full 3-D MHD magnetosheath model remains to be completed. The initial case studies with the new 3-D MHD model were constrained by availability of supercomputer time, and

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planetary applications for which the various magnetosheath models have now been successfully applied is remarkable. If John were asked to select which of the model developments that he was most proud of and impressed by, I believe he would have chosen those associated with the space weather forecasting model and the 3-D MHD model. The former because of the remarkable insights that it provides to terrestrial magnetosheath physical phenomena, and the latter because of its ability to provide quantitative con-rmation of the basis underlying the accuracy of the GDCF modeling concepts. We are fortunate indeed to be recipients of this legacy of his ideas. Acknowledgements The author wishes to express his thanks to Prof. C.T. Russell of the University of California in Los Angeles for the invitation and encouragement to provide this review of John Spreiter’s unique contributions to magnetosheath modeling. Financial support for the work was provided by the IR& D program at RMA Aerospace, Inc.

References

Fig. 20. Comparison of the variation of Bz magnetic -eld component along the stagnation streamline from the GDCF model with cutoH and the 3-D MHD model for various Alfven Mach numbers for perpendicular IMF cone angle.

involved only a limited number of solutions. However, with the full 3-D MHD magnetosheath interaction model now in hand, a number of interesting opportunities for magnetosheath studies now exist. For example, it would be a straightforward task to proceed through a systematic sequence of 3-D MHD solutions parameterized by sonic Mach number, Alfven Mach number, and IMF cone angle, archive these results into a computational library, and then develop a correction procedure that would rapidly proceed through the catalogued library to correct any GDCF solution result to be a very close approximation of the full 3-D MHD level solution at a fraction of the computational cost of determining that solution from scratch. 4. Concluding remarks The studies described here have amply con-rmed the validity and accuracy of Spreiter’s approach to magnetosheath modeling and have provided fundamental insights into the physics of planetary magnetoionopause boundaries, bow shock, and magnetosheath regions. The spectrum of the

Bauske, R., Nagy, A.F., Gombosi, T.I., De Zeeuw, D.L., Powell, K.G., Luhmann, J.G., 1998. A three-dimensional MHD study of solar wind mass loading processes at Venus: eHects of photoionization, electron impact ionization, and charge exchange. J. Geophys. Res. 103, 625– 638. Belotserkovskii, O.M., Breus, T.K., Krymskii, A.M., Ya Mitnitskii, V., Nagy, A.F., Gombosi, T.I., 1987. The eHect of hot oxygen corona on the interaction of the solar wind with Venus. Geophys. Res. Lett. 14, 503–506. Crooker, N.U., Luhmann, J.G., Spreiter, J.R., Stahara, S.S., 1985a. Magnetopause merging site asymmetries. J. Geophys. Res. 90, 341– 346. Crooker, N.U., Luhmann, J.G., Russell, C.T., Smith, E.J., Spreiter, J.R., Stahara, S.S., 1985b. Magnetic -eld draping against the dayside magnetopause. J. Geophys. Res. 90, 3505–3510. Engle, I.M., Beard, D.B., 1980. Idealized Jovian magnetosphere shape and -eld. J. Geophys. Res. 85, 579–592. Lepping, R.P., Burlaga, L.F., Lazarus, A.J., Vasyliunas, V.M., Szabo, A., Steinberg, J.T., Ness, N.F., Krimigis, S.M., 1992. Neptune’s polar cusp region: observations and magnetic -eld analysis. J. Geophys. Res. 97, 8135–8144. Luhmann, J.G., Walker, R.J., Russell, C.T., Crooker, N.U., Spreiter, J.R., Stahara, S.S., 1984a. Patterns of potential magnetic -eld merging sites on the dayside magnetopause. J. Geophys. Res. 89, 1739–1742. Luhmann, J.G., Walker, R.J., Russell, C.T., Crooker, N.U., Spreiter, J.R., Stahara, S.S., Williams, D.J., 1984b. Mapping the magnetosheath -eld between the magnetopause and the bow shock: implications for magnetospheric particle leakage. J. Geophy. Res. 89, 6829–6834. Luhmann, J.G., Russell, C.T., Spreiter, J.R., Stahara, S.S., 1985. Evidence for mass-loading of the Venus magnetosheath. Adv. Space Res. 5, 307–311. Luhmann, J.G., Warniers, R.J., Russell, C.T., Spreiter, J.R., Stahara, S.S., 1986. A gas dynamic magnetosheath -eld model for unsteady interplanetary -elds: application to the solar wind interaction with Venus. J. Geophys. Res. 91, 3001–3010.

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S.S. Stahara / Planetary and Space Science 50 (2002) 421–442

McGary, J.E., Pontius Jr., D.H., 1994. MHD simulations of boundary layer formation along the dayside Venus ionopause due to mass loading. J. Geophys. Res. 99, 2289–2300. Mihalov, J.D., Spreiter, J.R., Stahara, S.S., 1982. Comparison of gas dynamic model with steady solar wind 8ow around Venus. J. Geophys. Res. 87, 10,363–10,371. Moore, K.R., Thomas, V.A., McComas, D.J., 1991. Global hybrid simulation of the solar wind interaction with the dayside Venus. J. Geophys. Res. 96, 7779–7791. Nagy, A.F., Cavens, T.E., Yee, J.H., Stewart, A.I.P., 1986. Hot oxygen atoms in the upper atmosphere of Venus. J. Geophys. Res. 8, 629–633. Ness, N.F., Acuna, M.H., Burlaga, L.F., Connerney, J.E.P., Lepping, R.P., Neubauer, F.M., 1989. Magnetic -elds at Neptune. Science 246, 1473–1478. Richardson, J.D., Stahara, S.S., Siscoe, G.L., Spreiter, J.R., Szabo, A., 1994. The magnetosheath of Neptune: model and observations. J. Geophys. Res. 99, 14,789–14,797. Russell, C.T., 1985. Planetary bow shocks. in: Tsurutani, B.T., Stone, R.G. (Eds.), Collisionless Shocks in the Heliosphere: Review of Current Research. AGU Monograph 35, American Geophysical Union, Washington DC, pp. 109 –130. Russell, C.T., Luhmann, J.G., Spreiter, J.R., Stahara, S.S., 1984. The magnetic -eld of Mars: implications from gas dynamic modeling. J. Geophys. Res. 89, 2997–3003. Slavin, J.A., Holzer, R.E., 1981. Solar wind 8ow about the terrestrial planets 1. Modeling bow shock position and shape. J. Geophys. Res. 86, 11401–11418. Slavin, J.A., Holzer, R.E., Spreiter, J.R., Stahara, S.S., Chaussee, D.S., 1983. Solar wind 8ow about the terrestrial planets 2. Comparison with gas dynamic theory and implications for solar–planetary interactions. J. Geophys. Res. 88, 19–35. Slavin, J.A., Holzer, R.E., Spreiter, J.R., Stahara, S.S., 1984. Planetary Mach cones: theory and observation J. Geophys. Res. 89, 2708–2714. Slavin, J.A., Smith, E.J., Spreiter, J.R., Stahara, S.S., 1985. Solar wind 8ow about the outer plants: gasdynamic modeling of the Jupiter and Saturn bow shock. J. Geophys. Res. 90, 6275–6286. Song, P., Russell, C.T., Gombosi, T.I., Spreiter, J.R., Stahara, S.S., Zhang, X.X., 1999. On the processes in the terrestrial magnetosheath, 2. Case study. J. Geophys. Res. 104, 345–356. Spreiter, J.R., Alksne, A.Y., 1969. Plasma 8ow around the magnetosphere. Rev. Geophys. 7, 11–15.

Spreiter, J.R., Briggs, B.R., 1962. Theoretical determination of the form of the boundary of the solar corpuscular stream produced by interaction with the magnetic dipole -eld of the Earth. J. Geophys. Res. 67, 37–51. Spreiter, J.R., Rizzi, A.W., 1974. Aligned magnetohydrodynamic solution for solar wind 8ow past the Earth’s magnetosphere. Acta Astronaut. 1, 15–55. Spreiter, J.R., Stahara, S.S., 1980a. A new predictive model for determining solar wind–terrestrial planet interactions. J. Geophys. Res. 85, 6769–6777. Spreiter, J.R., Stahara, S.S., 1980b. Solar wind past Venus: theory and comparison J. Geophys. Res. 85, 7715–7738. Spreiter, J.R., Stahara, S.S., 1985. Magnetohydrodynamic and gasdynamic theories for planetary bow waves. in: Tsurutani, B.T., Stone, R.G. (Eds.), Collisionless Shocks in the Heliosphere: Review of Current Research. AGU Monograph, 35, American Geophysical Union, Washington DC, pp. 86 –107. Spreiter, J.R., Stahara, S.S., 1994. Gasdynamic and magnetohydrodynamic modeling of the magnetosheath: a tutorial Adv. Space Res. 14, (7)5– (7)19. Spreiter, J.R., Summers, A.L., Alksne, A.Y., 1966. Hydromagnetic 8ow around the magnetosphere. Planet. Space Sci. 14, 223–253. Spreiter, J.R., Alksne, A.Y., Summers, A.L., 1968. External aerodynamics of the magnetosphere. In: Carovillano, R.L., McClay, J.F., Radoski, H.R. (Eds.), Physics of the Magnetosphere. D. Reidel Pub. Co., Dordrecht, pp. 304–378. Spreiter, J.R., Summers, A.L., Rizzi, A.W., 1970a. Solar wind 8ow past non-magnetic planets—Venus and Mars. Planet Space Sci. 18, 1281– 1299. Spreiter, J.R., Marsh, M.C., Summers, A.L., 1970b. Hydromagnetic aspects of solar wind 8ow past the moon. Cosmic Electrodyn. 1, 5–50. Stahara, S.S., Molvik, G.A., Spreiter, J.R., 1987. A new computational model for prediction of mass loading phenomena for solar wind interactions with cometary and planetary ionospheres, Pap. AIAA-87-1410, Am. Inst. of Aero. And Astro. Stahara, S.S., Spreiter, J.R., Rachiele, R., Slavin, J.A., 1989. A three-dimensional gasdynamic model for solar wind 8ow past nonaxisymmetric magnetospheres: application to Jupiter and Saturn J. Geophys. Res. 94, 13,353–13,365. Stahara, S.S., Rachiele, R.R., Molvik, G.A., Spreiter, J.R., 1993. Development of a preliminary solar wind transport magnetosheath forecast model. PL-TR-93-2169. Szabo, A., Lazarus, A.J., Lepping, R.L., Ness, N.F., 1991. Magnetopause and cusp observations at Neptune. J. Geophys. Res. 96, 19,149–19,153.