Modeling Magnetosphere-Solar Wind Interactions with Basic Fluid Dynamics Alexander Freed

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1 Modeling Magnetosphere-Solar Wind Interactions with Basic Fluid Dynamics Alexander Freed Abstract The solar wind has a huge effect on the shape of Earth s magnetic field. This project took a basic fluid dynamics approach to modeling part of this effect by applying the classic fluid dynamics example of a uniform potential flow over a circular cylinder. Instead of dealing with the electromagnetic interactions using Maxwell s Equations, the solar wind s pressure was treated as coming from a laminar, inviscid, and incompressible flow. That pressure was then compared with the magnetic pressure of the existing offset, tilted dipole (OTD) model to incorporate the solar wind s effect. The results of the solar wind s effect are discussed below. I. BACKGROUND Earth s magnetosphere is essential to life as it is currently understood. It protects Earth from harmful charged particles being constantly emitted from the Sun. If the buffer zone created by the planetary magnetic field suddenly disappeared, the thousands of satellites orbiting Earth would most likely fail, rendering dependent technology like global positioning systems unusable. A basic first order approximation to modeling the magnetic field of Earth is to use an offset, tilted dipole (OTD) model. OTD models are extremely useful for quick approximations close to Earth but fail much farther away because of external magnetic fields. 1 Essentially, this modeling approach is like taking the magnetic field of a giant bar magnet going through the rotational axis of Earth, translating its center from the physical center, and then tilting it both longitudinally and latitudinally. A graphical representation of the OTD model used in this project is shown in Figure 1. Following the above steps is also how a coordinate change from a geocentric to magnetocentric system is made. In geocentric coordinates, the origin is at the physical center of Earth, the x-axis points in the direction of zero degrees longitude and latitude, the z-axis points through the northern rotational axis, and the y-axis obeys the right hand rule. By offsetting and tilting these axes, geocentric coordinates are converted into magnetocentric. The largest of the external magnetic fields comes from the Sun. Along with its own magnetic field, the Sun also emits a constant flow of ions known as the solar wind, and that is responsible for most of the contorting of Earth s magnetic field lines. The resulting magnetic pressure compresses the terrestrial magnetic field lines on the dayside to about 10 Earth radii (R E ) and greatly elongates them on the opposite side to the point where they resemble a sheet of field lines. 1 Conceptually, it is straightforward to visualize solar wind as a fluid made up of ions flowing from the Sun to Earth, and many comparisons have been made. 2 Thus, using fluid mechanics to help describe how the solar wind deforms Earth s magnetosphere is a reasonable approach. Today, it is usually applied using magnetohydrodynamics, or MHD, by combining the Navier-Stokes equations with Maxwell s equations. 3 As a more basic approach, this project attempted to apply the classic fluid mechanics problem of a twodimensional uniform flow over a circular cylinder to a basic three-dimensional OTD model of Earth s magnetosphere with the intention of increasing the overall accuracy. For this to be done without the use of computational fluid dynamics or empirical models, many simplifications were used to characterize the solar wind. First, the solar wind s flow was assumed to be steady and uniform, so the general characteristics of the interactions could be studied. Then the flow was treated as inviscid and incompressible, 4 allowing the use of an exact, analytical form of Bernoulli s equation, described later in Equation 1. Also, Earth was modeled as a two-dimensional cylinder for the solar wind calculations because in polar coordinates, Earth is simply a circle of radius one centered at the origin. For the sake of simplicity, time dependence, local or otherwise, was neglected. Having a time independent model allowed the project to focus on the main interaction of the solar wind and planetary magnetosphere, which is an important first step before trying to deal with the rotations of the Earth and Sun. II. THEORY For steady, incompressible, and inviscid flows, Bernoulli s equation can be used in the form of Equation 1 to relate the pressure P of a flow to its density ρ and kinetic energy, represented by its velocity v. (The gravitational potential energy is

2 assumed to be negligible for the distances used in this project.) Equation 1 Although this equation is generally only true along a unique streamline, the fact that all flows are also irrotational means Equation 1 holds true for any point in the flow. 5 Applying this equation to solar wind, a third coordinate system is needed. The Sun-Earth coordinate system is defined here, so the solar wind flows in the x direction and is coincident with the line connecting the physical centers of the two bodies (the Sun-Earth line), as shown in Figure 2. (Note: The +x-axis of the Sun-Earth coordinate system points in the opposite direction of the traditional Sun-Earth coordinate system.) This means the direction of flow points away from the Sun at the origin, Earth s physical center. The z-axis points in the direction described by Earth s orbit around the Sun, and the y- axis is defined by the right-hand rule. Since the solar wind s flow is uniform, the constant in Equation 1 can be replaced with average values for pressure and velocity outside the magnetosphere, P and v. Equation 2 shows this when solved for the solar wind pressure P sw. Equation 2 The velocity can be broken down into the radial velocity v r and transverse velocity v θ by using the velocity potential equations for a steady, uniform flow traveling in the +x direction (in Sun-Earth coordinates) over a cylinder represented by a doublet, an infinitely close source and sink. These have been related to the radius of the cylinder a based on position in standard polar coordinates, 6 defined by r and θ, as shown in Equations 3 and 4. Equation 3 Equation 4 It is important to note that this angular coordinate is actually used to describe latitude in Sun-Earth coordinates. Accounting for that, converting to Cartesian coordinates, and combining Equations 3 and 4 to get the velocity, the pressure can be described by Equation 5. Equation 5 Note that r is just the magnitude of the Cartesian position vector. For the solar wind pressure to be integrated into the OTD model, a way to compare it to the magnetic field strength of the OTD model was needed. Two different methods were attempted. The first method directly compared the solar wind s effective magnetic pressure to the OTD model s, using the magnetic pressure equation shown in Equation 6. To account direction, a unit vector along the x-axis of the Sun-Earth coordinate system was multiplied by the magnetic field strength scalar B. Equation 6 The other attempt converted the OTD model s magnetic field strength as a pressure to the solar wind s. This allowed scalar values of pressure to be compared as opposed to the vector quantities. Direction was intended to be incorporated to determine the fraction of the solar wind s pressure that would compress (or expand) Earth s OTD field, but time restraints prevented its full implementation. Results for this method are not provided due to complications with the program. To find the fraction of pressure of the solar wind frac sw that would act normal to the curve of the magnetic field line, two consecutive points along the field line trace would need to be utilized. The position vectors of these points, and, could then be used to get the normal vector to the curve, as shown in Equation 7. Equation 7 After normalizing this vector, the fraction of pressure from the solar wind could be calculated using the unit vector of the solar wind s direction: Equation 8. Equation 8 Finally, the resulting fractional amount of magnetic field strength frac B could be found by comparing the magnetic pressure including solar wind and the OTD model s calculated magnetic pressure P B, shown in Equation 9. Equation 9 III. METHODS

3 Dealing with so many coordinate systems, a major consideration was making sure the new model used the correct system for each step. The original OTD program required input to be in magnetocentric System III-like coordinates, which is a spherical coordinate system with latitude for its polar angle and west longitude for its azimuthal angle. It would then output its graphical information in geocentric coordinates to make the tilt of the magnetosphere apparent. When incorporating solar wind, a third coordinate system, Sun-Earth coordinates, was needed. This system had to take Earth s orientation into account because the original did not account for the tilt of Earth s rotational axis with respect to the ecliptic plane. To accommodate that orientation, a new angle was defined that expressed the degree of rotational separation of Earth s geocentric x-axis from the Sun-Earth line. Knowing the spin axis is never perpendicular or parallel to the orbital plane, the geocentric x-axis could be projected on that plane, so the separation angle α could be measured as shown in Figure 2. With the given separation angle, the solar wind s directional unit vector <1, 0, 0> (in Sun-Earth coordinates) could be converted into magnetocentric coordinates. This was done by first rotating the unit vector by the negative of the separation and obliquity (axial tilt) angles, the latter being ε 0 =23.44 ± Then the program offset and tilted that vector from magnetocentric to geocentric coordinates to match the output generated by the program. With the solar wind direction in magnetocentric coordinates, a unit vector was made and multiplied by the solar wind s equivalent magnetic field strength, so the scalar field could be converted into a vector field. This made it possible to directly add the magnetic field strength of the solar wind to the OTD model. The final major change to the program was updating the magnetic field strength subroutine. Originally, the coordinate input data were kept in magnetocentric coordinates because the OTD equations required this. For the solar wind calculations, however, that same input had to be in Sun-Earth coordinates, meaning the entire process described above needed to be done in reverse. This involved a similar, but new, rotation matrix. Everything was programmed in except for the offsetting because it causes the magnetic field line calculations to fail for unknown reasons. As a result, a small error in the output is expected. The new rotation consisted of two threedimensional Cartesian rotation matrices multiplied together. The first rotation was a polar rotation about the y-axis by the obliquity angle. After, the vector needed to be rotated azimuthally by the separation angle. Combining these matrices involves multiplying them in reverse order to preserve the correct order of rotations. This combined matrix was then simplified using the fact that the obliquity angle was constant before it was added into the program to cut down on the run time. IV. RESULTS All the constant values used can be found in Table 1. The combined and calculated values were found so the program would run faster. To get density, a proton density of 7 ± 1 cm -3 (assumed error) was multiplied by the mass of a proton and the conversion to get m -3 from cm -3. All other assumed error was based on variation from Internet searches for values. Those sources were not used for the actual value; actual values were taken from Russell. 3 The cosine and sine of the obliquity angle were calculated for the rotation matrix used with the solar wind unit vector. For error calculations, all error was assumed to be independent and random. The field line tracing programs read magnetocentric azimuthal angles as input; the radial and latitudinal coordinates are kept constant as initial input for the varying the azimuthal angle. The input can be edited to make Earth s geocentric x-axis point in any angle azimuthally from the Sun-Earth line by projecting it onto the ecliptic and then rotating it. The output is a text file containing all the points along the field line of the initial position in Cartesian, geocentric coordinates. Figures 3-5 show the modified OTD model against the original from different angles. These figures have a unit sphere at their origins to represent Earth. The solar wind is flowing in the direction the green magnetic field line extending from the planet. Table 1. Constant Values Solar Wind Value Variable Density ρ 12 2* (kg m -3 ) Pressure outside magnetosphere P (npa) Velocity outside magnetosphere U (km s -1 ) Obliquity ε 0 ( ) Error (*=Assumed) 2 1* * ρu

4 (kg m -1 s -2 ) 10-9 cos(ε 0 ) sin(ε 0 ) In Figure 3, the green line extends out of the page perpendicularly. Figures 4 and 5 are just rotated versions of Figure 3, each rotated 45 azimuthally from the previous one. The initial points for each modified field line were the same as for the original ones, starting at a distance of 3 R E and latitude of 50. Figure 3 shows a minute amount of extra tilt from the leftmost modified field line from the original. In Figures 4 and 5, the extra tilt increases. Another notable characteristic is how the modified lines seem to have increasingly elongated southern halves as they get farther from the green line, which is parallel to the solar wind. This comes to an extreme for the magnetocentric azimuthal angles within about 35 of the antiparallel field line. Here the field line trace fails and extends until the tracing program is manually stopped. The cyan line shows this in Figure 6. These unexpected results may be due to the solar wind s pressure field. Figure 7 shows the logarithmic scale of the pressure in the x-z plane in geocentric coordinates. Instead of the Earth s radius being used for the radius of the cylinder, the magnetopause distance (15 R E ) was used because that is where the solar wind (ideally) stops penetrating the magnetosphere. The most notable feature is the elongated ring of high pressure at about ±9 R E on the x-axis and extending about twice that far in the z- axis. Inside that, the pressure seems to return to the ambient value before dropping to a region also elongated along the vertical axis. However, once past this region, the rest are elongated slightly along the horizontal axis. V. DISCUSSION From the above graphs there are some expected and unexpected results. Because the solar wind is uniform, it was a success that the field line extending in the direction of flow stayed in the same plane as the corresponding field line from the original model. Also, the fact that the modified field lines tilt the most when they were perpendicular to the solar wind s direction matches intuition. In Figure 7, there are both kinds of results. First, the pressure is expected to be highest on the side closest and farthest from the oncoming flow in the case of a uniform flow over a cylinder, as shown in Figure 8. 8 This is especially true for the closest side, where the bow shock of the magnetosphere would appear. Also, the regions inside about 3 R E being elongated along the horizontal axis is not a surprising result because that is well within the dipole region of the magnetosphere, where the solar wind has much less of an effect. What is somewhat unexpected, however, is how the pressure is the same inside and outside the bow shock region. This might be explained by fact that the magnetopause is a region just inside the bow shock where the solar wind does not enter the magnetosphere, but further investigation of this issue is needed to reach a definitive conclusion. As for other unexpected results, there were many. Possibly the biggest one is that the modified field lines do not trace back to the planet for a 75 window centered on the field line that should be extending into the solar wind. This is unexpected because it is the region that should be most dipolar because the solar wind, based on observation, compresses the field lines instead of extending them out into a sheet of sorts like on the opposite side. 9 It is possible, however, the solar wind could be flowing in the opposite direction due to some faulty code. If this is the case and the (green) reference field line in Figure 3 is opposing the solar wind, the graphs make a bit more sense. In Figures 4 and 5, the reference field line is much smaller than the original model s field line. It can also be seen in Figure 5 that the field line 45 to the right is also compressed. The last field line, 135 from the reference line, is much larger in the southern half than its pair. Such a result is expected, although most likely for the northern half as well, since the solar wind travels in the same direction the field lines extend from Earth on that side of the planet. Another possibility for the unexpected results is the fact that Earth was treated as a twodimensional cylinder instead of a sphere. Modeling Earth this way means field lines traced on the sides of the planet facing away from the Sun-Earth line should not be accurate due to earth being modeled as a cylinder. If a field line is being traced at a height less than half an Earth radius from the center of the planet, the solar wind pressure (Equation 5) may become a negative value and result in an imaginary value for the magnetic field strength of the solar wind (Equation 6). Future work on this project would certainly include a thorough testing of the code to see what is actually going on. This would hopefully solve the problem of why the program will not work when trying to offset position vector during the conversion to Sun-Earth coordinates. Once that is fixed, a follow up project would be to finish the second method of comparing the solar wind pressure with the OTD model s magnetic pressure. Other work might also include viscous forces in the fluid dynamics. Because these were neglected, the horizontal pressure on the dipole field will be equal on the day and night side, which is physically impossible due to boundary

5 layer behavior, as d Alembert s paradox describes. 5 This may be another reason for the unexpected results. Although there are some good qualitative results from the OTD model including solar wind, there are too many inconsistencies with theory at this point to use this solar wind interaction model to estimate the magnetospheric environment around Earth. A lot of work is needed to debug the program, and then a check of the new model against a generally accepted one is essential before this model can be used. Until that happens, this model will not be suitable to approximate Earth s magnetosphere. VI. ACKNOWLEDGEMENTS Thank you Dr. Linda Winkler for all your help and patience through everything leading up to and including this project. Thank you Dr. Craig for your support and advising. of Oklahoma, Norman, OK. ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic =fl&chap_sec=07.4&page=theory 8 Fiedler, B. (Artist). (2009). Flow past a cylinder. pressure field and velocity vectors. [Web]. Retrieved from File:Flow_past_a_cylinder._Pressure_field_and_v elocity_vectors.png 9 Roederer, J. G. (1970). Dynamics of geomagnetically trapped radiation. Vol. 2. Springer Verlag: New York. VII. REFERENCES 1 Walt, M. (2005). Introduction to geomagnetically trapped radiation. Cambridge UP: New York City, NY. 2 Nishida, H., & Abe T. (2010). Magnetohydrodynamic analysis of the interaction of magnetized plasma flow with a perfect-conducting object. Physics of Plasmas, 17(5). 3 Russell, C. T. (n.d.). The solar wind interaction with the Earth s magnetosphere: a tutorial. Informally published manuscript, Department of Earth and Space Sciences, University of California Los Angeles, Los Angeles, CA. com/viewer?url=http%3a%2f%2fdawn.ucla.edu %2Fssc%2Ftutorial%2Fsolwind_interact_magsph ere_tutorial.pdf 4 (n.d.). Smaller scales: five physical processes-waves and turbulence. php?content=solar_wind/proc/p1.html#top 5 Fox, R. W., Pritchard, P. J., & McDonald, A. T. (2009). Introduction to fluid mechanics. 7 th ed. John Wiley & Sons, Inc.: Hoboken, NJ. 6 Ngo, C. C., & Gramoll, K. (n.d.). Multimedia engineering fluids. Online Text, School of Aerospace & Mechanical Engineering, University 7 Bizouard, C. (2010, March 29). Useful constants.

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7 APPENDIX A. FIGURES Figure 1. An example of the OTD model in geocentric coordinates. The B-field lines are started from 5 R E away at 40 S every 45 W in magnetocentric coordinates. The red line represents the line started at 0 W, the orange line at 45, and so on. Figure 2. This is an illustration of the geocentric coordinate system and the separation angle α between the projection of the x-axis of Earth X E and the x-axis of the Sun-Earth coordinate system X Sun-Earth. X E, its projection, and Z E are coming out of the page at an angle, X Sun-Earth is in the plane of the page, and Y E is going into the page at an angle.

8 Figure 3. The (green) modified magnetic field line has little to no tilt besides what was already present in the (blue) original field line right next to it.

9 Figure 4. The (red) modified field line is tilted more than the (blue) original one extending out of the page.

10 Figure 5. The difference in tilt is greatest in this graph when compared with Figures 3 and 4. It should be noted that each pair of field lines is spaced approximately 45 from its neighboring pair.

11 Figure 6. The new model fails to complete the field line trace for a field line antiparallel to the solar wind direction.

12 Figure 7. This is the logarithmic pressure field of the solar wind only. The flow is in the +x direction, and the units of the axes are meters. Regions outside of about 3 R E (the horizontal edges of the green field) are elongated along the z-axis, while the regions inside that distance elongate along the x-axis.

13 Figure 8. This graph of the pressure and velocity fields for a left-to-right (in the +x direction) flow past an infinite cylinder from Fiedler, 2009 shows high regions of pressure on the left and right side of the cylinder and low regions above and below it. This figure should be compared with Figure 7. The horizontal and vertical axes are the x and z axes, respectively, and are in units of cylinder radii.