A New Method for Solving the MHD Equations in the Magnetosheathdiscussions Open Access Open Access Atmospheric Atmospheric C

Home , Magnetopause, Magnetosheath

EGU Journal Logos (RGB) Open Access Open Access Open Access Ann. Geophys., 31, 419–437, 2013 www.ann-geophys.net/31/419/2013/ Advances in Annales Nonlinear Processes doi:10.5194/angeo-31-419-2013 © Author(s) 2013. CC Attribution 3.0 License.Geosciences Geophysicae in Geophysics Open Access Open Access Natural Hazards Natural Hazards and Earth System and Earth System Sciences Sciences A new method for solving the MHD equations in the magnetosheathDiscussions Open Access Open Access Atmospheric Atmospheric C. Nabert1, K.-H. Glassmeier1,2, and F. Plaschke3 Chemistry Chemistry 1Institut fur¨ Geophysik und extraterrestrische Physik, Technische Universitat¨ Braunschweig, Germany 2Max Planck Institute for Solar System Research,and Katlenburg-Lindau, Physics Germany and Physics 3Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA, USA Discussions Open Access Open Access Correspondence to: C. Nabert ([email protected])Atmospheric Atmospheric

Received: 29 November 2012 – Revised: 13Measurement February 2013 – Accepted: 13 February 2013 – Published:Measurement 5 March 2013 Techniques Techniques Discussions Abstract. We present a new analytical method to derive 1 Introduction Open Access Open Access steady-state magnetohydrodynamic (MHD) solutions of the magnetosheath in different levels of approximation. With Biogeosciences BiogeosciencesThe magnetosheath is the flow region of the solar wind this method, we calculate the magnetosheath’s density, ve- Discussions locity, and magnetic field distribution as well as its geome- around the Earth. Its characteristics, for example its thick- try. Thereby, the solution depends on the geomagnetic dipole ness or magnetic field distribution, depend on the solar wind conditions. In regions without reconnection, the earthward Open Access moment and solar wind conditions only. To simplify the rep- Open Access boundary of the flow region, the magnetopause, is defined as resentation, we restrict our model to northward IMF with the Climate solar wind flow along the stagnation streamline. The sheath’sClimatea boundary with vanishing normal flow velocity. It is characterized as a small region where a transition from magne- geometry, with its boundaries, bow shock and magnetopause,of the Past of the Past is determined self-consistently. Our model is stationary and tosheath plasma density and temperature to magnetosphericDiscussions time relaxation has not to be considered as in global MHD conditions occurs. Very often the magnetopause is identified by an abrupt change in the magnetic field, which is related Open Access simulations. Our method uses series expansion to transfer Open Access the MHD equations into a new set of ordinary differential to a region with spatially confined electricEarth current. SystemChapman equations. The number of equations is relatedEarth to the levelSystem of and Ferraro (1930) were the first to speculate about the ex- istence of such a magnetopause current layer.Dynamics Without a so- approximation considered including different physicalDynamics processes. These equations can be solved numerically; however, lar wind magnetic field, all current is locatedDiscussions at the magne- an analytical approach for the lowest-order approximation is topause, corresponding to a field-free magnetosheath with a jump of the magnetic field at the magnetopause. Therefore, Open Access also presented. This yields explicit expressions, not only for Open Access the flow and field variations but also for the magnetosheathGeoscientificthe flow within the magnetosheath canGeoscientific be treated in terms of thickness, depending on the solar wind parameters.Instrumentation Results hydrodynamics (e.g., Spreiter et al.Instrumentation, 1966). are compared to THEMIS data and offer a detailed expla- A sharp current layer, however, does not always exist at Methods andthe magnetopause as demonstrated in Fig.Methods1, showing and obser- nation of, e.g., the pile-up process and theData corresponding Systems Data Systems plasma depletion layer, the bow shock and magnetopause ge- vations made on-board the THEMIS-C spacecraft. The five spacecraft of the THEMIS mission were launchedDiscussions in 2007 Open Access ometry, the magnetosheath thickness, and the flow decelera- Open Access (Angelopoulos, 2008) and provide a wealth of plasma and tion. Geoscientific Geoscientificmagnetic field observations suitable for magnetosheath and Keywords. Magnetospheric physics (Magnetopause, cusp, magnetopause studies (e.g.,Model Glassmeier Development et al., 2008; Plaschke and boundary layers; Magnetosheath)Model – Development Space plasma et al., 2009; Zhang et al., 2009). On 29 OctoberDiscussions 2009 physics (Kinetic and MHD theory) the THEMIS-C spacecraft traversed the magnetosheath almost along the stagnation streamline. Measurements by Open Access Open Access Hydrology andthe ESA plasma instrument (McFaddenHydrology et al., 2008 and) allow for a clear identification of the magnetopause crossing at Earth System08:25 UT based on abrupt changes ofEarth the plasma System density and Sciencestemperature to magnetospheric conditions.Sciences Magnetic field Discussions Open Access Published by Copernicus Publications on behalf of the EuropeanOpen Access Geosciences Union. Ocean Science Ocean Science Discussions Open Access Open Access Solid Earth Solid Earth Discussions Open Access Open Access

The Cryosphere The Cryosphere Discussions 420 C. Nabert et al.: Method for solving the MHD equations

Bow Shock Magnetopause distribution in the magnetosheath) and/or require solutions 80 Smooth Increase of more complicated partial differential equations. All these 60

z models solve differential equations derived from the MHD

B 40

[nT] 20 Continuous equations. Global MHD simulations are another investiga- 0 tion method which solves the complete set of MHD equa- 100 0 tions directly to obtain a magnetosheath solution as shown -100 by Wu (1984), Wu (1992), Ogino et al. (1992), Siscoe et al. x -200

[km/s] u (Ion) -300 (2002) and Wang et al. (2004), among others. 40 Simple and very reduced models can provide insight 3 30 Sharp Jumps 20 into the basic physical processes underlying a phenomenon.

N (Ion) [1/cm ] 10 However, more complex models include more processes and 0 can show how the phenomenon is embedded into a com- 107 plex physical environment. One should be aware, however, 6 10 that numerical effects can strongly influence any result when 105 104 solving more complex differential equations. Numerical dif- 29.10.2009 06000700 0800 Time [h] fusion near the bow shock or magnetopause is large in global THEMIS C MHD simulations, as noted by Wu et al. (1981). Subsequent Fig. 1. A low-shear magnetopause transition on 29 October 2009. developments on the numerical schemes try to reduce the in- The upper panel shows Bz (GSM) magnetic field observations. The fluence of diffusion; however, it is still a difficult task (Toth, second panel displays the (ion) velocity in x-direction (GSM), and 2000). Only a detailed comparison of simple and complex the third one shows the density of the plasma ions. The last panel approaches provides a complete understanding of the phe- depicts the logarithmic ion temperature. nomenon considered. Here, we present a new analytical method to solve the MHD equations in different orders of approximation, from observations by the FGM (Auster et al., 2008), however, re- a lowest-order approach to the complex, full MHD solution. veal a magnetic pile-up throughout the entire magnetosheath; Our approach is able to classify the MHD models introduced i.e., the field slowly increases. At the magnetopause, the above with respect to different levels of approximation. We magnetosheath field smoothly adapts to the magnetospheric show that the different results in the density distribution of field. Thus, there is no confined magnetopause current but Lees (1964) and Zwan and Wolf (1976) can be referred back currents distributed over the entire magnetosheath. Simi- to different levels of approximation used. The method is able lar pile-up observations under low-shear conditions were to calculate the fluid properties, such as density, pressure, reported by, e.g., Crooker et al. (1979), Paschmann et al. velocity, and magnetic field, as well as the magnetosheath’s (1993), Phan et al. (1994), or Farrugia et al. (1997). In these geometry bounded by the bow shock and magnetopause. In earlier reported cases, however, the magnetic pile-up region the lowest-order approach, analytical solutions are obtained, is usually less extended than in the present case. yielding explicit expressions for the density and field distri- A first model to calculate the magnetic pile-up in the mag- bution, as well as for the magnetosheath thickness and its netosheath was presented by Lees (1964). In this model, the dependence on the solar wind magnetization. magnetohydrodynamic (MHD) equations are restricted to ax- The magnetosheath thickness and the related bow shock isymmetric flows, and the velocity tangential to the stagna- distance have been a major topic of investigation for decades, tion streamline is assumed to be a known function. Although as reviewed by Petrinec (2002). However, only empirical re- these restrictions strongly limit the scope of the theory, basic lations have been derived (Spreiter et al., 1966) and modified aspects of the magnetic pile-up could be investigated. A more by Farris and Russell (1994) to take the solar wind magnetic detailed theory was developed by Zwan and Wolf (1976), the field into account. We will compare our analytically derived so-called depletion layer model. They considered a flux tube expression with the empirical relations. moving through the magnetosheath in the MHD approach. Finally, our method is applied to two magnetosheath tran- Zwan and Wolf (1976) concluded that the plasma in the flux sitions observed by THEMIS: a transition with a confined tube is squeezed out near the magnetopause, leading to a magnetopause current layer and the transition shown in Fig. 1 lowered density and an enhanced magnetic field strength. exhibiting a strong magnetic pile-up. These observations are Note that this consideration leads to a somewhat different compared to MHD magnetosheath solutions in different or- density behavior compared to Lees (1964). Another theoret- ders of approximation. ical approach is the magnetic string model by Erkaev et al. (1988). In this model, the MHD equations are solved in a new coordinate system, specially designed to take advantage of the frozen-in magnetic field. However, both latter models also rely on additional assumptions (e.g., the pressure

Ann. Geophys., 31, 419–437, 2013 www.ann-geophys.net/31/419/2013/ C. Nabert et al.: Method for solving the MHD equations 421

Earth‘s Dipole Outflow Solar Wind MHD Equations MP Description Conditions Parameters z BS

Ansatz Rankine-Hugoniot Relations Solar Additional Boundary Magnetopause ODEs Conditions at Wind Conditions Bow Shock x x MS x Steady-State MHD Solution E Geomagnetic Fig. 2. Overview of the interplay between different aspects of our Dipole model. Each color represents a part discussed in a separate section in this chapter. The top row displays the input for our method.

2 Theory Fig. 3. The incident solar wind in x-direction with its field along the z-direction on the left side and the Earth with its dipole field on the Physical quantities in the steady-state magnetosheath, such right side. The origin of the Cartesian coordinate system (x,y,z) as density, pressure, velocity, or magnetic field, can be de- shall be at the nose of the bow shock (BS). termined by the stationary MHD equations. They depend on three spatial coordinates (x,y,z). We choose the x-axis to be tions read along the stagnation streamline and the y- and z-axis to be perpendicular to it. The quantities are expanded into power ∇ · (ρ u) = 0, (1) series with respect to y and z around the stagnation streamline: f (x,y,z) = f0(x)+f1(x)y +f2(x)z+f3(x)y z+.... Substituting such an ansatz for each quantity in the MHD 1 equations and equating coefficients of the tangential orders ρ (u · ∇)u + ∇p − (∇ × B) × B = 0, (2) µ (1, y, z, y z, . . . ), a system of ordinary differential equa- 0 tions (ODEs) depending only on the normal direction x is obtained. It is suitable to modify our ansatz with respect to ∇ × (u × B) = 0, (3) the magnetosheath geometry. Consequently, our system of ODEs and the corresponding solutions depend on bow shock and magnetopause geometry parameters. These parameters ∇ · = are calculated by taking the geomagnetic field into account. B 0, (4) Boundary conditions (inflow conditions) for our system of ODEs are determined at the bow shock. The shocked values are referred back analytically to the solar wind conditions p = k ργ . (5) using the Rankine–Hugoniot relations. Note that the MHD equations require further boundary conditions (in global Here ρ denotes mass density, u the fluid’s bulk velocity, p MHD simulations called outflow conditions) far away along the thermal gas pressure (assumed to be isotropic), B the −7 2 the y- and z-direction. The complete set of ODEs using the magnetic field, µ0 = 4π×10 N/A the vacuum permeabil- series expansion ansatz to infinite order is not easy to handle. ity, and γ the ratio of specific heats. Note that instead of the A finite expansion order leads to a finite system of ODEs, full energy conservation law, an adiabatic law (Eq. 5) is used which remains underdetermined. But we can use the outflow with the proportionality constant k. The adiabatic law is valid conditions to close the system. Finally, a solution of the mag- within the magnetosheath, but not through the bow shock due netosheath flow and field quantities as well as the bow shock to entropy changes. and magnetopause geometry can be calculated depending on The super-magnetosonic solar wind flow causes a bow solar wind conditions only. The interplay of the different shock in front of the Earth. A Cartesian coordinate system parts of the model are displayed in Fig. 2. is used with origin at the bow shock’s nose (subsolar point). The x-axis points towards the Earth, which is located at xE 2.1 Deriving ODEs from MHD (see Fig. 3). The bow shock distance to the Earth, that is, xE, will be determined later as part of the solution. We choose the Plenty of phenomena of the plasma flow in the magne- incident solar wind to flow along the x-axis. The z-axis is nor- tosheath are described by the ideal MHD theory (e.g., Siscoe mal to the ecliptic pointing northward, and the y-axis com- et al., 2002). The time-independent (stationary) MHD equa- pletes the right-hand triad. For simplification, we choose the www.ann-geophys.net/31/419/2013/ Ann. Geophys., 31, 419–437, 2013 422 C. Nabert et al.: Method for solving the MHD equations solar wind magnetic field to be parallel (northward or positive) along the z-axis. Furthermore, the Earth’s magnetic field 2 2 is represented by a dipole, its moment being anti-parallel to p = p0 + p20 y + p02 z , (12) the z-axis with magnitude M = 8 × 1015 Tm3. The components of the geomagnetic field BE are given by (e.g., Ogino, 2 3 1993) Bx = Bx01 z + Bx21 y z + Bx03 z , (13) −3z1x B = M, (6) E,x r5 By = By11 y z, (14) −3y z B = M, (7) E,y r5 2 2 Bz = Bz0 + Bz20 y + Bz02 z , (15) −2z2 + 1x2 + y2 B = M, (8) E,z r5 u = u + u y2 + u z2, (16) p x x0 x20 x02 with the radial distance r = 1x2 + y2 + z2, and 1x = xE − x defining the distance from the Earth’s center (Fig. 3). Note that the solar wind magnetic field and the dipole field u = u y + u y z2 + u y3, (17) are parallel close to the dayside stagnation streamline, which y y10 y12 y30 excludes reconnection. The situation introduced above is highly symmetric. The 2 3 dipole field has a rotational symmetry with respect to the uz = uz01 z + uz21y z + uz03 z . (18) dipole moment axis, and the solar wind flow is perpendicular to this symmetry axis, leading to symmetric properties in the Note that the expansion only holds for the tangential direc- MHD flow around the dipole field. A detailed discussion of tions. Thus, the coefficients are arbitrary functions of x. Their the symmetry relations is presented in Appendix A. indices indicate the order in y and z; e.g., “20” means second We approximate the bow shock as well as the magne- order in y and zeroth order in z and “0” means zeroth order topause shapes by elliptic paraboloids (functional relation in y and z. 2 2 Due to the curved shape of the bow shock and magne- x = x0 + a y + b z with constants x0, a, and b). This corresponds to a second-order series expansion in accordance topause, coefficient functions are more conveniently depend- ˜ → ˜ ˜ with the symmetry considerations. The bow shock and mag- ing on x instead of x (e.g., ρ0(x) ρ0(x)), with x defined netopause parametrization are by the following relation: ˜ = X 2 X x 2 x cBS,t t , (9) x =x ˜ + cBS,t + 1ct t , (19) t=y,z t=y,z xMS where 1c = c − c denotes the difference between X 2 t MP,t BS,t x = xMS + cMP,t t , (10) the magnetopause curvature and the bow shock curvature. t=y,z Each value for x˜ corresponds to an elliptic paraboloid de- respectively, where xMS denotes the magnetopause distance fined by Eq. (19). In particular, x˜ = 0 corresponds to the from the bow shock’s nose (which is the magnetosheath bow shock parametrization (Eq. 9) and x˜ = xMS to the mag- thickness along the stagnation streamline, x-axis). The con- netopause parametrization (Eq. 10). Therefore, the mod- stant parameters cBS,t for the bow shock and cMP,t for the ified ansatz describes the physical quantities on elliptic magnetopause represent curvatures in the respective tangen- paraboloids parametrized by x˜, starting at the bow shock tial direction t = y or t = z. These geometric parameters are for x˜ = 0 and finishing at the magnetopause at x˜ = xMS (see determined by the interaction of the magnetosheath plasma Fig. 4). Note that the modification of the ansatz does not af- with the dipole field as we will show later. fect its symmetry because x˜ depends on y2 and z2 only and, The flow and field variables in the magnetosheath are thus, x(y,z)˜ is independent of the sign of y and z. This can expanded into Taylor series with respect to the y- and z- be seen solving Eq. (19) with respect to x˜. Thus, the ansatz direction around the stagnation streamline (y = z = 0). Due still satisfies the symmetry relations given in Table A1. to the symmetry relations summarized in Table A1 (Ap- Substituting our ansatz into the MHD system (1)–(5) pendix A), several expansion terms vanish. This leads to the yields a set of ordinary differential equations (ODEs) by following ansatz expanded to the third order: equating the coefficients of zeroth order, 2 2 0 ρ = ρ0 + ρ20 y + ρ02 z , (11) (ρ0ux0) + ρ0 uy10 + uz01 = 0, (20)

Ann. Geophys., 31, 419–437, 2013 www.ann-geophys.net/31/419/2013/ C. Nabert et al.: Method for solving the MHD equations 423

x˜ with c˜t = c ,t + 1ct . Derivatives with respect to x˜ are z BS xMS marked with a prime. To simplify the reading of the equations, we set µ0 = 1. Equations obtained from equating co- efﬁcients of the second order can be found in Appendix B. We expressed the series expansion of the MHD quantities in new coordinates x,y,z˜ . However, as the MHD equations are explicitly expressed in Cartesian coordinates, the chain x rule was applied to compute the derivatives with respect to 0 x˜, e.g., ∂xp = p ∂xx˜. The factor ∂xx˜ can be calculated from Eq. (19):

X 1ct 2 4 ∂xx˜ = 1 − t + O(t ). (30) t=y,z xMS ~ Magnetopause Bow Shock 0

0 0 Bx01 + 2Bz02 + By11 − 2c˜zBz0 = 0, (24) 2.2 Solar wind at the bow shock boundary First-order derivatives of the coefﬁcient functions with re- 0 2 spect to x˜ in the above ODEs require knowledge of corre- ρ0ux0uy10 + ρ0uy10 + 2(p20 + Bz0Bz20) − By11Bz0 (25) 0 0 sponding boundary values at one point x˜0 = 0, i.e., ρ0(x˜0), −2c˜y p + Bz0B = 0, 0 z0 ρ20(x˜0), ρ02(x˜0),. . . . We choose to set these values at x˜0 = 0 (bow shock) using shocked solar wind parameters. These are related to the solar wind via Rankine–Hugoniot relations ρ u u0 + ρ u2 + 2p + B2 − B B0 (26) 0 x0 z01 0 z01 02 x01 x01 z0 (e.g., Petrinec and Russell, 1997): −2c˜ p0 = 0, z 0 ρ uξ = 0, (31)

γ −1 p20 = k γ ρ20 ρ , (27) 0 Bξ = 0, (32)

γ −1 " 2 # p02 = k γ ρ02 ρ , (28) 2 Bτ 0 ρ uξ + p + = 0, (33) 2µ0

B u + (B u + B u ) y11 x0 2 z02 x0 z0 x02 (29) Bξ −Bx01 uy10 + 2uz01 + 2c˜zBz0uy10 = 0, ρ uξ uτ − Bτ = 0, (34) µ0 www.ann-geophys.net/31/419/2013/ Ann. Geophys., 31, 419–437, 2013 424 C. Nabert et al.: Method for solving the MHD equations

The higher-order coefficient functions in the series expansion dominate far away from the stagnation streamline (e.g., − = 2 2 uξ Bτ uτ Bξ 0, (35) ρ(x,y˜ → ∞,z → ∞) ≈ ρ20(x)y˜ + ρ02(x)z˜ ). We suggest setting the highest-order coefficients to their bow shock boundary values. This condition is similar to common out- " ! # flow boundary condition in simulations (e.g., Ogino, 1993). ρ γ B2 B u2 + p + τ u − ξ B u = 0. (36) Different, specific choices of explicit functional relations to − ξ τ τ 2 γ 1 µ0 µ0 close the system of ODEs lead basically to the models by Lees (1964) and Zwan and Wolf (1976), as discussed below. The squared brackets [...] indicate that the quantity therein is conserved across the shock. The subscript ξ denotes the 2.4 Conditions from the geomagnetic field as an normal component and τ tangential components with respect obstacle to the bow shock. Solving the Rankine–Hugoniot relations with respect to the bow shock geometry (Eq. 9) and the cho- Despite the boundary conditions introduced above, the sys- sen solar wind conditions, the shocked values are obtained. tem of ODEs still contains five undetermined geometric pa- Power series expansion in y- and z-direction of this solution rameters: the bow shock curvatures cBS,y and cBS,z, the mag- and equating coefficients with our ansatz determine the coef- netopause curvatures cMP,y and cMP,z, and the magnetosheath ficient functions of the ansatz at x˜ = 0. Analytical solutions thickness xMS. To determine these, inner boundary condi- for the zeroth-order coefficients can be found in, e.g., Siscoe tions are necessary. (1983). A brief summary of his work and a detailed analyti- First, the flow should be tangential with respect to the cal approach for the the higher-order coefficient values up to magnetopause, not allowing flow to penetrate into the mag- the second order are given in Appendix C. netosphere (e.g., Biernat et al., 1999). This stems from the As a result, the velocity coefficients are given by definition of the magnetopause as earthward flow boundary. The flow direction at the magnetopause can be cal- ˜ = = i+j i j ul i j (x 0) fu(i,j)2 cBS,ycBS,z1u, (37) culated using magnetopause surface coordinates. The normal magnetopause vector is derived from the magnetopause where 1u = uSW − ux0(x˜ = 0), l ∈ {x,y,z}, i and j are la- parametrization (10): beled as in Eqs. (16)–(18), and fu(i,j) is defined by (sign- function) 1 T ξ = 1,−2c y,−2c z , (41) MP n MP,y MP,z + + ∈ { } MP = 1, i j 1 fu(i,j) − + ∈ { } . (38) q 1, i j 2,3 = + P 2 2 where nMP 1 t=y,z 4cMP,t t normalizes the vector. The magnetic field coefficients are given by: A vanishing normal flow velocity through the magnetopause yields ξ MP · u(x˜ = xMS,y,z) = 0. This expression holds for ˜ = = i+j i j the y- and z-direction, so using the velocity ansatz (16)–(18) Bl i j (x 0) fB (i,j)2 cBS,ycBS,z1B, (39) yields in which ux0(x˜ = xMS) = 0, (42) +1, i + j ∈ {1,2} f (i,j) = , (40) B −1, i + j ∈ {3} ˜ = where 1B = BSW −Bz0(x˜ = 0), l ∈ {x,y,z}, and i and j are ux20(x xMS) cMP,y = , (43) as in Eqs. (13)–(15). The boundary values for the density 2uy10(x˜ = xMS) and pressure coefficients ρ20, ρ02, p20, and p02 are approximately zero. ux02(x˜ = xMS) cMP,z = . (44) 2.3 Outflow boundary conditions 2uz01(x˜ = xMS) Since the MHD system (1)–(5) contains partial derivatives However, at the magnetopause surface the flow velocity van- with respect to all three space dimensions, boundary condi- ishes along the magnetic field direction (→ uz01(x˜ = xMS) = tions are needed for three linearly independent planes. In the 0) as pointed out by Siscoe et al. (2002). Thus, in the pres- previous section, boundary conditions were defined for the ence of magnetic fields Eq. (44) is no longer valid. Instead, bow shock plane; these are commonly called inflow condi- we use the assumption that magnetic fields of different ori- tions in global MHD simulations. Here the remaining set of gin cannot mix (Frozen-in theorem). This yields the vanish- outflow boundary conditions are defined, used to close the ing of any normal magnetic field component at the magne- system of ODEs. topause boundary: ξ MP · B(x˜ = xMS,y,z) = 0 (e.g., Biernat

Ann. Geophys., 31, 419–437, 2013 www.ann-geophys.net/31/419/2013/ C. Nabert et al.: Method for solving the MHD equations 425 et al., 1999). This requires the stagnation streamline. If the solar wind is along the stagnation streamline (i.e., the x-direction) with a perpendicular, Bx01(x˜ = xMS) northward IMF, we can directly apply the ODEs presented cMP,z = . (45) 2Bz0(x˜ = xMS) here. The system of ODEs related to the zeroth order is given by Eqs. (20)–(23). The second-order approach is given by Secondly, a restricting condition arises from the Earth’s Eqs. (20)–(28) and (B1)–(B7) as presented in Sect. 2.1 and dipole field itself. The total magnetic field of our MHD so- Appendix B. For solar wind conditions violating the required lution (B) should by a superposition of the magnetic fields symmetry conditions, ansatz (11)–(18) has to be replaced by generated by magnetosheath currents (Bj ) and the Earth’s an arbitrary series expansion. However, the derivation of the magnetic field (BE). Since the Earth’s dipole field is curl- corresponding ODEs is analogous. Further, higher-order ap- free (see Eqs. 6–8), the curl of the total field yields the current proximations require higher-order series expansions. We set distribution in the magnetosheath jMS only: the coefficient functions of the highest-order constant at their postshock values to close the system of equations (i.e., for ∇ × B = ∇ × Bj = µ0jMS. (46) the second-order approach presented, uy30, uy12, uz03, uz21, B , and B are constant). The boundary conditions for the The magnetic field generated by these currents can be cal- x30 x21 ODEs (ρ(x˜ = 0),p(x˜ = 0),u(x˜ = 0), and B(x˜ = 0)) are de- culated with Biot-Savart’s law. The superposition of this termined at the bow shock by solving the Rankine–Hugoniot magnetosheath-current-field and the Earth’s magnetic field relations for the solar wind conditions (ρ ,p ,u , and necessarily has to match the total field from our MHD so- SW SW SW B ) with respect to the shock geometry. However, we can lution. If this is not satisfied, geometric parameters and the SW also use the analytical approach presented in Appendix C magnetopause distance to the Earth’s center have to be mod- yielding equations for zeroth-order coefficients (Eqs. C1– ified accordingly. C4) and for the higher-order coefficients (Eqs. C19 and C27). A more simple approach to take the geomagnetic field into The second-order terms of density and pressure vanish in this account uses a pressure condition at the magnetopause which approach. is valid in the hydrodynamic limit. Mead and Beard (1964) The Rankine–Hugoniot relations as well as the ODEs re- pointed out that the tangential components of the total mag- quire knowledge of the geometric parameters x , c , netic field determine the (calculated) total pressure behind MS MP,y c , c , and c . As an initial choice, we can use the an- the magnetopause p : MP,z BS,y BS,z tot,MP alytical expressions (D13), (D24), (D28), (D25), and (D29) 2 with the magnetopause distance (Eq. D23). They are related ξ MP × BE + Bj ptot,MP = . (47) to solar wind conditions via Eqs. (C1)–(C4). The solution 2µ0 of our system has to satisfy the inner boundary conditions given by Eqs. (42), (43), (45), and (48). The latter condi- With respect to Mead and Beard (1964), a good first approx- tion holds for the y- and z-direction. The geometric param- imation for the right-hand side yields: eters need to be determined self-consistently; i.e., the initial 2 choice has to be modified until the conditions are satisfied. (f BE × ξ MP) p = , (48) Note that higher-order expansions of the bow shock and mag- tot,MP 2µ 0 netopause curvatures contain more parameters and lead to with f = 2.44. This expression is valid near the stagna- more inner boundary conditions. The zeroth- and first-order tion point and determines the magnetopause distance to the systems are not able to determine the geometric parameters Earth’s center. Variations in y- and z-direction give additional self-consistently. conditions for determination of two curvature parameters. Each system of equations together with the boundary and Hence, the conditions of vanishing normal flow and field closure conditions represents a distinct approach to obtain component and the condition for the Earth’s magnetic field a magnetosheath solution, characterized by the order of the system and corresponding level of approximation. determine the geometric parameters xMS, cMP,y, cMP,z, cBS,y and cBS,z. The differential equations together with the closure conditions for the highest-order coefficients and the inner boundary conditions determine a unique solution for given 3 Relations to other models solar wind parameters. Our zeroth-order equations provide a model similar to Lees 2.5 Application of the method (1964). Extending Lees’s first considerations, we present an analytical approach for the solution of the equations, To calculate magnetosheath solutions with our method, first by which the physical quantities (density, pressure, mag- an appropriate order of approximation has to be chosen. For netic field, and velocity) and the magnetosheath geometry example, the second-order model gives a good approxima- (bow shock, magnetopause, and magnetosheath thickness) tion of the dayside magnetosheath up to several RE beside are obtained as a function of solar wind conditions only. The www.ann-geophys.net/31/419/2013/ Ann. Geophys., 31, 419–437, 2013 426 C. Nabert et al.: Method for solving the MHD equations complete derivation is presented in Appendix D. In the fol- 5 lowing section, we focus on an important result of this calculation: the analytical expression for the magnetosheath thick- 4 ness. Furthermore, we show that the models by Lees (1964),

[RE] Zwan and Wolf (1976), and Erkaev et al. (1996) are included 3

MS in our new method for different orders of approximation. x 2

D 3.1 Analytical expression for the magnetosheath 1 thickness

As presented in Appendix D, an analytical expression of the 1 2 3 4 5 6 7 8 magnetosheath thickness can be found in the zeroth-order r [mP /cm³] approximation that is solely dependent on solar wind con- SW ditions. This expression is given by 4 1xMP xMS = , (49) 4 + mBS (gu − 1) − 1 3

5 [RE] where gu = uSW/ux0(0), the subsolar velocity jump at the MS 2

bow shock, which is introduced in Appendix C (Eq. C3) as D a function of solar wind conditions. Furthermore, the mag- 1 netopause distance to the Earth’s center 1xMP is given as a function of solar wind conditions in Eq. (D23), and mBS is a measure for the solar wind magnetization given by 0 2 4 6 8 BSW [nT] 1 mBS = 1 − . (50) 1 + γ p0(0) Fig. 5. Magnetosheath thickness of our model by Eq. (49) (blue 2 pmag(0) curves) in comparison to the model by Farris and Russell (1994) 2 (red curves). The upper plot shows the density dependence of Here pmag(0) = B (0)/2µ0 is the postshock magnetic pres- z0 = −1 = × 5 sure and γ = 5/3 is the ratio of specific heats. The magne- the sheath thickness for uSW 400 km s , TSW 2 10 K, and BSW = 0nT. The lower panel offers the magnetic field depen- tosheath thickness is proportional to the magnetopause dis- −1 5 dence for uSW = 400 km s , TSW = 2 × 10 K, and ρSW = 8.4 × tance to the Earth’s center. A typical value for the mag- −21 −3 −3 10 kg m = 5 mP cm . netopause distance is about 10RE; therewith, the magnetosheath thickness results in 2.3RE in the hydrodynamic limit (m = 1) for high Mach numbers (g = 4). The mag- BS u solar wind’s density. The relations are similar; however, an netosheath thickness increases with solar wind magnetic offset of about 20 % is observed. In the lower panel, both re- field. It should be noted that the sheath thickness may be- lations’ dependence on solar wind magnetic field are shown. come negative in Eq. (49). This occurs for very low Mach Eq. (49) yields a steeper increase in the sheath’s thickness. number conditions, for which conditions our approximation However, for tyical solar wind conditions both relations pro- is not valid. A very often used empirical relation based on vide a similar magnetosheath thickness, in accord with ob- hydrodynamic considerations (Spreiter et al., 1966) is servations. Thus, Eq. (49) is a suitable analytical expression 1 for the magnetosheath thickness as a function of solar wind xMS = 1.1 1xMP. (51) conditions. gu This equation offers the same proportionality between mag- 3.2 The Lees approach netosheath thickness and magnetopause distance. Farris and Russell (1994) modified this relation and obtained Lees (1964) presented a first model considering the effects of a solar wind magnetic field on the magnetosheath on the 2 (γ − 1)M + 2 stagnation streamline. Starting from ideal MHD and con- x = 1.1 1,SW 1x , (52) MS + 2 − MP sidering the same solar wind conditions, he derives equa- (γ 1)(M1,SW 1) tions similar to our zeroth-order differential equations (20)– where M1,SW is the solar wind magnetosonic Mach number (23). Note that these equations do not depend on the bow (see also, Bennett et al., 1997). Figure 5 shows a comparison shock and magnetopause geometry. Neglecting the magnetic between this relation and Eq. (49). The upper panel shows the shear (→ Bx01Bz0 = 0) and assuming an axisymmetric flow magnetosheath thickness as a function of the unmagnetized (i.e., uy10 = uz01) in our zeroth-order system leads to Lees’s

Ann. Geophys., 31, 419–437, 2013 www.ann-geophys.net/31/419/2013/ C. Nabert et al.: Method for solving the MHD equations 427 equations. As noted, a relation to close the system is re- The MHD equations can be simplified assuming the total quired. He suggested determining the divergence function pressure given in the entire magnetosheath, which leads to uy10 as part of the solution along the magnetopause away a set of two-dimensional PDEs that describes thin magnetic from the stagnation streamline without going into details. flux tubes. Similar to our method, a parametrized bow shock Lees’s model is a zeroth-order approach to the flow problem; is used. The bow shock and the magnetopause curvatures we extended his considerations by calculating analytical so- are calculated self-consistently using magnetopause bound- lutions presented in Appendix D. As shown in our approach, ary conditions. These conditions are a vanishing normal ve- higher-order differential equations allow for a self-consistent locity and magnetic field component, and they assume that determination of the divergence functions influencing the so- the pressure satisfies the Newtonian approximation (Petrinec lution. and Russell, 1997). The magnetic string approach was used to investigate the magnetic barrier region, which is equivalent 3.3 The depletion model to the depletion layer region. It was applied to the magnetosheath region of Earth (Farrugia et al., 1997), Venus (Bier- The depletion model by Zwan and Wolf (1976) investi- nat et al., 1999), and Jupiter (Erkaev et al., 1996). Further gates, in more detail, density and magnetic field variations considerations with respect to anisotropic pressure are pre- on the stagnation streamline and also in its vicinity. Starting sented by Erkaev et al. (2000). from the time-dependent ideal MHD equations, they derive a system of two-dimensional partial differential equations describing the properties of a magnetic flux tube moving 4 Application to THEMIS observations through the magnetosheath. Approximations such as WBK and Taylor expansion restrict the calculations to the vicinity We apply our method to two different scenarios: The hydro- of the stagnation streamline. Although the model by Zwan dynamic magnetosheath transition – i.e., the solar wind mag- and Wolf (1976) depends on several limiting assumptions, netic field is zero – and the magnetic pile-up transition as the picture of the pile-up process was deepened. They con- shown in Fig. 1, where the solar wind is significantly magne- clude that during northward IMF plasma is squeezed out of tized. flux tubes close to the magnetopause, inducing a density decrease corresponding to a magnetic field increase in order to 4.1 Hydrodynamic transition maintain pressure balance. The region of density decrease, called the depletion region, is narrower than predicted by Without a solar wind magnetic field, we expect a field-free Lees’s simple model. In the model of Zwan and Wolf (1976), magnetosheath and results comparable to the hydrodynamic initial postshock divergence values also need to be deter- calculations by, e.g., Spreiter et al. (1966). We choose the −1 mined, similarly to our model. However, they obtain these solar wind velocity to be uSW = 310 km s , the solar wind −20 −3 values from numerical hydrodynamic calculations after Spre- density ρSW = 1.34 × 10 kg m , the solar wind mag- iter et al. (1966). To close their system, the functional relation netic field BSW = 0.02nT, and the solar wind temperature 5 of the pressure is assumed to match the hydrodynamic simu- TSW = 1.75×10 K. These conditions agree with actual solar lations, too. If we applied the same pressure condition to our wind conditions during THEMIS C’s crossing of the mag- model, we would achieve closure of our first-order system netosheath region on 24 August 2008. The probe traversed (20)–(28). This suggests a correspondence of our first-order the magnetosheath about 5.5RE in y-direction and about system with the model of Zwan and Wolf (1976) despite the 2RE in z-direction away from the stagnation streamline approaches in solving the MHD system of equations being (RE = 6371km). Solar wind measurements were obtained by rather different (we solve only ODEs). Note that the second- THEMIS B, which was closely following THEMIS C. order velocity coefficient functions of our model do not con- We apply our method to this event using the analytical tribute to the first-order equations and thus do not need to be (zeroth-order) approach presented in Appendix D, the full determined. zeroth-order approach with the equations solved numerically, and the second-order approach. The results of the calcula- 3.4 The magnetic string approach tions and the THEMIS C observations on 24 August 2008 are displayed in Fig. 6. The numerical zeroth-order ap- The magnetic string approach by Erkaev et al. (1988) trans- proach yields the smaller magnetosheath thickness. The an- fers the stationary ideal MHD equations into a different set alytical and second-order approach (solutions on the stagna- of partial differential equations (PDEs), using so-called ma- tion streamline) yield comparable thicknesses. The observed terial coordinates (e.g., Erkaev et al., 2003). These coordi- magnetosheath thickness, however, is about 0.4RE larger. nates are similar to frozen-in coordinates, with directions This difference is explained by the distance of the probe’s or- along the flow velocity, the magnetic field, and the electric bit to the stagnation streamline, reflected in better agreement field (Pudovkin and Semenov, 1977b). These coordinates are of the solution of the second-order system on the probe’s or- preferably used in ideal MHD due to the frozen-in theorem. bit (black) to observations as shown in Fig. 6. www.ann-geophys.net/31/419/2013/ Ann. Geophys., 31, 419–437, 2013 428 C. Nabert et al.: Method for solving the MHD equations

8 40 20

z0 B [nT] 0 6 3 40 30 p 20

[m /cm ]

0 10

r 0 E 4 100 80 y [R ] 60 40

x0 20 2 u [km/s] 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

X[R]MS E 0 Fig. 6. Magnetosheath solutions on the stagnation streamline for the analytical (green), the numerical zeroth order (red), and the second 012345 order approach (blue). The second-order solution on the probe’s or- x [RE ] bit (close to but not exactly on the stagnation streamline) is shown in black, the corresponding THEMIS data of 24 August 2008 in grey. Fig. 7. Magnetosheath streamlines with the velocity obtained by solving the second-order approach. The parabolic bow shock (red) and the magnetopause (green) are delineated. The analytical magnetic ﬁeld diverges at the magnetopause, due to the linear approximation of the velocity’s x- 8 component. A slight density increase is apparent in all solutions but the analytical one. This results from the conver- sion of dynamic pressure, neglected only in the analytical calculations, into gas pressure. However, the measured den- 6 sity increases at a higher rate than that given by the models, which is attributed to a density increase in the solar wind

density of about 15 % during the observation time. All cal- E 4 culated velocities decrease nearly linearly. On the stagnation

y [R ] 33 streamline the velocity’s x-component vanishes, but close to it a ﬁnite value remains, vanishing after the magnetopause. 2 31

This behavior is in accord with the measured data. p

The second-order approach reveals the velocity distribu- [m /cm³] tion within the three-dimensional magnetosheath geometry. 29 The corresponding streamlines in the xy-plane are displayed 0 in Fig. 7. The density distribution in the xy-plane is shown 01234 5 in Fig. 8, which shows a decreasing density away from the x [RE ] stagnation streamline. Although these density variations ap- pear to be small, corresponding pressure gradients affect the Fig. 8. Magnetosheath density distribution obtained by solving the solution significantly. second-order approach. The parabolic bow shock (red) and the mag- The results presented are in agreement with the THEMIS netopause (green) are delineated. observations as well as numerical calculations by Spre- iter et al. (1966) over a large part of the dayside magnetosheath. Our three-dimensional solution of the second-order are compared to results of our method using different orders approach with self-consistently computed bow shock and of approximation. The data shown are just a part of the com- magnetopause geometries yields results that best match the plete magnetosheath observation. About 1.5h before the bow observations of all models considered. shock crossing shown in the figure, two further bow shock crossings were registered by the THEMIS C spacecraft. Dur- 4.2 Northward IMF transition ing the whole time interval, the solar wind velocity was about 325 km s−1, the temperature 2.1 × 105 K, and the solar wind Without magnetic reconnection, the IMF piles up in the mag- magnetic field between 5.0nT and 5.5nT. Note that the magnetosheath against the magnetopause. THEMIS C observa- netic field is purely northward with minor deviations of less tions on 29 October 2009 (Fig. 1), showing such a pile-up, than 45◦ (in the last part of the transition even less than 10◦)

Ann. Geophys., 31, 419–437, 2013 www.ann-geophys.net/31/419/2013/ C. Nabert et al.: Method for solving the MHD equations 429

3 6 4.6 4 æ p 4.4

[m /cm ]

2 E æ æ SW 4.2 r 0 40 æ æ 5.5 nT

MS 4.0

3 30 X[R] æ æ 20 3.8 p 4.5 nT æ æ [m /cm ] 10 3.6 æ æ

r 0 æ 0400 0500 0600 0700 0800 UT 3.4

Fig. 9. The solar wind density measured by THEMIS B and the 16.5 æ density measured by THEMIS C crossing the bow shock several æ

times. The absolute values of the solar wind density do not ﬁt well. E 16.0 This might be explained by a non-vanishing spacecraft potential; æ 15.5 æ 5.5 nT

however, the relative variations can be considered. Note that the BS density jump measured by THEMIS C at about 05:30 UT is related X [R ] æ

D æ to a mode change of the ESA plasma instrument of the spacecraft 15.0 æ (McFadden et al., 2008). 4.5 nT æ 14.5 æ æ

æ and the satellite’s trajectory is located close to the stagna- 4 5 6 7 8 9 3 tion streamline (distance less than 2RE) as required for our rSW [mp /cm ] method in the approximations presented here. However, the solar wind density varied signiﬁcantly during the observation Fig. 10. Calculations of the bow shock distance from the Earth’s time, which violates the assumed stationarity of our model center and the magnetopause thickness as a function of the solar with the consequences discussed below. Time-shifted solar wind density, using the second-order approach. Two different solar wind (ion) density measurements (observed by THEMIS B) wind magnetizations, 4.5 nT and 5.5 nT, were considered. and corresponding THEMIS C density measurements, both obtained by the on-board ESA plasma instrument (McFad- den et al., 2008), are shown in Fig. 9. Immediately after the 50 ﬁrst two bow shock crossings at 04:30 UT and 04:50 UT, the 40 30

z0 solar wind density increases, causing an inward motion of B [nT] 20 bow shock and magnetopause. Consequently, the THEMIS C spacecraft reenters the solar wind at about 05:00 UT. 3 20

The bow shock distance to the Earth’s center and that of p 10 the magnetosheath thickness of the second-order solution are [m /cm ] 0 0 shown in Fig. 10. Both quantities decrease with increasing r density. This is also consistent with the results of our an- 80 60 alytical approach (Eqs. 49 and D23). Additionally, Fig. 10 40 20 shows that the effect of the magnetic ﬁeld variability during x0

u [km/s] 0 the event discussed is negligible when compared to changes 0 1 2 3 4 in density. X[R]MS E As can be seen in Fig. 9, a solar wind density decrease preceded the bow shock crossing of THEMIS C at 06:05 UT. Fig. 11. Magnetosheath solutions on the stagnation streamline for This causes an outward motion of the bow shock. Therefore, the analytical (green), the numerical zeroth-order (red), and the the velocity behind the bow shock shown in Fig. 1 is about second-order approach (blue). 20 km s−1 lower than expected by the Rankine–Hugoniot relations (Eqs. 31–36), which are only valid for a stationary situation. Consequently, the bow shock and magnetopause netopause surface can flow in two dimensions at the surface, position and the magnetosheath thickness varied during the whereas the magnetic field can only escape in one direction, first part of the observation. For this reason, the length scales because field lines are already one-dimensional. Therefore, of the observations might not be comparable to our calcula- more plasma escapes than magnetic field in front of the mag- tions. The results of our analytical approach, our zeroth-order netopause, leading to a pile-up of the magnetic field. Note as well as our second-order approach are presented in Fig. 11. that the pile-up region exists in the hydrodynamic case above, In all cases, a magnetic pile-up is predicted. As explained too; however, the pile-up region is located closely to the mag- by Siscoe et al. (2002), the magnetosheath plasma at the magnetopause. www.ann-geophys.net/31/419/2013/ Ann. Geophys., 31, 419–437, 2013 430 C. Nabert et al.: Method for solving the MHD equations

Again, the magnetic field of the analytical approach di- Table A1. Symmetries of the situation considered. Axisymme- verges at the magnetopause due to the linear velocity ap- try of a quantity f (x,y,z) to the xy-plane means f (x,y,−z) = proximation made. Furthermore, the numerical zeroth-order f (x,y,z), point symmetry means f (x,y,−z) = −f (x,y,z), anal- approach shows the most-extended sheath thickness. This be- ogous with respect to the xz-plane. havior is associated with a slow density decrease, which con- trasts with the density drop at the magnetopause in the other Phys. quantity Symmetry to xy Symmetry to xz calculations. The solution of the zeroth-order system is com- Bx axisym. pointsym. parable to that of the Lees (1964) model, which exhibits the By pointsym. pointsym. same slow density decrease, whereas the second-order solu- Bz axisym. axisym. tion is similar to the solution by Zwan and Wolf (1976). The ux axisym. axisym. pressure gradients in the second-order approach enhance the uy pointsym. axisym. plasma flow away from the stagnation streamline close to the uz axisym. pointsym. magnetopause. Consequently, the depletion layer thickness, ρ axisym. axisym. which is given by the density decrease at the magnetopause, p axisym. axisym. differs. It can be shown that the often-used empirical magnetopause model by Shue et al. (1998) and the bow shock model by Bennett et al. (1997) provide approximately the We apply our method to THEMIS observations. First, a same curvatures as our theoretical approach. Furthermore, hydrodynamic transition is examined featuring the typical the second-order approach offers a well-known behavior: Chapman-Ferraro picture of a jump in the magnetic field at The divergence of the velocity along the magnetic field drops the magnetopause. The second transition investigated shows at the magnetopause. This is related to the fact that the stag- an extended magnetic pile-up region resulting from a sig- nation point is extended to a line along the magnetic field at nificantly magnetized solar wind. Model results in terms of the magnetopause as discussed by Pudovkin and Semenov streamline distribution, the density contours, the magnetic (1977a). field behavior, and the magnetosheath geometry validate our approach. We are currently exploring the possibility to generalize 5 Conclusions our approach to situations at other planetary bodies such as comets (Glassmeier et al., 2007) or asteroids (Auster et al., A method to calculate the properties of the magnetosheath 2010). as part of the solar wind–geomagnetic field interaction by solving the stationary MHD equations for different levels Appendix A of approximation is presented. The solutions are calculated without considering time relaxation into a stationary state, Symmetry relations as required, for instance, by MHD simulations. A power series ansatz is introduced which transfers the stationary The symmetry relations of the MHD variables with respect MHD equations into a set of ordinary differential equations to the xy- and xz-coordinate planes are summarized in Ta- (ODEs). The number of equations is determined by the order ble A1. of approximation considered in the ansatz. In the process, for In the following, we sketch the derivation of these rela- simplicity, we restricted the approach to northward IMF and tions. In Sect. 2.1, we assume the solar wind to flow along the a solar wind along the stagnation streamline. The postshock x-axis with its magnetic field along the z-axis and the Earth’s values, used as boundary conditions for the ODEs, are re- dipole field given by Eqs. (6)–(8). Both the undisturbed solar ferred back to solar wind conditions via Rankine–Hugoniot wind and the dipole field satisfy the symmetry relations. The relations. Furthermore, typical outflow boundary conditions time evolution of the flow is governed by the time-dependent are chosen to determine a unique solution. The Earth’s dipole MHD equations, which read (e.g., Wu, 1992) field yields additional restrictions for the solution determining the magnetosheath’s geometry. ρ˙ + ∇ · (ρ u) = 0, (A1) A full analytical solution of our zeroth-order approach is presented, which results in an analytical expression of the magnetosheath thickness (Eq. 49). The models by Lees 1 (1964), Zwan and Wolf (1976), and Erkaev et al. (1988) ρ u˙ + ρ (u · ∇)u + ∇p − (∇ × B) × B = 0, (A2) are classified by different orders of approximation with re- µ0 spect to our method, revealing similarities and differences and providing a more detailed insight into magnetosheath phenomena. B˙ − ∇ × (u × B) = 0, (A3)

Ann. Geophys., 31, 419–437, 2013 www.ann-geophys.net/31/419/2013/ C. Nabert et al.: Method for solving the MHD equations 431

0 (Bx01uz01 − Bz0ux20 − Bz02ux0) − Bz02uy10 (B5) 1cz 0 ∇ · B = 0, (A4) −Bz0uy12 + By11uz01 + (Bz0ux0) = 0, xMS

! ! 0 + 0 + + 0 ρu2 γ 1 ρ0 (ux0ux20) ρ20ux0ux0 2ρ0ux20uy10 p20 (B6) ˙+∇· + p u − (u × B) × B = 0. (A5) + 0 − − 2 γ − 1 µ0 (Bz0Bz20) Bx21Bz0 Bx01Bz20 0 1cy Here ρ denotes mass density, u the fluid’s bulk velocity, p −2c˜yρ0uy10ux0 − Bx01Bz0 = 0, xMS the thermal gas pressure assumed to be isotropic, B the mag- −7 2 netic field, µ0 = 4π × 10 N/A the vacuum permeability, 2 2 0 + 0 + + 0 and = ρu /2 + B /(2µ0) + p/(γ − 1) the energy density ρ0 (ux02ux0) ρ02ux0ux0 2ρ0ux02uz01 p02 (B7) 0 with γ as the ratio of specific heats. Time derivatives are pre- +(Bz0Bz02) − 3Bx03Bz0 − Bx01Bz02 sented in dot notation. The time-dependent MHD equations 0 0 1cz conserve the symmetry relations in time. For example, ap- −2c˜z ρ0uz01u − Bz0B − Bx01Bz0 = 0. x0 z02 x plying the relations, the divergence term of the first MHD MS Eq. (A1) is axisymmetric with respect to the xy-plane as a function of z and axisymmetric with respect to the xz-plane Appendix C as a function of y. Thus, Eq. (A1) requires the density to be axisymmetric in time. Because the initial situation (solar Solving the Rankine–Hugoniot relations at the bow shock wind and Earth’s magnetic field) satisfies the symmetry relations and these relations are conserved in time, the solution The MHD moment values of the shocked solar wind plasma in its final stationary state needs to satisfy the symmetries, at the bow shock’s position at (x˜ = 0) are ρ(x˜ = 0,y,z), too. p(x˜ = 0,y,z), u(x˜ = 0,y,z), and B(x˜ = 0,y,z). Via the Rankine–Hugoniot relations (31)–(36), they are related to the solar wind conditions introduced in Sect. 2.1: ρSW, pSW, T T Appendix B uSW = (uSW,0,0) , and BSW = (0,0,BSW) . On the stagnation streamline, i.e., y = z = 0, only zeroth-order coeffi- Second-order equations cients of our ansatz (11)–(18) remain and only a solar wind flow normal to the bow shock surface has to be considered. The ansatz (11)–(18) is expanded up to the third order. Sub- This simple scenario results in the following solution of the stituting this ansatz in the MHD equations, we derived the Rankine–Hugoniot relations: zeroth- and first-order equations as presented in Sect. 2.1. Analogously, the second-order equations are obtained by ρSW = gρ ρ0(x˜ = 0), (C1) equating coefficients of the second order: 0 (ρ0ux20 + ρ20ux0) + ρ0 3uy30 + uz21 (B1) 0 pSW = gp p0(x˜ = 0), (C2) +ρ20 3uy10 + uz01 − 2c˜y ρ0uy10 1cy 0 − (ρ0ux0) = 0, xMS uSW = gu ux0(x˜ = 0), (C3) 0 (ρ0ux02 + ρ02ux0) + ρ0 uy12 + 3uz03 (B2) 0 +ρ02 uy10 + 3uz01 − 2c˜z (ρ0uz01) BSW = gB Bz0(x˜ = 0). (C4) 1cz 0 − (ρ0ux0) = 0, xMS Explicit analytical expressions for gρ, gp, gu, and gB depending on the sonic and Alfvenic´ Mach number of the solar 0 wind can be found in, e.g., Siscoe (1983). In the limit of high Bx01uy10 − By11ux0 + 2 Bz02uy10 + Bz0uy12 (B3) Mach numbers, gρ = 1/4, gp → 0, gu = 4, and gB = 1/4. 0 −2By11uz01 − 2c˜z Bz0uy10 = 0, To determine the boundary values at x˜ = 0 for the higher- order coefficient functions, we need to consider the situation beside the stagnation streamline. This requires taking −(B u + B u )0 − 3B u + B u (B4) z0 x20 z20 x0 z20 y10 z0 y30 the bow shock geometry into account, and, consequently, we 0 1cy 0 introduce curvilinear bow shock coordinates. Using the bow +2c˜y Bz0uy10 + (Bz0ux0) = 0, xMS shock parametrization (Eq. 9), the normal vector ξ BS and two www.ann-geophys.net/31/419/2013/ Ann. Geophys., 31, 419–437, 2013 432 C. Nabert et al.: Method for solving the MHD equations linearly independent tangential vectors τ BS,1 and τ˜ BS,2 with z respect to the shock are determined: BS 1 = − − T ux ξ BS 1, 2cBS,y y, 2cBS,z z , (C5) utSW SW nyz a uSW 1 T a τ BS,1 = 2cBS,y y, 1, 0 , (C6) ny

1 T τ˜ BS,2 = 2cBS,z z, 0, 1 , (C7) nz q q x = + P 2 2 = + 2 2 where nyz 1 t=y,z 4cBS,t t , ny 1 4cBS,y y , q Fig. C1. Projection of the solar wind velocity vector with respect to = + 2 2 and nz 1 4cBS,z z are normalization functions. It is curvilinear bow shock coordinates. convenient to orthonormalize the tangential vector τ˜ BS,2 with respect to the other coordinate vectors ξ and τ to sim- BS BS,1 Note that variations in tangential direction (of the compo- plify the following calculations: nents with respect to the shock’s surface), given by derivatives with respect to α, require the opposite. As a conse- τ˜ BS,2 − τ BS,1 · τ˜ BS,2 τ BS,1 τ BS,2 = . (C8) quence of the negligible normal magnetic field component, τ˜ − τ · τ˜ τ BS,2 BS,1 BS,2 BS,1 the Rankine–Hugoniot relations require continuity of the tangential velocity components of the solar wind across the bow The three coordinate vectors ξ , τ , and τ form an BS BS,1 BS,2 shock (Siscoe, 1983): orthonormal system at the bow shock’s surface. First, we consider the shocked velocity at the bow shock 2cBS,y uSW y plane, which can be expressed by uτ,BS,1 = uSW · τBS,1 = , (C14) ny X u(x˜ = 0,y,z) = uξ,BS ξ BS + uτ,BS,i τ BS,i. (C9) i=1,2 2cBS,z uSW z uτ,BS,2 = uSW · τBS,2 = . (C15) ny nyz The coefficient functions uξ,BS, uτ,BS,1, and uτ,BS,2 need to be determined. Consider the xz-plane; parametrization of the Relation (C3) holds for a normal solar wind velocity with bow shock curve by introducing the angle α(z) (see Fig. C1) a tangential magnetic field. Because the magnetic field is allows for the representation of the solar wind velocity’s and nearly tangential close to the subsolar point, we extent the magnetic field’s component normal to the bow shock as scope of this relation to the vicinity of the stagnation streamline using the normal component of the velocities: uξ,SW = cos(α(z))uSW ≈ uSW, (C10) uSW · ξ BS = gu u(x˜ = 0,y,z) · ξ BS. (C16) With the solar wind flowing along the x-axis, Eqs. (C3), (C5), Bξ,SW = sin(α(z))BSW ≈ α(z)BSW, (C11) and (C9) yield and the tangential components as ux0(x˜ = 0) uξ,BS = . (C17) nyz uτ,SW = sin(α(z))uSW ≈ α(z)uSW, (C12) Substituting the coefficients (C14), (C15), and (C17) and the coordinate vectors ξ BS, τ BS,1, and τ BS,2 in Eq. (C9), the initial velocity writes Bτ,SW = cos(α(z)) BSW ≈ BSW. (C13) 1u nyz T u(x˜ = 0,y,z) = uSW − 1, 2cBS,y y, 2cBS,z z , Here, Taylor expansion up to the first order in α was used. nyz 1u Apparently, the solar wind’s normal velocity and tangential (C18) magnetic field remain approximately constant close to the stagnation streamline (small α), whereas the normal mag- where 1u = uSW − ux0(x˜ = 0). Expanding this relation into netic field component and the tangential velocity vanish. a Taylor series with respect to y and z around the stagnation

Ann. Geophys., 31, 419–437, 2013 www.ann-geophys.net/31/419/2013/ C. Nabert et al.: Method for solving the MHD equations 433 streamline and equating coefficients with the ansatz (16)– Appendix D (18) yields values for the coefficient functions that can be expressed as Zeroth-order approach ˜ = = i+j i j ul i j (x 0) fu(i,j)2 cBS,ycBS,z1u, (C19) To obtain a first approach to the physics and the correspond- where l ∈ {x,y,z}, the indices i and j label the coefficient ing solutions of the MHD equations, the zeroth-order approach with its Eqs. (20)–(23) is solved analytically. The co- functions as in the ansatz (16)–(18), and fu(i,j) denotes the sign-function: efficient functions of the first order are set to constant postshock values: Bx01 = Bx01(x˜ = 0), uy10 = uy10(x˜ = 0), and + + ∈ { } 1, i j 1 uz01 = uz01(x˜ = 0). The latter two expressions are called di- fu(i,j) = . (C20) −1, i + j ∈ {2,3} vergence parameters describing the amount of flow diversion The magnetic field can be treated in a similar manner. Its in the respective tangential direction. The system is solved postshock values are expressed as follows: for arbitrary solar wind conditions. X B(x˜ = 0,y,z) = Bξ,BS ξ BS + Bτ,BS,i τ BS,i. (C21) D1 General solutions i=1,2 For simplification, we neglect magnetic shear, meaning mag- The normal component of the magnetic field is always con- netic fields in z-direction only, and consequently B (x˜ = tinuous through the bow shock, which yields x01 0) = 0. The shock front decelerates and compresses the flow; −2BSWcBS,zz dynamic pressure is converted into gas and magnetic pres- Bξ,BS = . (C22) nyz sure. Most often, in the magnetosheath, the dynamic pressure term in the momentum equation of zeroth order (22) can be Similar to the velocity calculation, we extend the validity of neglected (the minor effects of the dynamic pressure are dis- relation (C4) to the vicinity of the stagnation streamline: cussed in Sect. 4.1). This yields BSW · τ BS,i = gB B(x˜ = 0,y,z) · τ BS,i, (C23) 0 1 γ + 0 = ∈ { } k ρ0 Bz0 Bz0 0, (D1) where i 1,2 . With the solar wind magnetic field along the µ0 z-axis, Eqs. (C4), (C6), (C8), and (C21) give one for which the zeroth-order adiabatic law (Eq. 23) has been Bτ,BS,1 = 0, (C24) used. Integration gives

B2 B (0)2 k ργ + z0 = kρ (0)γ + z0 , (D2) Bz0(0)ny 0 0 B = . (C25) 2µ0 2µ0 τ,BS,2 u z with the integration constant on the right side to be deter- Finally, the postshock magnetic field is mined at the bow shock. The zeroth-order continuity Eq. (20) 1B and the zeroth-order Frozen-in theorem (21) can be written B(x˜ = ,y,z) = c z,− c c y z, 0 2 2 BS,z 4 BS,y BS,z (C26) as nyz 0 2 2 2 2 !T u = −(uy10 + uz01) + , (D3) Bz0(0) + 4BSWc y + 4BSWc z x0 BS,y BS,z , 1B 0 where 1B = BSW − Bz0(x˜ = 0). The field boundary values ux0 = −uy10 − δ, (D4) on the stagnation streamline result from the derivatives of = − = Eqs. (13)–(15) at (x˜ = 0,y = 0,z = 0) and of Eq. (C26): where ux0∂xρ0/ρ0 and δ ux0∂xBz0/Bz0. The northward magnetic field and the density are positive functions. = i+j i j Bl i j (0) fB (i,j)2 cBS,ycBS,z1B, (C27) Furthermore, and δ are always positive because the magnetic field increases in the magnetosheath, corresponding to in which a decreasing density as given by Eq. (D1). The positive +1, i + j ∈ {1,2} and δ, and Eqs. (D3) and (D4) also provide lower and upper fB (i,j) = , (C28) −1, i + j ∈ {3} bounds for the velocity decrease: where l ∈ {x,y,z}, and the indices i and j label the coeffi- 0 −(uy10 + uz01) < u < −uy10. (D5) cient functions as in the ansatz (13)–(15). x0 Note that Eqs. (C19) and (C27) satisfy the Frozen-in con- Expanding the velocity in x˜ around the bow shock reads servation Eq. (29) as expected. Under the assumptions ap- ∞ plied in these calculations, the boundary values for the den- X i ux0 = ux0(0) + ai x˜ . (D6) sity and pressure coefficients ρ20, ρ02, p20, and p02 are zero. i=1 www.ann-geophys.net/31/419/2013/ Ann. Geophys., 31, 419–437, 2013 434 C. Nabert et al.: Method for solving the MHD equations

Note that the argument of the coefficient functions is with re- After substitution of Eq. (D11), we obtain spect to x˜, so u (0) = u (x˜ = 0) and is not explicitly noted x0 x0 − further. First, we take only the first order in x˜ into account: ux0(0) xMS = , (D13) uy10 + mBS uz01 ux0 = ux0(0) + a x.˜ (D7) with Here a = a1 for the expansion coefficient was used. An up- 1 per bound for the error of the linear approach can be esti- mBS = 1 − . (D14) mated using Eq. (D5). However, an almost-linear decrease of 1 + γ p0(0) 2 pmag(0) the velocity along the stagnation streamline is also in accord with global MHD simulation results by, e.g., Wu (1992) or The parameter mBS is a measure for the solar wind magneti- Wang et al. (2004), and also agrees with the observations dis- zation. For the unmagnetized solar wind, we obtain mBS = 1. played in Fig. 1. Note that in the hydrodynamic limit the den- Note that the magnetosheath expression results from ideal sity is constant (incompressible fluid) because of Eq. (D1). MHD equations, neglecting magnetic shear (Bx01(0) = 0) This leads to = 0 with the consequence of a linear veloc- and the dynamic pressure terms, and using a series ansatz T ity decrease with a = −uy10 −uz01. Substituting the velocity for the velocity (u = (ux0(0) + a x,˜ uy01(0), uz10(0)) ). ansatz (D7) in the zeroth-order continuity Eq. (20) and the Higher-order contributions of the velocity to the magne- Frozen-in theorem (21) gives tosheath thickness can also be obtained. For a second-order ansatz for the velocity in x˜-direction, similar calculations u +u +a − y10 z01 ux0 a yield the second-order expansion coefficient: ρ0 = ρ0(0) , (D8) ux0(0) 2 + 2 2γ pmag(0)p0(0) pmag(0) p0(0) uz01 a2 = . (D15) p ( ) + γp ( )3 u ( ) u +a 2 mag 0 0 0 x0 0 − y10 ux0 a B = B (0) . (D9) In the limit p (0) >> p (0) the expression is approxi- z0 z0 u (0) 0 mag x0 mated by: An expression for the expansion parameter a is obtained by p (0) u2 substituting the results for ρ0 and Bz0 in the simplified mo- mag z01 a2 ≈ 2γ . (D16) mentum equation (D2): p0(0) ux0(0)

 u +u +a γ − y10 z01 Consistently, a2 vanishes in the limit of hydrodynamics (i.e., ux0 a = k ρ0(0)  (D10) pmag(0) 0) and a linear velocity decrease remains. ux0(0) D2 Boundary conditions  u +a 2 − y10 1 ux0 a + Bz0(0)  The divergence parameters are set to their bow shock values, 2µ0 ux0(0) which are given by Eq. (C19): 2 γ Bz0(0) = = kρ0(0) + . uy10 2cBS,y1u, (D17) 2µ0 To solve this equation, both sides are expanded into Taylor series around x˜ = 0. Equating coefﬁcients of the lowest non- uz01 = 2cBS,z1u. (D18) vanishing order yields

  The geometry parameters cBS,y and cBS,z need to be deter- 1 mined. a = −uy10 − uz01 1 − . (D11) + γ p0(0) The dynamic pressure of the solar wind is completely con- 1 p ( ) 2 mag 0 verted into gas and magnetic pressure in the magnetosheath. = 2 However, pressure is not a conserved quantity and, thus, the Here pmag(0) Bz0(0)/2µ0 denotes the postshock magnetic pressure and γ = 5/3 the ratio of specific heats. solar wind pressure differs by a factor K from the magne- This result is used to obtain an expression for the mag- topause pressure on the stagnation streamline: netosheath thickness x . The magnetopause is defined MS K ρ u2 = p , (D19) by a vanishing flow velocity on the stagnation streamline SW SW tot,MP (ux(xMS) = 0). Equation (D7) thus yields where the total magnetopause pressure ptot,MP is given by u (0) Eq. (48). Here K ≈ 0.89 holds for a broad range of solar x = x0 . (D12) MS a wind conditions (Kivelson and Russell, 1995).

Ann. Geophys., 31, 419–437, 2013 www.ann-geophys.net/31/419/2013/ C. Nabert et al.: Method for solving the MHD equations 435

The pressure balance equation above is valid at the stag- where 1u = uSW − ux0(0) was used, as defined above. nation point (x˜ = xMS,y = 0,z = 0) only. In its vicinity, only The velocity divergence within the xz-plane remains to be the velocity component normal to the local magnetopause is determined. Using Eq. (48) to describe the magnetospheric important. The dynamic pressure with respect to the magne- pressure within the xz-plane leads to topause normal (Eq. 41) thus reads 2 2 22 2 2 2z − 1x − 6cMP,z 1x z f M 2 ptot,MP,xz = , (D27) u 2 5 µ p = ρ u2 = ρ SW . 1 + 4c z2 1x2 + z2 2 0 dyn,ξ SW SW,ξ 2 (D20) MP,z nMP which yields This expression holds for both the y- and z-directions. Here- with, the pressure equilibrium writes 1 cMP,z = . (D28) 21xMP K p = p , (D21) dyn,ξ tot,MP Assuming, again, the same functional expression for the bow where Eq. (48) was used and f = 2.44. Note that by setting shock curvature gives K = 1 the pressure relation becomes equivalent to the one 1 1 cBS,z = = , (D29) used by Mead and Beard (1964). 21xBS 2(1xMP + xMS) Using the magnetopause parametrization (10), the Earth’s dipole field (Eqs. 6–8), the magnetopause normal vector and using Eq. (D18) the divergence parameter uz01 reads (Eq. 41), and the magnetospheric pressure (Eq. 48) in the uSW − ux0(0) uz01 = . (D30) xy-plane yield 1xMP + xMS f 2 M2 Altogether, after some tedious calculations, Eqs. (D26) and p = . tot,MP,xy 3 (D22) (D30) provide suitable estimates for flow divergence just af- 2µ0 − 2 2 + 2 (1xMP cMP,y y ) y ter the bow shock. The numerator is determined by the velocity jump across the bow shock, and the denominator by the Substituting ptot,MP in Eq. (D19) by this pressure relation, the bow shock’s distance to the Earth’s center. The ratio of both magnetopause distance 1xMP at the stagnation point (y = 0) is obtained, given by the well-know expression (e.g., Pu- divergence parameters is dovkin et al., 1998) 4 u = u . (D31) y10 5 z01 ! 1 f 2 M2 6 1x = . This equation describes the asymmetry of the divergence in MP 2 (D23) 2µ0KρSWuSW y- and z-direction in the approach discussed. The expressions for the divergence still depend on the The geomagnetic dipole moment is given by M = 8 × magnetosheath thickness. With Eqs. (C3), (D13), (D26), and 15 3 10 Tm . The general pressure balance (D21) along with (D30) we get Eqs. (D20) and (D22) is solved using Taylor expansion with 1x respect to y around the stagnation point up to the second MP xMS = . (D32) 4 + − − order. Equating coefficients allows for determination of the 5 mBS (gu 1) 1 magnetopause curvature: √ This relation for the magnetosheath thickness depends on −3 + 21 0.4 the solar wind conditions only. The velocity jump at the cMP,y = ≈ . (D24) 41xMP 1xMP bow shock gu (Eq. C3), the magnetopause distance form the Earth’s center 1xMP (Eq. D23), and the solar wind magneti- This expression now enables us to estimate the parabolic co- zation mBS (Eq. D14) are explicit functions of the solar wind efficient cBS,y, which we need to know for an estimate of the parameters. Equations (C1)–(C4) are used to refer the bow postshock divergence. We assume the same functional ex- shock conditions ρ0(0), p0(0), Bz0(0), and ux0(0) back to pression for the curvature of the bow shock as derived for the solar wind conditions. Hence, Eqs. (D26) and (D30) can also magnetopause above: be written as explicit functions of these solar wind parame- 0.4 0.4 ters. cBS,y = = . (D25) 1xBS 1xMP + xMS D3 Application The divergence parameter u is now calculated with y10 The flow velocity, the density, and the magnetic field are Eq. (D17): given by Eqs. (D7), (D8), and (D9), with the expansion co- uSW − ux0(0) efficient given by Eq. (D11). The postshock boundary con- uy10 = 0.8 , (D26) ditions u (0), ρ (0), B (0), and p (0) are related to the 1xMP + xMS x0 0 z0 0 www.ann-geophys.net/31/419/2013/ Ann. Geophys., 31, 419–437, 2013 436 C. Nabert et al.: Method for solving the MHD equations solar wind conditions via Eqs. (C1–C4). Furthermore, the di- J. Geophys. Res., 99, 17681–17689, doi:10.1029/94JA01020, vergence parameters uy10 and uz01 were set constant to their 1994. postshock boundary values given by the relations (D26) and Farrugia, C. J., Erkaev, N. V., Biernat, H. K., Lawrence, G. R., and Elphic, R. C.: Plasma depletion layer model for low Alfven Mach (D30). This requires the magnetopause distance 1xMP determined by Eq. (D23) and the magnetosheath thickness by number: Comparison with ISEE observations, J. Geophys. Res., Eq. (D32). 102, 11315–11324, doi:10.1029/97JA00410, 1997. Glassmeier, K.-H., Boehnhardt, H. , Koschny, D., Kuehrt, E., and Richter, I.: The ROSETTA Mission: Flying towards the Acknowledgements. This work was financially supported by the Origin of the Solar System, Space Sci. Rev., 128, 1–21, German Ministerium fur¨ Wirtschaft und Technologie and the doi:10.1007/s11214-006-9140-8, 2007. Deutsches Zentrum fur¨ Luft- und Raumfahrt under contracts Glassmeier, K.-H., Auster, H.-U., Constantinescu, D., Fornacon, K.- 50OC1001, 50OC1102, and 50QP1001. We acknowledge NASA H., Narita, Y., Plaschke, F., Angelopoulos, V., Georgescu, E., contract NAS5-02099 and V. Angelopoulos for use of data from the Baumjohann, W., Magnes, W., Nakamura, R., Carlson, C. W., THEMIS Mission. Specifically: C. W. Carlson and J. P. McFadden Frey, S., McFadden, J. P., Phan, T., Mann, I., Rae, I. J., and for use of ESA data. Vogt, J.: Magnetospheric quasi-static response to the dynamic Topical Editor L. Blomberg thanks S. Fujita and one anonymous magnetosheath: A THEMIS case study, Geophys. Res. Lett., 35, referee for their help in evaluating this paper. L17S01, doi:10.1029/2008GL033469, 2008. Kivelson, M. 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