Magnetohydrodynamic Shocks Revisited: Magnetically Constraining the Upstream Solar Wind Condition

Magnetohydrodynamic (MHD) shocks are commonly found in various space and astrophysical systems. One often encounters the problem that only the magnetic field data are available when studying interplanetary shocks and planetary bow shocks in the solar system, and not the plasma data due to instrumental and/or telemetry limitations. We aim to reveal the mathematical structure of the MHD Rankine–Hugoniot relation under the condition that magnetic field changes across the shock are known, which can be devised into a diagnosis tool to constrain the shock parameters. The perturbative solution is already known for the Rankine–Hugoniot equation. The solution is combined with the condition of the de Hoffmann–Teller frame for the vanishing convective electric field, and we obtain a two-dimensional matrix equation for the density jump and the Alfvén Mach number. The lesson is that the shock parameters (density jump and Alfvén Mach number) are analytically obtained by inverting the MHD Rankine–Hugoniot equation under the condition that the plasma beta is either a priori set or known. Further numerical studies indicate that the presented shock jump estimation works particularly well for quasi-perpendicular shocks. Moreover, a test case with the Earth bow shock crossing data supports our theoretical development. The presented method opens the door to determining or constraining the shock parameters even if the data are limited to the magnetic field only.


Introduction
In the research field of in situ observations of the planetary magnetic and plasma environment using orbiting or flybypassing spacecraft, one often encounters the problem that only the magnetic field data are available for detailed analyses while the plasma data (ions and electrons) are limited in many aspects, such as the time resolution (e.g., Venus Express mission at Venus), the energy range (e.g., Galileo mission at comet P/Halley), the field of view (e.g., MESSENGER mission at Mercury), and/or in case of an unexpected failure of the instrument (e.g., ion detector on Cluster 2 at Earth).A conventional approach is that one determines the shock normal direction from the magnetic field data by applying the coplanarity theorem, indicating that the upstream field, the downstream field, and the field jump occur in the same plane and there is no rotation of the field around the shock normal direction.By revisiting the magnetohydrodynamic(MHD) jump conditions, the so-called Rankine-Hugoniot relation, we noticed that the density jump and the Alfvén Mach number can be determined from the magnetic field data given that the adiabatic constant (or polytropic index) γ and the plasma parameter beta are known or set in the analysis, which we call the magnetic diagnosis of the shock condition.

Magnetic Diagnosis Formula
The MHD Rankine-Hugoniot relation can be reduced to a cubic algebraic equation.The solutions of the cubic equation are obtained by Zhuang & Russell (1981) using Cardano's formula and by Grabbe & Cairns (1995) using a perturbation method.In the perturbative approach, the cubic equation is solved to the firstorder and the second-order accuracy for the jump X: The jump variable represents the density jump from the downstream side to the upstream, and is equivalent to the velocity jump from the upstream side to the downstream when the mass flux conservation (continuity equation) across the shock is used, where ρ (u) is the upstream mass density, ρ (d) the downstream mass density, u n u the upstream flow velocity normal to the shock, and u n u the downstream velocity normal to the shock, In the first-order closure, the jump variable is analytically obtained (Zhuang & Russell 1981;Grabbe & Cairns 1995) as where M A is the Alfvén Mach number, M s the sonic Mach number, γ the adiabatic constant (assumed to be 5/3), and θ (u)  the angle of the upstream magnetic field direction to the shock normal direction.The Alfvén Mach number and the sonic Mach number are defined, respectively, as The Alfvén Mach number refers to the Alfvén speed m r = V B A 0 determined by the magnetic field magnitude B, the permeability of free space μ 0 , and the mass density ρ.
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The sonic Mach number refers to the sound speed g r = c p s , which is a function of the adiabatic constant γ, the thermal pressure p, and the mass density.It is useful to introduce the plasma parameter beta as where the sonic Mach number is expressed as a function of the plasma parameter beta and the Alfvén Mach number as gives a constraint between the jump X and the Alfvén Mach number M A when the plasma beta is known, and is a valid expression in the normal incident frame (the upstream flow is normal to the shock front) as well as in the proper frame introduced by de Hoffmann & Teller (1950, hereafter HT), in which the flow velocity is aligned with the magnetic field.We find out that Equation (7) is conveniently solved in the HT frame.The ratio of the tangential to the normal components (or the angle from the normal direction) is the same between the flow velocity and the magnetic field in the HT frame (indicating no convective or motional electric field): Equation (9) holds on both sides of the shock.The tangential component of the momentum balance is given as Note that the mass flux (ρu n ) and the normal component of the magnetic field (B n ) are conserved quantities across the shock.We now normalize Equation (10) to the normal component of the flow in the downstream region, u n d ( ) .The left-hand side of Equation (10) is normalized as where the jump variable X is introduced with the help of Equation (1) in the second term.The right-hand side of (10) is normalized as where Equations (1) and (9) are used.By combining Equation (11) and Equation (13), we obtain the normalized form of Equation (10) as Equations ( 7) and (14) form a closed set of linear equations for the jump variable X and the inverse square of Alfvén Mach number - M A 2 u ( ) in the upstream region, given that the plasma beta is known or set in the analysis.We arrange Equations ( 7) and ( 14) in a matrix form with two variables (the density jump and the Alfvén Mach number) as The coefficient c is Equation ( 15) serves as an analysis tool to determine the jump variable and the Alfvén Mach number for the magnetic field jump across the shock under two caveats.First, the matrix in Equation (15) undergoes a singular behavior at a certain value of beta.The singularity occurs when the determinant of the matrix vanishes, that is, In the case of singular behavior, the jump condition (Equation ( 7)) and the momentum balance in the HT frame (Equation ( 14)) are identical to each other.Second, the value of beta must be known or set.From the experimental point of view, the jump condition (Equation ( 15)) gives a constraint to the jump variable and the Alfvén Mach number by solving Equation (15) in a wider range of beta.

A Test against the In Situ Measurement
The notion of MHD Rankine-Hugoniot inversion is tested against in situ shock crossing data recorded by the Cluster-1 Fluxgate Magnetometer (Balogh et al. 2001) and the Hot Ion Analyzer (Rème et al. 2001) as a proof of concept.The left panels in Figure 1 display the overview of the magnetic field and plasma data of Cluster-1 Earth's bow shock crossing on 2002 February 12 in a time series format: magnetic field component tangential B t (in black) and normal B ]n (in gray) to the shock front determined by the magnetic field coplanarity across the shock, the ion bulk velocity tangential u t and normal u n to the shock front, the ion number density, and the ion temperature parallel (in black) and perpendicular to the mean magnetic field.The shock transition is identified at about 16:10 UT with a precursor foreshock wave activity from 15:50 UT to 16:10 UT.The shock represents an oblique shock with a magnetic field angle of121.38°to the shock normal in the upstream region (taken from an interval of 15:30-15:45 UT) and99.34°in the downstream region (16:13-16:22 UT).The ion beta is about 1.02 in the solar wind.The Alfvénic Mach number is about M A ≈ 7.03 in the solar wind.The density jump across the shock is about X ≈ 0.234.
The MHD Rankine-Hugoniot inversion method (Equation ( 15)) is applied using the magnetic field angles in the upstream and downstream regions, and the density jump and the Alfvén Mach number are numerically evaluated as a function of plasma beta in the range from 0.1 to 10 (Figure 1, right panels).The density jump is insensitive to the value of beta and remains nearly constant at about 0.28.The Alfvén Mach number is moderately sensitive to the value of beta and exhibits an increasing sense of Mach number at higher beta.The estimated Alfvén Mach number is about 8.0 at the observed value of beta (β ≈ 1.02), which roughly agrees with the estimation using the magnetic field and plasma data (about 7.0).Even if the value of beta is not known, the MHD Rankine-Hugoniot inversion method gives a reasonable estimate of the density jump and Alfvén Mach number for a given range of beta.

Range of Plasma Beta
Figure 2 exhibits a histogram of near-Earth solar wind estimated by OMNI propagation to the Earth bow shock position for the years 2012-2023.One sees that the beta values vary significantly, yet the most probable values are located in the interval from β = 0.5 and β = 4.Even though our Rankine-Hugoniot inversion method cannot uniquely or exactly determine the shock variables (the density jump and the Alfvén Mach number), one may still reasonably constrain the variables to a most likely range using the statistical knowledge on the beta.
For quasi-perpendicular and oblique shock configurations, Equation (15) yields solutions independent of the solar wind plasma beta.As expected, the jump parameter X and Alfvénic Mach number M A increase with the plasma beta and/or increasing downstream shock normal angle.
In the case of a quasi-parallel shock, however, the determinant in Equation (21) vanishes for β < 1.2 and the matrix in Equation (15) becomes singular (dashed-dotted line in right panels).The lower limit of the plasma beta shows the limitation of the first-order solutions of Equation (3).A physical explanation for this lower limit might be that in the special case of parallel  MHD shocks, the magnetic field does not affect the shock and the bow shock becomes identical to a hydrodynamic shock of neutral fluids.In such a case, the shock is entirely characterized by the sonic Mach number M s and can only emerge when > M 1 s 2 .Following Equation (8), this implies that βγ/2 > 1 (β > 1.2 for γ = 5/3) needs to be fulfilled to create a fast magnetosonic shock.With an increasing upstream shock normal angle, the Alfvénic Mach number becomes the dominating parameter for the shock formation, and the limit of the plasma beta is shifted toward lower plasma betas and subsequently vanishes for θ (u) 35°.It should be mentioned that Equation (15) also yields results for plasma betas below the lower limit (here β < 1.2).However, these solutions are not physically meaningful for planetary bow shocks.
In practice, θ (d) is directly given by the downstream magnetic field measurements.By assuming a reasonable range for the solar wind plasma beta it is therefore possible to approximate the jump parameter X and the Alfvénic Mach number M A directly from Equation (15).
It should be mentioned that the accuracies of X and M A depend on the accuracies of the determined shock normal angles θ (u) and θ (d) , which, in turn, depend on (1) the accuracy of the magnetic field measurements, which is expected to be in the range between 0.1 and 1 nT, depending on the solar wind observation time (see, e.g., Plaschke 2019).For an accuracy of 0.15 nT, which can be achieved on less than 40 hr of solar wind observations, the error of the magnetic field direction remains below1.7°forambient fields that are larger than 5 nT (e.g., typical interplanetary magnetic field condition at 1 au).Hence, measurement errors typically have a minimal impact; (2) the correct determination of the shock normal direction.Commonly, the minimum variance analysis (MVA; Sonnerup & Cahill 1967) technique is used to derive the shock normal direction in the case of single spacecraft measurements.The accuracy of the normal direction depends on the ratio between the intermediate (λ 2 ) to minimum (λ 3 ) eigenvalue and the number of measured field vectors (N) used in the analysis (Sonnerup & Scheible 1998).The MVA is considered to yield reliable results for an eigenvalue ratio of λ 2 /λ 3 > 8 (Knetter et al. 2004).Under such a criterion (and, e.g., N = 20 data points across the boundary), the error in the shock normal direction remains below ∼5°.
It may also be recalled that X and M A depend on the particular value of the polytropic index γ.The value of γ is characteristic of the type of the thermodynamical process acting on the space plasma.Zhuang & Russell (1981) theoretically examined the jump conditions across Earthʼs bow shock and obtained a simple approximate formula for the expected thickness of the magnetosheath, which sensitively depends on the value of γ.Comparing the theoretical estimate of their model with the measured magnetosheath thicknesses of satellite observations, they concluded that γ is toward 2 (which is also a typical value for the polytropic index under adiabatic conditions in case of an atomic gas with 2 degrees of freedom).However, later spacecraft data analyses showed that the value of γ is actually closer to 5/3 (e.g., Farris et al. 1991), indicating an adiabatic process of an atomic gas with 3 degrees of freedom.Here we adopted the same value from Grabbe & Cairns (1995) and set γ = 5/3.Nevertheless, γ can be selected arbitrarily (i.e., appropriate to the particular plasma condition) to solve Equation (15).
To assess an error estimate for X and M A , we apply the Monte Carlo method of error propagation that includes the uncertainties of the magnetic field measurements, the angles of the magnetic field to the shock normal (Δθ (u) , Δθ (d) ± 7°) and the setting of the adiabatic index (5/3 Δγ 2), for the statistical distribution of the plasma beta between 0 beta 10, as shown in Figure 2. The results of the Monte Carlo experiment reveal that the variance coefficients (relative standard deviations) of X and M A become larger for increasing θ (d) and decreasing θ (u) but still remain below 10% and 20%, respectively.
Figure 3. Alfvénic Mach number, M A , and jump parameter, X, as a function of the solar wind plasma beta under various shock normal angles of the downstream magnetic field θ (d) (solid lines) in case of (a) quasi-perpendicular (left panels), (b) oblique (middle panels) and (c) quasi-parallel (right panels) shock configurations.For the quasi-parallel shock, when plasma beta β < 1.2, the matrix in Equation (15) becomes singular and does not yield a physical meaningful solution.

Summary and Outlook
Our work begins with a rearrangement of the MHD Rankine-Hugoniot equation using the perturbative solution and the de Hoffmann-Teller frame.The rearrangement yields a two-dimensional system of linear algebraic equations for the density jump and the Alfvén Mach number for a given set of magnetic field jump and plasma beta.When applicable to a fast shock, our theory offers a method to constrain the values of those shock parameters using the magnetic field profile across the shock.Numerical evaluation studies indicate that our shock jump estimation works particularly well for quasi-perpendicular shocks.Furthermore, a test case with the Earth bow shock crossing data supports our theoretical development.This opens the door to developing a data analysis tool for determining or constraining the shock parameters even if the data are limited to the magnetic field only.
An immediate application includes the MHD shocks in the solar system (interplanetary shocks and planetary bow shocks) in which only the magnetic field data are often available and not the plasma data.An observational test against the Earth bow shock crossing data demonstrates that our theoretical method yields reasonable results.To further improve the quality of the shock parameter estimation, advanced knowledge of the plasma beta profile is necessary.A different approach would be the Mach number estimation method proposed by Gedalin et al. (2022), which takes kinetic plasma physical processes into account.Furthermore, our method can potentially be applied to the shock phenomena on the solar surface (associated with the solar eruptions) in which the magnetic field data are available using the Zeeman effect, by exploiting the fact that the spectral lines split into several components in the presence of a static magnetic field.
and, e.g., Δθ (u) , Δθ (d) ≈ 7°(see Section 4), and Δβ is determined by a reasonable range around the expected solar wind plasma beta.As Δθ (u) and Δθ (d) are given as the square root of the variance, the errors in Equations (A1) and (A2) should be understood in an rms sense.

Figure 1 .
Figure 1.Left: Cluster-1 Earth bow shock crossing on 2002 February 12. Magnetic field (top panel) and ion bulk velocity (second panel) are given in the shocknormal coordinates (tangential component in black and normal component in gray).The ion number density is plotted in the third panel.The ion temperature (bottom panel) is given with respect to the mean magnetic field direction (parallel component in black and perpendicular component in gray).Right: Alfvén Mach number (top panel) and density/velocity jump (bottom panel) derived from Equation (15) (black lines) as a function of plasma beta and that from the direct measurement of magnetic field and ions at β = 1.0 for reference (diamond symbols).

Figure 2 .
Figure 2. Histogram of plasma beta (for ions) in the near-Earth solar wind obtained by the OMNI 5 minutes data propagated to the location of the Earth bow shock.