INVESTIGATIONS OF HYDRODYNAMIC AND MAGNETOHYDRODYNAMIC EQUATIONS ACROSS THE BOW SHOCK AND ALONG THE OUTER EDGE OF PLANETARY OBSTACLES

S. M. Petrinec¹ and C. T. Russell²

¹ Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229, Japan
² Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90095-1567

Originally published in:
Adv. Space Res., 20, 743-746, 1997.

ABSTRACT

Understanding the interaction of planetary obstacles with the solar wind is fundamental to the entire field of space physics. However, some details of magnetohydrodynamics and even hydrodynamics lead to confusion and the use of incorrect assumptions. One area of confusion involves the position of a bow shock in front of a planetary obstacle; especially as the upstream magnetosonic Mach number approaches unity. Another area concerns the Newtonian approximation along the surface of an obstacle to the solar wind. In particular, confusion arises when the obstacle itself is defined as a boundary across which pressure must be balanced. Again, problems with the Newtonian approximation at the magnetopause are most evident for low upstream Mach number. We investigate the analytic hydrodynamic and MHD formulations across the bow shock and along the boundary of obstacles to the solar wind flow, and offer better analytic approximations.

THE STANDOFF DISTANCE OF THE BOW SHOCK

The standoff distance of a detached bow shock from a blunt obstacle has been of interest from the earliest studies of supersonic aerodynamic flow (Laitone and Pardee, 1947, Nagamatsu, 1949, and many others). The position and shape of the bow shock is such as to allow all of the shocked fluid to flow between the shock and the obstacle. Since these relations contain no inherent length scales, however, the position and shape of the shock is extremely difficult to derive from gasdynamic or hydrodynamic relations. Instead, assumptions and approximations are used to estimate the curvature of the shock front, the form of the stream functions, and the vorticity and distribution of pressure downstream from the shock. Thus, progress in the theoretical understanding of the shape and size of a detached shock wave in front of a blunt obstacle is closely linked to the success of computer simulations and laboratory experiments.

It has long been known (Hayes, 1955) that the thickness of the sheath along the stagnation streamline divided by the obstacle standoff distance is proportional to the density ratio across the shock at high upstream Mach numbers. Seiff (1962) deduced a constant of proportionality for the case of a shock wave in front of a sphere , based on previous experimental results. Increased scatter in the experimental results for low density ratios (low upstream Mach numbers), however, suggested that this relation may not always be valid.

Spreiter et al. (1966) used a Rankine-Hugoniot relation to rewrite the relation of Seiff (1962) in terms of the upstream sonic Mach number. The coefficient 0.78 was also replaced with 1.1, based on results of wind tunnel experiments using an ellipsoidal model with the approximate shape of the magnetopause. The resulting relation agrees well with spacecraft observations at high Mach numbers, but it was cautioned that this formulation only be used for upstream sonic Mach numbers () greater than 5. An extrapolation to = 1 reveals a ratio of shock to obstacle standoff distances of 2.1; a value with no physical basis, and is contrary to the expectation that the gasdynamic shock retreats infinitely far from the obstacle as the Mach number approaches unity (Landau and Lifshitz, 1959). Farris and Russell (1994) approached this shortcoming from a different perspective. It was noted that a simple relation involving the downstream sonic Mach number resulted in the same asymptotic value as the density ratio across the shock at high upstream Mach numbers. This relation also causes the ratio of bow shock to obstacle standoff distances to retreat to infinity as the Mach number decreases towards unity, and is written as:

Eq. 1.

where the magnetosonic Mach number has replaced the sonic Mach number, D_OB is the distance from the focus of the ellipsoid to its nose, and D_BS is the distance from the ellipsoid focus to the shock subsolar position (it is also noted that the average radius of curvature of the obstacle is an important parameter in the placement of the bow shock (see Farris and Russell (1994) for explicit relations between the radius of curvature, obstacle eccentricity, and obstacle standoff distance)). Though this conjecture is very concise and attractive, no rigorous proof has yet been developed.

The position of the shock subsolar point from the obstacle standoff position can also be slightly influenced by the upstream magnetic field. The solution for the density ratio across the bow shock was solved for arbitrary by Russell and Petrinec (1996), and couples the 'switch-on' shock solution to the third real solution when = in the appropriate regions. Russell and Petrinec then used the conjecture of Farris and Russell (1994) to determine the shock subsolar position (Figure 1). This work was done to correct the claim by Cairns and Grabbe (1994) that there exists a factor of ~4 change in the shock standoff distance from the ob-stacle at low Alfven Mach numbers between = and other angles. The error with this result is that specific values of were inserted into the cubic equation derived from the Rankine-Hugoniot relations before they were solved. This produces 3 real roots at = , as this procedure decouples the 2 equivalent 'switch-on' shock solutions from the third solution. The 'switch-on' shock solution was discarded by Cairns and Grabbe (1994), leading to an incorrect estimate of the shock distance from the obstacle at low Alfven Mach numbers. While it is true that the upstream magnetic field can influence the shock subsolar position, the differences are not as large as had been claimed for low Alfven Mach numbers by Cairns and Grabbe.

Fig. 1. The ratio of the bow shock to magnetopause subsolar positions, for a) low plasma . b) high plasma . (adapted from Russell and Petrinec (1996))

It should be noted that when the upstream magnetic field is neither aligned with nor perpendicular to the upstream velocity, the velocity downstream from the subsolar bow shock obtains a tangential component (this also occurs for the 'switch-on' shock). Under these conditions the streamline which passes through the shock subsolar point is not the streamline that stagnates at the magnetopause nose (Walters, 1964). This does not change the calculated density ratio across the shock, but could cause small changes in any relations used to determine the distance of the shock from an obstacle.

As noted by Grabbe and Cairns (1995), several contradictory studies exist regarding the dependence of the distance between the bow shock and obstacle standoff distances on the upstream Alfven Mach number. Computational solutions by Spreiter and Rizzi (1974) found a decrease in the standoff distance ratio between the bow shock and obstacle with decreasing Alfven Mach number (for = ). This claim has received observational support from an empirical study by Peredo et al. (1995). However, this result contrasts with the computational study of Cairns and Lyon (1995) ( = and ) and theoretical studies by Cairns and Grabbe (1994), Grabbe and Cairns (1995), Farris and Russell (1994), and Russell and Petrinec (1996) (the latter two using magnetosonic Mach number). Additional studies are needed to better understand the exact role of the magnetic field on the bow shock position.

PRESSURE BALANCE ALONG THE MAGNETOPAUSE - HYDRODYNAMIC RELATIONS

The magnetopause is at its most basic definition the boundary across which the pressure of the magnetosheath is balanced by the pressure produced by the Earth's intrinsic magnetic field (and a very small contribution from the interior thermal plasma pressure of the magnetosphere). While total pressure is balanced across the bow shock and across the magnetopause in the equilibrium state, the total pressure at the magnetopause is not equal to the total pressure across the bow shock, since the flow parameters within the magnetosheath must change to satisfy the hydrodynamic relations and deflect the plasma flow around the obstacle.

In general, the magnetic field of the plasma precludes us from fully understanding analytically the plasma flow parameters in the magnetosheath region in terms of the solar wind parameters. However, much can be understood from consideration of the stagnation streamline in the hydrodynamic regime, since it lies closest to the magnetopause. The Rankine-Hugoniot relations across the shock are written as:

Eq. 2.

Eq. 3.

where is the solar wind sonic Mach number and M_s is the downstream Mach number. Bernoulli's equation and the adiabatic flow condition are used between the downstream side of the bow shock and the obstacle stagnation position to determine the stagnation thermal pressure. By replacing Eq. 2 for P and using the Mach number relation defined by Eq. 3 and the definition of the upstream sonic Mach number, we then obtain the following relation between the stagnation thermal pressure and the solar wind dynamic pressure (Landau and Lifshitz, 1959; Spreiter et al., 1966):

Eq. 4.

The thermal pressure along the obstacle surface is then determined from the Newtonian approximation, and the density and velocity are determined from the adiabatic condition and Bernoulli's equation, respectively:

Eq. 5.

Eq. 6.

Eq. 7.

where Q is used to indicate the uncertainty in the Newtonian approximation. In Eqs. 5-7, defines the angle between the upstream flow velocity vector and the normal to the obstacle. We next examine these relations, using different functions in place of Q.

The formulation is simple and useful for pressure balance at the high Mach number dayside magnetopause. It has been used by numerous authors, with various values for k. A proper understanding of the value of k was first considered in magnetospheric calculations by Spreiter et al. (1966) (see Eq. 4), and subsequently used in later studies (though not always). This relation breaks down as approaches , however, because the exterior pressure approaches zero. This implies that either the magnetotail radius never reaches an asymptotic value far downtail or the total pressure interior to the magnetopause becomes zero far downtail; both of which are contrary to observations. In addition, the magnetosheath velocity along the magnetopause surface from Eq. 7 exceeds the solar wind velocity as approaches (Figure 2). This cannot happen in a hydrodynamic flow, as no source exists for the additional kinetic energy (Spreiter et al., 1966). Thus, this relation clearly is a poor approximation for the far downtail region.

The relation has been used by several authors, and provides reasonably accurate solutions for large upstream Mach numbers. Often the solar wind magnetic field pressure is also added to the right hand side, and is replaced with P_static. This equation again is an approximation. The solar wind pressure contributions are added to provide a finite external pressure to the magnetosphere as approaches . Here we consider the case without a solar wind magnetic field. While the magnetosheath thermal pressure and density are now non-zero at = , the stagnation thermal pressure now does not agree with that derived in Eq. 4. Even worse, while the speed remains less than the solar wind speed far downtail for = 1.1, it is also imaginary in the subsolar region and over much of the dayside magnetosphere.

We propose that the relation is the simplest formulation which satisfies the hydrodynamic requirements both at the stagnation position and in the far downtail region. This also appears to be the only solution for which parameters vary monotonically along the obstacle. Thus at a minimum it should be used for boundaries which are defined by a balance of pressure. This has been used in earlier aerodynamic studies (Linnell, 1957; Daskin and Feldman, 1957), but not in magnetospheric applications.

Fig. 2. Thermal pressure, mass density, and velocity along an obstacle surface for an upstream sonic Mach number of 1.1 and = 5/3, and is normalized to the values upstream from the shock.