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Basic Physics of the Bow Shock

The solar wind is both super-Alfvenic and supersonic, with Alfven Mach number tex2html_wrap_inline770 and sonic Mach number tex2html_wrap_inline772 both typically tex2html_wrap_inline774 . What happens when the solar wind encounters an obstacle such as a planet's magnetic field, atmosphere or solid surface? In each case the flow must be deflected around the obstacle, either by Lorentz forces or by collisions (mediated by electromagnetic forces). Information on the flow's deflection, however, cannot reach the upstream plasma via the standard (small amplitude) MHD waves - since the flow is super-Alfvenic and supersonic. Accordingly the deflection must be accomplished by an MHD shock wave.

Figure 13.1 illustrates the relevant plasma regions: the bow shock, which slows, compresses, heats and deflects the solar wind flow; the magnetosheath, in which the shocked solar wind is deflected further around the obstacle and eventually accelerated back up to the solar wind speed (downstream from Earth), eventually merging back into the solar wind where the shock weakens and disappears; the magnetopause, which separates the magnetospheric plasma from the shocked solar wind plasma; and the foreshock upstream of the shock but downstream of the tangent magnetic field lines.

Figure 13.1: Schematic of Earth's bow shock, magnetosheath, and magnetopause [Cravens, 1997].

The characteristic shapes and boundaries are qualitatively familiar from the bow waves of ships, bullets, supersonic jets etc.

Qualitatively, a bow shock is just a nonlinearly steepened bow wave. An important aspect of this point is that the propagation speed of many wave modes depends on the wave amplitude (the MHD fast and slow modes and the whistler mode being examples); modes for which the propagation speed increases rapidly enough with wave amplitude can steepen nonlinearly (due to the intense parts of a wave packet piling up) and form shocks and other nonlinear structures such as solitons. One basic difference between solitons and shocks is that shock transitions depend intrinsically on dissipation and an increase in energy across the shock, contrary to the situation for solitons.

The position and shape of the bow shock depend on the ram pressure


Alfven and sonic Mach numbers, and the angle tex2html_wrap_inline776 between the solar wind velocity and magnetic field in the solar wind. This can be seen by writing down the MHD equations in conservative form and then manipulating them. In conservative form the MHD equations can be written:






Normalising throughout by tex2html_wrap_inline778 , tex2html_wrap_inline780 , tex2html_wrap_inline782 , etc., in time-invariant systems these equations can be rewritten as






From Eqs (13.6) - (13.9) it can be seen that the overall ram pressure tex2html_wrap_inline784 becomes a scale factor for the system and that the equations are otherwise only functions of tex2html_wrap_inline786 , tex2html_wrap_inline788 , tex2html_wrap_inline790 , and the angular variables. This is very important because it means that widely applicable results can be obtained using an understanding built up from a few calculations with different tex2html_wrap_inline792 , tex2html_wrap_inline710 , tex2html_wrap_inline796 , and angular variables.

The position and shape of Earth's bow shock, as well as the nature of the shock transition, depend on tex2html_wrap_inline792 , tex2html_wrap_inline710 , tex2html_wrap_inline796 , and the angle tex2html_wrap_inline776 between the upstream flow velocity and magnetic field [e.g., Spreiter et al., 1966; Cairns and Lyon, 1995]. The overall scale of the bow shock - magnetopause system is obtained by balancing the solar wind ram pressure against the magnetic pressure of Earth's dipole at the magnetopause, i.e.,


That is, the standoff distances for the magnetopause and bow shock (and by extension their transverse scales) vary as tex2html_wrap_inline806 . Thus, as expected from intuition, higher ram pressures compress the magnetosphere and bring the bow shock closer to Earth, while lower tex2html_wrap_inline792 lead to the opposite effects.

The effects of variations in tex2html_wrap_inline710 and tex2html_wrap_inline796 are also relatively easy to see qualitatively. For lower Mach number the wave speed is larger relative to the flow speed, so that a wave will travel further upstream in a given nonlinear steepening time, implying that the shock will be found further upstream. Thus, for high tex2html_wrap_inline710 the shock is closer to Earth and for low tex2html_wrap_inline710 it moves further from Earth. Changes in shock shape are also easy to understand based on the concept of the Mach cone (Figure 13.2), which is the locus of

Figure 13.2: A disturbance moves out isotropically at tex2html_wrap_inline706 but is carried downstream at tex2html_wrap_inline708 . The tangent to the circle from the initial point defines the Alfven Mach cone.

the maximum extent of a disturbance moving away from the obstacle at the wave speed. The Alfven Mach cone angle is given by


Thus, the shock shape becomes blunter for smaller tex2html_wrap_inline710 and more swept back for larger tex2html_wrap_inline710 . This is consistent with simple intuition for a slower and faster flow, respectively.

Figure 13.3 illustrates the changes in the shock's shape and location with variations in tex2html_wrap_inline710 .

Figure 13.3: Changes in the shock's shape and location with variations in tex2html_wrap_inline710 [unpublished work by Cairns and Lyon, 1996]: tex2html_wrap_inline712 (solid curves), 4.9 (dashed), 1.9 (dotted), and 1.4 (fancy).

Note that the shock's nose moves sunward as tex2html_wrap_inline710 decreases while the shock's shape becomes increasingly blunt, as predicted from Eq. (13.11).

One implication of the solar wind's time-varying characteristics and the dependence of the bow shock's location and shape on tex2html_wrap_inline792 , tex2html_wrap_inline710 , and tex2html_wrap_inline796 is that the bow shock is almost always moving; it is a dynamic object. Times when it is almost stationary are rare and therefore important for studies of the shock's intrinsic structure.

next up previous
Next: Fast Mode Nature of Up: Earth's Bow Shock and Previous: Earth's Bow Shock and

Iver Cairns
Thu Sep 9 09:54:58 EST 1999