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contents/shock.qmd

@@ -183,8 +183,7 @@ clearly sub-magnetosonic!). Only above the critical Mach number the downstream f
 
 The determination of the critical Mach number poses an interesting problem. The finite magnetic field compression ratio sets an upper limit to the rate of resistive dissipation that is possible in an MHD shock. Plasmas possess several dissipative lengths, depending on which dissipative process is considered. Any nonlinear wave that propagates in the plasma should steepen as long, until its transverse scale approaches the longest of these dissipative scales. Then dissipation sets on and limits its amplitude.
 
-Thus, when the wavelength of the fast magnetosonic wave approaches the resistive length, the magnetic field decouples from the wave by resistive dissipation, and the wave speed becomes the sound speed downstream of the shock ramp. The condition for the critical Mach number is then given by $v_{n2} = c_{2s}$. Similarly, for the slow-mode shock, because of its different dispersive properties, the resistive critical-Mach number is defined by the condition $v_{1n} = c_{1s}$[^slow_critical]. Since these quantities depend on wave angle, they have to be solved numerically. Prior studies showed that critical fast-mode Mach number varies between 1 and 3, depending on the upstream plasma parameters and flow angle to the magnetic field. It is usually called _first_ critical Mach number, because there is theoretical evidence in simulations for a second critical Mach number, which comes
-into play when the shock structure becomes time dependent, whistlers accumulate at the shock front and periodically cause its reformation. The dominant dispersion is then the whistler dispersion. An approximate expression for this second or whistler critical Mach number is
+Thus, when the wavelength of the fast magnetosonic wave approaches the resistive length, the magnetic field decouples from the wave by resistive dissipation, and the wave speed becomes the sound speed downstream of the shock ramp. The condition for the critical Mach number is then given by $v_{n2} = c_{2s}$. Similarly, for the slow-mode shock, because of its different dispersive properties, the resistive critical-Mach number is defined by the condition $v_{1n} = c_{1s}$[^slow_critical]. Since these quantities depend on wave angle, they have to be solved numerically. Prior studies showed that critical fast-mode Mach number varies between 1 and 3, depending on the upstream plasma parameters and flow angle to the magnetic field. It is usually called _first_ critical Mach number, because there is theoretical evidence in simulations for a second critical Mach number, which comes into play when the shock structure becomes time dependent, whistlers accumulate at the shock front and periodically cause its reformation. The dominant dispersion is then the whistler dispersion. An approximate expression for this second or whistler critical Mach number is
 $$
 M_{2c} \propto \left( \frac{m_i}{m_e} \right)^{1/2}\cos\theta_{Bn}
 $$
@@ -561,8 +560,7 @@ From MHD or double adiabatic theory, parallel shocks are more special in that th
 
 However, as have been indicated in @sec-switch-shocks, this does not cover the real physics involved into parallel shocks which must be treated on the basis of kinetic theory and with the simulation tool at hand. These shocks possess an extended foreshock region with its own extremely interesting dynamics for both types of particles, electrons and ions, reaching from the foreshock boundaries to the deep interior of the foreshock. Based mostly on kinetic simulations, the foreshock is the region where dissipation of flow energy starts well before the flow arrives at the shock. This dissipation is caused by various instabilities excited by the interaction between the flow and the reflected particles that have escaped to upstream from the shock. Interaction between these waves and the reflected and accumulated particle component in the foreshock causes wave growth and steeping, formation of shocklets and pulsations and causes continuous reformation of the quasi-parallel shock that differs completely from quasi-perpendicular shock reformation. It is the main process of maintaining the quasi-parallel shock which by its nature principally turns out to be locally nonstationary and, in addition, on the small scale making the quasi-parallel shock close to becoming quasi-perpendicular for the electrons. This process can be defined as turbulent reformation, with transient phenonmena like hot flow anomalies, foreshock bubbles, and the generation of electromagnetic radiation. Foreshock physics is important for particle acceleration.
 
-The turbulent nature implies that the quasi-parallel shock transition is less sharp than the quasi-perpendicular shock transition and thus less well defined; there exists an extended turbulent foreshock instead of a shock foot. This foreshock consists of an electron and an ion foreshock. The main population is a diffuse ion component. The turbulence in the foreshock is generated by the reflected and accelerated foreshock particle populations. An important point in quasi-parallel shock physics is the *reformation* of the shock which works completely differently from quasi-perpendicular shocks; here it is provided by upstream low-frequency electromagnetic waves excited by the diffuse ion component. Steeping of these waves during shockward propagation and addition of the large amplitude waves at the shock transition reforms the shock front. The old shock front is expelled
-downstream where it causes downstream turbulence. During the reformation process the shock becomes locally quasi-perpendicular for the ions supporting particle reflection.
+The turbulent nature implies that the quasi-parallel shock transition is less sharp than the quasi-perpendicular shock transition and thus less well defined; there exists an extended turbulent foreshock instead of a shock foot. This foreshock consists of an electron and an ion foreshock. The main population is a diffuse ion component. The turbulence in the foreshock is generated by the reflected and accelerated foreshock particle populations. An important point in quasi-parallel shock physics is the *reformation* of the shock which works completely differently from quasi-perpendicular shocks; here it is provided by upstream low-frequency electromagnetic waves excited by the diffuse ion component. Steeping of these waves during shockward propagation and addition of the large amplitude waves at the shock transition reforms the shock front. The old shock front is expelled downstream where it causes downstream turbulence. During the reformation process the shock becomes locally quasi-perpendicular for the ions supporting particle reflection.
 
 ### Turbulent Reformation