Add RD/TD check

Author: henry2004y

contents/shock.qmd

@@ -131,14 +131,16 @@ The categories of the solution of @eq-RH_planar_sol are shown in @tbl-shocks-dis
 
 1. The contact discontinuity (CD) is a special case of tangential discontinuity (TD) in which we assume $\lfloor U_t\rceil = 0$, i.e., the tangential velocity (and so the velocity) is continuous, but not the density and other thermodynamic variables.
 Since in a CD the thermal pressure remains constant, any change in density must be compensated by a change in temperature. However, a temperature jump is quickly disspated by electron heat conduction, which hints that CD do not persist long. 
-TDs are often observed in the solar wind.
+TDs are often observed in the solar wind. The Hot Flow Anomaly (HFA) requires a TD and a normal electric field pointing towards the discontinuity.
 
 2. The Earth's magnetopause (@sec-magnetopause) is generally a tangential discontinuity. When there is no flux rope been generated, the magnetopause can be treated as the surface of pressure balance between magnetic pressure, ram pressure and thermal pressure. However, when reconnection triggers flux rope generation, it may become a rotational discontinuity (RD) (TO BE CONFIRMED!).
 
 3. Intermediate (Alfvénic) shocks are incompressive and isentropic. The rotational discontinuity is a special case of the intermediate shock. The tangential velocity relation $\lfloor \mathbf{U}_t - \frac{\mathbf{B}_t}{\sqrt{\mu_0 \rho}}\rceil=0$ can be derived from @eq-RH_planar_sol2 assuming the _Walen relation_ holds, $U_n = B_n / \sqrt{\mu_0 \rho}$, i.e. this is an Alfvénic shock. All thermodynamic quantities are continuous across the shock, but the tangential component of the magnetic field can rotate. RD is frequently observed in fast solar wind. Intermediate shocks in general however, unlike rotational discontinuities, can have a discontinuity in the pressure.
 
 4. Fast- and slow-mode shocks are compressive and are associated with an increase in entropy. Fast/slow shocks have increasing/decreasing magnetic pressure from the upstream to the downstream of the shock. For example, the Earth's bow shock is a fast, supercritical shock (See criticality in @sec-criticality).
 
+5. RD and TD are similar. One easy way to distinguish these two are performing a MVA analysis, get the normal magnetic field component, and see if $B_N$ (not the jump!) is zero. If $B_N = 0$, then it is a TD ("tangential" means no normal component); otherwise it is a RD.
+
 The solutions can also be summarized in [the context of Riemann problem](https://pmocz.github.io/shocks/theory.html) or visually in @fig-summary-mhd-shocks-discontinuities.
 
 ![Some properties of MHD shocks and discontinuities. The thick vertical line is the shock surface. The shock normal is dotted. The full arrowed lines are magnetic field lines that are refracted through the shock surface. The dashed arrowed lines are velocity vectors. Region 1 is upstream, 2 is downstream. Reprinted from @de1998complex.](../images/Summary-MHD-shocks-and-discontinuities.png){#fig-summary-mhd-shocks-discontinuities}