Looking At Electrostatic Shocks In 'Cold' Space Plasmas (1)

 We look in this post at what is termed cold, electron-ion streaming plasma, which gives rise to nonlinear electrostatic waves. These are important in multiple arenas of space physics, including:  the solar wind, the Earth’s magnetosphere,  and the auroral acceleration region.  Also in association with shocks and turbulence in these domains. Owing to potentially large amplitude of electric fields E  within these structures, the effects on particle heating, scattering, or acceleration can be important.

Our planet's magnetosphere has been of particular interest since a 2017 publication (in The Journal Of Geophysical Research) revealing jets of ionospheric cold plasma discovered at the magnetopause which may actually play a larger role in magnetic reconnection than previously thought.

The big advantage in working in the electrostatic domain of plasma physics is the relative simplicity.  For example, the putative basis in the wave frame is the 1D Vlasov equation, e.g.

v ¶ f _s / ¶ x  - q _s /m _s [¶ j/ ¶ x  · ¶ f _s/ ¶ v]  = 0

With general solution:

f _s  =  f _s  (v _x²   +    2q _s j  /m _s )

Where f _s   is any function of the constant of the motion for particles species s.  And q _s ,  m _s  refer respectively to the charges and masses for plasma species s. With no magnetic field present, the only Maxwell equation we need is Poisson's  e.g.

 ¶ ² F / ¶ x²    =   - 4 p e (n _e – n _i) =     - 4 p r

 Which indicates a shock or double layer type of potential structure.  For example, one obtains the graph below as one double layer solution to the potential equation:

                  Solution to the potential equation for cold streaming (e, i) plasma, showing the electric potential against distance (x)

And further when the charge density is plotted against distance x:

                                                           Charge density plotted against distance, x. 

For which we have: E =  4 p  ò -_¥ ^x r _c dx

This answers the question of what type of charge distribution is needed to support the given type of potential.  When applied to electrostatic shocks our aim is to get the Poisson equation into a dimensionless form.   The easiest initial solution we note is the one for solitons whereby: 

n  = n _o    for  J = 0

And:   v_eo    =   v _io

The zero subscripts here refer to the condition of a zero current density J, since the latter requires the presence of a magnetic field.  Then also we will have the electron velocity in this domain equal to the ion velocity. Most importantly given zero current there is no net flow of ions, electrons.  

We consider instead the more realistic  case there is a net flow and specified as shown in the diagram below:

We are looking here at the respective particle populations in the 'upstream'  and downstream situations, such that:

Upstream:  transmitted ions  n _it    yielding   n _it    downstream.

Upstream:  reflected ions n _ir  yielding  none downstream ( n _ir  =  0 )

Upstream:  transmitted electrons  n _e    yielding  n _e   downstream.

We can now look at the applicable distribution function:

Where all particles with v  <   v_r   will be reflected. Note the shaded upward partial concave section at lower left denotes the mirror image portion of reflected particles.  The changes to the distribution function result from the relative changes of proportions of the particle populations and the possible presence of shocks.  We already briefly examined shocks before in the context of collisionless shocks associated with the ion-acoustic instability, i.e.

• Why Investigate Magnetized Collisionless Shocks?

And I note here there is a carryover in the mathematical approach, including use of analogous symbols, e.g. for electron temperature. T_e .

In the current case of collisionless electrostatic shocks we let:

½  mv_ro  ² =  e j _max 

½  mv ² +   e j   = const.   <     e j _max

Then we may write the reflected velocity:

   v _r    = Ö (2e (j _m  -  j )/ m _i

At this point we introduce a geometric simplification to represent the shock front, such as shown below:

And we can now write:

n _it = n _ot  / Ö 1  - 2 e j/  m_i v_o  ² 

Where:   n _e = n _o exp [e j/ T_e ]

For the reflected ion distribution function:

f _ir   =     { a        [v]  <   v_r

               { 0    otherwise

where a is the normalization constant

Further:

 n _ir =    2 ò ^{v r} ₀    f  _ir    dr

n _ir =    2a  Ö (2e (j _m  - j )/ m _i

n_ro =  n _ir (j = 0)  =    2a  Ö (2e j _m  / m _i 

Where: a =      (1/ Ö (2e j _m )    n_ro / 2   

Then:

n _ir = Ö (j _m   -  j )/ j _m                        upstream side

n _ir  = 0                                                 downstream

And:

n _e  (j = 0)  =  n _o    =    n _i (j = 0)  

n _o    =   n _ot      +    n _or 

We seek to frame the shock in terms of the potential  j , so write:

F     =  e j  / T _e 

And the sonic Mach number for the shock:

M _s   =     v_o   / (T _e / m _e )    

See Also:

Cold Ionospheric Ions in the Magnetic Reconnection Outflow Region - Li - 2017 - Journal of Geophysical Research: Space Physics - Wiley Online Library

And:

 https://doi.org/10.1029/2017EO084289.

In the next instalment we will obtain the dimensionless Poisson equation