Changes between Version 1 and Version 2 of u/johannjc/scratchpad28


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Timestamp:
12/21/20 16:19:31 (4 years ago)
Author:
Jonathan
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  • u/johannjc/scratchpad28

    v1 v2  
    1 == Hall MHD ==
     1== Generalized Equations ==
     2We start with a Generalized Ohm's law
    23
    3 || Generalized Ohm's law || $\frac{m_e}{n e^2} \frac{\partial J}{\partial t} - \frac{1}{ne}\nabla \cdot \mathbf{P_e} = \mathbf{E} + \frac{1}{c} \mathbf{v} \times \mathbf{B} - \frac{1}{nec} \mathbf{J} \times \mathbf{B} - \frac{\mathbf{J}}{\sigma}$ ||
    4 || Ideal MHD || $0 = \mathbf{E} + \frac{1}{c} \mathbf{v}\times \mathbf{B}$ ||
    5 || Resistive MHD || $ 0 = \mathbf{E} + \frac{1}{c} \mathbf{v} \times \mathbf{B} - \frac{\mathbf{J}}{\sigma}$ ||
    6 || Hall MHD || $0 = \mathbf{E} + \frac{1}{c} \mathbf{v} \times \mathbf{B} - \frac{1}{nec} \mathbf{J} \times \mathbf{B}$ ||
     4$\frac{m_e}{n e^2} \frac{\partial J}{\partial t} - \frac{1}{ne}\nabla \cdot \mathbf{P_e} = \mathbf{E} + \mathbf{v} \times \mathbf{B} - \eta\mathbf{J} - \frac{1}{ne} \mathbf{J} \times \mathbf{B} $
     5
     6And then under certain conditions, we can ignore various terms yielding various approximations
     7
     8|| Resistive + Hall MHD || $\mathbf{E} =  -\mathbf{v} \times \mathbf{B} + \eta \mathbf{J} +  \frac{1}{ne} \mathbf{J} \times \mathbf{B}$ ||
     9|| Resistive MHD || $ \mathbf{E} = -\mathbf{v} \times \mathbf{B} + \eta \mathbf{J}$ ||
     10|| Ideal MHD || $\mathbf{E} = - \mathbf{v}\times \mathbf{B}$ ||
     11
     12=== Electromagnetic energy ===
     13We also need Poynting's theorem for the change in electro-magnetic energy
     14
     15$\frac{\partial e}{\partial t} = - \nabla \cdot \mathbf{S} - \mathbf{J}\cdot \mathbf{E} $
     16
     17which is just the net loss of energy due to diverging electromagnetic Poynting flux $\mathbf{S} = \mathbf{E} \times \mathbf{B}$ minus the work done on the charge distribution $\mathbf{J} \cdot \mathbf{E} $
     18
     19=== Induction equation ===
     20
     21$\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \mathbf{E}$
     22
     23=== Ampere's equation ===
     24(without Maxwell's correction for time varying electric field)
     25
     26$\mathbf{J} = \nabla \times \mathbf{B}$ and
     27
     28=== Lorentz force law ===
     29$\frac{\partial \rho \mathbf{v}}{\partial t}=\mathbf{J} \times \mathbf{B}$
     30
     31=== Final set of equations ===
     32
     33$\frac{\partial \rho \mathbf{v}}{\partial t} = \mathbf{J} \times \mathbf{B} = \left (\nabla \times \mathbf{B} \right) \times \mathbf{B} = \nabla \cdot \left [ \mathbf{BB}-\frac{B^2}{2} \mathbf{I} \right ]$
     34
     35$\frac{\partial e}{\partial t} = -\nabla \cdot \left ( \mathbf{E}  \times \mathbf{B} \right) - \left ( \nabla \times \mathbf{B} \right) \cdot \mathbf{E} $
     36
     37$\frac{\partial B}{\partial t} = \nabla \times \left ( \mathbf{v} \times \mathbf{B} \right) $
    738
    839
    9 Combining these with the induction equation
     40== Ideal MHD ==
     41For ideal MHD we substitute
    1042
    11 $\frac{\partial \mathbf{B}}{\partial t} = -c \nabla \times \mathbf{E}$
     43$\mathbf{E} = -\mathbf{v} \times \mathbf{B} $
     44
     45into the equations above and after some vector calculus, manage to write momentum and energy as a total divergence (and the field as a curl)
     46
     47$\frac{\partial \rho \mathbf{v}}{\partial t} =  \left (\nabla \times \mathbf{B} \right) \times \mathbf{B} = \nabla \cdot \left [ \mathbf{BB}-\frac{B^2}{2} \mathbf{I} \right ]$
     48
     49$\frac{\partial e}{\partial t} = \nabla \cdot \left ( \left (\mathbf{v} \times \mathbf{B} \right ) \times \mathbf{B} \right) + \left ( \nabla \times \mathbf{B} \right) \cdot \left ( \mathbf{v} \times \mathbf{B} \right) = \nabla \cdot \left [ \mathbf{B} \left ( \mathbf{B} \cdot \mathbf{v} \right) - \frac{B^2}{2} \mathbf{v} \right]$
     50
     51$\frac{\partial B}{\partial t} = \nabla \times \left ( \mathbf{v} \times \mathbf{B} \right) $
     52
     53== Resistive MHD ==
     54
     55$ \mathbf{E} = -\mathbf{v} \times \mathbf{B} +  \eta \mathbf{J} =  -\mathbf{v} \times \mathbf{E} + \eta \nabla \times \mathbf{B}$
     56
     57into the equations above and after some vector calculus, manage to write momentum and energy as a total divergence (and the field as a curl)
     58
     59$\frac{\partial \rho \mathbf{v}}{\partial t} =  \left (\nabla \times \mathbf{B} \right) \times \mathbf{B} = \nabla \cdot \left [ \mathbf{BB}-\frac{B^2}{2} \mathbf{I} \right ]$
     60
     61$\frac{\partial e}{\partial t} = \nabla \cdot \left [ \mathbf{B} \left ( \mathbf{B} \cdot \mathbf{v} \right) - \frac{B^2}{2} \mathbf{v}  - \left ( \eta \nabla \times \mathbf{B} \right ) \times \mathbf{B} \right]$
     62
     63$\frac{\partial B}{\partial t} = \nabla \times \left ( \mathbf{v} \times \mathbf{B} \right) - \nabla \times \left ( \eta \nabla \times \mathbf{B} \right ) $
     64
     65Note the energy equation is missing the additional $\mathbf{J} \cdot \mathbf{\eta \mathbf{J}}$ term which would correspond to the loss in magnetic energy due to ohmic dissipation.  However this loss in magnetic energy is balanced by a gain in thermal energy - so those terms cancel in the final energy equation.
     66
     67
     68== Hall+Resistive MHD ==
     69
     70$\mathbf{E} =  -\mathbf{v} \times \mathbf{B} + \eta \mathbf{J} +  \frac{1}{ne} \mathbf{J} \times \mathbf{B}$
     71
     72
     73$\frac{\partial \rho \mathbf{v}}{\partial t} =  \left (\nabla \times \mathbf{B} \right) \times \mathbf{B} = \nabla \cdot \left [ \mathbf{BB}-\frac{B^2}{2} \mathbf{I} \right ]$
     74
     75$\frac{\partial e}{\partial t} = \nabla \cdot \left [ \mathbf{B} \left ( \mathbf{B} \cdot \mathbf{v} \right) - \frac{B^2}{2} \mathbf{v}  - \left ( \eta \nabla \times \mathbf{B} \right ) \times \mathbf{B} - \left ( \frac{1}{ne} \left ( \nabla \times \mathbf{B} \right) \times \mathbf{B} \right) \times \mathbf{B} \right]$
     76
     77$\frac{\partial B}{\partial t} = \nabla \times \left ( \mathbf{v} \times \mathbf{B} \right) - \nabla \times \left ( \eta \nabla \times \mathbf{B} \right ) - \nabla \times \left ( \frac{1}{ne} \left( \nabla \times \mathbf{B} \right) \times \mathbf{B} \right) $
     78
     79
     80We can also combine the Hall and Resistive terms
     81
     82$\frac{\partial \rho \mathbf{v}}{\partial t} =  \left (\nabla \times \mathbf{B} \right) \times \mathbf{B} = \nabla \cdot \left [ \mathbf{BB}-\frac{B^2}{2} \mathbf{I} \right ]$
     83
     84$\frac{\partial e}{\partial t} = \nabla \cdot \left [ \mathbf{B} \left ( \mathbf{B} \cdot \mathbf{v} \right) - \frac{B^2}{2} \mathbf{v}  - \left (  \eta \nabla \times \mathbf{B} + \left ( \frac{1}{ne} \left ( \nabla \times \mathbf{B} \right) \times \mathbf{B} \right) \right ) \times \mathbf{B} \right]$
     85
     86$\frac{\partial B}{\partial t} = \nabla \times \left ( \mathbf{v} \times \mathbf{B} \right) - \nabla \times \left ( \eta \nabla \times \mathbf{B} + \frac{1}{ne} \left( \nabla \times \mathbf{B} \right) \times \mathbf{B} \right) $
     87
     88Or defining $\mathbf{E}' = \eta \nabla \times \mathbf{B} + \frac{1}{ne} \left( \nabla \times \mathbf{B} \right) \times \mathbf{B} $
     89
     90$\frac{\partial \rho \mathbf{v}}{\partial t} =  \left (\nabla \times \mathbf{B} \right) \times \mathbf{B} = \nabla \cdot \left [ \mathbf{BB}-\frac{B^2}{2} \mathbf{I} \right ]$
     91
     92$\frac{\partial e}{\partial t} = \nabla \cdot \left [ \mathbf{B} \left ( \mathbf{B} \cdot \mathbf{v} \right) - \frac{B^2}{2} \mathbf{v}  - \mathbf{E}' \times \mathbf{B} \right]$
     93
     94$\frac{\partial B}{\partial t} = \nabla \times \left ( \mathbf{v} \times \mathbf{B} \right) - \nabla \times \mathbf{E}' $
     95
     96 
     97
     98
     99This along with
    12100
    13101we arrive at
    14102
    15103|| Ideal MHD || $\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left [  \mathbf{v}\times \mathbf{B} \right ]$ ||
    16 || Resistive MHD || $ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left [ \mathbf{v} \times \mathbf{B} - c \frac{\mathbf{J}}{\sigma} \right ]$ ||
     104|| Resistive MHD || $ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left [ \mathbf{v} \times \mathbf{B} - \frac{\mathbf{J}}{\sigma} \right ]$ ||
    17105|| Hall MHD || $\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left [  \mathbf{v} \times \mathbf{B} - \frac{1}{ne} \mathbf{J} \times \mathbf{B} \right ]$ ||
     106
     107In addition, Poynting's theorem states that the electromagnetic energy goes as
     108
     109$\frac{\partial u}{\partial t} = - \nabla \cdot \mathbf{S} - \mathbf{J}\cdot \mathbf{E} $
     110
     111
     112$\frac{\partial u}{\partial t} = - \nabla \cdot \left (\mathbf{E} \times \mathbf{B}\right) - \mathbf{J}\cdot \mathbf{E} $
     113
     114
     115
     116In Ideal MHD we have $\mathbf{E} = \mathbf{v} \times \mathbf{B}$
     117
     118
     119
     120
     121
    18122
    19123Now for Hall MHD, we can write the induction equation as 
    … …  
    24128$ \mathbf{v}_\mbox{H} = \frac{\mathbf{J}}{ne} $
    25129
    26 Note the Hall velocity is always parallel to the current and therefore perpendicular to the field (using Ampere's law without Maxwell's addition ignoring time varying electric fields) and as a result it does no work on the magnetic field and does not need any additional energy terms.
     130Note the Hall velocity is always parallel to the current and therefore perpendicular to the field (using Ampere's law without Maxwell's addition ignoring time varying electric fields) and as a result the corresponding energy term involving $\mathbf{v_H} \cdot \mathbf{B} = 0$.
    27131
    28132
    … …  
    34138
    35139 $\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \left [ \left ( \eta \nabla \times \mathbf{B} \right ) \right ]$
     140
     141So it is likewise a
     142
     143
    36144
    37145== Discretization ==