1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Collisionless plasma, conservation of energy

  1. Jun 21, 2009 #1

    In collisionless plasma physics, when you integrate the Vlasov equation for the energy you find two equations :

    one for thermal energy of species s:

    [tex]\frac{\partial u_s}{\partial t} + \nabla\cdot\left(\mathbf{q}_s + \mathbf{v}_s u_s + \vec{\vec{P}}\cdot\mathbf{v}_s \right) = \left(\nabla\cdot\vec{\vec{P}}\right)\cdot\mathbf{v}_s[/tex]

    and one for the convection kinetic energy :

    [tex]\frac{\partial }{\partial t}\frac{n_sm_s\mathbf{v}_s}{2} + \nabla\left(\frac{n_sm_s v_s^2\mathbf{v}_s}{2}\right) = n_s\mathbf{v}_s\cdot\mathbf{E} -\left(\nabla\cdot\vec{\vec{P}}\right)\cdot\mathbf{v}_s [/tex]

    considering that the electromagnetic energy equation is :

    [tex]\frac{\partial B^2/2\mu_0}{\partial t} + \nabla\cdot\left(\frac{\mathbf{E}\times\mathbf{B}}{\mu_0}\right) = -\mathbf{j}\cdot\mathbf{E}[/tex]

    [tex]n_s = \int f_s\left(\mathbf{r},\mathbf{w},t\right)[/tex]
    [tex]\mathbf{v}_s = \int \mathbf{w}f_s\left(\mathbf{r},\mathbf{w},t\right)[/tex]
    [tex]\mathbf{q}_s = \int\left(\mathbf{w}_s-\mathbf{v}_s\right)^2\left(\mathbf{w}_s-\mathbf{v}_s\right) f_s\left(\mathbf{r},\mathbf{w},t\right) d\mathbf{w}[/tex]
    [tex]\mathbf{\vec{\vec{P}}}_s = \int\left(\mathbf{w}_s-\mathbf{v}_s\right)\otimes\left(\mathbf{w}_s-\mathbf{v}_s\right) f_s\left(\mathbf{r},\mathbf{w},t\right)d\mathbf{w}[/tex]

    My questions are :

    1/ It appears that the loss of electromagnetic enery is gained by the convection energy, why ? I've always said "joule heating" so my intuition would have led me to say that the electromagnetic should be given to thermal energy...

    2/ the term [tex]\left(\nabla\cdot\vec{\vec{P}}\right)\cdot\mathbf{v}_s[/tex] appears in both kinetic energy equations as a source term, and does not appear when we sum these two equation... Therefore I interpret this term as a transfert between convection and thermal energy. My point of view is that thermal energy can lead to bulk motion (expansion for example) and thus creation of convection kinetic energy. Am I right ?

    3/ Does it mean that this term is the ONLY source of thermal energy ? And I can't understand how convection energy can be transfered to thermal energy.

    4/ can somebody help me understand physically the three different heat flux terms ? I think I understand the second one [tex]\mathbf{v}_s u_s[/tex], I see it as the convection of thermal energy by the flow. The first one, and most of all the third one appears to me more obscure...

    5/ If [tex]\mathbf{v}_s=0[/tex] (no mean velocity), the convection energy equation says nothing... I can't understand that. If a charged fluid is at rest, if I put an electric field, the fluid will move according to the coulomb force, and kinetic energy will be created. Why isn't it said by this equation ?

    6/ In the electromagnetic energy equation, the term [tex]-\mathbf{j}\cdot\mathbf{E}[/tex] can be positive (decelerated particles), does it mean that decelerating particles actually gives energy to the fields ? I can't see that...
    Last edited: Jun 21, 2009
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted