# Collisionless plasma, conservation of energy

1. Jun 21, 2009

### Heimdall

Hi,

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

one for thermal energy of species s:

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

and one for the convection kinetic energy :

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

considering that the electromagnetic energy equation is :

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

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

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 $$\left(\nabla\cdot\vec{\vec{P}}\right)\cdot\mathbf{v}_s$$ 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 $$\mathbf{v}_s u_s$$, 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 $$\mathbf{v}_s=0$$ (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 $$-\mathbf{j}\cdot\mathbf{E}$$ 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