I'm reading through an Introduction to Plasma Physics by Francis F. Chen, and in the simplified derivation for plasma oscillation in 1D, the book quotes the electron equations of motion as:

$$mn_e\left[\frac{\partial\mathbf{v}_e}{\partial t}+(\mathbf{v}_e\cdot\mathbf{\nabla})\mathbf{v}_e\right]=-en_e\mathbf{E}$$

This looks like the usual form of $m\frac{d\mathbf{v}_e}{dt}=\mathbf{F}=-e\mathbf{E}$, but with an additional $(\mathbf{v}_e\cdot\mathbf{\nabla})\mathbf{v}_e$ term. In this case the term is just $v_e^{(x)}\frac{\partial\mathbf{v}_e}{\partial x}$. It looks familiar, but I can't recall where this originates from. Other deviations I've read do not include it, and it is later ignored when considering the case when $\left|\mathbf{v}\right|<<1$. From this assumption, I would think special relativity but I haven't found a mention of it yet.

Does anyone know where this extra term comes from?

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    $\begingroup$ Isn't that just the material derivative? $\endgroup$ – Kyle Kanos May 25 '19 at 1:15
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    $\begingroup$ Also known as the convective derivative: $\frac{d}{dt}=\frac{\partial}{\partial t} + v \cdot \nabla$ - the total time rate of change of a quantity moving instantaneously with the velocity $v$. $\endgroup$ – Cinaed Simson May 27 '19 at 7:42
  • $\begingroup$ Thank you! I forgot to account for the fact that the force would change with position. It's been a while. $\endgroup$ – Levi Hall May 29 '19 at 18:23

In one dimension, we have: $\frac{dv_e}{dt}=\frac{\partial v_e}{\partial t}+\frac{\partial v_e}{\partial x}\frac{dx}{dt}=\frac{\partial v_e}{\partial t}+v_e\frac{\partial v_e}{\partial x}$

In 3 dimensions, the x derivative "becomes" a gradient. Thus we get: $ \frac{d\vec{v_e}}{dt}=\frac{\partial \vec{v_e}}{\partial t}+(\vec{v_e}\cdot\vec{\nabla})\vec{v_e}$


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