Divergence and curl of the vector field

\[ \nabla \cdot \vec{F} = \sum_i \frac{\partial F_i}{\partial x_i} \]
It is so called divergence, because it measures the tendency of a vector field to diverge (positive divergence) or converge (negative divergence). In particular, a vector field is said to be incompressible (or solenoidal) if its divergence is zero.

Examples:
$\vec{F} = (x,y,0)$, $\vec{G} = (x,-y,0)$ and $\vec{H}=(-x,-y,0)$, we have
\[ \nabla \cdot \vec{G} = 1-1 = 0 \]
\[ \nabla \cdot \vec{F} = 2 \]
\[ \nabla \cdot \vec{G} = -2 \]
The arrows on the plot of $\vec{F}$ diverge and on the plot of $\vec{H}$ converge.

\[ \nabla \times \vec{F} = (\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z})\vec{i} +  (\frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x})\vec{j} + (\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y})\vec{k}  \]
The curl of a vector measures its tendency to rotate. A vector field is said to be irrotational if its curl is the zero vector.

MHD plasma wave

sound wave in air
\[ \nabla p = (\frac{\gamma p}{\rho}) \nabla \rho = v_s^2 \nabla \rho \]
where $v_s$ is the adiabatic sound speed:
\[ v_s = dp/d\rho = (\frac{\gamma p}{\rho})^{1/2} = (\frac{\gamma k_B T}{m})^{1/2} \]

Alfven wave
\[ v_A = \frac{tension}{density} = (\frac{B^2/\mu_0}{\rho})^{1/2} \]

(Figure 1.1 is from Analysis and gyrokinetic simulation of MHD Alfvén wave interactions )

Ion acoustic and magnetoacoustic wave
For motion of the particles and propagation of the wave along the field, there will be no field perturbation since the particles are free to move in this direction. These wave will therefore be compressional waves, called ion acoustic wave propagation at velocity
\[ v_S = (\frac{\gamma_e k_B T_e + \gamma_i k_B T_i}{m_i})^{1/2} \]
along the field line.



In the direction perpendicular to $\vec{B}$, a new type of longitudinal oscillation is made possible by the magnetic restoring force (magnetic pressure). This is the magnetosonic wave that involves the compression and rarefaction of the magnetic field line of force as well as the plasma.
\[ v_M = \frac{\nabla(p+B^2/2/\mu_0)}{\nabla \rho} = v_s^2 + \frac{d(p+B^2/2/\mu_0)}{d\rho}  \]
Since the particles are tied to file lines, $B/\rho = B_0/\rho_0$ and we have
\[ v_M^2 = v_s^2 + v_A^2 \]



Reference:
chapter 7 MHD PLASMA WAVES
http://people.physics.anu.edu.au/~jnh112/AIIM/c17/chap07.pdf