## FANDOM

167 Pages

Let us consider light shining through a turbulent plasma. We assume a Kolmogorov velocity spectrum, so the relation between the size $l$ and velocity $\delta v$ of an eddy is given by

$\frac{\delta v}{a} \approx \left(\frac{l}{L}\right)^{1/3}$

where $a$ is the speed of sound and $L$ is the scale in which the turbulence is stirred. We assume that the fluctuation in thermal velocity is comparable to the fluctuation in bulk velocity, so the fluctuation in temperature $\delta T$ is given by

$\frac{\delta T}{T} \approx \frac{\delta v}{a}$

Assuming uniform pressure and an ideal gas equation of state, the fluctuation in number density $\delta n$is given by

$\frac{\delta n}{n} \approx \frac{\delta T}{T}$

The variation in density causes a variation in the wavenumber, which we can calculate using the plasma dispersion relation

$c^2 k^2 = \omega^2 - \omega_p^2$

where $\omega_p^2 \approx q^2 n/m$ is the plasma frequency, $q$ is the elementary charge and $m$ is the mass of the electron. In the limit of a frequency much larger than the plasma frequency $\omega \gg \omega_p$

$\frac{\delta k}{k} \approx \frac{\omega_p^2}{c^2 k^2} \frac{\delta n}{n}$

The phase difference due to a single eddy is given by

$\Delta \phi_1 \approx \delta k \cdot l$

If the plasma screen has a thickness $z$, then a beam passing through interacts with $z/l$ eddies. Since each eddy changes the phase by about the same amount but in a random direction, the net change in phase is given by

$\Delta \phi \approx \sqrt{\frac{z}{l}} \Delta \phi_1 \approx \frac{\sqrt{z} \omega_p^2 l^{5/6}}{c^2 k L^{1/3}}$

The phase difference increases with the size of the eddy. There is some critical eddy size $R_d$ where the beams passing on different sides of an eddy are incoherent

$R_d \approx \frac{c^{12/5} k^{6/5} L^{2/5}}{\omega_p^{12/5} z^{3/5}}$

If this length scale is much larger than the Fresnel scale $R_f \approx \sqrt{d/k}$, where $d$ is the distance to the screen (which is usually assumed to be comparable to the thickness of the screen $d \approx z$), then the effect of scintillation is small. This regime is often called weak scintillation. The relative change in flux in this regime is comparable to the phase difference at the Fresnel scale

$\frac{\delta f}{f} \approx \Delta \phi \left(l=R_f\right) \approx \frac{z^{4/3} \omega_p^2}{c^2 L^{1/3} k^{17/12}}$

If, on the other hand, the Fresnel scale is much larger than the diffractive scale $R_f \gg R_d$, then instead of a single luminous disc, the image breaks up into multiple bright spots of size $R_d$. This regime is called strong scintillation. The fluctuations in this regime are strong enough to deflect the light beams, so that the bright spots appear at larger angles. The last eddy the beam encounters deflects it by an angle $\left(k R_d\right)^{-1}$. Multiplying by the distance to the screen yields a length scale $R_r \approx R_f^2/R_d$ in which all the bright spots lie, which is larger than both $R_f$ and $R_d$.