Alfven speed

 

rigidity and larmor radius

 

From:https://w3.iihe.ac.be/~aguilar/PHYS-467/PA3.pdf

Collision Energy

 The kinetic energy between two colliding particles can be expressed in terms of center of mass and relative velocity by

\[ E_k = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2v_2^2 = \frac{1}{2} M v_c^2 + \frac{1}{2} \mu v_r^2 , \]

a sum of a center-of-mass term and a relative momentum term.

where $M$, $v_c$, $\nu$, and $v_r$ are given by

\[ M = (m_1+m_2) ; \]

\[ \mathbf{v_c} = \frac{m_1 \mathbf{v_1} + m_2 \mathbf{v_2}}{m_1 +m_2} ;\]

\[ \mu = \frac{m_1 m_2}{m_1 + m_2} ;\]

\[ \mathbf{v_r} = \mathbf{v_1} - \mathbf{v_2} .\]

$\frac{1}{2} \mu v_r^2$ is called collision energy.

amu is short for atomic mass unit. Protons have a positive electrical charge of one (+1) and a mass of 1 atomic mass unit (amu), which is about 1.67×10−27 kilograms.

Reference:

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Physical_Chemistry_for_the_Biosciences_(Chang)/02%3A_Properties_of_Gases/2.8%3A_Molecular_Collisions_and_the_Mean_Free_Path

https://web.chem.ox.ac.uk/teaching/Physics%20for%20CHemists/Mechanics/Collisions.html

https://www.angelo.edu/faculty/kboudrea/periodic/structure_mass.htm

The derivation of sound speed

Consider the sound wave propagating through a pipe with cross-section area $A$. In time interval $dt$ it moves through a tube of length $dz=vdt$. In the steady state, the mass flow rate $dm/dt=\rho v A$ must be the same at the ends of the tube, therefore the mass flux $j=\rho v = const. \to v d\rho=-\rho dv$. The pressure-gradient force provides the acceleration and apply the Newton's second law,

\[ \rho \frac{dv}{dt} =-\frac{dP}{dz} \]

\[\to dP= -\rho\frac{dv}{dt}dz = -\rho \frac{dz}{dt}dv =(-\rho \frac{dv}{dt})v = v^2d\rho .\]

And therefore,

\[ v^2=\frac{dP}{d\rho} ,\]

where, $P$ is the gas pressure, $\rho$ is the density.

Reference:

https://en.wikipedia.org/wiki/Speed_of_sound#:~:text=The%20speed%20of%20sound%20is,a%20mile%20in%204.7%20s.

The solution of diffusion equation

Reference:

http://folk.ntnu.no/leifh/teaching/tkt4140/._main034.html

Kolmogorov spectrum

Turbulence is a time-dependent, stochastic flow in fluids.

Energy spectrum is the energy distributed in the wavenumber space $E(k)=\frac{dE}{dk}$.

$dE=E(k)dk$ is the energy of a particular wavenumber and $\int E(k)dk$ is the total energy.

Energy self-similarly cascades through the series of scales known as the inertial range.

Figure is from http://www.dartmouth.edu/~cushman/courses/engs250/Kolmogorov.pdf

A turbulent flow is composed by "eddies" of different sizes. The large eddies are unstable and eventually break up originating smaller eddies, and the kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. These smaller eddies undergo the same process, giving rise to even smaller eddies which inherit the energy of their predecessor eddy, and so on. In this way, the energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy into internal energy

Cascade means that the energy is being transferred from one scale to another without dissipation.

The characteristic velocity on scale $l$ is $u_l$.

Assuming the energy cascade rate $\varepsilon=\frac{u_l^2}{t_c}$ and the cascading timescale $t_c$ is a dynamic time $l/u_l$, we get

\[ \varepsilon \sim u_l^3/l , \\ u_l \sim (\varepsilon l)^{1/3} \sim \varepsilon^{1/3}k^{-1/3} .\]

Since $E(k)k \sim u_l^2$, we have

\[ E(k) \sim \varepsilon^{2/3}k^{-5/3} .\]

Reference:

MHD turbulence

https://en.wikipedia.org/wiki/Turbulence

http://www.dartmouth.edu/~cushman/courses/engs250/Kolmogorov.pdf