Download Phys 784 - WVU Plasma Physics

Phys 784 - MHD
Homework # 1
Due Wednesday, January 22, 2014
Reminders: Show your work! Include references on your submitted version. Write legibly!
1. Practice with Gaussian Units - Characteristic parameters in the solar wind near
1 AU are a number density of n = 10 cm−3 , a temperature of 10 eV, and a magnetic
field strength of 5 nT.
(a) Assuming the solar wind is an ideal gas, calculate the gas pressure p and magnetic
field B in Gaussian units. (Don’t calculate p in SI then convert; do the whole
calculation in Gaussian units.)
(b) Calculate the magnetic energy density of this magnetic field in cgs units. What
is the ratio of pressure to magnetic energy density for these parameters? (This is
known as the plasma β; we will see it more later.)
2. Practice with Index Notation
(a) Write the equations of ideal-MHD (in cgs units) in index notation, namely the continuity equation, the momentum equation, the induction equation, and Ampère’s
Law. (Don’t worry about the pressure equation.) If you need more guidance that
what we did in class on writing cross products in index notation, please see any
math methods book. A Wikipedia site (ugh!) actually has some information.
(b) Use the identity
ijk klm = δil δjm − δim δjl ,
where ijk is the Levi-Civita symbol and δij is the Kroenecker delta, to prove the
vector identity
A × (B × C) = B(A · C) − C(A · B)
using index notation. Try to remind yourself how much algebra it takes to prove
this directly!
(c) Write the jth component of the following expressions in index notation: (i) (A·∇)B,
(ii) ∇ · τ , (iii) ∇2 B. Here, A and B are vectors and τ is a second rank tensor.
(d) Simplify the following expressions: (i) ijk δjk , (ii) δjk δjk , and (iii) δij δjk .
3. Scaling Analyses in “Geophysical Fluid Dynamics”
In class, we discussed the Coriolis “force” and when it is important. Its contributions are
often arise in Geophysical Fluid Dynamics (GFD). The form of Newton’s law discussed
in class is commonly used with additional terms to include dissipation through viscosity
and the effects of a pressure gradient. The governing equation including these terms
can be written as
dv
1
= −g − Ω2 R − 2Ω × v − ∇p + ν∇2 v,
dt
ρ
where ν is the viscosity. Define a characteristic length scale L, a characteristic velocity
scales V , and a characteristic pressure difference as ∆P .
(a) Use a scaling argument to determine when the viscous term is negligible compared
to the Coriolis term. The square root of the ratio of the viscous force to the Coriolis
force is called the “Ekman number” Ek- write it in terms of the appropriate scales.
Thus, for small Ekman number, the effect of viscosity can be ignored relative to
the Coriolis force, and vice versa for large Ekman number.
(b) Use a scaling argument to determine when the viscous term is negligible compared
to the centrifugal term. The square of the ratio of the centrifugal force to the viscous
force is called the “Taylor number” T a - write it in terms of the appropriate scales.
Thus, for large Taylor number, the effect of viscosity can be ignored relative to the
centrifugal force, and vice versa for small Taylor number.
(c) Consider a weather system with wind speeds of V ∼ 10 m/s, a length scale of
L ∼ 1, 000 miles, and a pressure difference of ∆P ∼ 1, 000 Pa. For horizontal
motions near the equator, we ignore gravity and the centrifugal term. In this
regime, which two terms in the governing equation are dominant (and thus, should
be retained in an analysis of this system)? (Use ν ∼ 10−5 m2 /s and ρ ∼ 1 kg/m3 .)
4. Scaling Analysis
Consider a mass m attached to an ideal anharmonic spring. The governing equation is
m
d2 x
= −kx − x3 ,
dt2
where k and are constants and x is the amount of stretching from equilibrium of the
spring. Suppose the mass is released from rest from an initial amplitude A.
(a) Do a scaling analysis to determine the condition on system parameters which allows
the anharmonic term (the x3 term) to be neglected.
(b) Normalize the equations to get a non-dimensional equation of
d2 X
= −X − 0 X 3 .
dT 2
What is 0 in terms of , k, m, and A? Rewrite your solution from (a) in normalized
units.
(c) If we neglect the X 3 term, the result is simple harmonic motion, which is exactly
solvable as a cosine. The plots below give the numerically integrated solution of
the full equation in gray and the exact solution of the harmonic oscillator obtained
by neglecting the x3 term in black. Results are shown below for 0 = 0.2 (on the
left) and 0 = 3 (on the right). Interpret the results of the plots in light of your
result to (a) and (b). Discuss the agreement in each case.
(d) For the plot on the right, the period of the oscillations for the anharmonic oscillator
clearly is different than the exact harmonic oscillator with 0 = 0. Use a scaling
argument to estimate the relative time scale of the anharmonic oscillator for 0 = 3,
and confirm the result agrees with your calculation.
5. Velocity Fields
Consider the velocity field v = 12 ωrθ̂ in polar coordinates, where ω is a constant.
(a) Sketch some representative velocity field vectors in the xy plane.
(b) Is the flow divergent? Is the flow rotational? Confirm your responses by calculating
∇ · v and ∇ × v. What kind of motion does this flow describe?
(c) Calculate the equation of the streamlines and sketch a few representative streamlines. (Do this in polar coordinates; careful with the setup!)
6. Convective Derivative - Physics
The temperature T at a point 50 km north of a weather station is 3 ◦ C cooler than at
the station. If the wind is blowing from the northeast at 20 m/s and the air is being
heated by solar radiation at the rate of 1 ◦ C / hr, then what is the local rate of change
of the temperature at the station? Include a sketch!
(Answer - The absolute value is 2.05 ◦ C / hr. I’ll let you figure out the sign.)