Phys 782 - Computer Simulation of Plasmas Homework # 4 (Project #1) Due Wednesday, October 22, 2014 Reminders: Show your work! As appropriate, include references on your submitted version. Write legibly! Instructions - Do not do every problem!!! Choose one problem to do and talk to Paul about it. No problem is to be done by more than one student in the class. Problems will be reserved on a first come, first served basis. Treat the provided instructions as a guideline, but be open to taking the problem in a different direction (in consultation with Paul). This one problem is expected to take you approximately 1.5 times a usual homework set. Therefore, you should start early (!). This project is weighted higher than the previous homeworks, so be clear in your responses and take pride in your work - this is a great opportunity to learn some really cool physics! 1. Test Particle Simulations of Velocity Shear due to Spatially Varying Electric Fields - This problem explores systems with a non-uniform electric field in a uniform magnetic field; the goal is to show that finite Larmor radius effects self-consistently introduce a density gradient. This is important because the density gradient can excite drift waves and instabilities, which may be important at the Earth s magnetopause and in the magnetotail plasma sheet. Consider an electric field of the form ( x E(r) = E 0 [2 + tanh ˆx, L)] where E 0 and L are constants, with a magnetic field where B 0 is a constant. B(r) = B 0 ẑ, (a) Predict the direction and magnitude (in real units, not normalized) of the E B drift for particles immersed in this field far from x = 0 (for both positive and negative x). (b) In non-normalized units, calculate the Larmor radius of ions for arbitrary (nonrelativistic) perpendicular speeds (ignoring the presence of the electric field) in terms of B 0. (c) Normalize the fields so that L = 2, B 0 = 1, and E 0 = 0.25. Add an option to the test particle code that was provided in class to update this combination of electric and magnetic fields. (d) Modify the code to allow an option for multiple particles instead of just a single one. To do so, put a loop around the existing code and make appropriate changes to the way the code does output. Put the initial conditions inside the loop over particles so they can be different for each particle. (e) Perform simulations for a particle distribution with initial positions between 10 < x < 10 in steps of 1 and multiple initial velocities v y between -2.5 and 2.5 in steps of 0.25. (Thus, there will be 441 particles total.) Confirm this range of speeds includes high enough values that there are particles with Larmor radius larger than L. Calculate the density by making a histogram of the final x position of particles. Show that the density near the gradient in electric field is a function of x.
2. Test Particle Simulations of a Magnetic Mirror/Converging Magnetic Fields - An example of a magnetic mirror field in cylindrical (r, θ, z) coordinates is with where B 0 and L are constants. B(r) = 1 ψ θ, (1) 2π ( ) ψ(r) = B 0 πr 2 1 + z2, L 2 (a) By dotting both sides of Eq. (1) with ψ, show that magnetic fields lines are confined to surfaces of constant ψ. (b) Plot a collection of lines of constant ψ in the r z plane for a collection of initial r values at fixed z (using the contour function in idl). It should reveal that the magnetic fields have the right structure to produce magnetic mirroring. (c) Calculate the magnetic field in rectangular coordinates (you can leave it in terms of r and θ, but have the components be ˆx, ŷ, and ẑ). (d) Normalize this magnetic field so that z is in units of L and the strength of the magnetic field for z = 0 is 1. Add this magnetic field configuration (with no electric field) to the test particle code given out in class. (e) Modify the test particle code to allow an option for multiple particles instead of just a single one. To do so, put a loop around the existing code and make appropriate changes to the way the code does output. Put the initial conditions inside the loop over particles so they can be different for each particle. (f) Run the simulation for electrons (q/m < 0) that start in the equatorial plane (z = 0) at (x, y, z) = (2, 0, 0) with a distribution of initial parallel v 0 and perpendicular v 0 velocities in steps of 0.1 from 0.5 < v y0 < 0.5 and 0.5 < v z < 0.5. (Thus, there will be 121 particles - note, you should print out data less often by increasing skip to 100 or so to avoid printing too much data.) Show the initial distribution of particles on a plot of v vs. v. Repeat for the final velocities of the same particles after enough time stepping that the particles bounce (200,000 time steps per particle). (g) Modify the idl code to read in the data for multiple particles. At each time step, use the µ analysis portion of the code to find the maximum and minimum of z using the max(z) and min(z) commands. Then, after this has been calculated for every particle, plot the initial speeds v 0 vs. v 0 for all particles that did not go outside z = 1. Repeat for the particles that did escape to z 1. What do you find for the distribution of particles that did not escape? (h) The expected result is that particles initially within a cone in v 0 v 0 space escape. (See, any plasma textbook, such as Bellan s Fundamentals of Plasma Physics, for the theory.) Analytically calculate the loss cone angle for this system and compare it to the numerical results you obtained. (i) How does v 0 compare to v 0 for particles that escape past z = 1? Why would this be useful to the operation of scanning electron microscopes?
3. Test Particle Simulations of RF Capacitively Coupled Plasmas (Fermi Acceleration) - Consider a 1D parallel plate capacitor with a plasma in the interior 0 < x < d. When the plate is charged, a thin sheath is created near the plate. The electric field at the capacitor plates is made to oscillate in the radio frequency (RF) range. This makes the thickness of the sheath s change in time. Here, we consider the single particle dynamics of electrons in this system. (a) The simplest way to estimate the sheath thickness s is to assume that ions have a uniform density n i and are stationary on relevant time scales and that there are no electrons present. (Note, this is not a very accurate model...) Integrate the 1D Poisson equation (analytically) and show that the associated electric field is E x (x) = en i ɛ 0 x + E 0, where e is the proton charge and E 0 is the electric field at x = 0. Thus, the electric field goes from E 0 (which is negative) at the surface linearly up to zero; the thickness of the sheath s is therefore s = ɛ 0 E 0 /en i. Thus, the electric field can be written as [ E x (x) = E 0 1 x ] s for 0 < x < s. Similarly, the sheath at the other plate has an electric field of [ E x (x) = E 0 1 d x ] s for d s < x < d. The minus sign enforces that the two sheaths are out of phase. The electric field for s < x < d s is zero. Now, the charge on the capacitor is made to oscillate in time. We model this as s evolving in time as s = s 0 (1 + cos ωt)/2, where ω is the frequency and s 0 is the maximum sheath thickness. (b) Normalize this electric field so that E 0 = 1 and d = 1 and code it into the test particle code given in class (with no magnetic field). Add a feature to the code where the simulation will stop if the particle goes outside 0 < x < d. Let s 0 = d/4. (c) Put an electron at x = d/2, and give it some initial speed in the x direction. Show that as you vary the initial speed, there are some values for which the particle rebounds at a lower speed and others where it rebounds with a higher speed. By plotting the sheath size as a function of time when the particle hits, determine what causes the particle to get a kick. (This is an example of what is called Fermi acceleration, which is important in many settings in plasma physics and astrophysics!) (d) For a particle that gained energy from the interaction, put particles in different initial positions and with different initial speeds that allow the particle to interact with the same sheath in the same way. Calculate the energy gain as a function of initial speed. You should find that the energy gain is proportional to a power of the velocity; what power is it? This specifies the order of the Fermi acceleration. (e) Modify the test particle code to allow an option for multiple particles instead of just a single one. To do so, put a loop around the existing code and make appropriate changes to the way the code does output. Put the initial conditions inside the loop over particles so they can be different for each particle. By plotting trajectories of multiple particles, find one that interacts sequentially with multiple sheaths. Electrons of this type gain a lot of energy, which may help generate the plasma!
4. Test Particle Simulations in a Reversed Field (Speiser Orbits) - There has been effort to understand aspects of dynamics in the Earth s magnetotail from single particle motion. The magnetotail consists of a current sheet with current in the y direction. The Earth s dipole magnetic field effectively adds a normal magnetic field in the z direction. We investigate single particle dynamics in simplified versions of these fields. Consider a reversing magnetic field of the form ( z ) B(z) = B 0 tanh ˆx, w where B 0 and w are constants. This magnetic field is associated with a current sheet in the y direction (it is called a Harris sheet ), which itself is associated with an electric field in the y direction. In a steady state, the electric field is uniform: where E 0 is a constant. E = E 0 ŷ, (a) Predict what electrons (q/m < 0) positioned at z = ±3w and released from rest would do in these fields. (b) Normalize the fields so that B 0 = 1 and E 0 = 0.1 and lengths are measured in terms of w. Code these electric and magnetic fields into the test particle code provided in class. (c) Run the simulation for a particle starting at rest at (x, y, z) = (0, 0, 3w). Describe the motion for z > 1 [make a quantitative comparison to part (a)!]. What qualitatively changes for z < 1? What is happening physically? (d) How does the kinetic energy of the particle change in time for z < 1? Is this perpendicular or parallel energy? Based on the physical situation, calculate the rate of change analytically and compare it to your numerical results (quantitatively!). (e) Interpret your results in terms of the magnetic moment µ (using the provided idl code). How much does µ vary for z > 1? Where does µ conservation strongly break down? How does this compare to expectations? (f) Now add a small normal magnetic field, ( z ) B(z) = B 0 tanh ˆx + B 1 ẑ, w where B 1 is small compared to B 0. Perform simulations for a few different (small) values of B 1. For what strength does B 1 begin to qualitatively change the trajectories and energy gains? What happens after this change? What do you conclude the impact is on particle acceleration due to the presence of a normal magnetic field? Interpret the results in terms of magnetic moment conservation. This is important for understanding the onset of the tearing instability (the linear phase of magnetic reconnection) in the magnetotail. It also suggests that the current sheet immediately around a reconnection site is not expected to give rise to much particle acceleration (for the case of anti-parallel magnetic fields). (g) For the case with the normal field B 1 that changed the energy gains, add in a so-called guide field in the y direction: ( z ) B(z) = B 0 tanh ˆx + B 1 ẑ + B g ŷ, w where B g is a constant. How does the particle energization change with a guide field of 1? Is the energization perpendicular or parallel to the magnetic field?
5. Test Particle Simulations in Oscillating Electric Fields with Magnetic Fields (Chaotic Trajectories) - We have already studied wave-particle interactions for particles in a traveling sinusoidal electric field; now we add in the effect of a uniform magnetic field perpendicular to the electric field. Consider an electric field of the form E(r, t) = E 0 cos(kx ωt)ˆx, where E 0, k, and ω are constants, and a uniform magnetic field B(r, t) = B 0 ẑ, where B 0 is a constant. The behavior of single particles in these fields is surprisingly rich considering the simplicity of the fields. (a) Using Newton s second law, find the equation of motion describing the evolution of the x coordinate of the particles. Normalize the result with distances to 1/k and times to 1/Ω = m/qb, showing that the result is d 2 x dt 2 + x = α cos(x νt ), where primes denote normalized quantities and ν = ω/ω. What is α in terms of the physical parameters of the system? This equation has no exact solution - numerical simulations can help! (b) Code the normalized electric and magnetic fields into the test particle stepper given in class. (c) In the limit of ν = 0 and small α = 0.5, plot a phase space portrait (v x vs. x, as we did in Homework 2) for a number of particles starting from rest at different initial x positions (-15 to 15 in increments of 5). Repeat for large α 43 with initial x positions (-25 to 25 in increments of 5) starting from rest and for particles initially at x = 0 but with v x ranging from 0 to 10 in increments of 2. (d) Now, let ν be non-zero, but small (ν = 0.5). Repeat part (c). How do the phase space portraits differ? (e) What happens to the closed orbits in the high α, non-zero ν case? You should find that they get smaller as they move to the right. It turns out this is key for this process producing irreversible heating - as the closed orbits shrink, particles near the separatrix gain energy. Interestingly, when we studied just the traveling electric field with no magnetic field, we showed that the particles gain energy, but the gain is reversible. It is surprising that one can get irreversible heating merely by adding in a uniform magnetic field! (f) Look at the particle trajectories for some of the particles for some of the parameters you have used. In particular, fix ν and start with small α, and gradually increase it. Notice that the trajectories are quasi-periodic for small α, but then undergo a period doubling route to chaos. Explore this transition.
6. Test Particle Simulations in Stochastic Magnetic Fields - Magnetic fields in tokamaks can become stochastic, meaning they become space filling. It has been proposed that stochastic fields enhance transport, which is usually not desirable for tokamak operation. Here, we investigate this, but we do so in a simplified rectangular geometry. Consider a magnetic field of the form ( ) ( ) ( y 2πx 2πx B(r) = B g ẑ + B 0 tanh ˆx + B 10 cos ŷ + B 11 cos + 2πz ) ŷ L y /2 L x L x L z on a rectangular periodic domain of size L x L y L z. (a) Write a field line tracing code (normalize the magnetic fields to B 0 ). Note that dx dz = B x B z and dy dz = B y B z are first order ODEs, so you can solve them using Euler, Runge-Kutta, etc. Assume periodic boundary conditions, so that if the field line leaves one side of the domain it comes in at the other side. (If a coordinate goes outside, say, x = L x, then subtract L x from it; do this for each of the six faces of the domain.) (b) Carry out the field line tracing for a number (about 25) of magnetic field lines at equally spaced y values for a domain of size 5.12 2.56 20.48 with a step size of dz = 0.32. (There is nothing special about these parameters - just choosing something!) A common way to plot the results is with a Poincaré plot - every time the magnetic field passes through z = 0, make a point on a plot. After many traversals of the domain in z, a pattern emerges. Make this plot for B g = 2, B 10 = 0.05, B 11 = 0.0 and B g = 2, B 10 = 0.0, B 11 = 0.05. (c) For these two parameters, you should see an island -type shape in the Poincaré plot. Show (numerically, not analytically) that the half thickness of the island is approximately equal to B 10 0.224. Also show that the case with B 10 0 has the island centered at y = 0, while the case with B 11 0 has the island centered at y 11 = (L y /2) tanh 1 (B g L x /L z ) 0.703. (This is the q = 1 rational surface. ) (d) Now make Poincaré plots when both modes are present. Repeat part (b) for the following parameters: (i) B 0 = 2, B 10 = B 11 = 0.05, (ii) B 0 = 2, B 10 = B 11 = 0.15, (iii) B 0 = 2, B 10 = B 11 = 0.3, and (iv) B 0 = 2, B 10 = B 11 = 0.5. (e) You should have found that the field lines are mostly regular for some parameters and space filling over some range of y for other parameters. The condition that stochastic fields dominate is called the Chirikov condition. This happens when the islands overlap; for the case here where B 10 = B 11, this occurs when the width B10 exceeds y 11. Show analytically that this is not too different from your results. (f) Now, code these magnetic fields (with zero electric field) into the test particle code and implement an option for periodic boundary conditions in the code. (g) For a good number of particles, start them at various locations in the domain with a speed of 0.1 in various directions for case (i) and (iii) from part (d). (You may want to modify the test particle code to allow for multiple particles by putting a loop around it and changing how the code does output.) What fraction of them travel a significant distance in y for each case? What does this imply about transport in a tokamak with stochastic fields? For a particle that travels a significant distance in y, how far has it traveled in z? What does this imply about transport in a tokamak with stochastic fields?
7. Test Particle Simulations of the Ponderomotive Force in Laser-Plasma Interactions - A charged particle interacting with a wave made up of two high-frequency components of comparable frequency feels an effective force at their beat frequency; this is the ponderomotive force. We investigate this physics here. Consider an electric field of the form E(r, t) = E 0 cos(k 1 x ω 1 t)ẑ + E 0 cos(k 2 x ω 2 t + φ)ẑ, where E 0, k 1, k 2, ω 1, ω 2 and φ are constants. (a) If these are electromagnetic (light) waves, what is the appropriate form for the magnetic field? (b) Normalize the equations so that E 0 = B 0 = 1 and choose k 1 = k 2 = 2π. Code these electric and magnetic fields into the test particle code provided in class. (c) Run a test particle simulation for an electron starting from rest. Choose ω 1 /k 1 = 1 and φ = 0.05, and make the frequencies of the two components differ by a small amount: ω 1 = ω 2 + 0.01. On a time scale of the waves ω 1, show that the particle gains and loses energy in the z direction quasi-periodically; this is called the quiver motion and does not yield any net acceleration of the particles. (d) Show that on a time scale of the beat frequency between the two waves, you get unidirectional motion in the x direction. This is due to the ponderomotive force. Qualitatively, how does the x position change in time? Confirm the relation to the beat frequency by choosing a different ω 1 ω 2. (e) Repeat part (c) for different field strengths; use twice, four times, and eight times as much. How does the peak in v x change with field strength? (f) Repeat part (c) for different frequencies. How does the peak v x change with frequency? (g) Investigate the theory of the ponderomotive force and confirm that the numerical results are consistent with the theory. The ponderomotive force is a very effective way to accelerate particles - an off-the-shelf laser can accelerate electrons to hundreds of MeV! 8. Devise Your Own Project - Using any of the tools we have discussed so far (ODE solvers, test particle simulations, diffusion and parabolic PDE solvers, relaxation of elliptic PDEs, Grad-Shafranov, convection solvers and hyperbolic PDEs) devise your own project. Obviously, you will have to talk to me about it and ensure it is both challenging enough and not too challenging.