Quizzes · PHY653B

Answers hidden — work each one first. Individual concept pages carry their own question sets; this page is for cross-cutting practice, and most questions are about judgement rather than recall, because that is what the course is for.

Conceptual

Q1. Rank Vlasov, PIC and MHD by cost, and state one physical effect each one can capture that the next cannot.

Answer

Cost per unit physics: MHD ≪ PIC ≪ Vlasov (at equal spatial resolution and dimension). Vlasov captures fine velocity-space structure with no noise — filamentation, weak damping rates — that PIC's sampling error buries. PIC captures full kinetic physics in high dimension — Weibel, trapping, tails — that MHD's closure discarded entirely. MHD captures long-timescale global dynamics that neither kinetic method can afford to reach.

Q2. Your code shows perfect energy conservation and beautiful convergence, and disagrees with experiment. What is the most likely explanation?

Answer

A model error, not an implementation error. Verification (convergence, conservation) tests whether you are solving the equations right; it says nothing about whether they are the right equations. A closure that discarded the relevant physics, a missing collision term, or the wrong geometry will all survive verification untouched. → Verification and validation

Q3. Why does the Weibel instability require an electromagnetic PIC code, while two-stream does not?

Answer

Two-stream is electrostatic and longitudinal — the growing quantity is \(E_\parallel\), obtained from Poisson. Weibel's growing quantity is a transverse magnetic field generated by current filaments, which requires the full Maxwell solver. An electrostatic code has no \(\mathbf{B}\) to grow.

Q4. A colleague reports a growth rate with \(R^2 = 0.997\) and no error bar. What do you ask?

Answer

(i) Amplitude or energy — i.e. is this \(\gamma\) or \(2\gamma\)? (ii) What fit window, and does the slope move when you change it? (iii) Where is the noise floor, and did the mode clear it? (iv) Does the rate survive refinement in grid, time step and marker count? \(R^2\) is nearly meaningless over several decades of \(\ln\mathcal{E}\). → Growth-rate fitting

Q5. In what sense is LBM's viscosity a prediction?

Answer

Nothing in the algorithm computes a viscous stress. You choose a relaxation time \(\tau\); Chapman–Enskog then predicts that the emergent fluid obeys Navier–Stokes with \(\nu = c_s^2(\tau - \tfrac12)\). Measuring \(\nu\) from a Poiseuille fit and comparing is a genuine falsifiable test of the theory, not a consistency check of the code.

Computational

Q1. A Vlasov run uses \(v_{\max} = 5\), \(N_v = 128\), \(k\lambda_D = 0.5\). How many damping times fit before recurrence?

Answer

\(\Delta v = 10/128 = 0.0781\); \(T_R = 2\pi/(0.5\times0.0781) = 161\). The damping time is \(1/0.15336 = 6.5\), so about 25 damping times — ample.

Q2. What \(\tau\) gives kinematic viscosity \(\nu = 0.01\) in lattice units, and is it safe?

Answer

\(\tau = \nu/c_s^2 + \tfrac12 = 3(0.01) + 0.5 = 0.53\). Above the stability floor of 0.5 but with little margin — workable with care, and a candidate for an MRT scheme if it misbehaves.

Q3. A PIC run at 64 markers per cell gives a growth rate of 0.043. At 256 markers it gives 0.041. At 1024, 0.0405. Is this converged?

Answer

Plausibly. The changes are 4.7% then 1.2%, roughly halving as \(N\) quadruples — consistent with a \(1/\sqrt{N}\) noise contribution decaying away toward an asymptote near 0.040. Report \(\gamma \approx 0.040\) with an uncertainty covering the residual drift, and say how you got it.

Q4. A second-order MHD scheme gives \(p_{\rm obs} = 1.0\) on Brio–Wu and \(p_{\rm obs} = 2.0\) on a smooth Alfvén wave. Bug?

Answer

No — expected. \(L_1\) convergence is limited to first order at a discontinuity for any scheme. The smooth test confirms the formal order; Brio–Wu tests structure, not order.

Q5. Ignition-scale reasoning aside, why does a weakly damped Landau case need a finer velocity grid than a strongly damped one?

Answer

Weak damping means a long run to accumulate enough e-foldings to fit, and a long run must finish before \(T_R = 2\pi/(k\Delta v)\) — so \(\Delta v\) must be small. The resolution requirement is set by the physics being measured, not by the equation being solved. → Recurrence planning

Multiple choice

1. The \(-\tfrac12\) in \(\nu = c_s^2(\tau-\tfrac12)\) arises from — A) an empirical fit · B) the discreteness of the LBM time step, computed exactly · C) the bounce-back boundary · D) the Mach-number truncation ??? success "Answer" B. Taylor-expanding the discrete update produces a negative numerical viscosity \(c_s^2\Delta t/2\), which is subtracted. A truncation error turned into part of the model.

2. Recurrence in a Vlasov code can be eliminated by — A) refining \(\Delta v\) · B) reducing \(\Delta t\) · C) it cannot be eliminated, only postponed or masked by dissipation · D) using a better interpolator ??? success "Answer" C. \(T_R \propto 1/\Delta v\) grows without bound but is never infinite on a finite grid. Filtering or collisions hide it at the cost of reversibility.

3. Mismatched deposit and interpolate shape functions in PIC cause — A) a CFL violation · B) a spurious self-force, broken momentum conservation and artificial heating · C) loss of charge conservation only · D) nothing measurable ??? success "Answer" B. Matched shapes make the self-force cancel identically; mismatched ones leave a residual that accelerates each marker by its own field.

4. Gyrokinetics cannot describe — A) drift-wave turbulence · B) zonal flows · C) ion cyclotron resonance heating · D) ITG modes ??? success "Answer" C. Cyclotron-frequency physics is averaged away by the ordering. A model limitation, not a resolution limitation.

5. A non-zero \(\nabla\cdot\mathbf{B}\) is especially damaging because it — A) violates energy conservation · B) produces an unphysical force parallel to \(\mathbf{B}\) · C) makes the scheme unstable · D) changes the Alfvén speed ??? success "Answer" B. The term \(-\mathbf{B}(\nabla\cdot\mathbf{B})/\mu_0\) acts along \(\mathbf{B}\), a direction in which magnetic forces should vanish identically.

6. Zonal flows matter because they — A) transport heat efficiently · B) are linearly unstable · C) shear apart the turbulent eddies that generate them, regulating transport · D) damp the ITG mode linearly ??? success "Answer" C. They carry no radial flux themselves but reduce the turbulence's radial correlation length — the dominant saturation mechanism.

7. The most informative single verification test is — A) that the error is small · B) the observed order of accuracy from a resolution scan · C) energy conservation · D) agreement with another code ??? success "Answer" B. Almost every implementation bug degrades the convergence rate, whether or not it makes the error visibly large at one resolution.