ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence October 5, 2016, Axel Brandenburg

Handout 12b: Ray tracing

Ray tracing means that one solves the eikonal equation, which is a nonlinear partial differential equation that emerges in problems of wave propagation, when the wave equation is approximated using the WKB theory. Here one follows a ray path r = r(t) and its direction k = k(t) by solving the eikonal equations,

\frac{dr}{dt} = v_{g},  (1)
\frac{dk}{dt} = −∇ω,  (2)

where v_{g} = ∇_{k} ω is the group velocity and ω = ω(r, k) is the known dispersion relation of the wave.

1 Sound waves

Take as an example a polytropic layer with c_{s} = (z_{0} − z)g, so

∇ω = −\frac{gk}{2c_{s}} z  (3)

but this we do numerically. Without magnetic fields, the Doppler-shifted frequency is ω = U ·k +\sqrt{c_{s}^{2}k^{2}}, so the group velocity is v_{g} = ∇_{k} ω = U + c_{s}k̂.

Figure 1: Rays launched from one point at the surface.

Figure 2: Focussing rays (launched from the focussing point).

2 Fast magnetosonic waves

In the presence of magnetic fields, one can use the dispersion relation for fast magnetosonic waves, ω = U · k + ω_{ms}. Here,

ω_{ms}^{2} = \frac{1}{2}k^{2}(c_{ms}^{2} + c_{m}^{2})  (4)

with c_{ms}^{2} = c_{s}^{2} + v_{A}^{2}, v_{A}^{2} = v_{A}^{2} = B^{2}/μ_{0}ρ, and

c_{m}^{4} = c_{ms}^{4} − 4c_{s}^{2}(v_{A} · k)^{2}/k^{2}.  (5)

The group velocity is v_{g} = ∇_{k} ω and can be written as

v_{g} = U + v_{m}k̂ + v_{m}^{∥}k̂_{∥},  (6)

with

v_{m} = \frac{ω_{ms}}{k} (1 + c_{n}^{2} \frac{k_{∥}^{2}}{ω_{ms}^{2}} ) , v_{m}^{∥} = −c_{n}^{2} \frac{k_{∥}}{ω_{ms}} .  (7)

Note that

v_{m} = \frac{ω_{ms}}{k} [1 + \frac{c_{s}^{2}}{c_{m}^{2}} \frac{(v_{A} · k)^{2}}{ω_{ms}^{2}} ] ,  (8)

and

v_{m}^{∥}k̂_{∥} = −\frac{c_{s}^{2}}{c_{m}^{2}} \frac{v_{A}k}{ω_{ms}} v_{A}.  (9)
c_{m}^{4} = (v_{A}^{2} − c_{s}^{2})^{2} + 4v_{A}^{2}c_{s}^{2} k_{⊥}^{2}/k^{2}  (10)

so v_{g} = ∇_{k} ω = U + \frac{1}{2}ω_{ms}^{−1}∇_{k} ω_{ms}^{2} with^{1}

∇_{k} ω_{ms}^{2} = (c_{ms}^{2} + c_{m}^{2}) k + 2c_{n}^{2}(k_{⊥} − kk_{⊥}^{2}/k^{2})  (13)

where

c_{n}^{2} = 2c_{s}^{2}v_{A}^{2}/c_{m}^{2}  (14)

so

v_{g} = ∇_{k} ω = U + (\frac{ω_{ms}}{k^{2}} − \frac{c_{n}^{2}}{ω_{ms}} \frac{k_{⊥}^{2}}{k^{2}} ) k + \frac{c_{n}^{2}}{ω_{ms}} k_{⊥}  (15)

or

v_{g} = U + v_{m}k̂ + v_{m}^{⊥}k̂_{⊥},  (16)

with

v_{m} = \frac{ω_{ms}}{k} (1 − c_{n}^{2} \frac{k_{⊥}^{2}}{ω_{ms}^{2}} ) , v_{m}^{⊥} = c_{n}^{2} \frac{k_{⊥}}{ω_{ms}} .  (17)

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∇_{k} ω_{ms}^{2} = k \{c_{ms}^{2} + [(v_{A}^{2} − c_{s}^{2})^{2} + 4v_{A}^{2}c_{s}^{2} \frac{k_{⊥}^{2}}{k^{2}} ]^{1/2}\} + \frac{\frac{1}{2} k^{2}}{2 [(v_{A}^{2} − c_{s}^{2})^{2} + 4v_{A}^{2}c_{s}^{2} \frac{k_{⊥}^{2}}{k^{2}} ]^{1/2}} \frac{8v_{A}^{2}c_{s}^{2}}{k^{4}} (k^{2}k_{⊥} − k_{⊥}^{2}k) (11)
∇_{k} ω_{ms}^{2} = k \{c_{ms}^{2} + [(v_{A}^{2} − c_{s}^{2})^{2} + 4v_{A}^{2}c_{s}^{2} \frac{k_{⊥}^{2}}{k^{2}} ]^{1/2}\} + \frac{2v_{A}^{2}c_{s}^{2}(k_{⊥} − kk_{⊥}^{2}/k^{2})}{[(v_{A}^{2} − c_{s}^{2})^{2} + 4v_{A}^{2}c_{s}^{2} \frac{k_{⊥}^{2}}{k^{2}} ]^{1/2}}  (12)

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