Ray Tracing

Let us now generalize the preceding analysis so that we can deal with pulse propagation though a three-dimensional magnetized plasma.

A general wave problem can be written as a set of $n$ coupled, linear, homogeneous, first-order, partial-differential equations, which take the form (Hazeltine and Waelbroeck 2004)

$${\bf M}(\,{\rm i}\,\partial/\partial t, -{\rm i}\,\nabla, {\bf r}, t)\,{\mbox{\boldmath$\psi$}} = {\bf0}.$$ (6.87)

The vector-field ${\mbox{\boldmath $\psi$}}({\bf r}, t)$ has $n$ components (e.g., ${\mbox{\boldmath $\psi$}}$ might consist of ${\bf E}$, ${\bf B}$, ${\bf j}$, and ${\bf V}$) characterizing some small disturbance, and ${\bf M}$ is an $n\times n$ matrix characterizing the undisturbed plasma.

The lowest order WKB approximation is premised on the assumption that ${\bf M}$ depends so weakly on ${\bf r}$ and $t$ that all of the spatial and temporal dependence of the components of ${\bf\psi}({\bf r}, t)$ is specified by a common factor $\exp(\,{\rm i}\,\phi)$. Thus, Equation (6.87) reduces to

$${\bf M}(\omega, {\bf k}, {\bf r}, t)\, {\mbox{\boldmath$\psi$}} = {\bf0},$$ (6.88)

where

$${\bf k} \equiv \nabla\phi,$$ (6.89)

$$\omega \equiv - \frac{\partial\phi}{\partial t}.$$ (6.90)

In general, Equation (6.88) has many solutions, corresponding to the many different types and polarizations of waves that can propagate through the plasma in question, all of which satisfy the dispersion relation

$${\cal M}(\omega,{\bf k}, {\bf r}, t) = 0,$$ (6.91)

where ${\cal M} \equiv {\rm det}({\bf M})$. As is easily demonstrated (see Section 6.2), the WKB approximation is valid provided that the characteristic variation lengthscale and variation timescale of the plasma are much longer than the wavelength, $2\pi/k$, and the period, $2\pi/\omega$, respectively, of the wave in question.

Let us concentrate on one particular solution of Equation (6.88) (e.g., on one particular type of plasma wave). For this solution, the dispersion relation (6.91) yields

$$\omega = {\mit\Omega}({\bf k}, {\bf r}, t):$$ (6.92)

that is, the dispersion relation yields a unique frequency for a wave of a given wavevector, ${\bf k}$, located at a given point, $({\bf r},\, t)$, in space and time. There is also a unique ${\mbox{\boldmath $\psi$}}$ associated with this frequency, which is obtained from Equation (6.88). To lowest order, we can neglect the variation of ${\mbox{\boldmath $\psi$}}$ with ${\bf r}$ and $t$. A general pulse solution is written

$${\mbox{\boldmath$\psi$}}({\bf r}, t) = \int F({\bf k})\,{\mbox{\boldmath$\psi$}}\,{\rm e}^{\,{\rm i}\, \phi}\,d^3{\bf k},$$ (6.93)

where (locally)

$$\phi = {\bf k}\cdot{\bf r} - {\mit\Omega}\,t,$$ (6.94)

and $F({\bf k})$ is a function that specifies the initial structure of the pulse in ${\bf k}$-space.

The integral (6.93) averages to zero, except at a point of stationary phase, where $\nabla_{\bf k} \phi=0$. (See Section 6.7.) Here, $\nabla_{\bf k}$ is the ${\bf k}$-space gradient operator. It follows that the (instantaneous) trajectory of the pulse matches that of a point of stationary phase:

$$\nabla_{\bf k}\phi = {\bf r} - {\bf v}_g\,t=0,$$ (6.95)

where

$${\bf v}_g = \frac{\partial{\mit\Omega}}{\partial {\bf k}}$$ (6.96)

is the group-velocity. Thus, the instantaneous velocity of a pulse is always equal to the local group-velocity.

Let us now determine how the wavevector, ${\bf k}$, and the angular frequency, $\omega$, of a pulse evolve as the pulse propagates through the plasma. We start from the cross-differentiation rules [see Equations (6.89) and (6.90)]:

$$\frac{\partial k_\alpha}{\partial t} + \frac{\partial\omega}{\partial r_\alpha} =0,$$ (6.97)

$$\frac{\partial k_\beta}{\partial r_\alpha} - \frac{\partial k_\alpha}{\partial r_\beta} = 0.$$ (6.98)

Here, $\alpha$ and $\beta$ run from 1 to 3, and denote Cartesian components. Equations (6.92), (6.97), and (6.98) yield [making use of the Einstein summation convention (Riley 1974)]

$$\frac{\partial k_\alpha}{\partial t} + \frac{\partial{\mit\Omega}... ... k_\alpha}{\partial r_\beta} +\frac{\partial{\mit\Omega}}{\partial r_\alpha}=0,$$ (6.99)

or

$$\frac{d{\bf k}}{dt}\equiv\frac{\partial {\bf k}}{\partial t} + ({\bf v}_g\cdot\nabla)\, {\bf k} = -\nabla{\mit\Omega}.$$ (6.100)

In other words, the variation of ${\bf k}$, as seen in a frame co-moving with the pulse, is determined by the spatial gradients in ${\mit\Omega}$.

Partial differentiation of Equation (6.92) with respect to $t$ gives

$$\frac{\partial\omega}{\partial t} = \frac{\partial{\mit\Omega}}{\... ...ac{\partial\omega}{\partial r_\beta} + \frac{\partial{\mit\Omega}}{\partial t},$$ (6.101)

which can be written

$$\frac{d\omega}{dt} \equiv \frac{\partial\omega}{\partial t} + ({\bf v}_g\cdot\nabla)\, \omega = \frac{\partial {\mit\Omega}}{\partial t}.$$ (6.102)

In other words, the variation of $\omega$, as seen in a frame co-moving with the pulse, is determined by the time variation of ${\mit\Omega}$.

According to the previous analysis, the evolution of a pulse propagating though a spatially and temporally non-uniform plasma can be determined by solving the ray equations:

$$\frac{d{\bf r}}{dt} = \frac{\partial{\mit\Omega}}{\partial {\bf k}},$$ (6.103)

$$\frac{d{\bf k}}{dt} = -\nabla{\mit\Omega},$$ (6.104)

$$\frac{d\omega}{d t} = \frac{\partial{\mit\Omega}}{\partial t}.$$ (6.105)

The previous equations are conveniently rewritten in terms of the dispersion relation (6.91) (Hazeltine and Waelbroeck 2004):

$$\frac{d{\bf r}}{dt} =-\frac{\partial{\cal M}/\partial {\bf k} } {\partial{\cal M}/\partial\omega},$$ (6.106)

$$\frac{d{\bf k}}{dt} = \frac{\partial{\cal M}/\partial{\bf r} } {\partial{\cal M}/\partial\omega},$$ (6.107)

$$\frac{d\omega}{d t} =-\frac{\partial{\cal M}/\partial t } {\partial{\cal M}/\partial\omega}.$$ (6.108)

Incidentally, the variation in the amplitude of the pulse, as it propagates through the plasma, can only be determined by expanding the WKB solutions to higher order. (See Exercises 3 and 4.)