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Next: Ionospheric Radio Wave Propagation Up: Wave Propagation Through Inhomogeneous Previous: Pulse Propagation

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)

$\displaystyle {\bf M}(\,{\rm i}\,\partial/\partial t, -{\rm i}\,\nabla, {\bf r}, t)\,{\mbox{\boldmath$\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

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


$\displaystyle {\bf k}$ $\displaystyle \equiv \nabla\phi,$ (6.89)
$\displaystyle \omega$ $\displaystyle \equiv - \frac{\partial\phi}{\partial t}.$ (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

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

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

that is, the dispersion relation yields a unique frequency for a wave of a given wave-vector, $ {\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

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

where (locally)

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

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


$\displaystyle {\bf v}_g = \frac{\partial{\mit\Omega}}{\partial {\bf k}}$ (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)]:

$\displaystyle \frac{\partial k_i}{\partial t} + \frac{\partial\omega}{\partial r_i}$ $\displaystyle =0,$ (6.97)
$\displaystyle \frac{\partial k_j}{\partial r_i} - \frac{\partial k_i}{\partial r_j}$ $\displaystyle = 0.$ (6.98)

Equations (6.92), (6.97), and (6.98) yield [making use of the Einstein summation convention (Riley 1974)]

$\displaystyle \frac{\partial k_i}{\partial t} + \frac{\partial{\mit\Omega}}{\pa...
...\frac{\partial k_i}{\partial r_j} +\frac{\partial{\mit\Omega}}{\partial r_i}=0,$ (6.99)


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

$\displaystyle \frac{\partial\omega}{\partial t} = \frac{\partial{\mit\Omega}}{\...
...}\frac{\partial\omega}{\partial r_j} + \frac{\partial{\mit\Omega}}{\partial t},$ (6.101)

which can be written

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

$\displaystyle \frac{d{\bf r}}{dt}$ $\displaystyle = \frac{\partial{\mit\Omega}}{\partial {\bf k}},$ (6.103)
$\displaystyle \frac{d{\bf k}}{dt}$ $\displaystyle = -\nabla{\mit\Omega},$ (6.104)
$\displaystyle \frac{d\omega}{d t}$ $\displaystyle = \frac{\partial{\mit\Omega}}{\partial t}.$ (6.105)

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

$\displaystyle \frac{d{\bf r}}{dt}$ $\displaystyle =-\frac{\partial{\cal M}/\partial {\bf k} } {\partial{\cal M}/\partial\omega},$ (6.106)
$\displaystyle \frac{d{\bf k}}{dt}$ $\displaystyle = \frac{\partial{\cal M}/\partial{\bf r} } {\partial{\cal M}/\partial\omega},$ (6.107)
$\displaystyle \frac{d\omega}{d t}$ $\displaystyle =-\frac{\partial{\cal M}/\partial t } {\partial{\cal M}/\partial\omega}.$ (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.)

next up previous
Next: Ionospheric Radio Wave Propagation Up: Wave Propagation Through Inhomogeneous Previous: Pulse Propagation
Richard Fitzpatrick 2016-01-23