The asymptotic solution of the ion-damped acoustic-gravity wave equation

Abstract

In his classic work Hydrodynamics, Horace Lamb devoted a significant amount of effort to the mathematical analysis of atmospheric waves, i.e. waves in a compressible medium under the influence of a local gravitational field and a variable temperature (and hence sound speed) profile. In so doing, he derived equations for both the divergence and the curl of the velocity field, yielding expressions of considerable mathematical beauty and complexity. Eight decades later, Derek Moore and Edward Spiegel extended Lamb’s analysis to include an arbitrary external applied force in the equations of motion. By a suitable choice of such force, the governing wave equations for a wide variety of Lorenz force/Coriolis force-induced wave motions can be derived. However, they chose to investigate the radiation field resulting from the application of a concentrated vertical force. In so doing, they were able to utilize an important theorem by James Lighthill concerning the asymptotic radiation field from a source with compact support, using the concept of a wavenumber surface. This paper has two main components: the first is to extend the work of Moore and Spiegel to ion-damped acoustic-gravity waves in the ionospheric F-region, based on the seminal work of C. H. Liu and K. C. Yeh. The second part is a consequence of the first: the corresponding wavenumber surface becomes complex, and so Lighthill’s method has to be modified to account for the effects of this. Furthermore, a significant inconsistency in the formulation of the physical problem by Liu and Yeh has been corrected and the corresponding derivations have been reformulated. Some graphical information has also been provided in special cases to illustrate the comparative effects of damping on the asymptotic behaviour of the acoustic-gravity radiation field. A final feature of the paper is that the equations derived are very general and can provide the basis for investigation of more realistic atmospheric temperature profiles in future work.

Appendix: The complex refractive index and surfaces

In [8] the (complex) refractive index n is defined by

$$\begin{aligned} n=\frac{kc}{\omega }, \end{aligned}$$

(12.1)

and in the case of axial symmetry here (vertical magnetic field) its square is given by

$$\begin{aligned} n^{2}=\frac{N}{D}\equiv \frac{\left[ 1-\left( \omega _{a}/\omega \right) ^{2}\right] \left[ 1-i(\nu /\omega )\right] }{1-\left( \omega _{B} /\omega \right) ^{2}\sin ^{2}\phi -i(\nu /\omega )\cos ^{2}\phi }. \end{aligned}$$

(12.2)

Recall that in the notation of this paper $\phi$ is the polar angle of the ${\mathbf {k}}$-vector (but denoted by $\theta$ in [8]). Since in the N / D shorthand $n^{2}$ may be rearranged as

$$\begin{aligned} n^{2}={\text {Re}}~\left( n^{2}\right) +i{\text {Im}}~\left( n^{2}\right) \equiv \frac{N_{r}-iN_{i}}{D_{r}-iD_{i}}=\frac{N_{r}D_{r} +N_{i}D_{i}}{\left| D\right| ^{2}}+i\left( \frac{N_{r}D_{i} -N_{i}D_{r}}{\left| D\right| ^{2}}\right) . \end{aligned}$$

(12.3)

We focus attention on the real part of this expression and eliminate the angle $\phi$ in terms of the components of the (now real) wavevector ${\mathbf {k}}$, using $k_{h}=\left| {\mathbf {k}}\right| \sin \phi$, etc. After considerable algebraic rearrangement, the following result is obtained:

$$\begin{aligned} \left( \omega ^{2}k^{2}-\omega _{B}^{2}k_{h}^{2}\right) \left[ c^{2}\left( \omega ^{2}k^{2}-\omega _{B}^{2}k_{h}^{2}\right) -\omega ^{2}\left( \omega ^{2}-\omega _{a}^{2}\right) \right] =\nu ^{2}\omega ^{2}k_{z}^{2}\left( \omega ^{2}-\omega _{a}^{2}-c^{2}k_{z}^{2}\right) . \end{aligned}$$

(12.4)

The imaginary part of $n^{2}$ yields the same left-hand side as the real part, namely

$$\begin{aligned} \left( \omega ^{2}k^{2}-\omega _{B}^{2}k_{h}^{2}\right) \left[ c^{2}\left( \omega ^{2}k^{2}-\omega _{B}^{2}k_{h}^{2}\right) -\omega ^{2}\left( \omega ^{2}-\omega _{a}^{2}\right) \right] =\nu \omega k_{h}^{2}\left( \omega ^{2}-\omega _{a}^{2}\right) \left( \omega ^{2}-\omega _{B}^{2}\right) -\nu ^{2}\omega ^{2}k_{z}^{4}c^{2}. \end{aligned}$$

(12.5)

Note that the second factor on the left-hand side of both equations is identical to Eq. (6.2) when $\nu =0.$ Since therefore these latter equations contain an additional factor even for $\nu =0$ the presence of spurious solutions is possible, and this approach is pursued no further.