The approximations we make are the usual in the literature (Hines, 1991; Chimonas, 1997), specifically, the atmosphere is assumed to be nonrotating characterized by a constant buoyancy frequency,

, and there is a stationary background wind

. A gravity wave is propagating upward with

the horizontal and vertical wavenumbers. Since we are interested in short vertical wavelengths $m H \gg 1$ and $m \gg k$ can be assumed. Therefore, the dispersion relation of these waves can be expressed as,

$\begin{displaymath} \Omega^2 = \left[\omega - U(z) k \right]^2= \left(\frac{N k}{m(z)} \right)^2 \end{displaymath}$ (1)

where $\Omega$ is the intrinsic frequency and $\omega$ is the ground-based frequency.

The wave solution is harmonic in

and in the

direction the WKB approximation is used, thus, the horizontal wind perturbation is given by (Lindzen, 1981),

$\begin{displaymath} u(x,z,t)= u_a \left(\frac{m(z)}{m_0}\right)^{1/2} \exp i (\omega t + k x - \int^z_0 m(z') dz') \end{displaymath}$ (2)

represents the vertical wavenumber of the wave in absence of the background wind which it is thought to occur at

. So this single wave can be interpreted as an incident monochromatic wave with initial amplitude

and initial vertical wavenumber $m_0=\frac{N k}{\omega}$ . Fig. 1 shows a profile of this wave.

**Figure 1:** The horizontal wind perturbation profile generated by the propagation of a single wave in a weak shear wind. The horizontal velocity is expressed in units of , the initial amplitude.
$\includegraphics[width=3.1in,bb= 20 0 301 248]{fig/fig1a.eps}$

**Figure 2:** Power spectrum of the horizontal wind perturbation belonging to the profile shown in Fig. 1 (dashed line). The straight line represents the theoretical prediction (3) for high vertical wavenumbers
$\includegraphics[width=3.1in,bb= 20 0 301 248]{fig/fig2a.eps}$

If the wind shear and horizontal phase speed are both positive, Doppler effects will diminish intrinsic the wave frequency until the wave attains the critical level condition $\Omega = 0$ . As the intrinsic frequency goes to zero, $m \rightarrow \infty$ and therefore the horizontal perturbation (2) is unbounded at critical level conditions.

A profile of this kind of single waves in a vertical interval from the reference level

to the critical level

will have a broad power spectrum with non-zero spectral amplitudes from

to $m=\infty$ , but note that at each altitude there is a unique vertical wavenumber

. The shape of this power spectrum is determined by two main factors, the amplitude versus vertical wavenumber dependence and the vertical wavenumber versus altitude dependence. An expression for the power spectrum in the high vertical wavenumbers range for a nearly linear wind background can be obtained analytically (see Appendix A),

$\begin{displaymath} PS=\frac{2 \mu u_a^2}{L m_0 m} \end{displaymath}$ (3)

That means the refraction by the background wind produces profiles which have power spectra with a -1 power law (Fig. 2). This law resembles the law used by Hines but because his formulation did not have an explicit

dependence it is not clear how to compare them. Apparently what Hines called power spectrum was a representation of a neighborhood of a height

while we work with a long interval height.

Of course, a superposition of waves is also a solution of the wave equation, in particular, taking into account that the atmosphere has not an upper limit, a continuous spectrum satisfies both the equations and the boundary conditions, thus, it is possible represent the general solution as,

$\begin{displaymath} u(x,z,t)= \int \int u_a(m_0,k) \left(\frac{m(z)}{m_0}\right)^{1/2} \exp i (\omega t + k x - \int^z_0 m(z') dz') dm_0 d k \end{displaymath}$ (4)

The dispersion relation gives a relationship between the variables $(\omega, m_0, k)$ . Thus only two of these variables are independent and are necessary and sufficient to generate the complete wavenumber space. We have chosen

and

while $\omega$ will be given by

$\begin{displaymath} \omega=\frac{N k}{m_0} \end{displaymath}$ (5)

Single waves can be represented by delta functions in

and

variables. To compare a profile resulting from (4) with the characteristic single wave profile (Fig. 1), let us calculate the profile resulting from a uniform spectrum in

and monochromatic,

, in horizontal wavenumber,

$\begin{displaymath} \widetilde u(k,m_0)=\left\{\begin{array}{ll} \frac{u_0}{\Del... ...} \le m_0 \le m_{02} 0 & {\rm otherwise} \end{array} \right. \end{displaymath}$ (6)

which is propagating in a constant wind shear region. Each wave belonging to this spectrum has a critical level at a different height. In fact, there is a altitude range from $z_{c2}=\mu m_{02}^{-1}$ to $z_{c1}=\mu m_{01}^{-1}$ where each wave attains its critical level. The integral (4) can be performed analytically, for

and

yielding the following profile

$\begin{displaymath} u(z)= - \frac{u_0 \mu}{z (1/2-i \mu) \Delta m_0} \left[ \lef... ... \mu}-\left(1 - z \mu^{-1} m_{01} \right)^{1/2- i \mu} \right] \end{displaymath}$ (7)

In the neighborhood of the critical layer the behavior of the broad spectrum profile is different from that of a single wave profile because of interference effects between different components, as it can be seen in Fig. 3, if $z \rightarrow z_{c2}$ the profile goes to

. This fact lead us to think that in this picture a convective instability does not necessarily arise before the critical level.

**Figure 3:** Horizontal wind perturbation profile generated by the propagation of a broad wave spectrum (7) in a shear wind $d_zU=5 m/s (10km)^{-1}$ .Tthe horizontal velocity is given in units of , the initial amplitude.
$\includegraphics[width=3.1in,bb= 20 0 301 248]{fig/fig03a.eps}$

**Figure 4:** Power spectrum of the horizontal wind perturbation belonging to the profile shown in Fig. 3 (dashed line). The continuous line represents the theoretical prediction (16) for high vertical wavenumbers. Vertical lines indicates the initial spectrum.
$\includegraphics[width=3.1in,bb= 20 0 301 248]{fig/fig04a.eps}$

An estimation of the power spectrum at large wavenumbers ( $m>m_{02}$ ) resulting from (7) can be calculated in a way similar to single waves whose asymptotic behavior gives (see Appendix B),

$\begin{displaymath} PS(m)=\frac{2 u_0^2 \mu}{L \Delta m_0^2 (1/4+\mu^2)} \frac{m_{02}^3+m_{01}^3}{m^3} \end{displaymath}$ (8)

The analyzed height interval must contain at least the first critical level in order to give a power spectrum containing wavenumbers large enough to assure a contribution to the spectral tail. Note that the asymptotic behavior has changed dramatically with respect to single waves. In this case the power law is

, in the place of

, a consequence of the profile behavior when $z \rightarrow z_{c2}$ .

Fig. 4 shows the power spectrum obtained by Fourier of the profile shown in Fig. 3 and the theoretical prediction (16). There is a very good agreement between them as long as $m>m_{02}$ .