In this work we analyze the power spectrum resulting from a broad gravity wave spectrum propagating upwards in a shearing background. Starting from the gravity wave solutions we show that the power spectrum of a continuous (in wavenumber space) wave packet has a -3 power law in the spectral tail while, as it is known, a monochromatic wave propagating in a shearing background has a -1 power law. The energy exchange with other parts of the spectrum maintains the amplitude within the tail in the range of the observed amplitudes.

Observations of energy distribution of horizontal winds and temperature irregularities show an apparent universal behavior in the short vertical wavelength end of the spectrum. The observed shape appears to be independent of the altitude, place, and time (Van Zandt, 1982; Allen and Vincent, 1995). This fact has suggested the presence of physical processes which can act over the wave field leading it to a saturated form.

One of the most accepted theories associates the saturation of the waves belonging the spectral tail to interactions between different components of the wave field, specifically, Hines (1991) suggests that the Doppler shifting that suffers a wave propagating upon the wave field is the dominant process in the determination of the high wavenumber amplitudes.

However, the so developed theory can not reproduce the

power law in the vertical wavenumber of the observed spectra. The asymptotic behavior obtained by Hines has a $m^{-1}$ dependence. He argued that the high wavenumber part of the spectrum may be influenced by dissipative processes which could diminish the amplitudes.

The

power law is directly related to the result obtained for a single gravity wave propagating in a shear flow (Pulido and Caranti, 2000). Hines used a similar law for a spectrum of gravity waves but to the best of our knowledge there is no work in the literature showing that these results for single waves can be extrapolated to continuous gravity wave spectra.

It is important to bear in mind the difference between the physical spectrum which is propagating upward, that we call spectrum, and the power spectrum resulting from the Fourier transform in a height interval which we call power spectrum (PS). For example, a single wave has a monochromatic spectrum in a fixed altitude while has a broad power spectrum when a height interval is analyzed.

In this preliminary work we show that when a continuous spectrum of gravity waves is propagating in a shear flow there are interference effects between the different components of the spectrum which alter the behavior observed for single waves, and a -3 power law results for the spectral tail. Our final objective is to include in the approach presented here the nonlinear advective interactions which are necessary to saturate the vertical power spectrum but not to give the observed shape of it as it is shown here.

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 $m>m_{02}$ .

In the middle atmosphere, the wind perturbations are composed of wave packets coming from several tropospheric sources, such as mountains, convective and frontal activity, breaking of mountain waves, shear instabilities, geostrophic adjustment of the jet stream, which have different characteristic propagation speeds.

To model this situation, we assume that a superposition of wave packets is found in the region of interest, each packet of the form (6) having a range of horizontal wave speeds between $c_2=N m_{02}^{-1}$ and $c_1=N m_{01}^{-1}$ .

In this representation a broad spectrum in the initial vertical wavenumber has also a broad spectrum in frequency through the dispersion relation, while in the horizontal wavenumber the spectrum is monochromatic. Note that

remains unaltered during the whole wave propagation. An extension of this formulation to include packets in

can also be made as long as the wave spectra are separable. At the present stage of development we have chosen to present only the monochromatic scenario where the main physical process are clearer specially for conceptual purposes.

Each generated profile contains 15 wave packets, 10 terminating inside of the interest region and 5 terminating above this region. The exact place of termination is random within an interval. In this way we determine the parameter $m_{02}$ .

From the rest of the free parameters of the wave packets, the amplitude

, and the initial phase are taken randomly within a reasonable interval, in a way similar to Chimonas, 1997 while the width of the spectrum which has also been taken at random within an interval ( $\lambda_z=400 m$ to $\lambda_z=1000 m$ ) in order to produce a reasonable perturbations. Note that it is impossible to isolate a wave packet in an observed wind perbutation profile in order to determine the spectral width. In future work it would be possible to estimate this parameter in a somewhat indirect way from profiles, if their effects on the observed vertical power spectrum through Doppler shifting are taken into account.

Figures 5, 7 and 9 show characteristic wind profiles for background conditions given by $N=0.01 s^{-1}$ and $d_z U=( 30, 10, 2) m s^{-1} (10 km)^{-1}$ , (

), respectively. There is a high correspondence between the numerically generated profiles and actual profiles.

For the weakest shear case the PS goes to the -3 predicted tail for higher wavenumbers because it has higher initial wavenumbers and the asymptotic behavior is satisfied for $m>m_{0}$ (See (16)). The spectral knee has also a mean shear dependence, the higher the shear the greater the characteristic wavenumber. The quantitative values may be strongly altered in a more realistic picture (interactions between components, time dependences, etc.).

In this work we have shown that there is interference effects between the components of continuous wave spectra which change the behavior of the profiles specially in the neighborhood of the critical level. This fact produces profound differences with the monochromatic case when the power spectrum is analyzed; a

power law arises in the spectral tail which is in accord with observed spectra.

While for a single wave propagating towards the critical level the overturning condition occurs always before the critical level for a broad spectrum this condition does not seem to be necessarily satisfied. So this work ignores the dissipative effects because taking into account the preliminary results shown here they do not appear to be essential in the determination of the wave termination.

The wind irregularites simulations reproduce the observed perturbations and the main aspects of the observed spectra. The observed spectral slope has been obtained through mean wind - gravity wave interacctions.

In a near future we will address a generalization of the results for any background wind abiding WKB theory. In this way the effects of the advective interaction between the components of a broad spectrum can be also analyzed. Note that the interactions of the wave field play a double role in determining the power spectrum from a Doppler shift point of view. As the background wind, a component propagating in a wave field is Doppler shifted to the spectral tail with a

power law. Besides, these interactions also saturate the amplitudes. The saturation originates in the correlation between energy content of the wave field and the energy eliminated or deposited into the critical layer. Clearly, the greater the wave field energy the larger the energy deposited.

It is important to mention that the analysis we have done in this work is for a fixed time; the way in which the measurements are performed. Conclusions must not be extrapolated to a time dependent scenario.

A proof of the PS resulting from a single wave propagating upon any background wind abiding WKB can be found in Pulido and Caranti (2000), here we just show the linear background case in order to compare results and procedures with the broad spectrum case.

The solution (2) of the wave equation for a wave propagating in an environment where the wind changes linearly with height at a particular time and a fixed horizontal position can be expressed as

$\begin{displaymath} u(z)=u_a (1- m_0 \mu^{-1} z)^{-1/2- i \mu} \end{displaymath}$ (9)

$\begin{displaymath} \frac{L}{2} \int PS dm = \int u(z) u^*(z) dz \end{displaymath}$ (10)

$\begin{displaymath} \int PS dm =u_a^2 \int (1- m_0 \mu^{-1} z)^{-1} dz \end{displaymath}$ (11)

$\begin{displaymath} \int PS dm= \frac{2 u_a^2}{L} \int \frac{\mu}{m_0 m} dm \end{displaymath}$ (12)

Then, as the extremes of the integration interval are free, the integrands must be equal,

$\begin{displaymath} PS=\frac{2 \mu u_a^2}{L m_0 m}. \end{displaymath}$ (13)

The same mathematical procedure as in Appendix A is followed here. Starting from the Parseval's theorem (10), and taking into account the expression for the perturbation (4) it yields two oscillatory functions where interference terms can be neglected. It is easy to see that these terms go to 0 as $m \gg m_{02}$ . On the other hand these terms are unbounded at $m=m_{01},m_{02}$ . So the expression yields

$\begin{displaymath} \int PS dm=\frac{2 u_a^2 \mu}{L \Delta m_0^2 (1/4+\mu^2)} \... ...}{z^2} dz + \int \frac{(1- m_{01} \mu^{-1} z)}{z^2} dz \right] \end{displaymath}$ (14)

Now a variable change from

to $m=\left({1/m_{0i}- \mu^{-1} z} \right)^{-1}$ is performed. This mathematical transformation is only related to the oscillatory terms. So we find

$\begin{displaymath} \int PS dm = \frac{2 u_a^2 \mu}{L \Delta m_0^2 (1/4+\mu^2)}... ...m_{01}^3}{m^3} \left(1-\frac{m_{01}}{m}\right)^{-2}\right] dm \end{displaymath}$ (15)

$\begin{displaymath} PS=\frac{2 u_a^2 \mu}{L \Delta m_0^2 (1/4+\mu^2)} \left[\fra... ...rac{m_{01}^3}{m^3} \left(1-\frac{m_{01}}{m}\right)^{-2}\right] \end{displaymath}$ (16)

This expresion gives an approximation to the amplitudes in the high wavenumber part of the spectrum.

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The vertical wavenumber power spectrum resulting from the propagation of a gravity wave spectrum

Manuel Pulido and Giorgio Caranti

Abstract