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Numerical simulations

The equation (3) can not be solved analytically so profiles were carried out numerically. There are two parameters which can alter the qualitative behavior in the wave termination profiles, the Richardson number and the initial phase. The latter is chosen in order to find the lowest height of discontinuity where (7) is fulfilled and its derivative is also equal to 0. Therefore, the only free parameter is $Ri$, and here we present two cases: a weak shear case, $Ri=1600$ and a strong shear case, $Ri=6$.

Fig 1 shows the altitude vs the new space coordinate for the $Ri=1600$ case, note that as the wave amplitude increases the transformation becomes multivalued.

 

Figure 4: As in Fig 2 for $Ri=6$
\includegraphics[width=3.1in,bb= 20 0 301 248]{/home/pulido/igw/nolin//fig/scfig04.eps}
Figure 5: As in Fig 3 for $Ri=6$
\includegraphics[width=3.1in,bb= 20 0 301 248]{/home/pulido/igw/nolin//fig/scfig05.eps}

The linear and nonlinear profiles are very similar for small wave amplitudes compared with the threshold (6), however differences appear when the amplitudes are comparable to the threshold. The horizontal wind profile is asymmetric, showing peaks in the minima (the shear has jumps) while the maxima have fatter ``bumps''. The maximum amplitude of the wave is the same that in the linear solution (Fig. 2). On the other hand, the temperature profile presents the classic jumps (Fig 3) which remind the shape of the shock waves (Lighthill, 1978; Einaudi, 1969).

Linear waves propagating on a background with strong shear have in short distances important amplitude and vertical wavenumber changes near the critical layer. So nonlinear waves upon a strong wind shear show differences in the profile just in the neighborhood of the contact surface (Figs. 4 and 5).

The shape of the temperature profiles seems to agree rather well with the abrupt termination proposed by Chimonas (1997) to model wind irregularities. As he has shown this shape appears to have a power spectra with the same behavior that the observed ones.

The power spectra of the nonlinear waves has associated two spectral laws, one for the low vertical wavenumber where contributions come from the wave far of the Hodges condition while the high vertical wavenumber part of the spectra is dominated by the internal leakage (Pulido and Caranti, 2000a). This mathematical artifact can not be diminished by windowing because wave terminations occur inside the analyzed interval and not at the extremes.

Comparison of the spectrum derived from the temperature profile with the horizontal wind spectrum shows that the low wavenumber part are similar. On the other hand, the spectral tail for the temperature spectrum is dominated by the leakage of zero order discontinuities with a $-2$ power law while for the horizontal wind spectrum the tail has a $-4$ power law (Fig. 6) which results from first order discontinuities. Of course as it has already been noted elsewhere [Pulido and Caranti, 2000a], for a superposition of waves the spectrum has a power law between $-4$ to $-2$.

The knee in Fig. 6 represents the transition value between the wave generated spectrum and the spectral tail due to internal leakage and it is found at,


\begin{displaymath} m_c=m(z_d)=\frac{\mu}{z_c-z_d}\approx\left( \frac{N^2 m_0}{u_a^2} \right)^{1/3} \end{displaymath} (8)

where it was assumed that for high Ri, $\mu\approx Ri^{1/2}$. $m_c$ is not independent of the initial parameters $u_a$, $m_0$. The amplitude at $m_c$ is (note that the nonlinear distortion does not alter the amplitude),


\begin{displaymath} u(z_d)^2=\frac{u_a^2 m(z_d)}{m_0}\approx\frac{N^2}{m^2} \end{displaymath} (9)

Clearly, this amplitude is independent of initial values.

Figure 6: PS of the horizontal wind perturbation (dashed line) and the temperature perturbation PS (dotted line), straight lines are the best fit with a $-3.35$ and $-2.25$ slope respectively.
\includegraphics[width=4.1in,bb= 20 0 301 248]{/home/pulido/igw/nolin//fig/scfig06.eps}

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