Previous: Introduction. Next: The altitude of the Hodges condition Up: Ext. Abst.

 

Nonlinear wave propagation

To introduce the effects of the wave nonlinearities we use the Riemann invariants method. A new independent variable is used and the equations are transformed to this new reference frame, where the dependent variables can be assumed linear [Lighthill, 1978].

We assume a gravity wave propagating in a nonrotating atmosphere with a constant buoyancy frequency $N_0$ and a linear wind background $u_0= d_zu_0~ z$, then the Richardson number is $Ri=\frac{N_0^2}{d_zu_0^2}$. When the wave is propagating towards the critical level the vertical wavenumber $m$ goes to infinity and therefore $m \gg k$ and $m H \gg 1$ are satisfied in the proximity to the critical level $z_c$. The linear solution yields for the vertical velocity,


\begin{displaymath} w_1=w_a (1- z_c^{-1} z)^{1/2-i \mu} \exp i (\omega t + k x) \end{displaymath} (1)

it has an amplitude $w_a$ and vertical wavenumber $m_0$ at $z=0$, $\mu=(N^2 d_zu_0^{-2}-4^{-1})^{1/2}$, $\omega$ is the ground-based frequency, the intrinsic frequency is $\Omega=\omega - k u_0$, and the critical level is at $z_c=Ri^{1/2} m_0^{-1}$. Note that linear quantities are represented by subscript $1$ and mean quantities by $0$ subscript.

In the absence of dissipation the characteristic surfaces of gravity waves are surfaces where the entropy and potential temperature are constant. The characteristic surfaces $\phi$ are given by,


\begin{displaymath} \partial_t \phi + u \partial_x \phi + w \partial_z \phi = 0 \end{displaymath} (2)

A similar equation is also satisfied by the potential temperature. At zero order the surfaces are planes at fixed altitudes. While, if the potential temperature perturbations are taken into account, the characteristic surfaces (in first order) are given by


\begin{displaymath} \phi=z - s - \frac{w_1}{i \Omega}=0 \end{displaymath} (3)

with satisfy (2) at first order.

The procedure shown here is similar to the developed by Einaudi (1969) for sound waves and applied to gravity waves by Teitelbaum and Sidi (1976, 1979). The independent variable $z$ is now changed to $s$. We again emphasize that isentropic surfaces are the characteristic surfaces for the gravity wave equations under the assumptions mentioned above. This means that the dependent variables expressed as functions of the new variable will be a better approximation to the nonlinear problem. This procedure is shown in Whitham (1974), called there the nonlinearization technique. When it is applied to Burgers equation it can be seen that the exact non linear solution is the same that the one obtained by this method.

Transforming the equations to the new variable we note that the linear version of the equations in $s$ are the same as the original linear equations in $z$ leading to the same solution (1) but in $s$.

Previous: Introduction. Next: The altitude of the Hodges condition Up: Ext. Abst.