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The altitude of the Hodges condition

Now we want to get the threshold wave amplitude that is necessary to generate an instability. The convective linear instability is achieved at places where the linear buoyancy frequency produced by the temperature perturbations is equal to the background buoyancy frequency, $N_0^2=N_1^2$, being,


\begin{displaymath} N_1^2= \frac{g}{\theta_0} \partial_z \theta_1 - \frac{g \theta_1}{\theta_0^2} N_0^2 \end{displaymath} (4)

Taking into account the polarization relation between potential temperature and vertical velocity which is obtained from the entropy conservation equation (we have adiabatic motions), we find


\begin{displaymath} \theta_1= \frac{-w_1}{i \Omega} d_z \theta_0 \end{displaymath} (5)

As it has been assumed that $N_0$ is constant in the region of interest, the threshold amplitude must satisfy


\begin{displaymath} \partial_z \left\vert\frac{w_1}{\Omega}\right\vert=1 \end{displaymath} (6)

On the other hand, the variable transformation (3) must be an injective function, but the wave amplitude increases as the wave propagates towards the critical level and before the wave reaches it the variable transformation becomes multivalued and the characteristic surface intersect other characteristic surfaces. The threshold value is given by


\begin{displaymath} \frac{dz}{ds}= 1 + \partial_s \left(\frac{w_1}{i \Omega}\right)=0 \end{displaymath} (7)

When this relation is satisfied a discontinuity surface appears. These are the same sort of discontinuities admitted in longitudinal waves which are called contact surfaces and separate two regions of different density and temperature, while the perpendicular velocity to the surface and the pressure are continuous.

As in first order, derivatives with respect to $s$ are equal to derivatives with respect to $z$, what we have shown above is that the discontinuity develops at the place where the threshold amplitude (6) is satisfied. The energy exchange with the background will occur at these places where heat conduction between the two sides of the surface discontinuity is large and therefore these surfaces as well as the wave itself will be quickly dissipated. The developed treatment would soon become unrealistic and does not allow us to predict the behavior after the discontinuity has appeared.

Figure 1: Coordinate transformation $z=z(s)$ (3) for the low mean shear case ($Ri=1600$), straight lines show the first discontinuity level. After that, around the next maximum, the transformation becomes multivalued. Both axes are normalized by the critical level height
\includegraphics[width=4in,bb= 20 0 301 248]{/home/pulido/igw/nolin/fig/scfig01.eps}

 

Figure 2: Horizontal wind perturbation. Continuous (dashed) line represents the nonlinear(linear) profile. The horizontal straight line indicates the height of the contact surface. The velocity axis is normalized by the initial amplitude ($u_a$).
\includegraphics[width=3.1in,bb= 20 0 301 248]{/home/pulido/igw/nolin/fig/scfig02.eps}
Figure 3: Temperature perturbation. Continuous (dashed) line represents the nonlinear (linear) profile. The horizontal straight line indicates the height of the contact surface. Temperatures are normalized by the initial amplitude ($T_a$).
\includegraphics[width=3.1in,bb= 20 0 301 248]{/home/pulido/igw/nolin/fig/scfig03.eps}

Previous: Nonlinear wave propagation Next: Numerical simulations Up: Ext. Abst.