The convective instability in a nonlinear gravity wave approach

Manuel Pulido and Giorgio Caranti

Grupo de Física de la Atmósfera
FaMAF, Universidad Nacional de Córdoba (Argentina)
E-mail: pulido@roble.fis.uncor.edu


FIGURES

Abstract

A nonlinear theoretical approach is used to study a gravity wave which becomes convectively unstable. Under this approach a better understanding of the physical processes is obtained. We see that the energy exchange with the background occurs in very thin layers where contact surfaces appear in the solution. The results show that the observed invariance of the spectrum can not be explained by the linear instability theory at least in its present stage of development. The simulated profiles with this model present discontinuities generated by physical nonlinear processes that lead naturally to a spectral tail produced by internal leakage.

Introduction

In a linear approach gravity waves propagating in a shear flow are strongly attenuated as they approach their critical levels. At this place the horizontal background velocity equals the horizontal phase speed and momentum of the waves will be transfered to the background flow [Booker and Bretherton, 1967].

Before the waves reach the critical level they have attained the Hodges condition where the mean buoyancy frequency ($N_0$) is equal to the buoyancy frequency perturbation generated by the wave. There the flow becomes convectively unstable [Hodges, 1967].

There are several speculations about how gravity waves break in the region between the first Hodges threshold and the critical level:

It is important to recognize that nonlinear distortions of the profile are crucial for a correct treatment of wave breaking. In this work we show that between the first instability condition and the critical level a contact surface arises. The temperature profile shows a discontinuity across this surface where fast conduction is expected resulting in a quick attenuation of the wave.

The consequences of these results are in disagreement with Dewan and Good's model, there is no harmonic wave form above the instability and the spectral tail amplitudes are due to internal leakage coming from the discontinuities in the temperature and wind profiles.

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 $s$.

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}

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}

Discussion

Using the nonlinear technique applied by Teitelbaum and Sidi (1976, 1979) to gravity shear waves we have shown that nonlinear terms becomes important when the amplitude of the wave is equal to the threshold given by the linear convective instability.

Nonlinear wave amplitudes are bounded everywhere even at the place where the transformation of coordinate fails. Moreover, the nonlinear maximum amplitude is equal to the linear maximum amplitude. What nonlinearities do is to ``move'' the maxima and minima to the same altitude which generate jumps in the density and temperature profiles. These surfaces where temperature jumps are located are not crossed by the flow, they move with it and because of that the wave should be quickly dissipated.

Therefore wave breaking is induced at the Hodges condition and it is unrealistic to assume a harmonic wave form above this condition altitude as it has been suggested by Dewan and Good. Temperature profiles fit better Chimonas' model because the wave is strongly attenuated at the contact surface.

From a spectral point of view, the power spectrum of a height interval can be divided into two parts one which has the contribution of the refraction and the other, the spectral tail, is composed by internal leakage. Therefore leakage is not only produced by Fourier transform contamination but also by physical processes involved in the wave termination.

Future work should take into account the dissipation effects and a time dependent scenario in order to give more realistic wave terminations and to address how the energy exchange with the surrounding is taking place. We think that these preliminary results shown in this work give a good starting point to improve the theoretical knowledge of these processes.

The conclusions extracted from this treatment are only valid for monochromatic waves and cannot be extrapolated to a broad spectrum where interference effects can alter the profile near the critical level and the presence of linear convective instabilities are not assured there, as it is shown in Pulido and Caranti (2000b).

 

References

Booker and Bretherton, 1967
Booker J. and F. Bretherton, 1967: The critical layer for internal gravity waves in a shear flow. J. Fluid. Mech., 27, 513-539.
Chimonas, 1997
Chimonas, G., 1997: Waves and the middle atmosphere wind irregularities. J. Atmos. Sci., 54, 2115-2128.
Dewan and Good, 1986
Dewan, E. M. and Good R. E., 1986: Saturation and the ``universal'' spectrum for vertical profiles of horizontal scalar winds in the atmosphere. J. Geophys. Res., 91, 2742-2748.
Fritts, 1985
Fritts, D. C., 1985: Gravity wave saturation in the middle atmosphere: A review of theory and observations. Rev. Geophys., 22, 275-308.
Hodges, 1967
Hodges R., Jr., 1967: Generation of turbulence in the upper atmosphere by internal waves. J. Geophys. Res., 72, 3455-3458.
Lighthill, 1978
Lighthill, J., 1978: Waves in fluids . Cambridge University Press, Cambridge.
Pulido and Caranti, 2000a
Pulido M. and G. Caranti, 2000a: Spectral tail of a gravity waves train propagating in a shearing background. J. Atmos. Sci., 57, 1473-1478.
Pulido and Caranti, 2000b
Pulido M. and G. Caranti, 2000b: The vertical wavenumber power spectrum resulting from the propagation of a gravity wave spectrum. (This session)
Sato and Yamada, 1994
Sato K. and M. Yamada, 1994: Vertical structure of atmospheric gravity waves revealed by the wavelet analysis. J. Geophys. Res., 99 , 20623-20631.
Teitelbaum and Sidi, 1975
Teitelbaum H. and C. Sidi, 1975: Formation of discontinuities in the atmospheric gravity waves. J. Atmos. Terr. Phys., 38, 413-421.
Teitelbaum and Sidi, 1979
Teitelbaum, H. and C. Sidi, 1979: Discontinuity formation due to gravity wave propagation in a shear layer. Phys. Fluids, 22, 209-213.
Whitham, 1973
Whitham G., 1973: Linear and nonlinear waves. Wiley-interscience.

Back to

Session 1 : Stratospheric Processes and their Role in Climate Session 2 : Stratospheric Indicators of Climate Change
Session 3 : Modelling and Diagnosis of Stratospheric Effects on Climate Session 4 : UV Observations and Modelling
AuthorData
Home Page