Modification of gravity waves propagating across the tropopause (GW-TP) : Results

Peter Spichtinger, Universität Mainz; Andreas Dörnbrack, DLR Oberpfaffenhofen; Rupert Klein, Freie Universität Berlin

PhD students:
Vera Bense, Uni Mainz; Sonja Gisinger, DLR Oberpfaffenhofen; Christopher Pütz, Freie Universität Berlin

The tropopause region has a potentially crucial impact on vertically propagating gravity waves. In this region static stability (N) and horizontal wind (U) change significantly, leading to wave propagation or (partial) reflection depending on the pertinent flow regime. In addition, non-linear effects, such as wave generation by shear instabilities, are likely to arise. Because of the strong gradients of N and U, the tropopause region has its own dynamical characteristics. Thus, we expect that gravity waves induce variations in the tropopause region. However, since detailed observations and—in particular--systematic theoretical investigations are lacking thus far, current understanding of these processes is limited. As a consequence, solid parameterisations for propagating waves through the tropopause are not available.


GW-LCYCLE 1&2 and DEEPWAVE – gravity wave measurements campaigns in Scandinavia and New Zealand

In the framework of the DFG project MS-GWaves and the related BMBF project ROMIC („Role of the middle Atmosphere in Climate“) extensive field campaigns were conducted in Scandinavia and New Zealand from 2013 to 2016. In accordance with the objectives of the projects, the excitation, propagation and dissipation of gravity waves was studied by airborne, balloon-borne, and ground-based measurements from the surface up to 90 km altitude.
The main focus of our studies in MS-GWaves was the modification of gravity waves propagating across the tropopause. For this purpose, the meteorological conditions during the DEEPWAVE campaign were analysed thoroughly. Based on these results, the tropopause inversion layer (TIL) was characterized by means of ECMWF IFS data and radiosonde measurements. Fig. 1 exemplarily shows profiles of the squared Brunt-Väisälä frequency N² from radiosonde data during a time period from 28 to 30 June 2014.
The strength of the TIL is represented by a peak in N² at around 10 to 12 km altitude. Peak values of N² weakened towards the end of the selected observational period as a trough approaching New Zealand lowered the tropopause and attenuated N² in the cyclonic region in accordance with previous findings from mid-latitudes. Figure 2 shows the difference between the N²-maximum and the tropospheric N²-minimum providing an objective measure of the TIL strength for the winter months of the DEEPWAVE period. An analysis of the time evolution of the TIL strength and the magnitude of the vertical energy fluxes associated with gravity waves across the TIL simulated with the WRF model was performed to elucidate the impact of the TIL on the vertical propagation of gravity waves. Both quantities are negatively correlated (Fig. 3), i.e. the stronger the TIL the less gravity wave energy was able to propagate into the stratosphere. The hydrostatic reflection coefficient r≈(N_max-N_T)/(N_T+N_max ) computed from the mean tropospheric and the maximum Brunt-Väisälä frequency at the TIL explains up to 36 % of the reduction of the vertical energy flux while a reduction of up to 77 % was found in the WRF simulations. This difference might be explained by non-hydrostatic gravity wave reflection due to vertical wind shear and at the enhanced static stability of the TIL.

Figure 1 Vertical profiles of the squared Brunt-Vaisala frequency (N²) from radiosondes during an observational period of the DEEPWAVE campaign. The TIL weakened during the course of this period due to a trough approaching New Zealand.
Figure 2 Difference between the maximum of N² at the TIL and the tropospheric minimum of N² showing the strength of the TIL during the DEEPWAVE campaign.
Figure 3 Vertical energy fluxes of the WRF model simulations at 4 km (grey) and 12 km (black) altitude during the DEEPWAVE campaign. (Data provided by C. Kruse)

Theoretical examinations of the gravity wave – tropopause interaction (Rupert Klein & Christopher Pütz, FU Berlin):

Our group is responsible for the theoretical analysis within the GW-TP subproject of MS-Gwaves. The main focus lies on the propagation of gravity waves through the tropopause. The latter is characterised by sharp changes in the stratification and also by strong jet winds. Above and below the tropopause, we have almost uniform stratification. The evolution of gravity waves can be described with the Euler equations. But since they cannot be solved analytically, we have to make assumptions to simplify the equations. The vertical extent of the tropopause is small compared to the density scale height, which gives a justification for the use of the Boussinesq approximation, in which the density changes are small compared to its characteristic value.
We developed a method to compute the transmission of gravity waves through a finite region of non-uniform stratification in a stationary atmosphere. It is based on an approximate solution of the Taylor-Goldstein equation, which can be derived from the linearised Boussinesq equations.
With the method, we are able to compute a transmission coefficient for gravity waves, which is defined as the ratio of the vertical wave energy fluxes below and above the region of non-uniform stratification. It makes use of the fact that plane wave solutions exist in uniform stratification and models the atmosphere as a multi-layer fluid where each layer is uniformly stratified. The solutions are matched at the interfaces in a physically meaningful way and in the end, we can relate incident and transmitted wave. Further, the limit of increasing number of layers is investigated and we obtain a reformulation of the Taylor-Goldstein equation. This equation can not be solved analytically, but numerically, giving a solution to the Taylor-Goldstein equation, in which it is possible to distinguish between the two branches of the dispersion relation, namely upward and downward travelling waves. Hence, we are also able to compute a transmission coefficient for a smoothly stratified layer from this procedure.

Moreover, it can be shown that the multi-layer solution converges to the limit solution quadratically with the number of layers. The results we obtain for some test cases are in accordance with several existing results, but give deeper and more general insights on the interaction of gravity waves propagating through non-uniform stratification.

Further, we succeeded to apply the method to vertically confined wave packets. For Gaussian initial conditions, i.e., a plane wave whose amplitude is modulated by a Gaussian curve, we are not only able to compute a transmission coefficient, but have the possibility to visualise the wavepacket evolution through a multi-layer atmosphere with analytically correct transmission and reflection.

Mathematically, the idea behind the computation of the transmission coefficient is of practical use. The intermediate steps can be taken in order to create a finite-element method that numerically solves the aforementioned Taylor-Goldstein equation. This is a very efficient and accurate procedure.


Work done at University of Mainz:

In the first phase we investigated the impact of tropopause layer characteristics as e.g. stratification or change of winds on the propagation (transmission and absorption) of GWs, using the research model EULAG in 2D and 3D setups with high vertical resolution. Especially the impact of strong TIL has been investigated. The propagation of GWs through the tropopause region is crucially changed for strong TILs; on the other hand, propagating GWs also have an impact on the tropopause region and can also affect the properties of the TIL in a crucial way.
For first investigations, vertically propagating monochromatic waves excited by flow over periodic topography were used. The excited wave lengths were chosen to be in a “realistic” range, as compared to observations, which were available from other parts of the project. In addition, the properties of the tropopause region were varied. For instance, the thickness of the tropopause layer and its stratification were varied, also including TILs of different strengths (i.e. large maxima in Brunt-Vaisala frequency between tropospheric and stratospheric values.) In addition, the environmental wind conditions were varied. For instance, the position of a strong jet relative to the tropopause layer or the TIL, its maximum values and the vertical extension of the jet region were changed in different sensitivity studies.

We could obtain several results about the propagation of waves through the tropopause region and the resulting wave spectra in the stratosphere. Generally, the propagation and the resulting spectra are crucially dependent on the environment conditions. Investigations with prescribed wave spectra were also carried out, leading to similar results as for monochromatic waves. Currently, an idealized TIL is used for deriving systematically differences in wave propagation for different scenarios in the parameter space of stratification. In a second step we carried out simulations based on observations as derived in the DEEPWAVE campaign. In case of a very strong TIL, there is a preferencing in wave length propagating through the tropopause; it seems that waves with horizontal wave lengths in order of 30 kilometres propagate through the tropopause without much loss of energy.
In one special case of a moderate TIL combined with a critical level (i.e. ue~0m/s) close to the maximum of the TIL (Fig. 4), we could observe secondary wave generation at the top of the tropopause layer. In fact, the breaking wave seems to generate a transient wave in the stratosphere. Additionally, the environmental conditions as wind and stratification are drastically changed; the critical level is extended downwards and the TIL is sharpened, leading to higher values in N2 (Fig. 4, right panel). Thus, we can conclude that there is a strong interaction of tropopause dynamics (e.g. TIL) and propagating GWs. In current work, we study the interaction of TIL and propagating GWs in idealized setups, results are probably available within the next few months (Bense & Spichtinger, in preparation).
We also investigated the transmission and reflection of vertically propagating waves in order to compare with theoretical work, as carried out in the subproject GW-TP-T, see below. Similar to the linear theory we could find wave tunneling (Pütz et al., in preparation).


Figure 4: Example of secondary wave generation for breaking waves at a critical level. (left: environmental wind, middle: vertical wind, right: stratification). LES simulation with EULAG.