Transient Tropopause Waves

a. Research flights

The research flights considered in this study were conducted over southern Scandinavia in the afternoon and evening of 28 January 2016. A detailed overview of the coordinated airborne observations by the DLR Falcon and the German research aircraft HALO is provided by Gisinger et al. (2020) in their section 2.2.1. Here, Table 1 lists the times and coordinates of the six cross-mountain flight legs used in this study. The naming of the individual legs is the same as in Gisinger et al. (2020), who also provide a graphical overview of the flights in their Figs. 1 and 2.

Table 1.

Times and locations of the two segments of the DLR Falcon research flights RF07 and RF08 as well as the four segments HL1–HL4 of the HALO research flight on 28 Jan 2016. Letters (a)–(f) refer to the profiles of the vertical displacements and vertical velocities along the respective flight legs as shown in Figs. 4 and A1. The simultaneous flight legs are in bold and correspond to the red lines in Figs. 4, 6, and A1.

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Simultaneous observations on board the two aircraft were conducted on a flight path near 61°N from east to west where the DLR Falcon flew at 9.8-km altitude and HALO flew 2 km below from 1736 UTC until 1840 UTC: These are the research flight legs RF08-FL1 and HL1, respectively, highlighted in bold in Table 1. The RF07-FL2 research flight preceded these simultaneous flights, and HL2 followed them on the same route and flight levels assigned to each aircraft. The operation of the remote sensing instrument Gimballed Limb Observer for Radiance Imaging of the Atmosphere (GLORIA) required two high-level HALO legs at ∼13-km altitude (HL3 and HL4) for probing the mountain waves over southern Scandinavia (Krisch et al. 2020).

b. Numerical integrations

The magnitude of the gravitational acceleration is given by g = (0, 0, −g). The Coriolis parameter is given as f ≡ 2Ω, where Ω denotes a constant angular velocity of the rotating reference frame. The differential operator of the material derivative in Eqs. (1)–(3) is defined as d/dt = ∂/∂t + u ⋅ ∇, where ∇ symbolizes the gradient operator. The operator ∇ ⋅ in Eq. (3) refers to the divergence.

Equation (2) is written in perturbation form, i.e., θ′ = θ − θ_e and π′ = π − π_e; see Smolarkiewicz et al. (2019). To facilitate the numerical solution, the auxiliary ambient states, sometimes called environmental states (u_e, θ_e), are assumed to be known particular solutions of Eqs. (1) and (2). Generally, ambient states can be time dependent. In this paper, only z-dependent profiles (u_e, υ_e, θ_e, ρ_e) are prescribed that automatically satisfy Eqs. (1) and (2). The ambient pressure p_e is computed from the ideal gas law equation [Eq. (4)] using an exponentially decreasing density ρ_e with a density scale height of 6500 m. For all numerical simulations, the initial conditions for the horizontal velocity components are identical to u_e and υ_e and θ′and w are set to zero.

The typical size of the computational domain in x, y, and z directions is 1008 km × 64 km × 70 km with spatial resolutions of 250 or 500 m. The lower boundary is free slip, and in the x direction, open boundaries with an additional Rayleigh damping of a width of 40 km on the inflow and outflow sides are used. In the y direction, cyclic boundaries are applied. Details for the individual numerical experiments can be found in Tables 3–6.

Equations (1)–(4) were implemented in the established geophysical flow solver EULAG (Prusa et al. 2008) as outlined by Smolarkiewicz et al. (2014). EULAG offers a variety of options to investigate the sensitivity of the solutions on numerical settings. Here, we differentiate between physical and numerical sensitivity studies. The physical sensitivity is studied by varying the amplitude damping of the vertically propagating mountain waves to keep the numerical solution nearly linear (section 5a). In addition, the smoothness of the orography is varied and the strength of the TIL is enhanced in separated numerical integrations (sections 5b and 5c). In section 5d, the flow fields at t = 0 are generated by a potential flow initialization and Eqs. (1)–(4) are integrated using the anelastic approximation; see Smolarkiewicz and Dörnbrack (2008), Smolarkiewicz et al. (2014) for details. Furthermore, we vary the stopping criteria of the elliptic solver (Smolarkiewicz et al. 1997), the distribution of processors used for the domain decomposition of the parallelized solver and test different options of the compiler (section 5d).

c. Global NPW data

Operational analyses of the Integrated Forecasting System (IFS) of the ECMWF are used to provide meteorological data to characterize the atmospheric environment; these data will be denoted by HRES IFS. The model fields were taken from the esuite of the IFS cycle 41r2 that became operational in March 2016. The IFS is discretized horizontally using a spherical harmonic expansion and a cubic-octahedral grid (Hólm et al. 2016). The TCo1279 resolution of the IFS cycle 41r2 uses 1279 total horizontal wavenumbers in the spherical harmonic expansion, resulting in an average 9-km horizontal resolution. In the vertical direction, 137 hybrid model levels cover the atmosphere from the surface up to the model top at 0.01 hPa. The IFS cycle 41r2 is the same as used for the most recent reanalysis data ERA5 (Hersbach et al. 2020). As ERA5, however, has a smaller resolution (TCo639), the operational analyses are used instead for the selected time period.

a. Airborne observations

Figure 4 shows the vertical displacements $\eta ={\int }_W^x{w}^{\prime }(x)/U(x)dx$, where the perturbations of the vertical wind w′ are determined from the in situ measurements as suggested by Smith et al. (2016) and W denotes the westernmost waypoints of the six cross-mountain legs. The w profiles as observed by the DLR Falcon and HALO are added as Fig. A1 in the appendix, where the individual measurement systems are also referenced. As all flights legs were nearly parallel to the zonal wind U (see Fig. 2b), the different η profiles in Fig. 4 reveal the observed large-scale and small-scale flow response in the upper troposphere, near the tropopause, and in the lowermost stratosphere. Based on the visual impression of these measurements and as a practical convention, the cross-mountain legs are divided into three segments: (i) an upstream segment from the westernmost waypoints to 7.5°E, (ii) a ridge segment from 7.5° to 11.5°E, and (iii) a downstream segment east of 11.5°E; see the vertical dashed lines in the upper panel of Fig. 4.

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(top) Vertical displacements η for the six cross-mountain legs of the research flights of DLR Falcon and HALO as listed in Table 1. The line colors correspond to the ones used in Fig. A1. The vertical dashed lines mark the boundaries between the upstream, ridge, and downstream segments. (bottom) ASTER orography at around 61°N underneath FL2 of RF07.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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In addition to the ubiquitous presence of gravity waves with ${\lambda }_x\lesssim 10\hspace{0.17em}\text{km}$, the lower η profiles show a mountain wave, in particular in the HALO leg HL1 [profile (c) in Fig. 4], whose horizontal wavelength can be estimated to be ∼6°. This corresponds to ${\lambda }_x\approx 330\hspace{0.17em}\text{km}$ and is in very good quantitative agreement with the wavelength of the mountain wave as estimated from the simulated T′ values of the IFS, as shown in section 3. The horizontally longwave is also discernible in both stratospheric legs flown by HALO along about 60°N [profile (e) in Fig. 4] and along ∼61°N [profile (f) in Fig. 4]. However, their amplitude is smaller at the higher flight levels. Since both legs are displaced in the north–south direction, the apparent coherence of the η profiles supports the quasi two dimensionality of the large-scale wave response in agreement with the previous findings from the IFS (cf. Fig. 3).

The η profiles in the upstream segments of the DLR Falcon [profiles (a) and (b) in Fig. 4] as well as of the HALO legs [profiles (c) and (d) in Fig. 4] are very similar. Here, it should be noted that the time difference of the observations was ∼4 h in the case of the DLR Falcon but only a few minutes by the HALO research flights. This pronounced agreement of the η profiles suggests a high degree of steadiness in the upstream flow. Over the mountains, however, the η profiles deviate due to mountain waves that alter both the horizontal and vertical wind fields with time. The vertical displacements calculated for two DLR Falcon legs differ markedly in the ridge and downstream segments, indicating a transient airflow induced by the mountain waves.

The only simultaneous observations were performed by DLR Falcon [RF08-FL1 at 9.8-km altitude, profile (b)] and by HALO [HL1 at 7.9-km altitude, profile (c)] and are marked in red in Figs. 4 and A1. Visual impression suggests irregular, low-amplitude oscillations west of 6.5°E, which become more distinct wave signals over the fjords at ∼7.5°E in the upstream segment. Over the mountains, horizontally longer waves with larger w amplitudes up to ∼2 m s⁻¹ dominate up to 12°E (Fig. A1). Here, the signatures of these longer waves are clearly visible in the lower leg HL1. Finally, further east, horizontally shorter waves with considerable amplitudes >3 m s⁻¹ predominate.

Overall, it is difficult to relate individual short-wave patterns in the two simultaneous legs, suggesting a pronounced vertical change in their wave characteristics over the ∼2 km between the two flight levels. In addition, the short-wave response is very transient when comparing the DLR Falcon legs at different times [profiles (a) and (b) in Figs. 4 and A1]. It is hardly possible to identify a wave pattern with the same phase at a specific location. This means that transient gravity waves probably existed during the 3–4 h between measurements. An analogous conclusion can be drawn by comparing the two HALO legs HL1 and HL2, which are separated by a maximum of 1 h at the eastern waypoints. Here, some similarities in the wave phases between 6.5° and 8.5°E and around 11°E can be observed. Nevertheless, both profiles (c) and (d) in Figs. 4 and A1 demonstrate a pronounced unsteadiness in the short-wave response. Profiles (e) and (f) in Figs. 4 and A1 show the uppermost HALO observations along legs HL2 and HL3: the larger amplitudes and longer horizontal wavelengths of the short gravity waves are the most obvious features of these measurements, which cover almost all segments. Again, as with the lower legs, there are no concurrent phase matches of these short gravity waves in the measurements along these legs.

On the DLR Falcon, a scanning DWL also measured the wind below the aircraft (Witschas et al. 2017). The observations of the vertical wind for the flight legs RF07-FL2 and RF08-FL1 are depicted in Fig. 5 and those of the horizontal wind along the RF08-FL1 leg in Fig. A2 of the appendix. A common feature of both DLR Falcon flight legs is a band of nearly continuous observations from 9.5 km down to ∼8-km altitude (Figs. 5a,b), below which dense clouds prevented further lidar measurements. In regions of lower cloud cover, as occurred upstream of the main ridge, the laser beam was able to penetrate to lower altitudes.

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(a) Observed vertical velocity (m s⁻¹; color shaded, nadir-pointing DWL, 1-s resolution data smoothed with a 10-s filter) and IFS potential temperature (K; contour lines; 2-K intervals) interpolated along the flight track RF07-FL2. (b) Observed vertical velocity (m s⁻¹; color shaded; forward–backward scanning DWL; 5-s resolution data smoothed with a 10-s filter) and IFS potential temperature (K; contour lines; 2-K intervals) interpolated along the flight track RF08-FL1. ASTER orography underneath the flight path in black. The reddish-brown line marks the IFS orography.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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The DWL data are the same as those used for Figs. 8 and 9b of Gisinger et al. (2020) and for the momentum flux analysis of Witschas et al. (2023). Moreover, lines of constant potential temperature calculated from the IFS-HRES data and spatiotemporally interpolated to the flight paths are superimposed here. First, these plots show that the DLR Falcon in situ flight observations [Figs. 4 and A1, profiles (a) and (b)] were made directly in the TIL and the lowermost stratosphere, where the static stability (densely packed isentropes in Fig. 5) is very high. Second, the DWL vertical winds show nearly vertical phase lines, indicating evanescent modes trapped in the TIL and upper troposphere.

Figure 5a shows narrow wave trains just beneath the flight level with small amplitudes in the upstream segment during RF07-FL2. Their depth is limited to ∼1 km, and there is no obvious connection to lower levels. During RF08-FL1 and about 4 h later, the DWL reveals gravity waves extending to lower altitudes; see Fig. 5b. The individual up- and downdrafts of the gravity waves are now broader, and the few measurements in the lower troposphere indicate that they are not directly connected to the orographic sources in the fjord region of the upstream segment; see the updrafts at around 6°–7°E. In the ridge segment, the waves extend lower into the troposphere, and especially over the highest mountains, the waves appear to be related to the forcing from the flow over the individual peaks; see the region around 8.5°E in Figs. 5a and 5b. Their amplitudes are larger, and the horizontal wavelength is longer than that of the upstream waves. Further downstream, the individual up- and downdrafts in the TIL and upper troposphere become thinner, i.e., their horizontal wavelength shrinks and the vertical extent of these apparently horizontally compressed wave trains is difficult to estimate. However, the downward decreasing amplitude of the observations in the downstream segment near the easternmost waypoint of RF07-FL2 indicates that they probably do not extend to lower levels. Indeed, the thermal stratification as shown by the IFS isentropes suggests that their vertical extent is related to the existence of stably stratified layers (cf. also patterns in the upstream segment of RF07-FL2). The horizontal wind observations by the DWL along RF08-FL1 reveal a long mountain wave (Fig. A2) corresponding to a potential temperature minimum directly over the ridge section. The position and the amplitude are reasonably well reproduced by the HRES IFS. Its phase and wavelength also correspond nicely to the coarse-grain GLORIA measurements as analyzed by Krisch et al. (2020).

b. Energy spectra

The spectral power of the vertical wind as measured by both aircraft is shown in Figs. 6 and 7. The binned energy spectra of the two DLR Falcon legs show maxima between λ_x ≈ 4 and 7 km (red and black solid lines in Fig. 6). The lower HALO legs HL1 and HL2 (dashed lines in Fig. 6) exhibit a broader distribution with smaller maximum amplitudes between λ_x ≈ 10 and 20 km. The upper HALO legs HL3 and HL4 also peak in this wavelength range but reach spectral amplitudes 5–8 times higher than the maxima at 7.8-km altitude. This altitude dependence is also reflected in the variances σ_w averaged over the legs, which increase from the upper troposphere (HL1 and HL2) and TIL region (RF07-FL2 and RF08-FL1) toward the uppermost HALO levels of legs HL3 and HL4 in the lower stratosphere. In addition to the 1-Hz data as used for the legs shown in Fig. 6, the spectra computed with the 10-Hz data for RF07-FL2 and RF08-FL1 show a similar shape (dotted curves), confirming the previous analysis.

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Binned energy spectra of the vertical velocity from research flights of DLR Falcon and HALO. (a) RF07-FL2, (b) RF08-FL1, (c) HL1, (d) HL2, (e) HL3, and (f) HL4. The red legs mark simultaneous measurements, the legs marked with dashed lines are taken at the same flight leg and altitude, and the solid gray lines mark the upper-level legs of HALO.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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Binned energy spectra of the vertical velocity measured by the DWL from DLR Falcon research flights (a) RF07-FL2 and (b) RF08-FL1 at selected vertical levels beneath the flight path. Their gray shading becomes darker with increasing altitude. The solid black lines in (a) and solid red lines in (b) are the w spectra from the DLR Falcon in situ measurements at 9.8-km altitude. The dashed red line in (b) is from the simultaneous HALO in situ flight at 7.8-km altitude along HL1.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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Below the DLR Falcon flight level, the vertical dependency of the spectral energy content between 7.9- and 9.1-km altitude was analyzed using the DWL observations of w at selected levels; see Fig. 7. The spectral distribution of the earlier flight RF07-FL2 is characterized by enhanced amplitudes in the range from λ_x ≈ 4 to 30 km (Fig. 7a). Individual spectral peaks are located at λ_x ≈ 5 and 10 km. The maximum values of the DWL observations are up to a factor 4 smaller than the values calculated from the in situ measurements at flight level (black line in Fig. 7a). This drop in amplitude and in σ_w from the DLR Falcon flight level down to an altitude of 8–9 km could indicate evanescent gravity waves trapped in the TIL.

The spectral distribution for the second flight RF08-FL1 is slightly different: the amplitude has increased and the energy is shifted to longer λ_x in accordance with the visual impression of Fig. 5. At 9.1-km altitude, short modes with λ_x ≈ 8 km are clearly separated from somewhat longer modes at λ_x ≈ 12 km. At the lower vertical levels, only these longer gravity waves dominate and the distribution is unimodal, a finding that is confirmed by the nearly simultaneous in situ observations from HALO at 7.8-km altitude along HL1 (red dashed line in Fig. 7b). As in the previous DLR Falcon flight leg RF07-FL2, the spectral distribution in the TIL is dominated by gravity waves between λ_x ≈ 3 and 8 km.

Together with the visual impression of Fig. 5, the results presented in Figs. 6 and 7 provide a consistent picture: horizontally short modes with λ_x < 10 km are mainly present in a layer from ∼8- to 10-km altitude. Their λ_x decrease and their amplitudes increase upward. Very short modes with λ_x < 5 km occur mainly in the ridge and downstream segments. Above the TIL, the two HALO observations indicate the existence of horizontally longer modes with λ_x > 10 km. Below the TIL, it is difficult to obtain a coherent picture from the DWL plots alone. However, the few instances where the DWL could observe almost the entire troposphere suggest that the gravity waves have horizontal wavelengths ${\lambda }_x\gtrsim 20\hspace{0.17em}\text{km}$ associated with the flow across individual mountain peaks.

c. Wave energy fluxes

In general, the wave energy flux vector determines in which direction and at which rate energy is being transported by the wave motion (Dutton 1976, section 12.4.3). Furthermore, the Eliassen–Palm relation provides a functional relation between the vertical flux of wave energy EF_z and the negative scalar product of the vectors of the horizontal wind U and the momentum flux $MF=({\text{MF}}_x,{\text{MF}}_y)=(\overline{\rho }⟨u^{\prime }{w}^{\prime }⟩,\overline{\rho }⟨{\upsilon }^{\prime }{w}^{\prime }⟩)$ as

$${\text{EF}}_z\text{=}-MF\cdot \mathbf{U}.$$(7)

According to Eliassen and Palm (1961), EF_z and −MF ⋅ U should be directly proportional for linearly propagating, stationary, nondissipative mountain waves. Figure 8 shows the wave energy fluxes from both DLR Falcon legs for L = 20 km. The EF_z and −MF ⋅ U oscillate around zero and achieve local maxima and minima of over 5 W m⁻² for distances x > 250 km, i.e., over the mountains and downstream. For both legs, these independently determined fluxes are largely in phase and correlation coefficients of 0.745 and 0.875 were computed, respectively. However, while the Eliassen–Palm relation holds in the first 250 km, the amplitude and phase difference between EF_z and −MF ⋅ U increase over the mountains and downstream. This deviation indicates transient and nonlinear processes that violate the assumptions for which Eq. (7) was derived, which is consistent with the 3D results of Woods and Smith (2011). The alternating sequence of positive and negative vertical energy fluxes indicates that upward- and downward-propagating short waves occur at the same time. Averaged over the entire cross-mountain leg, the vertical energy fluxes for L = 20 km for both DLR Falcon legs are negative and amount to −1.15 and −1.53 W m⁻², respectively, which is another indication that downward-propagating short gravity waves can be detected; see Table 2.

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(a),(b) Vertical energy fluxes EF_z as well as −MF ⋅ U [red and black lines in (a) and (b)] and (c),(d) horizontal energy fluxes EF_x and EF_y [black and blue curves in (c) and (d)] along DLR Falcon research flights (left) RF07-FL2 and (right) RF08-FL1 calculated from the flight level in situ data using moving averages of L = 20 km.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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Table 2.

Averaged energy fluxes for the two legs of the DLR Falcon research flights RF07 and RF08 as well as the four legs HL1–HL4 of the HALO research flight on 28 Jan 2016. Letters (a)–(f) refer to the lines in Table 1 as well as to profiles in Figs. 4 and A1. The simultaneous flight legs are in bold. The two comma-separated numbers refer to average lengths of L = 20 and 300 km, respectively.

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The zonal energy flux EF_x is always negative indicating that the detected waves are propagating against the mean westerly flow at this altitude (cf. Fig. 3b). The meridional component EF_y is smaller and fluctuates around zero. In the nearly zonal flow, one would expect that a positive EF_z correlates with negative EF_x for stationary, upward-propagating mountain waves. This clear correspondence valid for linearly upward-propagating waves cannot be observed for short gravity waves as detected along the both DLR Falcon flight legs as shown in Fig. 8.

Analysis of the horizontally longer waves with filter length L = 300 km gives a small positive vertical energy flux EF_z of 0.28 W m⁻² for the first DLR Falcon leg, a likewise small but negative value of −0.14 W m⁻² for the second DLR Falcon leg; see Table 2. Here, the zonal wave energy fluxes EF_x are clearly negative, and their absolute values are larger than their short-wave counterparts. As before, the meridional flux values EF_y are smaller and the sign change between the two legs of the DLR Falcon might be related to the gradually turning wind at this level; see Fig. 2b. These results indicate that the longer mountain waves propagated upward, but that this wave propagation gradually weakened or ceased until the combined DLR Falcon and HALO flights.

The simultaneous HALO flight HL1 also revealed negative energy fluxes EF_z for both filter lengths; see Table 2. This quantitative evaluation of vertical energy fluxes suggests that upward- and downward-propagating waves coexist near the tropopause and also in the upper troposphere where HALO flew. Concordantly, both HALO legs HL3 and HL4 in the lower stratosphere yield downward-propagating short waves with EF_z ≤ −1 W m⁻². Surprisingly, the contribution of longer waves does not deliver a systematic picture of predominantly upward-propagating mountain waves, again suggesting temporal variability in the airborne observations; see Table 2. Altogether, it must be noted that the EF_z is small compared to strong mountain-wave events as found over New Zealand’s Southern Alps (Smith et al. 2016) or over the Southern Andes (Dörnbrack et al. 2022b).

d. Interim summary and discussion

Combining the airborne observations with the IFS analysis presented in the previous sections reveals that horizontally longer mountain waves propagate predominantly upward. Their energy and momentum fluxes are small. However, the most striking feature of the airborne observations is the occurrence of internal gravity waves shorter than 10 km near the tropopause. Not exclusively, but the shortest modes can be found clustered in the lee of the Scandinavian mountains. These short modes are absent both further down in the upper troposphere and further up in the lower stratosphere.

It could not be proved incontrovertibly that these waves are stationary mountain waves. Instead, the analysis to this point, especially, the observed phase differences of η and w along RF07-FL2 and RF08-FL1, suggests that these short waves are transient and unsteady. It should be noted that the same diagnostic tools were earlier applied to determine upward-propagating mountain waves uniquely, e.g., Smith et al. (2008, 2016), Bramberger et al. (2017), Dörnbrack et al. (2022b). The above interim conclusions are supported by the earlier studies of Gisinger et al. (2020), Krisch et al. (2020). There, the low values of the observed momentum fluxes are interpreted as a sign for evanescent and trapped waves in agreement with the findings regarding the energy fluxes of this study. However, the question where these short waves come from and how they were excited has not been answered yet. An attempt to answer this question will be made in the next sections.

Based on linear wave theory, however, can we understand these results? Atmospheric airflow over complex terrain excites a broad spectrum of mountain waves (Smith and Kruse 2017). First, we assume that the short waves in the TIL result directly from the airflow across the rugged Scandinavian terrain (see the profile of the orography in the lower panel of Fig. A1). For sake of simplicity, let us consider a single harmonic mode with horizontal wavenumber k = 2π/λ_x, where λ_x is either 5 or 7 km corresponding to the short waves found in the DLR Falcon measurements. To allow for vertically propagating modes, a uniform flow of u₀ = 20 m s⁻¹ would require buoyancy frequencies N larger than 0.025 or 0.018 s⁻¹ for these 5- or 7-km waves (Durran 1990, Eq. 4.23). If one increases u₀ by 10 m s⁻¹, the limiting buoyancy frequencies would also increase to 0.038 or 0.027 s⁻¹, respectively. These values are much larger in comparison with the N values of ∼0.01 s⁻¹ characteristic for the upstream conditions on this day and as shown in Fig. 2c. Therefore, it is highly unlikely that these short waves will propagate through the troposphere to the TIL, provided that wind and thermal stratifications are uniform.

But there are the evanescent modes. Let us again assume uniform zonal velocities u₀ of 20 or 30 m s⁻¹ and a constant tropospheric N = 0.01 s⁻¹. Then, evanescent modes of horizontal wavenumber k exist if u₀k > N and the corresponding wave amplitude decays proportional to an attenuation factor e^−μz with the vertical decay rate $\mu =\sqrt{k^2-N^2/u_0^2}$ (Durran 1990, Eq. 4.23). In the lower troposphere, the airflow shall excites evanescent modes at wavenumbers k₁ = 2 π/5 km and k₂ = 2 π/7 km. How much does the amplitude attenuate over a vertical distance of 10 km to the TIL? The amplitude of the 5-km wave has attenuated to less than 0.001% of its original value; for the longer wave, it is 0.06% for u₀ = 20 m s⁻¹ and 0.02% for u₀ = 30 m s⁻¹. This result suggests that short evanescent waves are unlikely to affect TIL significantly under the above assumptions.

However, the decrease of the Scorer parameter $\mathcal{l}$ as shown in Fig. 2d reflects the strong tropospheric shear during the time of the observations. As discussed by Schoeberl (1985), stationary mountain waves reach their turning level when the intrinsic frequency ω = −u₀k equals the buoyancy frequency N, or in other words when their vertical wavelength (wavenumber) increases (decreases) (e.g., Fritts and Alexander 2003, section 3.2.2). There, only waves with λ_x > λ_c may propagate freely further upward; all other shorter waves are being reflected and are trapped or ducted and might create a resonant leewave train. The short, high-frequency gravity waves have intrinsic frequencies near N. In contradiction to the case with constant wind, the wave amplitude only decays gradually in the duct resulting in large vertical velocities. As a further result, these waves have little or no associated vertical flux of horizontal momentum (Fritts and Alexander 2003; Smith and Kruse 2017).

Wurtele’s one-layer Couette-flow model for constant stability and constant shear (Queney et al. 1960, p. 64) can be easily applied to estimate the free propagating modes and the number of discrete resonant modes. The dashed line in Fig. 2a indicates a mean shear of 40 m s⁻¹ (30 km)⁻¹ = 10 m s⁻¹ (7.5 km)⁻¹. The tabulated numerical solutions of Wurtele’s linear wave equation for depths of 6, 8, 10, and 12 km over which the wind increases to 10 m s⁻¹ provide the minimum wavelengths of the free waves. For a shear of 10 m s⁻¹ (8 km)⁻¹, the shortest wavelength is 10 km and the number of resonant modes is 6. The numerical solutions also reveal that increasing shear makes λ_x larger and decreases the number of resonant modes. Therefore, the consideration of realistic shear values also supports the hypothesis that only longer horizontal modes can propagate to the TIL.

It is obvious that one could extend linear approaches to two-layer and three-layer models including the stratosphere and the effect of the TIL (e.g., Danielsen and Bleck 1970; Gossard and Hooke 1975). However, the resulting equations become complicated too and most of the time require numerical integration. Therefore, we now turn to the numerical integration of the full nonlinear equations [Eqs. (1)–(4)] as they allow a large number of tests to investigate the sensitivity of the solutions.

a. Experiment A: Impact of downward-propagating gravity waves

The first sensitivity experiment A is dedicated to the question “Are downward-propagating gravity waves essential for the generation of horizontally short modes in the TIL?” To this end, two 3D simulations with different vertical profiles of imposed internal stratospheric friction are contrasted. Table 3 summarizes the setup of the numerical simulations. The reference simulation, using D_abs = 9.6 km in Eq. (5), allows wave breaking in the middle atmosphere and contains downward-propagating modes. These modes are either reflected mountain waves or waves generated by secondary sources in the breaking region. The second simulation, using D_abs = 12.8 km, keeps the amplitudes of the upward-propagating mountain waves finite and prevents wave breaking. As a result, only upward-propagating modes exist.

Table 3.

Sensitivity experiment A: Impact of downward-propagating gravity waves. Here, Δx, Δy, and Δz are the grid increments in the three spatial directions, and Δt is the time step. The values of n, m, and l indicate the number of grid cells and nt indicates the number of time steps used in the integration. NPX and NPY are the number of processors for the horizontal domain decomposition in x and y directions used in the numerical integrations. The numbers epp0 and epp1 refer to the requested accuracy of the pressure solver at t = 0, and for the subsequent time steps, respectively, see Smolarkiewicz et al. (1997). The variable EXPER = 0 stands for the default numerical coding of the pressure solver. The variable D_abs denotes the vertical damping scale in Eq. (5). For all numerical simulations, upstream and ambient profiles are taken from T21 L137 as average from 1200 to 1800 UTC and taken at 61°N and 1°E (see Fig. 2).

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Figure 9 compares the results of the two numerical simulations. In the top row, the reference simulation shows the horizontal and vertical wind fields at the center line y = 0 after 12-h simulation time. The atmospheric flow is dominated by upward-propagating mountain waves whose phase fronts are tilted into the wind. In addition, localized wave-breaking regions (i.e., overturning isentropes and strongly reduced horizontal wind) are visible above the TIL and in the stratosphere. In contrast, the simulation with enhanced stratospheric damping exclusively reveals upward-propagating mountain waves with no wave-breaking regions as shown in the lower row of Fig. 9. This simulation with the imposed finite-amplitude response does not yield short waves near the TIL.

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(left) Horizontal and (right) vertical wind along the centerline y = 0 of the 3D computational domain from (a),(b) the reference run and (c),(d) the run without wave breaking after 12-h simulation time. The black lines denote the potential temperature in logarithmic scaling, and the shaded area at the bottom of each panel is the orography. For the settings of the numerical parameters, refer to Table 3.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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Figure 10, showing the vertical wind from the reference simulation at 22 h, focuses on the shallow altitude region of interest, encompassing the troposphere and lower stratosphere up to 20-km altitude. With the adjusted color bar, the appearance of horizontally short waves over and downstream of the mountains becomes evident. Localized areas of wave breaking can be detected in the lower stratosphere. The short waves are trapped below the stratospheric wave-breaking region in a wave duct along the TIL.

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Vertical wind along the centerline y = 0 of the 3D computational domain from the reference run after 22-h simulation time. Thin black lines denote the potential temperature in logarithmic scaling, and the shaded area at the bottom is the orography. The horizontal red line denotes the altitude at which the Hovmöller plots are shown in Fig. 12. The vertical red lines denote the positions of the plots shown in Fig. 11. For the settings of the numerical parameters, refer to Table 3.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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The gradual formation of the stratospheric wave duct above and downstream of the mountains is demonstrated in Fig. 11 by showing the squared vertical wavenumber:

$$m^2\approx \frac{N^2}{u^2}-\frac{1}{u}\frac{d^{2}u}{d{z}^2}-k^2(k\hspace{0.17em}\text{=}\hspace{0.17em}2\pi /{\lambda }_x),$$(8)

for stationary mountain waves (e.g., Achatz 2022, Eq. 7.76) with a horizontal wavelength λ_x = 12 km. The small negative m² values indicate that those waves are evanescent in the troposphere. A similar plot for λ_x = 15 km (displace all curves in Fig. 11 to the right by 0.15 km⁻², which leads to positive but small m² values, i.e., long vertical wavelengths) explains the almost vertical phase lines of the simulated primary mountain waves with ${\lambda }_x\gtrsim 20\hspace{0.17em}\text{km}$ in the troposphere (Fig. 10). These numerical simulation results are consistent with the airborne DWL measurements over Scandinavia: Wagner et al. (2017) and Witschas et al. (2017) have demonstrated the existence of these trapped mountain waves with near-vertical phase lines throughout the troposphere caused by the flow over the underlying mountains.

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Vertical m² profiles for λ_x = 12 km according to Eq. (8) at x positions along the centerline y = 0 of the 3D computational domain from the reference run; see Fig. 10. Solid black line: t = 0, and dashed black line: t = 24 h. The gray shaded area covers the temporal variability of the 1-min m² profiles during the simulation.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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In and above the TIL, the upstream m² is positive and mountain waves with ${\lambda }_x\gtrsim 12\hspace{0.17em}\text{km}$ propagate freely (Fig. 11a). Above the mountains, the stratospheric wave breaking increases the variability of m² near and above the TIL (Fig. 11b). Furthermore, the decreasing static stability N² in the wave-breaking region at z > 10 km leads to m² falling to negative values in the course of the simulation (Fig. 11c). Simultaneously, N² below 10 km is increased leading to larger positive m² values. This vertically limited layer with m² > 0 gradually forms the thermal wave duct in which the short gravity waves are trapped and propagate downstream. As mentioned above, Fig. 11 shows the profiles for λ_x = 12 km, and a similar plot for λ_x = 15 km only reveals a few instants when m² < 0. This means only short waves with ${\lambda }_x\lesssim 12\hspace{0.17em}\text{km}$ can be guided along the wave duct.

Figure 10 is a snapshot taken from an animation of the entire numerical simulation over 24 h, which is provided by Dörnbrack (2024). When viewing this animation, both the formation of the stratospheric wave duct and the quasi-periodic vertical displacements of the tropopause due to the temporal modulation of strength and depth of the primary mountain waves, which lead to the temporal variability of m², become apparent (Fig. 11b). Wave packets with short horizontal wavelengths are emitted from this location and propagate downstream along the stratospheric wave duct.

Figure 12 displays the vertical velocity at 10-km altitude in the spirit of traditional Hovmöller plots. The comparison between the upper and lower panel of Fig. 12 reveals the distinct difference between the numerical simulations allowing/preventing stratospheric wave breaking: the finite-amplitude simulation produces steady mountain waves whose phase lines are stationary and which can be linked with the mountain ranges underneath (cf. Fig. 10). The flow response is completely different in the reference simulation, where the waves break in the stratosphere: temporal oscillations of the amplitude of the vertical wind appear in the phase lines associated with the main mountain ranges; see x < 100 km or near x = 0. In addition, gravity waves with λ_x < 10 km appear mostly in the lee of the mountains for x > 80 km. The phase lines of these short modes propagate upstream, i.e., their ground-based phase speed is negative. However, groups of these short waves propagate downstream with a phase velocity of ∼22 m s⁻¹ as estimated from the dashed line in Fig. 12.

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Distance–time sections of the vertical wind component at 10-km altitude from (top) the reference run and (bottom) the run without wave breaking. For the numerical settings, see Table 3.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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Figure 13 shows the vertical flux of wave energy EF_z at simulation times 12, 18, and 24 h. While the run without simulated wave breaking shows only positive, i.e., upward wave energy fluxes, the reference run shows regions (especially in the lower and middle stratosphere) where the wave energy propagates downward. At time t = 18 h, it becomes apparent how EF_z oscillates between positive and negative values in the wave duct. This oscillation in the sign of EF_z is the signature of downstream-traveling wave packets. It must be noted that the regions with EF_z < 0 extend up to ∼20-km altitude, a fact that could explain the observed negative vertical energy fluxes at higher flight levels; see discussion in section 4c. In contrast, the finite amplitude simulation has already reached a quasi-steady state after 4 h, persisting until the end of the numerical integration at t = 24 h as shown in the right column of Fig. 13. Obviously, the EF_z magnitudes of the finite-amplitude simulation are smaller due to the imposed damping according to Eq. (5). It should be noted that with longer filter lengths L = 50 km, only a few, very sporadic downward-propagating waves occur in the reference run (not shown).

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Vertical energy fluxes EF_z after 12-, 18-, and 24-h simulation time for (left) the reference run and (right) the run without wave breaking. The EF_z is calculated with a 20-km smoothing window.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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Figure 14 quantifies which spatial scales are involved in the different processes by presenting the binned power spectra of the vertical velocity and the vertical flux of the specific horizontal momentum u′w′ at z = 5, 10, and 20 km, respectively. The temporally averaged vertical velocity energy spectra reveal power on horizontal scales from ∼7 to 100 km (top row in Fig. 14). Within the wave duct at 10-km altitude, the singular peak at λ_x ≈ 10 km occurs exclusively in the reference run. The run without wave breaking reveals a strongly attenuated response at λ_x ≤ 30 km, i.e., all short modes are absent here. It is important to note that the λ_x ≈ 10 km peak is neither present in the troposphere nor in the lower stratosphere. There, the amplitudes of the vertical wind become smaller and the energy shifts to longer horizontal modes (Fig. 14, top row).

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(top) Binned power spectra of the vertical velocity and (bottom) the vertical flux of specific horizontal momentum u′w′ at 5-, 10-, and 20-km altitude from the reference run (solid lines) and from the run without wave breaking (dashed lines) as temporal average from 12- to 24-h simulation time. For the numerical settings, see Table 3.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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Gravity waves with λ_x < 10 km along the wave duct do not transport horizontal momentum vertically upward, i.e., their momentum flux amplitudes are small, lower middle panel in Fig. 14. Positive and negative values at λ_x ≈ 10 km are associated with the upward- and downward-propagating modes as discussed before. Their time-averaged values are small compared to the specific momentum fluxes of the momentum-carrying waves at longer horizontal wavelengths λ_x > 20 km. These are the upward-propagating mountain waves; see the bottom line of Fig. 14.

Figure 15 presents the binned power spectra at an altitude of 30 km to illustrate the excitation of short secondary modes due to wave breaking. The results of the selected wave-breaking periods (thin solid lines) are contrasted with the 12-h temporal averages (thick solid lines). Short modes with ${\lambda }_x\lesssim 10\hspace{0.17em}\text{km}$ are visible for the vertical velocities (Fig. 15a). The coexistence of both positive and negative values of the specific momentum flux and the vertical wave energy flux indicates the power of these secondary gravity waves that propagate upward and downward (Figs. 15b and 15c). It is evident that the power of these short secondary waves is considerably lower than that of the longer, momentum-carrying waves in agreement with the 3D results of Woods and Smith (2011). It is therefore unlikely that these short modes will reach the wave duct at 10 km given the downward increase in density. The negative horizontal wave energy flux for λ_x > 20 km indicates the dominance of primary mountain waves that propagate upward and against the mean flow (Fig. 15d).

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(a) Binned power spectra of the vertical velocity, (b) the vertical flux of specific horizontal momentum, and (c) the vertical and the zonal fluxes of wave energy, (d) at 30-km altitude from the reference run (thick solid lines) and from the run without wave breaking (thick dashed lines) as temporal average from 12- to 24-h simulation time. The thin black lines are the spectra averaged over 2-h periods beginning at 12.5- and 15.5-h simulation time, respectively. For the numerical settings, see Table 3.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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The conclusion from this physical sensitivity experiment A is as follows: Wave breaking modifies the stratospheric flow and the propagation conditions for the permanently excited primary mountains waves. In addition, the local body forces exerted by the breaking waves are a secondary source of gravity waves. However, it is the quasi-periodic modulation of the flow near the TIL and in the lower stratosphere that excites the short gravity waves. In the course of the numerical simulations, a thermal wave duct is formed, thereby strengthening the static stability of the TIL and favoring the occurrence of short trapped modes. These waves transport no horizontal momentum upward, they are transient, and they propagate in wave packets downstream.

b. Experiment B: Effects of orographic ruggedness

Although the results of the previous sections suggest that the short trapped waves are due to the interaction of the modified stratospheric airflow with the TIL and the downstream formation of a thermal wave duct, the following sensitivity experiment B tests whether the rugged terrain could still be a source for the observed short waves. The orography z_ter at 61°N as shown by the black solid line in Fig. 16 depicts a compact landmass extending over nearly 650 km in west–east direction. There are deep notches and gaps in z_ter near the west coast of Norway. These steep orographic features result from the elongated and meandering Sognefjord. The cliffs around the Sognefjord rise almost vertically out of the water and reach heights of 1 km and more. The other curves in Fig. 16 present smoothed profiles filtered with boxcar average lengths of 10, 15, and 40 km, respectively. The left panel of Fig. 17 presents the binned power spectra of the four different orographic profiles. Indeed, the high-resolution orography without smoothing contains short modes between 5- and 20-km zonal wavelength. The smoothed curves increasingly lose spectral power in the small and longer modes.

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Profiles of the digital orographies used for the sensitivity experiment B. The black line marks the orography along 61°N interpolated onto a 250-m grid and smoothed over three grid points as used for the reference run. The numbers on the colored lines denote the boxcar average lengths used to smooth the orography.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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(left) Binned power spectra of the digital orographies used for the sensitivity experiment B. (right) Binned power spectra of the vertical velocity at z = 10 km altitude. The reference run, here, refers to the same numerical settings as presented before but with the increased spatial resolution of Δx = Δy = 250 m. For the numerical settings, see Table 4. Same line coding as in Fig. 16.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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The numerical simulations are summarized in Table 4. They are identical to the reference simulation of the previous subsection except that the numerical resolution has been increased by a factor of 2, i.e., Δx = Δy = 250 m, and the time step is Δt = 1 s. The right panel of Fig. 17 shows the binned power spectra of the simulated vertical velocity at z = 10 km altitude averaged from 12- to 24-h simulation time. A comparison of the amplitude of the power spectra here with that of the reference simulation in Fig. 14 shows that the increase in spectral power by around 50% is due to the increased numerical resolution.

Table 4.

Sensitivity experiment B: effects of orographic rudgeness. The names in the first column refer to the line legends in Fig. 17. Refer to Table 3 for explanation of the used parameter names. For all numerical simulations, upstream and ambient profiles are taken from T21 L137 as average from 1200 to 1800 UTC and taken at 61°N and 1°E (see Fig. 2).

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However, the preservation of the spectral power for the smoothed terrain (10-km filter width; blue lines in Fig. 17) is the most important result of this sensitivity experiment B. If longer orographic modes are further attenuated by applying greater boxcar average lengths, the momentum of the upward-propagating modes becomes too low to excite stratospheric wave breaking and the short waves along the wave duct disappear. Consequently, it is the primary upward-propagating mountain waves with ${\lambda }_x\gtrsim 20\hspace{0.17em}\text{km}$ that modify the stratospheric airflow leading to the trapped modes as described in section 5a.

c. Experiment C: Impact of the TIL strength

An examination of Figs. 10 and 11c already reveals that the static stability along the TIL in the downstream segment is strongly increased due to the wave breaking in the lower stratosphere. For sensitivity experiment C, the vertical profile of the buoyancy frequency N(z) as computed from the upstream profiles (see Fig. 2c) is modified by adding a function $f(z)=\mathrm{\Delta }N\exp [-{(z-z_{\text{TP}})}^2/{\sigma }_z^2]$ whereby z_TP = 9 km and σ_z = 1.5 km. The amplitude ΔN is either 0.010 s⁻¹ (red curve in Fig. 18) or 0.015 s⁻¹ (blue curve in Fig. 18). From the modified N(z) profiles, the ambient θ_e profiles are computed numerically. Other parameters of the numerical setting are provided in Table 5.

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Vertical profiles of the buoyancy frequency N as used for the sensitivity experiment C with varying values of the strength of the TIL. The N profiles are used to compute the background potential temperature profiles Θ_e(z).

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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Table 5.

Sensitivity experiments C: Impact of the TIL strength. The function $f(z)=\exp \{-{(z-z_{\text{TP}})}^2/{\sigma }_z^2\}$ describes the modification of the TIL whereby z_TP = 9 km and σ_z = 1.5 km. The vertical profiles of the buoyancy frequencies N(z) are shown in Fig. 18. For all numerical simulations, upstream and ambient profiles are taken from T21 L137 as average from 1200 to 1800 UTC and taken at 61°N and 1°E (see Fig. 2).

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Figure 19 demonstrates the impact of increasing the upstream TIL strength on the short-wave response. The common feature of the sensitivity experiment C is that a stronger TIL (higher peak N value) leads to shorter gravity waves within the TIL. Figure 19 also demonstrates another interesting finding in comparison to the measured amplitudes of the trapped waves during the DLR Falcon flights along legs RF07-FL2 and RF08-FL1: The amplitude of the numerically produced waves now compares quantitatively very well with the observations. Furthermore, the wave response at 12-km altitude is still large and agrees well with the HALO observations along track HL3 (right panel in Fig. 19).

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Sensitivity of the binned vertical variance spectra within the wave duct as function of TIL strength. The simulation results are shown at 10- and 12-km altitude as temporal averages from 9- to 15-h simulation time. For the numerical settings, see Table 5. The airborne observations from flight legs RF07-FL2, RF08-FL01, and HL3 are superimposed (cf. Fig. 6).

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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d. Experiment D: Impact of numerical settings

Eight further numerical simulations form the final sensitivity experiment D that investigates the dependence of the solutions on selected numerical settings. The parameters of the different runs are summarized in Table 6. Figure 20 groups the results of this sensitivity experiment in three panels, which in turn show the binned power spectra of the simulated vertical velocity at an altitude of 10 km. The black solid line, which can be seen in all panels, is the reference run attaining a spectral amplitude of 7 m² s⁻² at λ_x ≈ 10 km.

Table 6.

Sensitivity experiment D: Impact of numerical settings. Parameters that are varied are in bold, and all other parameters remain the same as in the reference run. The names in the first column refer to the line legends in Fig. 20. For all numerical simulations, upstream and ambient profiles are taken from T21 L137 as average from 1200 to 1800 UTC and taken at 61°N and 1°E; see Fig. 2. Refer to Table 3 for explanation of the used parameter names.

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Sensitivity of the binned vertical variance spectra within the wave duct as a function of numerical settings; for details, see Table 6. Results are shown at 10-km altitude as temporal averages from 12- to 24-h simulation time. In all panels, the black lines (which are normally overlaid by the other lines) refer to the reference run.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0037.1

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Initially, the numerical integrations were modified to a potential flow initialization (red curve in Fig. 20a) and to an integration of the anelastic equations (Smolarkiewicz and Dörnbrack 2008) (blue curve in Fig. 20a). Both model options increase the spectral amplitude by either 8% or 29%. Thus, the chosen numerical initialization has almost no effect of the amplitude of the trapped waves. The small difference between the integration of the fully compressible equations and the sound-proofed equation confirms former numerical (Smolarkiewicz and Dörnbrack 2008) and theoretical findings (Klein et al. 2010). It must be noted that the run time is only 42% for the anelastic integration compared to the integration of the compressible equations.

The second group of numerical simulations concerns the accuracy of the iterative pressure solver. The variables epp0 and epp1 in Fig. 20b and Table 6 refer to the dimensionless norm of the divergence residual and denote how accurately the flow field is made divergence free (Smolarkiewicz et al. 1997, Eq. 4). Two different stopping criteria are applied for the first time step (epp0) and for all other ones (epp1). The accuracy of the pressure solver has a marginal effect on the spectral amplitude (increase by ∼12%). Likewise, another option to optimize the performance of pressure solver (EXPER = 3) reduces the spectral amplitude just by 7% (Figure 20b). Indeed, with this optimization, the runtime is only 82% compared to the reference run.

Figure 20(c) presents the results for simulations using a different domain decomposition of EULAG, i.e., different number of processors in the horizontal directions. In addition, a compiler statement for vectorization was activated, which was not used in all other runs. The spectral amplitudes vary by only ±15%. For the run with 288 × 16 processors, the runtime is reduced to 1/3; for the run with 96 × 4 processors, the runtime increases by a factor of 2.75.

All numerical changes in the settings have only a marginal impact on the spectral amplitude of the vertical wind component at an altitude of 10 km. The presented numbers must be related to the almost complete extinction of the short waves (amplitude reduction to 2%) using the stratospheric damping as presented in Fig. 14 of section 5a.