# On the Interpretation of Gravity Wave Measurements by Ground-Based Lidars

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Gravity Wave Signatures in Ground-Based Lidar Data

## 3. Basic Considerations

#### 3.1. Plane Boussinesq Waves

#### 3.2. Wave Packets

#### 3.3. Doppler Shift

#### 3.4. Illustrations

#### 3.4.1. Plane Waves in an Atmosphere with Zero Wind

#### 3.4.2. Wave Packets in an Atmosphere with Zero Wind

- (a)
- the observed phase lines belong to a spatially- and temporally-localized wave packet,
- (b)
- the observed phase lines allow the graphical determination of the vertical wavelength ${\lambda}_{z}$ and of the ground-based frequency ω,
- (c)
- the observed waves obey a dispersion relation for internal Boussinesq wave like Equation (3),
- (d)
- the background stratification N is nearly constant over an altitude range of at least one vertical wavelength ${\lambda}_{z}$ and can be computed from the observation or meteorological data and
- (e)
- there is no mean wind in the atmosphere.

#### 3.4.3. Doppler-Shifted Wave Packets

## 4. Idealized Numerical Simulations

#### 4.1. Archetypal Regimes of Vertically-Propagating Internal Gravity Waves

- (i)
- non-hydrostatic wave regime, the
- (ii)
- hydrostatic “nonrotating” wave regime and the
- (iii)
- hydrostatic “rotating” wave regime.

#### 4.2. Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Setup of the Numerical Simulations

## References

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^{1.}from B. Dylan: “One of us must know (sooner or later)”; Blonde On Blonde (1966).^{2.}H. Grogger, Simulation of Deep Gravity Wave Propagation Using EULAG, Master Thesis, University of Innsbruck, 2017.

**Figure 1.**Gravity wave-induced temperature perturbations observed by Rayleigh lidar on 15/16 December 2015 above Sodankylä, Finland. The temperature measurements are filtered with a 3-h running mean to highlight signatures of gravity waves with periods in the range of approximately 1 to 5 h. The temporal and vertical resolutions are 30 min and 1 km, respectively.

**Figure 2.**Horizontal (

**a**) and vertical (

**b**) components of the phase velocities ${c}_{px}$ and ${c}_{pz}$ (red lines) and group velocities ${c}_{gx}$ and ${c}_{gz}$ (blue lines). (

**c**) Horizontal (red lines) and vertical (blue lines) phase velocities ${c}_{Px}$ and ${c}_{Pz}$. All curves are drawn for fixed values of m > 0 (upper row) and m < 0 (lower row). Vertical wavelengths ${\lambda}_{z}$ = π, 2π, 3π, 4π, 5π and 6π km are represented by lines from dotted to solid, respectively. The wavenumber vectors ${\overrightarrow{\mathbf{k}}}_{\mathrm{A}}$, ${\overrightarrow{\mathbf{k}}}_{\mathrm{B}}$, ${\overrightarrow{\mathbf{k}}}_{\mathrm{C}}$ and ${\overrightarrow{\mathbf{k}}}_{\mathrm{D}}$ denote the four possible orientations of the wave number vector in the respective area of the phase angle φ, as described in Section 3.1.

**Figure 3.**Spatial snapshots of the vertical displacements $\xi (x,z,t)$ of horizontally- and vertically-propagating plane waves for the four wavenumber vectors ${\overrightarrow{\mathbf{k}}}_{\mathrm{A}}$ (

**a**), ${\overrightarrow{\mathbf{k}}}_{\mathrm{B}}$ (

**b**), ${\overrightarrow{\mathbf{k}}}_{\mathrm{C}}$ (

**c**) and ${\overrightarrow{\mathbf{k}}}_{\mathrm{D}}$ (

**d**) at t = 24 min, respectively. The values of the horizontal and vertical wavenumber components are k = $\pm 2\phantom{\rule{0.166667em}{0ex}}\pi /{\lambda}_{x}$ with ${\lambda}_{x}$ = 8 km and m = $\pm 2\phantom{\rule{0.166667em}{0ex}}\pi /{\lambda}_{z}$ with ${\lambda}_{z}$ = 4 km, respectively. Red and blue contour lines refer to ± 0.95 times the wave amplitude and illustrate phase lines. The dashed black lines refer to the horizontal position where the vertical time series shown in Figure 4 are recorded.

**Figure 4.**Vertical time series of the vertical displacements $\xi (x,z,t)$ of horizontally- and vertically-propagating plane waves for the four wavenumber vectors ${\overrightarrow{\mathbf{k}}}_{\mathrm{A}}$ (

**a**), ${\overrightarrow{\mathbf{k}}}_{\mathrm{B}}$ (

**b**), ${\overrightarrow{\mathbf{k}}}_{\mathrm{C}}$ (

**c**) and ${\overrightarrow{\mathbf{k}}}_{\mathrm{D}}$ (

**d**) recorded at the positions marked in Figure 3. The values of the horizontal and vertical wavenumber components are k = $\pm 2\phantom{\rule{0.166667em}{0ex}}\pi /{\lambda}_{x}$ with ${\lambda}_{x}$ = 8 km and m = $\pm 2\phantom{\rule{0.166667em}{0ex}}\pi /{\lambda}_{z}$ with ${\lambda}_{z}$ = 4 km, respectively. Red and blue contour lines refer to ±0.95 times the wave amplitude and illustrate phase lines. The vertical black lines refer to the time when the plots shown in Figure 3 are drawn.

**Figure 5.**Spatial snapshots of the vertical displacements $\xi (x,z,t)$ of horizontally- and vertically-propagating wave packets with four different wavenumber vectors ${\overrightarrow{\mathbf{k}}}_{\mathrm{A}}$ (

**a**), ${\overrightarrow{\mathbf{k}}}_{\mathrm{B}}$ (

**b**), ${\overrightarrow{\mathbf{k}}}_{\mathrm{C}}$ (

**c**) and ${\overrightarrow{\mathbf{k}}}_{\mathrm{D}}$ (

**d**) at t = 24 min, respectively. The wave packets are propagating downward (${c}_{gz}$ < 0) in the top row and upward (${c}_{gz}$ > 0) in the bottom row. Red and blue contour lines refer to ±0.95 times the wave amplitude and illustrate phase lines. The dashed black lines refer to the horizontal position where the time series shown in Figure 6 are recorded.

**Figure 6.**Vertical time series of the vertical displacements $\xi (x,z,t)$ of horizontally- and vertically-propagating wave packets with four different wavenumber vectors ${\overrightarrow{\mathbf{k}}}_{\mathrm{A}}$ (

**a**), ${\overrightarrow{\mathbf{k}}}_{\mathrm{B}}$ (

**b**), ${\overrightarrow{\mathbf{k}}}_{\mathrm{C}}$ (

**c**) and ${\overrightarrow{\mathbf{k}}}_{\mathrm{D}}$ (

**d**) recorded at the positions marked in Figure 5, respectively. The wave packets are propagating downward (${c}_{gz}$ < 0) in the top row and upward (${c}_{gz}$ > 0) in the bottom row. Red and blue contour lines refer to ±0.95 times the wave amplitude and illustrate phase lines. The vertical black lines refer to the time when the plots shown in Figure 5 are drawn.

**Figure 7.**Spatial snapshots of the vertical displacements $\xi (x,z,t)$ of horizontally- and vertically-propagating wave packets with wavenumber vectors ${\overrightarrow{\mathbf{k}}}_{\mathrm{D}}$ at t = 60 min. The wave frequencies are Doppler shifted by U = $+{c}_{Px}$ (

**a**), U = 0 (

**b**), U = $-{c}_{Px}$ (

**c**) and U = $-2\phantom{\rule{0.166667em}{0ex}}{c}_{Px}$ (

**d**), respectively. Red and blue contour lines refer to ± 0.95 times the wave amplitude and illustrate phase lines. The dashed vertical black lines refer to the horizontal positions x = −12 km, 0, 12 km where the vertical time series shown in Figure 8 are recorded. The blue, red, and black crosses refer to the vertical positions z = 24.6 km, 29.6 km and 34.6 km where the time series of Figure 9 are recorded.

**Figure 8.**Vertical time series of the vertical displacements $\xi (x,z,t)$ of horizontally- and vertically-propagating wave packets with the wavenumber vector ${\overrightarrow{\mathbf{k}}}_{\mathrm{D}}$ recorded at the horizontal positions marked in Figure 7. The wave frequencies are Doppler shifted by U $=\phantom{\rule{0.166667em}{0ex}}+{c}_{Px}$ (

**a**), U $=\phantom{\rule{0.166667em}{0ex}}$0 (

**b**), U =$-{c}_{Px}$ (

**c**) and U =$-2{c}_{Px}$ (

**d**), respectively. Red and blue contour lines refer to ±0.95 times the wave amplitude and illustrate phase lines. The vertical black lines refer to the time t = 60 min where the altitude-distance plots are shown in Figure 7.

**Figure 9.**Time series of the vertical displacements $\xi (x,z,t)$ of horizontally- and vertically-propagating wave packets with the wavenumber vector ${\overrightarrow{\mathbf{k}}}_{\mathrm{D}}$ recorded at the three positions as marked by colored crosses in Figure 7. The wave frequencies are Doppler shifted by U = $+{c}_{Px}$ (

**a**), U = 0 (

**b**), U = $-{c}_{Px}$ (

**c**) and U = $-2\phantom{\rule{0.166667em}{0ex}}{c}_{Px}$ (

**d**), respectively. The vertical black lines refer to the time t = 60 min where the altitude-distance plots are shown in Figure 7.

**Figure 10.**Spatial snapshots

**left**(

**a**,

**c**,

**e**) and vertical times series

**right**(

**b**,

**d**,

**f**) of the potential temperature perturbations ${\mathrm{\Theta}}^{\prime}$ for the three different wave regimes. (a,b) non-hydrostatic wave regime; (c,d) hydrostatic nonrotating wave regime; (e,f) hydrostatic rotating wave regime. The right panels in (b,d,f) depict time-averaged ${\mathrm{\Theta}}^{\prime}$-profiles computed over the period from the time indicated by the vertical line until the end time of the respective panels. The spatial snapshots are taken at t = 125 min (a), t = 10 h (c) and t = 5 d (e). The vertical time series are recorded at x = 10 km (b), x = 0 (d) and x = 500 km (f), as indicated by the vertical dashed lines in (a,c,e). The amplitude of the surface topography is exaggerated by a factor of 10 in (a,c,e).

**Figure 11.**Vertical time series of the potential temperature perturbations ${\mathrm{\Theta}}^{\prime}$ from wave regime (iii) directly above the mountain (

**a**) at x = 0 and (

**b**) at x = 500 km for two different periods. The right panels depict the time-averaged $|{\mathrm{\Theta}}^{\prime}{|}^{2}$-profiles over the respective periods.

**Figure 12.**Vertical time series of the potential temperature perturbations ${\mathrm{\Theta}}^{\prime}$ from wave regime (i) at x = 30 km for (

**a**) ${\tau}_{z}$ = 900 s, (

**b**) ${\tau}_{z}$ = 600 s, (

**c**) ${\tau}_{z}$ = 270 s and (

**d**) ${\tau}_{z}$ = 30 s. The right panels depict the time-averaged ${\mathrm{\Theta}}^{\prime}$-profiles over the respective periods.

**Table 1.**The characteristics of plane Boussinesq gravity waves for the four wavenumber vectors ${\overrightarrow{\mathbf{k}}}_{\mathrm{A}}$ (A), ${\overrightarrow{\mathbf{k}}}_{\mathrm{B}}$ (B), ${\overrightarrow{\mathbf{k}}}_{\mathrm{C}}$ (C) and ${\overrightarrow{\mathbf{k}}}_{\mathrm{D}}$ (D) with horizontal and vertical wavenumber components are k = $\pm 2\phantom{\rule{0.166667em}{0ex}}\pi /{\lambda}_{x}$ with ${\lambda}_{x}$ = 8 km and m = $\pm 2\phantom{\rule{0.166667em}{0ex}}\pi /{\lambda}_{z}$ with ${\lambda}_{z}$ = 4 km, respectively. The background stratification is N = 0.01 s${}^{-1}$, and the quantities in the table are computed using Equation (4) for φ, Equation (3) for $\widehat{\omega}$, Equation (9) for ${c}_{Px}$ and ${c}_{Pz}$, Equation (6) for ${c}_{px}$ and ${c}_{pz}$ and Equations (12) and (13) for ${c}_{gx}$ and ${c}_{gz}$, respectively. The units are m s${}^{-1}$ for the phase and group velocities and s${}^{-1}$ for the wave frequency, respectively.

Case | k | m | φ | $\widehat{\mathit{\omega}}$ | ${\mathit{c}}_{\mathit{Px}}$ | ${\mathit{c}}_{\mathit{Pz}}$ | ${\mathit{c}}_{\mathit{px}}$ | ${\mathit{c}}_{\mathit{pz}}$ | ${\mathit{c}}_{\mathit{gx}}$ | ${\mathit{c}}_{\mathit{gz}}$ |
---|---|---|---|---|---|---|---|---|---|---|

A | − | + | −63.4 | 0.0045 | −5.69 | +2.84 | −1.14 | +2.28 | −4.56 | −2.28 |

B | + | + | +63.4 | 0.0045 | +5.69 | +2.84 | +1.14 | +2.28 | +4.56 | −2.28 |

C | − | − | +63.4 | 0.0045 | −5.69 | −2.84 | −1.14 | −2.28 | −4.56 | +2.28 |

D | + | − | −63.4 | 0.0045 | +5.69 | −2.84 | +1.14 | −2.28 | +4.56 | +2.28 |

**Table 2.**Parameters for the numerical simulations: mountain width L, spatial increments $\Delta x$ and $\Delta z$ in the horizontal and vertical directions, time step $\Delta t$, thickness $\delta {x}_{ab}$ and time scale ${\tau}_{x}$ of the horizontal and altitude ${z}_{ab}$ and time scale ${\tau}_{z}$ of the vertical absorbers.

Run | L/km | $\mathbf{\Delta}\mathit{x}$/m | $\mathbf{\Delta}\mathit{z}$/m | $\mathbf{\Delta}\mathit{t}$/s | $\mathit{\delta}{\mathit{x}}_{\mathit{ab}}$/km | ${\mathit{\tau}}_{\mathit{x}}$/s | ${\mathit{z}}_{\mathit{ab}}$/km | ${\mathit{\tau}}_{\mathit{z}}$/s |
---|---|---|---|---|---|---|---|---|

(i) | 1 | 100 | 100 | 5 | 24 | 1800 | 38 | 900 |

(ii) | 10 | 1000 | 100 | 5 | 240 | 300 | 38 | 900 |

(iii) | 100 | 5000 | 100 | 60 | 1200 | 300 | 38 | 3600 |

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**MDPI and ACS Style**

Dörnbrack, A.; Gisinger, S.; Kaifler, B.
On the Interpretation of Gravity Wave Measurements by Ground-Based Lidars. *Atmosphere* **2017**, *8*, 49.
https://doi.org/10.3390/atmos8030049

**AMA Style**

Dörnbrack A, Gisinger S, Kaifler B.
On the Interpretation of Gravity Wave Measurements by Ground-Based Lidars. *Atmosphere*. 2017; 8(3):49.
https://doi.org/10.3390/atmos8030049

**Chicago/Turabian Style**

Dörnbrack, Andreas, Sonja Gisinger, and Bernd Kaifler.
2017. "On the Interpretation of Gravity Wave Measurements by Ground-Based Lidars" *Atmosphere* 8, no. 3: 49.
https://doi.org/10.3390/atmos8030049