# Modulation of Atmospheric Nonisothermality and Wind Shears on the Propagation of Seismic Tsunami-Excited Gravity Waves

## Abstract

**:**

## 1. Introduction

## 2. Modeling

#### 2.1. Mean-Field Properties

**Figure 1.**Altitude profiles of mean-field properties. (

**a**) Mass density ${\rho}_{0}$ (pink) and pressure ${p}_{0}$ (blue); (

**b**) density gradient $\mathrm{d}{\rho}_{0}/\mathrm{d}z$ (pink) and pressure gradient $\mathrm{d}{p}_{0}/\mathrm{d}z$ (blue); (

**c**) density scale height ${H}_{\rho}$ (blue) and pressure scale height ${H}_{p}$ (pink); (

**d**) density scale number ${k}_{\rho}$ (pink), pressure scale number ${k}_{p}$ (black) and temperature scale number ${k}_{T}$ (blue); (

**e**) temperature ${T}_{0}$ (blue) and sound speed C (pink); (

**f**) temperature gradient $\mathrm{d}{T}_{0}/\mathrm{d}z$; (

**g**) zonal (eastward) wind U (blue) and meridional (northward) wind V (pink); (

**h**) zonal wind gradient $\mathrm{d}U/\mathrm{d}z$ (blue) and meridional wind gradient $\mathrm{d}V/\mathrm{d}z$ (pink); in (e), a dashed line in red is given as a reference to show an ideal atmosphere, which is isothermal at all altitudes.

#### 2.2. Generalized Dispersion Relation

**Figure 2.**Vertical profiles of input parameters in Equation (11). (

**a**) Richardson number ${R}_{i}$ (blue) and ${R}_{I}$ (pink); (

**b**) ${k}_{gT}^{2}$ (black), ${k}_{g}^{2}$ (pink) and ${k}_{G}^{2}$ (blue); (

**c**) buoyancy frequencies ${\omega}_{B}$ (thin blue) and ${\omega}_{b}$ (dash blue) and cut-off frequencies ${\omega}_{A}$ (thin pink) and ${\omega}_{a}$ (dash pink); (

**d**) $\mathrm{cos}\theta $ (blue) and ${\omega}_{\mathbf{v}}$ (pink); (

**e**) the four periods, ${\tau}_{B}$ (thin blue), ${\tau}_{b}$ (dash blue), ${\tau}_{A}$ (thin pink) and ${\tau}_{a}$ (dash pink), corresponding to the four frequencies in the upper right panel.

## 3. Results

**Figure 3.**Imaginary vertical wavenumber, ${m}_{i}$ (1/10 km), of different tsunami-excited wave modes propagating in an atmosphere. Case 1: Hines’ locally-isothermal and shear-free model; Case 2: the extended Hines’ model under nonisothermal conditions; Case 3: inertio-acoustic-gravity waves under nonisothermal conditions; Case 4: acoustic-gravity waves under nonisothermal and wind shear conditions. In Cases 1–3, ${m}_{i}$-curves are superimposed upon each other (in blue); in Case 4, the ${m}_{i}$-band fluctuates upon those of Cases 1–3 (in pink). As a reference, a red straight line is shown in the figure to represent the result of ${m}_{i}$ for an ideal atmosphere, which is isothermal at all altitudes in response to the constant ${T}_{0}$ in Figure 1.

#### 3.1. Case 1: Hines’ Locally-Isothermal and Shear-Free Model

**Figure 4.**Squared real vertical wavenumber, ${m}_{r}^{2}$ (1/km${}^{2}$), of different tsunami-excited wave modes propagating in an atmosphere. Case 1: (

**a**) Hines’ locally-isothermal and shear-free model; Case 2: (

**b**) the extended Hines’ model under nonisothermal conditions; Case 3: (

**c**) inertio-acoustic-gravity waves under nonisothermal conditions; Case 4: (

**d**) acoustic-gravity waves under non-isothermal and wind shear conditions. Note that there exists a “quasi-straight line” of ${m}_{r}^{2}=0$ in every panel throughout all altitudes at ∼4 min in the tsunami period. This line separates the acoustic waveband of <4 min in wave periods from the gravity waveband of >4 min in wave periods. In (a), there are three straight lines in red, which are located at ${z}_{1}$∼ 12 km, ${z}_{2}$ ∼ 50 km and ${z}_{3}$ ∼ 90 km, respectively, to separate the space into three regions; in (b), there are three additional straight lines in blue, which are located at ${z}_{1}^{\prime}$∼ 4.5 km, ${z}_{2}^{\prime}$∼ 75 km and ${z}_{3}^{\prime}$∼ 110 km, respectively.

#### 3.2. Case 2: Extended Hines’ Model under Non-Isothermal Condition

#### 3.3. Case 3: Inertio-Acoustic-Gravity Waves under Nonisothermal Condition

#### 3.4. Case 4: Acoustic-Gravity Waves under Nonisothermal and Wind-Shear Conditions

#### 3.5. Case 5: IAG Waves under Nonisothermal and Wind Shear Conditions

**Figure 5.**Imaginary and squared real vertical wavenumbers, ${m}_{i}$ (1/10 km) and ${m}_{r}^{2}$ (1/km${}^{2}$), in Case 5 of IAG waves under non-isothermal and wind shear conditions. (

**a**) ${m}_{i}$-envelop; and (

**b**) ${m}_{r}^{2}$-envelop. Note that due to the negligible f-effect, (a) gives the pink band in Figure 3; while (b) has no difference from (d) in Figure 4.

**Figure 6.**Contours of the “damping factor”, β (or, alternatively, the error, $\mathcal{E}$) versus${T}_{ts}$ and z. (

**a**) special case with $\theta =0$; and (

**b**) generalized case with $\theta \ne 0$.

#### 3.6. Influence of Phase Speed ${V}_{ph}$

**Figure 7.**Dependence of ${m}_{i}$ and ${m}_{r}^{2}$ on phase speed ${V}_{ph}$ at a characteristic wave period of ${T}_{ts}=33.3$ min. (

**a**) ${m}_{r}^{2}$ in Case 1; (

**b**) ${m}_{r}^{2}$ in Case 2 (or 3); (

**c**) ${m}_{i}$ in Case 4 (or 5); and (

**d**) ${m}_{r}^{2}$ in Case 4 (or 5).

## 4. Summary and Discussion

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Ma, J.Z.G. Modulation of Atmospheric Nonisothermality and Wind Shears on the Propagation of Seismic Tsunami-Excited Gravity Waves. *J. Mar. Sci. Eng.* **2016**, *4*, 4.
https://doi.org/10.3390/jmse4010004

**AMA Style**

Ma JZG. Modulation of Atmospheric Nonisothermality and Wind Shears on the Propagation of Seismic Tsunami-Excited Gravity Waves. *Journal of Marine Science and Engineering*. 2016; 4(1):4.
https://doi.org/10.3390/jmse4010004

**Chicago/Turabian Style**

Ma, John Z. G. 2016. "Modulation of Atmospheric Nonisothermality and Wind Shears on the Propagation of Seismic Tsunami-Excited Gravity Waves" *Journal of Marine Science and Engineering* 4, no. 1: 4.
https://doi.org/10.3390/jmse4010004