## 1. Introduction

Nonlinear internal gravity waves and undular bores in the atmosphere have been intensively studied over several decades; see, for instance, the reviews by Smith (1988) and Rottman and Grimshaw (2001). But only in the last couple of decades have there been analogous observations and consequent theoretical studies in the mesosphere. For instance, Dewan and Picard (1998, 2001) drew attention to a “spectacular gravity wave event” at a height of around 85 km observed from Hawaii as reported by Taylor et al. (1995), and they identified it as a mesospheric bore. This was followed by several other similar observations, notably over Colorado by She et al. (2004); over Antarctica by Nielsen et al. (2006), Bageston et al. (2011), and Stockwell et al. (2011); over northern China by Li et al. (2013); and over the southwestern United States by Smith et al. (2003).

There have been several theoretical presentations reported, essentially based on the concept of wave trapping in a mesosphere duct, and consequent nonlinear evolution exploiting extant theories of undular bores in the context of the nonlinear shallow water equations (e.g., Dewan and Picard 1998, 2001; Snively and Pasko 2003; Seyler 2005). The studies by Laughman et al. (2009, 2011) combined full numerical simulations with an application of a Benjamin–Davis–Acrivos–Ono (BDAO) model, sometimes referred to as the BDO or BO model. The BDAO equation was developed in the pioneering work of Benjamin (1967) and Davis and Acrivos (1967) for the description of internal solitary waves riding on a shallow stratified layer beneath a deep homogeneous layer. The application by Laughman et al. (2009, 2011) requires extension to a stratified duct embedded between two deep homogeneous layers. Here, we extend that model to include weak stratification in the adjoining deep layers, which then allows for internal gravity waves to be emitted from the duct by the nonlinear waves propagating along the duct. We also consider more general initial conditions and in particular those that lead to the formation of undular bores.

In section 2, we present a derivation of an extended BDAO model, together with an estimate of the decay rate of an internal solitary wave propagating along the duct because of the emission of outwardly propagating internal gravity waves. Then, in section 3, we describe the numerical methods used for the BDAO equation and for some analogous simulations of the full Navier–Stokes equations, followed by some preliminary numerical results in section 4. We conclude in section 5.

## 2. Theoretical model

### a. Derivation

We consider a two-dimensional duct with strong stratification, embedded between two deep layers with weak stratification, extending the derivation of Grimshaw (1981a), which was for a shallow layer beneath a deep layer. For simplicity, we assume also that the deep layers have uniform but weak stratification and that there is no background wind. Some effects of background winds for ducted waves and bores near the mesopause are considered in Fechine et al. (2009), Bageston et al. (2011), and Snively et al. (2007). Further, also for simplicity and for compatibility with the fully nonlinear numerical simulations, we assume incompressible flow. Although this assumption applies in situations when the vertical displacements are much smaller than the density scale height, the present model could be extended to remedy this and take some account of compressibility while still excluding acoustic waves, for instance, as in Grimshaw (1981b), or by replacing the incompressible condition with an anelastic approximation (e.g., Durran and Arakawa 2007; Sutherland 2010; Vallis 2006).

*p*is the perturbation pressure,

*g*is gravity,

*ρ*is the perturbation density. In the Boussinesq approximation,

*ψ*such that

*N*, see Sutherland (2010) and Eckermann et al. (1998).

Figure 1 shows the model setup. The region of the duct, *h* = 5 km, and

*z*. The boundary conditions are that either

*m*is a constant and

*A*is the amplitude at the upper boundary of the duct.

*ϕ*and

*ϕ*at

*ϕ*at

*z*,

*k*, and

*c*, and then the boundary conditions at

*m*and the dispersion relation expressing

*c*in terms of

*k*.

*ε*such that

*ε*, in which it will be sufficient to keep only the leading

*ε*. At leading order,

*A*defined in Eq. (11). This gives

*T*is a slow time relative to the

*ξ*reference frame. Then Eq. (21) becomes a linear differential operator acting on

*f*to obtain

*A*. The nonlinear coefficient is

### b. Decay estimates

*A*and

## 3. Numerical methods

We have carried out two sets of simulations. The first set is based on the model evolution equation [Eq. (23)], referred to herein as the “model,” and the second set is based on the two-dimensional Navier–Stokes equations in the Boussinesq approximation, referred to herein as the “full system.”

### a. Model evolution equation

The model evolution equation [Eq. (23)] for *x* and a Runge–Kutta time step. The examples below were computed with 1024 grid points in *x*. The equations for the vertical dependence [i.e., Eq. (13) for

### b. Full system

*P*is the pressure,

*ρ*is the density,

*θ*is the potential temperature,

*g*is gravity,

*ν*is the kinematic viscosity, and

*κ*is the thermal diffusivity. The overbar indicates a background quantity, with the zero subscript for a representative value. We set

*g*= 9.55 m s

^{−2}and

The full system is solved by a pseudospectral method (Spalart et al. 1991) with a third-order Runge–Kutta routine for the time step. The computational domain is periodic in *x* and has reflecting boundaries in *z*. The grid has 1000 points in *x*, 2500 points in *z*, and a time step of

### c. Duct parameters

*N*is

We set *h* = 5 km. In the following examples, *ξ* reference frame *V* given by Eq. (28).

As noted in section 3d, the full-system initialization has an envelope function to ensure that the vertical velocity is zero at the upper and lower boundaries. For the BDAO initial condition, this led to some initial perturbations well outside the duct, which then radiated small-amplitude waves. To minimize this radiation, we set

### d. Initial conditions

*x*. The former has initial vertical displacement [Laughman et al. (2011), their Eq. (17a)]:

The initialization of the full system is described by Laughman et al. (2011) and is followed here. For instance, the initial vertical velocity is *V* given in Eq. (28); and

The model equation [Eq. (23)] requires only the initialization of

Figure 3 shows the initial condition for *u* is close to zero at the duct boundaries

*A*is required to be real valued. However, when deriving the linearized part of the model equation [Eq. (7)],

*A*was the amplitude of the Fourier component

*ψ*in region U is

*A*to obtain the proper phase for propagation in regions U and L. The way we do this involves the Hilbert transform

*H*(

*A*) (Hahn 1996), applied in the

*x*direction, where this is defined by its Fourier transform

*A*defined by Eq. (25). In region U, we write,

The computational procedure is to integrate the model equation [Eq. (23)] numerically for *A*, which are always real valued.

## 4. Numerical results

First, we show the case where the initial condition is the BDAO solitary wave equation [Eq. (45)] with

Overall, there is very good agreement with the theoretical predictions from the BDAO model and between the model and full-system simulations. One difference is in the transients generated in the full system and seen in Fig. 5 at

For the full system in Fig. 4, there is wave reflection evident at the top and bottom of the plot, near

Figure 6 shows that the amplitude in the full system initially increases by a small amount before following the same decay rate as that for the model and as predicted theoretically. This initial small increase is part of the response to the initial conditions, though the exact reason in unclear. It also occurs for the sinusoid initial condition (see Fig. 9, discussed below). An initial increase is suggested by the theory of Grimshaw (1981c), which contains higher-order terms that are not included in our theoretical model.

The maximum value for *t* = 108 min.

We next show the case when the initial condition is a sinusoidal wave [Eq. (47)]. A typical result is plotted in Figs. 7 and 8, which show the formation of an undular bore, albeit decaying because of radiation. Indeed, the effect of the radiation is to prevent the formation of rank-ordered waves in the undular bore, since, as the leading larger waves form, so do they begin to decay. We note that a similar simulation for a smaller initial amplitude, one-third of that shown in Fig. 7, did not show the formation of an undular bore and instead dispersed and decayed essentially by linear dynamics in both the full system and the model. Laughman et al. (2011, section 4.3) discuss the full-system solutions for other initial sinusoid amplitudes and other values of

The bore waves in Fig. 7 have a horizontal wavelength of about *t* = 58 min.

The decay rates are shown in Fig. 9 based on both the maximum amplitude and an integrated measure of the disturbance amplitude. For the model simulations, we see initially that the sinusoidal disturbance decays linearly according to the decay law [Eq. (39)], but when the undular bore begins to form around

In contrast, for the full system, the same decay rates are rather variable. After an initial adjustment in amplitude, the early stages do exhibit evidence of sinusoidal decay until

Figures 7 and 8 show that, compared to the model solution, the full-system solution has larger trapped waves inside the duct (by a factor of about 2 in Fig. 8) and smaller radiated waves outside the duct, particularly at the short scales of the developing bore. This difference persists for longer times [

## 5. Discussion and conclusions

In this work, we have extended the weakly nonlinear BDAO model of a mesospheric duct described by Laughman et al. (2009, 2011) to include weak stratification in the adjoining deep layers. This then allows for internal gravity waves to be emitted from the duct by the nonlinear waves propagating along the duct. We have considered both the case when essentially a single solitary wave forms in the duct and a sinusoidal initial condition, which leads to the formation of an undular bore in the duct. In both cases, the nonlinear waves in the duct decay by radiation, in good agreement with the theoretical predictions from our extended BDAO model. The model results were compared with analogous simulations of the fully nonlinear system, and there is good overall agreement with the model. The large discrepancy between the model and the full-system decay rate shown in Fig. 9 is due in part to the reflection of the radiated waves from the rigid upper and lower boundaries of the computational domain in the full system. This highlights the value of an analytic model capable of properly handling the waves radiated from the duct and the need for a full-system model with radiation boundary conditions.

Our main conclusion is that, while nonlinear solitary waves can form in the duct, and although they do decay by radiation, they can survive as significant structures over sufficiently long periods so as to be observable. For the results presented here, our calculations indicate that the generated waves remain significant for *t* > 95 min, which certainly allows them to be observable. For example, Fig. 4 of Nielsen et al. (2006) displays the evolution of an observed mesospheric bore over a time period of 90 min with horizontal wavelength of about 20–30 km, similar to the bore wavelength shown here in Fig. 7.

Our main interest has been in the investigation of how weak stratification outside the duct affects the nonlinear waves in the duct. But we also considered a generic sinusoidal initial condition and demonstrated that this deformed into an undular bore, the leading waves of which become solitary waves and then decay by radiation. The more general issue of the energy source for the nonlinear waves and undular bores that can propagate in the duct remains for future work, but it has been suggested that a likely candidate could be vertically propagating gravity waves generated in the troposphere, which become trapped and break in the duct, generating a mixed region source for the horizontally propagating duct waves (e.g., Snively and Pasko 2003).

## Acknowledgments

SDE was supported by the Chief of Naval Research through the Naval Research Laboratory’s base 6.1 research program. We acknowledge Dr. Joseph Werne as the primary architect of the NWRA triple code described as the “full system” in section 3b. Funding for BL was provided by NSF grant AGS-1242943.

## REFERENCES

Bageston, J. V., C. M. Wrasse, P. P. Batista, R. E. Hibbins, D. C. Fritts, D. Gobbi, and V. F. Andrioli, 2011: Observation of a mesospheric front in a thermal-doppler duct over King George Island, Antarctica.

,*Atmos. Chem. Phys.***11**, 12 137–12 147, doi:10.5194/acp-11-12137-2011.Benjamin, T. B., 1967: Internal waves of permanent form in fluids of great depth.

,*J. Fluid Mech.***29**, 559–592, doi:10.1017/S002211206700103X.Boyd, J. P., 2001:

*Chebyshev and Fourier Spectral Methods.*2nd ed. Dover, 688 pp.Davis, R. E., and A. Acrivos, 1967: Solitary internal waves in deep water.

,*J. Fluid Mech.***29**, 593–607, doi:10.1017/S0022112067001041.Dewan, E. M., and R. H. Picard, 1998: Mesospheric bores.

,*J. Geophys. Res.***103**, 6295–6305, doi:10.1029/97JD02498.Dewan, E. M., and R. H. Picard, 2001: On the origin of mesospheric bores.

,*J. Geophys. Res.***106**, 6295–6305, doi:10.1029/2000JD900697.Durran, D. R., and A. Arakawa, 2007: Generalizing the Boussinesq approximation to stratified compressible flow.

,*C. R. Mec.***335**, 655–664, doi:10.1016/j.crme.2007.08.010.Eckermann, S. D., D. E. Gibson-Wilde, and J. T. Bacmeister, 1998: Gravity wave perturbations of minor constituents: A parcel advection methodology.

,*J. Atmos. Sci.***55**, 3521–3539, doi:10.1175/1520-0469(1998)055<3521:GWPOMC>2.0.CO;2.Fechine, J., and Coauthors, 2009: First observation of an undular mesospheric bore in a Doppler duct.

,*Ann. Geophys.***27**, 1399–1406, doi:10.5194/angeo-27-1399-2009.Grimshaw, R., 1981a: Evolution equations for long nonlinear internal waves in stratified shear flows.

,*Stud. Appl. Math.***65**, 159–188.Grimshaw, R., 1981b: Solitary waves in a compressible fluid.

,*Pure Appl. Geophys.***119**, 780–797, doi:10.1007/BF01131255.Grimshaw, R., 1981c: A second-order theory for solitary waves in deep fluids.

,*Phys. Fluids***24**, 1611–1618, doi:10.1063/1.863583.Hahn, S. F., 1996:

*Hilbert Transforms in Signal Processing.*Artech House, 460 pp.Jorge, M. C., A. A. Minzoni, and N. F. Smyth, 1999: Modulation solutions for the Benjamin–Ono equation.

,*Physica D***132**, 1–18, doi:10.1016/S0167-2789(99)00039-1.Laughman, B., D. C. Fritts, and J. Werne, 2009: Numerical simulation of bore generation and morphology in thermal and Doppler ducts.

,*Ann. Geophys.***27**, 511–523, doi:10.5194/angeo-27-511-2009.Laughman, B., D. C. Fritts, and J. Werne, 2011: Comparisons of predicted bore evolutions by the Benjamin–Davis–Ono and Navier–Stokes equations for idealized mesopause thermal ducts.

,*J. Geophys. Res.***116**, D02120, doi:10.1029/2010JD014409.Li, Q., J. Xu, J. Yue, X. Liu, W. Yuan, B. Ning, S. Guan, and J. P. Younger, 2013: Investigation of a mesospheric bore event over northern China.

,*Ann. Geophys.***31**, 409–418, doi:10.5194/angeo-31-409-2013.Maslowe, S. A., and L. G. Redekopp, 1980: Long nonlinear waves in stratified shear flows.

,*J. Fluid Mech.***101**, 321–348, doi:10.1017/S0022112080001681.Matsuno, Y., V. S. Shchesnovich, A. M. Kamchatnov, and R. A. Kraenkel, 2007: Whitham method for the Benjamin–Ono–Burgers equation and dispersive shocks.

*Phys. Rev. E*,**75**, 016307, doi:10.1103/PhysRevE.75.016307.Nielsen, K., M. J. Taylor, R. G. Stockwell, and M. J. Jarvis, 2006: An unusual mesospheric bore event observed at high latitudes over Antarctica.

*Geophys. Res. Lett.*,**33**, L07803, doi:10.1029/2005GL025649.Pereira, N. R., and L. G. Redekopp, 1980: Radiation damping of long, finite-amplitude internal waves.

,*Phys. Fluids***23**, 2182, doi:10.1063/1.862913.Rottman, J. W., and R. Grimshaw, 2001: Atmospheric internal solitary waves.

*Environmental Stratified Flows*, R. Grimshaw, Ed., Topics in Environmental Fluid Mechanics, Vol. 3, Kluwer, 61–88, doi:10.1007/0-306-48024-7_3.Seyler, C. E., 2005: Internal waves and undular bores in mesospheric inversion layers.

,*J. Geophys. Res.***110**, D09S05, doi:10.1029/2004JD004685.She, C. Y., T. Li, B. P. Williams, and T. Yuan, 2004: Concurrent OH imager and sodium temperature/wind lidar observation of a mesopause region undular bore event over Fort Collins/Platteville, Colorado.

*J. Geophys. Res.*,**109**, D22107, doi:10.1029/2004JD004742.Smith, R. K., 1988: Traveling waves and bores in the lower atmosphere: The “Morning Glory” and related phenomena.

,*Earth-Sci. Rev.***25**, 267–290, doi:10.1016/0012-8252(88)90069-4.Smith, S. M., M. J. Taylor, G. R. Swenson, C. Y. She, W. Hocking, J. Baumgardner, and M. Mendillo, 2003: A multidiagnostic investigation of the mesospheric bore phenomenon.

,*J. Geophys. Res.***108**, 1083, doi:10.1029/2002JA009500.Snively, J., and V. P. Pasko, 2003: Breaking of thunderstorm-generated gravity waves as a source of short-period ducted waves at mesopause altitudes.

,*Geophys. Res. Lett.***30**, 2254, doi:10.1029/2003GL018436.Snively, J., V. P. Pasko, M. Taylor, and W. Hocking, 2007: Doppler ducting of short-period gravity waves by midlatitude tidal wind structure.

*J. Geophys. Res.*,**112**, A03304, doi:10.1029/2006JA011895.Spalart, P. R., R. D. Moser, and M. M. Rogers, 1991: Spectral methods for the Navier–Stokes equations with one infinite and two periodic directions.

,*J. Comput. Phys.***96**, 297–324, doi:10.1016/0021-9991(91)90238-G.Stockwell, R. G., M. J. Taylor, K. K. Nielsen, and M. J. Jarvis, 2011: The evolution of a breaking mesospheric bore wave packet.

*J. Geophys. Res.*,**116**, D19102, doi:10.1029/2010JD015321.Sutherland, B. R., 2010:

*Internal Gravity Waves.*Cambridge University Press, 377 pp.Taylor, M. J., D. N. Turnbull, and R. P. Lowe, 1995: Spectrometric and imaging measurements of a spectacular gravity wave event observed during the ALOHA-93 Campaign.

,*Geophys. Res. Lett.***22**, 2848–2852, doi:10.1029/95GL02948.Vallis, G., 2006:

*Atmospheric and Oceanic Fluid Dynamics.*Cambridge University Press, 745 pp.