scholarly journals Generalized modulation theory for strongly nonlinear gravity waves in a compressible atmosphere

2020 ◽  
Vol 6 (1) ◽  
pp. 97-112
Author(s):  
Mark Schlutow ◽  
Erik Wahlén
Keyword(s):  
Gravity Waves ◽  
Resonance Condition ◽  
Stationary Solutions ◽  
Mean Flow ◽  
Modulation Equations ◽  
Strongly Nonlinear ◽  
Modulational Stability ◽  
Background Fields ◽  
Compressible Atmosphere ◽  
Nonlinear Gravity

Abstract This study investigates strongly nonlinear gravity waves in the compressible atmosphere from the Earth’s surface to the deep atmosphere. These waves are effectively described by Grimshaw’s dissipative modulation equations which provide the basis for finding stationary solutions such as mountain lee waves and testing their stability in an analytic fashion. Assuming energetically consistent boundary and far-field conditions, that is no energy flux through the surface, free-slip boundary, and finite total energy, general wave solutions are derived and illustrated in terms of realistic background fields. These assumptions also imply that the wave-Reynolds number must become less than unity above a certain height. The modulational stability of admissible, both non-hydrostatic and hydrostatic, waves is examined. It turns out that, when accounting for the self-induced mean flow, the wave-Froude number has a resonance condition. If it becomes 1/ 1 / 2 1/\sqrt 2 , then the wave destabilizes due to perturbations from the essential spectrum of the linearized modulation equations. However, if the horizontal wavelength is large enough, waves overturn before they can reach the modulational stability condition.

scholarly journals On strongly nonlinear gravity waves in a vertically sheared atmosphere

2020 ◽  
Vol 6 (1) ◽  
pp. 63-74
Author(s):  
Mark Schlutow ◽  
Georg S. Voelker
Keyword(s):  
Gravity Waves ◽  
Lower Layer ◽  
Vertical Shear ◽  
Horizontal Wind ◽  
Necessary Condition ◽  
Mean Flow ◽  
Individual Layer ◽  
Strongly Nonlinear ◽  
The Mean ◽  
The Stability

Abstract We investigate strongly nonlinear stationary gravity waves which experience refraction due to a thin vertical shear layer of horizontal background wind. The velocity amplitude of the waves is of the same order of magnitude as the background flow and hence the self-induced mean flow alters the modulation properties to leading order. In this theoretical study, we show that the stability of such a refracted wave depends on the classical modulation stability criterion for each individual layer, above and below the shearing. Additionally, the stability is conditioned by novel instability criteria providing bounds on the mean-flow horizontal wind and the amplitude of the wave. A necessary condition for instability is that the mean-flow horizontal wind in the upper layer is stronger than the wind in the lower layer.

scholarly journals Modulational Stability of Nonlinear Saturated Gravity Waves

2019 ◽  
Vol 76 (11) ◽  
pp. 3327-3336
Author(s):  
Mark Schlutow
Keyword(s):  
Reynolds Number ◽  
Gravity Waves ◽  
Mean Flow ◽  
Order Unity ◽  
Direct Wave ◽  
Modulation Equations ◽  
Flow Acceleration ◽  
Saturation Region ◽  
Lee Waves ◽  
Pressure Scale

Abstract Stationary gravity waves, such as mountain lee waves, are effectively described by Grimshaw’s dissipative modulation equations even in high altitudes where they become nonlinear due to their large amplitudes. In this theoretical study, a wave-Reynolds number is introduced to characterize general solutions to these modulation equations. This nondimensional number relates the vertical linear group velocity with wavenumber, pressure scale height, and kinematic molecular/eddy viscosity. It is demonstrated by analytic and numerical methods that Lindzen-type waves in the saturation region, that is, where the wave-Reynolds number is of order unity, destabilize by transient perturbations. It is proposed that this mechanism may be a generator for secondary waves due to direct wave–mean-flow interaction. By assumption, the primary waves are exactly such that altitudinal amplitude growth and viscous damping are balanced and by that the amplitude is maximized. Implications of these results on the relation between mean-flow acceleration and wave breaking heights are discussed.

scholarly journals Spectral stability of nonlinear gravity waves in the atmosphere

2019 ◽  
Vol 5 (1) ◽  
pp. 12-33 ◽  
Author(s):  
Mark Schlutow ◽  
Erik Wahlén ◽  
Philipp Birken
Keyword(s):  
Gravity Waves ◽  
Blow Up ◽  
Plane Waves ◽  
Traveling Wave Solutions ◽  
Spectral Stability ◽  
Modulation Equations ◽  
Background Density ◽  
Instability Growth ◽  
Nonlinear Modulation ◽  
Nonlinear Gravity

AbstractWe apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from Wentzel-Kramers-Brillouin theory. First, we establish that plane waves, which represent exact solutions to the inviscid Boussinesq equations, are spectrally stable with respect to their nonlinear modulation equations under the same conditions as what is known as modulational stability from weakly nonlinear theory. In contrast to Boussinesq, the pseudo-incompressible regime does fully account for the altitudinal varying background density. Second,we show for the first time that upward-traveling non-plane wave fronts solving the inviscid nonlinear modulation equations, that compare to pseudo-incompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be ill-posed as the spectra allow for arbitrarily large instability growth rates. Third, a regularization is found by including dissipative effects. The corresponding nonlinear traveling wave solutions have localized amplitude. As a consequence of the nonlinearity, envelope and linear group velocity, as given by the derivative of the frequency with respect to wavenumber, do not coincide anymore. These waves blow up unconditionally by embedded eigenvalue instabilities but the instability growth rate is bounded from above and can be computed analytically. Additionally, all three types of nonlinear modulation equations are solved numerically to further investigate and illustrate the nature of the analytic stability results.

On strongly nonlinear gravity waves in a vertically sheared atmosphere

2021 ◽  
Author(s):  
Georg Sebastian Voelker ◽  
Mark Schlutow
Keyword(s):  
Gravity Wave ◽  
Gravity Waves ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Wave Drag ◽  
Mean Flow ◽  
Flow Strength ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
The Mean

<p>Internal gravity waves are a well-known mechanism of energy redistribution in stratified fluids such as the atmosphere. They may propagate from their generation region, typically in the Troposphere, up to high altitudes. During their lifetime internal waves couple to the atmospheric background through various processes. Among the most important interactions are the exertion of wave drag on the horizontal mean-flow, the heat generation upon wave breaking, or the mixing of atmospheric tracers such as aerosols or greenhouse gases.</p><p>Many of the known internal gravity wave properties and interactions are covered by linear or weakly nonlinear theories. However, for the consideration of some of the crucial effects, like a reciprocal wave-mean-flow interaction including the exertion of wave drag on the mean-flow, strongly nonlinear systems are required. That is, there is no assumption on the wave amplitude relative to the mean-flow strength such that they may be of the same order.</p><p>Here, we exploit a strongly nonlinear Boussinesq theory to analyze the stability of a stationary internal gravity wave which is refracted at the vertical edge of a horizontal jet. Thereby we assume that the incident wave is horizontally periodic, non-hydrostatic, and vertically modulated. Performing a linear stability analysis in the vicinity of the jet edge we find necessary and sufficient criteria for instabilities to grow. In particular, the refracted wave becomes unstable if its incident amplitude is large enough and both mean-flow horizontal winds, below and above the edge of the jet, do not exceed particular upper bounds.</p>

A high-order spectral method for the study of nonlinear gravity waves

1987 ◽  
Vol 184 ◽  
pp. 267-288 ◽  
Author(s):  
Douglas G. Dommermuth ◽  
Dick K. P. Yue
Keyword(s):  
Gravity Waves ◽  
Mode Coupling ◽  
Arbitrary Order ◽  
Travelling Wave ◽  
Computational Effort ◽  
Zakharov Equation ◽  
Long Time ◽  
Nonlinear Free Surface ◽  
High Order Spectral Method ◽  
Nonlinear Gravity

We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number (N = O(1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M, and the convergence with N and M is exponentially fast for waves up to approximately 80% of Stokes limiting steepness (ka ∼ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

Atmospheric general circulation and waves simulated by a Venus AORI GCM with topographical and radiative forcings

2021 ◽  
Author(s):  
Masaru Yamamoto ◽  
Takumi Hirose ◽  
Kohei Ikeda ◽  
Masaaki Takahashi
Keyword(s):  
Gravity Waves ◽  
General Circulation ◽  
Circulation Model ◽  
Stationary Wave ◽  
Static Stability ◽  
Small Scale ◽  
Mean Flow ◽  
Short Period ◽  
Low Latitudes ◽  
Thermal Tides

<p>General circulation and waves are investigated using a T63 Venus general circulation model (GCM) with solar and thermal radiative transfer in the presence of high-resolution surface topography. This model has been developed by Ikeda (2011) at the Atmosphere and Ocean Research Institute (AORI), the University of Tokyo, and was used in Yamamoto et al. (2019, 2021). In the wind and static stability structures similar to the observed ones, the waves are investigated. Around the cloud-heating maximum (~65 km), the simulated thermal tides accelerate an equatorial superrotational flow with a speed of ~90 m/s<sup></sup>with rates of 0.2–0.5 m/s/(Earth day) via both horizontal and vertical momentum fluxes at low latitudes. Over the high mountains at low latitudes, the vertical wind variance at the cloud top is produced by topographically-fixed, short-period eddies, indicating penetrative plumes and gravity waves. In the solar-fixed coordinate system, the variances (i.e., the activity of waves other than thermal tides) of flow are relatively higher on the night-side than on the dayside at the cloud top. The local-time variation of the vertical eddy momentum flux is produced by both thermal tides and solar-related, small-scale gravity waves. Around the cloud bottom, the 9-day super-rotation of the zonal mean flow has a weak equatorial maximum and the 7.5-day Kelvin-like wave has an equatorial jet-like wind of 60-70 m/s. Because we discussed the thermal tide and topographically stationary wave in Yamamoto et al. (2021), we focus on the short-period eddies in the presentation.</p>

scholarly journals ENSO Modulation of the QBO: Results from MIROC Models with and without Nonorographic Gravity Wave Parameterization

2019 ◽  
Vol 76 (12) ◽  
pp. 3893-3917 ◽  
Author(s):  
Yoshio Kawatani ◽  
Kevin Hamilton ◽  
Kaoru Sato ◽  
Timothy J. Dunkerton ◽  
Shingo Watanabe ◽  
...  
Keyword(s):  
Gravity Wave ◽  
Gravity Waves ◽  
El Niño ◽  
El Nino ◽  
Horizontal Resolution ◽  
La Niña ◽  
Mean Flow ◽  
Quasi Biennial Oscillation ◽  
La Nina ◽  
Zonal Wavenumber

Abstract Observational studies have shown that, on average, the quasi-biennial oscillation (QBO) exhibits a faster phase progression and shorter period during El Niño than during La Niña. Here, the possible mechanism of QBO modulation associated with ENSO is investigated using the MIROC-AGCM with T106 (~1.125°) horizontal resolution. The MIROC-AGCM simulates QBO-like oscillations without any nonorographic gravity wave parameterizations. A 100-yr integration was conducted during which annually repeating sea surface temperatures based on the composite observed El Niño conditions were imposed. A similar 100-yr La Niña integration was also conducted. The MIROC-AGCM simulates realistic differences between El Niño and La Niña, notably shorter QBO periods, a weaker Walker circulation, and more equatorial precipitation during El Niño than during La Niña. Near the equator, vertical wave fluxes of zonal momentum in the uppermost troposphere are larger and the stratospheric QBO forcing due to interaction of the mean flow with resolved gravity waves (particularly for zonal wavenumber ≥43) is much larger during El Niño. The tropical upwelling associated with the Brewer–Dobson circulation is also stronger in the El Niño simulation. The effects of the enhanced tropical upwelling during El Niño are evidently overcome by enhanced wave driving, resulting in the shorter QBO period. The integrations were repeated with another model version (MIROC-ECM with T42 horizontal resolution) that employs a parameterization of nonorographic gravity waves in order to simulate a QBO. In the MIROC-ECM the average QBO periods are nearly identical in the El Niño and La Niña simulations.

scholarly journals Ageostrophic instability in rotating, stratified interior vertical shear flows

2014 ◽  
Vol 755 ◽  
pp. 397-428 ◽  
Author(s):  
Peng Wang ◽  
James C. McWilliams ◽  
Claire Ménesguen
Keyword(s):  
Energy Source ◽  
Kinetic Energy ◽  
Potential Energy ◽  
Gravity Waves ◽  
Shear Flows ◽  
Vertical Shear ◽  
Mean Flow ◽  
Efficient Energy Transfer ◽  
The Mean ◽  
Inertia Gravity Waves

AbstractThe linear instability of several rotating, stably stratified, interior vertical shear flows $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\overline{U}(z)$ is calculated in Boussinesq equations. Two types of baroclinic, ageostrophic instability, AI1 and AI2, are found in odd-symmetric $\overline{U}(z)$ for intermediate Rossby number ($\mathit{Ro}$). AI1 has zero frequency; it appears in a continuous transformation of the unstable mode properties between classic baroclinic instability (BCI) and centrifugal instability (CI). It begins to occur at intermediate $\mathit{Ro}$ values and horizontal wavenumbers ($k,l$) that are far from $l= 0$ or $k = 0$, where the growth rate of BCI or CI is the strongest. AI1 grows by drawing kinetic energy from the mean flow, and the perturbation converts kinetic energy to potential energy. The instability AI2 has inertia critical layers (ICL); hence it is associated with inertia-gravity waves. For an unstable AI2 mode, the coupling is either between an interior balanced shear wave and an inertia-gravity wave (BG), or between two inertia-gravity waves (GG). The main energy source for an unstable BG mode is the mean kinetic energy, while the main energy source for an unstable GG mode is the mean available potential energy. AI1 and BG type AI2 occur in the neighbourhood of $A-S= 0$ (a sign change in the difference between absolute vertical vorticity and horizontal strain rate in isentropic coordinates; see McWilliams et al., Phys. Fluids, vol. 10, 1998, pp. 3178–3184), while GG type AI2 arises beyond this condition. Both AI1 and AI2 are unbalanced instabilities; they serve as an initiation of a possible local route for the loss of balance in 3D interior flows, leading to an efficient energy transfer to small scales.

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