Variational treatment of inertia–gravity waves interacting with a quasi-geostrophic mean flow

2016 ◽  
Vol 809 ◽  
pp. 502-529 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Variational Principle ◽  
Gravity Waves ◽  
Potential Vorticity ◽  
Electromagnetic Waves ◽  
Three Dimensional ◽  
Classical Electrodynamics ◽  
Mean Flow ◽  
The Third ◽  
Variational Treatment ◽  
Inertia Gravity Waves

The equations for three-dimensional hydrostatic Boussinesq dynamics are equivalent to a variational principle that is closely analogous to the variational principle for classical electrodynamics. Inertia–gravity waves are analogous to electromagnetic waves, and available potential vorticity (i.e. the amount by which the potential vorticity exceeds the potential vorticity of the rest state) is analogous to electric charge. The Lagrangian can be expressed as the sum of three parts. The first part corresponds to quasi-geostrophic dynamics in the absence of inertia–gravity waves. The second part corresponds to inertia–gravity waves in the absence of quasi-geostrophic flow. The third part represents a coupling between the inertia–gravity waves and quasi-geostrophic motion. This formulation provides the basis for a general theory of inertia–gravity waves interacting with a quasi-geostrophic mean flow.

scholarly journals A Formulation of Unified Three-Dimensional Wave Activity Flux of Inertia–Gravity Waves and Rossby Waves

2013 ◽  
Vol 70 (6) ◽  
pp. 1603-1615 ◽  
Author(s):  
Takenari Kinoshita ◽  
Kaoru Sato
Keyword(s):  
Wave Propagation ◽  
Gravity Waves ◽  
Rossby Waves ◽  
Three Dimensional ◽  
Wave Activity ◽  
Storm Tracks ◽  
Mean Flow ◽  
Wave Activity Flux ◽  
Inertia Gravity Waves ◽  
Dimensional Wave

Abstract A companion paper formulates the three-dimensional wave activity flux (3D-flux-M) whose divergence corresponds to the wave forcing on the primitive equations. However, unlike the two-dimensional wave activity flux, 3D-flux-M does not accurately describe the magnitude and direction of wave propagation. In this study, the authors formulate a modification of 3D-flux-M (3D-flux-W) to describe this propagation using small-amplitude theory for a slowly varying time-mean flow. A unified dispersion relation for inertia–gravity waves and Rossby waves is also derived and used to relate 3D-flux-W to the group velocity. It is shown that 3D-flux-W and the modified wave activity density agree with those for inertia–gravity waves under the constant Coriolis parameter assumption and those for Rossby waves under the small Rossby number assumption. To compare 3D-flux-M with 3D-flux-W, an analysis of the European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Re-Analysis (ERA-Interim) data is performed focusing on wave disturbances in the storm tracks during April. While the divergence of 3D-flux-M is in good agreement with the meridional component of the 3D residual mean flow associated with disturbances, the 3D-flux-W divergence shows slight differences in the upstream and downstream regions of the storm tracks. Further, the 3D-flux-W magnitude and direction are in good agreement with those derived by R. A. Plumb, who describes Rossby wave propagation. However, 3D-flux-M is different from Plumb’s flux in the vicinity of the storm tracks. These results suggest that different fluxes (both 3D-flux-W and 3D-flux-M) are needed to describe wave propagation and wave–mean flow interaction in the 3D formulation.

scholarly journals A Formulation of Three-Dimensional Residual Mean Flow Applicable Both to Inertia–Gravity Waves and to Rossby Waves

2013 ◽  
Vol 70 (6) ◽  
pp. 1577-1602 ◽  
Author(s):  
Takenari Kinoshita ◽  
Kaoru Sato
Keyword(s):  
Gravity Waves ◽  
General Circulation ◽  
Rossby Waves ◽  
Three Dimensional ◽  
Stokes Drift ◽  
Eastern Region ◽  
Mean Flow ◽  
Southern Andes ◽  
Horizontal Structure ◽  
Inertia Gravity Waves

Abstract The three-dimensional (3D) residual mean flow is expressed as the sum of the Eulerian-mean flow and the Stokes drift. The present study derives formulas that are approximately equal to the 3D Stokes drift for the primitive equation (PRSD) and for the quasigeostrophic equation (QGSD) using small-amplitude theory for a slowly varying time-mean flow. The PRSD has a broad utility that is applicable to both Rossby waves and inertia–gravity waves. The 3D wave activity flux whose divergence corresponds to the wave forcing is also derived using PRSD. The PRSD agrees with QGSD under the small-Rossby-number assumption, and it agrees with the 3D Stokes drift derived by S. Miyahara and by T. Kinoshita et al. for inertia–gravity waves under the constant-Coriolis-parameter assumption. Moreover, a phase-independent 3D Stokes drift is derived under the QG approximation. The 3D residual mean flow in the upper troposphere in April is investigated by applying the new formulas to the European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Re-Analysis (ERA-Interim) data. It is observed that the PRSD is strongly poleward (weakly equatorward) upstream (downstream) of the storm track. A case study was also made for dominant gravity waves around the southern Andes in the simulation by a gravity wave–resolving general circulation model. The 3D residual mean flow associated with the gravity waves is poleward (equatorward) in the western (eastern) region of the southern Andes. This flow is due to the horizontal structure of the variance in the zonal component of the mountain waves, which do not change much while they propagate upward.

On non-dissipative wave–mean interactions in the atmosphere or oceans

1998 ◽  
Vol 354 ◽  
pp. 301-343 ◽  
Author(s):  
OLIVER BÜHLER ◽  
MICHAEL E. McINTYRE
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Potential Vorticity ◽  
Three Dimensional ◽  
Global Scale ◽  
Finite Amplitude ◽  
The Body ◽  
Mean Flow ◽  
Boussinesq Model ◽  
Flow Changes

Idealized model examples of non-dissipative wave–mean interactions, using small-amplitude and slow-modulation approximations, are studied in order to re-examine the usual assumption that the only important interactions are dissipative. The results clarify and extend the body of wave–mean interaction theory on which our present understanding of, for instance, the global-scale atmospheric circulation depends (e.g. Holton et al. 1995). The waves considered are either gravity or inertia–gravity waves. The mean flows need not be zonally symmetric, but are approximately ‘balanced’ in a sense that non-trivially generalizes the standard concepts of geostrophic or higher-order balance at low Froude and/or Rossby number. Among the examples studied are cases in which irreversible mean-flow changes, capable of persisting after the gravity waves have propagated out of the domain of interest, take place without any need for wave dissipation. The irreversible mean-flow changes can be substantial in certain circumstances, such as Rossby-wave resonance, in which potential-vorticity contours are advected cumulatively. The examples studied in detail use shallow-water systems, but also provide a basis for generalizations to more realistic, stratified flow models. Independent checks on the analytical shallow-water results are obtained by using a different method based on particle-following averages in the sense of ‘generalized Lagrangian-mean theory’, and by verifying the theoretical predictions with nonlinear numerical simulations. The Lagrangian-mean method is seen to generalize easily to the three-dimensional stratified Boussinesq model, and to allow a partial generalization of the results to finite amplitude. This includes a finite-amplitude mean potential-vorticity theorem with a larger range of validity than had been hitherto recognized.

scholarly journals Dynamic Potential Vorticity Initialization and the Diagnosis of Mesoscale Motion

2004 ◽  
Vol 34 (12) ◽  
pp. 2761-2773 ◽  
Author(s):  
Álvaro Viúdez ◽  
David G. Dritschel
Keyword(s):  
Gravity Waves ◽  
Potential Vorticity ◽  
Mass Density ◽  
Three Dimensional ◽  
Velocity Fields ◽  
Practical Applications ◽  
Dynamic Potential ◽  
Highly Nonlinear ◽  
Mesoscale Motion ◽  
Inertia Gravity Waves

Abstract A new method for diagnosing the balanced three-dimensional velocity from a given density field in mesoscale oceanic flows is described. The method is referred to as dynamic potential vorticity initialization (PVI) and is based on the idea of letting the inertia–gravity waves produced by the initially imbalanced mass density and velocity fields develop and evolve in time while the balanced components of these fields adjust during the diagnostic period to a prescribed initial potential vorticity (PV) field. Technically this is achieved first by calculating the prescribed PV field from given density and geostrophic velocity fields; then the PV anomaly is multiplied by a simple time-dependent ramp function, initially zero but tending to unity over the diagnostic period. In this way, the PV anomaly builds up to the prescribed anomaly. During this time, the full three-dimensional primitive equations—except for the PV equation—are integrated for several inertial periods. At the end of the diagnostic period the density and velocity fields are found to adjust to the prescribed PV field and the approximate balanced vortical motion is obtained. This adjustment involves the generation and propagation of fast, small-amplitude inertia–gravity waves, which appear to have negligible impact on the final near-balanced motion. Several practical applications of this method are illustrated. The highly nonlinear, complex breakup of baroclinically unstable currents into eddies, fronts, and filamentary structures is examined. The capability of the method to generate the balanced three-dimensional motion is measured by analyzing the ageostrophic horizontal and vertical velocity—the latter is the velocity component most sensitive to initialization, and one for which a quasigeostrophic diagnostic solution is available for comparison purposes. The authors find that the diagnosed fields are closer to the actual fields than are either the geostrophic or the quasigeostrophic approximations. Dynamic PV initialization thus appears to be a promising way of improving the diagnosis of balanced mesoscale motions.

scholarly journals Spontaneous Generation of Inertia–Gravity Waves during the Merging of Two Baroclinic Anticyclones

2008 ◽  
Vol 38 (1) ◽  
pp. 213-234 ◽  
Author(s):  
Enric Pallàs-Sanz ◽  
Álvaro Viúdez
Keyword(s):  
Gravity Waves ◽  
Three Dimensional ◽  
Vertical Motion ◽  
Rate Of Change ◽  
Coherent Motion ◽  
Spontaneous Generation ◽  
Mean Flow ◽  
Radiation Theory ◽  
Deep Layers ◽  
Inertia Gravity Waves

Abstract The spontaneous generation and propagation of short-scale inertia–gravity waves (IGWs) during the merging of two initially balanced (void of IGWs) baroclinic anticyclones is numerically investigated. The IGW generation is analyzed in flows with different potential vorticity (PV) anomaly, numerical diffusion, numerical resolution, vortex aspect ratio, and background rotation. The vertical velocity and its vertical derivative are used to identify the IGWs in the total flow, while the unbalanced flow (the waves) is diagnosed using the optimal PV balance approach. Spontaneous generation of IGWs occurs in all the cases, primarily as emissions of discrete wave packets. The increase of both the vortex strength and vortex extent isotropy enhances the IGW emission. Three possible indicators, or theories, of spontaneous IGW generation are considered, namely, the advection of PV, the material rate of change of the horizontal divergence, and the three-dimensional baroclinic IGW generation analogy of Lighthill sound radiation theory. It is suggested that different mechanisms for spontaneous IGW generation may be at work. One mechanism is related to the advection of PV, with the IGWs in this case having wave fronts similar to the PV isosurfaces in the upper layers, and helical patterns in the deep layers. Trapped IGWs are ubiquitous in the vortex interior and have annular wave front patterns. Another mechanism is related to the spatially coherent motion of preexisting IGWs, which eventually cooperate to produce mean flow, in particular larger-scale horizontal divergence, and therefore larger-scale vertical motion, which in turns triggers the emission of new IGWs.

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.

On the mean motion induced by internal gravity waves

1969 ◽  
Vol 36 (4) ◽  
pp. 785-803 ◽  
Author(s):  
Francis P. Bretherton
Keyword(s):  
Gravity Waves ◽  
Wave Energy ◽  
Wave Train ◽  
Three Dimensional ◽  
Internal Gravity Waves ◽  
Wave Number ◽  
Mean Flow ◽  
Mean Velocity ◽  
Mean Motion ◽  
The Mean

A train of internal gravity waves in a stratified liquid exerts a stress on the liquid and induces changes in the mean motion of second order in the wave amplitude. In those circumstances in which the concept of a slowly varying quasi-sinusoidal wave train is consistent, the mean velocity is almost horizontal and is determined to a first approximation irrespective of the vertical forces exerted by the waves. The sum of the mean flow kinetic energy and the wave energy is then conserved. The circulation around a horizontal circuit moving with the mean velocity is increased in the presence of waves according to a simple formula. The flow pattern is obtained around two- and three-dimensional wave packets propagating into a liquid at rest and the results are generalized for any basic state of motion in which the internal Froude number is small. Momentum can be associated with a wave packet equal to the horizontal wave-number times the wave energy divided by the intrinsic frequency.

scholarly journals A New Method to Estimate Three-Dimensional Residual-Mean Circulation in the Middle Atmosphere and Its Application to Gravity Wave–Resolving General Circulation Model Data

2013 ◽  
Vol 70 (12) ◽  
pp. 3756-3779 ◽  
Author(s):  
Kaoru Sato ◽  
Takenari Kinoshita ◽  
Kota Okamoto
Keyword(s):  
Gravity Wave ◽  
Gravity Waves ◽  
General Circulation ◽  
Three Dimensional ◽  
Circulation Model ◽  
Mean Flow ◽  
Flux Divergence ◽  
Mean Circulation ◽  
Residual Mean Circulation ◽  
Wave Induced

Abstract A new method is proposed to estimate three-dimensional (3D) material circulation driven by waves based on recently derived formulas by Kinoshita and Sato that are applicable to both Rossby waves and gravity waves. The residual-mean flow is divided into three, that is, balanced flow, unbalanced flow, and Stokes drift. The latter two are wave-induced components estimated from momentum flux divergence and heat flux divergence, respectively. The unbalanced mean flow is equivalent to the zonal-mean flow in the two-dimensional (2D) transformed Eulerian mean (TEM) system. Although these formulas were derived using the “time mean,” the underlying assumption is the separation of spatial or temporal scales between the mean and wave fields. Thus, the formulas can be used for both transient and stationary waves. Considering that the average is inherently needed to remove an oscillatory component of unaveraged quadratic functions, the 3D wave activity flux and wave-induced residual-mean flow are estimated by an extended Hilbert transform. In this case, the scale of mean flow corresponds to the whole scale of the wave packet. Using simulation data from a gravity wave–resolving general circulation model, the 3D structure of the residual-mean circulation in the stratosphere and mesosphere is examined for January and July. The zonal-mean field of the estimated 3D circulation is consistent with the 2D circulation in the TEM system. An important result is that the residual-mean circulation is not zonally uniform in both the stratosphere and mesosphere. This is likely caused by longitudinally dependent wave sources and propagation characteristics. The contribution of planetary waves and gravity waves to these residual-mean flows is discussed.

scholarly journals Formulation of Three-Dimensional Quasi-Residual Mean Flow Balanced with Diabatic Heating Rate and Potential Vorticity Flux

2019 ◽  
Vol 76 (3) ◽  
pp. 851-863
Author(s):  
Takenari Kinoshita ◽  
Kaoru Sato ◽  
Kentaro Ishijima ◽  
Masayuki Takigawa ◽  
Yousuke Yamashita
Keyword(s):  
Potential Vorticity ◽  
Diabatic Heating ◽  
Three Dimensional ◽  
Wave Activity ◽  
Stationary Waves ◽  
Vertical Flow ◽  
Mean Flow ◽  
Material Transport ◽  
Mean State ◽  
Wave Activity Flux

Abstract Three-dimensional (3D) quasi-residual mean flow is derived to diagnose 3D dynamical material transport associated with stationary planetary waves. The 3D quasi-residual mean vertical flow does not include the vertical flow due to tilting of the potential temperature caused by stationary waves, which is apparent but not seen in the mass-weighted isentropic mean state. Thus, the quasi-residual mean vertical flow is balanced with the term of diabatic heating rate. The 3D quasi-residual mean horizontal flow is balanced with the sum of the forcing due to transient wave activity flux divergence and the forcing associated with fluctuation of the potential vorticity due to stationary waves (defined as the effective Coriolis forcing). The zonal mean of the effective Coriolis forcing corresponds to the divergence of stationary wave activity flux. Thus, the zonal mean of derived 3D quasi-residual mean flow is exactly equal to the traditional residual mean flow. To demonstrate the usefulness of this quasi-residual mean flow, we analyze material transport of atmospheric sulfur hexafluoride (SF6) by using an atmospheric chemistry transport model. Comparison between the derived 3D quasi-residual mean flow and traditional residual mean flow shows that the zonal mean of advection of SF6 associated with the 3D quasi-residual mean flow derived is almost equal to that of the traditional residual mean flow. Next, it is confirmed that the horizontal structure of advection of SF6 associated with the 3D quasi-residual mean flow is balanced with the transport because of the nonlinear, nonconservative effects of disturbances. This relation is similar to the results for traditional residual mean flow in the zonal-mean state.

scholarly journals Squall Lines and Convectively Coupled Gravity Waves in the Tropics: Why Do Most Cloud Systems Propagate Westward?

2012 ◽  
Vol 69 (10) ◽  
pp. 2995-3012 ◽  
Author(s):  
Stefan N. Tulich ◽  
George N. Kiladis
Keyword(s):  
Gravity Waves ◽  
Wrf Model ◽  
Zonal Flow ◽  
Mean Flow ◽  
Wave Speeds ◽  
Squall Lines ◽  
Beta Plane ◽  
Pure Gravity ◽  
The Tropics ◽  
Inertia Gravity Waves

Abstract The coupling between tropical convection and zonally propagating gravity waves is assessed through Fourier analysis of high-resolution (3-hourly, 0.5°) satellite rainfall data. Results show the familiar enhancement in power along the dispersion curves of equatorially trapped inertia–gravity waves with implied equivalent depths in the range 15–40 m (i.e., pure gravity wave speeds in the range 12–20 m s−1). Here, such wave signals are seen to extend all the way down to zonal wavelengths of around 500 km and periods of around 8 h, suggesting that convection–wave coupling may be important even in the context of mesoscale squall lines. This idea is supported by an objective wave-tracking algorithm, which shows that many previously studied squall lines, in addition to “2-day waves,” can be classified as convectively coupled inertia–gravity waves with the dispersion properties of shallow-water gravity waves. Most of these disturbances propagate westward at speeds faster than the background flow. To understand why, the Weather Research and Forecast (WRF) Model is used to perform some near-cloud-resolving simulations of convection on an equatorial beta plane. Results indicate that low-level easterly shear of the background zonal flow, as opposed to steering by any mean flow, is essential for explaining the observed westward-propagation bias.

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