Range of validity of an extended WKB theory for atmospheric gravity waves: one-dimensional and two-dimensional case

2013 ◽  
Vol 729 ◽  
pp. 330-363 ◽  
Author(s):  
Felix Rieper ◽  
U. Achatz ◽  
R. Klein
Keyword(s):  
Wave Packet ◽  
Gravity Wave ◽  
Large Scale ◽  
Equations Of Motion ◽  
Potential Temperature ◽  
Multiple Scale ◽  
Wave Number ◽  
Momentum Balance ◽  
Mean Flow ◽  
Governing Parameter

AbstractA computational model of the pseudo-incompressible equations is used to probe the range of validity of an extended Wentzel–Kramers–Brillouin theory (XWKB), previously derived through a distinguished limit of a multiple-scale asymptotic analysis of the Euler or pseudo-incompressible equations of motion, for gravity-wave packets at large amplitudes. The governing parameter of this analysis had been the scale-separation ratio $\varepsilon $ between the gravity wave and both the large-scale potential-temperature stratification and the large-scale wave-induced mean flow. A novel feature of the theory had been the non-resonant forcing of higher harmonics of an initial wave packet, predominantly by the large-scale gradients in the gravity-wave fluxes. In the test cases considered a gravity-wave packet is propagating upwards in a uniformly stratified atmosphere. Large-scale winds are induced by the wave packet, and possibly exert a feedback on the latter. In the limit $\varepsilon \ll 1$ all predictions of the theory can be validated. The larger $\varepsilon $ is the more the transfer of wave energy to the mean flow is underestimated by the theory. The numerical results quantify this behaviour but also show that, qualitatively, XWKB remains valid even when the gravity-wave wavelength approaches the spatial scale of the wave-packet amplitude. This includes the prevalence of first and second harmonics and the smallness of harmonics with wave number higher than two. Furthermore, XWKB predicts for the vertical momentum balance an additional leading-order buoyancy term in Euler and pseudo-incompressible theory, compared with the anelastic theory. Numerical tests show that this term is relatively large with up to $30\hspace{0.167em} \% $ of the total balance. The practical relevance of this deviation remains to be assessed in future work.

Mountain wave parametrization in a transient gravity wave model

2020 ◽  
Author(s):  
Fanni Dora Kelemen ◽  
Jan Weinkaemmerer ◽  
Gergely Bölöni ◽  
Ulrich Achatz
Keyword(s):  
Boundary Condition ◽  
Gravity Wave ◽  
Large Scale ◽  
Lower Boundary ◽  
Wave Model ◽  
Wave Number ◽  
Spectral Space ◽  
Mean Flow ◽  
Mountain Wave ◽  
Lower Boundary Condition

<p>Orography induced gravity waves are investigated in a Multi Scale Gravity Wave Model (MS-GWaM) over idealized topography. MS-GWaM is a prognostic gravity-wave model, which parametrizes both the propagation and dissipation of subgrid-scale GWs. It is a Lagrangian ray-tracer model, which applies WKB-theory and calculates the propagation of ray volumes in spectral space. Its novelty is that not only the dissipative effect, but also the non-dissipative effects due to direct wave-mean flow interaction are captured. In our conceptual studies we investigate mountain wave generation, which is induced via a time-dependent large-scale wind encountering a prescribed topography. The framework used in our experiments is the PincFloit model, which integrates the pseudo-incompressible equations. We use it both in low resolution with MS-GWaM and in high resolution LES mode as a wave resolving reference. In the reference LES simulations the idealized topography, a mountain chain, is represented with an immersed boundary method. In the MS-GWaM experiments there is no resolved topography, but its effect is modelled as a lower boundary condition. The lower boundary condition is represented by initializing ray volumes with wave number and wave action density depending on the mountain characteristics and the large scale wind speed, based on the assumption that the flow follows the terrain. In the wave resolving reference experiments the flow does not strictly follow the terrain, but other instabilities (rotor formation, boundary layer separation) arise around the mountains. These processes decrease the available momentum transported by GWs, which was initially not accounted for in MS-GWaM. Thus an overestimation of wind deceleration was found in the MS-GWaM parametrization compared to the wave resolving simulation. To correct for this overestimation, an effective mountain height is introduced into Ms-GWaM, which is calculated by a scaling function between mountain height and flow properties using the Froude number.</p>

Reducing Instrumentation Errors Caused by Circumferential Flow-Field Variations in Multistage Axial Compressors

2020 ◽  
Vol 142 (9) ◽  
Author(s):  
M. Chilla ◽  
G. Pullan ◽  
S. Gallimore
Keyword(s):  
Flow Field ◽  
Large Scale ◽  
Sampling Error ◽  
Leading Edge ◽  
Stagnation Pressure ◽  
Wave Number ◽  
Stagnation Temperature ◽  
Sampling Errors ◽  
Mean Flow ◽  
Axial Compressors

Abstract The effects of blade row interactions on stator-mounted instrumentation in axial compressors are investigated using unsteady numerical calculations. The test compressor is an eight-stage machine representative of an aero-engine core compressor. For the unsteady calculations, a 180-deg sector (half-annulus) model of the compressor is used. It is shown that the time-mean flow field in the stator leading edge planes is circumferentially nonuniform. The circumferential variations in stagnation pressure and stagnation temperature, respectively, reach 4.2% and 1.1% of the local mean. Using spatial wave number analysis, the incoming wakes from the upstream stator rows are identified as the dominant source of the circumferential variations in the front and middle of the compressor, while toward the rear of the compressor, the upstream influence of the eight struts in the exit duct becomes dominant. Based on three circumferential probes, the sampling errors for stagnation pressure and stagnation temperature are calculated as a function of the probe locations. Optimization of the probe locations shows that the sampling error can be reduced by up to 77% by circumferentially redistributing the individual probes. The reductions in the sampling errors translate to reductions in the uncertainties of the overall compressor efficiency and inlet flow capacity by up to 50%. Recognizing that data from large-scale unsteady calculations are rarely available in the instrumentation phase for a new test rig or engine, a method for approximating the circumferential variations with single harmonics is presented. The construction of the harmonics is based solely on the knowledge of the number of stators in each row and a small number of equispaced probes. It is shown how excursions in the sampling error are reduced by increasing the number of circumferential probes.

scholarly journals Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 1. Pseudomomentum-based theory

1995 ◽  
Vol 290 ◽  
pp. 67-104 ◽  
Author(s):  
Paul J. Kushner ◽  
Theodore G. Shepherd
Keyword(s):  
Conservation Laws ◽  
Potential Vorticity ◽  
Large Scale ◽  
Equations Of Motion ◽  
Lower Boundary ◽  
Wave Activity ◽  
Eddy Correlation ◽  
Mean Flow ◽  
Local Conservation ◽  
Stability Theorems

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics.In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms.The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent.This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii).In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

scholarly journals Tropical Cyclogenesis Associated with Rossby Wave Energy Dispersion of a Preexisting Typhoon. Part II: Numerical Simulations*

2006 ◽  
Vol 63 (5) ◽  
pp. 1390-1409 ◽  
Author(s):  
Tim Li ◽  
Xuyang Ge ◽  
Bin Wang ◽  
Yongti Zhu
Keyword(s):  
Large Scale ◽  
Wave Train ◽  
Deep Convection ◽  
Potential Temperature ◽  
Stable Stratification ◽  
Temperature Fields ◽  
Tropical Cyclogenesis ◽  
Mean Flow ◽  
Energy Dispersion ◽  
Vorticity Generation

Abstract The cyclogenesis events associated with the tropical cyclone (TC) energy dispersion are simulated in a 3D model. A new TC with realistic dynamic and thermodynamic structures forms in the wake of a preexisting TC when a large-scale monsoon gyre or a monsoon shear line flow is present. Maximum vorticity generation appears in the planetary boundary layer (PBL) and the vorticity growth exhibits an oscillatory development. This oscillatory growth is also seen in the observed rainfall and cloud-top temperature fields. The diagnosis of the model output shows that the oscillatory development is attributed to the discharge and recharge of the PBL moisture and its interaction with convection and circulation. The moisture–convection feedback regulates the TC development through controlling the atmospheric stratification, raindrop-induced evaporative cooling and downdraft, PBL divergence, and vorticity generation. On one hand, ascending motion associated with deep convection transports moisture upward and leads to the discharge of PBL moisture and a convectively stable stratification. On the other hand, the convection-induced raindrops evaporate, leading to midlevel cooling and downdraft. The downdraft further leads to dryness and a reduction of equivalent potential temperature. This reduction along with the recharge of PBL moisture due to surface evaporation leads to reestablishment of a convectively unstable stratification and thus new convection. Sensitivity experiments with both a single mesh (with a 15-km resolution) and a nested mesh (with a 5-km resolution in the inner mesh) indicate that TC energy dispersion alone in a resting environment does not lead to cyclogenesis, suggesting the important role of the wave train–mean flow interaction. A proper initial condition for background wind and moisture fields is crucial for maintaining a continuous vorticity growth through the multioscillatory phases.

Developing turbulent boundary layers with system rotation

1985 ◽  
Vol 157 ◽  
pp. 405-448 ◽  
Author(s):  
J. H. Watmuff ◽  
H. T. Witt ◽  
P. N. Joubert
Keyword(s):  
Skin Friction ◽  
Boundary Layers ◽  
Large Scale ◽  
Turbulent Boundary Layers ◽  
Momentum Balance ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Mean Velocity ◽  
Secondary Circulations ◽  
Spatially Periodic

Measurements are presented for low-Reynolds-number turbulent boundary layers developing in a zero pressure gradient on the sidewall of a duct. The effect of rotation on these layers is examined. The mean-velocity profiles affected by rotation are described in terms of a common universal sublayer and modified logarithmic and wake regions.The turbulence quantities follow an inner and outer scaling independent of rotation. The effect appears to be similar to that, of increased or decreased layer development. Streamwise-energy spectra indicate that, for a given non-dimensional wall distance, it is the low-wavenumber spectral components alone that are affected by rotation.Large spatially periodic spanwise variations of skin friction are observed in the destabilized layers. Mean-velocity vectors in the cross-stream plane clearly show an array of vortex-like structures which correlate strongly with the skin-friction pattern. Interesting properties of these mean-flow structures are shown and their effect on Reynolds stresses is revealed. Near the duct centreline, where we have measured detailed profiles, the variations are small and there is a reasonable momentum balance.Large-scale secondary circulations are also observed but the strength of the pattern is weak and it appears to be confined to the top and bottom regions of the duct. The evidence suggests that it has minimally affected the flow near the duct centreline where detailed profiles were measured.

scholarly journals Brief communication "On one mechanism of low frequency variability of the Antarctic Circumpolar Current"

2011 ◽  
Vol 18 (3) ◽  
pp. 361-365 ◽  
Author(s):  
O. G. Derzho ◽  
B. de Young
Keyword(s):  
Rossby Wave ◽  
Antarctic Circumpolar Current ◽  
Large Scale ◽  
Wave Train ◽  
Low Frequency ◽  
Wave Number ◽  
Mean Flow ◽  
Zonal Wave Number ◽  
Circumpolar Current ◽  
The Antarctic

Abstract. In this paper we present a simple analytical model for low frequency and large scale variability of the Antarctic Circumpolar Current (ACC). The physical mechanism of the variability is related to temporal and spatial variations of the cyclonic mean flow (ACC) due to circularly propagating nonlinear barotropic Rossby wave trains. It is shown that the Rossby wave train is a fundamental mode, trapped between the major fronts in the ACC. The Rossby waves are predicted to rotate with a particular angular velocity that depends on the magnitude and width of the mean current. The spatial structure of the rotating pattern, including its zonal wave number, is defined by the specific form of the stream function-vorticity relation. The similarity between the simulated patterns and the Antarctic Circumpolar Wave (ACW) is highlighted. The model can predict the observed sequence of warm and cold patches in the ACW as well as its zonal number.

On three-dimensional internal gravity wave beams and induced large-scale mean flows

2015 ◽  
Vol 769 ◽  
pp. 621-634 ◽  
Author(s):  
T. Kataoka ◽  
T. R. Akylas
Keyword(s):  
Gravity Wave ◽  
Large Scale ◽  
Evolution Equations ◽  
Three Dimensional ◽  
Internal Gravity Wave ◽  
Nonlinear Effects ◽  
Beam Width ◽  
Finite Amplitude ◽  
Horizontal Line ◽  
Mean Flow

The three-dimensional propagation of internal gravity wave beams in a uniformly stratified Boussinesq fluid is discussed, assuming that variations in the along-beam and transverse directions are of long length scale relative to the beam width. This situation applies, for instance, to the far-field behaviour of a wave beam generated by a horizontal line source with weak transverse dependence. In contrast to the two-dimensional case of purely along-beam variations, where nonlinear effects are minor even for beams of finite amplitude, three-dimensional nonlinear interactions trigger the transfer of energy to a circulating horizontal time-mean flow. This resonant beam–mean-flow coupling is analysed, and a system of two evolution equations is derived for the propagation of a small-amplitude beam along with the induced mean flow. This model explains the salient features of the experimental observations of Bordes et al. (Phys. Fluids, vol. 24, 2012, 086602).

scholarly journals Numerical simulation of mountain waves over the southern Andes, Part 2: Momentum fluxes and wave/mean-flow interactions

2021 ◽  
Author(s):  
David C. Fritts ◽  
Thomas S. Lund ◽  
Kam Wan ◽  
Han-Li Liu
Keyword(s):  
Gravity Wave ◽  
Gravity Waves ◽  
Acoustic Waves ◽  
Large Scale ◽  
Mean Flow ◽  
Mountain Wave ◽  
Southern Andes ◽  
Mountain Waves ◽  
Momentum Fluxes ◽  
Flow Interactions

AbstractA companion paper by Lund et al. (2020) employed a compressible model to describe the evolution of mountain waves arising due to increasing flow with time over the Southern Andes, their breaking, secondary gravity waves and acoustic waves arising from these dynamics, and their local responses. This paper describes the mountain wave, secondary gravity wave, and acoustic wave vertical fluxes of horizontal momentum, and the local and large-scale three-dimensional responses to gravity breaking and wave/mean-flow interactions accompanying this event. Mountain wave and secondary gravity wave momentum fluxes and deposition vary strongly in space and time due to variable large-scale winds and spatially-localized mountain wave and secondary gravity wave responses. Mountain wave instabilities accompanying breaking induce strong, local, largely-zonal forcing. Secondary gravity waves arising from mountain wave breaking also interact strongly with large-scale winds at altitudes above ~80km. Together, these mountain wave and secondary gravity wave interactions reveal systematic gravity-wave/mean-flow interactions having implications for both mean and tidal forcing and feedbacks. Acoustic waves likewise achieve large momentum fluxes, but typically imply significant responses only at much higher altitudes.

Reducing Instrumentation Errors Caused by Circumferential Flow Field Variations in Multi-Stage Axial Compressors

2019 ◽  
Author(s):  
M. Chilla ◽  
G. Pullan ◽  
S. Gallimore
Keyword(s):  
Flow Field ◽  
Large Scale ◽  
Sampling Error ◽  
Leading Edge ◽  
Stagnation Pressure ◽  
Wave Number ◽  
Stagnation Temperature ◽  
Sampling Errors ◽  
Mean Flow ◽  
Axial Compressors

Abstract The effects of blade row interactions on stator-mounted instrumentation in axial compressors are investigated using unsteady numerical calculations. The test compressor is an 8-stage machine representative of an aero-engine core compressor. For the unsteady calculations, a 180deg sector (half-annulus) model of the compressor is used. It is shown that the time-mean flow field in the stator leading edge planes is circumferentially non-uniform. The circumferential variations in stagnation pressure and stagnation temperature respectively reach 4.2% and 1.1% of the local mean. Using spatial wave number analysis, the incoming wakes from the upstream stator rows are identified as the dominant source of the circumferential variations in the front and middle of the compressor, while towards the rear of the compressor, the upstream influence of the eight struts in the exit duct becomes dominant. Based on three circumferential probes, the sampling errors for stagnation pressure and stagnation temperature are calculated as a function of the probe locations. Optimization of the probe locations shows that the sampling error can be reduced by up to 77% by circumferentially redistributing the individual probes. The reductions in the sampling errors translate to reductions in the uncertainties of the overall compressor efficiency and inlet flow capacity by up to 50%. Recognizing that data from large-scale unsteady calculations is rarely available in the instrumentation phase for a new test rig or engine, a method for approximating the circumferential variations with single harmonics is presented. The construction of the harmonics is based solely on the knowledge of the number of stators in each row and a small number of equi-spaced probes. It is shown how excursions in the sampling error are reduced by increasing the number of circumferential probes.

Parabolized Stability Equation Based Analysis of Noise From an Axisymmetric Hot Jet

Volume 1 ◽  
2004 ◽  
Author(s):  
Ray-Sing Lin ◽  
Ramons A. Reba ◽  
Satish Narayanan ◽  
Nathan S. Hariharan ◽  
Fabio P. Bertolotti
Keyword(s):  
Large Scale ◽  
Near Field ◽  
Sound Field ◽  
Wave Number ◽  
Navier Stokes ◽  
Mean Flow ◽  
Noise Sources ◽  
Stability Equation ◽  
Low Frequencies ◽  
Potential Core

Noise generated from large-scale wave-like eddies in a Mach 0.9 hot axisymmetric jet is studied. The mean jet flow is computed using a Reynolds-Averaged Navier-Stokes solver with the k-ω turbulence model. Spatial development of near-field pressure perturbations is computed using a 3D Parabolized Stability Equation (PSE) method, and the far-field noise radiated from these convective instabilities is obtained by solving the wave equation. The 3D PSE method developed here allows the effects of strong azimuthal mean-flow variations to be captured and analyzed. Results show that large-scale wave-like eddies, or instability waves, travel slightly above sonic speed for the first few jet diameters, and that the dominant noise sources are concentrated near the edge of the time-averaged potential core. Good agreement between computed results and experiment are found for relative sound pressure levels and directivity. Also, at the low frequencies considered, the sound field associated with azimuthal wave-number m = 0 shows better agreement with experimental data than with (the most amplified) m = 1.

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