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.

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.

Ageostrophic dynamics of an intense localized vortex on a β-plane

2001 ◽  
Vol 443 ◽  
pp. 351-376 ◽  
Author(s):  
G. M. REZNIK ◽  
R. GRIMSHAW
Keyword(s):  
Gravity Waves ◽  
Potential Vorticity ◽  
Rapid Development ◽  
Water Model ◽  
Wave Radiation ◽  
Main Role ◽  
Fast Time ◽  
Translation Speed ◽  
Continuous Potential ◽  
Inertia Gravity Waves

We consider the non-stationary dynamics of an intense localized vortex on a β-plane using a shallow-water model. An asymptotic theory for a vortex with piecewise-continuous potential vorticity is developed assuming the Rossby number to be small and the free surface elevation to be small but finite. Analogously to the well-known quasi-geostrophic model, the vortex translation is produced by a secondary dipole circulation (β-gyres) developed in the vortex vicinity and consisting of two parts. The first part (geostrophic β-gyres) coincides with the β-gyres in the geostrophic model, and the second (ageostrophic β-gyres) is due to ageostrophic terms in the governing equations. The time evolution of the ageostrophic β-gyres consists of fast and slow stages. During the fast stage the radiation of inertia–gravity waves results in the rapid development of the β-gyres from zero to a dipole field independent of the fast time variable. Correspondingly, the vortex accelerates practically instantaneously (compared to the typical swirling time) to some finite value of the translation speed. At the next slow stage the inertia–gravity wave radiation is insignificant and the β-gyres evolve with the typical swirling time. The total zonal translation speed induced by the geostrophic and ageostrophic β-gyres tends with increasing time to the speed of a steadily translating monopole exceeding (not exceeding) the drift velocity of Rossby waves for anticyclones (cyclones). This cyclone/anticyclone asymmetry generalizes the well-known finding about the greater longevity of anticyclones compared to cyclones to the case of non-stationary evolving monopoles. The influence of inertia–gravity waves upon the vortex evolution is analysed. The main role of these waves is to provide a ‘fast’ adjustment to the ‘slow’ vortex evolution. The energy of inertia–gravity waves is negligible compare to the energy of the geostrophic β-gyres. Yet another feature of the ageostrophic vortex evolution is that the area of the potential vorticity patch changes in the course of time, the cyclonic patch contracting and the anticyclonic one expanding.

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.

An explicit potential-vorticity-conserving approach to modelling nonlinear internal gravity waves

2002 ◽  
Vol 458 ◽  
pp. 75-101 ◽  
Author(s):  
ÁLVARO VIÚDEZ ◽  
DAVID G. DRITSCHEL
Keyword(s):  
Gravity Waves ◽  
Potential Vorticity ◽  
Three Dimensional ◽  
Internal Gravity Wave ◽  
Numerical Approach ◽  
Internal Gravity Waves ◽  
Static Stability ◽  
Three Dimensions ◽  
Fluid Particles ◽  
Monge Ampere

This paper discusses a potential-vorticity-conserving approach to modelling nonlinear internal gravity waves in a rotating Boussinesq fluid. The focus of the work is on the pseudo-plane motion (motion in the x, z-plane), for which we present a broad range of numerical results. In this case there are two material coordinates, the density and the y-component of the velocity in the inertial frame of reference, which are related to the x and z displacements of fluid particles relative to a reference configuration. The amount of potential vorticity within a fluid region bounded by isosurfaces of these material coordinates is proportional to the area within this region, and is therefore conserved as well. Two new potentials, defined in terms of the displacements and combining the vorticity and density fields, are introduced as new dependent variables. These potentials entirely govern the dynamics of internal gravity waves for the linearized system when the basic state has uniform potential vorticity. The final system of equations consists of three prognostic equations (for the potential vorticity and the Laplacians of the two potentials) and one diagnostic equation, of Monge–Ampère type, for a third potential. This diagnostic equation arises from the nonlinear definition of potential vorticity. The ellipticity of the Monge–Ampère equation implies both inertial and static stability. In three dimensions, the three potentials form a vector, whose (three-dimensional) Laplacian is equal to the vorticity plus the gradient of the perturbation density.Numerical simulations are carried out using a novel algorithm which directly evolves the potential vorticity, in a Lagrangian manner (following fluid particles), without diffusion. We present results which emphasize the way in which potential vorticity anomalies modify the characteristics of internal gravity waves, e.g. the propagation of internal wave packets, including reflection, refraction, and amplification. We also show how potential vorticity anomalies may generate internal gravity waves, along with the subsequent ‘geostrophic adjustment’ of the flow to a ‘balanced’ wave-less state. These examples, and the straightforward extension of the theoretical and numerical approach to three dimensions, point to a direct and accurate means to elucidate the role of potential vorticity in internal gravity wave interactions. As such, this approach may help a better understanding of the observed characteristics of internal gravity waves in the oceans.

Damping of inertial motions by parametric subharmonic instability in baroclinic currents

2014 ◽  
Vol 743 ◽  
pp. 280-294 ◽  
Author(s):  
Leif N. Thomas ◽  
John R. Taylor
Keyword(s):  
Kinetic Energy ◽  
Numerical Simulations ◽  
Gravity Waves ◽  
Potential Vorticity ◽  
Wind Shear ◽  
Finite Amplitude ◽  
Instability Criterion ◽  
Thermal Wind ◽  
Parametric Subharmonic Instability ◽  
Inertia Gravity Waves

AbstractA new damping mechanism for vertically-sheared inertial motions is described involving an inertia–gravity wave that oscillates at half the inertial frequency, $f$, and that grows at the expense of inertial shear. This parametric subharmonic instability forms in baroclinic, geostrophic currents where thermal wind shear, by reducing the potential vorticity of the fluid, allows inertia–gravity waves with frequencies less than $f$. A stability analysis and numerical simulations are used to study the instability criterion, energetics, and finite-amplitude behaviour of the instability. For a flow with uniform shear and stratification, parametric subharmonic instability develops when the Richardson number of the geostrophic current nears $Ri_{PSI}=4/3+\gamma \cos \phi $, where $\gamma $ is the ratio of the inertial to thermal wind shear magnitude and $\phi $ is the angle between the inertial and thermal wind shears at the initial time. Inertial shear enters the instability criterion because it can also modify the potential vorticity and hence the minimum frequency of inertia–gravity waves. When this criterion is met, inertia–gravity waves with a frequency $f/2$ and with flow parallel to isopycnals amplify, extracting kinetic energy from the inertial shear through shear production. The solutions of the numerical simulations are consistent with these predictions and additionally show that finite-amplitude parametric subharmonic instability both damps inertial shear and is itself damped by secondary shear instabilities. In this way, parametric subharmonic instability opens a pathway to turbulence where kinetic energy in inertial shear is transferred to small scales and dissipated.

scholarly journals Temperature Dependence of Density and Viscosity of Biobutanol-Gasoline Blends

2021 ◽  
Vol 11 (7) ◽  
pp. 3172
Author(s):  
Daniel Trost ◽  
Adam Polcar ◽  
Dorin Boldor ◽  
Divine Bup Nde ◽  
Artur Wolak ◽  
...  
Keyword(s):  
Octane Number ◽  
Mass Density ◽  
Three Dimensional ◽  
Coefficient Of Determination ◽  
Multivariate Models ◽  
Research Octane Number ◽  
Practical Applications ◽  
Negative Temperatures ◽  
Local Sources ◽  
Fluid Properties

Butanol seems to be an eligible fuel for compensating for the increasing fuel consumption. Biobutanol could be produced from local sources in the place of use. Its properties show similar results to gasoline, so biobutanol could be added as a biocomponent into fuels. Important properties, in the case of blending biobutanol into gasoline, are its fluid properties and their dependence on the temperature. Therefore, in this paper, the volumetric mass density and viscosity of the selected ratios between biobutanol and gasoline (0, 5, 10, 85, 100 vol.%) were tested over the temperature range from −10 °C up to 40 °C. Gasolines with a 95 Research Octane Number (RON 95) and with a 98 Research Octane Number (RON 98) were used. It was observed that as the temperature increased, the viscosity and volumetric mass density of the samples decreased nonlinearly. Four mathematical models were used for modelling the viscosity. The accuracy of models was evaluated and compared according to the coefficient of determination R2 and sum of squared estimate of errors (SSE). The results show that blends with 5 vol.% and 10 vol.% of biobutanol promise very similar fluid properties to pure gasoline. In contrast, a blend with 85 vol.% of biobutanol shows different fluid properties from gasoline, especially in negative temperatures, a lot. For practical applications, mathematical polynomial multivariate models were created. Using these models, three-dimensional graphs were constructed.

scholarly journals Diffusion of inertia-gravity waves by geostrophic turbulence

2019 ◽  
Vol 869 ◽  
Author(s):  
Hossein A. Kafiabad ◽  
Miles A. C. Savva ◽  
Jacques Vanneste
Keyword(s):  
Energy Spectrum ◽  
Gravity Waves ◽  
Wave Energy ◽  
Large Scale ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Constant Frequency ◽  
Geostrophic Turbulence ◽  
Wave Energy Spectrum ◽  
Inertia Gravity Waves

The scattering of inertia-gravity waves by large-scale geostrophic turbulence in a rapidly rotating, strongly stratified fluid leads to the diffusion of wave energy on the constant-frequency cone in wavenumber space. We derive the corresponding diffusion equation and relate its diffusivity to the wave characteristics and the energy spectrum of the turbulent flow. We check the predictions of this equation against numerical simulations of the three-dimensional Boussinesq equations in initial-value and forced scenarios with horizontally isotropic wave and flow fields. In the forced case, wavenumber diffusion results in a $k^{-2}$ wave energy spectrum consistent with as-yet-unexplained features of observed atmospheric and oceanic spectra.

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