A Barotropic Model of the Angular Momentum–Conserving Potential Vorticity Staircase in Spherical Geometry

2008 ◽  
Vol 65 (4) ◽  
pp. 1105-1136 ◽  
Author(s):  
Timothy J. Dunkerton ◽  
Richard K. Scott
Keyword(s):  
Angular Momentum ◽  
Asymptotic Analysis ◽  
Potential Vorticity ◽  
Spherical Geometry ◽  
Small Scale ◽  
Mean Flow ◽  
Conservation Of Energy ◽  
Barotropic Model ◽  
Global Conservation ◽  
Westerly Jets

Abstract An idealized analytical model of the barotropic potential vorticity (PV) staircase is constructed, constrained by global conservation of absolute angular momentum, perfect homogenization of PV in mixing zones between (prograde) westerly jets, and the requirement of barotropic stability. An imposed functional relationship is also assumed between jet speed and latitudinal separation using a multiple of the “dynamical Rossby wave” Rhines scale inferred from the strength of westerly jets. The relative simplicity of the barotropic system provides a simple relation between absolute angular momentum and PV (or absolute vorticity). A family of solutions comprising an arbitrary number of jets is constructed and is used to illustrate the restriction of jet spacing and strength imposed by the constraints of global conservation of angular momentum and barotropic stability. Asymptotic analysis of the theoretical solution indicates a limiting ratio of jet spacing to the dynamical Rhines scale equal to the square root of 6, meaning that westerly jets are spaced farther apart than predicted by the dynamical Rhines scale. It is inferred that an alternative “geometrical” Rhines scale for jet spacing can be obtained from conservation of absolute angular momentum on the sphere if the strength of zonal jets is known from other considerations. Numerical simulations of the full (nonaxisymmetric) equations reveal a pattern of zonal jet evolution that is consistent with our construction of ideal PV staircases in spherical geometry (which can be considered as limiting cases), as well as with the asymptotic analysis of a geometrical Rhines scale. The evolution of the PV staircase originating from an upscale cascade of energy in the barotropic model is therefore seen to depend on conservation of energy (for the strength of jets) and conservation of absolute angular momentum (for the spacing and number of jets). Further analysis of the numerical results confirms a “Taylor identity” relating the flux of eddy potential vorticity to mean-flow acceleration. Eddy fluxes are responsible for the occasional transitions between mode number as well as for maintaining the sharp westerly jets against small-scale dissipation. Suggestions are made for extending the theoretical model to PV staircases that are asymmetric between hemispheres or with latitudinal variation of amplitude, as modeled in the shallow-water system.

scholarly journals A Framework for Parameterizing Eddy Potential Vorticity Fluxes

2012 ◽  
Vol 42 (4) ◽  
pp. 539-557 ◽  
Author(s):  
David P. Marshall ◽  
James R. Maddison ◽  
Pavel S. Berloff
Keyword(s):  
Stress Tensor ◽  
Potential Vorticity ◽  
Large Scale ◽  
Eddy Diffusivity ◽  
Linear Instability ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Conservation Of Energy ◽  
The Mean ◽  
Energy And Momentum

Abstract A framework for parameterizing eddy potential vorticity fluxes is developed that is consistent with conservation of energy and momentum while retaining the symmetries of the original eddy flux. The framework involves rewriting the residual-mean eddy force, or equivalently the eddy potential vorticity flux, as the divergence of an eddy stress tensor. A norm of this tensor is bounded by the eddy energy, allowing the components of the stress tensor to be rewritten in terms of the eddy energy and nondimensional parameters describing the mean shape and orientation of the eddies. If a prognostic equation is solved for the eddy energy, the remaining unknowns are nondimensional and bounded in magnitude by unity. Moreover, these nondimensional geometric parameters have strong connections with classical stability theory. When applied to the Eady problem, it is shown that the new framework preserves the functional form of the Eady growth rate for linear instability. Moreover, in the limit in which Reynolds stresses are neglected, the framework reduces to a Gent and McWilliams type of eddy closure where the eddy diffusivity can be interpreted as the form proposed by Visbeck et al. Simulations of three-layer wind-driven gyres are used to diagnose the eddy shape and orientations in fully developed geostrophic turbulence. These fields are found to have large-scale structure that appears related to the structure of the mean flow. The eddy energy sets the magnitude of the eddy stress tensor and hence the eddy potential vorticity fluxes. Possible extensions of the framework to ensure potential vorticity is mixed on average are discussed.

Variations on a beta-plane: derivation of non-traditional beta-plane equations from Hamilton's principle on a sphere

2011 ◽  
Vol 674 ◽  
pp. 174-195 ◽  
Author(s):  
PAUL J. DELLAR
Keyword(s):  
Angular Momentum ◽  
Potential Vorticity ◽  
Spherical Geometry ◽  
Rotation Vector ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Rotating Sphere ◽  
Averaged Equations ◽  
Beta Plane ◽  
Cartesian Geometry

Starting from Hamilton's principle on a rotating sphere, we derive a series of successively more accurate β-plane approximations. These are Cartesian approximations to motion in spherical geometry that capture the change with latitude of the angle between the rotation vector and the local vertical. Being derived using Hamilton's principle, the different β-plane approximations each conserve energy, angular momentum and potential vorticity. They differ in their treatments of the locally horizontal component of the rotation vector, the component that is usually neglected under the traditional approximation. In particular, we derive an extended set of β-plane equations in which the locally vertical and locally horizontal components of the rotation vector both vary linearly with latitude. This was previously thought to violate conservation of angular momentum and potential vorticity. We show that the difficulty in maintaining these conservation laws arises from the need to express the rotation vector as the curl of a vector potential while approximating the true spherical metric by a flat Cartesian metric. Finally, we derive depth-averaged equations on our extended β-plane with topography, and show that they coincide with the extended non-traditional shallow-water equations previously derived in Cartesian geometry.

Inertial nonlinear equilibration of equatorial flows

1997 ◽  
Vol 331 ◽  
pp. 345-371 ◽  
Author(s):  
BACH LIEN HUA ◽  
DENNIS W. MOORE ◽  
SYLVIE LE GENTIL
Keyword(s):  
Angular Momentum ◽  
Steady State ◽  
Potential Vorticity ◽  
Normal Forms ◽  
Vertical Shear ◽  
Small Scale ◽  
Usual Condition ◽  
Zonal Flows ◽  
Taylor Flow ◽  
Nonlinear Normal

We explore the nature of inertial equilibration of equatorial flows in the presence of mean meridional and vertical shears of the basic state, with oceanic applications in mind. The study is motivated by the observational evidence that the subthermocline equatorial mean circulation displays nearly zero Ertel potential vorticity away from the equator, when taking into account the non-traditional horizontal component of the Earth rotation. This observed state precisely verifies the marginal condition for inertial instability: a linear analysis for the equatorial β-plane confirms that the usual condition of instability, namely that Ertel potential vorticity should be of opposite sign to the vertical Coriolis parameter, remains valid even when the traditional approximation is relaxed. Analytical linear normal modes reveal that a meridional shear of the basic state leads to a vertical stacking of equatorially-trapped zonal flows of alternate signs, with a new centre of symmetry located at the dynamical equator. A vertical shear of the basic state causes a meridional stacking of extra-equatorial zonal flows.In an inviscid framework, a two-dimensional formulation is ill-posed and we resort to non-hydrostatic viscous simulations to determine the nonlinear normal forms of the system. The influence of a small-scale eddy diffusivity and a large-scale Rayleigh damping on the equilibrated vertical scale is determined numerically. The nonlinear equilibration occurs through a steady-state bifurcation from a basic state without jets to another steady state with secondary jets of alternate signs. The final state corresponds to eastward jets located on the geographic equator, while westward jets are located near the dynamical equator. These results are consistent with in situ observations of equatorial deep jets.The analogy between the equatorial meridional shear flow and the cylindrical Couette–Taylor flow with an axial density stratification is detailed. There is a strong similarity in the general symmetries and nonlinear normal forms of the two problems. Similarly to the homogeneous Couette–Taylor flow, the gap width between the two cylinders is important for determining the axial scale of the secondary flow through the Reynolds number. For the equatorial problem, an upper bound for the height scale of inertial jets is such that the corresponding equatorial radius of deformation times √2 fits between the geographic and dynamic equators.One of our main conclusions is that the raisond’être of the observed region of zero Ertel potential vorticity is to facilitate angular momentum exchanges between the two hemispheres and inertial deep jets are the byproducts of this angular momentum mixing.

scholarly journals Circum-nuclear molecular disks: Role in AGN fueling and feedback

2019 ◽  
Vol 15 (S359) ◽  
pp. 312-317
Author(s):  
Francoise Combes
Keyword(s):  
Black Hole ◽  
Angular Momentum ◽  
High Resolution ◽  
Large Scale ◽  
Active Galaxy ◽  
Small Scale ◽  
Massive Black Hole ◽  
Molecular Outflows ◽  
Molecular Outflow

AbstractGas fueling AGN (Active Galaxy Nuclei) is now traceable at high-resolution with ALMA (Atacama Large Millimeter Array) and NOEMA (NOrthern Extended Millimeter Array). Dynamical mechanisms are essential to exchange angular momentum and drive the gas to the super-massive black hole. While at 100pc scale, the gas is sometimes stalled in nuclear rings, recent observations reaching 10pc scale (50mas), may bring smoking gun evidence of fueling, within a randomly oriented nuclear gas disk. AGN feedback is also observed, in the form of narrow and collimated molecular outflows, which point towards the radio mode, or entrainment by a radio jet. Precession has been observed in a molecular outflow, indicating the precession of the radio jet. One of the best candidates for precession is the Bardeen-Petterson effect at small scale, which exerts a torque on the accreting material, and produces an extended disk warp. The misalignment between the inner and large-scale disk, enhances the coupling of the AGN feedback, since the jet sweeps a large part of the molecular disk.

Turbulent flow between counter-rotating concentric cylinders: a direct numerical simulation study

2008 ◽  
Vol 615 ◽  
pp. 371-399 ◽  
Author(s):  
S. DONG
Keyword(s):  
Turbulent Flow ◽  
Large Scale ◽  
Outer Cylinder ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Taylor Vortex ◽  
Small Scale ◽  
Mean Flow ◽  
Concentric Cylinders ◽  
High Reynolds

We report three-dimensional direct numerical simulations of the turbulent flow between counter-rotating concentric cylinders with a radius ratio 0.5. The inner- and outer-cylinder Reynolds numbers have the same magnitude, which ranges from 500 to 4000 in the simulations. We show that with the increase of Reynolds number, the prevailing structures in the flow are azimuthal vortices with scales much smaller than the cylinder gap. At high Reynolds numbers, while the instantaneous small-scale vortices permeate the entire domain, the large-scale Taylor vortex motions manifested by the time-averaged field do not penetrate a layer of fluid near the outer cylinder. Comparisons between the standard Taylor–Couette system (rotating inner cylinder, fixed outer cylinder) and the counter-rotating system demonstrate the profound effects of the Coriolis force on the mean flow and other statistical quantities. The dynamical and statistical features of the flow have been investigated in detail.

scholarly journals Jet Formation and Evolution in Baroclinic Turbulence with Simple Topography

2010 ◽  
Vol 40 (2) ◽  
pp. 257-278 ◽  
Author(s):  
Andrew F. Thompson
Keyword(s):  
Southern Ocean ◽  
Potential Vorticity ◽  
Baroclinic Instability ◽  
Length Scale ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Jet Formation ◽  
Meridional Transport ◽  
Jet Behavior ◽  
Jet Variability

Abstract Satellite altimetry and high-resolution ocean models indicate that the Southern Ocean comprises an intricate web of narrow, meandering jets that undergo spontaneous formation, merger, and splitting events, as well as rapid latitude shifts over periods of weeks to months. The role of topography in controlling jet variability is explored using over 100 simulations from a doubly periodic, forced-dissipative, two-layer quasigeostrophic model. The system is forced by a baroclinically unstable, vertically sheared mean flow in a domain that is large enough to accommodate multiple jets. The dependence of (i) meridional jet spacing, (ii) jet variability, and (iii) domain-averaged meridional transport on changes in the length scale and steepness of simple sinusoidal topographical features is analyzed. The Rhines scale, ℓβ = 2πVe/β, where Ve is an eddy velocity scale and β is the barotropic potential vorticity gradient, measures the meridional extent of eddy mixing by a single jet. The ratio ℓβ /ℓT, where ℓT is the topographic length scale, governs jet behavior. Multiple, steady jets with fixed meridional spacing are observed when ℓβ ≫ ℓT or when ℓβ ≈ ℓT. When ℓβ < ℓT, a pattern of perpetual jet formation and jet merger dominates the time evolution of the system. Zonal ridges systematically reduce the domain-averaged meridional transport, while two-dimensional, sinusoidal bumps can increase transport by an order of magnitude or more. For certain parameters, bumpy topography gives rise to periodic oscillations in the jet structure between purely zonal and topographically steered states. In these cases, transport is dominated by bursts of mixing associated with the transition between the two regimes. Topography modifies local potential vorticity (PV) gradients and mean flows; this can generate asymmetric Reynolds stresses about the jet core and can feed back on the conversion of potential energy to kinetic energy through baroclinic instability. Both processes contribute to unsteady jet behavior. It is likely that these processes play a role in the dynamic nature of Southern Ocean jets.

scholarly journals Dynamical and Physical Processes Leading to Tropical Cyclone Intensification under Upper-Level Trough Forcing

2013 ◽  
Vol 70 (8) ◽  
pp. 2547-2565 ◽  
Author(s):  
Marie-Dominique Leroux ◽  
Matthieu Plu ◽  
David Barbary ◽  
Frank Roux ◽  
Philippe Arbogast
Keyword(s):  
Angular Momentum ◽  
Tropical Cyclone ◽  
Potential Vorticity ◽  
Weather Prediction ◽  
Three Dimensional ◽  
Vertical Wind Shear ◽  
Thick Layer ◽  
Limited Area ◽  
Southwest Indian Ocean ◽  
Upper Level

Abstract The rapid intensification of Tropical Cyclone (TC) Dora (2007, southwest Indian Ocean) under upper-level trough forcing is investigated. TC–trough interaction is simulated using a limited-area operational numerical weather prediction model. The interaction between the storm and the trough involves a coupled evolution of vertical wind shear and binary vortex interaction in the horizontal and vertical dimensions. The three-dimensional potential vorticity structure associated with the trough undergoes strong deformation as it approaches the storm. Potential vorticity (PV) is advected toward the tropical cyclone core over a thick layer from 200 to 500 hPa while the TC upper-level flow turns cyclonic from the continuous import of angular momentum. It is found that vortex intensification first occurs inside the eyewall and results from PV superposition in the thick aforementioned layer. The main pathway to further storm intensification is associated with secondary eyewall formation triggered by external forcing. Eddy angular momentum convergence and eddy PV fluxes are responsible for spinning up an outer eyewall over the entire troposphere, while spindown is observed within the primary eyewall. The 8-km-resolution model is able to reproduce the main features of the eyewall replacement cycle observed for TC Dora. The outer eyewall intensifies further through mean vertical advection under dynamically forced upward motion. The processes are illustrated and quantified using various diagnostics.

Eddy-mediated Hadley cell expansion due to axisymmetric angular momentum adjustment to greenhouse gas forcings

2021 ◽  
Author(s):  
Nicholas A. Davis ◽  
Thomas Birner
Keyword(s):  
Angular Momentum ◽  
Greenhouse Gas ◽  
Cell Expansion ◽  
Energy Transport ◽  
Baroclinic Instability ◽  
Temperature Response ◽  
Equilibrium Temperature ◽  
Hadley Cell ◽  
Mean Flow ◽  
Fully Coupled

AbstractThe poleward expansion of the Hadley cells is one of the most robust modeled responses to increasing greenhouse gas concentrations. There are many proposed mechanisms for expansion, and most are consistent with modeled changes in thermodynamics, dynamics, and clouds. The adjustment of the eddies and the mean flow to greenhouse gas forcings, and to one another, complicates any effort toward a deeper understanding. Here we modify the Gray Radiation AND Moist Aquaplanet (GRANDMA) model to uncouple the eddy and mean flow responses to forcings. When eddy forcings are held constant, the purely axisymmetric response of the Hadley cell to a greenhouse gas-like forcing is an intensification and poleward tilting of the cell with height in response to an axisymmetric increase in angular momentum in the subtropics. The angular momentum increase drastically alters the circulation response compared to axisymmetric theories, which by nature neglect this adjustment. Model simulations and an eddy diffusivity framework demonstrate that the axisymmetric increase in subtropical angular momentum – the direct manifestation of the radiative-convective equilibrium temperature response – drives a poleward shift of the eddy stresses which leads to Hadley cell expansion. Prescribing the eddy response to the greenhouse gas-like forcing shows that eddies damp, rather than drive, changes in angular momentum, moist static energy transport, and momentum transport. Expansion is not driven by changes in baroclinic instability, as would otherwise be diagnosed from the fully-coupled simulation. These modeling results caution any assessment of mechanisms for circulation change within the fully-coupled wave-mean flow system.

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>

Extraction Techniques and Analysis of Turbulence Quantities From In-Cylinder Velocity Data

1989 ◽  
Vol 111 (3) ◽  
pp. 466-478 ◽  
Author(s):  
A. E. Catania ◽  
A. Mittica
Keyword(s):  
Direct Injection ◽  
Detailed Comparison ◽  
Ensemble Average ◽  
Small Scale ◽  
Velocity Data ◽  
Mean Flow ◽  
Reciprocating Engine ◽  
Reduction Methods ◽  
The Mean ◽  
Correlation And Spectral Analysis

In addition to the frequently used statistical ensemble-average, non-Reynolds filtering operators have long been proposed for nonstationary turbulent quantities. Several techniques for the reduction of velocity data acquired in the cylinder of internal combustion reciprocating engines have been developed by various researchers in order to separate the “mean flow” from the “fluctuating motion,” cycle by cycle, and to analyze small-scale engine turbulence by statistical methods. Therefore a thorough examination of these techniques and a detailed comparison between them would seem to be a preliminary step in attempting a general study of unconventional averaging procedures for reciprocating engine flow application. To that end, in the present work, five different cycle-resolved data reduction methods and the conventional ensemble-average were applied to the same in-cylinder velocity data, so as to review and compare them. One of the methods was developed by the authors. The data were acquired in the cylinder of a direct-injection automotive diesel engine, during induction and compression strokes, using an advanced hot-wire anemometry technique. Correlation and spectral analysis of the engine turbulence, as determined from the data with the different procedures, were also performed.

Sign in / Sign up

Export Citation Format

Share Document