Geostrophic adjustment in a closed basin with islands

AbstractWe consider the geostrophic adjustment of a density-stratified fluid in a basin of constant depth on an $f$-plane in the context of linearized theory. For a single vertical mode, the equations are equivalent to those for a linearized shallow-water theory for a homogeneous fluid. Associated with any initial state there is a unique steady geostrophically adjusted component of the flow compatible with the initial conditions. This steady component gives the time average of the flow and is analogous to the adjusted flow in an unbounded domain without islands. The remainder of the response consists of superinertial Poincaré and subinertial Kelvin wave modes and expressions for the energy partition between the modes in arbitrary basins again follow directly from the initial conditions. The solution for an arbitrary initial density distribution released from rest in a circular domain is found in closed form. When the Rossby radius is much smaller than the basin radius, appropriate for the baroclinic modes, the interior adjusted solution is close to that of the initial state, except for small-amplitude trapped Poincaré waves, while Kelvin waves propagate around the boundaries, carrying, without change of form, the deviation of the initial height field from its average.

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The Evolution of Young Supernova Remnants

Symposium - International Astronomical Union ◽

10.1017/s0074180900033751 ◽

1983 ◽

Vol 101 ◽

pp. 119-124

Author(s):

A. C. Fabian ◽

W. Brinkmann ◽

G. C. Stewart

Keyword(s):

Density Distribution ◽

Stellar Wind ◽

Supernova Remnants ◽

Initial Conditions ◽

Numerical Models ◽

Variable Mass ◽

Initial Density ◽

X Ray ◽

Cassiopeia A ◽

Initial Density Distribution

Einstein X-ray observations of the young supernova remnants Cassiopeia A (Murray et al. 1980) and Tycho (Seward, Gorenstein and Tucker 1982) indicate that the swept-up mass does not much exceed that of the observed ejecta. The initial density distribution of the ejecta and surrounding material is then important in determining the X-ray structure and evolution. Some aspects of this behaviour have been dealt with in previous numerical (e.g. Gull 1973; Itoh 1977; Jones, Smith and Straka 1981) and analytical (e.g. Chevalier 1982a,b) studies. We present here results obtained from numerical models covering a wider range of initial conditions. In particular, we consider the effect of a constant stellar wind from the progenitor star on the expansion of the remnant. We have previously suggested that variable mass loss from SN1006 may explain its warm filled interior (Fabian, Stewart and Brinkmann 1982).

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Nonlinear geostrophic adjustment in the presence of a boundary

Journal of Fluid Mechanics ◽

10.1017/s0022112002002148 ◽

2002 ◽

Vol 471 ◽

pp. 257-283 ◽

Cited By ~ 14

Author(s):

G. M. REZNIK ◽

R. GRIMSHAW

Keyword(s):

Time Scales ◽

Potential Vorticity ◽

Total Mass ◽

Initial Conditions ◽

Slow Component ◽

Slow Motion ◽

Wave Profile ◽

Kelvin Waves ◽

Kelvin Wave ◽

Geostrophic Adjustment

The process of nonlinear geostrophic adjustment in the presence of a boundary (i.e. in a half-plane bounded by a rigid wall) is examined in the framework of a rotating shallow water model, using an asymptotic multiple-time-scale theory based on the assumed smallness of the Rossby number ε. The spatial scale is of the order of the Rossby scale. Different initial states are considered: periodic, ‘step’-like, and localized. In all cases the initial perturbation is split in a unique way into slow and fast components evolving with characteristic time scales f−1 and (εf)−1, respectively. The slow component is not influenced by the fast one, at least for times t [les ] (fε)−1, and remains close to geostrophic balance. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance and Kelvin waves confined near the boundary.The theory provides simple formulae allowing us to construct the initial profile of the Kelvin wave, given arbitrary initial conditions. With increasing time, the Kelvin wave profile gradually distorts due to nonlinear-wave self-interaction, the distortion being described by the equation of a simple wave. The presence of Kelvin waves does not prevent the fast–slow splitting, in spite of the fact that the frequency gap between the Kelvin waves and slow motion is absent. The possibility of such splitting is explained by the special structure of the Kelvin waves in each case considered.The slow motion on time scales t [les ] (εf)−1 is governed by the well-known quasigeostrophic potential vorticity equation for the elevation. The theory provides an algorithm to determine initial slow and fast fields, and the boundary conditions to any order in ε. For the periodic and step-like initial conditions, the slow component behaves in the usual way, conserving mass, energy and enstrophy. In the case of a localized initial disturbance the total mass of the lowest-order slow component is not conserved, and conservation of the total mass is provided by the first-order slow correction and the Kelvin wave.On longer time scales t [les ] (ε2f)−1 the slow motion obeys the so-called modified quasi-geostrophic potential vorticity (QGPV) equation. The theory provides initial and boundary conditions for this equation. This modified equation coincides exactly with the ‘improved’ QGPV equation, derived by Reznik, Zeitlin & Ben Jelloul (2001), in the step-like and localized cases. In the periodic case this equation contains an additional term due to the Kelvin-wave self-interaction, this term depending on the initial Kelvin wave profile.

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On the Geostrophic Adjustment of an Isolated Lens: Dependence on Burger Number and Initial Geometry*

Journal of Physical Oceanography ◽

10.1175/2010jpo4476.1 ◽

2011 ◽

Vol 41 (4) ◽

pp. 725-741 ◽

Cited By ~ 7

Author(s):

Grant A. Stuart ◽

Miles A. Sundermeyer ◽

Dave Hebert

Keyword(s):

Numerical Simulations ◽

Analytical Solutions ◽

Laboratory Experiments ◽

Initial Conditions ◽

Initial Density ◽

Initial Pressure ◽

Length Scale ◽

Geostrophic Adjustment ◽

Length Scales ◽

Horizontal Length

Abstract Geostrophic adjustment of an isolated axisymmetric lens was examined to better understand the dependence of radial displacements and the adjusted velocity on the Burger number and the geometry of initial conditions. The behavior of the adjustment was examined using laboratory experiments and numerical simulations, which were in turn compared to published analytical solutions. Three defining length scales of the initial conditions were used to distinguish between various asymptotic behaviors for large and small Burger numbers: the Rossby radius of deformation, the horizontal length scale of the initial density defect, and the horizontal length scale of the initial pressure gradient. Numerical simulations for the fully nonlinear time-dependent adjustment agreed both qualitatively and quantitatively with analogous analytical solutions. For large Burger numbers, similar agreement was found in laboratory experiments. Results show that a broad range of final states can result from different initial geometries, depending on the values of the relevant length scales and the Burger number computed from initial conditions. For Burger numbers much larger or smaller than unity, differences between different initial geometries can readily exceed an order of magnitude for both displacement and velocity.

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Handling Initial Conditions in Vector Fitting for Real Time Modeling of Power System Dynamics

Energies ◽

10.3390/en14092471 ◽

2021 ◽

Vol 14 (9) ◽

pp. 2471

Author(s):

Tommaso Bradde ◽

Samuel Chevalier ◽

Marco De Stefano ◽

Stefano Grivet-Talocia ◽

Luca Daniel

Keyword(s):

Dynamical Systems ◽

Power System ◽

System Dynamics ◽

Real Time ◽

Initial Conditions ◽

Test System ◽

Initial State ◽

Power System Dynamics ◽

Vector Fitting ◽

Time Modeling

This paper develops a predictive modeling algorithm, denoted as Real-Time Vector Fitting (RTVF), which is capable of approximating the real-time linearized dynamics of multi-input multi-output (MIMO) dynamical systems via rational transfer function matrices. Based on a generalization of the well-known Time-Domain Vector Fitting (TDVF) algorithm, RTVF is suitable for online modeling of dynamical systems which experience both initial-state decay contributions in the measured output signals and concurrently active input signals. These adaptations were specifically contrived to meet the needs currently present in the electrical power systems community, where real-time modeling of low frequency power system dynamics is becoming an increasingly coveted tool by power system operators. After introducing and validating the RTVF scheme on synthetic test cases, this paper presents a series of numerical tests on high-order closed-loop generator systems in the IEEE 39-bus test system.

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EFFECTS OF INITIAL INHOMOGENEOUS DENSITY DISTRIBUTION AND VOID SHAPE ON POWDER FORGING OF POROUS METAL

Modern Physics Letters B ◽

10.1142/s021798491341008x ◽

2013 ◽

Vol 27 (19) ◽

pp. 1341008

Author(s):

TAIQING DENG ◽

LIANXI HU ◽

YU SUN ◽

XIAOYA LIU

Keyword(s):

Density Distribution ◽

Initial Density ◽

Porous Metal ◽

Major Axis ◽

Average Density ◽

Semi Major Axis ◽

Punch Force ◽

Inhomogeneous Density ◽

Void Shape ◽

Initial Density Distribution

The deformation behavior during axisymmetric upsetting of sintered metals has been studied based on the finite-element method. The investigation on the effects of the initial density distribution, void shape and die friction on the density distribution and punch force during deformation have been conducted. It was found that under low-friction conditions, the initial density distribution affects the deformation geometry and the density distribution. However, the effect of the initial density distribution was found to be negligible under high-friction conditions. The initial density distribution did not affect the punch force or the average density, regardless of the friction conditions. When the force is perpendicular to semi-major axis of elliptical void, it is not only good for densification but also decrease the punch force in forging of porous metal.

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Evolvement law of a macroscopic traffic model accounting for density-dependent relaxation time

Modern Physics Letters B ◽

10.1142/s0217984917502918 ◽

2017 ◽

Vol 31 (31) ◽

pp. 1750291 ◽

Cited By ~ 19

Author(s):

Yu-Qing Wang ◽

Xing-Jian Chu ◽

Chao-Fan Zhou ◽

Bin Jia ◽

Sen Lin ◽

...

Keyword(s):

Relaxation Time ◽

Traffic Flow ◽

Flow Model ◽

Initial Density ◽

Traffic Model ◽

Traffic Flow Model ◽

Density Dependent ◽

Macroscopic Traffic Flow ◽

Boundary Perturbations ◽

Initial Density Distribution

In this paper, a modified macroscopic traffic flow model is presented. The term of the density-dependent relaxation time is introduced here. The relation between the relaxation time and the density in traffic flow is presented quantitatively. Besides, a factor R depicting varied properties of traffic flow in different traffic states is also introduced in the formulation of the model. Furthermore, the evolvement law of traffic flow with distinctly initial density distribution and boundary perturbations is emphasized.

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Spectral analysis of forced turbulence in a non-neutral plasma

Journal of Plasma Physics ◽

10.1017/s0022377817000423 ◽

2017 ◽

Vol 83 (3) ◽

Cited By ~ 3

Author(s):

S. Chen ◽

G. Maero ◽

M. Romé

Keyword(s):

Initial Conditions ◽

Initial Density ◽

Final States ◽

Self Similarity ◽

Subsequent Formation ◽

Electron Plasmas ◽

Particle In Cell ◽

Fluid Turbulence ◽

Inviscid Incompressible Fluid ◽

Forced Turbulence

The paper investigates the dynamics of magnetized non-neutral (electron) plasmas subjected to external electric field perturbations. A two-dimensional (2-D) particle-in-cell code is effectively exploited to model this system with a special attention to the role that non-axisymmetric, multipolar radio frequency (RF) drives applied to the cylindrical (circular) boundary play on the insurgence of azimuthal instabilities and the subsequent formation of coherent structures preventing the relaxation to a fully developed turbulent state, when the RF fields are chosen in the frequency range of the low-order fluid modes themselves. The isomorphism of such system with a 2-D inviscid incompressible fluid offers an insight into the details of forced 2-D fluid turbulence. The choice of different initial density (i.e. fluid vorticity) distributions allows for a selection of conditions where different levels of turbulence and intermittency are expected and a range of final states is achieved. Integral and spectral quantities of interest are computed along the flow using a multiresolution analysis based on a wavelet decomposition of both enstrophy and energy 2-D maps. The analysis of a variety of cases shows that the qualitative features of turbulent relaxation are similar in conditions of both free and forced evolution; at the same time, fine details of the flow beyond the self-similarity turbulence properties are highlighted in particular in the formation of structures and their timing, where the influence of the initial conditions and the effect of the external forcing can be distinguished.

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Waves and turbulence on a beta-plane

Journal of Fluid Mechanics ◽

10.1017/s0022112075001504 ◽

1975 ◽

Vol 69 (3) ◽

pp. 417-443 ◽

Cited By ~ 733

Author(s):

Peter B. Rhines

Keyword(s):

Weak Interaction ◽

Initial Conditions ◽

Computer Experiments ◽

Zonal Flow ◽

Similarity Solutions ◽

Large Reynolds Number ◽

Two Dimensional ◽

Homogeneous Fluid ◽

Restoring Force ◽

Model Equations

Two-dimensional eddies in a homogeneous fluid at large Reynolds number, if closely packed, are known to evolve towards larger scales. In the presence of a restoring force, the geophysical beta-effect, this cascade produces a field of waves without loss of energy, and the turbulent migration of the dominant scale nearly ceases at a wavenumber kβ = (β/2U)½ independent of the initial conditions other than U, the r.m.s. particle speed, and β, the northward gradient of the Coriolis frequency.The conversion of turbulence into waves yields, in addition, more narrowly peaked wavenumber spectra and less fine-structure in the spatial maps, while smoothly distributing the energy about physical space.The theory is discussed, using known integral constraints and similarity solutions, model equations, weak-interaction wave theory (which provides the terminus for the cascade) and other linearized instability theory. Computer experiments with both finite-difference and spectral codes are reported. The central quantity is the cascade rate, defined as \[ T = 2\int_0^{\infty} kF(k)dk/U^3\langle k\rangle , \] where F is the nonlinear transfer spectrum and 〈k〉 the mean wavenumber of the energy spectrum. (In unforced inviscid flow T is simply U−1d〈k〉−1/dt, or the rate at which the dominant scale expands in time t.) T is shown to have a mean value of 3·0 × 10−2 for pure two-dimensional turbulence, but this decreases by a factor of five at the transition to wave motion. We infer from weak-interaction theory even smaller values for k [Lt ] kβ.After passing through a state of propagating waves, the homogeneous cascade tends towards a flow of alternating zonal jets which, we suggest, are almost perfectly steady. When the energy is intermittent in space, however, model equations show that the cascade is halted simply by the spreading of energy about space, and then the end state of a zonal flow is probably not achieved.The geophysical application is that the cascade of pure turbulence to large scales is defeated by wave propagation, helping to explain why the energy-containing eddies in the ocean and atmosphere, though significantly nonlinear, fail to reach the size of their respective domains, and are much smaller. For typical ocean flows, $k_{\beta}^{-1} = 70\,{\rm km} $, while for the atmosphere, $k_{\beta}^{-1} = 1000\,{\rm km}$. In addition the cascade generates, by itself, zonal flow (or more generally, flow along geostrophic contours).

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Study of the influence of initial-state fluctuations on hydrodynamic simulations

EPJ Web of Conferences ◽

10.1051/epjconf/202024506005 ◽

2020 ◽

Vol 245 ◽

pp. 06005

Author(s):

Marcin Słodkowski ◽

Patryk Gawryszewski ◽

Dominik Setniewski

Keyword(s):

Initial Conditions ◽

Graphics Processing Unit ◽

Heavy Ion ◽

Processing Unit ◽

Initial State ◽

Final State ◽

Hydrodynamic Simulation ◽

Hydrodynamic Simulations ◽

Cartesian Coordinate ◽

Ion Collision

In this work, we are focusing on assessing the contribution of the initial-state fluctuations of heavy ion collision in the hydrodynamic simulations. We are trying to answer the question of whether the hydrodynamic simulation retains the same level of fluctuation in the final-state as for the initial stage. In another scenario, the hydrodynamic simulations of the fluctuation drowns in the final distribution of expanding matter. For this purpose, we prepared sufficient relativistic hydrodynamic program to study A+A interaction which allows analysing initial-state fluctuations in the bulk nuclear matter. For such an assumption, it is better to use high spatial resolution. Therefore, we applied the (3+1) dimensional Cartesian coordinate system. We implemented our program using parallel computing on graphics cards processors - Graphics Processing Unit (GPU). Simulations were carried out with various levels of fluctuation in initial conditions using the average method of events coming from UrQMD models. Energy density distributions were analysed and the contribution of fluctuations in initial conditions was assessed in the hydrodynamic simulation.

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The Influence of Particle Concentration on the Formation of Settling-Driven Gravitational Instabilities at the Base of Volcanic Clouds

Frontiers in Earth Science ◽

10.3389/feart.2021.640090 ◽

2021 ◽

Vol 9 ◽

Author(s):

Allan Fries ◽

Jonathan Lemus ◽

Paul A. Jarvis ◽

Amanda B. Clarke ◽

Jeremy C. Phillips ◽

...

Keyword(s):

Volcanic Ash ◽

Particle Concentration ◽

Initial Conditions ◽

Initial Density ◽

Fluid Phase ◽

Dimensionless Number ◽

Narrow Region ◽

Gravitational Instabilities ◽

Volcanic Clouds ◽

Particle Imaging

Settling-driven gravitational instabilities observed at the base of volcanic ash clouds have the potential to play a substantial role in volcanic ash sedimentation. They originate from a narrow, gravitationally unstable region called a Particle Boundary Layer (PBL) that forms at the lower cloud-atmosphere interface and generates downward-moving ash fingers that enhance the ash sedimentation rate. We use scaled laboratory experiments in combination with particle imaging and Planar Laser Induced Fluorescence (PLIF) techniques to investigate the effect of particle concentration on PBL and finger formation. Results show that, as particles settle across an initial density interface and are incorporated within the dense underlying fluid, the PBL grows below the interface as a narrow region of small excess density. This detaches upon reaching a critical thickness, that scales with (ν2/g′)1/3, where ν is the kinematic viscosity and g′ is the reduced gravity of the PBL, leading to the formation of fingers. During this process, the fluid above and below the interface remains poorly mixed, with only small quantities of the upper fluid phase being injected through fingers. In addition, our measurements confirm previous findings over a wider set of initial conditions that show that both the number of fingers and their velocity increase with particle concentration. We also quantify how the vertical particle mass flux below the particle suspension evolves with time and with the particle concentration. Finally, we identify a dimensionless number that depends on the measurable cloud mass-loading and thickness, which can be used to assess the potential for settling-driven gravitational instabilities to form. Our results suggest that fingers from volcanic clouds characterised by high ash concentrations not only are more likely to develop, but they are also expected to form more quickly and propagate at higher velocities than fingers associated with ash-poor clouds.

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