The free oscillations of fluid on a hemisphere bounded by meridians of longitude

1970 ◽  
Vol 266 (1174) ◽  
pp. 193-223 ◽  
Keyword(s):  
Normal Modes ◽  
Ocean Basin ◽  
Planetary Waves ◽  
Full Account ◽  
Precise Calculation ◽  
Plane Analysis ◽  
Asymptotic Expressions ◽  
Beta Plane ◽  
External Forces ◽  
Baroclinic Modes

A precise calculation is presented of the normal modes of oscillation of an ocean of uniform depth which is bounded by two meridians of longitude separated by an angle of 180°. The calculation takes full account of the horizontal divergence of the motion, and so is applicable to both barotropic and baroclinic modes of oscillation.At small values of the parameter (defined fully in § 1) the calculation yields both the familiar gravity waves and also the nondivergent planetary waves computed in an earlier paper (Longuet-Higgins 1966). At large, positive values of e , corresponding to baroclinic waves, new types of oscillation appear in which the flux of energy is concentrated near the equator, the circuit being completed by Kelvin waves along the meridianal boundaries. The calculated frequencies are compared with asymptotic expressions derived from a recent beta-plane analysis by D. W. Moore. Solutions are also found corresponding to negative values of e . These must be included in a complete calculation of the response of the ocean to external forces. At small values of e these solutions resemble the planetary waves. At large (negative) values of e they represent almost-inertial motions concentrated near the poles, having a phase-velocity towards the east and an amplitude modulated so as to vanish at the boundaries. The calculations are relevant to the real ocean in so far as they show the kinds of oscillation that might be expected in any ocean basin including any section of the equator (or including a pole). They also indicate the degree of accuracy to be expected in computing the frequencies of the normal modes by beta-plane methods.

Two-layer quasi-geostrophic singular vortices embedded in a regular flow. Part 2. Steady and unsteady drift of individual vortices on a beta-plane

2007 ◽  
Vol 584 ◽  
pp. 203-223 ◽  
Author(s):  
GREGORY REZNIK ◽  
ZIV KIZNER
Keyword(s):  
Exact Solution ◽  
Initial Instant ◽  
Symmetric Field ◽  
Dipole Component ◽  
Beta Plane ◽  
Flow Part ◽  
Baroclinic Modes ◽  
Regular Flow ◽  
Singular Vortex

Drift of individual β-plane vortices confined to one layer of a two-layer fluid under the rigid-lid condition is considered. For this purpose, the theory of two-layer quasi-geostrophic singular vortices is employed. On a β-plane, any non-zonal displacement of a singular vortex results in the development of a regular flow. An individual singular β-plane vortex cannot be steady on its own: the vortex moves coexisting with a regular flow, be the drift steady or not. In this paper, both kinds of drift of a singular vortex are considered. A new steady exact solution is presented, a hybrid regular–singular modon. This hybrid modon consists of a dipole component and a circularly symmetric rider. The dipole is regular, and the rider is a superposition of the singular vortex and a regular circularly symmetric field. The unsteady drift of a singular vortex residing in one of the layers is considered under the condition that, at the initial instant, the regular field is absent. The development of barotropic and baroclinic regular β-gyres is examined. Whereas the barotropic and baroclinic modes of the singular vortex are comparable in magnitudes, the baroclinic β-gyres attenuate with time, making the trajectory of the vortex close to that of a barotropic monopole on a β-plane.

Wind-driven Oscillations in the Meridional Overturning Circulation near the equator

2021 ◽  
Author(s):  
Adam Blaker ◽  
Michael Bell ◽  
Joel Hirschi ◽  
Amy Bokota
Keyword(s):  
Gravity Waves ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Ocean Basin ◽  
Ocean Model ◽  
Equatorial Region ◽  
Meridional Overturning Circulation ◽  
Overturning Circulation ◽  
Meridional Overturning ◽  
Inertia Gravity Waves

<p>Numerical model studies have shown the meridional overturning circulation (MOC) to exhibit variability on near-inertial timescales, and also indicate a region of enhanced variability on the equator. We present an analysis of a set of integrations of a global configuration of a numerical ocean model, which show very large amplitude oscillations in the MOCs in the Atlantic, Indian and Pacific oceans confined to the equatorial region. The amplitude of these oscillations is proportional to the width of the ocean basin, typically about 100 (200) Sv in the Atlantic (Pacific). We show that these oscillations are driven by surface winds within 10°N/S of the equator, and their periods (typically 4-10 days) correspond to a small number of low mode equatorially trapped planetary waves. Furthermore, the oscillations can be well reproduced by idealised wind-driven simulations linearised about a state of rest. Zonally integrated linearised equations of motion are solved using vertical normal modes and equatorial meridional modes representing Yanai and inertia-gravity waves. Idealised simulations capture between 85% and 95% of the variance of matching time-series segments diagnosed from the NEMO integrations. Similar results are obtained for the corresponding modes in the Atlantic and Indian Oceans. Our results raise questions about the roles of inertia-gravity waves near the equator in the vertical transfer of heat and momentum and whether these transfers will be explicitly resolved by ocean models or need to be parametrised.</p>

scholarly journals On the Observability of Oceanic Gyres

2014 ◽  
Vol 44 (9) ◽  
pp. 2498-2523 ◽  
Author(s):  
Olivier Marchal
Keyword(s):  
Symmetric Matrix ◽  
Normal Modes ◽  
Time Dependent ◽  
Closed Basin ◽  
Beta Plane ◽  
Sverdrup Balance ◽  
Stratified Ocean ◽  
Oceanic Gyres ◽  
Variable Influence ◽  
Degree Of Observability

Abstract This study examines the observability of a stratified ocean in a square flat basin on a midlatitude beta plane. Here, “observability” means the ability to establish, in a finite interval of time, the time-dependent ocean state given density observations over the same interval and with no regard for errors. The dynamics is linearized and hydrostatic, so that the motion can be decomposed into normal modes and the observability analysis is simplified. An observability Gramian (a symmetric matrix) is determined for the flows in an inviscid interior, in frictional boundary layers, and in a closed basin. Its properties are used to establish the condition for complete observability and to identify optimal data locations for each of these flows. It is found that complete observability of an oceanic interior in time-dependent Sverdrup balance requires that the observations originate from the westernmost location at each considered latitude. The degree of observability increases westward due to westward propagation of long baroclinic Rossby waves: data collected in the west are more informative than data collected in the east. Likewise, the best locations for observing variability in the western (eastern) boundary layer are near (far from) the boundary. The observability of a closed basin is influenced by the westward propagation and the boundaries. Optimal data locations that are identified for different resolutions (0.01 to 1 yr) and lengths of data records (0.2 to 20 yr) show a variable influence of the planetary vorticity gradient. Data collected near the meridional boundaries appear always less informative, from the viewpoint of basin observability, than data collected away from these boundaries.

Generating Traveling Vibration Waves in Finite Structures

2008 ◽  
Author(s):  
Ran Gabai ◽  
Izhak Bucher
Keyword(s):  
Traveling Waves ◽  
Traveling Wave ◽  
Normal Modes ◽  
Wave State ◽  
Naturally Occurring ◽  
Finite Structures ◽  
Applied Forces ◽  
Small Model ◽  
External Forces

This work is concerned with a method to generate pure traveling vibration waves in finite structures. Progressing elastic deformations, i.e. waves, are not common in forced vibrating structures since a structure is naturally vibrating in its, naturally occurring, normal modes that are usually referred to standing waves. This makes the generation of traveling waves in a structure a challenging task. In this work, external excitation is applied to the structure in order to create traveling waves in one dimensional structures. Based on a model of the structure and its boundaries, it is possible to calculate, theoretically, the required excitation, in order to generate a pure traveling wave in the structure. This calculation has little merit in practice since small model uncertainties and boundary dynamics effects may alter the generated waves such that they are far from being traveling waves. An iterative in situ method to tune the applied forces, until the desired traveling wave are formed, is presented. This method relays on estimating the current wave-vibrations state in a structure from measurements and tune the external forces toward a pure traveling wave state. An experimental validation of the theory is presented to support the theory.

Public Right

2020 ◽  
pp. 274-318
Author(s):  
Helga Varden
Keyword(s):  
Human Nature ◽  
Vulnerable Populations ◽  
Active Participation ◽  
Public Debate ◽  
Public Institutions ◽  
Full Account ◽  
And Gender ◽  
External Forces ◽  
Public Right

This chapter engages complexities concerning systemic justice in relation to sex, love, and gender. It shows how philosophical ideas in Kant’s account of public right in combination with his full account of human nature, yields a position that can take on systemic issues of dependency (including the state’s right and duty to fight poverty) and oppression (including through public laws protecting sexual or gendered minorities). In addition, I show how Kant’s account of different kinds of external forces people may find themselves subjected to—“barbaric,” “anarchic,” “despotic,” and “republican”—help us capture the moral complexity facing oppressed and vulnerable populations in different legal-political circumstances. Finally, I argue that the ultimate aim for states is to establish a legal-political whole characterized by the citizens governing themselves wisely through active participation in public debate and public institutions.

scholarly journals Time-Dependent Response to Cooling in a Beta-Plane Basin

2006 ◽  
Vol 36 (11) ◽  
pp. 2185-2198 ◽  
Author(s):  
Joseph Pedlosky
Keyword(s):  
Steady State ◽  
Vertical Motion ◽  
Ocean Basin ◽  
Time Dependent ◽  
Western Boundary ◽  
Boundary Current ◽  
Layer Model ◽  
Periodic Forcing ◽  
Beta Plane ◽  
Two Layer Model

Abstract The time-dependent response of an ocean basin to the imposition of cooling (or heating) is examined in the context of a quasigeostrophic, two-layer model on the beta plane. The focus is on the structure and magnitude of the vertical motion and its response to both a switch-on forcing and a periodic forcing. The model employed is a time-dependent version of an earlier model used to discuss the intensification of sinking in the region of the western boundary current. The height of the interface of the two-layer model serves as an analog of temperature, and the vertical velocity at the interface consists of a cross-isopycnal velocity modeled in terms of a relaxation to a prescribed interface height, an adiabatic representation of eddy thickness fluxes parameterized as lateral diffusion of thickness, and the local vertical motion of the interface itself. The presence of time dependence adds additional dynamical features to the problem, in particular the emergence of low-frequency, weakly damped Rossby basin modes. If the buoyancy forcing is zonally uniform the basin responds to a switch-on of the forcing by coming into steady-state equilibrium after the passage of a single baroclinic Rossby wave. If the forcing is nonuniform in the zonal direction, a sequence of Rossby basin modes is excited and their decay is required before the basin achieves a steady state. For reasonable parameter values the boundary layers, in which both horizontal and vertical circulations are closed, are quasi-steady and respond to the instantaneous state of the interior. As in the steady problem the flow is sensitive to small nonquasigeostrophic mass fluxes across the perimeter of the basin. These fluxes generally excite basin modes as well. The basin modes will also be weakly excited if the beta-plane approximation is relaxed. The response to periodic forcing is also examined, and the sensitivity of the response to the structure of the forcing is similar to the switch-on problem.

scholarly journals Large-Scale Modes in a Rotating Atmosphere with Radiative–Convective Instability and WISHE

2005 ◽  
Vol 62 (11) ◽  
pp. 4084-4094 ◽  
Author(s):  
Zeljka Fuchs ◽  
David J. Raymond
Keyword(s):  
Gravity Waves ◽  
Large Scale ◽  
Normal Modes ◽  
Moist Convection ◽  
Mean Flow ◽  
Beta Plane ◽  
Additional Mode ◽  
Diabatic Processes ◽  
Relaxation Time Scale ◽  
The One

Abstract A highly simplified parameterization of diabatic processes is applied to linearized equations on a equatorial beta plane. The diabatic processes include moist convection, cloud–radiation interactions (CRI), and wind-induced surface heat exchange (WISHE). The precipitation rate is assumed to increase linearly as the vertically averaged saturation deficit decreases. The modeled modes are Matsuno’s normal modes, that is, Kelvin waves, mixed Rossby–gravity waves, Rossby waves, and inertio–gravity waves, and an additional mode called here a slow moisture mode. All of the Matsuno modes are damped and remain stable even when CRI and WISHE are turned on. Their phase speeds do not vary much from Matsuno’s adiabatic values except for very long wavelength Kelvin and Rossby modes, for which the phase speeds are reduced compared to the adiabatic values. The slow moisture modes are stationary and unstable under CRI, while WISHE causes them to propagate. Under CRI and WISHE together the slow moisture modes are unstable and eastward propagating for long wavelengths and slowly moving relative to the mean flow for short wavelengths. The dispersion relations of the slow moisture modes are one of nearly constant or decreasing frequency with increasing wavenumber. The most important model parameter is the tropospheric moisture relaxation time scale, which is chosen to be 1 day. The model failed to explain the observed phase speeds of convectively coupled Matsuno modes. Following Mapes, the authors suggest that other dynamics, more realistic than the one including only the first baroclinic mode, may be responsible for these modes.

scholarly journals Dynamics in coalescing critical layers

2001 ◽  
Vol 449 ◽  
pp. 115-139 ◽  
Author(s):  
N. J. BALMFORTH ◽  
C. PICCOLO ◽  
O. M. UMURHAN
Keyword(s):  
Normal Modes ◽  
Critical Layer ◽  
Matched Asymptotic Expansion ◽  
Physical Parameters ◽  
Inviscid Fluid ◽  
Critical Levels ◽  
Governing Equations ◽  
Vortical Structures ◽  
Beta Plane ◽  
Critical Layers

This article continues an exploration of instabilities of jets in two-dimensional, inviscid fluid on the beta-plane. At onset, for particular choices of the physical parameters, the normal modes responsible for instability have critical levels that coalesce along the axis of the jet. Matched asymptotic expansion (critical layer theory) is used to derive a reduced model describing the dynamics of these modes. Because the velocity profile is locally parabolic in the vicinity of the critical levels the dynamics is richer than in standard critical layer problems. The model captures the inviscid saturation of unstable modes, the excitation of neutral Rossby waves, and the decay of disturbances when there are no discrete normal modes. Inviscid saturation occurs when the vorticity distribution twists up into vortical structures that take the form of either a pair of ‘cat's eye’ patterns straddling the jet axis, or a single row of vortices. The addition of weak viscosity destroys these cat's eye structures and causes the critical layer to spread diffusively. The results are compared with numerical simulations of the governing equations.

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