Nonlinear and time-dependent equivalent-barotropic flows

2019 ◽  
Vol 871 ◽  
pp. 925-951 ◽  
Author(s):  
Luis Zavala Sansón
Keyword(s):  
Wind Stress ◽  
Vertical Structure ◽  
Fluid Velocity ◽  
Bottom Topography ◽  
Bottom Friction ◽  
Time Dependent ◽  
Western Slope ◽  
Topographic Effects ◽  
Specific Level ◽  
Variable Topography

Some oceanic and atmospheric flows may be modelled as equivalent-barotropic systems, in which the horizontal fluid velocity varies in magnitude at different vertical levels while keeping the same direction. The governing equations at a specific level are identical to those of a homogeneous flow over an equivalent depth, determined by a pre-defined vertical structure. The idea was proposed by Charney (J. Met., vol. 6 (6), 1949, pp. 371–385) for modelling a barotropic atmosphere. More recently, steady, linear formulations have been used to study oceanic flows. In this paper, the nonlinear, time-dependent model with variable topography is examined. To include nonlinear terms, we assume suitable approximations and evaluate the associated error in the dynamical vorticity equation. The model is solved numerically to investigate the equivalent-barotropic dynamics in comparison with a purely barotropic flow. We consider three problems in which the behaviour of homogeneous flows has been well established either experimentally, analytically or observationally in past studies. First, the nonlinear evolution of cyclonic vortices around a topographic seamount is examined. It is found that the vortex drift induced by the mountain is modified according to the vertical structure of the flow. When the vertical structure is abrupt, the model effectively isolates the surface flow from both inviscid and viscous topographic effects (due to the shape of the bottom and Ekman friction, respectively). Second, the wind-driven flow in a closed basin with variable topography is studied (for a flat bottom this is the well-known Stommel problem). For a zonally uniform, negative wind-stress curl in the homogeneous case, a large-scale, anticyclonic gyre is formed and displaced southward due to topographic effects at the western slope of the basin. The flow reaches a steady state due to the balance between topographic,$\unicode[STIX]{x1D6FD}$, wind-stress and bottom friction effects. However, in the equivalent-barotropic simulations with abrupt vertical structure, such an equilibrium cannot be reached because the forcing effects at the surface are enhanced, while bottom friction effects are reduced. As a result, the unsteady flow is decomposed as a set of planetary waves. A third problem consists of performing simulations of the wind-driven flow over realistic bottom topography in the Gulf of Mexico. The formation of the so-called Campeche gyre is explored. It is found that such circulation may be consistent with the equivalent-barotropic dynamics.

Model for the three-dimensional structure of wind-driven and tidal circulation in Bass Strait

1983 ◽  
Vol 34 (1) ◽  
pp. 121 ◽  
Author(s):  
CB Fandry
Keyword(s):  
Wind Stress ◽  
Bottom Topography ◽  
Grid Point ◽  
Three Dimensional ◽  
Sea Surface ◽  
Time Dependent ◽  
Dimensional Structure ◽  
Return Flow ◽  
Surface Slope ◽  
Anticlockwise Direction

Earlier models of the circulation in Bass Strait have been extended to include vertical structure. Time- dependent circulation fields in Bass Strait, induced by wind driving at the surface and tidal oscillations along open-sea boundaries, are computed at a number of selected depths. The original two-dimensional model is combined with an analytical solution of the Ekman equations, which at each grid point provides an expression for the time-dependent flow at any depth in terms of a convolution integral over the sea- surface slope and wind stress. This model should be applicable to winter conditions when the strait is well mixed vertically and hence the dynamical effects of density stratification negligible. The predicted wind-induced circulation fields are highly depth dependent, with equilibrium surface currents in the central Bass Strait basin flowing in a direction approximately 45� to the left of the wind. At lower levels, currents are controlled by pressure gradient forces due to the sea-surface slope and friction. Significant upwelling and downwelling motions along the Victorian and Tasmanian coastlines can be inferred from these circulation fields. In the deep water off the continental shelf, currents in the upper 100 m are dominated by the (Ekman) drift current which rotates in an anticlockwise direction with increasing depth, such that the wind drift at the surface is accompanied by a measure of return flow at depth. Tidal currents are predicted in the absence of wind stress, but include the effects of bottom topography. Considerable variation with depth is found and the distinctive features are explained in terms of the relative importance of Coriolis force, bottom friction, and water depth. Comparison with the few existing observations reveals that the present model is producing realistic results.

scholarly journals Effect of an oscillating time-dependent pressure gradient on Dean flow: transient solution

2020 ◽  
Vol 9 (1) ◽  
Author(s):  
Basant K. Jha ◽  
Dauda Gambo
Keyword(s):  
Pressure Gradient ◽  
Initial Conditions ◽  
Fluid Velocity ◽  
Outer Cylinder ◽  
Time Dependent ◽  
Navier Stokes ◽  
Dean Flow ◽  
Continuity Equations ◽  
Riemann Sum ◽  
Flow Formation

Abstract Background Navier-Stokes and continuity equations are utilized to simulate fully developed laminar Dean flow with an oscillating time-dependent pressure gradient. These equations are solved analytically with the appropriate boundary and initial conditions in terms of Laplace domain and inverted to time domain using a numerical inversion technique known as Riemann-Sum Approximation (RSA). The flow is assumed to be triggered by the applied circumferential pressure gradient (azimuthal pressure gradient) and the oscillating time-dependent pressure gradient. The influence of the various flow parameters on the flow formation are depicted graphically. Comparisons with previously established result has been made as a limit case when the frequency of the oscillation is taken as 0 (ω = 0). Results It was revealed that maintaining the frequency of oscillation, the velocity and skin frictions can be made increasing functions of time. An increasing frequency of the oscillating time-dependent pressure gradient and relatively a small amount of time is desirable for a decreasing velocity and skin frictions. The fluid vorticity decreases with further distance towards the outer cylinder as time passes. Conclusion Findings confirm that increasing the frequency of oscillation weakens the fluid velocity and the drag on both walls of the cylinders.

scholarly journals The Behavior of Jet Currents over a Continental Slope Topography with a Possible Application to the Northern Current

2005 ◽  
Vol 35 (5) ◽  
pp. 790-810 ◽  
Author(s):  
M. M. Flexas ◽  
G. J. F. van Heijst ◽  
R. R. Trieling
Keyword(s):  
High Speed ◽  
Bottom Topography ◽  
Laboratory Model ◽  
Topographic Effects ◽  
Topographic Slope ◽  
Eddy Formation ◽  
Slow Flows ◽  
Bottom Depth ◽  
Source Sink ◽  
Northern Current

Abstract The Northern Current is a slope current in the northwest Mediterranean that shows high mesoscale variability, generally associated with meander and eddy formation. A barotropic laboratory model of this current is used here to study the role of the bottom topography on the current variability. For this purpose, a source–sink setup in a cylindrical tank placed on a rotating table is used to generate an axisymmetric barotropic current. To study inviscid topographic effects, experiments are performed over a topographic slope and also over a constant-depth setup, the latter being used as a reference for the former. With the aim of obtaining a fully comprehensive view of the vorticity balance at play, the flow may be forced in either azimuthal direction, leading to a “westward” prograde current (similar to the Northern Current) or an “eastward” retrograde current. For slow flows, eastward and westward currents showed similar patterns, dominated by Kelvin–Helmholtz-type instabilities. For high-speed flows, eastward and westward currents showed very different behavior. In eastward currents, the variability is observed to concentrate toward the center of the jet and shows strong meandering formation. Westward currents, instead, showed major variability toward the edges of the jet, together with a strong variability over the uppermost slope, which has been associated here with a topographic Rossby wave trapped over the shelf break. The differences between eastward and westward jets are explained through the balance between shear-induced and topographically induced vorticity at play in each case. Moreover, a model of jets over a beta plane is successfully applied here, allowing its extension to any ambient potential vorticity gradient caused either by latitudinal or bottom depth changes.

On the Role of Bottom Pressure Torques in Wind-Driven Gyres

2021 ◽  
Vol 51 (5) ◽  
pp. 1441-1464
Author(s):  
Andrew L. Stewart ◽  
James C. McWilliams ◽  
Aviv Solodoch
Keyword(s):  
Wind Stress ◽  
Bottom Friction ◽  
Bottom Pressure ◽  
Wind Stress Curl ◽  
Previous Theory ◽  
Model Experiments ◽  
Basin Scale ◽  
Vorticity Budget ◽  
Gyre Circulation

AbstractPrevious studies have concluded that the wind-input vorticity in ocean gyres is balanced by bottom pressure torques (BPT), when integrated over latitude bands. However, the BPT must vanish when integrated over any area enclosed by an isobath. This constraint raises ambiguities regarding the regions over which BPT should close the vorticity budget, and implies that BPT generated to balance a local wind stress curl necessitates the generation of a compensating, nonlocal BPT and thus nonlocal circulation. This study aims to clarify the role of BPT in wind-driven gyres using an idealized isopycnal model. Experiments performed with a single-signed wind stress curl in an enclosed, sloped basin reveal that BPT balances the winds only when integrated over latitude bands. Integrating over other, dynamically motivated definitions of the gyre, such as barotropic streamlines, yields a balance between wind stress curl and bottom frictional torques. This implies that bottom friction plays a nonnegligible role in structuring the gyre circulation. Nonlocal bottom pressure torques manifest in the form of along-slope pressure gradients associated with a weak basin-scale circulation, and are associated with a transition to a balance between wind stress and bottom friction around the coasts. Finally, a suite of perturbation experiments is used to investigate the dynamics of BPT. To predict the BPT, the authors extend a previous theory that describes propagation of surface pressure signals from the gyre interior toward the coast along planetary potential vorticity contours. This theory is shown to agree closely with the diagnosed contributions to the vorticity budget across the suite of model experiments.

Wind effects on the water in a narrow two-layered lake. Part I. Theoretical analysis. Part II. Analysis of observations from Windermere. Part III. Application of the theory to Windermere

1966 ◽  
Vol 259 (1102) ◽  
pp. 391-430 ◽  
Keyword(s):  
Wind Stress ◽  
Water Surface ◽  
Water Movement ◽  
Lower Layer ◽  
Longitudinal Section ◽  
Bottom Friction ◽  
Bottom Current ◽  
Numerical Application ◽  
Warm Surface Water ◽  
Temperature Observations

This paper presents a theoretical study of water movement in a long narrow lake subject to wind action during the summer season of thermal stratification. A model basin of uniform depth and width, consisting of two homogeneous layers of slightly different density, is considered. The motion of the water is assumed to be two dimensional in the vertical longitudinal section; geostrophic effects are ignored. The top and bottom layers in the model respectively represent the relatively warm surface water and the colder bottom water in the natural lake. Hydrodynamical equations are formulated in terms of the currents in the upper and lower layers, the elevation of the interface between the layers, and the elevation of the water surface. Solutions are sought to determine the dynamic response of the basin to an instantaneous rise in the wind stress applied tangentially over the surface. Three cases are considered corresponding to different frictional conditions at the bottom of the basin: (i) bottom friction zero, (ii) bottom friction proportional to the depth mean of the horizontal current in the lower layer, (iii) bottom current zero. It is assumed that internal friction is zero at the interface between the layers (this interface corresponds to the thermocline boundary in reality). Results obtained show that in the motion of the water there are ordinary and internal seiches characteristic of the two-layered model, together with a wind-driven circulation in the top layer. The theory is applied to determine vertical oscillations of the thermocline in an actual lake (Windermere) at one station, in response to a succession of wind pulses representing actual wind conditions over the lake. The oscillations thus obtained from theory compare satisfactorily with those derived from temperature observations taken in the lake. Depth-mean currents in the lake are deduced from theory, but there are no current measurements against which these values may be tested. The paper is divided into three parts. Part I deals with the development of the theory. Part II gives an account of actual physical conditions in Windermere, describing the analysis of temperature observations taken in the lake (yielding thermocline movements) and the analysis of wind records (yielding corresponding values of wind stress over the water surface). Part III is concerned with the numerical application of the theory to Windermere (under conditions described in part II), and gives general conclusions resulting from the entire work.

scholarly journals Boussinesq modeling of surface waves due to underwater landslides

2013 ◽  
Vol 20 (3) ◽  
pp. 267-285 ◽  
Author(s):  
D. Dutykh ◽  
H. Kalisch
Keyword(s):  
Surface Waves ◽  
Shallow Water ◽  
Bottom Topography ◽  
Equations Of Motion ◽  
Boussinesq Equations ◽  
Time Dependent ◽  
Viscous Drag ◽  
Underwater Landslide ◽  
Landslide Mass ◽  
Run Up

Abstract. Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion that govern the evolution of the barycenter of the landslide mass include various dissipative effects due to bottom friction, internal energy dissipation, and viscous drag. The surface waves are studied in the Boussinesq scaling, with time-dependent bathymetry. A numerical model for the Boussinesq equations is introduced that is able to handle time-dependent bottom topography, and the equations of motion for the landslide and surface waves are solved simultaneously. The numerical solver for the Boussinesq equations can also be restricted to implement a shallow-water solver, and the shallow-water and Boussinesq configurations are compared. A particular bathymetry is chosen to illustrate the general method, and it is found that the Boussinesq system predicts larger wave run-up than the shallow-water theory in the example treated in this paper. It is also found that the finite fluid domain has a significant impact on the behavior of the wave run-up.

Numerical analysis of time-dependent stagnation point flow of Oldroyd-B fluid subject to modified Fourier’s law

2021 ◽  
pp. 2150187
Author(s):  
Foukeea Qasim ◽  
Tian-Chuan Sun ◽  
S. Z. Abbas ◽  
W. A. Khan ◽  
M. Y. Malik
Keyword(s):  
Differential Equation ◽  
Fluid Flow ◽  
Stagnation Point ◽  
Numerical Solutions ◽  
Fluid Velocity ◽  
Nonlinear Partial Differential Equation ◽  
Thermal Relaxation ◽  
Stagnation Point Flow ◽  
Time Dependent ◽  
Parameter Values

This paper aims to investigate the time-dependent stagnation point flow of an Oldroyd-B fluid subjected to the modified Fourier law. The flow into a vertically stretched cylinder at the stagnation point is discussed. The heat flux model of a non-Fourier is intended for the transfer of thermal energy in fluid flow. The study is carried out on the surface heating source, namely the surface temperature. The developed nonlinear partial differential equation for regulating fluid flow and heat transport is transformed via appropriate similarity variables into a nonlinear ordinary differential equation. The development and analysis of convergent series solutions were considered for velocity and temperature. Prandtl number numerical values are computed and investigated. This study’s findings are compared to the previous findings. By making use of the bvp4c Matlab method, numerical solutions are obtained. Besides, high buoyancy parameter values are found to increase the fluid velocity for the stimulating approach. By improving the thermal relaxation time parameter values, heat transfer in the fluid flow decreases. The temperature field effects are displayed graphically.

Island in the Stream: Oceanography and Fisheries of the Charleston Bump

2001 ◽  
Keyword(s):  
South Carolina ◽  
Surface Temperature ◽  
Gulf Stream ◽  
Fast Moving ◽  
Bottom Topography ◽  
Sea Surface ◽  
Time Dependent ◽  
New Method ◽  
Sea Surface Temperatures ◽  
Charleston Bump

<em>Abstract.</em>—The animation of daily composites of sea surface temperatures (SST) from a National Oceanic and Atmospheric Administration Geostationary Operational Environmental Satellite (GOES) provides a new method for the detection of dynamics at the surface of the ocean. By rapidly viewing the daily SST composites of hourly images, it is possible for the human eye to separate the fast moving residual clouds from the slowly moving SST patterns associated with ocean currents, eddies, and upwelling. Although each individual daily composite is still partly cloud covered, the rapid display provides the appearance of continuity of the SST patterns. The GOES SST animations were used during 1998 and 1999 to monitor the time dependent deflection of the Gulf Stream due to a rise in bottom topography southeast of Charleston, South Carolina, locally known as the Charleston Bump. Examples of the sea surface temperature animations of the Gulf Stream appear at the website: http:// www. goes .noaa.gov

Alpha Beta Discs

1983 ◽  
Vol 72 ◽  
pp. 55-67
Author(s):  
G.T. Bath ◽  
A.C. Edwards ◽  
V.J. Mantle
Keyword(s):  
Mass Transfer ◽  
Vertical Structure ◽  
Accretion Discs ◽  
Time Dependent ◽  
Simple Method ◽  
Dwarf Nova ◽  
Physical Conditions ◽  
Alpha Beta ◽  
Symbiotic Binaries

Following earlier work of Lynden-Bell & Pringle (1974) and Lightman (1974a, 1974b), Bath & Pringle (1981) have presented a simple method for studying the time-dependent evolution of viscous accretion discs. These models are axisymmetrlc, with the vertical structure reduced to integrated averages of local physical conditions. Published work examines models of dwarf nova eruptions driven by mass transfer bursts (Bath & Pringle 1981 – Paper I), eruptions produced by global viscous changes within the disc (Bath & Pringle 1982a Paper II), and the time-dependent properties of giant discs in symbiotic binaries (Bath & Pringle 1982b – Paper III).

scholarly journals The vertical structure of oceanic Rossby waves: a comparison of high-resolution model data to theoretical vertical structures

Ocean Science ◽  
2012 ◽  
Vol 8 (1) ◽  
pp. 19-35 ◽  
Author(s):  
F. K. Hunt ◽  
R. Tailleux ◽  
J. J.-M. Hirschi
Keyword(s):  
Vertical Structure ◽  
Rossby Waves ◽  
Topographic Effects ◽  
Model Data ◽  
Mean Flow ◽  
Sign Reversal ◽  
Resolution Model ◽  
High Resolution Model ◽  
The Waves ◽  
Model Structures

Abstract. Tests of the new Rossby wave theories that have been developed over the past decade to account for discrepancies between theoretical wave speeds and those observed by satellite altimeters have focused primarily on the surface signature of such waves. It appears, however, that the surface signature of the waves acts only as a rather weak constraint, and that information on the vertical structure of the waves is required to better discriminate between competing theories. Due to the lack of 3-D observations, this paper uses high-resolution model data to construct realistic vertical structures of Rossby waves and compares these to structures predicted by theory. The meridional velocity of a section at 24° S in the Atlantic Ocean is pre-processed using the Radon transform to select the dominant westward signal. Normalized profiles are then constructed using three complementary methods based respectively on: (1) averaging vertical profiles of velocity, (2) diagnosing the amplitude of the Radon transform of the westward propagating signal at different depths, and (3) EOF analysis. These profiles are compared to profiles calculated using four different Rossby wave theories: standard linear theory (SLT), SLT plus mean flow, SLT plus topographic effects, and theory including mean flow and topographic effects. Our results support the classical theoretical assumption that westward propagating signals have a well-defined vertical modal structure associated with a phase speed independent of depth, in contrast with the conclusions of a recent study using the same model but for different locations in the North Atlantic. The model structures are in general surface intensified, with a sign reversal at depth in some regions, notably occurring at shallower depths in the East Atlantic. SLT provides a good fit to the model structures in the top 300 m, but grossly overestimates the sign reversal at depth. The addition of mean flow slightly improves the latter issue, but is too surface intensified. SLT plus topography rectifies the overestimation of the sign reversal, but overestimates the amplitude of the structure for much of the layer above the sign reversal. Combining the effects of mean flow and topography provided the best fit for the mean model profiles, although small errors at the surface and mid-depths are carried over from the individual effects of mean flow and topography respectively. Across the section the best fitting theory varies between SLT plus topography and topography with mean flow, with, in general, SLT plus topography performing better in the east where the sign reversal is less pronounced. None of the theories could accurately reproduce the deeper sign reversals in the west. All theories performed badly at the boundaries. The generalization of this method to other latitudes, oceans, models and baroclinic modes would provide greater insight into the variability in the ocean, while better observational data would allow verification of the model findings.

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