scholarly journals Steady-State Dynamics of a Density Current in an f-Plane Nonlinear Shallow-Water Model

2010 ◽  
Vol 67 (2) ◽  
pp. 500-514 ◽  
Author(s):  
Giovanni A. Dalu ◽  
Marina Baldi
Keyword(s):  
Steady State ◽  
Friction Coefficient ◽  
Shallow Water ◽  
Degrees Of Freedom ◽  
Lower Layer ◽  
Water Model ◽  
Coriolis Parameter ◽  
Six Degrees Of Freedom ◽  
Rayleigh Friction ◽  
Layer Flow

Abstract The authors study the nonlinear dynamics of a density current generated by a diabatic source in a rotating and a nonrotating system, both in the presence and in the absence of frictional losses, using a steady-state hydrostatic shallow-water model and producing solutions as a function of the Coriolis parameter and of the Rayleigh friction coefficient. Results are presented in the range of the parameter values that are relevant for shallow atmospheric flows as sea–land breezes and as cold pool outflows. In the shallow-water approximation, single-layer flows and two-layer flows with a lid have three degrees of freedom, and their steady-state dynamics are governed by three ordinary differential equations (ODEs), whereas two-layer flows bounded by a free surface have six degrees of freedom, and their dynamics are governed by six ODEs. It is shown that in the limit case of frictionless flow, the problem has an explicit analytical solution, and in the presence of friction, the system for a one-layer flow and for a two-layer flow bounded by a lid can be reduced to two algebraic equations, plus one second-order ordinary differential equation, which can be integrated numerically. Results show that the maximum runout length of the current occurs when the Rayleigh friction coefficient in the lower layer is on the order of the Coriolis parameter. This length is larger when the upper layer is deeper than the lower layer, but it shortens when the friction coefficient of the upper layer is smaller than that in the lower layer. In addition, the relative error of the solution to the linearized equations is computed. This error, which is enhanced when the width of the forcing is smaller than the Rossby radius, is sizable when the friction coefficient is smaller than the Coriolis parameter. In addition, by comparing the nonlinear solution with a lid (three degrees of freedom) to the nonlinear solution with a free surface as an upper boundary (six degrees of freedom), it is shown that the solution with the lid overestimates the geopotential for low values of the friction coefficient and it underestimates the geopotential for large values of this coefficient. The error, sizable when the two layers have a comparable depth, rapidly decreases when the upper layer becomes deeper than the lower layer; accordingly, a rigid lid can be safely adopted only when the depth of the upper layer is twice the depth of the lower layer, or deeper.

Instabilities of coupled density fronts and their nonlinear evolution in the two-layer rotating shallow-water model: influence of the lower layer and of the topography

2013 ◽  
Vol 716 ◽  
pp. 528-565 ◽  
Author(s):  
Bruno Ribstein ◽  
Vladimir Zeitlin
Keyword(s):  
Shallow Water ◽  
Nonlinear Evolution ◽  
Lower Layer ◽  
Phase Locking ◽  
Water Model ◽  
Shallow Water Model ◽  
Wave Instability ◽  
Rotating Shallow Water ◽  
New Generation ◽  
Rotating Shallow Water Equations

AbstractWe undertake a detailed analysis of linear stability of geostrophically balanced double density fronts in the framework of the two-layer rotating shallow-water model on the $f$-plane with topography, the latter being represented by an escarpment beneath the fronts. We use the pseudospectral collocation method to identify and quantify different kinds of instabilities resulting from phase locking and resonances of frontal, Rossby, Poincaré and topographic waves. A swap in the leading long-wave instability from the classical barotropic form, resulting from the resonance of two frontal waves, to a baroclinic form, resulting from the resonance of Rossby and frontal waves, takes place with decreasing depth of the lower layer. Nonlinear development and saturation of these instabilities, and of an instability of topographic origin, resulting from the resonance of frontal and topographic waves, are studied and compared with the help of a new-generation well-balanced finite-volume code for multilayer rotating shallow-water equations. The results of the saturation for different instabilities are shown to produce very different secondary coherent structures. The influence of the topography on these processes is highlighted.

A Quasi-three-dimensional Finite-volume Shallow Water Model for Green Water on Deck

2013 ◽  
Vol 57 (03) ◽  
pp. 125-140
Author(s):  
Daniel A. Liut ◽  
Kenneth M. Weems ◽  
Tin-Guen Yen
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Time Domain ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Green Water ◽  
Water Model ◽  
Ship Motions ◽  
Shallow Water Model

A quasi-three-dimensional hydrodynamic model is presented to simulate shallow water phenomena. The method is based on a finite-volume approach designed to solve shallow water equations in the time domain. The nonlinearities of the governing equations are considered. The methodology can be used to compute green water effects on a variety of platforms with six-degrees-of-freedom motions. Different boundary and initial conditions can be applied for multiple types of moving platforms, like a ship's deck, tanks, etc. Comparisons with experimental data are discussed. The shallow water model has been integrated with the Large Amplitude Motions Program to compute the effects of green water flow over decks within a time-domain simulation of ship motions in waves. Results associated to this implementation are presented.

Ship Motions in Shallow Water

1970 ◽  
Vol 14 (04) ◽  
pp. 317-328 ◽  
Author(s):  
E. O. Tuck
Keyword(s):  
Shallow Water ◽  
Water Depth ◽  
Incident Wave ◽  
Degrees Of Freedom ◽  
Added Mass ◽  
Exciting Force ◽  
Ship Motions ◽  
Wave System ◽  
Six Degrees Of Freedom ◽  
Damping Coefficients

The problem discussed concerns small motions of a ship, in all six degrees of freedom, but at zero speed of advance, due to an incident wave system in shallow water of depth comparable with the ship's draft. The problem is completely formulated for an arbitrary ship, and is partially solved for the case when the ship is slender and the wavelength much greater than the water depth. Sample numerical computations of heave, pitch, and sway added mass and damping coefficients and the sway exciting force are presented.

scholarly journals Two Different Aperiodic Phases of Wind-Driven Ocean Circulation in a Double-Gyre, Two-Layer Shallow-Water Model

2006 ◽  
Vol 36 (7) ◽  
pp. 1265-1286 ◽  
Author(s):  
Tomonori Matsuura ◽  
Mitsutaka Fujita
Keyword(s):  
Power Spectrum ◽  
Shallow Water ◽  
Metastable State ◽  
Ocean Circulation ◽  
Lower Layer ◽  
Water Model ◽  
Barotropic Instability ◽  
Shallow Water Model ◽  
Gyre Circulation ◽  
Double Gyre

Abstract A two-layer shallow-water model is used to investigate the transition of wind-driven double-gyre circulation from laminar flow to turbulence as the Reynolds number (Re) is systematically increased. Two distinctly different phases of turbulent double-gyre patterns and energy trajectories are exhibited before and after at Re = 95: deterministic and fully developed turbulent circulations. In the former phase, the inertial subgyres vary between an asymmetric solution and an antisymmetric solution and the double-gyre circulations reach the aperiodic solution mainly due to their barotropic instability. An integrated kinetic energy in the lower layer is slight and the generated mesoscale eddies are confined in the upper layer. The power spectrum of energies integrated over the whole domain at Re = 70 has peaks at the interannual periods (4–7 yr) and the interdecadal period (10–20 yr). The loops of the attractors take on one cycle at those periods and display the blue-sky catastrophe. At Re = 95, the double-gyre circulation reaches a metastable state and the attracters obtained from the three energies form a topological manifold. In the latter, as Re increases, the double-gyre varies from a metastable state to a chaotic state because of the barotropic instability of the eastward jet and the baroclinic instability of recirculation retrograde flow, and the eastward jet meanders significantly with interdecadal variability. The generated eddies cascade to the red side of the power spectrum as expected in the geostrophic turbulence. The main results in the simulation may indicate essential mechanisms for the appearance of multiple states of the Kuroshio and for low-frequency variations in the midlatitude ocean.

scholarly journals The shape of submarine levees: exponential or power law?

2009 ◽  
Vol 619 ◽  
pp. 367-376 ◽  
Author(s):  
V. K. BIRMAN ◽  
E. MEIBURG ◽  
B. KNELLER
Keyword(s):  
Steady State ◽  
Shallow Water ◽  
Power Law ◽  
Sediment Supply ◽  
Water Model ◽  
Navier Stokes ◽  
Entrainment Rate ◽  
Quasi Steady State ◽  
Shallow Water Model ◽  
Power Law Decay

Field observations indicate that the height of submarine levees decays with distance from the channel either exponentially or according to a power law. This investigation clarifies the flow conditions that lead to these respective shapes, via a shallow water model for the overflow currents that govern the levee formation. The model is based on a steady state balance of sediment supply by the turbidity current, and sediment deposition onto the levee, with the settling velocity and the entrainment rate appearing as parameters. It demonstrates that entrainment of ambient fluid is the determining factor for the levee shape. For negligible entrainment rates, levee shapes tend to exhibit exponential profiles, while constant rates of entrainment or detrainment result in power law shapes. Interestingly, whether a levee has an exponential or a power law shape is determined by kinematic considerations only, viz. the balance laws for sediment mass and fluid volume. We find that the respective coefficients governing the exponential or power law decay depend on the settling speeds of the sediment grains, which in turn is a function of the grain size. Two-dimensional, unsteady Navier–Stokes simulations confirm the emergence of a quasi-steady state. The depositional behaviour of this quasi-steady state is consistent with the predictions of the shallow water model, thus validating the assumptions underlying the model, and demonstrating its predictive abilities.

scholarly journals Influence of condensation and latent heat release upon barotropic and baroclinic instabilities of vortices in rotating shallow water f -plane model

2018 ◽  
Author(s):  
LMD
Keyword(s):  
Heat Release ◽  
Shallow Water ◽  
Latent Heat ◽  
Large Scale ◽  
Lower Layer ◽  
Passive Scalar ◽  
Water Model ◽  
Moist Convection ◽  
Latent Heat Release ◽  
Rotating Shallow Water

Analysis of the influence of condensation and related latent heat release upon developing barotropic and baroclinic instabilities of large-scale low Rossby-number shielded vortices on the f - plane is performed within the moist-convective rotating shallow water model, in its barotropic (one-layer) and baroclinic (two-layer) versions. Numerical simulations with a high-resolution well-balanced finite-volume code, using a relaxation parameterisation for condensation, are made. Evolution of the instability in four different environments, with humidity (i) behaving as passive scalar, (ii) subject to condensation beyond a saturation threshold, (iii) sub-ject to condensation and evaporation, with two different parameterisations of the latter, are inter-ompared.The simulations are initialised with unstable modes determined from the detailed linear stability analysis in the “dry” version of the model. In a configuration corresponding to low-level mid-latitude atmospheric vortices, it is shown that the known scenario of evolution of barotropically unstable vortices, consisting information of a pair of dipoles (“dipolar breakdown”) is substantially modified by condensation and related moist convection, especially in the presence of surface evaporation. No enhancement of the instability due to precipitation was detected in this case. Cyclone-anticyclone asymmetry with respect to sensitivity tothe moist effects is evidenced. It is shown that inertia-gravity wave emission during the vortex evolution is enhanced by the moist effects. In the baroclinic configuration corresponding to idealised cut-off lows in the atmosphere, it is shown that the azimuthal structure of the leading unstable mode is sensitive to the details of stratification. Scenarios of evolution are completely different for different azimuthal structures, one leading to dipolar breaking, and another to tripole formation. The effects of moisture considerably enhance the perturbations in the lower layer, especially in the tripole formation scenario.

SIMULATION OF A SPILLED OIL SLICK WITH A SHALLOW WATER MODEL WITH FREE BOUNDARY

2007 ◽  
Vol 17 (03) ◽  
pp. 393-409 ◽  
Author(s):  
B. DI MARTINO ◽  
P. ORENGA ◽  
M. PEYBERNES
Keyword(s):  
Free Boundary ◽  
Shallow Water ◽  
Lower Layer ◽  
Water Model ◽  
Shallow Water Model ◽  
New Approach ◽  
Ale Formulation ◽  
Special Basis ◽  
Spilled Oil ◽  
Discretized Problem

In this paper we present a new approach to describe the behaviour of a pollutant slick at the sea surface. To this end, we consider that the pollutant and the water are immiscible and we propose a two-layer model where the lower layer corresponds to the water and the upper layer represents the pollutant. Since the dimension of the pollutant slick is generally much smaller than the domain occupied by the sea, we propose to compute the motion of the pollutant with a shallow water model with free boundary only in the domain occupied by the pollutant. To discretize in time the problem with free boundary, we use an ALE formulation coupled with the characteristic method. Then, to solve the space discretized problem, we approximate the pollutant velocity by using a Galerkin method with a special basis which verifies the boundary conditions and simplifies significantly the resolution. Finally we test this work in a real situation: the dam of Calacuccia (Corsica).

scholarly journals Application of the Synchronized B-Grid Staggering for Solution of the Shallow-Water Equations on the Spherical Icosahedral Grid

2019 ◽  
Vol 147 (7) ◽  
pp. 2485-2509 ◽  
Author(s):  
Hiroaki Miura
Keyword(s):  
Shallow Water ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Velocity Fields ◽  
Second Order ◽  
Water Model ◽  
Attractive Alternative ◽  
Shallow Water Model ◽  
Grid Model ◽  
Geostrophic Balance

Abstract A shallow-water model using the hexagonal synchronized B grid (SB grid) is developed on the spherical icosahedral grid. The SB grid adopts the same variable arrangement as the ZM grid, but does not suffer from a computational mode problem of the ZM grid since interactions in the extra degrees of freedom of velocity fields through the nonlinear terms are excluded. For better representations of the geostrophic balance, a quadratic reconstruction of fluid height inside hexagonal/pentagonal cells is used to configure the gradient with the second-order accuracy. When nongeostrophic motions are more dominant than geostrophic ones, smaller-scale noises arise. To prevent a decoupling of the velocity fields, a hyperviscosity is added to force velocities adjacent to each other to evolve synchronously. Some standard tests are performed to examine the SB-grid shallow-water model. The model is almost second-order accurate if both the initial conditions and the surface topography are smooth and if the influence of the hyperviscosity is small. The SB-grid model is superior to a C-grid model regarding the convergence of error norms in a steady-state geostrophically balanced flow test, while it is inferior to that concerning conservation of total energy in a case of flow over an isolated mountain. An advantage of the SB-grid model is that both accuracy and stability are weakly sensitive to whether a grid optimization is applied or not. The SB grid is an attractive alternative to the conventional A grid and is competitive with the C grid on the spherical icosahedral grid.

Interaction of nonlinear Ekman pumping, near-inertial oscillations and geostrophic turbulence in an idealized coupled model

2021 ◽  
Author(s):  
Yanxu Chen ◽  
David Straub ◽  
Louis-Philippe Nadeau
Keyword(s):  
Shallow Water ◽  
Coupled Model ◽  
Water Model ◽  
Ekman Pumping ◽  
Shallow Water Model ◽  
Inertial Oscillations ◽  
Pressure Fields ◽  
Interior Flow ◽  
Layer Flow ◽  
Geostrophic Turbulence

AbstractA new coupled model is developed to investigate interactions among geostrophic, Ekman and near-inertial (NI) flows. The model couples a time-dependent nonlinear slab Ekman layer with a two-layer shallow water model. Wind stress forces the slab layer and horizontal divergence of slab-layer transport appears as a forcing in the continuity equation of the shallow water model. In one version of the slab model, self-advection of slab-layer momentum is retained and in another it is not. The most obvious impact of this explicit representation of the surface-layer dynamics is in the high-frequency part of the flow. For example, near-inertial oscillations are significantly stronger when self-advection of slab-layer momentum is retained, this being true both for the slab-layer flow itself and for the interior flow that it excites. In addition, retaining the self-advection terms leads to a new instability, which causes growth of slab-layer near-inertial oscillations in regions of anticyclonic forcing and decay in regions of cyclonic forcing. In contrast to inertial instability, it is the sign of the forcing, not that of the underlying vorticity that determines stability. High-passed surface pressure fields are also examined and show the surface signature of unbalanced flow to differ substantially depending on whether a slab-layer model is used and, if so, whether self-advection of slab-layer momentum is retained.

scholarly journals Instability of Lenticular Vortices: Results from Laboratory Experiments, Linear Stability Analysis and Numerical Simulations

Fluids ◽  
2021 ◽  
Vol 6 (11) ◽  
pp. 380
Author(s):  
Noé Lahaye ◽  
Alexandre Paci ◽  
Stefan G. Llewellyn Smith
Keyword(s):  
Stability Analysis ◽  
Numerical Simulations ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Laboratory Experiments ◽  
Lower Layer ◽  
Water Model ◽  
Layer Potential ◽  
Rotating Shallow Water ◽  
Layer Flow

The instability of surface lenticular vortices is investigated using a comprehensive suite of laboratory experiments combined with numerical linear stability analysis as well as nonlinear numerical simulations in a two-layer Rotating Shallow Water model. The development of instabilities is discussed and compared between the different methods. The linear stability analysis allows for a clear description of the origin of the instability observed in both the laboratory experiments and numerical simulations. While global qualitative agreement is found, some discrepancies are observed and discussed. Our study highlights that the sensitivity of the instability outcome is related to the initial condition and the lower-layer flow. The inhibition or even suppression of some unstable modes may be explained in terms of the lower-layer potential vorticity profile.

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