scholarly journals Multilayer shallow-water model with stratification and shear

2021 ◽  
Vol 67 (3 May-Jun) ◽  
pp. 351
Author(s):  
F. J. Beron-Vera
Keyword(s):  
Shallow Water ◽  
Baroclinic Instability ◽  
Single Layer ◽  
Vertical Shear ◽  
Water Model ◽  
Type Model ◽  
Inhomogeneous Layer ◽  
Good Representation ◽  
Water Type ◽  
Dissipation Energy

The purpose of this paper is to present a shallow-water-type model with multiple inhomogeneous layers featuring variable linear velocity vertical shear and startificaion in horizontal space and time. This is achieved by writing the layer velocity and buoyancy fields as linear functions of depth, with coefficients that depend arbitrarily on horizontal position and time. The model is a generalization of Ripa's (1995) single-layer model to an arbitrary number of layers. Unlike models with homogeneous layers the present model is able to represent thermodynamics processes driven by heat and freshwater fluxes through the surface or mixing processes resulting from fluid exchanges across contiguous layers. By contrast with inhomogeneous-layer models with depth-independent velocity and buoyancy, the model derived here can sustain explicitly at low frequency a current in thermal wind balance (between the vertical vertical shear and the horizontal density gradient) within each layer. In the absence of external forcing and dissipation, energy, volume, mass, and buoyancy variance constrain the dynamics; conservation of total zonal momentum requires in addition the usual zonal symmetry of the topography and horizontal domain.  The inviscid, unforced model admits a formulation suggestive of a generalized Hamiltonian structure, which enables the classical connection between symmetries and conservation laws via Noether's theorem.  A steady solution to a system involving one Ripa-like layer and otherwise homogeneous layers can be proved formally (or Arnold) stable using the above invariants. A model configuration with only one layer has been shown previously to provide: a very good representation of the exact vertical normal modes up to the first internal mode; an exact representation of long-perturbation (free boundary) baroclinic instability; and a very reasonable representation of short-perturbation (classical Eady) baroclinic instability. Here it is shown that substantially more accurate overall results with respect to single-layer calculations can be achieved by considering a stack of only a few layers. A similar behavior is found in ageostrophic (classical Stone) baroclinic instability by describing accurately the dependence of the solutions on the Richardson number with only two layers.

scholarly journals Quasi-geostrophic shallow-water doubly-connected vortex equilibria and their stability

2013 ◽  
Vol 723 ◽  
pp. 40-68 ◽  
Author(s):  
Hanna Płotka ◽  
David G. Dritschel
Keyword(s):  
Shallow Water ◽  
Minimum Distance ◽  
Structural Changes ◽  
Single Layer ◽  
Relative Equilibria ◽  
Water Model ◽  
Equilibrium Configurations ◽  
Horizontal Size ◽  
Boundary Of Stability ◽  
Asymmetric Wave

AbstractWe examine the form, properties, stability and evolution of doubly-connected (two-vortex) relative equilibria in the single-layer $f$-plane quasi-geostrophic shallow-water model of geophysical fluid dynamics. Three parameters completely describe families of equilibria in this system: the ratio $\gamma = L/ {L}_{D} $ between the horizontal size of the vortices and the Rossby deformation length; the area ratio $\alpha $ of the smaller to the larger vortex; and the minimum distance $\delta $ between the two vortices. We vary $0\lt \gamma \leq 10$ and $0. 1\leq \alpha \leq 1. 0$, determining the boundary of stability $\delta = {\delta }_{c} (\gamma , \alpha )$. We also examine the nonlinear development of the instabilities and the transitions to other near-equilibrium configurations. Two modes of instability occur when $\delta \lt {\delta }_{c} $: a small-$\gamma $ asymmetric (wave 3) mode, which is absent for $\alpha \gtrsim 0. 6$; and a large-$\gamma $ mode. In general, major structural changes take place during the nonlinear evolution of the vortices, which near ${\delta }_{c} $ may be classified as follows: (i) vacillations about equilibrium for $\gamma \gtrsim 2. 5$; (ii) partial straining out, associated with the small-$\gamma $ mode, where either one or both of the vortices get smaller for $\gamma \lesssim 2. 5$ and $\alpha \lesssim 0. 6$; (iii) partial merger, occurring at the transition region between the two modes of instability, where one of the vortices gets bigger, and (iv) complete merger, associated with the large-$\gamma $ mode. We also find that although conservative inviscid transitions to equilibria with the same energy, angular momentum and circulation are possible, they are not the preferred evolutionary path.

scholarly journals Determination of the Boundaries of Flooded Zones Based on Hydrodynamic Modeling

2020 ◽  
pp. 12-22
Author(s):  
Tatyana Dyakonova ◽  
Vladislava Krivko ◽  
Ekaterina Agafonnikova ◽  
Anna Klikunova ◽  
Evgeniy Sokolovskiy ◽  
...  
Keyword(s):  
Shallow Water ◽  
Spatial Data ◽  
Single Layer ◽  
Hydrodynamic Modeling ◽  
Water Levels ◽  
Water Model ◽  
Inundation Maps ◽  
Cadastral Maps ◽  
The Republic ◽  
State Registration

The paper discusses the problem of determining the boundaries of flooding zones, flooding of territories. The paper describes technique for constructing cadastral maps of flooded zones for various values of the estimated availability of water levels, which is based on hydrodynamic and geoinformation modeling. For creating digital elevation models (DEM) of the conformation, we use spatial data obtained from a large number of sources. We applied the described technique to construct cadastral maps for a number of settlements in the Volgograd region of the Russian Federation. As a result, we get a set of xml-files prepared for registration with the Federal Agency for State Registration, Cadastre and Cartography. For the Republic of Belarus, the problem of flooding the territories is also urgent. The floodplain areas of the Pripyat River are most prone to flooding. We built a DEM for the area in the vicinity of Petrikov city and carried out hydrodynamic modeling. As a result, the inundation maps were obtained. The method proposed in this work is based on a mathematical model of shallow water in a single-layer approximation. This model is successfully applied in simulation of flooding for territories by flood waters. Solving the problem of determining the boundaries of flooding zones requires a transition to a two-layer shallow water model that takes into account the interaction of surface and ground waters.

scholarly journals Simply-connected vortex-patch shallow-water quasi-equilibria

2014 ◽  
Vol 743 ◽  
pp. 481-502 ◽  
Author(s):  
H. Płotka ◽  
D. G. Dritschel
Keyword(s):  
Shallow Water ◽  
Single Layer ◽  
Water Model ◽  
Shallow Water Model ◽  
Geophysical Fluid Dynamics ◽  
Vortex Patch ◽  
Horizontal Size ◽  
The Stability ◽  
Simply Connected ◽  
Boundary Of Stability

AbstractWe examine the form, properties, stability and evolution of simply-connected vortex-patch relative quasi-equilibria in the single-layer $f$-plane shallow-water model of geophysical fluid dynamics. We examine the effects of the size, shape and strength of vortices in this system, represented by three distinct parameters completely describing the families of the quasi-equilibria. Namely, these are the ratio $\gamma = L/L_D$ between the horizontal size of the vortices and the Rossby deformation length; the aspect ratio $\lambda $ between the minor to major axes of the vortex; and a potential vorticity (PV)-based Rossby number $\mathit{Ro}= q^{\prime }/f$, the ratio of the PV anomaly $q^{\prime }$ within the vortex to the Coriolis frequency $f$. By defining an appropriate steadiness parameter, we find that the quasi-equilibria remain steady for long times, enabling us to determine the boundary of stability $\lambda _c=\lambda _c(\gamma ,\mathit{Ro})$, for $0.25 \leq \gamma \leq 6$ and $\delimiter "026A30C \mathit{Ro}\delimiter "026A30C \leq 1$. By calling two states which share $\gamma ,\delimiter "026A30C \mathit{Ro}\delimiter "026A30C $ and $\lambda $ ‘equivalent’, we find a clear asymmetry in the stability of cyclonic ($\mathit{Ro}> 0$) and anticyclonic ($\mathit{Ro}<0$) equilibria, with cyclones being able to sustain greater deformations than anticyclones before experiencing an instability. We find that ageostrophic motions stabilise cyclones and destabilise anticyclones. Both types of vortices undergo the same main types of unstable evolution, albeit in different ranges of the parameter space, (a) vacillations for large-$\gamma $, large-$\mathit{Ro}$ states, (b) filamentation for small-$\gamma $ states and (c) vortex splitting, asymmetric for intermediate-$\gamma $ and symmetric for large-$\gamma $ states.

Potential vorticity redistribution by localised transient forcing in the shallow-water model

2018 ◽  
Vol 852 ◽  
pp. 199-225 ◽  
Author(s):  
Michael C. Haigh ◽  
Pavel S. Berloff
Keyword(s):  
Shallow Water ◽  
Ocean Circulation ◽  
Potential Vorticity ◽  
Single Layer ◽  
Current Theory ◽  
Water Model ◽  
Eddy Flux ◽  
Shallow Water Model ◽  
Background Flow ◽  
Uniform Background

This study is motivated by the need to develop stochastic parameterisations for representing the effects of mesoscale oceanic eddies in non-eddy-resolving and eddy-permitting ocean circulation models. A necessary logical step on the way to such parameterisations is the understanding of flow responses to spatially stationary and localised, time-dependent ‘plunger’ forcings intended to represent transient eddy flux divergences. Specifically, this study develops an understanding of the plunger-induced convergence of potential vorticity (PV) fluxes using the linearised single-layer shallow-water model. Time-periodic solutions are obtained and the ‘footprint’, defined as the time-mean, quasi-linear PV flux convergence, quantifies the cumulative PV redistribution induced by the plunger. Using the footprint, the equivalent eddy flux (EEF) is defined such that it succinctly quantifies the extent of the PV redistribution, and its dependencies on the forcing latitude and the background flow are examined in detail. For a uniform background flow the EEF is positive for all forcing latitudes, corresponding to net-poleward PV flux convergence, as expected by current theory of $\unicode[STIX]{x1D6FD}$-plane Rossby waves. The EEF also has a robust dependence on the direction and magnitude of a uniform background flow, which is a useful quality for the EEF to provide a basis for a parameterisation of eddy PV fluxes. We also examine the PV redistribution due to forcing on top of a Gaussian jet background flow and find that forcing proximity to the jet core is the primary factor in determining whether the jet is sharpened or broadened.

Centrifugal, barotropic and baroclinic instabilities of isolated ageostrophic anticyclones in the two-layer rotating shallow water model and their nonlinear saturation

2014 ◽  
Vol 762 ◽  
pp. 5-34 ◽  
Author(s):  
Noé Lahaye ◽  
Vladimir Zeitlin
Keyword(s):  
Shallow Water ◽  
Initial Conditions ◽  
Vertical Shear ◽  
Water Model ◽  
Shallow Water Model ◽  
Nonlinear Saturation ◽  
Centrifugal Instability ◽  
Wide Range ◽  
Nonlinear Mechanism ◽  
Rotating Shallow Water

AbstractInstabilities of isolated anticyclonic vortices in the two-layer rotating shallow water model are studied at Rossby numbers up to two, with the main goal to understand the interplay between the classical centrifugal instability and other ageostrophic instabilities. We find that different types of instabilities with low azimuthal wavenumbers exist, and may compete. In a wide range of parameters, an asymmetric version of the standard centrifugal instability has larger growth rate than the latter. The dependence of the instabilities on the parameters of the flow, i.e. Rossby and Burger numbers, vertical shear and the ratios of the layers’ thicknesses and densities, is investigated. The zones of dominance of each instability are determined in the parameter space. Nonlinear saturation of these instabilities is then studied with the help of a high-resolution finite-volume numerical scheme, by using the unstable modes identified from the linear stability analysis as initial conditions. Differences in nonlinear development of the competing centrifugal and ageostrophic barotropic instabilities are evidenced. A nonlinear mechanism of axial symmetry breaking during the saturation of the centrifugal instability is displayed.

Hurricane Predictability Analysis with Singular Vectors in the Multiresolution Global Shallow Water Model

2021 ◽  
Author(s):  
Xiaoxu Tian ◽  
Kayo Ide
Keyword(s):  
Shallow Water ◽  
Environmental Flows ◽  
Single Layer ◽  
Hurricane Sandy ◽  
Water Model ◽  
Time Interval ◽  
Variable Resolution ◽  
Singular Vectors ◽  
Predictability Analysis ◽  
Computational Resources

AbstractIn this study, the tangent linear and adjoint (TL/AD) models for the Model for Prediction Across Scales (MPAS) Shallow Water (SW) component are tested and demonstrated. Necessary verification check procedures of TL/AD are included to ensure that the models generate correct results. The TL/AD models are applied to calculate the singular vectors (SVs) with a 48-hour optimization time interval (OTI) under both the quasi-uniform resolution (UR) and smoothly variable resolution (VR) meshes in the cases of Hurricanes Sandy (2012) and Joaquin (2015). For the global domain, the VR mesh with 30210 grid cells uses slightly fewer computational resources than the UR mesh with 40962 cells. It is found that at the points before Hurricanes Sandy and Joaquin made sharp turns, the leading SV from the VR experiment show sensitivities in both areas surrounding the hurricane and those relatively far away, indicating the significant impacts from the environmental flows. The leading SVs from the UR experiments are sensitive to only areas near the storm. Forecasts by the nonlinear SWmodel demonstrate that in the VR experiment, Hurricane Sandy has a northwest turn similar to the case in the real world while the storm gradually disappeared in the UR experiment. In the case of Hurricane Joaquin, the nonlinear forecast with the VR mesh can generate a track similar to the Best Track, while the storm became falsely dissipated in the forecast with the UR mesh. These experiments demonstrate, in the context of SW dynamics with a single layer and no physics, the track forecasts in the cases of Hurricanes Sandy and Joaquin with the VR mesh are more realistic than the UR mesh. The SV analyses shed light on the key features that can have significant impacts on the forecast performances.

A two-layer shallow flow model with two axes of integration with application to submarine avalanches and generated tsunamis

2021 ◽  
Author(s):  
Enrique D. Fernandez-Nieto ◽  
François Bouchut ◽  
Juan M. Delgado-Sanchez ◽  
Anne Mangeney ◽  
Gladys Narbona-Reina
Keyword(s):  
Shallow Water ◽  
Granular Layer ◽  
Water Layer ◽  
Type Model ◽  
Water Type ◽  
Time Step ◽  
Cartesian Coordinates ◽  
Local Coordinates ◽  
Averaged Model ◽  
Granular Surface

&lt;p&gt;There exits in the literature many approaches that has been used to model submarine avalanches (See [5]). These models are mainly based on the pioneer work of Savage and Hutter (SH) [4] that is a shallow water type model for aerial avalanches, which is written in local coordinates, in order to simulate the tangential velocity to the bottom. A depth-averaged SH model over a general bottom with curvature was introduced in [1]. An extension to submarine avalanches is developed in [2]. In this paper the same local coordinate system is used for the two layers. Nevertheless, using a local coordinates the model would prescribe the perturbation at the surface at a wrong placement. In [3] a bilayer depth-averaged model for submarine avalanches is presented with cartesian coordinates for the water layer and local coordinates for the avalanche. The drawback is that the seabed deformation is considered as an input data for the water layer equations, then no interaction between the two fluids are taken into account and it is necessary to do an interpolation of the granular surface at each time step of the numerical simulation. In this work we present firstly the details of the proposed model, a coupled two-layer shallow water system where we consider local coordinates for the granular layer and cartesian coordinates for the fluid one. The main difference with other models that adopt the same stragie is that any interpolation of the granular surface is required. Moreover, the velocity of the granular layer has an explicit influence on the mass and momentum conservation laws of the fluid layer. Secondly, several numerical tests will be presented.&lt;/p&gt;&lt;p&gt;References&lt;/p&gt;&lt;p&gt;[1] F. Bouchut, E.D. Fern&amp;#225;ndez-Nieto, A. Mangeney, and P.Y. Lagr&amp;#233;e. On new erosion models of Savage-Hutter type for avalanches. Acta Mechanica, 199(1):181--208, 2008.&lt;br&gt;[2] E.D. Fern&amp;#225;ndez-Nieto, F. Bouchut, D. Bresch, M.J. Castro D&amp;#237;az, and A. Mangeney. A new Savage-Hutter type model for submarine avalanches and generated tsunami. Journal of Computational Physics, 227(16):7720--7754, 2008.&lt;br&gt;[3] P.H. Heinrich, A. Piatanesi, and H. H&amp;#233;bert. Numerical modelling of tsunami generation and propagation from submarine slumps: the 1998 papua new guinea event. Geophysical Journal International, 145(1):97--111, 2001.&lt;br&gt;[4] S. B. Savage and K. Hutter. The dynamics of avalanches of granular materials from initiation to runout. part I: Analysis. Acta Mechanica, 86(1):201&amp;#8211;223, 1991.&lt;br&gt;[5] S. Yavari-Ramshe and B. Ataie-Ashtiani. Numerical modeling of subaerial and submarine landslide-generated tsunami waves-recent advances and future challenges. Landslides, 13(6):1325&amp;#8211;1368, 2016.&lt;/p&gt;

scholarly journals The Influence of Topography on the Stability of the Norwegian Atlantic Current off Northern Norway

2018 ◽  
Vol 48 (11) ◽  
pp. 2761-2777 ◽  
Author(s):  
Peygham Ghaffari ◽  
Pål Erik Isachsen ◽  
Ole Anders Nøst ◽  
Jan Erik Weber
Keyword(s):  
Shallow Water ◽  
Continental Slope ◽  
Bottom Topography ◽  
Baroclinic Instability ◽  
Mesoscale Eddy ◽  
Water Model ◽  
Barotropic Instability ◽  
Northern Norway ◽  
The Stability ◽  
Atlantic Current

AbstractThe steep continental slope off the Lofoten–Vesterålen islands of northern Norway appears to be the source of the most intense mesoscale eddy field in all of the Nordic Seas. Here we use linearized two-layer shallow-water equations to study the stability of the Norwegian Atlantic Current in this region. The study extends previous works that used linearized quasigeostrophic vertical mode equations to examine the effects of bottom topography on baroclinic instability here. We find evidence of baroclinic instability in the stacked shallow-water model but also of barotropic instability that is associated with the upper-layer lateral shear. The calculations give an indication that growth rates of barotropic instability may be comparable to or larger than those of baroclinic instability over the steepest parts of the continental slope.

The instability of vertically curved fronts in a two-layer shallow water model

2020 ◽  
Vol 32 (12) ◽  
pp. 124117
Author(s):  
M. W. Harris ◽  
F. J. Poulin ◽  
K. G. Lamb
Keyword(s):  
Shallow Water ◽  
Water Model ◽  
Shallow Water Model
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