Shallow Water Numerical Model Research Articles

Revisiting the Rossby–Haurwitz wave test case with contour advection

This paper re-examines a basic test case used for spherical shallow-water numerical models, and underscores the need for accurate, high resolution models of atmospheric and ocean dynamics. The Rossby–Haurwitz test case, first proposed by Williamson et al. [D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob, P.N. Swarztrauber, A standard test set for numerical approximations to the shallow-water equations on the sphere, J. Comput. Phys. (1992) 221–224], has been examined using a wide variety of shallow-water models in previous papers. Here, two contour-advective semi-Lagrangian (CASL) models are considered, and results are compared with previous test results. We go further by modifying this test case in a simple way to initiate a rapid breakdown of the basic wave state. This breakdown is accompanied by the formation of sharp potential vorticity gradients (fronts), placing far greater demands on the numerics than the original test case does. We also go further by examining other dynamical fields besides the height and potential vorticity, to assess how well the models deal with gravity waves. Such waves are sensitive to the presence or not of sharp potential vorticity gradients, as well as to numerical parameter settings. In particular, large time steps (convenient for semi-Lagrangian schemes) can seriously affect gravity waves but can also have an adverse impact on the primary fields of height and velocity. These problems are exacerbated by a poor resolution of potential vorticity gradients.

2D Shallow-Water Model Using Unstructured Finite-Volumes Methods

This paper presents a two-dimensional (2D) shallow-water numerical model, which is based on the resolution of the Saint–Venant equation using the unstructured finite-volumes method, combined with Green’s theorem technique. The model has been validated by several benchmarks. The numerical results obtained from the model are in good agreement with the analytical or experimental ones. The paper also presents an application of this model to flood diversion from the Red River into a water-retention zone for the purpose of reducing flood threat at Hanoi, capital of Vietnam.

Reflection and Transmission of Equatorial Rossby Waves*

Abstract The interaction of equatorial Rossby waves with a western boundary perforated with one or more narrow gaps is investigated using a shallow-water numerical model and supporting theory. It is found that very little of the incident energy flux is reflected into eastward-propagating equatorial Kelvin waves provided that at least one gap is located within approximately a deformation radius of the equator. Because of the circulation theorem around an island, the existence of a second gap off the equator reduces the reflection of short Rossby waves and enhances the transmission of the incident energy into the western basin. The westward energy transmitted past the easternmost island is further reduced upon encountering islands to the west, even if these islands are located entirely within the “shadow” of the easternmost island. A localized patch of wind forcing was also used to generate low-frequency Rossby waves for cases with island configurations representative of the western equatorial Pacific. For both idealized islands and a coastline based on the 200-m isobath, the amount of incident energy reflected into Kelvin waves depends on both the duration of the wind event and the meridional decay scale of the anomalous winds. For wind events of 2-yr duration with a meridional decay scale of 700 km, the reflected energy is 37% of the incident flux, and the energy transmitted into the Indian Ocean is approximately 10% of the incident flux, very close to that predicted by previous theories. For shorter wind events or winds confined more closely to the equator the reflected energy is significantly less.

Time‐dependent response of the tropical atmosphere to a fixed sea surface temperature anomaly

The time‐dependent response of the tropical atmosphere to a fixed, localized sea surface temperature (SST) anomaly is explored using a highly simplified nonlinear shallow water numerical model. The work builds on that by Webster [1972] and Gill [1980]. The present model has three layers and is formulated on an equatorial beta plane in an atmosphere initially in radiative‐convective equilibrium. Observations suggest that the tropical atmosphere is only marginally unstable to moist convection. Accordingly, the convective parameterization in the model assumes that clouds consume the convective available potential energy at approximately the same rate as large‐scale processes generate it. The convective parameterization allows for both shallow nonprecipitating and deep precipitating clouds. The convection scheme allows the shallow convection to condition the atmosphere for further deep convection, which is an important factor controlling deep convection in the tropics. The model calculations show that shallow convection moistens the middle troposphere, providing favorable conditions for the development of deep convection. In contrast, radiative cooling and drying of the middle troposphere act to suppress deep convection. In the model, these competing processes modulate the deep convection over the localized SST anomaly with a period of about 30 days. The convective heating also excites large‐scale, equatorially trapped normal modes. The response ranges from a steady state flow similar to that by Gill to a periodic generation of Rossby‐Kelvin wave couplets and finally to a transition to chaotic behavior depending on the strength of the forcing.

Rotating Hydraulics and Upstream Basin Circulation*

The flow in a source-fed f-plane basin drained through a strait is explored using a single-layer (reduced gravity) shallow-water numerical model that resolves the hydraulic flow within the strait. The steady upstream basin circulation is found to be sensitive to the nature of the mass source (uniform downwelling, localized downwelling, or boundary inflow). In contrast, the hydraulically controlled flow in the strait is nearly independent of the basin circulation and agrees very well with the Gill-theory solution obtained using the strait geometry and the numerically determined average potential vorticity in the strait entrance region. This Gill solution, however, gives a unique value of the upstream boundary layer flux splitting that does not agree with any of the full numerical solutions. The coupled basin–strait system is shown to select an average overflow potential vorticity corresponding to the Gill solution with maximum fluid depth on the strait boundaries. This state also corresponds to one of maximal upstream basin potential energy. This result is robust to changes in the basin geometry, strait characteristics, the dissipation parameter (linear drag), and the net mass flux. The nonunique relation between basin conditions and overflow transport is significant with regard to deep overflow transport monitoring. It is shown that the potential vorticity selection leads to overflow, or “weir,” transport relations that are well approximated by the zero potential vorticity theory. However, accurate estimates of the transport can only be obtained if conditions within the strait entrance region, and not the basin, are used.

Rotating Hydraulics and Upstream Basin Circulation*

Abstract The flow in a source-fed f-plane basin drained through a strait is explored using a single-layer (reduced gravity) shallow-water numerical model that resolves the hydraulic flow within the strait. The steady upstream basin circulation is found to be sensitive to the nature of the mass source (uniform downwelling, localized downwelling, or boundary inflow). In contrast, the hydraulically controlled flow in the strait is nearly independent of the basin circulation and agrees very well with the Gill-theory solution obtained using the strait geometry and the numerically determined average potential vorticity in the strait entrance region. This Gill solution, however, gives a unique value of the upstream boundary layer flux splitting that does not agree with any of the full numerical solutions. The coupled basin–strait system is shown to select an average overflow potential vorticity corresponding to the Gill solution with maximum fluid depth on the strait boundaries. This state also corresponds to one of maximal upstream basin potential energy. This result is robust to changes in the basin geometry, strait characteristics, the dissipation parameter (linear drag), and the net mass flux. The nonunique relation between basin conditions and overflow transport is significant with regard to deep overflow transport monitoring. It is shown that the potential vorticity selection leads to overflow, or “weir,” transport relations that are well approximated by the zero potential vorticity theory. However, accurate estimates of the transport can only be obtained if conditions within the strait entrance region, and not the basin, are used.

Wind‐ and buoyancy‐forced upper ocean circulation in two‐strait marginal seas with application to the Japan/East Sea

The wind‐ and buoyancy‐forced upper ocean circulation in a marginal sea connected to the open ocean through two straits is investigated using idealized numerical and analytical models. The study is motivated by the Japan/East Sea (JES) and other marginal seas found along the western North Pacific. It is shown that for anticyclonic wind stress curl and atmospheric cooling in the marginal sea, the inflow transport branches into eastern and western boundary currents, in qualitative agreement with observed branching of the Tsushima Current in the southern JES. The eastern boundary current arises because wind forcing in the open ocean, and a circulation integral around the island that separates the marginal sea from the open ocean, maintains the temperature on the island to be warmer than that found in the interior of the marginal sea. Buoyancy forcing in the marginal sea plays a key role in maintaining the eastern boundary current. The dynamics that control the water mass transformation and downwelling are described and related to the model parameters. The largest heat loss to the atmosphere is found in the eastern boundary current. The exchange rates with the open ocean, downwelling within the marginal sea, and current structure within the marginal sea predicted by a linear analytic theory compare closely with results from a shallow water numerical model.

Significance of non-rotational wind stress in the seasonal variation of continental torque

The physical mechanism by which seasonally varying atmospheric wind stress exerted on thesea surface is communicated to the solid earth as oceanic pressure torque (continental torque)and bottom frictional torque is investigated with a linear shallow-water numerical model ofbarotropic oceans. The model has a realistic land’ocean distribution and is driven by a seasonallyvarying climatic wind stress. A novel way to decompose the wind stress into rotational andnon-rotational components is devised. The rotational component drives ocean circulations asclassical theories of wind-driven circulations demonstrate. The non-rotational component doesnot produce ocean circulations within the framework of a barotropic shallow-water model, butbalances with the pressure gradient force due to surface displacement in the steady state. Basedon this decomposition, it is shown that most of the continental torque which plays a majorrole in producing the seasonal variation of length of day (LOD) is caused by the non-rotationalcomponent of the wind stress. Both continental torque due to the wind-driven circulationproduced by the rotational component of the wind stress and the bottom frictional torque areof minor importance.

An analytical solution for stratified tidal lagoons: Application to the validation of a numerical open boundary condition

An analytical solution is presented for the case of a stratified, tidally forced lagoon. This solution, especially its energy and mass balances, is useful for the validation of numerical shallow water models under stratified, tidally forced conditions. The utility of the analytical solution for validation is demonstrated for a simple finite difference numerical model. The energy, mass, and their balances from the numerical model are shown to converge to the analytical solution with a power law dependence as spatial and temporal resolution are increased. The power law convergence of the numerical results to the analytical solution validates both the finite difference scheme and the open boundary conditions used for the tidal forcing. The open boundary conditions work well because they are consistent with the characteristics of the analytical solution.

VALIDATION OF NUMERICAL SHALLOW WATER MODELS FOR STRATIFIED SEICHES

A new analytical solution is presented for the case of a stratified seiche. This solution, especially its energetics, is useful for the validation of numerical shallow water models under stratified conditions. The utility of the analytical solution for validation is shown by using it to validate a simple finite difference numerical model. A comparison of the energetics of the numerical and analytical solutions reveals that the model results converge rapidly to the analytical solution with increasing resolution, such that a grid size of 30×30 would appear adequate for validation. In addition to properly resolving the spatial features, good temporal resolution is also necessary for validation, i.e use of a Courant number (Cr) less than one. For example, owing to the numerical dispersion of the present model, using Cr=5/4 rather than Cr =1/4 for the 50×50 grid resulted in 3·6 times larger RMS errors of model versus analytical barotropic available potential energy. This new analytical solution should be applied to a test suite of such validation tools before using such numerical models to simulate the more realistic geophysical flows encountered in lakes, bays, harbours and semi-enclosed seas under stratified conditions. © 1997 by John Wiley & Sons, Ltd.

Simulation of Stratospheric Vortex Erosion Using Three Different Global Shallow Water Numerical Models

ABSTRACT Simulation of planetary wave breaking and polar vortex erosion in the stratosphere has been carried out using three different global shallow water numerical models. The models are an Eulerian spectral model, a semi-Lagrangian finite difference model based on a vector discretization of the momentum equation and a semi-Lagrangian finite difference model based on the potential vorticity and divergence equations. The results of the integrations indicate advantages of the potential vorticity-based model in the vortex erosion problem.

The effect of wind‐wave enhancement of bottom stress on the circulation induced by tropical cyclones on continental shelves

The effect of increased bottom friction due to wind‐wave current interaction on continental shelves is examined with a two‐dimensional shallow water numerical model using an adaptation of the Grant and Madsen [1979] bottom‐boundary layer theory. In our formulation we allow for the wind‐wave amplitude to depend on the wind stress and bottom topography. We find that under strong tropical cyclones, the Grant and Madsen theory may break down in very shallow water where the wind‐wave amplitudes can become quite large. Since this is a region where the wind‐generated waves are particularly likely to be breaking, we introduce a wave‐breaking criterion into our bottom friction formulation. The effects of this wind‐wave‐induced bottom friction law on the wind‐driven circulation are then examined.

A global shallow-water numerical model based on the semi-Lagrangian advection of potential vorticity

A global shallow-water numerical model based on the semi-Lagrangian advection of potential vorticity

Variability in a homogeneous global ocean forced by barometric pressure

The nature of the oceanic response to pressure loading is explored using a constant-density, shallow-water numerical model driven by atmospheric pressure fields from the European Centre for Medium Range Weather Forecasts. The model has realistic bottom topography and coastlines and is run for 1 year (1986) on a global domain. Meridional gradients in mean sea-level are generally large (10–20 cm over 20–30°), particularly in high southern latitudes. Sea-level variability is strong in mid- and high latitudes (typical standard deviations of 10–15 cm), but weakens towards the equator. Results indicate a significant contribution of pressure-driven fluctuations to the observed large-scale sea-level variability in mid- and high latitudes, away from western boundary regions. Pressure-induced velocity signals are, in contrast, generally small compared with other types of variability. The validity of the inverted barometer approximation is found to be strongly dependent on frequency and geographical location. Globally, the approximation is not reliable for periods shorter than approximately 2 days, but failure at longer periods occurs over extensive regions (e.g. the tropical Atlantic and Pacific, and the Southern Ocean). Nonisostatic contributions to the sea-level variability are substantial in many areas, including the tropics, the high-latitude North Atlantic, the Gulf of Mexico, and several other boundary regions. The dynamical signals are partly associated with the excitation of several high-frequency normal modes. Some of these features have a spatial structure and period very similar to normal modes calculated by Platzman and collaborators. Their presence in the model indicates that atmospheric pressure forcing is a possible mechanism for normal mode excitation.

The 1987?88 Alaskan Bight tsunamis: Deep ocean data and model comparisons

Excellent deep ocean records have been obtained of two tsunamis recently generated in the Alaskan Bight on 30 November 1987 and 6 March 1988, providing the best available data set to date for comparison with tsunami generation/propagation models. Simulations have been performed with SWAN, a nonlinear shallow water numerical model, using source terms estimated by a seafloor deformation model based on the rectangular fault plane formalism. The tsunami waveform obtained from the model is quite sensitive to the specific source assumed. Significant differences were found between the computations and observations of the 30 November 1987 tsunami, suggesting inadequate knowledge of the source characteristics. Fair agreement was found between the data and the model for the first few waves of the 6 March 1988 tsunami. Model estimates of the seismic moment and total slip along the fault plane are also in fair agreement with those derived from the published Harvard centroid solution for the 6 March 1988 event, implying that the computed seafloor deformation does bear some similarity to the actual source.

Barotropic motions and the exchange of angular momentum between the oceans and solid Earth

The role of the ocean in transferring angular momentum between the atmosphere and the solid Earth is examined for periods longer than a day. Changes in the total angular momentum of the ocean can occur through the action of external torques such as surface wind stresses and continental torques (i.e., body forces acting on lateral boundaries), which represent momentum exchange with the atmosphere and the solid earth respectively. Analysis of these external torques suggests that in general, the related angular momentum exchanges are mostly associated with barotropic variability in the ocean. Solutions for a mid‐latitude basin, obtained with a shallow water numerical model forced by simple periodic zonal winds, illustrate how the momentum exchanged with the atmosphere is quickly transmitted to the solid Earth through the continental torque, with the bottom friction torque being negligible. For wind forcing confined to the middle of the ocean basin, both free waves and forced variability with large horizontal decay scales can participate in the interaction with the boundaries. The fast time scales involved in oceanic barotropic adjustment permit a rapid equilibration between the forcing regions and the coastal boundaries. The angular momentum transfer mechanism between the ocean and the solid Earth is consistent with the observed coherence at no discernible time lag between atmospheric and solid Earth angular momentum variations.

A simple model describing the nonlinear dynamics of the dusk/dawn asymmetry in the high‐latitude thermospheric flow

A simple nonlinear, axisymmetric, shallow‐water numerical model has been used to study the asymmetry in the neutral flow between the dusk and dawn sides of the auroral oval. Our results indicate that the Coriolis force and the curvature terms are nearly in balance on the evening side and require only a small pressure gradient to effect adjustment. The pressure gradient on the morning side has to balance both the Coriolis force and the curvature terms. The result is smaller neutral velocities near dawn and larger velocities near dusk than would be the case for a linearized treatment. A consequence is that more gravity wave energy is produced on the morning side than on the evening side.