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Y. Tony Song
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Y. Tony Song

Abstract

A Jacobian formulation of the pressure gradient force for use in models with topography-following coordinates is proposed. It can be used in conjunction with any vertical coordinate system and is easily implemented. Vertical variations in the pressure gradient are expressed in terms of a vertical integral of the Jacobian of density and depth with respect to the vertical computational coordinate. Finite difference approximations are made on the density field, consistent with piecewise linear and continuous fields, and accurate pressure gradients are obtained by vertically integrating the discrete Jacobian from sea surface.

Two discrete schemes are derived and examined in detail: the first using standard centered differencing in the generalized vertical coordinate and the second using a vertical weighting such that the finite differences are centered with respect to the Cartesian z coordinate. Both schemes achieve second-order accuracy for any vertical coordinate system and are significantly more accurate than conventional schemes based on estimating the pressure gradients by finite differencing a previously determined pressure field.

The standard Jacobian formulation is constructed to give exact pressure gradient results, independent of the bottom topography, if the buoyancy field varies bilinearly with horizontal position, x, and the generalized vertical coordinate, s, over each grid cell. Similarly, the weighted Jacobian scheme is designed to achieve exact results, when the buoyancy field varies linearly with z and arbitrarily with x, that is, b(x,z) = b 0(x) + b 1(x)z.

When horizontal resolution cannot be made fine enough to avoid hydrostatic inconsistency, errors can be substantially reduced by the choice of an appropriate vertical coordinate. Tests with horizontally uniform, vertically varying, and with horizontally and vertically varying buoyancy fields show that the standard Jacobian formulation achieves superior results when the condition for hydrostatic consistency is satisfied, but when coarse horizontal resolution causes this condition to be strongly violated, the weighted Jacobian may give superior results.

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Y. Tony Song
and
Yi Chao

Abstract

An embedded bottom boundary layer (EBBL) scheme is developed to improve the bottom topographic representation in z-coordinate ocean general circulation models. The EBBL scheme is based on the combined techniques of an embedded topography-following slab, an explicit turbulent bottom boundary layer (BBL), and a generalized pressure gradient formulation. The coupling between the interior z-level model and the EBBL model is achieved by exchanging entrainment/detrainment and pressure gradients at the bottom layer surface, which allows temporal and spatial variations.

The EBBL is implemented into one of the most widely used z-coordinate models, the Modular Ocean Model (MOM). A test problem with a source of dense water on a slope is used. The new EBBL produces significantly more realistic plume spreading than the existing BBL scheme of Killworth and Edwards and is comparable to the results from a topography-following coordinate model (SCRUM), which is believed to be more suitable for such a problem. Calculation of the momentum budget demonstrates that the improved representation of the downslope pressure gradient formulation plays an important role in the simulations of dense slope flows.

Sensitivity experiments with different grid sizes, model parameters, and density contrast between the cold source water and the warm interior water are carried out to test the robustness of the EBBL scheme. In contrast to the BBL model of Killworth and Edwards, which tends to diffuse too much dense water along isobaths, the EBBL model allows dense water to sink across isobaths through a very thin bottom layer into the deep ocean. Even in the coarser-resolution case (1/4° and 15 levels) the EBBL produces more realistic deep water than the existing BBL with higher resolution (1/8° and 30 levels), and at only one-eighth the computational cost. It is therefore concluded that the EBBL scheme presented here is cost effective and robust to model resolution and mixing parameters, and should be easily implemented in any nontopography-following coordinate ocean model.

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Zhigang Xu
and
Y. Tony Song

Abstract

This paper proposes an effective approach on how to predict tsunamis rapidly following a submarine earthquake by combining a real-time GPS-derived tsunami source function with a set of precalculated all-source Green's functions (ASGFs). The approach uses the data from both teleseismic and coastal GPS networks to constrain a tsunami source function consisting of both sea surface elevation and horizontal velocity field, and uses the ASGFs to instantaneously transfer the source function to the arrival time series at the destination points. The ASGF can take a tsunami source of arbitrary geographic origin and resolve it as fine as the native resolution of a tsunami propagation model from which the ASGF is derived. This new approach is verified by the 2011 Tohoku tsunami using data measured by the Deep-Ocean Assessment and Reporting of Tsunamis (DART) buoys.

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Y. Tony Song
and
Daniel G. Wright

Abstract

A new formulation of the pressure gradient force for use in models with topography-following coordinates is proposed and diagnostically analyzed in Part I. Here, it is shown that important properties of the continuous equations are retained by the resulting numerical schemes, and their performance in prognostic simulations is examined. Numerical consistency is investigated with respect to global energy conservation, depth-integrated momentum changes, and the representation of the bottom pressure torque. The performances of the numerical schemes are tested in prognostic integrations of an ocean model to demonstrate numerical accuracy and long-term integral stability. Two typical geometries, an isolated tall seamount and an unforced basin with sloping boundaries, are considered for the special case of no external forcing and horizontal isopycnals to test numerical accuracy. These test problems confirm that the proposed schemes yield accurate approximations to the pressure gradient force. Integral consistency conditions are verified and the energetics of the “advective elimination” of the pressure gradient error () is considered.

A large-scale wind-driven basin with and without topography is used to test the model’s long-term integral performance and the effects of bottom pressure torque on the transport in western boundary currents. Integrations are carried out for 10 years in each case and results show that the schemes are stable, and the steep topography causes no obvious numerical problems. A realistic meandering western boundary current is well developed with detached cold cyclonic and warm anticyclonic eddies as it extends across the basin. In addition, the results with topography show earlier separation and enhanced transport in the western boundary currents due to the bottom pressure torque.

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Samuel S. P. Shen
,
Gregory P. Behm
,
Y. Tony Song
, and
Tangdong Qu

Abstract

This paper provides a spectral optimal gridding (SOG) method to make a dynamically consistent reconstruction of water temperature for the global ocean at different depth levels. The dynamical consistency is achieved by using the basis of empirical orthogonal functions (EOFs) derived from NASA Jet Propulsion Laboratory (JPL) non-Boussinesq ocean general circulation model (OGCM) output at ¼° resolution from 1958 to 2013. A convenient singular value decomposition (SVD) method is used to calculate the EOFs, in order to enable efficient computing for a fine spatial grid globally. These EOFs are used as explainable variables and are regressed against the sparsely distributed in situ ocean temperature data at 33 standard depth levels. The observed data are aggregated onto a 1° latitude–longitude grid at each level from the surface to the 5500-m layer for the period 1950–2014. Three representative temperature reconstruction examples are presented and validated: two 10-m-layer (i.e., the second layer from the surface) reconstructions for January 2008 and January 1998, which are compared with independent sea surface temperature (SST) observations; and one 100-m-layer reconstruction for January 1998, which shows a strong cold anomaly El Niño signal in the western tropical Pacific up to −5°C from 150°E to 140°W. The SOG reconstruction can accurately locate the El Niño signal region in different ocean layers. The SOG reconstruction method is shown reliable and yields satisfactory accuracy even with sparse data. Validation and error analysis indicate that no systematic biases exist in the observed and reconstructed data.

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