Non-hydrostatic pressure in σ-coordinate ocean models

Numerical ocean modelling is computationally very demanding. Traditionally, the hydrostatic approximation has been applied to reduce the computational burden. This approximation is valid in large scale studies with coarse grid resolution. With faster computers and gradually smaller grid sizes, we may expect that more studies will be performed with non-hydrostatic ocean models. In recent papers several methods for including non-hydrostatic pressure in σ-coordinate models have been suggested. In this paper the sensitivity of the non-hydrostatic pressure field, the velocity fields, and the density fields to changes in the method for computing non-hydrostatic pressure in σ-coordinate ocean models is addressed.The first test case used involves the propagation and breaking of an internal wave at an incline in a tank. The other test case concerns tidally driven flow over a sill in a stratified fjord. The results from our numerical exercises suggest that the velocity and density fields are very robust to the model choices investigated here. The differences between the model results are of the same order as the uncertainty due to the internal pressure gradient error, and they are smaller than an estimate of the uncertainty due to subgrid scale closure.     

A non-hydrostatic ocean model with a scalable multigrid Poisson solver

A non-hydrostatic ocean model using an effective Poisson solver is developed. The Poisson solver is a combination of the multigrid method, the Krylov-subspace method, and the sparse approximate inverse. Its numerical cost only linearly increases with total number of computational cells, and it also has high parallel computing efficiency. The numerical cost of the non-hydrostatic model described in the present paper remains only twice of that of a hydrostatic model, even with non-smooth topography and with a huge number of computational grid cells on massively parallelized computer systems. Therefore, it has a potential to expand the applicability of non-hydrostatic ocean models. We also present the preliminary result of the high-resolution non-hydrostatic experiment on Ice Shelf Water overflow in the southern Weddell Sea, which shows good agreement with observations in terms of the pathway of dense water and velocity field.     

Effects of hydrostatic approximation and resolution on the simulation of convective adjustment

A two-dimensional non-hydrostatic ocean model and a hydrostatic version of the same model are used to simulate convective adjustment, without the use of an instantaneous adjustment parameterization. The model geometry is a domain on the vertical plane of width 40 km and depth 500 m. Model results for four cases are examined: hydrostatic and non-hydrostatic, at 0.1 and 1 km spatial resolution. The convectively adjusted stable state obtained in all four cases are qualitatively similar; thus the hydrostatic approximation does not eliminate convective adjustment. The details of the simulated convective plumes depend on resolution and whether the hydrostatic approximation is made. The adjusted state has significant stratification which cannot be captured by the conventional instantaneous adjustment or diffusion-based parameterizations. We also compare the results to the case when an instantaneous adjustment parameterization is used.     

A non-hydrostatic algorithm for free-surface ocean modelling

An original implementation of a non-hydrostatic, free-surface algorithm based on a pressure correction method is proposed for ocean modelling. The free surface is implemented through an explicit scheme combined with a mode-spitting method but the depth-averaged velocity and the position of the free surface are updated at each non-hydrostatic iteration. The vertical momentum equation is also integrated up to the surface enabling a natural and accurate treatment of the surface layer. The consistent specification of the numerical schemes provides balanced transfers of potential and kinetic energy. This algorithm is well-suited for implementation as a non-hydrostatic kernel on originally hydrostatic free-surface ocean models such as Symphonie (http://poc.obs-mip.fr/pages/research_topics/modelling/symphonie/symphonie.htm) for which it has originally been developed.Energy balances associated with the propagation of short surface waves and solitary waves are presented for two dedicated well-documented configurations over closed domains. The buoyancy flux, the work rate of the pressure force together with the power of the advective terms are evaluated and discussed for the generation and the propagation of these two types of waves. The dissipation rate is in particular shown to be several orders of magnitude smaller than the work rates of the hydrostatic and non-hydrostatic pressure forces confirming the necessity for the exchanges of energy to be numerically balanced. The algorithm is subsequently applied to the complex generation of non-linear solitary internal waves by surface tides over Georges Bank, in the Gulf of Maine. The generation and the propagation of the observed non-linear and non-hydrostatic features in this region are correctly reproduced.     

大洋环流模式中Boussinesq近似与静压近似问题的讨论

改进环流模式中的Boussinesq近似与静压近似是当前海洋环流模式的主要研究方向之一。虽然目前已提出一些在模式中包含非Boussinesq效应和静压效应的方法,但是仍有很多关键问题亟待解决。系统地回顾了改进这2个基本近似的理论分析和数值模式的研究进展,介绍了不同方法的理论基础及其数值分析结果,并分析了这些方法中所存在的缺陷和问题,指出了开展进一步研究需要首先分析的几个关键问题。     

Modeling vertical motion at ocean fronts: Are nonhydrostatic effects relevant at submesoscales?

Through a suite of three-dimensional, high-resolution numerical modeling experiments, we examine the role of nonhydrostatic effects on O(1 km) submesoscale processes at ocean fronts, with particular focus on the vertical velocity field. Several differences between nonhydrostatic and hydrostatic models are pointed out using a framework that enables precise comparison, but it is difficult to identify categorical differences between the model solutions at the grid resolutions afforded. The instantaneous vertical velocity structure is sensitive to the model choice and, even more so, to grid resolution, but the average vertical flux is similar in both hydrostatic and nonhydrostatic cases.When a frontal region with horizontal density gradients is perturbed by wind, a profusion of submesoscale, O(1 km), secondary circulation features develops in the upper 50 m. Narrow, elongated cells of intense up- and down-welling are found to occur close to the surface, overlying broader regions of weaker up- and down-welling associated with the mesoscale meanders of the baroclinically unstable front. The submesoscale down-welling is considerably stronger than up-welling and is concentrated in 1–2 km width filaments within which velocities can attain magnitudes as high as 200 m day⁻¹. The submesoscale features are found to be robust at horizontal grid resolutions varying between 1 and 0.25 km and exist even in the hydrostatic model. Submesoscale circulation is difficult to observe or resolve in coarser resolution circulation models, but is likely to play a significant role in the exchange of energy and properties between the surface ocean and thermocline. Possible mechanisms for the generation of these features are investigated in a follow-on paper.     

Numerical Experiments of Isolated Convection under Polynya

The dense water formation process under polynya or lead is examined by numerical experiments using a three-dimensional non-hydrostatic model. Many numerical experiments on isolated convection in an initially homogeneous fluid have been performed for different sets of external parameters, in order to investigate a relationship between the convection process and the external parameters. The main focus is on the situation in which the horizontal length scale of disk-shaped buoyancy forcing (radius R) is comparable with the total water depth (H). The two dynamical regimes described in previous work—the baroclinically unstable convection and the baroclinically stable convection—are confirmed in the experiments. A horizontal shift of a convective chimney is important to a density anomaly in baroclinically stable convection. For the stable range, as R is reduced R/H < 0.7, a new regime is found, called “single-plume convection”, in which multiple convective plumes do not fully develop, and the density anomaly scale has nearly no dependency on R. This change of dependency on R is consistent with that derived by scaling analysis. The non-hydrostatic component is more significant than the hydrostatic one in the single-plume convection. The information obtained is useful for parameterizing dense water formation under ice cover in a numerical model with a large grid size; i.e., the newly formed water has a density anomaly independent of the polynya size smaller than the water depth, while the anomaly increases as the size exceeds the depth.     

Effects of rotation on self-resonant internal gravity waves in the ocean

The Resonant Triad Model (RTM) developed in (Ibragimov, 2007), is used to study the Thorpe’s problem (Thorpe, 1997) on the existence of self-resonant internal waves, i.e., the waves for which a resonant interaction occurs at second order between the incident and reflected internal waves off slopes. The RTM represents the extension of the McComas and Bretherton’s three wave hydrostatic model (McComas and Bretherton, 1977) which ignores the effects of the earth’s rotation to the case of the non-hydrostatic analytical model involving arbitrarily large number of rotating internal waves with frequencies spanning the range of possible frequencies, i.e., between the maximum of the buoyancy frequency (vertical motion) and a minimum of the inertial frequency (horizontal motion). The present analysis is based on classification of resonant interactions into the sum, middle and difference interaction classes. It is shown in this paper that there exists a certain value of latitude, which is classified as the singular latitude, at which the coalescence of the middle and difference interaction classes occurs. Such coalescence, which apparently had passed unnoticed before, can be used to study the Thorpe’s problem on the existence of self-resonant waves. In particular, it is shown that the value of the bottom slope at which the second-order frequency and wavenumber components of the incident and reflected waves satisfy the internal wave dispersion relation can be approximated by two latitude-dependent parameters in the limiting case when latitude approaches its singular value. Since the existence of a such singular latitude is generic for resonant triad interactions, a question on application of the RTM to the modeling of enhanced mixing in the vicinity of ridges in the ocean arises.     

Algorithm for non-hydrostatic dynamics in the Regional Oceanic Modeling System

A non-hydrostatic algorithm for the Regional Oceanic Modeling System (ROMS) is proposed. It is based on a decomposition technique for hydrostatic and non-hydrostatic pressure. The algorithm has a pressure-correction scheme with split-explicit time-stepping for baroclinic and barotropic vertical modes with a free surface. The algorithm implementation requires solving a Poisson equation for a non-hydrostatic pressure that has a non-symmetric matrix in discrete form. The efficiency of a different class of solvers and preconditioners were tested. The algorithm is successfully implemented with several examples where non-hydrostatic effects are important. These include standing external gravity waves; strongly nonlinear internal wave generation and transformation; stratified shear instability and its associated mixing; and nonlinear internal tidal generation over a ridge. The corresponding changes in the pre-processing and post-processing infrastructure in the existing hydrostatic ROMS code were performed to implement parallel elliptic solvers and a new set of dynamical equations.     

An optical model for the remote sensing of coloured dissolved organic matter in coastal/ocean waters

An optical model is developed for the remote sensing of coloured dissolved organic matter (CDOM) in a wide range of waters within coastal and open ocean environments. The absorption of CDOM (denoted as a_g) is generally considered as an exponential form model, which has two important parameters – the slope S and absorption of CDOM at a reference wavelength a_g(λ₀). The empirical relationships for deriving these two parameters are established using in-situ bio-optical datasets. These relationships use the spectral remote sensing reflectance (R_rs) ratio at two wavelengths R_rs(670)/R_rs(490), which avoids the known atmospheric correction problems and is sensitive to CDOM absorption and chlorophyll in coastal/ocean waters. This ratio has tight relationships with a_g(412) and a_g(443) yielding correlation coefficients between 0.77 and 0.78. The new model, with the above parameterization applied to independent datasets (NOMAD SeaWiFS match-ups and Carder datasets), shows good retrievals of the a_g(λ) with regression slopes close to unity, little bias and low mean relative and root mean square errors. These statistical estimates improve significantly over other inversion models (e.g., Linear Matrix-LM and Garver-Siegel-Maritorena-GSM semi-analytical models) when applied to the same datasets. These results demonstrate a good performance of the proposed model in both coastal and open ocean waters, which has the potential to improve our knowledge of the biogeochemical cycles and processes in these domains.     

Natural air–sea flux of CO2 in simulations of the NASA-GISS climate model: Sensitivity to the physical ocean model formulation

Results from twin control simulations of the preindustrial CO₂ gas exchange (natural flux of CO₂) between the ocean and the atmosphere are presented here using the NASA-GISS climate model, in which the same atmospheric component (modelE2) is coupled to two different ocean models, the Russell ocean model and HYCOM. Both incarnations of the GISS climate model are also coupled to the same ocean biogeochemistry module (NOBM) which estimates prognostic distributions for biotic and abiotic fields that influence the air–sea flux of CO₂. Model intercomparison is carried out at equilibrium conditions and model differences are contrasted with biases from present day climatologies. Although the models agree on the spatial patterns of the air–sea flux of CO₂, they disagree on the strength of the North Atlantic and Southern Ocean sinks mainly because of kinematic (winds) and chemistry (pCO₂) differences rather than thermodynamic (SST) ones. Biology/chemistry dissimilarities in the models stem from the different parameterizations of advective and diffusive processes, such as overturning, mixing and horizontal tracer advection and to a lesser degree from parameterizations of biogeochemical processes such as gravitational settling and sinking. The global meridional overturning circulation illustrates much of the different behavior of the biological pump in the two models, together with differences in mixed layer depth which are responsible for different SST, DIC and nutrient distributions in the two models and consequently different atmospheric feedbacks (in the wind, net heat and freshwater fluxes into the ocean).     

Non-hydrostatic Kelvin waves in the ocean

A theory of the coastal Kelvin wave is presented in which the pressure is assumed not to be hydrostatic. The results show that the non-hydrostatic Kelvin wave is dispersive and that the e^–1 decay distance of the wave amplitude from the coast decreases as the wave length becomes shorter. Similar conclusions can be drawn on the equatorial Kelvin wave.     

The solubility of CaCO3 in seawater at 2°C based upon in-situ sampled pore water composition

Analyses of the concentration product (Ca²⁺) × (CO₃²⁻) in the pore waters of marine sediments have been used to estimate the apparent solubility products of sedimentary calcite (K′_SPc) and aragonite (K′_SPa) in seawater. Regression of the data gives the relation In K′^P_SPc = 1.94 × 10⁻³ δP − 14.59 The 2°C, 1 atm value of K′_SPc is, then, 4.61 × 10⁻⁷ mol² l⁻². The pressure coefficient yields a at 2°C of −43.8 cm³ atm⁻¹. A single station where aragonite is present in the sediments gives a value of K′_SPa = 9.2 × 10⁻⁷ (4°C, 81 atm). The calcite data are very similar to those determined experimentally by Ingle et al. (1973) for K′_SPc at 2°C and 1 atm. The calculated is also indistinguishable from the experimental results of Ingle (1975) if is assumed to be independent of pressure.     

A Hybrid Finite-Volume and Finite Difference Scheme for Depth-Integrated Non-Hydrostatic Model

A depth-integrated, non-hydrostatic model with hybrid finite difference and finite volume numerical algorithm is proposed in this paper. By utilizing a fraction step method, the governing equations are decomposed into hydrostatic and non-hydrostatic parts. The first part is solved by using the finite volume conservative discretization method, whilst the latter is considered by solving discretized Poisson-type equations with the finite difference method. The second-order accuracy, both in time and space, of the finite volume scheme is achieved by using an explicit predictor-correction step and linear construction of variable state in cells. The fluxes across the cell faces are computed in a Godunov-based manner by using MUSTA scheme. Slope and flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purpose. Wave breaking is treated as a shock by switching off the non-hydrostatic pressure in the steep wave front locally. The model deals with moving wet/dry front in a simple way. Numerical experiments are conducted to verify the proposed model.     

Estimation of baroclinic tide energy available for deep ocean mixing based on three-dimensional global numerical simulations

The global distributions of the major semidiurnal (M₂ and S₂) and diurnal (K₁ and O₁) baroclinic tide energy are investigated using a hydrostatic sigma-coordinate numerical model. A series of numerical simulations using various horizontal grid spacings of 1/15–1/5° shows that generation of energetic baroclinic tides is restricted over representative prominent topographic features. For example, nearly half of the diurnal (K₁ and O₁) baroclinic tide energy is excited along the western boundary of the North Pacific from the Aleutian Islands down to the Indonesian Archipelago. It is also found that the rate of energy conversion from the barotropic to baroclinic tides is very sensitive to the horizontal grid spacing as well as the resolution of the model bottom topography; the conversion rate integrated over the global ocean increases exponentially as the model grid spacing is reduced. Extrapolating the calculated results in the limit of zero grid spacing yields the estimate of the global conversion rate to be 1105 GW (821, 145, 102, 53 GW for M₂, S₂, K₁, and O₁ tidal constituents, respectively). The amount of baroclinic tide energy dissipated in the open ocean below a depth of 1000 m, in particular, is estimated to be 500–600 GW, which is comparable to the mixing energy estimated by Webb and Suginohara (Nature 409:37, 2001) as needed to sustain the global overturning circulation.     

Verification of an operational ocean circulation-surface wave coupled forecasting system for the China's seas

An operational ocean circulation-surface wave coupled forecasting system for the seas off China and adjacent areas(OCFS-C) is developed based on parallelized circulation and wave models. It has been in operation since November 1, 2007. In this paper we comprehensively present the simulation and verification of the system, whose distinguishing feature is that the wave-induced mixing is coupled in the circulation model. In particular, with nested technique the resolution in the China's seas has been updated to(1/24)° from the global model with(1/2)°resolution. Besides, daily remote sensing sea surface temperature(SST) data have been assimilated into the model to generate a hot restart field for OCFS-C. Moreover, inter-comparisons between forecasting and independent observational data are performed to evaluate the effectiveness of OCFS-C in upper-ocean quantities predictions, including SST, mixed layer depth(MLD) and subsurface temperature. Except in conventional statistical metrics, non-dimensional skill scores(SS) is also used to evaluate forecast skill. Observations from buoys and Argo profiles are used for lead time and real time validations, which give a large SS value(more than 0.90). Besides, prediction skill for the seasonal variation of SST is confirmed. Comparisons of subsurface temperatures with Argo profiles data indicate that OCFS-C has low skill in predicting subsurface temperatures between 100 m and 150 m. Nevertheless, inter-comparisons of MLD reveal that the MLD from model is shallower than that from Argo profiles by about 12 m, i.e., OCFS-C is successful and steady in MLD predictions. Validation of 1-d, 2-d and 3-d forecasting SST shows that our operational ocean circulation-surface wave coupled forecasting model has reasonable accuracy in the upper ocean.     

静水压力变化对NTC热敏电阻器电学特性影响

负温度系数(negative temperature coefficient,NTC)热敏电阻器由于在窄温区内具有极高的灵敏度,在海洋领域中有着广泛的应用,但深海应用中固有的高静水压力,对器件的测温准确度造成不利影响。为了更好地认识这种现象,以珠状NTC热敏电阻器为研究对象,利用海洋环境高压模拟装置,开展了静水压力作用对器件电学特性影响的研究,固定环境温度25℃和0℃,压力范围0~60 MPa,步长5 MPa。结果显示,随着静水压力的增大,NTC热敏电阻器的电阻值减小;环境温度25℃和0℃时,5~60 MPa的静水压力造成的漂移分别为-0.11%~-2.81%、-0.19%~-2.78%,等效于温度漂移分别为0.024~0.624℃、0.036~0.535℃。这为热敏电阻器在深海温度准确测量的提升提供了指引,有利于在海洋测温领域中更好的应用。     

On the importance of resolving the western boundary layer in wind-driven ocean general circulation models

The western boundary layer (WBL) plays a fundamental role in basin-scale wind-driven ocean circulations. In idealized ocean models with flat bottom topography, this layer is required not only to balance the interior Sverdrup transport to close the gyre circulation, but also to dissipate the vorticity imposed by the wind-stress curl. The width of the WBL in Munk-type models is estimated to be δ_M(A_H/β)^1/3, where A_H and β are horizontal eddy viscosity and the meridional derivative of the Coriolis parameter respectively. For commonly used values of A_H, the boundary-layer width δ_M ranges from 30 to less than 200 km in the mid-latitude ocean. This scale is often poorly resolved in large-scale climate models.This paper intends to demonstrate some consequences when the western boundary layer is not adequately resolved. It is found that coarse resolution models reach equilibrium states by distorting some important dynamics in order to dissipate wind-imposed vorticity. In three-dimensional models, for instance, very strong spurious upwelling and downwelling can occur along the WBL. In models of two-dimensional flow, however, spurious recirculations may develop near the boundary. These false features can be removed when the boundary layer is better resolved. We propose a method in which a spatially varying A_H is used to broaden the WBL without affecting mixing in the interior. The method improves the model results considerably.     

Modeling of ocean dynamics with large variations in sea level

The formulation and the algorithm of solving an ocean model for the prediction and assimilation of the observed data which makes it possible to reconstruct the circulation in the deep-water parts of the sea and at a shallow water shelf, as well as to describe the large time–space variability in the surface level, are considered. The model uses a vertical hybrid σ–z coordinate system: the several upper tens of meters of the ocean are described in the σ-coordinate system and the rest of the water column is described in the z coordinates. Such hybridization extends the possibilities of models for reconstructing thermo-hydrodynamic processes in different sea basins and the World Ocean. The differential formulation of the model in the σ–z coordinate system is presented; the simplified records of several operators that are allowable in the case of a small thickness of the ocean σ-layer are described. The construction of a computational grid, approximation of the bottom topography on it, and discretization of equations and boundary conditions of the models are considered; an approach to describing the bottom friction at shallow waters is offered. The results of the comparative experiments in the z and σ–z coordinate models are analyzed.     

基于浮游植物吸收的海洋初级生产力模型的不确定性分析

Satellite-derived phytoplankton pigment absorption(a_(ph)) has been used as a key predictor of phytoplankton photosynthetic efficiency to estimate global ocean net primary production(NPP). In this study, an a_(ph)-based NPP model(Ab PM) with four input parameters including the photosynthetically available radiation(PAR), diffuse attenuation at 490 nm(K_d(490)), euphotic zone depth(Z_(eu)) and the phytoplankton pigment absorption coefficient(a_(ph)) is compared with the chlorophyll-based model and carbon-based model. It is found that the Ab PM has significant advantages on the ocean NPP estimation compared with the chlorophyll-based model and carbonbased model. For example, Ab PM greatly outperformed the other two models at most monitoring sites and had the best accuracy, including the smallest values of RMSD and bias for the NPP estimate, and the best correlation between the observations and the modeled NPPs. In order to ensure the robustness of the model, the uncertainty in NPP estimates of the Ab PM was assessed using a Monte Carlo simulation. At first, the frequency histograms of simple difference(δ), and logarithmic difference(δ~(LOG)) between model estimates and in situ data confirm that the two input parameters(Z_(eu) and PAR) approximate the Normal Distribution, and another two input parameters(a_(ph) and K_d(490)) approximate the logarithmic Normal Distribution. Second, the uncertainty in NPP estimates in the Ab PM was assessed by using the Monte Carlo simulation. Here both the PB(percentage bias), defined as the ratio of ΔNPP to the retrieved NPP, and the CV(coefficient of variation), defined as the ratio of the standard deviation to the mean are used to indicate the uncertainty in the NPP brought by input parameter to Ab PM model. The uncertainty related to magnitude is denoted by PB and the uncertainty related to scatter range is denoted by CV.Our investigations demonstrate that PB of NPP uncertainty brought by all parameters with an annual mean of5.5% covered a range of –5%–15% for the global ocean. The PB uncertainty of Ab PM model was mainly caused by a_(ph); the PB of NPP uncertainty brought by a_(ph) had an annual mean of 4.1% for the global ocean. The CV brought by all the parameters with an annual mean of 105% covered a range of 98%–134% for global ocean. For the coastal zone of Antarctica with higher productivity, the PB and CV of NPP uncertainty brought by all parameters had annual means of 7.1% and 121%, respectively, which are significantly larger than those obtained in the global ocean. This study suggests that the NPPs estimated by Ab PM model are more accurate than others, but the magnitude and scatter range of NPP errors brought by input parameter to Ab PM model could not be neglected,especially in the coastal area with high productivity. So the improving accuracy of satellite retrieval of input parameters should be necessary. The investigation also confirmed that the SST related correction is effective for improving the model accuracy in low temperature condition.