A method to reduce the spin-up time of ocean models

The spin-up timescale in large-scale ocean models, i.e., the time it takes to reach an equilibrium state, is determined by the slow processes in the deep ocean and is usually in the order of a few thousand years. As these equilibrium states are taken as initial states for many calculations, much computer time is spent in the spin-up phase of ocean model computations. In this note, we propose a new approach which can lead to a very large reduction in spin-up time for quite a broad class of existing ocean models. Our approach is based on so-called Jacobian–Free Newton–Krylov methods which combine Newton’s method for solving non-linear systems with Krylov subspace methods for solving large systems of linear equations. As there is no need to construct the Jacobian matrices explicitly the method can in principle be applied to existing explicit time-stepping codes. To illustrate the method we apply it to a 3D planetary geostrophic ocean model with prognostic equations only for temperature and salinity. We compare the new method to the ‘ordinary’ spin-up run for several model resolutions and find a considerable reduction of spin-up time.     

A time-domain method to evaluate the use of moving weights to reduce the roll motion of a ship

A method to evaluate the use of actively controlled moving weights on board ships to reduce roll motion is developed. The weights can simulate in principle anti-roll-tank systems, or they can be considered a possible anti-roll device in their own right. The ship, the moving weight, and the control device are considered components of a single dynamic system. The full eight-degree-of-freedom set of coupled governing equations for the complete dynamic system is derived. And a three-degree-of-freedom non-linear approximation for the roll motion only (MOTSIM) is derived from these eight equations. The reduced set of equations is used to determine the influence of various parameters and to evaluate control strategies. A PID controller is developed to command the position of the weight and a servomechanism model is used to predict its actual position. Then, the moving-weight system is incorporated into LAMP (Large–Amplitude–Motion Program), a computer code that integrates the governing equations of the sea and the motion of the ship interactively and simultaneously and predicts the motion of the ship in the time domain. A comparison of the results from the two simulations shows that there is fairly good correlation between the simple and complex models, but the simple model is a little optimistic in predicting the effectiveness of the moving-weight system. The results predict that the moving-weight system can be an excellent roll-suppressing device when the moving weight is as small as 1% of the displacement of the ship and the maximum distance the weight moves is as small as 15% of the half-beam.     

On the identification of ship coupled heave–pitch motions using neural networks

This paper outlines a procedure for the derivation of the differential equations describing the free response of a heaving and pitching ship from its stationary response to random waves. The coupled heave–pitch motion of a ship in random seas is modelled as a multi-dimensional Markov process. The partial differential equation describing the transition probability density function, known as the Fokker-Planck equation, for this process is derived. The Fokker-Planck equation is used to derive the random decrement equations for the coupled heave–pitch motion. The parameters in these equations are then identified using a neural network approach. The method is validated using numerical simulations and experimental results. The experimental data was obtained using an icebreaker ship model heaving and pitching in random waves. It is shown that the method produces good results when the system is lightly damped. An extension for using this method to identify couple heave–pitch motion in realistic seas is suggested.     

Distinguishing coupled ocean–atmosphere interactions from background noise in the North Pacific

When considering physical mechanisms for decadal-timescale climate variability in the North Pacific, it is useful to describe in detail the expected response of the ocean to the chaotic atmospheric forcing. The expected response to this white-noise forcing includes strongly enhanced power in the decadal frequency band relative to higher frequencies, pronounced changes in basin-wide climate that resemble regime shifts, preferred patterns of spatial variability, and a depth-dependent profile that includes variability with a standard deviation of 0.2–0.4°C over the top 50–100 m. Weak spectral peaks are also possible, given ocean dynamics. Detecting coupled ocean–atmosphere modes of variability in the real climate system is difficult against the spectral and spatial structure of this ‘null-hypothesis’ of how the ocean and atmosphere interact, especially given the impossibility of experimentally decoupling the ocean from the atmosphere. Turning to coupled ocean–atmosphere models to address this question, a method for identifying coupled modes by using models of increasing physical complexity is illustrated. It is found that a coupled ocean–atmosphere mode accounts for enhanced variability with a time scale of 20 years/cycle in the Kuroshio extension region of the model's North Pacific. The observed Pacific Decadal Oscillation (PDO) has many similarities to the expected noise-forced response and few similarities to the model's coupled ocean–atmosphere variability. However, model deficiencies and some analyses of observations by other workers indicate that the possibility that part of the PDO arises from a coupled ocean–atmosphere mode cannot be ruled out.     

Application of a coupled discontinuous–continuous Galerkin finite element shallow water model to coastal ocean dynamics

A coupled discontinuous–continuous Galerkin (DG–CG) shallow water model is compared to a continuous Galerkin generalized wave-continuity equation (GWCE) based model for the coastal ocean, whereby local mass imbalance typical of GWCE-based solutions is eliminated using the coupled DG–CG approach. Two mass imbalance indicators for the GWCE-based model are presented and analyzed. The indicators motivate discussion on the suitability of using a GWCE-based model versus the locally conservative coupled DG–CG model. Both realistic and idealized test problems for tide, wind, and wave-driven circulation form the basis of the study. For the problems studied, coupled DG–CG solutions retain the robustness of well-documented solutions from GWCE-based models and also capture the dynamics driven by small-scale, highly advective processes which are problematic for GWCE-based models. Issues associated with the coupled DG–CG model are explored, including increased cost due to increased degrees of freedom, the necessary application of slope limiters, as well as the actual coupling process.     

Parametric vertical coordinate formulation for multiscale, Boussinesq, and non-Boussinesq ocean modeling

Two physical parameters are introduced into the basic ocean equations to generalize numerical ocean models for various vertical coordinate systems and their hybrid features. The two parameters are formulated by combining three techniques: the arbitrary vertical coordinate system of Kasahara [Kasahara, A., 1974. Various vertical coordinate systems used for numerical weather prediction. Mon. Weather Rev. 102, 509–522], the Jacobian pressure gradient formulation of Song [Song, Y.T., 1998. A general pressure gradient formation for ocean models. Part I: Scheme design and diagnostic analysis. Mon. Weather Rev. 126 (12), 3213–3230], and a newly introduced parametric function that permits both Boussinesq (volume-conserving) and non-Boussinesq (mass-conserving) conditions. Based on this new formulation, a generalized modeling approach is proposed. Several representative oceanographic problems with different scales and characteristics––coastal canyon, seamount topography, non-Boussinesq Pacific Ocean with nested eastern Tropics, and a global ocean model––have been used to demonstrate the model’s capabilities for multiscale applications. The inclusion of non-Boussinesq physics in the topography-following ocean model does not incur computational expense, but more faithfully represents satellite-observed ocean-bottom-pressure data. Such a generalized modeling approach is expected to benefit oceanographers in solving multiscale ocean-related problems by using various coordinate systems on the same numerical platform.     

A four-dimensional validation of a coupled physical–biological model of the Arabian Sea

In this paper, we use a coupled biological/physical model to synthesize and understand observations taken during the US JGOFS Arabian Sea Process Study (ASPS). Its physical component is a variable-density, -layer model; its biological component consists of a set of advective–diffusive equations in each layer that determine nitrogen concentrations in four compartments, namely, nutrients, phytoplankton, zooplankton, and detritus. Solutions are compared to time series and cruise sections from the ASPS data set, including observations of mixed-layer thickness, chlorophyll concentrations, inorganic nitrogen concentrations, particulate nitrogen export flux, zooplankton biomass, and primary production. Through these comparisons, we adjust model parameters to obtain a “best-fit” main-run solution, identify key biological and physical processes, and assess model strengths and weaknesses.Substantial improvements in the model/data comparison are obtained by: (1) adjusting the turbulence-production coefficients in the mixed-layer model to thin the mixed layer; (2) increasing the detrital sinking and remineralization rates to improve the timing and amplitude of the model's export flux; and (3) introducing a parameterization of particle aggregation to lower phytoplankton concentrations in coastal upwelling regions.With these adjustments, the model captures many key aspects of the observed physical and biogeochemical variability in offshore waters, including the near-surface DIN and phytoplankton P concentrations, mesozooplankton biomass, and primary production. Nevertheless, there are still significant model/data discrepancies of P for most of the cruises. Most of them can be attributed to forcing or process errors in the physical model: inaccurate mixed-layer thicknesses, lack of mesoscale eddies and filaments, and differences in the timing and spatial extent of coastal upwelling. Relatively few are clearly related to the simplicity of the biological model, the model's overestimation of coastal P being the most obvious example. Overall, we conclude that future efforts to improve biogeochemical models of the Arabian Sea should focus on improving their physical component, ensuring that it represents the ocean's physical state as closely as possible. We believe that this conclusion applies to coupled biogeochemical modeling efforts in other regions as well.     

Euler–Lagrange equations for the spectral element shallow water system

We present the derivation of the discrete Euler–Lagrange equations for an inverse spectral element ocean model based on the shallow water equations. We show that the discrete Euler–Lagrange equations can be obtained from the continuous Euler–Lagrange equations by using a correct combination of the weak and the strong forms of derivatives in the Galerkin integrals, and by changing the order with which elemental assembly and mass averaging are applied in the forward and in the adjoint systems. Our derivation can be extended to obtain an adjoint for any Galerkin finite element and spectral element system.We begin the derivations using a linear wave equation in one dimension. We then apply our technique to a two-dimensional shallow water ocean model and test it on a classic double-gyre problem. The spectral element forward and adjoint ocean models can be used in a variety of inverse applications, ranging from traditional data assimilation and parameter estimation, to the less traditional model sensitivity and stability analyses, and ensemble prediction. Here the Euler–Lagrange equations are solved by an indirect representer algorithm.     

JONSWAP's parameters: Sorting out the inconsistencies

The original prognostic equations for the JONSWAP-spectrum contained inconsistencies. A subsequent paper (Hasselmann et al., 1976, J. phys. Oceanogr.6, 200–208) attempted to regularise the situation. This paper shows that there were still inconsistencies in the prognostic equations giving overestimations of the first moment of the spectrum and consequently the significant wave height. The prognostic equations are reworked systematically and results presented. It is shown that variable σ_a, σ_b and γ must be used to achieve consistent results     

Assimilation of the data of observations and adaptive prediction of the natural processes

We propose a dynamical model for the prediction of random components of natural processes. The model is based on the system concept of adaptive balance of causes (ABC-model) and contains dynamic equations for the coefficients of influence adapted to the correlations existing in the predicted processes. To improve the accuracy of predictions, we consider two possible schemes of assimilation of the data of observations in the equations of the ABC-model, namely, the Kolmogorov and Kalman schemes. Both schemes are oriented toward the application of sample correlation coefficients for the prediction of time series of measurements and, hence, take into account the nonstationarity of actual natural processes. We present some examples of prediction of the simulated time series clarifying the algorithms of assimilation of the data of observations. A conclusion is made that the methods of systems modeling and adaptive prediction of random processes by the ABC-method are quite promising.     

Phase-decoupled refraction–diffraction for spectral wave models

Conventional spectral wave models, which are used to determine wave conditions in coastal regions, can account for all relevant processes of generation, dissipation and propagation, except diffraction. To accommodate diffraction in such models, a phase-decoupled refraction–diffraction approximation is suggested. It is expressed in terms of the directional turning rate of the individual wave components in the two-dimensional wave spectrum. The approximation is based on the mild-slope equation for refraction–diffraction, omitting phase information. It does therefore not permit coherent wave fields in the computational domain (harbours with standing-wave patterns are excluded). The third-generation wave model SWAN (Simulating WAves Nearshore) was used for the numerical implementation based on a straightforward finite-difference scheme. Computational results in extreme diffraction-prone cases agree reasonably well with observations, analytical solutions and solutions of conventional refraction–diffraction models. It is shown that the agreement would improve further if singularities in the wave field (e.g., at the tips of breakwaters) could be properly accounted for. The implementation of this phase-decoupled refraction–diffraction approximation in SWAN shows that diffraction of random, short-crested waves, based on the mild-slope equation can be combined with the processes of refraction, shoaling, generation, dissipation and wave–wave interactions in spectral wave models.     

An unsteady wave driver for narrowbanded waves: modeling nearshore circulation driven by wave groups

In this paper, we derive an unsteady refraction–diffraction model for narrowbanded water waves for use in computing coupled wave–current motion in the nearshore. The end result is a variable coefficient, nonlinear Schrödinger-type wave driver (describing the envelope of narrow-banded incident waves) coupled to forced nonlinear shallow water equations (describing steady or unsteady mean flows driven by the short-wave field). Comparisons with experimental data show that good accuracy can be obtained for cases of nonbreaking wave transformation. Numerical simulations show that the interaction of wave groups with longshore topographic nonuniformities generates strong edge wave resonances, providing a generating mechanism for low-order edge waves.     

Spatial and temporal structure of Tropical Pacific interannual variability in 20th century coupled simulations

Tropical Pacific interannual variability is examined in nine state-of-the-art coupled climate models, and compared with observations and ocean analyses data sets, the primary focus being on the spatial structure and spectral characteristics of El Niño-Southern Oscillation (ENSO). The spatial patterns of interannual sea surface temperature (SST) anomalies from the coupled models are characterized by maximum variations displaced from the coast of South America, and generally extending too far west with respect to observations. Thermocline variability is characterized by dominant modes that are qualitatively similar in all the models, and consistent with the “recharge oscillator” paradigm for ENSO. The meridional scale of the thermocline depth anomalies is generally narrower than observed, a result that can be related to the pattern of zonal wind stress perturbations in the central-western equatorial Pacific. The wind stress response to eastern equatorial Pacific SST anomalies in the models is narrower and displaced further west than observed. The meridional scale of the wind stress can affect the amount of warm water involved in the recharge/discharge of the equatorial thermocline, while the longitudinal location of the wind stress anomalies can influence the advection of the mean zonal temperature gradient by the anomalous zonal currents, a process that may favor the growth and longer duration of ENSO events when the wind stress perturbations are displaced eastwards. Thus, both discrepancies of the wind stress anomaly patterns in the coupled models with respect to observations (narrow meridional extent, and westward displacement along the equator) may be responsible for the ENSO timescale being shorter in the models than in observations. The examination of the leading advective processes in the SST tendency equation indicates that vertical advection of temperature anomalies tends to favor ENSO growth in all the CGCMs, but at a smaller rate than in observations. In some models it can also promote a phase transition. Longer periods tend to be associated with thermocline and advective feedbacks that are in phase with the SST anomalies, while advective tendencies that lead the SST anomalies by a quarter cycle favor ENSO transitions, thus leading to a shorter period.     

Prediction of natural processes by the method of adaptive balance of causes

The problem of estimation of the future values of processes is studied as a problem of their adaptation to the known data of observations in the past. The method of adaptive balance of causes (ABC-method) is used for the construction of a dynamic model of the coefficients of influence. This model enables one to compute the current values of these coefficients according to the current correlation matrix determined as a result of reanalysis of the observed processes. We propose an ABC-model with variable coefficients of influence guaranteeing the optimal (from the viewpoint of accuracy) prediction of natural processes and present an example of its application.Translated from Morskoi Gidrofizicheskii Zhurnal, No. 5, pp. 53–63, September–October, 2004.This revised version was published online in May 2005 with corrections to cover date.     

A three-dimensional hydrodynamic model of estuarine circulation with an application to Southampton Water, UK

A three-dimensional hydrodynamic model has been developed to simulate water mass circulation in estuarine systems. This model is based on the primitive equation in Cartesian coordinates with a terrain-following structure, coupled with a Mellor–Yamada 2.5 turbulence scheme. A fractional-step method is applied and the subset of equations is solved with finite volume and finite element methods. A dry–wet process simulates the presence of the tidal flat at low water. River inputs are introduced using a point-source method. The model was applied to a partially mixed, macrotidal, temperate estuary: Southampton Water, UK. The model is validated by comparisons with sea surface elevation, ADCP measurements and salinity data collected in 2001. The mean spring range 2(M₂ + S₂) and the mean neap range 2(M₂ − S₂) are modelled with an error relative to observation of 12 and 16%, respectively. The unique tidal regime of the system with the presence of the ‘young flood stand’ corresponding to the slackening conditions occurring at mid flood and ‘double high water’ corresponding to an extension of the slackening conditions at high tide is accurately reproduced in the model. The dynamics of the modelled mean surface and bottom velocity closely match the ADCP measurements during neap tides (rms of the difference is 0.09 and 0.01 m s⁻¹ at the bottom and at the surface, respectively), whereas at spring the difference is greater (rms of the difference is 0.25 and 0.20 m s⁻¹ at bottom and surface, respectively). The spatial and temporal variation of the degree of stratification as indicated by salinity distributions compares well with observations.     

Numerical simulation of the hydrodynamic behaviour of submerged plane nets in current

The aim of this paper is to investigate the shape and tension distribution of fishing nets in current. A numerical model is developed, based on lumped mass method to simplify the net. The motion equation is set up for each lumped mass. The Runge–Kutta–Verner fifth-order and sixth-order method is used to solve these simultaneous equations, and then the displacement and tension of each lumped mass are obtained. In order to verify the validity of the numerical method, model tests have been carried out. The results by the numerical simulation agree well with the experimental data.     

SELFE: A semi-implicit Eulerian–Lagrangian finite-element model for cross-scale ocean circulation

Unstructured-grid models grounded on semi-implicit, finite-volume, Eulerian–Lagrangian algorithms, such as UnTRIM and ELCIRC, have enjoyed considerable success recently in simulating 3D estuarine and coastal circulation. However, opportunities for improving the accuracy of this type of models were identified during extensive simulations of a tightly coupled estuary–plume–shelf system in the Columbia River system. Efforts to improve numerical accuracy resulted in SELFE, a new finite-element model for cross-scale ocean modeling. SELFE retains key benefits, including computational efficiency of existing semi-implicit Eulerian–Lagrangian finite-volume models, but relaxes restrictions on grids, uses higher-order shape functions for elevation, and enables superior flexibility in representing the bathymetry. Better representation of the bathymetry is enabled by a novel, “localized” vertical grid that resembles unstructured grids. At a particular horizontal location, SELFE uses either S coordinates or SZ coordinates, but the equations are consistently solved in Z space. SELFE also performs well relative to volume conservation and spurious oscillations, two problems that plague some finite-element models. This paper introduces SELFE as an open-source code available for community use and enhancement. The main focus here is on describing the formulation of the model and on showing results for a range of progressively demanding benchmark tests. While leaving details to separate publications, we also briefly illustrate the superior performance of SELFE over ELCIRC in a field application to the Columbia River estuary and plume.     

Evaluation of Vector Winds Observed by NSCAT in the Seas around Japan

In order to validate wind vectors derived from the NASA Scatterometer (NSCAT), two NSCAT wind products of different spatial resolutions are compared with observations by buoys and research vessels in the seas around Japan. In general, the NSCAT winds agree well with the wind data from the buoys and vessels. It is shown that the root-mean-square (rms) difference between NSCAT-derived wind speeds and the buoy observations is 1.7 ms^–1, which satisfies the mission requirement of accuracy, 2 ms^–1. However, the rms difference of wind directions is slightly larger than the mission requirement, 20°. This result does not agree with those of previous studies on validation of the NSCAT-derived wind vectors using buoy observations, and is considered to be due to differences in the buoy observation systems. It is also shown that there are no significant systematic trends of the NSCAT wind speed and direction depending on the wind speed and incidence angle. Comparison with ship winds shows that the NSCAT wind speeds are lower than those observed by the research vessels by about 0.7 ms^–1 and this bias is twice as large for data observed by moving ships than by stationary ships. This result suggests that the ship winds may be influenced by errors caused by ship's motion, such as pitching and rolling.