Efficient higher-order finite-difference schemes for parabolic models

Implicit finite-difference schemes for use in parabolic equation models are developed. Like the familiar Crank-Nicolson scheme, which has hitherto been used almost exclusively for the solution of these equations, these schemes are unconditionally stable and use a computational molecule of only six points on two “time” levels. However, they are accurate to a higher order than the Crank-Nicolson scheme, thus allowing the solution grid to be coarser and the solution time to be (approximately) halved. Examples of computations on constant depth are shown, in which significant reductions in time and grid-point density are achieved, for two different parabolic models. The schemes are then extended to refraction and diffraction, and are shown to have a similar effect in this more general case too. It is recommended that finite-difference schemes based on these higher-order (or Hermitian) methods replace the more commonly used Crank-Nicolson scheme in all physical domain parabolic equation models, but especially in minimax (wide-angle) equation models.     

C网格嵌套技术及其在海洋波动传播模拟中的应用

采用粗细嵌套的ArakawaC网格模拟Klevin波和Rossby波沿赤道传播的过程，研究在粗细网格嵌套边界产生数值振荡和反射的原因及其消减方法，得到合理的并能用于复杂海洋模式的ArakawaC网格下的嵌套方案。数值实验结果表明：波形在粗细网格边界产生数值振荡的原因是波形在不同分辨率下的形态有差异，在粗网格下波形趋向于平坦化，而细网格下趋向于锐化。采用双向嵌套和粗细网格交界处加松弛的方法可以有效地消减数值振荡和反射。     

Upwelling into the thermocline of the Pacific Ocean

The established “island rule” and the recently introduced “separation formula” are combined to yield an analytical expression for the total upwelling into the thermocline in the Pacific. The combination of the two is achieved with the use of a hybrid model containing a stratified upper layer, a thick (slowly moving) homogenous intermediate layer and an inert lower layer. Both the upper and the intermediate layers are subject to diabatic cooling and heating (which need not be specified) and there is an exchange of mass between the two active layers. An attempt is made to examine the above analytical (hybrid) model numerically. Ideally, this should be done with a complete two-and-a-half layer model (with upwelling and downwelling), but such a model is much too complex for process-oriented studies (due to the required parameterization of vertical mixing). Consequently, we focus our attention on verifying that the separation formula and the island rule are consistent with each other in a much simpler, layer-and-a-half model (without upwelling). We first verified that the new “separation formula” provides a reasonable estimate of the wind-induced transport in an island-free basin. We then compare the wind-induced transport predicted by the separation formula and the island rule in an idealized basin containing an island. We show that in these idealized situations the two methods give results that are consistent with each other and the numerics. We then turned to an application of the (hybrid) two-and-a-half layer model to the Pacific where, in contrast to the idealized layer-and-a-half models (where the two methods address the same water mass), the two methods address two different water masses. While the separation formula addresses only thermocline water (σ_θ<26.20), the island rule addresses all the water down to 27.5σ_θ (i.e., both the upper and intermediate layer). This is why the application of the two methods to the Pacific gives two different results — an application of the formula gives zero warm water transport whereas an application of the island rule gives 16 Sv. Namely, the difference between the amount predicted by the island rule (16 Sv) and the amount predicted by the separation formula (zero) enters the Pacific as intermediate water and is then somehow upwelled into the thermocline. The upwelling should take place north of the southern western boundary currents separation (40°S).     

Verification studies for a z-coordinate primitive-equation model: Tidal conversion at a mid-ocean ridge

We document the accuracy and convergence of solutions for a z-coordinate primitive-equation model of internal tide generation and propagation. The model, which is based on MOM3 numerics, is linearized around a state of rest to facilitate comparison with analytic estimates of baroclinic generation at finite-amplitude topography in a channel forced by barotropic tidal flow at its boundaries. Unlike the analytical model, the numerical model includes mixing of both buoyancy and momentum, and several definitions of “baroclinic conversion” are possible. These are clarified by writing out the energetics of the linearized equations in terms of barotropic kinetic energy, baroclinic kinetic energy, and available potential energy. The tidal conversion computed from the model, defined as the rate of conversion of barotropic kinetic energy into available potential energy, agrees well with analytical predictions. A comparison of different treatments of bottom topography (full-cells, partial-cells, and ghost-cells) indicates that the partial-cell treatment is the most accurate in this application. Convergence studies of flow over a smooth supercritical ridge show that the dissipation along tidal characteristics is, apparently, an integrable singularity. When the ocean bottom is not smooth, the accuracy and convergence of the model depend on the power spectrum of the topography. A numerical experiment suggests that the power spectrum of the resolved topography must roll off faster than k⁻² to obtain convergent results from a linear numerical model of this type.     

Fourier filtering and coefficient tapering at the North Pole in OGCMs

This article considers how some of the measures used to overcome numerical problems near the North Pole affect the ocean solution and computational time step limits. The distortion of the flow and tracer contours produced by a polar island is obviated by implementing a prognostic calculation for a composite polar grid cell, as has been done at NCAR. The severe limitation on time steps caused by small zonal grid spacing near the pole is usually overcome by Fourier filtering, sometimes supplemented by the downward tapering of mixing coefficients as the pole is approached; however, filtering can be expensive, and both measures adversely affect the solution. Fourier filtering produces noise, which manifests itself in such effects as spurious static instabilities and vertical motions; this noise can be due to the separate and different filtering of internal and external momentum modes and tracers, differences in the truncation at different latitudes, and differences in the lengths of filtering rows, horizontally and vertically. Tapering has the effect of concentrating tracer gradients and velocities near the pole, resulting in some deformation of fields. In equilibrium ocean models, these effects are static and localised in the polar region, but with time-varying forcings or coupling to atmosphere and sea ice it is possible that they may seriously affect the global solution. The marginal stability curve in momentum and tracer time-step space should have asymptotes defined by diffusive, viscous, and internal gravity wave stability criteria; at large tracer time steps, tracer advection stability may become limiting. Tests with various time-step combinations and a flat-bottomed Arctic Ocean have confirmed the applicability of these limits and the predicted effects of filtering and tapering on them. They have also shown that the need for tapering is obviated by substituting a truncation which maintains a constant time step limit rather than a constant minimum wave number over the filtering range.     

Assessment of water exchange between a discharge region and the open sea – A comparison of different methodological concepts

Two different methods of estimating the water exchange through the Baltic coastal region of Laxemar have been used, consisting of particle trajectories and passive tracers. Water is traced from and to a small discharge region near the coast. The discharge material in this region is treated as zero-dimensional particles or tracers with neutral buoyancy. The real discharge material could be a leakage of radio-nuclides through the sea floor from an underground repository of nuclear waste.Water exchange rates between the discharge region and the model domain are estimated using both forward and backward trajectories as well as passive tracers. The Lagrangian trajectories can account for the time evolution of the water exchange while the tracers give one average age per model grid box. Water exchange times such as residence time, age and transient times have been calculated with trajectories but only the average age (AvA) for tracers. The trajectory calculations provide a more detailed time evolution than the tracers.On the other hand the tracers are integrated “on-line” simultaneously in the sea circulation model with the same time step while the Lagrangian trajectories are integrated “off-line” from the stored model velocities with its inherent temporal resolution, presently 1 h. The sub-grid turbulence is parameterised as the Laplacian diffusion for the passive tracers and with an extra stochastic velocity for trajectories. The importance of the parameterised sub-grid turbulence for the trajectories is estimated to give an extra diffusion of the same order as the Laplacian diffusion by comparing the Lagrangian dispersions with and without parameterisation. The results of the different methods are similar but depend on the chosen diffusivity coefficient with a slightly higher correlation between trajectories and tracers when integrated with a lower diffusivity coefficient.     

On the use of current meter data to assess the realism of ocean model simulations

The evaluation of ocean simulations against observed datasets is essential to assess their realism and to guide model development, but often remains qualitative, and ignores certain datasets. This paper presents a three-dimensional, quantitative comparison of a 1/6° Atlantic numerical simulation (CLIPPER) with the WOCE current meter dataset in terms of mean velocity and eddy kinetic energy. Our metrics reveal the good behaviour of CLIPPER open boundary conditions and forcing with respect to full-depth current records. Due to its still moderate resolution, however, the model globally underestimates the observed mean speeds and eddy activity. This discrepancy is barely noticeable at low latitudes but increases toward the poles, probably since the poleward decrease of the Rossby radius exceeds that of the horizontal grid step. At least in this eddy-admitting regime, it is suggested that the numerics of geopotential-coordinate models like ours dissipate mean and eddy momentum at depth and adversely affect current–topography interactions.     

On the propeller efficiency at spatially non-homogeneous inflow

The thrust produced by a marine propeller operating in a spatially non-homogeneous flow is reduced by inhomogenities depending on the angular co-ordinate. The loss of efficiency caused by this reduction of thrust is to a certain extent compensated by increasing the design thrust, a procedure called “unsteady wake adaption”. Numerical calculations show improvements of propeller efficiency which are dependent on the degree of inhomogenity of the flow fielf.     

σ-Coordinates hydrodynamic numerical model for coastal and ocean three-dimensional circulation

A three-dimensional finite-difference hydrodynamic model has been developed using σ-coordinate for the vertical dimension. An explicit scheme for temporal integration and a staggered grid for spatial discretization have been adopted. The model has been tested against analytical or literature cases for wind and tide induced circulation. Results are in good agreement both with analytical solutions under idealised conditions and with results from the model of Shankar et al. (1996).     

高程异常模型的已知点框架约束

在确保GPS/水准点成果可靠的前提下,用其对重力法计算所得的高程异常模型进行框架约束,以削弱重力系统与高程和坐标系统间的系统差,实现系统转换,提高最终模型的精度。通过对不同离散数据网格化方法的比较,给出相对计算快捷、精度高的框架约束方法。     

Transverse vibrations of circular annular plates with edges elastically restrained against rotation, used in acoustic underwater transducers

Simply supported or clamped boundary conditions are rather ideal situations difficult to satisfy from a physical viewpoint. This paper considers a more “moderate” restriction: the case of edges elastically restrained against rotation for which no exact solution appears in the open literature. Eigenvalues corresponding to a wide range of the intervening geometric and mechanical parameters are determined. Good agreement is obtained with frequency coefficients determined two decades ago by means of a variational method. Obviously the problem is of basic interest in many ocean engineering applications: from the design of certain underwater acoustic transducers to pump and compressor elements passing through the design of naval vehicles and ocean structures.     

Behavior of marine pipelines under seismic faults

A new study investigates an unburied offshore “snaked” pipeline behavior under various types of seismic faults. The snaking of the pipeline is caused by the thermal/pressure expansion and soil friction. The snaking takes place at a certain distance from the pipeline's unrestrained end and gradually increases towards the restraint. It is shown that longitudinal seismic faults have less effect on a straight pipeline than a snaked pipeline. The new seismic analysis demonstrates that an increase of ground displacement causes a very small change in bending and longitudinal stresses. The new approach results in a safe, subsea pipeline construction and operation with a significant cost reduction.     

High-resolution sea ice in long-term global ocean GCM integrations

The resolution of the sea-ice component of a coarse-resolution global ocean general circulation model (GCM) has been enhanced to about 22 km in the Southern Ocean. The ocean GCM is designed for long-term integrations suitable for investigations of the deep-ocean equilibrium response to changes in southern hemisphere high-latitude processes. The space and time scales of the high-resolution sea-ice component are commensurate with those of the resolution of satellite passive-microwave sea-ice data. This provides the opportunity for a rigorous evaluation of simulated sea-ice characteristics. It is found that the satellite-derived continuous high ice concentration of the interior winter ice pack can only be captured when vertical oceanic mixing is modified in a way that less local, intermittent convection occurs. Furthermore, the width and the variability of the coastal polynyas around the Antarctic continent and its ice shelves are best captured when some form of ice-shelf melting is accounted for. The width of the wintertime ice edge is reasonably reproduced, while its variability remains underestimated, closely following the coarse-grid pattern of the ocean model due to its high dependence on ocean temperature. Additional variability besides daily winds, e.g. in form of idealized tidal currents, improves the temporal and spatial ice-edge variability, while leads in the interior ice pack become more abundant, more in line with the fine-scale satellite-derived texture. The coast- or ice-shelf line is described on the fine grid based on satellite passive-microwave data. This method requires parts of a coarse coastal ocean grid cell to be covered by an inert layer of “fast ice” or “ice shelf”. Reasonable long-term global deep-ocean properties can only be achieved when these areas are not inert, i.e. are exposed to heat flux and ice growth, or when the vertical mixing parameterization allows for excessive open-ocean convection. The model area exposed to cold high-latitude atmospheric conditions thus being most decisive for a realistic representation of the long-term deep-ocean properties, suggests that high-latitude coastlines are definitely in need of being represented at high resolution, including ice sheets and their effects on the heat and freshwater flux for the ocean.     

The differences between a wing and a planing plate in two-dimensional flow

The normal force coefficient on a flat planing surface having arbitrary heave and pitch motion in two-dimensional flow is compared with the lift coefficient of a thin wing in an infinite fluid. Despite the totally different derivations, they are found to be identical (at large Froude numbers and low trim angles and allowing for the wing's interaction with twice as much fluid) at low reduced frequencies. For higher frequency motions, the wing's angle of attack induced lift and its pitch and heave damping are less than those of a planing surface, but the acceleration terms remain identical. The differences at the higher reduced frequencies are due to the fact that, in invisad irrotational flow, the planning plate cannot leave a vortex wake, whereas a wing does.It seems to follow that the “virtual mass” planing hull analysis can be applied to “quasi-static” problems involving wings and bodies in an infinite fluid without the slenderness restriction originally imposed by Jones (1946). Certainly, it is remarkable that the so called “quasi-steady” forces on a two-dimensional wing can be obtained in a few lines of elementary analysis. On the other hand, the method fails entirely when used to compute the pitching moment on a two-dimensional plate, even though it has been found to give good results for the three-dimensional case (Payne, 1981c).This work is offered as a very incomplete study of an intriguing relationship between two very different bodies of analysis. Much more work will need to be done before the relationship between the two approaches will be fully understood.     

A hybrid spectral/finite-difference large-eddy simulator of turbulent processes in the upper ocean

A three-dimensional numerical model for large-eddy simulation (LES) of oceanic turbulent processes is described. The numerical formulation comprises a spectral discretization in the horizontal directions and a high-order compact finite-difference discretization in the vertical direction. Time-stepping is accomplished via a second-order accurate fractional-step scheme. LES subgrid-scale (SGS) closure is given by a traditional Smagorinsky eddy-viscosity parametrization for which the model coefficient is derived following similarity theory in the near-surface region. Alternatively, LES closure is given by the dynamic Smagorinsky parametrization for which the model coefficient is computed dynamically as a function of the flow. Validation studies are presented demonstrating the temporal and spatial accuracy of the formulation for laminar flows with analytical solutions. Further validation studies are described involving direct numerical simulation (DNS) and LES of turbulent channel flow and LES of decaying isotropic turbulence. Sample flow problems include surface Ekman layers and wind-driven shallow water flows both with and without Langmuir circulation (LC), generated by wave effects parameterized via the well-known Craik–Leibovich (C–L) vortex force. In the case of the surface Ekman layers, the inner layer (where viscous effects are important) is not resolved and instead is parameterized with the Smagorinsky models previously described. The validity of the dynamic Smagorinsky model (DSM) for parameterizing the surface inner layer is assessed and a modification to the surface stress boundary condition based on log-layer behavior is introduced improving the performance of the DSM. Furthermore, in Ekman layers with wave effects, the implicit LES grid filter leads to LC subgrid-scales requiring ad hoc modeling via an explicit spatial filtering of the C–L force in place of a suitable SGS parameterization.     

Implementation of variable time stepping in an ocean general circulation model

Ocean general circulation models (OGCMs) which represent the governing equations on a finite difference grid require shorter time steps with increasing resolution. Thus, until now, in the absence of filtering, the time step length has been determined by the smallest grid spacing within the model domain. Here we present a method for reducing the time step length (and increasing the number of time steps taken) at selected points in the grid, so as to minimise the computational cost of integrating the OGCM, whilst achieving numerical stability throughout the model domain without filtering. This variable time stepping method can be used to overcome numerical constraints associated with the convergence of longitude–latitude grids at the poles, and also to allow efficient integration of model domains with variable resolution. Examples of the computational saving are given.     

On extracting current profiles from vertically integrated numerical models

A transformation method is presented by which current profiles (of tidal or wind-induced origin) can be extracted at any horizontal position and moment in time from a vertically integrated, two-dimensional, hydrodynamic numerical model. An arbitrary vertical variation of eddy viscosity can be included in the method, which can incorporate a no-slip bottom boundary condition. The technique assumes that the sea is homogeneous.The method is used to improve the representation of bottom stress within the two-dimensional model, whereby the bottom stress is no longer related simply to the depth-mean current as in the “conventional” two-dimensional, vertically integrated model.Idealized calculations for a range of eddy viscosity profiles, show that elevations, current profiles, and time series of current extracted from this “enhanced” two-dimensional numerical model are in good agreement with currents obtained from a full three-dimensional model.     

Experimental and numerical hydrodynamic analysis of a stepped planing hull

This work addresses the experimental and numerical study of a stepped planing hull and the related fluid dynamics phenomena typically occurring in the stepped hull in the unwetted aft body area behind the step. In the last few years, the interest in high-speed planing crafts, with low weight-to-power ratios, has been increasing significantly, and, in such context, naval architects have been orienting toward the stepped hull solution. Stepped planing hulls ensure good dynamic stability and seakeeping qualities at high speeds. This is mainly due to the reduction of the wetted area, which is caused by the flow separation occurring at the step. This paper presents the experimental results of towing tank tests in calm water on a single-step hull model, which is the first model of a new systematic series. The same flow conditions are analyzed via Reynolds Averaged Navier-Stokes (RANS) and Large Eddy Simulations (LES), with different moving mesh techniques (overset/chimera and morphing grid), performed at different model speeds. The numerical results are in accordance with experimental data, and overset/chimera grid is found to be the best approach between the analyzed ones. The flow patterns obtained numerically through LES on a refined grid appear similar to the ones observed in towing tank investigations through photographic acquisitions. These flow patterns are dominated by a rather complex 3D arrangement of vortices originating from air spillage at both sides of the step. The understanding of these phenomena is important for the effectiveness of stepped hull designs.     

The discrete methods for free vibration analyses of an immersed beam carrying an eccentric tip mass with rotary inertia

In this paper, a beam without contact with water is called the “dry” beam and the one in contact with water is called the “wet” beam. For a partially (or completely) immersed uniform beam carrying an eccentric tip mass possessing rotary inertia, the conventional analytical (closed-form) solution is achieved by considering the inertial forces and moments of the tip mass and rotary inertia as the boundary conditions at the tip end of the beam. However, it has been found that the approximate solution for the last problem may be achieved by two techniques: Method 1 and Method 2. In Method 1, the basic concept is the same as the conventional analytical method; but in Method 2, the tip end of the beam is considered as a free end, while the inertial forces and moments induced by the tip mass and rotary inertia are considered as the external loads applied at the tip end of the beam. The main differences between the formulation of Method 1 and that of Method 2 are: In Method 1, the “normal” shapes of the “dry” beam are functions of the frequency-dependent boundary conditions but the external loads at the tip end are equal to zero; On the contrary, in Method 2, the “normal” mode shapes of the “dry” beam are determined based on the zero boundary conditions at the tip end of the beam but the external loads at the tip end due to the inertial effects of the tip mass and rotary inertia must be taken into consideration for the free vibration analysis of the “wet” beam. Numerical results reveal that the approximate solution obtained from Method 2 are very close to that from Method 1 if the tip mass moment of inertia is negligible. Besides, the two approximate solutions are also very close to the associated analytical (closed-form) solution or the finite element solution. In general, it is hoped that there exist several methods for tackling the same problem so that one may have more choices to incorporate with the specified cases. It is believed that the two approximate methods presented in this paper will be significant from this point of view.