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Advanced Ocean Modelling: Using Open-Source Software
Abstract and Figures
This book introduces the reader to advanced methods used in the computer-based modelling of fluid processes. This includes nonhydrostatic processes such as breaking internal waves and density-driven convection, but the model code is also used to simulate an El-Nio event! The book contains 25 practical exercises, using freely available Open-Source software suites, which are widely used by the scientific community. In this book, the art of hydrodynamic modelling is made available and transparent to a wider readership. An attractive byproduct of the book is that results are animations rather than still images. Model codes and animation scripts for all exercises are supplied on a website. The reader can adopt model codes for own independent studies. © Springer-Verlag Berlin Heidelberg 2010. All rights are reserved.
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Chapters (4)
This chapter introduces the reader to one-dimensional water-column models using fixed vertical levels. Such a model is applied to study the dynamics of surface and bottom Ekman layers in the ocean.
This chapter introduces the reader to nonhydrostatic finite-difference solvers of the Navier-Stokes equations. For a start, the ocean is treated as as vertical slice. Flow and gradients of variables normal to this plane are assumed to vanish, and the Coriolis force is ignored. Exercises address deep-water (short) surface gravity waves, bottom-attached density-driven currents, internal waves, instabilities of vertical shear flows, lee waves, double-diffusive instability, double-diffusive layering and free convection.
Abstract This chapter introduces the reader to 2.5-dimensional modelling in a vertical ocean slice which allows for inclusion of the Coriolis force. Exercises address geostrophic adjustment of density fronts, coastal upwelling and Ekman pumping. The last exercise of this chapter explains the curiosity that winds can create flows in the ocean running opposite to the wind direction.
This chapter introduces the reader to fully three-dimensional hydrodynamic level models. Exercises address isolated oceanic eddies, eddy formation in a strait, density-driven exchange flows through a strait, coastal upwelling in three dimensions, the thermohaline deep circulation in the ocean, and, as a highlight, equatorially trapped Kelvin waves accompanying El-Niño events.
... The adopted coordinate system is set as x-axis pointing upslope (positive), y-axis across the slope, and a z-axis perpendicular to the planar ice base, with its origin set at the channel apex. All the governing equations are discretized in finite-difference form, the Coriolis terms are treated using a semi-implicit scheme 64 , the nonlinear terms are discretized using the Total Variation Diminishing (TVD) schemes 64 , and the nonhydrostatic pressure is solved using the Successive Over Relaxation (S.O.R.) method 64 . ...
... The adopted coordinate system is set as x-axis pointing upslope (positive), y-axis across the slope, and a z-axis perpendicular to the planar ice base, with its origin set at the channel apex. All the governing equations are discretized in finite-difference form, the Coriolis terms are treated using a semi-implicit scheme 64 , the nonlinear terms are discretized using the Total Variation Diminishing (TVD) schemes 64 , and the nonhydrostatic pressure is solved using the Successive Over Relaxation (S.O.R.) method 64 . ...
... The adopted coordinate system is set as x-axis pointing upslope (positive), y-axis across the slope, and a z-axis perpendicular to the planar ice base, with its origin set at the channel apex. All the governing equations are discretized in finite-difference form, the Coriolis terms are treated using a semi-implicit scheme 64 , the nonlinear terms are discretized using the Total Variation Diminishing (TVD) schemes 64 , and the nonhydrostatic pressure is solved using the Successive Over Relaxation (S.O.R.) method 64 . ...
Growing evidence has confirmed the critical role played by basal channels beneath Antarctic ice shelves in both ice shelf stability and freshwater input to the surrounding ocean. Here we show, using a 3D ice shelf-ocean boundary current model, that deeper basal channels can lead to a significant amplification in channelized basal melting, meltwater channeling, and warming and salinization of the channel flow. All of these channelized quantities are also modulated by channel width, with the level of modulation determined by channel height. The explicit quantification of channelized basal melting and the meltwater transport in terms of channel cross-sectional shape is potentially beneficial for the evaluation of ice shelf mass balance and meltwater contribution to the nearshore oceanography. Complicated topographically controlled circulations are revealed to be responsible for the unique thermohaline structure inside deep channels. Our study emphasizes the need for improvement in observations of evolving basal channels and the hydrography inside them, as well as adjacent to the ice front where channelized meltwater emerges.
... Of particular interest are applications to non-hydrostatic modeling. As a primer, we consider the vertical ocean-slice model with the inclusion of density effects, Kämpf [6], Lai et al. [8]. To solve these non steady PDE equations, we develop a pressure projection (PP) method for solution, see Almgren et al [1]. ...
... In this context, element I is denoted by Iá nd the outer normal n by n´. A classical numerical flux is that of Godunov given by (6) F˚¨n´:" F pu´q¨n´`|A|pu´´u˚q. ...
... A density driven flow with variable bottom topography. As a final example, let us consider a closed channel problem solved by Finite Differences in Kämpf [6]. The closed channel is initially composed of two vertical layers of water with constant but distinct density. ...
Element Method. The Finite Volume Method guarantees local and global mass conservation. A property not satisfied by the Finite Volume Method. On the down side, the Finite Volume Method requires non trivial modifications to attain high order approximations unlike the Finite Volume Method. It has been contended that the Discontinuous Galerkin Method, locally conservative and high order, is a natural progression for Coastal Ocean Modeling. Consequently, as a primer we consider the vertical ocean-slice model with the inclusion of density effects. To solve these non steady Partial Differential Equations, we develop a pressure projection method for solution. We propose a Hybridized Discontinuous Galerkin solution for the required Poisson Problem in each time step. The purpose, is to reduce the computational cost of classical applications of the Discontinuous Galerkin method. The Hybridized Discontinuous Galerkin method is first presented as a general elliptic problem solver. It is shown that a high order implementation yields fast and accurate approximations on coarse meshes.
... A non-hydrostatic method may be used to explain the mechanism of the Lee wave, as well as the internal wave. Numerical experiments were carried out based on Kämpf (2010). This study has a domain model with a horizontal boundary of 2,000 × 2,000 km and a uniform depth of 250 m (see Fig. 1(a)). ...
... The formation of Ekman and Kelvin waves is used to compare experiments theoretically. During the El Nino phase, Kelvin waves are thought to migrate into the eastern Pacific Ocean (Kämpf, 2010;Capotondi et al., 2019). Meanwhile, the Ekman layer (D E ) can be approximated using Eq. 5 (Kämpf, 2010). ...
... During the El Nino phase, Kelvin waves are thought to migrate into the eastern Pacific Ocean (Kämpf, 2010;Capotondi et al., 2019). Meanwhile, the Ekman layer (D E ) can be approximated using Eq. 5 (Kämpf, 2010). ...
BACKGROUND AND OBJECTIVES: El Niño-Southern Oscillation is known to affect the marine and terrestrial environment in Southeast Asia, Australia, northern South America, and southern Africa. There has been much research showing that the effects of El Niño-Southern Oscillation are extensive. In this study, a simulation of an El Niño event is carried out, which is ideal in the vertical layer of the Pacific Ocean (0-250 meters). The fast Fourier transform is used to process the vertical modeling data so that the results can accurately represent El Niño. METHODS: A non-hydrostatic 3-dimensional numerical model is used in this research. To separate the signal produced and obtain the quantitative difference of each sea layer, the simulation results are analyzed using the fast Fourier transform. Winds blow from the west to the east of the area in perfect El Niño weather, with a reasonably high wind zone near the equator (forming a cosine). Open fields can be found on the north and south sides, while closed fields can be found on the west and east sides. Density is uniform up to a depth of 100 meters, then uniformly increases by 1 kilogram per cubic meter from 100 to 250 meters. FINDINGS: The results of the model simulation show that one month later (on the 37th day), the current from the west has approached the domain's east side, forming a complete coastal Kelvin wave. The shape of coastal Kelvin waves in the eastern area follows a trend that is similar to the OSCAR Sea Surface Velocity plot data obtained from ERDDAP in the Pacific Ocean in October 2015. In this period, the density at a depth of 0-100 meters is the same, while the density at the depth layer underneath is different. CONCLUSION: Strong winds could mix water masses up to a depth of 100 meters, implying that during an ideal El Niño, the stratification of the water column is influenced by strong winds. The eastern domain has the highest sea level amplitude, resulting in perfect mixing up to a depth of 100 m, while wind effect is negligible in the lower layers. The first layer (0-50 m) and the second layer (50-100 m) have the same density and occur along the equator, according to FFT. The density is different and much greater in the third layer (100-150 m).
... This research was conducted by simulating a two-dimensional non-hydrostatic numerical model. This model was developed by Kämpf [17] from the Navier-Stokes equation with equations of advection and pressure gradient due to seawater density. The basic equation used in the model is as follows [18]: ...
... The stability equation for the Brunt-Vaisala frequency stratification [17] is as follows: ...
The instability of the density of seawater causes the formation of internal waves in the sea. This phenomenon is difficult to observe visually but can be studied with a hydrodynamic model approach. This study investigates the internal waves caused by underwater obstructions (seamounts) and the stratified density with a two-dimensional marine hydrodynamic model. Four scenarios are simulated in studying internal waves. Two scenarios used one barrier with uniform density and stratified density (based on the Brunt Vaisala stability frequency N 2 = 5 x 10-4 s-2). Meanwhile, the other two scenarios use two barriers with uniform density and stratified density (based on the Brunt Vaisala stability frequency N 2 = 5 x 10-4 s-2). Based on the simulation results, it is known that the density conditions have a significant effect on the dynamics of ocean currents. In hydrodynamic simulations of one and two barriers with varying or stratified densities, current resonances and density resonances are formed underwater topography. Meanwhile, in uniform density, the currents formed the rotor, cavity, turbulence, and resonance with the underwater topography. Thus the current dynamics are stronger in the case of uniform density than in the stratified density. It has implications for differences in the mixing of suspended matter in the sea. So this study can be useful in the study of sediment transport, upwelling, the thermocline layer, energy from internal waves, and the distribution of plankton or fish larvae.
... I was the first scientist who comprehensively studied the oceanography of these inverse estuaries with three-dimensional hydrodynamic models Kämpf et al., 2010). I am also an expert in hydrodynamic modelling having published two textbooks in this field (Kämpf, 2009;Kämpf, 2010). I also contributed as expert adviser to the assessment of the viability and environmental risks of seawater desalination in Australia including an assessment and recommendations for the Port Lincoln desalination project (Kämpf, 2021). ...
... 1−4 The convection produces column-like downwelling plumes that induce counter-current flow and vigorous chaotic vertical mixing. 5 When this phenomenon occurs in a porous medium, additional complications occur, which are associated with the discrete and irregular connectivity of the pore network. Spatial variations in the porosity and permeability of the medium yield highly heterogeneous fluid velocity fields and generate complex concentration distributions of the solute. ...
Dissolution trapping is one of the primary mechanisms of carbon dioxide (CO2) storage in deep saline aquifers. The determination of the realized rates of CO2 dissolution requires an understanding of the mixing process that takes place following the emplacement of CO2 into the formation. Owing to the difficulty of reproducing the time-dependent convective process in porous media, experiments so far have largely focused on 2D systems (e.g., Hele-Shaw cells) and used analogue fluid pairs with properties that differ from the subsurface CO2/brine system. Here, we present a novel experimental approach to investigate the evolution of the convective mixing process in 3D porous media (homogeneous packings of glass beads) using X-ray computed tomography (CT). We explore a range of Rayleigh numbers (Ra = 3000–55000) and observe directly the mixing structures that arise upon dissolution. We compute from the images the temporal evolution of the spatial moments of the concentration distribution, including the cumulative dissolved mass, the location of the center of mass, and the standard deviation of the concentration field. The scalings of the spatial moments suggest an impact of hydrodynamic dispersion on the longitudinal mixing. We propose a simplified representation of the mixing process by analogy with the 1D advection–dispersion model. This enables the estimation of the bulk advective velocity and the effective longitudinal dispersion coefficient for each bead packing. These estimates suggest that the presence of the finger pattern and the counter-current flow structure enhance the longitudinal spreading of the solute by roughly 1 order of magnitude compared to unidirectional dispersion of a single-solute plume.
Visualising fluid flow in porous media using optical techniques is challenging due to the inability to see through the medium. Here, we present an experimental methodology based on shadowgraphy to investigate the dynamic spreading of convective plumes in saturated transparent porous media made of glass beads. The saturated porous medium can be tuned transparent by matching the refractive index of the solid glass beads to that of the saturating fluid mixture. The proposed technique allows to investigate the essential elements of convective mixing within a porous medium using miscible fluids. We also describe a method to determine the velocity of convective plumes as they propagate. Our experimental results show that the density difference achieved during convection significantly affects the convective front velocity of the plumes. This is significant because it allows to quantitatively predict the intensity of convective mixing in porous media from the speed of the convective front.
Using a three-dimensional coupled physical–biological model, this paper explores the effect that short-lived wind events lasting a few days in duration have on the creation of phytoplankton blooms in island wakes. Findings show that wind-induced coastal upwelling creates initial nutrient enrichment and phytoplankton growth near the island, whereas an oscillating flow, typical of island wakes, expels patches of upwelled water, including its nutrient and phytoplankton loads, into the ambient ocean. Dependent on the wind direction, a short-lived wind event can create one or more plankton patches with diameters of the order of the island diameter. Phytoplankton continues to grow within floating patches, each forming an individual marine ecosystem. While the ecological features of island wakes are well documented, this study is the first that describes the significance of short-lived, transient wind-driven upwelling in the process.
На примере сейсмического события, обладающего магнитудой и механизмом глубокофокусного землетрясения у берегов Фиджи 19.08.2018, теоретически исследована связь параметров источника цунами с глубиной землетрясения. Показано, что сильные глубокофокусные землетрясения способны создавать обширные области косейсмических деформаций дна, размер которых сопоставим с баротропным радиусом деформации Россби. Волны цунами, формируемые такими источниками, обладают необычайно большой длиной и поэтому подвержены влиянию силы Кориолиса. Это влияние может варьироваться от существенного в высоких широтах до малозначительного в приэкваториальных районах. Методом численного моделирования исследовано влияние силы Кориолиса на слабые волны цунами, вызванные Фиджийским землетрясением 2018 г.
Using the example of a seismic event with the same magnitude and mechanism as for the 2018 Fiji deep-focus earthquake the relationship between the parameters of a tsunami source and the depth of an earthquake was theoretically investigated. It is shown that strong deep-focus earthquakes are capable of creating extensive areas of co-seismic deformations for the ocean bottom, the size of which is comparable to the barotropic Rossby deformation radius. Tsunami waves generated by such sources have an unusually large length, and therefore they are subject to the influence of the Coriolis force. The influence of the Coriolis force on the weak tsunami waves caused by the 2018 Fiji earthquake has been studied by numerical simulation. It was found that for a virtual sea-level station located to the south of the source at latitude of 40S, the amplitude of the leading tsunami wave under the influence of the Coriolis force decreased by ≈ 30%. For virtual stations located at the equator or at latitude of 20S, taking into account the Coriolis force changes the amplitude of the waves by no more than 10%. For stations located in the equatorial zone, the effect of delaying of the manifestation of differences in sea-level fluctuations, calculated with and without Coriolis force, as compared with the entry of the leading tsunami wave was discovered. A physical interpretation of the observed delay effect is proposed.
A new method to compute the transport of density in baroclinic numerical models without artificial diffusivity is presented. As a critical application, the salt intrusion into a schematic estuary of the size of the Elbe estuary is studied.
The model is based on the Navier Stokes equation solved numerically by an established method in the Eulerian framework. The transport of density is computed in a Lagrangian is done in a convenient way using the computed velocities. In contrast to the established tracer particle methods, each particle represents a certain amount of water of definite properties (salinity, temperature, etc.). The mean property of any Eulerian grid cell is the average of all Lagrangian particles being in the cell. Using this concept, four types of transport can be modelled completely independent of each other:
“large scale” transport by mean velocities
“medium scale” displacements by turbulence
buoyancy
“small scale” exchange of properties by irreversible mixing.
The sharp density gradient of the brackish water zone is reproduced qualitatively. Its dependence on the tide and on the fresh water input can be demonstrated, too.
Suggestions that the ongoing greenhouse buildup might induce a shut-down of the ocean's thermohaline circulation raise the questions as to how Earth's climate would change if such an event were to occur. The answer preferred by the popular press is that conditions akin to those that characterized the Younger Dryas - the last kiloyear cold snap - would return. But this extreme scenario is an unlikely one, for models suggest that in order to force a conveyor shutdown, Earth would have to undergo a 4 to 5 °C greenhouse warming. Hence, the conditions at the onset of the shutdown would be very different from those that preceded the Younger Dryas. Thus, it is unlikely that new climate conditions would be nearly so severe. Unfortunately, because no atmospheric model to date has been able to create the observed large and abrupt changes in climate state of Earth's atmosphere, we lack even the crudest road map. However, as was the case for each of the abrupt changes recorded in Greenland's ice, if the conveyor were to shut down, climate would likely flicker for several decades before locking into its new state. The consequences to agricultural production of these flickers would likely be profound.
We review what is known about the convec-tive process in the open ocean, in which the properties of large volumes of water are changed by intermittent, deep-reaching convection, triggered by winter storms. Observational, laboratory, and modeling studies reveal a fascinating and complex interplay of convective and geostrophic scales, the large-scale circulation of the ocean, and the prevailing meteorology. Two aspects make ocean convection interesting from a theoretical point of view. First, the timescales of the convective process in the ocean are sufficiently long that it may be modified by the Earth's rotation; second, the convective process is localized in space so that vertical buoyancy transfer by upright convection can give way to slantwise transfer by baroclinic instability. Moreover, the convec-tive and geostrophic scales are not very disparate from one another. Detailed observations of the process in the Labrador, Greenland, and Mediterranean Seas are de-scribed, which were made possible by new observing technology. When interpreted in terms of underlying dynamics and theory and the context provided by labo-ratory and numerical experiments of rotating convec-tion, great progress in our description and understand-ing of the processes at work is being made.