The purpose of this book is to provide an introduction to numerical modelling of the ocean and the atmosphere. It originates from courses given at Stockholm University and is intended to serve as a textbook for students in meteorology and oceanography with a background in mathematics and physics. Focus is on numerical schemes for the most commonly used equations in oceanography and meteorology as well as on the stability, precision and other properties of these schemes. Simple equations capturing the properties of the primitive equations employed in models of the ocean and atmosphere will be used. These model equations are solved numerically on a grid by discretisation, the derivatives of the differential equations being replaced by finite-difference approximations. The focus will be on the basic numerical methods used for oceanographic and atmospheric modelling. These models are based on the Navier-Stokes equations (including the Coriolis effect) and a tracer equation for heat in both the atmosphere and ocean and tracer equations for humidity and salt in the atmosphere and ocean, respectively. A coupled atmospheric and oceanic general circulation model represents the core part of an Earth System climate model. The book starts by presenting the most common types of partial differential equations and finite difference schemes used in meteorology and oceanography. Subsequently the limitations of these numerical schemes as regards stability, accuracy, presence of computational modes and accuracy the computationally determined phase speed are discussed. The shallow-water equations are discretised for different spatial grids and friction and diffusion terms are introduced. Hereafter implicit and semi-implicit schemes are discussed as well as the semi-Lagrangian technique. Coordinates for atmospheric as well as oceanic models are presented as well as a highly simplified 3D model. A brief description is given of how some atmospheric general circulation models use spectral methods as "horizontal coordinates". Finally, some "pen-and-paper" theoretical exercises and a number of GFD computer exercises are given.

This study deals with the perturbative solutions of the constant-potential-vorticity hydraulic problem, where the lowest-order solution corresponds to that for a non-rotating parabolic channel. The analytical structure of the series expansions and the corresponding numerical solutions are presented for flow through a horizontally constricted passage, and it is demonstrated how the series solution can be used to bridge the gap between non-rotating and rotating hydraulic models. It was found that the perturbative expansions converged close to the branch-point that corresponds to critical flow in rotating channel. This was not the case for the sub- and supercritical solution branches, where the perturbative series proved to diverge for discrete values of the expansion parameter, and hence one focus of the study is on improving the convergence here. The location and character of the dominant singularities affecting the series convergence were determined from analysis of the expansion coefficients using Domb-Sykes plots, where after the convergence was improved using Euler transformation rather than by direct Pade summation.

In an earlier study dealing with a nonlinear fluid oscillator governed by two autonomous ODEs, the solutions were found to display some aberrant characteristics adjacent to the boundaries of the oscillatory regime in parameter space. It was argued that this behaviour indicated the presence of canards. In the present study it is formally proved that this indeed is the case, and some numerical examples illustrating the phenomenon as well as its effects are presented.

A 70-day data set from bottom-mounted ADCPs on the two sides of the Faroe-Bank Channel was analysed using the recorded flow variance and echo intensity in the deeper reaches of the passage as proxi ...

The oscillatory behavior of an asymmetrically forced thermosyphon constituted by two connected vessels has been subjected to an asymptotically valid analysis using the vessel-volume ratio as expansion parameter. Due to the structure of the governing equations, the problem could not be dealt with using standard techniques; instead a phase-plane analysis was conducted. The analytically determined corrections to the previously established lowest-order discontinuous results proved to be useful even for comparatively large values of the expansion parameter. The relationship between these asymptotically valid corrections and the physics underlying the relaxation oscillation as well as the behavior of the system for strong thermal forcing is discussed. The study is concluded by an overview of some specific inconsistencies associated with the discontinuous lowest-order analysis and how these were alleviated by the asymptotically valid corrections.

The characteristics of a thermally forced connected-vessel thermosyphon operating in an oscillatory mode have been determined using analytical techniques, the outcome of which is compared with results obtained by numerical integration of the governing equations. From a previous investigation it was known that adequate phase–plane representations of the limit cycles associated with oscillations could be obtained if the vessel-volume ratio was sufficiently small. This study aims at demonstrating how this constraint on the vessel volumes can be relaxed by prescribing a Mandelstam condition, that is, by postulating that the total heat content of the system remains conserved during the rapid phases of the oscillation. It was concluded that incorporating this Mandelstam condition in the analysis had the highly beneficial consequence that good analytical results could be obtained for much larger values of the vessel-volume ratio than those previously permitted.

The climatic conditions over the Arctic Ocean are strongly influenced by the inflow of warm Atlantic water conveyed by the Norwegian Atlantic Slope Current (NwASC). Based on sea surface height (SSH) data from altimetry, we develop a simple dynamical measure of the NwASC transport to diagnose its spatio-temporal variability. This supports a dynamical division of the NwASC into two flow regimes; the Svinoy Branch (SvB) in the southern Norwegian Sea, and the Fram Strait Branch (FSB) west of Spitsbergen. The SvB transport is well correlated with the SSH and atmospheric variability within the Nordic Seas, factors that also affect the inflow to the Barents Sea. In contrast, the FSB is influenced by regional atmospheric conditions around Svalbard and northern Barents Sea. Using a composite analysis, we further relate anomalous strong SvB flow events to temperature fluctuations along the core of Atlantic water. A warm composite anomaly is found to propagate northward, with a tendency to amplify enroute, after these events. A roughly 12 months delayed temperature signal is identified in the FSB. However, also in the Lofoten Basin interior a delayed temperature signal is found, which appears to originate from the NwASC. This study suggests that hydrographic anomalies both upstream from the North Atlantic, and locally generated in the Norwegian Sea, are important for the oceanic heat and salt transport that eventually enters into the Arctic.

An overview of the deep-water flow from the Nordic Seas through the Faroe-Bank Channel and the Denmark Strait into the North Atlantic proper is given. These fluxes are of considerable importance for the global thermohaline circulation, and it is outlined how they can be modeled on the basis of rotating hydraulic theory for zero-as well as finite-potential-vorticity flow. The hydraulic framework is also shown to be useful for analyzing various dynamic features characterizing the deep-water flow.

This thesis deals with the dynamics and circulation in the northern North Atlantic and the Nordic Seas, processes of crucial importance for the mild climate of Scandinavia and Northern Europe. High-resolution ADCP scans of currents from Greenland to Scotland in the top 400 m demonstrate that the Reykjanes Ridge is a very effective separator of flow towards the Nordic and Labrador Seas, respectively. It was found that the meridional overturning circulation has weakened by ~1.7 Sv (1 Sv = 106 m3 s-1) during the 18-year period when altimetric data were available. This trend may be an effect of the Atlantic Multidecadal Oscillation, but is certainly not due to the North Atlantic Oscillation (NAO). By studying the circulation in the Faroe-Shetland Channel, which is an important choke point for the global thermohaline circulation, it was concluded that the contraction of the Norwegian-Sea gyre during low NAO periods plays an important role for disturbing the flow pattern. This specifically affects the regional ocean climate by leading to an accumulation of warm and saline Atlantic waters in the channel. During high NAO phases the circulation is strongly topographically controlled. The Norwegian Atlantic Slope Current (NwASC) is the main flow branch linking the North Atlantic to the Arctic and Barents Sea. It was found that the NwASC is largely coherent over seasonal to interannual time-scales. However, on shorter time-scales the coherency of the flow shows a sustained and pronounced weakening downstream of Lofoten. Intense eddy-shedding from the slope into the Lofoten Basin damps the coherent structure of the flow. The eddies take about two months to propagate to and to merge with the semi-permanent anticyclonic vortex above the deepest part of the Lofoten Basin. These results have implications for how flow/hydrographic anomalies are transferred through the Nordic Seas towards the Arctic. Anomalous transports of warm water into the Arctic and Barents Sea via the NwASC are found to be driven by a combination of the NAO and the other two leading modes of atmospheric variability in the North Atlantic. The results reported in the thesis may be of importance for achieving a correct representation of the heat conveyed polewards in climate models.

Abstract. Previous investigations concerning the design of an eddy-permitting LGM oceanic simulation are here extended with focus on whether this type of simulation is capable of improving the numerical results with regard to the available paleo-proxy reconstructions. Consequently, an eddy-permitting and two coarse-grid simulations of the same LGM period are confronted with a dataset from the Multiproxy Approach for the Reconstruction of the Glacial Ocean Sea Surface Temperatures (MARGO SSTs) and a number of sea-ice reconstructions. From a statistical analysis it was found that the eddy-permitting simulation does not significantly improve the SST representation with regard to the paleo-reconstructions. The western boundary currents are better resolved in the high-resolution experiment than in the coarse simulations, but, although these more detailed SST structures yield a locally improved consistency between modelled predictions and proxies, they do not contribute significantly to the global statistical score. As in the majority of the PMIP2 simulations, the modelled sea-ice conditions are still inconsistent with the paleo-reconstructions, probably due to the choice of the model equilibrium.

Most state-of-the-art climate models include a coarsely resolved oceanic component, which hardly captures detailed dynamics, whereas eddy-permitting and eddy-resolving simulations are developed to reproduce the observed ocean. In this study, an eddy-permitting and a coarse resolution numerical experiment are conducted to simulate the global ocean state for the period of the Last Glacial Maximum (LGM, similar to 26 500 to 19 000 yr ago) and to investigate the improvements due to taking into account the smaller spatial scales. The ocean state from each simulation is confronted with a data set from the Multiproxy Approach for the Reconstruction of the Glacial Ocean (MARGO) sea surface temperatures (SSTs), some reconstructions of the palaeo-circulations and a number of sea-ice reconstructions. The western boundary currents and the Southern Ocean dynamics are better resolved in the high-resolution experiment than in the coarse simulation, but, although these more detailed SST structures yield a locally improved consistency between model predictions and proxies, they do not contribute significantly to the global statistical score. The SSTs in the tropical coastal upwelling zones are also not significantly improved by the eddy-permitting regime. The models perform in the mid-latitudes but as in the majority of the Paleo-climate Modelling Intercomparison Project simulations, the modelled sea-ice conditions are inconsistent with the palaeo-reconstructions. The effects of observation locations on the comparison between observed and simulated SST suggest that more sediment cores may be required to draw reliable conclusions about the improvements introduced by the high resolution model for reproducing the global SSTs. One has to be careful with the interpretation of the deep ocean state which has not reached statistical equilibrium in our simulations. However, the results indicate that the meridional overturning circulations are different between the two regimes, suggesting that the model parametrizations might also play a key role for simulating past climate states.

Abstract. Most state-of-the-art climate models include a coarsely resolved oceanic component, which has difficulties in capturing detailed dynamics, and therefore eddy-permitting/eddy-resolving simulations have been developed to reproduce the observed World Ocean. In this study, an eddy-permitting numerical experiment is conducted to simulate the global ocean state for a period of the Last Glacial Maximum (LGM, ~ 26 500 to 19 000 yr ago) and to investigate the improvements due to taking into account these higher spatial scales. The ocean general circulation model is forced by a 49-yr sample of LGM atmospheric fields constructed from a quasi-equilibrated climate-model simulation. The initial state and the bottom boundary condition conform to the Paleoclimate Modelling Intercomparison Project (PMIP) recommendations. Before evaluating the model efficiency in representing the paleo-proxy reconstruction of the surface state, the LGM experiment is in this first part of the investigation, compared with a present-day eddy-permitting hindcast simulation as well as with the available PMIP results. It is shown that the LGM eddy-permitting simulation is consistent with the quasi-equilibrated climate-model simulation, but large discrepancies are found with the PMIP model analyses, probably due to the different equilibration states. The strongest meridional gradients of the sea-surface temperature are located near 40° N and S, this due to particularly large North-Atlantic and Southern-Ocean sea-ice covers. These also modify the locations of the convection sites (where deep-water forms) and most of the LGM Conveyor Belt circulation consequently takes place in a thinner layer than today. Despite some discrepancies with other LGM simulations, a glacial state is captured and the eddy-permitting simulation undertaken here yielded a useful set of data for comparisons with paleo-proxy reconstructions.

Most state-of-the-art climate models include a coarsely resolved oceanic component, which hardly captures detailed dynamics, whereas eddy-permitting and eddy-resolving simulations are developed to reproduce the observed ocean. In this study, an eddy-permitting and a coarse resolution numerical experiment are conducted to simulate the global ocean state for the period of the Last Glacial Maximum (LGM, ~26 500 to 19 000 yr ago) and to investigate the improvements due to taking into account the smaller spatial scales. The ocean state from each simulation is confronted with a data set from the Multiproxy Approach for the Reconstruction of the Glacial Ocean (MARGO) sea surface temperatures (SSTs), some reconstructions of the palaeo-circulations and a number of sea-ice reconstructions. The western boundary currents and the Southern Ocean dynamics are better resolved in the high-resolution experiment than in the coarse simulation, but, although these more detailed SST structures yield a locally improved consistency between model predictions and proxies, they do not contribute significantly to the global statistical score. The SSTs in the tropical coastal upwelling zones are also not significantly improved by the eddy-permitting regime. The models perform in the mid-latitudes but as in the majority of the Paleoclimate Modelling Intercomparison Project simulations, the modelled sea-ice conditions are inconsistent with the palaeo-reconstructions. The effects of observation locations on the comparison between observed and simulated SST suggest that more sediment cores may be required to draw reliable conclusions about the improvements introduced by the high resolution model for reproducing the global SSTs. One has to be careful with the interpretation of the deep ocean state which has not reached statistical equilibrium in our simulations. However, the results indicate that the meridional overturning circulations are different between the two regimes, suggesting that the model parametrizations might also play a key role for simulating past climate states.

The technically important problem of two-dimensional inviscid standing-wave motion of a fluid in a semi-cylindrical vessel is examined using a variety of the Boussinesq wave equation as well as the directed fluid sheet technique. The associated eigenvalue problems are dealt with by employing perturbative procedures and series expansions. The predictions were compared with the outcome of laboratory tank experiments and also with ‘exact' results. The Boussinesq equation perturbative scheme based on an approximate representation of the vessel geometry not only gave the best results, but was also the most straightforward to apply. Furthermore, it yielded practically useful explicit formulae for the frequencies at which sloshing may be initiated. A limitation is, however, that this formalism is only applicable to systems with comparatively small fill ratios.

It is demonstrated how a hyperbolic-tangential shelf bathymetry can be used to mimic classical convex and concave bottom profiles employed for previous investigations of coastally trapped waves. This bathymetry permits a transformation of the governing wave equation which makes it possible to use Frobenius series expansions for solving the associated eigenvalue problems which yield the dispersion relations.

When modelling is used for investigating estuarine systems, a choice generally has to be made between applying simple mass-balance considerations or using a process-resolving three-dimensional (3-D) numerical circulation model. In the present investigation of the Gulf of Finland, a gradually mixed estuary in the Baltic Sea, it is demonstrated how Lagrangian-trajectory analysis applied to the output from a 3-D model minimizes the disadvantages associated with both of the modelling techniques referred to above. This formalism made it possible to demonstrate that the main part of the Gulf is dominated by water originating from the Baltic proper, and that the most pronounced mixing with fresh water from the river Neva takes place over a limited zone in the inner part of the Gulf. Dynamical insights were furthermore obtained by using the Lagrangian formalism to construct overturning stream-functions for the two source waters.

In a recent paper Girton et al., due to what appears to be a misunderstanding, stated that a critical-flow analysis of the deep-water transport through the Faroe Bank Channel had been undertaken by Lake et al. on the basis of rotating hydraulic theory for a channel of parabolic cross section. In fact, this quoted investigation dealt with a rectangular passage. In the present comment it is demonstrated how the use of parabolic bathymetry leads to significant improvements of the Froude number results.

A nonlinear oscillator, based on two connected vessels which are thermally forced around the maximum‐density temperature of the working fluid, is investigated. For a suitable choice of parameters, the system executes self‐sustained relaxation oscillations. This process has been examined for a large volume difference between the vessels, in which case the solution makes almost discontinuous jumps between the two branches of the slow manifold of the problem. It was found that in this limit, a lowest‐order analysis does reasonable justice to the periodic behaviour of the system.

A linear shallow water model was used to study different harmonic oscillations in the Baltic Sea. The model was initialized using a linear sea surface slope from east to west and was hereafter run without forcing. In our results we could identify three different local oscillatory modes: one in the Gulf of Finland with the two distinct periods 23 and 27 h, one in the Danish Belt Sea with a less distinct period in the range 23–27 h, and one in the Gulf of Riga with the period 17 h. The most pronounced mode is that in the Gulf of Finland. No clear indications of basin‐wide seiches in the Baltic could be found from our simulations. These results were further corroborated by a frequency analysis of sea level observations from the Baltic. This shows an amplification of the K1 and O1 tidal modes in the Gulf of Finland but not of the M2 and S2 modes. No such amplification was seen in the rest of the Baltic Sea. On the basis of our model simulations we propose that the sea level oscillations of the Baltic be regarded as an ensemble of weakly coupled local oscillations. Each oscillator corresponds to a “gulf mode” or “harbor mode” in a particular bay or subbasin. These are not proper eigenmodes since their energy gradually leaks out to the rest of the Baltic Sea, resulting in radiation damping. Nevertheless, their resonance may in fact be sharper than that of the proper basin‐wide eigenmodes.