This thesis is dedicated to the study of flows on a network. In the first part of the work, we describe notation and give the necessary results from graph theory and operator theory that will be used in the rest of the thesis. Next, we consider the flow of particles between vertices along an edge, which occurs instantaneously, and this flow is described by a system of first order ordinary differential equations. For this system, we extend the results of Perthame [48] to arbitrary nonnegative off-diagonal matrices (ML matrices). In particular, we show that the results that were obtained in [48] for positive off diagonal matrices hold for irreducible ML matrices. For reducible matrices, the results in [48], presented in the same form are only satisfied in certain invariant subspaces and do not hold for the whole matrix space in general. Next, we consider a system of transport equations on a network with Kirchoff-type conditions which allow for amplification and/or absorption at the boundary, and extend the results obtained in [33] to connected graphs with no sinks. We prove that the abstract Cauchy problem associated with the flow problem generates a strongly continuous semigroup provided the network has no sinks. We also prove that the acyclic part of the graph will be depleted in finite time, explicitly given by the length of the longest path in the acyclic part. / Thesis (Ph.D)-University of KwaZulu-Natal, Durban, 2013.

Fragmentation equations occur naturally in many real world problems, see [ZM85, ZM86, HEL91, CEH91, HGEL96, SLLM00, Ban02, BL03, Ban04, BA06] and references therein. Mathematical study of these equations is mostly concentrated on building existence and uniqueness theories and on qualitative analysis of solutions (shattering), some effort has be done in finding solutions analytically. In this project, we deal with numerical analysis of fragmentation equation with transport. First, we provide some existence results in Banach and Hilbert settings, then we turn to numerical analysis. For this approximation and interpolation theory for generalized Laguerre functions is derived. Using these results we formulate Laguerre pseudospectral method and provide its stability and convergence analysis. The project is concluded with several numerical experiments. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2012.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2005. / Heuristics for obtaining upper bounds on crossing numbers of small graphs in the plane, and a heuristic for obtaining lower bounds to the crossing numbers of large graphs in the plane are presented in this thesis. It is shown that the two-page book layout framework is effective for deriving general upper bounds, and that it may also be used to obtain exact results for the crossing numbers of graphs. The upper bound algorithm is based on the well-kown optimization heuristics of tabu search, genetic algorithms and neural networks for obtaining two-page book layouts with few resultant edge crossings. The lower bound algorithm is based on the notion of embedding a graph into another graph, and, to the best knowledge of the author, it is the first known lower bound algorithm for the corssing number of a graph. It utilizes Dijkstra's shortest paths algorithm to embed one graph into another, in such a fashion as to minimize edge and vertex congestion values. The upper bound algorithms that were developed in this thesis were applied to all non-planar complete multipartite graphs of orders 6-13. A catalogue of drawings of these graphs with low numbers of crossings is provided in the thesis. Lower bounds on the crossing numbers of these graphs were also computed, using lowerbounds that are known for a number of complete multipartite graphs, as well as the fact that lower bounds on the crossing numbers of the subgraphs of a graph G, are lower bounds on the crossing number of G. A reference implementation of the Garey-Johnson algorithm is supplied, and finally, it is shown that Szekely's algorithm for computing the independent-odd crossing number may be converted into a heuristic algorithm for deriving upper bounds on the plane crossing number of a graph. This thesis also provides a thorough survey of results known for the crossing number of a graph in the plane. The survey especially focuses on algorithmic issues that have been proposed by researchers in the field of crossing number research.

Thesis (MSc)--University of Stellenbosch, 2004. / ENGLISH ABSTRACT: Motivation for the study of protection strategies for graphs is rooted in antiquity and has evolved as a subdiscipline of graph theory since the early 1990s. Using, as a point of departure, the notions of weak Roman domination and secure domination (where protection of a graph is required against a single attack) an initial framework for higher order domination was introduced in 2002 (allowing for the protection of a graph against an arbitrary finite, or even infinite, number of attacks). In this thesis, the theory of higher order domination in graphs is broadened yet further to include the possibility of an arbitrary number of guards being stationed at a vertex. The thesis firstly provides a comprehensive survey of the combinatorial literature on Roman domination, weak Roman domination, secure domination and other higher order domination strategies, with a view to summarise the state of the art in the theory of higher order graph domination as at the start of 2004. Secondly, a generalised framework for higher order domination is introduced in two parts: the first catering for the protection of a graph against a finite number of consecutive attacks, and the second concerning the perpetual security of a graph (protection of the graph against an infinite number of consecutive attacks). Two types of higher order domination are distinguished: smart domination (requiring the existence of a protection strategy for any sequence of consecutive attacks of a pre–specified length, but leaving it up to a strategist to uncover such a guard movement strategy for a particular instance of the attack sequence), and foolproof domination (requiring that any possible guard movement strategy be a successful protection strategy for the graph in question). Properties of these higher order domination parameters are examined—first by investigating the application of known higher order domination results from the literature, and secondly by obtaining new results, thereby hopefully improving current understanding of these domination parameters. Thirdly, the thesis contributes by (i) establishing higher order domination parameter values for some special graph classes not previously considered (such as complete multipartite graphs, wheels, caterpillars and spiders), by (ii) summarising parameter values for special graph classes previously established (such as those for paths, cycles and selected cartesian products), and by (iii) improving higher order domination parameter bounds previously obtained (in the case of the cartesian product of two cycles). Finally, a clear indication of unresolved problems in higher order graph domination is provided in the conclusion to this thesis, together with some suggestions as to possibly desirable future generalisations of the theory. / AFRIKAANSE OPSOMMING: Die motivering vir die studie van verdedigingstrategie¨e vir grafieke het sy ontstaan in die antieke wˆereld en het sedert die vroe¨e 1990s as ’n subdissipline in grafiekteorie begin ontwikkel. Deur gebruik te maak van die idee van swak Romynse dominasie en versterkte dominasie (waar verdediging van ’n grafiek teen ’n enkele aanval vereis word) het ’n aanvangsraamwerk vir ho¨er– orde dominasie (wat ’n grafiek teen ’n veelvuldige, of selfs oneindige aantal, aanvalle verdedig) in 2002 die lig gesien. Die teorie van ho¨er–orde dominasie in grafieke word in hierdie tesis verbreed, deur toe te laat dat ’n arbitrˆere aantal wagte by elke punt van die grafiek gestasioneer mag word. Eerstens voorsien die tesis ’n omvangryke oorsig van die kombinatoriese literatuur oor Romynse dominasie, swak Romynse dominasie, versterkte dominasie en ander ho¨er–orde dominasie strategie ¨e, met die doel om die kundigheid betreffende die teorie van ho¨er–orde dominasie, soos aan die begin van 2004, op te som. Tweedens word ’n veralgemeende raamwerk vir ho¨er–orde dominasie bekendgestel, en wel in twee dele. Die eerste deel maak voorsiening vir die verdediging van ’n grafiek teen ’n eindige aantal opeenvolgende aanvalle, terwyl die tweede deel betrekking het op die oneindige sekuriteit van ’n grafiek (verdediging teen ’n oneindige aantal opeenvolgende aanvalle). Daar word tussen twee tipes h¨oer–orde dominasie onderskei: intelligente dominasie (wat slegs die bestaan van ’n verdedigingstrategie vir enige reeks opeenvolgende aanvalle vereis, maar dit aan ’n strateeg oorlaat om ’n suksesvolle bewegingstrategie vir die verdediging teen ’n spesifieke reeks aanvalle te vind), en onfeilbare dominasie (wat vereis dat enige moontlike bewegingstrategie resulteer in ’n suksesvolle verdedigingstrategie vir die betrokke grafiek). Eienskappe van hierdie ho¨er–orde dominasie parameters word ondersoek, deur eerstens die toepasbaarheid van bekende ho¨er–orde dominasie resultate vanuit die literatuur te assimileer, en tweedens nuwe resultate te bekom, in die hoop om die huidige kundigheid met betrekking tot hierdie dominasie parameters te verbreed. Derdens word ’n bydrae gelewer deur (i) ho¨er–orde dominasie parameterwaardes vas te stel vir sommige spesiale klasse grafieke wat nie voorheen ondersoek is nie (soos volledig veelledige grafieke, wiele, ruspers en spinnekoppe), deur (ii) parameterwaardes wat reeds bepaal is (soos byvoorbeeld di´e vir paaie, siklusse en sommige kartesiese produkte) op te som, en deur (iii) bekende ho¨er–orde dominasie parametergrense te verbeter (in die geval van die kartesiese produk van twee siklusse). Laastens word ’n aanduiding van oop probleme in die teorie van ho¨er–orde dominasie in die slothoofstuk van die tesis voorsien, tesame met voorstelle ten opsigte van moontlik sinvolle veralgemenings van die teorie.

Thesis (MSc (Applied mathematics))--University of Stellenbosch, 2011. / Includes bibliography. / Please refer to full text to view abstract.

Thesis (MSc.)--University of Stellenbosch, 2000. / ENGLISH ABSTRACT: The notion of a classical graph theoretic Ramsey number is generalized by assuming that both the original graph whose edges are arbitrarily bicoloured and the monochromatic subgraphs to be forced are complete, balanced, multipartite graphs, instead of complete graphs as in the standard definition. Some small multipartite Ramsey numbers are found, while upper- and lower bounds are established for others. Analytic arguments as well as computer searches are used. / AFRIKAANSE OPSOMMING: Die klassieke grafiek-teoretiese definisie van ’n Ramsey getal word veralgemeen deur te aanvaar dat beide die oorspronklike grafiek, waarvan die lyne willekeurig met twee kleure gekleur word en die gesogte subgrafieke almal volledige, gebalanseerde, veelledige grafieke is, anders as in die standaard definisie. Klein veelledige Ramsey getalle word gevind, terwyl bo- en ondergrense vir ander daargestel word. Analitiese argumente en rekenaarsoektogte word gebruik.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: In this thesis we consider the problem of building a dense and consistent map of a mobile robot’s environment that is updated as the robot moves. Such maps are vital for safe and collision-free navigation. Measurements obtained from a range sensor mounted on the robot provide information on the structure of the environment, but are typically corrupted by noise. These measurements are also relative to the robot’s unknown pose (location and orientation) and, in order to combine them into a world-centric map, pose estimation is necessary at every time step. A SLAM system can be used for this task. However, since landmark measurements and robot motion are inherently noisy, the pose estimates are typically characterized by uncertainty. When building a map it is essential to deal with the uncertainties in range measurements and pose estimates in a principled manner to avoid overconfidence in the map. A literature review of robotic mapping algorithms reveals that the occupancy grid mapping algorithm is well suited for our goal. This algorithm divides the area to be mapped into a regular lattice of cells (squares for 2D maps or cubes for 3D maps) and maintains an occupancy probability for each cell. Although an inverse sensor model is often employed to incorporate measurement uncertainty into such a map, many authors merely state or depict their sensor models. We derive our model analytically and discuss ways to tailor it for sensor-specific uncertainty. One of the shortcomings of the original occupancy grid algorithm is its inability to convey uncertainty in the robot’s pose to the map. We address this problem by altering the occupancy grid update equation to include weighted samples from the pose uncertainty distribution (provided by the SLAM system). The occupancy grid algorithm has been criticized for its high memory requirements. Techniques have been proposed to represent the map as a region tree, allowing cells to have different sizes depending on the information received for them. Such an approach necessitates a set of rules for determining when a cell should be split (for higher resolution in a local region) and when groups of cells should be merged (for lower resolution). We identify some inconsistencies that can arise from existing rules, and adapt those rules so that such errors are avoided. We test our proposed adaptive occupancy grid algorithm, that incorporates both measurement and pose uncertainty, on simulated and real-world data. The results indicate that these uncertainties are included effectively, to provide a more informative map, without a loss in accuracy. Furthermore, our adaptive maps need far fewer cells than their regular counterparts, and our new set of rules for deciding when to split or merge cells significantly improves the ability of the adaptive grid map to mimic its regular counterpart. / AFRIKAANSE OPSOMMING: In hierdie tesis beskou ons die probleem om ’n digte en konsekwente kaart van ’n mobiele robot se omgewing te bou, wat opgedateer word soos die robot beweeg. Sulke kaarte is van kardinale belang vir veilige, botsingvrye navigasie. Metings verkry vanaf ’n sensor wat op die robot gemonteer is, verskaf inligting rakende die struktuur van die omgewing, maar word tipies deur ruis vervorm. Hierdie metings is ook relatief tot die robot se onbekende postuur (posisie en oriëntasie) en, om hulle saam te voeg in ’n wêreldsentriese kaart, is postuurafskatting nodig op elke tydstap. ’n SLAM stelsel kan vir hierdie doeleinde gebruik word. Aangesien landmerkmetings en die beweging van die robot inherent ruiserig is, word die postuurskattings gekarakteriseer deur onsekerheid. Met die bou van ’n kaart moet hierdie onsekerhede in afstandmetings en postuurskattings op ’n beginselvaste manier hanteer word om te verhoed dat te veel vertroue in die kaart geplaas word. ’n Literatuurstudie van karteringsalgoritmes openbaar die besettingsroosteralgoritme as geskik vir ons doel. Die algoritme verdeel die gebied wat gekarteer moet word in ’n reëlmatige rooster van selle (vierkante vir 2D kaarte of kubusse vir 3D kaarte) en onderhou ’n besettingswaarskynlikheid vir elke sel. Alhoewel ’n inverse sensormodel tipies gebruik word om metingsonsekerheid in so ’n kaart te inkorporeer, noem of wys baie outeurs slegs hulle model. Ons herlei ons model analities en beskryf maniere om sensorspesifieke metingsonsekerheid daarby in te sluit. Een van die tekortkominge van die besettingsroosteralgoritme is sy onvermoë om onsekerheid in die postuur van die robot na die kaart oor te dra. Ons spreek hierdie probleem aan deur die opdateringsvergelyking van die oorspronklike besettingsroosteralgoritme aan te pas, om geweegde monsters van die postuuronsekerheidsverdeling (verskaf deur die SLAM stelsel) in te sluit. Die besettingsroosteralgoritme word soms gekritiseer vir sy hoë verbruik van geheue. Tegnieke is voorgestel om die kaart as ’n gebiedsboom voor te stel, wat selle toelaat om verskillende groottes te hê, afhangende van die inligting wat vir hulle verkry is. So ’n benadering noodsaak ’n stel reëls wat spesifiseer wanneer ’n sel verdeel (vir ’n hoër resolusie in ’n plaaslike gebied) en wanneer ’n groep selle saamgevoeg (vir ’n laer resolusie) word. Ons identifiseer teenstrydighede wat kan voorkom as die huidige reëls gevolg word, en pas hierdie reëls aan sodat sulke foute vermy word. Ons toets ons voorgestelde aanpasbare besettingsroosteralgoritme, wat beide metings- en postuuronsekerheid insluit, op gesimuleerde en werklike data. Die resultate dui daarop dat hierdie onsekerhede op ’n effektiewe wyse na die kaart oorgedra word sonder om akkuraatheid prys te gee. Wat meer is, ons aanpasbare kaarte benodig heelwat minder selle as hul reëlmatige eweknieë. Ons nuwe stel reëls om te besluit wanneer selle verdeel of saamgevoeg word, veroorsaak ook ’n merkwaardige verbetering in die vermoë van die aanpasbare roosterkaart om sy reëlmatige eweknie na te boots.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Range images are images that store at each pixel the distance between the sensor and a particular point in the observed scene, instead of the colour information. They provide a convenient storage format for 3-D point cloud information captured from a single point of view. Range image segmentation is the process of grouping the pixels of a range image into regions of points that belong to the same surface. Segmentations are useful for many applications that require higherlevel information, and with range images they also represent a significant step towards complete scene reconstruction. This study considers the segmentation of range images into planar surfaces. It discusses the theory and also implements and evaluates some current approaches found in the literature. The study then develops a new approach based on the theory of graph cut optimization which has been successfully applied to various other image processing tasks but, according to a search of the literature, has otherwise not been used to attempt segmenting range images. This new approach is notable for its strong guarantees in optimizing a specific energy function which has a rigorous theoretical underpinning for handling noise in images. It proves to be very robust to noise and also different values of the few parameters that need to be trained. Results are evaluated in a quantitative manner using a standard evaluation framework and datasets that allow us to compare against various other approaches found in the literature. We find that our approach delivers results that are competitive when compared to the current state-of-the-art, and can easily be applied to images captured with different techniques that present varying noise and processing challenges. / AFRIKAANSE OPSOMMING: Dieptebeelde is beelde wat vir elke piksel die afstand tussen die sensor en ’n spesifieke punt in die waargenome toneel, in plaas van die kleur, stoor. Dit verskaf ’n gerieflike stoorformaat vir 3-D puntwolke wat vanaf ’n enkele sigpunt opgeneem is. Die segmentasie van dieptebeelde is die proses waarby die piksels van ’n dieptebeeld in gebiede opgedeel word, sodat punte saam gegroepeer word as hulle op dieselfde oppervlak lê. Segmentasie is nuttig vir verskeie toepassings wat hoërvlak inligting benodig en, in die geval van dieptebeelde, verteenwoordig dit ’n beduidende stap in die rigting van volledige toneel-rekonstruksie. Hierdie studie ondersoek segmentasie waar dieptebeelde opgedeel word in plat vlakke. Dit bespreek die teorie, en implementeer en evalueer ook sekere van die huidige tegnieke wat in die literatuur gevind kan word. Die studie ontwikkel dan ’n nuwe tegniek wat gebaseer is op die teorie van grafieksnit-optimering wat al suksesvol toegepas is op verskeie ander beeldverwerkingsprobleme maar, sover ’n studie op die literatuur wys, nog nie gebruik is om dieptebeelde te segmenteer nie. Hierdie nuwe benadering is merkbaar vir sy sterk waarborge vir die optimering van ’n spesifieke energie-funksie wat ’n sterk teoretiese fondasie het vir die hantering van geraas in beelde. Die tegniek bewys om fors te wees tot geraas sowel as die keuse van waardes vir die min parameters wat afgerig moet word. Resultate word geëvalueer op ’n kwantitatiewe wyse deur die gebruik van ’n standaard evalueringsraamwerk en datastelle wat ons toelaat om hierdie tegniek te vergelyk met ander tegnieke in die literatuur. Ons vind dat ons tegniek resultate lewer wat mededingend is ten opsigte van die huidige stand-van-die-kuns en dat ons dit maklik kan toepas op beelde wat deur verskeie tegnieke opgeneem is, alhoewel hulle verskillende geraastipes en verwerkingsuitdagings bied.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Fluid flow through porous media plays an important role in a variety of contexts of which filtration is one. Filtration efficiency of fibrous filters depends on the micro-structural characterization of these porous materials and is reflected in the permeability there-of. Compression of fibrous porous media has a significant effect on the permeability. Experimental data indicate that the permeability varies generally with more than an order of magnitude over the narrow porosity range in which the compression takes place. Relative to the amount of experimental studies regarding this phenomenon, there is a scarcity of geometric models in the literature that can account for the effect of compression on the permeability of a fibrous porous medium. Within the context of existing geometric porescale models based on rectangular geometry, a new model is presented and an existing model improved to predict the effect of one-dimensional compression in the streamwise direction. In addition, without compromising on a commitment to mathematical simplicity, empirical data of a non-woven fibrous porous medium was used to highlight the effect of model geometry on its predictive capability. Different mathematical expressions for the relationship between compression and porosity were considered. The permeability is expressed explicitly in terms of the fibre diameter and the compression fraction and implicitly in terms of the porosity. The porosity is incorporated through the relationship between the linear dimensions of the geometric model. The general applicability of the model(s) was validated by making use of data on airflow through a soft fibrous porous material as well as through glass and nylon fibres. The permeability predictions fall within the same order of magnitude as the experimental data. Given the mathematical simplicity of the model(s), the prediction capability is satisfactory. Attention is drawn to assumptions made and model restrictions within the analytical modelling procedure. A general predictive equation is presented for the permeability prediction in which a solid distribution factor is introduced. The proposed models serve as basis for further adaptation and refinement towards prediction capability. / AFRIKAANSE OPSOMMING: Vloei van vloeistowwe deur poreuse media speel ’n belangrike rol in ’n verskeidenheid kontekste waarvan filtrasie een is. Die filtrasie doeltreffendheid van vesel filters hang af van die mikro-strukturele karakterisering van hierdie poreuse materiale en word gereflekteer in die permeabiliteit. Kompressie van veselagtige poreuse media het ’n beduidende effek op die permeabiliteit. Eksperimentele data dui aan dat die verandering in permeabiliteit gewoonlik oor meer as ’n orde grootte strek oor die klein porositeitsinterval waarin die kompressie plaasvind. Relatief tot die aantal eksperimentele studies rakende hierdie verskynsel, is daar ’n tekort aan geometriese modelle in die literatuur wat die effek van kompressie op die permeabiliteit van veselagtige poreuse media in ag kan neem. Binne die konteks van bestaande geometriese kanaal-skaal modelle gebasseer op reghoekige geometrie, is ’n nuwe model voorgestel en ’n bestaande model verbeter om die effek van een-dimensionele kompressie in die stroomsgewyse rigting te voorspel. Sonder om die verbintenis tot wiskundige eenvoud prys te gee, is empiriese data van ’n nie-geweefde veselagtige poreuse medium gebruik om die effek van die geometrie van ’n model op sy voorspellingsvermo¨e uit te lig. Verskillende wiskundige uitdrukkings is oorweeg vir die verband tussen kompressie en porositeit. Die permeabiliteit is eksplisiet uitgedruk in terme van die veseldiameter en die kompressie breukdeel en implisiet in terme van die porositeit. Die porositeit is ge-inkorporeer deur die verhouding tussen die lineêre dimensies van die geometriese model. Die algemene toepaslikheid van die model(le) is gestaaf deur gebruik te maak van data oor lugvloei deur ’n sagte veselagtige poreuse materiaal sowel as deur glas en nylon vesels. Die voorspellings van die permeabiliteit val binne dieselfde groote orde as die eksperimentele data. Gegee die wiskundige eenvoud van die model(le), is die voorspellingsvermoë bevredigend. Aandag is gevestig op aannames wat gemaak is en modelbeperkings binne die analitiese modellerings prosedure. ’n Algemene voorspellingsvergelyking is voorgestel vir die voorspelling van die permeabiliteit waarin ’n vaste stof distribusie faktor geinkorporeer is. Die voorgestelde modelle dien as basis vir verdere aanpassing en verfyning van voorspellingsvermoë.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Shallow water equations (SWEs) are a set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid. They are widely applicable in the domain of fluid dynamics. To meet the needs of engineers working on the area of fluid dynamics, a method known as spectral/hp element method has been developed which is a scheme that can be used with complicated geometry. The use of discontinuous Galerkin (DG) discretisation permits discontinuity of the numerical solution to exist at inter-element surfaces. In the DG method, the solution within each element is not reconstructed by looking to neighbouring elements, thus the transfer information between elements will be ensured through the numerical fluxes. As a consequence, the accuracy of the method depends largely on the definition of the numerical fluxes. There are many different type of numerical fluxes computed from Riemann solvers. Four of them will be applied here respectively for comparison through a 2D Rossby wave test case. / AFRIKAANSE OPSOMMING: Vlakwatervergelykings (SWEs) is ’n stel hiperboliese parsiële differensiaalvergelykings wat die vloei onder ’n oppervlak wat druk op ’n vloeistof uitoefen beskryf. Hulle het wye toepassing op die gebied van vloeidinamika. Om aan die behoeftes van ingenieurs wat werk op die gebied van vloeidinamika te voldoen is ’n metode bekend as die spektraal /hp element metode ontwikkel. Hierdie metode kan gebruik word selfs wanneer die probleem ingewikkelde grenskondisies het. Die Diskontinue Galerkin (DG) diskretisering wat gebruik word laat diskontinuïteit van die numeriese oplossing toe om te bestaan by tussenelement oppervlakke. In die DG metode word die oplossing binne elke element nie gerekonstrueer deur te kyk na die naburige elemente nie. Dus word die oordrag van informasie tussen elemente verseker deur die numeriese stroomterme. Die akkuraatheid van hierdie metode hang dus grootliks af van die definisie van die numeriese stroomterme. Daar is baie verskillende tipe numeriese strometerme wat bereken kan word uit Riemann oplossers. Vier van hulle sal hier gebruik en vergelyk word op ’n 2D Rossby golf toets geval.