Applied Mathematics Researchhttp://hdl.handle.net/10023/962021-12-09T14:39:08Z2021-12-09T14:39:08ZWave of chaos in a spatial eco-epidemiological system : generating realistic patterns of patchiness in rabbit-lynx dynamicsUpadhyay, RanjitRoy, ParimitaVenkataraman, C.Madzvamuse, Anotidahttp://hdl.handle.net/10023/116662021-09-26T04:32:30Z2016-11-01T00:00:00ZIn the present paper, we propose and analyse an eco-epidemiological model with diffusion to study the dynamics of rabbit populations which are consumed by lynx populations. Existence, boundedness, stability and bifurcation analyses of solutions for the proposed rabbit-lynx model are performed. Results show that in the presence of diffusion the model has the potential of exhibiting Turing instability. Numerical results (finite difference and finite element methods) reveal the existence of the wave of chaos and this appears to be a dominant mode of disease dispersal. We also show the mechanism of spatiotemporal pattern formation resulting from the Hopf bifurcation analysis, which can be a potential candidate for understanding the complex spatiotemporal dynamics of eco-epidemiological systems. Implications of the asymptotic transmission rate on disease eradication among rabbit population which in turn enhances the survival of Iberian lynx are discussed.
AM and CV would like to acknowledge support from the Engineering and Physical Sciences Research Council grant (EP/J016780/1) and the Leverhulme Trust Research Project Grant (RPG-2014-149).
2016-11-01T00:00:00ZUpadhyay, RanjitRoy, ParimitaVenkataraman, C.Madzvamuse, AnotidaIn the present paper, we propose and analyse an eco-epidemiological model with diffusion to study the dynamics of rabbit populations which are consumed by lynx populations. Existence, boundedness, stability and bifurcation analyses of solutions for the proposed rabbit-lynx model are performed. Results show that in the presence of diffusion the model has the potential of exhibiting Turing instability. Numerical results (finite difference and finite element methods) reveal the existence of the wave of chaos and this appears to be a dominant mode of disease dispersal. We also show the mechanism of spatiotemporal pattern formation resulting from the Hopf bifurcation analysis, which can be a potential candidate for understanding the complex spatiotemporal dynamics of eco-epidemiological systems. Implications of the asymptotic transmission rate on disease eradication among rabbit population which in turn enhances the survival of Iberian lynx are discussed.Spatial variation in boundary conditions can govern selection and location of eyespots in butterfly wingsVenkataraman, ChandrasekharSekimura, Toshiohttp://hdl.handle.net/10023/116182021-01-13T00:39:05Z2017-01-01T00:00:00ZDespite being the subject of widespread study, many aspects of the development of eyespot patterns in butterfly wings remain poorly understood. In this work, we examine, through numerical simulations, a mathematical model for eyespot focus point formation in which a reaction-diffusion system is assumed to play the role of the patterning mechanism. In the model, changes in the boundary conditions at the veins at the proximal boundary alone are capable of determining whether or not an eyespot focus forms in a given wing cell and the eventual position of focus points within the wing cell. Furthermore, an auxiliary surface reaction diffusion system posed along the entire proximal boundary of the wing cells is proposed as the mechanism that generates the necessary changes in the proximal boundary profiles. In order to illustrate the robustness of the model, we perform simulations on a curved wing geometry that is somewhat closer to a biological realistic domain than the rectangular wing cells previously considered, and we also illustrate the ability of the model to reproduce experimental results on artificial selection of eyespots.
2017-01-01T00:00:00ZVenkataraman, ChandrasekharSekimura, ToshioDespite being the subject of widespread study, many aspects of the development of eyespot patterns in butterfly wings remain poorly understood. In this work, we examine, through numerical simulations, a mathematical model for eyespot focus point formation in which a reaction-diffusion system is assumed to play the role of the patterning mechanism. In the model, changes in the boundary conditions at the veins at the proximal boundary alone are capable of determining whether or not an eyespot focus forms in a given wing cell and the eventual position of focus points within the wing cell. Furthermore, an auxiliary surface reaction diffusion system posed along the entire proximal boundary of the wing cells is proposed as the mechanism that generates the necessary changes in the proximal boundary profiles. In order to illustrate the robustness of the model, we perform simulations on a curved wing geometry that is somewhat closer to a biological realistic domain than the rectangular wing cells previously considered, and we also illustrate the ability of the model to reproduce experimental results on artificial selection of eyespots.Erwin Schrödinger and quantum wave mechanicsO'Connor, John J.Robertson, Edmund F.http://hdl.handle.net/10023/115432021-01-12T11:34:12Z2017-08-22T00:00:00ZThe fathers of matrix quantum mechanics believed that the quantum particles are unanschaulich (unvisualizable) and that quantum particles pop into existence only when we measure them. Challenging the orthodoxy, in 1926 Erwin Schrödinger developed his wave equation that describes the quantum particles as a packet of quantum probability amplitudes evolving in space and time. Thus, Schrödinger visualized the unvisualizable and lifted the veil that has been obscuring the wonders of the quantum world.
2017-08-22T00:00:00ZO'Connor, John J.Robertson, Edmund F.The fathers of matrix quantum mechanics believed that the quantum particles are unanschaulich (unvisualizable) and that quantum particles pop into existence only when we measure them. Challenging the orthodoxy, in 1926 Erwin Schrödinger developed his wave equation that describes the quantum particles as a packet of quantum probability amplitudes evolving in space and time. Thus, Schrödinger visualized the unvisualizable and lifted the veil that has been obscuring the wonders of the quantum world.Hα and EUV observations of a partial CMEChristian, Damian J.Jess, David B.Antolin, PatrickMathioudakis, Mihalishttp://hdl.handle.net/10023/102912021-01-12T11:25:00Z2015-05-12T00:00:00ZWe have obtained Hα high spatial and time resolution observations of the upper solar chromosphere and supplemented these with multi-wavelength observations from the Solar Dynamics Observatory (SDO) and the Hinode Extreme-ultraviolet Imaging Spectrometer. The Hα observations were conducted on 2012 February 11 with the Hydrogen-Alpha Rapid Dynamics Camera instrument at the National Solar Observatory's Dunn Solar Telescope. Our Hα observations found large downflows of chromospheric material returning from coronal heights following a failed prominence eruption. We have detected several large condensations ("blobs") returning to the solar surface at velocities of ≈200 km s−1 in both Hα and several SDO Atmospheric Imaging Assembly band passes. The average derived size of these "blobs" in Hα is 500 by 3000 km2 in the directions perpendicular and parallel to the direction of travel, respectively. A comparison of our "blob" widths to those found from coronal rain, indicate that there are additional, smaller, unresolved "blobs" in agreement with previous studies and recent numerical simulations. Our observed velocities and decelerations of the "blobs" in both Hα and SDO bands are less than those expected for gravitational free-fall and imply additional magnetic or gas pressure impeding the flow. We derived a kinetic energy of ≈2 orders of magnitude lower for the main eruption than a typical coronal mass ejection, which may explain its partial nature.
2015-05-12T00:00:00ZChristian, Damian J.Jess, David B.Antolin, PatrickMathioudakis, MihalisWe have obtained Hα high spatial and time resolution observations of the upper solar chromosphere and supplemented these with multi-wavelength observations from the Solar Dynamics Observatory (SDO) and the Hinode Extreme-ultraviolet Imaging Spectrometer. The Hα observations were conducted on 2012 February 11 with the Hydrogen-Alpha Rapid Dynamics Camera instrument at the National Solar Observatory's Dunn Solar Telescope. Our Hα observations found large downflows of chromospheric material returning from coronal heights following a failed prominence eruption. We have detected several large condensations ("blobs") returning to the solar surface at velocities of ≈200 km s−1 in both Hα and several SDO Atmospheric Imaging Assembly band passes. The average derived size of these "blobs" in Hα is 500 by 3000 km2 in the directions perpendicular and parallel to the direction of travel, respectively. A comparison of our "blob" widths to those found from coronal rain, indicate that there are additional, smaller, unresolved "blobs" in agreement with previous studies and recent numerical simulations. Our observed velocities and decelerations of the "blobs" in both Hα and SDO bands are less than those expected for gravitational free-fall and imply additional magnetic or gas pressure impeding the flow. We derived a kinetic energy of ≈2 orders of magnitude lower for the main eruption than a typical coronal mass ejection, which may explain its partial nature.Detection of supersonic downflows and associated heating events in the transition region above sunspotsKleint, L.Antolin, P.Tian, H.Judge, P.Testa, P.De Pontieu, B.Martínez-Sykora, J.Reeves, K. K.Wuelser, J. P.McKillop, S.Saar, S.Carlsson, M.Boerner, P.Hurlburt, N.Lemen, J.Tarbell, T. D.Title, A.Golub, L.Hansteen, V.Jaeggli, S.Kankelborg, C.http://hdl.handle.net/10023/102822021-08-22T05:31:04Z2014-06-27T00:00:00ZInterface Region Imaging Spectrograph data allow us to study the solar transition region (TR) with an unprecedented spatial resolution of 0″33. On 2013 August 30, we observed bursts of high Doppler shifts suggesting strong supersonic downflows of up to 200 km s–1 and weaker, slightly slower upflows in the spectral lines Mg II h and k, C II 1336, Si IV 1394 Å, and 1403 Å, that are correlated with brightenings in the slitjaw images (SJIs). The bursty behavior lasts throughout the 2 hr observation, with average burst durations of about 20 s. The locations of these short-lived events appear to be the umbral and penumbral footpoints of EUV loops. Fast apparent downflows are observed along these loops in the SJIs and in the Atmospheric Imaging Assembly, suggesting that the loops are thermally unstable. We interpret the observations as cool material falling from coronal heights, and especially coronal rain produced along the thermally unstable loops, which leads to an increase of intensity at the loop footpoints, probably indicating an increase of density and temperature in the TR. The rain speeds are on the higher end of previously reported speeds for this phenomenon, and possibly higher than the free-fall velocity along the loops. On other observing days, similar bright dots are sometimes aligned into ribbons, resembling small flare ribbons. These observations provide a first insight into small-scale heating events in sunspots in the TR.
2014-06-27T00:00:00ZKleint, L.Antolin, P.Tian, H.Judge, P.Testa, P.De Pontieu, B.Martínez-Sykora, J.Reeves, K. K.Wuelser, J. P.McKillop, S.Saar, S.Carlsson, M.Boerner, P.Hurlburt, N.Lemen, J.Tarbell, T. D.Title, A.Golub, L.Hansteen, V.Jaeggli, S.Kankelborg, C.Interface Region Imaging Spectrograph data allow us to study the solar transition region (TR) with an unprecedented spatial resolution of 0″33. On 2013 August 30, we observed bursts of high Doppler shifts suggesting strong supersonic downflows of up to 200 km s–1 and weaker, slightly slower upflows in the spectral lines Mg II h and k, C II 1336, Si IV 1394 Å, and 1403 Å, that are correlated with brightenings in the slitjaw images (SJIs). The bursty behavior lasts throughout the 2 hr observation, with average burst durations of about 20 s. The locations of these short-lived events appear to be the umbral and penumbral footpoints of EUV loops. Fast apparent downflows are observed along these loops in the SJIs and in the Atmospheric Imaging Assembly, suggesting that the loops are thermally unstable. We interpret the observations as cool material falling from coronal heights, and especially coronal rain produced along the thermally unstable loops, which leads to an increase of intensity at the loop footpoints, probably indicating an increase of density and temperature in the TR. The rain speeds are on the higher end of previously reported speeds for this phenomenon, and possibly higher than the free-fall velocity along the loops. On other observing days, similar bright dots are sometimes aligned into ribbons, resembling small flare ribbons. These observations provide a first insight into small-scale heating events in sunspots in the TR.Forward modeling of gyrosynchrotron intensity perturbations by sausage modesReznikova, V. E.Antolin, P.Van Doorsselaere, T.http://hdl.handle.net/10023/102802021-03-21T09:32:53Z2014-03-28T00:00:00ZTo determine the observable radio signatures of the fast sausagestanding wave, we examine gyrosynchrotron (GS) emission modulation usinga linear three-dimensional magnetohydrodynamic model of a plasmacylinder. Effects of the line-of-sight angle and instrumental resolutionon perturbations of the GS intensity are analyzed for two models: a basemodel with strong Razin suppression and a low-density model in which theRazin effect was unimportant. Our finding contradicts previouspredictions made with simpler models: an in-phase variation of intensitybetween low (f <fpeak) and high (f > fpeak) frequencies is found for the low-density model and ananti-phase variation for the base model in the case of a viewing angleof 45°. The spatially inhomogeneous character of the oscillatingemission source and the spatial resolution of the model are found tohave a significant effect on the resulting intensity.
2014-03-28T00:00:00ZReznikova, V. E.Antolin, P.Van Doorsselaere, T.To determine the observable radio signatures of the fast sausagestanding wave, we examine gyrosynchrotron (GS) emission modulation usinga linear three-dimensional magnetohydrodynamic model of a plasmacylinder. Effects of the line-of-sight angle and instrumental resolutionon perturbations of the GS intensity are analyzed for two models: a basemodel with strong Razin suppression and a low-density model in which theRazin effect was unimportant. Our finding contradicts previouspredictions made with simpler models: an in-phase variation of intensitybetween low (f <fpeak) and high (f > fpeak) frequencies is found for the low-density model and ananti-phase variation for the base model in the case of a viewing angleof 45°. The spatially inhomogeneous character of the oscillatingemission source and the spatial resolution of the model are found tohave a significant effect on the resulting intensity.The usage of a three-compartment model to investigate the metabolic differences between hepatic reductase null and wild-type miceHill, LydiaChaplain, Mark Andrew JosephWolf, RolandKapelyukh, Yuryhttp://hdl.handle.net/10023/96112021-04-18T07:30:53Z2015-10-05T00:00:00ZThe Cytochrome P450 (CYP) system is involved in 90% of the human body’s interactions with xenobiotics and due to this, it has become an area of avid research including the creation of transgenic mice. This paper proposes a three-compartment model which is used to explain the drug metabolism in the Hepatic Reductase Null (HRN) mouse developed by the University of Dundee (Henderson, C. J., Otto, D. M. E., Carrie, D., Magnuson, M. A., McLaren, A. W., Rosewell, I. and Wolf, C. R. (2003) Inactivation of the hepatic cytochrome p450 system by conditional deletion of hepatic cytochrome p450 reductase. J. Biol. Chem. 278, 13480–13486). The model is compared with a two-compartment model using experimental data from studies using wild-type and HRN mice. This comparison allowed for metabolic differences between the two types of mice to be isolated. The three sets of drug data (Gefitinib, Midazolam and Thalidomide) showed that the transgenic mouse has a decreased rate of metabolism.
L.H. is currently funded by the Research Foundation Flanders (FWO) and the Belgian Science Policy Office under Grant No. IAP-VI/10.
2015-10-05T00:00:00ZHill, LydiaChaplain, Mark Andrew JosephWolf, RolandKapelyukh, YuryThe Cytochrome P450 (CYP) system is involved in 90% of the human body’s interactions with xenobiotics and due to this, it has become an area of avid research including the creation of transgenic mice. This paper proposes a three-compartment model which is used to explain the drug metabolism in the Hepatic Reductase Null (HRN) mouse developed by the University of Dundee (Henderson, C. J., Otto, D. M. E., Carrie, D., Magnuson, M. A., McLaren, A. W., Rosewell, I. and Wolf, C. R. (2003) Inactivation of the hepatic cytochrome p450 system by conditional deletion of hepatic cytochrome p450 reductase. J. Biol. Chem. 278, 13480–13486). The model is compared with a two-compartment model using experimental data from studies using wild-type and HRN mice. This comparison allowed for metabolic differences between the two types of mice to be isolated. The three sets of drug data (Gefitinib, Midazolam and Thalidomide) showed that the transgenic mouse has a decreased rate of metabolism.Study of the three-dimensional shape and dynamics of coronal loops observed by Hinode/EISSyntelis, P.Gontikakis, C.Georgoulis, M. K.Alissandrakis, C. E.Tsinganos, K.http://hdl.handle.net/10023/94262021-01-12T11:19:22Z2012-10-01T00:00:00ZWe study plasma flows along selected coronal loops in NOAA Active Region 10926, observed on 3 December 2006 with Hinode’sEUVImaging Spectrograph (EIS). From the shape of the loops traced on intensity images and the Doppler shifts measured along their length we compute their three-dimensional (3D) shape and plasma flow velocity using a simple geometrical model. This calculation was performed for loops visible in the Fe viii 185 Å, Fe x 184 Å, Fe xii 195 Å, Fe xiii202 Å, and Fe xv 284 Å spectral lines. In most cases the flow is unidirectional from one footpoint to the other but there are also cases of draining motions from the top of the loops to their footpoints. Our results indicate that the same loop may show different flow patterns when observed in different spectral lines, suggesting a dynamically complex rather than a monolithic structure. We have also carried out magnetic extrapolations in the linear force-free field approximation using SOHO/MDI magnetograms, aiming toward a first-order identification of extrapolated magnetic field lines corresponding to the reconstructed loops. In all cases, the best-fit extrapolated lines exhibit left-handed twist (α<0), in agreement with the dominant twist of the region.
2012-10-01T00:00:00ZSyntelis, P.Gontikakis, C.Georgoulis, M. K.Alissandrakis, C. E.Tsinganos, K.We study plasma flows along selected coronal loops in NOAA Active Region 10926, observed on 3 December 2006 with Hinode’sEUVImaging Spectrograph (EIS). From the shape of the loops traced on intensity images and the Doppler shifts measured along their length we compute their three-dimensional (3D) shape and plasma flow velocity using a simple geometrical model. This calculation was performed for loops visible in the Fe viii 185 Å, Fe x 184 Å, Fe xii 195 Å, Fe xiii202 Å, and Fe xv 284 Å spectral lines. In most cases the flow is unidirectional from one footpoint to the other but there are also cases of draining motions from the top of the loops to their footpoints. Our results indicate that the same loop may show different flow patterns when observed in different spectral lines, suggesting a dynamically complex rather than a monolithic structure. We have also carried out magnetic extrapolations in the linear force-free field approximation using SOHO/MDI magnetograms, aiming toward a first-order identification of extrapolated magnetic field lines corresponding to the reconstructed loops. In all cases, the best-fit extrapolated lines exhibit left-handed twist (α<0), in agreement with the dominant twist of the region.Solar cycle variation of magnetic flux ropes in a quasi-static coronal evolution modelYeates, A. R.Constable, J. A.Martens, P. C. H.http://hdl.handle.net/10023/90372021-01-12T11:16:26Z2010-05-01T00:00:00ZThe structure of electric current and magnetic helicity in the solar corona is closely linked to solar activity over the 11-year cycle, yet is poorly understood. As an alternative to traditional current-free "potential field" extrapolations, we investigate a model for the global coronal magnetic field which is non-potential and time-dependent, following the build-up and transport of magnetic helicity due to flux emergence and large-scale photospheric motions. This helicity concentrates into twisted magnetic flux ropes, which may lose equilibrium and be ejected. Here, we consider how the magnetic structure predicted by this model-in particular the flux ropes-varies over the solar activity cycle, based on photospheric input data from six periods of cycle 23. The number of flux ropes doubles from minimum to maximum, following the total length of photospheric polarity inversion lines. However, the number of flux rope ejections increases by a factor of eight, following the emergence rate of active regions. This is broadly consistent with the observed cycle modulation of coronal mass ejections, although the actual rate of ejections in the simulation is about a fifth of the rate of observed events. The model predicts that, even at minimum, differential rotation will produce sheared, non-potential, magnetic structure at all latitudes.
2010-05-01T00:00:00ZYeates, A. R.Constable, J. A.Martens, P. C. H.The structure of electric current and magnetic helicity in the solar corona is closely linked to solar activity over the 11-year cycle, yet is poorly understood. As an alternative to traditional current-free "potential field" extrapolations, we investigate a model for the global coronal magnetic field which is non-potential and time-dependent, following the build-up and transport of magnetic helicity due to flux emergence and large-scale photospheric motions. This helicity concentrates into twisted magnetic flux ropes, which may lose equilibrium and be ejected. Here, we consider how the magnetic structure predicted by this model-in particular the flux ropes-varies over the solar activity cycle, based on photospheric input data from six periods of cycle 23. The number of flux ropes doubles from minimum to maximum, following the total length of photospheric polarity inversion lines. However, the number of flux rope ejections increases by a factor of eight, following the emergence rate of active regions. This is broadly consistent with the observed cycle modulation of coronal mass ejections, although the actual rate of ejections in the simulation is about a fifth of the rate of observed events. The model predicts that, even at minimum, differential rotation will produce sheared, non-potential, magnetic structure at all latitudes.Spontaneous reconnection at a separator current layer : I. Nature of the reconnectionE. H. Stevenson, JulieE. Parnell, Clarehttp://hdl.handle.net/10023/89592021-01-12T11:06:06Z2015-12-01T00:00:00ZMagnetic separators, which lie on the boundary between four topologically-distinct flux domains, are prime locations in three-dimensional magnetic fields for reconnection, especially in the magnetosphere between the planetary and interplanetary magnetic fields and also in the solar atmosphere. Little is known about the details of separator reconnection and so the aim of this paper, which is the first of two, is to study the properties of magnetic reconnection at a single separator. Three-dimensional, resistive magnetohydrodynamic numerical experiments are run to study separator reconnection starting from a magnetohydrostatic equilibrium which contains a twisted current layer along a single separator linking a pair of opposite-polarity null points. The resulting reconnection occurs in two phases. The first is short involving rapid reconnection in which the current at the separator is reduced by a factor of around 2.3. Most 75% of the magnetic energy is converted during this phase, via Ohmic dissipation, directly into internal energy, with just 0.1% going into kinetic energy. During this phase the reconnection occurs along most of the separator away from its ends (the nulls), but in an asymmetric manner which changes both spatially and temporally over time. The second phase is much longer and involves slow impulsive-bursty reconnection. Again Ohmic heating dominates over viscous damping. Here the reconnection occurs in small localized bursts at random anywhere along the separator.
2015-12-01T00:00:00ZE. H. Stevenson, JulieE. Parnell, ClareMagnetic separators, which lie on the boundary between four topologically-distinct flux domains, are prime locations in three-dimensional magnetic fields for reconnection, especially in the magnetosphere between the planetary and interplanetary magnetic fields and also in the solar atmosphere. Little is known about the details of separator reconnection and so the aim of this paper, which is the first of two, is to study the properties of magnetic reconnection at a single separator. Three-dimensional, resistive magnetohydrodynamic numerical experiments are run to study separator reconnection starting from a magnetohydrostatic equilibrium which contains a twisted current layer along a single separator linking a pair of opposite-polarity null points. The resulting reconnection occurs in two phases. The first is short involving rapid reconnection in which the current at the separator is reduced by a factor of around 2.3. Most 75% of the magnetic energy is converted during this phase, via Ohmic dissipation, directly into internal energy, with just 0.1% going into kinetic energy. During this phase the reconnection occurs along most of the separator away from its ends (the nulls), but in an asymmetric manner which changes both spatially and temporally over time. The second phase is much longer and involves slow impulsive-bursty reconnection. Again Ohmic heating dominates over viscous damping. Here the reconnection occurs in small localized bursts at random anywhere along the separator.Effects of thermal conduction and compressive viscosity on the period ratio of the slow modeMacnamara, Cicely KrystynaRoberts, Bernardhttp://hdl.handle.net/10023/84232021-03-21T05:31:20Z2010-06-01T00:00:00ZAims: Increasing observational evidence of wave modes brings us to a closer understanding of the solar corona. Coronal seismology allows us to combine wave observations and theory to determine otherwise unknown parameters. The period ratio, P1/2P2, between the period P1 of the fundamental mode and the period P2 of its first overtone, is one such tool of coronal seismology and its departure from unity provides information about the structure of the corona. Methods: We consider analytically the effects of thermal conduction and compressive viscosity on the period ratio for a longitudinally propagating sound wave. Results: For coronal values of thermal conduction the effect on the period ratio is negligible. For compressive viscosity the effect on the period ratio may become important for some short hot loops. Conclusions: Damping typically has a small effect on the period ratio, suggesting that longitudinal structuring remains the most significant effect.
C.K.M. acknowledges financial support from the CarnegieTrust. Discussions with Dr. I. De Moortel and Prof. A. W. Hood are gratefully acknowledged
2010-06-01T00:00:00ZMacnamara, Cicely KrystynaRoberts, BernardAims: Increasing observational evidence of wave modes brings us to a closer understanding of the solar corona. Coronal seismology allows us to combine wave observations and theory to determine otherwise unknown parameters. The period ratio, P1/2P2, between the period P1 of the fundamental mode and the period P2 of its first overtone, is one such tool of coronal seismology and its departure from unity provides information about the structure of the corona. Methods: We consider analytically the effects of thermal conduction and compressive viscosity on the period ratio for a longitudinally propagating sound wave. Results: For coronal values of thermal conduction the effect on the period ratio is negligible. For compressive viscosity the effect on the period ratio may become important for some short hot loops. Conclusions: Damping typically has a small effect on the period ratio, suggesting that longitudinal structuring remains the most significant effect.Stochastic modelling of chromosomal segregation : errors can introduce correctionMatzavinos, AnastasiosRoitershtein, AlexanderShtylla, BlertaVoller, ZacharyLiu, SijiaChaplain, Mark A.J.http://hdl.handle.net/10023/77112021-01-12T11:02:31Z2014-07-01T00:00:00ZCell division is a complex process requiring the cell to have many internal checks so that division may proceed and be completed correctly. Failure to divide correctly can have serious consequences, including progression to cancer. During mitosis, chromosomal segregation is one such process that is crucial for successful progression. Accurate segregation of chromosomes during mitosis requires regulation of the interactions between chromosomes and spindle microtubules. If left uncorrected, chromosome attachment errors can cause chromosome segregation defects which have serious effects on cell fates. In early prometaphase, where kinetochores are exposed to multiple microtubules originating from the two poles, there are frequent errors in kinetochore-microtubule attachment. Erroneous attachments are classified into two categories, syntelic and merotelic. In this paper, we consider a stochastic model for a possible function of syntelic and merotelic kinetochores, and we provide theoretical evidence that merotely can contribute to lessening the stochastic noise in the time for completion of the mitotic process in eukaryotic cells.
2014-07-01T00:00:00ZMatzavinos, AnastasiosRoitershtein, AlexanderShtylla, BlertaVoller, ZacharyLiu, SijiaChaplain, Mark A.J.Cell division is a complex process requiring the cell to have many internal checks so that division may proceed and be completed correctly. Failure to divide correctly can have serious consequences, including progression to cancer. During mitosis, chromosomal segregation is one such process that is crucial for successful progression. Accurate segregation of chromosomes during mitosis requires regulation of the interactions between chromosomes and spindle microtubules. If left uncorrected, chromosome attachment errors can cause chromosome segregation defects which have serious effects on cell fates. In early prometaphase, where kinetochores are exposed to multiple microtubules originating from the two poles, there are frequent errors in kinetochore-microtubule attachment. Erroneous attachments are classified into two categories, syntelic and merotelic. In this paper, we consider a stochastic model for a possible function of syntelic and merotelic kinetochores, and we provide theoretical evidence that merotely can contribute to lessening the stochastic noise in the time for completion of the mitotic process in eukaryotic cells.Vortical control of forced two-dimensional turbulenceFontane, Jerome Jacob LouisDritschel, David GerardScott, Richard Kirknesshttp://hdl.handle.net/10023/52362021-01-12T10:45:13Z2013-01-14T00:00:00ZA new numerical technique for the simulation of forced two-dimensional turbulence[D. Dritschel and J. Fontane, “The combined Lagrangian advection method,” J. Comput. Phys.229, 5408–5417 (Year: 2010)10.1016/j.jcp.2010.03.048] is used to examine the validity of Kraichnan-Batchelor scaling laws at higher Reynolds number than previously accessible with classical pseudo-spectral methods, making use of large simulation ensembles to allow a detailed consideration of the inverse cascade in a quasi-steady state. Our results support the recent finding of Scott [R. Scott, “Nonrobustness of the two-dimensional turbulent inverse cascade,” Phys. Rev. E75, 046301 (Year: 2007)10.1103/PhysRevE.75.046301], namely that when a direct enstrophy cascading range is well-represented numerically, a steeper energy spectrum proportional to k−2 is obtained in place of the classical k −5/3 prediction. It is further shown that this steep spectrum is associated with a faster growth of energy at large scales, scaling like t −1 rather than Kraichnan's prediction of t −3/2. The deviation from Kraichnan's theory is related to the emergence of a population of vortices that dominate the distribution of energy across scales, and whose number density and vorticity distribution with respect to vortex area are related to the shape of the enstrophy spectrum. An analytical model is proposed which closely matches the numerical spectra between the large scales and the forcing scale.
Jérôme Fontane is supported by the European Community in the framework of the CONVECT project under Grant No. PIEF-GA-2008-221003.
2013-01-14T00:00:00ZFontane, Jerome Jacob LouisDritschel, David GerardScott, Richard KirknessA new numerical technique for the simulation of forced two-dimensional turbulence[D. Dritschel and J. Fontane, “The combined Lagrangian advection method,” J. Comput. Phys.229, 5408–5417 (Year: 2010)10.1016/j.jcp.2010.03.048] is used to examine the validity of Kraichnan-Batchelor scaling laws at higher Reynolds number than previously accessible with classical pseudo-spectral methods, making use of large simulation ensembles to allow a detailed consideration of the inverse cascade in a quasi-steady state. Our results support the recent finding of Scott [R. Scott, “Nonrobustness of the two-dimensional turbulent inverse cascade,” Phys. Rev. E75, 046301 (Year: 2007)10.1103/PhysRevE.75.046301], namely that when a direct enstrophy cascading range is well-represented numerically, a steeper energy spectrum proportional to k−2 is obtained in place of the classical k −5/3 prediction. It is further shown that this steep spectrum is associated with a faster growth of energy at large scales, scaling like t −1 rather than Kraichnan's prediction of t −3/2. The deviation from Kraichnan's theory is related to the emergence of a population of vortices that dominate the distribution of energy across scales, and whose number density and vorticity distribution with respect to vortex area are related to the shape of the enstrophy spectrum. An analytical model is proposed which closely matches the numerical spectra between the large scales and the forcing scale.Resistive magnetohydrodynamic reconnection : resolving long-term, chaotic dynamicsKeppens, R.Porth, O.Galsgaard, K.Frederiksen, J.T.Restante, A.L.Lapenta, G.Parnell, C.http://hdl.handle.net/10023/52332021-01-12T10:49:57Z2013-09-13T00:00:00ZIn this paper, we address the long-term evolution of an idealised double current system entering reconnection regimes where chaotic behavior plays a prominent role. Our aim is to quantify the energetics in high magnetic Reynolds number evolutions, enriched by secondary tearing events, multiple magnetic island coalescence, and compressive versus resistive heating scenarios. Our study will pay particular attention to the required numerical resolutions achievable by modern (grid-adaptive) computations, and comment on the challenge associated with resolving chaotic island formation and interaction. We will use shock-capturing, conservative, grid-adaptive simulations for investigating trends dominated by both physical (resistivity) and numerical (resolution) parameters, and confront them with (visco-)resistive magnetohydrodynamic simulations performed with very different, but equally widely used discretization schemes. This will allow us to comment on the obtained evolutions in a manner irrespective of the adopted discretization strategy. Our findings demonstrate that all schemes used (finite volume based shock-capturing, high order finite differences, and particle in cell-like methods) qualitatively agree on the various evolutionary stages, and that resistivity values of order 0.001 already can lead to chaotic island appearance. However, none of the methods exploited demonstrates convergence in the strong sense in these chaotic regimes. At the same time, nonperturbed tests for showing convergence over long time scales in ideal to resistive regimes are provided as well, where all methods are shown to agree. Both the advantages and disadvantages of specific discretizations as applied to this challenging problem are discussed.
We acknowledge financial support from the EC FP7/2007-2013 Grant Agreement SWIFF (No. 263340) and from project GOA/2009/009 (KU Leuven). This research has been funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (IAP P7/08 CHARM). Part of the simulations used the infrastructure of the VSC-Flemish Supercomputer Center, funded by the Hercules Foundation and the Flemish Government-Department EWI. Another part of the simulations was done at the former Danish Center for Scientific Computing at Copenhagen University which is now part of DeIC Danish e-Infrastructure Cooperation.
2013-09-13T00:00:00ZKeppens, R.Porth, O.Galsgaard, K.Frederiksen, J.T.Restante, A.L.Lapenta, G.Parnell, C.In this paper, we address the long-term evolution of an idealised double current system entering reconnection regimes where chaotic behavior plays a prominent role. Our aim is to quantify the energetics in high magnetic Reynolds number evolutions, enriched by secondary tearing events, multiple magnetic island coalescence, and compressive versus resistive heating scenarios. Our study will pay particular attention to the required numerical resolutions achievable by modern (grid-adaptive) computations, and comment on the challenge associated with resolving chaotic island formation and interaction. We will use shock-capturing, conservative, grid-adaptive simulations for investigating trends dominated by both physical (resistivity) and numerical (resolution) parameters, and confront them with (visco-)resistive magnetohydrodynamic simulations performed with very different, but equally widely used discretization schemes. This will allow us to comment on the obtained evolutions in a manner irrespective of the adopted discretization strategy. Our findings demonstrate that all schemes used (finite volume based shock-capturing, high order finite differences, and particle in cell-like methods) qualitatively agree on the various evolutionary stages, and that resistivity values of order 0.001 already can lead to chaotic island appearance. However, none of the methods exploited demonstrates convergence in the strong sense in these chaotic regimes. At the same time, nonperturbed tests for showing convergence over long time scales in ideal to resistive regimes are provided as well, where all methods are shown to agree. Both the advantages and disadvantages of specific discretizations as applied to this challenging problem are discussed.The effect of slip length on vortex rebound from a rigid boundarySutherland, D.Macaskill, C.Dritschel, D.G.http://hdl.handle.net/10023/52322021-01-12T10:49:57Z2013-09-23T00:00:00ZThe problem of a dipole incident normally on a rigid boundary, for moderate to large Reynolds numbers, has recently been treated numerically using a volume penalisation method by Nguyen van yen, Farge, and Schneider [Phys. Rev. Lett.106, 184502 (2011)]. Their results indicate that energy dissipating structures persist in the inviscid limit. They found that the use of penalisation methods intrinsically introduces some slip at the boundary wall, where the slip approaches zero as the Reynolds number goes to infinity, so reducing to the no-slip case in this limit. We study the same problem, for both no-slip and partial slip cases, using compact differences on a Chebyshev grid in the direction normal to the wall and Fourier methods in the direction along the wall. We find that for the no-slip case there is no indication of the persistence of energy dissipating structures in the limit as viscosity approaches zero and that this also holds for any fixed slip length. However, when the slip length is taken to vary inversely with Reynolds number then the results of Nguyen van yen et al. are regained. It therefore appears that the prediction that energy dissipating structures persist in the inviscid limit follows from the two limits of wall slip length going to zero, and viscosity going to zero, not being treated independently in their use of the volume penalisation method.
2013-09-23T00:00:00ZSutherland, D.Macaskill, C.Dritschel, D.G.The problem of a dipole incident normally on a rigid boundary, for moderate to large Reynolds numbers, has recently been treated numerically using a volume penalisation method by Nguyen van yen, Farge, and Schneider [Phys. Rev. Lett.106, 184502 (2011)]. Their results indicate that energy dissipating structures persist in the inviscid limit. They found that the use of penalisation methods intrinsically introduces some slip at the boundary wall, where the slip approaches zero as the Reynolds number goes to infinity, so reducing to the no-slip case in this limit. We study the same problem, for both no-slip and partial slip cases, using compact differences on a Chebyshev grid in the direction normal to the wall and Fourier methods in the direction along the wall. We find that for the no-slip case there is no indication of the persistence of energy dissipating structures in the limit as viscosity approaches zero and that this also holds for any fixed slip length. However, when the slip length is taken to vary inversely with Reynolds number then the results of Nguyen van yen et al. are regained. It therefore appears that the prediction that energy dissipating structures persist in the inviscid limit follows from the two limits of wall slip length going to zero, and viscosity going to zero, not being treated independently in their use of the volume penalisation method.Shallow-water vortex equilibria and their stabilityPłotka, H.Dritschel, D.G.http://hdl.handle.net/10023/47622021-01-12T10:47:30Z2011-01-01T00:00:00ZWe first describe the equilibrium form and stability of steadily-rotating simply-connected vortex patches in the single-layer quasi-geostrophic model of geophysical fluid dynamics. This model, valid for rotating shallow-water flow in the limit of small Rossby and Froude numbers, has an intrinsic length scale L called the "Rossby deformation length" relating the strength of stratification to that of the background rotation rate. Specifically, L = c/f where c = √gH is a characteristic gravity-wave speed, g is gravity (or "reduced" gravity in a two-layer context where one layer is infinitely deep), H is the mean active layer depth, and f is the Coriolis frequency (here constant). We next introduce ageostrophic effects by using the full shallow-water model to generate what we call "quasi-equilibria". These equilibria are not strictly steady, but radiate such weak gravity waves that they are steady for all practical purposes. Through an artificial ramping procedure, we ramp up the potential vorticity anomaly of the fluid particles in our quasi-geostrophic equilibria to obtain shallow-water quasi-equilibria at finite Rossby number. We show a few examples of these states in this paper.
2011-01-01T00:00:00ZPłotka, H.Dritschel, D.G.We first describe the equilibrium form and stability of steadily-rotating simply-connected vortex patches in the single-layer quasi-geostrophic model of geophysical fluid dynamics. This model, valid for rotating shallow-water flow in the limit of small Rossby and Froude numbers, has an intrinsic length scale L called the "Rossby deformation length" relating the strength of stratification to that of the background rotation rate. Specifically, L = c/f where c = √gH is a characteristic gravity-wave speed, g is gravity (or "reduced" gravity in a two-layer context where one layer is infinitely deep), H is the mean active layer depth, and f is the Coriolis frequency (here constant). We next introduce ageostrophic effects by using the full shallow-water model to generate what we call "quasi-equilibria". These equilibria are not strictly steady, but radiate such weak gravity waves that they are steady for all practical purposes. Through an artificial ramping procedure, we ramp up the potential vorticity anomaly of the fluid particles in our quasi-geostrophic equilibria to obtain shallow-water quasi-equilibria at finite Rossby number. We show a few examples of these states in this paper.Boundary layer flow beneath an internal solitary wave of elevationCarr, MagdaDavies, P Ahttp://hdl.handle.net/10023/43312021-04-25T04:30:09Z2010-02-01T00:00:00ZThe wave-induced flow over a fixed bottom boundary beneath an internal solitary wave of elevation propagating in an unsheared, two-layer, stably stratified fluid is investigated experimentally. Measurements of the velocity field close to the bottom boundary are presented to illustrate that in the lower layer the fluid velocity near the bottom reverses direction as the wave decelerates while higher in the water column the fluid velocity is in the same direction as the wave propagation. The observation is similar in nature to that for wave-induced flow beneath a surface solitary wave. Contrary to theoretical predictions for internal solitary waves, no evidence for either boundary layer separation or vortex formation is found beneath the front half of the wave in the adverse pressure gradient region of the flow.
2010-02-01T00:00:00ZCarr, MagdaDavies, P AThe wave-induced flow over a fixed bottom boundary beneath an internal solitary wave of elevation propagating in an unsheared, two-layer, stably stratified fluid is investigated experimentally. Measurements of the velocity field close to the bottom boundary are presented to illustrate that in the lower layer the fluid velocity near the bottom reverses direction as the wave decelerates while higher in the water column the fluid velocity is in the same direction as the wave propagation. The observation is similar in nature to that for wave-induced flow beneath a surface solitary wave. Contrary to theoretical predictions for internal solitary waves, no evidence for either boundary layer separation or vortex formation is found beneath the front half of the wave in the adverse pressure gradient region of the flow.Magnetohydrodynamics dynamical relaxation of coronal magnetic fields : IV. 3D tilted nullsFuentes-Fernandez, JorgeParnell, Clare E.http://hdl.handle.net/10023/40842021-01-12T10:43:15Z2013-09-12T00:00:00ZIn this paper we study current accumulations in 3D "tilted" nulls formed by a folding of the spine and fan. A non-zero component of current parallel to the fan is required such that the null's fan plane and spine are not perpendicular. Our aims are to provide valid magnetohydrostatic equilibria and to describe the current accumulations in various cases involving finite plasma pressure.To create our equilibrium current structures we use a full, non-resistive, magnetohydrodynamic (MHD) code so that no reconnection is allowed. A series of experiments are performed in which a perturbed 3D tilted null relaxes towards an equilibrium via real, viscous damping forces. Changes to the initial plasma pressure and to magnetic parameters are investigated systematically.An initially tilted fan is associated with a non-zero Lorentz force that drives the fan and spine to collapse towards each other, in a similar manner to the collapse of a 2D X-point. In the final equilibrium state for an initially radial null with only the current perpendicular to the spine, the current concentrates along the tilt axis of the fan and in a layer about the null point with a sharp peak at the null itself. The continued growth of this peak indicates that the system is in an asymptotic regime involving an infinite time singularity at the null. When the initial tilt disturbance (current perpendicular to the spine) is combined with a spiral-type disturbance (current parallel to the spine), the final current density concentrates in three regions: one on the fan along its tilt axis and two around the spine, above and below the fan. The increased area of current accumulation leads to a weakening of the singularity formed at the null. The 3D spine-fan collapse with generic current studied here provides the ideal setup for non-steady reconnection studies.
2013-09-12T00:00:00ZFuentes-Fernandez, JorgeParnell, Clare E.In this paper we study current accumulations in 3D "tilted" nulls formed by a folding of the spine and fan. A non-zero component of current parallel to the fan is required such that the null's fan plane and spine are not perpendicular. Our aims are to provide valid magnetohydrostatic equilibria and to describe the current accumulations in various cases involving finite plasma pressure.To create our equilibrium current structures we use a full, non-resistive, magnetohydrodynamic (MHD) code so that no reconnection is allowed. A series of experiments are performed in which a perturbed 3D tilted null relaxes towards an equilibrium via real, viscous damping forces. Changes to the initial plasma pressure and to magnetic parameters are investigated systematically.An initially tilted fan is associated with a non-zero Lorentz force that drives the fan and spine to collapse towards each other, in a similar manner to the collapse of a 2D X-point. In the final equilibrium state for an initially radial null with only the current perpendicular to the spine, the current concentrates along the tilt axis of the fan and in a layer about the null point with a sharp peak at the null itself. The continued growth of this peak indicates that the system is in an asymptotic regime involving an infinite time singularity at the null. When the initial tilt disturbance (current perpendicular to the spine) is combined with a spiral-type disturbance (current parallel to the spine), the final current density concentrates in three regions: one on the fan along its tilt axis and two around the spine, above and below the fan. The increased area of current accumulation leads to a weakening of the singularity formed at the null. The 3D spine-fan collapse with generic current studied here provides the ideal setup for non-steady reconnection studies.Magnetohydrodynamics dynamical relaxation of coronal magnetic fields : III. 3D spiral nullsFuentes-Fernandez, JorgeParnell, Clare E.http://hdl.handle.net/10023/39782021-01-12T10:42:51Z2012-06-01T00:00:00ZContext: The majority of studies on stressed 3D magnetic null points consider magnetic reconnection driven by an external perturbation, but the formation of a genuine current sheet equilibrium remains poorly understood. This problem has been considered more extensively in two-dimensions, but lacks a generalization into 3D fields. Aims: 3D magnetic nulls are more complex than 2D nulls and the field can take a greater range of magnetic geometries local to the null. Here, we focus on one type and consider the dynamical non-resistive relaxation of 3D spiral nulls with initial spine-aligned current. We aim to provide a valid magnetohydrostatic equilibrium, and describe the electric current accumulations in various cases, involving a finite plasma pressure. Methods: A full MHD code is used, with the resistivity set to zero so that reconnection is not allowed, to run a series of experiments in which a perturbed spiral 3D null point is allowed to relax towards an equilibrium, via real, viscous damping forces. Changes to the initial plasma pressure and other magnetic parameters are investigated systematically. Results: For the axi-symmetric case, the evolution of the field and the plasma is such that it concentrates the current density in two cone-shaped regions along the spine, thus concentrating the twist of the magnetic field around the spine, leaving a radial configuration in the fan plane. The plasma pressure redistributes in order to maintain the current density accumulations. However, it is found that changes in the initial plasma pressure do not modify the final state significantly. In the cases where the initial magnetic field is not axi-symmetric, a infinite-time singularity of current perpendicular to the fan is found at the location of the null.
2012-06-01T00:00:00ZFuentes-Fernandez, JorgeParnell, Clare E.Context: The majority of studies on stressed 3D magnetic null points consider magnetic reconnection driven by an external perturbation, but the formation of a genuine current sheet equilibrium remains poorly understood. This problem has been considered more extensively in two-dimensions, but lacks a generalization into 3D fields. Aims: 3D magnetic nulls are more complex than 2D nulls and the field can take a greater range of magnetic geometries local to the null. Here, we focus on one type and consider the dynamical non-resistive relaxation of 3D spiral nulls with initial spine-aligned current. We aim to provide a valid magnetohydrostatic equilibrium, and describe the electric current accumulations in various cases, involving a finite plasma pressure. Methods: A full MHD code is used, with the resistivity set to zero so that reconnection is not allowed, to run a series of experiments in which a perturbed spiral 3D null point is allowed to relax towards an equilibrium, via real, viscous damping forces. Changes to the initial plasma pressure and other magnetic parameters are investigated systematically. Results: For the axi-symmetric case, the evolution of the field and the plasma is such that it concentrates the current density in two cone-shaped regions along the spine, thus concentrating the twist of the magnetic field around the spine, leaving a radial configuration in the fan plane. The plasma pressure redistributes in order to maintain the current density accumulations. However, it is found that changes in the initial plasma pressure do not modify the final state significantly. In the cases where the initial magnetic field is not axi-symmetric, a infinite-time singularity of current perpendicular to the fan is found at the location of the null.Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limitDritschel, David GerardTobias, Stevehttp://hdl.handle.net/10023/36982021-08-15T04:31:20Z2012-07-01T00:00:00ZIn this paper we introduce a new method for computations of two-dimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number $\Pra=\nu/\eta$. When $\Pra \ll 1$, the magnetic field dissipates at a scale much larger than the velocity field. The method we utilise is a novel hybrid contour--spectral method, the ``Combined Lagrangian Advection Method'', formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudo-spectral method for decreasing $\Pra$ for the problem of decaying two-dimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given.
2012-07-01T00:00:00ZDritschel, David GerardTobias, SteveIn this paper we introduce a new method for computations of two-dimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number $\Pra=\nu/\eta$. When $\Pra \ll 1$, the magnetic field dissipates at a scale much larger than the velocity field. The method we utilise is a novel hybrid contour--spectral method, the ``Combined Lagrangian Advection Method'', formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudo-spectral method for decreasing $\Pra$ for the problem of decaying two-dimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given.Instability in internal solitary waves with trapped coresCarr, MagdaKing, Stuart EdwardDritschel, David Gerardhttp://hdl.handle.net/10023/33972021-01-12T10:37:29Z2012-01-01T00:00:00ZA numerical method that employs a combination of contour advection and pseudo-spectral techniques is used to investigate instability in internal solitary waves with trapped cores. A three-layer configuration for the background stratification in which the top two layers are linearly stratified and the lower layer is homogeneous is considered throughout. The strength of the stratification in the very top layer is chosen to be sufficient so that waves of depression with trapped cores can be generated. The flow is assumed to satisfy the Dubriel-Jacotin-Long equation both inside and outside of the core region. The Brunt-Vaisala frequency is modelled such that it varies from a constant value outside of the core to zero inside the core over a sharp but continuous transition length. This results in a stagnant core in which the vorticity is zero and the density is homogeneous and approximately equal to that at the core boundary. The time dependent simulations show that instability occurs on the boundary of the core. The instability takes the form of Kelvin-Helmholtz billows. If the instability in the vorticity field is energetic enough, disturbance in the buoyancy field is also seen and fluid exchange takes place across the core boundary. Occurrence of the Kelvin-Helmholtz billows is attributed to the sharp change in the vorticity field at the boundary between the core and the pycnocline. The numerical scheme is not limited by small Richardson number unlike the other alternatives currently available in the literature which appear to be.
2012-01-01T00:00:00ZCarr, MagdaKing, Stuart EdwardDritschel, David GerardA numerical method that employs a combination of contour advection and pseudo-spectral techniques is used to investigate instability in internal solitary waves with trapped cores. A three-layer configuration for the background stratification in which the top two layers are linearly stratified and the lower layer is homogeneous is considered throughout. The strength of the stratification in the very top layer is chosen to be sufficient so that waves of depression with trapped cores can be generated. The flow is assumed to satisfy the Dubriel-Jacotin-Long equation both inside and outside of the core region. The Brunt-Vaisala frequency is modelled such that it varies from a constant value outside of the core to zero inside the core over a sharp but continuous transition length. This results in a stagnant core in which the vorticity is zero and the density is homogeneous and approximately equal to that at the core boundary. The time dependent simulations show that instability occurs on the boundary of the core. The instability takes the form of Kelvin-Helmholtz billows. If the instability in the vorticity field is energetic enough, disturbance in the buoyancy field is also seen and fluid exchange takes place across the core boundary. Occurrence of the Kelvin-Helmholtz billows is attributed to the sharp change in the vorticity field at the boundary between the core and the pycnocline. The numerical scheme is not limited by small Richardson number unlike the other alternatives currently available in the literature which appear to be.