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The dynamics of modulated point vortex pairs is investigated on a rotating sphere, where modulation is chosen to reflect the conservation of angular momentum (potential vorticity). In this setting the authors point out a qualitati...
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The dynamics of modulated point vortex pairs is investigated on a rotating sphere, where modulation is chosen to reflect the conservation of angular momentum (potential vorticity). In this setting the authors point out a qualitative difference between the full spherical dynamics and the one obtained in a -plane approximation. In particular, dipole trajectories starting at the same location evolve to completely different directions under these two treatments, despite the fact that the deviations from the initial latitude remain small. This is a strong indication for the mathematical inconsistency of the traditional -plane approximation. At the same time, a consistently linearized set of equations of motion leads to trajectories agreeing with those obtained under the full spherical treatment. The -plane advection patterns due to chaotic advection in the velocity field of finite-sized vortex pairs are also found to considerably deviate from those of the full spherical treatment, and quantities characterizing transport properties (e.g., the escape rate from a given region) strongly differ.
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In Grohs and Obermeier (2015), the authors discussed the existence of numerically feasible solvers for advection equations that run in optimal computational complexity. In this paper, we complete the last remaining requirement to ...
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In Grohs and Obermeier (2015), the authors discussed the existence of numerically feasible solvers for advection equations that run in optimal computational complexity. In this paper, we complete the last remaining requirement to achieve this goal - by showing that ridgelets, on which the solver is based, approximate functions with line singularities (which may appear as solutions to the advection equation) with the best possible approximation rate (up to arbitrarily small 1 > 0) if the functions decay sufficiently fast, and with a slight penalty if not.
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A new mass conservative flux method is presented for the numerical solution of the stochastic collection equation. The method consists of a two-step procedure. In the first step the mass distribution of drops with mass x' that hav...
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A new mass conservative flux method is presented for the numerical solution of the stochastic collection equation. The method consists of a two-step procedure. In the first step the mass distribution of drops with mass x' that have been newly formed in a collision process is entirely added to grid box k of the numerical grid mesh with x(k) less than or equal to x' less than or equal to x(k+1). In the second step a certain fraction of the water mass in grid box k is transported to k + 1. This transport is done by means of an advection procedure. Different numerical test runs are presented in which the proposed method is compared with the Berry-Reinhardt scheme. These tests show a very good agreement between the two approaches. In various sensitivity studies it is demonstrated that the flux method remains numerically stable for different choices of the grid mesh and the integration time step. Since a time step of 10 s may be used without significant loss of accuracy, the flux method is numerically very efficient in comparison to the Berry-Reinhardt scheme. [References: 16]
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Hyperbolic points and their unsteady generalization - hyperbolic trajectories - drive the exponential stretching that is the hallmark of nonlinear and chaotic flow. In infinite-time steady or periodic flows, the stable and unstabl...
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Hyperbolic points and their unsteady generalization - hyperbolic trajectories - drive the exponential stretching that is the hallmark of nonlinear and chaotic flow. In infinite-time steady or periodic flows, the stable and unstable manifolds attached to each hyperbolic trajectory mark fluid elements that asymptote either towards or away from the hyperbolic trajectory, and which will therefore eventually experience exponential stretching. But typical experimental and observational velocity data are unsteady and available only over a finite time interval, and in such situations hyperbolic trajectories will move around in the flow, and may lose their hyperbolicity at times. Here we introduce a way to determine their region of influence, which we term a hyperbolic neighbourhood, that marks the portion of the domain that is instantaneously dominated by the hyperbolic trajectory. We establish, using both theoretical arguments and empirical verification from model and experimental data, that the hyperbolic neighbourhoods profoundly impact the Lagrangian stretching experienced by fluid elements. In particular, we show that fluid elements traversing a flow experience exponential boosts in stretching while within these time-varying regions, that greater residence time within hyperbolic neighbourhoods is directly correlated to larger finite-time Lyapunov exponent (FTLE) values, and that FTLE diagnostics are reliable only when the hyperbolic neighbourhoods have a geometrical structure that is 'regular' in a specific sense.
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In this note, we investigate the regularity of the extremal solution u* for the semilinear elliptic equation -Δu+c(x)·u=Λf(u) on a bounded smooth domain of R{double-struck}~n with Dirichlet boundary condition. Here f is a posit...
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In this note, we investigate the regularity of the extremal solution u* for the semilinear elliptic equation -Δu+c(x)·u=Λf(u) on a bounded smooth domain of R{double-struck}~n with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a?(0,~∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.
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Although the moisture feedback has been well known to be essential in the Madden-Julian Oscillation (MJO) dynamics, whether its pre-moistening effect plays a key role in exciting the onset of primary MJO events, as has been confir...
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Although the moisture feedback has been well known to be essential in the Madden-Julian Oscillation (MJO) dynamics, whether its pre-moistening effect plays a key role in exciting the onset of primary MJO events, as has been confirmed in the successive initiation, remains elusive. In this study, using a hybrid coupled climate model that has a good fidelity in simulating the intraseasonal variability, we develop a new framework of methodology to investigate the nonlinear excitation of primary MJO event, of which the key achievement is the successful implementation of the conditional nonlinear optimal perturbation (CNOP). In an application of this new framework, the CNOP-type moisture perturbations are calculated for the pre-chosen non-MJO reference states and generally favor a moistening in the equatorial region while drying in the poleward. Comparisons of the model simulation with observation give credibility to the existence of moisture signals several weeks before some primary MJO events. A suite of numerical experiments confirms that the CNOPs of moisture can contribute to the excitation and propagation of strong primary MJO events while random perturbations cannot. The moisture budget analysis further reveals the central importance of the horizontal moisture advection, especially the nonlinearly upscaled moisture transports associated with the high-frequency disturbances on the quasi-3-4-day and 6-8-day synoptic time scales, in supporting the nonlinear excitation of the primary MJO events. The subgrid-scale processes of evaporation, condensation and eddy transport of moisture are found to be critical for the pre-moistening effect in the boundary layer as well. This study directly supports the vital importance of the moisture perturbations, which are characterized by a particular pattern concentrated at low levels, to the nonlinear growth and propagation of the primary MJO events.
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The intertropical convergence zone (ITCZ) is one of the most striking features of Earth's climate system, often forming a narrow band of convection over many oceanic regions, especially in eastern ocean basins. It is not well unde...
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The intertropical convergence zone (ITCZ) is one of the most striking features of Earth's climate system, often forming a narrow band of convection over many oceanic regions, especially in eastern ocean basins. It is not well understood why the ITCZ is so thin; however, a recent study highlighted that classical Ekman balance is not obeyed near the equator and nonlinear horizontal wind advection can localize ITCZ boundary layer vertical motion so that it becomes very narrow and intense. In this study, we use a similar model but with more realistic forcings from the Year of Tropical Convection (YOTC) reanalysis, focusing on the eastern Pacific Ocean ITCZ. The model is a zonally symmetric, slab (subcloud) boundary layer numerical model on the sphere, which can be considered the simplest "dry" model of the ITCZ. Due to the slab model's simplicity, simulations are conducted at a range of resolutions, from 1 degrees to 1 km. The slab model dynamical fields are in general agreement with the YOTC dynamical fields and precipitation estimates from the Tropical Rainfall Measuring Mission for one summer and two spring ITCZ cases. We find that Ekman balance is indeed violated within 10 degrees-15 degrees of the equator and nonlinear horizontal wind advection is crucial to understanding the preferential location, width, and intensity of the eastern Pacific ITCZ. Additionally, it appears that these boundary layer processes involved in ITCZ intensification and narrowing are dependent on model resolution such that present-day general circulation models likely cannot sufficiently resolve them.
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This paper designs three quasi-geostrophic barotropic models with a radial/horizontal grid length being 2 km, one in the polar coordinates, one on a stationary typhoon circulation condition and another on a non-stationary typhoon ...
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This paper designs three quasi-geostrophic barotropic models with a radial/horizontal grid length being 2 km, one in the polar coordinates, one on a stationary typhoon circulation condition and another on a non-stationary typhoon circulation condition in the Cartesian coordinates, to investigate the effects of azimuthal and radial linear advections, and nonlinear advection on the inward propagation of mesoscale vorticity and the changes of typhoon intensity. Results show that the azimuthal linear advection may result in the formation of spiral vorticity bands; the radial linear advection in a certain parameter set is able to transfer vorticity inwards, leading to a slight enhancement of typhoon; the nonlinear advection of perturbation vorticity on a stationary typhoon circulation condition may transfer more vorticities inwards, thus resulting in a distinct enhancement of typhoon; and the nonlinear advection on a non-stationary typhoon circulation condition possesses duality, i. e. on the one hand, the advection increases the vorticity of inward propagation, thus favorable to the intensification of typhoon, and on the other hand, in the inward propagation process of vorticity the originally concentric and axisymmetric structure of typhoon basic flow is damaged, and a complex flow pattern forms, which in turn tends to weaken the circulation of typhoon. At last the paper discusses the possible applications of those results in typhoon intensity prediction.
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The nonlinear dynamics of one-dimensional systems of hydrodynamic type is examined based on the Lagrangian description. The formation of essentially nonlinear, time-dependent structures within a compressible advection problem is c...
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The nonlinear dynamics of one-dimensional systems of hydrodynamic type is examined based on the Lagrangian description. The formation of essentially nonlinear, time-dependent structures within a compressible advection problem is considered. The conditions leading to formation of new regular as well as singular collapse-type structures and wavelet structures in the density profile are derived.
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The split-operator approach proposes to divide the numerical solution of the governing flow and constituent transport equations into a series of successive steps, one of them being the numerically most challenging advection step. ...
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The split-operator approach proposes to divide the numerical solution of the governing flow and constituent transport equations into a series of successive steps, one of them being the numerically most challenging advection step. This paper presents tests of characteristics and finite difference methods used in the numerical solution of one-dimensional linear and nonlinear advection equations that describe the transport and flow advection steps, thus demonstrating their accuracy, oscillations and damping features. The studied methods were incorporated into the split-operator approach and evaluated as a part of the complete solution, which led to the development of numerical models implementing an algorithm that can endure larger time steps without losing much accuracy. This algorithm was tested on both one- and two-dimensional examples, for complex computational domains that predominate in natural watercourses.
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