A nonlocal logistic equation with nonlinear advection term

Multivalued dynamics of non-autonomous reaction–diffusion equation with nonlinear advection term

This study analyzes horizontal and vertical wind-driven circulation responses in small semienclosed bays, the associated offshore dynamic conditions, and the relative importance of each term in the momentum balance equations using a multiplatform observational system. The observational platform consists of three ADCPs and a land-based radar monitoring the velocity field within the bay and in the contiguous offshore area. The wind-driven patterns in the bay can switch from a barotropic cyclonic or anticyclonic circulation to a two-layer baroclinic mode response as a function of the wind regime (its direction and magnitude). For the baroclinic mode, the vertical location of the inflection point in the velocity profile can vary according to the proximity of the boundary current to the entrance of the bay. The influence of offshore combined meteorological and marine conditions on the inner-bay dynamics is evidenced under moderate to strong wind conditions and is almost nonexistent under negligible wind. The momentum balance analysis as well as the nondimensional numbers evidence the impact of wind stress, coastline shape, stratification, and the nonlinear advective terms. Advection can be at the same order of magnitude as pressure gradient, Coriolis, or wind stress terms and can be greater than the bottom stress terms. The nonlinear terms in the momentum equations are frequently neglected when analyzing wind-driven circulation by means of in situ data or analytical models.

Extratropical cyclones (ETCs) are three-dimensional synoptic systems in the middle and high latitudes. Previous studies on ETC propagation have typically focused on cyclones identified at a single level. However, more recent studies have found that ETCs have diverse vertical structures and cyclones with different vertical extents always exhibit distinct characteristics and surface impacts. In this work, we study the movement of wintertime (December–February) extratropical cyclones by classifying North Pacific ETCs into deep cyclones, shallow low-level cyclones, and shallow upper-level cyclones, based on reanalysis data from 1979 to 2019. Applying a Lagrangian perspective, we track the cyclones at different vertical levels to investigate the different characteristics and mechanisms for the propagation of deep and shallow ETCs. A potential vorticity (PV) tendency analysis of cyclone-tracking composites reveals that, for deep cyclones, the diabatic heating at 850 hPa and the horizontal advection by the stationary flow at 500 hPa are the main contributors to the poleward movement. For shallow cyclones, the nonlinear advection terms play a dominant role in their meridional motion, advecting shallow low-level cyclones poleward but shallow upper-level cyclones equatorward. A piecewise PV inversion analysis suggests that the nonlinear advection by winds induced from upper-level PV anomalies is responsible for the different performance of nonlinear advection terms for shallow low-level and upper-level cyclones. These findings further our understanding of the mechanisms and variations of cyclone propagation. Significance Statement Extratropical cyclones (ETCs) can be identified at different levels in the troposphere. These mobile low pressure cyclonic storms are the main sources of synoptic variability in the extratropics and often bring severe or even disastrous weather. Previous studies on ETC movement have typically been restricted to cyclones identified at a single level. Our study, by identifying ETCs at multiple levels, classifies cyclones into deep, shallow low-level, and shallow upper-level cyclones. For deep cyclones, their poleward movement is found to be a result of diabatic heating at lower levels and dominated by the stationary circulation at upper levels. For shallow cyclones, nonlinear advection determines whether they propagate poleward or equatorward. These findings further our understanding of the mechanisms of cyclone propagation and imply that the movement of deep and shallow cyclones may undergo different changes with different weather and climate impacts in the future, given the enhanced diabatic heating under global warming, which deserves further investigation.

In this study, we investigate the sensitivity of the equatorial circulation simulated with the high‐resolution World Ocean Circulation Experiment (WOCE) Community Modeling Effort (CME) model to the specification of eddy viscosity. In one of two simulations the coefficient of vertical viscosity is tripled. We focus on the dynamics associated with the anomalous eastward surface flow found in the simulation results. We find, in agreement with other studies, that close to the western boundary, the equatorial undercurrent is dominated by a large positive pressure gradient. In the top layer, for the low‐viscosity case, the balance in the western part of the basin is everywhere between the horizontal pressure gradient and vertical viscosity with the advective terms playing a secondary role. In the high‐viscosity zonal momentum budget analysis the advective terms are less significant in the surface layers in the west. Toward the east the advective terms become more important in the low‐viscosity experiment, whereas in the high‐viscosity case the balance is still predominantly between the horizontal pressure gradient and vertical diffusion terms. It is found that in the CME, horizontal viscosity plays a very minor role in the zonal momentum balance. Changes in the vertical viscosity lead to changes in the nonlinear advective terms. These, in turn, contribute to the unusual eastward surface flow.

Existence and regularity of nonlinear advection problems

Numerical model simulations of continental shelf flows off northern California

A Spectral Model for Two-Dimensional Incompressible Fluid Flow in a Circular Basin

The present work is concerned with a study of numerical schemes for solving two-dimensional time-dependent incompressible free-surface fluid flow problems. The primitive variable flow equations are discretized by the finite difference method. A projection method is employed to uncouple the velocity components and pressure, thus allowing the solution of each variable separately (a segregated approach). The diffusive terms are discretized by Implicit Backward and Crank- Nicolson schemes, and the non-linear advection terms are approximated by the high order upwind VONOS (Variable-Order Non-oscillatory Scheme) technique. In order to improved numerical stability of the schemes, the boundary conditions for the pressure field at the free surface are treated implicitly, and for the velocity field explicitly. The numerical schemes are then applied to the simulation of the Hagen-Poiseuille flow, and container filling problems. The results show that the semi-implicit techniques eliminate the stability restriction in the original explicit GENSMAC method.

Usually two kinds of transition to turbulent fluid flow are distinguished, the sudden onset of turbulence that occurs, for instance, in pipe or channel flows and the transition to chaotic fluid flow after several bifurcations for which Taylor vortex flow and thermal convection in boxes heated from below provide typical examples. Both kinds of transition have in common that the amplitude of the disturbance velocity by which the turbulent state differs from the basic or laminar state must be sufficiently strong such that the nonlinear advection terms cause the interaction of several modes of the system. The exact ways in which these modes are defined and in which their interaction takes place may differ in various systems and should not concern us here.

A depth‐integrated ocean‐current model is developed to simulate currents and sea surface elevations on the Australian continental shelf. the nonlinear advective terms are included and the equations of motion are solved using an efficient split‐explicit finite difference scheme, which yields accurate and stable results. the model predictions are compared with observations over a four‐day period of both surface elevations and currents in Bass Strait during the passage of a typical cold front in April 1984.

In this paper, a hybrid compact-CIP scheme is proposed to solve Korteweg-de Vries-Burgers equation.The nonlinear advective terms are computed based on the classical constrained interpolation profile (CIP) method, which is coupled with a high-order compact scheme for third-order derivatives in Korteweg-de Vries-Burgers equation.The strong stability preserving third-order Runge-Kutta time discretizations is adopted in this work.A test case is presented to demonstrate the high-resolution properties of the proposed compact-CIP scheme.

Long climate simulations with the Met Office Hadley Centre General Circulation Model show weak El Niño-Southern Oscillation (ENSO) amplitude asymmetry between El Niño and La Niña phases compared with observations. This lack of asymmetry is explored through the framework of a perturbed parameter experiment. Two key hypotheses for the lack of asymmetry are tested. First, the possibility that westerly wind burst activity is biased is explored. It is found that the observed difference in wind burst activity during El Niño and La Niña tends to be underestimated by the model. Secondly, the warming due to subsurface non-linear advection is examined. While the model exhibits non-linear dynamic warming during both La Niña and El Niño, and thus a contribution to ENSO asymmetry, it is shown to be consistently underestimated in comparison with ocean reanalyses. The non-linear zonal advection term contributes most to the deficiency and the simulation of the anomalous zonal currents may be playing a key role in its underestimation. Compared with the ocean reanalyses, the anomalous zonal currents associated with ENSO are too weak in the vicinity of the equatorial undercurrent and the surface wind driven zonal currents extend too deep.

Abstract. In this study, we construct a seven-dimensional Lorenz model (7DLM) to discuss the impact of an extended nonlinear feedback loop on solutions' stability and illustrate the hierarchical scale dependence of chaotic solutions. Compared to the 5DLM, the 7DLM includes two additional high wavenumber modes that are selected based on an analysis of the nonlinear temperature advection term, a Jacobian term (J(ψ, θ)), where, ψ and θ represent the streamfunction and temperature perturbations, respectively. Fourier modes that represent temperature in the 7DLM can be categorized into three major scales as the primary (the largest scale), secondary, and tertiary (the smallest scale) modes. Further extension of the nonlinear feedback loop within the 7DLM can provide negative nonlinear feedback to stabilize solutions, thus leading to a much larger critical value for the Rayleigh parameter (rc ∼ 116.9) for the onset of chaos, as compared to an rc of 42.9 for the 5DLM as well as an rc of 24.74 for the 3DLM. The rc is determined by an analysis of ensemble Lyapunov exponents (eLEs) with a Prandtl number (σ) of 10. To examine the dependence of rc on the value of the Prandtl number, a linear stability analysis is performed near the nontrivial critical point using a wide range of the Rayleigh parameter (40 ≤ r ≤ 195) and the Prandtl number (5 ≤ σ ≤ 25). Then an eLE analysis is conducted using selected values of the Prandtl number. The linear stability analysis is done by solving for the analytical solutions of the critical points, by linearizing the 7DLM with respect to the analytical solutions, and by calculating the eigenvalues of the linearized system. Within the range of (5 ≤ σ ≤ 25), the 7DLM requires a larger rc for the onset of chaos than the 5DLM. In addition to the negative nonlinear feedback illustrated and emulated by the quasi-equilibrium state solutions for high wavenumber modes, the 7DLM reveals the hierarchical scale dependence of chaotic solutions. For chaotic solutions with r = 120, the Pearson correlation coefficients (PCCs) between the primary and secondary modes (i.e., Z and Z1) and between the secondary and tertiary modes (i.e., Z1 and Z2) are 0.988 and 0.998, respectively. Here, Z, Z1, and Z2 represent the time-varying amplitudes of the primary, secondary, and tertiary modes, respectively. High PCCs indicate a strong linear relationship among the modes at various scales and a hierarchy of scale dependence. Future work will be undertaken to examine how higher-dimensional LMs may produce a larger critical value for the Rayleigh parameter for the onset of chaos and reveal stronger hierarchical scale dependence.

The Arctic Oscillation (AO) is the leading climate mode of sea level pressure (SLP) anomalies during cold season in the Northern Hemisphere. To a large extent, the atmospheric climate anomalies associated with positive and negative phases of the AO are opposite to each other, indicating linear impact. However, there is also significant nonlinear relationship between the AO and other winter climate variability. We investigate nonlinear impacts of the AO on surface air temperature (SAT) using reanalysis data and a multimillennial long climate simulation. It is found that SAT response to the AO, in terms of both spatial pattern and magnitude, is almost linear when the amplitude of the AO is moderate. However, the response becomes quite nonlinear as the amplitude of the AO becomes stronger. First, the pattern shift in SAT depends on AO phase and magnitude, and second, the SAT magnitude depends on AO phase. In particular, these nonlinearities are distinct over the North America and Eurasian Continent. Based on the analyses of model output, we suggest that the nonlinear zonal advection term is one of the critical components in generating nonlinear SAT response, particularly over the North America.

The propagation of tides in the Fraser River is discussed, with particular attention to the nonlinear interactions that characterize tides in such a system. The tidal regime is reviewed, and then the distortion of the tidal wave as it propagates upstream is discussed. The seasonal variation in tidal constituents can be related to the seasonally varying magnitude of the Fraser River discharge. The distortion in the tidal wave is related to the production of overtides through the nonlinear advective and friction terms. By considering the astronomical constituents of the tide, expressions are derived for the production of such overtides as the M4 and are quantitatively related to the relative magnitudes of tidal and mean velocity contributions.