Nonlinear Advection Terms Research Articles

Causal dependencies and Shannon entropy budget: Analysis of a reduced‐order atmospheric model

AbstractThe information entropy budget and the rate of information transfer between variables is studied in the context of a nonlinear reduced‐order atmospheric model. The key ingredients of the dynamics are present in this model; namely, the baroclinic instability, the instability related to the presence of an orography, the dissipation related to the surface friction, and the large‐scale meridional imbalance of energy. For the parameters chosen, the solutions of this system display a chaotic dynamics reminiscent of the large‐scale atmospheric dynamics in the extratropics. The detailed information entropy budget analysis of this system reveals that the linear rotation terms play a minor role in the generation of uncertainties compared with the orography and the surface friction. Additionally, the dominant contribution comes from the nonlinear advection terms, and their decomposition in synergetic (covariability) and single (impact of each single variable on the target one) components reveals that for some variables the covariability dominates the information transfer. The estimation of the rate of information transfer based on time series is also discussed, and an extension of the Liang's approach to nonlinear observables is proposed.

A blow-up result of a free boundary problem with nonlinear advection term

We study a free boundary problem for the reaction-diffusion equation with nonlinear advection term u t = u xx − u u x + u p ( p > 1 ) on [ 0 , h ( t ) ] . This model describes species that live in an uninhabitable area and tries to spread into a new area, the free boundary h ( t ) represents such a spreading front. By considering the initial data σϕ , we have two sharp results. There is some σ ∗ > 0 , blow up happens when σ > σ ∗ , vanishing happens when σ < σ ∗ and the transition case happens when σ = σ ∗ . Additionally, we also have another trichotomy result: the solution is either blow up or converging to a small stationary state or converging to a big stationary state.

A Scale‐Dependent Analysis of the Barotropic Vorticity Budget in a Global Ocean Simulation

AbstractThe climatological mean barotropic vorticity budget is analyzed to investigate the relative importance of surface wind stress, topography, planetary vorticity advection, and nonlinear advection in dynamical balances in a global ocean simulation. In addition to a pronounced regional variability in vorticity balances, the relative magnitudes of vorticity budget terms strongly depend on the length‐scale of interest. To carry out a length‐scale dependent vorticity analysis in different ocean basins, vorticity budget terms are spatially coarse‐grained. At length‐scales greater than 1,000 km, the dynamics closely follow the Topographic‐Sverdrup balance in which bottom pressure torque, surface wind stress curl and planetary vorticity advection terms are in balance. In contrast, when including all length‐scales resolved by the model, bottom pressure torque and nonlinear advection terms dominate the vorticity budget (Topographic‐Nonlinear balance), which suggests a prominent role of oceanic eddies, which are of km in size, and the associated bottom pressure anomalies in local vorticity balances at length‐scales smaller than 1,000 km. Overall, there is a transition from the Topographic‐Nonlinear regime at scales smaller than 1,000 km to the Topographic‐Sverdrup regime at length‐scales greater than 1,000 km. These dynamical balances hold across all ocean basins; however, interpretations of the dominant vorticity balances depend on the level of spatial filtering or the effective model resolution. On the other hand, the contribution of bottom and lateral friction terms in the barotropic vorticity budget remains small and is significant only near sea‐land boundaries, where bottom stress and horizontal viscous friction generally peak.

Inter-scale Causality Relations in Wall Turbulence

We investigate the inter-scale energy transfer in the buffer layer between different streamwise and spanwise lengthscales. In a periodic channel, inter-scale transfer of momentum in the periodic directions is only possible through wavenumber interactions in the non-linear advection term. The aim of this study is to map the inter-scale interactions which transfer energy through advection to different lengthscales of interest. The non-linear advection term has nine contributions, which correspond to three different advecting and advected velocity components. Because the flow is anisotropic, each of these contributions can be viewed as portraying a different physical mechanism. Maps of inter-scale transfer are therefore constructed for each of the nine contributions to the advection term in order to understand the relative importance of advecting and advected velocity components in wall turbulence. As a preliminary exploration, two maps of the inter-scale energy transfer in the buffer layer are presented for Reτ = 180, one for transfer to a ‘target’ lengthscale dynamically relevant in the near-wall cycle, λx+=188 , λz+=94 and one for transfer to a ‘target’ lengthscale in the dissipative range, λx+=42 , λz+=21 . In the first case, we observe that the transfer of energy is dominated by transfer into streamwise-velocity (u) by spanwise-elongated structures of spanwise-velocity (w) that advect streamwise-elongated structures of streamwise-velocity (u). This is nothing but the process of streak meandering induced by spanwise-elongated (w) structures. In the second case, the inter-scale transfer can be split into three dominant mechanisms. The first and second mechanisms are observed across many of the nine contributions to the advection term. Velocity structures with size similar or larger than streamwise streaks either advect (mechanism 1) or are advected by (mechanism 2) velocity structures with lengthscales close to the target one. The third mechanism is only significant for streamwise (u) velocity structures advecting streamwise (u) and spanwise (w) velocity, and involves structures with an intermediate scale which is in-between the dissipation scale and the near-wall cycle dynamic lengthscale, but still in the dissipation range. In this mechanism, intermediate-scale streamwise-velocity structures advect intermediate-scale streamwise- and spanwise-velocity structures.

Fully higher-order coupling of finite element and level set methods for two-phase flow with a new explicit projection method

This work presents the development of a higher-order finite element method for modeling unsteady, incompressible and inviscid two-phase flows using the level set method. A new explicit projection method is introduced to solve the incompressible Euler equation, where the pressure and the velocity fields are solved separately. To implement high order temporal schemes, the Hodge decomposition in this projection method is not applied to the velocity but to its time derivative. A discontinuous Galerkin discretization is used to solve both the level set and the nonlinear advection terms present in the Euler equations, while a Galerkin discretization is used to solve the pressure Poisson equation. The present method is a fully high order numerical approach, both in space and temporal dimensions. Numerical results provided by several 2D benchmark test cases in naval hydrodynamics (i.e hydrostatic equilibrium, sloshing and dam break) validate the computational code, illustrating the power of the proposed method. Compared to the CFD codes based on low order VOF/finite-volume methods, the present approach demonstrates greater accuracy on coarse meshes, leading to reduced CPU time.

Physics-informed neural network for solution of forward and inverse kinematic wave problems

The one-dimensional kinematic wave (KW) model (KWM) is widely used in surface water and water quality hydrology as well as for modeling the movement of traffic on long highways. The KW model involves a first-order nonlinear partial differential equation which is solved numerically for realistic geometric representation, input, and initial and boundary conditions. This paper developed a method to solve the partial differential equation using a neural network structure, called the physics-informed neural network (PINN). The PINN was applied to solve the one-dimensional nonlinear KWM for forward and inverse problems. To enhance the accuracy and convergence and to suppress the spurious oscillation at the steep front, a technique with redistributed collocation points as an improved algorithm of PINN was employed. To deal with the fractional order of the nonlinear advection term, a stop-gradient technique was proposed. For validation, the KWM for steady and unsteady overland flow, and open channel flow without lateral inflow was analyzed for the forward problem. For the inverse problem, the unknown Manning coefficient was correctly computed, demonstrating the capability of the proposed algorithm for the identification of KWM parameters.

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

In this paper, we investigate a reaction–diffusion population model with a nonlinear advection term and a time-dependent force given by the equation ut−Δu+α→⋅∇up=f(u)+h(t)in(τ,∞)×Ω,subject to the boundary condition u=0 on (τ,∞)×∂Ω. Here, Ω⊂RN with N≥1 is a bounded domain with smooth boundary, τ∈R, α→=(α1,…,αN) is a given advective direction and p>1. The presence of the nonlinear advection term α→⋅∇up introduces technical difficulties in the analysis, leading to a scenario where the uniqueness of weak solutions cannot be guaranteed. Consequently, the equation generates a multi-valued nonautonomous dynamical system. In this context, we establish the existence of minimal pullback attractors, considering universes of bounded and tempered sets. Moreover, we explore the relationships between these pullback attractors. Finally, we prove the upper semicontinuity of pullback attractors with respect to the advective vector α→.

Different Propagation Mechanisms of Deep and Shallow Wintertime Extratropical Cyclones over the North Pacific

Abstract 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.

On the theory of fast projection methods for high-order Navier-Stokes solvers

We propose a new framework for the design of fast-projection solvers for the Navier-Stokes equations using Runge-Kutta integrators. The framework uses the full nonlinear equations along with rooted trees and exposes the need to track non-linear advection terms for fourth-order and higher integrators as those introduce irreducible elementary derivatives in the truncation error. The framework is then used to derive approximations to the pressure projection by replacing those with suitable approximations that do not affect formal order of accuracy. The proposed pseudo-pressure approximations are easy-to-implement with some of the variants including parametric coefficients that can be exploited to optimize fast-projection solvers for stability or other desirable properties. In addition, we report approximations for adaptive timestepping. Both the parametric and adaptive timestepping results are being reported for the first time in the literature. The proposed approximations are verified with computations of the full Navier-Stokes equations on two benchmark problems with steady and unsteady boundary conditions and fixed and adaptive timestepping. All calculations agree with the theoretical results and produce the correct formal order of accuracy. The proposed framework lays the foundation for future work on fast-projection methods and can be used to reproduce existing results in the literature.

On the Observed Wind-Driven Circulation Response in Small Semienclosed Bays

Abstract 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.

High accuracy compact difference and multigrid methods for two-dimensional time-dependent nonlinear advection-diffusion-reaction problems

We focus on high-accuracy and stable numerical methods for the time-dependent nonlinear advection-diffusion-reaction problems in this work. A novel fourth-order fully implicit compact difference scheme is derived, in which we make use of the fourth-order compact formulas to discretize the diffusion terms, the fourth-order Padé formulas to compute the nonlinear advection terms, and the fourth-order backward differencing formula to approximate the time derivative term. Convergence and stability of the new scheme are analysed by the energy method. On the basis of the novel scheme, a multigrid algorithm is established such that a time advancement algorithm for solving these nonlinear problems are obtained. We provide various numerical experiments to verify the performances of the proposed scheme including the reliability, stability and computational efficiency. And the results obtained by our method show it is more accurate than most same kind schemes reported in the literature.

Seasonality of the Migrating Semidiurnal Tide in the Tropical Upper Mesosphere and Lower Thermosphere and Its Thermodynamic and Momentum Budget

AbstractThis work uses the Specified Dynamics‐Whole Atmosphere Community Climate Model with Ionosphere/Thermosphere eXtension (SD‐WACCM‐X) to determine and explain the seasonality of the migrating semidiurnal tide (SW2) components of tropical upper mesosphere and lower thermosphere (UMLT) temperature, zonal wind, and meridional wind. This work also quantifies aliasing due to SW2 in satellite‐based tidal estimates. Results show that during equinox seasons, the vertical profiles of tropical UMLT temperature SW2 and zonal‐wind SW2’s amplitudes have a double‐peak structure while they, along with meridional‐wind SW2, have a single‐peak structure in their amplitudes in June solstice. Hough mode reconstruction reveals that a linear combination of five SW2 Hough modes cannot fully reproduce these features. Tendency analysis reveals that for temperature, the adiabatic term, nonlinear advection term, and linear advection term are important. For the winds, the classical terms, nonlinear advection term, linear advection term, and gravity wave drag are important. Results of our alias analysis then indicate that SW2 can induce an ∼60% alias in zonal‐mean and DW1 components calculated from sampling like that of the Thermosphere–Ionosphere–Mesosphere Energetics and Dynamics satellite and the Aura satellite. This work concludes that in situ generation by wave–wave interaction and/or by gravity waves plays significant roles in the seasonality of tropical UMLT temperature SW2, zonal‐wind SW2, and meridional‐wind SW2. The alias analysis further adds that one cannot simply assume that SW2 in the tropical UMLT is negligible.

Wind work at the air-sea interface: a modeling study in anticipation of future space missions

Abstract. Wind work at the air-sea interface is the transfer of kinetic energy between the ocean and the atmosphere and, as such, is an important part of the ocean-atmosphere coupled system. Wind work is defined as the scalar product of ocean wind stress and surface current, with each of these two variables spanning, in this study, a broad range of spatial and temporal scales, from 10 km to more than 3000 km and hours to months. These characteristics emphasize wind work's multiscale nature. In the absence of appropriate global observations, our study makes use of a new global, coupled ocean-atmosphere simulation, with horizontal grid spacing of 2–5 km for the ocean and 7 km for the atmosphere, analyzed for 12 months. We develop a methodology, both in physical and spectral spaces, to diagnose three different components of wind work that force distinct classes of ocean motions, including high-frequency internal gravity waves, such as near-inertial oscillations, low-frequency currents such as those associated with eddies, and seasonally averaged currents, such as zonal tropical and equatorial jets. The total wind work, integrated globally, has a magnitude close to 5 TW, a value that matches recent estimates. Each of the first two components that force high-frequency and low-frequency currents, accounts for ∼ 28 % of the total wind work and the third one that forces seasonally averaged currents, ∼ 44 %. These three components, when integrated globally, weakly vary with seasons but their spatial distribution over the oceans has strong seasonal and latitudinal variations. In addition, the high-frequency component that forces internal gravity waves, is highly sensitive to the collocation in space and time (at scales of a few hours) of wind stresses and ocean currents. Furthermore, the low-frequency wind work component acts to dampen currents with a size smaller than 250 km and strengthen currents with larger sizes. This emphasizes the need to perform a full kinetic budget involving the wind work and nonlinear advection terms as small and larger-scale low-frequency currents interact through these nonlinear terms. The complex interplay of surface wind stresses and currents revealed by the numerical simulation motivates the need for winds and currents satellite missions to directly observe wind work.

Traveling waves solutions for a cooperative system with nonlinear advection and nonlinear KPP term

In this work, a characterization of Traveling Waves (TW) solutions to a cooperative system formulated with an order four operator, with a nonlinear advection and a Fisher–KPP reaction term is introduced. The order four operator induces a set of instabilities in the proximity of the critical points. This property can be understood from a biological perspective to describe the synchronizing effects between both species even when the diffusion is not regular. This phenomenon reflects the capability of a cooperative species system to propagate along the media while keeping cooperation even for instability regions. In addition, given a certain advection coefficient, it is possible to assess a TW-velocity for which the described instabilities do not occur. This TW-velocity has been sharply assessed. The techniques developed along this study are based on a mix of analytical and numerical approaches.

Parallel-Computing Two-Way Grid-Nested Storm Surge Model with a Moving Boundary Scheme and Case Study of the 2013 Super Typhoon Haiyan

This study presents a numerical tool for calculating storm surges from offshore, nearshore, and coastal regions using the finite-difference method, two-way grid-nesting function in time and space, and a moving boundary scheme without any numerical filter adopted. The validation of the solitary wave runup on a circular island showed the perfect matches between the model results and measurements for the free surface elevations and runup heights. After the benchmark problem validation, the 2013 Super Typhoon Haiyan event was selected to showcase the storm surge calculations with coastal inundation and flood depths in Tacloban. The catastrophic storm surges of about 8 m and wider, storm-induced inundation due to the Super Typhoon Haiyan were found in the Tacloban Airport, corresponding to the findings from the field survey. In addition, the anti-clockwise, storm-induced currents were explored inside of Cancabato Bay. Moreover, the effect of the nonlinear advection terms with the fixed and moving shoreline and the parallel efficiency were investigated. By presenting a storm surge model for calculating storm surges, inundation areas, and flood depths with the model validation and case study, this study hopes to provide a convenient and efficient numerical tool for forecasting and disaster assessment under a potential severe tropical storm with climate change.

The logistic equation with nonlinear advection term

We study the logistic equation with nonlinear advection term −Δu+α→⋅∇up=λu−u2 in Ω with u=0 on ∂Ω, where Ω⊂RN, N≥1, is a bounded domain with smooth boundary, α→=(α1,…,αN) is a flow satisfying suitable condition, p>1 and λ∈R. The classical logistic equation differs from ours by the inclusion of the nonlinear advection term α→⋅∇up, meaning that the velocity of transport movement of the species depends on its density and may be higher or lower, depending on u and the size of p and |α→|. From the mathematical point of view, the inclusion of this term brings technical difficulties in the analysis, especially because it has no definite sign. Depending on λ, by the method of subsolution and supersolution and Picone Identity we show the existence and uniqueness of a positive solution. We also study the behavior of the solutions with respect to α→ and as p→1+.

Oscillatory solutions and smoothing of a higher-order p-Laplacian operator

&lt;abstract&gt;&lt;p&gt;The goal of this paper was to provide a general analysis of the solutions to a higher-order p-Laplacian operator with nonlinear advection. Generally speaking, it is well known that any solution to a higher-order operator exhibits oscillations. In the present study, an advection term is introduced. This will allow us to analyze smoothing conditions in the solutions. The study of existence and uniqueness is based on a variational approach. Solutions are analyzed with an energy formulation initially discussed by Saint-Venant and extended in the works by Tikhonov and Täklind. This variational principle is supported by the definition of generalized norms under Hilbert-Sobolev spaces, enabling focus on the oscillating properties of solutions. Afterward, the paper introduces an analysis to characterize the traveling wave kind of solutions together with their characterization to understand the oscillations. Finally, a numerical exploration focuses on the smoothing conditions by the action of the nonlinear advection term. As a main finding to report: There exist a traveling wave speed ($ \lambda $) and an advection coefficient ($ c^* $) for which the profile's first minimum is almost positive, and such positivity holds beyond the first minimum.&lt;/p&gt;&lt;/abstract&gt;

ENSO Amplitude Asymmetry in Met Office Hadley Centre Climate Models

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.

Nonlinear amplification in hydrodynamic turbulence

Using direct numerical simulations performed on periodic cubes of various sizes, the largest being $8192^3$ , we examine the nonlinear advection term in the Navier–Stokes equations generating fully developed turbulence. We find significant dissipation even in flow regions where nonlinearity is locally absent. With increasing Reynolds number, the Navier–Stokes dynamics amplifies the nonlinearity in a global sense. This nonlinear amplification with increasing Reynolds number renders the vortex stretching mechanism more intermittent, with the global suppression of nonlinearity, reported previously, restricted to low Reynolds numbers. In regions where vortex stretching is absent, the angle and the ratio between the convective vorticity and solenoidal advection in three-dimensional isotropic turbulence are statistically similar to those in the two-dimensional case, despite the fundamental differences between them.

Tropical Cyclone Intensification Simulated in the Ooyama-type Three-layer Model with a Multilevel Boundary Layer

AbstractThe first successful simulation of tropical cyclone (TC) intensification was achieved with a three-layer model, often named the Ooyama-type three-layer model, which consists of a slab boundary layer and two shallow water layers above. Later studies showed that the use of a slab boundary layer would produce unrealistic boundary layer wind structure and too strong eyewall updraft at the top of TC boundary layer and thus simulate unrealistically rapid intensification compared to the use of a height-parameterized boundary layer. To fully consider the highly height-dependent boundary layer dynamics in the Ooyama-type three-layer model, this study replaced the slab boundary layer with a multilevel boundary layer in the Ooyama-type model and used it to conduct simulations of TC intensification and also compared the simulation with that from the model version with a slab boundary layer. Results show that compared with the simulation with a slab boundary layer, the use of a multilevel boundary layer can greatly improve simulations of the boundary-layer wind structure and the strength and radial location of eyewall updraft, and thus more realistic intensification rate due to better treatments of the surface layer processes and the nonlinear advection terms in the boundary layer. Sensitivity of the simulated TCs to the model configuration and to both horizontal and vertical mixing lengths, sea surface temperature, the Coriolis parameter, and the initial TC vortex structure are also examined. The results demonstrate that this new model can reproduce various sensitivities comparable to those found in previous studies using fully physics models.