The recently developed short-time linear response algorithm, which predicts the response of a nonlinear chaotic forced-dissipative system to small external perturbation, yields high precision of the response prediction. However, the computation of the short-time linear response formula with the full rank tangent map can be expensive. Here, a numerical method to potentially overcome the increasing numerical complexity for large scale models with many variables by using the reduced-rank tangent map in the computation is proposed. The conditions for which the short-time linear response approximation with the reduced-rank tangent map is valid are established, and two practical situations are examined, where the response to small external perturbations is predicted for nonlinear chaotic forced-dissipative systems with different dynamical properties.

The authors consider the multidimensional aggregation equation ∂ _t ρ-div(ρ▿K* ρ) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better), and review recent results on this problem concerning well-posedness of nonnegative solutions and finite time blowup in multiple space dimensions depending on the behavior of the kernel at the origin. The problem with bounded initial data, data in L ^p ∩ L ¹, and measure solutions are also considered.

Amplitude equations governing the nonlinear resonant interaction of equatorial baroclinic and barotropic Rossby waves were derived by Majda and Biello and used as a model for long range interactions (teleconnections) between the tropical and midlatitude troposphere. An overview of that derivation is presented and geared to readers versed in nonlinear wave theory, but not in atmospheric sciences. In the course of the derivation, two other sets of asymptotic equations are presented: the long equatorial wave equations and the weakly nonlinear, long equatorial wave equations. A linear transformation recasts the amplitude equations as nonlinear and linearly coupled KdV equations governing the amplitude of two types of modes, each of which consists of a coupled tropical/midlatitude flow. In the limit of Rossby waves with equal dispersion, the transformed amplitude equations become two KdV equations coupled only through nonlinear fluxes. Four numerical integrations are presented which show (i) the interaction of two solitons, one from either mode, (ii) and (iii) the interaction of a soliton in the presence of different mean wind shears, and (iv) the interaction of two solitons mediated by the presence of a mean wind shear.

The authors construct self-similar solutions for an N-dimensional transport equation, where the velocity is given by the Riezs transform. These solutions imply nonuniqueness of weak solution. In addition, self-similar solution for a one-dimensional conservative equation involving the Hilbert transform is obtained.

The author reviews briefly some of the recent results on the blow-up problem for the incompressible Euler equations on ℝ ^N, and also presents Liouville type theorems for the incompressible and compressible fluid equations.

In this paper, the Tricomi problem and the generalized Tricomi problem for a quasilinear mixed type equation are studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to these problems is proved. The method developed in this paper can be used to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.

Lateral energy exchange between the tropics and the midlatitudes is a topic of great importance for understanding Earth’s climate system. In this paper, the authors address this issue in an idealized set up through simple shallow water models for the interactions between equatorially trapped waves and the barotropic mode, which supports Rossby waves that propagate poleward and can excite midlatitude teleconnection patterns. It is found here that the interactions between a Kelvin wave and a fixed meridional shear (mimicking the jet stream) generates a non-trivial meridional velocity and meridional convergence in phase with the upward motion that can attain a maximum of about 50%, which oscillates on frequencies ranging from one day to 10 days. When, on the other hand, the barotropic flow is forced by slowly propagating Kelvin waves a complex flow pattern emerges; it consists of a phase-locked barotropic response that is equatorially trapped and that propagates eastward with the forcing Kelvin wave and a certain number of planetary Rossby waves that propagate westward and toward the poles as seen in nature. It is suggested here that the poleward propagating waves are to some sort of multi-way resonant interaction with the phase locked response. Moreover, it is shown here that a numerical scheme with dispersion properties that depend on the direction perpendicular to the direction of propagation, namely the 2D central scheme of Nessyahu and Tadmor, can artificially alter significantly the topology of the wave fields and thus should be avoided in climate models.

The ensemble technique has been widely used in numerical weather prediction and extended-range forecasting. Current approaches to evaluate the predictability using the ensemble technique can be divided into two major groups. One is dynamical, including generating Lyapunov vectors, bred vectors, and singular vectors, sampling the fastest error-growing directions of the phase space, and examining the dependence of prediction efficiency on ensemble size. The other is statistical, including distributional analysis and quantifying prediction utility by the Shannon entropy and the relative entropy. Currently, with simple models, one could choose as many ensembles as possible, with each ensemble containing a large number of members. When the forecast models become increasingly complicated, however, one would only be able to afford a small number of ensembles, each with limited number of members, thus sacrificing estimation accuracy of the forecast errors.

To uncover connections between different information theoretic approaches and between dynamical and statistical approaches, we propose an (ɛ, τ)-entropy and scale-dependent Lyapunov exponent—based general theoretical framework to quantify information loss in ensemble forecasting. More importantly, to tremendously expedite computations, reduce data storage, and improve forecasting accuracy, we propose a technique for constructing a large number of “pseudo” ensembles from one single solution or scalar dataset. This pseudo-ensemble technique appears to be applicable under rather general conditions, one important situation being that observational data are available but the exact dynamical model is unknown.

The simulation of wave phenomena in unbounded domains generally requires an artificial boundary to truncate the unbounded exterior and limit the computation to a finite region. At the artificial boundary a boundary condition is then needed, which allows the propagating waves to exit the computational domain without spurious reflection. In 1977, Engquist and Majda proposed the first hierarchy of absorbing boundary conditions, which allows a systematic reduction of spurious reflection without moving the artificial boundary farther away from the scatterer. Their pioneering work, which initiated an entire research area, is reviewed here from a modern perspective. Recent developments such as high-order local conditions and their extension to multiple scattering are also presented. Finally, the accuracy of high-order local conditions is demonstrated through numerical experiments.

Under the assumptions that the initial density ρ ₀ is close enough to 1 and ρ ₀ − 1 ∈ H ^s+1(ℝ²), u ₀ ∈ H ^s(ℝ²) ∩ Ḣ^−ε(ℝ²) for s > 2 and 0 < ε < 1, the authors prove the global existence and uniqueness of smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with the viscous coefficient depending on the density of the fluid. Furthermore, the L ² decay rate of the velocity field is obtained.

Homogenization theory provides a rigorous framework for calculating the effective diffusivity of a decaying passive scalar field in a turbulent or complex flow. The authors extend this framework to the case where the passive scalar fluctuations are continuously replenished by a source (and/or sink). The basic structure of the homogenized equations carries over, but in some cases the homogenized source can involve a non-trivial coupling of the velocity field and the source. The authors derive expressions for the homogenized source term for various multiscale source structures and interpret them physically.

The author presents a simple approach to both regularity and singularity theorems for free boundaries in classical obstacle problems. This approach is based on the monotonicity of several variational integrals, the Federer-Almgren dimension reduction and stratification theorems, and some simple PDE arguments.