Multiscale Asymptotics Research Articles

On the interactions between planetary geostrophy and mesoscale eddies

Abstract Multiscale asymptotics are used to derive three systems of equations connecting the planetary geostrophic (PG) equations for gyre-scale flow to a quasigeostrophic (QG) equation set for mesoscale eddies. Pedlosky (1984), following similar analysis, found eddy buoyancy fluxes to have only a small effect on the large-scale flow; however, numerical simulations disagree. While the impact of eddies is relatively small in most regions, in keeping with Pedlosky’s result, eddies have a significant effect on the mean flow in the vicinity of strong, narrow currents. First, the multiple-scales analysis of Pedlosky is reviewed and amplified. Novel results of this analysis include new multiple-scales models connecting large-scale PG equations to sets of QG eddy equations. However, only introducing anisotropic scaling of the large-scale coordinates allows us to derive a model with strong two-way coupling between the QG eddies and the PG mean flow. This finding reconciles the analysis with simulations, viz. that strong two-way coupling is observed in the vicinity of anisotropic features of the mean flow like boundary currents and jets. The relevant coupling terms are shown to be eddy buoyancy fluxes. Using the Gent-McWilliams parameterization to approximate these fluxes allows solution of the PG equations with closed tracer fluxes in a closed domain, which is not possible without mesoscale eddy (or other small-scale) effects. The boundary layer width is comparable to an eddy mixing length when the typical eddy velocity is taken to be the long Rossby wave phase speed, which is the same result found by Fox-Kemper and Ferrari (2009) in a reduced gravity layer.

Multiscale tip asymptotics in hydraulic fracture with leak-off

This paper is concerned with an analysis of the near-tip region of a fluid-driven fracture propagating in a permeable saturated rock. The analysis is carried out by considering the stationary problem of a semi-infinite fracture moving at constant speedV. Two basic dissipative processes are taken into account: fracturing of the rock and viscous flow in the fracture, and two fluid balance mechanisms are considered – leak-off and storage of the fracturing fluid in the fracture. It is shown that the solution is characterized by a multiscale singular behaviour at the tip, and that the nature of the dominant singularity depends both on the relative importance of the dissipative processes and on the scale of reference. This solution provides a framework to understand the interaction of representative physical processes near the fracture tip, as well as to track the changing nature of the dominant tip process(es) with the tip velocity and its impact on the global fracture response. Furthermore, it gives a universal scaling of the near-tip processes on the scale of the entire fracture and sets the foundation for developing efficient numerical algorithms relying on accurate modelling of the tip region.

New multi-scale models on mesoscales and squall lines

Squall lines are coherent turbulent traveling waves on scales of order 100 km in the atmosphere that emerge in a few hours from the interaction of strong vertical shear and moist deep convection on scales of order 10 km. They are canonical coherent structures in the tropics and middle latitudes reflecting upscale conversion of energy from moist buoyant sources to horizontal kinetic energy on larger scales. Here squall lines are introduced through high resolution numerical simulations which reveal a new self-similarity with respect to the shear amplitude. A new multi-scale model on mesoscales which allows for large vertical shears, appropriate for squall lines, is developed here through systematic multi-scale asymptotics. Mathematical and numerical formulations of the new multi-scale equations are utilized to illustrate both new mathematical and physical phenomena captured by these new models. In particular, non-hydrostatic Taylor-Goldstein equations govern the upscale transports of momentum and temperature from the order 10 km microscales to the order 100 km mesoscales; surprisingly, upright single mode convective heating without tilts can lead to significant upscale convective momentum transport from the microscales to the mesoscales due to the strong shear. The multi-scale models developed here should be especially useful for dynamic parameterizations of upscale transports as well as for new theory in three-dimensions with a transverse shear component, where contemporary theoretical understanding is meager.

Multiscale Asymptotic Method for Maxwell's Equations in Composite Materials

In this paper we discuss the multiscale analysis of Maxwell's equations in composite materials with a periodic microstructure. The new contributions in this paper are the determination of higher-order correctors and the explicit convergence rate for the approximate solutions (see Theorem 2.3). Consequently, we present the multiscale finite element method and derive the convergence result (see Theorem 4.1). The numerical results demonstrate that higher-order correctors are essential for solving Maxwell's equations in composite materials.

A Reexamination of the Classical PKN Model of Hydraulic Fracture

This article reexamines the classical PKN model of hydraulic fracture Perkins and Kern (J. Pet. Tech. Trans. AIME, 222:937–949 (1961)) and Nordgren (J. Pet. Tech. 253:306–314 (1972)) using novel approaches, which have recently been developed to tackle this class of problems that are characterized by a moving boundary and strong non-linearities in the governing equations. First, we demonstrate, using scaling arguments only, that a PKN hydraulic fracture has two limiting time asymptotic behaviors: storage-dominated at small time, and leak-off-dominated at large time. Next, we investigate the multiscale nature of the tip asymptotics and its implication for the construction of a robust and efficient numerical algorithm. In particular, we show that in the storage-dominated regime the tip aperture w behaves according to w ~ x 1/3 (where x is the distance from the tip), and in the leak-off-dominated regime according to w ~ x 3/8. However, the solution in the leak-off-dominated regime has a boundary layer structure at the tip, with w ~ x 3/8 acting as an intermediate asymptote that matches an inner solution with a w ~ x 1/3 asymptote to the outer (global) solution. Finally, we describe an efficient numerical algorithm with the multiscale tip asymptotics embedded in the tip element, which is used to compute the transient solution that connects the small- and large-time asymptotes.

Experimental validation of the tip asymptotics for a fluid-driven crack

This paper provides experimental confirmation of the opening asymptotes that have been predicted to develop at the tip of fluid-driven cracks propagating in impermeable brittle elastic media. During propagation of such cracks, energy is dissipated not only by breaking of material bonds ahead of the tip but also by flow of viscous fluid. Theoretical analysis based on linear elastic fracture mechanics and lubrication theory predicts a complex multiscale asymptotic behavior of the opening in the tip region, which simplifies either as 1 2 or as 2 3 power law of the distance from the tip depending on whether the dominant mechanism of energy dissipation is bond breaking or viscous flow. The laboratory experiments entail the propagation of penny-shaped cracks by injection of glycerin or glucose based solutions in polymethyl methacrylate (PMMA) and glass specimens subjected to confining stresses. The full-field opening is measured from analysis of the loss of intensity as light passes through the dye-laden fluid that fills the crack. The experimental near-tip opening gives excellent agreement with theory and therefore confirms the predicted multi-scale tip asymptotics.

On intrinsic oscillation in radiation-affected diffusion flames

A linear stability analysis is conducted to study the onset of near-limit flame oscillation with radiative heat loss in 1-D chambered planar flames using multi-scale activation-energy asymptotics. The oscillatory instability near the radiation-induced extinction limit at large Damköhler numbers is identified, in additional to the one near the kinetic limit at small Damköhler numbers. It is shown that radiative loss assumes a similar role as varying the thermal diffusivity of the reactants. Thus, flame oscillation near the radiative limit is still thermal-diffusive in nature although it may develop under unity Lewis numbers. The unstable range of Damköhler numbers near the radiative limit shows quite similar parametric dependence on the Lewis numbers of reactants, Le F and Le O, the stoichiometry, ϕ, and the radiative loss as that near the kinetic limit. They both increase monotonically with Le O and ϕ and increase then decrease with Le F. Increasing radiative loss extends the parameter range under which flame oscillations may develop. However, they show different dependence on the temperature difference between the supplying reactants. Unless radiative loss approaches its maximum value the system can sustain, flame oscillation near the radiative limit is only possible within a limited range of Δ T, whereas it is promoted monotonically with decreasing Δ T near the kinetic limit. Furthermore, while radiative loss shows small effect on the nondimensional oscillation frequency, the dimensional frequency of flame oscillations near the radiative limit can be substantially smaller than that near the kinetic limit.

Preface to the special issue on “Theoretical Developments in Tropical Meteorology”

During the 13th Atmosphere–Ocean Fluid Dynamics Conference of the American Meteorological Society in Breckenridge in 2001, I was invited to participate as an external consultant in the NSF-Focused Research Group (FRG) on “The weak temperature gradient equations for tropical atmosphere dynamics”, which several of the authors in this special issue were planning to propose at the time. The FRG, which was designed to forge cooperations amongst theoretical meteorologists and applied mathematicians, was granted and during the subsequent 4 years I witnessed a number of intriguing new theoretical developments in tropical meteorology pursued within the FRG and its associated research groups. A range of different facets of the ubiquitous moisture-related scale interactions within the tropical atmosphere were addressed successfully, and it seemed only natural to propose a special issue on “Theoretical Developments in Tropical Meteorology” of Theoretical and Computational Fluid Dynamics to the editor in chief, Prof. Yussuf Hussaini, who judged it to be an important and timely endeavour. Tropical moist processes originate mostly in the atmospheric boundary layer near the ocean surface. Convective, well-mixed, and cloud-topped boundary layers play a particularly important role in driving and conditioning bulk tropospheric motions. The papers by Stevens, Neggers et al., Sobel and Neelin, and Khouider and Majda specifically consider the parameterization of moist boundary layer processes and their influence on the large scale, bulk tropospheric dynamics. A fruitful approach to constructing approximate reduced models for bulk tropospheric flows is vertical mode decomposition. The idea being that only very few dominant vertical modes need to be accounted for in order to capture the essence of the large scale motions. In the tropics, nonlinear moist processes associated with a number of essentially different cloud regimes participate strongly in setting up these large scale flows, making such vertical mode decompositions a highly nontrivial task. The papers by Sobel and Neelin, Khouider and Majda, Stechman and Majda, Biello and Majda, Bretherton et al., Burns et al., and Zhou and Sobel address various facets of this overall problem by considering vertical mode interactions, analyzing the importance of various approximations introduced often in addition to the vertical mode-related order reduction, and by analyzing in detail the structure of solutions supported by such low-order models. Deep moist convection, which covers vertical scales comparable with the height of the troposphere, is of central importance for a wide range of phenomena in the near-equatorial atmosphere. It arises when the atmosphere is unstable under the added buoyancy from latent heat release and leads to deep vertical motions occurring on horizontal scales of about 1 km. As a consequence, the net effect of deep convection on meso(∼200 km) to planetary scales is of an inherently multiscale nature. The papers by Pauluis et al., Peters and Bretherton, and Klein and Majda address these effects through techniques of direct computational modelling and of systematic multiscale asymptotics.

Multiscale Asymptotic Analysis and Numerical Simulation for the Second Order Helmholtz Equations with Rapidly Oscillating Coefficients Over General Convex Domains

The multiscale asymptotic analysis and numerical simulation for the second order Helmholtz equations with rapidly oscillating coefficients over general convex domains are discussed in this paper. A multiscale asymptotic analysis formulation for this problem is presented by constructing properly the boundary layer. A multiscale numerical algorithm and a postprocessing technique are given. Finally, numerical results show that the method presented in this paper is effective and reliable.

A low Mach number scheme based on multi-scale asymptotics

The low Mach number regime is characterized by a large discrepancy between the flow velocity and the speed of sound, leading to physical effects on different length scales and of different orders of magnitude. A single time scale, multiple space scale asymptotic analysis provides detailed insight into the limit behavior of solutions of the compressible Euler equations as the Mach number tends to zero. This analysis shows that "the pressure" splits up into three parts with different physical meanings. This knowledge is then used to develop a numerical scheme including multiple pressure variables to account for the different effects. The numerical method is a semi-implicit predictor-corrector algorithm. In the predictor step, the asymptotic equations are used to guess the global and large scale effects. Then the corrector step can be viewed as an incompressible solver with compressibility effects acting as source terms.

Diffusion approximation for billiards with totally accommodating scatterers

We study the 2D motion of independent point particles colliding with a periodic array of circular obstacles. The interaction between the particles and the obstacles is described by a total accommodation reflection law. Assuming that the array of scatterers has finite horizon, the density of particles is approximated by the solution of a diffusion equation in the long-time and large-scale regime. The proof relies on a multiscale asymptotics and gives the order of approximation.

Asymptotics of arbitrary order for a thin elastic clamped plate, I. Optimal error estimates

This paper is the first of a series of two, where we study the asymptotics of the displacement in a thin clamped plate made of a rigid monoclinic material, as the thickness of the plate tends to O. The combination of a polynomial Ansatz (outer expansion) and of a boundary layer Ansatz (inner expansion) yields a complete multi-scale asymptotics of the displacement and leads to optimal error estimates in energy norm. We investigate the polynomial Ansatz in Part I, and the boundary layer Ansatz in Part II. If c denotes the small parameter in the geometry, we first construct the algorithm for an infinite Ansatz involving only even powers of c, which is a natural extension of the usual Kirchhoff-Love Ansatz. The boundary conditions of the clamped plate being only satisfied at the order 0, we try to compensate for them by boundary layer terms: we rely on a result proved in Part II giving necessary and sufficient conditions for the exponential decay of such terms. In order to fulfill these conditions, the constructive algorithm for the boundary layer terms has to be combined with an odd polynomial Ansatz. The outcome is a two-scale asymptotics involving all nonnegative powers of c, the in-plane space variables Xu, the transverse scaled variable X3 and the quickly varying variable r / c where r is the distance to the clamped part of the boundary.

Lagrangian rings. Multiscale asymptotics of a spectrum near resonance

Lagrangian rings. Multiscale asymptotics of a spectrum near resonance