Embedded Domains Research Articles

Stochastic superparameterization in quasigeostrophic turbulence

In this article we expand and develop the authors' recent proposed methodology for efficient stochastic superparameterization algorithms for geophysical turbulence. Geophysical turbulence is characterized by significant intermittent cascades of energy from the unresolved to the resolved scales resulting in complex patterns of waves, jets, and vortices. Conventional superparameterization simulates large scale dynamics on a coarse grid in a physical domain, and couples these dynamics to high-resolution simulations on periodic domains embedded in the coarse grid. Stochastic superparameterization replaces the nonlinear, deterministic eddy equations on periodic embedded domains by quasilinear stochastic approximations on formally infinite embedded domains. The result is a seamless algorithm which never uses a small scale grid and is far cheaper than conventional SP, but with significant success in difficult test problems.Various design choices in the algorithm are investigated in detail here, including decoupling the timescale of evolution on the embedded domains from the length of the time step used on the coarse grid, and sensitivity to certain assumed properties of the eddies (e.g. the shape of the assumed eddy energy spectrum). We present four closures based on stochastic superparameterization which elucidate the properties of the underlying framework: a ‘null hypothesis’ stochastic closure that uncouples the eddies from the mean, a stochastic closure with nonlinearly coupled eddies and mean, a nonlinear deterministic closure, and a stochastic closure based on energy conservation. The different algorithms are compared and contrasted on a stringent test suite for quasigeostrophic turbulence involving two-layer dynamics on a β-plane forced by an imposed background shear.The success of the algorithms developed here suggests that they may be fruitfully applied to more realistic situations. They are expected to be particularly useful in providing accurate and efficient stochastic parameterizations for use in ensemble-based state estimation and prediction.

Efficient stochastic superparameterization for geophysical turbulence

Efficient computation of geophysical turbulence, such as occurs in the atmosphere and ocean, is a formidable challenge for the following reasons: the complex combination of waves, jets, and vortices; significant energetic backscatter from unresolved small scales to resolved large scales; a lack of dynamical scale separation between large and small scales; and small-scale instabilities, conditional on the large scales, which do not saturate. Nevertheless, efficient methods are needed to allow large ensemble simulations of sufficient size to provide meaningful quantifications of uncertainty in future predictions and past reanalyses through data assimilation and filtering. Here, a class of efficient stochastic superparameterization algorithms is introduced. In contrast to conventional superparameterization, the method here (i) does not require the simulation of nonlinear eddy dynamics on periodic embedded domains, (ii) includes a better representation of unresolved small-scale instabilities, and (iii) allows efficient representation of a much wider range of unresolved scales. The simplest algorithm implemented here radically improves efficiency by representing small-scale eddies at and below the limit of computational resolution by a suitable one-dimensional stochastic model of random-direction plane waves. In contrast to heterogeneous multiscale methods, the methods developed here do not require strong scale separation or conditional equilibration of local statistics. The simplest algorithm introduced here shows excellent performance on a difficult test suite of prototype problems for geophysical turbulence with waves, jets, and vortices, with a speedup of several orders of magnitude compared with direct simulation.

MDM: A Mode Diagram Modeling Framework

Periodic control systems used in spacecrafts and automotives are usually period-driven and can be decomposed into different modes with each mode representing a system state observed from outside. Such systems may also involve intensive computing in their modes. Despite the fact that such control systems are widely used in the above-mentioned safety-critical embedded domains, there is lack of domain-specific formal modelling languages for such systems in the relevant industry. To address this problem, we propose a formal visual modeling framework called mode diagram as a concise and precise way to specify and analyze such systems. To capture the temporal properties of periodic control systems, we provide, along with mode diagram, a property specification language based on interval logic for the description of concrete temporal requirements the engineers are concerned with. The statistical model checking technique can then be used to verify the mode diagram models against desired properties. To demonstrate the viability of our approach, we have applied our modelling framework to some real life case studies from industry and helped detect two design defects for some spacecraft control systems.

A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem

We consider a transmission wave equation in two embedded domains in , where the speed is a1 > 0 in the inner domain and a2 > 0 in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strongly convex and a1 > a2. As a consequence of this inequality, uniqueness and Lipschitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient from a single time-dependent Neumann boundary measurement.

Analytic Element Modeling of Embedded Multiaquifer Domains

An analytic element approach is presented for the modeling of multiaquifer domains embedded in a single-aquifer model. The inside of each domain may consist of an arbitrary number of aquifers separated by leaky layers. The analytic element solution is obtained through a combination of existing single-aquifer and multiaquifer analytic elements and allows for the analytic computation of head and leakage at any point in the aquifer. Along the boundary of an embedded multiaquifer domain, the normal flux is continuous everywhere; continuity of head across the boundary is met exactly at collocations points and approximately, but very accurately, in between. The analytic element solution compares well with an existing exact solution. A hypothetical example with a river intersecting two embedded domains illustrates the practical application of the proposed approach.

Study and control of phase morphology in liquid crystal polyester–poly(alkylene terephthalate) blends

AbstractScanning electron microscopy (SEM) reveals that the morphologies of poly(hexamethylene terephthalate) (PHMT) and poly(p‐oxybenzoate co‐ethylene terephthalate) (LCP6/4) blends are biphasic. When LCP6/4 is the minor component, liquid crystalline domains are embedded in the matrix. The reaction in the melt between the polymers apparently increases the adhesion of the matrix to the embedded domains. As the reaction proceeds, the chemical composition of the matrix progressively becomes modified. High reaction temperatures, small particle size, mixing coupled with particle dispersion and high LCP6/4 content in the mixture promote the melt reaction and decrease the PHMT content in the polymer formed. At 260°C, the coaddition of di‐n‐octadecyl phosphite (DNOP) at a 2% weight fraction with PHMT and LCP6/4 inhibits the reaction between the polymers for 2 h.

Ferromagnetic random-bond Ising model: Metastable states and complexity of the energy surface.

Metastable states of the ferromagnetic random-bond Ising model are produced by simulated quenches from infinite to zero temperature. Two-dimensional systems (square lattice) show many domains: the many-valley picture of the energy surface of spin glasses applies to these unfrustrated systems as well. The number of the domains, which serves as a rough measure of the complexity of the energy surface in two dimensions (but not in three), is only 30% greater for a broad bond distribution (uniform from 0 to 1) than for a narrow one (uniform from 0.499 to 0.501). We conclude that the uniform-bond model also has a complex energy surface; for a square lattice the surface has many terraces (or steps) rather than many valleys, but for the honeycomb lattice it has a many-valley structure. The large domains in our two-dimensional systems have holes on many length scales and are highly ramified, with total perimeter proportional to the number of sites; their dimensions from the capacity, information, and radius-of-gyration definitions are 1.82\ifmmode\pm\else\textpm\fi{}0.06, 1.84\ifmmode\pm\else\textpm\fi{}0.04, and 1.83\ifmmode\pm\else\textpm\fi{}0.07, respectively. In three dimensions (simple cubic lattice), the initial concentrations of both up and down spins exceed the percolation threshold and the metastable states are dominated by two large spanning domains; only a few, small, embedded domains are found.