Papers by Boualem Khouider
Linear well posedness of regularized equations of sea-ice dynamics
Journal of Mathematical Physics, May 1, 2023
The viscous–plastic equations (VPE) of Hibler [J. Geophys. Res. 82(27), 3932–3938 (1977)] are wid... more The viscous–plastic equations (VPE) of Hibler [J. Geophys. Res. 82(27), 3932–3938 (1977)] are widely adopted and used in Earth system models to represent sea-ice drift due to surface winds, ocean currents, and internal stresses. However, it has been reported by various investigators, at least in one space dimension, that both Hibler’s original equations and their variant using a pressure replacement are ill posed in divergent flow regimes. Especially, Guba et al. [J. Phys. Oceanogr. 43(10), 2185–2199 (2013)] shows that both variants are ill-posed when the flow divergence exceeds a minimum threshold and their results seem to extend to two dimensions when a tensile cut-off is used. In particular, Hibler uses a Heaviside function cut-off for the viscosity coefficients of the VPE’s to avoid a singularity at infinity. Lemieux et al. [J. Comput. Phys. 231(17), 5926–5944 (2012)] regularized the Heaviside function by a hyperbolic tangent for numerical efficiency. Here, we show that, for periodic data, the linearized one-dimensional regularized VPE’s, in which the Heaviside function is replaced with a hyperbolic tangent, is well posed in the case of Hibler’s original equations. Moreover, we prove that the linearization procedure, for the regularized equations, is consistent, in the sense that the residual converges to zero that the perturbation of the solutions goes to zero, in suitable norms.
Theoretical and Computational Fluid Dynamics, Dec 15, 2012
This paper presents a study of the diurnal cycle of tropical precipitation and its interaction wi... more This paper presents a study of the diurnal cycle of tropical precipitation and its interaction with convectively coupled waves in the context of simple models with crude vertical resolution. One and two baroclinic mode models are tested in both the context of a one-column model and the context of full spatial dependency that permits waves to propagate and interact with the diurnal cycle. It is found that a one baroclinic mode model is capable of reproducing a realistic diurnal cycle of tropical precipitation both over land and over the ocean provided an adequate switch function is used to mimic the congestus preconditioning mechanism that operates in the multicloud model of Khouider and Majda. However, a full two baroclinic mode multicloud model is needed to capture the interaction of convectively coupled tropical waves with the diurnal cycle. In a more conventional mass-flux parameterization framework, both one and two baroclinic mode models fail to capture the diurnal cycle of tropical precipitation.
Fluid mechanics and its applications, 2002
An idealized model for turbulent ames is introduced. It consists of a scalar advection-reaction-d... more An idealized model for turbulent ames is introduced. It consists of a scalar advection-reaction-di usion equation that describes the interaction of a thin ame with a turbulent-like ow eld acting on two separate scales. Rigorous asymptotic results as well as a ordable and reliable direct numerical simulations are available to describe the e ective large scale behavior of the idealized front a strategy is proposed to use this framework to validate closure models for more realistic turbulent ames. Examples are given to illustrate the strategy as applied to premixed ames.
Multiscale Organization in Simulated Shear Parallel Mesoscale Convective Systems
33rd Conference on Hurricanes and Tropical Meteorology, Apr 18, 2018
AGU Fall Meeting Abstracts, Dec 19, 2014
The representation of the Madden-Julian oscillation (MJO) is still a challenge for numerical weat... more The representation of the Madden-Julian oscillation (MJO) is still a challenge for numerical weather prediction and general circulation models (GCMs) due to the inadequate treatment of convection and the associated interactions across scales by the underlying cumulus parameterizations. One new promising direction is the use of the stochastic multicloud model (SMCM) that have been designed specifically to capture the missing variability due to unresolved processes of convection and their impact on the large scale flow. The SMCM specifically models the area fractions of the three cloud types (congestus, deep and stratiform) that characterize organized convective systems on all scales. The SMCM captures the stochastic behavior of these three cloud types via a judiciously constructed Markov birthdeath process using a particle interacting lattice model. The SMCM has been successfully applied for convectively coupled waves in a simplified primitive equation model and validated against radar data of tropical precipitation. In this work, we use for the first time the SMCM in a GCM. We build on previous work of coupling the High-Order Methods Modeling Environment (HOMME) NCAR-GCM to a simple multicloud model. We tested the new SMCM-HOMME model in the parameter regime considered previously and found that the stochastic model drastically improves the results of the deterministic model. Clear MJO-like structures with many realistic features from nature are reproduced by SMCM-HOMME in the physically relevant parameter regime including wave trains of MJO's that organize intermittently in time. Also one of the caveats of the deterministic simulation of requiring a doubling of the moisture background is not required anymore.
Implementation of the Stochastic Multicloud Model in the NCEP Climate Forecast System version 2 (CFSv2)
2015 AGU Fall Meeting, Dec 18, 2015
Journal of the Atmospheric Sciences, May 1, 2021
The barrier effect of the Maritime Continent (MC) in stalling or modifying the propagation charac... more The barrier effect of the Maritime Continent (MC) in stalling or modifying the propagation characteristics of the MJO is widely accepted. The strong diurnal cycle of convection over the MC is believed to play a dominant role in this regard. This hypothesis is studied here, with the help of a coarse-resolution atmospheric general circulation model (AGCM). The dry dynamical core of the AGCM is coupled to the multicloud parameterization piggybacked with a dynamical bulk boundary layer model. A set of sensitivity experiments is carried out by systematically varying the strength of the MC diurnal flux to assess the impact of the diurnal convective variability on the MJO propagation. The effects of deterministic and stochastic diurnal forcings on MJO characteristics are compared. It is found that the precipitation and zonal wind variance, on the intraseasonal time scales, over the western Pacific region decreases with the increase in diurnal forcing, indicating the blocking of MC precipitation. An increase in precipitation variance over the MC associated with the weakening of precipitation variance over the west Pacific is evident in all experiments. The striking difference between deterministic and stochastic diurnal forcing experiments is that the strength needed for the deterministic case to achieve the same degree of blocking is almost double that of stochastic case. The stochastic diurnal flux over the MC seems to be more detrimental in blocking the MJO propagation. This hints at the notion that the models with inadequate representation of organized convection tend to suffer from the MC-barrier effect.
Multiscale Modeling & Simulation, 2007
An idealized model for turbulent premixed flames is introduced. It consists of a scalar advection... more An idealized model for turbulent premixed flames is introduced. It consists of a scalar advection-reaction-diffusion equation that describes the interaction of a thin flame with a turbulentlike flow field acting on two separate scales. Rigorous asymptotic results as well as affordable and reliable direct numerical simulations are available to predict the effective large scale behavior of the idealized front. This framework is used as part of a strategy to validate a modeling approach for the large scale simulations of turbulent premixed flame fronts. The relevance to practical computations is demonstrated by addressing three important issues regarding closure models for more realistic turbulent flames: scaling regimes in the parameterization of the flame speed; relationship between the front at large scales and the resolved reaction zone in a direct simulation; accuracy and efficiency of the large scale flamelet approach.
SIAM Journal on Numerical Analysis, 2002
Turbulence enhances the speed of propagation of a premixed flame front. According to the Majda-So... more Turbulence enhances the speed of propagation of a premixed flame front. According to the Majda-Souganidis model, the procedure to predict this enhancement involves computing the effective Hamiltonian in a small-scale nonlinear cell-problem. We first discuss how to transform this problem into computing the steady-state solution of a system of conservation laws whose vector solution represents the gradient of the eigenfunction associated with the effective Hamiltonian. Theoretical arguments as well as numerical evidence are presented to emphasize the importance of enforcing the constraint that the vector solution must effectively be the gradient of a scalar function. We introduce a scheme that satisfies this constraint exactly by relying on staggered grids for the gradient components. Also discussed is the issue of selecting a time integrator to achieve fast convergence to a steady state. Validation is performed by examining convergence under grid refinement and by comparison with analytical results when available.
Communications in Mathematical Sciences, 2015
We present a new method for particle image velocimetry, a technique using successive laser images... more We present a new method for particle image velocimetry, a technique using successive laser images of particles immersed in a fluid to measure the velocity field of the fluid flow. The main idea is to recover this velocity field via the solution of the L 2 -optimal transport problem associated with each pair of successive distributions of tracers. We model the tracers by a network of Gaussian-like distributions and derive rigorous bounds on the approximation error in terms of the model's parameters. To obtain the numerical solution, we employ Newton's method, combined with an efficient spectral method, to solve the Monge-Ampère equation associated with the transport problem. We present numerical experiments based on two synthetic flow fields, a plane shear and an array of vortices. Although the theoretical results are derived for the case of a single particle in dimensions one and two, the results are valid in R d , d ≥ 1. Moreover, the numerical experiments demonstrate that these results hold for the case of multiple particles, provided the Monge-Ampère equation is solved on a fine enough grid.
Research Square (Research Square), Sep 22, 2022
The way in which moist convection interacts with large scale flows is a major contemporary resear... more The way in which moist convection interacts with large scale flows is a major contemporary research issue. Organized mesoscale systems are, in particular, important for the interactions between convection and the ambient shear. Here we present numerical simulations of mesoscale systems evolving in a background shear using the Research and Weather Forecasting (WRF) model. We are particularly interested in the long time integration, allowing the systems to repeatedly develop and die and effectively interact with the background shear. Starting with a typical African and equatorial jet-shear, the simulated solution goes through various phases or stages. First, a transient state, consisting of scattered squall -like systems that are aligned perpendicular to the background shear, develops and then evolves into a regime of multiscale mesoscale systems with large stratiform anvils. During the latter period the background wind changes substantially through the effect of both up scale and down scale convective momentum transport (CMT). At this stage, the systems become aligned parallel to the wind shear, with elongated stratiform anvils in which meso-beta scale convective cells evolve and propagate in the shear direction, relative to the stratiform anvils. These results are reminiscent of the development of shear parallel mesoscale convective systems observed for instance in the Eastern Pacific ITCZ and corroborate recent theoretical results Article Title obtained with a simple multi-cloud model. As such they have important implications for the parameterization of CMT in climate models.
Evaluation of Mean State in NCEP Climate Forecast System (Version 2) Simulation Using a Stochastic Multicloud Model Calibrated With DYNAMO RADAR Data
Earth and Space Science, Aug 1, 2021
Stochastic parameterizations are continuously providing promising simulations of unresolved atmos... more Stochastic parameterizations are continuously providing promising simulations of unresolved atmospheric processes for global climate models (GCMs). One of the stochastic multi‐cloud model (SMCM) features is to mimic the life cycle of the three most common cloud types (congestus, deep, and stratiform) in tropical convective systems. To better represent organized convection in the Climate Forecast System version 2 (CFSv2), the SMCM parameterization is adopted in CFSv2 (SMCM‐CTRL) in lieu of the pre‐existing revised simplified Arakawa–Schubert (RSAS) cumulus scheme and has shown essential improvements in different large‐scale features of tropical convection. But the sensitivity of the SMCM parameterization from the observations is yet to be ascertained. Radar data during the Dynamics of the Madden‐Julian Oscillation (DYNAMO) field campaign is used to tune the SMCM in the present manuscript. The DYNAMO radar observations have been used to calibrate the SMCM using a Bayesian inference procedure to generate key time scale parameters for the transition probabilities of the underlying Markov chains of the SMCM as implemented in CFS (hereafter SMCM‐DYNAMO). SMCM‐DYNAMO improves many aspects of the mean state climate compared to RSAS, and SMCM‐CTRL. Significant improvement is noted in the rainfall probability distribution function over the global tropics. The global distribution of different types of clouds, particularly low‐level clouds, is also improved. The convective and large‐scale rainfall simulations are investigated in detail.
A stochastic multicloud convective parameterization in the NCEP Climate Forecast System (CFSv2) : implementation and calibration
AGUFM, Dec 1, 2017
A coarse grained stochastic particle interacting system for tropical convection
AGU Fall Meeting Abstracts, Dec 1, 2012
Implementation of the SMCM in a Global Climate Model
Mathematics of planet earth, 2019
Here, we discuss the implementation of the stochastic multicloud model (SMCM) presented in the pr... more Here, we discuss the implementation of the stochastic multicloud model (SMCM) presented in the previous chapter in comprehensive climate models. First, we look at the case of an aquaplanet setting using the HOMME atmosphere only, dynamical core used in Chapter 9 and then consider, in Chapter 12, the case of a more involved state-of-the-art ocean-atmosphere coupled model, used in actual-operational climate predictions, namely, the National Centers for Environmental Predictions (NCEP)’s Climate Forecasting System (CFS). As portrayed by the results presented below important improvements in terms of the simulation of the MJO and convective coupled waves as well as the monsoon variability are achieved through the addition of stochasticity to the HOMME-MCM aquaplanet simulations presented in Chapter 9. Moreover, the implementation of the SMCM in CFS resulted in unprecedented improvements in terms of the simulation of the tropical modes of atmospheric variability on synoptic and intra-seasonal scales in coarse resolution GCMs.
Mathematics of planet earth, 2019
Springer's Mathematics of Planet Earth collection provides a variety of well-written books of a v... more Springer's Mathematics of Planet Earth collection provides a variety of well-written books of a variety of levels and styles, highlighting the fundamental role played by mathematics in a huge range of planetary contexts on a global scale. Climate, ecology, sustainability, public health, diseases and epidemics, management of resources and risk analysis are important elements. The mathematical sciences play a key role in these and many other processes relevant to Planet Earth, both as a fundamental discipline and as a key component of cross-disciplinary research. This creates the need, both in education and research, for books that are introductory to and abreast of these developments.
Theoretical and Computational Fluid Dynamics, Jul 28, 2012
The atmospheric variability in the tropics is primarily driven by convective heating. Observation... more The atmospheric variability in the tropics is primarily driven by convective heating. Observations revealed that convection in the tropics is organized into a hierarchy of multiscale convective systems ranging from the individual cloud cells to planetary scale disturbances that are nested within each other like Russian dolls. Current global climate models simulate very poorly these convectively coupled waves due in part to inadequate treatment of organized convection by the underlying cumulus parameterizations. Here, we present idealized simulations of convectively coupled equatorial waves (CCWs) using the weather research and forecast model in a horizontally limited domain consisting of a 4,500 km-wide square centered at the equator at moderate horizontal resolution of 10 km. We attempted and compared various configuration options, including switching on and off the cumulus parameterization (CP) and nesting a fine resolution 3.33 km domain, a 2,000 km-wide square, in the middle of the domain. It turns out that the results without a CP are much superior than those using a CP. While the cases without a CP resulted in a coherent eastward propagating CCW, which has many common features with observed convectively coupled Kelvin waves, the cumulus parameterization tends to destroy both the coherence of the propagating waves, even in the case with a nested domain, and reduces dramatically the variability. A primary demonstration on how such results could be used to show evidence of energy exchange, through momentum transport, between small-scale circulation due to mesoscale convection and the propagating synoptic scale wave will be reported is also presented. Convectively coupled equatorial waves • Kelvin waves • Convective momentum transport • Cumulus parameterization • WRF • Nested domain Communicated by R. Klein.
Ima Journal of Applied Mathematics, Jun 28, 2013
We present an extension of the numerical method of Loeper and Rapetti (2005, Numerical solution o... more We present an extension of the numerical method of Loeper and Rapetti (2005, Numerical solution of the Monge-Ampère equation by a Newton's algorithm. C.R. Acad. Sci. Paris, I, 319-324) for the Monge-Ampère equation to non-uniform target densities and adopt it to solve the optimal transport problem with quadratic cost. The method employs a damped Newton algorithm to solve the Monge-Ampère equation. We show that the algorithm converges for sufficiently large damping coefficients, for the case where the source and target densities are sufficiently smooth, periodic and bounded away from zero. At each Newton iteration, we solve a non-constant coefficient linear partial differential equation. To improve the efficiency of the procedure, we use an analytically preconditioned fast Fourier transform method coupled with GMRES (Strain, J. (1994) Fast spectrally-accurate solution of variable-coefficients elliptic problems. Proc. Amer. Math. Sci., 122,[843][844][845][846][847][848][849][850] to solve this equation, as opposed to a more straightforward approach based on a second-order finite-difference discretization combined with biconjugate gradient used in the original LOEPER and RAPETTI paper. Finally, we present some numerical experiments in image processing to demonstrate the efficiency of the method.
Nonlinearity, Nov 16, 2012
Clouds in the tropics can organize the circulation on planetary scales and profoundly impact long... more Clouds in the tropics can organize the circulation on planetary scales and profoundly impact long range seasonal forecasting and climate on the entire globe, yet contemporary operational computer models are often deficient in representing these phenomena. On the other hand, contemporary observations reveal remarkably complex coherent waves and vortices in the tropics interacting across a bewildering range of scales from kilometers to ten thousand kilometers. This paper reviews the interdisciplinary contributions over the last decade through the modus operandi of applied mathematics to these important scientific problems. Novel physical phenomena, new multi-scale equations, novel PDEs, and numerical algorithms are presented here with the goal of attracting mathematicians and physicists to this exciting research area.
A simple multi-cloud convective parametrization for tropical superclusters
17th Conference on Climate Variability and Change/15th Conference on Atmospheric and Oceanic Fluid Dynamics/13th Conference on Middle Atmosphere, Jun 16, 2005
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Papers by Boualem Khouider