David Gurarie | Case Western Reserve University (original) (raw)

Papers by David Gurarie

Research paper thumbnail of Long range dynamics of shallow water: renormalization, modulation and long cycles

Long-range dynamics of rotating shallow water (RSW) in the low Rossby-Froude regime, Ro = Fr ll1,... more Long-range dynamics of rotating shallow water (RSW) in the low Rossby-Froude regime, Ro = Fr ll1, exhibits multiscale structure with oscillations on different scales, from fast (gravity), to slower ``eddy turnover" , and yet slower ``long weather cycles". We search for an effective theory, that would ``average" fast oscillations on each scale, to produce higher level ``slow evolution". The principal source of fast gravity waves - dominant linear dispersion, could be eliminated by passing to the amplitude equations. In nonlinear systems, however, it does not remove oscillations completely, but transplants them to nonlinear terms. We implement the Bogoliubov-Mitropolskii averaging (BM) to produce renormalized system (RN-RSW), made of the resonant quadratic part of RSW, plus order(Ro) - cubic, and O(Ro^2) - quartic corrections. Renormalized system evolves on the first slow scale. Next we conduct the detailed analysis of RN-RSW for a single 9D-Lorenz-type triad. The triad system allows to implement second renormalization (from ``first slow" to ``second slow" time), based on its 5 adiabatic invariants: two conserved integrals of QGS-oscillator (a 3D subsystem, solvable in Jacobi elliptic functions), and three wave-intensities. The adiabatic invariants evolve on the second slow scale, and describe slow modulation of the basic QGS (Jacobi) parameters: modulus, period, magnitude. The off-shot of our two-step renormalization (BM followed by ``adiabatic averaging") are ``modulated oscillations" of the vortical and gravity modes. We verify the modulation phenomena by numeric simulations of (i) complete RSW-triad, vs. (ii) renormalized system (RN-RSW), vs. (iii) its ``modulated (adiabatic) approximation". All three show good qualitative agreement in their gross features. The analysis of adiabatic system explains some long range phases of the RSW-dynamics, like nonlinear ``relaxation", and ``intensification" regimes, and pinpoints ``modulation" as the principal long-range effect of the wave-vortex interaction in conservative RSW-triads.

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Research paper thumbnail of Large scale structures andf spectra of the beta plane turbulence

Evolution of two-dimensional fluids on beta plane or rotating sphere stirred by small-scale sourc... more Evolution of two-dimensional fluids on beta plane or rotating sphere stirred by small-scale sources leads to formation of large-scale zonal jets. Rhines (1975) proposed a theory of the beta plane turbulence, and mechanism for equilibration of the inverse cascade through Rossby wave radiation. He also introduced the basic parameters of zonal structure (Rhines scale). Subsequent studies confirmed some of Rhines' predictions (relations between jet-number and Rhines scale). They also lead to speculations about zonal (jet) spectra, and scaling laws in the inverse cascade range. We conducted numerical study of the beta-plane turbulence for various forcing and dissipation regimes. It revealed the important role of frictional dissipation in equilibration of the inverse cascade. We discuss the resulting energy spectra and transfers, parameterization of zonal structure, and the mechanisms that sustain it. The key feature observed in our study is the organization of vorticity into frontal ("saw tooth") bands with strong eastward, and broad weak westward jets. We shall examine its implications for spectral laws, inverse cascade, coherence and long term evolution of turbulence.

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Research paper thumbnail of Spectra, Scaling and Zonal Jets of Beta Plane Turbulence

The Rhines theory of the beta-plane turbulence predicts the inverse cascade into zonal modes, hen... more The Rhines theory of the beta-plane turbulence predicts the inverse cascade into zonal modes, hence an array of eastward­westward jets. Subsequent studies corroborated these findings, and claimed to observe the k-5 scaling law of the zonal energy spec- tra. We study numerically the beta-plane turbulence and make some theoretical pre- dictions for different types of forcing-dissipation. We explore in particular, the role of friction in halting the inverse cascade, the scaling laws, the energy spectra, and the physical space structures. The dominant feature of the beta-plane turbulent flows forced at small scales are regular arrays of zonal jets, and sharp peaks in energy spec- tra in the low k range (provided the dissipation is reasonably low). We discuss the connections of zonal jets to the zonal energy spectra, and show based on numerical simulations that the k-5 scaling law for spectral envelope is not universal. We also demonstrate that beta-plane turbulence has memory to initial conditions.

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Research paper thumbnail of Random topography in geophysical models

ABSTRACT

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Research paper thumbnail of Turbulent transport of passive tracers and the onset of diffusivity

The paper reviews some basic results of turbulent transport of passive tracers by a random Gaussi... more The paper reviews some basic results of turbulent transport of passive tracers by a random Gaussian velocity field with short temporal correlation. Our emphasis is on the early “transport” phase of the process and the onset of diffusivity. We study the tracer mean-field and fluctuations via the reduced Fokker-Planck equation. The early (transport) phase exhibits steepening of gradients, fractalization of isocontours and exponential growth of statistical moments (fluctuations). We estimate the effective parameters of the process and ...

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Research paper thumbnail of Asymptotic inverse spectral problem for anharmonic oscillators with odd potentials

Inverse Problems, 1989

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Research paper thumbnail of 2-sphere Schrodinger operators with odd potentials

Inverse Problems, 1990

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Research paper thumbnail of Long-Range Dynamics of a Shallow Water Triad: Renormalization, Modulation, and Cyclogenesis

Journal of The Atmospheric Sciences, 2002

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Research paper thumbnail of A transitive Banach * -algebra of compact operators with no nontrivial projections

Integral Equations and Operator Theory, 1983

An example of a transitive Banach*-algebra of compact operators is constructed, which has no nont... more An example of a transitive Banach*-algebra of compact operators is constructed, which has no nontrivial projections.

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Research paper thumbnail of Representation of Compact Groups of Banach Algebras

Transactions of The American Mathematical Society, 1984

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Research paper thumbnail of The role of geometry on the equilibrium configurations of two dimensional, inviscid channel flows

The effect of aspect ratios on the equilibria of channel flows with vorticity is studied. Numeric... more The effect of aspect ratios on the equilibria of channel flows with vorticity is studied. Numerical simulations of two dimensional, inviscid channel flows with no slip boundary conditions are performed. Starting with random initial conditions, the flows undergo self-organization and attain equilibrium with special arrangements of vortices depending on the aspect ratio (ratio of channel width to streamwise period). An inviscid, semi-Lagrangian code is employed, where interpolation mimics the effects of mixing. To study the role of geometry, the aspect ratio of the channel is varied from 0.1 to 1.0 (from very narrow to relatively wide channel). In particular, three or more pairs of vortex dipoles per period appear when the aspect ratio is very small (˜0.1), but there is only one such pair for larger aspect ratio (˜0.5). More exotic configurations, such as triangular diploes and highly asymmetric dipole structures, are found for still larger values of aspect ratios. The relationship between vorticity and stream function is studied through scatter plots. Three types of relations are identified, namely, multi-sinh curves, symmetric sinh curve and highly shifted sinh curve.

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Research paper thumbnail of A stochastic model of immune-modulated malaria infection and disease in children

Mathematical Biosciences, 2007

We develop a simple three-state stochastic description of individual malaria infections that rela... more We develop a simple three-state stochastic description of individual malaria infections that relates dynamics of disease and immune status to age and previous exposure, under different intensities of transmission. We apply the resulting individual-based community models to examine the effects of drug treatment and vaccination on the frequency and severity of disease in ensembles of children. The several broad qualitative similarities between our results and field observations include potential rebound effects following intervals of drug treatment.

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Research paper thumbnail of A new approach to modelling schistosomiasis transmission based on stratified worm burden

Parasitology, 2010

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Research paper thumbnail of Equilibrium states for quasigeostrophic flows with random topography

Physica D-nonlinear Phenomena, 1996

We study the 2-D quasigeostrophic flows in the presence of complex bottom topography. The meaning... more We study the 2-D quasigeostrophic flows in the presence of complex bottom topography. The meaning of randomness of the bottom topography is discussed and the proper model is offered. In the zeroth order approximation this model yields a quasigeostrophic equation with random Gaussian profile. We study the Gaussian stationary ensembles and derive explicit relations between the correlation functions and spectral densities of solution fields (velocity/stream function) and the topography. Statistic ensembles play important role in the 2-D hydrodynamic and were subject of numerous studies (see Kraichnan and Montgomery (1980; Isichenko (1992)and references there). Our results are consistent with the earlier works (Kraichnan et al.), but the method is different. We work directly with the moment equations and utilize their symmetries, rather than the Gibbsian energy/enstrophy ensembles on the phase space. We hope the scope of our technique could be extended to other cases, particularly a nonzero base current and (turbulent) viscocity dissipation.

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Research paper thumbnail of Diffusing Passive Tracers in Random Incompressible Flows: Statistical Topography Aspects J. Stat. Physics 84(1996), 797-836

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Research paper thumbnail of Soliton Propagation in Optical Fibers with Random Parameters

We study a system of nonlinear Schrödinger equations that models propagation of optical pulses in... more We study a system of nonlinear Schrödinger equations that models propagation of optical pulses in a monomode fiber. It includes linear terms (second and third order dispersions, and attenuation) as well as nonlinear terms (cross-phase modulation,and Raman scattering). The Whitham variational (averaging) method is used to reduce the nonlinear partial differential equations to an ordinary differential system for a finite number of soliton parameters: distance between pulses, phase frequency, width and amplitude. When the random medium coefficients are turned on the reduced ODE’s becomes a stochastic system. We derive the corresponding Fokker-Planck equation and discuss its solutions in special cases. The stationary Fokker-Planck solution (equilibrium ensemble) gives the expected mean values and correlations of soliton parameters over large spatial scales and allows us to analyze the long-term effects of the random fiber on the 2-pulse system.

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Research paper thumbnail of Radial bounds for Schrödinger operators in euclidean domains

We extend study of regularity properties of elliptic operators (c.f. [2]) to second order operato... more We extend study of regularity properties of elliptic operators (c.f. [2]) to second order operators on domains bounded by finite numbers of hyperplanes. Previous results for Euclidean space and the symmetry of the domains are exploited to obtain resolvent bounds. Corollaries include semigroup generation, essential self-adjointness, and regularity of eigenfunction expansions for such operators. The present work provides basic results aimed at extending regularity information for partial differential operators (especially with singular coefficients) to a general class of operators in domains with boundary. In one dimension these results encompass a body of work in Sturm-Liouville theory on the half-line.

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Research paper thumbnail of Finite propagation speed and kernels of strictly elliptic operators

International Journal of Mathematics and Mathematical Sciences, 1985

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Research paper thumbnail of Short-Time Correlation Approximations for Diffusing Tracers in Random Velocity Fields: AF unctional Approach

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Research paper thumbnail of Age and Risk-Targeted Control of Schistosomiasis - Associated Morbidity Among Children and Adult Age Groups

The Open Tropical Medicine Journal, 2008

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Research paper thumbnail of Long range dynamics of shallow water: renormalization, modulation and long cycles

Long-range dynamics of rotating shallow water (RSW) in the low Rossby-Froude regime, Ro = Fr ll1,... more Long-range dynamics of rotating shallow water (RSW) in the low Rossby-Froude regime, Ro = Fr ll1, exhibits multiscale structure with oscillations on different scales, from fast (gravity), to slower ``eddy turnover" , and yet slower ``long weather cycles". We search for an effective theory, that would ``average" fast oscillations on each scale, to produce higher level ``slow evolution". The principal source of fast gravity waves - dominant linear dispersion, could be eliminated by passing to the amplitude equations. In nonlinear systems, however, it does not remove oscillations completely, but transplants them to nonlinear terms. We implement the Bogoliubov-Mitropolskii averaging (BM) to produce renormalized system (RN-RSW), made of the resonant quadratic part of RSW, plus order(Ro) - cubic, and O(Ro^2) - quartic corrections. Renormalized system evolves on the first slow scale. Next we conduct the detailed analysis of RN-RSW for a single 9D-Lorenz-type triad. The triad system allows to implement second renormalization (from ``first slow" to ``second slow" time), based on its 5 adiabatic invariants: two conserved integrals of QGS-oscillator (a 3D subsystem, solvable in Jacobi elliptic functions), and three wave-intensities. The adiabatic invariants evolve on the second slow scale, and describe slow modulation of the basic QGS (Jacobi) parameters: modulus, period, magnitude. The off-shot of our two-step renormalization (BM followed by ``adiabatic averaging") are ``modulated oscillations" of the vortical and gravity modes. We verify the modulation phenomena by numeric simulations of (i) complete RSW-triad, vs. (ii) renormalized system (RN-RSW), vs. (iii) its ``modulated (adiabatic) approximation". All three show good qualitative agreement in their gross features. The analysis of adiabatic system explains some long range phases of the RSW-dynamics, like nonlinear ``relaxation", and ``intensification" regimes, and pinpoints ``modulation" as the principal long-range effect of the wave-vortex interaction in conservative RSW-triads.

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Research paper thumbnail of Large scale structures andf spectra of the beta plane turbulence

Evolution of two-dimensional fluids on beta plane or rotating sphere stirred by small-scale sourc... more Evolution of two-dimensional fluids on beta plane or rotating sphere stirred by small-scale sources leads to formation of large-scale zonal jets. Rhines (1975) proposed a theory of the beta plane turbulence, and mechanism for equilibration of the inverse cascade through Rossby wave radiation. He also introduced the basic parameters of zonal structure (Rhines scale). Subsequent studies confirmed some of Rhines' predictions (relations between jet-number and Rhines scale). They also lead to speculations about zonal (jet) spectra, and scaling laws in the inverse cascade range. We conducted numerical study of the beta-plane turbulence for various forcing and dissipation regimes. It revealed the important role of frictional dissipation in equilibration of the inverse cascade. We discuss the resulting energy spectra and transfers, parameterization of zonal structure, and the mechanisms that sustain it. The key feature observed in our study is the organization of vorticity into frontal ("saw tooth") bands with strong eastward, and broad weak westward jets. We shall examine its implications for spectral laws, inverse cascade, coherence and long term evolution of turbulence.

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Research paper thumbnail of Spectra, Scaling and Zonal Jets of Beta Plane Turbulence

The Rhines theory of the beta-plane turbulence predicts the inverse cascade into zonal modes, hen... more The Rhines theory of the beta-plane turbulence predicts the inverse cascade into zonal modes, hence an array of eastward­westward jets. Subsequent studies corroborated these findings, and claimed to observe the k-5 scaling law of the zonal energy spec- tra. We study numerically the beta-plane turbulence and make some theoretical pre- dictions for different types of forcing-dissipation. We explore in particular, the role of friction in halting the inverse cascade, the scaling laws, the energy spectra, and the physical space structures. The dominant feature of the beta-plane turbulent flows forced at small scales are regular arrays of zonal jets, and sharp peaks in energy spec- tra in the low k range (provided the dissipation is reasonably low). We discuss the connections of zonal jets to the zonal energy spectra, and show based on numerical simulations that the k-5 scaling law for spectral envelope is not universal. We also demonstrate that beta-plane turbulence has memory to initial conditions.

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Research paper thumbnail of Random topography in geophysical models

ABSTRACT

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Research paper thumbnail of Turbulent transport of passive tracers and the onset of diffusivity

The paper reviews some basic results of turbulent transport of passive tracers by a random Gaussi... more The paper reviews some basic results of turbulent transport of passive tracers by a random Gaussian velocity field with short temporal correlation. Our emphasis is on the early “transport” phase of the process and the onset of diffusivity. We study the tracer mean-field and fluctuations via the reduced Fokker-Planck equation. The early (transport) phase exhibits steepening of gradients, fractalization of isocontours and exponential growth of statistical moments (fluctuations). We estimate the effective parameters of the process and ...

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Research paper thumbnail of Asymptotic inverse spectral problem for anharmonic oscillators with odd potentials

Inverse Problems, 1989

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Research paper thumbnail of 2-sphere Schrodinger operators with odd potentials

Inverse Problems, 1990

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Research paper thumbnail of Long-Range Dynamics of a Shallow Water Triad: Renormalization, Modulation, and Cyclogenesis

Journal of The Atmospheric Sciences, 2002

Bookmarks Related papers MentionsView impact

Research paper thumbnail of A transitive Banach * -algebra of compact operators with no nontrivial projections

Integral Equations and Operator Theory, 1983

An example of a transitive Banach*-algebra of compact operators is constructed, which has no nont... more An example of a transitive Banach*-algebra of compact operators is constructed, which has no nontrivial projections.

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Research paper thumbnail of Representation of Compact Groups of Banach Algebras

Transactions of The American Mathematical Society, 1984

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Research paper thumbnail of The role of geometry on the equilibrium configurations of two dimensional, inviscid channel flows

The effect of aspect ratios on the equilibria of channel flows with vorticity is studied. Numeric... more The effect of aspect ratios on the equilibria of channel flows with vorticity is studied. Numerical simulations of two dimensional, inviscid channel flows with no slip boundary conditions are performed. Starting with random initial conditions, the flows undergo self-organization and attain equilibrium with special arrangements of vortices depending on the aspect ratio (ratio of channel width to streamwise period). An inviscid, semi-Lagrangian code is employed, where interpolation mimics the effects of mixing. To study the role of geometry, the aspect ratio of the channel is varied from 0.1 to 1.0 (from very narrow to relatively wide channel). In particular, three or more pairs of vortex dipoles per period appear when the aspect ratio is very small (˜0.1), but there is only one such pair for larger aspect ratio (˜0.5). More exotic configurations, such as triangular diploes and highly asymmetric dipole structures, are found for still larger values of aspect ratios. The relationship between vorticity and stream function is studied through scatter plots. Three types of relations are identified, namely, multi-sinh curves, symmetric sinh curve and highly shifted sinh curve.

Bookmarks Related papers MentionsView impact

Research paper thumbnail of A stochastic model of immune-modulated malaria infection and disease in children

Mathematical Biosciences, 2007

We develop a simple three-state stochastic description of individual malaria infections that rela... more We develop a simple three-state stochastic description of individual malaria infections that relates dynamics of disease and immune status to age and previous exposure, under different intensities of transmission. We apply the resulting individual-based community models to examine the effects of drug treatment and vaccination on the frequency and severity of disease in ensembles of children. The several broad qualitative similarities between our results and field observations include potential rebound effects following intervals of drug treatment.

Bookmarks Related papers MentionsView impact

Research paper thumbnail of A new approach to modelling schistosomiasis transmission based on stratified worm burden

Parasitology, 2010

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Research paper thumbnail of Equilibrium states for quasigeostrophic flows with random topography

Physica D-nonlinear Phenomena, 1996

We study the 2-D quasigeostrophic flows in the presence of complex bottom topography. The meaning... more We study the 2-D quasigeostrophic flows in the presence of complex bottom topography. The meaning of randomness of the bottom topography is discussed and the proper model is offered. In the zeroth order approximation this model yields a quasigeostrophic equation with random Gaussian profile. We study the Gaussian stationary ensembles and derive explicit relations between the correlation functions and spectral densities of solution fields (velocity/stream function) and the topography. Statistic ensembles play important role in the 2-D hydrodynamic and were subject of numerous studies (see Kraichnan and Montgomery (1980; Isichenko (1992)and references there). Our results are consistent with the earlier works (Kraichnan et al.), but the method is different. We work directly with the moment equations and utilize their symmetries, rather than the Gibbsian energy/enstrophy ensembles on the phase space. We hope the scope of our technique could be extended to other cases, particularly a nonzero base current and (turbulent) viscocity dissipation.

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Research paper thumbnail of Diffusing Passive Tracers in Random Incompressible Flows: Statistical Topography Aspects J. Stat. Physics 84(1996), 797-836

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Soliton Propagation in Optical Fibers with Random Parameters

We study a system of nonlinear Schrödinger equations that models propagation of optical pulses in... more We study a system of nonlinear Schrödinger equations that models propagation of optical pulses in a monomode fiber. It includes linear terms (second and third order dispersions, and attenuation) as well as nonlinear terms (cross-phase modulation,and Raman scattering). The Whitham variational (averaging) method is used to reduce the nonlinear partial differential equations to an ordinary differential system for a finite number of soliton parameters: distance between pulses, phase frequency, width and amplitude. When the random medium coefficients are turned on the reduced ODE’s becomes a stochastic system. We derive the corresponding Fokker-Planck equation and discuss its solutions in special cases. The stationary Fokker-Planck solution (equilibrium ensemble) gives the expected mean values and correlations of soliton parameters over large spatial scales and allows us to analyze the long-term effects of the random fiber on the 2-pulse system.

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Research paper thumbnail of Radial bounds for Schrödinger operators in euclidean domains

We extend study of regularity properties of elliptic operators (c.f. [2]) to second order operato... more We extend study of regularity properties of elliptic operators (c.f. [2]) to second order operators on domains bounded by finite numbers of hyperplanes. Previous results for Euclidean space and the symmetry of the domains are exploited to obtain resolvent bounds. Corollaries include semigroup generation, essential self-adjointness, and regularity of eigenfunction expansions for such operators. The present work provides basic results aimed at extending regularity information for partial differential operators (especially with singular coefficients) to a general class of operators in domains with boundary. In one dimension these results encompass a body of work in Sturm-Liouville theory on the half-line.

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Research paper thumbnail of Finite propagation speed and kernels of strictly elliptic operators

International Journal of Mathematics and Mathematical Sciences, 1985

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Short-Time Correlation Approximations for Diffusing Tracers in Random Velocity Fields: AF unctional Approach

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Age and Risk-Targeted Control of Schistosomiasis - Associated Morbidity Among Children and Adult Age Groups

The Open Tropical Medicine Journal, 2008

Bookmarks Related papers MentionsView impact