Leonid Piterbarg - Academia.edu (original) (raw)
Uploads
Papers by Leonid Piterbarg
Applied Mathematical Modelling, 2015
a b s t r a c t A simple stochastic system is considered modeling Lagrangian motion in a vicinity... more a b s t r a c t A simple stochastic system is considered modeling Lagrangian motion in a vicinity of a hyperbolic stationary point in two dimensions. We address the dependence of the Finite Size Lyapunov Exponent (FSLE) k on the diffusivity D and the direction of the initial separation h. It is shown that there is an insignificant difference between the curves k ¼ kðhÞ for pure dynamics (D ¼ 0) and for infinitely large noise (D ¼ 1). For small D a well known boundary layer asymptotic is employed and compared with numerical results.
Journal of Physical Oceanography, 2000
An inversion technique for estimating the terms of the oceanic near-surface heat transport is ext... more An inversion technique for estimating the terms of the oceanic near-surface heat transport is extended to include the vertical heat flux at the bottom of the surface mixed layer. The mixed-layer heat balance equation uses a conventional parameterization of the vertical heat flux via entrainment into the mixed layer of interior fluid during the mixed layer deepening. A heat conservation equation defined here for the sea temperature anomalies, deviations from the annual cycle, is driven by stochastic atmospheric forcing, thereby becoming essentially a stochastic partial differential equation. This equation is reduced to the regression estimator aimed on inversion of the sea temperature time series for the unknowns: vertical entrainment velocity, horizontal velocity and diffusivity, feedback factor, and atmospheric forcing parameter. The inversion scheme also involves the velocity divergence norm. The regression estimator is applied to the time series of vertical profiles of temperature anomalies compiled from the Comprehensive Ocean-Atmosphere Data Sets and the World Ocean Atlas 1994 on a 10-day mean basis over the spatial grid of 1Њ latitude ϫ 2Њ longitude in two regions of the North Pacific: 1) near the ocean western boundary east of Japan and 2) between the Hawaiian islands and California. The inversion is implemented for winter data (December-March) from 1965 to 1990. The entrainment velocities are found to be of the order of 10 Ϫ5 m s Ϫ1 . The entrainment effect is particularly significant in the Kuroshio-Oyashio frontal zone. The model skill is also substantially enhanced in this region: the inversion yields more realistic features of the Kuroshio transport as compared with the authors' previous study, which neglected the vertical heat flux.
Physics Letters a, Mar 1, 2008
Poisson bracket for dipoles of ideal 2D hydrodynamics reduces to Zakharov-Faddeev-Gardner bracket... more Poisson bracket for dipoles of ideal 2D hydrodynamics reduces to Zakharov-Faddeev-Gardner bracket by introducing new variables: coordinates of the vorticity extrema and geodesical distances from the poles to the vorticity lines. The new Hamiltonian equations include the well-known two point-vortex system and a decoupled pair of monopoles as asymptotical cases.
Siam Journal on Applied Mathematics, Jul 27, 2006
A stochastic model is proposed for multiparticle Lagrangian motion in the upper ocean. The model ... more A stochastic model is proposed for multiparticle Lagrangian motion in the upper ocean. The model is based on hydrodynamics equations with random forcing, includes a few well interpreted and well estimated parameters, and implies a common description of the one-particle motion via a Langevin equation for the particle velocity. The dependence of the top Lyapunov exponent on the model parameters is studied as a part of a Lagrangian predictability problem. In particular, it is found that the Coriolis effect can radically improve the prediction of a Lagrangian particle position based on observations of other particles.
Theor Probab Appl Engl Tr, 1987
Fuzzy Sets and Systems, May 1, 2011
We consider a problem of estimating an unknown location parameter from several small biased sampl... more We consider a problem of estimating an unknown location parameter from several small biased samples. The biases and scale parameters of the samples are not known as well. For the case of two samples a fuzzy estimator based on a triangular membership function is introduced and studied. In particular, it is shown that its asymptotic bias is less than that of the weighted mean for the majority of key parameters in the problem. For small samples the fuzzy estimator is compared with the weighted mean and weighted median for a bunch of distributions. The main conclusion is that the fuzzy estimator performs better in most of the scenarii, however its advantage is subtle except for a few cases. Similar conclusions are obtained for the case of three information sources. The theoretical and simulation results for two samples might serve as a guidance for choosing a particular estimation method from the discussed ones based on preliminary information on relations between unknown parameters.
Theory of Probability and Mathematical Statistics
Russian Meteorology and Hydrology
Journal of Geophysical Research Atmospheres
We describe a heat anomaly transport in the upper ocean mixed layer in the Kuroshio extension reg... more We describe a heat anomaly transport in the upper ocean mixed layer in the Kuroshio extension region and the subtropical gyre of the northwest Pacific. Emphasis is on behavior in the cool season (December-March) during the Asian Winter Monsoon. The heat anomaly transport is estimated by applying an inversion technique to the stochastic partial differential equation for the heat anomaly balance of advection, diffusion, stabilizing feedback, and atmospheric forcing. The inversion consists of (1) derivation of statistical parametric model from the heat anomaly balance equation; (2) fitting the derived statistical model to the sea surface temperature (SST) anomaly covariances; and (3) calculation of the heat anomaly net advection velocity, horizontal diffusion coefficient, feedback factor and atmospheric forcing correlation from the parameters of the evaluated statistical model. The inversion was applied to the Comprehensive Ocean-Atmosphere Data Set Compressed Marine Reports SST dam, averaged at 1 ø latitude x 2' longitude boxes on a 10-day mean basis from 1965 to 1990. The estimates of the net advection velocity are consistent in magnitude and direction with the general circulation in the surface layer of the Northwest Pacific in winter. SST anomalies are transported to the west at -0.15 m s -• in the northern part of the North Equatorial Current. Between 21 ø and 29øN in the recirculating region, SST anomalies propagate westward with the mean velocity less than 0.1 m s '•. South and east of Honshu the observed pattern of the SST anomaly transport agrees broadly with the circulations of the Kuroshio current and its extension and the Oyashio current. South of Honshu, the eastward transport is about 200-300 km wide; its absolute velocity is up to 0.2 m s -•. One branch of the transport separates from the coast near the large meander path of the Kuroshio current and follows the east-southeast direction. The second separation from the coast occurs south of Hokkaido. Over the analysis domain the estimates of the diffusion coefficient are in the range of 3x 103 to 6x 103 m 2 s -•. The higher values of the diffusion coefficient confirm the enhancement of the mesoscale eddy processes near the subtropical convergence zone. The analysis supports Hasselmann's (1976) theory in which generation of midlatitude SST anomalies lasting the dominant timescale of atmospheric processes is primarily attributed to the short period stochastic weather forcing. However, the analysis indicates that the inertia of SST anomalies to their "memory" of earlier winds can not be neglected in the vicinity of the western boundary and in the tropics. 1. Introduction Since the late 1950s, sea surface temperature (SST) anomalies have been regarded as one of the key elements of climate variations [Bjerknes, 1959; Namias, 1959]. During the past three decades many publications have described the generation and evolution of SST anomalies (c.f. review of Frankignoul [1985]). They considered a heat budget of the top layer of the ocean, atmospheric forcing of the sea, feedbacks, and multiple timescale interactions in the coupled oceanatmosphere system. The concept of the uniform mixed layer [Kraus and Turner, 1967, Niiler and Kraus, 1977] played a major role in the formulation of a model for the upper sea heat anomaly balance. Statistical studies of the global SST Copyright 1995 by the American Geophysical Union. Paper number 94JC03041. 0148-0227/95/94JC-03041 $05.00 anomaly variability in terms of empirical orthogonal functions were originated by Davis [1976]. Numerical simulations of SST anomalies with ocean general circulation models were initiated by Haney et al. [1978]. At the beginning of the 1980s it was generally accepted that the SST anomaly behavior differs between the tropics and midlatitudes. The tropical SST anomalies were suggested to be generated by the large-scale ocean-atmosphere feedback processes [see Philander, 1990]. White et al. [1985] and Pazan et al. [1986] examined heat content redistribution in the tropical western Pacific during E1 Nifio-Southern Oscillation events. The heat content redistribution was shown to be associated with wind-driven baroclinic Rossby and Kelvin wave activity. Recent numerical experiments indicate that there are at least two classes of ocean-atmosphere modes in tropics. In the first class of modes, SST and surface wind variations can be in phase, but other oceanic parameters, for example, thermocline depth variations, have a phase lag that represents the inertia of the ocean and its "memory" of earlier 4845 4846 OSTROVSKII AND PITERBARG: HEAT ANOMALY TRANSPORT IN NW PACIFIC winds [Philander et al., 1992]. In numerical models that capture this class of modes the ocean response to the wind is of the "delayed oscillator" type, and the simulated Southern Oscillation can be made irregular by introducing highfrequency modes such as atmospheric "weather" forcing. The second class of ocean-atmosphere modes in the tropics is characterized by phase differences between SST and surface wind fluctuations [Lau et al., 1992].
Russian Meteorology and Hydrology
Izvestiya Atmospheric and Oceanic Physics
Izvestiya Atmospheric and Oceanic Physics
Russian Journal of Mathematical Physics
The heat diffusion and transport equation for a random velocity field is considered. An averaging... more The heat diffusion and transport equation for a random velocity field is considered. An averaging principle for the temperature field is derived. Expressions for the effective tensor of heat conductivity are obtained. In the case of short time correlations, equations for the mean temperature field, its correlation function, and equations for statistical moments of higher orders are derived. The temporal evolution of the temperature field and its gradients is investigated.
The IMA Volumes in Mathematics and its Applications, 1996
ABSTRACT
Open Journal of Applied Sciences, 2013
A method is suggested for estimating unknown velocities by combining their sparse measurements wi... more A method is suggested for estimating unknown velocities by combining their sparse measurements with observations of a tracer on a fine grid advected by the underlined velocity field. The dependence of the estimation error on a coarseness parameter and parameters of the flow in question is investigated numerically using synthetic velocity fields typical for real oceanic circulation. In an advanced version of the estimation procedure uncertainty in the transport equation forcing is modeled via a fuzzy sets approach. We also compare the method with a traditional interpolation which is in contrast to the developed procedure unable to capture the flow details.
Applied Mathematical Modelling, 2015
a b s t r a c t A simple stochastic system is considered modeling Lagrangian motion in a vicinity... more a b s t r a c t A simple stochastic system is considered modeling Lagrangian motion in a vicinity of a hyperbolic stationary point in two dimensions. We address the dependence of the Finite Size Lyapunov Exponent (FSLE) k on the diffusivity D and the direction of the initial separation h. It is shown that there is an insignificant difference between the curves k ¼ kðhÞ for pure dynamics (D ¼ 0) and for infinitely large noise (D ¼ 1). For small D a well known boundary layer asymptotic is employed and compared with numerical results.
Journal of Physical Oceanography, 2000
An inversion technique for estimating the terms of the oceanic near-surface heat transport is ext... more An inversion technique for estimating the terms of the oceanic near-surface heat transport is extended to include the vertical heat flux at the bottom of the surface mixed layer. The mixed-layer heat balance equation uses a conventional parameterization of the vertical heat flux via entrainment into the mixed layer of interior fluid during the mixed layer deepening. A heat conservation equation defined here for the sea temperature anomalies, deviations from the annual cycle, is driven by stochastic atmospheric forcing, thereby becoming essentially a stochastic partial differential equation. This equation is reduced to the regression estimator aimed on inversion of the sea temperature time series for the unknowns: vertical entrainment velocity, horizontal velocity and diffusivity, feedback factor, and atmospheric forcing parameter. The inversion scheme also involves the velocity divergence norm. The regression estimator is applied to the time series of vertical profiles of temperature anomalies compiled from the Comprehensive Ocean-Atmosphere Data Sets and the World Ocean Atlas 1994 on a 10-day mean basis over the spatial grid of 1Њ latitude ϫ 2Њ longitude in two regions of the North Pacific: 1) near the ocean western boundary east of Japan and 2) between the Hawaiian islands and California. The inversion is implemented for winter data (December-March) from 1965 to 1990. The entrainment velocities are found to be of the order of 10 Ϫ5 m s Ϫ1 . The entrainment effect is particularly significant in the Kuroshio-Oyashio frontal zone. The model skill is also substantially enhanced in this region: the inversion yields more realistic features of the Kuroshio transport as compared with the authors' previous study, which neglected the vertical heat flux.
Physics Letters a, Mar 1, 2008
Poisson bracket for dipoles of ideal 2D hydrodynamics reduces to Zakharov-Faddeev-Gardner bracket... more Poisson bracket for dipoles of ideal 2D hydrodynamics reduces to Zakharov-Faddeev-Gardner bracket by introducing new variables: coordinates of the vorticity extrema and geodesical distances from the poles to the vorticity lines. The new Hamiltonian equations include the well-known two point-vortex system and a decoupled pair of monopoles as asymptotical cases.
Siam Journal on Applied Mathematics, Jul 27, 2006
A stochastic model is proposed for multiparticle Lagrangian motion in the upper ocean. The model ... more A stochastic model is proposed for multiparticle Lagrangian motion in the upper ocean. The model is based on hydrodynamics equations with random forcing, includes a few well interpreted and well estimated parameters, and implies a common description of the one-particle motion via a Langevin equation for the particle velocity. The dependence of the top Lyapunov exponent on the model parameters is studied as a part of a Lagrangian predictability problem. In particular, it is found that the Coriolis effect can radically improve the prediction of a Lagrangian particle position based on observations of other particles.
Theor Probab Appl Engl Tr, 1987
Fuzzy Sets and Systems, May 1, 2011
We consider a problem of estimating an unknown location parameter from several small biased sampl... more We consider a problem of estimating an unknown location parameter from several small biased samples. The biases and scale parameters of the samples are not known as well. For the case of two samples a fuzzy estimator based on a triangular membership function is introduced and studied. In particular, it is shown that its asymptotic bias is less than that of the weighted mean for the majority of key parameters in the problem. For small samples the fuzzy estimator is compared with the weighted mean and weighted median for a bunch of distributions. The main conclusion is that the fuzzy estimator performs better in most of the scenarii, however its advantage is subtle except for a few cases. Similar conclusions are obtained for the case of three information sources. The theoretical and simulation results for two samples might serve as a guidance for choosing a particular estimation method from the discussed ones based on preliminary information on relations between unknown parameters.
Theory of Probability and Mathematical Statistics
Russian Meteorology and Hydrology
Journal of Geophysical Research Atmospheres
We describe a heat anomaly transport in the upper ocean mixed layer in the Kuroshio extension reg... more We describe a heat anomaly transport in the upper ocean mixed layer in the Kuroshio extension region and the subtropical gyre of the northwest Pacific. Emphasis is on behavior in the cool season (December-March) during the Asian Winter Monsoon. The heat anomaly transport is estimated by applying an inversion technique to the stochastic partial differential equation for the heat anomaly balance of advection, diffusion, stabilizing feedback, and atmospheric forcing. The inversion consists of (1) derivation of statistical parametric model from the heat anomaly balance equation; (2) fitting the derived statistical model to the sea surface temperature (SST) anomaly covariances; and (3) calculation of the heat anomaly net advection velocity, horizontal diffusion coefficient, feedback factor and atmospheric forcing correlation from the parameters of the evaluated statistical model. The inversion was applied to the Comprehensive Ocean-Atmosphere Data Set Compressed Marine Reports SST dam, averaged at 1 ø latitude x 2' longitude boxes on a 10-day mean basis from 1965 to 1990. The estimates of the net advection velocity are consistent in magnitude and direction with the general circulation in the surface layer of the Northwest Pacific in winter. SST anomalies are transported to the west at -0.15 m s -• in the northern part of the North Equatorial Current. Between 21 ø and 29øN in the recirculating region, SST anomalies propagate westward with the mean velocity less than 0.1 m s '•. South and east of Honshu the observed pattern of the SST anomaly transport agrees broadly with the circulations of the Kuroshio current and its extension and the Oyashio current. South of Honshu, the eastward transport is about 200-300 km wide; its absolute velocity is up to 0.2 m s -•. One branch of the transport separates from the coast near the large meander path of the Kuroshio current and follows the east-southeast direction. The second separation from the coast occurs south of Hokkaido. Over the analysis domain the estimates of the diffusion coefficient are in the range of 3x 103 to 6x 103 m 2 s -•. The higher values of the diffusion coefficient confirm the enhancement of the mesoscale eddy processes near the subtropical convergence zone. The analysis supports Hasselmann's (1976) theory in which generation of midlatitude SST anomalies lasting the dominant timescale of atmospheric processes is primarily attributed to the short period stochastic weather forcing. However, the analysis indicates that the inertia of SST anomalies to their "memory" of earlier winds can not be neglected in the vicinity of the western boundary and in the tropics. 1. Introduction Since the late 1950s, sea surface temperature (SST) anomalies have been regarded as one of the key elements of climate variations [Bjerknes, 1959; Namias, 1959]. During the past three decades many publications have described the generation and evolution of SST anomalies (c.f. review of Frankignoul [1985]). They considered a heat budget of the top layer of the ocean, atmospheric forcing of the sea, feedbacks, and multiple timescale interactions in the coupled oceanatmosphere system. The concept of the uniform mixed layer [Kraus and Turner, 1967, Niiler and Kraus, 1977] played a major role in the formulation of a model for the upper sea heat anomaly balance. Statistical studies of the global SST Copyright 1995 by the American Geophysical Union. Paper number 94JC03041. 0148-0227/95/94JC-03041 $05.00 anomaly variability in terms of empirical orthogonal functions were originated by Davis [1976]. Numerical simulations of SST anomalies with ocean general circulation models were initiated by Haney et al. [1978]. At the beginning of the 1980s it was generally accepted that the SST anomaly behavior differs between the tropics and midlatitudes. The tropical SST anomalies were suggested to be generated by the large-scale ocean-atmosphere feedback processes [see Philander, 1990]. White et al. [1985] and Pazan et al. [1986] examined heat content redistribution in the tropical western Pacific during E1 Nifio-Southern Oscillation events. The heat content redistribution was shown to be associated with wind-driven baroclinic Rossby and Kelvin wave activity. Recent numerical experiments indicate that there are at least two classes of ocean-atmosphere modes in tropics. In the first class of modes, SST and surface wind variations can be in phase, but other oceanic parameters, for example, thermocline depth variations, have a phase lag that represents the inertia of the ocean and its "memory" of earlier 4845 4846 OSTROVSKII AND PITERBARG: HEAT ANOMALY TRANSPORT IN NW PACIFIC winds [Philander et al., 1992]. In numerical models that capture this class of modes the ocean response to the wind is of the "delayed oscillator" type, and the simulated Southern Oscillation can be made irregular by introducing highfrequency modes such as atmospheric "weather" forcing. The second class of ocean-atmosphere modes in the tropics is characterized by phase differences between SST and surface wind fluctuations [Lau et al., 1992].
Russian Meteorology and Hydrology
Izvestiya Atmospheric and Oceanic Physics
Izvestiya Atmospheric and Oceanic Physics
Russian Journal of Mathematical Physics
The heat diffusion and transport equation for a random velocity field is considered. An averaging... more The heat diffusion and transport equation for a random velocity field is considered. An averaging principle for the temperature field is derived. Expressions for the effective tensor of heat conductivity are obtained. In the case of short time correlations, equations for the mean temperature field, its correlation function, and equations for statistical moments of higher orders are derived. The temporal evolution of the temperature field and its gradients is investigated.
The IMA Volumes in Mathematics and its Applications, 1996
ABSTRACT
Open Journal of Applied Sciences, 2013
A method is suggested for estimating unknown velocities by combining their sparse measurements wi... more A method is suggested for estimating unknown velocities by combining their sparse measurements with observations of a tracer on a fine grid advected by the underlined velocity field. The dependence of the estimation error on a coarseness parameter and parameters of the flow in question is investigated numerically using synthetic velocity fields typical for real oceanic circulation. In an advanced version of the estimation procedure uncertainty in the transport equation forcing is modeled via a fuzzy sets approach. We also compare the method with a traditional interpolation which is in contrast to the developed procedure unable to capture the flow details.