A. Chertock, A. Kurganov, M. Lukáčová-Medvid’ová, P. Spichtinger, B. Wiebe
Abstract We develop a stochastic Galerkin method for a coupled Navier-Stokes-cloud system that models dynamics of warm clouds. Our goal is to explicitly describe the evolution of uncertainties that arise due to unknown input data, such as model parameters and initial or boundary conditions. The developed stochastic Galerkin method combines the space-time approximation obtained by a suitable finite volume method with a spectral-type approximation based on the generalized polynomial chaos expansion in the stochastic space. The resulting numerical scheme yields a second-order accurate approximation in both space and time and exponential convergence in the stochastic space. Our numerical results demonstrate the reliability and robustness of the stochastic Galerkin method. We also use the proposed method to study the behavior of clouds in certain perturbed scenarios, for examples, the ones leading to changes in macroscopic cloud pattern as a shift from hexagonal to rectangular structures.
{"title":"Stochastic Galerkin method for cloud simulation","authors":"A. Chertock, A. Kurganov, M. Lukáčová-Medvid’ová, P. Spichtinger, B. Wiebe","doi":"10.1515/mcwf-2019-0005","DOIUrl":"https://doi.org/10.1515/mcwf-2019-0005","url":null,"abstract":"Abstract We develop a stochastic Galerkin method for a coupled Navier-Stokes-cloud system that models dynamics of warm clouds. Our goal is to explicitly describe the evolution of uncertainties that arise due to unknown input data, such as model parameters and initial or boundary conditions. The developed stochastic Galerkin method combines the space-time approximation obtained by a suitable finite volume method with a spectral-type approximation based on the generalized polynomial chaos expansion in the stochastic space. The resulting numerical scheme yields a second-order accurate approximation in both space and time and exponential convergence in the stochastic space. Our numerical results demonstrate the reliability and robustness of the stochastic Galerkin method. We also use the proposed method to study the behavior of clouds in certain perturbed scenarios, for examples, the ones leading to changes in macroscopic cloud pattern as a shift from hexagonal to rectangular structures.","PeriodicalId":106200,"journal":{"name":"Mathematics of Climate and Weather Forecasting","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117118896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Juliane Rosemeier, Manuel Baumgartner, P. Spichtinger
Abstract Clouds are important components of the atmosphere. As it is usually not possible to treat them as ensembles of huge numbers of particles, parameterizations on the basis of averaged quantities (mass and/or number concentration) must be derived. Since no first-principles derivations of such averaged schemes are available today, many alternative approximating schemes of cloud processes exist. Most of these come in the form of nonlinear differential equations. It is unclear whether these different cloud schemes behave similarly under controlled local conditions, and much less so when they are embedded dynamically in a full atmospheric flow model. We use mathematical methods from the theory of dynamical systems and asymptotic analysis to compare two operational cloud schemes and one research scheme qualitatively in a simplified context in which the moist dynamics is reduced to a system of ODEs. It turns out that these schemes behave qualitatively differently on shorter time scales, whereas at least their long time behavior is similar under certain conditions. These results show that the quality of computational forecasts of moist atmospheric flows will generally depend strongly on the formulation of the cloud schemes used.
{"title":"Intercomparison of Warm-Rain Bulk Microphysics Schemes using Asymptotics","authors":"Juliane Rosemeier, Manuel Baumgartner, P. Spichtinger","doi":"10.1515/mcwf-2018-0005","DOIUrl":"https://doi.org/10.1515/mcwf-2018-0005","url":null,"abstract":"Abstract Clouds are important components of the atmosphere. As it is usually not possible to treat them as ensembles of huge numbers of particles, parameterizations on the basis of averaged quantities (mass and/or number concentration) must be derived. Since no first-principles derivations of such averaged schemes are available today, many alternative approximating schemes of cloud processes exist. Most of these come in the form of nonlinear differential equations. It is unclear whether these different cloud schemes behave similarly under controlled local conditions, and much less so when they are embedded dynamically in a full atmospheric flow model. We use mathematical methods from the theory of dynamical systems and asymptotic analysis to compare two operational cloud schemes and one research scheme qualitatively in a simplified context in which the moist dynamics is reduced to a system of ODEs. It turns out that these schemes behave qualitatively differently on shorter time scales, whereas at least their long time behavior is similar under certain conditions. These results show that the quality of computational forecasts of moist atmospheric flows will generally depend strongly on the formulation of the cloud schemes used.","PeriodicalId":106200,"journal":{"name":"Mathematics of Climate and Weather Forecasting","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132992695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nikolas Porz, M. Hanke, Manuel Baumgartner, P. Spichtinger
Abstract The representation of cloud processes inweather and climate models is crucial for their feedback on atmospheric flows. Since there is no general macroscopic theory of clouds, the parameterization of clouds in corresponding simulation software depends crucially on the underlying modeling assumptions. In this study we present a new model of intermediate complexity (a one-and-a-half moment scheme) for warm clouds, which is derived from physical principles. Our model consists of a system of differential-algebraic equations which allows for supersaturation and comprises intrinsic automated droplet activation due to a coupling of the droplet mass- and number concentrations tailored to this problem. For the numerical solution of this system we recommend a semi-implicit integration scheme, with effcient solvers for the implicit parts. The new model shows encouraging numerical results when compared with alternative cloud parameterizations, and it is well suited to investigate model uncertainties and to quantify predictability of weather events in moist atmospheric regimes.
{"title":"A model for warm clouds with implicit droplet activation, avoiding saturation adjustment","authors":"Nikolas Porz, M. Hanke, Manuel Baumgartner, P. Spichtinger","doi":"10.1515/mcwf-2018-0003","DOIUrl":"https://doi.org/10.1515/mcwf-2018-0003","url":null,"abstract":"Abstract The representation of cloud processes inweather and climate models is crucial for their feedback on atmospheric flows. Since there is no general macroscopic theory of clouds, the parameterization of clouds in corresponding simulation software depends crucially on the underlying modeling assumptions. In this study we present a new model of intermediate complexity (a one-and-a-half moment scheme) for warm clouds, which is derived from physical principles. Our model consists of a system of differential-algebraic equations which allows for supersaturation and comprises intrinsic automated droplet activation due to a coupling of the droplet mass- and number concentrations tailored to this problem. For the numerical solution of this system we recommend a semi-implicit integration scheme, with effcient solvers for the implicit parts. The new model shows encouraging numerical results when compared with alternative cloud parameterizations, and it is well suited to investigate model uncertainties and to quantify predictability of weather events in moist atmospheric regimes.","PeriodicalId":106200,"journal":{"name":"Mathematics of Climate and Weather Forecasting","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133050296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In making weather and climate predictions, the goal is often not to predict the instantaneous, local value of temperature, wind speed, or rainfall; instead, the goal is often to predict these quantities after averaging in time and/or space-for example, over one day or one week. What is the impact of spatial and/or temporal averaging on forecasting skill?Here this question is investigated using simple stochastic models that can be solved exactly analytically. While the models are idealized, their exact solutions allow clear results that are not affected by errors from numerical simulations or from random sampling. As a model of time series of oscillatory weather fluctuations, the complex Ornstein-Uhlenbeck process is used. To furthermore investigate spatial averaging, the stochastic heat equation is used as an idealized spatiotemporal model for moisture and rainfall. Space averaging and time averaging are shown to have distinctly different impacts on prediction skill. Spatial averaging leads to improved forecast skill, in line with some forms of basic intuition. Time averaging, on the other hand, is more subtle: it may either increase or decrease forecast skill. The subtle effects of time averaging are seen to arise from the relative definitions of the time averaging window and the lead time. These results should help in understanding and comparing forecasts with different temporal and spatial averaging windows.
{"title":"Spatial and Temporal Averaging Windows and Their Impact on Forecasting: Exactly Solvable Examples","authors":"Ying Li, S. Stechmann","doi":"10.1515/mcwf-2018-0002","DOIUrl":"https://doi.org/10.1515/mcwf-2018-0002","url":null,"abstract":"Abstract In making weather and climate predictions, the goal is often not to predict the instantaneous, local value of temperature, wind speed, or rainfall; instead, the goal is often to predict these quantities after averaging in time and/or space-for example, over one day or one week. What is the impact of spatial and/or temporal averaging on forecasting skill?Here this question is investigated using simple stochastic models that can be solved exactly analytically. While the models are idealized, their exact solutions allow clear results that are not affected by errors from numerical simulations or from random sampling. As a model of time series of oscillatory weather fluctuations, the complex Ornstein-Uhlenbeck process is used. To furthermore investigate spatial averaging, the stochastic heat equation is used as an idealized spatiotemporal model for moisture and rainfall. Space averaging and time averaging are shown to have distinctly different impacts on prediction skill. Spatial averaging leads to improved forecast skill, in line with some forms of basic intuition. Time averaging, on the other hand, is more subtle: it may either increase or decrease forecast skill. The subtle effects of time averaging are seen to arise from the relative definitions of the time averaging window and the lead time. These results should help in understanding and comparing forecasts with different temporal and spatial averaging windows.","PeriodicalId":106200,"journal":{"name":"Mathematics of Climate and Weather Forecasting","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133354349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The Dynamic State Index (DSI) is a scalar diagnostic field that quantifies local deviations from a steady and adiabatic wind solution and thus indicates non-stationarity aswell as diabaticity. The DSI-concept has originally been developed through the Energy-Vorticity Theory based on the full compressible flow equations without regard to the characteristic scale-dependence of many atmospheric processes. But such scaledependent information is often of importance, and particularly so in the context of precipitation modeling: Small scale convective events are often organized in storms, clusters up to “Großwetterlagen” on the synoptic scale. Therefore, a DSI index for the quasi-geostrophic model is developed using (i) the Energy-Vorticity Theory and (ii) showing that it is asymptotically consistent with the original index for the primitive equations. In the last part, using meteorological reanalysis data it is demonstrated on a case study that both indices capture systematically different scale-dependent precipitation information. A spin-off of the asymptotic analysis is a novel non-equilibrium time scale combining potential vorticity and the DSI indices.
{"title":"Scale Dependent Analytical Investigation of the Dynamic State Index Concerning the Quasi-Geostrophic Theory","authors":"A. Müller, P. Névir, R. Klein","doi":"10.1515/mcwf-2018-0001","DOIUrl":"https://doi.org/10.1515/mcwf-2018-0001","url":null,"abstract":"Abstract The Dynamic State Index (DSI) is a scalar diagnostic field that quantifies local deviations from a steady and adiabatic wind solution and thus indicates non-stationarity aswell as diabaticity. The DSI-concept has originally been developed through the Energy-Vorticity Theory based on the full compressible flow equations without regard to the characteristic scale-dependence of many atmospheric processes. But such scaledependent information is often of importance, and particularly so in the context of precipitation modeling: Small scale convective events are often organized in storms, clusters up to “Großwetterlagen” on the synoptic scale. Therefore, a DSI index for the quasi-geostrophic model is developed using (i) the Energy-Vorticity Theory and (ii) showing that it is asymptotically consistent with the original index for the primitive equations. In the last part, using meteorological reanalysis data it is demonstrated on a case study that both indices capture systematically different scale-dependent precipitation information. A spin-off of the asymptotic analysis is a novel non-equilibrium time scale combining potential vorticity and the DSI indices.","PeriodicalId":106200,"journal":{"name":"Mathematics of Climate and Weather Forecasting","volume":"420 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115248844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from Wentzel-Kramers-Brillouin theory. First, we establish that plane waves, which represent exact solutions to the inviscid Boussinesq equations, are spectrally stable with respect to their nonlinear modulation equations under the same conditions as what is known as modulational stability from weakly nonlinear theory. In contrast to Boussinesq, the pseudo-incompressible regime does fully account for the altitudinal varying background density. Second,we show for the first time that upward-traveling non-plane wave fronts solving the inviscid nonlinear modulation equations, that compare to pseudo-incompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be ill-posed as the spectra allow for arbitrarily large instability growth rates. Third, a regularization is found by including dissipative effects. The corresponding nonlinear traveling wave solutions have localized amplitude. As a consequence of the nonlinearity, envelope and linear group velocity, as given by the derivative of the frequency with respect to wavenumber, do not coincide anymore. These waves blow up unconditionally by embedded eigenvalue instabilities but the instability growth rate is bounded from above and can be computed analytically. Additionally, all three types of nonlinear modulation equations are solved numerically to further investigate and illustrate the nature of the analytic stability results.
{"title":"Spectral stability of nonlinear gravity waves in the atmosphere","authors":"M. Schlutow, E. Wahlén, P. Birken","doi":"10.1515/mcwf-2019-0002","DOIUrl":"https://doi.org/10.1515/mcwf-2019-0002","url":null,"abstract":"Abstract We apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from Wentzel-Kramers-Brillouin theory. First, we establish that plane waves, which represent exact solutions to the inviscid Boussinesq equations, are spectrally stable with respect to their nonlinear modulation equations under the same conditions as what is known as modulational stability from weakly nonlinear theory. In contrast to Boussinesq, the pseudo-incompressible regime does fully account for the altitudinal varying background density. Second,we show for the first time that upward-traveling non-plane wave fronts solving the inviscid nonlinear modulation equations, that compare to pseudo-incompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be ill-posed as the spectra allow for arbitrarily large instability growth rates. Third, a regularization is found by including dissipative effects. The corresponding nonlinear traveling wave solutions have localized amplitude. As a consequence of the nonlinearity, envelope and linear group velocity, as given by the derivative of the frequency with respect to wavenumber, do not coincide anymore. These waves blow up unconditionally by embedded eigenvalue instabilities but the instability growth rate is bounded from above and can be computed analytically. Additionally, all three types of nonlinear modulation equations are solved numerically to further investigate and illustrate the nature of the analytic stability results.","PeriodicalId":106200,"journal":{"name":"Mathematics of Climate and Weather Forecasting","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125364179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Data assimilation for multi-scale models is an important contemporary research topic. Especially the role of unresolved scales and model error in data assimilation needs to be systematically addressed. Here we examine these issues using the Ensemble Kalman filter (EnKF) with the two-level Lorenz-96 model as a conceptual prototype model of the multi-scale climate system. We use stochastic parameterization schemes to mitigate the model errors from the unresolved scales. Our results indicate that a third-order autoregressive process performs better than a first-order autoregressive process in the stochastic parameterization schemes, especially for the system with a large time-scale separation.Model errors can also arise from imprecise model parameters. We find that the accuracy of the analysis (an optimal estimate of a model state) is linearly correlated to the forcing error in the Lorenz-96 model. Furthermore, we propose novel observation strategies to deal with the fact that the dimension of the observations is much smaller than the model states. We also propose a new analog method to increase the size of the ensemble when its size is too small.
{"title":"Data Assimilation in a Multi-Scale Model","authors":"Guannan Hu, C. Franzke","doi":"10.1515/mcwf-2017-0006","DOIUrl":"https://doi.org/10.1515/mcwf-2017-0006","url":null,"abstract":"Abstract Data assimilation for multi-scale models is an important contemporary research topic. Especially the role of unresolved scales and model error in data assimilation needs to be systematically addressed. Here we examine these issues using the Ensemble Kalman filter (EnKF) with the two-level Lorenz-96 model as a conceptual prototype model of the multi-scale climate system. We use stochastic parameterization schemes to mitigate the model errors from the unresolved scales. Our results indicate that a third-order autoregressive process performs better than a first-order autoregressive process in the stochastic parameterization schemes, especially for the system with a large time-scale separation.Model errors can also arise from imprecise model parameters. We find that the accuracy of the analysis (an optimal estimate of a model state) is linearly correlated to the forcing error in the Lorenz-96 model. Furthermore, we propose novel observation strategies to deal with the fact that the dimension of the observations is much smaller than the model states. We also propose a new analog method to increase the size of the ensemble when its size is too small.","PeriodicalId":106200,"journal":{"name":"Mathematics of Climate and Weather Forecasting","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133215251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this study we investigate a covariate-based stochastic approach to parameterize unresolved turbulent processes within a standard model of the idealised, wind-driven ocean circulation. We focus on vertical instead of horizontal coarse-graining, such that we avoid the subtle difficulties of horizontal coarsegraining. The corresponding eddy forcing is uniquely defined and has a clear physical interpretation related to baroclinic instability.We propose to emulate the baroclinic eddy forcing by sampling from the conditional probability distribution functions of the eddy forcing obtained from the baroclinic reference model data. These conditional probability distribution functions are approximated here by sampling uniformly from discrete reference values. We analyze in detail the different performances of the stochastic parameterization dependent on whether the eddy forcing is conditioned on a suitable flow-dependent covariate or on a timelagged covariate or on both. The results demonstrate that our non-Gaussian, non-linear methodology is able to accurately reproduce the first four statistical moments and spatial/temporal correlations of the stream function, energetics, and enstrophy of the reference baroclinic model.
{"title":"Covariate-based stochastic parameterization of baroclinic ocean eddies","authors":"N. Verheul, J. Viebahn, D. Crommelin","doi":"10.1515/mcwf-2017-0005","DOIUrl":"https://doi.org/10.1515/mcwf-2017-0005","url":null,"abstract":"Abstract In this study we investigate a covariate-based stochastic approach to parameterize unresolved turbulent processes within a standard model of the idealised, wind-driven ocean circulation. We focus on vertical instead of horizontal coarse-graining, such that we avoid the subtle difficulties of horizontal coarsegraining. The corresponding eddy forcing is uniquely defined and has a clear physical interpretation related to baroclinic instability.We propose to emulate the baroclinic eddy forcing by sampling from the conditional probability distribution functions of the eddy forcing obtained from the baroclinic reference model data. These conditional probability distribution functions are approximated here by sampling uniformly from discrete reference values. We analyze in detail the different performances of the stochastic parameterization dependent on whether the eddy forcing is conditioned on a suitable flow-dependent covariate or on a timelagged covariate or on both. The results demonstrate that our non-Gaussian, non-linear methodology is able to accurately reproduce the first four statistical moments and spatial/temporal correlations of the stream function, energetics, and enstrophy of the reference baroclinic model.","PeriodicalId":106200,"journal":{"name":"Mathematics of Climate and Weather Forecasting","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128285297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract For dealing with dynamical instability in predictions, numerical models should be provided with accurate initial values on the attractor of the dynamical system they generate. A discrete control scheme is presented to this end for trailing variables of an evolutive system of ordinary differential equations. The Influence Sampling (IS) scheme adapts sample values of the trailing variables to input values of the determining variables in the attractor. The optimal IS scheme has affordable cost for large systems. In discrete data assimilation runs conducted with the Lorenz 1963 equations and a nonautonomous perturbation of the Lorenz equations whose dynamics shows on-off intermittency the optimal IS was compared to the straightforward insertion method and the Ensemble Kalman Filter (EnKF). With these unstable systems the optimal IS increases by one order of magnitude the maximum spacing between insertion times that the insertion method can handle and performs comparably to the EnKF when the EnKF converges. While the EnKF converges for sample sizes greater than or equal to 10, the optimal IS scheme does so fromsample size 1. This occurs because the optimal IS scheme stabilizes the individual paths of the Lorenz 1963 equations within data assimilation processes.
{"title":"Influence sampling of trailing variables of dynamical systems","authors":"P. Krause","doi":"10.1515/mcwf-2017-0003","DOIUrl":"https://doi.org/10.1515/mcwf-2017-0003","url":null,"abstract":"Abstract For dealing with dynamical instability in predictions, numerical models should be provided with accurate initial values on the attractor of the dynamical system they generate. A discrete control scheme is presented to this end for trailing variables of an evolutive system of ordinary differential equations. The Influence Sampling (IS) scheme adapts sample values of the trailing variables to input values of the determining variables in the attractor. The optimal IS scheme has affordable cost for large systems. In discrete data assimilation runs conducted with the Lorenz 1963 equations and a nonautonomous perturbation of the Lorenz equations whose dynamics shows on-off intermittency the optimal IS was compared to the straightforward insertion method and the Ensemble Kalman Filter (EnKF). With these unstable systems the optimal IS increases by one order of magnitude the maximum spacing between insertion times that the insertion method can handle and performs comparably to the EnKF when the EnKF converges. While the EnKF converges for sample sizes greater than or equal to 10, the optimal IS scheme does so fromsample size 1. This occurs because the optimal IS scheme stabilizes the individual paths of the Lorenz 1963 equations within data assimilation processes.","PeriodicalId":106200,"journal":{"name":"Mathematics of Climate and Weather Forecasting","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115993560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Diffusion ofwater vapor is the dominant growth mechanism for smallwater droplets and ice crystals in clouds. In current cloud models, Maxwell’s theory is used for describing growth of cloud particles. In this approach the local interaction between particles is neglected; the particles can only grow due to changes in environmental conditions, which are assumed as boundary conditions at infinity. This assumption is meaningful if the particles are well separated and far away from each other. However, turbulent motions might change the distances between cloud particles and thus these particles are no longer well separated leading to direct local interactions. In this study we develop a reference model for investigating the direct interaction of cloud particles in mixed-phase clouds as driven by diffusion processes. The model is numerically integrated using finite elements. Additionally, we develop a numerical method based on generalized finite elements for including moving particles and their direct interactions with respect to diffusional growth and evaporation. Several idealized simulations are carried out for investigating direct interactions of liquid droplets and ice particles in a mixed-phase cloud. The results show that local interaction between cloud particles might enhance life times of droplets and ice particles and thus lead to changes in mixed-phase cloud life time and properties.
{"title":"Local Interactions by Diffusion between Mixed-Phase Hydrometeors: Insights from Model Simulations","authors":"Manuel Baumgartner, P. Spichtinger","doi":"10.1515/mcwf-2017-0004","DOIUrl":"https://doi.org/10.1515/mcwf-2017-0004","url":null,"abstract":"Abstract Diffusion ofwater vapor is the dominant growth mechanism for smallwater droplets and ice crystals in clouds. In current cloud models, Maxwell’s theory is used for describing growth of cloud particles. In this approach the local interaction between particles is neglected; the particles can only grow due to changes in environmental conditions, which are assumed as boundary conditions at infinity. This assumption is meaningful if the particles are well separated and far away from each other. However, turbulent motions might change the distances between cloud particles and thus these particles are no longer well separated leading to direct local interactions. In this study we develop a reference model for investigating the direct interaction of cloud particles in mixed-phase clouds as driven by diffusion processes. The model is numerically integrated using finite elements. Additionally, we develop a numerical method based on generalized finite elements for including moving particles and their direct interactions with respect to diffusional growth and evaporation. Several idealized simulations are carried out for investigating direct interactions of liquid droplets and ice particles in a mixed-phase cloud. The results show that local interaction between cloud particles might enhance life times of droplets and ice particles and thus lead to changes in mixed-phase cloud life time and properties.","PeriodicalId":106200,"journal":{"name":"Mathematics of Climate and Weather Forecasting","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128847800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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