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- Author or Editor: Rex J. Fleming x
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Abstract
It is suggested that when one more properly views meteorological prediction as an initial value problem couched in random or stochastic terms, there is an additional way for the wind and the mass fields to adjust to a state of balance. A simplified spectral model is used to isolate the gravity wave energy and to show how it becomes dispersed through phase space. Calculations show that this non-Gaussian adjustment cannot be described without the inclusion of third moments. Monte Carlo and stochastic dynamic calculations show that the entire ensemble of points which describe a system in phase space tend to a state of balance which is consistent with a solution to the stochastic balance equation. Based upon the results, the implications of stochastic models in dealing with gravity waves in meteorological predictions is assessed.
Abstract
It is suggested that when one more properly views meteorological prediction as an initial value problem couched in random or stochastic terms, there is an additional way for the wind and the mass fields to adjust to a state of balance. A simplified spectral model is used to isolate the gravity wave energy and to show how it becomes dispersed through phase space. Calculations show that this non-Gaussian adjustment cannot be described without the inclusion of third moments. Monte Carlo and stochastic dynamic calculations show that the entire ensemble of points which describe a system in phase space tend to a state of balance which is consistent with a solution to the stochastic balance equation. Based upon the results, the implications of stochastic models in dealing with gravity waves in meteorological predictions is assessed.
Abstract
A widely used relative humidity (RH) sensor in atmospheric science is based upon a capacitive device that outputs voltage as a linear function of RH and then is corrected by an empirically determined polynomial expression, which is only a function of temperature. Based upon results of a dry bias in these measurements and upon a recent physical–mathematical model of how these devices may work, a new correction formula is derived, which is a function of temperature and RH. The new formula reproduces the old one near 40% RH and corrects the dry bias for RH values less than 40%.
Abstract
A widely used relative humidity (RH) sensor in atmospheric science is based upon a capacitive device that outputs voltage as a linear function of RH and then is corrected by an empirically determined polynomial expression, which is only a function of temperature. Based upon results of a dry bias in these measurements and upon a recent physical–mathematical model of how these devices may work, a new correction formula is derived, which is a function of temperature and RH. The new formula reproduces the old one near 40% RH and corrects the dry bias for RH values less than 40%.
ON STOCHASTIC DYNAMIC PREDICTION
I. The Energetics of Uncertainty and the Question of Closure
Abstract
In this, the first part of a two-part study, we examine the rationale of the stochastic dynamic approach to numerical weather prediction. Advantages of the stochastic dynamic method are discussed along with problems associated with the method. This method deals with the initial uncertainty by considering an infinite ensemble of initial states in phase space, relative frequencies within the ensemble being proportional to probability densities. The evolution of this ensemble in time, given by the stochastic dynamic equation set, is based upon the original deterministic hydrodynamic equation set. One may consider the latter set as a subset of the former. Insight into the nature of these equations is obtained by deriving the energy transformations associated with them. A simple baroclinic model is used to isolate the energy concepts and relations. The energetics yield qualitative and quantitative information on the nature of the growth of uncertainty. It is found that the baroclinic instability mechanism is responsible for most of the error growth as would be expected.
Previous predictability studies have considered that the simulation of the forces governing the atmosphere has been perfect. The effects of imperfect forcing can be viewed with the stochastic dynamic equations by adding another dimension to phase space for each parameter considered to be uncertain. The effect of the inclusion of this imperfect forcing is shown by the new energetic relations that result, and by numerical calculation of the changes in the growth of uncertainty.
The stochastic dynamic equations are faced with the same mathematical problem of “closure” found in analytical treatments of homogeneous isotropic turbulence; that is, an approximation concerning higher order moments must be made to close the system. A number of closure schemes are studied and it is found that the third moments, which are individually small, should nevertheless be retained. It is shown in the equations and verified by numerical calculations that the third moments do not affect energy conservation but affect energy conversion between uncertain components, with the eventual result of altering the forecast of the mean. An eddy-damped third moment scheme is found to give extremely accurate results when compared to Monte Carlo calculations.
Abstract
In this, the first part of a two-part study, we examine the rationale of the stochastic dynamic approach to numerical weather prediction. Advantages of the stochastic dynamic method are discussed along with problems associated with the method. This method deals with the initial uncertainty by considering an infinite ensemble of initial states in phase space, relative frequencies within the ensemble being proportional to probability densities. The evolution of this ensemble in time, given by the stochastic dynamic equation set, is based upon the original deterministic hydrodynamic equation set. One may consider the latter set as a subset of the former. Insight into the nature of these equations is obtained by deriving the energy transformations associated with them. A simple baroclinic model is used to isolate the energy concepts and relations. The energetics yield qualitative and quantitative information on the nature of the growth of uncertainty. It is found that the baroclinic instability mechanism is responsible for most of the error growth as would be expected.
Previous predictability studies have considered that the simulation of the forces governing the atmosphere has been perfect. The effects of imperfect forcing can be viewed with the stochastic dynamic equations by adding another dimension to phase space for each parameter considered to be uncertain. The effect of the inclusion of this imperfect forcing is shown by the new energetic relations that result, and by numerical calculation of the changes in the growth of uncertainty.
The stochastic dynamic equations are faced with the same mathematical problem of “closure” found in analytical treatments of homogeneous isotropic turbulence; that is, an approximation concerning higher order moments must be made to close the system. A number of closure schemes are studied and it is found that the third moments, which are individually small, should nevertheless be retained. It is shown in the equations and verified by numerical calculations that the third moments do not affect energy conservation but affect energy conversion between uncertain components, with the eventual result of altering the forecast of the mean. An eddy-damped third moment scheme is found to give extremely accurate results when compared to Monte Carlo calculations.
ON STOCHASTIC DYNAMIC PREDICTION
II. Predictability and Utility
Abstract
The stochastic dynamic equations, as investigated in part I of this two-part study, can be applied to any time-dependent set of differential equations which are, at most, nonlinear quadratic. In this study, they are used to explore various aspects of the question of atmospheric predictability.
The growth of uncertainty due to ill-defined initial conditions in the nonlinear advection field is viewed by considering a simple barotropic model. A wave number is defined to be “unpredictable” when the “uncertain” energy associated with that wave becomes as large as the “certain” energy associated with it. The predictability of wave number 12 is used as a reference point and as an arbitrary minimum requirement for useful synoptic forecasts. It is found that, based upon the average root-mean-square vector error in the wind field today, such a wave number has a predictability value of about 1.5 days. If this error could be reduced by a factor of 4 (i.e., down to 1 m/s), this value would be approximately 5 days. Using a stochastic barotropic model with 2,015 degrees of freedom, it is found that any initial energy spectra for the certain or uncertain eddy kinetic energy will give essentially the same predictability values. This is because the complete nonlinearity is accounted for in the stochastic dynamic equation set and the dynamics of the two-dimensional fluid tend to drive any initial spectrum into approximately a – 3 power law in some averaged sense—as expected from theory.
It is shown that the rather pessimistic predictability values, based solely upon error growth due to uncertain advection and instability processes, are considerably lengthened (at least in the largest scales) when additional forcing and dissipation terms are included in the mathematical models. However, such additional forces can never be simulated perfectly and the qualitative effect of these imperfections is shown by calculations with a simple baroclinic model having heating and friction. Based upon arguments presented, the author speculates that in 10 yr the projected uncertainties in the physics and the uncertainties arising from the computational wave number cutoff will still restrict the predictability of wave number 12 to within 5–7 days. It is shown how the eventual application of the stochastic dynamic equations to more complicated models can replace such speculation with more concrete evidence.
The globally averaged value of predictability considered above is very general and it is shown how the utility of the stochastic dynamic set can provide more meaningful information to the user. Only one aspect of this utility is shown (the growth of the phase error of a wave with time), but the stochastic set of equations gives the “believability” of each variable at every point in the space-time domain.
Abstract
The stochastic dynamic equations, as investigated in part I of this two-part study, can be applied to any time-dependent set of differential equations which are, at most, nonlinear quadratic. In this study, they are used to explore various aspects of the question of atmospheric predictability.
The growth of uncertainty due to ill-defined initial conditions in the nonlinear advection field is viewed by considering a simple barotropic model. A wave number is defined to be “unpredictable” when the “uncertain” energy associated with that wave becomes as large as the “certain” energy associated with it. The predictability of wave number 12 is used as a reference point and as an arbitrary minimum requirement for useful synoptic forecasts. It is found that, based upon the average root-mean-square vector error in the wind field today, such a wave number has a predictability value of about 1.5 days. If this error could be reduced by a factor of 4 (i.e., down to 1 m/s), this value would be approximately 5 days. Using a stochastic barotropic model with 2,015 degrees of freedom, it is found that any initial energy spectra for the certain or uncertain eddy kinetic energy will give essentially the same predictability values. This is because the complete nonlinearity is accounted for in the stochastic dynamic equation set and the dynamics of the two-dimensional fluid tend to drive any initial spectrum into approximately a – 3 power law in some averaged sense—as expected from theory.
It is shown that the rather pessimistic predictability values, based solely upon error growth due to uncertain advection and instability processes, are considerably lengthened (at least in the largest scales) when additional forcing and dissipation terms are included in the mathematical models. However, such additional forces can never be simulated perfectly and the qualitative effect of these imperfections is shown by calculations with a simple baroclinic model having heating and friction. Based upon arguments presented, the author speculates that in 10 yr the projected uncertainties in the physics and the uncertainties arising from the computational wave number cutoff will still restrict the predictability of wave number 12 to within 5–7 days. It is shown how the eventual application of the stochastic dynamic equations to more complicated models can replace such speculation with more concrete evidence.
The globally averaged value of predictability considered above is very general and it is shown how the utility of the stochastic dynamic set can provide more meaningful information to the user. Only one aspect of this utility is shown (the growth of the phase error of a wave with time), but the stochastic set of equations gives the “believability” of each variable at every point in the space-time domain.
Abstract
The practical limit of atmospheric predictability is a function of the uncertainties in the initial conditions, the uncertainties in the parameterization of the external forces, and the uncertainties involved with discretizing the continuous atmospheric variables. The stochastic dynamic method of treating time-dependent differential equations, when applied to the relevant meteorological equations, is capable of treating the above sources of error and the atmospheric predictability problem in an explicit manner. New calculations emphasizing the effects of random forcing on predictability are presented. While it has been convenient to express predictability as a function of wavenumber in rather general energy terms, it is shown how this expression of predictability is related to a new definition based upon the degree of belief one has in the phase-position of atmospheric waves. Finally, an attempt is made to isolate the uncertainties associated with the computational cutoff in physical space by the incorporation of an artificial viscosity which is considered as a nonlinear stochastic external force.
Abstract
The practical limit of atmospheric predictability is a function of the uncertainties in the initial conditions, the uncertainties in the parameterization of the external forces, and the uncertainties involved with discretizing the continuous atmospheric variables. The stochastic dynamic method of treating time-dependent differential equations, when applied to the relevant meteorological equations, is capable of treating the above sources of error and the atmospheric predictability problem in an explicit manner. New calculations emphasizing the effects of random forcing on predictability are presented. While it has been convenient to express predictability as a function of wavenumber in rather general energy terms, it is shown how this expression of predictability is related to a new definition based upon the degree of belief one has in the phase-position of atmospheric waves. Finally, an attempt is made to isolate the uncertainties associated with the computational cutoff in physical space by the incorporation of an artificial viscosity which is considered as a nonlinear stochastic external force.
Abstract
A low-order general circulation model contains all the elements of baroclinic instability, including differential heating to drive the mean zonal shear flow against dissipation. Simulations exhibit vacillation ending in fixed-point solutions and chaotic solutions with significant amplifications of the baroclinic cycles compared to those of vacillation. The chaos sensitivity to initial conditions, covering a broad landscape of initial values, demands analysis of why the chaos occurs and its impact on subsequent storm intensity.
Three attractors found in the dynamic system are important. One attractor is the stable fixed-point solution—the ultimate destination of a vacillation trajectory. A second attractor represents an unstable zonal solution. Though this dynamic system is bound, some trajectories get extremely close to the unstable, but strongly attracting, zonal solution. It is while traversing such a trajectory that the buildup of available potential energy is such to allow subsequent explosive baroclinic instability to develop.
The roots of the characteristic matrix of the dynamic system are examined at every time step. A single critical value of one of the roots is found to be the cause of the chaos for a given value of the differential heating H. The system becomes more stable with increased values of H; vacillation is stronger and more prominent, and the critical value for chaos increases with H. When chaos does occur, it is stronger and more explosive.
Abstract
A low-order general circulation model contains all the elements of baroclinic instability, including differential heating to drive the mean zonal shear flow against dissipation. Simulations exhibit vacillation ending in fixed-point solutions and chaotic solutions with significant amplifications of the baroclinic cycles compared to those of vacillation. The chaos sensitivity to initial conditions, covering a broad landscape of initial values, demands analysis of why the chaos occurs and its impact on subsequent storm intensity.
Three attractors found in the dynamic system are important. One attractor is the stable fixed-point solution—the ultimate destination of a vacillation trajectory. A second attractor represents an unstable zonal solution. Though this dynamic system is bound, some trajectories get extremely close to the unstable, but strongly attracting, zonal solution. It is while traversing such a trajectory that the buildup of available potential energy is such to allow subsequent explosive baroclinic instability to develop.
The roots of the characteristic matrix of the dynamic system are examined at every time step. A single critical value of one of the roots is found to be the cause of the chaos for a given value of the differential heating H. The system becomes more stable with increased values of H; vacillation is stronger and more prominent, and the critical value for chaos increases with H. When chaos does occur, it is stronger and more explosive.
The use of commercial aircraft for obtaining weather and climate change related information is beginning to accelerate at a rapid pace. A brief history of the use of commercial aircraft for these purposes is provided along with a discussion of the factors that are responsible for the current growth. A major federal program to provide profiles of winds, temperatures, and water vapor is described, along with a description of the new formats and information that will be available to the scientific community. Further details on the water vapor measurements, those expected this year and potential future upgrades, are provided. The advanced technologies that are now available on the aircraft, new advances in in situ and remote sensing, and an entrepreneurial spirit of some package carriers will combine to provide new kinds of measurements via commercial aircraft. A brief review of these factors and a vision of future environmental measurements is provided.
The use of commercial aircraft for obtaining weather and climate change related information is beginning to accelerate at a rapid pace. A brief history of the use of commercial aircraft for these purposes is provided along with a discussion of the factors that are responsible for the current growth. A major federal program to provide profiles of winds, temperatures, and water vapor is described, along with a description of the new formats and information that will be available to the scientific community. Further details on the water vapor measurements, those expected this year and potential future upgrades, are provided. The advanced technologies that are now available on the aircraft, new advances in in situ and remote sensing, and an entrepreneurial spirit of some package carriers will combine to provide new kinds of measurements via commercial aircraft. A brief review of these factors and a vision of future environmental measurements is provided.
Abstract
Stochastic dynamic prediction provides information on the variances and covariances of the predicted meteorological fields as well as the expected values of the fields themselves. It is shown that this new information can be depicted in a variety of graphical formats that illustrate various aspects of the certainty and uncertainty of the predictions and demonstrate the value of specific information on uncertainty.
Abstract
Stochastic dynamic prediction provides information on the variances and covariances of the predicted meteorological fields as well as the expected values of the fields themselves. It is shown that this new information can be depicted in a variety of graphical formats that illustrate various aspects of the certainty and uncertainty of the predictions and demonstrate the value of specific information on uncertainty.
