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Abstract
The analytic solutions in time to the highly truncated (low-order) spectral vorticity equation, involving the nonlinear interaction of one planetary wave with an arbitrary zonal flow, have been investigated for a wide variety of initial conditions which span the range of atmospheric observations. These conditions include both a characteristic mean wintertime zonal jet, a simulated double jet, all planetary waves from wavenumbers 1–12, and various wave configurations for wavenumber 3. The results show wide variability in solutions from configurations which are almost independent in time to those which describe highly elliptic oscillations. There is no systematic dependence of nonlinearity on total energy amplitude of the system nor do small initial perturbations necessarily imply quasi-linearity. The solutions described for this simple model exemplify the extreme complexity of large-scale atmospheric turbulence.
Abstract
The analytic solutions in time to the highly truncated (low-order) spectral vorticity equation, involving the nonlinear interaction of one planetary wave with an arbitrary zonal flow, have been investigated for a wide variety of initial conditions which span the range of atmospheric observations. These conditions include both a characteristic mean wintertime zonal jet, a simulated double jet, all planetary waves from wavenumbers 1–12, and various wave configurations for wavenumber 3. The results show wide variability in solutions from configurations which are almost independent in time to those which describe highly elliptic oscillations. There is no systematic dependence of nonlinearity on total energy amplitude of the system nor do small initial perturbations necessarily imply quasi-linearity. The solutions described for this simple model exemplify the extreme complexity of large-scale atmospheric turbulence.
Abstract
The vertical levels used in atmospheric models are selected for a variety of reasons, but the selection process has not been systematized. The study presented herein represents an attempt to do so. An atmospheric model is linearized about a state of rest and vertical modes are determined for the vertical structure equation-an ordinary differential equation-by solving a difference form of that equation. Since the solutions of the differential equation do not correspond to the solutions of the difference equation, the distribution of points used for the difference equation (the vertical levels) is adjusted until both sets of solutions coalesce. This distribution is considered the optimum set of model levels.
To test the impact of such a distribution on a numerical integration, the NCAR CCMOB is integrated using both its standard levels and the levels determined above. Comparisons of the integrations show that the solutions separate as time evolves, despite the fact that the initial conditions for the separate integrations are as similar as possible. The results suggest that care in selecting vertical levels is essential to successful integration and that perhaps a start on finding a systematic way of choosing levels has been made.
Abstract
The vertical levels used in atmospheric models are selected for a variety of reasons, but the selection process has not been systematized. The study presented herein represents an attempt to do so. An atmospheric model is linearized about a state of rest and vertical modes are determined for the vertical structure equation-an ordinary differential equation-by solving a difference form of that equation. Since the solutions of the differential equation do not correspond to the solutions of the difference equation, the distribution of points used for the difference equation (the vertical levels) is adjusted until both sets of solutions coalesce. This distribution is considered the optimum set of model levels.
To test the impact of such a distribution on a numerical integration, the NCAR CCMOB is integrated using both its standard levels and the levels determined above. Comparisons of the integrations show that the solutions separate as time evolves, despite the fact that the initial conditions for the separate integrations are as similar as possible. The results suggest that care in selecting vertical levels is essential to successful integration and that perhaps a start on finding a systematic way of choosing levels has been made.
Abstract
A fine-mesh, limited-area, finite-difference model has been developed both to predict precipitation over a limited geographic region and to be utilized in experiments for precipitation modification. The model is primitive with 15 vertical levels, shows many features in common with current models of its type, but lacks resolution of the boundary layer. Lateral boundary conditions are specified when needed from a data set which also provides initial conditions and comparisons for the forecasts. Utilizing one integration as a control, a number of experiments were performed under model modifications and compared to the control. In all cases, modification did not substantially alter the flow field over a 24 h period, but precipitation forecasts were altered. Regions of marginal precipitation showed almost no precipitation in an experiment where lack of freezing nuclei was incorporated, whereas the experiment based on cloud seeding showed significant increase in total precipitation. The addition of carbon black heating to the model did not show substantial changes in precipitation. Modified initial conditions based on poor (coarse grid) resolution had a significant effect on precipitation prediction.
Abstract
A fine-mesh, limited-area, finite-difference model has been developed both to predict precipitation over a limited geographic region and to be utilized in experiments for precipitation modification. The model is primitive with 15 vertical levels, shows many features in common with current models of its type, but lacks resolution of the boundary layer. Lateral boundary conditions are specified when needed from a data set which also provides initial conditions and comparisons for the forecasts. Utilizing one integration as a control, a number of experiments were performed under model modifications and compared to the control. In all cases, modification did not substantially alter the flow field over a 24 h period, but precipitation forecasts were altered. Regions of marginal precipitation showed almost no precipitation in an experiment where lack of freezing nuclei was incorporated, whereas the experiment based on cloud seeding showed significant increase in total precipitation. The addition of carbon black heating to the model did not show substantial changes in precipitation. Modified initial conditions based on poor (coarse grid) resolution had a significant effect on precipitation prediction.
Abstract
Although atmospheric prediction models appear to yield results similar to observation, both their detailed predictive capability and their time-averaged forecasts depend on space truncation. Such dependence may be readily studied with a spectral model because of the ease of modifying truncation. A simple, two-level, quasi-geostrophic, forced general circulation model was represented in spectral form and nine cases of different truncation were integrated for the same forcing, starting with initial conditions generated from a state of rest. The truncations ranged from six to sixteen meridional waves, from five to ten degrees of freedom with latitude, and the models were integrated for about 60 days with finite-amplitude nonlinearity. Considering the kinetic energy in the vertical mean flow, and separating this energy into zonal and eddy, the results show that the general circulation may be predicted with as few as twelve planetary waves and eight latitudinal degrees of freedom, whereas detailed prediction for a period of 15–20 days requires at least sixteen planetary waves and eight to ten latitudinal degrees of freedom. The broad variation in solutions for different truncations observed in this study implies that care must be taken in selecting space truncation for any physical model chosen for integration.
Abstract
Although atmospheric prediction models appear to yield results similar to observation, both their detailed predictive capability and their time-averaged forecasts depend on space truncation. Such dependence may be readily studied with a spectral model because of the ease of modifying truncation. A simple, two-level, quasi-geostrophic, forced general circulation model was represented in spectral form and nine cases of different truncation were integrated for the same forcing, starting with initial conditions generated from a state of rest. The truncations ranged from six to sixteen meridional waves, from five to ten degrees of freedom with latitude, and the models were integrated for about 60 days with finite-amplitude nonlinearity. Considering the kinetic energy in the vertical mean flow, and separating this energy into zonal and eddy, the results show that the general circulation may be predicted with as few as twelve planetary waves and eight latitudinal degrees of freedom, whereas detailed prediction for a period of 15–20 days requires at least sixteen planetary waves and eight to ten latitudinal degrees of freedom. The broad variation in solutions for different truncations observed in this study implies that care must be taken in selecting space truncation for any physical model chosen for integration.
Abstract
The impact of gravity modes in atmospheric model predictions is assessed quantitatively by comparing integrations with a normal mode initialized primitive equation model and its corresponding pseudogeostrophic form to document some generally accepted presumptions. Analysis with a linear system yields horizontally scale-dependent differences in the Rossby frequencies derived from the two conditions (initialized and geostrophic) as well as differences in the initial divergence required. Nonlinear calculations indicate that initializing gravity modes in the primitive system does not affect the forecast of the Rossby modes. However, integration with the initialized primitive equation model shows differences in both the Rossby and gravity modes after five days when compared to the corresponding pseudogeostrophic model results, differences which depend both on horizontal scale and vertical mode. Comparison results become more similar if the geostrophic model is converted to balanced form. As the integration time is extended, the modal amplitudes predicted by the initialized primitive equation and geostrophic model rapidly become different. Yet the statistics of energy in the shorter scales for both these experiments during days 20–30 of the integrations are remarkably similar. Inclusion of forcing in the model showed changes in details of the response but not its fundamental character.
Abstract
The impact of gravity modes in atmospheric model predictions is assessed quantitatively by comparing integrations with a normal mode initialized primitive equation model and its corresponding pseudogeostrophic form to document some generally accepted presumptions. Analysis with a linear system yields horizontally scale-dependent differences in the Rossby frequencies derived from the two conditions (initialized and geostrophic) as well as differences in the initial divergence required. Nonlinear calculations indicate that initializing gravity modes in the primitive system does not affect the forecast of the Rossby modes. However, integration with the initialized primitive equation model shows differences in both the Rossby and gravity modes after five days when compared to the corresponding pseudogeostrophic model results, differences which depend both on horizontal scale and vertical mode. Comparison results become more similar if the geostrophic model is converted to balanced form. As the integration time is extended, the modal amplitudes predicted by the initialized primitive equation and geostrophic model rapidly become different. Yet the statistics of energy in the shorter scales for both these experiments during days 20–30 of the integrations are remarkably similar. Inclusion of forcing in the model showed changes in details of the response but not its fundamental character.
Abstract
A general numerical integration formula is presented that generates many of the commonly used one-dimensional finite-difference schemes. A number of these schemes are tested on a simple wave equation; three implicit and three explicit are chosen for further analysis with a nonlinear set of equations with known solutions. A seventh method of the implicit type not requiring iteration is also tested. A transformation is developed that allows the removal of linear terms from the nonlinear equations, thereby avoiding truncation of the linear terms. The results of the analysis show that energy components may have large errors when the total energy shows essentially none, and phase errors may be quite serious without indication from linear analysis. By treating the uncoupled linear terms exactly (no truncation), significant improvement in the numerical solutions ensues. The multilevel implicit schemes give superior results and are to be recommended if computing time is not a criterion. Great care must be taken in interpreting the linear stability criterion. The critical truncation increment should be considerably reduced to avoid significant truncation errors, especially for long time integrations.
Abstract
A general numerical integration formula is presented that generates many of the commonly used one-dimensional finite-difference schemes. A number of these schemes are tested on a simple wave equation; three implicit and three explicit are chosen for further analysis with a nonlinear set of equations with known solutions. A seventh method of the implicit type not requiring iteration is also tested. A transformation is developed that allows the removal of linear terms from the nonlinear equations, thereby avoiding truncation of the linear terms. The results of the analysis show that energy components may have large errors when the total energy shows essentially none, and phase errors may be quite serious without indication from linear analysis. By treating the uncoupled linear terms exactly (no truncation), significant improvement in the numerical solutions ensues. The multilevel implicit schemes give superior results and are to be recommended if computing time is not a criterion. Great care must be taken in interpreting the linear stability criterion. The critical truncation increment should be considerably reduced to avoid significant truncation errors, especially for long time integrations.
Abstract
A procedure is outlined which adjusts the initial conditions for any prediction model of a planetary fluid such that no motions of the fluid will evolve with high-frequency, gravity-type time scales despite the model's nonlinearity. Any model which can be characterized by a reasonably small Rossby number may be balanced by the method. The technique requires the determination of the normal modes of the linear part of the equations to be integrated—finite difference or spectral—and proceeds by an expansion technique to build up higher order, nonlinear adjustments to the initial state.
Abstract
A procedure is outlined which adjusts the initial conditions for any prediction model of a planetary fluid such that no motions of the fluid will evolve with high-frequency, gravity-type time scales despite the model's nonlinearity. Any model which can be characterized by a reasonably small Rossby number may be balanced by the method. The technique requires the determination of the normal modes of the linear part of the equations to be integrated—finite difference or spectral—and proceeds by an expansion technique to build up higher order, nonlinear adjustments to the initial state.
Abstract
Growth of incipient precipitation particles by collision and coalescence with cloud droplets is one of the primary mechanisms of natural rain. Comparison of previous research shows wide divergence between various theoretical and laboratory values of collision efficiency and coalescence efficiency. In an effort to obtain additional laboratory measurements of droplet coalescence, high-speed photographs were taken of colliding droplets at the breakup point in a Rayleigh jet. With 700-micron diam droplets, less than 30 per cent of the collisions result in coalescence under no field condition. At fields of about 40 v per cm, the coalescence was about 100 per cent under all conditions of field.
Abstract
Growth of incipient precipitation particles by collision and coalescence with cloud droplets is one of the primary mechanisms of natural rain. Comparison of previous research shows wide divergence between various theoretical and laboratory values of collision efficiency and coalescence efficiency. In an effort to obtain additional laboratory measurements of droplet coalescence, high-speed photographs were taken of colliding droplets at the breakup point in a Rayleigh jet. With 700-micron diam droplets, less than 30 per cent of the collisions result in coalescence under no field condition. At fields of about 40 v per cm, the coalescence was about 100 per cent under all conditions of field.
