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Abstract
Continuous partial reflection of linear hydrostatic gravity waves that propagate through a stratified shear flow is examined. The complex reflection coefficient R satisfies a Riccati equation, which is a first-order nonlinear differential equation. It is shown that |R|<1 since critical levels and overreflection are not considered. In this case the conservation of wave action flux may be expressed as a relationship between |R| and E l −1, where E is the wave energy and l a characteristic inverse vertical length scale of the background state.
It is demonstrated that R for a layered model represents a limiting solution of the Riccati equation. A general solution is also derived, under the assumption that the characteristic woe l is directly proportional to the inverse scale height of the characteristic impedance associated with a stratified shear flow. It is shown that the vanishing of |R| at a specific level is analogous to the vanishing of |R| in a three layer model, when the characteristic impedances in the top and the bottom layers satisfy a matching condition. Finally, various properties of the reflection coefficient are displayed for a particular background state. The extension of the theory to encompass other types of wave motion is indicated.
Abstract
Continuous partial reflection of linear hydrostatic gravity waves that propagate through a stratified shear flow is examined. The complex reflection coefficient R satisfies a Riccati equation, which is a first-order nonlinear differential equation. It is shown that |R|<1 since critical levels and overreflection are not considered. In this case the conservation of wave action flux may be expressed as a relationship between |R| and E l −1, where E is the wave energy and l a characteristic inverse vertical length scale of the background state.
It is demonstrated that R for a layered model represents a limiting solution of the Riccati equation. A general solution is also derived, under the assumption that the characteristic woe l is directly proportional to the inverse scale height of the characteristic impedance associated with a stratified shear flow. It is shown that the vanishing of |R| at a specific level is analogous to the vanishing of |R| in a three layer model, when the characteristic impedances in the top and the bottom layers satisfy a matching condition. Finally, various properties of the reflection coefficient are displayed for a particular background state. The extension of the theory to encompass other types of wave motion is indicated.
Abstract
The Hoskins-Bretherton model of frontogenesis employed here represents the counterpart of the two-dimensional Eady problem expressed in geostrophic coordinate space. The fundamental characteristics of the model solution are shown to be derivable from the properties of the nonlinear one-dimensional advection equation and the linearized Eady problem. Detailed comparisons are made between the predictions of this model and the analysis of an intense frontal zone presented by Sanders. Qualitative agreement is found in details of the horizontal wind field and potential temperature distributions. The major discrepancy occurs in the vertical velocity field: the most intense vertical velocities occur at midlevel in the model and are significantly smaller in magnitude than the rising narrow jet above the analyzed zone of maximum cyclonic relative vorticity. The presence of this jet is responsible for the most significant frontogenetical properties of the front associated with vertical tilting of potential isotherms and isopleths of the horizontal velocity component parallel to the frontal zone. In contrast, ageostrophic convergence and horizontal distortion of potential isotherms make the largest contribution to frontogenesis in the model.
Ekman-layer pumping is introduced into the model to simulate the vertical velocity jet. Yet this feature is not sufficient to increase the contribution of vertical tilting to frontogenesis because the vertical gradients of potential temperature and geostrophic velocity are weaker in this case.
Trajectories of the air motion tend to show the pattern of upgliding warm air ahead of the frontal zone with relatively stagnant cold air to the rear. In general, the model is able to provide qualitative agreement with gross features of this frontal situation. Discrepancies seem to be associated with the absence of a realistic boundary layer formulation and mesoscale mixing processes in the model.
Abstract
The Hoskins-Bretherton model of frontogenesis employed here represents the counterpart of the two-dimensional Eady problem expressed in geostrophic coordinate space. The fundamental characteristics of the model solution are shown to be derivable from the properties of the nonlinear one-dimensional advection equation and the linearized Eady problem. Detailed comparisons are made between the predictions of this model and the analysis of an intense frontal zone presented by Sanders. Qualitative agreement is found in details of the horizontal wind field and potential temperature distributions. The major discrepancy occurs in the vertical velocity field: the most intense vertical velocities occur at midlevel in the model and are significantly smaller in magnitude than the rising narrow jet above the analyzed zone of maximum cyclonic relative vorticity. The presence of this jet is responsible for the most significant frontogenetical properties of the front associated with vertical tilting of potential isotherms and isopleths of the horizontal velocity component parallel to the frontal zone. In contrast, ageostrophic convergence and horizontal distortion of potential isotherms make the largest contribution to frontogenesis in the model.
Ekman-layer pumping is introduced into the model to simulate the vertical velocity jet. Yet this feature is not sufficient to increase the contribution of vertical tilting to frontogenesis because the vertical gradients of potential temperature and geostrophic velocity are weaker in this case.
Trajectories of the air motion tend to show the pattern of upgliding warm air ahead of the frontal zone with relatively stagnant cold air to the rear. In general, the model is able to provide qualitative agreement with gross features of this frontal situation. Discrepancies seem to be associated with the absence of a realistic boundary layer formulation and mesoscale mixing processes in the model.
Abstract
The nonlinear evolution of unstable two-dimensional Eady waves is examined by means of a two-layer version of the Hoskins and Bretherton (1972) model. The upper layer is characterized by a higher static stability than the lower layer. Two types of unstable solutions are realized: the relatively long-wave solution has a vertical structure that extends throughout the vertical depth of the fluid and is the counter-part of the solution for a single layer system, while the shorter wave is essentially confined to the lower fluid layer. Model parameters, lower layer depth and static stability difference are chosen such that the two waves have comparable growth rates. The solution is determined by means of a Stokes expansion and terminated at second-order in the amplitude. The nonlinear interaction process between these growing baroclinic waves is then related to the wave interaction process described by the one-dimensional advection equation. Finally, an interpretation is proposed to explain disparate observations of cyclogenesis in polar air streams.
Abstract
The nonlinear evolution of unstable two-dimensional Eady waves is examined by means of a two-layer version of the Hoskins and Bretherton (1972) model. The upper layer is characterized by a higher static stability than the lower layer. Two types of unstable solutions are realized: the relatively long-wave solution has a vertical structure that extends throughout the vertical depth of the fluid and is the counter-part of the solution for a single layer system, while the shorter wave is essentially confined to the lower fluid layer. Model parameters, lower layer depth and static stability difference are chosen such that the two waves have comparable growth rates. The solution is determined by means of a Stokes expansion and terminated at second-order in the amplitude. The nonlinear interaction process between these growing baroclinic waves is then related to the wave interaction process described by the one-dimensional advection equation. Finally, an interpretation is proposed to explain disparate observations of cyclogenesis in polar air streams.
Abstract
A model of quasi-geostrophic uniform potential vorticity flow, previously examined by Blumen (1978a,b), is considered. The total depth-integrated energy and the available potential energy on level boundaries are conserved by the motion. Nonlinear interactions between three different scales of motion are examined. The linear system is first analyzed to determine the normal modes of the model. There are two sets of normal modes, corresponding to two different unstable growth rates. It is then shown that if normal mode initial conditions are specified for the nonlinear initial-value problem, the two conservation principles may be combined to yield a single constraint on the nonlinear interactions that occur between three scales of motion. The properties of normal mode initial conditions are also used to cast this constraint into a relatively simple form that is appropriate during the initial stages of the finite amplitude motion.
Numerical integrations of the basic set of equations reveal that the solutions are quite sensitive to the initial conditions. When normal mode initial conditions corresponding to the largest unstable growth rate are used, the simpler constraint continues to apply past the initial stages of growth. Analytical confirmation of this result is also provided. Nonlinear motions, associated with the other set of normal mode initial conditions, are also examined. The initial stages of the motion are similar to those above, but then the solutions tend to become aperiodic and the simpler form of the constraint on scale interactions does not apply. Extension of the range of integration over a broader range of initial conditions is suggested by these results.
Abstract
A model of quasi-geostrophic uniform potential vorticity flow, previously examined by Blumen (1978a,b), is considered. The total depth-integrated energy and the available potential energy on level boundaries are conserved by the motion. Nonlinear interactions between three different scales of motion are examined. The linear system is first analyzed to determine the normal modes of the model. There are two sets of normal modes, corresponding to two different unstable growth rates. It is then shown that if normal mode initial conditions are specified for the nonlinear initial-value problem, the two conservation principles may be combined to yield a single constraint on the nonlinear interactions that occur between three scales of motion. The properties of normal mode initial conditions are also used to cast this constraint into a relatively simple form that is appropriate during the initial stages of the finite amplitude motion.
Numerical integrations of the basic set of equations reveal that the solutions are quite sensitive to the initial conditions. When normal mode initial conditions corresponding to the largest unstable growth rate are used, the simpler constraint continues to apply past the initial stages of growth. Analytical confirmation of this result is also provided. Nonlinear motions, associated with the other set of normal mode initial conditions, are also examined. The initial stages of the motion are similar to those above, but then the solutions tend to become aperiodic and the simpler form of the constraint on scale interactions does not apply. Extension of the range of integration over a broader range of initial conditions is suggested by these results.
Abstract
A dynamical/statistical approach to initialization that is compatible with the dynamics of potential vorticity conservation is proposed. This approach consists of combining weighted assimilation, which minimizes the analysis error by means of linear regression, with a dynamical constraint imposed by this conservation principle. As a consequence, the initial analysis is shown to be optimal and dynamically compatible with the forecast model used in the present study.
Two situations that contribute to error growth in numerical prediction models are considered. 1) differences in the phase propagation speed of the model disturbance relative to that of the true or control state and 2) distortion of the initial error field by nonlinear wave interactions. In each case results obtained with the proposed dynamical/statistical initialization are compared with results from weighted assimilation using uncorrelated observational errors and from initialization by direct use of error-contaminated observations. These comparisons demonstrate the theoretical advantage of using an initialization scheme that is compatible with the model dynamics. However, it is pointed out that practical aspects involving additional computations and use of data from mixed observing systems have not been taken into account.
Abstract
A dynamical/statistical approach to initialization that is compatible with the dynamics of potential vorticity conservation is proposed. This approach consists of combining weighted assimilation, which minimizes the analysis error by means of linear regression, with a dynamical constraint imposed by this conservation principle. As a consequence, the initial analysis is shown to be optimal and dynamically compatible with the forecast model used in the present study.
Two situations that contribute to error growth in numerical prediction models are considered. 1) differences in the phase propagation speed of the model disturbance relative to that of the true or control state and 2) distortion of the initial error field by nonlinear wave interactions. In each case results obtained with the proposed dynamical/statistical initialization are compared with results from weighted assimilation using uncorrelated observational errors and from initialization by direct use of error-contaminated observations. These comparisons demonstrate the theoretical advantage of using an initialization scheme that is compatible with the model dynamics. However, it is pointed out that practical aspects involving additional computations and use of data from mixed observing systems have not been taken into account.
Abstract
Predictability experiments are carried out with a divergent barotropic model that describes the evolution of quasi-geostrophic planetary waves and high-frequency gravity-inertia waves. Error growth, relative to a model-determined control state, is initiated by an initialization procedure that is not compatible with the model equations. An analysis of error growth due to improper representation of the physics incorporated in prediction models is also carried out with the present model. The error growth rate and the range of predictability determined from these experiments, based on a simple triad solution of the nonlinear forecast equation, compare very well with the results from experiments carried out with multi-level numerical models. The mechanism of predictability decay by nonlinear energy exchange is shown to differ from the corresponding mechanism discussed by Lorenz and Leith, which is based on a model of two-dimensional turbulence.
Abstract
Predictability experiments are carried out with a divergent barotropic model that describes the evolution of quasi-geostrophic planetary waves and high-frequency gravity-inertia waves. Error growth, relative to a model-determined control state, is initiated by an initialization procedure that is not compatible with the model equations. An analysis of error growth due to improper representation of the physics incorporated in prediction models is also carried out with the present model. The error growth rate and the range of predictability determined from these experiments, based on a simple triad solution of the nonlinear forecast equation, compare very well with the results from experiments carried out with multi-level numerical models. The mechanism of predictability decay by nonlinear energy exchange is shown to differ from the corresponding mechanism discussed by Lorenz and Leith, which is based on a model of two-dimensional turbulence.
Abstract
The divergent barotropic model presented in Part I is used to investigate reduction of rms forecast errors by periodic updating with model-produced observations. Results show that an asymptotic error level is reached in about 2 days. This rapid adaptation reflects the initial balancing provided to the data at each update. Asymptotic rms forecast errors are increasing functions of both the updating period and the observation error, but the asymptotic error level is shown to be independent of the initial error. These results are in basic agreement with experiments carried out with various numerical models. Error reduction by statistically optimal assimilation of data is expected to yield results similar to those obtained in a previous study by the author.
Abstract
The divergent barotropic model presented in Part I is used to investigate reduction of rms forecast errors by periodic updating with model-produced observations. Results show that an asymptotic error level is reached in about 2 days. This rapid adaptation reflects the initial balancing provided to the data at each update. Asymptotic rms forecast errors are increasing functions of both the updating period and the observation error, but the asymptotic error level is shown to be independent of the initial error. These results are in basic agreement with experiments carried out with various numerical models. Error reduction by statistically optimal assimilation of data is expected to yield results similar to those obtained in a previous study by the author.
Abstract
Truncation error and, possibly, inadequate parameterization of physical processes, cause the propagation speed of atmospheric disturbances to be generally underestimated by numerical atmospheric models. Updating meteorological variables in a model with atmospheric data may improve its predictive capability, but a significant root mean square error will remain. The contribution to this error, due to differences in phase between atmospheric and model disturbances, is analyzed by means of a simple linear model that permits gravity-interia wave propagation superposed on a more slowly evolving quasi-geostrophic flow. Model pressure or wind variables are updated with control data, designed to simulate real data assimilation. Under this circumstance, the model fields of pressure or wind will always differ from the control fields. As the number of updates increases, this difference approaches an asymptotic error that depends only on the characteristic spatial scale of the wave disturbance and the difference in phase between the model and control disturbances. For scales of motion characteristic of mid-latitude synoptic-scale flow, this asymptotic error is essentially reached after four to seven updates with the control field. The asymptotic error level will be increased, however, if the phase error varies with time in a more or less random manner or if the disturbance flow has a spatially varying amplitude. As a corollary, when phase errors exist between the observed and model states, it is shown that asynoptic data assimilation, carried out on a random basis, increases the asymptotic error by the addition of random noise error. Some of the results are in agreement with Williamson's numerical experiments, while others have not been tested.
Error reduction appears to be attainable, for mid-latitude flow, if truncation error associated with the principal energy bearing modes can be controlled. However, it does not appear that updating tropical flow will yield significant error reduction because energy is distributed over a broader spectral range and, consequently, truncation error would be more difficult to control.
Abstract
Truncation error and, possibly, inadequate parameterization of physical processes, cause the propagation speed of atmospheric disturbances to be generally underestimated by numerical atmospheric models. Updating meteorological variables in a model with atmospheric data may improve its predictive capability, but a significant root mean square error will remain. The contribution to this error, due to differences in phase between atmospheric and model disturbances, is analyzed by means of a simple linear model that permits gravity-interia wave propagation superposed on a more slowly evolving quasi-geostrophic flow. Model pressure or wind variables are updated with control data, designed to simulate real data assimilation. Under this circumstance, the model fields of pressure or wind will always differ from the control fields. As the number of updates increases, this difference approaches an asymptotic error that depends only on the characteristic spatial scale of the wave disturbance and the difference in phase between the model and control disturbances. For scales of motion characteristic of mid-latitude synoptic-scale flow, this asymptotic error is essentially reached after four to seven updates with the control field. The asymptotic error level will be increased, however, if the phase error varies with time in a more or less random manner or if the disturbance flow has a spatially varying amplitude. As a corollary, when phase errors exist between the observed and model states, it is shown that asynoptic data assimilation, carried out on a random basis, increases the asymptotic error by the addition of random noise error. Some of the results are in agreement with Williamson's numerical experiments, while others have not been tested.
Error reduction appears to be attainable, for mid-latitude flow, if truncation error associated with the principal energy bearing modes can be controlled. However, it does not appear that updating tropical flow will yield significant error reduction because energy is distributed over a broader spectral range and, consequently, truncation error would be more difficult to control.
Abstract
A simple theoretical approach is formulated, based on part I of this study, in which inadequate modelling of physical processes and the use of numerical algorithms are assumed to introduce random phase errors between model predictions of large-scale atmospheric disturbances and the true state of the atmosphere. An attempt is made to prevent excessive growth of these errors by updating the model predictions with error-contaminated observations, available either periodically or aperiodically. It is demonstrated that the root mean square prediction error can be controlled by updating,, when the technique of weighted assimilation is employed. This well-known technique uses both predicted and observed values of an atmospheric variable to form an estimate of the true state. In general, this estimate is a better data source than direct replacement by an observed value. However, the results show that when observation errors are relatively small, weighted assimilation is essentially equivalent to replacement by the observed variable. When the model prediction errors are relatively small, significant improvement over replacement by the observed value is attained. These results are displayed for various model and observation errors and for different length scales of the wave disturbance.
A critique of the present results and inherent difficulties that are met in application to numerical weather prediction are discussed.
Abstract
A simple theoretical approach is formulated, based on part I of this study, in which inadequate modelling of physical processes and the use of numerical algorithms are assumed to introduce random phase errors between model predictions of large-scale atmospheric disturbances and the true state of the atmosphere. An attempt is made to prevent excessive growth of these errors by updating the model predictions with error-contaminated observations, available either periodically or aperiodically. It is demonstrated that the root mean square prediction error can be controlled by updating,, when the technique of weighted assimilation is employed. This well-known technique uses both predicted and observed values of an atmospheric variable to form an estimate of the true state. In general, this estimate is a better data source than direct replacement by an observed value. However, the results show that when observation errors are relatively small, weighted assimilation is essentially equivalent to replacement by the observed variable. When the model prediction errors are relatively small, significant improvement over replacement by the observed value is attained. These results are displayed for various model and observation errors and for different length scales of the wave disturbance.
A critique of the present results and inherent difficulties that are met in application to numerical weather prediction are discussed.
Abstract
The momentum flux by small-amplitude gravity waves produced by steady-state flow over a three- dimensional circular mountain in an isothermal plane rotating atmosphere is investigated. There is an upward transfer of momentum normal to the basic current by external-type gravity-inertia waves. This momentum transfer yields a flux convergence of momentum primarily in the lowest kilometer of the atmosphere. In contrast, the component of momentum parallel to the basic current is transported downward by internal-type gravity waves. This flux is independent of height and is essentially independent of the earth's rotation. Computed values of this surface drag are comparable with estimates of the frictional drag over ordinary terrain. The dependence of the various drag coefficients on atmospheric and mountain-shape parameters is also presented.
Abstract
The momentum flux by small-amplitude gravity waves produced by steady-state flow over a three- dimensional circular mountain in an isothermal plane rotating atmosphere is investigated. There is an upward transfer of momentum normal to the basic current by external-type gravity-inertia waves. This momentum transfer yields a flux convergence of momentum primarily in the lowest kilometer of the atmosphere. In contrast, the component of momentum parallel to the basic current is transported downward by internal-type gravity waves. This flux is independent of height and is essentially independent of the earth's rotation. Computed values of this surface drag are comparable with estimates of the frictional drag over ordinary terrain. The dependence of the various drag coefficients on atmospheric and mountain-shape parameters is also presented.
