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## Abstract

Geostrophic adjustment and frontogenesis are examined by means of a two-dimensional, inviscid, rotating and nonlinear fluid model that satisfies the condition of zero potential vorticity. The fluid is bounded top and bottom by level, rigid lids. The initial state is one of no motion, but an unbalanced horizontal temperature gradient is prescribed. The subsequent motion is represented as the sum of an inertial oscillation, with the frequency of the local Coriolis frequency *f,* and an evolving geostrophic flow. When a nondimensional parameter *a,* a Rossby number, satisfies *a* < 1, the gradient of the evolving geostrophic flow increases (frontogenesis) during the period 0 < *t* â©½ *Ï€*/*f*; the gradient decreases during the period *Ï€*/*f* < *t* â©½ 2*Ï€*/*f* (frontolysis). When *a* â‰¥ 1, the relative vorticity of the evolving geostrophic flow becomes infinite: a discontinuity forms at the top and bottom boundaries during the period 0 < *t* â©½ *Ï€*/*f.* There is an equipartition of energy between the inertial oscillation and the geostrophic flow, and nonlinear interactions occur between them. An exact (Fourier) spectral representation of the solution on the bottom boundary is used to display the kinetic energy spectrum and the transfer of energy through the spectrum at the time that the discontinuity forms. Applications of the model to oceanic and to atmospheric frontogenesis and to restratification of the surface mixed layer, following a storm, are noted.

## Abstract

Geostrophic adjustment and frontogenesis are examined by means of a two-dimensional, inviscid, rotating and nonlinear fluid model that satisfies the condition of zero potential vorticity. The fluid is bounded top and bottom by level, rigid lids. The initial state is one of no motion, but an unbalanced horizontal temperature gradient is prescribed. The subsequent motion is represented as the sum of an inertial oscillation, with the frequency of the local Coriolis frequency *f,* and an evolving geostrophic flow. When a nondimensional parameter *a,* a Rossby number, satisfies *a* < 1, the gradient of the evolving geostrophic flow increases (frontogenesis) during the period 0 < *t* â©½ *Ï€*/*f*; the gradient decreases during the period *Ï€*/*f* < *t* â©½ 2*Ï€*/*f* (frontolysis). When *a* â‰¥ 1, the relative vorticity of the evolving geostrophic flow becomes infinite: a discontinuity forms at the top and bottom boundaries during the period 0 < *t* â©½ *Ï€*/*f.* There is an equipartition of energy between the inertial oscillation and the geostrophic flow, and nonlinear interactions occur between them. An exact (Fourier) spectral representation of the solution on the bottom boundary is used to display the kinetic energy spectrum and the transfer of energy through the spectrum at the time that the discontinuity forms. Applications of the model to oceanic and to atmospheric frontogenesis and to restratification of the surface mixed layer, following a storm, are noted.

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## Abstract

The stability of a parallel flow, exhibiting both vertical and horizontal shear in a Boussinesq fluid, is investigated by a linear analysis. The effect of the earth's rotation is not considered and the disturbances are assumed to be hydrostatic, adiabatic and inviscid. A theorem, restricting the range of the complex phase speed *c*=*c _{r}
*+

*ic*is exhibited. Then neutral wave solutions are found when the basic flow is represented by

_{i}*Å«*(

*y*,

*z*) = (

*a*+

*z*) tanh

*y*and

*c*=

_{r}*c*= 0. It is concluded that these neutral waves are adjacent to unstable waves (

_{i}*c*= 0,

_{r}*c*> 0), which owe their existence to inertial instability associated with the inflection point in the profile of tanh

_{i}*y*. The effect of the vertical shear is passive, only modifying the characteristics of the instability but not being responsible for its occurrence.

## Abstract

The stability of a parallel flow, exhibiting both vertical and horizontal shear in a Boussinesq fluid, is investigated by a linear analysis. The effect of the earth's rotation is not considered and the disturbances are assumed to be hydrostatic, adiabatic and inviscid. A theorem, restricting the range of the complex phase speed *c*=*c _{r}
*+

*ic*is exhibited. Then neutral wave solutions are found when the basic flow is represented by

_{i}*Å«*(

*y*,

*z*) = (

*a*+

*z*) tanh

*y*and

*c*=

_{r}*c*= 0. It is concluded that these neutral waves are adjacent to unstable waves (

_{i}*c*= 0,

_{r}*c*> 0), which owe their existence to inertial instability associated with the inflection point in the profile of tanh

_{i}*y*. The effect of the vertical shear is passive, only modifying the characteristics of the instability but not being responsible for its occurrence.

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## Abstract

Steady, two-dimensional, hydrostatic, nonlinear mountain waves are examined within the context of Long's model. Both uniform and periodic upstream flows are considered. The well-known condition for a hydrostatic wave to break (convective instability), under uniform upstream conditions, is reviewed and a reinterpretation provided. Long's wave solution appropriate for periodic upstream conditions is introduced, and shown to satisfy the same wave-breaking condition that is appropriate for uniform upstream flow: overturning is associated with convective instability. Moreover, there is no obvious relationship between wave overturning and the upstream distribution of either the static stability or the Richardson number. In essence, the physical process of wave breaking, associated with this particular solution, is decoupled from details of the upstream profiles. However, the levels at which breaking occurs, and profiles of streamline displacements, are both affected by upstream conditions.

## Abstract

Steady, two-dimensional, hydrostatic, nonlinear mountain waves are examined within the context of Long's model. Both uniform and periodic upstream flows are considered. The well-known condition for a hydrostatic wave to break (convective instability), under uniform upstream conditions, is reviewed and a reinterpretation provided. Long's wave solution appropriate for periodic upstream conditions is introduced, and shown to satisfy the same wave-breaking condition that is appropriate for uniform upstream flow: overturning is associated with convective instability. Moreover, there is no obvious relationship between wave overturning and the upstream distribution of either the static stability or the Richardson number. In essence, the physical process of wave breaking, associated with this particular solution, is decoupled from details of the upstream profiles. However, the levels at which breaking occurs, and profiles of streamline displacements, are both affected by upstream conditions.

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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*
^{âˆ’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*
^{âˆ’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.

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## Abstract

The Hoskins and Bretherton two-dimensional semigeostrophic and uniform potential vorticity model is modified by the incorporation of momentum diffusion in a thin layerâ€”the frontal transition zone. The derived solutions are valid for an extended period following the critical time that the inviscid solution for the cross-frontal geostrophic velocity *v* becomes discontinuous. This discontinuous behavior is removed by momentum diffusion. The evolution of frontal development is described until equilibration is attained, a quasi-steady state exists, or until decay occurs.

Principal features of the solutions are the lifting of the warm sector above the ground, and the interplay between unstable growth of the baroclinic Eady wave and momentum diffusion that acts as a dissipative mechanism. The semigeostrophic ageostrophic circulation is characterized by a broad clockwise cell. The narrow counterclockwise direct circulation, that encompasses the frontal zone before *v* becomes discontinuous, is not described by semigeostrophic model dynamics when the front has equilibrated. Similarities and differences between results obtained in primitive equation numerical model experiments, presented by both Williams and by Nakamura and Held, are discussed and analyzed. Nakamura and Held find a change in the vertical structure of the baroclinic wave, that becomes prominent as equilibration is reached. This feature does not emerge as a characteristic of the present model solutions. It is concluded that ageostrophic effects that have been omitted in the semigeostrophic formulation are responsible for this discrepancy between the model results. However, the lifting of the warm air sector above the ground, the widening of the frontal transition zone with time and the magnitudes of the velocities predicted by the primative equation model are all replicated by the semigeostrophic model solutions. Means to control the excessive velocity amplitudes, that are common to all the two-dimensional models, are discussed.

## Abstract

The Hoskins and Bretherton two-dimensional semigeostrophic and uniform potential vorticity model is modified by the incorporation of momentum diffusion in a thin layerâ€”the frontal transition zone. The derived solutions are valid for an extended period following the critical time that the inviscid solution for the cross-frontal geostrophic velocity *v* becomes discontinuous. This discontinuous behavior is removed by momentum diffusion. The evolution of frontal development is described until equilibration is attained, a quasi-steady state exists, or until decay occurs.

Principal features of the solutions are the lifting of the warm sector above the ground, and the interplay between unstable growth of the baroclinic Eady wave and momentum diffusion that acts as a dissipative mechanism. The semigeostrophic ageostrophic circulation is characterized by a broad clockwise cell. The narrow counterclockwise direct circulation, that encompasses the frontal zone before *v* becomes discontinuous, is not described by semigeostrophic model dynamics when the front has equilibrated. Similarities and differences between results obtained in primitive equation numerical model experiments, presented by both Williams and by Nakamura and Held, are discussed and analyzed. Nakamura and Held find a change in the vertical structure of the baroclinic wave, that becomes prominent as equilibration is reached. This feature does not emerge as a characteristic of the present model solutions. It is concluded that ageostrophic effects that have been omitted in the semigeostrophic formulation are responsible for this discrepancy between the model results. However, the lifting of the warm air sector above the ground, the widening of the frontal transition zone with time and the magnitudes of the velocities predicted by the primative equation model are all replicated by the semigeostrophic model solutions. Means to control the excessive velocity amplitudes, that are common to all the two-dimensional models, are discussed.

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## Abstract

Momentum diffusion has been introduced into a semigeostrophic Eady-wave frontal model by Blumen (Part I). This model is used to determine the kinetic energy and enstrophy dissipations within a frontal zone that extends from the ground to a midtropospheric level. The largest amount of kinetic energy dissipation occurs when a relatively small nondimensional eddy viscosity coefficient is used, and the front attains an equilibrated stateâ€”a balance between unstable growth and momentum diffusion. The magnitude of kinetic energy dissipation ranges from about 50 to 250 W m^{âˆ’2} for parameter values that characterize surface-based fronts. These values are comparable to the 75 W m^{âˆ’2} determined by Kennedy and Shapiro from observations in an upper-level front, but are about one to two orders of magnitude larger than previous estimates of kinetic energy dissipated locally in clear-air turbulence zones and in the planetary boundary layer. An estimate of global kinetic energy dissipation in the planetary boundary layer is provided. A comparison establishes that fronts may make a relatively large contribution to the dissipation occurring during the life cycle of a cyclone, but the global contribution is less than that associated with the planetary boundary layer.

Frontal equilibration is characterized by a balance between production and dissipation of enstrophy. However, as frontolysis sets in, the dissipation of enstrophy becomes the dominant feature. Finally, it is noted that the physical process associated with the cascade of energy and enstrophy to dissipative scales differs from the cascade process described by the theory of homogeneous turbulence, and a different spectral decay law is realized.

## Abstract

Momentum diffusion has been introduced into a semigeostrophic Eady-wave frontal model by Blumen (Part I). This model is used to determine the kinetic energy and enstrophy dissipations within a frontal zone that extends from the ground to a midtropospheric level. The largest amount of kinetic energy dissipation occurs when a relatively small nondimensional eddy viscosity coefficient is used, and the front attains an equilibrated stateâ€”a balance between unstable growth and momentum diffusion. The magnitude of kinetic energy dissipation ranges from about 50 to 250 W m^{âˆ’2} for parameter values that characterize surface-based fronts. These values are comparable to the 75 W m^{âˆ’2} determined by Kennedy and Shapiro from observations in an upper-level front, but are about one to two orders of magnitude larger than previous estimates of kinetic energy dissipated locally in clear-air turbulence zones and in the planetary boundary layer. An estimate of global kinetic energy dissipation in the planetary boundary layer is provided. A comparison establishes that fronts may make a relatively large contribution to the dissipation occurring during the life cycle of a cyclone, but the global contribution is less than that associated with the planetary boundary layer.

Frontal equilibration is characterized by a balance between production and dissipation of enstrophy. However, as frontolysis sets in, the dissipation of enstrophy becomes the dominant feature. Finally, it is noted that the physical process associated with the cascade of energy and enstrophy to dissipative scales differs from the cascade process described by the theory of homogeneous turbulence, and a different spectral decay law is realized.

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## 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.

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## 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.

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## 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.

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## 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.