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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.
Abstract
Nonlinear features of the geostrophic adjustment process in a one-dimensional barotropic atmosphere are investigated by means of a perturbation expansion in the Froude number. The initial unbalanced velocity field is a continuous (nonconstant) even function of the spatial coordinate. The steady-state solution shows the southward shift of the axes of maximum geostrophic velocity and zero pressure, first found by Rossby. In addition, the geostrophic fields are asymmetric about their respective axes.
The nonlinear oscillation of the whole current system approaches the inertial period and decays like t −½ as time t→∞. However, this oscillation continues for a significantly longer time, before approximate geostrophic balance is reached, than the “adjustment time” determined from a linear analysis. A possible shortcoming in the quasi-geostrophic approximation, used in some large-scale dynamical models, is indicated by this result.
Abstract
Nonlinear features of the geostrophic adjustment process in a one-dimensional barotropic atmosphere are investigated by means of a perturbation expansion in the Froude number. The initial unbalanced velocity field is a continuous (nonconstant) even function of the spatial coordinate. The steady-state solution shows the southward shift of the axes of maximum geostrophic velocity and zero pressure, first found by Rossby. In addition, the geostrophic fields are asymmetric about their respective axes.
The nonlinear oscillation of the whole current system approaches the inertial period and decays like t −½ as time t→∞. However, this oscillation continues for a significantly longer time, before approximate geostrophic balance is reached, than the “adjustment time” determined from a linear analysis. A possible shortcoming in the quasi-geostrophic approximation, used in some large-scale dynamical models, is indicated by this result.
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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=cr +ici is exhibited. Then neutral wave solutions are found when the basic flow is represented by ū(y,z) = (a + z) tanhy and cr = ci = 0. It is concluded that these neutral waves are adjacent to unstable waves (cr = 0, ci > 0), which owe their existence to inertial instability associated with the inflection point in the profile of tanhy. 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=cr +ici is exhibited. Then neutral wave solutions are found when the basic flow is represented by ū(y,z) = (a + z) tanhy and cr = ci = 0. It is concluded that these neutral waves are adjacent to unstable waves (cr = 0, ci > 0), which owe their existence to inertial instability associated with the inflection point in the profile of tanhy. The effect of the vertical shear is passive, only modifying the characteristics of the instability but not being responsible for its occurrence.
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
The geostrophic coordinate transformation is a contact transformation that preserves the correspondence between the slopes of geopotential surfaces ϕ in physical space and the slopes of the surfaces ϕ, the maps of ϕ, in transformed space. The transformation back to physical space may be accomplished by integration along characteristic surfaces. This technique may be used to determine the time and place where a discontinuity would form as a function of the initial conditions. A model solution is used to illustrate properties of the transformation.
Abstract
The geostrophic coordinate transformation is a contact transformation that preserves the correspondence between the slopes of geopotential surfaces ϕ in physical space and the slopes of the surfaces ϕ, the maps of ϕ, in transformed space. The transformation back to physical space may be accomplished by integration along characteristic surfaces. This technique may be used to determine the time and place where a discontinuity would form as a function of the initial conditions. A model solution is used to illustrate properties of the transformation.
Abstract
The Hoskins and Bretherton two-dimensional semigeostrophic and uniform potential vorticity model is modified by the incorporation of momentum diffusion in a thin layer—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 layer—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
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 state—a balance between unstable growth and momentum diffusion. The magnitude of kinetic energy dissipation ranges from about 50 to 250 W m−2 for parameter values that characterize surface-based fronts. These values are comparable to the 75 W m−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 state—a balance between unstable growth and momentum diffusion. The magnitude of kinetic energy dissipation ranges from about 50 to 250 W m−2 for parameter values that characterize surface-based fronts. These values are comparable to the 75 W m−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
The two-dimensional, semigeostrophic and uniform potential vorticity Eady model is considered. An unstable baroclinic wave develops large velocity and temperature gradients in a narrow zone. Momentum diffusion and wave dispersion are incorporated into the model to prevent the ultimate development of a discontinuity in the alongfront geostrophic velocity υ(υx = ∞). Diffusion and dispersion act to reduce the amplitude of the growing baroclinic wave, and these processes also act to expand the width of the frontal zone, where the maximum velocity gradient is located. Explicit relationships are derived that reveal how these processes are dependent on two parameters: ε, the nondimensional eddy diffusion coefficient, and λ the ratio of a dispersion coefficient μ to ε 2. The total dissipation of kinetic energy D is separated into two parts,D 1andD 2:D 1 provides the dissipation that is largely confined to the relatively narrow frontal zone, and D 2 = D − D 1 provides the dissipation that is associated with the decaying waves that trail behind the front. These evaluations are carried out for a range of parameter values (ε, λ). Results show that the dissipation is not confined exclusively to the frontal zone but that D 2 ≲ D 1 when λ is large. Limitations of the present model development are associated with the excessive growth of the unstable Eady wave in the absence of dissipation and the lack of fine-scale measurements that may be used to design a dynamical model of the frontal zone.
Abstract
The two-dimensional, semigeostrophic and uniform potential vorticity Eady model is considered. An unstable baroclinic wave develops large velocity and temperature gradients in a narrow zone. Momentum diffusion and wave dispersion are incorporated into the model to prevent the ultimate development of a discontinuity in the alongfront geostrophic velocity υ(υx = ∞). Diffusion and dispersion act to reduce the amplitude of the growing baroclinic wave, and these processes also act to expand the width of the frontal zone, where the maximum velocity gradient is located. Explicit relationships are derived that reveal how these processes are dependent on two parameters: ε, the nondimensional eddy diffusion coefficient, and λ the ratio of a dispersion coefficient μ to ε 2. The total dissipation of kinetic energy D is separated into two parts,D 1andD 2:D 1 provides the dissipation that is largely confined to the relatively narrow frontal zone, and D 2 = D − D 1 provides the dissipation that is associated with the decaying waves that trail behind the front. These evaluations are carried out for a range of parameter values (ε, λ). Results show that the dissipation is not confined exclusively to the frontal zone but that D 2 ≲ D 1 when λ is large. Limitations of the present model development are associated with the excessive growth of the unstable Eady wave in the absence of dissipation and the lack of fine-scale measurements that may be used to design a dynamical model of the frontal zone.
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.
