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Abstract
The Hoskins-Bretherton (HB) model is adopted to study two-dimensional frontogenesis in unsteady basic shear flows. The solutions exhibit the nonlinear evolution of an unstable Eady wave up to the formation of a frontal discontinuity. This development is described by the HR solution for a steady shear flow with time reinterpreted by means of a coordinate transformation. The computations are carried out by means of a relatively simple but hightly accurate approximation to the exact solution. The results show that typical wintertime variability of midlatitude zonal flows may either retard or accelerate frontal development compared to frontogenesis in a steady basic flow. Rapid frontal development is also associated with relatively rapid frontal movement and a more intense ageostrophic circulation. In contrast, a prolonged period of development is associated with relatively slow movement and a weaker ageostrophic circulation. The effect of different time-varying basic flows is examined and the results interpreted in relation to atmospheric frontogenesis.
Abstract
The Hoskins-Bretherton (HB) model is adopted to study two-dimensional frontogenesis in unsteady basic shear flows. The solutions exhibit the nonlinear evolution of an unstable Eady wave up to the formation of a frontal discontinuity. This development is described by the HR solution for a steady shear flow with time reinterpreted by means of a coordinate transformation. The computations are carried out by means of a relatively simple but hightly accurate approximation to the exact solution. The results show that typical wintertime variability of midlatitude zonal flows may either retard or accelerate frontal development compared to frontogenesis in a steady basic flow. Rapid frontal development is also associated with relatively rapid frontal movement and a more intense ageostrophic circulation. In contrast, a prolonged period of development is associated with relatively slow movement and a weaker ageostrophic circulation. The effect of different time-varying basic flows is examined and the results interpreted in relation to atmospheric frontogenesis.
Abstract
The Physical process responsible for short-wave baroclinic instability (zonal wavenumbers>10) is examined by means of a linearized two-layer Eady model. The static stability is uniform but different in each layer and the wind shear is uniform throughout both layers. Analysis of the unstable growth rates reveals that the instability is associated with the delta function distribution of potential vorticity at one boundary and at the interface between the two layers. This interpretation complements the interpretation of the unstable modes of a multi-layer model by Staley and Gall (1977). However, the present analysis also demonstrates how the short- and long-wave baroclinic instabilities depend on the relative layer depths as well as on the jump in static stability between the two layers. The effect of a jump in zonal wind shear is shown to be analogous to a jump in static stability in the present model. Finally, some implications of modeling atmospheric flows by multi-layered models, exhibiting discontinuities in potential vorticity, are pointed out.
Abstract
The Physical process responsible for short-wave baroclinic instability (zonal wavenumbers>10) is examined by means of a linearized two-layer Eady model. The static stability is uniform but different in each layer and the wind shear is uniform throughout both layers. Analysis of the unstable growth rates reveals that the instability is associated with the delta function distribution of potential vorticity at one boundary and at the interface between the two layers. This interpretation complements the interpretation of the unstable modes of a multi-layer model by Staley and Gall (1977). However, the present analysis also demonstrates how the short- and long-wave baroclinic instabilities depend on the relative layer depths as well as on the jump in static stability between the two layers. The effect of a jump in zonal wind shear is shown to be analogous to a jump in static stability in the present model. Finally, some implications of modeling atmospheric flows by multi-layered models, exhibiting discontinuities in potential vorticity, are pointed out.
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
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.
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
A model of inertial oscillations that may occur with, and be modulated by, deformation frontogenesis is formulated. The deformation parameter is α ∼ 10−5 s−1 and the Coriolis parameter is f ∼ 10−4 s−1. This timescale separation, distinguished by the ratio α/f ∼ 10−1, provides the basis for application of a two-timescale analysis that separates the frontal evolution from the inertial frequency oscillations. To lowest order, the inertial oscillations do not influence frontogenesis, described by the classical Hoskins and Bretherton model. The frontal evolution, characterized by the alongfront geostrophic wind, does, however, provide an amplitude modulation of the inertial wind oscillation and of the temperature that also undergoes an oscillation at the inertial frequency. Parameter values are chosen to illustrate frontal contraction and translation characteristics that can distort the wind hodograph from circular motion. Ground-level temperature traces also exhibit unusual attributes, such as an initial temperature increase with a cold frontal passage, that can be associated with the relative phase of the oscillation compared to the leading edge of the front. Lack of adequate observations for verification purposes and neglect of the boundary layer provide two important limitations.
Abstract
A model of inertial oscillations that may occur with, and be modulated by, deformation frontogenesis is formulated. The deformation parameter is α ∼ 10−5 s−1 and the Coriolis parameter is f ∼ 10−4 s−1. This timescale separation, distinguished by the ratio α/f ∼ 10−1, provides the basis for application of a two-timescale analysis that separates the frontal evolution from the inertial frequency oscillations. To lowest order, the inertial oscillations do not influence frontogenesis, described by the classical Hoskins and Bretherton model. The frontal evolution, characterized by the alongfront geostrophic wind, does, however, provide an amplitude modulation of the inertial wind oscillation and of the temperature that also undergoes an oscillation at the inertial frequency. Parameter values are chosen to illustrate frontal contraction and translation characteristics that can distort the wind hodograph from circular motion. Ground-level temperature traces also exhibit unusual attributes, such as an initial temperature increase with a cold frontal passage, that can be associated with the relative phase of the oscillation compared to the leading edge of the front. Lack of adequate observations for verification purposes and neglect of the boundary layer provide two important limitations.
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
Atmospheric cold fronts observed in the boundary layer represent relatively sharp transition zones between air masses of disparate physical characteristics. Further, wavelike features and/or eddy structures are often observed in conjunction with the passage of a frontal zone. The relative merits of using both global and local (with respect to the span of a basis element) transforms to depict cold-frontal features are explored. The data represent both tower and aircraft observations of cold fronts. An antisymmetric wavelet basis set is shown to resolve the characteristics of the transition zone, and associated wave and/or eddy activity, with a relatively small number of members of the basis set. In contrast, the Fourier transformation assigns a significant amplitude to a large number of members of the basis set to resolve a frontal-type feature. In principle, empirical orthogonal functions provide an optimal decomposition of the variance. The observed transition zone, however, has to be phase aligned and centered to yield optimal results, and variance may not be the optimum norm to depict a front. It is concluded that the wavelet or local transform provides a superior representation of frontal phenomena when compared with global transform methods. Further, the local transform offers the potential to provide some physical insight into wave and/or eddy structures revealed by the data.
Abstract
Atmospheric cold fronts observed in the boundary layer represent relatively sharp transition zones between air masses of disparate physical characteristics. Further, wavelike features and/or eddy structures are often observed in conjunction with the passage of a frontal zone. The relative merits of using both global and local (with respect to the span of a basis element) transforms to depict cold-frontal features are explored. The data represent both tower and aircraft observations of cold fronts. An antisymmetric wavelet basis set is shown to resolve the characteristics of the transition zone, and associated wave and/or eddy activity, with a relatively small number of members of the basis set. In contrast, the Fourier transformation assigns a significant amplitude to a large number of members of the basis set to resolve a frontal-type feature. In principle, empirical orthogonal functions provide an optimal decomposition of the variance. The observed transition zone, however, has to be phase aligned and centered to yield optimal results, and variance may not be the optimum norm to depict a front. It is concluded that the wavelet or local transform provides a superior representation of frontal phenomena when compared with global transform methods. Further, the local transform offers the potential to provide some physical insight into wave and/or eddy structures revealed by the data.
Abstract
The baroclinic instability of a two-dimensional uniform potential vorticity flow above a relatively thin viscous boundary layer is examined. The disturbance field is constrained by the geostrophic momentum approximation, and boundary layer dynamics are incorporated by prescribing the vertical velocity, derived by Wu and Blumen, at the bottom boundary of the inviscid layer. Characteristics of the instability and frontogenetical properties of the model are delineated by comparison with the results obtained using Ekman boundary layer dynamics to prescribe the vertical velocity at the boundary.
It is established that the unstable growth rates, phase speeds and qualitative aspects of the frontogenetical process are not significantly different from results obtained using Ekman boundary layer dynamics. However, significant modifications to the vertical velocity field at the lower boundary occur when the amplitude of the relative vorticity at the lower boundary attains a value equal to about f, the Coriolis parameter. In comparison with the vertical velocity field associated with Ekman layer dynamics, 1) the upward motion is smaller in cyclonic regions and larger in anticyclonic regions and 2) a broader band of relatively high values of upward motion exists. These features are interpreted in terms of the physical properties of the modified boundary layer dynamics.
Abstract
The baroclinic instability of a two-dimensional uniform potential vorticity flow above a relatively thin viscous boundary layer is examined. The disturbance field is constrained by the geostrophic momentum approximation, and boundary layer dynamics are incorporated by prescribing the vertical velocity, derived by Wu and Blumen, at the bottom boundary of the inviscid layer. Characteristics of the instability and frontogenetical properties of the model are delineated by comparison with the results obtained using Ekman boundary layer dynamics to prescribe the vertical velocity at the boundary.
It is established that the unstable growth rates, phase speeds and qualitative aspects of the frontogenetical process are not significantly different from results obtained using Ekman boundary layer dynamics. However, significant modifications to the vertical velocity field at the lower boundary occur when the amplitude of the relative vorticity at the lower boundary attains a value equal to about f, the Coriolis parameter. In comparison with the vertical velocity field associated with Ekman layer dynamics, 1) the upward motion is smaller in cyclonic regions and larger in anticyclonic regions and 2) a broader band of relatively high values of upward motion exists. These features are interpreted in terms of the physical properties of the modified boundary layer dynamics.
Abstract
A system of equations that describe motions in the boundary layer are derived. This system doors from the Ekman boundary layer equations by the inclusion of inertial terms through the geostrophic momentum approximation. The equations are solved, subject to the condition that the, horizontal motions approach the ageostrophic interior flow at the top of the boundary layer. The vertical velocity field at the top of the boundary layer is also determined. Interpretations of results are provided.
A model of a circular vortex is to display characteristics of the velocity field in the boundary layer. In comparison with the Ekman boundary layer solution: 1) the magnitude of the horizontal and vertical velocities are relatively higher in an anticyclonic vortex and relatively lower in a cyclonic vortex, and 2) the depth of the boundary layer, which is a function of the vortex radius and the Rossby number (Ro ≤0.3), is higher in an anticyclonic vortex and lower in a cyclonic vortex than the constant Ekman layer depth. Comparison with other studies and the model's limitations are also presented.
Abstract
A system of equations that describe motions in the boundary layer are derived. This system doors from the Ekman boundary layer equations by the inclusion of inertial terms through the geostrophic momentum approximation. The equations are solved, subject to the condition that the, horizontal motions approach the ageostrophic interior flow at the top of the boundary layer. The vertical velocity field at the top of the boundary layer is also determined. Interpretations of results are provided.
A model of a circular vortex is to display characteristics of the velocity field in the boundary layer. In comparison with the Ekman boundary layer solution: 1) the magnitude of the horizontal and vertical velocities are relatively higher in an anticyclonic vortex and relatively lower in a cyclonic vortex, and 2) the depth of the boundary layer, which is a function of the vortex radius and the Rossby number (Ro ≤0.3), is higher in an anticyclonic vortex and lower in a cyclonic vortex than the constant Ekman layer depth. Comparison with other studies and the model's limitations are also presented.
