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William Blumen

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.

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William Blumen

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 = DD 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 2D 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.

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Nimal Gamage
and
William Blumen

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.

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William Blumen
and
Rongsheng Wu

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.

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Rongsheng Wu
and
William Blumen

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.

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Vern Ostdiek
and
William Blumen

Abstract

Potential temperature and wind profiles obtained during a case of nighttime low-level deformation frontogenesis are examined. The winds in the lowest kilometer over a region in the central United States are shown to be controlled by thermal wind shear in a stable layer above 400 m, in accordance with the Hoskins–Bretherton semigeostrophic frontogenesis model, and by surface-drag-generated shear in the nearly neutral layer below. In this lower layer, the wind profiles are shown to be in good agreement with a simple baroclinic Ekman–Taylor model. The opposite shear in the two layers produces a low-level jet that appears in soundings taken hundreds of kilometers apart. The agreement of the observed profiles with these models is revealed only after a height-dependent inertial oscillation in both layers is removed from the rapidly evolving hourly wind data.

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Vern Ostdiek
and
William Blumen

Abstract

A low-level deformation frontogenesis event that occurred during the Stormscale Operational and Research Meteorology-Fronts Experiment Systems Test (STORM-FEST) field program is analyzed in the context of semigeostrophic theory. The observed evolution and vertical structure of the potential temperature and alongfront wind fields are compared to that predicted by both numerical and analytical solutions of the semigeostrophic equations initialized at the onset of the deformation frontogenesis. The model solutions provide relatively accurate predictions of the surface potential temperature distribution 5 h later, when the frontogenesis ended. The point along the front with the steepest potential temperature gradient is observed to move closer to the point with the highest relative vorticity by an amount that is in rough agreement with the model prediction. Vertical profiles of potential temperature from soundings show a nearly mixed layer below ∼400 m that cannot be predicted by the inviscid solutions, but there is good agreement with inviscid theory above this level. The observed profiles of alongfront wind are characterized by a low-level jet with maximum speed at the level of the inversion, and the vertical shear below the jet maximum is opposite that predicted by the thermal wind equation. The semigeostrophic model does appear to depict this frontogenesis event in the upper layer, while the lower layer is dominated by surface drag and shear-induced turbulent mixing.

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William Blumen
and
R. T. Williams

Abstract

Unbalanced frontogenesis is studied in a two-dimensional, Boussinesq, rotating fluid that is constrained between two rigid, level surfaces. The potential vorticity is zero. The initial state is unbalanced because there is no motion and the potential temperature is given by the error function of x. An analytic solution is derived based on the neglect of the barotropic pressure gradient. The solution procedure uses momentum coordinates to obtain nonlinear solutions. When the initial Rossby number (Ro) is less than 1.435 the horizontal wind components display an inertial oscillation. During the first part of the inertial period (0 < ft < π) the isentropes develop a tilt and frontogenesis occurs, while in the second part (π < ft < 2π) the isentropes return to a vertical orientation and frontolysis brings the temperature gradient back to its original value at ft = 2π. For larger values of Ro a frontal discontinuity forms before ft = π.

The importance of the barotropic pressure gradient is determined in a scale collapse problem with a constant potential temperature and no rotation. In this case the inclusion of the barotropic pressure gradient increases the time before the discontinuity forms.

Numerical solutions of the original problem with rotation show that the presence of the barotropic pressure gradient term increases the critical Rossby number from 1.435 to about 1.55. Otherwise the complete solutions are very similar to the analytic solutions, except that the isentropes are no longer straight and the vorticity shows evidence of strong vertical advection by a small-scale vertical jet. Further, shorter timescales are expected with unbalanced fronts as compared with balanced fronts.

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William Blumen
and
Richard G. Hendl

Abstract

Observations of ionospheric disturbances by various investigators have led to the suggestion that auroral energy may be coupled to atmospheric wave motions through joule heating. A linear model of internal gravity-wave generation by joule heating in the region of the auroral electrojet (100–150 km above the earth's surface) is investigated. Heat conduction, viscosity and reflection of wave energy by atmospheric inhomogeneities are not considered. The computed value of the upward wave-energy flux from the source region is of order 0.1–1 erg cm−2 sec−1 and is of sufficient magnitude to be of importance in the energetics of the F region. Shortcomings of the present model are discussed, with emphasis on how the physical features which have been neglected might affect the present results.

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Brian D. Gross
and
William Blumen

Abstract

Steady, three-dimensional, inviscid flow over orography is examined by means of a semi-geostrophic model. A nonuniform basic current, represented by a deformation flow, is employed. A constant Coriolis parameters ƒ and uniform potential vorticity (constant Brunt-Väisälä frequency N) characteristic this model. A nondimensional mountain height ε/D ≲ 0.5, based on the deformation depth D ∼ 3 × 103 m, and a Rossby number Ro ≲ 0.3, based on the mountain breadth L ≳ 3.5 × 105 m, provide constraints on the flow field. Analytic solutions are represented in geostrophic coordinate space as the sum of the deformation flow and an anticyclonic mountain vortex. Although the two solutions are independent in geostrophic coordinate space, these flows are coupled nonlinearly in the transformation to physical coordinate space.

A solution is presented for flow over an isolated mountain. The decomposition of the physical space solution into fields of translation, rotation, divergence, and deformation forms the basis of the present analysis. The principal features associated with the solution are a region of relatively strong cyclonic vorticity in the lee of the mountain, accompanied by a region of convergence, and a region of weaker cyclonic vorticity on the windward slope, accompanied by a region of divergence. It is the ageostrophic component of the vorticity that provides these cyclonic centers, which are associated with enhanced deformation upstream and downstream of the peak. Further, the lee-side cyclonic vorticity enhancement is associated with the advection of geostrophic deformation, a feature of semi-geostrophic models that is absent in quasi-geostrophic models. By displacing the basic current's axis of dilatation into the lee of the obstacle, a deformation advection pattern is established that enhances the lee-side cyclonic vorticity center. The uniform flow solution is characterized by a single band of cyclonic vorticity north of the peak. This pattern is also established by the advection of geostrophic deformation. The possible relevance of the present model results to physical mechanisms that promote the initiation of lee cyclogenesis is discussed.

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