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Paul Michael

Abstract

The ability to estimate horizontal advective tendencies of environmental variables from measurements at a finite set of observation points has been evaluated. The observation points include a central point plus from three to six boundary points at a man distance of R 0. Two methods of estimation were considered: either by a numerical approximation to the flux line integral, or by integrating a quadratic fit to the field function (the latter if there are five of more boundary points). Errors arise from random instrument errors (or turbulent fluctuations) and because of truncation errors. The latter results from a mismatch between the spatial distribution of the field being considered and the assumptions underlying the approximation algorithm. Both types of errors were considered. Random errors were considered using standard theory for the propagation of errors. Terms of the Fourier series were used as test functions to study truncation errors. The standard against which estimates were evaluated was multipoint numerical integrations.

For truncation errors, the significant result is that the use of a triangle of boundary points yields estimates close to 10% of the exact value only for a wavelength greater than about 20R 0; if cells have radius of 100 km, that would he a wavelength of 2000 km; for a square, the estimates are better than 10% at a wavelength of about 7R 0 (700 km); and for a pentagon, the estimates are less than 10% for the smallest nonaliasing wavelength. For this application, the use of a quadratic fit added little to accuracy; for a symmetrical army of points, the quadratic term does not contribute to the advective tendency. When one considers joint random and truncation errors. The general result is that truncation errors are more important than random errors for small wavelengths, and the reverse for large wavelengths.

The results indicate that there is a substantial gain in going from three to four or five boundary points. The improvement for each further increment is less dramatic. It is recommended that simple algorithms be augmented by the use of remotely sensed finescale observations of surrogate and by occasional periods of observations at higher spatial density.

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Paul A. Hirschberg
and
J. Michael Fritsch

Abstract

Top-down height tendency reasoning is explained and examined. This approach uses the assumption of a stratospheric level of insignificant dynamics (LID)—where height and pressure tendencies are considered negligible—to simplify the understanding of cyclone-scale hydrostatic height (pressure) tendency in the troposphere. Quasigeostrophic analytic model results confirm the existence of such a LID for scales less than approximately 5000 km. An examination of a height tendency equation with the LID assumption shows that there must be net integrated local warming (cooling) between the LID and any level below the LID where heights are falling (rising). The local temperature tendency, which from the thermodynamic equation results from advection, diabatic heating, and the product of vertical motion and static stability, reflects the combined actions of all thermodynamic and dynamic processes that together promote hydrostatic height change in isobaric coordinates. In particular, the important dynamic effects of mass-diverging secondary circulations are implicitly contained in the local temperature tendency.

New observational evidence and analytic model simulations supporting the top-down approach for understanding height tendency are also provided. The analytic model simulations show that isolated layers of equivalent diabatic heating and temperature advection do not produce equivalent dynamic responses in the vertical-motion field and height tendency fields. This result is used to explain observations that temperature advections in the upper troposphere /lower stratosphere are associated with larger lower-tropospheric height tendencies than equivalent temperature advections in the lower troposphere.

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Paul A. Hirschberg
and
J. Michael Fritsch

Abstract

An analytic quasigeostrophic model is used to examine the sensitivity of type B cyclogenesis to the vertical structure of the troposphere given a particular stratospheric temperature configuration. It is found that there is an optimal tropospheric configuration that produces the largest negative height tendency at the center of the 1000-mb model cyclone. Based on the response of the 1000-mb height tendencies, altering the baroclinicity in the model planetary boundary layer (PBL) does not significantly affect the instantaneous quasigeostrophic dynamics of the deep atmosphere. Rather, the PBL temperature anomalies affect the development of lower-tropospheric model lows by hydrostatically shifting or steering the cyclone centers to locations beneath more (or less) favorable deep atmospheric quasigeostrophic conditions for development.

Diagnostic analyses of three individual stratospheric-tropospheric model configurations are also performed to examine the dynamics that drive the height (pressure) tendency field. Generally, the analytic model findings confirm previous observational and numerical investigations of height tendency mechanisms and support the notion of a stratospheric level of insignificant dynamics. In the optimal development case, the 1000-mb low is located almost directly underneath the region of strongest 200-mb temperature advection associated with a tropopause undulation (potential vorticity anomaly). This strong lower-stratospheric warm advection instantaneously overwhelms adiabatic cooling in the stratosphere and troposphere so that there are height falls over and downstream of the 1000-mb low. When the static stability is lowered in the troposphere and raised in the stratosphere to realistic “warm-sector” values, the vertical motion increases, and the local warming in the stratosphere and cooling in the troposphere decrease. The reduced tropospheric cooling results in larger net local column warming that intensifies the 1000-mb height falls. The intensified vertical circulation also acts to amplify the tropopause undulation. As the amplitude of the undulation increases, characteristics of the occlusion process can be identified.

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Paul A. Hirschberg
and
J. Michael Fritsch

Abstract

No abstract available.

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Paul A. Hirschberg
and
J. Michael Fritsch

Abstract

The hypothesis that the development of extratropical cyclones is influenced by the evolution of tropopause undulations is described and examined. These undulations exhibit large temperature and potential vorticity anomalies, and are often observed prior to and during surface cyclogenesis. Typically, an undulation has a half wavelength of approximately 2000 km and a vertical amplitude of over 200 mb. Warm and cold temperature anomalies which lie respectively over the low and under the high portions of the undulation, are often embedded within strong upper-level flow, so that large temperature advections are found upstream and over developing cyclones. A case analysis of a cyclone event indicates that the distributions of tropospheric height change and vorticity change can be strongly sensitive to the undulation-related temperature changes in the lower stratosphere, especially near 200 mb.

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Paul A. Hirschberg
and
J. Michael Fritsch

Abstract

A case study of a developing cyclone is used to show that the three-dimensional distribution of height change during development can be strongly sensitive to temperature changes which occur in the lower stratosphere in association with an evolving tropopause undulation. Quantitative analysis of the storm with a geopotential height tendency equation indicates that a synergistic process developed between the stratosphere and troposphere, whereby the vertical motion pattern maintained and intensified the upper-level temperature anomalies while the subsequent upper-level temperature advection led to an enhanced vertical circulation. Using the results of this diagnostic study, a conceptual model is constructed. The conceptual model is based on the hydrostatic and wind-field adjustments that occur as tropopause undulations propagate over favored regions of tropospheric warm advection and less stable air.

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Paul D. Reasor
and
Michael T. Montgomery

Abstract

Using brightness temperatures from channels 3 and 4 of the Microwave Sounding Unit (MSU) as approximations to mean-layer temperatures, the geostrophic winds at 50 mb can be computed through a “bottom-up” approach. When this method is applied at high latitudes during austral winter and spring, it is found that accurate descriptions of the seasonal evolution and interannual variability of the lower-stratospheric circumpolar vortex are obtained. Variations in early-spring vortex strength from year to year appear to relate well to variations in the timing of the first large late-winter wavenumber one event in the lower stratosphere. Since wave forcing of the mean flow in the lower stratosphere is known to be weak, the variability in vortex strength may result from variations in wave-induced subsidence through the downward control principle.

Previous studies have demonstrated a biennial harmonic in both extratropical wave forcing and the mean flow, suggesting a link with the equatorially confined quasi-biennial oscillation (QBO). This study attempts to find a similar signal in the strength of the lower-stratospheric austral circumpolar vortex. It is first found that during the easterly (westerly) phase of the QBO large-amplitude wavenumber one in MSU channel 4, brightness temperature generally occurs earlier (later) in the season than normal. Subsequently, for most years of the study when the QBO is in its easterly (westerly) phase, the circumpolar vortex is observed to be weaker (stronger) than average.

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Paul A. Hirschberg
and
J. Michael Fritsch

Abstract

A new five-layer, quasigeostrophic model of baroclinic development is utilized to examine the initial-value problem sensitivity of extratropical cyclogenesis to the variation of stratospheric thermal and geopotential configurations associated with tropopause undulations. Previous studies have suggested that such undulations or potential vorticity anomalies can influence both the structure and evolution of lower-tropospheric cyclones. A series of experiments with the five-layer model are performed to evaluate the sensitivity of the model height, vertical motion, and height tendency patterns to various stratospheric temperature and geopotential distributions. It is found that idealized tropospheric baroclinic systems do not show typically observed characteristics unless certain stratospheric temperature, geopotential, and wind anomaly configurations associated with tropopause undulations are present. Furthermore, for given tropospheric patterns, there are particular lower-stratospheric configurations that optimize the development of model lower-tropospheric cyclones. These stratospheric configurations are functions of 1) the value of the lower-stratospheric temperature anomaly, 2) the amplitude of the tropopause undulation, and 3) the horizontal location of the undulation relative to the tropospheric temperature anomalies. Finally, both the rate of cyclogenesis and the amplification of the tropopause undulation increase if tropospheric static stability is reduced.

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Paul A. Hirschberg
and
J. Michael Fritsch

Abstract

A five-layer analytic model of quasigeostrophic flow is developed. The model provides exact analytic solutions to the nonlinear quasigeostrophic omega and vorticity equations for various atmospheric temperature and geopotential structures. These solutions yield instantaneous three-dimensional fields of vertical motion and geopotential tendency given some finite-amplitude flow. Hence, unlike traditional eigenvalue analyses that provide time-dependent solutions for simple linearized flows, the five-layer model yields nonlinear diagnostic solutions to initial-value problems.

It is demonstrated that the five-layer model can reproduce many of the disturbance characteristics that are deduced from more traditional analyses of baroclinic instability. It is also shown that, because of its flexible vertical temperature structure specification, it can simulate complex temperature and geopotential structures in the atmosphere. The flexible specification of the total temperature and geopotential structure makes the five-layer model an attractive means for comparing theory with observations. Additionally, the versatility and simplicity of the five-layer model make it a potentially useful research and pedagogical tool.

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Paul D. Reasor
and
Michael T. Montgomery

Abstract

The vertical alignment of an initially tilted geostrophic vortex is shown here to be captured by linear vortex Rossby wave dynamics when the vortex cores at upper and lower levels overlap. The vortex beta Rossby number, defined as the ratio of nonlinear advection in the potential vorticity equation to linear radial advection, is less than unity in this case. A useful means of characterizing a tilted vortex flow in this parameter regime is through a wave–mean flow decomposition. From this perspective the alignment mechanism is elucidated using a quasigeostrophic model in both its complete and linear equivalent barotropic forms. Attention is focused on basic-state vortices with continuous and monotonically decreasing potential vorticity profiles.

For internal Rossby deformation radii larger than the horizontal scale of the tilted vortex an azimuthal wavenumber 1 quasi mode exists. The quasi mode is characterized by its steady cyclonic propagation, long lifetime, and resistance to differential rotation, behaving much like a discrete vortex Rossby wave. The quasi mode traps disturbance energy causing the vortex to precess, or corotate, and thus prevents alignment. For internal deformation radii smaller than the horizontal vortex scale, the quasi mode disappears into the continuous spectrum of vortex Rossby waves. Alignment then proceeds through the irreversible redistribution of potential vorticity by the sheared vortex Rossby waves. Further decreases in the internal deformation radius result in a decreased dependence of vortex evolution on initial tilt magnitude, consistent with a reduction of the vortex beta Rossby number.

These results are believed to have relevance to the problem of tropical cyclone (TC) genesis. Cyclogenesis initiated through the merger and alignment of low-level convectively generated positive potential vorticity within a weak incipient vortex is captured by quasi-linear dynamics. A potential dynamical barrier to TC development in which the quasi mode frustrates vertical alignment can be identified using the linear alignment theory in this case.

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