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Richard Grotjahn

Abstract

The observed vertical structures of the trough axes for 27 extratropical cyclones are presented. This study is motivated by results from a simple theoretical model. Two observing times during the cyclone life cycle are shown: prior to development and during the “mature” but still amplifying stage. Prior to development, upper and lower troughs are present and separate, each has little or no tilt, the upper one is typically prominent down to 4-km elevation, and the separation between the lower and the upper features varies depending on where the approaching upper trough happens to be at the observing time. At the mature stage, upper and lower features are connected, a uniform tilt typically develops through the entire troposphere, the tilt is typically due west with height, and the tilt may have a preferred slope. An empirical orthogonal function (EOF) analysis finds that two modes account for more than 97% of the variance. The equivalent barotropic EOF has the most variance by far, though the fractional amount diminishes over time as this EOF also extends further downward. The first baroclinic EOF increases fractional amplitude in compensation.

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Richard Grotjahn

Abstract

All terms of the frictionless, nonlinear, vorticity equation are examined. Traditional scale analysis provides one of several justifications for using the quasigeostrophic (QG) system of equations to model extratropical cyclones. Analysts of observations have long known that some of the other terms (non-QG) are individually comparable to terms kept in quasigeostrophy. While the non-QG terms are not small, they are assumed to have a large degree of cancellation and so are still neglected in sum. The distributions, magnitudes, and possible cancellations of vorticity equation terms are examined. Analyzed data composites for 15 cases of mature, developing, extratropical cyclones are used.

These results lead us to conclude that several commonly neglected terms are neither especially small nor do they cancel. The way each term contributes to the redistribution, advection, of amplification of vorticity is discussed. In sum, cyclone growth is greater at all levels, especially at low levels, in the full set of terms compared to the QG terms.

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Richard Grotjahn and Cris Castello

Abstract

An earlier article deduced a doubling of the scale of the sea level pressure pattern for lows as they developed in the North Pacific. Scale here refers to horizontal extent of the low. This study uses a different technique to estimate scale change in the upper troposphere. The prior study used wavelets; here circular averaging is used on several fields, with primary emphasis on the geostrophic kinetic energy (gKE) field.

The technique herein confirms the earlier result that sea level pressure (SLP) scale increases. When applied to the 300-hPa level, the trough extent does not change scale significantly. The average scale has radius of about 1200 km at sea level and 1700 km at 300 hPa. During development the average radius of maximum gKE changes little at the surface but decreases at upper levels. The maximum gKE is typically located 600–1100 km from the 300-hPa low center, 450–650 km from the SLP low center. Composite maps of gKE are shown during different stages in cyclone development at both levels. Consistency between the results presented here and the conventional view of jet streak migration around an upper low is mentioned. Some implications for theoretical work are mentioned.

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Richard Grotjahn and James J. O'Brien

Abstract

The errors introduced by the use of various numerical schemes for solving mathematical models have generally been only vaguely determined previously by numerical modelers. A method for a more quantitative analysis of the inaccuracies is outlined. The error associated with some simple schemes is analyzed for several linear hyperbolic systems representative of typical problems in meteorology and oceanography. Results of previous studies of phase velocity inaccuracies are confirmed and form a basis for an extension of the analysis to group velocities. Significant angular and magnitude errors are found in the group velocity. Directional errors of 180° are found for some waves. Since the group velocity is the propagation speed of the energy, such errors may have severe consequences in a numerical model. When analysis was made of complex systems of equations, results found for simple systems reappeared. Thus, studies of simple systems may provide useful indications of behavior in more complex problems where the analysis may have to be limited. Only the long waves, i.e., those resolved by many grid points, are represented with any reasonable accuracy.

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Richard Grotjahn, Daniel Hodyss, and Cris Castello

Abstract

Wavelet transforms in the longitudinal and latitudinal directions are applied to sea level pressure data for 12 extratropical cyclones. Each low is tracked over time from a stage of small amplitude to a stage of large amplitude. The wavelet transform provides a quantitative, localized estimate of the size of the low pressure. Separate one-dimensional transforms are taken in the longitudinal and latitudinal directions; these are averaged to reduce scale variations created as circular asymmetries rotate around a low center.

On average, the size of the lows increases such that the diameter doubles over a 4-day period. These results pass a standard “f test” with greater than 99% confidence. Some implications for theoretical studies are included.

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