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Mark J. Laufersweiler and Hampton N. Shirer

Abstract

Some latent heating effects on horizontal wavenumber selection and cell circulation patterns in stratocumulus- topped boundary layers are investigated via a study of two-dimensional shallow moist Boussinesq convection. No-entrainment and zero moisture flux assumptions are used to develop a simple latent heating representation; accordingly, upward and downward motions below a uniform cloud base are assumed to be dry adiabatic and those above cloud base are moist adiabatic. A nonlinear nine-coefficient low-order spectral model is developed in which representations of all nonlinear and linear terms of the original system of partial differential equations are retained.

In dry Rayleigh-Bénard convection, the first convective steady solution branching from the conductive state is assumed to occur at the smallest value of the critical Rayleigh number and the resulting flow field is dominated by one horizontal wavenumber. In this case, the circulation patterns are symmetric because the horizontal wavelengths of the updraft and the downdraft within a single cell are equal; these preferred wavelengths are given by the above minimization calculation. In moist convection, owing to the vertically asymmetric effects of latent heating, the minimum values of the critical Rayleigh number are smaller than those for dry convection. Moreover, these latent heating effects cause the activation of two new additional linear terms in the moist spectral model. Because the branching convective solution is dominated now by two horizontal wavenumbers. the resulting circulation patterns are asymmetric. The horizontal widths of the updraft and the downdraft of the convective cell are not equal, but the preferred wavelengths of the cell couplets are given by the above minimization procedure. A linear stability analysis of the conductive solution to the low-order model is compared with that for the original partial differential system. and the low-order model is shown to produce excellent approximations of both the critical Rayleigh number and the preferred wavelengths.

Cell circulation patterns for a few selected cases are shown, and qualitative differences in the steady convective solutions for deep and shallow clouds are demonstrated. As the amount of latent heating is increased when the cloud is deepened. the horizontal asymmetry of the circulation within a convective couplet increases correspondingly. When stratocumulus clouds fill ⅓ to ½of the domain, the initial circulation is elevated from the ground and occurs primarily within the cloud itself., in a sense, the circulation within the cloud layer is detached from that in the subcloud layer. As the value of the Rayleigh number increases, the circulation gradually fills the domain, thereby linking the two layers; only circulation patterns that fill the domain occur when shallow clouds are present.

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Hampton N. Shirer and John A. Dutton

Abstract

The dynamics of two-dimensional, shallow moist convection is examined with the use of a six-component spectral model. Latent heating effects are incorporated by assuming that upward motion is moist adiabatic and that downward motion is dry adiabatic. The resulting nondimensional system of equations has the same form as that for Bénard convection, with the moist effects included by replacing the Rayleigh number with a modified form.

The six-coefficient model contains a wide variety of multiple solutions with as many as 12 time-independent convective states and one conductive state occurring simultaneously. Temporally periodic solutions are also indicated, and some are found numerically that branch from stationary solutions at critical values of the external parameters. Only some of the solutions are linearly stable and hence observable, and we give a summary of the possible branching orders of these solutions. We find that the development of moist convection proceeds in the model via one of several available sequences of distinct transitions of the flow regime to increasingly complex structures.

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Hampton N. Shirer, Christian J. Fosmire, Robert Wells, and Laurentia Suciu

Abstract

The correlation dimension D is commonly used to quantify the chaotic structure of atmospheric time series. The standard algorithm for estimating the value of D is based on finding the slope of the curve obtained by plotting ln C(r) versus ln r, where C(r) is the correlation integral and r is the distance between points on the attractor. An alternative, probabilistic method proposed by is extended and tested here. This method is based on finding the sample means of the random variable (r/ρ)p[ln(r/ρ)]k, expressed as the conditional expected value E((r/ρ)p[ln(r/ρ)]k : r < ρ), for p and k nonnegative numbers.

The sensitivity of the slope method and of the extended estimators D pk(ρ) for approximating D is studied in detail for three ad hoc correlation integrals and for integer values of p and k. The first two integrals represent the effects of noise or undersampling at small distances and the third captures periodic lacunarity, which occurs by definition when the ratio C(x ρ)/C(ρ) fails to converge as ρ approaches zero. All the extended estimators give results that are superior to that produced by the most commonly used slope method. Moreover, the various estimators exhibit much different behavior in the two ad hoc cases: noise-contaminated signals are best diagnosed using D 11(ρ), and lacunar signals are best studied using D 0k(ρ), with k as large as possible in magnitude. Therefore, by using a wide range of values of p and k, one can infer whether degradation arising from noise or arising from lacunarity is more pronounced in the time series being studied, and hence, one can decide which of the estimates most efficiently approximates the correlation dimension for the series.

These ideas are applied to relatively coarse samplings of the Hénon, Lorenz convection, and Lorenz climate attractors that in each case are obtained by calculating the distances between pairs of points on two trajectories. As expected from previous studies, lacunarity apparently dominates the Hénon results, with the best estimate of D, D = 1.20 ± 0.01, given by the case D 03(ρ). In contrast, undersampling or noise apparently affects the Lorenz convection and climate attractor results. The best estimates of D are given by the estimator D 11(ρ) in both cases. The dimension of the convection attractor is D = 2.06 ± 0.005, and that of the climate attractor is D = 14.9 ± 0.1. Finally, lagged and embedded time series for the Lorenz convection attractor are studied to identify embedding dimension signatures when model reconstruction is employed.

In the last part of this study, the above results are used to help identify the best possible estimate of the correlation dimension for a low-frequency boundary layer time series of low-level horizontal winds. To obtain such an estimate, Lorenz notes that an optimally coupled time series must be extracted from the data and then lagged and embedded appropriately. The specific kinetic energy appears to be more closely coupled to the underlying low-frequency attractor, and so more nearly optimal, than is either individual wind component. When several estimates are considered, this kinetic energy series exhibits the same qualitative behavior as does the lagged and embedded Lorenz convective system time series. The series is either noise contaminated or undersampled, a result that is not surprising given the small number of time series points used, for which the best estimate is given by D 11(ρ). The obtained boundary layer time series dimension estimate, 3.9 ± 0.1, is similar to the values obtained by some other investigators who have analyzed higher-frequency boundary layer time series. Although this time series does not contain as many points as might be required to accurately estimate the dimension of the underlying attractor, it does illustrate the requirement that in any estimate of the correlation dimension, a function of the measured variables must be chosen that is strongly coupled to the attractor.

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Dustin J. Swales, George S. Young, Todd D. Sikora, Nathaniel S. Winstead, and Hampton N. Shirer
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Richard A. Mason, Hampton N. Shirer, Robert Wells, and George S. Young

Abstract

Bursts in the kinematic vertical transports of heat and horizontal momentum in a moderately convective marine atmospheric surface layer are studied by applying the variable interval time averaging (VITA) detection method to principal components analysis (PCA)–decomposed datasets obtained from the Floating Instrumentation Platform (FLIP) moored vessel during the 1995 April–May Pacific Marine Boundary Layer (PMBL) experiment. For convective plumes, a well-defined dimensionless relationship is shown to exist between the vertical transports of heat and horizontal momentum; this relationship cannot be easily deduced if PCA and VITA are not both applied.

PCA decomposes a dataset using correlations within that dataset instead of bandpass filtering it to retain energy in a predetermined range of scales; PCA thus respects all scales contributing to the phenomena retained in the dataset. Subsequent use of cross-spectral techniques to group the PCA-decomposed dataset into coherent structure types leads to, among other types of coherent structures, PCA-derived plumes. The VITA method is applied to a decomposed dataset in order to identify updrafts (bursts) and downdrafts (sweeps) in the time series of correlated variables by searching the signal for events that satisfy user-specified criteria. With proper use of PCA, surface-layer plumes can be reassembled in a way that yields the same transport relationships no matter which of the two different detecting variables is used.

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Todd D. Sikora, George S. Young, Hampton N. Shirer, and Rick D. Chapman

Abstract

Kilometer-scale mottling seen on real and synthetic aperture radar imagery of the sea surface can be linked to the presence of microscale cellular convection (thermals) spanning the marine atmospheric boundary layer. In the current study, it is hypothesized that the typical scale of the mottling, found via standard Fourier spectral analysis, can be used to estimate the depth of the convective marine atmospheric boundary layer (z i) using a modified form of traditional mixed-layer similarity theory for these thermals’ aspect ratio. The hypothesis linking the typical scale of mottling to z i is substantiated using previously published boundary layer results and supporting meteorological and oceanographic data from a number of case studies.

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Dustin J. Swales, George S. Young, Todd D. Sikora, Nathaniel S. Winstead, and Hampton N. Shirer

Abstract

The synthetic aperture radar ocean surface signature of atmospheric internal gravity waves in the vicinity of a synoptic-scale warm front is examined via a classic Kelvin–Helmholtz velocity profile with a rigid lower boundary and a sloping interface. The horizontal distance that the waves extend from the surface warm front is consistent with a bifurcation along the warm frontal inversion from unstable to neutral solutions. Similarity theories are derived for the wave span and the location of maximum growth rate relative to the surface front position. The theoretical maximum wave growth rate is demonstrated to occur near this bifurcation point and, hence, to explain the observed pattern of wave amplitude. Finally, a wave crest-tracing procedure is developed to explain the observed acute orientation of waves with respect to the surface warm front.

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