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Daniel Hodyss

Abstract

A new framework is presented for understanding how a nonnormal probability density function (pdf) may affect a state estimate and how one might usefully exploit the nonnormal properties of the pdf when constructing a state estimate. A Bayesian framework is constructed that naturally leads to an expansion of the expected forecast error in a polynomial series consisting of powers of the innovation vector. This polynomial expansion in the innovation reveals a new view of the geometric nature of the state estimation problem. It is shown that this expansion in powers of the innovation provides a direct relationship between a nonnormal pdf describing the likely distribution of states and a normal pdf determined by powers of the forecast error. One implication of this perspective is that when state estimation is performed on a nonnormal pdf it leads to state estimates based on the mean to be nonlinear functions of the innovation. A direct relationship is shown between the degree to which the state estimate varies with the innovation and the moments of the distribution. These and other implications of this new view of ensemble state estimation in nonlinear systems are illustrated in simple scalar systems as well as on the Lorenz attractor.

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Daniel Hodyss

Abstract

A practical data assimilation algorithm is presented that explicitly accounts for skewness in the prior distribution. The algorithm operates as a global solve (all observations are considered at once) using a minimization-based approach and Schur–Hadamard (elementwise) localization. The central feature of this technique is the squaring of the innovation and the ensemble perturbations so as to create an extended state space that accounts for the second, third, and fourth moments of the prior distribution. This new technique is illustrated in a simple scalar system as well as in a Boussinesq model configured to simulate nonlinearly evolving shear instabilities (Kelvin–Helmholtz waves). It is shown that an ensemble size of at least 100 members is needed to adequately resolve the third and fourth moments required for the algorithm. For ensembles of this size it is shown that this new technique is superior to a state-of-the-art ensemble Kalman filter in situations with significant skewness; otherwise, the new algorithm reduces to the performance of the ensemble Kalman filter.

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Daniel Hodyss and Eric Hendricks

Abstract

This paper explores the hypothesis that a tropical cyclone (TC) may produce baroclinic waves through the divergent circulation that arises from its low-level inflow and upper-level outflow. The model setting is a quasigeostrophic (QG) two-layer fluid in which the effect of the tropical cyclone is parameterized through a source term on the QG potential vorticity equation. Equations predicting the spectral subset of baroclinic waves that are excited through linear resonance are derived. The near-TC pattern of the baroclinic waves in the streamfunction field typically takes the form of a ridge–trough couplet whose phase with respect to the TC varies with the speed and direction of the TCs motion vector. The predictions from the linearized theory are verified in two ways: 1) fully nonlinear simulations are shown and 2) comparison is made to the observed upper-level ridge–trough couplets produced by recurving TCs in the Navy’s Operational Global Prediction System (NOGAPS). The implications of this work for the predictability of downstream impacts from recurving TCs are briefly described.

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Daniel Hodyss and Terrence R. Nathan

Abstract

A theory is presented that addresses the connection between low-frequency wave packets (LFWPs) and the formation and decay of coherent structures (CSs) in large-scale atmospheric flow. Using a weakly nonlinear evolution equation as well as the nonlinear barotropic vorticity equation, the coalescence of LFWPs into CSs is shown to require packet configurations for which there is a convergent group velocity field. These LFWP configurations, which are consistent with observations, have shorter wave groups with faster group velocities upstream of longer wave groups with slower group velocities. These wave group configurations are explained by carrying out a kinematic analysis of wave focusing, whereby a collection of wave groups focus at some point in space and time to form a large amplitude wave packet having a single wave front. The wave focusing and the subsequent formation of CSs are enhanced by zonal variations in the background flow, while nonlinearity extends the lifetimes of the CSs. These results are discussed in light of observed blocking formation in the Atlantic–European and South Pacific regions.

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Craig H. Bishop and Daniel Hodyss

Abstract

An adaptive ensemble covariance localization technique, previously used in “local” forms of the ensemble Kalman filter, is extended to a global ensemble four-dimensional variational data assimilation (4D-VAR) scheme. The purely adaptive part of the localization matrix considered is given by the element-wise square of the correlation matrix of a smoothed ensemble of streamfunction perturbations. It is found that these purely adaptive localization functions have spurious far-field correlations as large as 0.1 with a 128-member ensemble. To attenuate the spurious features of the purely adaptive localization functions, the authors multiply the adaptive localization functions with very broadscale nonadaptive localization functions. Using the Navy’s operational ensemble forecasting system, it is shown that the covariance localization functions obtained by this approach adapt to spatially anisotropic aspects of the flow, move with the flow, and are free of far-field spurious correlations. The scheme is made computationally feasible by (i) a method for inexpensively generating the square root of an adaptively localized global four-dimensional error covariance model in terms of products or modulations of smoothed ensemble perturbations with themselves and with raw ensemble perturbations, and (ii) utilizing algorithms that speed ensemble covariance localization when localization functions are separable, variable-type independent, and/or large scale. In spite of the apparently useful characteristics of adaptive localization, single analysis/forecast experiments assimilating 583 200 observations over both 6- and 12-h data assimilation windows failed to identify any significant difference in the quality of the analyses and forecasts obtained using nonadaptive localization from that obtained using adaptive localization.

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Daniel Hodyss and William F. Campbell

Abstract

The main goal of this work is to present a new square root ensemble generation technique that is consistent with a recently developed extension of Kalman-based linear regression algorithms such that they may perform nonlinear polynomial regression (i.e., includes a quadratically nonlinear term in the mean update equation) and that is applicable to ensemble data assimilation in the geosciences. Along the way the authors present a unification of the theories of square root and perturbed observation (sometimes referred to as stochastic) ensemble generation in data assimilation algorithms configured to perform both linear (Kalman) regression as well as quadratic nonlinear regression. The performance of linear and nonlinear regression algorithms with both ensemble generation techniques is explored in the three-variable Lorenz model as well as in a nonlinear model configured to simulate shear layer instabilities.

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David R. Ryglicki and Daniel Hodyss

Abstract

A deeper analysis of possible errors and inconsistencies in the analysis of vortex asymmetries owing to the placement of centers of tropical cyclones (TCs) in mesoscale models is presented. Previous works have established that components of the 2D and 3D structure of these TCs—primarily radial wind and vertical tilt—can vary greatly depending on how the center of a model TC is defined. This work will seek to expand the previous research on this topic, but only for the 2D structure. To be specific, this work will present how low-wavenumber azimuthal Fourier analyses can vary with center displacement using idealized, parametric TC-like vortices. It is shown that the errors associated with aliasing the mean are sensitive primarily to the difference between the peak of vorticity inside the radius of maximum winds and the average vorticity inside the core. Tangential wind and vorticity aliasing occur primarily in the core; radial wind aliasing spans the whole of the vortex. It is also shown that, when adding low-wavenumber asymmetries, the aliasing is dependent on the placement of the center relative to the location of the asymmetries on the vortex. It is also shown that the primary concern for 2D analysis when calculating the center of a TC is correctly resolving azimuthal wavenumber 0 tangential wind, because errors here will alias onto all higher wavenumbers, the specific structures of which are dependent on the structure of the mean vortex itself.

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Daniel Hodyss and David S. Nolan

Abstract

A linear anelastic-vortex model is derived using assumptions appropriate to waves on vortices with scales similar to tropical cyclones. The equation set is derived through application of a multiple-scaling technique, such that the radial variations of the thermodynamic fields are incorporated into the reference state. The primary assumption required for the model is that the horizontal variations in the thermodynamic variables describing the reference state are appreciably longer than the waves on the vortex. This new version of the anelastic system makes no approximation to the requirements for hydrostatic and gradient wind balance, or the buoyancy frequency, in the core of the vortex. A small but measurable improvement in the performance of the new equation set is demonstrated through simulations of gravity waves and vortex–Rossby waves in a baroclinic vortex.

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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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David R. Ryglicki, Daniel Hodyss, and Gregory Rainwater

Abstract

The interactions between the outflow of a tropical cyclone (TC) and its background flow are explored using a hierarchy of models of varying complexity. Previous studies have established that, for a select class of TCs that undergo rapid intensification in moderate values of vertical wind shear, the upper-level outflow of the TC can block and reroute the environmental winds, thus reducing the shear and permitting the TC to align and subsequently to intensify. We identify in satellite imagery and reanalysis datasets the presence of tilt nutations and evidence of upwind blocking by the divergent wind field, which are critical components of atypical rapid intensification. We then demonstrate how an analytical expression and a shallow water model can be used to explain some of the structure of upper-level outflow. The analytical expression shows that the dynamic high inside the outflow front is a superposition of two pressure anomalies caused by the outflow’s deceleration by the environment and by the environment’s deceleration by the outflow. The shallow water model illustrates that the blocking is almost entirely dependent upon the divergent component of the wind. Then, using a divergent kinetic energy budget analysis, we demonstrate that, in a full-physics TC, upper-level divergent flow generation occurs in two phases: pressure driven and then momentum driven. The change happens when the tilt precession reaches left of shear. When this change occurs, the outflow blocking extends upshear. We discuss these results with regard to prior severe weather studies.

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