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Chris Snyder
,
Thomas Bengtsson
,
Peter Bickel
, and
Jeff Anderson

Abstract

Particle filters are ensemble-based assimilation schemes that, unlike the ensemble Kalman filter, employ a fully nonlinear and non-Gaussian analysis step to compute the probability distribution function (pdf) of a system’s state conditioned on a set of observations. Evidence is provided that the ensemble size required for a successful particle filter scales exponentially with the problem size. For the simple example in which each component of the state vector is independent, Gaussian, and of unit variance and the observations are of each state component separately with independent, Gaussian errors, simulations indicate that the required ensemble size scales exponentially with the state dimension. In this example, the particle filter requires at least 1011 members when applied to a 200-dimensional state. Asymptotic results, following the work of Bengtsson, Bickel, and collaborators, are provided for two cases: one in which each prior state component is independent and identically distributed, and one in which both the prior pdf and the observation errors are Gaussian. The asymptotic theory reveals that, in both cases, the required ensemble size scales exponentially with the variance of the observation log likelihood rather than with the state dimension per se.

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Christopher Davis
,
Chris Snyder
, and
Anthony C. Didlake Jr.

Abstract

Tropical cyclone formation over the eastern Pacific during 2005 and 2006 was examined using primarily global operational analyses from the National Centers for Environmental Prediction. This paper represents a “vortex view” of genesis, adding to previous work on tropical cyclone formation associated with tropical waves. Between 1 July and 30 September during 2005 and 2006, vortices at 900 hPa were tracked and vortex-following diagnostic quantities were computed. Vortices were more abundant during periods of an enhanced “Hadley” circulation with monsoon westerlies around 10°N in the lower troposphere. This zonally confined Hadley circulation was significantly stronger during the genesis of developing vortices. Developing vortices were stronger at the outset, with a deeper potential vorticity maximum, compared to nondeveloping vortices. This implies that developing disturbances were selected early on by favorable synoptic-scale features.

The characteristic time-mean reversal of the meridional gradient of absolute vorticity in the lower troposphere was found to nearly vanish when the aggregate contribution of strong vortices was removed from the time-mean vorticity. This finding implies that it is difficult to unambiguously attribute development to a preexisting enhancement of vorticity on the synoptic scale. The time-mean enhancement of cyclonic vorticity primarily results from the accumulated effect of vortices. It is suggested that horizontal deformation in the background state helps distinguish developing vortices from nondevelopers, and also biases the latitude of development poleward of the climatological ITCZ axis.

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Altuğ Aksoy
,
David C. Dowell
, and
Chris Snyder

Abstract

The quality of convective-scale ensemble forecasts, initialized from analysis ensembles obtained through the assimilation of radar observations using an ensemble Kalman filter (EnKF), is investigated for cases whose behaviors span supercellular, linear, and multicellular organization. This work is the companion to , which focused on the quality of analyses during the 60-min analysis period. Here, the focus is on 30-min ensemble forecasts initialized at the end of that period. As in , the Weather Research and Forecasting (WRF) model is employed as a simplified cloud model at 2-km horizontal grid spacing. Various observation-space and state-space verification metrics, computed both for ensemble means and individual ensemble members, are employed to assess the quality of ensemble forecasts comparatively across cases. While the cases exhibit noticeable differences in predictability, the forecast skill in each case, as measured by various metrics, decays on a time scale of tens of minutes. The ensemble spread also increases rapidly but significant outlier members or clustering among members are not encountered. Forecast quality is seen to be influenced to varying degrees by the respective initial soundings. While radar data assimilation is able to partially mitigate some of the negative effects in some situations, the supercell case, in particular, remains difficult to predict even after 60 min of data assimilation.

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Sangil Kim
,
R. M. Samelson
, and
Chris Snyder

Abstract

Estimates of three components of an uncertainty budget for a coastal ocean model in a wind-forced regime are made based on numerical simulations. The budget components behave differently in the shelf regime, inshore of the 200-m isobath, and the slope-interior regime, between the 200-m isobath and a fixed longitude (126°W) that is roughly 150 km offshore. The first of the three budget components is an estimate of the uncertainty in the ocean state given only a known history of wind stress forcing, with errors in the wind forcing estimated from differences between operational analyses. It is found that, over the continental shelf, the response to wind forcing is sufficiently strong and deterministic that significant skill in estimating shelf circulation can be achieved with knowledge only of the wind forcing, and no ocean data, for wind fields with these estimated errors. The second involves initial condition error and its influence on uncertainty, including both error growth with time from well-known initial conditions and error decay with time from poorly known initial conditions but with well-known wind forcing. The third component is that of boundary condition error and its influence on the interior solutions, including the dependence of that influence on the specific location along the boundary of the boundary condition error. Boundary condition errors with amplitude comparable to the root-mean-square variability at the boundary lead eventually to errors equal to the root-mean-square variability in the slope-interior regime, and somewhat smaller errors in the shelf regime. Covariance estimates based on differences of the wind-forced solutions from the ensemble mean are not dramatically different from those based on the full fields, and do not show strong state dependence.

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Altuğ Aksoy
,
David C. Dowell
, and
Chris Snyder

Abstract

The effectiveness of the ensemble Kalman filter (EnKF) for assimilating radar observations at convective scales is investigated for cases whose behaviors span supercellular, linear, and multicellular organization. The parallel EnKF algorithm of the Data Assimilation Research Testbed (DART) is used for data assimilation, while the Weather Research and Forecasting (WRF) Model is employed as a simplified cloud model at 2-km horizontal grid spacing. In each case, reflectivity and radial velocity measurements are utilized from a single Weather Surveillance Radar-1988 Doppler (WSR-88D) within the U.S. operational network. Observations are assimilated every 2 min for a duration of 60 min and correction of folded radial velocities occurs within the EnKF. Initial ensemble uncertainty includes random perturbations to the horizontal wind components of the initial environmental sounding. The EnKF performs effectively and with robust results across all the cases. Over the first 18–30 min of assimilation, the rms and domain-averaged prior fits to observations in each case improve significantly from their initial levels, reaching comparable values of 3–6 m s−1 and 7–10 dBZ. Representation of mesoscale uncertainty, albeit in the simplest form of initial sounding perturbations, is a critical part of the assimilation system, as it increases ensemble spread and improves filter performance. In addition, assimilation of “no precipitation” observations (i.e., reflectivity observations with values small enough to indicate the absence of precipitation) serves to suppress spurious convection in ensemble members. At the same time, it is clear that the assimilation is far from optimal, as the ensemble spread is consistently smaller than what would be expected from the innovation statistics and the assumed observation-error variance.

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Xuguang Wang
,
Chris Snyder
, and
Thomas M. Hamill

Abstract

Hybrid ensemble–three-dimensional variational analysis schemes incorporate flow-dependent, ensemble-estimated background-error covariances into the three-dimensional variational data assimilation (3DVAR) framework. Typically the 3DVAR background-error covariance estimate is assumed to be stationary, nearly homogeneous, and isotropic. A hybrid scheme can be achieved by 1) directly replacing the background-error covariance term in the cost function by a linear combination of the original background-error covariance with the ensemble covariance or 2) through augmenting the state vector with another set of control variables preconditioned upon the square root of the ensemble covariance. These differently proposed hybrid schemes are proven to be equivalent. The latter framework may be a simpler way to incorporate ensemble information into operational 3DVAR schemes, where the preconditioning is performed with respect to the background term.

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Steven M. Cavallo
,
Judith Berner
, and
Chris Snyder

Abstract

Accurate predictions in numerical weather models depend on the ability to accurately represent physical processes across a wide range of scales. This paper evaluates the utility of model time tendencies, averaged over many forecasts at a given lead time, to diagnose systematic forecast biases in the Advanced Research version of the Weather Research and Forecasting (WRF) Model during the 2010 North Atlantic hurricane season using continuously cycled ensemble data assimilation (DA). Erroneously strong low-level heating originates from the planetary boundary layer parameterization as a consequence of using fixed sea surface temperatures, impacting the upward surface sensible heat fluxes. Warm temperature bias is observed with a magnitude 0.5 K in a deep tropospheric layer centered 700 hPa, originating primarily from the Kain–Fritsch convective parameterization.

This study is the first to diagnose systematic forecast bias in a limited-area mesoscale model using its forecast tendencies. Unlike global models where relatively fewer time steps typically encompass a DA cycling period, averaging all short-term forecast tendencies can require potentially large data. It is shown that 30-min averaging intervals can sufficiently represent the systematic model bias in this modeling configuration when initializing forecasts from an ensemble member that is generated using a DA system with an identical model configuration. However, the number of time steps before model error begins to dominate initial condition (IC) errors may vary between modeling configurations. Model and IC error are indistinguishable in short-term forecasts when initialized from the ensemble mean, a global analysis from a different model, and an ensemble member using a different parameterization.

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Richard Rotunno
,
David J. Muraki
, and
Chris Snyder

Abstract

Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophic theory also represents a leading-order theory in the sense that it is derivable from the primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophic theory, and the centrality of potential vorticity, the authors have recently developed a systematic asymptotic framework from which balanced, next-order corrections in Rossby number can be obtained. The approach is illustrated here through numerical solutions pertaining to unstable waves on baroclinic jets. The numerical solutions using the full primitive equations compare well with numerical solutions to our equations with accuracy one order beyond quasigeostrophic theory; in particular, the inherent asymmetry between cyclones and anticyclones is captured. Explanations of the latter and the associated asymmetry of the warm and cold fronts are given using simple extensions of quasigeostrophic– potential-vorticity thinking to next order.

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David J. Muraki
,
Chris Snyder
, and
Richard Rotunno

Abstract

Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophy also represents a leading-order theory in the sense that it is derivable from the full primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophy, and the centrality of potential vorticity, a systematic asymptotic framework is developed from which balanced, next-order corrections in Rossby number are obtained. The simplicity of the approach is illustrated by explicit construction of the next-order corrections to a finite-amplitude Eady edge wave.

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Chris Snyder
,
Riwal Plougonven
, and
David J. Muraki

Abstract

Previous simulations of dipole vortices propagating through rotating, stratified fluid have revealed small-scale inertia–gravity waves that are embedded within the dipole near its leading edge and are approximately stationary relative to the dipole. The mechanism by which these waves are generated is investigated, beginning from the observation that the dipole can be reasonably approximated by a balanced quasigeostrophic (QG) solution. The deviations from the QG solution (including the waves) then satisfy linear equations that come from linearization of the governing equations about the QG dipole and are forced by the residual tendency of the QG dipole (i.e., the difference between the time tendency of the QG solution and that of the full primitive equations initialized with the QG fields). The waves do not appear to be generated by an instability of the balanced dipole, as homogeneous solutions of the linear equations amplify little over the time scale for which the linear equations are valid. Linear solutions forced by the residual tendency capture the scale, location, and pattern of the inertia–gravity waves, although they overpredict the wave amplitude by a factor of 2. There is thus strong evidence that the waves are generated as a forced linear response to the balanced flow. The relation to and differences from other theories for wave generation by balanced flows, including those of Lighthill and Ford et al., are discussed.

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