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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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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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Rebecca E. Morss
,
Chris Snyder
, and
Richard Rotunno

Abstract

Results from homogeneous, isotropic turbulence suggest that predictability behavior is linked to the slope of a flow’s kinetic energy spectrum. Such a link has potential implications for the predictability behavior of atmospheric models. This article investigates these topics in an intermediate context: a multilevel quasigeostrophic model with a jet and temperature perturbations at the upper surface (a surrogate tropopause). Spectra and perturbation growth behavior are examined at three model resolutions. The results augment previous studies of spectra and predictability in quasigeostrophic models, and they provide insight that can help interpret results from more complex models. At the highest resolution tested, the slope of the kinetic energy spectrum is approximately at the upper surface but −3 or steeper at all but the uppermost interior model levels. Consistent with this, the model’s predictability behavior exhibits key features expected for flow with a shallower than −3 slope. At the highest resolution, upper-surface perturbation spectra peak below the energy-containing scales, and the error growth rate decreases as small scales saturate. In addition, as model resolution is increased and smaller scales are resolved, the peak of the upper-surface perturbation spectra shifts to smaller scales and the error growth rate increases. The implications for potential predictive improvements are not as severe, however, as in the standard picture of flows exhibiting a finite predictability limit. At the highest resolution, the model also exhibits periods of much faster-than-average perturbation growth that are associated with faster growth at smaller scales, suggesting predictability behavior that varies with time.

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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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Riwal Plougonven
,
David J. Muraki
, and
Chris Snyder
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Robert G. Nystrom
,
Chris Snyder
, and
Mohamad El Gharamti

Abstract

In this study, a one-step-ahead ensemble Kalman smoother (EnKS) is introduced for the purposes of parameter estimation. The potential for this system to provide new constraints on the surface-exchange coefficients of momentum (Cd ) and enthalpy (Ck ) is then explored using a series of observing system simulation experiments (OSSEs). The surface-exchange coefficients to be estimated within the data assimilation system are highly uncertain, especially at high wind speeds, and are well known to be important model parameters influencing the intensity and structure of tropical cyclones in numerical simulations. One major benefit of the developed one-step-ahead EnKS is that it allows for simultaneous updates of the rapidly evolving model state variables found in tropical cyclones using a short assimilation window and a long smoother window for the parameter updates, granting sufficient time for sensitivity to the parameters to develop. Overall, OSSEs demonstrate potential for this approach to accurately constrain parameters controlling the amplitudes of Cd and Ck , but the degree of success in recovering the truth model parameters varies throughout the tropical cyclone life cycle. During the rapid intensification phase, rapidly growing errors in the model state project onto the parameter updates and result in an overcorrection of the parameters. After the rapid intensification phase, however, the parameters are correctly adjusted back toward the truth values. Last, the relative success of parameter estimation in recovering the truth model parameter values also has sensitivity to the ensemble size and smoothing forecast length, each of which are explored.

Significance Statement

Large uncertainty in the surface-exchange coefficients of momentum and heat/moisture exists for hurricane conditions. This is a problem because the numerical weather model predictions of hurricane intensity and storm structure are sensitive to the surface-exchange coefficient values used. In this study we use data assimilation, or the relationships estimated between the surface-exchange coefficients and forecasted observations, to constrain uncertainty in the model’s surface-exchange coefficient values. More specifically, an approach to limit both the rapidly growing errors associated with the hurricane itself and the hurricane’s accumulated response to the surface-exchange coefficient values is presented. Overall, this approach has potential to accurately estimate the surface-exchange coefficients, but the success depends on the number of forecast realizations used and how rapidly the hurricane is changing.

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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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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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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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Kathryn R. Fossell
,
David Ahijevych
,
Rebecca E. Morss
,
Chris Snyder
, and
Chris Davis

Abstract

The potential for storm surge to cause extensive property damage and loss of life has increased urgency to more accurately predict coastal flooding associated with landfalling tropical cyclones. This work investigates the sensitivity of coastal inundation from storm tide (surge + tide) to four hurricane parameters—track, intensity, size, and translation speed—and the sensitivity of inundation forecasts to errors in forecasts of those parameters. An ensemble of storm tide simulations is generated for three storms in the Gulf of Mexico, by driving a storm surge model with best track data and systematically generated perturbations of storm parameters from the best track. The spread of the storm perturbations is compared to average errors in recent operational hurricane forecasts, allowing sensitivity results to be interpreted in terms of practical predictability of coastal inundation at different lead times. Two types of inundation metrics are evaluated: point-based statistics and spatially integrated volumes. The practical predictability of surge inundation is found to be limited foremost by current errors in hurricane track forecasts, followed by intensity errors, then speed errors. Errors in storm size can also play an important role in limiting surge predictability at short lead times, due to observational uncertainty. Results show that given current mean errors in hurricane forecasts, location-specific surge inundation is predictable for as little as 12–24 h prior to landfall, less for small-sized storms. The results also indicate potential for increased surge predictability beyond 24 h for large storms by considering a storm-following, volume-integrated metric of inundation.

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