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Chris Snyder
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Laura Slivinski and Chris Snyder

Abstract

Particle filtering methods for data assimilation may suffer from the “curse of dimensionality,” where the required ensemble size grows rapidly as the dimension increases. It would, therefore, be useful to know a priori whether a particle filter is feasible to implement in a given system. Previous work provides an asymptotic relation between the necessary ensemble size and an exponential function of , a statistic that depends on observation-space quantities and that is related to the system dimension when the number of observations is large; for linear, Gaussian systems, the statistic can be computed from eigenvalues of an appropriately normalized covariance matrix. Tests with a low-dimensional system show that these asymptotic results remain useful when the system is nonlinear, with either the standard or optimal proposal implementation of the particle filter. This study explores approximations to the covariance matrices that facilitate computation in high-dimensional systems, as well as different methods to estimate the accumulated system noise covariance for the optimal proposal. Since may be approximated using an ensemble from a simpler data assimilation scheme, such as the ensemble Kalman filter, the asymptotic relations thus allow an estimate of the ensemble size required for a particle filter before its implementation. Finally, the improved performance of particle filters with the optimal proposal, relative to those using the standard proposal, in the same low-dimensional system is demonstrated.

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Chris Snyder and Fuqing Zhang

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Assimilation of Doppler radar data into cloud models is an important obstacle to routine numerical weather prediction for convective-scale motions; the difficulty lies in initializing fields of wind, temperature, moisture, and condensate given only observations of radial velocity and reflectivity from the radar. This paper investigates the potential of the ensemble Kalman filter (EnKF), which estimates the covariances between observed variables and the state through an ensemble of forecasts, to assimilate radar observations at convective scales. In the basic experiment, simulated observations are extracted from a reference simulation of a splitting supercell and assimilated using the EnKF and the same numerical model that produced the reference simulation. The EnKF produces accurate analyses, including the unobserved variables, after roughly 30 min (or six scans) of radial velocity observations. Additional experiments, in which forecasts are made from the ensemble-mean analysis, reveal that forecast errors grow significantly in this simple system, so that the ability of the EnKF to track the reference solution is not simply because of stable system dynamics. It is also found that the covariances between radial velocity and temperature, moisture, and condensate are important to the quality of the analyses, as is the initialization chosen for the ensemble members prior to assimilating the first observations. These results are promising, especially given the ease of implementing the EnKF. A number of important issues remain, however, including the initialization of the ensemble prior to the first observation, the treatment of uncertainty in the environmental sounding, the role of error in the forecast model (particularly the microphysical parameterizations), and the treatment of lateral boundary conditions.

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Riwal Plougonven and Chris Snyder

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The spontaneous generation of inertia–gravity waves in idealized life cycles of baroclinic instability is investigated using the Weather Research and Forecasting Model. Two substantially different life cycles of baroclinic instability are obtained by varying the initial zonal jet. The wave generation depends strongly on the details of the baroclinic wave’s development. In the life cycle dominated by cyclonic behavior, the most conspicuous gravity waves are excited by the upper-level jet and are broadly consistent with previous simulations of O’Sullivan and Dunkerton. In the life cycle that is dominated by anticyclonic behavior, the most conspicuous gravity waves even in the stratosphere are excited by the surface fronts, although the fronts are no stronger than in the cyclonic life cycle. The anticyclonic life cycle also reveals waves in the lower stratosphere above the upper-level trough of the baroclinic wave; these waves have not been previously identified in idealized simulations. The sensitivities of the different waves to both resolution and dissipation are discussed.

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Yongsheng Chen and Chris Snyder

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Observations of hurricane position, which in practice might be available from satellite or radar imagery, can be easily assimilated with an ensemble Kalman filter (EnKF) given an operator that computes the position of the vortex in the background forecast. The simple linear updating scheme used in the EnKF is effective for small displacements of forecasted vortices from the true position; this situation is operationally relevant since hurricane position is often available frequently in time. When displacements of the forecasted vortices are comparable to the vortex size, non-Gaussian effects become significant and the EnKF’s linear update begins to degrade. Simulations using a simple two-dimensional barotropic model demonstrate the potential of the technique and show that the track forecast initialized with the EnKF analysis is improved. The assimilation of observations of the vortex shape and intensity, along with position, extends the technique’s effectiveness to larger displacements of the forecasted vortices than when assimilating position alone.

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Joshua P. Hacker and Chris Snyder

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In situ surface layer observations are a rich data source that could be more effectively utilized in NWP applications. If properly assimilated, data from existing mesonets could improve initial conditions and lower boundary conditions, leading to the possibility of improved simulation and short-range forecasts of slope flows, sea breezes, convective initiation, and other PBL circulations.

A variance–covariance climatology is constructed by extracting a representative column from real-time mesoscale forecasts over the Southern Great Plains, and used to explore the potential for estimating the state of the PBL by assimilating surface observations. A parameterized 1D PBL model and an ensemble Kalman filter (EnKF) approach to assimilation are used to test this potential. Analysis focuses on understanding how effectively the EnKF can spread the surface observations vertically to constrain the state of the PBL model. Results confirm that assimilating surface observations can substantially improve the state of a modeled PBL. Experiments to estimate the moisture availability parameter through the data assimilation system show that the EnKF is a viable tool for parameter estimation, and may help mitigate model error in forecasting and simulating the PBL.

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Chris Snyder and Richard S. Lindzen

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In this study, the free-shear problem, a minimal version of baroclinic, quasi-geostrophic wave-CISK, is analyzed. The basic state consists of a zonal flow, unbounded above and below, with constant vertical shear and Brunt-Väisälä frequency and zero meridional gradient of the potential vorticity; and convective heating is parameterized in terms of the convergence below an arbitrary level. Because of the sensitivity to the vertical distribution of the parameterized heating typical of wave-CISK models, a simple thermodynamic constraint on the heating profile is used to broadly identify appropriate parameter regimes. The unstable waves in the free-shear problem grow rapidly and share many structural characteristics with dry baroclinic waves, although the dynamical process associated with dry baroclinic instability is absent; consideration of the potential vorticity dynamics of the unstable modes illustrates how heating may act as a dynamical surrogate for potential vorticity gradients. Although highly idealized, the free-shear problem also explains much of the behavior of more general wave-CISK models.

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Thomas M. Hamill and Chris Snyder

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A hybrid ensemble Kalman filter–three-dimensional variational (3DVAR) analysis scheme is demonstrated using a quasigeostrophic model under perfect-model assumptions. Four networks with differing observational densities are tested, including one network with a data void. The hybrid scheme operates by computing a set of parallel data assimilation cycles, with each member of the set receiving unique perturbed observations. The perturbed observations are generated by adding random noise consistent with observation error statistics to the control set of observations. Background error statistics for the data assimilation are estimated from a linear combination of time-invariant 3DVAR covariances and flow-dependent covariances developed from the ensemble of short-range forecasts. The hybrid scheme allows the user to weight the relative contributions of the 3DVAR and ensemble-based background covariances.

The analysis scheme was cycled for 90 days, with new observations assimilated every 12 h. Generally, it was found that the analysis performs best when background error covariances are estimated almost fully from the ensemble, especially when the ensemble size was large. When small-sized ensembles are used, some lessened weighting of ensemble-based covariances is desirable. The relative improvement over 3DVAR analyses was dependent upon the observational data density and norm; generally, there is less improvement for data-rich networks than for data-poor networks, with the largest improvement for the network with the data void. As expected, errors depend on the size of the ensemble, with errors decreasing as more ensemble members are added. The sets of initial conditions generated from the hybrid are generally well calibrated and provide an improved set of initial conditions for ensemble forecasts.

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Chris Snyder and Thomas M. Hamill

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Leading Lyapunov exponents and vectors are calculated for a turbulent baroclinic jet in a quasigeostrophic model with O(105) degrees of freedom. The leading exponent is close to 0.4 day−1, and the unstable subspace has dimension between 30 and 40. The leading Lyapunov vectors exhibit a strong correlation of their potential vorticity (PV) with the PV gradients of the unperturbed flow. These perturbations do not, however, appear to be instabilities of smaller scale on the turbulent flow. Instead, they share the scales of the flow itself (at least if measured along PV contours) and often simply represent local phase shifts or displacements of existing features in the flow. Singular vectors constrained to the subspace of Lyapunov vectors are also calculated. Maximum amplification factors over 2 days are, on average, about 6, 7.5, and 9 (compared to the factor of 2 implied by the leading exponent) for subspaces of the leading 20, 35, and 60 Lyapunov vectors, respectively.

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Jeffrey S. Whitaker and Chris Snyder

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The effects of spherical geometry on the nonlinear evolution of baroclinic waves are investigated by comparing integrations of a two-layer primitive equation (PE) model in spherical and Cartesian geometry. To isolate geometrical effects, the integrations use basic states with nearly identical potential vorticity (PV) structure.

Although the linear normal modes are very similar, significant differences develop at finite amplitude. Anticyclones (cyclones) in spherical geometry are relatively stronger (weaker) than those in Cartesian geometry. For this basic state, the strong anticyclones on the sphere are associated with anticyclonic wrapping of high PV in the upper layer (i.e., high PV air is advected southward and westward relative to the wave). In Cartesian geometry, large quasi-barotropic cyclonic vortices develop, and no anticyclonic wrapping of PV occurs. Because of their influence on the synoptic-scale flow, spherical geometric effects also lead to significant differences in the structure of mesoscale frontal features.

A standard midlatitude scale analysis indicates that the effects of sphericity enter in the next-order correction to β-plane quasigeostrophic (QG) dynamics. At leading order these spherical terms only affect the PV inversion operator (through the horizontal Laplacian) and the advection of PV by the nondivergent wind. Scaling arguments suggest, and numerical integrations of the barotropic vorticity equation confirm, that the dominant geometric effects are in the PV inversion operator. The dominant metric in the PV inversion operator is associated with the equatorward spreading of meridians on the sphere, and causes the anticyclonic (cyclonic) circulations in the spherical integration to become relatively stronger (weaker) than those in the Cartesian integration.

This study demonstrates that the effects of spherical geometry can be as important as the leading-order ageostrophic effects in determining the structure of evolution of dry baroclinic waves and their embedded mesoscale structures.

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