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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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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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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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Chris Snyder and Gregory J. Hakim

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Singular vectors (SVs) have been applied to cyclogenesis, to initializing ensemble forecasts, and in predictability studies. Ideally, the calculation of the SVs would employ the analysis error covariance norm at the initial time or, in the case of cyclogenesis, a norm based on the statistics of initial perturbations, but the energy norm is often used as a more practical substitute.

To illustrate the roles of the choice of norm and the vertical structure of initial perturbations, an upper-level wave with no potential vorticity perturbation in the troposphere is considered as a typical cyclogenetic perturbation or analysis error, and this perturbation is then decomposed by its projection onto each energy SV. All calculations are made, for simplicity, in the context of the quasigeostrophic Eady model (i.e., for a background flow with constant vertical shear and horizontal temperature gradient). Viewed in terms of the energy SVs, the smooth vertical structure of the typical perturbation, as well as its evolution, results from strong cancellation between the growing and decaying SVs, most of which are highly structured and tilted in the vertical.

A simpler picture, involving less cancellation, follows from decomposition of the typical perturbation into SVs using an alternative initial norm, which is based on the relation between initial norms and the statistics of initial perturbations together with the empirical assumption that the initial perturbations are not dominated by interior potential vorticity. Differences between the energy SVs and those based on the alternative initial norm can be understood by noting that the energy norm implicitly assumes initial perturbations with second-order statistics given by the covariance matrix whose inverse defines the energy norm. Unlike the “typical” perturbation, perturbations with those statistics have large variance of potential vorticity in the troposphere and fine vertical structure.

Finally, a brief assessment is presented of the extent to which the upper wave, and more generally the alternative initial norm, is representative of cyclogenetic perturbations and analysis errors. There is substantial evidence supporting deep perturbations with little vertical structure as frequent precursors to cyclogenesis, but surrogates for analysis errors are less conclusive: operational midlatitude analysis differences have vertical structure similar to that of the perturbations implied by the energy norm, while short-range forecast errors and analysis errors from assimilation experiments with simulated observations are more consistent with the alternative norm.

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Ryan D. Torn and Chris Snyder

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With the growing use of tropical cyclone (TC) best-track information for weather and climate applications, it is important to understand the uncertainties that are contained in the TC position and intensity information. Here, an attempt is made to quantify the position uncertainty using National Hurricane Center (NHC) advisory information, as well as intensity uncertainty during times without aircraft data, by verifying Dvorak minimum sea level pressure (SLP) and maximum wind speed estimates during times with aircraft reconnaissance information during 2000–09. In a climatological sense, TC position uncertainty decreases for more intense TCs, while the uncertainty of intensity, measured by minimum SLP or maximum wind speed, increases with intensity. The standard deviation of satellite-based TC intensity estimates can be used as a predictor of the consensus intensity error when that consensus includes both Dvorak and microwave-based estimates, but not when it contains only Dvorak-based values. Whereas there has been a steady decrease in seasonal TC position uncertainty over the past 10 yr, which is likely due to additional data available to NHC forecasters, the seasonal TC minimum SLP and maximum wind speed values are fairly constant, with year-to-year variability due to the mean intensity of all TCs during that season and the frequency of aircraft reconnaissance.

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Michael L. Waite and Chris Snyder

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The atmospheric mesoscale kinetic energy spectrum is investigated through numerical simulations of an idealized baroclinic wave life cycle, from linear instability to mature nonlinear evolution and with high horizontal and vertical resolution (Δx ≈ 10 km and Δz ≈ 60 m). The spontaneous excitation of inertia–gravity waves yields a shallowing of the mesoscale spectrum with respect to the large scales, in qualitative agreement with observations. However, this shallowing is restricted to the lower stratosphere and does not occur in the upper troposphere. At both levels, the mesoscale divergent kinetic energy spectrum—a proxy for the inertia–gravity wave energy spectrum—resembles a −5/3 power law in the mature stage. Divergent kinetic energy dominates the lower stratospheric mesoscale spectrum, accounting for its shallowing. Rotational kinetic energy, by contrast, dominates the upper tropospheric spectrum and no shallowing of the full spectrum is observed. By analyzing the tendency equation for the kinetic energy spectrum, it is shown that the lower stratospheric spectrum is not governed solely by a downscale energy cascade; rather, it is influenced by the vertical pressure flux divergence associated with vertically propagating inertia–gravity waves.

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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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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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