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Thomas M. Hamill
and
Chris Snyder

Abstract

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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Thomas M. Hamill
and
Chris Snyder

Abstract

A method for determining adaptive observation locations is demonstrated. This method is based on optimal estimation (Kalman filter) theory; it determines the observation location that will maximize the expected improvement, which can be measured in terms of the expected reduction in analysis or forecast variance. This technique requires an accurate model for background error statistics that vary both in space and in time. Here, these covariances are generated using an ensemble Kalman filter assimilation scheme. A variant is also developed that can estimate the analysis improvement in data assimilation schemes where background error statistics are less accurate.

This approach is demonstrated using a quasigeostrophic channel model under perfect-model assumptions. The algorithm is applied here to find the supplemental rawinsonde location to add to a regular network of rawinsondes that will reduce analysis errors the most. The observation network is configured in this experiment so there is a data void in the western third of the domain. One-hundred-member ensembles from three data assimilation schemes are tested as input to the target selection procedure, two variants of the standard ensemble Kalman filter and a third perturbed observation (3DVAR) ensemble. The algorithm is shown to find large differences in the expected variance reduction depending on the observation location, the flow of the day, and the ensemble used in the adaptive observation algorithm. When using the two variants of the ensemble Kalman filter, the algorithm defined consistently similar adaptive locations to each other, and assimilation of the adaptive observation typically reduced analysis errors significantly. When the 3DVAR ensemble was used, the algorithm picked very different observation locations and the analyses were not improved as much.

The amount of improvement from assimilating a supplemental adaptive observation instead of a fixed observation in the middle of the void depended on whether the observation was assimilated sporadically or during every analysis cycle. For sporadic assimilation, the adaptive observation provided a dramatic improvement relative to the supplemental fixed observation. When an adaptive observation was regularly assimilated every cycle, the improvement was smaller.

For the sporadic assimilation of an adaptive observation, targeting based simply on the maximum spread in background forecasts provided similar target locations and similar analysis improvements to those generated with the full algorithm. The improvement from the regular assimilation of an adaptive observation based on the spread algorithm was no larger than when observations from a fixed target in the middle of the void were regularly assimilated.

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F. Zhang
,
Chris Snyder
, and
Richard Rotunno

Abstract

A mesoscale model is used here to investigate the possible sources of forecast error for the 24–25 January 2000 snowstorm along the east coast of the United States. The primary focus is the quantitative precipitation forecast out to lead times of 36 h. The success of the present high-resolution control forecast shows that the storm could have been well forecasted with conventional data in real time. Various experiments suggest that insufficient model grid resolution and errors in the initial conditions both contributed significantly to problems in the forecast. Other experiments, motivated by the possibility that the forecast errors arose from the operational analysis poorly fitting one or two key soundings, test the effects of withholding single soundings from the control initial conditions. While no single sounding results in forecast changes that are more than a small fraction of the error in the operational forecast, these experiments do reveal that the detailed mesoscale distribution of precipitation in the 24- or 36-h forecast can be significantly altered even by such small changes in the initial conditions. The experiments also reveal that the forecast changes arise from the rapid growth of error at scales below 500 km in association with moist processes. The results presented emphasize the difficulty of forecasting precipitation relative to, say, surface pressure and suggest that the predictability of mesoscale precipitation features in cases of the type studied here may be limited to less than 2–3 days.

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Joshua P. Hacker
and
Chris Snyder

Abstract

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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F. Zhang
,
Chris Snyder
, and
Juanzhen Sun

Abstract

The ensemble Kalman filter (EnKF) uses an ensemble of short-range forecasts to estimate the flow-dependent background error covariances required in data assimilation. The feasibility of the EnKF for convective-scale data assimilation has been previously demonstrated in perfect-model experiments using simulated observations of radial velocity from a supercell storm. The present study further explores the potential and behavior of the EnKF at convective scales by considering more realistic initial analyses and variations in the availability and quality of the radar observations. Assimilation of simulated radial-velocity observations every 5 min where there is significant reflectivity using 20 ensemble members proves to be successful in most realistic observational scenarios for simulated supercell thunderstorms, although the same degree of success may not be readily expected with real observations and an imperfect model, at least with the present EnKF implementation. Even though the filter converges toward the truth simulation faster from a better initial estimate, an experiment with the initial estimate of the supercell displaced by 10 km still yields an accurate estimate of the storm for both observed and unobserved variables within 40 min. Similarly, radial-velocity observations below 2 km are certainly beneficial to capturing the storm (especially the detailed cold pool structure), but in their absence the assimilation scheme can still achieve a comparably accurate estimate of the state of the storm given a slightly longer assimilation period. An experiment with radar observations only above 4 km fails to assimilate the storm properly, but, with the addition of a hypothetical surface mesonet taking wind and temperature observations, the EnKF can again provide a good estimate of the storm. The supercell can also be successfully assimilated in the case of radar observations only below 4 km (such as those from the ground-based mobile radars). More frequent observations can help the storm assimilation initially, but the benefit diminishes after half an hour. Results presented here indicate that the vertical resolution and the uncertainty of observations, for the typical range of most of the observational radars, would have little impact on the overall performance of the EnKF in assimilating the storm.

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David J. Muraki
and
Chris Snyder

Abstract

A new class of exact vortex dipole solutions is derived for surface quasigeostrophic (sQG) models. The solutions extend the two-dimensional barotropic modon to fully three-dimensional, continuously stratified flow and are a simple model of localized jets on the tropopause. In addition to the basic sQG dipole, dipole structures exist for a layer of uniform potential vorticity between two rigid boundaries and for a dipole in the presence of uniform background vertical shear and horizontal potential temperature gradient. In the former case, the solution approaches the barotropic Lamb dipole in the limit of a layer that is shallow relative to the Rossby depth based on the dipole’s radius. In the latter case, dipoles that are bounded in the far field must propagate counter to the phase speed of the linear edge waves associated with the surface temperature gradient.

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F. Zhang
,
Chris Snyder
, and
Richard Rotunno

Abstract

In a previous study by the authors, it was shown that the problematic numerical prediction of the 24–25 January 2000 snowstorm along the east coast of the United States was in some measure due to rapid error growth at scales below 500 km. In particular they found that moist processes were responsible for this strong initial-condition sensitivity of the 1–2-day prediction of mesoscale forecast aspects. In the present study they take a more systematic look at the processes by which small initial differences (“errors”) grow in those numerical forecasts. For initial errors restricted to scales below 100 km, results show that errors first grow as small-scale differences associated with moist convection, then spread upscale as their growth begins to slow. In the context of mesoscale numerical predictions with 30-km resolution, the initial growth is associated with nonlinearities in the convective parameterization (or in the explicit microphysical parameterizations, if no convective parameterization is used) and proceeds at a rate that increases as the initial error amplitude decreases. In higher-resolution (3.3 km) simulations, errors first grow as differences in the timing and position of individual convective cells. Amplification at that stage occurs on a timescale on the order of 1 h, comparable to that of moist convection. The errors in the convective-scale motions subsequently influence the development of meso- and larger-scale forecast aspects such as the position of the surface low and the distribution of precipitation, thus providing evidence that growth of initial errors from convective scales places an intrinsic limit on the predictability of larger scales.

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Michael L. Waite
and
Chris Snyder
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Michael L. Waite
and
Chris Snyder

Abstract

The role of moist processes in the development of the mesoscale kinetic energy spectrum is investigated with numerical simulations of idealized moist baroclinic waves. Dry baroclinic waves yield upper-tropospheric kinetic energy spectra that resemble a −3 power law. Decomposition into horizontally rotational and divergent kinetic energy shows that the divergent energy has a much shallower spectrum, but its amplitude is too small to yield a characteristic kink in the total spectrum, which is dominated by the rotational part. The inclusion of moist processes energizes the mesoscale. In the upper troposphere, the effect is mainly in the divergent part of the kinetic energy; the spectral slope remains shallow (around − ) as in the dry case, but the amplitude increases with increasing humidity. The divergence field in physical space is consistent with inertia–gravity waves being generated in regions of latent heating and propagating throughout the baroclinic wave. Buoyancy flux spectra are used to diagnose the scale at which moist forcing—via buoyant production from latent heating—injects kinetic energy. There is significant input of kinetic energy in the mesoscale, with a peak at scales of around 800 km and a plateau at smaller scales. If the latent heating is artificially set to zero at some time, the enhanced divergent kinetic energy decays over several days toward the level obtained in the dry simulation. The effect of moist forcing of mesoscale kinetic energy presents a challenge for theories of the mesoscale spectrum based on the idealization of a turbulent inertial subrange.

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Chris Snyder
and
Thomas M. Hamill

Abstract

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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