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Richard Rotunno and Jian-Wen Bao

Abstract

It is universally agreed that cyclogenesis in midlatitudes occurs through baroclinic conversion of the potential energy available from an initial state. The mechanical process by which that conversion takes place is a perennial subject of discussion. At least as far back as the 1950s, it was recognized that in any practical forecast problem, the initial condition is influential. Observational research continues to confirm the prevalence of tropopause-level perturbations preceding surface cyclogenesis. The observations also suggest that the growing disturbances have time-varying vertical structures. Relating these observations to the classical linear theory of baroclinic instability is not immediately obvious since, in the latter, the precise form of the initial condition is not important, and the theory predicts cyclogenesis with a fixed-in-time vertical structure. These differences between theory and observations are but a few of the many that have been recognized and treated in modified theories of baroclinic instability. We attempt herein to draw a closer connection between the modified theories and observations by performing a case study using a hierarchy of models of decreasing complexity.

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Tomislava Vukićević and Jian-Wen Bao

Abstract

The authors show that the linear approximation errors in the presence of a discontinuous convective parameterization operator are large for a small number of grid points where the noise produced by the convective parameterization is largest. These errors are much smaller for “smooth convective” points in the integration domain and for the nonconvective regions. Decreasing of the amplitude of initial perturbations does not reduce the errors in noisy points. This result indicates that the tangent linear model solution is erroneous in these points due to the linearization that does not include the linear variations of regime changes (i.e., due to use of standard method).

The authors then show that the quality of local four-dimensional variational (4DVAR) data assimilation results is correlated with the linearization errors: Slower convergence is associated with large errors. Consequently, the 4DVAR assimilation results are different for different convective points in the integration domain. The negative effect of linearization errors is not, however, significant for the cases that are studied. Erroneous points slightly degrade 4DVAR results in the remaining points. This degradation is reflected in decreased monotonicity of the cost function gradient reduction with iterations.

These results suggest that there is a probability for locally bad 4DVAR assimilation results when using standard adjoints of discontinuous parameterizations. In practice, when using for example observations, this is unlikely to cause errors that are larger than errors associated with other approximations and uncertainties in the data assimilation integrations such as the linear approximation errors and the uncertainties associated with the background and model errors statistics. This conclusion is similar to the conclusions of prior 4DVAR assimilation studies that use the standard adjoints but unlike in these studies the results in the current study show that 1) the linearization errors are nonnegligible for small-amplitude initial perturbations and 2) the assimilation results are locally and even globally affected by these errors.

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Jian-Wen Bao and Ronald M. Errico

Abstract

A regional adjoint modeling system is modified to determine the sensitivities of data assimilation and forecast results with respect to perturbations of the nudging fields and coefficients. A generalized linear system is used to explain the sensitivities both mathematically and physically. A linearized shallow-water model is utilized to show that the dynamics determining the sensitivities can be well described in terms of the dynamics of geostrophic adjustment, with the added effects of dissipation and nudging terms. The purpose of the study is to reveal the dynamics responsible for the sensitivities of assimilated fields and forecasts to a given observed variable, and thus to gain insight into what kinds of information are most (or least) effectively assimilated by the nudging method.

The results of the adjoint study reveal that the nudging terms contribute significantly to the prognostic tendencies, even if the values of the nudging coefficient are smaller than those commonly used. When either all dynamic fields or only wind fields are nudged, the assimilation result is much more sensitive to the analyzed data at a later time. The sensitivity of the variance of the difference between the assimilation result and the analyzed data at the final time within various bands of horizontal and vertical spatial scales shows that little scale interaction is evident in this study.

The qualitative comparison of the sensitivity results for nudging only wind or temperature or both are apparently well explained by referring to results of a sensitivity analysis for a nudged, linear shallow-water model. The latter results indicate that nudging high-frequency gravity waves toward an analysis that varies on a much slower timescale had little effect on the final assimilation fields, aside from damping. The same was not true for either rotational modes or slowly propagating inertial-gravitational modes. The sensitivity analysis of the shallow-water model also explains why nudging temperature alone does not produce desirable results.

All the results indicate that the advection is being overwhelmed by the nudging even when the value of the nudging coefficient is half as large as commonly used, but geostrophic and dissipative adjustment are acting effectively. For larger values of the nudging coefficients, the effects of advection are diminished more.

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Sara A. Michelson and Jian-Wen Bao

Abstract

The sensitivity of the Weather and Research Forecasting (WRF) model-simulated low-level winds in the Central Valley (CV) of California to uncertainties in the atmospheric forcing and soil initialization is investigated using scatter diagrams for a 5-day period in which meteorological conditions are typical of those associated with poor-air-quality events during the summer in the CV. It is assumed that these uncertainties can be approximated by two independent operational analyses. First, the sensitivity is illustrated using scatter diagrams and is measured in terms of the linear regression of the output from two simulations that differ in either the atmospheric forcing or the soil initialization. The spatial variation of the sensitivity is then investigated and is linked to the dominant low-level flows within the CV. The results from this case study suggest that the WRF-simulated low-level winds in the northern CV [i.e., the Sacramento Valley (SV)] are more sensitive to the uncertainties in the atmospheric forcing than to those in the soil initialization in the typical weather conditions during the summer that are prone to poor air quality in the CV. The simulated low-level winds in the southernmost part of the San Joaquin Valley (SJV) are more sensitive to the uncertainties in the soil initialization than they are in the SV. In the northern SJV, the simulated low-level winds are overall more sensitive to the uncertainties in the large-scale upper-level atmospheric forcing than to those in the soil initialization. This spatial variation in sensitivity reflects the important roles that the large-scale forcing, specified by the lateral boundary conditions and the local forcing associated with the soil state, play in controlling the low-level winds in the CV.

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Jian-Wen Bao and Ying-Hwa Kuo

Abstract

Numerical models that are used in four-dimensional data assimilation (FDDA) involve on-off switches associated with physical processes. Mathematically these on–off switches are represented by first-order discontinuous functions or step functions. In the development of the adjoint for the variational FDDA, the numerical models must be linearized. While insight has been gained into how to handle the on–off switches represented by first-order discontinuous functions, it is still unclear how to deal with the switches represented by step functions when the model equations are linearized. In this study, the calculus of variations is applied to under-stand how to treat step functions in the development of the adjoint. It is shown that in theory, if adding small perturbations to the initial state does not change the grid points in a forecast model where switching occurs, there is no difficulty in dealing with both first-order discontinuous points and the discontinuous points represented by step functions. However, in practice, first-order discontinuous points are much easier to deal with than those described by step functions.

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David J. Stensrud and Jian-Wen Bao

Abstract

Variational and nudging data-assimilation schemes are examined within the framework of a model initialization problem using the Lorenz three-component model of Rayleigh–Benard convection. Since the intent of this study is to explore what factors influence the abilities of the two assimilation techniques to produce accurate initial conditions, identical twin experiments are conducted for various lengths of the data-assimilation window when the flow is both study state and chaotic. These experiments illustrate that the location of the model solution in phase space is an important consideration when applying either data-assimilation scheme. Decision points, when the model “chooses” which stationary point to orbit, are found to affect the ability of the assimilation techniques to find an accurate initial condition. For chaotic flow, variational assimilation produces a better initial condition when the assimilation window is short, while nudging produces a better initial condition when the assimilation window is long. This is due to both the increasing complexity of the cost-function shape as the assimilation window is lengthened and the longer time period over which nudging can operate. When using a variational technique, extending the assimilation window past a certain length may be detrimental.

Admittedly, it is uncertain to what extent these results can be generalized to more complicated models of the atmosphere, since we have examined only one particular model and variants of the assimilation methods used may lead to different results with the same model. Studies using mesoscale and small-scale models, however, show similar behaviors. These behaviors suggest that decision points are present in these more complicated models. Thus, chaotic low-order models that include decision points may be useful in exploring the characteristics of data-assimilation techniques.

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Chong-Jian Qiu, Jian-Wen Bao, and Qin Xu

Abstract

The significance of mass sinks (or sources) due to precipitation (or evaporation) is examined using numerical experiments performed with the Pennsylvania State University-National Center for Atmospheric Research Mesoscale Model. The results show that the effect of mass sinks (or sources) can have a significant impact on numerical simulations of heavy precipitation. When this effect is ignored, as is commonly done in most global and regional weather prediction models, precipitation is reduced in the model.

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Sara A. Michelson, Irina V. Djalalova, and Jian-Wen Bao

Abstract

A season-long set of 5-day simulations between 1200 UTC 1 June and 1200 UTC 30 September 2000 are evaluated using the observations taken during the Central California Ozone Study (CCOS) 2000 experiment. The simulations are carried out using the fifth-generation Pennsylvania State University–NCAR Mesoscale Model (MM5), which is widely used for air-quality simulations and control planning. The evaluation results strongly indicate that the model-simulated low-level winds in California’s Central Valley are biased in speed and direction: the simulated winds tend to have a stronger northwesterly component than observed. This bias is related to the difference in the observed and simulated large-scale, upper-level flows. The model simulations also show a bias in the height of the daytime atmospheric boundary layer (ABL), particularly in the northern and southern Central Valley. There is evidence to suggest that this bias in the daytime ABL height is not only associated with the large-scale, upper-level bias but also linked to apparent differences in the surface forcing.

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David J. Stensrud, Jian-Wen Bao, and Thomas T. Warner

Abstract

Two separate numerical model ensembles are created by using model configurations with different model physical process parameterization schemes and identical initial conditions, and by using different model initial conditions from a Monte Carlo approach and the identical model configuration. Simulations from these two ensembles are investigated for two 48-h periods during which large, long-lived mesoscale convective systems develop. These two periods are chosen because, in some respects, they span the range of convective forecast problems routinely handled by operational forecasters.

Calculations of the root-mean-square error, equitable threat score, and ranked probability score from both ensembles indicate that the model physics ensemble is more skillful than the initial-condition ensemble when the large-scale forcing for upward motion is weak. When the large-scale forcing for upward motion is strong, the initial-condition ensemble is more skillful than the model physics ensemble. This result is consistent with the expectation that model physics play a larger role in model simulations when the large-scale signal is weak and the assumptions used within the model parameterization schemes largely determine the evolution of the simulated weather events.

The variance from the two ensembles is created at significantly different rates, with the variance in the physics ensemble being produced two to six times faster during the first 12 h than the variance in the initial-condition ensemble. Therefore, within a very brief time period, the variance from the physics ensemble often greatly exceeds that produced by the initial-condition ensemble. These results suggest that varying the model physics is a potentially powerful method to use in creating an ensemble. In essence, by using different model configurations, the systematic errors of the individual ensemble members are different and, hence, this may allow one to determine a probability density function from this ensemble that is more diffuse than can be accomplished using a single model configuration.

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Yimin Ma, Noel E. Davidson, Yi Xiao, and Jian-Wen Bao

Abstract

In high-wind conditions, sea spray, in conjunction with a generally decreasing drag coefficient for increasing winds, greatly modulates surface heat and momentum fluxes. It has been suggested that the process can be particularly important for the prediction of tropical cyclones (TCs), yet its robust application in operational forecast systems has remained elusive. A sea spray inclusion scheme and a modified algorithm for momentum exchange have been implemented in the Australian Bureau of Meteorology’s current operational TC model. Forecasts for a limited sample of TCs demonstrate that the revised parameterizations improve initialized and forecast intensities, while mostly maintaining track prediction skill. TC Yasi (2011) has been studied for impacts of the revised parameterization on rapid intensification (RI). Compared with the conventional bulk air–sea exchange parameterization, the revised version simulates a cooler and moister region near the surface in the eyewall/eye region, adjusts the RI evolution by an earlier and stronger subsidence in the eye, and simulates a stronger radial pulsating of the eye and eyewall convection on relatively short time scales. The inclusion of the new scheme enhances RI features characterized by eyewall ascent, radial convergence, and inertial stability inside the radius of azimuthal-mean maximum wind over low- to midlevels, and by a ringlike radial distribution of relative vorticity above the boundary layer. In addition, it allows a higher maximum intensity wind speed based on Emanuel’s maximum potential intensity theory. It is shown that, as expected, this is mainly because of a larger ratio of enthalpy and momentum exchange coefficients.

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