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Chungu Lu
and
Yuanfu Xie

Abstract

The computational modes associated with a centered finite-differencing scheme in space are studied. The existence and impact of these computational modes in a numerical solution are demonstrated with the use of theoretical analyses and numerical experiments.

The results show that the computational modes due to a spatial discretization can have a detrimental effect on the numerical solution in situations where flows are evolved near shock (or having large spatial derivative). The numerical diffusion can reduce the impact of the computational modes, but can also impose an adverse effect on the physical modes.

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Ning Wang
and
Chungu Lu

Abstract

The two-dimensional continuous wavelet transform (2D CWT) has become an important tool to examine and diagnose nonstationary datasets on the plane. Compared with traditional spectral analysis methods, the 2D CWT provides localized spectral information of the analyzed dataset. It also has the advantage over the 2D discrete wavelet transform (DWT) in that it covers the domain of the analyzed data with a continuous analysis from which detailed, shift-invariant spectral information of different positions and orientations can be obtained. In this paper, a brief introduction of the 2D CWT and some of the most common wavelet mother functions are given, and some practical issues arising from the implementation and applications of the 2D CWT are discussed. The 2D CWT is applied to several test functions to illustrate the effects of the transforms. To demonstrate its practical application, the 2D CWT is used to analyze a set of meteorological data obtained from a numerical model stimulation.

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Xiaohui Shi
,
Chungu Lu
, and
Xiangde Xu

Abstract

Using the daily maximum air temperature and mean humidity observations at 394 surface weather stations across China, the changes in the annual number of days of high temperature weather (HTW), high humidity weather (HHW), and sultry weather (STW) in China over the period 1961–2004 are studied. The results indicate that there were considerable spatial differences and temporal variability of HTW, HHW, and STW across China. Under a climatic mean condition, a notable feature is that southeastern China is the region of collocation of high values of the annual number of days of HTW, HHW, and STW, as well as the region of the most significant variabilities of these parameters. About 55% of the stations in China have increasing trends of the annual number of days of HTW. Most stations in China show decreasing trends of the annual number of days of HHW and are mainly located either in the area south of 30°N or in northern and northeastern China. The stations with increasing trends of the annual number of days of STW are mainly located in northern China, while the stations that have decreasing trends are primarily located in southern China. The analysis results suggest that the variability of the annual number of days of STW corresponds mainly to HTW, and less to HHW. The change in the East Asian monsoon may be responsible for the changes of these statistics in extreme weather in China.

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Chungu Lu
and
Gerald L. Browning

Abstract

The impact of observational and model errors on four-dimensional variational (4DVAR) data assimilation is analyzed for a general dynamical system. Numerical experiments with both the barotropic vorticity equation and the shallow water system are conducted. It is shown from the analysis and the numerical experiments that when there are random errors in observations or in model parameterizations, the 4DVAR assimilation method can suppress these errors; however, when the errors are systematic or biased, the 4DVAR assimilation method tends to either converge to the erroneous observations or introduce the model error into the data analysis, or both.

For a multiple-timescale fluid dynamical system, such as the shallow water equations with fluid depth corresponding to the external mode, the skewness in the system can amplify the errors, especially in the fast variable (e.g., the geopotential or height field).

Forecasts using the assimilated initial condition with an imperfect model indicate that the forecasts may or may not be improved, depending upon the nature of the model and observational errors, and the length of the assimilation and forecast periods.

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Chungu Lu
and
Steven E. Koch

Abstract

Spectral and structure function analyses of horizontal velocity fields observed in the upper troposphere and lower stratosphere during the Severe Clear Air Turbulence Collides with Air Traffic (SCATCAT) field program, conducted over the Pacific, were carried out in an effort to identify the scale interactions of turbulence and small-scale gravity waves. Because of the intermittent nature of turbulence, these analyses were conducted by clearly separating out the cases when turbulence did or did not occur in the data. In the presence of turbulence, transitional power spectra from k −2 to k −5/3 were found to be associated with gravity waves and turbulence, respectively. The second-order structure function analysis was able to translate these spectral slopes into r and r2/3 scaling, consistent with the Monin and Yaglom conversion law, in physical space, which presented clearer pictures of scale interactions between turbulence and gravity waves. The third-order structure function analysis indicated the existence of a narrow region of inverse energy cascade from the scales of turbulence up to the gravity waves scales. This inverse energy cascade region was linked to the occurrence of Kelvin–Helmholtz instability and other wave-amplifying mechanisms, which were conjectured to lead to the breaking of small-scale gravity waves and the ensuing generation of turbulence. The multifractal analyses revealed further scale breaks between gravity waves and turbulence. The roughness and intermittent properties were also calculated for turbulence and gravity waves, respectively. Based on these properties, turbulence and gravity waves in a bifractal parameter space were mapped. In this way, their physical and statistical attributes were clearly manifested and understood.

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Chungu Lu
and
John P. Boyd

Abstract

The effects of divergence on low-frequency Rossby wave propagation are examined by using the two-dimensional Wentzel–Kramers–Brillouin (WKB) method and ray tracing in the framework of a linear barotropic dynamic system. The WKB analysis shows that the divergent wind decreases Rossby wave frequency (for wave propagation northward in the Northern Hemisphere). Ray tracing shows that the divergent wind increases the zonal group velocity and thus accelerates the zonal propagation of Rossby waves. It also appears that divergence tends to feed energy into relatively high wavenumber waves, so that these waves can propagate farther downstream. The present theory also provides an estimate of a phase angle between the vorticity and divergence centers. In a fully developed Rossby wave, vorticity and divergence display a π/2 phase difference, which is consistent with the observed upper-level structure of a mature extratropical cyclone. It is shown that these theoretical results compare well with observations.

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Chungu Lu
and
Gerald L. Browning

Abstract

Mathematical issues arising when applying four-dimensional variational (4DVAR) data assimilation to limited-area problems are studied. The derivation of the adjoint system for the initial-boundary value problem for a general hyperbolic system using the standard variational approach requires that the inflow adjoint variables at an open boundary be zero. However, in general, these “natural” boundary conditions will lead to a different solution than that provided by the global assimilation problem. The impact of using natural boundary conditions when there are errors (on the boundary) in the initial guess on the assimilated initial conditions is discussed.

A proof of the uniqueness of the solution for both forward and adjoint equations in the presence of open boundaries at each iteration of the minimization procedure is provided, along with an assessment of the convergence of numerical solutions.

Numerical experiments with a simple advection equation support the theoretical analyses. Numerical results show that if observational data are perfect, 4DVAR data assimilation using a limited-area model can produce a reasonable initial condition. However, if there are errors in the observational data at the open boundaries and if natural boundary conditions are assumed, boundary errors can contaminate the assimilated solutions.

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Michael T. Montgomery
and
Chungu Lu

Abstract

To understand the nature of coupling between a hurricane vortex and asymmetries in its near-core region, it is first necessary to have an understanding of the spectrum of free waves on barotropic vortices. As foundation for upcoming work examining the nonaxisymmetric initial-value problem in inviscid and swirling boundary layer vortex flows, the complete spectrum of free waves on barotropic vortices is examined here.

For a variety of circular vortices in gradient balance the linearized momentum and continuity equations are solved as a matrix eigenvalue problem for perturbation height and wind fields. Vortex eigensolutions are found to fall into two continuum classes. Eigenmodes with frequencies greater than the advective frequency for azimuthal wavenumber n are modified gravity–inertia waves possessing nonzero potential vorticity in the near-core region. Eigenmodes whose frequencies scale with the advective frequency comprise both gravity–inertia waves and Rossby–shear waves. Linearly superposing the Rossby–shear waves approximates the sheared disturbance solutions. For wavenumbers greater than a minimum number, Rossby–shear waves exhibit gravity wave characteristics in the near-vortex region. Although such eigenstructure changes are not anticipated by traditional scaling analyses using solely external flow parameters, a criterion extending Rossby’s characterization of “balanced” and “unbalanced” flow to that of azimuthal waves on a circular vortex is developed that correctly predicts the observed behavior from incipient vortices to hurricane-like vortices. The criterion is consistent with asymmetric balance theory. Possible applications of these results to the wave-mean-flow dynamics of geophysical vortex flows are briefly discussed.

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Chungu Lu
and
Gerald L. Browning

Abstract

The impact of observational errors on objective analyses is investigated with mathematical analyses, analytical examples, and real data experiments. Cases with observational errors at one or more stations are considered. It is found that in the presence of observational errors, the analysis error in an objective analysis scheme generally consists of two parts: the signal fitting error and the noise contamination error. Although every objective analysis scheme has its own procedure(s) to control the two errors, the procedures to suppress the noise contamination error in one and two dimensions are shown to be relatively ineffective. It is shown that the extension of an objective analysis method to more dimensions significantly reduces the noise contamination.

Based on these results, higher dimensional versions of the least squares polynomial fitting (LSPF) methods and the Barnes scheme are examined. In both analytic and real data experiments, the 3D and 4D LSPF methods and the 3D Barnes scheme show an enhanced ability to filter observational noise.

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Chungu Lu
and
Gerald L. Browning

Abstract

The impact of discontinuous model forcing on the initial conditions obtained from 4DVAR data assimilation is studied with mathematic analyses, idealized numerical examples, and more realistic meteorological cases. The results show that a discontinuity in a parameterization, like a model bias, can introduce a systematic error in the assimilated initial fields. However, the most detrimental effect of a model discontinuity is the retention of roughness in the assimilated initial fields, although in some cases the 4DVAR procedure provides some smoothing effect. The obvious consequences of this roughness is that it will introduce spurious modes in the ensuing forecast, and derivatives of the assimilated initial data will be unrealistically large, which can lead to large errors in data analysis. The smoothing effect on the initial conditions with the addition of artificial diffusion to the constraining model is also studied. Possible solutions to the problem of 4DVAR data assimilation with discontinuous model forcing are discussed.

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