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Hung-Chi Kuo and R. T. Williams

Abstract

The accuracy of a numerical model is often scale dependent. Large spatial-scale phenomena are expected to be numerically solved with better accuracy, regardless of whether the discretization is spectral, finite difference, or finite element. The purpose of this article is to discuss the scale-dependent accuracy associated with the regional spectral model variables expanded by sine–cosine series. In particular, the scale-dependent accuracy in the Chebyshev-tau, finite difference, and sinusoidal- or polynomial-subtracted sine–cosine expansion methods is considered. With the simplest examples, it is demonstrated that regional spectral models may possess an unusual scale-dependent accuracy. Namely, the numerical accuracy associated with large-spatial-scale phenomena may be worse than the numerical accuracy associated with small-spatial-scale phenomena. This unusual scale-dependent accuracy stems from the higher derivatives of basic-state subtraction functions, which are not periodic. The discontinuity is felt mostly by phenomena with large spatial scale. The derivative discontinuity not only causes the slow convergence of the expanded Fourier series (Gibbs phenomenon) but also results in the unusual scale-dependent numerical accuracy. The unusual scale-dependent accuracy allows large-spatial-scale phenomena in the model perturbation fields to be solved less accurately.

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Hung-Chi Kuo and R. T. Williams

Abstract

The choice of an appropriate spectral spatial discretization is governed by considerations of accuracy and efficiency. The purpose of this article is to discuss the boundary effects on regional spectral methods. In particular, we consider the Chebyshev τ and sinusoidal- or polynomial-subtracted sine–cosine expansion methods. The Fourier and Chebyshev series are used because of the orthogonality and completeness properties and the existence of fast transforms. The rate of convergence of expansions based on Chebyshev series depends only on the smoothness of the function being expanded, and not on its behavior at the boundaries. The sinusoidal- or polynomial-subtracted sine–cosine expansion Tatsumi-type methods do not, in general, possess the exponential- convergence property. This is due to the fact that the higher derivatives of the polynomial- or sinusoidal- subtracted function are not periodic in a model with time-dependent boundary conditions. The discontinuity in derivatives causes the slow convergence of the expanded series (Gibbs phenomenon). When a large disturbance is near the boundary so that derivative discontinuities in the expanded function are large, the Tatsumi-type method causes not only erroneous numerical values in the outgoing boundary, but also spurious oscillations in the incoming boundary region. When the wave is away from the boundary, low resolution in the Tatsumi-type method converges exponentially, just as with the Chebyshev τ method. High-resolution solutions of the Tatsumi-type method do not, however, yield high accuracy due to the discontinuity in higher derivatives of the expanded function.

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Hung-chi Kuo and Wendell A. Nuss

Abstract

The quasigeostrophic geopotential tendency equation is derived using P vectors in both pressure and entropy coordinates. This vector form of the geostrophic forcing in the geopotential tendency equation is similar to the Q-vector form of the ω equation. It is shown that the horizontal components of P are the advection of geostrophic momentum and the vertical component of P is the horizontal temperature advection. The P vectors are shown to be related to Q and C vectors, as well as the ageostrophic circulation. The three-dimensional pseudocurl of the P vector gives the C vector that equals the ageostrophic pseudovorticity in the quasigeostrophic model. The horizontal components of the pseudocurl of P are perpendicular and proportional to the Q. The horizontal divergence of the P vector is the geostrophic absolute vorticity advection while the three-dimensional divergence of the P vector is the geostrophic quasigeostrophic potential vorticity advection. The ageostrophic wind can be partitioned into the P vector (geostrophic advective) and isallobaric wind contributions.

A numerical simulation of an idealized cyclone is used to characterize the distribution of the P vectors and P-vector diagnostics in order to demonstrate their potential application to the diagnosis of synoptic-scale circulations. The distribution of the P vectors clearly indicates the advections of geostrophic momentum and temperature that characterize cyclogenesis. An examination of the P vectors and the isallobaric wind demonstrates that the P vectors provide insight into the ageostrophic circulation of the cyclone. Diagnoses of the three-dimensional P-vector divergence and curl are shown to produce useful depictions of cyclonic vortex spinup and the propagation of both the large- and smaller-scale features of the system. These diagnostics can be interpreted from a variety of perspectives, including the height tendency and the advection of quasigeostrophic potential vorticity. The use of P vectors to diagnose synoptic-scale circulations appears to provide potentially useful insights into the dynamics of synoptic-scale disturbances not readily obtained from other diagnoses.

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Hung-chi Kuo and R. T. Williams

Abstract

We explore the use of semi-Lagrangian methods in a situation where the spatial scale of the flow collapses to zero during the time integration. The inviscid Burgers equation is used as the test model because it is the simplest equation that allows scale collapse (shock formation), and because it has analytic solutions. It is shown that despite the variable manner in which the gradient of the wind field approaches infinity in the neighborhood of the shock, the semi-Lagrangian method allows the error to be localized near the steep slope region. Comparisons with second-order finite difference and tau methods are provided. Moreover, the semi-Lagrangian method gives accurate results even with larger time steps (Courant number greater than 2 or 4) than are possible with the Eulerian methods. The semi-Lagrangian method, along with other recently developed numerical methods, is useful in simulating the development Of steep gradients or near discontinuities in a numerical model. Some applications of the semi-Lagrangian method are discussed.

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Satoki Tsujino and Hung-Chi Kuo

Abstract

The inner-core dynamics of Supertyphoon Haiyan (2013) undergoing rapid intensification (RI) are studied with a 2-km-resolution cloud-resolving model simulation. The potential vorticity (PV) field in the simulated storm reveals an elliptical and polygonal-shaped eyewall at the low and middle levels during RI onset. The PV budget analysis confirms the importance of PV mixing at this stage, that is, the asymmetric transport of diabatically generated PV to the storm center from the eyewall and the ejection of PV filaments outside the eyewall. We employ a piecewise PV inversion (PPVI) and an omega equation to interpret the model results in balanced dynamics. The omega equation diagnosis suggests eye dynamical warming is associated with the PV mixing. The PPVI indicates that PV mixing accounts for about 50% of the central pressure fall during RI onset. The decrease of central pressure enhances the boundary layer (BL) inflow. The BL inflow leads to contraction of the radius of the maximum tangential wind (RMW) and the formation of a symmetric convective PV tower inside the RMW. The eye in the later stage of the RI is warmed by the subsidence associated with the convective PV towers. The results suggest that the pressure change associated with PV mixing, the increase of the symmetric BL radial inflow, and the development of a symmetric convective PV tower are the essential collaborating dynamics for RI. An experiment with 500-m resolution shows that the convergence of BL inflow can lead to an updraft magnitude of 20 m s−1 and to a convective PV tower with a peak value of 200 PVU (1 PVU = 10−6 K kg−1 m2 s−1).

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Satoki Tsujino, Kazuhisa Tsuboki, and Hung-Chi Kuo

Abstract

Typhoons with long-lived concentric eyewalls (CEs) are more intense than those with short-lived CEs. It is important for more accurate prediction of typhoon intensity to understand the maintenance mechanism of the long-lived CEs. To study the mechanism of the long-term maintenance of CEs, a numerical experiment of Typhoon Bolaven (2012) is performed using a nonhydrostatic model with full physics. Two aspects of the maintenance of simulated CEs are investigated: the maintenance of the inner eyewall and the contraction of the outer eyewall. To examine the maintenance of the inner eyewall, the equivalent potential temperature budget and air parcel trajectories of the simulated inner eyewall are calculated. The results show that the entropy supply to the inner eyewall is sufficient to maintain the inner eyewall after secondary eyewall formation (SEF). During the early period after SEF, entropy is supplied by an axisymmetric inflow, and later it is supplied by nonaxisymmetric flows of the outer eyewall. To examine the contraction of the outer eyewall, the potential vorticity (PV) budget of the outer eyewall is diagnosed. The result reveals that the negative contribution to the contraction of the outer PV peak (i.e., the outer eyewall) in the early period is the negative PV generation due to axisymmetric advection and diabatic heating just inside of the outer PV peak. In the later period, the negative PV generation due to nonaxisymmetric structure is important for the prevention of contraction. The present study reveals that the structure of the outer eyewall plays important roles in the maintenance of long-lived CEs.

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Hung-Chi Kuo, Chih-Pei Chang, and Ching-Hwang Liu

Abstract

This study examines the convection and rapid filamentation in Typhoon Sinlaku (2008) using the Naval Research Laboratory (NRL) P-3 aircraft data collected during the Tropical Cyclone Structure 2008 (TCS-08) and The Observing System Research and Predictability Experiment (THORPEX) Pacific Asian Regional Campaign (T-PARC) field experiments. The high-resolution aircraft radar and wind data are used to directly compute the filamentation time, to allow an investigation into the effect of filamentation on convection. During the reintensification stage, some regions of deep convection near the eyewall are found in the vorticity-dominated area where there is little filamentation. In some other parts of the eyewall and the outer spiral rainband region, including areas of upward motion, the filamentation process appears to suppress deep convection. However, the magnitude of the suppression differs greatly in the two regions. In the outer spiral band region, which is about 200 km from the center, the suppression is much more effective, such that the ratio of the deep convective regime occurrence over the stratiform regime varies from around 50% (200%) for filamentation time shorter (longer) than 24 min. In the eyewall cloud region where the conditions are conducive to deep convection, the filamentation effect may be quite limited. While effect of filamentation suppression is only about 10%, it is still systematic and conspicuous for filamentation times shorter than 19 min. The results suggest the possible importance of vortex-scale filamentation dynamics in suppressing deep convection and organizing spiral bands, which may affect the development and evolution of tropical cyclones.

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Li-Huan Hsu, Hung-Chi Kuo, and Robert G. Fovell

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This paper examines the effect of topographically phase-locked convection on the motion of typhoons across the island of Taiwan. Data for 84 typhoons that reached Taiwan’s eastern coast from 1960 to 2010 are analyzed, with motions compared to the long-term average overland translation speed. For 61 continuous-track typhoons among all cases, 77% of the slow-moving tropical cyclones (TCs) made landfall on the northern end of Taiwan’s eastern coast, while 60% of the fast storms had southeastern coastal landfalls.

This geographic asymmetry with respect to typhoon translation speeds widened after landfall, as the slow movers typically decelerated during the overland period, whereas the faster TCs sped up. In particular, the average overland duration was 16 h for the slow class, compared to only 3 h for the fast-moving typhoons. The combination of slower translation with longer duration for the northern class of TCs led to large rainfall on the southwestern slope of the island’s Central Mountain Range.

Weather Research and Forecasting model experiments are used to study the effect of convection on storm motion over a mountainous island resembling Taiwan. The authors find that the topographically phase-locked convection acts to slow down (speed up) the northern (southern) landfalling typhoons. The model results also suggest that a positive feedback mechanism exists for the slow storms, in which the convective heating pattern forced by topography acts to reduce the TC motion, leading to even more prolonged precipitation and heating, yielding further speed reductions.

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Yu-Ming Tsai, Hung-Chi Kuo, and Wayne H. Schubert

Abstract

This paper analyzes synoptic-scale trough-thinning processes using a filamentation time diagnostic. The filamentation time diagnostic is derived from the potential vorticity equation expressed in spherical coordinates in the horizontal plane and the isentropic coordinate in the vertical direction. The diagnostic is an accurate measure of stirring processes under the condition of “slowly varying velocity gradients.” Trough-thinning processes are analyzed for one tropical example and two midlatitude examples. The results indicate that the filamentation time for the tropical trough-thinning event is generally longer than those for the midlatitude trough events. In addition to the effects of stretching and shearing deformation, the filamentation time diagnostic contains the effects of divergence. For the calculation of filamentation time on isentropic surfaces in spherical coordinates, it is acceptable to ignore the curvature effects in the tropics; however, in both the midlatitudes and the tropics, isentropic divergence effects should be retained for improved accuracy. Combining an analysis of cross potential vorticity contour flows on isentropic surfaces with the filamentation time analysis gives a more complete description of the dynamics. The results show that the filamentation time diagnostic can serve as a useful aid in the analysis and prediction of trough thinning and cutoff-low formation.

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