Recent Developments in Computing Streamfunction and Velocity Potential in a Limited Domain

CAO Jie, XU Qin, and GAO Shou-Ting

1Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China

2Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman 73072, USA

3National Severe Storms Laboratory, Norman 73072, USA

Recent Developments in Computing Streamfunction and Velocity Potential in a Limited Domain

CAO Jie1,2, XU Qin3, and GAO Shou-Ting1

1Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China

2Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman 73072, USA

3National Severe Storms Laboratory, Norman 73072, USA

After first reviewing historical and current difficulties in solving streamfunction and velocity potential in a limited domain, and describing recent developments in obtaining accurate solutions in a limited domain with arbitrary shape, a newly proposed approach is introduced and its application to a torrential rain event is reported. The results show that the newly developed method has advantages in capturing mesoscale information, compared with horizontal winds.

streamfunction, velocity potential, limited domain, mesoscale meteorology

1　Historical problems

Streamfunction and velocity potential, together with their associated vorticity and divergence, are efficient prognostic variables in weather and climate analysis (Hollingsworth and Lonnberg, 1986; Parrish and Derber, 1992; Daley and Barker, 2001; Xu and Gong, 2003; Xu et al., 2006, 2007; Zhao et al., 2006). Physically, the streamfunction plays an important role in large-scale motion since it represents the rotational flow. The velocity potential, meanwhile, represents the ageostrophic wind, which is the essence of vertical motion. Thus, it is a small but significant factor in mesoscale meteorology. Mathematical approaches to obtaining streamfunction and velocity potential globally, with periodic or no boundary conditions, are straightforward, and have been proven to be quite successful through wide usage in global numerical prediction models (Haltiner and Williams, 1980).

It is known that the boundary conditions are of vital importance for mesoscale meteorology. Several methods have been proposed with the hope of obtaining accurate streamfunction and velocity potential in a limited domain (Sangster, 1960; Hawkins and Rosenthal, 1965; Shukla and Saha, 1974; Stephens and Johnson, 1978; Bijlsma et al., 1986). Unfortunately, however, these methods either lack accuracy or fail in convergence. Lynch (1988, 1989) first found that the lack of accuracy and difficulty of convergence resulted from the non-uniqueness of the calculated streamfunction and velocity potential in a limiteddomain. Chen and Kuo (1992a, b) reexamined this problem mathematically and physically, with special consideration of this issue, and proved that the sum of the streamfunction and velocity potential components of the wind is unique. Besides, they designed Fourier sine/cosine series expansion methods after separating the total wind into internal and external components. Ren et al. (2013) tested the accuracy of both relaxation methods (Richardson method and accelerated Liebmann method) and the Chen and Kuo cosine series expansion method, and found that the solutions from the latter approach are very satisfying in a rectangular domain. Applications of this spectral method in assisting the identification of divergent motion in weather systems have also been performed (Zhou et al., 2008; Zhou and Cao, 2010; Deng et al., 2012; Ren et al., 2013). Spectral methods have been used successfully in global models with accurate solutions (Machenhauer, 1979). However, they still have some shortcomings, such as the solutions are easily influenced by observed values at each grid point, not to mention the difficulty in choosing appropriate basis functions (Tatsumi, 1986; Fulton and Schubert, 1987; Chen and Kuo, 1992a).

Since revisiting this historic problem is a welcome means to raise awareness within the meteorological community about the importance of the interaction between divergent and rotational components of the atmospheric flow and their maintenance mechanisms (J. F. Chou, 2011, personal communication), we next present a brief review of current issues and recent developments in this arena.

2　Current issues and recent developments

With the rapid development of numerical prediction models in recent years, the adoption of irregular grids instead of rectangular ones has become increasingly popular. However, this introduces an unavoidable problem for mesoscale meteorology, which mainly concerns weather events in a limited domain of arbitrary shape. Besides, irregular inner boundaries emerge when the analysis surface of pressure, for example, is lowered and intercepted by the land or ocean surface (Xu et al., 2011). These issues represent new complications for solving streamfunction and velocity potential in a limited domain.

Instead of solving the two Poisson equations with coupled boundary conditions directly, Li et al. (2006) de-signed a minimization approach suitable for irregular domains. By applying a regularization constraint, the non-uniqueness problem is solved. However, it is accompanied by suppression of the solutions and loss of accuracy. Xu et al. (2011) revisited the non-uniqueness problem theoretically with rigorous mathematical treatments and explored the use of classical integral formulae for limited domains of arbitrary shape by minimizing the difference between the domain-integrated kinetic energy of the original horizontal velocity and that of the reconstructed one. Numerical discretization schemes (Cao and Xu, 2011) have been designed based on the integral formulae derived for computing the internally and externally induced solutions of streamfunction and velocity potential of a data-hole free domain. The internal part is only determined by the distribution of vorticity and can be expressed by the divergence inside the domain, while the external one is determined by the boundary conditions. The method was then further generalized to domains in the presence of inner data-holes of arbitrary shape. Two automated algorithms were designed to facilitate the computations of the boundary integrals and the extended Cauchy integral method for constructing the externally induced solution. Test experiments showed the accuracy and efficiency of each scheme relative to others.

3　The newly developed method and its application

Cao et al. (2014) extended the Helmholtz theorem to divide the horizontal wind (v) into purely rotational (vr), purely divergent (vd), and harmonic deformational flow (vh) (v = k × ∇(ψr+ ψh) + ∇(χd+ χh) = k × ∇ψr+ ∇χd+ (k × ∇ψh+ ∇χh) ≡ vr+ vd+ vh; the subscripts r, d, and h represent rotational, divergent, and harmonic deformational components respectively; ψ and χ are the streamfunction and velocity potential respectively). Compared to the partitioned streamfunction and velocity potential wind components obtained from the harmonic-cosine method of Chen and Kuo (1992b), this new method can be used to separate the harmonic deformational wind from the divergent and rotational flow. In our previous paper, we focused on checking the accuracy by performing idealized experiments and used a heavy rainfall event in North China that occurred on 21 July 2012 as a simple example to show the physical meaning or potential use of extracting the deformational field. In this paper, we show more aspects by analyzing the partitioned wind components with the original GFS reanalysis data.

As illustrated by Sun et al. (2013), this torrential rain event was a combined result of favorable synoptic circulations, sources of water vapor, mesoscale convective systems, and topographic effects. The ‘northwest' vortex, which generated at 0600 UTC 21 July and developed eastwards at 1200 UTC and 1800 UTC, was found to be the dominant dynamic system responsible for the torrential rain event. Figure 1 shows the horizontal wind and its rotational components during the process. The rotational part showed better signals of the ‘northwest' vortex. Besides, it showed the enhancements of the low-level jet in North China.

The frontogensis first occurred northeast of (40°N, 117°E) at 0600 UTC, when the warm advection was not strong. Seen from the angle between the stretching axis of winds and isotherms, both the divergent component (Fig. 2c) and deformational one (Fig. 2d) clearly showed the frontogenesis, while the total wind was not clear (Fig. 2a). At 1200 UTC, the frontogenesis moved to 39-40°N, which was the same place as the encounter with the warm and cold advection. The total wind (Fig. 3a) and the divergent (Fig. 3c) and deformational (Fig. 3d) components all showed this frontogenesis. At 1800 UTC, when the frontogenesis was around (39°N, 115°E), only the deformational component (Fig. 4d) had signals. In short, the partitioned wind components offer better representations for diagnosing deformations.

4　Conclusion

This study revisited the historical problems in solving Poisson equations under coupled boundary conditions in a limited domain. Due to irregular inner and outer boundaries, previous methods have encountered difficulties. Recent developments in this area include the integral method proposed by Xu et al. (2011), and an extended method to separate the harmonic deformational component by Cao et al. (2014). Detailed analysis of the partitioned wind components and their relations with developments in a torrential rain event was carried out using the newly developed method. More numerical simulations are needed for testing and analyzing the application of this method in real case studies, particularly in terms of considering its limitations with respect to certain types of real flow fields. With the development of these accessible and accurate approaches to separate the divergent, rotational, and harmonic deformational flows in atmospheric circulation, follow-up studies will continue to explore the basic dynamics and maintenance mechanisms of the divergent and rotation components, especially in mesoscale meteorology.

Acknowledgements. The authors are thankful to the two anonymous reviewers for their constructive comments. This research was supported by the National Basic Research Program of China (Grant No. 2012CB417201), the National Science and Technology Support Program (Grant No. GYHY201406001), the National Natural Science Foundation of China (Grant No. 41205033), and the Key Project of the Key Laboratory of Atmosphere and Environments on the Plateau in Sichuan Province (Grant No. PAEKL-2014-C1). Dr. Jie CAO is thankful to the Innovation Society for Young Sciences, Chinese Academy of Sciences.

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10.3878/AOSL20150041.

28 April 2015; revised 7 September 2015; accepted 24 September 2015; published 16 November 2015

CAO Jie, caojie@mail.iap.ac.cn