Hindawi Advances in Meteorology Volume 2017, Article ID 6272158, 15 pages https://doi.org/10.1155/2017/6272158

Research Article A New Vortex Initialization Scheme Coupled with WRF-ARW

Jimmy Chi Hung Fung1,2 and Guangze Gao1

1 Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2Division of Environment, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Correspondence should be addressed to Jimmy Chi Hung Fung; [email protected]

Received 21 August 2016; Revised 29 October 2016; Accepted 20 November 2016; Published 3 January 2017

Academic Editor: Anthony R. Lupo

Copyright © 2017 J. C. H. Fung and G. Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The ability of numerical simulations to predict typhoons has been improved in recent decades. Although the track prediction is satisfactory, the intensity prediction is still far from adequate. Vortex initialization is an efficient method to improve the estimations of the initial conditions for typhoon forecasting. In this paper, a new vortex initialization scheme is developed and evaluated. The scheme requires only observational data of the radius of maximum wind and the max wind speed in addition to the global analysis data. This scheme can also satisfy the vortex boundary conditions, which means that the vortex is continuously merged into the background environment. The scheme has a low computational cost and has the flexibility to adjust the vortex structure. Itwas evaluated with 3 metrics: track, center sea-level pressure (CSLP), and maximum surface wind speed (MWSP). Simulations were conducted using the WRF-ARW numerical weather prediction model. Super and severe typhoon cases with insufficiently strong initial MWSP were simulated without and with the vortex initialization scheme. The simulation results were compared with the 6-hourly observational data from Hong Kong Observatory (HKO). The vortex initialization scheme improved the intensity (CSLP and MWSP) prediction results. The scheme was also compared with other initialization methods and schemes.

1. Introduction the initial vortices provided by the large-scale analysis are often too weak or misplaced [5]. For example, for the super To reduce the damage caused by tropical cyclones (TC), typhoon Genevieve (2014) at 12:00 (UTC) on 9 Aug 2014, the it is very important to predict their track and intensity center sea-level pressure (CSLP) derived from FNL (NCEP correctly. In recent decades, track prediction has been steadily final analyses) data is around 993 hPa, whereas the observa- improved [1]. However, there has been little improvement tional CSLP from HKO is 940 hPa. in predicting the intensity of typhoons [1]. During 1989– Vortex initialization is an efficient method to provide 2012, TC intensity errors from the best available models have −1 better estimates of the initial conditions for numerical simu- decreased at 1%-2% yr at 24, 48, and 72 h [2]. lation of tropical cyclones. The advantages of the use of bogus As mentioned by Marks et al. [3], the primary reason for vortices are the ability to customize the vortex structure and the slow progress in improvement of intensity prediction is low computational cost, compared with dynamical initializa- that the intensity depends on the inner-core dynamics, tion and variational data assimilation methods [6]. whereas the track prediction depends more on the large-scale In Kwon and Cheong [7], an initialization method with an environments. The inner-core dynamics are mainly asso- idealized 3D bogus vortex is developed for track and intensity ciated with vortex structure, intensity, and vortex size. It predictions. In their scheme, a 3D axisymmetric bogus vortex has been suggested that the accurate representation of the is designed empirically in terms of analytic functions. The inner-core structure of tropical cyclones in the initial con- bogus vortex is used as a disturbance field and merged into ditions is as important as the representation of the large- the environmental field. The surface pressure profile ofthe scale environment [4]. To improve the representation of bogus vortex is given by an exponent decay function. The the initial vortex, a bogus vortex is often applied because temperature is derived from the 3D geopotential by using the 2 Advances in Meteorology hydrostatic balance equation. The relative humidity is given specific humidity is not adjustable, which means that the by an empirical function, which is not dynamically consistent samespecifichumidityfieldmustbeusedforallthetropical with the other variables. In our scheme, this problem is cyclones. Therefore, the scheme may be lacking flexibility to overcome by solving the temperature and the humidity simulate different tropical cyclones. simultaneously (Equation (50)). In Rappin et al. [6], a highly configurable vortex initializa- Reed and Jablonowski [8] used analytic functions to tion scheme is developed, which can be easily manipulated derive the initial conditions. The initial axisymmetric vortex to generate different initial vortex structures. The vortex is in hydrostatic and gradient wind balance. The first step is construction starts with parameterizations of the radial wind to provide the analytic background specific humidity profile: structure. The tangential wind 𝑉(𝑟) of the MR (modified Rankine) vortex is given by 𝑞 (𝑧) 𝑟 2 {𝑉max ,𝑟≤RMW; 𝑧 𝑧 RMW {𝑞 (− ) [− ( ) ], 0≤𝑧≤𝑧; (1) 𝑉 (𝑟) = { 𝛼 (5) = 0 exp 𝑧 exp 𝑧 𝑡 {𝑉 (RMW) ,𝑟> , { 𝑞1 𝑞2 { max 𝑟 RMW {𝑞𝑡,𝑧𝑡 <𝑧, where 𝛼 is the decaying factor, RMW is the radius of where 𝑧𝑡 issetas15km,whichapproximatesthetropopause maximum wind, 𝑉max isthetangentialvelocityatRMW,and height; 𝑞0 is the specific humidity at the surface, which is 𝑟 is the distance from the TC center. −1 set as 21 g kg ;and𝑞𝑡 is the specific humidity in the upper The vertical structure of tangential wind speed follows −8 −1 atmosphere, which is set as 10 gkg . 𝑧𝑞1 = 3000 mand Stern and Nolan [9]: 𝑧 = 8000 󵄨 󵄨𝛾 𝑞2 m are two constants to control the specific 󵄨𝑧−𝑧 󵄨 humidity profile. The virtual temperature profile is given by 𝑉 (𝑟,) 𝑧 =𝑉(𝑟) (−󵄨 max󵄨 ), exp 𝛾𝐿𝛾 (6)

{𝑇V0 − Γ𝑧, 0 ≤ 𝑧𝑡 ≤𝑧 ; where 𝛾 is set as 2 and 𝐿 is set as 3175 m below 𝑧max and 𝑇V (𝑧) = { (2) 𝑇 ,𝑧<𝑧, 𝑧 𝑧 { V𝑡 𝑡 4762.5 m above max [9]. max is the height of the boundary layer top. Finally, the hydrostatic balance and gradient wind where 𝑇V0 is the virtual potential temperature at the surface, balance are used to determine temperature and pressure. One and Γ is the virtual temperature lapse rate, which is set as advantageofthisschemeisthatitmakesuseof(6),which −1 0.007 K m . 𝑇V𝑡 =𝑇V0 −Γ𝑧𝑡 isthevirtualtemperatureinthe considers the vertical structure of tangential wind speed. By upper atmosphere. The background temperature profile can using this equation, the 3D wind field can be derived from then be derived as a 2D wind structure (e.g., the MR profile) directly, which makes the scheme highly configurable. The equations for 𝑇 (𝑧) 𝑇 (𝑧) = V . (3) constructing horizontal and vertical profiles introduced in 1 + 0.608𝑞 (𝑧) Rappin et al. [6] may also be helpful in designing new schemes. In our new scheme, we further improve the wind The background pressure profile 𝑝(𝑧) is computed using the construction by making the decaying factor and RMW as hydrostatic equation and the ideal gas law. To obtain the functions of 𝑧. perturbation profile of temperature, an empirical exponent In this paper, a vortex initialization scheme is developed decayfunctionisusedtomodifythepressureperturbation. for the weather research and forecasting (WRF) model with Then, by the hydrostatic equation and the ideal gas law, the the advanced research WRF (ARW) core. Further, in our temperature is derived. Finally, the tangential velocity V𝑇(𝑟, 𝑧) scheme, the variables satisfy geophysical formulas, which is derived by the gradient wind balance, which is given by ensures that the variables are dynamically consistent. The analytic formulations of this scheme enable it to be conducted 𝑓 𝑟 𝑐 efficiently. The problems in other schemes mentioned above V𝑇 (𝑟,) 𝑧 =− 2 canalsobesolvedbyourscheme.Noticethattheseproblems 1/2 (4) are mainly associated with two aspects: (1) the variables 𝑓2𝑟2 𝑅 𝑇 (𝑟,) 𝑧 𝑟 𝜕𝑝 (𝑟,) 𝑧 +( 𝑐 + 𝑑 V ) , shouldbedynamicallyconsistentwitheachother;(2)theTCs 4 𝑝 (𝑟,) 𝑧 𝜕𝑟 constructed by a scheme should be adjustable and realistic. The vortex initialization problem can be viewed as fol- where 𝑓𝑐 =2Ωsin(𝜑𝑐) is the Coriolis parameter at the (𝑢, V,𝑝,𝑇,𝑞, .) −5 −1 lows: given the data sets of etc from global anal- constant latitude 𝜑𝑐 and Ω = 7.292 × 10 s is the rotational ysis data (e.g., FNL and GFS (global forecast system) data), ∘ ∗ ∗ ∗ ∗ ∗ speed of Earth. 𝜑𝑐 is set to be 10 . 𝑟 is distance to the TC center. how can we correct them to new data (𝑢 , V ,𝑝 ,𝑇 ,𝑞 , etc.)? −1 −1 𝑅𝑑 = 287.04 Jkg K is the ideal gas constant for dry air. Here, 𝑢, V,𝑝,𝑇,𝑞 represent EW-wind, NS-wind, pressure, 𝑇V and 𝑝 arevirtualtemperatureandpressure.Theanalytic temperature, and specific humidity, respectively; “∗”isused formulation of this scheme enables it to be applied in various to represent corrected data throughout this paper. In most weather and climate models. However, the specific humidity situations, we need to provide some extra data besides the is constructed as a horizontally homogeneous variable, which global analysis data to build the new vortex. For example, we may not be realistic for the tropical cyclones. Also, the may need the max wind speed and RMW.Although providing Advances in Meteorology 3 moredatamayhelpustoachievebetterinitialfields,itis In practice, the vortex boundary is defined as the contour difficult to obtain observational data on the ocean. In our line of a sea-level pressure value (e.g., 1013 hPa). After size ∗ scheme, we aim to construct a 3D vortex that can represent correction, the new vortex is called 𝑢1 -vortex. basic features of typhoon’s structure. The extra inputs are The second step is “intensity correction” [13]. As men- vortex location, max wind speed, and the radius of the max tioned by Lord [14], the vortex initialization is usually wind. carried out by first implanting a synthetic vortex into the The rest of the paper is organized as follows: in Section 2, large-scale analysis. Similarly, the intensity correction in the vortex initialization scheme is derived; in Section 3, this paper is conducted by superposing an axisymmetric ∗ numericalsimulationsforsuperandseveretyphoonsinthe vortex onto 𝑢1 -vortex. The axisymmetric vortex is called 𝑢2- ∗ northwestern Pacific are conducted to evaluate the perfor- vortex. When adding 𝑢2-vortex to 𝑢1 -vortex, the intensity mance of the new scheme; the simulation results are discussed is adjusted through the intensity correction factor. In the in Section 4; and in Section 5, conclusions are given. next subsection, the issues related to the computation of the intensity correction factor are addressed. 𝑢 2. The New Vortex Initialization Scheme In the axisymmetric 2-vortex, the tangential velocity is 𝑉(𝑟, 𝑧),andtheradialvelocityiszero.𝑉(𝑟, 𝑧) is defined as 2.1. 2D Wind Correction. The goal in this paper is to follows: denote 𝑉𝑚(𝑧) as the max tangential wind speed at construct a new 3D vortex. Construction of one vertical layer height 𝑧 in the axisymmetric vortex, which is the same as the 𝑢∗ is first discussed, which is started from wind construction. maxwindmagnitudeintheoriginalvortexand 1 -vortex. The wind construction can be disaggregated as size correc- 𝑉(𝑟, 𝑧) is the tangential velocity on the circle of radius 𝑟 (the tion, intensity correction, and computation of the intensity center of the circle is the TC center). 𝑉(𝑟, 𝑧) isdefinedinthe correction factor. form of an MR vortex: 𝑉 (𝑟,) 𝑧 2.1.1. Size and Intensity Correction. The first step is “size 𝑟 correction” [13]. If the max wind speed and the RMW are 𝑉 (𝑧) , 0≤𝑟≤𝑟∗ (𝑧) ; { 𝑚 ∗ 𝑚 computed by the FNL data directly, the max wind speed is { 𝑟𝑚 (𝑧) { 𝛼(𝑧) (10) generally smaller than the observational data, and the RMW = 𝑟∗ (𝑧) {𝑉 (𝑧) ( 𝑚 ) ,𝑟∗ (𝑧) <𝑟≤𝑅∗ (𝑧) ; is generally too large. The typhoon size can be corrected { 𝑚 𝑚 𝑚 ∗ { 𝑟 𝑟 𝑟 ∗ by compressing the model grids. and are the distances {0, 𝑟 > 𝑅 (𝑧) , from a grid point to the typhoon center, before and after size 𝑚 correction. The size correction is conducted by where 𝛼(𝑧) is the profile decay rate, which is determined later. ∗ ∗ ∗ ∗ For variable 𝑎, define 𝑢1 (𝑟 ,𝜃,𝑧)=𝑢(𝑟, 𝜃,) 𝑧 ,(if 𝑟 ≤𝑅𝑚); 2𝜋 ∗ ∗ ∗ ∗ (7) 1 V1 (𝑟 ,𝜃,𝑧)=V (𝑟, 𝜃,) 𝑧 ,(if 𝑟 ≤𝑅𝑚), ⟨𝑎⟩ = ∫ 𝑎 (𝑟, 𝜃,) 𝑧 𝑑𝜃. (11) 2𝜋 0 ∗ ∗ where 𝑢1 and V1 are the wind components after size correc- 𝜃 𝑧 The physical meaning of ⟨𝑎⟩ is the azimuthal mean of variable tion. is the azimuthal angle and is the height from the 𝑎 ground. This means that the wind speed with distance 𝑟 is . By using this symbol, define ∗ moved to the position with distance 𝑟 .Inparticular,the ∗ 𝑉𝑚 (𝑧) =𝑉(𝑟𝑚,𝑧) wind speeds with distance 𝑟𝑚 (RMW) and 𝑅𝑚 (vortex size) are ∗ ∗ moved to new positions with distance 𝑟𝑚 and 𝑅𝑚,respectively. ∗ = √⟨𝑢2 (𝑟 (𝑧) ,𝜃,𝑧)+V2 (𝑟 (𝑧) , 𝜃, 𝑧)⟩, The relationship between 𝑟 and 𝑟 is given by the assumption 𝑚 𝑚 (12) of linear compression as follows: 𝑉(𝑅∗ ,𝑧)=√⟨𝑢2 (𝑅 (𝑧) ,𝜃,𝑧)+V2 (𝑅 (𝑧) ,𝜃,𝑧)⟩. 𝑑𝑟∗ 𝑚 𝑚 𝑚 = 𝑎 + 𝑏𝑟, 𝑑𝑟 Then, the decay rate can be deduced: ∗ ∗ 𝑟 (𝑟𝑚)=𝑟𝑚, ∗ (8) 𝑉(𝑟𝑚,𝑧)=𝑉𝑚 (𝑧) , 𝑟∗ (𝑅 )=𝑅∗ , 𝑚 𝑚 ∗ 𝛼(𝑧) ∗ 𝑟𝑚 ∗ 𝑉(𝑅 ,𝑧)=𝑉 (𝑧) ( ) , 𝑟 (0) =0. 𝑚 𝑚 ∗ 𝑅𝑚 (13)

It can deduced that 𝑉𝑚 (𝑧) 𝛼 (𝑧) = ∗ ∗ . log𝑅𝑚(𝑧)/𝑟𝑚(𝑧) ∗ 𝑟∗ 𝑅2 −𝑟2 𝑅∗ 𝑉(𝑅𝑚,𝑧) 𝑎= 𝑚 𝑚 𝑚 𝑚 , 𝑅 𝑟 (𝑅 −𝑟 ) 𝑚 𝑚 𝑚 𝑚 Azimuthal means are used to compute 𝛼(𝑧) because this is (9) ∗ ∗ more representative than the use of single-point estimations. 𝑅𝑚𝑟𝑚 −𝑅𝑚𝑟𝑚 𝑏=2 . Then, 𝑢2(𝑟, 𝜃, 𝑧) = −𝑉(𝑟, 𝑧) sin(𝜃), V2(𝑟, 𝜃, 𝑧) = 𝑉(𝑟, 𝑧) cos(𝜃). 𝑅 𝑟 (𝑅 −𝑟 ) 𝑚 𝑚 𝑚 𝑚 Here, 𝜃 is the angle between 𝑥-axis and the radial directional 4 Advances in Meteorology

vector; 𝑢2 and V2 are the EW- and NS-wind components of the max wind speed at RMW is intensified to be the same as 𝑢2-vortex. In the final step, the wind is corrected by theobservationalmaxwind: 2 ∗ ∗ ∗ 2 𝑉obs (𝑧) =(𝑢 (𝑟𝑚 (𝑧) ,𝜃𝑚 (𝑧) , 𝑧)) ∗ ∗ 𝑢 (𝑟, 𝜃,) 𝑧 =𝑢1 (𝑟, 𝜃,) 𝑧 +𝛽(𝑟,) 𝑧 𝑢2 (𝑟, 𝜃,) 𝑧 ; +(V∗ (𝑟∗ (𝑧) ,𝜃∗ (𝑧) , 𝑧))2 , ∗ ∗ (14) 𝑚 𝑚 V (𝑟, 𝜃,) 𝑧 = V1 (𝑟, 𝜃,) 𝑧 +𝛽(𝑟,) 𝑧 V2 (𝑟, 𝜃,) 𝑧 , 2 ∗ ∗ ∗ 𝑉obs (𝑧) =(𝑢1 (𝑟𝑚 (𝑧) ,𝜃𝑚 (𝑧) ,𝑧) (15) ∗ ∗ ∗ ∗ 2 where 𝑢 and V are the functions of corrected wind and 𝛽 is +𝛽1 (𝑧) 𝑢2 (𝑟𝑚 (𝑧) ,𝜃𝑚 (𝑧) ,𝑧)) the intensity correction factor. ∗ ∗ ∗ +(V1 (𝑟𝑚 (𝑧) ,𝜃𝑚 (𝑧) ,𝑧)

∗ ∗ 2 2.1.2. Computation of the Intensity Correction Factor. In +𝛽1 (𝑧) V2 (𝑟𝑚 (𝑧) ,𝜃𝑚 (𝑧) ,𝑧)) , this subsection, the function of 𝛽 (the intensity correction ∗ factor) is determined. To simplify notations, let 𝛽1(𝑧) = where 𝜃𝑚(𝑧) is the corresponding angle at RMW with height ∗ ∗ 𝛽(𝑟𝑚(𝑧), ,𝑧) 𝛽2(𝑧) = 𝛽(𝑅𝑚(𝑧), ,and𝑧) 𝜆(𝑧) 2= (𝛽 (𝑧) − 𝑧.Theverticalprofileof𝑉obs(𝑧) is specified later. Then the ∗ ∗ 𝛽1(𝑧))/(𝑅𝑚(𝑧) −𝑚 𝑟 (𝑧)).Thefollowingequationshowsthat following equation can be deduced:

󵄨 ∗ ∗ √ 2 2 2 ∗ ∗ 2 󵄨 −𝑢 𝑢2 − V V2 + 𝑉 (𝑢 + V )−(𝑢 V2 − V 𝑢2) 󵄨 𝛽 (𝑧) = 1 1 obs 2 2 1 1 󵄨 . 1 2 2 󵄨 (16) 𝑢2 + V2 󵄨 󵄨 ∗ ∗ 󵄨(𝑟𝑚(𝑧),𝜃𝑚(𝑧),𝑧)

The next step is to deduce 𝛽2(𝑧),whichistheintensity This can be expanded as ∗ correction factor at 𝑅𝑚(𝑧). The reasoning is based on the fact ∗ ∗ ∗ 2 that the typhoon vortex should be merged with the envi- (𝑢1 (𝑅𝑚,𝜃𝐵,𝑧)+𝛽2 (𝑧) 𝑢2 (𝑅𝑚,𝜃𝐵,𝑧)) ronment continuously. In other words, the vortex boundary ∗ ∗ ∗ 2 values should be close to those of the large-scale background. +(V1 (𝑅𝑚,𝜃𝐵,𝑧)+𝛽2 (𝑧) V2 (𝑅𝑚,𝜃𝐵,𝑧)) (19) Mathematically, on the vortex boundary of a typhoon, all of 2 ∗ 2 ∗ the variables should be continuous. Now, consider a radial =𝑢 (𝑅𝑚,𝜃𝐵,𝑧)+V (𝑅𝑚,𝜃𝐵,𝑧). direction with the corresponding angle 𝜃𝑚.Bytheboundary continuity, Therefore, ∗ 2 (𝑢 𝑚(𝑅 ,𝜃𝐵,𝑧)+𝛽2 (𝑧) 𝑢2 (𝑅𝑚,𝜃𝐵,𝑧))

∗ ∗ ∗ 2 lim 𝑢 (𝑟,𝑚 𝜃 ,𝑧)= lim 𝑢 (𝑟,𝑚 𝜃 ,𝑧). ∗− ∗+ (17) +(V (𝑅𝑚,𝜃𝐵,𝑧)+𝛽2 (𝑧) V2 (𝑅𝑚,𝜃𝐵, 𝑧)) (20) 𝑟→𝑅𝑚 𝑟→𝑅𝑚 2 ∗ 2 ∗ =𝑢 (𝑅𝑚,𝜃𝐵,𝑧)+V (𝑅𝑚,𝜃𝐵,𝑧).

∗ ∗ ∗ ∗− 𝑢 (𝑟, 𝜃 ,𝑧)= 𝑢 (𝑅 ,𝜃 ,𝑧) 𝑢 𝑢 𝛽 Note that lim𝑟→𝑅𝑚 𝑚 𝑚 𝑚 ,whichisthe Because and 2 areknown,theaboveequationfor 2 can be EW-windcomponentatthevortexboundaryafter solved numerically. Here, an analytic approximation for 𝛽2 is ∗ correction. On the other hand, lim𝑟→𝑅∗+ 𝑢 (𝑟,𝑚 𝜃 ,𝑧) = introduced, which can make the computations more efficient. ∗ 𝑚 ∗+ (𝑢 (𝑟, 𝜃 ,𝑧) + 𝛽 𝑢 (𝑟, 𝜃 , 𝑧)) = lim𝑟→𝑅𝑚 1 𝑚 2 2 𝑚 ∗ ∗ ∗ ∗ 𝑢2 (𝑅𝑚,𝜃𝐵,𝑧)=−𝑉(𝑅𝑚,𝑧)sin (𝜃𝐵) ∗+ 𝑢 (𝑟, 𝜃 ,𝑧) = ∗+ 𝑢(𝑟, 𝜃 ,𝑧) = 𝑢(𝑅 ,𝜃 ,𝑧) lim𝑟→𝑅𝑚 1 𝑚 lim𝑟→𝑅𝑚 𝑚 𝑚 𝑚 . 𝑢(𝑅∗ ,𝜃 ,𝑧) Recall that 𝑚 𝑚 is the EW-wind component at the √ 2 2 =− 𝑢 (𝑅𝑚,𝑧)+V (𝑅𝑚,𝑧)sin (𝜃𝐵) vortex boundary before correction. The same arguments can be applied to V. Therefore, at the boundary of the vortex, the ≈−√𝑢2 (𝑅 ,𝜃 ,𝑧)+V2 (𝑅 ,𝜃 ,𝑧) (𝜃 ) wind magnitude after the change should be equal to the wind 𝑚 𝐵 𝑚 𝐵 sin 𝐵 (21) magnitude before the change. Choose point 𝐵0 on the vortex 𝐵 √ 2 2 boundary before size correction. After size correction, 0 =− 𝑉𝐵 +𝑈𝐵 sin (𝜃𝐵)≈−𝑉𝐵 sin (𝜃𝐵) becomes 𝐵.Considerthewindspeedat𝐵: ≈−𝑉𝐵 sin (𝜃𝐵)+𝑈𝐵 cos (𝜃𝐵)

∗2 ∗ ∗2 ∗ =𝑢(𝑅𝑚,𝜃𝐵,𝑧), 𝑢 (𝑅𝑚,𝜃𝐵,𝑧)+V (𝑅𝑚,𝜃𝐵,𝑧) (18) where 𝑉𝐵 is the tangential wind at 𝐵, which is counterclock- =𝑢2 (𝑅∗ ,𝜃 ,𝑧)+V2 (𝑅∗ ,𝜃 ,𝑧). 𝑚 𝐵 𝑚 𝐵 wise. 𝑈𝐵 istheradialwindat𝐵,whichisoutward.Inthe Advances in Meteorology 5 aboveapproximations,twofactsareapplied:(1)thewind’s typhoon,theradialwindabove3kmisverysmall(typically −1 magnitude changes a little as 𝜃 changes; and (2) the radial less than 1 m s ). Notice that new parameters introduced wind is small, compared with the tangential wind. Similarly, 𝑉obs(𝑧max) 𝛾 𝐿 ∗ here are , ,and (aheightrelatedtothedecay V2(𝑅𝑚,𝜃𝐵,𝑧)=V(𝑅𝑚,𝜃𝐵,𝑧). Therefore, rate). 𝛾 is set as 2 in Stern and Nolan [9]. By estimating the observational data, Stern and Nolan [9] set parameter 𝐿 as 𝑢2 (𝑅∗ (𝑧) ,𝜃 ,𝑧)+V2 (𝑅∗ (𝑧) ,𝜃 ,𝑧) 3175 m below 𝑧max and 4762.5 m above 𝑧max. 𝛽 (𝑧) = √ 𝑚 𝐵 𝑚 𝐵 −1. (22) 2 𝑢2 (𝑅 (𝑧) ,𝜃 ,𝑧)+V2 (𝑅 (𝑧) ,𝜃 ,𝑧) Stern and Nolan [15] also confirmed that the absolute 𝑚 𝐵 𝑚 𝐵 angular momentum at the RMW for different heights was 𝑉 (𝑧) 𝛽 almost a constant. Now use obs to represent the tangential In practice, azimuthal means are used to estimate 2,and wind in reverse: theapproximationformulaisasfollows: ∗ 1 ∗ 2 𝑀 (𝑧) =𝑉obs (𝑧) 𝑟𝑚 (𝑧) + 𝑓0𝑟𝑚 (𝑧) , ⟨𝑢2 (𝑅∗ (𝑧) ,𝜃,𝑧)+V2 (𝑅∗ (𝑧) , 𝜃, 𝑧)⟩ 2 𝛽 (𝑧) = √ 𝑚 𝑚 2 2 2 ∗ 1 ∗ 2 ⟨𝑢 (𝑅𝑚 (𝑧) ,𝜃,𝑧)+V (𝑅𝑚 (𝑧) , 𝜃, 𝑧)⟩ (23) 𝑀(𝑧 )=𝑉 (𝑧 )𝑟 (𝑧 )+ 𝑓 𝑟 (𝑧 ) , (27) max obs max 𝑚 max 2 0 𝑚 max −1. 𝑀 (𝑧) =𝑀(𝑧max). The above estimation is deduced under an assumption of Therefore, wind continuity. Further conclusions, considering the conti- 1 nuity of other variables, are demonstrated in later sections. 𝑟∗ (𝑧) = [−𝑉 (𝑧) + √𝑉2 (𝑧) +2𝑀(𝑧 )𝑓]. 𝛽(𝑟, 𝑧) 𝑚 obs obs max 0 (28) Finally, the formula of is given. Because of sparse data, 𝑓0 let 𝛽 be a piecewise linear function: The 3D wind can be corrected from (26) and (28). 𝛽 (𝑟,) 𝑧

𝛽 (𝑧) ,0≤𝑟≤𝑟∗ (𝑧) ; 2.3. 3D Pressure Correction. The CSLP is an important { 1 𝑚 { (24) metric, which is always used to evaluate the quality of = 𝛽 (𝑧) +𝜆(𝑧) (𝑟 − 𝑟∗ (𝑧)), 𝑟∗ (𝑧) <𝑟≤𝑅∗ (𝑧) ; { 1 𝑚 𝑚 𝑚 tropical cyclone simulations. In this section, the gradient { 0, 𝑟 > 𝑅∗ (𝑧) . wind balance is used to deduce the pressure from the wind. { 𝑚 This method is especially efficient when the wind fields are ∗ constructed analytically. By the gradient wind balance of the By following all the steps above, the 2D wind fields 𝑢 and ∗ governing equation for the axisymmetric component, the V can be constructed if the observational max wind speed following approximation can be deduced: is provided. In later sections, the vertical profile of 𝑉obs is demonstrated and the 3D wind field is deduced. 1 𝜕𝑝 𝑉2 = +𝑓𝑉, 𝜌 𝜕𝑟 𝑟 0 (29) 2.2. 3D Wind Correction. The inputs to correct the vertical ∗ ∗ layer at height 𝑧 are 𝑟𝑚(𝑧), 𝑅𝑚(𝑧),and𝑉obs(𝑧).Oncethese where 𝑉 isconstructedwiththesamemaxwindspeedand three profiles are provided, the 3D wind fields can be vortex boundary value as the original vortex. Therefore, its 𝑅∗ (𝑧) = 𝑅 (𝑧) ∗ corrected. In practice, 𝑚 𝑚 ,whichcanbededuced profile is used as the axisymmetric part of 𝑢1 -vortex. Because ∗ ∗ from the continuous vortex boundary conditions in later 𝑢 =𝑢1 +𝛽𝑢2, (1+𝛽)𝑉 is used as the axisymmetric part of the 𝑟∗ (𝑧) ∗ sections. In this section, the focus is on solving 𝑚 and corrected vortex. 𝑉 corresponds to 𝑢1 ,and𝛽𝑉 corresponds to 𝑉 (𝑧) obs . In Rappin et al. [6] and Stern and Nolan [9], the 𝛽𝑢2.Thesetwoprofileshavethesamedirectionsonthesame vertical wind structure is constructed as grids, so they can be superimposed. The sea-level pressure 󵄨 󵄨𝛾 󵄨𝑧−𝑧 󵄨 correction step is based on the gradient wind stream function 𝑉 (𝑟,) 𝑧 =𝑉(𝑟) (−󵄨 max󵄨 ), 𝜓(𝑟, 𝑧), which satisfies exp 𝛾𝐿𝛾 (25) 𝜕𝜓 𝑉2 = +𝑉, where 𝑉(𝑟) isthetangentialwindspeedat𝑟 and 𝑧max, 𝑉(𝑟, 𝑧) (30) 𝜕𝑟 𝑟𝑓0 is the tangential wind speed at 𝑟 and 𝑧, 𝑧max is the height of 𝐿 𝛾 the maximum wind, is the vertical depth scale, and is where 𝑓0 is the Coriolis parameter at the TC center. Then, the parameter that controls the decay rate. This profile is an important result in Stern and Nolan [15]. Change (25) to 𝑟 𝑉2 𝑟 𝑉2 𝜓=∫ ( +𝑉)𝑑𝑟=∫ ( +𝑉)𝑑𝑟. (31) 󵄨 󵄨𝛾 ∞ 𝑟𝑓 𝑅∗ 𝑟𝑓 󵄨𝑧−𝑧 󵄨 0 𝑚 0 𝑉 (𝑧) =𝑉 (𝑧 ) (−󵄨 max󵄨 ) . obs obs max exp 𝛾 (26) 𝛾𝐿 The new gradient wind stream function is

The reasons for this are as follows: (1)theradialwindis 𝑟 ((1 + 𝛽) 𝑉)2 𝜓∗ = ∫ [ +(1+𝛽)𝑉]𝑑𝑟. smaller than the tangential wind, so the magnitude of the 𝑟𝑓 (32) horizontal wind is dominated by the tangential wind; (2)ina ∞ 0 6 Advances in Meteorology

∗ Let 𝑃𝑒 be the environmental sea-level pressure. The profiles 𝜓 (𝑟,) 𝑧 =0, 𝑟>𝑅𝑚 (𝑧) , (36c) oftheenvironmentalpressurearegivenbythefollowing (1 + 𝛽 (𝑧))2 𝑉 (𝑧)2 𝑟2 equations: 𝜓∗ (𝑟,) 𝑧 = 1 𝑚 ∗ 2 2𝑓0 (𝑟𝑚 (𝑧)) 2 2 Θ=Θ󸀠 + 300, (1 + 𝛽 (𝑧))𝑉 (𝑧) 𝑟2 (1 + 𝛽 (𝑧)) 𝑉 (𝑧) + 1 𝑚 − 1 𝑚 ∗ (37a) 2𝑟𝑚 (𝑧) 2𝑓0 𝑝 −2/7 0 1 ∗ ∗ ∗ 𝑇𝑘 =Θ( ) , − (1 + 𝛽 (𝑧))𝑉 (𝑧) 𝑟 (𝑧) +𝜓 (𝑟 (𝑧) ,𝑧), 𝑝 2 1 𝑚 𝑚 𝑚 (33) ∗ 𝑟<𝑟𝑚 (𝑧) ; 𝑇V =𝑇𝑘 (1 + 𝛿𝑞) , 2𝛼(𝑧) 󵄨 𝑉 (𝑧)2 (𝑟∗ (𝑧)) 𝜆 (𝑧)2 𝑟2 𝑔𝑧 󵄨 𝜓∗ (𝑟,) 𝑧 ={ 𝑚 𝑚 [ 𝑝 (𝑧) =𝑃 (− )󵄨 , 2𝛼(𝑧) 𝑒 𝑒 exp 󵄨 𝑓0𝑟 2−2𝛼(𝑧) 𝑅𝑑𝑇V 󵄨𝑅∗ (𝑧) 𝑚 2𝜆 (𝑧) 𝑟(1+𝛽 (𝑧) −𝜆(𝑧) 𝑟∗ (𝑧)) + 1 𝑚 1−2𝛼(𝑧) Θ󸀠 where is the perturbed virtual temperature, which can be ∗ 2 Δ𝑝(𝑟, 𝜃, 𝑧)= (1 + 𝛽 (𝑧) −𝜆(𝑧) 𝑟 (𝑧)) obtained from the outputs of WRF-ARW. Let − 1 𝑚 ]+𝑉 (𝑧) 𝑝(𝑟, 𝜃, 𝑧) −𝑝 (𝑧) Δ𝑝∗(𝑟, 𝜃, 𝑧)∗ =𝑝 (𝑟, 𝜃, 𝑧) −𝑝 (𝑧) 𝑚 𝑒 and 𝑒 2𝛼 (𝑧) (37b) be the pressure perturbations before and after correction, ∗ 𝛼(𝑧) 1−𝛼(𝑧) 𝜆 (𝑧) 𝑟 respectively. Assuming that 𝜌 and 𝑓0 are constants, then, ⋅(𝑟 (𝑧)) 𝑟 ( 𝑚 2−𝛼(𝑧) 𝑟 ∗ 󵄨 𝑟 𝑟 󵄨 1 𝜕𝑝 𝜕𝜓 1+𝛽1 (𝑧) −𝜆(𝑧) 𝑟𝑚 (𝑧) 󵄨 ∫ 𝑑𝑟 = ∫ 𝑓 𝑑𝑟, + )}󵄨 , 0 1−𝛼(𝑧) 󵄨 ∞ 𝜌 𝜕𝑟 ∞ 𝜕𝑟 󵄨 ∗ 𝑅𝑚(𝑧) 1 𝑟∗ (𝑧) ≤𝑟≤𝑅∗ (𝑧) ; Δ𝑝 = 𝑓 𝜓, 𝑚 𝑚 𝜌 0 (34) ∗ ∗ 𝜓 (𝑟,) 𝑧 =0, 𝑟>𝑅𝑚 (𝑧) . (37c)

1 ∗ ∗ 𝑝∗(𝑟, 𝜃, 𝑧)∗ =[𝜓 (𝑟, 𝑧)/𝜓(𝑟, 𝑧)][𝑝(𝑟, 𝜃,𝑧)−𝑝 (𝑧)]+ Δ𝑝 =𝑓0𝜓 . Finally, 𝑒 𝜌 𝑝𝑒(𝑧) is the desired pressure profile.

This shows that the pressure perturbation is proportional 2.4. 3D Temperature and Humidity Correction. In this sub- to the gradient wind stream function. The new pressure section, a method to construct a 3D virtual temperature field 𝑇∗ perturbation is ( V ) is first introduced [13]. The key technique is to divide the total virtual temperature field into an environmental field ∗ andavortexfield.Theenvironmentalfieldprovidesthebase ∗ 𝜓 Δ𝑝 =Δ𝑝 . (35) state, and the vortex field provides the perturbations. From 𝜓 thehydrostaticequationandtheidealgaslaw,thefollowing equation is obtained: The explicit formulations are 𝜕𝑝 𝑝 =− 𝑔, (38) 𝜕𝑧 𝑅𝑑𝑇V 𝑉 (𝑧)2 𝑟2 𝑉 (𝑧) 𝑟2 𝑉 (𝑧)2 1 𝜓 (𝑟,) 𝑧 = 𝑚 + 𝑚 − 𝑚 − 𝑅 = 287.04 −1 −1 𝐻 ∗ 2 ∗ where 𝑑 Jkg K .Let betheheightofthe 2𝑓 (𝑟 (𝑧)) 2𝑟𝑚 (𝑧) 2𝑓0 2 0 𝑚 (36a) top of the typhoon vortex. 𝑧0 is a fixed height. Integrate the ∗ ∗ (𝑥, 𝑦) × [𝑧 ,𝐻] ⋅𝑉𝑚 (𝑧) +𝜓(𝑟𝑚 (𝑧) ,𝑧), 𝑟<𝑟𝑚 (𝑧) ; above equation along the segment 0 .Inthe 2 ∗ 2𝛼(𝑧) environment, 1 𝑉𝑚 (𝑧) (𝑟 (𝑧)) 𝜓 (𝑟,) 𝑧 =− 𝑚 𝑝 𝑔 𝐻 2𝛼 (𝑧) 𝑓 𝑟2𝛼(𝑧) 0 𝑑𝑧 0 ln = ∫ , (39) 𝑝𝑡 𝑅𝑑 𝑧 𝑇 (𝑧) ∗ 𝛼(𝑧) 0 V 𝑉𝑚 (𝑧) (𝑟𝑚 (𝑧)) 1 + + 𝑝 ,𝑝 𝑧 (1−𝛼(𝑧)) 𝑟𝛼(𝑧)−1 2𝛼 (𝑧) where 0 𝑡 are the pressures at 0 and the top of the compu- 2𝛼(𝑧) tational domain, respectively. 𝑇V is the virtual temperature of 𝑉 (𝑧)2 (𝑟∗ (𝑧)) ⋅ 𝑚 𝑚 (36b) the environment. In the total field, ∗ 2𝛼(𝑧) 𝑓0 (𝑅𝑚 (𝑧)) 𝐻 𝑝0 +Δ𝑝0 𝑔 𝑑𝑧 ∗ 𝛼(𝑧) ln = ∫ , (40) 𝑉 (𝑧) (𝑟 (𝑧)) 𝑝 𝑅 𝑧 𝑇 (𝑧) +Δ𝑇 (𝑧) − 𝑚 𝑚 , 𝑡 𝑑 0 V V (1−𝛼(𝑧)) (𝑅∗ (𝑧))𝛼(𝑧)−1 𝑚 where “Δ” signifies perturbation in the vortex, and the ∗ ∗ 𝑟𝑚 (𝑧) ≤𝑟≤𝑅𝑚 (𝑧) ; subscript “0” signifies the value at 𝑧0. Δ𝑇V is the enhancement Advances in Meteorology 7 madetotheoriginalenvironmentbythevortexpart.Thenthe The corrected humidity is equation derived in the total field is modified as ∗ ∗ 17.67 ⋅ 243.5 (𝑇 −𝑇) 𝑝 Δ𝑝 𝑔 𝐻 𝑑𝑧 𝑞 = exp [ ]𝑞. (49) 0 ⋅(1+ 0 )= ∫ (𝑇∗ − 29.66)(𝑇 − 29.66) ln 𝑝 𝑝 𝑅 𝑡 0 𝑑 𝑧0 𝑇V (𝑧) +Δ𝑇V (𝑧) ∗ (41) Equations (46) and (49) are two equations about 𝑞 and 𝑔 𝐻 1 Δ𝑇 𝑇∗ 𝑇∗ 𝑞 ≈ ∫ (1 − V )𝑑𝑧. .Combinethemandnoticethat V and are known: 𝑅 𝑑 𝑧0 𝑇V 𝑇V 1 𝑇∗ 17.67 ⋅ 243.5 (𝑇∗ −𝑇) ( V −1)=𝑞 [ ]. Use this equation and the equation for the environment, and 𝛿 𝑇∗ exp (𝑇∗ − 29.66)(𝑇 − 29.66) (50) an approximation can be obtained for the perturbation of the virtual temperature: The above pointwise equation is solved on each grid point Δ𝑝 𝑔 𝐻 Δ𝑇 within the vortex. Now, the 3D temperature and humidity (1 + 0 )≈− ∫ V 𝑑𝑧. fields are corrected. ln 2 (42) 𝑝 𝑅 𝑧 0 𝑑 0 𝑇V [1 + (Δ𝑝 )/𝑝 ] ≈ (Δ𝑝 )/𝑝 2.5. Vortex Boundary Conditions. Pressure, temperature, and Because ln 0 0 0 0, specific humidity should also be continuously merged into Δ𝑝 𝑔 𝐻 Δ𝑇 the environment. In other words, at the vortex boundary, the 0 =− ∫ V 𝑑𝑧. 2 (43) following equations should be valid: 𝑝 𝑅 𝑧 0 𝑑 0 𝑇V ∗ ∗ lim 𝑝 (𝑟, 𝜃,) 𝑧 =𝑝(𝑅𝑚,𝜃,𝑧), 𝑟→𝑅∗− After correction, 𝑚 ∗ 𝐻 ∗ ∗ ∗ Δ𝑝 𝑔 Δ𝑇 lim 𝑇 (𝑟, 𝜃,) 𝑧 =𝑇(𝑅𝑚,𝜃,𝑧), 0 =− ∫ V 𝑑𝑧. 𝑟→𝑅∗− (51) 𝑝 𝑅 2 (44) 𝑚 0 𝑑 𝑧0 𝑇 V 𝑞∗ (𝑟, 𝜃,) 𝑧 =𝑞(𝑅∗ ,𝜃,𝑧). lim∗− 𝑚 ∗ ∗ 𝑟→𝑅𝑚 Define Γ(𝑟, 𝑧) =𝜓 (𝑟, 𝑧)/𝜓(𝑟, 𝑧).RecallthatΔ𝑝0 = Γ(𝑟,0 𝑧 )Δ𝑝0. Therefore, Thecontinuityofvirtualtemperaturecanalsobededuced Γ(𝑟,𝑧 )Δ𝑝 𝑔 𝐻 Γ(𝑟,𝑧 )Δ𝑇 from the continuity of temperature and humidity: 0 0 =− ∫ 0 V 𝑑𝑧, 2 ∗ ∗ ∗ 𝑝0 𝑅𝑑 𝑧 0 𝑇V 𝑇V =𝑇 (1 + 𝛿𝑞 )󳨀→𝑇(1+𝛿𝑞)=𝑇V, (45) (52) 𝐻 Δ𝑇∗ 𝐻 Γ(𝑟,𝑧 )Δ𝑇 (𝑟 󳨀→ 𝑅∗−). ∫ V 𝑑𝑧 = ∫ 0 V 𝑑𝑧. 𝑚 2 2 𝑧0 𝑇 𝑧0 𝑇 V V Equation (51) is equivalent to 𝑧 Δ𝑇∗(𝑧) = Γ(𝑟, 𝑧 )Δ𝑇 (𝑧) 𝑧∈[𝑧,𝐻] For a fixed 0, V 0 V for any 0 𝜓∗ (𝑟,) 𝑧 Δ𝑇∗(𝑧) = Γ(𝑟, 𝑧)Δ𝑇 (𝑧) is one simple solution. In particular, V V . lim =1. (53) ∗ 𝑟→𝑅∗− 𝜓 (𝑟,) 𝑧 Now, 𝑇V is known. Between virtual temperature and temper- 𝑚 ature, they are connected by ∗ Because lim𝑟→𝑅∗− 𝜓 (𝑟, 𝑧) = 0, lim𝑟→𝑅∗− 𝜓(𝑟, 𝑧), =0 ∗ 𝑚 𝑚 ∗ 𝑇V (𝑟, 𝜃,) 𝑧 𝑇 (𝑟, 𝜃,) 𝑧 = , ∗ ∗ ∗ (46) 𝜓 (𝑟,) 𝑧 (𝜕𝜓 /𝜕𝑟) (𝑟,) 𝑧 1+𝛿𝑞 (𝑟, 𝜃,) 𝑧 = lim∗− lim∗− 𝑟→𝑅𝑚 𝜓 (𝑟,) 𝑧 𝑟→𝑅𝑚 (𝜕𝜓/𝜕𝑟) (𝑟,) 𝑧 𝛿 = 0.608 𝑞∗/𝑒∗ =𝑞/𝑒 where . During the corrections, 𝑠 𝑠 is 2 2 𝑒 𝑒 (𝑇) (1 + 𝛽) 𝑉 /𝑟𝑓0 +(1+𝛽)𝑉 assumed, where and 𝑠 are vapor pressure and saturation = lim 𝑟→𝑅∗− 2 vapor pressure. By definition, 𝑞 = (0.622𝑒)/(𝑝 − 0.378𝑒), 𝑚 𝑉 /𝑟𝑓0 +𝑉 𝑞∗ =(𝑒∗/𝑒 )𝑞 which is the specific humidity. Then, 𝑠 𝑠 .Fromthe 2 2𝛼 𝛼 (54) (1 + 𝛽 ) 𝑉2 (𝑟∗ /𝑅∗ ) +𝑓𝑅∗ (1 + 𝛽 )𝑉 (𝑟∗ /𝑅∗ ) definition of saturation vapor pressure, = 2 𝑚 𝑚 𝑚 0 𝑚 2 𝑚 𝑚 𝑚 𝑉2 (𝑟∗ /𝑅∗ )2𝛼 +𝑓𝑅∗ 𝑉 (𝑟∗ /𝑅∗ )𝛼 𝑇 − 273.16 𝑚 𝑚 𝑚 0 𝑚 𝑚 𝑚 𝑚 𝑒 𝑇 = 6.112 (17.67 ), 𝑠 ( ) exp 2 2 ∗ ∗ 2𝛼 ∗ ∗ ∗ 𝛼 𝑇 − 29.66 (𝛽2 +2𝛽2)𝑉𝑚 (𝑟𝑚/𝑅𝑚) +𝑓0𝑅𝑚𝛽2𝑉𝑚 (𝑟𝑚/𝑅𝑚) (47) =1+ . 𝑇∗ − 273.16 𝑉2 (𝑟∗ /𝑅∗ )2𝛼 +𝑓𝑅∗ 𝑉 (𝑟∗ /𝑅∗ )𝛼 𝑒∗ (𝑇∗) = 6.112 (17.67 ). 𝑚 𝑚 𝑚 0 𝑚 𝑚 𝑚 𝑚 𝑠 exp 𝑇∗ − 29.66 Therefore, 𝛽2 =0is a necessary and sufficient condition to It is obtained that make 𝑒∗ (𝑇∗) 17.67 ⋅ 243.5 (𝑇∗ −𝑇) 𝜓∗ (𝑟,) 𝑧 𝑠 = [ ]. =1. exp ∗ (48) lim∗− (55) 𝑒𝑠 (𝑇) (𝑇 − 29.66)(𝑇 − 29.66) 𝑟→𝑅𝑚 𝜓 (𝑟,) 𝑧 8 Advances in Meteorology

From the continuity of wind, it can be deduced that Table 1: Simulation periods.

Name Start time (UTC) End time (UTC) Length (h) ⟨𝑢2 (𝑅∗ (𝑧) ,𝜃,𝑧)+V2 (𝑅∗ (𝑧) , 𝜃, 𝑧)⟩ 𝛽 (𝑧) = √ 𝑚 𝑚 −1 Wutip 00:00, 28 Sep, 2013 12:00, 30 Sep, 2013 60 2 2 2 ⟨𝑢 (𝑅𝑚 (𝑧) ,𝜃,𝑧)+V (𝑅𝑚 (𝑧) , 𝜃, 𝑧)⟩ (56) Fitow 12:00, 2 Oct, 2013 12:00, 6 Oct, 2013 96 Krosa 00:00, 30 Oct, 2013 00:00, 3 Nov, 2013 96 =0. Genevieve 12:00, 9 Aug, 2014 12:00, 11 Aug, 2014 48 ∗ Phanfone 00:00, 1 Oct, 2014 00:00, 5 Oct, 2014 96 𝑅𝑚(𝑧) =𝑚 𝑅 (𝑧) can guarantee the above equations. The continuity of other variables can be deduced from the Nuri 00:00, 1 Nov, 2014 00:00, 5 Nov, 2014 96 continuity of pressure and temperature. In conclusion, the vortex boundary conditions for all of the variables require ∗ that 𝑅𝑚(𝑧) =𝑚 𝑅 (𝑧). The variables to be compared are track, CSLP, and maximum surface wind speed (MWSP). The observational data were 3. Evaluations of the Vortex available every 6 hours. Therefore, the simulated data are Initialization Scheme compared with observations every 6 hours. Consider the time series of observational data 𝑂𝑖 and simulated data 𝑆𝑖, 1≤𝑖≤ 3.1. A Real Case Example: Super Typhoon Phanfone (2014). 𝑁. 𝑁 isthenumberoftimepointstobecompared,thatis, The super typhoon Phanfone (2014) is used as an example (length of simulation)/6 + 1. The definition of absolute error to illustrate the effects of the vortex initialization scheme on (AE) in this paper for 𝑖-th time point is |𝑆𝑖 −𝑂𝑖|,whichis the initial conditions. The initial time is 00:00:00 (UTC) on slightly different from the traditional definition𝑆 ( 𝑖 −𝑂𝑖). The 1 Oct, 2014. In Figures 1–4, the surface wind speed WSP𝑠, mean absolute error (MAE) is defined as surface pressure 𝑝𝑠, surface temperature 𝑇𝑠, and surface 𝑞 water vapor mixing ratio 𝑠 before and after initialization are 1 𝑁 󵄨 󵄨 compared in the parent domain (D01), the middle domain = ∑ 󵄨𝑆 −𝑂󵄨 . MAE 𝑁 󵄨 𝑖 𝑖󵄨 (57) (D02), and the inner domain (D03). The parent domain 𝑖=1 (D01) has a horizontal resolution of 27 km, with 199 × 172 grids; the middle nest domain (D02) has a resolution of The MAE is affected equally by all of the time points during 9km, with 301 × 223grids;andtheinnerdomain(D03) thesimulationperiodandcanbeusedasavariabletoevaluate has a resolution of 3 km, with 469 × 370 grids. The domains a simulation’s overall performance. The improvements of the are shown in Figure 5. The variables in the inner domain initialization scheme are evaluated by are identically interpolated back to the middle domain and no − init 𝑟∗ (𝑧 ) = AE AE × 100 , the parent domain. The modified RMW 𝑚 max is set as improvement of AE no % 90 km (the sensitivity of this parameter is discussed later); AE 𝑉 (𝑧 ) theobservationalmaxwindspeed obs max is set as 60 kt. improvement of MAE (58) After initialization, strong wind around the typhoon center no init can be observed in Figure 1(f). Before initialization, the eye MAE − MAE = × 100%, wall of the typhoon is not formed (Figure 1(e)). At the initial MAEno time,theobservationalCSLPis975hPa,whereastheCSLP before initialization is higher than 990 hPa (Figure 2(e)), where “no” signifies no initialization and “init” signifies which needs to be intensified. As shown in Figures 3 and 4, initialization. In the figures and tables, the units for track, −1 thetemperatureandmoisturefieldsattheinitialtimeareboth CSLP, and MWSP are km, hPa, and m s ,respectively. intensified simultaneously. 3.3. WRF Configurations. Numerical simulations were con- 3.2. Typhoon Cases and Evaluation Metrics. Simulations were ducted by WRF-ARW v3.5.1. As an example, the domains for conductedforsevereandsupertyphooncasesinthenorth- super typhoon Phanfone (2014) are illustrated in Figure 5. western Pacific during 2013-2014. The simulated cases are For the other cases, the domains are horizontally shifted by severe typhoon Wutip (2013), severe typhoon Fitow (2013), changing the reference latitude and reference longitude in severe typhoon Krosa (2013), super typhoon Genevieve the WRF preprocessing system (WPS). For each case, D03 is (2014), super typhoon Phanfone (2014), and super typhoon designed to make the initial vortex located at its right-bottom Nuri(2014),becausetheinitialMWSPderivedfromFNL corner, which helps the inner domain to cover the tracks as is not strong enough for these cases. The cases are listed far as possible. All of the domains have 39 vertical layers. in chronological order, and the simulation periods are also The Lambert projection method is applied. The time steps are listed in Table 1. The simulations start half a day before 120s,40s,and40/3sforD01,D02,andD03,respectively.To the typhoon stage. The duration for most simulations is 96 maintain stability, the time step in seconds must be 6×Δ𝑥 hours, except for Genevieve (for which the observational data or less, where Δ𝑥 isthegridsizeinkm.Thecurrentsettings starts from the 180th meridian) and Wutip (which ends with fortimestepscansatisfythestabilityconstraintandcontrol dissipation). The simulation results are compared with the the computational cost at the same time. The pressure at the observational data from the Hong Kong Observatory (HKO). top of the domains is set as 50 hPa, which is enough to reach Advances in Meteorology 9

30 30 40 40 35 25 35 25 30 30 20 20 25 25

20 15 20 15 Latitude Latitude 15 15 10 10 10 10

5 5 5 5 0 0 0 0 110 120 130 140 150 160 170 110 120 130 140 150 160 170 Longitude Longitude (a) (b) 30 30 26 26 24 25 24 25 22 22 20 20 20 20 18 18 15 15 16 16 Latitude Latitude 14 10 14 10 12 12 5 10 10 5 8 8 0 0 130 135 140 145 150 155 130 135 140 145 150 155 Longitude Longitude (c) (d) 30 30 22 22 21 25 21 25 20 20 19 20 19 20 18 18 17 15 17 15 Latitude Latitude 16 16 10 10 15 15 14 5 14 5 13 13 12 0 12 0 138 140 142 144 146 148 150 138 140 142 144 146 148 150 Longitude Longitude (e) (f)

Figure 1: Phanfone: (a) WSP𝑠 before initialization in D01 (27 km); (b) WSP𝑠 after initialization in D01 (27 km); (c) WSP𝑠 before initialization in D02 (9 km); (d) WSP𝑠 after initialization in D02 (9 km); (e) WSP𝑠 before initialization in D03 (3 km); and (f) WSP𝑠 after initialization in D03 (3 km). The unit in these figures is m/s.

the tropopause height for tropical cyclone simulations. Two- RRTMG schemes [17]. The Grell-Freitas cumulus scheme [18] way nesting is used for all of the simulations. Time varying is applied in D01 and D02. sea surface temperature, sea-ice, vegetation fraction, and The initial conditions and boundary conditions are gen- albedo are used. No data assimilations are used. For physical erated by the FNL data (6-hourly, 1-degree resolution). The parameterizations, the YSU planetary boundary layer scheme initial conditions can be further modified by the vortex is used. The microphysics scheme is the WSM6 scheme [16]. initialization scheme, if the initial MWSP is not strong The long-wave and short-wave radiation schemes are the enough. The configurations of all the six cases simulated with 10 Advances in Meteorology

1010 1010 40 40

35 1005 35 1005 30 30 1000 1000 25 25

20 995 20 995 Latitude Latitude 15 15 990 990 10 10

5 985 5 985 0 0 980 980 110 120 130 140 150 160 170 110 120 130 140 150 160 170 Longitude Longitude (a) (b) 1010 1010 26 26 24 1005 24 1005 22 22 20 1000 20 1000 18 18 995 995 16 16 Latitude Latitude 14 990 14 990 12 12 985 985 10 10 8 8 980 980 130 135 140 145 150 155 130 135 140 145 150 155 Longitude Longitude (c) (d) 1010 1010 22 22 21 1005 21 1005 20 20 19 1000 19 1000 18 18 995 17 995 17 Latitude 16 Latitude 16 990 990 15 15 14 985 14 985 13 13 12 980 12 980 138 140 142 144 146 148 150 138 140 142 144 146 148 150 Longitude Longitude (e) (f)

Figure 2: Phanfone: (a) 𝑝𝑠 before initialization in D01 (27 km); (b) 𝑝𝑠 after initialization in D01 (27 km); (c) 𝑝𝑠 before initialization in D02 (9 km); (d) 𝑝𝑠 after initialization in D02 (9 km); (e) 𝑝𝑠 before initialization in D03 (3 km); and (f) 𝑝𝑠 after initialization in D03 (3 km). The unit in these figures is hPa. the vortex initialization scheme are summarized in Table 2. 4. Results and Analysis Although the observational RMW is currently unavailable, the input parameter RMW in the initialization scheme has For Wutip, Fitow, Krosa, Genevieve, Phanfone, and Nuri, a low sensitivity. As shown in Figure 6, the super typhoon comparisons between without and with initialization are Phanfone (2014) was simulated with three RMW values (80, made for track, CSLP, and MWSP. The main metric to 90, and 100 km). From the time series of track, CSLP, and evaluate the simulations is the MAE, which is defined in (57). MWSP,it can be observed that the simulation results are close If the FNL data for one case can provide sufficiently strong to each other for different RMW values. initial MWSP but cannot provide a low enough CSLP, the Advances in Meteorology 11

31 31 40 40

35 30.5 35 30.5 30 30 30 30 25 25

20 29.5 20 29.5 Latitude 15 Latitude 15 29 29 10 10

5 28.5 5 28.5 0 0 28 28 110 120 130 140 150 160 170 110 120 130 140 150 160 170 Longitude Longitude (a) (b) 31 31 26 26 24 30.5 24 30.5 22 22 20 30 20 30 18 18 29.5 29.5 16 16 Latitude Latitude 14 29 14 29 12 12 28.5 28.5 10 10 8 8 28 28 130 135 140 145 150 155 130 135 140 145 150 155 Longitude Longitude (c) (d) 31 31 22 22 21 30.5 21 30.5 20 20 19 30 19 30 18 18 29.5 29.5 17 17 Latitude 16 Latitude 16 29 29 15 15 14 28.5 14 28.5 13 13 12 28 12 28 138 140 142 144 146 148 150 138 140 142 144 146 148 150 Longitude Longitude (e) (f)

Figure 3: Phanfone: (a) 𝑇𝑠 before initialization in D01 (27 km); (b) 𝑇𝑠 after initialization in D01 (27 km); (c) 𝑇𝑠 before initialization in D02 (9 km); (d) 𝑇𝑠 after initialization in D02 (9 km); (e) 𝑇𝑠 before initialization in D03 (3 km); and (f) 𝑇𝑠 after initialization in D03 (3 km). The ∘ unit in these figures is C. vortex initialization scheme will still not be used because 4.1. Analysis of Case-Specific Results. In Table 3, the improve- our scheme uses the gradient wind balance to deduce the ments of track, CSLP, and MWSP made by the vortex pressure field from the wind field. If the wind field from initialization scheme are illustrated for each case. The vortex FNL is stronger than the observation, the intensity correction initialization scheme has little impact on the track predic- factor will be negative, which further increases the center tion results. This is because the track is largely dependent pressure. In the following subsections, we analyze the case- on the large-scale analysis, while vortex initialization can specific results and summarize the overall performance. onlymodifythevortexpart.ForFitowandGenevieve,the 12 Advances in Meteorology

22.5 22.5 40 40 35 35 22 22 30 30

25 21.5 25 21.5 20 20 Latitude 15 21 Latitude 15 21 10 10 5 20.5 5 20.5 0 0 20 20 110 120 130 140 150 160 170 110 120 130 140 150 160 170 Longitude Longitude (a) (b) 22.5 22.5 26 26 24 24 22 22 22 22 20 20 21.5 21.5 18 18 16 16 Latitude Latitude 21 21 14 14 12 12 20.5 20.5 10 10 8 8 20 20 130 135 140 145 150 155 130 135 140 145 150 155 Longitude Longitude (c) (d) 22.5 22.5 22 22 21 21 22 22 20 20 19 19 21.5 21.5 18 18 17 17 Latitude Latitude 16 21 16 21 15 15 14 20.5 14 20.5 13 13 12 20 12 20 138 140 142 144 146 148 150 138 140 142 144 146 148 150 Longitude Longitude (e) (f)

Figure 4: Phanfone: (a) 𝑞𝑠 before initialization in D01 (27 km); (b) 𝑞𝑠 after initialization in D01 (27 km); (c) 𝑞𝑠 before initialization in D02 (9 km); (d) 𝑞𝑠 after initialization in D02 (9 km); (e) 𝑞𝑠 before initialization in D03 (3 km); and (f) 𝑞𝑠 after initialization in D03 (3 km). The unit in these figures is g/kg. scheme increases the MAE of track by 4.6% and 3.4%, averaged MAEs of track, CSLP, and MWSP are improved by respectively. For Wutip and Nuri, the scheme results in 4.0%, 17.6%, and 14.5%, respectively (Table 4). In Table 4, larger improvements in the MAE of track. In general, the averaged AEs and MAEs are reported. With the excep- vortex initialization scheme improves the CSLP and MWSP tion of 24 h track, 48 h track, and 72 h CSLP, the scheme prediction results. The scheme reduces the MAE of CSLP by improves all of the variables (Table 4). Table 5 presents 7.6%–30.2% and the MAE of MWSP by 7.9%–42.3%. comparisons between our scheme and some other schemes. Inthecolumn“ourscheme,”avalueinboldistheminimum 4.2. Analysis of Overall Performance. In this section, the ofitsrow(whichmeansthatourschemeprovidesthe averaged terms are analyzed, with the averages taken for best performance of the corresponding metric among all allthecases.Throughthevortexinitializationscheme,the the schemes). Advances in Meteorology 13

Table 2: Configurations in the vortex initialization scheme. Table 4: Averaged terms and improvements of track, CSLP, and MWSP by the vortex initialization scheme. Max Radius of observational Averaged AE Without With Name max wind Initial location Improvement wind speed terms initialization initialization (km) (kt) 24 h track 58.2 69.7 −19.8% ∘ ∘ Wutip 60 90 16.4 N, 114.1 E 48 h track 99.3 103.4 −4.2% ∘ ∘ Fitow 60 90 18.9 N, 129.9 E 72 h track 178.3 163.4 8.4% ∘ ∘ Krosa 60 90 16.2 N, 129.8 E 96 h track 365.6 326.3 10.8% ∘ ∘ Genevieve 100 75 22.7 N, 177.3 E MAE of track 119.0 114.2 4.0% ∘ ∘ Phanfone 60 90 16.9 N, 143.8 E 33.6% ∘ ∘ 24 h CSLP 13.9 9.2 Nuri 60 90 13.0 N, 133.9 E 48 h CSLP 15.8 15.0 4.8% 72 h CSLP 12.8 12.9 −0.7% 96 h CSLP 14.0 14.0 0.1% Table 3: Improvements of track, CSLP, and MWSP by the vortex MAE of 14.1 11.6 17.6% initialization scheme. CSLP Improvements Improvements Improvements 24 h MWSP 6.6 6.5 2.1% Name of track of CSLP of MWSP 48 h MWSP 8.1 7.0 13.5% Wutip 13.8% 30.2% 42.3% 72 h MWSP 7.3 6.8 7.0% Fitow −4.6% 7.6% 21.9% 96 h MWSP 5.0 3.2 35.2% Krosa 6.8% 21.0% 13.2% MAE of 6.7 5.7 14.5% Genevieve −3.4% 18.7% 8.6% MWSP Phanfone 5.9% 14.0% 7.9% Nuri 10.7% 18.7% 10.3% 60

50 In Xiao et al. [10], the averaged track error of the bogus 40 data assimilation (BDA) scheme is around 110 km, whereas D01 the averaged MAE of track simulated by our scheme is 30 114.2 km; the averaged CSLP error is around 17 hPa (our scheme: 11.6 hPa, as shown in Table 5); and the averaged D02 20 03

Latitude D MWSP is around 9 m/s (our scheme: 5.7 m/s). The scheme in Kwon and Cheong [7] decreases the 48 h track error to 10 173.9 km on average (our scheme: 103.4 km); the 48 h error for CSLP is improved from 21.2 hPa to 10.0 hPa on average 0 (our scheme: 15.0 hPa); and the 24 h CSLP error is improved from 37.3hPa to 11.2 hPa on average (our scheme: 9.2 hPa). −10 In Hsiao et al. [11], the track errors are 208, 95, and 312 km 110 120 130 140 150 160 170 for the with-relocation runs at 24, 48, and 72 h, respectively Longitude (our scheme: 69.7, 103.4, and 163.4 km at 24, 48, and 72 h). In Cha and Wang [12], for the MAEs averaged for all forecasts, Figure 5: WRF domains for the super typhoon Phanfone (2014) D01 the dynamic initialization scheme reduces the absolute track (27 km); D02 (9 km); and D03 (3 km). error to 103.3 km (our scheme: 114.2 km) and reduces the absolute MWSP error to 7.3m/s (our scheme: 5.7 m/s). TC vortex with only little observational data. The derivation 5. Summary and Conclusions of the scheme is mainly based on the dynamical balance constraints. Therefore, the scheme is highly analytic, which In this paper, a new vortex initialization scheme is developed makes it satisfy the vortex boundary conditions and with low and tested. Through this scheme, a new framework of initial computational cost. Some highlights of this work include (1) vortex construction is introduced, which starts from 3D determination of the intensity correction factor at the vortex wind construction. The other variables, such as pressure, boundary 𝛽2 (23); (2) using the absolute angular momentum ∗ temperature, and humidity, are also constructed on the basis at the RMW to derive the RMW function 𝑟𝑚(𝑧) (28); (3) of dynamical and thermodynamical equations. explicit analytic formulations of the gradient wind stream The scheme requires only observational data for the cen- functions ((36a), (36b), (36c), (37a), (37b), and (37c)), which ter location and max wind speed in addition to global analysis are useful for improving the computational speed of pres- data. One advantage of this scheme is that it can estimate a 3D sure construction; (4) the construction of temperature and 14 Advances in Meteorology

Table 5: Comparisons of our scheme with other schemes.

Metrics\schemes Xiao et al. [10] Kwon and Cheong [7] Hsiao et al. [11] Cha and Wang [12] Our scheme 24 h track — — 208 — 69.7 48 h track — 173.9 95 — 103.4 72 h track — — 312 — 163.4 MAE of track 110 — — 103.3 114.2 24 h CSLP — 11.2 — — 9.2 48 h CSLP — 10.0 — — 15.0 MAE of CSLP 17 — — — 11.6 MAE of MWSP 9 — — 7.3 5.7

40 990 980 35 970 30 960 25 950

Latitude 940 20 Pressure (hPa) Pressure 930 15 920 10 910 120 125 130 135 140 145 150 0 10 20 30 40 50 60 70 80 90 100

Longitude Forecast time (h) RMW =80km RMW = 100 km RMW =80km RMW = 100 km RMW =90km Obs RMW =90km Obs (a) (b) 55 )

−1 50 45 40 35 30 25

Max surface wind speed (mMax s 20 0 10 20 30 40 50 60 70 80 90 100

Forecast time (h) RMW =80km RMW = 100 km RMW =90km Obs (c)

Figure 6: Phanfone: simulation results with RMW = 80, 90 and100km(a)track;(b)CSLP;(c)MWSP.

humidity is unified in (50); and (5) discussion of the vortex weakness of the scheme is about the RMW, which cannot be boundary continuity (a necessary and sufficient condition obtained from observations currently. of vortex boundary continuity under the framework of this The scheme was evaluated via simulations for severe and scheme is given in Section 2.5). super typhoons in the northwestern Pacific. The scheme can Some future work includes improving the parameters, for provide a better prediction of typhoon intensity: the averaged example, 𝛾 and 𝐿 introduced in (6), which is the estimation MAEs of CSLP and MWSP are improved by 17.6% and 14.5%, for vertical profile of tangential wind speed. One potential respectively. Compared with other methods and initialization Advances in Meteorology 15 schemes,theschemecanprovidethebestperformanceforthe [7] I. H. Kwon and H. B. Cheong, “Tropical cyclone initialization averaged MAEs of CSLP and MWSP. with a spherical high-order filter and an idealized three- dimensional bogus vortex,” Monthly Weather Review,vol.138, Acronyms no. 4, pp. 1344–1367, 2010. [8] K. A. Reed and C. Jablonowski, “Ananalytic vortex initialization AE: Absolute error technique for idealized tropical cyclone studies in AGCMs,” CSLP: Center sea-level pressure Monthly Weather Review,vol.139,no.2,pp.689–710,2011. FNL: NCEP final analyses [9]D.P.SternandD.S.Nolan,“Ontheverticaldecayrateofthe GFS: Global forecast system maximum tangential winds in tropical cyclones,” Journal of the Atmospheric Sciences,vol.68,no.9,pp.2073–2094,2011. HKO: Hong Kong Observatory MAE: Mean absolute error [10]Q.Xiao,L.Chen,andX.Zhang,“EvaluationsofBDAscheme using the advanced research WRF (ARW) model,” Journal of MR: Modified Rankine AppliedMeteorologyandClimatology,vol.48,no.3,pp.680– MWSP: Maximum surface wind speed 689, 2009. NCEP: National Centers for Environmental [11] L.-F. Hsiao, C.-S. Liou, T.-C. Yeh et al., “A vortex relocation Prediction scheme for tropical cyclone initialization in advanced research PBL: Planetary boundary layer WRF,” Monthly Weather Review, vol. 138, no. 8, pp. 3298–3315, RMW: Radius of max wind 2010. TC: Tropical cyclone [12] D.-H. Cha and Y. Wang, “A dynamical initialization scheme for UTC: Coordinated Universal Time real-time forecasts of tropical cyclones using the WRF model,” WRF/ARW: Weather research and Monthly Weather Review,vol.141,no.3,pp.964–986,2013. forecasting/advanced research WRF [13] S. G. Gopalakrishnan, Q. Liu, T. Marchok et al., Hurricane WPS: WRF preprocessing system Weather Research and Forecasting (HWRF) Model Scientic Doc- WSP: Surface wind speed. umentation, Edited by L. Bernardet, Monthly Weather Review, 2010. Competing Interests [14] S. J. Lord, “Abogusing system for vortex circulations in the National Meteorological Center global forecast model,” in The authors declare that there are no competing interests Proceedings of the 19th Conference on Hurricanes and Tropical regarding the publication of this paper. Meteorology, pp. 328–330, American Meteorological Society, 1991. Acknowledgments [15] D. P.Stern and D. S. Nolan, “Reexamining the vertical structure of tangential winds in tropical cyclones: observations and The authors appreciate the assistance of the HKO, which theory,” Journal of the Atmospheric Sciences,vol.66,no.12,pp. provided the meteorological data. This work was supported 3579–3600, 2009. by NSFC/RGC, Grant N HKUST631/05; NSFC-FD, Grant [16] S. Y. Hong and J. O. J. Lim, “The WRF single-moment 6- U1033001; and the RGC, Grant 16300715. class microphysics scheme (WSM6),” Journal of the Korean Meteorological Society,vol.42,no.2,pp.129–151,2006. [17] M. J. Iacono, J. S. Delamere, E. J. Mlawer, M. W. Shephard, S. References A. Clough, and W. D. Collins, “Radiative forcing by long-lived greenhouse gases: calculations with the AER radiative transfer [1]R.L.Elsberry,T.D.B.Lambert,andM.A.Boothe,“Accuracy models,” Journal of Geophysical Research Atmospheres, vol. 113, of Atlantic and Eastern North Pacific tropical cyclone intensity no. 13, Article ID D13103, 2008. forecast guidance,” Weather and Forecasting,vol.22,no.4,pp. [18] G. A. Grell and S. R. Freitas, “Ascale and aerosol aware stochas- 747–762, 2007. tic convective parameterization for weather and air quality [2]M.DeMaria,C.R.Sampson,J.A.Knaff,andK.D.Musgrave, modeling,” Atmospheric Chemistry and Physics Discussions,vol. “Is tropical cyclone intensity guidance improving?” Bulletin of 13, no. 9, pp. 5233–5250, 2013. the American Meteorological Society,vol.95,no.3,pp.387–398, 2014. [3]F.D.Marks,L.K.Shay,G.Barnesetal.,“Landfallingtropical cyclones: forecast problems and associated research opportuni- ties,” Bulletin of the American Meteorological Society,vol.79,no. 2, pp. 305–323, 1998. [4]R.A.Anthes,Tropical Cyclones: Their Evolution, Structure and Effects, Meteorological Monographs, American Meteorological Society, 1982. [5] X. Zou and Q. Xiao, “Studies on the initialization and simulation of a mature hurricane using a variational bogus data assimila- tion scheme,” Journal of the Atmospheric Sciences,vol.57,no.6, pp. 836–860, 2000. [6] E. D. Rappin, D. S. Nolan, and S. J. Majumdar, “A highly configurable vortex initialization method for tropical cyclones,” Monthly Weather Review,vol.141,no.10,pp.3556–3575,2013. International Journal of Journal of Ecology Mining

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