Advances in Meteorology

Volume 2017 (2017), Article ID 6272158, 15 pages

https://doi.org/10.1155/2017/6272158

## A New Vortex Initialization Scheme Coupled with WRF-ARW

^{1}Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong^{2}Division 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; kh.tsu@gnufjam

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

Academic Editor: Anthony R. Lupo

Copyright © 2017 Jimmy Chi Hung Fung and Guangze 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.

#### Abstract

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. It was 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

To reduce the damage caused by tropical cyclones (TC), it is very important to predict their track and intensity correctly. In recent decades, track prediction has been steadily improved [1]. However, there has been little improvement in predicting the intensity of typhoons [1]. During 1989–2012, TC intensity errors from the best available models have decreased at 1%-2% at 24, 48, and 72 h [2].

As mentioned by Marks et al. [3], the primary reason for the slow progress in improvement of intensity prediction is that the intensity depends on the inner-core dynamics, whereas the track prediction depends more on the large-scale environments. The inner-core dynamics are mainly associated with vortex structure, intensity, and vortex size. It has been suggested that the accurate representation of the inner-core structure of tropical cyclones in the initial conditions is as important as the representation of the large-scale environment [4]. To improve the representation of the initial vortex, a bogus vortex is often applied because the initial vortices provided by the large-scale analysis are often too weak or misplaced [5]. For example, for the super typhoon Genevieve (2014) at 12:00 (UTC) on 9 Aug 2014, the center sea-level pressure (CSLP) derived from FNL (NCEP final analyses) data is around 993 hPa, whereas the observational CSLP from HKO is 940 hPa.

Vortex initialization is an efficient method to provide better estimates of the initial conditions for numerical simulation of tropical cyclones. The advantages of the use of bogus vortices are the ability to customize the vortex structure and low computational cost, compared with dynamical initialization and variational data assimilation methods [6].

In Kwon and Cheong [7], an initialization method with an idealized 3D bogus vortex is developed for track and intensity predictions. In their scheme, a 3D axisymmetric bogus vortex is designed empirically in terms of analytic functions. The bogus vortex is used as a disturbance field and merged into the environmental field. The surface pressure profile of the bogus vortex is given by an exponent decay function. The temperature is derived from the 3D geopotential by using the hydrostatic balance equation. The relative humidity is given by an empirical function, which is not dynamically consistent with the other variables. In our scheme, this problem is overcome by solving the temperature and the humidity simultaneously (Equation (50)).

Reed and Jablonowski [8] used analytic functions to derive the initial conditions. The initial axisymmetric vortex is in hydrostatic and gradient wind balance. The first step is to provide the analytic background specific humidity profile:where is set as 15 km, which approximates the tropopause height; is the specific humidity at the surface, which is set as 21 g ; and is the specific humidity in the upper atmosphere, which is set as . m and m are two constants to control the specific humidity profile. The virtual temperature profile is given bywhere is the virtual potential temperature at the surface, and is the virtual temperature lapse rate, which is set as 0.007 K . is the virtual temperature in the upper atmosphere. The background temperature profile can then be derived asThe background pressure profile is computed using the hydrostatic equation and the ideal gas law. To obtain the perturbation profile of temperature, an empirical exponent decay function is used to modify the pressure perturbation. Then, by the hydrostatic equation and the ideal gas law, the temperature is derived. Finally, the tangential velocity is derived by the gradient wind balance, which is given bywhere is the Coriolis parameter at the constant latitude and is the rotational speed of Earth. is set to be . is distance to the TC center. J kg^{−1} K^{−1} is the ideal gas constant for dry air. and are virtual temperature and pressure. The analytic formulation of this scheme enables it to be applied in various weather and climate models. However, the specific humidity is constructed as a horizontally homogeneous variable, which may not be realistic for the tropical cyclones. Also, the specific humidity is not adjustable, which means that the same specific humidity field must be used for all the tropical cyclones. Therefore, the scheme may be lacking flexibility to simulate different tropical cyclones.

In Rappin et al. [6], a highly configurable vortex initialization scheme is developed, which can be easily manipulated to generate different initial vortex structures. The vortex construction starts with parameterizations of the radial wind structure. The tangential wind of the MR (modified Rankine) vortex is given bywhere is the decaying factor, RMW is the radius of maximum wind, is the tangential velocity at RMW, and is the distance from the TC center.

The vertical structure of tangential wind speed follows Stern and Nolan [9]:where is set as and is set as 3175 m below and 4762.5 m above [9]. is the height of the boundary layer top. Finally, the hydrostatic balance and gradient wind balance are used to determine temperature and pressure. One advantage of this scheme is that it makes use of (6), which considers the vertical structure of tangential wind speed. By using this equation, the 3D wind field can be derived from a 2D wind structure (e.g., the MR profile) directly, which makes the scheme highly configurable. The equations for constructing horizontal and vertical profiles introduced in Rappin et al. [6] may also be helpful in designing new schemes. In our new scheme, we further improve the wind construction by making the decaying factor and RMW as functions of .

In this paper, a vortex initialization scheme is developed for the weather research and forecasting (WRF) model with the advanced research WRF (ARW) core. Further, in our scheme, the variables satisfy geophysical formulas, which ensures that the variables are dynamically consistent. The analytic formulations of this scheme enable it to be conducted efficiently. The problems in other schemes mentioned above can also be solved by our scheme. Notice that these problems are mainly associated with two aspects: (1) the variables should be dynamically consistent with each other; (2) the TCs constructed by a scheme should be adjustable and realistic.

The vortex initialization problem can be viewed as follows: given the data sets of from global analysis data (e.g., FNL and GFS (global forecast system) data), how can we correct them to new data ? Here, represent EW-wind, NS-wind, pressure, temperature, and specific humidity, respectively; “” is used to represent corrected data throughout this paper. In most situations, we need to provide some extra data besides the global analysis data to build the new vortex. For example, we may need the max wind speed and RMW. Although providing more data may help us to achieve better initial fields, it is difficult to obtain observational data on the ocean. In our scheme, we aim to construct a 3D vortex that can represent basic features of typhoon’s structure. The extra inputs are vortex location, max wind speed, and the radius of the max wind.

The rest of the paper is organized as follows: in Section 2, the vortex initialization scheme is derived; in Section 3, numerical simulations for super and severe typhoons in the northwestern Pacific are conducted to evaluate the performance of the new scheme; the simulation results are discussed in Section 4; and in Section 5, conclusions are given.

#### 2. The New Vortex Initialization Scheme

##### 2.1. 2D Wind Correction

The goal in this paper is to construct a new 3D vortex. Construction of one vertical layer is first discussed, which is started from wind construction. The wind construction can be disaggregated as size correction, intensity correction, and computation of the intensity correction factor.

###### 2.1.1. Size and Intensity Correction

The first step is “size correction” [13]. If the max wind speed and the RMW are computed by the FNL data directly, the max wind speed is generally smaller than the observational data, and the RMW is generally too large. The typhoon size can be corrected by compressing the model grids. and are the distances from a grid point to the typhoon center, before and after size correction. The size correction is conducted bywhere and are the wind components after size correction. is the azimuthal angle and is the height from the ground. This means that the wind speed with distance is moved to the position with distance . In particular, the wind speeds with distance (RMW) and (vortex size) are moved to new positions with distance and , respectively. The relationship between and is given by the assumption of linear compression as follows:It can deduced thatIn practice, the vortex boundary is defined as the contour line of a sea-level pressure value (e.g., 1013 hPa). After size correction, the new vortex is called -vortex.

The second step is “intensity correction” [13]. As mentioned by Lord [14], the vortex initialization is usually carried out by first implanting a synthetic vortex into the large-scale analysis. Similarly, the intensity correction in this paper is conducted by superposing an axisymmetric vortex onto -vortex. The axisymmetric vortex is called -vortex. When adding -vortex to -vortex, the intensity is adjusted through the intensity correction factor. In the next subsection, the issues related to the computation of the intensity correction factor are addressed.

In the axisymmetric -vortex, the tangential velocity is , and the radial velocity is zero. is defined as follows: denote as the max tangential wind speed at height in the axisymmetric vortex, which is the same as the max wind magnitude in the original vortex and -vortex. is the tangential velocity on the circle of radius (the center of the circle is the TC center). is defined in the form of an MR vortex:where is the profile decay rate, which is determined later. For variable , defineThe physical meaning of is the azimuthal mean of variable . By using this symbol, defineThen, the decay rate can be deduced:

Azimuthal means are used to compute because this is more representative than the use of single-point estimations. Then, , . Here, is the angle between -axis and the radial directional vector; and are the EW- and NS-wind components of -vortex. In the final step, the wind is corrected bywhere and are the functions of corrected wind and is the intensity correction factor.

###### 2.1.2. Computation of the Intensity Correction Factor

In this subsection, the function of (the intensity correction factor) is determined. To simplify notations, let , , and . The following equation shows that the max wind speed at RMW is intensified to be the same as the observational max wind:where is the corresponding angle at RMW with height . The vertical profile of is specified later. Then the following equation can be deduced:

The next step is to deduce , which is the intensity correction factor at . The reasoning is based on the fact that the typhoon vortex should be merged with the environment continuously. In other words, the vortex boundary values should be close to those of the large-scale background. Mathematically, on the vortex boundary of a typhoon, all of the variables should be continuous. Now, consider a radial direction with the corresponding angle . By the boundary continuity,Note that , which is the EW-wind component at the vortex boundary after correction. On the other hand, . Recall that is the EW-wind component at the vortex boundary before correction. The same arguments can be applied to . Therefore, at the boundary of the vortex, the wind magnitude after the change should be equal to the wind magnitude before the change. Choose point on the vortex boundary before size correction. After size correction, becomes . Consider the wind speed at :This can be expanded asTherefore,Because and are known, the above equation for can be solved numerically. Here, an analytic approximation for is introduced, which can make the computations more efficient.where is the tangential wind at , which is counterclockwise. is the radial wind at , which is outward. In the above approximations, two facts are applied: (1) the wind’s magnitude changes a little as changes; and (2) the radial wind is small, compared with the tangential wind. Similarly, . Therefore,

In practice, azimuthal means are used to estimate , and the approximation formula is as follows:

The above estimation is deduced under an assumption of wind continuity. Further conclusions, considering the continuity of other variables, are demonstrated in later sections. Finally, the formula of is given. Because of sparse data, let be a piecewise linear function:

By following all the steps above, the 2D wind fields and can be constructed if the observational max wind speed is provided. In later sections, the vertical profile of is demonstrated and the 3D wind field is deduced.

##### 2.2. 3D Wind Correction

The inputs to correct the vertical layer at height are , , and . Once these three profiles are provided, the 3D wind fields can be corrected. In practice, , which can be deduced from the continuous vortex boundary conditions in later sections. In this section, the focus is on solving and . In Rappin et al. [6] and Stern and Nolan [9], the vertical wind structure is constructed aswhere is the tangential wind speed at and , is the tangential wind speed at and , is the height of the maximum wind, is the vertical depth scale, and is the parameter that controls the decay rate. This profile is an important result in Stern and Nolan [15]. Change (25) toThe reasons for this are as follows: () the radial wind is smaller than the tangential wind, so the magnitude of the horizontal wind is dominated by the tangential wind; () in a typhoon, the radial wind above 3 km is very small (typically less than 1 m s^{−1}). Notice that new parameters introduced here are , , and (a height related to the decay rate). is set as in Stern and Nolan [9]. By estimating the observational data, Stern and Nolan [9] set parameter as 3175 m below and 4762.5 m above .

Stern and Nolan [15] also confirmed that the absolute angular momentum at the RMW for different heights was almost a constant. Now use to represent the tangential wind in reverse:Therefore,The 3D wind can be corrected from (26) and (28).

##### 2.3. 3D Pressure Correction

The CSLP is an important metric, which is always used to evaluate the quality of tropical cyclone simulations. In this section, the gradient wind balance is used to deduce the pressure from the wind. This method is especially efficient when the wind fields are constructed analytically. By the gradient wind balance of the governing equation for the axisymmetric component, the following approximation can be deduced:where is constructed with the same max wind speed and vortex boundary value as the original vortex. Therefore, its profile is used as the axisymmetric part of -vortex. Because , is used as the axisymmetric part of the corrected vortex. corresponds to , and corresponds to . These two profiles have the same directions on the same grids, so they can be superimposed. The sea-level pressure correction step is based on the gradient wind stream function , which satisfieswhere is the Coriolis parameter at the TC center. Then,The new gradient wind stream function isLet be the environmental sea-level pressure. The profiles of the environmental pressure are given by the following equations:where is the perturbed virtual temperature, which can be obtained from the outputs of WRF-ARW. Let and be the pressure perturbations before and after correction, respectively. Assuming that and are constants, then,

This shows that the pressure perturbation is proportional to the gradient wind stream function. The new pressure perturbation is

The explicit formulations are

Finally, is the desired pressure profile.

##### 2.4. 3D Temperature and Humidity Correction

In this subsection, a method to construct a 3D virtual temperature field () is first introduced [13]. The key technique is to divide the total virtual temperature field into an environmental field and a vortex field. The environmental field provides the base state, and the vortex field provides the perturbations. From the hydrostatic equation and the ideal gas law, the following equation is obtained:where . Let be the height of the top of the typhoon vortex. is a fixed height. Integrate the above equation along the segment . In the environment,where are the pressures at and the top of the computational domain, respectively. is the virtual temperature of the environment. In the total field,where “” signifies perturbation in the vortex, and the subscript “0” signifies the value at . is the enhancement made to the original environment by the vortex part. Then the equation derived in the total field is modified asUse this equation and the equation for the environment, and an approximation can be obtained for the perturbation of the virtual temperature:Because ,After correction,Define . Recall that . Therefore,For a fixed , for any is one simple solution. In particular, . Now, is known. Between virtual temperature and temperature, they are connected bywhere . During the corrections, is assumed, where and are vapor pressure and saturation vapor pressure. By definition, , which is the specific humidity. Then, . From the definition of saturation vapor pressure,It is obtained thatThe corrected humidity is

Equations (46) and (49) are two equations about and . Combine them and notice that and are known:The above pointwise equation is solved on each grid point within the vortex. Now, the 3D temperature and humidity fields are corrected.

##### 2.5. Vortex Boundary Conditions

Pressure, temperature, and specific humidity should also be continuously merged into the environment. In other words, at the vortex boundary, the following equations should be valid:The continuity of virtual temperature can also be deduced from the continuity of temperature and humidity:Equation (51) is equivalent toBecause ,Therefore, is a necessary and sufficient condition to makeFrom the continuity of wind, it can be deduced that can guarantee the above equations. The continuity of other variables can be deduced from the continuity of pressure and temperature. In conclusion, the vortex boundary conditions for all of the variables require that .

#### 3. Evaluations of the Vortex Initialization Scheme

##### 3.1. A Real Case Example: Super Typhoon Phanfone (2014)

The super typhoon Phanfone (2014) is used as an example to illustrate the effects of the vortex initialization scheme on the initial conditions. The initial time is 00:00:00 (UTC) on 1 Oct, 2014. In Figures 1–4, the surface wind speed , surface pressure , surface temperature , and surface water vapor mixing ratio before and after initialization are compared in the parent domain (D01), the middle domain (D02), and the inner domain (D03). The parent domain (D01) has a horizontal resolution of 27 km, with 199 172 grids; the middle nest domain (D02) has a resolution of 9 km, with 301 223 grids; and the inner domain (D03) has a resolution of 3 km, with 469 370 grids. The domains are shown in Figure 5. The variables in the inner domain are identically interpolated back to the middle domain and the parent domain. The modified RMW is set as 90 km (the sensitivity of this parameter is discussed later); the observational max wind speed is set as 60 kt. After initialization, strong wind around the typhoon center can be observed in Figure 1(f). Before initialization, the eye wall of the typhoon is not formed (Figure 1(e)). At the initial time, the observational CSLP is 975 hPa, whereas the CSLP before initialization is higher than 990 hPa (Figure 2(e)), which needs to be intensified. As shown in Figures 3 and 4, the temperature and moisture fields at the initial time are both intensified simultaneously.