Evaluations on Profiles of the Eddy Diffusion Coefficients through Simulations of Super Typhoons in the Northwestern Pacific

The modeling of the eddy diffusion coefficients (also known as eddy diffusivity) in the first-order turbulence closure schemes is important for the typhoon simulations, since the coefficients control the magnitude of the sensible heat flux and the latent heat flux, which are energy sources for the typhoon intensification. Profiles of the eddy diffusion coefficients in the YSU planetary boundary layer (PBL) scheme are evaluated in the advanced researchWRF (ARW) system.Three versions of the YSU scheme (original, K025, and K200) are included in this study. The simulation results are compared with the observational data from track, center sealevel pressure (CSLP), and maximum surface wind speed (MWSP). Comparing with the original version, the K200 improves the averagedmean absolute errors (MAE) of track, CSLP, andMWSP by 6.0%, 3.7%, and 23.1%, respectively, while the K025 deteriorates the averaged MAEs of track, CSLP, and MWSP by 25.1%, 19.0%, and 95.0%, respectively. Our results suggest that the enlarged eddy diffusion coefficients may be more suitable for super typhoon simulations.


Introduction
The PBL is important for the typhoon intensification since the turbulent mixing in the PBL affects the momentum and heat in the typhoons.However, it is still not possible to fully resolve the diffusion processes within the boundary layer due to the limitations of physical models and grid resolution.Therefore, the boundary layer parameterizations are required to model the sub-grid-scale effects.Among various boundary layer schemes, the first-order turbulence closure schemes are widely used in the tropical cyclone studies and the weather research and forecasting (WRF) model.Comparing with higher-order turbulence closure schemes, the firstorder schemes are computationally cheaper.The simplest first-order schemes are based on the local-K approach.However, the eddy transportation in the planetary boundary layer is mainly conducted by large eddies, which should be represented by bulk properties of the PBL, rather than local properties.Therefore, the nonlocal first-order schemes were developed to resolve this problem and keep the simplicity at the same time.
The eddy diffusivity' modeling is fundamental to the firstorder turbulence closure PBL schemes.We briefly go through the history of the eddy diffusivity' developments.In the study of O'brien, the eddy diffusivity for momentum () is parameterized as where  is the von Kármán's constant,  * is the friction velocity, and () is the stability function [1].In Brost and Wyngaard, the eddy diffusivity for momentum   is represented by where ℎ is the boundary layer height and  is the Monin-Obukhov length [2]. 2

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In Troen and Mahrt,   is parameterized as where  = 0.4 and   is the wind profile function at the top of the surface layer [3].They further assumed that where  * is the convective velocity scale and  = 0.1 is the fraction of the surface layer height and the boundary layer height.The Prandtl number in Troen and Mahrt is modeled as where  ℎ is the eddy diffusivity for temperature and moisture and   is the eddy diffusivity for momentum [3]. ℎ and   are the profile functions for ,  and , V, respectively.Based on Troen and Mahrt, in Hong and Pan, a boundary layer diffusion package was implemented into the NCEP medium-range forecast model [3,4].The turbulence diffusion equations for a prognostic variable  can be expressed by where   is the eddy diffusivity for  and   is the countergradient term for , which was introduced by Deardorff [5].
can be , V (horizontal wind components),  (potential temperature), and  (water vapor mixing ratio).Same as Troen and Mahrt, the eddy diffusivity for momentum is formulated as where  is the profile shape exponent taken as 2 and   is the mixed-layer velocity scale [3].The counter-gradient terms for  and  are given by where     is the corresponding surface flux for  and .The boundary layer height is given by where Rib cr is the critical bulk Richardson number, (ℎ) is the horizontal wind speed at ℎ,  V is the virtual potential temperature at the lowest model level,  V (ℎ) is the virtual potential temperature at ℎ, and   is the temperature near surface.In Hong and Pan, the Prandtl number is a constant within the whole mixed boundary layer [4].It is given by The nonlocal scheme can improve the capability on representing the large eddy turbulence within a well-mixed boundary layer, and it is still computationally cheap as other first-order turbulence closure schemes.In Hong et al., a revised vertical diffusion package is developed on the basis of Noh et al. [6,7].The turbulence diffusion equations for the prognostic variables can be expressed by where (    ) ℎ is the flux at the inversion layer.The Prandtl number is parameterized as where Pr 0 is the Prandtl number at the top of the surface layer.The profile functions  ℎ and   are given for different conditions.For unstable and neutral conditions, For the stable condition, Comparing with the model proposed by Troen and Mahrt, the new model has the following main features: (1) incorporation of an explicit entrainment term into the heat fluxes; (2) the heat fluxes above the boundary layer height are also parameterized; (3) a profile of the Prandtl number is used, in contrast to the constant value; and (4) nonlocal mixing of momentum is also included [3].
Gopalakrishnan et al. started to use flight-level observations to modify the eddy diffusivity in the first-order planetary boundary layer schemes [8].In Zhang and Drennan, they investigated the vertical eddy diffusivity in the surface wind regime between 18 and 30 m s −1 over the ocean [9].They showed that the magnitudes of the eddy diffusion coefficients for momentum and latent heat fluxes are comparable, whereas the magnitude of the eddy diffusion coefficient for sensible heat flux is much smaller.The authors noted that the data were limited to wind speeds less than 30 m s −1 .However, it is quite common to have wind speed higher than 30 m s −1 in the super typhoons.
In Gopalakrishnan et al., they studied the impacts of modifying   and  ℎ to be 25% or 50% of the coefficients' original values [8].To the best of their knowledge, this is the first time that flight-level observations are used as the basis to provide an improvement to the existing boundary layer parameterization schemes.They found that reductions of   and  ℎ to 25% of their original values (for later references, this version is named as GFS-K025 in this article) produced eddy diffusion coefficients which were consistent with observations.It was also found that the GFS-K025 induced stronger hurricane intensity in the idealized frameworks, compared with the original GFS scheme and the GFS-K050.Since it is likely to underestimate the typhoon intensity in numerical models, it is possible that the GFS-K025 can provide better real-time predictions.In Gopalakrishnan et al., the boundary layer scheme used is the Global Forecast System (GFS) scheme [8].The GFS scheme is a prior version of the YSU scheme, which is based on Hong and Pan [4].The major difference between these two schemes is that the fluxes in the GFS scheme do not contain an entrainment component.
The eddy diffusivity   in the YSU scheme and the   in the GFS scheme have the same formulation: However, the mixed-layer velocity scales (  ) are modeled differently: In Gopalakrishnan et al., they pointed out that a reduction of diffusion would lead to a reduction in the dissipation of the angular momentum in the boundary layer, which would further lead to stronger spin-up and enhanced moisture convergence [8].The enhanced latent heat flux provides better thermo conditions for tropical cyclones to develop and intensify.However, we also noticed that while the reduction of   could reduce the sink of momentum and angular momentum, the reduction of  ℎ would also reduce the sources of heat and moisture and weaken the tropical cyclones.It might still be difficult to determine whether the eddy diffusivity coefficients should be enlarged or decreased in the PBL schemes.To make an attempt at addressing this issue, we conduct simulations for the super typhoons in 2014 to evaluate 3 versions of the YSU scheme (original YSU, K025, and K200) in this article.
The YSU scheme is used because it is a state-of-the-art first-order turbulence closure scheme.In the K025 and the K200, the eddy diffusion coefficients are modified to be 25% and 200% of their original values, respectively.The rest of this article is organized as follows: in Section 2, methods and numerical simulations are introduced; in Section 3, simulation results are analyzed; and in Section 4, conclusions are provided.

Methods and Numerical Simulations
In this section, we introduce the designs of the K025 and the K200, the simulated super typhoon cases, the WRF configurations, and the evaluation metrics.

K-Profiles:
Original, K025, and K200.The eddy diffusivity for momentum (  ) is parameterized in (15).The eddy diffusivity for heat and moisture ( ℎ ) is calculated by   and the Prandtl number (Pr).We introduce a new parameter  to control the magnitudes of   and  ℎ : For the original YSU scheme,  = 1; for the K025,  = 0.25; and for the K200,  = 2.In the K025 and the K200, the eddy diffusion coefficients are modified to be 25% and 200% of their original values, respectively.1.

Super Typhoon
There are 39 vertical layers in all the three domains.The map projection method used for the simulations is the Lambert projection.The time steps are 120 s, 40 s, and 40/3 s for D01, D02, and D03, respectively.The current settings for the time steps can satisfy the stability constraints and control the computational cost at the same time.The pressure at the top of the computational domains is set as 50 hPa.Two-way nesting has been used for all the simulations.Time varying (6-hourly) sea surface temperature has been used.
The YSU planetary boundary layer scheme is used.The long wave and short wave radiation schemes are the RRTMG schemes.The Grell-Freitas cumulus scheme is applied in D01 and D02.The microphysics scheme is the WRF Singlemoment 6-class (WSM6) scheme.The surface layer scheme is the MM5 similarity scheme.The land surface model is the Unified Noah Land Surface Model.The initial conditions and the boundary conditions are generated by the FNL (NCEP Final Analyses) data.

Evaluation Metrics.
The simulation results are compared with the observational data from the HKO.The variables to be compared are track, CSLP, and MWSP.The observational data is 6-hourly.Therefore, the simulated data are compared with observations at  = 0, 6, 12, 18, 24, . . ., h.Consider time series of the observational data   and the simulated data   , 1 ≤  ≤ , where  is the number of time points compared; that is,  = (length of simulation)/6 + 1.The definition of absolute error (AE) in this article for the th time point is |  −   |, which is slightly different from the traditional definition (  −   ).The mean absolute error (MAE) is defined as The MAE is affected by all the time points during the simulation period equally, which can be used as a variable to evaluate a simulation's overall performance.The improvements made by a new K-profile are evaluated by where "original" represents the original YSU scheme and "new" represents a new version of YSU scheme with a modified K-profile.The AEs are averaged for all the cases at  = 24, 48, 72 and 96 h, which are named as the averaged AE at .The averaged MAE is defined as the weighted-averaged MAE for all the cases similarly, with the weight equal to the number of compared time points for each case.In the figures and tables, the units for track, CSLP, and MWSP are km, hPa, and m s −1 , respectively.

Case-Specific Analysis.
To evaluate the three versions of YSU, we compare the simulated track, CSLP, and MWSP with their corresponding observational data.The computation of AE and MAE follows the definitions in Section 2.4.The cases are listed in chronological order.with the observational track and is better than the tracks simulated by the original YSU and the K025 in the middle of the simulation period.As shown in Table 3, at  = 48 h, the AE of track simulated by the K200 is 31.7 km, while the AE of track with the K025 is 134.5 km.In Figures 3(d  the K025.The MAEs of CSLP simulated by the 3 versions are close to each other.However, the MAE of MWSP simulated with the K025 is much larger than the other 2 versions (Table 3), which is 17.0 m s −1 .As shown in Figure 3(e), the version of K025 cannot simulate the intensification correctly.4(a), 4(b), and 4(c), the track simulated by the K025 is closer to the observational track and is better than the tracks simulated by the K200 and the original YSU.In Figures 4(d) and 4(e), the CSLP simulated by the K025 is the best among all the 3 versions, and the

Case 3: Genevieve. In Figures
MWSP simulated by the K200 is the best.As shown in Table 4, the MAE of MWSP simulated by the K200 is 4.2 m s −1 , which is lower than the MAE of MWSP simulated by the K025 (8.8 m s −1 ).

Case 4: Phanfone.
In Figure 5 and Table 5, it is observed that the K025 performs better than the K200 in terms of track.The K025 can improve the MAE of track by 38.6% compared with the original YSU, whereas the K200 can improve the MAE of track by 23.0%.In terms of CSLP, the K025 performs slightly better than the K200 (Table 5).The K025 can improve the MAE of CSLP by 1.9 hPa compared with the original YSU, while the K200 can improve the MAE of CSLP by 1.4 hPa.In terms of MWSP, the K200 is much better than the K025.The MAE of MWSP simulated by the K025 is 16.9 m s −1 ; the MAE of MWSP simulated by the K200 is 2.9 m s −1 ; and the MAE of MWSP simulated by the original YSU is 4.9 m s −1 (Table 5).

Case 5:
Vongfong.In Table 6, the MAE of track simulated by the K200 is 121.0 km, which is lower than the MAE of track simulated by the K025 (222.2 km) and the original YSU (151.9 km).As shown in Figure 6(d), the CSLP time series simulated by the 3 versions are close to each other.In terms of MWSP, the K200 and the original YSU can perform better than the K025 (Figure 6(e)).
3.1.6.Case 6: Nuri.For the super typhoon Nuri (2014), the CSLP time series simulated by the K200 and the original YSU are very close (Figure 7(d)).As shown in Table 7, the MAE of track, CSLP, and MWSP simulated by the K025 are the worst.The MAEs of track, CSLP, and MWSP simulated by the original YSU and the K200 are close to each other (Table 7).As shown in Table 8, at  = 48 h, the AE of CSLP simulated by the original YSU, the K025, and the K200 are 50.1,57.5, and 47.8 hPa, respectively.In Figure 8(e), the MWSP simulated by the K200 is the strongest among all the 3 versions, which maybe resulted from larger heat and moisture fluxes.3.1.8.Summary.In general, for all the 7 super typhoons, the simulations by the K200 always provide the best MWSP prediction, in the sense of lowest MAE.For track and CSLP, we use the averaged AEs and MAEs to evaluate the overall performance of these 3 versions.It is observed that the intensity simulated with larger eddy diffusivity is likely to be stronger.

Overall Performance.
The averaged AEs at  = 24, 48, 72, 96 h are listed in Table 9.In Table 9, a number in bold means that the corresponding K-profile can provide the best performance for that averaged AE or MAE.It shows that the K200 provides the best performance for all the AEs and MAEs, except for the averaged AE of 48 h-CSLP.For all the averaged MAEs, the K200 can always provide the best performance.For the averaged AEs of 24 h-track, 48 htrack, 72 h-track, and 96 h-track, the K200 reduces/improves them by 3.6-8.5%,whereas the K025 increases/deteriorates them by 9.6-39.1%.The K200 improves the averaged AEs of 24 h-CSLP and 96 h-CSLP by 23.1% and 20.8%, respectively.However, the K025 deteriorates the averaged AEs of 24 h-CSLP and 96 h-CSLP by 40.3% and 54.0%, respectively.For the averaged AEs of 24 h-MWSP, 48 h-MWSP, 72 h-MWSP, and 96 h-MWSP, the K200 improves them by 16.1-50.1%,whereas the K025 deteriorates them by 78.1-105.9%.
Comparing with the original version, the K200 improves the averaged MAEs of track, CSLP, and MWSP by 6.0%, 3.7%, and 23.1%, respectively.However, the K025 deteriorates the averaged MAEs of track, CSLP, and MWSP by 25.1%, 19.0%, and 95.0%, respectively.To test the sensitivity of the enlarged eddy diffusion coefficients, the   and  ℎ are also modified to be 3 times their original values in a new version of K-profile, which is named as K300.We use the super typhoon Phanfone (2014) to give an example.In Figure 9, the track, CSLP, and MWSP of 4 different versions (original YSU, K025, K200, and K300) are compared.It can be observed in Figure 9(a) that the track simulated by the original YSU is the worst and the track simulated by the K025 is the best.The tracks simulated by the K200 and the K300 are close to each other.From Figure 9  it is observed that the CSLP time series simulated by the original YSU, the K200, and the K300 are close to each other, especially during the last 2 days of the simulation period.In Figure 9(c), the MWSP time series simulated by the K200 and the K300 are close to each other during the whole simulation period.These results indicate that the typhoon intensity may not be sensible to the magnitudes of enlarged eddy diffusion coefficients.

Conclusions
In this article, WRF simulations for super typhoons are used to evaluate 3 versions of the YSU scheme.For all the averaged MAEs, the K200 can always provide the best performance.For all the super typhoon cases, the simulations with the K200 can always provide the best MWSP prediction.
Comparing with the original version, the K200 improves the averaged MAEs of track, CSLP, and MWSP by 6.0%, 3.7%, and 23.1%, respectively, whereas the K025 deteriorates the averaged MAEs of track, CSLP, and MWSP by 25.1%, 19.0%, and 95.0%, respectively.Our results suggest that the enlarged eddy diffusion coefficients may be more suitable for super typhoon simulations, because that larger eddy diffusion coefficients for heat and moisture can induce larger sensible and latent heat fluxes.
However, we shall also notice that this study is limited to the YSU PBL scheme and the WRF-ARW model.The modifications to the eddy diffusion coefficients also cannot be applied to the higher-order closure models, for example, Wyngaard and Coté [10], and Enger [11].Besides, the study is also limited to the simple choices of the parameter .In the

3. 1 . 1 .
Case 1: Neoguri.In Figures2(a), 2(b), and 2(c), the tracks simulated by the original YSU, the K025, and the K200 are similar.In Figures2(d) and 2(e), it can be observed that the CSLP simulated with the K200 and the original YSU are close to each other.The MWSP simulated with the K200 agrees well with the observational MWSP.It is observed that the MWSP becomes stronger as the eddy diffusion coefficients become larger.As illustrated in Table2, the MAE of MWSP for the K200 is 2.4 m s −1 , which is much smaller compared with the MAE of MWSP simulated with the K025 (17.3 m s −1 ).3.1.2.Case 2:Rammasun.In Figures3(a), 3(b), and 3(c), it is observed that the track simulated by the K200 agrees well

Figure 2 :
Figure 2: Case 1-Neoguri: (a) track simulated by the original YSU; (b) track simulated by the K025; (c) track simulated by the K200; (d) comparisons of center sea-level pressure; and (e) comparisons of maximum surface wind speed.
) and 3(e), it is observed that the intensity variables (CSLP and MWSP) simulated by the original YSU and the K200 are relatively closer to each other, compared with the variables simulated by

Figure 3 :
Figure 3: Case 2-Rammasun: (a) track simulated by the original YSU; (b) track simulated by the K025; (c) track simulated by the K200; (d) comparisons of center sea-level pressure; and (e) comparisons of maximum surface wind speed.

3. 1 . 7 .
Case 7: Hagupit.During the first 2 days of the simulations, the CSLP time series are close to each other (Figure 8(d)) and have large AEs comparing with observations.

Figure 4 :
Figure 4: Case 3-Genevieve: (a) track simulated by the original YSU; (b) track simulated by the K025; (c) track simulated by the K200; (d) comparisons of center sea-level pressure; and (e) comparisons of maximum surface wind speed.

Figure 5 :
Figure 5: Case 4-Phanfone: (a) track simulated by the original YSU; (b) track simulated by the K025; (c) track simulated by the K200; (d) comparisons of center sea-level pressure; and (e) comparisons of maximum surface wind speed.

Figure 6 :
Figure 6: Case 5-Vongfong: (a) track simulated by the original YSU; (b) track simulated by the K025; (c) track simulated by the K200; (d) comparisons of center sea-level pressure; and (e) comparisons of maximum surface wind speed.

Figure 7 :
Figure 7: Case 6-Nuri: (a) track simulated by the original YSU; (b) track simulated by the K025; (c) track simulated by the K200; (d) comparisons of center sea-level pressure; and (e) comparisons of maximum surface wind speed.

Figure 8 :
Figure 8: Case 7-Hagupit: (a) track simulated by the original YSU; (b) track simulated by the K025; (c) track simulated by the K200; (d) comparisons of center sea-level pressure; and (e) comparisons of maximum surface wind speed.

Table 1 :
Simulation periods and durations for super typhoon cases in 2014.