Adjusting the Wind Stress Drag Coefficient in Storm ... - AMS Journals

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Adjusting the Wind Stress Drag Coefficient in Storm Surge Forecasting Using an Adjoint Technique SHIQIU PENG State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China, and Department of Marine, Earth and Atmospheric Sciences, North Carolina State University, Raleigh, North Carolina

YINENG LI State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China

LIAN XIE Department of Marine, Earth and Atmospheric Sciences, North Carolina State University, Raleigh, North Carolina (Manuscript received 23 February 2012, in final form 11 November 2012) ABSTRACT A three-dimensional ocean model and its adjoint model are used to adjust the drag coefficient in the calculation of wind stress for storm surge forecasting. A number of identical twin experiments (ITEs) with different error sources imposed are designed and performed. The results indicate that when the errors come from the wind speed, the drag coefficient is adjusted to an ‘‘optimal value’’ to compensate for the wind errors, resulting in significant improvements of the specific storm surge forecasting. In practice, the ‘‘true’’ drag coefficient is unknown and the wind field, which is usually calculated by an empirical parameter model or a numerical weather prediction model, may contain large errors. In addition, forecasting errors may also come from imperfect model physics and numerics, such as insufficient resolution and inaccurate physical parameterizations. The results demonstrate that storm surge forecasting errors can be reduced through data assimilation by adjusting the drag coefficient regardless of the error sources. Therefore, although data assimilation may not fix model imperfection, it is effective in improving storm surge forecasting by adjusting the wind stress drag coefficient using the adjoint technique.

1. Introduction The upper-ocean circulation and sea surface waves are mainly driven by sea surface winds. Particularly, the strong winds of tropical cyclones (TC) inevitably induce severe storm surge. Through the wind forcing (wind stress) on the sea surface due to the friction, the momentum is transferred from the atmosphere to the ocean. Therefore, a successful storm surge modeling greatly depends on an accurate estimate of the wind stress (Doyle 2002; Moon 2005; Xie et al. 2008).

Corresponding author address: Shiqiu Peng, State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, 164 West Xingang Road, Guangzhou 510301, China. E-mail: [email protected] DOI: 10.1175/JTECH-D-12-00034.1 Ó 2013 American Meteorological Society

The most common formula employed to calculate wind stress is the quadratic formulation with respect to the wind speed, that is, t 5 ra Cd jvjv,

(1)

v 5 va 2 vo , where ra is the air density, Cd is the drag coefficient, va is the wind velocity at 10-m height above the surface, and vo is the velocity of ocean surface currents (the effect of ocean surface currents is negligible when va vo , i.e., v ’ va ). It is obvious that the values of wind stress depend on both the wind speed and the drag coefficient Cd . Theoretically, Cd is a function of the wind speed, the sea state, and the atmospheric stability (Monin and Obukhov

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1953; Charnock et al. 1955). In practice, however, Cd has often been set either as a constant (Jones and Davies 1998; Konishi et al. 1985, 1986) or using an empirical formula that is a linear function of wind speed (Sheppard 1958; Smith 1980; Large and Pond 1981; Wu 1980, 1982), as shown: Cd 5

ra (a 1 bjvj) , rw

(2)

where ra and rw are the air density and water density, respectively, and a and b are empirical parameters. Since Sheppard (1958) first proposed the linear formula for Cd in the middle of the last century, many scientists have made large efforts in estimating the values of parameters a and b and have come to significantly different results (Deacon and Webb 1962; Miller 1964; Zubkovskii and Kravchenko 1967; Brocks and Krugermeyer 1970; Sheppard et al. 1972; Wieringa 1974; Kondo 1975; Smith and Banke 1975; Smith 1980; Wu 1980; Large and Pond 1981; Donelan 1982; Geernaert et al. 1987; Yelland and Taylor 1996). Among all the formulas, the most widely used nowadays are those of Wu (1980), Smith (1980), and Large and Pond (1981). After extensive research, Wu (1980) provided the values for a and b under the condition of wind speed less than 15 m s21: (a, b) 5 (0.8, 0.065), and later extended it to the situation of strong wind, such as in a hurricane (Wu 1982). Smith (1980) analyzed wind speed, temperature, and wave height observed at a fixed platform, and obtained the values of a and b at the wind speed range of 6–22 m s21: (a, b) 5 (0.61, 0.063). Based on the data over the deep-ocean area and the assumption of unlimited wind fetch, Large and Pond (1981) found (a, b) 5 (0.49, 0.065) in the wind speed range of 10–26 m s21 and Cd was held constant in the range of 4–10 m s21. Although some recent studies have shown the nonlinear dependence of Cd on wind speed at the sea surface

FIG. 1. Topography of South China Sea, the model domain, and the best track of Typhoon Ketsana (2009). The locations 1–6 are referred to in other figures.

when it exceeds 30 m s21 (Emanuel 2003; Powell et al. 2003; Donelan et al. 2004), the linear parameterization [Eq. (2)] is still widely used in storm surge forecasting. Therefore, we still employ the formula of Cd by Large and Pond (1981) in this study with a correction under the large wind speed (i.e., Cd is set to be unchanging with the wind velocity when the wind speed is larger than 30 m s21). We will demonstrate that even with this simple linear parameterization, obtaining an optimal value for Cd can lead to improvements in storm surge forecasting. One of the efficient ways to estimate the wind stress drag coefficient is through fitting the model output to the observations using adjoint technique (Derber 1987;

TABLE 1. The design of ITEs. Here v0 and x0 are the wind field calculated by the Holland model and the ICs with a resolution of 1/58 3 1/58 after a 24-h spinup; vHR and xHR are the same as v0 and x0 , respectively, but with a higher resolution of 1/108 3 1/108; and vWM is the 0 0 0 wind field from the output of WRF model. Set of Expt I

II

III

Expt name

Error source

Adjustment of Cd

First guess of v

First guess of IC

Resolution (8)

CTRL_V2 DA_V2 CTRL_V1 DA_V1 NAT_1 CTRL_WM DA_WM NAT_2 CTRL_RS DA_RS NAT_2

Wind Wind Wind Wind — Wind Wind — Resolution Resolution —

No Yes No Yes — No Yes — No Yes —

0:9*v0 0:9*v0 1:1*v0 1:1*v0 v0 vWM 0 vWM 0 vHR 0 v0 v0 vHR 0

x0 x0 x0 x0 x0 xHR 0 xHR 0 xHR 0 x0 x0 xHR 0

1/5 1/5 1/5 1/5 1/5 1/10 1/10 1/10 1/5 1/5 1/10

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FIG. 2. Comparison between the Joint Typhoon Warning Center (JTWC) data and the WRF model output: (a) the track and (b) the minimum sea level pressure (SLP, unit: hPa).

Le Dimet and Talagrand 1986). Yu and O’Brien (1991) estimated the wind stress drag coefficient and the vertical profile of eddy viscosity coefficient by assimilating the observational data into a simple 1D Ekman ocean model using adjoint technique. Zhang et al. (2002, 2003) used a 2D Princeton Ocean Model (POM) and its adjoint model to estimate the wind stress drag coefficient by assimilating pseudo-observations of subtidal water level and coastal tidal elevation. Chen et al. (2008) performed an idealized numerical experiment on an estuary to derive the wind stress drag coefficient using a 2D adjoint model based on an unstructured grid with finite volume. The results of all these studies indicate that the adjoint data assimilation is an effective approach for estimating the wind stress drag coefficient and other parameters. However, all these studies used 1D or 2D models with simplified physics. Some previous studies

have shown that 3D storm surge models can improve the storm surge forecasting considerably, compared to 1D or 2D models (Xie et al. 2004; Peng et al. 2006), since the 3D models can take into account the nonlinear processes such as the bottom friction and tide–current–wave interaction (although they have relatively smaller impacts on the storm surge compared to the wind forcing). Although 3D models require more computational resources, it will not be a big problem with the rapid development of the computer technology. Therefore, it is worth to explore the effects of optimizing the wind stress drag coefficient in storm surge simulation in the framework of 3D ocean models. Although a nongradient simple method may also work for optimizing only a few parameters, the adjoint approach is more effective and necessary when considering the spatial variation of Cd or optimizing both the initial conditions and Cd simultaneously

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FIG. 3. Schematic diagram of experiment setup, where Cd0 denotes the initial value of the * is the optimal value from data assimwind stress drag coefficient for the control run, and Cd0 ilation. The long dashed line represents the model integration with Cd0 , the short dashed line is * , and the empty dot is the observations. the one with Cd0

(that means the dimension of the control variables will be a large number). When we employ the adjoint method for optimizing parameters or initial conditions (ICs) in the framework of 3D ocean models, the corresponding 3D adjoint model should be used for the optimization process. The reasons are as follows. First, the adjoint method for optimizing parameters or ICs is model dependent; that is, if you use a 3D model to make a forecast, then you should use its 3D adjoint model for optimizing the model parameters or ICs. Second, the wind stress drag coefficient (Cd) we try to optimize is not only case dependent and temporally/spatially various but also model dependent. In the framework of 2D models, although the 2D adjoint model of a 2D forward model could be sufficient for obtaining an ‘‘optimal’’ Cd for the 2D model, such an optimal Cd may be not suitable or optimal to a 3D forward model, and vice versa. Actually, as a subsequent study of our previous study, which demonstrates the feasibility of optimizing the initial conditions in improving storm surge forecasts using the 3D adjoint model of Princeton Ocean Model developed by Peng and Xie (2006) and Peng et al. (2007), this study aims to demonstrate the feasibility of adjusting Cd in improving storm surge forecasts using the same 3D adjoint model, a preparation for our future work that will combine the two, that is, optimize both the initial conditions and Cd simultaneously. Besides the uncertainties in the wind stress drag coefficient, there is still a number of uncertainties that may cause considerable systematic bias in storm surge forecasts, such as the wind speed forecasting, the resolution, and the simplification or parameterization of the physical/ dynamical processes in a storm surge model, etc. In state of the art, these uncertainties are inevitable or cannot be removed completely because of imperfect forecasting of the storm track or its intensity, limited computer resource, and poor understanding of physical/dynamical processes of the ocean, etc. Ensemble forecasting is a practical way to offset the systematic bias in storm surge forecasts caused by these uncertainties (Stamey et al.

2007; Flowerdew et al. 2007; Kazuo et al. 2010). In addition to the ensemble forecasting, adjusting an appropriate parameter in the storm surge model could be another practical solution to the problem (Peng et al. 2007). In the study of Peng et al. (2007), the maximum wind radius in the calculation of wind stress based on the empirical Holland model was chosen as the parameter to be adjusted. However, when wind fields are obtained from a numerical weather model output rather than using an empirical formula that is the function of the maximum wind radius, it is invalid to adjust the maximum wind radius. As mentioned above, Cd plays an important role in the calculation of wind stress, but its magnitude still has considerable uncertainties, especially under the circumstance of high wind speed. Therefore, in this study, we explore whether adjusting the wind stress drag coefficient through four-dimensional variational data assimilation (4DVAR) is an effective and feasible way to reduce

FIG. 4. The best track of Typhoon Hagupit (2008). The stations S1–S4 are referred to in other figures.

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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY TABLE 2. The design of real case experiments.

Expt name

Starting time

Adjustment of Cd

Resolution

CTRL_RL1 DA_RL1 CTRL_RL2 DA_RL2

0000 UTC 23 Sep 2008 0000 UTC 23 Sep 2008 0600 UTC 23 Sep 2008 0600 UTC 23 Sep 2008

No Yes No Yes

1/108 1/108 1/108 1/108

the systematic bias in storm surge forecasts caused by various uncertainties. In section 2, the Princeton Ocean Model and its adjoint model are briefly introduced. Section 3 describes the experimental setup design, including the selection of the storm surge case and model domain. The results are presented in section 4. Conclusions and discussion are given in the last section. The verification of the POM4DVAR system using Cd as the control variable is given in the appendix.

2. The Princeton Ocean Model and its adjoint model The models used in this study are the POM, 2002 version (Blumberg and Mellor 1987; Mellor 2004 and its adjoint model (Peng and Xie 2006; Peng et al. 2007). POM is a three-dimensional ocean model with primitive equations, embedded in a second-moment turbulence closure model (the level 2.5 Mellor–Yamada scheme; Mellor and Yamada 1982). The main features of POM include vertically terrain-following sigma coordinates, horizontally curvilinear orthogonal coordinates with Arakawa C-grid differencing scheme, a free surface, and time-splitting step for external (fast) and internal (slow) modes. The external mode is two dimensional and uses a short time step based on the Courant–Friedrichs– Lewy (CFL) condition and the external wave speed, whereas the internal mode is three dimensional and uses a long time step based on the CFL condition and the internal wave speed. In addition, while the horizontal time differencing is explicit, the vertical differencing is

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implicit, which eliminates time constraints for the vertical coordinate and allows for the use of fine vertical resolution in the surface and bottom boundary layers. The model state variables include 2D current velocity UA and VA, 3D current velocity U and V, temperature T, salinity S, surface elevation h, and turbulent kinetic energy, doubled q, and turbulent length scale l. Readers are referred to Blumberg and Mellor (1987) and Mellor (2004) for a full description of POM. In this study, either the empirical Holland model (Holland 1980) or a numerical weather prediction model is employed to calculate the wind speed for the surface wind stress. The tangential wind speed from the empirical Holland model, which is based on the balance between the pressure gradient and centrifugal forces, can be expressed as Vc 5 [AB( pn 2 pc ) exp(2A/rB )/ra rB ]1/2 A 5 RB MW ,

(3) (4)

where A and B are the scaling parameters; pn and pc are the ambient and central pressure of the storm, respectively; ra is the air density; r is the distance from the storm center; and RMW is the radius of the maximum wind (RMW, the distance between the center of a cyclone and its band of strongest winds). Empirically, B lies between 1 and 2.5. From (3) and (4), we can see that the accuracy of wind speed calculated by the Holland model is greatly dependent on the accuracy of predicted storm track and intensity. The tangent linear and adjoint models of the 3D POM with turbulence closure scheme were developed by Peng and Xie (2006). Because of the high nonlinearity and discontinuity of vertical turbulence, a smoothing approximation was made in the linearization of the Mellor– Yamada turbulence scheme. As addressed in the first section, the purpose of this study is to reduce the bias in storm surge forecasts caused by various uncertainties through adjusting Cd . Therefore, in this study, parameters

FIG. 5. Variation of (a) the cost function and (b) its gradient with respect to the number of iterations for experiments DA_V2 and DA_V1.

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FIG. 6. The errors of 24-h forecasted water level from (a) NAT_1, (b) CTRL_V2, (c) DA_V2, and (d) CTRL_V1, (unit: m).

a and b of Cd are chosen as the control variables in the adjoint model. When changing the control variables from initial conditions to parameters a and b, minor modification in the adjoint codes and corresponding correctness check are needed. The correctness test of the POM-4DVAR coding with parameters a and b being the control variables is given in the appendix. It

shows that, in the idealized circumstances, the ‘‘true’’ values of the parameters a and b can be recovered by data assimilation based on the modified POM-4DVAR system. The cost function with parameters a and b being the control variables is defined as a misfit between the model and the observations, that is,

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FIG. 7. Time evolutions of water level for different experiments at the sites (a) 2, (b) 4, and (c) 6 starting at 1800 UTC 26 Sep 2009 (unit: m).

J(a, b) 5

ðt

1

H[x(t), t] dt ,

(5)

t0

where x represents the vector of model state variables and H[x(t), t] is a scalar function measuring the distance between x(t) and observations at time t. To find the optimal values for a and b, the minimization of cost function is performed. It is achieved by obtaining its gradient with respect to the control variables a and b by integrating

the adjoint model of POM backward in time. The limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton minimization algorithm (Liu and Nocedal 1989) is employed to obtain the optimal control variables.

3. Experiment setup The model domain is set to cover an area of 08;308N, 998;1258E (Fig. 1) with a horizontal resolution of

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FIG. 8. RMSEs of (a) water level (unit: m) and (b) currents (m s21) averaged over the entire domain for different experiments during a 24-h simulation period starting at 1800 UTC 26 Sep 2009. 1/ 58

3 1/ 58 (1/ 108 3 1/ 108 for high-resolution experiments) and four vertical levels. In this study, the external (2D) velocity uses closed lateral boundary condition, while the internal velocity (3D) employs the radiation scheme for its lateral boundaries. For storm surge, only the wind stress is included as the upper boundary condition (the heat flux is omitted since its impact on storm surge is negligible, and the temperature and salinity are fixed since they also have little impact on storm surge). And the 1-minute gridded elevations/bathymetry for the world (ETOPO1) (Marks and Smith 2006) is used for the bathymetry in this study.

a. Identical twin experiments To address whether the model errors from surface wind stress and other sources except Cd can be corrected by adjusting Cd using the adjoint data assimilation approach, we perform identical twin experiments (ITEs) first. The storm surge case of Typhoon Ketsana (2009) was chosen for ITEs. As shown in Fig. 1, Ketsana (2009) formed as a tropical storm over the western Pacific at 0000 UTC 26 September 2009, and intensified to become a strong tropical storm at 2100 UTC 26 September and a typhoon at 0200 UTC 28 September with a low pressure of 960 hPa and a maximum wind speed of 46 m s21. After crossing the Philippine Islands and entering the South China Sea, Ketsana (2009) swept all the way westward until landing at the east coast of Vietnam at 0700 UTC 29 September 2009 (Fig. 1). It caused at least 272 deaths and large damage in the Philippines, Vietnam, and China. The tidal station located at Yongxin Island

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FIG. 9. SDEs of (a) water level (unit: m) and (b) currents (m s21) averaged over the entire domain for different experiments during a 24-h simulation period starting at 1800 UTC 26 Sep 2009.

(indicated in Fig. 1) reported a maximum water level increment of 2 m. For ITEs, we assume that the surface wind speed is much larger than the ocean surface current speed, that is, v ’ va in Eq. (1). We design three sets of ITEs as listed in Table 1. Each set of ITEs includes three types of experiments, that is, a control run (starting with ‘‘CTRL’’). in which no adjustment is made in Cd ; a data assimilation experiment (starting with ‘‘DA’’), in which Cd is adjusted through 4DVAR; and a ‘‘nature’’ run (starting with ‘‘NAT’’), whose output is taken as the true value for assimilation and comparison. In the first set of ITEs, the errors of storm surge forecasts are assumed to come from wind speed by using a smaller or larger value than v0 calculated by the Holland model, and water-level ‘‘observations’’ (obtained from the output of the NAT run) over all grid points are assimilated into the model to adjust Cd . In the second set, we assume the surface wind to be the error source, too. However, more accurate surface wind fields forecasted by a numerical weather model are used instead of the ones calculated by the Holland model, which may have considerable uncertainties. Besides taking into account that there are usually only a few stations along the coast to collect the water-level data in practice, we consider a more realistic circumstance by assimilating water-level observations only from a limited number of coastal sites (1–5). In the third set, the errors of storm surge forecasts are assumed to come from the lower resolution, which may impose errors in the initial conditions and the integration process, while the truth is generated by a higher-resolution

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FIG. 10. Vectors and absolute values of wind stress bias of (a) CTRL_V2, (b) CTRL_V1, (c) DA_V2, and (d) DA_V1 with respect to the truth (NAT_1) at 1800 UTC 27 Sep 2009 (unit: m2 s22).

model. Similar to the second set of ITEs, only the five coastal sites (1–5) are used for the assimilation. For all experiments in all sets of ITE, Cd0 is computed with (a0 , b0 )5(0.49, 0.065) according to Large and Pond (1981) and is taken as the initial value of the wind stress drag coefficient; Cd0 is adjusted in the 4DVAR data assimilation experiment of each set, and no adjustment is made in Cd for the control run of each set. In the second set of ITE, the Weather Research and Forecasting (WRF) model (Skamarock et al. 2007) is employed to provide the surface winds for the control run and the data assimilation experiment. A two-nested-domain configuration

with horizontal resolutions of 72 and 24 km and 28 vertical levels is set for the WRF model, and the National Centers for Environmental Prediction (NCEP, www.ncep.noaa.gov) final analysis (FNL) data are used to provide initial conditions and boundary conditions for the outer domain. Although the wind fields forecasted from WRF may be more accurate than those calculated from the Holland model, they might also have a considerable bias because of the uncertainties in the prediction of TC track and intensity (Fig. 2). For the nature run of each set, the ocean surface wind speed V is calculated using the Holland model (3), in

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FIG. 12. As in Fig. 11, but for 24-h wind stress (unit: N m22).

FIG. 11. Time evolutions of 24-h water level from NAT_2, CTRL_WM, and DA_WM at the sites (a) 2, (b) 4, and (c) 6 starting at 1800 UTC 26 Sep 2009 (unit: m).

which RMW is set to 50 km, pn is set to 1010 hPa, and the best track and central pressure of Ketsana (2009) obtained from satellite analysis data are provided by the National Meteorological Centre of the China Meteorological Administration (NMC CMA, www.nmc.gov. cn/). For the first set of ITEs, (a0 , b0 ) and x0 and v0 are used to perform the nature run (denoted as NAT_1), where x0 and v0 represent the vectors of ICs (obtained from a 24-h spinup with a resolution of 1/ 58 3 1/ 58) and

a wind field (calculated by the Holland model) at t0 ; for the second and the third sets, (a0 , b0 ) and higherand vHR are used to perform the nature resolution xHR 0 0 and vHR are the run (denoted as NAT_2), where xHR 0 0 same as x0 and v0 except with a higher resolution of 1/ 108 3 1/ 108. The results from NAT_1 or NAT_2 are taken as the truth to which those from the other experiments are compared. For the first set, the 3-h (1800–2100 UTC 26 September) water-level outputs from NAT_1 over all the ocean grid points are taken as pseudo-observations and assimilated into the model to adjust Cd . For the second and the third sets, the 3-h (1800–2100 UTC 26 September) water-level outputs from NAT_2 at

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FIG. 13. The errors of 24-h forecasted water level from (a) CTRL_WM and (b) DA_WM compared to the truth (NAT_2) (unit: m).

sites 1–5 are assimilated. As schematically displayed in Fig. 3, before performing all experiments, a 24-h spinup of the POM with a ‘‘cold start’’ at 1800 UTC 25 September is carried out. A 3-h (1800–2100 UTC 26 September) window in which the pseudo-observations are assimilated every 10 min is adopted to find the optimal Cd in each data assimilation experiment of all the sets. A 24-h forward model run starting at 1800 UTC 26 September with either the original Cd or the adjusted Cd is performed for the control run or the data assimilation experiments.

b. Real case experiment To test whether the proposed method is efficient in practice, we apply this method to the surge forecasting for the storm of Hagupit (2008) by using real data. Typhoon Hagupit (2008) formed on 14 September 2008 in the western Pacific and moved westward toward the Philippines (Fig. 4). It entered the South China Sea on 21 September 2008 and intensified when moving northwest to China, and made landfall near Maoming in Guangdong Province of China at 0645 local time (LT) 24 September. It destroyed 14 333 houses and cost $824 million (U.S. dollars) in damages. The experiment setup is given in Table 2. The observations of water level were collected from four stations along the coastline in Guangdong Province of China (Fig. 4). The surface winds are from the WRF model output. The model settings are the same as those for the second set of ITEs described in section 3a. Similar to the notation of ITEs, CTRL_RL1/CTRL_RL2 and DA_RL1/DA_RL2 denote the control run and data assimilation experiments,

respectively, where ‘‘1’’ and ‘‘2’’ represent different a starting time of the model as shown in Table 2. For DA_RL1/DA_RL2, the observed water-level data of two stations (S1 and S3) are assimilated to adjust Cd, while those of other two stations are used for verification. A 24h spinup is made before the control run or data assimilation experiments. A 3-h cycle within which the waterlevel data are assimilated every one hour is employed to get the optimal Cd in DA_RL1/DA_RL2. A 24-h forecast starting at 0000 UTC 23 September (for DA_RL1) or 0600 UTC 23 September (for DA_RL2) with the corresponding optimal Cd is made to compare with the control run without adjusting Cd.

4. Results a. Results from the first set of ITEs Figure 5 shows the variations of cost function and its gradient values with the iteration number of minimization for experiments DA_V2 and DA_V1. The cost

TABLE 3. The optimal values of a and b obtained from different 4DVAR experiments. Expt name

Error source

Starting time

DA_V2 DA_V1 DA_WM DA_RS DA_RL1 DA_RL2

Wind Wind Wind Resolution All All

1800 UTC 26 Sep 2009 1800 UTC 26 Sep 2009 1800 UTC 26 Sep 2009 1800 UTC 26 Sep 2009 0000 UTC 23 Sep 2008 0600 UTC 23 Sep 2008

Optimal Optimal a b 0.3875 0.5019 0.171 0.7101 0.5938 0.516

0.1028 0.043 35 0.0098 0.1038 0.081 55 0.0771

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FIG. 14. As in Fig. 13, but from (a) CTRL_RS and (b) DA_RS.

function and its gradient value decrease by about two to four orders within the first three iterations, In practice, if G drops by more than one order, then the solution is already a good result, which is sometimes difficult to attain (De Pondeca and Zou 2001). The obtained optimal parameters (a, b) of Cd at the end of the minimization procedure are (0.3875, 0.1028) for DA_V2 and (0.5019, 0.0434) for DA_V1 (Table 2). Figure 6 displays the spatial distribution of the errors of the 24-h forecasted water level from CTRL_V2, CTRL_V1, DA_V2, and DA_V1 compared to NAT_1 (the ‘‘true’’). It is obvious that CTRL_V2 (or CTRL_V1) underestimates (or overestimates) the water level. After adjusting Cd by the assimilation of water-level observations, either the underestimated or overestimated water level along the coast is corrected significantly. Figure 7 shows the time evolution of the water level at sites 2, 4, and 6 along the coast (indicated in Fig. 1). The errors in the forecasts of water level increase with time for both in CTRL_V2 and CTRL_V1. However, the assimilation of water level observations to adjust the parameters of Cd is able to reduce these errors effectively up to at least 24 h. The improvements in storm surge forecasting by data assimilation can be further seen clearly in the root-mean-square errors (RMSEs) and standard deviation errors [SDEs, following the definition of Oke et al. (2002)] averaged over the whole model domain (Figs. 8 and 9). Here, RMSE and SDE are defined as

2

n

6å 4 XRMSE 5 i51 n 2

n

3

1 /2 Xe2,i 7 5

,

(6) 31/2 2

n

31/2

6 å (Xm,i 2 X m ) 7 6 å (Xt,i 2 X t ) 7 4 5 4 5 2 i51 , XSDE 5 i51 n n (7) 2

2

respectively, where Xe,i (Xe,i 5 Xm,i 2 Xt,i ) is the ith error of the modeled variable Xm,i compared to the true value Xt,i (pseudo-observations or observations); X m and X t are the mean value of Xm and Xt , respectively; and n is the sample volume of variable X. It is obvious that the forecasts of both water level and surface currents are improved by the data assimilation. The improvements of the storm surge forecasts can be attributed to the correction of wind stress bias caused by the erroneous wind speed (CTRL_V2 or CTRL_V1) through the optimal Cd obtained by 4DVAR, as shown in Fig. 10. For most operational storm surge forecasting systems, systematic bias may exist in the calculation of the surface wind fields using the Holland model or other empirical models. The results of this set of ITEs indicate that the systematic bias in the surface wind fields can be offset by using an optimal wind stress drag coefficient obtained through 4DVAR data assimilation under idealized situations.

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FIG. 16. RMSEs of (a) water level (unit: m) and (b) currents (m s21) averaged over the entire domain for different experiments during a 24-h simulation period starting at 1800 UTC 26 Sep 2009.

FIG. 15. Time evolutions of water level from NAT_2, CTRL_RS, and DA_RS at sites (a) 2, (b) 4, and (c) 6 starting at 1800 UTC 26 Sep 2009 (unit: m).

CTRL_WM and DA_WM with respect to the ‘‘true’’ (NAT_2). Large bias with the maximum of about 0.35 m is seen along the coast of south China in CTRL_WM, which is reduced greatly in DA_WM. The improvements are seen not only in the area near sites 1–5, where the ‘‘observed’’ water level is assimilated to adjust the wind drag coefficient but also in other areas where no observed water level is assimilated, such as the regions of the Beibu Gulf and the coast of eastern Vietnam. In the practice of the real-time storm surge forecasts, the numerical weather

b. Results from the second set of ITEs The optimal parameters (a, b) of Cd after 4DVAR are (0.171, 0.0098). Because the intensity of the TC is overestimated by the WRF model (Fig. 2), which results in an overestimated wind speed, Cd is adjusted to a smaller value (0.171, 0.0098) to offset the overestimated wind speed. The time evolutions of water level and wind stress from NAT_2, CTRL_WM, and DA_WM at sites 2, 4, and 6 (indicated in Fig. 1) during a 24-h simulation period are shown in Figs. 11 and 12, respectively. Please note that site 6 is an independent validation point that is not included in the data assimilation. Compared to the ‘‘true’’ (NAT_2), the water level at sites 2, 4, and 6 becomes extremely overestimated at the end of the 24-h simulation in CTRL_WM but is greatly corrected in DA_WM. The improvements of water-level prediction are attributed to the improvements of the wind stress by using the optimal Cd in DA_WM (Fig. 12). Figure 13 shows the bias of 24-h predicted water level from

FIG. 17. As in Fig. 16, but for SDEs.

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FIG. 18. Time evolutions of water level from observations, CTRL_RL1, and DA_RL1 at the stations (a) S1, (b) S2, (c) S3, and (d) S4 starting at 0000 UTC 23 Sep 2008 (unit: m).

model is usually used to make the forecast of wind fields. However, because of large uncertainties in the prediction of typhoon intensity and track, the forecasted wind fields may be underestimated (or overestimated). The results of this set of ITEs indicate that adjusting the wind stress drag coefficient through 4DVAR is a practical solution to correct the errors of wind stress caused by the inaccurate prediction of TC intensity or track and thus can improve the storm surge forecasts effectively.

c. Results from the third set of ITEs As presented in Table 3, the optimal parameters (a, b) of the wind drag coefficient after 4DVAR are (0.7101, 0.1038). Figure 14 shows the 24-h forecasted water-level fields for NAT_2 (the ‘‘true’’), CTRL_RS, and DA_RS. The errors of the water level caused by lower resolution are corrected effectively by adjusting Cd through 4DVAR. Figure 15 shows the time evolution of the water level from NAT_2, CTRL_RS, and DA_RS at sites 2, 4, and 6 (indicated in Fig. 1) during a 24-h simulation period. For all of the three sites, the bias of the water level increases with time in CTRL_RS and is corrected in DA_RS. The RMSE and SDE of the water level and surface

current averaged over the entire domain further display the improvements of forecasts of water level or surface currents by data assimilation (Figs. 16 and 17). These results indicate that the errors of storm surge forecasts caused by lower resolution of the model can be reduced by adjusting the parameters of wind stress drag coefficient.

d. Results from the real case experiments As presented in Table 2, the optimal parameters (a, b) of the wind stress drag coefficients for DA_RL1 (starting at 0000 UTC 23 September) and DA_RL2 (starting at 0600 UTC 23 September) are (0.5935, 0.081 55) and (0.516, 0.0771), respectively. Figures 18 and 19 show the time evolution of 24-h water level at sites 1–4 for CTRL_ RL1/CTRL_RL2 and DA_RL1/DA_RL2. It is found that the simulated water-level values from DA_RL1/ DA_RL2 are closer to the observations than those from CTRL_RL1/CTRL_RL2 most of the time during the 24-h period. The RMSE and the maximum values of the water level for each experiment are presented in Table 4. The errors of the 24-h forecasted water level are reduced effectively in DA_RL1/DA_RL2. The maximum

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FIG. 19. As in Fig. 18, but from observations, CTRL_RL2 and DA_RL2 at the stations (a) S1, (b) S2, (c) S3 and (d) S4 starting at 0600 UTC 23 Sep 2008 (unit: m).

values of the water level from DA_RL1/DA_RL2 are closer to the observed ones than those from CTRL_ RL1/CTRL_RL2, too. These results indicate that adjusting Cd through 4DVAR is a feasible and practical solution to improve the storm surge forecasts in a real situation. It should be noted, however, that in a real situation, the improvements of storm surge forecasts by adjusting Cd thought 4DVAR are much less than those in the twin experiments. The primary reasons could be 1) the real observations are too sparse; 2) the real observations may contain large errors; and 3) the minimization process does not converge because of a number of factors, such as model instability, too many local minimum points, and nonpositive definite features of the background error covariance.

5. Conclusions and discussion In this study, we employ the adjoint technique to adjust the parameters of wind stress drag coefficient Cd in the three-dimensional POM for improving storm surge forecasts. The identical twin experiments are

performed by assigning different error sources. The twin experimental results indicate that it is an efficient and practical way to reduce errors in storm surge forecasting by optimizing the value of Cd using adjoint technique, regardless of the error sources. The results of the real case experiment further confirm that the proposed method can improve the storm surge forecasts to a certain extent in real situations.

TABLE 4. The RMSE and maximum values of the 24-h forecasted water level at stations S1–S4 compared to the observations (units: m). Set of Expt RMSE

Maximum water level

Expt name

S1

S2

S3

S4

CTRL_RL1 DA_RL1 CTRL_RL2 DA_RL2 CTRL_RL1 DA_RL1 OBS CTRL_RL2 DA_RL2 OBS

1.195 0.434 6.874 4.469 1.219 1.606 1.837 1.068 1.312 1.959

0.548 0.358 0.665 0.203 1.256 1.465 1.523 1.340 1.553 1.523

1.224 0.754 0.751 0.535 0.601 0.702 0.915 0.621 0.692 0.915

1.060 0.879 0.900 0.861 0.560 0.648 0.845 0.642 0.720 0.845

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The errors of storm surge forecasts may come from several sources: 1) the true value of Cd is usually unknown no matter what method is used to parameterize it; 2) there may be a systematic bias in wind speed forecasting, whether using a parameter model or a numerical weather prediction model; and 3) the storm surge model is not perfect and may contain representative bias due to many known or unknown sources, such as insufficient resolution. While it is straightforward but difficult to improve the resolution of wind field forecasting models or storm surge models, our experimental results from either idealized cases or real cases imply that adjusting the wind stress drag coefficient could be an alternative and practical way to reduce storm surge forecasting errors caused by inaccurate wind forecasting or the imperfect storm surge model itself, regardless what is the physical meaning of Cd . One should be aware that the ‘‘optimal’’ value of Cd obtained here by 4DVAR is not a universal one; in contrary, it is only valid and applicable to the specific storm surge case. Furthermore, for each specific storm surge case, it could also vary temporally and spatially rather than being constant. This can be achieved by implementing 4DVAR repeatedly at short time intervals (for instance, every 3 or 6 h) and by using spatially varied wind stress drag coefficients as a control variable in the 4DVAR procedure. On the other hand, only Cd is adjusted in this study, and it is expected that adjusting both Cd and initial conditions simultaneously may provide more improvements in the storm surge forecasts, which will be our future work. Finally, it should be pointed out that the adjustment of Cd under the linear framework may have its limitations. The purpose of this study is to demonstrate the feasibility of using an adjoint approach to adjust Cd in order to reduce storm surge forecast errors. For this purpose, the linear framework for Cd parameterization used in this study is appropriate. It should also be pointed out that although adjusting Cd through data assimilation is a practical way to improve storm surge forecasts, this approach should not and cannot be used to identify storm surge model deficiencies or to fix such deficiencies. Acknowledgments. This work was jointly supported by the Innovation Key Program of the Chinese Academy of Sciences (Grant KZCX2-EW-208), the National High Technology Research and Development Program of China (863 Program) (Grant 2010AA012304), the National Natural Science Foundation of China (Grant 41076009), the Ministry of Science and Technology of the People’s Republic of China (MOST) (Grant 2011CB403504), and the Hundred Talents Program of the Chinese Academy of Sciences.

TABLE A1. The results of the TLM check. Iterations

a

1 2 3 4 5 6 7 8 9 10 11 12 13

1.00 3 10 1.00 3 101 1.00 3 1021 1.00 3 1023 1.00 3 1025 1.00 3 1027 1.00 3 1029 1.00 3 10211 1.00 3 10213 1.00 3 10215 1.00 3 10217 1.00 3 10219 1.00 3 10221

Coefficient C 3

0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.999 1.002

748 875 988 999 000 000 000 000 000 002 998 899 872

196 468 377 67 799 912 923 912 819 003 473 294 431

APPENDIX Correctness Test of the POM-4DVAR for Adjusting Cd To test the correctness of the tangent linear model (TLM), a coefficient C is defined as C5

kM(x 1 adx) 2 M(x)k , akMT (dx)k

(A1)

where M is the nonlinear model, MT is the tangent linear model, x is the state variable, dx is the perturbation of x, and a is the varying coefficient. According to the Taylor expression, we should have lim C 5 lim

a/0

a/0

kM(x 1 adx) 2 M(x)k ’ 1:0: akMT (dx)k

(A2)

Table A1 shows the result of the TLM check based on Eq. (A2). When using a 64-bit compiler, six-digit precision (in this case, six digits of ‘‘9’’ or ‘‘0’’) is considered enough to confirm the correctness of the TLM coding. From Table A1, we can conclude that the TLM coding is correct.

TABLE A2. The design of numerical experiments. The asterisk means (a0, b0) 5 (0.49, 0.065). Expt name

First guess of (a, b)*

First guess of v

First guess of ICs

DA_Cd2 DA_Cd1 NAT_1

0.5*(a0, b0) 2*(a0, b0) (a0, b0)

v0 v0 v0

x0 x0 x0

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FIG. A1. Variation of (a) values and (b) gradient of the cost function with respect to the number of iterations for experiments DA_Cd2 and DA_Cd1.

To test the correctness of the adjoint model, the following equation should satisfy: MT (x)MT (x) 5 xMA [MT (x)],

(A3)

where MA is the adjoint model and other variables are the same as in Eq. (A2). In our testing case, the value of left-hand side of Eq. (A3) is 42 045.333 953 763 2, while the right-hand side is 42 045.333 953 762 8. In a 64-bit compiler, we can conclude that the adjoint model coding is correct. To verify whether the true wind drag coefficient can be retrieved in an idealized situation using the adjoint data assimilation approach, ITEs are designed, as shown in Table A2. The ocean surface wind speed v0 is calculated using the Holland model (3), in which RMW is set to 50 km, pn is set to 1010 hPa, and the track positions and central pressure of Ketsana (2009) are provided by the NMC CMA; Cd0 5(a0 , b0 )5(0.49, 0.065) by Large and Pond (1981) is taken as the ‘‘true’’ and used to perform the nature run (denoted as NAT_1). The model forecasting errors are assumed to come from the wind stress drag coefficient by setting the first guess of (a, b) to a

half or a twice the ‘‘true’’ (a0 , b0 ). All experiments use the same initial conditions x0 and surface wind field v0 . The 3-h (1800–2100 UTC 26 September) water level outputs from NAT_1 over all the ocean grid points are taken as pseudo-observations and assimilated into the model to adjust Cd . Fig. A1 displays the variation of the cost function and the gradient values with the iteration numbers of the minimization procedure for experiments DA_Cd2 and DA_Cd1. It is found that for both experiments, both the cost function and its gradient decrease dramatically by more than three orders within the first seven iterations, indicating that the minimization procedure converges very fast. Fig. A2 shows the optimal values of parameters a and b at each iteration for experiments DA_Cd2 and DA_Cd1. It is found that both a and b reach their true values (i.e., a 5 0.49 and b 5 0.065) after the fourth iteration. Therefore, the adjoint model of the POM with parameters of the wind drag coefficient being control variables was correctly coded, and the POM-4DVAR approach is able to obtain the true values of the wind stress drag coefficient at the idealized situation (i.e., everything is perfect except the erroneous wind stress drag coefficient).

FIG. A2. Retrieved values of (a) a and (b) b at each iteration of the 4DVAR minimization procedure for experiments DA_Cd2 and DA_Cd1. The solid line denotes the true values of a and b.

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PENG ET AL. REFERENCES

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