Parameterization of typhoon-induced ocean cooling using temperature equation and machine learning algorithms: an example of typhoon Soulik (2013)

Abstract

This study proposed three algorithms that can potentially be used to provide sea surface temperature (SST) conditions for typhoon prediction models. Different from traditional data assimilation approaches, which provide prescribed initial/boundary conditions, our proposed algorithms aim to resolve a flow-dependent SST feedback between growing typhoons and oceans in the future time. Two of these algorithms are based on linear temperature equations (TE-based), and the other is based on an innovative technique involving machine learning (ML-based). The algorithms are then implemented into a Weather Research and Forecasting model for the simulation of typhoon to assess their effectiveness, and the results show significant improvement in simulated storm intensities by including ocean cooling feedback. The TE-based algorithm I considers wind-induced ocean vertical mixing and upwelling processes only, and thus obtained a synoptic and relatively smooth sea surface temperature cooling. The TE-based algorithm II incorporates not only typhoon winds but also ocean information, and thus resolves more cooling features. The ML-based algorithm is based on a neural network, consisting of multiple layers of input variables and neurons, and produces the best estimate of the cooling structure, in terms of its amplitude and position. Sensitivity analysis indicated that the typhoon-induced ocean cooling is a nonlinear process involving interactions of multiple atmospheric and oceanic variables. Therefore, with an appropriate selection of input variables and neuron sizes, the ML-based algorithm appears to be more efficient in prognosing the typhoon-induced ocean cooling and in predicting typhoon intensity than those algorithms based on linear regression methods.

Appendix. Algorithms and data

1.1 Algorithms

1.1.1 Temperature equation (TE-based) algorithm I

This algorithm is derived from the linear primitive temperature budget equation, which can be written as follows:

$$ \underset{\mathrm{DT}}{\underbrace{\frac{\partial T}{\partial t}}}=\underset{MIX}{\underbrace{\frac{\partial }{\partial z}\left( K\frac{\partial T}{\partial z}\right)}\ }\underset{ A DV}{\underbrace{-\left( u\frac{\partial T}{\partial x}+ v\frac{\partial T}{\partial y}+ w\frac{\partial T}{\partial z}\right)}\ }\underset{HDIF}{\underbrace{+\frac{\partial }{\partial x}\left({A}_h\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left({A}_h\frac{\partial T}{\partial y}\right)}}\underset{HFX}{\ \underbrace{+\frac{Q}{h{\rho}_0{C}_p}.}} $$

(5)

The left-hand side term is the rate of temperature change. The right-hand side terms represent vertical mixing (MIX), advection (ADV), horizontal diffusion (HDIF), and heat flux (HFX), respectively. The horizontal diffusion term can be neglected because it has been shown to be very small compared to other terms under TC conditions (Price 1981; Uhlhorn and Shay 2012, 2013; Wei et al. 2014a). Although we recognized the importance of surface heat fluxes, the strength of their influence remains uncertain (Price 1981; Vincent et al. 2012). To avoid uncertain errors, we do not consider heat flux forcing in any of the algorithms. Thus, Eq. (1) can be simplified as follows:

$$ \underset{\mathrm{DT}}{\underbrace{\frac{\partial T}{\partial t}}}=\underset{MIX}{\ \underbrace{K\frac{\partial^2 T}{\partial {Z}^2}} -}\underset{ADV}{\underbrace{\left( u\frac{\partial T}{\partial x}+ v\frac{\partial T}{\partial y}+ w\frac{\partial T}{\partial z}\right)}} $$

(6)

where K is the vertical diffusivity (assumed constant) and $K\frac{\partial }{\partial Z}$ represents the strength of the vertical mixing, which is proportional to typhoon wind stress. Then, a regression algorithm of SSTC can be derived as follows:

$$ \mathrm{SSTC}=\underset{MIX}{\underbrace{a_1\times wnd}}+\underset{ADV}{\underbrace{b_1\times \mathit{\operatorname{curl}}\left(\tau \right)}}+{c}_{1,} $$

(7)

where the vertical mixing term (MIX) is represented simply by wind speed (wnd) and the ADV-induced upwelling (ADV) is represented by wind curl (curl(τ)). This algorithm considers the two most important ocean processes only, which are assumed to be dominated by winds. Weighting coefficients, a ₁, b ₁, and c ₁, can be determined by a statistical fitting from data, and the resultant weighting coefficients are given in Table 1 of Section 2, below.

1.1.2 Temperature equation (TE-based) algorithm II

This algorithm is based on the concept of algorithm I, but is empirically upgraded to include subsurface ocean factors that might influence the surface SSTC structure (Lin et al. 2008; Wei et al. 2014a; Liu and Wei 2015). Algorithm II can be derived from the following ocean equations:

$$ \frac{du}{dt}= fv+{\tau}_y; $$

(8)

$$ \frac{dv}{dt}=- fu+{\tau}_x; $$

(9)

$$ \mathrm{CUR}={\left({u}^2+{v}^2\right)}^{1/2}; $$

(10)

$$ \mathrm{DIV}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}; $$

(11)

$$ \mathrm{SSTC}=\underset{WND}{\underbrace{\left({a}_2\times \mathrm{CUR}+{b}_2\times \mathrm{DIV}+{c}_2\right)}}\times \underset{OCN}{\underbrace{\left({T}_{100 m}-\mathrm{SST}\right)/\mathrm{SSH}}\times {d}_2}. $$

(12)

where u and v are ocean current velocities, τ _x  and τ _y are wind stress acting on the ocean surface in the x and y directions, f is the Coriolis parameter, CUR and DIV are ocean current speeds and divergence, SSH is sea surface height, and T _100 m is ocean temperature at 100-m depth.

During strong storm events, the contributions from horizontal diffusion and pressure gradient are negligible (Uhlhorn and Shay 2012, 2013; Wei et al. 2014), and therefore, they are not included in the algorithm. The dynamic ocean response to typhoon winds can be regarded as a transient solution of wind forcing (Eqs. S8–9). As known, the dominant ocean processes governing the typhoon-induced ocean cooling is vertical mixing and upwelling. First, to resolve the wind-induced mixing, different from the TE-based algorithm I using wind speeds as a proxy, we here used the ocean surface current speed (Eq. S10), which represents a solution of combined wind-driven currents and inertial motions (Eqs. S8–9). Second, the upwelling process is related to flow divergence, which can be represented by DIV in this algorithm (Eq. S11). Therefore, the wind-induced cooling (WND) can be determined by combining the vertical mixing (CUR) and the upwelling (DIV), in which the partition between CUR (75%) and DIV (25%) given in Table 1 is set empirically based on our previous study (Wei et al. 2014a) and is generally consistent with previous observational and modeling studies (Price 1981; Jacob et al. 2000; Jullien et al. 2012). Third, this algorithm also considers ocean factors by including (T _100 m − SST) / SSH to reflect impacts of subsurface ocean temperature and warm/cold eddies. All weighting coefficients are given in Table 1.

1.1.3 Machine learning (ML-based) algorithm

This algorithm originates from the concept of machine learning, which dates back to the late 1950s (Rosenblatt 1957, 1958) but has recently become a popular technique used widely for artificial intelligence, data mining, and speech and image recognition (Hinton et al. 2006; LeCun et al. 2015; Silver et al. 2016). Generally speaking, the machine learning technique is a study of computer algorithms that improves itself automatically, discovering general rules from large datasets and making data-driven predictions. In this study, this concept was used to derive a nonlinear SSTC algorithm from a set of atmospheric and oceanic training data.

The proposed machine algorithm included three layers: the input layer, the hidden layer, and the output layer (Fig. 4 in the text). To be consistent with TE-based algorithm II, the input layer included the same five variables: typhoon 10-m winds (U ₁₀ and V ₁₀), ocean temperature at a depth of 100 m (T _100 m), SSH, and SST. The hidden layer consisted of two layers of neural elements; each of them was a linear combination of the input variables. A weighting matrix connecting these neural elements was determined by machine training and improved itself by a backpropagation (BP) algorithm (Rumelhart et al. 1986), on the basis of which the machine algorithm can be derived as follows:

$$ {N}_1={a}_3\times \left[\begin{array}{c} U10\\ {}\begin{array}{c} V10\\ {}\mathrm{SST}\\ {}\mathrm{SSH}\\ {} T100\end{array}\end{array}\right]+{\mathrm{b}}_3 $$

(13)

$$ {N}_2={c}_3\times {N}_1+{d}_3 $$

(14)

$$ X=\frac{2}{1+{e}^{-2 N}}-1 $$

(15)

$$ \mathrm{SSTC}={e}_3\times \mathrm{X}+{f}_3 $$

(16)

$$ E=\frac{1}{2}\sum {\left(\mathrm{SSTC}- Obs.\right)}^2 $$

(17)

where a₃, b₃, c₃, and d₃ are weighting matrices that connect the input variables to the neurons N_1,2; X performs a normalization of N to [0 1]; Eq. S16 connects the neurons to output SSTC. Equation S17 calculates the errors between diagnostic and observed SSTC. Based on the BP algorithm, the error matrix E is propagated backward to the neurons, which, in turn, adjust its weighting matrices. Through iterations on backward propagation of E, the neural network learns by itself to achieve an optimum weighting function and a minimum error E. This learning process repeats until the error is zero or reaches a prescribed threshold. During the learning process, 70% of samples are used for training, in which the Levenberg–Marquardt training algorithm is adopted to adjust the weighting coefficients, because it requires less time but more memory than other training algorithms, such as Bayesian regularization and scaling conjugate gradients (Moré 1977; Argyros 2004; Özgür 2009; Dongre et al. 2012). Thirty percent of the remaining samples are then used for validating the network performance. For the typhoon case in this study, we eventually reach a correlation coefficient between diagnostic and observed SSTC by 0.89 and a threshold of error by 0.001. The resultant algorithm and weighting matrices (given in Table 1) were then implemented into the WRF model to update the SST conditions.

1.2 Data sources and preprocessing

1.2.1 Data sources

The data used to determine the weighting coefficients in the algorithms can be categorized into oceanic and atmospheric variables. We assumed that the oceanic fields involved in the algorithms, such as SSH, T _100 m, and MLD, change slowly compared to wind fields during the storm passage, and therefore, these data were initially taken from real-time ocean states from satellite-derived or reanalysis datasets and kept unchanged during the learning process. SSH data were derived from Archiving, Validation, and Interpretation of Satellite Oceanographic (AVISO) dataset (http://www.aviso.oceanobs.com), with a resolution of 0.25° × 0.25°; T _100 m and MLD were obtained from the HYCOM reanalysis, which provided global oceanic reanalysis data with a resolution of 1/12° from 1992 to 2012, using the Navy Coupled Ocean Data Assimilation system for data assimilation (http://hycom.org/dataserver/glb-reanalysis). As the initial SST conditions for the WRF model, the SST maps were derived from the Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI), which provides cloud-penetrating daily SST data (http://pmm.nasa.gov/TRMM) at a 0.25° × 0.25° resolution. For the atmospheric data, the intensity and track of typhoon Soulik (2013) were obtained from the best-track data supplied by the Japan Meteorological Agency (JMA, http://www.jma.go.jp/jma/indexe.html). As shown, the three algorithms all involve high-resolution wind fields to estimate SSTC; in this study, we attempted to use wind data from the coupled simulation of Soulik for the following reasons: (i) The JMA data or other satellite-derived wind fields cannot provide accurate, instantaneous, and high-resolution winds in time and space required by the algorithm; (ii) as shown, for the case of Soulik, the simulated results from the coupled model are able to reproduce reasonably well the observed storm intensity, track, and storm-induced SSTC (Fig. 2). Therefore, although the model data are unreal, to be process-oriented, we think that they can be used as a benchmark to verify the effectiveness of the new-proposed algorithms; and (iii) the coupled model not only produces typhoon surface winds but also resolves three-dimensional ocean processes, which can be used as a basis for understanding oceanic response to typhoons, and for developing a practical scheme.

1.2.2 Data preprocessing

Since the atmospheric and oceanic variables used for the algorithms involved different units and scales, it was inappropriate to use them directly, as this may have caused the fitting to deviate from physical constraints. For example, the contributions of those variables with large amplitudes may have been overestimated by regressions or machine training, which could be invalid from a physical prospective. To constrain the statistical fitting and training, we carried out the following two-step preprocessing for the data. First, to be consistent in data resolution, all input variables are interpolated to a grid of 0.25° × 0.25°, which is the same resolution of satellite-derived SST and SSH maps. To unify the contributions of variables with different units on the resultant SSTC, all variables were normalized with their maxima within values between 0.5 and 1.5. This range also guaranteed nonzero values in the denominator. By doing so, the resultant SSTC was also normalized and nondimensional, and therefore, an inverse normalization was carried out to obtain the actual values of SSTC.

Second, to make the resultant SSTC physically distinct, this study focused on the open oceans only, where typhoon-induced ocean processes dominate, for which the driving mechanisms have been well established in previous studies. We realized that there is an abundance of real-time observations available near coastal areas during typhoon’s approach; however, these data are not suitable to verify these “open-ocean” algorithms, because these algorithms did not consider any land and topography effects near coastal areas, where the physical mechanisms of SSTC are far more complex than those in the open oceans. To focus on the data in the open oceans, from the model-gridded data, we filtered out those if the wind speed at the model grid point is less than 10 m/s. By doing so, we spatially excluded the outer range of the storm and temporally excluded the spinning-up and decaying-down periods. We also excluded the data near the coasts (within 500 km of the storm center). Such data processing enabled scenarios to be dominated by open-ocean processes, thus facilitating an examination and verification of the individual budget terms diagnosed from the algorithms (Fig. 3). However, we noted that this configuration might cause certain errors in estimating SSTC when the model storm approaches towards the coasts.