本文以水深平均之淺水方程式 (Shallow-Water Equations, SWEs) 為基礎，發展高階且高解析度的淺水長波模式，模擬東沙環礁之水動力情形。本文控制方程式採用變數(η, qx, qy)，與前人研究使用的變數(η, u, v)相比，其變數(η, qx, qy)在不論是在處理入流邊界條件、精確度、守恆性質上更合乎物理特性。在淺水模式中對時間離散使用θ-方法，其中使用高階的九點四邊形元素並採用有限元素最小平方法將空間離散，求解其殘餘值。設計一駐波通過平底床渠道案例比較九點四邊形元素與三點三角形元素的解析解與數值解之偏差，其中顯示高階元素的偏差值小於低階元素，代表高階元素之精準度是較高的。使用θ-方法進行模式敏感度分析，採用精準度較高九點四邊形元素進行分析，其中顯示在θ= 0.67時，其數值解與解析解偏差值是最小的。
本文應用高階淺水模式模擬東沙環礁水域之流場，其網格精度約為120-5000 m。其結果顯示隨著漲潮與退潮的更迭，在環礁邊緣之南北水道各形成方向相反之渦漩，在退潮時，南北水道分別出現逆時針與順時針之渦漩；在漲潮時，南北水道分別出現順時針與逆時針之渦漩，東沙環礁水域呈現之計算流場與研究量測之流場吻合。另外與POM模式之計算結果做比較，其結果顯示此兩者模式計算出的一般特徵是相似的。

A high-order and high-resolution shallow-water model (SWM) is based on the depth-averaged and nonlinear shallow-water equations (SWEs) which is developed and applied for simulating the ocean circulation of Dongsha waters. The surface ele-vation (η) and specific discharge per unit width in x and y direction (qx, qy) are used as variables in the SWM instead of those used in the previous study. The (η, qx, qy) based on SWM is more accurate as well as reasonable to deal with the inflow boundary con-ditions. It takes the advantage to have better conservative properties than that of (η, u, v) variables. High-order 9-node quadrilateral element is employed for the space inter-polation and θ-method is used for time integration. The developed model is first ap-plied to a standing wave propagating in a flat bottom channel. The results of 9-node quadrilateral element SWM are compared with that of the low-order 3-node triangular element SWM, and the better accuracy is found. Sensitivity study of the θ-method is also performed. It is shown that θ = 0.67 is the optimal value for both 9-node SWM.
A high-resolution SWM, with spatial resolution 120-5000 m, is then applied to study the hydrodynamics of the Dongsha waters. Computed results show detailed ocean circulation inside the Dongsha Atoll. Computed results are in good agreement with that of the field measurements at ebb and flood tides, respectively. It is interest-ing to note that there is a pair of periodically moving clockwise/counterclockwise vor-tices along the edge of northern and southern entrance channels. General feathers of ocean circulation of Dongsha waters by SWM is also consistent with the results com-puted by POM (Princeton Ocean Model).

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