西方邊界流模擬—切比雪夫配置法與沉浸邊界法整合

Western Boundary Current Intensification Modeling with Chebyshev and Immerse Boundary Methods

張光宇(Kuang-Yu Chang)

指導教授 ： 郭鴻基

共同指導教授 ： 曾于恆(Yu-Heng Tseng)

國立臺灣大學/理學院/大氣科學研究所/碩士(2010年)

https://doi.org/10.6342/NTU.2010.00134

摘要

摘要
本研究以物理海洋中經典的西方邊界流(Western boundary current) 為例, 首次嘗試整合切比雪夫配置法與虛網格沉浸邊界法。西方邊界流現象是由於海洋中的波動能量在海洋西邊界不斷地累積而產生, 在狹窄的西方邊界流區域內有著整個海洋中的最大流速, 藉以維持海洋流體運動的質量與絕對渦度守恆。雖然西方邊界流生成理論於過去已有相當好的理解, 但在數值模擬中, 海洋的複雜邊界會大幅提高模擬困難度, 本研究希望能夠擷取具有指數收斂與快速轉換等性質的切比雪夫配置法(Chebyshev collocation method) 以及在模式中能藉由邊界強迫機制有效處理複雜邊界問題的虛網格沉浸邊界法(Ghost cell immersed boundary method) 兩者之優點, 建立出一套同時具有高精確度以及高效率的計算方法。
研究中利用海洋渦度方程式設計一系列正壓模式進行模擬, 模擬結果顯示切比雪夫配置法以及虛網格沉浸邊界法的使用皆可有效地掌握西方邊界流運動,且數值穩定解具有一致性。由行星渦度梯度產生的羅士比波(Rossby wave) 將於海洋西方邊界造成能量累積, 為瞬態(Transient state) 西方邊界流生成的物理機制, 在近似對稱的海洋中造成高度不對稱性; 而非線性動力則扮演將波動能量重新分佈的角色, 有效地將能量由平均流向西處往向東處傳遞。除單一定義域的模擬外, 我們更藉由分解模式定義域的方式整合各種數值方法, 研究結果顯示子定義域合併處之網格點配置對耦合模式有顯著的影響; 合併處網格點完全重疊時, 子定義域間可透過分解定義域的方式相互提供邊界條件,藉以成功地整合切比雪夫配置法與虛網格沉浸邊界法。然而, 合併處網格點不完全重疊時, 子定義域間邊界條件之交換須仰賴數值內差的使用; 雖可於各子定義域中有效地提供邊界條件, 但將受限於子定義域間網格點解析度之差異導致合併處質量通量不連續, 於耦合定義域產生數值上的不連續, 進而影響積分預報的穩定性。研究結果顯示, 耦合模式之二階中差分法搭配切虛網格沉浸邊界法子定義域與比雪夫配置法子定義域間大小比值須為六分之一以上方可有效運作, 兩者間相互給定邊界條件時所使用的內差方法精確度亦將對積分穩定性造成影響; 且藉由控制子定義域間質量通量量值與數值方法間精確度達一致將可抑制數值不穩定產生, 有效延長耦合模式穩定預報時間。

關鍵字

西方邊界流 ； 切比雪夫配置法 ； 沉浸邊界法 ； 分解定義域

並列摘要

ABSTRACT
By simulating the famouse Western Boundary Current (WBC), this thesis first integrate Chebyshev collocation method with Ghost Cell Immersed BoundaryMethod (GCIBM) into a powerful coupled model. WBC intensification is caused by wave energy accumulation at the western boundary of the ocean basins, causing a narrow region which has higher meridional velocity than anywhere else. Complex boundary geometry may also strongly affect the WBC which is difficult to handle in the numerical model. In order to solve this problem and increase flexibility, we use GCIBM to implicitly provide the boundary conditions for the numerical model. GCIBM is an advanced numerical method, based on the use of artificial forcing to represent the effect of the realistic boundaries on the flow. To accurately resolve the wave dynamic, we employee the Chebyshev Collocation method with exponential solution convergence rate. Combing GCIBM with Chebyshev Collocation method, we present high accuracy and flexibility barotropic model of different complexity to simulate the WBC intensification in the β plane ocean.
The numerical results show that both Chebyshev Collocation method and central difference with GCIBM can provide a reasonable WBC. The width of the WBC agrees well with the theoretical one. It seems that during the transient state Rossby waves play an important role in the generation of WBC, causing the asymmetric property in the nearly symmetric ocean, while the nonlinear dynamics would redistribute the wave energy of the ocean. Besides single domain modeling, we use domain decomposition to integrate Chebyshev collocation method with GCIBM. The resluts sugesst that the arrangments of grids at the embedding regions are essential to the coupled model; Chebyshev collocation method and GCIBM would work efficeintly through domain decomposition with the embedding grids matching exactly between the subdomains. However, as long as the embedding grids don’t match each other; interpolation schemes are used in the two-way embedding algorithms, which would cause the mass flux inconsistency between the subdomains due to the difference grid resolutions. This numerical inconsistency would lead to error accumulation at the embedding regions, restricting the effective prediction time steps in the coupled models. Our research suggest that size ratio between GCIBM and Chebyshev collocation method subdomains should be larger than onesixth for the couplde model to run effectively; furthermore, we notice that both mass flux tuning and solution cnovergence rate matching between the subdomains could enlonggate the effective prediction time steps in the coupled models.

並列關鍵字

Western boundary current ； Chebyshev collocation method ； Immersed boundary method ； Domain decomposition

參考文獻

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