كاربست روش ‌طيفي براي حل عددي معادلات آب كم‌‌عمق منطقه محدود

Application of the spectral method to solve the limited-area shallow-water equations

در اين پژوهش، روش ‌شبه‌طيفي براي حل عددي معادلات آب كم‌عمق دوبُعدي غيرخطي در منطقه محدود به كار گرفته مي‌شود. روش شبه‌طيفي بر استفاده از توانايي روش طيفي در برآورد مشتقات فضايي تابع‌هايي كه به قدر كافي هموارند، استوار است. در مدل منطقه محدود ساخته شده برمبناي طرحواره داراي پايستاري آنستروفي پتانسيلي سادورني براي معادلات آب كم‌عمق يا بسيط فشارورد، به‌جاي تفاضل مركزي از روش طيفي برمبناي تابع‌هاي فوريه براي برآورد مشتقات فضايي استفاده مي‌شود. براي كاربست روش شبه‌طيفي در منطقه محدود كه در هيچ‌يك از دو راستاي شرقي- غربي و شمالي- جنوبي حوزه محاسباتي دوره‌اي نيست، از الگوي گسترش و واهلش هاگن و مكنهاور استفاده مي‌شود. براي پايداري محاسباتي يك ميرايي صريح به‌صورت پخش عددي با ضرايب متفاوت براي معادلات تكانه و ژيوپتانسيل به الگوريتم افزوده مي‌شود. براي به‌روزرساني كميت‌هاي مدل در هر گام زماني از طرحواره زماني ليپ‌فراگ به همراه پالايه روبرت- آسلين استفاده مي‌شود. نتايج براي يك مورد پيش‌بيني 48 ساعته با استفاده از روش طيفي در تفكيك‌هاي فضايي 150، 75 و 5/37 كيلومتر عرضه و مقايسه مي‌شوند. جواب‌ها در تفكيك 150 كيلومتر به‌دليل اثر پخش عددي و نيز ميرايي خود ميدان زمينه بسيار هموارند و اين ميرايي زياد باعث جدايي جواب‌ها از جواب‌هاي متناظر الگوريتم تفاضل متناهي مي‌شود. با اين‌حال در تفكيك‌هاي 75 و 5/37 كيلومتر حركت الگوهاي مقياس همديدي و حتي زيرهمديدي داراي نمايشي رضايت‌بخش هستند. محاسبه خطاي روش طيفي از ديدگاه نبود توازن، حاكي از گسيل بي‌توازني نسبتاً بزرگي از منطقه گسترش به داخل حوزه محاسباتي است.

The pseudo-spectral method is used to solve the shallow-water equations in a limited-area domain. The spectral computation of spatial derivatives in the pseudo-spectral method makes it in effect equivalent to a Galerkin spectral-transform method commonly used in numerical modeling of fluid flows. Based on the potential-enstrophy conserving scheme introduced by Sadourny in 1975, the limited-area model is constructed by replacing the second-order finite-difference computation of spatial derivatives with the spectral method using Fourier basis functions. Since the limited-area domain is not periodic in the east-west and north-south directions, in order to apply the spectral method, the domain has to be made periodic using an extension zone as proposed by Haugen and Machenhauer. The background field obtained using the "Global Forecast System" (GFS) data is extended periodically and matched to the limited-area field across a relaxation zone which is introduced in order to reduce the adverse effects of the artificial extension of the field variables. In this way, the prognostic variables of the limited-area model including the velocity components and geopotential height are made periodic. To alleviate false generation of imbalance in the extension zone, two actions are taken. First, because of its large variation across the main limited-area domain, the geopotential height was decomposed to a zonal mean and a perturbation and the latter field was made periodic. Second, in the extension zone the velocity components were constructed using geostrophic approximation. Even with these two actions, the amount of imbalance penetrating the computation domain was sufficient to cause computational instability. In order to further control the generation of imbalance and reach computational stability, an explicit damping in the form of numerical diffusion was added to the equations of momentum and geopotential height with different diffusion coefficients for momentum and height. For time-stepping, a three-time-level leapfrog scheme with Robert-Asselin filter to remove the computational mode was used. For a case, previously examined in literature, related to 1st of February 2003, results for the 48-hour prediction using the spectral algorithm in 150, 75 and 37.5 km resolutions are presented, compared and assessed using two norms, one measuring the deviation from the actual GFS fields and the other measuring the maintenance of balance during the 48-hour integrations. To determine balanced fields, the first-order implicit normal-mode initialization procedure is used. The spatial resolution refers to the grid spacing at latitude. At 150 km resolution, the predicted fields are excessively damped due to a combination of the numerical diffusion required for stability and the smoothness of the background fields used in the relaxation process. The excessive damping causes a significant departure of the fields from the results for the second-order finite-difference algorithm. At 75 and 37.5 km resolutions, however, the synoptic and even sub-synoptic scale motions are represented sufficiently well. The assessment of balance maintenance shows that a large amount of imbalance is emitted from the extension zone to the computational domain, making the limited-are spectral algorithm much less balance-preserving compared with the corresponding second-order finite-difference algorithm.