潮波模型的优化开边界方法研究

数值模式是潮波研究的一种有利手段，但在研究中会面临各种具体问题，包括开边界条件的确定、底摩擦系数和耗散系数的选取等。数据同化是解决这些问题的一种途径，即利用有限数量的潮汐观测资料对潮波进行最优估计，其根本目的是迫使模型预报值逼近观测值，使模式不要偏离实际情况太远。本文采用了一种优化开边界方法，沿着数值模型的开边界优化潮汐水位信息，目的是设法使数值解在动力约束的意义下接近观测值，获得研究区域的潮汐结果。边界值由指定优化问题的解来定，以提高模拟区域的潮汐精度，最优问题的解是基于通过开边界的能量通量的变化，处理开边界处的观测值与计算值之差的最小化。这里提供了辐射型边界条件，由Reid 和Bodine（本文简称为RB）推导，我们将采用的优化后的RB方法（称为ORB）是优化开边界的特殊情况。 本文对理想矩形海域（ E- E， N- N， 分辨率 ）进行了潮波模拟，有东部开边界，模式采用ECOM3D模式。对数据结果的误差分析采用，振幅平均偏差，平均绝对偏差，平均相对误差和均方根偏差四个值来衡量模拟结果的好坏程度。 需要优化入开边界的解析潮汐值本文采用的解析解由方国洪《海湾的潮汐与潮流》（1966年）方法提供，为验证本文所做的解析解和方文的一致，本文做了其第一个例子的关键值a,b,z,结果与其结果吻合的相当好。但略有差别，分析的可能原因是两法在具体迭代方案和计算机保留小数上有区别造成微小误差。另外，我们取m=20，得到更精确的数值，我们发现对前十项的各项参数值，取m=10,m=20各项参数略有改进。当然我们可以获得m更大的各项参数值。 同时为了检验解析解的正确性讨论m和l变化对边界值的影响，结果指出，增大m，m=20时，u的模最大在本身u1或u2的模的6%；m=100时，u的模最大在本身u1或u2的模的4%；m再增大，m=1000时，u的模最大在本身u1或u2的模的4%，改变不大。当l<1时， =0处u的模最大为2。当l=1时， =0处u的模最大为0.1，当l>1时，l越大，u的模越小，当l=10时，u的模最大为0.001，可以认为为0。 为检验该优化方法的应用情况，我们对理想矩形区域进行模拟，首先将本文所采用的优化开边界方法应用于30m的情况，在开边界优化入开边界得出模式解，所得模拟结果与解析解吻合得相当好，该模式解和解析解在整个区域上，振幅平均绝对偏差为9.9cm,相位平均绝对偏差只有4.0 ，均方根偏差只有13.3cm，说明该优化方法在潮波模型中有效。 为验证该优化方法在各种条件下的模拟结果情况，在下面我们做了三类敏感性试验： 第一类试验：为证明在开边界上使用优化方法相比于没有采用优化方法的模拟解更接近于解析解，我们来比较ORB条件与RB条件的优劣，我们模拟用了两个不同的摩擦系数，k分别为：0,0.00006。 结果显示，针对不同摩擦系数，显示在开边界上使用ORB条件的解比使用RB条件的解无论是振幅还是相位都有显著改善，两个试验均方根偏差优化程度分别为84.3%，83.7%。说明在开边界上使用优化方法相比于没有采用优化方法的模拟解更接近于解析解，大大提高了模拟水平。上述的两个试验得出， k=0.00006优化结果比k=0的好。 第二类试验，使用ORB条件确定优化开边界情况下，在东西边界加入出入流的情况，流考虑线性和非线性情况，结果显示，加入流的情况，潮汐模拟的效果降低不少，流为1Sv的情况要比5Sv的情况均方根偏差相差20cm,而不加流的情况只有0.2cm。线性流和非线性流情况两者模式解相差不大，振幅，相位各项指数都相近, 说明流的线性与否对结果影响不大。 第三类试验，不仅在开边界使用ORB条件，在模式内部也使用ORB条件，比较了内部优化和不优化情况与解析解的偏差。结果显示，选用不同的k,振幅都能得到很好的模拟，而相位相对较差。另外，在内部优化的情况下，考虑不同的k的模式解， 我们选用了与解析解相近的6个模式解的k,结果显示，不同的k,振幅都能得到很好的模拟，而相位较差。 总之，在开边界使用ORB条件比使用RB条件好，振幅相位都有大幅度改进，在加入出入流情况下，流的大小对模拟结果有影响，但线形流和非线性流差别不大。内部优化的结果显示，模式采用不同的k都能很好模拟解析解的振幅。

Numerical model is an advantaged measurement to research tide, but it will be confronted with many material problems, such as specify open boundaries、bottom friction coefficient、dissipation coefficient and so on. Data assimilation is an approach to solve these problems, using restricted numerous tide observation data to optimize estimate for tide field, the ultimate purpose is put the mode predict data close to the observation data, avoid the mode data far from the real data too much. This paper introduces an optimized assimilation approach; an optimization approach is derived for assimilating tidal height information along the open boundaries for a numerical model. The aim is trying to make the observation data close to the momentum restriction resolution, to simulate the tide of study area. The boundary values are determined from the solution of the special optimization problem: minimization of the difference between the model and reference boundary values, to enhance tide precision of simulation areas. The approach is then extended so that similar data along transects inside a model domain can also be optimally assimilated. It shows the well-known radiation-type boundary conditions introduced by Reid and Bodine; the optimized RB (ORB) solution is the special cases of the derived optimized conditions. Based on the ECOM3D model with resolution 1/12º×1/12º, this work presents a model study on an ideal square area with only an open boundary on the east side, which locates between E- E and N- N. We try to evaluate the performance of the model by comparing the swing mean deviation, mean absolute deviation, mean relative deviation and variances with analytic solution. The analytic resolution of the tides and tidal streams, which is needed in the assimilation model, is get by an approximate method first presented in paper “Tides and tidal streams in gulfs” by Fang Guohong and Wang Shuren in 1966. We compared the key parameters, say a, b and z of the first case study preferred in this paper and found that our results match well with the original authors’. But they have nuance, the causation maybe thin differentiate between the two methods about iterative scheme and different reserve decimal digits. Besides, we choose m=20, obtain more precise numerical value, we found that each parameters of the first ten items, with m=10 and m=20, have little ameliorate. We also can obtain each parameters of large m. To test the validity of analytical resolution, we discuss the effect on the boundary value of changing m and l; the results show that, when augmenting m with m=20, the largest module of u is 6% of u1 or u2’s module; with m=100, the value is 4%; and with 1000, the value is 4%, the change is small. When l<1, =0, the largest module of u is 2. when l=1,the value is 0,1,when l>1,the larger l, the less the module of u. when l=10,the value is 0.001,it can be seen zero. It tests the conclusion of FANGGUOHONG (1966). In order to test the performance of the optimal methodology, we conduct model study on the ideal square area. First, we apply the above preferred optimal open boundary method to case study of 30-meter depth. After assimilating only the analytic solutions on the open boundary into the model, the results in the whole area agree quite well with the corresponding analytic solutions with mean absolute deviation is 9.9cm, mean relative deviation is 4.0 and variances is 13.3cm, which reveals a good applicability of this optimal method on the tidal model. To further investigate the performance of this optimal assimilation method on different conditions, we conduct 3 kinds of sensitive tests: The first kind of test: for testifying this method with the best-** method more close to resolution than without it, we have compared the advantage between the conditions of ORB and RB, two rub coefficients have been used in the model, k will be equal to 0 and 0.00006 separately. The conclusion shows that, with different rub coefficient, there are both advantages in phase and swing on open boundary with ORB condition, the prior grade of covariance in the two standard results is 84.3% and 83.7% respectively. This shows that the model can be better with the previous method, and get another level. It is concluded that the simulation with k=0.00006 better than k=0. The second kind of test: using the condition of ORB and add the current with in and out, we consider current of linearity and nonlinearity instances. The results show that the precision of tide simulation depress when current adds. Compare 1Sv to 5Sv, the variances deviation is 20cm, while 0.2cm of no current. Linearity and nonlinearity current have little influence for the modle, swing and phase are close. It can be said that no matter linearity or nonlinearity, he influence is small. The third kind of test: using ORB condition not only in OBCs, but also in interior, compares the different between interior assimilation and nonassimilation to analyze solution. It shows that, using different k, swing have good simulation, but phase have not. Besides, in case of interior assimilation, consider mode solution of different k, we choose 6 k which close to the analyze solution. It is show that, swing improved much, but phase have not. From the above analysis we find that tide model results with optimal open boundary assimilation method, with a significant error decreasing, agree better with the analytic solution than that without using this method; we also find that, with the same boundary condition, parameters shifting of the depth, in and out flow or the bottom friction will also induce a change on the model results. The interior assimilation’s result show that, the mode adopts different k can simulate swing good.