1.College of Meteorology and Oceanography, National University of Defense Technology, Nanjing 211101, China 2.State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, State Oceanic Administration, Hangzhou 310012, China
Abstract:In this paper, we propose a new method of numerical differentiation to determine the height of the top layer of the atmospheric boundary layer. In this method, a regularization technique is used to convert the problem of calculating the differential of the curve of the corners into the problem of finding the minimum value of the objective function. The two-parameter model function method is used to select the regularization parameters. Finally, the maximum gradient method is used to determine the top height of the boundary layer. Firstly, the effectiveness of the new method is validated through two numerical experiments. The experimental results show that as the noise of the occultation data increases, the fluctuation of the height of the boundary layer top obtained by the difference method and the numerical differentiation method combined with the L curve scheme increases. And the height obtained by the two-parameter model function method is very stable, which shows that the new method can filter the noise well, thereby retaining the main information about the profile. Then, based on the COSMIC angle data in January, April, July and October 2007-2011, the new method is used to analyze the seasonal characteristics of the height of the global oceanic and atmospheric boundary layer, compared with the seasonal distribution obtained by “zbalmax” with the occultation data. The results show that the seasonal distribution characteristics of the two data are very consistent: the height of the boundary layer is higher in the area where the sea surface temperature is higher than that in the surrounding sea area; on the contrary, the height of the boundary layer top is lower. In the sea area where the warm current passes, the height of the boundary layer is higher; in the sea area where the cold current passes, the height of the boundary layer is lower. Keywords:boundary layer height/ regularization method/ two-parameter model function method/ bend angle gradient
确定正则化双参数的计算流程如图2所示. 因为在正则化方法中物理量往往是无量纲的, 为了使正则化中的量纲一致, 将$\alpha, \beta $做了以下处理: 将$\alpha $扩大$500 \times {\rm{std}}\left( {{b}} \right)$倍, 将$\beta $扩大${\rm{std}}\left( {{b}} \right)$倍, 这里std(·)表示取标准差. 图 2 双参数模型函数法流程图 Figure2. Flow chart of two-parameter model function method.
3.数值实验本节比较了三种确定边界层顶高度的方法, 分别是模型函数法、差分法和L曲线方法. 所用数据是2011年1, 4, 7, 10月份的COSMIC弯角资料. 在处理之前先对数据进行质量控制, 然后借助spline插值法将资料插值到间隔为10 m的网格点上. 双参数模型函数法的两个初始参数$\alpha, \beta $都取为1, 根据弯角的量级为${10^{ - 2}}$, 故取$\delta = {10^{ - 2}}$, 收敛参数$\varepsilon = {10^{ - 4}}$. 由差分法、L曲线法以及模型函数法计算出来的边界层顶高度分别记为${H_{\rm{d}}}, {H_{\rm{L}}}, {H_{\rm{M}}}$. 掩星探测资料不可避免地会带有误差, 为了比较在存在噪声情况下几种方法的表现, 本节将$[ - \delta, + \delta ]$上的随机误差添加到弯角资料, 由于在边界层中弯角的量级为${10^{ - 2}}$, 故令$\delta \in [0.0001, 0.01]$, 在带有噪声的弯角资料的基础上, 通过与差分法和L曲线法相比较, 检验模型函数法在确定边界层顶高度上是否具有准确性. 图3选取了廓线atmPrf_C001.2011.182.05.19.G18_2013.3520_nc, 并且分别添加随机误差$ \delta = $$ 0.0025, 0.005, 0.0075, 0.01$, 比较了差分法和数值微分结合模型函数方法得到的边界层顶高度. 从结果可以看出, 图3(a)是由带有误差$\delta = 0.0025$的弯角得到的弯角梯度, 差分法得到的弯角梯度廓线在一定水平范围内呈密集的锯齿状, 其最小梯度在1.4 km左右, 数值微分方法结合模型函数法得到的弯角廓线比较光滑, 其最小梯度在1.2 km左右, 由图3(b)—(d)可知, 随着弯角资料误差的增大, ${H_{\rm{M}}}$始终稳定在1.2 km左右, 而${H_{\rm{d}}}$由1.4 km (图3(a)和图3(b))变为3.7 km(图3(d)), 变化幅度超过2 km, 说明当掩星资料中存在观测误差时, 模型函数法能够很好地过滤噪声, 保留主要信息, 得到的边界层顶高度的结果更加稳定. 图 3 由差分法和模型函数法得到的弯角梯度廓线(BA表示弯角) (a)弯角廓线添加随机误差$\delta = 0.0025$, 边界层顶高度${H_{\rm{M}}} = 1.2\;{\rm{km}}, {H_{\rm{d}}} = 1.4\;{\rm{km}}$; (b)弯角廓线添加随机误差$\delta = 0.005$, 边界层顶高度${H_{\rm{M}}} = 1.1\;{\rm{km}}, {H_{\rm{d}}} = 1.4\;{\rm{km}}$; (c)弯角廓线添加随机误差$\delta = 0.0075$, 边界层顶高度${H_{\rm{M}}} = 1.2\;{\rm{km}}, {H_{\rm{d}}} = 1.8\;{\rm{km}}$; (d)弯角廓线添加随机误差$\delta = 0.01$, 边界层顶高度HM = 1.2 km, Hd = 3.7 km Figure3. Angle gradient profile obtained by the difference method and the model function method using the bending angle gradient profile (BA represents the bending angle): (a) Bending angle profile with uniform random error $\delta = 0.0025$, boundary layer top height ${H_{\rm{M}}} = 1.2\;{\rm{km}}, {H_{\rm{d}}} = 1.4\;{\rm{km}}$; (b) bending angle profile with uniform random error $\delta = 0.005$, boundary layer top height ${H_{\rm{M}}} = 1.1\;{\rm{km}}, {H_{\rm{d}}} = 1.4\;{\rm{km}}$; (c) bending angle profile with uniform random error $\delta = 0.0075$, boundary layer top height ${H_{\rm{M}}} = 1.2\;{\rm{km}}, {H_{\rm{d}}} = 1.8\;{\rm{km}}$; (d) bending angle profile with uniform random error $\delta = 0.01$, boundary layer top height ${H_{\rm{M}}} = 1.2\;{\rm{km}}, {H_{\rm{d}}} = 3.7\;{\rm{km}}$.
图4给出了三种方法(模型函数法、差分法、L曲线法)分别在两条添加误差后的弯角廓线(atmprf_C001.2001.182.00.22.G23_2013.3520_nc, atmPrf_C001.2011.182.00.39.G20_2013.3520_nc)上的表现, 以结果的标准差衡量方法的稳定度. 由图4(a)可以看出, ${H_{\rm{M}}}$比较稳定, 保持在3.15 km左右, 而${H_{\rm{d}}}$和${H_{\rm{L}}}$随着误差的增大, 波动越来越大, 其中, ${H_{\rm{M}}}, {H_{\rm{d}}}, {H_{\rm{L}}}$的稳定度分别为0.013, 0.44和0.61; 同样由图4(b)也可以看出, ${H_{\rm{M}}}$依然是比较稳定, 保持在4.55 km左右, 而${H_{\rm{d}}}$和${H_{\rm{L}}}$随着误差的增大, 波动也越来越大, 其中${H_{\rm{M}}}, {H_{\rm{d}}}, {H_{\rm{L}}}$的稳定度分别为0.02, 0.89和1.19. 由以上分析可以看出, 弯角资料的噪声越大, ${H_{\rm{d}}}$振荡程度就越强, ${H_{\rm{L}}}$振荡的程度次之, 而${H_{\rm{M}}}$的稳定性则比较好, 说明对于含有噪声的弯角资料, 模型函数法也能稳定地得到边界层顶高度, 所以模型函数法在确定边界层顶高度时, 准确性能得到保证. 图 4 由差分法、L曲线法和模型函数法得到的边界层顶高度 (a) 三种方法基于廓线1得到的边界层顶高度, Htrue = 3.15 km, std(HM) = 0.013, std(HL) = 0.44, std(HM) = 0.61; (b) 三种方法基于廓线2得到的边界层顶高度, Htrue = 4.55 km, std(HM) = 0.020, std(HL) = 0.89, std(HM) = 1.19 Figure4. Height of the boundary layer obtained by the three methods: (a) Three methods to get the height of the boundary layer top based on profile 1, Htrue = 3.15 km, std(HM) = 0.013, std(HL) = 0.44, std(HM) = 0.61; (b) three methods to get the height of the boundary layer top based on profile 2, Htrue = 4.55 km, std(HM) = 0.020, std(HL) = 0.89, std(HM) = 1.19
4.全球边界层顶高度的季节特征本节应用模型函数法, 对海洋上边界层顶高度的季节特征进行分析. 图5展示了2007—2011年的1, 4, 7, 10四个月份年平均的边界层顶高度, 可以看出, 1月份南半球的边界层顶达到最高, 尤其是中高纬度海洋上的边界层顶高度达到2 km以上; 4月份南半球的边界层顶高度开始降低, 南美洲与南极大陆南设得兰群岛之间的德雷克海峡附近降低的程度最为明显, 而北极附近海域的边界层顶高度也降低, 原因在于北极的海冰覆盖面积在3月份达到高峰, 4月份海冰的面积比1月份多[40], 所以4月份下垫面温度更低, 相应的边界层顶高度也在4月份比1月份低; 七月份南半球中高纬度的边界层顶高度在四个月份中最低, 北半球的低纬度地区边界层顶高度明显增高, 如墨西哥湾地区、阿拉伯地区以及日本海地区, 而受西太平洋副热带高压下沉气流影响, 我国东南沿海、孟加拉湾和菲律宾附近海域边界层顶高度明显降低; 10月份南半球中高纬边界层顶高度略有回升, 北极海域边界层顶高度开始降低, 而此时我国东南沿海、孟加拉湾和菲律宾附近海域边界层顶高度明显升高. 图 5 用模型函数法得到的海洋5年平均的边界层顶高度(所用资料为2007—2011年1, 4, 7, 10四个月份的掩星弯角的资料) (a) 1月份平均边界层顶高度; (b) 4月份平均边界层顶高度; (c) 7月份平均边界层顶高度; (d) 10月份平均边界层顶高度 Figure5. The 5-year average boundary layer height of the ocean obtained by the model function method, the data used is the bending angle profile of the four months of 2007?2011 in January, April, July, and October: (a) The average height of the boundary layer in January; (b) the average height of the boundary layer in April; (c) the average height of the boundary layer in July; (d) the average height of the boundary layer in October.
从以上现象分析可以得知, 随着太阳直射点的南北移动, 海洋上的边界层顶高度呈现明显的季节变化特征, 边界层顶高度与海面的温度密切相关, 当海面的温度相对周围平均温度越低时, 边界层顶高度越低, 反之, 边界层顶高度就越高. 但是四个月份共同的特征是, 在寒流流经的区域, 边界层顶高度普遍偏低, 比如巴西寒流和加利福尼亚寒流, 暖流流经的区域边界层顶普遍较高, 比如赤道暖流. 图6展示了利用掩星自带的zbalmax的资料求得的边界层顶高度的季节分布, 通过与图5相比较, 可以看出模型函数法得到的边界层顶高度略高, 但两者边界层顶高度的时空分布十分一致, 说明模型函数法得到的季节特征具有准确性. 图 6 海洋5年平均的边界层顶高度(所用资料为2007—2011年1, 4, 7, 10四个月份的掩星自带zbalmax的资料) (a) 1月份平均边界层顶高度; (b) 4月份平均边界层顶高度; (c) 7月份平均边界层顶高度; (d) 10月份平均边界层顶高度 Figure6. The 5-year average boundary layer height of the ocean obtained by the model function method, the data used is the zbalmax provided by CDAAC of the four months of 2007?2011 in January, April, July, and October: (a) The average height of the boundary layer in January; (b) the average height of the boundary layer in April; (c) the average height of the boundary layer in July; (d) the average height of the boundary layer in October.