A Data Assimilation Algorithm: the Paradigm of the 3D Leray-α Model of Turbulence
A Data Assimilation Algorithm: the Paradigm of the 3D Leray-α Model of Turbulence
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DOI：
10.1017/9781108367639.006
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发表时间：
2019-05
期刊：
影响因子：
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通讯作者：
A. Farhat;E. Lunasin;E. Titi
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文献类型：
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作者：
A. Farhat;E. Lunasin;E. Titi
.在本文中，我们调查的各种实现的一个新的数据同化（降尺度）算法的基础上空间粗网格测量。作为范例，我们展示了该算法在3D Leray- α亚网格尺度湍流模型中的应用。最重要的是，我们使用这种范式表明，它并不总是必要的，一个必须收集粗网格测量的所有状态变量，这是在底层的进化系统，以恢复相应的精确的参考解决方案。具体地说，我们证明了在三维Leray- α湍流模型的情况下，仅使用三维速度场的任何两个分量的粗网格观测而不使用第三分量的任何信息构造的算法的解，在时间上以指数速率收敛到三维Leray- α模型的相应精确参考解。这项研究作为一个补充，我们最近的工作简化连续资料同化的二维Navier-Stokes方程。值得注意的是，类似的结果也建立了三维粘性行星地转环流模型，其中我们表明，粗网格测量的温度单独是足够的恢复，通过我们的数据同化算法，完整的解决方案，即三个组成部分的速度矢量场和温度。因此，这证明了三维行星地转模型的Charney猜想;即，大空间尺度温度的历史足以确定模型的所有其他量（状态变量）。
. In this paper we survey the various implementations of a new data assimilation (downscaling) algorithm based on spatial coarse mesh measurements. As a paradigm, we demonstrate the application of this algorithm to the 3D Leray- α subgrid scale turbulence model. Most importantly, we use this paradigm to show that it is not always necessary that one has to collect coarse mesh measurements of all the state variables, that are involved in the underlying evolutionary system, in order to recover the corresponding exact reference solution. Specifically, we show that in the case of the 3D Leray- α model of turbulence the solutions of the algorithm, constructed using only coarse mesh observations of any two components of the three-dimensional velocity field , and without any information of the third component, converge, at an exponential rate in time, to the corresponding exact reference solution of the 3D Leray- α model. This study serves as an addendum to our recent work on abridged continuous data assimilation for the 2D Navier-Stokes equations. Notably, similar results have also been recently established for the 3D viscous Planetary Geostrophic circulation model in which we show that coarse mesh measurements of the temperature alone are sufficient for recovering, through our data assimilation algorithm, the full solution; viz. the three components of velocity vector field and the temperature. Consequently, this proves the Charney conjecture for the 3D Planetary Geostrophic model; namely, that the history of the large spatial scales of temperature is sufficient for determining all the other quantities (state variables) of the model.