Difficulty of data assimilation arises from a large difference between the sizes of a state vector to be determined, that is, the number of spatiotemporal mesh points of a discretized numerical model and a measurement vector, that is, the amount of measurement data. Flow variables on a large number of mesh points are hardly defined by spatiotemporally limited measurements, which poses an underdetermined problem. In this study we conduct the sensitivity analysis of two and threedimensional vortical flow fields within a framework of data assimilation. The impact of measurement strategy, which is evaluated by the sensitivity of the 4DVar cost function with respect to measurements, is investigated to effectively determine a flow field by limited measurements. The assimilation experiment shows that the error defined by the difference between the reference and assimilated flow fields is reduced by using the sensitivity information to locate the limited number of measurement points. To conduct data assimilation for a long time period, the 4DVar data assimilation and the sensitivity analysis are repeated with a short assimilation window.
The use of measurement data to improve a numerical prediction is known as a data assimilation method in meteorological and oceanographic communities [
The present authors have been studying the applicability of data assimilation methods in aeronautical researches. Numerical simulations of atmospheric turbulences such as clear air turbulence and aircraft wake turbulence were performed by the 4DVar method coupled with aeronautical computational fluid dynamics (CFD) codes [
Predictability in systems with the large degree of freedom is a topic of concern in numerical weather prediction (NWP) in conjunction with data assimilation [
The idea of targeted observation has been studied in meteorological community to improve the NWP. The targeted observation has a large potential to use recently available measurements from aircraft (the Aircraft Meteorological DAta Relay, AMDAR) and unmanned aerial vehicles (UAVs) which can be considered as “flying sensors.” One of the major approach of the sensitivity analysis aiming for targeted observations is an adjointbased method because the major operational weather centers employ the adjointbased 4DVar data assimilation system [
The present study is an attempt to investigate the impact of measurement strategy in a data assimilation system by using a sensitivity analysis method. We refer to the preceding work of the sensitivity analysis conducted by Daescu and Navon [
For flow simulation we employ incompressible NavierStokes equations:
The equations are discretized by the fully conservative fourthorder central difference scheme [
The objective of the 4DVar data assimilation is to obtain an initial flow state which reproduces corresponding measurements during a certain time period (assimilation window) [
Schematic of data assimilation based on the 4DVar data assimilation.
The difference between measurements (usually measurements have spatiotemporally less information compared to a prediction model) and corresponding numerical results evaluated by conducting the numerical simulation over a period of time is represented as a cost function with respect to an initial flow state
To obtain the gradient of
In this study we investigate the impact of measurements on a retrieved flow field within a framework of the 4DVar data assimilation. We refer to the preceding work of Daescu [
Using (
The development of linear and adjoint codes can be done by stepbystep processes in the following way. First the derived linear code is checked by
The validation of the adjoint code based on small state perturbations.





















Strong scaling of the gradient computation.
Number of processors  Wallclock time [sec] 

8  1476.56 
16  638.72 
32  277.10 
64  184.39 
128  93.93 
256  109.62 
512  42.15 
We consider a flow field defined by a vortex pair in two and threedimensional computational domains. The computational setting of the twodimensional case is as follows. A flow field is defined by a pair of LambOseen vortices which are characterized by vortex circulation
Computational domain for the twodimensional case, where mesh lines and initial vortex positions of the reference flow field are shown.
Computational domain for the threedimensional case, where the vortex position is shown by the isosurface of vorticity magnitude. (a) A reference flow field with a sinusoidal perturbation of the vortex positions along vortex centerline and (b) an initial flow field before data assimilation.
In the twodimensional case, a numerical experiment is conducted first by generating a reference flow field starting from the conditions mentioned above. From the reference flow field, we acquire pseudomeasurements based on the following strategies; that is, velocity components of all mesh points are extracted as measurements (referred to as 2DF), velocity components on every second mesh point in both
Another procedure of the numerical experiment is considered in two and threedimensional vortex evolutions for a long time period of one vortex reference time
Figure
Cost function and global error with different number of measurement points.
Figure
Cost function and global error with adaptive measurement based on the measurement sensitivity.
Figure
Distribution of measurement points with adaptive measurement at (a) 1st, (b) 6th, and (c) 16th 4DVar iteration in 2DHA, where the measurement points are colored by the magnitude of the measurement sensitivity at those locations.
Distribution of measurement points with adaptive measurement at (a) 1st, (b) 6th, and (c) 16th 4DVar iteration in 2DQA, where the measurement points are colored by the magnitude of the measurement sensitivity at those locations.
In this section, we again consider a twodimensional case, but for a longer time evolution considering multiple assimilation windows during that period. Figure
Cost function and global error with and without adaptive measurement in the twodimensional cases for a long time period.
The 2DLF shows a drastic reduction of the cost function, where the values of the cost function and the global error are the same in this case. The 2DLHA shows the degraded reduction of both cost function and global error. Even so, the global error keeps deceasing during the evolution of a vortex pair. 2DLHA indicates the improved error reduction compared to 2DLH.
Twodimensional evolution of a vortex pair is well evaluated by the evolution of vortex positions because a vortex pair descends due to selfinduced velocity; therefore, the velocity distribution needs to be correctly reproduced to obtain correct vortex positions in time. Figure
Time history of vertical positions of a descending vortex pair.
Figure
Distribution of measurement points with adaptive measurement at (a)
In this section, we consider the threedimensional evolution of a vortex pair. As in the previous section, we use an iterative procedure for a long time evolution of one vortex reference time. Figure
Cost function and global error with and without adaptive measurement in the threedimensional cases.
Figure
The evolution of the reference flow field shown by the isosurface of vorticity magnitude at (a)
Figure
Distribution of measurement points with adaptive measurement at (a)
In this study we conducted the sensitivity analysis of a vortical flow field within a framework of the 4DVar data assimilation. The idea of adaptive/targeted observation in meteorology which aims to effectively determine a flow state by limited measurements was employed in fluid dynamic problems where unsteady vortical flows of much smaller scales are of interest. We firstly investigated a twodimensional flow field defined by a pair of vortices which descends due to selfinduced advection velocity. The amount of measurement points affected the convergence of the cost function as well as that of the global error against the reference flow field. The measurement strategy based on the measurement sensitivity effectively redistributed measurement points near vortices. This resulted in the further reduction of the global error. Then, the same configuration with a longer time period was investigated by repeating 4DVar data assimilation with a short assimilation window, where the impact of adaptive measurement was also confirmed. In the threedimensional configuration of a vortex pair, the reproduction of the flow field becomes difficult because there exist threedimensional mechanisms leading to flow instabilities such as the Crow instability. The redistribution of measurement points based on the measurement sensitivity successfully reduced the global error against the reference flow field. During the data assimilation iterations, the measurement points were focused near the position of the vortex reconnection which helps to reproduce the onset of the vortex reconnection.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the anonymous reviewer for the constructive comments, which helped us to improve the paper. The authors also thank the Advanced Fluid Information Research Center at Institute of Fluid Science, Tohoku University, for the computational resources used in this study.