Stability analysis of multi-layer film flow on an inclined plane is technologically very important in the making of products such as photosensitive materials which requires multilayers of thin films. In this work, stability analysis of one layer, two layer and generally multi-layer (n layer) flow on an inclined plane is made by solving a set of Orr-Sommerfeld equations as eigenvalue problem with respect to wave speed c and wave number a. Analysis is made, not under the assumption that wavelength is long, but that it can be any value. As for numerical method, Galerkin finite element is used. In the method, all the eigenvalues can be calculated, and the shape of eigen function which corresponds to an eigenvalue can also be calculated. As for instability modes, there are three modes, namely, surface mode, interface mode and shear mode. Surface mode is the one which is related with free surface and strongly affected by surface tension. Interface mode is the one which is related with liquid-liquid interface and is the most dominant mode when interfacial tension is very small or negligible. Shear mode is the one which is caused by shear stress when Reynolds number is large. As for surface related modes, further investigation shows that there are two types. Namely, one is a mode that propagates downward and has been studied in previous works elsewhere. The other is a mode which propagates upward if wave number a is large (namely, wavelength is short). These two modes change from shallow water wave to deep water wave and finally to capillary wave, as wave number a increases.
Data assimilation has been recently considered as a key and essential component of understanding of oceanic phenomena, development of parameterization process in numerical ocean modeling, mapping out of observation strategies and forecasting of ocean state. Variational adjoint method is introduced as one of the data assimilation method in this introductory review. The basic concept of the variational adjoint method is to combine observations via variational method (estimation theory or Lagrange's multiplier method) with a dynamical model. The method is applied to a parameter estimation in a simple model of an unsteady Ekman flow in physical oceanography. Recent developments of the method are also introduced briefly.