Adaptive Grid Generation in Inverse Atmospheric Transport Problems
Introduction
Inverse atmospheric modeling (often referred to as adjoint modeling) plays a crucial role in atmospheric sciences. In complex modeling environments, adjoint models allow for accurate data assimilation. In contaminant dispersion modeling inverse methods allow for back tracking of measured constituents to their sources.
Idea
Adaptive mesh refinement allows for high local resolution. Therefore, we anticipate a good representation of point sources or finescale phenomena (like filamentation or mixing/stirring phenomena), when combining adaptive mesh refinement with inverse modeling techniques.
Example
In order to give a first simple example, we consider the (linear) advection equation and derive the adjoint. It turns out that the adjoint of the advection operator is the advection operator with inverted wind component:
 start from the advection operator: %\[ \mathcal{D}(\cdot):= \partial_t(\cdot)+\nabla\cdot(\mathbf{v}(\cdot)), \]%
 now, the adjoint formulation: %\[ \langle \mathcal{D}\phi, \psi \rangle = \langle \phi, \mathcal{D}^\ast\psi \rangle, \]%
 finally, the resulting adjoint operator: %\[ \mathcal{D}^\ast(\cdot)= \partial_t(\cdot) \mathbf{v}\cdot\nabla(\cdot). \]%
Thus, starting from a given distribution of an atmospheric tracer, we can now compute the source of its emmission. The pictures below, show the initial distribution and the corresponding source (the TUM logo) together with the corresponding adaptively refined meshes.
Figures
 Figure 1  The wind field

 Figure 2  The initial situation (measured tracer distribution)

 Figure 3  The final situation (source of tracer emmission)

Animation
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