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1-D viscous Burgers' equation

We have then considered a very simple nonlinear geophysical model. The evolution model is the viscous Burgers' equation over a one-dimensional cyclic domain:

$\displaystyle \frac{\partial X}{\partial t} + \frac{1}{2} \frac{\partial (X^2)}{\partial s} - \nu \frac{\partial^2X}{\partial s^2} = 0,$ (3.14)

where $ X$ is the state variable, $ s$ represents the distance in meters around the $ 45^o$ N constant-latitude circle, and $ t$ is the time. The sampling of the observations provide a spatial and temporal density similar to the longitudinal distribution of the mid-latitude radiosonde network. The period of the domain, the diffusion coefficient, and the length of the assimilation period also make the situation as realistic as possible.

Note that this system is nonlinear and the viscosity makes it irreversible. However, it is possible to stabilize the backward resolution with the nudging term. The numerical and convergence results, as well as the comparison with the variational scheme, are detailed in [21]. Some other numerical experiments and comparisons are detailed in [19] in a slightly different situation (i.e. different physical parameters).

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