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I was reading an article which states :

The linear-stability analysis for this system can be performed in complete generality; but it will be best for purposes of this review to go directly to what is called the "quasistationary" approximation. We are looking for a linear equation of motion for the interfacial position, $z (\text{interface}) = \zeta(x,t)$

Can anyone explain what does the author mean by "quasistationary approximation".

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There are systems where distinct, interacting processes take place simultaneously, which can be challenging to study. When those processes, though, take place at different enough time scales, a possible approach is treat the slow process as stationary, and (at least at first) consider only the dynamics of the fast process: that's the quasistationary / quasistatic approximation.

This approximation is very common in thermodynamics, where the changes under consideration in the system (say, the melting of ice) are usually supposed to be slow compared to the thermalization time scale - allowing one to consider that the system changes from equilibrium state to equilibrium state as it evolves in time.

In the paper cited in the OP, the specific form of quasistationary approximation that is used is made explicit a few lines bellow the quotation:

it turns out that the interface moves so slowly that it remains effectively stationary during the time needed for relaxation of the diffusion field. Thus it seems reasonable to solve the problem approximately by, first, solving the time-independent diffusion equation (3.6) subject to the thermodynamic boundary condition (3.4) on the quasistationary interface $\zeta(x, t)$, and then inserting this result into the continuity condition (3.2) to find an explicit expression for $\partial \zeta /\partial t$.

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