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I'm reading this paper, where atop pg 3, the author mentions the "quasi-static" assumption for earth displacements which leads to the mechanical deformation of a fluid-solid system to be governed by:

$$\nabla \sigma + \rho_h g = 0$$.

What exactly does "quasi-static" mean in this context?

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I believe that the author means that the equation does not describe a dynamical situation. After any change in the system, the density and the stress is assumed to come into equilibrium very fast (compared to the time-scale of the phenomena the author is trying to study). The dynamics of how this equilibrium occurs is neclected.

You could think of a weight suspended on a spring. If mass is added slowly to the weight then it is a good approximation to simply assume that the lenght of the spring is at the changed equilibrium position. Do it quickly however and the lenght will osccilate.

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