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We analyse the limit of stable solutions to the Ginzburg-Landau (GL)equations when $\varepsilon$, the inverse of the GL parameter, goes to zero andin a regime where the applied magnetic field is of order $|\log \varepsilon |$whereas the total energy is of order $|\log \varepsilon|^2$. In order to dothat we pass to the limit in the second inner variation of the GL energy. Themain difficulty is to understand the convergence of quadratic terms involvingderivatives of functions converging only weakly in $H^1$. We use an assumptionof convergence of energies, the limiting criticality conditions obtained bySandier-Serfaty by passing to the limit in the first inner variation andproperties of limiting vorticities to find the limit of all the desiredquadratic terms. At last we investigate the limiting stability condition wehave obtained. In the case with magnetic field we study an example of anadmissible limiting vorticity supported on a line in a square $\Omega=(-L,L)^2$and show that if $L$ is small enough this vorticiy satisfies the limitingstability condition whereas when $L$ is large enough it stops verifying thatcondition. In the case without magnetic field we use a result ofIwaniec-Onninen to prove that every measure in $H^{-1}(\Omega)$ satisfying thefirst order limiting criticality condition also verifies the second orderlimiting stability condition.

Stability conditions for mean-field limiting vorticities of the Ginzburg-Landau equations in 2D