  We analyse the limit of stable solutions to the Ginzburg-Landau (GL)
equations when $\varepsilon$, the inverse of the GL parameter, goes to zero and
in 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 do
that we pass to the limit in the second inner variation of the GL energy. The
main difficulty is to understand the convergence of quadratic terms involving
derivatives of functions converging only weakly in $H^1$. We use an assumption
of convergence of energies, the limiting criticality conditions obtained by
Sandier-Serfaty by passing to the limit in the first inner variation and
properties of limiting vorticities to find the limit of all the desired
quadratic terms. At last we investigate the limiting stability condition we
have obtained. In the case with magnetic field we study an example of an
admissible 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 limiting
stability condition whereas when $L$ is large enough it stops verifying that
condition. In the case without magnetic field we use a result of
Iwaniec-Onninen to prove that every measure in $H^{-1}(\Omega)$ satisfying the
first order limiting criticality condition also verifies the second order
limiting stability condition.
