# Normal oscillations of ideal stratified fluid with a free surface completely covered with the elastic ice.

Let a rigid immovable vessel be partially ﬁlled with an ideal incompressible stratiﬁed ﬂuid.

We assume that in an equilibrium state the density of a ﬂuid is a function of the vertical variable

$x_3$, i.e., $\rho_0 = \rho_0(x_3)$. In this case the gravitational ﬁeld with constant acceleration $\overrightarrow{g} = −g\overrightarrow{e_3}$ acts

on the ﬂuid, here $g > 0$ and $\overrightarrow{e_3}$ is unit vector of the vertical axis $Ox_3$, which is directed opposite

to $\overrightarrow{g}$. Let $\Omega$ be the domain ﬁlled with a ﬂuid in equilibrium state, $S$ be rigid wall of the vessel

adherent to the ﬂuid, Γ be a free surface completely covered with the elastic ice.

Let us consider the basic case of stable stratiﬁcation of the ﬂuid on density:

$$ 0 < N_{min}^2 \leq N^2(x_3) \leq N_{max}^2 =: N_0^2 < \infty, $$

$$N^2(x_3) := − \frac{g\rho_0^{\prime}(x_3)}{\rho_0(x_3)},\ \rho_0(0) > 0,$$

where $N^2(x_3)$ is square frequency of buoyancy.

The initial boundary value problem is reduced to a Cauchy problem

$$ \mathscr{A} \frac{d^2x}{dt^2} + \mathscr{C} x = f(t), x(0) = x^0,\ x^{\prime}(0) = x^1,$$

$$ 0 \ll \mathscr{A} = \mathscr{A}^∗ \in \mathscr{L(H ),\ C = C^∗} \geq 0. $$

in some Hilbert space $\mathscr{H}$ .

The spectrum of normal oscillations, basic properties of eigenfunctions and other questions

are studied.

**Keywords:** stratiﬁcation eﬀect in ideal ﬂuids, diﬀerential equation in Hilbert space, normal

oscillations, spectral problem, eigenvalues, Riesz basis.