Internal waves under an ice cover

Abstract

A theoretical model for propagation of internal waves under an ice cover is developed. The sea water is considered to be inviscid, non-rotating, and incompressible and the Brunt-Väisälä frequency is supposed to be constant. The ice is considered of uniform thickness, with constant values of Young's modulus, Poisson's ratio, density and compressive stress in the ice. The boundary conditions are such that the normal velocity at the bottom is zero and, at the undersurface of the ice, the linearized kinematical and dynamic boundary conditions are satisfied. We present and analyze explicit solutions for the internal waves under the ice cover and the dispersion equations. It is shown that when the frequency is near, but smaller than the Brunt-Väisälä frequency the ice deflections can be considerable. The theoretical results are compared with experimental data for the Arctic regions. Keywords : ice deflections; internal waves; Brunt-Väisälä frequency 1 Introduction The influence of the ice cover on the propagation of internal waves is practically important, but poorly studied. It is a common assumption that the rigid lid approximation that filters out the surface mode and adequately describes the properties of internal waves should also be valid in the case, where the free surface is substituted by an ice cover. Since the vertical velocity at the surface is zero in the approximation, the internal waves cannot be recorded on the basis of fluctuations of the ice cover. Such a conclusion, however, contradicts the data from previous observations [Czipott et al. (1991); Smirnov (1972); Smirnov, Savchenko (1972); Smirnov et al (2002)]. For example, oscillations of the ice cover recorded on the "Severnyi Polyus-20" ice drifting station in 1970 [Smirnov (1972)] cannot be related to surface flexural-gravity waves. They were interpreted as manifestations of internal waves in conditions with clearly manifested interleaving of waters in the Arctic Basin. Therefore, it is necessary to develop a more specific theory of internal waves applied to the waves in a stratified ocean covered with ice. In this paper we develop a theory of internal waves under an ice cover at constant Brunt-Väisälä frequency. 2 Formulation of the problem Let us consider water motion under an ice cover, which is considered to be a thin elastic plate of constant thickness floating at the sea surface. We will consider wave motions as small deviations from the hydrostatic equilibrium state at zero background currents in the ocean. Then, the linearized system of equations of motion for non-rotating stratified fluid is written as [Pedlosky (2003)]: 0 1 P t       u ; 0

Surface manifestations of the waves in the ocean covered with ice

“…Measurements of internal waves in a shallow fjord in Spitsbergen showed that fluctuations of temperature and velocity with a period of approximately 10 minutes and amplitude of internal waves about 1 m correlate with the fluctuations of the ice cover of the same period with an amplitude of a 5-8 millimeters [Marchenko et al, 2011]. These results agree with the theory [Muzylev, 2008]. One can distinguish a spectral peak at a frequency corresponding to a period of 10 min on the spectrum of bottom pressure measured with an SBE-39 pressure gauge on a submarine slope at a depth of 8 m.…”

Mass transport induced by internal Kelvin waves beneath shore‐fast ice

“…In the present problem we consider tidally generated waves. For the semidiurnal tide M 2 we have ω = 1.4 × 10 −4 s −1 , which is much smaller than the buoyancy frequency for a continuous stratification normally found in Arctic waters, where N may range from 10 −3 to 5 × 10 −2 s −1 , according to Muzylev [2008]. Hence, the rigid‐lid approximation works well for the continuously stratified case.…”

“…The presence of an ice cover will generally affect the propagation of gravity waves (see, e.g., Liu and Mollo ‐ Christensen [1988] in the case of surface waves). For internal waves in a continuously stratified ocean under ice, Muzylev [2008] showed that, because of the elastic properties of the ice, nonzero surface deflections occur for the lowest internal mode if the wave frequency ω is close to the buoyancy frequency N . However, when ω / N ≪ 1, corresponding to long waves, the rigid‐lid approximation (negligible vertical deflection under the ice) is very well fulfilled.…”

Short-Period Internal Waves under an Ice Cover in Van Mijen Fjord, Svalbard

“…In the "rigid lid" approximation, the vertical velocity at the surface is assumed to be equal to zero; therefore, due to the kinematic condition, the internal waves cannot cause any vertical displacements of the ice cover. Such conclusion is, however, inconsistent with the experimental data obtained for relatively deep parts of the Arctic Ocean [3][4][5][6] and also contradicts the theoretical results [7,8], according to which internal waves may be reflected in the fluctuations of the ice cover at frequencies comparable with the Brunt-Väisälä frequency. Scientists from the Shirshov Institute of Oceanology (Russian Academy of Sciences) and The University Centre in Svalbard (UNIS) carried out marine studies in the shallow Van Mijen Fjord in Svalbard to perform experimental tests of the theoretical conclusion concerning the possible influence of short internal waves on the fluctuations of the compact ice cover.…”

“…If the processes inside the ice cover are ignored the main equations and boundary conditions of the ice cover should be similar to the equations and conditions in the situation when the sea surface is free of ice. The only exception is the dynamic condition that may be expressed under the constant ice thickness h in the following form [7][8][9][10][11]:…”