Phase Structure of Internal Gravity Waves in the Ocean with Shear Flows

Abstract

Purpose. The description of the internal gravity waves dynamics in the ocean with background fields of shear currents is a very difficult problem even in the linear approximation. The mathematical problem describing wave dynamics is reduced to the analysis of a system of partial differential equations; and while taking into account the vertical and horizontal inhomogeneity, this system of equations does not allow separation of the variables. Application of various approximations makes it possible to construct analytical solutions for the model distributions of buoyancy frequency and background shear ocean currents. The work is aimed at studying dynamics of internal gravity waves in the ocean with the arbitrary and model distributions of density and background shear currents. Methods and Results. The paper represents the numerical and analytical solutions describing the main phase characteristics of the internal gravity wave fields in the stratified ocean of finite depth, both for arbitrary and model distributions of the buoyancy frequency and the background shear currents. The currents are considered to be stationary and horizontally homogeneous on the assumption that the scale of the currents' horizontal and temporal variability is much larger than the characteristic lengths and periods of internal gravity waves. Having been used, the Fourier method permitted to obtain integral representations of the solutions under the Miles – Howard stability condition is fulfilled. To solve the vertical spectral problem, proposed is the algorithm for calculating the main dispersion dependences that determine the phase characteristics of the generated wave fields. The calculations for one real distribution of buoyancy frequency and shear flow profile are represented. Transformation of the dispersion surfaces and phase structures of the internal gravitational waves’ fields is studied depending on the generation parameters. To solve the problem analytically, constant distribution of the buoyancy frequency and linear dependences of the background shear current on depth were used. For the model distribution of the buoyancy and shear flow frequencies, the explicit analytical expressions describing the solutions of the vertical spectral problem were derived. The numerical and asymptotic solutions for the characteristic oceanic parameters were compared. Conclusions. The obtained results show that the asymptotic constructions using the model dependences of the buoyancy frequency and the background shear velocities’ distribution, describe the numerical solutions of the vertical spectral problem to a good degree of accuracy. The model representations, having been applied for hydrological parameters, make it possible to describe qualitatively correctly the main characteristics of internal gravity waves in the ocean with the arbitrary background shear currents.

Internal ring waves in a three-layer fluid on a current with a constant vertical shear

“…Requiring the determinant of this linear system to vanish, and non-dimensionalising the variables as in section 1, we obtain the long-wave speeds as the roots of the quartic equation for c, which exactly coincides with the equation (10) previously obtained in section 3 as a reduction of the angular adjustment equation (7). For brevity, the same symbols are used for non-dimensional quantities here.…”

“…Unlike the generalised Burns' condition, the form of the equation depends on the choice of stratification and current, and is obtained as part of solution of the spectral problem (modal equations). We note that a useful discussion of the linear internal waves in a general setting can be found in [10]. This angular adjustment equation extends the results obtained in [24] for a two-layer fluid.…”

“…At the transition, the curve H − a 2 intersects the horizontal axis only once at a = a * , and we have the parabolic regime with a ∈ (−∞, a * ]. We observe that the values of γ p1 , γ p2 can be found in explicit form by setting c = 0 in equation (10), which yields a biquadratic equation for γ. Differentiating b 2 = H − a 2 with respect to a and using (11), we obtain…”

“…where the coefficients a 0 , a 2 , a 4 are precisely those found in (8). When γ > 0, the critical values γ − , γ + between which long waves are unstable can be found by computing the roots of the discriminant, with respect to c, to the quartic equation (10). This yields a polynomial equation for γ (of degree 12) whose roots can be computed numerically.…”