| We have applied two supplementary methods to solve the linearized problem of 3D disturbances generation in a viscous exponentially stratified liquid by a oscillating surface taking into account diffusivity effects. The source of disturbances is a part of vertical circular cylinder performing infinitesimal vertical or torsion periodic oscillations. We study the problem with both large and small Schmidt numbers. To calculate the fluid motion in the body vicinity we have applied the well-known method of singular boundary layers functions. The 3D spectral expansion technique is also used to calculate set of boundary layers on the solid surface and wave beams far from the source. We have found that in the general case produced set motions consist of emitting internal wave beams and three types of boundary layers even if the Schmidt number is equal to unity. The first one is a periodic Stokes boundary layer existing even in a homogeneous fluid. Its thickness is defined by kinematic viscosity and the frequency of the source oscillation. Two other layers, which are named internal viscous and diffusion boundary layers, are characterized by a more complex structure. Their thickness is defined besides intrinsic (Stokes) length scales by a similar scale including diffusivity coefficient, some derivative length scales and geometrical factors. In a limiting case of a weak viscosity and even more weak diffusivity the set of singular components of motion includes two types of split boundary layers. The ratio of their transverse sizes is proportional to the square root from the Schmidt number. The wave part of the solution describes internal gravity waves propagating along a wave cone as in an ideal fluid. To calculate wave fields far from the source, which satisfy right boundary conditions, we have used the spectral expansion method taking into account all set of regular and singular roots of the dispersion relation. The constructed exact solutions also describe the set of the same boundary layers and emitting 3D periodic internal wave beams. The transverse structure of the beam is defined by the ratio of the source size to intrinsic combined internal length scale. If the characteristic size of the generator is smaller than the combined internal length scale the beam is unimodal. In opposite case near the source the wave beam is bimodal. We have calculated the distance from the source where the bimodal beam transforms into the unimodal one. Constructed solutions present in the form adopted for comparison with the laboratory data. Given schlieren images demonstrate sections of wave cones and flows produced separating boundary currents from the periodically oscillating disks and rectangular in a continuously stratified fluid. Results of laboratory experiments a reasonable agreement with the calculations. |
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