unstable mode. Furthermore, velocity in the plume was measured by the PIV and a
theoretical linear stability analysis was performed.
Note that the procedure of measuring mean velocity profiles was different in the
experiments by Bondur et al. [10], Ezhova et al. [20]. Linear stability analysis
requires a basic flow. In the first case, the authors averaged the instantaneous
profiles in the reference frame moving with the plume, while in the second case the
standard averaging procedure was applied in a fixed reference frame. The average
profiles in these reference frames are different. The velocity profiles in a reference
frame moving with the plume resemble “top hat” profiles (see Fig. 5), while the
standard procedure in a fixed reference frame leads to the Gaussian-shaped profiles
(Fig. 9). In the latter case, unstable axisymmetic mode cannot develop in the
quasi-parallel framework, because the necessary condition of instability is not
satisfied for this mode. However, the analysis accounting that the flow is not
parallel shows that the axisymmetric mode is unstable even in the case of the
Gaussian-shaped profiles; addition of a counterflow causes absolute instability
similar to the previous case. Theoretically predicted frequencies of an axisymmetric
mode are close to those experimentally measured and to the frequencies of IW.
Additional confirmation of self-sustained regimes of oscillating fountains in
two-layer stratified fluid was obtained in the numerical simulations by Druzhinin
and Troitskaya [19]. The authors demonstrated that laminar fountains oscillate in a
two-layer stratification, emitting internal waves in the form of concentric circles at
small Froude numbers replaced by spirally propagating internal waves at larger
Froude numbers. The result was explained in the framework of the mode competition (competition of modes developing on the fountain in the pycnocline).
Finally, we comment on the influence of stratification on the generation of
internal waves. Ansong and Sutherland [1] did not find correlation between the
frequency of plume oscillations and internal waves in the experiments with the
Fig. 9 Profiles of the average vertical (left) and radial (right) jet velocity in self-similar
coordinates [20]
Surface Manifestations of Internal Waves Induced …
75
theoretical linear stability analysis was performed.
Note that the procedure of measuring mean velocity profiles was different in the
experiments by Bondur et al. [10], Ezhova et al. [20]. Linear stability analysis
requires a basic flow. In the first case, the authors averaged the instantaneous
profiles in the reference frame moving with the plume, while in the second case the
standard averaging procedure was applied in a fixed reference frame. The average
profiles in these reference frames are different. The velocity profiles in a reference
frame moving with the plume resemble “top hat” profiles (see Fig. 5), while the
standard procedure in a fixed reference frame leads to the Gaussian-shaped profiles
(Fig. 9). In the latter case, unstable axisymmetic mode cannot develop in the
quasi-parallel framework, because the necessary condition of instability is not
satisfied for this mode. However, the analysis accounting that the flow is not
parallel shows that the axisymmetric mode is unstable even in the case of the
Gaussian-shaped profiles; addition of a counterflow causes absolute instability
similar to the previous case. Theoretically predicted frequencies of an axisymmetric
mode are close to those experimentally measured and to the frequencies of IW.
Additional confirmation of self-sustained regimes of oscillating fountains in
two-layer stratified fluid was obtained in the numerical simulations by Druzhinin
and Troitskaya [19]. The authors demonstrated that laminar fountains oscillate in a
two-layer stratification, emitting internal waves in the form of concentric circles at
small Froude numbers replaced by spirally propagating internal waves at larger
Froude numbers. The result was explained in the framework of the mode competition (competition of modes developing on the fountain in the pycnocline).
Finally, we comment on the influence of stratification on the generation of
internal waves. Ansong and Sutherland [1] did not find correlation between the
frequency of plume oscillations and internal waves in the experiments with the
Fig. 9 Profiles of the average vertical (left) and radial (right) jet velocity in self-similar
coordinates [20]
Surface Manifestations of Internal Waves Induced …
75