are trapped by the QG component; dispersion of the inertial oscillations packet
occurs on times t ∼ ðfRo
2
Þ
− 1 . Without the TA the “non-traditional” terms in the
amplitude equation result in a meridional dispersion of the inertial oscillations on
much shorter times t ∼ ðfRoÞ
− 1 and in doing so the terms provide an effective
energy radiation from the initial perturbation domain.
Another feasible effect of the “non-traditional” terms is energy exchange
between the slow QG component and inertial oscillations in the homogeneous layer
at times Oðf
− 1 Ro
− 2
Þ. The possibility of such exchange was demonstrated in the
barotropic case. In the case of SNS fluid, one can speculate that the QG energy is
transferred from the stratified layer to the homogeneous one and then to the fast
inertial oscillations. It would be useful to elucidate the existence and efficiency of
this mechanism using a non-hydrostatic numerical model without the TA.
Inertial oscillations with horizontal scale L ≤ L R cannot exist in the stratified
upper layer. To prevent their penetration from the lower layer to the upper one, a
non-stationary boundary layer develops in the upper layer near the interface at large
times. The boundary layer is a result of joint impact of internal modes with large
vertical wavenumbers whose frequencies are close to the inertial frequency f. Thus
the near interface domain is characterized by large vertical gradients of the horizontal velocities that can result in strong mixing and instability here.
In geostrophic adjustment with gravity waves (surface or internal) the inertial
oscillations arise only if the dominating scale L of the initial perturbation exceeds
the corresponding Rossby scale L R , i.e., L ≫ L R (cf. [24, 29]). In the presence of
gyroscopic waves the “shorter” inertial oscillations with the scales H ≪ L ≤ L R are
possible. The significant vertical velocities of the near-inertial oscillations observed
by van Haren and Millot [27] in the practically barotropic deep Western Mediterranean Sea can be related to this property of the gyroscopic waves. We note that
some other regions of the deep ocean are also characterized by very weak stratification, as for example, the Canada Basin in the Arctic Ocean [26], or the Pacific
Ocean near 179° E [6].
Acknowledgements This work was supported by the Russian Science Foundation grant no.
14-50-00095 (section “Barotropic Model”), the Ministry of Education and Science of Russian
Federation grant no. 14.W03.31.0006 (section “Stably-Neutrally Stratified Fluid”), and the
Russian Foundation for Basic Research grant no. 17-05-00094 (analysis of slow evolution).
References
1. Aref, H. (1984). Stirring by chaotic advection. Journal of Fluid Mechanics, 143, 1–21.
2. Balmforth, N. J., Llewellyn Smith, S. G., & Young, W. R. (1998). Enhanced dispersion of
near-inertial waves in an idealized geostrophic flow. Journal of Marine Research, 56, 1–40.
3. Balmforth, N. J., & Young, W. R. (1999). Radiative damping of near-inertial oscillations in
the mixed layer. Journal of Marine Research, 57, 561–584.
4. Brekhovskikh, L. M., & Goncharov, V. (1994). Mechanics of continua and wave dynamics.
Berlin: Springer.
330
G. M. Reznik
occurs on times t ∼ ðfRo
2
Þ
− 1 . Without the TA the “non-traditional” terms in the
amplitude equation result in a meridional dispersion of the inertial oscillations on
much shorter times t ∼ ðfRoÞ
− 1 and in doing so the terms provide an effective
energy radiation from the initial perturbation domain.
Another feasible effect of the “non-traditional” terms is energy exchange
between the slow QG component and inertial oscillations in the homogeneous layer
at times Oðf
− 1 Ro
− 2
Þ. The possibility of such exchange was demonstrated in the
barotropic case. In the case of SNS fluid, one can speculate that the QG energy is
transferred from the stratified layer to the homogeneous one and then to the fast
inertial oscillations. It would be useful to elucidate the existence and efficiency of
this mechanism using a non-hydrostatic numerical model without the TA.
Inertial oscillations with horizontal scale L ≤ L R cannot exist in the stratified
upper layer. To prevent their penetration from the lower layer to the upper one, a
non-stationary boundary layer develops in the upper layer near the interface at large
times. The boundary layer is a result of joint impact of internal modes with large
vertical wavenumbers whose frequencies are close to the inertial frequency f. Thus
the near interface domain is characterized by large vertical gradients of the horizontal velocities that can result in strong mixing and instability here.
In geostrophic adjustment with gravity waves (surface or internal) the inertial
oscillations arise only if the dominating scale L of the initial perturbation exceeds
the corresponding Rossby scale L R , i.e., L ≫ L R (cf. [24, 29]). In the presence of
gyroscopic waves the “shorter” inertial oscillations with the scales H ≪ L ≤ L R are
possible. The significant vertical velocities of the near-inertial oscillations observed
by van Haren and Millot [27] in the practically barotropic deep Western Mediterranean Sea can be related to this property of the gyroscopic waves. We note that
some other regions of the deep ocean are also characterized by very weak stratification, as for example, the Canada Basin in the Arctic Ocean [26], or the Pacific
Ocean near 179° E [6].
Acknowledgements This work was supported by the Russian Science Foundation grant no.
14-50-00095 (section “Barotropic Model”), the Ministry of Education and Science of Russian
Federation grant no. 14.W03.31.0006 (section “Stably-Neutrally Stratified Fluid”), and the
Russian Foundation for Basic Research grant no. 17-05-00094 (analysis of slow evolution).
References
1. Aref, H. (1984). Stirring by chaotic advection. Journal of Fluid Mechanics, 143, 1–21.
2. Balmforth, N. J., Llewellyn Smith, S. G., & Young, W. R. (1998). Enhanced dispersion of
near-inertial waves in an idealized geostrophic flow. Journal of Marine Research, 56, 1–40.
3. Balmforth, N. J., & Young, W. R. (1999). Radiative damping of near-inertial oscillations in
the mixed layer. Journal of Marine Research, 57, 561–584.
4. Brekhovskikh, L. M., & Goncharov, V. (1994). Mechanics of continua and wave dynamics.
Berlin: Springer.
330
G. M. Reznik