Shear Flow Instability over a Finite Time Interval

M. V. Kalashnik; Калашник М. В.

doi:10.31857/S0002351523020037

Shear Flow Instability over a Finite Time Interval

Authors: Kalashnik M.V.¹^,2^,3
Affiliations:
1. Obukhov Institute of Atmospheric Physics RAS
2. Institute of Physics of the Earth. O.Yu. Schmidt RAS
3. Research and Production Association Typhoon
Issue: Vol 59, No 2 (2023)
Pages: 165-172
Section: Articles
URL: https://journals.rcsi.science/0002-3515/article/view/136907
DOI: https://doi.org/10.31857/S0002351523020037
EDN: https://elibrary.ru/HOZDEJ
ID: 136907

Cite item

Full Text

Open Access
Restricted Access

Access granted
Restricted Access

Subscription Access

Abstract
About the authors
References
Supplementary files
Statistics

Abstract

Within the framework of a discrete quasi-geostrophic model with two vertical levels, the problem of linear stability of the flow of a stratified rotating fluid with constant vertical and horizontal velocity shifts is solved. It is shown that taking into account the horizontal shear leads to a qualitative change in the dynamics of unstable wave disturbances. The main feature is related to the effect of temporary exponential growth of unstable perturbations, i.e. growth over a finite time period. This effect manifests itself in the alternation of stages of smooth oscillating behavior (in time) with stages of exponential (explosive) growth of finite duration. A kinematic interpretation of the effect of temporal exponential growth is given, which is associated with the passage of a time-dependent perturbation wave vector through the region of exponential instability that exists in the absence of a horizontal shear. It is shown that mathematically this effect is described by solutions of a second-order differential equation containing turning points. Asymptotic solutions of the equation are given for weak horizontal shifts.

Keywords

baroclinic instability, horizontal and vertical velocity shear, growth rate, unstable modes

About the authors

M. V. Kalashnik

Obukhov Institute of Atmospheric Physics RAS; Institute of Physics of the Earth. O.Yu. Schmidt RAS; Research and Production Association Typhoon

Author for correspondence.
Email: kalashnik-obn@mail.ru
Russia, 119017, Moscow, Pyzhevsky per., 3,; Russia, 123242, Moscow, Bolshaya Gruzinskaya str., 10; Russia, 249038, Kaluga obl., Obninsk, Pobedy str., 4,

References

Гилл А. Динамика атмосферы и океана. М.: Мир, 1986. Т. 2. 415 с.
Калашник М.В. Линейная динамика волн Иди в присутствии горизонтального сдвига // Известия РАН. Физика атмосферы и океана. 2009. Т. 45. № 6. С. 764–774.
Калашник М.B., Курганский М.В., Чхетиани О.Г. Бароклинная неустойчивость в геофизической гидродинамике // Успехи физических наук. 2022. Т. 192. № 10. С. 1110–1144. https://doi.org/10.3367/UFNr.2021.08.039046
Кочин Н.Е., Кибель Н.А., Розе Н.В. Теоретическая гидромеханика. Ч. 1. М.: Физматгиз, 1963. 530 с.
Ламб Г. Гидродинамика. Л.: Гостехиздат, 1947. 1084 с.
Лайтхилл Дж. Волны в жидкостях. М.: Мир, 1981. 600 с.
Ле Блон П., Майсек Л. Волны в океане. М.: Мир, 1981. Т. 1. 480 с., Т. 2. 365 с.
Монин А.С. Теоретические основы геофизической гидродинамики. Л.: Гидрометеоиздат, 1988. 433 с.
Найфэ А. Методы возмущений. М.: Мир, 1976. 456 с.
Уизем Дж. Линейные и нелинейные волны. М.: Мир, 1977. 622 с.
Фабер Т.Е. Гидроаэродинамика. М.: Постмаркет, 2001. 560 с.
Barcilon A., Bishop C.H. Nonmodal development of baroclinic waves undergoing horizontal shear deformation // J. Atmos. Sci. 1998. V. 55. № 4. P. 3583–3597.
Bishop C.H. On the behavior of baroclinic waves undergoing horizontal shear deformation I: The “RT” phase diagram // Quart. J. Roy. Met. Soc. 1993. V. 119. P. 221–240.
Chagelishvili G.D., Khujadze G.R., Lominadze J.G., Rogava A.D. Acoustic Waves in Unbounded Shear Flows // Physics of Fluids. 1997. V. 9. P. 1955–1965.
Chagelishvili G.D., Tevzadze A.G., Bodo G., Moiseev S.S. Linear mechanism of wave emergence from vortices in smooth shear flows // Phys. Rev. Letters. 1997. V. 79. № 17. P 3178–3181.
Eady E.T. Long waves and cyclone waves // Tellus. 1949. № 1(3). P. 33–52.
Held I.M., Pierrehumbert R.T., Garner S.T., Swanson K.L. Surface quasi-geostrophic dynamics // J. Fluid Mech. 1995. V. 282. P. 1–20.
Kalashnik M.V., Chkhetiani O.G., Kurgansky M.V. Discrete SQG models with two boundaries and baroclinic instability of jet flows // Phys. Fluids. 2021. V. 33. P. 076608. https://doi.org/ Submitted: 14 May 2021. Accepted: 04 July 2021. Published Online: 20 July 2021.https://doi.org/10.1063/5.0056785
Kalashnik M.V., Mamatsashvili G.R., Chagelishvili G.D., Lominadze J.G. Linear dynamics of non-symmetric perturbations in geostrophic flows with a constant horizontal shear // Quart. J. Roy. Met. Soc. 2006. V. 132. № 615. P. 505–518.
Pedlosky J., Geophysical Fluid Dynamics (Springer-Verlag, Berlin/New York, 1987). P. 710.
Phillips N.A. Energy transformation and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model // Tellus. 1954. V. 6. P. 273–283.