Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics. (English) Zbl 1222.86003
Summary: We show that Lie group analysis of differential equations provides the exact solutions of two-dimensional stratified rotating Boussinesq equations which are a basic model in geophysical fluid dynamics. The exact solutions are obtained as group invariant solutions corresponding to the translation and dilation generators of the group of transformations admitted by the equations. The comparison with the previous analytic studies and experimental observations confirms that the anisotropic nature of the wave motion allows to associate these invariant solutions with uni-directional internal wave beams propagating through the medium. It is also shown that the direction of internal wave beam propagation is in the transverse direction to one of the invariants which corresponds to a linear combination of the translation symmetries. Furthermore, the amplitudes of a linear superposition of wave-like invariant solutions forming the internal gravity wave beams are arbitrary functions of that invariant. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams that have different general profiles along the obtained invariant and propagating in the transverse direction are considered. The behavior of the invariant solutions near the critical latitude is illustrated.
MSC:
| 86A05 | Hydrology, hydrography, oceanography |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76B55 | Internal waves for incompressible inviscid fluids |
| 76B70 | Stratification effects in inviscid fluids |
| 76E20 | Stability and instability of geophysical and astrophysical flows |
References:
| [1] | Bell, T. H., Lee waves in stratified flows with simple harmonic time dependence, J Fluid Mech, 67, 705-722 (1975) · Zbl 0296.76010 |
| [2] | Bluman, G. W.; Anco, S. C., Symmetry and integration methods for differential equations (2002), Springer: Springer New York · Zbl 1013.34004 |
| [3] | Bluman, G. W.; Kumei, S., Symmetries and differential equations (1989), Springer: Springer New York · Zbl 0698.35001 |
| [4] | Cantwell, B. J., Introduction to symmetry analysis (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1082.34001 |
| [5] | Cohen, A., An introduction to the Lie theory of one-parameter groups with applications to the solution of differential equations (1911), D.C. Heath: D.C. Heath New York · JFM 42.0708.02 |
| [6] | Dalziel, S. B.; Hughes, G. O.; Sutherland, B. R., Whole field density measurements by synthetic schlieren, Exp Fluids, 28, 322-337 (2000) |
| [7] | Dauxois, T.; Young, W. R., Near-critical reflection of internal waves, J Fluid Mech, 390, 271-295 (1999) · Zbl 0972.76018 |
| [8] | Dickson, L. E., Differential equations from the group standpoint, Ann Math, 25, 287 (1924) · JFM 50.0297.01 |
| [9] | (Ibragimov, N. H., CRC handbook of lie group analysis of differential equations. CRC handbook of lie group analysis of differential equations, Applications in engineering and physical sciences, vol. 2 (1995), CRC Press: CRC Press Boca Raton (FL)) · Zbl 0864.35002 |
| [10] | (Ibragimov, N. H., CRC handbook of lie group analysis of differential equations. CRC handbook of lie group analysis of differential equations, New trends in theoretical development and computational methods, vol. 3 (1996), CRC Press: CRC Press Boca Raton (FL)) · Zbl 0864.35003 |
| [11] | Ibragimov NH. Transformation groups applied to mathematical physics. Moscow: Nauka; 1983. English transl., Dordrecht: Reidel; 1985.; Ibragimov NH. Transformation groups applied to mathematical physics. Moscow: Nauka; 1983. English transl., Dordrecht: Reidel; 1985. · Zbl 0529.53014 |
| [12] | Ibragimov, N. H., Elementary Lie group analysis and ordinary differential equations (1999), Wiley: Wiley Chichester · Zbl 1047.34001 |
| [13] | Ibragimov, R. N., Resonant triad model for studying evolution of the energy spectrum among a large number of internal waves, Commun Nonlinear Sci Numer Simul, 13, 593-623 (2008) · Zbl 1155.35412 |
| [14] | Ibragimov, R. N., Oscillatory nature and dissipation of the internal waves energy spectrum in the deep ocean, Eur Phys J, 40, 315-334 (2007) |
| [15] | Ibragimov, R. N., Generation of internal tides by an oscillating background flow along a corrugated slope, Phys Scripta, 78, 065801 (2008) |
| [16] | Ibragimov, R. N., Stationary surface waves in a circular liquid layer with constant gravity, Quest Math, 23, 1, 1-12 (2000) · Zbl 0969.76011 |
| [17] | Ibragimov, R. N., Shallow water theory and solution of the problem on the atmospheric motion around a celestial body, Phys Scripta, 61, 391-395 (2000) · Zbl 1063.85502 |
| [18] | Javam, A.; Imberger, J.; Armfield, S. W., Numerical study of internal wave-wave interactions in a stratified fluid, J Fluid Mech, 415, 65-87 (2000) · Zbl 0986.76015 |
| [19] | Kistovich, A. V.; Chashechkin, Y. D., Nonlinear interactions of two dimensional packets of monochromatic internal waves, Izv Atmos Ocean Phys, 27, 12, 946-951 (1991) |
| [20] | Khatiwala, S., Generation of internal tides in an ocean of finite depth: analytical and numerical calculations, Deep-See Res I, 50, 3-21 (2003) |
| [21] | Lam, F. P.; Mass, L. M.; Gerkema, T., Spatial structure of tidal and residual currents as observed over the shelf break in the Bay of Biscay, Deep-See Res I, 51, 10751096 (2004) |
| [22] | Lie S. Vorlesungen uber Differentialgleichungen Mit Bekannten Infinitesimalen Transformationen. (Bearbeited und herausgegeben von Dr. G. Scheffers), B.G. Teubner, Leizig. 1891.; Lie S. Vorlesungen uber Differentialgleichungen Mit Bekannten Infinitesimalen Transformationen. (Bearbeited und herausgegeben von Dr. G. Scheffers), B.G. Teubner, Leizig. 1891. |
| [23] | Lombard, P. N.; Riley, J., On the breakdown into turbulence of propagating internal waves, Dyn Atmos Oceans, 23, 345-355 (1996) |
| [24] | Nash, J. D.; Kunze, E.; Lee, C. M.; Sanford, T. B., Structure of the baroclinic tide generated at Keana Ridge, Hawaii, J Phys Oceanogr, 36, 1123-1135 (2006) |
| [25] | Nigam, S.; Held, I., The influence of a critical latitude on topographically forced stationary waves in a barotropic model, J Atmos Sci, 40, 554-571 (1983) |
| [26] | Olver, P. J., Applications of Lie groups to differential equations (1986), Springer: Springer New York · Zbl 0588.22001 |
| [27] | Ovsyannikov, L. V., Group properties of differential equations Siberian branch (1962), USSR Academy of Sciences: USSR Academy of Sciences Novosibirsk |
| [28] | Ovsyannikov LV. Group analysis of differential equations. Moscow: Nauka; 1978. English transl., Ames WF, editor. New York: Academic Press; 1982.; Ovsyannikov LV. Group analysis of differential equations. Moscow: Nauka; 1978. English transl., Ames WF, editor. New York: Academic Press; 1982. · Zbl 0484.58001 |
| [29] | Stephani, H., Differential equations: their solution using symmetries (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0704.34001 |
| [30] | Tabaei, A.; Akylas, T. R., Nonlinear internal gravity wave beams, J Fluid Mech, 482, 141-161 (2003) · Zbl 1057.76010 |
| [31] | Tabaei, A.; Akylas, T. R.; Lamb, K., Nonlinear effects in reflecting and colliding internal wave beams, J Fluid Mech, 526, 217-243 (2005) · Zbl 1065.76034 |
| [32] | Teoh, S. G.; Imberger, J.; Ivey, G. N., Laboratory study of the interactions between two internal wave rays, J Fluid Mech, 336, 91 (1997) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
