The Barotropic Instability of the Oceanic Jet Currents

Transcription

1 Hydrology 017; 5(5): doi: /j.hyd ISSN: (Print); ISSN: (Online) The Barotropic Instability of the Oceanic Jet Currents Alexander Alekseyevich Solovyev 1, Ditry Alexandrovich Solovyev, * 1 Departent of Geography, M. V. Loonosov Moscow State University, Moscow, Russia Sea-Air Interaction and Cliate Laboratory, P. P. Shirshov Institute of Oceanology RAS, Moscow, Russia Eail address: geosolgu@ail.ru (A. A. Solovyev), solovev@ocean.ru (D. A. Solovyev) * Corresponding author To cite this article: Alexander Alekseyevich Solovyev, Ditry Alexandrovich Solovyev. The Barotropic Instability of the Oceanic Jet Currents. Hydrology. Vol. 5, No. 5, 017, pp doi: /j.hyd Received: February 1, 017; Accepted: March 8, 017; Published: Noveber 1, 017 Abstract: In this article we present the results of the oratory experients with shift currents in the hoogenous shallow water generated by differentially rotating eleents of the vessel botto: central axisyetric disk and outer coupled rings. Based on the experiental results, we explain the generation of vortex rings due to the developent of barothropic or shift instability of currents. We show that the surface of the differentially rotating liquid in the horizontal rate shift zone creates the conditions for generating chains of vortices rotating clockwise. When changing the values of rate shift and the rotation rate of the whole syste, the foration of odes of instability of vortex structures is observed, the conditions for which were quantified on the oratory current stability diagra in Rossby and Ekan nubers. To interpret the experiental data we considered the solutions of the equations for quasi-two-diensional geophysical currents in the for of eleentary waves of the current function disturbance. We estiated the paraeters of the perturbations developent at the eanders of the different parts of Gulf Strea using the calculation of the increent of experiental curves for neutral stability. The evaluation results provide a basis for the developent of realistic approaches to understanding the processes of generation and evolution of synoptic vortices in the eanders of intense oceanic jet currents. Keywords: Jet Currents in the Ocean, Barotropic Instability, Modeling of Currents, Laboratory Experient 1. Introduction Due to the developent of the general atosphere and ocean circulation theory the particular attention is paid to the studies of the ipact of both baroclinic and barotropic processes caused by (respectively) vertical and horizontal rate shifts, on the developent of the instability of largescale oceanic currents [1, ]. The research of eandering processes of large oceanic currents leading to the foration of ring vortices fro large eanders was carried out in soe works [3 8]. We considered the proble of specifying the echanis of the foration of rings due to the baroclinic instability of oceanic jet currents, the developent of which is possible only in the rotating liquid stratified by density [6]. The typical size of vortices associated with the local baroclinic Rossby radius of deforation exceed it in several ties [7]. Jet currents belong to the class of otions in geological environents where horizontal shifts can have a significant ipact on the foration of the barotropic instability of hydrodynaic processes in the ocean. First of all we should consider the vortex rings in the Gulf Strea type systes of jet currents [8]. The shift instability occurring during the horizontal rate steps ay lead to the foration of vortices, the radius of which is typically lesser than Rossby radius of deforation [7]. These vortices are observed on the outer periphery of Gulf Strea jet. Soe contribution to the establishent of the physical echaniss of the doinant ipact on the dynaics of ocean currents is ade by the oratory hydrodynaic experients [3 9, 10]. Researching the odels with rotating liquids in the wide range of wave nubers and frequencies provides the dynaic siilitude with the ocean-scale processes [3, 5]. The oratory observations of the analogues of oceanic processes, like subscale and esoscale vortices, and energetic estiates of the baroclinic instability of various scale vortices revealed both forward and back energy transfer cascades [11]. Of equal interest for understanding the dynaics of the developent of large-scale oceanic

2 78 Alexander Alekseyevich Solovyev and Ditry Alexandrovich Solovyev: The Barotropic Instability of the Oceanic Jet Currents processes is the proble of the stability of the syste of two opposed jet currents in the rotating layer of shallow water and its solution based on the experiental research of jet currents with horizontal rate shift in the oratory odel, where the barothropic instability is presented in its pure for. Our subject was the experiental research of jet currents with horizontal rate shift of the oratory ring odel, in which the barotropic instability presents in the pure for. In this work we consider the ipact of the barotropic effects on the eandering of the oceanic jet currents and ring foration based on the oratory experients with differentially rotating shallow liquid.. Materials and Experiental Setup To siulate the barotropic instability of jet currents we used cylindrical vessel in which the jet current was generated by the syste of rings at the botto, rotating relative to each other and to the vessel [1]. The horizontal rate shift was created by the independent rotation of the disk and two rings placed on the flat botto of the vessel (see Fig. 1). Figure 1. General view of the experiental setup. Disk and rings were placed precisely at the sae level. The axes of the disk and rings were placed outside the vessel so that any part of the vessel botto was availe for the rotating water. Vessel height was 5 c, diaeter of the disk 0 c, central ring width 10 c, outer ring width 3 c. The experiental setup allowed cobining different rotation odes of the botto segents. During the experients with the barotropic instability of jet currents the disk was rotating with the rate ω 1 which was different fro the rates ω and ω 3 of the central and outer rings, respectively. The angular rates of the rings were equal. Disk and coupled rings rotated in the sae counterclockwise direction. We observed the condition of the sall thickness of the undisturbed water Н о = 0.05 with respect to the typical horizontal scale of the current. The rotating rate ω 0 of the structures generated at the rate shifts reained lesser than the rotating rate Ω of the whole syste. gh The ratio of the Obukhov-Rossby radius r* = o to the Ω size of the vortex structures a was close to 1 (r*/a 1). An iportant feature of these experients was the condition 1 4 < a, where = H E is the thickness of the Proudan- o ν Stewartson botto layer; E = Ekan nuber; ν ΩH o kineatic viscosity of the water. The effect β was provided by the dependency of the effective thickness H of the layer of rotating water in the radial direction. The profile of the current in the experients corresponded to the stepwise rate shift that was created on the end surface of the vessel at the border where the disk contacts the ring. By setting different rotating rates for the center and periphery the ctationary for and aplitude of the shift was defined. To characterize currents we used, besides the Ekan nuber, the Rossby ω shift nuber Ro =, in which the rate shift ω was Ω defined by the difference between the rotating rates of the disk and the coupled rings. Figure. Photos of the current structure under the shift rotation of the shallow water. Left (a) is three-ode wave contour bounding the wave syste on the edge of the rate shift zone (right (b)).

3 Hydrology 017; 5(5): We found that various stable current line patterns are fored on the liquid surface (Fig. ). In the central part they were foring various figures with different syetry indexes fro the visualizing particles. At the periphery zone of the rotating shallow water the crests and troughs of the waves were foring [13]. Their axes were oriented along the radius or at soe angle to it. The wave syste was a chain of vortices with clockwise rotation. Their nuber was equal to the nuber of sides of the polygonal structures rotating around the vessel axis. When the values of the rate shift ω and the rotating rate of the whole syste Ω was changed synchronously and discretely, we observed two different scenarios of the instability developent. Increasing the shift intensity resulted in reducing the nuber of vortices; increasing the whole rotating rate and, respectively, decreasing the width of the shift zone resulted in the increasing the nuber of vortices while keeping the nondiensional wave nuber. According to the results of the easureents we calculated Ro and E nubers for supercritical conditions and constructed a oratory current stability graph (Fig. 3) [8]. Ψ(x,y,z) ik(x ct) = φ(y,z) e (1) Where φ (y,z) is wave aplitude; k wave nuber; c c = r + ici coplex phase velocity. Value i i = ck is the increent coefficient for the iaginary part of the coplex frequency which characterizes the aount of the exponential change of the wave aplitude during its propagation. To ake a transition fro the oratory experients to full-scale currents we propose the following calculation schee. The nondientional nuber in the odel is defined by the equation [10]: = k π α, where k= is k 0 r r1 radial wave nuber; k0= aziuthal wave nuber; n r + r1 ode nuber of radial odes, ode nuber of aziuthal odes. Take ode nuber of radial odes equal to n=1. In our case, with radiuses of the rings r=0.3, 1 r =0.1, the wave nuber equals α =0.15. Now proceed fro the horizontal rates and horizontal rate shifts to the rate value u of the ain jet current using the following equation: α Curves u f ( α ) u (ω1 ω )r1 = = aren t illustrative enough for practical evaluation of the perturbation increase degree. More useful to copare oratory and full-scale data are stability diagras in the for of faily of curves u = f(l), where the average flow rate across the current is ultiplied by the transition scale factor u=50 u. The diensional length of the perturbation wave L considering the oratory and full-scale scales equates the following: Figure 3. Diagra of the stability of shift perturbations of differentially rotating shallow water at Н о = Nubers show the wave nubers of the asyetric rotation odes. To define quantitative characteristics of the stability threshold for such conditions with different nondiensional wave nuber values we used inverse power dependence between Rossby nuber and Ekan nuber which characterizes the experiental situation rather adequately. 3. Results and Discussion Analysis of the results of oratory experients was carried out only relating to Gulf Strea. To describe the oratory experients, just like for the full-scale currents, we considered the solutions of the equation for quasi-twodiensional geophysical currents in the for of eleentary wave in the for of the current function perturbation: 100 L = ( k) () α The revaluation of the neutral stability curves obtained in the oratory tests to the full-scale ones with isolines in the for of increase factors was perfored using the following equation: ν = (3) H To evaluate the values of horizontal turbulent viscosity 8 c coefficient we choose the value equal to 10. During sec oratory tests we took the value of this coefficient equal to c 10. Considering the ratio between oratory and fullscale scales is 10, using Eq. 3 calculate the sec 7 value:

4 80 Alexander Alekseyevich Solovyev and Ditry Alexandrovich Solovyev: The Barotropic Instability of the Oceanic Jet Currents full 10 ν 10 = = (4) Н 4 6 Н we can see that the value of the doinating wavelength doesn t depend on the horizontal rate shift. Fro the isolines faily = сonst (see Fig. 4) characterizing the instability developent in the jet current Figure 4. Neutral curves shown as isolines of increase factors (their values are ultiplied by the nuerical factor 10 6 ). Now proceed to the evaluation of the perturbation developent paraeters caused by the eanders in the different parts of Gulf Strea. According to the definition of the increase factor, the perturbation aplitude should increase twice at the tie T equal to T =l n (5) 1 where. [ t ] = c Consider as an exaple two cross-sections ' ' of Gulf Strea with coordinates (Tab. 1): 68 8 w,38 50 n ' ' and w,3819 n [13]. Value was defined using the easured values of u, L. Table 1. Paraeters of barotropic instability in eandering of the Gulf Strea. Paraeters Cross-sections 1 Cross-sections L, k , sec Т, day 3 7 This result satisfactorily confored with the observed data in [14], where it was noted that the eander aplitude in Gulf Strea increases twice during about 10 days. The calculations and observations were in copliance also for the wavelength as for wavelength [5]. The scale spectru of its data has its peak near 00 k. Soe overestiated values of the factor, and hence low values of T are due to the fact that in the calculations the turbulent viscosity coefficient was taken with excess. This soewhat increased the value of the stabilizing factor. 4. Conclusion The results of experients with the resolution of shallow water give the following conclusions. Barotropic instability of currents is a source of generation of vortex rings fro eander jet currents in the ocean with diensions close to the Obukhov-Rossby deforation radius. The occurrence of a different nuber of vortex rings is possible under conditions of shear instability of current, because of a change in the shear rate and the velocity of the jet current. The estiates show the paraeters of the developent of barotropic perturbations caused by the eandering of various sections of the Gulf Strea. The evaluation results presented here provide a basis for a ore realistic perspective on the features of the foration of eanders in ocean currents. Acknowledgeents The research was funded by the grant of the Russian Scientific Fund (project No ).

5 Hydrology 017; 5(5): References [1] G. Korotaev, Structure, Dynaics And Energetics of the Synoptic Instability of the Ocean, Sevastopol, [] A. Monin and G. Zhikharev, Oceanic Vortices, UFN, vol. 5, no. 160, pp. 1 47, [3] V. V. Alekseev, S. V. Kisileva, and S. S. Lappo, Laboratory odels of the physical processes in the atosphere and ocean. Moscow: Nauka, 005. [4] Ø. Thie, J. Berntsen, and B. Gjevik, Developent of eddies in an idealised shelf slope area due to an along slope barotropic jet, Cont. Shelf Res., vol. 6, no. 1, pp , 006. [5] K. Wyrtki, L. Magaard, and J. Hager, Eddy energy in the oceans, J. Geophys. Res., vol. 81, no. 15, pp , [6] K. N. Fedorov, The physical nature and structure of oceanic fronts. Leningrad: Gidroeteoizdat, [7] V. M. Kaenkovich, M. N. Koshlyakov, and A. S. Monin, Synoptic eddies in the ocean. Leningrad: Gidroeteoizdat, 198. [8] G. Stoel, Gulf Strea: Physical And Dynaical Description. Moscow: Inostrannaya Literatura, [9] F. Dolzhanskiy, V. Kryov, and D. Manin, Stability And Vortice Structures of the Quazi-two-diensional Shift Currents, UFN, vol. 7, no. 160, pp. 1 47, [10] H. Greenspen, Theory of Rotating Fluids. Leningrad: Gidroeteoizdat, [11] Y. Zhang and Y. D. Afanasyev, Baroclinic turbulence on the polar β-plane in the rotating tank: Down to subesoscale, Ocean Model., vol. 107, pp , 016. [1] V. M. Lushin and C. C. Lappo, Geohydraulic odel, А. с , [13] D. Soloviev, A. Soloviev, and K. Pavlovskiy, Diagra of the Wave Stability in the Differentially Rotating Shallow Water, in Currents And Structures in Fluids. Proceedings of the International Conference., 005, pp [14] T. Lee and P. Cornillon, Propagation of Gulf Strea eanders, J. Phys. Ocean., vol. 6, pp. 5 41, 1996.