Billant, P., Chomaz, J.-M.: Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid.. Billant, P., Chomaz, J.-M.: Theoretical analys
Trang 12 Stability of Quasi Two-Dimensional Vortices 53such as the pairing of same sign vortices or the creation of vorticity filaments Theyshould therefore modify the phenomenology of the turbulence In particular for largeRossby number and small Froude number, the pairing of two vortices is unstable tothe generalized zigzag instability and, while approaching each other the vorticesshould form thinner and thinner layers resulting in an energy transfer to smallerlength scales and not larger scales as in the 2D turbulence This effect of the zigzaginstability would then explain the result of Lindborg [32] who shows, processingturbulence data collected during airplane flights and computing third-order moment
of the turbulence, that the energy cascade in the horizontal energy spectra is directand not reverse as it was conjectured by Lilly [31]
2.7 Appendix: Local Approach Along Trajectories
In this appendix, we investigate the instability of a 2D steady basic flow
character-ized by a velocity UBand a pressure PBin the inviscid case The normal vector of
the flow field is denoted ez The 3D perturbations are denoted(u, p) In a frame,
rotating at the angular frequency = e z, the linearized Euler equation read as
u = exp {iφ(x, t)/ξ} a(t) + O(δ), (2.28)
p = exp {iφ(x, t)/ξ} (π(t) + O(δ)), (2.29)
where ξ is a small parameter Substitution in the linearized Euler equations leads
to a set of differential equations for the amplitude and the wave vector k = ∇φ
evolving along the trajectories of the basic flow:
withL B = ∇UB the velocity gradient tensor, the superscript T denoting the
trans-position,I the unity tensor, and C the Coriolis tensor:
Trang 2The stability is analyzed looking for the behavior of the velocity amplitude a
Fol-lowing Lifschitz and Hameiri [33], the flow is unstable if there exists a trajectory on
which the amplitude a is unbounded at large time.
2.7.1 Centrifugal Instability
For flows with closed streamlines, centrifugal instability can be understood by sidering spanwise perturbations, as they have been shown to be the most unsta-ble in centrifugal instability studies (see [2] or [48] for a detailed discussion) If
con-k(t = 0) = k0e z, (2.31) yields k(t) = k0e z; the base flow, evolving in a plane perpendicular to ez, does not impose tilting or stretching along z direction The
incompressibility equation (2.35) gives az = 0
In the plane of the steady base flow, the trajectories (streamlines in the steadycase) may be referred to as streamline function valueψ The two vectors, U B and
∇ψ, provide an orthogonal basis The amplitude equation in the plane perpendicular
to ez may be expressed in this new coordinate system, and following [48], (2.32)becomes
with V = |UB (x)|, Vthe lagrangian derivative d
dt V = UB ∇V and R the local
algebraic curvature radius defined by [48]
Trang 32 Stability of Quasi Two-Dimensional Vortices 55the existence of diverging solutions of (2.36) for a closed (non-circular) streamlinewithδ negative requires further mathematical analyses.
2.7.2 Hyperbolic Instability
In this section, we focus again on a spanwise perturbation characterized by a
wavenumber k perpendicular to the flow field The contribution of the pressure
disappears in that case and the instability is called pressureless Equation (2.32)reduces to
with the local strain and ζ the vorticity of the base flow For || greater than
|ζ2| the streamlines are hyperbolic The integration of (2.38) is straightforward The
perturbed velocity amplitude a has an exponential behavior, exp(σt), with σ
as discussed in Sect 2.3.2
2.7.3 Elliptic Instability
The basic flow is defined by the constant gradient velocity tensor (2.39) but with
|ζ2| greater than || Trajectories are found to be elliptical with an aspect ratio or
Trang 456 J.-M Chomaz et al.The wave vector solution of (2.31),
k = k0(sin(θ) cos(Q(t − t0)), E sin(θ) sin(Q(t − t0)), cos(θ)) , (2.43)
with k0 and t0integration constants, is easily obtained (see, for example, Bayly [3]).The wave vector describes an ellipse parallel to the (x, y) plane, with the same
eccentricity as the trajectories ellipse, but with the major and minor axes reversed.The stability is investigated with Floquet theory, looking for the eigenvalues of themonodromy matrixM(T ), which is a solution of
integrated over one period T The basic flow lies in the (x, y) plane, so that one
of the eigenvalues of the monodromy matrix is m3 = 1 The average of the trace
with m1,2 either complex conjugate indicating stable flow, or real and inverse for
unstable flow Generally, m1 ,2 are obtained by numerical integration of (2.44) and(2.45) Note, however, that the following two cases may be tackled analytically
Following Waleffe [50], we derive from (2.32) the equation for the rescaled
compo-nent of the velocity along the z-axis denoted q:
Trang 52 Stability of Quasi Two-Dimensional Vortices 57
q = a3 |k|2
with k// the component of the wavenumber on the(x, y) plane For small strain,
this equation leads to a Mathieu equation with a rescaled time, t = ζ t,
d2q
dt 2 + α + 2bsin t q = 0, (2.50)where
Ro + 32
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5 Billant, P., Chomaz, J.-M.: Theoretical analysis of the zigzag instability of a vertical columnar
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rotating fluid Bull Am Phys Soc 45(9), 175 In: Proceeding of the 53rd Meeting of the
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Trang 9by the instability or by the change of direction of ocean currents, by geostrophicadjustment after convection, or via topographic influences (e.g., lee eddies behindislands) But nearly all oceanic vortices share the common properties, which may
be used to define them:
Oceanic vortices are coherent structures, with a dominant horizontal motion andclosed fluid circulation in their core The predominance of horizontal motion is due
to the importance of the Coriolis force and of the buoyancy effects in the dynamics
of vortices and to their small aspect ratio Vortex lifetimes are usually much longerthan the timescale of their spinning motion In general, mixing and ventilation affectoceanic vortices on timescales much larger than the turnover time But such mixingprocesses can also have drastic consequences when they reach the vortex core: thenthe vortex collapses Since mixing is generally slow and since their flow pattern isquasi-circular, oceanic vortices retain in their core a water mass characteristic oftheir region of formation Thus, oceanic vortices, which drift over long distances,participate in the transport of heat, momentum, chemical tracers, and biologicalspecies across ocean basins and contribute to the mixing of oceanic water masses.The present review will first concentrate on oceanic observations of vortices, onthe description of their physical characteristics, and on the salient features of theirdynamics The second part of this review will present dynamical models often used
to represent vortex motion, evolution, and interactions and recent findings on thesesubjects
X Carton ( B)
Laboratoire de Physique des Oceans, Universite de Bretagne Occidentale, Brest, France,
xcarton@univ-brest.fr
Carton, X.: Oceanic Vortices Lect Notes Phys 805, 61–108 (2010)
DOI 10.1007/978-3-642-11587-5_3 Springer-Verlag Berlin Heidelberg 2010 c
Trang 1062 X Carton
3.1 Observations of Oceanic Vortices
3.1.1 Different Types of Oceanic Vortices
First, we will present the large, surface-intensified “rings” of the major westernboundary currents which have been historically identified and studied first; then, wewill describe smaller (mesoscale) vortices, identified later on the eastern boundary
of the oceans, but of importance for the large-scale fluxes of heat and salt Some ofthese mesoscale eddies are concentrated at depth (for instance, in the thermocline)and thus their identification and study have been more recent
3.1.1.1 Large Rings
In general, wind-induced currents are intensified at the western boundaries ofthe ocean and detach at mid-latitudes to form intense, horizontally and verticallysheared, eastward jets, prone to barotropic and baroclinic instabilities These insta-bilities cause these jets to meander, the occlusion of the meanders resulting in theformation of so-called rings or synoptic eddies These rings, and in particular thewarm-core rings which are surface intensified, have long since been identified in thevicinity of the Gulf Stream, of the Kuroshio, of the Agulhas Current, or of the NorthBrazil Current, to name a few [140, 76] Rings were so called because the originalcurrent circles on itself, so that the velocity maximum is then located on a ring thatencircles and isolates a core with a trapped water mass
Gulf Stream Rings
The Gulf Stream is the fastest current in the North Atlantic Ocean; it detaches fromthe American coast at Cape Hatteras to enter the Atlantic basin as an intense, quasi-zonal jet Its peak velocity is on the order of 1.5 m/s, the jet width is about 80 km,and it is intensified above the main thermocline (roughly the upper 800 m of theocean); below, the jet velocity is usually less than 0.1 m/s; Fig 3.2 also showsthat the isotherms dive by 600 m across the Gulf Stream (see also [131]) As astrongly sheared current, the Gulf Stream is unstable and forms meanders whichcan grow, occlude, and detach from the jet, forming anticyclonic/cyclonic rings onits northern/southern flanks ([166]; see Fig 3.1) Since the Gulf Stream separatesthe warm waters of the Sargasso Sea from the cold waters of the Blake Plateau (seeFig 3.2), cyclonic rings carry these cold waters and anticyclonic rings the warmwaters Cold-core rings are usually wider than warm-core rings (250 versus 150 km
in diameter, [117]), and they extend down to 4000 m depth whereas warm-core ringsare concentrated above the thermocline [137] The maximum orbital velocity of therings is comparable to the peak velocity of the Gulf Stream (about 1 or 1.5 m/s);
it lies at a 30–40 km distance from the vortex center and decreases exponentiallybeyond [116] In warm-core rings, intense velocities are still found at mid-depth, as
in the Gulf Stream itself (e.g., 0.5 m/s at 500 m depth, [131])