• Afanasyev, Y., 2003: Spontaneous emission of gravity waves by interacting vortex dipoles in a stratified fluid: Laboratory experiments. Geophys. Astrophys. Fluid Dyn., 97 , 7995.

    • Search Google Scholar
    • Export Citation
  • Ahlnäs, K., 1987: Multiple dipole eddies in the Alaska Coastal Current detected with Landsat Thematic Mapper Data. J. Geophys. Res., 92 , 1304113047.

    • Search Google Scholar
    • Export Citation
  • Berson, D., , and Z. Kizner, 2002: Contraction of westward-travelling nonlocal modons due to the vorticity filament emission. Nonlinear Proc. Geophys., 20 , 115.

    • Search Google Scholar
    • Export Citation
  • Carton, X., 2001: Hydrodynamical modeling of oceanic vortices. Surv. Geophys., 22 , 179263.

  • de Ruijter, W. P. M., , H. M. van Aken, , E. J. Beier, , J. R. E. Lutjeharms, , R. P. Matano, , and M. W. Schouten, 2004: Eddies and dipoles around South Madagascar: Formation, pathways and large-scale impact. Deep-Sea Res. I, 51 , 383400.

    • Search Google Scholar
    • Export Citation
  • Dritschel, D. G., , and A. Viúdez, 2003: A balanced approach to modelling rotating stably-stratified geophysical flows. J. Fluid Mech., 488 , 123150.

    • Search Google Scholar
    • Export Citation
  • Eames, I., , and J. B. Flór, 1998: Fluid transport by dipolar vortices. Dyn. Atmos. Oceans, 28 , 93105.

  • Fedorov, K. N., , and A. I. Ginzburg, 1986: Mushroom-like currents (vortex dipoles) in the ocean and in a laboratory tank. Ann. Geophys. B-Terr. P., 4 , 507516.

    • Search Google Scholar
    • Export Citation
  • Ginzburg, A. I., , and K. N. Fedorov, 1984: The evolution of a mushroom-formed current in the ocean. Dokl. Akad. Nauk SSSR, 274 , 481484.

    • Search Google Scholar
    • Export Citation
  • Ikeda, M., , L. A. Mysak, , and W. J. Emery, 1984: Observation and modeling of satellite-sensed meanders and eddies off Vancouver Island. J. Phys. Oceanogr., 14 , 321.

    • Search Google Scholar
    • Export Citation
  • Johannessen, J. A., , E. Svendsen, , O. M. Johannessen, , and K. Lygre, 1989: Three-dimensional structure of mesoscale eddies in the Norwegian Coastal Current. J. Phys. Oceanogr., 19 , 319.

    • Search Google Scholar
    • Export Citation
  • Kloosterziel, R. C., , G. F. Carnevale, , and D. Philippe, 1993: Propagation of barotropic dipoles over topography in a rotating tank. Dyn. Atmos. Oceans, 19 , 65100.

    • Search Google Scholar
    • Export Citation
  • Mied, R. P., , J. C. McWilliams, , and G. J. Lindermann, 1991: The generation and the evolution of a mushroom-like vortices. J. Phys. Oceanogr., 21 , 489510.

    • Search Google Scholar
    • Export Citation
  • Millot, C., 1985: Some features of the Algerian Current. J. Geophys. Res., 90 , 71697176.

  • Pallàs-Sanz, E., , and A. Viúdez, 2007: Three-dimensional ageostrophic motion in mesoscale vortex dipoles. J. Phys. Oceanogr., 37 , 84105.

    • Search Google Scholar
    • Export Citation
  • Sansón, L. Z., , G. J. F. van Heijst, , and N. A. Backx, 2001: Ekman decay of a dipolar vortex in a rotating fluid. Phys. Fluids, 13 , 440451.

    • Search Google Scholar
    • Export Citation
  • Stern, M. E., 1975: Minimal properties of planetary eddies. Euro. J. Mar. Res., 33 , 113.

  • Viúdez, A., 2006: Spiral patterns of inertia-gravity waves in geophysical flows. J. Fluid Mech., 562 , 7382.

  • Viúdez, A., , and D. G. Dritschel, 2003: Vertical velocity in mesoscale geophysical flows. J. Fluid Mech., 483 , 199223.

  • Viúdez, A., , and D. G. Dritschel, 2004: Optimal potential vorticity balance of geophysical flows. J. Fluid Mech., 521 , 343352.

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    Potential vorticity jumps (PV jump value δϖϖ ≅ 0.0436) on isopycnal il = 65 (z = 0) at (a) t = 0, (b) t = 5Tip, (c) t = 6Tip, and (d) t = 7Tip. An asterisk marks the ϖ center of each vortex. The thick line, normal to the line joining the vortices (thin line) and located at half the distance between them, has a length δl = 2c. This line, hereinafter the along-dipole line, and its midpoint (black square symbol) are included for reference. The horizontal extent is x, y ∈ [−π, π]c.

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    Horizontal distribution of w on the plane iz = 54 (z = −0.54): w ∈ [−2.1, 2.2] × 10−3. Contour interval is Δw = 0.25 × 10−3: zero contour omitted and time in Tip. The vortex locations and the along-dipole line are included as defined in Fig. 1. The horizontal extent is x, y ∈ [−2.5, 2.5]c.

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    Time series of the spatial average of the kinetic (EK), potential (EP), and total energy (ET) of the total flow. For comparison purposes the plots represent EX ≡ 〈EX〉 − , where is the time average of the spatial average 〈EX〉(t). The time averages and standard deviations are = (1115.1 ± 1.8) × 10−4, = (443.4 ± 1.8) × 10−4, and = (1558.5 ± 0.7) × 10−4. The vertical axis is in units of 10−4; the horizontal axis is time in Tip.

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    The dipole’s speed of displacement U(t) as a function of time t (in Tip). The time average Ũ ≃ 0.219 (solid line) and ± one standard deviation (dashed lines) are included.

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    As in Fig. 2 but for wi: wi ∈ [−4.7, 5.0] × 10−4 and Δwi = 10−4.

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    Space–time (r, t) distributions of (a) uit(r, t) (Δuit = 5 × 10−4), (b) uin(r, t) (Δuin = 5 × 10−4), (c) wi(r, t) (Δwi = 5 × 10−5), and (d) Di(r, t) (ΔDi = 5 × 10−5). The vertical axis is the distance r/c on the along-dipole line (z = −0.54). The horizontal axis is time in Tip.

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    Space–time (z, t) distributions of (a) uit(z, t) (Δuit = 5 × 10−4), (b) uin(z, t) (Δuin = 5 × 10−4), (c) wi(z, t) (Δwi = 5 × 10−5), and (d) Di(z, t) (ΔDi = 5 × 10−5) on the vertical line located at point r = c (black square location in Fig. 1). The vertical and horizontal axes are depth (z) and time in Tip, respectively.

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    Time series of the slope b(t) in the linear fitting (∂uit/∂r)(t) = a(t) + b(t)(∂wi/∂z)(t) for t ∈ [5, 28]Tip [where a(t) is negligible]. The time averaged b ≃ −1 (for t ∈ [6, 28]Tip) and plus/minus one standard deviation are included.

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    Vertical distributions of the time averages ur(r, z) (ur < 0, dotted contours, Δur = 0.02) and uz(r, z) (uz > 0, solid contours, Δuz = 0.2) in the moving along-dipole vertical section. The time averages range from t = 5Tip to t = 21Tip. The thick solid line joins the spatial locations where uz/ur ≅ −c.

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    As in Fig. 9 but for |cur + uz| (solid contours, Δ = 0.2) and wi(r, z) (solid and dotted wave contours, Δwi = 5 × 10−5). The thick solid line is the contour ut = U.

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    Vertical distributions of wi(r, z, t) (wave packet) and ut(r, z, t) − U(t) (circular solid lines) on the (moving) plane of the along-dipole line (thin horizontal line at a depth z = −0.54) as defined in Fig. 1: Δwi = 5 × 10−5. Contours of ut(r, z, t) − U(t) are {1, 1/2, 1/3, 1/4} and ut(r, z, t) − U(t) = 0 (circular dotted line). The dotted vertical line marks the location of the dipole’s center. The solid vertical line crossing the along-dipole line at the black square location (Fig. 1) is included for reference. Horizontal and vertical extents are δr = 2.2c and z ∈ [0, −2.2], respectively: time in Tip.

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    Horizontal distribution of w at z = −1.18 (iz = 161) and t = 7Tip in a numerical simulation with 5123 grid points of a dipole with ϖ+ = 1.8, ϖ = −0.8, A±x = 1.4, A±y = A±z = 1.2, and ef = 200: x, y ∈ 2πc[−1, 1] and w ∈ [−4, 4] × 10−3.

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    (a) The hodograph of [uit(r), uin(r)] × 103 on the along-dipole line at z = −0.54. The time average is from t = 5 to 39 Tip, and the bars mean one standard deviation. (b) As in (a) but for [wi(r), D̃i(r)] × 2 × 104.

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    The (time mean) unbalanced vertical velocity in the material description ŵi(r0, z0, t) at z0 = −0.74. Vertical lines of the same pattern are separated by Δt = (1/2Tip so that a wave with a period Δt has a frequency of ω = 2f. Time is in Tip, and the vertical axis is in units of 10−4. The symbols * indicate equal spatial intervals Δr. The time average [see (11)] is from t = 5Tip to 39Tip.

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    The distribution of ŵi(r0, Z, t) (solid and dotted wave contours, Δŵ = 2.5 × 10−5), and r(r0, z, t)/c (dashed contours, Δ(r/c) = 0.2). Plus symbols mark the locations of the data points r(r0, zi, ti) used to obtain the gridded field rijr(r0, i · Δz, j · Δt).

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    The distribution of wi(r, z) (solid and dotted wave contours) Δwi = 2.5 × 10−5), and t(r0, r, z) (solid thin contours, Δt = 2/2Tip). Plus symbols mark the locations of the data points t(r0, ri, zi) used to obtain the gridded field tijt(r0, i · Δr, j · Δz).

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    Time series of the spatial average of the total energy of the total flow ET, the balanced flow ETb, the unbalanced flow ETi × 5 × 102, and the interaction ETc. The plots represent EX ≡ 〈EX〉 − , where the time averages and standard deviations are = (1558.5 ± 0.8) × 10−4, = (1557.4 ± 0.9) × 10−4, = (0.005 ± 0.001) × 10−4, and = (1.1 ± 0.2) × 10−4. The vertical axis is in units of 10−4; horizontal axis is time in Tip.

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    As in Fig. 11 but with Δwi = 10−5 and z ∈ [−π, −π + 2.2], that is, the four vertical sections correspond to the rear part below the dipole, in the same location relative to the moving along-dipole line: time in Tip.

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    Vertical velocity w on plane iz = 54 and t = 10Tip in five different numerical simulations explained in the text: w ∈ {[−0.21, 0.22], [−0.19, 0.22], [−1.3, 1.0], [−1.6, 1.9], [−1.5, 2.0]} × 10−2.

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    Vertical distributions of w(x, y0, z) on plane y0 = 0 for the isolated frontal wave packet simulation with initial time t0 = 11Tip: x ∈ [−π, π], z ∈ [−π, 0], w ∈ {[−2.5, 2.5], [−1.9, 1.7], [−1.6, 1.6], [−1.7, 1.5], [−1.2, 1.3]} × 10−4; time in Tip.

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    Frontal wave packet (a) ϖ(z = 0) and (b) w(z = −0.54) (w ∈ [−3.0, 4.0] × 10−3) and (c) w (w ∈ [−2.6, 3.2] × 10−3), (d) ϖ(z = 0), and (e) w(z = −0.54) (w ∈ [−3.9, 4.3] × 10−3). Time is t = 8Tip.

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The Stationary Frontal Wave Packet Spontaneously Generated in Mesoscale Dipoles

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  • 1 Institut de Ciències del Mar, CSIC, Barcelona, Spain
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Abstract

Three-dimensional numerical simulations of rotating, statically and inertially stable, mesoscale flows show that wave packets, with vertical velocity comparable to that of the balanced flow, can be spontaneously generated and amplified in the frontal part of translating vortical structures. These frontal wave packets remain stationary relative to the vortical structure (e.g., in the baroclinic dipole, tripole, and quadrupole) and are due to inertia–gravity oscillations, near the inertial frequency, experienced by the fluid particles as they decelerate when leaving the large speed regions. The ratio between the horizontal and vertical wavenumbers depends on the ratio between the horizontal and vertical shears of the background velocity. Theoretical solutions of plane waves with varying wavenumbers in background flow confirm these results. Using the material description of the fields it is shown that, among the particles simultaneously located in the vertical column in the dipole’s center, the first ones to experience the inertia–gravity oscillations are those in the upper layer, in the region of the maximum vertical shear. The wave packet propagates afterward to the fluid particles located below.

Corresponding author address: Álvaro Viúdez, Institut de Ciències del Mar, CSIC, P. Marítim de la Barceloneta, 37–49 08003 Barcelona, Spain. Email: aviudez@cmima.csic.es

Abstract

Three-dimensional numerical simulations of rotating, statically and inertially stable, mesoscale flows show that wave packets, with vertical velocity comparable to that of the balanced flow, can be spontaneously generated and amplified in the frontal part of translating vortical structures. These frontal wave packets remain stationary relative to the vortical structure (e.g., in the baroclinic dipole, tripole, and quadrupole) and are due to inertia–gravity oscillations, near the inertial frequency, experienced by the fluid particles as they decelerate when leaving the large speed regions. The ratio between the horizontal and vertical wavenumbers depends on the ratio between the horizontal and vertical shears of the background velocity. Theoretical solutions of plane waves with varying wavenumbers in background flow confirm these results. Using the material description of the fields it is shown that, among the particles simultaneously located in the vertical column in the dipole’s center, the first ones to experience the inertia–gravity oscillations are those in the upper layer, in the region of the maximum vertical shear. The wave packet propagates afterward to the fluid particles located below.

Corresponding author address: Álvaro Viúdez, Institut de Ciències del Mar, CSIC, P. Marítim de la Barceloneta, 37–49 08003 Barcelona, Spain. Email: aviudez@cmima.csic.es

1. Introduction

Coherent vortical structures, like the vortex dipole or tripole, are possible features in mesoscale dynamics (e.g., Ikeda et al. 1984; Ginzburg and Fedorov 1984; Carton 2001). Mesoscale dipoles have been repeatedly observed from satellite imagery (Millot 1985; Fedorov and Ginzburg 1986; Ahlnäs et al. 1987; Johannessen et al. 1989), and recently from hydrographic data south of Madagascar (de Ruijter et al. 2004). Theoretically, Stern (1975) derived an exact dipolar solution, for nondivergent barotropic flow on the β plane, called the modon (e.g., Berson and Kizner 2002, and references therein). Laboratory experiments in rotating tanks show that barotropic dipoles can be generated from an impulsive jet (Kloosterziel et al. 1993), become important transporters of fluid (Eames and Flór 1998), and finally decay, for example, due to bottom friction effects (Sansón et al. 2001). Numerically, the generation of oceanic dipoles has been investigated within the two-layer, f-plane, shallow-water theory (Mied et al. 1991).

Unsteady dipoles in a balanced flow may also produce the spontaneous generation of inertia–gravity waves (IGWs) as observed experimentally in unsteady vortex collisions (Afanasyev 2003). The velocity and density perturbations of the IGWs spontaneously generated by these unsteady mesoscale flows are usually of small magnitude compared to the initially balanced (void of waves), generating flow, and often this IGW emission occurs sporadically, when wave packets are emitted during a finite period of time. However, in a recent numerical study of the small-amplitude spiral wave patterns caused by the IGWs spreading away from the vortical flow in the baroclinic dipole (Viúdez 2006) it was noticed that there is a wave packet, of large vertical velocity, that remains stationary, trapped in the frontal part of the translating dipole. The characteristics of this wave packet, named here the frontal wave packet because it amplifies in the frontal part of translating vortical structures, are numerically and theoretically investigated in this paper.

The numerical models used (described in section 2a) simulate the three-dimensional flow in a rotating reference frame. The initial conditions of the main simulation (section 2b) correspond to a baroclinic dipole composed by two potential vorticity (PV) vortices. The dipolar total flow (section 3a) remains statically and inertially stable and spontaneously generates the frontal wave packet. The unbalanced flow (section 3b) is extracted from the total flow, and the stationary frontal wave packet (section 3c) and other transient waves (section 3d) are then analyzed. It is found that the wave packet (a phenomenon primarily observed in the spatial description) is, in fact, due to inertia–gravity oscillations, close to the inertial frequency, experienced by the fluid particles (thus, a phenomenon better explained in the material description). The frontal wave is a robust phenomenon occurring in different dipole configurations (section 3e), is highly dependent on the balanced flow (section 3f), and occurs also in other coherent vortex structures, as in the baroclinic tripole and quadrupole (section 3g).

2. Numerical models and initial conditions

a. The numerical models

The three-dimensional baroclinic, stably stratified, volume-preserving, nonhydrostatic flow, under the f-plane and Boussinesq approximations, is simulated using a triply periodic numerical model (Dritschel and Viúdez 2003) initialized with the PV initialization approach (Viúdez and Dritschel 2003). This initialization approach largely avoids the initial generation of IGWs due to the initial velocity and density conditions and makes it easier to observe the spontaneous generation of IGWs. The PV is represented by contours on isopycnals. We use, unless otherwise specified, a 1283 grid, with 128 isopycnals, in a domain of vertical extent Lz = 2π (which defines the unit of length) and horizontal extents Lx = Ly = cLz, where c ≡ 10 is the ratio of the mean Brunt–Väisälä to the Coriolis frequency c ≡ ε−1N/f. The mean buoyancy period (Tbp ≡ 2π/N) is set as the unit of time (thus N ≡ 2π). One inertial period (Tip ≡ 2π/f ) equals 10Tbp. The initialization time tI = 5Tip and the time step δt = 0.01. The initialization time is the minimum time required for the fluid to reach its initial perturbed state with minimal generation of IGWs.

The vertical displacement D of isopycnals is D(x, t) ≡ zd(x, t), where d(x, t) ≡ (ρ(x, t) − ρ0)/ϱz is the depth that an isopycnal located at x at time t has in the reference density configuration defined by ρ0 + ϱzz, where ρ is the mass density, and ρ0 > 0 and ϱz < 0 are constants that do not need to be specified in the Boussinesq approximation. The mean Brunt–Väisälä frequency N ≡ −α0gϱz, where α0 ≡ 1/ρ0. Static instability occurs when the stratification number Dz ≡ ∂D/∂z > 1. The Rossby number Rζ/f, and the Froude number Fωh/N, where ωh and ζ are the horizontal and vertical components of the relative vorticity ω, respectively, and N is the total Brunt–Väisälä frequency. The PV anomaly ϖ ≡ Π − 1, where Π ≡ (ω/f + k) · d is the dimensionless total PV. The state variables are the components of the vector potential φ ≡ (φ, ψ, ϕ) from which both the three-dimensional velocity u = −f × φ and D = −ε2 · φ are obtained.

The primary model (ABϖ) integrates the two equations for the rate of change of the dimensionless horizontal ageostrophic vorticity Ah = (A, B) ≡ ωh/fc2hD ≡ (ωhωgh)/fωh/f, where ωgh is the horizontal geostrophic vorticity, and a third equation for the explicit material conservation of ϖ on isopycnals (dϖϖ/dt = 0). The horizontal potentials φh = (φ, ψ) are recovered from the inversion of Ah = ∇2φh, while the vertical potential ϕ is obtained from the inversion of the definition of the PV anomaly ϖ ≡ 1 − Π ≡ 1 − (ω/f + k) · (kD).

To avoid the generation of grid-scale noise, a biharmonic hyperdiffusion term − μ4qAh, where qχ ≡ (∂χ/∂x, ∂χ/∂y, ε∂χ/∂z) is the gradient operator in the vertically stretched space, is added to the equation for dAh/dt. The coefficient μ is chosen by specifying the damping rate (e-folding, ef = 50) of the largest wavenumber in spectral space per Tip.

The secondary numerical model (ABC) is the full pseudospectral version of the hybrid ABϖ algorithm and uses the same grid-based procedures except that it omits those involving the PV contours. The prognostic variables are A′ ≡ (A, B, C) ≡ ω/fc2D. Hence, there is no PV inversion, but only a Poisson equation for all three components of the vector potential, ∇2φ = A′, which is inverted spectrally. All of the parameter settings are the same except for ef . The results of the ABC model are used to verify that the frontal wave packet is robust and independent of the explicit conservation of PV inherent to the ABϖ model.

b. Initial conditions

The baroclinic dipole is simulated as two ellipsoids of oppositely signed ϖ (Fig. 1). The number of initial PV contours in the middle isopycnal (il = 65) of each vortex is nc = 20, ϖ varying from ϖ ≅ 0 (outermost surface) to extrema ϖ± = ±0.85 (vortex cores). The ϖ jump is fixed for all contours δϖϖ = |ϖ±|/nc. The outermost ϖ ellipsoidal layer has horizontal major and minor semiaxes (A±x, A±y) = (1.4, 1.2)c, and vertical semiaxes (A±z, Az) = (1.2, 0.52). Further details on the PV configuration of an ellipsoid are given in Viúdez and Dritschel (2003). The initial distance between vortex centers is 2A+y + 0.01c. The ϖ ellipsoids are defined on the isopycnal space (or physical space with flat isopycnals). During the initialization period (0 ≤ ttI) the isopycnals of the anticyclone (cyclone) stretch (shrink) so that at t = tI the vortices have similar vertical extents.

3. Numerical results and theoretical development

a. The total flow

The PV anomalies cause a moderately large ageostrophic flow both static and inertially stable. The time average, and standard deviations, from t = 5Tip to 50Tip, of the extreme R in the domain {Rmin(t), Rmax(t)} are {〈Rmin〉, 〈Rmax〉} = {−0.73, 0.48} (±{0.6, 6} × 10−3), while 〈Dzmax〉 = 0.47 (±0.7 × 10−3). The vortices soon deform from their initial elliptical PV configuration and start their eastward propagation as a dipole (Fig. 1).

We focus on the vertical velocity w since, as is typical of mesoscale balanced flows, w is 10−3–10−4 times smaller than the horizontal velocity uh, which is here O(1). The horizontal distribution of w (Fig. 2) displays features of three different spatial scales. First, on the larger dipole scale, w has the quadrupolar, time-independent pattern typical of a stationary translating balanced dipole, with w < 0 (w > 0) on the right (left) of the dipole’s head. Second, on the smaller vortex scale this quadrupolar pattern is modified by several time-dependent relative extrema, of smaller magnitude, produced by the phase oscillations, or dipole’s heading, of the vortices, in this case, predominantly by the anticyclone (Pallàs-Sanz and Viúdez 2007). These two features are quasigeostrophic (QG) balanced phenomena that can be diagnosed obtaining the QG w from the density field by solving the QG omega equation. Third, there is a finer, wave-scale, non-QG phenomenon located on the dipole’s head that, despite being of smaller magnitude, is clearly noticeable in the total w. This wave, here called the frontal wave, is the subject of this study and its characteristics are described in the next sections, where the unbalanced flow is extracted from the total flow.

The phase oscillations of the baroclinic vortices mentioned above seem to be the oscillation mode proper of a stable state of a physical system having internal structure. Though a complete explanation of the perturbation and restoring force that keeps the dipole stable (i.e., as a coherent pair of vortices) is not provided here, we notice that these oscillations imply interchange between the kinetic EK and potential EP energy of the total flow, where
i1520-0485-38-1-243-e1
The time series of the domain average of the kinetic 〈EK〉, potential 〈EP〉, and total energy 〈ET〉 = 〈EK〉 + 〈EP〉 (Fig. 3) show that the dipole’s heading implies a cyclic conversion between kinetic and potential energy with a period of about 5Tip. The dipole’s heading has also an effect in the dipole’s velocity U(t) (Fig. 4), which has a time mean U ≅ 0.22, but experiences small oscillations of amplitude 0.1 approximately every 5Tip.

b. Diagnosis of the unbalanced flow

The balanced vector potential φb = (φb, ψb, ϕb) is diagnosed using the optimal PV balance (OPVB) approach (Viúdez and Dritschel 2004), and the balanced quantities are derived from this. From a given PV field, the OPVB approach diagnoses a flow having only those IGWs that have been spontaneously generated during the process of acquiring its own PV (during a time interval set equal to tI = 5Tip). Thus, if the frontal wave is not spontaneously generated locally at the dipole’s head in a short time period (i.e., shorter than tI), but either is due to IGWs spontaneously generated elsewhere that travel to the dipole’s head from where does not propagate or is due to the amplification during a time scale larger that tI of IGWs generated locally at the dipole’s head, then the OPVB flow will not contain most of the frontal wave, which will remain, almost entirely, in the unbalanced vector potential φiφφb. The unbalanced velocity and vertical displacement of isopycnals, obtained directly from φi through the usual relations ui = −f × φi and Di = −ε2 · φi, are described below.

c. The frontal wave packet

1) Unbalanced velocity and vertical displacement

The horizontal distributions of the unbalanced vertical velocity wi (Fig. 5) show the frontal waves as a wave packet localized at the dipole’s head with wimax ≃ 5 × 10−4, that is, about 4 times smaller than wmax at this depth. Here |wi| is larger in the anticyclonic side, suggesting that a large part frontal wave in this case is generated in the anticyclone. The wave amplitude grows during the first inertial periods after initialization (from t ≃ 5Tip to ≃ 8Tip) until it becomes approximately stationary relative to the translating dipole.

Let the tangent and normal unit vectors relative to the along-dipole line be t and nk × t, respectively. The tangent and normal components of uh are utuh · t and unuh · n, respectively. The stationarity of the frontal wave and the relations between the velocity components and density anomaly can be appreciated from the distributions of uit, uin, wi, and Di as functions of (r, t), and (z, t), where r is the distance on the along-dipole line from the dipole’s center (Figs. 6 and 7). The frontal wave packet remains mostly stationary with no appreciable decay during the time shown. The time oscillations of these wave amplitudes have a period of ≃5Tip and seem to be related to the vortex phase oscillations mentioned earlier. The wave generation occurs in the first 5Tip after initialization, starting at short r and propagating forward. It can be also appreciated that, in the quasi-steady state, the horizontal (along dipole) and vertical wavenumbers, |k| and |m|, respectively, increase with r and |z|; that is, the wavelength decreases as the distance from the dipole’s center increases.

Numerical experiments carried out with the ABC model initialized with the balanced fields φb(x, t0) at times t0 > 5Tip show the generation of the wave packet irrespective of the initial time t0. The wave front is therefore independent of the idealized initial conditions (ellipsoidal PV surfaces) of the dipole.

Apart from the time variability associated to the dipole’s heading, it seems that, after the wave packet generation, the dipole’s flow, the extreme Rossby numbers, and the wave packet remain stationary, so there is no time-dependent geostrophic adjustment due to the wave packet.

2) Inertia–gravity wave solutions in background flow

The relations between the unbalanced quantities uit, uin, wi, and Di are locally similar to those of IGW solutions, close to the inertial frequency, in a background flow. Consider the momentum, mass, and volume conservation equations for the disturbances ũ and in the total flow u,
i1520-0485-38-1-243-e2
where dX/dt ≡ ∂X/∂t + u · X is the material derivative. We seek plane wave complex solutions for ũ = (ũ, υ̃, ) and in the (r, z) plane, with no n dependence, of the form
i1520-0485-38-1-243-e3
where the phase θ(x, z, t). The ambiguity in the usual notation, where the same symbols are used to denote functions both in the spatial (x, y, z, t) and material description, is extended now since the same symbols are used also to denote a function of variables (r, n, z, t), including the spatial Cartesian coordinates in the orthogonal reference frame (t, n, k) moving with the dipole. To avoid an excess of notation and since distinction in only needed when time operations, like time differentiation or averaging, are involved, the usual convention is followed here and the symbol ∂X/∂t is used for the time derivative of X(x, y, z, t), dX/dt is used for the time derivative of X in the material description, and symbol DυX ≡ ∂X/∂t + υX/∂r is the more general rate of change of X relative to an observer moving with velocity υ along the r axis.
For wavenumbers k(x, z, t), l = 0, and m(x, z, t),
i1520-0485-38-1-243-eq1
and for the local (or absolute) frequency ωl(x, z, t),
i1520-0485-38-1-243-eq2
The isochoric (volume preserving) condition in the along-dipole plane,
i1520-0485-38-1-243-e4
is very well satisfied in the frontal wave packet as deduced from the time series of the slope b(t) in the linear fitting (∂uit/∂r)(t) = a(t) + b(t) (∂wi/∂z)(t) (Fig. 8). Thus, gradients of the wave quantities in the direction are small (l = 0). The vortex phase oscillations, with a period ∼5Tip, are also noticeable in the time evolution of b(t).
Equations (2) admit plane wave solutions where the frequency relative to the fluid particle ωp (or intrinsic) satisfies the dispersion relation:
i1520-0485-38-1-243-eq3
Neglecting the advection by the total vertical velocity the total rates of change of ũ and D̃ in (2) can be expressed as
i1520-0485-38-1-243-eq4
Consistency of the plane wave solutions (which must have constant coefficients) requires that ωp, and therefore the ratio Mm/k, be constant. Defining the frequency of the waves relative to the moving dipole,
i1520-0485-38-1-243-eq5
we obtain the kinematical relations:
i1520-0485-38-1-243-eq6
The above expression is a particular case of the invariance of the wavenumber k to arbitrarily moving rigid frames of reference,
i1520-0485-38-1-243-e5
valid for any velocity υU.
Stationarity of the wave packet relative to the moving dipole (ωd = 0) implies therefore that
i1520-0485-38-1-243-e6
Since ωp = 0, and × θ = 0 implies that kz ≡ ∂k/∂z = mr ≡ ∂m/∂r = Mkr, and defining the velocity gradient (ur, uz) ≡ (∂ut/∂r, ∂ut/∂z), we obtain from (6) the vector relation
i1520-0485-38-1-243-eq7
which implies that
i1520-0485-38-1-243-e7
that is, the ratio m/k is given by the aspect ratio of the vertical and horizontal shears of the along-dipole background horizontal velocity.

In the reference case the ratio uz/ur is equal to −c = −N/f, the ratio between the horizontal and vertical scales of the dipole. The horizontal and vertical gradients of the background velocity ut on the along-dipole plane (Fig. 9) have a similar structure after a reflection along the axis r(z) = −cz and a change of sign. Therefore, the condition for the existence of the plane wave (ur = −cuz) is approximately satisfied on the along-dipole plane on the line r(z) = −cz, inside the dipole’s separatrix [i.e., ut(r, z, t) > U(t)], where the frontal wave develops (Fig. 10).

3) Wave numbers, frequency, and velocity components

With the above results it follows that
i1520-0485-38-1-243-e8
which means that the fluid particle experiences inertia–gravity oscillations with frequency ωp = 2f as it passes across the frontal wave packet. In the along-dipole line the velocity components of the waves satisfy the relations
i1520-0485-38-1-243-e9
That is, the wave horizontal velocity vector uih of a fixed fluid particle rotates anticyclonically with time.
Therefore, from (6) and (8)
i1520-0485-38-1-243-e10
The mean dipole’s speed U ≅ 0.22 (Fig. 4) while O(ut) = 1 (Fig. 11). Thus utU > 0 implies k < 0; that is, at a fixed time uhi rotates anticyclonically with increasing r, as can be inferred from Fig. 6.

Since |k|, |m| → ∞ as utU, the increase of the wavenumbers is always limited by the numerical grid size. However, the larger wavelengths of the frontal wave are well resolved with the current resolution (1283 grid points), and are observed also with 2563 and 5123 grid point simulations (Fig. 12). The imbalanced fields were not extracted in these cases because the OPVB is very costly with these high spatial resolutions. The increase of |k| and m happens because the horizontal distance completed in a period by a fixed fluid particle, which is undergoing quasi-inertial oscillations, decreases as the background flow decreases with increasing r and |z|.

The numerical results approximately satisfy relations (9). The time average on the along-dipole line in the dipole’s reference frame of the wave quantities (Fig. 13) show that (r, z0) ≅ −i2 (r, z0) and (r, z0) ≅ −i2f(r, z0), especially the rotation, anticyclonic for the vector [uit(r), uin(r)] and cyclonic for [wi(r), i(r)], and the ratios |uit|/|uin| ∼ 2 > 1, |wi|/|i| ∼ 2f = 2π2/10 ≅ 0.89 < 1. Relations (9) also imply that the ratio between horizontal and vertical wave velocity components, which is obtained from the volume conservation on the along-dipole line neglecting the n dependence, is uit/wic = 10, which can be inferred in the numerical results shown in Figs. 6 and 7.

The results above also imply ωl < 0; that is, at a fixed location the wave horizontal velocity uhi rotates cyclonically with time. Thus, an anchored current meter located at some point on the along-dipole line will measure cyclonic horizontal velocity perturbations as the dipole passes along, which are in fact related to the anticyclonic quasi-inertial oscillations experienced by the fluid particle.

The fields shown in the horizontal and vertical distributions are only those in the lower half of the domain (z ∈ [0, −π]) because, owing to the initial PV symmetry ϖ(xh, z) = ϖ(xh, −z), it results that u(xh, z) = u(xh, −z), υ(xh, z) = υ(xh, −z), w(xh, z) = −w(xh, −z), D(xh, z) = −D(xh, −z), and the same relations hold for the balanced and unbalanced components so that there is one frontal wave in each half of the domain. The frontal wave in the lower half corresponds to m > 0, implying that, at a fixed time, uhi rotates cyclonically (anticlockwise) with increasing z (as observed in Fig. 7). In the upper half m < 0, so that at a fixed time uhi rotates anticyclonically (clockwise) with increasing z (not shown).

4) The wave packet in the material description

A mean value of the wave vertical velocity ŵi(R, Z, t) as a function of the fluid particle (R, Z) and time t, that is, wi in the material description, can be obtained from ui(r, z, t), and wi(r, z, t) in the following way. First, the time averages ut(r, z) ≡ 〈ut(r, z, t)〉 and wi(r, z) ≡ 〈wi(r, z, t)〉 are computed on the along-dipole line. Then, the time t taken for the fluid particle (R, Z) = (r0, z0), located at r = r0 = 0 and z = z0 at t = 0, to move a distance r on the along-dipole line with a velocity ut(r, z) is computed as
i1520-0485-38-1-243-e11
Last, the unbalanced vertical velocity ŵi experienced by the fluid particle moving on the along-dipole line, as a function of t and at a depth z = z0, can be visualized in a scatterplot of wi(rj, z0) versus t(r0, rj, z0) and joining values at consecutive locations rj. This is equivalent to invert t(r0, rj, z0), to get the r location of the fluid particle (r0, z0) at time t, rj(r0, z0, t), and composite it with wi(rj, z0), that is, ŵi = wirj. The results (Fig. 14) show the amplification and decay of the amplitude of ŵi as the fluid particle moves on the along-dipole line, and that the oscillatory motion has an intrinsic frequency ωp2f.

More information can be obtained from the two-dimensional (Z, t) distribution ŵi(r0, Z, t), that is, as a function of Z, together with r(r0, Z, t) (Fig. 15). Since the vertical displacements of the fluid particle are small relative to the horizontal displacement, we can consider the depth z as a material invariant and think of ŵi(r0, Z, t) as the vertical velocity in the material description with material variables r0 = 0 and Z(z) = z. The distribution ŵi(r0, Z, t) in Fig. 15 can be interpreted as the time series of wi measured by Lagrangian floats moving at constant z and initially deployed along a vertical array at the dipole’s center. The distribution ŵi(r0, Z, t) shows that, among the particles located at r = 0 at a given time, the first ones to experience the inertia–gravity oscillations are those in the upper layer, in the region of the maximum vertical shear uz. The wave packet propagates afterward to the fluid particles located below. The time interval between points of equal phase is the intrinsic period of the IGWs Tp = 2π/ωp = 1/2 Tip. The wave fronts in the material description tend to be vertically aligned, which means that the wave motion of the fluid particles located at r = 0 at a given time approximately have the same phase (phase synchronization). This is especially true between t = 1.5Tip and t = 2.5Tip. The waves beyond that time have very small wavenumbers and are not well resolved numerically (grid points are shown in Fig. 15).

This phase synchronization is a result of (10) and (11). If (10) is approximately satisfied from r = 0 to r = r′, then the relation between time t and the wave phase θ is a linear one:
i1520-0485-38-1-243-e12
The above result follows also from the integration of /dt = −ωp, with ωp constant. Thus, if the initial phase θ(r0, z) = θ0 (const), then ωpt = −θ, that is, θ is in the direction of t (in the direction of the x axis in Fig. 15).

One might think of Fig. 15 also as the representation reciprocal to plotting wi(r, z) together with t(r0, r, z) (Fig. 16). Both representations contain the same information though the downward propagation (relative to the fluid particles) and the IGW phase synchronization is observed better in Fig. 15. The distribution ŵi(r0, Z, t) may be interpreted as a space contraction of wi(r, z) (depending on z, with larger contraction in the upper layers) so that contours of t(r0, r, z) in Fig. 16 transform into vertical parallel lines. Reciprocally, the distribution wi(r, z) may be interpreted as a time dilation of ŵi(r0, Z, t) (depending on Z, with larger dilation in the upper layers) so that contours of r(r0, Z, t) in Fig. 15 transform into vertical parallel lines.

5) Energy budget

The budget of the total energy may be expressed as
i1520-0485-38-1-243-e13
where 〈ETb〉 ≡ 〈u2b〉 + N2〈D2b〉 and 〈ETi〉 ≡ 〈u2i〉 + N2D2i〉 are the domain averages of the total energy of the balanced and unbalanced flow, respectively, and 2〈ub · ui〉 + 2N4DbDi〉 are the interaction terms. The energy of the balanced flow 〈ETb〉 (Fig. 17) has a behavior similar to that of the total flow 〈ET〉 (displaying several extrema but with an overall decrease due to numerical diffusion) except between t = 5Tip and t = 7Tip, where 〈ETb〉 decreases when 〈ET〉 increases. The 〈ETi〉 increases as a result of the development of the wave packet. Since, in order of magnitude, (〈ET〉 − 〈ETb〉)/〈ETi〉 ∼200, the largest part of the energy deficit 〈ET〉 − 〈ETb〉 is transferred to the interaction term 〈ETc〉. The 〈ETi〉 and 〈ETc〉 experience small-amplitude oscillations with the same period of 5Tip as the dipole’s heading.

d. Transient wave packets

The tz distributions of the unbalanced variables (Fig. 7) show the amplification of the frontal wave packet during the first inertial periods as well as the downward propagation of a wave packet, different from the stationary frontal wave packet, at lower levels (z < −1). The vertical group velocity of this transient wave packet decreases as it approaches the critical level ut = U. The wave becomes progressively horizontal, and therefore it is observed better in (uit, uin) than in wi and Di, until it finally no longer propagates vertically and its amplitude, relative to the moving dipole, slowly decays. These transient, downward propagating waves do, however, leave the dipole’s rear (or, equivalently, are left behind as the dipole propagates forward) and, before arriving to the bottom layer, form a small-amplitude wave tail left behind by the dipole (Fig. 18). The origin of these transient waves seems to be related to the local variability due to the initial development of the frontal wave packet. Once the frontal wave packet becomes stationary, the transient waves are no longer generated.

e. Robustness of the frontal wave packet

A series of different numerical experiments were carried out to verify that the generation of the frontal wave packet is a robust phenomenon in dipoles where (7) is physically possible. Figure 19 shows w in five different cases. The reference case (Fig. 19a) is included for comparison. The case, Fig. 19b, has vortices with a larger number of PV contours (nc = 60, proving that the frontal wave is independent of nc), and an anticyclone with larger vertical extent (Az = 0.7), resulting in a flow with Rmin = −0.75, Rmax = 0.48, Fmax = 0.44, and Dzmax = 0.39. The anticyclone is stronger, relative to the case, Fig. 19a, and this causes an increase in the horizontal (cross dipole) extent of the frontal wave packet in the anticyclonic side of the dipole. The case in Fig. 19c is as in Fig. 19b, but with vortices of equal vertical extent (A±z = A±y = 1.2) and larger ϖ+ = 2 (Rmin = −0.91, Rmax = 0.87, Fmax = 0.60, Dzmax = 0.34). The increase of cyclonic vorticity causes an increase in both U and in the trajectory curvature of the dipole. In this case the cyclone experiences phase oscillations (as the anticyclone does in the previous cases) so that waves also develop in the cyclone as stationary spiral waves (better noticed in the next cases). The case in Fig. 19d is as in Fig. 19b but with larger ϖ± = {−0.95, 2.1}, which causes a flow at the margin of the static and inertial stability (Rmin = −1.0, Rmax = 0.87, Fmax = 0.98, Dzmax = 0.93). In this case U is larger and the amplitude of the frontal wave increases in relation to the background mesoscale w.

The case in Fig. 19e has the same dipole configuration as in Fig. 19b but was simulated using the ABC model (with ef = 100), initialized with the potential φ (x, t0) provided by case (d) at t0 = 5Tip. The results are very consistent with the case in Fig. 19d, suggesting that the stationary frontal wave is a robust phenomenon, independent of the explicit conservation of PV (e.g., PV contour advection on isopycnals and PV inversion). The ABϖ model, and the PV initialization approach, was however used to obtain the balanced initial conditions at t0 = 5Tip.

f. Evolution of an isolated frontal wave packet

It is plausible that the small-amplitude waves spontaneously generated by the otherwise balanced dipole may amplify and cause the large-amplitude frontal wave packet. Thus, the frontal wave continually depends on the dipole’s flow. Intuitively, in absence of the dipole, the frontal wave packet would move backward, in a direction oppositive to the dipole’s velocity. To verify this behavior the evolution of the isolated frontal wave packet is simulated by initializing the ABϖ model only with the unbalanced potential φi(x, t0) and ϖ = 0 everywhere, that is, only the frontal wave packet is present initially. In the initial conditions (t0 = 11 Tip) the frontal wave packet and the deeper tail wave are clearly observed (Fig. 20).

As expected, as time evolves the wave packet rapidly moves backward and downward, with group velocity perpendicular to the wavenumber vector, and intersects the tail wave, which is more stationary in the fixed reference frame than the wave packet above, causing interference patterns. The wave packet continues its downward propagation and spreading, rapidly arriving at the domain’s lower layers and interfering with the upward propagating wave packet of the upper half (z ∈ [0, π], not shown). Thus, the unbalanced flow behaves as an isolated internal wave packet, which supports the fact that the OPVB approach is correctly extracting most of the wave packet from the balanced flow.

g. Frontal wave packets in a mesoscale tripole and quadrupole

The dipole is not the only vortical structure able to support a frontal wave packet. Other examples of coherent vortical structures generating frontal wave packets are the stable mesoscale tripole and quadrupole. These structures may be thought of as consisting of a set of partial dipoles.

An anticyclonically rotating tripole is simulated here with three ellipsoidal vortices: a central anticyclone with ϖ = −0.85 and two outer cyclones with ϖ+ = 1.5, and semiaxes A+x/c = 1, Ax/c = 1, and A±y/c = A±z = 0.8 in the initial reference configuration. The tripole remains coherent (Fig. 21a) and is static and inertially stable with Rmin = −0.89, Rmax = 0.73, Fmax = 0.56, and Dzmax = 0.38. Soon after the initialization time (t = 5 Tip) two stationary (relative to the rotating tripole reference frame) wave packets are generated at the frontal part of the rotating vortices (Fig. 21b). A stable cyclonic tripole of similar characteristics generates the frontal wave as well (Fig. 21c).

An anticyclonic quadrupole, simulated with three outer cyclones (Fig. 21d) remains coherent and stable with Rmin = −0.96, Rmax = 0.75, Fmax = 0.49, and Dzmax = 0.30. In this case three stationary wave packets are generated at the frontal sides of the rotating vortices (Fig. 21c). These frontal waves are very similar to the one studied in detail in the single dipole, and are in fact the same phenomenon, being always located where the flow decelerates, in the frontal part of the dipolelike vortical structures.

4. Concluding remarks

We have seen that wave packets of large-amplitude vertical velocity, relative to the background vertical velocity, can be generated in the frontal part of translating vortical structures in rotating, statically and inertially stable, mesoscale geophysical flows. The frontal wave packets remain stationary relative to the translating vortical structures and are due to the inertia–gravity oscillations of intrinsic frequency ωp = 2f experienced by the fluid particles as they decelerate when leaving the large speed regions. Consistent with these quasi-inertial oscillations, the magnitude of the horizontal and vertical wavenumbers decrease as the fluid speed decreases, the ratio between the horizontal and vertical wavenumbers remains however constant and equal to the ratio between the horizontal and vertical shears of the background horizontal velocity.

It is suggested, though not proved, that, among the wavenumber components of the small-amplitude waves spontaneously generated by vortex phase oscillations (dipole’s heading), only those components with wavenumbers k satisfying the relation of stationarity relative to the moving vortical structure (ωlUk = 0) are amplified and cause the frontal wave packet in the region where −ur/uz ∼ −k/mf/N. Thus, the dipole interior behaves like a cavity resonator for IGWs in the background flow.

Further theoretical work is therefore needed to understand the physical mechanism that excites and maintains with finite amplitude these inertia–gravity components. Whatever the cause, it seems that the persistence of this frontal wave, spontaneously generated and continually supported by the vortical flow, is a phenomenon beyond the range of application of the strict separation between balanced and unbalanced dynamics.

Acknowledgments

I thank two anonymous reviewers for their positive comments, which significantly improved the original manuscript. Partial support for this research has come from the Spanish Ministerio de Educación y Ciencia (Grant CGL2005-01450/CLI), and the U.K. Engineering and Physical Sciences Research Council (Grant XEP294). Computer resources and technical assistance provided by the Barcelona Supercomputing Center-Centro Nacional de Supercomputación is also acknowledged.

REFERENCES

  • Afanasyev, Y., 2003: Spontaneous emission of gravity waves by interacting vortex dipoles in a stratified fluid: Laboratory experiments. Geophys. Astrophys. Fluid Dyn., 97 , 7995.

    • Search Google Scholar
    • Export Citation
  • Ahlnäs, K., 1987: Multiple dipole eddies in the Alaska Coastal Current detected with Landsat Thematic Mapper Data. J. Geophys. Res., 92 , 1304113047.

    • Search Google Scholar
    • Export Citation
  • Berson, D., , and Z. Kizner, 2002: Contraction of westward-travelling nonlocal modons due to the vorticity filament emission. Nonlinear Proc. Geophys., 20 , 115.

    • Search Google Scholar
    • Export Citation
  • Carton, X., 2001: Hydrodynamical modeling of oceanic vortices. Surv. Geophys., 22 , 179263.

  • de Ruijter, W. P. M., , H. M. van Aken, , E. J. Beier, , J. R. E. Lutjeharms, , R. P. Matano, , and M. W. Schouten, 2004: Eddies and dipoles around South Madagascar: Formation, pathways and large-scale impact. Deep-Sea Res. I, 51 , 383400.

    • Search Google Scholar
    • Export Citation
  • Dritschel, D. G., , and A. Viúdez, 2003: A balanced approach to modelling rotating stably-stratified geophysical flows. J. Fluid Mech., 488 , 123150.

    • Search Google Scholar
    • Export Citation
  • Eames, I., , and J. B. Flór, 1998: Fluid transport by dipolar vortices. Dyn. Atmos. Oceans, 28 , 93105.

  • Fedorov, K. N., , and A. I. Ginzburg, 1986: Mushroom-like currents (vortex dipoles) in the ocean and in a laboratory tank. Ann. Geophys. B-Terr. P., 4 , 507516.

    • Search Google Scholar
    • Export Citation
  • Ginzburg, A. I., , and K. N. Fedorov, 1984: The evolution of a mushroom-formed current in the ocean. Dokl. Akad. Nauk SSSR, 274 , 481484.

    • Search Google Scholar
    • Export Citation
  • Ikeda, M., , L. A. Mysak, , and W. J. Emery, 1984: Observation and modeling of satellite-sensed meanders and eddies off Vancouver Island. J. Phys. Oceanogr., 14 , 321.

    • Search Google Scholar
    • Export Citation
  • Johannessen, J. A., , E. Svendsen, , O. M. Johannessen, , and K. Lygre, 1989: Three-dimensional structure of mesoscale eddies in the Norwegian Coastal Current. J. Phys. Oceanogr., 19 , 319.

    • Search Google Scholar
    • Export Citation
  • Kloosterziel, R. C., , G. F. Carnevale, , and D. Philippe, 1993: Propagation of barotropic dipoles over topography in a rotating tank. Dyn. Atmos. Oceans, 19 , 65100.

    • Search Google Scholar
    • Export Citation
  • Mied, R. P., , J. C. McWilliams, , and G. J. Lindermann, 1991: The generation and the evolution of a mushroom-like vortices. J. Phys. Oceanogr., 21 , 489510.

    • Search Google Scholar
    • Export Citation
  • Millot, C., 1985: Some features of the Algerian Current. J. Geophys. Res., 90 , 71697176.

  • Pallàs-Sanz, E., , and A. Viúdez, 2007: Three-dimensional ageostrophic motion in mesoscale vortex dipoles. J. Phys. Oceanogr., 37 , 84105.

    • Search Google Scholar
    • Export Citation
  • Sansón, L. Z., , G. J. F. van Heijst, , and N. A. Backx, 2001: Ekman decay of a dipolar vortex in a rotating fluid. Phys. Fluids, 13 , 440451.

    • Search Google Scholar
    • Export Citation
  • Stern, M. E., 1975: Minimal properties of planetary eddies. Euro. J. Mar. Res., 33 , 113.

  • Viúdez, A., 2006: Spiral patterns of inertia-gravity waves in geophysical flows. J. Fluid Mech., 562 , 7382.

  • Viúdez, A., , and D. G. Dritschel, 2003: Vertical velocity in mesoscale geophysical flows. J. Fluid Mech., 483 , 199223.

  • Viúdez, A., , and D. G. Dritschel, 2004: Optimal potential vorticity balance of geophysical flows. J. Fluid Mech., 521 , 343352.

Fig. 1.
Fig. 1.

Potential vorticity jumps (PV jump value δϖϖ ≅ 0.0436) on isopycnal il = 65 (z = 0) at (a) t = 0, (b) t = 5Tip, (c) t = 6Tip, and (d) t = 7Tip. An asterisk marks the ϖ center of each vortex. The thick line, normal to the line joining the vortices (thin line) and located at half the distance between them, has a length δl = 2c. This line, hereinafter the along-dipole line, and its midpoint (black square symbol) are included for reference. The horizontal extent is x, y ∈ [−π, π]c.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 2.
Fig. 2.

Horizontal distribution of w on the plane iz = 54 (z = −0.54): w ∈ [−2.1, 2.2] × 10−3. Contour interval is Δw = 0.25 × 10−3: zero contour omitted and time in Tip. The vortex locations and the along-dipole line are included as defined in Fig. 1. The horizontal extent is x, y ∈ [−2.5, 2.5]c.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 3.
Fig. 3.

Time series of the spatial average of the kinetic (EK), potential (EP), and total energy (ET) of the total flow. For comparison purposes the plots represent EX ≡ 〈EX〉 − , where is the time average of the spatial average 〈EX〉(t). The time averages and standard deviations are = (1115.1 ± 1.8) × 10−4, = (443.4 ± 1.8) × 10−4, and = (1558.5 ± 0.7) × 10−4. The vertical axis is in units of 10−4; the horizontal axis is time in Tip.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 4.
Fig. 4.

The dipole’s speed of displacement U(t) as a function of time t (in Tip). The time average Ũ ≃ 0.219 (solid line) and ± one standard deviation (dashed lines) are included.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 5.
Fig. 5.

As in Fig. 2 but for wi: wi ∈ [−4.7, 5.0] × 10−4 and Δwi = 10−4.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 6.
Fig. 6.

Space–time (r, t) distributions of (a) uit(r, t) (Δuit = 5 × 10−4), (b) uin(r, t) (Δuin = 5 × 10−4), (c) wi(r, t) (Δwi = 5 × 10−5), and (d) Di(r, t) (ΔDi = 5 × 10−5). The vertical axis is the distance r/c on the along-dipole line (z = −0.54). The horizontal axis is time in Tip.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 7.
Fig. 7.

Space–time (z, t) distributions of (a) uit(z, t) (Δuit = 5 × 10−4), (b) uin(z, t) (Δuin = 5 × 10−4), (c) wi(z, t) (Δwi = 5 × 10−5), and (d) Di(z, t) (ΔDi = 5 × 10−5) on the vertical line located at point r = c (black square location in Fig. 1). The vertical and horizontal axes are depth (z) and time in Tip, respectively.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 8.
Fig. 8.

Time series of the slope b(t) in the linear fitting (∂uit/∂r)(t) = a(t) + b(t)(∂wi/∂z)(t) for t ∈ [5, 28]Tip [where a(t) is negligible]. The time averaged b ≃ −1 (for t ∈ [6, 28]Tip) and plus/minus one standard deviation are included.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 9.
Fig. 9.

Vertical distributions of the time averages ur(r, z) (ur < 0, dotted contours, Δur = 0.02) and uz(r, z) (uz > 0, solid contours, Δuz = 0.2) in the moving along-dipole vertical section. The time averages range from t = 5Tip to t = 21Tip. The thick solid line joins the spatial locations where uz/ur ≅ −c.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 10.
Fig. 10.

As in Fig. 9 but for |cur + uz| (solid contours, Δ = 0.2) and wi(r, z) (solid and dotted wave contours, Δwi = 5 × 10−5). The thick solid line is the contour ut = U.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 11.
Fig. 11.

Vertical distributions of wi(r, z, t) (wave packet) and ut(r, z, t) − U(t) (circular solid lines) on the (moving) plane of the along-dipole line (thin horizontal line at a depth z = −0.54) as defined in Fig. 1: Δwi = 5 × 10−5. Contours of ut(r, z, t) − U(t) are {1, 1/2, 1/3, 1/4} and ut(r, z, t) − U(t) = 0 (circular dotted line). The dotted vertical line marks the location of the dipole’s center. The solid vertical line crossing the along-dipole line at the black square location (Fig. 1) is included for reference. Horizontal and vertical extents are δr = 2.2c and z ∈ [0, −2.2], respectively: time in Tip.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 12.
Fig. 12.

Horizontal distribution of w at z = −1.18 (iz = 161) and t = 7Tip in a numerical simulation with 5123 grid points of a dipole with ϖ+ = 1.8, ϖ = −0.8, A±x = 1.4, A±y = A±z = 1.2, and ef = 200: x, y ∈ 2πc[−1, 1] and w ∈ [−4, 4] × 10−3.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 13.
Fig. 13.

(a) The hodograph of [uit(r), uin(r)] × 103 on the along-dipole line at z = −0.54. The time average is from t = 5 to 39 Tip, and the bars mean one standard deviation. (b) As in (a) but for [wi(r), D̃i(r)] × 2 × 104.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 14.
Fig. 14.

The (time mean) unbalanced vertical velocity in the material description ŵi(r0, z0, t) at z0 = −0.74. Vertical lines of the same pattern are separated by Δt = (1/2Tip so that a wave with a period Δt has a frequency of ω = 2f. Time is in Tip, and the vertical axis is in units of 10−4. The symbols * indicate equal spatial intervals Δr. The time average [see (11)] is from t = 5Tip to 39Tip.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 15.
Fig. 15.

The distribution of ŵi(r0, Z, t) (solid and dotted wave contours, Δŵ = 2.5 × 10−5), and r(r0, z, t)/c (dashed contours, Δ(r/c) = 0.2). Plus symbols mark the locations of the data points r(r0, zi, ti) used to obtain the gridded field rijr(r0, i · Δz, j · Δt).

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 16.
Fig. 16.

The distribution of wi(r, z) (solid and dotted wave contours) Δwi = 2.5 × 10−5), and t(r0, r, z) (solid thin contours, Δt = 2/2Tip). Plus symbols mark the locations of the data points t(r0, ri, zi) used to obtain the gridded field tijt(r0, i · Δr, j · Δz).

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 17.
Fig. 17.

Time series of the spatial average of the total energy of the total flow ET, the balanced flow ETb, the unbalanced flow ETi × 5 × 102, and the interaction ETc. The plots represent EX ≡ 〈EX〉 − , where the time averages and standard deviations are = (1558.5 ± 0.8) × 10−4, = (1557.4 ± 0.9) × 10−4, = (0.005 ± 0.001) × 10−4, and = (1.1 ± 0.2) × 10−4. The vertical axis is in units of 10−4; horizontal axis is time in Tip.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 18.
Fig. 18.

As in Fig. 11 but with Δwi = 10−5 and z ∈ [−π, −π + 2.2], that is, the four vertical sections correspond to the rear part below the dipole, in the same location relative to the moving along-dipole line: time in Tip.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 19.
Fig. 19.

Vertical velocity w on plane iz = 54 and t = 10Tip in five different numerical simulations explained in the text: w ∈ {[−0.21, 0.22], [−0.19, 0.22], [−1.3, 1.0], [−1.6, 1.9], [−1.5, 2.0]} × 10−2.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 20.
Fig. 20.

Vertical distributions of w(x, y0, z) on plane y0 = 0 for the isolated frontal wave packet simulation with initial time t0 = 11Tip: x ∈ [−π, π], z ∈ [−π, 0], w ∈ {[−2.5, 2.5], [−1.9, 1.7], [−1.6, 1.6], [−1.7, 1.5], [−1.2, 1.3]} × 10−4; time in Tip.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

Fig. 21.
Fig. 21.

Frontal wave packet (a) ϖ(z = 0) and (b) w(z = −0.54) (w ∈ [−3.0, 4.0] × 10−3) and (c) w (w ∈ [−2.6, 3.2] × 10−3), (d) ϖ(z = 0), and (e) w(z = −0.54) (w ∈ [−3.9, 4.3] × 10−3). Time is t = 8Tip.

Citation: Journal of Physical Oceanography 38, 1; 10.1175/2006JPO3692.1

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