Vertical velocities in quasigeostrophic laboratory vortices

TL;DR

This study tests the quasi‑geostrophic ω‑Equation as a tool to infer vertical velocity $w$ from horizontal fields in rotating, stratified vortices, by comparing predictions to direct laboratory measurements. Using a rotating table, density‑stratified water, PIV, and LIF, the authors generate multiple anticyclonic vortices and compute the ω‑Equation source term to diagnose $w$; they test two vortex models (Gaussian and shielded) to illustrate the dependence on geometry and frontogenesis. Across both models and the experiments, the ω‑Equation predicts vertical motions near vortex edges but underestimates their magnitude by a factor of 3–5, while two independent direct measurements yield $w$ around $1 imes 10^{-4}$ m s$^{-1}$. Motivated by this discrepancy, the authors derive a diffusive extension of the ω‑Equation incorporating momentum and scalar diffusion, showing enhanced vertical velocities in a Gaussian vortex model and highlighting viscosity as a key factor in ageostrophic vertical exchange with potential oceanographic relevance.$N^2 \nabla_h^2 w + f^2 \partial^2 w/\partial z^2 = 2 \frac{g}{\rho_0} \nabla_h \cdot (\nabla_h u_g \cdot \nabla_h \rho) + \frac{g}{\rho_0} (\nu - \kappa) \nabla^2 \nabla_h^2 \rho$, and noting $Sc = \nu/\kappa \approx 700$ for salt diffusion in seawater.

Abstract

In the present study, we test the predictions of the ω-Equation against laboratory experiments with direct measurements of the vertical velocity w. Our results are further completed through the use of theoretical models of oceanic vortices, with the aim of helping oceanographers in better quantifying regions of upwelling and downwelling in the ocean. Using a rotating table and density stratification, we investigate non-axisymmetric surface vortices. The predicted vertical velocities calculated from the ω-Equation are relatively small (|w| ~ 20 μm/s) and primarily appear at the vortex edges, where the vorticity sign changes, acting to restore flow stratification. However, our estimates of w, obtained from the divergence of the horizontal velocity field measured by PIV, are five times larger. This discrepancy is further confirmed by direct particle tracking measurements, which indicate a magnitude of approximately 100 μm/s for w. To address this inconsistency, we incorporate dissipative terms into the ω-Equation to assess the role of viscous diffusion in enhancing internal recirculation in the vortex and thus vertical velocity magnitude. This hypothesis is favorably tested on a Gaussian vortex model.

Figures (8)

• Figure 1: For $\textup{Ro} = -0.3$, $r_m = 2 \times 10^{-2}$ m.s$^{-1}$, $f = 2$ s$^{-1}$ and $\rho_0 = 1005$ kg.m$^{-3}$: diagnosis of the vertical velocity w of a Gaussian vortex model with the $\omega-$equation at $\tilde{z} \simeq -0.002$ (eq. \ref{['eq:omega_equation']}). (a) Horizontal velocity field $\vec{u_g}$ for $\beta = 1$ (axisymmetric configuration). (b) Horizontal velocity field $\vec{u_g}$ for $\beta = 1.4$ (similar horizontal aspect ratio as Viudez's study viudez2018two). (c,d) Vertical velocity w resulting from the $\omega-$equation for the axisymmetric vortex and the elliptical vortex. (e,f) Isocontours of density and vorticity for both axisymmetric and elliptical vortices. In panel (e), both fields are parallel and no vertical velocity is generated as can be seen in panel (c). On the contrary, in panel (f), both field contours intersect and vertical velocities are generated as shown in panel (d).
• Figure 2: For $\textup{Ro} = -0.3$, $r_m = 2 \times 10^{-2}$ m, $f = 2$ s$^{-1}$ and $b = 3$. (a) Shielded vortex velocity field below the free-surface. (b) Vertical velocity w diagnosed with the $\omega-$Equation from the shielded vortex model. (c) Isocontours of density and vorticity.: their intersections correspond to where vertical velocities arise.
• Figure 3: For $\textup{Ro} = -0.3$, $r_m = 2 \times 10^{-2}$ m.s$^{-1}$, $f = 2$ s$^{-1}$ and $b = 3$. (a) Vertical velocity w diagnosed with the $\omega-$Equation from the shielded vortex model for $(x,y)>0$. (b) Fluid density derived from the thermal wind balance and the velocity field (eq. \ref{['eq:shielded_velocity2']}). (c) Profile of vertical velocity (right axis) superimposed with the vorticity profile (left axis) along $x = y$ (shown as a dotted line in (a)). (d) Profile of vertical velocity (right axis) superimposed with the density profile (left axis) along $x = y$ (shown as a dotted line in (b)).
• Figure 4: (a) Sketch of the experimental set-up. A tank, filled with salted water seeded with silver coated PIV particles, is placed on a rotating table of rotation frequency $\Omega$. Pure water dyed with fluoresceine is injected with a peristaltic pump at the center of the free surface. A translating 488 nm laser sheet illuminates an horizontal plane of interest at fixed $z$. Two cameras mounted with bandpass blue filter and highpass green filter respectively used for PIV and LIF capture the experiment from above. (b) Process of vortex generation : the radial movement of the injection paired with the Coriolis force generates an anticyclonic vortex. (c) Raw image of Laser Induced Fluorescence : in bright, we visualize the dyed fresh water of the anticyclone surrounded by salted water in dark. (d) Superimposition of 50 raw images of PIV particles.
• Figure 5: Measured (a) horizontal density field $\rho$ and (b) horizontal geostrophic velocity $\vec{u_g}$ magnitude at a fixed vertical position $z_0 = -1$ mm in the vortex for $\textup{Ro} \simeq -0.15$. (c) Corresponding vorticity field $\vec{\omega} \cdot \vec{z} = \vec{\nabla}_h \times \vec{u_g}$. (d) Experimental Rossby number $\textup{Ro} = \omega_c / 2f$ as a function of dimensionless time $t \nu / R^2(t=0)$ where $R^2(t=0)$ is the characteristic radius of the vortex estimated at $t = 0$, compared to the theory of Facchini and Le Bars facchini2016lifetime (bottom axis : normalized by the viscous time, top axis : number of table rotations).