# An experimental investigation of a boundary current instability

Emmanuelle Thivolle-Cazat, Joël Sommeria

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Emmanuelle Thivolle-Cazat, Joël Sommeria. An experimental investigation of a boundary current instability. International symposium on shallow flows, Jun 2003, Delft, Netherlands. 10.1201/9780203027325. hal-00261116

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# An experimental investigation of a boundary current instability

E.Thivolle-Cazat & J. Sommeria

Laboratoire des Ecoulements Géophysiques et Industriels – Coriolis, Grenoble Cedex, France

**ABSTRACT:** This study deals with the instability of a surface boundary current maintained along a vertical lateral wall by the Coriolis force. The flowing water floats over a homogeneous denser layer. Experiments have been performed in the large “Coriolis” turntable (Grenoble, France). High resolution measurements of the horizontal velocity field, obtained by Particle Image Velocimetry (PIV), at different depths, bring consistent results to explain the mechanism of instability. We observe the formation of a counter-current in the homogeneous lower layer. Instability first develops in this lower layer, in the form of small vortices generated in the viscous layer near the wall. This is essentially a barotropic instability. In a second stage, a baroclinic instability develops, leading to the meandering of the surface current. This interpretation is supported by measurements of the growth rates and phase difference between both layers, compared with the simple two-layer model of baroclinic instability.

## 1 INTRODUCTION

Boundary currents are commonly produced by different mechanisms. The Atlantic water entering the Mediterranean Sea is deflected to the right by the Coriolis force, forming the Algerian current. Due to its low density (salinity), this current remains at the surface. Other boundary currents are generated by wind stress, like in the Gulf of Alaska, and are associated with upwelling effects.

Such boundary currents are maintained along the coast by the Coriolis force. They generally become unstable, producing meanders and detachment of vortices away from the coast. Eventually, these instabilities control the horizontal mixing of the current. They play therefore a major role in the distribution of the bio-geochemical components and their transfer between different basins.

Such instabilities are commonly observed in nature, e.g. in the Gulf of Alaska (Thomson et al. 1998) with seasonal alongshore winds. Such phenomena have also been observed in the Mediterranean Sea with the Algerian current (Millot 1999). Other studies in laboratory, on density fronts, have been carried out by many authors e.g. Chia et al. (1982) who modeled the formation of cyclonic eddies by baroclinic instability at upwelling fronts, or by Holford & Dalziel (1996) and Bouruet-Aubertot & Linden (2002). Obaton (1994) performed a systematic study in order to determine the parameters which lead to instability. In the present study, we reproduce similar experiments for an unstable case, with quantitative

measurements of the velocity fields in the lower and upper layers.

The instability of boundary currents is generally classified as a baroclinic instability. It can be indeed described by the classical Phillips model involving a uniform velocity in each layer (in the frame of the quasi-geostrophic approximation). However, a necessary condition for baroclinic instability is that the transverse gradient of potential vorticity is of opposite sign in both layers. Since the lower layer is initially at rest, its potential vorticity is uniform and it should remain so after the onset of the surface current. Therefore another mechanism is necessary to trigger the instability. In our experiments, the potential vorticity is obtained in both layers from Particle Image Velocimetry measurements. This will allow us to analyze in details the mechanism of instability.

This article is divided into six parts. In part 2, we present the experimental setup of our study and in part 3, we present the relevant parameters. In part 4, we present some qualitative and quantitative observations in both layers. We analyze the results giving estimation of growth rate and phase difference between layers in part 5 and compare them with theoretical results. We conclude in part 6.

## 2 EXPERIMENTAL SETUP

Laboratory experiments have been performed in the large Coriolis rotating tank (LEGI, France). The large dimensions of the turntable (13 m diameter)
<figure><img src="image_2.png" /><figcaption>Figure 1. Sketch of the experimental setup.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2. Vertical cross section of the current along the wall.</figcaption></figure>

provide large Reynolds and very small Rossby numbers and allow then relevant comparisons with oceanic flows.

Initially, an homogeneous water layer, with thickness H = 60 cm is at rest in the tank rotating with a period T = 50 s. A surface boundary current is produced along a straight wall corresponding to a diameter of the tank (Fig. 1). It is introduced by an injector with a triangular cross section (Fig. 2). The height h₀ and width L₀ of this triangle are adjusted to initiate a geostrophically balanced current. The steady

flow rate Q₀ is imposed by a volumetric pump, starting at an initial time t=0. The injector is filled with glass tubes to favor an homogeneous and laminar initial velocity. The density ρ₁ and ρ₀, respectively in the lower and upper layers, are controlled by salinity. We define g' = g(ρ₀ - ρ₁)/(ρ₁) the reduced gravity. We repeated many times the experiment to adjust the experimental setup. The occurrence of instability remained the same in terms of time and place of the first instability onset. However, the number of means and their amplitude differs a little because of initial perturbations and small residual currents.

Measurements of horizontal velocity fields have been realized by PIV at two different depths (1 cm and 21 cm from the surface) with high resolution in time and space, in a 2.5 m × 2.5 m domain. The first position provides the velocity field in the upper layer, while the second level remains always inside the lower layer. To do so, pliolite particles of density 1025 g/l are introduced in both layers, visualized thanks to a laser sheet and tracked by a digital camera. The PIV process is described in Fincham & Spedding (1997) and Fincham & Delerce (2000). In addition, the interface position is controlled using ultrasonic probes downstream from the PIV domain and density profiles are realized every two minutes using conductivity probes upstream from the PIV domain. Temperature probes, located next to the conductivity probes are used to check the uniformity of the temperature (realized within 0.5°C).

## 3 PHYSICAL PARAMETERS

Once the current is established, its dynamic depends on 7 physical parameters: the velocity U₀, the width L₀ and the initial height at the wall h₀ of the current, the Coriolis parameter f = 4π/T (where T is the rotation period of the tank), the height of the lower layer H, the viscosity ν and the reduced gravity g'. These parameters allow to form the following non dimensional parameters:

$$ - \text{the Burger number: } Bu = \frac{g' h_0}{f^2 L_0^2} = \left(\frac{rd}{L_0}\right)^2. $$

This number characterizes the competition between stratification and Coriolis effects. It is the ratio of the Rossby deformation radius r₉ over the current width L₀.

- the depth ratio: h₀/H.

- the aspect ratio: h₀/L₀

- the Froude number: $Fr = \frac{U_0}{\sqrt{g'h_0}}$.

- the Ekman number: $Ek = \frac{1}{\nu h_0}$.

The Ekman number quantifies the influence of viscous friction which tends to stabilize the current.
In our case, it is possible to write down two other relations which characterize the flow in a geostrophic and hydrostatic equilibrium:

- the geostrophic equilibrium: $L_0fU_0 = g'h_0$ (1)

- the propagation velocity: $U_0 = C\sqrt{g'h_0}$ (2)

In our experiments, we do not impose $U_0$ but the flow rate:

$$Q_0 = \frac{h_0 L_0 U_0}{2} \qquad (3)$$

From (1), (2) and (3), it results that:

$$- U_0 = C(2fg'Q_0)^{1/4} \qquad (4)$$

$$- h_0 = \left( \frac{2fQ_0}{g'} \right)^{1/2} \qquad (5)$$

$$- L_0 = \frac{1}{C} \left( \frac{2Q_0g'}{f^3} \right)^{1/4} \qquad (6)$$

These relations have been experimentally checked by Obaton (1994). We observe that the current head propagates at a velocity given by (4) with a coefficient $C_h = 0.9$. This is in agreement with Obaton (1994). However, we find that the velocity of the established parallel current, behind the head, is smaller with a coefficient $C = 0.5$. For each experiment, we adjust the injector height and width to satisfy (5) and (6). In the absence of such an adjustment, the same values $h_0$ and $L_0$ tend to be reached by the jet, after some transient oscillations.

Because of relation (1), the Burger number is given: $Bu = C^2 = 0.25$. Furthermore, using (2), the Froude number is given: $Fr = C = 0.5$ (note that the Rossby number $Ro = U_0/L_0f = Bu^{1/2}Fr$ is also of order unity). Therefore, the dynamics depends on three non-dimensional parameters: $h_0/L_0$, Ek and $h_0/H$. We are interested in small slopes $h_0/L_0$, like in oceanic cases. In this limit, consistent with the hydrostatic approximation, the dynamics should not depend on this slope anymore. We want to minimize the Ekman number, to approach inertial regimes. We have been led to adjust the values of the physical parameters at $h_0 = 9$ cm, $L_0 = 40$ cm, $H = 60$ cm, $Q_0 = 0.5$ l/s, $f = 0.25$ s⁻¹ and $g' = 3.10^{-2}$. This corresponds to an unstable case, with $h_0/L_0 = 0.225$, $E_k = 4.6 \cdot 10^{-4}$ and $h_0/H = 0.15$. Obaton (1994) indeed observed instabilities for $E_k < 1.5 \cdot 10^{-3}$ and $h_0/H > 0.08$.

## 4 OBSERVATIONS

The time evolution of the boundary current can be separated in two phases for both layers of fluid. In the

<figure><img src="image_4.png" /><figcaption>Figure 3. Photos of instability growth of the surface current along the straight wall for $t = 19$T, 25T, 26.5T and 30T (T: rotation period of the tank).</figcaption></figure>

upper one (the surface current), after the arrival of the head, the current stays laminar, adjusting to a width of about $3/4L_0$, during about 20 periods of rotation and then becomes unstable as can be seen in Figure 3. During the time of the experiment (~80T), we see the successive development of three similar instabilities. As one instability crosses the measurement field, moving with the current and growing, a new one starts its growth. In the lower layer, initially at rest, water is then driven in the direction of current propagation. Just after the arrival of the head, a very thin current moving in the opposite direction compared to the upper layer starts to propagate. This current propagation is periodically interrupted by small cyclonic and anticyclonic vortices moving along the wall in the direction of the surface current. After about 20 periods of rotation, these small-scale instabilities start to grow in phase with the upper layer. This corresponds to a probable development of baroclinic instability.

### 4.1 Growth of the instability

To try to explain how the instability develops in the boundary current, we represent the velocity fields obtained from PIV in the surface current and in the lower layer. The Figures 4a, b show respectively these velocity fields for the upper and the lower layer for time $t = 11$T. The wall is located on the left and the
current propagate from the top to the bottom of the field along the wall. For the representations, we used a regular grid of 50 × 50 vectors. In the upper layer, the current is still laminar adjusting to a width of 3/4L₀ with a velocity of about 2.5–3 cm/s.

In the lower layer, the current moves in the oppo-
site direction with a much smaller velocity of about
0.5 cm/s and is interrupted by small vortices near the
wall. The presence of such instabilities in the lower
layer tends to indicate that instability starts in this
layer. In a latter time, for t = 25T, as can be seen in the
Figures 4c, d, instabilities are growing in both layers.

In the upper one, an anticyclone is located near the
wall inside a meander of the main current and a cyclone

goes with this meander outside of the main current. In
the lower layer, we clearly see a dipole formed of an
anticyclone followed by a cyclone near the wall. These
dipoles are moving away from the measurements field
in the direction of the propagation of the current.

4.2 Velocity as a function of time

Figure 5 represents the longitudinal velocity u in the
current as a function of time. The maximum value
in velocity for time t ~ 2T corresponds to the arrival
of the head in the measurement field although the
average velocity of the head is a little bit smaller
(dash-dotted line). The velocity in the jet tends then to

<figure><img src="image_5.png" /><figcaption>Figure 4. Velocity fields in the upper (a), (c) and lower layer (b), (d) for time t = 11T and t = 25T.</figcaption></figure>
stabilize around 2.3 cm/s. The oscillations in velocity correspond to the jet detachment during the successive growth of instabilities. We plotted in Figure 6 the transverse velocity $v$ as a function of time, in one point located 20 cm away from the wall and 2 m away from the injector in order to get an overview of the different instabilities with the time. This figure shows a succession of four maxima of transverse velocity which should correspond to the end of instability growth. These peaks of transverse velocity are growing almost in phase in both layers. We distinguished the first peak from the others. In fact, as we looked at the experiments, we only saw three instabilities leading to a current meander in the upper layer (the last three

<figure><img src="image_6.png" /><figcaption>Figure 6. Transverse velocity $v$ (in cm/s) as a function of time (in rotation periods) for a point located at $y = 20$ cm away from the wall and $x = 2$ m away from the injector.</figcaption></figure>

ones for $t = 25$T, 45T and 60T). The first maximum in transverse velocity at $t = 11$T appears as a weak oscillation along the wall (compared to current velocity) in the first layer and as small vortices in the lower layer. This corresponds to the vector fields shown in Figure 4a, b. Observed at a fixed point, this instability saturates at a small amplitude (corresponding to the first velocity maximum in Figure 5), and it remains hardly visible in the upper current itself. In the following part, we shall look at these instabilities more precisely in terms of growth rate and phase differences.

# 5 ANALYSIS

## 5.1 Thickness of the current and of the interface

As we impose a condition of geostrophic equilibrium at the exit of the injector, it is necessary to control if the height imposed there, is the one measured at the place of the measurements. Profiles of density are realized by means of the conductivity probes, every two minutes upstream of the PIV field. An example of density profile across the interface air/upper layer and upper layer/lower layer is given in Figure 7. It shows first the uniform density in each layer. Secondly, it allows to determine the interface thickness between the two layers of fluid. The tangential fit function used gives an interface thickness of 0.2 cm. The comparison with two-layer models is then relevant.

Finally, density profiles give also the current thickness as a function of time (Fig. 8). A constant value, a little smaller than $h_0$ is found in less than 10 periods of rotation. The geostrophic equilibrium is still satisfied at this point of the current and we can use it as a

<figure><img src="image_7.png" /><figcaption>Figure 7. Density profile (in g/l) measured by a conductivity probe located at $x = 51$ cm downstream from the injector and $y = 3$ cm away from the wall, as a function of the depth (in cm). The solid line corresponds to the experimental results and the dotted one to a tangential fit function.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 8. Current depth (in cm) measured by a conductivity probe located at x = 51 cm downstream from the injector and y = 3 cm away from the wall as a function of time (in rotation periods).</figcaption></figure>

condition in the following part, to calculate potential vorticity profiles.

## 5.2 Potential vorticity conservation

Assuming that the lower layer is at rest at the beginning of the experiment, potential vorticity is uniform in this layer. A growth of baroclinic instability implies a presence of opposite potential vorticity gradients in both layers.

To determine the profile of potential vorticity without a measure of the depth of the current, it is necessary to assume that geostrophic equilibrium is satisfied after the injector (in the field of measurement). We controlled this point in part 5.1 showing that the depth of the current remains constant during the experiment in one point outside of the measurement field. We assume it is still true when the current crosses the PIV field. As the opposite current in lower layer, is very thin along the wall (~1/10 of L₀ which means less than 5 cm), we increase the resolution of vector fields from 10 000 to 100 000 vectors. We define potential vorticity in the surface current as followed with $q = ω + f/h$ and in the lower layer with $q = (ω + f)/(H - h)$. ω represents the curl of the transverse velocity and h the depth of the current. h is calculated with a geostrophic equilibrium assumption: $h = \int 2Ω/g'(U_1 - U_2)dy + h_0$. U₁ and U₂ represent respectively the velocity in the upper and lower layer. In Figure 9, profiles of velocity u are represented as a function of the distance to the wall for the upper and the lower layer for t = 16T.

On the first graph representing the upper layer, the velocity is negative and has a maximum value of about 3 cm/s in the center of the current jet. In the lower layer, there is an opposite current. The velocity has a

<figure><img src="image_9.png" /><figcaption>Figure 9. Velocity in the direction of the current propagation (in cm/s) for upper and lower layer as a function of the distance to the wall (in cm). The solid lines correspond to measurements form CIV and the dashed one to the theoretical profile assuming geostrophic equilibrium with potential vorticity conservation.</figcaption></figure>

positive value of about 0.2 cm/s in its maximum. It also decreases as in the upper layer, near the wall, because of the viscous layer. We represented with a dashed line, the theoretical velocity profile assuming geostrophic equilibrium with potential vorticity conservation. We estimate then a thickness of the viscous layer to a value of 3–4 cm. This configuration of opposite velocity in upper and lower layer leads to opposite gradients of potential vorticity which are necessary for baroclinic instability. The velocity shear in the lower layer can lead to barotropic instability as well.

## 5.3 Phase and growth rate of instability

Once we have seen that we have opposite gradients of potential vorticity (characteristic of baroclinic instability), we try to estimate values of growth rate in each layer and phase difference between both layers. We plot the transverse velocity maximum 50 cm away from the wall as a function of time. Figure 10 represents the position of the maximum in transverse velocity for both layers during the growth phase of the instabilities. As seen before, we distinguish two phases. Positions between both layers are the same for instability at time t = 10T and different for the following instabilities. In that case, the interval between both layers remains fairly constant during each event. The lower layer (dashed line) is in advance compared to the upper one (solid line). Comparing the form of the transverse velocity to a sinus function, the position difference between the minimum and the maximum values in the transverse velocity is equivalent to a half wavelength λ of a sinus function. x_min representing the
<figure><img src="image_10.png" /><figcaption>Figure 10. Position of transverse velocity maximum $x_0$ (in cm) as a function of time (in rotation periods). The dashed line represents the lower layer and the solid one the upper layer.</figcaption></figure>

position of the minimum and $x_{max}$ the position of the maximum, it comes: $x_{min} - x_{max} = \lambda/2 = \pi/k$ with k the wave number. Multiplying the difference between both positions by k, we obtain an estimation of the phase difference. We find $\Delta\phi = 0$ for the first instability and $\Delta\phi = 1.1 \pm 0.2$ rad for the following ones. We estimate also the growth rate of these instabilities in both layers. For the first instability, we find $0.001 \pm 0.0005 \text{ s}^{-1}$ in both layers. For the following ones, we find $0.006 \pm 0.0005 \text{ s}^{-1}$ for the upper layer and $0.002 \pm 0.0005 \text{ s}^{-1}$ for the lower layer. We give also an estimation of the velocity of the perturbation. We find an intermediate value between the velocity in the upper and lower layer which is the shear condition necessary for baroclinic instability. We obtain a value of $2.2 \pm 0.5 \text{ cm/s}$.

From these results, we conclude that we have two phases in the development of the instability: a first barotropic (with equal growth rate in both layers and no phase difference between layers) and a second one baroclinic (with constant phase difference and growth rate of same order in both layers). The smaller growth rate in lower layer may be due to the fact, that instability in this layer has already finished its growth.

## 5.4 Comparison with the two-layer model of Phillips for baroclinic instability

We compare the growth rate and phase difference in the experiment with the theoretical predictions from the linear instability results of Phillips (1954).

In this model, the two-layer flow of inviscid fluid is constrained in an infinitely long, vertically sided channel between $y = 0$ and $y = R$, which is rotating about a vertical axis with rotation rate $1/2$ f. The velocities in each layer $U_1$ and $U_2$ are independent of y but differ

in magnitude. The wave number is represented by K and the phase speed by c which can be decomposed in real part $c_r$ and imaginary part $c_i$. The growth rate is given by the imaginary part $c_i$ of the wave speed:

$$ \delta m = Kc_i = K \times \frac{Us(4F_1F_2 - K^4)^{1/2}}{2(K^2 + F_1 + F_2)} \quad \text{where} \\ Us = U_1 - U_2, \quad Fn = \frac{f^2 R^2}{g' Hn} \quad \text{and} \quad \delta = \frac{H_1}{H_1 + H_2} $$

We also consider the phase difference which is characteristic of baroclinic instability. If we take the tangent of the ratio of wave amplitude in upper layer with wave amplitude in lower layer (Pedlovsky 1979), we obtain the following expression:

$$ \tan \varphi = \frac{ci \times Us \times F_2}{|c - U_2|^2 (K^2 + F_2) - F_2 \times Us \times cr - U_2} $$

where $c_r$ is given by: $c_r = (U_1 + U_2)K^2/(2(K^2 + F_2/\delta))$. With the conditions of the experiments, $\delta = 0.15$, $Us = 0.025$ and $F = 0.6$, we obtain with the theory a growth rate equal to $0.004 \text{ s}^{-1}$ and a phase difference of $1.1$ rad. These values corresponds to the ones calculated in the experiment. This leads to the conclusion that a comparison with the simple two-layer model of Phillips for baroclinic instability is relevant.

# 6 CONCLUSION

In this paper, we showed that the current instability is initiated by a barotropic instability occurring in the viscous layer of the counter-current in the lower layer. Indeed there is no phase difference between the upper layer and the lower. In a second stage, a baroclinic instability develops and lead to the detachment of a meander in the boundary current. This second stage is characterized by a constant phase difference of perturbations between the two layers (the lower layer is in advance compared to the upper one). Values found for the growth rate and the phase difference in this baroclinic phase is found in reasonable agreement with the classical Phillips model for a two-layer system.

# REFERENCES

* Bouruet-Aubertot, P. & Linden, P.F. 2002. The influence of the coast on the dynamics of upwelling fronts: Part 1. Laboratory experiments. *Dyn. Atmos. Oceans*. 36: 153–173.

* Chia, F., Grifiths, R.W. & Linden, P.F. 1982. Laboratory experiments on fronts. Part 2: the formation of cyclonic eddies at upwelling fronts. *Geophys. Astrophys. Fluid Dyn.* 19: 189–206.

* Fincham, A.M. & Delerce, G. 2000. Advanced optimization of correlation imaging Velocimetry algorithms. *Experiments in Fluids*. Suppl: 13–22.
Fincham, A.M. & Spedding, G.R. 1997. Low cost, high resolution DPIV for measurement of turbulent fluid flow. *Experiments in Fluids* 23: 449-462.

Holford, J.M. & Dalziel, S.B. 1996. Measurements of layer depth during baroclinic instability in a two-layer flow. *Appl. Sc. Res.* 56: 191-207.

Millot, C. 1999. Circulation in the Western Mediterranean Sea. *J. Marine Sys.* 20: 423-442.

Obaton, D. 1994. Etude expérimentale de la stabilité d'un courant côtier de gravité-Application au courant algérien. PhD Thesis of Joseph-Fourier university in Grenoble.

Pedlovsky, J. 1979. *Geophysical Fluid Dynamics*. New-York: Springer Verlag.

Phillips, N.A. 1954. Energy transformations and meridional circulations associated with simple baroclinic waves in a two level quasi-geostrophic model. *Tellus* 6: 273-286.

Richardson, P.L., Bower, A.S. & Zenk, W. 2000. A census of meddies tracked by floats. *Prog. Oceanog.* 45: 209-250.

Thomson, R.E. & Gower, J.F.R. 1998. A basin-scale oceanic instability event in the Gulf of Alaska. *J. Geophys. Res.* 103(C2): 3033-3040.