Fluctuating mesoscale frontal features: structures and manifestations in the real ocean

Transcription

1 Fluctuating mesoscale frontal features: structures and manifestations in the real ocean Angelo Rubino Universität Hamburg Hamburg, 25

2 Content 1 Introduction Structure of fluctuating mesoscale frontal features Analytical solutions for circular, warm-core eddies of the nonlinear reducedgravity, shallow-water equations Warm-core eddies studied by laboratory experiments and numerical modeling The decay of stable frontal warm-core eddies in a layered frontal model Near-inertial oscillations of geophysical surface frontal currents Fluctuating mesoscale frontal features in the real ocean Surface fronts in the Rhine outflow area Intraseasonal fluctuations in the Southwestern Arabian Sea and their relations to the dynamics of the Great Whirl Outlook Acknowledgement References

3 1 Introduction The oceanic dynamics is composed of very different, mostly interacting processes whose time (space) scales span from seconds (millimeters) to millennia (global scale). To such a huge variability mesoscale phenomena contribute considerably (see, e.g., Charney 1971; Robinson 1983; Olson et al. 1985; Esenkov and Cushman-Roisin 1999; Stammer and Wunsch 1999; Stammer et al. 21). Having spatial scales ranging from a few to hundreds of kilometers and temporal scales ranging from hours to many weeks, the mesoscale variability represents a crucial link between small-scale phenomena and larger-scale circulation patterns. Ocean mixing largely affects pollutant dispersal, strongly influences marine productivity, and crucially contributes to determine global climate (Gregg 1987); it mainly results from breaking internal gravitational waves (see, e.g., Munk and Wunsch 1998; Alford 23). Together with the semidiurnal internal tides, near inertial internal disturbances represent the major source of energy available for the production of mixing and turbulence in the interior ocean (Munk and Wunsch 1998; Alford 2). In the transmission of these disturbances toward the abyss, oceanic mesoscale features and their oscillations play a fundamental role (Kunze 1985; Garrett 21; Alford 23; Rubino et al. 23a; Xing and Davies 24a; Xing and Davies 24b). The influence of frontal, warm-core eddies like those generated by the meanders of the Gulf Stream, but also like the smaller-scale fresh-water lenses often existing in coastal regions of fresh water influence, on the larger-scale ocean circulation involves an important transfer of energy and physical, chemical, and biological properties across frontal zones and also profoundly affects oceanic mixing (see, e.g., Olson 1991; Hessner et al. 21). The Great Whirl develops as strong anticyclonic eddies in the southwestern Arabian Sea every year in response to the onset of the Southwest Monsoon. It contributes substantially to the upwelling along the northern Somali coast and to the eastward transport of cold water into the interior Arabian Sea and hence strongly affects the heat balance of the Arabian Sea (Schott et al. 22). At high latitudes, smaller mesoscale frontal eddies, which detach from the core of convectively generated oceanic chimneys, fundamentally contribute to the horizontal spreading of newly formed dense water (Marshall and Schott 1999) and may exert a profound influence on convective preconditioning (Gascard et al. 22; Budéus et al. 24; Wadhams et al. 24; Androssov et al. 24). In determining the exchange flow between different basins, mesoscale frontal activity, often related to internal waves and vortices, may play an important role (see, e.g., Käse et al. 23; Rubino et al. 23b; Brandt et al. 24). In the approaches to sea straits, mesoscale undular activity may reflect (and serve as an indicator for) aspects of the larger-scale circulation and variability (see, e.g., Armi and Farmer 1988; Brandt et al. 1999; Pierini and Rubino 21; Rubino et al. 21; Izquierdo et al. 21; Brandt et al. 24). The examples quoted above are certainly not exhaustive of the countless existing oceanic interactions in which mesoscale oceanic features and their fluctuations are involved. They clearly indicate that a better understanding of the dynamics of mesoscale oceanic features and their fluctuations is an essential step toward a deeper comprehension of the oceanic dynamics and variability. This relevance explains why mesoscale phenomena are commonly analyzed in their (very) different aspects. Their intrinsic dynamics, their interaction with the exterior 2

4 ocean, their short-term as well as long-term variability, as well as their activity in key regions like, e.g., sea straits or coastal regions of fresh water influence have been actively investigated during the last years. The present summary paper is focused on investigations, carried out by the author, aimed at elucidating aspects of the intrinsic dynamics and of the interaction with the exterior ocean of fluctuating mesoscale oceanic frontal features such as frontal vortices and coastal currents and at describing aspects of their manifestations in the real ocean. These investigations were carried out using different methods: frontal surface vortices were investigated analytically and numerically and produced in laboratory experiments; fluctuations associated with their activity were observed in current and temperature data; their surface signatures were detected in Synthetic Aperture Radar (SAR) data as well as in TOPEX/POSEIDON sea level anomaly data. Finally, mesoscale surface frontal currents were investigated analytically as well as numerically. The paper is divided in two parts. The first one is based on the following individual contributions: Rubino, A., P. Brandt, and K. Hessner, 1998b: Analytical solutions for circular eddies of the reduced-gravity, shallow-water equations. J. Phys. Oceanogr., 28, ; Rubino, A., K. Hessner, and P. Brandt, 22: The decay of stable frontal warm-core eddies in a layered frontal model. J. Phys. Oceanogr., 32, ; Rubino, A. and P. Brandt, 23: Warm-core eddies studied by laboratory experiments and numerical modeling, J. Phys. Oceanogr., 33, ; Rubino, A., S. Dotsenko, and P. Brandt, 23a: Near-inertial oscillations of geophysical surface frontal currents, J. Phys. Oceanogr., 33, In this part analytical and numerical methods as well as experimental facilities are employed to elucidate the nature of mesoscale surface frontal vortices and coastal currents of special forms in the reduced-gravity and two-active-layer environment. A new class of pulsating mesoscale vortices has been discovered as analytical solutions of the non-stationary, nonlinear reduced-gravity shallow-water equations on the f-plane (Rubino et al. 1998b). It constitutes an important extension of previous solutions (Cushman-Rosin 1987; Rogers 1989) to more arbitrary shapes and velocity structures and paves the way to further developments (see, e.g., Dotsenko and Rubino 23). The detectability of the fluctuations investigated theoretically as well as the degree of realism which can be ascribed to the new solutions are investigated experimentally (Rubino and Brandt 23) using the rotating tank of LEGI/Coriolis (Grenoble, France). Owing to its 13 m diameter, this experimental facility allows for the production of vortices having aspect ratios as well as Rossby and Burger numbers typical of geophysical warm-core eddies. It has been thus demonstrated that the new solutions rather than previous solutions describe spatial and temporal structure of mesoscale frontal vortices emerging in laboratory experiments from the impulsive release of a buoyant cylinder (Rubino and Brandt 23). These findings are consistent with recent, accurate in situ observations of the velocity structure of oceanic mesoscale vortices (Gascard 24). Analytical solutions of unsteady nonlinear equations describing geophysical phenomena are quite rare and, apart their intrinsic value and fascination, they can be used to test quantitatively the accuracy of numerical models. To this extent it has been shown for the first time that the numerical simulation of both near-inertial as well as long-term dynamics of frontal surface vortices is feasible using frontal numerical models equipped with techniques 3

5 for the treatment of movable lateral boundaries (Rubino et al. 22). New analytical solutions of the reduced-gravity, nonlinear shallow-water equations on the f- plane describing fluctuating mesoscale coastal features have been found. They describe a class of superinertially pulsating frontal surface currents. In a linear context, disturbances superimposed on such currents are standing waves within the bounded region. Their frequencies are inertial/superinertial for the first mode/higher modes. In the same frame, a zeroth mode, referring to the superposition of an inertial wave on a background vorticity field would formally yield subinertial frequencies (Rubino et al. 23a). These results have profound implications for the study of the transformation of near-inertial motion in an active ambient ocean. In the upper ocean, the energy of the wind excites inertial waves having similar spatial scales as the generating atmospheric fronts. Such waves need to experience a variation in their frequency in order to be able to rapidly transfer their energy downward. Interactions of inertial waves with inhomogeneities of the ambient vorticity field are able to produce such a frequency shift (Kunze 1985). But, on the other hand, many of the same inhomogeneities possess intrinsic modes of near-inertial oscillations, which would possibly generate near-inertial waves in an active ambient ocean (Alford 23; Rubino et al. 23a). The second part of the work is based on the following individual contributions: Hessner, K., A. Rubino, P. Brandt, and W. Alpers, 21: The Rhine outflow plume studied by synthetic aperture radar images and numerical modelling, J. Phys. Oceanogr., 31, ; Brandt, P., M. Dengler, A. Rubino, D. Quadfasel, and F. Schott, 23: Intraseasonal variability in the southwestern Arabian Sea and its relation to the seasonal circulation, Deep-Sea Res. II, 5, In this part aspects of the dynamics of fluctuating frontal mesoscale vortical features as observed in the real ocean are analyzed. The numerical methods developed for the academic simulations of frontal vortices have demonstrated to constitute a powerful tool for investigating mesoscale frontal features of the real ocean. In fact a numerical model similar to that used by Rubino et al. (22) has been employed to reproduce surface frontal features emerging at the mouth of the river Rhine, in the European North Sea (Hessner et al. 21). Here, strong frontal features separating sea water and river plume were detected in different Synthetic Aperture Radar (SAR) imageries acquired by the First and Second European Remote Sensing Satellites (ERS-1 and ERS-2). The analysis of SAR images suggests that their form and location are mainly linked to the semidiurnal tidal phase in the outflow region. The numerical simulations corroborate this conjecture. They allow for a description of the position of the river Rhine buoyant plume within a tidal cycle (Hessner et al. 21). Finally, strong intra-seasonal fluctuations observed in the southwestern Arabian Sea, an oceanic region where the Great Whirl is present during the Southwest Monsoon, are analyzed using TOPEX/POSEIDON altimeter data and in situ current and temperature data. Instabilities of the flow in the transition region between the Southern Gyre and the Great Whirl are identified as likely sources of these fluctuations (Brandt et al. 23). The overall view of the ocean emerging from these investigations serves to corroborate the awareness that a zoo of oceanic frontal fluctuating mesoscale features exists, whose dynamics may play a larger role than believed in the past in determining the observed equilibrium and variability of the World Ocean. Moreover, from the work presented here it strongly emerges that process-oriented studies in a simplified context can significantly contribute to the comprehension of oceanic phenomena as they allow processes at work in them to be readily diagnosed, and hence constitute a valuable instrument to face the huge 4

6 complexity inherent in the dynamics of the real ocean. 5

7 2 Structure of fluctuating mesoscale frontal features 2.1 Analytical solutions for circular, warm-core eddies of the nonlinear reduced-gravity, shallow-water equations Among the most energetic features characterizing the mesoscale circulation in the nearsurface ocean are frontal, warm-core eddies. These anticyclonic vortices consist of rotating masses of anomalous (lighter than the ambient) water that, at the sea surface, are separated from the surrounding water by a closed frontal line. In the last three decades, owing particularly to the advent of satellite oceanography, the existence of frontal, warm-core eddies was revealed in all of the World Ocean, their most famous representatives being perhaps the warm-core rings released from Gulf Stream meanders (see, e.g., Saunders 1971; Joyce 1984). Typically these eddies are characterized by almost circular surface areas, surface radii of 2-4 times the internal Rossby radius of deformation, maximum thicknesses of 3-6 m, and maximum horizontal velocities of 1-2 m/s (Saunders 1971; Joyce 1984). As frontal, warm-core eddies may contribute substantially to the transport of temperature, salt, momentum, as well as chemical and biological properties across frontal zones, they are thought to exert a significant influence on the heat, energy, and ecological equilibrium of large oceanic regions (see, e.g., Olson 1991). The large amount of existing observational data showing aspects of the dynamics of frontal, warm-core eddies finds its counterpart in a great number of laboratory experiments and theoretical studies devoted to understanding the general nature of this phenomenon (see, e. g., Csanady 1979; Pavia and Cushman-Roisin 1988, Rogers 1989; Olson 1991; Rubino et al. 1998b; Rubino et al. 22; Rubino and Brandt 23). One of the most commonly used theoretical models for the study of frontal, warm-core eddies consists in the nonlinear, reduced-gravity, shallow-water equations on an f-plane (Csanady 1979; Pavia and Cushman- Roisin 1989; Ochoa et al. 1998; Rubino et al. 22; Rubino and Brandt 23). The degree of realism which can be achieved by using these equations is limited: they don t allow for important oceanic processes such as baroclinic instabilities, interactions between eddy and barotropic or baroclinic currents or features of the bottom topography, or energy radiation via internal wave emission. Nevertheless, by solving these equations, many fundamental characteristics of the eddy dynamics can be studied. Moreover, they admit analytical, nonstationary solutions which describe the temporal and spatial evolution of a special class of frontal, warm-core eddies (Cushman-Roisin 1987; Rogers 1989, Rubino et al. 1998b). The fundamental representative of such class is the pulson (Cushman-Roisin 1987). It is characterized by a paraboloidic shape and horizontal velocity components which are linear functions of the horizontal coordinates. This solution consists substantially of a pulsation of the eddy in which contractions and deepenings, expansions and shoalings alternate during an exact inertial period. Actually, observed frontal, warm-core eddies often differ remarkably from the pulson either in their velocity field or in their shape (see, e.g., Evans et al. 1985). Assuming circular symmetry, the unsteady nonlinear, reduced-gravity, shallow-water equations for a rotating system expressed in cylindrical coordinates are: t v vrv + vr + + fvr =, (1) r r v ϑ ϑ ϑ 6

8 vr t 2 v vϑ v fvϑ g h r + r + ' =, (2) r r r h hvr hvr + + =. (3) t r r Here, h is the thickness of the upper layer, f is the (constant) Coriolis parameter, g is the reduced gravity, t is the time, and v ϑ and v r represent the azimuthal and radial velocity components of the upper layer along the ϑ and r coordinates, respectively. Positive v r are directed from the origin of the coordinate system, located at the eddy center, toward the eddy periphery, and positive v ϑ are directed counterclockwise. Solutions of (1)-(3) are searched which represent unsteady circular, frontal, warm-core eddies with the following velocity field and shape: vϑ = n i L i r 2 1, (4) i= 1 v = r Kr, (5) h = 2n 1 i A i r 2, (6) i= where the coefficients L i, K, and A i are functions of time only and n 1 is the order of the system. Inserting (4)-(6) into (1)-(3) yields the following series of ordinary differential and algebraic equations: dl dt i + 2i KL δ fk =, i = 1...n, (7) i i1 i dk 2 δi 1 L jli j+ 1 + δi 1 K + f Li + 2ig' Ai =, i = 1...2n-1, (8) dt j= 1 da dt i + 2( i + 1) KA =, i =...2n-1, (9) i where δ i1 is the Kronecker delta and L i = for i > n. The general solutions of (7)-(9) are: ~ f Li Li = δi + i, γ sin ft + ϕ i = 1...n, (1) [ ( )] 7

9 K = fγ 2 1 cos ( ft + ϕ) ( ft ϕ) [ + γ sin + ], (11) A i = ~ A [ 1 + γ sin( ft + ϕ) ] i i+ 1, i =...2n-1, (12) where γ, ϕ, and A ~ are arbitrary constants of integration. The range of applicability of these ~ parameters is: γ < 1, ϕ < 2π, and A >. The last condition is required for the description of eddies whose fluid circulates inside of a surface frontal line. The coefficients ~ A i and L ~ i are related to each other by: ( 1 γ ) 2 2 i f 2ig' A ~ = δ 1 L ~ L ~ 4 i i + j i j+ 1 j= 1, i = 1...2n-1, (13) with L ~ i = for i > n. Together with conditions (13), the solutions of (1)-(3), which describe circular, frontal, warm-core eddies, are: v ϑ n f ~ r = r L i 2 i= 1 1 2i 1 i [ + γ sin( ft + ϕ) ] ( ft ϕ) ( ft ϕ) f v = γ cos + r 2 1 [ + γ sin + ], (14) r, (15) h = 2n 1 A ~ i i= r 2i [ 1 + γ sin( ft + ϕ) ] i+ 1. (16) In analogy to the circular pulson solution, these solutions describe circular eddies which alternatively contract and deepen, expand and shallow during an inertial period T = 2π / f. Warm-core eddies whose azimuthal velocities even largely differ from a linear function of the radius and/or whose shapes even largely differ from a paraboloid exist as analytical solutions of (1)-(3). 8

10 Fig. 2.1: Interface depth h (a), azimuthal velocity component v ϑ (b), and radial velocity component v r (c) of a circular, frontal, warm-core eddy solution of (1) (3) (n=4) for the times t =, t = 1 4T, t = 1 2T, and t = 3 4T ( T = 2π f ) as function of the radius r (from Rubino et al. 1998b). One example is depicted in Fig Here, the depth of the interface separating the eddy from the ambient water (Fig. 2.1a), the azimuthal (Fig. 2.1b), and the radial velocity components of the eddy (Fig. 2.1c) are shown as functions of the radius for four different times of an inertial period. In this case the maximum steepness of the interface depth is not located at the eddy periphery, as it would be in the case of the pulson. Accordingly, the associated azimuthal velocity decreases from its maximum, located inside of the eddy, toward the periphery, where it is almost zero (Fig. 2.1b). Thus, due to this characteristic of the new solutions, a more realistic description of eddy dynamics is possible than it is possible by using the previous circular pulson solution (Cushman-Roisin 1987), which predicts unrealistically large azimuthal velocities at the eddy periphery. Furthermore, a central region (core) of the eddy exists, where the interface depth is almost constant (Fig. 2.1a). Accordingly, the associated azimuthal velocity is, in this region, almost zero (Fig. 2.1b). This characteristic of our solutions corresponds to a frequently observed characteristic of frontal warm-core eddies (warm-core eddies having an almost motionless core) which also cannot be described by the pulson solution. The presented solutions could be used to infer a possible internal structure of frontal, warmcore eddies from surface measurements. For example, remote sensing techniques give the possibility to measure the spatial extent of frontal eddies. If this information, together with the surface eddy velocity structure is provided, the coefficient L ~ i can be calculated by fitting the ~ measured velocity by a polynomial. Thus, using (13), the coefficients A i ( i 1) can be calculated, which describe the spatial structure of the eddy. The center depth ~ A can thus be obtained by letting the measured and calculated eddy surface radius coincide. In this way, the whole stationary structure of the eddy can be determined. 9

11 2.2 Warm-core eddies studied by laboratory experiments and numerical modeling In the frame of the reduced-gravity approximation, it is of great importance to investigate which equilibrium emerges from the impulsive release of a motionless cylinder of buoyant fluid in a deep environment on an f-plane (see, e.g., Csanady 1979). In fact, by the analysis of such equilibrium valuable information about the resulting vortex oscillations and structure of its shape and velocity as well as about the experimental occurrence and robustness of the different solutions considered in the previous section (in particular first-order versus higherorder solutions) could be retrieved. The experimental results presented here (Rubino and Brandt 23) were obtained using the rotating tank of LEGI/Coriolis (Grenoble, France), which, owing to its 13 m diameter, allows for the investigation of vortices having aspect ratios as well as Rossby and Burger numbers typical of geophysical warm-core eddies. The spatial structure of the vortex radial and tangential velocity components was analyzed using the experimental results as well as numerical simulations carried out by means of a nonlinear, reduced-gravity frontal model (Rubino et al. 22; see also next section for further details about the model). The vortices were produced experimentally in a system brought to solid body rotation by rapidly lifting a 2 m radius bottomless cylinder containing fresh water immersed in a salty ambient fluid. Time series of interface displacement were measured using a set of interface followers located at different distances from the vortex center. Moreover, time series of the vortex radial and tangential velocity were measured using ultrasonic Doppler probe that have a precision of about.1 mm/s for velocities in the range -2 cm/s. The parameters characterizing the experiment to which I will refer in the present section are: 3 relative density difference ρ ρ =.925 1, initial eddy thickness H = 1 cm, initial eddy radius L = 2 m, total water depth D = 86 cm, and rotation period of the tank T = 12 s. Note that for this kind of parameters the reduced-gravity assumption is valid for the description of the two-layer vortex evolution during at least the first 1 inertial periods (Rubino et al. 22). Given the above mentioned vortex parameters, the vortex Burger g ρ ρ H number is Bu = =.2, which belong to the range of Burger numbers typical for 2 2 f L warm-core eddies in the ocean. 5 1 t= Interface Position [cm] t=1/4 T t=1/2 T Distance from Vortex Center [m] 1 t=3/4 T Fig. 2.2: Interface depth at the initial stage of the experiment as well as at three phases (t=1/4t; t=1/2t; t=3/4t, where T is the inertial period) after the lift of the bottomless cylinder (stars). Also included are the simulated curves. Note that the tank bottom is at z = -86 cm (from Rubino and Brandt 23).

12 In Fig. 2.2 the interface depth at the initial stage of the experiment as well as at three successive phases after the lift of the bottomless cylinder as obtained experimentally and as simulated numerically are depicted. After the cylinder is lifted, a strong vortex expansion occurs (Fig. 2.2, t = 1/ 4T and t = 1/ 2T ), characterized by a strong increase of the vortex radius and a strong decrease in the vortex thickness. This phase is followed by a remarkable vortex contraction (Fig. 2.2, t = 3/ 4T ) connected with a deepening of the vortex core. The temporal evolution of the spatial structure of the vortex radial velocities is assessed using a linear fit. In Fig. 2.3 the time series of the radial velocity at 1.2 m, 1.6 m, and 2.1 m from the vortex center (top panel), the slope of the calculated fit (middle panel) and the norm of the obtained residuals (bottom panel) are presented. Strong inertial oscillations are found, after an initial adjustment phase of O(1 inertial period), in the alternating sign of the slope. These are still well visible after 1 inertial periods (Fig. 2.3). The norm of the residuals, instead, rapidly becomes small, suggesting a rapid evolution of the spatial structure of the vortex radial velocity toward linearity. Velocity [cm/s] Radial Velocity Slope of Linear Fit 1.2 m 1.6 m 2.1 m Slope [1/s].5 3 Norm of Residuals of Linear Fit Norm [cm/s] Nondimensional Time t/t Fig. 2.3: Time series of the radial velocity at three positions (1.2 m; 1.6 m; 2.1 m) within the produced vortex (top panel), of the slope of the linear fit to the spatial structure of the radial velocity (middle panel), and of the norm of the residuals of the linear fit (bottom panel). The figure is from Rubino and Brandt (23). This experimental result is consistent with the results of our numerical simulations carried out using a nonlinear frontal model similar to that of Rubino et al. (22). In fact the norms of the residuals of the simulated vortex radial velocity becomes small as time elapses (Fig. 2.4, top panel). This fact indicates a tendency of the simulated radial velocity toward linearity. Instead, the evolution of the structure of the simulated vortex tangential velocity is quite different: from its norm it can in fact be evinced that the spatial structure of this velocity component rapidly attains a quasi-periodic state that is far from linearity (Fig. 2.4, top panel). It also significantly deviates from quadraticity (Fig. 2.4, middle panel), while a much better agreement with a cubic structure is found (Fig. 2.4, bottom panel). 11

13 6 4 Linear Fit Tangential Velocity Radial Velocity 2 6 Quadratic Fit Norm [cm/s] Cubic Fit Nondimensional Time t/t Fig. 2.4: Simulated norm of residuals to the linear (top panel), quadratic (middle panel), and cubic fit (bottom panel) of radial (dashed lines) and tangential velocity (solid lines). The figure is from Rubino and Brandt (23). This deviation from linearity means that the vortex rotation deviates from solid body rotation, and hence that the vortex relative vorticity is in general not constant within the vortex core. Our results indicate that, indeed, vortices characterized by a velocity structure consistent with the analytical solution proposed by Rubino et al. (1998b) rather than with previous solutions emerge by the impulsive release of a motionless cylinder of buoyant fluid in a deep environment on an f-plane. 2.3 The decay of stable frontal warm-core eddies in a layered frontal model Analytical solutions of nonlinear equations describing aspects of the non-stationary dynamics of geophysical fluids like those described analytically in Sect. 2.1 and reproduced experimentally in Sect. 2.2, apart their intrinsic value and fascination stressed above, offer an ideal possibility to test quantitatively and systematically the accuracy of numerical models (Cushman-Roisin 1987; Sun et al. 1993). Such an opportunity is very valuable, especially in the case of numerical simulations of frontal, warm-core eddies, where one is faced with the treatment of a surface front. Its numerical description is in fact non-trivial, as it requires the implementation of special algorithms allowing for the free horizontal eddy expansion and contraction. A continuously stratified model could thus be thought as the most suitable tool for describing realistically the dynamics of localized water masses having an outcropping interface as well as their long-term behavior including their frictional and non-frictional decay. However, an accurate description of the temporal and spatial evolution of a surface front requires an often unsustainably high computational effort as it necessitates a high horizontal and vertical resolution over large areas of the integration domain (Esenkov and Cushman-Roisin 1999). For this reason more efficient simulation methods have been proposed which derive, substantially, from two different approaches to the problem of describing horizontal discontinuities in the water density. Pavia and Cushman-Roisin (1988) adapted to oceanography the particle-in-cell method: the frontal line can be traced here as the envelope of the particles contained in the surface layer. However, no attempt was made, using this 12

14 numerical technique, to quantify the eddy decay due to numerical energy dissipation as well as due to energy dissipation caused by frictional effects, nor a test elucidating the capability of such a model of reproducing existing nonstationary analytical solutions that describe frontal vortices was performed. The second approach consists in the development of numerical models equipped with techniques for the treatment of movable lateral boundaries (Backhaus 1976; Burchard 1995; Rubino et al. 1998a; Rubino et al. 22). In the frontal, layered model presented here for the quantification of the eddy long-term evolution as well as for the description of aspects of the eddy energy dissipation such a technique allows for the description of the temporal and spatial evolution of a localized layer with an outcropping interface on the top of a dynamically passive or active environment. The pulson can be considered as the solution of the first order (n=1) of the set of solutions presented in Sect It is a nonlinear, oscillating, frontal anticyclonic surface eddy whose thickness h has the form of a circular paraboloid and whose horizontal velocity components u and v are linear functions of the horizontal coordinates x and y: 2 H 2 2 h = α H α ( x + y ), (21) R 2 1 f g H u = f x ' αβ α γ 2 f R 2 2 y, (22) f g H v = x f y f R α 1 γ 2 8 ' αβ, (23) where α = 1 and β = γ cos( ft + ϕ). (24) 1 + γ sin( ft + ϕ) Here H represents the mean thickness of the pulson at its center (which, in our notation, coincides with the origin of the coordinate system), R its mean radius at the sea surface, and γ and ϕ the strength and phase of the oscillations that characterize the pulson evolution. The basic experiment of our simulations was performed for a pulson characterized by H = 4 m, R = 12 km, γ =.1, f = s -1, g' = ms -2, A h =, c dil =, c diq =, λ =, and ϕ = -π/2 for t =. The horizontal grid steps x and y and the time step t were chosen to be 12 m and 36 s respectively. The numerical simulations were carried out over 15 inertial periods T = 2π/f as, for a pulson characterized by the above mentioned parameters, this time is of the same order as the time needed for nonstationary frontal processes to develop in the frame of the frontal geostrophic dynamics (Cushman-Roisin 1986; Pavia and Cushman-Roisin 1988). In order to illustrate the long-term evolution of the simulated pulson we compared the temporal evolution of the analytical pulson radius R at the sea surface: R R = α, (25) of the analytical pulson center depth H: H = α H, (26) of the analytical potential energy density E p : E p = 1 g' α H, (27) 3 of the analytical kinetic energy density E k : 13

15 1 8 g H 4g H Ek = R f ' ' γ α f R f R 2 2, (28) and of the analytical total energy density E = E p + E k : 1 8g H E = R f ' γ 12 2 f R 2 (29) with the temporal evolution of the corresponding quantities as obtained by using the numerical model. Figure 2.5 shows the temporal evolution of the normalized pulson thickness at its center (Fig. 2.5a), of the normalized pulson radius at the sea surface (Fig. 2.5b), and of the normalized pulson potential, kinetic, and total energy densities (Fig. 2.5c) as calculated analytically and as calculated by using our numerical model. The simulation results show that our numerical model is able to simulate with a very good accuracy the main characteristics of the pulson evolution. From the time series of pulson radius, center depth, as well as potential and kinetic energy density, the presence of inertial oscillations is evident. In particular, potential and kinetic energy density pulsate with the same amplitude and are exactly in phase opposition. Also a predicted asymmetry of the analytical pulson solution is captured by our numerical model: the period spent by the pulson in its shallow-wide state is larger than the period spent by the pulson in its deep-narrow state. A deviation between analytical and numerical solution can be noted in each of the curves presented in Fig However, after 15 inertial periods more than 92% of the initial total energy density is conserved by our numerical model. Assuming an exponential decay of the total energy density associated with the simulated pulson, we may derive an e-folding time, $T, for the total energy density of the simulated pulson. For the present simulation it results $T 192 d. This means that the lifetime of our numerical pulson is comparable with the lifetimes of surface frontal eddies observed in the World Ocean (see, e.g., Olson 1991). 1,15 / H H / R R E / E 1,1 1,5 1,,95,9 1,1 1,5 1,,95,9 1,2 1,,8,6,4,2, E/ E E E p/ E E k / t/ T a) b) c) 14

16 Fig. 2.5: Time series of the normalized pulson thickness at its center (a), of the normalized pulson radius at the sea surface (b), and of the normalized pulson potential, kinetic, and total energy densities E p, E k, and E (c) for the basic experiment described in the text. Solid lines denote the results of our numerical simulation, dotted lines are obtained analytically. Note that E represents the (analytical) pulson total energy density (from Rubino et al. 22). In order to elucidate aspects of the frictional decay of stable frontal warm-core eddies, we discuss how interfacial (linear as well as quadratic) friction, harmonic horizontal momentum diffusion, as well as linear ambient-water entrainment contribute to the eddy decay. The results of many simulations show that, while numerical dissipation, harmonic eddy viscosity, and linear entrainment represent negligible contributions to this decay of the warm-core eddy, linear and quadratic interfacial friction are able to considerably speed down eddy swirl and oscillations. / H H 1,1 1,,9,8,7,6 1,4 1,3 a) R / R 1,2 1,1 1,,9 1,,8 E/ E b) E / E,6,4,2, E E p/ E E k / t/ T c) Fig. 2.6: Time series of the normalized pulson thickness at its center (a), of the normalized pulson radius at the sea surface (b), and of the normalized pulson potential, kinetic, and total energy densities E p, E k, and E (c) for the experiment including quadratic interfacial friction (from Rubino et al. 22). Figure 2.6 shows the temporal evolution of the normalized vortex thickness at its center (Fig. 2.6a), of the normalized vortex radius at the sea surface (Fig. 2.6b), and of the normalized vortex potential, kinetic, and total energy densities (Fig. 2.6c) as calculated by using the numerical model for a vortex evolving under the influence of an interfacial friction obeying a quadratic drag law. The aperiodicity induced in the vortex evolution by the use of a value of the drag coefficient (c diq =3 1-4 ) belonging to the range of values commonly used in ocean modeling for sharp and stable interfaces (see, e.g., Csanady 1979) is remarkable: the inertial pulsations are very strongly attenuated, their presence being hardly recognizable after 5 inertial periods. The energy dissipation drops off rapidly once the motion slows. The e-folding time for (total) energy dissipation is in this experiment is only $ T 22 d. 15

17 ,1 2 T / T, C diq x Fig. 2.7: Dissipation rate and e-folding time for energy decay due to quadratic interfacial friction as a function of the drag coefficient (from Rubino et al. 22). In Fig. 2.7 the e-folding time for energy decay due to quadratic interfacial friction is illustrated for 11 different experiments carried out by varying the value of the drag coefficient. For realistic values of the drag coefficients we obtain e-folding times which are in agreement with the result obtained by Csanady (1979). A comparison between the results obtained using linear and quadratic interfacial friction that yield approximately the same e- folding time shows that, in general, quadratic friction induces a smaller decrease in the vortex thickness at its center and a larger increase in the vortex radius than linear friction. This is due to the fact that linear friction is more efficient than quadratic friction near the vortex axis, while the opposite is true near the vortex periphery. Moreover it appears that quadratic friction is more efficient than linear friction in damping the vortex pulsation. However, for realistic values of the drag coefficients, both parameterizations induce a strong energy decay. Thus, the results of these simulations suggest that interfacial friction is the main responsible of the attenuation of the vortex pulsation. This, on its turn, suggests that in a deep, stratified ocean (on the f-plane) the vortex decay due to the radiation of internal waves excited by the vortex pulsation can be efficient only episodically. Energy inputs to the eddy from the atmosphere or from the ambient ocean (in the form of wind stress curl providing momentum to the near-surface ocean, heat fluxes increasing the density contrast between surface layer and deep ocean, or vortex-currents interactions) may rejuvenate the eddy pulsation dynamics, as they may be able to considerably disturb the vortex equilibrium and hence to cause nearinertial perturbations to develop. 2.4 Near-inertial oscillations of geophysical surface frontal currents Almost half of the energy contained in the oceanic internal waveband belongs to near-inertial waves (Munk 1981). As these disturbances, mostly generated in the upper ocean by the action of the wind, can transfer a considerable part of their energy downward, they represent one of the major sources of energy available for the production of mixing and turbulence in the interior ocean (see, e.g., Kunze 1985; Garrett 21; Alford 23). In this energy transmission from the near-surface layers to the abyss, near-inertial equatorward propagation as well as interaction with oceanic mesoscale features play a fundamental role (see, e.g., Kunze 1985; Chant 21; Garrett 21). On the f-plane, the transformation of oceanic near-inertial disturbances having similar spatial scales as the generating atmospheric fronts into smallerscale near-inertial waves, which are able to rapidly propagate their energy downward, is 16

18 attributed to the interactions of these long near-inertial waves with background inhomogeneities in the flow field like those associated with mesoscale fronts and vortices (see, e.g., Kunze 1985; Chant 21). On the other hand, the same coherent mesoscale frontal features involved in the transformation of the spatial scales of oceanic near-inertial internal waves are known to possess intrinsic near-inertial modes of oscillations (Cushman-Roisin 1987; Pavia and Cushman-Roisin 1988; Rubino et al. 1998b; Rubino et al. 22; Rubino and Brandt 23). Thus, if on the one hand mesoscale features do influence the pre-existing, wind-generated large-scale near-inertial wave field, on the other hand they may act, according to their intrinsic modes of oscillations, as generators of smaller-scale near-inertial waves when, as a response to larger-scale disturbances, they are forced to oscillate in a stratified ocean. Extending our knowledge on possible intrinsic oscillations inherent in oceanic mesoscale frontal features may thus contribute to a better understanding of the intricate dynamic interactions leading to the observed near-inertial wave field in the ocean. As a condition, a deeper comprehension of the intrinsic oscillations of such frontal features in a simplified context may be important. In the previous three sections we discussed the pulsating, inertial structure of nonlinear circular surface vortices; here we will concentrate on a special class of nonlinear surface frontal currents. The plane motion of a surface frontal layer on an f-plane in the frame of the nonlinear, reduced-gravity, frictional shallow-water model is governed, in a nondimensional form, by the equations: u u h v v h ( hu) + u v = su, + u + u = sv, + =, (3) t x x t x t x where x is the horizontal coordinate, t the time, and (u,v) and h the components of the horizontal velocity and the thickness of the surface layer respectively. The first two equations of (3) include a linear Rayleigh friction with a constant friction coefficient s. The scales for x, t, and s are the internal Rossby radius of deformation Ro = f 1 c, the inertial period f 1 (divided by 2π), and the inertial frequency f respectively. The value 17 + c = g' h is the phase velocity of the linear internal gravitational long waves, g' = g(1 ρ 1 /ρ 2 ) is the reduced gravity, g is the acceleration due to gravity, ρ 1 and ρ 2 the densities of the fluid within the front and in its surroundings respectively, and h + the maximum surface layer thickness at the initial stage. This thickness is scaled by h +, while the horizontal velocity components by c. It is assumed that h> within the strip x 1 <x<x 2, while h = at x = x 1 (t) and x = x 2 (t). We assume now that the fields be characterized by the following horizontal structure: u=a (t)+a 1 (t)x, v=b (t)+b 1 (t)x, h=c (t)+c 1 (t)x+c 2 (t)x 2. (31) The model (3), (31) is applicable to unsteady surface frontal coastal currents constrained by a straight wall if the fields (31) are written as u=a(t)x, v=b(t)x, h=c(t)x 2 +D(t). (32) The unknown functions in (32) satisfy the initial-value problem: 2 A& = sa + B 2C A, B& = A sb AB, C& = 3AC, (33) D& = AD,

19 A ( ) = A, B() = B, C() = C, D() = D with C < and D >., The initial-value problem (33) can be solved analytically (see Frei 1993; Shapiro 1996; Rubino et al. 23a). Consider a surface frontal coastal current emerging from a motionless initial state, i.e. A =B =, C <. In this case the exact solution of (33) describes surface frontal coastal currents that oscillates always superinertially. As stressed above, the same coherent mesoscale frontal features involved in the transformation of the spatial scales of oceanic near-inertial internal waves possess intrinsic modes of near-inertial oscillations. Kunze (1985) found that inertial waves interacting with larger scale, anticyclonic geostrophic mesoscale features experience a decrease in their frequency, i.e., they become subinertial. One can ask whether our results are in contradiction to the results of Kunze (1985). To answer this question, we analyze small-amplitude disturbances superimposed on a surface frontal current in geostrophic equilibrium: u =, v = V(x) = dh/dx, h = H(x) (34) within the strip x 1 <x<x 2, where H is the thickness of the surface frontal layer (with H(x 1,2 ) = ) and V the alongfront, geostrophic velocity. We impose small perturbations u 1, v 1, and h 1 on (34): u = u 1 (x,t), v = v 1 (x,t) + V(x), h = h 1 (x,t)+h(x). From (3) we obtain the following system of linearized equations: u1 h v1 dv v1 =, u1 = t x t dx 1 h t u x Vu1 + H = (35). (36) We now assume that the perturbed fields have the form h iωt 1 = Z( x) e, iωt u = U ( 1 1 x) e, and iωt = V ( 1 1 x) e. Substituting these expressions in (35) and (36) leads to the boundary-value v problem for the unknown variable Z: d dx 2 ω 1 H dv ( x) dx dz dx + Z =, x x x, (37) 1 2 x 1, 2 Z ( ) <, (38) where ω 2 is the eigenvalue to be found. The boundary conditions (38) are required to ensure that the solution is bounded at the outcropping lines x = x 1,2, which are singular points for (37). Note that, in our linear assumption, these boundaries have fixed positions. For surface frontal currents with parabolic cross sections in geostrophic equilibrium we obtain H = 1 γx 2 1/ 2, V = 2γx, x 1 = x2, x2 = γ. A coordinate transformation according to x = ξ/γ 1/2 allows to rewrite (37) and (38) as d 2 dz (1 ξ ) + λz =, 1 ξ 1, Z ( ± 1) <, (39) dξ dξ 18

20 where λ = (ω γ)/γ. Nontrivial solutions of (39) can be expressed in terms of Legendre polynomials for λ = n(n+1), n =,1,. Hence, for n >, all cross-front disturbances are standing waves of the form h 2 n [(1 γx ) ] exp( iω t), ω = 1+ [ n( + 1) 2]γ n d 1 = c n n n n n, (4) dx where c n are arbitrary constants, and x 1/ 2 γ. The form of the frontal interface is asymmetric (symmetric) with respect to the current axis, if n is odd (even). The frequency ω n is exactly inertial for n =1 and superinertial for n >1. Note that, in the case of surface frontal coastal currents, only symmetric modes (even n) are allowed. For the lowest symmetric mode of oscillations (n = 2), the disturbance is parabolic. For n = we obtain h 1 dv = const, ω = 1 2γ 1 (41) 2 dx 1 + for <γ<.5. The frequency of the disturbance is thus subinertial and corresponds to the approximate dispersion relation derived by Kunze (1985). The solution (41) simply results from (35), for vanishing gradient of the perturbed thickness, but cannot fulfill the continuity equation (36), and hence it cannot be considered as a simple mode of oscillation of the surface frontal current. The solution presented here for n = as well as that presented by Kunze (1985) can be seen as a superposition of inertial oscillations on a background vorticity field. In a linear context, disturbances superimposed on the surface frontal current are thus standing waves within the bounded region, whose frequencies are inertial/superinertial for the first mode/higher modes. A zeroth mode, referring to the superposition of an inertial wave on a background vorticity field, and hence resembling the situation investigated by Kunze (1985), would formally yield subinertial frequencies. Note that the approach used in our investigation, i.e., the reduced-gravity assumption, does not allow to study vertical propagation of nearinertial activity from the upper to the lower ocean. Such propagation would crucially depend on the properties of an active ambient ocean like, e.g., density and velocity distribution. Despite the simplifications inherent in the reduced-gravity approximation, a deeper comprehension of the intrinsic oscillations of geophysical frontal features in a such a simplified context may represent a prerequisite for better understanding the intricate dynamics leading to the observed near-inertial wave field in the ocean. In the upper ocean, the energy of the wind excites inertial waves having similar spatial scales as the generating atmospheric fronts. Such waves need to experience a variation in their frequency in order to be able to rapidly transfer their energy downward. Together with their equatorward propagation (Garrett 21), interactions of inertial waves with inhomogeneities of the ambient vorticity field are able to produce such a frequency shift (Kunze 1985). But, on the other hand, mostly depending on their specific geometry, many of the same inhomogeneities possess intrinsic modes of near-inertial oscillations, which would possibly generate near-inertial waves in an active ambient ocean. This suggests that, on an f-plane, the zoo of existing fluctuating surface frontal mesoscale features may play a larger role than believed in the past in the observed rapid propagation of the energy of the wind toward the abyss. 19

21 3 Fluctuating mesoscale frontal features in the real ocean 3.1 Surface fronts in the Rhine outflow area Oceanic surface fronts can be defined as regions where a maximum in the horizontal gradient of one or more physical, chemical, or biological characteristics of the surface water exists (McClimans 1988). The occurrence of these surface features, which seem to be ubiquitous in the world ocean, can be caused by a great variety of physical processes like, e.g., differential tidal mixing in shelf areas, variability of wind stress, coastal upwelling, or freshwater injections on the top of a heavier ambient ocean (Fedorov 1986). Different reasons contribute to determine the importance of studying oceanic surface fronts: often their presence reveals regions where two different water masses meet. Moreover they can contribute to the determination of the ecological equilibrium of large oceanic regions, as they are often linked to an increased concentration of phytoplankton or pollutants. As oceanic surface fronts can be linked to different phenomena occurring at the sea surface like, e.g., variations in water velocity, salinity, temperature, passive tracers or color, accumulation of floating debris or breaking of surface waves, different measurement techniques can be employed for their detection. Among them, remote sensing has proved to be a powerful tool for the study of oceanic surface fronts because, detecting their signatures synoptically, it can deliver a detailed description of their temporal and spatial evolution often contributing significantly to the understanding of their dynamics achieved using in-situ measurement techniques. Through the analysis of remote sensing data showing sea surface manifestations of oceanic flow features in general, and oceanic surface frontal features in particular, information can be obtained on their spatial and temporal variability which can lead to the formulation of conjectures about different aspects of their dynamical structure. Often the information retrieved from these data can be corroborated by carrying out numerical simulations and/or laboratory experiments. One of the major sources of difficulty in the numerical simulation of frontal processes is the treatment of the surface area separating the different water masses. One could consider threedimensional primitive equation models as the most adequate tools for the simulation of these fronts. However, a detailed description of the entire complexity inherent in surface frontal dynamics is largely inhibited by its often unsustainably high computational effort, as well as by the sensitivity of the obtained solutions to the parameterizations adopted for taking into account turbulent processes. For these reasons, more efficient, process modeling strategies have been developed in the last three decades, which are still considered among the most valid tools for the description of non-stationary oceanic flow features including frontal processes. In this context, e.g., O Donnell and Garvine (1983), O Donnel (199), and McCreary et al. (1997) developed time-dependent layered models, enabling the assessment of the potential implications of plume fronts on the larger scale dynamics. Indeed aspects of the dynamics of different mesoscale oceanic frontal features can be also described accurately using even more simplified frontal models that neglect the influence of mixing processes near the surface front on the interior dynamics as they refer to an immiscible (or only weakly miscible) fluid and simulate surface fronts by simply including techniques for the treatment of movable lateral boundaries (see, e.g., Pavia and Cushman-Roisin 1988; Rubino et al. 1998a; Esenkov and Cushman-Roisin 1999; Rubino et al. 22). 2

22 Near the river Rhine mouth, the interaction of the fluvial currents of the river Rhine with the predominantly semidiurnal tide of the European North Sea is responsible for the existence of a discontinuous river outflow which causes a periodic generation of freshwater plumes (see, e.g., van Alphen et al. 1988; Simpson and Souza 1995). Figure 3.1 shows the horizontal distribution of sea surface temperature (SST) in the Dutch coastal area as measured by the Advanced Very High Resolution Radiometer (AVHRR) of the National Oceanic and Atmospheric Administration (NOAA) on 29 October 1995 (Dech 1996). Visible are, among other things, two distinct isolated water masses of riverine origin, located north of the Rhine mouth, about 3 km offshore. Fig. 3.1: horizontal distribution of SST in the Dutch coastal area as measured by the AVHRR at October In SAR images, the boundaries between riverine and sea water are usually associated with surface frontal zones that are often visible as narrow, elongated areas of enhanced and/or enhanced/reduced radar backscatter (Ruddick et al. 1994; Vogelzang et al. 1997). The observed variability in the structure of the NRCS of the frontal lines may be due to a variable surface convergence or shear near the front (Johannesen et al. 1991), to the variable spatial orientation of the front with respect to the radar look direction and/or to the wind direction (Brandt et al. 1999), and to the presence in the frontal area of natural or anthropogenic surfactants. Here we study the dynamics of the Rhine surface front in order to infer characteristics of the dynamics of the Rhine outflow plume. The study (Hessner et al. 21) is based on an analysis of SAR data acquired by the European Remote Sensing satellites ERS-1 and ERS-2 over the Rhine outflow area and on an analysis of the results of numerical simulations of the hydrodynamics of the Rhine outflow region carried out with a two-layer frontal model. The analysis of the SAR data consists in the recognition and interpretation of sea surface patterns 21

23 associated with the Rhine surface front. radar flight look 5 km NRCS [db] wind tidal phase Rhine discharge 11 ms -1 HW - 2:7 h 243 m 3 s -1 Fig. 3.2: ERS SAR image acquired at October 1993 over the Rhine outflow area. The white arrow marks the position of the river mouth. The gray scale is proportional to the measured normalized radar cross-section (NRCS). The Rhine surface front is imaged as an almost semicircular line. Note that wind speed, tidal phase, and river discharge were measured at Hoek van Holland (from Hessner et al. 21). The analysis of the results of the numerical simulations consists mainly in the determination of the temporal and spatial evolution of the Rhine surface front for different river discharges, semidiurnal tidal amplitudes, and residual currents. In order to perform such determination, the Rhine surface front is identified by recognizing and interpreting surface patterns obtained by inserting the simulated surface velocity field into a simple radar-imaging model which relates the modulation of the backscattered radar power to the surface velocity convergence in radar look direction. In the following an analysis of 41 precision-processed ERS SAR images (i.e., SAR images having a nominal resolution of 25 m) acquired over the Rhine outflow area from 1992 to 1998 is presented. For each of the 41 SAR images analyzed, we determined acquisition time, semidiurnal and neap-spring tidal phases, wind speed and direction, and mean water discharge of the river Rhine. In particular, in 36 images of our data set, roughness patterns were 22

24 recognized which can be interpreted as sea surface manifestations of the Rhine surface front. Our analysis of the ERS SAR images gives no indication that the occurrence of the Rhine surface front varies significantly with season, as it should happen if its imaging mechanism were strongly connected with biologic activity. An example illustrating typical sea surface patterns visible on SAR images of the Rhine outflow area is shown in Fig The imaged area is 25 km 25 km. In the lower right corner of this image part of the Dutch coast including the Rhine mouth (white arrow) is visible. In the center of the image, a sea surface pattern associated with the Rhine surface front can be delineated as a narrow, elongated region (frontal line) enclosing a large surface area in which patches of slightly reduced normalized radar cross-section (NRCS) dominate. The frontal line is generally characterized by enhanced NRCS, their maximum values occurring at its northeastern edge. However, we note that the enhancement of the NRCS is smaller in the southern and especially in the western part, where the line seems partly imaged as a zone of enhanced/reduced NRCS. The analysis of the 36 SAR images showing sea surface manifestations of the Rhine surface front indicates that, independently of tidal state, wind condition, river discharge, and residual currents, the frontal lines manifest themselves on ERS SAR images as bands dominated by an enhancement of the NRCS. However, in some cases, small portions of the frontal lines are imaged as narrow zones of enhanced/reduced radar backscatter. The form and the location of the radar signatures of the Rhine surface front vary significantly. Our analysis indicates that semidiurnal tidal phase, river discharge, and wind-induced currents possibly exert a significant role in such a variability. Their combined effect may explain the large variability observed in the radar signatures of the Rhine surface front. However, from the analysis of our data set it emerges also that different recurrent radar signatures of the Rhine surface front exist, the most frequent ones being (1) almost strait lines located near the river mouth, (2) nearly semicircular lines located several km off the coast, and (3) slightly curved lines located further offshore. The recurrence of these patterns appears to be mainly linked to the semidiurnal tidal phase in the Rhine outflow region. Numerical simulations of the hydrodynamics of the Rhine outflow region area were performed in order to produce surface frontal features associated with the Rhine outflow plume. Using a simple radar-imaging model (see, e.g., Alpers 1985) the simulated surface velocity field is used to obtain theoretical radar signatures of the Rhine outflow front. The numerical model used in our study is a frontal two-layer model, similar to that of Rubino et al. (22) discussed in Sect It solves the hydrostatic, nonlinear shallow-water equations on an f-plane. The model is a frontal model. In fact in this model a special technique for the numerical treatment of movable lateral boundaries allows for the description of the temporal and spatial evolution of a localized layer with an outcropping interface in a dynamically active environment. The domain, on which the numerical simulations carried out in the present investigation were performed, is composed of an interior region, which includes the Rhine outflow area and of an outer region, in which a grid zooming was implemented. Within the interior region, which covers an area of 8 km 5 km, the spatial resolution is 5 m in both directions. This resolution was reduced in the outer region by successively increasing the grid step by a factor 1.5 in both directions. The model was forced by imposing in the upper water layer of its southeastern open boundary the river transport and by imposing in the lower water layer of its southwestern open boundary a transport U TR which represents the sum of the tidal transport U T and of the residual transport U R. Obviously, using such a simplified numerical model it is not possible to address the whole complexity characterizing the hydrodynamics of a region of freshwater influence like the Rhine outflow area, but it is nevertheless possible to elucidate aspects of the evolution of the Rhine surface front near the river mouth. 23

25 Flight look 5 km HW - 2 h u / x 1 [s ] r r Fig. 3.3: Surface velocity convergence in the direction coinciding with the radar look direction of the SAR images shown in Fig. 3.1 as calculated by the numerical model for high water minus two hours (from Hessner et al. 21). Figure 3.3 shows the surface velocity convergence in the direction coinciding with the radar look direction of the SAR image depicted in Fig. 3.1 as simulated by our numerical model for high water minus two hours. The northeastern edge of the simulated Rhine plume is linked to a narrow band of enhanced surface velocity convergence which, according to the radarimaging theory used in our investigation, results in a narrow band of enhanced NRCS in a theoretical radar image. Note that such band is located near the model frontal line. The proximity of these two regions is a characteristic that we observed in all numerical simulations carried out using our numerical model. In a large portion of the plume area which is enclosed by the Rhine surface front, the surface velocity field is slightly divergent, which yields a region of reduced NRCS in a theoretical radar image. In order to elucidate the dependence of the form and of the location of the Rhine surface front on river discharge, amplitude of the semidiurnal tidal transport, and residual transport, we performed different numerical simulations by varying the values of these forcing parameters. The results indicate that, in the region near the river mouth, the form and the location of the Rhine surface front do not strongly depend on these parameters. This result appears to be independent on the phase of the semidiurnal tide. Our numerical investigation indicates thus that, as suggested by the analysis of the SAR data, the form and the location of the Rhine surface front are mainly linked to the phase of the semidiurnal tidal cycle in the outflow region and that they only weakly depend on river discharge, residual currents, and neap-spring tidal cycle. On this basis an attempt can be made to infer, from SAR images showing sea surface manifestations of the Rhine surface front which were acquired during different tidal cycles over the Rhine outflow area, a mean spatial and temporal evolution of the Rhine outflow plume. 24

26 Fig. 3.4: Schematic plot of the form and of the location of the Rhine surface front as inferred from 8 ERS SAR images acquired during different phases of different tidal cycles. The white lines refer to the surface manifestations of the Rhine surface front which can be associated with the evolution of the Rhine plume within a single semidiurnal tidal cycle. The gray lines refer to the surface manifestations of the Rhine surface front which can be associated with the evolution of the Rhine plume generated during the previous tidal cycle (from Hessner et al. 21). Figure 3.4 schematizes the form and the location of the Rhine surface front as inferred from 8 ERS SAR images acquired during different phases of different tidal cycles. Shortly after high water the Rhine outflow gradually starts and a buoyant plume begins to develop. At this time the tidal current is about maximum and is directed northeastward, along the coast. At the Rhine mouth a nearly straight front is visible which is aligned with the direction of the tidal current (line 1a). About 8 km northeast of the river mouth a second front is visible (line 1b), which is associated with the plume generated during the previous tidal cycle. About two hours after high water the front located in the vicinity of the river mouth has moved about 1 km offshore and its curvature has increased (line 2a). The front associated with the plume generated during the previous tidal cycle has moved northeastward, advected by the tidal current, and is now located approximately 12 km off the Rhine mouth (line 2b). About four hours after high water the tidal current turns from northeast to southwest. The buoyant plume has spread further offshore. Its front is now located about 4 km off the river mouth (line 3). Around low water the outflow at the river mouth is about maximum. The front is now visible as an almost semicircular line located in the vicinity of the river mouth (line 4). Note that a northeastward spreading of the buoyant plume is now inhibited by the presence of a southwestward tidal current. About half an hour after low water the buoyant plume has spread further offshore. At this time only its northwestern and coastal edges are visible as frontal lines (line 5). About three hours after low water only the part of the front located near the New Waterway quay is visible (line 6). Note that the form and the location of this portion of the Rhine surface front have not changed substantially during the previous three hours. About two hours before high water the tidal current turns again northeastward. Now the riverine water is free to spread northeastward and the front is visible as a nearly semicircular line (line 25

27 7). About one hour before high water the front has moved further northeastward, advected by the northeastward tidal current (line 8). 3.2 Intraseasonal fluctuations in the Southwestern Arabian Sea and their relations to the dynamics of the Great Whirl The Southern Gyre and the Great Whirl (GW) develop as strong anticyclonic eddies in the southwestern Arabian Sea every year in response to the onset of the Southwest Monsoon (Swallow and Fieux 1982). They are part of the Somali Current system, one of the most energetic regions of the World Ocean, where surface currents of more than 3 m/s have been observed (Swallow and Bruce 1966; Schott, 1983; Fischer et al. 1996). Both eddies, but in particular the GW, contribute substantially to the upwelling along the northern Somali coast and to the eastward transport of cold water into the interior Arabian Sea. As such, the GW is part of the upward branch of the shallow tropical-subtropical circulation cell in the Indian Ocean and strongly affects the heat balance of the Arabian Sea (Schott et al. 22). Most of the details concerning the generation, the persistence, and the decay or collapse of this eddy are still poorly understood (Schott and McCreary 21). Rapid changes such as a sudden northward propagation of the Southern Gyre and coalescence with the GW have been observed during several years (Evans and Brown 1981), but in other years this two-gyre system remained in place and slowly decayed at the end of the Southwest Monsoon season. Until now this short-term and interannual variability could not be related in any simple way to variability in observed forcing fields, nor were numerical models able to realistically simulate the observed complex flow field in the area (Wirth et al. 22). It appears, however, that intraseasonal variability plays a major role in the development of the seasonal cycle of the Somali Current. In a recent study, intraseasonal fluctuations emerged in a numerical ocean model of the Indian Ocean, which was forced with seasonal winds excluding periods shorter than 9 days (Sengupta et al. 21). These fluctuations were interpreted as the result of hydrodynamical instabilities in the region of the GW. Intraseasonal variability with planetary wave characteristics have thus been observed and modeled over the whole life cycle of the GW. It is, however, not clear how these fluctuations are generated and how they interact with the GW: do they contribute to the build-up of the GW, or do they lead to its decay and/or collapse? Here an analysis of TOPEX/POSEIDON altimeter data and current and temperature data is presented (Brandt et al. 23) showing intraseasonal fluctuations in the GW region. The latter were collected during the World Ocean Circulation Experiment in 1995 and 1996 at the mooring array ICM-7 in the central part of the Somaly basin. Here the sea-surface height anomaly (SSHA) from along-track data is used. Time-series of the along-track data were analyzed using a wavelet transform. 26

28 1.5 Angelo Rubino SSH Wavelet Energy between 52 6 E and 5 11 N 6 Period [days] Fig. 3.5: Period-time plot of normalized SSH wavelet energy averaged between 52-6 E and 5-11 N. Soli d lines marks the 95% significance level (from Brandt et al. 23). Figure 3.5 shows the average SSHA wavelet energy spectrum in a region close to the coast of Somalia between 52 E and 6 E and 5 N and 11 N. The adopted wavelet is a Morlet wavelet, i.e. a cos-function that is modulated by a Gaussian distribution. A considerable yearto-year variability in the strength of the fluctuations of the wavelet energy as well as in its distribution among the different periods is evident. In late summer and fall strong signals were found at periods between 3 and 5 days; at larger periods, significant signals were present only during some years. SSH Wavelet Energy 2 N MAM 2 N JJA 1 N 1 N.5 Eq. Eq. 2 N 5 E 6 E 7 E SON 2 N DJF 5 E 6 E 7 E 1 N N Eq. Eq. 5 E 6 E 7 E 27 5 E 6 E 7 E Fig. 3.6: Mean seasonal variation of the normalized SSH wavelet energy at the period range days. The light gray shaded areas mark the regions with a significance level higher than 95% (from Brandt et al. 23).

29 The spatial distribution of the SSHA wavelet energy at the period range between 38 and 45 days, corresponding to the central period range of the observed shorter-period fluctuations, is depicted in Fig. 3.6 for the different seasons of a mean annual cycle. High energy was found during summer and fall, while during winter and spring energy was relatively weak. Because significant SSHA wavelet energy signals at this period range are found only in the northern Somali basin we suggest a local generation mechanism for the observed waves. The in situ data were collected in the southwestern Arabian Sea from April 1995 to October 1996 by moorings carrying Aanderaa rotor current meters for measuring deep currents and temperatures. Here we use data obtained from the moorings K17-K21 deployed in the central Somali Basin, which were part of a triangular set-up specifically conceived to capture planetary waves propagating in this region. Temperature [ C] ~ ~ ~ ~ K19 K18 K17 K2 K21 Apr Jul Oct Jan Apr Jul Oct Fig. 3.7: Time series of temperature at 2 m water depth at the five moorings K17 - K21 (from Brandt et al. 23). The temperature records measured at the five moorings at 2 m water depth (Fig. 3.7) show large oscillations up to.4 C amplitude. Given the typical stratification in the area, such temperature variability corresponds to oscillation amplitudes of the isopycnals at this depth level of about 1m. From July 1995 to February 1996 these temperature fluctuations were very similar over the mooring array, indicating the presence of large-scale baroclinic waves. Characteristics of the observed waves can be quantified by calculating a local dispersion relation. For our estimate, we used the time-series of the meridional velocity at 4 m and 7 m water depth, and the 2 m temperature records. We selected these data sets because they were available for at least four of the five moorings. Periods were simply calculated as the temporal interval between two successive maxima or minima in the respective time-series. Phase differences and velocities were calculated by identifying corresponding extreme values in the different time-series. From our earlier analysis of the altimeter data, we know that the zonal phase speeds were of the order 2-3 cm/s, which narrows the range of possible solutions. Once periods and phase velocities were estimated, the corresponding wavelengths were calculated. Figure 3.8 summarizes these estimates. 28

30 a) T [days] Period K19 K18 K17 K2 K21 b) λ x [km] c x [cm s 1 ] c) Jul Oct Jan Apr Jul Oct Zonal Phase Velocity v 4m v 7m T 2m Jul Oct Jan Apr Jul Oct Zonal Wavelength v 4m v 7m T 2m d) c y [cm s 1 ] Jul Oct Jan Apr Jul Oct Meridional Phase Velocity v 4m v 7m T 2m e) λ y [km] Jul Oct Jan Apr Jul Oct Meridional Wavelength v 4m v 7m T 2m Jul Oct Jan Apr Jul Oct Fig. 3.8: Period (a), zonal phase velocity (b), and zonal wavelength (c), meridional phase velocity (d), and meridional wavelength (e) as calculated from meridional velocity time series at 4m and 7m water depth as well as from temperature time series at 2 m water depth. Zonal phase velocity and zonal wavelength are calculated using data from the three moorings K19, K2, and K21. The other parameters are calculated using data from all moorings (from Brandt et al. 23). 29

31 The fluctuation periods were smallest between July and August and increased steadily until January (Fig. 3.8a). The zonal phase velocity of the fluctuations, calculated from the K19-K21 records, was maximum in July (about 4 cm/s towards the west) and decreased strongly until October (about 15 cm/s; Fig. 3.8b). Such clear time-dependence cannot be identified in the evolution of the zonal wavelength, but there was also an indication of a decrease from 1 km to 5 km in the period from July to October (Fig. 3.8c). The meridional phase velocity of the fluctuations was calculated from the whole data set obtained from K17 K21. In general, the meridional phase velocity was directed southward, its magnitude being about 2 cm/s (Fig. 3.8d). Similar to the zonal wavelengths, the calculated meridional wavelengths ranged between 5 km and 1 km (Fig. 3.8e). A comparison between observed and theoretical dispersion relation suggests that the observed fluctuations emerged during July and evolved toward freely propagating Rossby waves. The observed values were close to the wavelength where the maximum frequency in the theoretical dispersion relation occurs. Here Rossby waves are characterized by a minimum in their zonal group velocity. Since our data suggest a general southward phase velocity, this corresponds to northward energy propagation. How are these disturbances generated? Do they interact (and how) with the GW? The local concentration of the energy in the northern Somali Basin means that remote forcing effects through the wind are not important in this context. Local wind fluctuations could also generate such fluctuations, but an analysis of the wind curl over the Arabian Sea failed to show such a concentration of fluctuation energy in the region. Although the wind curl wavelet energy in the period range between 38 and 45 days is relatively high during the Southwest Monsoon, it is mainly concentrated in the northern Arabian Sea. Slightly increased wavelet energy is also found at about 8 N and 6 E during June to August. However, the mismatch in the pattern of the SSHA and wind curl wavelet energy as well as the inferred northward energy propagation rules out the wind curl fluctuations as a dominant source of energy in the intraseasonal oceanic variability of the southwestern Arabian Sea. This conclusion is consistent with a study by Sengupta et al. (21), who found similar oceanic fluctuations using a numerical model of the Indian Ocean forced only with seasonal winds. They proposed hydrodynamical instabilities as the most likely generation mechanism. This seems reasonable, as the fluctuations develop with, but not before, the onset of the Southwest Monsoon, when the Southern Gyre and the GW are building up. A likely energy source for the fluctuations is the boundary between the Southern Gyre and the GW at around 4-5 N. Here, vertical shears of up to 3 m/s per 1 m have been observed already during relatively early stages of the monsoon (see, e.g., Leetmaa et al. 1982). The lower boundary of this energetic layer rises from more than 1 m depth in the Southern Gyre to the surface in the transition zone to the GW and to more than 2 m in its center. The associated reversal of the meridional gradient of potential vorticity allows baroclinic instabilities to develop. Length-scales of such baroclinic instabilities would then be 5 km or longer. 3

32 4 Outlook In the present summary paper different investigations, carried out by the author, were presented, which are aimed at elucidating aspects of the nature of fluctuating mesoscale oceanic frontal features such as frontal vortices and coastal currents and at describing aspects of their manifestations in the real ocean. These investigations were performed using different methods: frontal surface vortices were investigated analytically and numerically and produced in laboratory experiments using the large turntable of LEGI/Coriolis (Grenoble, France); fluctuations associated with their activity were observed in current and temperature data obtained from a mooring array deployed in the southwestern Arabian Sea; their surface signatures were detected in Synthetic Aperture Radar (SAR) imageries acquired by the First and Second European Remote Sensing Satellites (ERS- 1 and ERS-2) as well as in TOPEX/POSEIDON sea level anomaly data. Finally, mesoscale surface frontal currents were investigated analytically as well as numerically. The view of the ocean emerging from these investigations serve to corroborate the awareness that the zoo of frontal fluctuating mesoscale features existing in the ocean may play a larger role than believed in the past in determining the observed equilibrium and variability of the World Ocean. From the work presented it strongly emerges that processoriented studies in a simplified context can significantly contribute to the comprehension of oceanic phenomena as they allows processes at work in them to be readily diagnosed, and hence constitute a valuable instrument to face the huge complexity inherent in the dynamics of the real ocean. Much work remains to be done in this context, and one can only attempt to envisage possible developments which are more directly connected with his own activity. Analytical solutions of the nonlinear shallow-water equations (note that the solutions of Rubino et al. (1998b) were extended to describe barotropic oscillations in a paraboloidic basin by Dotsenko and Rubino (23)) describing circular vortices could be extended to a multilayer system. This would render the realism inherent in the solutions much larger, as rotating frontal features of a stratified ocean, often characterized by both cyclonic and anticyclonic vorticity fields, could be described by these solutions, which could also contribute to elucidate aspects of the energy transfer within the different vortex layers (Rubino and Dotsenko 24a). The structure of near-inertial oscillations of mesoscale, surface as well as intermediate frontal vortices in the presence of an active, homogeneous or stratified ambient ocean and/or on the β-plane is poorly investigated. 31

33 Fig. 4.1: Time series of the normalized pulson potential, kinetic, and total energy densities E p, E k, and E for a pulson evolving on a f-plane (f = 7*1-5 s -1 ) and for the same pulson evolving on a β-plane (β = m -1 s -1 ). Note that E represents the (analytical) pulson total energy density (from Rubino and Dotsenko 24b). Figure 4.1 shows the temporal evolution of the normalized pulson potential, kinetic, and total energy densities as calculated by using the numerical model described in Sect. 2.3 for the same, initially circular vortex evolving on a f-plane and on a β-plane. The variation of the Coriolis parameter with the latitude considered in the model on the β-plane induces a northsouth asymmetry within the vortex which destroys the initial circular vortex structure. As a result, exact inertial oscillations are no longer produced: the vortex develops a two-mode pulsation instead, which is much more rapidly attenuated than in the f-plane analogous (Rubino and Dotsenko 24b). The influence of frontal features like those investigated analytically in Sect. 2.4 in the transformation of wind-induced near-inertial waves is presently subject of intense study (see, e.g., Xing and Davies 23; 24a; 24b). In this context, 3D, very high resolution simulations of frontal features in a realistic oceanic environment are planned, which should clarify the role played by the intrinsic pulsations of such features in the downward energy transmission of wind-generated inertial disturbances. As mentioned in the introduction, at high latitudes convectively generated mesoscale frontal features are known to play an important role in preconditioning the upper ocean to deeppenetrating, open-ocean convection and are believed to contribute substantially to the mixing and spreading of the newly formed dense water in the ambient ocean. The formation of such mesoscale features is intimately connected to small-scale, plume activity (Marshall and Schott 1999) whose existence, on its turn, largely depends on the larger-scale atmospheric and oceanic circulation. The realistic simulation of the dynamics of such convectively generated oceanic mesosclae frontal features is thus only conceivable in the context of nested largescale/regional/submesoscale models. This new approach to the simulation of convective events in the Greenland Sea is one of the goals of the project Wechselwirkung Ozean/Atmosphäre im nördlichen Nordatlantik: Wassermassentransformation und -transporte, which is part of the German project Sonderforschungsbereich 512 Tiefdruckgebiete und Klimasystem des Nordatlantiks of the Deutsche Forschungsgemeinschaft (DFG). In Fig. 4.2 the model configuration utilized to perform realistic simulations of convective events induced by the presence of an observed, convectively generated mesoscale vortex in the Greenland Sea is schematised (Androssov et 32

34 Angelo Rubino al. 24) distance (km) Dim. 278*278 Min grid size = 125 m Max grid size = 4 km Greenland distance (km) Dim. 126*126 Min grid size = 1 km Max grid size = 32 km -1-5 Norway 5 distance (km) b) a) Fig. 4.2: The hierarchy of nested numerical models used for the realistic simulation of convective events triggered through an observed mesoscale chimney in the Greenland Sea. In the figure, a) refers to the REMO/MPI-OM global coupled general model and b) to the regional hydrostatic (lower panel) and non-hydrostatic (upper panel) models (from Androssov et al. 24). The large-scale oceanic and atmospheric dynamics is described using the REMO/MPI-OM global coupled general model (Maier-Reimer 1997; Jacob and Podzun 1997). On this basis a nested, very high resolution, 3D hydrostatic oceanic model simulates the oceanic dynamics of the Central Greenland Sea. The obtained results are then used to initialise and force a submesoscale, non-hydrostatic oceanic model for the description of the small-scale dynamics associated to open-ocean convection due to an observed, convectively generated oceanic chimney. 5 Acknowledgement I would like to express my gratitude to the colleagues of the Institut für Meereskunde of the University of Hamburg for their help and cooperation. I thank the coauthors of the studies presented here for their engaged work: a special thank is due to Peter Brandt for his longstanding cooperation and to Werner Alpers, Detlef Stammer, and Wilfried Zahel for their valuable advice. 33