Lateral Circulation

Generation of Circulation in Homogenous Layers

To characterize the spatial patterns of the horizontal currents use is customarily made of a variable called vorticity Z defined as the curl (a mathematical operation, denoted by the symbol Vx, involving spatial derivatives) of the horizontal velocity vector u, i.e.,

A flow field where a fluid particle describes counterclockwise (or cyclonic) loops will have a positive vor-ticity; a flow field with negative vorticity, in turn, will make a fluid particle to rotate clockwise (or anti-cyclonically). The magnitude of Z is equal to twice the angular velocity of a fluid parcel: the larger Z becomes the higher will be the speed at which the particle circulates. Here, the equation governing the evolution of the vorticity Z of the depth-averaged flow is horizontal velocity analyzed to understand the mechanisms generating circulation in HBs and SMLs. Ignoring advection and diffusion of vorticity, the governing equation for Z in a homogenous layer of thickness h(x, y, t) can be written as dZ f + Z\ (dh dh dh\ „ ft dr(VJ u+u dX+v d-y) +Vn «) ~Vxl Ph

The three terms on the right hand side of eq. 13 represent the sources and sinks of vorticity in a closed basin. The generation of cyclonic or anticyclonic vorticity depends on the balance of those terms, which represent:

1. generation of vorticity by temporal changes in the layer thickness, or by flow running into regions with larger or smaller thickness (e.g., a sloping bottom), which results in the stretching and/or squashing of the fluid columns (Figure 2.1);

2. flux of vorticity at the top boundary caused by spatial variations of the wind stress tw and/or the bottom bathymetry (Figure 2.2);

3. flux of vorticity at the bottom boundary caused by the curl of the ratio of the bottom stress (tb) to the depth of the basin.

Term 3 is the only sink of vorticity. All other terms are sources. Term (1) is associated with spatial and/or temporal changes (usually of oscillatory nature) in the thickness h of the homogenous layer. In a stratified water body, it is intricately linked to the existence of internal wave motions. Under the influence of oscillatory currents induced by internal waves, water parcels will describe open trajectories, with a small net displacement after each oscillation. The net basin-wide circulation created by Term (1), hence, is characterized by mean velocities (when averaged over several wave periods) that are much less than the instantaneous values. Such circulation is referred to as residual circulation, and it is cyclonic in nature. Residual circulations are defined in contrast with direct circulations, which are driven directly by spatially variable winds (Term 2). Water parcels in this type of circulations will move continuously in large gyral patterns, and the mean velocities of water parcels are similar in magnitude and direction to the instantaneous velocities. In contrast with the residual circulations, direct circulation can be either cyclonic or anticyclonic depending on the spatial variations of the wind over the lake surface (see Clear Lake in Table 2).

Topographic Gyres in Homogenous Basins

The evolution of Z in a HB acted upon by a suddenly imposed wind can be to first order, described as a result of the spatial variations of wind and the lake bathymetry (i.e. the second term in the right hand side of eqn [13]. This description is only valid for a period of time after the onset of the wind stress which must be short compared to f_1, so that Coriolis forces (see Fundamental Concepts) and bottom friction can be safely neglected. The extent to which circulation is controlled by topographic effects or the spatial variability of wind can be assessed using simple scaling arguments, by breaking the second source term in eqn [13] as follows vX ïHVxtw+xv(1

Here, the symbol A applied to a variable refers to changes in that variable's magnitude across the basin. The first term represents the changes in magnitude and/or direction of the wind stress over the lake. The second is the result of wind acting in a direction perpendicular to depth variations, and it is referred to as the topological moment. The spatial variability of wind stress will dominate over the topological moment when

'At1

Figure 2 Three dimensional section of a lake. (1) Conservation of volume and circulation of a fluid parcel undergoing squeezing or stretching. Vorticity decreases as a fluid parcel travels to shallower regions, and vice versa. (2) Vorticity induced by the curl of the wind stress tw.

In flat basins (A H « 0) the main source of circulation is the curl of the wind stres. In lakes of more realistic varying bathymetry, though, both wind stress and depth variations control the lake circulation. For the case of nearly uniform winds acting along the main axis of a narrow and elongated lake with sloping sides, the only source of circulation is the topological moment and the resulting circulation is characterized by in a double gyre pattern: to the right of the axis in the direction of the wind the curl of tw/h is positive and cyclonic vorticity is produced, whereas to the left anti-cyclonic vorticity is produced (Figure 3). Bottom friction (not included, so far in our analysis of circulation) tends to cancel the vorticity input by the wind, but only after the topographic gyres have become established.

Topographic Waves in Homogenous Bodies

As a consequence of the Coriolis forces acting on an initial double-gyre pattern, this will tend to rotate anticlockwise (see Figure 4). This mode of motion is referred to as topographic or vorticity wave. It is free, since it does not depend on the existence of external forcing (i.e., it only requires that the initial double gyre has been previously established), and has a characteristic frequency s which scales as s « Xflk [16]

Here l is a constant of order unity, l is a length scale characterizing the size of the sloping sides in a lake, and k is the wave-number of the wave, which for the gravest first mode, is given by k = 2p/P (P being the perimeter). For typical values f = 10~4 s-1, I = 104 m and k = 10-5 m-1, s is of the order 10~5 hence, s << f.

Even though the possiblity of vorticity waves occurring in lakes has been recognized in classical hydrodynamics, evidence of their existence in real lakes, revealed by the cyclonic rotation of the velocity vector at a frequency given by eqn [16], is sparse and weak (see Figure 4). This is, in part, due to the fact that winds are almost never absent and will, most likely, interact with existing topographic waves. Moderate and strong winds with enough impulse can change existing circulation patterns, and establish a new basin-scale topographic gyre. Furthermore,

Lake Circulation

Figure 3 Large-scale winter circulation in Lake Ontario, reconstructed from observations collected ad different stations (shown as black thin arrows). The numbers in the figure indicate the depth at which measurements were taken: 1 = 15m and 5 = 75 m. The size of the arrows shows the magnitude of the currents. The gray thick and curved arrows represent the circulation interpreted from the data. Black thick arrow indicates the predominant wind direction. From Beletsky D, Saylor JH, and Schwab DJ (1999) Mean Circulation in the Great Lakes. Journal of Great Lakes Research 25: 78-93. Reproduced with permission from the International Association for Great Lakes Research.

Figure 3 Large-scale winter circulation in Lake Ontario, reconstructed from observations collected ad different stations (shown as black thin arrows). The numbers in the figure indicate the depth at which measurements were taken: 1 = 15m and 5 = 75 m. The size of the arrows shows the magnitude of the currents. The gray thick and curved arrows represent the circulation interpreted from the data. Black thick arrow indicates the predominant wind direction. From Beletsky D, Saylor JH, and Schwab DJ (1999) Mean Circulation in the Great Lakes. Journal of Great Lakes Research 25: 78-93. Reproduced with permission from the International Association for Great Lakes Research.

Figure 4 Normalized stream function calculated with a2D-model of lake circulation in the Gulf of Riga, at (a) 16.5, (b) 17.5, (c) 18.5 and (d) 19.5 days after the start of simulations. Currents will be parallel to lines of equal stream function. Continuous lines show the cyclonic gyre while dashed lines show the anticyclonic gyre of the double-gyre. Bold arrows show the direction of wind. Adapted from Raudsepp U, Beletsky D, and Schwab DJ (2003) Basin-scale topographic waves in the Gulf of Riga. Journal of Physical Oceanography 33:1129-1140. Copyright 2003 American Meteorological Society. Reproduced with permission from the American Meteorological Society.

time-varying wind directions can also affect circulation patterns: cyclonically rotating winds have been shown in numerical experiments to reinforce the basin-scale topographic waves, while anticyclonically rotating winds tend to destroy the wave. In shallow areas, where the direct wind forcing and bottom friction dominate the vorticity balance, the topographic waves are not likely observed. Hence, only in the deepest points, evidence of topographic waves can be found.

Spatial Variability of Wind Forcing

The specific wind-driven circulation patterns that develop in the SML or HB are tightly linked to the spatial and temporal variations of the wind stress over the lake (see eqn [13]). While the time variability of wind stress at a single point in space can be characterized with high-resolution wind sensors (e.g., sonic anemometers), characterizing its spatial variability, though, has proved to be a difficult task. Considerable effort in the recent years has been devoted to characterize the spatial variability of the wind stress field over lakes. This is done either by applying dynamic models of atmospheric circulation or by measuring wind speed and direction in arrays of wind sensors located on and around the lake (see, for example, Figure 5 in Lake Kinneret). Bulk aerodynamic formulations are typically used to derive wind stress values from the wind speed. Technologies developed to characterize wind stress fields directly (scatterometry) have, so far, only applied to oceanic scales, given the very low resolution of existing sensors.

Studies conducted to characterize wind fields over lakes ranging in size from small to large demonstrate that a considerable degree of spatial variability exists both on synoptic and local scales. Synoptic scale variability of the wind field will only affect lakes of large dimensions (e.g., Great Lakes). On a local scale, factors such as spatial variations in the land surface thermal and/or moisture properties, surface roughness, or the topography can modify and even generate flows in the atmospheric boundary-layer. All of them are, most probably, at play over or in the immediate vicinity of all lakes. The most significant effect of the topography is the aerodynamic modification of ambient synoptic winds. The topographic features, existing around lakes will, among some other effects, cause the ambient wind to change direction (deflection effect) and will create areas of momentum

Internal Wave Modeling

Figure 5 Lake Kinneret, Israel with 10-, 20-, and 30-m depth contours; wind measurements over the lake and on the shore line. Average wind speed and frequency of occurrence in 108 direction bins, from which the wind is coming (meteorological convention), during days 170-183 are plotted as black lines and grey bars, respectively. Adapted from Laval B, Imberger J, and Hodges BR (2003) Modelling circulation in lakes: Spatial and temporal variations. Limnology and Oceanography 48(3): 983-994. Copyright 2003 by the American Society of Limnology and Oceanography, Inc. Reproduced with permission from the American Society of Limnology and Oceanography.

Figure 5 Lake Kinneret, Israel with 10-, 20-, and 30-m depth contours; wind measurements over the lake and on the shore line. Average wind speed and frequency of occurrence in 108 direction bins, from which the wind is coming (meteorological convention), during days 170-183 are plotted as black lines and grey bars, respectively. Adapted from Laval B, Imberger J, and Hodges BR (2003) Modelling circulation in lakes: Spatial and temporal variations. Limnology and Oceanography 48(3): 983-994. Copyright 2003 by the American Society of Limnology and Oceanography, Inc. Reproduced with permission from the American Society of Limnology and Oceanography.

deficit or wakes (sheltering effect). These effects are particularly relevant for lake applications, as they induce spatially variable wind fields over the lower levels in the regional topography where lakes are usually located.

Other Sources of Circulation

Shoreline irregularities (headlands, peninsulas, islands, bays, etc) or inflow/outflow features (i.e., open boundaries) can also act as sources of vorticity in the SML and HB (see Figure 6). The extent to which shoreline irregularities or inflow/outflow features can create vortex structures will depend on geometrical and hydraulic characteristics of the flow such as

• the length scale L of the features; in the case of the peninsulas this length scale, compared with the width of the lake determines the extent to which the flow is blocked and the peninsula effectively seperates the basin in sub-basins;

• their sharpness (i.e., whether the changes in shoreline direction are abrupt or smooth), which can by characterized be a radius of curvature Rw;

• the inflow velocity or, more generally, the spatial gradients of horizontal velocity induced by shoreline features:

• the rate at which momentum is transferred horizontally; and

• the Coriolis force, which represents the rate at which flow tend to veer in response to the Earth's rotation, which is characterized by the inertial frequency f.

For smooth features and small velocities gradients, currents will be diverted but they will tend to follow the shoreline, and no gyral patterns will develop.

Figure 6 Schematic representation of a shoreline feature where velocity gradients form. The magnitude of the undisturbed velocity is U; L represents a characteristic length scale for the feature and Rw characterizes its sharpness. This scheme could represent a portion of a river entering a lake, part of a peninsula, or a section of a bay.

For sharp features and large velocity gradients, gyral patterns will, most likely, develop. Dimensional analysis and laboratory experiments suggest that gyral patterns will develop near inflows or shoreline irregularities if the sharpness of the exit corner is larger than the inertial radius, i.e.,

The dividing line between gyre formation and shoreline attachment is not well established and may depend on details such as the actual current profile across in the main body or in the inflowing water. In any case, the behavior of currents near shoreline irregularities is controlled by the nonlinear nature of water motions.

The horizontal extent of the gyres generated near shoreline irregularities will most likely scale with the size of the irregularities, and in most cases, these gyres will have a only be local (not basin) scale (see Figure 7). In the case of peninsulas or islands that are large compared to the total width of the lake, the large-scale circulation can be modified to the extent that the basins on each side of the obstruction respond to wind as if they, effectively, were independent, with gyral structures similar to those described above for individual lakes. Vortices associated to river jets will only drive basin-scale gyral patterns for very large inflows. For the Upper Lake Constance, for example, it has been estimated that the observed basin-scale cyclonic gyres (12 000 m wide) can only be driven by extreme inflow events of about 1500 m3 s-1 from the Alpine Rhine.

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