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  • View in gallery

    Solution obtained for the (a) streamfunction, (b) alongshore current, and (c) lake/sea level and bottom pressure assuming a basic set of parameters: D = 1 kg m−3, 2b = 100 km, H = 100 m, Ay = 100 m2 s−1, Az = 0.01 m2 s−1, and kz = 0.01 m s−1. The alongshore current is positive when it is outflowing.

  • View in gallery

    Current vorticity in midbasin (−∂u/∂y at y = 0) as a function of depth and (a) the coefficient of bottom friction, (b) the coefficient of vertical friction, (c) the coefficient of lateral friction, and (d) basin halfwidth. The depth of the bottom is kept at 100 m. Positive values imply cyclonic vorticity, negative values anticyclonic vorticity.

  • View in gallery

    As in Fig. 2, but for a 10-m bottom depth.

  • View in gallery

    Solution obtained for the (a),(d) streamfunction; (b),(e) alongshore current; and (c),(f) lake/sea level and bottom pressure, assuming that the coefficient of the bottom friction equals (a)–(c) 10−6 and (d)–(f) 10−3 m s−1. Values of all other parameters are standard. The alongshore current is positive for an outflow.

  • View in gallery

    As in Fig. 4, but for the coefficient of vertical friction equal to (a)–(c) 10−6 and (d)–(f) 10−3 m2 s−1.

  • View in gallery

    As in Fig. 4, but for the coefficient of lateral friction equal to (a)–(c) 100 and (d)–(f) 104 m2 s−1.

  • View in gallery

    As in Fig. 4, but for basin halfwidth equal to (a)–(c) 3 and (d)–(f) 100 km.

  • View in gallery

    As in Fig. 4, but for basin depth equal to (a)–(c) 10 and (d)–(f) 1000 m.

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Cyclonic versus Anticyclonic Circulation in Lakes and Inland Seas

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  • 1 Department of Geophysics, Faculty of Science, University of Zagreb, Zagreb, Croatia
  • 2 University of Dubrovnik, Dubrovnik, Croatia
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Abstract

A simple diagnostic model, reproducing circulation in lakes and marginal seas in which low-density waters are found close to the coasts while high-density waters dominate the offshore areas, is developed. An explicit solution is obtained for the central transverse section of an elongated basin, assuming that the Boussinesq and hydrostatic approximations are valid and that the alongshore variability vanishes. The model reveals cyclonic circulation that may either extend throughout the vertical (type C) or may top anticyclonic circulation developed in the bottom layer (type C/A). With the amplitude of the imposed density anomaly being fixed, the flow type is controlled by the frictional processes and by the basin dimensions. In a typical basin, type C/A flow is supported by weak bottom and vertical friction and by moderate lateral friction, unlike type C flow, which is supported by moderate bottom and vertical friction and by weak lateral friction. Strong frictional influence, especially in the basin interior, suppresses the flow everywhere. The flow is also suppressed in a basin that is narrow O(1 km) and shallow O(10 m), even without the frictional control being too strong. A basin that is narrow and deep favors type C/A flow, whereas a basin that is wide and shallow tends to support type C flow. The theoretical findings are related to observations, particularly those originating from the Adriatic Sea where type C flow prevails but may occasionally be replaced by type C/A flow, as well as to previous modeling results.

Corresponding author address: Mirko Orlić, Dept. of Geophysics, Faculty of Science, University of Zagreb, Horvatovac bb, 10000 Zagreb, Croatia. Email: orlic@olimp.irb.hr

Abstract

A simple diagnostic model, reproducing circulation in lakes and marginal seas in which low-density waters are found close to the coasts while high-density waters dominate the offshore areas, is developed. An explicit solution is obtained for the central transverse section of an elongated basin, assuming that the Boussinesq and hydrostatic approximations are valid and that the alongshore variability vanishes. The model reveals cyclonic circulation that may either extend throughout the vertical (type C) or may top anticyclonic circulation developed in the bottom layer (type C/A). With the amplitude of the imposed density anomaly being fixed, the flow type is controlled by the frictional processes and by the basin dimensions. In a typical basin, type C/A flow is supported by weak bottom and vertical friction and by moderate lateral friction, unlike type C flow, which is supported by moderate bottom and vertical friction and by weak lateral friction. Strong frictional influence, especially in the basin interior, suppresses the flow everywhere. The flow is also suppressed in a basin that is narrow O(1 km) and shallow O(10 m), even without the frictional control being too strong. A basin that is narrow and deep favors type C/A flow, whereas a basin that is wide and shallow tends to support type C flow. The theoretical findings are related to observations, particularly those originating from the Adriatic Sea where type C flow prevails but may occasionally be replaced by type C/A flow, as well as to previous modeling results.

Corresponding author address: Mirko Orlić, Dept. of Geophysics, Faculty of Science, University of Zagreb, Horvatovac bb, 10000 Zagreb, Croatia. Email: orlic@olimp.irb.hr

1. Introduction

As a rule, surface circulation in Northern Hemisphere lakes and marginal seas is cyclonic or counterclockwise (Shtokman 1967; Emery and Csanady 1973). The pattern is observed in a number of North American and Eurasian lakes as well as in various inland seas: the Adriatic Sea, Baltic Sea, Bay of Fundy, Black Sea, Gulf of St. Lawrence, Japan Sea, Mediterranean Sea, Persian Gulf, etc. The only exception to the rule appears to be the Aral Sea in which the circulation of surface waters is anticyclonic (i.e., clockwise). The term “circulation” implies residual motion obtained by removing the variability related to tides and other high-frequency phenomena (primarily currents directly driven by wind, seiches, and inertial oscillations) from the current records. The most appropriate way of defining the circulation pattern would be to subject the simultaneously collected long time series to a low-pass filter with a cutoff period selected so as to correspond with the synoptic atmospheric forcing (a few days). The necessary data, however, are rarely available, and various other approaches have been used to deduce the prevailing cyclonic circulation at the surface of landlocked water bodies.

It appears that the surface cyclonic flow dominating the Northern Hemisphere basins was originally attributed either to coastal freshwater inflows and their deflection to the right by the Coriolis force or to the fast response of shallow coastal areas to thermal forcing and the alongshore currents supported by the resulting cross-shore gradients (Shtokman 1967; Emery and Csanady 1973). Subsequently, a number of other possible explanations have been proposed, with some of them invoking long-term effects of wind-driven transients. Shtokman pointed out that the wind fields above lakes and inland seas are most often characterized by the positive curl and that they impart cyclonic vorticity to the basins irrespective of the wind direction; he was also able to attribute the anomalous conditions in the Aral Sea to the negative wind curl influencing the lake dynamics. Emery and Csanady (1973) considered stratified water bodies and emphasized the importance of the positive wind stress curl, resulting from the upwelling (downwelling) on the left-hand (right hand) shore and the consequent crosswind differences of surface temperature and atmospheric stability. Wunsch (1973) invoked the second-order Lagrangian drift associated with large-amplitude internal Kelvin waves as a possible contributor to the net cyclonic flow. Bennett (1975) based his interpretation on the crosswind differences of stratification, with the low (high) stability of the water column at the upwelling (downwelling) shore supporting counterclockwise circulation even when the wind stress is uniform. Finally, Simons (1986) showed that the nonlinear interactions of topographic Rossby waves in homogeneous lakes and shallow seas can also support the surface cyclonic flow.

While some researchers explored novel interpretations of the surface cyclonic circulation, others maintained interest in the original explanation based on the low density of onshore waters as opposed to the high density of offshore areas and the alongshore flow supported by the density difference. As a result of these latter efforts, various processes contributing to the cross-shore density difference have been elucidated. Huang (1971), Oonishi (1975), and Davidson et al. (1998) considered the case of the temperature in a lake being everywhere higher than the temperature of maximum density, with a shallow coastal zone that is relatively warm after vertical heat diffusion extends to the bottom there. Huang additionally paid attention to the case when in a whole lake the temperature is lower than the temperature of maximum density, with the shallow coastal area being relatively cold due to the controlling influence of vertical mixing. Heaps (1972), Orlić (1996), and Lazar et al. (2007) allowed for the cross-shore salinity difference in inland seas, usually resulting from the coastal river inflows and evaporation dominating the open sea. Moreover, Hill (1993, 1996), Xing and Davies (2005), and Davies and Xing (2006) focused on an inland sea after the onset of stratification, with the cold water remaining trapped beneath the thermocline in the open sea and the warm water being subjected to strong tidal mixing in the shallow coastal area.

All of the modeling studies listed above are in agreement that the offshore increase in density supports a cyclonic circulation of surface waters, although they disagree somewhat on the flow intensity. The studies, however, are strongly polarized when it comes to the circulation of bottom waters: some predict cyclonic flow that extends to the bottom (Heaps 1972; Hill 1993, 1996; Davidson et al. 1998; Lazar et al. 2007), while others obtain anticyclonic circulation in the bottom layer as opposed to cyclonic circulation in the surface layer (Huang 1971; Oonishi 1975; Orlić 1996; Xing and Davies 2005; Davies and Xing 2006). It is rather difficult to interpret the diversity of results because the authors consider different basins and take widely differing dynamics into account. Transverse sections of most basins have dimensions close to 100 km × 100 m, but in one paper the basin considered is much shallower (Heaps 1972) and in another it is both narrower and shallower (Oonishi 1975). Internal friction is either not considered (Hill 1993) or is allowed for by employing a Rayleigh-type parameterization (Orlić 1996), a Boussinesq-type parameterization [vertical—Heaps (1972); Hill (1996), vertical and lateral—Huang (1971); Oonishi (1975); Davidson et al. (1998); Lazar et al. (2007)], or a turbulent closure scheme (Xing and Davies 2005; Davies and Xing 2006). As for the bottom, the authors impose various conditions there: free slip (Orlić 1996), linear slip (Heaps 1972; Lazar et al. 2007), quadratic slip (Davidson et al. 1998; Xing and Davies 2005; Davies and Xing 2006), and no slip (Huang 1971; Oonishi 1975; Hill 1993, 1996). Most authors consider steady-state conditions, and some (Oonishi 1975; Hill 1996; Xing and Davies 2005; Davies and Xing 2006) address transients. Attention is paid not only to linear dynamics but also to nonlinearity (Huang 1971; Oonishi 1975; Davidson et al. 1998; Xing and Davies 2005; Davies and Xing 2006).

As for the observations documenting circulation close to the bottom of lakes and inland seas, it appears that they are rather scanty. In the basin that is best known to us, the Adriatic Sea, most data point to the wintertime cyclonic circulation that develops in the shelf area almost independently from the dynamics prevailing in the southern, deeper part of the basin, since it is related to the local offshore density increase, and that extends to the bottom (Orlić et al. 1992). However, shipboard acoustic Doppler current profiler measurements suggest that on the Adriatic shelf the current vorticity may occasionally vary from cyclonic in the surface layer to anticyclonic in the bottom layer (Gačić et al. 1999). It would thus appear that in the Adriatic there are two types of flow—C (cyclonic from the surface to the bottom) and C/A (cyclonic in the surface layer, anticyclonic in the bottom layer)—and that the former may under some circumstances give way to the latter. Most of the models mentioned above, even when considered over a range of parameters (e.g., Heaps 1972; Lazar et al. 2007), do not suggest that the two flow types may be interchangeable. There are only two exceptions: Davidson et al. (1998) mention that in the absence of bottom friction anticyclonic circulation forms close to the bottom, while Xing and Davies (2005) show that an increase of bottom friction tends to suppress anticyclonic circulation in the bottom layer.

With the aim of revealing various possible conditions under which the type C flow may transform into type C/A flow and vice versa, we formulate a simple diagnostic model by assuming that density is greater in midbasin than close to the coasts, obtain an explicit solution, and explore the sensitivity of the solution to the choice of different parameters. The findings prove to be useful while interpreting the observations. The solution also helps to explain why the previous models, which allow for rather similar cross-basin density differences, so sharply differ when it comes to the flow type that they support.

In section 2 of the present paper the model is formulated and its solution is derived. In section 3 the dependence of the solution on basin dimensions and various friction-related parameters is considered. In section 4 an interpretation of the results is attempted. The findings are summarized and are related to observations and previous modeling results in section 5.

2. The model

a. Basic equations

A transverse section positioned in the central part of an elongated basin of constant width (2b) and depth (H) is considered. The axes of the coordinate system are placed alongshore (x), cross shore (y), and along the vertical (z) so that the domain of interest is defined by −byb and −Hz ≤ 0.

Response of the lake or sea to the temporarily invariant density gradient imposed across the basin is analyzed. The steady-state equations governing the motion and pressure fields may be written as (e.g., Majda 2003)
i1520-0485-39-9-2247-eq1
where v = (u, υ, w) is the velocity vector, k the unit vector in the z direction, p pressure, ρ density, f the Coriolis parameter (regarded as invariable), g = (0, 0, −g) the acceleration due to gravity, and F = (Fx, Fy, Fz) is friction.
By means of the Boussinesq approximation and assuming that there is no alongshore variability, cross-differentiation and linearization give for the perturbations
i1520-0485-39-9-2247-e1
where δ is the density anomaly, while ρ0 stands for a reference density. The density anomaly is assumed to be of the following form:
i1520-0485-39-9-2247-e2
where D is amplitude and the cosine function allows for low density of onshore waters and high density of offshore areas. The assumption that the density anomaly is uniform along the vertical appears to be acceptable under conditions of pronounced vertical mixing. More realistic cross-basin density distributions, as are observed in the Adriatic during winter (Lazar et al. 2007), may be expanded into a Fourier series. For strong surface fronts the Fourier series may exhibit some Gibbs phenomenon patterns, but these decrease in both amplitude and width as the number of terms in the expression increases, unless the distribution is a discontinuous function. Thus, it should be straightforward to apply the solution obtained under the present, simple forcing to the more general, but still depth-independent case.
We have parameterized frictional forces as follows:
i1520-0485-39-9-2247-eq2
where Ay and Az are coefficients of the lateral and vertical turbulent friction. To obtain an analytical solution of the problem we neglect Ay2w/∂y2 and Az2w/∂z2; that is, it is assumed that the hydrostatic approximation is valid.
Introducing ΔAAy2/∂y2 + Az2/∂z2, the system (1) can be written now as
i1520-0485-39-9-2247-e3
which has to be solved subject to the following boundary conditions:
i1520-0485-39-9-2247-e4
where kz is the bottom friction coefficient. Thus, free slip is assumed at the surface, no slip at the coast, and linear slip at the bottom. By neglecting cross-boundary flows the currents driven by the addition/removal of mass are filtered out, but they are expected to be several orders of magnitude smaller than the thermohaline flow (Orlić 1996).
The equation of continuity (3c) enables one to introduce the streamfunction Ψ:
i1520-0485-39-9-2247-eq3
Thus, the system (3) yields
i1520-0485-39-9-2247-e5
and the boundary conditions (4) are transformed into
i1520-0485-39-9-2247-e6
where Cz = kz/Az. By applying the operator f∂/∂z to the first equation in (5) and ΔA to the second, a single equation for the streamfunction is obtained:
i1520-0485-39-9-2247-e7
It has been taken into account that δ does not depend on depth.
The relation between the streamfunction and alongshore current follows from integration of equation (5b) and the use of boundary condition (6b):
i1520-0485-39-9-2247-e8
where
i1520-0485-39-9-2247-eq4
that is, G is equal to the rhs of Eq. (5b).

b. Explicit solution

To find the streamfunction we assume it to be of the form Ψ(y, z) = Y(y)Z(z). Although the variables in (7) cannot be separated, by assuming Y to be of the form
i1520-0485-39-9-2247-eq5
where λ = π/b, we ensure that the boundary conditions for Ψ at the coast are satisfied. This choice of Y also enables a separate equation for the function Z to be obtained (for more details see the appendix).
Thus, one obtains for the solution
i1520-0485-39-9-2247-e9
where ω1,2 = (Ayλ2 ± if )/Az, while the constants ci, i = 1, … , 6, are determined from the boundary conditions:
i1520-0485-39-9-2247-eq6
and E = ω2[ω2 sinh(ω2H) + Cz cosh(ω2H)].

The alongshore current u is then calculated by means of Eq. (8).

Besides the circulation, we also analyze the forces supporting it. More precisely, we explore the lake/sea surface profile and the pressure field, as well as their dependence on the friction coefficients.

By integrating the hydrostatic equation along the vertical and taking into account the equation of motion and the mass conservation law, one obtains for sea level
i1520-0485-39-9-2247-eq7
and for bottom pressure
i1520-0485-39-9-2247-eq8
which completes the solution (see the appendix).

In the rest of the paper, the difference pHpa will be considered since it adequately documents the bottom pressure variability.

3. Results

The solution obtained for the streamfunction (Ψ), alongshore current (u), and pressure field (ζ and pHpa) is illustrated in Fig. 1 for a basic set of parameters: D = 1 kg m−3, 2b = 100 km, H = 100 m, Ay = 100 m2 s−1, Az = 0.01 m2 s−1, and kz = 0.01 m s−1. As expected, transverse currents are characterized by upwelling that is developed close to the coasts and is compensated by downwelling over the open part of the basin. Upwelling occurs in the zone in which density is small, and downwelling develops in the area in which density is high. As for the alongshore current, for the present choice of parameters it implies cyclonic circulation over the greater part of the water column, corresponding with lake/sea level that is elevated at the coasts and depressed in the middle part of the basin. Only close to the bottom is there an indication of a different regime, related to the pressure distribution being opposite to that at the surface, but for the parameters selected the values are too small to be discernible.

To illustrate how the solution depends on the choice of parameters, we decided to vary the parameters over broad ranges exceeding those usually observed in lakes and inland seas. Only one parameter at a time was changed while the others were kept as originally selected. The only parameter that was not subjected to this scrutiny was D because the cross-shore density differences supporting the alongshore currents do not change much if various basins or different years in a particular basin are considered. Moreover, the solution is linear in D, implying that its variability is simply reflected in changes of all the dependent variables. The current vorticity in midbasin (i.e., −∂u/∂y at y = 0) has been depicted with the aim of demonstrating conditions under which the solution reproduces the two flow types (C and C/A).

Figure 2 shows the results obtained for H = 100 m. When the bottom friction coefficient is small, the type C/A flow occurs, with the level of no motion being positioned at middepth and the surface cyclonic circulation being exactly opposite to the bottom anticyclonic circulation (Fig. 2a). An increase in the coefficient brings about weakening of the bottom-layer anticyclonic flow and strengthening of the near-surface cyclonic flow. The latter attains a maximum for kz = 0.001 m s−1. A further increase of the coefficient results in a slight decrease of surface currents.

Dependence of the solution on the coefficient of vertical friction is rather similar (Fig. 2b). When the coefficient is small, type C/A flow is obtained, although the level of no motion is now positioned closer to the bottom and the cyclonic flow peaks at the sea surface while the anticyclonic flow attains a maximum at some distance above the bottom. An increase of the coefficient to Az = 0.001 m2 s−1 results in weakening of the anticyclonic flow and strengthening of the cyclonic flow. A further increase of the coefficient causes a decrease of flow in the whole water column.

The coefficient of lateral friction controls the solution in a completely different way (Fig. 2c). A small value of the coefficient supports the type C flow. With an increase of the coefficient—to a rather high value of Ay = 10 000 m2 s−1—type C/A flow develops, although the surface cyclonic circulation is much more pronounced than the near-bottom anticyclonic circulation, while the level of no motion is found at middepth. If the coefficient is increased even more—to values that may be realistic for the open ocean but not for a lake or an inland sea—the currents are reduced throughout the water column.

The basin halfwidth influences the solution in yet another way (Fig. 2d). When it equals 3 km, type C/A flow is pronounced with the cyclonic flow attaining a maximum at the lake/sea surface, the anticyclonic flow culminating at some distance above the bottom, and the level of no motion being positioned at middepth. With an increase of the basin width there is a tendency for type C flow to develop, although with considerably reduced speed. As expected, a decrease of the basin halfwidth below 3 km suppresses the flow everywhere.

By increasing the basin depth to H = 1000 m, the properties of the solution (not shown) remain similar except that the current vorticity is much greater and that the type C/A flow tends to extend over larger values of kz and Az and somewhat smaller values of Ay. A decrease in basin depth to H = 10 m (Fig. 3) results in a considerably reduced current vorticity, as well as in a shift of the type C/A flow toward smaller values of kz and Az, larger values of Ay, and to smaller basin halfwidths.

4. Discussion

With the aim of demonstrating the physical processes that control various flow regimes, cross-basin currents, alongshore currents, lake/sea surface profiles, and bottom-pressure distributions are plotted by selecting, for each parameter, two values that support the largest difference between flow types. Figure 4 thus shows the results for an extremely small value of the bottom friction coefficient (10−6 m s−1), practically implying free slip at the bottom, and a moderate value of the coefficient (10−3 m s−1). When the coefficient is small, the balance of forces is close to geostrophic, with the alongshore current being well developed in both surface and bottom layers. An increase of the coefficient results in a more pronounced departure from geostrophy in the bottom layer, manifested by an initial decrease of the alongshore current and a slight increase of the cross-basin current there. The latter implies an increase of the lake/sea surface slope, which strengthens the surface-layer currents, weakens the bottom-layer pressure gradient, and, consequently, reduces the bottom-layer currents. To summarize, an increase of the coefficient of bottom friction supports transformation of the type C/A flow to type C flow—a finding that parallels conclusions previously reached by Davidson et al. (1998) and Xing and Davies (2005).

In Fig. 5 the results obtained for small (10−6 m2 s−1) and moderate (10−3 m2 s−1) value of the coefficient of vertical friction are depicted. When the coefficient is small, so is the Ekman layer thickness; therefore, the influence of bottom friction is limited: the alongshore current freely develops in the surface layer and in much of the bottom layer. With an increase of the coefficient, the influence of bottom friction is felt to a greater height, the alongshore current in the bottom layer is suppressed, the cross-basin current there perceptibly intensifies, and the lake/sea surface slope increases. Consequently, currents strengthen in the surface layer and the pressure gradient decreases in the bottom layer, thus supporting weaker currents there. Overall, the effect is similar to that for varying the coefficient of bottom friction, with an increase of the coefficient bringing about transformation of type C/A flow to type C flow. There are, however, also some differences: the alongshore current under the type C/A flow regime does not extend to the bottom as it did before (Fig. 5b versus Fig. 4b) and a further increase of the coefficient reduces the alongshore flow considerably (as may be seen, e.g., by comparing Fig. 5e with Fig. 1b).

Figure 6 shows results for a small (100 m2 s−1) and a rather large (104 m2 s−1) coefficient of lateral friction. A small value of the coefficient implies a thin coastal boundary layer and a flow regime that appears to be geostrophically balanced in the surface layer, whereas it is dominated by the vertical and bottom frictional processes in the bottom layer. An increase of lateral friction, combined with the no-slip coastal boundary condition, results in a diminished alongshore current and augmented cross-basin current in the surface layer. The latter supports a decrease in the lake/sea surface slope and an increase in the near-bottom pressure gradient, which, in turn, implies stronger currents in the bottom layer—notwithstanding the strong control that bottom, vertical, and lateral friction exerts on processes close to the bottom. A further increase of the coefficient of lateral friction results in suppression of all currents. As stated before, the coefficient is allowed to reach values that are not realistic for a lake or a marginal sea in order to emphasize the dynamics underlying the solution. Unlike the other two friction-related parameters, a limited increase of the coefficient of lateral friction brings about transformation of type C flow to type C/A flow. More similar to the other parameters, a further increase of the coefficient suppresses the flow everywhere.

Figure 7 illustrates effects of the halfwidth being varied from 3 to 100 km, with the amplitude of the density anomaly held fixed. When the basin is narrow, the pressure gradient is pronounced and, consequently, both transverse and alongshore currents are well developed. In a wide basin the pressure gradient is reduced, even at the lake/sea surface where greater variability is counteracted by the greater basin width, the cross-basin currents are also diminished as are the alongshore currents—especially in the bottom layer. In a basin that is extremely narrow, lateral friction suppresses currents in the whole domain. Various basin widths may support either strong cross-basin currents and type C/A flow or weak transverse circulation and type C flow, which is reminiscent of the effect of lateral friction.

Finally, let us consider the effect of shallow (10 m) and deep (1000 m) basin depths on the flow regime (Fig. 8). In the former case the influence of bottom friction is felt over the whole water column, and cross-basin currents are weak, as are alongshore currents—more so in the bottom layer. An increase of basin depth implies that the pressure gradient and the cross-basin current intensify everywhere. The effect of bottom friction is now limited to the lower part of the basin and, therefore, an alongshore current freely develops in the surface layer, whereas weak, but still perceptible, alongshore flow occurs in the bottom layer. Overall, variable basin depths allow for a weak transverse circulation to be combined with the type C flow or a strong cross-basin circulation to occur along with type C/A flow, which again reminds one of the effect of lateral friction.

5. Conclusions

The present model clearly demonstrates that in lakes and inland seas in which low-density water is found close to the coasts while high-density water occupies the interior parts of the basin, cyclonic circulation may either extend throughout the vertical (type C flow) or may top an anticyclonic circulation that develops in the bottom layer (type C/A flow), depending on the control that frictional processes and basin dimensions exert on the flow. In typical basins the type C/A flow is supported by weak bottom and vertical friction and by moderate lateral friction, unlike the type C flow that is supported by moderate bottom and vertical friction and weak lateral friction. Strong frictional processes, especially those in the basin interiors, suppress the flow everywhere. Flow is also suppressed in basins that are narrow and shallow, even without the frictional control being too strong. Basins that are narrow and deep favor type C/A flow, whereas basins that are wide and shallow tend to support type C flow. In each particular case the flow is determined by all of the controlling factors, as represented by the explicit solution given in section 2.

The solution shows that, whenever density increases in an offshore direction in landlocked water bodies, a cyclonic circulation may be expected to occur at the surface—a finding that is not novel. The solution, however, may prove to be useful when interpreting the precise relationship between the cross-shore density gradient, as documented by in situ data, on one hand, and intensity of the surface circulation as determined by direct current measurements or the lake/sea surface slope as documented by satellite altimetry data, on the other hand. The solution may be even more directly applicable when interpreting the near-bottom circulation. As already mentioned, it has been observed that in the wintertime Adriatic Sea the type C flow prevails but occasionally it may be replaced by type C/A flow (Gačić et al. 1999). The present model, aimed at simulating steady-state conditions, cannot reproduce the transformation from one flow type to the other. Yet, it suggests that the moderate (weak) bottom and vertical friction and the weak (moderate) lateral friction support the type C (C/A) flow. It is well known that in the Adriatic Sea turbulence is pronounced and mesoscale activity is low when strong winds are blowing, and that the turbulence is reduced while the eddies are well developed after relaxation of some wind events. The two regimes may be accompanied by different levels of momentum exchange that would, in the bottom layer, support cyclonic circulation in the former case and anticyclonic circulation in the latter case. This finding deserves more detailed study.

The present model may also help to interpret the diversity of theoretical results previously obtained for circulation supported by cross-shore density differences in lakes and marginal seas. Thus, type C flow was simulated for the following reasons: Heaps (1972) considered a shallow basin subjected to strong mixing in the vertical only, Hill (1993) imposed a no-slip bottom-boundary condition on the frictionless fluid, Hill (1996) employed a no-slip condition at the bottom and retained vertical friction but not lateral friction, Davidson et al. (1998) initially assumed the bottom friction to be moderate while minimizing the effect of lateral friction, whereas Lazar et al. (2007) allowed for moderate vertical friction and did not impose any dynamic condition at the coasts. On the other hand, the type C/A flow was probably obtained because the basin considered was extremely narrow (Oonishi 1975), free slip was allowed at the bottom (Orlić 1996), and the effect of bottom friction was initially (Xing and Davies 2005) or consistently (Davies and Xing 2006) assumed to be small. Still puzzling is the type C/A flow arrived at by Huang (1971), who considered a basin having typical dimensions, rather pronounced bottom and vertical friction and less pronounced lateral friction, that is, conditions under which the present model would produce type C flow.

As is the case with all explicit solutions, the present one enables the influence of various controlling parameters to be easily explored, but at the same time is not completely realistic. One of the simplifying assumptions is that the basin has a flat bottom. As shown by Davies and Xing (2006), topographic variability, which is important when simulating the response of both the current and density fields to the external forcing, does not play an important role when modeling adjustment of currents to initially imposed densities. Since the present model is diagnostic, it may be expected that its dominant features are independent of topography.

Another basic assumption is that density is vertically uniform. An adverse effect of this assumption becomes obvious if one inserts a density anomaly (2) and streamfunction (9) into the density dispersion equation. The only possible balance is then between advection and horizontal diffusion, and—as pointed out by one of the reviewers—the coefficient of horizontal eddy diffusivity stemming from such an exercise is meaningless. If, however, one defines a density anomaly as δδ1(y) + δ2(z), where δ1 = D cos(πy/b), as before, whereas, for example, δ2 = d cos[π(z + H)/2H], it is possible to obtain reasonable coefficients of horizontal and vertical eddy diffusivities for the case dD, that is, when the solution in the present paper remains valid. It is interesting to note that if density varies even slightly along the vertical, the advection is primarily balanced by vertical diffusion, suggesting that buoyancy is diffused downward in the source areas and advected to the sink areas where it is diffused upward. Of course, proper modeling of this process cannot be based on an assumed density field but would require that both hydrodynamic and thermodynamic parts of the problem be addressed. While the modeling may prove rewarding, it falls outside the scope of the present diagnostic effort—which was to reproduce the current and pressure fields that correspond to a realistic density distribution.

The assumption that density is vertically uniform is acceptable for the basins subjected to strong vertical mixing, including the wintertime Adriatic Sea, but less so for basins in which a surface warm water pool is observed above a bottom cold water dome (Hill 1993, 1996; Xing and Davies 2005; Davies and Xing 2006). As demonstrated by Davies and Xing (2006), such stratification supports type C/A flow, with the surface-layer cyclonic flow reaching a maximum at some depth, not at the sea surface. The present model is not able to reproduce a subsurface current maximum because it does not allow for surface- and bottom-layer density gradients in opposite directions, but it is successful at simulating the basic flow type.

Yet another limitation is the parameterization of frictional forces, especially because the flow type was found to depend not only on the basin dimensions but also on frictional processes. On the other hand, all parameterizations have their shortcomings, so the best one can do is to treat carefully any property of a solution that sensitively depends on the particular choice of parameterization. Thus, for example, it would not be advisable to use the present solution to determine the level of no motion in lakes and marginal seas without previously showing that the vertical distribution of currents does not change perceptibly if subjected to a different frictional control. On the other hand, it is expected that the basic conclusions, regarding qualitative conditions under which a cyclonic circulation either occurs in the whole basin or only in the surface layer below which an anticyclonic circulation develops, would be preserved irrespective of the way frictional forces are parameterized.

As mentioned in the introduction, the surface cyclonic circulation in the Northern Hemisphere basins may be interpreted not only in terms of the alongshore flow supported by an onshore–offshore density difference but also by invoking a number of other mechanisms (Shtokman 1967; Emery and Csanady 1973; Wunsch 1973; Bennett 1975; Simons 1986). It would appear, however, that these alternative models—except possibly the one proposed by Wunsch (1973)—imply a cyclonic circulation that extends down to the bottom. If evidence is found in the future that the near-bottom anticyclonic circulation occasionally occurs in the other basins besides the Adriatic, the ability of a model to reproduce such flow may be decisive while discriminating it from the other models.

Acknowledgments

We thank anonymous reviewers for constructive criticism of the original version of the manuscript. The work presented in this paper was supported by the Ministry of Science, Education and Sports of the Republic of Croatia (Projects 119-1193086-3085 and 037-1193086-3226).

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APPENDIX

Derivation of the Solution

Current field

To find a solution to (7), 10 boundary conditions for the streamfunction are needed (4 at the coasts and 6 at the surface and bottom). Six of these are already given by (6), while the other four are derived from the boundary conditions for the alongshore current u and relations (8) and (5). These read
i1520-0485-39-9-2247-eqa1
Here, the properties of function δ were used again.
The streamfunction is supposed to be of the form Ψ(y, z) = Y(y)Z(z), while the function Y is chosen as B sinπy/b. Substituting the last relation into (7) we get
i1520-0485-39-9-2247-eqa2
where λ = π/b. The constant B is chosen such that Y = ∂δ/∂y; that is, B = −λD.
With this definition, the equation for Z reads
i1520-0485-39-9-2247-eqa3
Let us rewrite function Z as
i1520-0485-39-9-2247-eqa4
where ZP is a particular solution of the above equation:
i1520-0485-39-9-2247-eqa5
while ZH solves the homogeneous equation together with the appropriate boundary conditions. We suppose it to be of the form
i1520-0485-39-9-2247-eqa6
By solving the characteristic equation, one gets
i1520-0485-39-9-2247-eqa7
and the solution reads
i1520-0485-39-9-2247-eqa8

The boundary conditions for Ψ impose a system of linear equations determining constants ci, i = 1, … , 6.

Pressure field

Integrating the hydrostatic equation in the vertical, we get
i1520-0485-39-9-2247-ea1
where pa denotes atmospheric pressure and ζ denotes lake/sea level. To remove the unknown pressure p, we integrate the y component of the equation of motion from −b (left coast) to y, thus obtaining
i1520-0485-39-9-2247-eqa9
Combining the above two equations and taking z = −H, one obtains a relation for the lake/sea level:
i1520-0485-39-9-2247-ea2
The only unknown in the above expression is ζ−b—the lake/sea level at the left coast, which is determined by the mass conservation law,
i1520-0485-39-9-2247-eqa10
and integrating (A2) across the basin. Thus, we get
i1520-0485-39-9-2247-eqa11
implying
i1520-0485-39-9-2247-eqa12
By means of simple calculus, the above relation simplifies to
i1520-0485-39-9-2247-eqa13
The pressure field is now determined by (A1). Specifically, at the bottom we have
i1520-0485-39-9-2247-eqa14
which completes the solution.
Fig. 1.
Fig. 1.

Solution obtained for the (a) streamfunction, (b) alongshore current, and (c) lake/sea level and bottom pressure assuming a basic set of parameters: D = 1 kg m−3, 2b = 100 km, H = 100 m, Ay = 100 m2 s−1, Az = 0.01 m2 s−1, and kz = 0.01 m s−1. The alongshore current is positive when it is outflowing.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4068.1

Fig. 2.
Fig. 2.

Current vorticity in midbasin (−∂u/∂y at y = 0) as a function of depth and (a) the coefficient of bottom friction, (b) the coefficient of vertical friction, (c) the coefficient of lateral friction, and (d) basin halfwidth. The depth of the bottom is kept at 100 m. Positive values imply cyclonic vorticity, negative values anticyclonic vorticity.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4068.1

Fig. 3.
Fig. 3.

As in Fig. 2, but for a 10-m bottom depth.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4068.1

Fig. 4.
Fig. 4.

Solution obtained for the (a),(d) streamfunction; (b),(e) alongshore current; and (c),(f) lake/sea level and bottom pressure, assuming that the coefficient of the bottom friction equals (a)–(c) 10−6 and (d)–(f) 10−3 m s−1. Values of all other parameters are standard. The alongshore current is positive for an outflow.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4068.1

Fig. 5.
Fig. 5.

As in Fig. 4, but for the coefficient of vertical friction equal to (a)–(c) 10−6 and (d)–(f) 10−3 m2 s−1.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4068.1

Fig. 6.
Fig. 6.

As in Fig. 4, but for the coefficient of lateral friction equal to (a)–(c) 100 and (d)–(f) 104 m2 s−1.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4068.1

Fig. 7.
Fig. 7.

As in Fig. 4, but for basin halfwidth equal to (a)–(c) 3 and (d)–(f) 100 km.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4068.1

Fig. 8.
Fig. 8.

As in Fig. 4, but for basin depth equal to (a)–(c) 10 and (d)–(f) 1000 m.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4068.1

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