Analytical Model of Thermohaline Circulation in Land-Locked Basins: Analyzing the Impact of Friction on Circulation Reversal

Martin Lazar aDepartment of Electrical Engineering and Computing, University of Dubrovnik, Dubrovnik, Croatia

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Maja Bubalo bDepartment of Geophysics, Faculty of Science, University of Zagreb, Zagreb, Croatia

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Josip Begić cLebesgue, Inc., San Francisco, California

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Abstract

The paper investigates switches of circulation orientation in inland basins, either at the surface or near the bottom. The study is based on an analytical 2D model used to simulate thermohaline circulation in lakes and inland seas. The model allows different density profiles varying in both horizontal and vertical directions. By assuming some simplifications (such as steady state, vanishing of an alongshore variability, and flat bottom), we are able to obtain an explicit expression of the circulation in the central transverse section of an elongated basin. Starting from three typical density profiles (bottom dense water, surface light water, and a combination of the two), the model reveals different circulation types (cyclonic and anticyclonic surface circulation, either prevailing along the whole vertical column or accompanied by an opposite circulation in the bottom layer). In addition, we analyze the impact of friction coefficients and basin dimensions on the switch from one circulation type to another. The simplified assumptions turn out not to be limiting, as other studies have shown that they do not change the main flow characteristics. More importantly, the results obtained are in keeping with empirical findings, numerical simulations, and physical experiments studied elsewhere.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Martin Lazar, mlazar@unidu.hr

Abstract

The paper investigates switches of circulation orientation in inland basins, either at the surface or near the bottom. The study is based on an analytical 2D model used to simulate thermohaline circulation in lakes and inland seas. The model allows different density profiles varying in both horizontal and vertical directions. By assuming some simplifications (such as steady state, vanishing of an alongshore variability, and flat bottom), we are able to obtain an explicit expression of the circulation in the central transverse section of an elongated basin. Starting from three typical density profiles (bottom dense water, surface light water, and a combination of the two), the model reveals different circulation types (cyclonic and anticyclonic surface circulation, either prevailing along the whole vertical column or accompanied by an opposite circulation in the bottom layer). In addition, we analyze the impact of friction coefficients and basin dimensions on the switch from one circulation type to another. The simplified assumptions turn out not to be limiting, as other studies have shown that they do not change the main flow characteristics. More importantly, the results obtained are in keeping with empirical findings, numerical simulations, and physical experiments studied elsewhere.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Martin Lazar, mlazar@unidu.hr

1. Introduction

Basin circulation on a large time scale is significantly determined by global motion related to thermohaline properties (Rahmstorf 2007), with turbulence caused by local forcing (such as wind) having a secondary role. This paper analyzes reversal of thermohaline circulation in inland basins that might occur either at the surface or near the bottom.

In the Northern Hemisphere, surface circulation in seas and lakes is predominantly cyclonic (Shtokman 1967; Emery and Csanady 1973). Exceptions to this rule have been observed in the Aral Sea, in which the circulation of surface waters is anticyclonic (Sirjacobs et al. 2004), as well as in the Ionian Sea and Lake Erie, where surface circulation occasionally shifts from cyclonic to anticyclonic (Borzelli et al. 2009; Malanotte-Rizzoli et al. 2014; Beletsky et al. 2012). As for the circulation of bottom waters, it appears that observations are rather scanty. However, some experiments have shown that the bottom circulation of a basin can, in certain conditions, be opposite to the surface one (Gačić et al. 1999; Borzelli et al. 2009). This phenomenon has also been confirmed by theoretical and numerical studies (Huang 1971; Oonishi 1975; Orlić 1996; Xing and Davies 2005; Davies and Xing 2006; Orlić and Lazar 2009).

Orlić and Lazar (2009) developed a simple diagnostic model to reproduce thermohaline currents in lakes and marginal seas. The model reveals cyclonic surface circulation, and the authors explore conditions under which it either extends throughout the vertical, or tops anticyclonic flow in the bottom layer. However, the model assumes a vertically uniform density profile, which prevents study of some realistic phenomena (such as a bottom dome formation) that might have a significant impact on the type of the circulation, either at the surface or near the bottom (Borzelli et al. 2009; Beletsky et al. 2012). This work is a continuation of that study, in which we allow for vertical variations in density of basin water. Several types of density distributions are considered (dense bottom water, surface light water, and a combination of both), as well as the impact of friction coefficients on the switch from cyclonic to anticyclonic circulation (and vice versa).

The chosen density profiles are idealized versions of density profiles that can be encountered in real lakes and seas. Both bottom dense water and surface light water profiles with their accompanying circulations were observed in the Ionian Sea, where the switch between the two profiles is related to the strength of the Eastern Mediterranean Transient (EMT) (Borzelli et al. 2009). The case of light surface water was also observed in Lake Erie, whose summer thermal structure corresponds to a vertical density profile that has a minimum in the central part of the basin surface (Beletsky et al. 2012). The combination of bottom dense water and surface light water is similar to what was observed in the Yellow Sea (Yuan and Li 1993).

In this paper, we develop an explicitly solvable, parameter-dependent model of thermohaline circulation in landlocked basins run by different density distributions and subject to variable friction forces. The model is able to reproduce different basic types of thermohaline circulation. Its aim is to analyze and try to reveal answers to the following list of questions:

  1. How are different density profiles related to the types of circulation appearing in the basins?

  2. Under which conditions does one circulation type (either cyclonic or anticyclonic) prevail along the entire basin vertical?

  3. What triggers the switch of bottom circulation from cyclonic to anticyclonic and vice versa?

  4. Which conditions support the change in orientation of surface currents?

The explicit expression of the circulation turns out to be crucial in accomplishing this task. It allows one to easily explore the sensitivity of the model to different friction and density parameters, which in turn determine the main characteristics of basin circulation and its change from one type to another.

The paper is organized as follows. In the next section, we develop the model and derive the formula for its solution. Section 3 provides results for different density distributions and various values of the friction coefficients. In section 4 we discuss the findings and relate them to observations (both from real seas/lakes and from laboratory experiments) and previous modeling results. We also address the assumptions of the model and its possible generalizations. The final section provides answers to the questions posed above and finalizes the paper with concluding remarks.

2. The model

We develop a 2D model describing the response of a land-locked basin to a temporarily invariant density gradient. The basin is approximated by an elongated channel of constant width (2b) and depth (H). Axes of the coordinate system are placed alongshore (x), cross shore (y) and along the vertical (z). A transverse section positioned in the central part of the basin is considered, so that the domain of interest is defined by −byb and −Hz ≤ 0 (Fig. 1).

Fig. 1.
Fig. 1.

Schematic presentation of the considered basin. The domain of the model is restricted to the transverse section (shaded).

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0251.1

The steady-state equations governing the motion and pressure fields in the absence of external forcing may be written (e.g., Majda 2003)
(v)vfv×k=1ρp+g+Fvρ+ρv=0,
where v = (u, υ, w) is the velocity vector, k the unit vector in z direction, p pressure, ρ density, f Coriolis parameter (regarded as invariable), g = (0, 0, −g) acceleration due to gravity, and F = (Fx, Fy, Fz) friction.
By means of the Boussinesq approximation and assuming that there is no alongshore variability, the linearization of the above system gives
fυ=Fx,
fu=1ρ0py+Fy,
0=1ρ0pzρgρ0+Fz,
υy+wz=0,
where the density ρ = ρ0 + δ is split into a constant reference density ρ0 and density anomaly δ, which is assumed to be of the form
δ(y,z)=[1+cos(λy)]{cos[μ(z+H)]q}.
with λ = π/b and μ = π/(2H). The parameter q regulates the shape of the density distribution. In this paper we used q = 0, 1 and 2/2 which describe bottom dense water, surface light water and a combination of the two, respectively.
The frictional forces are parameterized as follows:
Fx=ΔAu,Fy=ΔAυ,Fz=ΔAw,
where ΔA=Ah(2/y2)+Aυ(2/z2) is the anisotropic Laplace operator, while Ah and Aυ are coefficients of lateral and vertical eddy viscosity, respectively.
By assuming the hydrostatic approximation is valid (i.e., by neglecting the terms Ay2w/∂y2 and Az2w/∂z2), the cross differentiation of the system (1) gives
fυ=ΔAu,
fuz=zΔAυ+gρ0δy,
υy+wz=0.
The boundary conditions for this problem are imposed as follows:
ub=ub=0,
υb=υb=0,
wH=w0=0,
(uz)0=(υz)0=0,
Az(uz)HkυuH=Az(υz)HkυυH=0,
where kυ is the bottom friction coefficient. Thus, free slip is assumed at the surface, a no slip boundary condition at the coast and linear slip at the bottom.
Remark 1: The method used in this article allows for more general boundary conditions to be applied. In particular, the wind forcing can be included by imposing surface boundary conditions of the form
ρ0Ah(uz)0=τx,ρ0Ah(υz)0=τy,
where τx and τy are wind stress components. However, the aim of the article is to analyze the response of a basin to an imposed density distribution in the absence of external forcing. We also neglect cross-boundary flows, but the currents driven by addition/removal of mass are expected to be several orders of magnitude smaller than the thermohaline flow (Orlić 1996).
The equation of continuity (1d) enables us to define the streamfunction Ψ(y, z):
υ=Ψz,w=Ψy,
which allows us to transform the equations of motion (2) to the following system:
fΨz=ΔAu,
fuz=gρ0δyΔA2Ψz2

a. The solution for the currents

By applying cross differentiation to system (4), a single equation for the streamfunction is obtained:
f22Ψz2+ΔA22Ψz2=gρ0ΔAδy
The corresponding boundary conditions stemming from (3) read as
Ψb=Ψb=0,(3Ψy2z)±b=0,ΨH=Ψ0=0,(2Ψz2)0=0,(4Ψz4)0=gρ0Aυ(δy)0,Aυ(2Ψz2)Hkυ(Ψz)H=0,Aυ(gρ03δz2yΔA4Ψz4)Hkυ(gρ02δzyΔA3Ψz3)H=0.
An explicit solution for the problem (5) + (6) can be found by assuming it to be of the form Ψ(y, z) = Y(y)Z(z). Although Eq. (5) does not allow for variable separation, by choosing Y(y) = sin(λy) we ensure all the boundary conditions at the coast (y = ±b) are fulfilled and the problem reduces to an ordinary differential equation for function Z:
A1Z(z)2AhAυλ2Z(iυ)(z)+Aυ2Z(υi)(z)=gρ0λ{λ2Ahq+A2cos[μ(z+H)]}.
where we use abbreviations A1=(f2+λ4Ay2) and A2=λ2Ah+μ2Aυ.
The solution is written in the form Z = ZH + ZP, where the particular solution is given by the relation
ZP(z)=gρ0λ{λ2Ayq2A1z2+A2Acos[μ(z+H)]},
with A=μ2(A1+2λ2μ2AhAυ+μ4Aυ2), while ZH stands for the solution to the homogeneous problem. By solving the corresponding characteristic equation, one gets that the solution for the streamfunction reads as
Ψ(y,z)=sin(λy)(Zp+c1+c2z+c3eω1z+c4eω2z+c5eω1z+c6eω2z),
where ω1,2=(Ahλ2 ± if)/Aυ, while the constants ci, i = 1, …, 6 are determined from the boundary conditions.
Once having obtained the explicit expression for the streamfunction, a formula for the alongshore current follows from integration of Eq. (4b) and the linear slip condition at the bottom (3e):
u(y,z)=AυkυfG(y,H)+1fHzG(y,z˜)dz˜,
where
G(y,z)=gρ0δ(y,z)yΔA2Ψ(y,z)z2

b. The solution for bottom pressure and lake/sea level

Apart from the circulation, we also analyze the forces supporting it. In particular, we explore the pressure field and the lake/sea level profile, as well as their dependence on the friction coefficients.

To obtain an expression for the lake/sea level, we first integrate the hydrostatic Eq. (1c) (with Fz = 0) along the vertical
p[y,ζ(y)]p(y,H)=pap(y,H)=Hζ(y)g[ρ0+δ(y,z˜)]dz˜,
where ζ denotes lake/sea level, while pa stands for the atmospheric pressure. Hereby we impose continuity of pressure across an air–sea interface and we neglect the surface tension (which has a significant impact on dynamics only on small scales; Cushman-Roisin and Beckers 2011).
To get rid of the unknown pressure p in the last equation, we integrate the y component of the equation of motion, (1b), from −b (left coast) to y (for z = −H), thus obtaining
p(y,H)p(b,H)=byρ0(Fyfu)Hdy˜.
By combining the last two equations, one gets
Hζ(y)g(ρ0+δ)dz˜=Hζ(b)g(ρ0+δ)bdz˜+byρ0(Fyfu)Hdy˜.
Integration of the last equation across the basin and the mass conservation law
bbHζ(y)g(ρ0+δ)dz˜dy˜=gρ02bH,
where the right-hand side stands for the mass of a nondisturbed basin, implies
Hζ(b)g(ρ0+δ)bdz˜=ρ0gH12bbbbyρ0(Fyfu)Hdy˜dy
Plugging the last expression into (10), we finally obtain
Hζ(y)g(ρ0+δ)dz˜=ρ0gH12bbbbyρ0(Fyfu)Hdy˜dy+byρ0(Fyfu)Hdy˜.
Using the expressions for δ, Ψ, and Fy, and performing integrations in the last line, we get a nonlinear, scalar equation for the lake/sea level:
ρ0ζ(y)+[1+cos(λy)]{1μsin{μ[ζ(y)+H]}q[ζ(y)+H]}=C11λgcos(λy)
where
C1=ρ0λ2AhZHρ0AυZH+ρ0Aυgλkυρ0(1q)λ2ρ0AυAhkυZH+ρ0Aυ2kυZH(iυ)
The last equation cannot be solved explicitly, and the value of ζ is obtained by numerical methods based on the Levenberg–Marquardt algorithm (in particular the function fsolve in MATLAB).

Once ζ is calculated, the pressure field at the bottom follows directly from (9).

3. Results

We illustrate the solutions for the streamfunction Ψ, alongshore current u, density profile δ, lake/sea level ζ, and bottom pressure p−H for various values of horizontal (Ah) and vertical viscosity (Aυ), bottom friction coefficient (kυ), and the parameter q that determines the density distribution. All the solutions and figures are obtained by means of the MATLAB program package.

a. Varying the parameter q and basin depth

In the first set of numerical simulations, we present the results for three types of the density distribution determined by the various values of the parameter q: bottom dense water (q = 0), surface light water (q = 1), and a combination of the two (q=2/2). For friction coefficients we use a basic set of values: Ah = 100 m2 s−1, Aυ = 0.01 m2 s−1, and kυ = 0.01 m s−1.

In the case of bottom dense water (q = 0, Fig. 2c), the density is constant at the surface and it increases toward the center bottom, where it is the highest. The streamfunction (Fig. 2a) describes both cross-shore and vertical velocity components (υ and w, respectively). Higher-density variations at the bottom cause stronger currents in that area, whereas closer to the surface the transverse circulation is weaker. The considered density distribution results in a bottom pressure profile obtaining its peak in the center. This kind of profile drives a bottom circulation directed onshore. The divergence of the flow is compensated by downwelling in the middle of the basin, while upwelling occurs near the coasts. The Coriolis force influences cross-shore currents, resulting in a surface cyclonic flow. However, its effect is negligible at the bottom due to the presence of bottom friction. As a result, cyclonic circulation prevails over the largest part of the water column, which is typical for most seas of the Northern Hemisphere (Fig. 2b). Positive density anomalies also cause lake/sea level to drop, resulting in a negative lake/sea level anomaly along the whole cross section, with the largest depression in the middle part of the basin (Fig. 2d). This kind of lake/sea level profile corresponds to a surface cross-shore current directed opposite to the bottom one.

Fig. 2.
Fig. 2.

Graphic representation of the solution for (a) streamfunction, (b) alongshore current, (c) density anomaly, and (d) lake/sea level and bottom pressure, with parameters Ah = 100 m2 s−1, Aυ = 0.01 m2 s−1, kυ = 0.01 m s−1 and the density distribution defined by q = 0. Black arrows in (a) denote flow direction.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0251.1

Light surface water (q = 1, i.e., density anomaly δ < 0, Fig. 3c) causes a different circulation to occur in the basin. Density is constant at the bottom of the basin, and it decreases toward the center of the basin surface where it is the lowest. The streamfunction (Fig. 3a) shows stronger currents in the near-surface area where density variations are the largest, accompanied by weaker currents in the deeper part of the basin where small density variations occur. Lower density in the central part of the basin results in bottom pressure having a minimum in that area. This kind of profile causes bottom circulation directed toward the center of the basin where the flow convergence causes upwelling. This is compensated by surface circulation directed toward the shore and by downwelling at the coasts. Cross-shore currents affected by Coriolis force result in a surface anticyclonic flow (Fig. 3b), while the bottom flow is constricted by the bottom friction. In this way the anticyclonic flow prevails over the largest part of the water column. Negative density anomalies cause lake/sea level to rise, reaching its maximum in the center of the basin (Fig. 3d), which supports the cross-shore surface circulation.

Fig. 3.
Fig. 3.

Graphic representation of the solution for (a) streamfunction, (b) alongshore current, (c) density anomaly, and (d) lake/sea level and bottom pressure, with parameters Ah = 100 m2 s−1, Aυ = 0.01 m2 s−1, kυ = 0.01 m s−1 and the density distribution defined by q = 1. Black arrows in (a) denote flow direction.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0251.1

A combination of the two density distribution profiles (bottom dense water and surface light water, q=2/2, Fig. 4c) changes the profile of the streamfunction Ψ, i.e., of the velocity components υ and w. The depth of the constant density level ρ = ρ0, which was either at the surface (q = 0) or at the bottom (q = 1) of the basin in the previous two cases, is now located in the middle of the basin, dividing it into two layers of opposite circulation types. The lower layer exhibits circulation similar to the one developed in the case of bottom dense water (q = 0), while the characteristics of the upper one essentially follow the pattern associated with the surface light water (q = 1). The density distribution dictates the flow directed toward the shore both at the surface and bottom, with a counter current directed toward the center of the basin in the strip around the constant density level. Vertical currents at the coasts cause downwelling (Fig. 4a) in the upper layer, accompanied by upwelling in the bottom layer. The opposite circulation occurs at the center of the basin, where upwelling occurs in the surface layer and downwelling in the lower one. The negative net density anomaly of the vertical column, supported by the upper layer upwelling in the center of the basin, causes lake/sea level to have a maximum in that area. Bottom pressure also achieves its maximum in the center of the basin, similar to the case of bottom dense water (q = 0), however, the maximum is not as high due to lower surface density (Fig. 4d). As for the alongshore currents, the cyclonic circulation prevails over the largest part of the water column (albeit weaker than in the case of q = 0). The influence of the surface lighter water is detected in the surface layer, where anticyclonic circulation occurs. On the other hand, bottom friction prevents a similar kind of circulation from developing at the bottom (Fig. 4b).

Fig. 4.
Fig. 4.

Graphic representation of the solution for (a) streamfunction, (b) alongshore current, (c) density anomaly, and (d) lake/sea level and bottom pressure, with parameters Ah = 100 m2 s−1, Aυ = 0.01 m2 s−1, kυ = 0.01 m s−1 and the density distribution defined by q=2/2. Black arrows in (a) denote flow direction.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0251.1

The previous three cases were modeled for a basin depth of 100 m. With an increase in basin depth the main circulation properties remain the same, with some exceptions near the bottom. Figure 5 depicts the solution for a depth of H = 1000 m and the case of bottom dense water, q = 0. The streamfunction is similar in shape, however, the values are much greater than for a shallow basin and the cross-shore current is stronger near the bottom of the deeper basin (Fig. 5a). Higher density in the central part of the basin (Fig. 5c) results once again in a bottom pressure profile obtaining its peak in the center (Fig. 5d). Its amplitude is an order of magnitude larger than in the case of a shallow basin, thus supporting the bottom circulation directed onshore. The Coriolis force influences cross-shore currents, resulting in a surface cyclonic flow and a bottom anticyclonic flow—the deeper basin allows the depth at which u = 0 to rise high enough (around 160 m from the bottom) for bottom circulation to form regardless of bottom friction (Fig. 5b). Positive density anomalies cause lake/sea level to drop on the whole cross section, with the largest depression in the middle part of the basin (an order of magnitude larger than for a shallow basin, Fig. 2d).

Fig. 5.
Fig. 5.

Graphic representation of the solution for (a) streamfunction, (b) alongshore current, (c) density anomaly, and (d) lake/sea level and bottom pressure, with parameters Ah = 100 m2 s−1, Aυ = 0.01 m2 s−1, kυ = 0.01 m s−1 and the density distribution defined by q = 0, but with a depth H = 1000 m. Black arrows in (a) denote flow direction.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0251.1

The intensification of the current magnitude with increase of basin depth can be understood by observing that our model incorporates the thermal wind effect. In particular, Eq. (2b), which relates vertical changes in velocity to the horizontal density gradient, is the thermal wind equation (up to the frictional effects). As the consequence, the velocity grows linearly with depth. Although it is known that the thermal wind might not be realistic, its concept is still useful because it explains the change of geostrophic flow with depth. Similarly, in this article, rather than reproducing exact circulation, we aim to analyze and explain the changes in the main features of the flow and their dependence on various parameters entering the model.

Similar changes in circulation also occur when analyzing the increase of the basin depth for other two density profiles (q = 1 and q=2/2), namely, preservation of main circulation characteristics accompanied by their intensification and the rise of u = 0 level.

b. Varying the friction coefficients

In the preceding section the friction coefficients were set to the fixed, basic set of values. However, these coefficients vary from basin to basin, change in time, and impose different levels of momentum exchange, which eventually influences circulation characteristics. For this reason, it is particularly important to check the sensitivity of the model results with respect to different values of the friction parameters.

The parameters are allowed to vary within the following intervals: kυ ∈ [10−4, 10−1] m s−1, Aυ ∈ [10−4, 10−1] m2 s−1, and Ah ∈ [1, 104] m2 s−1. These intervals extend somewhat beyond the ranges commonly observed in lakes and inland seas (Pedlosky 1987; Bowden 1983). Such choice is made in order to better analyze the dynamics and its sensitivity with respect to the friction parameters.

As we have seen, the presence of bottom friction significantly influences, or better to say, suppresses the development of currents in the bottom layer. If its value is significantly lowered (to kυ = 10−4 m s−1, which, according to Lazar et al. (2006), is still feasible for inland basins), then the model predicts the level u = 0 to move to a smaller depth and allows the circulation to develop below it. Figure 6 depicts the model results for a basin 100 m deep and the case of bottom dense water (q = 0). The streamfunction has a similar distribution as in Fig. 2, but the isolines are denser near the bottom of the basin where smaller bottom friction allows for stronger currents directed toward the coasts (Fig. 6a). The divergence of the flow is compensated by downwelling that occurs in the middle of the basin, accompanied by upwelling at the sides. This results in a lake/sea level profile similar to the case of larger friction, kυ = 0.01 m s−1, but with a higher amplitude. Coriolis force causes cyclonic circulation at the surface, and, due to the small value of the friction coefficient kυ, it also generates anticyclonic circulation near the bottom. However, it is important to notice that bottom pressure no longer attains its maximum in the center of the basin (Fig. 6d). This is a consequence of the lake/sea level profile—the profound difference in lake/sea level anomalies along the transverse section is large enough to overcome the effect of dense water and cause bottom pressure to have a minimum in the center of the basin. This would generally be associated with a bottom current directed toward the center of the basin, unlike the situation we have here. However, the circulation is not driven by pressure gradient only, but is a result of an interplay between the geostrophic balance and the momentum exchange among fluid layers, whose intensity is determined by friction coefficients. As already discussed, the attenuated friction allows for more pronounced bottom currents, which in turns allows for larger departure from geostrophic balance and finally results in the obtained circulation profile.

Fig. 6.
Fig. 6.

Graphic representation of the solution for (a) streamfunction, (b) alongshore current, (c) density anomaly, and (d) lake/sea level and bottom pressure, with parameters Ah = 100 m2 s−1, Aυ = 0.01 m2 s−1, kυ = 10−4 m s−1 and the density distribution defined by q = 0. Black arrows in (a) denote flow direction.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0251.1

As we have seen, the decrease of bottom friction resulted in the rise of u = 0 level and formation of an anticyclonic gyre below it. Similarly, we expect that variation of other friction coefficients might change some main circulation features. This motivates us to analyze the sensitivity of the no-motion level, at which a change from one type of circulation to another occurs, to these variations. In particular, we solve the equation u(y, z) = 0 in dependence of various friction coefficients. Since the equation u(y, z) = 0 cannot be solved explicitly, it was solved numerically, using MATLAB and implementing the Levenberg–Marquardt algorithm. A detailed description of the method is described in Press et al. (1992). The parameters are allowed to vary within the intervals given at the beginning of this subsection. The results presented here are obtained by keeping one friction coefficient fixed to the basic value and varying the other two within the mentioned intervals.

Figure 7 depicts results for the case of bottom dense water (q = 0). If we set Aυ = 0.01 m2 s−1 fixed (Fig. 7a), the depth of the no-motion level varies greatly with changes in horizontal viscosity Ah, and it is less influenced by changes in the bottom friction coefficient—once kυ is larger than 10−3 m s−1 bottom friction has almost no effect. However, if kυ is smaller than 10−3 m s−1, then the weaker bottom friction causes stronger bottom circulation and the level u = 0 rises to a higher depth.

Fig. 7.
Fig. 7.

Depth of the u = 0 level as a function of friction coefficients for (a) fixed vertical viscosity Aυ = 0.01 m2 s−1, (b) fixed bottom friction coefficient kυ = 0.01 m s−1, and (c) fixed horizontal viscosity Ah = 100 m2 s−1. The density distribution profile parameter is q = 0 (bottom dense water).

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0251.1

Setting the bottom friction coefficient kυ = 0.01 m s−1 and varying the viscosity parameters Ah and Aυ (Fig. 7b) produces similar results. The height of the u = 0 level is mostly influenced by the value of Ah, while a decrease in Aυ results in a rather small rise of the level. However, apart from the intensity of the influence, it is important to notice that change in Ah has an opposite effect than variation of other two friction parameters. In particular, rise of the no-motion level u = 0 is supported by stronger lateral friction and smaller values of kυ and Aυ. Indeed, when the latter ones are small, so is the Ekman layer thickness. Taking into account that the Ekman layer measures the reach of the bottom friction effect and that its thickness is proportional to vertical viscosity, small values of these two parameters mean that the influence of bottom friction is limited. This allows bottom circulation to develop, pushing the u = 0 level upward. On the other hand, an increase of lateral friction, combined with the no-slip boundary conditions at the coasts, suppresses the alongshore currents and leads to an enhanced cross-basin circulation at the surface. The latter pushes the water mass toward the center of the basin, thus reducing the lake/sea level slope. As a result, the pressure gradient near the bottom becomes more profound, leading to stronger transversal circulation in the bottom layer. This allows development, albeit of a weak intensity, of the alongshore flow in the bottom layer and rise of the no-motion u = 0 level.

For a fixed horizontal viscosity Ah = 0.01 m2 s−1 (Fig. 7c), variations of the coefficients Aυ and kυ result in smaller variations of the depth of u = 0 level than in the previous two cases—the values range from 80 to 95 m, as opposed to from 60 to 95 m before. Lowering both Aυ and kυ at the same time reduces the height of the Ekman layer. This allows stronger circulation to be formed in the bottom layer, as expected from the previous two figures.

In the case of light surface water (q = 1, Fig. 8), the distribution of u = 0 isolines is visually similar to the bottom dense water case, however, the range of results is larger, i.e., the u = 0 level can go up to the 40-m level. This is somehow already to be expected from the comparison of the Figs. 2 and 3. In the latter case, the circulation is dominantly developed in the upper part of the basin, where the density variations are largest. In particular, the onshore directed surface currents that support the anticyclonic flow are confined to a shallow, near-surface layer. This leaves enough space for the development of a bottom layer where, although of small intensity, the opposite type of the alongshore circulation occurs.

Fig. 8.
Fig. 8.

Depth of the u = 0 level as a function of friction coefficients for (a) fixed vertical viscosity Aυ = 0.01 m2 s−1, (b) fixed bottom friction coefficient kυ = 0.01 m s−1, and (c) fixed horizontal viscosity Ah = 100 m2 s−1. The density distribution profile parameter is q = 1 (surface light water).

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0251.1

The opposite phenomenon occurs when the two density distributions are combined (q=2/2). As already observed, two no-motion levels u = 0 might occur in this case. Figure 9 depicts variations of the deeper one, in consistency with the previous two figures. We note that the range of its depth is quite restricted and varies between 80 and 100 m. Unlike before, in this case, for a suitable combination of friction coefficients, it might occur that the bottom layer completely disappears. The reason for this is twofold. First, it is important to note that in this case the cross-sectional circulation is split into two horizontal layers, and therefore, depending on the friction coefficients, two levels of no-alongshore motion might appear (cf. Fig. 4). As a result, the bottom circulation is topped by two layers (rather than just one), which suppresses its evolution. Second, the dense bottom dome supporting the formation of the bottom layer with anticyclonic alongshore circulation is weaker than in the case q = 0. Consequently, the height of this layer is also lower, and for weak lateral friction and large values of Aυ and kυ the corresponding circulation does not develop at all.

Fig. 9.
Fig. 9.

Depth of the lower u = 0 level as a function of friction coefficients for (a) fixed vertical viscosity Aυ = 0.01 m2 s−1, (b) fixed bottom friction coefficient kυ = 0.01 m s−1, and (c) fixed horizontal viscosity Ah = 100 m2 s−1. The density distribution profile parameter is q=2/2 (a combination of bottom dense water and surface light water).

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0251.1

The variations of the upper u = 0 level are shown in Fig. 10. We observe that the bottom friction coefficient kυ has a rather limited impact, which is to be expected since we deal with the upper level that is far from the bottom. The influence of the lateral eddy viscosity is also significantly reduced when compared to the previous figures. However, the effect is the same as before. The increase of its value results in expansion of the surface layer, similar to the way it supports the rise of the lower no-motion level u = 0, while the central layer, where the alongshore circulation is cyclonic, shrinks. The depth of the upper u = 0 level is mostly influenced by the vertical friction coefficient Aυ. Its large values force the surface layer to expand, at the same time suppressing the lower two layers. Thereby the lowest one, which is directly influenced by the bottom friction, might even completely disappear, as we have observed in Fig. 9.

Fig. 10.
Fig. 10.

Depth of the upper u = 0 level as a function of friction coefficients for (a) fixed vertical viscosity Aυ = 0.01 m2 s−1, (b) fixed bottom friction coefficient kυ = 0.01 m s−1, and (c) fixed horizontal viscosity Ah = 100 m2 s−1. The density distribution profile parameter is q=2/2 (a combination of bottom dense water and surface light water).

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0251.1

4. Discussion

Currents and the type of circulation present in a basin are greatly affected by density distribution and frictional processes. This has been modeled previously by Orlić and Lazar (2009); however, their model did not encompass a vertically varying density anomaly and thus they were only able to reproduce a cyclonic-type flow in a basin, with an opposite flow forming in the bottom layer for suitable values of friction coefficients and specific basin dimensions. By allowing the density anomaly to vary along the vertical as well as in the cross-basin direction, the model presented here is capable of reproducing both cyclonic and anticyclonic surface circulation, while variation of parameters such as depth and friction coefficients allows the flow, usually of the opposite type, to develop in the bottom layer.

The three-dimensional prognostic model developed by Davies and Xing (2006) predicts that in the case of an isolated bottom dome (corresponding to the case q = 0), the surface circulation will be cyclonic, with anticyclonic flow forming in the bottom layer and downwelling/upwelling occurring in the center/at the edges of the basin. In the case of the surface frontal system (case q = 1), their calculations showed surface and near-bed circulations in opposite directions to those found for a bottom dome. They considered a basin with a horizontal bottom, of the same size as ours, and their results are in keeping with those presented here: similar circulation profiles are obtained, and the associated current speeds are of the same order of magnitude. In particular, they obtain the alongshore velocity of order 0.2 m s−1 for the bottom dome case and only slightly lower for the density profile characterized by the surface light water, which is similar to our findings. The same kind of agreements between two models are obtained in the case of a bottom dome suppressed by a pool of warmer surface water (case q=2/2).

Of course, some differences between numerical simulations of Davies and Xing (2006) and our results exist, but they are most likely due to the slightly different density profiles. In this work, the density profile is set, while in their case it is obtained from an initial temperature distribution by spinning down the model in the absence of the external forcing. The initial distribution was either prescribed (corresponding to an isolated bottom cold water dome or the surface pool of warm water above it) or generated by spinning up the model with external (thermal and wind) forcing for a 60-day period. However, in both cases, a near-steady state occurred after 10 days, thus justifying the steady-state assumption used in this paper. In addition, the density profiles obtained after the transient period are similar to those assumed in this manuscript. Our model, being an analytical one, is restricted to a rectangular basin, but the results presented in Davies and Xing (2006) show that changes in bottom topography have little effect on circulation driven by a prescribed density distribution. This leads us to conclude that the rectangular basin assumption is appropriate for representing main flow characteristics. It should be noted that their study is based on numerical modeling, while we restrict ourselves to analytical modeling, and yet similar results were obtained.

The modeled results are also in agreement with recent observations for the Ionian Sea. Moreover, they allow for a better understanding of phenomena underlying circulation dynamics, specifically the reversal of circulation type. The surface circulation in the Ionian shifts between anticyclonic and cyclonic, depending on the strength of the EMT (Borzelli et al. 2009). When the EMT is strong, there is an increase in communication between the Ionian and Aegean basins, Aegean deep water flows north along the eastern side of the Ionian basin and bottom circulation in the Ionian is cyclonic, accompanied by a stationary anticyclonic shear in the surface layer. When the EMT weakens, the circulation reverses completely, both at the surface and in the bottom layer. In that case the vertical density transect across the central Ionian Sea displays doming isopycnals, which corresponds to the case of bottom dense water (q = 0). For a deep enough basin (such as the Ionian Sea), our model allows an anticyclonic circulation to develop in the bottom layer (Fig. 5), in accordance with experimental findings. On the other hand, in the period of strong EMT, the observed density distribution shows a depression in the central part, which corresponds to the case of surface light water (q = 1), supporting anticyclonic surface circulation.

Borzelli et al. (2009) have attributed the change in circulation to the inversion of the bottom pressure gradient. However, as we have seen from the model results, the pressure gradient is not a decisive factor for determination of the type of flow to be developed. In particular, in certain circumstances the type of circulation remains resistant to the inversion of the bottom pressure gradient and is dominantly determined by the density distribution and frictional and basin parameters (Figs. 4 and 5). In addition, the bottom pressure profile cannot be estimated through the density distribution alone, as its impact might be annihilated by the anomalies in lake/sea level (Fig. 5). It rather seems that a particular type of circulation is determined by several factors: density profile, frictional forces, and basin characteristics, which produce a specific lake/sea level and bottom pressure profiles, that, in turn, establish a quasigeostrophic balance with circulation through a complex interplay.

Malanotte-Rizzoli et al. (2014) mentioned that further studies were needed to clarify the role of external (atmospheric) forcing in triggering the thermohaline circulation in the Eastern Mediterranean. The hypothesis that the change in surface circulation in the Ionian Sea can be explained solely by the change in density distribution was later confirmed by both laboratory and numerical experiments. Rubino et al. (2020) performed a laboratory experiment with a two-layer system whose dynamics is induced merely by an injection of dense water on a sloping bottom. Their findings are in fairly good agreement with observations, suggesting that changes in circulation can arise solely from extreme dense water formation events. However, they restrict the analysis and provide results only for the upper layer, leaving the processes in the bottom zone obscure. A similar laboratory experiment was performed by Gačić et al. (2021), where the authors analyze the entire vertical column. Their two-layer system examines the evolution of the potential vorticity and compared it with the one obtained for the Ionian Sea from measurements in the surface layer and model results in the bottom layer for a reversal event following an inflow of extremely dense cold water from the Adriatic Sea in 2012. Their findings show that evolution of the flow field in the Ionian Sea after the dense water outflow from the Adriatic is dynamically similar to the flow field in the rotating tank following a dense water injection. Considering that the injection was the only forcing in the model, i.e., there was no atmospheric forcing, they also conclude that inversions of the horizontal circulation at the surface are not wind induced but are due to inversions of the internal density gradients, which is consistent with the results presented in section 3. Meanwhile, the bottom currents measured in the central part of the basin remain rather suppressed and are not correlated with the circulation in the upper layer. The authors suggest that such behavior might be interpreted by the frictional influence, but the corresponding analysis (on dependence of circulation on frictional parameters) was not part of the study. They do not provide the spatial distribution of the density gradient after the transient phase and do not show the profile of the bottom pressure gradient, which restricts further comparison with the results of this paper.

Another interesting case is Lake Erie, as observations have shown a depressed thermocline in the summer with an anticyclonic circulation developing in the basin (Beletsky et al. 2012), which is the opposite of what occurs in most lakes (Shtokman 1967). This pattern was also obtained with a three-dimensional primitive equation numerical model intended to study Lake Erie’s summer circulation and thermal structure (Beletsky et al. 2013). Due to the depth of the thermocline and bottom friction, anticyclonic circulation (of order 0.1 m s−1) prevails in the entire lake. The depressed thermocline (deeper in the center of the basin, shallower at the edges) corresponds to the case of surface light water (q = 1), where our model predicts anticyclonic circulation in the surface layer. Once again, there is an agreement of both observations and numerical simulations with the results of the present analytical model.

There have also been several analytical studies focused on circulation in basins with cold water domes. Yuan and Li (1993) studied the circulation and formation of the cold water dome observed in the Yellow Sea, suppressed by a shallow front of surface light water. The dome develops in a stably stratified basin during the summer months. Their 2D analytical model produced a temperature distribution in an idealized, radially symmetric basin that was in good agreement with observations. The solution obtained with the model showed that the cold water dome remained mostly motionless, accompanied by upwelling in the upper, central part of the basin and downwelling at the edges, with cyclonic circulation (of the order of magnitude 0.1 m s−1) prevailing throughout the entire basin. This finding is similar to results of this study obtained for a combination of bottom dense water and surface light water (q=2/2). However, in our case the surface light water was deeper and therefore the cyclonic circulation in the uppermost layer of the basin changes to an anticyclonic one. Similarly, Hill (1996) used a one-dimensional, two-layer, quasigeostrophic model to show that dense water bottom domes in shallow seas are mostly static and accompanied by cyclonic surface circulation. His model showed that a stationary flow is reached rather quickly, which is, again, an indication that the steady state assumption is a sound one. He confirmed his analytical findings with observations from the Irish Sea. The study did not encompass changes in the vertical and thus there were no regions of upwelling or downwelling.

Another study of the Irish Sea (Xing and Davies 2001) was conducted using a high-resolution, three-dimensional baroclinic shelf sea model in order to reproduce the thermal stratification and associated density-driven circulation in the western Irish Sea during the summer months. The modeled formation of the dense water bottom dome and the accompanying cyclonic circulation are in good agreement with observations, but also with the results and assumptions of this study (the case of q = 0). In particular, the dynamics of the system, run without external forcing and driven by the density gradient, reaches a quasi-steady state after 3 days. The lake/sea level is depressed in the central part of the basin, while the near-bottom currents are of lower intensity and of the opposite type to the flow above.

The density distribution considered in the paper was modeled by a simple analytic function containing the first Fourier mode and a parameter q, where the latter allows for different density profiles: dense water at the bottom, surface light water, and a combination of both. More complicated and realistic density profiles can also be explored using Fourier expansion and the superposition principle. Thus, it should be straightforward to generalize the model developed under the present simplified forcing to a more general case. However, the purpose of this article was not to mimic a particular real situation as much as possible, but rather to provide an explanation and understanding of the phenomena responsible for the main circulation characteristics.

The density distribution is given, and the model is run under the steady state assumption. The consistency of the stationary flow with the transport of density through the obtained flow field can be explained by analyzing the density dispersion equation. The steady state requires balance of the advection and diffusion processes. Of course, proper modeling of these processes requires both the hydrodynamic and thermodynamic parts of the problem to 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 circulation is modeled under the assumption of no alongshore variability, which is a plausible simplification for an elongated basin or channel. Inclusion of the “transverse” boundary in the model would disturb that assumption. It would require development of a 3D model, which is beyond the scope of this article and the capacity of analytical modeling. However, we do not expect this would significantly change the flow pattern. It would close the circulation horizontally, producing either cyclonic or anticyclonic gyres, and would not affect its main features in the most part of the basin.

Similarly, although the model allows for wind forcing (cf. remark 1), this was not the subject of investigation in this study. This is partly because we are primarily interested in residual motion obtained after removing variations in high-frequency phenomena (tides, winds, etc.). The other reason is that we do not expect wind forcing to be essential for understanding the switch of circulation in the upper layer (Borzelli et al. 2009; Rubino et al. 2020), which is one of our main topics of interest. However, in the future it would be of interest to verify the solution obtained under a steady wind forcing and analyze its influence in combination with other parameters included in the model.

5. Conclusions

In this paper we have developed a simple diagnostic model that reproduces the thermohaline circulation in a landlocked basin driven by variations in density anomalies. By varying the density distribution as well as friction parameters and basin dimensions, the model predicts different types of circulation to occur, both at the surface and in the bottom layer. The obtained results suggest that the type of surface circulation is predominantly determined by the density profile and can be either cyclonic (in the case of bottom dense water, q = 0) or anticyclonic (in the case of surface light water, q = 1). Variation of the parameter q between these two values produces a combination of the two density distribution profiles, with the surface light water overlying the cold bottom dome. The resulting surface circulation in this case remains anticyclonic but is accompanied by an opposite flow in the middle layer.

The situation changes significantly when the circulation in the bottom layer is analyzed, as its development turns to be essentially related to frictional and basin parameters. In particular, a strong bottom friction and large values of vertical viscosity suppress the bottom flow, leading to a level of no longshore motion close to the bottom. In this case, for both extreme distribution profiles considered in the study (corresponding to q = 0 and q = 1), a surface type of circulation prevails along the entire vertical column, whose intensity decreases with depth. On the other hand, an increase in the basin depth and horizontal viscosity supports the rise of the no-motion level and development of a circulation below it, which might be of either type, depending on the particular density profile. Thus, a change in the density profile triggers a switch in the orientation of the surface currents, while a similar change in the bottom layer must be additionally supported by appropriate friction and basin parameters allowing bottom circulation to develop. It is interesting to note the role of the bottom pressure gradient, which turns out not to be a decisive factor for determining the type of flow that develops. In particular, the same type of bottom circulation can occur under opposite bottom pressure profiles and thus the prevalence of one type of circulation over the entire water column depends on the interplay between the horizontal density gradients, bottom friction, and vertical viscosity.

The model presented here successfully simulates different types of circulation occurring in landlocked water bodies. It reproduces the cyclonic circulation at the surface, which dominantly prevails in the Northern Hemisphere, but it is also capable of revealing anticyclonic surface circulation that occasionally occurs in some basins (e.g., Lake Erie, Ionian Sea). At the same time, it allows for different types of circulation in the bottom layer, which can follow the same direction as the surface flow, but might also take the reverse one (the Adriatic and Ionian Sea). Although the discussion and comparisons presented in this paper are limited to the Northern Hemisphere, this is not an intrinsic restriction of the study, and it can be generalized to the Southern Hemisphere by reversing the sign of the Coriolis force in the model.

The model allows for explicit expressions for basin currents, which enables the role and influence of various controlling parameters to be efficiently explored. This allows us to analyze conditions that support different types of circulations and to contribute to a better understanding of the underlying phenomena. At the same time, like all analytical models, it requires several assumptions that are not entirely realistic (e.g., steady state and flat bottom). However, other studies have shown that topography does not play an important role in modeling the response of currents to initially imposed densities, while a steady state is a common response, with a time delay, to a transient period caused by wind or some other external forcing (Hill 1996; Davies and Xing 2006). We also emphasize that the model, being analytical, does not necessarily lead to realistic values. Specially, it cannot be used for reproducing exact circulation in a particular basin.

Despite the imposed simplifications and limitations of the analytical modeling, the obtained main flow characteristics are in good agreement with numerical simulations as well as with empirical findings obtained in both idealized and real basins with different types of surface and bottom circulation. This indicates that the studied model, although based on an idealized and simplified situation, is a good tool for analyzing mean features of a circulation on a large time scale, as well as for detecting key factors responsible for its development and its particular form.

Acknowledgments.

The authors acknowledge M. Orlić for valuable discussions and comments. This work was supported by the Croatian Science Foundation through projects Control of Dynamical Systems (ConDyS, IP-2016-06-2468) and Middle Adriatic upwelling and downwelling (MAUD, IP-2018-01-9849). The paper was partially developed while the first author was a visiting researcher at the Chair of Dynamics, Control and Numerics (Alexander von Humboldt Professorship) at the Friedrich-Alexander-Universität Erlangen-Nürnberg with the support of the DAAD (Research Stays for University Academics and Scientists, 2021 programme).

Data availability statement.

No datasets were generated or analyzed during the current study.

REFERENCES

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Save
  • Beletsky, D., N. Hawley, Y. R. Rao, H. A. Vanderploeg, R. Beletsky, D. J. Schwab, and S. A. Ruberg, 2012: Summer thermal structure and anticyclonic circulation of Lake Erie. Geophys. Res. Lett., 39, L06605, https://doi.org/10.1029/2012GL051002.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beletsky, D., N. Hawley, and Y. R. Rao, 2013: Modeling summer circulation and thermal structure of Lake Erie. J. Geophys. Res. Oceans, 118, 62386252, https://doi.org/10.1002/2013JC008854.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Borzelli, G. L. E., M. Gačić, V. Cardin, and G. Civitarese, 2009: Eastern Mediterranean Transient and reversal of the Ionian Sea circulation. Geophys. Res. Lett., 36, L15108, https://doi.org/10.1029/2009GL039261.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bowden, B. F., 1983: Physical Oceanography of Coastal Waters. Ellis Horwood, 302 pp.

  • Cushman-Roisin, B., and J. M. Beckers, 2011: Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects. International Geophysics Series, Vol. 101, Academic Press, 875 pp.

    • Search Google Scholar
    • Export Citation
  • Davies, A. M., and J. Xing, 2006: Effect of topography and mixing parameterization upon the circulation in cold water domes. J. Geophys. Res., 111, C03018, https://doi.org/10.1029/2005JC003066.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Emery, K. O., and G. T. Csanady, 1973: Surface circulation of lakes and nearly land-locked sea. Proc. Natl. Acad. Sci. USA., 70, 9397, https://doi.org/10.1073/pnas.70.1.93.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gačić, M., M. Astraldi, and P. A. L. Violette, 1999: The Mediterranean Sea - Circulation, strait exchange and dense water formation processes - Preface. J. Mar. Syst., 20, 14.

    • Search Google Scholar
    • Export Citation
  • Gačić, M., and Coauthors, 2021: Impact of the dense water flow over the sloping bottom on the open-sea circulation: Laboratory experiments and the Ionian Sea (Mediterranean) example. Ocean Sci., 17, 975996, https://doi.org/10.5194/os-17-975-2021.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hill, A. E., 1996: Spin-down and the dynamics of dense pool gyres in shallow seas. J. Mar. Res., 54, 471486, https://doi.org/10.1357/0022240963213538.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Huang, J. C. K., 1971: The thermal current in Lake Michigan. J. Phys. Oceanogr., 1, 105122, https://doi.org/10.1175/1520-0485(1971)001<0105:TTCILM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lazar, M., M. Pavić, Z. Pasarić, and M. Orlić, 2006: Analytical modelling of wintertime coastal jets in the Adriatic Sea. Cont. Shelf Res., 27, 275285, https://doi.org/10.1016/j.csr.2006.10.007.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Majda, A., 2003: Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics, Vol. 9, American Mathematical Society and Courant Institute of Mathematical Sciences, 234 pp.

    • Search Google Scholar
    • Export Citation
  • Malanotte-Rizzoli, P., and Coauthors, 2014: Physical forcing and physical/biochemical variability of the Mediterranean Sea: a review of unresolved issues and directions for future research. Ocean Sci., 10, 281322, https://doi.org/10.5194/os-10-281-2014.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Oonishi, Y., 1975: Development of the current induced by the topographic heat accumulation (I): The case of the axisymmetric basin. J. Oceanogr. Soc. Japan, 31, 243254, https://doi.org/10.1007/BF02107439.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Orlić, M., 1996: An elementary model of density distribution, thermohaline circulation and quasigeostrophic flow in landlocked seas. Geofizika, 13, 6180.

    • Search Google Scholar
    • Export Citation
  • Orlić, M., and M. Lazar, 2009: Cyclonic versus anticyclonic circulation in lakes and inland seas. J. Phys. Oceanogr., 39, 22472263, https://doi.org/10.1175/2009JPO4068.1.

    • Crossref
    • Search Google Scholar
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  • Fig. 1.

    Schematic presentation of the considered basin. The domain of the model is restricted to the transverse section (shaded).

  • Fig. 2.

    Graphic representation of the solution for (a) streamfunction, (b) alongshore current, (c) density anomaly, and (d) lake/sea level and bottom pressure, with parameters Ah = 100 m2 s−1, Aυ = 0.01 m2 s−1, kυ = 0.01 m s−1 and the density distribution defined by q = 0. Black arrows in (a) denote flow direction.

  • Fig. 3.

    Graphic representation of the solution for (a) streamfunction, (b) alongshore current, (c) density anomaly, and (d) lake/sea level and bottom pressure, with parameters Ah = 100 m2 s−1, Aυ = 0.01 m2 s−1, kυ = 0.01 m s−1 and the density distribution defined by q = 1. Black arrows in (a) denote flow direction.

  • Fig. 4.

    Graphic representation of the solution for (a) streamfunction, (b) alongshore current, (c) density anomaly, and (d) lake/sea level and bottom pressure, with parameters Ah = 100 m2 s−1, Aυ = 0.01 m2 s−1, kυ = 0.01 m s−1 and the density distribution defined by q=2/2. Black arrows in (a) denote flow direction.

  • Fig. 5.

    Graphic representation of the solution for (a) streamfunction, (b) alongshore current, (c) density anomaly, and (d) lake/sea level and bottom pressure, with parameters Ah = 100 m2 s−1, Aυ = 0.01 m2 s−1, kυ = 0.01 m s−1 and the density distribution defined by q = 0, but with a depth H = 1000 m. Black arrows in (a) denote flow direction.

  • Fig. 6.

    Graphic representation of the solution for (a) streamfunction, (b) alongshore current, (c) density anomaly, and (d) lake/sea level and bottom pressure, with parameters Ah = 100 m2 s−1, Aυ = 0.01 m2 s−1, kυ = 10−4 m s−1 and the density distribution defined by q = 0. Black arrows in (a) denote flow direction.

  • Fig. 7.

    Depth of the u = 0 level as a function of friction coefficients for (a) fixed vertical viscosity Aυ = 0.01 m2 s−1, (b) fixed bottom friction coefficient kυ = 0.01 m s−1, and (c) fixed horizontal viscosity Ah = 100 m2 s−1. The density distribution profile parameter is q = 0 (bottom dense water).

  • Fig. 8.

    Depth of the u = 0 level as a function of friction coefficients for (a) fixed vertical viscosity Aυ = 0.01 m2 s−1, (b) fixed bottom friction coefficient kυ = 0.01 m s−1, and (c) fixed horizontal viscosity Ah = 100 m2 s−1. The density distribution profile parameter is q = 1 (surface light water).

  • Fig. 9.

    Depth of the lower u = 0 level as a function of friction coefficients for (a) fixed vertical viscosity Aυ = 0.01 m2 s−1, (b) fixed bottom friction coefficient kυ = 0.01 m s−1, and (c) fixed horizontal viscosity Ah = 100 m2 s−1. The density distribution profile parameter is q=2/2 (a combination of bottom dense water and surface light water).

  • Fig. 10.

    Depth of the upper u = 0 level as a function of friction coefficients for (a) fixed vertical viscosity Aυ = 0.01 m2 s−1, (b) fixed bottom friction coefficient kυ = 0.01 m s−1, and (c) fixed horizontal viscosity Ah = 100 m2 s−1. The density distribution profile parameter is q=2/2 (a combination of bottom dense water and surface light water).

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