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in the haline mode (Fig. 5.133a). However, the effect of wind-driven circulation on the meridional overturning rate is reversed in the energy-constraint model (Fig. 5.133b).

Note that when gyration is smaller than a critical value, the energy-constraint model has three steady solutions: a stable thermal mode, a stable haline mode, and an unstable haline mode; beyond this critical value the model has only one stable solution - a thermal mode. For the current parameters: p = 1 m/yr and e = 2.5 x 10-7 kg/m2/s, the critical value of the volumetric transport of the gyre is 3.1 Sv (Fig. 5.133b). Therefore, if the model starts from an initial state in the thermal mode and with a small wind-driven circulation, the solution will remain in the stable thermal mode. However, the story can be quite different if the model starts from an initial state of a stable haline mode with a weak wind-driven circulation. When wind-driven circulation is increased beyond the critical value, the haline mode is no longer a viable state and the system will be switched to the thermal mode through a catastrophic change.

Limitations of box models In spite of the fact that models have been widely used in various applications, there are some inherent limitations of box models. Most importantly, the number of boxes in the model is often rather limited, so they cannot really simulate the details of the general circulation. Although box models with a large number of boxes have been constructed and used to study the thermohaline circulation, such complicated box models are not very popular because it is much easier to construct a model based on partial differential equations and finite difference. In addition, owing to the nature of the highly truncated boxes, large numerical diffusion due to the upwind scheme used in box models arises. Excessive numerical diffusion in box models prevents their application to cases where more accurate results are desirable.

Another major limitation of most box models is due to the exclusion of the Coriolis force. Lacking the effect of rotation makes the box model less useful as a tool for understanding the circulation. However, this is not an intrinsic limit for box models. Maas (1994) discussed a three-dimensional box model in which the dynamical effect of rotation is included. In fact, the effect of rotation, even the so-called j-effect, can be included in a box model. For example, a 3 x 3 barotropic box model based on the C-grid can include the Coriolis force and can be used to demonstrate the j-effect; i.e., western intensification can be demonstrated in the model. Two rows to the east in the model play the role of the ocean interior, while the western row plays the role of the western boundary layer. However, the model is so highly truncated that the parameters used in this kind of model are far from realistic.

The water-wheel experiments were first carried out by Malkus and Howard at the Massachusetts Institute of Technology in the 1970s (Malkus, 1972). The experimental set-up is relatively simple; the simplest version is just a slightly tilted toy water-wheel with leaky paper cups suspended from its rim. Water is supplied from the tap on the top of the rim (Fig. 5.134).

The experiments show that when the water flow rate is slow, the top cups are not filled up enough to overcome the friction, since the cups are leaky. The wheel remains motionless, owing to friction. When the flow rate is large, the wheel starts to rotate, resulting in steady rotation in either direction. When the flow rate is even greater and larger than a certain

Fig. 5.134 Sketch of the water-wheel designed by Willem Malkus and Lou Howard at MIT in the 1970s.

5.4.3 Thermohaline circulation based on loop models

The water-wheel model

Fig. 5.134 Sketch of the water-wheel designed by Willem Malkus and Lou Howard at MIT in the 1970s.

Tap water threshold, the motion becomes chaotic, i.e., the wheel rotates clockwise and anticlockwise, with angular velocity changing with time in a chaotic way. The above experiments can be described in terms of the water-wheel equation, which will be detailed in connection with a so-called loop model in this section.

A loop model forced by evaporation and precipitation Loop models have been studied and used extensively in many fields, including many engineering applications; thus, a large number of papers have been published along these lines. The thermohaline circulation in the oceans can have regular or irregular oscillations over a wide spectrum in both temporal and spatial space, and loop models can be used as a one-dimensional idealization. These models can be formulated for thermal circulation, haline circulation, or thermohaline circulation.

In this section, we discuss a simple loop model of the haline circulation. The system can be formulated under two slightly different boundary conditions: the natural boundary condition and the virtual salt flux condition. A model forced by the natural boundary condition is shown in Figure 5.135. The model based on the virtual salt flux condition can be reduced to an ordinary differential equation system, the water-wheel equation (Dewar and Huang, 1995; Huang and Dewar, 1996).

Precipitation

Fig. 5.135 A loop oscillator for the haline circulation - a tube filled with salty water. The fresh water is passed through the skin of the tube.

Precipitation

Fig. 5.135 A loop oscillator for the haline circulation - a tube filled with salty water. The fresh water is passed through the skin of the tube.

The model consists of a water tube loop filled with salty water (Fig. 5.135). The radius of the loop is R and the radius of the tube is r, where R > r. In the following analysis, we assume that the flow and water properties are uniform in each cross-section. At the surface of the tube, freshwater is exchanged with the environment as precipitation (p) and evaporation (e):

where E is the amplitude of the freshwater flux. We now set out the equations describing this loop system.

Continuity

Under the Boussinesq approximations, continuity requires that the velocity convergence equals the freshwater flux.

In the following analysis, we use angular velocity, defined as w = u/R (5.238)

Thus, integrating Eqn. (5.237) leads to w = & + 2E sin 6/r (5.239)

where & is the mean angular velocity of the water circulating the loop, and the second term on the right-hand side is due to the accumulated water caused by the freshwater flux through the surface of the tube.

Salt conservation Similarly, the salt conservation equation is

where K is the coefficient of salt diffusivity. The contribution to salinity balance from evaporation and precipitation is included in the continuity equation, namely through the angular velocity relation, Eqn. (5.239); thus, such a model is under the natural boundary condition (NBC).

For the model subject to virtual salt flux (VSF) and relaxation boundary conditions, the corresponding salt conservation equations are

r R2

2r , * n k St + (&S)6 = — (S* - S) + R2 S66 (5.242)

where T is a relaxation coefficient, S is the average salinity, and S* is a specified reference salinity (traditionally chosen to be the observed mean salinity). Note that the net salt in the problem is guaranteed to be a constant if Eqns. (5.240) or (5.241) are used. No such guarantee exists if Eqn. (5.242) is used.

Momentum equation The momentum balance for an infinitely small sector of water tube is p0 (ut + uui) = —Pl — pg sin 0 — ep0u (5.243)

where l is the arc element along the axis of the tube, Pl is the pressure gradient, and e is the frictional coefficient. Integrating Eqn. (5.243) around the entire loop, all the gradient terms vanish and we obtain ttt = — ett — — f — sin 0 d0 (5.244)

2nR J0 P0

In this study we apply a linear equation of state p = —0 (1 + jS) (5.245)

Thus, Eqn. (5.244) can be written as gj ttt = —ett -jj-iS sin 0} (5.246)

where i} = /02n -d0 is an averaging operator.

Nondimensionalization Equations (5.240) and (5.246) are nondimensionalized using

where

T = V R/g j S (5.248) After dropping the primes, we obtain ttt = —att — iS sin 0} (5.249a)

where a-eT-£ ( R ^1/2 ^ 2ET _ 2E / R \1/2 _ KT K / 1 \1/2

are the nondimensional parameters of the model. The interpretation of T in Eqn. (5.248) is as a loop "flushing" time, as determined by buoyancy anomalies; thus, a represents the ratio of flushing time to viscous decay time, k the ratio of flushing time to a tube "filling" time (due to freshwater flux), and k the ratio of flushing time to salt diffusion time. In view of these interpretations, we are interested in the limit of small values of a, k, and k. Note that Eqn. (5.249b) implies a normalization condition for S, i.e., /02n Sd0 = 2n.

We now consider the solutions of the coupled nonlinear, integro-differential equation set (5.249). For the given initial conditions and parameter values, this set of equations can be solved by a Fourier spectral approach and a fourth-order Runge-Kutta algorithm in time stepping. An advantage of the spectral technique is that the torque in the momentum equation appears naturally as the coefficient of the lowest Fourier mode. Solutions from this simple model can help us understand the oscillations of more complicated climate models; hence, we will focus on steady solutions and limit cycle oscillations. The nature of the stationary solutions depends critically on the relative significance of the parameters a, k, and k .

When k = 0, the steady solution of Eqn. (5.249) is straightforward; i.e., for arbitrary a and k:

where

is the mean angular velocity.

A much more interesting case is that of non-zero k , because diffusion exists naturally and is also necessary for numerical stability. From the preceding discussion, the regime of most physical interest involves small values of a and k, and we shall exploit this by assuming that a « k ^ 1. When k is non-zero, the solutions for A and S can simply be obtained by employing a perturbation approach which makes use of these parametric restrictions. Indeed, a convenient classification of the solutions to Eqn. (5.249) can be found if we consider k larger than, comparable to, and smaller than a and k, respectively.

The solution of this loop model can be obtained by expanding the salinity in a Fourier series to

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