L

Fig. 5.17 Sketch of overflow through an idealized rectangular strait with width L in a rotating frame: a fast rotation, b slow rotation.

Fig. 5.18 Sketch of density structure related to overflow, a before cooling and b after cooling, with the heavy curve on the lower-right side indicating the sloping topography; c (complete mixing) and d (total penetration) depict two extreme scenarios of mixing between the newly formed deep water and the environment.

Fig. 5.18 Sketch of density structure related to overflow, a before cooling and b after cooling, with the heavy curve on the lower-right side indicating the sloping topography; c (complete mixing) and d (total penetration) depict two extreme scenarios of mixing between the newly formed deep water and the environment.

Dense water formation and overflow as a sink of GPE

At first glance, dense water formation and the overflow and gravity current flowing down-slope may appear to provide the energy driving the circulation. However, this is not the case. As shown in Figure 5.18, dense water formation, the associated mixing, and movement are associated with the loss of gravitational potential energy (GPE) for the mean state. It is true that dense water formation and movement afterward can partly transform GPE lost from the mean state into kinetic energy and thus drive the circulation. On the other hand, since mechanical energy of the mean state is reduced owing to dense water formation and its subsequent overflow, another source of mechanical energy is required in order to maintain the circulation in a steady state.

Assume that the depth of the ocean is D, which can be used as the reference level for the definition of gravitational potential energy; however, it is more convenient to use the base of the second water parcel, i.e., depth H + h, as the reference level (Fig. 5.18a). Note that, for a mass-conserving model, the choice of reference level does not affect the loss/gain of gravitational potential energy. However, for models based on the traditional Boussinesq approximations, the choice of reference level can affect the loss/gain of gravitational potential energy because of the artificial source/sink of mass existing in such models. The water parcel near the sea surface is labeled with a subscript 1, and the water parcel below it is labeled with a subscript 2. The total amount of GPE and its changes before and after the deepwater formation and descent can be calculated as follows.

1. In the initial state before cooling, the total GPE of these two water parcels are

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