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1.1.1 Free Surface and Rigid Lid Models

In reality, the ocean surface is free to deform under the influence of wind, heating and tidal forces. We find wind-driven waves and surges up to several meters high on its surface. These are typically short-lived, have short spatial scales and fast wave speeds. The surface elevations due to the large, quasi-permanent ocean currents, on the other hand, such as those associated with the Gulf Stream and its eddies, and mid-ocean gyres, are only fractions of a meter or so, and move very slowly. In order to avoid the severe limitation on the time step due to the fast gravity waves (to be discussed below), one puts a rigid lid on the ocean as this affects the large-scale motions only slightly. The first such model was formulated by Bryan [4]. This model has been recently reformulated by Killworth et al. [36] to retain the free surface by treating the fast modes separately. Models that treat the fast waves implicitly have been developed by Hurlburt et al. [31] in connection with ``layered'' models, by Madala and Piacsek [39] in connection with ``level'' models, and by Blumberg and Mellor [3] in connection with ``sigma-coordinate'' models [for definitions of ``layered'', ``level'' and ``sigma'' see the next section].