For cases where the ocean model is driven by a realistic atmospheric model, the atmospheric pressure pa may be determined. But since the lid pressure pl is actually an artifact of making the rigid lid approximation, it is generally not available. To address this fact, Bryan formed the barotropic vorticity equation in which case the lid and atmospheric pressures are eliminated. Through inverting the vorticity equation, the streamfunction is obtained. As a result, the barotropic velocities can then be determined through the relation (6.6). As will be seen in Section 6.6, the lid and atmospheric pressures will likewise not be needed for determining the baroclinic velocity. This algorithm has been used for solving the rigid lid in MOM up until the formulation of the surface pressure approach of Smith, Dukowicz, and Malone (1992) and Dukowicz, Smith, and Malone (1993) (Section 6.8). In the present section, a derivation of the barotropic vorticity equation is provided.