This paper presents a groundbreaking non-stationary model, intricately crafted using the fictitious domain technique, to delve into the complex dynamics of baroclinic ocean motion. This study marks a significant leap in our understanding of water mass interaction, shedding light on the profound impact of temperature and salt gradients on sea currents. The methodology uses modified Navier-Stokes equations for viscous, incompressible flow, considering advection, diffusion, and Coriolis force. The results of this study underscore the immediate and tangible implications of our research. The solutions unveiled the pivotal role of pressure and temperature differentiation in the genesis of ocean currents. The analysis demonstrated that by integrating nonlinear terms and detailed modeling of initial and boundary conditions, we can markedly improve the precision of water mass movement forecasts. This work underscores the urgent necessity for further research into dynamic ocean modeling to enhance our ability to predict climate change. This article introduces truly innovative approaches to numerical modeling, which hold immense potential for the future of the field. These approaches have the power to transform existing models of sea currents and pave the way for the development of more efficient methods for monitoring and predicting the state of the marine environment.