We analyze a recently developed nonconforming surface finite element which is used to discretize 2D vector-valued compressible geophysical flow problems in a 3D domain. The study is performed on a vector-valued Laplace problem, which results from modeling surface flows. In our approach, the flow is approximated via edge integration on local flat triangles using the nonconforming linear Crouzeix-Raviart element. The approach is numerically efficient and straightforward to implement. For this discretization we derive optimal error bounds and present an estimate for the geometrical error. Numerical experiments validate the theoretical results. Furthermore, we demonstrate that the non-conforming finite element discretization provides a stable realization of large-scale geophysical flow problem. A comparison with observation shows that we can simulate