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(2023) Lang, YandongThesisThe magnitude of the diffusivity that characterizes lateral mixing in the ocean is about 106 108 times larger than that of vertical mixing. The lateral direction is along the direction of the neutral tangent plane (same as the direction of the locally referenced potential density surface). However, due to the helical nature of the neutral trajectories (the normal vector of the neutral tangent plane is not curlfree), welldefined neutral surfaces do not exist. Welldefined but only approximately neutral surfaces have traditionally been chosen based on either (i) constructing a threedimensional density variable whose isosurface (the surface with a constant density value of the density variable) describes the lateral direction, or (ii) creating a twodimensional approximately neutral surfaces (ANS), which are normally more neutral than the isosurfaces of the threedimensional density variable A threedimensional neutral density variable is here derived called rSCV, which is an improvement on the neutral density rn of Jackett and McDougall (1997). Compared with rn, rSCV is independent of pressure and thus is insensitive to the ubiquitous vertical heaving motions of waves and eddies, and has similar neutrality as rn. The material derivatives (the rate of change of the density variables) of rSCV and rn have also been derived using numerical methods. The material derivative of rSCV is shown to be close to that of rn. Oceanographers have traditionally estimated the quality of an ANS by focusing on the fictitious vertical diffusion caused by lateral diffusion being applied in the wrong direction. This thesis shows that the spurious advection through an ANS is another important consideration that limits the accuracy and usefulness of an ANS. Because of this concern, a twodimensional approximately neutral surface is constructed called the Wu.ssurface, which minimizes the spurious diasurface advection through the surface. The spurious diasurface advection through the Wu.ssurface is more than a hundred times smaller than that on the most neutral ANS to date, however, the fictitious diapycnal diffusion on it is larger. Therefore, the Wu.s+s2surface is created to control both the spurious diasurface advection and the fictitious diapycnal diffusion on the surface. It is shown that minimizing the fictitious diffusion and the spurious diasurface advection is important for using such surfaces in inverse studies. Hence the Wu.s+s2surface is the best choice of surface for such studies.