Peter Korn
The pressure of a non-hydrostatic ocean model is globally coupled as the solution of a global elliptic problem, computed at every time step on the geometry of the world ocean, the computational costs of this computation renders global non- hydrostatic ocan modelling impossible and confines it to regional domains. We present AC/DC (Artificial Compressibility with Direct Column solve), which computes the pressure locally. Three elements constitute the method. The vertical part of the non-hydrostatic pressure follows from a tridiagonal solve within each column. The horizontal residual is relaxed by artificial compressibility, with an error of second order in the acoustic Froude number. The relaxation carries a pseudo-density that satisfies an exact flux-form continuity equation, and momentum, tracers and energy are weighted by it throughout. This weighting constitutive: it cancels the cubic,s ign-indefinite term that obstructs energy conservation under artificial compressibility, and it is the reason the density-weighted energy is conserved identically, tracer content is conserved with a uniformly bounded representation error, and the budget of tracer variance holds as a discrete identity. AC/DC performs no global communication of its own, and the cost of non-hydrostatic dynamics falls from hundreds of mesh sweeps per time step to two, a fixed factor of about $1.2$ in operation count and $1.3$ in runtime at any resolution. At this price the non-hydrostatic equations can be integrated on the global ocean. One equation set then spans all scales, reducing to the primitive equations wherever the flow is hydrostatic, with no interface and no switching. This increase also flow information: exact budgets separate physical dissipation and mixing from numerical contributions, and turn the mixing efficiency into a computed quantity of the simulation.