The physical foundation

NeuralPOM advances the three-dimensional hydrostatic primitive equations with the Boussinesq approximation on an Arakawa C grid and a terrain-following vertical coordinate. The neural corrector acts at endpoints of this physical evolution. The formulation follows the Princeton Ocean Model; the implementation and numerical choices are described in the manuscript, Section 2.1 and Blumberg & Mellor (1987).

Momentum and hydrostatic pressure

Let $x,y,z$ be eastward, northward, and upward coordinates, $t$ time, and ${\mathit{u}}_h=(u,v)$ horizontal velocity. Let $w$ be vertical velocity, $f$ the Coriolis parameter, $\hat{\mathit{k}}$ the upward unit vector, $p$ pressure, $\rho$ density, ${\rho }_0$ constant reference density, and $g$ gravitational acceleration. With horizontal gradient ${\mathrm{\nabla }}_h$, vertical viscosity $K_M$, and horizontal momentum mixing ${\mathit{F}}_h$, the physical-coordinate equations are

$$({\partial }_t+{\mathit{u}}_h\cdot {\mathrm{\nabla }}_h+w{\partial }_z){\mathit{u}}_h+f\hat{\mathit{k}}\times {\mathit{u}}_h=-\frac{1}{{\rho }_0}{\mathrm{\nabla }}_{h}p+{\partial }_z(K_M{\partial }_z{\mathit{u}}_h)+{\mathit{F}}_h,$$$${\partial }_{z}p=-\rho g,{\mathrm{\nabla }}_h\cdot {\mathit{u}}_h+{\partial }_{z}w=0.$$

Let $\eta$ denote free-surface elevation and ${\rho }^{\prime }=\rho -{\rho }_0$ the density anomaly. With spatially uniform atmospheric pressure $p_{\mathrm{a}\mathrm{t}\mathrm{m}}$, hydrostatic pressure is

$$p(z)=p_{\mathrm{a}\mathrm{t}\mathrm{m}}+{\rho }_{0}g(\eta -z)+g{\int }_z^{\eta }{\rho }^{\prime }(z^{\prime })\mathrm{d}{z}^{\prime }.$$

Its horizontal gradient contains the surface-slope contribution and a depth-dependent contribution from density. This connects tracer corrections to subsequent velocity evolution. The transformed pressure-gradient operators are implemented in momentum.py.

Tracers and density

For tracer $C$, representing temperature $T$ or salinity $S$, let $K_H$ be vertical diffusivity and $F_C$ horizontal tracer mixing. The interior transport equation is

$$({\partial }_t+{\mathit{u}}_h\cdot {\mathrm{\nabla }}_h+w{\partial }_z)C={\partial }_z(K_H{\partial }_{z}C)+F_C.$$

Surface and bottom fluxes enter through the selected case's boundary conditions. The model evaluates density from temperature, salinity, and pressure through the POM equation of state. The EOS reference is Mellor (1991); vertical mixing follows the separate Mellor–Yamada closure.

Follow the seabed with sigma coordinates

Let $H(x,y)>0$ be resting depth and $D=H+\eta$ the total water-column depth. Define

$$\sigma =\frac{z-\eta }{H+\eta },-1\le \sigma \le 0.$$

The seabed is $\sigma =-1$ and the free surface is $\sigma =0$. The numerical operators account for this coordinate transformation in transport and pressure gradients. The vertical arrays z and zz store sigma interfaces and midpoints; dz and dzz store their separations. The published coarse configuration has 15 active layers within 16 vertical storage levels. (Grid implementation, model contract)

Connect equations to the numerical model

Physical operation	Implementation family	Explanation
Momentum and density	momentum.py	Pressure gradients, vertical momentum mixing, EOS
Tracer transport	tracers.py	Advection and vertical diffusion
Horizontal momentum and turbulence	baroclinic.py	Internal-mode operators and mixing closure
Free-surface evolution	time_loop.py, implicit_barotropic.py	Explicit or semi-implicit external mode
Geometry	grid.py	Staggered grid and masks

The two core directories retain their own numerical implementation while sharing the hybrid state and correction interfaces. Continue with mode splitting to see why the fast surface mode receives special treatment.