SHOC-HOMME 3D Turbulence Coupling

This note documents the EAMxx 3D-turbulence path that couples SHOC and HOMME. It has two main pieces:

1. SHOC diagnoses horizontal eddy diffusivities and passes them to HOMME so the dycore can apply horizontal turbulent diffusion.
2. HOMME provides horizontal velocity-gradient information back to SHOC so SHOC can use a three-dimensional shear-production term in the TKE equation.

Horizontal Eddy Diffusivity Passed to HOMME

When ctl_nl::do_3d_turbulence=true, SHOC computes horizontal eddy diffusivities on the physics grid after the main SHOC update. The formulation is

\[ K_{H,h} = C_{kh,h}\,L_h\,\sqrt{\mathrm{TKE}}, \qquad K_{M,h} = C_{km,h}\,L_h\,\sqrt{\mathrm{TKE}}, \]

where the horizontal length scale is

\[ L_h = \sqrt{\Delta x\,\Delta y}. \]

In the current implementation, SHOC obtains \Delta x and \Delta y from the grid geometry and uses the runtime parameters shoc::coeff_kh_horiz and shoc::coeff_km_horiz, both of which default to 0.1.

After SHOC computes eddy_diff_heat_horiz and eddy_diff_mom_horiz, EAMxx forwards those fields to HOMME:

1. On the standard physics-grid path, the physics-to-dynamics remapper registers the SHOC fields with HOMME helper fields Kh_dyn and Km_dyn.
2. On pgN finite-volume physics grids, the same fields are supplied to GllFvRemap::run_fv_phys_to_dyn, which remaps them into HOMME's dynamics representation.
3. HOMME then binds the remapped fields into its derived-state turbulence slots m_turb_diff_heat and m_turb_diff_mom, which are consumed by the dycore's horizontal turbulent-diffusion machinery.

This split keeps the turbulence closure in SHOC while leaving the actual horizontal diffusion operator in HOMME, where the horizontal discretization and metric terms already live.

Three-Dimensional Shear Production of TKE

In the legacy one-dimensional SHOC path, shear production uses only vertical shear of the horizontal wind. In the 3D-turbulence path, SHOC instead forms the full local strain invariant and uses it in the TKE tendency.

The shear-production contribution is

\[ \begin{align} -P_s &= 2K_M \Bigg[ \left(\frac{\partial u}{\partial x}\right)^{2} + \left(\frac{\partial v}{\partial y}\right)^{2} + \left(\frac{\partial w}{\partial z}\right)^{2} \notag\\ &\quad + \tfrac{1}{2}\Big( \left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)^{2} + \left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\right)^{2} + \left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)^{2} \Big) \Bigg]. \end{align} \]

In code, SHOC stores the bracketed strain invariant as tke_shear_strain3d = 2 S_{ij}S_{ij} and then advances TKE with

\[ \text{shear production} = C_k K_M \left(2 S_{ij}S_{ij}\right), \]

with C_k = 0.1, consistent with the existing SHOC TKE closure.

The calculation is split across HOMME and SHOC:

1. After each HOMME dynamics step, HOMME computes horizontal derivatives of the local Cartesian velocity components on the dynamics grid.
2. Those derivatives are projected back to the local basis to form the six horizontal pieces of the velocity-gradient tensor: du/dx, du/dy, dv/dx, dv/dy, dw/dx, and dw/dy.
3. EAMxx remaps those six components onto the physics grid as tke_shear_strain3d_components.
4. Inside SHOC, vertical derivatives du/dz, dv/dz, and dw/dz are computed on interface levels and then interpolated to midpoint levels, matching SHOC's native staggered-grid treatment.
5. SHOC assembles the full symmetric strain tensor and forms the invariant 2 S_{ij}S_{ij}, which is then used in the TKE equation.

The same assembled 3D strain is also converted back to a midpoint shear magnitude for SHOC's cold-surface fallback in the eddy-diffusivity calculation, so the 3D path remains consistent with the rest of the closure.