workbench_idemix2: IDEMIX2 - Internal Wave Energy Parameterisation for FESOM2

[patrickscholz]

IDEMIX2: Internal Wave Energy Parameterisation for FESOM2

IDEMIX2 (Internal wave Dissipation, Energy and MIXing, version 2) is a prognostic parameterisation of internal gravity wave energy and its contribution to diapycnal mixing, based on Olbers & Eden (2013) and Eden et al. (2014). It is implemented here as an extension to the existing IDEMIX1/TKE mixing scheme in FESOM2.

Concept

Unlike IDEMIX1, which tracks a single vertically integrated internal wave energy field, IDEMIX2 resolves the horizontal propagation direction of internal waves through a spectral angular dimension (nfbin directional bins, default 50 interior + 2 ghost bins covering 0–2π). This captures anisotropic wave fields near generation sites such as rough topography and coastlines, and allows wave energy to be tracked as it propagates across ocean basins.

Wave energy compartments

Two spectral wave energy fields are evolved independently:

  Surface wind forcing                 M2 tidal generation
  (NIW, near-inertial)                 (rough topography)
          │                                    │
          ▼                                    ▼
  ┌───────────────┐                  ┌─────────────────┐
  │  E_niw(φ, x)  │                  │   E_M2(φ, x)    │
  │  Near-inertial│                  │   M2 tidal IW   │
  │  waves        │                  │   (~12.4 h)     │
  └───────┬───────┘                  └────────┬────────┘
          │  ω ≈ f (lat-dependent)            │  ω = ω_M2 (fixed)
          └──────────────┬────────────────────┘
                         │  wave-wave interaction
                         ▼
               ┌──────────────────┐
               │   E_iw(z, x)     │   background IW continuum
               │   (vertically    │   (GM spectrum)
               │    resolved)     │
               └────────┬─────────┘
                        │  dissipation → diapycnal mixing
                        ▼
               ┌──────────────────┐
               │   TKE scheme     │   iw_diss added as
               │   Kv, Av         │   energy source term
               └──────────────────┘

Timestep structure

Each call to calc_cvmix_idemix2 follows this sequence:

  ┌─────────────────────────────────────────────────────────────────┐
  │ 1. Node parameters                                              │
  │    cn (baroclinic wave speed), α_c, v0, c0, τ_M2, τ_niw,        │
  │    vertical structure functions  [loop: node, OMP parallel]     │
  ├─────────────────────────────────────────────────────────────────┤
  │ 2. Element group velocities                                     │
  │    u_M2(φ), v_M2(φ), w_M2(φ)  — spectral propagation speeds     │
  │    scatter elem → node (area-weighted)                          │
  ├─────────────────────────────────────────────────────────────────┤
  │ 3. Horizontal spectral integration  [for M2, then NIW]          │
  │                                                                 │
  │   E^n  ──▶  superbee horizontal advection  ──▶  flx_uv(φ,edge)  │
  │              │                                                  │
  │              ├──▶  reflective coastal BC                        │
  │              │     (redirect toward-coast flux into             │
  │              │      mirror spectral bin at interior node)       │
  │              ▼                                                  │
  │         superbee cross-spectral advection ──▶  flx_cs(φ,node)   │
  │              │                                                  │
  │              ▼                                                  │
  │         flux → divergence  +  reflect_src injection             │
  │              │                                                  │
  │              ▼                                                  │
  │         E^(n+1) = (E^n + dt·(Edivh + Edivs + forc))             │
  │                   / (1 + dt·τ)     [AB2 or Euler]               │
  ├─────────────────────────────────────────────────────────────────┤
  │ 4. Eiw vertical diffusion + implicit dissipation                │
  │    tridiagonal solve per node  [OMP parallel]                   │
  ├─────────────────────────────────────────────────────────────────┤
  │ 5. Eiw horizontal diffusion  (optional)                         │
  ├─────────────────────────────────────────────────────────────────┤
  │ 6. Wave-wave interaction  E_M2/niw ↔ E_iw                       │
  │    [loop: node, OMP parallel]                                   │
  ├─────────────────────────────────────────────────────────────────┤
  │ 7. iw_diss → TKE (Kv, Av)                                       │
  └─────────────────────────────────────────────────────────────────┘

Reflective coastal boundary condition

Wave energy reaching a coastline is redirected into the spectrally mirrored propagation direction at the last interior node, rather than leaking into land:

  LAND │ n_coast ──── edge ──── n_int (interior node)
       │
       │  toward-coast flux in bin φ_i  →  zeroed
       │                                       │
       │                                       ▼
       │                            energy re-injected at n_int
       │                            in mirror bin φ_k = edge normal
       │                            (net energy change = 0,
       │                             spectral redistribution only)

The mirror bin φ_k for each coast edge is pre-computed once at init from the edge geometry and stored in iwe2_refl_bin(edge).

Forcing inputs

Field	Source	Role
M2 tidal forcing	Observed anisotropic tidal generation (modes 1+2)	E_M2 surface source
NIW / surface forcing	Wind-derived near-inertial flux	E_niw surface source or E_iw surface BC
Bottom tidal forcing	Near-field tidal dissipation dataset	E_iw bottom source
Topographic HRMS / λ	Seafloor roughness fields	Scattering timescale τ