Wave-Mean Interaction

This page describes the two-way coupling between near-inertial waves and the balanced mean flow.

Physical Motivation

Why Waves Affect the Mean Flow

Near-inertial waves carry momentum and energy. When they:

Break or dissipate
Refract through vorticity gradients
Interact nonlinearly with the flow

...they can transfer energy and momentum to the balanced circulation.

Observational Evidence

Anticyclones contain enhanced NIW energy (chimney effect)
Wave dissipation correlates with mixing in anticyclones
Waves can energize the mesoscale eddy field

The Wave Feedback Term

Definition

Following Xie & Vanneste (2015), the wave-induced potential vorticity is:

\[q^w = \frac{i}{2f_0} J(B^*, B) + \frac{1}{4f_0} \nabla_h^2 |B|^2\]

where $B$ is the complex wave envelope with units of velocity (m/s) and $f_0$ is the Coriolis parameter.

Dimensional Equations

The model solves dimensional equations where $B$ has actual velocity amplitude. No additional scaling factors (like W2F) are needed.

Decomposition in Real/Imaginary Parts

Writing $B = B_R + i B_I$, the Jacobian term becomes:

\[\frac{i}{2f_0} J(B^*, B) = \frac{1}{f_0}\left(\frac{\partial B_R}{\partial y} \frac{\partial B_I}{\partial x} - \frac{\partial B_R}{\partial x} \frac{\partial B_I}{\partial y}\right)\]

And the wave intensity:

\[|B|^2 = B_R^2 + B_I^2\]

So the complete formula in spectral space is:

\[q^w = \frac{1}{f_0}\left( \frac{\partial B_R}{\partial y} \frac{\partial B_I}{\partial x} - \frac{\partial B_R}{\partial x} \frac{\partial B_I}{\partial y} \right) - \frac{k_h^2}{4f_0} (B_R^2 + B_I^2)\]

Note: In spectral space, $\nabla_h^2 \to -k_h^2$, so $+\frac{1}{4f_0}\nabla_h^2|B|^2 \to -\frac{k_h^2}{4f_0}|B|^2$.

How It Enters the QG Equation

The wave feedback modifies the effective PV used for streamfunction inversion:

\[q^* = q - q^w\]

Then $\psi$ is computed from $q^*$ via the elliptic inversion:

\[\nabla^2\psi + \frac{\partial}{\partial z}\left(\frac{f_0^2}{N^2}\frac{\partial\psi}{\partial z}\right) = q^*\]

Physical Interpretation

Term	Meaning
$J(B^*, B)$	Jacobian of complex wave field (wave momentum flux)
$\nabla_h^2\|B\|^2$	Horizontal curvature of wave energy density

The wave feedback represents:

Radiation stress from wave momentum flux
Form drag from wave-induced pressure fluctuations

Energy Exchange

Wave-to-Flow Transfer

Energy flows from waves to mean flow when:

\[\mathcal{E}_{w \to f} = -\int \psi \cdot J(\psi, q^w) \, dV\]

This can be positive or negative:

Positive: Waves energize the flow
Negative: Flow energizes waves (less common)

Conservation

In the inviscid limit, total energy is conserved:

\[\frac{d}{dt}(E_{flow} + E_{wave}) = 0\]

The wave feedback term merely redistributes energy.

Refraction Mechanism

How Eddies Focus Waves

Anticyclones (negative vorticity) trap waves:

Effective frequency is reduced: $f_{eff} = f_0 + \zeta/2$
Waves propagate toward lower effective frequency
Energy accumulates in anticyclone cores

The Chimney Effect

         Wind Forcing
              ↓
    ┌─────────────────────┐
    │   Surface Layer     │
    └─────────┬───────────┘
              │
    ┌─────────┼───────────┐
    │    ↙    ↓    ↘      │  ← Waves spread horizontally
    │   ↙     ↓     ↘     │
    └──↙──────┼──────↘────┘
       ↘      ↓      ↙
        ↘     ↓     ↙
         ↘    ↓    ↙        ← Anticyclone focuses waves
    ┌─────────┼───────────┐
    │         ↓           │
    │    Anticyclone      │  ← Enhanced dissipation
    │      (ζ < 0)        │
    └─────────────────────┘

Waves are funneled into anticyclones, enhancing deep mixing.

Implementation

Computing Wave Feedback

# In nonlinear.jl: compute_qw! (BR/BI form)
function compute_qw!(qwk, BRk, BIk, par, G, plans; Lmask=nothing)
    f0 = par.f₀  # Coriolis parameter

    # 1. Compute derivatives of BR and BI
    # BRx = ∂BR/∂x, BRy = ∂BR/∂y, etc.
    BRxk = im * kx .* BRk
    BRyk = im * ky .* BRk
    BIxk = im * kx .* BIk
    BIyk = im * ky .* BIk

    # 2. Transform to real space
    # ...

    # 3. Compute Jacobian term: (1/f₀)(BRy*BIx - BRx*BIy)
    qwr = (BRyr .* BIxr - BRxr .* BIyr) / f0

    # 4. Compute |B|² = BR² + BI²
    mag2 = BRr.^2 + BIr.^2

    # 5. Assemble in spectral space
    # qw = J_term - (kh²/4f₀)*|B|²  (note: ∇² → -kh² in spectral)
    qwk = fft(qwr) - (0.25/f0) * kh2 .* fft(mag2)

    # Note: No additional scaling needed - B has dimensional velocity units (m/s)
end

For the complex envelope form used in YBJ+ time stepping, use:

compute_qw_complex!(qwk, Bk, par, G, plans; Lmask=L)

Usage in Time Stepping

The wave feedback enters via q* = q - qw:

# After computing q at new time step
if wave_feedback_enabled
    compute_qw!(qwk, BRk, BIk, par, G, plans; Lmask=L)
    q_arr .-= qwk_arr  # q* = q - qw
end

# Complex B form (YBJ+ path)
if wave_feedback_enabled
    compute_qw_complex!(qwk, Bk, par, G, plans; Lmask=L)
    q_arr .-= qwk_arr  # q* = q - qw
end

# Then invert q* to get ψ
invert_q_to_psi!(state, grid; a=a_vec)

Enabling/Disabling

# With wave feedback (default)
params = default_params(Lx=500e3, Ly=500e3, Lz=4000.0; no_feedback=false, no_wave_feedback=false)

# Without wave feedback
params = default_params(Lx=500e3, Ly=500e3, Lz=4000.0; no_feedback=true)
# or
params = default_params(Lx=500e3, Ly=500e3, Lz=4000.0; no_wave_feedback=true)

Disabling is useful for:

Studying one-way wave-flow interaction
Isolating wave dynamics from flow effects
Computational efficiency when feedback is weak

Scaling Analysis

When is Feedback Important?

The feedback strength scales as:

\[\frac{|q^w|}{|q|} \sim \left(\frac{A_0}{U}\right)^2 \cdot \left(\frac{L_w}{L}\right)^2\]

where:

$A_0/U$: Wave-to-flow velocity ratio
$L_w/L$: Wave-to-eddy length ratio

Feedback matters when:

Strong wind forcing (large $A_0$)
Compact wave packets (small $L_w$)
Weak background flow (small $U$)

Typical Values

Scenario	Wave Feedback
Weak winds, strong eddies	Negligible
Storm forcing	Moderate (1-10%)
Tropical cyclone	Strong (10-50%)

Coupled Dynamics

Feedback Loop

┌──────────────┐         ┌──────────────┐
│              │         │              │
│   Eddies     │◄────────│    Waves     │
│   (ψ, q)     │         │   (A, B)     │
│              │         │              │
└──────┬───────┘         └──────┬───────┘
       │                        │
       │ Refraction             │ Feedback
       │ ∂ζ/∂t                  │ qw
       │                        │
       ▼                        ▼
┌──────────────────────────────────────┐
│                                      │
│       Wave-Mean Energy Exchange      │
│                                      │
└──────────────────────────────────────┘

Equilibration

The coupled system can reach statistical equilibrium where:

Wave generation (wind) balances dissipation
Energy flux from waves to flow balances eddy dissipation
Net energy is constant on average

Diagnostics

Monitoring Energy Exchange

# Compute wave feedback contribution
qw = compute_wave_pv(state.A, grid)

# Energy exchange rate
exchange_rate = compute_energy_exchange(state.psi, qw, grid, plans)

Typical Analysis

Track $E_{flow}$ and $E_{wave}$ over time
Compute their time derivatives
Compare with wave feedback term to verify energy conservation

Complete Energy Budget

Energy Components

The total energy of the QG-YBJ+ system consists of five components:

\[E_{total} = \underbrace{E_{KE}^{flow} + E_{PE}^{flow}}_{\text{Mean flow energy}} + \underbrace{E_{KE}^{wave} + E_{PE}^{wave} + E_{CE}^{wave}}_{\text{Wave energy}}\]

Mean Flow Kinetic Energy

\[E_{KE}^{flow} = \frac{1}{2} \int \int \int (u^2 + v^2) \, dx\, dy\, dz\]

In spectral space with dealiasing:

\[E_{KE}^{flow} = \frac{1}{2} \sum_{k_x, k_y, z} L(k_x, k_y) \cdot k_h^2 |\hat{\psi}|^2 - \frac{1}{2}|\hat{\psi}(k_h=0)|^2\]

where $L(k_x, k_y)$ is the dealiasing mask (2/3 rule) and the second term corrects for the zero-wavenumber mode.

Mean Flow Potential Energy

\[E_{PE}^{flow} = \frac{1}{2} \int \int \int \frac{f_0^2}{N^2} \left(\frac{\partial \psi}{\partial z}\right)^2 dx\, dy\, dz\]

In spectral space:

\[E_{PE}^{flow} = \frac{1}{2} \sum_{k_x, k_y, z} \frac{f_0^2}{N^2(z)} |\hat{b}|^2\]

where $b = \partial\psi/\partial z$ is the buoyancy from thermal wind balance.

Wave Kinetic Energy

\[E_{KE}^{wave} = \frac{1}{2} \int \int \int |B|^2 \, dx\, dy\, dz\]

In spectral space with $B = B_R + iB_I$:

\[E_{KE}^{wave} = \frac{1}{2} \sum_{k_x, k_y, z} (|\hat{B}_R|^2 + |\hat{B}_I|^2) - \frac{1}{2}|\hat{B}(k_h=0)|^2\]

Wave Potential Energy

From the YBJ+ formulation, the wave potential energy involves $C = \partial A/\partial z$:

\[E_{PE}^{wave} = \frac{1}{2} \int \int \int \frac{N^2}{2f_0^2} k_h^2 |C|^2 \, dx\, dy\, dz\]

In spectral space:

\[E_{PE}^{wave} = \frac{1}{2} \sum_{k_x, k_y, z} \frac{k_h^2}{2 a_{ell}} (|\hat{C}_R|^2 + |\hat{C}_I|^2)\]

where $a_{ell} = f_0^2/N^2$ is the elliptic coefficient.

Wave Correction Energy (YBJ+)

The YBJ+ equation introduces a higher-order correction:

\[E_{CE}^{wave} = \frac{1}{8} \int \int \int \frac{N^4}{f_0^4} k_h^4 |A|^2 \, dx\, dy\, dz\]

In spectral space:

\[E_{CE}^{wave} = \frac{1}{2} \sum_{k_x, k_y, z} \frac{k_h^4}{8 a_{ell}^2} (|\hat{A}_R|^2 + |\hat{A}_I|^2)\]

This term accounts for horizontal wave dispersion and becomes significant at small scales.

Energy Conservation Theorem

Theorem: In the inviscid limit (no dissipation), the total energy is conserved:

\[\frac{dE_{total}}{dt} = 0\]

Proof sketch:

The QG PV equation conserves mean flow energy in the absence of wave feedback
The YBJ+ equation conserves wave energy in the absence of mean flow
The wave feedback term $q^w$ transfers energy between waves and flow without dissipation
The refraction term $\frac{1}{2}B \cdot \zeta$ exchanges energy via wave-vorticity interaction

Energy Transfer Pathways

                    Wind Forcing
                         │
                         ▼
    ┌────────────────────────────────────────┐
    │           Wave Energy                  │
    │   E_KE^wave + E_PE^wave + E_CE^wave    │
    └────────────────┬───────────────────────┘
                     │
           Refraction │ Wave Feedback
           (B·ζ term) │ (q^w term)
                     │
                     ▼
    ┌────────────────────────────────────────┐
    │         Mean Flow Energy               │
    │        E_KE^flow + E_PE^flow           │
    └────────────────┬───────────────────────┘
                     │
                     ▼
              Viscous Dissipation

Energy Exchange Rate

The rate of energy transfer from waves to mean flow is:

\[\mathcal{P}_{w \to f} = -\int \psi \cdot J(\psi, q^w) \, dV\]

This can be computed diagnostically:

# Compute energy exchange rate
qw = compute_qw(state, grid, params, plans)
P_exchange = -sum(psi .* jacobian(psi, qw, grid, plans))

Energy Scales

Using the characteristic scales:

Velocity: $U$ (mean flow), $U_w$ (waves)
Length: $L$ (horizontal), $H$ (vertical)
Time: $1/f_0$

The energy ratio scales as:

\[\frac{E^{wave}}{E^{flow}} \sim \left(\frac{U_w}{U}\right)^2\]

Typical oceanic values for wave-to-flow energy ratio:

Gulf Stream region: $\sim 10^{-2}$ to $10^{-1}$
Open ocean: $\sim 10^{-3}$
After storm: $\sim 1$

Diagnostic Implementation

Energy diagnostics are automatically saved to separate files:

# Output files in diagnostic/ folder:
# - wave_KE.nc: E_KE^wave time series
# - wave_PE.nc: E_PE^wave time series
# - wave_CE.nc: E_CE^wave time series
# - mean_flow_KE.nc: E_KE^flow time series
# - mean_flow_PE.nc: E_PE^flow time series
# - total_energy.nc: All energies + totals

# Verify conservation
ds = NCDataset("output/diagnostic/total_energy.nc")
E_total = ds["total_energy"][:]
dE = (E_total[end] - E_total[1]) / E_total[1]
# Should be < 10^-6 for inviscid runs

See Diagnostics Guide for detailed usage.

References

Xie, J.-H., & Vanneste, J. (2015). A generalised-Lagrangian-mean model of the interactions between near-inertial waves and mean flow. J. Fluid Mech., 774, 143-169.
Wagner, G. L., & Young, W. R. (2016). A three-component model for the coupled evolution of near-inertial waves, quasi-geostrophic flow and the near-inertial second harmonic. J. Fluid Mech., 802, 806-837.
Asselin, O., & Young, W. R. (2019). An improved model of near-inertial wave dynamics. J. Fluid Mech., 876, 428-448.