Eddy–Wave Interactions

class: center, middle
<br/><br/><br/><br/>
.title[An inverse cascade walks into a wave field]<br/>
<br/><br/><br/>
.author[Jonathan M. Lilly]
.institution[The Planetary Science Institute, Tucson, Arizona]
<br/>
.institution[L. Hiron, J. Early, K. Raja, C. Wortham]
<br/>
.institution[Gordon Research Conference on Ocean Mixing]
.date[June 16, 2026]<br/>
<br/>
.note[Created with [{Liminal}](https://github.com/jonathanlilly/liminal) using [{Remark.js}](http://remarkjs.com/) + [{Markdown}](https://github.com/adam-p/markdown-here/wiki/Markdown-Cheatsheet) +  [{KaTeX}](https://katex.org)]

<!-- Structure: 
Discussion leaders get 15 minutes to introduce their session and the speakers.   The recommendation 10 minutes on the science, 5 minutes on your speakers.
Each speaker gets 30 minutes for their presentation followed by a 15 minute discussion chaired by the discussion leader.
After the 3rd speaker’s discussion slot, there’s a wider general discussion for 30 minutes about the session as a whole to think about challenges ahead. This also a good time for us to raise topics we'd like discussed in a wider context and propose discussion points rather than let it be entirely free-flowing.
Finally, the session concludes with 10 minutes for poster presenters to introduce themselves highlight their poster topics

We are Tuesday Morning at 9 am.  The schedule can be found here. The order is Maja, Manuel, Jonathan-->

---
class: center
##A geostrophic eddy in a tidal beam

Showing vorticity split into geostrophic and wave components.  
The eddy fragments into a field of smaller eddies.  
From Hiron, Early, Lilly, Raja, and Wortham (2026), submitted.

---
class: left
##Results

- Geostrophic energy cascades to smaller scales
- There is **no transfer of energy** between wave and geostrophic components
- Instead, increase in geostrophic kinetic energy is due to a **conversion from potential energy**
- Two of the four nonlinear flux terms are proven to be **identically zero** for this dynamical configuration 
- The role of the wave field in the inducing the geostrophic energy cascade is ** catalytic**: essential but non-exchanging

---
class: left
##A brief history of geostrophic turbulence

In three-dimensional turbulence, vortex stretching mediates the transfer of energy to small scales (Taylor, 1917).

In two-dimensional turbulence, the suppression of vortex stretching leads to an *inverse cascade* of energy (Kraichnan, 1967).

In planetary flows,  rotation induces two-dimensionality, and *geostrophic turbulence* emerges in which the inverse cascade eventually excites Rossby waves (Rhines, 1975, 1977).

**A cornerstone of theoretical oceanography is that geostrophically balanced energy cascades to larger scales.**

Many subsequently refinements,  e.g. Larichev and Held (1995), Smith and Vallis (2001), Thompson and Young (2006), Scott and Arbic (2007), Chen (2023).

Yet the basic picture that **geostrophic energy goes upscale** remained unchanged for a half century.  Until  the past few years!

---
class: center
##What role do waves play in this process?

Until recently, the answer was: not much, just a (relatively minor) energy sink.  Note the one-way arrow from eddies to waves!  
From Wunsch and Ferrari (2004).

---
class: left
##Recent work on eddy–wave interactions

It has long been known that internal waves are strongly affected by an ambient eddy field through refraction (Kunze, 1985), distortion (Young and Ben Jelloul, 1997), and other processes.

The reverse interaction, the shaping of the geostrophic eddy field by waves, was long thought to be minor, limited to loss of geostrophic energy through the generation of internal waves.

The story has gone a lot more complicated in recent years! In particular, there is one central question where different studies give opposing answers.

---
class: left
##Do  eddies get stronger or weaker?

Interaction with an internal wave field can result in geostrophic energy moving to *smaller* scales, effectively **reversing the direction of the inverse cascade** (Barkan et al., 2017).

A similar effect has now been seen by many authors with different types of models, e.g., Xie (2020), Thomas
and Daniel (2021), Herñandez-Dueñas et al. (2021), Barkan et al. (2021, 2024).

The result is a substantially *weaker* eddy field together with enhanced dissipation.

But other studies show the seemingly opposite result of geostrophic eddies becoming *intensified* due to eddy–wave interactions  (Thomas and Yamada, 2019; Thomas and Arun, 2020; Thomas and Daniel, 2020;  Cusack et al., 2020).

In a study of internal tides generated by rough bottom topography, Shakespeare (2023)
finds the azimuthal velocities of geostrophic eddies are *accelerated* due to a critical level effect.

Two different qualitative results for eddy–wave interactions!

---
class: left
##Why this is complicated

A plethora of different mechanisms have been proposed.

- *stimulated generation/loss of balance*, geostrophic energy → wave potential energy (Xie and Vanneste, 2015; Xie, 2020)
- *direct extraction*, geostrophic energy → wave field followed by forward cascade (Barkan et al., 2017)
- *critical level acceleration*, wave energy → geostrophic energy  (Shakespeare, 2023)
- *catalytic wave induction*, forward cascade of geostrophic energy but no wave–eddy energy exchange (Xie, 2020)
- *stimulated imbalance* or *stimulated cascades*, waves faciliate forward geostrophic cascade (Barkan et al., 2017, 2021)
- *barotropization*, waves facilitate  barotropic geostrophic energy growth (Thomas and Yamada, 2019; Thomas and Arun, 2020)

Which mechanisms dominate?  Are these really all different?

**Does the induced  geostrophic energy cascade require wave  → energy transfer?**

---
class: left
##Why this is complicated

In the literature there are two very different approaches for separating the flow and geostrophic components.

**Coarse-graining.** This is essentially relying on temporal and/or spatial scale of the flow to filter the fields into two portions.  It is only *approximate*, but easy to employ on realistic models.

See Aluie et al., 2017; Barkan et al., 2017, 2021, 2024; Delpech et al., 2024.

**Orthogonal decomposition.** This approach involves a projection of the flow fields onto a complete dynamical basis.  It is *exact*, but typically employed in reduced dynamics, e.g., two layer models.

See Bartello, 1995; Thomas and Yamada, 2019; Thomas and Arun, 2020; Thomas and Daniel, 2020, 2021.

This difference makes it hard to compare results between realistic and idealized models, and is an obstacle in sorting out the mechanisms.

---
class: left
##Approach

We follow the latter approach, extended towards more realistic dynamics, attempting to mimic Shakespeare (2023).

**Modeling setup**

- Geostrophic eddy +  continuously forced internal tidal beam
- Continuous stratification with constant $N_0$ 
- Flat, rigid, isopycnic upper and lower boundaries
- Horizontal periodicity
- Boussinesq dynamics  
- Comparable wave and eddy energy levels

**Analysis approach**

Unambiguous separation of all flow variables into a geostrophic portion and a wave portion at each point in space and time, using the wave–vortex decomposition of Early et al. (2021, 2026).

This gives the ability to define conserved quantities energy and enstrophy, separate these into wave and geostrophic portions, and track their fluxes throughout wavenumber–modal space.

---
class: left
##The Boussinesq equations of motion

In a constant stratification $N(z)=N_0$, the Boussinesq equations of motion are

<div>
\[\!\!\!\!\!\!\!\!\partial_t u = fv-\frac{1}{\rho_0} \partial_x p' -\underline{ \mathbf{u} \cdot \bm \nabla u} \]
\[\!\!\!\!\!\partial_t v = -fu -\frac{1}{\rho_0} \partial_y p' - \underline{ \mathbf{u} \cdot \bm \nabla v} \]
\[\partial_{t} w = -\eta N_0^2 -\frac{1}{\rho_0} \partial_z p' -\underline{ \mathbf{u} \cdot \bm \nabla w}  \]
\[\partial_{t} \eta = w- \underline{ \mathbf{u} \cdot \bm \nabla \eta} \hphantom{\quad\quad\quad\quad\quad\quad}  \]
\[\bm\nabla \cdot \mathbf{u} =0\hphantom{\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad}\]
</div>

where $\eta$ is a vertical displacement, and $p'$ is a deviation from a state of rest. Non-linear terms are underlined.

---
class: left
##The wave–vortex decomposition

Let $\left|\bm{\Psi}\right>(x,y,z,t)\equiv[u\,\, v  \,\, w \,\, \eta\,\,p]^T$ be a state vector representing the instantaneous state of the fluid.  Then the linear wave modes

<div>\[\left|\bm{\Psi}^\pm_\mathrm{w}\right>
= \frac{1}{\omega_\kappa^j \kappa}
\begin{bmatrix}
(k\omega_\kappa^j \mp f i \ell) F^j_\kappa(z) \\
(\ell \omega_\kappa^j \pm f i k)F^j_\kappa(z) \\
- i \kappa^2 \omega_\kappa^j h_\kappa^j G^j_\kappa(z) \\
\mp \kappa^2 h_\kappa^j G^j_\kappa(z)\\
\mp \rho_0 g \kappa^2 h_\kappa^j F^j_\kappa(z)
\end{bmatrix} e^{i k x + i \ell y\pm i\omega_\kappa^j t}
\]</div>

together with the geostrophic modes

<div>\[
\left|\bm{\Psi}_\mathrm{g}\right> =  \frac{-1}{\kappa^2 + \lambda_j^{-2}}
\begin{bmatrix}
- i \ell F^j_\textrm{g}(z) \\
i k F^j_\textrm{g}(z) \\
0\\
\frac{f}{g} G^j_\textrm{g}(z) 
\\
\rho_0 f F^j_\textrm{g}(z)
\end{bmatrix} e^{i k x + i \ell y}
\]</div>

form a complete basis for the  non-linear system  (Early et al, 2026),

<div>\[
\left|\bm{\Psi}\right>(x,y,z,t) = \sum_{jk\ell} \left(A_\pm^{jk\ell} \left|\bm{\Psi}^\pm_\mathrm{w}\right>+A_0^{jk\ell} \left|\bm{\Psi}_\mathrm{g}\right> \right)+\mathrm{c.c.}
\]</div>

---
class: center
##Continuously forced tidal beam

Coefficients of 10 M$\_2$ vertical modes are held fixed.  
Intrinsic frequencies $\tilde{\omega} =\bm{\omega} - \bm{\kappa} \cdot \mathbf{u} \leq f$ indicates critical layers.

---
class: center
##Three phases of energy evolution

Geostrophic kinetic energy is (I) stable, (II) increasing, (III) stable.  
Total geostrophic energy is essentially constant.  
Wave energy is (I) increasing, (II) stable, (III) rapidly increasing.

---
class: center
##Explicitly tracking energy fluxes

**No exchange of energy** between the geostrophic flow and waves.
Increase in geostrophic kinetic energy is due to a loss of geostrophic potential energy, i.e., a PE → KE conversion.

---
class: center
##Evolution of  wave and geostrophic spectra

Wavenumber–mode decomposition of wave & geostrophic energy.  
Phase I: full wave spectrum develops from forced modal points.
Phase II: geostrophic energy fluxes to smaller scales and higher modes,  and the zero mode (barotropization).

---
class: left
##Why is there no energy exchange?

The evolution of the geostrophic flow occurs in the absence of energy exchange with the wave field.  Why is there no exchange?

We will show two of the four non-linear terms that could potentially exchange energy are **identically zero**.

This is true not just for this particular configuration, but for Boussinesq dynamics in constant stratification with the chosen boundary conditions in general.

---
class: left
## Non-linear terms in the QGPV equation

The evolution of quasigeostrophic potential  vorticity is given by
<div>\[\partial_t q = -\partial_x\left(\mathbf{u} \cdot \bm \nabla v\right) +\partial_y\left( \mathbf{u} \cdot \bm \nabla u\right)+ f\partial_z\left( \mathbf{u} \cdot \bm \nabla \eta\right)\]</div>

and if we expand  the variables into wave and geostrophic components,
<div>\[\mathbf{u}=\mathbf{u}_{\mathrm{g}}+\mathbf{u}_{\mathrm{w}}\quad \quad\quad
\eta=\eta_{\mathrm{g}}+\eta_{\mathrm{w}}\]</div>

we obtain non-linear terms of the form

<div>\[\mathrm{g}\nabla \mathrm{g}\leadsto\mathrm{g},\quad \boxed{\mathrm{w}\nabla \mathrm{g}\leadsto\mathrm{g},} \quad \mathrm{g}\nabla \mathrm{w}\leadsto\mathrm{g},\quad \mathrm{w}\nabla \mathrm{w}\leadsto\mathrm{g}.
\]</div>

The this shorthand means, e.g., that wave component advects the geostrophic component which forces the geostrophic component.

Even more succinctly, we may write

<div>\[\mathrm{g}\mathrm{g}\mathrm{g},\quad\quad \mathrm{w}\mathrm{g}\mathrm{g}, \quad\quad \mathrm{g}\mathrm{w}\mathrm{g},\quad\quad \mathrm{w}\mathrm{w}\mathrm{g}\]</div>

where the order of the letters matters.

---
class: left
##A more compact form for QGPV evolution

Introduce what we term the QGPV *source vector*

<div>\[\bm{\chi} \equiv v \mathbf{i} - u \mathbf{j} - f\eta \mathbf{k},\quad\quad\quad q = \bm \nabla \cdot \bm{\chi},
\]</div>

the vector whose divergence is the QGPV. The geostrophic component of the vector is $\bm{\chi}_{\mathrm{g}} = \partial_x\psi \mathbf{i} +\partial_y\psi \mathbf{j} +\frac{f^2}{N_0^2} \partial_z\psi \mathbf{k}$.

This has appeared a few times  in the literature. See e.g. Muraki et al. (1999), Schneider et al. (2003),  Maddison and Marshall (2013).

The quasigeostrophic evolution equation then becomes

<div>\[\partial_t q = -\bm \nabla \cdot \left(\mathbf{u} \bm \nabla \bm{\chi}\right)\]</div>

which we can alternately expand as

<div>\[\partial_t q + \mathbf{u} \cdot \bm \nabla q=\boxed{- \mathrm{tr} \left(\bm\nabla\mathbf{u}\,\bm\nabla \bm{\chi}\right)} \]</div>

where "$\mathrm{tr}$" is the tensor trace. This non-linear term is responsible for *forcing*, as distinct from *advection*.

---
class: left
##The forcing term

The forcing term can be expanded in two different ways as

<div>\[\mathrm{tr} \left(\bm\nabla\mathbf{u}\,\bm\nabla \bm{\chi}\right) =\sum_{i=1}^{3}\sum_{j=1}^{3}\frac{\partial u^j}{\partial x^i} \frac{\partial \chi^i}{\partial x^j} = \left( \partial_x \mathbf{u} \right) \cdot \bm\nabla v - \left( \partial_y \mathbf{u} \right) \cdot \bm\nabla u - f\left( \partial_z \mathbf{u} \right) \cdot\bm\nabla\eta\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad= \left( \partial_x \bm{\chi} \right) \cdot \bm\nabla u+ \left( \partial_y \bm{\chi} \right) \cdot \bm\nabla v +\left( \partial_z \bm{\chi} \right) \cdot\bm\nabla w\]</div>

with the former generating terms that we would see in the usual for the QGPV evolution

<div>\[\partial_t q = -\partial_x\left(\mathbf{u} \cdot \bm \nabla v\right) +\partial_y\left( \mathbf{u} \cdot \bm \nabla u\right)+ f\partial_z\left( \mathbf{u} \cdot \bm \nabla \eta\right).\]</div>

Expanding into wave and geostrophic components, we have

<div>\[
\partial_t q +\overset{\mathrm{g}\mathrm{g}\mathrm{g}}{\overbrace{ \mathbf{u}_{\mathrm{g}} \cdot \bm \nabla q}} +\boxed{\overset{\mathrm{w}\mathrm{g}\mathrm{g}}{\overbrace{ \mathbf{u}_{\mathrm{w}} \cdot \bm \nabla q}}}= \hphantom{\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad}
\\\quad\quad\quad\quad \quad\quad\quad\quad \boxed{-\overset{\mathrm{w}\mathrm{g}\mathrm{g}}{\overbrace{\left\{ \mathrm{tr} \left(\bm\nabla\mathbf{u}_{\mathrm{w}}\,\bm\nabla \bm{\chi}_{\mathrm{g}}\right) \right\}}}}
-\overset{\mathrm{g}\mathrm{w}\mathrm{g}}{\overbrace{\mathrm{tr} \left(\bm\nabla\mathbf{u}_{\mathrm{g}}\,\bm\nabla \bm{\chi}_{\mathrm{w}}\right)}}-\overset{\mathrm{w}\mathrm{w}\mathrm{g}}{\overbrace{\mathrm{tr} \left(\bm\nabla\mathbf{u}_{\mathrm{w}}\,\bm\nabla \bm{\chi}_{\mathrm{w}}\right)}}\]</div>
 
where only wgg appears as both advection *and* forcing.

---
class: left
##The QGPV energy equation

As is standard, multiplying the QGPV evolution equation

<div>\[\partial_t q = -\bm \nabla \cdot \left(\mathbf{u} \bm \nabla \bm{\chi}\right)\]</div>

by $-\psi$ and integrating  leads to  an equation for the energy evolution

<div>\[\partial_t\mathcal{E}_{\mathrm{g}} = \frac{1}{V}\int_V \psi \bm \nabla \cdot\left(\mathbf{u}\bm \nabla \bm{\chi}\right)\mathrm{d} V\]</div>

where  the total energy $\mathcal{E}\_{\mathrm{g}}$ is the integral of an energy density $\varepsilon\_{\mathrm{g}}$,

<div>\[\mathcal{E}_{\mathrm{g}} \equiv\frac{1}{V}\int_V \varepsilon_{\mathrm{g}} \mathrm{d} V,\quad\quad\quad\varepsilon_{\mathrm{g}}\equiv \frac{1}{2}\left(\left\|\mathbf{u}_{\mathrm{g}}\right\|^2 + N_0^2\eta^2_{\mathrm{g}}\right). \]</div>

---
class: left
##The QGPV energy equation

Expanding again in terms of wave and geostrophic components, it may be shown that

<div>\[\partial_t\mathcal{E}_{\mathrm{g}} =
-\overset{\mathrm{g}\mathrm{w}\mathrm{g}}{\overbrace{\frac{1}{V}\int_V\left(\mathbf{u}_{\mathrm{g}}\bm \nabla \bm{\chi}_{\mathrm{w}}\right) \cdot \bm \nabla \psi \,\mathrm{d} V}}-\overset{\mathrm{w}\mathrm{w}\mathrm{g}}{\overbrace{\frac{1}{V}\int_V\left(\mathbf{u}_{\mathrm{w}}\bm \nabla \bm{\chi}_{\mathrm{w}}\right)\cdot  \bm \nabla \psi\,\mathrm{d} V}}\]</div>

such that the evolution depends on only two  non-linear triads, gwg and wwg. **Neither ggg nor wgg can change the  total energy**.

It is not at all surprising that ggg, that is, $ \mathrm{g}\nabla \mathrm{g}\leadsto\mathrm{g}$, vanishes, since this appeared in QGPV simply as advection, and since we know that in the absence of waves geostrophic energy would be conserved.

However it is not at all evident that wgg, that this $ \mathrm{w}\nabla \mathrm{g}\leadsto\mathrm{g}$, should also vanish, since this appeared as both an advection and a forcing.

**This is a new result that strongly constraints the exchange of energy between wave and geostrophic components in Boussinesq flows.** (Jeffrey Early's result.)

---
class: center
##Depth/time slice of energy

Phase II: Mid-depth PE is converted into near-surface KE.   
Isocontours of both energies deepen, a.k.a. barotropization.

---
class: center
##Vorticity snapshots every 14 days, part 1

Phase I: Eddy is relatively unperturbed.  
A cyclonic halo develops.  
Meanwhile, wave energy builds and a wave spectrum develops.

---
class: center
##Vorticity snapshots every 14 days, part 2

A tripole begins to emerges around day 207, beginning of Phase II.  
Phase II: intensification and elongation of the tripole.  
Phase III: tripole fragmentation into an smaller eddies.

---
class: center
##Initial eddy condition

Domain  750 $\times$ 750 $\times$ 2 km,  $N_0^2=2\times10^{-5}$ s$^{-2}$,  $45^\circ$N

Anticyclonic Gaussian eddy, azimuthal velocity $U_e=-15$ cm s$^{-1}$  
Vorticity zero crossing $L_e=80$ km,  penetration depth $H_e=0.42$ km

---
class: center
##Cross-section of emergent tripole, day 235

Two cyclonic satellite eddies develop on periphery.  
Penetration depth of cyclones is 2-3 times that of main eddy.

This matches the canonical mode-2, or elliptical instability, e.g. Griffiths and Linden (1981), Flierl (1988), and many others.

---
class: center
##Cross-section of distorted tripole, day 304

A deep cyclonic filament has developed underneath the main eddy.

The eddies fragments into smaller eddies.  But elliptical instability usually leads to a stable tripole or symmetric dipoles!

---
class: left
##What is the mechanism of the cascade?

We have the following observations about the induced cascade:

- Not associated with a wave → eddy energy transfer
- Associated with a geostrophic PE → KE conversion
- Resembles mode-2 or elliptical instability in the horizontal
- Apparent vortex stretching below eddy, thus overturning flow 
- Barotropization in the vertical

Question: doesn't this sound like baroclinic instability?

Could the interaction with the wavefield permit the PE → KE conversion of baroclinic  instability to proceed, even if the eddy is otherwise be too small (Griffiths and Linden, 1981) to be unstable?

If not, then in what ways is it different?

This can be easily settled by comparison with the evolution of an unforced, larger, baroclinically unstable eddy.

---
name: conclusions
class: left
## Questions

What exactly is this induced cascade? Probably not: **suppression** of the inverse cascade, **permissivity** of the usual forward cascade, or inverse cascade operating in a **reverse** direction.

How does the mechanism of the induced cascade differ from that of the inverse cascade?

How prevalent is the non-exchange of energy between waves and geostrophy?

What is the role of the relative energy levels of the geostrophic and wave components?

How do we explain the differences between these results and those of Shakespeare (2023)?  Isotropic vs. anisotropic wave field?

More widespread use of the wave–vortex decomposition can help answer these questions via definitive qualifications of fluxes.

It's important to bring potential energy into the discussion explicitly, so we can track conversion.