# Info Overload: A Cooperative Evacuation Game

MJ Johns<EMAIL_ADDRESS>0009-0002-4016-2517 University of California Santa
Cruz1156 High StSanta CruzCaliforniaUSA95064 ,  Rita Tesfay<EMAIL_ADDRESS>University of California Santa Cruz1156 High StSanta CruzCaliforniaUSA95064 ,
Mário Escarce Junior<EMAIL_ADDRESS>Lancaster UniversityLA1
4YWLancasterUK ,  Emmanuel Ezenwa Jr<EMAIL_ADDRESS>University of
California Santa Cruz1156 High StSanta CruzCaliforniaUSA95064 ,  Thomas
Maiorana<EMAIL_ADDRESS>University of California Davis1 Shields
AveDavisCaliforniaUSA95616 ,  Magy Seif El-Nasr<EMAIL_ADDRESS>University
of California Santa Cruz1156 High StSanta CruzCaliforniaUSA95064 ,  Edward
Melcer<EMAIL_ADDRESS>University of California Santa Cruz1156 High
StSanta CruzCaliforniaUSA95064 and  Katherine Isbister
<EMAIL_ADDRESS>University of California Santa Cruz1156 High
StSanta CruzCaliforniaUSA95064

(2024)

###### Abstract.

Info Overload is a minigame within a larger collection aimed at increasing
awareness and preparation for an evacuation in the event of a wildfire. The
game relies on experiential two-player cooperative gameplay and is played on a
mobile device (a phone or tablet).

Gamification, Multiplayer, Cooperative Play, Player Interaction

††copyright: rightsretained††doi: ††ccs: Human-centered computing Ubiquitous
and mobile computing

##  1\. Introduction

In the event of a disaster, such as wildfire, many neighborhoods and
communities in the path of destruction will be ordered to evacuate to save as
many lives as possible. While community leaders and experts provide as much
accurate direction as they can, the individuals evacuating are likely to face
unforeseen obstacles and the need to make quick decisions with limited or
conflicting information.

We have designed the Info Overload minigame – part of a suite of minigames
addressing wildfire preparation – to help people understand the evacuation
process and the challenging situations they may need to think about.

The game is cooperative for two players sharing one device (a phone or tablet)
to simulate the experience of evacuating with another person in the same
vehicle. In an effort to reduce congestion along evacuation routes, multicar
households are strongly encouraged to evacuate in one vehicle and leave the
other behind. The game emphasizes communication between the two players.

##  2\. Related Work

Some existing games address wildfire preparedness, including This is Fine
(Larakaa, 2020) which is a game about putting out fires within your home,
Disaster Master (Ready, 2024) which is an interactive comic about staying
alert for signs of wildfire, and Stop Disasters (playerthree and UNDRR, 2018)
which is a top-down simulation management game about creating safer
communities that can withstand various disasters (including wildfire) from a
city planning perspective.

What sets ours apart is the perspective of a first-person view within the car
during an active evacuation, the emphasis on cooperation and communication
between two players working together, and the real-time decision making based
on changing information. Outside of climate resilience, two entertainment-
focused games exemplify the real-time cooperative gameplay we explore in Info
Overload. Space Team (Henry Smith, 2012) is a local-multiplayer mobile game
where different players have different information which they must relay to
each other to accomplish collaborative tasks to control a spaceship. Keep
Talking and Nobody Explodes (Steel Crate Games, 2018) similarly involves
collaborative tasks, except in this case one player wears a VR headset while
the other players have a manual of instructions that must be relayed to help
the VR player diffuse a bomb.

##  3\. Design

Info Overload is designed to simulate the experience of two people evacuating
in a car during a wildfire, as can be seen in figure 1. The players each have
their own responsibilities and must work together to safely evacuate.

Figure 1. Screenshot of Gameplay

Player one is driving and has control of the steering wheel and brakes. When
they see brake lights ahead of them or emergency flashing lights in the
rearview mirror they should hold the brake button down to bring the car to a
stop. Decisions about when to turn require directions from player two.

Player two is in charge of triaging information and navigating to safety. They
have a phone with GPS as well as alert messages. They may also choose to turn
on the radio for additional information. The text alerts will give them
updated information about their route, such as unsafe roads or the need to
find temporary shelter if the fire is moving quickly. The map view has a
marker for their designated exit point, and they use the information from
alerts to decide the best places to turn, which they relay to player one.

The car is constantly moving forward (except when player one uses the brakes)
and can only turn at intersections. If the players reach a deadend or fail to
heed a warning about finding temporary shelter, they will be prompted to start
over as the ‘lose’ condition. If they safely reach the designated exit point,
they win the minigame.

Our objective with the design is to avoid overly harsh penalties or making the
players feel overwhelmed or upset, but to still convey the importance of
communicating with others, using caution during an evacuation, and paying
attention for updated information about their route.

##  4\. Playable Demo

Play the game here: https://ucsc-wildfire-games.itch.io/info-overload or by
scanning the QR code in Fig. 2.

Figure 2. QR Code to play the game demo

##  5\. Early Feedback

The game is still in early development, but we have had some opportunities to
discuss the design with community members and local experts at Participatory
Design workshops.

Figure 3. Initial storyboard shown at the first workshop Figure 4. Revised
storyboard used at second workshop

Participants suggested we incorporate observable contextual information, such
as stop lights being out or seeing smoke or fire in the distance. In general
they liked that we had a two-player experience so players could practice
working together, but they had some concerns about players being frustrated
with each other, particularly if they have different levels of skill.

Some workshop participants with fire training also pointed out that a lot of
this scenario in real life would be outside of the players control, and
cautioned us to avoid giving a false sense to the player of how much they
would realistically be able to do about it.

We also learned that while some communities rely on radio for real-time
updates, others do not, so we may need to consider regional variants for our
design.

##  6\. Conclusion and Future Work

Info Overload is a minigame to improve preparedness for wildfire evacuation by
simulating the experience of evacuating in a vehicle, allowing players to
practice communication and collaboration as well as triaging information
during a disaster. As we continue to develop the game we will seek community
feedback and insights from experts to ensure the information we present is
accurate and beneficial.

###### Acknowledgements.

This material is based upon work supported by the National Science Foundation
under Grant No. 2230636. We also would like to thank our collaborators Dr.
Kenichi Soga and Dr. Louise Comfort and their respective teams, and the
community members who offered valuable insights at our community workshops.

## References

  * (1)
  * Henry Smith (2012) Henry Smith. 2012.  Spaceteam (Game).  http://spaceteam.ca/
  * Larakaa (2020) Larakaa. 2020.  This is Fine (Game).  https://larakaa.itch.io/this-is-fine
  * playerthree and UNDRR (2018) playerthree and UNDRR. 2018.  Stop Disasters (game).  https://www.stopdisastersgame.org/
  * Ready (2024) Ready. 2024.  Disaster Master (game).  https://www.ready.gov/kids/games/data/dm-english/wildfire.html
  * Steel Crate Games (2018) Steel Crate Games. 2018.  Keep Talking and Nobody Explodes (Game).  https://keeptalkinggame.com/


#  Negative APE dissipation as the fundamental criterion for double diffusive
instabilities

R. Tailleux1<EMAIL_ADDRESS>1Department of Meteorology,
University of Reading, Reading RG6 6BB, UK

###### Abstract

The background potential energy (BPE) is the only reservoir that double
diffusive instabilities can tap their energy from when developing from an
unforced motionless state with no available potential energy (APE). Recently,
Middleton and Taylor linked the extraction of BPE into APE to the sign of the
diapycnal component of the buoyancy flux, but their criterion can only predict
diffusive convection instability, not salt finger instability. Here, we show
that the problem can be corrected if the sign of the APE dissipation rate is
used instead, making it emerge as the most fundamental criterion for double
diffusive instabilities. A theory for the APE dissipation rate for a two-
component fluid relative to its single-component counterpart is developed as a
function of three parameters: the diffusivity ratio, the density ratio, and a
spiciness parameter. The theory correctly predicts the occurrence of both salt
finger and diffusive convection instabilities in the laminar unforced regime,
while more generally predicting that the APE dissipation rate for a two-
component fluid can be enhanced, suppressed, or even have the opposite sign
compared to that for a single-component fluid, with important implications for
the study of ocean mixing. Because negative APE dissipation can also occur in
stably stratified single-component and doubly stable two-component stratified
fluids, we speculate that only the thermodynamic theory of exergy can explain
its physics; however, this necessitates accepting that APE dissipation is a
conversion between APE and the internal energy component of BPE, in contrast
to prevailing assumptions.

###### keywords:

negative dissipation, mixing, double-diffusion, salt fingers, diffusive
convection, thermohaline staircases, available potential energy

##  1 Introduction

Stratified binary fluids, in which the stratifying agents diffuse at different
rates, exhibit a range of intriguing phenomena not observed in single-
component stratified fluids. One such phenomenon are double diffusive
instabilities, which arise when one of the stratifying agents act to
destabilise the stratification. In the context of ocean mixing, this leads to
the emergence of new types of flows, including diffusive convection, salt
fingering, thermohaline intrusions, and thermohaline staircases (Turner, 1985;
Schmitt, 1994; Radko, 2013). Theoretical analysis categorises the different
dynamically relevant types of stratification based on the density ratio
$(R_{\rho})$ of the fluid: 1) the doubly stable case $(R_{\rho}<0)$, when both
temperature and salinity are stabilising; 2) salt fingers favourable
$(R_{\rho}>1)$, with salinity as the destabilising agent; 3) diffusive
convection favourable $(0<R_{\rho}<1)$, where temperature acts as the
destabiliser. Understanding how double diffusion modulates mixing in the
doubly stable case and characterising double diffusive instabilities in both
laminar and turbulent regimes are key questions of interest.

To date, linear stability analysis has been the primary approach for
identifying the physical parameters that control the development of double
diffusive instabilities from a quiescent sate. Such an analysis reveals, for
instance, that molecular viscosity and diffusion can severely restrict the
range of density ratios at which these instabilities can occur, viz.,

$\begin{split}\frac{Pr+\tau}{Pr+1}<R_{\rho}<1,&\qquad{\rm
for\,\,diffusive\,\,convection},\\\ 1<R_{\rho}<\frac{1}{\tau},&\qquad{\rm
for\,\,salt\,\,fingers},\end{split}$ (1)

e.g., Radko (2013), where $\tau=\kappa_{S}/\kappa_{T}$ represents the ratio of
salt diffusivity to temperature diffusivity, $Pr=\nu/\kappa_{T}$ denotes the
Prandtl number, and $\nu$ is the molecular viscosity. For example, using the
typical oceanic values $\tau\approx 0.01$ and $Pr\approx 7$ yield a reduced
density ratio range of $1<R_{\rho}<100$ for salt fingers and
$0.8076<R_{\rho}<1$ for diffusive convection.

While linear stability theory provides valuable insights into double diffusive
instabilities, other approaches are necessary to understand their energy
source and whether they can persist in a turbulent environment like the
oceans. In this regard, the theory of available potential energy (APE)
introduced by Lorenz (1955) and applied to the study of turbulent stratified
fluids by Winters et al. (1995) (referred to as W95 thereafter) appears to be
one of the most promising avenues of research. According to APE theory, the
energetics of a stratified fluid can be separated into two main budgets, one
for the total mechanical energy (APE+kinetic energy), and one for the
background potential energy (BPE). Physically, the BPE represents the
potential energy (PE) associated with the state of minimum potential energy
achievable from the actual state through an adiabatic (and isohaline in
seawater) mass re-arrangement. The generic form of these budget equations is

$\frac{d(APE+KE)}{dt}={\rm
Forcing}-\int_{V}\rho(\varepsilon_{k}+\varepsilon_{p})\,{\rm d}V$ (2)
$\frac{d(BPE)}{dt}=\int_{V}\rho(\varepsilon_{k}+\varepsilon_{p})\,{\rm d}V$
(3)

where $\varepsilon_{k}$ is the positive definite viscous dissipation rate, and
$\varepsilon_{p}$ the diffusive rate of APE dissipation. As is well known,
viscous dissipation is an irreversible process that converts KE into ‘heat’,
which in APE theory is played by the BPE.

In a single-component compressible fluid, Tailleux (2013c) (of which an
improved version is presented in Appendix A) established that the APE
dissipation rate $\varepsilon_{p}$ may be broken down into three components:

$\varepsilon_{p}=\varepsilon_{p,lam}+\varepsilon_{p,tur}+\varepsilon_{p,eos},$
(4)

with $\varepsilon_{p,lam}$ and $\varepsilon_{p,tur}$ representing the laminar
and turbulent parts of $\varepsilon_{p}$ respectively, and
$\varepsilon_{p,eos}$ representing the part of $\varepsilon_{p}$ arising from
the nonlinearities of the equation of state. To date, the APE dissipation rate
has been primarily defined and studied for single-component Boussinesq fluids
with a linear equation of state, for wich only $\varepsilon_{p,lam}$ and
$\varepsilon_{p,tur}$ subsist. As clarified further in the text,
$\varepsilon_{p,lam}$ and $\varepsilon_{p,tur}$ relate to the terms
$-\Phi_{i}$ and $\Phi_{d}$ in W95’s framework. Physically,
$\varepsilon_{p,tur}$ is the fundamental quantity for defining the turbulent
diapycnal diffusivity

$K_{\rho}=\frac{\varepsilon_{p,tur}}{N^{2}}=\frac{\Gamma\varepsilon_{k}}{N^{2}}$
(5)

e.g., Osborn & Cox (1972); Osborn (1980); Lindborg & Brethouwer (2008), often
quantified by the dissipation ratio or ‘mixing efficiency’
$\Gamma=\varepsilon_{p,tur}/\varepsilon_{k}$. Typically, $\varepsilon_{p,tur}$
is estimated from microstructure measurements using formula such as the widely
used expression:

$\varepsilon_{p,tur}=\frac{g\alpha\kappa_{T}|\nabla\theta^{\prime}|^{2}}{d\overline{\theta}/dz},$
(6)

(Oakey, 1982; Gargett & Holloway, 1984), where $\alpha$ is the thermal
expansion coefficient, $g$ is the acceleration due to gravity, and
$\overline{\theta}$ represents the mean potential temperature profile.
Consequently, $\varepsilon_{p}$, like the viscous dissipation rate, is
generally considered a positive quantity that also converts mechanical energy
into BPE or ‘heat’ (for further views on this, see Tailleux (2009)).

In a double diffusive fluid, double diffusive instabilities can develop even
in the absence of mechanical forcing. Within the framework of APE theory, this
is only possible if BPE, like APE, can become a source of kinetic energy, and
hence if we accept the notion that $\varepsilon_{p}$ can become negative. In
this paper, we derive a theoretical expression for the ratio ${\cal
F}=\varepsilon_{p}^{dd}/\varepsilon_{p}^{std}$ of the APE dissipation rates
for a double diffusive case over that for a simple fluid. We demonstrate that
$\varepsilon_{p}^{dd}$ is negative under the conditions corresponding to the
linear stability analysis for both salt finger and diffusive convection
regimes. Recently, Middleton & Taylor (2020) proposed to characterise double
diffusive instabilities in terms of the sign of the diapycnal component of the
molecular buoyancy flux, which is related to the turbulent part
$\varepsilon_{p,tur}$ of $\varepsilon_{p}$ (equivalently $\Phi_{d}$), rather
than $\varepsilon_{p}$. However, their criterion only succeeds in predicting
the occurrence of the diffusive convection instability but not the salt finger
instability, which suggests that it is the sign of the net APE dissipation
rate $\varepsilon_{p,lam}+\varepsilon_{p,tur}$ (equivalently
$\Phi_{d}-\Phi_{i}$) that represents the most fundamental criterion for double
diffusive instabilities. Additionally, we show that considering the APE
dissipation rate sheds light on other aspects of double diffusive
instabilities in turbulent regimes. For example, it correctly predicts the
regime associated with diffusive interleaving in the presence of density-
compensated isopycnal gradients of temperature and salinity, which are thought
to be essential for the formation of thermohaline staircases (Merryfield,
2000).

The remainder of this paper is organised as follows: Section 2 establishes the
general form of the local APE dissipation rate for a double diffusive fluid.
Section 3 outlines the conditions necessary for negative APE dissipation.
Section 4 provides a comparison of the local and APE frameworks. Section 5
discusses the practical issues related to the computation of the reference
temperature and salinity profiles used in our definition of spiciness.
Finally, Section 6 discusses the results and provides future perspectives.

##  2 Local APE theory and APE dissipation rate

Similarly as Middleton & Taylor (2020), we study the energetics of double
diffusive instabilities for a Boussinesq fluid with a linear equation of state
for density governed by the system of equations:

$\frac{D{\bf v}}{Dt}+\frac{1}{\rho_{\star}}\nabla\delta
p=-\frac{g\delta\rho}{\rho_{\star}}{\bf k}+\nu\nabla^{2}{\bf v},$ (7)
$\nabla\cdot{\bf v}=0,$ (8) $\frac{D\theta}{Dt}=-\nabla\cdot{\bf
J}_{\theta},\qquad\frac{DS}{Dt}=-\nabla\cdot{\bf J}_{S},$ (9)
$\rho=\rho_{\star}\left[1-\alpha(\theta-\theta_{\star})+\beta(S-S_{\star})\right],$
(10)

where $\rho$ is density, $p$ is pressure, $g$ is gravitational acceleration,
$S$ is salinity, $\theta$ is potential temperature, $\alpha$ is the thermal
expansion coefficient, $\beta$ is the haline contraction coefficients, while
$\rho_{\star}$, $S_{\star}$ and $\theta_{\star}$ are constant reference values
for $\rho$, $S$ and $\theta$ respectively. Moreover, $\delta p=p-p_{0}(z)$ and
$\delta\rho=\rho-\rho_{0}(z)$ denote the pressure and density anomalies
defined relative to the reference pressure and density profiles $p_{0}(z)$ and
$\rho_{0}(z)=-g^{-1}dp_{0}/dz$ characterising Lorenz reference state. Simple
diffusive laws are assumed for $\theta$ and $S$ so that ${\bf
J}_{\theta}=-\kappa_{T}\nabla\theta$ and ${\bf J}_{s}=-\kappa_{S}\nabla S$,
where $\kappa_{T}$ and $\kappa_{S}$ are the molecular diffusivities of heat
and salt respectively. The tracer equations (9) may be combined with the
equation of state (10) to form an equation for the Boussinesq buoyancy
$b_{bou}=g[\alpha(\theta-\theta_{\star})-\beta(S-S_{\star})]$,

$\frac{Db_{bou}}{Dt}=-\nabla\cdot{\bf J}_{b}$ (11)

where

${\bf J}_{b}=g(\alpha{\bf J}_{\theta}-\beta{\bf
J}_{s})=-g(\kappa_{T}\alpha\nabla\theta-\kappa_{S}\beta\nabla S)$ (12)

is the molecular buoyancy flux. Note that in the absence of salinity, ${\bf
J}_{b}=-\kappa_{T}\nabla b_{bou}$ is down the gradient of the Boussinesq
buoyancy, but not in the general case.

To understand how double diffusion affects APE dissipation, the local APE
framework (Tailleux, 2013b, 2018; Novak & Tailleux, 2018) is preferred over
the Winters et al. (1995)’s global APE framework used by Middleton & Taylor
(2020). Indeed, we show in Section 4 that defining the total APE of a fluid as
the volume integral of non-sign definite integrand makes the global APE
framework problematic in several respects; for instance, it introduces
unphysical terms in its budget. To avoid such difficulties, it is therefore
preferable to define the APE of a fluid as the volume integral of the APE
density

$e_{a}=e_{a}(S,\theta,z)=-\int_{z_{r}}^{z}b(S,\theta,z^{\prime})\,{\rm
d}z^{\prime},$ (13)

where $b(S,\theta,z)=-(g/\rho_{\star})(\rho(S,\theta)-\rho_{0}(z))$ is the
buoyancy defined relative to Lorenz reference density profile $\rho_{0}(z)$
characterising the notional reference state of minimum potential energy
obtainable from the actual state by means of an adiabatic and volume-
conserving re-arrangement. This definition of buoyancy has the advantage of
vanishing in Lorenz reference state of rest, which is not the case of the
Boussinesq buoyancy. To avoid any ambiguity, note that
$b(S,\theta,z^{\prime})=-(g/\rho_{\star})(\rho(S(x,y,z,t),\theta(x,y,z,t))-\rho_{0}(z^{\prime}))$
in the integral (13), that is, the values of $S$ and $\theta$ are held
constant as the parcel moves from $z_{r}$ to $z$, consistent with the idea of
an adiabatic and isohaline re-arrangement. As is well known, the APE density
(13) represents the work against buoyancy forces that a parcel needs to
counteract to move from its reference position at $z_{r}$, solution of the
level of neutral buoyancy equation

$b(S,\theta,z_{r})=0,$ (14)

to its actual position at $z$. For simplicity, we disregard the temporal
dependence of $\rho_{0}(z)$ as the APE dissipation only depends on spatial
gradients of the tracer fields. Inversion of Eq. (14) establishes that
$z_{r}=z_{r}(\rho)=z_{r}(S,\theta)$ is a material function of $S$ and $\theta$
and therefore a Lagrangian invariant following fluid parcels in the absence of
diffusive sources/sinks of heat and salt.

A local budget equation for the APE density is easily obtained by taking the
material derivative of (13), which yields

$\begin{split}\frac{De_{a}}{Dt}=&-b\frac{Dz}{Dt}+\underbrace{b(S,\theta,z_{r})}_{=0}\frac{Dz_{r}}{Dt}-(z-z_{r})\frac{Db_{ou}}{Dt}\\\
=&-bw-\nabla\cdot{\bf J}_{a}-\varepsilon_{p},\end{split}$ (15)

where ${\bf J}_{a}$ is the diffusive flux of APE and $\varepsilon_{p}$ the APE
dissipation rate, given by

${\bf J}_{a}=-(z-z_{r}){\bf J}_{b},$ (16) $\begin{split}\varepsilon_{p}=&{\bf
J}_{b}\cdot\nabla(z-z_{r})\\\
=&\underbrace{-g\left(\alpha\kappa_{T}\frac{\partial\theta}{\partial
z}-\beta\kappa_{S}\frac{\partial S}{\partial
z}\right)}_{\varepsilon_{p,lam}}+\underbrace{g(\alpha\kappa_{T}\nabla\theta-\beta\kappa_{S}\nabla
S)\cdot\nabla z_{r}}_{\varepsilon_{p,tur}}.\end{split}$ (17)

As indicated by (17), $\varepsilon_{p}$ can be decomposed into a ‘laminar’ and
‘turbulent’ components $\varepsilon_{p,lam}$ and $\varepsilon_{p,tur}$ whose
volume integrals

$\int_{V}\varepsilon_{p,lam}\rho_{\star}{\rm
d}V=-\Phi_{i},\qquad\int_{V}\varepsilon_{p,tur}\rho_{\star}{\rm d}V=\Phi_{d},$
(18)

can easily be shown to correspond to Winters et al. (1995) global energy
conversions $-\Phi_{i}$ and $\Phi_{d}$ in the global APE framework. So far,
the turbulent mixing community has generally assumed $\Phi_{i}$ and $\Phi_{d}$
to represent distinct types of energy conversions, the former with internal
energy (IE) and the latter with the background potential energy (BPE);
however, the analysis of such conversions in real fluids detailed in Section 4
clearly indicates that there is no physical basis for such a view, and that in
reality, $\varepsilon_{p}$, $\Phi_{d}$, and $\Phi_{i}$ all represent
conversions with internal energy as previously established by Tailleux (2009).
In other words, $\varepsilon_{p,lam}$ and $\varepsilon_{p,tur}$ are parts of
the same energy conversion.

Physically, Eq. (15) states that locally, $e_{a}$ can be modified through: 1)
conversion with kinetic energy via the buoyancy flux $bw$; 2) diffusive and
advective transport via ${\bf J}_{a}$ and advection; 3) ‘dissipation’ that can
be occasionally be negative associated with $\varepsilon_{p}$. Physically, our
local APE budget (15) is simpler in form than the one previously derived by
Scotti & White (2014) for a single-component fluid, although the two can be
verified to be mathematically equivalent. One of the reasons is due to Scotti
& White (2014) imposing the diffusive flux of APE density to be down-gradient,
viz,

${\bf J}_{a,sw14}=-\kappa\nabla e_{a}=\kappa(z-z_{r})\nabla b_{bou}+\kappa
b{\bf k}={\bf J}_{a}+\kappa b{\bf k},$ (19)

which differs from ${\bf J}_{a}$ by the last term, with the consequence of
adding extra terms in their budget equation. Physically, however, Scotti &
White (2014)’s approach is inconsistent with the analysis of the APE budget
for a real fluid (cf Section 4 or Tailleux (2009)), which reveals that
decomposing the potential energy into available and unavailable components
leads to a decomposition of the heat flux ${\bf J}_{q}=\Upsilon{\bf
J}_{q}+(1-\Upsilon){\bf J}_{q}$, with $\Upsilon=(T-T_{r})/T$ a Carnot-like
thermodynamic efficiency. Physically, it is the first term
${\bf}J_{a}=\Upsilon{\bf J}_{q}$ that represents the diffusive flux of APE
density, whose Boussinesq approximation is given by (16). Another reason to be
sceptical of (19) is that it cannot generalise to the case of a doubly-
diffusive two-component fluid, whereas our definition of ${\bf J}_{a}$ (16)
obviously does.

In their study, Middleton & Taylor (2020) derived an expression for the
diapycnal component of the buoyancy flux ${\bf J}_{b}\cdot\hat{\bf n}$ in
terms of three parameters, the diffusivity ratio $\tau=\kappa_{S}/\kappa_{T}$,
the gradient ratio $G_{\rho}=(\alpha|\nabla\theta|)/(\beta|\nabla S)$, and the
angle $\gamma$ between $\nabla\theta$ and $\nabla S$ such that
$\cos{\gamma}=\nabla\theta\cdot\nabla S/(|\nabla\theta||\nabla S|)$, with
$\hat{\bf n}=\nabla z_{r}/|\nabla z_{r}|$, and could similarly be used to
study $\varepsilon_{p}$. However, while $G_{\rho}$ and $\cos{\gamma}$ might be
easily diagnosed from the output of a numerical experiment, they cannot easily
be inferred from measured oceanic properties nor are they naturally related to
physical parameters known to play a key role in double diffusive instabilties
such as density-compensated thermohaline variations. For instance, in the
canonical salt finger experiment that they discuss in their Section 2.5,
Middleton & Taylor (2020) find that $\Phi_{d}$ becomes negative as the result
of the gradients of salinity growing more rapidly than temperature gradients,
thus resulting in the creation of density-compensated thermohaline variations
(usually referred to as ‘spice’). In oceanography, the term spice was perhaps
first used by Munk (1981) to quantify the range of possible $\theta/S$
behaviour of a seawater sample of given density, from warm and salty (spicy)
to fresh and cold (minty). Physically, the creation of spice is of
considerable dynamical importance because lateral stirring along isopycnal
surfaces is not opposed by restoring buoyancy forces and is therefore
considerably more efficient at dissipating tracer variances than vertical
stirring. To study the role of spice, Middleton et al. (2021) subsequently
reformulated Middleton & Taylor (2020)’s expression in terms of the
‘spiciness’ variable $\tau=\alpha\theta+\beta S$ (denoted $S_{p}$ in their
paper), which may be regarded as the ‘linear’ version of the spiciness
variables developed by Jacket & McDougall (1985) or Flament (2002) for
instance. Recently, the theory of spiciness was revisited by Tailleux (2021)
who argued that to be physically meaningful, spice should ideally vanish in a
spiceless ocean and hence that spice variables should be defined as isopycnal
anomalies, which is not the case of $\tau$ or of any related variable. In the
present case, this can be implemented in practice by decomposing $\theta$ and
$S$ as follows:

$\theta=\theta_{0}(z_{r})+\theta_{\xi},\qquad S=S_{0}(z_{r})+S_{\xi},$ (20)

with the reference profiles $\theta_{0}(z_{r})$ and $S_{0}(z_{r})$ defining
the temperature and salinity profiles of the assumed spiceless state, with
$z_{r}=z_{r}(\rho)$ denoting the reference position of a fluid parcel as
before. Provided that $\theta_{0}(z_{r})$ and $S_{0}(z_{r})$ are defined as
the (thickness-weighted) isopycnal mean temperature and salinity respectively,
they contain all necessary information to compute density
$\rho=\rho_{\star}[1-\alpha(\theta_{0}(z_{r})-\theta_{\star})-\beta(S_{0}(z_{r})-S_{\star})]$,
thus allowing them to be interpreted as the ‘active’ components of $\theta$
and $S$. In contrast, the isopycnal anomalies $\theta_{\xi}$ and $S_{\xi}$ are
density compensated $\alpha\theta_{\xi}=\beta S_{\xi}$ and therefore do not
contribute to density, thus allowing them to be interpreted as the passive
components of $\theta$ and $S$. Physically, defining the spiceless
stratification $\theta_{0}(z),S_{0}(z)$ in terms of thickness-weighted
isopycnal means can be justified on the ground that such a state is the one
that would be expected to result in the idealised limit of infinitely fast
(resp. infinitely slow) isopycnal (resp. diapycnal) mixing. How to estimate
these in practice are discussed in Section 5. As shown below, defining
spiciness in this way allows us to define a new set of parameters with which
to study the behaviour of $\varepsilon_{p}$, different from those used by
Middleton & Taylor (2020) and Middleton et al. (2021), and which we consider
to be more physical and more relevant.

In the following, $S_{\xi}$ and $\theta_{\xi}$ are combined into the variable
$\xi=\rho_{\star}(\alpha\theta_{\xi}+\beta S_{\xi}$) to define a single
measure of spiciness having the dimension of density, as is commonly done in
the literature. After some manipulation, it is possible to rewrite
$\varepsilon_{p}$ in the following form

$\begin{split}\varepsilon_{p}=&g\left(\kappa_{T}\alpha\frac{\partial\tilde{\theta}}{\partial
z_{r}}-\kappa_{S}\beta\frac{\partial\tilde{S}}{\partial
z_{r}}\right)\left(|\nabla z_{r}|^{2}-\frac{\partial z_{r}}{\partial
z}\right)+\frac{g(\kappa_{T}-\kappa_{S})}{2\rho_{\star}}\left(\nabla
z_{r}\cdot\nabla\xi-\frac{\partial\xi}{\partial z}\right)\\\
=&{\color[rgb]{0,0,0}K_{v0}}N_{0}^{2}(z_{r})\left[\frac{R_{\rho}-\tau}{R_{\rho}-1}+(1-\tau){\cal
M}\right]\left(1-\frac{\partial_{z}z_{r}}{|\nabla
z_{r}|^{2}}\right),\end{split}$ (21)

which depends on the following parameters:

$\begin{split}{\cal
M}\equiv\frac{g}{2\rho_{\star}N_{0}^{2}(z_{r})}\frac{(\nabla
z_{r}\cdot\nabla\xi-\partial_{z}\xi)}{|\nabla
z_{r}|^{2}-\partial_{z}z_{r}}\qquad&{\rm Spiciness\,\,parameter}\\\ \\\
{\color[rgb]{0,0,0}K_{v0}}\equiv\kappa_{T}|\nabla
z_{r}|^{2}=\kappa_{T}\left(\frac{d\rho_{0}}{\partial
z_{r}}(z_{r})\right)^{-2}|\nabla\rho|^{2}\qquad&{\rm
Effective\,\,diffusivity}\\\ \\\
N_{0}^{2}(z_{r})\equiv-\frac{g}{\rho_{\star}}\frac{d\rho_{0}}{dz_{r}}(z_{r})=g\left(\alpha\frac{d\theta_{0}}{dz_{r}}-\beta\frac{dS_{0}}{dz_{r}}\right)(z_{r})\qquad&{\rm
Reference\,\,N^{2}}\\\ \\\ R_{\rho}\equiv
R_{\rho}(z_{r})=\left.\alpha\frac{d\theta_{0}}{dz_{r}}\right/\beta\frac{dS_{0}}{dz_{r}}\qquad&{\rm
Density\,\,Ratio}\\\ \\\ \tau\equiv\frac{\kappa_{S}}{\kappa_{T}}\qquad&{\rm
Diffusivity\,\,Ratio}\end{split}$ (22)

Our choice of control parameters $R_{\rho}$ and ${\cal M}$ differ from those
of Middleton & Taylor (2020), who chose the gradient ratio
$G_{\rho}\equiv\alpha|\nabla\theta|/(\beta|\nabla S|)$ and the angle $\phi$
between $\nabla\theta$ and $\nabla S$ defined by
$\cos{\phi}=-\nabla\theta\cdot\nabla S/(|\nabla\theta||\nabla S|)$ instead.
The present approach is preferred here because $R_{\rho}$ is a more commonly
encountered parameter in the literature, while ${\cal M}$ is more naturally
connected to the physics of lateral thermohaline intrusions, whose importance
is well established. Moreover, $K_{v0}$ represents a local version of the
effective diffusivity concept originally proposed by Winters & d’Asaro (1996)
and Nakamura (1996) based on thermal diffusion, and is positive definite by
construction. $K_{v0}$ is different from the effective diffusivity $K_{\rm
eff}$ introduced in Section 5 and governing the evolution of the sorted
density profile. Note that although the term within square brackets goes to
infinity as $R_{\rho}\rightarrow 1$, $\varepsilon_{p}$ remains finite as
$N_{0}^{2}\rightarrow 0$ then.

##  3 APE dissipation and diffusive stability

###  3.1 Single component fluid

Even in the case of a single component fluid, the APE dissipation rate
$\varepsilon_{p}$, unlike the viscous dissipation rate $\varepsilon_{k}$, is
not guaranteed to be always positive definite. To see this, let us first set
${\cal M}=0$ and $\tau=1$ in (21), thus leading to

$\varepsilon_{p}\equiv\varepsilon_{p}^{\rm
std}={\color[rgb]{0,0,0}K_{v0}}N_{0}^{2}(z_{r})\left(1-\frac{\partial_{z}z_{r}}{|\nabla
z_{r}|^{2}}\right)=\kappa_{T}\left(|\nabla z_{r}|^{2}-\frac{\partial
z_{r}}{\partial z}\right)N_{0}^{2}.$ (23)

Since $\kappa_{T}$ and $N_{0}^{2}$ are positive by construction, (23) shows
that $\varepsilon_{p}$ can occasionally be negative provided that

${\cal F}\equiv|\nabla z_{r}|^{2}-\frac{\partial z_{r}}{\partial
z}=|\nabla_{h}z_{r}|^{2}+\left(\frac{\partial z_{r}}{\partial
z}\right)^{2}-\frac{\partial z_{r}}{\partial z}<0,$ (24)

where $\nabla_{h}$ denotes the horizontal gradient. If viewed as a quadratic
polynomial in $\partial z_{r}/\partial z$ with discriminant
$\Delta=1-4|\nabla_{h}z_{r}|^{2}|$, ${\cal F}$ can only be negative when
possessing two real roots, that is when $\Delta<0$, or equivalently when

$|\nabla_{h}z_{r}|<\frac{1}{2}.$ (25)

In that case, ${\cal F}$ and $\varepsilon_{p}^{\rm std}$ achieve their
maximally negative values for $\partial z_{r}/\partial z=1/2$,

$\left|\varepsilon_{p,min}^{\rm
std}\right|=\kappa_{T}N_{0}^{2}\left(\frac{1}{4}-|\nabla_{h}z_{r}|^{2}\right)<\frac{\kappa_{T}N_{0}^{2}}{4},$
(26)

which is bounded from above by $\kappa_{T}N_{0}^{2}/4$, which is laminar like
in character and therefore extremely small relative to turbulent values.
Nevertheless, because a negative $\varepsilon_{p}$ can potentially cause
perturbations to grow with time, it seems of interest to estimate the time
scale $\tau_{ape}=e_{a}/\varepsilon_{p}$ of the latter. Using the fact that
the APE density scales as $N_{0}^{2}\zeta^{2}/2$, where $\zeta=z-z_{r}$, this
leads to

$\tau_{ape}=\frac{N_{0}^{2}\zeta^{2}/2}{\kappa_{T}N_{0}^{2}/4}=\frac{2\zeta^{2}}{\kappa_{T}},$
(27)

For illustration, using $\kappa_{T}=10^{-7}{\rm m^{2}.s^{-1}}$ and
$\zeta=1\,{\rm cm}=10^{-2}{\rm m}$ yields $\tau=2\times
10^{-4}/10^{-7}=2\times 10^{3}{\rm s}\approx 33\,{\rm min}$, which is fast
enough to be observable in the laboratory, at least in principle. Eq. (25),
which may be rewritten as $|\nabla_{h}\zeta|<1/2$, also requires that the
horizontal scales of the unstable perturbations be larger than their vertical
scales. A priori, perturbations with the required characteristics cannot occur
spontaneously, they can only be created with some form of appropriate gentle
mechanical stirring. These considerations appear to be consistent with the
experimental observation of steps formation by Park et al. (1994), who
observed the smaller steps to grow faster than larger ones. However, these
conclusions are only speculative at this stage and more research is needed to
confirm or disprove the relevance of our results.

###  3.2 Binary fluid: Spiceless case ${\cal M}=0$

In the double diffusive case but in the absence of spiciness, Eq. (21) reduces
to:

$\varepsilon_{p}={\color[rgb]{0,0,0}K_{v0}}N_{0}^{2}(z_{r})\left(\frac{R_{\rho}-\tau}{R_{\rho}-1}\right)\left(1-\frac{\partial_{z}z_{r}}{|\nabla
z_{r}|^{2}}\right).$ (28)

This time, the sign of $\varepsilon_{p}$ is controlled by the sign of the
product of ${\cal F}$ times the density-ratio dependent term within
parentheses. Restricting attention to the case $\tau<1$ pertaining to
thermosolutal convection, Table 1 shows that $\varepsilon_{p}$ can be negative
for all possible values of $R_{\rho}$, although not necessarily in both
laminar and turbulent regimes.

| $-\infty<R_{\rho}\leq\tau$ | $\tau<R_{\rho}\leq 1$ | $1<R_{\rho}<+\infty$
---|---|---|---
${\cal F}>0$ | $\epsilon_{p}>0$ | $\epsilon_{p}<0$ | $\epsilon_{p}>0$
${\cal F}<0$ | $\epsilon_{p}<0$ | $\epsilon_{p}>0$ | $\epsilon_{p}<0$
Table 1: Sign of the APE dissipation rate $\varepsilon_{p}$ as a function of
the density ratio $R_{\rho}$ and of the sign of ${\cal F}=|\nabla
z_{r}|^{2}-\partial_{z}z_{r}$ in the absence of spiciness (${\cal M}=0)$, in
the case where $\tau=\kappa_{S}/\kappa_{T}<1$ characteristic of salty water.

####  3.2.1 Diffusive convection: $\tau<R_{\rho}<1$

In that case, $\varepsilon_{p}$ is negative only if ${\cal F}>0$, in which
case the diapycnal component of the buoyancy flux is associated with up-
gradient diffusion, as shown by Middleton & Taylor (2020). Note that the range
$\tau<R_{\rho}<1$ is associated with diffusive convection instability only for
inviscid flows. In reality, stability analysis shows that viscosity restricts
the range of instabilities to the range

$\frac{Pr+\tau}{Pr+1}<R_{\rho}<1,$ (29)

where $Pr=\nu/\kappa_{T}$ is the Prandtl number, with $\nu$ the molecular
viscosity. For typical oceanic conditions, $\tau\approx 0.01$ and ${\rm
Pr}\approx 7$, yields the restricted range $0.876<R_{\rho}<1$.

####  3.2.2 ‘Salt’ finger instability: $R_{\rho}>1$

According to classical linear stability analysis, salt fingers are expected to
develop in the density ratio range

$1<R_{\rho}<\frac{1}{\tau},$ (30)

e.g. Radko (2013). In that range, Table 1 shows that APE dissipation can only
be negative if ${\cal F}<0$, and hence that the appropriate expression for
$\varepsilon_{p}$ is

$\varepsilon_{p}=\kappa_{T}N_{0}^{2}\left(|\nabla z_{r}|^{2}-\frac{\partial
z_{r}}{\partial
z}\right)\left(\frac{R_{\rho}-\tau}{R_{\rho}-1}\right)=\kappa_{T}N_{0}^{2}{\cal
F}\left(\frac{R_{\rho}-\tau}{R_{\rho}-1}\right).$ (31)

Eq. (31) shows that negative APE dissipation in that case corresponds to the
amplification of the negative APE dissipation in a single component fluid
multiplied by the following amplification factor

$\frac{R_{\rho}-\tau}{R_{\rho}-1}>1.$ (32)

Since the time scale for the growth of an unstable perturbation is a priori
inversely proportional to $\varepsilon_{p}$, the expectation is that the same
perturbations in a single component and double diffusive fluid will grow
faster in the former than the latter. A key difference, however, is that in a
double diffusive fluid, diffusive instabilities are able to grow
spontaneously, whereas in a single component fluid their development requires
initiation by means of external mechanical stirring of the right kind.

Since ${\cal F}$ can only be negative for laminar-like conditions, it follows
that density-compensated thermohaline variations need to develop in order for
the APE dissipation to remain negative as the flow transitions to turbulence.
As it happens, this is precisely what appears to happen in the salt-finger
experiments of Middleton & Taylor (2020), and as was previously advocated by
Merryfield (2000).

Figure 1: The amplification factor $f(R_{\rho},\tau,{\cal M})$ (given by Eq.
(34)) as a function of the spiciness parameter ${\cal M}$ and density ratio
$R_{\rho}$, for $\tau=0.01$

###  3.3 Impact of spiciness anomalies on diffusive instability

Density-compensated thermohaline variations are a characteristic feature of
ocean stratification in all regions and at all depths (Tailleux, 2021).
Consequently, spiciness and a non-zero value of ${\cal M}$ are likely to be
representative of the general situation. Assuming a prevalence of the
turbulent regime, we can make the approximation $1-\partial_{z}z_{r}/|\nabla
z_{r}|^{2}\approx 1$, and the relevant expression for $\varepsilon_{p}$ in the
oceanic case appears as follows:

$\varepsilon_{p}={\color[rgb]{0,0,0}K_{v0}}N_{0}^{2}(z_{r})\left[\frac{R_{\rho}-\tau}{R_{\rho}-1}+(1-\tau){\cal
M}\right].$ (33)

This equation shows that the single-component APE dissipation rate
$\varepsilon_{p}^{std}={\color[rgb]{0,0,0}K_{v0}}N_{0}^{2}(z_{r})$ is
modulated by the factor:

$f(R_{\rho},\tau,{\cal M})=\frac{R_{\rho}-\tau}{R_{\rho}-1}+(1-\tau){\cal M}.$
(34)

To illustrate the effect of this factor, Figure 1 depicts
$f(R_{\rho},\tau,{\cal M})$ as a function of $R_{\rho}$ and ${\cal M}$ for
$\tau=\kappa_{S}/\kappa_{T}=0.01$. A notable difference from the spiceless
case is the possibility of negative $\varepsilon_{p}$ for a wide range of
stratification, depending on the values of ${\cal M}$:

${\cal M}<-\frac{1}{1-\tau}\frac{R_{\rho}-\tau}{R_{\rho}-1}.$ (35)

This opens up various scenarios, ranging from complete inhibition of mixing
($f=0$) to enhanced mixing ($f>1$) and even diffusive instability ($f<0$). In
the turbulent case, the value of ${\cal M}$ approximates to:

${\cal M}=\frac{g}{2\rho_{\star}N_{0}^{2}}\frac{(\nabla
z_{r}\cdot\nabla\xi-\partial_{z}\xi)}{|\nabla
z_{r}|^{2}-\partial_{z}z_{r}}\approx\frac{g}{2\rho_{\star}N_{0}^{2}}\frac{\nabla
z_{r}\cdot\nabla\xi}{|\nabla
z_{r}|^{2}}=-\frac{\nabla\xi\cdot\nabla\rho}{2|\nabla\rho|^{2}}.$ (36)

This reveals that for ${\cal M}$ to be negative, spiciness and density
gradients need to be positively correlated. Exploring the possibility and
magnitude of such a situation requires a way to express in terms of the large-
scale gradients of $\xi$ and $\rho$, which warrants further investigation
beyond the scope of this study.

##  4 Comparison of APE frameworks

Holliday & McIntyre (1981) and Andrews (1981) had already shown the
possibility of building Lorenz APE theory from a local principle by the time
W95 published their study in the mid-1990s. However, the local APE framework
was still rudimentary and had not been used for any concrete applications.
This situation has improved in the recent years, as the local APE framework
has progressed rapidly and achieved full maturity, being able to deal with a
fully compressible multi-component stratified fluid (Tailleux, 2018). In this
paper, we prefer the local APE framework to the global APE framework, because
it is objectively simpler and more physical, even though this is not
necessarily widely appreciated or acknowledged. The next subsection intends to
reveal some of the unphysical aspects of the global APE framework that are
commonly overlooked, but easily uncovered by the local APE framework.

We first start by reviewing the main resuls of the global APE framework as
first derived by W95. Following Lorenz (1955), W95 defined the APE of a fluid
as the potential energy of the actual state minus that of the adiabatically
sorted state

$E_{a}^{W95}=\int_{V}\rho gz\,{\rm d}V-\int_{V}\rho_{r}gz_{r}\,{\rm
d}V=\int_{V}\rho g(z-z_{r})\,{\rm d}V$ (37)

and showed that its budget equation may be written in the form

$\frac{dE_{a}^{W95}}{dt}=\underbrace{\int_{V}\rho gw\,{\rm
d}V}_{\Phi_{z}}\,\,\underbrace{-g\oint_{S}(z\rho-\psi){\bf v}\cdot{\bf n}{\rm
d}S}_{S_{APE}^{adv}}+\underbrace{\kappa g\oint_{S}(z-z_{r})\nabla\rho\cdot{\bf
n}\,{\rm d}S}_{S_{APE}^{diff}}-(\Phi_{d}-\Phi_{i})$ (38)

where $\Phi_{z}$ represents the conversion with kinetic energy,
$S_{ADV}^{adv}$ represents the advective flux of APE through the boundaries
enclosing the control volume considered, $S_{APE}^{dif}$ represents the
diffusive flux of APE through the same boundaries. The expressions for
$\Phi_{i}$ and $\Phi_{d}$ are given by

$\Phi_{i}=-\kappa gA(\overline{\rho}_{top}-\overline{\rho}_{bottom})$ (39)
$\Phi_{d}=-\kappa g\int_{V}\frac{\partial
z_{r}}{\partial\rho}|\nabla\rho|^{2}\,{\rm d}V$ (40)

where $A$ is the cross-section of the domain, assumed constant, the overbar
denoting surface-average, while the quantity $\psi$ entering the expression of
the advective flux is given by

$\psi=\int^{\rho}z_{r}(\tilde{\rho},t)\,{\rm d}\tilde{\rho}.$ (41)

For the most part, the derivation of (38) is straightforward, except for the
demonstration that

$\int_{V}\rho\frac{\partial z_{r}}{\partial t}\,{\rm d}V=0.$ (42)

which is non-trivial. A key point of this paper concerns the proper
interpretation of $\Phi_{i}$ and $\Phi_{d}$. Following W95, $\Phi_{d}$ and
$\Phi_{i}$ have been interpreted as representing two different types of energy
conversion, with BPE and IE respectively. The need for revisiting this
interpretation is explained below.

The global APE budget (38) is now compared with that obtained by integrating
the local APE density budget. This is done here for a two-constituent fluid
with time-dependent Lorenz reference state, for which the local APE density
takes the form

$e_{a}=e_{a}(S,\theta,z,t)=-\int_{z_{r}}^{z}b(S,\theta,\tilde{z},t)\,{\rm
d}\tilde{z}=\frac{g}{\rho_{\star}}\int_{z_{r}}^{z}[\rho(S,\theta)-\rho_{0}(\tilde{z},t)]\,{\rm
d}\tilde{z},$ (43)

with the global APE being now defined as the volume-integrated APE density

$E_{a}=\int_{V}\rho_{\star}e_{a}\,{\rm d}V.$ (44)

Because (43) implies that $\rho_{\star}e_{a}=\rho
g(z-z_{r})+p_{0}(z,t)-p_{0}(z_{r},t)$, it follows that $E_{a}$ can be regarded
as the sum of $E_{a}^{W95}$ plus an additional term related to the work
against the vertical reference pressure gradient as follows:

$E_{a}=E_{a}^{W95}+\int_{V}[p_{0}(z,t)-p_{0}(z_{r},t)]\,{\rm d}V.$ (45)

Eq. (45) shows that the validity of the global definition of APE $E_{a}^{W95}$
requires the second term pertaining to the work against the reference pressure
gradient to vanish. This is easily verified to be the case if the volume $V$
enclosing the fluid remains unchanged under the isochoric adiabatic re-
arrangement ${\bf x}\rightarrow{\bf x}_{r}$ used to construct $\rho_{0}(z,t)$,
as then $\partial(x,y,z)/\partial(x_{r},y_{r},z_{r})=1$, ${\rm d}V={\rm
d}V_{r}$, $V_{r}=V$, thus ensuring that

$\int_{V}p_{0}(z_{r},t)\,{\rm d}V=\int_{V_{r}}p_{0}(z_{r})\,{\rm
d}V_{r}=\int_{V}p_{0}(z,t)\,{\rm d}V.$ (46)

as requested, e.g., Roullet & Klein (2009), Tailleux (2013b) and Winters &
Barkan (2013) for instance. However, the identity $E_{a}=E_{a}^{W95}$ is not
expected to necessarily hold for more complex situations, although the precise
circumstances under which this happens have yet to be fully understood and
discussed in the literature.

For a time-dependent Lorenz reference state, the local APE budget (15) derived
earlier needs is easily shown to be given by

$\frac{\partial e_{a}}{\partial t}=-{\bf v}\cdot\nabla
e_{a}-bw-\nabla\cdot{\bf
J}_{a}-\varepsilon_{p}-\frac{g}{\rho_{\star}}\int_{z_{r}}^{z}\frac{\partial\rho_{0}}{\partial
t}(\tilde{z},t)\,{\rm d}\tilde{z}$ (47)

where ${\bf J}_{a}=-(z-z_{r}){\bf J}_{b}$ and $\varepsilon_{p}={\bf
J}_{b}\cdot\nabla(z-z_{r})$ as before. By taking the volume integral of (47),
the evolution equation for the global APE $E_{a}$ is easily shown to be

$\frac{dE_{a}}{dt}=\underbrace{-\int_{V}\rho_{\star}bw\,{\rm
d}V}_{-\Phi_{z}}\underbrace{-\oint_{S}\rho_{\star}e_{a}{\bf v}\cdot{\bf
n}dS}_{S_{APE}^{adv}}+\underbrace{\oint_{S}\rho_{\star}(z-z_{r}){\bf
J}_{b}\cdot{\bf n}\,{\rm
d}S}_{S_{APE}^{diff}}-\underbrace{\int_{V}\rho_{\star}\varepsilon_{p}\,{\rm
d}V}_{\Phi_{d}-\Phi_{i}},$ (48)

where the expressions for $\Phi_{i}$ and $\Phi_{d}$ are

$\begin{split}\Phi_{i}=&-\int_{V}\rho_{\star}\varepsilon_{p,lam}=\rho_{\star}gA\left[\alpha\kappa_{T}\Delta\theta-\beta\kappa_{S}\Delta
S\right]\\\ =&\rho_{\star}gA\kappa_{T}\beta\Delta
S\left[\frac{\alpha\Delta\theta}{\beta\Delta
S}-\frac{\kappa_{S}}{\kappa_{T}}\right]=\rho_{\star}gA\kappa_{T}\beta\Delta
S\left(R_{\rho}^{bulk}-\tau\right)\end{split}$ (49)
$\Phi_{d}=\int_{V}\rho_{\star}\varepsilon_{p,tur}=\int_{V}g(\alpha\kappa_{T}\nabla\theta-\beta\kappa_{S}\nabla
S)\cdot\nabla z_{r}\rho_{\star}\,{\rm d}V$ (50)

where $\Delta\theta=\overline{\theta}_{top}-\overline{\theta}_{bottom}$ and
$\Delta S=\overline{S}_{top}-\overline{S}_{bottom}$, while $R_{\rho}^{bulk}$
is a bulk density ratio. For canonical salt finger instability,
$1<R_{\rho}^{bulk}<1/\tau$, in which case (49) predicts $\Phi_{i}>0$ as
required to act as a source of APE. Note that the last term in (47) may be
written in the form $F(z,t)-F(z_{r},t)$, with $F$ such that $\partial
F/\partial z(z,t)=\partial\rho_{0}/\partial t(z,t)$. As a result, its volume
integral must vanish whenever the second term of (45) also vanishes for the
same reason, as explained above.

Winters & Barkan (2013) assumed that when $E_{a}$ equals $E_{a}^{W95}$, their
respective budgets should be equivalent and described by the equations for an
open domain derived previously by Winters et al. (1995). This view is clearly
incorrect, however, because the budgets of $E_{a}$ and $E_{a}^{W95}$ are
closely related to the local budgets of their integrands, but these are not
equivalent as established above. Physically, the non-equivalence of the
budgets for $E_{a}$ and $E_{a}^{WP5}$ is due to the integrand of $E_{a}^{W95}$
lacking a vital piece of information about the energetics of stratified fluids
compared to $E_{a}$, namely the fact that when a fluid parcel moves from its
reference position $z_{r}$ to its actual position, it not only has to work
against gravity, but also against the vertical reference pressure gradient.
More generally, Tailleux (2013b) shows that work against buoyancy forces
actually involves three terms: 1) one associated with changes in gravitational
potential energy; 2) one associated with changes in internal energy; 3) one
associated with work against the reference pressure gradient. Even if the
latter term integrates to zero over the whole fluid domain, it is still
necessary to retain it for correctly predicting the advective flux of APE
through the boundaries of open domains. Two key differences characterise the
non-equivalence of the budgets for $E_{a}$ and $E_{a}^{W95}$: 1) in W95, the
reversible conversion with kinetic energy $\Phi_{z}$ is associated with the
density flux $\rho gw$ but with the buoyancy flux $-bw$ in the local APE
framework; 2) in W95, the advective flux of APE through boundaries
$S_{APE}^{adv}$ is associated with the flux of the bizarre quantity
$g(z\rho-\psi)$ but with the flux of APE density $e_{a}$ in the local
framework, which makes much more sense. The unphysical character of the
advective flux of APE in W95 highlights the well known problematic character
of the global APE framework for regional studies, which was one of the key
motivations for the quest of a local available energy framework (Tailleux,
2013a).

The unphysical character of the advective flux of APE in Winters et al. (1995)
global APE budget for open domains is a problematic and can only be corrected
by using the local APE framework. The issue is vital, because recent studies
of local energetics have demonstrated the key importance of the lateral
advection of APE density for understanding important physical processes, such
as storm track dynamics (Novak & Tailleux, 2018; Federer et al., 2024),
tropical cyclone intensification (Harris et al., 2022), or the energetics of
the thermohaline circulation (Gregory & Tailleux, 2011; Sijp et al., 2012).
Importantly, it is worth noting that $E_{a}$ and $E_{a}^{W95}$ are also non-
equivalent numerically. Indeed, in the global framework, the APE of the fluid
is estimated as a small residual between two very large terms, which is ill-
conditioned and prone to large numerical errors. In the local framework,
however, the APE of a fluid is estimated as the sum of small non-negative
numbers, which is numerically robust and very accurate. As a result, the field
would greatly benefit from switching to the local APE framework, as the
continued use of the global APE framework only serves to perpetuate generally
unsound and inferior practices of limited usefulness.

##  5 Reference state variables

The practical implementation of the above theoretical results require the
computation of Lorenz reference density profile $\rho_{0}(z,t)$, as well as of
the associated isopycnal temperature and salinity averages necessary for
defining spiciness. Here, we provide the mathematical foundations for defining
the latter in terms of probability density functions (p.d.f.), which allows
one to compute these in practice without the use of sorting. These results
pertain to arbitrary domains and unify the p.d.f approaches of heng Tseng &
Ferziger (2001) and Saenz et al. (2015), while also making use of previous
ideas by Hochet & Tailleux (2019) and Hochet et al. (2021).

###  5.1 Reference position $z_{r}(\rho,t)$ and density profile
$\rho_{0}(z,t)$

The classical interpretation of Lorenz reference state as an isochoric re-
arrangement of fluid parcels is easy to understand and implement
algorithmically. However, it does not easily lend itself to mathematical
analysis. Here, we derive expressions for defining and constructing Lorenz
reference density profile $\rho_{0}(z,t)$ and its inverse function
$z_{r}(\rho,t)$ [such that $\rho_{0}(z_{r}(\rho,t),t)=\rho$ at all times] in a
way that greatly facilitates the study of its mathematical properties. Our
approach is not entirely new, as elements of it can be found in Winters et al.
(1995) or in the p.d.f. approaches of heng Tseng & Ferziger (2001), Saenz et
al. (2015) or Hochet & Tailleux (2019); Hochet et al. (2021), but is useful to
introduce the approaches discussed next. To that end, let us assume that the
domain containing the fluid is bounded at the top by a rigid lid at $z=0$ and
at its bottom by a spatially varying bottom topography at $z=-H(x,y)$. We
denote by $\hat{V}(z)$ the volume between an arbitrary depth $z$ and the
surface, and by $A(z)=-{\rm d}V/{\rm d}z(z)$ the cross-sectional area of the
domain at the depth $z$.

Figure 2: Mapping of physical space (a) onto reference state space (b).

Physically, the reference depth $z=z_{r}(\rho,t)$ of a fluid parcel must be
such that the volume of water between $z_{r}(\rho,t)$ and the surfafe, viz.,

$\hat{V}(z_{r})=\int_{z_{r}}^{0}A(z)\,{\rm d}z$ (51)

must equal the volume of water in physical space made up of all the fluid
parcels of densities less than $\rho$, viz.,

$V(\rho,t)=\int_{V(\rho({\bf x}^{\prime},t)\leq\rho)}{\rm
d}V^{\prime}=\int_{V}H(\rho-\rho({\bf x}^{\prime},t))\,{\rm d}V^{\prime}$ (52)

where $H$ denotes the Heaviside step function defined as in Winters et al.
(1995),

$H(\rho({\bf x},t)-\rho({\bf
x}_{0},t))=\left\\{\begin{array}[]{ll}0,&\rho({\bf x},t)<\rho({\bf
x}_{0},t)\\\\[2.0pt] \frac{1}{2},&\rho({\bf x},t)=\rho({\bf
x}_{0},t)\\\\[2.0pt] 1,&\rho({\bf x},t)>\rho({\bf
x}_{0},t)\end{array}\right..$

Equating the two volumes $\hat{V}(z_{r})=V(\rho,t)$ with $V(\rho,t)$ yields an
implicit equation for $z_{r}(\rho,t)$, which may be inverted to yield

$z_{r}(\rho,t)=\hat{V}^{-1}[V(\rho,t)],$ (53)

which in the case where $A(z)={\rm constant}$ reduces to

$z_{r}(\rho,t)=-\frac{V(\rho,t)}{A}$ (54)

as previously derived by Winters et al. (1995). As an illustration, Fig. 2 (a)
shows the constant Lorenz reference depth $z_{r}(\rho,t)=-1000\,{\rm m}$ for a
realistic ocean, with panel (b) showing the corresponding position in
reference state space. Construction of Lorenz reference state for a realistic
ocean has been discussed in Saenz et al. (2015); Tailleux (2016, 2021). Once
$z_{r}(\rho,t)$ has been determined for a number of target densities, the
reference density profile $\rho_{0}(z,t)$ may be obtained simply by solving
the implicit relation $\rho_{0}(z_{r}(\rho,t),t)=\rho$ at a number of discrete
reference depths, from which an interpolator may be constructed to infer the
reference density for other values. Alternatively, using the fact that
$\rho=\rho_{0}(z_{r},t)$ by definition, one may also solve the equation
$\hat{V}(z_{r})=V(\rho,t)$ in the form $\hat{V}(z_{r})=V(\rho_{0}(z_{r},t),t)$
to infer $\rho_{0}(z_{r},t)$ at a number of predetermined target reference
depths.

Note that by differentiating the equality
$\hat{V}(z_{r})=V(\rho,t)=V(\rho_{0}(z_{r},t),t)$ with respect to $z_{r}$, the
following useful expression for $\partial\rho_{0}/\partial z_{r}$ may be
obtained,

$\frac{\partial\rho_{0}}{\partial z}(z_{r},t)=-\left(\frac{\partial
V}{\partial\rho}\right)^{-1}A(z_{r}).$ (55)

The above expressions establish that the practical computation of
$z_{r}(\rho,t)$ and $\rho_{0}(z,t)$ can be achieved without resorting to a
sorting algorithm. Indeed, all one has to do is to define a number of target
densities $\rho_{i},{i=1,\cdots,N}$ at which to compute the corresponding
volumes $V(\rho_{i},t)$. Having defined the inverse function
$\hat{V}^{-1}(z)$, one may simply compute the corresponding target reference
depths $z_{r}(\rho_{i},t)$ at which $\rho_{0}(z,t)$ may be evaluated. Once all
quantities have been determined at the discrete target depths, interpolators
can be constructed to find values at other points.

###  5.2 Construction of isopycnal mean profiles

Physically, the isopycnally-averaged temperature and salinity profiles
$\theta_{0}(z,t)$ and $S_{0}(z,t)$ needed to isolate the passive spice
component need to be obtained as thickness-weighted isopycnal averages, as
illustrated in Fig. 3, as this is the only physically meaningful way to define
an isopycnal average (Young, 2012). Mathematically, $\theta_{0}(z,t)$ can be
defined as a solution of the following equation:

$\int_{z_{r}(\rho,t)}^{0}A(z)\theta_{0}(z,t)\,{\rm
d}z=\underbrace{\int_{V(\rho,t)}\theta({\bf x},t)\,{\rm d}V}_{H(\rho,t)}.$
(56)

Physically, the right-hand side of (56) represents the volume integrated
temperature over the body of water made up of fluid parcels with densities
less than $\rho$, which we denote $H(\rho,t)$. The right-hand side is the
volume integral of the thickness-weighted isopycnally averaged temperature
between $z_{r}(\rho,t)$ and the surface. An explicit expression for
$\theta_{0}(z,t)$ at $z=z_{r}$ may be obtained by differentiating (56) with
respect to $z_{r}$, which leads to

$-A(z_{r})\theta_{0}(z_{r},t)=\frac{\partial
H}{\partial\rho}\frac{\partial\rho_{0}}{\partial
z_{r}}(z_{r},t)\qquad\Rightarrow\qquad\theta_{0}(z_{r},t)=\left(\frac{\partial
V}{\partial\rho}\right)^{-1}\frac{\partial H}{\partial\rho}$ (57)

which made use of (55).

Figure 3: Schematics of an isopycnal surface (dashed-red line) bounded by two
nearby isopycnal surfaces (thick black lines) defining the volume of fluid
used to define the thickness-weighted average value of any arbitrary tracer
for the isopycanl surface considered.

Figure 4: (a) Latitude/depth section of Conservative Temperature along
$30^{\circ}W$ in the Atlantic ocean; (b) Global isopycnal mean (red line)
profile used for the spice/heave decomposition of Conservative Temperature
shown in (c) and (d). For comparison, horizontal mean profile (black line) is
also depicted. (c) Active heave component of Conservative Temperature along
same $30^{\circ}W$ section as in (a). (d) Passive spice component of
Conservative Temperature along same $30^{\circ}W$ section as in (a). Black
contour lines represent constant Lorenz reference density surfaces labelled in
terms of thermodynamic neutral density $\gamma^{T}$.

Eq. (57) is easily implemented in practice, at least for the type of data
considered here. As an illustration, Fig. 4 shows a decomposition of
temperature into active and passive components. Specifically, Fig. 4 (a) shows
a meridional section of Conservative Temperature along $30^{\circ}W$ in the
Atlantic Ocean. Panel (b) shows the thickness-weighted isopycnal temperature
average profile compared with the more standard horizontal mean profile. The
thickness-weighted averaged temperature is illustrated in Panel (c), while
Panel (d) illustrates the passive spice component, obtained as
$\theta_{spice}=\theta-\theta_{0}(z_{r},t)$. Our approach to defining spice,
based on Tailleux (2021), is strongly supported by the fact that the patterns
in panel (d) all appear to coincide with known ocean water massses in the
Atlantic Ocean, namely Antarctic Bottom Water (AABW), North Atlantic Deep
Water (NADW), Antarctic Intermediate Water (AAIW), and Mediterranean
Intermediate Water (MIW).

###  5.3 Evolution of the reference density profile

By definition, $\rho_{0}(z,t)$ must coincide with its thickness-weighted
isopycnal average and therefore satisfy (56), viz.,

$\int_{z_{r}}^{0}A(z)\rho_{0}(z,t)\,{\rm
d}z=\underbrace{\int_{V(\rho,t)}\rho({\bf x}^{\prime},t)\,{\rm
d}V^{\prime}}_{R(\rho,t)}.$ (58)

We now show that this equation can be manipulated to yield an exact expression
for the evolution equation satisfied by $\rho_{0}(z,t)$ as as well as for the
effective diffusivity first introduced by Winters & d’Asaro (1996) and
Nakamura (1996). For simplicity, we focus only on the case where the buoyancy
flux vanishes through all solid boundaries as well as the surface, but the
derivation can easily be extended to account for a non-zero surface buoyancy
flux.

To obtain the sought-for evolution equation for $\rho_{0}$, let us take the
time derivative of (58), which yields

$\int_{z_{r}}^{0}A(z)\frac{\partial\rho_{0}}{\partial t}(z,t)\,{\rm
d}z=\int_{V(\rho,t)}\frac{\partial\rho}{\partial t}({\bf x}^{\prime},t)\,{\rm
d}V^{\prime}.$ (59)

Now, using the fact that the evolution equation for density may be written as

$\frac{\partial\rho}{\partial t}=-{\bf
v}\cdot\nabla\rho+\nabla\cdot(\rho_{\star}{\bf J}_{b}/g),$ (60)

(59) may be rewritten as follows

$\begin{split}\int_{z_{r}}^{0}A(z)\frac{\partial\rho_{0}}{\partial
t}(z,t)\,{\rm d}z=&\int_{V(\rho,t)}[\nabla\cdot(\rho_{\star}{\bf
J}_{b}/g)-{\bf v}\cdot\nabla\rho]\,{\rm d}V^{\prime}\\\
=&\oint_{S_{\rho}}(\rho_{\star}/g){\bf J}_{b}\cdot{\bf n}\,{\rm
d}S=\oint_{S_{\rho}}(\rho_{\star}/g){\bf
J}_{b}\cdot\frac{\nabla\rho}{|\nabla\rho|}\,{\rm d}S\\\
=&\frac{\partial\rho_{0}}{\partial
z}(z_{r},t)\oint_{S_{\rho}}(\rho_{\star}/g)\frac{{\bf J}_{b}\cdot\nabla
z_{r}}{|\nabla\rho|}\,{\rm d}S\\\ =&-\frac{\partial\rho_{0}}{\partial
z}(z_{r},t)\oint_{S_{\rho}}(\rho_{\star}/g)\frac{\varepsilon_{p,tur}}{|\nabla\rho|}\,{\rm
d}S.\end{split}$ (61)

In the second line, $S_{\rho}$ refers to the isopycnal surface $\rho={\rm
constant}$, meaning that the outward normal unit vector ${\bf
n}=\nabla\rho/|\nabla\rho|=-\nabla z_{r}/|\nabla z_{r}|$ as schematically
illustrated in Fig. 2 (a). To go from the second to third line we used the
fact that as $\rho=\rho_{0}(z_{r},t)$, we have
$\nabla\rho=\partial\rho_{0}/\partial z_{r}(z_{r},t)\nabla z_{r}$. To go from
the third to fourth line, we used the fact that by definition ${\bf
J}_{b}\cdot\nabla z_{r}=-\varepsilon_{p,tur}$.

By defining the reference squared buoyancy profile as

$N_{0}^{2}(z_{r},t)=-\frac{g}{\rho_{\star}}\frac{\partial\rho_{0}}{\partial
z}(z_{r},t)$ (62)

and the locally-defined effective diffusivity as

$\kappa_{\rm eff}=\frac{\varepsilon_{p,turb}}{N_{0}^{2}(z_{r})},$ (63)

(61) may be rewritten as

$\int_{z_{r}}^{0}A(z)\frac{\partial\rho_{0}}{\partial t}(z,t)\,{\rm
d}z=-A(z_{r})K_{\rm eff}\frac{\partial\rho_{0}}{\partial
z_{r}}(z_{r},t)=\frac{\partial R}{\partial t},$ (64)

where the globally defined effective diffusivity is related to the locally-
defined one by

$K_{\rm eff}=\frac{1}{A(z_{r})}\oint_{S_{\rho}}\frac{\kappa_{\rm eff}}{|\nabla
z_{r}|}\,{\rm d}S.$ (65)

as previously established by Hochet et al. (2019). The final step of the
derivation consists in differentiating (64) with respect to $z_{r}$, which
yields

$\frac{\partial\rho_{0}}{\partial
t}(z_{r},t)=\frac{1}{A(z_{r})}\frac{\partial}{\partial
z_{r}}\left[A(z_{r})K_{\rm eff}\frac{\partial\rho_{0}}{\partial
z_{r}}\right](z_{r},t).$ (66)

Note that in practice, $K_{\rm eff}$ can also be determined by inverting the
diffusion equation (66) from the knowledge of the temporal variations of the
sorted density profile $\rho_{0}(z,t)$. Alternatively, one may combine Eqs.
(55) and (64) to verify that the globally-defined effective diffusivity may be
written as

$K_{\rm eff}(z_{r},t)=\frac{1}{A^{2}(z_{r})}\frac{\partial
V}{\partial\rho}(\rho,t)\frac{\partial R}{\partial t}(\rho,t)$ (67)

which in principle allows one to calculate $K_{\rm eff}$ using simple binning
techniques. The above results establish that the effective diffusivity only
depends on the turbulent part $\varepsilon_{p,tur}$ of $\varepsilon_{p}$,
consistent with established results.

##  6 Summary, discussion, and conclusions

Double diffusive instabilities are mysterious as they can seemingly develop
from a state with zero initial mechanical energy, without the need for any
external mechanical source(s) of energy (except for that of the initial
perturbation triggering the instability). In the context of APE theory, such
instabilities are possible only if the mechanical energy KE+APE can grow at
the expense of BPE. Since viscous dissipation $\varepsilon_{k}$ is always
positive, this can only happen when the APE dissipation rate $\varepsilon_{p}$
is negative. Therefore, it is more logical and physically plausible to
consider the sign of the APE dissipation rate $\varepsilon_{p}={\bf
J}_{b}\cdot\nabla(z-z_{r})$, rather than the sign of the diapycnal component
of the buoyancy flux proposed by Middleton & Taylor (2020), as the most
fundamental criterion for characterising double diffusive instabilities. In
the context of Winters et al. (1995) framework, this is equivalent to
associating the development of double diffusive instabilities with the sign of
$\Phi_{d}-\Phi_{i}$ rather than that of $\Phi_{d}$ only. This is justified by
the fact that $\Phi_{d}$ is only negative for diffusive convection instability
but not for salt finger instability, while $\Phi_{d}-\Phi_{i}$ is negative for
both.

While our current findings unequivocally acknowledge the existence and
necessity of negative APE dissipation in single- and two-component stratified
fluids, the precise underlying mechanisms remain elusive. The controversy
primarily revolves on the nature of background potential energy (BPE) and of
the energy conversion type between APE and BPE. The primary challenge stems
from the fact that, in fully compressible fluids, APE dissipation naturally
appears as a conversion between APE and the internal energy component of BPE,
akin to viscous dissipation (Tailleux, 2009, 2013c). However, this
interpretation is not the most intuitive within standard Bousinesq models, as
the BPE lacks an explicit link to internal energy, thereby complicating our
understanding of its energetics. Nevertheless, the crucial role of internal
energy in understanding the energetics of turbulent stratified fluids is
underscored by the term $gzD\rho/Dt$, which approximates the compressible work
$pD\upsilon/Dt$ in Boussinesq models (Young, 2010; Tailleux, 2012), implying a
conversion with internal energy. While some may find considering
thermodynamics as central to the understanding of the energetics of turbulent
stratified fluids unpalatable, it is notheworthy that, in the current state of
knowledge, there does not appear to be any other way to explain the occurrence
of negative APE dissipation in single-component fluids or doubly stable two-
component fluids. Indeed, existing explanations for the release of BPE into
APE by Smyth et al. (2005) and Ma & Peltier (2021) have linked it to the ‘APE’
associated with a destabilising stratifying tracer, but in addition to lacking
rigorous physical justification, this explanation clearly does not work for
simple or doubly stable stratified fluids. On the other hand, exergy theory in
thermodynamics offers a potentially more promising avenue. Exergy theory
posits that because internal energy is convex with respect to its canonical
variables (specific entropy, salinity, and specific volume for two-component
seawater), a portion of it — referred to as exergy, proportional to the
deviation from thermodynamic equilibrium — is available for conversions into
work (Tailleux, 2013a; Bannon, 2013; Bannon & Najjar, 2014) under diabatic
transformation, regardless of whether the stratifying tracers are stabilising
or destabilising. For single-component fluids, exergy is directly proportional
to temperature variance at leading order, providing a means to identify and
quantify the portion of BPE that can be converted into APE in simple and
doubly diffusive flows. However, this approach necessitates accepting APE
dissipation as a conversion between APE and the internal energy component of
BPE, thereby challenging the prevailing assumption. Pursuing this line of
enquiry is beyond the scope of this paper and is deferred to future studies.
We anticipate that this will require the development of a new class of
Boussinesq models with explicit and thermodynamically consistent internal
energy. The development of such a model, building upon the recent findings of
Tailleux & Dubos (2024), is currently under way and will be presented in a
subsequent study.
In this study, the local APE framework was preferred over Winters et al.
(1995) global APE framework owing to its greater simplicity, physical
justification, and flexibility. Despite its popularity, the global APE
framework has several unphysical aspects that can sometimes lead to erroneous
conclusions and/or interpretations. Examples include its unsuitability for
regional studies in open domains due to the unphysical character of its
advective fluxes, and the inability to meaningfully decompose the sign-
indefinite integrand $\rho g(z-z_{r})$ into mean and eddy components. These
issues, which tend to be overlooked in the turbulent mixing community, can
only be corrected by using the local APE framework. While the non-intuitive
character of the early formulations of the local APE theory by Andrews (1981),
Holliday & McIntyre (1981), and Shepherd (1993) made it somewhat difficult to
use or apply, the most recent formulations have greatly clarified it and
several applications have since demonstrated its usefulness and ease of use,
e.g., Roullet & Klein (2009); Zemskova et al. (2015); Novak & Tailleux (2018);
Harris & Tailleux (2018). Given that the local APE framework has made the
global APE framework largely obsolete, there is no longer any real
justification for continuing using the global framework, especially as it is
obviously unsuitable for the study of individual turbulent mixing events in
large domains such as that pertaining to the oceans. We hope that the
simplicity of our derivations, which we believe to be more intuitive and
transparent than those of Scotti & White (2014), will convince some readers to
switch frameworks.
For a single-component fluid, the Boussinesq APE dissipation rate
$\varepsilon_{p}={\bf J}_{b}\cdot\nabla(z-z_{r})$ can be shown to approximate
the following quantity

$\rho\varepsilon_{p}=\kappa c_{p}\nabla
T\cdot\nabla\left(\frac{T-T_{r}}{T}\right)$ (68)

(Tailleux, 2009, 2013c). Eq. (68) features a Carnot-like thermodynamic
efficiency $(T-T_{r})/T$ underscoring the inherent thermodynamic character of
turbulent stratified mixing and APE dissipation, which is hidden by the
Boussinesq approximation. Tailleux (2013c) demonstrated that (68) could be
broken down into three parts (see Appendix A for a simpler and clearer
derivation of the result): a laminar component $\varepsilon_{p,lam}$ related
to the term $-\Phi_{i}$ in W95; a turbulent term $\varepsilon_{p,turb}$
related to the term $\Phi_{d}$ in W95; and a term $\varepsilon_{p,eos}$ that
arises from the nonlinearities of the equation of state and has no equivalent
in the standard Boussinesq model and W95. As far as we understand the issue,
these components are indissociable parts of $\varepsilon_{p}$ and therefore
necessarily of the same type, thus implying that $\Phi_{d}$ and $\Phi_{i}$
should both be interpreted as conversions between APE and internal energy, in
contrast to the prevailing interpretation. In the literature, $\Phi_{i}$ or
$\varepsilon_{p,lam}$ are often neglected based on the assumption that they
are negligible in fully turbulent flows. However, this may not necessarily be
the case, as $\Phi_{i}$ appears to be a significant term in many published
studies. For example, $\Phi_{i}$ is comparable to $\Phi_{d}$ in the
simulations discussed by W95. The present theory predicts that $\Phi_{i}$
should also be dominant in the early stages of the canonical salt finger
instability experiment discussed by Middleton & Taylor (2020); it is therefore
unfortunate that the authors did not diagnose $\Phi_{i}$ in their experiments,
as this would undoubtedly have been useful to demonstrate its fundamental
importance. We hope to illustrate this behaviour in the future by using an
energy complete Boussinesq approximation currently under development and based
on the findings of Tailleux & Dubos (2024).

Like Middleton & Taylor (2020), we find that three physical parameters are
sufficient to fully describe the behaviour of $\Phi_{d}$ or $\varepsilon_{p}$,
but apart from the diffusivity ratio $\kappa_{S}/\kappa_{T}$, how to chose the
two remaining parameters is not unique. The parameters chosen by Middleton &
Taylor (2020) are strongly affected by turbulence and the micro-structures of
$\theta$ and $S$, so diagnosing them from observational data or environmental
parameters, or even linking them to parameters used in previous studies (such
as the density ratio), is not straightforward. To overcome some of these
difficulties, Middleton et al. (2021) subsequently reformulated the problem in
terms of density/spiciness variable, using a linear spiciness variable to
achieve a more physical and useful characterisation of $\Phi_{d}$. However,
this spiciness variable is arguably suboptimal for measuring spiciness because
it is not properly calibrated to vanish in a spiceless fluid. For this reason,
Tailleux (2021) recommended that spiciness be defined as an isopycnal anomaly
instead. On this basis, we propsoed a new set of physical parameters that we
used to characterise $\varepsilon_{p}$: a density ratio and a new spiciness
parameter more in line with the parameters used in previous studies. However,
estimating these parameters in numerical simulations or in the field requires
the computation of thickness-weighted isopycnal averages, which most non-
oceanographers are likely to be unfamiliar with. For this reason and to
facilitate the adoption of our ideas, we provided explicit mathematical
expressions and practical examples to show how to implement this form of
averaging in practice, valid even for a fully compressible and nonlinear
ocean, and illustrated it on a realistic oceanographic example with real data.

While the presence of double diffusive instabilities is determined by the sign
of the full APE dissipation rate
$\varepsilon_{p}=\varepsilon_{p,lam}+\varepsilon_{p,tur}$, the effective
diapycnal diffusivity controlling the evolution of the sorted density profile
is only determined by the turbulent part $\varepsilon_{p,tur}$. Consequently,
the conventional expressions for turbulent diapycnal diffusivity and
dissipation ratio should be formulated as $K_{v}=\varepsilon_{p,turb}/N^{2}$
and
$\Gamma=\varepsilon_{p}/\varepsilon_{k}=\varepsilon_{p,turb}/\varepsilon_{k}$
rather than in terms of the full $\varepsilon_{p}$. This is consistent with
the common practice, but it may not be necessarily obvious if one considers
that $\varepsilon_{p,lam}$ (equivalently $\Phi_{i}$) is part of the definition
of the net APE dissipation rate, which departs from the prevailing view that
only $\varepsilon_{p,turb}$ or $\Phi_{d}$ contributes to it. As a consequence,
not all double-diffusive instabilities can be expected to exhibit negative
effective diffusivities, which, for instance, are not present in the initial
stages of the canonical salt finger instability discussed by Middleton &
Taylor (2020). Our theory shows that double diffusion can enhance, reduce, or
even change the sign of $K_{v}$, whose consequences for the study of mixing in
the oceans need to be further investigated. The implications for the
theoretical understanding of thermohaline staircases is unclear, as recent
theories tend to emphasise the spatial variations (linked to the functional
dependence on some turbulent parameters such as the buoyancy Reynolds number)
of an otherwise positive effective diffusivity as the main causee, e.g., see
Radko (2005), Taylor & Zhou (2017) or Ma & Peltier (2022b, a) and references
therein for some recent discussion. Further research is clearly needed to
clarify the issue, but beyond the scope of this paper.

Declarations of interests. The author reports no conflict of interest.

##  Appendix A APE dissipation in compressible and Boussinesq fluids

In contrast to the concept of viscous dissipation, whose origin can be traced
back to the development of the Navier-Stokes equations in the early 19th
century, the concept of APE dissipation is comparatively much more recent, as
it only appears to have been developed in a somewhat ad-hoc way from a re-
scaling of the dissipation of temperature variance by Gargett & Holloway
(1984) and Oakey (1982). In particular, it was not obtained as part of a local
APE budget, even though the possibility to construct Lorenz (1955) APE theory
from a local principle had been established by Andrews (1981) and Holliday &
McIntyre (1981). Local balance equations for a local APE density was perhaps
first derived by Molemaker & McWilliams (2010). As far as we are aware, a
theoretical formulation of APE dissipation for a fully compressible fluid
clarifying how it relates to its Boussinesq counterpart was only developed
much more recently by Tailleux (2009) and Tailleux (2013c). Here, we revisit
our earlier derivation to provide a somewhat simpler treatment, in which
dependence of our expressions in terms of entropy is replaced by a dependence
on potential temperature. Concretely, the task here is to understand how to
relate the expression for the APE dissipation rate for a fully compressible
fluid

$\varepsilon_{p}=\kappa_{T}c_{p}\nabla
T\cdot\nabla\Upsilon=-\kappa_{T}c_{p}\nabla
T\cdot\nabla\left(\frac{T_{r}}{T}\right)$ (69)

to the terms $\Phi_{i}$ and $\Phi_{d}$ that appear in Winters et al. (1995)
global APE framework, where $T_{r}=T(\eta,p_{r})$ and $T=T(\eta,p)$ are the
in-situ temperature at the reference pressure $p_{r}$ and actual pressure $p$
respectively, with $\Upsilon=(T-T_{r})/T$ the thermodynamic efficiency.  As in
Tailleux (2013a), we take as our starting point the following thermodynamic
relation for the total differential of specific entropy $\eta$ regarded either
as a function of $(T,p)$ or $\theta$, viz.,

${\rm d}\eta=\frac{c_{p}}{T}{\rm d}T-\frac{\alpha}{\rho}{\rm
d}p=\frac{c_{p\theta}}{\theta}{\rm d}\theta,$ (70)

in which $\eta$ is the specific entropy, $\alpha$ is the thermal expansion
coefficient, $c_{p}$ the specific heat capacity at constant pressure,
$\theta=T(\eta,p_{a})$ the potential temperature, i.e., the temperature
reference to the mean surface atmospheric pressure $p_{a}$, and
$c_{p\theta}=c_{p}(\theta,p_{a})$. The thermodynamic relation (70) can be re-
arranged to yield an expression for the total differential of the logarithm of
the temperature $\ln{T}$, viz.,

$\frac{{\rm d}T}{T}=\frac{\alpha}{\rho c_{p}}{\rm
d}p+\frac{c_{p\theta}}{c_{p}}\frac{{\rm d}\theta}{\theta},$ (71)

thus allowing one to regard $T=T(\theta,p)$ as a function of potential
temperature and pressure. Because ${\rm dT}/T$ is an exact differential, Eq.
(71) implies the following Maxwell relationship (that is, the equality of the
cross derivatives):

$\frac{\partial}{\partial\theta}\left(\frac{\alpha}{\rho
c_{p}}\right)=\frac{c_{p\theta}}{\theta}\frac{\partial}{\partial
p}\left(\frac{1}{c_{p}}\right)$ (72)

which will prove useful in the following. For the cases of interest, the
reference temperature entering calculations are ‘potential temperature’ of
some kind referenced to an arbitrary pressure $p_{r}$. It follows that if we
denote $T_{r}=T(\theta,p_{r})$, then we may write

$\ln{\frac{T_{r}}{T}}=\int_{p}^{p_{r}}\frac{\alpha}{\rho
c_{p}}(\theta,\tilde{p})\,{\rm d}\tilde{p}$ (73)

Taking the gradient, we then obtain

$\frac{T}{T_{r}}\nabla\frac{T_{r}}{T}=-\frac{\alpha}{\rho
c_{p}}(\theta,p)\nabla
p+\left[\int_{p}^{p_{r}}\frac{\partial}{\partial\theta}\left(\frac{\alpha}{\rho
c_{p}}\right)(\theta,\tilde{p}){\rm d}\tilde{p}+\frac{\alpha}{\rho
c_{p}}(\theta,p_{r})\frac{dp_{r}}{d\theta}\right]\nabla\theta$ (74)

Using the above Maxwell relationship (72)

$\int_{p}^{p_{r}}\frac{\partial}{\partial\theta}\left(\frac{\alpha}{\rho
c_{p}}\right)(\theta,\tilde{p})\,{\rm
d}\tilde{p}=\frac{c_{p\theta}}{\theta}\int_{p}^{p_{r}}\frac{\partial}{\partial
p}\left(\frac{1}{c_{p}}\right)(\theta,\tilde{p})\,{\rm
d}\tilde{p}=\frac{c_{p\theta}}{\theta}\left(\frac{1}{c_{pr}}-\frac{1}{c_{p}}\right)$
(75)

so that (74) may be rewritten as follows

$\nabla\left(\frac{T_{r}}{T}\right)=-\frac{\alpha T_{r}}{\rho c_{p}T}\nabla
p+\frac{\alpha_{r}T_{r}}{\rho_{R}c_{pr}T}\nabla
p_{r}+\frac{c_{p\theta}T_{r}}{T\theta}\left(\frac{1}{c_{pr}}-\frac{1}{c_{p}}\right)\nabla\theta$
(76)

with $c_{pr}=c_{p}(\theta,p_{r})$, $\alpha_{r}=\alpha(\theta,p_{r})$,
$\rho_{r}=\rho(\theta,p_{r})$. Note here that the notation $Q=Q(\theta,p)$ is
an abuse of notations for the function $Q=Q(T(\theta,p),p)$, i.e., it is
computed from the expressions in terms of in-situ temperature and pressure in
which $T$ is replaced by its expression as a function of potential temperature
and pressure.  The above results may be used to express the non-viscous
nonconservation term $\varepsilon_{A}$ as the sum

$\varepsilon_{p}=\varepsilon_{p,lam}+\varepsilon_{p,tur}+\varepsilon_{p,eos}$
(77)

Moreover, if one use the fact that $p_{r}=p_{0}(z_{r}(\theta))$ so that
$\nabla p_{r}=-\rho_{r}gdz_{r}/d\theta\nabla\theta$, one may write

$\varepsilon_{p,lam}=\frac{\alpha T_{r}}{\rho T}\kappa_{T}\nabla T\cdot\nabla
p$ (78)
$\varepsilon_{p,tur}=\frac{c_{p}T_{r}\alpha_{r}g}{c_{pr}T}\frac{dz_{r}}{d\theta}\kappa_{T}\nabla
T\cdot\nabla\theta$ (79)
$\varepsilon_{p,eos}=-\frac{c_{p\theta}T_{r}}{c_{pr}T}\frac{(c_{p}-c_{pr})}{\theta}\kappa_{T}\nabla
T\cdot\nabla\theta$ (80)

Importantly, the term $\varepsilon_{p,lam}$ vanishes only at standard
thermodynamic equilibrium, characterised by uniform in-situ temperature $T$.
However, both $\varepsilon_{p,tur}$ and $\varepsilon_{p,eos}$ vanish both at
standard thermodynamic equilibrium, but also for the ‘turbulent’ thermodynamic
equilibrium characterised by uniform potential temperature $\theta$.

###  A.1 Link with Boussinesq approximation

The above expressions show that the exact counterpart of the $\Phi_{i}$ term
in Winters et al. (1995) is the following expression:

$\Phi_{i}^{exact}=-\int_{V}\frac{\alpha T_{r}}{T}\kappa_{T}\nabla T\cdot\nabla
p\,{\rm d}V$ (81)
$\Phi_{d}^{exact}=\int_{V}\frac{c_{p}T_{r}\rho\alpha_{r}g}{c_{pr}T}\frac{dz_{r}}{d\theta}\kappa_{T}\nabla
T\cdot\nabla\theta\,{\rm d}V$ (82)

In the Boussinesq approximation, $p\approx-\rho_{\star}gz$, $T_{r}\approx T$,
and $\alpha$ is treated as a constant. As a result,

$\Phi_{i}^{exact}\approx\int_{V}\rho_{\star}\alpha g\frac{\partial T}{\partial
z}\,{\rm d}V=\rho_{\star}\alpha g\left(\langle T\rangle_{top}-\langle
T\rangle_{bottom}\right)=\Phi_{i}$ (83)

As is well known, this term integrates to a term that depends on the
difference between the surface integrated temperature at the upper boundary
minus that at the lower boundary. For a single-component fluid, $\Phi_{i}$ is
generally positive for a stable stratification. In particular, $\Phi_{i}$ is
independent of the turbulent character of the fluid. It is important that this
is not the case of the exact expression. An important open question is whether
strong turbulence could potentially make $\Phi_{i}^{exact}$ significantly
different from $\Phi_{i}$.  Regarding the $\Phi_{d}$ term,

$\Phi_{d}^{exact}\approx\int_{V}\rho_{\star}\alpha
g\frac{dz_{r}}{d\theta}\kappa_{T}|\nabla\theta|^{2}\,{\rm
d}V=-\int_{V}g\frac{dz_{r}}{d\rho}\kappa_{T}|\nabla\rho|^{2}\,{\rm d}V$ (84)

assuming ${\rm d}\rho=-\rho_{\star}\alpha{\rm d}\theta$. The Boussinesq
approximation of $\Phi_{d}^{exact}$ is therefore potentially much more
accurate than that for $\Phi_{i}$.  Note that

$\varepsilon_{p,eos}\approx-\frac{c_{p}-c_{pr}}{\theta}\kappa_{T}\nabla
T\cdot\nabla\theta$ (85)

is related to a nonlinear equation of state. This term vanishes if the
pressure variations of $c_{p}$ can be neglected. As shown by the maxwell
relation 72, this is the case if

$\frac{\partial}{\partial\theta}\left(\frac{\alpha}{\rho c_{p}}\right)=0$ (86)

Therefore, the condition $\alpha/(\rho c_{p})={\rm constant}$ was taken as the
appropriate definition of a linear equation of state underlying the
construction of the Boussinesq approximation in Tailleux (2013c).

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