# Boiling stratified flow: a laboratory analogy for atmospheric moist
convection

Hao Fu1, 5<EMAIL_ADDRESS>Claudia Cenedese2  Adrien Lefauve3  and
Geoffrey K. Vallis4  1Department of the Geophysical Sciences, University of
Chicago, USA 2Physical Oceanography Department, Woods Hole Oceanographic
Institution, USA 3Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, UK 4Department of Mathematics and Statistics,
University of Exeter, UK 5Department of Earth System Science, Stanford
University, USA

###### Abstract

We present a novel laboratory experiment, boiling stratified flow, as an
analogy for atmospheric moist convection. A layer of diluted syrup is placed
below freshwater in a beaker and heated from below. The vertical temperature
profile in the experiment is analogous to the vapor mixing ratio in the
atmosphere while the vertical profile of freshwater concentration in the
experiment is analogous to the potential temperature profile in the
atmosphere. Boiling starts when the bottom of the syrup layer reaches the
boiling point, producing bubbles and vortex rings that stir the two-layer
density interface and bring colder fresh water into the syrup layer. When the
syrup layer at the beginning of the experiment is sufficiently thin and
diluted, the vortex rings entrain more cold water than needed to remove
superheating in the syrup layer, ending the boiling. When the syrup layer is
deep and concentrated, the boiling is steady since the entrained colder water
instantaneously removes the superheating in the bottom syrup layer. A theory
is derived to predict the entrainment rate and the transition between the
intermittent and steady boiling regimes, validated by experimental data. We
suggest that these dynamics may share similarities with the mixing and
lifecycle of cumulus convection.

###### keywords:

##  1 Introduction

Atmospheric convection is a particular type of fluid convection that involves
phase changes. As an ascending parcel cools by adiabatic expansion, water
vapor condenses and releases latent heat. This provides extra buoyancy that
makes parcels penetrate the troposphere, which has a stratification stable to
dry convection. As a result, in-cloud saturated parcels are unstable, but
clear-sky unsaturated parcels are stable, leading to conditional instability
(Bjerknes, 1938). When a cumulus cloud is deep enough, the re-evaporation of
liquid water in the dry atmosphere produces a downdraft. The downdraft brings
down dry air and shuts the convection. The updraft and downdraft couplet
renders a convective lifecycle (Byers & Braham, 1949; Feingold et al., 2010;
Dagan et al., 2018).

Although observation and numerical simulation have been the main tools for
studying clouds, efforts to simulate clouds in the laboratory have a long
history that continues to this day. At the cloud microphysics scale, particle-
laden flows with real droplets and ice interacting with turbulence have been
studied in cloud chambers (e.g., Shawon et al., 2021). The dynamics of the
whole cloud as an entity, namely cloud dynamics, is much harder to reproduce.
This is because the condensation due to lifting requires an apparatus as tall
as the scale height of the saturation vapor mixing ratio (the vapor mixing
ratio that makes the partial pressure of vapor reach the saturated vapor
pressure, i.e., a relative humidity of 100%), which is around 3 km (Wallace &
Hobbs, 2006). To study cloud dynamics in the laboratory, analogies must
therefore be found, with the awareness that no analogy is complete. There are
at least three perspectives on studying the nature of moist convection.

In the first perspective, an individual cloud is considered as a buoyant plume
or bubble. Possible internal sources of buoyancy include the heat released
from a chemical reaction (Turner, 1963), gas bubbles (Turner & Lilly, 1963),
heating coil (Narasimha et al., 2011), and radiation from a lamp (Zhao et al.,
2001). Although this setup cannot mimic the convective lifecycle, it suits
problems where the feedback to the heat source can be neglected.

In the second perspective, moist convection is treated as a hydrodynamic
instability, essentially an extension of the Rayleigh-Bénard convection
problem that includes moisture (e.g., Bénard, 1901; Rayleigh, 1916;
Chandrasekhar, 1961; Zhang et al., 2019; Fu, 2021). Models of conditional
instability qualitatively reproduced the length scale of a cloud system: with
narrow ascent and a wide descent (Kuo, 1961; Bretherton, 1987, 1988; Pauluis &
Schumacher, 2010; Vallis et al., 2019). Krishnamurti (1998) designed an
experiment of conditional instability using a temperature vertical gradient to
generate the buoyancy stratification and a pH vertical gradient to mimic the
vapor stratification. The selective radiative absorption of a pH indicator is
used to mimic the latent heat release.

The third perspective is based on the realization that moist convection as a
heat transfer mechanism tends to be in a quasi-equilibrium state (QE),
especially in the tropics where radiative cooling balances condensation
heating (Betts, 1973; Arakawa & Schubert, 1974; Emanuel et al., 1994; Yano &
Plant, 2012a). QE is a highly nonlinear state where conditional instability is
self-regulated, and it still lacks an experimental analog. Next, we will
briefly review the boundary layer quasi-equilibrium hypothesis (BLQE, Emanuel,
1989; Raymond, 1995). Whether convection can occur depends on the moisture in
the boundary layer. Convection induces mixing that brings up moist air and
brings down dry air. As a result, convection is a “valve” that releases the
extra moisture accumulated by surface evaporation. One open question we aim to
answer is: what controls the vertical mixing across the top of the boundary
layer, and how does it influence the adjustment to equilibrium? Mixing can be
undertaken by boundary layer convective cells (e.g., Lilly, 1968; Deardorff et
al., 1970; Arakawa & Schubert, 1974), and the transport by cumulus updraft and
downdraft. One particularly interesting question posed by Thayer-Calder &
Randall (2015) is how the cumulus updraft and downdraft produce turbulence at
the top of the boundary layer and may indirectly influence mixing.

An ideal experiment of cloud dynamics should reproduce both conditional
instability and QE. Conditional instability is a threshold behavior that
requires the boundary layer to be close to saturation. Boiling reproduces such
a threshold behavior, for it limits the water temperature around the boiling
point by absorbing the heat of vaporization and then mixing the superheated
water with the cold water above (Collier & Thome, 1994; Oresta et al., 2009).
To improve the analogy to moist convection, we must also reproduce the mostly
stable stratification in the atmosphere, which traps the moisture in the
boundary layer and facilitates its accumulation, making convection
intermittent. Our idea is thus to make a boiling stratified flow, essentially
coupling the two-layer Rayleigh-Bénard convection (Turner, 1965; Davaille,
1999) with the boiling effect. Although some studies have investigated the
boiling of two layers of immiscible fluids like water and oil (e.g., Mori,
1978, 1985; Greene et al., 1988; Takahashi et al., 2010; Filipczak et al.,
2011; Onishi et al., 2013; Kawanami et al., 2020), the boiling of two separate
layers of miscible fluids appears relatively novel.

In this paper, we propose a laboratory analogy that uses freshwater to mimic
the free troposphere and a thin layer of diluted syrup to mimic the
atmospheric boundary layer (Fig. 1). The geophysical questions we would like
to address with this experiment are:

  * •

How does moist convection favor the entrainment of dry air into the boundary
layer?

  * •

How does the dry air feed back on moist convection?

The water temperature mimics the atmospheric humidity while the boiling
temperature mimics the saturation vapor mixing ratio. The freshwater
concentration mimics the atmospheric potential temperature, defined as the
temperature of an air parcel that is adiabatically moved to the surface,
representing buoyancy. Translating the questions into the context of
laboratory experiments, we ask:

  * •

How do boiling plumes favor the entrainment of colder water into the syrup
layer?

  * •

How does the entrained colder water feedback on the boiling?

Such a self-regulating behavior is analogous to the lifecycle of moist
convection on Earth and giant planets (Feingold et al., 2010; Yano & Plant,
2012b; Li & Ingersoll, 2015). We stress that the experiment is only an
analogy, not an exact correspondence. For example, the buoyancy of a cumulus
cloud is gained from the latent heat release in condensation. In the
experiment, the buoyancy of a bubble is gained from the volume expansion in
vaporization. This paper focuses on the fluid mechanics of boiling stratified
flow and leaves a detailed comparative study with atmospheric convection for
future work.

The paper is organized as follows. Section 2 introduces the experimental
setup. Section 3 analyzes the flow evolution of the reference experiment,
which inspires the theory in section 4. Section 5 applies the theory to
understand experiments that sample the parameter space. Section 6 extends the
theory to study the transition between two types of boiling. Section 7
concludes the paper. The supplemental materials, including the experimental
videos, are deposited at https://doi.org/10.5281/zenodo.11222909.

Figure 1: Schematic of the boiling stratified flow as an analogy to
atmospheric convection. The left panel illustrates shallow cumulus clouds and
idealised profiles of the water vapor mixing ratio and potential temperature.
The right panel shows the experimental setup and the corresponding idealised
profiles of temperature and freshwater concentration.

##  2 The experiment

###  2.1 Experimental setup

The experimental setup is shown in Fig. 1. It consists of a 2000 ml beaker
filled with a layer of dark corn syrup water solution underlying a thicker
layer of fresh water, with a density ratio of $\rho_{s,max}/\rho_{w}=1.4$,
where $\rho_{s,max}\approx 1.4\times 10^{3}$ kg m-3 is the density of the pure
syrup and $\rho_{w}\approx 1\times 10^{3}$ kg m-3 is the density of the pure
water. The beaker is heated on an electric hot plate with adjustable heating
power. More details on the setup are given in Appendix A. When diluted, the
density of the syrup solution $\rho_{s}$ obeys:

$\displaystyle\rho_{s}=S\rho_{s,max}+(1-S)\rho_{w},\quad$ (1)

where $0\leq S\leq 1$ is the dimensionless syrup concentration:

$\displaystyle S\equiv\frac{\rho_{s}-\rho_{w}}{\rho_{s,max}-\rho_{w}}.$ (2)

The system is required to be statically stable at the onset of boiling. The
vertical gradient of syrup concentration stabilizes the two-layer
configuration against the destabilizing effect of the temperature gradient.
The buoyancy $b$ (unit: m s-2) is defined as:

$\displaystyle
b=g\left(\frac{\rho_{w}-\rho_{s}}{\rho_{w}}+\gamma_{T}T\right)=g\left(-\gamma_{s}S+\gamma_{T}T\right),$
(3)

where $T$ (unit: K) is temperature,
$\gamma_{s}\equiv(\rho_{s,max}-\rho_{w})/\rho_{w}=0.4$ is the syrup
concentration coefficient, $\gamma_{T}\approx 6\times 10^{-4}$ K-1 is the
volumetric thermal expansion coefficient of the solution (taken as the value
for 75∘C pure water, which is between the room temperature and the boiling
temperature), and $g=9.8$ m s-2 is the gravitational acceleration. At the
onset of boiling, the water layer temperature is $T_{w}\approx 30^{\circ}$C
(slightly above the 20∘C room temperature), while the syrup layer is around
the boiling temperature $T_{*}\approx 100^{\circ}$C. We let the characteristic
temperature difference be
$T_{*}-T_{w}=100^{\circ}\mathrm{C}-30^{\circ}\mathrm{C}=70^{\circ}\mathrm{C}$.
To make the system stable, $S$ must be above a minimum value $S_{min}$:

$\displaystyle
S_{min}\equiv\frac{\gamma_{T}\left(T_{*}-T_{w}\right)}{\gamma_{s}}\approx
0.11.$ (4)

The temperature in the experiment is analogous to the water vapor mixing ratio
in the atmosphere. Water vapor is not only a triggering factor of moist
convection but also a component of buoyancy that makes a parcel lighter (e.g.,
Yang, 2018), analogous to the temperature in the experiment that controls
boiling and influences buoyancy via thermal expansion. The quantity $(1-S)$ in
the experiment is the freshwater concentration. It is analogous to the
potential temperature in the atmosphere, which increases with height. Because
in our experiments, the initial concentration $S_{0}\gg S_{min}$, the buoyancy
from the syrup plays the dominant role.

We use syrup because it has a higher density and viscosity than water, both of
which suppress interfacial heat and mass transfer (Turner, 1986). The high
syrup viscosity is crucial in that it enables the lower layer to reach the
boiling point before the two-layer stratification is eroded by turbulence
(unlike what happens with a dense salt solution, as discussed in Appendix A).
The high viscosity does not have a direct analogy to the atmosphere, but it
might be thought of as an amplifier of density stratification.

The flow is recorded with a camera from a side view, providing an integrated
view of the flow field. Temperature is measured with thermocouples located at
$z=1$ cm, 3 cm, 5 cm, and 7 cm above the bottom of the beaker, though this
paper only discusses the $z=1$ cm and 5 cm data.

###  2.2 List of experiments

The experimenter can vary several parameters governing the system:

  1. 1.

The surface heating flux $F_{s}$, which is analogous to the solar heating in
the atmosphere.

  2. 2.

The syrup initial layer thickness $h_{0}$, which is analogous to the
atmospheric convective boundary layer thickness.

  3. 3.

The syrup concentration $S$, which is analogous to the strength of buoyancy
stratification near the top of the boundary layer.

  4. 4.

The water layer thickness, which is analogous to the tropospheric depth.

We decided to leave the investigation on the water layer thickness for future
work by keeping the initial freshwater amount fixed to around 10 cm
(corresponding to the 1400 ml scale on the beaker), much thicker than the
convective penetration height. We performed four groups of experiments,
varying $F_{s}$ (F1-F5), $h_{0}$ (T1-T7), $S$ (S1-S7), and $S$ and $h_{0}$
together (ST1-ST4), as shown in Table 1. Note that the labels (F3, S5), (T4,
S6), (T6, ST1), and (T7, ST2) correspond to the same experiments. Experiment
S3 (in bold) is the reference experiment that will be analyzed in detail.

We measured $S_{0}$ with a density meter and $h_{0}$ with the video (using a
pixel-to-length calibration). Note that the value of $S_{0}$ or $h_{0}$
reported in Table 1 within an experimental group (e.g., $S_{0}=$0.47, 0.48,
0.46, etc. for the F group) are not exactly equal due to small fluctuations
introduced in preparing the two-layer fluid.

Table 1:  A table of experimental parameters, which include heating voltage, surface heat flux $F_{s}$, initial syrup density $\rho_{s}$, initial syrup concentration $S_{0}$, and initial syrup thickness $h_{0}$. The post-boiling syrup thickness $h_{1}$ is shown in the rightmost column, with the diagnostic procedure introduced in Appendix B. For experiments in the steady boiling regime (section 5.3), their $h_{1}$ is denoted as “-”. Note that some experiments are shared by different groups. S3 is the reference experiment (in bold).  Name | Voltage (V) |  $F_{s}$ (kW m-2) |  $\rho_{s}$ ($\times 10^{3}$ kg m-3) | $S_{0}$ |  $h_{0}$ (cm) |  $h_{1}$ (cm)
---|---|---|---|---|---|---
F1 | 50 | 6.4 | 1.188 | 0.47 | 2.00 | 5.84
F2 | 60 | 9.3 | 1.190 | 0.48 | 2.10 | 5.45
F3 and S5 | 86 | 19.0 | 1.184 | 0.46 | 2.05 | 4.35
F4 | 100 | 25.7 | 1.205 | 0.51 | 1.93 | 4.35
F5 | 120 | 37.0 | 1.197 | 0.49 | 1.80 | 3.52
S1 | 86 | 19.0 | 1.070 | 0.18 | 1.76 | 1.98
S2 | 86 | 19.0 | 1.112 | 0.28 | 1.83 | 4.74
S3 | 86 | 19.0 | 1.124 | 0.31 | 1.91 | 5.48
S4 | 86 | 19.0 | 1.138 | 0.35 | 1.96 | 5.32
S5 and F3 | 86 | 19.0 | 1.184 | 0.46 | 2.05 | 4.35
S6 and T4 | 86 | 19.0 | 1.218 | 0.54 | 1.94 | 4.13
S7 | 86 | 19.0 | 1.272 | 0.68 | 1.89 | 4.04
T1 | 86 | 19.0 | 1.205 | 0.51 | 0.80 | 4.19
T2 | 86 | 19.0 | 1.203 | 0.51 | 1.20 | 3.90
T3 | 86 | 19.0 | 1.199 | 0.50 | 1.50 | 4.14
T4 and S6 | 86 | 19.0 | 1.218 | 0.54 | 1.94 | 4.13
T5 | 86 | 19.0 | 1.211 | 0.53 | 2.64 | 4.74
T6 and ST1 | 86 | 19.0 | 1.204 | 0.51 | 3.40 | 5.24
T7 and ST2 | 86 | 19.0 | 1.204 | 0.51 | 4.13 | -
ST1 and T6 | 86 | 19.0 | 1.204 | 0.51 | 3.40 | 5.24
ST2 and T7 | 86 | 19.0 | 1.204 | 0.51 | 4.13 | -
ST3 | 86 | 19.0 | 1.136 | 0.34 | 4.08 | 6.61
ST4 | 86 | 19.0 | 1.133 | 0.33 | 4.74 | -

##  3 Temporal evolution of boiling stratified flow

Figure 2 illustrates the basic physics of boiling stratified flow in the
reference experiment S3. It shows that the flow has three stages: the initial
two-layer stage, the boiling stage, and the post-boiling two-layer stage.

Figure 2: The flow snapshots at $t=650$ s, 750 s, and 850 s of the reference
experiment (S3), showing the initial two-layer stage (a), the boiling stage
(b), and the post-boiling two-layer stage (c).

Figure 3: Quantitative measurements of the reference experiment S3. (a) The
temperature time series at $z=5$ cm (blue line) and $z=1$ cm (red line). The
dashed black lines denote the snapshots in Fig. 2. Note that heating starts at
$t=0$. (b) The same as (a) but zooming into the boiling stage. (c) The zoom-in
time evolution of the video’s horizontally averaged green light pixel
intensity value. The dashed white line annotates the internal interface
between the bottom syrup layer and the middle mixed layer.

Before boiling starts, the syrup layer temperature gradually rises, and the
water layer temperature remains close to the initial temperature (Fig. 3a).
This is because the density stratification suppresses heat transfer by
suppressing eddy mixing (Turner, 1965). Boiling begins at around $t=700$ s by
which the syrup temperature reaches 100 ∘C (Fig. 3b). Bubbles are formed near
the beaker’s bottom and mostly quench before leaving the syrup layer (Fig.
2b). This is because the upper part of the syrup is still below the boiling
point. It can be viewed as a sub-cooled nucleate boiling phenomenon (Collier &
Thome, 1994, section 4.4.4), with the sub-cooling substantially amplified by
the two-layer stratification. Though the bubble quenches, its momentum can
drive a vortex ring that rises into the freshwater layer, mixes with
freshwater, and sinks to the interface, producing a middle mixed layer.

Boiling only lasts about 1 minute, during which the $z=1$ cm temperature drops
from 100∘C to around 60∘C and the $z=5$ cm temperature slightly rises (Fig.
3b). The bubble-induced mixing brings colder freshwater to the syrup layer and
quenches boiling. At $t=850$ s, the system still has a two-layer
stratification. However, the interface rises from the initial 2 cm to 6 cm,
the syrup layer is significantly diluted, and the interface is less sharp than
before boiling (Fig. 3c).

Next, we use the horizontally averaged video pixel intensity value to track
the interface height, denoted as $h$. The printed scale of the beaker is
excluded from the averaging region. The video records the pixel intensity
values of red, green, and blue light. We use the green light component because
it captures the interface position most clearly. Figure 3c shows boiling
significantly lifts the interface from the initial value $h_{0}$ to a post-
boiling value $h_{1}$. Thus, boiling can be viewed as a mixing event. We ask:

  * •

What controls the boiling duration time $\Delta t$?

  * •

What controls the interface’s rising rate $dh/dt$?

Knowing $\Delta t$ and $dh/dt$, we will be able to predict the net effect of
mixing:

$\displaystyle h_{1}-h_{0}\approx\Delta t\frac{dh}{dt}.$ (5)

A closer look at the green light pixel intensity value (Fig. 3c) shows an
internal interface between the bottom and middle syrup layers. The internal
interface splits from the outer interface at the onset of boiling and descends
to the bottom at the end of boiling (dashed white line in Fig. 3c). The vortex
rings carry up syrup from the bottom syrup layer, mix with the freshwater, and
deposit the mixture in the middle mixed layer. Thus, the bottom syrup layer
gets thinner and finally disappears, letting the relatively cold middle mixed
layer touch the bottom and quench the boiling. This suggests that $\Delta t$
is the time needed for vortex rings to remove the bottom syrup layer:

$\displaystyle\Delta t\approx\frac{h_{0}}{\overline{w_{+}}},$ (6)

where $\overline{w_{+}}$ (unit: m s-1) is the horizontally averaged syrup
volume flux across the internal interface, analogous to the mass flux of
cumulus convection at the cloud bottom (e.g., Arakawa & Schubert, 1974).

##  4 Theory: from vortex rings entrainment to the final layer thickness

This section builds a theoretical framework to understand how $h_{1}-h_{0}$
depends on the control parameters $F_{s}$, $h_{0}$, and $S_{0}$. Modeling
$dh/dt$ is equivalent to modeling the ensemble effect of vortex rings. The
mixing by individual vortex rings has been investigated by Olsthoorn & Dalziel
(2015). The boiling stratified flow provides a unique setup to study the
nonlinear interaction between successive vortex rings.

###  4.1 Escaped vs. trapped vortex rings

Figure 4: Examples of the two life paths of a vortex ring in the reference
experiment S3. The first row shows an escaped vortex ring, and the second row
shows a trapped vortex ring, both with a time interval of 0.17 s between
snapshots.

Figure 4 shows the two typical life paths of vortex rings: escaping and
trapping. For the escaping path, the bubble quenches in the syrup layer,
leaving a vortex ring that rises into the water layer and sinks back to the
interface. Note that some vortex rings completely dissipate in the freshwater
layer. Their wake, consisting of syrup, can sink back to the interface. For
the trapping path, the initial bubble has a similar size to the escaping case,
but the vortex ring crashes near the interface, producing a wide turbulent
patch. The two paths are conceptualized in Fig. 5. The escaping path has a
relatively long mixing length, characterized by the vortex ring’s penetration
depth $l$. By contrast, the trapping path has a shorter mixing length, with a
limited ability to bring down the colder water. Thus, we infer that the
escaped vortex rings are mainly responsible for the thickening of the middle
mixed layer.

Figure 5: A schematic diagram of two life paths of a vortex ring: escaping and
trapping.

We define the escape ratio $E$ to quantify the fraction of the vortex rings
that could escape the middle mixed layer and rise into the water layer. We
hypothesize that the stratified turbulence on the path of a vortex ring causes
trapping. The turbulence can be induced by the wake of ascending vortex rings
(Innocenti et al., 2021) or the baroclinic vorticity generated at the
interface (Olsthoorn & Dalziel, 2018). The turbulence could tilt the
orientation of the vortex ring, causing an oblique incidence onto the
interface. The experiments of Pinaud et al. (2021) showed that an oblique
incidence could significantly tilt the vortex ring and turn it horizontal due
to the interaction between the vortex ring and the baroclinically generated
vorticity at the interface. Because a smaller vortex ring is more easily
tilted by turbulence, and a thicker syrup layer ($h_{0}$) increases the chance
of tilting, we heuristically parameterize $E$ as:

$\displaystyle E=\exp\left(-\frac{C_{E}h_{0}}{R}\right),$ (7)

where $C_{E}$ is a nondimensional escaping parameter depending on the
turbulent kinetic energy in the syrup layer. In section 4.3, $C_{E}$ will be
shown to be equivalent to a drag coefficient.

We further hypothesize that $C_{E}$ is higher for a higher $F_{s}$ because
stronger surface heating reduces the time interval between vortex rings. The
turbulent wake of the current vortex ring could trap the next one. The
trapping leads to vortex ring breaking, causing stronger turbulence and,
therefore, a pileup of vortex rings. This hypothesis will be tested in section
5 where experiments with different $F_{s}$ are introduced.

###  4.2 The vortex ring penetration depth $l$

A hotter surface temperature generally increases the initial bubble radius $R$
in boiling (Barathula & Srinivasan, 2022). A bubble is highly buoyant but
quenches (condenses) quickly once it leaves the hot bottom. A hotter fluid
interior makes the bubble condense more slowly and yields a longer
acceleration path $h_{*}$ for buoyancy. Combining these arguments, we see that
$h_{*}$ should increase with the bubble radius. For simplicity, we assume:

$\displaystyle h_{*}\approx\beta R,$ (8)

where $\beta$ is a nondimensional bubble acceleration coefficient. The bubble
exerts pressure on the environment and accelerates the surrounding liquid. The
added mass theory (e.g., Falkovich, 2011) indicates that for a spherical
bubble, the surrounding liquid moving with the bubble has half of its volume
(Fig. 6). Ignoring the mass of vapor, the mean density of the bubble and the
surrounding liquid is approximately $(2/3)\rho_{w}$. The initial velocity of
the vortex ring, $w_{*}$, is estimated with a free-fall scaling:

$\displaystyle w_{*}\approx\left(2\frac{2}{3}gh_{*}\right)^{1/2},$ (9)

where $(2/3)g$ is the acceleration. After the bubble quenches, the moving
liquid turns into a vortex ring of radius $R$.

We now perform a force analysis of an escaped vortex ring and let its vertical
velocity be $w$. In the syrup layer, the vortex ring has neutral buoyancy,
only influenced by drag:

$\displaystyle\frac{dw}{dt}=-\frac{C_{D}}{R}w^{2},\quad w|_{t=0}=w_{*},$ (10)

where $C_{D}$ is the nondimensional drag coefficient. This drag
parameterization, taken from Maxworthy (1974), is also used in modeling the
drag of thermals in clouds (Romps & Charn, 2015; Romps & Öktem, 2015). For a
turbulent vortex ring, $C_{D}$ is a constant. For simplicity, we assume
$C_{D}$ to be constant in our model as well. When the vortex ring enters the
water layer, it is influenced by both drag and negative buoyancy:

$\displaystyle\frac{dw}{dt}=-\frac{C_{D}}{R}w^{2}-g\gamma_{s}S_{0}\approx-g\gamma_{s}S_{0}.$
(11)

For simplicity, we only consider the buoyancy effect after the vortex ring has
escaped the syrup layer because the vortex ring would not have a maximum
penetration height without considering buoyancy. The buoyancy is assumed to be
controlled by $S_{0}$ rather than temperature, as discussed in section 2.1.

Figure 6: A schematic diagram of the vortex ring initiation and development
processes. The syrup layer includes both the bottom and middle layers.

We can solve for the kinetic energy of the vortex ring when it crosses the
interface and the negative buoyancy work done in the water layer. They are
linked with the interface crossing velocity $w_{+}$, using (10) and (11):

$\displaystyle\frac{2}{3}gh_{*}\exp\left(-\frac{2C_{D}h_{0}}{R}\right)=\frac{w^{2}_{+}}{2}=g\gamma_{s}S_{0}l.$
(12)

Equation (12) yields an expression for the vortex ring penetration depth $l$:

$\displaystyle
l=\frac{2}{3}\frac{h_{*}}{\gamma_{s}S_{0}}\exp\left(-\frac{2C_{D}h_{0}}{R}\right).$
(13)

The theory predicts that a more diluted syrup (smaller $S_{0}$) would make the
vortex ring lighter and penetrate a longer distance. A thicker syrup layer
(higher $h_{0}$) induces more accumulated drag and reduces $l$.

The $l$ can be used to calculate the expansion rate of the vortex ring’s
volume $V\equiv\frac{4}{3}\pi R^{3}$:

$\displaystyle\frac{1}{V}\frac{dV}{dz}=\frac{3}{R}\frac{dR}{dz},$ (14)

where $R$ is temporarily treated as a variable. For a turbulent vortex ring,
the expansion rate of $R$ obeys:

$\displaystyle\frac{dR}{dz}=\alpha.$ (15)

Here, $\alpha$ is the nondimensional entrainment coefficient (Morton et al.,
1956), which is $O(10^{-2})$ (Maxworthy, 1974). Next, we calculate the volume
of the vortex ring when it sinks back to the interface, using the volume of
the vortex ring when it first crosses the interface $V_{+}$. The traveling
distance is $2l$, which includes an updraft part and a downdraft part.
Further, assuming $R$ only grows by a small amount in the freshwater layer, we
substitute (15) into (14) to get:

$\displaystyle V\approx V_{+}\left(1+6\alpha\frac{l}{R}\right),$ (16)

From now on, we return to assuming $R$ as a constant quantity. Next, we study
the rising rate of the interface, which is the collective effect of many
vortex rings.

###  4.3 The post-boiling syrup layer thickness $h_{1}$

The rising rate $dh/dt$, where $h$ is the syrup layer thickness, depends on
detrainment and entrainment across the syrup-water interface. Detrainment
denotes the mass leaving the syrup layer, and entrainment denotes the mass
entering the syrup layer. A vortex ring is detrained from the syrup layer
first and then entrained. The interface height $h$ obeys:

$\displaystyle\frac{dh}{dt}=\underbrace{-E\overline{w_{+}}}_{\text{detrain}}+\underbrace{E\overline{w_{+}}\frac{V}{V_{+}}}_{\text{entrain}}\approx
6\alpha\frac{l}{R}E\overline{w_{+}},$ (17)

where $\overline{w_{+}}$ is the horizontally averaged volume flux of vortex
rings from the bottom syrup layer to the middle mixed layer, and
$E\overline{w_{+}}$ is the flux from the middle mixed layer to the freshwater
layer. The $V/V_{+}$ is calculated with (16).

Our model is consistent with the experimental result of Olsthoorn & Dalziel
(2015), who studied successive vortex rings impinging onto a stratification
interface. Their experiments showed that the ratio of net entrainment to
detrainment, essentially the $6\alpha l/R$ factor in our formulation, is
proportional to $\mathrm{Ri^{-1}}$. Here, $\mathrm{Ri}$ is the bulk Richardson
number that obeys:

$\displaystyle\mathrm{Ri}\equiv
g\frac{\rho_{s}-\rho_{w}}{\rho_{w}}\frac{R}{w^{2}_{+}}=g\gamma_{s}S_{0}\frac{R}{w^{2}_{+}},$
(18)

which shows $\mathrm{Ri}\propto S_{0}$. Substituting the expression of $l$
(13) into (17), we get:

$\displaystyle\frac{dh}{dt}\propto 6\alpha\frac{l}{R}\propto
S_{0}^{-1}\propto\mathrm{Ri}^{-1},$ (19)

which is consistent with the $\mathrm{Ri}^{-1}$ scaling of their measured
entrainment rate.

Combining (5), (6), (7), (13), and (17), we obtain the expression of $h_{1}$:

$\displaystyle
h_{1}=h_{0}\left[1+\frac{4\alpha\beta}{\gamma_{s}S_{0}}\exp\left(-\frac{2C_{D}+C_{E}}{R}h_{0}\right)\right].$
(20)

Note that the mean vertical volume flux of vortex rings, $\overline{w_{+}}$
[see Eq. (6)], does not appear in the expression of $h_{1}$. This expression
has two unknown nondimensional parameters:

  1. 1.

$\alpha\beta$, the product of the entrainment parameter $\alpha$ and the
bubble acceleration parameter $\beta$.

  2. 2.

The effective drag coefficients $2C_{D}+C_{E}$, which represents the bulk
effect of the physical drag and the trapping by turbulence.

To close the theory of $h_{1}$, we still need to find an expression for the
vortex ring radius $R$, which is determined by the thermodynamics of boiling.

###  4.4 The bubble radius $R$

The bubble radius $R$ in this experiment depends on how superheated the syrup
layer temperature is (Barathula & Srinivasan, 2022). We let the syrup layer
temperature be $T$, and the boiling temperature be $T_{*}=100^{\circ}$C. When
$T\ll T_{*}$, there is no boiling, so $R=0$. When $T\gtrsim T_{*}$, the water
is superheated, and Narayan et al. (2020) showed that $R$ has an upper bound
with respect to $T-T_{*}$, which we take as $R_{m}$. There is still
uncertainty in the diameter-superheating relation. Chang & Ferng (2019)
reported a steady increase in merged bubble diameter with the superheated
temperature, though the diameter of isolated bubbles seems insensitive to
superheating. See Barathula & Srinivasan (2022) for a summary. For simplicity,
we acknowledge the upper bound and parameterize $R$ as an error function of
$T-T_{*}$:

$\displaystyle R\approx
R_{m}\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{T-T_{*}}{\delta
T_{*}}\right)\right].$ (21)

Here, $\delta T_{*}$ is the temperature range of the transition zone in which
$R$ is sensitive to temperature. See Fig. 7a for an illustration.

Figure 7: A schematic diagram that illustrates the parameterization of the
bubble radius $R$ as a function of $T$, which is ultimately linked to the
initial syrup concentration $S_{0}$.

The syrup temperature $T$ depends on the heat balance of the syrup layer,
which involves the surface heat flux $F_{s}$, the ventilation by the
interfacial heat transfer, and the vaporization and mixing caused by boiling.
For simplicity, we define this equilibrium temperature without considering
boiling:

$\displaystyle\frac{dT}{dt}\approx\underbrace{\frac{F_{s}}{\rho_{w}c_{w}h_{0}}}_{\text{surface
heating}}-\underbrace{\frac{T-T_{w}}{h_{0}}w_{e}}_{\text{ventilation}}\approx
0,$ (22)

where $\rho_{w}$ is the density of pure water, $c_{w}$ is the specific heat of
pure water [the volumetric heat capacity of $\rho_{w}c_{w}$ could
approximately represent that of syrup solution, see Table A1.8 of Mohos
(2017)], $w_{e}$ is the characteristic eddy vertical velocity at the syrup-
water interface, and $T_{w}$ is the water temperature that is approximately
equal to the room temperature. One might be concerned that $T$ could be
unrealistically large without considering the cooling by boiling mixing. We
argue this is not a serious problem because $T$ only controls the bubble
radius $R$, an error function of $T$ with an upper bound. The $R$ is only
sensitive to $T$ where the superheating (the surplus of liquid temperature
over the boiling point) is weak and boiling is not vigorous. Thus, an
overestimation of $T$ in the vigorously boiling regime should yield little
error in $R$.

Equation (22) shows that $T$ depends on $w_{e}$, with more efficient
ventilation reducing $T$. Here, we model $w_{e}$. The syrup-layer eddy is
driven by convection. It is analogous to Rayleigh-Bénard convection (RBC),
with the beaker’s bottom as the warm plate and the syrup-water interface as
the cold plate. In our setup, the Rayleigh number (Ra) and Nusselt number (Nu)
are defined as:

$\displaystyle\mathrm{Ra}\equiv\frac{g\gamma_{T}(T-T_{w})h_{0}^{3}}{\nu\kappa},\quad\mathrm{Nu}\equiv\frac{w_{e}}{\kappa/h_{0}},$
(23)

where $\nu$ is the kinematic viscosity and $\kappa$ is the thermal
diffusivity. The Ra represents the relative strength of convective instability
and the diffusive damping. The Nu represents the ratio of convective to
conductive heat transfer. For the regime where the heat transfer is diffusive
in the boundary layer of RBC (a thin layer attached to the beaker’s bottom)
and turbulent in the syrup interior, Nu obeys:

$\displaystyle\mathrm{Nu}\approx c\mathrm{Ra}^{1/3},$ (24)

where $c=0.085$ is an empirical constant (Turner, 1965). Substituting (23)
into (24), we obtain an expression of $w_{e}$:

$\displaystyle
w_{e}=c\mathrm{Ra}^{1/3}\frac{\kappa}{h_{0}}=c\left[\frac{g\gamma_{T}(T-T_{w})}{\nu\kappa}\right]^{1/3}\kappa.$
(25)

The $w_{e}$ depends on $T-T_{w}$, $\nu$, and $\kappa$. For syrup, $\kappa$ is
insensitive to $S_{0}$ (Table A1.9 of Mohos, 2017). However, $\nu$ is very
sensitive to $S_{0}$ and approximately obeys an exponential function (Table
A1.8 of Mohos, 2017):

$\displaystyle\nu\approx\nu_{w}\exp{\left(\frac{S_{0}}{S_{\nu}}\right)},$ (26)

where $\nu_{w}$ (unit: m2 s-1) is the reference kinematic viscosity of water
and $S_{\nu}$ is the critical syrup concentration for viscosity to change
significantly. Substituting (26) into (25), we get:

$\displaystyle w_{e}=w_{e,ref}\exp\left(-\frac{S_{0}}{3S_{\nu}}\right),\quad
w_{e,ref}\equiv
c\left[\frac{g\gamma_{T}(T-T_{w})}{\nu_{w}}\right]^{1/3}\kappa^{2/3},$ (27)

where $w_{e,ref}$ is a reference eddy velocity scale for $S_{0}=0$ (water).
Equation (27) indicates that a denser syrup suppresses heat transfer.

Equation (27) shows the heat transfer ability of RBC, and (22) shows the
requirement on $w_{e}$ to make the syrup-layer temperature steady when
neglecting boiling. Combining them and letting $T=T_{*}$ (the water boiling
temperature 100 ∘C) yields a critical $w_{e}$ and, therefore, a critical
$S_{0}$ for boiling, $S_{*}$:

$\displaystyle
S_{*}=3S_{\nu}\ln\left[\frac{\rho_{w}c_{w}(T_{*}-T_{w})w_{e,ref*}}{F_{s}}\right].$
(28)

where $w_{e,ref*}$ is the $w_{e,ref}$ at the boiling state:

$\displaystyle w_{e,ref*}\equiv
c\left[\frac{g\gamma_{T}(T_{*}-T_{w})}{\nu_{w}}\right]^{1/3}\kappa^{2/3}.$
(29)

For $S>S_{*}$, convective heat transfer is too weak to keep the syrup
temperature steady and below the boiling point, so boiling must occur.
Substituting (28) into (22), we obtain the relationship between the
supercritical syrup concentration ($S_{0}-S_{*}$) and the superheated
temperature ($T-T_{*}$):

$\displaystyle T-T_{*}$
$\displaystyle=\frac{F_{s}}{\rho_{w}c_{w}w_{e,ref}}\exp\left(-\frac{S_{0}}{3S_{\nu}}\right)-\left(T_{*}-T_{w}\right)$
(30)
$\displaystyle\approx\frac{F_{s}}{\rho_{w}c_{w}w_{e,ref*}}\exp\left(-\frac{S_{0}}{3S_{\nu}}\right)-\left(T_{*}-T_{w}\right)$
$\displaystyle=\left[\exp\left(\frac{S_{0}-S_{*}}{3S_{\nu}}\right)-1\right]\left(T_{*}-T_{w}\right)$
$\displaystyle\approx\frac{S_{0}-S_{*}}{3S_{\nu}}\left(T_{*}-T_{w}\right),$

where we have used $w_{e,ref}\approx w_{e,ref*}$ in the second line and Taylor
expansion in the fourth line. Equation (30) indicates a denser syrup increases
the superheating by increasing viscosity and suppressing heat transfer.

Substituting (30) into (21), we express $R$ as a function of $S_{0}$:

$\displaystyle
R=R_{m}\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{S_{0}-S_{*}}{\delta
S_{*}}\right)\right],\quad\delta S_{*}\equiv 3S_{\nu}\frac{\delta
T_{*}}{T_{*}-T_{w}},$ (31)

where $\delta S_{*}$ is the range of the initial syrup concentration in which
the bubble radius is sensitive to $S_{0}$ (Fig. 7b). Equation (31) indicates
that a denser syrup makes bubbles larger.

Equations (20) and (31) finally close the model for how $h_{1}$ depends on
$h_{0}$ and $S_{0}$:

$\displaystyle
h_{1}=h_{0}\left\\{1+\frac{4\alpha\beta}{\gamma_{s}S_{0}}\exp\left[-\frac{h_{0}}{h_{DE}}\frac{2}{1+\mathrm{erf}\left(\frac{S_{0}-S_{*}}{\delta
S_{*}}\right)}\right]\right\\}.$ (32)

Here, $h_{DE}$ is the vortex ring dissipation length scale:

$\displaystyle h_{DE}\equiv\frac{R_{m}}{2C_{D}+C_{E}},$ (33)

which represents the bulk effect of drag and trapping. The system has four
unknown parameters: $\alpha\beta$, $h_{DE}$, $\delta S_{*}$, and $S_{*}$. In
the next section, we apply this theory to understand the experiments with
varying $F_{s}$, $S_{0}$, and $h_{0}$.

##  5 Experimental validation of the theory

To test equation (32), this section analyzes the post-boiling interface height
$h_{1}$ diagnosed from horizontally averaged green light pixel value for
experiments with varying $F_{s}$, $S_{0}$, and $h_{0}$. The diagnostic method
for $h_{1}$ (relying on imaging) is introduced in Appendix B, and the results
are shown in Figs. 8 and 9.

Figure 8: Time evolution of the syrup layer thickness shown with the
horizontally averaged green light pixel value of the video. The first row
shows experiments F1-F5, where $F_{s}$ is changed by varying the heating
voltage. The solid black lines show the diagnosed height of the syrup-water
interface. The dashed red lines show the diagnosed $h_{1}$. The second row is
for experiments S1-S7 that change the initial syrup concentration $S_{0}$. The
third row is for experiments T1-T7 that change the initial syrup thickness
$h_{0}$. The T7 experiment (s) is in the steady boiling regime without a well-
defined $h_{1}$.

###  5.1 Experiments with varying $F_{s}$

The different surface heat fluxes ($F_{s}$) are analogous to different solar
radiative heating rates on the atmospheric lower boundary. The theory (32)
shows that $F_{s}$ influences $h_{1}$ in two competing mechanisms:

  1. 1.

A higher $F_{s}$ reduces the critical syrup concentration necessary to
initiate boiling, $S_{*}$ [Eq. (28)]. It should make bubbles larger and
increase $h_{1}$.

  2. 2.

A higher $F_{s}$ reduces the time interval between vortex rings and increases
the turbulence intensity in the syrup layer. It should increase $C_{E}$, trap
more vortex rings, reduce entrainment, and reduce $h_{1}$.

Figure 9: The post-boiling interface height $h_{1}$ of (a) experiments F1-F5
that vary the surface heat flux, (b) experiments S1-S7 that vary the initial
syrup concentration $S_{0}$, and (c) experiments T1-T7 that vary the initial
syrup thickness $h_{0}$. The blue shadings in (c) show the $h_{0}<0.5$ cm
regime where the post-boiling state lacks a clear interface and the $h_{0}>4$
cm steady boiling regime. The blue dots show the experimental result, and the
red lines show the theoretical prediction with the best-fit parameters.

Experiments show that $h_{1}$ decreases with $F_{s}$ (Fig. 9a). It indicates
that mechanism (ii) should play a more important role than mechanism (i). For
future work, we need to answer why this is the case by quantitatively modeling
how $C_{E}$ depends on $F_{s}$. It involves a careful analysis of vortex ring
interaction.

###  5.2 Experiments with varying $S_{0}$

The initial syrup concentration ($S_{0}$) is analogous to the atmospheric
stratification near the boundary layer top. The theory predicts that $S_{0}$
influences $h_{1}$ with two competing mechanisms:

  1. 1.

A higher $S_{0}$ increases the viscosity. It reduces the convective
ventilation of the syrup layer, enhances the superheating, and increases the
vortex ring radius $R$ [Eq. (31)]. A higher $R$ makes the vortex ring less
influenced by drag and trapping, makes the vortex ring penetrate deeper, and
increases $h_{1}$ [Eq. (32)].

  2. 2.

A higher $S_{0}$ makes the vortex rings more negatively buoyant, penetrate
less deep, and reduce $h_{1}$.

For the experimental results, $h_{1}$ first increases with $S_{0}$ and then
decreases, yielding an $S_{0}$ around 0.3 (Fig. 9b) that maximizes $h_{1}$. We
call it an optimal $S_{0}$. Thus, for the relatively dilute regime
($S_{0}\lesssim 0.3$), the radius effect dominates. For the relatively dense
regime ($S_{0}\gtrsim 0.3$), the buoyancy effect dominates.

The red lines of Fig. 9b show the quantitative prediction of $h_{1}$ using
(32). We use $h_{0}=2$ cm. The values of the four unknown parameters are
prescribed as $\alpha\beta=0.25$, $h_{DE}=1.35$ cm, $\delta S=0.05$,
$S_{*}=0.25$. This is a set of best-fit parameters, which makes the theory
agree well with the experiments. The sensitivity to the values of the four
parameters, perturbed with $\pm 20\%$ magnitude, is tested in the first row of
Fig. 10, showing the trend is robust. The optimal $S_{0}$ mainly depends on
$S_{*}$ and $\delta S_{*}$, with a higher $S_{*}$ and higher $\delta S_{*}$
shifting the optimal $S_{0}$ to larger values.

Figure 10: The first row shows the theoretical prediction of the $h_{1}$ vs.
$S_{0}$ relation with perturbed parameters. The experimental results are blue
circles, and the theoretical curves are solid red lines. (a) Varying $h_{DE}$.
(b) Varying $\delta S_{*}$. (c) Varying $\alpha\beta$. (d) Varying $S_{*}$.
The perturbation magnitude is $\pm 20\%$. The second row is the same as the
first but for the $h_{1}$ vs. $h_{0}$ relation. The blue shadings show the
$h_{0}<0.5$ cm and $h_{0}>4$ cm regimes where $h_{1}$ is not well-defined.

###  5.3 Experiments with varying $h_{0}$

The initial syrup thickness ($h_{0}$) represents the thickness of the
atmospheric boundary layer. The theory predicts that $h_{0}$ influences
$h_{1}$ with three competing mechanisms:

  1. 1.

Obviously, a higher $h_{0}$ makes $h_{1}$ higher given the same entrainment
amount ($h_{1}-h_{0}$).

  2. 2.

A higher $h_{0}$ increases the boiling duration time $\Delta t$ because it
takes longer to remove a thicker bottom syrup layer by detrainment [Eq. (6)].
This effect increases $h_{1}$.

  3. 3.

A higher $h_{0}$ increases the path for a vortex ring to be influenced by the
drag and turbulence in the syrup layer, reducing its escape ratio $E$ and
penetration depth $l$ [Eqs. (7) and (13)]. Thus, vortex rings entrain less and
should yield a lower $h_{1}$.

For the experimental results, $h_{1}$ slightly increases with $h_{0}$ (Fig.
9c), indicating that the three mechanisms roughly balance each other, and a
higher $h_{0}$ yields a less efficient dilution of the syrup layer by boiling.
Using the same set of parameters as in section 5.2 and the measured $h_{0}$,
the trend is captured by the theory (Fig. 9c). The theoretical trend is also
robust in the 20% range sensitivity tests (Fig. 10). Experiments with
$h_{0}\lesssim 0.5$ cm yield an overly dilute post-boiling state that directly
transitions to a well-mixed state. The post-boiling interface in our T1
experiment is marginally distinguishable. Experiments with $h_{0}\gtrsim 4$
cm, including our T7 experiment (Fig. 8s), are in a steady boiling regime
where boiling is continuous and the interfacial rising rate is steady. We name
the regime with a clear end-of-boiling state the intermittent boiling regime.

##  6 Transition between the intermittent and steady boiling regimes

###  6.1 Solving the transitional $h_{0}$

We observed a steady boiling regime for a relatively high $h_{0}$, a regime
which is not considered in the theory of section 4. Below, we extend the
theory to include the steady boiling regime and study the transition between
the intermittent and steady boiling regimes.

In the steady boiling regime, the thick syrup layer sufficiently dissipates
the vortex rings, reduces their penetrating depth, and limits the entrainment
of the colder water. As a result, entrainment is maintained at the minimum
rate that keeps the syrup temperature around $100^{\circ}$C:

$\displaystyle\frac{dh}{dt}=\frac{F_{s}}{\rho_{w}c_{w}\left(T_{*}-T_{w}\right)}.$
(34)

Latent heating does not appear in (34) because all bubbles condense in the
syrup layer. The latent heat absorption and release balances. Using
$F_{s}\approx 20$ kW m-2, $\rho_{w}c_{w}\approx 4\times 10^{6}$ J m-3 K-1, and
$T_{*}-T_{w}=70^{\circ}$C, we predict a 5.7 cm rise of the interface in 800 s,
which is close to the approximately 5 cm rise in 800 s shown in Fig. 8s.

Next, we study what controls the transitional $h_{0}$. If the net entrainment
rate by vortex rings [shown in Eq. (17)] is higher than that required to keep
the syrup temperature around $100^{\circ}$C, boiling should be intermittent:

$\displaystyle\mathrm{Intermittent\,\,when}:\quad
6\,\frac{l}{R}\,E\,\overline{w_{+}}\,>\frac{F_{s}}{\rho_{w}c_{w}\left(T_{*}-T_{w}\right)},$
(35)

where we have used (17) and (34). Here, we must solve for the mean detrainment
flux from the bottom syrup layer, $\overline{w_{+}}$, a quantity canceled out
in solving $h_{1}$. The $\overline{w_{+}}$ depends on the vaporization rate at
the bottom. Not all surface heating is used to vaporize water because the
turbulent heat transfer between the bottom syrup layer and the middle mixed
layers also cools the surface. We parameterize the ratio of vaporization
cooling rate to $F_{s}$ as a vaporization efficiency $\chi$, an unknown
parameter. We hypothesize that a lower vortex ring escape ratio $E$ enhances
the turbulent mixing within the syrup layer, i.e., across the two lowest
layers, and reduces $\chi$. The validation of this hypothesis is left for
future work. The flux of bubble number density, $N$ (unit: m-2 s-1), should
obey:

$\displaystyle N=\frac{F_{s}\ \chi}{\frac{4}{3}\pi R^{3}L_{v}\rho_{v}},$ (36)

where $L_{v}=2.5\times 10^{6}$ J kg-1 is the vaporization heat and
$\rho_{v}=0.6$ kg m-3 is the density of vapor. The added mass argument
introduced in section 4.2 indicates that the displaced volume of syrup around
a bubble is half its volume ($\frac{2}{3}\pi R^{3}$), hence

$\displaystyle\overline{w_{+}}=\frac{2}{3}\pi R^{3}N=\frac{F_{s}\
\chi}{2L_{v}\rho_{v}}.$ (37)

We need to constrain $\chi$ from experiments. Figure 3c shows that the
interface between the bottom syrup layer and the middle mixed layer drops from
2 cm to 0 cm in around 60 s, indicating $\overline{w_{+}}\approx 3.3\times
10^{-4}$ m s-1. To fit (37), the $\chi$ is constrained to be
$\chi=2L_{v}\rho_{v}\overline{w_{+}}/F_{s}\approx 0.05$, where we have used
$F_{s}\approx 20$ kW m-2. Thus, vaporization should play a minor role compared
to turbulent mixing in cooling the syrup layer’s bottom. In other words, the
mechanical removal of superheating by mixing is more important than the phase
change effect. Substituting the expressions for $\overline{w_{+}}$ (37), $R$
(31), $l$ (13), and $E$ (7) into (35), we obtain the critical $h_{0}$ for
transitioning to steady boiling:

$\displaystyle\mathrm{Steady\,\,when}:h_{0}>h_{\perp}(S_{0}),\quad
h_{\perp}(S_{0})=h_{DE}\ln\left[\frac{2\alpha\beta\chi}{\gamma_{s}S_{0}}\frac{\rho_{w}c_{w}\left(T_{*}-T_{w}\right)}{L_{v}\rho_{v}}\right],$
(38)

where $h_{\perp}(S_{0})$ is the transitional $h_{0}$ that is a function of
$S_{0}$. In deriving (38), we have assumed that the non-boiling regime
($S<S_{*}$) is sufficiently separated from the steady boiling regime by
letting $R=R_{m}$. Substituting in estimated values ($\alpha\beta=0.25$,
$\chi=0.05$, $\gamma_{s}=0.4$, $\rho_{w}c_{w}\approx 4\times 10^{6}$ J m-3
K-1, $T_{*}-T_{w}=70^{\circ}$C, $L_{v}=2.5\times 10^{6}$ J kg-1, and
$\rho_{v}=0.6$ kg m-3), we get $h_{\perp}(S_{0}=0.5)=4.25$ cm. It is close to
the experimental results where the critical $h_{0}$ lies between $h_{0}=3.40$
cm (T6) and $h_{0}=4.13$ cm (T7).

Equation (38) predicts that the critical initial thickness for steady boiling
$h_{\perp}$ is proportional to the dissipation length scale $h_{DE}$. The
proportional factor is higher for a smaller $S_{0}$ because a lighter syrup
penetrates deeper and entrains more. To verify the dependence of the critical
$h_{0}$ on $S_{0}$, we performed experiments ST1-ST4, as shown in Fig. 11. For
$S_{0}\approx 0.5$, the critical $h_{0}$ lies between $h_{0}=3.40$ cm (ST1)
and 4.13 cm (ST2). For $S_{0}\approx 0.35$, the critical $h_{0}$ lies between
$h_{0}=4.08$ cm (ST3) and 4.74 cm (ST4). Though 4.13 cm (ST2) is still
slightly above 4.08 cm (ST3), the former is clearly in the steady boiling
regime, and the latter is in the intermittent boiling regime, sufficiently
showing a strong influence of $S_{0}$. This confirms that a more dilute syrup
yields a higher critical $h_{0}$ for steady boiling. In summary, a smaller
$h_{0}$ and $S_{0}$ increase the entrainment in boiling and make it more
intermittent.

###  6.2 System evolution in phase space

Figure 11: The same as Fig. 8, but for experiments ST1-ST4 that vary both
$S_{0}$ and $h_{0}$ to study the critical $h_{0}$. ST2 and ST4 enter the
steady boiling regime at the first boiling onset. ST1 and ST3 enter it at the
second onset.

The above discussions are for the first onset of boiling. We can analyze the
second, third, and even $n^{th}$ onset using the same theoretical framework by
taking $h_{1}$ as the new initial condition $h_{0}$. Some experiments with
intermittent boiling at the first onset enter steady boiling after the syrup-
layer temperature recovers to the boiling point. They include T5, T6 (ST1),
and ST3, where either $S_{0}$ or $h_{0}$ is relatively large (Fig. 8). For
other experiments, the first onset of boiling makes the syrup layer too dilute
to restore enough heat and boil again (i.e., $S<S_{*}$).

We summarize the system evolution in the phase space of the instantaneous
syrup concentration $S$ and syrup-layer thickness $h$, as shown in Fig. 12.
Assuming the syrup layer is diluted by entraining freshwater and no syrup is
released into the freshwater layer, there should be conservation of the total
syrup in the syrup layer:

$\displaystyle S\,h=S_{0}\,h_{0},$ (39)

which sets the system’s trajectory in the phase space as an inverse
proportional function. The parameter space is divided into four regimes:

  1. 1.

The non-boiling one-layer regime ($S<S_{min}$), where a two-layer
configuration is convectively unstable.

  2. 2.

The non-boiling two-layer regime ($S_{min}<S<S_{*}$), where the convective
heat transfer prevents the syrup layer from boiling.

  3. 3.

The intermittent boiling regime ($S>S_{*}$ and $h<h_{\perp}$).

  4. 4.

The steady boiling regime ($S>S_{*}$ and $h>h_{\perp}$).

Here, the expressions of $S_{min}$ and $h_{\perp}$ are documented in (4) and
(38).

Figure 12: The system evolution in the $S$-$h$ phase space with four regimes.
Phase trajectories of experiments S1-S7, T1-T7, and ST1-ST4 are plotted. Blue
trajectories denote the experiments with intermittent boiling at the first
onset. Red trajectories denote the experiments with steady boiling at the
first onset, and no ending of the trajectory is set. The trajectories are
assumed to obey $h=h_{0}S_{0}/S$, with the dots denoting $h=h_{0}$ and the
circles denoting $h=h_{1}$. Generally, a smaller $h$ or $S$ enhances the
boiling entrainment.

##  7 Discussion and conclusions

This paper has introduced a novel experiment, boiling stratified flow, as an
analogy to vertical mixing induced by atmospheric moist convection. A thin
layer of syrup in a beaker, analogous to the atmospheric boundary layer, is
heated beneath a thick layer of freshwater, analogous to the free troposphere.
The temperature in the experiment is analogous to the atmospheric humidity,
and the threshold behavior imposed by the boiling point of the syrup solution
is analogous to the saturation vapor mixing ratio.

We found that when the initial syrup concentration $S_{0}$ and the syrup layer
thickness $h_{0}$ are relatively small (but $S_{0}$ is not too small, see Fig.
12), the system is in an intermittent boiling regime. The bubbles generated at
the bottom of the beaker quench on their way up and drive vortex rings that
penetrate the syrup-water interface, mix with water, and sink to the
interface, producing a middle mixed layer that lies above the bottom syrup
layer. The bottom syrup layer gradually diminishes due to the mass detrainment
by vortex rings. The relatively cold middle mixed layer then contacts the
bottom and ends boiling. Boiling is intermittent because more cold water is
entrained into the syrup layer than is needed to remove superheating.

We constructed a simple model of entrainment in a boiling event, which is
quantified with the interface growth $h_{1}-h_{0}$. Key quantities include the
escape ratio $E$, which measures the fraction of vortex rings that escape the
syrup layer against the disturbance by turbulence, and the vortex ring
penetration depth $l$, which determines the amount of freshwater a vortex ring
can entrain. The model explains the trend of experiments with varying surface
heat flux $F_{s}$, the initial syrup concentration $S_{0}$, and initial syrup
layer height $h_{0}$. The post-boiling interface height $h_{1}$ decreases with
increasing $F_{s}$ because the higher surface heating raises the bubble number
density and the turbulent strength in the syrup layer and traps more vortex
rings. The height $h_{1}$ is non-monotonic with $S_{0}$. For $S_{0}\lesssim
0.3$, $h_{1}$ increases with $S_{0}$ due to the stronger superheating and the
larger bubble radius. For $S_{0}\gtrsim 0.3$, $h_{1}$ decreases with $S_{0}$
due to the more negative buoyancy of the vortex ring. The height $h_{1}$ is
relatively insensitive to $h_{0}$ because a thicker syrup layer (higher
$h_{0}$) makes a vortex ring more susceptible to drag and trapping, reducing
its penetration depth $l$. This reduces $h_{1}-h_{0}$, keeping
$h_{0}+(h_{1}-h_{0})$ approximately constant. We quantitatively modeled the
dependence of $h_{1}$ on $S_{0}$ and $h_{0}$. When the four unknown parameters
take the best-fit values, the agreement with experiments is very good.

When $S_{0}$ and $h_{0}$ are relatively large, the entrainment rate drops to
the minimum value for removing superheating in the syrup layer. This yields a
steady regime where boiling is continuous. The overshooting vortex rings
continuously entrain cold water into the syrup layer. By matching the
intermittent and steady regimes theory, our transition curve predicts that
smaller $S_{0}$ raises the transitional $h_{0}$ to steady boiling, which
qualitatively agrees with experiments.

The experiments raised a number of questions that require further
investigation, for which direct numerical simulation might be helpful. First,
what determines the escape ratio $E$? How is this quantity influenced by the
interaction of vortex rings and surface heating? Second, what determines the
fraction of surface heating used for vaporization $\chi$? It depends on the
turbulent strength in the syrup layer and may indirectly depend on $E$. These
questions lead to a more fundamental question: how does the vertical eddy
diffusivity change with height, and how can it be parameterized?

Finally, let us note again that the boiling stratified flow does differ from
atmospheric convection in various ways. First, the direction of phase change
is the opposite of that of the atmosphere. In the experiment, buoyancy is
gained by vaporization and lost by condensation. There is latent heat
absorption when the bubble forms and release when the bubble quenches.
Although the temperature effect on buoyancy plays a minor role compared to the
bubble volume change, they make the analogy less straightforward. Second, the
bubble nucleation in the experiment is different from the initiation of
shallow non-precipitating cumuli in the atmosphere (Fig. 1), not to mention
the more complicated precipitating deep convection. Shallow cumuli are
produced by thermals in the convective boundary layer and have regular spatial
patterns (e.g., Stull, 1985; Öktem & Romps, 2021). There are also convective
cells in the syrup layer, but bubble formation in the experiment is more
likely associated with imperfections on the beaker’s surface. Third, the
viscosity of the syrup layer plays an important role in suppressing the
interfacial heat transfer, which strengthens the sub-cooled boiling. The
atmospheric boundary layer has a much higher Reynolds number than the syrup
layer, and atmospheric viscosity does not play such a special role. Despite
these limitations, we feel that the boiling stratified flow may be an
enlightening model for studying the lifecycle and mixing process of
atmospheric convection. In addition, the experiment is easy to set up and
could be used as an educational demonstration to illustrate the feedbacks
between convection, entrainment and stratification.

[Supplementary data] Illustrative experimental videos, raw experimental
videos, raw temperature records, and data postprocessing codes are deposited
at https://doi.org/10.5281/zenodo.11222909.

[Acknowledgements]We thank Yunjiao Pu for providing experimental support at
the kitchen stage of this project. We thank Jim McElwaine, Anders Jensen, and
Bruce Sutherland for critical experimental support. We thank Quentin Kriaa,
Keaton Burns, Wanying Kang, Detlef Lohse, Xi Zhang, Huazhi Ge, Zhiming Kuang,
Wojciech Grabowski, Hugh Morrison, and Morgan O’Neill for stimulating
discussions.

[Funding]We acknowledge the National Science Foundation and the Office of
Naval Research for their support of the 2023 WHOI Geophysical Fluid Dynamics
Summer Program, where the experiments were performed. AL is funded by a NERC
Independent Research Fellowship NE/W008971/1.

[Declaration of interests]The authors report no conflict of interest.

[Data availability statement]All the data is deposited in the supplemental
material.

##  Appendix A Experimental details

The beaker (model: Karter Scientific 213D20) has a volume of 2000 ml and a
diameter of approximately 130 mm. In all experiments, the volume of freshwater
is fixed to 1400 ml. The beaker is heated by an electric hot plate (model:
SUNAV-HP102-D2, 1500 W power for 110 V voltage). The heating power is
controlled by a voltage regulator (brand: VEVOR), which has a $\pm 2$ V
fluctuation. The dark corn syrup (brand: Golden Barrel) has a dextrose
equivalent of 42 and a density of $\rho_{s,max}\approx 1.4\times 10^{3}$ kg
m-3. The kinematic viscosity of syrup increases approximately exponentially
with its concentration [Table A1.8 of Mohos (2017)]. Having tried to replace
the syrup layer with a much less viscous nearly saturated sodium chloride
(NaCl) solution of density $1.15\times 10^{3}$ kg m-3, we found that the two-
layer stratification is eroded before boiling occurs.

The illumination for imaging is provided by a desk lamp diffused by a 3 mm
white acrylic sheet. The light transmitted through the beaker is recorded by
the camera of an iPhone 11. The temperature is recorded with K-type
thermocouples (model: NUZAMAS) plugged into a temperature recorder (model:
Gain Express). The temperature resolution is 0.1 ∘C, and the accuracy is
$\pm(1^{\circ}\mathrm{C}+0.3\%\mathrm{rdg})$ between 18 ∘C and 28 ∘C. The four
sensors are bound by heat-shrink tubes and fixed to a portable retort stand.

The experimental procedure has three steps.

Step 1: Prepare the solution. First, we add approximately 1400 ml of tap water
to the beaker. Then, we use an injector to add syrup to the bottom of the
beaker manually. The injecting process unavoidably causes mixing and dilutes
the syrup. As a rule, the injection stops when the syrup layer reaches the
required $h_{0}$ scale on the beaker, even if the injected syrup is less than
expected. Given a desired $h_{0}$, the volume of the syrup layer is known, but
due to mixing, we use less than 100% but typically $>75$% of the syrup. Once
the two-layer stratification is set, we use a portable resonant density meter
(model: Anton Paar, DMA 35) to measure the syrup density near the bottom of
the beaker.

Step 2: Start heating. We move the beaker onto the heating pad and put the
temperature sensor array into it. Then, we turn on the heating pad, whose
power is approximately fixed and controlled by the voltage regulator. The
surface heat flux is measured via temperature rising rate in a specially
designed experiment. It uses 86 V heating and 1000 ml of water, and the beaker
is sealed on top by a plastic membrane to insulate heat. The surface heat flux
at the bottom of the beaker is measured to be around 19 kW m-2. The heat flux
for different voltages is calculated with Ohm’s law based on this reference
value.

Step 3: Cool the device. After an experiment is finished, we cool down the
heating pad to be close to the room temperature before starting another
experiment. An experimental cycle takes around two hours.

It is worth noticing that the experiments are sensitive to the geometry of the
beaker. Ideally, we need a beaker whose bottom is not uniformly heated, which
permits local superheating. The concept of superheating denotes the case where
the liquid’s temperature exceeds the boiling point. For our beaker, heating is
strongest on a ring near the lateral boundary. This steady heating ring
produces large bubbles that mix efficiently, leading to intermittent boiling.
One consequence of the ring-intensified heating is that some boiling plumes
are observed to crash onto the lateral wall. We have tried a 3000 ml beaker
from another brand (model: ULAB, UBG1029) with a more uniform surface and an
electric kettle with a perfectly uniform metal surface (model: COSORI,
GK172-CO). They steadily produce tiny bubbles, a regime classified as steady
boiling in section 6. Because we are particularly interested in the
intermittent boiling regime with larger bubbles, which is more relevant to
cumulus convection, we decided to use the beaker with a more nonuniform
surface. Despite the sensitivity to the container, given the same beaker, the
results are repeatable and robust.

##  Appendix B Diagnostic method of the post-boiling interface height

The diagnosis of the post-boiling interface height $h_{1}$ has three steps.

Step 1: Identify the syrup-water interface, essentially the top of the middle
mixed layer (Fig. 5). The interface appears as a jump of the horizontally
averaged pixel intensity value in the vertical direction. To facilitate the
diagnosis of the jump, we vertically smooth the pixel value with a Gaussian
filter whose stencil spans 20 pixels at each time snapshot. The width of a
pixel, which depends on the distance of the camera to the beaker, is around
0.03 cm. The height where the vertical gradient of the smoothed pixel value is
the largest is identified as the interface. This operation renders a time
series of the interface height, $h(t)$, shown as the solid black line in Fig.
8.

Step 2: Identify the boiling start time. We make a temporal Gaussian filter on
$h(t)$ with a stencil of 20 snapshots (a time span of 101.8 s) and denote it
as $\widetilde{h}(t)$. The initial value of $\widetilde{h}(t)$ is taken as
$h_{0}$. The boiling start time is taken as the time by which
$\widetilde{h}(t)$ first rises above $1.1h_{0}$.

Step 3: Identify the boiling end time using $\widetilde{h}(t)$. We build a
moving window spanning 20 snapshots (101.8 s) and move it from the boiling
start time. The range (maximum minus minimum) of $\widetilde{h}(t)$ in the
window gradually decreases as the window approaches the post-boiling stage. We
let the time by which the range first drops below 0.3 cm as the boiling end
time and denoted it as $h_{1}$. The $h_{1}$ is shown as the dashed red line in
Fig. 8 and summarized in Fig. 9. The 0.3 cm threshold renders a reasonable
diagnosis of $h_{1}$ that agrees with visual inspection. There are two
exceptions. One is experiment S1 (Fig. 8f), where no significant boiling
occurs, and the system directly transitions to a well-mixed state after a long
time (about 1400 s). The $h_{1}$ is taken as $h_{0}$. The other is experiment
T7 (Fig. 8s), where boiling is steady, and no boiling end time is found.

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