NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-23-319-2016Intermittent heat instabilities in an air plumeLe MouëlJean-LouisKossobokovVladimir G.volodya@mitp.ruhttps://orcid.org/0000-0002-3505-7803PerrierFredericMoratPierreGeomagnetism, Institut de Physique du Globe de Paris, 1 rue Jussieu,
75238 Paris, CEDEX 05, FranceInstitute of Earthquake Prediction Theory and Mathematical Geophysics,
Russian Academy of Sciences, 84/32 Profsoyuznaya Street, 117997 Moscow,
Russian FederationVladimir G. Kossobokov (volodya@mitp.ru)30August201623431933018March201630March20166July201618July2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/23/319/2016/npg-23-319-2016.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/23/319/2016/npg-23-319-2016.pdf
We report the results of heating experiments carried out in an abandoned
limestone quarry close to Paris, in an isolated room of a volume of about
400 m3. A heat source made of a metallic resistor of power
100 W was installed on the floor of the room, at distance from the
walls. High-quality temperature sensors, with a response time of
20 s, were fixed on a 2 m long bar. In a series of 24 h heating
experiments the bar had been set up horizontally at different heights or
vertically along the axis of the plume to record changes in temperature
distribution with a sampling time varying from 20 to 120 s. When
taken in averages over 24 h, the temperatures present the classical
shape of steady-state plumes, as described by classical models. On the
contrary, the temperature time series show a rich dynamic plume flow with
intermittent trains of oscillations, spatially coherent, of large amplitude
and a period around 400 s, separated by intervals of relative
quiescence whose duration can reach several hours. To our knowledge, no
specific theory is available to explain this behavior, which appears to be a
chaotic interaction between a turbulent plume and a stratified environment.
The observed behavior, with first-order factorization of a smooth spatial
function with a global temporal intermittent function, could be a universal
feature of some turbulent plumes in geophysical environments.
Introduction
Thermal plumes, columns of hot fluid that rise above a
localized heat source, have received, like jets, a lot of attention (e.g.,
Turner, 1973). Numerous processes on Earth and in small-scale
environmental sites require the description of the effect of injecting heat
or matter from natural and/or industrial sources into a stationary organized
system, such as an ocean, a lake, or the atmosphere (Woods, 2010).
Models of plumes (and jets) have existed for decades. Models rely on a
turbulent entrainment of ambient fluid in a shear layer within the edges of
the plume and a hypothesis of complete similarity along the downstream axis
OZ of the plume. In these
contributions, the entrainment rate of ambient fluid is proportional to the
vertical velocity w (along OZ), with the same entrainment constant
αe along OZ. Such is the case in the classical model of
Morton et al. (1956), who used three conservation equations (for fluxes of
mass, momentum and buoyancy) to obtain the expressions of the temperature
difference, plume radius, and mean velocity along OZ, w(z). Fischer et
al. (1979) got the same expressions using essentially dimensional analysis.
In fact, this hypothesis of complete similarity is strong and debated, not
supported by all experiments. Recently, Crouzeix et al. (2003) resumed the
study of the similarity, using available experimental data and concluded in
favor of local states of partial self-similarity, in accordance with the
theoretical analyses (George Jr. et al., 1977; George, 1989) and evolving
along the z coordinate according to a universal route.
All those models are for stationary, time-independent jets or plumes. While
we will still consider the stationary states of plumes in confined
environments, we will now turn to the spatiotemporal structures, inside the
plumes, and in their immediate vicinity, and reveal a dynamically chaotic
plume. Thus, our main interest in the present paper is the fluctuations
around the mean values representing the stationary states. Our sampling in
time and space, however limited, allows us to describe the main features of
this dynamic flow, whose observation, to our knowledge, is unprecedented.
To demonstrate how the temperature time series disclose rich dynamics of a
plume in an enclosed stratified environment, we first describe the setup of
the heating experiments in an isolated room of the abandoned limestone quarry
in Sect. 2, then the results of a series of the experiments with different
configuration of the measurement device in Sect. 3, which show the
development of the plume flow with a clear presence of systematic
instabilities, expressed with temperature pulses of high amplitude,
interspersed with long periods of stability. We check that the observed
average temperatures are in a satisfactory agreement with classical models of
stationary plumes (Sect. 4), as well as with some features of self-similarity
(Sect. 5). The discussion and conclusion are given in Sect. 6, which compares
plume dynamics observed in other geophysical conditions and environments.
The heating experiments in the Vincennes quarryThe Vincennes quarry
Various thermodynamic experiments were carried out from February 2002 to
April 2005 in an abandoned quarry located in Vincennes, close to Paris
(Crouzeix et al., 2003). The Vincennes quarry has a surface of
32 000 m2, and its ceiling is at a depth varying from 14 to
20 m. The walls show a section of the different tabular layers that
had been exploited in a Lutetian formation, mostly limestone beds. The
quarry consists of corridors and rooms separated by pillars of different
sizes, most of them supporting the roof. Room shapes are roughly rectangular;
the height of their ceiling varies between 1.5 and 7 m (the distance
from the ceiling to the surface of fillings, about 2 m thick covering the
rock floor). The total air volume in the quarry, difficult to evaluate
precisely because of unexplored and collapsed sections, is estimated to be
around 80 000 m3. The quarry is connected to the ground surface by
a single large access pit with 4.56 m diameter (Perrier et al., 2002, 2005;
Perrier and Le Mouël, 2016).
Temperature had been measured in the quarry since 2001, giving in 2003–2005
an annual mean temperature in the range from 12 to 12.2 ∘C with a
seasonal variation of the order of 0.8 ∘C related to natural
ventilation through the access pit (Perrier et al., 2004; Perrier and Richon,
2010). At times smaller than a day, temperature variations are mostly due to
variations of atmospheric pressure (Perrier et al., 2001), and do not exceed
0.03 ∘C peak to peak in the absence of perturbations (visits,
heating experiments). As in most underground systems, relative humidity in
the atmosphere of the quarry is high (99.2–99.8 %).
Sketch of the experiment room S15 in the Vincennes quarry showing
the location of temperature setups AI, AII, AIII, and S, and of the
insulating wall installed in July 2003.
Among various observations of temperature, we took advantage of the
exceptional conditions of stability offered by such a large quarry to
conduct, among other types of experiments, a long series of measurements of
temperature on a plume set up in an isolated room of a volume of about
400 m3. This set of long duration heating experiments is the subject
of the present paper.
Configurations of the experiments carried out in March–June 2003.
Each was performed with a heat source of 100 W and lasted
24 h. AIII position indicates the device orientation: V for vertical
and Hz for horizontal at altitude z.
The room S15 selected for the heating experiments has a surface of about
24 m by 9 m (Fig. 1) with an average height of about
2 m (Fig. 2). Air exchanges with the rest of the quarry proceed
through two entrances of about 3 and 3.5 m wide (Fig. 1). In order to reduce
ventilation effects, the experiments were performed in the inner part of the
room at distance from the entrances. On 2 July 2003, this part of about
12 m by 9 m in surface was insulated with a wall made of
Styrofoam. However, most experiments reported in this paper were performed
before the installation of the partition wall. Nevertheless, the natural
ventilation of the section of room S15 used for the experiments is small,
with a ventilation rate estimated to be of the order of 2×10-6s-1 (Perrier and Richon, 2010). We presume that the air
exchanges in room S15 with such a low ventilation rate are negligible in
relation to the natural stratification of the air in its volume of about
400 m3. Having in mind the location and dimensions of a resulting
plume in our heating experiments, we may also consider the effect
of the distant walls negligible, while the ceiling acts as the limiting boundary in
the system. Naturally, the ceiling, walls, and the floor act as cooling
elements of the system where hot air rising from the heat source spreads
through the volume of the room.
Temperature measurements by calibrated thermistors
We will consider space and time variations of the temperature with an
amplitude as small as 0.01 ∘C, even a few 0.001 ∘C, so we
need very sensitive sensors. For this reason, and because the experiment
relies on temperature data, we will briefly describe the sensors and their
calibration. The three types of temperature sensors commonly used are
thermocouples, resistance temperature detectors (RTDs), and thermistors.
Thermistors are very sensitive, are cheap (one can buy and use many of them),
and present a weak noise. However, their response time can be somewhat long,
and must be known; they are not very stable on the long term, but we consider
rather short time constant relative variations (see below). A thermistor
is made of semi-conductor materials whose electric resistance R decreases
monotonously when the temperature increases. The relation between temperature
T and R is strongly nonlinear, often written in the form T-1=A+BlnR+C(lnR)2, where A, B, and C are coefficients to be
determined. In the present study, we measure temperature space and time
variations smaller than a few degrees, simultaneously at 10 or 12 locations
(see Sect. 2.3). What we need is an accurate intercalibration of n
thermistors, or n thermistors plus a unique reference one. In other words,
the relative calibration of a series of n thermistors consists in reducing
the value of the resistance Ri (i=1,2,…,n) of thermistor to
the value Rref of the reference thermistor at the same temperature.
For that all the thermistors are plunged in a bath of uniform temperature,
controlled by a thermometer with 0.001 ∘C accuracy. From a number of
measurements in the bath, we determine the coefficients of the simplified
relationship:
Rref=ai+biRi,i=1,2,…,n.
As a result, the temperature variations measured in the experiments, at all
locations i, after using relation in Eq. (1) are the same as if measured by the
reference thermistor within 0.005 ∘C (Crouzeix et al., 2003). The
response time of the thermistors is in the range of 15–20 s,
depending on the type. The sampling interval of the records is 20, 40, or
120 s (Table 1).
(a) Schematic cross section of the experimental room S15
showing the four setups (AI, AII, AIII, and S) along with the positions of
the temperature sensors (1–10 for each set-up); (b, c) photos of
the setup AIII in advance of the heating experiments A2 and A4 (Table 1).
Configuration of the measurement device of the experiment
Four setups instrumented with 10 thermistors each were installed in a part
of the room at distance from entrances. The sensors, 30 mm long and about
5 mm in diameter, had an intrinsic response time of about
20 s. The precision of relative temperature measurements was found to
be about 0.005 ∘C (Crouzeix et al., 2003). One of the setups, AIII,
was placed just above the heat source and, unlike the other three, which
remained at the same positions, was configured differently in 13
heating experiments from 10 March to 9 June. Specifically, the 2 m bar with
thermistors was placed vertically in experiments A1–A3 to evaluate the
effect of the source at different heights, and then horizontally in
experiments A4–A7 successively at a height of 1, 2, and 1.5 m (see
Fig. 2 and Table 1). To study the possible contribution of sideway
radiations, a screen was placed around the source in experiments A3 and A5–A7
(Table 1). Moreover, the sampling rate of measurements in AIII varied from
once in 20 s in the first four experiments to once in 120 s
in the last two. The setups AI and AII were used to measure the temperature
in vertical air columns away from the walls, while the sensors of setup S
used for the same purpose were placed at about 1.5 cm inside the rock
of the ceiling and wall and in the filling of the floor. These latter
experiments, dedicated to studying the reaction of the room to the heating,
are outside the scope of the present paper, in which we focus on the experiments
studying the plume itself.
Most of the heating experiments (including all those considered in the
present study) lasted for 24 h, from Monday at 00:00 to Tuesday
at 00:00, when the heat source was switched on, and were separated from the
previous one by 6 days, when the heat source was switched off. Thus, the
heat source is continuously on during the whole duration of the heating
period. The heat source was a metallic resistor of power 100 W with a
rectangular surface of 5 by 7 cm and height of 15 cm.
Measurements in the laboratory have shown that the source reaches its maximum
temperature within 600 s (i.e., 10 min). During the
experiments, it was located about 3.4 m from the nearest back wall of
the room (Fig. 1). Except for one experiment (A2, see Table 1), the source
was put directly on the floor as indicated in Fig. 2. The thermal stability
of the source was studied during dedicated experiments with thermistors
directly attached to the source. The source did not show any significant
periodic variations and none of the effects reported below can be attributed
to the source itself.
Results: temperature fluctuations in the plume
We now present the data which are at the core of the present paper. The
temperature variations are measured along the 2 m long bar AIII, by
10 sensors (Fig. 3). In experiments A5a, A5b, A7a, A7b, A6a, and A6b the bar
is maintained horizontally, centered above the heating source, at a height of
1 m for A5, 1.5 m for A7, and 2 m for A6 above the
floor, respectively (Table 1). In experiment A3 the bar is maintained along
the vertical of the source. These experiments were all performed with a
screen around the source, in order to cancel or reduce radiative heating.
A previous study of the thermal stratification induced in the same room S15
by the heating (Crouzeix et al., 2006a) had shown that temperature variations
in the environment outside the plume were small compared to temperature
variations in the plume itself; we will consider that the plume is in an
environment with a uniform temperature T0. In the following graphs the
temperatures are estimated from the reference temperature of the environment
T0, ΔT(t)=T(t)-T0.
The temperatures in ∘C at 1 m height 1 h before,
during, and 2 h after the A5a experiment. T1–T10 denote temperature
series of the 10 sensors on the AIII setup.
The horizontal recordings at 1.00 m
Figure 3 shows the recordings of temperature at a height of 1 m, made
from 11:00 on 7 April to 14:00 on 8 April 2003, 1 h before, during,
and 2 h after the heating experiment A5a, using a sampling interval
of 40 s. After a transient phase of about an hour from switching on
the heating source, the curves take the form of regular spikes of high
amplitude at sensor 5 (up to 1.5 ∘C), smaller but still high on
sensors 4 and 6. Note that the curves 4 and 6 are not always exactly
in proportion to each
other, which is presumably an effect of shifts of the plume, and that the
spikes are present with smaller and smaller amplitudes on all the curves.
The duration of the temperature peaks of instability is close to
360 s (i.e., 6 min). They appear either isolated, in pairs,
or in trains; when a peak is isolated, or is the last of a train, the
declining phase of instability is longer than 360 s. Between these
temperature instabilities phases of stability are present, sometimes longer
than 5 h (i.e., 18 000 s).
The temperatures 3 h before, during, and 3 h after
the A5a heating experiment in space–time domain: temperature color-coded in
∘C; on the vertical axis the locations of the sensors are shown on the
horizontal 2 m bar; temporal tick marks on the horizontal axis correspond to
3600 s= 1 h.
Figure 4 displays the temperature data from the same experiment in the form
of a color-coded contour space–time map (level lines) over the time interval
from 09:00 on 7 April to 15:00 on 8 April 2003. The color scale indicates the
amplitude of the measured temperatures. We observe more vividly the trains of
nearly regular strong heat pulses along with their spread along the
horizontal device. These heat pulses of instability should not be taken as
physical drops of air observed at a given time.
Let us zoom in on the train of instabilities of 7 April, from 18:00 to 20:00
(Fig. 5, upper panel). The peaks take the form of localized pulses, each
about 380 s. Look at the abrupt shift of the plume at about 18:30.
Another illustration comes from the same experiment A5a from 04:00 to at
06:00 on 8 April (Fig. 5, lower panel). One can see six isolated pulses with
peak-to-peak interevent times of 420, 480, 840, 960, and 1360 s
(which are suggestive of a likely “doubling of period”) followed by a train
of at least six of apparently connected pulses whose mean duration is about
320 s.
The temperatures in the space–time domain during 2 h of the
A5a heating experiment when a long train (upper panel) and a likely
“doubling of period” (first half of the lower panel) of strong heat pulses
are observed. Times of the start and finish are indicated for each
panel.
The temperatures in ∘C at 1.5 m height 1 h before,
during, and 2 h after the A7a experiment. T1–T10 denote
temperature series of the 10 sensors on the AIII setup.
The temperatures in space–time domain during 110 min of the
A7a heating experiment where a long “train” of strong heat pulses is
observed. The color-coding is the same as in Figs. 4 and 5.
The horizontal recordings at 1.50 m
A similar analysis of temperatures collected at the height of 1.50 m
was performed (experiments A7a on 26–27 May 2003 and A7b on 2–3 June
2003). Figure 6 shows the recordings from 11:00 on 26 May to 17:00 on 27 May.
Clearly, the general behavior of the recording is the same as at the height
of 1.00 m (Sect. 3.1), although the larger sampling rate of one per
120 s provides coarser curves. Nevertheless, the contrast between
active and quiet segments is smaller than in the case of temperatures at 1 m
height.
The temperatures in ∘C at 2 m height 1 h before,
during, and 3 h after the A6a experiment. T1–T10 denote
temperature series of the 10 sensors on the AIII setup.
A zoom on a train of the temperature peaks, in space–time representation, is
illustrated by Fig. 7. The mean peak-to-peak time for the eight pulses of the
train starting just before midnight (at about 23:58) is 480±40s. However, the individual durations of pulses range from 360 up
to about 600 s, which might be considered as a mixture of single and
double periods – or the first and the last heat pulses might be considered as
isolated ones. We point out again that in 2003 we were not able to make
simultaneous recordings, i.e., to perform a direct comparison of several
temperature recordings made at the same time at several positions above the
source.
The horizontal recordings at 2.00 m
Figure 8 shows the results of experiment A6a made on 21–22 April at the
height of 2 m. Again the recording presents periods with and without
peaks, with the largest amplitude at sensors 4 and 5 (in fact, very similar).
One observes a slow temperature increase from 13:00 to 19:53 on 21 April,
then a descent till 22:13, followed by a stable behavior till the end of the
heating phase at 12:00 on 22 April. At this 2 m height, the interaction of
the plume with the ceiling makes the situation somewhat more complex.
Figure 9 shows two zooms on the time intervals 18:53–20:33 on 21 April and
04:11–05:51 on 22 April. Despite the 120 s sampling, it is seen that, when
the oscillations are large enough, they are practically in phase at all
sensors: the temperature varies in the same way in the whole plume. In the
first interval we observe a train of 10 temperature pulses with duration of
about 410 s each, while in the second interval we observe two trains
of 5 and 4 pulses, respectively, with durations of about 440 and
425 s respectively. Figure 10 gives a representation of a long train
of 10 pulses from 19:12 to 20:20 on 21 April; their mean duration is 408±34s with a larger variability (from 240 to 600 s).
Moreover, we observe a companion series of instability pulses of heat
separated from the main one.
The temperatures during the A6a experiment heating in the periods
21 April 2003 at 18:50–20:40 (upper panel) and 22 April 2003 at 04:12–05:52
(lower panel): temperature is given in ∘C. T1–T10
denote temperature series of the 10 sensors on the AIII setup.
Temperatures in space–time domain during 110 min of the A6a
heating experiment where a long “train” of strong heat pulses is observed.
The color coding is the same as in Figs. 4 and 5. Note the presence of
concomitant pulses aside of the main series.
Temperatures before, during, and after the A3 heating experiment in
space–time domain: temperature color-coded in ∘C; on the vertical
axis the position of the sensors is represented on the vertical bar (from 10
at the bottom to 1 on the top).
Temperatures in space–time domain during the three 3 h
intervals (a) and 30 min(b) of the A3 heating
experiment.
Times of the 50 largest local maxima of the average temperature of
all the sensors on AIII bar (left), and the empirical cumulative distribution
function of their interevent times, in units of fractions of a day (right)
observed in the A5a, A7a, A6a, and A3 experiments. Time increases when going
up.
AIII device maintained vertically
In experiments A1, A2, and A3 (Table 1), the device AIII is held vertically,
with sensor T10 close to the heating source (placed on the floor). Results of
experiment A3 are displayed in Fig. 11, in the form of a space–time
presentation of the temperature level lines (as in Figs. 4, 5, 7, 10). At
each moment of time tk (tk=t0+20k in seconds) we report the
value of temperature T(hi,tk), hi being the height at location of
the ith sensor (i=1,2,…,10). A series of conspicuous
dilatations or contractions of the T sections appear all along the graph;
in general, the dilatations are sharp, the following contractions much
slower. Unfortunately, no simultaneous horizontal recording exists (a single
AIII device was available). Figure 12 shows 3 h (a) and
30 min (b) of zoom of Fig. 11.
As is evident from Fig. 11, in the first 4 min after switching on the
heat source at 12:00, the temperature T10 (i.e., the nearest to the source)
rises more slowly, while the temperatures T6–T8 grow faster than at any of the
other thermistors. At 12:09, the temperature T9 surpasses T6 and T7–T9 rise
to their maximal values of about 17–18 ∘C. At this moment of time T10
continues to grow steadily, while temperatures T7–T9 start falling down. At
12:16 T10 rises above all the others and by 12:20 it is 1.4–2.6 ∘C
higher than any of T1–T9. It appears that at about 12:30 the formation of a
plume proceeds to its final stage lasting for about 30 min followed
by rather regular dynamics with domination of T10 ∼ 16.5–17 and
T9 ∼ 15.4–16 ∘C. From 13:00 on, and till the end of heating
experiment, the average values of T10 and T9 are 16.9 ∘C with
σ=0.2 and 15.8 ∘C with σ=0.7∘C,
respectively. The difference in σ's allows for sporadic rise of T9
above T10, with cases exemplified in Fig. 12. In particular, one can
clearly observe rather quick propagation of the heat pulse through the entire
plume from the floor to the ceiling of the room S15 (Fig. 12b).
Joint study of the various time series
The intermittent nature of the various temperature signals is further studied
in Fig. 13. On the left side of the four panels in Fig. 13, the times of the
50 largest local maxima of the average temperature of all the sensors on the
AIII bar are shown for each of the three experiments performed at heights 1,
1.5, and 2 m (corresponding to experiments A5a, A7a, and A6a,
respectively) in horizontal position and the experiment (A3) in vertical
position. The empirical cumulative distribution function of the obtained 49
interevent times, expressed in fractions of a day, is plotted on the right
side of each panel. Table 2 summarizes a few statistics of these interevent
times, when split into two classes of short and long intervals by the unique
threshold of 720 s. Except for the bar in vertical position (first
line), the mean of the short times ranges from about 360 to 495 s,
while the mean of the long times is about 5 or more times larger, with the
largest ones lasting for a few hours. Overall, the statistics of the
interevent times is remarkably similar in all experiments.
Peak-to-peak interevent time statistics: ID identifies the 24 h
heating experiment; AIII indicates position of the 2 m bar: V for vertical
and Hz for horizontal at height z; Ns is the number of short
(Δt< 12 min) intervals; Nl is the number of long
(Δt> 12 min) intervals; E(Δts) is the average
duration of the short intervals, in s; E(Δtl) is the
average duration of the long intervals, in s; columns n1, n2, and
n3+ give the number of single, double, and multiple pulses,
correspondingly.
IDAIIINsNlE(Δts), sE(Δtl), smax(Δt), sn1n2n3+A3V2623227342118 8401536A5aH12920357374817 5601257A5bH12227395280120 84017104A7aH1.524254952400768020102A7bH1.52524432281511 2801946A6aH23019432218563601057A6cH23118449247375601346Averaged temperature profiles
Assuming a Gaussian shape for the horizontal profiles of temperature
difference ΔT(r,z)∼exp(-r2/b2(z)), r being the radial
distance from the maximum of ΔT(z) in the planes z= 1, 1.5, and
2 m, the “radius” b(z) can be estimated from the measurements; it
is found that this radius b(z) is larger and the temperature difference
ΔT(z) is several times lower than the values predicted by the Morton
et al. (1956) model. Horizontal profiles show that the mean plume is deviated
along the bar from the vertical of the source; it is likely that it also
deviates in the perpendicular horizontal direction (no data). Measured
temperatures should then be corrected before being compared with model
predictions. Phase changes of water can also be a cause for temperature
deviations being lower than predicted by the model, as well as some possible
inadequacy of the model.
Let us consider now in brief the average temperatures Th at height h
taken over the 24 h of each heating experiment. Vertical profiles
(Fig. 14a) present a negative gradient ΔTh=Th-T0 in the
lower part of the room, which turns positive in a layer about 50 cm thick
below the ceiling. On the horizontal profiles (Fig. 14b), maximal averaged
temperatures are observed on sensors 4 and 5, although the heat source is
located below sensor 6. As expected, the width of the plume becomes wider
when the AIII bar is placed higher, and the temperature profile presents the
classical bell-shaped form observed in thermal plumes generated by a point
source. A Gaussian form of the radial temperature profile T(r) in the plume
is often presupposed in stationary plume models, which focus on the variation
of mean width, velocity, and temperature versus the vertical coordinate (Morton
et al., 1956; Landau and Lifshitz, 1987; Tritton, 1988; Guyon et al., 2015).
(a) The difference ΔT=Th-T0 of the
average temperatures in 24 h during (Th) and 12 h before
(T0) a heating experiment with a vertical bar AIII (experiment, A3).
(b) The average temperatures on a horizontal bar AIII in
12 h before (blue), 24 h during (red), and 12 h after
a heating experiment at 2 m (top), 1.5 m (middle), and 1 m (bottom)
heights (experiments A6a, A6b, A7a, A7b, A5a, and A5b, correspondingly).
We will just make a few comments on our observations in the light of
stationary models. Plumes belong to the class of free convection flows,
maintained by temperature differences. Here we consequently follow the
qualitative rule of thumb proposed by numerous authors: “No velocity scale
is generated by the specification of a free convection parameter” (Tritton,
1988). Nevertheless, an estimate of the order of magnitude U of the
velocity can be obtained, with some caution, from the Navier–Stokes equation
for the steady state:
u∇u=-1ρ∇ρ+ν∇2u-gαΔTz,
where u is the velocity, z the upward vertical unit vector,
ρ the air density, ν its kinematic viscosity (0.15×10-4m2s-1), g the gravity acceleration
(∼ 10 ms-2), α the expansion coefficient (α∼T-1∼3×10-3K-1), and ΔT the temperature
anomaly. Taking the value of ΔT from Fig. 14b as 0.3 ∘C, we
get gαΔT∼10-2ms-1 (the buoyancy). From
u∇u∼U2/L∼gαΔT, it becomes U∼(gαΔTL)1/2, which gives U∼10-1ms-1.
For this estimate to be valid, we have to check a posteriori that the viscous
force is weak compared to the inertial one: u∇u/ν∇2u∼UL/ν∼ (gαΔTL3/ν2)1/2=Gr1/2∼21/2×104; Gr is the Grashof
number, which in the case of air has a large value. Note that the large value
of the ratio UL/ν (i.e., the Reynolds number) found here also suggests
turbulent flow in the plume (Turner, 1973).
The flux F of the buoyancy B=gΘ=gαΔT at height z
is
F(z)=g∫S(z)w(z)Θ(z)ds,
where w is the vertical mean velocity, and S the area of the section of
the plume. Let us take z= 1 m, w≈U≈10-1ms-1, and S(1m)≈10-2m2.
From Fig. 14b the mean of Θ in the section of the plume at the 1 m
altitude is ∫SΘ(1m)ds≈1.43×10-3m2. Then,
Fz=1m≈1.4×10-3m4s-3.
Let us compare this estimate of F with the flow of buoyancy delivered by the
source:
F0=gαPρCp-1,
where P is the power of the source (100 W), ρ the air density at
12.5 ∘C (1.236 kgm-3), and Cp the specific heat of air
at constant pressure (Cp= 1006 Jkg-1K-1). It the becomes
F0=2.4×10-3m4s-3.
The comparison between F0 and our estimate of F can be judged
satisfactory. This agreement suggests that the measured energy flux in the
plume itself corresponds to the energy delivered by the source; thus thermal
losses, such as large radiation losses or interactions with the walls
(Crouzeix et al., 2006a), can be neglected at the level of the plume.
Some global characteristics of the plume dynamics
We presented in Sect. 4 the average temperature profiles obtained from our
heating experiments as a representation of stationary plume models, which
raises some interesting questions (which will be touched upon below). The
interest and novelty of our study, however, relies in the higher-frequency
content of our time series of temperatures in the plume (Figs. 3–13). Let us
thus come back to the plume dynamics.
Consider T‾(t) being the average temperature recorded over the 10
sensors on the bar. Figure 15 shows the energy spectra of the series of the
empirical first derivative of the average temperature, (ΔT‾(t))/Δt, in experiments A6a and A6c at 2 m height (top),
A7a and A7b both at 1.5 m height (middle), and A5a and A5b both at 1 m
height (bottom). The maximum peaks appear at periods of about 478 and 467,
474 and 388, 395, and 467 s respectively; the average is
445 s and the standard deviation 42 s.
The power spectra, as a function of frequency in Hz, of the first
derivative of the average temperature during A6a, A6c (both with the bar at
2 m, top), A7a, A7b (both with the bar at 1.5 m, middle),
A5a, and A5b (both with the bar at 1 m, bottom). Each maximum is
supplied with the frequency and the power.
A snapshot from a movie (Carazzo et al., 2014, Supplement Movie S3)
recording an experiment simulating a volcanic vent in a stratified water
tank.
From the comparison of the temperature variations ΔT(t) registered
along the horizontal bar AIII disposed at the three heights of 1, 1.5, and
2 m, it appears that (as already pointed out above) the curves
ΔTi(t) relative to the different sensors (i=1,2,…,10),
to a first approximation, are proportional to each
other:
ΔTi(t)=aijΔTj(t),aij being parameters (constants). For example, taking j=5 (i.e., of
the sensor which according to Fig. 14b detects the largest variations,
presumably, next to the plume axis),
ΔTi(t)≈ai 5ΔT5(t)=biΔT5(t).
This observation is not trivial. First, it clearly confirms the significance
of variations of a few hundredths of a degree and the quality of the
calibration. Second, and more importantly, it demonstrates that the plume
varies in time grossly as a block. In other words, extrapolating a bit boldly
and going from discreet local measurement to a continuous spatial function,
we have
ΔTz(r,t)≈fz(r)θ(t).ΔTz is, at each z, the product of a space function fz(r) by
a time function θ(t). Taking a step further, we assimilate fz(r)
to exp(-r2/b2), bi depending on z according to a self-similar
law. Note again that we have only three altitudes z (1.0, 1.5,
and 2 m, respectively) available and no simultaneous recording at two
altitudes.
The causes of the observed instabilities remain unclear. The mean axis of the
plume could be affected by unstable motions and distortions, especially given
the fact that external influences cannot be completely ruled out before the
construction of the partition wall (Fig. 1). Complementary experiments
performed after the completion of the wall, however, indicated that the mean
barycenter of the plume can indeed move, but that this effect does not
dominate and that the observed instabilities of the plume are not a
consequence of slight plume motion, but rather large and intrinsic instabilities of
the plume itself.
Discussion and conclusions
In this paper, we report on the
temporal behavior of turbulent plumes in geophysical conditions, which still
remains poorly known and rarely studied. Our objective of studying these
temporal fluctuations from the quasi-stationary states proved to be well
justified, as we indeed observed a rich set of temporal behaviors for
temperatures in the plume itself and in the stratified environment in its
vicinity as well. The instabilities observed in our heating experiments as
pulses of high amplitude in air temperature time series, nevertheless, seem
to be a universal and familiar feature. For example, in experiments
simulating plumes with saline solutions in large tanks (Kaminski et
al., 2005; Carazzo et al., 2006, 2014), the recordings reveal the peculiar
temporal instabilities of the plumes, with the occurrence of transient large
voids, even close to the plume axis, and also rather high above the plume
source (Fig. 16).
Our experiments and our data sets, while able to reveal important aspects of
the instabilities, nevertheless, suffer from a number of limitations. First,
we made only temperature measurements. Second, the 10 sensors were attached
every 20 cm on a 2 m long bar. Because we had only one moveable bar,
no simultaneous recordings for different arrangements are available. Third,
air velocity was not simultaneously measured and our temperature measurements
could not be transferred into estimates of velocity. Fourth, the response
time of our sensors (20 s) did not allow us to access the probably
important higher-frequency part of the temporal effects. Finally, the
experimental room was subject to a small level of natural ventilation during
our experiments, and an influence on the dynamical regimes of the plumes
cannot be completely ruled out.
Despite that, our experiments indicate that, in a first approximation, the
temperatures ΔT(x,t) recorded versus time t along the horizontal
bar, at various positions x above the source, suggest essentially spatially
coherent trains of pulses arranged in a quasi-periodic manner, with durations
of 360–400 s, separated by intervals of stability, which can last up
to several hours. The local response function ΔT(x,t) thus appears
as the product of a smooth spatial function fz(r) by a nonlinear
mechanism θ(t), generating a chaotic regime (note, e.g., that doubling
of periods is observed). A Markovian process appears as an adequate
description of the function θ(t) (Iosifescu, 1980; Blanter et
al., 2006). It would be interesting to explain the value of the periods
observed during quasi-periodic regimes and that of the intermittent intervals
of stability (up to several hours). Clearly, the observed factorization of
ΔT(x,t) function is reminiscent of chaotic solutions of a system of
nonlinear differential equations mimicking the behavior of sometimes complex
actual dynamic systems (e.g., Morat et al., 1999). As an illustration, one
may think of one coordinate of the celebrated system of the Lorenz equations
(Lorenz, 1963). Such an exercise however cannot really be attempted for the
dynamical plume, due to the limitations of our observations and additional
important aspects of the problem. Indeed, the dynamic plume is the major
acting ingredient of a filling box with non-adiabatic boundaries (Linden et
al., 1990; Crouzeix et al., 2006b, c). During a 24 h experiment, heat is
accumulated at the ceiling and exchanged with the ceiling and side walls,
causing probably evaporation and condensation, and significant associated
heat transport by phase changes of water. The hot air thus cooled at the
ceiling is then fed again into the plume, with a circulation time which is
probably an important characteristic time of the plume dynamics, contributing
to the intermittent quasi-periodicity. Nevertheless, the plume integrates the
complexity of the various heat relaxation scheme and feedback into an overall
simple organization, with a first-order factorization of the spatial and
temporal variations. While numerical simulations that propose such a
first-order factorization have started to shed light on these mechanisms
(Hernandez, 2015), experiments remain necessary to establish important
properties.
Some aspects of this dynamics could be captured in the experiments performed
in water tanks alluded to above (Fig. 16) and also point to the intermittent
behavior of the reversals of the mean wind in Rayleigh–Bénard convection
(Sreenivasan et al., 2002; Sugiyama et al., 2010). In the case of
Rayleigh–Bénard convection, chaotic temporal variations arise despite
the constant boundary conditions, in the absence of a source of motion, for a
non-localized source of buoyancy. In the case of plumes in confined
environments, the plume or jet itself is a source of velocity in addition to
buoyancy (Hernandez, 2015; Lopez and Marques, 2013). Despite the variability
of conditions, the observed dynamical behavior seems to be remarkably similar
to the behavior of the flickering candle (Maxworthy, 1999) and could be
considered, tentatively, as universal. These temporal fluctuations bear also
astonishing similarities to the dynamics of smoke above a bonfire or the
structure of clouds. In more viscous media and geological timescales, the
episodic temporal structure of deep mantle plumes, which bear important
consequences in terms of the time structure of hot spot volcanism (Kumagai et
al., 2008), could also reflect similar fluctuations around the stationary
state. However, the apparent similarity of the situations may be due to the
limited spatial sampling in our experiment. More elaborated and dedicated
experiments are needed to study the temporal variations inside a turbulent
plume and also in its environment. In confined situations, indeed, the plume
dynamics might result from the interactions of the plume with its environment
and the various relaxation times that it can provide. Underground
environments offer a promising context where these poorly known aspects of
fundamental physics could be studied fruitfully, potentially providing useful
insights for situations of geophysical or industrial relevance.
Data availability
The original data on the experiments made in the Vincennes quarry is
available from Frederic Perrier (perrier@ipgp.fr).
Acknowledgements
The authors acknowledge Catherine Crouzeix for the invaluable unique
collection of data recorded in an underground quarry in Vincennes and thank
the two anonymous reviewers for their comments and suggestions that helped
improve the presentation of the results of the heat experiments in an isolated
room of the quarry.
Edited by: J. M. Redondo
Reviewed by: two anonymous referees
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