IMPACT OF VISITS ON MICROCLIMATE OF CAVES –  
AN ANALYTICAL APPROACH
VPLIV OBISKOV NA MIKROKLIMO JAM –  
ANALITIČNI PRISTOP
Jože RAKOVEC
1
Abstract  UDC 551.584.6:551.588
Jože Rakovec: Impact of visits on microclimate of caves – An 
analytical approach
Theoretical basis for describing natural steady state conditions 
in caves as well as for their changes in time that follow from the 
simple advection-diffusion equation, is given. The impacts of 
visits to caves – direct impacts due to anthropogenic emissions 
of heat and CO
2
, as well as indirect ones, such as illumination 
and possible drafts when opening the door to the cave – are 
estimated in dependence to the number of visitors and the size 
of the cave: the size with which the effects of the visits are below 
the detection threshold is estimated. It is shown that the sources 
cause linear responses, while the consequences of the exchange 
with the walls of the cave or with the exterior depend on time 
exponentially. Characteristic times for linear as well as for ex-
ponential responses are roughly estimated.
Key words: cave microclimate, analytical solution, heat trans-
port, carbone dioxide.
Izvleček UDK 551.584.6:551.588
Jože Rakovec: Vpliv obiskov na mikroklimo jam – analitični 
pristop 
Podane so teoretične podlage za opis naravnih ravnotežnih po-
gojev v jamah in za njihove časovne spremembe, kot izhajajo 
iz preproste advektivno-difuzijske enačbe. Ocenjeni so vplivi 
obiskov v jame - tako neposredno zaradi antropogenih emisij 
toplote in CO
2
, kot posrednih, kot so razsvetljevanje in možni 
prepihi pri odpiranju vrat v jamo. V elikost teh vplivov je ocenje-
na glede na število obiskovalcev in glede na velikost jame: ocen-
jena je velikost, pri kateri so učinki obiskov pod pragom zazna-
vanja. Pokazano je, da viri povzročajo linearne odzive, medtem 
ko so posledice izmenjav s stenami jame ali z zunanjostjo ek-
sponentno odvisne od časa. Grobo so ocenjeni karakteristični 
časi za linearno kot za eksponentne odzive so grobo ocenjeni.
Ključne besede: jamska mikroklima, analitična rešitev, prenos 
toplote, ogljikov dioksid.
1
 upokojeni profesor Univerze v Ljubljani, Fakulteta za matematiko in fiziko, Katedra za meteorologijo, Jadranska 19, 1000 Ljubljana,  
e-mail: joze.rakovec@fmf.uni-lj-si
Received/Prejeto: 10.04.2019 DOI: 10.3986/ac.v49i1.8721
ACTA CARSOLOGICA 49/1, 97-107, POSTOJNA 2020
COBISS: 1.01

JOŽE RAKOVEC
INTRODUCTION
Natural microclimatic conditions in caves are character-
ized by quasi-stationary values of temperature, humidity, 
CO
2
 concentration, etc. In closed caves air temperature 
is determined by the temperature of walls, CO
2 
concen-
trations are normally an order of magnitude higher than 
outside because of specific CO
2
 tranport mechanisms to 
caves. These comprise diffusive and advective transport 
(including the chimney effect) through fractures and 
openings of both, the gaseous, as well as the CO
2
 dissolved 
in water. For example rainwater enriches during percola-
tion through soil due to root respiration as well as from 
organic matter decay in deeper sources (e.g., Faimon & 
Lièbinská 2010; Prelovšek 2012; Ek & Godissart 2014; 
Sallsted et al. 2014). But Covington (2016) stresses also 
the importance of the air-water CO
2
 exchange in frac-
tures – so several several ways of CO
2
 transport into caves 
could generally be relevant. When percolating water 
reaches the cave the precipitation of calcite is triggered 
by the quick initial degassing of CO
2
, folowed by for ap-
prox. 3-times slower equilibration to the new CO
2
 con-
centration in the cave‘s atmosphere (Dreybrodt 2011). As 
regards the air advection with outside through counduits 
it increases the temperature in the warm part of the year 
and decreases it in winter, but always decreases the inner 
CO
2
 concentration (due to low outside concentration).
The further from the entrance a particular location 
in a cave is the weaker the influences from the outside. 
From general principles of diffusion-related processes we 
may conclude that deeper in the cave the amplitudes due 
to outer variations get smaller and phase shifts are larger. 
One borderline case is a widely open cave where close to 
its entrance outer conditions prevail. The other extreme is 
reached deep in a narrow cave where the changes caused 
by outer weather and/or seasonal variations become so 
small that could no more be detected – such a cave is 
effectively a closed cave. 
In tourist caves visitors also influence the 
microclimate: with their heat release and the release of 
CO
2
 and water vapour due to their metabolism. Indirect 
impact of visits could be also draughts, heat released by 
illumination. The more numerous the visitors are, the 
more frequent the visits are and the smaller the cave is, the 
greater is the impact. The recovery starts after visitors have 
left the cave or moved further into other parts of the cave. 
Since warm air tends to rise and cold air tends to 
sink it is reasonable to expect that in summer the warm 
outer air will normally not sink or mix downwards 
into concave cave volumes. Only very strong wind may 
produce enough turbulence to cause some downward 
mixing. On the other hand cold air during winter will 
tend to fill concave holes and caves. If a cave has more 
connections with the outer environment, the air moves 
into or out of it by draughts or breezes. 
The microclimate of caves has been studied by several 
authors. Monographic publications using deterministic 
approach are e.g., by Badino (1995), Lismonde (2002 a;b) 
or Gunn (2004). There are numerous papers in scientific 
journals about that, e.g., Dreybrodt et al. (2005), 
Milanolo & Gabrovšek (2009), Gregorič et al. (2014). The 
microclimate of caves was also a scope of an international 
school in karst sciences (Gabrovšek & Mihevc 2009) and 
of PhD theses (Fernández Cortés 2004). There are many 
reports on microclimate monitoring in caves, e.g., Šebela 
& Turk (2011), Prelovšek et al. (2018).
There are not many analytical descriptions of 
processes in caves depending on both time and space, 
but some of these start from basic principles (Badino 
1995; Dreybrodt et al. 2005) of energy conservation in 
heat exchange, mass conservation including sources and 
sinks. In this paper we consider a cave’s air equilibria and 
changes using only one simple principle: that the local 
value of a certain variable may change in time 1. due to 
advection of that variable, 2. due to sources and sinks and, 
3. in the case when variables are not homogeneous with 
the cave’s volume also due to mixing with neighbouring 
air. The equation appropriate for such an approach 
is the advection-diffusion equation called also scalar 
transport equation, see e.g., Bird et al. (2007) or Becker 
& Kaus (2016) and Covington (2015) for water-air CO
2
 
exchanges in karstic fractures. Based on such general 
approach analytical solutions will be studied for different 
processes in caves. In particular, we are going to describe 
equilibrium stationary cases as well as time changes due 
to visits including their recoveries to natural conditions.
ADVECTION-DIFFUSION EQUATION APPLIED TO CAVE MICROCLIMATE PROCESSES
Consider a constant air mass  in a cave with 
volume . The variables that characterize its 
state change when there are sources or sinks for that 
characteristic – for example heat sources/sinks change 
enthalpy , sources/sinks of CO
2
 change its con-
centration  For a constant mass  the 
Jože Rakovec: Impact of visits on microclimate of caves – An analytical approach
ACTA CARSOLOGICA 49/1 – 2020 98

IMPACT OF VISITS ON MICROCLIMATE OF CAVES – AN ANALYTICAL APPROACH
two individual changes are  and 
Sources for air’s enthalpy  are heat sources/
sinks 
 
inside the cave as well as heat exchanges with 
the cave’s walls. Adiabatic air expansion or compression 
influences temperature only if air moves upwards or 
downwards; phase changes water/vapour are neglected. 
Divergence of transport due to moving air  locally 
changes temperature; as in air 
 
only advection 
 due to breeze with velocity  contributes to local 
change. If there are important fluctuations of velocity 
 and/or of temperature 
 
, local temperature might 
be affected also by eddy (turbulent) transport  
The divergence of such non-advective eddy transport 
 is often simplified in Stokes’ manner as 
 
where  stands for eddy (turbulent) diffusivity being 
an order of magnitude greater than the molecular one. 
So the advection-diffusion equation for local change of 
temperature  takes the form:
 (1)
Analogue reasoning may be applied also for CO
2 
concen-
tration c:
 (2)
Different terms of such equations are for a cave schemati-
cally presented in Fig. 1a ( denoting either T or ), while 
the qualitative ranges and efficiencies of advection and 
diffusion transports in Fig. 1b. 
The time changes of air properties  are 
schematically presented in Fig. 1 a; in our case  stands 
for CO
2 
or for temperature. The sources/sinks in the 
cave   might be internal  (e.g. anthropogenic, 
cave organisms, technical equipment etc.), from surface 
through fractures  from some deeper sources (e.g. 
for CO
2
 from the decaying organic matter or deep CO
2
 
sources), due to echange with walls 
 
etc. The most 
Fig. 1a: Schematic presentation of 
different sources  – of matter (in 
our case CO
2
) or heat  of ad-
vection exchanges  and of 
diffusion processes  
t t+ ∆t t+2 ∆t
calm air
diffusion only
moderate breeze
advection & diffusion
stronger draft / chimney effect
stronger advection & diffusion
Fig. 1 b: Schematic presentation of range and efficiency of advection and diffusion, represented in three consecutive times. 
ACTA CARSOLOGICA 49/1 – 2020 99

important advection exchange  is with the outer 
environment through channels with various cross-
sections and open fractures connecting to the surface 
and/or advection exchanges with other parts of the 
cave through side channels, and also through fractures. 
Advection is commonly related to the chimney effect 
which drives the air upwards during winter period and 
downwards during summer. Diffusive dispersion  
is in calm air weaker (only molecular), while by moving 
air it might be also partly turbulent and thus stronger.
The transport processes in caves are advection and 
diffusion (Fig. 1b). If the air in a cave is calm (upper row), 
only diffusion dissipates properties (in our case CO
2
 
concentration and temperature) without displacement 
of their mass centres, until these are homogeneously 
distributed (right columns). With moderate air 
movement (middle row; breeze, light draft) is the mass 
centre of the diffused property displaced by advection 
and the diffusion is stronger, as there is some mixing 
in the air that moves. By stronger draft the advection 
conveys properties farther from the origin and as 
stronger air movement causes more turbulence, also the 
diffusion process is faster.
APPROPRIATE V ALUES AND SCALES FOR CAVES
Scale analysis is a method of roughly estimating the val-
ues of variables and the relative importance of different 
processes represented by different terms in equations. In 
our case two variables are considered: temperature T and 
CO
2
 concentration c. Their time changes depend on  
and  on air velocity  – with magnitude U, and on 
the size of the cave – say L. Additional influencing factors 
could be the degree of turbulence (affecting D
T
 and D
C
) 
,the shape of the cave: more or less round e.g. expressed 
for example by the ratio between its volume V and the 
walls’ area A.
THE SIZE OF A CAVE
Perturbations of natural conditions during visits depend 
on the number of visitors, their heat and CO
2
 releases 
and on cave properties (size, shape, openness,..). In tour-
ist caves the number of visitors per guided tour is usu-
ally up to 100 – higher numbers are accommodated by 
increasing the number of tours. Larger groups (around 
100) are frequent in the warm season, while in winter the 
number is generally lower – e.g. 30 per visit. Human heat 
emission rate is approx. 120 W per person, by stronger 
activity also more (e.g., Mukarami et al. 2000 or EngT ool-
Box1). So taking heat release rate of 150 W/person results 
in emissions around 15 kW for a group of 100 persons, 
or around 4.5 kW per smaller group of 30 persons. If the 
visitors stay in a particular part of a cave for 15 minutes, 
their released heat is 13.5 MJ and 4 MJ, resp. The reliably 
detectable temperature change is 0.1 °C and with 13.5 MJ 
an air mass of 135 10
3
 kg is heated for 0.1 °C – corre-
sponding to a volume of 135 10
3 
m
3
. In a round cave that 
would mean the diameter r ~ 30 m or size (perimeter)  
L = 2r ~ 60 m. In bigger caves the human heat release 
causes a temperature rise below this threshold.
Similar analysis, though more uncertain due to 
more complex processes, could be done for CO
2
. Each of 
n visitors emits  of CO
2
 per unit time. With relatively 
low human activity breathing gives around 6 10
-4
 kg CO
2
 
per person per minute while with normal work activity 
the emission is around 30 10
-4
 kg per person per minute 
(Prairie & Duarte, 2007; EngToolBox2). Thus in a case of 
100 visitors 4.5 kg of CO
2
 is emitted in 15 minutes. If a 
typical natural concentration of CO
2 
in a cave is around 
2000 ppm that would increase the concentration in the 
bigger cave for around 20 ppm; normally at the limit of 
detectability for many indoor sensors, as detectability 
is typically a few percent of the measuring range 
(CO2meter.com).
What about the possible indirect impacts of visits? 
To allow visits to closed caves, it is necessary to open an 
entrance – for 100 people typically for about 5 minutes. 
Let us assume a breeze through the open door of 1 m/s 
through the 5 m
2 
opening. That brings into the cave 
1500 m
3 
of outer air. Let us take that the outer air is 
10 °C warmer than the air in the cave. To increase the 
cave’s air temperature by a detectable 0.1 °C, 1 percent 
of outer air should mix with the air in the cave. We get 
the threshold volume of V = 150 10
3
 m
3
 – similar to the 
previous threshold estimate. And for CO
2
? The external 
concentration is much smaller – about 400 ppm (for 
example, the NOAA Trends). Mixing one part of the air 
with 400 ppm with 99 parts with 2000 ppm results in 
1984 ppm – after full mixing around 20 ppm decrease: 
a decrease similar to previously estimated increase due 
human breathing. Both changes initially prevail in the 
vicinity of the door, and later mix into the entire volume 
of the cave.
Another indirect impact could be illumination. 
Classical bulbs emit around 90% of their energy 
consumption as heat, so their emissions are comparable 
JOŽE RAKOVEC
ACTA CARSOLOGICA 49/1 – 2020 100

to that of a human (the consumption of LED light is 
negligible). If the number of (conventional) bulbs is 
comparable to the number of visitors, their impact could 
be included with a corresponding increase in the number 
of visitors.
Among indirect influences air intrusion from the 
outside prevails. The direct impacts of visitors are mostly 
negligible in caves of the size greater than 50 or 60 m – so 
we consider only smaller caves.
RATES OF TEMPERATURE CHANGES DUE TO 
HUMAN HEAT EMISSION
Let us assume two caves – a smaller and a larger one 
with the number of visitors per visit either n = 30 or 100. 
Assuming that all human heat released in 15 minutes is 
used to increase the air temperature the heating rates  
are as in Tab. 1.
Tab. 1: Temperature change rates  due emis-
sions of human heat of 150 W per person. 
L = 10 m L = 30 m
 n = 30 3.9 K/h 0.14 K/h
n = 100 12.9 K/h 0.48 K/h
The values in Tab. 3.2 are the upper limits, as the 
heat exchange with walls reduces these rates. This 
linear increase is therefore only appropriate at the very 
beginning of heating and for larger caves, where the time 
of the exchange with the walls is long – see later in 3.4! 
Also, visitors normally stay at a certain location less than 
15 minutes. 
TIME CHANGES DUE TO ADVECTION AND 
TURBULENT MIXING
In partly open caves advection from the outside with a 
breeze or draught may cause an important change. If we 
know the air velocity U, the size of the cave L and the 
temperature difference  then  can be used 
to roughly estimate the magnitude of 
 
and the 
inverse value of L/U roughly describes the time in which 
the advection process with velocity U affects a location at 
the distance L. Similarly for CO
2
: 
After an abrupt air intrusion into a certain location 
of an otherwise closed cave, the conditions first change 
only locally. Then the initial local disturbance is spread 
around with turbulent mixing: 
 
and  
The magnitudes can be evaluated as
  
and 
 The inverse values L
2
/D describe the time 
for the diffusion processes to affect locations L away. 
L
2
/D does not describe how quickly the perturbations 
at the origin are reduced (that will be assessed in section 
below).
The fact that both advection and diffusion estimates 
are dependent on  and  allows us to compare the 
time rates of both processes: U/L vs. D/L
2
. We again 
take one small and one larger cave: 
 
10 m, and 
 
30 m. Eddy diffusivity D is far from being constant: it 
depends on the spatial scale of the process, on the rate of 
turbulence which depends on velocity and on hydrostatic 
stability. Values of D
T
 in the boundary layer range from 
less than 1 m
2
/s to a few 1 m
2
/s (Pasquill 1962; Tennekes 
& Lumley 1972; Obukhov 1971). T o avoid all complicated 
arguments, we use a constant value D
T
 = 1 m
2
/s. For these 
values the timescale estimates are shown in Tab. 2.
Tab. 2: Timescales for effects of advection and eddy mixing to reach 
over a distance L for smaller and bigger caves with very weak and 
moderate airflow, for D
T
 = 1 m
2
/s.
timescale
L = 
10 m
L = 30 m
advective L/U 1.7 min 5 min 
U = 0.1 m/s
diffusive L
2
/D
T
1.7 min 15 min
advective L/U 0.17 min 0.5 min 
U = 1 m/s
diffusive L
2
/D
T
1.7 min 15 min
Diffusive timescales depend only on a cave’s size, 
while the advective ones depend on velocity and on size 
and are in all cases close to one minute. For a very weak 
air flow and for smaller spatial dimensions, the timescales 
of advection and mixing are similar. They differ the most 
in the case of stronger air movement in larger caves. 
Rates of changes due to both processes depend on 
the above timescales and on the differences between 
the values of temperature and CO
2
 concentration in a 
cave   and outside. Large temperature differences can be 
expected in summer and winter being positive in the 
first case, and negative in the other. Intrusion of the 
air from outside always essentially decreases the CO
2
 
concentration.
EXCHANGES WITH CAVES’ W ALLS
Temperature differences between the cave air and cave’s 
walls are of the order of 0.1 K. But the walls may still be 
considered as sources or sinks of heat and CO
2
 for the air 
in a cave:
  
and similarly for con-
centration c:  Here h
T
 and h
C 
are the 
overall heat and mass transfer coefficients between air 
and solid rock walls through a laminar boundary layer 
and further into the rock, and A is the area of the walls. 
IMPACT OF VISITS ON MICROCLIMATE OF CAVES – AN ANALYTICAL APPROACH
ACTA CARSOLOGICA 49/1 – 2020 101

For a multilayer system (laminar air boundary layer, rock, 
eventual cracks, … ) the overall transfer coefficient h
T
 de-
pends on properties of the layers: 
 
and 
may be much smaller than for  boundary layer alone  
( denotes the one for walls). Boundary layer transfer 
coefficient  depends on air velocity, on degree of tur-
bulence. According to different data (Strnad 1992; Gupta 
& Roy 2007; Chavez-Galán et al. 2014; EngT oolBox3) but 
none from caves – in calm air  ranges from around 1 
to around 10 Wm
-2
K
-1
. As in closed caves the air is very 
calm and not knowing the appropriate value we will for 
now apply the lower value of 1 Wm
-2
K
-1
. 
The sources/sinks of CO
2
 and the exchange with 
walls need some additional explanation. The main 
input of CO
2
 into caves is by CO
2 
enriched rainwater 
during percolation from the surface through soils rich 
in decomposed organic matter (Prelovšek 2012). Thus 
concentrations in caves are high, and under natural 
conditions rather constant. From wet walls CO
2 
is 
in general released into cave air (occasionally also 
readsorbed): the process depends on non-equilibrium of 
a system H
2
O – CO
2
 – CaCO
3
. If the CO
2
 concentration 
in cave air is lower than the equilibrium one, part of 
CO
2
 is released from the water on wet walls (being 
previously bound in HCO
3
-
) into the air resulting in 
calc-sinter formation. When the process is reversed the 
consumption of CO
2
 on walls dissolves calcite CaCO
3
 
from the rock resulting in HCO
3
-
 and water becomes 
corrosive (Dreybrodt et al. 2005).
Due to the much more complex CO
2
 exchange 
processes between the air in the caves and the caves 
(mostly wet, several chemical reactions of air-water-
rock), experimental data on CO
2
 exchange are scarce. 
However, from the Reynolds analogy which states that 
the exchange processes through the thin boundary layer 
of the air at the wall should be similar, one estimate for h
c
 
involving air density and heat capacity suggests its value 
to be above 1 10
-3 
ms
-1
, (Baehr & Stephan 2006, p. 85) 
and the other involving mass diffusivity (EngToolBox4) 
and heat conductivity below that value (Baehr & Stephan 
1998, p. 303; EngToolBox4 2020). Thus we estimate 
 
As  both timescale estimates  
and  are equal: for smaller cave (L = 10 m, 
V/A = 1.66 m) around ½ hour and for bigger (L = 30 m, 
V/A = 5 m) around 1.4 hour. 
TEMPERATURE AND CO
2
 IN CAVES
In caves that are wide open to the external environment, 
external daily and seasonal changes prevail over (much 
smaller) anthropogenic impacts. For such caves, only 
reduced amplitude and phase shift are worth to be con-
sidered. The deeper a location in the cave, the smaller is 
the amplitude, and the longer is the time lag. Processes 
connecting the cave with the external environment are 
advection and eddy (turbulent) diffusion, while the in-
fluence of the walls of the cave damps the external influ-
ences.
Another extreme example is a completely closed 
cave without any wind or air draught. In such caves, 
natural conditions are stable so the net exchanges with 
walls are zero. The only disruption (e.g., breeze, draught) 
occurs when an entrance to the cave is opened for visitors 
and some of the outer air can intrude into the cave. Once 
the door is closed, the air will calm down again. For 
the air as a whole in such a cave initially the diffusion 
dominates, and then the exchange with walls is the only 
process that leads to natural conditions. 
LOCATIONS IN OPEN CAVES
Wide open caves are affected by outer weather and/or 
seasonal changes. Changes in temperature outside are of-
ten close to sinusoidal:  
on daily and on seasonal scale. In the case of quiet weath-
er advection may be negligible; then the equation de-
scribing time and space characteristics is the well-known 
diffusion equation:
  (3a)
and
  
(3b),
for which local temperature variations T(t, x) at a dis-
tance x from the entrance are:
  (4a)
(see e.g., Pasquill 1974; Lismonde 2002b). The amplitude 
at the distance x decreases exponentially as  
while the time delay of the phase is   
JOŽE RAKOVEC
ACTA CARSOLOGICA 49/1 – 2020 102

The deeper the location, the smaller are the variations 
and with a larger phase shift. Daily and seasonal temper-
ature changes are commonly known, so it may be worth 
only mentioning that CO
2
 concentration in the external 
environment varies daily as well as annually, mainly due 
to vegetation (Chapman et al. 1954; Buchmann et al. 
1996; NOAA CO2).
IMPACT OF VISITS ON MICROCLIMATE OF CAVES – AN ANALYTICAL APPROACH
Fig. 2: Temperature increase dur-
ing the 15 minutes stay of n = 100 
or n = 30 visitors in a round closed 
cave of size L = 30 m or L = 100 m, 
followed by a decrease after they 
leave the cave for h
T
 = 1 Wm
-2
K
-1
. 
THE CLOSED CAVE
In a completely closed cave there is no air movement and no turbulence, so    and    If there 
are no heat sources T = const. Such temperature is the natural temperature of the closed cave T
nat 
, qeual to the tem-
perature of walls T
W
; similarly also for CO
2
 concentration:
   (4b)
If visitors enter the closed cave without any influence of the outside air (e.g., the entrance with double doors), 
the only sources of anthropogenic emissions are  and  . At the beginning the temperature disturbances increase 
almost linearly  but very soon the exchange with the walls that depends on temperature difference starts 
and the increase becomes slower. The two equations are:
   
(5a)
         and 
  
 
(5b)
Such linear nonhomogeneous equations are well known and using the initial  and the asymptotic  value  
(for ) the solution, with time t running from the start of the visit to its end at t
1
, is:
  
(6a)
and similarly for concentration c
 
 (6b)
ACTA CARSOLOGICA 49/1 – 2020 103

Maximum perturbation temperature depends 
on the number n of visitors and on duration t
1
 of their 
stay in the cave:  
The higher the number of visitors and the smaller the 
cave, the higher the temperature. When visitors leave 
the site, only the exchange with walls remains and the 
temperature starts to recover towards natural conditions: 
 An example of such 
an increase and decrease for a round cave is given in Fig. 
2.
Essential increases happen only in smaller caves 
with many visitors. Recovery back to natural conditions 
is quicker in smaller caves (
 
about 30 min), and slower 
in bigger ones: (
 
about 80 min). 
A SHORT DRAUGHT INTO A CLOSED CAVE
If a closed cave is arranged for visits, the entrance must 
open for at least a short time. When temperature dif-
ference between the cave and the external environment 
is big and by proper geometry of the cvave, the outside 
air may enter the cave (or the inner air may leave it, e.g., 
Gregorič et al. 2014), what causes local disturbance T
0 
near the entrance. After closing the door the perturba-
tion spreads around the entire cave by turbulent mix-
ing. We have already introduced the diffusion equa-
tion  Its solution for a point source is 
 From Fig. 3 we learn that 
for diffusivity D
T
 = 1 m
2
s
-1
 already after a short period of 
time – about 1 minute – the temperature is practically 
uniform throughout the cave. For larger distances (in 
bigger caves) the difference in temperature evens out in 
slightly longer time – but still only in a couple of minutes: 
e.g. at a distance of 30 m the maximum is reached after 
3.5 minutes (the increase is negligibly small, so that the 
line can not be distinguished from the abscissa in Fig.3).. 
A SIMPLIFIED GENERAL CASE
In partially open caves, both advective and diffuse ex-
changes with the external environment depend on the 
differences between the cave and the outside environ-
ment. Both effects could thus be expressed with one 
single term: for the temperature  
representing both exchanges; and similarly for CO
2
. To-
gether with exchanges with walls and with source, such 
simplified equations read as:
 
(7a)
 
  
(7b)
Taking K
T
= 0 and K
c
= 0 they describe also the cases 
of closed caves. When there are no visitors (n = 0), the 
constant natural temperature of a partly open cave is the 
weighted average between the temperature of walls and 
the outer temperature:
 
(8)
If there are some constant sources in caves, such 
as energy-using equipment or longer lasting visits 
such as social events, another higher value stationary 
temperature might be reached: 
 
(9)
As it takes a long time ( )
 
to reach this constant 
value, 
 
is the upper limit for such a temperature 
increase. 
JOŽE RAKOVEC
Fig. 3: Time evolution of pertur-
bations  during the first 
3 minutes at different distances r 
from the origin of perturbation at 
r=0 for diffusivity D
T
= 1 m
2
/s and 
for initial  = + 5 °C.
ACTA CARSOLOGICA 49/1 – 2020 104

CONCLUSIONS
A simple advection-diffusion equation is used to describe 
processes that govern temperature and CO
2 
equilibrium 
values in a cave, their time courses during visits, as well 
as exponential recoveries after visitors have left the cave. 
In closed caves natural conditions are reached by 
the equilibrium of heat and mass exchange between the 
cave’s walls and the air in it. In semi-open caves there 
is also some exchange with the outer environment. The 
exchange with outside air normally causes an increase 
of temperature during the warm part of the year and 
diminishes it during winter. For CO
2 
that is not the case: as 
the concentration in cave air is for an order of magnitude 
higher than outside, the eventual breeze or draught 
always diminishes the CO
2
 concentration. In semi-open 
caves natural conditions are the weighted average of the 
values at walls and of outer air values. The more a cave is 
closed, the more important is the exchange with walls, 
and the more it is open, the more the influences from 
the outside prevail. Therefore, in largely open caves the 
values are rather close to the outer ones (but still different 
from those – as it is still a cave). 
Anthropogenic heat and CO
2
 emissions increase 
temperature and CO
2
 concentration in caves. The 
amplitude of these increases depends on the number of 
visitors, their physical activity and the duration of their 
visit in the cave, on the cave’s size, its geometry (volume, 
walls) and in partially open caves also on exchanges with 
outside air. In caves bigger than L  50 m human impacts 
are negligible even for greater groups of visitors and even 
in closed caves.
The direct impacts of visits (anthropogenic heat 
and CO
2
 emissions) cause a linear increase in time, 
while exchanges with walls and with external air show 
exponential time dependencies. After the visits, when 
there are no more emissions, perturbations decrease 
exponentially.
Through the open door the outer air may break into 
the cave, causing a local disturbance near the entrance. 
This perturbation first spreads around the cave with a 
rather fast eddy diffusion, followed by a slower return to 
natural conditions.
Several variables are often monitored in caves. Using 
their time series some parameters can be evaluated, like 
typical times of recovery back to natural conditions, 
phase shifts, reductions of amplitudes, etc. 
ACKNOWLEDGEMENTS
Thanks to A. Kodre for suggestions regarding the style 
that essentially contributed to a shorter and clearer ar-
ticle and to Ž. Zaplotnik for reading the paper and check-
ing it for errors. The valuable comments of the reviewers 
are appreciated as well
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