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Espelunc@Espelunc@Espelunc@Espelunc@digitaldigitaldigitaldigital Publicación Científica Seriada No Periódica de la Sociedad Espeleológica de Cuba

ISSN 2072-5892

No. 10. Mayo, 2012, Ciudad de La Habana, Cuba

Apartado 6219, CP. 10600, Habana 6, Ciudad de La Habana, Cuba

e-mail: [email protected]

Director: L.F. Molerio LeónL.F. Molerio LeónL.F. Molerio LeónL.F. Molerio León

Hydrological controls in the development of the slopes Hydrological controls in the development of the slopes Hydrological controls in the development of the slopes Hydrological controls in the development of the slopes of the mogotes (hillstacks, conic karst, kegel karst, of the mogotes (hillstacks, conic karst, kegel karst, of the mogotes (hillstacks, conic karst, kegel karst, of the mogotes (hillstacks, conic karst, kegel karst, tower karst, turm karst) of Sierra de Los Organos, tower karst, turm karst) of Sierra de Los Organos, tower karst, turm karst) of Sierra de Los Organos, tower karst, turm karst) of Sierra de Los Organos,

CubaCubaCubaCuba....1111

L.F. Molerio León2

RESUMEN

Los mogotes (kegel karst, karst de torres, turm karst) de la sierra de Los Orgasnos, en la parte más occidental de Cubason

montañas típicamnete de paredes verticales. Estos relieves han sido tradicional pero erróneamente considerados por la

Geomorfología climática como la última fase de evolución del karst en los Trópicos. El control exo y endogenético de la verticalidad

de las paredes de los mogotes es ejercido por factores geológicos e hidrológicos. Los factores geológicos están bien documentadosy

se relaciona con la composición litológica de los mogotes y el dominante agrietamiento vertical.Los controles ghidrológicos han

sidomenos estudiados y en esta contribución se mencionan tres de ellos: las cuevas al pie, las cuevas subcutáneas abortadas pro

recesión de las paredes y las galerías fluviales y/o freáticas también abortadas a superficie por el retroceso de las paredes de los

mogotes. Una atención particular se ofrece en este artículo al modelo erosivo basado en la progresiva ampliación de las cuevas al

pie de las montañas por corrientes secundarias y su relación con los rasgos corrosivos asociados a las fases lacustres de evolución

de los poljes y dolinas cársicas de la Sierra de Los Organos. El modelo de recesión de escarpes descrito se basa en las

aproximaciones de Scheidegger.

Palabras clave: mogote, cueva al pie, modelo matemático, pendiente, carso, Cuba, espeleología.

ABSTRACT

The “mogotes” (hillstacks, conic karst, kegel karst, tower karst, turm karst) of the Sierra de Los Organos, at the westernmost part of

Cuba are typically carbonate steeped wall mountains, almost vertical, highly karstified. These landscapes have been traditionally

but erroneously considered by the climate geomorphology the last phase of the karst evolution in the Tropics. The exogenetic and

endogenetic control of the slope verticality of the mogotes is exerted by geologic and hydrologic factors. Geologic factors are well

documented and use to be related with the lithological composition of the mogotes and the huge vertical dominating jointing.

Hydrologic controls have been less studied and in this paper three main controls are mentioned: the foot caves, the subcutaneous

caves aborted by wall recession and the fluvial and/or phreatic galleries also aborted by wall recession. Particular attention is given

in this paper to the erosional model based in the progressive enhancement of foot caves by secondary currents and its relation with

the corrosional features associated to the lacustrine phases of the evolution of karst poljes in the Sierra de Los Organos. The model

of slope recession is described based on Scheidegger approaches.

Key words: mogote, foot cave,mathematical model, slope, karst, Cuba, speleology .

INTRODUCTION ........................................................................................................................................................................................................... 2 THE CAVES AT THE BASE OF THE MOGOTES (FÜSSHÖHLEN) ........................................................................................................................... 3 CONCEPTUAL MODEL OF LATERAL EROSION ...................................................................................................................................................... 4

Secondary currents ..................................................................................................................................................................................................... 4 Slope recession ......................................................................................................................................................................................................... 11 Volume reduction and block instability .................................................................................................................................................................... 15

FINAL REMARKS ........................................................................................................................................................................................................ 16 ACKNOWLEDGEMENTS ........................................................................................................................................................................................... 17 REFERENCES .............................................................................................................................................................................................................. 18

1 Manuscrito recibido en Diciembre, 2010. Revisado y aprobado en Marzo, 2012. 2 INVERSIONES GAMMA, S.A.

P.O. Box 6219, CP 10600, Habana 6, La Habana, Cuba E mail: [email protected]


No. 10, Mayo, 2012, La Habana, Cuba

2

INTRODUCTION

One of the most outstanding features of the “mogotes” (hillstacks, conic karst, kegel karst, tower karst,

turm karst) of the Sierra de Los Organos, at the westernmost part of Cuba (Fig. 1), are its vertical slopes

(Fig. 2). The steeped walls of the mogotes distinguish this landscape from any other positive morphology

of the karst land (Fig. 3). The slopes of the mogotes are very steep, almost vertical, with slopes steeper

than 70º, very close to the absolute verticality.

In most cases, verticality is associated with a clearly observed lithologic control due to the presence of

massive vertically jointed Jurassic black and grey carbonate rocks of the San Vicente Member of the

Guasasa Formation as was noted early by Lehmann and later very well commented by Panos and Steclc

(1968). When the mogotes walls are gentler, the slope is usually controlled by stratified carbonate rocks.

These more gentle slopes are also associated with interbedded terrigenous rocks or it becomes

apparent because masked by dejection cones (Acevedo and Molerio, 1982; Molerio, 1975, 1995, 2004;

Flores and Molerio, 1995; Farfán et al., 2005).

But lithological composition seems not to be the only control in the verticality of the mogotes wall. The

receding slopes are also controlled by several hydrological factors. One of them is associated with the

development of a typical concavity at the base of the mogotes, the so called “balcony” by Glazek (1968)

commonly associated –in Cuba- with a special corrosional feature, the “foot-caves” (“cuevas al pie” in

the Cuban terminology; “Füsshöhlen”, as named by Lehmann and its collaborators 50 years ago) formed

by lateral erosion of flows linked with the different stages of the hydrologic evolution of the karst

landscape.

Foot-caves are small caves developed parallel to the base of the mogotes, at the contact with the

surrounding floor of the valley, whose mayor axis is rather parallel to the mogotes walls and do not use

to develop more than a few dozens of meters through the interior of the mogotes (Fig. 3).

Fig. 1. Location map.

This paper discusses a theoretical approach to the explanation of the receding vertical walls of the

mogotes, traditionally (but erroneously) considered the last phase of the evolution of karst in the Humid



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Tropics as was pointed out by the authors linked with the so-called climate-approach of karst

development (see the analysis by Lehmann, 1953, 1954, 1960; Lehmann, Krommelbein and Lotschert,

1956; Gradzinsky and Radomski, 1963,

1968; Panos and Stlecl, 1968; Mangin

and Bakalowicz, 1990). Because the

lithologic control is evident, in this

paper we will focus the attention on

the less evident but abundant features

of the verticality control due to the

hydrological evolution of the karst

landscape.

Therefore, because of the strong

relation among the factors controlling

hydrologic evolution and the recession

of the slopes special attention will be

devoted in this paper to the role of the

diffuse and concentrated erosion at

the base of the hills with particular

emphasis in the effect of helicoidal

flow in the development of the Foot-

Caves and its relation with the slope

recession of the mogotes. Therefore, a

more general model of slope recession

of the mogotes, as a particular feature

of Karst Mountains, could be

approached.

Fig. 2. Steeped wall of the mogotes near Viñales.

Fig. 3. Isolated mogote (Photo by Ana M. Sardiñas).

THE CAVES AT THE BASE OF THE MOGOTES (FÜSSHÖHLEN)

Almost 50 years ago, in his classical 1954 paper, H. Lehmann (Lehmann, 1954) drew his attention to the

caves that developed at the base of the steeped walls of the mogotes. Lehmann named “füsshöhlen” to



4

this small caves developed at the contact between the base of the mogotes and the floor of the valley.

In the Cuban literature, Núñez (1967) considered this caves as a special case of speleogenesis associated

to the mogotes and later, in our typological study of Cuban karst (Molerio, 1975, 1980) they were

considered a particular type of cave due to lateral erosion, maintaining the denomination use by Núñez,

who literally translated the term “füsshohlen” into the Spanish “cuevas al pie” or Foot-caves accounting

that the component “Foot” its circumscribed to the “base” of the mogotes (Fig. 5). Glazek (1968) stated

that the balconies at the base of the mogotes of Viet Nam are a typical feature of horizontal water

corrosion developed at the level of the polje bottom.

These Foot-Caves should not be confused with the subcutaneous caves developed parallel to the

mogotes wall that eventually can outcrop by lateral erosion and form big chambers also at the base of

the mogotes or even at high altitudes linked with ancient cave levels. Neither should be included under

this term those fluvial and/or phreatic cave galleries that abort to the surface because of the mogotes

wall recession or by a sort of combined actions of cave enhancement and mogotes wall recession.

Nevertheless, subcutaneous caves and dismantled parallel galleries aborting at surface are also

hydrological controls of the recession of the mogotes walls contributing to its steepness.

Commonly, these Foot-Caves are the result of systematic lateral erosion that can develop a large but not

deep notch along the wall of the mogotes. In fact, they result from the progressive lateral erosion of

Glazek´s balconies. In other cases, the original Foot-cave could develop further and mostly horizontal

through the interior of the mogotes as a consequence of the Foot-cave hydrological role of capturing

surface or ground waters during the rainy season. As a matter of fact the Foot-caves performs and

important hydrological action during the lacustrine phase of the evolution of karst valleys (poljes and

uvalas).

This hydrologic behavior of this important feature can be presently observed when the karst valleys

surrounding the mogotes become flooded during the rainy season or during heavy or storm and

hurricane rains (Molerio, 1981; Molerio et al., 1983; Molerio and Flores, 2003). Therefore the erosional

model of the mogotes landscape should be improved adding to the commonly accepted fluvial and

speleogenetic control, the lacustrine flow controlling lateral helicoidal erosion.

CONCEPTUAL MODEL OF LATERAL EROSION

The lateral erosion model is explained in this paper accounting for the presence of secondary or crossed

currents controlling slope recession by free flowing water. Therefore two main aspects will be

discussed:

• The slope development by

water flow including the

helicoidal flows.

• The different models of

slope recession.

Secondary currents

These currents were originally semi

empirically studied more than a

century ago by Bousinnesq (1877).

This author establishes that the head

loss relations could be expressed by

(Scheidegger, 1991):

Fig. 4. Cueva de Jose Miguel (José Miguel Cave,

a typical Foot-Cave near Viñales valley)

Publicación Científica Seriada No Periódica


h

vH

2

1 β=∆

Where, β is a coefficient; v, is the velocity, and h, the depth

of the river in the case of rectilinear streams. For rivers with

curvature, the above relation is expressed

22 v

R

b

hH

α=∆

Where α is another coefficient; b, the river width, and R,

the curvature radio. According to this, the total head loss

could be expressed as:

+=∆+∆=∆R

b

hh

vHHH

αβ

2

21

This equation allows the correlation of depth in straight (h

+=

R

b

h

vv

h curvorecto

αββ 2

2

+=

R

bhh rectocurvo β

α1

It can be derived that the depth increa

increases more effective the secondary currents will be. The importance of this is that these secondary

currents are the responsible for the development of helicoidal flows and of meander featur

turn, in the case described here, helicoidally erosion the slope of the hill. As Scheidegger points out,

other authors have explained the origin of the secondary currents according the hypothesis of the

velocity potential. According to this, t

at the bottom of the channel is accounted, that distribution of forces

velocity potential accounts for the flow irrotationality which makes that hypothesis

The erosion at the base of the mogotes could be approached considering the presence of concentrated

flow bearing in mind that such problem is much more complex than described here. Following

Scheidegger, if the equation is linearized,

coordinates:

x

y

∂∂

−=ζ

The following equation, where a

Espelunc@Espelunc@Espelunc@Espelunc@digitaldigitaldigitaldigital blicación Científica Seriada No Periódica de la Sociedad Espeleológica de Cuba

La Habana, Cuba

is a coefficient; v, is the velocity, and h, the depth

of the river in the case of rectilinear streams. For rivers with

curvature, the above relation is expressed as:

is another coefficient; b, the river width, and R,

the curvature radio. According to this, the total head loss

+=

R

b

h

vv αβ

22

Fig. 5. Cueva de Los Tomates (Tomatoes Cave),

a typical Foot-cave close to Viñales Valley (after Núñez, 1967).

This equation allows the correlation of depth in straight (hrecto) and curved (hcurvo) channels according to:

It can be derived that the depth increases when the curvature radio diminishes. As the channel depth


currents are the responsible for the development of helicoidal flows and of meander featur



velocity potential. According to this, the velocity is greater at the convex side of the river but if, friction

at the bottom of the channel is accounted, that distribution of forces cannot exist. Moreover, the

velocity potential accounts for the flow irrotationality which makes that hypothesis little acceptable.



Scheidegger, if the equation is linearized, being x, the horizontal and y, the vertical (upward)

a is some constant, could be obtained:

5

Fig. 5. Cueva de Los Tomates (Tomatoes Cave),

cave close to Viñales Valley (after Núñez, 1967).

) channels according to:

ses when the curvature radio diminishes. As the channel depth


currents are the responsible for the development of helicoidal flows and of meander features which, in



he velocity is greater at the convex side of the river but if, friction

exist. Moreover, the

little acceptable.



being x, the horizontal and y, the vertical (upward)



6

2

22

xa

t ∂

∂=

∂∂ ζζ

Because of the non-linear character of the equation, Green and Wilts obtained:

2

222

xa

t ∂

∂′=∂∂ ζζ

Where,

ζ ′=′a

a

And the slope could be expressed, according to Scheidegger (1991):

ta

xerfcS

′−≅

41ζ

For the velocity, it holds as:

ta

xerfcvv

′≈

41

The height of the accumulation is then expressed as:

∫∞ ′−≅

x

dxta

xerfcy

4

And the slope itself is governed the following equation:

−==−

at

x

t

constSy

4exp

2

4

The above equation explains “a priori” the convexity of the debris that later on are dismantled. Further

discussion of the theoretical aspects could be found in Scheidegger (1991).

The erosion due to helicoidal flows should be considered now. The formation of seasonal lakes and

rivers is part of the evolution of karst valleys (poljes and uvalas); therefore, their contribution to the

erosion of the mogotes is outstanding. Particularly the role of these lakes seems to be more important

than it has been traditionally considered. In effect, the walls of the mogotes are commonly distinguished

by horizontal and almost continuous erosion lines at different levels and at the same altitude in the

valleys. In some cases they form notches but in others abandoned Foot-Caves controlled by

stratification are recognized (Figs. 6-7). These erosion lines could be considered as strand lines marking

the levels of ancient lakes. The presence of lacustrine sediments at the present bottom of dolines, poljes

and uvalas sustains this approach.

Shallow waters show a phase velocity of the wave that could be expressed as:

ghc =



7

The displacements ∆x and ∆z could be computed as:

( ) ( )δσσ

+++=∆ tmxhzmam

x coscosh

( ) ( )δσσ

+++=∆ tmxhzmam

z sinsinh

Fig. 6. Superimposed strand lines associated with abandoned and present Foot-caves at the Eastern

slope of the Sierra de Quemado.

Fig. 7. Superimposed abandoned Foot caves and subcutaneous galleries (Photo by J.L. Clinche)

These equations show that the wave movement is elliptical and its amplitude depends on the constant

a:

mham

A sinhσ

=

Scheidegger (1991) has pointed out that the potential flow theory becomes inapplicable if the drag force

at the bottom of the channel is considered. In fact, the potential flow theory states that the flow is

faster in the inner side of the channel but if the drag force is accounted the velocity will decrease from

the surface to the bottom in the vertical column of water. In this case, the individual fluid particles

moving each one at its own velocities (v) will be forced along their individual circular paths (with their

own radius of curvature, r) induced by a radial pressure loss (dp/dr) in a water column of density ρ,

given by:



8

r

v

dr

dp 2ρ=

But in slow moving waters, as in the case of the seasonal lakes formed at the valley’s bottom, the radial

pressure gradient (dp/dr) will not change along a vertical line. This means that the radial pressure loss

approaches very big values preventing the slow moving bottom fluid on the same curvature as the fast-

moving particles of the surface or close to the surface. This difference is enough to force inward the fluid

at the bottom of the channel onto paths showing a stronger curvature. When this phenomenon takes

place the onset of secondary currents gives rise to helicoidal flows superimposed to the mean flow of

the channel.

In spite of the known limitations of the explanation of the onset of secondary currents in river channels

no other comprehensive explanation has been definitely stated after that given by Einstein in 1926 and

later on by Einstein and Li in 1958 (fide, Scheidegger, 1991). Undoubtedly it seems to be very attractive

to explain the origin of the Foot-caves and the dissolution strand lines associated to the poljes

seasonally lakes.

In this case, the classical Navier-Stokes equations could be formulated in terms of the vorticity (ξ,η, ζ),

as for example:

z

v

y

w

∂∂

−∂∂

=ξ

With

0=∂∂

+∂∂

+∂∂

zyx

ζηξ

Being x, y, z the Cartesian coordinates and, u, v, w the velocity components, the Navier-Stokes equation

could be re-expressed as:

ξηξξ

vlapy

u

x

u

Dt

D+

∂∂

+∂∂

=

Here D/Dt is the total time derivative moving with the fluid and v is the kinematic viscosity.

Accounting for the mean and fluctuating velocity components:

uuu ′+=

Where,

0=∂

′∂=′t

uu

Scheidegger (1991) introduces them into the Navier-Stokes equations and, averaging, it can now be read

as:

( ) ( ) ( )wvy

wvz

wvzyt

′′∂

∂−′′

∂

∂+′−′

∂∂∂

=∂∂

2

2

2

222

2ζ



9

This shows that in the case of turbulent flow the x component of the vorticity not necessarily vanishes.

When this is the case, secondary currents may appear. The discussion whether isotropic turbulence

makes that the right hand terms add up to 0 is out of the scope of this paper. Nevertheless, turbulent

stresses could be considered even in the case of turbulent instability and the analysis should then

account for neglecting viscosity.

The Navier –Stokes equations in a rotating system continue to be the basis from which the equations of

motion for lakes could be derived (Wang, 2000). In this approach we subscribe the “Boussinesq and the

shallow water approximations (that) leads, after an appropriate scaling, to the classical hydrostatic

primitive equations, in which the density variations are neglected except in the buoyancy force, and the

vertical momentum equations reduces to a balance between buoyancy and vertical gradient of pressure”

(Wang, 2000:369).

Following this approach it can be stated, in the horizontal curvilinear coordinate and vertical s-

coordinate system the primitive equations can be written (Wang, 2000) for the Mass Balance:

0=

Ω

∂∂

+

∂∂

+

∂∂

mn

H

sv

m

Hu

n

H θθ θ

ηξ

For the Horizontal Momentum Balance, it holds:

=−

∂∂

−

∂∂

−

∂∂

+

∂∂

+

∂∂

+

∂∂

fvmn

HvH

mu

nvF

mn

H

sF

m

H

nF

n

Hu

mn

H

t

u

s

uu θθ

θη

θξ

θθ

ηξξ11

+

∂∂

+

∂∂

∂∂

+

∂∂

+

∂∂

+

∂∂

∂Φ∂

−∂Φ∂

−= vu

s

uu Fn

Hv

mn

H

tD

mnsD

m

H

nD

n

Hz

sn

Hξ

θθη

θξ

θθ

ξξξξ,

1

=+

∂∂

−

∂∂

+

∂∂

+

∂∂

+ fumn

HuH

mu

nvF

mn

H

sF

m

H v

s

v θθ

θη

θ

ηξη11

∂∂

+

∂∂

+

∂∂

+

∂∂

∂Φ∂

−∂Φ∂

−= v

s

vv Dmns

Dm

HD

n

Hz

sm

H 1η

θξ

θθ

ηξηη

For the Energy Balance, it holds:

=

∂∂

+

∂∂

+

∂∂

+

∂∂ T

s

TT Fmn

H

sF

m

HF

n

HT

mn

H

t

θη

θξ

θθ

ηξ

∂∂

+

∂∂

+

∂∂

= T

s

TT Dmns

Dm

HD

n

H 1η

θξ

θ

ηξ

In the above equations



10

( )wvus

zH ,,,

∂∂

≡θ are the velocity components in the (ξ, η, s) coordinate directions, respectively; ρ,

is the density; Φ the dynamic pressure (

o

p

ρ=Φ ) being p, pressure and ρ0 the reference density; f, is

the Coriolis parameter. Following Wang, the Modified Vertical Velocity, Ω, is defined as:

∂∂

−∂∂

−=Ωηξθ

znv

zmuw

H

1

The advection fluxes Fξϕ, Fη

ϕ and Fs

ϕ can be written for a representative variable ϕ (either u, v or T) as:

ϕ

ϕ

ϕ

ϕ

ϕη

ϕξ

Ω=

=

=

sF

vF

uF

;

;

The turbulent diffusion fluxes Dξϕ, Dη

ϕ, Ds

ϕ can be written as:

∂∂

∂∂

−∂∂

=s

z

HmvD H

ϕξξ

ϕ

θ

ϕϕξ

1

∂∂

∂∂

−∂∂

=s

z

HnvD H

ϕηξ

ϕ

θ

ϕϕη

1

ϕϕ

θ

ϕϕη

ϕξ

ϕ ϕν

ξξ sVsVs DHDsH

Dz

nDz

mD +≡∂∂

+∂∂

−∂∂

−=1

Being,

sHD

Dz

nz

mD

VsV

sH

∂∂

≡

∂∂

−∂∂

−≡

ϕν

ξξ

θ

ϕϕ

ϕη

ϕ

1

Where νHϕ, νV

ϕ are the horizontal and vertical momentum (if ϕ = u or v) or thermal diffusivities,

respectively (if ϕ = T).

Therefore (Wang, 2000) the balance equations of linear momentum in the directions ξ and ηcan be

written in the following general form:

QDmns

Dmns

Dm

HD

n

H

Fmn

H

sF

m

HF

n

H

mn

H

t

sVsH

s

+

∂∂

+

∂∂

+

∂∂

+

∂∂

=

=

∂∂

+

∂∂

+

∂∂

+

∂∂

ϕϕϕη

θϕξ

θ

ϕθϕη

θϕξ

θθ

ηξ

ηξϕ

11



11

Where ϕ may be either the water temperature T or a velocity component u or v, while Q represents the

remaining terms that can vanishes for the temperature equation and indicates the sum of the Coriolis

force.

Slope recession

The general models of denudation and slope recession of the mogotes depends on the most simple and

general cases found in Sierra de Los Organos:

• With respect to lithology:

o The mogotes are built by homogeneous (or quite homogeneous) rocks.

o The mogotes show lithological variations.

• With respect to debris accumulation and mogotes relicts:

o Debris accumulates at the bottom of the slopes.

o Slope material has been removed.

o No evidences of recession (karren fields or small mogotes) appear.

o Karren fields outcrops aligned parallel to the mogotes walls.

In this general approach free flowing water is the governing factor of the slope recession. The lithologic

composition and local tectonic features are local governing factors controlling the steepness and the

stability of the slopes. Free flowing water is then controlling the development of all the karst features

developed at the base of the mogotes (strand levels and Foot-Caves), at the interior (subcutaneous

caves) or allowing mixed effects of coalescence due to external (wall recession of the mogotes) and

internal (cave passage enlargement) side erosion. Debris can accumulate or not at the base of the

mogotes depending upon the organization of the flow at the valley bottom. Flow can remove the

material depending of its energy and of the block size.

The mathematical approach to the most appropriate recession model could not be uniform. Two

general approaches have been described in the literature (see Scheidegger, 1991 for references and a

more detailed explanation). But in the particular case of the evolution of the slope of the mogotes the

surrounding non karstic rocks where almost all of the rivers sources are is of special importance. It is

also significant that the most part of the rivers flowing into the mogotes and through its interior

develops the main karst systems. These rivers are mainly formed at the Southern non karstic belt of the

mountain range known as “Alturas de Pizarras del Sur” (Fig. 8). Therefore the evolution of these

mountains should be considered as the changing boundary conditions allowing the most important

energy source for the organized free flow systems entering the Sierra de Los Organos.

Cuban literature concerning the Sierra the Los Organos karst is not very abundant in relation with the

surrounding non karstic landscapes. Two mainly mountain ranges composed by argillaceous, sandstone

and shale rocks surrounds the mogotes of the Sierra de Los Organos by the North and South. They form

a low energy landscape known as “Alturas de Pizarras del Norte” and “Alturas de Pizarras del Sur”,

where the “pizarras” term concerns to its dominantly terrigenous nature. Local inhabitants call this

mountains “Cuchillas de Pinares” a name describing its morphology (Cuchillas = Knife shaped) and its

vegetation (Pinares = Pine forest), completely different of the internal limestone ridges of the mogotes.

Except in the case of the Sierra de Trepada de Francisco and another few cases, the most important

rivers of the Sierra de Los Organos flows from South to North or changes westward. Almost the half part

of the Sierra de Los Organos is developed within the Cuyaguateje river Basin (736,9 km2), an important

fluvial system of 99 km length responsible (with its tributaries) for the development of hundreds of cave

passages in the region. The most important cave systems of the region (and almost of Cuba) are

controlled by its evolution (Gran Caverna de Santo Tomás: 47 km; Sistema Cavernario Majaguas –

Cantera: 24 km; Gran Caverna de Fuentes: 17 km; Sistema Amistad: 11 km).

The Alturas de Pizarras (both del Norte and del Sur) is a low energy mountain range (the highest peak,

the Cerro de Cabras, has only 448 m above sea level, Fig. 9). The most important erosion processes here

are basically the seasonal lineal erosion due to mountain creeks (rill wash) and sheet flood acting upon



12

terrigenous rocks of Low Jurassic age. This rocks show a remarkable tectonic evolution associated where

intense folding –including overthrusts- and faulting took place.

Fig. 8. The dominating landscape of the “Alturas de Pizarras del Sur” or “Cuchillas de Pinares” (Photo

by Marjorie Condis)

Fig. 9. Cerro de Cabras, the highest peak of the “Alturas de Pizarras” (Foto by A. Núñez Jiménez,

courtesy of the Foundation “Antonio Núñez Jiménez” of Nature and Man).



13

Generally speaking the morphogenetic evolution of the mountain range is particularly young and mainly

associated with the relative tectonic stability that became after the Cuban Orogeny (Middle Eocene), the

so-called Post Laramic Peneplain by classic authors. Older landscape features associated with ancient

erosive events are difficult to identify and maybe they have been successively replaced by younger ones

due to the low coherence of the shales and sandstones.

The depth of the valleys within the

Cuchillas is between 50 and 100 meters

showing maxima of 250 m; while their

width ranges from 50 to 500 m. The

slopes of the hills are between 7 and 24º

in contrast with the steeped walls of the

mogotes (Fig. 10).

The Gerber equation describes the index

of erosive strength (λ) of these mountain

creeks in the limits of the disarticulation

(∆M):

h

M

∆∆

=λ

Fig. 10. General appearance of the slopes of the terrigenous

hills of the “Alturas de Pizarras (Photo by Marjorie Condis).

Expressing ∆h, the thickness of the debris, with a value of 5 meters, as measured in Cerro de Cabras,

very close to the Cuyaguateje river source, the distribution of values obtained for the Gerber equation

are summarized in Table 1.

Table 1. Gerber equation parameters for a sector of the Cerro de Cabras.

Vertical

disarticulation

(m)

Horizontal

disarticulation

(m)

∆∆∆∆M

(m2)

∆∆∆∆h

(m)

λλλλ1

(m)

λλλλ2

(m)

50 50-500 1250-12500 5 250 2500

100 50-500 2500-25000 5 500 5000

250 50-500 6250-62500 5 125 12500

In the limit:

0tan =+−= hxdx

dhαλ

αλα tantan −= xh

But as Scheidegger (1991) points out, Gerber Equation is a very simplified description of the problem

and does not account for the progressive enhancement of the valleys. A slightly different approach is

then proposed, and now λ could be expressed as:

hx

M

∆∆

=λ



14

Being,

xh1

tan

+=λ

α

Results of this modification are shown in Table 2 for the extreme values of the Cuchillas slopes.

Observation of Table 2 shows that λ remains constant for given slopes independently of the

disarticulation of the landscape and ∆M is typically proportional for constant magnitudes. Therefore, the

Gerber equation is useful to compare λ among different landscapes. For example for the Sierra del

Pesquero, at the Western part of the territory, λ ranges from 0,2744 (max) to 0,02745 (min).

Table 2. Extreme values for the Gerber modified (Scheidegger) Equation.

Parameters X=50 X= 100 X= 250

α1 (º) 7º 7º 7º

α2 (º) 24º 24º 24º

λ1 (m) 250 500 1250

λ2 (m) 2500 5000 12500

α1λ1 0,0245 0,0245 0,0245

α2λ1 0,08869 0,08886 0,08895

α1λ2 0,00245 0,00245 0,00245

α2λ2 0,00890 0,00890 0,00890

Another particular feature of the Alturas de Pizarras is the evolution of their slopes. In fact, the slopes of

these hills recede parallel without being linear. These means that a smoothing of their slopes should be

expected differentiating its evolution from that of the mogotes, as it actually happens.

The theory of the linear evolution of the slopes is essentially based in the variations of the slope

exposures to denudation. Two extreme cases could be examined:

• Slope denudation is proportional to the height of a certain point above a certain base level due

to the increase of rainfall with altitude.

• Denudation is proportional to the value of the slope accounting that as more steeped the slope

faster the base debris will be removed.

The natural model seems to be inadequate described mathematically. Scheidegger has pointed out that

“one really should allow for the fact that weathering acts normal to the slope so that the vertical

lowering is the represented by the vertical effect of the weathering action which is directly normally

against the slope” and presents a differential solution that exhibits the particular feature of being non-

linear. This partial hyperbolic differential equation has the form:

Φ

∂∂

+−=∂∂

2

1x

y

t

y

Applying this equation to our second case satisfactory results are obtained, particularly with respect to

the progressive smoothing of the slopes and its evident concave profile.

This equation is the base of the non-linear approach where Φ is derived from:

Φ−∂∂

constt

y



15

For three different cases, as described:

Case 1 Φ = 1 When the denudation is independent of the slope and proceeds at an equal

rate at any exposed portions of the slope.

Case 2 Φ = y For the case when the denudation is proportional to the height of the point

under consideration above a certain base level. A fact that could be justified

by the observation that in certain areas, precipitation increases with height.

Case 3 Φ =

xy∂

∂

When the denudation is proportional to the steepness of the slope a case that

is valid when the weathering is due to the exposure of the slope in which, as

was mentioned above, the steeper the slope, the faster the debris will be

removed. This implies that the steeper slopes will generally be more exposed

than the steeper ones and, for the mogotes evolution this is an important

fact.

For each case, the governing equations are the following:

Case 1, which could be approximated by the development of the slopes of the non karstic belt of the

Alturas de Pizarras, the governing equation is:

2

1

∂∂

+−=∂∂

x

y

t

y

Case 2, which could be approximated by the development of superimposed base levels in homogeneous

or lithologically heterogeneous mogotes:

2

1

∂∂

+−=∂∂

x

yy

t

y

Case 3, which could be the general case of the receding walls of lithological homogeneous mogotes:

2

1

∂∂

+∂∂

−=∂∂

x

y

x

y

t

y

The case of lithological variations should be considered briefly following Scheidegger (1991), who

introduced a function a(y) in the general equation for Case 3 (Values of a are summarized in Table 3),

and therefore it could be read as:

( )2

1

∂∂

+∂∂

−=∂∂

x

y

x

yya

t

y

The effect of the systematic tectonic uplift should also be added. Vertical movements causing fluvial

cutting and valley deepening as well as the remarkable superimposed cave levels are facts of cardinal

importance in the hydrologic evolution of Sierra de Los Organos.

Volume reduction and block instability

The conjugated action of erosion and dissolution promotes the deepening of the strand line or the

notch. Limestone dissolution becomes the kinetic factor to be considered. Kinetics of dissolution could

be described by a diffusion equation with mass transport of the type:



16

t

CvgradCDlapC

∂∂

=−

Where C is solute concentration, v, flow velocity, lap the Laplacian, and D the diffusivity factor that can

be taken as:

scmxD

25102 −=

Table 3. Values of a function in lithological variable slopes (summarized after Scheidegger, 1991):

Slope description a Range of y

Slope bank 1 0 ≤ y ≤ 0,9

0,1 0,9 ≤ y ≤ 1,0

Horizontal resistant layer 1 For all other values of y

0,1 0,4 ≤ y ≤ 0,5

Horizontal soft layer 1 Otherwise

0,1 0,4 ≤ y ≤ 0,5

Slope with soft bottom 1 0 ≤ y ≤ 0,1

0,1 0,1 ≤ y

Accounting for the restriction that this equation implies laminar parallel flow and, in the limit, the

solution is saturated at the wall.

The development of unstable morphologies leads to spontaneous mass movements (landslides,

breakdown, creeping) of parts of the mogotes controlled by the volumetric anisotropic reduction. In this

case, the maximum loads are defined as:

p23=σ

Where p is the loading stress. Therefore, the wall section falls down helped by the lost of cohesion due

to the enhancement of the dominating vertical joints. And the Critical Height of the Slope could be

successively obtained after the following equation:

Nsg

Chc ρ

=

FINAL REMARKS

The slope recession of the mogotes (hill stacks, conic karst, kegel karst, tower karst, turm karst) of the

Sierra de Los Organos could be explained after the non-linear theory of slope evolution accounting

whether their geologic structure exposes vertically jointed and faulted massive or bedded limestones.

This non linear model could be extended to the surrounding non karstic rocks where the main fluvial

systems traversing the karst land are originated. Although the variables account for different values in

each case, a particular recession model could be theoretically used to describe how the slopes could

recede. Therefore, a more general explanation could be approached for this outstanding

geomorphological feature combining the pure geological description with a more rigorous and

comprehensive mathematical model.

The mathematical models here described account for the important role of free flowing water. As karst

development is governed by the hydrologic history of the region a direct relation among karst landforms

and hydrology could be stressed. In the particular case of the mogotes, the most important factors

controlling the non linear model are hydrological by nature and as it should be expected in a karst

region, several morphological features could be identified as directly related with slope recession.



17

As the main hydrologic controls on slope recession of the mogotes are base erosion, subcutaneous

erosion and outcroppings of lateral caves, the associated morphologies are then:

• The foot – caves (füsshöhlen)

• The subcutaneous caves aborted by wall recession

• The fluvial and/or phreatic galleries also aborted by wall recession.

In fact, Foot-caves cave development seems to be rather the most common process related with the

mogotes recession. Associated with the lateral erosion due to the seasonal lakes that are part of the

evolution of karst valleys and to the permanent, seasonal or even episodic streams traversing the karst

lands; they form an extensive morphology that permanently contributes to excavate the base of the

mogotes and in turn contributing to its instability. The mechanisms of erosion could be perfectly

explained by the onset of secondary currents.

Obviously the receding wall can also intersect subcutaneous or fluvial and/or phreatic galleries or the

own enlargement of them could approach the mogotes wall contributing, in both cases to the instability

and recession of the wall.

In any case the continuous undercutting is also helped by more or less rapid mass movements on slopes.

Instability of part of the slope will cause its final collapse. Vertical jointing and faulting together with the

high values of the angles of internal friction and repose of these limestones thoroughly contribute to

this process of slope recession because it is already known that a direct relation holds for stability

factors as a function of slope for different angles of internal friction of the rocks.

ACKNOWLEDGEMENTS

The basic ideas of this paper were discussed more than 30 years ago when together with my colleagues

Mario Guerra and Ernesto Flores we were developing a systematic research on the hydrogeology of

Cuban karst mountains and particularly of the mogotes of the Sierra de Los Organos. Later on, as part of

my Geol BSc Thesis, detailed field work and theoretical discussions of the speleogenetic features of

Cuban mogotes was carried out with them. A systematic inventory of the different karst forms and their

hydrogeological role was also performed with these colleagues in particular environments of the Sierra

de Los Organos like the so-called sierras of Sumidero, Resolladero, El Pesquero, Quemado, Viñales,

Ancón, Pan de Azúcar and San Carlos. Surveying of ancient cave levels and strand lines of those localities

was systematically performed by our working group with the cooperation of Nélida Pérez Clavero,

Bárbara Pérez, Evelio Balado, Manuel Rivero Glean, Rafael Lavandero and Irela Martinez. Much of the

cave level correlations were performed by Ernesto Flores and morphogenetic and morphometric

analysis was basically done and tested by Mario Guerra. During the last years he has insisted in the

publication of those results. I am particularly indebted to Guerra and Flores support all these years.

Theoretical discussions of the hydrological evolution of the Sierra de Los Organos, including the analysis

of its physical and mathematical description were enriched with the opinions of J. J. Valdés and P.J.

Astraín. I am also particularly indebted to Prof. A. E. Scheidegger (Austria) for his inspiring ideas and to

its physical approach to theoretical geomorphology and hydrology.

Marjorie Condis and J.L. Clinche contribute with their photographs to this paper. Angel Graña and the

Foundation “Antonio Núñez Jimenez” of Nature and Man kindly contribute with the photo of Cerro de

Cabras.

My wife Ana and my daughters also share part of the field work all these years and made field sessions

more pleasant. Once again, I want to acknowledge their smiles and their systematic support.



18

REFERENCES Acevedo González, M. and L.F. Molerio León (1982): El Valle de San Carlos y sus Inmediaciones. Características de un Sistema Cársico para Propósitos de Simulación Matemática. Coloquio Internac. Hidrol. Cársica de la Región del Caribe, UNESCO, La Habana:248-256 Farfán González, H.; L. F. Molerio León; J. E. Díaz Sosa (2005): Aproximación al funcionamiento hidrológico del Valle de Viñales. Mem. 65 Congr. Soc. Espel. Cuba. http://www.sec1940.galeon.com Flores Valdés, Ernesto & L.F. Molerio León (1995): Patrones de Agrietamiento en la Sierra de Quemado, Pinar del Río, Cuba. Congr. Internac. LV Aniv. Soc. Espel. Cuba y Primera Reunión Iberoamericana, La Habana,:35-36 Glazek, J. (1968): Some observation on karst phenomena in North Vietnam. Proc. 4th Internatl. Congr. Speleol. In Yugoslavia, Ljubljana, T.III: 451-455 Gradzinski, R. & A. Radomski (1963): Types of Cuban caves and their dependence on Factors controlling karst development. Bull. Acad. Sci.Polon., Ser. Sci.Geol. et Geogr. (11) 2 Gradzinski, R. & A. Radomski (1968): Factors controlling karst development and cave types. Proc. 4th Internatl. Congr. Speleol. In Yugoslavia, Ljubljana, T.III: 457-461 Lehmann, H. (1953): Karst-Entwicklung in den Tröpen. Die Uns. In Wissenschaft und Technik, Frankfurt, (18):32-45 Lehmann, H. (1954): Der Tropische Kegelkarst auf den Groben Antillen. Erdkunde, 8. Lehmann, H. (1960): Las Áreas Cársicas del Caribe. Rev. Soc. Geog. de Cuba.(30) Lehmann, H., K. Krommelbein, W. Lotschert (1956): Karstmorphologische, geologische und botanische studien in der Sierra de Los Órganos auf Cuba. Erdkunde, 30 Mangin, A., M. Bakalowicz (1990): Le karst conique: sa genese a partir de l´exemple du karst du sud de la Chine. C.R. Acad.Sci. Paris, 310, II:301-307 Molerio León, L. F. (1975): Notas para una Tipología Geoespeleológica del Karst Cubano. Simp. XXXV Aniv. Soc. Espel. Cuba, La Habana, :65 Molerio León, L. F. (1980): Tipología Hidrogeológica del Carso Cubano. Inst. Hidroeconomía, La Habana, 44: Molerio León, L. F. (1981): Problemas Hidrogeológicos del Karst de Montaña de Cuba. Voluntad Hidráulica, La Habana XVIII(55):37-40 Molerio León, L. F.(1995): Field Trip Guide: Mogotes in the Viñales Valley, Pinar del Río Province, Cuba. Internatl. Geogr. Union (IGU) Conf. of Latin America and Caribbean Countries, La Habana, 38: Molerio León, L.F. (2004): Los mogotes del Valle de Viñales, Monumento Nacional, Pinar del Río, Cuba. Mapping, Revista Internac. Ciencias de la Tierra (98), Madrid, Noviembre,:12-22. Molerio León, L. F.; M. Guerra Oliva & E. Flores Valdés (1983): Geomorfología e Hidrogeología Cársica del Valle de Pan de Azúcar, Sierra de los Órganos, Pinar del Río. Voluntad Hidráulica, (62):23-36 Molerio León, L.F. & E. Flores Valdés (2003): Hidrogeología y geomorfología cársica de Valle Ancón, Pinar del Río, Cuba. Ing. Hidr. y Ambiental, La Habana XXIV, 3:3-9 Núñez Jiménez, A., V. Panos & O. Stelcl. (1964): Investigaciones carsológicas en Cuba. Acad. Cienc. Cuba 80: Núñez Jiménez, A. (1967): Clasificación Genética de las Cuevas de Cuba. Acad.Cienc.Cuba,Inst. Geogr., Depto. Espel., Edic. Prov.,La Habana, 224: Panos, V. & O. Stelcl (1968): Problems of the conical karst in Cuba. Proc. 4th Internatl. Congr. Speleol. In Yugoslavia, Ljubljana, T.III: 533-555 Scheidegger, A.E. (1991): Theoretical Geomorphology. Springer-Verlag, Wien, Austria, 434: Wang, Y. (2000): Comparing different numerical treatments of advection terms for wind-induced circulations in Lake Constance. In B. Straughan, R. Greve, H. Ehrentraut, Y. Wang (Eds.) Continuum mechanics and applications in Geophysics and the Environment. Springer, Berlin, Heidelberg, New York, Barcelona, Hong Kong, London, Milan, Paris, Singapore, Tokyo:368-393



19

Próximos artículos/Forthcoming papers

• Magnetismo ambiental en sedimentos cuaternarios del Sistema Cavernario Majaguas-Cantera, Pinar del Río,

Cuba, por I. I. Pedroso Herrera, L. Sagnott,, J.M. Pajón Morejón y M. J. Fundora Granda.

• Sistema de abastecimiento de agua en el karst de montaña de la Sierra de Soroa, Pinar del Río, Cuba, por I.

Gómez Carmona, R. Hernández Díaz y F. Márquez Montesino.

• Tipología ingeniero geológica del carso cubano, por L.F. Molerio León.

• Aguas terrestres y relieve en la cuenca subterránea Jaruco, La Habana, Cuba, por M.G. Guerra Oliva.

• Paisajes hidrológicos cársicos del tramo de Guira – Quivicán de la Cuenca Costera Sur de la Habana, Cuba, por

M. G. Guerra Oliva y O. E. Pérez López.

• Metodología de la investigación del carso, por L.F. Molerio León.



20

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