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Yizhaq et al 2007
PHYSICAL REVIEW LETTERS week ending
PRL 98, 188001 (2007) 4 MAY 2007
Why Do Active and Stabilized Dunes Coexist under the Same Climatic Conditions?
Hezi Yizhaq,1 Yosef Ashkenazy,1 and Haim Tsoar2
1
Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research,
Ben-Gurion University of the Negev, Sede Boqer Campus 84990 Israel
2
Department of Geography and Environmental Development Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
(Received 4 November 2006; revised manuscript received 13 February 2007; published 2 May 2007)
Sand dunes can be active (mobile) or stable, mainly as a function of vegetation cover and wind power.
However, there exists as yet unexplained evidence for the coexistence of bare mobile dunes and vegetated
stabilized dunes under the same climatic conditions. We propose a model for dune vegetation cover driven
by wind power that exhibits bistabilty and hysteresis with respect to the wind power. For intermediate
wind power, mobile and stabilized dunes can coexist, whereas for low (or high) wind power they can be
fixed (or mobile). Climatic change or human intervention can turn active dunes into stable ones and vice
versa; our model predicts that prolonged droughts with stronger winds can result in dune reactivation.
DOI: 10.1103/PhysRevLett.98.188001 PACS numbers: 45.70.ÿn, 05.45.ÿa
Sand dunes form an important, unique, and complex DP hU2 U ÿ Ut i; (1)
ecological [1] and physical [2] system. Approximately
20% [3] of desert areas are covered by sand dunes, some where U is the wind speed (in knots: 1 knot 0:514 m=s)
of which endanger human settlements, agricultural fields, measured at a height of 10 m and the average is over time;
roads, etc. (e.g., [4]). Sand dunes may be either active Ut is a minimal threshold velocity (12 knots) necessary
(mobile) or fixed, as determined primarily by vegetation for sand transport [8]. Still, the DP is a potential sand drift,
cover and wind power. Climate changes and human activ- with the actual sand drift further depending on the mean
ities may therefore transform fixed dunes into mobile grain diameter, the degree of surface roughness, the
dunes, and vice versa [5], and may even accelerate deserti- amount and type of vegetation or crust cover, the amount
fication processes. It is thus important to understand dune of moisture in the sand, and the uniformity of the wind
dynamics and their response to various climatic changes direction. According to Fryberger [8] wind energy can be
and human activities. classified as follows: DP < 200, low energy; 200 DP <
One of the most interesting phenomena related to sand 400, intermediate energy; and DP 400, high energy.
dunes is bistability; i.e., under the same climatic conditions A striking example of the dune bistability phenomenon,
(mainly wind power and precipitation) and in the same i.e., mobile barchan dunes and stabilized parabolic dunes
geographical area, it is possible to find both active and [Fig. 1(a)], is found in northeastern Brazil [10]. Here the
fixed dunes [Fig. 1(a)]. The underlying mechanism for this average annual rainfall is 2.4 m and the DP is over 1000. In
bistability is not yet clear. Recently, Tsoar [6] suggested spite of the extremely high rainfall, many dunes are active
that it is driven by the interaction between wind and due to the strong winds.
vegetation. This Letter will develop a simple, physically A vegetated dune can naturally become active when the
motivated model to explain the bistability and hysteresis of wind power is sufficiently high to cause the decay of
sand dunes. vegetation. Once active, only a climatic reversal to much
Dunes are driven by the wind [6,7], but vegetation cover- weaker winds which do not suppress vegetation growth,
ing them weakens this impact. On one hand, even the will allow reestablishment of vegetation and stabilization
action of very strong winds may be masked by vegetation of the dune. These dynamics describe hysteresis behavior.
leaving the dunes immobile. On the other hand, even Natural changes in windiness or forced changes in vegeta-
relatively weak winds may lead to sand mobility when tion cover (e.g., overgrazing, clear cutting, or forced stabi-
dunes are bare. Moreover, wind also affects the vegetation lization by planting) may shift the dune into a new state. An
cover; when wind becomes stronger the vegetation cover example of such a transition is the clear cutting of vegeta-
decreases, a fact that further enhances dune mobility, tion on dunes in North Holland that took place in
which in turn further reduces vegetation cover. This feed- December 1998 (DP 1570) [11], which led to the dunes
back mechanism underlines the model we propose below. remaining bare and mobile until today.
Wind power is usually expressed by aeolian geomor- In places with very weak wind power (low DP), the
phologists as a drift potential (DP) [6,8]. Theoretical and dunes have only one stable fixed vegetated state; here we
empirical studies have shown [2,8,9], that DP is propor- assume that the precipitation is above the threshold needed
tional to the potential sand volume that can be transported for vegetation growth (80 mm=yr, based on observations
by the wind trough a 1 m wide cross section per unit time in the Negev Desert). In these cases, the dunes can be
(in most cases, given as per year). DP is given by active due to either anthropogenic pressure or prolonged
0031-9007=07=98(18)=188001(4) 188001-1 © 2007 The American Physical Society
PHYSICAL REVIEW LETTERS week ending
PRL 98, 188001 (2007) 4 MAY 2007
tated, fixed dunes within two years, as dictated by the
area’s weak wind power and above-threshold precipitation.
We formulate here a general model of wind-vegetation-
dune interactions without going into the details of dune
types. We assume that wind power is the dominant factor
affecting dune mobility in areas where the annual precipi-
tation is above the threshold needed for vegetation devel-
opment (50 –80 mm per year [6]). The impact of vegetation
cover on sand transport involves complicated processes
(cf. [13,14]). Vegetation acts to reduce wind-based sand
transport as it extracts momentum from the air, reduces
wind velocity, masks the sand surface from direct wind
action, and traps sand particles. The sand transport depends
on the height, roughness, and concentration of plants (e.g.,
[15]). In the absence of a complete, accepted theory for the
coupling between sand transport and vegetation, we adopt
a simple modeling approach that is consistent with basic
physical principles [16].
The model we propose is
dv v
v 1 ÿ
dt vmax
ÿ " DP vc ÿ vv ÿ DP2=3 v: (2)
The dynamical variable is v, the vegetation areal cover
density, which values between 0 (bare dune) and vmax ,
where for completely vegetated dune vmax 1. The vege-
tation growth rate, v 1 ÿ v=vmax , is logistic
growth [production v multiplied by resource com-
petition 1 ÿ v=vmax ] [17] where ‘‘’’ is a ‘‘spontaneous’’
vegetation factor growth that describes an average growth
rate for even bare dune due to soil seed banks, underground
roots [18], and seed carried by wind, animals, etc. Logistic
models are often used to describe hysteresis phenomena of
ecological and economical systems [19].
The second term on the right-hand side of Eq. (2),
¸´
FIG. 1 (color online). (a) Lencois Maranhenses dunes in north- " DP vc ÿ vv, represents the effect of sand movement
eastern Brazil (43 W, 2 S); a field of active barchan dunes
on vegetation (such as root exposure and plant burial by
(upper right corner) coexist with fixed vegetated parabolic dunes.
Landsat Satellite image. [Landsat imagery courtesy of NASA sand) and since it is a mortality term it is proportional to v.
Goddard Space Flight Center and USGS Center for Earth Re- vc is a critical vegetation cover above which sand transport
sources Observations and Science.] (b) Satellite image (Google sharply decreases [20,21]. The Heaviside step function
Earth) of the Israel-Egypt border area showing high contrast in vc ÿ v, which is 1 when v < vc , while otherwise is
albedo between the bright active Egyptian dunes and the darker zero, reflects the fact that for high vegetation density, v >
fixed vegetated Israeli dunes. Arrows indicate the prevailing vc , the surface is effectively protected from wind action
wind direction. [15]. The speed of dune migration is proportional to the
sand flux, which is self proportional to DP. Thus a change
in dune height is also proportional to DP [2,8] and hence
drought. This occurs, for example, at the Israeli-Egyptian the erosion or accumulation of vegetation due to sand
border [Fig. 1(b)], where there is a clear visual differ- transport is proportional to DP. " is a proportionality
ence between the active dunes on the Egyptian side and constant that may have different values for different plants.
the vegetated and almost fully stabilized dunes on the The last term in Eq. (2), DP2=3 v, stands for a reduction
Israeli side [12]. This difference is attributed to wood in vegetation cover due to direct wind action that can
gathering, overgrazing, and trampling of the sand crust increase evapotranspiration and uproot, erode, or suppress
on the Egyptian side, practices that have been prohibited the growth of vegetation [22]. Generally, wind drag is pro-
on the Israeli side since the establishment of the border in portional to the square of the wind speed [2] while DP is
1982. Once this anthropogenic pressure ceased on the proportional to the cube of the wind speed [Eq. (1)]. Thus,
Israeli side, the dunes took on their natural state of vege- DP2=3 may represent wind drag on vegetation and hence
188001-2
PHYSICAL REVIEW LETTERS week ending
PRL 98, 188001 (2007) 4 MAY 2007
the vegetation growth suppression due to direct wind ac- higher values of vc (like the Australian dunes) can be
tion. This term is proportional to v, as it is basically a reactivated at lower DP values (weaker winds), whereas
mortality term. is a proportionality constant. The term, for dunes with lower values of vc (like the Kalahari dunes)
DP2=3 v, unlike the other terms in Eq. (2), acts even when stronger winds are needed for reactivate them. Following
the dune is maximally vegetated. Eq. (6), DP1 also decreases with increasing vc but more
Although it is possible to reduce the model (2) to contain slowly as compare to DP2 . Thus, as vc increases, the
only four independent parameters we choose to follow the hysteresis diagram shrinks and shifts to the left, mainly
seven original parameters to allow easier physical inter- since the fixed dune branch F becomes smaller; see
pretation. The parameter values that we use are: Fig. 3(a).
0:1 yrÿ1 which is a typical growth rate for dune vegetation Human intervention may cause active dunes to become
(see [17]); 0:2 is estimated as dv = starting from bare
dt
stable or stable dunes to turn active. The model shows that
dune and when DP 0, vmax 1 represents ideal vegeta- vegetated dunes with higher DP values are more vulnerable
tion growth conditions: vc 0:2 [7,23], " 0:001, and to activation as only a relatively small disturbance in
0:0008. Both " and can be estimated from field vegetation cover such as, clear cutting or overgrazing
experiments. may shift the system to the lower branch of activated
The proposed model [Eq. (2)] has two stable stationary dunes. The reactivation of some Netherlands coastal dunes
states, A and F, representing active and fixed dunes, re- described above can be regarded as an example of this
spectively. The vegetation cover density under which ac- transition, which is seen in Fig. 2(b). The graph shows
tive and fixed states may interchanging va;f is numerical solutions of Eq. (2) with different initial pertur-
q
va;f ÿÿa;f ÿ2 vmax ;
a;f (3) DP1 DP2
where 0.8
(a)
Israel-Eygpt border
ÿa;f vmax ÿ1 =vmax " DP vc ÿ v= F
0.6
Vegetation cover, v
DP2=3 = =2: (4) 0.4
The active dune state va is valid when v < vc and the fixed Netherlands
0.2 A vc
dune state vf , is valid otherwise. For DP 0, v vmax .
Figure 2(a) shows the stable stationary states as a func- 0
tion of DP. For a wide regime of DP values, both the fixed 100 1000 10 000
and active dune states coexist, indicating the hysteresis and Wind drift potential, DP
bistability of the model. This dune hysteresis may be Kraansvlak (Netherlands) Israel-Egypt border
described as follows: starting from very low DP, only the 1 (c) DP=50 DP=100
fixed dune solution vf exists; when the DP slowly in- 0.8 (b) DP=1570
0.6
creases, this solution persists until the very high DP 0.4 v0=0.3
DP2 is reaching at which point vf vc . The DP at this 0.2 v0=0.1 DP=200 vc
point is given by 0
0 20 40 60 80 100 20 40 60 80 100
DP 2=3 1 ÿ vc =vmax vc = vc : (5) Time Time
2
Beyond this point, the system switches to the active dune FIG. 2 (color online). (a) Stable states diagram showing vege-
state, A. When DP is then slowly decreased, the solution tation cover v vs DP. The gray area indicates the domain of
continues to be the active dune state until a very low DP bistability. The arrows indicate the hysteresis scenario, and the
dash-dot vertical lines indicate the transition from fixed to active
(DP1 ), at which point it switches to the fixed dune state.
dune states and vice versa. Solid lines indicate stable states.
DP1 can be approximated as Parameter values used are: vmax 1, vc 0:2, 0:001,
0:1, 0:0008, 0:2. For these parameters, DP1 139 and
DP 1=3 DP2=3 ="1=3 ÿ = 3":
1 2 (6)
DP2 2828. The vertical arrows indicate two examples shown
Thus, as DP2 decreases, DP1 also decreases. in panels b and c. (b) An example of transition from a fixed dune
The critical vegetation cover vc varies for different to an active dune state (Netherlands case). Time evolution (in
arbitrary time units) of vegetation cover v for DP 1570 for
geographical locations, from vc 0:14 for the Kalahari
different initial conditions (v 0:3 and v 0:1). The upper
desert [23] to vc 0:35 for Australian deserts [7]; vc is curve converges to the fixed dune F state, while the lower curve
sometimes difficult to measure and may have only mar- (that is associated with large disturbance) converges to the active
ginal accuracy. The various vc values found in nature may dune A state. (c) Same as b, but for different DP values, (Israel-
indicate different dynamic scenarios predicted by our Egypt border case), starting from v 0:01. For DP 50 and
model. More specifically, DP2 [Eq. (5)] decreases drasti- DP 100, the asymptotic state is the fixed dune state F,
cally with increasing vc , which means that dunes with whereas for DP 200, it is the active dune state A.
188001-3
PHYSICAL REVIEW LETTERS week ending
PRL 98, 188001 (2007) 4 MAY 2007
Active dunes (b) Active dunes a scenario. Figure 3(b) illustrates the impact of drought on
(a) DP1
10
3 dune activation. We simulate drought conditions by de-
Active & fixed dunes DP2 creasing the vegetation logistic growth rate in Eq. (2). As
DP
Active & fixed dunes becomes smaller, the hysteresis diagram shrinks, which
2
10 means that transition from the fixed to the active dune state
Fixed dunes Fixed dunes (DP2 ) occurs at weaker winds (smaller DP).
0.15 0.2 0.25 0.3 0.02 0.04 0.06 0.08 0.1 The model we suggest is very simple and does not
vc α definitively simulate the complicated process of dune
dynamics and the effects of wind direction variability.
FIG. 3. (a) Active to fixed dune transition point DP1 [Eq. (6)] Thus, our conclusions should be regarded as speculative.
and fixed to active dunes transition point DP2 [Eq. (5)] as a Nevertheless, the proposed model may shed light on the
function of critical vegetation cover vc . The various gray shad- dynamical scenarios underlining dune fixation and reacti-
ings represent different stability regimes. As vc increases the fix- vation processes.
active and active-fix dune transitions occur for weaker winds We thank E. Zaady, Y. Zarmi, and A. Zemel for helpful
(low DP) (b) Same as (a), but for different growth rates . As discussions and the Center for Complexity Science (Y. A.)
decreases fix-active and active-fix dune transitions occur for and the Israeli Ministry of the Environmental Protection
weaker winds (low DP), suggesting that drought conditions are
for financial support.
more favorable for transitions from fixed to mobile dune states
(desertification process). Parameter values are as in Fig. 2.
bations for DP 1570 (which is close to that of the Dutch [1] A. Danin, Plants of Desert Dunes (Springer, Berlin, 1996).
dune area). The asymptotic behavior of the system depends [2] R. A. Bagnold, The Physics of Blown Sands and Desert
on the initial state; if the initial vegetation cover is above Dunes (Chapman and Hall, London, 1941).
vc , v converges to the fixed dune branch F; otherwise it [3] K. Pye and H. Tsoar, Aeloian Sand and Sand Dunes
converges to the active dune branch A. (Unwin Hyman, London, 1990).
In places with very low wind power the dune has only [4] Z. B. Dong et al., J. Arid Environ. 57, 329 (2004).
[5] D. S. G. Thomas et al., Nature (London) 435, 1218 (2005).
one stable state, which is vegetated and fixed, suggesting
[6] H. Tsoar, Physica (Amsterdam) A357, 50 (2005).
that human intervention is likely to be the dominant factor [7] J. E. Ash and R. J. Wasson, Z. Geomorphologie 45, 7
causing the dunes to turn bare and become active. An (1983).
example of such a phenomenon is the dunes at the [8] S. G. Fryberger, U.S. Geol. Surv., Washington DC 1052,
Israeli-Egyptian border mentioned above (DP 50); see 137 (1979).
Fig. 1(b). Figure 2(c) shows the stabilization evolution for [9] J. E. Bullard, J. Sedimentary Res. 67, 499 (1997).
three different values of DP starting from a very low [10] N. Levin et al., Catena (to be published).
vegetation cover (v 0:01), such as that on the bare, [11] S. M. Arens et al., Geomorphology 59, 175 (2004).
active dunes on the Israeli-Egyptian border. For DP < [12] A. Meir and H. Tsoar, Human Ecology 24, 39 (1996).
DP1 , v converges quickly to the fixed, vegetated dune state, [13] N. Lancaster and A. Baas, Earth Surf. Processes
with the response curves seeming to reflect to actual data Landforms 23, 69 (1998).
´
[14] O. Duran and H. Herrmann, Phys. Rev. Lett. 97, 188001
[12,17]. However, our model predicts that with stronger
(2006).
winds (DP 200), the bare and active dunes should re- [15] S. A. Wolfe and W. C. Nickling, Prog. Phys. Geography
main active even when the anthropogenic pressure is re- 17, 50 (1993).
leased. In fact, rapid natural recovery of dune vegetation [16] A. C. W. Bass, Geomorphology 48, 309 (2002).
was also reported for parts of North Africa [3]. [17] C. H. Hugenholtz and S. A. Wolfe, Geomorphology 70, 53
There is a wealth of evidence for global climate change (2005).
over the past decades [24]. Global circulation models [18] S. M. Arens and L. H. W. T. Geelen, J. Coastal Research :
predict that the global warming trend will persist if the JCR/CERF 22, 1094 (2006).
‘‘business as usual’’ scenario [24] continues. Some studies [19] D. Ludwig et al., Ecology and Society 1, Art. 7 (1997).
have forewarned that the climate system will become even [20] H. Nishimori and H. Tanaka, Earth Surf. Processes
more extreme, with stronger atypical extreme climate Landforms 26, 1143 (2001).
[21] F. de Castro, Ecol. Mod. 78, 205 (1995).
events appearing more frequently (e.g., [24]). It is possible
[22] P. Hesp, Geomorphology 48, 245 (2002).
that some desert areas will experience more severe and [23] G. F. S. Wiggs et al., Earth Surf. Processes Landforms 20,
prolonged droughts with stronger and more frequent wind 515 (1995).
storms; this could lead to the activation of hitherto fixed [24] Intergovernmental Panel on Climate Change: The Scien-
dunes, as has been observed in the past decades (e.g., [25]), tific Basis (Cambridge University Press, Cambridge,
with severe damage to the environment and the quality of England, 2001).
life of millions of people. Our simple model supports such [25] L. Marin et al., Geomorphology 70, 163 (2005).
188001-4
PRL 98, 188001 (2007) 4 MAY 2007
Why Do Active and Stabilized Dunes Coexist under the Same Climatic Conditions?
Hezi Yizhaq,1 Yosef Ashkenazy,1 and Haim Tsoar2
1
Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research,
Ben-Gurion University of the Negev, Sede Boqer Campus 84990 Israel
2
Department of Geography and Environmental Development Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
(Received 4 November 2006; revised manuscript received 13 February 2007; published 2 May 2007)
Sand dunes can be active (mobile) or stable, mainly as a function of vegetation cover and wind power.
However, there exists as yet unexplained evidence for the coexistence of bare mobile dunes and vegetated
stabilized dunes under the same climatic conditions. We propose a model for dune vegetation cover driven
by wind power that exhibits bistabilty and hysteresis with respect to the wind power. For intermediate
wind power, mobile and stabilized dunes can coexist, whereas for low (or high) wind power they can be
fixed (or mobile). Climatic change or human intervention can turn active dunes into stable ones and vice
versa; our model predicts that prolonged droughts with stronger winds can result in dune reactivation.
DOI: 10.1103/PhysRevLett.98.188001 PACS numbers: 45.70.ÿn, 05.45.ÿa
Sand dunes form an important, unique, and complex DP hU2 U ÿ Ut i; (1)
ecological [1] and physical [2] system. Approximately
20% [3] of desert areas are covered by sand dunes, some where U is the wind speed (in knots: 1 knot 0:514 m=s)
of which endanger human settlements, agricultural fields, measured at a height of 10 m and the average is over time;
roads, etc. (e.g., [4]). Sand dunes may be either active Ut is a minimal threshold velocity (12 knots) necessary
(mobile) or fixed, as determined primarily by vegetation for sand transport [8]. Still, the DP is a potential sand drift,
cover and wind power. Climate changes and human activ- with the actual sand drift further depending on the mean
ities may therefore transform fixed dunes into mobile grain diameter, the degree of surface roughness, the
dunes, and vice versa [5], and may even accelerate deserti- amount and type of vegetation or crust cover, the amount
fication processes. It is thus important to understand dune of moisture in the sand, and the uniformity of the wind
dynamics and their response to various climatic changes direction. According to Fryberger [8] wind energy can be
and human activities. classified as follows: DP < 200, low energy; 200 DP <
One of the most interesting phenomena related to sand 400, intermediate energy; and DP 400, high energy.
dunes is bistability; i.e., under the same climatic conditions A striking example of the dune bistability phenomenon,
(mainly wind power and precipitation) and in the same i.e., mobile barchan dunes and stabilized parabolic dunes
geographical area, it is possible to find both active and [Fig. 1(a)], is found in northeastern Brazil [10]. Here the
fixed dunes [Fig. 1(a)]. The underlying mechanism for this average annual rainfall is 2.4 m and the DP is over 1000. In
bistability is not yet clear. Recently, Tsoar [6] suggested spite of the extremely high rainfall, many dunes are active
that it is driven by the interaction between wind and due to the strong winds.
vegetation. This Letter will develop a simple, physically A vegetated dune can naturally become active when the
motivated model to explain the bistability and hysteresis of wind power is sufficiently high to cause the decay of
sand dunes. vegetation. Once active, only a climatic reversal to much
Dunes are driven by the wind [6,7], but vegetation cover- weaker winds which do not suppress vegetation growth,
ing them weakens this impact. On one hand, even the will allow reestablishment of vegetation and stabilization
action of very strong winds may be masked by vegetation of the dune. These dynamics describe hysteresis behavior.
leaving the dunes immobile. On the other hand, even Natural changes in windiness or forced changes in vegeta-
relatively weak winds may lead to sand mobility when tion cover (e.g., overgrazing, clear cutting, or forced stabi-
dunes are bare. Moreover, wind also affects the vegetation lization by planting) may shift the dune into a new state. An
cover; when wind becomes stronger the vegetation cover example of such a transition is the clear cutting of vegeta-
decreases, a fact that further enhances dune mobility, tion on dunes in North Holland that took place in
which in turn further reduces vegetation cover. This feed- December 1998 (DP 1570) [11], which led to the dunes
back mechanism underlines the model we propose below. remaining bare and mobile until today.
Wind power is usually expressed by aeolian geomor- In places with very weak wind power (low DP), the
phologists as a drift potential (DP) [6,8]. Theoretical and dunes have only one stable fixed vegetated state; here we
empirical studies have shown [2,8,9], that DP is propor- assume that the precipitation is above the threshold needed
tional to the potential sand volume that can be transported for vegetation growth (80 mm=yr, based on observations
by the wind trough a 1 m wide cross section per unit time in the Negev Desert). In these cases, the dunes can be
(in most cases, given as per year). DP is given by active due to either anthropogenic pressure or prolonged
0031-9007=07=98(18)=188001(4) 188001-1 © 2007 The American Physical Society
PHYSICAL REVIEW LETTERS week ending
PRL 98, 188001 (2007) 4 MAY 2007
tated, fixed dunes within two years, as dictated by the
area’s weak wind power and above-threshold precipitation.
We formulate here a general model of wind-vegetation-
dune interactions without going into the details of dune
types. We assume that wind power is the dominant factor
affecting dune mobility in areas where the annual precipi-
tation is above the threshold needed for vegetation devel-
opment (50 –80 mm per year [6]). The impact of vegetation
cover on sand transport involves complicated processes
(cf. [13,14]). Vegetation acts to reduce wind-based sand
transport as it extracts momentum from the air, reduces
wind velocity, masks the sand surface from direct wind
action, and traps sand particles. The sand transport depends
on the height, roughness, and concentration of plants (e.g.,
[15]). In the absence of a complete, accepted theory for the
coupling between sand transport and vegetation, we adopt
a simple modeling approach that is consistent with basic
physical principles [16].
The model we propose is
dv v
v 1 ÿ
dt vmax
ÿ " DP vc ÿ vv ÿ DP2=3 v: (2)
The dynamical variable is v, the vegetation areal cover
density, which values between 0 (bare dune) and vmax ,
where for completely vegetated dune vmax 1. The vege-
tation growth rate, v 1 ÿ v=vmax , is logistic
growth [production v multiplied by resource com-
petition 1 ÿ v=vmax ] [17] where ‘‘’’ is a ‘‘spontaneous’’
vegetation factor growth that describes an average growth
rate for even bare dune due to soil seed banks, underground
roots [18], and seed carried by wind, animals, etc. Logistic
models are often used to describe hysteresis phenomena of
ecological and economical systems [19].
The second term on the right-hand side of Eq. (2),
¸´
FIG. 1 (color online). (a) Lencois Maranhenses dunes in north- " DP vc ÿ vv, represents the effect of sand movement
eastern Brazil (43 W, 2 S); a field of active barchan dunes
on vegetation (such as root exposure and plant burial by
(upper right corner) coexist with fixed vegetated parabolic dunes.
Landsat Satellite image. [Landsat imagery courtesy of NASA sand) and since it is a mortality term it is proportional to v.
Goddard Space Flight Center and USGS Center for Earth Re- vc is a critical vegetation cover above which sand transport
sources Observations and Science.] (b) Satellite image (Google sharply decreases [20,21]. The Heaviside step function
Earth) of the Israel-Egypt border area showing high contrast in vc ÿ v, which is 1 when v < vc , while otherwise is
albedo between the bright active Egyptian dunes and the darker zero, reflects the fact that for high vegetation density, v >
fixed vegetated Israeli dunes. Arrows indicate the prevailing vc , the surface is effectively protected from wind action
wind direction. [15]. The speed of dune migration is proportional to the
sand flux, which is self proportional to DP. Thus a change
in dune height is also proportional to DP [2,8] and hence
drought. This occurs, for example, at the Israeli-Egyptian the erosion or accumulation of vegetation due to sand
border [Fig. 1(b)], where there is a clear visual differ- transport is proportional to DP. " is a proportionality
ence between the active dunes on the Egyptian side and constant that may have different values for different plants.
the vegetated and almost fully stabilized dunes on the The last term in Eq. (2), DP2=3 v, stands for a reduction
Israeli side [12]. This difference is attributed to wood in vegetation cover due to direct wind action that can
gathering, overgrazing, and trampling of the sand crust increase evapotranspiration and uproot, erode, or suppress
on the Egyptian side, practices that have been prohibited the growth of vegetation [22]. Generally, wind drag is pro-
on the Israeli side since the establishment of the border in portional to the square of the wind speed [2] while DP is
1982. Once this anthropogenic pressure ceased on the proportional to the cube of the wind speed [Eq. (1)]. Thus,
Israeli side, the dunes took on their natural state of vege- DP2=3 may represent wind drag on vegetation and hence
188001-2
PHYSICAL REVIEW LETTERS week ending
PRL 98, 188001 (2007) 4 MAY 2007
the vegetation growth suppression due to direct wind ac- higher values of vc (like the Australian dunes) can be
tion. This term is proportional to v, as it is basically a reactivated at lower DP values (weaker winds), whereas
mortality term. is a proportionality constant. The term, for dunes with lower values of vc (like the Kalahari dunes)
DP2=3 v, unlike the other terms in Eq. (2), acts even when stronger winds are needed for reactivate them. Following
the dune is maximally vegetated. Eq. (6), DP1 also decreases with increasing vc but more
Although it is possible to reduce the model (2) to contain slowly as compare to DP2 . Thus, as vc increases, the
only four independent parameters we choose to follow the hysteresis diagram shrinks and shifts to the left, mainly
seven original parameters to allow easier physical inter- since the fixed dune branch F becomes smaller; see
pretation. The parameter values that we use are: Fig. 3(a).
0:1 yrÿ1 which is a typical growth rate for dune vegetation Human intervention may cause active dunes to become
(see [17]); 0:2 is estimated as dv = starting from bare
dt
stable or stable dunes to turn active. The model shows that
dune and when DP 0, vmax 1 represents ideal vegeta- vegetated dunes with higher DP values are more vulnerable
tion growth conditions: vc 0:2 [7,23], " 0:001, and to activation as only a relatively small disturbance in
0:0008. Both " and can be estimated from field vegetation cover such as, clear cutting or overgrazing
experiments. may shift the system to the lower branch of activated
The proposed model [Eq. (2)] has two stable stationary dunes. The reactivation of some Netherlands coastal dunes
states, A and F, representing active and fixed dunes, re- described above can be regarded as an example of this
spectively. The vegetation cover density under which ac- transition, which is seen in Fig. 2(b). The graph shows
tive and fixed states may interchanging va;f is numerical solutions of Eq. (2) with different initial pertur-
q
va;f ÿÿa;f ÿ2 vmax ;
a;f (3) DP1 DP2
where 0.8
(a)
Israel-Eygpt border
ÿa;f vmax ÿ1 =vmax " DP vc ÿ v= F
0.6
Vegetation cover, v
DP2=3 = =2: (4) 0.4
The active dune state va is valid when v < vc and the fixed Netherlands
0.2 A vc
dune state vf , is valid otherwise. For DP 0, v vmax .
Figure 2(a) shows the stable stationary states as a func- 0
tion of DP. For a wide regime of DP values, both the fixed 100 1000 10 000
and active dune states coexist, indicating the hysteresis and Wind drift potential, DP
bistability of the model. This dune hysteresis may be Kraansvlak (Netherlands) Israel-Egypt border
described as follows: starting from very low DP, only the 1 (c) DP=50 DP=100
fixed dune solution vf exists; when the DP slowly in- 0.8 (b) DP=1570
0.6
creases, this solution persists until the very high DP 0.4 v0=0.3
DP2 is reaching at which point vf vc . The DP at this 0.2 v0=0.1 DP=200 vc
point is given by 0
0 20 40 60 80 100 20 40 60 80 100
DP 2=3 1 ÿ vc =vmax vc = vc : (5) Time Time
2
Beyond this point, the system switches to the active dune FIG. 2 (color online). (a) Stable states diagram showing vege-
state, A. When DP is then slowly decreased, the solution tation cover v vs DP. The gray area indicates the domain of
continues to be the active dune state until a very low DP bistability. The arrows indicate the hysteresis scenario, and the
dash-dot vertical lines indicate the transition from fixed to active
(DP1 ), at which point it switches to the fixed dune state.
dune states and vice versa. Solid lines indicate stable states.
DP1 can be approximated as Parameter values used are: vmax 1, vc 0:2, 0:001,
0:1, 0:0008, 0:2. For these parameters, DP1 139 and
DP 1=3 DP2=3 ="1=3 ÿ = 3":
1 2 (6)
DP2 2828. The vertical arrows indicate two examples shown
Thus, as DP2 decreases, DP1 also decreases. in panels b and c. (b) An example of transition from a fixed dune
The critical vegetation cover vc varies for different to an active dune state (Netherlands case). Time evolution (in
arbitrary time units) of vegetation cover v for DP 1570 for
geographical locations, from vc 0:14 for the Kalahari
different initial conditions (v 0:3 and v 0:1). The upper
desert [23] to vc 0:35 for Australian deserts [7]; vc is curve converges to the fixed dune F state, while the lower curve
sometimes difficult to measure and may have only mar- (that is associated with large disturbance) converges to the active
ginal accuracy. The various vc values found in nature may dune A state. (c) Same as b, but for different DP values, (Israel-
indicate different dynamic scenarios predicted by our Egypt border case), starting from v 0:01. For DP 50 and
model. More specifically, DP2 [Eq. (5)] decreases drasti- DP 100, the asymptotic state is the fixed dune state F,
cally with increasing vc , which means that dunes with whereas for DP 200, it is the active dune state A.
188001-3
PHYSICAL REVIEW LETTERS week ending
PRL 98, 188001 (2007) 4 MAY 2007
Active dunes (b) Active dunes a scenario. Figure 3(b) illustrates the impact of drought on
(a) DP1
10
3 dune activation. We simulate drought conditions by de-
Active & fixed dunes DP2 creasing the vegetation logistic growth rate in Eq. (2). As
DP
Active & fixed dunes becomes smaller, the hysteresis diagram shrinks, which
2
10 means that transition from the fixed to the active dune state
Fixed dunes Fixed dunes (DP2 ) occurs at weaker winds (smaller DP).
0.15 0.2 0.25 0.3 0.02 0.04 0.06 0.08 0.1 The model we suggest is very simple and does not
vc α definitively simulate the complicated process of dune
dynamics and the effects of wind direction variability.
FIG. 3. (a) Active to fixed dune transition point DP1 [Eq. (6)] Thus, our conclusions should be regarded as speculative.
and fixed to active dunes transition point DP2 [Eq. (5)] as a Nevertheless, the proposed model may shed light on the
function of critical vegetation cover vc . The various gray shad- dynamical scenarios underlining dune fixation and reacti-
ings represent different stability regimes. As vc increases the fix- vation processes.
active and active-fix dune transitions occur for weaker winds We thank E. Zaady, Y. Zarmi, and A. Zemel for helpful
(low DP) (b) Same as (a), but for different growth rates . As discussions and the Center for Complexity Science (Y. A.)
decreases fix-active and active-fix dune transitions occur for and the Israeli Ministry of the Environmental Protection
weaker winds (low DP), suggesting that drought conditions are
for financial support.
more favorable for transitions from fixed to mobile dune states
(desertification process). Parameter values are as in Fig. 2.
bations for DP 1570 (which is close to that of the Dutch [1] A. Danin, Plants of Desert Dunes (Springer, Berlin, 1996).
dune area). The asymptotic behavior of the system depends [2] R. A. Bagnold, The Physics of Blown Sands and Desert
on the initial state; if the initial vegetation cover is above Dunes (Chapman and Hall, London, 1941).
vc , v converges to the fixed dune branch F; otherwise it [3] K. Pye and H. Tsoar, Aeloian Sand and Sand Dunes
converges to the active dune branch A. (Unwin Hyman, London, 1990).
In places with very low wind power the dune has only [4] Z. B. Dong et al., J. Arid Environ. 57, 329 (2004).
[5] D. S. G. Thomas et al., Nature (London) 435, 1218 (2005).
one stable state, which is vegetated and fixed, suggesting
[6] H. Tsoar, Physica (Amsterdam) A357, 50 (2005).
that human intervention is likely to be the dominant factor [7] J. E. Ash and R. J. Wasson, Z. Geomorphologie 45, 7
causing the dunes to turn bare and become active. An (1983).
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Israeli-Egyptian border mentioned above (DP 50); see 137 (1979).
Fig. 1(b). Figure 2(c) shows the stabilization evolution for [9] J. E. Bullard, J. Sedimentary Res. 67, 499 (1997).
three different values of DP starting from a very low [10] N. Levin et al., Catena (to be published).
vegetation cover (v 0:01), such as that on the bare, [11] S. M. Arens et al., Geomorphology 59, 175 (2004).
active dunes on the Israeli-Egyptian border. For DP < [12] A. Meir and H. Tsoar, Human Ecology 24, 39 (1996).
DP1 , v converges quickly to the fixed, vegetated dune state, [13] N. Lancaster and A. Baas, Earth Surf. Processes
with the response curves seeming to reflect to actual data Landforms 23, 69 (1998).
´
[14] O. Duran and H. Herrmann, Phys. Rev. Lett. 97, 188001
[12,17]. However, our model predicts that with stronger
(2006).
winds (DP 200), the bare and active dunes should re- [15] S. A. Wolfe and W. C. Nickling, Prog. Phys. Geography
main active even when the anthropogenic pressure is re- 17, 50 (1993).
leased. In fact, rapid natural recovery of dune vegetation [16] A. C. W. Bass, Geomorphology 48, 309 (2002).
was also reported for parts of North Africa [3]. [17] C. H. Hugenholtz and S. A. Wolfe, Geomorphology 70, 53
There is a wealth of evidence for global climate change (2005).
over the past decades [24]. Global circulation models [18] S. M. Arens and L. H. W. T. Geelen, J. Coastal Research :
predict that the global warming trend will persist if the JCR/CERF 22, 1094 (2006).
‘‘business as usual’’ scenario [24] continues. Some studies [19] D. Ludwig et al., Ecology and Society 1, Art. 7 (1997).
have forewarned that the climate system will become even [20] H. Nishimori and H. Tanaka, Earth Surf. Processes
more extreme, with stronger atypical extreme climate Landforms 26, 1143 (2001).
[21] F. de Castro, Ecol. Mod. 78, 205 (1995).
events appearing more frequently (e.g., [24]). It is possible
[22] P. Hesp, Geomorphology 48, 245 (2002).
that some desert areas will experience more severe and [23] G. F. S. Wiggs et al., Earth Surf. Processes Landforms 20,
prolonged droughts with stronger and more frequent wind 515 (1995).
storms; this could lead to the activation of hitherto fixed [24] Intergovernmental Panel on Climate Change: The Scien-
dunes, as has been observed in the past decades (e.g., [25]), tific Basis (Cambridge University Press, Cambridge,
with severe damage to the environment and the quality of England, 2001).
life of millions of people. Our simple model supports such [25] L. Marin et al., Geomorphology 70, 163 (2005).
188001-4