Interactive models

	stochastic forcing				high frequency stochastic forcing		low frequency stochastic forcing
$D_B$

Changing the parameters may lead to a small time delay before the figures adjust to the new values.

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About 5000 years ago, some regions of northern Africa shifted relatively abruptly (within centuries) from a surface covered mainly by grass to a desert state. The long term desertification tendancy during the Holocene was likely forced by a decrease in summer insolation. However, this astronomical forcing is gradual. Therefore, it cannot explain why the vegetation cover changed so abruptly. Several theories had been developed to account for this rapid transition. For instance, abrupt desertification may have its origin within a strong positive vegetation-climate feedback (Claussen et al., 1999). It may also have been induced by a non linear response of vegetation to low frequency climate variability (Liu et al., 2006).

Liu et al. (2006), following Brovkin et al. (1998), proposed a conceptual climate-vegetation model to study the relative importance of different processes which may have taken part in the change. This model includes only two dynamic variables: one for the vegetation and one for the climate. Equations are:

$${dV}/{dt}={[V_E(P)-V]}/{τ}$$

(1)

where $V$ is the vegetation cover, $P$ is the annual rainfall and $τ$ is the vegetation equilibrium time, taken as 5 years for the Sahara grasslands. $V_E(P)$ is the equilibrium vegetation cover:

$$V_E(P)=\{\,\cl"ma-join-align"{\table {1}, \, ,{\text"if " P>P_{C2}}; {{P-P_{C1}}/{D_C}}, \, , {\text"if " P_{C2}>P>P_{C1}}; {0}, \, , {\text"if " P<P_{C1}$$

(2)

where $P_{C1}$ and $P_{C2}$ are the lower and upper rainfall thresholds respectively ($P_{C1}$=270 mm/yr and $P_{C2}$=370 mm/yr). $D_C=P_{C2}-P_{C1}>0$ is the precipitation range for the transition of vegetation from the green state to the desert state.

The equilibrium atmospheric precipitation is assumed to include two terms:

$$P_E(V,t)=P_d(t)+D_BV$$

(3)

$D_BV$ represents the influence of the vegetation cover on precipitation with $D_B$ being the vegetation feedback coefficient. $P_d(t)$ is the backgroung precipitation in the absence of vegetation. This background precipitation takes into account the gradual decline in response to orbital forcing as follows:

$$P_d(t)=P_{d0}[1-{t+6500}/{T}]$$

(4)

where $P_{d0}$ is the background precipitation at 6500 yr BP and $T$ is the period of interest (3000 yr here).

Finally, the internal variability of rainfall is introduced through a stochastic forcing $P_N(t)$ added to $P_E(V,t)$ to obtain the annual rainfall used in (1) and (2):

$$P(V,t)=\text"max"\{P_E(V,t)+P_N(t),0\}$$

(5)

The model equations are integrated using the finite difference method. Two stochastic forcings are tested:
- a high frequency stochastic forcing following a normal distribution with a mean of 0 mm/yr and a standard deviation of 130 mm/yr.
- a low frequency stochastic forcing derived from the previous one by applying a 10-yr running mean.

The vegetation cover and the precipitation are shown over the last 6500 years for a specific value of the parameters. Analyzing the solution will help you to answer the questions in the following quiz.

$\left[1\right]$V. Brovkin, M. Claussen, V. Petoukhov, and A. Ganopolski, 1998. On the stability of the atmosphere-vegetation system in the Sahara/Sahel region. J. Geophys. Res., 103(D24):31613–31624.
$\left[2\right]$ Claussen M., C. Kubatzi, V. Brovkin and A. Ganopolski, 1999, Simulation of an abrupt change in Saharan vegetation in the mid-Holocene, Geophysical Research Letters, vol. 26, No. 14, 2037-2040.
$\left[3\right]$ Liu Z., Y. Wang, R. Gallimore, M. Notaro and I. C. Prentice, 2006, On the cause of abrupt vegetation collapse in North Africa during the Holocene: Climate variability vs. vegetation feedback, Geophysical Research Letters, vol. 33, L22709, doi:10.1029/2006GL028062.