Spatial interactions in a modified Daisyworld model: Heat diffusivity and greenhouse effects

In this work we investigate a modified version of the Daisyworld model, originally introduced by Lovelock and Watson to describe in a simple way the interactions between an Earth-like planet, its biosphere, and the incoming solar radiation. Here a spatial dependency on latitude is included, and both a variable heat diffusivity along latitudes and a simple greenhouse effect description are introduced in the model. We show that the spatial interactions between the variables of the system can locally stabilize the coexistence of the two vegetation types. The feedback on albedo is able to generate equilibrium solutions which can efficiently self-regulate the planet climate, even for values of the solar luminosity relatively far from the current Earth conditions.

Figure 1

Stationary solutions for (a) temperature $T\left(\theta \right)$, (b) white daisy population ${\alpha }_w\left(\theta \right)$, and (c) black daisy population ${\alpha }_b\left(\theta \right)$, as a function of latitude, from the numerical solution of Eq. (9), using $\kappa \left(\theta \right)=1$, $g\left(T\right)=1$, and three different values of the parameter $L$.

Figure 2

Contour plots of the stationary solutions for (a) $T\left(\theta \right)$, (b) white daisy population ${\alpha }_w\left(\theta \right)$, and (c) black daisy population ${\alpha }_b\left(\theta \right)$, in the plane $(L,\theta )$ using $\kappa \left(\theta \right)=1$ and $g\left(T\right)=1$.

Figure 3

Stationary solutions for (a) temperature $T\left(\theta \right)$, (b) white daisy population ${\alpha }_w\left(\theta \right)$, and (c) black daisy population ${\alpha }_b\left(\theta \right)$, as a function of latitude, by numerical solution of Eq. (9), using the profile (32) for $\kappa \left(\theta \right)$ and $g\left(T\right)=1$, and three different values of the parameter $L$.

Figure 4

Contour plots of the stationary solutions for (a) $T\left(\theta \right)$, (b) white daisy population ${\alpha }_w\left(\theta \right)$, and (c) black daisy population ${\alpha }_b\left(\theta \right)$, in the plane $(L,\theta )$ using the profile (32) for $\kappa \left(\theta \right)$ and $g\left(T\right)=1$.

Figure 5

Stationary solutions for (a) $T\left(\theta \right)$, (b) white daisy population ${\alpha }_w\left(\theta \right)$, and (c) black daisy population ${\alpha }_b\left(\theta \right)$, as a function of latitude, by numerical solution of Eq. (9), using the profile (32) for $\kappa \left(\theta \right)$ and (13) for $g\left(T\right)$, and three different values of the parameter $L$.

Figure 6

Contour plots of the stationary solutions for (a) $T\left(\theta \right)$, (b) white daisy population ${\alpha }_w\left(\theta \right)$, and (c) black daisy population ${\alpha }_b\left(\theta \right)$, in the plane $(L,\theta )$ using the profile (32) for $\kappa \left(\theta \right)$ and (13) for $g\left(T\right)$.

Figure 7

Same as Fig. 5, but with different initial values for vegetal coverages.

Figure 8

Same as Fig. 6, but with different initial values for vegetal coverages.

Figure 9

Contour plots of the stationary solutions for (a) $T\left(\theta \right)$, (b) white daisy population ${\alpha }_w\left(\theta \right)$, and (c) black daisy population ${\alpha }_b\left(\theta \right)$, in the plane $(k,\theta )$ for different initial conditions according to the value of $k$ and fixing $L=0.5$.

Figure 10

Contour plots of the stationary solutions for (a) $T\left(\theta \right)$, (b) white daisy population ${\alpha }_w\left(\theta \right)$, and (c) black daisy population ${\alpha }_b\left(\theta \right)$, in the plane $(k,\theta )$ for different initial conditions according to the value of $k$ and fixing $L=1.0$.

Figure 11

Contour plots of the stationary solutions for (a) $T\left(\theta \right)$, (b) white daisy population ${\alpha }_w\left(\theta \right)$, and (c) black daisy population ${\alpha }_b\left(\theta \right)$, in the plane $(k,\theta )$ for different initial conditions according to the value of $k$ and fixing $L=1.5$.