Lab 3: Surface Energy Balance and Biogeophysical Climate-Vegetation Interactions
Answer all the questions posted in this exercise and submit all your computer code. Please make sure all plots submitted should have a proper title, axis labels and (for maps) legend with the correct unit or labels. Remember to include your name and student ID.
In class and described in greater detail in Wallace & Hobbs Ch.10.3 and Bonan Ch.27.3, the concept of biogeophysical feedback and the role of life in stabilizing climate can be heuristically illustrated by a hypothetical daisyworld. We will examine this further in this lab exercise.
Let us imagine a planet without an atmosphere, but somehow one type of plant can grow on it, namely, white daisy. The daisy is completely reflective of solar radiation with an intrinsic albedo of one. The surface of the planet is otherwise black, completely absorptive of solar radiation with an intrinsic albedo of zero. Therefore, the planetary albedo (r) is simply equivalent to the fractional coverage of white daisy on the planet’s surface. The climate of the planet in terms of surface temperature (Ts, in °C) is governed by the surface energy balance equation:
��! " = (1 − �)�# where F0 (in W m–2 ) is the solar insolation (i.e., incoming solar radiative flux) and σ = 5.670×10–8 W m–2 K–4 is the Stefan-Boltzmann constant. The surface energy balance equation is shown as the blue curves in Fig. 1 for different values of F0.
The abundance of white daisy is governed by plant physiology. Let us assume that fractional daisy coverage (r) takes the form of a quadratic function between a minimum (Tmin) and maximum (Tmax) temperature, below and above which coverage would be zero. That is,
for Ts ≤ Tmin: � = 0 for Tmin < Ts < Tmax: � = ��! $ + ��! + � for Ts ≥ Tmax: � = 0
where a, b and c are quadratic coefficients that are dependent on Tmin, Tmax and the maximum possible fractional coverage, rmax. The physiological function is represented by the red curve as well as the parts of the x-axis that are below Tmin and above Tmax in Fig. 1.
Theoretically, a climate-vegetation equilibrium exists at any point where the surface energy balance curve and physiological curve intersects (e.g., as represented by the points T1, T2, P, Q, P’, T3 and T4). However, numerically (and often in reality), only an equilibrium that is stable can be achieved, because any small perturbation (due to numerical errors in the computer, or climate fluctuations in reality) on an unstable equilibrium would kick off a cascade of feedback that leads the system further away from the unstable equilibrium. A perturbation on an unstable equilibrium can generally lead two possible outcomes. Either the system may simply diverge and then settle into another equilibrium that is stable; or the system may endlessly oscillate between two different states. If the curves are drawn properly, whether an equilibrium is stable or unstable, as well as which of the two outcomes will arise from a perturbation on an unstable equilibrium, can indeed be diagnosed graphically from the shapes of the curves.
In this exercise, we will explore how a gradually increasing solar insolation would affect the climate-vegetation equilibrium in the daisyworld. To save you time, three functions are already provided to you in Lab03.R, as explained below. Please make sure you examine the code carefully to understand the algorithms and what they do.
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