David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling of Environmental Systems• The next portion of this course will examine the balance / flows / cycling of three quantities that are present in ecosystems:– Energy– Water–Nutrients• We will look at each of these at two scales:– Global– Ecosystem• Before we can build models of these phenomena, we need to have some background on the functioning of these systems with respect to these quantitiesDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Energy Propagation through Ecosystems• Understanding the amount of energy that coming into an ecosystem is key to understanding the functioning of ecological processes within that ecosystem• First, the amount of incoming solar radiationavailable to any ecosystem on the Earth’s surface changes with both the time of year and throughout the day• It changes with the time of year due to the fact that the latitude at which the Sun shines from directly above at noon changes from 23.5 degrees north to 23.5 degrees south through the course of the year:David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Seasons and Sun DeclinationSun shines from directly above at the EquatorSun shines from directly above at 23.5 degrees NSun shines from directly above at the EquatorSun shines from directly above at 23.5 degrees SBaker, D. 1978. The Larousse Guide to Astronomy. Larousse and Co., New York, p.73.David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Daily Sun Declination• On any date, the latitude (Sun declination) where the sun shines from directly above at noon can be calculated as:δ = 23.5 sin(2π(284+jday)/365) degreeswhere: jday is the day of the year, or the Julian date• Within a day, the amount of solar radiationthat an ecosystem receives is a function of the Sun’s position•At noon, when the Sun is highest in the sky, the surface receives the most energyDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Important Concepts for Calculating Energy Received by the Earth• Radiance: Refers to the amount of radiant energy coming from a given direction to a unit area of surface perpendicular to the direction of ray, measured as watts/m2/sr (a two-dimensional solid angle)• Irradiance: Refers to the total amount of radiant energy received on a unit area of surface (from all angles), measured as watts/m2• Insolation: The irradiance of a unit horizontal area, measured as watts/m2(As has been the case with all of our models, we need to define quantities precisely, if we want the model to produce results that are predictively accurate)David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Insolation as a F(x) of Date and Time• To calculate the insolation received on the Earth’s surface at a given moment:h0I0• Given solar radiance of I0shining onto the ground at a declination angle of h0, the insolation on the flat ground is:I = I0sin(h0)• To use this simple equation operationally, we need to be able to calculate sin(h0) as a f(x) of date and timeDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Insolation as a F(x) of Date and Time• The quantity sin(h0) can be calculated using the following equation:sin(h0)=sin(φ)sin(δ) + cos(φ)cos(δ)cos(ω)where: φ is the latitude of the locationδ is the Sun declinationω is hour angle• The hour angle is defined as 0 degrees at local noon• ω takes a negative value before noon and a positive value in the afternoon, with 15 degrees per houraway from noon (recall that the day is 24 hours long, and a full revolution of the Earth is 360 degrees, thus 360 degrees/24 hours = 15 degrees/hour)David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Solar Radiation Received by Bare Ground•For bare ground, the radiation that the surface receives is determined by the surface reflectance:RiRrRa = Ri–Rr• We know that Riis a function or R0(Sun position)• However, the portion of Rithat is reflected is purely a function of surface physical characteristics• The proportion reflected is called reflectance or albedo (α) and we can rewrite the above equation as:Ra= Ri (1-α)David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Radiation Interaction with a Single Leaf• To model the movement of energy (and water) in a terrestrial ecosystem, we will begin with the radiation interaction with a single leaf:RiRrLeafTissueRaRt• We can represent this with an equation that partitions the incident radiation into components that describe what happens to that radiation after it strikes the leaf :Ri= Ra/Ri + Rr/Ri+ Rt/Ri= α + ρ + τ =1David Tenenbaum – GEOG 110 – UNC-CH Fall 2005From a Single Leaf to a Canopy• Leaf physiology is such that different wavelengths of radiation tend to have different fates after striking a leaf:Coefficient UV PAR NIRρ (reflected) 0.3 0.09 0.51τ (through) 0.2 0.06 0.34α (absorbed) 0.5 0.85 0.15Horn, S.H. 1971. The Adaptive Geometry of Trees. Princeton Univ. Press.David Tenenbaum – GEOG 110 – UNC-CH Fall 2005From a Single Leaf to a Canopy• We can now start thinking about how to model radiation as it penetrates a vegetation canopy•What factors might influence how radiation would penetrate a canopy?– Leaf size– Leaf angle– Leaf position– Leaf surface characteristics– Leaf thickness– Tree type–Tree size– Tree position– Wavelength, etc.David Tenenbaum – GEOG 110 – UNC-CH Fall 2005From a Single Leaf to a Canopy• We would have a hard time building a model that takes into account all of the factors here … there are simply too many of them, and a few of them will be very hard to express mathematically• Instead, we will aim to use a model that captures the behavior in a predictively valid way (although it may not be structurally valid and may require some calibration). Some assumptions:– Leaves are randomly distributed throughout the canopy– Leaves are infinitely small (so we don’t have to worry about their configuration) {Taken together: A uniformly dense suspension of light-absorbing particles}David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Beer’s Law• Graphically, we can think of our simplification as:I0HI• We need to get from the insolation incident on top of the canopy (I0) to that which is transmitted through it• We will use an empirical equation known as Beer’s Law:I= I0 e-kLwhere I is the insolation transmittedk is the light extinction coefficientL is the leaf area index (LAI)David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Beer’s Law – Leaf Area Index• Leaf area