| Tree Equations and Alien Plants |
What physical laws govern the
structure, shape and form of trees?
Can we construct a mathematical model
and use it to predict the nature of alien
trees growing on other planets?
In this section we look at some key
equations describing plant/tree form
and function and consider what the
implications are for alien life.
structure, shape and form of trees?
Can we construct a mathematical model
and use it to predict the nature of alien
trees growing on other planets?
In this section we look at some key
equations describing plant/tree form
and function and consider what the
implications are for alien life.
Equation 1 is derived from Euler's column
equation and is for both solid and hollow
columns. The maximum allowable height of the
column before it buckles under its own weight
(ignoring additional stresses like wind forces)
is hcrit, EI is the flexural stiffness. Details are
given below. Equation one can actually be
simplified further, and this is done below when
we put it to use. E is the elastic modulus and I
is the second moment of area - not to be
confused with the second moment of inertia of
the mass, also called simply the moment of
inertia - moment of area is concerned with the
spatial geometry of an object in cross-section,
and gives us a measure of how spread out the
material, spatially, is from the central axis,
whereas the moment of inertia is concerned
specifically with the distribution of mass and
gives us a measure of how spread out the
mass is from the central axis. A wide cylinder
has a higher moment of area than a narrow
cylinder. Here we generalise to a hollow
cylinder with an outer radius, ro, and an inner
radius, ri.
Equation 2 models a branch like a cantilever
(a horizontal beam fixed at one end and
loaded at the other) and Eq. 2a gives the
maximum moment (turning force due to
loading of the cantilever) and eq. 2b gives the
maximum deflection as the cantilever bends
under its own weight. Again this ignores
wind-forces and real branches reduce loading
by being angled upwards.
More equations may be added to this list as
we develop the model. First of all let us look at
equation 1 and what it tells us.
equation and is for both solid and hollow
columns. The maximum allowable height of the
column before it buckles under its own weight
(ignoring additional stresses like wind forces)
is hcrit, EI is the flexural stiffness. Details are
given below. Equation one can actually be
simplified further, and this is done below when
we put it to use. E is the elastic modulus and I
is the second moment of area - not to be
confused with the second moment of inertia of
the mass, also called simply the moment of
inertia - moment of area is concerned with the
spatial geometry of an object in cross-section,
and gives us a measure of how spread out the
material, spatially, is from the central axis,
whereas the moment of inertia is concerned
specifically with the distribution of mass and
gives us a measure of how spread out the
mass is from the central axis. A wide cylinder
has a higher moment of area than a narrow
cylinder. Here we generalise to a hollow
cylinder with an outer radius, ro, and an inner
radius, ri.
Equation 2 models a branch like a cantilever
(a horizontal beam fixed at one end and
loaded at the other) and Eq. 2a gives the
maximum moment (turning force due to
loading of the cantilever) and eq. 2b gives the
maximum deflection as the cantilever bends
under its own weight. Again this ignores
wind-forces and real branches reduce loading
by being angled upwards.
More equations may be added to this list as
we develop the model. First of all let us look at
equation 1 and what it tells us.
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The answer we obtained for the maximum or critical height, h, of 85 m is only approximate. We have estimated values
for the radii and mass and a real tree is tapered to some degree and more-or-less conical not cylindrical (we return to
this later). However, the answer is about what i expected - real trees have safety factors of 4 or more, which means
that a wood-cutter must remove at least 3/4 of the wood from a cross-section of the trunk before a tree begins to
collapse under its own weight (assuming negligible winds). Oak trees are typically 30 m in height, so this gives us a
safety factor of 3 - about right. A safety factor of 4,incidentally, is much higher than commonly used in
human-engineered structures (where the safety factor is about 2) - trees have to cope with the unexpected, such as
high winds, frost damage, grazing and lightning damage; trees are built to last!
Next, we make this critical height a function of gravity. On Earth the force due to gravity is about 10 Newtons (9,8 N).
How tall might trees grow on planets with more or less gravity than this?
for the radii and mass and a real tree is tapered to some degree and more-or-less conical not cylindrical (we return to
this later). However, the answer is about what i expected - real trees have safety factors of 4 or more, which means
that a wood-cutter must remove at least 3/4 of the wood from a cross-section of the trunk before a tree begins to
collapse under its own weight (assuming negligible winds). Oak trees are typically 30 m in height, so this gives us a
safety factor of 3 - about right. A safety factor of 4,incidentally, is much higher than commonly used in
human-engineered structures (where the safety factor is about 2) - trees have to cope with the unexpected, such as
high winds, frost damage, grazing and lightning damage; trees are built to last!
Next, we make this critical height a function of gravity. On Earth the force due to gravity is about 10 Newtons (9,8 N).
How tall might trees grow on planets with more or less gravity than this?
Below we begin with Euler's column equation for an ideal column (an ideal column is one that is straight,
homogeneous and free from initial stress. This equation allows us to determine the critical load (P), or conversely the
maximum critical height (h) before a column permanently warps under its won weight - it assumes only self-loading
and so only looks at (vertical) compression stresses, and ignores, for example, wind forces which cause lateral
(sideways) forces to act on the column too. The elastic modulus is a measure of the stiffness of the material making
up the column, but the overall stiffness of the column also depends upon the geometry of the column, which is
included in the moment of area, I. In other words, we can estimate the maximum height of a tree before it collapses
under its own weight.
homogeneous and free from initial stress. This equation allows us to determine the critical load (P), or conversely the
maximum critical height (h) before a column permanently warps under its won weight - it assumes only self-loading
and so only looks at (vertical) compression stresses, and ignores, for example, wind forces which cause lateral
(sideways) forces to act on the column too. The elastic modulus is a measure of the stiffness of the material making
up the column, but the overall stiffness of the column also depends upon the geometry of the column, which is
included in the moment of area, I. In other words, we can estimate the maximum height of a tree before it collapses
under its own weight.
Note, some quoted variants on Euler's equation result in different coefficients, for example 1.95934 instead of 0.6168,
however, these equations are only estimates and we are more concerned with trends and will not worry too much
about the coefficient here. (Using 1.95934 gives a critical height of 103 m). Remember that trees strive to be tall in
order to compete for sunlight, however, there are limits, and from this result we might expect trees to be shorter on a
higher gravity planet, but not by that much - even when the gravity is 4o N (4 times stronger than on Earth) the
predicted height of trees is only halved.
Wood Density
Additionally, a tree could overcome this if it had less dense tissues. The effects of density on critical height are shown
below:
however, these equations are only estimates and we are more concerned with trends and will not worry too much
about the coefficient here. (Using 1.95934 gives a critical height of 103 m). Remember that trees strive to be tall in
order to compete for sunlight, however, there are limits, and from this result we might expect trees to be shorter on a
higher gravity planet, but not by that much - even when the gravity is 4o N (4 times stronger than on Earth) the
predicted height of trees is only halved.
Wood Density
Additionally, a tree could overcome this if it had less dense tissues. The effects of density on critical height are shown
below:
Tapered Trunks
A tree like balsa has a density as low as 380 kg/m^3 (critical h = 88 m), a redwood 450 kg/m^3 (critical height = 83 m),
whilst an ebony might have a density as high as 1120 kg/m^3 (critical height = 61 m). Now redwoods can reach heights of
over 115 m. This discrepancy can be explained partly by the fact that redwood trunks strongly taper, indeed all tree
trunks taper to a greater or lesser degree. A tapered stem reaches the same height by using less material (lower mass)
and also distributes stresses evenly throughout the column (in a cylindrical column, the stresses are greatest at the
base, remember that stress is force per unit area in this case). For a tapered trunk, the critical height increases. For a
trunk in which the topmost radius is half the basal radius (a truncated cone) the critical height increases by 26%, whilst a
greater taper may increase it by up to about a third or more. (The coefficient is increased).
Solid Trunks
The results for a solid cylindrical tree trunk are shown below:
A tree like balsa has a density as low as 380 kg/m^3 (critical h = 88 m), a redwood 450 kg/m^3 (critical height = 83 m),
whilst an ebony might have a density as high as 1120 kg/m^3 (critical height = 61 m). Now redwoods can reach heights of
over 115 m. This discrepancy can be explained partly by the fact that redwood trunks strongly taper, indeed all tree
trunks taper to a greater or lesser degree. A tapered stem reaches the same height by using less material (lower mass)
and also distributes stresses evenly throughout the column (in a cylindrical column, the stresses are greatest at the
base, remember that stress is force per unit area in this case). For a tapered trunk, the critical height increases. For a
trunk in which the topmost radius is half the basal radius (a truncated cone) the critical height increases by 26%, whilst a
greater taper may increase it by up to about a third or more. (The coefficient is increased).
Solid Trunks
The results for a solid cylindrical tree trunk are shown below:
At am Earth-like gravitational force of 10 N, being hollow for our tree (outer radius 0.5 m, inner radius 0.4 m) increased
the critical height by 18%. However, for a larger tree (outer radius 1.0 m, inner radius 0.9 m) the increase was 22%,
whilst for a narrow tree with outer radius 0.25 m and inner radius 0.15 m the difference is only 11%. Thus, for a constant
wall thickness, larger trees derive more benefit from being hollow. (If we scale the wall thickness proportionally to
increase in trunk diameter then there is no difference).
Being hollow is much more importance when we consider lateral wind forces. Increasing trunk diameter increases the
moment of area (I) and so increases flexural stiffness. Being hollow means that a trunk of given mass can have a larger
diameter - being hollow increases trunk stiffness whilst economising on material. Experiments have shown that high
winds are more likely to topple the tall, solid and thin younger trees than it is to topple the older, larger but more hollow
trees. Hollow trees may also allow wind to pass through them more easily. The alien tree in the picture at the top of the
page is adapted for high winds - the wind easily passes through the spaces in its trunk and through the slits in its
dome-like crown.
Stem Shape
Some plants, especially small green herbaceous plants, have square stems. For a given amount of material (constant
cross-sectional area) a square stem has a slightly higher moment of area and so is stiffer. This is an advantage for
these softer green stems. Many trees have elliptical stems and branches. This is because the tree responds to the
stresses acting upon it as it grows. In an ellipse one radius (the major radius) is larger than the other (minor) radius and
in these plant parts the long axis will be aligned with the direction that experiences greatest bending stresses -
increasing moment of area and stiffness whilst economising on the total mass of material used.
the critical height by 18%. However, for a larger tree (outer radius 1.0 m, inner radius 0.9 m) the increase was 22%,
whilst for a narrow tree with outer radius 0.25 m and inner radius 0.15 m the difference is only 11%. Thus, for a constant
wall thickness, larger trees derive more benefit from being hollow. (If we scale the wall thickness proportionally to
increase in trunk diameter then there is no difference).
Being hollow is much more importance when we consider lateral wind forces. Increasing trunk diameter increases the
moment of area (I) and so increases flexural stiffness. Being hollow means that a trunk of given mass can have a larger
diameter - being hollow increases trunk stiffness whilst economising on material. Experiments have shown that high
winds are more likely to topple the tall, solid and thin younger trees than it is to topple the older, larger but more hollow
trees. Hollow trees may also allow wind to pass through them more easily. The alien tree in the picture at the top of the
page is adapted for high winds - the wind easily passes through the spaces in its trunk and through the slits in its
dome-like crown.
Stem Shape
Some plants, especially small green herbaceous plants, have square stems. For a given amount of material (constant
cross-sectional area) a square stem has a slightly higher moment of area and so is stiffer. This is an advantage for
these softer green stems. Many trees have elliptical stems and branches. This is because the tree responds to the
stresses acting upon it as it grows. In an ellipse one radius (the major radius) is larger than the other (minor) radius and
in these plant parts the long axis will be aligned with the direction that experiences greatest bending stresses -
increasing moment of area and stiffness whilst economising on the total mass of material used.
Branches can be modelled using a modification of the column equation for a beam. The results are similar and as gravity
increases the branches are predicted to shorten in proportion to the shortening of the stem, so that the overall tree
shape is maintained.
increases the branches are predicted to shorten in proportion to the shortening of the stem, so that the overall tree
shape is maintained.
Water Transport
So far we have only considered the effects of gravity on tree shape. Also of importance is the need for a tree to move
water up the trunk from the soil, via the roots, into the leaves. The function of the leaves is to obtain sunlight and carbon
dioxide for photosynthesis. To achieve this, the leaves must be thin and plate-like (otherwise too many
non-photosynthesising cells would occupy the centre of the leaf where light cannot penetrate). They also need leaf pores
(stomata) to let the carbon dioxide gas pass from the atmosphere into the leaf. A consequence of this is that water will
also escape by evaporation from these pores (a process called transpiration) and this water must be replaced by water
moving up the plant from the roots in what is called the transpiration stream. However, plants exploit the transpiration
stream to carry mineral nutrients from the soil to the cells and transpiration also helps the sunbaked leaves to keep cool.
See also: Transport in plants
So far we have only considered the effects of gravity on tree shape. Also of importance is the need for a tree to move
water up the trunk from the soil, via the roots, into the leaves. The function of the leaves is to obtain sunlight and carbon
dioxide for photosynthesis. To achieve this, the leaves must be thin and plate-like (otherwise too many
non-photosynthesising cells would occupy the centre of the leaf where light cannot penetrate). They also need leaf pores
(stomata) to let the carbon dioxide gas pass from the atmosphere into the leaf. A consequence of this is that water will
also escape by evaporation from these pores (a process called transpiration) and this water must be replaced by water
moving up the plant from the roots in what is called the transpiration stream. However, plants exploit the transpiration
stream to carry mineral nutrients from the soil to the cells and transpiration also helps the sunbaked leaves to keep cool.
See also: Transport in plants
Equation 3 - The Transpiration equation for Stomatal Conductance
Using Fick's Law we can derive an equation to model diffusion across the stomata. Essentially, this is one-dimensional
diffusion across stomata modeled as narrow tubes. However, the diameter of the tube is also important and a number of
variants on this equation incorporate corrections to account for this. We use one such simple correction below. Note that
the variants given of this equation all give values of conductance within an order of magnitude of one-another and the
patterns and trends are essentially the same (all the ones I have seen differ only by a constant of proportionality).
Plants can open and close their stomata in response to a variety of internal and external conditions and also in
accordance to an internal clock. In addition, in the long term, plants can vary both the number (density of stomata per
square mm of leaf surface area) and the maximum size of the stomatal apertures. These characteristics differ greatly
between species, but are also capable of some adaptive variation within a species. For example, it has been shown that
as carbon dioxide levels have increased, both by natural means since the last peak glacial period of the Ice Age and
more recently from anthropogenic emissions, plants respond by reducing stomatal conductance. This is achieved by
having fewer, larger pores per leaf, as explained below:
Using Fick's Law we can derive an equation to model diffusion across the stomata. Essentially, this is one-dimensional
diffusion across stomata modeled as narrow tubes. However, the diameter of the tube is also important and a number of
variants on this equation incorporate corrections to account for this. We use one such simple correction below. Note that
the variants given of this equation all give values of conductance within an order of magnitude of one-another and the
patterns and trends are essentially the same (all the ones I have seen differ only by a constant of proportionality).
Plants can open and close their stomata in response to a variety of internal and external conditions and also in
accordance to an internal clock. In addition, in the long term, plants can vary both the number (density of stomata per
square mm of leaf surface area) and the maximum size of the stomatal apertures. These characteristics differ greatly
between species, but are also capable of some adaptive variation within a species. For example, it has been shown that
as carbon dioxide levels have increased, both by natural means since the last peak glacial period of the Ice Age and
more recently from anthropogenic emissions, plants respond by reducing stomatal conductance. This is achieved by
having fewer, larger pores per leaf, as explained below:
Equation 5 - The Growth Equation
A tree must invest a lot of resources in its stem, to maintain the dominant position of its leaves high in the canopy. The
trunk of large trees is not photosynthetic at all and so these materials do not bring direct gain in terms of growth. The
growth of a tree is driven by its leaves (supplemented by nutrients and water from the roots). The fastest growing plants
are almost all leaf. Duckweed is one of the fastest growing of all plants, and each plant consists of a little leaf and a tiny
root and the plant floats in the water. Single-celled algae (protoplants) are even faster growing. In contrast, a large tree
which must invest so many materials in supporting structures is very slow growing relative to its size, that is in terms of
percentage weight or mass increase.
The growth equation is:
A tree must invest a lot of resources in its stem, to maintain the dominant position of its leaves high in the canopy. The
trunk of large trees is not photosynthetic at all and so these materials do not bring direct gain in terms of growth. The
growth of a tree is driven by its leaves (supplemented by nutrients and water from the roots). The fastest growing plants
are almost all leaf. Duckweed is one of the fastest growing of all plants, and each plant consists of a little leaf and a tiny
root and the plant floats in the water. Single-celled algae (protoplants) are even faster growing. In contrast, a large tree
which must invest so many materials in supporting structures is very slow growing relative to its size, that is in terms of
percentage weight or mass increase.
The growth equation is:
This gives us the growth rate, R, which is the rate of increase in mass relative to the present mass. This is relative
growth (and obtained by logarithmic differentiation).
A young sapling grows more-or-less exponentially and relatively very fast. However, once a tree reaches its full height,
the trunk continues to grow by adding annual rings of new wood to the outside of the trunk, beneath the bark. Typically
the cross-sectional surface area (and hence volume and mass as height is now constant) of new wood added each year
remains approximately constant for a mature tree. (It is well known that it varies from year to year according to
environment, but the average rate is more-or-less fixed). In the end, their is not enough new wood added to encompass
the trunk and parts of the tree start to die back and eventually the tree enters decline.
Thus, for a mature tree, the rate of increase in mass per unit time (dM/dt) is essentially constant and the growth equation
becomes:
growth (and obtained by logarithmic differentiation).
A young sapling grows more-or-less exponentially and relatively very fast. However, once a tree reaches its full height,
the trunk continues to grow by adding annual rings of new wood to the outside of the trunk, beneath the bark. Typically
the cross-sectional surface area (and hence volume and mass as height is now constant) of new wood added each year
remains approximately constant for a mature tree. (It is well known that it varies from year to year according to
environment, but the average rate is more-or-less fixed). In the end, their is not enough new wood added to encompass
the trunk and parts of the tree start to die back and eventually the tree enters decline.
Thus, for a mature tree, the rate of increase in mass per unit time (dM/dt) is essentially constant and the growth equation
becomes:
The plot shows the growth rate (this time given the symbol G) as a function of mass. This relative growth declines, as the
yearly addition of mass represents a diminishing fraction of the tree's mass. (The units of mass in this plot are arbitrary).
Actual growth curves have been obtained for trees by taking measurements. For the yew tree, one of the longest tree
species (of which a number of specimens are dated to around 3000 to 5000 years of age) the following was obtained (by
measuring trees up to 1000 years of age):
yearly addition of mass represents a diminishing fraction of the tree's mass. (The units of mass in this plot are arbitrary).
Actual growth curves have been obtained for trees by taking measurements. For the yew tree, one of the longest tree
species (of which a number of specimens are dated to around 3000 to 5000 years of age) the following was obtained (by
measuring trees up to 1000 years of age):
This matches our picture of slowing relative growth, though in the case of the yew this growth can
continue for a remarkably long period of time!
continue for a remarkably long period of time!
This effect is also reflected in our conductance equation, shown below, in which conductance, G, is not only a
function of stomatal depth, d, and total leaf area accounted for by stomatal pores (stomatal density, n, multiplied by
the mean pore area of the stomata, a) but also of stomatal radius, r. If the total pore area is kept constant, then
stomatal conductance decreases as pore radius increases, as shown below:
function of stomatal depth, d, and total leaf area accounted for by stomatal pores (stomatal density, n, multiplied by
the mean pore area of the stomata, a) but also of stomatal radius, r. If the total pore area is kept constant, then
stomatal conductance decreases as pore radius increases, as shown below:
Of course, if the stomata close, then both the pore radius and the total area of pores reduces and conductance
naturally decreases, as shown below. Notice from the graph that the stomata affect the greatest relative changes on
conductance at very small pore radius. Most stomata are between 3 and 15 micrometres (0.003 and 0.015 mm) in
radius and so can rapidly regulate conductance by slight changes in diameter:
naturally decreases, as shown below. Notice from the graph that the stomata affect the greatest relative changes on
conductance at very small pore radius. Most stomata are between 3 and 15 micrometres (0.003 and 0.015 mm) in
radius and so can rapidly regulate conductance by slight changes in diameter:
Each species occupies a narrow part of such a conductance curve, with some species having intrinsically high
conductance (those from carbon dioxide poor atmospheres in which water is plentiful) whilst some have lower
conductance (those from carbon dioxide rich atmospheres, or regions of water shortage). The fact the increasing the
levels of carbon dioxide causes plants to adapt by reducing conductance (whilst still possibly increasing net
photosynthesis) reduces stomatal conductance, shows the importance of conserving water - stomata function primarily
to allow carbon dioxide to diffuse into the leaf, and although the transpiration stream in the xylem transports some
useful materials, like minerals from the roots, this function is secondary and xylem transport is in excess of that
required for these transport functions, serving primarily to replenish water lost by the leaves through their stomata.
However, i am not aware of any studies quantifying the importance of mineral transport - could plants without stomata
obtain sufficient nutrients without flow in the xylem?
Within a species, maximum pore area and stomatal density typically vary by 2-fold in response to environmental
conditions. Densities may vary from 100 to 1500 per square mm in different species, and maximum pore areas from 30
to 170 square micrometres.
conductance (those from carbon dioxide poor atmospheres in which water is plentiful) whilst some have lower
conductance (those from carbon dioxide rich atmospheres, or regions of water shortage). The fact the increasing the
levels of carbon dioxide causes plants to adapt by reducing conductance (whilst still possibly increasing net
photosynthesis) reduces stomatal conductance, shows the importance of conserving water - stomata function primarily
to allow carbon dioxide to diffuse into the leaf, and although the transpiration stream in the xylem transports some
useful materials, like minerals from the roots, this function is secondary and xylem transport is in excess of that
required for these transport functions, serving primarily to replenish water lost by the leaves through their stomata.
However, i am not aware of any studies quantifying the importance of mineral transport - could plants without stomata
obtain sufficient nutrients without flow in the xylem?
Within a species, maximum pore area and stomatal density typically vary by 2-fold in response to environmental
conditions. Densities may vary from 100 to 1500 per square mm in different species, and maximum pore areas from 30
to 170 square micrometres.
The graphs below show the shapes of curves of stomatal area as a function of stomatal density (now given the symbol x so
as not to confuse Mathcad, x = n) different conductance values. See how at higher conductances the curve shifts up and to
the right. Similar plots can be seen in many papers on plant physiology.
as not to confuse Mathcad, x = n) different conductance values. See how at higher conductances the curve shifts up and to
the right. Similar plots can be seen in many papers on plant physiology.
What about alien plants? We might expect those plants that live in a carbon dioxide rich atmosphere to have fewer larger
pores for carbon dioxide absorption. Carbon dioxide may also, however, have major effects on tree branching patterns
and leaf shape, as we shall explore qualitatively below (the diffusion equation).
pores for carbon dioxide absorption. Carbon dioxide may also, however, have major effects on tree branching patterns
and leaf shape, as we shall explore qualitatively below (the diffusion equation).
Equation 4 - Transport Equation (Poiseuille's Law)
One very important equation for plant transport is the equation for water potential (see transport in plants). Here we look
at the equation governing flow in xylem and phloem: Poiseuille's Equation for laminar, parabolic flow in a straight tube:
One very important equation for plant transport is the equation for water potential (see transport in plants). Here we look
at the equation governing flow in xylem and phloem: Poiseuille's Equation for laminar, parabolic flow in a straight tube:
This flow describes flow in xylem especially, but also bulk flow in phloem, quite accurately. However, in phloem, different
constituents of the sugary sap move at different rates, a process that can not be explained by bulk flow and Poiseuille
flow. The driving forces for fluid flow in the two systems is also quite different. In xylem, transpiration through the stomata
that creates a suction force that pulls sap up the stem along the xylem (the cohesion-tension theory of sap ascent) with a
small contribution from root pressure, as roots actively pump water toward the stem (an osmotically driven pump). In
herbs and perhaps small shrubs, root pressure alone may be sufficient. However, the suction is needed to lift water to
the top of tall trees, the tallest of which can approach 150 metres in height. Water flows along the xylem at an average
velocity of about 1 mm/s. The Reynold's number (Re) at such flow rates is very small, much less than one (about 0.05),
so that we have predominantly creeping viscous flow which is laminar and non-turbulent. (Note that the lack of
turbulence does not rule-out the presence of laminar and non-turbulent vortices which can occur at low Re < 10 in
complex geometries).
The pressure gradient set-up by gravity alone is about -0.01 MPa/m and the pressure gradient in the xylem must be able
to overcome this. The value of about -0.03 MPa/m for xylem, gives (upon integration over tree height) 4.5 MPa total
pressure for a 150 m tree. This is 45 times atmospheric pressure (45 atmospheres (atm)). For a more typical tree at
least 10 atm of pressure is needed, plus more to overcome resistances in the pathway, so 20-30 atm. It is remarkable
that evaporation of water through the stomata can generate such large pressures!
One problem that frequently occurs in xylem is embolism, air-bubble formation due to cavitation. When a column of water
is sucked up the stem of a tree, that column is placed under tension. Cohesion in the water (the tendency of water
molecules to adhere to one another) prevents the column breaking, however, they do sometimes break and then air
comes out of solution to form a bubble where the resulting vacuum would be at the breakage point. These bubbles are
very hard to shift and tend to block the vessel and stop fluid flow. The degree of cavitation depends on xylem tube
diameter and temperature, increasing for wider tubes and lower temperatures (at which water becomes more brittle). For
a typical tree, cavitation can reach 50% at pressures of 2-3 MPa for vessel diameters of 20-30 micrometres (fortunately
trees have methods to work around this). Conifers have much narrower vessels, less than 10 micrometres in diameter
which for at least one species exhibit 50% cavitation at pressures of 12 MPa. Xerophytes are also similarly resistant to
cavitation at high pressures. This suggests that cavitation is not the main limiting factor in tree height, but that
mechanical structural factors may be more limiting.
Effects of gravity
Trees growing on alien worlds with higher gravity will have a greater gravitational pressure to overcome when raising
water. For a fluid of constant density, this pressure (given by the equation of fluid statics) will double. This suggests that
most trees would struggle in g = 2 to g =3 or greater, though some may manage in g = 10.
Effects of fluid type
It is often taken for granted that water is the best solvent for biological processes. However, it is worth noting that the
active sites of most enzymes must exclude water in order to function - they are hydrophobic. This suggests that organic
solvents may work well chemically. Some inorganic solvents also have water-like properties, such as ammonia. If we
consider methane and ammonia, these fluids have viscosities about one-hundredth that of water. Since flow-rate,
according to Poiseuille's equation, is inversely proportional to viscosity, we might expect such fluids to ascend alien trees
rapidly (perhaps several cm/s). However, the cohesion forces in these molecules are weaker. Ammonia forms one
hydrogen bond per molecule, compared to two for water and methane is bonded only by very weak van der Waals
forces. Although methane is liquid at very low temperatures (so might be a solvent in organisms on very cold planets) we
might expect cavitation to be a real problem. Thus, despite the rapidity with which such fluids could be sucked up a tree,
their weak cohesion counters this as the tree would have to lower the suction pressure. Other organic solvents with
stronger cohesion might work better, such as halogenoalkanes.
constituents of the sugary sap move at different rates, a process that can not be explained by bulk flow and Poiseuille
flow. The driving forces for fluid flow in the two systems is also quite different. In xylem, transpiration through the stomata
that creates a suction force that pulls sap up the stem along the xylem (the cohesion-tension theory of sap ascent) with a
small contribution from root pressure, as roots actively pump water toward the stem (an osmotically driven pump). In
herbs and perhaps small shrubs, root pressure alone may be sufficient. However, the suction is needed to lift water to
the top of tall trees, the tallest of which can approach 150 metres in height. Water flows along the xylem at an average
velocity of about 1 mm/s. The Reynold's number (Re) at such flow rates is very small, much less than one (about 0.05),
so that we have predominantly creeping viscous flow which is laminar and non-turbulent. (Note that the lack of
turbulence does not rule-out the presence of laminar and non-turbulent vortices which can occur at low Re < 10 in
complex geometries).
The pressure gradient set-up by gravity alone is about -0.01 MPa/m and the pressure gradient in the xylem must be able
to overcome this. The value of about -0.03 MPa/m for xylem, gives (upon integration over tree height) 4.5 MPa total
pressure for a 150 m tree. This is 45 times atmospheric pressure (45 atmospheres (atm)). For a more typical tree at
least 10 atm of pressure is needed, plus more to overcome resistances in the pathway, so 20-30 atm. It is remarkable
that evaporation of water through the stomata can generate such large pressures!
One problem that frequently occurs in xylem is embolism, air-bubble formation due to cavitation. When a column of water
is sucked up the stem of a tree, that column is placed under tension. Cohesion in the water (the tendency of water
molecules to adhere to one another) prevents the column breaking, however, they do sometimes break and then air
comes out of solution to form a bubble where the resulting vacuum would be at the breakage point. These bubbles are
very hard to shift and tend to block the vessel and stop fluid flow. The degree of cavitation depends on xylem tube
diameter and temperature, increasing for wider tubes and lower temperatures (at which water becomes more brittle). For
a typical tree, cavitation can reach 50% at pressures of 2-3 MPa for vessel diameters of 20-30 micrometres (fortunately
trees have methods to work around this). Conifers have much narrower vessels, less than 10 micrometres in diameter
which for at least one species exhibit 50% cavitation at pressures of 12 MPa. Xerophytes are also similarly resistant to
cavitation at high pressures. This suggests that cavitation is not the main limiting factor in tree height, but that
mechanical structural factors may be more limiting.
Effects of gravity
Trees growing on alien worlds with higher gravity will have a greater gravitational pressure to overcome when raising
water. For a fluid of constant density, this pressure (given by the equation of fluid statics) will double. This suggests that
most trees would struggle in g = 2 to g =3 or greater, though some may manage in g = 10.
Effects of fluid type
It is often taken for granted that water is the best solvent for biological processes. However, it is worth noting that the
active sites of most enzymes must exclude water in order to function - they are hydrophobic. This suggests that organic
solvents may work well chemically. Some inorganic solvents also have water-like properties, such as ammonia. If we
consider methane and ammonia, these fluids have viscosities about one-hundredth that of water. Since flow-rate,
according to Poiseuille's equation, is inversely proportional to viscosity, we might expect such fluids to ascend alien trees
rapidly (perhaps several cm/s). However, the cohesion forces in these molecules are weaker. Ammonia forms one
hydrogen bond per molecule, compared to two for water and methane is bonded only by very weak van der Waals
forces. Although methane is liquid at very low temperatures (so might be a solvent in organisms on very cold planets) we
might expect cavitation to be a real problem. Thus, despite the rapidity with which such fluids could be sucked up a tree,
their weak cohesion counters this as the tree would have to lower the suction pressure. Other organic solvents with
stronger cohesion might work better, such as halogenoalkanes.
Equation 6 - The Diffusion equation and Branching Patterns
Trees branch repeatedly in order to position their leaves for optimum function, that is optimum light interception and carbon
dioxide absorption. The leaves themselves may also be branches or divided. Structural mechanics are also important -
branches have to be able to bear the weight and in particular against dynamic loadings, such as from high winds. Here we
will focus on the need for carbon dioxide absorption. The 3D diffusion equation is given below:
Trees branch repeatedly in order to position their leaves for optimum function, that is optimum light interception and carbon
dioxide absorption. The leaves themselves may also be branches or divided. Structural mechanics are also important -
branches have to be able to bear the weight and in particular against dynamic loadings, such as from high winds. Here we
will focus on the need for carbon dioxide absorption. The 3D diffusion equation is given below:
For many situations we need to use computers to solve this equation by numerical approximation. (We have done this for a
simple case using the 1D diffusion equation using the finite difference method). I am not about to attempt a 3D solution for
tree branching, however, similar studies have been conducted. In particular, models of sponge growth (insert ref) recreate
the branching, tree-like forms characteristic of certain sponges. The sponges tissues are filtering nutrients from the can not
obtain sufficient nutrition, as neighbouring areas have depleted the water in the local vicinity. This level of depletion, of
course, depends on nutrient density and the rate of nutrient replenishment, for example by water currents. These deprived
areas grow less whilst more rapidly growing adjacent areas grow out as finger-like projections. Over time, the sponge
adopts an optimum branching pattern. Similar principles would apply to corals and sea pens, etc. Sponges are particularly
plastic and modify their body forms as they grow in accordance to fluid flow and nutrient patterns. Similar principles may
help account for branching patterns and leaf shape in plants, although in plants we might expect the patterns to be more
tightly confined by genetic factors.
Computer algorithms, based on optimum resource utilisation, have produced a small number of branching patterns that are
equally viable (excluding structural mechanical considerations) and a number of these types account for most branching
patterns seen in nature. It is worth remembering, however, that some plants, even tree-sized plants, like certain cacti, so not
exhibit typical branching patterns, if they branch at all.
Leaf Shape and Size
An analysis of leaf morphology, size and distribution on a plant, are complicated by the fact that leaf morphology and size
are affected by mechanical factors (weight support, wind strength), light interception, carbon dioxide absorption, and
temperature.
We can distinguish between sun leaves and shade leaves on he same plant. Sun leaves, those exposed to sunlight, such
as at the top of the canopy, tend to be thicker. This increased thickness is due mostly to an increased number of layers of
photosynthetic palisade mesophyll cells of greater height (these cells are roughly cylindrical, see leaf structure). Thus, this
plasticity is clearly a photosynthetic adaptation, with sun leaves growing with a structure that maximises use of the extra light
for photosynthesis.
Sun leaves also tend to be more finely divided, very much more so in some plants, and may be quite branched. This has
been interpreted as a mechanism to break up the boundary layer to prevent overheating, and it might be, however it could
also be a response to wind stresses (divided leaves offer less resistance to wind and so are less likely to tear) or, in the
light of the sponge study, a response to carbon dioxide limitation.
Dehydration and cold also have a massive influence on leaf form. Consider the tough needles of conifers, narrow and with
tough waterproof cuticles. The narrowness of the needles probably reduces dynamic loadings from wind and snow, whilst
the cuticles reduce damage and loss of function from dehydration. The cylindrical form of some conifer leaves also reduces
dehydration (compared to thin flat leaves). Cacti, of course, have no leaves, though the stems may flatten and become
more leaf-like (phyllodes). This reduces water-loss.
Whether or not alien plants will be highly branched with divided leaves or with fewer larger leaves and less branching (as in
our picture above) will depend on several factors. Low carbon dioxide levels, high winds (dynamic loadings) and high
temperatures, perhaps all favour branching with finely divided leaves, unless the high winds become too dehydrating. High
carbon dioxide levels and low temperatures might favour less divided canopies. Thus, we might conclude that our alien tree
above exists on a cool planet with high carbon dioxide levels and perhaps low winds.
Leaf Colour
For a discussion of leaf colour see the plant FAQs.
simple case using the 1D diffusion equation using the finite difference method). I am not about to attempt a 3D solution for
tree branching, however, similar studies have been conducted. In particular, models of sponge growth (insert ref) recreate
the branching, tree-like forms characteristic of certain sponges. The sponges tissues are filtering nutrients from the can not
obtain sufficient nutrition, as neighbouring areas have depleted the water in the local vicinity. This level of depletion, of
course, depends on nutrient density and the rate of nutrient replenishment, for example by water currents. These deprived
areas grow less whilst more rapidly growing adjacent areas grow out as finger-like projections. Over time, the sponge
adopts an optimum branching pattern. Similar principles would apply to corals and sea pens, etc. Sponges are particularly
plastic and modify their body forms as they grow in accordance to fluid flow and nutrient patterns. Similar principles may
help account for branching patterns and leaf shape in plants, although in plants we might expect the patterns to be more
tightly confined by genetic factors.
Computer algorithms, based on optimum resource utilisation, have produced a small number of branching patterns that are
equally viable (excluding structural mechanical considerations) and a number of these types account for most branching
patterns seen in nature. It is worth remembering, however, that some plants, even tree-sized plants, like certain cacti, so not
exhibit typical branching patterns, if they branch at all.
Leaf Shape and Size
An analysis of leaf morphology, size and distribution on a plant, are complicated by the fact that leaf morphology and size
are affected by mechanical factors (weight support, wind strength), light interception, carbon dioxide absorption, and
temperature.
We can distinguish between sun leaves and shade leaves on he same plant. Sun leaves, those exposed to sunlight, such
as at the top of the canopy, tend to be thicker. This increased thickness is due mostly to an increased number of layers of
photosynthetic palisade mesophyll cells of greater height (these cells are roughly cylindrical, see leaf structure). Thus, this
plasticity is clearly a photosynthetic adaptation, with sun leaves growing with a structure that maximises use of the extra light
for photosynthesis.
Sun leaves also tend to be more finely divided, very much more so in some plants, and may be quite branched. This has
been interpreted as a mechanism to break up the boundary layer to prevent overheating, and it might be, however it could
also be a response to wind stresses (divided leaves offer less resistance to wind and so are less likely to tear) or, in the
light of the sponge study, a response to carbon dioxide limitation.
Dehydration and cold also have a massive influence on leaf form. Consider the tough needles of conifers, narrow and with
tough waterproof cuticles. The narrowness of the needles probably reduces dynamic loadings from wind and snow, whilst
the cuticles reduce damage and loss of function from dehydration. The cylindrical form of some conifer leaves also reduces
dehydration (compared to thin flat leaves). Cacti, of course, have no leaves, though the stems may flatten and become
more leaf-like (phyllodes). This reduces water-loss.
Whether or not alien plants will be highly branched with divided leaves or with fewer larger leaves and less branching (as in
our picture above) will depend on several factors. Low carbon dioxide levels, high winds (dynamic loadings) and high
temperatures, perhaps all favour branching with finely divided leaves, unless the high winds become too dehydrating. High
carbon dioxide levels and low temperatures might favour less divided canopies. Thus, we might conclude that our alien tree
above exists on a cool planet with high carbon dioxide levels and perhaps low winds.
Leaf Colour
For a discussion of leaf colour see the plant FAQs.