Supplementary Information for Predicting Chronic Climate-Driven Disturbances and Their Mitigation

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3. Additional theory linking size-dependent disturbance mortality to ecosystem stocks and
fluxes
a. Metabolic scaling theory linking plant communities and ecosystems
Here we extend metabolic scaling theory for forest structure and dynamics [125-126] to
link disturbance mortality to changes in ecosystem stocks and fluxes. Building from [127-128],
the total ecosystem phytomass Mtot (kg) can be quantified in terms of the average size-dependent
plant mass m(r) (kg) and the stand size distribution f(r) (m-1), such that
8/3 8/3 2 8/3 2/3 8/3 5/3 5/3
3
() () 5
max max max
min min min
rr r
tot m n n m n m max min
rr r
M
m r f r dr c r c r dr c c r dr c c r r
  

 

  (S1)
Here, , where cm (m kg-3/8) is a normalization constant relating stem radius r (m) to
plant mass (r = cmm3/8), and f(r) = dn/dr = cnr-2, where cn (m) is a size-corrected measure of the
number of individuals of a given size. When the stem radius of the smallest individual rmin (m) is
much smaller than the stem radius of the largest individual rmax (m) (rmin <<< rmax), the total
ecosystem biomass from Eqn. (S1) can be approximated as
8/3 5/3
3
5
tot n m max
M
cc r

 (S2)
Thus, Eqns. (S1) and (S2) show that the total ecosystem phytomass increases with the 5/3 power
of the largest stem radius and decreases with the 5/3 power of the smallest stem radius.
We can recast these relationships in terms of plant height h (m) by recognizing that
[130], where ch (m1/3) is a normalization constant relating stem height to stem radius.
We can then express the average size-dependent plant mass as , the stand size
distribution as , and 3/2 1/2
3
2h
dr c h dh

 which allows us to rewrite Eqn. (S1) as
8/3 8/3
() m
mr c r


3/2 3/2
h
rc h


8/3 4 4
() mh
mh c c h


33
() nh
f
hcch



8/3 4 4 3 3 3/2 1/2 5/2 8/3 3/2 5/2 8/3 5/2 5/2
33 3
() ()
22 5
max max max
min min min
rh h
tot m h n h h n h m n h m max min
rh h
Mmhfhdrcchcchchdhccchdhccchh
    

   

 
(S3)
Again, as above, when the height of the smallest individual hmin (m) is much smaller than the
height of the largest individual hmax (m) (hmin <<< hmax), the total ecosystem biomass from Eqn.
(S3) can be approximated as
5/2 8/3 5/2
3
5
tot n h m max
M
cc c h

 (S4)
Thus, Eqns. (S3) and (S4) show that the total ecosystem biomass increases with the 5/2 power of
height of the largest individual and decreases with the 5/2 power of height of the smallest
individual.
This theory can be further extended to ecosystem fluxes such as gross primary
production, net primary production, and canopy transpiration. We begin by considering the total
metabolic rate of all individuals in the stand Btot (W), which can be expressed in terms of the
stand size distribution as [128]

00 0
() ()
max max
min min
rr
tot n n max min n max
rr
B
B r f r dr bc dr bc r r bc r
 (S5)
where the metabolic normalization b0 (W m-2) relates whole-plant metabolic rate B(r) (W) and
stem radius as 2
0()bBrr

. Assuming that each individual’s gross production rate is directly
proportional to its whole-plant metabolic rate, we can express the total gross primary production
rate GPPtot (kg yr-1) as [128]


00 0
() ()
max max
min min
rr
tot n n max min n max
rr
GPP GPP A G r f r dr g c dr g c r r g c r   
 (S6)
where GPP (kg m-2 yr-1) is gross primary production, A (m2) is the stand area under
consideration, G(r) = g0r2 (kg yr-1) is the average individual gross production rate, and g0 (kg m-2
yr-1) is a gross production normalization constant. Similarly, the total net primary production rate
NPPtot (kg yr-1) can be characterized as [130]

00 0
() ()
max max
min min
rr
tot n n max min n max
rr
NPP NPPA Nrfrdrnc drncr r ncr   
 (S7)
where NPP (kg m-2 yr-1) is net primary production, N(r) = n0r2 (kg yr-1) is the average individual
net production rate, and n0 (kg m-2 yr-1) is a net production normalization constant. The canopy
transpiration rate can be derived in two ways. First, as shown in [128], Eq. (S5) can be extended
to give the total transpiration rate Etot (kg yr-1) as

111 1
00 0
() ()
max max
min min
rr
tot w wn wnmaxmin wnmax
rr
EEA Brfrdrbcdrbcrr bcr
   
 
    
 (S8)
where E (kg m-2 yr-1) is the transpiration rate per unit stand area, τ (s yr-1) is a time unit
conversion factor, /
ww
B
R

 (J kg-1) is the water-to-energy conversion factor, and w
R
(kg s-1) is
the supply rate of water for metabolism. Since the water-to-energy conversion factor may be
difficult to parameterize, we can also build from Eq. (S7) to derive a second expression for the
total transpiration rate

111 1
00 0
() ()
max max
min min
rr
tot n n max min n max
rr
EEA Nrfrdr ncdr ncr r ncr
 
 
    
 (S9)

where α (kg kg-1) is the water use efficiency of production. As a first approximation, α may be
considered constant on an annual basis. Typical values for α are compiled in [131]. As we see,
Eqns. (S5) - (S9) predict that rates of ecosystem production and transpiration will scale
isometrically with stem radius, increasing with the radius of the largest individual and decreasing
with the radius of the smallest individual.
We can recast the above relationships for ecosystem fluxes in terms of plant height by
recalling that [129], which allows us to express the stand size distribution as
, the average whole-plant metabolic rate as , the average whole-plant
gross production rate as , and the average individual net production rate as
, and 3/2 1/2
3
2h
dr c h dh

. Substituting these expressions into Eqns. (S5) - (S9), we
can rewrite these ecosystem fluxes as
3 3 3 3 3/2 1/2 3/2 1/2
00
3/2 3/2 3/2 3/2 3/2
00
33
() () 22
max max max
min min min
rh h
tot h n h h n h
rh h
nh max min nh max
B
B h f h dr b c h c c h c h dh b c c h dh
bcc h h bcc h
 

 



 
(S10)

3 3 3 3 3/2 1/2 3/2 1/2
00
3/2 3/2 3/2 3/2 3/2
00
33
() () 22
max max max
min min min
rh h
tot h n h h n h
rh h
nh max min nh max
GPP GPP A Gh f hdr gc hcch c h dh gcc h dh
gcc h h gcc h
 

  



 
(S11)

3/2 3/2
h
rc h


33
() nh
f
hcch

33
0
() h
Bh bc h


33
0
() h
Gh gc h


33
0
() h
Nh nc h



3 3 3 3 3/2 1/2 3/2 1/2
00
3/2 3/2 3/2 3/2 3/2
00
33
() () 22
max max ma x
min min min
rh h
tot h n h h n h
rh h
nh max min nh max
NPP NPPA Nhfhdr nchcch c hdh ncc hdh
ncc h h ncc h
 

  



 
(S12)
1 1 3 3 3 3 3/2 1/2 1 3/2 1/2
00
1 3/2 3/2 3/2 1 3/2 3/2
00
33
() () 22
max max max
min min min
rh h
tot w w h n h h w n h
rh h
wnh maxmin wnhmax
E E A Bh f hdr bc hcch c h dh bcc h dh
bcc h h bcc h
  
 

 
  



 
(S13)
and
1 1 3 3 3 3 3/2 1/2 3/2 1/2
00
1 3/2 3/2 3/2 1 3/2 3/2
00
33
() () 22
max max max
min min min
rh h
tot h n h h n h
rh h
nh max min nh max
E EA Nhfhdr nchcch c hdh ncc hdh
ncc h h ncc h



 
  



 
(S14)
Thus, Eqns. (S10) - (S14) predict that rates of ecosystem production and transpiration will scale
with the 3/2 power of plant height, increasing with the height of the largest individual and
decreasing with the height of the smallest individual.
a. Size-dependent disturbance mortality
i. Fire mortality
Fire mortality is size-dependent and generally kills the smallest trees. Fire mortality
results from an interaction of heat injuries to a plant’s roots, stem, and crown [132-135]. These

injuries generally comprise tissue necroses and sapwood dysfunction [135] that can limit primary
and secondary growth or adversely affect whole-plant carbon and water budgets. Since the
functional traits governing heat injuries in fire are all size-dependent (e.g. bark thickness,
sapwood depth, crown height), mechanistic models linking these injuries to whole-plant
functioning also predict size-dependent mortality [132-135]. Fire mortality is governed by the
interaction of multiple injuries and mortality mechanisms within whole-plant nutrient and water
budgets. Here, for simplicity, we consider a single mechanism of post-fire mortality (vascular
cambium necrosis), but the same general theory applies to other fire mortality mechanisms as
they are all size-dependent.
The minimum stem radius (m) that will be killed via vascular cambium necrosis in a fire
can be characterized as
(S15)
where xn is the depth of tissue necrosis in the stem (m) and a and b are fitted coefficients from
linear cambial depth allometries (e.g. bark thickness, xbark = aD + b). The depth of tissue necrosis
is given by an analytical solution for Fourier’s law of conduction [132]
(S16)
where α is the thermal diffusivity of bark (m2 s-1), t is the fire residence time (s), Tf is the fire
temperature (ºC), and Ta is the air temperature (ºC). The fire temperature Tf (ºC) scales with air
temperature and fireline intensity I (kW m-1) [132-135], such that
(S17)
2
n
min
x
b
ra



1/2 160
2erf
f
n
af
T
xt TT







1/3 2 /3
f
aa
TTI T

Thus, Eqns. (S15) - (S17) show how fire mortality, via rmin, can alter the size distribution of a
plant stand. Increases in air temperature and vapor pressure deficit projected under climate
change can influence variables such as fuel drying that control fireline intensity, which will in
turn influence size-dependent mortality and, ultimately, ecosystem stocks and fluxes.
ii. Drought mortality
Drought mortality occurs when the available supply of water to a plant cannot meet the
evaporative demand for water from the plant [137]. This can be characterized using the hydraulic
corollary of Darcy’s Law
(S18)
where G is the canopy-scale water conductance (mol m-2 s-1), hmax is the maximum plant height
that can be hydraulically supported (m), As is the conducting area (cm2), Al is the leaf area (m2),
ks is the specific conductivity (m s-1), η is the water viscosity (Pa s), ψs – ψl is the soil-to-leaf
water potential difference. The evaporative demand for water is quantified by the vapor pressure
deficit D (kPa), which increases exponentially with temperature such that
(S19)
where Hr is the relative humidity (dimensionless) and es(Ts) is the vapor pressure of sub-stomatal
air (kPa) evaluated at the sub-stomatal air temperature Ts (ºC), which is given by the Tetens
formula [138]
(S20)

max
s
ss l
l
Ak
hGAD






()1
s
sr
DeT H
17.502
( ) 0.611exp 240.97
s
ss
s
T
eT T





Eqs. (S18) - (S20) show that increases in temperature and decreases in relative humidity during
drought will drive increases in the vapor pressure deficit and, all else equal, a reduction in the
maximum plant height that can be hydraulically sustained.

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