Ice Stream Variability and Links to Climate

Transcription

1 Ice Stream Variability and Links to Climate The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Accessed Citable Link Terms of Use Robel, Alexander Abram Ice Stream Variability and Links to Climate. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. July 11, :47:15 AM EDT This article was downloaded from Harvard University's DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at (Article begins on next page)

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16 ( ) + ( ) + ( ) ( ) + ( ) + ( ) ( ) + ( ) + ( ) = = = = + (,, )

17 = = ( [( ) + ( ) + ( ) ] + ( ) + ( ) + ( ) ).

18

19 Figure 1.1: Surface ice velocities of the Antarctic Ice Sheet from Rignot et al. (2011a), derived from a composite of satellite radar interferometry, overlaid on a MODIS mosaic of the Antarctic Ice Sheet and colored on a logarithmic scale. Black lines indicate major ice divides.

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32 ... Table 2.1: Parameters used in this study (unless otherwise indicated) =, ( ) = =

33 > = = = [ + ] ( ) +, = = = + ( ) > =

34 = < < = + ( + ) [ ( ) ],, =

35 ( ) = ( ( )) = >, ( ) = < < = = <.

36 = = = > = = < <,

37 Velocity (m/yr) Velocity (m/yr) Velocity (m/yr) Velocity (m/yr) Surface Temperature = 15 C Years Surface Temperature = 20 C Years Surface Temperature = 22 C Years Surface Temperature = 35 C Years x 10 4 a b c d Till Water Content (m) Till Water Content (m) Till Water Content (m) Till Water Content (m) Figure 2.1: Characteristic numerical results for the ice stream model with parameters given by Table 2.1, geothermal heat flux of 0.07 and four different prescribed surface temperatures (see location in parameter space in Figure 2.2). In all panels, ice sliding velocity is a blue solid line and till water content is a red dashed line. (a) Steady-streaming with drainage; (b) Steady-streaming without drainage; (c) Weak binge-purge oscillation; (d) Strong binge-purge oscillation.

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39 Geothermal Heat Flux (W/m 2 ) Meters Ice Surface Temperature ( C) Figure 2.2: Ice thickness oscillation range (in meters) on a plane of the parameter space for which geothermal flux and ice surface temperature are varied and dimensionless parameter is constant. Rightmost white solid line is analytic approximation to stability boundary between steady-streaming (zero range region in top-left) and binge-purge (finite range region in bottom-right) modes (accurate to within the thickness of the line). Leftmost white solid line is location of the last appearance of binge-purge oscillations in numerical simulations. Both solid white line bound the region of hysteresis. White dashed line is boundary between steady-streaming with and without drainage. White stars indicate locations of characteristic examples in plotted in Figure 2.1. White ellipse marks approximate parameter regime of modern Siple Coast ice streams. Ellipse drawn using geothermal heat flux range estimates cited in Joughin et al. (2004) and a conservative range of mean air temperatures over Siple Coast ice streams from UWisc AMRC data found at amrc.ssec.wisc.edu/aws/.

40 Geothermal Heat Flux (W/m 2 ) Years Ice Surface Temperature ( C) Figure 2.3: Binge-purge oscillation period (in years) on a plane of the parameter space for which geothermal flux and ice surface temperature are varied and dimensionless parameter is constant.

41 = = = = +

42 = [][ ][ ] = [ ][ ] = [] = [][ ][ ] = [], [] = [ ] + ( + ) ( + ), ( + < ) + ( + ) +. +

43 = = ( ) ( ) = ( ) ( ).

44 Ice Thickness Oscillation Range (m) a Binge Purge Steady Streaming Ice Surface Temperature (C) Sliding Velocity (m/yr) b Warming Cooling Ice Surface Temperature (C) 23 Figure 2.4: (a) Bifurcation diagram. Each point represents a fixed point or limit cycle determined from simulations with a single prescribed ice surface temperature and numerous initial ice thicknesses (initial till water content was kept constant near the fixed point value). Filled points are stable. Open points are unstable. (b) Transient numerical simulation with slow (0.005 C/century) increase (red) and decrease (blue) in surface ice temperature. Both panels for prescribed geothermal heat flux of 0.07.

45 =.

46 > = ( / ) = [ ] = [ ( ) ( + ) ] +, > > ( + ) + ( + ) + + >

47 = [( )+ ( ) + ], > > ( + ) + ( + ) + + = = [], [] [] []

48 =[] [ ( ) + ( +) + + ( + ) ( +) ] +, = % (+) +

49 =[] = = ( ) ( ) [] % = +[], %

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51 =. =

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54 Sliding Velocity (m/yr) Porosity (%) Time (years) a b Ice Thickness (m) Basal Melt Rate (mm/yr) Time (years) Time (years) Time (years) c d Figure 2.5: Simulation replicating parameter regime of experiment 1 in Figure 2 of Bougamont et al. (2011). We used all parameters given in that study and estimate ice stream width to be 35 km and to be Pa s (corresponding to an average temperature of -7 C in the ice stream). (a) sliding velocity. (b) till porosity (note that till porosity = + ). (c) ice thickness. (d) basal melt rate..

55 Ice Thickness (m) a Years Ice Flux (Sv) b Years Figure 2.6: Heinrich event simulation with ice stream 800 km long, 90 km wide, catchment area of. km, 2 m thick effective till layer, geothermal heat flux of. and ice surface temperature of = C. (a) Ice thickness. (b) Instantaneous ice flux in units of sverdrups. Note that catchment area refers to surrounding ice field with thickness the same as ice stream trunk - presumably the ice stream draws ice from a much larger region.

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63 + + = + + = + + =. = + ( ) = + ( ), + + =. ( ) + =. = =

64 ( ) = (, )+ (, )+, (, ) =( = ;, ) = () [ ] = = ( ) ( )( ). = = ( = ) =

65 [ () = + ( + ) ] +. = = + (,, ) =, = + = = (, ) = + (,, ) + ( ) =, ( )= ( ),

66 + ()+ ( ) () = +, ( = + ) = ( = ) = = =, = ( + + ), = =

67 = = > = = >, < > = < < = = <.

68 = = +, = = [ ( )], =

69 . / /... Table 3.1: Parameters used for baseline simulation in this study (unless otherwise indicated).

70 = = = () = (, ) = () = =

71 GL Position (km) GL Position (km) Geothermal Heat Flux (W/m 2 ) a b Cooling Cooling Warming Warming Surface Temperature ( C) Geothermal Heat Flux (W/m 2 ) 0.09 c Surface Temperature ( C) Steady Streaming Thermal Oscillations 20 Figure 3.1: (a) Transient numerical simulation with slow ( Wm /century) decrease (blue dashed) and then increase (red solid) in geothermal heat flux. Surface temperature held constant at = C. Solid black line is the analytical stability boundary corresponding to solid black line in panel c. (b) Transient numerical simulation with slow (. C/century) decrease (blue dashed) and then increase (red solid) in ice surface temperature. Geothermal heat flux held constant at =. Wm. Solid black line is the analytical stability boundary corresponding to solid black line in panel c. (c) A summary of model results in a parameter space of ice surface temperature and geothermal heat flux. Blue crosses are steady-streaming simulations (stable fixed point). Red circles are oscillatory simulations (stable limit cycle). Solid black line is an analytically-derived stability boundary from chapter 2 with a correction for bed slope. See appendix B for details of parameter mapping and correction. Dashed black lines correspond to range of parameter variation for hysteresis simulations shown in panels a-b. All other parameters are specified in Table 3.1.

72 =

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74 a Ice Thickness (m) b Velocity (m/yr) σ σ time (kyr) time (kyr) c Till Water Content (m) d dh/dx x σ σ time (kyr) time (kyr) 3 Figure 3.2: Hovmöller diagrams of a single ice stream thermal oscillation. Transient initialization occurs from 0-10 kyr. All panels are a function of time on horizontal axis and stretched alongstream coordinate,, on vertical xis. (a) Ice thickness. (b) Basal horizontal velocity. (c) Till water content. (d) Ice thickness slope,. In panel a, dashed line indicates location of stagnant ice stream state snapshot corresponding to Figure 3a-b. Solid line indicates location of active ice stream state snapshot corresponding to Figure 3.3c-d.

75 z (m) z (m) x (km) C C x (km) a c Velocity (m/yr) Velocity (m/yr) b σ d σ Unfrozen till thickness (m) Void Ratio Figure 3.3: Snapshots of ice stream stagnant state ( =. kyr in Figure 3.2 and active state ( =. kyr in Figure 3.2) from thermal oscillatory regime run described in section 3. (a) Side-view of stagnant ice stream, contours indicate ice temperature. (b) Horizontal basal velocity (blue solid line) and unfrozen till thickness (red dashed line; void ratio is at lower consolidation threshold,, everywhere) of stagnant ice stream as a function of alongstream coordinate. Maximum available till layer thickness is 5 meters. (c) Side-view of active ice stream, contours indicate ice temperature. (d) Horizontal basal velocity (blue solid line) and till void ratio (red dashed line) of active ice stream as a function of alongstream coordinate.

76 =

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78 0 0.2 a Basal Velocity (m/yr) σ time (kyr) 0.2 b Longitudinal Stress (kpa) c Frictional Heat Flux (W/m 2 ) σ σ time (kyr) time (kyr) Stress (kpa) d Long Basal Lateral Driving Time (kyr) Heating (W/m 2 ) 0.4 Geothermal Frictional 0.35 Conductive e" Time (kyr) Figure 3.4: Mechanism of activation wave propagation. Panels a-c are Hovmöller diagrams with time on the horizontal axis and stretched alongstream coordinate,, on vertical axis. (a) Horizontal basal velocity. (b) Longitudinal stress. (c) Frictional heat flux. Panels d-e are plots of (d) force balance and (e) basal heat budget at a single location ( =.) with time on the horizontal axis. In panel e, we plot the negative of the conductive heat flux and split the vertical axis to highlight the difference in geothermal and vertical conductive heat flux. =..

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81 Activation Wave Speed (km/yr) Upstream Grid Spacing ( σ) Figure 3.5: Activation wave speed convergence with finer upstream horizontal grid spacing. X-axis is grid spacing measured in non-dimensional stretch coordinate units (in text, we refer to physical units of grid spacing corresponding to an ice stream with = km for simplicity). Activation wave speed is calculated by tracking the movement of the concave slope break referenced in section 3.1 from the grounding zone to a location / from ice divide.

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83 GL Position (km) a GZ Mass Balance (m 2 /yr) time (kyr) b Flowline net mass balance Effective grounding line flux x 10 6 u x in GZ (yr 1 ) Flux (m 2 /yr) time (kyr) 1 Figure 3.6: Time series of grounding line state over one thermal oscillation. (a) Solid black line is grounding line position. Dashed blue line is net mass balance over the grounding zone, defined as + ( = ) ( = ) with = and =.. (b) Solid black line is average velocity divergence over grounding zone. Dashed red line is the net mass balance for the entire ice stream flowline domain. Dotted red line is the effective grounding line flux: =[( ) ]( ). Time is on the x-axis for both panels.

84 [( ) ]( ). =,

85 =, + ( = ) ( = ) = =.

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87 Table 3.2: Compilation of results from parameter sensitivity experiments. Simulations marked with an X exhibit steady-streaming behavior. Other parameter values are the same as the baseline simulation and are listed in Table 3.1. Baseline simulation has amplitude 137 km and period 1382 years. =

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96 ( ) = (, )+ +,

97 . / /... Table 4.1: Parameters used for oscillatory simulations in this study (i.e. those described in the main text). () =, () = ()

98 x 0 z x g x L Figure 4.1: Schematic of bed configuration for model simulations. Solid black line indicates ice sheet profile. Brown shaded region is bedrock. is the horizontal position where the retrograde section begins. is the length of the retrograde section. indicates the range over which the grounding line migrates during thermal oscillations. () = ()+ (). () () = ( ) [ ( ) ] ( ) < + > +

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100 = + = = =. < <. = =

101 Figure 4.2: Comparison of equilibrium grounding line positions with varying accumulation rate. (a) Bed properties are static with = kpa everywhere. = (b) Bed properties dynamically evolve. =. (c) Bed properties dynamically evolve. =. In all panels = km and = km. Shading indicates range over which grounding line oscillates in equilibrium behavior. x marks indicate location of steady states for simulations plotted in Figure 4.4. =

102 =. / >. /. < <. /. < <. / =. /

103 = =. /

104 . < <.

105 GL Position (km) GL Position (km) GL Position (km) GL Position (km) a time (kyr) b time (kyr) c time (kyr) d time (kyr) Figure 4.3: Four representative examples of interaction between retrograde section and ice stream thermal oscillations. In all panels, solid line is simulated grounding line migration after transient initialization period ( < kyr), dashed line is grounding line migration in baseline simulation run without any retrograde section ( = km) and grey shaded area is extent of retrograde section. (a) Minimally-modified thermal oscillations with ice stream persistence on the retrograde section during stagnation ( = ). (b) Amplified thermal oscillations with part of active phase on retrograde section ( = ). (c) Suppressed thermal oscillations ( = ). (d) Thermal oscillations with reduced amplitude ( = ). All examples are initialized with = km. = + =

106 Figure 4.4: Transient evolution of ice stream grounding line position with bed properties kept static ( = everywhere). = km and = km and =. Solid line is initialized from = km and =. m/yr, decreasing to. m/yr over the first 300 years. Dashed line is initialized from = km and =. m/yr, increasing to. m/yr over the first 300 years.

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111 bxr x x 0 =700km,x g(t =0)=640km a Oscillation Amplitude (km) bxr x x 0 =700km,x g(t =0)=1150km b Oscillation Amplitude (km) L (km) L (km) 0 bxr x x 0 =800km,x g(t =0)=640km c Oscillation Amplitude (km) bxr x x 0 =800km,x g(t =0)=1150km d Oscillation Amplitude (km) L (km) L (km) 0 Figure 4.5: Amplitude of grounding line migration associated with thermal oscillations as a function of length () and slope ( ) of retrograde section. Simulations with zero oscillation amplitude are shaded in grey. x markers indicate simulations where the grounding line is on a retrograde slope at its minimum position during the stagnant phase. Circle markers indicate simulations where the grounding line is on a retrograde slope at its maximum position during the active phase. Panels (a) and (b) have a retrograde section starting at = km. Panels (c) and (d) have a retrograde section starting at = km. Panels (a) and (c) have initial ice stream grounding line position landward of retrograde section at ( = ) = km. Panels (b) and (d) have initial ice stream grounding line position seaward of retrograde section at ( = ) = km. In the baseline simulation the grounding line position oscillations between 690 km and 815 km.

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121 [ ( ) ] () =, = + =( ),

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125 Discharge (km 3 /yr) Ice Volume (km 3 ) 8 x a Time (kyr) c Time (kyr) Fraction of base temp erate Fraction of margin streams b Time (kyr) d Time (kyr) Figure 5.1: Evolution of important diagnostic quantities during ice sheet growth. (a) Ice sheet volume. (b) Percentage of base that is at pressure melting point. (c) Discharge due to calving at margin. (d) Percentage of margin in ice streams.

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127 Surface Mass Balance (km 3 /yr) Ice Volume (km 3 ) 8 x Time (kyr) Time (kyr) a c Surface Mass Balance (km 3 /yr) Ice Volume Change (km 3 ) x Time after forcing (kyr) Time after forcing (kyr) b d Discharge (km 3 /yr) Time (kyr) e Discharge (km 3 /yr) kyr 20 kyr 30 kyr 40 kyr 50 kyr Time after forcing (kyr) f Figure 5.2: Ice sheet response to climate forcing at various times during growth. (a) Ice volume. (b) Change in ice volume with reference to ice volume at time of forcing. (c) Surface mass balance. (d) Surface mass balance following forcing. (e) Discharge due to calving at ice sheet margin. (f) Discharge following forcing.

128 <

129 Distance from Margin (km) a Basal Velocity (m/yr) Distance from Margin (km) b Driving Stress (kpa) Time since forcing (kyr) Time since forcing (kyr) 0 Figure 5.3: Mechanism of ice stream acceleration. In both panels, the x-axis is time since climate forcing and the y-axis is the distance from ice sheet margin (radial distance upslope from margin). All quantities are averaged over a radial distance from the center of the domain. (a) Basal ice velocity. (b) Driving stress Surface Elevation (m) Surface Mass Balance (m/yr) Figure 5.4: Surface mass balance changes due to change in climate forcing. Crosses indicate SMB at a typical ice stream elevation. Open circles denote typical SMB in ice dome location, upstream of ice streams. Black line and markers indicate surface mass balance just before the change to climate forcing is applied. Blue dash line and markers indicate surface mass balance just after.

130 () () = ()+ (), = + ( ) = + (). / / ()

131 / <, ()

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133 Discharge (km 3 /yr) Mean Elevation (m) Flux through H =400m(km 3 /yr) a non stream stream Time since forcing (kyr) c non stream stream Time since forcing (kyr) e non stream stream Time since forcing (kyr) Surface Mass Balance (km 3 /yr) Area (km 2 ) Ice volume loss (km 3 ) b non stream stream Time since forcing (kyr) 5 x d non stream stream Time since forcing (kyr) x 106 f SMB non str SMB stream discharge non str discharge stream total Time since forcing (kyr) Figure 5.5: Decomposition of ice sheet diagnostics into ice stream and non-ice stream components. (a) Contribution to mass balance by discharge due to calving following forcing. (b) Surface mass balance following forcing. (c) Mean elevation following forcing. (d) Area following forcing. (e) Ice volume flux through the = m elevation contour following forcing. (f) Ice volume loss following forcing decomposed into components due to: surface mass balance in non-ice stream regions (black solid line), surface mass balance in ice stream regions (blue solid line), discharge from non-ice stream regions (black dashed line), discharge from ice stream regions (blue dashed line) and the total mass loss (orange line, as in Figure 5.2b). In all panels black line is in non-streaming regions defined as basal velocity less than 100 m/yr, blue line is in streaming regions defined as basal velocity greater than 100 m/yr.

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135 Equilibrium Line Altitude (m) a Time after forcing start (kyr) Surface Mass Balance (km 3 /yr) Time after forcing start (kyr) c Ice Volume Change (km 3 ) x b Time after forcing start (kyr) Discharge (km 3 /yr) Time after forcing start (kyr) d Figure 5.6: Ice sheet response in simulations with differing rates of changing climate forcing. (a) Specified equilibrium line altitude. (b) Change in ice volume following start of forcing change. (c) Surface mass balance following start of forcing change. (d) Discharge due to calving at the ice sheet margin following start of forcing. All curves in panels c and d are smoothed with a 100 year filter from raw model output to eliminate sub-centennial numerical noise.

136 > <

137 Surf. Mass Bal. (km 3 /yr) Discharge (km 3 /yr) Ice Volume (km 3 ) x 10 6 a Time after forcing (kyr) 0 b Time after forcing (kyr) km 10 km km 800 c 40 km km 10 Time after forcing (kyr) 20 km (ENH) Figure 5.7: Ice sheet response in simulations with differing horizontal model resolution. (a) Change in ice volume following forcing. (b) Discharge due to calving at the ice sheet margin following forcing. All curves in panels b are smoothed from raw model output to eliminate sub-centennial numerical noise.

138 =

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149 A =[]

150 =[] =[ ] =[ ] =[ ], [ ]= [], =( ). [ ]=[ ] = ( ( )), = [ ] [ ] [ ]= + ( + ) ( ) [ ], []

151 [( ) =, ]. =[][ ], [] [] = [ + ( ) ( + ) ] +. [] = []. = = + > = = > > ( ( )) = >

152 [( ) =, ], = [, ], = [][ ][ ] = [] = [ + ( ) ( + ) ] + = [ ][ ] = [] = [ + ( ) ( + ) ] + = [][ ][ ] = [] = [ + ( ) ( + ) ] +.

153 () = +,, = = = < > = ( ) + ( ) = = ( + ) ( )+ ( ) () + ( ) ( )

154 = + = [ + ( ) ( ) ] = [ ] () = [ ] () () ( + ) = () ( + )( ) = () ( + )+( + ) = = +.

155 ( ( ) + ) ( ) = + ( ) ( ) + ( ) = ( + = ( ) + + =. ) + ( + ) +. + (, ) (, ) = + = + =,,, = ( / ) ( + / )+/ ( + +/ ) ( + / ) +/ ( +/ ), = =

156 = = (, ) +, (, ) = = = [ ] ( ) ( ) ( )( )+( )( ), = = > ( < ) () = = () ( ) () ()

157 = ( )() ()() = ( ) = ( ) = () = () = ( ) ( )( ) = + ( )( ). / = ( /)( / ) = ( /) ( /) = ( /) = ( ) ( ) + ( ) [ ][ ( ) ( ) ] ( ) = / = /

158 (, )= (, ) (, )= ( + ) ( + (, )), + (, )= ( ) + (, )= ( ( )/ ) ( + (, )= / ). [ ( ) = + + ] ( ) ( ) = [ + ( ) + ] ( ) ( ) = ( ), = = = ( = ) = ( ) ( ) [ ]

159 <, < > ( = ) > < / = (), = () () () ( )= ( ) / + ( ) ( / ) [ ] + +(). + = () () = > = = + ( = ).

160 =() () =. > > <. >. > + =.

161 = + = + =, = ( )+ ( ) +. () = ( ) () + +, + +( + ) +( + + ) ( + + ) (+) =.

162 + ( ) ( + ) + ( ). [ +( ) ]+( + ) + ( + + ) + +( + ) + ( ) =. = + = ( + )+ + = = % %.%

163 ()

164 = < < / = > > > = / = / ( / ) = /,. = () = / / =(/( + )) ( /( + ) + ) + / ( + ) + + +,

165 = ( )=(/( + )) = > < < (, ) / > / = < () > < / = () = () () > () ( ) > > < /

166 + / >, = () = / = > < / < / < < / () >

167 () / = ( /(+) ) = ( /(+) ) = ( (+)/(+) < = () ( ) / = () = () < / /( ()) / = () /( + ()) ( = ( ) () () / /

168 = () = / = () =( ) / = = + = = = () = / = () = ( /(+ ) = ( /(+ ) = () = ( /(+) ) / /( + ( (+)/(+) ) / = /( + ( (+)/(+) ) ( ) ( ) ( /(+) ) ( /(+ ) /( + ( (+)/(+) )

169 = () = () / = ( ) = () / () / / + /( + ) + (/) + () < = / = () / () = / ( ) > / / < / =

170 > (/) = / / () (/) = ( ( )) = (/) = + [(/) / ] / = = / = / ( ), = / + /. =, =. = = / = = + / = +( )/ ( / ) = / + / + ( ( )/ ) = ( ) + + ( ).

171 > = (+)/( ) (+)/( ) /( ) [( )( )] /( ) = ( / +( )/ ) = = = / + / = / =(/) (/) / / = =[(/) (/) / ], =(/)[ (( ))] = = (+)/( ) (+)/( ) /( ) [ ( ) ] /( ) = (/) = = =(/) = / / ()

172 (, ) =( / ) + / =, (, ) (, )=, (, )=. > = (, ), = [( / )/ ] / > = (, ), = (, ), = (, ). < > >

173 = / ( ), ( )= /, = /. (, ) = + =. > ( ) [ ] / / ( ). = ( / ), =, = + ( / ),

174 =[ / ] + / = [( / )/ ] /( ) < = = = ( ) + + = = (+) + = + =

175 = ( ) +. () = ( ) () + +, ( + + ) + + ( + + ) =, ( ) [ + ] +(+) + +(+) =. [ ( + ) + + ] = = + ( +) +

176 ( + ) = = ( ( + ) +) + = ( ) + ( +) + + ( + ) ( +) + = =, () = +,

177 = + () = ( + ) ( + )+ () = ( + ) = ( ) () = = ( ) =

178 =, ( [ ( + ) = ( + )]) ( ) ( + ) = ( ) = ( ) ( ) + ( ) + ( ) = = ( ( ) ) () << ()

179 = ( )

180 h h a actual system trajectory γ β stable active branch unstable active branch h s stable stagnant branch u c u Figure A.1: Phase portrait of the reduced model.

181 B

182 = =, = = ( ) = (, )+ (, )+, (, ) = ( = ;, ) (, ) =,

183 = ( ) () = + + [ ( ) +]. + =. ( + ) + =,

184 + = = = = () () (,, ) = ( + ), = + ( ) = (, ) = + (,, ) + ( ) = + ) ] [( + =

185 + ( ) = + ()+ () = ( + ) ( ( + ) ) + () + ( + ) () + () + = () () ( ) ( ) ( ) + () ( + ) () + () =

186 + [ ( ) ] [( + { + + ( ) } ) ] + [ ( ) ] + =. { + ( ) }. = ( + () ()+ () ) (, = ) = =

187 = > = = > = = < < = = < = +, = [ ( )],

188 = = [ ] = = ( ) ( ). = = ( = ) = ( = + ) =

189 = = = = = ( ) (,...) =. [ ] ( ) (,...) =, ( ) = +.

190 ( = ) = ( = ) = = = = ( ). [ ] (,...) + =. [ () = ( ) (,...) + ] [ ( ) ( )= + = ( ) ( ( + +,...)+ ( +,...) ) ( ) ] + + +

191 + + [ + + (+ + [ (+ + )/ )/ ] [ ( ) ] [ + ( + ] ( )] ) + ( =, + ) + = [ ( + + ( ))] + +

192 +,, + +,+ + [ + +, + + +, + (+ + +, (+ + [ + +, + + ( +, + )/ )/ +,+ [ ( +,+ + +, [ + +, [ +, ( +,+ + ( ) + ] + +, + ( )] + +, + )/ ) ] + +,,+ + ( + )/ + ( ) ], + +,, + +, ( + + )/ ( )( ( ( )] + ) + ) ) = (,, +,+ ).

193 , + = > =, = > +,, = = <, < =, = < < > = > ( ) =. < ( ) =

194 + = ( ) +, + +, < > + +, = ( ) + +, + +, < > =.

195 [. ] = % = = =.

196 = ( ) [] [ ]= [] +. =[][ ] = + [] ( + ) [ [] ( + )]. [] [] [] [] () + +. = = [ + ( ) ( + ) ] +.

197 = = [ + ]. =. = [ + + ( )+ ]. =. +, %

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