The International Review of BIPOLAR DISORDERS NH Convention Centre Seville, Spain
J.M. Abril Department of Applied Physics (I); University of Seville (Spain)
description
Transcript of J.M. Abril Department of Applied Physics (I); University of Seville (Spain)
![Page 1: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/1.jpg)
J.M. AbrilDepartment of Applied Physics (I); University of Seville (Spain)
IAEA Regional Training Course on Sediment Core Dating Techniques. RAF7/008 Project
J.M. Abril, University of Seville
Lecture 3:Clasical dating models using 210Pb
210Pbex fluxes
Radionuclide profiles and inventories
Radiometric dating modelsCIC CF-CSR, CRS, CMZ-CSR , CD-CSR IMZ (*)-CSR
1
![Page 2: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/2.jpg)
2J.M. Abril, University of Seville2
![Page 3: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/3.jpg)
z
aw
222Rn
210Pb
137Cs
J.M. Abril, University of Seville
3
![Page 4: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/4.jpg)
y = 9,3 1 x - 1,87
R2 = 0,689*
0
10
20
30
40
50
60
70
1 2 3 4 5 6
ETo (mm/d)
222 R
n E
xh
ala
tio
n (
Bq
h-1
m-2
)
Abril et al. (JENVRAD, 2009)
222Rn exhalation depends, among other factors, on 226Ra content in soil, soil texture and structure, water content, and the forcing factors…
4J.M. Abril, University of Seville
4
![Page 5: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/5.jpg)
Author: Israel López, Univ. Huelva (Spain)J.M. Abril, University of Seville5
![Page 6: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/6.jpg)
J.M. Abril, University of Seville6
![Page 7: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/7.jpg)
Some global patterns for 210Pbex fallout
•Predominant west-east movement of air masses 210Pbex fallout is low in the western areas of the continents
•210Pbex fallout is higher in the North hemisphere
•210Pbex fallout is positively correlated with rainfall
Figures from P.G. Appleby, STUK-A145
J.M. Abril, University of Seville7
![Page 8: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/8.jpg)
Some reference values for annual fallout of excess 210Pb (Bq m-2 y-1) Some reference values for annual fallout of excess 210Pb (Bq m-2 y-1)
Global scale , F ~ 23-367 Bq m-2 y-1 (Robbins, 1978)
Tropical Australia , F ~ 50 Bq m-2 y-1
(Brunskill and Pfitzner, 2000)
Catchment concentration factor (normalization or focusing factor) : Z
Input (*) = ZF
Steady State Inventories Σ = ZF/λ
For 210Pb = ln2/T1/2 with T1/2 = 22.26 y.
Inputs and Inventories (Bq m-2 ) in sediments Inputs and Inventories (Bq m-2 ) in sediments
J.M. Abril, University of Seville8
![Page 9: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/9.jpg)
Z [cm]
210Pb
[Bq/kg]
226Ra
total210Pb (unsupported)
Radiometric dating with 210Pb: Basic aspects
If we assume that there is no Rn exhalation from the sediment, then the total activity of 210Pbtotal will be 210Pbtotal = 210Pbsupported + 210Pbunsupported
and 210Pbsupported = 226Ra activity
Supported fraction
J.M. Abril, University of Seville9
![Page 10: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/10.jpg)
Basic Concepts and definitionsBasic Concepts and definitions
z
aw
J.M. Abril, University of Seville10
![Page 11: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/11.jpg)
Compaction and bulk density
As depth increases in the sediment core, water pores are replaced by solids
Saturated porous media
V
ms
Bulk density
z
Vmw
ms
J.M. Abril, University of Seville11
![Page 12: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/12.jpg)
Practical measurement of bulk densities
w
s
s
s
w
wsw
mmVVV
w
s
s
w
s
s
s
w
w
s
mmm
m
1
mw
ms
m
Drying and gravimetric method
J.M. Abril, University of Seville12
![Page 13: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/13.jpg)
Practical measurement of bulk densities. Refinement
mw
ms,om
Drying and gravimetric method and loss by ignition
w
s,0
ms,i s,i
is
is
os
os
w
wisosw
mmmVVVV
,
,
,
,,,
is
is
os
os
w
w
isos
mmm
mm
,
,
,
,
,,
is
is
os
os
w
w
isos
mmm
mm
,
,
,
,
,,
J.M. Abril, University of Seville13
![Page 14: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/14.jpg)
Bulk density versus depth profiles in sediment cores
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 15 20
(g
/cm
3)
Depth [cm]
ze 1
ze 1
J.M. Abril, University of Seville14
![Page 15: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/15.jpg)
z
Mass thickness, Δm , and mass depth:, m
zm z
dzm0
'
[ g dry weight cm-2]
Δz
J.M. Abril, University of Seville15
![Page 16: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/16.jpg)
(Mass) Sedimentation rate : w
dt
dmw [ g dry weight cm-2 y-1]
Time versus m for constant w (*)
w
dmdt w
mt
Z
≈
ZiA (Zi, t)
A (Zi+1, t)
A (Zi-1, t)
w (Zi-1, t)
w (Zi+1, t)
J.M. Abril, University of Seville16
![Page 17: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/17.jpg)
Basic processes
Z
≈
ZiA (Zi, t)
A (Zi+1, t)
A (Zi-1, t)
w (Zi-1, t)
w (Zi+1, t)
J.M. Abril, University of Seville17
![Page 18: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/18.jpg)
In situations where the tracer is partially carried by pore water or in presence of selective and/or translocational bioturbation Eqs. has to be revisited
Fundamental equationsFundamental equations
BOUNDARY CONDITIONS
Mass conservation for a particle-associated radiotracer
Mass conservation for solids
J.M. Abril, University of Seville18
![Page 19: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/19.jpg)
[Bq L-2 T-1]
Constant Flux and Constant Sedimentation rate (CF-CSR)
Activity concentration at interface(non post-depositional mixing) Constant A0 w
FA 0
w
F incoming flux
sedimentation rate
J.M. Abril, University of Seville19
![Page 20: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/20.jpg)
(non post-depositional mixing)
Layer at time t=0
The sediment-water interface displaces upwards
Specific activity A0
time = 0
z=z(t)
time = t =m/w
toeAA
m=m(t)
J.M. Abril, University of Seville20
![Page 21: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/21.jpg)
m
Ln(A)
Validation:
Goldberg first validated the 210Pb dating method in varved sediments
Curve-fitting model , free parameters : Ao , w
wmt
eAA w
m
/0
Think about: Any implicit assumption concerning compaction?
J.M. Abril, University of Seville21
![Page 22: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/22.jpg)
Schweiz. Z. Hydrol. 49/3, 1987
ZF = 172 Bq m-2 y-1
EXAMPLE from a case study
J.M. Abril, University of Seville22
![Page 23: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/23.jpg)
Don't forget:
Estimated sedimentation rates, ages and dates have to be provided with the corresponding uncertainties.
Don't forget:
Estimated sedimentation rates, ages and dates have to be provided with the corresponding uncertainties.
Age : T(m) or T(z) , from m(z)/w
w , (mass) sedimentation rate
Dates or chronology: Year of sampling – Age
W = 0.115 ± 0.014 g cm-2 y-1
J.M. Abril, University of Seville23
![Page 24: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/24.jpg)
Associated uncertainties in 210Pb chronology
is
is
os
os
w
w
isos
mmm
mm
,
,
,
,
,,
is
is
os
os
w
w
isos
mmm
mm
,
,
,
,
,,
iii zm
i
imm
i
im2
2,
2, zrrii m
,r
,...),,( 321 xxxf
...,;,;, 332211 xxx
i
ji
f x
f2
General formulae for error propagation
G.F.
mm
J.M. Abril, University of Seville24
![Page 25: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/25.jpg)
bxaAxf
mx
ln)(
11
2
/1
2
2
2
RN
a
x
a
ab
2,
2, brrw w
bw
2,
2, wrmrt
w
mt t
t
Associated uncertainties in 210Pb chronology
J.M. Abril, University of Seville25
![Page 26: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/26.jpg)
Time resolution . Each sectioned layer in the core corresponds to a time interval Δt = dm/w
Remember: As the analytical method is homogenizing the material from each layer, it is not possible to solve other time marks within such an interval (e.g. two 137-Cs peaks).
Note for advanced students:
•Apply lineal regression taking into account the associated uncertainties in measurements
J.M. Abril, University of Seville26
![Page 27: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/27.jpg)
CAUTION !
•Estimation of the supported fraction is not a trivial task !
• 226Ra may be non uniform in depth and being different from the 210Pb baseline
•Settling particles can be depleted in 226Ra in the water column while enriched in
210Pb
Data from Axelsson and El-Daoushy, 1989Data from Axelsson and El-Daoushy, 1989
J.M. Abril, University of Seville27
![Page 28: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/28.jpg)
10
100
1000
10000
0 0.2 0.4 0.6 0.8 1
210 P
b (B
q/kg
)
Mass depth (g cm-2)
RedóGossenkollesee
1.- Many unsupported 210Pb profiles do not follow a simple exponential decay pattern
More complex models are required
PROBLEMS:
J.M. Abril, University of Seville28
![Page 29: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/29.jpg)
CIC model (Constant Initial Concentration)
w
F incoming flux
Activity concentration at interface(no post-depositional mixing) w
FAo
CIC model assumes constant Ao; Thus, changes in F must be compensated with changes in w.
Also , it assumes non post-depositional mixing
-Reasonable when F is associated with inputs of solids
sedimentation rate
J.M. Abril, University of Seville29
![Page 30: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/30.jpg)
CIC model can equally be formulated in terms of actual depth (z) or mass depth (m)
Chronology (one date per data point)
Alternative estimation of sedimentation rates (one per data point) – only for cores with high spatial resolution-
CAUTION !•Estimation of the initial concentration, Ao, is not a trivial task !
A0
A(m)
m
A
-Unknowns for CIC: Ao and wi (N+1; N= number of sections in the core)
- It is a “mapping” model
J.M. Abril, University of Seville30
![Page 31: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/31.jpg)
Schweiz. Z. Hydrol. 49/3, 1987
EXAMPLE from a case study
CF-CSR CIC
J.M. Abril, University of Seville31
ZF (recent) = 76 Bq m-2 y-1
![Page 32: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/32.jpg)
CRS model (Constant Rate of Supply)
w
F incoming flux
Initial concentration w
FAo
CRS model assumes constant F, independently of w. Ao can vary. Also assumes non post-depositional mixing.
-Reasonable when F is not coupled with inputs of matter
J.M. Abril, University of Seville32
![Page 33: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/33.jpg)
CRS model
Inventory under the horizon z
z
dzzzAz ')'()'()(
After a time t, the horizon now at z=0 will be located at depth z(t), and because of the radioactive decay.
tez 0)(tez 0)(
0
0 ')'()'()0( dzzzAz
At “geological” timescale the inventory is steady state; thus,
000
Fdt
dz 0F 0F
Z
z
J.M. Abril, University of Seville33
![Page 34: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/34.jpg)
)(
ln1
)( 0
zzt
)(
ln1
)( 0
zzt
CRS Chronology:
Once the chronology is established, sedimentation rates can be obtained for each two adjacent layers:
t
zw
dtwdzdm
t
zw
dtwdzdm
z
Alternatively, from the mass balance in the steady state inventory below depth z
CRS model
-Unknowns for CRS: F, wi (N+1; N= number of sections in the core)- It is a “mapping” model
J.M. Abril, University of Seville34
![Page 35: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/35.jpg)
CAUTION
•Check for completeness of inventories (sometimes it will be necessary to estimate the “missing” part of the total inventory)
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25 30
Uns
uppo
rted
Pb-
210
(pC
i/g)
Depth (cm)
MARINE SEDIMENT- GOTEBORG-
"data2"2*exp(-0.09*(x-9))
2
J.M. Abril, University of Seville35
![Page 36: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/36.jpg)
Schweiz. Z. Hydrol. 49/3, 1987
EXAMPLE from a case study
J.M. Abril, University of Seville36
![Page 37: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/37.jpg)
ZF = 170 Bq m-2 y-1
from CF-CSR w = 0.115 ± 0.014 g cm-2 y-1
J.M. Abril, University of Seville37
![Page 38: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/38.jpg)
Complete mixing zone model with constant sedimentation rate and constant flux.
Mixing ma
w
F
F
Aama
Radioactive decay
wAa Sediment growth
aa mw
FA
aa mw
FA
Steady-state mass balance
w
mm
aa
a
eAmmA
)( w
mm
aa
a
eAmmA
)(
Curve-fitting model , free parameters : Aa, w, ma
J.M. Abril, University of Seville38
![Page 39: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/39.jpg)
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25 30
Un
sup
po
rte
d P
b-2
10
(p
Ci/g
)
Depth (cm)
MARINE SEDIMENT- GOTEBORG-
"data2""cmz"
Example CMZ-1
mixing
ma=9.5 g cm-2; w=0,374 g cm-2 y-1ma=9.5 g cm-2; w=0,374 g cm-2 y-1
J.M. Abril, University of Seville39
![Page 40: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/40.jpg)
Acceleration or mixing?10
100
1000
10000
0 0.2 0.4 0.6 0.8 1
210 P
b (B
q/kg
)
Mass depth (g cm-2)
RedóGossenkollesee
2.- Many times unsupported 210Pb profiles can be equally explained by different models
210Pb chronologies must be validated against an independent dating method
PROBLEMS:
J.M. Abril, University of Seville40
![Page 41: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/41.jpg)
J.M. Abril, University of Seville41
Think about:
What other hypothesis are implicitly assumed in all the previous models ?
![Page 42: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/42.jpg)
Constant flux, CSR and constant difussion Model
Demonstration will be provided within lecture 6
Curve-fitting model , free parameters : ZF, km , w
Data: CF-CS-C DiffusionFit : CF-CSR Model
w 0,1 g cm^(-2) y^(-1)
km 6 g^2 cm^(-4) y^(-1)
ZF 200 Bq m^(-2) y^(-1)
w 0,49 g cm^(-2) y^(-1)ZF 200,6 Bq m^(-2) y^(-1)
J.M. Abril, University of Seville42
![Page 43: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/43.jpg)
J.M. Abril, University of Seville43
![Page 44: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/44.jpg)
J. N. Smith proposed a protocol for research journals for theacceptance of papers that rely on 210Pb dating to establish a sediment core geochronology:
‘‘The 210Pb geochronology must be validated using at least one independent tracer which separately provides an unambiguous time-stratigraphic horizon’’.
J.M. Abril, University of Seville44
![Page 45: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/45.jpg)
ZFo=10 mBq/(cm^2 y) , w=0.1+0.1 t/150 g/(cm^2 y) D=0
Examples generated with numerical solutions
Constat aceleration, constant diffusion or CF-CSR?
J.M. Abril, University of Seville45
![Page 46: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/46.jpg)
Effect of “episodic” changes in sedimentation rates?
J.M. Abril, University of Seville46
Examples generated with numerical solutions
![Page 47: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/47.jpg)
Numerical algorithm: MSOU
T= - 50 y sgt= 5 y
J.M. Abril, University of Seville47
λ=0
Ts =150 y
![Page 48: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/48.jpg)
J.M. Abril, University of Seville48
T= - 20 y sgt= 2 y
Ts =150 y
λ=0
Numerical algorithm: MSOU
![Page 49: J.M. Abril Department of Applied Physics (I); University of Seville (Spain)](https://reader035.fdocuments.net/reader035/viewer/2022070419/56815b1c550346895dc8d0cc/html5/thumbnails/49.jpg)
Periodic changes in w with T=7 y
When data are smooth enough to apply CSR models?
J.M. Abril, University of Seville49
Examples generated with numerical solutions