Download A benchmark study on the implementation of sea

A benchmark study on the implementation of sea-level equation
in GIA modelling: Specifications
Z. Martinec, May 14, 2017
Note: the deadline for submitting the results is July 31, 2017
The benchmark examples are carried out for the following Earth and load parameters for
various scenarios.
1. Viscoelastic/density structure
• model M3–L70–V01 (Spada et al., 2011, Table 3)
2. Ice load - spatial distribution
• Spherical cap (Spada et al., 2011, Table 4, col.1) of height h0 and radius α0 = 10◦ that
is centered at co-latitude ϑ0 and longitude ϕ0 . Table 1 summarizes three scenarios of ϑ0 ,
ϕ0 and h0 considered in the benchmark.
Table 1. Summary of the scenarios of the spatial distribution of a spherical cap ice load.
Scenario ϑ0
L1
0◦
L2
25◦
L3
25◦
ϕ0
0◦
75◦
75◦
h0 [m]
1500
1500
500
3. Ice load - time evolution
Table 2 summarizes two scenarios of time evolution of ice load considered in the benchmark.
Table 2. Summary of the scenarios of time evolution of ice load.
Scenario
T1
T2
Description
Heaviside loading of the height of load, h(t), and the radius of load,
α(t), for t ∈ (0, t0 ], t0 =10 kyrs, i.e., h(t) = h0 and α(t) = α0 .
A linear increase of h(t) and α(t) for t ∈ (0, t0 ] such that
h(0) = 0, h(t0 ) = h0 , α(0) = 0 and α(t0 ) = α0 . For t ∈ (t0 , t1 ],
t1 = 15 kyrs, h(t) = h0 and α(t) = α0 .
The final time tf of numerical experiments is tf = t0 and t1 for the time evolution scenarios
T1 and T2, respectively.
4. Solid Earth topography at t = 0:
• Circular exponential basin b(ϑ, ϕ) centered at co-latitude ϑb and longitude ϕb
b(ϑ, ϕ) = bmax − b0 exp{−ψ 2 /2σb2 }
where σb = 26◦ and
cos ψ = cos ϑ cos ϑb + sin ϑ sin ϑb cos(ϕ − ϕb )
Table 3 summarizes four scenarios of ϑb , ϕb , bmax and b0 considered in the benchmark.
Table 3. Summary of the scenarios of the spatial distribution of ocean basin at t = 0.
Scenario
B0
B1
B2
B3
ϑb
0◦
100◦
35◦
35◦
ϕb
0◦
320◦
25◦
25◦
bmax [m] b0 [m]
0
0
760
1200
760
1200
3800
6000
5. Sea-level equation
Table 4 summarizes four scenarios of the sea-level equation considered in the benchmark.
Table 4. Summary of the scenarios of the sea-level equation.
Scenario Description
SLE=0 Sea-level equation is not considered.
SLE=1 An equivalent water layer is spread over a fixed ocean region.
SLE=2 The ocean load is computed according to the sea-level equation,
but the ocean region is fixed. When ice reaches the ocean,
it is immediatelly calved.
SLE=3 The coast lines move according to the sea-level equation.
When ice reaches the ocean, it flows (= floating ice)
or continues to be grounded (= grounded ice).
6. Cut-off degrees
• jmin = 0 and jmax = 128, except for SLE=0 scenario where jmin = 2.
7. Other parameters:
• density of ice %ice = 931.
• density of sea-water %water = 1000.
8. Coordinate system: The center of mass (CM) frame.
9. No-rotating Earth: no rotational deformation, no rotational feedback.
10. The overview of the specifications of benchmark examples
Table 5 summarizes the specifications of 9 examples considered in the benchmark.
Table 5. Summary of the specifications of benchmark examples.
Specification
SLE
Load spatial
Load temporal
Ocean basin
A
0
L1
T1
B0
B
1
L1
T1
B1
Benchmark examples
C1 C2 D1 D2 E1
2
2
3
3
3
L1 L2 L2 L2 L2
T1 T1 T1 T2 T2
B1 B1 B1 B1 B2
E2 D3
3
2
L3 L3
T2 T2
B3 B3
F1
3
L3
T2
B3∗
B3∗ . . . solid Earth topography B3 is prescribed at the final time tf . The solid Earth
topography at the initial time t = 0 needs to be searched iteratively such that the solid
Earth topography B3 is adjusted at the final time tf .
11. The overview of results provided so far by the participants
Table 6 summarizes the results of benchmark examples provided so far by 13 participants.
Table 6. The summary of benchmark results provided by the participants.
Participant
ZM
VK
WW
YS/RR
GS/DM
KS/TM
BB
GA
VB
SB
A
x
x
x
x
x
x
x
x
x
B
x
x
x
Benchmark examples
C1 C2 D1 D2 E1
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
E2
x
x
D3
x
x
x
x
F1
x
Table 7. The solution methods.
Participant
ZM
VK
WW
YS/RR
GS/DM
KS/TM
BB
GA
VB
SB
Representation
temporal radial lateral
TD
FE
SH
TD
FE
SH
LD
MP
SH
LD
MP
SH
LD
MP
SH
LD
MP
SH
TD
FE
FE
LD
SH
SH
LD
MP
SH
?
?
?
TD=time domain, LD Laplace domain, MP=matrix propagator
11. Results
Figure 1. Grounded ice mass at t = 0.
Figure 2. Grounded/floating ice mass at t = tf .
Figure 3. Solid Earth topography b(ϑ, ϕ) at t = 0.
Figure 4. Ocean function t = 0.
Figure 5. Ocean function t = tf .
Figure 6. Time evolution of ice mass.
Figure 7. Time evolution of ocean-basin area.
Figure 8. Time evolution of three contributions to hU F .
Figure 9. Longitudinal and latitudinal cross-sections of ocean function through the center
of ocean basin at t = tf .
Figure 10. The ϑ cross-section of surface vertical displacement, the ϑ and ϕ components
of horizontal displacement, and the negative geoidal heights through the center of ice load
at t = tf .
Figure 11. The ϕ cross-section of surface vertical displacement, the ϑ and ϕ components
of horizontal displacement, and the negative geoidal heights through the center of ice load
at t = tf .
Figure 12. The ϑ cross-section of surface vertical displacement, the ϑ and ϕ components
of horizontal displacement, and the negative geoidal heights through the center of ocean
basin at t = tf .
Figure 13. The ϕ cross-section of surface vertical displacement, the ϑ and ϕ components
of horizontal displacement, and the negative geoidal heights through the center of ocean
basin at t = tf .
Figure 14. Top: The ϑ cross-section of sea-surface variations wrt hU F and SLE load
through the center of ocean basin at t = tf . Bottom: The ϕ cross-section of sea-surface
variations and SLE load through the center of ocean basin at t = tf .
Figure 15. Surface vertical displacement at t = tf .
Figure 16. SLE load at t = tf .
Figure 17. Sea-surface variations wrt hU F at t = tf .
Figures 10diff-13diff. The same as Figures 10-13, but for the differences between the
solution where the initial solid topography is changed such that the solid topography is
adjusted at final time tf and the solution where the solid Earth topography is not changed.