Robin Hogan andRobin Hogan and Jon ShonkJon Shonk

Implementation of Implementation of multiple regions in multiple regions in

Edwards-SlingoEdwards-Slingo

Shadows!Shadows!• Need separate

regions even in clear skies?

Downward short-wave flux (W m-2) Upward short-wave flux (W m-2)

Natural logarithm of IWC (g m-3)

Anomalous horizontal Anomalous horizontal transporttransport

• Homogenization of clear-sky fluxes:– Reflected radiation has more chance

to be absorbed -> TOA shortwave bias– Effect is very small in the longwave

• This problem can be solved in a way that makes the code more efficient

Single cloud

Two-stream Two-stream schemescheme

0.5

1 1 0.5 1

1 1 1.5 1

2 2 1.5 2

2 2 2.5 2

2.5

1

1

1

1

1

1

TOA

s S

F S

R T F S

T R F S

R T F S

T R F S

F S

F1.5+ F1.5

-

F0.5+ F0.5

-

F2.5+ F2.5

-

• Consider a two-layer atmosphere

• Solve a tridiagonal matrix problem to obtain the fluxes

Layer 1 reflection, transmission

and emission: R1, T1, S1+, S1

-

Layer 2 reflection, transmission

and emission: R2, T2, S2+, S2

-

TOA flux: STOA-

Surface emission and albedo: Ss+, s

• It is conceptually convenient to solve the system by– Working up from the surface calculating the albedo i and

upward emission Gi of the whole atmosphere below half-level i.

– Then working down from TOA, calculating the upwelling and downwelling fluxes from i and Gi.

Edwards-Slingo solutionEdwards-Slingo solution

0.50.5 0.5

0.5

1 1.5 1 1.51 1 1 1.5

1 1 1.5 2

2 2 2 2.5 2 2

2 2 2.52

1

1

1

1

1

1

TOA

s s

GFSF

G ST F

T R F S

T F S ST R F S

Two-regions: cloud and clear-Two-regions: cloud and clear-skysky

0

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

1

1

1

1

1

1

1

1

1

1

1

1

aa ab a b

ba bb a b

a b aa ab

a b ba bb

aa ab a b

ba bb a b

a b aa ab

a b ba bb

s

s

F

R R T T

R R T T

T T R R

T T R R

R R T T

R R T T

T T R R

T T R R

.5

0.5

0.5 1

0.5 1

1.5 1

1.5 1

1.5 2

1.5 2

2.5 2

2.5 2

2.5

2.5

aTOA

bTOA

a a

b b

a a

b b

a a

b b

a a

b b

as

bs

S

F S

F S

F S

F S

F S

F S

F S

F S

F S

F S

F S

a

a

b

b

Layer 1

Layer 2

• With 2 regions (a & b), matrix is denser– Latest Edwards-Slingo only has approximate

solvers SOLVER_MIX_DIRECT and SOLVER_TRIPLE for the 3-region version

– OK for upright convection but unacceptable errors introduced with realistic overlap

Note that the overlap coefficients have been omitted in

this example

New versionNew version

0

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

1

1

1

1

1

1

1

1

1

1

1

1

aa ab a b

ba bb a b

a b aa ab

a b ba bb

aa ab a b

ba bb a b

a b aa ab

a b ba bb

s

s

F

R R T T

R R T T

T T R R

T T R R

R R T T

R R T T

T T R R

T T R R

.5

0.5

0.5 1

0.5 1

1.5 1

1.5 1

1.5 2

1.5 2

2.5 2

2.5 2

2.5

2.5

aTOA

bTOA

a a

b b

a a

b b

a a

b b

a a

b b

as

bs

S

F S

F S

F S

F S

F S

F S

F S

F S

F S

F S

F S

a

a

b

b

Layer 1

Layer 2

• But some elements represent unwanted anomalous horizontal photon transport– Remove them and the problem can be

solved exactly, and in closer agreement with ICA

– Enables triple (and quadruple, quintuple…) regions to be implemented quite easily!

Rab

Rab is the reflection

from region a to region

b at the same level

Performance of new solverPerformance of new solver• In calculating upwelling flux at half-

level 1.5, downwelling flux sees albedo of whole atmosphere below, 1.5, which it reflects back into the same region– Much closer to the independent

column approximation!

0.5

1.51.5