1. Entrapment
An important mechanism for shortwave 3D cloud radiative effects and its inclusion in the SPARTACUS radiative transfer model
Robin Hogan, Mark Fielding (ECMWF)
Howard Barker (Environment & Climate Change Canada)
Najda Villefranque (University of Toulouse)

2. Frequently claimed, never backed up with evidence:
“Shortwave 3D effects of clouds are negligible because the sign of the effect depends on sun angle so cancels across the diurnal cycle”
“Longwave 3D effects of clouds are negligible because I don’t know of any papers that say otherwise”
European Centre for Medium-Range Weather Forecasts

3. 3D effects in the ECMWF radiation scheme “ecRad”
SPARTACUS solver: add extra terms to the two-stream equations:
𝑑 𝑣 𝑎 𝑑𝑧 = 𝛽 𝑒 − 𝛾 1 𝑎 𝑣 𝑎 + 𝛾 2 𝑎 𝑢 𝑎 + 𝑠 𝑎− − 𝑓 𝑎𝑏 𝑣 𝑎 + 𝑓 𝑏𝑎 𝑣 𝑏
Rate of exchange between regions is:
Write two-stream equations in matrix form including direct solar flux s:
where
Solution per layer in terms of matrix exponentials:
Solve for flux profile using matrix version of the Adding Method
SPARTACUS solver is 5.7 times slower than McICA (OK for research)
New terms representing exchange between regions
Length of cloud perimeter per unit gridbox area (related to cloud size)
𝑓𝑎𝑏= 𝐿𝑎𝑏 2𝑐𝑎
Fraction of gridbox occupied by clear skies (region a)
Speedy Algorithm for Radiative Transfer through Cloud Sides a ua va ub vb
Matrix exponential
Waterman (1981), Flatau & Stephens (1998) a b c a b c a
Hogan et al., Schafer et al. (JGR 2016)

4. SPARTACUS applied to cumulus cloud scene
Lower reflectance… why? ?
Compare independent column approximation with fully 3D
Excellent agreement between SPARTACUS and (far slower) Monte Carlo
Higher reflectance due to cloud edges
Hogan et al. (JGR 2016)

5. Mechanisms for 3D shortwave effects
What about multi-level clouds? Entrapment!
Upward trapping
Downward trapping
Upward escape
Downward escape ≈
Side illumination
Side escape =
Varnai & Davies (1999) Hogan & Shonk (2013)
European Centre for Medium-Range Weather Forecasts

6. Idealized calculation: what is the albedo of this scene?
R R* R 0
Surface albedo = 0
Reflectance of each cloud: R
No absorption so transmittance T = 1 – R
R/2
Independent column approximation
Reflectance of 2 non-absorbing clouds
Adding method with R* = 2R/(1+R)
Reflectance of scene
Weighted average Rscene = R/2 + R*/4 = R(1+R/2)/(1+R)
Optically thick limit: Rscene = 3/4
Horizontal transport in regions
Mean reflectance of layer = R/2
Reflectance of scene
Random overlap so apply adding method to mean reflectances: Rscene = 2(R/2)/(1+R/2) = 2R/(2+R)
Optically thick limit: Rscene = 2/3

7. Representing three extremes of “entrapment” in SPARTACUS
We need albedo matrix A at layer interfaces
𝛼 𝑏𝑎
No 3D effects requires matrix to be diagonal
Better approach: compute RMS horizontal migration distance of light paths beneath cloud
2016 SPARTACUS: full horizontal homogenization of radiation under clouds
𝐀= 𝛼 𝑎𝑎 𝛼 𝑏𝑎 𝛼 𝑎𝑏 𝛼 𝑏𝑏 = 𝛼 0 0 𝛼
𝐀= 𝛼/2 𝛼/2 𝛼/2 𝛼/2

8. Computing horizontal migration distances within SPARTACUS
xdiff(z)
ARM cumulus case
At each height z, estimate mean horizontal distance travelled by reflected light xdiff(z) and xdir(z)
Given cloud effective size, we can estimate fraction of radiation that is reflected into the base of another region
First comparison of mean horizontal distances to Monte Carlo shows promise but refinements needed (Najda Villefranque)
direct
diffuse z
xdir(z)
European Centre for Medium-Range Weather Forecasts

9. Broadband testing against Monte Carlo
Howard Barker has run Monte Carlo calculations on 59 varied scenes from Canadian and Met Office models at ~200 m resolution
Minimum entrapment
Computed entrapment
Maximum entrapment
Entrapment changes 3D effect by a factor of three for overhead sun!

10. 3D impact in 4x one-year coupled simulations with ECMWF model
3D effect on LW surface downwelling: +1.6 W m-2
Minimum (no) entrapment
SW surface: +0.1 W m-2
SW TOA: +0.3 W m-2
Computed entrapment
SW surface: +0.8 W m-2
SW TOA: +1.2 W m-2
Maximum entrapment
SW surface: +2.2 W m-2
SW TOA: +3.1 W m-2

11. European Centre for Medium-Range Weather Forecasts
Summary
Shortwave 3D radiative effects traditionally considered to be about transport through cloud sides
Significant biases when sun is low in the sky
Led to development of SPARTACUS solver in ECMWF radiation scheme “ecRad”
Near-cancellation in global impact between side illumination & side escape (~0.2 W m-2)
We find that horizontal transport within clear or cloudy regions has a larger effect
Systematically reduces scene albedo due to the “entrapment” mechanism
Added to SPARTACUS solver and being evaluated using Monte Carlo simulations
Estimate of global SW radiative effect: warming of around +1 W m-2
Remaining questions (see Mark Fielding’s presentation tomorrow)
Can we derive the effective cloud size from observations?
What is the impact of 3D radiative effects in long coupled climate simulations?
What is the relative importance of longwave and shortwave 3D effects?
European Centre for Medium-Range Weather Forecasts