coupling the continuity equation with the physics in the ifs … · 2016. 10. 5. · mass...

Coupling the continuity equation with the physicsin the IFS

(Work in slow progress)

Sylvie MalardelAnna Agusti-Panareda, Peter Bechtold

new continuity equation for conservation of dry mass instead of totalmass

net mass transport in mass flux scheme

Earth System Modelling@ECMWF ECMWF Workshop 2016

Mass conservation

Lagrangian/Eulerian mass conservation

DmDt = 0 or ∂m

∂t = −~u.~∇(m)

Equation for density

m = ρV =⇒ DρDt = −ρ 1

VDVDt = −ρD3

What if an ”unresolved” source of mass?DmDt = S =⇒ Dρ

Dt = −ρ 1V

DVDt + 1

V S

in an hydrostatic model, S need to be in hydrostatic balance,

in a compressible model, S has to be understood by the model as asource of mass and not as a source of volume.


Total mass conservation versus dry mass conservation

Current IFS

DρtDt

= −ρtD3

DρwDt

= −ρwD3 − Sϕ

⇒ DρdDt

= −ρdD3 + Sϕ

total mass

dry mass

total water

mas

s ch

ang

e fr

om

t=

0 (

kg

/m2

)

current IFS with total mass fixer

time (hour)

−5

−4

−3

−2

−1

0

1

2

3

4

5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

New continuity

DρtDt

= −ρtD3 − SϕDρwDt

= −ρwD3 − Sϕ

⇒ DρdDt

= −ρdD3

mas

s ch

ang

e fr

om

t=

0 (

kg

/m2

)

new continuity equation with dry mass fixer

dry mass

total water

total mass

time (hour)

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000


Total mass conservation versus dry mass conservation

In practice:

Current IFS

In the hydrostatic IFS, thanks to thehybrid pressure levels, the continuityequation is reduced to a 2D equationfor the surface hydrostatic pressure:

∂πs∂t

=

[−∫ top

surf

~∇.(~v ∂p∂η

)dη

]

New continuity

Add physics tendency to theequation for the surface pressure,but also in the diagnostic equationsfor ω and η̇ (the vertical velocityused for vertical advection).

Should the physics mass tendency affect all specific variables?

∂(ρtψ)

∂t= −~∇(ρtψ~u) + Sψ

⇓∂ψ

∂t= −~u.~∇ψ − ψ

ρt

(∂ρt∂t

+ ~u.~∇ρt + ρt ~∇(~u)

)+

Sψρt


Error (in %) in the computation of ”dry” mixing ratios oftracers (for example CO2) from specific ratios

Tracer in an explicit cumulonimbus: before/after

0°N/20°W 0°N/15°W 0°N/10°W 0°N/5°W 0°N/0°E 0°N/5°E 0°N/10°E 0°N/15°E 0°N/20°E

200

400

600

800

1000

-0.2

-0.2

-0.2

0.2

0

00

0

0

0 00

0

0

0

0

0

0

0

0 0

0

0.4

0.4

Cross section of Instantaneous moisture flux 20160531 00 step 120 Expver gj2m-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0°N/20°W 0°N/15°W 0°N/10°W 0°N/5°W 0°N/0°E 0°N/5°E 0°N/10°E 0°N/15°E 0°N/20°E

200

400

600

800

1000

0

0

0

0

0

0

0

0

00

0

0

0

0

0

0

Cross section of Instantaneous moisture flux 20160531 00 step 120 Expver gj2m-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Generalisation to a net sub-grid transport of total mass indeep convection parametrisations

HYMACS (Kuell, Gassmann and Bott, 2007), also Grell 3D in WRF

In the grey zone⇒ replace the parametrised compensating subsidence inthe convective column by an ”explicit” 3D subsidence computed by thedynamics.

Net mass advection inside the physics

∂(ρψ)

∂t )conv = −∂(Mu(ψu − ψ)

∂z= −[

∂(Muψu)

∂z+∂(−Muψ)

∂z]

⇓∂(ρψ)

∂t )conv = −∂(Muψu)

∂z

If the compensating subsidence is not parametrised by the convectionscheme ⇒ the dynamics is expected to produce the mass adjustment as aconsequence of the net mass transport by the physics;


MCs case : 3 June 2015, 00UTC, wind at 200hPa

IFS increment computedby 4DVAR dataassimilation.

30°N

40°N

50°N

30°N

40°N

50°N

80°W100°W120°W

80°W100°W120°W

Wednesday 03 June 2015 00 UTC ecmf t+12 VT:Wednesday 03 June 2015 12 UTC 200 hPa U component of wind/V component of wind

200hPa Wind differencebetween simulationswithout and with deepconvection scheme


How does the dynamics react to a parametrised net masstransport?

Academic simulation of a tracer transport by a single column updraft

entrainment

detrainment

constant mass flux

subgrid mixing of a passive tracer (notpart of the total mass, not part of thebuoyancy, unlike moisture)

passive tracer with same initial profile asmoisture

no wind, temperature and moisture fromKlemp and Wilhelmsom,78

only the concentration of traceur + totalmass in the new scheme are transportedby the mass flux (temperature andmoisture of parcels adjust instantaneouslyto the environment, not energeticallycorrect, but only for illustration purpose)

the vertical integral of the masstendencies in the column is zero.



Hydrostatic model,

Small planet, cubic grid, dx = 5 km, dt = 1 min

New mass flux scheme

⇒ change total mass with mass flux tendencies using same modificationswritten for total water tendencies (cf moist/dry mass conservation)



6h. accumulated tendencies for the specific ratio of a passive tracer

Conventional mass flux scheme

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of Forecast logarithm of surface roughness for heat 20160531 00 step 360 Expver gkgt0.0001 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 -0.01 -0.009 -0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 -0.0001

Physics +

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of Forecast albedo 20160531 00 step 360 Expver gkgt0.0001 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 -0.01 -0.009 -0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 -0.0001

Dynamics =

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of Logarithm of surface roughness length for heat 20160531 00 step 360 Expver gkgt0.0001 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 -0.01 -0.009 -0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 -0.0001


0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of Forecast logarithm of surface roughness for heat 20160531 00 step 360 Expver gkgt0.0001 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 -0.01 -0.009 -0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 -0.0001

Physics +

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of Forecast albedo 20160531 00 step 360 Expver gkgt0.0001 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 -0.01 -0.009 -0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 -0.0001

Dynamics =

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000




Non hydrostatic model,

Small planet, cubic, dx = 5 km, dt = 1 min

NH-spectral IFS

Hybrid mass vertical coordinate (πs is the hydrostatic surface pressure)Thermo equation for internal energy instead of enthalpyTwo more equations : q̂ = log( p̂+ππ ) and d4 related to the ”verticaldivergence”.


⇒ mass flux tendencies projected on NH pressure departuredp̂ = d(p − π) and temperature dT such as dθ = 0 (Kuell et al, 2007)



Wind after 1 k

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of U component of wind 20160531 00 step 60 Expver gki00.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0.05

H

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000


NH, 1ρ∂ρ∂t → dp̂, dT (dθ = 0)

Acc. tracer tend. after 6 hours

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of Logarithm of surface roughness length for heat 20160531 00 step 360 Expver gki00.0001 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 -0.01 -0.009 -0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 -0.0001

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000




Non-hydrostatic model



⇒ change total mass with mass flux tendencies as in hydrostatic model(physics tendency for hydrostatic surface pressure, ω and η̇)



Wind after 1 k

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of U component of wind 20160531 00 step 60 Expver gki00.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.3 0.325 0.35 0.375 0.4 0.425 0.45 0.475 0.5 -0.5 -0.475 -0.45 -0.425 -0.4 -0.375 -0.35 -0.325 -0.3 -0.275 -0.25 -0.225 -0.2 -0.175 -0.15 -0.125 -0.1 -0.075 -0.05 -0.025

Hydrostatic

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of U component of wind 20160531 00 step 60 Expver gki00.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.3 0.325 0.35 0.375 0.4 0.425 0.45 0.475 0.5 -0.5 -0.475 -0.45 -0.425 -0.4 -0.375 -0.35 -0.325 -0.3 -0.275 -0.25 -0.225 -0.2 -0.175 -0.15 -0.125 -0.1 -0.075 -0.05 -0.025

NH, 1ρ∂ρ∂t → dπ


0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000


Hydrostatic

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000


NH



Non hydrostatic model,

Small planet, cubic, dx = 500 m, dt = 5 s


⇒ mass flux tendencies projected as in hydrostatic model (physicstendency for hydrostatic surface pressure, ω and η̇)



Wind after 1 k

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of U component of wind 20160531 00 step 720 Expver glds0.0025 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 -0.05 -0.045 -0.04 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 -0.0025

Hydrostatic

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of U component of wind 20160531 00 step 720 Expver glds0.0025 0.005 0.0075 0.01 0.0125 0.015 0.0175 0.02 0.0225 0.025 0.0275 0.03 0.0325 0.035 0.0375 0.04 0.0425 0.045 0.0475 0.05 -0.05 -0.0475-0.045-0.0425 -0.04 -0.0375-0.035-0.0325 -0.03 -0.0275-0.025-0.0225 -0.02 -0.0175-0.015-0.0125 -0.01 -0.0075-0.005-0.0025

NH


0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of Logarithm of surface roughness length for heat 20160531 00 step 2160 Expver glds0.0001 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 -0.01 -0.009 -0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 -0.0001

Hydrostatic

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of Logarithm of surface roughness length for heat 20160531 00 step 2160 Expver glds0.0001 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 -0.01 -0.009 -0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 -0.0001

NH



Hydrostatic model



⇒ change total mass with mass flux tendencies as in hydrostatic model(physics tendency for hydrostatic surface pressure, ω and η̇)

+

∂ψ

∂t= −~u.~∇ψ − ψ

ρt

(∂ρt∂t

+ ~u.~∇ρt + ρt ~∇(~u)

)+

Sψρt



Wind after 1 k

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000


H, Ref

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000

Cross section of U component of wind 20160531 00 step 60 Expver gkgt0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0.05

H, extra terms


0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000


H, Ref

0.28°N/4°W 0.28°N/2°W 0.28°N/0°E 0.28°N/2°E 0.28°N/4°E

100

200

500

1000


H, extra terms


Summary

It is possible to conserve dry mass instead of total mass in thehydrostatic IFS. Neutral in term of scores, improve ”dry” mixingratios for tracers.

It is also possible to parametrise a net sub-grid mass transport (e.g.mass flux scheme) and let the dynamics do the compensatingsubsidence. The same solution works for both the hydrostatic and NHIFS.

But it is dangerous to play with mass!


coupling the continuity equation with the physics in the ifs … · 2016. 10. 5. · mass...

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