Resolved and sub-grid-scale transport in CAM5-FVPeter Hjort Lauritzen and Julio Bacmeister (NCAR) Resolved and sub-grid-scale transport in CAM5-FV July 11, 2012 8 / 23 Finite-volume

Resolved and sub-grid-scale transport in CAM5-FV

Peter Hjort Lauritzen and Julio Bacmeister

Atmospheric Modeling and Predictability Section (AMP)Climate and Global Dynamics Division (CGD)

NCAR Earth System Laboratory (NESL)National Center for Atmospheric Research (NCAR)

ACD group meeting

Peter Hjort Lauritzen and Julio Bacmeister (NCAR) Resolved and sub-grid-scale transport in CAM5-FV July 11, 2012 1 / 23

Motivation

Sometime in May Steven Massie and William Randel came into my visitor office in ACD and askedme: ‘How do tracers get moved around in CAM?’

This Monday Laura Pan said: ‘I have a lot of questions regarding how convection is represented inmodels like CAM/WACCM?’

The question I am going to address:

If you add a tracer to CAM-FV (with CAM5 physics), how is it ‘moved around’ both grid-scale andsub-grid-scale?


CAM5 process ‘flow chart’

Physical processes on tracers

‘Resolved’ scale transport (Lin and Rood, 1996)

Deep convective transport (Zhang and McFarlane, 1995; Neale et al., 2008)

Shallow convective transport (Park and Bretherton, 2009)

Turbulent transport (Park and Bretherton, 2009)

Scavenging through wet deposition (only for aerosols not trace gases)

Chemistry (for reactive tracers)

Fig. 1. A diagram illustrating interactions among various physics and dynamic processesin CAM5. Thick arrows denote a sequence of process-splitting at each time step with aflow of normal grid-mean state variables of ! = T, qv, ql, qi, u, v, "i, "c where "i and "c are themass and number concentrations of 15 interstitial (i.e., outside of cloud liquid droplets andice crystals) and stratus-borne aerosol species, respectively. Thin arrows denote the flowsof additional variables (TKE:Turbulent Kinetic Energy, PBLH:PBL top height, Kh:eddydi!usivity for heat and moisture, (A, ql, qi, nl, ni, rl, ri) are cloud fraction and condensatemass, number concentration and radius of cloud liquid droplets and ice crystals, subscriptsdp, sh, st, tot denotes deep cumulus, shallow cumulus, stratus, total clouds, overbar and hatdenote grid-mean and in-cloud average, respectively, QR,L is LW radiative cooling rate, (P,E)are precipitation production and evaporation, M is convective mass flux, (#!

PBL, q!v,PBL) are

convective excess of temperature and moisture within PBL), LHF and SHF are surfacelatent and sensible fluxes, and $x, $y are surface wind stresses. TMS is implicit turbulentmountain stress parameterized as a function of sub-grid variation of topography and grid-mean static stability at the lowest model interface above surface. CAM5 prints most of clouddiagnostics at the beginning of radiation scheme.

69

Figure from Park et al. (2012) - process split advancement of tendencies


Vertical coordinate

CAM-FV uses a Lagrangian (‘floating’) vertical coordinate ξ so that

dξ

dt= 0,

i.e. vertical surfaces are material surfaces (no flow across them).

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Figure shows ‘usual’ hybrid σ−p vertical coordinate η(ps , p)(where ps is surface pressure):

η(ps , p) is a monotonic function of p.

η(ps , ps) = 1

η(ps , 0) = 0

η(ps , ptop) = ηtop .

Boundary conditions are:dη(ps ,ps )

dt= 0

dη(ps ,ptop)

dt= ω(ptop) = 0


Vertical coordinate

CAM-FV uses a Lagrangian (‘floating’) vertical coordinate ξ so that

dξ

dt= 0,

i.e. vertical surfaces are material surfaces (no flow across them).

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Figure:

Set ξ = η at time tstart (black lines).

For t > tstart the vertical levels deform as they movewith the flow (blue lines).

To avoid excessive deformation of the vertical levels(non-uniform vertical resolution) the prognosticvariables defined in the Lagrangian layers ξ areperiodically remapped (= conservative interpolation)back to the Eulerian reference coordinates η (more onthis later).


Adiabatic frictionless equations of motion using Lagrangian verticalcoordinates

Assuming a Lagrangian vertical coordinate the hydrostatic equations of motion integrated over alayer can be written as

mass air:∂(δp)

∂t= −∇h · (~vhδp) ,

mass tracers:∂(δpq)

∂t= −∇h · (~vh qδp) ,

horizontal momentum:∂~vh

∂t= − (ζ + f )~k × ~vh −∇hκ−∇pΦ,

thermodynamic:∂(δpΘ)

∂t= −∇h · (~vhδpΘ)

where δp is the layer thickness, ~vh is horizontal wind, q tracer mixing ratio, ζ vorticity, f Coriolis,κ kinetic energy, Θ potential temperature. The momentum equations are written in vectorinvariant form.


Adiabatic frictionless equations of motion using Lagrangian verticalcoordinates

Assuming a Lagrangian vertical coordinate the hydrostatic equations of motion integrated over alayer can be written as

mass air:∂(δp)

∂t= −∇h · (~vhδp) ,

mass tracers:∂(δpq)

∂t= −∇h · (~vh qδp) ,

horizontal momentum:∂~vh

∂t= − (ζ + f )~k × ~vh −∇hκ−∇pΦ,

thermodynamic:∂(δpΘ)

∂t= −∇h · (~vhδpΘ)

The equations of motion are discretized using an Eulerian finite-volume approach.


Finite-volume discretization of continuity equation

A

δp

∆

Integrate the flux-form continuity equation horizontally over a control volume:

∂

∂t

∫∫Aδp dA = −

∫∫A∇h (~vhδp) dA, (1)

where A is the horizontal extent of the control volume. Using Gauss’s divergence theorem for theright-hand side of (1) we get:

∂

∂t

∫∫Aδp dA = −

∮∂Aδp ~v · ~n dA, (2)

where ∂A is the boundary of A and ~n is outward pointing normal unit vector of ∂A.



A

δp

∆

Integrate the flux-form continuity equation horizontally over a control volume:

∂

∂t

∫∫Aδp dA = −

∫∫A∇h (~vhδp) dA, (1)

where A is the horizontal extent of the control volume. Using Gauss’s divergence theorem for theright-hand side of (1) we get:

∂

∂t

∫∫Aδp dA = −

∮∂Aδp ~v · ~n dA, (2)

Right-hand side of (2) represents the instantaneous flux of mass through the vertical faces of thecontrol volume.



∂

∂t

∫∫Aδp dA = −

∮∂Aδp ~v · ~n dA. (3)

Discretize (3) in space

∆A∂δp

∂t= −

4∑f =1

[〈δp~v〉 · ~n∆`]f , (4)

where

δp = horizontal mean value of δp

~nf = unit vector normal to the f th cell face pointing outward

∆`f is the length of the face in question

~vf = instantaneous values of ~v at the cell face f

brackets represent averages in either λ or θ direction over the cell face.



∂

∂t

∫∫Aδp dA = −

∮∂Aδp ~v · ~n dA. (3)

Discretize (3) in space

∆A∂δp

∂t= −

4∑f =1

[〈δp~v〉 · ~n∆`]f , (4)

and integrate (4) over the time-step ∆tdyn

∆A δpn+1

= ∆A δpn −∆tdyn

4∑f =1

[〈δp~v〉 · ~n∆`

]f, (5)

where n is the time-level index and the double-bar refers to the time average over ∆tdyn.

Each term in the sum on the right-hand side of (6) represents the mass transported through oneof the four vertical control volume faces into the cell during one time-step (graphical illustrationon next page).


Finite-volume discretization of continuity equation: Tracking mass

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Lagrangian form

Eul

eria

n fl

ux f

orm

The yellow areas are ‘swept’ through the control volume faces during one time-step. The greyarea is the corresponding Lagrangian area (area moving with the flow with no flow through itsboundaries that ends up at the Eulerian control volume after one time-step). Black arrows showparcel trajectories.

Equivalence between Eulerian flux-form and Lagrangian form!



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Lagrangian form

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n fl

ux f

orm

Until now everything has been exact. How do we approximate the fluxes numerically?

In CAM-FV the Lin and Rood (1996) scheme is used which is a dimensionally split scheme(that is, rather than estimating the boundaries of the yellow areas and integrate over them,fluxes are estimated by successive applications of one-dimensional operators in eachcoordinate direction).



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Lagrangian form

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orm


(before showing equations for Lin and Rood (1996) scheme) What is the effectiveLagrangian area associated with the Lin and Rood (1996) scheme?



Figure: Red lines define boundary of exact Lagrangiancell for a special case with deformational, rotational anddivergent wind field. Blue colors is Lagrangian cellassociated with the Lin and Rood (1996) scheme. Darkblue shading weights integrated mass with 1 and lightblue shading weights integrated mass with 1/2. SeeMachenhauer et al. (2009) for details.


(before showing equations for Lin and Rood (1996) scheme) What is the effectiveLagrangian area associated with the Lin and Rood (1996) scheme?


The Lin and Rood (1996) advection scheme

δpn+1

= δpn

+ Fλ[

12

(δp

n+ f θ(δp

n))]

+ F θ[

12

(δp

n+ f λ(δp

n))],

where

Fλ,θ = flux divergence in λ or θ coordinate direction

f λ,θ = advective update in λ or θ coordinate direction



δpn+1

= δpn

+ Fλ[

12

(δp

n+ f θ(δp

n))]

+ F θ[

12

(δp

n+ f λ(δp

n))],

Figure: Graphical illustration of flux-divergence operator Fλ. Shaded areas show cell averagevalues for the cell we wish to make a forecast for and the two adjacent cells.



δpn+1

= δpn

+ Fλ[

12

(δp

n+ f θ(δp

n))]

+ F θ[

12

(δp

n+ f λ(δp

n))],

∆ t U* ∆west t U* east

u∗east/west

are the time-averaged winds on each face (more on how these are obtained later).

Fλ is proportional to the difference between mass ‘swept’ through east and west cell face.

f λ = Fλ +< δp >∆tdynD, where D is divergence.

On Figure we assume constant sub-grid-cell reconstructions for the fluxes.



δpn+1

= δpn

+ Fλ[

12

(δp

n+ f θ(δp

n))]

+ F θ[

12

(δp

n+ f λ(δp

n))],

east t U* ∆west t U* ∆

Higher-order approximation to the fluxes:

Piecewise linear sub-grid-scale reconstruction (van Leer, 1977): Fit a linear function usingneighboring grid-cell average values with mass-conservation as a constraint (i.e. area underlinear function = cell average.).



δpn+1

= δpn

+ Fλ[

12

(δp

n+ f θ(δp

n))]

+ F θ[

12

(δp

n+ f λ(δp

n))],




Piecewise parabolic sub-grid-scale reconstruction (Colella and Woodward, 1984): Fitparabola using neighboring grid-cell average values with mass-conservation as a constraint.Note: Reconstruction is C0 across cell edges.



δpn+1

= δpn

+ Fλ[

12

(δp

n+ f θ(δp

n))]

+ F θ[

12

(δp

n+ f λ(δp

n))],




Piecewise parabolic sub-grid-scale reconstruction (Colella and Woodward, 1984): Fitparabola using neighboring grid-cell average values with mass-conservation as a constraint.Note: Reconstruction is continuous at cell edges.

Reconstruction function may ‘over’- or ‘undershoot’ which may lead to unphysical and/oroscillatory solutions. Use limiters to render reconstruction function shape-preserving.



δpn+1

= δpn

+ Fλ[

12

(δp

n+ f θ(δp

n))]

+ F θ[

12

(δp

n+ f λ(δp

n))],

Advantages:

Inherently mass conservative (note: conservation does not necessarily imply accuracy!).

Formulated in terms of one-dimensional operators.

Preserves a constant for a non-divergent flow field (if the finite-difference approximation todivergence is zero).

Preserves linear correlations between trace species (if shape-preservation filters are notapplied)

Has shape-preserving options. Note: Since the Lin and Rood (1996) is dimensionally splitand the shape-preserving filters are applied along the coordinate axis tiny over-/under-shootmay be present in the traverse direction.


Free-stream preserving ‘super-cycling’ of tracers with respect to air ρ

Simply solving the tracer continuity equation for qδpn+1

using ∆ttrac will lead to inconsistencies.Why?

Continuity equation for air δp∂δp

∂t+∇ · (δp ~vh) = 0, (6)

and a tracer with mixing ratio q

∂(δp q)

∂t+∇ · (δp q ~vh) = 0, (7)

For q = 1 equation (7) reduces to (6). If this is satisfied in the numerical discretizations, thescheme is ‘free-stream’ preserving.

Solving (7) with q = 1 using ∆ttrac will NOT produce the same solution as solving (6) nspltrac

times using ∆tdyn!


Graphical illustration of ‘free stream’ preserving transport of tracers

Assume no flux through east cell wall.

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n

time

flow direction

ρ

ρ

Solve continuity equation for air ρ = δp together with momentum and thermodynamicsequations.




tn

ρ

u ∆

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n+1/4

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n

time

flow direction

ρ

ρ





u

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ρn+2/4

n+1/4∆t tn

ρ

u ∆

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n+1/4

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n

time

flow direction

ρ

ρ


Repeat ksplit times




∆t

ρ��

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n+3/4

un+2/4

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u

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ρn+2/4

n+1/4∆t tn

ρ

u ∆

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n+1/4

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n

time

flow direction

ρ

ρ


Repeat ksplit times




��

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t

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u ∆n+3/4

n+1ρ

∆t

ρ��

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n+3/4

un+2/4

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u

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ρn+2/4

n+1/4∆t tn

ρ

u ∆

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n+1/4

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n

time

flow direction

ρ

ρ


Repeat ksplit times




t∆un+3/4

n+1ρ��

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n+3/4

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n+2/4

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u

ρ

∆t

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ρn+2/4

n+1/4u ∆t

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ρ

tnu

n+1/4

∆

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ρ

n

flow direction

time

ρ��

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Repeat ksplit times

Brown area = average flow of mass through cell face.

Compute time-averaged value of q across brown area using Lin and Rood (1996) scheme:

< q >.

Forecast for tracer is: < q >×∑kspliti=1 δpn+i/ksplit

Yields ‘free stream’ preserving solution!


‘Resolved’ scale transport: FYI

We are ‘switching’ dynamical core

November release of CESM → CAM-SE (Spectral Elements)

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In default CAM-SE n = 3(polynomials of degree 3;4th-order accurate)

Note: On Figure n = 4

Aside: Ongoing DOE project (PI: J.-F. Lamarque)

Among goals: Evaluate CAM-SE as a transport model (Lauritzen and Thuburn, 2012; Lauritzenet al., 2012), add specified dynamics option to CAM-SE and investigate higher-order dynamics-chemistry coupling.


Assesing ‘accuracy’ of tracer transport (Rasch et al., 2006)

Test case setup by Rasch et al. (2006) designed to assess transport in region of atmosphere(LOW) strongly influenced by sub-grid scale transport processes (convection and boundarylayer processes),(HIGH) less strongly influenced by sub-grid scale processes; large role being played byresolved scale dynamics(MID) ‘in the middle’ !

smaller regions poleward of about 60° in the other twomodels. There are also significant differences in thelower tropospheric polar regions. While the contours ofFig. 3 are linear in the light and middle-gray shadedregions of the figure, they are approximately logarith-mic in the white and dark-gray shaded regions of thefigure. In light of this, one can see that the concentra-tion near the surface is a factor of 2 to 4 lower in the FVcore than either of the other two cores, highlighting thereduced vertical mixing in the FV core. Simulationsusing tracer MID produced results consistent with theLOW and HIGH tracers; for this reason we have omit-ted a discussion of those results.

b. Radon

Radon has many virtues as a tracer of air recently atthe surface (e.g., Liu et al. 1984). It has a short radio-active lifetime (half-life of 3.8 days), is thought to havea relatively uniform source, has low reactivity withother atmospheric constituents and is insoluble. Itsmain disadvantage is that it occurs at very low concen-trations in air (mixing ratios of order 1 part in 1020) and

so direct measurements are difficult to make rapidly.Nevertheless it has frequently been used as a metric forevaluating atmospheric models (Feichter and Crutzen1990; Jacob and Prather 1990; Genthon and Armen-gaud 1995; Jacob et al. 1997; Rasch et al. 2000)

We have employed a relatively standard scenario forour radon simulations (see, e.g., Jacob et al. 1997) byemploying uniform emissions of 1 atom cm!2 s!1 overall land masses equatorward of 60°. The model runswere initialized with a concentration of zero, run for10 yr, and averaged for the last 5 yr to produce thediagnostics used here.

Figure 4 shows some zonally averaged results. Be-cause of the very strong stratification of radon withaltitude (see bottom center panel showing the spectralT42 simulation), and the relatively small differences inthe simulation, we have chosen to compare the simula-tions in terms of the ratio of zonal averages for eachcase with respect to the spectral T42 simulation. Theupper left panel shows the ratio of the SLD simulationto the control; 20%–30% more radon is seen in theupper troposphere in the SLD simulation. As will be

FIG. 1. Initial conditions for (a) HIGH, (b) MID, and (c) LOW tracer.

1 JUNE 2006 R A S C H E T A L . 2247



LOW tracer (30 day simulation)

In the tropics: Rapid mixing between surface and tropopause

Subtropical subsidence region: Low values of tracer mixing ratio

Midlatitudes-polar regions: Broader mixing

Quantitative differences between models: FV shows steeper gradient between tropics andsubtropics; (day 30) gradient between low and high mixing ratio in subtropics is 2x higher!

seen in other tracers, this signature is consistent with areduced cross-tropopause exchange of the SLD com-pared to the spectral core. Less transport of radon intothe stratosphere results in higher concentrations in thetroposphere. The ratio of radon in these two simula-tions drops to 0.4 as one approaches the model top,

suggesting that transport processes operate more slowlyin the SLD core than the spectral core in the strato-sphere. There is little difference in the middle andlower troposphere between these two cases, where mix-ing is quite rapid and strongly controlled by the sub-grid-scale parameterizations.

FIG. 3. As in Fig. 2 but for HIGH tracer.

FIG. 2. Zonal averaged mixing ratio for days (left) 15 and (right) 30 for LOW tracer: (a) spectral, (b) semi-Lagrangian, and (c)finite volume.

2248 J O U R N A L O F C L I M A T E VOLUME 19



HIGH tracer (let the model evolve for 30 days):

Subtropical features seen in LOW tracer also in HIGH tracer

Substantial differences in mixing in the the upper tropospheric polar region: FV core haspreserved initial gradient much more strongly

seen in other tracers, this signature is consistent with areduced cross-tropopause exchange of the SLD com-pared to the spectral core. Less transport of radon intothe stratosphere results in higher concentrations in thetroposphere. The ratio of radon in these two simula-tions drops to 0.4 as one approaches the model top,

suggesting that transport processes operate more slowlyin the SLD core than the spectral core in the strato-sphere. There is little difference in the middle andlower troposphere between these two cases, where mix-ing is quite rapid and strongly controlled by the sub-grid-scale parameterizations.

FIG. 3. As in Fig. 2 but for HIGH tracer.

FIG. 2. Zonal averaged mixing ratio for days (left) 15 and (right) 30 for LOW tracer: (a) spectral, (b) semi-Lagrangian, and (c)finite volume.

2248 J O U R N A L O F C L I M A T E VOLUME 19


Deep convection


PBL, q!v,PBL) are


69

Figure from Park et al. (2012)


Deep convection scheme - schematic (Washington and Parkinson, 2005)

Simple model of convective cloud:(ignoring entrainment, increased buoyancy due to T & q differences between cloud core updrafts and compensating downdrafts outside cloud, ...)

parcel, say with 50% relative humidity, near surface starts to rise, say from strong heating→ parcel cools at approximately dry adiabatic lapse rate (9.8 K/km−1)

assume the parcel does not mix (i.e. water vapor content remains the same): no entrainment

as the parcel rises it continues to cool → it can hold less water → relative humidity increases

when reaching 100% relative humidity the parcel saturates:parcel has reached Lifting Condensation Level (LCL)

from LCL and upward the parcel cools less rapidly (latent heat release of condensation orfusion releases heat): 5-6 K/km−1

at some point well above LCL the parcel is cooler than the environmental air (stops rising -top of cloud); however, buoyancy forces and the parcels upward momentum can make thecloud extend above the cross-over point.

Simple model ignores many important processes: entrainment, buoyancy due to updraft-downdraftdifferences, ...


Deep convection scheme - schematic

Consider a model grid cell with area ∆A (typical scale 100km) with several deep convective towers

There is a lot going on sub-grid-scale:

updrafts, downdraft, entrainment, detrainment, condensation, evaporation, ...

What we know is the cell-averaged model state within that grid cell: (T , q, u, v ,P, ....)What the parameterization should give us:

∂T

∂t= FT (T , q, u, v ,P, ....) (8)

∂q

∂t= Fq(T , q, u, v ,P, ....) (9)



Ensemble plume approach (Arakawa and Schubert, 1974)

Ensemble of convective updrafts and associated saturated downdrafts exist whenever the at-mosphere is conditionally unstable in the lower troposphere → effectively the model ‘sees‘ one‘ensemble column’ of convection

Among the assumptions are:

no tilting of the ‘ensemble’ convective tower

area of ‘ensemble’ convective tower ∆Ac << ∆A→ qe = q



‘Ensemble’ convective tower may span many vertical levels (even entire column)



Consider ‘ensemble plume’ in one layer:




Processes represented:

Mu : mass flux of ‘ensemble’ updraft defined at model layer interfaces






Md : mass flux of ‘ensemble’ downdraft defined at model layer interfaces






Md : mass flux of ‘ensemble’ downdraft defined at model layer interfaces

Ex , x = u, d : entrainment rate of environmental air associated with updrafts anddowndrafts, respectively (defined at layer centers)




Processes represented:Mu : mass flux of ‘ensemble’ updraft defined at model layer interfacesMd : mass flux of ‘ensemble’ downdraft defined at model layer interfacesEx , x = u, d : entrainment rate of environmental air associated with updrafts anddowndrafts, respectively (defined at layer centers)Dx , x = u, d : detrainment rate of ‘plume air’ associated with updrafts and downdrafts,respectively



...+=∂+⋅∇+∂ Qwsss zt ρρρ u DEawa czt −=∂+∂ ρρ

cccczct aQDssEsawas +−=∂+∂ ~ρρ...+!!∂−=∂+⋅∇+∂ swQswss zzt ρρρρ u

Thermodynamic-Equa1on- Plume/cloud-cont.-equa1on-

Grid-box-average-therm.-equa1on- Plume-therm.-equa1on-

Latent-hea1ng- Entrainment-Detrainment-

Cloud-areal-frac1on-

E

D

environ.-subsidence-- up

draB

-

Figure courtesy of J. Bacmeister (NCAR)



CAM5 deep convection scheme

The challenge of cloud researchers is to determine how nature decides which process dominatesunder different large-scale environmental conditions.

CAM5 deep convection scheme is a simplification of Arakawa and Schubert (1974) for large-scalemodels (Zhang and McFarlane, 1995) with modified momentum transport by Richter and Rasch(2008) and a modified dilute plume calculation following Raymond and Blyth (1992)

The details of how the deep convection scheme determines Mu , Md , Eu , Du , Ed , Eu is beyond thescope of this talk (local experts: S. Park, R. Neale, J. Bacmeister), i.e. assume that mass-fluxes,entrainment and detrainment rates are given!


Deep convective tracer transport

Convection is an effective way of mixing tracers in the vertical (e.g. Mahowald et al., 1995; Collinset al., 1999), e.g., convective updrafts can transport a tracer from the surface to the upper tropo-sphere on time scales of O(1h).

Steady state continuity equation for ‘bulk’ updraft mixing ratio ϕu

∂ (Muϕu)

∂p= Euϕe − Duϕu (8)

where

Mu is mass-flux at layer interfaces

ϕe mixing ratio of environment(in CAM: ϕe = ϕ; i.e. we assume that area of updraft <<grid cell area)

Eu and Du are entrainment/detraiment rates for the updrafts.

Solve (8) for ϕu

u

E φe

E φe

Mu

φu

Mu

φu

D φu

D φ

CAM5 subroutine convtran in physics/cam/zm conv.F90 file




Steady state continuity equation for ‘bulk’ downdraft mixing ratioϕd

∂ (Mdϕd )

∂p= Edϕe − Ddϕd (8)

where

Md is mass-flux at layer interfaces


Ed and Dd are entrainment/detraiment rates for thedowndrafts.

Solve (8) for ϕd

d

E φe

E φe

M φdd

M φdd

D φd D φ





Steady state continuity equation for ‘bulk’ downdraft mixing ratioϕd

∂ (Mdϕd )

∂p= Edϕe − Ddϕd (8)

where

Md is mass-flux at layer interfaces


Ed and Dd are entrainment/detraiment rates for thedowndrafts.

Solve (8) for ϕd

∂ϕ

∂t=

∂

∂p[Mu (ϕu − ϕ) + Md (ϕd − ϕ)] (9)

d

E φe

E φe

M φdd

M φdd

D φd D φ



How much mixing does deep convection do?

Use Rasch et al. (2006) transport test setup: day 0 , zonal average

(left) no deep convective transport of tracers - there is deep convective transport of water variables!, (middle) default, (right) difference











Use Rasch et al. (2006) transport test setup: day 1 , along Equator



Shallow convection


PBL, q!v,PBL) are


69



Turbulent mixing


PBL, q!v,PBL) are


69



Turbulent diffusion

Given vertical profile of eddy diffusion coefficient K(p):

∂ϕ

∂t=

∂

∂p

[K(p)

∂ϕ

∂p

](10)

Contrary to convective tracer transport turbulent diffusion is a local process!

t=0 t=T


This completes the ‘cycle’


PBL, q!v,PBL) are


69



References I

Arakawa, A. and Schubert, W. H. (1974). Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. Atmos. Sci., 31:674–701.

Colella, P. and Woodward, P. R. (1984). The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys., 54:174–201.

Collins, W., Stevenson, D. S., Johnson, C., and Derwent, R. G. (1999). Role of convection in determining the budget of odd hydrogen in the uppertroposphere. J. Geophys. Res., 104:26 927–26 941.

Lauritzen, P. and Thuburn, J. (2012). Evaluating advection/transport schemes using interrelated tracers, scatter plots and numerical mixing diagnostics.Quart. J. Roy. Met. Soc., 138(665):906–918.

Lauritzen, P. H. (2007). A stability analysis of finite-volume advection schemes permitting long time steps. Mon. Wea. Rev., 135:2658–2673.

Lauritzen, P. H., Skamarock, W. C., Prather, M. J., and Taylor, M. A. (2012). A standard test case suite for 2d linear transport on the sphere. Geo. Geosci.Model Dev. Discuss., 5:189–228.

Lin, S. J. and Rood, R. B. (1996). Multidimensional flux-form semi-Lagrangian transport schemes. Mon. Wea. Rev., 124:2046–2070.

Machenhauer, B., Kaas, E., and Lauritzen, P. H. (2009). Finite volume methods in meteorology, in: R. Temam, J. Tribbia, P. Ciarlet (Eds.), Computationalmethods for the atmosphere and the oceans. Handbook of Numerical Analysis, 14. Elsevier, 2009, pp.3-120.

Mahowald, N., Rasch, P. J., and Prinn, R. G. (1995). Cumulus parameterizations in chemical transport models. J. Geophys. Res., 100:26 173–26 189.

Nair, R. D., Levy, M. N., and Lauritzen, P. H. (2011). Emerging numerical methods for atmospheric modeling, in: P.H. Lauritzen, R.D. Nair, C. Jablonowski,M. Taylor (Eds.), Numerical techniques for global atmospheric models. Lecture Notes in Computational Science and Engineering, Springer, 80.

Neale, R. B., Richter, J. H., and Jochum, M. (2008). The impact of convection on ENSO: From a delayed oscillator to a series of events. J. Climate,21:5904–5924.

Park, S. and Bretherton, C. S. (2009). The university of washington shallow convection and moist turbulence schemes and their impact on climatesimulations with the community atmosphere model. J. Climate, 22:3449–3469.

Park, S., Bretherton, C. S., and Rasch, P. J. (2012). Global cloud simulation in the community atmosphere model, 5. J. Climate. in prep.

Rasch, P. J., Coleman, D. B., Mahowald, N., Williamson, D. L., Lin, S. J., Boville, B. A., and Hess, P. (2006). Characteristics of atmospheric transportusing three numerical formulations for atmospheric dynamics in a single GCM framework. J. Climate, 19:2243–2266.

Raymond, D. and Blyth, A. (1992). Extension of the stochastic mixing model to cumulonimbus clouds. J. Atmos. Sci., 49(21):1968–1983.

Richter, J. H. and Rasch, P. J. (2008). Effects of convective momentum transport on the atmospheric circulation in the community atmosphere model,version 3. J. Climate, 21(7):1487–1499.

van Leer, B. (1977). Towards the ultimate conservative difference scheme. IV: A new approach to numerical convection. J. Comput. Phys., 23:276–299.

Washington, W. M. and Parkinson, C. L. (2005). Introduction To Three-dimensional Climate Modeling. University Science Books.

Zhang, G. and McFarlane, N. (1995). Sensitivity of climate simulations to the parameterization of cumulus convection in the canadian climate centergeneral-circulation model. ATMOSPHERE-OCEAN, 33(3):407–446.


Documents

Resolved and sub-grid-scale transport in CAM5-FVPeter Hjort Lauritzen and Julio Bacmeister (NCAR) Resolved and sub-grid-scale transport in CAM5-FV July 11, 2012 8 / 23 Finite-volume