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Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Convection problem in a single column - modellingand numerical solutions
B. Cheng, M.J.P. Cullen, J. Norbury, M.R. Turner
Surrey, Oxford, Met Office
Maths Foresees General Assembly05th September 2016
Update of work from Turing Gateway Study Group 2015with O. Bokhove, A. Dedner, J.G. Esler, J. Vanneste
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Outline
1 Background
2 Model Approach
3 Lagrangian Form and Conservation Laws
4 Numerical Algorithm
5 Results
6 Conclusions
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
The problem of convection in a moist atmosphere is a hardproblem.The Met Office’s subroutine for convection used in weatherand climate models shows disagreement with physical data(systematically in some regions).The aim of this work is to understand and suggestimprovements to this subroutine.This problem was set at Turing Gateway Study GroupMeeting, Cambridge (2015).This report is online with a link from Maths Foresees website.
http://www.turing-gateway.cam.ac.uk/documents/mfsg-sept2015/Met%20Office%20Solutions%20Report.pdfhttps://arxiv.org/abs/1608.05245
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Outline
1 Background
2 Model Approach
3 Lagrangian Form and Conservation Laws
4 Numerical Algorithm
5 Results
6 Conclusions
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Met Office code is divided into a 3D grid with column cells,where each cell is effectively a vertical column of air.At each time-step the air needs to be adjusted and arrangedsuch that it is buoyantly stable.In the atmosphere buoyancy is an increasing function ofpotential temperature θ, hence need to ensure ∂θ
∂z ≥ 0 forlocal buoyant stability of the column before performing thenext time-step.In doing this adjustment we might cause some moisture tocondense.
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Consider a column of air with z ∈ [0, ztop]. ztop is above thetop of tropopause ≈ 10− 20km. z =pseudo-height.
Make Boussinesq approximation ∂w∂z = 0, w =vertical column
velocity.Then conservation of thermal energy and moisture give
∂θ
∂t + w ∂θ
∂z = −r(z) + QL,
∂q∂t + w ∂q
∂z = −QLL ,
q is specific humidity of air. r is radiative cooling which weset to zero.
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
The difficulty with this convection problem lies in the step likenonlinearity in QL.
QL ={−LDQsat
Dt if q = Qsat and DQsatDt < 0,
0 if q < Qsat or DQsatDt ≥ 0 .
Qsat(θ, z) denotes the saturation specific humidity and weapproximate it by
Qsat(θ, z) = Q0 exp (r [θ − βz − θPBL]) .
Q0, r and β are fitting parameters to the true form of Qsatand θPBL = θ(0).Could solve PDEs via usual discrete PDE methods, but this istough and usually does not determine the true solution. Herethe solution is not a classical smooth solution.
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Outline
1 Background
2 Model Approach
3 Lagrangian Form and Conservation Laws
4 Numerical Algorithm
5 Results
6 Conclusions
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Instead of Eulerian approach, consider Lagrangianapproach.Divide column into N equally sized parcels of height ∆z .For unsaturated (q < Qsat) or dry parcels (q = 0), PDEsbecome
DθDt = Dq
Dt = 0.
Hence θ and q are conserved. Thus dry problem isstraightforward and unique solution is θ arrangedmonotonically in z increasing (with mass conserved).
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Saturated parcels
For a saturated parcel q = Qsat.Adding combination of Eulerian equations to remove QL (withr(z) ≡ 0) leads to
DDt (θ + Lq) = 0,
Therefore for each parcel label A = 1, ..,N the quantity
θA+LqA = θ(zA(t), t)+Lq(zA(t), t) = θ(zA(0), 0)+Lq(zA(0), 0) = θMA
is conserved. θMA is the moist potential temperature.
This says that the parcel A when it ascends in the column, itdoes so along the moist adiabat. This moist adiabat isdifferent for each parcel.
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
z
θ
parcel A
q < Qsat
q = Qsat
Therefore the temperature of a saturated parcel (q = Qsat) asit ascends in the column is found by solving
θA + LQsat(θA, z) = θMA .
So as a parcel rises θA increases and qA decreases to conserveθM
A when the parcel is ascending along the moist adiabat.
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Outline
1 Background
2 Model Approach
3 Lagrangian Form and Conservation Laws
4 Numerical Algorithm
5 Results
6 Conclusions
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
We wish to construct a numerical algorithm which rearrangesthe air parcels in the column, subject to the conservation laws(air mass, moist potential temperature).We expect there to be multiple possible solutions to the moistconvection problem (dry problem is unique). We seek analgorithm which converts the maximum amount of condensedmoisture into latent heat for the parcel (condensed moisturethen leave the problem).
Write z̃i = ztop(i− 1
2 )N and define θ̃1, θ̃2,..., θ̃N , and q̃1, q̃2,...,
q̃N at z̃1, z̃2,..., z̃N .Tildes denote that these are now Eulerian positions in thecolumn, NOT Lagrangian labels.
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Algorithm
Start at z = z̃N
1- Temporarily move every parcel from their current height tothis new height.2- If the parcel is saturated and ascending, move from currentheight z̃j , say, condense excess moisture and increasetemperature. Find new potential temperature θ̂j solving
θ̂j + LQsat(θ̂j , z̃N)− θ̃j − Lq̃j = 0.
Hat on θ̂j denotes sorting phase variable. Then
q̂j =(θ̃j + Lq̃j − θ̂j
)/L
If the parcel is unsaturated or descending then θ̂j = θ̃j andq̂j = q̃j .
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
3- After every parcel has been temporally lifted to z̃N , theparcel with the largest potential temperature, from height z̃Jsay, is then installed at the Eulerian level z̃N with
θ̃1N = θ̂J , q̃1
N = q̂J .
The parcel from height z̃J is then eliminated from the sort,and the algorithm moves to height z̃N−1.4- Steps 1-3 are repeated until every parcel is assigned aheight.
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Outline
1 Background
2 Model Approach
3 Lagrangian Form and Conservation Laws
4 Numerical Algorithm
5 Results
6 Conclusions
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Dry convection
0
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300 320 340 360 380
θ0 (K)
z (m)
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300 320 340 360 380θ (K)
z (m)
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
0
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0 2000 4000 6000 8000 10000A
z (m)
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Moist convection
Initial profiles
0
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300 320 340 360 380
θ0 (K)
z (m)
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0 0.005 0.01 0.015 0.02 0.025
q0
z (m)
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Adjustment profiles, N = 100
0
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300 320 340 360 380θ (K)
z (m)
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0 0.005 0.01 0.015 0.02 0.025q
z (m)
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
N = 10000
0
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300 320 340 360 380θ (K)
z (m)
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0 0.005 0.01 0.015 0.02 0.025q
z (m)
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
0
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0 20 40 60 80 100A
z (m)
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z (m)
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Numerical convergence of scheme
10
10.05
10.1
10.15
10.2
10.25
0 10000 20000 30000N
qTot (m3)
2.94x1010
2.95x1010
2.96x1010
2.97x1010
2.98x1010
0 10000 20000 30000N
V (kgms−2)
qtot =∫ ztop
0q(s) ds, V =
∫ ztop
0sθ(s) ds,
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
Outline
1 Background
2 Model Approach
3 Lagrangian Form and Conservation Laws
4 Numerical Algorithm
5 Results
6 Conclusions
Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions
A column model for the adjustment of moist air, analyticaland numerical, to achieve local θ stability.We can generate weak, probabilistic solutions to the moistconvection problem. These would not be obtainable fromPDEs with the usual discrete PDE methods.Continuous problem can be formulated using a variationalapproach, maximising total (moist) potential energy.Algorithm seems to find global maximum of potential energy(i.e. max moisture condensed).Highly likely that other local maxima exist (i.e. otherrearrangements). Work underway on this.