Convection problem in a single column - modelling and ... · Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions Convection

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

1 Background

2 Model Approach

3 Lagrangian Form and Conservation Laws

4 Numerical Algorithm

5 Results

6 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

1 Background

2 Model Approach



5 Results

6 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.


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.


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

1 Background

2 Model Approach



5 Results

6 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).


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.


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

1 Background

2 Model Approach



5 Results

6 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.


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 .


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

1 Background

2 Model Approach



5 Results

6 Conclusions


Dry convection

0

2000

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300 320 340 360 380

θ0 (K)

z (m)

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z (m)


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z (m)


Moist convection

Initial profiles

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θ0 (K)

z (m)

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q0

z (m)


Adjustment profiles, N = 100

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z (m)

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z (m)


N = 10000

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z (m)

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z (m)


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z (m)

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z (m)


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

1 Background

2 Model Approach



5 Results

6 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.

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Convection problem in a single column - modelling and ... · Outline Background Model Approach Lagrangian Form and Conservation Laws Numerical Algorithm Results Conclusions Convection