Improving the mass conservation in the semi- Lagrangian scheme of the IFS/H

Improving the mass conservation in the semi-Lagrangian scheme of the IFS/HARMONIE model

T. Morales(1) and M. Hortal(2)

(1) Spanish Meteorological Agency (AEMet) (2) HIRLAM group

XXXIII Jornadas Científicas de la AME

Oviedo, 7 a 9 de abril de 2014

HIRLAM-B Programme

Objectives of the Programme !

Develop a LAM in order to provide a high quality operational short- and very-short-range deterministic and probabilistic analysis and forecasting system

! HARMONIE LAM developed by Aladin and HIRLAM groups ! Dynamic core of HARMONIE

! semi-Lagrangian (SL) advection

Outline:!

Motivation !

Short description of the ECMWF model Integrated Forecast System (IFS) Implementation of the semi-Lagrangian semi-implicit (SLSI) scheme in the ECMWF model

!Dry Air and Ozone mass conservation in IFS

Mass conservation of dry air Mass conservation of ozone

!Conclusions

Implementation of new numerical scheme for the SL advection with the purpose of improving the mass conservation of dry air and the remaining of the atmosphere: ozone, NOx, SO2. !

! The dynamic kernel of the global model, IFS (ECMWF), and the limited area model, HARMONIE (HIRLAM Consortium), is the same. Thus, initially carried out the modifications in the IFS, where the total mass should be conserved for later transferring the improvements made to HARMONIE LAM.

Integrated Forecast Model (IFS) Motivation

GM LAM

SLSI scheme

ECMWF HARMONIE

Outline:!

Motivation !


!Air and Ozone mass conservation in IFS


!Conclusions

Integrated Forecast Model (IFS) Integrated Forecast System (IFS)

Hydrostatic formulation !Equations: !• Prognostic equations for couple

variables!

• Hybrid vertical coordinate!

• Continuity eq. for dry air!

• Thermodynamic eq.!

• Time integration scheme: semi-Lagrangian semi-implicit (SLSI)

(�, ⇢, ⌘)

⌘(p, ps)

Variables: !Vertical coordinate:! !!!Equations!!!!! ! !

Dx

Dt= RHS x ⌘ q,O3, NO

x

, ..

(u, v, T )

Implementation of the SLSI scheme in the ECMWF model

D~r

Dt= ~V

rt+�tA � rt��t

r

2�t= V t

M

Three time levels SL scheme

Current SL scheme: Stable Extrapolation Two Time-Level Scheme (SETTLS)

SL advection

SL trajectory

D�

Dt= 0

�t+�tA � �t

D

�t= 0

Outline:!

Motivation !




!Conclusions

SL trajectory (SETTLS algorithm)

Mass Conservation of dry air

Continuity equations (SLSI discretization)

rt+�tA = rtD +

�t

2([2vt � vt��t]D + vt

A)

Lnp+sup =NLEVX

k=1

�Bj(Lnp�sup +�t{@Lnpsup

@t+ vkr(Lnpsup)}�) +

��t

prefsup

�tt{NLEVX

j=1

(�prefj Dj)}

A

= without quasi-monotone filter + !!!!!!!!

bi-cubic-linear interpolation


A

RM = Reference model

RM + without QM + bi-cubic-linear in A termRM

Mass Conservation of Dry Air


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%

0 1 2 3 4 5 6 7 8 9 10

Forecast Day

es oper 12UTC | Mean method: fairDate: 20091015 12UTC to 20091015 12UTCNHem Extratropics (lat 20.0 to 90.0, lon -180.0 to 180.0)

Anomaly correlation500hPa geopotential

CY37R1: T159L91 MR

CY38R1: T159L91 MR

T159L91

Comparative between CY37R2 and 38R2. Reference Model (RM).


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%

0 1 2 3 4 5 6 7 8 9 10

Forecast Day



CY37R2: RM

CY38R2: RM

CY38R2: RM without QM

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100

%

0 1 2 3 4 5 6 7 8 9 10

Forecast Day


Anomaly correlation500hPa geopotentialT159L91

CY38R2: RM

CY38R2: RM without QM

T159L91Mass Conservation of dry air

T511L91

Consistency of semi-Lagrangian scheme for the continuity equation:


Kinetic Energy Spectrum: Comparative between CY37R2 and 38R2.


T1279L91

~ 1000 mb

~ 500 mb

~ 200 mb


Outline:!

Motivation !




!Conclusions

The evolution of the ozone throughout simulation period is the combination of the continuity equation and the evolution of the ozone mixing ratio.

!• continuity equation ⟶ dry air mass !

• ozone mixing ratio evolution ⟶ rO3 =gr O3

Kg Dry Air

Mass Conservation of ozone

DX

Dt= sink and source terms | X ⌘ rO3

X+A = X�

D + sink and source terms

Mass Conservation of ozone ⟶ sink and source terms = 0 ( Forecast: 10 days)

X+A = value

X+A = X�

D

gr O3 = value ⇤ kg Dry Air

Error: kind of interpolation

Error: behavior of the continuity equation


32 points !interpolations

- Departure point SL trajectory

linear reduced gaussian grid


It is not possible !to apply the !quasi-monotone filter

- Departure point SL trajectory

- Value interpolated in the levels l, l+1, l+2 and l+3

tri-cubic-linear interpolation in the vertical

value of the field = weight * cubic inter + weight * linear inter


Ozone equation without sink and source terms.

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0 1 2 3 4 5 6 7 8 9Forecast Day



Reference Model

Laitili + B-S

laitlicl + B-S

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0 1 2 3 4 5 6 7 8 9Forecast Day

es oper 12UTC | Mean method: fairDate: 20091015 12UTC to 20091015 12UTCSHem Extratropics (lat -90.0 to -20.0, lon -180.0 to 180.0)


Reference Model

Laitili + B-S

laitlicl + B-S


tri-cubic interpolation + B-S

tri-cubic-lineal in the vertical + B-S

Mass conservation of ozone (only advection )


Conclusions !• It is better to continue using interpolation methods to calculate the value of the field at the departure point of the semi-Lagrangian trajectory. !

• Reduction of the aliasing in vorticity coming from the pressure gradient terms improves the mass conservation of dry air in the continuity equation. !

• Not to apply the quasi-monotone filter in the continuity equation in the CY38R2 improves more the mass conservation of dry air. !

• Apply a cubic-linear interpolation after 32-points interpolation without the quasi-monotone filter improves the conservation of mass of ozone throughout the simulation period.

Science

￼Improving the mass conservation in the semi- Lagrangian scheme of the IFS/H

Improving the mass conservation in the semi- Lagrangian scheme of the IFS/H