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Adiabatic formulation of the ECMWF model. Agathe Untch e-mail: [email protected] (office 11). Introduction. Step by step guide through the decisions to be taken / choices to be made when designing the adiabatic formulation of a global Numerical Weather Prediction (NWP) model.

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Adiabatic formulation of the ecmwf model l.jpg

Adiabatic formulation of the ECMWF model

Agathe Untch

e-mail: [email protected]

(office 11)

Adiabatic formulation of the ECMWF model


Introduction l.jpg
Introduction

  • Step by step guide through the decisions to be taken / choices to be made when designing the adiabatic formulation of a global Numerical Weather Prediction (NWP) model.

  • In the process we are “constructing” the dynamical core of the ECMWF operational NWP Model.

Adiabatic formulation of the ECMWF model


Introduction cont l.jpg
Introduction (cont.)

  • A numerical model has to be:

    • stable

    • accurate

    • efficient

  • No compromise possible on stability!

  • The relative importance given to accuracy versus efficiency depends on what the model is intended for.

    • For example:

      • an operational NWP model has to be very efficient to allow the running of all applications (data-assimilation, forecasts, ensemble prediction system) in a tight daily schedule.

      • a research model might not have to be so efficient but can’t compromise on accuracy.

Adiabatic formulation of the ECMWF model


Introduction cont4 l.jpg
Introduction (cont.)

  • Essential to the performance of any NWP or climate-prediction model are

    a.) the form of the continuous governing equations (approximated or full Euler equations?)

    b.) boundary conditions imposed (conservation properties depend on these).

    c.) the numerical schemes chosen to discretize and integrate the governing equations.

Adiabatic formulation of the ECMWF model


Euler equations for a moist atmosphere on a rotating sphere l.jpg
Euler Equations for a moist atmosphere on a rotating sphere

3D momentum equation

Continuity equation

Thermodynamic equation

Humidity equation

Transport equations of

various physical/chemical

species

Equation of state

Adiabatic formulation of the ECMWF model


Slide6 l.jpg

specific volume

total time derivative

Notations:

q specific humidity

L latent heat

Xi mass mixing ratios of physical or chemical species

(e.g. aerosols, ozone)

g gravity = gravitation g* + centrifugal force

Spherical geopotential approximation is made: neglect Earth’s

oblateness (~0.3%). => spherical geometry assumed!

Adiabatic formulation of the ECMWF model


Slide7 l.jpg

Momentum equations in spherical coordinates :

With

Euler Equations in spherical coordinates

x

Adiabatic formulation of the ECMWF model


Continuity equation in spherical coordinates l.jpg
Continuity equation in spherical coordinates

With

With

Thermodynamic equation in spherical coordinates

Adiabatic formulation of the ECMWF model


Shallow atmosphere approximation l.jpg
Shallow atmosphere approximation

1. Replace r by the mean radius of the earth a and

by

,

where z is height above mean sea level.

3. Neglect all metric terms not involving

4. Neglect the Coriolis terms containing

(resulting from

).

the horizontal component of

a

a

a

z

2. Neglect vertical and horizontal variations in g.

a

Adiabatic formulation of the ECMWF model


Euler equations in shallow atmosphere approximation l.jpg
Euler Equations in shallow atmosphere approximation

is horizontal velocity

n is the tracer for the

hydrostatic approximation:

n=0 => vertical momentum

equation = hydrostatic eq.

For n=1 we refer to these

equations as

Non-hydrostatic equations

in Shallow atmosphere

Approximation (NH-SA)

Adiabatic formulation of the ECMWF model


Slide11 l.jpg

Choice of predicted variables

Predicted variables:

?

or

=>

(Allows enforcement of mass conservation)

(Thermodyn. Computations are simpler)

Combine continuity, thermodynamic & gas

equation to obtain a prognostic equation for p.

Form of NH-SA equations more

commonly used in meteorology:

Adiabatic formulation of the ECMWF model


Hydrostatic approximation n 0 l.jpg
Hydrostatic Approximation (n=0)

  • Benefits from hydrostatic approximation

    • Vertical momentum equation becomes a diagnostic relation (=> one prognostic variable (w) less!)

    • Vertically propagating acoustic waves are eliminated (these are the fastest waves in the atmosphere, causing the biggest stability problems in numerical integrations!)

  • Drawbacks of hydrostatic approximation

    • Not valid for short horizontal scales (for mesoscale phenomena)

    • Short gravity waves are distorted in the hydrostatic pressure field.

  • Operational version of the ECMWF model is a hydrostatic model.

    • operational horizontal resolution ~25km (T799), so hydrostatic approximation is (still) OK.

Adiabatic formulation of the ECMWF model


Hydrostatic shallow atmosphere equations h ydrostatic p rimitive e quations hpe l.jpg
Hydrostatic shallow atmosphere equations(Hydrostatic Primitive Equations (HPE))

  • pis monotonic function ofz and

    can be used as vertical coordinate.

    (Eliassen (1949))

Adiabatic formulation of the ECMWF model


Slide14 l.jpg

Potential temperature (isentropic coordinate):

  • good coordinate where atmosphere is stably stratified

  • (potential temperature increases monotonic with z).

  • adiabatic flow stays on isentropic surfaces (2D flow)

  • good coord. in stratosphere, not very good in troposphere

Choice of vertical coordinate

Most commonly used vertical coordinates:

Height above mean sea level z: - most natural vertical coordinate

Pressure p (isobaric coordinate):

- has advantages for thermodynamic calculations

- makes the continuity equation a diagnostic relation

in the hydrostatic system

- can be extended for use in non-hydrostatic models

Adiabatic formulation of the ECMWF model


Slide15 l.jpg

Choice of vertical coordinate (cont.)

Generalized vertical coordinate s:

(Kasahara (1974), Staniforth & Wood (2003))

Any variable s which is a monotonic single-valued

function of height zcan be used as a vertical coordinate.

Coordinate transformation rules (from z to any vertical coordinate s):

Adiabatic formulation of the ECMWF model


Slide16 l.jpg

Hydrostatic relation between p and z:

=>

with geopotential

Pressure p as vertical coordinate

in the hydrostatic system

Coordinate transformation rules (from z to p):

Adiabatic formulation of the ECMWF model


Hydrostatic primitive equations with pressure as vertical coordinate l.jpg
Hydrostatic Primitive Equations with pressure as vertical coordinate

Pressure gradient replaced by

geopotential gradient (at constant

pressure).

pressure vertical velocity

Continuity eq. is a diagnostic eq. in p-coordinates.

  • Number of prognostic variables

    reduced to 3 (horizontal winds & T)!

Geopotential computed from

hydrostatic equation.

Adiabatic formulation of the ECMWF model


Slide18 l.jpg

For every s for which

=> continuity is diagnostic eq.!

Hydrostatic pressure as vertical coordinate

for a non-hydrostatic shallow atmosphere model

Introduced by Laprise (1991)

In the hydrostatic system with pressure as vertical coordinate the

continuity equation is a diagnostic equation.

The idea is to find a vertical coordinate for the NH system which

makes the continuity equation a diagnostic equation.

Continuity equation in generalized vertical coordinate s:

(Kasahara(1974))

Adiabatic formulation of the ECMWF model


Slide19 l.jpg

Choose

and denote with the coordinate s for which

i.e.

For

is the weight of a column of air (of unit area)

above a point at height z, i.e. hydrostatic pressure.

Hydrostatic pressure as vertical coordinate

for a non-hydrostatic shallow atmosphere model (cont.)

Adiabatic formulation of the ECMWF model


Slide20 l.jpg

Hydrostatic pressure as vertical coordinate

for a non-hydrostatic shallow atmosphere model (cont.)

=>

D3

inZ

Adiabatic formulation of the ECMWF model


Boundary conditions l.jpg
Boundary conditions

Governing equations have to be solved subject to boundary conditions.

  • The lower boundary of the atmosphere (surface of the earth) is

  • a material boundary (air parcel cannot cross it!)

  • velocity component perpendicular to surface has to vanish

    (e.g. at a flat and rigid surface vertical velocity w = 0)

Unfortunately, the topography of the earth is far from flat, making it

quite tricky to apply the lower boundary condition.

Solution: Use a terrain-following vertical coordinate.

For example: traditional sigma-coordinate

(Phillips, 1957)

Adiabatic formulation of the ECMWF model


Terrain following vertical coordinate l.jpg
Terrain-following vertical coordinate

The easiest case is a flat and rigid boundary where the boundary

condition simply is:

e.g.

at the bottom &

at the top

We are looking for a vertical coordinate “s” which makes it easy

to apply the condition of zero velocity normal to the boundary, even

for very complex boundaries like the earth’s topography.

at the boundary.

Therefore, we look to create a vertical coordinate which makes the

upper and lower boundaries “flat”. That is, s is constant following

the shape of the boundary (i.e. the boundary is a coordinate surface).

Adiabatic formulation of the ECMWF model


Terrain following vertical coordinate cont l.jpg
Terrain-following vertical coordinate(cont.)

is a monotonic single-valued function of hydrostatic

pressure and also depends on surface pressure in

such a way that

(Where

is the pressure at the top boundary.)

this is the traditional sigma-coordinate of Phillips (1957)

For

The ECMWF model uses a terrain-following vertical coordinate

based on hydrostatic pressure.

The principle will be explained based on hydrostatic pressure :

A simple function that fulfils these conditions is

Adiabatic formulation of the ECMWF model


Sigma coordinate l.jpg
Sigma-coordinate

(First introduced by Phillips (1957))

Drawback:

Influence of topography is felt

even in the upper levels far away

from the surface.

Remedy:

Use hybrid sigma-pressure

coordinates

x

Adiabatic formulation of the ECMWF model


Hybrid vertical coordinate l.jpg
Hybrid vertical coordinate

First introduced by Simmons and Burridge (1981).

The difference to the sigma coordinate is in the way the monotonic

relation between the new coordinate and hydrostatic pressure is

defined:

The functions A and B can be quite general and allow to design a

hybrid sigma-pressure coordinate

where the coordinate surfaces are sigma surfaces near the ground,

gradually become more horizontal with increasing distance from the

surface and turn into pure pressure surfaces in the stratosphere(B=0).

In order that the top and bottom boundaries are coordinate surfaces

(=> easy application of boundary condition), A and B have to fulfil:

at the surface

at the top

Adiabatic formulation of the ECMWF model


Slide26 l.jpg

Comparison of

sigma-coordinates & hybrid η-coordinates

Coordinate surfaces over a hill for

sigma-coordinate

η-coordinate

Adiabatic formulation of the ECMWF model


Slide27 l.jpg

Non-hydrostatic equations in hybrid vertical coordinate

prognostic continuity eq.

Adiabatic formulation of the ECMWF model


Hydrostatic primitive equations in hybrid vertical coordinate l.jpg
Hydrostatic Primitive Equations in hybrid ηvertical coordinate

In addition to the

geopotential gradient term, a

pressure gradient term again!

Continuity equation is prognostic

again because the (hydrostatic)

pressure is not the vertical

coordinate anymore.

Adiabatic formulation of the ECMWF model


Slide29 l.jpg

Hydrostatic equations of the ECMWF operational model(incorporating moisture)

Adiabatic formulation of the ECMWF model


Slide30 l.jpg

virtual temperature

gas constant of dry air,

gas constant of water vapour

specific heat of dry air at constant pressure

specific heat of water vapour at constant pressure

p-coordinate vertical velocity

Notations:

q specific humidity

Xmass mixing ratio of physical or chemical species

(e.g. aerosols, ozone)

contributions from physical parametrizations

horizontal diffusion terms

Adiabatic formulation of the ECMWF model


Slide31 l.jpg

with boundary conditions:

B from def. of vert. coord.

From the continuity equation

we can derive (by vertical integration) the following equations:

Needed for the energy-conversion

term in the thermodynamic equation

Needed for the semi-Lagrangian

advection

Prognostic equations for surface

pressure

Adiabatic formulation of the ECMWF model


Slide32 l.jpg

Prognostic equations

of the ECMWF hydrostatic model

These equations are discretized and integrated in the ECMWF model.

Adiabatic formulation of the ECMWF model


Discretisation in the ecmwf model l.jpg
Discretisation in the ECMWF Model

  • Space discretisation

    • In the horizontal: spectral transform method

    • In the vertical: cubic finite-elements

  • Time discretisation

    • Semi-implicit semi-Lagrangian two-time-level scheme

Adiabatic formulation of the ECMWF model


Horizontal discretisation l.jpg
Horizontal discretisation

Options for discretisation are:

  • in grid-point space only (grid-point model)

    • finite-difference, finite volume methods

  • (in spectral space only)

  • in bothgrid-point and spectral space and transform back and forth between the two spaces (spectral transform method, spectral model)

    • Gives the best of both worlds:

      • Non-local operations (e.g. derivatives) are computed in spectral space (analytically)

      • Local operations (e.g. products of terms) are computed in

        grid-point space

    • The price to pay is in the cost of the transformations between the two spaces

  • in finite-element space(basis functions with finite support)

Adiabatic formulation of the ECMWF model


Slide35 l.jpg

Horizontal discretisation (cont.)

ECMWF model uses the spectral transform method

Representation in spectral space in terms of spherical harmonics:

Ideally suited set of basis functions for spherical geometry (eigenfunctions of

the Laplace operator).

m: zonal wavenumber

n: total wavenumber

λ= longitude

μ= sin(θ) θ: latitude

Pnm: Associated Legendre functions of the first kind

Adiabatic formulation of the ECMWF model


The horizontal spectral representation l.jpg
The horizontal spectral representation

Triangular truncation

(isotropic)

Spherical harmonics

Fourier functions

associated Legendre polynomials

FFT (fast Fourier transform)

using

NF 2N+1

points (linear grid)

(3N+1 if quadratic grid)

Legendre transform

by Gaussian quadrature

using NL (2N+1)/2

“Gaussian” latitudes (linear grid)

((3N+1)/2 if quadratic grid)

No “fast” algorithm available

Triangular truncation:

n

N

m

m = -N

m = N

Adiabatic formulation of the ECMWF model


Slide37 l.jpg

Horizontal discretisation (cont.)

Representation in grid-point space is on the reduced Gaussian grid:

Gaussian grid: grid of Guassian quadrature points (to facilitate

accurate numerical computation of the integrals

involved in the Fourier and Legendre transforms)

- Gauss-Legendre quadrature in latitude:

Grid-points in latitude are the zeros of the Legendre polynomial

of order NG

Gaussian latitudes

NG (2N+1)/2 for the linear grid.

NG (3N+1)/2 for the quadratic grid.

- Gauss-Fourier quadrature in longitude:

Grid-points in longitude are equidistantly spaced (Fourier) points

2N+1 for linear grid

3N+1 for quadratic grid

Adiabatic formulation of the ECMWF model


The gaussian grid l.jpg
The Gaussian grid

Reduced grid

Full grid

About 30% reduction in number of points

• Associated Legendre functions are very small near the poles for large m

Adiabatic formulation of the ECMWF model


T799 t1279 l.jpg
T799 T1279

25 km grid-spacing

( 843,490 grid-points)

Current operational

resolution

16 km grid-spacing

(2,140,704 grid-points)

Future operational

resolution (from end 2009)

Adiabatic formulation of the ECMWF model


Spectral transform method l.jpg
Spectral transform method

Grid-point space

-semi-Lagrangian advection

-physical parametrizations

Inverse FFT

FFT

Fourier Space

Fourier Space

Spectral space

-horizontal gradients

-semi-implicit calculations

-horizontal diffusion

Inverse LT

LT

FFT: Fast Fourier Transform, LT: Legendre Transform

Adiabatic formulation of the ECMWF model


Slide41 l.jpg

Horizontal discretisation (cont.)

Advantages of the spectral representation:

a.) Horizontal derivatives are computed analytically

=> pressure-gradient terms are very accurate

=> no need to stagger variables on the grid

b.) Spherical harmonics are eigenfunctions of the the

Laplace operator

=> Solving the Helmholtz equation (arising from

the semi-implicit method) is straightforward.

=> Applying high-order diffusion is very easy.

Disadvantage: Computational cost of the Legendre transforms is

high and grows faster with increasing horizontal

resolution than the cost of the rest of the model.

Adiabatic formulation of the ECMWF model


Comparison of cost profiles at different horizontal resolutions l.jpg
Comparison of cost profilesat different horizontal resolutions

Adiabatic formulation of the ECMWF model


Cost of legendre transforms l.jpg
Cost of Legendre transforms

T2047

T1279

T799

T511

Adiabatic formulation of the ECMWF model


Profile for t2047 on ibm p690 768 cpus l.jpg
Profile for T2047on IBM p690+ (768 CPUs)

Legendre

Transforms

~17% of

total cost

of model

Physics

~36% of

total cost

Adiabatic formulation of the ECMWF model


Slide45 l.jpg

L91

L60

Vertical discretisation

Variables are discretized on

terrain-following pressure

based hybrid η-levels.

Vertical resolution of the operational ECMWF model:

91 hybrid η-levels

resolving the atmosphere

up to 0.01hPa (~80km)

(upper mesosphere)

Adiabatic formulation of the ECMWF model


Slide46 l.jpg

Vertical discretisation (cont.)

Choices: - finite difference methods

- finite element methods

Operational version of the ECMWF model uses a cubic

finite-element (FE) scheme based on cubic B-splines.

No staggering of variables, i.e. all variables are held on the same

vertical levels. (Good for semi-Lagrangian advection scheme.)

Inspection of the governing equations shows that there are only

vertical integrals (no derivatives) to be computed (if advection is

done with semi-Lagrangian scheme).

Adiabatic formulation of the ECMWF model


Slide47 l.jpg

Prognostic equations

of the ECMWF hydrostatic model

Reminder: slide 32

These equations are discretized and integrated in the ECMWF model.

Adiabatic formulation of the ECMWF model


Slide48 l.jpg

with boundary conditions:

B from def. of vert. coord.

Reminder: slide 31

From the continuity equation

we can derive (by vertical integration) the following equations:

Needed for the energy-conversion

term in the thermodynamic equation

Needed for the semi-Lagrangian

advection

Prognostic equations for surface

pressure

Adiabatic formulation of the ECMWF model


Vertical integration in finite elements l.jpg

Aji

Bji

Vertical integration in finite elements

can be approximated as

Basis sets

Applying the Galerkin method with test functions tj =>

Adiabatic formulation of the ECMWF model


Vertical integration in finite elements50 l.jpg
Vertical integration in finite elements

Including the transformation from grid-point (GP) representation to

finite-element representation (FE)

and the projection of the result from FE to GP representation

one obtains

Matrix J depends only on the choice of the basis functions and the level spacing.

It does not change during the integration of the model, so it needs to be computed

only once during the initialisation phase of the model and stored.

Adiabatic formulation of the ECMWF model


Slide51 l.jpg

Cubic B-splines for regular spacing of levels

(Prenter (1975))

Adiabatic formulation of the ECMWF model


Slide52 l.jpg

Cubic B-splines as

basis elements

Basis elements

for the

representation

of the integral

F

Basis elements

for the represen-

tation of the

function to

be integrated

(integrand)

f

Adiabatic formulation of the ECMWF model


Benefits from using the finite element scheme in the vertical l.jpg
Benefits from using the finite-element scheme in the vertical

  • High order accuracy (8th order for cubic elements)

    • Very accurate computation of the pressure-gradient term in conjunction with the spectral computation of horizontal derivatives

    • More accurate vertical velocity for the semi-Lagrangian trajectory computation

      • Improved ozone conservation

  • Reduced vertical noise in the stratosphere

  • No staggering of variables required in the vertical: good for semi-Lagrangian scheme because winds and advected variables are represented on the same vertical levels.

  • Adiabatic formulation of the ECMWF model


    Slide54 l.jpg

    Discretisation in time

    Decisions to be taken:

    • Discretize on

    • three time-levels (e.g. leapfrog scheme)

    • - produce a computational mode (time-filtering needed)

    • two time-levels

    • - more efficient than three-time-level schemes

    • - (less stable)

    How to discretize the right-hand sides of the equations in time:

    - explicitly

    - implicitly

    - semi-implicitly

    How to treat the advection:

    - in Eulerian way

    - in semi-Lagrangian way

    Adiabatic formulation of the ECMWF model


    Slide55 l.jpg

    Discretization in time (cont.)

    • Operational version of the ECMWF model uses

    • Two-time-level scheme

    • Semi-implicit treatment of the right-hand sides

    • Semi-Lagrangian advection

    Adiabatic formulation of the ECMWF model


    Slide56 l.jpg

    Time discretisation of the right-hand sides

    Discretisation of the right-hand side (RHS) of the equations:

    • RHS taken at the centre of the time interval: explicit

    • (second order) discretisation. Stability is subject

    • to CFL-like criterion

    • - RHS average of its value at initial time and at final time:

    • implicit discretisation (generally stable)

    • => leads to a difficult system to solve (iterative solvers)

    • treat only some linearized terms of RHS implicitly

    • (semi-implicit discretisation)

    Adiabatic formulation of the ECMWF model


    Slide57 l.jpg

    For compact notation define:

    “implicit correction term”

    Semi-implicit time integration

    Notations:

    X : advected variable

    RHS: right-hand side of the equation

    L: part of RHS treated implicitly

    Superscripts:

    “0” indicates value for explicit discretiz,

    “-” indicates value at start of time step

    “+” indicates value at end of time step

    =>

    L=RHS => implicit scheme

    L= part of RHS => semi-implicit

    Benefit: slowing-down of the waves too fast for the explicit CFL cond.

    Drawback: overhead of having to solve an elliptic boundary-value prob.

    Adiabatic formulation of the ECMWF model


    Slide58 l.jpg

    Semi-implicit time integration (cont.)

    Choice of which terms in RHS to treat implicitly is guided by the knowledge of which waves cause instability because they are too fast (violate the CFL condition) and need to be slowed down with an implicit treatment.

    In a hydrostatic model, fastest waves are horizontally propagating

    external gravity waves (long surface gravity waves), Lamb waves

    (acoustic wave not filtered out by the hydrostatic approximation)

    and long internal gravity waves. => implicit treatment of the

    adjustment terms.

    L= linearization of part of RHS (i.e. terms supporting the fast modes)

    => good chance of obtaining a system of equations in the variables

    at “+” that can be solved almost analytically in a spectral model.

    Adiabatic formulation of the ECMWF model


    Slide59 l.jpg

    Semi-implicit time integration (cont.)

    semi-implicit corrections

    semi-implicit

    equations

    Adiabatic formulation of the ECMWF model


    Slide60 l.jpg

    Semi-implicit time integration (cont.)

    semi-implicit

    equations

    Reference state for

    linearization:

    ref. temperature

    ref. surf. pressure

    Where:

    => lin. geopotential for X=T

    => lin. energy conv. term for X=D

    Adiabatic formulation of the ECMWF model


    Slide61 l.jpg

    ref. temperature

    ref. surf. pressure

    unity operator

    Semi-implicit time integration (cont.)

    semi-implicit

    equations

    Reference state for

    linearization:

    By eliminating in above system all but one of the unknowns (D+)

    =>

    operator acting only on the vertical

    Adiabatic formulation of the ECMWF model


    Slide62 l.jpg

    One equation for each

    In spectral space (spherical harmonics space):

    because

    Semi-implicit time integration (cont.)

    Vertically coupled set of

    Helmholtz equations.

    Coupling through

    Uncouple by transforming to the eigenspace of this matrix gamma

    (i.e. diagonalize gamma). Unity matrix “I” stays diagonal. =>

    Once D+ has been computed, it is easy to compute the other

    variables at “+”.

    Adiabatic formulation of the ECMWF model


    Slide63 l.jpg

    Semi-Lagrangian advection

    Semi-Lagrangian (SL) schemes are more efficient & more stable

    than Eulerian advection schemes.

    Coupling SL advection with semi-implicit treatment of the fast

    modes results in a very stable scheme where the timestep can

    be chosen on the basis of accuracy rather than for stability.

    Disadvantage: Lack of conservation of mass and tracer

    concentrations. (More difficult to enforce

    conservation than in Eulerian schemes)

    Adiabatic formulation of the ECMWF model


    Semi lagrangian advection cont l.jpg

    A

    x x x x

    x x x x

    x x x x

    M

    x

    D

    *

    Semi-Lagrangian advection (cont)

    All equations are of this (Lagrangian) form:

    (See slide 32)

    Three time-level scheme:

    Centred second order accurate scheme

    Ingredients of semi-Lagrangian advection are:

    1.) Computation of the departure point (tajectory computation)

    2.) Interpolation of the advected fields at the departure location

    Adiabatic formulation of the ECMWF model


    Slide65 l.jpg

    Prognostic equations

    of the ECMWF hydrostatic model

    Reminder: slide 32

    These equations are discretized and integrated in the ECMWF model.

    Adiabatic formulation of the ECMWF model


    Semi lagrangian advection cont66 l.jpg

    with

    Extrapolation in time to middle of time interval

    Semi-Lagrangian advection (cont)

    Two-time-level second order accurate schemes :

    Unstable! => noisy forecasts

    Forecast of temperature

    at 200 hPa

    (from 1997/01/04)

    Adiabatic formulation of the ECMWF model


    Slide67 l.jpg

    With

    and

    Forecast 200 hPa T

    from 1997/01/04

    using SETTLS

    Stable extrapolating two-time-level semi-Lagrangian

    (SETTLS):

    Taylor expansion

    to second order

    Adiabatic formulation of the ECMWF model


    Interpolation in the semi lagrangian scheme l.jpg
    Interpolation in the semi-Lagrangian scheme

    y

    x

    x

    x

    x

    x

    with the weights

    x

    x

    x

    x

    x

    x

    x

    x

    x

    x

    x

    x

    x

    x

    x

    x

    x

    ECMWF model uses

    quasi-monotonequasi-cubic Lagrange interpolation

    Cubic Lagrange interpolation:

    Number of 1D cubic interpolations

    in two dimensions is 5,

    in three dimensions 21!

    To save on computations:

    cubic interpolation only for nearest

    neighbour rows, linear interpolation

    for rest => “quasi-cubic interpolation”

    => 7 cubic + 10 linear in 3 dimensions.

    Adiabatic formulation of the ECMWF model


    Interpolation in the semi lagrangian scheme cont l.jpg
    Interpolation in the semi-Lagrangian scheme (cont)

    interpolated cubically

    x

    x

    x

    x

    x

    Quasi-monotone interpolation:

    x: interpolation point

    quasi-monotone

    procedure:

    x

    x: grid points

    Quasi-monotone interpolation is used in the horizontal for all variables

    and also in the vertical for humidity and all “tracers” (e.g. ozone, aerosols).

    Has a detrimental effect on conservation, but prevents unphysical

    negative concentrations.

    Adiabatic formulation of the ECMWF model


    Slide70 l.jpg

    Modified continuity & thermodynamic equations

    Accuracy of cubic interpolation is much reduced when the field

    to be interpolated is rough (e.g. surface pressure over orography)

    Idea by Ritchie & Tanguay (1996): Subtract a time-independent

    term from the surface pressure which “contains” a large part of the

    orographic influence on surface pressure, advect the rest (smoother term)

    and treat the advection of the “rough term” with the right-hand side

    of the continuity equation [RHS].

    Continuity equation

    => Reduced mass loss/gain during a forecast.

    Adiabatic formulation of the ECMWF model


    Slide71 l.jpg

    Modified continuity & thermodynamic equations(cont)

    Similar idea for thermodynamic equation:

    (Hortal &Temperton (2001))

    Thermodynamic equation

    with

    Approximation to the change of T with height in the standard atmosphere.

    Reduces noise levels over orography in all fields, but in particular in

    vertical velocity.

    Adiabatic formulation of the ECMWF model


    Semi lagrangian advection on the sphere l.jpg

    Z

    V

    j

    j

    j

    A

    i

    x

    A

    Y

    D

    D

    M

    i

    i

    X

    Semi-Lagrangian advection on the sphere

    Momentum eq. is discretized in vector form (because a vector is continuous across

    the poles, components are not!)

    Interpolations at departure point are done

    for components u & v of the velocity vec-

    tor relative to the system of reference local

    at D. Interpolated values are to be used at A,

    so the change of reference system from D to

    A needs to be taken into account.

    Trajectories are arcs of great circles if

    constant (angular) velocity is assumed

    for the duration of a time step.

    Tangent plane projection

    Trajectory calculation

    Adiabatic formulation of the ECMWF model


    Treatment of the coriolis term l.jpg
    Treatment of the Coriolis term

    In three-time-level semi-Lagrangian:

    • treated explicitly with the rest of the RHS

    In two-time-level semi-Lagrangian:

    Extrapolation in time to the middle of the trajectory leads to

    instability (Temperton (1997))

    Two stable options:

    • Advective treatment:

    (Vector R here

    is the position

    vector.)

    • Implicit treatment :

    • Helmholtz eqs partially coupled for

      individual spectral components =>

      tri-diagonal system to be solved.

    Adiabatic formulation of the ECMWF model


    Summary of the adiabatic formulation of the operational ecmwf atmospheric model l.jpg
    Summary of the adiabatic formulation of the operational ECMWF atmospheric model

    Hydrostatic shallow-atmosphere equations with pressure-based hybrid vertical coordinate

    • Two-time-level semi-Lagrangian advection

      • SETTLS (Stable Extrapolation Two-Time-Level Scheme)

      • Quasi-monotone quasi-cubic Lagrange interpol. at departure point

      • Linear interpolation for trajectory computations and RHS terms

      • Modified continuity & thermodynamic equations to advect smoother fields (net of the orographic roughness)

    • Semi-implicit treatment of linearized adjustment terms & Coriolis terms

    • Cubic finite elements for the vertical integrals

    • Spectral horizontal Helmholtz solver (and derivative computations)

    • Uses the linear reduced Gaussian grid

    Adiabatic formulation of the ECMWF model


    Slide75 l.jpg

    Thank you very much ECMWF atmospheric model

    for your attention

    Adiabatic formulation of the ECMWF model


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