ecmwf governing equations 1 slide 1 governing equations i: reference equations at the scale of the...

ECMWFGoverning Equations 1 Slide 1

Governing Equations I: reference equations at the scale of the continuum

by Sylvie Malardel (room 10a; ext. 2414)after Nils Wedi (room 007; ext. 2657)

Thanks to Clive Temperton, Mike Cullen and Piotr Smolarkiewicz

Recommended reading: An Introduction to Dynamic Meteorology, Holton (1992)An Introduction to Fluid Dynamics, Batchelor (1967)Atmosphere-Ocean Dynamics, Gill (1982)Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Durran (1999)

Illustrations from “Fondamentaux de Meteorologie”, Malardel (Cepadues Ed., 2005)


Overview

Introduction : from molecules to “continuum”

Eulerian vs. Lagrangian derivatives

Perfect gas law

Continuity equation

Momentum equation (in a rotating reference frame)

Thermodynamic equation

Spherical coordinates

“Averaged” equations in numerical weather prediction models

(Note : focus on “dry” (no water phase change) equations)


Note: Simplified view !

From molecules to continuum

N

0

contLl

Newton’s second law

Boltzmann equations

Navier-Stokes equations

Euler equationsindividual particles statistical distribution continuum

Mean free path

number of particles kinematic viscosity~1.x10-6 m2s-1, water~1.5x10-5 m2s-1, air

Mean value of the parameter

Molecular fluctuations

Mean value at the scale of the continuum

Continuous variations

contfree Ll


Perfect gas law

The pressure force on any surface element containing M

The temperature is defined as

The link between the pressure, the temperature and the

density of molecules in a perfect gas :

pressure

s velocitiemolecule of variance"lstatistica" '

molecule individualan of mass

volumeofunit per molecules ofnumber /with

'3

1

2

2

p

u

m

VNr

npdsndsurmFp

2'3

1u

k

mT

constantBoltzmann : 10.38.1 23k

rkTp


Perfect gas law : other forms

NkTpV

11.kgJ.K287air dry for

/ with

d

gas*

gasgasgas

R

/MRmkRTRp

1-1-*

*

*

K.J.mol314.8 and .

and molesin expressed

molecules ofnumber the with

R

kR

nTnRpV

23Avogadro 106.022Ν

Ν

or

or


Fundamental physical principles

Conservation of mass

Conservation of momentum

Conservation of energy

Consider budgets of these quantities for a control volume

(a) Control volume fixed relative to coordinate axes

=> Eulerian viewpoint

(b) Control volume moves with the fluid and always contains the same particles

=> Lagrangian viewpoint


Eulerian versus Lagrangian

Eulerian : Evolution of a quantity inside a fixed box

Lagrangian : evolution of a quantity following the particules in their motion


Eulerian vs. Lagrangian derivatives

Particle at temperature T at position at time moves to in time .

Temperature change given by Taylor series:

i.e.,

),,( 000 zyx t0

),,( 000 zzyyxx t

T

...)()()()(

zz

Ty

y

Tx

x

Tt

t

TT

0limt

dT T T T dx T dy T dz

dt t t x dt y dt z dt

,,, wdt

dzv

dt

dyu

dt

dx

z

Tw

y

Tv

x

Tu

t

T

dt

dT

dt

dT

then Let

dT TT

dt t

v

is the rate of change following the motion.

total derivative

local rate of change

advection t

T

is the rate of change

at a fixed point.


Sources vs. advection

Tdt

dT

t

T

.V

TQ

Tdt

dT

t

T

.

.

V

V

Q

dt

dT

t

T

Tt

T

.V


Mass conservation

Inflow at left face is . Outflow at right face is

Difference between inflow and outflow is per unit volume.

Similarly for y- and z-directions.

Thus net rate of inflow/outflow per unit volume is

= rate of increase in mass per unit volume

= rate of change of density

=> Continuity equation (Eulerian point of view)

zyu zyxux

u ])([

)( u

x

)]()()([ wz

vy

ux

( ) v

( )t

v

x

z

yEulerian budget :

Mass flux on left face

Mass flux on right face


Mass conservation (continued)

Continuity equation, mathematical transformation

lnd

dt v

Continuity equation : Lagrangian point of view

... vvvt

By definition, mass is conserved in a Lagrangian volume: 0)(

dt

d

dt

md

0

.

V

dt

d

dt

d


Momentum equation : frames

D

Dtav

F (1)

We want to express this in a reference frame which rotates with the earth

“fixed” star

Rotating frame


Momentum equation : velocities = angular velocity of earth

.

A

For any vector

r

Dt

DA

Dt

DA

dt

dA

dt

dA

A

= position vector relative to earth’s centre

AAA

dt

d

Dt

D

A

Dt

Dra v

dt

drv

or )(tA

)( dttA

Evolution in absolute frame

Evolution in rotating frame

e

rav

vv


Momentum equation : accelerations

)( rdt

d

dt

d

Dt

Da

aa vvv

vvv

forces "absolute"onaccelerati lcentripetaonaccelerati Coriolis

)(2 Fvvv

rdt

d

Dt

D aThen from (1):

In practice: Fvv

force lcentrifugaforce Coriolis

)(2 rdt

d


Momentum equation : Forces

Forces - pressure gradient, gravitation, and friction

Where = specific volume (= ), = pressure,

= sum of gravitational and centrifugal force,

= molecular friction,

= vertical unit vector

k

F

2d

p gdt

kv

v

/1 p

g

gMagnitude of varies by ~0.5% from pole to equator and by ~3% with altitude (up to 100km).

kk


Spherical polar coordinates (in the same rotating frame)

: = longitude, = latitude, = radial distance

Orthogonal unit vectors: eastwards, northwards, upwards.

As we move around on the earth, the orientation of the coordinate system changes:

),,( r r

i j k

u v w i j kv

tand du uv uw

dt dt r r

v

i

2

tandv u vw

dt r r

j

2 2dw u v

dt r

k

Extra terms due tothe spherical curvature


Components of momentum equation

p

rr

uw

r

uvwv

dt

du

cos

1tancos2sin2

p

rr

vw

r

uu

dt

dv 1tan sin2

2

rg

r

p

r

vuu

dt

dw

1

cos222

rw

r

v

r

u

tdt

d

cos

with


The “shallow atmosphere” approximation

parameter Coriolis

sin2 f

“Shallowness approximation” –

For consistency with the energy conservation and angular momentum conservation, some terms have to be neglected in the full momentum equation. This approximation is then valid only if these terms are negligible.

p

aa

uvv

dt

du

cos

1tansin2

p

aa

uu

dt

dv 1tansin2

2

rg

z

p

dt

dw

1

zw

a

v

a

u

tdt

d

cos

with

arazzar suppose and Earth theof radius:a


Energy conservation : Thermodynamic equation (total energy conservation – macroscopic kinetic energy equation)

First Law of Thermodynamics:

where I = internal energy,

Q = rate of heat exchange with the surroundings

W = work done by gas on its surroundings by compression/expansion.

For a perfect gas, ( = specific heat at constant volume),

WQdt

dI

TcvI cv

dt

dpW

)/1(

dT dQ pv dt dtc


Thermodynamic equation (continued)

Alternative forms:

p

dT RTc Q

dt p

dt

dp

R=gas constant and

pRT

Eq. of state:

Rcc vp


Thermodynamic equation (continued)

Alternative forms:

QTdt

dpc

/

0

pR c

Tpp

where

The potential temperature is conserved in a Lagrangian “dry” and adiabatic motion

is the potential temperature


Where do we go from here ?

So far : We derived a set of evolution equations based on 3 basic conservation principles valid at the scale of the continuum : continuity equation, momentum equation and thermodynamic equation.

What do we want to (re-)solve in models based on these equations?

grid

sca

lere

solv

ed s

cale

(?)

The scale of the grid is much bigger than the

scale of the continuum


Observed spectra of motions in the atmosphere

Spectral slope near k-3 for wavelengths >500km. Near k-5/3 for shorter wavelengths.

Possible difference in larger-scale dynamics.

No spectral gap!


Scales of atmospheric phenomena

Practical averaging scales do not correspond to a physical scale separation.

If equations are averaged, there may be strong interactions between resolved and unresolved scales.


“Averaged” equations : from the scale of the continuum to the mean grid size scale

The equations as used in an operational NWP model represent the evolution of a space-time average of the true solution.

The equations become empirical once averaged, we cannot claim we are solving the fundamental equations.

Possibly we do not have to use the full form of the exact equations to represent an averaged flow, e.g. hydrostatic approximation OK for large enough averaging scales in the horizontal.


“Averaged” equations

The sub-grid model represents the effect of the unresolved scales on the averaged flow expressed in terms of the input data which represents an averaged state.

The mean effects of the subgrid scales has to be parametrised.

The average of the exact solution may *not* look like what we expect, e.g. since vertical motions over land may contain averages of very large local values.

The averaging scale does not correspond to a subset of observed phenomena, e.g. gravity waves are partly included at TL799, but will not be properly represented.


Demonstration of the averaging effect

Use high resolution (10km in horizontal) simulation of flow over Scandinavia

Average the results to a scale of 80km

Compare with solution of model with 40km and 80km resolution

The hope is that, allowing for numerical errors, the solution will be accurate on a scale of 80km

Compare low-level flows and vertical velocity cross-section, reasonable agreement

Cullen et al. (2000) and references therein


High resolution numerical solution

Test problem is a flow at 10 ms-1 impinging on Scandinavian

orography

Resolution 10km, 91 levels, level spacing 300m

No turbulence model or viscosity, free-slip lower boundary

Semi-Lagrangian, semi-implicit integration scheme with 5 minute

timestep


Low-level flow

10km resolution

40km resolution


Low-level flow

40km resolution

10km resolution averaged to 80km


Cross-section of potential temperature


x-z vertical velocity

40km resolution

10km resolution averaged to 80km


Conclusion

Averaged high resolution contains more information than lower resolution runs.

The better ratio of comparison was found approximately as

dx (averaged high resol) ~ ·dx (lower resol) with ~1.5-2

The idealised integrations suggest that the predictions represent the averaged state well (despite hydrostatic assumption for example).

The real solution is much more localised and more intense.

ecmwf governing equations 1 slide 1 governing equations i: reference equations at the scale of the...

Documents

ecmwf governing equations

advection slide

motion slide

rotating frame slide

reference equations

lagrangian viewpoint

right face slide

wave equations