ecmwf governing equations 2 slide 1 governing equations ii: classical approximations and other...

ECMWFGoverning Equations 2 Slide 1

Governing Equations II:classical approximations and other systems of equations

Thanks to Clive Temperton, Mike Cullen and Piotr Smolarkiewicz

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


The continuous set of dry adiabatic equations

TRp d V. dt

d

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

dt

dp

p

TRQ

dt

dTc d

pd


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


“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.


Overview

Introduction

Scale analysis of momentum equations

Geostrophic and hydrostatic relations

IFS hydrostatic equations

Map projections and alternative spherical coordinates

Shallow-water equations

Isopycnic/isentropic equations


Introduction

Classical (and educational) approach: Simplify the governing

equations BEFORE numerics is introduced (e.g. by scale

analysis, hydrostatic approximation, Boussinesq or Anelastic

approximation) depending on the context.

Solutions to the governing equations have three important

propagation speeds: acoustic waves (speed of sound), gravity

waves (gravity-wave speed), and advective motion (wind

speed), which affect the time-step that can be used in

numerical procedures (constraint : cdt<dx). The equations

may be filtered through the choice of approximate governing

equations BEFORE OR through appropriate numerical

treatments AFTER. Ex: Explicit anelastic NH reasearch model

vs. Fully compressible SI SL NH model.


IntroductionRemove singularities (i.e. Pole problem) by choosing

appropriate set of equations (e.g. in IFS we use the vector

form of the equations for semi-Lagrangian advection; for

limited-area applications the Pole is often rotated to another

location).

Add mapping transformations to the equations for

convenience of presentation and accuracy in limited-area

models.

Choose a generalized vertical coordinate for a proper

treatment of the boundaries or better treatment of

conservative variables or easier interpretation of results etc.

Change pronostic variables to make the “numerics” easier or

more accurate or more stable.


Introduction

Use simpler sets of equations as a first approach :

Shallow water equations are a useful tool to test a new

dynamical core, as they represent a single vertical mode

but a comprehensive set of horizontal solutions.

Adiabatic tests before introducing full “physics”

Single column models, 2D vertical plane model, 3D cartesian

models…. f-plane, beta-plane models


Scale analysis

Example :

Typical observed values for mid-latitude synoptic systems:

U ~ 10 ms-1

W~ 10-2 ms-1

L ~ 106 m

~ 103 m2s-2

f0 ~ 10-4 s-1

a ~ 107 m

H ~ 104 m

p

(f 2 sin )


Scale analysis (continued)

UW/L f0U U2/a 0p H g

10-7 10-3 10-5 1010

U2/L f0U f0W U2/a UW/a p L10-4 10-3 10-6 10-810-5 10-3

p

rr

uw

r

uvwv

dt

du

cos

1tancos2sin2

p

rr

vw

r

uu

dt

dv 1tan sin2

2

gr

p

r

vuu

dt

dw

1cos2

22



Consequences if you want to resolve synoptic motions in the mid-latitudes:

Quasi-Geostrophic balance : accelerations du/dt, dv/dt are “small” differences between two large terms

Allow to drop Coriolis and metric terms which depend on w

Make the hydrostatic approximation

Assume a shallow atmosphere with radius r = a + z ~ a

gg fuy

pfufv

x

pfv

1

and 1

1

2

note



10 3Ufo

If you want to resolve smaller scales:, ex.

If you want to cover other latitudes, ex:

BUT 10/ 22 LU A small scale circulation may be far from the geostrophic balance.The wind may be very ageostrophic.

m10L 4

16s10 f

s10 -15Ufo AND 10/ 42 LUNear the equator, synoptic scale motion may be strongly ageostrophic


Hydrostatic balance

The pressure gives “the weight” of the atmosphere above

This is true for a very wide range of meteorological scales

gz

p

1


Hydrostatic approximation

),,(')(

),,(')(

zyxz

zyxpzpp

ref

ref

residue" NH"

''1

0

12 dz

dp

z

pg

z

p

dt

dw ref

refref

ref

ref

gz

p

dt

dw

refref

''1

UW/L

10-7

)/( Hp

10-1 10-1For synoptic, at mid-latitude

gz

p'

'

The vertical acceleration is still very negligible compared with the residual force terms when the hydrostatic balance has been removed


Hydrostatic approximation : consequences

Filter of isotropic acoustic waves : acoustic pressure perturbations are not related to the “weight” of the atmospheric column, then they are not described anymore.

w is obtained diagnostically from the continuity equation,

in agreement with an instantaneous mass reorganisation to fulfil the hydrostatic balance


Use p as a vertical coordinate or any other pressure type coordinate (terrain following : sigma, hybrid)

Hydrostatic approximation : consequences

Hypsometric equation : the thickness between 2 isobars is proportional to the mean temperature in the layer between these 2 isobars

2

112 ln

p

p

g

TRzzz

The geopotential of a layer is obtained thanks to the integration of the hydrostatic equation

gz

soos pBpApp /


Validity of hydrostatic approximation

For internal gravity wave :

km101 ,,

yxyx

z LL

L

m

k

Toward the smaller scales :

force grad. pres. hor.acc. hor.

)/(/ LpU

Hydrostatic approximation if :

1)/(/2

2

force grad. pres. vert.acc. ver.

L

HHpW


Hydrostatic vs. Non-hydrostatic

Horizontal divergence for a flow past a 3D - mountain on the sphere ( r = a/100 ) with a T159L91 IFS simulation

hydrostatic non-hydrostatic


Hydrostatic vs. Non-hydrostatic

Hydrostatic waves only Hydrostatic + non-hydrostatic waves


Anelastic approximation

What is “anelastic approximation”?

Neglect the elasticity of the atmosphere which is responsible

for the accoustic wave propagation.

How to do that ?

Modify the continuity equation in order to neglect the “quick”

response of the density to compression

0 uu refdt

d The air is still compressible in the sense that its density may change, for exemple in a vertical motion, but it will change passively, without “elastic” reaction or oscillations.law) gas(perfect diagnostic

pronosticfully


Anelastic approximation : consequences

Balance between horizontal and vertical mass fluxes

The anelastic approximation may be useful if you need to take into account the NH effect and you don’t have very sophisticated numerics to treat the sound waves.


Primitive (hydrostatic) equations in IFS

forMomentum equations

Sub-grid model :“physics”

Numerical diffusion



Thermodynamic equation

Moisture equation

Note: virtual temperature Tv instead of T from the equation of state.



Continuity equation

dp

pp

t hs

s

1

0.

1)ln( v

Vertical integration of the continuity equation in hybrid coordinates


One word about water species….Phase changes are treated inside the “physics” (P terms)

But the pronostic water species have a weight. They are included

in the full density of the moist air and in the definition of the

“specific” variables. It does some “tricky” changes in the

equations. For ex. :

Pronostic water species should be advected. They are then also

treated by the dynamics.

k

ilvdk mmmmmVm scondensate pronostic

... with /

)1(1or )1(1with

or or

v

avv

v

avd

vdvdd

M

MqTT

M

MqRR

TRpRTpRvTTRp Perfect gas equation

mmq kk /


Map projections

Invented to have an angle preserving mapping from the sphere onto a plane for convenience of display.

Hence idea to perform computations already in transformed coordinates.

Map factor: 2

( ) , = latitude1 sin

m

Wind components in the model are then usually not the real zonal and meridional winds.


Rotated spherical coordinates

Move pole so that area of interest lies on the equator such that system gives more uniform resolution.

Limited-area gridpoint models: HIRLAM, Ireland, UK Met. Office….; Côté et al. MWR (1993)

Move pole to area of interest, then “stretch” in the new “north-south” direction to give highest resolution over the area of interest.

Global spectral models – Arpege/IFS: Courtier and Geleyn, QJRMS Part B (1988)

Ocean models have sometimes two poles in the continents to give uniform resolution over the ocean of interest.

( , )


Courtier and Geleyn (1988)


Shallow water equations

Useful for (hydrostatic) dynamical core test cases before full implementation.

Route to interpret isentropic or isopycnal models.

eg. Williamson et. al., JCP Vol 102, p. 211-224 (1992)

Further reading: Gill (1982)

hcsteo

oop Free surface


Shallow water equationsAssume constant density + free surface at z=h(x,y)

horizontal pressure force independent of height:

Horizontal wind independent of height as :

Use only the horizontal motion equation at the ground, where w=0

fluide) sible(incompres 0.0 udt

d o

zx

hg

x

phzgpzp oooo

)()(

fuy

hg

dt

dvfv

x

hg

dt

du

;

y

uv

x

uu

t

u

dt

du



Boundary conditions: w=0 at z=0 and free surface following the motion at the top (dh/dt=w). Integrating the continuity equation we obtain:

)0(0)(

hy

v

x

u

dt

dh

hw

dz

yxD

y

v

x

udz

z

w hz

z

hz

z

00

),(



In component form in Cartesian geometry:

(1)

(2)

(3)



Deriving an alternative form:

Vorticity:

Divergence:

Kinetic Energy



The advection term in the velocity equation may be transform into the Lamb form; the vector product of vorticity with velocity is called the Lamb vector (useful for generalised form of Bernoulli equation):

In spherical geometry:


Isopycnal/isentropic coordinates : representation of a stratified fluid as a superposition of “shallow water” models (model levels = material surface)

defines depth between “shallow water layers”

(momentum)

(continuity)

(thermo)

(hydrostatic)


More general isentropic-sigma equationsKonor and Arakawa (1997);

terrain-following

ecmwf governing equations 2 slide 1 governing equations ii: classical approximations and other...

Documents

averaged equations

ecmwf governing equations

exact equations

fundamental equations

set of evolution equations

systems of equations

continuum slide

mean grid size scale