ECMWFGoverning Equations 1 Slide 1

Governing Equations I

by Clive Temperton (room 124) and Nils Wedi (room 128)

ECMWFGoverning Equations 1 Slide 2

Overview

Introduction

Fundamental physical principles

Eulerian vs. Lagrangian derivatives

Continuity equation

Thermodynamic equation

Momentum equation (in rotating reference frame)

Spherical coordinates

Recommended: An Introduction to Dynamic Meteorology, Holton (1992)An introduction to fluid dynamics, Batchelor (1967)

ECMWFGoverning Equations 1 Slide 3

Equations

N 0

Newton’s second law

Boltzmann equations

Navier-Stokes equations

Euler equations

individual particles statistical distribution continuum

0l

0l

Note: Simplified view !

Mean free path

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

ECMWFGoverning Equations 1 Slide 4

Continuum assumption

All macroscopic length (and time) scales are to be taken large compared to the molecular scales of motion.

Mean free path length l of molecules in atmosphere:

Surface ~ 10-7 m

16 km ~ 10-6 m

100 km ~ 0.1 m

135 km ~ 15 m

In ocean:

~ 10-9 m

ECMWFGoverning Equations 1 Slide 5

Fundamental physical principles

Conservation of mass

Conservation of energy

Conservation of momentum

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

ECMWFGoverning Equations 1 Slide 6

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

ECMWFGoverning Equations 1 Slide 7

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 per unit volume is

= rate of increase in mass per unit volume

= rate of change of density

=> Continuity equation (N.B. Eulerian point of view)

zyu zyxux

u ])([

)( ux

)]()()([ wz

vy

ux

( ) v

( )t

v

x

z

y

ECMWFGoverning Equations 1 Slide 8

Thermodynamic equation

First Law of Thermodynamics:

where I = internal energy,

Q = rate of addition of heat (energy),

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

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

Alternative forms:

WQdt

dI

TcvI cv

dt

dpW

)/1(

dT dQ pv dt dtc

p

dT RTc Q

dt p dt

dp

QTdt

dcp

0

Tpp

cR p/

Note: Lagrangian point of view.whereor equivalent ,

(R=gas constant) and

pRT

Eq. of state:

ECMWFGoverning Equations 1 Slide 9

Momentum equation

Newton’s Second Law in fixed frame of reference:

N.B. use D/Dt to distinguish the total derivative in the fixed frame of reference.

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

= angular velocity of earth, = velocity relative to earth,

=position vector relative to earth’s centre.

Orthogonal unit vectors: in fixed frame,

in rotating frame.

D

Dtav

F

rav v

v

r

, , i j k

, ,i j k

(1)

(2)

ECMWFGoverning Equations 1 Slide 10

Momentum equation (continued)

For any vector ,

in fixed frame

in rotating frame.

v

x y zv v v i j kv

x y zv v v i j kv

yx z

yx zx y z

dvdv dv

dt dt dtdvdv dv d d d

v v vdt dt dt dt dt dt

i j k

= i j k + i j k

Dv

Dt(fixed frame)

(rotating frame)

Now = velocity of due to its rotation = ,etc.d

dti i i

ECMWFGoverning Equations 1 Slide 11

Momentum equation (continued)

x y z

D dv v v

Dt dt i j k

v v

( x y z

dv v v )

dt i j k

v

D d

Dt dt

v vv

Reminders: (a) is the total derivative in the rotating system.

(b) Eq. (3) is true for any vector .

dt

d

v

(3)

ECMWFGoverning Equations 1 Slide 12

Momentum equation (continued)

D d

Dt dt a a

a

v vv

[ ] ( )D d

Dt dt r rav

v v

( )d d

dt dt

rr

vv

2 ( )d

dt r

vv

2 ( )d

dt r F

vv

Coriolis centrifugal

and finally using Newton’s Law [Eq. (1)],

Now substituting from Eq. (2),

In particular:

ECMWFGoverning Equations 1 Slide 13

Momentum equation (continued)

Forces - pressure gradient, gravitation, and friction

Where = specific volume (= ), = pressure,

= sum of gravitational and centrifugal force,

= friction.

Magnitude of varies by ~0.5% from pole to equator and by ~3% with altitude

(up to 100km).

F

2d

p gdt

kv

v

/1 p

g

g

ECMWFGoverning Equations 1 Slide 14

Spherical polar coordinates

: = 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

ECMWFGoverning Equations 1 Slide 15

Components of momentum equation

p

rr

uw

r

uvwv

dt

du

cos

1tancos2sin2

p

rr

vw

ruu

dt

dv 1tansin2

2

rg

r

p

rvuu

dt

dw 1

cos222

rw

a

v

a

u

tdt

d

cos

“Shallowness approximation” – take r = a = constant,

where a = radius of earth.

with