forces Coriolis f. deviation to the RHS in q high - - PDF document

SLIDE 1

Applied

PhD

Talks

@ HW Surface Semi

• geo strophic

Equations

Stefania Lisei

02/05/18

Geophysical

Fluid

Dynamics

:

fluid

dynamics

• f naturally

Occur ing

flows

,

such

as

• ceans

a

atmosphere

Dynamics

dominated

by

:

Pressure

gradient

Rotation

f. µ

Coriolis forces

⑤←

deviation

to q

the

RHS

in

/

high pressure

horizontal N

. atmosphere .

f=2R

sine

pressure

gradient

⇒ f-

• along

equator

SLIDE 2

Geographic

behave

:

Coriolis

force

deviates pressure

gradient

by

It

, so

dynamics

along

isobars

Limitation

:

friction

!

gn

Friction weakens

Coriolis force

high

pr%

before

⑤

cyclones

a

①

anticyclones

We need

a

model for atmosphere

&

• ceans

with better

approximation

than

apostrophe balance

:

• quasi
• apostrophe

equations

(

QG )

: easier

but

less precise ,

• semi
• goo strophic

equations

I

SG ) : more

complex

I fully

nonlinear )

but better

approximation for

some

dynamics

I eg fronts

formation )

.

SLIDE 3

Semi

• goo strophic

equations

Iincompressible

, in viscid

)

:

I ft the

. D)

igtffuz

fu )at

to

=

• ff

t =

fzp

I

= Ig t

Ia

✓ → ages strophic

full

wind

velocity

thestrophic

div

I

= wind

fig

=

f- dyp

, 2×10)

to

= pressure ,

f= buoyancy

anomaly

I ft the

. D) t =

• f

=

Coriolis

frequency

, assumed

constant

Ig

is

the solution

in

the

genestrophic

balance

.

Important quantity

: Rossby

number Ro

.

Ro

measures

how

strong

the

effect

• f

Earth

rotation

is

for

the

dynamics

.

Ro

= ¥

I

dimensionless )

.

Ro

=

corresponds

to

apostrophe

behave

I

no

friction I QG

a SQ have

physical meaning

when Ro

is small .

SLIDE 4

Some references

:

• Eliasson

' 48 :

into

SG .

• Hoskins

A

Bretherton

' 72 : SG is

useful for

large-scale

dynamics

a

front genesis

• Hoskins

' 75 :

intro

SSG

and

apostrophe

coordinates

.

• Cullen

A

Purser

' 84 :

results

• n

Lagrangian

space

.

• Cullen

9

Shuts

I 87 : stability principle

• Cullen

' 06 :

book

• n

henge

scale

atmospheric

loonie flows

.

• Benemou

A Brenier

' 98 :

week solutions

• f

SG in

apostrophe

coordinates

through Optimal Transport

.

• Ambrosio

, Colombo , De Philipp is , Figolli ' 12 , ' 16 : week

solutions

• n

special

domains

.

• Bad

in A Regine 46 : numeric

• n

SSG .

SLIDE 5

Surface

semi

• Geestrophic

Equations

Hoskins

I

' 75 )

intro

word

tnensf

• n

D= IR2 x

lol )

think

that

)

=

III.

3151

Use

X= IX. 42 )

for guest

coordinates

.

A

new

stream function

I

is

def

• n

X sit .

DX

IH1H

=

Dap

IHIIXI

't )

→ new eg

for E.

Assuming

constant PV t

b.

as

ftttw

.

0/0=0

• n

IR2

¥

THIN Issa Given f.

define

formally

TITI

=

4 if

4--5/2=1

and

DE

=

2xx$2yy§

• l2xy$ )

'

am

SLIDE 6

Need

to

define

rigorously

T

.

Thm I

L ,

Wilkinson

' 18 ) .

There

exists a constant

Go

st .

given

EC

"

CITY

smell ,

there

exists a

solution

EE0Y

• f

IBVP ) .

Moreover

, § is

unique

• n

a small ball .

Idea

• f proof

: BFPT t

elliptic

theory

for the linear BVP

Thm

It

, Wilkinson ' 18) :

For

smell foe

• d. TTY

It

>

• sit

there exists a

unique

classical solution

to ISSG )

• n

IT4

6II

Notes written on Good notes