# Numerical Modeling of Weather and Climate

## Full text

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### The shallow water equations represent a homogeneous layer of incompressible fluid of constant density, which is confined above by a free surface. The free surface represents a discontinuous density interface, and it implies stratification effects. The one-dimensional shallow-water system is sketched in Fig.3.1.1. We make two important assumptions: First, the interaction with the overlying layer of fluid (e.g. air) is neglected. Second, it is assumed that the horizontal velocity is independent of height, i.e.

∂u/∂z=0. The latter assumption restricts

*

!

g* =g"

!

"

=

\$

u

u

!

g*=g"

Δ

(4)

*

*

o

*

o

(5)

nj

nj+1/ 2

### Thus, the grid has the following structure:

ui!1/ 2 ui+1/ 2 ui+3/ 2

Hi!1 Hi Hi+1 Hi+2

H[i-1] H[i] H[i+1] H[i+2]

u[i] u[i+1] u[i+2]

!

ui+1 / 2

!

ui+1 / 2

!

1

2"t

### [

un+1j+1/ 2#un#1j+1/ 2

+ u2"xnj+1/ 2

### [

unj+3 / 2#unj#1/ 2

+ 1

"x

## [ (

Hnj+1+hj+1

#

Hnj +hj

= 0

n+1j

n#1j

nj+1

nj+1

nj#1

nj#1

nj

nj#1/ 2

nj+1/ 2

n+1

n+1

(6)

!

Hu

!t + !

u Hu

!x + H!

h+H

!x =0

!

"(H2/2) "x

a b

2

2

ab

### I

NTEGRATION OF THE CONTINUITY EQUATION

### The continuity equation (3.1.6) is integrated using the first-order flux form of the upstream scheme (see Schär 2006, chapter 6.3).

!

Hin+1 " Hin

#t + Fi+1/ 2n+1/ 2 " Fi"1/ 2n+1/ 2

#x = 0

### with the upstream fluxes estimated as

!

Fi+1 / 2n+1 / 2 =

ui+1/ 2n Hin+1 for ui+1/ 2n " 0 ui+1/ 2n Hin for ui+1/ 2n # 0

\$ %

&

### On the computer, this is implemented as

!

Fi+1/ 2n+1/ 2 = Max

0,ui+1/ 2n

Hin + Min

0,ui+1/ 2n

Hi+1n

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### – 3.7 – I

NTEGRATION OF THE

OMENTUM EQUATION

!

"q

"t + "

uq

"x + H"

h+H

"x =0

in

in

in"1/ 2

i+1/ 2n

in+1

in

i+1/ 2n+1/ 2

i"1/ 2n+1/ 2

in

i+1

i"1

## ( ) + ( H

i+1n

i"1n

### where, analogous to (3.1.25), the upstream fluxes

!

Qi+1/ 2n+1/ 2 = Max

0,ui+1 / 2n

qin + Min

0,uin+1 / 2

qi+1n

!

qin+1

### has been computed, one diagnoses the “advective velocities” on the staggered grid as

!

uin+1+1 / 2 = qin+1+qin+1+1

Hin+1+Hi+1n+1

(8)

g

g

!1

y

x

xx

yy

2

2

z

z

o

o

1/2

= "o g

#

#z

t

g

t

g

t

Dg

D t = !

!t + ug !

!x + vg !

!y

(9)

t

xx

yy

2

2

z

z

t

z

o

t

g

g

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!u

!x

"

# \$

\$

%

&

' '

p

+ !v

!y

"

# \$

\$

%

&

' '

p

+!

!p=0

= Dp Dt

!

!t +!

u

!x +!

v

!y +!

w

!z =0

s

s

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:= p

ps

s

### The deformation of σ -coordinates that results from underlying topography decays with height. Sigma-coordinates are also suited for oceanic models. However, to avoid collapsing computational surfaces near coast lines, vertical walls must be introduced (Fig.3.3.1d). This implies neglecting the continental shelf.

! = 0

! = 1 p = ps p = 0

! = const

p = const z = const

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(c)

(b)

(d)

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!

(x,y, ˜ z )

### The implementation of an atmospheric or oceanic model in generalized coordinates

!

( ˜ x , ˜ y , ˜ z )

### involves two steps. In a first step, the governing equations need to be transformed into the new coordinate system. In a second step, a discretization is introduced, for instance by using grid increments

!

("x ˜ ,"˜ y ,"˜ z ) in the new coordinate directions.

### We consider the transformation of a Cartesian coordinate system (x,y,z) into a (generally non- Cartesian) coordinate system

!

( ˜ x , ˜ y , ˜ z )

### and it can readily be generalized to an arbitrary (curvilinear) coordinate system

!

( ˜ x , ˜ y , ˜ z )

D Dt

### According to (3.3.10b), the vertical wind in

!

( ˜ x , ˜ y , ˜ z )=(x,y,p)

### and measured in [hPa/s]. Note that it is negative (positive) for ascending (descending) motions. Similarly, the vertical wind in isentropic coordinates

!

( ˜ x , ˜ y , ˜ z )=(x,y,

)

. := D

D t

. "0

D

D t"0

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(x,y,z)

!

D

D t =

t + D x

D t

x

#

\$ % &

' (

y,z

+ D y

D t

y

#

\$ % &

' (

x,z

+ D z D t

z

#

\$ % &

' (

x,y

=

t + u

x

#

\$ % &

' (

y,z

+ v

y

#

\$ % &

' (

x,z

+ w

z

#

\$ % &

' (

x,y

### and in the generalized

!

( ˜ x , ˜ y , ˜ z )

!

D

D t =

t + Dx ˜

D t

x ˜

#

\$ % &

' (

y ˜ ,˜ z

+ Dy ˜ D t

y ˜

#

\$ % &

' (

x ˜ ,˜ z

+ Dz ˜ D t

z ˜

#

\$ % &

' (

x ˜ , ˜ y

=

t + ˜ u

x ˜

#

\$ % &

' (

y ˜ ,˜ z

+ ˜ v

y ˜

#

\$ % &

' (

x ˜ ,˜ z

+ ˜ w

z ˜

#

\$ % &

' (

x ˜ , ˜ y

(x,y,z)

### and

!

( ˜ x , ˜ y , ˜ z )=(x,y,p)

!

" "x

p:=

" "x

y,p

!

" "p:=

" "p

x,y

y,z

y ˜ z

x ˜ z

x ˜ , ˜ y

!

s=s(x,y,z)

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### At each location (x,y), the function s(z) must be strictly monotonic in z to guarantee an invertible coordinate system. To address the associated transformation

!

(x,y,z)"(x,y,s), we

!z !s

o

o

o

!

(

x#0,

y=0,

s)

s

s

z

z

s

z

### Equations (3.3.20) and (3.3.21) are completely general. We have not made any assumptions, not even that (x,y,z) is a Cartesian system. It follows that s and z can be arbitrarily exchanged with each other, or replaced by other symbols (e.g. by p or θ ). The only important assumption is that the transformation

!

s=s(x,y,z) is invertible and continuous.

Du

Dt ! fv = !1

#p

#x

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Dv

Dt + fu = !1

#p

#y

!

f =2"sin(

### #

) denotes the Coriolis-parameter with

!

"=2

24h

!

D

Dt = "

"t+u "

"x +v "

"y +w "

"z

u=(u,v,w)

!

=Dp Dt

!

D

Dt = "

"t+u "

"x

#

\$ % &

' (

p

+v "

"y

#

\$ % &

' (

p

+

"

"p

x

#

\$ %

%

&

' ( (

p

=

x

#

\$ %

%

&

' ( (

z

+

z

z

x

#

\$ %

%

&

' ( (

p

p

x

"

# \$

\$

%

&

' '

p

=

p

x

"

# \$

\$

%

&

' '

z

+

p

z

z

x

"

# \$

\$

%

&

' '

p

(

p

x

"

# \$

\$

%

&

' '

z

=)

p

z

z

x

"

# \$

\$

%

&

' '

p

p

z="g

p

p

(16)

= Dp/ Dt

=g z

!

ps(x,y,t)

p=

RT

p

p

DT

Dt =

cp +Q

cp

!

!p=#1

Du

D t !f v =! "

"x

\$

% &

&

' ( ) )

p

+Fx

(17)

D v

Dt +f u =! "

"y

\$

% &

&

' ( ) )

p

+Fy

p

p

x

y

s

s

s

s

!

(p=0)=0 .

!

### "

(p= ps)=#ps #t+vs\$%hps

### where

!

vs =(us,vs) denotes the horizontal velocity at the surface. Equation (3.4.9) ensures

s

s

### (as the respective air parcel must remain at the surface) and that

!

"ps "p=0 (as the surface pressure ps

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=T pref p

"

#

\$ %

&

R cp

ref

p

.

.

=

(z)

=

(p)

!

!p <0

(19)

### 3.5.2 Isentropic form of governing equations (A) A

DVECTIVE FORM OF MOMENTUM EQUATION

p

v=(u,v)

p

(

(

.

!

!x

p

'

p

'

=RT/p

p

'

'

RT p

!p

!x

"

# \$ %

&

' (

= RT

pref

T

"

# \$ %

&

'

cp/R

pref !

!x

"

# \$ %

&

' (

T

"

# \$ %

&

'

cp R

= ! cpT

!x

"

#

\$ \$

%

&

' ' (

(20)

'

(

p

p

### (B) I

SENTROPIC MASS DENSITY AND CONTINUITY RELATION

### The mass between two isentropic surfaces, i.e. !

dz, can be derived from the hydrostatic

dz = "1 g

#p

#zdz = "1 g

#p

#z

#z

#

d

= "1 g

#p

#

d

!

:= #1 g

\$p

\$

dz =

d

)

)

.

### (C) H

YDROSTATIC RELATION

ln

= lnT " R

cplnp + R

cplnpref

cpT

=cp

"T

"

# RT

p

"p

"

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!p !z="g

p

p

p

p

ref

R cp

### (D) F

LUX FORM OF MOMENTUM EQUATIONS

)

)

.

)

### (E) B

OUNDARY CONDITIONS

s

t

t

t

s

s

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k = K+1/2

k = 1/2

### vertical section

k = K–1/2 k = K–3/2

k = 3/2 k = K k = K–1

k = 1

j–2 j–1/2

j–1 j+1/2

j j+3/2

j+1

i–1 i–1/2

i i+1/2

i+1 i+3/2

i+2

u v v, !, M u

M, !, ", ", p

" = "s+(k–1/2)#"

", ", p

### .

" = "s

"t = "s+(k–1/2)#"

="s+K#"

k = 5/2 k = 2

. "0

f =0

!

s=const

i+1/2, k

i, k

i, k

i, k+1/2

### (A) P

ROGNOSTIC TIME STEP FOR ISENTROPIC DENSITY AND MOMENTUM

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'

(

k

s

n+1

n+1

### (B) D

IAGNOSIS OF PRESSURE

n+1

n+1

n+1

i,kn+1

i,kn+1+1/2

i,kn+1"1/2

### g #\$ , which can be cast into the form

!

pi,k"1 / 2n+1

=pi,k+1 / 2n+1 +g#

i,kn+1

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