Numerical simulation of a climate energy balance model with continents distribution

Numerical simulation of a climate energy balance model

with continents distribution.

Lourdes Tello & Arturo Hidalgo

Universidad Politécnica de Madrid.

Some processes involved in global climate models:

Some processes involved in global climate models:

w

w

w

w

w

(-1,-H) Pole S (-1,0)

Pole N (1,0)

(1,-H) Equator

(0,0)

Solar radiation

Cooling Cooling

Physical problem

upwelling

THE MODEL (Based on Watts-Morantine [1990])

•

The model represents the evolution of temperature within an ocean of

depth H.

•

Spatial variables (x,z): x = sin(latitude) and

–z

(depth).

•

Spatial domain

•

Boundary:



1,1

 

0,

H



  





1 0 1 H 

   



























0 1 1

( , )

:

0

(

( , )

:

1

, )

:

:

(

1

, )

H

x z

x z

x z

z

z

x

z

x

x

H





 





 

 

 

 

 

 

 







_H ₀

_-1 1

(-1,-H) (1,-H)

(-1,0) (1,0)

•The model considers the average temperature over each parallel as the unknown.

The governing equation for the ocean interior is a heat equation with advective transport (DOM)

Mathematical model

U: temperature,

w

: vertical velocity,

K

_v

: vertical diffusivity,

K

_H

: horizontal diffusivity,

R: radius of the Earth.

2 2

(

(1

)

)

0 in (0, )

,

1... .



H













t x x V zz z i

K

U

x U

K U

U

T

i

N

0,

(0, )

x v z H

xU

K U

on

T

w





 

Boundary condition for upper boundary: Energy Balance Model (EBM)

 

0 0 2 2 ₂ 2

0

1

(1

)

(

(

,

)

)

p

p H

t x x V x

x

DK

_U

Du

x

u

u

Bu

C

K

xu

QS x

u

z

o

c

T

R

n

w











_



_



_



 





 







Boundary condition for ocean bottom

u: temperature,

w

: velocity,

K

_v

: vertical diffusivity,

K

_H0

: horizontal diffusivity,

D: thickness mixed layer,



: density,

c: specific heat coeff.,



(u): coalbedo,

Q: solar constant,

Bu+C: cooling term,

-10ºC (u)

(u)=0.24

(u)=0.69

u

1) We consider the case p=3 (Stone, 1972)

2) We use the coalbedo,



(u), (Budyko model)

2 2

(

(1

)

)

0 in

(0, ),



H











t x x V zz z

K

U

x U

K U

U

T

R

w

0 in

(0, ),

x V z H

xU

K U

T

w





 





0 2 3/ 2

2 0

1

(1

)

( ) ( )

on

(0, ),

H

t x x _x V x

DK

_U

Du

x

u u

K

xu

Bu

C

QS x

u

z

c

R

T

w

















 



 

 

1

0,

0,

,

x

U



on

 

_

T

 

1

0,

0,

,

x

U



in

 

T

0

0 0

( , ,0)

( , ),

,

( ,0)

( ),

.









U x z

U x z

in

u x

u x

in

Final system

v

K

U

D

z





Represents the coupling atmosphere-ocean in the sense of analyzing the influence of the ocean temperature in the atmosphere.

In this work we shall show results with and without this term.

Mathematical model

Some references about global climate EBM models with or without

deep ocean effect :

•Watts&Morantine (1990),

•Xu (1990),

•Hetzer (1990),

• Kim, North & Huang (1992),

•Díaz (1993),

•Schmidt (1994),

•Díaz-Hernández-Tello (1997),

•Arcoya-Díaz-Tello (1998),

•Hetzer (2000),

•Díaz-Tello (2007),

•Bermejo et al (2008),

•Hidalgo-Tello (2011,2013,2014,2015),

We rewrite this problem as advection-reaction-diffusion equations, both for the upper boundary EBM and for the DOM.







, ( , ),

( , )



( , ( , ),

( , 0, ))

t x _x

U

u

f x u x t u x t

x u x t

x

t

z











EBM:





0 2 3/2

2

, ( , ),

_x

( , ) :



K

H

(1



)

_x

( , )

_x

( , )



w

( , )

f x u x t u x t

x

u x t u x t

xu x t

D

R

with the flux:

1

( , ,

x u

U

) :

(

C

Q

S x

( ) ( )

u

(

x

_x

B u x t

) ( , )

K

_V

U

)

n

D

c

z





w

w





_

_{ }

_

_

_

_







and the source term:

DOM:

( , , )

_t



( ( ,

_x

( , , )))

_x



( ( ( , , ),

_z

( , , )))

_z

 

( , ( , , )),

U x z t

F x U x z t

G U x z t U x z t

x U x z t

with the fluxes :







2



2

,

_x

( , , ) :



K

H

1



_x

( , , ),

F x U

x z t

x U

x z t

R

( ( , , ),

_z

( , , )) :



_{V z}

( , , ) -

( , , ),

G U x z t U x z t

K U x z t

wU x z t

and the source term:

( , ( , , )) :

x U x z t

_z

U x z t

( , , ).





w

Numerical approach: finite volume method with Weighted Essentially Non-Oscillatory (WENO) reconstruction in space and third-order Runge-Kutta TVD

for time integration.

For each time step, we compute a numerical solution of the EBM model equation for each cell

then we use as a Dirichlet boundary condition for the DOM to obtain

2 2

(

H

(1

)

)

0 in

(0, )

t x x V zz z

K

U

x U

K U

U

T

R

w













 

0 2 ₂

2 0

2

0

(1

)

(

( )

,

)

p

p H

t x x V x

x

DK

_U

Du

x

u

u

Bu

C

K

xu

QS x

n

T

z

o

u

R

w









_



_



_



 









 



1 n i

u

 1 n i

Upper boundary:



₀

Domain:



(1,-H)

(1,0)

(-1,0)

(-1,-H)



1/2 1/2



( )

1

( )

( ( )),

 











i

i i i i

i

du t

f

f

t

l u t

dt

x



The finite volume framework

 

1/2 1/2

1

( )

,

 







i i x i i x

u t

u x t dx

x



 







1/2

x

,

1/2

,

,

1/2

,





 

i i x i

f

f

u x

t u

x

t

where

We integrate the equation dividing by the length of the control volume to obtain the following ordinary differential equation (ODE)

integral average of the unknown,

right intercell numerical flux,

1/ 2

1

( )

( , ,

)

i i x

U

t

x u

dx

















integral average of the source term.

x_i-1/2 x_i+1/2

The finite volume framework

We discretize the 2D domain [-1,1] [0,-H] in N_xN_z control volumes of area x_iz_j x_i=x_i+1/2-x_i-1/2, z_j=z_j+1/2-z_j-1/2

x

_i

(x

i+1/2

, z

j-1/2

)

(i,j)

(x

_i-1/2

, z

_j-1/2

)

(x

_i-1/2

, z

_j+1/2

)

(x

_i+1/2

, z

_j+1/2

)

z

_j

(1,-H)

(1,0)

(-1,0)











 







1/ 2 1/ 2 1/ 2 1/ 2 1/ 2, , 1

1/ 2 1/ 2

1/ 2 1

/ 2 / 2

,

, ,

,

,

,

,

,

1

,

1

,

         













j j i i i j j i z

i x i

z

x

j z j

x j i

F

z

G

x

F x

U

x

z t

dz

G U x z

t U

x z

t

dx

Spatial integral average of intercell fluxes,

1/ 2 _{1/ 2}

1

( , ( , ,

( )

))

,

j _i

z _x

t

x U x z t

dx

dz





_



















 



integral average of the source term.

The finite volume framework

where









,

1/2, 1/2, , 1/2 , 1/2 , ,

1

1

i j

i j i j i j i j i j i j

i j

dU

F

F

G

G

L

dt





x











z







  

1/ 2 _{1/ 2}

1/ 2 1/ 2

,

1

( , , )

j _i j i z x i j

i j z x

U

U x z t dx dz

x z

   









_



_

 

 

_

_

We integrate the equation dividing by the area of the control volume to obtain the following ordinary differential equation (ODE)

,1 ,2 ,1 ,1

1 ,2 ,2

3

1

1

(

),

(

),

4

4

4

1

2

2

(

).

3

3

3

k n n k n k k

n n k k

u

u

tl u

u

u

u

tl u

u



u

u

tl u



 





 





 

Runge Kutta TVD

 

 

,1

,2 ,1 ,1

1 ,2 ,2

(

),

3

1

1

,

4

4

4

1

2

2

.

3

3

3

k n n

k n k k

n n k k

U

U

tL U

U

U

U

tL U

U



U

U

tL U



 





 





 

EBM:

WENO reconstruction

EBM

For an order of accuracy r we have r candidate stencils each one of them with r cells {S_i-r+1, S_i-r+2,…,S_i}, {S_i-r+2, S_i-r+3,…,S_i+1}, … , {S_i, S_i+1,…,S_i+r-1}





( ),

0,

,

1

l

p x

l

r

For each stencil we consider a (r-1)th degree interpolating polynomial

1

( )

( ),

0

1,

0

1

k

l k

S k

p x dx

u t

l

r

k

r

x



  

  





Each one of the polynomials considered must be conservative:

1) For intercell fluxes

x

_i

(x

i+1/2

, z

j-1/2

)

(i,j)

(x

_i-1/2

, z

_j-1/2

)

(x

_i-1/2

, z

_j+1/2

)

(x

_i+1/2

, z

_j+1/2

)

z

_j

(0,0)

(1,0)

(1,1)

(0,1)

1/ 2 1/ 2







 



 

i i

j j

x

x

x

z

z

z





ξ

η

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1 1/ 2 1 0 1 1 1/ 2, 1/ 2, , 1 / 2 1 0 /

2 1 2

/

1

1

ˆ

ˆ

(

, ,

)

(1, ,

)

(1,

,

)

ˆ

ˆ

(

, ,

)

(0, ,

)

(0,

1

,

)

( ,

,

)

            

































j j j j i z _NG

n n z n

i k k

k z

y

NG

n n z n

i k k

k y i j j i j j i j i n j x

F

z

F

F x

z t dz

F

t d

F

t

F x

z t dz

F

t d

F

t

G x z

t d

x

x

z

G

















1/ 2 1/ 2 1/ 2

, 1/ 2

1 1 0 1 1/ 2 1 0

ˆ

_{( ,1,}

₎

ˆ

₍

_,1,

₎

ˆ

ˆ

( ,

,

)

( , 0,

)

(

, 0

1

,

)

      

























i i i x _NG

n x n

k k

i

k

x _NG

n n x n

j k

i x k

j k

G

t d

G

t

G x z

t dx

t d

G

G

G

t

x

















Integrals are approximated using Gaussian numerical quadrature

(1,1)

(0,1)

η

1 2 ... NG

We must compute a unique flux at each control volume interface

, , 1, , 1/ 2, 1/ 2, 1/ 2,

, , , 1, 1/ 2, , 1/ 2 , 1/ 2

1

(

)

2

1

(

)

2



  



  









i j RIGHT i j LEFT

i j i j i j

i j UP i j DOWN

i j i j i j

F

F

G

G

G

F

REMARKS:

1)When considering advective terms, other types of flux averaging

or Riemann problem solutions must be introduced: Force, Rusanov, Osher, Roe… 2) In the diffusive averaging of the flux a term that accounts for the jump

If we use 2 integration points (NG=2) the values are: , , 0 1 0 0

1

3

3







 

x z x z









1/ 2 _{1/ 2}

1/ 2 1/ 2

1 , 1 0 0 1 1

1

( , ,

)

1

( ,

( )

(

,

,

,

)

)

 _    

























_

_



















_



 











 

 

 

j _i j i z _x n z x NG NG x n i j i j n i j i j z n

k l k l

k l

t

S x z t

dx dy

x z

S x z t

x d

d

G

t

z

x z









 

(0,0)

(1,0)

(1,1)

(0,1)

ξ

η

1 2 ... NG

In order to obtain the solution and gradients at Gaussian points we

need to perform a reconstruction procedure. This give rise to a piecewise polyonomial function whose restriction to cell T_ij is the polynomial:

1 1 , 1 1

ˆ

( , ,

)

( )

( ) ( )

   



M

 

M

n k l n

ij ij k l

k l

w

 

t

w

t

   

And its gradient

, 1 1 , 1 1

ˆ ( ) ' ( ) ( )

ˆ ( ) ( ) ' ( )

   









_



_

_



_{ }

_

_







_



_

_











 

ij

k l n M M

ij k l

k l n

k l ij k l

ij

w

w

t

w

t

w

   

   





Dimension-by-Dimension WENO reconstruction

which must be conservative:

1 _{1 1} 1 1

, 1 1 0 0 0 1 0

ˆ

( , ,

)

( )

( ) ( )

   











n n M

 

M k



n

i i

l n

ij ij k l

k l

, ,

and

   

     



i s e



j s e

s x s y

ij pj ij qj

p i s e q j s e

S

T

S

T

Dimension-by-Dimension WENO reconstruction

Substencils for obtaining nonlinear reconstruction operator

If we consider 3 cells in the stencils: e=1 and s=-e for the left-sided stencil s=0 for the central stencil and s=e for the right-sided stencil.

The total stencils are given by the union of the substencils

, ,

and





x s x z s z

ij ij ij ij

s s

i-3 i-2 i-1 i i+1 i+2 i+3

M=even (M=2)

M=odd (M=3)

Three stencils

Four stencils

M+1 cells in each stencil

j R i R

s,x s,z

ij ej ij ei

e i L e j L

I ,

I

 

   





WENO RECONSTRUCTION

s=1 s=2

s=3

s=0

s=1

s=2

x ij

S

z ij

S

x

_x

z

z

ij

T

T

_ij



x

x

_{i1/ 2}

x

_

x

_

1/ 2



j

z

1/ 2



j

z

1/ 2



j

z

1/ 2



j

z

Let us denote the one-dimensional polynomial WENO reconstruction in x direction as:

Dimension-by-Dimension WENO reconstruction

ˆ ( )

_l n



_x

(

_pjn

),

_pjn



_ijx

w t

R T

T

S

And the one-dimensional polynomial WENO reconstruction in z direction as:

ˆ ( )

_l n



_z

(

_iqn

),

_iqn



_ijz

w t

R T

T

S

Application of the reconstruction operator in x-direction to the cell averages yields the coefficients of the reconstruction polynomial in x-direction.

n pj

T

,0

(

),

ˆ (

_ijl

t

n

)



R T

_{x pj}n

T

_pjn



S

_ijx

w

Since the reconstruction operator in direction acts on averages in z-direction, it can be applied to each single coefficient of the reconstruction

polynomial in x-direction.

(

n

)

y iq

R T

, ,0

(

), 0

,

ˆ

_ijl m

( )

n



_z

ˆ

_iql

( )

n

 



_iqn



_ijz

In this way we obtain all the necessary coefficients of the 2D tensor-product reconstruction polynomial

Dimension-by-Dimension WENO reconstruction

1 1 , 1 1

ˆ

( , ,

)

( )

( ) ( )

 

 



M

 

M

n k l n

ij ij k l

k l

w

 

t

w

t

   

Remarks:

This way to proceed is different to the classical point-wise WENO approach, since entire polynomials are obtained instead of piecewise values (as in the original WENO Of Jiang and Shu).

Some references of polynomial WENO reconstruction:

“High order space-time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems“, M. Dumbser, A. Hidalgo, O. Zanotti

Computer Methods in Applied Mechanics and Engineering, 268, 359-387 (2014)

























1 1 1 1 1 1 1 1

(1,

,

);

ˆ 1,

,

0,

,

(0,

,

)

;

ˆ

,1,

,1,

, 0,

, 0,

       

















n n

k k k k

n n n n

k k

M M

n n

k ij k ij

k k

M M

k ij k ij

k k

k k

u

t

u

t

u

t

t

w

t

w

t

w

u

t

w

t

















w

w

w

w

























1 1 0 0 1 1 0 0

ˆ

1,

,

ˆ

0,

,

ˆ

,1,

,1,

ˆ

(1,

,

);

, 0,

,

(

0

0,

,

)

;

,

       

































r r n n

k ij k ij

k k

r r

k ij k ij

k

n n

k k k k

n n n n

k k

k

k k

w

t

w

t

w

w

u

t

u

t

u

t

t

u

t

t

w

w

w







w











Parameter Scaled Value

K_H 0.049

K_H0 0.55510-3

K_V 0.0125

C, B 190, 2 c,  1, 1

Q 340

D 60

Numerical example without latent heat:



(U)=U

Physical parameters:

10(x 0.75)(x 0.75)

(x, z) W(x)

(0.1 10 x 0.75 )(0.1 10 x 0.75 )

 

 

   

w

Space and time discretization:





1

x 2 / 60; z 1/ 60



 

 

 

 

2 1

S 1 P (x)

2  

w=0.1

w=-0.1 w=-0.1

w=0.1

Initial condition:

2 2 2

( , , 0)

1

8

x z

80

x

6

0

U x z



e

 



e





Numerical example

DOM solution. Output time = 1.0

0



_



v

K

U

D

n

WITH influence of deep ocean on atmosphere.



_

v

K

U

WITHOUT influence of deep ocean

2 2

(

(1

)

)

0 in (0, )

,

1... .



H













t x x V zz z i

K

U

x U

K U

U

T

i

N

R

w



Final system

Mathematical model with land-sea distribution





0 2 0 1 2 2 / 2

(1

)

1

( ) ( , )

in (0, )

,

 









_

















 







i M V i p H _p

t x x x

x

DK

Du

x

u

u

wxu

Bu

C

R

QS x

x u

U

z

c

K

T











2 2 / 2

1 1

0

0 0

0

in (0, )

,

1... .

(1

)

0

in

1,

0

in ((0, )

)

((0, )

),

(0, , )

( , )

in

,

(0, , 0)

( )

in

 













 













i

x V z H

p p x x i i x i

wxU

K U

T

i

N

x

u

u

x

U

T

T

U

x z

U x z

U

x

u x







90



67



7.5



55



90



Temperature in the deep ocean for t=5.

A) First ocean; B) Second Ocean.

Solution for t =5.

Full green line

:

land-sea model

;

dotted red line: only

Continental model

;

dash-dotted blue line: Only Ocean model





2





₂

2 2

1

1

( , , )

10

1









x

x

z

U t x z

t

Validation of the numerical scheme

with land-sea distribution





0

2 2

2 2 3/ 2

2

1

(

(1

)

)

,

in (0, )

,

1, 2

0

in (0, )

,

1, 2

(1

)

1

(

( , , )

)





































_

_



 













i H

t x x V zz z i

i

x V z H

H

t x x _x V x

i

K

U

x U

K U

U

T

i

R

wxU

K U

T

i

DK

_U

Du

x

u u

K

wxU

C

Bu

z

R

QS x

c

t x z



w



















 

 

 

 









 

 

 



2 3/ 2

1 1

2 _{2 2 2 2 2}

2

2 _{2 2 2 2}

2

0

( , )

(1

)

0

in

1,

0

in ((0, )

)

((0, )

),

(0, , )

200

1

0.8

0.4

0.4

1

,

,

( , )

1

(0, )

200

1

0.8

0.4

0.4

1

in

.









 



































x x i i x

x u

x

u u

x

U

T

T

U

x z

x

x

x

x

x

z

u

x

x

x

x

x

t

x

x









2

2

1

|| ||

( ( )

( ))







Nx n



n

L i i

i

x

u t

u t



Validation of the numerical scheme

with land-sea distribution

Other results (latent heat)

Multiple solutions and numerical analysis to the dynamic and stationary models coupling a delayed energy balance model involving latent heat and

discontinuous albedo with a deep ocean

J. I. Díaz, A. Hidalgo, L. Tello

WITH LATENT HEAT (t=5)

Conclusions and

further research

•We have obtained the numerical solution of a 1D energy balance model with nonlinear diffusion, coupled with a 2D deep ocean model in a rectangular domain.

•The method used is a finite volume method with 3rd order Runge-Kutta TVD. •It has been obtained the evolution of the temperature in the deep ocean and also in the surface, due to the combination of melting ice, heating-cooling of the surface of the ocean.

•The results show the thermostatic effect of the ocean.

•The effect of the land-sea distribution has been considered in the problem. •A verification of the accuracy of the scheme has been carried out solving an auxiliary problem with known analytical solution.