Paläozeanographische Modellierung

André Paul

Email: apaul@marum.de

Raum: GEO 5510, Tel.: 218 65450

• “The feature, which runs parallel to the

contour of zero wind stress curl some 5 -

10 degrees north of it, is called the

Subtropical Front.” (Tomczak and Godfrey,

1994)

Tomczak and Godfrey (1994), after Sverdrup et al. (1942)

Tomczak and Godfrey (1994)

Speer et al. (2000)

What is a model?

Models are• smaller than reality

(finite number of processes, reduced size of “phase space”)

• simpler than reality

(description of processes is idealized)

• closed, whereas reality is open

(infinite number of external, unpredictable forcing factors is reduced to a few specified factors)

(Hans von Storch)

Basics of numerical models

1. State variables

2. Fundamental equations

3. Parameterization

4. Discretization

5. Numerical solution

State variables

• Many variables can be thought of as a

“concentration“ or “property per unit

volume“.

• Fluxes then have dimensions of “property

per unit time and area”.

Examples of state variables

• Ocean

– Temperature

– Salinity

– Pressure

– Current velocity

• Atmosphere

– Temperature

– Density

– Humidity

– Cloud water content

– Pressure

– Wind velocity

Fundamental equations

• Conservation of momentum(horizontal) velocity (winds, currents)

• Conservation of mass (“principle of continuity”)vertical velocity, humidity, salinity

• Conservation of energy (“first law of thermodynamics”)temperature

• Equation of statedensity (air, sea water)

Parameterization in climate models

• Sub-gridscale processes, or processes

that cannot be derived from „first

principles“, must be parameterized

– e.g. thundercloud formation, soil moisture

transfer in the atmosphere, eddies and

convection in the ocean

• Beispiele für Parametrisierungen in CLISIM: – Ost-West-Druckgradient (als proportional zum

Nord-Süd-Druckgradienten angenommen)

– Wärmezufuhr an der Meeresoberfläche (als proportional zur Abweichung von einer Referenztemperatur oder “restoring temperature” angenommen)

To find a numerical solution to the fundamental equations on a digital

computer, they must be discretized in space and time.

Discretization

Most common in ocean models:

• “Finite difference” method in time

• “Finite difference” or “finite volume”

method in space

[Figure 3-30 from Ruddiman (2001)]

Discretization in space for a three-dimensional ocean model

• In CLISIM gibt es verschiedene “versetzte” oder “gestaffelte Gitter” für – Temperatur und Salzgehalt (“tracer” “T-

Gitterzellen”)

– horizontale Geschwindigkeit (an den nördlichen und südlichen Grenzflächen der T-Gitterzellen definiert) und

– vertikale Geschwindigkeit (am Boden der T-Gitterzellen definiert).

• In CLISIM ist der Zeitschritt t so gewählt, dass 40 Zeitschritte einem Modelljahr entsprechen.

Numerical solution

• Must be implemented as computer code

(mostly in Fortran)

• Must satisfy stability criteria

Numerical solution

u

xt

“No transport faster than one grid cell per timestep”

Example of stability criterion for many explicit time-stepping schemes: Courant-Friedrich-Levy (CFL) criterion

Puts severe constraint on time step and determines duration of model simulation

Initialization with T and S

Calculation of density field

Calculation of new velocities

Calculation of new T and S fields

Run completed?

End of run

T and S at sea surface(or heat and fresh-

water fluxes)

Wind stress at sea surface

No

Yes

Model output

Flow diagram for an ocean model

Zonally-averaged ocean circulation models

• Based on zonally-averaged primitive

equations

• Solved in zonally-averaged ocean basins

(only latitude and depth are resolved)

Wright and Stocker (1991, 1995)

Zonally-averaged ocean circulation models: geometry

No longitudinal resolution within

basins!

Stocker and Wright (1995)

Zonally-averaged ocean circulation models : example output

Pacific Atlantic

Salinity

Overturning

• Stromfunktion der Meridionalzirkulation im Atlantischen Ozean:– Massenfluss in der Deckschicht vom

Südatantischen Ozean in den Nordatlantischen Ozean wird durch eine Gegenströmung in der Tiefe kompensiert

Vertical-meridional streamfunction:A measure of the meridional overturning

circulation

• Vertical-meridional streamfunction:

A measure of the meridional overturning circulation

Common unit of is a Sverdrup with 1 Sv = 106 m3

s-1.

Streamlines are lines of constant values.

Rule: Volume transport between any two

streamlines = difference between corresponding

streamfunction values, where:

volume transport = velocity × cross-sectional area