Paläozeanographische Modellierung André Paul Raum: GEO 5510, Tel.:

The feature, which runs parallel to the contour of zero wind stress curl some 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 state density (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 verschiedeneversetzte 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 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 Common unit of is a Sverdrup with 1 Sv = 10 6 m 3 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