Dynamics Gerrit Lohmann

1. Dynamics Gerrit Lohmann Dynamics for the atmosphere-ocean system Theory, numerical models & statistical data analysis Concepts of flow, energetics, vorticity, wave motion Atmosphere: extratropical synoptic scale systems Oceanic wind driven and thermohaline circulation

2. Dynamics • 26.Oct 2006- 8. Feb 2007: Thursday, 13-15 + 15-16, Room S3032Literature: • Holton, J.R., Introduction to Dynamical Meteorology, Academic PressGill, A., Atmosphere-Ocean Dynamics, Academic PressDutton, J.A., The Ceaseless Wind, DoverOlbers, D.J., Ocean Dynamics, Script, University of BremenCushman-Roisin, B., ENVIRONMENTAL FLUID MECHANICSCushman-Roisin, B. & Beckers, J.-M., Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects http:engineering.dartmouth.edu/%7EBenoit_R_Roisin/books/GFD.html • Lohmann, G, Mathematical modeling, Script, Unversity of Bremen, 2006 • http://www.awi.de/Modelling/Paleo/teaching.html • http://www.awi-bremerhaven.de/Modelling/Paleo/lessons/VL_WS2006/WS_0607_DynamicsI.html • How to get the Credit Points / Schein? • Homework / minutes (5 pages including figures) • Practical work within the lessons/ Projects with matlab and R • Help of Norel Rimbu & Thomas Laepple • Written exam: 26. Feb 2007, 10-12

3. Dynamics Preliminary Schedule: • 26 Oct: Intro, diff. Eq. & warming up • 2 Nov: Equations of motion • 9 Nov: Diffusion-Advection • 16 Nov: Simplified A-O Equations (NR) • 23 Nov: continued • 30 Nov: Examples • 7 Dec: Waves • 14 Dec:Vorticity • 21 Dec: continued

4. Language: • Position (x,y,z) • Velocity (u,v,w) • Pressure p • Density rho • Temperatur T • Salinity S • … Operators: d/dt Nabla …

5. Picture • Skales, dimensions: L, T, M, K • Derived: V, A, F, Pressure

6. Units

7. Variables

8. Northward Heat Transport

9. (residual method, TOA radiation 1985-89 and ECMWF/NMC atmos obs) Error est.: ± 9% at mid-latitude; Bryden est 2.0 ±0.42 pW at 24N Global meridional heat transport divides roughly equally into 3 modes: 1. atmosphere (dry static energy) 2. ocean (sensible heat)3. water vapor/latent heat transportThe three modes of poleward transport are comparable in amplitude, and distinct in character (sensible heat flux divergence focused in tropics, latent heat flux divergence focus in the subtropics) 90S 90N

10. Ocean Circulation Methaphor conversion surface deep

11. Stommel (1961) Box Model Heat, freshwater fluxes

12. Wind

13. Currents

14. Operators • f(x,y,z,t)

15. Vertical Gradients • dT/dz • Atmosphere • Ocean

16. Atlantic Ocean Water masses (schematic) N S Source: Dietrich, Kalle, Kraus, Siedler

17. Ocean: vertical gradients

18. "Euler" is pronounced "oiler.„ • Leonhard Euler, 1707-1783, Swiss mathematician who worked in Berlin and St. Petersburg. • L. Euler, 1762:Lettre de M. Euler à M. de Lagrange, Recherches sur la propagation des ébranlements dans une milieu élastique, • Misc. Taur. (1760-1761), 1-10 Opera(2) 10, 255-263 • Oeuvres de Lagrange14, 178-188 • J. L. D'Alembert, 1752:Essai d'une Nouvelle Théorie de la Resistance des Fluides, Paris Euler

19. Vector field In most cases the fluid is considered to be a continuum, whereas for rarefied gases one needs to take into account the behaviour of molecules in a statistical way (-> Boltzmann equation)

20. Language Remember, vector fields are the mathematical language that we will use to describe the dynamics of the atmosphere-ocean system. You must learn how to speak and interpret this language! dynamics

21. Euler scheme: Streamline A streamline can be considered as the path traced by an imaginary massless particle dropped into a steady fluid flow described by the field. The construction of this path consists in the solving an ordinary differential equation for successive time intervals. In this way, we obtain a series of points pk, 0<k<n which allow visualizing the streamline. The differential equation is defined as follows : (dp)/(dt) = v(p(t)), p(0) = p0 where p(t) is the position of the particle at time t, v is a function which assigns a vector value at each point in the domain (possibly by interpolation), and p0 is the initial position. The position after a given interval T is given by : p(t + T) = p(t) + \int_t^{t+T} v(p(t)) dt Several numeric methods have been proposed to solve this equation.

22. Euler integrator This algorithm approximates the point computation by this formula pk+1 = pk + hv(pk) where h specifies the integration step. The streamline is then constructed by successive integration.

23. Ocean: Flow

24. Ocean: Trajectories

25. Experiments

26. Experiments

27. Fundamental Laws • ?

28. http://anusf.anu.edu.au/~sjm651/meng/thesis/node2.html

29. Conservation of mass

30. Fluid dynamics To construct a model for fluid motion one looks at properties of matter in a control volume. These properties of a fluid are often expressed in terms of mass, momentum and energy. To arrive at a mathematical model of fluid motion one looks at the properties of mass, momentum and energy in a control volume and considers the following physical laws:

31. Rate of mass In = Rate of mass Out rho1 A1 v1 = rho2 A2 v2

32. Conservation of mass An expression for the conservation of mass is arrived at by considering the rate of change of mass in the control volume; that is, the flow of mass in and the flow of mass out of our control volume. In differential form this is where is the density and is the velocity vector. If the flow is incompressible this becomes:

33. The continuity equation relates the partial derivative of the density with respect to time to the gradient of the momentum density. It amounts to conservation of mass, saying (in integrated form) that mass cannot enter or leave a volume in space without flowing across the boundaries of that volume.

34. Euler‘s Equation is the equivalent of Newton's second law for fluids, including the effect of neighboring fluids and the force of gravity. In it, the vector v is the velocity field a a point in space. v is not the velocity of a particular parcel of fluid, except when it is at that point in space. • The left side of Euler's equation is the full derivative of the velocity, comprised of the "convective derivative" (the spatial component) and the partial derivative of the velocity field with respect to time. The right side includes the force on a unit volume due to the pressure gradient, grad P, and the force on a unit volume due to the gravitational field, g. The force of neighboring fluid points towards lower pressure; this is the origin of the minus sign on the pressure gradient term.

35. DynamicsExercise 1, 26. October 2006 1) From the weather chart in today's newspaper or internet site of your choice, identify the horizontal extent of a major atmospheric feature at mid-latitudes and the associated wind speed. From these length and velocity scales, determine a time scale. 2) The potential temperature in the atmosphere is defined as \beqn \Theta = T (p_0/p)^{R/c_p} \eeqn With $ p_0=$ const. Calculate the vertical temperature gradient \beqn \gamma = - \frac{dT}{dz} \eeqn What is the result when assuming the hydrostatic equillibrium $$\frac{dp}{dz} = -g \rho $$ with $ g = 9.81 m/s^2 $ ? What is the condition for which the the potential temperature is constant in the vertical? 3) Given f(x,y,z,t). What is the definition of partial derivatives for this variales? What is the definition of nabla, Laplace, divergence, total (substantial) derivative, total differential? 4) Please list the fundamental and derived quantities for the ocean and atmosphere? What are the typical length and time scales for the ocean and atmosphere?