Dynamics Gerrit Lohmann

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

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 Practical: Laptop?

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)

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.

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.

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

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:

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

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:

The continuity equation mass cannot enter or leave a volume in space without flowing across the boundaries of that volume.

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.

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?

Prime features of the climate system

Advection • Advection ``carries'' any pollution with the flow.

Dynamical EquationsCartesian system Mass budget A necessary statement in uid mechanics is that mass be conserved. That is, any imbalance between convergence and divergence in the three spatial directions must create a local compression or expansionof the fluid.

Dynamical Equationscartesian system Momentum budget For a fluid, Isaac Newton's second law „mass times acceleration equals the sum of forces“ is better stated per unit volume with density replacing mass. In the absence of rotation, the resulting equations are called the Navier-Stokes equations. real forces and apparent forces

real forces and apparent forces • apparent forces which account for the acceleration (the change in direction) of our frame of reference and which accounts for the observed motions of air/water parcels. • real forces - those which exist regardless of our frame of reference

Material derivative Because the acceleration in a fluid is not counted as the rate of change in velocity at a fixed location but as the change in velocity of a fluid particle as it moves along with the flow,the time derivatives in the acceleration components, du/dt, dv/dt, dw/dt consist of both the local time rate of change and the so-called advective terms: „Material derivative“

Pressure • Air moves because of an imbalance in the forces acting on the air molecules. In the warm column, the pressure at the level of 5,500 meters was greater than originally (600 hPa rather than 500 hPa). In the cold column, the pressure at the level of 5,500 meters was less than originally (400 hPa rather than 500 hPa).

Motion The difference in pressure between the two columns initiated air molecules to begin flowing from the regions of higher pressure in the warm column to the lower pressure in the cold column. If we consider this process on a hemispheric scale, on an ideal planet (only the pressure gradient forces and gravity acting, no Coriolis force, Centrifugal Force or friction) and the surface is the same substance with no mountains, we should see a similar pattern like

Formalism of a rotating frame Capital letters: inertial framework of reference Rotation

Formalism of a rotating frame

Dynamical EquationsCartesian system The preceding equations assume a Cartesian system of coordinates and thus hold only if the dimension of the domain under consideration is much shorter than the earth's radius. On Earth, a length scale not exceeding 1000 km is usually acceptable. The neglect of the curvature terms is in some ways analogous to the distortion introduced by mapping the curved earth's surface onto a plane.

Numerics du/dt

Dynamics Gerrit Lohmann