Time: Tues/Thurs, 3:00 pm. -4:15 a.m.

ATMS 500 2008 (Middle latitude) Synoptic Dynamic Meteorology Prof. Bob Rauber 106 Atmospheric Science Building PH: 333-2835 e-mail: rauber@atmos.uiuc.edu Time: Tues/Thurs, 3:00 pm. -4:15 a.m.

Synoptic dynamics of extratropical cyclones Our goal: Examine the fundamental dynamical constructs required to examine the behavior of extratropical cyclones: Martin, J. “Mid-latitude atmospheric dynamics” Wiley -Kinematic properties of the horizontal wind field -Fundamental and apparent forces -Mass, Momentum and Energy -Governing equations (Conservation of momentum, mass, energy, eqn of state) -Equations of motion and their application -Balanced flow -Isentropic flow -Circulation, vorticity, Potential vorticity -Quasi-Geostrophic theory -Ageostrophic flow -QG diagnostic tools ( equation, Q vectors) -Frontogenesis -Vertical circulation about fronts -Semi-geostrophic theory -Potential vorticity diagnostics

Synoptic dynamics of extratropical cyclones Potential additional topics Extratropical Cyclones -Wave dynamics -Jetstream and Jetstreak dynamics -Subtropical and Polar front jets -Cyclogenesis and explosive cyclogenesis -Cyclone lifecycle -Pacific/Continental/East Coast cyclone structure and dynamics -Fronts and frontal dynamics -Occlusions

COURSE WEBSITE http://www.atmos.uiuc.edu/courses/atmos500-fa08/ • Posted on this website are: • All PowerPoint files I use in class • Any Papers I will review • Syllabus GRADES A. Mid-term exam (25% of grade) B. Final exam (25% of grade) C. Derivation Notebooks (26% of grade) D. Homework (24% of grade)

In this course we will use Système Internationale (SI) units

Review of basic mathematical principles Constant: A quantity that has a single value Scalar: A quantity that is described completely by it’s magnitude Examples: temperature, pressure, relative humidity, volume, snowfall Vector: A quantity that requires more than one value to describe it completely Examples: position, wind, vorticity

Vector Calculus If: Then: The magnitude of is given by:

Adding Vectors Adding vectors is commutative Adding vectors is associative

Subtracting Vectors Simply adding the negative of the vector

Multiplying a Scalar and a Vector Multiplying two Vectors to obtain a scalar (the scalar or dot product) Carrying out multiplication gives nine terms Commutative and distributive properties of scalar product

Multiplying two Vectors to obtain a vector (the curl or cross product) Magnitude of cross product Where quantity on the RHS of the equation is called the determinant Properties of the cross product It is not commutative rather It is not associative

Derivatives of vectors and scalars

Special mathematical operator we will employ extensively The del operator: Used to determine the gradient of a scalar quantity Used to determine the divergence of a vector field Used to determine the rotation or curl of a vector field

Special mathematical operator we will employ extensively The Laplacian Operator The advection operator

The Taylor Series Expansion A continuous function can be represented about the point x = 0 by a power series of the form • Provided certain conditions are true. These are: • The polynomial expression passes through the point (0, f(0)) • f(x) is differentiable at x = 0 • The first n derivatives of the polynomial match the first n derivatives of f(x) at x = 0. For these conditions to be met, we must chose the “a” coefficients properly

(1) Let’s substitute x = 0 into the above equation Take first derivative of (1) and substitute x = 0 into the result Take second derivative of (1) and substitute x = 0 into the result Take third derivative of (1) and substitute x = 0 into the result Carrying out all derivatives

To determine the value of a function at a point x, near x0, this function can be generalized to give We will use this function often, except that we will ignore the higher order terms This is equivalent to assuming that the function changes at most linearly In the small region between x and x0

Centered difference approximation to derivates Consider two points, x1 and x2 in the near vicinity of point x0 Use Taylor expansion to estimate value of f(x1) and f(x2)

Subtracting equations gives: Ignoring higher order terms Estimate of first derivative Adding equations gives: Estimate of second derivative

Temporal changes of a continuous variable expand differential: divide by dt Note that the position derivatives are the wind components

We can write this in vector form: Let’s let Q be temperature T and switch around the equation:

ADVECTION OF T local change in temperature At a point x, y, z the import of temperature To x,y,z by the flow the change in temperature following an air parcel is warm advection: warm air is transported toward cold air ADVECTION ALWAYS IMPLIES A GRADIENT EXISTS IN THE PRESENCE OF A WIND ORIENTED AT A NON-NORMAL ANGLE TO THE GRADIENT

Time: Tues/Thurs, 3:00 pm. -4:15 a.m.