Circulation and Vorticity
Circulation is scalar: Macroscopic measure of rotation for finite area of the
fluid
Vorticity is vector: Microscopic measure of rotation at any point in the fluid.
Circulation
rr
r
C V .dl = V cos .dl
C > 0 for counterclockwise i

Component Equations in Spherical
Coordinates
Spherical co-ordinate system is the most natural
one to use for writing the momentum equation (9)
Co-ordinate axes are
: longitude
: latitude
z
: geometrical height
x = a cos
y=a
(10)
d
dz
;v = a
and w =
Henc

Sigma coordinate in the Vertical
Advantages of isobaric coordinates
1) Meteorological data normally referred to isobaric surfaces.
2) Continuity equation has a simple form.
3) Density does not appear explicitly in the equations used for NWP.
4) Vertical s

Richardson Factory (1992)
Picture Credit
:
A. Lannerback
Growth of NWP due to the factors:
1. Theoretical research by Rossby, Petterssen,
Bjerkness, Charney, Eady, Eliassen etc. led to
some direct application to practical problems:
Radical departure from

Weather observations and
Forecasting
Values of meteorological parameters and
conditions at a future time constitute weather
forecasting:
Temperature
Wind
Humidity
Rainfall
Snowfall
Cyclones
Important sources of data in India:
a) Surface observations
b)

ThermodynamicDiagrams
The primary function of a thermodynamic diagram is
to provide a graphical display of the lines
representing the major kinds of process to which air
may be subjected, namely isobaric, isothermal, dry
adiabatic and pseudo adiabatic. Th

TheParcelMethod
Itisasimpleapproachtounderstandthestability
in the atmosphere without the formal use of
mathematics.
Consider the vertical motions of an individual
parcelofairwiththesimplifiedassumptions
1. No compensating motions occur in the
environment

TheParcelMethod
It is a simple approach to understand the stability
in the atmosphere without the formal use of
mathematics.
Consider the vertical motions of an individual
parcel of air with the simplified assumptions
1. No compensating motions occur in t

Pressure as vertical coordinate
From the hydrostatic relation, it is clear that there exists a single valued
monotonic relationship between pressure and height in each vertical column
of the atmosphere. Hence one can use pressure as the independent vertic

Continuity equation
z
x
u (u )
x
2
y
u
u +
(u ) x
x
2
z
x
y
x
Net rate of mass flow into the volume
due to u component
x
x
u (u ) u (u ) yz
2
2
x
x
= (u )xyz
x
total rate of mass flow into the volume element
x (u ) y (v ) z (w )xyz
()
= U xyz
Rate of m

Circulation and Vorticity
Circulation is scalar: Macroscopic measure of rotation for finite area of the
fluid
Vorticity is vector: Microscopic measure of rotation at any point in the fluid.
Circulation
rr
r
C V .dl = V cos .dl
C > 0 for counterclockwise i

A portion of the absolute circulation is due to the rotation of the earth
about its axes. Let us compute this term Ce
rrr
We have, U = r
e
rr
r
Ce = U e .dl = U e .ndA
A
(
by applying the Stokes theorem
)
rr rr
Ue dl = Ue ndA
A
Here A is the area enclose

1
00:00:00,199 -> 00:00:03,238
Announcer: Previously on
AMC's The Walking Dead.
2
00:00:03,358 -> 00:00:05,859
Rick: That exit sends
them right at us.
3
00:00:05,960 -> 00:00:07,561
We force them west here.
4
00:00:07,661 -> 00:00:08,762
Away from the com

5.
Brownian motion
Section 1. The de nition and some simple properties. Section 2. Visualizing Brownian motion. Discussion and demysti cation of some strange and scary pathologies. Section 3. The re ection principle. Section 4. Conditional distribution of

Numerical Method
Finite difference scheme
Taylors series
x 3
x 2
+ f ' ' ' (x)
+ .(1)
f(x + x) = f(x) + f ' (x) x + f ' ' (x)
3!
2!
x 2
x 3
f ' ' ' (x)
+ .(2)
f(x x) = f(x) f ' (x) x + f ' ' (x)
2!
3!
where x > 0
f(x + x) f(x)
+R
f ' (x) =
from (1),
x
f(

Numerical Method
Finite difference scheme
Taylors series
x 3
x 2
+ f ' ' ' (x)
+ .(1)
f(x + x) = f(x) + f ' (x) x + f ' ' (x)
3!
2!
x 2
x 3
f ' ' ' (x)
+ .(2)
f(x x) = f(x) f ' (x) x + f ' ' (x)
2!
3!
where x > 0
f(x + x) f(x)
+R
f ' (x) =
from (1),
x
f(

Solenoidal term in the
circulation theorem
dp
If the fluid is barotropic (density a function of pressure alone) this term
becomes the closed line integral of an exact differential, and so is zero. This
argument is good for any given moment at which the fl

Twisting or tilting term: Second term in the RHS of equation (3)
represents vertical vorticity which is generated by the tilting of
horizontally oriented components of vorticity into the vertical by a non
uniform vertical motion field.
Solenoidal term: I

Use of Vorticity and Divergence equations
in NWP
The horizontal components of wind u&v are not
pure scalars due to nearly spherical shape of the
earths surface and singularities at the poles.
Those are pseudo-scalars.
Whereas vertical component of vortici

Relationship between circulation and vorticity (vertical component )
rr
lim U kdA
rr
U A0 A
=k
A
rr
r
Use Stokes theorem U kdl = U dl
A
r
lim U dl
C
= A0
= lim
A0 A
A
Physical interpretation of Vorticity
Circulation about the infinitesimal contour
V
C =

Relationship between circulation and vorticity (vertical component )
rr
lim U kdA
rr
U A0 A
=k
A
rr
r
Use Stokes theorem U kdl = U dl
A
r
lim U dl
C
= A0
= lim
A0 A
A
Physical interpretation of Vorticity
Circulation about the infinitesimal contour
V
C =

Scale Analysis
(1)Estimates magnitudes of various terms in governing equations for
particular type of motion.
(2)Systematically simplifies the equations by neglecting smaller
terms.
(3) Filters unwanted scales of motions.
Types of disturbances in the atmo

Scale Analysis
(1)Estimates magnitudes of various terms in governing equations for
particular type of motion.
(2)Systematically simplifies the equations by neglecting smaller
terms.
(3) Filters unwanted scales of motions.
Types of disturbances in the atmo

Total differentiation of a vector in a rotating system
Consider an arbitrary vector
)
in an inertial frame
i , j, k
r)
)
)
A = i A x + j A y + k A z (1)
(
)
r
A
Consider the rotating frame with
r
angular velocity
r
The vector A is represented by
r)
)
)
A

Optimization Methods in Finance
(EPFL, Fall 2010)
Lecture 8: Fundamental Theorem of Asset Pricing
10.11.2010
Lecturer: Prof. Friedrich Eisenbrand
Scribe: Parmeet Singh Bhatia
Financial background
In the last lecture we have been through basics of arbitrag

BROWNIAN MOTION
A tutorial
Krzysztof Burdzy
University of Washington
A paradox
f : [0,1] R,
sup | f (t ) | <
t[ 0 ,1]
P ( f (t ) < Bt < f (t ) + , 0 < t < 1)
11
2
c( ) exp ( f (t ) dt
2
0
(*)
(*) is maximized by f(t) = 0, t>0
The most likely (?!?) sha

4 Brownian motion
4.1 De nition of the process
Our discrete models are only a crude approximation to the way in which stock markets actually move. A better model would be one in which stock prices can change at any instant. As early as 1900 Bachelier, in

Brownian Motion
Draft version of May 25, 2008
Peter Mrters and Yuval Peres
o
1
Contents
Foreword
7
List of frequently used notation
9
Chapter 0. Motivation
13
Chapter 1. Denition and rst properties of Brownian motion
1. Paul Lvys construction of Brownian