09 - math and equations
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Math Review and the EquationsMath Review and the Equations
of Motionof MotionATMS-303
Fall 2010
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Atmospheric VariablesAtmospheric Variables
y Most atmospheric variables (temperature,pressure, moisture, etc) vary in all threespatial dimensions (x, y, z) and time (t)
y
Example: Temperature We write this as T(x, y, z, t)
Or T(x, y, p, t) if pressure is vertical coordinate
y Cannot be treated as a constant in
differentiation or integration unlesssimplifying assumptions are made Variables assumed constant in one more
directions or in time
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Example: Temperature (x, y)Example: Temperature (x, y)
XXyy
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Example: Temperature (z)Example: Temperature (z)
zz
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Example: Temperature (t)Example: Temperature (t)
TimeTime
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DifferentiationDifferentiation
yWe can differentiate these variables with
respect to (w.r.t.) x, y, z, or t
y A partial derivative expresses how Tchanges in one dimension with all other
dimensions kept constant
yWritten as
x
T
x
x
y
T
x
x
z
T
x
x
t
T
x
x
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Example: Temperature (x, y)Example: Temperature (x, y)
xxx
y : y, z, t remain
constant How temperature
changes in x directiononly
y : x, z, t remainconstant
xx
x
yx
x
yx
x
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Time DerivativesTime Derivatives
y Partial derivative :
Keep x, y, z constant
How does temperature change at a fixed point?
Also called Eulerian derivative
y But what if you are moving?
Example: Driving from Champaign to Chicago
Temperature can change due to weather, solarheating, etc
Also expect temperature change betweenChampaign and Chicago at any fixed time
t
T
x
x
t
T
x
x
y
T
x
x
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Total Time DerivativeTotal Time Derivative
y Must use total time derivative to expresstime rate of change of temperature followinga moving object Examples: A car, an air parcel
yWritten as ory Expressed as
y u, v, w are components of velocity vectory Remember, the total derivative is w.r.t. time
only!y Also called Lagrangian derivative
dtdT
DtDT
z
Tw
y
Tv
x
Tu
t
T
dt
dT
x
x
x
x
x
x
x
x!
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Chain RuleChain Rule
y Remember to apply the chain rule (productrule) when differentiating products ofvariables
y Example: Take z-derivative of Ideal Gas Law
y
Often simplify this be assuming that densityis constant Called incompressibility or Boussinesq
Approximation
RTz
pz
Vx
x!
x
x
z
TR
zRT
z
p
x
x
x
x!
x
xV
V
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VectorsVectors
y Vectors define quantities that have bothmagnitude and direction
y Example: Position (x, y, z) x = East-west (zonal) direction
y = North-south (meridional) direction z = Up-down (vertical) direction
y Example: Velocity (u, v, w) u = Zonal velocity
v = Meridional velocity w = Vertical velocity
y Scalar: A quantity with magnitude but no direction Examples: Pressure, temperature, moisture
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Vector MagnitudeVector Magnitude
y For any vector a, its magnitude is given by
y Can calculate direction of 2-
Dvectors withinverse tangent function
y Be careful of signs and angles Draw it out Tangent has period of 180 Wind direction is direction wind is blowing from
y Can also obtain x and y components ofvectors given magnitude and angle
2
3
2
2
2
1aaaa !
T
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Adding or Subtracting VectorsAdding or Subtracting Vectors
y Add vectors head to tail
y Must split vectors into components
x, y, z, components
Add components if pointing in same direction
Subtract components if pointing in oppositedirections
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Unit VectorsUnit Vectors
y Often use unit vectors (i, j, k) to denote
direction
i = x direction
j = y direction
k = z direction
y Magnitudes of unit vectors are one, so
they do not affect magnitudey Example: Velocity vector
kwjviuv !T
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Dot ProductDot Product
y Consider two vectors a and b
y Can define dot product ab as
y Result is scalar
Dot product of a unit vector with itself is oney Dot product of two perpendicular
vectors is always zero. Why?
kajaiaa 321
!T
kbjbibb 321
!T
332211babababa !
TT
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Dot ProductDot Product
y The dot product is also given by
y Note:E is angle between two vectors
y Note: A scalar times a vector is just the
scalar times each component of the
vector
EcosbabaTTTT
!
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Cross ProductCross Product
y Consider two vectors a and b
y Can define cross product a x b as
y Result is a vectory Cross product of two parallel vectors is
always zeroy Cross product of any two vectors is always
perpendicular to both vectorsy Remember right-hand rule!
kajaiaa 321
!T
kbjbibb 321
!T
kbabajbabaibababa )()()(122131132332
!vTT
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Cross ProductCross Product
y Easy way to remember: 3 x 3 determinant!
y Can write first two columns to right of
third
y Down and right Positive
y Down and left Negative
321
321
bbb
aaa
kji
ba !vTT
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Vector DifferentiationVector Differentiation
y To differentiate a vector, take the
appropriate derivative of eachcomponent:
y Remember, unit vectors are constant
kx
wj
x
vi
x
u
x
v x
x
x
x
x
x!
x
xT
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The Del Operator GradientThe Del Operator Gradient
y The vector operator is defined by
y
This is an operator; it has no value (justlike differentials (d/dt)
y Gradient: Del operator applied to a scalar
Result is always a vector, directedperpendicular to isopleths toward highervalues
kz
jy
ix
x
x
x
x
x
x|
kz
jy
ix
x
x
x
x
x
x!
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The Del Operator GradientThe Del Operator Gradient
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The Del Operator DivergenceThe Del Operator Divergence
y Divergence:
Del operator applied to a vector
Result is always a scalar
y The divergence of the wind field yields a
scalar known as the divergence
Note:Divergence = -1 x Convergence
y Important: The del operator isdistributive, but not commutative
zw
yv
xuv
x
xx
xx
x!T
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The Total DerivativeThe Total Derivative
y Recall that the total derivative is
y Can rewrite this using the dot product
and del operators as
y Can think of v dot del as operator:
Not equal to divergence!
z
Tw
y
Tv
x
Tu
t
T
dt
dT
x
x
x
x
x
x
x
x!
Tvt
T
dt
dT
x
x!
Tvv
t
v
dt
vd TTTT
x
x!
zw
yv
xuv
x
x
x
x
x
x!
T
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AdvectionAdvection
y Advection The transport of a quantityby the wind If the wind is zero, advection is zero
If the spatial derivative is zero, advection iszero
y Advection =
y Moisture advection =
y Advection in x direction =
y Total time derivative = Local timederivative + Advection
zw
yv
xuv
x
x
x
x
x
x!
T
z
qw
y
qv
x
quqv
x
x
x
x
x
x!
T
xqux
x
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TheThe LaplacianLaplacian 22
y Two consecutive applications of the deloperator
y Scalars: The divergence of the gradient
Always returns a scalar
y
Vectors: The gradient of the divergence Always returns a vector
2
2
2
2
2
2
2
z
T
y
T
x
TT
x
x
x
x
x
x!
kz
w
y
v
x
u
zj
z
w
y
v
x
u
yi
z
w
y
v
x
u
xv
2
!
T
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The Del Operator CurlThe Del Operator Curl
yWritten out this is
wv
zyx
kji
vx
x
x
x
x
x!v
T
kyx
vj
xzi
z
v
yv
x
x
x
x
x
x
x
x
x
x
x
x!vT
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Momentum Equation: Vector FormMomentum Equation: Vector Form
y LHS: Acceleration of wind
y RHS1: Pressure gradienty RHS2: Coriolis acceleration
y RHS3: Gravity
y RHS4: Friction
y RHS is the four fundamental forces at workin the atmosphere
Fkgvkfpdt
vdTT
T
v! )(1
V
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Momentum Equations:ComponentMomentum Equations:Component
FormForm
y Coriolis parameter:y
F = frictiony p = pressurey V = densityy g = gravity
xFfvx
p
dt
du
x
x!
V
1
yFfuy
p
dt
dv
x
x!
V
1
zFgz
p
dt
dw
x
x!
V
1
-14s10sin2
};! Nf
xFfvx
puv
t
u
x
x!
x
x
V
1T
yFfuy
pvv
t
v
x
x!
x
x
V
1T
zFgz
pwv
t
w
x
x!
x
x
V
1T
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The Four ForcesThe Four Forces
y Pressure gradient force (PGF)
y Coriolis force Apparent force due to rotation of earth
y Frictiony Gravity
y Note:Centripetal force is not a unique force.It is the force needed for an object to movein a circular path. The force must be suppliedby a physical force.
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Pressure Gradient ForcePressure Gradient Force
y Pressure gradient force always directedperpendicular to isobars toward lowerpressure Why? Negative sign
y Unitsy Air molecules want to flow from where
there is greater pressure to where there isless pressure
Nature abhors a vacuum!y Increasing the pressure gradient yields dv/dt
>0 and faster flow
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Pressure Gradient ForcePressure Gradient Force
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Pressure Gradient ForcePressure Gradient Force
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Rotation of the EarthRotation of the Earth
y Viewed from NorthPole, earth appears to
rotate counter-clockwise
y Analogous to merry-
go-round
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MerryMerry--GoGo--Round ExampleRound Example
y If merry-go-round is NOT moving, ball doesnot appear to be deflected
y If merry-go-around is moving, ball will still
travel in straight path as seen from abovey Ball will appear to be deflected to its right
from rotating platform Catcher sees ball move to his left appears to be
due to external force Catcher actually rotates to his right out of the
way of ball
y Same effect occurs on rotating earth
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The Coriolis ForceThe Coriolis Force
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CoriolisCoriolis Force and LatitudeForce and Latitude
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TheThe CoriolisCoriolis ForceForce
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CoriolisCoriolis ForceForce SummarySummary
y Deflects objects to the RIGHT in NH
y Deflects objects to the LEFT in SH
y Changes the direction, not the speed of
motion
y Force is proportional to speedy Maximum at Poles, zero at Equator
Distance from axis of rotation does not change atEquator
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FrictionFriction
y Molecular friction
Between air molecules and the earths surface
Between air molecules (viscosity)
Not very important
y Eddies
Small circulations that mix the effect of
friction in the atmosphere Boundary layer
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Eddy ViscosityEddy Viscosity
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GravityGravity
y Force acting toward the center of the
earth
y Small components in x and y directions
Earth is not a perfect sphere
Centrifugal force
y For our purposes, treat gravity as a
constant force acting downward Vertical equation of motion only