Transcript Lecture #2: Review of Dynamic Meteorology

Dynamic Meteorology:
A Review
Advanced Synoptic
M. D. Eastin
Total Vs. Partial Derivatives
Total Derivatives
• The rate of change of something following a fluid element is called
the Lagrangian rate of change
d
D
dt Dt
• Example: How temperature changes following an air parcel as is moves around
Partial Derivatives
• The rate of change of something at a fixed point is called
the Eulerian rate of change




t
x
y
p
• Example: The temperature change at a surface weather station
Euler’s Relation
• Shows how a total derivative can be decomposed into a local rate of change
and advection terms
DT T
T
T
T

u
v
w
Dt
t
x
y
z
Advanced Synoptic
M. D. Eastin
Vectors
Scalar:
Has only a magnitude (e.g. temperature)
Vector:
Has a magnitude and direction (e.g. wind)

Usually represented in bold font (V) or as (V )
Unit Vectors:
Represented by the letters i, j, k
Magnitude is 1.0
Point in the x, y, and z (or p) directions
Total Wind Vector:
Defined as V = ui + vj + wk, where u, v, w are the scalar
components of the zonal, meridional, and vertical wind
Vector Addition/Subtraction: Simply add the scalars of each component together
V1+V2 = (u1+u2)i + (v1+v2)j + (w1+w2)k
Vector Multiplication:
Dot Product: Defined as the product of the magnitude of the vectors
Results in a scalar
V1•V2 = u1u2+v1v2+w1w2
i•i = j•j = k•k = 1
The dot product of any unit vector with another = 0
Advanced Synoptic
M. D. Eastin
Vectors
Vector Multiplication:
Cross Product: Results in a third vector that points perpendicular to the first two
Follows the “Right Hand Rule”
Often used in meteorology when rotation is involved (e.g. vorticity)
V1 x V2 = i(v1w2 – v2w1)+j(u2w1-u1w2)+k(u1v2-u2v1)
Differential “Del” Operator:
Definition:
i



j k
x
y
z
Del multiplied by a scalar (“gradient” of the scalar):
a  i
a
a
a
j k
x
y
z
Dot product of Del with Total Wind Vector (“divergence”):
 V 
Advanced Synoptic
u v w
 
x y z
M. D. Eastin
Vectors
Differential “Del” Operator:
Cross product of Del with Total Wind Vector (“vorticity”):
 w v   u w 
 v u 
  V  i
   j 
  k   
 y z   z x 
 x y 
Note: The third term is rotation in the horizontal plane about the vertical axis
This is commonly referred to “relative vorticity” (ζ)
We can arrive at this by taking the dot product with the k unit vector
  k V 
v u

x y
Dot product of Del with itself (“Laplacian” operator)
    2
If we apply the Laplacian to a scalar:
 2a  2a  2a
  a   a  2  2  2
x
y
z
2
Advanced Synoptic
M. D. Eastin
Vectors
Euler’s Relation Revisited:
If we dot multiply the gradient of a scalar (e.g. Temperature) with the total wind vector
we get the advection of temperature by the wind:
V  T  u
T
T
T
v
w
x
y
z
Recall, the total derivative of temperature can be written as (in scalar form)
DT T
T
T
T

u
v
w
Dt
t
x
y
z
Or as (in vector form) upon substituting from above:
DT T

 V  T
Dt
t
Advanced Synoptic
M. D. Eastin
Equations of Motion
The equations of motion describe the forces that act on an air parcel in a
three-dimensional rotating system → describe the conservation of momentum
Fundamental Forces:
Pressure Gradient Force (PGF) → Air parcels always accelerate down the pressure
gradient from regions of high to low pressure
Gravitational Force (G) → Air parcels always accelerate (downward) toward the Earth’s
center of mass (since the Earth’s mass is much greater
than an air parcel’s mass)
Frictional Force (F) → Air parcels always decelerate due to frictional drag forces both
within the atmosphere and at the boundaries
Apparent Forces (due to a rotating reference frame):
Centrifugal Force (CE) → Air parcels always accelerate outward away from their
axis of rotation
Coriolis Force (CF) → Air parcels always accelerate 90° to the right of their current
direction (in the Northern Hemisphere)
Advanced Synoptic
M. D. Eastin
Equations of Motion
The equation of motion for 3D flow can be written symbolically as:
DV
 CE  PGF  G  CF  F
Dt
Normally, this equation is decomposed into three equations:
Du uv tan  uw 1 p



 2v sin   2w cos   Frx
Dt
a
a  x
Dv  u 2 tan  vw 1 p



 2u sin   Fry
Dt
a
a  y
Dw u 2  v 2 1 p


 g  2u cos   Frz
Dt
a
 z
What are each of these terms?
Advanced Synoptic
M. D. Eastin
Equations of Motion
The equations of motion for 3D flow:
Du uv tan  uw 1 p



 2v sin   2w cos   Frx
Dt
a
a
 x
Dv  u 2 tan  vw 1 p



 2u sin   Fry
Dt
a
a  y
Dw u 2  v 2 1 p


 g  2u cos   Frz
Dt
a
 z
where:
Total Derivative of Wind
Pressure Gradient Force
Gravitational Force
Frictional Force
Curvature Terms
Coriolis Force
Are all of these terms significant? Can we simplify the equations?
Advanced Synoptic
M. D. Eastin
Equations of Motion
Scale Analysis:
• Method by which to determine which terms in the equations can be neglected:
[Neglect terms much smaller than other terms (by several orders of magnitude)]
• Use typical values for parameters in the mid-latitudes on the synoptic scale
Horizontal velocity (U)
Vertical velocity (W)
Horizontal Length (L)
Vertical Height (H)
Angular Velocity (Ω)
Time Scale (T)
Frictional Acceleration (Fr)
Gravitational Acceleration (G)
Horizontal Pressure Gradient (∆p)
Vertical Pressure Gradient (Po)
Air Density (ρ)
Coriolis Effect (C)
≈ 10 m s-1
≈ 10-2 m s-1
≈ 106 m
≈ 104 m
≈ 10-4 s-1
≈ 105 s
≈ 10-3 m s-2
≈ 10 m s-2
≈ 103 Pa
≈ 105 Pa
≈ 1 m3 kg-1
≈1
(u,v)
(w)
(dx,dy)
(dz)
(Ω)
(dt)
(Frx, Fry, Frz)
(g)
(dp/dx, dp/dy)
(dp/dz)
(ρ)
(2sinφ, 2cosφ)
Using these values, you will find that numerous terms can be neglected…..
Advanced Synoptic
M. D. Eastin
Equations of Motion
The “simplified” equations of motion for synoptic-scale 3D flow:
Du u
u
u
u
1 p

u
v
w

 fv
Dt
t
x
y
z
 x
Dv v
v
v
v
1 p

u v w  
 fu
Dt t
x
y
z
 y
0
where:
1 p
g
 z
f = 2ΩsinΦ and Φ is the latitude
This set of equations is often called the “primitive equations” for large-scale motion
Note: The total derivatives have been decomposed into their local and advective terms
The vertical equation of motion reduces to the hydrostatic approximation – vertical
velocity can NOT be predicted using the vertical equation of motion – other
approaches must be used
Advanced Synoptic
M. D. Eastin
Mass Continuity Equation
The continuity equation describes the conservation of mass in a 3D system
• Mass can be neither created or destroyed
• Must account for mass in synoptic-scale numerical prediction
Mass Divergence Form:
1   u   v   w



 t
x
y
z
Interpretation: Net mass change
is equal to the 3-D convergence
of mass into the column
Velocity Divergence Form:
1 D u v w




 Dt x y z
Form commonly used by
numerical models to predict
density changes with time
Scale Analysis results in:
u v w
0


x y z
Advanced Synoptic
Form commonly used by
observational studies to
identify regions of vertical motion
M. D. Eastin
Mass Continuity Equation
If we isolate the vertical velocity term on one side:
 u v 
w
   
z
 x y 
OR
w
   Vh 
z
Thus, changes in the vertical velocity can be induced
from the horizontal convergence/divergence fields
Example:
Convergence near the surface
(low pressure) leads to upward
motion that increases with height
L
Advanced Synoptic
Divergence near the surface
(e.g. high pressure) leads to
downward motion increasing
with height
M. D. Eastin
Thermodynamic Equation
The thermodynamic equation describes the conservation of energy in a 3D system
Begin with the First Law of Thermodynamics:
DQ  c p DT  1  Dp
After some algebra….
DT
1 Dp 1 DQ


Dt
c p Dt c p Dt
Decomposed into local and advective components:
T
T
T
T
1 Dp 1 DQ
u
v
w


t
x
y
z c p Dt c p Dt
What are each of these terms?
T
T
T
T
1 Dp 1 DQ
u
v
w


t
x
y
z c p Dt c p Dt
Advanced Synoptic
Local change in temperature
Advection of temperature
Adiabatic temperature
change due to expansion
and contraction
Diabatic temperature
change from condensation,
evaporation, and radiation
M. D. Eastin
Isobaric Coordinates
Advantages of Isobaric Coordinates:
• Simplifies the primitive equations
• Remove density (or mass) variations that are difficult to measure
• Upper air maps are plotted on isobaric surfaces
Characteristics of Isobaric Coordinates:
• The atmosphere is assumed to be in hydrostatic balance
• Vertical coordinate is pressure → [x,y,p,t]
• Vertical velocity (ω)

Dp
Dt
ω > 0 for sinking motion
ω < 0 for rising motion
• Euler’s relation in isobaric coordinates
D 



 u  v 
Dt t
x
y
p
What are the primitive equations in isobaric coordinates?
Advanced Synoptic
M. D. Eastin
Isobaric Coordinates
Primitive Equations (for large-scale flow) in Isobaric Coordinates:
Du u
u
u
u
z

u
v

 g
 fv
Dt
t
x
y
p
x
Dv v
v
v
v
z

u v

 g
 fu
Dt t
x
y
p
y
z
RT
g

p
p
u v 


0
x y p
DT T
T
T
T
RT
1 DQ

u
v



Dt
t
x
y
p
pc p c p Dt
p  RT
Zonal Momentum
Meridional Momentum
Hydrostatic Approximation
Mass Continuity
Thermodynamic
Equation of State
See Holton Chapter 3 for a complete description of the transformations
We will be working with (starting from) these equations most of the semester!!!
Advanced Synoptic
M. D. Eastin
Hypsometric Equation
What it means:
The thickness between any two pressure levels is proportional
to the mean temperature within that layer
Warmer layer → Greater thickness
Pressure decrease slowly with height
Colder layer → Less thickness
Pressure decreases rapidly with height
Derivation: Integrate the Hydrostatic Approximation between two pressure levels
z2  z1 
Advanced Synoptic
RT  p1 
ln  
g  p2 
M. D. Eastin
Hypsometric Equation
Application:
Can infer the mean vertical structure of the atmosphere:
• Location/structure of pressure systems
• Location/structure of jet streams
• Precipitation type (rain/snow line)
500-mb Heights – 0600 UTC 22 Jan 2004
1000-500-mb Thickness – 0600 UTC 22 Jan 2004
From Lackmann (2011)
Advanced Synoptic
M. D. Eastin
Geostrophic Balance
Recall the horizontal momentum equations:
Du
z
 g
 fv
Dt
x
Dv
z
 g
 fu
Dt
y
• Scale analysis for large-scale (synoptic) motions above the surface reveals that the total
derivatives are one order of magnitude less than the PGF and CF.
• Neglect the total derivatives and do some algebra….
vg 
g z
f x
ug  
g z
f y
• The PGF exactly balances the CF
• There are no accelerations acting
on the parcel (once balance is achieved)
Advanced Synoptic
M. D. Eastin
Geostrophic Balance
Pressure
Gradient
Force
Coriolis
Force
Advanced Synoptic
Geostrophic
Wind
M. D. Eastin
Thermal Wind
What is Means:
Derivation:
The vertical shear of the geostrophic wind over a layer
is directly proportional to the horizontal temperature
(or thickness) gradient through the layer
Differentiate the geostrophic balance equations with respect to pressure
and apply the hydrostatic approximation
vg
p

R  T 


fp  x  p
ug
p

R  T 


fp  y  p
Characteristics:
• Relates the temperature field to the wind field
• Describes how much the geostrophic wind will change with height (pressure)
for a given horizontal temperature gradient
• The thermal wind is the vector difference between the two geostrophic winds above
and below the pressure level where the horizontal temperature gradient resides
• The thermal wind always blows parallel to the mean isotherms (or lines of constant
thickness) within a layer with cold air to the left and warm air to the right
Advanced Synoptic
M. D. Eastin
Thermal Wind: Application
The thermal wind can be used to diagnose the mean horizontal temperature advection
within a layer of the atmosphere
Warm Air Advection (WAA)
(within a layer)
Cold
V850
Vtherm
Warm
V500
Geostrophic winds turn
clockwise (or “veer”)
with height through
the layer
Advanced Synoptic
Cold Air Advection (CAA)
(within a layer)
V500
V850
Cold
Warm
Vtherm
Geostrophic winds turn
counterclockwise
(or “back”) with height
through the layer
M. D. Eastin
Thermal Wind: Application
International Falls, MN
• Winds turn counterclockwise (“back”)
with height between 850 and 500 mb
• We should expect CAA within the layer
 Note that CAA appears to be
occurring at both 850 and 500 mb
500 mb
Buffalo, NY
• Winds turn clockwise (“veer”) with
height between 850 and 500 mb
• We should expect WAA within the layer
850 mb
Advanced Synoptic
M. D. Eastin
Thermal Wind: Application
Minneapolis / Saint Paul (MSP)
We can infer WAA and CAA
with a single sounding from
the vertical profile of wind
direction
Winds are backing with
height → CAA
Winds are veering with
height → WAA
Advanced Synoptic
M. D. Eastin
Surface Pressure Tendency
What it means:
The net divergence (convergence) of mass out of (in to) a column
of air will lead to a decrease (increase) in surface pressure
Derivation: Integrate the Continuity Equation (in isobaric coordinates) through the
entire depth of the atmosphere and apply boundary conditions
s
 u v 
ps
     dp
t
x y 
0
p
Characteristics:
• Provide qualitative information concerning the movement (approach) of pressure systems
• Difficult to apply as a forecasting technique since small errors in wind (i.e. divergence)
field can lead to large pressure tendencies
• Also, divergence at one level is usually offset by convergence at another level
Note: Temperature changes in the column do not have a direct effect on the surface
pressure – they change the height of the pressure levels, not the net mass
Advanced Synoptic
M. D. Eastin
Circulation and Vorticity
Circulation:
Vorticity:
The tendency for a group of air parcels to rotate
If an area of atmosphere is of interest, you compute the circulation
The tendency for the wind shear at a given point to induce rotation
If a point in the atmosphere is of interest, you compute the vorticity
Planetary Vorticity: Vorticity associated with the Earth’s rotation
f  2 sin 
Relative Vorticity:
Vorticity associated with 3D shear in the wind field
  v   u  
 v u 
  V  i
   j 
  k   
 y p   p x 
 x y 
Only the vertical component of vorticity (the k component)
is of interest for large-scale (synoptic) meteorology
 
v u

x y
Absolute Vorticity: The sum of relative and planetary vorticity
   f
Advanced Synoptic
M. D. Eastin
Circulation and Vorticity
Circulation:
Vorticity:
The tendency for a group of air parcels to rotate
If an area of atmosphere is of interest, you compute the circulation
The tendency for the wind shear at a given point to induce rotation
If a point in the atmosphere is of interest, you compute the vorticity
500-mb Heights
Absolute Vorticity (η = ζ + f)
From Lackmann (2011)
Advanced Synoptic
M. D. Eastin
Circulation and Vorticity
Vorticity Types:
   f
 
v u

x y
f  2 sin 
Absolute Vorticity (η = ζ + f)
 
v u

x y
Relative Vorticity (ζ)
From Lackmann (2011)
Advanced Synoptic
M. D. Eastin
Circulation and Vorticity
Vorticity Types:
Positive Vorticity: Associated with cyclonic (counterclockwise) circulations in
the Northern Hemisphere
Negative Vorticity: Associated with anticyclonic (clockwise) circulations in
the Northern Hemisphere
Advanced Synoptic
M. D. Eastin
Circulation and Vorticity
Vorticity Types:
Shear Vorticity: Associated with gradients along local straight-line wind maxima
Curvature Vorticity: Associated with the turning of flow along a stream line
Shear Vorticity
Curvature Vorticity
+
_
+
From Lackmann (2011)
Advanced Synoptic
M. D. Eastin
Vorticity Equation
Describes the factors that alter the magnitude of the absolute vorticity with time
Derivation: Start with the horizontal momentum equations (in isobaric coordinates)
Take
u
u
u
u
z
u
v

 g
 fv
t
x
y
p
x
Zonal Momentum
v
v
v
v
z
u  v 
 g
 fu
t
x
y
p
y
Meridional Momentum


of the meridional equation and subtract
of the zonal equation
x
y
After use of the product rule, some simplifications, and cancellations:
 u v    u  v 




f
u
v

v
    f 
   


t
x
y
p
y
 x y   y p x p 
Advanced Synoptic
M. D. Eastin
Vorticity Equation
What do the terms represent?
 u v 
  u  v 




f
 u
v

v
   f 
   


t
x
y
p
y
 x y 
 y p x p 
Local rate of change of relative vorticity
~10-10
Horizontal advection of relative vorticity
~10-10
Vertical advection of relative vorticity
~10-11
Meridional advection of planetary vorticity
~10-10
Divergence Term
~10-9
Tilting Terms
~10-11
What are the significant terms? → Scale analysis and neglect of “small” terms yields:
 u v 



f
 u
v
v
   f 
 
t
x
y
y

x
y 

Advanced Synoptic
M. D. Eastin
Vorticity Equation
Physical Explanation of Significant Terms:



 u
v
t
x
y
v
 u v 
f
   f 
 
y

x
y 

Horizontal Advection of Relative Vorticity
• The local relative vorticity will increase (decrease) if greater (less) relative vorticity is
advected toward the location → Positive (cyclonic) vorticity advection (CVA / PVA)
→ Negative (anticyclonic) vorticity advection (AVA / NVA)
• CVA / PVA often leads to a decrease in surface pressure (intensification of surface lows)
Meridional Advection of Planetary Vorticity
• The local relative vorticity will decrease (increase) if the local flow is southerly (northerly)
due to the advection of planetary vorticity (minimum at Equator; maximum at poles)
Divergence Term
• The local relative vorticity will increase (decrease) if local convergence (divergence) exists
Advanced Synoptic
M. D. Eastin
Vorticity Equation
Physical Explanation: Horizontal Advection of Relative Vorticity



 u
v
t
x
y
Relative Vorticity (ζ)
v
 u v 
f
   f 
 
y

x
y 

Relative Vorticity Advection
From Lackmann (2011)
Advanced Synoptic
M. D. Eastin
Quasi-Geostrophic Theory
Most meteorological forecasts:
• Focus on Temperature, Winds, and Precipitation (amount and type)**
• Are largely a function of the evolving synoptic-scale weather patterns
Quasi-Geostrophic Theory:
• Makes further simplifying assumptions about the large-scale dynamics
• Diagnostic methods to estimate: Changes in large-scale surface pressure
Changes in large-scale temperature (thickness)
Regions of large-scale vertical motion
• Despite the simplicity, it provides accurate estimates of large-scale changes
• Will provide the basic analysis framework for remainder of the semester
Next Time……
Advanced Synoptic
M. D. Eastin
Summary
Important Dynamic Meteorology (METR 3250) Concepts:
•
•
•
•
•
•
•
•
•
•
•
Advanced Synoptic
Total / Partial Derivatives and Vector Notation
Equation of Motion (Components and Simplified Terms)
Mass Continuity Equation
Thermodynamic Equation
Isobaric Coordinates and Equations
Hypsometric Equation
Geostrophic Balance
Thermal Wind
Surface Pressure Tendency
Circulation and Vorticity
Vorticity Equation
M. D. Eastin