1. Dynamic Meteorology:
A Review
Advanced Synoptic
M. D. Eastin

2. Total Vs. Partial Derivatives
Total Derivatives
The rate of change of something following a fluid element is called
the Lagrangian rate of change
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
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
Advanced Synoptic
M. D. Eastin

3. Vectors V1+V2 = (u1+u2)i + (v1+v2)j + (w1+w2)k V1•V2 = u1u2+v1v2+w1w2
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 ( )
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
Vector Multiplication:
Dot Product: Defined as the product of the magnitude of the vectors
Results in a scalar
The dot product of any unit vector with another = 0
V1+V2 = (u1+u2)i + (v1+v2)j + (w1+w2)k
V1•V2 = u1u2+v1v2+w1w2
i•i = j•j = k•k = 1
Advanced Synoptic
M. D. Eastin

4. V1 x V2 = i(v1w2 – v2w1)+j(u2w1-u1w2)+k(u1v2-u2v1)
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)
Differential “Del” Operator:
Definition:
Del multiplied by a scalar (“gradient” of the scalar):
Dot product of Del with Total Wind Vector (“divergence”):
V1 x V2 = i(v1w2 – v2w1)+j(u2w1-u1w2)+k(u1v2-u2v1)
Advanced Synoptic
M. D. Eastin

5. Vectors Differential “Del” Operator:
Cross product of Del with Total Wind Vector (“vorticity”):
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
Dot product of Del with itself (“Laplacian” operator)
If we apply the Laplacian to a scalar:
Advanced Synoptic
M. D. Eastin

6. 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:
Recall, the total derivative of temperature can be written as (in scalar form)
Or as (in vector form) upon substituting from above:
Advanced Synoptic
M. D. Eastin

7. 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

8. Equations of Motion
The equation of motion for 3D flow can be written symbolically as:
Normally, this equation is decomposed into three equations:
What are each of these terms?
Advanced Synoptic
M. D. Eastin

9. Equations of Motion The equations of motion for 3D flow:
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

10. 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) ≈ 10 m s-1 (u,v)
Vertical velocity (W) ≈ 10-2 m s-1 (w)
Horizontal Length (L) ≈ 106 m (dx,dy)
Vertical Height (H) ≈ 104 m (dz)
Angular Velocity (Ω) ≈ 10-4 s-1 (Ω)
Time Scale (T) ≈ 105 s (dt)
Frictional Acceleration (Fr) ≈ 10-3 m s-2 (Frx, Fry, Frz)
Gravitational Acceleration (G) ≈ 10 m s-2 (g)
Horizontal Pressure Gradient (∆p) ≈ 103 Pa (dp/dx, dp/dy)
Vertical Pressure Gradient (Po) ≈ 105 Pa (dp/dz)
Air Density (ρ) ≈ 1 m3 kg-1 (ρ)
Coriolis Effect (C) ≈ 1 (2sinφ, 2cosφ)
Using these values, you will find that numerous terms can be neglected…..
Advanced Synoptic
M. D. Eastin

11. Equations of Motion
The “simplified” equations of motion for synoptic-scale 3D flow:
where: 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

12. 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:
Velocity Divergence Form:
Scale Analysis results in:
Interpretation: Net mass change
is equal to the 3-D convergence
of mass into the column
Form commonly used by
numerical models to predict
density changes with time
Form commonly used by
observational studies to
identify regions of vertical motion
Advanced Synoptic
M. D. Eastin

13. Mass Continuity Equation
If we isolate the vertical velocity term on one side: OR
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
Divergence near the surface
(e.g. high pressure) leads to
downward motion increasing
with height L
Advanced Synoptic
M. D. Eastin

14. Thermodynamic Equation
The thermodynamic equation describes the conservation of energy in a 3D system
Begin with the First Law of Thermodynamics:
After some algebra….
Decomposed into local and advective components:
What are each of these terms?
Local change in temperature
Advection of temperature
Adiabatic temperature
change due to expansion
and contraction
Diabatic temperature
change from condensation,
evaporation, and radiation
Advanced Synoptic
M. D. Eastin

15. 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)
ω > 0 for sinking motion
ω < 0 for rising motion
Euler’s relation in isobaric coordinates
What are the primitive equations in isobaric coordinates?
Advanced Synoptic
M. D. Eastin

16. Isobaric Coordinates
Primitive Equations (for large-scale flow) in Isobaric Coordinates:
See Holton Chapter 3 for a complete description of the transformations
We will be working with (starting from) these equations most of the semester!!!
Zonal Momentum
Meridional Momentum
Hydrostatic Approximation
Mass Continuity
Thermodynamic
Equation of State
Advanced Synoptic
M. D. Eastin

17. 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
Advanced Synoptic
M. D. Eastin

18. 1000-500-mb Thickness – 0600 UTC 22 Jan 2004
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
mb Thickness – 0600 UTC 22 Jan 2004
From Lackmann (2011)
Advanced Synoptic
M. D. Eastin

19. Geostrophic Balance Recall the horizontal momentum equations:
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….
The PGF exactly balances the CF
There are no accelerations acting
on the parcel (once balance is achieved)
Advanced Synoptic
M. D. Eastin

20. Geostrophic Balance Pressure Gradient Force Geostrophic Wind Coriolis
Advanced Synoptic
M. D. Eastin

21. Thermal Wind
What is Means: The vertical shear of the geostrophic wind over a layer
is directly proportional to the horizontal temperature
(or thickness) gradient through the layer
Derivation: Differentiate the geostrophic balance equations with respect to pressure
and apply the hydrostatic approximation
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

22. Warm Air Advection (WAA) Cold Air Advection (CAA)
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 Air Advection (CAA)
(within a layer)
Cold
Vtherm
V500
V850
Warm
V850
Cold
V500
Vtherm
Warm
Geostrophic winds turn
clockwise (or “veer”)
with height through
the layer
Geostrophic winds turn
counterclockwise
(or “back”) with height
through the layer
Advanced Synoptic
M. D. Eastin

23. 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
Buffalo, NY
Winds turn clockwise (“veer”) with
height between 850 and 500 mb
We should expect WAA within the layer
500 mb
850 mb
Advanced Synoptic
M. D. Eastin

24. Minneapolis / Saint Paul (MSP)
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

25. 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
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

26. Circulation and Vorticity
Circulation: The tendency for a group of air parcels to rotate
If an area of atmosphere is of interest, you compute the circulation
Vorticity: 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
Relative Vorticity: Vorticity associated with 3D shear in the wind field
Only the vertical component of vorticity (the k component)
is of interest for large-scale (synoptic) meteorology
Absolute Vorticity: The sum of relative and planetary vorticity
Advanced Synoptic
M. D. Eastin

27. Absolute Vorticity (η = ζ + f)
Circulation and Vorticity
Circulation: The tendency for a group of air parcels to rotate
If an area of atmosphere is of interest, you compute the circulation
Vorticity: 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

28. Absolute Vorticity (η = ζ + f) Relative Vorticity (ζ)
Circulation and Vorticity
Vorticity Types:
Absolute Vorticity (η = ζ + f)
Relative Vorticity (ζ)
From Lackmann (2011)
Advanced Synoptic
M. D. Eastin

29. Circulation and Vorticity
Vorticity Types:
Positive Vorticity: Associated with cyclonic (counterclockwise) circulations in
the Northern Hemisphere
Negative Vorticity: Associated with anticyclonic (clockwise) circulations in
Advanced Synoptic
M. D. Eastin

30. 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

31. 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 of the meridional equation and subtract of the zonal equation
After use of the product rule, some simplifications, and cancellations:
Zonal Momentum
Meridional Momentum
Advanced Synoptic
M. D. Eastin

32. Vorticity Equation What do the terms represent?
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:
Advanced Synoptic
M. D. Eastin

33. Vorticity Equation Physical Explanation of Significant Terms:
Horizontal Advection of Relative Vorticity
The local relative vorticity will increase (decrease) if positive (negative) relative vorticity is
advected toward the location → Positive Vorticity Advection (PVA) and
→ Negative Vorticity Advection (NVA)
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

34. Relative Vorticity (ζ) Relative Vorticity Advection
Vorticity Equation
Physical Explanation: Horizontal Advection of Relative Vorticity
Relative Vorticity (ζ)
Relative Vorticity Advection
From Lackmann (2011)
Advanced Synoptic
M. D. Eastin

35. 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

36. Summary Important Dynamic Meteorology (METR 3250) Concepts:
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
Advanced Synoptic
M. D. Eastin