synoptic scale balance equations using scale analysis (to identify the dominant ‘forces at...
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Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and manipulating the equations of motion we can arrive at:
geostrophic balancedeviations from geostrophic balance (curvature and
friction) hydrostatic balancehypsometric equationthermal wind equationQuasigeostrophic omega equation
The space and time scales of motion for a particular type of system are the characteristic distances and times traveled by air parcels in the system (or by molecules for molecular scales).
DLA Fig.10.2
Horizontal Momentum Equation
Synoptic Scale:U ≈ 10 m/sW ≈ 10-2 m/sL ≈ 106 mH ≈ 104 mT = L/U ≈ 105 sR ≈ 107 mfo ≈ 10-4 1/sPo ≈ 1000 hPa 1 Pa = kg/(ms2)ρ ≈ 1 kg/m3
Example Scale Analysis
geostrophic balance
Forces Acting on the Atmosphere – Pressure Gradient Force
DLA Fig. 7.5
causes a net force on air, directed toward lower
pressure
Forces Acting on the AtmosphereCoriolis Force
to the right of motion in the NH
strength determined by:1.latitude2.speed of motion
DLA Fig. 7.7A
Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and manipulating the equations of motion we can arrive at:
geostrophic balancedeviations from geostrophic balance (curvature and
friction) hydrostatic balancehypsometric equationthermal wind equationQuasigeostrophic omega equation
Forces Acting on the Atmosphere Centripetal Force & Gradient Wind Balance
DLA Fig. 7.13
force pointing away the center around which an object is turning
centripetal acc = - centrifugal force
(difference beteeen PGF and
COR)
Winds and Heights at 500 mb
Geostrophic Approximation: Strengths and Weaknesses – curved flow
Geostrophic Winds at 500 mb (determined using analyzed Z and geostrophic equations)
Geostrophic Approximation: Strengths and Weaknesses
Winds - Geostrophic Winds = Ageostrophic Winds (What’s Missing From Geostrophy)
Geostrophic, Gradient, and Real Winds
Vg is too weak
Vg is too strong
Forces Acting on the AtmosphereFriction
DLA Fig. 7.14
DLA Fig. 7.15
Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and manipulating the equations of motion we can arrive at:
geostrophic balancedeviations from geostrophic balance (curvature and
friction) hydrostatic balancehypsometric equationthermal wind equationQuasigeostrophic omega equation
Vertical Momentum Equation
Synoptic Scale:U ≈ 10 m/sW ≈ 10-2 m/sL ≈ 106 mH ≈ 104 mT = L/U ≈ 105 sR ≈ 107 mfo ≈ 10-4 1/sPo ≈ 1000 hPa 1 Pa = kg/(ms2)ρ ≈ 1 kg/m3
Example Scale Analysis
hydrostatic balance
DLA Fig. 7.6
Hydrostatic Balance
air parcel in hydrostatic balance experiences no net force in the vertical
Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and manipulating the equations of motion we can arrive at:
geostrophic balancedeviations from geostrophic balance (curvature and
friction) hydrostatic balancehypsometric equationthermal wind equationQuasigeostrophic omega equation
Geopotential, Geopotential Height, and the Hyposmetric Equation
Hypsometric Equation
We arrive at the hypsometric equation by using scale analysis (hydrostatic balance) and by combining the hydrostratic equation and the equation of state
The hypsometric equation:1. provides a quantitative measure of the geometric distance between 2 pressure
surfaces – it is directly proportional to the temperature of the layer2. Shows that the gravitational potential energy gained when raising a parcel is
also proportional to the temperature of the layer
We can quantitatively see what we intuitively know: a warm layer will be thicker than a cool layer
Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and manipulating the equations of motion we can arrive at:
geostrophic balancedeviations from geostrophic balance (curvature and
friction) hydrostatic balancehypsometric equationthermal wind equationQuasigeostrophic omega equation
Thermal Wind - Concepts
• Horizontal T gradients horizontal p gradients vertical variations in winds (e.g. geostrophic winds)
A non-zero horizontal T gradient leads to vertical wind shear
• Thermal wind (VT) describes this vertical wind shear:→ not an actual wind → it represents the difference between the geostrophic wind at 2
vertical levels
→ specifically, VT relates the horizontal T gradient to the vertical wind shear
Thermal Wind - Concepts
• VT is therefore a useful tool for analyzing the relationship between T, ρ, p and winds
• VT also provides information about T advection (backing vs. veering)
The Thermal Wind Equation
• VT is derived by combining the hypsometric equation and the geostrophic equation
• Note similarity to geostrophic wind, except T replaces Φ
• VT ‘blows’ parallel to isotherms, with low T on the left
Spatial relationships between horizontal T and thickness gradients, horizontal p gradient, and vertical geostrophic wind gradient.
Thermal Wind
H, Fig. 3.8
warmcold
vT is positivevg increases w/ height
Thermal Wind – Climatological Averages
WH Figure 1.11
North Southy
Thermal Wind – Extratropical Cyclone
Vertical cross section from Omaha, NE to Charleston, SC. WH Figure 3.19
NW
SE
we can apply the same logic to the
instantaneous picture in an extratropical
cylcone
Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and manipulating the equations of motion we can arrive at:
geostrophic balancedeviations from geostrophic balance (curvature and
friction) hydrostatic balancehypsometric equationthermal wind equationQuasigeostrophic omega equation
Term B – Relationship of Upper Level Vorticity to Divergence / Convergence
DLA Fig. 8.31
following air parcel motion:- divergence occurs where ζa is decreasing- convergence occurs where ζa is increasing
Omega Equation – Derivation
quasigeostrophic vorticity equation
quasigeostrophic thermodynamic equation
(1)
(2)
quasigeostrophic relative vorticity can be expressed as the Laplacian of geopotential
(3)
plug (3) into (1) (4)
re-arrange (2) (5)
Omega Equation – Derivation
the QG Omega Equation is a diagnostic equation used to determine rising and sinking motion based solely on the 3D
structure of the geopotential
• no wind observations necessary• no info regarding vorticity tendency• no T structure• downside: higher order derivates
Omega Equation – Derivation
AB C
Rising/Sinking A ≅ - signLHS ≅ - ω
+ RHS = rising motion- RHS = sinking motion
Differential Vorticity Advection
+ B = + vorticity adv. rising
- B = - vorticiy adv. sinking
Thickness Advection+ C = warm adv.
rising - C = cold adv.
sinking
H Fig. 6.11500 mb Height
1000 mb Height
Term B – Differential Vorticity Advection
PVA the column is coolingthere is very little temperature
advection above the L center the only way for the layer to cool is
via adiabatic cooling (rising)
PVA
Above Surface L
H Fig. 6.11500 mb Height
1000 mb Height
Term B – Differential Vorticity Advection
NVAthe column is warming
there is very little temperature advection above the H center
the only way for the layer to warm is via adiabatic warming (sinking)
NVA
Above Surface H
Term B – Differential Vorticity Advection
the ageostrophic circulation (rising/sinking) predicted in the previous slides maintains a hydrostatic T field (T and thickness are proportional) in the presence of differential
vorticity advection
without the vertical motion, either the vorticity changes at 500 mb could not remain geostrophic or the T changes in
the 1000-500 mb layer would not remain hydrostatic
H Fig. 6.11500 mb Height
1000 mb Height
Term C – Thickness Advection
WAA anticyclonic vorticity must increase at the 500 mb ridge,
vorticity advection cannot produce additional anticyclonic vorticity
divergence is required (rising)
WAA
At the 500 mb Ridge
H Fig. 6.11500 mb Height
1000 mb Height
CAA
CAA
cyclonic vorticity must increase at the 500 mb trough, vorticity
advection cannot produce additional cyclonic vorticity
convergence is required (sinking)
At the 500 mb Trough
Term C – Thickness Advection
the predicted vertical motion pattern is exactly that required to keep the upper-level vorticity field
geostrophic in the presence of height changes caused by the thermal advection
Term C – Thickness Advection