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 balance deviations from geostrophic balance (curvature and friction) hydrostatic balance hypsometric equation thermal wind equation Quasigeostrophic omega equation
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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.
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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
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).
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
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