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

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/s W ≈ m/s L ≈ 10 6 m H ≈ 10 4 m T = L/U ≈ 10 5 s R ≈ 10 7 m f o ≈ /s P o ≈ 1000 hPa 1 Pa = kg/(ms 2 ) ρ ≈ 1 kg/m 3 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 Atmosphere Coriolis Force to the right of motion in the NH strength determined by: 1.latitude 2.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 balance deviations from geostrophic balance (curvature and friction) hydrostatic balance hypsometric equation thermal wind equation Quasigeostrophic omega equation

Forces Acting on the Atmosphere Centripetal Force & Gradient Wind Balance DLA Fig 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 V g is too weak V g is too strong

Forces Acting on the Atmosphere Friction DLA Fig 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 balance deviations from geostrophic balance (curvature and friction) hydrostatic balance hypsometric equation thermal wind equation Quasigeostrophic omega equation

Vertical Momentum Equation Synoptic Scale: U ≈ 10 m/s W ≈ m/s L ≈ 10 6 m H ≈ 10 4 m T = L/U ≈ 10 5 s R ≈ 10 7 m f o ≈ /s P o ≈ 1000 hPa 1 Pa = kg/(ms 2 ) ρ ≈ 1 kg/m 3 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 balance deviations from geostrophic balance (curvature and friction) hydrostatic balance hypsometric equation thermal wind equation Quasigeostrophic 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 layer 2.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 balance deviations from geostrophic balance (curvature and friction) hydrostatic balance hypsometric equation thermal wind equation Quasigeostrophic 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 (V T ) describes this vertical wind shear: → not an actual wind → it represents the difference between the geostrophic wind at 2 vertical levels → specifically, V T relates the horizontal T gradient to the vertical wind shear

Thermal Wind - Concepts V T is therefore a useful tool for analyzing the relationship between T, ρ, p and winds V T also provides information about T advection (backing vs. veering)

The Thermal Wind Equation V T is derived by combining the hypsometric equation and the geostrophic equation Note similarity to geostrophic wind, except T replaces Φ V T ‘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 v T is positive v g increases w/ height

Thermal Wind – Climatological Averages WH Figure 1.11 NorthSouth y

Thermal Wind – Extratropical Cyclone Vertical cross section from Omaha, NE to Charleston, SC. WH Figure 3.19 NWNW 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 balance deviations from geostrophic balance (curvature and friction) hydrostatic balance hypsometric equation thermal wind equation Quasigeostrophic omega equation

Term B – Relationship of Upper Level Vorticity to Divergence / Convergence DLA Fig following air parcel motion: - divergence occurs where ζ a is decreasing - convergence occurs where ζ a 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 A B C Rising/Sinking A ≅ - sign LHS ≅ - 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 mb Height 1000 mb Height Term B – Differential Vorticity Advection PVA the column is cooling there 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 mb Height 1000 mb Height Term B – Differential Vorticity Advection NVA the 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 mb layer would not remain hydrostatic

H Fig 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 mb Height 1000 mb Height 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