Synoptic Scale Balance Equations Using scale analysis (to identify the dominant ‘forces at work’) and manipulating the equations of motion we can arrive

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





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





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





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





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





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)


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


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


1000 mb Height


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


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


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


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


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


Documents

Synoptic Scale Balance Equations Using scale analysis (to identify the dominant ‘forces at work’) and manipulating the equations of motion we can arrive