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AOSS 321, Winter 2009 Earth Systems Dynamics
Lecture 132/19/2009
Christiane Jablonowski Eric Hetland
[email protected] [email protected]
734-763-6238 734-615-3177
Today’s class
• Equations of motion in pressure coordinates• Pressure tendency equation• Evolution of an idealized low pressure system• Vertical velocity
Generalized vertical coordinate
x
z Constant surface s0
Δz
Δx
p1 p2
p3
The derivation can be further generalized for any verticalcoordinate ‘s’ that is a single-valued monotonic function of height with ∂s /∂z ≠ 0.
Pressure values
Vertical coordinate transformations
x
z
p0
p0+Δp
Δz
Δx
Pressure gradient along ‘s’:
€
∂p
∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟z
=∂p
∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟p
−∂p
∂z
∂z
∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟p
= − −ρg( )∂z
∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟p
= ρ∂gz
∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟p
= ρ∂Φ
∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟p
For s = p:(pressure levels)
€
p3 − p1
Δx=
p3 − p2
Δz
Δz
Δx+
p2 − p1
Δx
∂p
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟s
=∂p
∂z
∂z
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟s
+∂p
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟z
= 0
What do we do with the material derivative when using p in the vertical?
€
DT
Dt
⎛
⎝ ⎜
⎞
⎠ ⎟p
=∂T
∂t
⎛
⎝ ⎜
⎞
⎠ ⎟p
+ u∂T
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟p
+ v∂T
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
+ ω∂T
∂p
DT
Dt=
∂T
∂t+ u
∂T
∂x+ v
∂T
∂y+ ω
∂T
∂p
By definition:
ω===Dt
Dpv
Dt
Dyu
Dt
Dx,,
Total derivative DT/Dt on constant pressure surfaces:
(subscript omitted)
Our approximated horizontal momentum equations (in p coordinates)
€
du
dt
⎛
⎝ ⎜
⎞
⎠ ⎟p
= −∂Φ
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟p
+ fv
dv
dt
⎛
⎝ ⎜
⎞
⎠ ⎟p
= −∂Φ
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
− fu
Dr v h
Dt
⎛
⎝ ⎜
⎞
⎠ ⎟p
+ f r k ×
r v h = −∇ pΦ
No viscosity, no metric terms, no cos-Coriolis terms
Subscript h: horizontal Subscript p: constant p surfaces!Sometimes subscript is omitted, tells you that this is on p surfaces
Thermodynamic equation
(in p coordinates)
€
cv
DT
Dt+ p
Dα
Dt= J
(cv + R)DT
Dt− α
Dp
Dt= J
c p = cv + R
c p
∂T
∂t+ u
∂T
∂x+ v
∂T
∂y+ ω
∂T
∂p
⎛
⎝ ⎜
⎞
⎠ ⎟− αω = J
Two forms of the thermodynamic equation
use
Thermodynamic equation(in p coordinates)
€
c p
∂T
∂t+ u
∂T
∂x+ v
∂T
∂y+ ω
∂T
∂p
⎛
⎝ ⎜
⎞
⎠ ⎟−αω = J
∂T
∂t+ u
∂T
∂x+ v
∂T
∂y+ ω
∂T
∂p−
α
c p
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟=
J
c p
∂T
∂t+ u
∂T
∂x+ v
∂T
∂y+ ω
∂T
∂p−
RT
pc p
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟=
J
c p
Equation of state
Thermodynamic equation(in p coordinates)
€
∂T
∂t+ u
∂T
∂x+ v
∂T
∂y+ ω
∂T
∂p−
RT
pc p
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟=
J
c p
∂T
∂t+ u
∂T
∂x+ v
∂T
∂y− Spω =
J
c p
with Sp =RdT
pc p
−∂T
∂p
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Sp is the static stability parameter.
Static stability parameter
€
Sp =RdT
pc p
−∂T
∂p
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= −
T
Θ
∂Θ
∂p
(lecture 11, slide 14) we get
With the aid of the Poisson equation we get:
Plug in hydrostatic equation :
€
T
Θ
∂Θ
∂z= Γd − Γ
€
Sp = (Γd − Γ) /(ρg)
Using
€
∂p
∂z= −ρg
€
Sp =T
Θ
∂Θ
∂z
1
ρg
The static stability parameter Sp is positive (statically stable atmosphere) provided that the lapse rate of the air is less than the dry adiabatic lapse rate d.
Continuity equation
€
Dρ
Dt= −ρ∇ •
r v
∂u
∂x+
∂v
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
+∂ω
∂p= 0
Let’s think about this derivation!
in z coordinates:
in p coordinates:
Continuity equation:Derivation in p coordinates
Consider a Lagrangianvolume: V= x y z
xy
z
Apply the hydrostatic equation p= -gz to express the volume element as V= - x y p/(g)
The mass of this fluid element is:M = V= - x y p/(g) = - x y p/g
Recall: the mass of this fluid element is conserved following the motion (in the Lagrangian sense): D(M)/Dt = 0
Continuity equation:Derivation in p coordinates
€
DδM
Dt= 0
1
δM
DδM
Dt= 0
−g
δxδyδp
D δxδyδp( )Dt
1
−g= 0
1
δxδyδp
Dδx
Dtδyδp +
Dδy
Dtδxδp +
Dδp
Dtδxδy
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
1
δxδ
Dx
Dt
⎛
⎝ ⎜
⎞
⎠ ⎟+
1
δyδ
Dy
Dt
⎛
⎝ ⎜
⎞
⎠ ⎟+
1
δpδ
Dp
Dt
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
δu
δx+
δv
δy+
δω
δp= 0
∂u
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟p
+∂v
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
+∂ω
∂p= 0
Thus:
Take the limit x,y,p 0
Continuity equation inp coordinates
Product rule!
Continuity equation(in p coordinates)
€
∂u
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟p
+∂v
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
+∂ω
∂p= 0
This form of the continuity equation contains no reference to the density field and does not involve time derivatives.
The simplicity of this equation is one of the chief advantages of the isobaric system.
€
Recall : ω =Dp
Dt
Hydrostatic equation(in p coordinates)
€
∂p
∂z= −ρg
∂z
∂p= −
1
ρg= −
α
g
∂(gz)
∂p= −α
∂Φ
∂p= −α = −
RdT
pHydrostatic equation inp coordinates
Start with
The hydrostatic equation replaces/approximates the 3rd momentum equation (in the vertical direction).
Approximated equations of motion in pressure coordinates
(without friction, metric terms, cos-Coriolis terms,with hydrostatic approximation)
€
Dr v h
Dt+ f
r k ×
r v h = −∇ pΦ
∂Φ
∂p= −
RdT
p
(∂u
∂x+
∂v
∂y)p +
∂ω
∂p= 0
∂T
∂t+ u
∂T
∂x+ v
∂T
∂y− Spω =
J
c p
p = ρRdT
€
Material derivative : D( )Dt
=∂( )∂t
+ u∂( )∂x
+ v∂( )∂y
+ ω∂( )∂p
Let’s think about growing and decaying disturbances.
€
∂u
∂x+
∂v
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
+∂ω
∂p= 0
dω = −∂u
∂x+
∂v
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
dp
Mass continuity equation in pressure coordinates:
Let’s think about growing and decaying disturbances
€
dω = −∂u
∂x+
∂v
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
dp
dωω( psfc )
ω( p= 0)
∫ = −∂u
∂x+
∂v
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
dppsfc
p= 0
∫
ω( p = 0) − ω(psfc ) = −∂u
∂x+
∂v
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
dppsfc
p= 0
∫
ω( psfc ) = −∂u
∂x+
∂v
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
dpp= 0
psfc
∫ = − ∇ p •r v h( ) dp
p= 0
psfc
∫
Formally links vertical wind and divergence.
= 0 (boundary condition)
Integrate:
€
Since r v g =
1
ρf
r k ×∇p it follows
r v g • ∇p = 0
ω =∂p
∂t+
r v a • ∇p − wgρ
Let’s think about growing and decaying disturbances.
€
ω( psfc ) = −∂u
∂x+
∂v
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟p
dpp= 0
psfc
∫ = − ∇ p •r v h( ) dp
p= 0
psfc
∫
€
ω =Dp
Dt=
∂p
∂t+ (
r v h • ∇p) + w
∂p
∂z
ω =∂p
∂t+ (
r v g • ∇p +
r v a • ∇p) − wgρ
€
use r v h =
r v g +
r v a
Recall:
use hydrostatic equation
€
rv a • ∇pBoth and w are nearly zero at the surface (note:
not true in the free atmosphere away from the surface)
Surface pressure tendency equation
€
∂psfc
∂t≈ − (
∂u
∂x+
∂v
∂y)p dp
p= 0
psfc
∫ = − ∇ p •r v h( ) dp
p= 0
psfc
∫
Convergence (divergence) of mass into (from) column above the surface will increase (decrease) surface pressure.
At the surface:
€
ω( ps) ≈∂psfc
∂t
It follows:
Possible development of a surface low.
Earth’s surface
pressure surfaces
Possible development of a surface low.
Earth’s surface
pressure surfaceswarming
Possible development of a surface low.
Earth’s surface
pressure surfaces
warming
warming increases thickness
aloft(hypsometric
equation)
Possible development of a surface low.
Earth’s surface
Pressure gradient force is created,
mass diverges up here,
causes surface pressure to fall
warmingWarming increases
thickness aloft
Possible development of a surface low.
Earth’s surface
mass diverges up here
warming
warming increases thickness
LOW
generates surface low here
Possible development of a surface low.
Earth’s surface
mass diverges up here
warming
LOW
generates low here
H H
and highs here
Possible development of a surface low.
Earth’s surface
mass diverges up here
warming
LOW
pressure gradient initiates
convergence down hereH H
Possible development of a surface low.
Earth’s surface
mass diverges up here
warming
LOW
pressure gradient initiates
convergence down here
H H
Continuity Eqn.:upward
motion
• Increase in thickness from heating at mid levels.
• Pushes air up, creates pressure gradient force• Air diverges at upper levels to maintain mass
conservation.• This reduces mass of column, reducing the
surface pressure. Approximated by pressure tendency equation.
• This is countered by mass moving into the column, again conservation of mass. Convergence at lower levels.
A simple conceptual model of a low pressure system
A simple conceptual model of a low pressure system
• Pattern of convergence at low levels and divergence at upper levels triggers rising vertical motions.
• Rising motions might form clouds and precipitation.
• The exact growth (or decay) of the surface low will depend on characteristics of the atmosphere.
• Link of upper and lower atmosphere.
€
ω =Dp
Dt=
∂p
∂t+
r v h • ∇p + w
∂p
∂z
ω =∂p
∂t+
r v g • ∇p +
r v a • ∇p − wgρ
ω =∂p
∂t+
r v a • ∇p − wgρ
with the help of scale analysis
ω ≈ −wgρ
Vertical motions in the free atmosphere:The relationship between w and ω
= 0 hydrostatic equation
≈ 10 hPa/d≈ 1m/s 1Pa/km≈ 1 hPa/d
≈ 100 hPa/d
