Fundamental Equation in z coordinate
p = ρ RT or pα = RT
Du uv tan φ uw
1 ∂p
−
+
=−
+ fv − f 'w + Fx
Dt
a
a
ρ ∂x
Dv u 2 tan φ vw
1 ∂p
+
+
=−
− fu + Fy
Dt
a
a
ρ ∂y
Dw u 2 + v 2
1 ∂p
−
=−
− g + f 'u + Fz
Dt
a
ρ ∂z
Dρ
+ ρ∇ ⋅ U = 0
Dt
DT
Dp
cp
−α
= L(C − E) + Q rad + Q diff
Dt
Dt
(for full moist model, equation of water vapor is also included)
f --- 2Ωsin φ
f ' --- 2Ω cos φ
C—condensation
E—evaporation
6 equations
6 unknowns: u, v, w, p, ρ, T
4 dimensions x, y, z, t
SCALE ANALYSIS
Equations above describe motions of the atmosphere at all scales from mm to 1000s km.
If we are interested in the motions that matter to our day-to-day weather (i.e.,
phenomenology described by “synoptic” weather map), we may want to eliminate or
filter out the unwanted type of motions, such as sound waves. Elimination of terms on
scaling considerations not only has the advantage of simplifying the mathematics, but
also elimination of the unwanted motions which may corrupt the numerical solution for
the wanted motion (e.g., the first attempt of Richardson on numerical weather forecast
failed because of the equations he used contain the acoustic and gravity waves).
To determine the dominant balance of terms in the equations, and apply approximations
to simplify their solutions, we define the following characteristics scales of the field
variables based on observed values from midlatitude synoptic systems. These estimates
of scales are then substituted into the equations to determine which terms are of the same
(larger) order of magnitude---the balance of leading orders.
Note that scale analysis in general only tells “what” but not “why”.
1
U  10m s −1
−2
W  10 m s
horizontal velocity
−1
vertical velocity
L  10 m
horizoantl length
H  10 4 m
depth scale (vertial length)
δ P/ρ  10 3 m 2 s −2
horizonal pressure fluctuation scale
T  L / U  10 5 s
advective time scale
f  10 −4 s −1
planetary vorticity,
time scale of earth rotation,
Coriolis acceleration on relative motion
6
Table 2.1 shows the characteristic magnitude of each term in the equation set above.
1. Geostrophic approximation
It is apparent from Table 2.1 that for midlatitude synoptic scale disturbances the Coriolis
force and the pressure gradient force are in approximate balance. Retaining only these
two terms in the equations gives as a first approximation – the geostrophic relationship
1 ∂p
1 ∂p
− fv ≈ −
;
− fu ≈
(2.22)
ρ ∂x
ρ ∂y
which is just a diagnostic relationship.
To predict time evolution, we need keep the higher order term---the time tendency term
and inertial terms. For example,
2
Du ∂u
∂u
∂u ⎧ ∂u ⎫
1 ∂p

+u
+ v + ⎨w ⎬ ≈ −
+ fv
Dt ∂t
∂x
∂y ⎩ ∂z ⎭
ρ ∂x


U2 /L
10 −4
f0 U
10 −3
U
 1.
f0 L
By analogy to the geostrophic approximation, it is possible to define horizontal
geostrophic wind Vg = iug + jvg , which satisfies (2.22) identically. In vectorial form,
For the geostrophic relationship to hold, Ro ≡
1
∇p
(2.23)
ρf
It should be kept in mind that (2.23) always defines the geostrophic wind, but only for
large-scale motions away from equator should (2.22) apply as an approximation to the
actual horizontal wind field. The typical accuracy of (2.22) is within 10-15% of the actual
wind.
Vg ≡ k ×
Homework: show ∇p ⊥ Vg or ∇p ⋅ Vg = 0
2. Hydrostatic approximation
Similar scale analysis leads to, to a high degree of accuracy, the hydrostatic equilibrium,
that is, the pressure at any point is simply equal the weight of the of a unit cross-section
column of air above that point.
However, the above analysis is somehow misleading. It is not sufficient to show merely
that the vertical acceleration is small compared to g. Because only that part of pressure
field that varies horizontally is directly coupled to the horizontal velocity field, it is
actually necessary to show that the horizontally varying pressure component is itself in
hydrostatic equilibrium with the horizontally varying density field. To do this it is
conventional to first define a standard pressure p0 (z) , which is the horizontally averaged
pressure at each height, and a corresponding standard density ρ0 (z) , defined as such so
that p0 (z) and ρ0 (z) are in exact hydrostatic balance:
3
Therefore, for synoptic motion, vertical accelerations are negligible and the vertical
velocity cannot be determined from the vertical momentum equation.
Can be transformed to p-coordinate, because p is a monotonic and single-valued function
of z.
3. Scale analysis of the Continuity Equation
1 Dρ
+ ∇⋅U = 0
ρ Dt
(2.31)
Following the technique developed above, and assuming that ρ '/ ρ0  1 , we
approximate the continuity equation (2.31) as
4
1 ⎛ ∂ρ '
⎞ w d ρ0
+ U ⋅ ∇ρ '⎟ +
+ ∇⋅U ≈ 0
⎜⎝
⎠ ρ0 dz 
ρ ∂t


 
 

A
ρ' U
ρ0 L
10 −7 s −1
B
W
H
C
10 −1
U
L
10 −6 s −1 10 −6 s −1
A. For synoptic scale motion: ρ '/ ρ0 ~ 10 −2
B. For motion in which the depth scale H is comparable to the density scale height,
d ln ρ0 / dz  H −1  10km .
∂u ∂v ∂w
+ +
, for synoptic motions the first two terms tend to cancel each
∂x ∂y ∂z
other, giving rise to a sum that is an order smaller.
C. ∇ ⋅ U =
Thus, terms B and C are each an order of magnitude larger than term A, and to a first
approximation, terms B and C balance each other in the continuity equation.
∂u ∂v ∂w
d
+ +
+ w (ln ρ0 ) = 0
∂x ∂y ∂z
dz
or in vectorial form
(2.34)
∇ ⋅ ( ρ0 U) = 0
This means that for synoptic scale motions the mass flux computed using the basic state
density is nondivergent. For purely horizontal flow, the atmosphere behaves as though it
were an incompressible fluid. This is similar to, but not the same as the idealization of
incompressibility used in fluid mechanics. However, an incompressible fluid has density
constant following the motion:
Dρ
=0
Dt
Thus, by (2.31), velocity divergence vanishes ( ∇ ⋅ U = 0 , used in Boussinesq sets of
equations1) in an incompressible fluid, which is not the same as (2.34).
1
The Boussinesq approximation ignores all variations of density of a fluid in the
momentum equations, except when associated with the gravitational (or buoyancy) term.
Boussinesq sets of equations use ∇ ⋅ U = 0 , but this should absolutely not allow one to go
back and use (2.31) to say that Dρ = 0 ; the evolution of density is given by the
Dt
thermodynamic equation in conjunction with an equation of state.
However, using p-coordinates, the Boussinesq equations for atmospheric circulation do
not need these approximation in continuity equations. Let the oceanographers to worry
about this issue.
5
It is also worthy of note that equation (2.34) is where the eponymous ‘anelastic
approximation’ arises: the elastic compressibility of the fluid is neglected (by assuming
T ~ U / L , in which time scale the sound waves do not dominate --- A term drops out of
the equation), and this serves to eliminate sound waves. This approximation is also
referred to as ‘weak Boussinesq approximation’. Because if we let ρ0 (z) be a constant,
the anelastic equations become identical to the Boussinesq equations.
6
GOVERNING EQUATION OF MOTION IN PRESSURE COORDINATES
1. Horizontal momentum equations
Recall equation (1.25) and (1.26),
1 ∂p
∂Φ
=
ρ ∂x z ∂x
p
1 ∂p
∂Φ
=
ρ ∂y z ∂y
p
In vectorial form, let the horizontal velocity V = iu + jv , the horizontal momentum
equation in isobaric coordinate can be written as
DV
(3.2)
+ fk × V = −∇ p Φ
Dt
where ∇ p is the horizontal gradient operator applied with pressure held constant. (some
nuances about the observation of the horizontal wind on isobaric surface)
In p-coor, variables are function of x, y, p,t , and total derivative
D
∂
∂
∂
∂
= +u + v +ω
Dt ∂t
∂x
∂y
∂p
Dp
where ω ≡
is the pressure change following the motion, which plays the same role in
Dt
p-coor as w does in z-coor.
2. Continuity equation
It is possible to transform the continuity equation (2.31) from z-coor to p-coor. But I
leave it to you. However, it is simpler to directly derive the isobaric form by considering
a Lagrangian control volume δV = δ xδ yδ z and applying the hydrostatic equation
δ p = − ρ gδ z to express the volume element as δV = −δ xδ yδ p / ( ρ g) . The mass of this
fluid element, which is conserved following the motion, is then
δ M = ρδV = −δ xδ yδ z / g , thus
1 D
g
D ⎛ δ xδ yδ p ⎞
(δ M ) =
=0
δ M Dt
δ xδ yδ p Dt ⎜⎝
g ⎟⎠
chain rule and change order of the differential operators
⇒
1 ⎛ Dx ⎞ 1 ⎛ Dy ⎞ 1 ⎛ Dp ⎞
δ⎜
δ⎜
⎟ + δ⎜
⎟+
⎟ =0
δ x ⎝ Dt ⎠ δ y ⎝ Dt ⎠ δ p ⎝ Dt ⎠
or
δ u δ v δω
+
+
=0
δx δy δ p
7
Taking the limit δ x, δ y, δ p → 0 and observing that δ x and δ y are evaluated at constant
pressure, we derive the continuity equation in the isobaric system:
⎛ ∂u ∂v ⎞
∂ω
⎜⎝ ∂x + ∂y ⎟⎠ + ∂p = 0
p
This form of the continuity equation contains no reference to the density field and does
not involve time derivatives---the chief advantages of the isobaric coordinates.
3. Thermodynamic Energy Equation
From (2.42)
DT
Dp
−α
= J,
Dt
Dt
we have the following written in isobaric system:
⎛ ∂T
∂T
∂T
∂T ⎞
cp ⎜
+u
+v
+ω
− αω = J
⎝ ∂t
∂x
∂y
∂p ⎟⎠
This may be further, by combining the last two terms of the lhs, written as
∂T
∂T
∂T
J
+u
+v
− S pω =
∂t
∂x
∂y
cp
where
RT ∂T
T ∂θ
(3.7, derive this)
Sp ≡
−
=−
c p p ∂p
θ ∂p
is the static stability parameter for the isobaric system. Using (2.49) and the hydrostatic
equation, (3.7) may be rewritten as
S p = (Γ d − Γ) / ρ g
cp
Thus, S p is positive provided that the lapse rate is less than dry adiabatic.
Scale analysis of
Dp ∂p
∂p
=
+ V ⋅ ∇p + w
Dt ∂t
∂z
shows that it is quite a good approximation to let
ω = − ρ gw
ω≡
Put together, we now have
8
Fundamental Equation in p-coordinate
p = ρ RT or pα = RT
∂u
∂u
∂u
∂u uv tan φ uω
f 'ω ∂Φ
+u
+ v +ω
=
+
+ fv +
−
+ Frx
∂t
∂x
∂y
∂p
a
ρ ga
ρg
∂x
∂v
∂v
∂v
∂v
u 2 tan φ vω
+u + v +ω
=−
+
− fu
∂t
∂x
∂y
∂p
a
ρ ga
∂Φ
= −α
∂p
−
∂Φ
+ Fry
∂y
∂u ∂v ∂ω
+ +
=0
∂x ∂y ∂p
⎛ ∂T
⎞
∂T
∂T
cp ⎜
+u
+v
− αω ⎟ = L(C − E) + Q rad + Q diff
⎝ ∂t
⎠
∂x
∂y
OR
∂T
∂T
∂T
L
+u
+v
= Sω + (C − E) + qrad + qdiff
∂t
∂x
∂y
cp
qrad = Q rad / c p
qdiff = Q diff / c p
where
T ∂θ
S=−
(static stability)
θ ∂p
∂r
∂r
∂r
∂r
+u + v +ω
= Sources − Sinks
∂t
∂x
∂y
∂p
…after the scale ananlysis of the first round…
9
p = ρ RT or pα = RT
∂u
∂u
∂u
∂u
∂Φ
+u
+ v +ω
= fv −
+ Frx
∂t
∂x
∂y
∂p
∂x
∂v
∂v
∂v
∂v
∂Φ
+u + v +ω
= − fu −
+ Fry
∂t
∂x
∂y
∂p
∂y
∂Φ
RT
= −α = −
∂p
p
∂u ∂v ∂ω
+ +
=0
∂x ∂y ∂p
⎛ ∂T
⎞
∂T
∂T
cp ⎜
+u
+v
− αω ⎟ = L(C − E) + Q rad + Q diff
⎝ ∂t
⎠
∂x
∂y
OR
∂T
∂T
∂T
L
+u
+v
= Sω + (C − E) + qrad + qdiff
∂t
∂x
∂y
cp
qrad = Q rad / c p
qdiff = Q diff / c p
where
T ∂θ
S=−
(static stability)
θ ∂p
(see page 147, equations (6.1-4))
10
Geostrophic wind
fVg = −k × ∇ p Φ
∂Φ
∂x
∂Φ
fug = −
∂y
fvg =
Thermal Wind
∂
1 ∂ ⎛ ∂Φ ⎞
vg =
∂p
f ∂x ⎜⎝ ∂p ⎟⎠
∂
1 ∂ ⎛ ∂Φ ⎞
ug = −
∂p
f ∂y ⎜⎝ ∂p ⎟⎠
∂Φ
RT
= −α = −
using hydrostatic equation:
, then,
∂p
p
∂vg
∂vg
R ⎛ ∂T ⎞
p
=
=− ⎜
⎟
∂p ∂ ln p
f ⎝ ∂x ⎠ p
p
or in vectorial form
∂Vg
∂ug
∂p
=
∂ug
∂ ln p
=
R ⎛ ∂T ⎞
f ⎜⎝ ∂y ⎟⎠ p
R
k × ∇ pT
thermal wind
(3.30)
∂ ln p
f
Designating the thermal wind vector by VT , we may integrate (3.30) from pressure level
p0 to p1 ( p1 < p0 ) to get
R p1
VT ≡ Vg ( p1 ) − Vg ( p0 ) = − ∫ k × ∇ pT d ln p
(3.31)
f p0
Letting T denote the mean temperature in the layer between pressure level p0 and p1 ,
than the thermal wind relation can be written as:
⎛p ⎞ 1
R
g
(3.35)
VT = k × ∇ T ln ⎜ 0 ⎟ = k × ∇(Φ1 − Φ 0 ) = k × ∇ZT
f
f
⎝ p1 ⎠ f
The equivalence of these equations can be verified readily by integrating the hydrostatic
equation vertically from p0 to p1 .
=−
(
)
With (3.35), it is therefore possible to obtain a reasonable estimate of the horizontal
temperature advection and its vertical dependence at a given location solely from data on
the vertical profile of the wind of a single sounding. In the meantime, the geostrophic
wind at any level can be estimated from the mean temperature field, provided that the
geostrophic velocity is known at a single level.
Counterclockwise turning with height  cold advection
Clockwise turning (veering) with height  warm advection
11
12
13
Homework A: read page 73 carefully and derive the hypsometric equation
⎛p ⎞
Φ1 − Φ 0 ≡ gZT = R T ln ⎜ 0 ⎟
⎝ p1 ⎠
(3.34)
Homework B: Use the knowledge of page 73 and the hypsometric equation above to
estimate the expected tilt of 500 hPa pressure surface between the tropics and the pole
based on the following conditions and assumptions: (i) T only varies only merdionally,
i.e., isothermal in vertical; (ii) surface pressure ps = 1000 hPa; (iii) equator-to-pole
equator
= 40K .
temperature difference ΔT pole
And draw picture to demonstrate the results schematically.
Hints: use equation Δz
equator
pole
=
equator
RΔT pole
g
⎛ p ⎞
ln ⎜ s ⎟ and diagram below as clues.
⎝ p500 ⎠
14
Barotropic flow
“Barotropic” defined by ρ = ρ( p) , so the isobaric surface is also the surface of constant
density. For an ideal gas, the isobaric surface will be also isothermal for the barotropic
atmosphere. Thus
∇ pT = 0 in a barotropic ideal-gas atmosphere
and
∂Vg
R
= − k × ∇ pT = 0
∂ ln p
f
and geostrophic wind is independent of p (vertical coordinate).
Barotropy produces a very strong constraint on the motions and forms basis of much of
theoretical understanding.
If ρ = ρ( p,T ) , then the flow is baroclinic.
15