Final Review
1
Conservation Relations in Lagrangian form
• Mass:
dM
=0
dt
(1)
dP
= FB + FS
dt
(2)
dE
= Q̇ − Ẇ
dt
(3)
• Linear mo:
• Energy
2
Material derivative
D
∂
=
+u·∇
Dt
∂t
3
(4)
Velocity gradient tensor
G=
G
∂u
∂y
∂v
∂y
∂u
∂x
∂v
∂x
(5)
1
1
1
1
G + GT + G − GT
2
2
2
2
1
e
+
r
2
=
=
(6)
(7)
e ≡ strain rate tensor
r ≡ rotation rate tensor
⎡
e=⎣
⎡
r=⎣ 1
2
0
∂v
∂x
−
∂u
∂y
∂u
∂x
∂u
∂y
+
−
∂v
∂x
∂v
∂x
1
2
∂u
∂y
∂v
∂x
+
(8)
⎤
⎦
(9)
∂v
∂y
−
∂u
∂y
⎤
⎦=
0
0
ωz
−ωz
0
(10)
M ≡ Linear uncertainty propagator
(t) = M(0)
(11)
[V, Λ] = eig(MT M)
[U, Λ] = eig(MM )
T
1
(12)
(13)
Figure 1: (fig:ReviewLinUncerProp) Linear uncertainty propagator.
4
Reynolds Transport Theorem
∂c
dC
d c dV =
c u · dA
=
dV +
dt
dt
∂t
V
V
(14)
A
≡ extensive (sum of parts)
C
c ≡ intensive (stuff per volume)
5
Divergence (or Gauss) Theorem
∇ · Q dV =
V
Q · dA
(15)
A
Divergence on left, flux on right.
6
Another form of the RTT
dC
dt
7
=
∂c
+ ∇ · (c u) dV
∂t
(16)
V
Yet Another form of the RTT
dC
dt
=
Dc
+ c∇ · u dV
Dt
(17)
V
(18)
Use of RTT to go from Lagrange conservation laws (e.g.
8
dM
dt
= 0) to Eulearian (e.g.
Dρ
Dt +ρ∇·u
= 0)
Continuity
No approximation:
Dρ
+ ρ∇ · u = 0
Dt
(19)
∇·u=0
(20)
∇·u=0
(21)
Boussinesq:
Euler (but can have compressible Euler):
2
9
Momentum
No approximation:
ρ
Du
Dt
= ρg − ∇p + µ∇2 u +
µ
∇(∇ · u)
3
(22)
Boussinesq:
ρ0
Du
Dt
= ρ g − ∇p + µ∇2 u
(23)
Euler:
ρ
10
Du
Dt
= ρg − ∇p
(24)
Heat
No approximation: (Jim – There are a couple approximations in there ;)
ρcv
DT
Dt
= −p(∇ · u) + φ − ∇ · q
(25)
Boussinesq:
Dt
Dt
11
k 2
∇ T
pcp
=
(26)
State
Atmosphere:
p
= ρRT
(27)
Ocean:
ρ = ρ0 (1 − λT (T − T0 ) + λ0 (S − S0 ))
DS
= f ∇2 S
Dt
12
(28)
(29)
Entropy
No approximation and Boussinesq:
µ > 0, k > 0
13
(30)
Bernoulli
• Steady Flow (inviscid barotropic)
1
dp
u · u + gz +
=
2
ρ
constant along streamlines and vortex lines
3
(31)
• Steady irrotational flow (inviscid barotropic):
1
dp
u · u + gz +
=
2
ρ
constant everywhere
(32)
• Irrotational flow (inviscid barotropic):
∂φ 1
+ u · u + gz +
∂t
2
dp
ρ
u = ∇φ
(33)
(34)
Remember, ∇ · u ⇒ ∇ · ∇φ = 0 ⇒ ∇2 φ = 0.
14
Solid body rotation
uθ
ωz
= Ω0 r
= 2Ω0
Λ = ωz πr2
15
16
(35)
(36)
(37)
Point Vortex
uθ
=
ωz
=
Γ
2πr
0 (Except at location of point vortex)
(38)
(39)
Circulation
Γ=
u · ds
(40)
(∇ × u) · dA
(41)
c
17
Stoke’s Theorem
u · ds =
c
18
A
Kelvin’s circulation theorem (non-rotating)
DΓ
=0
Dt
Inviscid, barotropic, only conservative body forces.
4
(42)
19
Helmholtz vortex theorems
1. Vortex lines move with the fluid
2. Circulation of a vortex tube is constant along its length
3. A vortex tube can only end at a solid boundary, or form a closed loop
4. The circulation of a vortex tube is constant in time
20
Vorticity equation (non-rotating)
Incompressible, barotropic:
Dω
Dt
= ω · ∇u + ν∇2 ω
(43)
Incompressible, baroclinic:
Dω
Dt
21
1
∇ρ × ∇p + ν∇2 ω
ρ2
(44)
Rotating frame
uf ixed
af ixed
22
= ω · ∇u +
= urot + Ω × r
= arot + 2Ω × urot + Ω × (Ω × r)
= arot + 2Ω × urot − Ω2 R
(45)
(46)
(47)
Momentum equation (rotating)
Du
Dt
ν
1
= − ∇p + (g + Ω2 R) − 2Ω × u + ν∇2 u + ∇(∇ · u)
p
3
ν
1
= − ∇p − g − 2Ω × u + ν∇2 u + ∇(∇ · u)
p
3
(48)
(49)
Centrifugal force sucked into gravity term.
23
Coriolis force
• Turns stuff to the right (N.H.)
• Does no work
24
Vorticity equation (rotating earth-centric coordinates)
Dω
1
= (ω + 2Ω) · ∇u + 2 ∇ρ × ∇p + ν∇2 ω
Dt
ρ
5
(50)
25
Kelvin’s circulation theorem (rotating)
DΓa
= 0,
Dt
26
(ω + 2Ω) · dA
Γa =
(51)
A
Simplified momentum equations
1 ∂p
∂2u
+ fv + ν 2
ρ ∂x
∂z
1 ∂p
∂2v
= −
− fu + ν 2
ρ ∂y
∂z
1 ∂p
0 = −
−g
ρ ∂z
Du
Dt
Dv
Dt
= −
(52)
(53)
(54)
Know scaling arguments that got us here.
27
Inviscid, simplified momentum equations
Du
Dt
Dv
Dt
1 ∂p
+ fv
ρ ∂x
1 ∂p
= −
− fu
ρ ∂y
∂p
0 = −
− pg
∂z
28
= −
(57)
(58)
β plane approximation
f = f0 + βy = 2Ω sin φ0 +
30
(56)
f plane approximation
f = f0 = 2Ω sin φ0
29
(55)
2Ω cos φ0
y
r
(59)
Balances and flows from simplified, inviscid mo. eq.
1. Inertial oscillations:
−
u2θ
= f uθ
r
1
p
1
p
∂p
∂x
∂p
∂y
(60)
2. Geostrophy
6
= fv
(61)
= −f u
(62)
3. Gradient wind (can be less than or greater than geostrophic wind):
−
u2θ
r
uθ
1 ∂p
+ f uθ
p ∂r
1 ∂p
= f (R0 , 2 , )
ρf r ∂r
= −
(63)
(64)
• Describes cyclonic/anticyclonic highs/lows
• High pressure
∂p
∂r
less than low pressure
∂p
∂r
• High pressure systems have gentle winds in center
4. Cyclostrophic wind
u2θ
1 ∂p
=
r
ρ ∂r
(65)
5. Isallobaric wind
u
31
∂u
∂x
∂u
∂t
= fv
(66)
= fv
(67)
1 ∂p
− f v − νH ∇2 u = 0
ρ ∂x
(68)
Geostrophy plus friction
Three way balance, cross-isobaric flow.
32
Balances and flows from simplified vorticity equation
• Taylor-Proudman (barotropic):
2Ω · ∇u = 0
∂u
∂v
⇒
=
=0
∂z
∂z
⇒ vertical rigidity
(69)
(70)
(71)
• Thermal wind (baroclinic):
1
∇ρ × ∇p = 0
ρ2
∂u
g
⇒
= − k × ∇ρ
∂z
fρ
∂u
gρ0 α
⇒
=
k × ∇T
∂z
fρ
2Ω · ∇u +
This last step was made based on the relationship between p and T .
7
(72)
(73)
(74)
33
Ekman Layer
• Non-rotating boundary layers grow, Ekman layer does not.
• Mass transport:
τ ×k
f
(75)
1
∇×τ ·k
ρf
(76)
MEk =
• Ekman spiral
• Ekman pumping and suction
WEk =
34
Sverdrup Transport
• Explained using Kelvin’s circulation theorem: (Jim – You appear to have two ρ in your
notes)
v=
1
(∇ × τ ) · k
hρ
(77)
• Wind-driven circulation sensitive to curl of wind stress
• Return flow in the western boundary current
35
Shallow water equations
Du
Dt
Dv
Dt
∂h
∂t
36
∂h
+ fv
∂x
∂h
= −g
− fu
∂y
∂u ∂v
=0
+ h
+
∂x ∂y
= −g
(78)
(79)
(80)
Wave kinematics
k=
• Phase speed =
• Group speed =
2π
,
λx
l=
2π
,
λy
ω=
2π
T
(81)
ω
k
∂ω
∂k
(carries information).
• Dispersion relation relates ω, k, l, and provides graphical information about phase and group
speed.
8
Figure 2: (fig:ReviewWaveDispRel) Wave dispersion relationship. Positive slope means positive
group speed. Negative slope means negative group speed. Slope of line connecting a point on the
curve to the origin gives phase speed.
37
Shallow water potential vorticity equation
D
Dt
ωz + f
h
=0
(82)
Obtained from cross-differentiating, subtracting and simplifying the shallow water equations. Note:
D
fixed depth barotropic vorticity equation is Dt
(ωz + f ) = 0.
38
Potential vorticity
ωz + f
= constant
h
Potential vorticity is conserved following the flow. “Flow over a mountain” example.
39
(83)
Shallow water gravity waves without rotation
• Obtained from non-rotating shallow water equations
• ω = g h̄K = cK
• Non-dispersive
40
Inertia-gravity waves
• Obtained from rotating shallow water equations
• ω = 0, ± f02 + c2 K 2
• Dispersive, except in limit of large K
• Behave like non-rotating gravity waves at large K, ω = cK
• Behave like inertial oscillations at small k, ω = f0
41
Kelvin waves
• Obtained from rotating shallow water equations with v = 0 (transverse velocity) and a lateral
boundary
• ω = cK
• Non-dispersive
9
42
Constant depth, barotropic Rossby waves
D
• Obtained from barotropic vorticity equation ( Dt
(ωz + f ) = 0)
• Dispersion relationship (see figure):
ω=−
βk
k 2 + l2
(84)
• Dispersive
• Phase and group speeds can be in opposite directions
Figure 3: (fig:ReviewBarotropicRossWave) Constant depth, barotropic Rossby wave dispersion
relationship.
43
Shallow water Rossby waves
• Obtained from conservation of potential vorticity
• ω = − k2 +lβk
2+
1
R2
d
, Rd =
C
f0
• Dispersive
• Phase and group speeds can be in opposite directions
• Rd sets a length scale that forces a maximum phase speed
Figure 4: (fig:ReviewShallowWaterRossWave) Shallow water Rossby wave dispersion relationship.
10