Concepts: Buoyancy and Vorticity (IPO-6, 12) Buoyancy
Archimedes Principle
Weight
density of the block
W gV!
!
=
=
!
!
Buoyancy Force
density of the water
BF gV!
!
=
=
If W> block sinks
If W< block rises
B
B
F
F
! !
! !
" >
" <
!
!
A Canonical Ocean Profile of Temperature (T), Density (ρ)
20m-100m
1km
4km
Mixed Layer
PycnoclineThermocline
T ρ
But , , )T S p! != (
Equation of State
0P
!"#
"Note that which shows that seawater is compressible!
The compressibility of seawater allows sound to be propagated at a speedp
c!
"=
"
In this equation p is taken relative to atmospheric pressure
0 0
03
-3
3 3 3
( ) ( )
1027 10 S=35psu
kg kg kg =.15 =.78 = 4.5 10
(m )( ) (m )( ) (m )( )
o
o
a T T b S S kp
where
kgT C
m
a b kC psu decibar
! !
!
0
0
" = " " + " +
= =
#
1500sec
mc !
pc
!
"=
"
Homework ProblemHow much does pressure contribute to seawater density at the bottom of an ocean of depth 4 km?
0 0
03
-3
3 3 3
0 0
-3
4
( ) ( ) p in dbars
1027 10 S=35psu
kg kg kg =.15 =.78 = 4.5 10
(m )( ) (m )( ) (m )( )
( ) ( ) z in meters
kg= 4.5 10
m
o
o
a T T b S S kp
where
kgT C
m
a b kC psu decibar
a T T b S S kz
k
! !
!
! !
!
0
0
0
" = " " + " +
= =
#
" = " " + " +
#
0 0 2
2 2
3
3
N( ) ( ) p in
m
Using
10 101500
4.5 10 sec
pa T T b S S k
g
p kc c
p g
g mc
k
!!
!
! !
!
0
0
"
0
0
"
" = " " + " +
$ $= % = %$ $
#= = =
#
Addendum
22
2
2 2
2
2
0 0
N ( )
Proof
Given ( )
in the equation of state
( ) ( )
g g g g T Sa b
z z c z z
g gT Sa b k
z z c z z c
gkUse c k
p g c
a T T b S S kz
!
!
" "
" " "
" " ""
""
"
" "
0 0 0
0 0
# 0
0
0
$ $ $ $= # = # # = #
$ $ $ $
$ $ $ $= # = # + + #
$ $ $ $
$= = % =$
# = # # + # +
Addendum
Potential Temperature , θ, and Density,
σ Notation
where potential temperature
( , , )
( , ,0)
( , , ) 1000
( , ,0) 1000
t
T S P
T S
T S P
S
! !
!
" #
" #
" #
# !
=
= $1000
= $1000
= $
= $
“potential”.
Adiabatic change (noheat exchanged)
!"
T
θ
!"
ρz
z+δz
)W z gV!"= (
)BF z z gV!" #= ( +
22 2
0 2
0 0 0
25 2
2
{ ) )}
{ ) )}but
where ( ) { } ( )
4.4 10 sec
net B
net
F F W z z z gV
z z z
z z
g g g g T SF zgV N zV N a b
z z z c z z
g
c
! !
! ! !
! !
" # "
" " # "
#
" " "# " #
" " "
$ $
= $ = ( + $ (
% ( + $ (&
%
% % % % %= = $ = $ = $ + = $
% % % % %
& '
Concept of Buoyancy Frequency N
Homework ProblemCalculate the and draw the density profile , ρ(z) and buoyancy profile N(z) for the temperature profile shown. Assume a constant salinity of S = 35.5 psu. Repeat your calculation for S = 36 psu. Comment on how the N profile changes when the salinity changed.
100m
1km
4km
20o
T C=
8o
T C=
2o
T C=
z =0m
Vorticity
Concept of Conservation of angular momentum
Definition of Vorticityω= 2 x angular velocityω units of 1/sec
10
Vorticity velocity angular velocity =
2 radius
v
2 r
!"
=
= =
Component form
v
v
x
y
z
w
y z
u w
z x
u
x y
!
!
!
" "= #" "
" "= #" "
" "= #" "
u! ="#! !
But vorticity is a vector! (why?)
r
vα
Most important for large scale flow >~10km)
Counter clockwise ω > 0Clockwise ω < 0
2
4
( )
but ( )
( )
I Amr
m V r H
I A Hr
! !
" " #
! #"
2
=
= =
$ =
1!
1 1 1 1 1 2
1 2
1 2
Using &
I I V V
H H
! !
! !
= =
" =
Conservation of Angular MomentumFollowing a Fluid Parcel (Case of f =0)
2
2
vorticity
Moment of Inertia
V Volume H r
I Amr
!
"= =
=
1H
1r
1 1 2 2I I! !=
2H
2!
2r
Conservation of Potential Vorticity 0f !
1 1 2 2
1 2
2
{ } 0
{( ) } 0
f f
H H
d f
dt H
df N
dt
! !
!
!
+ +=
+=
+ =
= relative vortivity
2 sin( planetary vortivity
total vortivity
potential vorticity
f
f
f
H
!
"
!
!
= # ) =
+ =
+$ = =
Terminlogy
ω2>ω1
ω1
Homework: von Arx (1962) has suggested that to understandconservation of potential vorticity we consider what happens to abarrel of water as it moves around the earth. Take the shaded circles asthe initial rest position ( no relative vorticity , ω = 0 ) of a barrel ofwater. Assume that the shape and volume of the barrel is constant, i.e.H, r constant. Using the concept of conservation of potential vorticityexplain what happens to the relative vorticity and its direction ofrotation (clockwise, counterclockwise) as it moves along paths #1,2,3,4,5,6,7.
#1#2
#3
#4
#5
#7#6 H
!
r
relative vorticity of the flow
relative vorticity from input from wind stress
relative vorticity of WBC b
!
"
"
"
=
=
=
Why are there Western boundary Currents (WBC)?
Answer: Conservation of Potential Vorticity (PV)
ω
ω
ωb
ωτ
ωω
ωτ
No WBC Unbalanced PV in western side
WBC Balanced PV both east and west
ωτ
Homework: Why does Barotropic flow over asub-sea ridge turned equatorward to conservepotential vorticity.
From Dietrich, et al. (1980).
Example of PV Conservation: Barotropic Flow over a Sub-Sea Ridge
