Basic dynamics The equations of motion and continuity Scaling Hydrostatic relation Boussinesq approximation Geostrophic balance in ocean’s interior
Dec 29, 2015
Basic dynamics
The equations of motion and continuity Scaling
Hydrostatic relation
Boussinesq approximation
Geostrophic balance in ocean’s interior
Newton’s second law in a rotating frame.(Navier-Stokes equation)
The Equation of Motion
FgVpdtVd
rrrrr++×Ω−∇−= 21
ρ
dtVdr
: Acceleration relative to axis fixed to the earth.
p∇−ρ1 : Pressure gradient force.
Vrr
×Ω−2 : Coriolis force, where sraddayrad 510292.724.365
112 −⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ×=+=Ω π
⎟⎟⎠
⎞⎜⎜⎝
⎛ ×Ω×Ω−= Rggrrrrr
0: Effective (apparent) gravity.
VFrr
2∇≈ν : Friction. molecular kinematic viscosity.sm2610−≅ν
g0=9.80m/s2
1sidereal day =86164s
1solar day = 86400s
Gravity: Equal Potential Surfaces• g changes about 5%
9.78m/s2 at the equator (centrifugal acceleration 0.034m/s2, radius 22 km longer)
9.83m/s2 at the poles) • equal potential surface
normal to the gravitational vectorconstant potential energythe largest departure of the mean sea surface from the “level” surface is
about 2m (slope 10-5) • The mean ocean surface is not flat and smooth
earth is not homogeneous
In Cartesian Coordinates:
xFwvxp
dtdu +Ω−Ω+
∂∂−= ϕϕρ cos2sin21
yFuyp
dtdv +Ω−
∂∂−= ϕρ sin21
zFguzp
dtdw +−Ω+
∂∂−= ϕρ cos21
zuw
yuv
xuu
tu
dtdu
∂∂+
∂∂+
∂∂+
∂∂=
where
Accounting for the turbulence and averaging within T: ∫+
−
=2
2
)(1
),(
Tt
Tt
dttuT
Ttu
( ) ( )( ) ( ) vuvuvuvuvuvuvvuuuv ′′+=′′+′+′+=′+′+=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂∂+
∂∂+
∂∂++
∂∂−=
∂∂+
∂∂+
∂∂+
∂∂
2
2
2
2
2
21
z
u
y
u
x
ufvxp
zuw
yuv
xuu
tu νρ
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂′′∂+
∂′′∂+
∂′′∂−=
∂′∂′−
∂′∂′−
∂′∂′−=
x
wu
x
vu
x
uu
x
uvx
uvx
uuxF
0=∂′∂+
∂′∂+
∂′∂
zw
yv
xu
Given the zonal momentum equation
If we assume the turbulent perturbation of density is small
ρρ ≅i.e.,
The mean zonal momentum equation is
Where Fx is the turbulent (eddy) dissipation
If the turbulent flow is incompressible, i.e.,
Eddy Dissipation
xuAuu xxx ∂∂=′′−=τ
yuAvu yxy ∂∂=′′−=τ
zuAwu zxz ∂∂=′′−=τ
Ax=Ay~102-105 m2/sAz ~10-4-10-2 m2/s
>> sm2610−≅ν
Reynolds stress tensor and eddy viscosity:
Where the turbulent viscosity coefficients are anisotropic.
,
2
2
2
2
2
2
z
uAy
uAx
uAzyx
Fzyx
xzxyxxx
∂∂+
∂∂+
∂∂=
∂∂+
∂∂+
∂∂= τττ
Then
yuAvu yxy ∂∂=′′−=τ
xvAuv xyx ∂
∂=′′−=τyxxy ττ ≠
(incompressible)
Reynolds stress has no symmetry:
A more general definition:
xvA
yuA xyxy ∂
∂+∂∂=τ yxxy ττ =
0=∂∂+
∂∂+
∂∂
zw
yv
xu
2
2
2
2
2
2
z
uAy
uAx
uAzyx
Fzyx
xzxyxxx
∂∂+
∂∂+
∂∂=
∂∂+
∂∂+
∂∂= τττ
if
Continuity Equation
Mass conservation law 0=⋅∇+ Vdtd r
ρρ
In Cartesian coordinates, we have
0=∂∂+∂
∂+∂∂+∂
∂+∂∂+∂
∂+∂∂
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
zw
yv
xu
zwyvxut ρρρρρ
( ) ( ) ( ) 0=∂
∂+∂
∂+∂
∂+∂∂
zw
yv
xu
tρρρρor
For incompressible fluid, 0=dtdρ
0=∂∂+∂
∂+∂∂=⋅∇ z
wyv
xuV
r
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂∂
∂∂≡∇
yxH ,),( vuVH ≡If we define and
0=∂∂+⋅∇zwVHH
r, the equation becomes
Scaling of the equation of motion• Consider mid-latitude (ϕ≈45o) open ocean
away from strong current and below sea surface. The basic scales and constants:
L=1000 km = 106 mH=103 mU= 0.1 m/sT=106 s (~ 10 days)2Ωsin45o=2Ωcos45o≈2x7.3x10-5x0.71=10-4s-1
g≈10 m/s2
ρ≈103 kg/m3
Ax=Ay=105 m2/sAz=10-1 m2/s
• Derived scale from the continuity equation
W=UH/L=10-4 m/s
0=∂∂+∂
∂+∂∂=⋅∇ z
wyv
xuV
r0~
HW
LU +
Scaling the vertical component of the equation of motion
111011101110105103101110111011101010 −+−+−+−−+Δ−=−+−+−+−H
Pz
2
2
2
2
2
2cos21
zwA
ywA
xwAgu
zp
zww
ywv
xwu
tw
zyx ∂∂+
∂∂+
∂∂+−Ω+
∂∂−=
∂∂+
∂∂+
∂∂+
∂∂ ϕρ
21
25
2543
2101010101010
HW
LW
LWU
HP
HW
LUW
LUW
TW z −−− +++−+Δ=+++
1010 3 =Δ−HPz PaHPz
74 1010 ==Δ
gzp ρ−=∂∂
Hydrostatic Equation
accuracy 1 part in 106
Boussinesq ApproximationConsider a hydrostatic and isentropic fluid
€ ∂p∂z=−ρg€
dpdρ=c2€ ∂ρ∂z=−ρgc2
€ ρz()=ρoexp−gd′ z c20z∫ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥€
HS=c2g~200km>>H~1kmLocal scale height
€ dρdt=1c2dpdt=−ρgc2dzdt=−ρgc2w€
∂u∂x+∂v∂y+∂w∂z=gc2w€
O∂w∂z()Ogwc2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=OWH()OWHS ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=HSH>>1€ ∂u∂x+∂v∂y+∂w∂z=0
The motion has vertical scale small compared with the scale height
Boussinesq approximationDensity variations can be neglected for its effect
on mass but not on weight (or buoyancy).
ρρ ′>>oρρρ ′+= o ( ) pzpp o ′+=Assume that where , we have
gzp
oo ρ−=
∂∂
where gzp ρ′−=∂′∂
2
2
2
2
2
21zuA
yuA
xuAfv
xp
zuw
yuv
xuu
tu
zyxo ∂
∂+∂∂+∂
∂++∂′∂−=∂
∂+∂∂+∂
∂+∂∂
ρ
2
2
2
2
2
21zvA
yvA
xvAfu
yp
zvw
yvv
xvu
tv
zyxo ∂
∂+∂∂+∂
∂+−∂′∂−=∂
∂+∂∂+∂
∂+∂∂
ρ
0=∂∂+∂
∂+∂∂
zw
yv
xu
gzp ρ′−=∂′∂
ϕsin2Ω=f
Then the equations are
where
wϕcos2Ω−
(1)
(2)
(3)
(4)(The termis neglected in (1) for energy consideration.)
Geostrophic balance in ocean’s interior
Scaling of the horizontal components
2
2
2
2
2
2cos21
zuA
yuA
xuAwfv
xp
zuw
yuv
xuu
tu
zyx ∂∂+
∂∂+
∂∂+Ω−+
∂∂−=
∂∂+
∂∂+
∂∂+
∂∂ ϕρ
25
25
25443
222101010101010
L
U
L
U
L
UWUL
PL
UL
UL
UTU h −−−−−− +++−+
Δ−=+++
810810810810510910810810810710 −+−+−+−−−+Δ−=−+−+−+− Ph
3103103103101410310310310210 −+−+−+−−+Δ−=−+−+−+− Ph
Zero order (Geostrophic) balancePressure gradient force = Coriolis force
01 =+∂∂− fvxp
ρ
01 =−∂∂− fuyp
ρ yp
fu
∂∂−= ρ
1
xp
fv
∂∂=ρ
1
410=Δ Ph (accuracy, 1% ~ 1‰)
Re-scaling the vertical momentum equation
ρρρ ′+= o ( ) pzpp o ′+=
Since the density and pressure perturbation is not negligible in the vertical momentum equation, i.e.,
gzp
oo ρ−=
∂∂
gz
pg
z
p
z
p
z
p
z
p
z
p
z
p
z
p
z
p
z
p
z
p
oo
o
o
o
o
o
oo
o
oo
o
o
ρρ
ρ
ρρ
ρρρ
ρ
ρρρ
ρρρ
′−
∂′∂
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂′
−∂′∂
+∂∂
−≈⎟⎠
⎞⎜⎝
⎛∂′∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛ ′−−≈
⎟⎠
⎞⎜⎝
⎛∂′∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛ ′+
−=⎟⎠
⎞⎜⎝
⎛∂′∂
+∂∂
′+−=
∂∂
−
1
11
1
1
111
, , and
The vertical pressure gradient force becomes
Taking into the vertical momentum equation, we have
2
2
2
2
2
2cos21
zwA
ywA
xwAug
zp
zww
ywv
xwu
tw
zyx
o
o ∂∂+∂
∂+∂∂+Ω+′−∂
′∂−=∂∂+∂
∂+∂∂+∂
∂ ϕρρ
ρ
410~ =Δ′Δ PPz h
21
25
25423
2101010101010
HW
LW
LWU
HP
HW
LUW
LUW
TW z −−−− ++++−′Δ−=+++ δρ
H
P
z
p z ′Δ∂′∂~ δρρ ~′If we scale , and assume
1110111011105102102101110111011101010 −+−+−+−+−−−=−+−+−+− δρ
3/1 mkg=δρ
then
gzp ρ′−=∂′∂
and
(accuracy ~ 1‰)
Geopotential Geopotential is defined as the amount of work done to
move a parcel of unit mass through a vertical distance dz
against gravity is
dpgdzd α−==
(unit of : Joules/kg=m2/s2).
( ) ( ) ∫ ∫−=∫ ===−=2
1
2
1
2
1
12
z
z
p
pdpgdz
z
zdzzzz α
The geopotential difference between levels z1 and z2 (with pressure p1 and p2) is
Dynamic height
Given δαα += p,0,35 , we have
where ∫=Δ2
1
,0,35
p
pdppstd α is standard geopotential distance (function of p only)
∫=Δ2
1
p
pdpδ is geopotential anomaly. In general, 310~Δ
Δstd
( ) ( ) Δ−Δ−=∫ ∫−−==−=std
p
p
p
pdpdppppp p
2
1
2
1
,0,3512 δα
Δ is sometime measured by the unit “dynamic meter” (1dyn m = 10 J/kg). which is also called as “dynamic distance” (D)
Note: Though named as a distance, dynamic height (D) is still a measure of energy per unit mass.
∫=−=Δ2
112 10
1 p
pdpDDD δ Units: δ~m3/kg, p~Pa, D~ dyn m
Geopotential and isobaric surfacesGeopotential surface: constant , perpendicular to gravity, also referred to as
“level surface”
Isobaric surface: constant p. The pressure gradient force is perpendicular to the isobaric surface.
In a “stationary” state (u=v=w=0), isobaric surfaces must be level (parallel to geopotential surfaces).
In general, an isobaric surface (dashed line in the figure) is inclined to the level surface (full line).
In a “steady” state ( ),
the vertical balance of forces is
() ()()
()igi
iin
pinp tan
cos
sincos)sin( =∂
∂=∂∂
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛αα
0=∂∂=
∂∂=
∂∂
tw
tv
tu
np∂∂α
ginp =∂∂ )(cosα
The horizontal component of the pressure gradient force is
Geostrophic relationThe horizontal balance of force is
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛=Ω igV tansin21
ϕwhere tan(i) is the slope of the isobaric surface. tan (i) ≈ 10-5 (1m/100km) if V1=1 m/s at 45oN (Gulf Stream).
In principle, V1 can be determined by tan(i). In practice, tan(i) is hard to measure because
(1) p should be determined with the necessary accuracy
(2) the slope of sea surface (of magnitude <10-5) can not be directly measured (probably except for recent altimetry measurements from satellite.) (Sea surface is a isobaric surface but is not usually a level surface.)
Calculating geostrophic velocity using hydrographic data
⎟⎠⎞
⎜⎝⎛=Ω
11tansin2 igVϕ
⎟⎠⎞
⎜⎝⎛=Ω
22tansin2 igVϕ
The difference between the slopes (i1 and i2) at two levels (z1 and z1) can be determined from vertical profiles of density observations.
Level 1:
Level 2:
⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ −=−Ω
2121tantansin2 iigVVϕ
Difference:
⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞
⎜⎝⎛
−−−=
−=
−=
−=−Ω
4231
2121
2121
22
22
11
1121
sin2
zzzzL
g
AABBL
g
CCBBL
gCA
CB
CA
CBgVVϕi.e.,
because A1C1=A2C2=L and B1C1-B2C2=B1B2-C1C2
because C1C2=A1A2
Note that z is negative below sea surface.
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ ∫−∫=−−− dpdp
Lzzzz
L
gp
p
A
p
p
B
2
1
2
1
4231
1 δδ
( ) dpdpzzgp
pA
p
pp
∫∫ +=−2
1
2
1
,0,3542 δα
( ) dpdpzzgp
pB
p
pp
∫∫ +=−2
1
2
1
,0,3531 δα
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ Δ−Δ
Ω=−
ABDD
LVV
ϕsin2
1021
Since
and
,
we have
The geostrophic equation becomes