Lecture Guidelines for GEOF110

Chapter 5 ( 2 hours) Chapter 6 (2 Hours)

Ilker Fer

Guiding for blackboard presentation.Following Pond & Pickard, Introductory Dynamical Oceanography

GEOF110 Guidelines / 3 2

Static Stability

Atmosphere Ocean

Sketch: Incompressible fluid. Parcel 0 is in hydrostatic equilibrium.

Atmosphere: 0

0

0

stablez

neutralz

unstablez

0

0

0

stablez

neutralz

unstablez

Ocean:

“Parcel” or “particle” : we mean a (infinitesimal) fluid sample of uniform T and composition in a continuous environment

GEOF110 Guidelines / 3 3

, , ,S T p

, , ,S T T p p 2 2 2, , ,S T p p

, , ,S T pLevel 1

z

Level 2z+z

PARCEL(P)

ENVIRONMENT(W)

Hydrostatic:

: Adiabatic temperature gradient

ad

ad

dTT p

dp

p g z

dTT g z z

dp

Parcel at level 1 is displaced to level 2 with environment in situ properties given on the figure.

2 2

2 2

Vertical acceleration released is:

z

F V g V g Vg

Vg gFa

m V

At level 2, the restoring force on the parcel with volume V is:

F= buoyant upthrust – weight

Archimedes: Buoyant upthrust = weight of displaced volume (=g2V)

GEOF110 Guidelines / 3 4

2 ( )

( )

W

P

z z z zz

z z z zz

2

1

11

W Pz

P

g z zg z z

a

zz

(salinity constant)

Pressure difference for the particle and the environment is equal

W W

P P

P W

S T pz z

z S z T z p z

pz z

z T p z

pp z p

zp

z

P W

p

z

z

W P

W W

g S Tz

S

g S T p pa z

S z T z p z T p z

z T z

GEOF110 Guidelines / 3 5

Stability of water column for z unit length

[E]=1/

1

mz

S T

S z T

aE

g

z

E > 0 : STABLE parcel displaced vertically will tend to return to its position. Due to its inertia, overshoot equilibrium position oscillate (with frequency N)E = 0 : NEUTRAL vertically displaced parcel will remain thereE < 0 : UNSTABLE overturning

2

Buoyancy Frequency

[N]=radians/sN gE

GEOF110 Guidelines / 3 6

In the upper 1 km of ocean E range between 100-1000x10-8 1/m, largest values at the pycnocline, in the upper a few 100 m.

In deep ocean, deep trenches, E is close to 1x10-8 1/m.

0

0

1Deep Ocean

S TE

S z T z

T

z

i.e., in situ T increases about (0.1 – 0.2) K / km

2

1Exact relation using - density: E=

where ( , , ) 1500 /

is the speed of sound in seawater.

gin situ

z C

C f S T p m s

This is OK using eq. of state and an accurate relation for C.

2

1 1If =0 ; neutral ;

W P W ad ad

p gE g

z z z p z p C

Use of in-situ density would look very stably stratified in neutral stability:

compensates for compressibility

These are comparable

GEOF110 Guidelines / 3 7

, , , ,

0

, ,

- density: 1000

1

1

t t p s p s t p

s p t pt t

In situ

S TE

S z T z

S T

S S z T T z

Note: ( , )t

t t t

f T S

S T

z S z T z

, ,

, ,

, , ,

Re-arrange E:

1

1

1

s p t pt t t

t p s pt t

t p s p t pt t

S S T TE

S z S z T z T T z

T S

z T T z S z

T S

z T z S z T T

/

1

T

t

z

This is not very convenient, nor practical (e.g. /S taken while holding in situ values of T and P).

GEOF110 Guidelines / 3 8

, , , ,

0

, ,

Using potential temperature:

- density: 1000

1 1

1

p s p s p

s p p

In situ

S T SE

S z T z S z z

S

S S z z

, ,1 1 p s p S

z z S z z

NOTE: In a stable environment, an inviscid fluid parcel displaced a small distance vertically will go through simple harmonic oscillations described by

22

20

dN

dt

22

1 = g [N]=radians/

Buo

s

Use gE / 2 for cycles/s

yancy Frequenc

= z

y

[ H ]

t

gN gE

z C

g g

z z

GEOF110 Guidelines / 3 9

Double Diffusion

Double-diffusion occurs when the density variations are caused by two different components with different rates of diffusion.

In seawater heat and salt have different molecular rates of diffusion: KH / KS 100

zT S T S

Salt FingerDouble-Diffusive Convection

Recall- diffusion and mixing is from high concentration of stuff to low concentration, i.e. downgradient. Double-Diffusion gives UPGRADIENT density flux.

GEOF110 Guidelines / 3 10

Equation of Motion (Ch. 6)

Newton's 2nd law:

More convenient: /

F ma

a F m

Acceleration due to

resultant force acting per unit mass

Forces: PressureCoriolisGravity (gravitational acceleration, gf)Other (e.g., tidal, friction)

Will derive the terms, and then give the complete equation

GEOF110 Guidelines / 3 11

Pressure force on a surface element δA with outward normal vector n:

i.e., directed towards the surface element.

Derivation of the Pressure Term (Pressure Gradient Force)

The net force δFp in the x-direction:

pF p An

PxF p y zi p p y zi

p y zi

p px y zi Vi

x x

Using δm = δV = 1/ δV 1

1Pressure gradient force per unit mass

pF p m

p

p

p p pF i j k V

x y z

p V

Using all 3-D:

GEOF110 Guidelines / 3 12

Source: J.H.E.Weber

GEOF110 Guidelines / 3 13

Source: J.H.E.Weber

GEOF110 Guidelines / 3 14

GEOF110 Guidelines / 3 15

2 fe

dvp v r g F

dt

ff

dvp g F

dt

Eq. of motion relative to fixed axes:

Eq. of motion relative to rotating Earth:

For our use: Absolute frames is fixed subscript f Relative frames is Earth subscript e No translation of Earth ao = 0 velocity relative to Earth v

2fe

dva v r

dt

= 2 radians per sidereal day = 7.292x10-5 radians/s

Conversion from fixed to Earth frames:

r = Distance to the centre of Earth

GEOF110 Guidelines / 3 16

Gravitation and Gravity

1 22g

mmF G

r

m1 m2

r

, directed along the line connecting masses. G : Gravitational constant

Gravitation is the attractive force between two masses:

Equator

r

fg

2

g Ef

F Gmg

m r

In our case, mass of Earth, mE. Gravitational force on body m, per unit mass:

r

Centripetal acceleration :to move a body at distance r from the center of Earth to circulate about Earth’s axis with .

Acceleration due to gravity: fg g r

fg

Note, g does not point to Earth’s center of mass due to centripetal acc. Equatorial bulge. Earth’s surface not spherical

g=f(latitude)Max at poles

Min. near eq.

Variation about 0.5% Assume g = 9.8 m/s2

GEOF110 Guidelines / 3 17

Equation of motion

1( ) 2 sin 2 cos +

1( ) 2 sin +

1( ) + 2 cos +

x

y

z

du px v w F

dt x

dv py u F

dt y

dw pz u g F

dt z

12

dv vv v p v g F

dt t

Pressure GravityCoriolis Other forces/m

2 fe

dvp v r g F

dt

Coriolis parameter : 2 sinf 2 cos is very small

2 cos is small (for dyn. oceanogr.)

w

u

1vv v p fk v gk F

t

For convenience, move the coordinate to Earth’s surface.

In Cartesian:

cos sinj k

GEOF110 Guidelines / 3 18

Coordinate Systems

Cartesian Coordinate System: The standard convention in geophysical fluid mechanics is x is to the east, y is to the north, and z is up.

f-Plane is a Cartesian coordinate system in which the Coriolis parameter is assumed constant. It is useful for describing flow in regions small compared with the radius of the Earth (up to 100 km).

β-plane is a Cartesian coordinate system in which the Coriolis force is assumed to vary linearly with latitude. It is useful for describing flow over areas as large as ocean basins.

Spherical coordinates are used to describe flows that extend over large distances and in numerical calculations of basin and global scale flows.

0 ; /f f y f y