Law of Motion (Physics):

� if measured in an appropriate frame of reference (one that does NOT rotate)

Sum of all forces = time-rate of change of (mass*velocity)

F = m * a

Need to address both, but first,what forces act on the fluid?

� applies to particles (Lagrangian view)

Forces acting on geophysical fluid• Gravity• Pressure gradient force• Friction (dissipation) (viscous force)

Towards center of Earth

Forces acting on geophysical fluid• Gravity• Pressure gradient force• Friction (dissipation) (viscous force)

Example of pressure gradients

daily surface

pressure map (in the atmosphere, we can simply measure the pressure at the surface)

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Pressure gradient force in ocean?

• Small deviations of sea surface drive all flows - usually much less than 1 meter height. Pressure gradient is very difficult to measure directly.

1 meter height

Equivalent to pressure of 1 dbar, since water density is ~ 1000 kg/m3

100 kilometers width

Gulf Stream example

Pressure gradient is directed from low to high. Calculate size.

Pressure gradient force is directed from high to low (water pushed towards lower pressure).

(rest of water column: 5 km deep)1

met

er h

eigh

t

HIGHLOW

Compute acceleration due to PGF

• Take the example of the Gulf Stream and compute the velocity after 1 year of acceleration.

• You’ll find it’s ridiculously large (compared with the observed 1 m/sec). Why can such a large pressure gradient be maintained without large velocities? (Earth’s rotation - Coriolis to be discussed later)

Friction, Viscosity, and Turbulence

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Turbulent: acceleration >> friction

An unsolved problem of physics

On his death-bed Physics Nobel Prize winner Werner Heisenberg is reported to have said: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence?I really believe he will have an answer for the first."

http://www.eng.auburn.edu/users/thurobs/Turb.html

Shear stress is the force Fapplied tangentially to a surface A(think winds over the ocean),e.g., =F/A

For water and air it is found experimentally that

the stress is proportional to the strain u/z (change of velocity in the direction of F perpendicular to A)

~ u/z

z

x

u

How does this accelerate the flow?

Acceleration occurs ifthe stress at the upper surface

differs from

the stress at the lower surface

Eddy diffusivity and eddy viscosity

• Molecular viscosity and diffusivity are extremely small

• We know from observations that the ocean behaves as if diffusivity and viscosity are much larger than molecular

• The ocean has lots of turbulent motion (like any fluid)

• Turbulence acts on larger scales of motion like a viscosity - think of each random eddy or packet of waves acting like a randomly moving molecule carrying its property/mean velocity/information

Eddy diffusivity and viscosity

Gulf Stream (top) and Kuroshio (bottom): meanders and makes rings (closed eddies) that transport properties to a new location

Eddy diffusivity and viscosityExample of surface drifter tracks: dominated to the eye by variability (they can be averaged to make a very useful mean circulation)

Law of Motion (Physics):

� if measured in an appropriate frame of reference (one that does NOT rotate)

Sum of all forces = time-rate of change of (mass*velocity)

F = m * a

� applies to particles (Lagrangian view)

What about flow fields where velocity does not depend on tracking particles all the time?

Lagrangian View:Observations

Salinity

Fresh riverine water turns right at the coast. From Muenchow (1992)

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Model Predictions of

Winds (pink)Ocean Currents (black)Surface Salinity (color)

From John Wilkin, Rutgers U.

Eulerian View

Fresh riverine waters turn right at the coast.

Advection of velocity --> field accelerations

Particle acceleration Du/DtEquals

Local plus field acceleration:

zero local acceleration,butparticles change velocity(passing through above field)

Du/Dt = u/t + (u u/x + v u/y + w u/z)

local + field acceleration

strong flow

to gradient

--> large

weak flow

to gradient

--> small

strong flow

|| to gradient

--> zero

grad

ient

High temperature or velocity

Low temperature or velocity

Advection (Eulerian perspective)

• Move “stuff” - temperature, salinity, oxygen, velocity, etc.

• By moving stuff, we might change the value of the stuff at the next location. We only change the value if there is a gradient of the stuff (difference from one point to the next)

• Proportional to velocity

• Proportional to gradient in the same direction as the velocity, e.g., v T/y is the advection of temperature in the y-direction

Advection:

Law of Motion (Physics):

� if measured in an appropriate frame of reference (one that does NOT rotate)

Sum of all forces = time-rate of change of (mass*velocity)

F = m * a

Lets walk the plank …

… Coriolis and Centrifugal accelerations

… ficticious forces in a rotating reference (earth)

Movie time:

1. Merry-go-round (fast and slow)acceleration = coriolis+centrifugal

2. Idealized motion of merry-go-roundacceleration = coriolis+centrifugal

3. Coriolis simulation without frictionacceleration = coriolis

4. Coriolis simulation with frictionacceleration=coriolis+friction

Rotating coordinates• The Earth is rotating. We measure things relative to this “rotating reference frame”.

• Quantity that tells how fast something is rotating:

Angular speed or angular velocity = angle/second

360° is the whole circle, but express angle in radians (2 radians = 360°)

For Earth: 2 / 1 day = 2 / 86,400 sec = 0.707 x 10-4 /sec

Also can show = v/R where v is the measured velocity and R is the radius to the axis of rotation (therefore v = R)

R

Rotating coordinates

• Vector that expresses direction of rotation and how fast it is rotating:

vector pointing in direction of thumb using right-hand rule, curling fingers in direction of rotation

R

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Centripetal and Centrifugal forces(now looking straight down on the rotating plane)

Centripetal force is the actual force that keeps the ball “tethered” (here it is the string, but it can be gravitational force)

Centrifugal force is the pseudo-force (apparent force) that one feels due to lack of awareness that the coordinate system is rotating or curving

centrifugal acceleration = 2R

Effect of centrifugal force on earth and ocean

Radius:

Equatorial 6,378.135 km

Polar 6,356.750 km

Mean 6,372.795 km

(From wikipedia entry on Earth)

The ocean is not 20 km deeper at the equator, rather the earth itself is deformed. We bury the centrifugal force term in the gravity term (which we call “reduced gravity”), and ignore it henceforth.

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Coriolis effect

Inertial motion:motion in a straight line relative to the fixed stars

Coriolis effect: apparent deflection of that inertially moving body just due to the rotation of you, the observer.

Coriolis effect deflects bodies (water parcels, air parcels) to the right in the northern hemisphere and to the left in the southern

hemisphere

Coriolis force

Additional terms in momentum equations, at latitude

x-momentum equation: -sinv = -f v

y-momentum equation: sinu = f u

f is the “Coriolis parameter”. It depends on latitude.

Movie time:

1. Merry-go-round (fast and slow)acceleration = coriolis+centrifugal

2. Idealized motion of merry-go-roundacceleration = coriolis+centrifugal

3. Coriolis simulation without frictionacceleration = coriolis

4. Coriolis simulation with frictionacceleration=coriolis+friction

Complete force balance with rotation

Three equations:

Horizontal (x) (west-east)

acceleration + advection + Coriolis =

pressure gradient force + viscous term

Horizontal (y) (south-north)

acceleration + advection + Coriolis =

pressure gradient force + viscous term

Vertical (z) (down-up)acceleration +advection (+ neglected very small Coriolis) =

pressure gradient force + effective gravity (including centrifugal force) +

viscous term

Momentum Equation in Cartesian co-ordinates (x,y,z)

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X:

Y:

Z:

Acceleration = p-gradient + friction + gravity

local acceleration

field acceleration (or advection or nonlinear advection)

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Momentum Equations in Spherical co-ordinates

Radial:

Latitude:

Longitude:

Acceleration = pressure gradient + friction + gravity