Transcript rate of change

OCG 501 Class 6
September 28, 2004


Momentum flux across the sea surface
Chapter 5 Dynamics of Ocean Currents

Concepts of Fluid Mechanics




Continuum Hypothesis
Newton’s Second Law
Eulerian and Lagrangian Descriptions of Flow
Forces on a Fluid Element
1
4.7 Momentum flux across the sea surface





Tangential Stress, τ
Tangential component of the force on a
surface
Force per unit area (N m-2=Pascal)
Example 1: the stress your hand exerts on a
table top as you slide your hand across it.
Example 2: Stress exerted by the wind on
the water.
2
4.7 Momentum flux across the sea surface



Momentum is transferred by turbulent (eddy)
motion in the atmospheric boundary layer.
A profile of the wind in the atmospheric BL.
Momentum flux is proportional to the
existence of a shear stress in the medium
u
x  .
z
3
4.7 Momentum flux across the sea surface

Close to the boundary the stress
becomes constant:
u
 constant
z

Detailed measurements of the vertical
gradient of the horizontal velocity in this
layer allow us to compute the
(horizontal) wind stress
u
x  .
z
4
4.7 Momentum flux across the sea surface


Although difficult to measure, the Reynolds
Stress formulation is the most satisfactory
formulation.
The vertical transfer of u-directed momentum
(the stress in the x-direction) is given by
u
 x    a u' w'  .
z
5
4.7 Momentum flux across the sea surface

Stress based on empirical observation
gives us a functional relation:
 
  au' w'iˆ  au' w' ˆj  aCD | U10 | U10



U10: wind velocity measured at 10 m.
CD: drag coefficient:

typical value of 1 to 2 x 10-3 for the
atmosphere.
6
4.7 Momentum flux across the sea surface

CD depends on

Stability (N2) of the atmosphere:



Air speed:


CD increases with air speed
Size of the surface waves (i.e., how long the wind has been
blowing):


CD decreases if the water is colder than the air
CD increases if the water is warmer than the air
CD increases with increasing wave height
A typical value for the wind stress:

*
 
  aCD | U10 | U10

*
  1kgm3 103 10ms 1 


2
 0.1Nm 2 .
7
4.7 Momentum flux across the sea surface


Wind stress may be measured from satelliteborne instruments.
This type of drag law can be used in any
boundary layer.


For example, the stress exerted by the water
flowing above a solid bed (e.g., a river or bottom
current in the ocean).
The appropriate value for CD must be determined
for each situation, but it is still typically of order
10-3 in the ocean.
8
Chapter 5

Dynamics of Ocean Currents


5.1 Concepts of Fluid Mechanics
5.2 Forces on a Fluid Element (Parcel)
9
5.1 Concepts of Fluid
Mechanics

5.1 Continuum Hypothesis

A fluid parcel is VERY large compared to the
molecules or molecular spacing.




There are a VERY large number of molecules in a fluid
parcel.
Mean free path of molecules is microscopic
Fluid properties change continuously as the size of
a fluid parcel changes.
We ignore the discrete molecular structure and
focus on a continuous distribution, a continuum
(a macroscopic approach).
10
5.1 Concepts of Fluid
Mechanics

5.1.2 Newton’s Second Law of Motion

Newton’s second law is:

F
 
 m
Force
mass

a
(5.1)
acceleration

Acceleration is the rate of change of velocity:

Newton’s Second Law can be rewritten as:



 du d 2 x
a
 2
dt dt


du F

dt
m
(5.2)
The velocity of an object remains constant (at zero or any
other value), unless an unbalanced force acts on the
object.
11
5.1 Concepts of Fluid
Mechanics



Newton’s Second Law applies to an inertial coordinate systems.
Newton’s Second Law is at the heart of the motion of fluid in
the ocean.
In Fluid Mechanics, we write Newton’s Second Law in terms of
the mass per unit volume, or density:





Denote the parcel volume by V and its mass by m.
Definethe force per unit volume as
 F
F
(5.3)
V
With density
defined by ρ=m/V, Newton’s Second Law becomes:


du F / V 1 

 F
(5.4)
dt
m /V

This equation means that the velocity of a parcel of water changes
only if a force is applied to it.
Because ρu is momentum per unit volume, Equation (5.4) is called
the momentum equation.
12
5.1 Concepts of Fluid
Mechanics

5.1.3 Eulerian and Lagrangian Notations

The acceleration defined by Newton’s Second Law
is the Lagrangian acceleration



Acceleration of a fluid parcel
This is not the same acceleration you would measure at
a fixed point in the fluid, the Eulerian acceleration.
With Lagrangian dynamics, we



examine the forces on a parcel of water while following the
parcel,
treat each parcel of water like a particle and examine the
forces on it.
With Eulerian dynamics,


most often we see changes in the properties of a fluid
(e.g., its velocity or salinity) at a fixed location.
This is an Eulerian rate of change.
13
5.1 Concepts of Fluid
Mechanics

5.1.3 Eulerian and Lagrangian Notations

The Substantial Derivative ?



Consider a 1-d fluid with a velocity field defined by

u( x , t )
The velocity at (xo ,to) is:

u ( xo , to )
And the velocity at (xo+ ∆x , to+ ∆t) is:

u ( xo  x, to  t )
Using the Taylor series expansion:


u
u
u ( xo  x, to  t )  u ( xo , to ) 
t 
x  ( higher order terms).
t
x

14
5.1 Concepts of Fluid
Mechanics

5.1.3 Eulerian and Lagrangian Notations

The Substantial Derivative ?


u
u
u ( xo  x, to  t )  u ( xo , to ) 
t 
x  (higher order terms)
t
x

Change in velocity per unit time between (xo , to) and (xo+ ∆x , to+ ∆t):
u( xo  x, to  t )  u( xo , to ) u t u x


 (higher order terms)
t
t t t t

Now, as ∆x  0 and ∆t  0,
u( xo  x, to  t )  u( xo , to ) du

t
dt


So, as x  0 and t  0,
and
x
dx

t
dt
* du  u  u dx .
dt t x dt
15
5.1 Concepts of Fluid
Mechanics

5.1.3 Eulerian and Lagrangian Notations

The Substantial Derivative ?
du u u dx


dt t x dt


This holds for an infinitesimal change in x and t.
Consider a special change in x and t, one following a fluid parcel.

In this case:
dx
u
dt


And our equation reduces to:
du u
u

u
dt t
x
This is called the substantial derivative and is written:
Du du

Dt dt
Du u
u

u
Dt t
x

or

We often write D/Dt interchangeably with d/dt.
16
5.1 Concepts of Fluid
Mechanics

5.1.3 Eulerian and Lagrangian Notations

The Substantial Derivative is
Du u
u

u
Dt t
x


This is the acceleration of a parcel of fluid: “Lagrangian
acceleration”.
More generally
D 

 u
Dt t
x


The Lagrangian term dt is the rate of change experienced by a given
tagged water parcel.
The Eulerian term t is the local rate of change at a fixed point.
du


(5.5)
u
t
is what you get with a current meter at a fixed point in space
The advective term is u ux which converts between the Eulerian and
Lagrangian rates of changes.
17
5.1 Concepts of Fluid
Mechanics

5.1.3 Eulerian and Lagrangian Notations

Steady Flow in a Pipe






So,

Consider the steady ( t  0 everywhere) flow in an incompressible fluid in a
narrowing pipe.
A water parcel enters the pipe with velocity u1.
And leaves it with velocity u2.
u2 > u1, since the pipe narrows.
The parcel clearly accelerates as it moves into the narrower region,
but
the local acceleration is zero.
Du
u
u .
Dt
x
18
5.1 Concepts of Fluid
Mechanics

5.1.3 Eulerian and Lagrangian Notations

The substantial derivative extended to three dimensions
Du u
u
u
u

u v w
Dt t
x
y
z
Dv u
v
v
v

u v w
Dt t
x
y
z
Dw u
w
w
w

u
v
w
Dt t
x
y
z

In vector notation, these three component equations are
represented as




Du u

 ( u   )u .
Dt t


Eulerian description of flows when using current meters.
Lagrangian descriptions of flows when dealing with drift cards or
bottles, bottom drifters, drogues, free drifting buoys, etc.
19
5.2 Forces on a Fluid Element
(Parcel)

Next time


Forces on a fluid element (parcel)
And more.
20