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Momentum flux across the sea surface
OCG Class 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

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.

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

4.7 Momentum flux across the sea surface
Close to the boundary the stress becomes constant: Detailed measurements of the vertical gradient of the horizontal velocity in this layer allow us to compute the (horizontal) wind stress

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

4.7 Momentum flux across the sea surface
Stress based on empirical observation gives us a functional relation: U10: wind velocity measured at 10 m. CD: drag coefficient: typical value of 1 to 2 x 10-3 for the atmosphere.

4.7 Momentum flux across the sea surface
CD depends on Stability (N2) of the atmosphere: CD decreases if the water is colder than the air CD increases if the water is warmer than the air Air speed: CD increases with air speed Size of the surface waves (i.e., how long the wind has been blowing): CD increases with increasing wave height A typical value for the wind stress: *

4.7 Momentum flux across the sea surface
Wind stress may be measured from satellite-borne 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.

Chapter 5 Dynamics of Ocean Currents 5.1 Concepts of Fluid Mechanics
5.2 Forces on a Fluid Element (Parcel)

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).

5.1 Concepts of Fluid Mechanics
5.1.2 Newton’s Second Law of Motion Newton’s second law is: Acceleration is the rate of change of velocity: Newton’s Second Law can be rewritten as: The velocity of an object remains constant (at zero or any other value), unless an unbalanced force acts on the object.

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 With density defined by ρ=m/V, Newton’s Second Law becomes: 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.

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.

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 The velocity at (xo ,to) is: And the velocity at (xo+ ∆x , to+ ∆t) is: Using the Taylor series expansion:

5.1 Concepts of Fluid Mechanics
5.1.3 Eulerian and Lagrangian Notations The Substantial Derivative ? Change in velocity per unit time between (xo , to) and (xo+ ∆x , to+ ∆t): Now, as ∆x  0 and ∆t  0, So, as x  0 and t  0, *

5.1 Concepts of Fluid Mechanics
5.1.3 Eulerian and Lagrangian Notations The Substantial Derivative ? 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: And our equation reduces to: This is called the substantial derivative and is written: or We often write D/Dt interchangeably with d/dt.

5.1 Concepts of Fluid Mechanics
5.1.3 Eulerian and Lagrangian Notations The Substantial Derivative is This is the acceleration of a parcel of fluid: “Lagrangian acceleration”. More generally The Lagrangian term is the rate of change experienced by a given tagged water parcel. The Eulerian term is the local rate of change at a fixed point. is what you get with a current meter at a fixed point in space The advective term is which converts between the Eulerian and Lagrangian rates of changes.

5.1 Concepts of Fluid Mechanics
5.1.3 Eulerian and Lagrangian Notations Steady Flow in a Pipe Consider the steady ( 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. So,

5.1 Concepts of Fluid Mechanics
5.1.3 Eulerian and Lagrangian Notations The substantial derivative extended to three dimensions In vector notation, these three component equations are represented as 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.

5.2 Forces on a Fluid Element (Parcel)
Next time Forces on a fluid element (parcel) And more.

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