Unit 5: Equations of Motion

1. Unit 5:Equations of Motion Introductory Physical Oceanography (MAR 555) - Fall 2009 Prof. G. Cowles

2. Key Concepts: • Newton’s Second Law • Free Body Diagrams • Coordinate Systems: Inertial and Non-Inertial • Review: Flux Divergence – Substantial Derivative • Forces on a Fluid • Equations of Motion • Scaling and Key Non-dimensional Parameters Lot’s of Equations - Physical Basis of Each Term - Why it takes on a particular form

3. Newton’s Second Law: Sum of forces equals the rate of change of momentum For solid objects of a fixed mass Forces are Vectors, Acceleration is a Vector, Mass is a Scalar!! Sum of Forces in direction x = mass times acceleration in direction x

4. Free Body Diagram (1D): Forces Acting on the Block: 1.) Force of Gravity = -mg 2.) Force of Floor (normal force) = N If the block is static (rate of change of momentum=0) then the sum of forces = 0 and thus we have that the normal force = the weight of the block N=mg This kind of analysis is known as “statics” and is fundamental to structural mechanics

5. Static Free Body Diagram (Vector): Block is Static: sum of forces in each direction should equal 0 Solve for Normal (F3) and tangential (F2) forces Note: could have worked in a more natural coordinate system (horizontal-vertical) but would have required more work for same solution.

6. Dynamic Free Body Diagram: Spring-Mass N y Fs = -kx x mg Sum Forces in Y Sum Forces in X Governing Equation General Solution Natural Frequency

7. Free Body Diagram: Rotating Mass V Force Balance V T=? Ball is moving at constant speed, how can it be accelerating?

8. Free Body Diagram: Rotating Mass V Centrifugal Force (felt by your hand) Centripetal Force (felt by the ball) V Velocity is a vector. In this case the magnitude of velocity is constant but the direction is constant changing. This is an acceleration.

9. Reference Frames and 2nd Law Pt 1: Linear Skating Rink: Summing Forces in the Ice Plane Bird’s Eye View, Fixed to Rink Puck at 1 m/s relative to rink Newton’s Law Works Zamboni View Puck at 1 m/s relative to rink Zamboni at 0.5 m/s Relative to Rink Yes, Newton’s Law Works Zamboni View Puck at 0 m/s relative to rink Zamboni accelerating at 0.1g Relative to Rink No, relative to zamboni, puck is Accelerating at -0.1g What Happened? In the last example we tried to balance forces from the Point of a view of a Non-Inertial (accelerating) reference frame One Solution: Add a new term

10. Non-Inertial Ref Frame 2: Rotating Plane Spinning Record, time t=0, ball set in motion along y-axis y y Inertial coordinate system (x,y) x y’ x x’ y’ x’ INERTIAL NON-INERTIAL From DJ’s Point of View, marble Rolls in straight line along y. This Is reasonable as no forces are acting In the x-y plane. Point of View of gerbil sitting on the record, marble is curving to the right. youtube Video: http://www.youtube.com/watch?v=49JwbrXcPjc (Ignore discussion of centrifugal force)

11. Non-Inertial Ref Frame 3: Rotating Spheroid Instead of a marble we will use a pendulum oscillating from a frictionless bearing Pendulum over Equator, No deflection Pendulum above Pole: Same as record player! Perceived deflection depends on latitude!! Poles: Same as our rotating plane example Equator: No deflection Mid-Latitude: Somewhere in between Wiki images

12. Foucault Pendulum You can actual build such a pendulum. The key is to minimize damping by using a large mass (~ bowling ball) and a long stiff wire. (Note: AGU S2010-Portland) Interesting simulations: http://en.wikipedia.org/wiki/File:Pendule_de_Foucault.jpg http://www.youtube.com/watch?v=wlhHWYKswik http://www.youtube.com/watch?v=FcNmNafQL10&feature=related Wiki images

13. What is the Point of this Discussion? • We will be deriving our equations of motion in an Earth-Attached coordinate system • This system is NON-INERTIAL • This messes with Newton’s 2nd Law: Objects appear to accelerate (remember this can mean a direction change with no speed change) without any apparent forces • Remember back to our linear example (zamboni). We could add a term to Newton’s 2nd Law to account for the acceleration of our reference frame relative to an inertial reference frame. This additional term accounted for the perceived acceleration of the puck from the point of view of the zamboni driver. • We will do the same to our equations of motion for the ocean. We will have the usual sum of forces = change in momentum but we will add some extra terms to account for the fact that we are deriving these equations in a non-inertial reference frame. • Taken together, the extra terms are called the Coriolis force (which we will write as an acceleration). • In some scenarios, these extra terms are among the biggest terms in the equation. They will lead to some very counterintuitive behavior, particular if you are already familiar with fluid mechanics from an engineering point of view. Welcome to geophysical fluid dynamics.

14. Review: Conservation of Mass Sum of the fluxes into a control volume = 0

15. Review: Conservation of Scalar (e.g. salt) Sum of the fluxes of Salt into a control volume = Rate of Change of Salt in the C.V.

16. Review: Total Derivative

17. New: Conservation of Momentum For a metal block or fixed chunk of the ocean For a fixed control volume For the ocean How? Conservation of Mass Expand and invoke This expression represents three equations for (u,v,w) in terms of the forces in units of Force/(unit mass), e.g. units of acceleration. Let’s look at what these forces are that can lead to changes in the momentum of the fluid

18. Forces on the Fluid Two types of Forces: 1.) Body forces, act on the mass of water in the control volume i.) Gravity ii.) Coriolis (an apparent force) iii.) other: Magnetohydrodynamic (think Red October) 2.) Surface forces, act on the surface of the control volume iv.) Pressure Stresses v.) Viscous Stresses (a.k.a. friction)

19. Gravity Gravitational force on the block of water Gravitational force/unit mass on the water But Forces are VECTORS!! Gravity: Acts in negative z direction** Magnitude: Direction: **strictly speaking, this is not exactly true due local anomalies Recall: geoid

20. Pressure Pressure always acts normal to a surface. For our cube, only the pressure on the left and right faces can contribute to a change in x-momentum Positive pressure (p > 0) implies a negative force when dotted with outward unit normal vector Force on Left Face Right Face Sum Force in x We want Force/unit mass Shrink Volume to a Point

21. Pressure We can do the same in the other two directions. Remember, the force is normal to the surface, so the y-direction forces are only on the y-normal faces, z-direction on the z-normal faces. In Vector Form, we can write: Note the Minus Sign: If pressure is the only force, the fluid will want to move in the direction of decreasing pressure.

22. Coriolis Recall: Zamboni Case, we were able to apply Newton’s 2nd Law by adding an apparent acceleration related to the kinematics of the non-inertial reference frame We will do the same here. We wish to derive our governing equations in a reference frame attached to the spinning Earth. This requires an additional apparent force: Real Forces Apparent Note: the term is proportional to the angular velocity of the Earth, If we want this term to be zero when our coordinate system is inertial, then this rotation rate must be the inertial (relative to stars, a.k.a. sidereal) rotation rate. Note: the term is proportional to velocity field See Supplementary Materials on Website for a Coriolis Treatise

23. Coriolis by Direction Let’s break down the Coriolis by direction We can use scaling arguments to show that the z component is tiny compared to the other terms in the z equation (for example gravity) and we will ignore it. However, based on energy arguments, this requires us to eliminate the second Coriolis term in the x-equation. This allows us to define a new parameter for mid-latitudes

24. Coriolis Acceleration: Approximations “Tangent Planes” • full latitude dependence: • f-plane approximation: • b-plane approximation: Sundermeyer Lecture Notes

25. Coriolis Implications and Myths VERY IMPORTANT Implications: Northern Hemisphere – Currents deflect to Right • Northward Current is deflected East • Southward Current is deflected West • Eastward Current is deflected South • Westward Current is deflected North Coriolis strength • Deflection acts at right angles of instantaneous direction • Strength of Coriolis is proportional to the instantaneous velocity magnitude • Strength of Coriolis dependent on latitude (0 at Equator) • Typical mid-latitude value for Coriolis acceleration : 1e-5 m/s2 • This should seem quite small and we will see later the circumstance in which it is important. Myth Coriolis influences the direction of flow in a toilet or sink.

26. Friction • The friction force is the most complex term in the equation because the stress on a fluid (think of our box) is a tensor (a quantity dependent on more than one direction – think vector on steroids). • Fundamentally it should be approached from a mechanical point of view, linking the stress on a the fluid with the rate of strain of the fluid • We will visit this concept when we discuss boundary layers and mixing later in the semester. The Friction Term from a Diffusion point of view Q Flow Qleft Rectangular Opening, Area A m2 Left Room Right Room Qright This is a FLUX

27. Friction, cont’d U-momentum can change in the box due to diffusion of u z – use Vertical Eddy Viscosity (x,y) direction, use Horizontal Eddy Viscosity

28. Incompressible Navier-Stokes Equations (a.k.a. Non-Hydrostatic Equations) Difficult to solve: No prognostic equation for pressure Fundamentally 4 equations, but we have 7 unknowns (p,u,v,w,ρ,Ah,km)

29. Incompressible Navier-Stokes Equations (a.k.a. Non-Hydrostatic Equations) Fundamentally 4 equations, but we have 7 unknowns (p,u,v,w,ρ,Ah,km) For density, we can use an equation of state. Now we will need prognostic equations for T and S And finally a means to calculate the horizontal and vertical eddy viscosity and diffusivity using the flow variables. Typically this is done using a turbulence model (e.g. Mellor and Yamada)

30. Equation Simplification via Scale Analysis Recipe (recall discussion on calculating distance from UMD to UMB) 1.) Determine order of magnitude for each term based on problem of interest 2.) Toss terms that are negligible in size compared to others Scale Analysis has dynamic implications: examples 1.) If frictional terms are negligible: fluid can be considered invisicid 2.) If Coriolis is negligible: earth’s rotation not important The ratio of two particular parameters form important non-dimensional numbers will give us immediate information of about the flow regime and guide us to the right set of governing equations for the problem.

31. The Continuity Equation Scaling: We have learned that the continuity tells us that the ratio of the vertical velocity to the horizontal velocity goes as the ratio of the depth to a horizontal length scale Plug in some numbers for a particular problem: Ocean Basin These tiny vertical velocities are a consequence of how thin the ocean is (like a sheet of paper)

32. Scaling the w-momentum equation Acc + Adv + Adv + Adv = PG + Grav + Diff + Diff + Diff We can ignore all but the pressure gradient and gravity – AT THESE SCALES • Note: • Advection terms all same order of magnitude • Friction terms all same order of magnitude

33. Hydrostatic Assumption So what? This means we can solve for the pressure at a point (X,Y,Z) anywhere just knowing the vertical density distribution. With our hydrostatic approximation, we have: We now have a diagnostic equation for pressure

34. Hydrostatic Primitive Equations (HPEs) Conservation of Mass Conservation of x-Mom Conservation of y-Mom Hydrostatic (formerly w-momentum) Conservation of Temp Conservation of Salt Equation of State Turbulence Closure 9 Equations for 9 Unknowns: Note Units are Acceleration (L^2/T)

35. Hydrostatic Primitive Equations (HPEs) Coupled Nonlinear Partial Differential Equations Ill-posed (hydrostatic assumption messes with nature) What can I do? Use an Ocean Model Simplify • ROMS • FVCOM • POM • BOM • SUNTANS • UNTRIM • HYCOM Beta Plane Linearize or Drop Advection term Use Simple Geometries: Squares/Circles Solve steady-state problem Assume Horizontally-Infinite Domain

36. Scaling the (u/v)-momentum equations Basin Scale We have no easy basis for scaling the horizontal PG Acc + Adv = PG + Coriolis+ Friction - Pressure Gradient must be Order 1 to Balance Coriolis

37. Summary of Scale Analysis for Ocean Basin At these Basinwide scales, friction, advection (nonlinear) and acceleration are negligible. These simplified equations are accurate to about 1% Rather than rescaling with a new set of parameters each time I start a new problem, is there a faster way to see when I can drop specific terms? Yes

38. Scaling: Flow Classification – Rossby How do my Nonlinear terms compare to my Coriolis? Nonlinear Term This is the Rossby number Coriolis Flow Regimes: • Basin Scale Flows • Geostrophic Nonlinear Terms negl. Unity Rossby • 1000-10000 km • Flow through a pipe • Flow in a narrow estuary • Flow around a ship Rotation can be ignored:

39. Scaling: Flow Classification – Ekman How do my Frictional terms compare to my Coriolis? Frictional Term This is the Horizontal Ekman number Coriolis Flow Regimes: Friction negligible • Interior (10-3) Both terms important • Gulf Stream E=1 (Unity Ekman) is useful for defining a length scale (by solving for L) that provides the limit of frictional influence (e.g. Ekman depth). We will see this again more formally later in the semester.

40. Scaling: Flow Classification – Reynolds How do my Nonlinear terms compare to my Frictional? Nonlinear This is the Reynolds Number Friction Most widely known non-dimensional parameter in fluid mechanics Flow Regimes: • Creeping Flow (Stokes flow) • phytoplankton to first hatchers • Flow is reversible – weird stuff. Nonlinear Terms negl. Nonlinear Terms important • 1000-10000 km Flow effectively inviscid • Many flows outside the BBL

41. Other Important Non-Dimensional Params Froude • Ratio of Advection Speed to Gravity Wave Speed • Analogous to Mach number in compressible flow Weber • Relative importance of inertia and surface tension • Important for capillary waves (influence SAR) Prandtl • Ratio of momentum viscosity to thermal diffusivity • Depends only on fluid state (no length / velocity / time scales) Aspect Ratio • Ratio of length scales • e.g., span/chord for wind planforms • e.g. H/L for the ocean (we have already seen this) + Grashof, Rayleigh, etc.

42. Key Concepts: • Newton’s Second Law • Free Body Diagrams • Coordinate Systems: Inertial and Non-Inertial • Review: Flux Divergence – Substantial Derivative • Forces on a Fluid • Equations of Motion • Scaling and Key Non-dimensional Parameters