# 3. Rotational Dynamics Flashcards

1
Q

Rotational Reference Frames

A

-R denotes a rotational reference frame, e.g. a room rotating around a central axis -I indicates an inertial reference frame

2
Q

Rotation Vector

A

-the rotation vector is Ω_ -this is generally set to Ω_ = Ωez^ -where Ω is the rate of rotation, and Ω>0 indicates anticlockwise rotation

3
Q

Period of Rotation

A

τ = 2π/Ω

4
Q

Centrifugal Force Description

A

-proportional to distance from the axis of rotation -e.g. when you are in a car and it goes around a bend, your momentum is in the tangential direction so you experience a radial force

5
Q

Coriolis Force Description

A

-associated with motion -if a person in a rotational reference frame throws a ball, the ball deflects to follow the path of rotation

6
Q

Rotaional Reference Frame

Position

A

to describe position x_(t), we choose a basis {ej_} for j=1,2,3

-in R, choose {ej~_(t)} fixed in the room:

ej~_ = P[Ωt] ei_

-so that:

x_(t) = xj~(t) ej~_(t)

7
Q

Inertial Reference

Position

A
• to describe position x_(t), we choose a basis {ej_} for j=1,2,3
• in I, {ej_} is fixed in time:

x_(t) = xj(t) ej_

8
Q

Relationship Between Position in Inertial and Rotational Reference Frames

A

-the relationship between the two sets of coordinates is: xj(t) ej_ = xj~(t) ej~(t)

9
Q

Rotation Matrix

A

-P[Ωt] is a 3x3 matrix describing rotation around the z-axis by Ωt -with entries cos(Ωt), -sin(Ωt), 0 in the first row, sin(Ωt), cos(Ωt), 0 in the second row and 0,0,1 in the third row

10
Q

Rotational Basis Vectors

A

e1~_ = cosθ e1_ - sinθ e2_ e2~_ = sinθ e1_ + cosθe2_ e3~_ = e3_

11
Q

Inertial Basis Vectors

A

e1_ = cosθ e1~_ + sinθ e2~_ e2_ = -sinθ e1~_ + cosθ e2~_ e3_ = e3~_

12
Q

Inertial Reference Frame

Velocity

A

dx_/dt |I = dxj/dt ej_

13
Q

Rotational Reference Frame

Velocity

A

dx_/dt |R = dxj~/dt ej~_

-the apparent velocity

14
Q

Relationship Between Velocity in Inertial and Rotational Reference Frames

A

dx_/dt|I = dx_/dt|R + Ω_ x x_

15
Q

Relationship Between Acceleration in Inertial and Rotational Reference Frames

A

d²x_/dt²|I = d²x_\dt²|R + 2Ω_xdx_/dt|R + Ω_x(Ω_xx_)

16
Q

Equation of Motion in the Inertial Reference Frame

A

m d²x_/dt²|I = F_

17
Q

Equation of Motion in the Rotational Reference Frame

A

m d²x_/dt²|R = F_ - 2m Ω_ x dx_/dt|R - mΩ_ x (Ω_ x x_)

-the secon term on the RHS represents the Coriolis force and the third term the Centrifugal force

18
Q

Ficticious Forces in the Rotational Reference Frame

A

-in moving to the rotational reference frame, ficticious forces arise which keep track of the fact that we are in a rotating reference frame

19
Q

Properties of the Centrifugal Force

A

-if Ω_ = Ω ez^ and x_ = r er^ + z ez^

Fu_ = mΩ²r er^

• this is always positive, as expercted since the centrifugal force points outwards
• a conservative force

Fu_ = -∇(1/2 mΩ²r²)

20
Q

Coriolis Force Properties

A

Fc_ . x_ = 0

• so Fc_ is perpendicular to the direction of motion
• no work done
• deflecction is energetically free
21
Q

Coriolis Dominated Dynamics

A

|Fu_|/|Fc_| ~ Ωr/U << 1

• Fu_ negligible and Fc_ dominant
• Fu_ can also be balanced out in the governing equation:

–by gravity in a planetary context

–by the reaction force from the outer wall in a tank

22
Q

Coriolis Dominated

Equation

A

Fu_ = 0

=>

mx’‘_ = -2mΩ_ x x’_ + F_

23
Q

Coriolis Dominated

Solutions

A

F_ = (0,F)

=>
x = a/4Ω² [2Ωt - sin(2Ωt)]

y = a/4Ω² [1 - cos(2Ωt)]

24
Q

Coriolis Dominated

Graphs

A
• on small scales, lots of oscillations
• on a large scale, this smooths out to Coriolis deflection

evolution = drift at 90’ to applied force + inertial oscillation

25
Q

Coriolis Dominated

Drift

A

-drift satisfies the pure coriolis equations with zero acceleration (zero inertia)

=>

y = constant

x = at/2Ω = drift

26
Q

Coriolis Dominated

Drift Dominated

A
• for t >> 1/Ω, drift is dominant
• cf geostrophic balance in fluid dynamics
27
Q

Fluid Mechanics in a Rotating Frame

Governing Equations

A

ρDu_/Dt = -∇p + ∇(1/2 ρΩ²r²) - 2ρΩ_xu_ + F_

-taking F_=0

=>

ρDu_/Dt = -∇(p - 1/2 ρΩ²r²) - 2ρΩ_xu_

∇.u_ = 0

28
Q

Fluid Mechanics in a Rotating Frame

Losing Fu_

A
• for certain situations, the predictions of the governing equations are the same with or without Fu_
• this is the case for closed boundries where the reaction force from the solid boundary balances the centrifugal force
29
Q

Reduced Pressure

A

-for rotating fulids, p is replaced in the governing equation by P, where:

P = p - 1/2 ρΩ²r²

30
Q

Relative and Absolute Vorticity in Rotating and Inertial Reference Frames

A
• imagine a circular room rotating with angular velocity Ω
• inside the room is an object rotating with angular velocity ζ/2
• an observer in the rotating reference frame observes the rotation vector of the object as ζ/2
• an inertial observer outside of the rotating room observers the rotation vector of the object as ζ/2 + Ω, which is the actual rate of rotation
• the angular momentum of the object as observed by the inertial observer needs to be conserved whereas the angular momentum as observed by the rotational observer is not conserved
31
Q

Reative Vorticity

Definition

A

ζ_ = ∇_ x u_

32
Q

Absolute Vorticity

Definition

A

q_ = ζ_ + 2Ω_

33
Q

Vorticity Equation

A

Dq_/Dt = q_ . ∇_u_

• where Dq_/Dt is the rate of change of q_ for a fluid element
• and q_ . ∇_u_ represents vortex stretching
34
Q

Ballerina Effect

A
• say, q_=q ez^
• then the vorticity equation becomes:

q_ . ∇_u_ = q ∂w/∂z = Dq/Dt

-rate of change of absolute vorticity = rate of extension

35
Q

Inertial Wave

Definition and Governing Equations

A

-fluid mechanical analogue of inertial oscillations for particles (also called inertial oscsillations)

∂u/∂t = - 1/ρ ∂p/∂x - 2Ωv

∂v/∂t = - 1/ρ ∂p/∂y + 2Ωu

∂w/∂t = - 1/ρ ∂p/∂z

-and incompressibility:

∂u/∂x + ∂v/∂y + ∂w/∂z = 0

36
Q

The Rossby Number

Definition

A

inertia ~ ρU²/L

Coriolis ~ ρΩU

-take the ratio:

inertia / Coriolis = U/ΩL = Ro

-where Ro is the Rossby Number

37
Q

The Rossby Number

Interpretation

A

Ro << 1 => Coriolis dominant, geostrophic flow, inertial oscillations negligible, planetary scales

Ro >> 1 => Inertia dominant, cyclostrophic flow, everyday sclaes

-e.g. for the ocean Coriolis is dominant over inertia, for the atmosphere Coriolis is also dominant but less so and for Jupiter the result is similar

38
Q

Geostrohpic Balance

Definition and Governing Equations

A

-Coriolis dominated fluid dynamics so inertial term, ρDu_/Dt, can be set to 0

=>

2ρ Ω_ x u_ = -∇_P + F_

-this gives three equations, x, y, z, as well as the incompressibility condition

39
Q

Geostrophic Balance

Results

A
• if the pressure field is known, then u and v can be calculated directly
• the horizontal divergence (∂u/∂x + ∂v/∂y) = 0, so using incompressibility:

∂w/∂z = -(∂u/∂x + ∂v/∂y) = 0

• w is independent of z
• also horizontal divergence = 0 => existence of streamfunction Ψ such that:

u = -∂Ψ/∂y, v = ∂Ψ/∂x

-where Ψ=Ψ(x,y)

40
Q

Geostrophic Balance

Pressure and Streamfunction

A

-using the governing equations,

Ψ = P / 2Ωρ

=> P is a streamfunction with constant

-isobars are streamlilnes since Ψ=const. <=> P=const.

41
Q

Cyclonic vs Antiscyclonic Weather Systems

A
• when pressure is low, you get anticlockwise rotation, a cyclonic weather system
• when pressure is high, you get clockwise rotation, an anticyclonic weather system
42
Q

Coriolis and Latitude

A
• would expect greatest Coriolils effect at poles and zero at equator
• this is generally true but dynamics at the pole are dominated by other mechanisms
43
Q

Taylor - Proudman Theorem

A
• for a geostrophic system, ∂w/∂z=0
• calculating ∂u/∂z from the governing equations

=>

∂u/∂z = 0

• and similarly, ∂v/∂y=0
• a geostrophich flow does not depend on z
44
Q

Taylor Column

A
• by the Taylor-Proudman theorem, a geostrophic flow does not depende on z
• so if there is an obstacle at ground level z=0, for example an island then the motion of the flow around this obstacle will be replicated at every value of z as if the flow is movin around a virtual obstacle
• the resulting structures are called Taylor columns
45
Q

Ekman Layers

Definition

A
• even if flow is geostrophic in the upper atmosphere, further down the ground exerts friction
• model this by incorporating a viscous stress due to vertical shear (cf. inertial boundary layer, Blassius)
46
Q

Ekman Layers

Governing Equations and Boundary Conditions

A

0 = -∇_P - 2ρΩ_ x u_ + ν∂²u_/∂z²

• gives three equations:
• 2Ωv = -1/ρ ∂p/∂x + ν∂²u_/∂z²

2Ωu = -1/ρ ∂p/∂y + ν∂²v_/∂z²

∂p/∂z = -ρg

-boundary conditions:

u ~ uo(x,y) and v~vo(x,y) as z->∞, i.e. known geostrophic flow in the upper atmosphere

u=v=0 at z=0, no-slipi boundary condition

47
Q

Ekman Layers

Solutions for u and v

A

-integrate the z equation for:

P = -ρgz + po(x,y)

• substitute into each of x and y and apply the geostrophic limit boundary condition
• let U=u+iv, this gives a second order equation for U
• solve with particular integral U=Uo (Uo=uo+ivo) and complimentary function (U=Be^[-z√[2Ωi/ν]]
• plotting v against u gives an Ekman spiral
48
Q

Ekman Layers

Solution for w

A

-using incompressibility:

∂w/∂z = - (∂u/∂x + ∂v/∂y)

-integrate with respect to z

=>

w = ζo(x,y) [1 - e^(z/L) (sin(z/L)+cos(z/L))]

• where ζo(x,y) is the relative vorticity of the geostrophic flow at z->∞
• in the limit z->∞, w ~ Lζo/2 ≠ 0
• so the upper atmosphere is moving up and down

w>0 if ζo>0 => low pressure weather system

w<0 if ζo<0 => high pressure weather system

49
Q

Ekman Layer

Implications for Weather

A
• when there is low pressure, moisture is driven upwards to colder air where is condenses forming clouds and possibly precipitation
• when there is high pressure, moisture staus at ground level, clear skies
50
Q

Fluid Dynamics on a Rotating Planet

Ω_

A

2Ω_ = 2Ωcosθ ey^ + 2Ωsinθ ez^

where 2Ωcosθ = γ and 2Ωsinθ = f

51
Q

Fluid Dynamics on a Rotating Planet

Governing Equations

A

Du/Dt - fv = -1/ρ ∂p/∂x

Dv/Dt + fu = -1/ρ ∂p/∂y

Dw/Dt = -1/ρ ∂p/∂z - g

52
Q

Fluid Dynamics on a Rotating Planet

f

A
• often interested in a single zone, θ=θo (i.e. a latitude) so let θ^<<θo and θ = θo + θ^
• then f = 2Ωsinθ = 2Ωsin(θo + θ^)
• use double angle formula and linearise

=>

f = 2Ωsinθo + 2Ωθ^cosθo

f = f + βy

-where θ^ = y/R

53
Q

Fluid Dynamics on a Rotating Planet

fo

A

fo>0 in the Northern hemisphere

fo<0 in the Southern hemisphere

54
Q

Fluid Dynamics on a Rotating Planet

f-plane and β-plane approximations

A
• if f≈fo , f-plane approx.
• if f≈fo+βy , β-plane approx - important for Rossby waves
55
Q

Fluid Dyanmics on a Rotating Planet

Vorticity

A

q_ = f_ + ζ_

• where q_ is the absolute vorticity
• can show:

Dq_/Dt = q_ . ∇u_

56
Q

Rossby Waves

Outline

A
• focus on 2D, w=0
• incompressibility still holds so can introduce the streamfunction
• vorticity equation reduces to Dq_/Dt=0
• consider contours of q at rest, in the perturbed state, these straight lines become sinusoidal waves
• at peaks, y>0, ζ=-βy < 0 => clockwise rotation
• at troughs, y<0, ζ=-βy > 0 => anticlockwise rotation
• the contour is advected as a translation to the west
57
Q

Rossby Waves

Deriving the Dispersion Relation

A
• sub in for absolute vorticity q_ = fo + βy + ∇²Ψ, this gives the non-linear Rossby-wave equation

∂/∂t(∇²Ψ) + β ∂Ψ∂/x = 0

-try 2D wave ansatz

ω = -βk / [k² + l²]

58
Q

Rossby Waves

Phase Velocity and Group Velocity

A
• consider case where l=0, then ω = -β/k
• phase velocity, cp = ω/k = -β/k²
• group velocity, cg = ∂ω/∂k = β/k²

cp ≠ cg => dispersive waves, but speeds are equal, cp<0 always and cg>0 always