3. Rotational Dynamics Flashcards
Rotational Reference Frames
-R denotes a rotational reference frame, e.g. a room rotating around a central axis -I indicates an inertial reference frame
Rotation Vector
-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
Period of Rotation
τ = 2π/Ω
Centrifugal Force Description
-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
Coriolis Force Description
-associated with motion -if a person in a rotational reference frame throws a ball, the ball deflects to follow the path of rotation
Rotaional Reference Frame
Position
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)
Inertial Reference
Position
- 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_
Relationship Between Position in Inertial and Rotational Reference Frames
-the relationship between the two sets of coordinates is: xj(t) ej_ = xj~(t) ej~(t)
Rotation Matrix
-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
Rotational Basis Vectors
e1~_ = cosθ e1_ - sinθ e2_ e2~_ = sinθ e1_ + cosθe2_ e3~_ = e3_
Inertial Basis Vectors
e1_ = cosθ e1~_ + sinθ e2~_ e2_ = -sinθ e1~_ + cosθ e2~_ e3_ = e3~_
Inertial Reference Frame
Velocity
dx_/dt |I = dxj/dt ej_
Rotational Reference Frame
Velocity
dx_/dt |R = dxj~/dt ej~_
-the apparent velocity
Relationship Between Velocity in Inertial and Rotational Reference Frames
dx_/dt|I = dx_/dt|R + Ω_ x x_
Relationship Between Acceleration in Inertial and Rotational Reference Frames
d²x_/dt²|I = d²x_\dt²|R + 2Ω_xdx_/dt|R + Ω_x(Ω_xx_)
Equation of Motion in the Inertial Reference Frame
m d²x_/dt²|I = F_
Equation of Motion in the Rotational Reference Frame
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
Ficticious Forces in the Rotational Reference Frame
-in moving to the rotational reference frame, ficticious forces arise which keep track of the fact that we are in a rotating reference frame
Properties of the Centrifugal Force
-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 ‘radial gravity’
- a conservative force
Fu_ = -∇(1/2 mΩ²r²)
Coriolis Force Properties
Fc_ . x_ = 0
- so Fc_ is perpendicular to the direction of motion
- no work done
- deflecction is energetically free
Coriolis Dominated Dynamics
|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
Coriolis Dominated
Equation
Fu_ = 0
=>
mx’‘_ = -2mΩ_ x x’_ + F_
Coriolis Dominated
Solutions
F_ = (0,F)
=>
x = a/4Ω² [2Ωt - sin(2Ωt)]
y = a/4Ω² [1 - cos(2Ωt)]
Coriolis Dominated
Graphs
- 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
Coriolis Dominated
Drift
-drift satisfies the pure coriolis equations with zero acceleration (zero inertia)
=>
y = constant
x = at/2Ω = drift
Coriolis Dominated
Drift Dominated
- for t >> 1/Ω, drift is dominant
- cf geostrophic balance in fluid dynamics
Fluid Mechanics in a Rotating Frame
Governing Equations
ρDu_/Dt = -∇p + ∇(1/2 ρΩ²r²) - 2ρΩ_xu_ + F_
-taking F_=0
=>
ρDu_/Dt = -∇(p - 1/2 ρΩ²r²) - 2ρΩ_xu_
∇.u_ = 0
Fluid Mechanics in a Rotating Frame
Losing Fu_
- 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
Reduced Pressure
-for rotating fulids, p is replaced in the governing equation by P, where:
P = p - 1/2 ρΩ²r²
Relative and Absolute Vorticity in Rotating and Inertial Reference Frames
- 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
Reative Vorticity
Definition
ζ_ = ∇_ x u_
Absolute Vorticity
Definition
q_ = ζ_ + 2Ω_
Vorticity Equation
Dq_/Dt = q_ . ∇_u_
- where Dq_/Dt is the rate of change of q_ for a fluid element
- and q_ . ∇_u_ represents vortex stretching
Ballerina Effect
- 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
Inertial Wave
Definition and Governing Equations
-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
The Rossby Number
Definition
inertia ~ ρU²/L
Coriolis ~ ρΩU
-take the ratio:
inertia / Coriolis = U/ΩL = Ro
-where Ro is the Rossby Number
The Rossby Number
Interpretation
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
Geostrohpic Balance
Definition and Governing Equations
-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
Geostrophic Balance
Results
- 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)
Geostrophic Balance
Pressure and Streamfunction
-using the governing equations,
Ψ = P / 2Ωρ
=> P is a streamfunction with constant
-isobars are streamlilnes since Ψ=const. <=> P=const.
Cyclonic vs Antiscyclonic Weather Systems
- when pressure is low, you get anticlockwise rotation, a cyclonic weather system
- when pressure is high, you get clockwise rotation, an anticyclonic weather system
Coriolis and Latitude
- 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
Taylor - Proudman Theorem
- 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
Taylor Column
- 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
Ekman Layers
Definition
- 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)
Ekman Layers
Governing Equations and Boundary Conditions
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
Ekman Layers
Solutions for u and v
-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
Ekman Layers
Solution for w
-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
Ekman Layer
Implications for Weather
- 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
Fluid Dynamics on a Rotating Planet
Ω_
2Ω_ = 2Ωcosθ ey^ + 2Ωsinθ ez^
where 2Ωcosθ = γ and 2Ωsinθ = f
Fluid Dynamics on a Rotating Planet
Governing Equations
Du/Dt - fv = -1/ρ ∂p/∂x
Dv/Dt + fu = -1/ρ ∂p/∂y
Dw/Dt = -1/ρ ∂p/∂z - g
Fluid Dynamics on a Rotating Planet
f
- 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
Fluid Dynamics on a Rotating Planet
fo
fo>0 in the Northern hemisphere
fo<0 in the Southern hemisphere
Fluid Dynamics on a Rotating Planet
f-plane and β-plane approximations
- if f≈fo , f-plane approx.
- if f≈fo+βy , β-plane approx - important for Rossby waves
Fluid Dyanmics on a Rotating Planet
Vorticity
q_ = f_ + ζ_
- where q_ is the absolute vorticity
- can show:
Dq_/Dt = q_ . ∇u_
Rossby Waves
Outline
- 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
Rossby Waves
Deriving the Dispersion Relation
- start with the reduced vorticity equation, Dq_/Dt=0
- sub in for absolute vorticity q_ = fo + βy + ∇²Ψ, this gives the non-linear Rossby-wave equation
- linearise about rest state
∂/∂t(∇²Ψ) + β ∂Ψ∂/x = 0
-try 2D wave ansatz
ω = -βk / [k² + l²]
Rossby Waves
Phase Velocity and Group Velocity
- 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
