Introduction to Astrophysical Gas Dynamics

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Introduction to Astrophysical Gas Dynamics
Part 6

• Bram Achterberg
• a.achterberg_at_astro.uu.nl
• http//www.astro.uu.nl/achterb/aigdppt

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Rotation

• Two aspects of rotation in fluid dynamics
• Vorticity swirling motions within a flow as
• a dynamical entity long-lived structures
• due to Kelvins circulation theorem
• Large-scale rotation
• - rotating frame-of-reference
• - Coriolis - and centrifugal forces

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Applications
Meteorology Cyclones Tornados

• Astrophysics
• Jupiters Great Red Spot
• Accretion Disks

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Hurricane Katrina (august 2005)
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Vortex Shedding
Flow direction
Obstacle
Fluctuating lift force
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Jupiters Great Red Spot
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Smoke ring from volcanic vent on Mnt. Etna
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Definition vorticity
Vorticity field is the rotation (curl) of the
velocity field
Vorticity field is divergence-free closed field
lines
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Vorticity in component form
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Equation of motion for vorticity
Step 1 take curl of equation of motion
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Equation of motion for vorticity
Step 1 take curl of equation of motion
Step 2 use vector identity
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Step 3 some more manipulation
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Step 3 some more manipulation
Equation of motion for vorticity
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Equation of motion for vorticity
Yet another vector identity
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Equation of motion for vorticity
Yet another vector identity
Another form of the vorticity equation
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Vorticity equation
Mass conservation
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Vorticity equation
Mass conservation
Final best form of the equation
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Interpretation of the vorticity equation
Vortex stretching
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Interpretation of the vorticity equation
Vortex stretching
Vorticity generation
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Interpretation of the vorticity equation
Vortex stretching
Vorticity generation
Ideal gas law
Condition for vorticity generation
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Velocity at each point equals fluid velocity
Definition of tangent vector
Equation of motion of tangent vector
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Vortex Stretching
Equation of motion for curve carried by flow
Vorticity equation without generation term
Conclusion vortex lines are carried by the flow
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Definition vortex line
Vortex lines are the field lines of the vorticity
field
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Definition vortex line
Vortex lines are the field lines of the vorticity
field
Vortex lines are carried by the flow
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Vortex tubes and the circulation theorem
Definition of circulation integral
dr is carried by the flow!
Use of Stokes theorem!
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Circulation number of vortex lines piercing
surface O vortex flux
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Change of circulation
Deformation of surface
Change of vorticity
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Deformation of surface-element carried passively
by the flow
Definition of surface-element
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Deformation of surface-element carried passively
by the flow
Definition of surface-element
Change of the two vector-elements carried by flow
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Deformation of surface-element carried passively
by the flow
Definition of surface-element
Change of the two vector-elements carried by flow
Use of chain rule
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Smart choice
Use determinant form of cross-product
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Smart choice
Use determinant form of cross-product
Write out determinants
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And now for some horrific algebra
Add and subtract the same term!
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Equation for surface change
Divergence effect of isotropic
compression/expansion
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Equation for surface change
Divergence effect of isotropic
compression/expansion
New animal velocity gradient tensor effect of
surface warping
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Change of circulation
Deformation of surface
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Change of circulation
Deformation of surface
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Use vorticity equation
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Use vorticity equation
Kelvins theorem
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Important consequence for barotropic fluid with
?P ??
Circulation is conserved!
Stretching the tube increases Vorticity!
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Alternative derivation
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Stokes Theorem
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Change of circulation (1)
Vorticity equation of motion (2)
12 together yield Kelvins Circulation Theorem
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(Fluid)dynamics in a rotating frameand
curvilinear coordinates
Introduction a overview over linear
vector-algebra
Vector in terms of its components
Orthonormal base vectors
Vector components as a scalar product
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Change of a vector field(difference vector)
Change of the components
Change of the base vectors
Components of the difference vector
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Change of base-vectors
(i1, 2, 3)
Example cylindrical polar coordinates
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Cylindrical Polars
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Distances
In three dimensions
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Gradient of a function
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Surfaces and volumes
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Derivative of a vector
Definition of V-grad
j-th component
Change of components
Change of unit vectors
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Example fluid acceleration in circular polar
coordinates
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Summary so far
Definition vorticity
Kelvins circulation theorem Vortices in ideal
fluids are long-lived
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Rotating coordinates
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Rotating coordinates
Equation of motion for unit vectors
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Rate-of-change of a vector
Rate-of-change of unit vector
Rate-of-change of an arbitrary vector
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Interpretation
Rate-of-change in Inertial Frame
Rate-of-change in Rotating Frame
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Interpretation
Rate-of-change in Inertial Frame
Rate-of-change in Rotating Frame
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Applications
Basic relation for any vector
Apply to velocity
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Basic relation for any vector
Apply to acceleration
Put in relation between velocities
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Basic relation for any vector
Apply to acceleration
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Summary
Acceleration as seen by a rotating observer
Coriolis force term
Centrifugal force term
Euler force
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Illustration Coriolis Force
Ball moves with constant velocity in the
inertial (laboratory) frame NO FORCE!!
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Illustration Coriolis Force
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Fluid equation in a rotating frame
Step1 use relation between acceleration in
inertial and rotating frames (? constant)
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Fluid equation in a rotating frame
Re-order terms
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Fluid equation in a rotating frame
Use definition of comoving time-derivative
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Effective gravity for rotationalong z-axis
true gravity centrifugal force
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Application cyclones
- ?P
L
Vh
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Cyclone Mechanism
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Approximate balance between the Coriolis force
and the pressure force
Gradient in horizontal plane!
Component of ? in vertical direction
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Approximate balance between the Coriolis force
and the pressure force
Take vector product with vertical unit vector
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Approximate balance between the Coriolis force
and the pressure force
Use property of double vector product
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Geostrophic Flow(Coriolis term dominates!)
Flow is to lowest order- along isobars!
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Equation for vorticity
Vorticity in rotating frame
Vorticity equation
Influence of Coriolis force!
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Definition absolute vorticity
Relative vorticity
Planetary vorticity
Alternative form vorticity equation for ?
constant
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Vorticity equation
Thermal wind equation
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Ideal gas law
Incompressible flow
Thermal wind equation
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Ideal gas law
Incompressible flow
Thermal wind equation
Density gradient mostly vertical due to gravity
Atmospheric scale-height
Temperature gradient Equator-to-pole!
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Example of Thermal WindGlobal Eastward
Circulation!
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More extreme example Jupiters cloud bands
Great Red Spot
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Shallow water theory
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Equations of motion in horizontal plane
Shallow water assumption
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Vertical direction hydrostatic equilibrium
Weight/unit area of overlying layer
Atmospheric pressure
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Barometric formula
Horizontal pressure gradients
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Barometric formula
Horizontal pressure gradients
substitute into eqn. of motion
Variations in depth drive the motions in the
horizontal plane!
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Height variations and mass conservation
Volumehorizontal area x depth
Surface change law two-dimensional volume-chang
e law!
2D divergence
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Constant-density flow
Surface-change law
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Equation for layer depth
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Summary the shallow water equations
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Application I water waves
Water waves are surface waves, leading to
varying depth
Assume that unperturbed flow is at rest Small
velocity perturbations
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Linarized equations
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Standard approach seek plane-wave solutions
Result set of three linear algebraic equations
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Solution if determinant 3x3 matrix vanishes
Determinant
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Solution if determinant 3x3 matrix vanishes
Determinant
Dispersion relation
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Solution if determinant 3x3 matrix vanishes
Determinant
Dispersion relation
Wave frequency
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Physical interpretation
Compare
1. Sound waves in rotating cylinder
2. Shallow-water waves
Pressure at bottom unperturbed layer
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Shallow-water vorticity
Velocity in horizontal plane
Vorticity associated with horizontal motions
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Shallow-water approximation
Exact (3D) velocity
Constant-density flow
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z-component of vorticity equation
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z-component of vorticity equation
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Conservation of the potential vorticity
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Finally the Great Red Spot
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Merging of like-signed vortices
(Dye visualization, TU Delft)
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