1. Prof. dr. A. Achterberg, Astronomical Dept
Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit
Gas Dynamics, Lecture 5 (The Solar Wind; Waves: theoretical introduction) see:

2. Theoretical results for steady flow that we will use today (from last Lecture):
Bernoulli’s Law:

3. Solar wind
velocity as
measured by
Ulysses satellite

4. The Parker Model Assumptions: The wind is steady and adiabatic
The flow is spherically symmetric
Neglect effect of magnetic fields and rotation star

6. There must be a
sonic radius where
flow speed = sound speed

7. Basic Equations: steady, spherically symmetric flow
Conservation of mass in
steady flow
Bernoulli: conservation
of energy
Entropy is constant:
Adiabatic Flow
Gravitational potential
of a single star

9. Mass conservation: continuity equation
Steady flow in radial direction:

10. General approach: use “constants of motion” : stream lines are KNOWN!

11. General approach: use “constants of motion” : stream lines are KNOWN!

12. Aim: to convert all the equations into a single equation for the velocity V(r):
Step 1: calculate density change

13. Step 2: Calculate velocity change

14. Step 3: combine velocity and density results:
Adiabatic sound speed

15. Step 4: covert result into a differential equation

16. Parker’s equation for a spherical stellar wind:
Special velocity:
sound speed (“Mach One”)
Special radius:
critical radius

17. Mathematical Interlude: singularities in differential equations (1)

18. Mathematical Interlude: singularities in differential equations (2a)
1. Problems arise if solution curve passes through
point where
That is the same as saying:

19. Mathematical Interlude: singularities in differential equations (2b)
1. Problems arise if solution curve passes through
point where
2. Slope of curve changes sign if solution curve passes through
a point where ; if B is monotonic,
THEN Y(x) has a SINGLE maximum or a minimum!

20. Mathematical Interlude: singularities in differential equations (3)
SPECIAL CASE: CRITICAL POINT SOLUTION THROUGH
ONLY IN THIS CASE IS A MONOTONIC SOLUTION Y(x) POSSIBLE!

21. Mathematical Interlude: singularities in differential equations (4)
Formal solution near critical point :

22. Mathematical Interlude: singularities in differential equations (5)
Formal solution near critical point :

23. Solution space Parker Eqn: diagram

24. Solution space for Parker’s Equation
Accelerating wind solution:
V > 0 and dV/dr > 0!
Solution should remain
regular at all radii!

25. Solution space for Parker’s Equation
Critical Point Condition:

26. Wind and Breeze Solutions
Special case:
Isothermal Wind with
constant temperature

27. Accretion
Solution

28. Bondi Accretion
Critical Point Condition:

29. Isothermal Bondi Accretion

30. Similar flows:
Laval Nozzle (jet engines)

31. Basic equations:

32. Similar flows: (2)
2. Astrophysical jets:

33. Stellar Winds and Jets: similarities and differences
Steady flow Steady flow
Large opening angle Small opening angle
Parker-equation Parker-type equation
Flow geometry known Pressure known

34. Waves

35. Simple (linear) waves Properties:
Small perturbations of velocity, density and pressure
Periodic behavior (“sines and cosines”) in space and time
No effect of boundary conditions

36. Small amplitude waves: 1. fundamental approach
Wave amplitude is small: position of a fluid element can
be ‘decomposed’ as:
unperturbed
position
small
displacement

37. Small amplitude waves: 2. The plane wave assumption
Small displacement exhibits a sinusoidal behavior
in space and time:
plane wave representation
Phase factor
Complex amplitude
vector

38. Small amplitude waves: 3. Phase, wave vector, wavelength and frequency
Alternative formulation of plane waves:
angular
frequency
wave
period
wave vector
wavelength

39. Waves, wavelength and the wave vector

40. Small amplitude waves 4. what does ‘small amplitude’ mean?
4. Wave amplitude is small in the following sense:
|a| is much smaller than the wavelength λ;
|a| is much smaller than gradient scale of the flow;
Density and pressure and temperature variations remain small:

41. Mathematical technique:
Perturbation analysis:
Expand fundamental equations in displacement ξ(x,t);
Neglect all terms of order ξ2 and higher!
Express density and pressure variations in terms of
ξ(x,t); Neglect all terms of order ξ2 and higher!
Find equation of motion for ξ(x,t) where only terms
linear in ξ(x,t) appear;
- Substitute plane wave assumption.

42. Perturbation analysis: simple mechanical example
Small-amplitude
motion;
Valid in the vicinity
of an equilibrium
position;

43. Perturbation analysis: fundamental equations
Equilibrium position:

44. Perturbation analysis: motion near x = 0
Taylor expansion near
x = 0:
General case:

45. Perturbation analysis: like a harmonic oscillator
Equation of motion
near x=0:
“spring constant”

46. Perturbation analysis: fundamental solutions

47. Mathematical preliminaries
Aim:
1) Construct a generally valid method for perturbation analysis
in fluids or gases;
2) Express all perturbations in terms of the displacement vector
(x,t) and its derivatives.
In the end you should only see a linear equation with
things like:

48. Who measures what variation in a wave
Who measures what variation in a wave? Lagrangian and Eulerian variations
Two fundamental types of observer in fluid mechanics:
Observer fixed to coordinate system measures the Eulerian perturbation:
Observer moving with the flow
measures the Lagrangian perturbation

49. Lagrangian labels: useful mathematical concept
are carried along
by the flow

50. Lagrangian labels: useful mathematical concept
Conventional choice:
position x0 of a fluid-element
at some fixed reference time t0
As always:

51. Re-interpretation of time-derivatives:
At a fixed position
Comoving with the flow

52. Re-interpretation of time-derivatives + Re-interpretation of perturbations:
At a fixed position
Comoving with the flow
Lagrangian and Eulerian perturbations:
Lagrangian:
Eulerian:

53. Important consequence: Commutation Relations for derivatives!
All at a fixed position in
the coordinate grid
Moving with
the flow

54. Relation between Lagrangian and Eulerian perturbations
Stay at old position!
Follow the fluid to
new position!
Unpertubed value
at old position!

55. Relation between Lagrangian and Eulerian perturbations (2)

56. Relation between Lagrangian and Eulerian perturbations (3)

57. Relation between Lagrangian and Eulerian perturbations (4)

58. Final result for small perturbations:
Small change induced
by ξ in Q at fixed position
Effect of position shift ξ

59. Almost trivial example of these rules (1):

60. Almost trivial example of these rules (2):
Formal calculation:

61. Perturbation analysis: general approach (example: sound waves)

62. Application: velocity perturbation due to small-amplitude wave (1)
Commutation
Rules

63. Application: velocity perturbation due to small-amplitude wave (2)
Commutation
Rules
Definition of the
comoving derivative
(V = unpertubed velocity!):

64. Eulerian and Lagrangian velocity perturbations:
General relation between the
two kinds of perturbations:

65. Summary: velocity perturbations (1)
Special simple case: stationary unperturbed fluid that has V = 0:

66. Summary: velocity perturbations (2)
Another special case: unperturbed fluid has uniform velocity V ≠ 0:

67. Density perturbation: 1D case
Mass conservation:

68. Density perturbation (2)

69. Density perturbation (2)

70. Density perturbation (3)

71. Density perturbation (4)

72. Generalization results from 1D to 3D:
One dimension: Three dimensions:

73. Pressure perturbation:
Adiabatic flow:
entropy conservation
if you move with the flow
From general relation between Lagrangian and Eulerian
perturbations ΔP and δP:

74. Summary: changes in fluid quantities induced by a small fluid displacement ξ(x,t):

75. Perturbation analysis: general approach

76. Linear sound waves in a homogeneous, stationary gas
Main assumptions:
Unperturbed gas is uniform: no gradients in density,
pressure or temperature (P = ρ = 0);
Unperturbed gas is stationary: without the presence
of waves the velocity vanishes (V = 0);
The velocity, density and pressure perturbations
associated with the waves are small

77. Immediate consequence: perturbations are “simple”:
Velocity associated
with the wave:
Density perturbation
associated with the wave
Pressure perturbation
associated with the wave:

78. Immediate consequence: perturbations are “simple”:
Velocity associated
with the wave:
Density perturbation
associated with the wave
Pressure perturbation
associated with the wave:
This is KINEMATICS, not DYNAMICS!

79. Aim: to derive the DYNAMICS of the problem!
To derive a linear equation of motion for the
displacement vector (x,t) by linearizing the
equation of motion for the gas.
Method:
Take the Lagrangian variation of the equation
of motion.

80. Perturbing the Equation of Motion
To find the equation of motion governing small
perturbations you have to perturb the equation
of motion!

81. Unperturbed gas is
uniform and at rest:
Apply a small displacement

82. Unperturbed gas is
uniform and at rest:
Apply a small displacement
Because the unperturbed
state is so simple, the linear
perturbations in density,
pressure and velocity are
also simple!

83. Effect of linear perturbations on the equation of motion: fluid acceleration
Use commutation rules again:

84. Effect of linear perturbations on the equation of motion: pressure force
Use commutation rules again:
I have used:

85. Effect of linear perturbations on the equation of motion
What do we know at this point:

86. Finally: the equation of motion for ξ(x,t):

87. Grand finale: an equation for plane sound waves!

88. Grand finale: an equation for plane sound waves!