The Structure, Stability, and Dynamics
of Self-Gravitating Systems
Joel E. Tohline
tohline@rouge.phys.lsu.edu
[ Download PDF file dated
]
Joel E. Tohline
tohline@rouge.phys.lsu.edu
[ Download PDF file dated
]
Among the principal governing equations, we have included the
[Equation I.A.1]
BT87, Appendix 1.E, Eq. (1E-6)
Tassoul
'78, § 3.3, Eq. (38)
| Replacing the Lagrangian (or total ) time-derivative on the left-hand-side of this equation [I.A.1] by the Eulerian (or partial ) time-derivative via the relationship [VI.M.14] defined in an accompanying appendix, we readily derive the |
¶tv + (v×Ñ)v = - (1/r)ÑP - ÑF.
[Equation I.A.2]
=
Landau & Lifshitz '75, Chapter I, Eq. (2.4)
BT87, Appendix 1.E, Eq. (1E-8)
| Multiplying the standard Lagrangian representation of Euler's Equation [I.A.1] through by the mass density r produces the relation, |
| Combining this with the standard Lagrangian representation of the Continuity Equation [I.B.1], we derive, |
| Note that if, at any location in space, the gradient in the gas pressure P and the gradient in the gravitational potential F conspire to make the right-hand-side of this expression equal to zero, this form of Euler's equation may be cast into a generic Lagrangian Conservative Form [I.B.5], |
in which the momentum of the gas is identified as a conserved quantity. This is simply a mathematical way of expressing the physical concept that any particle of gas that happens to be passing through that location in space will instantaneously experience no acceleration and, hence, its (linear) momentum will remain unchanged as it passes through that location.
For any two vectors A and B, we can utilize the following
Letting A = B = v, this identity implies,
where the fluid vorticity
| This relationship [I.A.7] may be combined with the standard Eulerian representation of Euler's equation [I.A.2] to derive |
¶tv + z ´ v = - (1/r)ÑP - ÑF - (1/2) Ñ( v2 ) .
[Equation I.A.9]
At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with a uniform (i.e., constant in time) angular velocity Wf. In order to transform Euler's equation from the inertial reference frame to such a rotating reference frame, the operator d/dt which denotes Lagrangian time-differentiation in the intertial frame must everywhere be replaced as follows:
[Equation I.A.10]
Notice that here we have used the color orange to
distinguish time-differentiation in the rotating frame of reference from
time-differentiation in the inertial frame. Throughout this H_Book we
frequently will adopt this "marking" technique to clearly identify variables
or operators that are associated with the rotating frame of reference in
situations where confusion with inertial-frame variables or operators might
arise.
Performing this transformation implies, for example, that
[Equation I.A.11]
and
[Equation I.A.12]
Rotating Reference Frame
d/dt ® d/dt +
Wf ´ .
vinertial = v +
Wf ´ x,
=
BT87, Appendix 1.D, Eq. (1D-38)
Tassoul '78
, § 3.3, Eq. (50)
Dvinertial
=
Dv +
2 Wf ´ v +
Wf ´ ( Wf ´ x )
=
Dv +
2 Wf ´ v -
Ñ[ ( 1/2 ) | Wf ´ x |
2 ] .
=
Tassoul '78
, § 3.3, Eq. (51)
| [If we were to allow Wf to be a function of time, one additional term involving the time-derivative of Wf also would appear on the right-hand-side of this last expression [I.A.12] (cf., BT87, expression 1D-42). Throughout this H_Book, we will restrict our flow analyses to either the inertial reference frame or to rotating frames in which Wf is time-invarient, so this additional term will not appear.] |
| Using this last expression [I.A.12] in conjunction with the standard representations of Euler's Equation [I.A.1 & I.A.2] given above, we derive |
or,
¶tv +
(v×Ñ)v = -
(1/r)ÑP -
ÑF -
2 Wf ´ v -
Wf ´ ( Wf ´ x ),
[Equation I.A.13b]
¶tv
+
( z + 2 Wf )
´ v
= -
(1/r)ÑP -
ÑF -
( 1/2 ) Ñ(
v2
) -
Wf ´ ( Wf ´ x ).
[Equation I.A.13c]
(Eulerian Representation)
(in terms of the Fluid Vorticity)
| Following along the lines of the discussion presented in Appendix 1.D, § 3 of Binney and Tremaine (1987), in a rotating coordinate system the Lagrangian representation of Euler's equation [I.A.13a] may be written in the form, |
where,
[Equation I.A.16]
| When comparing this form of Euler's equation [I.A.15] to the standard Lagrangian representation that is valid in the inertial reference frame [I.A.1], we see that in a rotating coordinate system material moves as if it were subject to two fictional accelerations which traditionally are referred to as the |
|
Coriolis Acceleration aCoriolis º - 2 Wf ´ v [Equation I.A.17]
|
and the
|
Centrifugal Acceleration aCentrifugal º - Wf ´ ( Wf ´ x ) = Ñ[ ( 1/2 ) | Wf ´ x |2 ] . [Equation I.A.18]
|
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