Download 1 8. CONSERVATION LAWS The general form of a conservation law

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Transcript
8. CONSERVATION LAWS
The general form of a conservation law is
!U ijk...
!t
="
!
Fmijk... !!!,
!xm
(8.1)
where U ijk... is a tensor of rank N , and Fmijk... is a tensor or rank N + 1 . Integrating over
the volume and applying Gauss’ theorem, we have
!
U ijk...dV = !
" dSm Fmijk... !!!,
!t V"0
S0
(8.2)
which expresses the conservation of the volume integral of U ijk... . The tensor Fmijk... is the
flux of U ijk... in the direction of xm ; the surface integral expresses the total flux through
the bounding surface.
For example, if N = 0 we have the scalar conservation law
!U
= "# $ F!!!,
!t
(8.3)
where U is a scalar and F is a vector. We have already encountered one of these in the
continuity equation, which expresses the conservation of mass. If N = 1 we have the
vector conservation law
!U
= "# $F!!!,
!t
(8.4)
where U is a vector and F is a second rank tensor (or, equivalently, dyad). The quantity
Fij is the flux of U i in the x j direction. This can be continued ad infinitum (or ad
nauseum).
The MHD equations can be written in the form of conservation laws that express the
physical principles of conservation of mass, momentum and energy.
As mentioned above, we already have the law of conservation of mass:
!"
= #$ % "V!!!.!
!t
(8.5)
The quantity !V is called the mass flux.
We now obtain the law of conservation of momentum. This will be of the form
!"V
= #$ % T!!!,
!t
(8.6)
where is the momentum per unit volume and T is called the total stress tensor. We have
seen that the equation of motion (Newton’s law) is
1
% "V
(
!'
+ V # $V * = +$ # ( pI + , ) + J - B!!!.
& "t
)
(8.7)
The first term on the right hand side is already in conservation form. We transform the
left hand side according to
!
"V "!V
"! "!V
=
#V
=
+ V $ %!V!!!,
"t
"t
"t
"t
(8.8)
and
! " #VV = V " !#V + #V " !V!!!.
(8.9)
We have used the continuity equation to obtain the second equality in Equation (8.8).
Then we have
!"V
+ # $ "VV = %# $ ( pI % & ) + J ' B!!!.
!t
(8.10)
Consider the second term on the right hand side. Using Cartesian tensor notation,
( J ! B)i =
1
" jki " jlm Bk #l Bm !!!,
µ0
!!!!!!!!!!!!!=
1
( Bl !l Bi " Bm !i Bm )!!!.
µ0
(8.11)
(
)
Now Bl !l Bi = !l ( Bl Bi ) , since !l Bl = 0 ( ! " B = 0 ), and Bm !i Bm = !l B 2" li / 2 , so we
have
( J ! B)i =
1 %
1
(
"l ' Bl Bi # B 2$ li * !!!,
)
µ0 &
2
(8.12)
or, in coordinate-free vector notation,
J!B=
1
1 (
%
" # ' BB $ B 2I* !!!.
µ0 &
2 )
(8.13)
The final result is in the conservation form of Equation (8.6), with
T = !VV "
#
1
1 2&
BB + % p +
B I( " )!!!.
µ0
2 µ0
$
'
(8.14)
Note that T is symmetric, as is required of a stress tensor. The component Tij is the total
flux of the j th of momentum in the direction xi .
The tensor !VV is called the Reynolds’ stress; !ViVj is the rate at which the
momentum component !Vj is carried through surface element dSi by velocity
We recognize P = pI ! " from our previous discussion as the
1 "1 2
%
hydrodynamic stress tensor. The tensor TM =
$# B I ! BB'& is called the Maxwell
µ0 2
component Vi .
2
stress tensor. Note that the magnetic field transports (or carries) momentum. The
quantity p + B 2 / 2 µ0 is the total pressure. The second contribution is called the
magnetic pressure; the magnetic field resists compression, just like the fluid pressure.
The tensor BB / µ0 is called the hoop stress. We will see that it resists shearing motions.
The total energy is the sum of the kinetic, magnetic, and internal energies,
u=
1 2 B2
!V +
+ !e!!!.
2
2 µ0
(8.15)
To obtain the law of conservation of energy, we work in the Lagrangian frame. We first
consider the rate of change of kinetic energy. The scalar product of V with the equation
of motion is
!
d " 1 2%
$ V ' = (V ) *P + V ) J + B!!!.
dt # 2 &
(8.16)
Using the Lagrangian form of the continuity equation, the identity
V ! "P = " ! (P ! V ) # P : "V , and interchanging terms in the second term on the right
hand side, this becomes
d " 1 2% 1 2
$ !V '& + !V ( ) V = *( ) (P ) V ) + P : (V * J ) V + B!!!.
dt # 2
2
(8.17)
We can use the resistive form of Ohm’s law to eliminate V ! B in favor of J and E .
Then writing !V 2" # V = " # ( !V 2 V) $ V # " !V 2 , and using the definition of P ,
(
)
d " 1 2%
"1 2 %
" 1 2%
$# !V '& + ( ) $# !V V '& * V ) ( $# !V '& =
dt 2
2
2
* ( ) (P ) V ) + p( ) V * + : (V + J ) E * , J !!!.
(8.18)
2
If we now transform to the Eulerian frame, the last term on the right hand side cancels,
and we obtain the final expression for the rate of change of kinetic energy:
! # 1 2&
,# 1 2
& /
%$ "V (' = )* + .%$ "V I + P(' + V 1
!t 2
2
-!##
#"###
$0
(8.19)
Flux through surface
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+ p* + V + J%
+E )
!"$
PV work
EM work
2
: *V
!"$
Viscous dissipation
)
3%
J2
! .
Resistive dissipation
Note that this is not yet a conservation law. The last 4 terms act as sources and sinks of
kinetic energy. They must show up elsewhere.
The calculation of the rate of change of magnetic energy is relatively
B!
straightforward.
Taking
the
induction
equation,
and
using
B ! " # E = " ! ( E # B ) + E ! " # B along with Ampére’s law, we find
3
,1
/
! " B2 %
= () * . E + B 1 ( J%
* E !!!.
$
'
!t # 2 µ0 &
-!µ#
0
0
EM work
"#$
(8.20)
Poynting flux
We already have an expression for the rate of change of internal energy:
!
eV ) # p$ % V # $
%q + &
: $V + '
J 2 !!!.
( "e) = #$ % (!""
!"#
$
$
# !"# Heat
!t
Viscous heating
flow
Resistive heating
PV work
(8.21)
Flux
The law of conservation of energy is obtained by adding Equations (8.19), (8.20),
and (8.21). All the sources and sinks cancel, and the result is
3 ,&
6
!u
1
1
/
)
= "# $ 4 .( %e + %V 2 + I + P 1 $ V +
E 2 B + q 7!!!!!.
*
!t
2
µ0
0
5 -'
8
(8.22)
Another conserved quantity is the magnetic flux density,
!B
= "# $ E!!!.
!t
(8.23)
We remark that this can be written in the form of Equation (8.1) as
!Bi
= "# ijk ! j Ek = "! j # ijk Ek = "! j # jki Ek != "! j Fji !!!.
!t
(
)
(
)
(8.24)
The tensor Fji is formally the flux of Bi in the direction of x j . Note that Fji is
completely anti-symmetric, i.e., Fij = ! ikj Ek = "! jki Ek = "Fji by the properties of ! jki . For
the case of ideal MHD, then Ek = !" kmnVm Bn , so Fij = BiVj ! B jVi and
!B
= "# $ ( BV " VB )!!!.
!t
(8.25)
Recall that B is a pseudo-vector; Equation (8.25) is an example of a pseudo-vector
conservation law. The conserved quantity is the magnetic flux through an area element
d! = B " dS , which is a true scalar (since both B and dS = dl1 ! dl 2 are pseudo-vectors).
The appropriate integral theorems are
d!
= " & # $ E % dS = " !
E % dl !!!,
&
dt
S
C
(8.26)
d
BdV = " ! dS # E !!!.
dt V!
S
(8.27)
and
Note that the evolution of B , the volume average of B , is completely determined by the
tangential component of E in the bounding surface.
4