293
4.1 Neighborhood of an Extremum
In the neighborhood of a particular extremum (steady state) denoted by TJO), TJO), . .. ,
we may write (in function space) T(p,) = T(O) + rf>(Ji.), or the deviation may be written in
terms of its spectral components rf>o, rf>2, . ... Expanding F[T] about the local extremum,
we obtain
(25)
where the subscript zero denotes evaluation at the extremum. The terms linear in
locally a quadratic in rPn. The matrix elements
(26)
are the structure constants for the quadratic geometric surface, F(To, T2, • •• ). If all eigenvalues of Nnm are positive, the surface is concave upward; if one or more of the eigenvalues
are negative, the surface is locally a saddle point. We proceed to show that these eigenvalues are the stability eigenvalues studied earlier.
First note that if the temperature field is allowed to be a function of time, then by
following the approach for the zero dimensional models we have
(27)
That is, the time derivative is given by the gradient in this multidimensional space. For
infinitesimal departures from steady state we set Tn( t) = T~q + rPn€-)..t above, expand
about
- L::Nnmrf>m.
(28)
m
The latter equation concludes the proof that the local geometrical structure constants of
F(To, T2, • •• ) yield the stability eigenvalues for that particular steady state.
Finally, as a conclusion to this section, consider the time behavior of the value of
F(To, T2, • •• ) when the point (To, T2, ••• ) is governed by the time-dependent energy balance
equation
dF
L:: aF .
dt
aT Tn
n
n
= - L::(Tn)2
(29)
n
where we inserted the equation of motion. This latter result is the multidimensional analog
of one-dimensional slope-stability problem. It has a corresponding interpretation: initial
departures of the state (To, T2 , ••• ) from a local extremum of F lead to a trajectory of the
system point such that F decreases. The point will continue down the gradient of F until
a local extremum is found. Clearly, saddle points are unstable-the system point can leak
out into some neighboring basin.
4.1 Neighborhood of an Extremum
In the neighborhood of a particular extremum (steady state) denoted by TJO), TJO), . .. ,
we may write (in function space) T(p,) = T(O) + rf>(Ji.), or the deviation may be written in
terms of its spectral components rf>o, rf>2, . ... Expanding F[T] about the local extremum,
we obtain
(25)
where the subscript zero denotes evaluation at the extremum. The terms linear in
locally a quadratic in rPn. The matrix elements
(26)
are the structure constants for the quadratic geometric surface, F(To, T2, • •• ). If all eigenvalues of Nnm are positive, the surface is concave upward; if one or more of the eigenvalues
are negative, the surface is locally a saddle point. We proceed to show that these eigenvalues are the stability eigenvalues studied earlier.
First note that if the temperature field is allowed to be a function of time, then by
following the approach for the zero dimensional models we have
(27)
That is, the time derivative is given by the gradient in this multidimensional space. For
infinitesimal departures from steady state we set Tn( t) = T~q + rPn€-)..t above, expand
about
- L::Nnmrf>m.
(28)
m
The latter equation concludes the proof that the local geometrical structure constants of
F(To, T2, • •• ) yield the stability eigenvalues for that particular steady state.
Finally, as a conclusion to this section, consider the time behavior of the value of
F(To, T2, • •• ) when the point (To, T2, ••• ) is governed by the time-dependent energy balance
equation
dF
L:: aF .
dt
aT Tn
n
n
= - L::(Tn)2
(29)
n
where we inserted the equation of motion. This latter result is the multidimensional analog
of one-dimensional slope-stability problem. It has a corresponding interpretation: initial
departures of the state (To, T2 , ••• ) from a local extremum of F lead to a trajectory of the
system point such that F decreases. The point will continue down the gradient of F until
a local extremum is found. Clearly, saddle points are unstable-the system point can leak
out into some neighboring basin.