AIDJEX Bulletin #40 - Polar Science Center - University of Washington

in which there is shearing but no extending since these lie at the origin
of the horizontal axis that is a measure of,normal stretching.
is shown as the angle from the x-axis to point c and appears as 2Cp in the
Mohr's circle.
axis and are at +(Cp-y).
The angle Cp
Points c and c' are symmetric about the principal stretching
Geometrically, it is clear that when IDI I > DII
the circle does not intersect the origin, and so there cannot be directions
of no stretching. It is also clear that when lDIl = DII the two roots for
4 coincide providing only one direction of no stretching. These special
cases occur when the equations are elliptic and parabolic, respectively.
The General Non-Normal Flow Rule Case
When advection is included in the momentum equation, the eigenvectors
are somewhat different from the simplest uncoupled system. Our attention
returns to equations (59) and (60) to define the eigenvectors R a = 1, 2,
-a
We note first that the eigenvectors associated with the veZocity
..., 4.
characteristics (a = 3,4) are unchanged by including advection. These are
defined by det C = 0 and for distinct roots (non-normal flow rule) require
-a
that det c 0. As before, we obtain R = 9. Then S satisfies (59) or
-a -a -a
(67) and is given by (78). The equation governing the solution along these
characteristics remains the same (93) so that the velocity components still
satisfy the simple relationships.
as given in (94).
The roots are also unchanged and remain
A similar consideration of the equations governing solutions along
stress characteristics shows changes. The stress characteristics (a = 1,2)
are defined by det ?? = 0 and det # 0. Thus, from equation (60) we find
-a
the eigenvectors R to be unchanged from the uncoupled case and given by
-a
equations (73) and (74). But by (59) we find that S is no longer zero.
-a
Instead, we have
since C is nonsingular. Substituting into equation (66) and evaluating
--a
all coefficients provide the equations governing solutions along these
characteristics. Tt is seen that the forcing function and coefficient of
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