As before the associated energy equation may be obtained by forming the combination:
This gives
In view of (5.1), the coefficients of the operations ( 
) commute wit the operator.
Moreover, in view of (5.3), terms like 
can be expressed
as 
.
Accordingly, (5.15) takes the form
where
and
This reduces to (4.21) for the case 
(for which
).
The term 
can redistribute the energy 
but cannot
cause any net change of the total energy, at least for a closed
system.
In the general case 
is normal to the surfaces of 
and 
is normal to the surfaces of 
and have
different slopes.
If the space/time average of 
is positive it will tend to increase .
When this occurs the basic state is said to be baroclinically
unstable; it can occur even if 
is positive and hence
is a dynamic instability related to the non-linearity of
the basic equations.
That and differ is a necessary but not a sufficient
condition. If the instability occurs, the source of the energy
increase of the perturbations must be from the basic state and
in particular from the potential energy associated with
the tilted surfaces.
A second possible source of instability is related to the gradient
of the basic state velocity field.
The term 
when represented in Cartesian indicial form is
where
The time average of 
represents a general
Reynold's stress associated with the perturbations; it may
include a contribution by wave modes as well as by turbulent
vortex modes.
The term 
is the deformation tensor of the
basic flow regime which contains both shear and normal components.
For the basic flow under consideration, it is primarily horizontal.
Let 
be the magnitude of 
and let s and n denote
distance along and to the left of . Then, since
is assumed negligible, the
dominant terms in are
where
the lateral flux of longitudinal momentum, and
the vertical flux of longitudinal momentum.
If 
is positive then a shear instability
is said to exist.
This may be due to the vertical shear 
(Kelvin-Helmholtz instability) or the horizontal shear
or both.
In a barotropic fluid in the absence of viscous effects, only
the horizontal shear can exist and if 
is positive then a barotropic instability
is said to exist.
In general, the source of perturbation energy due to shear
instability is from the kinetic energy of the basic state.
Obviously if either shear instability or baroclinic instability exists, then the energy of the basic state must change, which implies that it cannot remain in steady state. This implies in turn that the coefficients in the perturbation equations are slowly changing functions of time and that the terms responsible for the instability are undergoing an evolution in time, and may (under certain circumstances) vacillate from an unstable to a stable state.