The linearized MND-model
The MND model is
here, and the independent variables in it
are A(x,t), m(x,t), U(x,t), W(x,t) and G(x,t). The parameters, a and d,
are constants, w2 is gmG2/ápñ, and ápñ
is given by the analog of the caloric equation of state. These are accompanied by the
externally specified variables, Qr(x,t) and B(x,t). In a state of rest where
U=W=0 and G,A and m are the constants, g, _A and
_m, it follows that Qr is zero, and B is a constant
that can be set equal to zero in general. Thus, the basis of the linearization is
| U | ® | U | , |
W | ® | W | , |
G | ® | g+G | , |
A | ® | _A+A | , |
m ® | _m+m | , |
Qr | ® | Qr | , |
B | ® | B | , |
with _w2 = gg2/R_T,
d = 1-1/g and a = 1+d.
The linearized equations are here in
eqs1.
In this writing of the linear model, the independent parameters are g,
g, _m and _T, with
_B=0 and
_Qr=0, and the externally prescribed inputs
are B(x,t) and Qr(x,t).
For the case where B = 0 and Qr = 0, the homogeneous, linear system can be
written as
The introduction of the operators, kQ = -iQx and wQ = iQt then
gives the dispersion relation,
For this, the convenient dimensionless frequency and wave number are
v = w/_w
and k = (R_T/g)k = k_Z,
and the further relation,
a/g = 1-d2, is used in the derivation of
The result is shown
here. The case where the two branches almost touch
one another is an unphysical example where g is very slightly greater than one.
The essential features of this dispersion relation are:
On the upper branch,
-
|
w| ® _w as k ® 0,
-
(
w/k)2 ® gR_T as |k| ® ¥.
-
These are soundwaves that
will have to be filtered out of circulation models.
On the lower branch,
-
(
w/k)2 ® aR_T as k ® 0,
-
|
w| ® _w
___Öa/g
as |k| ® ¥.
-
These are edgewaves (including mountain waves) that may or may not have to be filtered out of various circulation
models. Their phase velocities decrease from the maximum _ slightly
less than soundspeed at k = 0
_ to zero as k
® ¥.
Ultimately, the waves with short wavelengths and high frequencies
will have to be filtered out of the
numerical algorithms for the nonlinear circulation models. Some hints
about how that can be done follow from considerations of analytic results
for the linearized models. A scheme that does not work, for
example, is to set W=0 (G=0 here, G=g in nonlinear models). Given the
observation that |G|/g (or |G-g|/g) is seldom more than a few percent, it
is an attractive approximation that leads to a considerable simplification
of both linear and nonlinear models. But _ the linear model's
dispersion relation is
The unacceptable properties of that are:
- The waves are neither soundwaves nor edgewaves, and
- there is no attenuation whatever of the high frequency waves.
The challenge is to find ways to suppress the upper branch of the dispersion
relation, without making qualitative changes of the lower one.
The crudest of schemes that actually works is to set
Gtt = 0 in the second of the pair of equations for U and G
(here). The resulting equation for U is
and the dispersion relation is
The longwave limit (k ® 0) is correct, as expected, and the shortwave
limit (|k| ® ¥) gives a limiting frequency that is too large
by the factor 7/2, but it is not qualitatively wrong.
The nonhomogeneous, nonlinear version of this filtering of fast waves is to
neglect the term proportional to DG/Dt in
A different approach to filtering the fast waves in both cases is to introduce
a fictitious relaxation time t and to replace G by G + tDG/Dt in
the first equation and W by W + tDW/Dt in the second. The linearized
version of this (G ® g + ^Gexp{st±ikx},
etc.) has as its dispersion relation for free waves (Qr = 0 and B = 0),
After the substitution, s = r±iw, the real and imaginary parts of the result are separately
zero, and it follows easily (details later) that when t ¹ 0 the only zero of r
is where w and k are both zero. Thus the damping of sound waves, which occurs for
t > 0, takes place for all waves, but the rate of damping approaches zero for the long
wave limit of edge waves. In that limit, where both s and k approach zero, the
expansion of s in powers of k follows from successive approximations of
The first iteration after
s = ±iÖ___aR_Tk is
(coming soon)
The serendipitous result is that the damping of waves we might prefer to have undamped
approaches zero as O(k4), rather than O(k2).
(More on the other limiting cases to follow _ maybe.)
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