The MD model
In the local tangent space at the latitude f and longitude l,
the z-axis is normal to the geoid, the x-axis is directed eastward and the
y-axis is directed northward. The coriolis parameter is
¦(y)=2Wsinf. As conservation laws,
the equations of nondissipative gasdynamics are:
eqs1
Though seldom seen in fluid dynamics texts, the second of these is a key element of
some shallow water and shallow atmosphere approximations.
The mechanical equations are accompanied by thermal equations, and the simplest of
thermal models is a perfect gas, where R is a constant in the
equation of state, p=rRJ, and cv is a constant in the
equation for the specific internal energy, e=cvJ.
(Note that J has been introduced for the temperature; T will be
used presently for the mean
value of J.) The total energy density (per unit mass) that appears in the
last of the conservation laws is
| e(x,y,z,t) | = |
1
_
2 |
(u2+v2+w2) |
+ | gz | + | cvJ. |
Also in that equation, k is a thermal conductivity, and qr is the radiant
heat source (per unit mass).
The MD model is a shallow layer approximation that makes use of the equations
of motion for quantities Q(x,y,t) that are mean values of other quantities q(x,y,z,t).
The definitions of terms are:
p, r and J are zero at z=H(x,y,t),
áqñ is the integral of q(x,y,z,t)dz, from B(x,y,t) to H,
m(x,y,t) is árñ (not a mean value),
Q(x,y,t) is árqñ/m (mass-weighted mean values),
~q(x,y,z,t) is q-Q (deviations that will be neglected),
J(x,y,z,t) is T(x,y,t)+~J (special notation).
Details of the derivation of approximate conservation laws for
m(x,y,t) and
the mean fields will be given elsewhere (in a simpler context). The set of
two-dimensional conservation laws that is inherited from the
three-dimensional ones is:
eqs2
The terms that look like Reynolds stress have been left in the Z-equation just
to show where many of the terms left out of MD are; as displayed, the
Z-equation is exact. When those deviations are neglected too,
the results are almost perfect two-dimensional copies of the three-dimensional
equations. The
term pB is pressure at z=B(x,y,t), and pH is zero.
The lumped heat source in the
energy equation is just the integral of the r.h.s. of the 3-d energy equation.
If heat conduction is assumed to be unimportant, then Qr is the
mean value of qr.
And now, all that remains is to reinvent a small part of gasdynamics, in a
rotating flatland approximation of rotating sphereland and with a few
new wrinkles.
To begin with strict analogies,
and, when deviations are neglected in the evaluation of
mE=áreñ,
Further analogies that are almost perfect appear in the rewriting of the field equations
as transport equations in which D/Dt is the directional derivative,
¶/¶t+U¶/¶x+V¶/¶y:
eqs3
The new symbol G, which is pB/m, will be called the
effective gravity.
The first of the wrinkles comes now, as a consequence of neglecting deviations in
general, and setting ~w=0, in particular. From
the vertical momentum equation of gasdynamics, it follows then that
The wrinkle is not that there is an equation for pz, but that the
pressure is hydrostatic in the accelerated coordinate system. In fact,
G seldom differs from g by more than a few percent, but replacement of G by g
gives a theory whose waves are neither soundwaves nor gravity waves, with a
dispersion relation that is qualitatively wrong in both cases. As already
noted, pH=0 implies pB=mG, and from the
first moment of the pressure equation,
ázpzñ + mGZ = 0 |
® |
ápñ = mG(Z-B) = mRT. |
After Z has been eliminated, the coupled transport equations for W and G are
The last term in the second equation is DB/Dt, and the notation has been introduced
to distinguish a term that is to be evaluated from others that contain
D/Dt. The quantity G2/RT is a shallow atmosphere approximation
of the square of the Brunt-Väisälä frequency; the corresponding time-scale is
a very short one (minutes), and it will become necessary to
filter effects connected with it out of circulation models.
From here, it is a matter of carrying out the familiar process, which is to use the
evaluations of Dm/Dt, DU/Dt and DV/Dt to convert the equation for the total
energy density E to an equation for the thermal energy density, cvT.
The wrinkle is that equations for DZ/Dt (eliminated) DW/Dt and DG/Dt are
included in the derivation. The
result can be expressed in several ways, and the one that has been chosen is to
eliminate DT/Dt. Instead, we look for the analog of
the caloric equation of state, which in gasdynamics is
p=arg where g=cp/cv.
Details will be given elsewhere; the result is
eqs4
The parameters are
For a diatomic gas, a is 9/7 and d is 2/7.
(All the equations on the popups are in eqns.dvi.)
A few notes of caution:
- This approximation still has sound waves that will have to be filtered
out of any reasonable circulation model. Depending on the application, the internal
(gravity) waves and the inertial waves, due to the presence of
¦, may have to
be filtered out, as well.
- This is a barotropic model. At least the first moments or some other treatment
of z-dependence of the various fields will be needed for the description of the
baroclinic instabilities that figure prominently in weather patterns.
- Sooner or later, the perfect, dry atmosphere will have to be abandoned too.
(Much more to follow.)