A vertical structure model



When equations for (d/dt)lnr and (d/dt)Ñ·u are used in a vertical structure model, it contains soundwaves that will have to be filtered out of most circulation models. One way to do that is to introduce an artificial dissipation, proportional to some positive r(x,y,z,t) in

( d
__
dt		 +r ) Ñ·u = Ñ·( du
__
dt		 ) -ux·Ñu -uy·Ñv -uz·Ñw .


A different approach will be examined in this section. Chances are it will have soundwaves too, and ways to filter them out will have to be found. The case to be considered is the simplest one, where Note: J(x,z,t) is the temperature; its mean value is T(x,t).

The Euler equations for compressible flow are

rt+ (ru)x+ (rw)z= dr
__
dt		 +r(ux+wz) = 0.
dz
__
dt		 =w  and  du
__
dt		 +1
_
r		 px=0  and  dw
__
dt		 +1
_
r		 pz + g=0

In these and other equations like them, the subscripts, t,x and z, denote partial derivatives, and dq/dt=qt+uqx+wqz. Note the artificial transport equation for z ; it plays a crucial role in the determination of the pressure at the base of the atmosphere, where z=B(x,t).

The plan here is to use mass-weighted mean values and moments of 1, z, u, and w to give an approximate description of what the circulation, u(x,z,t) and w(x,z,t), would be if only we knew two of r(x,z,t), p(x,z,t) and J(x,z,t). There are several ways to proceed and they do give different results. In this, the thermodynamic fields are J(x,z,t), r(x,z,t) and p=rRJ. That leaves open the question of whether to use equations for (d/dt)lnr and (d/dt)(ux+wz) or

dw
__
dt		 +g+RJ pz
__
p		 =0


for the determination of the z-dependence of pressure. Either approach can be used, but not both at the same time, and they define slightly different sequences of approximations. The one to be pursued here is the second one, where

(lnp)z+ ·w+g
_____
RJ		 =0 .


The notation,  ·w (for dw/dt), is meant to imply the expansion in orthogonal polynomials

dw
__
dt		 =S ·wk(x,t)jk(x,~z,t) ,


where jo=1, j1=~z, and we desperately hope j2 will never be needed. Equations for the mean value and moments of dw/dt can all be found, and that plus the equations for the mean value and moments of J gives a local approximation, dependent upon x and t, where (lnp)z is a rational function of z. In the familiar example where Jz is a negative constant and  ·w is zero, the result is a polytropic equilibrium. It takes little more than a glance at the generic results to see that the evaluation of  ·wk for k>1 is insanely difficult. But there is a pony here, so bear with me.

From the Euler equation for w it follows that the moments of dw/dt can be evaluated in terms of moments of pz/r, and those turn out to be pretty easy. Let


pz/r = v(x,z,t) = Svkjk(x,~z,t) ,

where, as in the generic results,

B(x,t) < z < H(x,t) ,

árqñ(x,t)=   H
 ó
 õ
B		rq(x,z,t)dz ,

árñ=m , árzñ=mZ , ~z=z-Z ,


and the polynomials {jk} are orthogonal, with árjijjñ=0 for i¹j.

The general structure of all this can be seen by looking at the first two moments of v. First, let it be noted that from

rdq
__
dt		 +P=rs   with   dq
__
dt		 = qt+ uqx+ wqz


and the kinematic boundary conditions at z=H and z=B, it follows that

m D
__
Dt		 árjk2ñqk
_______
m		 +ár~ujkqñx +ájkPñ =árjk2ñsk +árqjk'(w -Zt-uZx)ñ


with DQ/Dt=Qt+UQt. The cases where jo=1 with q=1 and q=Z give

Dm
__
Dt		 +mUx=0   and   DZ
__
Dt		 +1
_
m		 (ár~z2ñu1)x =W .

These two are exact, and common to all orders of approximation. The first two moments of v are

mvo= árvñ=ápzñ =p(x,H,t) - p(x,B,t) = -pB(x,t) ,
and
ár~z2ñv1 =á~zpzñ =(Z-B)pB - ápñ =(Z-B)pB - mRT .


(The evaluation of v2 is not particularly difficult — just ugly.)

(CAUTION ! This is a mess, and it needs at least one more serious revision.)

These are accompanied by transport equations for moments of u, w and J, including

DW
__
Dt		 +1
_
m		 ár~u~wñx + g =G   with  pB =mG

and
m D
__
Dt		 ár~z2ñw1
_____
m		 +ár~z~uwñx + mG(Z-B)= mRT + árw(w-Zt-uZx)ñ .


Perhaps not the handsomest of ponies, but there it is: The equation for DZ/Dt defines W, and the equation for Dw1/Dt provides the relation between Z, G, and T that closes the circulation model. In the simplest case, where ~u and ~w are both arbitrarily set equal to zero, W=DZ/Dt, Z=B+RT/G and

DW
__
Dt		 + g = G   and   W = DB
__
Dt		 +RT
__
G		 ( DT
___
TDt		- DG
___
GDt		 ) .


In the model where

u(x,z,t)®U(x,t)+~zu1(x,t)   and   w(x,z,t)®W(x,t)+~zw1(x,t)


the approximation of dw/dt is

dw
__
dt		 = DW
__
Dt		 + ~zu1Wx + ~z( Dw1
___
Dt		 + ~zu1w1x) + w1 d~z
__
dt		 ,


and the approximation of d~z/dt is

d~z
__
dt		 = 1
_
m		 (ár~z2ñu1)x + ~z(w1-u1Zx) .


This is enough information to evaluate the numerator of

(lnp)z+ g+·wo+~z·w1
_________
R(T+~zJ1)		 =0 .


Finding the equations for T(x,t) and J1 involves other choices that are discussed here.