It was not long before the one-parameter model was replaced by a two-parameter model, since the vertical motion which is essential for any type of development in the atmosphere could not be computed with a single-level model. A two-parameter model allows not only the vorticity to vary in the vertical but also the thickness between two isobaric surfaces which depends upon the horizontal distribution of temperature. To derive the model, we divide the atmosphere into two discrete layers, bounded by pressure surfaces marked 0, 2 and 4, as shown in Fig. 17.1.

We now apply the vorticity equation (17.4.2) at levels 1 and 3 representing isobaric surfaces 250 mb and 750 mb which mark the middle of the upper and the lower layer respectively. To do this, we have to evaluate the divergence term da/dp at each of these pressure surfaces using finite-difference approximation to the vertical derivatives. Thus,

(da/dp)1 ~ (a - ®o)/Ap, (da/dp)3 ~ (a - ®2)Ap where the suffixes denote the designated levels and Ap is the pressure interval between the levels 0-2, 1-3, and 2-4. If we now assume that a = 0 at the top and the bottom of the atmosphere, we can write the vorticity equations as d(V2¥i)/dt = -(kxVy1)-V(V2y1 + f) + (f0/Ap)ffl2 (17.5.4)

d(V2V3)/dt = -(kxVy3)-V(V2y3 + f) - (f0/Ap)ffl2 (17.5.5)

Next, we write the thermodynamic energy equation (17.4.9) for level 2. After evaluating d\y/dp by the finite-difference (¥3 - ¥i )/Ap, the equation is d(¥t -¥3)/dt = -(kxVv2)-V(Vx - V3) + (° Ap^/fo (17.5.6)

where ¥2 = (¥1 + ¥3)/2 is the stream function at 500 mb, which is not a predicted variable but determined by the predicted values of ¥1 and ¥3. Using this value of ¥2 in terms of ¥1 and ¥3, we obtain a closed set of equations in ¥1, ¥3, and a2.

We now eliminate a2 between (17.5.4) and (17.5.6) to obtain two equations in ¥1 and ¥3 alone.

First we add (17.5.4) and (17.5.5) to eliminate a2 and obtain

dV2(Vi + y3)/dt = — (kxV^i)^V(V2Vi + f) — (kxVy3>V(VV3 + f) (17.5.7)

We now introduce a length scale X—1, where = f02/{o(Ap)2}.

We next subtract (17.5.5) from (17.5.4) and add the result to —2X2 times (17.5.6) to obtain d{(V2 — 2^2)(¥1 — ¥3)}/dt = — (kxV¥1)-V(VV + f) + (kxV¥3)^V(V2¥3 + f)

The physical interpretation of (17.5.7) is simple. It states that the local rate of change of the vertically-averaged vorticity between 250 mb and 750 mb is given by the vertical average of the horizontal advections of vorticity at the two pressure surfaces. Thus, it represents the barotropic part of the flow vorticity. The baroclinic part is represented by (17.5.8) which states that thickness tendency at a point is determined partly by the difference in vorticity advections between 250 mb and 750 mb and partly by the thermal advection by the mean nondivergent wind between the two pressure surfaces.

We can derive an expression for vertical motion m2 by combining (17.5.4) and (17.5.6) and eliminating the time tendencies. For this, we first operate on (17.5.6) with V2 and add the result to the difference between (17.5.4) and (17.5.5). Rearranging the terms, we obtain

— (kxV^1).V(V2^1 + f) + (kxVy3>V(VV3 + f)] (17.5.9)

The two-level model as introduced here has not been found to be very useful in NWP, since it tended to produce stronger baroclinic development than observed. However, it remains a useful tool for the analysis of physical processes occurring in baroclinic disturbances. Multi-level baroclinic models and higher horizontal resolution used in later years led to improved forecasts.

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