Thus, at some time during the formation process, the bottom must be melting to compensate for the deposi-

tional growth but the process cannot be steady.

Let us fix the upper surface so that it remains stationary, as shown in Figure 14. Then it would appear

that a steady flow of material is moving at a constant velocity *U *and the original soil surface seems to be

moving downward at a steady velocity. The energy equation is transformed such that a convective term ex-

ists. The equations for regions 1 and 2 are

2

αi

-*U * i - i = 0

(25)

The boundary and initial conditions are exactly the same as those of the heterogenetic case (see Appendix

C for details of the syngenetic equations).

The heat balance integral form of the energy equations is

{ρ1c1 T1( x, *t*)dx + ρ2c2

∫ T2 ( x, *t*)dx - ρ1lX + (ρ2c2 - ρ1c1)Tf X

∫

0

(26)

-ρ2c2 ( X + δ)*T*o + ( X + δ) = -*k*1 1

[

]

+ *k*2G - ρ1c1U∆*T*1 - ρ2c2U ∆*T *+ *G*( X + δ) .

2

We note that this is identical to the relation for heterogenetic growth, eq 5, except for the two additional

terms on the right-hand side of eq 26. Carrying out the integrations and making eq 26 nondimensional,

leads to the following result.

σ

τ = ∫ K1dσ

(27)

0

1

σ*g*′

2

* C*21 σ + φ σβ′

(

)

- *C*21σ β + 1 -

1 -

6*g *

3

3

(28)

1 1

{

]}

[

- 2 + *k*21 - ψ 1 + 1 / *S*T + *C*21 φ + σ(β + 1)

σ*g*

where

13