Hillslope process-response models based on the continuity equation

M. J. Kirkby

目錄介紹模式架構 ◇連續方程式與運輸條件 ◇特徵形式結論

介紹 本篇的研究目的是檢視一系列的邊坡過程發展反應模式是否能合理應用，並具體說明實地測量的經驗公式運用於斜坡過程模式。

連續性方程式是任何邊坡反應模式的基礎，土石運輸速率是連續性方程的重要形式，該研究的關鍵點是在地勢起伏的變化。

解連續性方程式除了關鍵的坡度和與流域距離的變化量外，還需要其他條件：

(1) 邊坡剖面的初始形態 (2) 邊界條件

(A) The continuity equation:

M is the rate of mechanical lowering, D is the rate of chemical lowering, y is the elevation, t is time elapsed.

t

yDM

模式架構 ◇連續方程式與運輸條件

(B) The relationship between mechanical lowering and mechanical transport:

indicates the vector divergence, S is the vector sediment transport.

SM

(C) The relationship between rate of lowering and soil thickness:

z is the soil depth W is the rate of lowering of the soil-bedrock

interface

Wt

y

t

z

slope development

物理和化學的降低速率與岩石土壤界線的關係式可表示如下：

土壤發育程度遠小於物理降低速率，因此本文只考慮物理搬運能力與邊坡發展關係

WM WD )1(

DM

M

μ表示岩石不風化的比例

(D) The relationship between the rate of weathering, S, and the transporting capacity of the process, C.

(i) Transport limited removal: C = S 堆積(ii) Weathering limited removal: C>>S 沖刷(iii) Erosion-limited removal:

)( SCkt

y

(E) The boundary conditions. (i) At x=0, S=0 and y is a maximum (ii) At x=x1 , y is a function of time alone

(x1 fixed in horizontal position at distance)

(F) The initial form will usually be taken to be a straight slope.

(G) The transport law or process law is specified by the transport capacity, C.

a is the area drained per unit contour length f(a) is a positive function of a n is a constant exponent describing the influence

of increasing gradient.

n

t

yafC

)(

in the case where the exponent of slope n = I, and for a fixed base level, that there exists a solution to the continuity equation of the form

U, V are functions of x alone T is a function of time alone.

T(t) V(x). U(x)y

模式推導 ◇特徵形式

(a) The continuity equation is taken in the form

0

t

yS

x

S

WM

S

x

SSM

Wt

y

t

z

── 近似解

T(t) V(x). U(x)y

0

t

yS

x

S

boundary conditions

Applying the inequality

20

10

2

1

0 2

11

2

1cos

2

xVx

xVeV

x

00 xVV

tan1 xxxU

m 1≧ ，凹坡 m 0≦ ，凸坡

同一研究區域的邊坡剖面假設都遵循一樣的過程定律，則可由坡度和與流域的距離推導出運輸能力的關係式。

k

VdxVC

x

0

結論 本文試圖探討模式與邊坡演化過程之間的關係，如連續性方程式與經驗條件之關係。

連續方程式可描述地形的變遷，可求得地形的剖面。

謝謝玲聽