Solving the Vialov Equation of Glaciology in Terms of Elementary Functions

Abstract

Very few exact solutions are known for the non-linear Vialov ordinary differential equation describing the longitudinal profiles of alpine glaciers and ice caps under the assumption that the ice deforms according to Glen’s constitutive relationship. Using a simple, yet wide, class of models for the accumulation rate of ice and Chebysev’s theorem on the integration of binomial differentials, many new exact solutions of the Vialov equations are obtained in terms of elementary functions.

Appendices

Appendices

1.1 Exact Solutions of the Vialov Equation for Some Rational Values of r

In the range of parameters $\left( p, q, r \right)$ in which the Chebysev theorem is satisfied, computer algebra easily provides the integral (6) for the choice (10) of c(x). Here some of these integrals and the corresponding longitudinal glacier profiles are reported for various values of the parameter r listed in Eq. (10) (in decreasing order), which correspond to values of $m_0$ reported in Eq. (15).

$$\begin{aligned} r= & {} \frac{4}{3} \,, \end{aligned}$$

(A.1)

$$\begin{aligned} V(x)= & {} \frac{9\left( a+b \,x^{4/3}\right) ^{4/3}}{16b} +D \,, \end{aligned}$$

(A.2)

$$\begin{aligned} h(x)= & {} h_0 \left[ \left( a+b \,x^{4/3}\right) ^{4/3}+D\right] ^{3/8} \,, \end{aligned}$$

(A.3)

where D is, as usual, an integration constant. For $D=0$ one obtains

$$\begin{aligned} h(x) = h_0 \left( a+b \,x^{4/3}\right) ^{3/8} \end{aligned}$$

(A.4)

and, if also $a=0$, one obtains again $h(x)=h_0 \sqrt{x}$. Another possibility is

$$\begin{aligned} r= & {} \frac{2}{3} \,, \end{aligned}$$

(A.5)

$$\begin{aligned} V(x)= & {} \frac{9\left( a+b \, x^{2/3}\right) ^{1/3} \left( -3a^2+abx^{2/3}+4b^2 x^{4/3} \right) }{56b^2} +D \,, \end{aligned}$$

(A.6)

$$\begin{aligned} h(x)= & {} h_0 \left[ D_0 +\left( a+b \, x^{2/3}\right) ^{1/3} \left( -3a^2+abx^{2/3}+4b^2 x^{4/3} \right) \right] ^{3/8}\,, \end{aligned}$$

(A.7)

where $D_0$ is another constant. Other possibilities are

$$\begin{aligned} r= & {} \frac{4}{9} \,, \end{aligned}$$

(A.8)

$$\begin{aligned} V(x)= & {} \frac{27}{560b^3} \Bigg ( a+b \, x^{4/9}\Bigg )^{1/3} \nonumber \\&\times \Bigg ( 9a^3 -3 a^2 b x^{4/9} +2 a b^2 x^{8/9} +14 b^3 x^{4/3} \Bigg ) +D \,, \end{aligned}$$

(A.9)

$$\begin{aligned} h(x)= & {} h_0 \Bigg [ D_0 + \Bigg ( a+b \, x^{4/9}\Bigg )^{1/3}\nonumber \\&\times \Bigg ( 9a^3 -3 a^2 b x^{4/9} +2 a b^2 x^{8/9} +14 b^3 x^{4/3} \Bigg )\Bigg ]^{3/8} \end{aligned}$$

(A.10)

$$\begin{aligned} r= & {} \frac{1}{3} \,, \end{aligned}$$

(A.11)

$$\begin{aligned} V(x)= & {} \frac{9}{1820b^4} \Bigg ( a+b \, x^{1/3}\Bigg )^{1/3} \nonumber \\&\times \Bigg ( -81a^4 +27 a^3 b x^{1/3} -18 a^2 b^2 x^{2/3} +14 ab^3 x +140 x^{4/3} \Bigg ) \nonumber \\&+D \,, \end{aligned}$$

(A.12)

$$\begin{aligned} h(x)= & {} h_0 \Bigg [ D_0 + \Bigg ( a+b \, x^{1/3}\Bigg )^{1/3}\nonumber \\&\cdot \Bigg ( -81a^4 +27 a^3 b x^{1/3} -18 a^2 b^2 x^{2/3} +14 ab^3 x +140 x^{4/3} \Bigg ) \Bigg ]^{3/8} \,; \end{aligned}$$

(A.13)

$$\begin{aligned} r= & {} \frac{4}{15} \,, \end{aligned}$$

(A.14)

$$\begin{aligned} V(x)= & {} \frac{9}{5824 b^5} \Bigg ( a+b \, x^{4/15}\Bigg )^{1/3} \Bigg ( 243a^5 -81a^4 b x^{4/15} +54 a^3 b^2 x^{8/15} \nonumber \\&-42 a^2 b^3 x^{4/5}+35 a b^4 x^{16/15} +455 b^5 x^{4/3} \Bigg ) +D \,, \end{aligned}$$

(A.15)

$$\begin{aligned} h(x)= & {} h_0 \Bigg [ D_0+ \Bigg ( a+b \, x^{4/15}\Bigg )^{1/3} \Bigg ( 243a^5 -81a^4 b x^{4/15} +54 a^3 b^2 x^{8/15} \nonumber \\&-42 a^2 b^3 x^{4/5} +35 a b^4 x^{16/15} +455 b^5 x^{4/3} \Bigg ) \Bigg ]^{3/8}; \end{aligned}$$

(A.16)

$$\begin{aligned} r= & {} \frac{2}{9} \,, \end{aligned}$$

(A.17)

$$\begin{aligned} V(x)= & {} \frac{27}{55328b^6} \Bigg ( a+b \, x^{2/9}\Bigg )^{1/3} \nonumber \\&\times \Bigg ( -729a^6 +243 a^5 b x^{2/9} -162 a^4 b^2 x^{4/9} +126 a^3b^3 x^{2/3} \nonumber \\&-105 a^2b^4 x^{8/9} +91ab^5 x^{10/9} +1456 b^6x^{4/3} \Bigg ) +D, \end{aligned}$$

(A.18)

$$\begin{aligned} h(x)= & {} h_0 \Bigg [ D_0 + \Bigg ( a+b \, x^{2/9}\Bigg )^{1/3} \nonumber \\&\times \Bigg ( -729a^6 +243 a^5 b x^{2/9} -162 a^4 b^2 x^{4/9} +126 a^3b^3 x^{2/3} \nonumber \\&-105 a^2b^4 x^{8/9} +91ab^5 x^{10/9} +1456 b^6x^{4/3} \Bigg ) \Bigg ]^{3/8} , \end{aligned}$$

(A.19)

$$\begin{aligned} r= & {} \frac{4}{21} \,, \end{aligned}$$

(A.20)

$$\begin{aligned} V(x)= & {} \frac{9}{173888 b^7} \Bigg ( a+b \, x^{4/21}\Bigg )^{1/3} \nonumber \\&\times \Bigg ( 6561 a^7 -2187 a^6 b x^{4/21} +1458 a^5 b^2 x^{8/21} -1134 a^4 b^3 x^{4/7} \nonumber \\&+945 a^3 b^4 x^{16/21} -819 a^2 b^5 x^{20/21} +728 ab^6 x^{8/7} +13832 b^7 x^{4/3} \Bigg ) +D,\nonumber \\ \end{aligned}$$

(A.21)

$$\begin{aligned} h(x)= & {} h_0 \Bigg [ D_0+ \Bigg ( a+b \, x^{4/21}\Bigg )^{1/3} \nonumber \\&\times \Bigg ( 6561 a^7 -2187 a^6 b x^{4/21} +1458 a^5 b^2 x^{8/21} -1134 a^4 b^3 x^{4/7} \nonumber \\&+945 a^3 b^4 x^{16/21} -819 a^2 b^5 x^{20/21} +728 ab^6 x^{8/7} +13832 b^7 x^{4/3} \Bigg ) \Bigg ]^{3/8}, \end{aligned}$$

(A.22)

$$\begin{aligned} r= & {} \frac{1}{6} \,, \end{aligned}$$

(A.23)

$$\begin{aligned} V(x)= & {} \frac{9}{543400 b^8} \Bigg ( a+b \, x^{1/6}\Bigg )^{1/3} \nonumber \\&\times \Bigg ( -19683 a^8 +6561 a^7 b x^{1/6} -4374 a^6 b^2 x^{1/3} +3402 a^5 b^3 \sqrt{x} \nonumber \\&-2835 a^4 b^4 x^{2/3} +2457 a^3 b^5 x^{5/6} -2184 a^2 b^6 x \nonumber \\&+1976 a b^7 x^{7/6} +43472 b^8 x^{4/3} \Bigg ) +D, \end{aligned}$$

(A.24)

$$\begin{aligned} h(x)= & {} h_0 \Bigg [ D_0+ \Bigg ( a+b \, x^{1/6}\Bigg )^{1/3} \Bigg ( -19683 a^8 +6561 a^7 b x^{1/6} \nonumber \\&\times -4374 a^6 b^2 x^{1/3} +3402 a^5 b^3 \sqrt{x} \nonumber \\&\, -2835 a^4 b^4 x^{2/3} +2457 a^3 b^5 x^{5/6} -2184 a^2 b^6 x \nonumber \\&\times +1976 a b^7 x^{7/6} +43472 b^8 x^{4/3} \Bigg ) \Bigg ]^{3/8}{.} \end{aligned}$$

(A.25)