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GEOPHYSICS
Kh. N. ZEITUNYAN
ON A NONLINEAR THEORY OF UNSTEADY SLOPE WIND
(Presented by Academician L. I. Sedov, 19 IV 1960)
In the present work we consider the plane unsteady problem of a wind arising over an inclined thermally homogeneous surface. Introduce a curvilinear coordinate system \((x, z)\) associated with the relief (the \(x\)-axis is directed along the line of relief, \(z\) upward). For our problem a typical system of equations will be \((1)\)
\[ \begin{gathered} \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + w\frac{\partial u}{\partial z} = \lambda \sin\alpha\,\vartheta + \frac{\partial}{\partial z}\,\nu\,\frac{\partial u}{\partial z} \quad \left(\lambda=\frac{g}{T}\right); \\[4pt] \frac{\partial \vartheta}{\partial t} + u\frac{\partial \vartheta}{\partial x} + w\frac{\partial \vartheta}{\partial z} = -\mu \sin\alpha\,u + \frac{\partial}{\partial z}\,\chi\,\frac{\partial \vartheta}{\partial z} \quad \left(\left(\mu=\gamma_a+\frac{dT}{dz}\right)\right); \\[4pt] \frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}=0. \end{gathered} \tag{1} \]
Here \(t\) is time; \(u, w\) are the components of the wind velocity, respectively, along the axes \(x, z\); \(\vartheta\) is the deviation of the temperature from \(T\), its value in the resting atmosphere; \(g\) is the acceleration due to gravity; \(\gamma_a\) is the dry-adiabatic gradient; \(\alpha=\alpha(x)\) is the angle of inclination of the lines of relief (the slope) to the horizontal surface; \(\nu, \chi\) are the coefficients of vertical turbulent viscosity and thermal conductivity, respectively.
Put \(\chi=\nu=\text{const}\). The conditions of the problem will be
\[ \begin{gathered} u=\vartheta=0 \quad \text{for } t=0 \text{ and for } z=\infty; \\ u=w=0,\quad \vartheta=f(t) \quad \text{for } z=0, \end{gathered} \tag{2} \]
where \(f(t)\) is the prescribed temperature of the slope surface.*
Let us now proceed to the solution of the problem, first putting \(\lambda=\text{const}\), \(\mu=\text{const}\). In order to reduce the problem to ordinary differential equations, we shall use a device analogous to that employed by A. A. Dorodnitsyn.
Denote
\[ \xi=\frac{x}{L},\qquad \zeta=\frac{1}{2}\sqrt[4]{\frac{\lambda\sin\alpha_0\theta_{00}}{\nu^2 L}}\, \frac{z}{\tau},\qquad \tau=\sqrt[4]{\frac{\lambda\sin\alpha_0\theta_{00}}{L}}\,\sqrt{t}, \]
where \(L, \alpha_0\), and \(\theta_{00}\) are, respectively, a characteristic length, characteristic slope, and characteristic value of the slope temperature.
If now \(f(t)\) is represented in the form
\[ f(t)=\theta_{00}F(t)=\theta_{00}\sum_{n=0}^{\infty} F_n \tau^{n+2}, \tag{3} \]
* The solution obtained below can easily be generalized to the case when \(\vartheta_{z=0}\), being a function of the coordinates and time, is determined from the condition of heat balance at the slope surface.
where \(F_n\) are numerical coefficients, one may seek the solution of system (1) in the form*
\[ \vartheta=\theta_{00}\sum_{n=0}^{\infty}\theta_n\tau^{n+2},\qquad u=\sqrt{\lambda\sin\alpha_0 L\theta_{00}}\sum_{n=0}^{\infty}U_n\tau^{n+4}, \tag{4} \]
\[ w=2\sqrt[4]{\frac{\nu^2}{L}\lambda\sin\alpha_0\,\theta_{00}}\, \sum_{n=0}^{\infty}W_n\tau^{n+5}, \]
where \(\theta_n\), \(U_n\), and \(W_n\) are functions depending on \(\xi\) and \(\zeta\) and satisfying the infinite system
\[ \Lambda_{n+4}(U_n)=-4\chi\theta_n+ 4\sum_{k=0}^{n-6}\left( U_k\frac{\partial U_{n-6-k}}{\partial\xi} + W_k\frac{\partial U_{n-6-k}}{\partial\zeta} \right); \tag{5} \]
\[ \Lambda_{n+2}(\theta_n)=4\Gamma\chi U_{n-4}+ 4\sum_{k=0}^{n-6}\left( U_k\frac{\partial\theta_{n-6-k}}{\partial\xi} + W_k\frac{\partial\theta_{n-6-k}}{\partial\zeta} \right); \tag{6} \]
\[ \frac{\partial U_n}{\partial\xi}+\frac{\partial W_n}{\partial\zeta}=0, \tag{7} \]
where the notation** is
\[ \Lambda_m(y_n)=y_n''+2\zeta y_n'-2m y_n,\qquad \chi=\frac{\sin\alpha}{\sin\alpha_0},\qquad \Gamma=\frac{\mu}{\theta_{00}}\sin\alpha_0 L. \tag{8} \]
The system (5), (6), (7) must be solved under the boundary conditions
\[ U_n=\theta_n=0\quad \text{for } \xi=\infty;\qquad U_n=W_n=0,\quad \theta_n=F_n\quad \text{for } \xi=0, \tag{9} \]
which are a consequence of (2).
From (6), for \(n<4\), we obtain for \(\theta_n\) the homogeneous equation
\[ \Lambda_{n+2}(\theta_n)=\theta_n''+2\zeta\theta_n'-2(n+2)\theta_n=0. \tag{10} \]
Under the boundary conditions (9), equation (10), as is known \((^2)\), has the solution
\[ \theta_n=F_n L_{n+2}(\zeta). \tag{11} \]
The functions \(L_m\) were introduced in \((^3)\) and have the form
\[ L_m(\zeta)=\frac{A_m}{m!}\int_{\infty}^{\zeta}(\zeta-\zeta')^m\exp(-\zeta'^2)\,d\zeta', \tag{12} \]
\[ (A_m=2mA_{m-2},\qquad A_0=-2/\sqrt{\pi},\qquad A_1=2,\qquad L_m(0)=1,\qquad L_m(\infty)=0). \]
Next, up to \(n<6\), for \(\theta_n\) and \(U_n\) one must solve an inhomogeneous equation of the form
\[ \Lambda_m(y_n)=\sum_{k=0}^{r}a_k L_k, \tag{13} \]
* The convergence of the series depends on the decrease of the coefficients \(\theta_n\), \(U_n\), and \(W_n\) and is not rigorously justified by us. However, the series for small moments of time (at least up to \(\tau\approx1\)) apparently converge asymptotically.
** A prime denotes derivatives with respect to \(\zeta\). We shall see that the system (5), (6), (7) depends on \(\xi\) only parametrically; therefore in the left-hand sides of equations (5) and (6) we have an ordinary linear operator.
where the coefficients \(a_k\) depend on \(\xi\). Taking into account that
\[ \Lambda_m(a_k L_k)=2a_k(k-m)L_k, \]
we obtain the solution of (13), under the boundary conditions (9), in the form
\[ y_n=F_nL_m+\sum_{k=0}^{r}\frac{a_k}{2(k-m)}(L_k-L_m). \tag{14} \]
Beginning with \(n\geqslant 6\), nonlinear terms appear in (5) and (6). For the problem under consideration, the right-hand side of the nonhomogeneous equation (for \(n\geqslant 6\)) will be combinations of functions \(L_m\) with lower indices. For example, beginning with \(n=6\) up to \(n=11\), it is necessary to solve a nonhomogeneous equation of the form
\[ \Lambda_m(y_n)=\sum_{i,j} b_{i,j}L_iL_j, \tag{15} \]
where \(b_{i,j}\) depend on \(\xi\).
Using the properties of the functions \(L_m\), one can establish that
\[ \Lambda_m\left(\frac{b_{i,j}}{2}\left[ \frac{A_iA_j}{A_{i+1}A_{j+1}}L_{i+1}L_{j+1} +\sum_{k=2}^{r}2^{k-1}(r-1)(r-2)\ldots(r-k+1) \frac{A_iA_j}{A_{i+k}A_{j+k}}L_{i+k}L_{j+k} \right]\right)=b_{i,j}L_iL_j, \]
where \(*\) \(2r=m-(i+j)\).
Then the solution of (15), under the boundary conditions (9), will be
\[ y_n=F_nL_m+\sum_{i,j}\frac{b_{i,j}}{2}\left\{ \frac{A_iA_j}{A_{i+1}A_{j+1}}(L_{i+1}L_{j+1}-L_m)+ \right. \]
\[ \left. +\sum_{k=2}^{r}2^{k-1}(r-1)(r-1)\ldots(r-k+1) \frac{A_iA_j}{A_{i+k}A_{j+k}}(L_{i+k}L_{j+k}-L_m)\right\}. \tag{16} \]
Thus, using the solutions (11), (14), and (16), one can determine \(\theta_n\), \(U_n\), and \(W_n\) up to and including \(n=11\)**. In order not to encumber the calculations, we shall consider the solution of the system (5), (6), (7), under the boundary conditions (9), for the first 10 indices \((n\leqslant 9)\). Then we finally obtain
\[ \theta_n=F_n\theta_{n,0}-\Gamma\chi^2F_{n-4}\theta_{n,1} +\Gamma^2\chi^4F_{n-8}\theta_{n,2} -\sum_{k=0}^{n-6}F_{n-6-k}F_k\frac{\partial\chi}{\partial\xi}\theta_{n,k+3}; \tag{17} \]
\[ U_n=\chi\left\{F_nU_{n,0}-\Gamma\chi^2F_{n-4}U_{n,1} +\Gamma^2\chi^4F_{n-8}U_{n,2} -\sum_{k=0}^{n-6}F_{n-6-k}F_k\frac{\partial\chi}{\partial\xi}U_{n,k+3}\right\}; \tag{18} \]
\[ W_n=-\frac{\partial\chi}{\partial\xi} \{F_nW_{n,0}-3\Gamma\chi^2F_{n-4}W_{n,1} +4\Gamma^2\chi^3F_{n-8}W_{n,2}\} + \]
\[ +\sum_{k=0}^{n-6}F_{n-6-k}F_k \frac{\partial}{\partial\xi}\left(\chi\frac{\partial\chi}{\partial\xi}\right)W_{n,k+3} \quad (F_{-n}\equiv0,\ n\leqslant9), \tag{19} \]
\[ \text{* We note that for the problem under consideration the condition } m-(i+j)=2r \text{ is always satisfied.} \]
\[ \text{** The determination of further terms of the expansions presents no fundamental or technical difficulties if the right-hand sides of the corresponding equations are approximated by simple combinations of the functions } L_m. \]
where \(\theta_{n,j}, U_{n,j}\), and \(W_{n,j}\) \((j=0,\ldots,6)\) are functions of only one argument \(\zeta\), which are combinations of the functions \(L_m\); they are universal in character and can be calculated in advance. The expressions for these functions as functions of \(\zeta\) have been obtained by us explicitly; we do not present them here for lack of space. All these functions have been tabulated by us. Note that all \(\theta_{n,j}\) and \(U_{n,j}\) tend to zero both for \(\zeta=0\)
Fig. 1. Functions \(\theta_{n,j}\) as functions of \(\zeta\)
Fig. 2. Functions \(U_{n,j}\) and \(W_{n,j}\) as functions of \(\zeta\)
(except for \(\theta_{n,0}\), which at \(\zeta=0\) is equal to unity), and for \(\zeta \to \infty\). \(W_{n,j}=0\) for \(\zeta=0\), while for \(\zeta \to \infty\) all \(W_{n,j}\) assume various constant values, as follows from the conditions of our problem. Graphs of some of the functions \(\theta_{n,j}, U_{n,j}\), and \(W_{n,j}\) are given in Figs. 1 and 2.
Using the formulas given above, we have calculated examples describing the development of a slope wind over time and with height at various points of the slope.
In conclusion, I express my sincere gratitude to Corresponding Member of the Academy of Sciences of the USSR I. A. Kibel’ for his attention to the present work.
Institute of Applied Geophysics
Academy of Sciences of the USSR
Received
19 IV 1960
REFERENCES
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- A. M. Mkhitaryan, Izv. AN ArmSSR, Ser. Fiz.-Mat. Nauk, No. 1 (1955).
- L. N. Gutman, Sborn. Inst. Mekh. AN SSSR, 15 (1953).