Symmetry Reductions of Equations for Solute Transport in Soil

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<ul><li><p>Nonlinear Dynamics 22: 1527, 2000. 2000 Kluwer Academic Publishers. Printed in the Netherlands.</p><p>Symmetry Reductions of Equations for Solute Transport in Soil</p><p>P. BROADBRIDGE, J. M. HILL, and J. M. GOARDSchool of Mathematics and Applied Statistics, University of Wollongong, Northfields Ave., Wollongong 2522,Australia</p><p>(Received: 15 March 1999; accepted: 15 April 1999)</p><p>Abstract. Solute transport in saturated soil is represented by a nonlinear system consisting of a FokkerPlanckequation coupled to Laplaces equation. Symmetries, reductions and exact solutions are found for two dimensionaltransient solute transport through some nontrivial wedge and spiral steady water flow fields. In particular, themost general complex velocity potential is determined, such that the solute equation admits a stretching group oftransformations that would normally be possessed by a point source solution.</p><p>Keywords: Solute transport, saturated soil, dispersion, FokkerPlanck.</p><p>1. Introduction</p><p>One of the most serious environmental problems of the late 20th century is the chemicalcontamination and salinisation of soil. For example, in Australias Murray-Darling basin, wecan directly observe vast tracts of irrigated land that have been ruined by rising salt. Thetime scales of these regional processes is of the order of several decades. Therefore, for thepurposes of environmental management, it takes too long to experimentally determine theoutcomes of agricultural and industrial practices. Predictions must be made by mathemat-ical modelling. A full theory of solute transport will require understanding of complicatedmicroscopic processes (e.g. [1]). However coarser-scaled macroscopic models will remainimportant for economically predicting field-scale phenomena. The favoured models are sys-tems of conservation laws, expressed locally as partial differential equations for water andsolute concentrations. In practical problems, it is normal to solve these by approximate nu-merical methods. However, we face the serious problem that available numerical packageshave significant disagreement in their predictions of solute dispersion [2]. Therefore, exactsolutions are very important, not only because they provide direct unambiguous insight butalso because they are valuable bench tests for numerical schemes.</p><p>Here, we concentrate on macroscopic deterministic models based on local conservationlaws (see, e.g., [3, 4]).The solute flux density J is the sum of three components,</p><p>J D 2D0rc 2De.v/rc C cV; (1)due to molecular diffusion, dispersion and convection respectively. The volumetric Darcianwater flux V.r; t/ is a smoothly varying representative flux, averaged over a region that islarge compared to an individual grain or pore. The dispersion term makes some allowance for</p></li><li><p>16 P. Broadbridge et al.</p><p>extra mixing within the tortuous interconnected flow paths. The dispersion coefficient De isfound to be an increasing function of water speed, which is given by v D jVj=2; where 2 isthe volumetric water concentration in the soil. In many experiments, this function is found tobe a power law, De D D1vm; with D1 &gt; 0 and 1 m 2 [5, 6]. When we combine (1) withthe mass conservation law@.c2/</p><p>@tCr:J D 0; (2)</p><p>we obtain the convection-dispersion equation,</p><p>@.c2/</p><p>@tD r:.2D.v/rc/ r:.cV/; (3)</p><p>where D.v/ D D0 CDe.v/:Most of the existing analytical solutions have assumed a uniform water flow, with V and</p><p>D constant [7]. In this case, (3) reduces to the standard convection-diffusion equation withconstant coefficients. For the remainder of this article, we shall be dealing with more generaltwo-dimensional steady saturated water flows. These satisfy 2 D 2s where 2s is the watercontent at saturation. By Darcys law, V D Ksr, where is the total hydraulic pressurehead and Ks is the hydraulic conductivity at saturation [4, 8]. Now the equation of continuityr:V D 0 implies Laplaces equationr2 D 0: (4)</p><p>In this case, (3) reduces to@c</p><p>@tD r:.D.v/rc/C kr:rc; (5)</p><p>where k D Ks=2s and v D jkrj. Equations (4) and (5) for dependent variables and c,form a nonlinear system, since the dependent variables c and are coupled nonlinearly in (5).In practice, we treat (5) as a linear equation, for which (4) provides the variable coefficients.</p><p>In Section 2, we find similarity solutions for the case of D D D0 (constant). This mayrepresent convection-diffusion in an open body of flowing liquid solvent. It may also representconvection-dispersion in soil wherein D is taken to be an average value of dispersivity. InSection 3, we consider symmetry reductions for the caseD D D1vm. Neglect of the moleculardiffusion coefficient D0 is a common assumption since in experiments, diffusion is often aminor effect compared to dispersion.</p><p>2. Reductions for Convection-Diffusion</p><p>In a model of convection-diffusion, we assume that D.v/ in (5) is constant. Futhermore, bychoosing a length scale s and a time scale ts so that 2s =ts D D, we may rescale D to 1.Hence, we consider@c</p><p>@tD r2c C kr:rc; (6)</p><p>together with Laplaces equation (4). Wherever possible, it will be assumed that k and havebeen rescaled so that jrj is of order 1. Thence k may be interpreted as a Pclet number. Veryfew exact solutions are known, except for the case of steady unidimensional water flow, V =</p></li><li><p>Symmetry Reductions of Equations for Solute Transport in Soil 17constant. Notable exceptions are some cases with point sources of water in the plane ([6] andreferences therein), and continuous point water sources bounded within a right angled sector[10].</p><p>If we look for Lie point symmetries of the entire system (4) and (6), then we will findnothing more than rescaling of c, translations in t , and translations and rotations in .x; y/.These are the only conformal maps that leave both (4) and (6) invariant. However, we mayfind symmetries that leave the single equation (6) invariant, for a given special solution .x; y/of Laplaces equation. This may lead to useful reductions and solutions of (6) even if .x; y/is not itself an invariant solution of Laplaces equation. For this purpose, we could carry outa symmetry classification of the single equation (6), treating .x; y/ as a free coefficientfunction. Given the class of functions .x; y/ which lead to extra symmetries, we could laterrestrict these to be solutions of Laplaces equation. The only point symmetries for the generaldiffusion-convection equation are combinations of translations in t , rescaling of c and linearsuperposition. The symmetry generating vector fields are linear combinations of 01 D @t ,02 D c@c, and .x; y; t/@c , where c D .x; y; t/ is a general solution of the original linearequation (6). The program DIMSYM [9] finds that special cases may possibly arise when anyof the following 17 functions are linearly dependent:xxy y y; xxy y; xxy x y; xxy x; </p><p>2xy y; </p><p>2xy ; xy y; xy x; xxx y y; xxx y;</p><p>xx y; xxx x y; xxx x; 2xx y; </p><p>2xx; xx x; and 1:</p><p>The full classification of these special cases would be a major task and we have not completedit. Apart from the case of radial flow from a point source, which is already well known,we have found some interesting special cases which allow reduction to familiar ordinarydifferential equations.</p><p>First, for incompressible and irrotational flow with stream function .x; y/, we revisit thestrained flow restricted to a right angled sector, D xy; D .y2 x2/=2: (7)In this case the bounding streamline D 0 is piecewise linear, with polar angles D</p><p>=4: Then the finite part (excluding linear superposition) of the Lie algebra of infinitesimalpoint symmetry generators is nine dimensional, compared to two dimensional in the genericcase. The additional basis vectors of the symmetry algebra may be chosen to be03 D k .x; y/c@c C @ where @ D x@y y@x;</p><p>04 D ektck</p><p>2.x y/@c C @x</p><p>;</p><p>05 D ektck</p><p>2.x y/@c C @y</p><p>;</p><p>06 D ekt Tck.x C y/@c C 2@xU;07 D ekt Tck.x C y/@c C 2@yU;</p><p>08 D e2ktkx@x C ky@y C @t </p><p>k2c</p><p>2.x2 C y2/C k2xyc C kc</p><p>@c</p><p>;</p><p>09 D e2ktkx@x ky@y C @t </p><p>k2c</p><p>2.x2 C y2/ k2xyc kc</p><p>@c</p><p>:</p></li><li><p>18 P. Broadbridge et al.</p><p>A wide variety of symmetry reductions is possible. In particular, we have found that 03leads to explicit solutions satisfying standard boundary conditions. These invariant solutionsare of the form</p><p>c D e.1=2/kxyF .R; t/; (8)where R denotes r2 D x2 C y2, and F satisfies</p><p>4RFRR C 4FR k2R4 F D Ft : (9)</p><p>At the boundary streamlines, there is no normal convection and neither is there normaldiffusion, since it is clear from (8) that with D tan1.y=x/, @c=@ D 0 at D =4.Hence, we automatically satisfy the physical boundary conditions that both water and soluteare contained within the bounds of the wedge.</p><p>If we apply the additional boundary conditions that F.R/ is analytic as R! 0 and F ! 0as R ! 1, as well as general initial condition F D g.kR=2/, then we obtain the seriessolution</p><p>F D1XnD1</p><p>Bn entkR=4Ln.kR=2/; (10)</p><p>where n D .2nC 1/k, Ln is the nth Laguerre polynomial, and</p><p>Bn D0@ 1Z</p><p>0</p><p>eu=2Ln.u/g.u/ du</p><p>1A.0@ 1Z0</p><p>eu.Ln.u//2 du</p><p>1A :We have constructed the smooth solution with initial condition F D 1 for r &lt; 0:2 and</p><p>F D 0 for r &gt; 0:2. As expected, the solute concentration contours spread diffusively in analmost circular fashion at early times t &lt; 0:1 but at later times they are elongated along thedominant direction of water flow , D =4. The continuous solute line source solutionfor this problem was given by Maas [10] and the instantaneous source solution was given byZoppou and Knight [11]. These solutions could be used as Greens functions in an alternativeapproach to the solution of initial value problems and boundary-initial problems.</p><p>We can also solve boundary value problems on the finite region .r; / 2 T";AU T=4; =4U. Consider the boundary conditionsF."2/ D ; F .A2/ D 0;</p><p>as well as the previously mentioned no-flow conditions at D =4. By (8), this correspondsto almost uniform concentration c at a thin circular discharge pipe r D ", and completeremoval of solute (c D 0) at an outer surface r D A. The steady state solution satisfies</p><p>R2F 00.R/C RF 0.R/ k2</p><p>16R2F D 0: (11)</p><p>It follows that the steady solution satisfying the prescribed boundary conditions is</p><p>F1.R/ D C1I0.kR=4/C C2K0.kR=4/; (12)</p></li><li><p>Symmetry Reductions of Equations for Solute Transport in Soil 19where I0 and K0 are the modified Bessel functions of order zero,</p><p>C1 D I0.kA2=4/</p><p>I0.kA2=4/K0.k"2=4/ I0.k"2=4/K0.kA2=4/ ;</p><p>C2 D K0.kA2=4/</p><p>I0.kA2=4/K0.k"2=4/ I0.k"2=4/K0.kA2=4/ :</p><p>Now we solve for the transient eF.R; t/ D F.R; t/ F1.R/, which now satisfies a linearequation (9), as well as homogeneous boundary conditions eF D 0 at R D "2; A2. Using theseparation of variables technique, we obtain the solution</p><p>eF D 1XnD1</p><p>Bn entkR=4Fn.kR=2/;</p><p>Fn.u/ D M.n; 1; u/ U.n; 1; u/M.n; 1; k"2=2/=U.n; 1; k"2=2/; (13)where M.a; b; c/ and U.a; b; c/ are the standard confluent hypergeometric functions (see[12, p. 504]) and the eigenvalues n D T1 n=kU=2 are the solutions of the equationM.; 1; k"2=2/U.; 1; kA2=2/ U.; 1; k"2=2/M.; 1; kA2=2/ D 0: (14)</p><p>The series coefficients are given by</p><p>Bi DkA2=2Zk"2=2</p><p>g.2u=k/Fi.u/ eu du=kA2=2Zk"2=2</p><p>Fi.u/2 eu du; (15)</p><p>where g.R/ is the initial value for eF.R/.Figure 1 displays the solute contours, plotted by MAPLE [13], for the case of " D 0:05,</p><p> D 1, k D 10 and A D 1, with initial condition g.R/ D 55k2.R "2/.R A2/.Note that neither water nor solute flows across the bounding streamlines at polar angles</p><p> D =4;3=4. We are principally interested in the sector 2 T=4; =4U. However,the solute concentration contours may be extended smoothly and asymmetrically into theneighbouring sectors, wherein other solutions may be constructed from the series (13). Fromthe boundary conditions, the outer contour must be the circle r D A. The figure displays theinterior region r &lt; 0:7, wherein the initial condition is highly irregular. Since the smoothsteady state solution is stable, information on the initial condition will eventually be washedaway. With this high value of Pclet number, the initial irregularity is still evident at timet D 0:1 even though the contours have been elongated in the direction of the water flow. Attime t D 0:5, the contours show little evidence of the initial conditions.</p><p>For a water flow constrained to a more general wedge sector 2 T0; =nU, with n 6D 0 thecomplex velocity potential is</p><p>! D C i D .x C iy/n; (16)and in this case, Equation (6) still has the symmetry generator</p><p>03 D 12nk .x; y/c@c C @ : (17)</p></li><li><p>20 P. Broadbridge et al.</p><p>Figure 1. Contours of equally spaced concentration contours, for the initial-boundary problem described in thetext: (a) t D 0:1, (b) t D 0:5.</p><p>Invariant solutions of Equation (6) for convection-diffusion in the general wedge are of theform</p><p>c D e1=2 krn cos.n/F .r; t/; (18)where F.r; t/ is any solution of the linear equation</p><p>Ft D Frr C 1rFr 14k</p><p>2n2r2n2F: (19)</p><p>Since the Ft term has a constant coefficient, clearly we may reduce (19) to an ordinarydifferential equation by the Laplace transform in t , or by separation of variables.</p><p>In each of the above cases, the water speed is given by</p><p>v D knrn1;which is constant on any circle about the origin. However, the streamlines are hyperbolic inshape. The standard point-source water flow, given by D log.r/, may be considered asa nontrivial n D 0 member of the above sequence, except that the streamlines are linear. Inaddition, we have the n D 1 member given by C i D exp.x C iy/. In this case, eachstreamline enters asymptotically from one plate, changes direction and exits asymptoticallyalong another plate. Beyond time translation and rescaling of c, this flow also gives rise toadditional exotic symmetries, which are generated by</p><p>03 D Tktc ex sin.y/ ycU@c C 2t@y; 04 D Tkc ex sin.y/U@c C 2@y:</p><p>3. Reductions for Convection-Dispersion</p><p>In the general case, in Equation (5) the dispersion coefficient D.v/ may be any positivevalued function of water speed v D jkrj. However, we focus on the case D D vp. From</p></li><li><p>Symmetry Reductions of Equations for Solute Transport in Soil 21both experiment and theory, the dispersion coefficient is usually perceived to take the power-law form with p 2 T1; 2U. During the preliminary symmetry analysis of (5), we considera general function .x; y/. This leads to complicated determining relations. However, theconvection-dispersion equation can be considerably simplified after changing independentspatial variables from (x; y) to the streamline coordinates .; / (e.g., see [4]). The solutetransport equation (5) then transforms to</p><p>1v2@c</p><p>@tD @@</p><p>vp@c</p><p>@</p><p>C @@ </p><p>vp@c</p><p>@ </p><p>C @c@; (20)</p><p>where v D .2x C 2y/1=2 and .x; y/ and .x; y/ are such thatx D y; y D x: (21)</p><p>Of particular practical interest are the scale-invariant solutions that would have the same in-variance properties as those of the point source solution, should it exist. For an instantaneouspoint source at time zero,</p><p>c.x; y; 0/ D c0.x/.y/; (22)where c0 is some constant and denotes the Dirac delta function. In terms of independentvariables .; / this condition becomes</p><p>c.; ; 0/ D c0v2./. /; (23)arising from</p><p>dx dy D @.x; y/@.; /</p><p> d d D d d v2 ;and the conditionZZ</p><p>c.x; y; 0/ dx dy D c0:</p><p>Now for the two special cases</p><p> C...</p></li></ul>


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