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Hydrology 510 Quantitative Methods in Hydrology New Mexico Tech Hyd 510 Hydrology Program Quantitative Methods in Hydrology Hydrology 510 Quantitative Methods in Hydrology Motive Preview: Example of a function and its limits Consider the solute concentration, C [ML-3], in a confined aquifer downstream of a continuous source of solute with fixed concentration, C0, starting at time t=0. Assume that the solute advects in the aquifer while diffusing into the bounding aquitards, above and below (but which themselves have no significant flow). (A homology to this problem would be solute movement in a fracture aquitard (diffusion controlled) Flow aquifer Solute (advection controlled) aquitard (diffusion controlled) x with diffusion into the porous-matrix walls bounding the fracture: the so-called “matrix diffusion” problem. The difference with the original problem is one of spatial scale.) An approximate solution for this conceptual model, describing the space-time variation of concentration in the aquifer, is the function: 1/ 2 x x D C x D C(x,t) = C0 erfc or = erfc B vt − x vB C B v(vt − x) 0 where t is time [T] since the solute was first emitted x is the longitudinal distance [L] downstream, v is the longitudinal groundwater (seepage) velocity [L/T] in the aquifer, D is the effective molecular diffusion coefficient in the aquitard [L2/T], B is the aquifer thickness [L] and erfc is the complementary error function. Describe the behavior of this function. Pick some typical numbers for D/B2 (e.g., D ~ 10-9 m2 s-1, typical for many solutes, and B = 2m) and v (e.g., 0.1 m d-1), and graph the function vs. time at a one more locations, x, and vs. space at one or more times, t. Later we’ll examine derivatives and integrals of this function and its parent, which includes details of concentrations in the aquitard. And later still we’ll look at its origin, through solutions of PDEs. -1- v.1.1, F2008 New Mexico Tech Hyd 510 Hydrology Program Quantitative Methods in Hydrology Review of calculus (This is review material, taken and expanded from Carol Ash and Robert B. Ash, The Calculus Tutoring Book, IEEE Press, New York, 1986) Functions Introduction to functions “A function can be thought of as an input-output machine.” Given a particular input, x, the function f(x) is the corresponding output. Functions are usually denoted by single letters. We’ll often used f and g in this review to denote functions. “If the function g produces the output 3 when the input is 2, we write g(2)=3.” Mathematicians represent this process by a mapping table or diagram, as shown here. TABLE MAPPING DIAGRAM Input Output 2 3 2 3 8 4 8 4 9 4 9 10 -1 10 -1 In hydrology the machine can represent a model of a process. Then, for example, x could be location in an aquifer and f(x) spatially variable hydraulic conductivity as a function of location. Or for streamflow, x could be time and f(x) stream discharge as a function of time at a stream gauge location. Or x could be a state, such as pressure or temperature, and f(x) a state-dependent property, such as fluid density or viscosity which are functions of pressure or temperature (this is called an equation of state, or EOS). We often consider forcings as an input x, such as precipitation or solar radiation. For precipitation over a watershed as in input, streamflow at the outlet is a typical output, while for radiation over a land-surface plot as an input, evapotranspiration from that plot back to the atmosphere is a typical output. Inputs can be properties (or parameters), location, time, or forcings. Outputs can be states, fluxes, other properties (or parameters), location, or (travel, residence, or arrival) times. Input, x Output, f(x) Machine or Model, f The input, x, is called an independent variable while the output, f(x), is a called a dependent variable.1 Mathematicians say that “f maps x to f(x), and call f(x) the value of the function at x. The set of inputs x is called the domain of f and the set of outputs is called the range.” 1 In this review of calculus we assume one input and one output. Later we’ll extend the review to multivariate calculus with more that one input and often more than one output. In hydrologic applications the multivariate case is the norm. -2- New Mexico Tech Hyd 510 Hydrology Program Quantitative Methods in Hydrology Formally, a function f(x) is not allowed to send one input to more than one output. Not a function Consider the domain of x, between the limits a and b. The set of all x such that a ≤ x ≤ b is denoted by mathematicians by [a,b] and is called a closed interval. The set of all x such that a < x < b is denoted by mathematicians by (a,b) and is called an open interval. “Similarly we use [a,b) for the set of x where a ≤ x < b, (a,b] for a < x ≤ b, [a,∞) for x ≥ a, and (-∞,a] for x ≤ a, and (-∞,a) for x < a. In general, the square bracket, and the solid dot in … the figure below … , means that the endpoint belongs to the set; a parentheses, and the small circle in … the figure …, means that the end point does not belong to the set. The notation (-∞,∞) refers to the set of all real numbers.” [a,b] (a,b) [a,∞) (-∞,b) Issues to review on your own: Equations v. functions One-to-one functions Increasing and decreasing functions Elementary functions (all of which are common functions in hydrology; also, see the elfun directory in Matlab) Type Examples Constant function f(x) = 2 for all x, g(x) = -π for all x Power function x-1, x0.995, x, x2, x2.7, x12 Trigonometric functions sine, cosine, tangent, secant Inverse trigonometric sin-1 x, cos-1 x, tan-1 x Functions Exponential functions 2x, (1/4)-x, 104, and especially ex, where e = 2.71828 … Logarithmic functions log2 x, log10 x, and especially loge x = ln x Trigonometric functions Commonly encountered in, e.g., -3- New Mexico Tech Hyd 510 Hydrology Program Quantitative Methods in Hydrology -problems with periodic forcing (e.g., diurnal, seasonal, decadal) -cylindrical and spherical coordinates (e.g., radial well hydraulics, intra-particle spherical diffusion and adsorption) -geometric definition of object shapes -certain (finite) spatial domain problems (e.g, Fourier series solutions) Definitions of sine, cosine, and tangent y (x,y) y x y sinθ sinθ = , cosθ = , tanθ = = (1) r θ r r x cosθ x where (x,y) = Cartesian coordinates (r,θ) = radial coordinates Radius r is always positive, but the signs of x and y depend on the quadrant, thus the signs of the trig functions also depend on quadrant: y x sign of sin θ sign of cos θ sign of tan θ Degrees v. radians We measure angles in both degrees and radians. Recall that an angle θ of 180º = π radians. More generally, number of radians π = (2)2 number of degrees 180 Examples of important angle and related trig functions: Degrees Radians sin cos tan Degrees Radians sin cos tan 0º 0 0 1 0 30º π/6 1/2 √3/4 √1/3 90º π/2 1 0 none 45º π/4 √1/2 √1/2 1 180º π 0 -1 0 60º π/3 √3/4 1/2 √3 270º 3π/2 -1 0 none 360º 2π 0 1 0 2 The equation numbers refer to the numbers in Ash and Ash (1986). -4- New Mexico Tech Hyd 510 Hydrology Program Quantitative Methods in Hydrology We prefer to use radians and, in some cases, we must use radians, such as s=rθ r measuring the arc length along a circle. Let s equal the arc length, a θ fraction of the circumference. The circumference is 2πr, therefore the arc length is s= θ r, (5) where θ is in radians. Reference angle Trig tables list sin θ, cos θ, tan θ for 0º < θ <90º. To find trig functions for other angles use signs given in the box above, plus reference angles. For example, if θ is in the second quadrant (upper left quadrant) then the reference angle is 180º - θ. In particular, if θ is 150º then the reference angle is 30º. Etc. Right angle trigonometry opposite leg adjacent leg hypotenuse sinθ = cosθ = (6a,b) opposite hypotenuse hypotenuse opposite leg θ tanθ = (6c) adjacent leg adjacent Graphs of sin x, cos x, and tan x Exercise: Use Matlab to graph these three functions from x= -4π to +4π Graphs of a sin(bx + c) You should be familiar with: -defns. of amplitude (a), period (2π /b), frequency (b), and phase lag (c) (for example, the phase lag describes a shift (in radians) of the peak). -applications to > harmonic motion > (earth and ocean) tides > approximations to diurnal, seasonal, and other periodic temporal signals (applications: temperature, evapotranspiration, spring discharge) Application: A monitoring well on Cape Cod, Massachusetts, located 700m from the Atlantic Ocean coast, observes that water table elevations fluctuate over time. The water level data is fit to the model a sin(bx + c) where a is the amplitude (one-half the total range) of the water level fluctuation, the period is 12 hours, x is time (in hours), and the observed phase lag is 3 hours (compared to the local ocean tide; where for a lag of 3 hours c is expressed in radians = π /2).
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