Block Matrix Sinc-Galerkin Solution of the Wind-Driven Current Problem

Block Matrix Sinc-Galerkin Solution of the Wind-Driven Current Problem Sanoe Koonprasert and Kenneth L. Bowers Department of Mathematical Sciences, Montana State University Bozeman, MT 5977, USA Abstract A block matrix formulation is presented for the Sinc-Galerkin technique applied to the wind-driven current problem from oceanography. The block matrix form is used to determine an approximate solution for the coupled system of differential equations describing the wind-driven current problem. The approximate solution determined by this new Sinc-Galerkin scheme is compared to the approximate solution for the original complex-valued Sinc-Galerkin approach to ocean current models. The approximate solution is also compared to exact solutions to illustrate the exponential convergence rate of the method. Key words: Block matrix, Sinc-Galerkin method, wind-driven currents Introduction In recent years, the complex-valued Sinc-Galerkin method for wind-driven current models has been applied to the model formulated as a complex-valued ordinary differential equation. However, a Sinc-Galerkin technique can also be applied to a formulation of the model that consists of two real-valued coupled ordinary differential equations. The block matrix depends on the sinc basis functions, the differential equations, and the boundary conditions. The complex-valued Sinc-Galerkin method results in a smaller discrete system but requires complex arithmetic at all steps. The block formulation eliminates the need for any complex-valued calculations at the expense of a slightly larger matrix system. In this paper, we perform a numerical study on a model of wind-driven currents in the ocean. This model is found in []. There the Sinc-Galerkin technique was applied to a model describing wind-driven subsurface currents in coastal regions and semi-enclosed seas where the vertical eddy viscosity coefficient is Preprint submitted to Elsevier Science 3 July 23

x q z Sea surface χ z = τ = τ w cosχˆx + sinχŷ y τ z = ρa v z dq / z z = D, Seabed A v z, eddy viscosity Fig.. The general physical model of the depth-dependent eddy viscosity oceanography problem represented as a continuously differentiable function of depth. Our goal is to construct the block matrix that represents this Sinc-Galerkin formulation and use it for solving a coupled linear system. 2 Governing Equations To develop a mathematical model, we first construct a right-handed coordinate system with the vertical coordinate z directed positive downward from the free surface, and with x and y directed northward and eastward, respectively. A plane at z = D corresponds to the impermeable boundary at the seabed. For the purpose of illustrating the Sinc-Galerkin method, we employ several of the simplifying assumptions invoked in one-dimensional wind-drift studies - the ocean depth, D, and ocean mass density, ρ, are assumed constant, and the effects of tides, inertial terms, free surface slope, and variations in atmospheric pressure are neglected. Currents are driven by a tangential surface wind stress the surface is at z = of magnitude τ w, represented as τ = τ w cosχˆx + sinχŷ, with χ being the angle between the positive x -axis and the wind direction in Figure. Here ˆx and ŷ represent unit vectors in the directions of the positive x -axis and the positive y -axis, respectively. In this model, called a specified eddy viscosity model, internal frictional stresses are parameterized as τ z = ρa v z dq /, where the specified effective vertical eddy viscosity coefficient A v z is a continuously differentiable function of z, D. Here q z = U z ˆx + V z ŷ represents the horizontal wind-drift current which is the difference between the total velocity and the geostrophic current in [2]. The Coriollis force acting on the moving water is balanced by a horizontal pressure gradient force. 2

Under the present assumptions, the conservation of linear momentum equations express a balance between the Coriollis force and the internal friction associated with turbulence. The wind-drift current q is determined by solving the boundary-value problem d A v z dq = fẑ q, < z < D, where the stress condition at the sea surface, z =, is the tangential surface wind stress ρa v dq = τ w cosχˆx + sinχŷ 2 while at the seabed, z = D, the frictional stress is linearly proportional to the current, hence ρa v D dq D = k f ρq D. 3 Here f 2Ω sin θ is the Coriollis parameter at latitude θ where Ω = 7.29 5 rad s is the angular speed of rotation of the earth. Though π/2 < θ < π/2, for the rest of our work we assume we are in the northern hemisphere and hence < θ < π/2 f >. The parameter k f is the linear slip bottom stress coefficient. If q D =, 3 is called a zero-velocity no-slip condition. From, we can simplify the boundary-value problem as follows d A v z dq = d A v z du z ˆx + d A v z dv z ŷ = fẑ q = fẑ [U z ˆx + V z ŷ ] = f U z ŷ V z ˆx, < z < D. Then it can be written in component form as and d A vz du z = fv z, < z < D 4 d A vz dv z = fu z, < z < D. 5 The stress condition at the sea surface, z =, separates as ρa v du = τ w cosχ, ρa v dv = τ w sinχ. 6 3

At the seabed, z = D, the frictional stress separates as ρa v D du D = k f ρu D, ρa v D dv D = k f ρv D. 7 To nondimensionalize the model equations we begin with a measure of near surface turbulent eddy viscosity, A A v, and define a nominal upperlayer Ekman depth by D E 2A /f. Also define a current speed in units of U = τ w D E /ρa = 2τ w /ρ A f. U is the natural velocity scale in an infinitely deep sea with uniform eddy viscosity in the steady-state. A nondimensional form of the equations of motion can be expressed with the introduction of the nondimensional variables z z, A v z A vz D A v, qz q z Uzˆx + V zŷ 8 U together with two nondimensional constants a depth ratio κ and a bottom friction parameter σ given by κ D f = D, σ A A v = A v D. 9 D E 2A k f D k f D We apply 8 and 9 to nondimensionalize the equations 4 and 5. For 4, we have D which results in d A A v z du Uz = fu V z, < z <, D d A v z duz Similarly, the result from 5 is d A v z = 2κ 2 V z, < z <. dv z = 2κ 2 Uz, < z <. The surface boundary conditions in 6 are nondimensionalized to produce which leads to ρ A du U = τ w cosχ, ρ A du V D D = τ w sinχ, du = κ cosχ, dv = κ sinχ. 2 4

The seabed conditions in 7 are also nondimensionalized and become U + σ du In the special no-slip case, we can set σ = in 3. dv =, V + σ =. 3 We first transform the nonhomogenous boundary conditions to homogeneous boundary conditions by using the linear transformations Uz = uz+κ + σ z cosχ, V z = vz+κ + σ z sinχ. 4 The first derivative of each transformation in 4 yields duz = duz κ cosχ, dv z = dvz so the resulting boundary-value problem for uz satisfies κ sinχ, d A v z du + d A vzκ cosχ = 2κ 2 [vz + κ + σ z sinχ], < z <, which gives d A v z du + κ cosχa vz = 2κ 2 vz 2κ 3 + σ z sinχ, Similarly the boundary-value problem for vz satisfies < z <. 5 d A v z dv + κ sinχa v z = 2κ2 uz + 2κ 3 + σ z cosχ, < z <. 6 The surface boundary conditions 2 become du =, dv while the seabed boundary conditions 3 become = 7 u + σ du =, v + σ dv =. 8 5

For the purpose of illustrating the exposition of the Sinc-Galerkin technique, we define Luz d A v z du, Lvz d A v z dv. 9 Then 5 and 6 are now given by the coupled u and v equation systems where Luz + 2κ 2 vz = F z, < z <, 2 Lvz 2κ 2 uz = F 2 z, < z <, 2 F z = 2κ 3 + σ z sinχ κ cosχa v z, 22 F 2 z = 2κ 3 + σ z cosχ κ sinχa v z. 23 The surface boundary conditions are du =, dv =. 24 The seabed boundary conditions are u + σ du =, v + σ dv =. 25 3 The Sinc Function and the Sinc-Galerkin Method We first review sinc function properties, sinc quadrature rules, and the Sinc- Galerkin method. These are discussed thoroughly in [3] and [4]. The sinc function is defined for all z C by sincz sinπz πz, if z, if z =. For h > and k =, ±, ±2, ±3,... the translated sinc functions with evenly spaced nodes are given by 6

z kh Sk, hz sinc h sinπ z kh h π z kh h, if z kh, if z = kh. The Sinc-Galerkin procedure for the coupled linear system in 2-2 begins by selecting composite sinc functions appropriate to the interval, as the basis functions for the expansion of approximate solutions for the current components uz and vz. We introduce the conformal mapping function z w = φz = ln, 26 z which is a conformal mapping from D E, the eye-shaped domain in the z plane, onto the infinite strip in the w-plane, D S, where { z D E = z = x + iy : arg z < d π }, 2 { D S = w = u + iv : v < d π }. 2 iy φ iv w plane d z plane x u d D E φ D S Fig. 2. The relationship between the eye-shaped domain, D E, and the infinite strip, D S This is shown in Figure 2. The basis functions are derived from the composite translated sinc functions, φz kh S k z Sk, h φz = sinc, 27 h for z D E. These are shown in Figure 3 for real values, x. 7

.8 Sinc basis on, k = k = k =.6 Sk, h φx.4.2 -.2 -.4..2.3.4.5.6.7.8.9 x Fig. 3. Three adjacent members Sk, h φx when k =,, and h = π 8 of the mapped sinc basis on the interval, For the Neumann or radiation boundary conditions 24-25, the sinc basis functions in 27 do not have a derivative when z tends to or. Thus we modify the sinc basis functions as Sk, h φz φ z sinc φz kh h. 28 φ z These are shown in Figure 4 for real values, x. Note that the derivatives of the modified sinc basis functions are defined as z approaches or. We also add boundary basis functions that are Hermite polynomials. Thus the derivatives at z = and for these basis functions are defined. These Hermite polynomials are given by B z = 2z + z 2, B z = zz 2 + σ3 2zz 2. 29 Sinc rules for a special class of functions BD E have been developed. A discussion of the properties of functions in BD E is found in [3] and [4]. Definition Let φ : D E D S be a conformal map of D E to D S with inverse ψ. Let Γ = {ψu D E : < u < } =,. Then BD E is the class of functions F which are analytic in D E, satisfy F z, t ±, ψt+l where L = { } iv : v < d π 2, and on the boundary of DE, denoted D E, satisfy N F F z <. D E 8

.2.25.2 Sinc basis on, k = k = k = Sk, h φx/φ x.5..5 -.5 -...2.3.4.5.6.7.8.9 x Sk, h φx Fig. 4. Three adjacent members when k =,, and h = π φ x 8 of the modified sinc basis on the interval, The proofs of the following theorems are found in [3] and [4]. Theorem If φ F BD E then for all z Γ, F z = F z j Sj, h φz + E F j= where E F πφz sin h 2πi D E φ wf wdw φw φz sinπφw/h. Theorem 2 If F BD E then F z = h j= F z j φ z j + I F where with I F i 2 F zκφ, hz D E sinπφz/h [ ] iπφz κφ, hz = exp sgniφz. h The infinite quadrature rule appearing in Theorem 2 can be evaluated directly, but in general it must be truncated to a finite sum for the Sinc-Galerkin method. The following theorem indicates the conditions under which exponential convergence results. 9

Theorem 3 Let F BD E and φ be a conformal map with constants α, β, and C so that exp α φz, z Γ a F z φ z C exp β φz, z Γ b where Γ a {z Γ : φz = x, } =,, 2 [ Γ b {z Γ : φz = x [, } = 2,. Then the sinc trapezoidal quadrature rule is F z = h N j= M Hence, make the selections F z j + Oexp αmh + Oexp βnh φ z j + Oexp 2πd/h. 3 N = [ ] α β M +, h = 2πd/αM, where [ ] denotes the greatest integer, and the exponential order of the sinc trapezoidal quadrature rule in 3 is Oexp 2πdαM. Corollary An important special case of Theorem 3 occurs when the integrand has the form GzSl, h φz. Due to the interpolation Sl, h φz j = Sl, hjh = δ jl, the sinc quadrature rule is a weighted point evaluation to the order of the method, GzSl, h φz = h Gz l + Oexp 2πd/h. 3 φ z l In the Sinc-Galerkin technique, we first assume the approximate solutions for uz and vz in 2-2, subject to the mixed conditions 24 and 25, are represented by

where choosing M = N gives u a z = c N B z + u h z + c N+ B z, 32 v a z = d N B z + v h z + d N+ B z, 33 u h z = N j= N c j S j z φ z, v hz = N j= N d j S j z φ z. 34 Both approximate solutions clearly satisfy the boundary conditions 24-25. The m = 2N + 3 coefficients { } N+ c j and the m coefficients { } N+ d j j= N j= N are determined by orthogonalizing the residual Lu a z + 2κ 2 v a z F z and Lv a z 2κ 2 u a z F 2 z with respect to the sinc basis functions { } N+ S j in 27. The inner product F, G = F zgzwz, j= N uses a weight function wz = / φ z = z z. So this yields the discrete Sinc-Galerkin system, for j = N,..., N + Lua + 2κ 2 v a F, S j =, 35 Lva 2κ 2 u a F 2, S j =. 36 This leads to c N LB, S j + Lu h, S j + c N+ LB, S j + 2κ 2 v a, S j = F, S j, 37 d N LB, S j + Lv h, S j + d N+ LB, S j + 2κ 2 u a, S j = F2, S j. 38 With the sinc quadrature rule in 3 we evaluate the inner products in 37-38. First, we evaluate LB, S j and LB, S j as LB, S j = LB, S j = LB z S jz LB z S jz φ z h LB z j, 39 φ z j 3/2 φ z h LB z j. 4 φ z j 3/2 The nodal points z j = e jh / + e jh. The numerators in these terms are

LB z = d = 6 A v z db z = 6 d A v z 2z + A vzz z A v zz z LB z = d A v z db z = d A v z2 3z + 6σ 6σzz A v z2 + 6σ 6z 2z + A v z2 3z + 6σ 6σzz =,. The inner products Lu h, S j and Lv h, S j in 37-38 are evaluated by integrating by parts twice. Thus since Lu h z = N c k k= N the inner product involving Lu h is Lu h, S j = N c k k= N A v Sk φ A v z Sk z, 4 φ z, S j. 42 The inner product is evaluated by integrating by parts twice to get where Sk A v φ B T = A v z, S j = Sk z φ z A v z Sk z φ z S j z φ z S k z = B T φ z A vz S jz φ z S k z φ z A v z S jz φ z, + A φ v z S kz S jz. z φ z φ z Sz 43 B T tends to zero as shown in []. z Since φz = ln, then z φ z = z z. This leads to the result 2

φ z φ z in 43 are = 2 2z, φ z 3/2 φ z =. The derivative 4 S jz = ds jz φ z + S j z φ z dφ φ z dsj z = dφ + 2 2zS jz φ z, 44 and S jz φ z = d2 S j z φ z 3/2 + ds jz φ z dφ 2 dφ φ z + φ z z 2φ 2 φ z 3/2 + S j z φ z d 2 S j z = dφ 2 4 S jz φ z 3/2. 45 Applying the sinc quadrature rule, 43 becomes Sk A v φ, S j = S k z { h + h S k z d 2 φ z A S j z vz dφ 2 4 S jz φ z 3/2 dsj z + 2 2zS jz φ z φ z A v z h 2 δ2 jk + 4 δ jk dφ { h δ jk + 2 2z k δ jk } A v z k φ z k /2 } A v z k φ z k 3/2. So Lu h, S j N k= N { h 2 δ2 jk + N k= N + 4 δ jk } A v z k φ z k /2 c k 46 { h δ jk + } 2 2z k δ jk A v z k φ z k 3/2 c k. Here the r th derivative of Sj, h φz, with respect to φ, evaluated at the 3

nodal point z k is denoted by Similarly Lv h, S j N k= N h r δr jk dr dφ [Sj, h φz] r z=zk. 47 { h 2 δ2 jk + N k= N + 4 δ jk } A v z k φ z k /2 d k 48 { h δ jk + } 2 2z k δ jk A v z k φ z k 3/2 d k. Next, applying the sinc quadrature rule 3, the inner products u a, S j and v a, S j are u a, S j = v a, S j = u a z S jz φ z h φ z j u az 3/2 j, 49 v a z S jz φ z h φ z j v az 3/2 j, 5 where the solutions u a z j and v a z j are given by u a z j = c N B z j + c j φ z j + c N+B z j, 5 v a z j = d N B z j + d j φ z j + d N+B z j. 52 Last, applying the sinc quadrature rule 3, the inner products F, S j and F 2, S j are F, S j = F 2, S j = F z S jz φ z h φ z j F z 3/2 j, 53 F 2 z S jz φ z h φ z j F 2z 3/2 j, 54 where F and F 2 are given in 22-23. Substituting 39, 4, 46, 48, 5, 52, 53, and 54 into 37-38, the result will be linear equations whose solutions give the coefficients for u a z and v a z. For N j N +, 4

LB z j φ z j 3/2 c N + + N k= N N k= N { h 2 δ2 jk + 4 δ jk } A v z k φ z k /2 c k { h δ jk + } 2 2z k δ jk A v z k φ z k 3/2 c k 55 + LB z j φ z j 3/2 c N+ + 2κ 2 φ z j 3/2 v a z j = F z j φ z j 3/2 and LB z j φ z j 3/2 d N + + N k= N N k= N { h 2 δ2 jk + 4 δ jk } A v z k φ z k /2 d k { h δ jk + } 2 2z k δ jk A vz k φ z k 3/2 d k 56 + LB z j φ z j 3/2 d N+ 2κ 2 φ z j 3/2 u a z j = F 2z j φ z j 3/2. Introduce the column vectors c and d that represent the coefficients of the approximate solutions in 32-33, c [c N c N c N c N+ ] T, d [d N d N d N d N+ ] T, and define F [F z N F z N F z N F z N+ ] T, F 2 [F 2 z N F 2 z N F 2 z N F 2 z N+ ] T, u a [u a z N u a z N u a z N u a z N+ ] T, v a [v a z N v a z N v a z N v a z N+ ] T, where F z and F 2 z are given in 22-23. The expressions in 47 for each j and k can be stored in a matrix I r = [δ r jk ]. For r =,, 2, we have, if k = j δ jk [Sj, h φz] z=z k =, if k j, 57 5

, if k = j δ jk h d dφ [Sj, h φz] z=z k = k j, if k j, k j π 2, if k = j 3 δ 2 jk h2 d2 dφ 2 [Sj, h φz] z=z k = 2 k j k j 2, if k j. The following matrices will be some examples for I, I, I 2. Given N j N +, m = 2N + 3, the m m, square matrices I, I, I 2 are given by... I =..... = I, I =...... m 2 m................... 2 2, 58.......... m m... 2 π2 2 2... 2 m 3 2 2 m 2. 2......... I 2 = 2......... 2 2 2 2 2. 59.......... 2 2 m... 2 2 π2 m 2 2 2 3 When N j N, we remove the first and last columns of I, I, and I 2 in 58-59, to arrive at the m n, n = 2N +, non-square matrices...... I z =..........., I z =... m 2 2 m 2............. 2...., 6.. 2....... m... m 2 2 6 6

I z 2 = 2 m 2 m 2 2 2 2... 2 2....... π 2 3 2 2...... 2 2 2...... 2 2 2....... π 2 2 m 2 m 2 2... 2 2 2 2 The n n square diagonal matrix D n f is written as 3. 62 fz N... D n f = fz. 63... fz N From 55 and 56 we arrive at the discrete system B b c + 2κ 2 D m E φ 3/2 b d = D m F φ 3/2 B b d 2κ 2 D m E φ 3/2 b c = D m F φ 3/2 2, where B b is defined by the m m bordered matrix The non-square m n matrix B b [ a N A ns a N+ ]. 64 { A ns h 2 I2 z + 4 I z } D n Av φ /2 { + h I z + 2 I z D n 2z } D n A v φ 3/2 65 and the m column vectors a N and a N+ have j th component, N j N +, [a N ] j LB z j φ z j, [a N+] 3/2 j LB z j. 66 φ z j 3/2 7

The m m evaluator matrix E b is defined by [ E b b N I D n φ b N+ ] 67 and the m column vectors b N and b N+ have j th component [b N ] j B z j, [b N+ ] j B z j. 68 This leads to the coupled linear discrete system AX = C 69 where the 2m 2m block matrix m = 2N + 3 B b 2κ 2 D m A = 2κ 2 D m φ Eb B 3/2 b φ Eb 3/2 7 and the 2m column vectors X = c d D m φ F 3/2, C =. 7 D m φ F 3/2 2 It follows that the solutions for u a and v a at the nodal points are the elements of this column vector X that are computed from the 2m 2m block evaluator matrix u a E b O c =, 72 O d v a where O is an m m zero matrix. These solutions are related to the approximate nondimensional current components evaluated at the nodal points as E b U a z j = u a z j + κ + σ z j cosχ, V a z j = v a z j + κ + σ z j sinχ. 8

4 Numerical Testing: Constant Eddy Viscosity We illustrate the accuracy of the Sinc-Galerkin method when applied to the constant eddy viscosity model formulated as a system of real-valued coupled ordinary differential equations. Since the governing equations and variables were nondimensionalized, the only operative constants in 2-25 are κ, σ, and χ. In relating these parameters to the constants of nature, we adopt the following nominal values: f =. s appropriate to temperate northern latitudes, sea water density ρ = 3 kg m 3, and air density ρ air =.25 kg m 3. Surface wind stress is assumed to be related to the square of the wind speed W w in m s by τ w = C D ρ air W 2 w, 73 where the dimensionless parameter C D.2 for W w < 2 m s, thereafter increasing linearly to about.25 at gale force winds W w 3 m s [5]. In keeping with [6] and [7], the linear slip bottom stress coefficient k f is assigned a value of.2 m s for comparison with other work and to dramatize the change in current speed over the water column. In practice, a value of k f lower by an order of magnitude may be preferred. Field evidence suggests that the near-surface value of the vertical eddy viscosity is related to the wind speed. Carter [8] has suggested that, if the wind is not fetch-limited and the sea state is fully developed, then A v in units of m 2 s is given by A v.34 4 W 3 w. 74 With the parameters and relationships above, and keeping in mind our desire to compare results with those in [9], we choose our constant eddy viscosity to be A v z.2 m 2 s 75 with τ w = 2/ =.44 N m 2. Since D E = 2A v/f = 2 m, we then have κ = D /D E = D /2. In keeping with [9], we will use D = m and hence κ = 5, which it will be throughout. The numerical results are compared to the exact solution W z = U [Uz+ iv z] where Uz is given by and V z is given by Uz = RW c z cosχ IW c z sinχ 76 V z = RW c z sinχ + IW c z cosχ. 77 Here RW c z and IW c z denote the real and imaginary parts of W c z, 9

respectively, and W c z = κ iσcosh κ i z + sinh κ i z. i[cosh κ i + κ iσsinh κ i] The results of the Sinc-Galerkin approximations U a z j and V a z j were compared with the exact solutions for Uz j and V z j at the sinc grid points S with h = π/ 2N given by S = { z j = φ jh = e jh /e jh + : j = N,..., N + }. 78 These results were then multiplied by the natural velocity scale U to give a dimensional representation of the velocities. All numerical simulations were run on a SUN BLADE with MATLAB Version 6.. To illustrate the performance of the method, the maximum absolute errors are reported as and where the units are m s. U S = max {U U a z j Uz j }, N j N+ V S = max {U V a z j V z j }, N j N+ E S = max { U S, V S }, 79 Throughout, comparable graphs of eddy viscosity functions and velocity components are shown on the same scale. The horizontal projections of the Ekman spirals are also shown on the same scale. This way visual comparisons of these various quantities are readily made. Example Linear stress condition at the seabed For this example we choose χ = 45 o and for the linear stress condition at the seabed we have σ = A v D /k f D =.. We find the approximate solution U a z and V a z, respectively, using the coupled discrete system of size 2m 2m m = 2N + 3, given in 69 where U U a z = Ua z and U V a z = Va z. The errors are given in Table and illustrate the classic exponential convergence typical of Sinc-Galerkin methods. Figure 5 graphically depicts the numerical convergence of the Sinc-Galerkin method to the exact solution with the linear stress bottom boundary condition, as N is repeatedly doubled in size. The horizontal projection of the Ekman spiral for N = 64 is indistinguishable from the true solution shown as the solid line. 2

N 2m h U S m s V S m s E S m s PSfrag replacements -.2 4 22..8e-3 7.478e-4.8e-3 8 38.785 2.496e-4.293e-4 2.496e-4 6 7.555 2.762e-5.32e-5 2.762e-5 32 34.393 8.985e-7 4.24e-7 8.985e-7 64 262.278 5.775e-9 2.77e-9 5.775e-9 28 58.96 4.53e-2.937e-2 4.53e-2 Table Errors for Example constant eddy viscosity on the sinc grid S with the linear stress bottom condition for σ =., χ = 45 o, κ = 5, D = m, D E = 2 m Sinc-Galerkin Ekman spiral projections for increasing N Northward current component U a m/s..5.5 -. -.5 -.2 -.25 -.3 -.35 N = 4 N = 8 N = 6 N = 32 N = 64 True -.4 -.2.2.4.6.8..2 Eastward current component V a m/s Fig. 5. Sinc-Galerkin Ekman spiral projections for Example with increasing N for constant eddy viscosity with the linear stress bottom boundary condition for σ =., χ = 45 o, κ = 5, D = m, D E = 2 m Example 2 No-slip condition at the seabed We set σ = and χ = 45 o and solve for approximate solutions U a z and V a z, respectively, by using the coupled discrete system in 69. The reported errors are shown in Table 2 and are very similar to those for Example. The accuracy is no different for the Neumann condition in this example than it is for the mixed condition in Example. The horizontal projection of the Ekman spirals is shown in Figure 6. This figure portrays the exponential convergence of the method. 2

N 2m h U S m s V S m s E S m s PSfrag replacements -.2 4 22..7e-3 7.5e-4.7e-3 8 38.785 2.483e-4.297e-4 2.483e-4 6 7.555 2.752e-5.36e-5 2.752e-5 32 34.393 8.957e-7 4.255e-7 8.957e-7 64 262.278 5.756e-9 2.78e-9 5.756e-9 28 58.96 4.73e-2.975e-2 4.73e-2 Table 2 Errors for Example 2 constant eddy viscosity on the sinc grid S with the zerovelocity bottom condition for σ =, χ = 45 o, κ = 5, D = m, D E = 2 m Sinc-Galerkin Ekman spiral projections for increasing N Northward current component U a m/s..5.5 -. -.5 -.2 -.25 -.3 N = 4 N = 8 N = 6 N = 32 N = 64 True -.35 -.4 -.2.2.4.6.8..2 Eastward current component V a m/s Fig. 6. Sinc-Galerkin Ekman spiral projections for Example 2 with increasing N for the case of constant eddy viscosity with no-slip bottom boundary condition for σ =, χ = 45 o, κ = 5, D = m, D E = 2 m 5 Numerical Testing: Variable Eddy Viscosity The following examples show the results for the system of real-valued coupled ordinary differential equations with either a decreasing or quadratic eddy viscosity function given by 8 or 8. An eddy viscosity which decreases quadratically from the value of A v =.2 m 2 s to the minimum value of 22

PSfrag replacements A v D =.25 m 2 s is given by A vz =.2[.75z ] 2, < z < D = m. 8 A graph of this A vz is shown in Figure 7 where it is contrasted with the constant eddy viscosity of A v z.2 m 2 s. A quadratic eddy viscosity Decreasing eddy viscosity function 2 Decreasing A v z Constant A v z 3 Depth z m 4 5 6 7 8 9..2.3.4.5.6.7.8.9. Eddy viscosity function A vz m 2 s Fig. 7. Eddy viscosity functions A v z =.2 [.75z ] 2 m 2 s and A v z.2 m 2 s function that ranges from A v =.2m 2 s back to A vd =.2 m 2 s is given by A vz =.2[ +.2z.z ], < z < D = m, 8 which is shown in Figure 8. The model with the decreasing and quadratic eddy viscosity functions are solved by the coupled discrete system in 69 of size 2m 2m m = 2N + 3. Also the results from the coupled system scheme are compared to the results in [] which used a complex-valued ordinary differential equation. Example 3 For this example in a sea of depth D = m, the parameters are chosen to be σ =., χ = 45 o, and since D E = 2 m κ = 5. The decreasing eddy viscosity function A v z is given by 8. We find the approximate solutions U a z and V a z by using the coupled discrete system in 69. The results comparing the constant eddy viscosity and the decreasing eddy viscosity are shown in Figure 9 and Figure for N = 32. Figure 9 contrasts the Ekman spiral projections while Figure depicts the velocity components Ua z and Va z from the surface to the sea bottom. The convergence of 23

Quadratic eddy viscosity function Quadratic A v z Constant A v z 2 3 Depth z m 4 5 6 7 8 9..2.3.4.5.6.7.8.9. Eddy viscosity function A vz m 2 /s Fig. 8. Eddy viscosity functions A v z =.2[ +.2z.z ] m 2 s and A v z.2 m 2 s PSfrag replacements the Sinc-Galerkin Ekman spiral projections for the decreasing eddy viscosity function in 8 is illustrated in Figure. -.2 Sinc-Galerkin Ekman spiral projections.2.2 Northward current component U a m/s. -. -.2 -.3 Ekman spiral projection for quadratic A v z Ekman spiral projection for constant A v z -.4 -.2.2.4.6.8..2 Eastward current component V a m/s Fig. 9. Sinc-Galerkin Ekman spiral projections for Example 3 with both constant and decreasing eddy viscosity functions and linear stress bottom boundary condition for σ =., χ = 45 o, κ = 5, D = m, D E = 2 m 24

Sinc-Galerkin velocity components 2 3 PSfrag replacements -.2 Depth z m 4 5 6 7 8 9 U a for decreasing A v z V a for decreasing A v z U for constant A v z V a for constant A v z -. -.8 -.6 -.4 -.2 Velocity components U a and V a.2.4.6.8. m/s Fig.. Sinc-Galerkin northward and eastward calculated velocity profiles for Example 3 with constant and decreasing eddy viscosity functions and linear stress bottom boundary condition for σ =., χ = 45 o, κ = 5, D = m, D E = 2 m Sinc-Galerkin Ekman spiral projections for increasing N Northward current component U a m/s..5.5 -. -.5 -.2 -.25 -.3 -.35 N = 4 N = 8 N = 6 N = 32 N = 64 True -.4 -.2.2.4.6.8..2 Eastward current component V a m/s Fig.. Sinc-Galerkin Ekman spiral projections for Example 3 with increasing N for the case of the decreasing eddy viscosity function and linear stress bottom boundary condition for σ =., χ = 45 o, κ = 5, D = m, D E = 2 m Example 4 With the quadratic eddy viscosity function in 8, the parameters are again chosen to be σ =., χ = 45 o, and κ = 5. The approximate solutions U a z and V a z are illustrated by graphs comparing the constant eddy viscosity with the quadratic eddy viscosity in Figure 2 and Figure 3 for N = 32. The convergence of the Sinc-Galerkin Ekman spiral projections 25

PSfrag replacements for the quadratic eddy viscosity function in 8 is illustrated in Figure 4. -.2 The approximations are calculated from the 2m 2m coupled discrete system in 69..2 PSfrag replacements Northward current component U a m/s.2. -. -.2 -.3 Sinc-Galerkin Ekman spiral projections Ekman spiral projection for quadratic A v z Ekman spiral projection for constant A v z -.4 -.2.2.4.6.8..2 Eastward current component Va m/s Fig. 2. Sinc-Galerkin Ekman spiral projections for Example 4 with both constant and quadratic eddy viscosity functions and linear stress bottom boundary condition for σ =., χ = 45 o, κ = 5, D = m, D E = 2 m Sinc-Galerkin velocity components 2 3 Depth z m 4 5 6 7 8 9 U a for quadratic A v z V a for quadratic A v z U a for constant A v z V a for constant A v z -. -.8 -.6 -.4 -.2 Velocity components U a and V a.2.4.6.8. m/s Fig. 3. Sinc-Galerkin northward and eastward calculated velocity profiles for Example 4 with constant and quadratic eddy viscosity functions and linear stress bottom boundary condition for σ =., χ = 45 o, κ = 5, D = m, D E = 2 m 26

Sinc-Galerkin Ekman spiral projections for increasing N Northward current component U a m/s..5.5 -. -.5 -.2 -.25 -.3 -.35 N = 4 N = 8 N = 6 N = 32 N = 64 True -.4 -.2.2.4.6.8..2 Eastward current component V a m/s Fig. 4. Sinc-Galerkin Ekman spiral projections for Example 4 with increasing N for the case of the quadratic eddy viscosity function and linear stress bottom boundary condition for σ =., χ = 45 o, κ = 5, D = m, D E = 2 m Next, we want to compare the approximate solutions U a z and V a z for the complex velocity system in [] and the new coupled system introduced here. To illustrate the comparative performance of both methods, the maximum absolute errors between their respective numerical approximations are reported as U C = max {U U z j U 2 z j }, N j N+ V C = max {U V z j V 2 z j }, N j N+ E C = max { U C, V C }, where U z and V z are computed by the complex velocity system in [] and U 2 z and V 2 z are computed by the coupled system in 69. Example 5 This example compares the solution introduced here to that computed by the complex velocity system in []. The example used is that described in Example. All comparisons show a similarity greater than the accuracy of the method reported in Table 27

N m 2m U C V C E C 4 22.5e-5 8.63e-6.5e-5 8 9 38 9.49e-6.93e-5.93e-5 6 35 7 6.279e-6 6.27e-5 6.27e-5 32 67 34.3e-5 2.9e-5 2.9e-5 64 3 262 5.65e-5.55e-4.55e-4 28 259 58.463e-4.758e-4.758e-4 Table 3 Comparison between the approximate solutions for the complex velocity system and the coupled system by using the same sinc grid size for Example for the case of constant eddy viscosity with linear stress bottom boundary condition for σ =., χ = 45 o, κ = 5, D = m, D E = 2 m Example 6 This example compares the solution introduced here to the complex velocity system in []. The comparison used Example 4. Again the results are extremely similar. Note that the method used here requires none of the complex arithmetic necessary in the work of []. N m 2m U C V C E C 4 22.884e-6 9.49e-6 9.49e-6 8 9 38 3.768e-6.93e-5.93e-5 6 35 7 2.98e-6 3.579e-5 3.579e-5 32 67 34.38e-5 2.26e-5 2.26e-5 64 3 262 4.969e-5.68e-4.68e-4 28 259 58 4.458e-4 2.3e-4 4.458e-4 Table 4 Comparison between the approximate solutions for the complex velocity system and the coupled system on the same sinc grid for Example 4 for the decreasing eddy viscosity A vz =.2 +.2z.z with linear stress bottom boundary condition for σ =., χ = 45 o, κ = 5, D = m, D E = 2 m 28

References [] D. F. Winter, J. Lund, and K. L. Bowers. Wind-driven currents in a sea with a variable eddy viscosity calculated by a Sinc function Galerkin technique. Internat. J. Numer. Methods Fluids, 33:4-73, 2. [2] J. Brown, A. Colling, D. Park, J. Phillips, D. Rothery, and J. Wright. Ocean Circulation, The Open University, Keynes, 989. [3] J. Lund and K. L. Bowers. Sinc Methods for Quadrature and Differential Equations, SIAM: Philadelphia, 992. [4] F. Stenger. Numerical Methods Based on Sinc and Analytic Functions, Springer- Verlag, New York, 993. [5] W. G. Large and S. Pond. Open ocean momentum flux measurements in moderate to strong winds. J. Phys. Oceanogr, :324-336, 98. [6] A. M. Davies and A. Owen. Three-dimensional numerical sea model using Galerkin method with a polynomial basis set. Appl. Math. Modeling, 3:42-428, 979. [7] N. S. Heaps. On the numerical solution of the three-dimensional hydrodynamical equations for tides and storm surges. Mem. Soc. Sci. Liege., Ser. 6, :43-8, 97. [8] D. J. T. Carter. Estimation of wave spectra from wave height and period. Inst. Oceanogr. Sci. Rep, 35, 982. [9] C. E. Naimie. A Turbulent Boundary Layer Model for the Linearized Shallow Water Equations, NUBBLE USER S MANUAL Release.. Technical Report NML-96-, Dartmouth College, July 3, 996. 29