- Email: [email protected]
Analytical solutions for storm tide codes
Coastal Engineering 46 (2002) 213 – 231 www.elsevier.com/locate/coastaleng
Analytical solutions for storm tide codes Rodney J. Sobey * Department of Civil and Environmental Engineering, University of California, 539 Dacis Hall, Berkeley, CA 94720, USA Received 29 August 2001; received in revised form 24 May 2002; accepted 4 June 2002
Abstract A sequence of analytical solutions explore aspects of response patterns expected from numerical codes for storm tides in one- and two-dimensional basins. Complete analytical details of the solutions are provided, together with specific suggestions for an associated set of analytical benchmark tests. Illustrations of predicted response patterns provide the basis for a discussion of many significant physical aspects and their representation in discrete numerical codes. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Analytical solution; Benchmark problems; Storm tide; Long wave; Numerical code; Shallow water
1. Introduction Numerical modeling has evolved to become the tool of choice in studies of storm tide influences in lakes and coastal basins. It has become sufficiently routine that questions of model credibility are often overlooked. The trend in modern codes has been increasing complexity in the physical processes and increasing length and complexity in the code and in the graphical interfaces. While numerical models have the potential to significantly enhance our predictive capability, their increasing size and complexity must also enhance the opportunity for both error (inappropriate numerical formulation, coding errors, inadequate grid resolution,. . .) and misuse. As a measure of physical and code credibility in rational model evaluation, a sequence of well-defined
*
Tel.: +1-510-642-3162; fax: +1-510-642-7483. E-mail address: [email protected] (R.J. Sobey).
analytical benchmark problems is proposed. These benchmark problems are analytical in the sense that each problem has an exact analytical solution. Analytical solutions alone have absolute credibility. A numerical code must be modified to exactly match the context of an analytical solution. But then the numerical and analytical solutions should match exactly. Any differences can be attributed to the code. This paper will introduce a sequence of analytical benchmark problems that are appropriate for numerical codes that focus on storm tides and closely related problems. Six application problems explore the underlying physical process and the interaction with the operational context of a numerical model. Numerical models of storm tide circulation are driven by a combination of internal meteorological forcing and boundary conditions. Internal meteorological forcing is the unique feature of storm tides. Boundary forcing is familiar from problems associated with astronomical tide circulation. But storm tides introduce some rather different challenges, as these flows are often
0378-3839/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 3 9 ( 0 2 ) 0 0 0 9 3 - 5
214
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
associated with transient boundary forcing. Detailed attention is given to response patterns associated with schematic meteorological forcing and a range of common boundary conditions. For each problem, the complete analytical solution is given for both water surface elevation g(xa, t) and flow qa(xa, t), in a manner immediately suitable for coding. The problems include the response to a sudden uniform wind on a closed and on a partially open basin, the response to a moving front across a closed and across a partially open basin, and the steady-state open coast storm tide forced by a spatially varied wind.
2. Field equations for storm tides For coastal basins (see Fig. 1), the incompressible flow mass and momentum conservation equations are the storm tide equations, B Bqx Bqy ðh þ gÞ þ þ ¼0 Bt Bx By 2 Bqx B qx B qx qy þ þ f C qy Bx h þ g By h þ g Bt B ps Rsx Rbx gþ ¼ gðh þ gÞ þ Bx qg q q
Bqy B qx qy þ Bx h þ g Bt
q2y
ps qg
þ
Rsa q
ð2Þ
where surface pressure ps(xa, t) and surface shear stress Rsa(xa, t) completely define the surface stress structure. The water surface will respond to surface shear directly and to horizontal spatial gradients of the surface pressure. Note in particular that meteorological forcing is predominantly internal to a solution domain. For typical astronomical tide, flood and tsunami propagation problems, the forcing is predominantly external to a solution domain, and imposed through the open boundary conditions. But open boundary conditions remain crucially important in storm tide problems, to represent meteorological influence beyond a truncated solution domain and to represent simultaneous tides. A conceptually useful approximation is available from a linearized form of these equations. With the following assumptions: advective accelerations are negligible, Coriolis accelerations are negligible, AgAbh0, and bed shear can be linearized to Rba/q = kqa, where k is a linear friction factor.
ð1Þ
in which g(xa, t) is the local water surface elevation and qa(xa, t) are the local depth-integrated flows (see
Fig. 1. A coastal basin.
B Fa ¼ gðh þ gÞ Bxa
!
B þ þ fC qx By h þ g Rsy Rby B ps gþ ¼ gðh þ gÞ þ By qg q q
Fig. 2), fC = 2X sin U is the Coriolis parameter, ps is the atmospheric pressure at the water surface, Rs is the surface wind shear, and Rb is the bed shear. (x, y) = xa, a = 1 and 2, is the local horizontal position and t is time. The meteorological forcing is the vector
Eq. (1) may be approximated as Bg Bqx Bqy þ þ ¼0 Bt Bx By
ð3aÞ
Bqx Bg kqx þ Fx ¼ gh Bx Bt
ð3bÞ
Bqy Bg kqy þ Fy ¼ gh By Bt
ð3cÞ
These are the linearized storm tide equations. Recall that the meteorological forcing, Eq. (2), includes both
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
215
Fig. 2. Definition sketches for storm tide equations.
surface pressure and surface wind shear. The Coriolis accelerations are linear terms, but their retention would preclude some useful analytical solutions. Except for the advective and Coriolis accelerations, these linearized equations retain all the complicated hyperbolic physics of storm tide evolution in a coastal basin. In addition, the linearization does not invalidate a numerical algorithm choice that was based on the complete Eq. (1). A numerical solution to Eqs. (3a) – (3c) imposes almost identical challenges.
arized Eq. (3b), k must be constant. Suitable predictive equations for k would be 8 f > > hqi > > 8h20 > > > < g k ¼ C2 h2 hqi > 0 > > > gn2 > > > : 7=3 hqi h0
for Darcy Weisbach model for Chezy model for Manning model ½SI units ð5Þ
3. Linear friction coefficient While linear friction is certainly a compromise, it must be recalled that quadratic friction also is not entirely satisfactory. The utility of the linear approximation will be enhanced by realistic estimates of k. Equating the linear and Darcy –Weisbach, Chezy or Manning estimates give 8 f > > AqA for Darcy Weisbach model > > > 8ðh þ gÞ2 > > < g k ¼ C2 ðh þ gÞ2 AqA for Chezy model > > > > gn2 > > > AqA for Manning model ½SI units : ðh þ gÞ7=3 ð4Þ in which f, C and n are the Darcy– Weisbach, Chezy and Manning friction factors, respectively. In the line-
where hqi is a discharge scale that is spatially and temporally averaged over the local flow. Suitable estimators for hqi are Z T Z LZ L 1 AqA dx dy dt; L2 T 0 0 0
1=2 Z T Z LZ L 1 2 q dx dy dt L2 T 0 0 0 Z T Z LZ L AqAq2 dx dy dt or 0Z T0Z L0Z L ð6Þ q2 dx dy dt 0
0
0
The first and second are absolute mean and RMS values. The third balances the total work done by the boundary shear from the linear and quadratic approximations. If the tidal flow is periodic with constant amplitude qˆ throughout, these estimators become 2ˆq=
216
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
pffiffiffi p; qˆ = p and 8ˆq=ð3pÞ , respectively. Adopting the latter gives 8 f qˆ > for Darcy Weisbach model > > > 3ph20 > > > < 8gqˆ for Chezy model ð7Þ k¼ 3pC2 h20 > > > 2 > > 8gn qˆ > > for Manning model ½SI units : 7=3 3ph0 4. Code modifications Analytical solutions can be used for numerical code evaluation in either of two ways. (1) Make no changes to the numerical code. Average values for h0 and k would be adopted. The analytical and numerical solutions should have trend agreement, but they will not be identical. Such a comparison is valuable, but not absolute. (2) Modify the numerical code to be a solution to the linearized equations. The numerical code would be modified to be a numerical solution of the linearized equations. A comparison of analytical and numerical solutions should then be absolute. The necessary code modifications certainly provide the opportunity for coding error, but access to an exact solution should facilitate the identification of any such errors. Both modes have value. The latter is absolute, and must be preferred. But the code modifications must be carefully undertaken. Four changes are necessary: (i) Omit the advective acceleration term (B/Bxa) (( qaqb)/(h + g)) in the momentum equation. This term is strongly nonlinear (quadratic in qa), and is often a problem in numerical codes like finite difference and finite element that approximate Eq. (1) as simultaneous linear algebraic equations in the nodal g’s and qa’s. In most situations, this term is a small contributor to the momentum balance, and many codes have an existing option to exclude the advective acceleration. (ii) Omit the Coriolis acceleration term ( fCqy, fCqx) in the momentum equation. This term is linear, and is not a problem in numerical codes. But the time scale for geostrophic influences forced by the Coriolis acceleration is the inertial period 2p/AfCA. The duration of storm tide forcing in coastal
basins is often much shorter than this, and geostrophic influences are not strong. Latitude U would be an input parameter to any numerical code, and the Coriolis acceleration is easily omitted by setting U to F 90j. Alternatively, fC could be set to zero directly. (iii) Change the friction term from Rba/q to kqa in the momentum equation. Many codes have an optional choice of friction formula, Darcy –Weisbach, Chezy or Manning. An additional option would not be difficult to include. (iv) Change the gravity term from g (h + g)(Bg/Bxa) to gh0(Bg/Bxa) in the momentum equations. The nonlinearity here is moderately weak, as Bg/ Bxa will be the more rapidly varying part. Typical finite difference and finite element codes would locally linearize this term, by adopting a locally constant approximation to (h(x, t) + g(x, t)). It would not be difficult to modify this code segment to always return a globally constant h0 in place of local estimates.
5. Storm tides in one-dimensional basins For the special case of a narrow one-dimensional basin, qy = Fy = 0, qx = q, and Fx = F. Eqs. (3a) –(3c) become Bg Bq þ ¼0 Bt Bx Bq Bg ¼ gh0 kq þ F Bt Bx
ð8Þ
The general analytical solution for g(x, t) of Eq. (8) is gðx; tÞ ¼ mx þ b þ a1 expðlxÞcosðkx xt þ /1 Þ þ a2 expðþlxÞcosðkx þ xt þ /2 Þ " l X gˆ n þ kfˆn =2 fˆn expðkt=2Þcosxn t þ þ xn n¼0 Z t 1 expðkt=2Þsinxn t þ 0 xn expðkðt sÞ=2Þsinxn ðt sÞ/ˆ n ðsÞds Z t 1 expðkðt sÞ=2Þ þ 0 xn # sinxn ðt sÞwn ðsÞds Xn ðxÞ
ð9Þ
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2n C 2 ðk=2Þ2
l " X þ fˆn expðkt=2Þcosxn t n¼1
gˆ n þ kfˆn =2 þ expðkt=2Þsinxn t xn
þ
#
sinxn ðt sÞwn ðsÞds Xn ðxÞ
ð10bÞ
for the forced and free modes, respectively; C=(gh0)1/2. These forms are special cases of the same generalized dispersion relationship. Sobey (2002) gives the complete details, including the definition of the solution parameters m, b, a1, /1, a2, /2, bn and the functions wn(t), and Xn(x) from the boundary conditions, the ˆ definition of the functions /n(t) from the meteorological forcing, and the definition of the parameters ˆfn and gˆn from the initial conditions. Terms 1 through 4 are contributed by the tidal boundary conditions. Terms 1 and 2 describe the steady gradually varied flow profile. Terms 3 and 4 are the tidal forced modes. Modes in the boundary conditions will appear in the response. Simultaneous tidal forcing could easily be included in the following discussion, but is not, to focus attention on the meteorological forcing. Within the summation, terms 5 and 6 are contributed by the initial conditions. Term 7 is contributed by the meteorological forcing, and term 8 is contributed by the residual (non-tidal) boundary conditions, such as tsunami penetration. Free mode responses at the (xn, bn) eigenmodes are excited by the initial conditions, by the meteorological forcing, and by nontidal transient boundary forcing. The dispersion relationship, Eqs. (10a) and (10b), relates space and time periodicities in both the free and forced modes. Without friction, it has the classical long wave form x = Ck. The general analytical solution for q(x, t) of Eq. (8) is qðx; tÞ ¼ b þ a1 expðlxÞcosðkx xt þ /1 Þ þ a2 expðþlxÞcosðkx þ xt þ /2 Þ
t
1 expðkðt sÞ=2Þ 0 xn sinxn ðt sÞ/ˆ n ðsÞds Z t 1 þ expðkðt sÞ=2Þ 0 xn
The dispersion relationship, relating space and time periodicities, has two forms " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #1=2 2 x 1 þ 1 þ ðk=xÞ k¼ ð10aÞ C 2 xn ¼
Z
217
ð11Þ
in which b, a1, /1, a2, /2, ˆfn, gˆn, wn and Xn(x) are redefined in relation to q. But they must be carefully coordinated to be exactly consistent with the conditions imposed for the g(x, t) solution. The dispersion relationship remains unchanged, Eqs. (10a) and (10b). Again, complete details are given in Sobey (2002). Solution of Eq. (8) is an initial, boundary value problem. The solutions for g and q change with both the initial conditions and the boundary conditions.
6. Termination of free mode summations The free modes in the Eq. (9) analytical solution for g and Eq. (11) analytical solution for q involve an infinite summation of eigenmode contributions. The free mode summations may be expanded and written as ) gVðx; tÞ l " X ¼ expðkt=2Þ a1n cosxn t n¼1 qVðx; tÞ #
þ a2n sinxn t Xn ðxÞ
ð12Þ
where 1 a1n ¼ fˆn xn
a2n
Z
t
"
#
expðks=2Þsinxn s /ˆ n ðsÞ þ wn ðsÞ ds
0
Z t " 1 gˆ n þ kfˆn =2 ¼ þ expðks=2Þcosxn s /ˆ n ðsÞ xn xn 0 #
þ wn ðsÞ ds
ð13Þ
Both the /ˆ n(t) and wn(t) functions would have finite durations, so that the time integrals in Eq. (13) would be terminated at the larger of these durations.
218
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
Fig. 3. Narrow rectangular basins.
The free modes have frequencies xn and amplitudes "
An ¼ a21n þ a22n
#1=2
ð14Þ
The frequencies are the same for the g and q solutions, but neither the magnitude nor the units of the amplitudes. The actual eigenmode amplitudes depend on the initial conditions, the meteorological forcing and the transient boundary conditions, but their magnitudes will generally decrease with increasing mode number, though not always monotonically. How far it is necessary to take the summations will be problem dependent. For the present purposes, convergence of the summation is assumed at M terms when AM 1 þ AM V0:01 2A1
These problems are intended as a supplement, not a substitute, for a sequence of numerical benchmark tests. It is anticipated that the analytical problems identified here will be of primary benefit in initial model development, and in the evaluation of userspecific variations or subsequent versions that introduce new physical, geometrical, numerical or graphical capabilities. Each test has a limited objective, seeking to focus on crucial problems in relative isolation. Each of the following problems have been given an ST identifier, suggesting their relevance to codes principally intended for storm tide-related problems. The Class I (one-dimensional) problems assume a schematic one-dimensional basin, either closed at both ends (Fig. 3a) or partially open (Fig. 3b), open at x = 0 but closed at x = L. h0 and k are known.
ð15Þ 8. ST1: Sudden uniform wind on a closed basin
The terminal M will be reported with each application.
7. Analytical benchmark tests A sequence of analytical benchmark tests have been designed to spotlight some physically and numerically significant response patterns that are expected to be within the predictive capabilities of depth-integrated models of storm tide gradually varied, unsteady flow responses to meteorological forcing. Two classes of problems are addressed: (I) unsteady flow in one-dimensional basins (II) steady flow at an open coast. Consideration is necessarily limited to contexts for which analytical solutions are possible.
Problem ST1, outlined in Table 1, is perhaps the simplest wind tide problem. It is the sudden imposition of a uniform meteorological forcing to a narrow basin of constant depth h0 that is closed at both ends, Fig. 3a. The forcing is Fðx; tÞ ¼ F0 HðtÞ
ð16Þ
where H(t) is the Heaviside or unit step function. The magnitude of the meteorological forcing in Table 1 corresponds to a moderate wind speed of order 25 m/s. In the analytical solution for g, the initial and boundary conditions are Bg gðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx;0Þ ð17Þ Bg F0 Bg F0 ¼ 2; ¼ 2 Bx ðx¼0;tÞ C Bx ðx¼L;tÞ C
R.J. Sobey / Coastal Engineering 46 (2002) 213–231 Table 1 ST1: Sudden uniform wind on a closed basin xF
xL
h0
0
5000 m
2 m 1.5 10 3 m2/s2
F0
IC g(x, 0) = 0 BC q(xF, t) = 0 F F(x, t) = F0H(t)
tF
DtOutput tL
0
60 s
2h
q(x, 0) = 0 q(xL, t) = 0
The meteorological forcing is /(x, t) = BF/Bx = 0, and forcing is imposed only through the boundary conditions. For the forced mode, boundary condition matching gives m = F0/C2 and b = mL/2. For the free modes, the eigenvalues and eigenfunctions are 1=2 1 b0 ¼ 0; X0 ðxÞ ¼ L 1=2 np 2 ; Xn ðxÞ ¼ bn ¼ cosbn x; n ¼ 1; 2; 3; . . . L L ð18Þ The modal coefficients are Z
L
0
F0 C2
and the meteorological forcing is /(x, t)= + BF/ Bt = F0d(t). For the forced mode, b = 0. For the free modes, the eigenvalues do not change, but the eigenfunctions are 1=2 2 Xn ðxÞ ¼ sinbn x; n ¼ 1; 2; 3; . . . ð22Þ L The modal coefficients are ˆfn = wn = 0, and
k = 10 4 s 1.
fˆn ¼
219
x
L 2
Xn ðxÞdx;
gˆ n ¼ 0
ð19Þ
Also /ˆn = wn = 0. The analytical solution is F0 L gðx; tÞ ¼ 2 x þ fˆ0 X0 þ expðkt=2Þ 2 C " # l X ˆn k f sinxn t Xn ðxÞ fˆn cosxn t þ 2xn n¼1 ð20Þ The ˆf0 is zero in this case, so that the n = 0 mode could be omitted. The forced mode scales as F0L/C2, as do the free modes. In the analytical solution for q, the initial and boundary conditions are qðx; 0Þ ¼ f ðxÞ ¼ 0; qðx ¼ 0; tÞ ¼ 0;
Bq ¼ gðxÞ ¼ F0 Bt ðx;0Þ qðx ¼ L; tÞ ¼ 0
ð21Þ
gˆ n ¼
Z
L
F0 Xn ðxÞdx;
/ˆ n ¼ F0 dðtÞ
0
Z
L
Xn ðxÞdx
0
ð23Þ but the /ˆn contribution to the free modes becomes identically zero after time integration. The analytical solution for q(x, t) is then qðx; tÞ ¼ expðkt=2Þ
l
X gˆ n n¼1
xn
sinxn t Xn ðxÞ
ð24Þ
This solution scales as F0L/C. Eq. (18) lists the discrete eigenmodes or resonant modes of the closed basin. The wave lengths of the eigenmodes are 2p/bn, where the bn are the wave numbers. The dispersion relationship, Eq. (10b), relates the discrete frequencies xn to the wave numbers and the basin friction. In this particular case, the meteorological forcing F is introduced through the boundary conditions for the forced mode, and through the initial conditions for the free modes. It is immediately clear from Eqs. (20) and (24) that the free or resonant modes of the basin have a very fundamental role in the response. The solution describes an unsteady and spatially varied response that results from the superposition of free standing wave modes that are oscillatory in both space and time. The waves are exponentially damped in time, but at a time scale Tf ¼
2 k
ð25Þ
that is expected to be rather slow. The specific response is illustrated in Fig. 4.The initial transients in g and q are the eigenmode responses. The periods Tn = 2p/xn of the eigenmodes
220
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
Fig. 4. Sudden imposition of a uniform wind stress on a closed basin.
are 0.63, 0.31, 0.21,. . . h for n = 1, 2, 3, . . .. The primary mode T1 dominates Fig. 4, though the important role of numerous higher modes is guaranteed by the sharp saw-tooth-style response, especially for g. Eight (M = 8) eigenmodes were included to achieve the Eq. (15) convergence. The saw-tooth wave response reflects the initial step excitation; in the Eqs. (20) and (24) analytical solution, it is represented as the summation over the M eigenmodes. The initial transient response decays slowly with time at time scale Tf = 5.6 h. This trend can just be seen at the short 2-h duration of Fig. 4, most clearly in the g evolution. The steady-state wind tide, for tHTf = 2/k, is
¯ gðxÞ ¼
F0 C2
x
L 2
This is a lightly damped system, and the initial transients significantly overshoot the steady-state response. This is a repeating observation on storm tides. The peak response is most likely associated not with the steady forced mode but with the free mode response excited by the typically transient character of the meteorological forcing.
9. ST2: Moving pulse across a closed basin Problem ST2, outlined in Table 2, is a schematic representation of an intense storm moving across a closed basin. An intense moving storm can be represented (Chrystal, 1908) as the schematic moving pulse (see Fig. 5)
ð26Þ Fðx; tÞ ¼ F0 ½Hðx þ a VtÞ Hðx VtÞ
ð28Þ
The maximum sustained setup is g¯ max ¼
F0 L 2C 2
at x = L.
ð27Þ
in which F0 is the magnitude of a rectangular pulse in meteorological forcing of width a that is moving across the basin at speed V. The leading edge is moving along the path x = Vt, the trailing edge along
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
221
Table 2 ST2: Moving pulse across a closed basin xF
xL
h0
F0
tF
DtOutput
tL
0
5000 m
2m
1.5 10 3 m2/s2
0
60 s
3h
IC BC F
g(x, 0) = 0 q(xF, t) = 0 F(x, t) = F0[H(x + a Vt) H(x Vt)]
q(x, 0) = 0 q(xL, t) = 0
a = 1500 m, V = 1 m/s, k = 10 4 s 1.
x = Vt a. The magnitude of the meteorological forcing in Table 2 again corresponds to a moderate wind speed of order 25 m/s. In the analytical solution for g, the initial and boundary conditions are Bg gðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx;0Þ Bg F0 h a i ð29Þ ¼ 2 HðtÞ H t ; Bx ðx¼0;tÞ C V
Bg F0 L Lþa ¼ H t H t Bx ðx¼L;tÞ C 2 V V
and the meteorological forcing is BF Bx ¼ F0 ½dðx þ a VtÞ dðx VtÞ
/ðx; tÞ ¼
ð30Þ
The boundary conditions are entirely transient, so that the forced mode is identically zero. For the free modes, the eigenvalues and eigenfunctions are
/ˆ n ðtÞ ¼
¼
1=2 Z L
2 F0 dðx VtÞ L 0 dðx þ a VtÞ cosbn x dx 1=2
2 L F0 cosbn Vt HðtÞ H t L V cosbn ðVt aÞ
a Lþa H t H t V V
ð33Þ
and
1=2 2 wn ðtÞ ¼ L
L Lþa n F0 ð1Þ H t H t V V h i a ð34Þ HðtÞ H t V The analytical solution for g(x, t) is then l Z t X 1 gðx; tÞ ¼ expðkðt sÞ=2Þ 0 xn n¼0 n o sinxn ðt sÞ /ˆ n ðsÞ þ wn ðsÞ ds Xn ðxÞ ð35Þ 2
1=2 1 b0 ¼ 0; X0 ðxÞ ¼ L ð31Þ 1=2 np 2 ; Xn ðxÞ ¼ bn ¼ cosbn x; n ¼ 1; 2; 3; . . . L L
This solution scales as F0L/C . In the analytical solution for q, the initial and boundary conditions are Bq qðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx; 0Þ ð36Þ qðx ¼ 0; tÞ ¼ 0;
qðx ¼ L; tÞ ¼ 0
and the meteorological forcing is The modal coefficients are fˆn ¼ gˆ n ¼ 0
BF ¼ F0 V ½dðx þ a VtÞ Bt dðx VtÞ
/ðx; tÞ ¼ þ ð32Þ
ð37Þ
222
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
1=2
2 L F0 V sinbn Vt HðtÞ H t ¼ L V
a Lþa sinbn ðVt aÞ H t H t V V
Fig. 5. Moving pulse.
ð40Þ The boundary conditions are identically zero, so that the forced mode is again identically zero. For the free modes, the eigenvalues do not change, but the eigenfunctions are 1=2 2 Xn ðxÞ ¼ sinbn x; n ¼ 1; 2; 3; . . . ð38Þ L
and wn(t) = 0. The analytical solution for q(x, t) is then qðx; tÞ ¼
l Z X
t
1 expðkðt sÞ=2Þ x n 0 n¼0 sinxn ðt sÞ/ˆ n ðsÞds Xn ðxÞ
ð41Þ
The modal coefficients are fˆn ¼ gˆ n ¼ 0;
/ˆ n ðtÞ ¼
1=2 Z L 2 F0 V ½dðx VtÞ L 0 dðx þ a VtÞsinbn x dx
ð39Þ
This solution scales as F0LV/C2. There is no steady-state wind setup, and the entire response is free mode transients. These respond at the resonant standing wave modes (bn, xn), which are initially strong, and decay slowly with friction through the term exp( kt/2). As is usual, the dominant free mode is the first, with wave number b1 and frequency x1.
Fig. 6. Moving pulse over a closed basin.
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
The specific response is illustrated in Fig. 6. M = 6 eigenmodes were included to achieve the Eq. (15) convergence. The front moves across the basin in (L + a)/V = 1.8 h and forces a significant initial transient that reflects the propagation path and width of the moving pulse. But the magnitude of the initial transient drops rapidly after passage of the pulse. The residual transients have a dominant periodicity at the first resonant mode, T1 = 2p/x1 = 0.6 h. The residual transients decay at the frictional time scale Tf = 2/k = 5.6 h slowly toward quiescence. Again, this is a lightly damped system. The front has passed in 1.8 h, but the steady state has not been reached at 3 h.
gives m = F0/C2 and b = 0. For the free modes, the eigenvalues and eigenfunctions are 1 1=2 n p 2 2 ; Xn ðxÞ ¼ bn ¼ sinbn x; L L n ¼ 1; 2; 3; . . . The modal coefficients are Z L F0 fˆn ¼ x Xn ðxÞdx; C2 0
10. ST3: Sudden uniform wind on a partially open basin
The meteorological forcing is /(x, t) = BF/Bx = 0. For the forced mode, boundary condition matching
xF
xL
h0
0
5000 m
2 m 1.5 10 3 m2/s2
IC g(x, 0) = 0 BC g(xF, t) = 0 F F(x, t) = F0H(t) k = 10 4 s 1.
F0
tF
DtOutput tL
0
60 s
q(x, 0) = 0 q(xL, t) = 0
3h
gˆ n ¼ 0
ð44Þ
F0 x þ expðkt=2Þ C2 " # l X ˆn k f sinxn t Xn ðxÞ ð45Þ fˆn cosxn t þ 2xn n¼1
The forced mode and the free modes again scale as F0L/C2. In the analytical solution for q, the initial and boundary conditions are Bq qðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ F0 Bt ðx;0Þ ð46Þ Bq ¼ 0; qðx ¼ L; tÞ ¼ 0 Bx ðx¼0;tÞ and the meteorological forcing is /(x, t)= + BF/ Bt = F0d(t). For the forced mode, b = 0. For the free modes, the eigenvalues do not change, but the eigenfunctions are 1=2 2 Xn ðxÞ ¼ cosbn x; n ¼ 1; 2; 3; . . . ð47Þ L The modal coefficients are fˆn = wn = 0, and Z L Z L F0 Xn ðxÞdx; /ˆ n ¼ F0 dðtÞ Xn ðxÞdx gˆ n 0
Table 3 ST3: Sudden uniform wind on a closed basin
ð43Þ
Also /ˆn = wn = 0. The analytical solution is gðx; tÞ ¼
Problem ST3, outlined in Table 3, is a variation on problem ST1. It is the sudden imposition of a uniform meteorological forcing to a narrow basin of constant depth h0 that is open at one end and closed at the other end, Fig. 3b. The forcing is Eq. (28). The magnitude of the meteorological forcing in Table 3 again corresponds to a moderate wind speed of or- der 25 m/s. This problem provides the opportunity to investigate different boundary conditions at xF, which in turn determines the character of the free mode responses. In the analytical solution for g, the initial and boundary conditions are Bg gðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx;0Þ ð42Þ Bg F0 gðx ¼ 0; tÞ ¼ 0; ¼ Bx ðx¼L;tÞ C 2
223
ð48Þ
0
but the /ˆn contribution to the free modes becomes identically zero after time integration. The analytical solution for q(x, t) is then l
X gˆ n qðx; tÞ ¼ expðkt=2Þ sinxn t Xn ðxÞ ð49Þ xn n¼1 This solution scales as F0L/C.
224
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
Eq. (43) lists the discrete eigenmodes or resonant modes of the partially open basin. The wave lengths of the eigenmodes are 2p/bn, where the bn are the wave numbers. The dispersion relationship, Eq. (10b), relates the discrete frequencies xn to the wave numbers and the basin friction. In this particular case, the meteorological forcing F is introduced through the boundary conditions for the forced mode, and through the initial conditions for the free modes. It is immediately clear from Eqs. (45) and (49) that the free or resonant modes of the basin once again have a very fundamental role in the response. The solution describes an unsteady and spatially varied response that results from the superposition of free standing wave modes that are oscillatory in both space and time. The waves are exponentially damped in time, but at time scale Tf, that is expected to be rather slow. The specific response is illustrated in Fig. 7. The initial transients in g and q are the eigenmode responses. The periods Tn = 2p/xn of the eigenmodes are 1.26, 0.42, 0.25,. . . h for n = 1, 2, 3,. . .. These differ from problem ST1 because of the different boundary conditions. The primary mode T1 dominates
Fig. 7, though the important role of numerous higher modes is again guaranteed by the sharp saw-toothstyle response, especially for g. Six (M = 6) eigenmodes were included to achieve the Eq. (15) convergence. The saw-tooth wave response reflects the initial step excitation; in the Eqs. (45) and (49) analytical solution, it is represented as the summation over the M eigenmodes. The initial transient response decays slowly with time at time scale Tf = 5.6 h. This trend can just be seen at the short 3-h duration of Fig. 7, most clearly in the g evolution. The steady-state wind tide, for tHTf = 2/k, is ¯ gðxÞ ¼
F0 x C2
ð50Þ
The maximum sustained setup is F0 L ð51Þ C2 at x = L. This is a lightly damped system, and the initial transients once again significantly overshoot the steady-state response. The peak response is most
g¯ max ¼
Fig. 7. Sudden imposition of a uniform wind stress on a partially closed basin.
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
225
Table 4 ST4: Moving pulse across a partially open basin xF
xL
h0
F0
tF
DtOutput
tL
0
5000 m
2m
1.5 10 3 m2/s2
0
60 s
3h
IC BC F
g(x, 0) = 0 g(xF, t) = 0 F(x, t) = F0[H(x + a Vt) H(x Vt)]
q(x, 0) = 0 q(xL, t) = 0
a = 1500 m, V = 1 m/s, k = 10 4 s 1.
likely associated not with the steady forced mode but with the free mode response excited by the typically transient character of the meteorological forcing.
/ˆ n ðtÞ ¼
dðx þ a VtÞsinbn x dx 1=2
2 L ¼ F0 sinbn Vt HðtÞ H t L V
a sinbn ðVt aÞ H t V Lþa H t ð55Þ V
11. ST4: Moving pulse across a partially open basin Problem ST4, outlined in Table 4, is a schematic representation of an intense moving storm across a partially open basin. The schematic moving pulse is represented in Fig. 5 and Eq. (28). The meteorological forcing is a rectangular pulse of width a that is moving across the basin at speed V. The magnitude of the meteorological forcing in Table 4 again corresponds to a moderate wind speed of order 25 m/s. In the analytical solution for g, the initial and boundary conditions are Bg gðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx;0Þ
gðx ¼ 0; tÞ ¼ 0;
Bg F0 L Lþa ¼ H t H t Bx ðx¼L;tÞ C 2 V V ð52Þ and the meteorological forcing is Eq. (30). The boundary forcing is entirely transient, so that the forced mode is identically zero. For the free modes, the eigenvalues and eigenfunctions are 1 1=2 n p 2 2 ; Xn ðxÞ ¼ bn ¼ sinbn x; L L n ¼ 1; 2; 3; . . .
ð53Þ
The modal coefficients are fˆn ¼ gˆ n ¼ 0
ð54Þ
1=2 Z L 2 F0 ½dðx VtÞ L 0
and
1=2
2 L F0 ð1Þn H t L V Lþa H t ð56Þ V The analytical solution for g(x, t) is then l Z t X 1 expðkðt sÞ=2Þ gðx; tÞ ¼ x n 0 n¼0 n o sinxn ðt sÞ /ˆ n ðsÞ þ wn ðsÞ ds Xn ðxÞ wn ðtÞ ¼
ð57Þ This solution scales as F0L/C2. In the analytical solution for q, the initial and boundary conditions are Bq qðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx;0Þ ð58Þ Bq ¼ 0; qðx ¼ L; tÞ ¼ 0 Bx ðx¼0;tÞ and the meteorological forcing is Eq. (37). The boundary conditions are identically zero, so that the forced mode is again identically zero. For the free modes, the eigenvalues do not change, but the eigenfunctions are 1=2 2 Xn ðxÞ ¼ cosbn x; n ¼ 1; 2; 3; . . . ð59Þ L
226
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
The modal coefficients are fˆn ¼ gˆ n ¼ 0; 1=2 Z L 2 /ˆ n ðtÞ ¼ F0 V ½dðx VtÞ L 0
ð60Þ
dðx þ a VtÞcosbn x dx 1=2
2 L ¼ F0 V cosbn Vt HðtÞ H t L V " a cosbn ðVt aÞ H t V ## Lþa H t ð61Þ V and wn(t) = 0. The analytical solution for q(x, t) is then l Z t X 1 expðkðt sÞ=2Þ qðx; tÞ ¼ 0 xn n¼0 sinxn ðt sÞ/ˆ n ðsÞds Xn ðxÞ ð62Þ This solution scales as F0LV/C2. There is no steady-state wind setup, and the entire response is free mode transients. These respond
at the resonant modes (bn, xn), which are initially strong, and decay slowly with friction through the term exp( kt/2). As is usual, the dominant free mode is the first, with wave number b1 and frequency x1. The specific response is illustrated in Fig. 8. M = 6 eigenmodes were included to achieve the Eq. (15) convergence. The moving pulse translates across the basin in (L + a)/V = 1.8 h and forces a significant initial transient that reflects the propagation path and width of the moving pulse. But the magnitude of the initial transient rapidly drops after passage of the pulse. The residual transients have a dominant periodicity at the first resonant mode, T1 = 2p/x1 = 0.6 h. The residual transients decay at the frictional time scale Tf = 2/ k = 5.6 h slowly toward quiescence. Again, this is a lightly damped system. The front has passed in 1.8 h, but the steady state (quiescence) has not been reached at 3 h. Reid (1956) has addressed a variation on this problem, in which the depth is linearly varying with x, but friction is neglected. He presents some approximate solutions, using the graphical method of characteristics.
Fig. 8. Moving pulse over a partially open basin.
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
227
12. Open coast storm tides
the field equation and the boundary conditions defines the following problem in Fourier transform space:
At open coasts, spatial variations in storm forcing in the longshore direction can be an important influence. In a land-falling hurricane, for example, there are regions of intense onshore winds to one side of the path (right in northern hemisphere) and offshore winds to the other side. Analytical solutions for steady-state wind tides at an open coast provide some schematic measure of the impact of longshore spatial structure of storm forcing. Dean and Pearce (1972) have formulated a schematic but useful open coast problem that is amenable to analytical solution. The context is sketched in Fig. 9. The shelf has a uniform slope, h(x, y) = h0 S0x, with no longshore variation. The storm forcing is steady, and represented as
d2 H S0 dH b2 H ¼ 0 dx2 h dx dH FðbÞ H ¼ 0; ¼ dx ghL x¼0 x¼L
Fx ¼ Fx ðyÞ;
Fy ¼ 0
ð63Þ
Two distinct forcing patterns, sketched in Fig. 9c and d, are considered. The Dean and Pearce (1972) presentation is rather brief. Their figures could not be reproduced from their equations. But they could be reproduced from the following re-working of their solution. The field equations are the steady form of Eqs. (3a) – (3c): Bqx Bqy þ ¼0 Bx By Bg 0 ¼ gh kqx þ Fx Bx Bg kqy þ Fy 0 ¼ gh By
in which the Fourier transform of g(x, y) is Z þl Hðb; xÞ ¼ gðy; xÞexpðbyÞdy
ð67Þ
ð68Þ
l
and F(b) is the Fourier transform of forcing Fx( y), being FðbÞ ¼ 2F0
sinbb 1 cosbb and FðbÞ ¼ l 2F0 ð69Þ b b
respectively, for forcing patterns I and II. The solution to Eq. (67) in Fourier transform space is Hðb; xÞ ¼ 8 > < Fðb ¼ 0Þ ðx0 LÞln x0 for b ¼ 0 ghL x0 x > : AðbÞI0 ðbðx0 xÞÞ þ BðbÞK0 ðbðx0 xÞÞ for b > 0 ð70Þ
ð64aÞ ð64bÞ ð64cÞ
in which I0(z) and K0(z) are the modified Bessel functions of the first and second kind of order zero, F(b = 0) is 2F0b and 0 respectively for forcing patterns I and II, and AðbÞ ¼
K0 ðbx0 Þ FðbÞ B and BðbÞ ¼ I0 ðbx0 Þ ghL CðbÞ
ð71Þ
which can be written in the wave equation form
ð66Þ
where x0 = h0/S0 and C(b)=[(K0(bx0)/I0(bx0))I1 (b (x0 L)) + K1(b(x0 L))]. Knowing H(b; x), g(x, y) is recovered from the inverse Fourier transform Z þl 1 gðx; yÞ ¼ Hðb; xÞexpðbyÞdb ð72Þ 2p l
where hL = h(L, y) = h0 S0L. The infinite solution domain in y and the localized meteorological forcing Fx( y) suggests a Fourier transform solution. Taking the Fourier transform of
The depth-integrated flows are then available from Eqs. (64b) and (64c) as
1 Bg gh Bg gh þ F x ; qy ¼ ð73Þ qx ¼ k Bx k By
B2 g B2 g S0 Bg ¼0 þ Bx2 By2 h Bx The boundary conditions are Bg Fx gð0; yÞ ¼ 0; ¼ Bx ðL;yÞ ghðL; yÞ
ð65Þ
228
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
Fig. 9. Spatially structured forcing at open coast.
All flows are scaled by F0/k, directly with the magnitude of the forcing and inversely with the friction factor. Application of the field solutions for g, qx and qy would take advantage of the inverse FFT algorithm (in which negative wave numbers are completed by the Hermitian property). From Eq. (72), Bg/Bx and Bg/By are the inverse Fourier transforms of dH/dx and ıbH, respectively. The maximum sustained setup would be expected at the shore, x = L. The shore profile is Z þl 1 gL ðyÞ ¼ Hðb; x ¼ LÞexpðbyÞdb ð74Þ 2p l Eq. (72) for the symmetric-in-y forcing pattern I (Fig. 9c) predicts the maximum setup as Z þl 1 gmax ¼ gðL; 0Þ ¼ Hðb; LÞdb ð75Þ 2p l The spatially uniform (b = 0) contribution to this result is identical to the maximum setup at the coast (Eq.
(51)) where there is no spatial structure to the forcing. The spatial attenuation in the forcing will moderate this result. The simple 1-D theory provides an upperbound estimate to the scale of the response. The maximum depth-integrated flow would be expected at x = 0. For the symmetric-in-y forcing pattern I again, Eq. (73) predicts the maximum depth-integrated flow to be qmax ¼ qx ð0; 0Þ ¼
F0 k
ð76Þ
at the shelf line. The direct response to both the forcing and the boundary friction is evident.
13. ST5: Steady spatially limited onshore wind at open coast Problem ST5 is outlined in Table 5, and corresponds to forcing pattern I in Fig. 9. The magnitude of
Table 5 ST5: Steady spatially limited onshore wind at open coast xF
xL
h0
0
100 km
BC
g(xF, y) = 0, qx(xL, y) = 0, qy(x, yF) = 0, qy(x, yL) = 0 Fy(x, t) = 0, Fx(x, t) = F0[H(x + b) H(x b)]
! F
b = 25 km, k = 10 4 s 1.
100 m
hL 5m
yF 100 km
yL + 100 km
DxOutput
F0 1.5 10
3
2 2
m /s
100 m
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
229
Fig. 10. Steady open-coast storm tide for ST5.
the forcing again corresponds to a wind speed of about 25 m/s. Spatial variations away from the maximum are illustrated in Fig. 10 for this forcing pattern. The limited spatial structure (Fig. 9c) of the forcing leads to attenuation of the forcing along the coast away from the peak setup, gmax at (L, 0). The spatial scale of this attenuation corresponds to b, the spatial scale of the forcing. There is an associated steady-state circulation with symmetric gyres about x = 0, and flow toward the coast for b < y < + b, and away from the coast beyond this band.
14. ST6: Steady spatially limited and reversing wind at open coast Problem ST6 is outlined in Table 6, and corresponds to forcing pattern II (Fig. 9d). The steady-state storm tide is shown in Fig. 11. This situation reflects the expected response to a landfalling southern-hemisphere hurricane. To the left of the storm track, b < y < 0, winds are onshore, there is an onshore flow and a setup at the coast. To the right of the storm track, 0 < y < + b, winds are offshore, there is an offshore flow and a set-down at the coast.
Table 6 ST6: Steady spatially limited and reversing wind at open coast xF
xL
h0
5m
yF 100 km
0
100 km
BC ! F
g(xF, y) = 0, qx(xL, y) = 0, qy(x, yF) = 0, qy(x, yL) = 0 Fy(x, t) = 0, Fx(x, t) = F0[H(x + b) 2H(x) + H(x b)]
b = 25 km, k = 10 4 s 1.
100 m
hL
yL + 100 km
DxOutput
F0 1.5 10
3
2 2
m /s
100 m
230
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
Fig. 11. Steady open-coast storm tide for ST6.
The limited spatial structure leads to a strong flow gyre with the active band over b < y < + b, and weaker associated gyres to the left and right. There is also the expected g attenuation away from regions of peak setup and set-down.
15. Conclusions Analytical solutions have a useful role in rational evaluation of numerical codes for storm tide responses. Analytical solutions alone have absolute credibility. They provide a measure of physical and code credibility that is not otherwise available to numerical codes. The necessary code modifications are outlined. A sequence of six analytical benchmark problems are outlined:
ST1 Sudden uniform wind on a closed basin ST2 Moving pulse across a closed basin ST3 Sudden uniform wind on a partially open basin
ST4 Moving pulse across a partially open basin ST5 Steady spatially limited onshore wind at open coast ST6 Steady spatially limited and reversing wind at open coast
Each case includes the full details of the analytical solution, illustration of the predicted response pattern, and a discussion of the significant physical and numerical aspects. Collectively, these analytical solutions provide the framework for a wide-ranging confirmation of numerical codes for storm tides.
References Chrystal, G., 1908. An investigation of the seiches of Loch Earn by the Scottish Lake Survey: Part V. Mathematical appendix on the effect of pressure disturbances upon the seiches in a symmetric parabolic lake. Transactions of the Royal Society of Edinburgh 46, 499 – 516. Dean, R.G., Pearce, B.R., 1972. Storm tide response of idealized
R.J. Sobey / Coastal Engineering 46 (2002) 213–231 continental shelves: Effects of steady wind fields of limited lateral extent. Technical Report 9, Department of Coastal and Oceanographic Engineering, University of Florida. Reid, R., 1956. Approximate response of water level on a sloping shelf to a wind fetch which moves towards shore. Technical
231
Memorandum 83, U.S. Army Corps of Engineers, Beach Erosion Board. Sobey, R.J., 2002. Analytical solution of non-homogeneous wave equation. Coastal Engineering Journal 44 (1), 1 – 24.
Analytical solutions for storm tide codes Rodney J. Sobey * Department of Civil and Environmental Engineering, University of California, 539 Dacis Hall, Berkeley, CA 94720, USA Received 29 August 2001; received in revised form 24 May 2002; accepted 4 June 2002
Abstract A sequence of analytical solutions explore aspects of response patterns expected from numerical codes for storm tides in one- and two-dimensional basins. Complete analytical details of the solutions are provided, together with specific suggestions for an associated set of analytical benchmark tests. Illustrations of predicted response patterns provide the basis for a discussion of many significant physical aspects and their representation in discrete numerical codes. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Analytical solution; Benchmark problems; Storm tide; Long wave; Numerical code; Shallow water
1. Introduction Numerical modeling has evolved to become the tool of choice in studies of storm tide influences in lakes and coastal basins. It has become sufficiently routine that questions of model credibility are often overlooked. The trend in modern codes has been increasing complexity in the physical processes and increasing length and complexity in the code and in the graphical interfaces. While numerical models have the potential to significantly enhance our predictive capability, their increasing size and complexity must also enhance the opportunity for both error (inappropriate numerical formulation, coding errors, inadequate grid resolution,. . .) and misuse. As a measure of physical and code credibility in rational model evaluation, a sequence of well-defined
*
Tel.: +1-510-642-3162; fax: +1-510-642-7483. E-mail address: [email protected] (R.J. Sobey).
analytical benchmark problems is proposed. These benchmark problems are analytical in the sense that each problem has an exact analytical solution. Analytical solutions alone have absolute credibility. A numerical code must be modified to exactly match the context of an analytical solution. But then the numerical and analytical solutions should match exactly. Any differences can be attributed to the code. This paper will introduce a sequence of analytical benchmark problems that are appropriate for numerical codes that focus on storm tides and closely related problems. Six application problems explore the underlying physical process and the interaction with the operational context of a numerical model. Numerical models of storm tide circulation are driven by a combination of internal meteorological forcing and boundary conditions. Internal meteorological forcing is the unique feature of storm tides. Boundary forcing is familiar from problems associated with astronomical tide circulation. But storm tides introduce some rather different challenges, as these flows are often
0378-3839/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 3 9 ( 0 2 ) 0 0 0 9 3 - 5
214
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
associated with transient boundary forcing. Detailed attention is given to response patterns associated with schematic meteorological forcing and a range of common boundary conditions. For each problem, the complete analytical solution is given for both water surface elevation g(xa, t) and flow qa(xa, t), in a manner immediately suitable for coding. The problems include the response to a sudden uniform wind on a closed and on a partially open basin, the response to a moving front across a closed and across a partially open basin, and the steady-state open coast storm tide forced by a spatially varied wind.
2. Field equations for storm tides For coastal basins (see Fig. 1), the incompressible flow mass and momentum conservation equations are the storm tide equations, B Bqx Bqy ðh þ gÞ þ þ ¼0 Bt Bx By 2 Bqx B qx B qx qy þ þ f C qy Bx h þ g By h þ g Bt B ps Rsx Rbx gþ ¼ gðh þ gÞ þ Bx qg q q
Bqy B qx qy þ Bx h þ g Bt
q2y
ps qg
þ
Rsa q
ð2Þ
where surface pressure ps(xa, t) and surface shear stress Rsa(xa, t) completely define the surface stress structure. The water surface will respond to surface shear directly and to horizontal spatial gradients of the surface pressure. Note in particular that meteorological forcing is predominantly internal to a solution domain. For typical astronomical tide, flood and tsunami propagation problems, the forcing is predominantly external to a solution domain, and imposed through the open boundary conditions. But open boundary conditions remain crucially important in storm tide problems, to represent meteorological influence beyond a truncated solution domain and to represent simultaneous tides. A conceptually useful approximation is available from a linearized form of these equations. With the following assumptions: advective accelerations are negligible, Coriolis accelerations are negligible, AgAbh0, and bed shear can be linearized to Rba/q = kqa, where k is a linear friction factor.
ð1Þ
in which g(xa, t) is the local water surface elevation and qa(xa, t) are the local depth-integrated flows (see
Fig. 1. A coastal basin.
B Fa ¼ gðh þ gÞ Bxa
!
B þ þ fC qx By h þ g Rsy Rby B ps gþ ¼ gðh þ gÞ þ By qg q q
Fig. 2), fC = 2X sin U is the Coriolis parameter, ps is the atmospheric pressure at the water surface, Rs is the surface wind shear, and Rb is the bed shear. (x, y) = xa, a = 1 and 2, is the local horizontal position and t is time. The meteorological forcing is the vector
Eq. (1) may be approximated as Bg Bqx Bqy þ þ ¼0 Bt Bx By
ð3aÞ
Bqx Bg kqx þ Fx ¼ gh Bx Bt
ð3bÞ
Bqy Bg kqy þ Fy ¼ gh By Bt
ð3cÞ
These are the linearized storm tide equations. Recall that the meteorological forcing, Eq. (2), includes both
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
215
Fig. 2. Definition sketches for storm tide equations.
surface pressure and surface wind shear. The Coriolis accelerations are linear terms, but their retention would preclude some useful analytical solutions. Except for the advective and Coriolis accelerations, these linearized equations retain all the complicated hyperbolic physics of storm tide evolution in a coastal basin. In addition, the linearization does not invalidate a numerical algorithm choice that was based on the complete Eq. (1). A numerical solution to Eqs. (3a) – (3c) imposes almost identical challenges.
arized Eq. (3b), k must be constant. Suitable predictive equations for k would be 8 f > > hqi > > 8h20 > > > < g k ¼ C2 h2 hqi > 0 > > > gn2 > > > : 7=3 hqi h0
for Darcy Weisbach model for Chezy model for Manning model ½SI units ð5Þ
3. Linear friction coefficient While linear friction is certainly a compromise, it must be recalled that quadratic friction also is not entirely satisfactory. The utility of the linear approximation will be enhanced by realistic estimates of k. Equating the linear and Darcy –Weisbach, Chezy or Manning estimates give 8 f > > AqA for Darcy Weisbach model > > > 8ðh þ gÞ2 > > < g k ¼ C2 ðh þ gÞ2 AqA for Chezy model > > > > gn2 > > > AqA for Manning model ½SI units : ðh þ gÞ7=3 ð4Þ in which f, C and n are the Darcy– Weisbach, Chezy and Manning friction factors, respectively. In the line-
where hqi is a discharge scale that is spatially and temporally averaged over the local flow. Suitable estimators for hqi are Z T Z LZ L 1 AqA dx dy dt; L2 T 0 0 0
1=2 Z T Z LZ L 1 2 q dx dy dt L2 T 0 0 0 Z T Z LZ L AqAq2 dx dy dt or 0Z T0Z L0Z L ð6Þ q2 dx dy dt 0
0
0
The first and second are absolute mean and RMS values. The third balances the total work done by the boundary shear from the linear and quadratic approximations. If the tidal flow is periodic with constant amplitude qˆ throughout, these estimators become 2ˆq=
216
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
pffiffiffi p; qˆ = p and 8ˆq=ð3pÞ , respectively. Adopting the latter gives 8 f qˆ > for Darcy Weisbach model > > > 3ph20 > > > < 8gqˆ for Chezy model ð7Þ k¼ 3pC2 h20 > > > 2 > > 8gn qˆ > > for Manning model ½SI units : 7=3 3ph0 4. Code modifications Analytical solutions can be used for numerical code evaluation in either of two ways. (1) Make no changes to the numerical code. Average values for h0 and k would be adopted. The analytical and numerical solutions should have trend agreement, but they will not be identical. Such a comparison is valuable, but not absolute. (2) Modify the numerical code to be a solution to the linearized equations. The numerical code would be modified to be a numerical solution of the linearized equations. A comparison of analytical and numerical solutions should then be absolute. The necessary code modifications certainly provide the opportunity for coding error, but access to an exact solution should facilitate the identification of any such errors. Both modes have value. The latter is absolute, and must be preferred. But the code modifications must be carefully undertaken. Four changes are necessary: (i) Omit the advective acceleration term (B/Bxa) (( qaqb)/(h + g)) in the momentum equation. This term is strongly nonlinear (quadratic in qa), and is often a problem in numerical codes like finite difference and finite element that approximate Eq. (1) as simultaneous linear algebraic equations in the nodal g’s and qa’s. In most situations, this term is a small contributor to the momentum balance, and many codes have an existing option to exclude the advective acceleration. (ii) Omit the Coriolis acceleration term ( fCqy, fCqx) in the momentum equation. This term is linear, and is not a problem in numerical codes. But the time scale for geostrophic influences forced by the Coriolis acceleration is the inertial period 2p/AfCA. The duration of storm tide forcing in coastal
basins is often much shorter than this, and geostrophic influences are not strong. Latitude U would be an input parameter to any numerical code, and the Coriolis acceleration is easily omitted by setting U to F 90j. Alternatively, fC could be set to zero directly. (iii) Change the friction term from Rba/q to kqa in the momentum equation. Many codes have an optional choice of friction formula, Darcy –Weisbach, Chezy or Manning. An additional option would not be difficult to include. (iv) Change the gravity term from g (h + g)(Bg/Bxa) to gh0(Bg/Bxa) in the momentum equations. The nonlinearity here is moderately weak, as Bg/ Bxa will be the more rapidly varying part. Typical finite difference and finite element codes would locally linearize this term, by adopting a locally constant approximation to (h(x, t) + g(x, t)). It would not be difficult to modify this code segment to always return a globally constant h0 in place of local estimates.
5. Storm tides in one-dimensional basins For the special case of a narrow one-dimensional basin, qy = Fy = 0, qx = q, and Fx = F. Eqs. (3a) –(3c) become Bg Bq þ ¼0 Bt Bx Bq Bg ¼ gh0 kq þ F Bt Bx
ð8Þ
The general analytical solution for g(x, t) of Eq. (8) is gðx; tÞ ¼ mx þ b þ a1 expðlxÞcosðkx xt þ /1 Þ þ a2 expðþlxÞcosðkx þ xt þ /2 Þ " l X gˆ n þ kfˆn =2 fˆn expðkt=2Þcosxn t þ þ xn n¼0 Z t 1 expðkt=2Þsinxn t þ 0 xn expðkðt sÞ=2Þsinxn ðt sÞ/ˆ n ðsÞds Z t 1 expðkðt sÞ=2Þ þ 0 xn # sinxn ðt sÞwn ðsÞds Xn ðxÞ
ð9Þ
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2n C 2 ðk=2Þ2
l " X þ fˆn expðkt=2Þcosxn t n¼1
gˆ n þ kfˆn =2 þ expðkt=2Þsinxn t xn
þ
#
sinxn ðt sÞwn ðsÞds Xn ðxÞ
ð10bÞ
for the forced and free modes, respectively; C=(gh0)1/2. These forms are special cases of the same generalized dispersion relationship. Sobey (2002) gives the complete details, including the definition of the solution parameters m, b, a1, /1, a2, /2, bn and the functions wn(t), and Xn(x) from the boundary conditions, the ˆ definition of the functions /n(t) from the meteorological forcing, and the definition of the parameters ˆfn and gˆn from the initial conditions. Terms 1 through 4 are contributed by the tidal boundary conditions. Terms 1 and 2 describe the steady gradually varied flow profile. Terms 3 and 4 are the tidal forced modes. Modes in the boundary conditions will appear in the response. Simultaneous tidal forcing could easily be included in the following discussion, but is not, to focus attention on the meteorological forcing. Within the summation, terms 5 and 6 are contributed by the initial conditions. Term 7 is contributed by the meteorological forcing, and term 8 is contributed by the residual (non-tidal) boundary conditions, such as tsunami penetration. Free mode responses at the (xn, bn) eigenmodes are excited by the initial conditions, by the meteorological forcing, and by nontidal transient boundary forcing. The dispersion relationship, Eqs. (10a) and (10b), relates space and time periodicities in both the free and forced modes. Without friction, it has the classical long wave form x = Ck. The general analytical solution for q(x, t) of Eq. (8) is qðx; tÞ ¼ b þ a1 expðlxÞcosðkx xt þ /1 Þ þ a2 expðþlxÞcosðkx þ xt þ /2 Þ
t
1 expðkðt sÞ=2Þ 0 xn sinxn ðt sÞ/ˆ n ðsÞds Z t 1 þ expðkðt sÞ=2Þ 0 xn
The dispersion relationship, relating space and time periodicities, has two forms " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #1=2 2 x 1 þ 1 þ ðk=xÞ k¼ ð10aÞ C 2 xn ¼
Z
217
ð11Þ
in which b, a1, /1, a2, /2, ˆfn, gˆn, wn and Xn(x) are redefined in relation to q. But they must be carefully coordinated to be exactly consistent with the conditions imposed for the g(x, t) solution. The dispersion relationship remains unchanged, Eqs. (10a) and (10b). Again, complete details are given in Sobey (2002). Solution of Eq. (8) is an initial, boundary value problem. The solutions for g and q change with both the initial conditions and the boundary conditions.
6. Termination of free mode summations The free modes in the Eq. (9) analytical solution for g and Eq. (11) analytical solution for q involve an infinite summation of eigenmode contributions. The free mode summations may be expanded and written as ) gVðx; tÞ l " X ¼ expðkt=2Þ a1n cosxn t n¼1 qVðx; tÞ #
þ a2n sinxn t Xn ðxÞ
ð12Þ
where 1 a1n ¼ fˆn xn
a2n
Z
t
"
#
expðks=2Þsinxn s /ˆ n ðsÞ þ wn ðsÞ ds
0
Z t " 1 gˆ n þ kfˆn =2 ¼ þ expðks=2Þcosxn s /ˆ n ðsÞ xn xn 0 #
þ wn ðsÞ ds
ð13Þ
Both the /ˆ n(t) and wn(t) functions would have finite durations, so that the time integrals in Eq. (13) would be terminated at the larger of these durations.
218
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
Fig. 3. Narrow rectangular basins.
The free modes have frequencies xn and amplitudes "
An ¼ a21n þ a22n
#1=2
ð14Þ
The frequencies are the same for the g and q solutions, but neither the magnitude nor the units of the amplitudes. The actual eigenmode amplitudes depend on the initial conditions, the meteorological forcing and the transient boundary conditions, but their magnitudes will generally decrease with increasing mode number, though not always monotonically. How far it is necessary to take the summations will be problem dependent. For the present purposes, convergence of the summation is assumed at M terms when AM 1 þ AM V0:01 2A1
These problems are intended as a supplement, not a substitute, for a sequence of numerical benchmark tests. It is anticipated that the analytical problems identified here will be of primary benefit in initial model development, and in the evaluation of userspecific variations or subsequent versions that introduce new physical, geometrical, numerical or graphical capabilities. Each test has a limited objective, seeking to focus on crucial problems in relative isolation. Each of the following problems have been given an ST identifier, suggesting their relevance to codes principally intended for storm tide-related problems. The Class I (one-dimensional) problems assume a schematic one-dimensional basin, either closed at both ends (Fig. 3a) or partially open (Fig. 3b), open at x = 0 but closed at x = L. h0 and k are known.
ð15Þ 8. ST1: Sudden uniform wind on a closed basin
The terminal M will be reported with each application.
7. Analytical benchmark tests A sequence of analytical benchmark tests have been designed to spotlight some physically and numerically significant response patterns that are expected to be within the predictive capabilities of depth-integrated models of storm tide gradually varied, unsteady flow responses to meteorological forcing. Two classes of problems are addressed: (I) unsteady flow in one-dimensional basins (II) steady flow at an open coast. Consideration is necessarily limited to contexts for which analytical solutions are possible.
Problem ST1, outlined in Table 1, is perhaps the simplest wind tide problem. It is the sudden imposition of a uniform meteorological forcing to a narrow basin of constant depth h0 that is closed at both ends, Fig. 3a. The forcing is Fðx; tÞ ¼ F0 HðtÞ
ð16Þ
where H(t) is the Heaviside or unit step function. The magnitude of the meteorological forcing in Table 1 corresponds to a moderate wind speed of order 25 m/s. In the analytical solution for g, the initial and boundary conditions are Bg gðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx;0Þ ð17Þ Bg F0 Bg F0 ¼ 2; ¼ 2 Bx ðx¼0;tÞ C Bx ðx¼L;tÞ C
R.J. Sobey / Coastal Engineering 46 (2002) 213–231 Table 1 ST1: Sudden uniform wind on a closed basin xF
xL
h0
0
5000 m
2 m 1.5 10 3 m2/s2
F0
IC g(x, 0) = 0 BC q(xF, t) = 0 F F(x, t) = F0H(t)
tF
DtOutput tL
0
60 s
2h
q(x, 0) = 0 q(xL, t) = 0
The meteorological forcing is /(x, t) = BF/Bx = 0, and forcing is imposed only through the boundary conditions. For the forced mode, boundary condition matching gives m = F0/C2 and b = mL/2. For the free modes, the eigenvalues and eigenfunctions are 1=2 1 b0 ¼ 0; X0 ðxÞ ¼ L 1=2 np 2 ; Xn ðxÞ ¼ bn ¼ cosbn x; n ¼ 1; 2; 3; . . . L L ð18Þ The modal coefficients are Z
L
0
F0 C2
and the meteorological forcing is /(x, t)= + BF/ Bt = F0d(t). For the forced mode, b = 0. For the free modes, the eigenvalues do not change, but the eigenfunctions are 1=2 2 Xn ðxÞ ¼ sinbn x; n ¼ 1; 2; 3; . . . ð22Þ L The modal coefficients are ˆfn = wn = 0, and
k = 10 4 s 1.
fˆn ¼
219
x
L 2
Xn ðxÞdx;
gˆ n ¼ 0
ð19Þ
Also /ˆn = wn = 0. The analytical solution is F0 L gðx; tÞ ¼ 2 x þ fˆ0 X0 þ expðkt=2Þ 2 C " # l X ˆn k f sinxn t Xn ðxÞ fˆn cosxn t þ 2xn n¼1 ð20Þ The ˆf0 is zero in this case, so that the n = 0 mode could be omitted. The forced mode scales as F0L/C2, as do the free modes. In the analytical solution for q, the initial and boundary conditions are qðx; 0Þ ¼ f ðxÞ ¼ 0; qðx ¼ 0; tÞ ¼ 0;
Bq ¼ gðxÞ ¼ F0 Bt ðx;0Þ qðx ¼ L; tÞ ¼ 0
ð21Þ
gˆ n ¼
Z
L
F0 Xn ðxÞdx;
/ˆ n ¼ F0 dðtÞ
0
Z
L
Xn ðxÞdx
0
ð23Þ but the /ˆn contribution to the free modes becomes identically zero after time integration. The analytical solution for q(x, t) is then qðx; tÞ ¼ expðkt=2Þ
l
X gˆ n n¼1
xn
sinxn t Xn ðxÞ
ð24Þ
This solution scales as F0L/C. Eq. (18) lists the discrete eigenmodes or resonant modes of the closed basin. The wave lengths of the eigenmodes are 2p/bn, where the bn are the wave numbers. The dispersion relationship, Eq. (10b), relates the discrete frequencies xn to the wave numbers and the basin friction. In this particular case, the meteorological forcing F is introduced through the boundary conditions for the forced mode, and through the initial conditions for the free modes. It is immediately clear from Eqs. (20) and (24) that the free or resonant modes of the basin have a very fundamental role in the response. The solution describes an unsteady and spatially varied response that results from the superposition of free standing wave modes that are oscillatory in both space and time. The waves are exponentially damped in time, but at a time scale Tf ¼
2 k
ð25Þ
that is expected to be rather slow. The specific response is illustrated in Fig. 4.The initial transients in g and q are the eigenmode responses. The periods Tn = 2p/xn of the eigenmodes
220
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
Fig. 4. Sudden imposition of a uniform wind stress on a closed basin.
are 0.63, 0.31, 0.21,. . . h for n = 1, 2, 3, . . .. The primary mode T1 dominates Fig. 4, though the important role of numerous higher modes is guaranteed by the sharp saw-tooth-style response, especially for g. Eight (M = 8) eigenmodes were included to achieve the Eq. (15) convergence. The saw-tooth wave response reflects the initial step excitation; in the Eqs. (20) and (24) analytical solution, it is represented as the summation over the M eigenmodes. The initial transient response decays slowly with time at time scale Tf = 5.6 h. This trend can just be seen at the short 2-h duration of Fig. 4, most clearly in the g evolution. The steady-state wind tide, for tHTf = 2/k, is
¯ gðxÞ ¼
F0 C2
x
L 2
This is a lightly damped system, and the initial transients significantly overshoot the steady-state response. This is a repeating observation on storm tides. The peak response is most likely associated not with the steady forced mode but with the free mode response excited by the typically transient character of the meteorological forcing.
9. ST2: Moving pulse across a closed basin Problem ST2, outlined in Table 2, is a schematic representation of an intense storm moving across a closed basin. An intense moving storm can be represented (Chrystal, 1908) as the schematic moving pulse (see Fig. 5)
ð26Þ Fðx; tÞ ¼ F0 ½Hðx þ a VtÞ Hðx VtÞ
ð28Þ
The maximum sustained setup is g¯ max ¼
F0 L 2C 2
at x = L.
ð27Þ
in which F0 is the magnitude of a rectangular pulse in meteorological forcing of width a that is moving across the basin at speed V. The leading edge is moving along the path x = Vt, the trailing edge along
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
221
Table 2 ST2: Moving pulse across a closed basin xF
xL
h0
F0
tF
DtOutput
tL
0
5000 m
2m
1.5 10 3 m2/s2
0
60 s
3h
IC BC F
g(x, 0) = 0 q(xF, t) = 0 F(x, t) = F0[H(x + a Vt) H(x Vt)]
q(x, 0) = 0 q(xL, t) = 0
a = 1500 m, V = 1 m/s, k = 10 4 s 1.
x = Vt a. The magnitude of the meteorological forcing in Table 2 again corresponds to a moderate wind speed of order 25 m/s. In the analytical solution for g, the initial and boundary conditions are Bg gðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx;0Þ Bg F0 h a i ð29Þ ¼ 2 HðtÞ H t ; Bx ðx¼0;tÞ C V
Bg F0 L Lþa ¼ H t H t Bx ðx¼L;tÞ C 2 V V
and the meteorological forcing is BF Bx ¼ F0 ½dðx þ a VtÞ dðx VtÞ
/ðx; tÞ ¼
ð30Þ
The boundary conditions are entirely transient, so that the forced mode is identically zero. For the free modes, the eigenvalues and eigenfunctions are
/ˆ n ðtÞ ¼
¼
1=2 Z L
2 F0 dðx VtÞ L 0 dðx þ a VtÞ cosbn x dx 1=2
2 L F0 cosbn Vt HðtÞ H t L V cosbn ðVt aÞ
a Lþa H t H t V V
ð33Þ
and
1=2 2 wn ðtÞ ¼ L
L Lþa n F0 ð1Þ H t H t V V h i a ð34Þ HðtÞ H t V The analytical solution for g(x, t) is then l Z t X 1 gðx; tÞ ¼ expðkðt sÞ=2Þ 0 xn n¼0 n o sinxn ðt sÞ /ˆ n ðsÞ þ wn ðsÞ ds Xn ðxÞ ð35Þ 2
1=2 1 b0 ¼ 0; X0 ðxÞ ¼ L ð31Þ 1=2 np 2 ; Xn ðxÞ ¼ bn ¼ cosbn x; n ¼ 1; 2; 3; . . . L L
This solution scales as F0L/C . In the analytical solution for q, the initial and boundary conditions are Bq qðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx; 0Þ ð36Þ qðx ¼ 0; tÞ ¼ 0;
qðx ¼ L; tÞ ¼ 0
and the meteorological forcing is The modal coefficients are fˆn ¼ gˆ n ¼ 0
BF ¼ F0 V ½dðx þ a VtÞ Bt dðx VtÞ
/ðx; tÞ ¼ þ ð32Þ
ð37Þ
222
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
1=2
2 L F0 V sinbn Vt HðtÞ H t ¼ L V
a Lþa sinbn ðVt aÞ H t H t V V
Fig. 5. Moving pulse.
ð40Þ The boundary conditions are identically zero, so that the forced mode is again identically zero. For the free modes, the eigenvalues do not change, but the eigenfunctions are 1=2 2 Xn ðxÞ ¼ sinbn x; n ¼ 1; 2; 3; . . . ð38Þ L
and wn(t) = 0. The analytical solution for q(x, t) is then qðx; tÞ ¼
l Z X
t
1 expðkðt sÞ=2Þ x n 0 n¼0 sinxn ðt sÞ/ˆ n ðsÞds Xn ðxÞ
ð41Þ
The modal coefficients are fˆn ¼ gˆ n ¼ 0;
/ˆ n ðtÞ ¼
1=2 Z L 2 F0 V ½dðx VtÞ L 0 dðx þ a VtÞsinbn x dx
ð39Þ
This solution scales as F0LV/C2. There is no steady-state wind setup, and the entire response is free mode transients. These respond at the resonant standing wave modes (bn, xn), which are initially strong, and decay slowly with friction through the term exp( kt/2). As is usual, the dominant free mode is the first, with wave number b1 and frequency x1.
Fig. 6. Moving pulse over a closed basin.
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
The specific response is illustrated in Fig. 6. M = 6 eigenmodes were included to achieve the Eq. (15) convergence. The front moves across the basin in (L + a)/V = 1.8 h and forces a significant initial transient that reflects the propagation path and width of the moving pulse. But the magnitude of the initial transient drops rapidly after passage of the pulse. The residual transients have a dominant periodicity at the first resonant mode, T1 = 2p/x1 = 0.6 h. The residual transients decay at the frictional time scale Tf = 2/k = 5.6 h slowly toward quiescence. Again, this is a lightly damped system. The front has passed in 1.8 h, but the steady state has not been reached at 3 h.
gives m = F0/C2 and b = 0. For the free modes, the eigenvalues and eigenfunctions are 1 1=2 n p 2 2 ; Xn ðxÞ ¼ bn ¼ sinbn x; L L n ¼ 1; 2; 3; . . . The modal coefficients are Z L F0 fˆn ¼ x Xn ðxÞdx; C2 0
10. ST3: Sudden uniform wind on a partially open basin
The meteorological forcing is /(x, t) = BF/Bx = 0. For the forced mode, boundary condition matching
xF
xL
h0
0
5000 m
2 m 1.5 10 3 m2/s2
IC g(x, 0) = 0 BC g(xF, t) = 0 F F(x, t) = F0H(t) k = 10 4 s 1.
F0
tF
DtOutput tL
0
60 s
q(x, 0) = 0 q(xL, t) = 0
3h
gˆ n ¼ 0
ð44Þ
F0 x þ expðkt=2Þ C2 " # l X ˆn k f sinxn t Xn ðxÞ ð45Þ fˆn cosxn t þ 2xn n¼1
The forced mode and the free modes again scale as F0L/C2. In the analytical solution for q, the initial and boundary conditions are Bq qðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ F0 Bt ðx;0Þ ð46Þ Bq ¼ 0; qðx ¼ L; tÞ ¼ 0 Bx ðx¼0;tÞ and the meteorological forcing is /(x, t)= + BF/ Bt = F0d(t). For the forced mode, b = 0. For the free modes, the eigenvalues do not change, but the eigenfunctions are 1=2 2 Xn ðxÞ ¼ cosbn x; n ¼ 1; 2; 3; . . . ð47Þ L The modal coefficients are fˆn = wn = 0, and Z L Z L F0 Xn ðxÞdx; /ˆ n ¼ F0 dðtÞ Xn ðxÞdx gˆ n 0
Table 3 ST3: Sudden uniform wind on a closed basin
ð43Þ
Also /ˆn = wn = 0. The analytical solution is gðx; tÞ ¼
Problem ST3, outlined in Table 3, is a variation on problem ST1. It is the sudden imposition of a uniform meteorological forcing to a narrow basin of constant depth h0 that is open at one end and closed at the other end, Fig. 3b. The forcing is Eq. (28). The magnitude of the meteorological forcing in Table 3 again corresponds to a moderate wind speed of or- der 25 m/s. This problem provides the opportunity to investigate different boundary conditions at xF, which in turn determines the character of the free mode responses. In the analytical solution for g, the initial and boundary conditions are Bg gðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx;0Þ ð42Þ Bg F0 gðx ¼ 0; tÞ ¼ 0; ¼ Bx ðx¼L;tÞ C 2
223
ð48Þ
0
but the /ˆn contribution to the free modes becomes identically zero after time integration. The analytical solution for q(x, t) is then l
X gˆ n qðx; tÞ ¼ expðkt=2Þ sinxn t Xn ðxÞ ð49Þ xn n¼1 This solution scales as F0L/C.
224
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
Eq. (43) lists the discrete eigenmodes or resonant modes of the partially open basin. The wave lengths of the eigenmodes are 2p/bn, where the bn are the wave numbers. The dispersion relationship, Eq. (10b), relates the discrete frequencies xn to the wave numbers and the basin friction. In this particular case, the meteorological forcing F is introduced through the boundary conditions for the forced mode, and through the initial conditions for the free modes. It is immediately clear from Eqs. (45) and (49) that the free or resonant modes of the basin once again have a very fundamental role in the response. The solution describes an unsteady and spatially varied response that results from the superposition of free standing wave modes that are oscillatory in both space and time. The waves are exponentially damped in time, but at time scale Tf, that is expected to be rather slow. The specific response is illustrated in Fig. 7. The initial transients in g and q are the eigenmode responses. The periods Tn = 2p/xn of the eigenmodes are 1.26, 0.42, 0.25,. . . h for n = 1, 2, 3,. . .. These differ from problem ST1 because of the different boundary conditions. The primary mode T1 dominates
Fig. 7, though the important role of numerous higher modes is again guaranteed by the sharp saw-toothstyle response, especially for g. Six (M = 6) eigenmodes were included to achieve the Eq. (15) convergence. The saw-tooth wave response reflects the initial step excitation; in the Eqs. (45) and (49) analytical solution, it is represented as the summation over the M eigenmodes. The initial transient response decays slowly with time at time scale Tf = 5.6 h. This trend can just be seen at the short 3-h duration of Fig. 7, most clearly in the g evolution. The steady-state wind tide, for tHTf = 2/k, is ¯ gðxÞ ¼
F0 x C2
ð50Þ
The maximum sustained setup is F0 L ð51Þ C2 at x = L. This is a lightly damped system, and the initial transients once again significantly overshoot the steady-state response. The peak response is most
g¯ max ¼
Fig. 7. Sudden imposition of a uniform wind stress on a partially closed basin.
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
225
Table 4 ST4: Moving pulse across a partially open basin xF
xL
h0
F0
tF
DtOutput
tL
0
5000 m
2m
1.5 10 3 m2/s2
0
60 s
3h
IC BC F
g(x, 0) = 0 g(xF, t) = 0 F(x, t) = F0[H(x + a Vt) H(x Vt)]
q(x, 0) = 0 q(xL, t) = 0
a = 1500 m, V = 1 m/s, k = 10 4 s 1.
likely associated not with the steady forced mode but with the free mode response excited by the typically transient character of the meteorological forcing.
/ˆ n ðtÞ ¼
dðx þ a VtÞsinbn x dx 1=2
2 L ¼ F0 sinbn Vt HðtÞ H t L V
a sinbn ðVt aÞ H t V Lþa H t ð55Þ V
11. ST4: Moving pulse across a partially open basin Problem ST4, outlined in Table 4, is a schematic representation of an intense moving storm across a partially open basin. The schematic moving pulse is represented in Fig. 5 and Eq. (28). The meteorological forcing is a rectangular pulse of width a that is moving across the basin at speed V. The magnitude of the meteorological forcing in Table 4 again corresponds to a moderate wind speed of order 25 m/s. In the analytical solution for g, the initial and boundary conditions are Bg gðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx;0Þ
gðx ¼ 0; tÞ ¼ 0;
Bg F0 L Lþa ¼ H t H t Bx ðx¼L;tÞ C 2 V V ð52Þ and the meteorological forcing is Eq. (30). The boundary forcing is entirely transient, so that the forced mode is identically zero. For the free modes, the eigenvalues and eigenfunctions are 1 1=2 n p 2 2 ; Xn ðxÞ ¼ bn ¼ sinbn x; L L n ¼ 1; 2; 3; . . .
ð53Þ
The modal coefficients are fˆn ¼ gˆ n ¼ 0
ð54Þ
1=2 Z L 2 F0 ½dðx VtÞ L 0
and
1=2
2 L F0 ð1Þn H t L V Lþa H t ð56Þ V The analytical solution for g(x, t) is then l Z t X 1 expðkðt sÞ=2Þ gðx; tÞ ¼ x n 0 n¼0 n o sinxn ðt sÞ /ˆ n ðsÞ þ wn ðsÞ ds Xn ðxÞ wn ðtÞ ¼
ð57Þ This solution scales as F0L/C2. In the analytical solution for q, the initial and boundary conditions are Bq qðx; 0Þ ¼ f ðxÞ ¼ 0; ¼ gðxÞ ¼ 0 Bt ðx;0Þ ð58Þ Bq ¼ 0; qðx ¼ L; tÞ ¼ 0 Bx ðx¼0;tÞ and the meteorological forcing is Eq. (37). The boundary conditions are identically zero, so that the forced mode is again identically zero. For the free modes, the eigenvalues do not change, but the eigenfunctions are 1=2 2 Xn ðxÞ ¼ cosbn x; n ¼ 1; 2; 3; . . . ð59Þ L
226
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
The modal coefficients are fˆn ¼ gˆ n ¼ 0; 1=2 Z L 2 /ˆ n ðtÞ ¼ F0 V ½dðx VtÞ L 0
ð60Þ
dðx þ a VtÞcosbn x dx 1=2
2 L ¼ F0 V cosbn Vt HðtÞ H t L V " a cosbn ðVt aÞ H t V ## Lþa H t ð61Þ V and wn(t) = 0. The analytical solution for q(x, t) is then l Z t X 1 expðkðt sÞ=2Þ qðx; tÞ ¼ 0 xn n¼0 sinxn ðt sÞ/ˆ n ðsÞds Xn ðxÞ ð62Þ This solution scales as F0LV/C2. There is no steady-state wind setup, and the entire response is free mode transients. These respond
at the resonant modes (bn, xn), which are initially strong, and decay slowly with friction through the term exp( kt/2). As is usual, the dominant free mode is the first, with wave number b1 and frequency x1. The specific response is illustrated in Fig. 8. M = 6 eigenmodes were included to achieve the Eq. (15) convergence. The moving pulse translates across the basin in (L + a)/V = 1.8 h and forces a significant initial transient that reflects the propagation path and width of the moving pulse. But the magnitude of the initial transient rapidly drops after passage of the pulse. The residual transients have a dominant periodicity at the first resonant mode, T1 = 2p/x1 = 0.6 h. The residual transients decay at the frictional time scale Tf = 2/ k = 5.6 h slowly toward quiescence. Again, this is a lightly damped system. The front has passed in 1.8 h, but the steady state (quiescence) has not been reached at 3 h. Reid (1956) has addressed a variation on this problem, in which the depth is linearly varying with x, but friction is neglected. He presents some approximate solutions, using the graphical method of characteristics.
Fig. 8. Moving pulse over a partially open basin.
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
227
12. Open coast storm tides
the field equation and the boundary conditions defines the following problem in Fourier transform space:
At open coasts, spatial variations in storm forcing in the longshore direction can be an important influence. In a land-falling hurricane, for example, there are regions of intense onshore winds to one side of the path (right in northern hemisphere) and offshore winds to the other side. Analytical solutions for steady-state wind tides at an open coast provide some schematic measure of the impact of longshore spatial structure of storm forcing. Dean and Pearce (1972) have formulated a schematic but useful open coast problem that is amenable to analytical solution. The context is sketched in Fig. 9. The shelf has a uniform slope, h(x, y) = h0 S0x, with no longshore variation. The storm forcing is steady, and represented as
d2 H S0 dH b2 H ¼ 0 dx2 h dx dH FðbÞ H ¼ 0; ¼ dx ghL x¼0 x¼L
Fx ¼ Fx ðyÞ;
Fy ¼ 0
ð63Þ
Two distinct forcing patterns, sketched in Fig. 9c and d, are considered. The Dean and Pearce (1972) presentation is rather brief. Their figures could not be reproduced from their equations. But they could be reproduced from the following re-working of their solution. The field equations are the steady form of Eqs. (3a) – (3c): Bqx Bqy þ ¼0 Bx By Bg 0 ¼ gh kqx þ Fx Bx Bg kqy þ Fy 0 ¼ gh By
in which the Fourier transform of g(x, y) is Z þl Hðb; xÞ ¼ gðy; xÞexpðbyÞdy
ð67Þ
ð68Þ
l
and F(b) is the Fourier transform of forcing Fx( y), being FðbÞ ¼ 2F0
sinbb 1 cosbb and FðbÞ ¼ l 2F0 ð69Þ b b
respectively, for forcing patterns I and II. The solution to Eq. (67) in Fourier transform space is Hðb; xÞ ¼ 8 > < Fðb ¼ 0Þ ðx0 LÞln x0 for b ¼ 0 ghL x0 x > : AðbÞI0 ðbðx0 xÞÞ þ BðbÞK0 ðbðx0 xÞÞ for b > 0 ð70Þ
ð64aÞ ð64bÞ ð64cÞ
in which I0(z) and K0(z) are the modified Bessel functions of the first and second kind of order zero, F(b = 0) is 2F0b and 0 respectively for forcing patterns I and II, and AðbÞ ¼
K0 ðbx0 Þ FðbÞ B and BðbÞ ¼ I0 ðbx0 Þ ghL CðbÞ
ð71Þ
which can be written in the wave equation form
ð66Þ
where x0 = h0/S0 and C(b)=[(K0(bx0)/I0(bx0))I1 (b (x0 L)) + K1(b(x0 L))]. Knowing H(b; x), g(x, y) is recovered from the inverse Fourier transform Z þl 1 gðx; yÞ ¼ Hðb; xÞexpðbyÞdb ð72Þ 2p l
where hL = h(L, y) = h0 S0L. The infinite solution domain in y and the localized meteorological forcing Fx( y) suggests a Fourier transform solution. Taking the Fourier transform of
The depth-integrated flows are then available from Eqs. (64b) and (64c) as
1 Bg gh Bg gh þ F x ; qy ¼ ð73Þ qx ¼ k Bx k By
B2 g B2 g S0 Bg ¼0 þ Bx2 By2 h Bx The boundary conditions are Bg Fx gð0; yÞ ¼ 0; ¼ Bx ðL;yÞ ghðL; yÞ
ð65Þ
228
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
Fig. 9. Spatially structured forcing at open coast.
All flows are scaled by F0/k, directly with the magnitude of the forcing and inversely with the friction factor. Application of the field solutions for g, qx and qy would take advantage of the inverse FFT algorithm (in which negative wave numbers are completed by the Hermitian property). From Eq. (72), Bg/Bx and Bg/By are the inverse Fourier transforms of dH/dx and ıbH, respectively. The maximum sustained setup would be expected at the shore, x = L. The shore profile is Z þl 1 gL ðyÞ ¼ Hðb; x ¼ LÞexpðbyÞdb ð74Þ 2p l Eq. (72) for the symmetric-in-y forcing pattern I (Fig. 9c) predicts the maximum setup as Z þl 1 gmax ¼ gðL; 0Þ ¼ Hðb; LÞdb ð75Þ 2p l The spatially uniform (b = 0) contribution to this result is identical to the maximum setup at the coast (Eq.
(51)) where there is no spatial structure to the forcing. The spatial attenuation in the forcing will moderate this result. The simple 1-D theory provides an upperbound estimate to the scale of the response. The maximum depth-integrated flow would be expected at x = 0. For the symmetric-in-y forcing pattern I again, Eq. (73) predicts the maximum depth-integrated flow to be qmax ¼ qx ð0; 0Þ ¼
F0 k
ð76Þ
at the shelf line. The direct response to both the forcing and the boundary friction is evident.
13. ST5: Steady spatially limited onshore wind at open coast Problem ST5 is outlined in Table 5, and corresponds to forcing pattern I in Fig. 9. The magnitude of
Table 5 ST5: Steady spatially limited onshore wind at open coast xF
xL
h0
0
100 km
BC
g(xF, y) = 0, qx(xL, y) = 0, qy(x, yF) = 0, qy(x, yL) = 0 Fy(x, t) = 0, Fx(x, t) = F0[H(x + b) H(x b)]
! F
b = 25 km, k = 10 4 s 1.
100 m
hL 5m
yF 100 km
yL + 100 km
DxOutput
F0 1.5 10
3
2 2
m /s
100 m
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
229
Fig. 10. Steady open-coast storm tide for ST5.
the forcing again corresponds to a wind speed of about 25 m/s. Spatial variations away from the maximum are illustrated in Fig. 10 for this forcing pattern. The limited spatial structure (Fig. 9c) of the forcing leads to attenuation of the forcing along the coast away from the peak setup, gmax at (L, 0). The spatial scale of this attenuation corresponds to b, the spatial scale of the forcing. There is an associated steady-state circulation with symmetric gyres about x = 0, and flow toward the coast for b < y < + b, and away from the coast beyond this band.
14. ST6: Steady spatially limited and reversing wind at open coast Problem ST6 is outlined in Table 6, and corresponds to forcing pattern II (Fig. 9d). The steady-state storm tide is shown in Fig. 11. This situation reflects the expected response to a landfalling southern-hemisphere hurricane. To the left of the storm track, b < y < 0, winds are onshore, there is an onshore flow and a setup at the coast. To the right of the storm track, 0 < y < + b, winds are offshore, there is an offshore flow and a set-down at the coast.
Table 6 ST6: Steady spatially limited and reversing wind at open coast xF
xL
h0
5m
yF 100 km
0
100 km
BC ! F
g(xF, y) = 0, qx(xL, y) = 0, qy(x, yF) = 0, qy(x, yL) = 0 Fy(x, t) = 0, Fx(x, t) = F0[H(x + b) 2H(x) + H(x b)]
b = 25 km, k = 10 4 s 1.
100 m
hL
yL + 100 km
DxOutput
F0 1.5 10
3
2 2
m /s
100 m
230
R.J. Sobey / Coastal Engineering 46 (2002) 213–231
Fig. 11. Steady open-coast storm tide for ST6.
The limited spatial structure leads to a strong flow gyre with the active band over b < y < + b, and weaker associated gyres to the left and right. There is also the expected g attenuation away from regions of peak setup and set-down.
15. Conclusions Analytical solutions have a useful role in rational evaluation of numerical codes for storm tide responses. Analytical solutions alone have absolute credibility. They provide a measure of physical and code credibility that is not otherwise available to numerical codes. The necessary code modifications are outlined. A sequence of six analytical benchmark problems are outlined:
ST1 Sudden uniform wind on a closed basin ST2 Moving pulse across a closed basin ST3 Sudden uniform wind on a partially open basin
ST4 Moving pulse across a partially open basin ST5 Steady spatially limited onshore wind at open coast ST6 Steady spatially limited and reversing wind at open coast
Each case includes the full details of the analytical solution, illustration of the predicted response pattern, and a discussion of the significant physical and numerical aspects. Collectively, these analytical solutions provide the framework for a wide-ranging confirmation of numerical codes for storm tides.
References Chrystal, G., 1908. An investigation of the seiches of Loch Earn by the Scottish Lake Survey: Part V. Mathematical appendix on the effect of pressure disturbances upon the seiches in a symmetric parabolic lake. Transactions of the Royal Society of Edinburgh 46, 499 – 516. Dean, R.G., Pearce, B.R., 1972. Storm tide response of idealized
R.J. Sobey / Coastal Engineering 46 (2002) 213–231 continental shelves: Effects of steady wind fields of limited lateral extent. Technical Report 9, Department of Coastal and Oceanographic Engineering, University of Florida. Reid, R., 1956. Approximate response of water level on a sloping shelf to a wind fetch which moves towards shore. Technical
231
Memorandum 83, U.S. Army Corps of Engineers, Beach Erosion Board. Sobey, R.J., 2002. Analytical solution of non-homogeneous wave equation. Coastal Engineering Journal 44 (1), 1 – 24.