The coastal-trapped wave paddle and open boundary conditions

Ocean Modelling 12 (2006) 224–236 www.elsevier.com/locate/ocemod

The coastal-trapped wave paddle and open boundary conditions John F. Middleton

*

School of Mathematics, University of New South Wales, Sydney 2052, Australia Received 12 July 2005; received in revised form 9 September 2005; accepted 12 September 2005 Available online 6 October 2005

Abstract The problem of the specification and interpretation of backward open boundary conditions (b.c.
s) is examined in the context of regional numerical models of wind-forced shelf circulation. (Backward here is opposite to the direction of coastal-trapped wave (CTW) propagation.) In particular, we examine the problem where the effect of the wind stress vanishes at a geographical origin. This origin (y = 0) is the point at which CTWs are generated and its inclusion or otherwise, can have an important effect on the shelf circulation. Due to computational costs however, the origin may be chosen to lie well outside the site (y = Y) of the backward open b.c. of the regional model. In order to simulate the wind-forcing over the omitted region [0, Y], previous studies have used coastal sea level data to drive a first mode CTW paddle at the backward boundary. Using linear theory, we show that if only one mode is present, then the sum of the (local) wind and paddle forced solutions obtained for y > Y is exact. That is, the total solution is identical to that obtained if the origin were included. Where higher modes are present, a first mode CTW paddle may lead to errors in the shelf break circulation and upwelling. However, using results for the CODE region, we show that these errors decrease with frequency and that relative to the first mode, may be 20% or less. A first mode paddle might be used in storm surge and models of coastal circulation. These results are used to discuss the nature and design of regional models of wind-forced shelf circulation. Ó 2005 Published by Elsevier Ltd.

1. Introduction At low frequencies, the alongshore component of wind stress acts to drive a cross-shelf Ekman flux. At the coast, this flux is returned back across the shelf leading the vortex stretching and the generation of coastaltrapped waves (CTWs). A key feature of CTW theory is the concept of the geographical origin (e.g., Middleton and Cunningham, 1984; Chapman, 1987). Such an origin may be defined as the point or region where this return cross-shelf flow is absent. Such a situation may arise where the alongshore winds vanish or where there is an abrupt change in the coastline orientation or shelf topography. An example of the former is given by Middleton and Leth (2004) where the alongshore component of the mean summer winds off Chile vanish at about 26°S. Origins that arise from changes in coastline orientation include the West Florida shelf (Mitchum and Clarke, 1986) and the Australian shelf near Cape Leeuwin.

*

Tel.: +61 2 9385 7069; fax: +61 2 9385 7123. E-mail address: [email protected]

1463-5003/$ - see front matter Ó 2005 Published by Elsevier Ltd. doi:10.1016/j.ocemod.2005.09.001

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Further examples include Hudson Strait (Labrador shelf; Middleton and Wright, 1991) and Bass Strait (east Australian shelf; Middleton, 1988, 1991). (These straits are also sites of CTW generation by mechanisms other than the alongshore wind stress. A related example is the equatorial region where disturbances can generate CTWs on the shelves of the Americas, (Brink, 1982).) Changes in shelf topography also give rise to geographical origins including the canyon to the south (35.8°N) of Pigeon Point on the Californian shelf (Chapman, 1987). Presumably, CTW scattering and dissipation within the canyon here eliminates much of the CTW energy generated backward of this region. Backward here is in the opposite direction to CTW propagation. A further example arises from the abrupt widening of the Great Barrier Reef shelf near Heron Is, Australia, (Middleton and Cunningham, 1984). Here much of the CTW energy that is generated backward of Heron Is. is likely steered along isobaths, seaward of the shelf break and thus away from the wide shelf lagoon. The importance of these geographical origins is that they serve as a starting point for the generation of forward propagating CTWs. These waves lead to a circulation that increases in magnitude along the shelf and that is asymptotic to that directly forced by the winds: the alongshore scale is set by friction. Middleton and Cunningham (1984), Mitchum and Clarke (1986) and Chapman (1987) all showed that the identification and incorporation of the geographical origin was crucial to the successful prediction of current and sea level observations using linear CTW theory and space-time dependent wind fields. Here we are concerned with a similar problem that arises in regional (non-linear) numerical models of regional shelf circulation. Indeed, for steady winds, we have shown (Middleton and Leth, 2004), that the incorporation of the geographical origin into the numerical model domain is necessary to get the correct circulation and degree of upwelling. For realistic wind-forcing, a similar problem arises. Since very-high resolution numerical models may be restricted in domain size, the backward boundary condition may be placed at some y = Y that lies forward of the natural geographical origin of the region, at y = 0. That is, the geographical origin is excluded from the model domain y > Y as shown in Fig. 1. The question then is ‘‘what is an appropriate backward boundary condition?’’. In their linear CTW analysis, Mitchum and Clarke (1986) suggested that the backward boundary condition might be closed by using local sea level data to drive a first mode ‘‘CTW paddle’’. The paddle here is assumed to simulate the wind-forced CTW energy that is generated backward of the boundary condition. Indeed, in their numerical modelling studies, Middleton and Black (1994) and Evans and Middleton (1998) made this assumption. In addition, the interior domains of their models were forced with realistic winds and results found to be superior to those obtained in the absence of the CTW paddle. Martinez and Allen (2005), also adopted a mode 1 paddle in their numerical study of the circulation of the Gulf of California. Related examples also include the regional studies of Thompson et al. (2003) and Fan et al. (2005) who respectively adopted a linear and exponential cross-shelf distribution of coastal sea level observations.

geographical origin

CTW paddle

y=0

y=Y

y

shelf

x

regional model domain

Fig. 1. A schematic of the coordinate system and shelf region. The shelf increases with depth with the offshore coordinate x. A geographical origin is assumed at y = 0 and for the southern hemisphere shelf adopted, the CTWs propagate in the positive y-direction. The domain of a regional numerical model is assumed to lie forward of y = Y (or y 0 > 0). At y = Y, the coastal sea level solution (or data) is used to drive a CTW paddle along the open boundary of the regional model.

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The underlying assumption of most of these regional studies is that the use of local winds and a CTW paddle (based on coastal sea level and one mode) will reproduce the circulation that would be found were it possible to include the geographical origin. In this note we will examine this assumption. The problem is an important one as it goes to the heart of regional models of weather-band shelf circulation, storm surges and upwelling. In Sections 2 and 3, we present the linear equations, CTW modes and the solutions for forcing by a periodic wind stress where a geographical origin is present. In Section 4, we obtain linear CTW solutions that correspond to the circulation of a regional numerical model for which the backward boundary condition is set forward (y = Y) of the geographical origin (y = 0). The solution consists of a first mode CTW paddle solution and a local wind-forced solution which add to give the correct solution if the first mode is present. In Section 5, we investigate the effects of higher modes and the discussion in Section 6 focuses on regional numerical model design. 2. The equations of motion We take y to be the alongshore direction and x to be the cross-shelf direction (Fig. 1) such that the depth of the shelf is h(x) and h0 = h(0) is non-zero. Making the long-wave and hydrostatic approximations, the equations of motion may be written (e.g., Clarke and Van Gorder, 1986) as  fv ¼ px =q

ð1Þ

sz vt þ fu ¼ py =q þ q 0 ¼ pz =q þ g

ð3Þ

r u¼0

ð4Þ

ð2Þ

and q0t ¼ woq0 =oz where density q(x, y, z, t) = q 0 + q0 + q*, q 0 (x, y, z, t) denotes the perturbation, q0(z) a depth dependent average and q* an overall average. The wind stress in the y-direction, s only depends on time and the alongshore coordinate: sw = sw(y, t). At the sea floor, the bottom stress is assumed to be linear and of the form sb(x, y, t) = q*rvb, where r is a friction parameter and vb the bottom velocity. Boundary conditions are that p ! 0 as x ! 1. Following Mitchum and Clarke (1986), the effective coast occurs where the water depth is equal to three Ekman layer depths. The coastal location here, say x = b, only effects the determination of the CTW eigenfunctions Fn(x, z). Without loss of generality, we take b = 0. Now following Clarke and Van Gorder (1986), the pressure p is expanded in terms of the inviscid eigenfunctions of the system as X p¼ /n ðy; tÞF n ðx; zÞ ð5Þ n

where the /n satisfy the forced wave equation X c1 anm /m ¼ bn sw ðy; tÞ n o/n =ot þ o/n =oy þ r

ð6Þ

m

The bn are wind-coupling coefficients and are given by Z 0 1 bn ¼ F n ð0; zÞ dz F n ð0; 0Þ h0 h0 where Parseval
s theorem yields X h0 b2n ¼ 1

ð7Þ

ð8Þ

n

The anm in (6), are frictional coupling coefficients. In the notation of Clarke and Van Gorder (1986), these would be replaced by anm/r. To proceed, we will consider the uncoupled wave equation that results from (6) by putting the diagonal terms (anm) to zero:

J.F. Middleton / Ocean Modelling 12 (2006) 224–236

c1 n o/n =ot þ o/n =oy þ /n =Ln ¼ bn sw ðy; tÞ

227

ð9Þ

where Ln = (rann)1 > 0 is the frictional length-scale of the nth CTW mode. Note also that we take cn > 0 in (9) so that the equation corresponds to that for the southern hemisphere example considered below (Fig. 1). Brink and Allen (1978) showed that (9) is asymptotically correct (their (2.13a)) for a barotropic ocean and where bottom friction is small: that is E1/2 = r/(jfjh0)  1 where E is the Ekman number. For stratified shelves, we are unaware of any equivalent derivation of (9) although we note that Clarke and Van Gorder (1986) do find the solutions to the coupled and uncoupled equations to be qualitatively similar for the West Florida shelf. However, for the purposes of the analysis here, illustrative solutions to (9) are readily obtained and will shed light on the CTW paddle and open boundary conditions for regional numerical models. Moreover, as we will see, the solutions to the coupled equation share the same properties and implications for regional models. 3. Solutions for a periodic wind stress Now assume that the wind stress is given by a travelling wave, that vanishes for y < 0. For a given frequency x, the wavenumber is m and the stress may be written as sw ðy; tÞ ¼ s0 H ðyÞ cosðmy  xtÞ

ð10Þ

where H = H(y) is the Heaviside step function. The amplitude s0 may be a function of wavenumber and frequency, so that (10) represents a single Fourier constituent of the wind spectra. In addition, we assume the shelf to be in the southern-hemisphere, so that any CTWs generated will propagate in the positive y-direction. Since the wind stress vanishes for y 6 0, the site y = 0, is a geographical origin for the generation of CTWs and the boundary condition is /n(y, 0) = 0 at y 6 0 (e.g., Clarke and Van Gorder, 1986, p. 1017). Without loss of generality, the solution to (9) and the wind stress (10) is   /n ðy; tÞ ¼ H ðyÞP n Re eiðmyxtþln Þ  ey=Ln eiðkn yxtþln Þ ð11Þ or /n ðy; tÞ ¼ H ðyÞP n ½cosðmy  xt þ ln Þ  ey=Ln cosðk n y  xt þ ln Þ

ð12Þ

where 2

P n ¼ bn s0 Ln =½1 þ ðLn ½k n  mÞ  cosðln Þ

ð13Þ

tanðln Þ ¼ ðk n  mÞLn

ð14Þ

and

The kn = x/cn are the wavenumbers of the CTWs and note, that as j(m  kn)Lnj ! 0, ln ! 0 and cos(ln) = 1. As j(m  kn)Lnj ! 1, ln ! p/2 and cos(ln) 1/j(m  kn)Lnj. The solution (11) or (12) consists of two parts. The first part is the component that is directly forced by the local wind. The second part consists of a CTW mode that frictionally decays over a scale Ln from the geographical origin at y = 0. The phase speed cn = x/kn decreases with increasing mode number as indicated by the results for the Californian shelf presented in Table 1. These results (Chapman, 1987), also show that the frictional scale Ln decreases with mode number so that far from the geographical origin (y/Ln  1), the CTW component in (11) and (12) will vanish. The amplitude of the overall response in (11) and (12) is proportional to bn. For relatively medium to wide shelves, the coupling coefficients bn (and thus /n) generally decrease with mode number n (Table 1). However, for stratified flows and narrow shelves, the second mode can be larger than the first and b2 > b1 (e.g., Middleton, 1988). In addition, the amplitude (13) shows that the response is that of damped resonance with largest response where the wavelength of the wind stress matches that of the CTW modes and m = kn. Since wind spectra are generally red, and kn increases with mode number n, the amplitude will be larger for the gravest modes. We will discuss this further in Section 5 below.

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Table 1 The mode number, phase speed cn and frictional length Ln scale for the CODE region, Chilean shelf and the Labrador shelf Mode

cn (m/s)

h0 b2n

Ln (km)

EPn

ESn

CODE region 1 2 3

3.55 1.77 1.00

0.18 0.09 0.03

1670 1053 1031

1.0 0.45 0.25

1.0 0.32 0.10

Chilean shelf 1 2 3

2.63 1.0 0.56

0.35 0.17 0.09

770 330 204

1.0 0.30 0.13

1.0 0.20 0.07

16.0 2.37

0.24 0.07

12,500 5882

1.0 0.25

1.0 0.14

Labrador shelf 1 2

Also presented are the non-dimensional wind-coupling coefficients h0 b2n and the (low frequency) scale factors (relative to mode 1), for the modal amplitudes (errors) EPn and amplitudes of coastal sea level ESn.

4. The CTW paddle and a regional numerical model We now seek to use these idealised solutions to interpret numerical solutions that might be found in a regional numerical of wind-forced shelf circulation. In particular, we will assume that the regional model is forced using local wind data (i.e., the idealised form (10) above), but the domain does not include the geographical origin at y = 0. Moreover, we also assume that solutions or data for coastal sea level are used to drive a CTW paddle at the backward open boundary of the model (y = Y > 0). The CTW paddle is adopted as an attempt to include the wind-forced signal that is in reality generated between y = 0 and y = Y, but which is excluded by the numerical domain. Now for arguments sake, we shall assume the coastal response is made up only of the first CTW mode (n = 1), and drop the subscript n in the following analysis. Now since sea level g(x, y, t) is equal to p(x, y, 0, t)/(q*g), we have from (12) that gðx; y; tÞ ¼ NF ð0; xÞ½cosðmy þ hÞ  ey=L cosðky þ hÞ

ð15Þ

where h = l  xt and N = Pn/(q*g) for n = 1. 4.1. The CTW paddle—regional solution Now at the backward boundary of the regional model (y = Y), the coastal sea level solution or ‘‘data’’ (gD) is from (15) given by gD ðtÞ ¼ gP ðY ; tÞ ¼ NF ð0; 0Þ½cosðmY þ hÞ  eY =L cosðkY þ hÞ

ð16Þ

Now this sea level solution (or data) can be used to drive a CTW paddle at y = Y by assuming that all energy resides in the first mode and by varying sea level (and alongshore velocity v(x, y, z, t), etc.) across the shelf as gP ðx; Y ; tÞ ¼ gD ðtÞF ðx; 0Þ=F ð0; 0Þ

ð17Þ

Numerically, the application of this paddle will propagate a mode 1 CTW into the model domain (y > Y) as a 0 0 free damped wave: gP ¼ gD ðtÞeiky ey =L , where y 0 = y  Y. Mathematically, this free wave corresponds to the solution to the unforced wave Eq. (9), but with sea level prescribed by (16) at y = Y. The solution for y > Y may written in real form as 0

gP ðx; y; tÞ ¼ NF ðx; 0Þ½cosðmY þ ky 0 þ hÞ  eY =L cosðky þ hÞey =L

ð18Þ

and is readily verified using complex arithmetic. Note, that the first term on the right side of (18) is erroneous since the wind forced component now propagates into the interior as a mode 1 CTW by the paddle at y 0 = 0.

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4.2. Wind-forced solutions—regional solution The sea level and alongshore velocity are completely prescribed by the CTW paddle at the open boundary at y = Y. The wind stress (10) (or data), is only applied for y > Y or y 0 > 0 and therefore leads to CTW solutions that have an artificial geographical origin at y = Y or y 0 = 0. The wind stress is again given by (10) and may be rewritten as sw ¼ s0 ½cosðmY Þ cosðmy 0  xtÞ  sinðmY Þ sinðmy 0  xtÞ

ð19Þ

In analogy with (10) and (12), the solution for sea level may be determined for each of the cosine and sine components of the wind stress (19). Adding these together, the solution is 0

gW ðx; y; tÞ ¼ NF ðx; 0Þ½cosðmy þ hÞ  ey =L cosðmY þ ky 0 þ hÞ

ð20Þ

and at y 0 = 0, gW = 0 as required. However, due to the artificial geographical origin imposed by the boundary at y 0 = 0, the second term on the right side of (20) is erroneous and consists of a CTW propagating into the domain. The regional model solution gM is sum of the paddle (18) and wind-forced solution (20) and the erroneous terms cancel to yield the correct solution (15). Thus, provided that the response is dominated by one mode, coastal sea level and local winds can be used to obtain the correct solutions that effectively include the natural geographical origin. In the following section, we will examine the relative contributions of the first three modes. The above results are generalised in Appendix A to solutions of the fully coupled wave Eq. (6), and more generally, to linear systems with unique solutions. That is, the paddle and local wind-forced solutions are again incorrect, but add to give the correct solution that would have been obtained had the geographical origin been included. The implications of the above results for regional numerical models are important. While such models contain non-linear and diffusive terms, they should at least be able to reproduce linear CTW dynamics with weak bottom friction. Thus, provided mode 1 is dominant, a CTW paddle and local winds might sensibly be used to drive a fully non-linear regional model. The solutions so obtained would then approximate those that would have been obtained had the model domain been larger and the geographical origin included. The linear solutions above also illustrate a second important feature of numerical solutions that does not appear to have been reported elsewhere. As shown by (20), the wind-forced solutions alone are incorrect due to the erroneous CTW term that propagates from the artificial geographical origin at y = Y. This term and the error will only vanish at points in the domain far from the backward boundary, y 0 /L  1. The artificial origin at this backward boundary may be expected for conditions that include, clamped, a solid wall and possibly zero-gradient and radiation. Thus, numerical wind-forced solutions obtained using these boundary conditions may be expected to be in error if the real geographical origin lies outside of the model domain. To correct the solutions, a CTW paddle would need to be applied at the backward boundary and for all modes. 5. The role of the higher modes: the size of the errors As noted, and a first mode CTW paddle and local winds will only provide exact solutions if the higher modes can be ignored. To explore this further, we consider the response of all modes but where the paddle is based on the assumption of a mode 1 CTW alone at y = Y. In this case, the solution for sea level within the regional model is given by X gM ðx; y; tÞ ¼ N 1 F 1 ð0; xÞ½cosðmy þ h1 Þ  ey=L1 cosðk 1 y þ h1 Þ þ N n Gn F 1 ðx; 0Þ½cosðmY þ k 1 y 0 þ hn Þ e

Y =Ln

0

cosðk n y þ k 1 y þ hn Þe

y 0 =L1

þ

X

2 0

N n F n ðx; 0Þ½cosðmy þ hn Þ  ey =Ln cosðmY þ k n y 0 þ hn Þ

2

where Gn = Fn(0, 0)/F1(0, 0) = bn/b1 and hn(m, x) = ln(m, x)  xt. The first mode part of the solution (the first line) is reproduced exactly (see (15)). The third line corresponds to the response to local winds and now includes all modes. The second line corresponds to the paddle solution

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J.F. Middleton / Ocean Modelling 12 (2006) 224–236

However, there are now errors in the cross-shelf amplitude of the higher modes since the modal structures Fn(x, 0) have been replaced by GnF1(x, 0). Since F1 may be zero near the shelf break (while the Fn are not), these errors may effect the alongshore and cross-shore velocity fields and upwelling. Errors also arise from the replacement of the higher modal wavenumbers (kn) and frictional decay scales (Ln) with those of the first mode. All of these errors will be small provided y 0 /L1 > 2 in which case all information about the backward boundary condition and paddle is small at y 0 > 2L1: the solution here is correct and responds only to the local winds. In this case, the paddle solution need not be applied at all—only local winds. This scenario is easily checked by both calculation of the frictional length scales from linear theory using the Brink and Chapman (1985) suite of programs. In addition, it would seem sensible to also validate this assumption by applying a mode 1 CTW paddle alone at the backward boundary so as to verify that the CTW modes do decay to near zero for the region of interest. The size of the domain will necessarily be greater than 2L1 and this may be prohibitive. On the other hand, canyons and other scattering regions may lead to effective frictional length scales that are much shorter than those predicted by linear CTW theory for uniform shelves. Further discussion is made in Section 6. 5.1. Spectra A second way in which the errors may be small is if the amplitude Pn of the higher modes are relatively small due to the contribution from the wind stress spectrum. To evaluate these terms, we consider first the net and time averaged alongshore energy flux of the exact solution (11). Using the normalisation of eigenfunctions (Fn(x, z)) adopted by Chapman (1987) and Brink and Chapman (1985), this flux may be written as X X Cn ¼ /n /n =d ð21Þ C¼ n

n

where the * denotes the complex conjugate and d = 4q*jfj a factor that shall be dropped for convenience. Using the complex form (11) of the solution for /n, we obtain, Cn ¼ T n ðm; xÞH n ðm; xÞS s ðm; xÞ

ð22Þ

where the transfer function T n ðm; xÞ ¼ P 2n =s20 and is given by 2 2

T n ¼ h0 b2n L2n =½h0 ½1 þ ½Ln ðm  k n Þ  cos2 ðln Þ

ð23Þ

H n ¼ 1 þ e2y=Ln  2ey=Ln cosð½m  k n yÞ

ð24Þ

with and S s ðm; xÞ ¼ hs0 ðm; xÞs0 ðm; xÞi the spectra of the alongshore component of wind stress (see below): the h i denote band or block averaging of the Fourier coefficients s0(m, x). To facilitate the analysis, consider the frequency spectra of the energy flux that is defined by Z þ1 Cn ðxÞ ¼ Cn ðm; xÞ dm ð25Þ 1

In addition, this is directly related to the auto-spectra of coastal sea level S n ðxÞ ¼ hgn gn i ¼ h0 Cn ðxÞh0 b2n =ðq gh0 Þ2

ð26Þ 2

where gn = /nbn/(q*g). From (23), the spectra Cn(x) and Sn(x) are thus proportional to h0 b2n L2n and ½h0 b2n Ln  so that the latter will be generally dominated by the first mode. 5.2. The CODE region: an example To evaluate the higher mode terms using the spectra above, we use the CTW modal results determined by Chapman (1987) and the wind spectra estimated by Halliwell and Allen (1987). The former are listed in Table 1.

J.F. Middleton / Ocean Modelling 12 (2006) 224–236

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The wind stress spectral model S s ðm; xÞ ¼ r2w kw T w expðkw jmj  T w jxjÞ=2

ð27Þ

was fitted to their (wintertime) results (Figs. 7 and 8), to obtain r2w ¼ 0:36 Pa2 , kw = 541 km and Tw = 0.8 d. Results for Ss(k, x) are presented in Fig. 2a for a typical mid-range frequency of (10 d)1. (The model is onesided in frequency and results must be converted to the units of cpd and cpkm to be directly comparable with that of Halliwell and Allen (1987).) To estimate the contribution of the higher modes, we consider the exact solution for the spectra far from the geographical origin y/Ln  1. In this case Hn = 1. Now, to understand the integrated results in frequency space, it is first worth examining the transfer function Tn that is plotted below the wind stress spectra in Fig. 2. As shown, Tn acts to preferentially sample the wind stress spectra at the modal wavenumbers kn = x/cn which increase with n. Since the wind spectra is red, the higher modal amplitudes will be relatively smaller. In addition, the amplitude of the transfer function decreases as h0 b2n L2n so that again, the higher modal amplitudes will be relatively small. The net effect is that the modal amplitudes will be smaller at larger frequencies and higher mode numbers. We note also that as x ! 0, the only modal dependence of the transfer function is through the constant h0 b2n L2n and the factor of (mLn)2 in the denominator of Tn (see (23)). Thus, at low frequencies, the constant should provide a scaling for the energy flux spectra. The Pn should scale with the square root, ðh0 b2n L2n Þ1=2 . The errors relative to the first mode are then of order EP n ¼ bn Ln =b1 L1

ð28Þ

and are listed in Table 1. Consider first the results for the sea level spectra Sn shown in Fig. 3a. As expected, the first mode dominates and the modal amplitudes are smaller at higher frequencies. The contribution by the wavenumber component of the wind stress spectra, leads to sea level spectra that are all much steeper than the wind stress frequency

50

(a)

Sτ (m,ω)

40

ω = 0.1

30 20 10 0 –5

0

5

10

15 –4

x 10 9

x 10

T(m,ω)

15

1

(b)

ω = 0.1

10

2

5

3 0 -5

0

5

m (cpkm)

10

15 –4

x 10

Fig. 2. (a) The wavenumber-frequency wind spectra Ss(m, x) at x = 0.1 cpd and for the CODE region (units Pa2 km d). (b) The transfer function Tn(m, x) for the first three modes and at x = 0.1 cpd, (units m).

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J.F. Middleton / Ocean Modelling 12 (2006) 224–236

spectrum shown in Fig. 3a. Results for the energy flux spectra Cn are proportional to S n ðxÞ=ðh0 b2n Þ and show similar results. Estimates of the importance of the higher modes are presented in Fig. 3b as fraction of that of the first 1=2 mode for P n =P 1 C1=2 n =C1 . At the lowest frequency shown (period 100 d), the second mode is of order 50% of the first. This would indicate that the neglect of the second mode in the CTW paddle will lead to serious errors. The third mode magnitude is relatively smaller and both scale with EPn as indicated by the horizontal dashed lines. At the highest frequency (period 5 d), the second mode amplitude is smaller and about 27% of the first. The errors at this period might be acceptable in the CTW paddle. 1=2 Results are also shown in Fig. 3b for the contribution made by modes 2 and 3 to coastal sea level, S 1=2 n =S 1 . The contributions here scale as ES n ¼ b2n Ln =b2n L1

ð29Þ

and are also listed in Table 1. At low frequencies, the mode 2 contribution is around 35% while at higher frequencies it drops to 20%. This result is important as it shows that regional models with a mode 1 CTW paddle might reasonably reproduce variations in coastal sea level, and to a lesser extent, alongshore velocity. Applications here could include storm surge models as well as models of the nearshore circulation around headlands and embayments. Finally, we note that for other regions such as the Chilean shelf (Middleton and Leth, 2004), the frictional decay scales Ln may decrease faster with mode number than appears the case for the CODE region. The first mode will thus be more dominant as indicated by the results in Table 1. In addition, for wide shelves, such as Saglek Bank on the Labrador shelf (Middleton and Wright, 1991), the first mode coupling coefficient can be relatively larger, so that again the first mode can be relatively more important (Table 1).

0.1

(b)

(a)

0.09

0.5 0.08 Sτ / 3

0.07

0.06

2

n

S (ω)

0.4

0.05

0.3 2

1

0.04

0.2 0.03

0.02

3 0.1 2

0.01 3 3 0

0

0.05

0.1

ω (cpd)

0.15

0.2

0

0

0.05

0.1

ω (cpd)

0.15

0.2

Fig. 3. (a) The sea level spectra Sn(x) for the modes labelled. (units m2 d). The frequency wind spectra Ss(x) is also indicated, (units Pa2 d). 1=2 (b) The dashed curves are C1=2 and estimates of the modal amplitudes Pn(x)/P1(x) for modes 2 and 3. The horizontal dashed lines n =C1 1=2 denote the scale estimates EPn from Table 1. The solid curves are S 1=2 and estimates of the coastal sea level amplitudes gn(x)/g1(x). n =S 1 The solid horizontal lines are the scale estimates ESn from Table 1.

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6. Discussion and regional model design We have investigated the use of local winds and a CTW paddle to drive the circulation in regional numerical models of shelf circulation. This approach was originally suggested by Mitchum and Clarke (1986), is still used (e.g., Martinez and Allen, 2005), although no formal analysis of it has been given. Indeed, we have shown that when one mode (say the first), dominates the circulation, linear theory shows that the circulation so produced will be exactly that which would be obtained had the natural geographical origin been included in the model domain. The implication here is that the backward boundary of the model need only extend to a site where coastal sea level data is available. Higher resolution can then be focused on the region of interest. However, where other modes are present, the assumption of a first mode CTW paddle will introduce errors into the regional model solution and notably near the shelf break. For the CODE region, we have shown how the errors may be simply estimated using linear theory and estimates of the alongshore wind stress spectra. Relative to the first mode, and at low frequencies, the errors are of order EPn = (bnLn/b1L1) < 0.45. For coastal sea level the errors are smaller ES n ¼ b2n Ln =b21 L1 < 0:32 since both the wind-coupling coefficients bn and frictional scales Ln decrease with mode number n. In addition, the errors are smaller at higher frequencies since the wind spectrum is red in wavenumber space. The first mode thus dominates the solutions for coastal sea level (and to a lesser extent alongshore velocity). These results may be expected to apply to other regions although the details will depend on the CTW properties and the wind spectrum. For the Chilean shelf, frictional effects are larger for the higher modes (Ln/L1 is smaller), and the error estimates are smaller (Table 1). For the wide Labrador shelf, the first mode is more dominant and errors will also be smaller (Table 1). Depending on the region, storm surge models and models of coastal circulation might well use a first model paddle with some reliability. The CTW paddle is readily implemented and may allow for significant savings in computational overhead. However, the errors that may arise over the shelf break should be estimated. This can be done using the linear analysis above or through numerical sensitivity studies that incorporate higher mode CTW paddles. It should also be noted that the use of inviscid CTW eigenfunctions to obtain the total (exact) wind-forced response may be inefficient (e.g., Lopez and Clarke, 1989). The wind stress is expanded as X sw ðy; tÞ ¼ sw ðy; tÞ bn F n ðx; 0Þ n

and this results in the solution for the bn in (7). From Parseval
s theorem (8) and the results for h0 b2n in Table 1, it is clear that the first three modes contribute only 30% (63%) of Parseval
s sum for the CODE (Chilean) regions and a significant part of the solution and variability may remain in the higher modes. A more precise estimate would involve a detailed analysis using the asymptotic properties of the CTW modes and is beyond the scope of this work. In addition, the assumption of weak friction made in the analysis formally requires that the periods of the first three modes be much shorter than 2pLn/cn or 34 d, 43 d and 75 d for the CODE region. Formally, the analysis is not valid at these or longer periods. As we have shown, this is where the wind stress energy and higher modes and errors are largest. At these low frequencies, one way to construct a more accurate and efficient CTW paddle might be to expand the steady frictional solution in terms of the arrested topographic waves (Csanady, 1978) or viscous eigenfunctions of the steady problem (e.g., Webster, 1985; Power et al., 1989). Middleton and Leth (2004) used this approach in their solutions for the Chilean shelf circulation driven by steady winds from a geographical origin. The first mode here accounted for 81% of the sum in Parseval
s theorem and of the cross-shelf transport and a much larger fraction than that of 35% given by the inviscid mode, (Table 1). These comments highlight general problems with CTW theory and suggest that where possible, numerical solutions should be obtained and the geographical origin included. This may not always be possible and we now discuss some aspects of regional model design. The first point is that the presence of the nearest geographical origin needs to be identified. As noted in the introduction, such an origin may result from abrupt changes in coastline orientation (e.g., straits, bends in the coast) or shelf topography (e.g., canyons). The identification

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of such an origin could be aided by coarse resolution model simulations using idealised and realistic winds. The second point is that the geographical origin may also be a significant source of free CTWs as demonstrated for Bass and Hudson Straits (Middleton, 1988; Middleton and Wright, 1991). Such sources should be included. Let us now consider three cases based on the relative location of the shelf region of interest to the geographical origin. The relevant scale here is the frictional spin-down scale of the first mode and this scale should be determined using both linear CTW theory (Brink and Chapman, 1985) as well as from regional model numerical solutions. The latter can better allow for CTW scattering and more realistic effects of friction. Case (1) y/L1 1: The region of interest lies near the geographical origin. In this case, the geographical origin is included in the model domain. An example here would be a regional model of the shelf circulation for the CODE region. Chapman (1987) identified the canyon to the south of Pigeon Point as a geographical origin and it only lies several hundred kilometers backward of the CODE region. If the geographical origin is also a source of free CTWs then these must be incorporated and allowance made for the correct modal distribution. The choice here will be to use either a CTW paddle for each of the dominant modes or to use an additional numerical model that focuses on the source. An example here is Bass Strait which acts as a geographical origin and as a source of free waves for the east Australian shelf (Middleton, 1988). The simplest and most accurate way to model the wind-forced circulation on the shelf would be to construct two models. The first would model the weather-band circulation within the strait (which is largely barotropic) and would adopt a simplified shelf geometry several hundred kilometers from the strait (e.g., Middleton and Black, 1994). This model could then be used to drive a second model of the shelf circulation that would have a simplified strait geometry. Case (2) y/L1  1: The region of interest lies far from the geographical origin. We assume here that the geographical origin cannot be included in the regional model domain. In reality, the circulation here will consist of only that directly forced by the wind since any CTW components will have decayed to near zero. However, the introduction of a solid wall or clamped type backward boundary will generate spurious CTWs. There are two options: Option (a): The backward boundary (y = Y) of the regional model can be extended backwards so that y 0 /L1 2. No paddle need be applied and the erroneous free CTW component of the wind-forced circulation (generated at the backward boundary) should have decayed for the regions of interest: y 0 2L1. This can be demonstrated by applying a mode 1 paddle at the backward boundary condition. Option (b): The backward boundary condition (y = Y) of the regional model cannot be extended. Assuming sea level data is available at this boundary, a CTW paddle could be adopted. The sensitivity of results to the paddle can be determined along with the expected linear solutions and errors outlined above. With sufficient data, the first two or more dominant modes and corresponding CTW paddles might also be determined. A second approach would be to develop a separate regional numerical model for the region Y  2L1 < y < Y. In this case, the solutions for the field variables at y = Y implicitly correspond to the infinite sum of all modal CTW paddles. Specification of field variables at the backward boundary of the regional model then corresponds to a specification of the infinite sum of CTW paddles. Information propagates as free waves and into the domain interior as described above. The errors associated with this solution, cancel with any errors in the locally wind-forced solution. The solution is then exactly that required and is as if a much larger domain was included. The approach here is perhaps not as contrived as it may appear. Each of the two regional models can have different grid resolutions and purposes. The purpose of the model of the backward region Y  2L1 < y < Y is to provide the boundary conditions at y = Y. The purpose of the model for y > Y may be related to process studies or realistic simulations/hindcasts of storm surges, upwelling or nearshore circulation. Acknowledgement I thank Bill McLean for useful discussion regarding the Appendix.

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Appendix A. Linear systems and unique solutions Let L[Æ] denote a set of differential operators that includes the total solution for pressure (e.g., Clarke and Van Gorder, 1986, Eqs. (2.1) and (2.2)) or the coupled modal-wave Eq. (6). We assume the solution variable is sea level g(x, y, t) which must satisfy L½g ¼ sw ðy; tÞ

ð30Þ

for y P 0, with gðx; 0; tÞ ¼ 0

ð31Þ

and sw(y, t) the wind forcing term. Now (30) also applies to the regional model domain (y P Y) and the solution there, denoted gM, must satisfy L½gM  ¼ sw ðy; tÞ

ð32Þ

for y P Y, with gM ðx; y; tÞ ¼ gP ðx; Y ; tÞ ¼ gðx; Y ; tÞ

ð33Þ

at y = Y. Now provided the solutions to (30) and (31) are unique, then the regional solution gM that is forced by the paddle gP and local winds sw(y, t) for y P Y must be equal to the ‘‘correct’’ solution g that satisfies (30) and (31). In addition, if the system is linear, then this regional solution gM can also be obtained as the sum of the paddle-forced solution (L[gP] = 0 with gP = g(x, Y, t) at y = Y) and local wind-forced solution (L[gW] = s(y, t) with gW = 0 at y = Y). The solution gM is equal to that which would have been obtained had the geographical origin been included in the model domain. Finally we note that for the sinusoidal time-dependence adopted in Sections 3 and 4, (eixt), that the coupled wave Eq. (6) reduces to a set of linear ordinary differential equations which have unique solutions. References Brink, K.H., 1982. A comparison of long coastal trapped wave theory with observations off Peru. J. Phys. Oceanogr. 12, 897–913. Brink, K.H., Allen, J.S., 1978. On the effect of bottom friction on barotropic motion over the continental shelf. J. Phys. Oceanogr. 8, 919– 923. Brink, K.H., Chapman, D.C., 1985. Programs for computing properties of coastal-trapped waves and wind-driven motions over the continental shelf and slope. Tech. Rep. 85-17, Wood Hole Oceanographic Institution, 99p. Chapman, D.C., 1987. Application of wind-forced long, coastal-trapped wave theory along the Californian Coast. J. Geophys. Res. 92, 1798–1816. Clarke, A.J., Van Gorder, S., 1986. A method for estimating wind-driven frictional time-dependent, stratified shelf and slope water flow. J. Phys. Oceanogr. 16, 1013–1102. Csanady, G.T., 1978. The arrested topographic wave. J. Phys. Oceanogr. 8, 47–62. Evans, S.R., Middleton, J.F., 1998. A regional model of shelf circulation near Bass Strait: a new upwelling mechanism. J. Phys. Oceanogr. 28, 1439–1457. Fan, Y., Brown, W.S., Yu, Z., 2005. Model simulations of the Gulf of Maine response to storm forcing. J. Geophys. Res., 110. doi:10.1029/2004JC002479. Halliwell, G.R., Allen, J.S., 1987. Wavenumber frequency domain properties of coastal sea level response to alongshore wind stress along the west coast of North America. J. Geophys. Res. 92, 11761–11788. Lopez, M., Clarke, A.J., 1989. The wind-driven shelf and slope water flow in terms of a local and remote response. J. Phys. Oceanogr. 19, 1091–1101. Martinez, J.A., Allen, J.S., 2005. A modeling study of coastal-trapped wave propagation in the Gulf of California, Part I: Response to remote forcing. J. Phys. Oceanogr. 34, 1313–1331. Middleton, J.F., 1988. Long shelf waves generated by a coastal flux. J. Geophys. Res. 93 (C9), 10724–10730. Middleton, J.F., 1991. Coastal-trapped wave scattering into and out of straits and bays. J. Phys. Oceanogr. 21, 681–694. Middleton, J.H., Cunningham, A., 1984. Wind-forced continental shelf waves from a geographical origin. Cont. Shelf. Res. 3, 215–231. Middleton, J.F., Wright, D.G., 1991. Coastal-trapped waves on the Labrador shelf. J. Geophys. Res. 96, 2599–2617. Middleton, J.F., Black, K.P., 1994. The low frequency circulation in and around Bass Strait: a numerical study. Cont. Shelf Res. 14, 1495– 1521. Middleton, J.F., Leth, O., 2004. Wind-forced setup of upwelling, geographical origins and numerical models: the role of bottom drag. J. Geophys. Res. 109, C12019. doi:10.1029/2003JC002126.

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Mitchum, G.T., Clarke, A.J., 1986. Evaluation of frictional, wind-forced long-wave theory along the West Florida Shelf. J. Phys. Oceanogr. 16, 1029–1037. Power, S.B., Middleton, J.H., Grimshaw, R.H., 1989. Frictionally modified Continental Shelf waves and the sub-inertial response to wind and deep-ocean forcing. J. Phys. Oceanogr. 19, 1486–1506. Thompson, K.R., Sheng, J., Smith, P.C., Cong, L., 2003. Prediction of surface currents and drifter trajectories on the inner Scotian Shelf. J. Geophys. Res., 108. doi:10.1029/2001JC001119. Webster, I., 1985. Frictional continental shelf waves and the circulation response of a continental shelf to wind forcing. J. Phys. Oceanogr. 15, 855–864.