Variability in a fjord-like coastal estuary I: Quantifying the circulation using a formal multi-tracer inverse approach Olivier G.J. Riche a, * , Rich Pawlowicz b a School of Earth and Ocean Sciences, University of Victoria, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada b Department of Earth and Ocean Sciences, University of British Columbia, 6339 Stores Rd., Vancouver, BC V6T 1Z4, Canada article info Article history: Received 20 July 2013 Accepted 16 November 2013 Available online 28 November 2013 Keywords: estuary dynamics river discharge seasonal variability estuarine flow scaling inverse methods box model abstract During 2002e2005, a comprehensive set of observations covering physical, biological, radiative and atmospheric parameters was obtained from the southern Strait of Georgia (SoG), Western Canada by the STRATOGEM program. Monthly time series of estuarine layer transports over 2002e2005 were estimated using a time-dependent 2-box model in a formal inverse approach. The formal inverse approach builds up upon a so-called “pseudo-inverse” and the Singular Value Decomposition methodology. The trans- ports are then consistent with the temperature and salinity fields, as well as riverine freshwater inflow (R) and detailed atmospheric heat fluxes. Uncertainty was analyzed by resampling observations using bootstrap methods. Analysis of these time series suggests that the SoG estuarine circulation is not very sensitive to the seasonal changes of R. Comparison of the surface layer transport (U1) and R yields the first observational relationship between the SoG estuarine circulation and R. The analysis of this relationship shows that U1 has a fractional form as R to the power of 1/n with n < 1. Such fractional relationship shows that the flows change only slightly with the freshet. A 5-fold change in R results only in a 40% change in U1. However, freshwater range and uncertainty in the data prevented us from clearly determining the fraction n. Analysis of the transports in light of the residuals in the mass, heat and salt budgets suggests that our inversion procedure works properly and improves on the SVD inverse procedure. Analysis of the transports sensitivity to inversion parameters shows that the transports are close to both the a priori and true transports and that they are dependent on both a priori information and data. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction The first step in any attempt to understand the dynamics of an estuarine system must be an estimate of the circulation and mass transports within the system. These transports govern the distri- bution of nutrients and/or pollutants in the system, and their ex- change with shelf waters, and hence also have an important controlling effect on biological activity in the estuary. This study attempts to objectively quantify the estuarine transports in the Strait of Georgia (SoG), a fjord-like estuarine system located in British Columbia, Canada (Fig. 1). A subsequent paper will deal with the implications of the physical circulation on biological produc- tivity in this region. In spite of the importance of transport estimates, obtaining quantitative values from direct measurements of the circulation in estuaries (including this one) are often impractical as current measurements capable of adequately resolving spatial variations are difficult to obtain, even over a few key transects (Godin et al., 1981). Instead, various formalisms are often used in order to esti- mate these transports from the measurements of different scalars. The simplest approach is the diagnostic use of quasi-steady state mass and salt budgets based on “Knudsen” relations (Dyer, 1973). In this approach the estuarine circulation is directly linked to fresh- water inflow rates and the increasing salinity of the outflow as it moves seaward. The lack of explicit time dependence in the Knudsen relations may be a problem, but as most datasets are not comprehensive enough to address this issue the lack of methodological rigour is perhaps irrelevant. Many estuarine studies concern themselves solely with the mean transport (Austin, 2002; Pawlowicz et al., 2007) or just assume a quasi steady state (England et al., 1996; Mackas and Harrison, 1997; Savenkoff et al., 2001; Pawlowicz, 2001). Box models can be used to extend the Knudsen approach by calculating not only horizontal flows, but also vertical flows (Gordon et al., 1996), adding time-dependence and combining * Corresponding author. E-mail addresses: [email protected](O.G.J. Riche), [email protected](R. Pawlowicz). Contents lists available at ScienceDirect Estuarine, Coastal and Shelf Science journal homepage: www.elsevier.com/locate/ecss 0272-7714/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ecss.2013.11.018 Estuarine, Coastal and Shelf Science 137 (2014) 1e13
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lable at ScienceDirect
Estuarine, Coastal and Shelf Science 137 (2014) 1e13
Contents lists avai
Estuarine, Coastal and Shelf Science
journal homepage: www.elsevier .com/locate/ecss
Variability in a fjord-like coastal estuary I: Quantifying the circulationusing a formal multi-tracer inverse approach
Olivier G.J. Riche a,*, Rich Pawlowicz b
a School of Earth and Ocean Sciences, University of Victoria, 3800 Finnerty Road, Victoria, BC V8P 5C2, CanadabDepartment of Earth and Ocean Sciences, University of British Columbia, 6339 Stores Rd., Vancouver, BC V6T 1Z4, Canada
a r t i c l e i n f o
Article history:Received 20 July 2013Accepted 16 November 2013Available online 28 November 2013
Keywords:estuary dynamicsriver dischargeseasonal variabilityestuarine flow scalinginverse methodsbox model
* Corresponding author.E-mail addresses: [email protected] (O.G.J. Ri
(R. Pawlowicz).
0272-7714/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.ecss.2013.11.018
a b s t r a c t
During 2002e2005, a comprehensive set of observations covering physical, biological, radiative andatmospheric parameters was obtained from the southern Strait of Georgia (SoG), Western Canada by theSTRATOGEM program. Monthly time series of estuarine layer transports over 2002e2005 were estimatedusing a time-dependent 2-box model in a formal inverse approach. The formal inverse approach buildsup upon a so-called “pseudo-inverse” and the Singular Value Decomposition methodology. The trans-ports are then consistent with the temperature and salinity fields, as well as riverine freshwater inflow(R) and detailed atmospheric heat fluxes. Uncertainty was analyzed by resampling observations usingbootstrap methods.
Analysis of these time series suggests that the SoG estuarine circulation is not very sensitive to theseasonal changes of R. Comparison of the surface layer transport (U1) and R yields the first observationalrelationship between the SoG estuarine circulation and R. The analysis of this relationship shows that U1has a fractional form as R to the power of 1/nwith n < 1. Such fractional relationship shows that the flowschange only slightly with the freshet. A 5-fold change in R results only in a 40% change in U1. However,freshwater range and uncertainty in the data prevented us from clearly determining the fraction n.
Analysis of the transports in light of the residuals in the mass, heat and salt budgets suggests that ourinversion procedure works properly and improves on the SVD inverse procedure. Analysis of thetransports sensitivity to inversion parameters shows that the transports are close to both the a priori andtrue transports and that they are dependent on both a priori information and data.
� 2013 Elsevier Ltd. All rights reserved.
1. Introduction
The first step in any attempt to understand the dynamics of anestuarine system must be an estimate of the circulation and masstransports within the system. These transports govern the distri-bution of nutrients and/or pollutants in the system, and their ex-change with shelf waters, and hence also have an importantcontrolling effect on biological activity in the estuary. This studyattempts to objectively quantify the estuarine transports in theStrait of Georgia (SoG), a fjord-like estuarine system located inBritish Columbia, Canada (Fig. 1). A subsequent paper will deal withthe implications of the physical circulation on biological produc-tivity in this region.
In spite of the importance of transport estimates, obtainingquantitative values from direct measurements of the circulation inestuaries (including this one) are often impractical as current
measurements capable of adequately resolving spatial variationsare difficult to obtain, even over a few key transects (Godin et al.,1981). Instead, various formalisms are often used in order to esti-mate these transports from the measurements of different scalars.The simplest approach is the diagnostic use of quasi-steady statemass and salt budgets based on “Knudsen” relations (Dyer, 1973). Inthis approach the estuarine circulation is directly linked to fresh-water inflow rates and the increasing salinity of the outflow as itmoves seaward.
The lack of explicit time dependence in the Knudsen relationsmay be a problem, but as most datasets are not comprehensiveenough to address this issue the lack of methodological rigour isperhaps irrelevant. Many estuarine studies concern themselvessolely with the mean transport (Austin, 2002; Pawlowicz et al.,2007) or just assume a quasi steady state (England et al., 1996;Mackas and Harrison, 1997; Savenkoff et al., 2001; Pawlowicz,2001).
Box models can be used to extend the Knudsen approach bycalculating not only horizontal flows, but also vertical flows(Gordon et al., 1996), adding time-dependence and combining
Fig. 1. Geography of the Salish Sea. The dashed squares indicate the sampling area of the STRATOGEM and JEMS programs (see Section 2.2). Insert, physical fluxes and processes inthe box model. The left-hand boxes represent the SoG, and the right-hand box the Haro Strait (HS). The thick arrows represent the transports in and out the SoG (U1, U2,W1 andW2).In the model, the surface SoG water enters the surface of HS (U1) while the deep HS water enters the deep SoG (U2). The arrow thickness approximates the relative magnitude oftransports. The upper left arrow represents the freshwater inflow (R). The thick wavy arrow represents the turbulent and radiative heat fluxes (F). FSW is the shortwave componentof F (thin wavy arrow) that penetrates deeper into the SoG than the longwave component.
O.G.J. Riche, R. Pawlowicz / Estuarine, Coastal and Shelf Science 137 (2014) 1e132
additional tracer budgets (usually temperature (Roson et al., 1997;Pawlowicz and Farmer, 1998) and sometimes oxygen (Pawlowiczet al., 2007) and nutrients (Savenkoff et al., 2001)), and/or addingmore boxes (Burrard Inlet Environmental Action Program, 1996;Gordon et al., 1996). In spite of this increasing complexity thebox-model approach can still provide a more flexible and lessresource-consuming approach than high-resolution 3D-numericalmodels for estimating the circulation. Extended box models can beanalyzed in various ad-hoc ways (Pawlowicz et al., 2007) or inmoreformal inverse approaches (Savenkoff et al., 2001).
Prognostic approaches are also used. These range from box-models which incorporate simplified and tuned dynamics (Liet al., 1999), to fully 3D numerical modelling (Masson andCummins, 2004). However, the tuning required in prognostic ap-proaches in order to make results “match” observations of scalarvariables is somewhat ad-hoc, and it is not always obvious whetheror not particular details of the implementation are aiding ordetracting from the match. In many of these models, freshwaterinflow rates are again a fundamental forcing factor.
Theoretical analyses of estuarine circulations, developed usingidealized numerical models, have also been proposed in whichestuary dynamics are modelled by scaling arguments (Chatwin,1976; MacCready, 1999; Geyer et al., 2000; Hetland and Geyer,2004; MacCready and Geyer, 2010). In particular, the estuarineflow is again scaled to the freshwater flow and the tidal flow. Oneinteresting conclusion is that vertical turbulent mixing (modelledas tidal mixing in these studies) tends to limit the estuarineresponse at high flow levels (MacCready and Geyer, 2010). Theestuary response tends not to be proportional to the freshwater
flow, but rather scales with a small fractional power of the fresh-water flow (MacCready and Geyer, 2010). Thus the magnitude ofthe estuarine circulation may be insensitive to changes in fresh-water inflow when the inflow is “large”. However, these ideas havenot been tested in real systems.
The aim of this paper is then two-fold. First, to estimate theseasonal variations in estuarine transport in the Strait of Georgia. Acomprehensive new dataset, comprising 3 years of monthly ob-servations, is used in this analysis. Results therefore provide thefirst accurate description of the seasonal transport changes andtheir phasing, and of their interannual variability, in the SoG. Theseestimates are made using a formal inverse procedure, which allowsfor the estimation of uncertainties associated with numericalvalues obtained. Uncertainties are obtained using a bootstrapprocess (Efron and Tibshirani, 1993; Pawlowicz, 2001) whichinherently models the structure of the data with minimal need forassuming independence and ‘Gaussian’ statistics, and for arbitrarilyseparating measured variables into those with associated errors(the “forcing” terms), and those assumed to be known exactly(those describing the scalar fields). In addition, sensitivity studiesare carried out to determine the effect of assumptions inherent inthe box-model formalism.
Second, we analyze these estimates in relation to theoreticalscalings to determine the effect of freshwater flow on the trans-ports. A lack of sensitivity to freshwater inflow rates is found.However, in contrast to previous theoretical studies, actual trans-ports are used in the comparisons and not the output of idealizednumerical models. Our analysis thus provides geophysical evidencefor these scalings.
O.G.J. Riche, R. Pawlowicz / Estuarine, Coastal and Shelf Science 137 (2014) 1e13 3
The rest of this paper will first introduce the sampling and theobservations (Section 2.2), the formal inverse approach and thebox-model (Sections 3.1 and 3.2), and their processing and boot-strapping (Section 3.3), as well as the effect of unobserved parts ofthe system to calculate the box-model inputs (Section 3.3). In theResults section (Section 4) wewill show the box-model outputs anddiscuss their seasonal variability. We will also discuss these resultsin the light of budget residuals and sensitivity analyses (Section 5).We will examine the sensitivity of the transports to the separationdepth (between the surface and deep SoG boxes) and the inversionparameters (Section 6). In the next section (Section 6) we will dealwith the relationship between the estuarine flow and the fresh-water flow, the main forcing of the estuarine circulation and theimplications of the Fraser River variability for the variability of SoGcirculation. We will contrast the scaling found for the estuarineflow response to the freshwater flow with recent literature on thephysics of estuaries. Finally, we will summarize the important re-sults of this work (Section 6).
2. Site and observations
2.1. The Strait of Georgia
The SoG is about 200 km long by 30 km wide, bounded to thewest by Vancouver Island, and to the north by long and constrictedchannels (Fig. 1). Most of the exchange with the open ocean isgenerally thought to occur through the wider and deeper passagesto the south, through the Gulf and San Juan Islands, and the Strait ofJuan de Fuca (Thomson, 1981, 1994; LeBlond, 1983). The Strait is asmuch as 400 m deep, but is separated from the ocean by sills ofdepth 100 m (Thomson, 1994; Davenne and Masson, 2001). TheStrait of Georgia is relatively unique among large estuaries becausethe major source of freshwater inflow, the Fraser River, is one of theonly free-flowing large (longer than 1000 km) rivers left in theworld (WWF (2006), Figs. 3e4 and Appendix 1). There is a strongseasonal component to its flow, with summer flows being almostan order of magnitude larger than winter flows, so that changes inforcing occur over time scales relatively long compared to theresidence time of at least the surface layer (Pawlowicz et al., 2007).
The overall circulation of the Strait of the Georgia (SoG) resultsfrom fluxes of seawater and freshwater. The most striking andrecognizable feature of SoG, the silty plume of the Fraser River,seasonally discharges a large volume of freshwater into the SoG.The surface inflow of freshwater contributes to both the surfaceheight gradient and the flow shear that drives the estuarine cir-culation. Deep inflow of dense oceanic seawater, mainly comingfrom Haro Strait through Boundary Pass, further enhances thevertical salinity gradient and the estuarine circulation. Observa-tions and numerical modelling suggest that the intermediate waterlayer is in a geostrophic balance and circulates cyclonically (Staceyet al., 1987, 1991; Marinone and Pond, 1996). During Fall, episodicpulses of denser oceanic seawater, called deep water renewals, sinkto the bottom of the SoG and slowly mixes upward. Weak verticalturbulent mixing, except for the northern and southern passages,slowly erodes the gradients built by the fresh- and seawater fluxes.Energy for turbulent mixing is mainly supplied by tides, at semi-diurnal and fortnightly periods, river flow shear and winter windmixing.
2.2. Observations
The observations used here were mostly obtained from theSTRAit Of GEorgia Monitoring (STRATOGEM) program. Using ahovercraft (CCGH Siyay), the STRATOGEM program regularlysampled 9 stations evenly distributed through the southern half of
the Strait (Fig. 1) over a 240-km long track in about 9 h. The surveyswere scheduled near the beginning of everymonth fromApril 2002to June 2005, with more frequent sampling around the springbloom in March/April: a total of 47 cruises.
CTD (conductivityetemperatureedepth) casts providedcontinuous vertical profiles of physical variables (pressure, tem-perature and conductivity) and biogeochemical variables (chloro-phyll-a, dissolved oxygen, fluorescence and photosyntheticallyavailable radiation). Vertical profiles were obtained at the front ofthe hovercraft (limiting ship mixing) to sample at high verticalresolution the entire water column down to within 15 m of thebottom. Salinity measurements are obtained on the PracticalSalinity Scale PSS-78 (UNESCO, 1981a,b, 1983), but they have beenconverted into Absolute Salinities (using the TEOS-10 standard (IOCet al., 2010) and assuming the salinity anomaly dSA ¼ 0. The verticalprofiles of salinity and temperature are accurate to 0.01 g kg�1 and�0.003 �C, respectively.
The northern Strait was not sampled by STRATOGEM. However,a regular series of observations was carried out by the Institute ofOcean Sciences, Sidney BC (Masson and Cummins, 2004; Masson,2006) over both the northern and southern Strait 4 times a year(16 cruises over the same time period and we shall compare thetwo datasets (Section 6)).
The JEMS (Joint Effort to Monitor the Strait of Juan de Fuca)program provided complementary observations in Haro Strait (HS)at the southern boundary of the Strait of Georgia. The JEMS pro-gram collected continuous profiles at three hydrographic stationsin HS (Fig. 1). Based on the available JEMS information (Newtonet al., 2002), the methods of sampling and the accuracy of themeasurements are similar to those of STRATOGEM.
Additional observations were required to quantify freshwater(FW) sources into the SoG. Freshwater inflow and temperatureestimates are based on the Fraser River discharge RFraser (see Figs 1.4and 6.4 in Riche (2011)) measured at Hope by Environment Canada(water station 08MF005). The total freshwater inflow into the StraitR is estimated as (Pawlowicz et al., 2007).
R ¼ 1165þ 1:66� RFraser�unit : m3s�1
�: (1)
The surface air/sea heat fluxes into the Strait (Fig. 2a) wereestimated using data from various weather stations at the HalibutBank buoy, at Vancouver International Airport, and at the Univer-sity of British Columbia climate station at Totem Field. Turbulent(sensible and latent heats) and radiative (LW) fluxes were calcu-lated using the methods provided by Pawlowicz et al. (2001). Cal-culations required the wind speed, the cloudiness, the relativehumidity and airesea temperatures. These were obtained fromweather observations at either Vancouver International Airportstation (LW radiation) or Halibut Bank buoy (sensible and latentheats). The downwelling shortwave (SW) heat flux was measuredby a pyranometer located 10 m above the ground at Totem Field.
3. Methods
Physical tracers and their transports in an estuary are linkedthrough kinematic equations for the conservation of mass, salt andheat. If transports are known then tracer distributions can be pre-dicted. This is denoted the “forward” problem. The determinationof transports can come from the solution of dynamical equations(Chatwin, 1976; Li et al., 1999; MacCready and Geyer, 2010), how-ever the results will depend somewhat on the parameterizationrequired to simplify issues related to turbulent mixing.
In contrast, an “inverse” problem involves a computation oftransports that can reproduce observations of tracers. The advan-tage of an inverse problem is that transports can be determined
O.G.J. Riche, R. Pawlowicz / Estuarine, Coastal and Shelf Science 137 (2014) 1e134
without the need to make assumptions about the form of turbulentmixing. The results can then be used as an independent test ofdynamical theories. However, inverse problems are often under-determined, and so the estimation of errors and biases in results iscritical to their interpretation. For complex problems, it is easiest toformulate the system as a matrix equation, for which an inverse (ormore typically a pseudo-inverse) can be computed using a least-squares (or singular value decomposition) methodology (Wunsch,1996, 2006).
3.1. Formal inverse approach
A simple way to idealize the circulation in partially and highlystratified estuaries is to use a 2-box simplification (MacCready and
Fig. 2. Panel a) surface heat budget. The net heat flux is the sum of the shortwave (SW), long(A) and freshwater (O) time series. In both panels, the left axis indicates the box averages oright axis indicates the surface FW input rate (m3 s�1). Panels d) 0e30 m and e) 30e400 mindicates the box averages of the temperature (�C). In panel d), the right axis indicates the saverage in the SoG (A) and 0e200 m in Haro Strait (O). The study of phase lags between thPanels f) and g) comparison of STRATOGEM and IOS SoG salinity box averages. Panels h) a
Geyer, 2010). In the 2-box model, e.g. surface and deep boxes, theSoG estuarine circulation can be described by a set of 4 transports(insert, Fig. 1): surface seaward (U1), deep landward (U2), down-ward (W1) and upward (W2) transports. The separation depth be-tween the surface and deep boxes is defined as the depth abovewhich surface brackish water is expelled from the Strait, and belowwhich denser salt water has flowed into the Strait from the oceanthrough Haro Strait.
The governing equations are obtained by a volume integration ofthe conservation equations of mass, salt and heat in each box. Themain assumptions are: that the largest external sources of mass andheat are due to the advective fluxes and air-sea heat fluxes, thatvertical turbulent mixing dominates purely diffusive exchanges(Pawlowicz, 2001), that tracers are well-mixed within each box,
wave (LW), latent and sensible heat fluxes. Panels b) 0e30 m and c) 30e400 m salinityf the salinity (g kg�1). Each marker is a box average from a single cruise. In panel b), thetemperature (A) and surface net heat flux (O) time series. In both panels, the left axisurface net heat flux (�10 W m�2). In panels c) and e), the left axis indicates 30e400 me SoG and Haro Strait deep temperatures have been studied by Pawlowicz et al. (2007).nd i), similar to f) and g) but for temperature.
Table 2Inversion Parameters. Note that W and S are both diagonal matrices with diagonalelements described in this table. The elements in S are all equal to s. U01,U02,W01 andW02 are the a priori values, approximately equal to the average of the pseudo-inverse.
O.G.J. Riche, R. Pawlowicz / Estuarine, Coastal and Shelf Science 137 (2014) 1e13 5
and the flux of correlated variations of flow and tracers are smallrelative to the mean advective fluxes. However, we do not assume aquasi-steady budget, so that rates of change of observed tracers areincluded. Further detail on the volume integration can be found inRiche (2011).
The volume integrated mass, heat and salt budgets can bewritten into the following algebraic equations (Eqs. (2)e(7)) wherethe solutions are U1 (seaward surface outflow), U2 (landward deepinflow), W1 (downward component of the turbulent mixing ex-change), and W2 (upward component of the turbulent mixingexchange þ net upward estuarine entrainment), as described ininsert, Fig. 1:
Mass budget:
�U1 �W1 þW2 ¼ �Rþ ε1 (2)
U2 þW1 �W2 ¼ ε2 (3)
Heat budget:
ðT2 � T1ÞW2 ¼ V1vT1vt
þ ðT1 � TRÞR� ar0Cp
FSW
Z d
0ke�kzdz
� ar0Cp
ðF � FSWÞ þ ε3 (4)
ðTH�T1ÞU2þðT1�T2ÞW2 ¼ V2vT2vt
� ar0Cp
FSW
Z N
dke�kzdzþ ε4
(5)
Salt budget:
ðS2 � S1ÞW2 ¼ V1vS1vt
þ S1Rþ ε5 (6)
ðSH � S1ÞU2 þ ðS1 � S2ÞW2 ¼ V2vS2vt
þ ε6 (7)
Volumes of the upper and lower boxes are taken to be constantand given by V1 and V2 respectively (Table 1). The quantities Ti, Si,Ui,Wi, εj, R, F, FSW and k variables are time-dependent scalars. The ratesof change of relevant parameters are explicitly included in theright-hand-side of the equations. The quantities R, F, FSW and krespectively represent, the freshwater inflow from rivers, the netsurface heat flux and its shortwave (SW) component (see Fig. 2a)and a light attenuation coefficient (in m�1). In Eqs. (4) and (5), F andFSW fluxes are converted into units of �C m3 s�1 by the factor a/(r0Cp) where a is the SoG surface area (Table 1), r0 a referencedensity and Cp the water specific heat capacity. A single-band lightattenuation of FSW, the SW component of F, or “blue” component isassumed (Kara et al., 2005). It penetrates deeper than the longwavecomponent and some fraction can potentially enter the lower box.It decays according to k, the light attenuation (over the character-istic distance k�1, average 3.6 m), based on an estimated 1%photosynthetically available radiation (PAR) level in the SoG (at anaverage depth of 15 m), using STRATOGEM data. Each equation alsocontains a residual term εj, which models the results of
where H1 (¼0.99 for d ¼ 30 m) and H2 (¼0.01 for d ¼ 30 m) are theintegral coefficients appearing in Eqs. (4) and (5), respectively.
In the standard inverse approach, the residuals are assumed toarise from the observations b and model misfit. However, obser-vational data also appears in the Amatrix, and special techniques tobe discussed below are required to determine the effects ofobservational uncertainties there. Ignoring these for now, the so-called optimal inverse solution bu minimizes an objective function J:
J ¼ εTW�2
εþ zTS�2z (12)
where W and S are weighting diagonal matrices (Wunsch, 2006), zis the difference between the solution u and the a priori mean uA,all defined in Table 2. The minimum of the objective function Jrepresents a compromise solution that minimizes the equationresiduals while also penalizing solutions that are far from the apriori constraints. The solution bu that minimizes Eq. (12) can bewritten (see Wunsch, 1996, pp 127 and 167):
bu ¼ uA þ�ATW�2A þ S�2
��1�ATW�2
��b� AuA
�(13)
O.G.J. Riche, R. Pawlowicz / Estuarine, Coastal and Shelf Science 137 (2014) 1e136
It is computationally easier to find this solution by finding thesingular value decomposition of the scaled A matrix:
A0 ¼ W�1A ¼ U0L0V0T (14)
and substituting into Eq. (13) to find:
bu ¼ uA þ V0�L0TL0 þ S�2
��1�L0TU0TW�1
��b� AuA
�(15)
This inverse procedure has been devised to improve and buildon a “pseudo-inverse” procedure that directly inverts Au ¼ b þ ε
with the SVD methodology (Wunsch, 1996) and provides asolution:
buSVD ¼ VL�1UTb (16)
where matrix L is the matrix of the singular values of A, L�1
the diagonal rectangular matrix whose non-zero diagonal co-efficients are the inverse of the singular values (matrices definedWunsch, 1996, p. 149), and U and V the matrix of the singularvectors. The solution bu (Eq. (15)) tapers the coefficients of thesingular vectors V when there are very small singular values
Fig. 3. Transport estimates (U1 and E) and their errors ðsbi Þ. Panel a) transports U1 and E; parepresented by the strips (see legend). Panel b), transport M is the line inside the strips, errright hand side, represent the overall mean (x) and the error (s), respectively over 47 surv
(associated with unstable solutions) of the matrix A. It alsoconstrains the optimal solution to keep values closer to the apriori solution uA. For further detail on the mathematics involvedwith the SVD, we refer the readers to the general theory inWunsch (1996, 2006) and Riche (2011).
Table 2 shows the actual values of the inversion parameters (s,ui
and uA). Horizontal and vertical transports in the vector bu werescaled by s, 5 � 104 m3 s�1, as this is known to be a reasonable scalefor transport (Godin et al., 1981; Li et al., 1999; Pawlowicz et al.,2001; Masson and Cummins, 2004; Pawlowicz et al., 2007). Thescaling coefficients ui are obtained by estimating the residuals ofthe conservation equations assuming that the magnitude of thetransports were about 5 � 104 m3 s�1, and the sources and forcingsat their absolute maximum value. The a priori estimates of uA el-ements ðuA
i Þ are consistent with previous estimates, but are alsoapproximately equal to the average of the “pseudo-inverse” buSVDfor the whole time series. The uA
i components reflect the pattern ofthe estuarine circulation in the SoG where vertical turbulent mix-ing M(¼W01) is usually smaller than the other transports. Theentrainment E(¼W02 �W01) and the landward inflow U02 are closeand the seaward outflow U01 is greater because of the contributionof the river inflow R (maximum at about 104 m3 s�1).
nel b), transport M. The transports are the lines inside the strips, error ranges ðsbi Þ areor range ðsbi Þ is represented by the strips (see legend). The marker and the bar, at theeys.
O.G.J. Riche, R. Pawlowicz / Estuarine, Coastal and Shelf Science 137 (2014) 1e13 7
The optimal inverse solution is then computed for all 47 surveysto provide a time series of transports. These estimates are formallyindependent of one another (except weakly through the determi-nation of time derivatives in b, which are found by fitting a parabolaover a 5-survey window). However, the results should show a de-gree of autocorrelation arising purely from auto-correlation in thetime series for observed temperatures, salinities, and river and heatflux forcings.
A critical test of the optimal solutions is that the equation re-siduals εj obtained by substituting the solution back into the orig-inal equations are consistent with a priori estimates of their size.However, this procedure by itself tells us little about the effects ofuncertainties in A, and the degree to which these propagatethrough to estimates of u. In order to account for these un-certainties, the optimal inverse procedure for finding the transporttime series is then embedded in a larger process that implements astatistical bootstrap (Efron and Tibshirani, 1993; Pawlowicz, 2001).In brief, the observational information discussed in the next sectionis “resampled with replacement” and used to provide “bootstrapreplicate” A and b matrices. We computed 200 such replicates foreach of the 47 surveys. These replicate matrices then form a sta-tistical population of budget equations (Eqs. (8)e(11)) that reflectsthe uncertainties arising from the finite number of observationsmade in the field program.
For each bootstrap replicate an optimal inverse (Eq. (15)) iscomputed, giving rise to a statistical population of estimates of thetime series of bu. The spread of this population then represents thepropagation of sampling uncertainty through the non-linearoptimal inversion procedures into our solutions. An estimate ofthe spread in these bootstrapped replicate transports is used as ourbootstrapped estimate of the solution uncertainty.
3.2. Box-model idealization
In applying this formalism to the Strait the most criticalparameter is the separation depth between upper and lower boxes.The separation depth roughly corresponds to the lower boundary ofthe pycnocline, belowwhich the water column contains mainly saltwater fromHaro Strait (HS). A detailed analysis of the depth profilesof the temperature and the salinity (based on STRATOGEM data, notshown) suggests that the dense salt water enters the SoG below15 m in summer (below 30 m in winter). During winter time, ver-tical mixing induced by sea surface wind and water cooling con-vection (Thomson, 1994; Pawlowicz et al., 2007) can bring thelower boundary of the pycnocline between 30 and 50 m.
However, the ultimate purpose of the box model is to be used toquantify fluxes, sinks and sources of macronutrients (phosphate,nitrate, silicic acid) and dissolved oxygen in biogeochemical bud-gets of the surface SoG. Most of the time, the depths of the 1% PAR(Photosynthetically Available Radiation) level, a proxy for theeuphotic zone depth, and the chlorophyll-a fluorescencemaximum, a proxy for phytoplankton biomass, are above, and oftenwell above, 30 m (based on STRATOGEM data, not shown). Thus,biological considerations suggest that the separation depth shouldnot be deeper than 30 m. The separation depth is therefore chosento be 30 m as a compromise between the needs of physical andbiological investigations. However, it will be important to test thesensitivity of the results to this choice.
The deep SoG box receives dense salt water from HS, which isrepresented as a single well-mixed box. In HS, the water column isusually relatively uniform compared to the SoG. In our idealization,the SoG’s only connection to the open ocean is through thesouthern entrance. The northern entrance can be neglected despitehaving currents of similar magnitude because of the cross-sectionalarea is much smaller. The cross-sectional areas of the northern
passages are only 7% of the areas of the southern passages(Waldichuk, 1957). In Section 6, we will further discuss the validityof this assumption in our box-model.
3.3. Processing of observations
To determine the best average of the observations from STRA-TOGEM over the surface and deep boxes, it is necessary to respectthe SoG geomorphology and to use all available data. In order to getthe box averages for various tracers, a mean vertical profile is firstfound by averaging all station data at a particular depth. Thenvolume weighted averages were computed using the SoG hypsog-raphy (see Fig. 3.2 in Riche (2011)). A similar method was used tocalculate the HS salinity and temperature conditions of theinflowing seawater.
Salinities in the surface box (Fig. 2b) vary through the year,primarily affected by the inflow of freshwater, and are well-correlated with the freshwater inflow rate with a lag of less thana month. Waters are freshest in June and July, and saltiest in winter.Salinities in the deeper water also vary seasonally, although thevariation has a much smaller range, and freshest waters tend tooccur in April (Fig. 2c).
The seasonal cycle of temperature in the surface waters variesalmost sinusoidally (Fig. 2d), and is clearly related (albeit with adelay of a month or two) with the net air/sea heat flux (Fig. 2a andd). Coldest temperatures in the surface box occur in January andFebruary, and in the deeper box in March and April (Fig. 2e). Thephase relationships between the seasonal cycles of these differentparameters are obviously related to the circulation; this has beendiscussed extensively in Pawlowicz et al. (2007).
Although the STRATOGEM data was obtained only in thesouthern half of the SoG, the observations are a good representa-tion of the Strait as a whole. Similar box-averages performed forquarterly surveys over the whole SoG (Fig. 2fei) are not verydifferent. The whole-SoG average salinity is slightly saltier than ourSTRATOGEM estimate in 2002 when the Fraser River plume in thesouthern SoG was especially prominent, and deep salinities overthe whole Strait slightly fresher than our STRATOGEM estimates in2003. The advantage of using the STRATOGEM data is that it pro-vides more detailed resolution in time, catching the timing of peaksmore precisely, and allowing for a better estimation of the time-derivative terms in b of Eq. (11).
The derivative scheme is based on a 5-point parabolic fit whichreasonably represents the actual time derivative of the tracers. Thereason for this choice is to remove the sampling noise that appearsfrom survey to survey in the salinity and temperature time-derivatives. Time derivatives, at the beginning and the end of thetracer time series, are handled by applying a 2- or 3-point time-difference scheme.
The standard optimal inverse procedure requires uncertaintyestimates, the estuarine transports Ui and Wi in vector bu and theresiduals εj (Wunsch, 1996). However, we also wish to account foruncertainty in box averages Si and Ti in matrix A. More sophis-ticated procedures known as the total least-squares and the totalinversion (Wunsch, 2006) can be used to take into account thesmall changes in both b and A. However, these procedures aresomewhat complex. Instead, the use of non-parametric andparametric bootstraps (Efron and Tibshirani, 1993) enables one totake advantage of the existing theoretical knowledge aboutoptimal inversions. We use a non-parametric bootstrap for thesalinity and temperature observations, and a parametric boot-strap for the external sources and forcings. We assumed thefollowing relative errors: 2.5%, 10% and 15% for HS observations,river inflow and heat fluxes respectively. The bootstrap is used togenerate 200 bootstrap replicates to calculate statistics. We
O.G.J. Riche, R. Pawlowicz / Estuarine, Coastal and Shelf Science 137 (2014) 1e138
computed an upper and lower deviation from the estimatesbased on the 16 and 50th percentiles, and the 50 and 84th per-centiles, respectively.
4. Results
Based on our observations and the formal inversion procedure,we obtained 47 transport estimates quantifying the whole SoGestuarine circulation over 3 years (Fig. 3aeb). These include theseaward (U1) and net upward entrainment (E ¼ W2 � W1) trans-ports, as well as the mixing exchange (M ¼ W1). The mixing ex-change is the amount of turbulent mixing that occurs between thetwo boxes (i.e. symmetric exchange of mass that changes tracerdistributions but results in no net transport), while the upwardentrainment is a unidirectional flow of mass. The estimates of U2(landward transport) and W2 (total upwelling), and the corre-sponding errors are not shown because U2 is not significantlydifferent from E (¼W2 � W1 x U2 according to Eq. (3)) and there islittle change in W2 time series (average of 6.2 � 104 m3 s�1 withseasonal variability of �0.4 � 104 m3 s�1).
The most noticeable aspect of these transports is that, althoughthe freshwater inflow has a large seasonal variability compared toits mean, the seasonal variation in the other estuarine circulationparameters is relatively small compared to their mean values. Meantransports for the whole time series are (4.5 � 0.1) � 104 m3 s�1 forU1 and (4.0 � 0.1) � 104 m3 s�1 for E. However, the individualtransport estimates range between 3.4 � 104 m3 s�1 and6.2�104m3 s�1 forU1, and 2.8� 104m3 s�1 and 5�104m3 s�1 for E(each with an uncertainty of around �5 � 103 m3 s�1. A relativelycrude statistical analysis of variance using an F-test (i.e. not takinginto account any autocorrelation in the time series) suggests thatonly U1 exhibits a significant seasonal variation, but even this sig-nificance is only marginal (p ¼ 0.06). However, it is visually clearthat there is a correlation between U1 and R, with both increasing insummer and decreasing in winter.
In contrast, the magnitude of the seasonality of the verticalmixing exchange M is as large as its average value. This variationis not statistically significant given the size of the uncertainty inindividual estimates, but a visual anti-correlation can be seenwith freshwater inflow, with smaller M when R is large and vice-versa.
Oscillations, of large magnitude and of approximate two-monthperiod, are visible around MarcheApril 2003 and 2004 (andpossibly around May 2005) when the sampling period usually is aweek or two weeks. The model could be aliasing a fortnightly tidalsignal in the transports and the box averages. However, an analysisof the phase between these oscillations and the tidal current(estimated at Active Pass, east side of Vancouver Island) or the seasurface level (at Sand Head) in MarcheApril 2003 and 2004 showno clear relationship with the fortnightly tides. Instead, these os-cillations could also be associated with the model misfit defined inSection 3.1 (Eqs. (4)e(7)) when the temperature and salinity aver-ages at the separation depth are approximated by the box averages(Sakov and Parslow, 2003). Before the freshet period, the approxi-mation error is the largest and it could propagate to the estimatedtransports.
Finally, interannual variations may be described by annual av-erages (see Fig. 4.11 in Riche (2011)). The estuarine circulation U1 ismuch larger than the river inflow R, and does vary slightly fromyear to year, although the means for the first and last year areprobably biased slightly by being based on estimates for only part ofthe year. The upward transport W2 is the largest of all the param-eters. Although the estuarine circulation must be driven by thefresh inflow, the sensitivity of the circulation to changes in riverinflow is small.
5. Analysis of the budget residuals and the sensitivity of thetransports
5.1. Budget residuals
Fig. 4aec show the residuals ε1, ., ε6 of the budget equations(Eqs. (2)e(7)) and the associated bootstrap variation. In the inverseprocedure, the weighting scales, u1,., u6, are the a priori values ofε1,., ε6. The weighting scales have been defined in Section 3.1 andTable 2 so that all the equations could be ranked against the ab-solute magnitude of each other and be scaled to have the sameweight in the optimized inverse procedure. In addition, theui’s givean idea of the size of the residuals when all the source and forcingterms in the budget equations take on large values. If the actualresiduals (when x ¼ bx) are smaller than the ui’s, one can expectthat the inversion procedure worked reasonably well.
An analysis of the sum of the squared and scaled residuals (lastrow in Table 3), one of the two quantities that are minimized in theinverse procedure (see Section 3 and Eq. (12)), indicates that theinformation in the observations is useful and leads to animprovement of the estimated transports relative to the a prioritransports. The overall appearance of the residuals (Fig. 4aec andTable 3) indicates that the actual residuals (εj’s) are smaller than orclose to the a priori residuals (ui’s). This suggests that the residuals(εj’s) are consistent. Thus, although the residuals for upper layermass and salt and lower layer heat do exhibit small annual cycleswhich are significant with respect to their bootstrapped variability,these variations are not significant with respect to our a prioriuncertainties.
In summary, the optimal solution provides an improvementover the a priori solution. It minimizes the residuals of the massbudgets and the bottom salt budget further than the a priori so-lution (Table 3).
5.2. Solution sensitivity
The trade-off in the objective function J between minimizing ε
and minimizing z (equation residuals and difference between apriori transports and estimated transports, respectively; see Eq.(12)) is equivalent to the trade-off between bias and variability inthe estimated transports. The trade-off is set by the weightingmatrices W and S (see Section 3.1). The objective function can berewritten:
J ¼ εTW�2
εþ s�2zTz (17)
Note that scaling down s and scaling up the coefficients of Whave equivalent effect. Although the resulting J would be different,only the relative change of J’s size is relevant.
For the sake of simplicity, but without loss of generality, we canset ad¼ s�1 as the default trade-off parameter and scale it with g, anarbitrary scale, so that the new trade-off parameter is a ¼ gad. Thescale g represents the influence of the trade-off because it repre-sents the relative weight between the norm of the matrices W andS. We can get the default trade-off back by setting g ¼ 1.
Varying g not only affects J, but also all the transports (Fig. 5).When g is large, transports tend to converge towards the a prioritransports (U01, U02, W01 and W02, Table 2). When g is small,transports tend towards the “pseudo-inverse” transports (Eq. (16))which have the largest variability (vertical bars in Fig. 5) of all thepossible solutions. The curves in Fig. 5 are roughly horizontal. Thismeans either that the inversion and data provide little informationon the transports or that the a priori transports are close to the truetransports. The next sensitivity test is designed to show that it is thelatter case.
Fig. 4. Residuals of the mass, salt and heat equations of the SoG box model. Although there is a seasonal variability in b), its magnitude is well below the a priori scale of 1.5 � 105.
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We replace the a priori transports uA with bu, where b is amultiplicative scale factor used to alter the a priori estimates. Whenb is large, the estimated transports are also larger, but are not aslarge as the a priori estimates (Fig. 6). Conversely, if we reduce b, wefind that the estimated transports also decrease, but they tend toremain larger than the a priori estimates.
Thus the transports are not strictly determined by the a prioritransports, but also by the information in the data. There is someinformation in the inversion and in the data that enable us to es-timate the true transports. In such situations, the optimal inverseprocedure will provide transport estimates that lie somewherebetween the true and the a priori transports. Thus, in Fig. 6 when b
is small enough, i.e. a priori much smaller than true transports, thetransports are larger than the a priori. Conversely, when b is largeenough, i.e. a priori much larger than true transports, the transportsare smaller than the a priori. But, in the latter case, the a priori, trueand estimated transports are close. In the default case (b ¼ 1), the apriori, true and estimated transports are close to each other, but theestimated transports retained some seasonal variability.
Finally, the basic equations used in our budget contain time-derivative terms for the rate of change of temperature andsalinity. Although these are straightforward to include in thedevelopment, they do imply a degree of correlation betweendifferent surveys. However, we can investigate their effect bycomparing solutions with and without these terms included. Adetailed investigation (see Riche (2011)) suggests that the inclusionof the time-derivative terms resolve the seasonal variability of the
estuarine circulation. Results are shown on Fig. 7 in the case of thesurface seaward transport (U1) because it characterizes the estua-rine response to the freshwater inflow (R). U1 is also more conve-nient to distinguish the effects of the inclusion of the time-derivatives. The reason for this is that the magnitude of U1 sea-sonal variability is large (mean of 4.5�104m3 s�1 and an amplitudeof �1.4 � 104 m3 s�1) and statistically significant compared to thethree other transports U2, M and W2 (more detail on the statisticsand the test can be found in Riche (2011)).
With the time-derivative terms, the summer maximum (winterminimum) is large (small) and in phase with the summer freshetpeak (winter low discharge) although spring oscillations occur(particularly large in spring 2003), most likely because of similarlarge spring oscillations in the surface salinity (see Fig. 2bec andSection 4). Without the time-derivative terms, the transports aresmoother without spring oscillations (circles and thin line in Fig. 7).However, winter water transports tend to be as large as summertransports and out of phase with winter low river inflow. One-month minimum transports occur during spring and fall and areout of phase with the river inflow minimum.
6. Discussion and conclusions
Using the inverse procedure, a variety of relationships have beendetermined between flow in different parts of this estuarine system(schematically shown in Fig. 8). Surface salinity (S1) covariesstrongly with the freshwater inflow (R). Salinity drops to its annual
Table 3A priori and estimated values of the residuals of the conservation equations in theSoG box model. The last line provides the average and maximum sums of thesquared scaled residuals with x ¼ bx and x ¼ xA. Values are percents of theweighting scales, except for the numbers on the last row (*) which are fractions of 1.
Equation A priori residuals Residuals with x ¼ bx(in % of uj)
Residuals with x ¼ xA
(in % of uj)
Average Maximum Average Maximum
Top mass u1 ¼ 1.6 � 104 3.1 11 14 69Bottom
massu2 ¼ 1 � 104 1.1 4.4 10 10
Top heat u3 ¼ 2.7 � 105 10.5 28 11 31.5Bottom
heatu4 ¼ 1.5 � 105 30 72 27.5 71
Top salt u5 ¼ 4.7 � 105 10 25 9 29Bottom
saltu6 ¼ 3.5 � 104 8 33 73 251
Sum ofscaled
Squaredεi’s (*)
Pmi¼1ðεi=uiÞ2
0.16 0.54 1.1 6.9
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low in July, lagging by less than a month behind the Peak Freshet(see Fig. 2b). The surface temperature peaks in August (soon afterthe net heat flux peaks in July). High temperatures thus approxi-mately coincide with low surface salinities and contribute, to the2nd order, to the enhancement of the summer stratification (seeFig. 2a and d). Incoming deep water also has changing character-istics, with saltiest water entering in the fall. However, the verticalstratification is mostly sustained by freshwater inflow R (see Fig. 3a).The enhanced summer stratification is found to coincide with aminimum in the rate of turbulent mixing (M). However, even in
Fig. 5. Circulation sensitivity to the trade-off parameter a ¼ g s�1. Overall averages anderrors of U1, E, U2, M and W2 over 47 surveys are the vertical lines and markers.
summer the turbulent mixing is not negligible with respect to thehorizontal transports (about 1/5 of U1). Surprisingly, the seasonalchange of outflow U1 is significantly smaller than its averagemagnitude (see Fig. 3a), and this coefficient of variation is muchsmaller than the corresponding value for freshwater inflow R. Thatis, large changes in forcing only produce relatively small changes inthe estuarine circulation. The deep inflow U2 (or equivalently theentrainment E) is even more steady, with no statistically significantseasonal variation.
The monthly estimates of estuarine circulation (U1, surfaceoutflow) and freshwater forcing (R) are highly correlated, with acorrelation coefficient of 0.6 � 0.16 (Fig. 9). However, a linearrelationship between the two, although generally consistent with areasonable assumption that increased freshwater inflow results inincreased estuarine circulation, does not pass through or even veryclose to the origin, even though one would naturally expect theestuarine circulation to disappear when no fresh inflow is present.The variation in seaward transport U1 is much smaller relative to itsmean compared the variation in freshwater inflow relative to itsmean, both in terms of the seasonal variability (Fig. 3a), but also interms of interannual variability (see Fig. 4.11 in Riche (2011)).Freshwater inflow varies by a factor of 5, but U1 varies by a factor ofless than 2.
Here the relationship between U1 and R is clearly consistentwith a small fractional power (Fig. 8). It is difficult to determine theexact fraction for two reasons. First, the scatter is large, and thereare no values at very low inflows where the fractional powerrelationship has a steep variation. Second, there is a tendency forthe inverse to pull values towards the a priori estimate(5 � 104 m3 s�1), although our sensitivity analysis (Section 5.2) andexperience suggests that this effect would not be strong enough to
Fig. 6. Circulation sensitivity to the a priori parameter b. b is the ratio between thenew a priori average b xAi and the corresponding default value xAi , (Table 2, U01, U02,W01, and W02). Overall averages and errors of U1, E, U2, M and W2 over 47 surveys arethe vertical lines and markers.
Fig. 7. Effect of the time-derivative terms on the estuarine response to freshwater, U1 the surface seaward transport. This figure highlights the changes on the transport U1 with thetime-derivative terms (squares and thick line) and without them (circles and thin line), and its phase with respect to the seasonal cycle of the river inflow, R (triangles and thin line).
O.G.J. Riche, R. Pawlowicz / Estuarine, Coastal and Shelf Science 137 (2014) 1e13 11
disguise a linear relationship. Thus, as far as the authors are aware,this is the first time that fractional power relationship has beenquantitatively determined from observations in an estuary.
Although the use of an optimal inverse procedure does tend toshift solutions towards their a priori values (and hence reducesvariability), a “stiffness” (as coined by Geyer (2010)) in the estua-rine response is characteristic of all inverse solutions that havebeen investigated using the dataset and hence appears to be arobust result. Two factors are important. First, summer increases infreshwater inflow should result in stronger current shear, whichwould promote entrainment and hence proportionately greaterestuarine circulation. However, the increases in freshwater inflowalso increase the stratification, which acts to inhibit mixing withdeeper waters and hence inhibits entrainment. This second factor
Fig. 8. Schematic of the Flow Relationships in the Strait of Georgia Box Model. We are only sThe notations used here are identical to the notations found in the rest of this paper, see Figrespectively.
would result in a smaller estuarine circulation, and hence a flatteror stiffer response curve.
It is not necessarily clear in advance which effect will dominate,but the transport estimates obtained here suggest that the secondfactor is very important. We cannot say whether the balance seenhere is a peculiarity of the SoG and not typical of estuaries ingeneral. However, in previous theoretical studies of estuarine cir-culation, the deduced response to freshwater forcing often appearsin the form U1fR1=n with n equal to 3, 5 or 7 (Chatwin, 1976;MacCready, 1999; Monismith et al., 2002; Hetland and Geyer,2004; MacCready and Geyer, 2010). The exponent n represents ameasure of the influence of the length of the estuary and theincreasing influence of the tidal mixing on the estuarine transports,with larger n and “stiffer” estuaries resulting from increased
howing one year. But, there is evidently interannual variability in the actual time series.s. 1 and 2, in particular s1 and s2 are residence times in the surface and bottom layers,
Fig. 9. Surface seaward transport, U1, plotted with respect to freshwater inflow. Thedashed lines represent theoretical curves aR1/n (with a ¼ 2.68 � 103 in the case ofn ¼ 3), the straight dash line is the best fit of R to illustrate the lack of proportionalitybetween U1 and R. Note that there is also a possible linear approximation based on themass conservation (Eqs. (2) and (3)) U1 E þ R with E¼(4 � 0.3) � 104 m3 s�1. The twolow values for R z 12 � 103 m3 s�1 are associated with the large spring oscillationsdiscussed in Section 4.
O.G.J. Riche, R. Pawlowicz / Estuarine, Coastal and Shelf Science 137 (2014) 1e1312
geometric variation along the estuary and increasing influence ofstratification on mixing (Geyer, 2010), and n closer to 1 corre-sponding to shorter estuaries dominated by tidal dispersion(MacCready and Geyer, 2010). These works also suggest that n ¼ 3could be considered the intermediate exponent between shortestuaries and stiff estuaries and perhaps a default value formodelling estuarine response. Here the relationship between U1and R is clearly consistent with a small fractional power (Fig. 8),although the scatter is large enough that the exact fraction cannotbe easily determined. As far the authors are aware, this is the firsttime that such a relationship has been quantitatively determinedfrom observations in an estuary.
Note that relationships involving such a small fractional powerare not inconsistent with the obvious constraint that in the limit ofno freshwater inflow the estuarine circulation must also vanish. Inthis case, if the freshwater inflow was to be reduced in winter thanquite large decreases in the estuarine circulation might occur.Conversely, the power relationship might be better determined ifeven larger flows occur during summer freshets, since the flowduring those times will have an important effect on the power inany fit.
However, neither of these situations is likely to occur, at least inthe Strait of Georgia. Examining Fraser River variability over theperiod 1912e2008 (data not shown, but available in Riche (2011))suggests that the STRATOGEM observational period containedconditions representative of both historical high and low flows.Although the flow in 2003 is typical of average conditions, the 2002summer freshet and 2004/2005 winter discharge are representa-tive of long-term maximums for those times of year (see Figs 1.4and 6.4 in Riche (2011)). On the other hand, observed winterminima in 2002/2003 and 2003/2004 are very close to historicallow levels, as is the flow during the summer of 2004 (see Figs 1.4and 6.4 in Riche (2011)).
Projections of the Fraser River discharge and precipitations inthe SoG (Morrison et al., 2002; Jakob et al., 2003) suggest a long-
term shift of the discharge profile from a freshet dominated to arainfall dominated (lower and earlier freshet and more frequentshort duration rainfalls year around). This shift would furtherreduce the range of the seasonal variability of the FW inflow intothe SoG and as a likely consequence further reduce the seasonalvariability of the estuarine circulation.
The relatively small variations of W2, the total upwellingtransport (overall seasonality magnitude se of 0.4 � 104 m3 s�1 andaverage transport of 6.2 � 104 m3 s�1), suggests a relatively con-stant upward transport. This particular property of W2 is useful toestimate the overall residence time in the top and bottom layers ofthe SoG. Given the relative invariance ofW2 and its largemagnitudewith respect to R, these residence times are also relatively constantfor the SoG box model. Eqs. (4)e(7) show that the time rate of heatand salt outward advection leads to a residence time s ¼ volume/outflow: in the top box (T1 and S1) s1 ¼ V1/(W2 þ R) ¼ 33 � 2 days(minimumemaximum range: 28e37 days), in the bottom box (T2and S2) s2 ¼ V2/W2 ¼ 170 � 11 days (minimumemaximum range:142e197). These values are consistent with previous estimates(England et al., 1996; Pawlowicz et al., 2007). The residence timesthen indicate the lag in the observation of the SoG propertiesrelative to those of a changing source. A direct estimate of lag timeswas used in Pawlowicz et al. (2007) to infer residence times, underthe assumption that these were relatively constant. The analysis ofthis paper confirms those assumptions.
Since the main forcing of the SoG estuarine circulation can varyquickly during the Fraser River freshet, it is natural to wonderwhether the SoG is in a quasi-steady balance, and to ask over whattime scales this is true. The deeper waters, with a residence time ofhalf a year, will be in quasi-steady equilibrium with interannualchanges in the source water of the intermediate depth PacificOcean. The surface waters will be in a quasi-steady equilibriumwith seasonal changes in forcing (heat and freshwater fluxes), butwill not respond to short-term events like storms. However, it isimportant to emphasize that these conclusions would hold only foran overall average of surface water properties. The Fraser Riverplume, which is an important part of the surface waters of the SoG,has a much shorter residence time of only a few days (Halversonand Pawlowicz, 2008), and hence is significantly affected by theseshort time scale events.
In the case of the theoretical scalings in exponent n ¼ 3 and 5,Geyer (2010) suggested a rule of thumb to verify the validity of thecoupled equations and the corresponding scalings. Any changewithin characteristic time scale near or above the residence time ofthe estuary can be considered slow enough. This suggests thatperhaps, the scaling is not as accurate during the sometimes rapidchanges that occur in high flow periods as it is at low flow. This is atleast consistent with the increased scatter at high inflow (Fig. 8).However, MacCready (2007) examined the adjustment time of es-tuaries using a time-dependent version of Hansen and Rattray(1965) and Chatwin (1976)’s coupled equations. The results fromhis analytical solutions and numerical model, although different byapproximately a factor from 1 to 4 (depending on the estuaryconsidered: San Francisco Bay, Delaware Bay, and Hudson River,Table 2 (MacCready, 2007)) suggest that the time adjustment ofexchange-dominated estuaries is faster at high flow (order of103 m3 s�1) than at low flow (order of 102 m3 s�1), so the issue mustremain unresolved.
It would therefore be necessary to study other estuaries to gainfurther knowledge on the U1 ¼ f(R) relationship. Future studiescould focus on estuaries with similar features of that the SoG, butwith smaller or larger magnitude of transports and freshwater.However studies of other systems would also enable investigationof the diversity of fractional relationships (Geyer, 2010; Jay, 2010;MacCready and Geyer, 2010) across different estuaries and further
O.G.J. Riche, R. Pawlowicz / Estuarine, Coastal and Shelf Science 137 (2014) 1e13 13
understand the link between exponent and the dynamics of a givenestuary.
Finally, the existence of this rather simple scaling for the estu-arine circulation has significant implications for further investiga-tion and monitoring of the biological variability of this estuarinesystem. For example, it suggests that measurements of biologicalparameters alone will be sufficient for monitoring the large-scalevariability of the Strait, and that advective loss terms in particularcan be deduced from measurements of tracers in the SouthernStrait in conjunction with river flow alone.
Acknowledgement
Funding for the STRATOGEM program was provided by theNatural Sciences and Engineering Research Council of Canada un-der grant 246274. We would like to thank three anonymous re-viewers and the editor of ECSS. Their comments helped us greatlyimproving the original manuscript.
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