Low frequency variations in currents on the southern continental shelf of the Caspian Sea

Dynamics of Atmospheres and Oceans 87 (2019) 101095

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Low frequency variations in currents on the southern continental shelf of the Caspian Sea ⁎

T

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Mina Masouda, , Rich Pawlowiczb, , Masoud Montazeri Namina a b

School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran Dept. of Earth, Ocean and Atmospheric Sciences, University of British Columbia, 2207 Main Mall, Vancouver, B.C., Canada V6T 1Z4

A R T IC LE I N F O

ABS TRA CT

Keywords: Coastal trapped waves Caspian Sea Continental shelf Spectral analysis Current meter observation CTW model

The Caspian Sea (CS) is the largest enclosed basin in the world, located inside the Eurasian continent in the Northern Hemisphere. Although there have been few studies of the dynamics of the coastal zone in the CS, observations show that oscillations with periods from 2–3 days to 1–3 weeks dominate. These oscillations are presumed to be related to the synoptic variability of direct wind impact and to coastally-trapped waves (CTW). Here, we describe and interpret current meter observations on the continental margins of the southern CS from 2012 to 2014 to identify and characterize CTW there. Time series analysis provides evidence for both remote and locally wind-forced eastward traveling signals with time lags consistent with CTW theory. A wind-forced model with two CTW modes is able to reproduce the structure, amplitudes, and phases of observed alongshore current fluctuations, explaining half of the variance at frequencies less than 1 cpd. Remote forcing effects are present at all times, but are most striking when the local winds are weak, as in summer. The CTW calculations also suggest that the source region for the remote forcing may extend farther north along the west coast of the CS.

1. Introduction Observations in large lakes and oceans often exhibit strong, low frequency variability in currents and stratification near coasts (Csanady, 1981). In mid-latitudes, this variability is strongly coupled to synoptic winds and can be explained in terms of the properties of coastal trapped waves (Pizarro and Shaffer, 1998, CTWs;). CTWs are therefore one of the major components of the mesoscale/subinertial variability in sea level and alongshore current; they make important contributions to coastal circulation, exchange, and mixing in continental shelf and slope regions (Huthnance, 1995). CTWs typically have amplitudes on the order of 10 cm, periods longer than the inertial period, and wavelengths of tens to hundreds of kilometers, depending on the continental shelf width and slope (Mysak, 1980). CTWs, through their propagation, spread the ocean's response in one direction only, like classical Kelvin waves. In the Northern (Southern) hemisphere they travel with their right (left) side against the coast (Brink, 1991). The Caspian Sea (CS) is the largest enclosed basin in the world, with a length of over 1030 km and width of approximately 310 km, located inside the Eurasian continent in the Northern Hemisphere (Fig. 1a). The CS is divided into 3 basins. The northern basin is very shallow (maximum depth less than 20 m). Central and southern basins with maximum depths greater than 700 m are separated by a sill of depth about 170 m. A narrow and distinct shelf exists along the southern and western edges of the southern basin. Since tides in the CS are very weak (Medvedev et al., 2016, 2017), it is possible that most of the lower frequency variability in the CS, especially in this shelf area, may be due to CTW.

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Corresponding authors. E-mail addresses: [email protected] (M. Masoud), [email protected] (R. Pawlowicz).

https://doi.org/10.1016/j.dynatmoce.2019.05.004 Received 13 December 2018; Received in revised form 6 May 2019; Accepted 22 May 2019 Available online 01 June 2019 0377-0265/ © 2019 Elsevier B.V. All rights reserved.

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Fig. 1. (a) The Caspian Sea and the study area located in southern basin of the CS. Shown in the inset are the locations of current meter moorings and variance ellipses for barotropic currents at these locations. Depth contours at 10, 50, 100, 200, 500 and 800 m are also shown. The dashed gray lines are transects used for extracting shelf profiles. Topography is derived from the ETOPO2 dataset (Center, 2006); the mean water level in the CS is at 28 meters below mean sea level. (b) Continental shelf profiles at current meter locations. (c) Current data measurements dataset. (d) Sea level dataset.

Various field measurements have been used to investigate currents in the northern and the central basins of the CS (Klevtsova, 1967; Tsytsarev, 1967; Bondarenko, 1993; Tuzhilkin et al., 1997; Tuzhilkin and Kosarev, 2005; Ibrayev, Özsoy et al., 2010; Gunduz, 2014; Bohluly et al., 2018). Although there have been few studies on the dynamics of the coastal zone in the CS, observations show that oscillations with periods from 2–3 days to 1–3 weeks dominate (Baidin and Kosarev, 1986). These oscillations are presumed to be related to the synoptic variability of direct wind impact and to CTWs (Kosarev, 1990). In the first attempt to specifically investigate CTW in the CS, Bondarenko (1994) analyzed field measurements in the middle and the northern basins and showed that there is no direct relation between wind and currents even in extreme winds, but showed that CTW were present with current speeds of 0.15–0.2 m s−1 and periods of about 6 days (Bondarenko (1994); as summarized by Ghaffari and Chegini (2010)). However, only a few measurements have been carried out in the southern basin, which has the most developed shelf (Zaker et al., 2007, 2011; Ghaffari et al., 2013). Ghaffari and Chegini (2010), using current measurements at one station in the southeastern part of the CS suggested that the propagation of long period waves could be described in terms of CTWs that are remotely generated. However, a detailed study of long-period variability propagating along the continental shelf of the CS as CTW is still lacking. In late 2012 to early 2014, a series of field studies were conducted in the southern basin of the CS (Fig. 1c and d) to monitor the area and provide data to evaluate numerical models. Acoustic Doppler Current Profilers (ADCP) equipped with pressure gauges to measure sea level were deployed at a number of locations on the shelf. Here, we describe and interpret the results obtained from analysis of these observations to identify and characterize coastally trapped waves on the continental margins of the southern CS. 2. Material and methods 2.1. The study area The CS is an enclosed basin in which freshwater from rivers (primarily the Volga) enters and evaporates (Fig. 1a). The northern basin (North CS) with maximum depth about 20 m is very shallow with numerous islands and extinct river channels. The southern (South CS) and the central basins of the CS (Middle CS) have maximum water depths of 1025 and 788 m respectively, and are 2

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separated by a sill with a maximum depth of about 170 m (Kosarev, 2005). The Middle CS is asymmetric with narrow shelf and a steep slope in its western part, and wide shelf and a gentle slope in its eastern part. The southern and southwestern sides of the South CS are bounded by a narrow “continental shelf” (Fig. 1a). The depth increases gently from the coast to approximately 50–100 m near a shelf break about 10–30 km offshore, after which it sharply increases to 800 m over another 20 km offshore (Fig. 1a and b). However, this increase is more gradual at the far ends. A broad combined continental shelf and slope, more than 300 km wide, exists on the east side of the South CS north of AmirAbad, and a 150 km wide shelf/slope region is located on the western side at Astara. Atmospheric processes govern the CS circulation. The impact of the wind in the form of fluxes of momentum and relative vorticity generates the three-dimensional general circulation in the CS (Tuzhilkin and Kosarev, 2005). Northerly or southeasterly winds are prominent over the CS during most of the year. In the South CS, where strong winds are rarely observed, the mean annual wind speed is 3–4 m s−1, and the occurrence rate of weak winds (wind speed less than 5 m s−1) reaches 90%. South CS storms (wind speed greater than 15 m s−1) occur not more than 20–30 days per year (Kosarev, 2005). The general circulation in the CS is mostly cyclonic based on both indirect estimates of currents (floats and dynamical methods), and diagnostic simulations by numerical hydrodynamic models (Bondarenko, 1993; Ibrayev, Özsoy et al., 2010; Gunduz, Özsoy, 2014; Lebedev, 2018) although sometimes anticyclonic circulation is observed in some regions of CS. The currents in deep layers (below 20–50 m) have a stable cyclonic character during the whole year (Zyryanov, 2015). Currents in the north are weak with magnitude of only a few centimeters per second; currents in the south are stronger with a strength of 10 s of centimeters per second. 2.2. Data and data processing A current measurement program using Nortek AWAC 600 kHz Acoustic Doppler Current Profilers (ADCP) was carried out at five stations in the South CS over a period of about 16 months from late 2012 to early 2014 (Fig. 1c). Here we use ADCP data for the period of December 2012 to December 2013 when coverage is most complete. Each ADCP was deployed on the bottom and measured currents in 0.5 m bins at 10 min intervals. Current velocity variations on the shelf follow almost the same trend at all depths and appear essentially in phase, with amplitude decreasing slightly with depth throughout the water column (only shown for Astara station in Fig. 2b). Since the flow fields at all stations show a barotropic behavior, the depth average currents are used for further data analysis. ADCP instruments were sometimes equipped with Acoustic Surface Tracking (AST) sensors to measure water level. Some of the gaps in the water level records are filled with data from tide gauges installed at Anzali, Noshahr and AmirAbad stations from May 2013 to February 2014 (Fig. 1d). These sea level records are adjusted for the effect of the atmospheric pressure on sea level according to the hydrostatic approximation to eliminate the assumed in-phase effect of atmospheric pressure variations on sea level (Cutchin and Smith, 1973). Meteorological parameters used are from a Weather Research and Forecasting (WRF) model (Ghader et al., 2014), interpolated to the location of ADCP measurement stations. The WRF model is configured in two nests with 0.3° and 0.1° horizontal grid resolution, respectively. The domain includes the Caspian Sea in a latitude-longitude projection; the finer (inner) domain covers the Caspian Sea. The outer and inner domains have 42 × 52 and 94 × 124 grid points, respectively. Initial and boundary conditions for the WRF model simulations are provided by the 6-hourly ERA-Interim reanalysis data of the European Centre for Medium-Range Weather Forecasts (ECMWF). The 10 m elevation wind velocity from the model is transformed to wind stress here following Large and Pond (1981). 3. Results 3.1. Observations The mean flow on the shelves is predominantely in an alongshore direction from west to east except at Roodsar where the mooring is located inshore of a known anticyclonic eddy that exists behind Cape Sefidrood (Table 1). Variance ellipses are highly elongated, with the variability also mostly aligned alongshore, even at Roodsar (Fig. 1a and Table 1). Variability with amplitudes of more than 0.1 m s−1 is larger than the mean flows which range from 0.012 to 0.086 m s−1. Subtracting the mean, we compute the principal axes for each record and consider only the major axis component, taking this to be the alongshore direction. Fluctuations in alongshore current speed generally decrease from west to east, although they are largest at Noshahr (Table 1). The spectral energy in the current records is relatively flat at low frequencies, but decreases at about f−2 for frequencies f ≳ 0.2 cpd (Fig. 2a). Wind spectra are very similar in shape, but also have clear narrow peaks at 1 and 2 cpd related to a sea breeze system in the southern CS. This sea breeze peak is visible in across-shore velocity spectra (not shown) but is insignificant in alongshore velocity spectra. To remove sea breeze effects, we filter the data with a 25-hour moving average filter; this preserves about 65–82% of the variance depending on the station. The local low frequency alongshore stress is usually weak with only a few strong events at all sites. These mostly occurred during the cold seasons from December 2012 to April 2013 and October 2013 to early January 2014, with far fewer in other seasons (Fig. 2, parts b to f (i)). The most intense event in alongshore wind stress can be seen in early October 2013 at all stations except at AmirAbad, located in the most eastern part of the study area. There are also some other, weaker, events in wind stress in late June and early July 2013 at Roodsar and AmirAbad. The minimum values of sea level height are observed in the winter, and the maximum values are in summer (Fig. 2, parts b to f, 3

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(caption on next page)

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Fig. 2. (a) Spectral estimates of wind and current energy using a multi-taper technique. (b–f) Time series of low frequency wind stress, sea level and alongshore current at (b) Astara, (c) Anzali, (d) Roodsar, (e) Noshahr and (f) AmirAbad stations (Fig. 1a). Currents are shown at three depths for Astara, but only barotropic currents are shown elsewhere. At each location we show (i) Low frequency hourly wind stress, (ii) Low frequency hourly sea level and (iii) Low frequency hourly alongshore current. Grey arrows show the intense events in low frequency sea level and current data associated with the strong bursts in wind stress as discussed in the text.

Table 1 The Location, water depth (m), mean velocity (m s−1), direction of mean flow, magnitude of major and minor axis of variance ellipses (m s−1) and direction of major axis of ADCP measurement. Station

Astara

Anzali

Roodsar

Noshahr

AmirAbad

Longitude (°E) Latitude (°N) Water depth (m) Mean speed (m s−1) Mean Direction (degrees) Magnitude major axis (m s−1) Magnitude minor axis (m s−1) Direction major axis (degrees)

49.05 38.37 31 0.044 161.37 0.144 0.035 175.74

49.45 37.49 10.5 0.086 98.64 0.130 0.036 93.78

50.35 37.23 32 0.013 324.98 0.121 0.027 158.54

51.39 36.7 10.5 0.056 99.14 0.154 0.038 85.14

53.41 36.91 13.7 0.012 76.65 0.108 0.029 75.6

(ii)). This seasonal variation is controlled by the amounts of the river inflow, mostly from the Volga River (Tuzhilkin and Kosarev, 2005). The low frequency current and sea level time series are characterized by a continual but irregular occurrence of events with a duration of few days (Fig. 2, parts b to f, (ii) and (iii)). The more intense bursts of low frequency current and sea level are responses to similar bursts in the wind stress. However there are also low frequency current and sea level fluctuations that have no apparent association with the local wind stress. These bursts of sea level change and current motion correlate well with one another along the southern CS basin. Low frequency alongshore current speeds of about 0.4 and 0.6 m s−1 are recorded in late December 2012, late June 2013, early October 2013 and December 2013 subsequent to strong wind events at Astara, Anzali and Noshahr stations. There are events in low frequency alongshore current which occurred in late June and early July 2013 corresponding to strong wind episodes at Roodsar and AmirAbad stations. However, wind stress is often near zero in this area, while low frequency alongshore current and sea level fluctuations are still observed, suggesting a significant non-local forcing. Variations in low frequency current data with magnitude up to 0.4 m s−1 and sea level of up to 0.3 m can be observed from late January to June 2013 and from late July to September 2013 but they do not have a clear relation with any local wind events. Overall, the visual correlation between low frequency winds and local currents is generally low. Strong currents and sea level changes are evident in calm wind conditions, but some wind events cause strengthening of the currents. Unfiltered time series of wind and current data in the alongshore direction have correlations of 0.15–0.5 during the measurement period (not shown). However, the zero-lag correlation increases to 0.4–0.7 when low-pass-filtered alongshore wind and current are considered and a maximum correlation 0.5–0.7 occurs at a time lag of 4–13 h. This suggests the alongshore low frequency local wind forcing drives between about 25% and 50% of the variance in low frequency current velocities in the alongshore direction. Time lags and correlations are now investigated using coherence (Fig. 3). A multi-taper method is used to preserve lowest frequencies (Percival and Walden, 1993). To derive the phase lag δϕ as a function of frequency ω, current velocity time series are used for pairs of adjacent stations along the South CS. The resulting phase lags corresponding to statistically significant coherence are then converted to mean time delays Td using δϕ = 2πωTd. The wavelength L corresponding to the phase lag and the speed of signal propagation cp between pairs of stations separated by a distance D can be calculated from L = 2πD/δϕ and cp = D/Td, respectively. At lowest frequencies (less than 0.1 cpd), the variabilities are mostly coherent and in phase, albeit with large error bars. Eastward lags exist for coherence at higher frequencies. The average estimated time lags between adjacent ADCP stations are in the range of about 6–20 h implying propagation from west to east with phase speeds of 1.5–4 m s−1 (Fig. 3). To investigate the relations with both local and remote winds, coherence is calculated between currents at AmirAbad with winds at all locations (Fig. 4a). Coherence with the remote winds is of the same order or higher than with local winds, with time lags (computed only for frequencies of significant coherence) of 0–20 h, with winds leading. Similar results are found for other stations (not shown). In contrast, the wind data are highly coherent along the study area (Fig. 4b). However, although the phase of traveling current signals progresses from west to east (Fig. 3), the wind system propagates both westwards and eastwards towards Noshahr with an average time lag of −2 to 6 h between pairs of wind stations (Fig. 4b). Thus, the patterns of propagation direction and speed for wind fluctuations are completely different to those for currents. 3.2. Application of CTW theory The stratification in the CS (Fig. 5) depends on both temperature and salinity, and varies seasonally with warm salty waters in a summer mixed layer and fresher, less warm surface waters in winter, so that the density contrast with deeper waters is somewhat greater in summer (Jamshidi, 2017). The summer water column is almost isohaline, so that density profiles are largely governed by 5

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Fig. 3. Squared coherence and spectral phase of alongshore current data measurements between adjacent stations (a) Astara-Anzali, (b) AnzaliRoodsar, (c) Roodsar-Noshahr and (d) Noshahr-AmirAbad stations; (i) Coherence squared and phase lag in degrees. The red dotted and black dashed lines are the 95% significant coherence level and zero phase degree, respectively. Positive phase shows the first station leads the second one. (ii) Time lags in hours. Grey area shows the 95% confidence interval. The hatched area between 0 and 0.1 cpd frequency range corresponds to the lowest frequency area where the error bars of phase lag are extremely large.

temperature, but in winter a shallow halocline and a deeper thermocline both affect density. The continental shelf waters are located in the surface mixed layer which extends down to almost 100 m in outer parts of the shelf (Fig. 5). The relative importance of stratification and bottom topography in the study area can be further understood by calculating the Burger number S = NH/fL (where H and L are depth at the shelf break and shelf width, respectively, N is the buoyancy frequency and f is the Coriolis parameter) (Brink, 1991). A small S value shows a weak influence of stratification on subinertial motions, indicating behavior is more similar to barotropic continental shelf waves, while large S implies a baroclinic Kelvin wave structure. The Burger number at south and southwest parts of the CS with narrow shelves (Anzali, Roodsar and Noshahr) is around 2 to 3 in winter and 9 to 13 in summer. Further west and east where shelves are wider (Astara and AmirAbad), the Burger number is about 0.9 to 1.2 in winter and 4 to 5.5 in summer. Therefore, we expect that waves would be affected mainly by stratification and approach baroclinic Kelvin waves in summer, with more baroclinic structure at narrower shelves. In winter, topography is much more important for wave dynamics, especially in the narrow shelf areas. Results of the analysis above suggest that both stratification and topography have to be considered in understanding the details of shelf-wave propagation, and encouraged us to apply a full CTW model to the coastal zone of the South CS. With this model we can determine the fraction of observed low frequency currents that are due to such shelf waves, and can also separate the locally and remotely forced components.

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Fig. 4. (a) Coherence and time lag of alongshore current data measurements with alongshore local and remote wind at AmirAbad. (b) Coherence and time lag of alongshore wind stress at pairs of stations; (i) Coherence squared and (ii) Time lags in hours corresponding to coherence more than 0.5. The positive time lag shows the first station lead to second one. The red dotted and black dashed lines are coherence squared equal to 0.5 and zero time lag. The hatched area between 0 and 0.1 cpd frequency range corresponds to the lowest frequency area where the error bars of phase lag are extremely large.

Fig. 5. Temperature, salinity and density profiles (Jamshidi, 2017) in (a) Summer and (b) Winter. Profiles of squared buoyancy frequency (N2) in (c) Summer and (d) Winter.

3.2.1. Theory and definition CTW theory has been discussed by Gill and Clarke (1974), Allen (1975), Wang and Mooers (1976), Clarke (1977), Huthnance (1978), Brink (1982a), Brink (1982b), Brink (1991) and Brink (2018) and we give here only a brief overview. For a coordinate system aligned with x in the offshore direction and y in the alongshore direction, the linearized equations governing an inviscid ocean under constant rotation are:

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ut − fv

=

vt + fu

=

Pz = u x + vy + wz = ρt =

−Px ; ρ0 −Py ; ρ0 − ρg ; 0; − wρ0z ,

(1)

where u, v and w are the cross-shore, alongshore and vertical velocities, ρ(x, y, z, t) is the perturbation density from a rest state ρ0(z), P is the perturbation pressure, and f and g are the Coriolis parameter and acceleration due to gravity, respectively. With a free surface, no normal flow at the bottom and at a coastal wall, and coastal trapping boundary conditions, Eqs. (1) reduce to

Pxxt + Pyyt + (f 2 + ∂tt)(

Pz )zt = 0 N2

(2)

with the following set of boundary conditions:

Pzt + g −1N 2Pt = 0 at z = 0; (f 2 + ∂tt) Pzt + N 2h x (Pxt + fPy ) = 0 at z = −h (x ); Pxt + fPy = 0 at x = 0; P→0 as x → − ∞ ,

(3)

2

where N (z) is the buoyancy frequency and h(x) is the local depth of the ocean. The last condition specifies trapped coastal behavior. In the usual mathematical approach a separated wavelike response of the form

P = P˜ (x , z )exp[i (ly + ωt )],

(4)

is assumed, where l is the alongshore wavenumber and ω is the frequency, and Eqs. (2) and (3) are converted into a two-dimensional eigenvalue problem with eigenfunctions Fn(x, z) for pressure (and associated shapes for the different velocity components) as well as a dispersion relation that gives wave speeds cn(ω). Using the long-wave approximation, the pressure field in a system forced by alongshore winds τ(y) (assumed constant in the x direction) and subject to weak vorticity damping through an Ekman suction with a bottom resistance coefficient r, can then be expanded in terms of these free-wave modes (Brink, 1982b):

P (x , y, z , t ) =

∑ Fn (x , z ) ϕn (y, t ),

(5)

n

where mode amplitudes ϕn are now governed by a set of approximately uncoupled simple wave equations (Clarke, 1977; Brink, 1982b)

cn−1 (ϕn )t − (ϕn ) y + an ϕn ≈ bn τ ,

(6)

The cn are the (constant) long-wave speeds for each mode, an are frictional damping coefficients, and bn wind coupling coefficients. If the local depth profile h(x) varies only slowly in the y direction, and the frictional decay length an−1 is much larger than a characteristic wave length, the coefficients of Eq. (6) vary slowly enough that a solution can be integrated by the method of characteristics to give

ϕn (0, t ) = ϕn (Y , t −

∫0

Y

dy′ − ∫Y an (y′)dy′ )e 0 + c (y′)

∫0

Y

e− ∫0

y

a (y′)dy ′

bn (y ) τ (y, t −

∫0

y

dy′ )dy c (y′)

(7)

The first term in Eq. (7) describes a CTW propagating along the coast from an upstream boundary at y = Y, damped by friction. The second term describes the response of the wave amplitude function ϕn to past forcing along the modeled propagation path for 0 < y < Y, also reduced in magnitude by friction. As we shall see, the long-wave approximation is not entirely valid for our waves, which are slightly dispersive over the frequency range of interest. We have also developed an alternative solution for along-coast propagation, based on a frequency-domain approach, which allows us to take into account the effects of dispersion. Details are provided in Appendix A. To compute the eigenfunctions and coupling coefficients the most recent version (Brink, 2018) of the Brink and Chapman (1985) numerical code was used. The South CS was first divided into five segments, each roughly centered on a location where ADCP measurements were available. For each segment, a cross-shore bottom topography was determined (Fig. 1b) ending in a vertical coastal wall of 5-m depth. The bottom friction coefficient r was set to 1.5×10−4 m s−1. The free CTW modes, with their dispersion relations, frictional coefficients, and wind coefficients, were calculated using a grid of 30 vertical by 120 horizontal points in winter and summer and the N2 profiles shown in Fig. 5c, d. Along-coast propagation for each mode was then determined by numerically solving Eq. (7), for which a numerical discretization was described by Gill and Clarke (1974), and/or Eq. (11). 3.2.2. Characteristics of CTW The spatial structures for the first and second modes have almost the same characteristics in all segments, with nearly vertical isopleths and a barotropic structure in winter, and a more baroclinic structure in summer (Fig. 6). The first mode in summer has 8

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Fig. 6. Calculated alongshore velocity structure for (a) First mode in summer, (b) First mode in winter, (c) Second mode in summer and (d) Second mode in winter for each station in Fig. 1a, (i) Astara, (ii) Anzali, (iii) Roodsar, (iv) Noshahr and (v) AmirAbad. Shaded contours are negative.

nearly horizontal isopleths intersecting the shelf slope at a depth of about 400 m in the three middle locations (Fig. 6a). However the zero crossing is deeper and isopleths are almost vertical for the first mode in summer at Astara and AmirAbad. Within the shallow shelf zone, velocities are roughly independent of depth, with nearly barotropic structure near the coast for the first and second mode at all stations in both winter and summer cases (Figs. 6 and 7a). The largest currents in these modes also occur in the shelf zone from the coast to about 10-25 km offshore near the shelf break at depths less than 100 m (Figs. 6 and 7a). Mode 1 velocities continue to decay seawards, but mode 2 velocities have a zero-crossing at about 50 km. Across-isobath and vertical velocities (not shown) peak at the region of the steepest slope at depths of around 40 to 100 m, and are typically two orders of magnitude smaller than along-isobath velocities. The spatial variation of the coefficients in Eq. (6) are shown graphically in Fig. 7. CTW mode 1 in winter has a phase speed of about 1.4 m s−1 in regions with narrow shelves (Anzali, Roodsar and Noshahr), with larger phase speeds at the eastern and western end of our array (4.3 m s−1 at AmirAbad and 2.1 m s−1 at Astara) where shelves are wider (Fig. 7b). Mode 2 phase speeds in winter are about 50% as large as mode 1 phase speeds, but the spatial contrast is not as large, with speeds at AmirAbad and Astara only slightly larger than speeds over the narrower shelves. In summer, mode speeds at all locations increase without changing this basic spatial pattern (Fig. 7b). The wind coupling and frictional coefficients which are related to continental shelf width plus shelf rise width and stratification for the first mode increase from western part of the region (Astara) toward southwestern part of the CS (Anzali and Roodsar) and then decrease again farther east (Noshahr and AmirAbad) (Fig. 7c and d). Despite the narrow continental shelf at Noshahr, the wind coupling coefficient is dramatically decreased, because the bottom continues to slope for a very long distance offshore (more than 150 km) in this region. Therefore, wind coupling and frictional effects for mode 1 CTW are more pronounced at Anzali and Roodsar where the continental shelf width is narrow, and also are stronger in winter than summer. The second modes are forced more efficiently by wind, with a higher wind coupling coefficient in areas with wider shelves (Astara and AmirAbad) than for the first mode. This suggests that the effect of local wind is more pronounced for second mode waves near these stations. Generally, wind coupling coefficients and frictional decay have more pronounced effects on higher modes than on lower modes (Fig. 7c and d). 3.2.3. Evidence of wave propagation on the shelf Dispersion curves were computed for frequencies covering the range 0 to 2.5 × 10−5 (rad s−1) (periods greater than 3 days) in both summer and winter for each segment (Fig. 8a). In summer, all dispersion curves for the two first modes are almost straight lines in the range of forcing frequencies. That is, the phase speeds are constant, consistent with the notion that these are long, nondispersive waves, with their propagation speed determined by the stratification and topography. However, dispersion curves are noticeably bent for the first and second mode in winter so the long wave assumption may not be as valid. The wavelengths corresponding to the dominant frequency range are of order 200-2000 km. Although this approaches the circumference of the CS, frictional effects, especially in the North CS, will likely prevent any significant propagation around the whole 9

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Fig. 7. (a) Calculated alongshore velocity structure for (i) First mode in summer, (ii) First mode in winter, (iii) Second mode in summer and (iv) Second mode in winter, all using data for the segment centered at Anzali. The structure functions near the shelf zone for other segments are similar. Properties of the first two free CTW modes for each station in Fig. 1a; (b) Phase speed of CTW (m s−1), (c) Wind coupling coefficient (s1/2 cm−1/2) and (d) Frictional coefficient (m−1).

basin. Computed theoretical time lags obtained from the phase speed of the first mode at each station in winter and summer are compared with observed time lags between all adjacent stations (Fig. 3) as well as non-adjacent stations. Empirically-determined average phase lags calculated from the coherence analysis are cumulatively summed to give phase delays across the whole region (Fig. 8b). The cumulative time lag is about 50 ± 10 h from Astara to AmirAbad. On the other hand, the calculated long-wave phase speeds predict a delay of 45 h in summer and 76 h in winter between these two endpoints. The calculated and empirical phase lags agree quite well as far as Noshahr, but over the final segment correlations show a faster propagation than we have calculated. However, note that AmidAbad is actually in a “corner” of the South CS (Fig. 1a). The bathymetric profile used in our calculations extends approximately WNW into the basin, whereas that at Noshahr extends Northwards. Thus it is entirely feasible that the 1-D propagation model will overestimate the propagation time between these two stations.

3.2.4. Comparison of CTW simulations with observational data Finally, we carry out a comprehensive comparison using full numerical solutions of Eqs. (7) and (11) for alongshore currents at AmirAbad, using as inputs the observed alongshore currents at Astara to force the first mode only and the alongshore wind time series at intermediate points to force all modes. Predicted time series for three to four month periods are made according to coefficients calculated using either summer and winter conditions as appropriate. Several simulation scenarios for alongshore current at 10

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Fig. 8. (a) Dispersion curve of the first and second modes in winter and summer for each station in Fig. 1a. (b) Comparison of theoretical time lag in winter and summer with observational time lag between each station with others along the case study. Time lag based on only current data at adjacent stations shown with the black dash line which is corresponding to Fig. 3. The bars in phase spectral indicate the 95% of confidence interval.

AmirAbad are considered. These are (1) remote current at Astara and all coastal winds (a full input scenario), (2) only the coastal winds, (3) only the local wind at AmirAbad, (4) only remote current at Astara without winds (no wind input), (5) remote current at Astara and coastal winds except at AmirAbad (a remote forcing scenario) and (6) remote winds (Astara, Anzali, Roodsar and Noshahr) only. This allows a clear evaluation of whether components of the predicted currents are wind-driven between the western part of the study area (Astara) and our measurement point (AmirAbad), locally drive, or propagating in from the west (upsream of Astara). In order to investigate the number of effective CTW modes, simulations of the above scenarios are performed using up to four modes. We also run cases with both the standard long-wave (LW) approximation and also the full nonlinear dispersion relation (NLD). The correlation and percent ratio of standard deviation of simulation data to standard deviation of observational data are used as quantitative measures of the fit between simulated and observed data. The correlation measures the degree to which peaks and trough appear at the same time, and the ratio of standard deviations measures the degree to which the overall magnitude of the response is correct. The correlation with observations is only slight larger when modes three and mode four are added (not shown), indicating that just two modes are adequate. Note that sensitivity to the frictional coefficient also tends to increase with mode number, indicating greater uncertainty for higher mode parameters. In general, predicted and observed amplitudes tend to be well-matched and the model predicts many events reasonably well (Fig. 9). During summer, solutions using only the first mode capture most features (Fig. 9a and b). Adding the second mode improves the results only slightly. There is little difference between the LW and NLD solutions. In winter, adding the second mode significantly improves comparisons (Fig. 9c and d). This is because the wind coupling coefficient at AmirAbad station is much larger, as are frictional coefficient at all stations for this mode, which leads to damping the remotely forced current contribution but emphasizing local aspects. Using the NLD solution also improves the match to observations, e.g., in late February and mid April, compared to the LW solution. The timing of peaks and troughs are well matched but amplitudes are sometimes a little small. Standard deviations of predicted time series using only the first mode in the full input scenario are smaller than observed amplitudes in both summer and winter. More quantitatively, with the LW (NLD) model, using two modes increases the ratio of standard deviation of predicted time series to observations from about 53% (57%) to 67% (71%) in summer and from 35% (34%) to 95% (79%) in winter (Fig. 10). Under the remote forcing scenario, more than about 40% of the alongshore current signal at AmirAbad is remotely generated in both summer and winter (Fig. 10). In summer, using the LW (NLD) solution, a greater percentage of the signal, about 52% (62%) is generated by the combination of remote wind forcing and CTW propagating along the coast after entering the study area at Astara. In winter, the remote forcing scenario predicts about 40% (46%) of the signal although due to spatial correlation in wind, the local wind scenario explains nearly 80% (56%) of the signal in alongshore velocity. Forcing the first mode by local winds only accounts for no more than 6% of the amplitude of the entire predicted time series in summer and winter. The small size of the wind-driven contribution appears to be due mainly to the relatively weak wind stress fluctuations (typically 0.02 dyn cm−2). However, adding the second mode using the LW (NLD) solution increases the percentage ratio for local wind forcing from 3% (3%) to 23% (19%) and from 6% (5%) to 79% (56%) in summer and winter, respectively. The frictional decay scale of the second mode is relatively short so that only local winds are important for its amplitude. There is a degree of sensitivity to bottom friction, but this tends to control the ratio of standard deviations and not the correlation. Varying bottom friction by an order of magnitude from 0.5 ×10−4 to 5 ×10−4 m s−1 (not shown) changes the correlation coefficient (ratio of standard deviation) in the two first modes solution by about 5% (29%) and 13% (117%) in summer and winter, respectively. The percentage change is higher for modes 3 and 4.

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Fig. 9. Comparing low-passed, alongshore flow measured at AmirAbad with CTW model simulations (full input scenario) using both LW and NLD methods; (a) Only first mode in summer, (b) First two modes in summer, (c) Only first mode in winter and (d) First two modes in winter.

4. Discussion and conclusions ADCP current velocities and surface heights observed between early December 2012 to early December 2013 along the shelf of the South CS are analyzed together with WRF-wind data and sea level to investigate the propagation of low frequency variations in ocean conditions. We concentrate most attention on current measurements because the theory for shelf waves predicts that prominent horizontal current fluctuations accompany rather modest sea level fluctuations, and also because of the existence of large gaps in our sea level records, as measured by different instruments (ADCP and tide gauges). We find that mean currents are consistent with a cyclonic circulation in the South CS, with barotropic eastward flow of 0.01–0.1 m s−1 all along the coast except at Roodsar where the mooring location is inshore of a known eddy some 20 km across that occurs as flow separates behind a headland. These coastal flows are in the same direction but slightly weaker than the flows of up to 0.2 m s−1 previously found by Zaker et al. (2011). Also in agreement with previous studies (Ghaffari and Chegini, 2010; Zaker et al., 2011) we find that spectral energy levels are highest at low frequencies (with periods of more than about 1 day), and that there is a strong tendency towards eastward propagation of variations at these low frequencies. Here our observations show these signals take about 50 h to propagate from Astara to AmirAbad. In contrast, wind energy, which is similarly concentrated at low frequencies, is correlated across the whole region with phase lags of less than 6 h, and the “latest” winds occur at Noshahr. Going further, we applied a model based on the theory for coastally-trapped long waves and find that model predictions are able to mostly reproduce low frequency alongshore current fluctuations in both winter and summer (Figs. 9 and 10). As dispersion relations are noticeably nonlinear in winter, we also develop a solution using a full nonlinear dispersion relation solution, applicable for both dispersive and nondispersive waves. The results of both methods have nearly similar correlation coefficients and the ratios of standard deviation in summer. However, the dispersive solution better predicts the CTW signal in winter. The existence of CTW, which has previously been found in many oceanic coastal areas (Brink, 1982a, e.g.,][; Battisti and Hickey, 1984; Pizarro and Shaffer, 1998; Amol et al., 2012; Connolly et al., 2014) thus also holds true in this isolated non-oceanic basin. Due to the absence of tides in the CS, with the LW (NLD) solution, the sum of mode 1 and mode 2 CTW variations predicts 67% (71%) and 95% (79%) of the measured current standard deviation over low frequencies (less than 1 cpd) in summer and winter, respectively. The squared correlation of the simulation data with both methods is about 0.5 in summer and 0.4 in winter, so the CTW explains 40% to 50% of the variance in the South CS. According to model results, during both summer and winter, more than about 32% of low frequency alongshore current variations in the southeast of the CS are generated by propagating CTW which enter our study area at Astara (Fig 10). This signal propagates eastward about 450 km to AmirAbad as a mode one CTW, arriving about two to three days later. In both summer and winter, more than 40% of the contribution to the alongshore current signal at low frequencies off AmirAbad is generated remotely. However, there 12

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Fig. 10. The correlation coefficient and ratio of standard deviation of simulation to standard deviation of observational data are shown; (a) LW solutions in summer, (b) NLD solutions in summer, (c) LW solutions in winter and (d) NLD solutions in winter. The bars for correlation indicate the 95% confidence interval. Simulations are performed with; (1) Full input, (2) Coastal winds only, (3) Local wind only, (4) No wind input, (5) Remote forcing and (6) Remote winds only scenarios.

are seasonal variations to these percentages. In winter, there is less of a contribution from remote forcing to alongshore current at AmirAbad (about 40%) than in summer (> 52%). On the other hand, the local wind scenario predicts a greater percentage of the low frequency alongshore current in winter (> 56%) than in summer (< 23%) (Fig 10). The seasonal difference is partly due to seasonal differences in the large scale wind field which has significantly higher strength during winter months (Fig. 2, parts b to f). The number of modes need to be summed to achieve accurate results for simulating CTWs varies substantially among studies. While Clarke and Van Gorder (1986), suggested that seven or more modes typically need to be used to obtain accurate results, other researchers show that using only first or two first modes are adequate for simulating CTWs (Brink, 1982a; Battisti and Hickey, 1984; Chapman, 1987; Pizarro and Shaffer, 1998; Leth and Middleton, 2006; Liao and Wang, 2018). In this study, adding the second mode improves the results (increasing correlations about 20%) in both summer and winter. However, there is only a slight change in the correlation coefficient when adding three modes and higher (about 5% in summer and −10% in winter). Also, similar to studies by Chapman (1987) and, Lopez and Clarke (1989), we find the higher modes are more sensitive to uncertainties in the input quantities which indicates greater uncertainty for higher mode parameters. Thus higher order modes are difficult to obtain accurately in the forced CTW method for estimating the alongshore shelf velocity and any results which depend on high modes in the CS may be questionable. Overall, the first wave mode is dominant, and about half of sea level and alongshore velocity fluctuations are governed by remote forcing. The higher-order modes are governed more by local forcing because the wind coupling and the frictional decay 13

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length scales decrease with increasing mode number, damping the remotely forced contributions. Therefore, adding more modes to our calculations will increase the locally forced current substantially, but will not have a comparable impact on the remotely forced current. Thus, when using higher modes, wave propagation becomes less important, similar to other case studies (Lopez and Clarke, 1989; Leth and Middleton, 2006; Connolly et al., 2014). In our observational results, barotropic behavior is seen in the shelf zone throughout the year (Fig. 2b) again consistent the features of our model eigenfunctions (Figs. 6 and 7a). However, offshore of the shelf break the modal structure shows baroclinic behavior in summer, with the surface in-phase with shelf currents but deeper waters out of phase. Previous studies in the southern part of the CS in summer 2003 also showed barotropic behavior near the shelf (Ghaffari and Chegini, 2010). Furthermore, characteristics of the currents on the southeastern part of the CS during August 2003 to April 2004 showed uniform alongshore currents across the shelf both horizontally and vertically (Zaker et al., 2011). However, no observations exist to show whether the baroclinic structure we predict exists further offshore. The first mode displays a maximum amplitude at the coast, a rapid decrease toward the shelf break, and an asymptotic decrease toward zero over the slope, which are typical CTW characteristics (Huthnance, 1992; Brink, 1991; Schulz et al., 2012). Conversely, the second mode has a zero crossing on the shelf break at about 50 km offshore, similar to findings in other continental shelf areas of the world ocean, such as in northern California and northern Oregon (Battisti and Hickey, 1984). The locations of ADCPs in this study are near to the coast at almost one third of the distance to the shelf break. We suggest that if future arrays contained more resolution in the offshore direction, they would be better able to discriminate between CTW modes. This might be particularly important in determining the upstream condition at Astara. Installing another ADCP there would be useful for obtaining initial condition for the second mode, which we set to zero here. Model results show that increasing stratification in summer increases the phase speed of all modes, with mode isopleths tilting offshore from near-vertical with weak stratification to near-horizontal with strong stratification (Figs. 6, 7a, b, and 8a). At AmirAbad with the broad shelf, the first mode is correspondingly fast and stratification effects tend to be weak. These results show that the phase speed and modal structure of CTW depend on shelf width and stratification, as also observed on different shelves in other parts of the world ocean (Mysak, 1980; Brink, 1991; Schulz et al., 2012; Woodham et al., 2013). However, the exact details of the seasonal cycle of stratifications in the South CS is not well-known, and more observations throughout the year would be useful in understanding the transition from summer to winter and vice versa. More than 32% of the South CS signal propagates into the area from north of Astara. It can be concluded that there is a considerable traveling signal which originates in other parts of CS and enters the study area. In west coast of the Middle CS, the known existence of strong winds, downwelling in summer and upwelling in winter (Tuzhilkin and Kosarev, 2005) could generate a CTW signal which propagates toward the South CS. It is also interesting to speculate on the fate of the CTW after they reach the most eastern part of the study area. Existence of a much wider shelf past AmirAbad in the east part of the CS might continue to guide energy as topographic waves with higher phase speed. Conversely the wide shallow region might act as a sink of wave energy due to higher frictional losses. The coherence and delay between current data observations illustrate a traveling eastward oscillation with phase speeds consistent with the CTW theory (Figs. 3 and 8). However, mean flows have a strong effect on qualitative properties and stability of CTWs, especially flows comparable in speed with a wave phase speed (Huthnance, 1992). Although our mean flows with speeds of order 0.1 m s−1 are somewhat smaller than the calculated phase speeds of order 1 m s−1 in the South CS, omitting the effect of mean current in calculating theoretical phase speed (time lag) could be another reason for discrepancies between the model and observations. Mean current effects are suggested to be considered for future investigation of CTW in the CS. In summary, although CTWs have been demonstrated to represent a major mode of flow variability on the South CS shelf, there are important unresolved questions: in particular, what is the extent and structure of the wind-generated response over CS as a whole? What is the effect of CTWs on shelf exchange, water mass transport, mixing and circulation in the southern CS, and by extension more widely along the east margins. Finally, the question of where and how the energetic fluctuations that propagate coherently along the South CS from Astara are first energized and generated, and what will happen after waves propagate into the eastern part of the CS is unresolved. Acknowledgements We are grateful to Kenneth Brink (Woods Hole Oceanographic Institution) for providing the numerical CTW code and assisting us in getting it running. and to the six(!) reviewers, whose comments greatly improved this work. We thank the Ports and Maritime Organization of Iran for providing the current meter and sea level data measurements and the Ministry of Science, Research and Technology of Iran for providing a scholarship to the first author. Support for RP is provided by the Natural Sciences and Engineering Research Council of Canada under Discovery Grant 2016-03783. Appendix A. Dispersive CTW propagation For dispersive waves, the assumption of constant phase speed is not valid and Eq. (6) cannot be derived. However, the standard approach can be generalized to account for dispersion; since this does not seem to have been done before we describe the procedure here. Starting from equations (2) and (3), take the Fourier transform-in-time and separate variables, i.e. let Pˆ (x , y, z , ω) =  {P (x , y, z , t )} = P′ (x , z , ω)exp(ily). Solve the resulting eigenvalue problem for the mode shapes Fn′ (x , z , ω) and the 14

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dispersion relations ln = ln(ω) or cn = cn(ω), where n is the mode number. Since this is done independently for each ω the procedures for solving the eigenvalue problem are identical to those of the standard approach where the separation P(x, y, z, t) = P′(x, z) exp (iωt + ily) is used. Next, to solve the forced problem in the frequency domain with a wind stress τ (t , y ) = −1{T (y, ω)} , assume a modal decomposition: ∞

Pˆ (x , y, z , ω) =

∑ Fn′ (x , z, ω)Φn (y, ω)

(8)

n=0

with mode amplitudes Φn (y, ω) =  {ϕn (y, t )} , and proceed onwards in the standard way. In general, the mode- and wind-coupling ′ and bn′ respectively, which are integrals over the mode shapes, will also be functions of ω. Although large variations are terms anm discussed in Brink (2006), the changes are weak in our case. Following the usual mathematical development, if the coupling is weak, it is reasonable to assume that cross-coupling terms between different modes will be even smaller, so we eventually derive an equivalent frequency-domain version of Eq. (6) for the alongshore propagation of each mode, independent of other modes:

iω ′ Φn ≈ bn′ T Φn − (Φn) y + a nn cn

(9)

If we consider a coastline segment 0 < y < Y of length Y, over which the wind forcing is coherent (constant in space but not in time) with bathymetry also independent of y, so that the coefficients in the equation are independent of y, and the upstream condition Φ(Y , ω) =  {ϕ (Y , t )} is known, then the mode amplitude at the downstream end of a segment when y = 0 is exactly:

b′ {τ (t )} ϕ (0, t ) = −1⎧ {ϕ (Y , t )} e−i lY − a′Y + [1 − e−i lY − a′Y ] ⎫ ⎨ ⎬ i l + a′ ⎩ ⎭

(10)

(where we drop the mode numbers to simplify the notation). Note that built-in to this approach is the fact that propagation must be along the y axis towards −∞ only, so that l(ω) = ω/c is a well-defined function. That is, group and phase velocity must also be restricted to propagation towards −∞ only. Although Eq. (7) is valid for slow variations along a coastline, in practice it is solved numerically by breaking up a coastline into separate segments, each centered on a transect of known bathymetry and surface wind (Gill and Clarke, 1974). However, if we break up a slowly-varying coastline into a number of segments of length ΔYi identified by subscripts i, i = 1, …, N, approximating the conditions in each segment with their own (spatially constant) wind τi (t ) = −1{Ti (ω)} and their own dispersion curve ci(ω), then we can chain our solution (10) across the segments, with the mode amplitude at the end of one segment being the initial condition for the next:

Φi + 1 (ω) = Φi (ω) e−iωΔYi / ci− ai′ ΔYi +

bi′ Ti (ω) [1 − e−iωΔYi / ci− ai′ ΔYi] iω/ ci + ai′

(11)

−1{ΦN + 1 (ω)} .

If the ci, ai′, and bi′ are all independent of frequency in the ith so that our final downstream mode amplitude ϕ (0, t ) = segment (i.e. nondispersive propagation), then we have merely found an alternative numerical method of solving Eq. (7). In practice there is a small difference between solutions computed using the two different numerical schemes, with the frequency-domain approach providing slightly smoother results since the second term in eq. (10) acts to smooth the wind input in a segment. However, we can also solve for dispersive propagation in the frequency-domain approach by letting ci vary with ω in Eq. (10) or (11). We have checked to make sure that differences we ascribe due to dispersion in our analysis are larger than those that arise from the different discretizations.

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