Observations of nonlinear momentum fluxes over the inner continental shelf

1 Nonlinear momentum fluxes over the inner continental shelf are examined using moored 2 observations from multiple years at two different locations in the Middle Atlantic Bight. 3 Inner shelf dynamics are often described in terms of a linear alongshore momentum bal4 ance, dominated by frictional stresses generated at the surface and bottom. In this study, 5 observations over the North Carolina inner shelf show that the divergence of the cross6 shelf flux of alongshore momentum is often substantial relative to the wind stress during 7 periods of strong stratification. During upwelling at this location, offshore fluxes of along8 shore momentum in the surface layer partially balance the wind stress and reduce the role 9 of the bottom stress. During downwelling, onshore fluxes of alongshore momentum re10 inforce the wind stress and increase the role of bottom stress. Over the New England 11 inner shelf, nonlinear terms have less of an impact in the momentum balance and exhibit 12 different relationships with the wind forcing. Differences between locations and time pe13 riods are explained by variations in bottom slope, latitude, vertical shear and cross-shelf 14 exchange. Over the New England inner shelf, where moored density data are available, 15 variations in vertical shear are explained by a combination of thermal wind balance and 16 wind stress. An implication of this study is that cross-shelf winds can potentially influence 17 the alongshore momentum balance over the inner shelf, in contrast with deeper locations 18 over the middle to outer shelf. 19 Keywords— Momentum balance, Nonlinear, Momentum flux, Coastal dynamics, Upwelling 20 dynamics, Downwelling dynamics, Thermal wind balance, Inner shelf 21


Introduction
The dynamics of the inner continental shelf govern exchange between shallower wa-23 ters in the surf zone and deeper waters over the middle to outer shelf. The inner shelf 24 is often dynamically defined as a region where the surface and bottom boundary layers 25 interact and turbulent stresses are present throughout the entire water column (Mitchum 26 and Clarke, 1986; Lentz, 1995;Lentz and Fewings, 2012). The inner shelf region is also 27 characterized by cross-shelf mass transport that is reduced from the theoretical Ekman 28 transport expected for deeper water Kirincich et al., 2005). The off-29 shore extent of the inner shelf is strongly influenced by stratification, which inhibits tur-30 bulent mixing and restricts the region of reduced cross-shelf transport to shallower depths 31 . Unlike the middle to outer shelf, cross-shelf winds often drive signifi-  The alongshore momentum balance is frequently used as a framework for understand-37 ing the dynamics of coastal regions, including the inner shelf. The depth-averaged balance 38 over the inner shelf is often characterized as being dominated by the frictional terms, wind 39 stress and bottom stress, with secondary contributions from local acceleration and along-40 shore pressure gradients (Hickey, 1989;Lentz et al., 1999;Lentz and Fewings, 2012). The 41 alongshore pressure gradient has also been shown to be important in balancing the wind 42 stress at some locations, particularly at locations near alongshore variations in bathymetry 43 and coastline (Kirincich and Barth, 2009b;Fewings and Lentz, 2010). However, the po-44 tential impact of additional nonlinear terms in the alongshore momentum balance is not 45 well known and is often neglected for simplicity (Lentz and Fewings, 2012). If nonlinear 46 terms are significant, neglecting them could lead to misinterpretation of the magnitude of 47 stresses at the bottom or in the interior of the water column, which are often uncertain evidence for this type of mass balance at both inner-shelf locations examined in this study 129 when the effects of wave-driven transport are included . However, this  As a starting point for developing a simplified two-dimensional momentum balance 146 for the inner shelf, a three-dimensional balance that includes the effects of wave breaking 147 is first considered. The x coordinate is defined as positive offshore, and the y coordinate 148 is oriented alongshore (Fig. 1b,c). Following Uchiyama et al. (2010), a depth-averaged 149 alongshore momentum balance that includes the effects of both breaking and unbroken 150 surface waves can be written in a flux-divergence form, where η is sea level, h is bottom depth, D = η + h is the total thickness of the water waves, the alongshore momentum balance in equation (1) can be simplified as The left-hand side of equation (3) Including transport due to Stokes drift is important because it often exceeds the wind-166 driven transport in the surface layer at locations inshore of the 20-m isobath at MVCO and 167 FRF . Over the Martha's Vineyard inner shelf, 15-30% of the cross- 168 shelf heat flux during summer is associated with Stokes drift (Fewings and Lentz, 2011). 169 However, Stokes drift is only one component of the wave-driven circulation. The Stokes-

170
Coriolis force induces an Eulerian wave-driven flow which tends to cancel the Stokes drift 171 in the limit of weak eddy viscosity (Xu and Bowen, 1994;Lentz et al., 2008). To assess the

176
This nonlinear term is present in a simplified two-dimensional framework, but it is not 177 present in one-dimensional models of the water column.

178
The present study primarily uses information from cross-shelf current meter arrays where z s is the depth of the first zero crossing in the vertical profile of u L . The terms in 190 the alongshore momentum balance are integrated over this same surface layer.

191
Consistent with the depth-averaged balance described above in Section 2.a, the sur-192 face layer momentum balance considered in this study neglects alongshore variations in 193 currents and waves, as well as wave dissipation in the surf zone. The alongshore momen-194 tum balance integrated over the upper layer is given by where w L is the Lagrangian vertical velocity and τ iy | z=zs is the interior turbulent stress at 196 the base of the surface layer. The Lagrangian vertical velocity at the base of the surface 197 layer can be determined from conservation of volume, Like the depth-averaged momentum balance, the left hand side of the surface layer 199 momentum balance in equation (6)

359
The relationship between alongshore wind stress and cross-shelf transport differs be- layer transport U s , consistent with upwelling circulation (Fig. 3a). Reversals to onshore 363 and downwelling favorable wind stress at this site are typically associated with onshore 364 U s . The relationship between wind forcing and cross-shelf transport is more complex at 365 MVCO, where offshore U s is typically present during a wide range of wind conditions 366 (Fig. 3b). Offshore U s at this site is observed during onshore and upwelling-favorable 367 wind forcing, as well as reversals to offshore and downwelling favorable conditions. Weak 368 onshore U s is present some of the time during onshore and downwelling favorable wind 369 stress, and during events in which wind stress is directly onshore with no alongshore component. However, the magnitude of onshore U s at MVCO is not as high as that observed 371 during similar wind conditions at FRF.

372
A subset of the FRF time series from 2013 is now used to describe wind forcing 373 and circulation patterns at time scales of days-weeks (Fig. 4). In the next section, it 374 will be shown that these circulation patterns influence the momentum balance through the The timing of the low-salinity events indicates that the alongshore pressure gradient con-

468
The strength and direction of vertical shear in the alongshore current, however, is sensi-469 tive to the alongshore wind stress . Variability of vertical shear in 470 response to alongshore wind stress and cross-shore density gradients will be examined in 471 greater detail in Section 2.d.

472
The magnitude of the nonlinear term relative to the wind stress also differs between 473 locations. The regression slope of 0.15 between the two terms at MVCO during June- 474 August is smaller than those obtained at FRF during upwelling favorable winds ( (Fig. 6, 475 Table 1). Comparing the standard deviations of the two terms also shows that the nonlinear 476 term has a greater contribution to the momentum balance at FRF than MVCO (Fig. 7).

477
There are also consistent differences between seasons at each location ( Fig. 7). At both locations, the relative importance of the nonlinear term decreases during the months of 479 January-March when the water column is more weakly stratified (Fig. 2).
where A = κu * h/6, and u * = |τ s |/ρ o is a friction velocity based on the magnitude

583
Like the geostrophic shear, the hypothetical wind-supported shear alone cannot com-584 pletely explain the observations (Fig. 9b). There is a significant correlation during periods previous studies at the same locations (Fig. 11).  (Fig. 11a). The most common wind pattern at FRF in North Carolina is upwelling fa-629 vorable and offshore (Fig. 2c). This type of wind forcing is strongest before the passage 630 of an atmospheric front and is also associated with the strongest surface heat fluxes from 631 the atmosphere to the ocean (Austin and Lentz, 1999). As observed by Lentz (2001), the magnitude of offshore U s increases with water depth but typically remains less than the 633 theoretical deep water Ekman transport U Ek at sites where water depth h < 10 m (Fig.   634 4b, 8a). The positive divergence of cross-shelf surface transport ∂U s /∂x is consistent with layer. This flux increases with offshore distance as offshore transport increases and as the 658 surface-to-bottom velocity difference increases for similar ∂v/∂z. This mechanism is sim-659 ilar to that described by Lentz and Chapman (2004) for upwelling conditions at mid-shelf locations. However, the dynamics are modified over the inner shelf because U s changes 661 with offshore distance and there is no clear separation between the surface and bottom 662 boundary layers.

663
During downwelling-favorable and onshore wind forcing, the circulation patterns are 664 essentially reversed over the inner shelf at FRF (Fig. 11b). This type of wind forcing com-  The circulation patterns associated with downwelling winds at FRF (Fig. 11b) lead to 684 a positive momentum flux divergence. This nonlinear term in the depth-averaged momen-685 tum balance has the same positive sign during both upwelling and downwelling (Fig. 6a).
The approximation in equation (10) was used by Lentz and Chapman (2004) to estimate 781 the momentum flux divergence from single moorings at mid-shelf sites. Over the inner 782 shelf, the observed variability of the depth-integrated momentum flux is consistent with 783 a monotonic decrease towards the coast (Fig. 12). Standard deviations of the depth- ing only (Fig. 13a). The approximation is strongly correlated with the estimates made  The scaling analysis assumes linear vertical profiles of u L and v, where v s is the surface velocity at z = 0. The linear cross-shelf velocity profile in equa-812 tion (11) describes a two-layer flow that satisfies two-dimensional mass balance,ū L = 0.

813
With the vertical structure of the velocity given by equations (11) and (12), the vertically-814 integrated cross-shelf flux of alongshore momentum is Three further simplifications are made about the cross-shelf structure of the circulation.

816
First, the ratio of the surface transport U s to the theoretical Ekman transport U Ek is propor- stratified conditions (Fig. 10a). It is therefore possible that a turbulent thermal wind 842 balance, in which geostrophic shear is modified by turbulent stresses, often applies over 843 the inner shelf where the water depth and boundary layer thickness are comparable.

844
To test whether the scaling in equation (13)  -2.1 × 10 −2 and 1.2 × 10 −2 s −1 at MVCO (Fig. 9) but vary over a much larger range from 908 -7.0 × 10 −2 to 7.0 × 10 −2 s −1 at FRF (Fig. 4c). Based on regression slopes between  . The low salinity signature of the plume is evident in Fig. 4f and the unresolved pressure gradient contributes to unresolved variance in the momentum 968 balance analysis. In contrast, at MVCO, the alongshore pressure gradient largely balances 969 the local wind stress, likely due to the effects of topography (Fewings and Lentz, 2010).      Figure 4: Time series at FRF during the 45-day period 13 June-28 July 2013. a) Crossshore (red) and alongshore (blue) components of wind stress. b) Cross-shore surface transport, U S , at the 8 m (black), 6 m (dark gray) and 5 m (light gray) sites. c) Vertical shear in alongshore currents, ∂v/∂z, at the 6 m and 8 m sites. d) Selected terms in the depthintegrated alongshore momentum balance, averaged between the 6 m and 8 m sites: wind stress (τ sy /ρ o D, blue) and nonlinear advection (1/D∂/∂x η −h (u L v)dz, red). e) Selected terms in the depth-integrated alongshore momentum balance, averaged between the 6 m and 8 m sites: wind stress (τ sy /ρ o D, blue) and logarithmic bottom stress (τ by /ρ o D, black). f) Practical salinity, S P , from CTD casts at a depth of 1 m.    Regression slopes are shown with 95% confidence intervals. b) As in panel a, for wind stress term vs. nonlinear terms at FRF. c) As in panel a, for wind stress term vs. Coriolis term at MVCO. d) As in panel a, for wind stress term vs. nonlinear terms at MVCO, and with one linear regression fit for all data points.  depth-averaged momentum balance -nonlinear