Initial time dependence of wind- and density-driven Lagrangian residual velocity in a tide-dominated bay

Abstract

The nonlinear effect of the summer southeast wind and density on the 3D structures of the full Lagrangian residual velocity (LRV) was quantified for a generally nonlinear system, using Jiaozhou Bay (JZB), China as the test site. In the tidally energetic JZB, the basic patterns of the wind- and density-driven full LRVs were found to be consistent with semi-analytical solutions but highly dependent on initial times. The wind-driven full LRVs at different tidal phases flowed similarly downwind over shoals and upwind in the deep region; however, the main branches could migrate across nearly half of the bay. A density-driven, clockwise flow was dominant in the western inner bay at low tide, but it almost disappeared at high tide. The effect of density generally enhanced the outward flow in the surface layer and inward flow in the bottom layer. Along-trajectory integrated momentum balances indicated that viscosity was the main factor responsible for the time dependence of the wind-driven full LRVs, while viscosity, barotropic and baroclinic pressure gradients were the main drivers of the intra-tidal variations in the density-driven full LRVs. Generally, the summer wind and density had opposing effects, although their influence was weaker than that of the tide and could not change the patterns of the tide-driven full LRVs. When analysing the effects of wind and density on the coastal circulation in JZB, both the 3D structures and the possibility of a high initial tidal phase dependency should be considered.

Appendices

Appendix 1. Definitions of residual velocities and their interrelations

A full LRV $\left({\overset{\rightharpoonup }{u}}_L\right)$ is defined as the net replacement per tidal cycle (Zimmerman 1979; Feng et al. 2008) and can be calculated as shown in Eq. (7):

$$ {\overset{\rightharpoonup }{u}}_L\left({\overset{\rightharpoonup }{x}}_0,{t}_0\right)=\frac{{\overset{\rightharpoonup }{\xi}}_{nT}\left({t}_0+ nT;{\overset{\rightharpoonup }{x}}_0,{t}_0\right)}{nT}, $$

(7)

where ${\overset{\rightharpoonup }{u}}_L\left({\overset{\rightharpoonup }{x}}_0,{t}_0\right)$ represents the full LRV with the initial position vector (${\overset{\rightharpoonup }{x}}_0$) and the initial intra-tidal process-independent time (t₀) for an arbitrary water parcel to be tracked; ${\overset{\rightharpoonup }{\xi}}_{nT}$ denotes the net displacement of an arbitrarily labelled water parcel over n tidal periods, T; $\overset{\rightharpoonup }{\xi}\left(t;{\overset{\rightharpoonup }{x}}_0,{t}_0\right)={\int}_{t_0}^t\left(\overset{\rightharpoonup }{u}\left({\overset{\rightharpoonup }{x}}_0+\overset{\rightharpoonup }{\xi },{t}^{\prime}\right)d{t}^{\prime}\right)$ denotes the displacement of an arbitrarily labelled water parcel with initial position ${\overset{\rightharpoonup }{x}}_0$ from time t₀ to time t; and t^′ represents a time scale related to tidal residual currents.

The ERV (${\overset{\rightharpoonup }{u}}_E$) can be obtained using a fixed current velocity metre, averaged over the tidal cycle as shown in Eq. (8):

$$ {\overset{\rightharpoonup }{u}}_E=\left\langle \overset{\rightharpoonup }{u}\right\rangle, $$

(8)

where <> indicates the tidal cycle mean function. The LRV $\left({\overset{\rightharpoonup }{u}}_L\right)$ has the inherent ability to include complete information on coastal subtidal circulation (Zimmerman 1979; Cheng and Casulli 1982), while the ERV (${\overset{\rightharpoonup }{u}}_E$) represents the 0th order of the LRV $\left({\overset{\rightharpoonup }{u}}_L\right)$ (Feng et al. 1986, 2008). The discrepancy between LRV $\left({\overset{\rightharpoonup }{u}}_L\right)$ and ERV (${\overset{\rightharpoonup }{u}}_E$) depends on the degree of nonlinearity in the applicable hydrodynamics.

Hydrodynamics in coastal waterbodies, such as tidal estuaries and shallow bays, usually feature nonlinear effects that are relatively stronger than those in marginal seas or open oceans. From marginal seas and open oceans to coastal waters, water depth, h, decreases with topography, while water elevations (i.e., usually tidal elevation), η, increase due to wave shoaling effects. Thus, κ increases, indicating a stronger nonlinearity. In addition, irregular coastlines enhance the flow-field gradient, which also results in a stronger nonlinearity; hence, discrepancies between LRV $\left({\overset{\rightharpoonup }{u}}_L\right)$ and ERV (${\overset{\rightharpoonup }{u}}_E$) become significant.

Regarding weakly nonlinear hydrodynamics systems, the LRV can be simplified into its first-order form, also called the MTV (Feng et al. 1986, 1990, 2008). The MTV $\left({\overset{\rightharpoonup }{u}}_M\right)$ can be calculated using Eq. (9):

$$ {\overset{\rightharpoonup }{u}}_M={\overset{\rightharpoonup }{u}}_E+{\overset{\rightharpoonup }{u}}_S, $$

(9)

where ${\overset{\rightharpoonup }{u}}_S$ denotes the Stokes drift (SD). On some occasions, the ERV (${\overset{\rightharpoonup }{u}}_E$) can show a pattern fairly similar to that of the MTV $\left({\overset{\rightharpoonup }{u}}_M\right)$, owing to a small SD $\left({\overset{\rightharpoonup }{u}}_S\right)$. Hence, ERV (${\overset{\rightharpoonup }{u}}_E$) is a widely used parameter in coastal oceanography (e.g., Delhez 1996; Winant 2004; Klingbeil et al. 2019). In some other cases, however, the SD $\left({\overset{\rightharpoonup }{u}}_S\right)$ is not negligible, resulting in a clear discrepancy between the ERV (${\overset{\rightharpoonup }{u}}_E$) and MTV $\left({\overset{\rightharpoonup }{u}}_M\right)$ (e.g., Feng et al. 1986; Jiang and Feng 2011).

However, in a generally nonlinear hydrodynamic system, the coastal circulation has to be described in terms of full LRVs$\left({\overset{\rightharpoonup }{u}}_L\right)$ rather than MTVs $\left({\overset{\rightharpoonup }{u}}_M\right)$ or ERVs (${\overset{\rightharpoonup }{u}}_E$). A full LRV $\left({\overset{\rightharpoonup }{u}}_L\right)$ can be expressed as the sum of the ERV (${\overset{\rightharpoonup }{u}}_E$), SD (${\overset{\rightharpoonup }{u}}_S$), Lagrangian drift velocity (${\overset{\rightharpoonup }{u}}_{ld}$), and a higher order of extension (Feng et al. 1986, 2008), as shown in Eq. (10):

$$ {\overset{\rightharpoonup }{u}}_L={\overset{\rightharpoonup }{u}}_E+{\overset{\rightharpoonup }{u}}_S+\upkappa {\overset{\rightharpoonup }{u}}_{ld}+O\left({\upkappa}^2\right), $$

(10)

The MTV $\left({\overset{\rightharpoonup }{u}}_M\right)$ cannot be used as a substitute for the LRV$\left({\overset{\rightharpoonup }{u}}_L\right)$, because of the importance of the absent high-order terms, such as the Lagrangian drift velocity $\left({\overset{\rightharpoonup }{u}}_{\mathrm{ld}}\right)$ (Feng et al. 1986, 2008; Jiang and Feng 2011). Therefore, the full LRV$\left({\overset{\rightharpoonup }{u}}_L\right)$ must be used to describe the mass transport trend. The distinction between weakly and generally nonlinear hydrodynamic fields is that the variations in the full LRV with the initial tidal phase are significant, while the ERV and MTV do not (e.g., Feng et al. 2008; Ju et al. 2009).

Appendix 2. Definitions of dimensionless parameters

To analyse the effects of wind and density qualitatively, the Ekman (E_k), Kelvin (K_e), and Wedderburn (W) numbers, which are three key parameters associated with the flow pattern and dynamics of wind- and density-driven flows, were calculated. These parameters have been widely used in coastal studies to examine the controlling dynamics of flow patterns (e.g., Valle-Levinson 2008; Li and Li 2011; Jia and Li 2012).

The Ekman number, E_k, compares the friction with Coriolis effects, as shown in Eq. (11):

$$ {E}_k=\frac{A_z}{\left(f{h}^2\right)}, $$

(11)

where A_z denotes the flow eddy viscosity. At each horizontal grid point, we estimated A_z as the depth-averaged value of the eddy viscosity calculated using the model. The Coriolis parameter is denoted by f.

The Kelvin number, K_e, can be obtained as shown in Eq. (12):

$$ {K}_e=\frac{B}{R_i}, $$

(12)

where B refers to the basin width (Garvine 1995). R_i is given by (g^,h)^1/2/f, where g^, = g∆ρ^′/ρ₀ denotes the reduced gravity, g represents the gravitational acceleration, ∆ρ^′ indicates the contrast between the buoyant water density and density in the bottom layer, and ρ₀ is a reference water density. The basin width determines whether the Earth’s rotational effect on the density-driven or wind-driven water exchange is appreciable (Pritchard 1952; Valle-Levinson 2008).

The relative importance of wind-driven and gravitational circulations can be quantified using the Wedderburn number (Monismith 1986) (Eq. 13):

$$ W=\frac{L{\tau}_{\mathrm{wx}}}{\varDelta \rho \mathrm{g}{{\mathrm{h}}_{\mathrm{mean}}}^2}, $$

(13)

where τ_wx denotes the along-channel wind, L represents the basin length, ∆ρ stands for the density change over L, and h_mean is the averaged depth. Wind-driven circulation is dominant when W > 1, while gravitational circulation is dominant when W < 1 (Geyer 1997).

Appendix 3. Lagrangian momentum balance analysis

We integrated arbitrary momentum terms (denoted as ψ) along particle trajectories, as ${\int}_{\overset{\rightharpoonup }{\xi }}\psi d\overset{\rightharpoonup }{\xi }$, where $d\overset{\rightharpoonup }{\xi }$ denotes a piecewise trajectory over one model time step. The difference among the momentum terms of the tide-wind-density, tide-wind, and tide-density systems could be regarded as the effect of wind and density in the tide-wind-density system. The POM employs a sigma coordinate in the vertical direction; thus, the momentum equations are as shown in Eqs. (14)– (16) (Mellor 2004):

$$ \frac{\partial UD}{\partial t}+\overset{nonlinear\ advection}{\overbrace{\frac{\partial {U}^2D}{\partial x}+\frac{\partial UVD}{\partial y}+\frac{\partial U\upomega}{\mathrm{\partial \upsigma }}}}\overset{Coriolis\ force}{\overbrace{- fVD}}+\overset{barotropic\ pressure\ gradient\ }{\overbrace{gD\frac{\mathrm{\partial \upeta }}{\partial x}}}+\kern1em \overset{baroclinic\ pressure\ gradient}{\overbrace{\frac{g{D}^2}{\uprho_o}{\int}_{\upsigma}^0\left[\frac{\partial {\uprho}^{\prime }}{\partial x}-\frac{\upsigma^{\prime }}{D}\frac{\partial D}{\partial x}\frac{\partial {\uprho}^{\prime }}{\partial {\upsigma}^{\prime }}\right]d{\upsigma}^{\prime }}}=\overset{viscosity}{\overbrace{\frac{\partial }{\mathrm{\partial \upsigma }}\left[\frac{A_z}{D}\frac{\partial U}{\mathrm{\partial \upsigma }}\right]+\frac{\partial }{\partial x}\left(2h{A}_M\frac{\partial U}{\partial x}\right)+\frac{\partial }{\partial y}\left({h\mathrm{A}}_{\mathrm{M}}\left(\frac{\mathrm{\partial U}}{\mathrm{\partial y}}+\frac{\mathrm{\partial V}}{\mathrm{\partial x}}\right)\right)}}, $$

(14)

The surface boundary condition can be expressed as shown in Eq. (15):

$$ \frac{A_z}{D}\left(\frac{\mathrm{\partial U}}{\mathrm{\partial \upsigma }},\frac{\mathrm{\partial V}}{\mathrm{\partial \upsigma }}\right)=\overset{wind\ stress}{\overbrace{-\left(< wu(0)>,< wv(0)>\right)}},\upsigma \to 0, $$

(15)

while the bottom boundary condition can be described as in Eq. (16):

$$ \frac{A_z}{D}\left(\frac{\partial U}{\mathrm{\partial \upsigma }},\frac{\partial V}{\mathrm{\partial \upsigma }}\right)=\overset{bottom\ friction}{\overbrace{C_z{\left[{U}^2+{V}^2\right]}^{1/2}\left(U,V\right)}},\upsigma \to -1, $$

(16)

where x, y, and z are conventional Cartesian coordinates; U, V, and ω represent the velocities in the sigma coordinate; D = h + η represents the total water depth; $\sigma =\frac{z-\eta }{h+\eta }$ represents the sigma coordinate, which ranges from 0 at the surface to -1 at the bottom; and ρ_o and ρ^′ denote the reference water density and density perturbation, respectively; A_M represents the horizontal viscosity coefficient, while wu(0) and wv(0) stand for wind momentum fluxes at the surface; C_z represents the bottom friction coefficient; and the integrated momentum terms are divided by the total water depth, D, along each trajectory.