Scaling Estimates for the DSW Salinity with Wind and Buoyancy Forcing

Figure 2: Horizontal distributions of salinity anomaly near the bottom for (a)Ua=0 ms⁻¹, (b) 5 ms⁻¹ and (c) 9 ms⁻¹.

Figure 3: Vertical streamfunction ψ (contour) and salinity anomalies (color) for Ua =9 ms⁻¹. In ψ, positive and negative values are drawn by solid and dashed lines.

We consider salinity budget within a rectangular polynya adjacent to a straight coastal boundary with a constant salinity flux at the surface. Here, a quasi-steady state is assumed in which the surface

where over-bar and prime denote along-shore mean and perturbation components, respectively. Here, S denotes salinity anomaly from the initial value (=32.5 PSU) and u and v denote along-shore and cross-shore oceanic velocities, respectively.

Equilibrium Salinity Anomaly due to the Baroclinic Eddies

First, we assume that the surface salinity flux is balanced by the offshore salinity flux associated with baroclinic eddies. Supposing negligible salinity flux due to the mean flow, eq. 6 can be approximated by

UeSeLxH =FsLxb, (7)

whereUe andSe denote the velocity and the equilibrium salinity anomaly associated with eddies. Lx is an along-shore length of polynya, andH is a typical ocean depth.

In terms of the thermal wind balance,Ue can be scaled by the geostrophic current velocityUr along the front, as

Ue=CeUr=Ce

gHγ

ρwfWSe, (8)

where f is the Coriolis parameter, g the gravity acceleration. W is the offshore scale of the density front and assumed to be equal to the internal Rossby radius (R =

ΔρgH

ρ_w /f =5∼10 km). Ce is an efficiency constant viewed as the ratio betweenUeandUr. Y. Ikumi (Personal Communication) examined dependency of Ce on the offshore bottom slope α, and presented Ce =0.07 as the most appropriate constant forα=0.001. Thus, we adopt this value here. In eq. 8, we also assumed linear relation between density anomaly Δρand the salinity anomalySe (i.e., Δρ=γS×10⁻³, whereγ is 0.9).

Substituting eq. 8 into eq. 7, the equilibrium salinity anomaly determined by baroclinic eddies is derived as

Se= ρwfW bFs

gH²Ceγ ₁/2

. (9)

Equilibrium Salinity Anomaly due to the Wind-Induced Circulations

Next, an equilibrium salinity anomaly associated with the along-shore wind is discussed.

In general, the along-shore wind stress yields the along-shore coastal jet (Csanady, 1977; Michtum and Clarke, 1986), the surface Ekman flow accompanied by downwelling/upwelling and the internal compensation flow (hereafter, Ekman compensation flow; ECF) (Gill and Schumann, 1974; Mitsudera and Hanawa, 1987). Assuming that the surface salinity input is balanced by the lateral salinity flux due to these wind-induced currents, eq. 6 can expressed by

UwSwLyH+VESwLxH=FsbLx, (10) where Sw is the equilibrium salinity anomaly determined by wind, Uw the along-shore velocity of the coastal current, and VE the cross-shore velocity of ECF. Ly is an offshore scale of the along-shore jet, andLyb in general.

At first, the surface Ekman flow and its compensation flow are considered. The along-shore wind is assumed to be downwelling-favorable, which yields the onshore, surface Ekman flow in the northern hemisphere. Since salinity anomaly outside of polynya is principally zero, there is no salinity anomaly inflow by the onshore surface Ekman flow. In contrast, ECF occurring due to the mass conservation can export the saline water beneath the forcing region through the internal layer (Fig. 3). The offshore transport due to ECF is expressed by,

VEH = τa

ρwf, (11)

opposing to the onshore Ekman transport at the surface (=-τa/ρwf).

Next, the coastal jet is considered. When the easterly wind stress is imposed on the ocean, vertically integrated momentum balance is written by (c.f., Gill and Schumann, 1974)

fv = −g∂η

velocities. In the along-shore direction, −ruis a parameterization of the bottom friction. Assuming a horizontally uniformτa and no net cross-shore transport (i.e.,v=0), eq. 12 yields

The empirical constantr for the bottom stress is chosen from the comparison with the numerical cal-culation. From comparison with numerical results,r=1.5×10⁻⁴ m⁻¹s⁻¹ is the most appropriate to the estimateUw.

Substituting eq. 14 and eq. 11 into eq. 10, the equilibrium salinity anomalySwassociated with the along-shore wind is written by According to eq. 15,Swis inversely proportional toτa.

Equilibrium Salinity Anomaly due to the Baroclinic Eddies and the Wind-Induced Circu-lations

Finally, we estimate the equilibrium salinity anomaly in which both the eddies and the wind-driven currents (i.e., the coastal jet and ECF) contribute equivalently. Supposing a quasi-steady state, the salinity budget beneath the polynya is roughly approximated by

UeLxHS^∗+UwLyHS^∗+VELxHS^∗=FsLxb, (16) whereS^∗is the equilibrium salinity associated with wind and eddies. Substituting eq. 14, eq. 11 and eq.

8 intoUw, VE andUe, respectively, a following quadratic equation is derived:

S^∗

From eq. 17, the non-dimensional salinity anomaly, normalized bySe, is derived as S^∗

That is, the relative importance between eddies and wind varies on the basis ofUc. Uc can be expressed explicitly by

Uc⁴ = ρwbCeγgFsfH²

ρ²aCa²W(αLy+H)²(fHLy+rLx)²

(fH²Ly+rLx(αLy+H))², (22)

−1

The bottom salinity distributions forUa =0, 5, 9 ms⁻¹are displayed in Figure 2. For the case ofUa =0 (Fig. 2a), salinity front near the edge of forcing region becomes baroclinically unstable since around day 10. The baroclinic eddies develop rapidly as the density front enhances, and carry away the dense water offshore. The mature eddies finally become 20 to 30 km in size (middle and bottom panels of Fig.

2a). When there is no wind forcing, the salinity anomaly beneath polynya reaches 0.6∼0.8 PSU at the maximum. For the case ofUa =5 ms⁻¹, where Ua is almost same asUc, eddy activities are significantly moderated, in which saline water with more than 0.5 PSU ofSe is carried away to the west of polynya (Fig. 2b). As a result, the maximum salinity anomaly is typically 0.2 PSU less than that ofUa =0. In the case of Ua =9 ms⁻¹, the salinity increase over the shelf is remarkably small and 0.2 PSU at most (Fig. 2c). This is caused by the wind-driven advection in which fluid particle travels over the shelf with cross-shore velocity as well as along-shore velocity. In addition, interestingly, there is almost no eddy activity over the shelf for the case ofUa=9 ms⁻¹.

Fig. 3 shows a vertical section of salinity anomaly and streamfunction whenUa=9 ms⁻¹, corresponding to Fig. 2c. It can be seen that the onshore surface Ekman current occurs within 10 m from the surface due to the easterly wind. Then, the strong downwelling occurs near the coastal boundary, accompanied by ECF. From the salinity distributions, we see that large fraction of salinity is carried offshore by the ECF through the internal and bottom layers, which enhances the density stratification near the edge of the front. In light of this, we infer that the along-shore wind should relax the inclined isopycnal surface by strengthening the vertical stratification, and consequently stabilizes the density front as seen in Fig.

2c.

Fig. 6 of the original paper shows the salinity anomaly (averaged over an area extending 15 km from the coast) as a function of time. According to the figure, initially the salinity anomaly linearly increases with time, and subsequently it breaks away from the linear increase approaching each quasi-steady value.

We emphasize that the equilibrium anomaly also decreases with increasing along-shore wind speedUa. Additionally, the time to reach the equilibrium salinity anomaly strongly depends on Ua when Ua >5 ms⁻¹. That is, the equilibrium time becomes short when Ua increases as long as Ua is greater than a certain point. The time dependency is consistent with that on the ambient current velocity discussed in Chapman (2000).

The theoretical estimates for the equilibrium salinity anomaly are tested by the numerical calculations,

Figure 4: Non-dimensional relationship between the quasi-steady salinity anomalyS^∗and the along-shore wind speedUa. Curves are based on the theoretical estimates of eq. 9 (thin), eq. 15 (dashed), and eq.

19 (blue).

and it is depicted in Fig. 4. From the figure, the numerical results have a good agreement with the theoretical estimatesS^∗ that includes the effects of both eddies and wind. According to the theoretical curve, there is only a slight influence of winds on S^∗ for Ua <0.5Uc (i.e., S^∗ ∼ Se). After that, the equilibrium salinity gradually decreases with increasing Ua, and reaches Sc (=0.6Se) at Ua =Uc. For Ua>2Uc,S^∗ asymptotically reduces to zero overlaying with the curve ofSw. Besides, it is also suggested that if no salinity flux due to ECF were included in the estimate (solid curve in gray), 1.7 times of Uc

would be required forUa to yield the same amount of salinity transport.

The time to reach the equilibrium salinity is further discussed here. First, a strong wind case is considered where eddy transport may be ignored. A salinity equation of a certain water column, without effects of mixing and diffusion, can be written in a Lagrangian form as

dS

The particle traveling under constantFsachieves following salinity anomaly in timet: S(t) = −

Eq. 24 means that the salinity anomaly continues to increase as long as it is beneath the polynya. In other words, the particle with large velocity tends to get only a little salt. As shown in Fig. 2c, the particle traverses the shelf for largeUa. Therefore, either along-shelf or cross-shelf direction that particle gets across the imposed region in shorter period of time determines the residence time beneath polynya and the maximum salinity anomaly. Defining the advective time scale asTw= _U^Lx

w andTE = _V^b b=15 km. Consequently, the residence timeT is determined by the across-shore velocity of particle, and T = TE =8.7 days for the parameters. The salinity increase achieved during the period is estimated around 0.15 PSU, and approximately coincides with the numerical calculations.

Chapman (2000) compared the eddy developing time Te = f W b ad-vection time associated with the along-shelf current and found a critical ambient velocity. Similarly, we derive the critical wind speed as the advection time here. SinceTE < Twin this case, comparison between TeandTE leads to condition thanUc to compensate that of the coastal jet. Eventually, 6.7 ms⁻¹ of Ua is required for the standard parameters, which is indeed slightly greater thanUc.