a. Initial plume scales

First, we develop the scalings for the initial geometry (width and depth) and along-shelf transport of the plume, assuming the plume has reached a steady state where the cross-shore momentum balance is geostrophic and the velocity at the bottom (and thus the bottom stress) is zero (Chapman and Lentz 1997; Lentz and Helfrich 2002). The width of the plume front W_γ can be approximated by the scale (Gill 1976)

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in which is the reduced gravity associated with the front and ρ_s is the surface density at x = W_α (Fig. 1). To develop our theory in terms of the overall density difference found in the plume, we take advantage of the constant lateral density difference assumption to rewrite g′_γ as

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where g′ = gΔρ_c/ρ_o is the reduced gravity of the plume. Replacing Eq. (3) in Eq. (2) and solving for W_γ, we obtain a quadratic equation with solution (only the positive root gives a positive width)

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where λ² = g′h_p/f² is an internal radius of deformation squared. Notice, however, that the similarly defined deformation radius in Eq. (2) is a more physically relevant scale for this problem.

Another important property associated with a buoyant plume is the along-shelf transport and how it is related to the plume geometry and buoyancy deficit. We can start with the alongshore vertical shear in the plume, given by

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Integrating the equation above and assuming that the alongshore velocity is zero at the bottom for the area A_α [i.e., υ(x, h_α) = 0] and that the velocity vanishes along the frontal boundary for the area A_γ [i.e., υ(x, h_γ) = 0], we obtain

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which, integrated over the area of the plume, results in the along-shelf transport, given by

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where h_α(x) = h_c + αx is the bottom depth and h_γ(x) = h_p (W_p − x)/W_γ is the depth of the front offshore of W_α, assuming a triangular shape (Fig. 1). We can readily solve for the depth of the plume:

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or

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Note that the chosen bathymetry, with a vertical wall h_c at the coast, results in a more complex expression that requires the details of the plume geometry, instead of simply its transport and reduced gravity. This bathymetry, however, was chosen to accurately judge the numerical model results that we will present in the next section. However, here and throughout the manuscript we also present expressions with h_c set to zero, as they are more relevant for discussion and application to observational studies. Note also that the expression for the plume depth for this problem (with h_c = 0 and constant stratification) differs only by a factor of ⅓ from the estimation of the depth for a plume with a narrow front (Lentz 2004; Avicola and Huq 2003; Pimenta et al. 2011).

b. Response to downwelling-favorable wind stress forcing

We now turn our attention to the response of the plume to the onset of wind stress forcing. We assume that the wind is constant and downwelling favorable. In this simplified theory, the wind stress acts to modify the plume geometry and buoyancy anomaly through two mechanisms considered here sequentially. First, it creates a surface mixed layer that quickly entrains ambient water into the plume and steepens the isopycnals within the mixed layer (Fig. 2, top). In a second stage, the spinup of a cross-shelf Ekman circulation advectively steepens the interior isopycnals. Entrainment of ambient water into the plume after the initial formation of the mixed layer is neglected in the theory. The advective steepening is assumed to continue until the isopycnals are essentially vertical (Fig. 2). Below, we derive expressions for the final scales that depend only on the initial conditions and the strength of the wind forcing.

Schematic of the theoretical evolution of the response to downwelling-favorable wind stress. Initially ambient water is entrained into the plume in a layer of thickness δ_e that can be (top) shallower or (middle) deeper than the initial plume depth h_p. The entrainment expands the initial frontal area to A_γe. (bottom) Advective steepening leads to vertical isopycnals and the final depth h_p*, width W_p*, and density anomaly ρ_o − ρ_c*. In this final state A_o* = A_γe.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Schematic of the theoretical evolution of the response to downwelling-favorable wind stress. Initially ambient water is entrained into the plume in a layer of thickness δ_e that can be (top) shallower or (middle) deeper than the initial plume depth h_p. The entrainment expands the initial frontal area to A_γe. (bottom) Advective steepening leads to vertical isopycnals and the final depth h_p*, width W_p*, and density anomaly ρ_o − ρ_c*. In this final state A_o* = A_γe.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Schematic of the theoretical evolution of the response to downwelling-favorable wind stress. Initially ambient water is entrained into the plume in a layer of thickness δ_e that can be (top) shallower or (middle) deeper than the initial plume depth h_p. The entrainment expands the initial frontal area to A_γe. (bottom) Advective steepening leads to vertical isopycnals and the final depth h_p*, width W_p*, and density anomaly ρ_o − ρ_c*. In this final state A_o* = A_γe.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Initially, the wind stress forcing results in the rapid formation of a mixed layer whose depth can be approximated as (Pollard et al. 1972; McWilliams et al. 2009)

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where δ_e is the depth of the wind-forced Ekman layer and N² = g′/(γW_p) is the initial plume stratification. The formation of this layer leads to the entrainment of ambient water into the plume (Fig. 2). Thus, if we consider the area of the frontal portion of the plume prior to the onset of wind forcing, that is,

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we can write an expression for the area offshore of W_α immediately after the mixed layer has formed:

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where

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is the bottom depth at the outer edge of the plume (x = W_p). Thus, when the wind-generated mixed layer is shallower than the initial plume depth, the entrained area is only a function of the Ekman layer depth and the initial isopycnal slope, and therefore a weaker isopycnal slope leads to relatively more ambient water entrained into the plume. For the same plume stratification and depth prior to the onset of the wind, a steeper bottom slope results in stronger mixing of the plume if the wind is strong enough, that is, for δ_e > h_p. Together with the wind stress intensity, the local bathymetry below the plume front determines the total amount of ambient water that can be initially entrained into the plume. The theory predicts that the maximum entrainment will occur when the wind stress is τ_m = ρ_o(h_c + αW_p)²(Nf). Increasing the wind intensity above this value would not result in additional entrainment of ambient water, although, as we will see later, stronger winds do affect the plume in other, key ways. However, note that

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so that, when h_c ≪ h_p, α/γ ≈ W_γ/W_α and so Eq. (14) suggests that for bottom-trapped plumes (W_γ/W_α ≪ 1), h_f ≈ h_p, and thus the potential for significant entrainment relative to the total area of the plume is reduced, even for strong winds.

In the final state, where the isopycnals are nearly vertical (Fig. 2), the final area offshore of the initial value of W_α is

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in which is the final depth of the plume. Hereafter, we use asterisk superscripted quantities for all plume scales in the final state. Now, the final depth is given by

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where A_e is the total amount of ambient water entrained into the plume during the wind event. If the evolution of the plume after the initial mixing is dominated by advective steepening of the isopycnals and there is no further entrainment of water into the plume (i.e., A_e = A_γe − A_γ), it follows from volume conservation that A_γe = A_o*. The final depth of the plume can then be derived by equating Eqs. (16) and (12) and solving for h_p*:

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and, as hinted above, for W_γ/W_α ≪ 1 (bottom-trapped plumes) h_p* ≈ h_p even for strong winds. Thus, in this model the depth and width of bottom-trapped plumes are relatively insensitive to downwelling-favorable wind forcing. The fundamental reason is that the depth change that is attainable through entrainment and advective steepening is small relatively to the overall depth of a bottom-trapped plume.

To understand the evolution of the density field, we can use buoyancy conservation for the initial and final states of the plume. First we integrate (ρ₀ − ρ) over the initial plume area to obtain

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In the final state, where the isopycnals are nearly vertical and W_γ* ≈ 0, the density can be written as ρ_* = ρ_c* + (Δρ_c*/W_p*)x and the above equation is reduced to

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Using the above equation and the equation for the final depth of the plume we can estimate the final density difference from the initial conditions:

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where W_p* ≈ W_α* = (h_p* − h_c)/α is the final width of the plume. As hinted by the equation for the final depth, the change in density difference is also dependent on the initial geometry of the plume. For bottom-trapped plumes, h_p ≈ h_p* and thus the density anomaly does not change significantly, even for strong winds. Taken together, the above equations can be used to predict the final geometry and buoyancy difference of the plume using the initial and forcing conditions.

An important part of the response is the acceleration of the along-shelf flow. In the final state, the effect of wind stress forcing on the along-shelf transport can be conceptualized as having two components (Whitney and Garvine 2005; Lentz and Largier 2006). First, there is the spinup of a geostrophic, barotropic alongshore flow, which can be approximated as

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where A_p* = 1/2W_p*(h_p* + h_c) is the final cross-sectional area of the plume and r is a linear bottom drag coefficient. The wind also mixes and deepens the plume, which results, overall, in an increased baroclinic transport. Assuming a 1½-layer model for the later component (see section 2a), the final transport is

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where u_w/u_g can be thought as a ratio of a wind-driven barotropic velocity [u_w = τ/(rρ_o)] and a baroclinic velocity scale [](for a similar reasoning, see Whitney and Garvine 2005). When the winds are relatively weak, the transport can be approximated by the final depth and density anomaly.

A reasonable time scale for the steepening process is given by

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which reduces to t_a = A_γ/(2U_e) for α/γ ≪ 1, where U_e = τ/(ρf) is the Ekman transport in the cross-shelf direction. This scale represents the time it takes for the cross-shore Ekman circulation to steepen the isopycnals, which depends on both the frontal (γ) and bottom (α) slopes. For relatively strong winds (or small plumes) this time scale can be of the same order as the mixed layer spinup [Eq. (10)], namely, 1/f. To incorporate this time dependence into the theory, we assume a linear increase in the mixed layer depth for times shorter than the mixed layer spinup time, such that

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It is important to note that buoyant plumes can have a sharp frontal structure separating the ambient water from the fresh coastal water and is not constantly stratified as assumed here. The role that this variability plays in the transport and propagation properties of the plumes is explored in Pimenta et al. (2011), although at least some relatively small (Lentz and Largier 2006) and large (Moffat et al. 2008) coastal plumes can be approximated this way. Moreover, a key underlying assumption is that the along-shelf variability of the plume structure will not change significantly during the event, an assumption that should be tested carefully when contrasting the theory presented here against observations.

a. Numerical model description and setup

To test of the theory developed in the previous section, we use results from a series of numerical model runs. The Regional Ocean Modeling System (ROMS) is a widely used, primitive equation, open source numerical model. The model is terrain following, includes a free surface, and is hydrostatic (Shchepetkin and McWilliams 2005). Here we run the model in a two-dimensional configuration using periodic boundary conditions in the along-shelf direction. We define x as the cross-shelf direction, positive offshore, and z the vertical dimension, with zero at the undisturbed surface and positive upward (Fig. 1). The model domain is 50 km in the cross-shelf direction and has a horizontal resolution of 0.5 km. The coastal bathymetry is approximated by a small (h_c = 5 m) vertical wall at the coast and constant bottom slope α, which is either 10⁻³ (gentle slope) or 5 × 10⁻³ (steep slope). A wall is used at the offshore boundary. Making the domain wider while maintaining the horizontal resolution did not change the results in any significant way. In the vertical, the setup includes 25 levels, which results in an interior vertical resolution of ~2 m or better within the plume for all model runs. Vertical resolution is higher near the surface and bottom. The model is run using the Mellor–Yamada turbulence closure scheme (Mellor and Yamada 1982), and the bottom stress is parameterized using a linear formulation with a coefficient r = 5 × 10⁻⁴ m s⁻¹.

The model is initialized with a plume with constant stratification [Eq. (1)], a density anomaly Δρ_c = ρ_o − ρ_c, width W_α, and plume depth h_p. The frontal width W_γ is calculated using Eq. (4), and a velocity field in geostrophic balance with the density field is used to initialize the alongshore velocity field. The temperature in the model is set to 10°C everywhere, and the buoyancy anomaly of the plume is therefore accomplished by setting relatively freshwater with respect to an ambient salinity of 32 psu. The Coriolis parameter is set to f = 10⁻⁴ s⁻¹. To fully explore the theory developed in section 2, we conducted 120 model runs, spanning a wide range of initial and forcing conditions for the fundamental parameters of the system, W_γ/W_α and δ_e/h_p (Table 1). Prior to simulating the response to wind stress forcing, the model is run unforced for several days, which allows transients to decay. On day 10 of each model run, a constant, downwelling-favorable wind stress is applied for 10 days, a reasonable choice given our focus on synoptic-scale forcing variability.

Table 1.

Initial conditions used to run the numerical model and forced with wind stress τ = −0.01, −0.05, −0.10, −0.50, and −1.00 N m⁻².

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Given our choice of initial conditions, the model runs result in 24 distinct initial states, which we use to test the theoretical predictions for the initial scales (section 2a). Figure 3 illustrates the initial conditions of salinity (top panels) and alongshore velocity (bottom panels) for a bottom-trapped, intermediate, and surface-trapped plume. For each run, the initial geometry of the plume is obtained by using an isohaline whose value is the ambient salinity (32) minus 10% of the overall salinity difference of the plume. That is, if the initial surface salinity anomaly (ambient salinity minus the surface salinity at the coast) is 5, the reference isohaline for the entire run would be 31.50 psu. Overall, the reference isohaline ranged from 31.05 to 31.90 psu. The volume transport of the plume is calculated by integrating the alongshore velocity component shoreward of the location of that reference isohaline. The density anomaly is calculated as the difference between ambient and surface density at the coast. Figure 4 shows a comparison between the predicted and modeled plume width and depth for day 9 of all model runs, prior to wind forcing. The scale for the width [Eq. (4)] is shown multiplied by a constant c = 1.2. After several days of adjustment, the plume has reached a geostrophic balance in the cross-shelf momentum balance, which results, as expected, in the good agreement between the theory and the model output.

Initial salinity and alongshore velocity for (left) a bottom-trapped (run 13), (middle) an intermediate (run 11), and (right) a surface-trapped (run 12) plume.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Initial salinity and alongshore velocity for (left) a bottom-trapped (run 13), (middle) an intermediate (run 11), and (right) a surface-trapped (run 12) plume.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Initial salinity and alongshore velocity for (left) a bottom-trapped (run 13), (middle) an intermediate (run 11), and (right) a surface-trapped (run 12) plume.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Initial (left) plume depth and (right) frontal area width from the numerical model (vertical axis) vs the theoretical predictions (horizontal axis, with subscript ith). The theoretical width is scaled by a constant c = 1.2.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Initial (left) plume depth and (right) frontal area width from the numerical model (vertical axis) vs the theoretical predictions (horizontal axis, with subscript ith). The theoretical width is scaled by a constant c = 1.2.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Initial (left) plume depth and (right) frontal area width from the numerical model (vertical axis) vs the theoretical predictions (horizontal axis, with subscript ith). The theoretical width is scaled by a constant c = 1.2.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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b. Evolution of key plume scales during a wind event

To illustrate the basic structure of the response of a plume to downwelling-favorable wind stress forcing, we show, in Fig. 5, a case forced by relatively weak wind stress forcing. By day 9, the plume is 8.7 m deep, with a surface buoyancy anomaly of 5.3 kg m⁻³. The frontal width is approximately 8.7 km, which results in a width ratio of W_γ/W_α = 2.4. The wind forcing is −0.01 N m⁻², which, given the plume stratification, means that the forcing ratio δ_e/h_p is about 0.14. Figure 5 shows the evolution of the wind-driven event over several days.

Evolution of the potential density (σ_θ), along-shelf velocity (υ), streamfunction (ψ) and normalized stress (τ_i/τ, where τ_i is the internal stress) for model run 5 (left) as the wind forcing (τ = −0.01 N m⁻²) begins, (middle) 24 h after, and (right) 96 h after. Gray (black) solid streamlines indicate clockwise (anticlockwise) circulation in the z–x plane. A vertical dotted line at x = 10 km is added for reference.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Evolution of the potential density (σ_θ), along-shelf velocity (υ), streamfunction (ψ) and normalized stress (τ_i/τ, where τ_i is the internal stress) for model run 5 (left) as the wind forcing (τ = −0.01 N m⁻²) begins, (middle) 24 h after, and (right) 96 h after. Gray (black) solid streamlines indicate clockwise (anticlockwise) circulation in the z–x plane. A vertical dotted line at x = 10 km is added for reference.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Evolution of the potential density (σ_θ), along-shelf velocity (υ), streamfunction (ψ) and normalized stress (τ_i/τ, where τ_i is the internal stress) for model run 5 (left) as the wind forcing (τ = −0.01 N m⁻²) begins, (middle) 24 h after, and (right) 96 h after. Gray (black) solid streamlines indicate clockwise (anticlockwise) circulation in the z–x plane. A vertical dotted line at x = 10 km is added for reference.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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After the wind stress starts, the surface quickly becomes mixed throughout the domain, as evidenced by the nearly vertical isopycnals at the surface (Fig. 5, top). A surface Ekman layer spins up and the surface water starts to be advected shoreward. Offshore of the plume front, the surface and bottom Ekman layers become increasingly distinct from each other as the bottom deepens. Onshore of the front, the Ekman layers are thinner as a consequence of the plume stratification. As a result of the steepening isopycnals, which increases the geostrophic shear, and the increase of the surface slope, the alongshore flow accelerates significantly over the duration of the event.

The development of the cross-shelf circulation offshore of the plume resembles the response expected for an unstratified shelf, with surface and bottom Ekman layers clearly distinguishable at the offshore edge of the right panels of Fig. 5 (see in particular the streamfunction) but merging together onshore. The cross-shore divergence of the Ekman transport leads to downwelling just offshore of the plume front, which consist by this point of nearly vertical isopycnals. The resulting along-shelf transport has increased considerably. As for the cross-shelf circulation, the vertical shear within the plume causes a response that is clearly distinct compared to the region offshore of the plume (or what it would be in the absence of stratification). The region offshore of the plume shows a weak recirculation with a sense opposite to the circulation offshore of the front, which reinforces the surface convergence (divergence) at the surface (bottom) on the offshore edge of the plume. The conditions for the formation of this reverse circulation near the shore are explored in the discussion.

Figure 6 shows the evolution of the key scales of the plume during the event. Clearly, the largest changes in the plume buoyancy anomaly and transport occur as the plume deepens and narrows; that is, when the initial mixing occurs, the barotropic wind response is spun up and the plume undergoes advective steepening. Because the entrainment of ambient water into the plume increases its volume, the steepening process results in more deepening and less narrowing than with advective steepening alone. Although the theory only includes the evolution up to the point where the isopycnals are nearly vertical (just after day 13 in the run shown in Fig. 6), it is clear that this state is not completely steady, and the plume continues to slowly widen, deepen, and become more mixed (the transport is reduced slightly throughout the event) as frontal processes continue to entrain ambient water along the front. However, and as assumed in the theory, the major changes in plume geometry and other properties occur during the early mixing and steepening process.

(top) Evolution of the front of the plume (S = 31.95 psu) for case 5 forced with a wind stress τ = −0.01 N m⁻². The darkest line corresponds to the time just prior to the unset of the wind, and the increasingly lighter ones are snapshots of the isohaline every 12 h. Also shown are (middle) the plume depth and width and (bottom) the reduced gravity and transport.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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(top) Evolution of the front of the plume (S = 31.95 psu) for case 5 forced with a wind stress τ = −0.01 N m⁻². The darkest line corresponds to the time just prior to the unset of the wind, and the increasingly lighter ones are snapshots of the isohaline every 12 h. Also shown are (middle) the plume depth and width and (bottom) the reduced gravity and transport.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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(top) Evolution of the front of the plume (S = 31.95 psu) for case 5 forced with a wind stress τ = −0.01 N m⁻². The darkest line corresponds to the time just prior to the unset of the wind, and the increasingly lighter ones are snapshots of the isohaline every 12 h. Also shown are (middle) the plume depth and width and (bottom) the reduced gravity and transport.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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The evolution of the scales during the event shows that, after the initial spinup of the Ekman circulation, the plume narrows at progressively slower rates. This can be readily understood by considering that, as the plume isopycnals are advected onshore, the mixed layer thickens and thus the average velocity within the Ekman layer is reduced. This behavior is analogous to the offshore motion of a downwelling front on a shelf with constant slope and starting from rest (Austin and Lentz 2002), with the bottom slope in that case replaced by the frontal slope here.

c. Comparing the theoretical predictions to the model results

The theory derived in section 2 provides predictions for the final depth, width, density anomaly, and along-shelf transport for a plume forced by constant wind. Model runs spanning a wide range of initial and forcing conditions (Table 1) were analyzed to test these predictions. Initial conditions were taken at t = 9.75 days, and final scales are taken at t = 17 days. Six runs that do not reach the final state in those seven days (i.e., t_a > 7 days) when forced with the weakest stress (0.01 N m⁻²) are not considered in the analysis. After discarding those runs, 114 remain for comparison with the theory.

Comparison for theoretical predictions against numerical model results is shown in Fig. 7. The predictions are, overall, reasonable (Table 2). For event time scales t_a shorter than the spinup period t_i = 1/f, the theory tends to do well. When the time it takes for the isopycnals to become vertical due to advection is long compared to the spinup time (black circles in Fig. 7), the theoretical predictions comparison to the model results are worse. The predicted changes for the density difference are poorer than for other plume properties, an indication of the importance of entrainment processes not considered in our simplified model.

Comparison between the numerical model output and theoretical predictions for (top left) the total deepening, (top right) change in the along-shelf transport, (bottom left) change in density anomaly, and (bottom right) total narrowing during an event. For all panels, open (solid) circles correspond to cases where t_a/t_i ≤ 1 (t_a/t_i > 1).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Comparison between the numerical model output and theoretical predictions for (top left) the total deepening, (top right) change in the along-shelf transport, (bottom left) change in density anomaly, and (bottom right) total narrowing during an event. For all panels, open (solid) circles correspond to cases where t_a/t_i ≤ 1 (t_a/t_i > 1).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Comparison between the numerical model output and theoretical predictions for (top left) the total deepening, (top right) change in the along-shelf transport, (bottom left) change in density anomaly, and (bottom right) total narrowing during an event. For all panels, open (solid) circles correspond to cases where t_a/t_i ≤ 1 (t_a/t_i > 1).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Table 2.

Correlation coefficients and slopes for the theory and numerical model results comparisons shown in Fig. 7, for t_a/t_i ≤ 1 (t_a/t_i > 1). All results have a p value <0.005.

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The dependency of the response to the initial geometry of the plume and the intensity of the wind forcing is illustrated in Fig. 8. The left panel shows the normalized plume depth as a function of α/γ. The curves are the theoretical predictions for δ_e/h_p = 0.06–0.2 (black curves) and 0.4–0.6 (gray curves). The dots and squares are the numerical model results whose initial conditions fall within each range of δ_e/h_p (the color of the symbols match each curve). As predicted by the theory, increasingly bottom-trapped plumes (α/γ ≪ 1) show little relative deepening, regardless of wind strength and initial stratification. As the plume becomes surface trapped, the deepening is significant, with the plume deepening to maximum values of more than twice its initial depth for the initial geometry and forcing conditions chosen here.

(left) Plume depth at t = 17 days normalized by initial value (at t = 9.5 days) as a function of the geometry ratio α/γ and (right) normalized plume width as a function of the width ratio W_γ/W_α. The black (gray) color corresponds to a range of forcing ratio (δ_e/h_p) of 0.06–0.2 (0.4–0.6). The factor ϵ = (1 − h_c/h_p)/(1 − h_c/h_p*) arises from having a vertical wall at the coast, and it ranges from 0.71 to nearly 1. The symbols correspond to the numerical model runs, and the solid lines correspond to the theory predictions (one curve for each extreme of each range of forcing ratio).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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(left) Plume depth at t = 17 days normalized by initial value (at t = 9.5 days) as a function of the geometry ratio α/γ and (right) normalized plume width as a function of the width ratio W_γ/W_α. The black (gray) color corresponds to a range of forcing ratio (δ_e/h_p) of 0.06–0.2 (0.4–0.6). The factor ϵ = (1 − h_c/h_p)/(1 − h_c/h_p*) arises from having a vertical wall at the coast, and it ranges from 0.71 to nearly 1. The symbols correspond to the numerical model runs, and the solid lines correspond to the theory predictions (one curve for each extreme of each range of forcing ratio).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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(left) Plume depth at t = 17 days normalized by initial value (at t = 9.5 days) as a function of the geometry ratio α/γ and (right) normalized plume width as a function of the width ratio W_γ/W_α. The black (gray) color corresponds to a range of forcing ratio (δ_e/h_p) of 0.06–0.2 (0.4–0.6). The factor ϵ = (1 − h_c/h_p)/(1 − h_c/h_p*) arises from having a vertical wall at the coast, and it ranges from 0.71 to nearly 1. The symbols correspond to the numerical model runs, and the solid lines correspond to the theory predictions (one curve for each extreme of each range of forcing ratio).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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The model results shown, however, do not fit well the expected dependency on the forcing ratio (δ_e/h_p). The theory presented was unsatisfactory to correctly account for the entrainment over the parameter considered. Reasons for the disagreement include the continuous entrainment of ambient water during the advective steepening process (particularly for weak winds); the entrainment process being less efficient that assumed (for strong winds); and the continuous evolution after the isopycnals first become nearly vertical, which the theory ignores. This issue is explored further in the discussion.

A similar picture arises from examination of the dependency of the normalized width (including a factor ϵ involving h_c) as a function of the width ratio W_γ/W_α. Bottom-trapped plumes suffer little change of width as a result of the wind forcing, whereas surface-trapped plumes undergo the largest relative narrowing. However, as expected from the analysis of evolution of the depth scale, the entrainment processes involved in these events cannot be accounted by the simple approximation used in our theory.

Overall, the theory provides a reasonable first-order prediction of the resulting scales from a downwelling-favorable wind event forcing a coastal buoyant plume. The initial conditions used in the numerical model are relevant for many midlatitude coastal buoyant plumes. A few issues of interest related to this work, as well as its shortcomings, are discussed in the next section.

a. Evolution of the entrainment

Although the theoretical model and numerical model results show reasonable agreement, the former assumes a simplified entrainment model that does not capture the complexity of the evolution of the plume during a wind event. Figure 9 illustrates some of the key mixing processes occurring during an event. The top three panels show the gradient Richardson number [Ri = N²/(du/dz² + dυ/dz²)] and the along- and cross-shore velocity shear evolution with time for the plume frontal region. Each property is measured along the frontal isohaline, and their projected distance over the cross-shelf axis is plotted on the y axis of these panels. As expected, much of the shear leading to critical values of the Richardson number occur primarily near the surface, with high values occurring near the bottom as the return flow spins up. Notice that, as the event progresses, the projected length of the isohaline is reduced as the mixing and Ekman circulation sharply steepen the density field.

Evolution of (top) the gradient Richardson number, (upper middle) along- and (lower middle) cross-shore velocity gradient squared, and (bottom) change of plume area and plume depth for run 5 forced with τ = −0.05 N m⁻². (top)–(bottom middle) The corresponding variables in the top three panels are shown as a function of the distance along the reference salinity for the run; that is, the height of the colored area represents the projected length of the isohaline at a given time (which is reduced as the plume narrows and deepens forced by the wind).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Evolution of (top) the gradient Richardson number, (upper middle) along- and (lower middle) cross-shore velocity gradient squared, and (bottom) change of plume area and plume depth for run 5 forced with τ = −0.05 N m⁻². (top)–(bottom middle) The corresponding variables in the top three panels are shown as a function of the distance along the reference salinity for the run; that is, the height of the colored area represents the projected length of the isohaline at a given time (which is reduced as the plume narrows and deepens forced by the wind).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Evolution of (top) the gradient Richardson number, (upper middle) along- and (lower middle) cross-shore velocity gradient squared, and (bottom) change of plume area and plume depth for run 5 forced with τ = −0.05 N m⁻². (top)–(bottom middle) The corresponding variables in the top three panels are shown as a function of the distance along the reference salinity for the run; that is, the height of the colored area represents the projected length of the isohaline at a given time (which is reduced as the plume narrows and deepens forced by the wind).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Figure 9 (bottom) shows the rate of change of the plume area as well as evolution of the depth. The largest change in area occurs shortly after onset of the wind, associated mainly with the increased surface shear, as assumed in the theory. Notice that the deepening does not occur at this stage but when the bottom return flow is developed. Clearly, there is entrainment of ambient water after the initial entrainment, and both across- and alongshore vertical shears seem to contribute to this process. Other events (not shown) show a similar pattern of high values of entrainment (as measured by areal changes) early in the process, which are associated with inertial-scale shear variability [for a similar discussion for the upwelling-favorable case, see Fong and Geyer (2001)]. Future inclusion of these more complex entrainment processes on the surface and bottom in the theory are likely to improve the predictions presented in the previous section.

b. Cross-shelf circulation

In the absence of buoyant water at the coast, the cross-shelf circulation during a downwelling-favorable wind event is characterized by onshore flow at the surface and a return flow at depth (Allen and Newberger 1996; Austin and Lentz 2002). As the surface and bottom boundary layers converge toward the coast, the cross-shelf circulation weakens. However, as shown in section 3b, the plume cross-shelf circulation during a downwelling-favorable wind event can result in oceanward flow at the surface and shoreward flow at the bottom. This creates surface convergence (and bottom divergence) of the circulation at the offshore edge of the plume.

This reversal of the near-shore circulation reveals the influence of the lateral density gradient and accompanying vertical geostrophic shear in the buoyancy and momentum balances. The stress associated with the geostrophic shear can induce an Ekman circulation in the opposite sense if it becomes larger than the surface wind stress. Otherwise, the circulation is qualitatively similar to the case where no buoyant water is found at the coast. A rough scaling for this geostrophic shear is, from the bulk formula for the surface stress, , where υ_g = ρ_o⁻¹g (Δρ_c*/W_p*)(h_p*/f) is an estimate of the geostrophic velocity during the event and C_d is a drag coefficient, here taken as 1.3 × 10⁻³.

Figure 10 illustrates this competition and its effect on the cross-shore circulation. For τ/τ_g > 1, the cross-shelf circulation as a top-to-bottom velocity difference Δu at x = W_p*/2 after seven days of forcing (so that negative values represent an anticlockwise circulation in the x–z plane) is qualitatively as expected for a wind-driven response in a homogeneous ocean. For τ/τ_g < 1, however, the circulation is reversed. The intensity of the wind required for a circulation reversal can be readily calculated from the theoretical estimates of the plume geometry and density anomaly.

Influence of the geostrophic shear on the cross-shelf circulation. (left) The top-to-bottom difference in the cross-shelf velocity (positive values indicate clockwise circulation in the x–z plane) as a function of the ratio of the geostrophic (τ_g) to wind (τ) stress for all events after seven days of forcing. Prior to the start of the wind (top right panel) the cross-shelf circulation is extremely weak. When τ < τ_g (middle right panel), there is strong surface convergence and oceanward (shoreward) surface (bottom) flow associated with the plume. A much weaker circulation in the same sense as offshore of the plume is evident in the model for τ > τ_g (bottom right panel).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Influence of the geostrophic shear on the cross-shelf circulation. (left) The top-to-bottom difference in the cross-shelf velocity (positive values indicate clockwise circulation in the x–z plane) as a function of the ratio of the geostrophic (τ_g) to wind (τ) stress for all events after seven days of forcing. Prior to the start of the wind (top right panel) the cross-shelf circulation is extremely weak. When τ < τ_g (middle right panel), there is strong surface convergence and oceanward (shoreward) surface (bottom) flow associated with the plume. A much weaker circulation in the same sense as offshore of the plume is evident in the model for τ > τ_g (bottom right panel).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Influence of the geostrophic shear on the cross-shelf circulation. (left) The top-to-bottom difference in the cross-shelf velocity (positive values indicate clockwise circulation in the x–z plane) as a function of the ratio of the geostrophic (τ_g) to wind (τ) stress for all events after seven days of forcing. Prior to the start of the wind (top right panel) the cross-shelf circulation is extremely weak. When τ < τ_g (middle right panel), there is strong surface convergence and oceanward (shoreward) surface (bottom) flow associated with the plume. A much weaker circulation in the same sense as offshore of the plume is evident in the model for τ > τ_g (bottom right panel).

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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c. Transient wind forcing events

In the absence of wind forcing, the initial plume geometry is critical to understand the dynamics of propagation along the coast (Lentz and Helfrich 2002). Surface-trapped plumes (W_γ/W_α ≫ 1) propagate faster along the coast than bottom-trapped ones (W_γ/W_α ≪ 1) with the same depth and reduced gravity. Here we have shown that the initial geometry of the plume is also key to understand its response to downwelling-favorable wind stress forcing. In all cases, the wind stress forcing results in mixing, steepening of the isopycnals, and an increase of the along-shelf volume transport. However, bottom-trapped plumes suffer relatively little deepening, narrowing, and entrainment of ambient water compared to surface-trapped ones. Overall, the effect of downwelling-favorable wind stress forcing is to make relatively surface-trapped plumes more bottom trapped.

As shown in the previous section, bottom-trapped plumes are less sensitive to the forcing than surface-trapped plumes. This is also evident after the wind event ends. Figure 11 (left) shows the alongshore transport of the plume for all numerical runs, after a 10-day wind event has ended and an additional 10 days have passed to allow for adjustment. The transport is normalized by its value prior to the start of the wind event. The time evolution of this normalized alongshore transport for a bottom-trapped and surface-trapped plume is also shown (Fig. 10, right). Surface-trapped plumes undergo stronger changes in geometry and density anomaly during the event, which results in a larger alongshore transport after the wind stops. Bottom-trapped plumes, on the other hand, show increased along-shelf transport during the event but relax to a state closely resembling the one prior to the start of wind forcing.

(left) Alongshore transport measured 10 days after the relaxation of downwelling-favorable wind stress. (right) The evolution of the alongshore transport for a surface-trapped (α/γ > 1, run 6 with τ = −0.05 N m⁻², solid line) and a bottom-trapped (α/γ < 1, run 7 with τ = −0.01 N m⁻², dashed line) plume. In both panels the transport is normalized by the transport prior to the wind event (Q_p). For the right plot constant downwelling-favorable wind stress was maintained between days 10 and 20.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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(left) Alongshore transport measured 10 days after the relaxation of downwelling-favorable wind stress. (right) The evolution of the alongshore transport for a surface-trapped (α/γ > 1, run 6 with τ = −0.05 N m⁻², solid line) and a bottom-trapped (α/γ < 1, run 7 with τ = −0.01 N m⁻², dashed line) plume. In both panels the transport is normalized by the transport prior to the wind event (Q_p). For the right plot constant downwelling-favorable wind stress was maintained between days 10 and 20.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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(left) Alongshore transport measured 10 days after the relaxation of downwelling-favorable wind stress. (right) The evolution of the alongshore transport for a surface-trapped (α/γ > 1, run 6 with τ = −0.05 N m⁻², solid line) and a bottom-trapped (α/γ < 1, run 7 with τ = −0.01 N m⁻², dashed line) plume. In both panels the transport is normalized by the transport prior to the wind event (Q_p). For the right plot constant downwelling-favorable wind stress was maintained between days 10 and 20.

Citation: Journal of Physical Oceanography 42, 7; 10.1175/JPO-D-11-015.1

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Finally, it is worth noting that the theory and numerical model results presented above make significant simplifications, which allows us to gain insight into key features of the response of a buoyant plume to wind forcing. However, the two-dimensional modeling approach does not allow us to consider interaction of the plume with the upstream region where the plume originates and where important processes take place (Garvine 2001; Horner-Devine 2009) or the possible role of along-shelf pressure gradients in the dynamics considered. We have also omitted the influence of tides in mixing and modifying the circulation of the plume (e.g., Horner-Devine et al. 2009). All of the above processes are likely to be important, individually or in combination, in most real systems, thus creating the potential for additional entrainment mechanisms and complicating the relatively simple circulation we have shown here. Our choice of a simple forcing function is reasonable when considering wind events of short duration (thus our focus in the response in the first few days after the start of the forcing), but the role of time-varying winds and cross-shore winds are a natural extension of the work presented here.