## Abstract

Two-dimensional (cross-shelf and depth) circulation by downwelling wind in the presence of a prograding front (with isopycnals that slope in the same direction as the topographic slope) over a continental shelf is studied using high-resolution numerical experiments. The physical process of interest is the cross-shelf circulation produced by northeasterly monsoon winds acting on the Kuroshio front over the East China Sea outer shelf and shelfbreak where upwelling is often observed. However, a general problem is posed and solved by idealized numerical and analytical models. It is shown that upwelling is produced shoreward of the front. The upwelling is maintained by (1) a surface bulge of negative vorticity at the head of the front; (2) bottom offshore convergence beneath the front; and (3) in the case of a surface front that is thin relative to water depth, also by upwelling due to the vorticity sheet under the front. The near-coast downwelling produces intense mixing due to both upright and slant-wise convection in regions of positive potential vorticity. The analytical model shows that the size and on-shore propagating speed of the bulge are determined by the wind and its shape is governed by a nonlinear advection–dispersion equation which yields unchanging wave-form solutions. Successive bulges can detach from the front under a steady wind. Vertical circulation cells develop under the propagating bulges despite a stable stratification. These cells can have important consequences to vertical exchanges of tracers and water masses.

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## Notes

- 1.
“Strong”, “large”, or “increased” will mean numerically large and negative PV while “weak,” “small”, or “decreased” will mean that the fluid moves towards a state of gravitational, inertial or symmetric instability, PV moves towards 0 or becomes positive.

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## Acknowledgments

We thank the two anonymous reviewers and editor Dr. Hua Wang for help in improving the manuscript. YLC received a fellowship from the Graduate Student Study Abroad Program (NSC97-2917-I-003-103) of the National Science Council of Taiwan. LYO is grateful to supports by the Minerals Management Service contract number M09PS20004.

## Author information

### Affiliations

### Corresponding author

## Additional information

This article is part of the Topical Collection on *2nd International Workshop on Modelling the Ocean 2010*

Responsible Editor: Hua Wang

## Appendices

### Appendix A. Model equations and boundary conditions

### (a) Model equations

The model solves the following 2-D incompressible fluid equations with hydrostatic and Bousinesq approximations:

Here, *u*, *v*, *w* are velocities in *x*, *y*, and *z* directions, respectively, *T* the potential temperature, *S* the constant salinity = 35‰, ρ is density calculated from the equation of state (A1f) with constant *S*, ρ_{o} a constant reference density, *p* the pressure, *f* is the constant Coriolis parameter (=6.2 × 10^{−5} s^{−1} corresponding to a latitude ≈25°N), and *g* is the acceleration of gravity. The *A*
_{
M
} is the horizontal eddy viscosity (=10 m^{2} s^{−1}) and *A*
_{
H
} the horizontal eddy diffusivity (=2 m^{2} s^{−1}). The *A*
_{
M
} is used to model sidewall friction in a very crude way; it also serves to stabilize the numerical model. The model is unstable for viscosity less than 0.5 m^{2} s^{−1}. We have repeated the basic case experiment with *A*
_{
M
} and *A*
_{
H
} reduced to 2 and 0.4 m^{2} s^{−1}, respectively, and obtained virtually identical results. The vertical eddy viscosity *K*
_{
M
} and diffusivity *K*
_{
H
} are calculated according to Mellor and Yamada’s (1982) level 2.5 turbulence model

where, *l* is the turbulence length scale, *q*
^{2}/*2* the turbulence kinetic energy, *S*
_{
M
} and *S*
_{
H
} are stratification-dependent stability functions, and \( \nu = {2} \times {1}{0^{{ - {5}}}}\,{{\text{m}}^{{2}}}\,{{\text{s}}^{{ - {1}}}} \) is a constant background viscosity. The *q*
^{2}/*2* is solved using an advection–diffusion equation similar to the heat Eq. A1e with sources and sinks representing dissipation as well as shear and buoyancy production (and destruction) of turbulence. A similar equation is used for the turbulence length scale *l*. As found by Allen and Newberger (1996; see their Fig. 15), the recirculation cells found here as a result of symmetric instability exist irrespective of the particular parameterizations of *K*
_{
M
} and *K*
_{
H
}. We have also confirmed this robustness of our results by repeating the basic experiment with constant *K*
_{
M
} and \( {K_H} = {2} \times {1}{0^{{ - {4}}}}\;{{\text{m}}^{{2}}}\,{{\text{s}}^{{ - {1}}}} \) (using the same method as Allen and Newberger (1996) in regions where the water column is statically unstable).

It is useful when examining the flow field to define a stream function *ψ* such that (A1a) is satisfied, ∂*ψ*/∂*z* = −*u*, ∂*ψ*/∂*x* = *w*; then

Thus defined, a positive (negative) *ψ* contour indicates anticlockwise (clockwise) circulation in the *xz* plane, looking northward.

### (b) Boundary conditions

Boundary conditions at the free surface, z = η, are

Where \( \left( {\tau_o^x,\tau_o^y = {\tau_{\text{o}}}} \right) \) is the kinematic wind stress vector (i.e., wind stress divided by water density), \( u_{\text{t}}^2 = - {{{\tau }}_{\text{o}}}|,{B_{\text{cb}}} \) is the Craig and Banner’s (1994) coefficient given by *B*
_{cb} = 15.8α_{cb}, with α_{cb} = 100, κ is the von Karman’s constant = 0.4, and \( {z_w} = {2} \times {1}{0^{{ - {5}}}}\left( {u_{\text{t}}^2/g} \right) \) (Stacey 1999; Terray et al. 2000). The surface boundary conditions (A4d) for *q*
^{2} and *l* are empirical parameterizations of input of turbulence by wind waves (Mellor and Blumberg 2004). They are found to improve the calculation of depths of surface mixed layer (hence the temperature); they differ from the original Mellor and Yamada’s (1982) conditions without wind waves, i.e., (*q*
^{2}, *l*) = (\( {B_1}^{{2/3}}{u_{\tau }}^2 \), 0) at *z* = 0, where *B*
_{1} = 16.6.

Boundary conditions at the ocean bottom, *z* = −*H*, are

In Eq. A5, *C*
_{
D
} is a drag coefficient, chosen so that the “law of the wall” logarithmic velocity profile is obtained near the ocean bottom, *z*
_{
o
} is the roughness parameter chosen to be =0.01 m, the “0.5Δσ_{
b
}
*H*” is the *z* value at 1/2 grid cell above the ocean’s bottom (i.e., at the velocity grid nearest the bottom), and the “*u*” and “*v*” are values evaluated at this near-bottom level.

Boundary conditions at the coast, *x* = 0, are:

Boundary conditions at the seaward open boundary, *x* = *x*
_{
L
}, are:

or,

Equation A7a ensures that no net volume flux enters or leaves the model domain. The first of Eq. A7b sets the gradient of the deviation of the *u* velocity from its depth-averaged value to zero. We find this condition to be simpler than a radiation condition which works well also. One-sided upwind differencing are indicated in Eq. A7c, in which the initial temperature is used for inflow *u*(*x*
_{
L
}, *z*, *t*) ≤ 0, and the temperature is advected for outflow *u*(*x*
_{
L
}, *z*, *t*) > 0. The effects of the additional term, *w∂T*/*∂z*, are small in the present case, but the term generally gives smoother *T*(*x*
_{
L
}, *z*, *t*) during outflow (Mellor 1989, personal correspondence).

### Appendix B. Daily averaged fields

We compare the daily averaged fields (shown here as Fig. 10) with the corresponding instantaneous fields of Fig. 3. In general, they are similar. The main difference is in the *w* fields which are more intense in the instantaneous plot especially at day 63 (compare Fig. 3b with Fig. 10b). Inertial motion occurs near the propagating front, resulting in oscillations of the vertical velocity which are smoothed out in the daily averaged plot. In the daily averaged plot, the stream function (ψ, Fig. 10b) shows a structure which is similar to the quasi-steady dynamics shown in Fig. 1. For example, two anti-clockwise circulation cells centered about the propagating front can be seen in Fig. 10b, in good agreement with the theoretical deductions of Section 2.

### Appendix C. Bulge dynamics

We first construct a simplified model (Fig. 11) that describes the development of the intrusive bulge (Fig. 4) of thickness δ(*x*,*t*) as a result of horizontal transport and divergence (and convergence) induced by the action of the up-front wind stress \( (\tau_{\text{o}}^y < 0) \) on the ocean’s near-surface fields. We then present additional numerical experiments that lend support to our theory and that also demonstrate that the process is robust.

In the theory below, the δ is assumed to be thin, δ/*H* << *1*, where *H* = water depth, and is infinitely elongated in the along-bulge direction *y*, or |∂/∂*y*| << |∂/∂*x*|. We consider only the main portion of the bulge (i.e., excluding the sharp variation near the bulge’s head) which is simpler to treat analytically but which we found in the numerical model is also where the bulge grows. Within the bulge, the fluid with temperature *T* = *T*
_{
b
}(*x*,*z*,*t*) is weakly stratified in the vertical and is also horizontally stratified:

where, β is a constant equal to the weak (to be defined precisely below) vertical temperature gradient in the bulge, and \( {T_{ - }}_{\delta } \) is the bulge’s temperature at its base *z* = −δ. The \( {T_{ - }}_{\delta } \) is related to the temperature *T*
_{
i
}(*z*) beneath the bulge where the fluid is assumed to be vertically stratified with a constant buoyancy frequency *N*
^{2}:

where \( \alpha = - (\partial \rho /\partial T)/{\rho_{\text{o}}} \) is the thermal expansion coefficient, and *T*
_{deep} is the uniform temperature for *z* < −*z*
_{deep}, a fixed depth; thus

The *T*
_{deep} and *z*
_{deep} are (constants) of secondary importance to the mathematical formulation but are included to emulate the numerical model result that the fluid is homogeneous deep beneath the bulge. By “weak” vertical stratification in the bulge, we mean then that:

i.e., that the bulge’s stratification is much weaker than the fluid’s stratification beneath the bulge.

The heat equation is:

where *Q* is the vertical heat flux. Integrate (C.5) through the bulge layer, we have for the first term:

after integration by parts and using (C.1). For the \( \int_{{ - \delta }}^0 {(u\partial T/\partial x)dz} \) term, we assume that within the bulge the time scale of motion is larger than the inertial time scale, so that from the linearized *y*-momentum equation:

we have

Substitute this approximation for *u* in the integral, use (C.1) and (C.2), and note that τ^{y} = *0* at *z* = −δ, we obtain:

For the \( \int_{{ - \delta }}^0 {(w\partial T/\partial z)dz} \) term, we approximate \( w \approx \left[ {w\left( {z = 0} \right) + w(z = - \delta )} \right]/{2} \), where

*v*
_{
o
} is the surface along-bulge (y-direct) velocity, and

is the vertical component of the absolute vorticity at the surface (Niiler, 1969); thus

The integral of ∂*Q*/∂*z* across the bulge is zero, since *Q* is assumed = *0* at *z* = *0* and *z* ≅δ. Collecting terms from (C.6a,b,c) and dropping terms *O*(αβ*g*/*N*
^{2})^{2} and higher, we have,

To evaluate *v*
_{
o
}, the along-bulge velocity *v* is assumed to be in geostrophic balance:

We first calculate *v* at *z* = −δ, either by finite-differencing the hydrostatic pressure at the base of the bulge and taking limit as Δ*x* ~ *0*, or evaluating the pressure at *z* = -δ by integrating the hydrostatic equation \( \partial (p/{\rho_{\text{o}}})/\partial z = \alpha gT \) across the bulge, using Eq. C.1, then (C.11):

Equation C.11 also gives the thermal wind in the bulge:

which upon integrating across the bulge gives, after substituting for \( v{|_{ - }}_{\delta } \) from (C.12):

Let

we obtain the following non-dimensionalized form of (C.10a, b):

where the primes have been dropped, and

Consider only non-zero δ, (C.15a) can then be rewritten as:

Another form is obtained by setting η = δ^{2} in (C.15a):

For β = 0 (no vertical stratification in the bulge) or uniform surface vorticity (\( {\partial^3}{\delta^2}/\partial {x^3} = 0 \)), the up-front wind advects an initial bulge shoreward. Equation C.15 suggests that the bulge’s head travels slowest and a “shock” would develop and “break backward.” This turns out to be an incorrect inference, for the change in “δ” at once results in non-zero ∂^{3}δ^{2}/∂*x*
^{3} which alters (in a nonlinear fashion) the balance between Ekman advection and vorticity-induced divergence. For a given wind, we show next that this balance admits a solitary bulge solution whose propagation speed and bulge amplitude depends on the wind stress.

### Solution to (C.15c)

We seek a solution of the following form for the constant wind case (τ_{
n
} = constant < 0):

Substituting into (C.15c) and integrating once:

where, *C*
_{1} is a constant. This may be integrated twice (set *G* = *dF*/*d*ξ, *d*
^{2}
*F*/*d*ξ^{2} = *d*(*G*
^{2}/*2*)/*dF*, etc.):

where the negative sign of √{..} is chosen, *C*
_{2} is a constant, and another integration constant is set zero, giving ξ ≤ 0. The lower limit has been conveniently chosen to be *F* = 1 where at ξ = 0, *dF*/*d*ξ is also required to vanish. This determines *C*
_{2} in terms of *C*
_{1} and *c*, so that Eq. C.20 becomes:

This gives a two-parameter (*c* and *C*
_{1}) family of solution which may be obtained numerically.

For real solution, the quantity in {..} must be positive and it may be shown that:

must be satisfied for some large but finite η_{o} (>*1*). The η_{o} is the zero of:

The η_{o} (and corresponding ξ_{o} < 0) is where *dF*/*d*ξ vanishes the “second time” (in addition to the first *dF*/*d*ξ = 0 at ξ = 0). Since *dF*/*d*ξ ∝ *v*
_{
o
} (see C.16), these are therefore locations where the along-front velocity also vanishes. We may identify the first zero at ξ = 0 to be the location where warm water from the main front first intrudes to form the bulge, here *v*
_{
o
} ≈ 0 (Fig. 4a, day 63 and later, *x* ≈ 35–40 km). The second zero then corresponds to the head of the bulge where η reaches a maximum and *v*
_{
o
} again vanishes. In the numerical model (Fig. 4a), there is a significant, nearly barotropic wind-driven interior current \( v \approx \tau_{\text{o}}^{\text{y}}/r \) (*r* = “friction coefficient”) shoreward of the bulge, so that the along-slope velocity does not vanish at the head of the bulge; nevertheless, the current reaches a minimum there. The bulge solution is not valid for ξ < ξ_{o} where we expect a “front” within which η rapidly decreases from η = η_{o} to η = 1.

Figure 12 plots the solution (C.21) for *C*
_{1} = 1 and various values of *c*. The stationary solution (*c* = 0; we omit the trivial solution when τ_{
n
} = 0) corresponds to an exact balance between onshore flux by Ekman advection and influx of upwelled water into the bulge. Solutions for other values of *c* indicate bulge growth for finite propagation speed. Each curve represents a permanent wave-form that translates to the left at the corresponding “*c*”—faster for a stronger wind stress τ_{
n
} (<0). The similarity of the analytical bulge shape with the numerical bulge is apparent (compare Figs. 4 and 12). Stronger wind (larger “*c*”) also increases the bulge’s amplitude, which is also seen in the numerical model. The growth is limited by the formation of front at the bulge head determined (approximately) by (ξ_{o}, η_{o}) by Eq. C.23, and indicated in Fig. 12 by the dotted line for each “*c*.” The backward-breaking wave solution without dispersion is not realized because as soon as spatial gradients develop, divergence (and upwelling, i.e., the dispersion term) prevents thinning of the bulge behind the head.

### Additional numerical experiments

We wish to demonstrate that the bulge solution as seen in the numerical experiments (e.g., Fig. 4) is independent of the coastal boundary (with which the bulge eventually interacts). We also wish to examine vertical motions generated by the propagating bulge. We therefore set up a numerical model in which the coast (*x* = 0) is replaced by an open boundary specified in a similar manner as the open-boundary condition Eq. A7 at the other end of the *xz* channel *x* = *x*
_{
L
}. Moreover, this new open boundary (*x* = 0) is placed sufficiently far (250 km) from the (main) front, so that (1) effects of inexact open-boundary conditions at *x* = 0 are minimized and (2) long-time (tens of days instead of ~5 days as in Fig. 4) evolution of the bulge may be simulated. Since our focus is near the surface, the channel is flat (*H* = constant). Two experiments were conducted—one with *H* = 350 m (NoWall350 experiment) and the other one with *H* = 1,000 m. Both give similar results and only the NoWall350 experiment is described here. The same initial front as in the BC experiment is used (except the bottom is now flat). A uniform vertical grid size = 1 m (i.e., 350 sigma levels) is used but the finest Δ*x* is 1 km for 0 km < *x* ≤ 300 km (instead of 0.25 km used in the BC experiment). The initial frontal edge is at *x* = 250 km. The up-front wind stress is applied as in the BC experiment (Section 3), except that it is kept constant after 5 days when \( \tau_{\text{o}}^{\text{y}} \cong {10^{{ - 4}}}{m^2}{s^{{ - 2}}} \), through *t* = 80 days.

Figure 13 shows (a) the *v* contours superimposed on color *T*, and (b) the stream function contours superimposed on color PV. In the absence of a coast, the steady wind forces upwelling under the front because of the large ∂*ζ*/∂*x* (>0) near the surface. The effect is analogous to a positive wind stress curl centered approximately at the frontal edge, and since the fluid is homogeneous west of the edge, a nearly barotropic flow with *v* < 0 is established there. A bulge develops and propagates westward in the first 5 days (Fig. 13 top panel) much as is anticipated from the BC experiment and the analytical solution. The bulge is thin but vertical recirculation cells develop beneath it. Since the wind stress is kept constant at \( \tau_{\text{o}}^{\text{y}} \cong {10^{{ - 4}}}{m^2}{s^{{ - 2}}} \) (after day 65), the bulge ceases to grow and its width is fixed for each \( \tau_{\text{o}}^{\text{y}} \) value (by C.23); it continues to propagate westward as a permanent wave-form predicted by the analytical model. The bulge head itself therefore forms a *frontlet* upon which the continued action of the wind generates yet another bulge with its own recirculation cells. This process is seen at day 70 in Fig. 13 when the second bulge forms; the formation of this second bulge is shown more clearly in Fig. 14 which plots only the upper 100 m of the temperature (color) and *v* contours from day 62 through 70. By day 80, the fourth bulge begins to form near the western open boundary (fourth row panels in Fig. 13).

The establishment of vertical recirculation cells that extend to deep layers beneath the bulges is a unique and interesting (time-dependent–tens of days) feature of the bulge solution. It is interesting because the passage of the bulge establishes a very stable near-surface fluid just beneath the bulge. This is seen clearly by the white contours of high (negative) PV behind the bulges in Fig. 13 which should be contrasted with the thick low PV fluid just ahead of the westernmost bulge. Stable stratification inhibits vertical motions, yet the bulge allows deep recirculation cells which can produce vertical exchanges of heat, salt and nutrients in the oceans.

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Chang, Y., Oey, L. Frontal circulation induced by up-front and coastal downwelling winds.
*Ocean Dynamics* **61, **1345–1368 (2011). https://doi.org/10.1007/s10236-011-0435-2

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### Keywords

- Ocean fronts
- Wind-front interaction
- Coastal downwelling
- Upfront winds
- Boluses
- Vertical cells
- Ekman transport modified by vorticity