## 1. Introduction

The interior ocean imposes pressure anomaly signals at multiple spatiotemporal scales on continental shelf seas, including the superinertial swell, tides, and subinertial mesoscale eddy impingement or basin-scale boundary currents. Subinertial (i.e., with the time scale greater than the local inertial period) dynamics are essential for the coastal sea level variations, shelf circulations, and cross-shelf exchanges (Huthnance 2004; Brink 2016; Hughes et al. 2019). However, unlike swells or tides, they are blocked by the steep shelf break, also known as the potential vorticity (PV) barrier, and therefore hardly reach the coast.

Csanady (1978) suggested that bottom friction is the key to breaking the PV barrier. A “heat-conduction analogy” was proposed in his classic arrested topographic wave (ATW) theory. This analogy states that the pressure anomaly imposed at the shelf edge diffuses shoreward (in a manner analogous to heat conduction) and propagates downshelf (in a manner analogous to temporal evolution), leaving an initial coastal region unaffected, i.e., the insulating region (Csanady and Shaw 1983). The insulating region is often large for a steep or wide shelf. Therefore, Wang (1982) argued that the pressure gradient driving the Mid-Atlantic Bight (MAB) circulation is unlikely to originate from the open ocean. As also reported by Hetland et al. (1999), the Loop Current affects the inner West Florida Shelf circulation only when it initially hits the shallow water region. Another consequence of insulation is that coastal tidal gauge stations often sense only a small fraction of the along-shelf sea level difference caused by the offshore boundary current. For instance, the Gulf Stream–induced 1-m northward drop in offshore sea level between the Florida Keys and Halifax is replaced by a smaller 0.2-m drop at the coast, with a ∼10° southward shift in latitude (Higginson et al. 2015; Wise et al. 2018). A similar phenomenon occurs in the East China Sea, where the meridional sea level difference caused by the Kuroshio Current (∼1 m) is reduced to ∼0.4 m along the coast and shifts southward notably (Lin et al. 2021).

However, subinertial processes vary strongly in spatiotemporal scales. Except for the large-scale ones described earlier, some processes are indeed subinertial, but have a time scale not much longer than the inertial period, or they are definitely of small Rossby number, but have a spatial scale comparable to the shelf width. For instance, the short-term (2–10 day) fluctuation of Gulf Stream transport induces coherent coastal sea level changes along the U.S. East Coast (Ezer 2016). In addition, the impingement of the warm-core ring on the MAB shelf break induces a fluctuation of the alongshore pressure gradient near the coast at a magnitude even close to its mean value, although the ring water does not really penetrate the shelf (Xu and Oey 2011). Indeed, as often seen from the Archiving, Validation and Interpretation of Satellite Oceanographic Data (AVISO, https://marine.copernicus.eu) sea level anomaly data (Fig. 1), eddy-like patterns can be frequently found on the MAB shelf or near the coast, with a size close to the rings that impinge on the shelf break.

The ATW theory or its extension on the *β* plane (e.g., Wise et al. 2018; Wu 2021) assumed a geostrophic balance in the cross-shelf direction, i.e., the semigeostrophic approximation. Under this approximation, the Laplacian of sea level *η* (i.e., ∇^{2}*η* ≡ *η _{xx}* +

*η*;

_{yy}*x*and

*y*are the cross-shelf and along-shelf directions, respectively) was reduced to

*η*only. Note, ∇

_{xx}^{2}

*η*represents the relative vorticity, and

*η*and

_{xx}*η*are the shears of along-shelf and cross-shelf velocities, respectively, for a geostrophic fluid. Dropping

_{yy}*η*, thus cross-shelf transport, is reasonable only when the cross-shelf scale is much smaller than the along-shelf scale, such that ∂

_{yy}^{2}/(∂

*y*

^{2}) ≪ ∂

^{2}/(∂

*x*

^{2}). However, this approximation is questionable if looking at a shelf that is

*wide*relative to the scale of the offshore forcing. For instance, the typical warm-core ring is

*O*(100) km in size, which is close to or even smaller than the MAB shelf width. When the subinertial forcing has a length scale close to the shelf width,

*η*has the same order of magnitude as

_{yy}*η*and therefore cannot be dropped. This issue was first pointed out by Hopkins (1982). Xu and Oey (2011) also emphasized that the cross-shelf transport (in associated with

_{xx}*η*) may pile up the sea level near the coast.

_{yy}Allowing time variation in the equations of motions allows the existence of coastal trapped waves. Semigeostrophic approximation was also made sometimes (e.g., Robinson 1964; Brink and Allen 1978; Pedlosky 1987; Wise et al. 2020). Presumption that the shelf wave propagates only in the along-shelf direction was also often adopted (e.g., Mysak 1980; Power et al. 1990; Chapman and Brink 1987). Solving the wave equation under this approximation, Chapman and Brink (1987) found that the shelf response is “nearly always barotropic” and decreases dramatically at very low frequency. Huthnance (2004) further emphasized that friction aids in signal transmission and that coastal gauges can be effective ocean monitors only at very large scales (typically thousands of kilometers).

However, in many circumstances the imposed pressure signal is neither steady (the ATW theory) nor propagating (the shelf wave theory). For instance, the interaction between a boundary current (e.g., the Kuroshio) and topographic irregularity may produce a finite-scale pressure signal fixed in location, but varying in time. Even for warm-core ring impingement in the MAB, only a small fraction of energy propagates freely to the downshelf (Chapman and Brink 1987), and at least in the initial stage, the imposed pressure anomaly varies in time, but is fixed in location. In this situation, the wave structure function can hardly be solved by the variable separation method, and the underlying physical characteristics remain unclear.

In this study, the shelf wave equation forced by a more general offshore forcing was solved analytically. With this method, the response of a wide shelf to finite-scale and nonpropagating offshore signals was investigated. In section 2, the development of the theory and the solution method will be presented in detail. The wave characteristics for a “standard” case will then be shown in section 3, and their dependency on different length scales, time scales, friction, and slope will be analyzed in section 4. The new solution will be compared with two previous theories in section 5. Finally, conclusions will be drawn in section 6.

## 2. Theory

### a. Formulation of the problem

*f*-plane, barotropic, and subinertial cases. The shelf is not necessarily a western boundary shelf, hence, the

*β*effect was not considered. Local wind effect was not considered too, which has been studied extensively. The barotropic approximation is acceptable because this study focuses on the response of sea level over the shelf, which by nature is barotropic (Chapman and Brink 1987). Some limitations of the above approximations will be discussed in section 6. The Rossby number was assumed to be small, thus the controlling equations are

*r*. The domain and coordinates can be seen in Fig. 2. The terms

*u*,

*υ*,

*h*, and

*η*are cross-shelf velocity, along-shelf velocity, water depth, and sea level, respectively. Assuming the time scale is longer than the inertial period, the length scale of offshore forcing is comparable to the shelf width, and the frictional Ekman number is small, the following vorticity equation can be derived by eliminating

*u*and

*υ*in (1):

*s*≡

*dh*/

*dx*is the slope. The derivation of (2) can be found in the appendix A. Equation (2) can be reduced to the equations used in the ATW theory (Csanady 1978), the free shelf wave theory (e.g., Mysak 1980), or the frictional shelf wave theory (e.g., Power et al. 1990; Wise et al. 2020) by dropping one or several terms such as tendency, friction, or

*η*. Because the interest of this study was in the dynamics on a wide shelf with forcing scale close to shelf width, all terms were retained. The

_{yy}*s*was assumed constant for simplicity. Throughout this study, the discussion is restricted to the

*f*plane, but including the

*β*term will not increase the mathematical complexity.

*i*

^{2}= −1,

*ω*≡ 2

*π*/

*T*, and

*T*is the time scale of the offshore signal. Substituting (3) into (2) results in an equation for the wave structure:

*y*should be set at +∞. In this case, because topographic wave propagates to the downshelf, surface disturbance must decay to zero when

^{p}*y*→ +∞. On the other hand, if the shelf is forced by an upshelf forcing,

*y*is finite and

^{p}*y*= +∞. Downshelf boundary condition is not set due to the propagation of topographic wave, which was similar to the arrested topographic wave theory (Csanady 1978).

^{p}Equation (4) is an elliptical equation with complex and nonconstant coefficients subject to the above boundary conditions. Equation (6b) is derived because the cross-shelf velocity *u* vanishes at the coast; for details see appendix B. The term *x* only, which can be solved with associated methods (e.g., Chapman and Brink 1987; Huthnance 2004). However, if *y*. In this case, *x* and *y* as a prior. Please also note, setting a clamped boundary condition

### b. Nondimensionalization and scaling analysis

*B*is the shelf width and

*L*is the length scale of the signal imposed at the shelf edge. This yield

*R*is the Rossby radius at the shelf break. Note that because (8) is linear in

_{d}*ϑ*=

*O*(1),

*ϑ*,

*σ*, and

*κ*can be interpreted in two ways. On one hand, it is the ratio between

*r*/(

*sf*) and

*B*. The former represents “conductivity” as suggested by Csanady (1978). It means the length scale at which the bottom friction can diffuse the pressure signal shoreward, in a manner analogous to heat conduction. Therefore,

*κ*is the conductivity normalized by shelf width, or

*shelf conductivity*in short. On the other hand, because

*sB*is simply the water depth

*H*at the shelf edge,

*κ*can be interpreted as the Ekman number at the shelf break. Because the processes of interest involve the transmission of an open ocean signal across the shelf, it is more reasonable to consider

*κ*as a conductivity, which represents an essential property of a shelf.

It is easy to see that terms associated with *ϑ* ∼ 1, because the subinertial approximation requires *σ* < 1 and *O*(1) for a reasonable shelf. This indicates that for an imposed signal with a scale close to the shelf width, the solution of (4) is essentially determined by two parameters associated with the forcing, *ϑ* and *σ*, and one parameter that essentially reflects a shelf property, *κ*. Whereas, for *ϑ* ≪ 1, the 3–5 terms in (8) could have the same order of magnitude. Therefore, a complete solution to (8) should keep all terms.

### c. Solution method

Equation (4) was solved in the dimensional form to maintain its intuitive clarity. The solution method is the integral transform detailed in Özısık (1993) and Johnston (1994), which converts the problem to a Sturm–Liouville problem and calculates the eigenvalues and eigenvectors altogether. Wise et al. (2020) introduced this method in their paper with some improvements. In this study, Wise et al.’s method was extended to include the

The following discussion includes rather lengthy, but necessary mathematics. Readers who are only interested in the final solution can go directly to (29).

*ψ*= cos(

_{j}*k*), with

_{j}x*j*= 1, 2, …. According to the Sturm–Liouville theorem, {

*ψ*} form a complete set of orthogonal basis functions such that

_{j}*δ*

_{i}_{,}

*is the Kronecker delta function and*

_{j}*ψ*=

^{N}*x*

_{in}/2. Note, the index "i" that appears in the subscript is not the imaginary unit. By substituting (16) into two to five terms of (11), multiplying the resulting equation with

*ψ*(

_{i}*x*), integrating it over

*x*and dividing it by

*ψ*, applying integration by parts on the first term, and then writing the result in the matrix form, the following second-order linear system can be obtained:

^{N}**Λ**

^{−1}, where

**Λ**was diagonal with each element

*λ*being an eigenvalue of

_{j}**q**

*of*

_{j}*y*) =

**Λ**

*y*)

^{−}^{1}and

^{−1}(

*y*) =

*y*)

^{−1}. Equation (28) can therefore be rewritten as

*K*in (29) is the number of integral kernel functions, or in other words, the number of wave modes. The solution becomes exact as

*K*→ +∞. The summation range in (29) is 2

*K*instead of

*K*because the size of

*K*× 2

*K*, and therefore it has 2

*K*eigenvalues.

For the second-order linear system (18), half of the eigenvalues have a positive real part (denoted as a positive eigenvalue for short, or *λ*^{+}), and half have a negative real part (denoted as a negative eigenvalue for short, or *λ*^{−}). The exponentials in (30) thus decay in +*y* and −*y* directions, respectively, for *λ*^{−} and *λ*^{+}. Physically, this arises because when the cross-shelf friction is included [see Eq. (1a)], the cross-shelf current *u* is slowed down due to the bottom friction, which decreases the magnitude of *dη*/*dy*. Consequently, *η* contours “diffuse” in both +*y* (upshelf) and −*y* (downshelf) direction, in associated with negative and positive eigenvalues, respectively. The contribution of eigenvalues is far more than diffusion, as we will see in sections 3 and 4.

### d. Offshore forcing

*hits*the shelf break,

*η*

_{0}was the amplitude and was normalized to 1 throughout the paper;

*l*≡ 2

*π*/

*L*, i.e., the “wavenumber” of the oceanic pressure anomaly (but not a true wavenumber, because it was not a propagating wave). To simplify the dynamics discussion and mathematical derivation, we defined a periodic offshore forcing in the

*y*direction in (−∞, +∞). To derive an explicit solution, we first set the upshelf boundary at

*y*=

*y*, temporarily, to obtain a solution that was dependent on

^{p}*y*; then let

^{p}*y*→ +∞, so as to get the final solution as follows:

^{p}### e. Structure of the solution

#### 1) Based on the forcing structure

*η*/∂

*t*term in (1c) or the

*η*term in (2). The ratio of this part to

*η*/∂

*t*term in (1c), is an appropriate approximation (e.g., Wise et al. 2020). For completeness, the

#### 2) Based on eigenvalue

*λ*):

## 3. Characteristics of shelf response to offshore signal: A standard case

The following discussion explores the nature of shelf response under various specified parameters, which are detailed in Table 1. To begin with, a standard case (C0) was configured, with the shelf width *B* of 200 km, the shelf break water depth *H* of 150 m, the friction *r* of 0.0005 m s^{−1}, and the Coriolis parameter *f* of 10^{−4} s^{−1}, corresponding to a wide and gently sloping shelf. The pressure signal time scale *T* was 15 days, and its along-shelf scale *L* was 200 km. The nondimensional parameters defined in (9) were *ϑ* = 1, *κ* ≈ 0.03, *σ* ≈ 0.05, and *K* in (29) was set to 20, which was sufficient to resolve the dynamics.

List of experiments.

### a. Wave characteristics

*T*, 0.125

*T*, 0.25

*T*, 0.375

*T*, and 0.5

*T*are shown (Figs. 3a–e). The current was calculated by (1a) and (1b) using the fact that the time scale (15 days in this case) was much longer than the Ekman spindown time (∼2.5 days in this case), such that

*α*≡

*r*/(

*fh*) is the local Ekman number. Clearly, the shelf responded strongly to the offshore signal. The pressure signal intruded into the shelf at an

*incidence angle*(defined in Fig. 1). Please note, the term “incidence” here means the shoreward prorogation of offshore signal after penetrating the shelf break, i.e., the sea surface height signal in 100–200 km in Fig. 3a. It does not refer to the impingement of interior ocean signal toward the shelf break. The downshelf-inclined incidence shoreward of the shelf edge is apparently associated with shelf slope and friction. Near the coast, there was a sea level setup with a magnitude of ∼0.1 of the signal amplitude (Fig. 3a). Note, hereinafter, the sea level

*η*was normalized by the amplitude of offshore signal

*η*

_{0}. The current was strong near the shelf edge, but became weak near the coast. The crests of sea level were nearly orthogonal to the coast and propagated along the coast. This is similar to the edge wave (Ursell 1952; Munk et al. 1956; Yankovsky 2009), although the latter is often at superinertial frequencies.

An isolated sea level high or low appeared when the offshore pressure anomaly returned to zero in 0.125–0.375*T* (Figs. 3b–d). Under geostrophic adjustment, this sea level anomaly caused cyclonic or anticyclonic rotation and behaves as an eddy. The eddy had a cross-shelf size of ∼0.5*B* and an along-shelf size of ∼0.5*L*. The eddy moved offshore gradually, indicating a reflection nature from the previous incident signal. Meanwhile, sea level in both the coastal region and the shelf edge was close to zero when the eddy was maturely developed (at time of 0.25*T*), indicating that the eddy had a ∼*π*/2 phase lag relative to the external signal. Then, at 0.5*T*, the eddy disappeared and the incident wave turned to the opposite phase (Fig. 3e). The above coastal setup and reflected eddy were similar to those observed in the MAB (see Fig. 1a; Xu and Oey 2011).

The horizontal wave structure function

### b. Wave decomposition

As described in section 2e, there are two ways to decompose the wave solution. First,

The terms *λ*^{+} and negative eigenvalues *λ*^{−}, respectively. The mode number was determined by counting the zero-crossings. The eigenvalues are shown in Fig. 5. For the positive eigenvalues, mode 1 for both *superinertial* edge wave modes (Ursell 1952; Yankovsky 2009) or the *subinertial f*-plane shelf wave modes (Pedlosky 1987), indicating the effect of sloping topography. The phases of positive eigenvalue modes are standing in general. However, for higher modes (e.g., mode 2) there was also a cross-shelf phase velocity around the internal zero-crossings (Figs. 4f,h), which was different from the inviscid *f*-plane shelf wave, but similar to the solution in Wise et al. (2020) who studied a frictional shelf wave. The *π*/2. The positive eigenvalue mode 1 for

The negative eigenvalue modes were essentially from the *T* when the offshore pressure anomaly returned to zero.

### c. Eigen values

*λ*

^{+}and

*λ*

^{−}are shown in Fig. 5. The ℜ(

*λ*

^{+}) and ℜ(

*λ*

^{−}) both increased in magnitude with mode number. The ℑ(

*λ*

^{+}) increased in magnitude from ∼0 for mode 1 to ∼−1.5 × 10

^{−5}for mode 10, whereas ℑ(

*λ*

^{−}) was ∼−4.2 × 10

^{−5}for mode 1 and dropped in magnitude to ∼−2.0 × 10

^{−5}for mode 10. The ℜ(

*λ*) is associated with a decay rate. The imaginary part ℑ(

*λ*) signifies the downshelf propagation velocity. This can be understood by observing the wavenumber in the

*y*direction:

*φ*is the phase associated with eigenvalue

_{j}*λ*and only the wave part

_{j}*λ*) < 0. Therefore, ℑ(

_{j}*λ*) determines the propagating nature of the wave in the −

*y*direction. If ℑ(

*λ*) ∼ 0, the wave is standing, which was the case for low

*λ*

^{+}modes (Fig. 5a). Otherwise, if ℑ(

*λ*) < 0, the wave propagates in the −

*y*direction. Hence, the eigenvalue shown in Fig. 5 explains why the wave associated with

*λ*

^{−}was propagating, but that associated with

*λ*

^{+}was standing.

One strange feature was that mode 5 with *λ*^{−} disappeared. Instead, there were two modes 4 (Fig. 5c). Such a disappearance of one or several modes, even mode 1, and a duplication of other modes was seen frequently even the parameters were changed.

## 4. Parameter dependency

This section investigates the dependency of shelf response on the length scale and time scale of the offshore forcing, and the friction and slope of the shelf. Based on previous analysis, times of 0*T* and 0.25*T* are associated with incident wave and the reflected eddy, respectively. The parameters considered are listed in Table 1.

### a. Incident wave

#### 1) Response to length scale

The incident wave varied strongly under different parameters. At short signal scale (100 km, case C1), the offshore signal propagated nearly orthogonal to the coast, i.e., at an incidence angle of ∼90° (Fig. 6a). The coastal sea level setup was much larger than that with the forcing scale of 200 km (case C0). Physically, this arises because for a given magnitude *η*_{0}, shorter length scale produces a larger along-shelf gradient of the sea surface level, thus a stronger cross-shelf current and a larger incidence angle. At larger signal scale (400 km, case C2), the incidence angle was much smaller, and the coastal setup was much weaker (Fig. 6b). This is understandable, as suggested by (8), when the along-shelf scale *L* is large compared with the shelf width *B*, *ϑ* (≡*B*/*L*) is small thus the term *ϑ*. In addition, the high setup is essentially determined by the

#### 2) Response to time scale

In Fig. 6b, the time scale of the offshore signal was decreased to 5 days (Case C3), and the incident wave characteristics became very different. The initial propagation of the pressure signal was to the upshelf. As the time scale approached the inertial period 1/*f*, Poincaré wave characteristics emerged, and the coastal setup increased dramatically (Fig. 7b). These results were intuitively correct because if the time scale were further decreased to tidal frequency, such a feature would be apparent. However, this result should be treated conservatively because the derivation of (2) assumed *σ* < 1 and in principle precluded the superinertial motions. Nevertheless, because the spatiotemporal spectra of ocean motion are continuous, it can be expected that relatively high-frequency subinertial motion (with *ω* only marginally less than *f*) should feature some characteristics similar to superinertial motions. However, when the period of offshore forcing was longer than ∼7 days, the coastal setup became stable. This can also be seen from Figs. 6d and 3a, where the incident waves with time scales of 15 and 60 days (case C4) were essentially the same.

#### 3) Response to friction and shelf slope

Decreasing the bottom friction (Fig. 6e, case C5) or increasing the shelf slope (Fig. 6h, case C8) at the same rate gave identical results, that the shoreward incidence of the offshore signal was strongly weakened. In contrast, increasing the bottom friction (Fig. 6f, case C6) or decreasing the shelf slope (Fig. 6g, case C7) both amplified the shoreward incidence. The role of friction in breaking the PV constraint and transmitting the oceanic signal to the coast was highlighted by Huthnance (2004). In the nondimensional analysis performed in this study, the solution of (2) is dependent on the nondimensional parameter *κ* ≡ *r*/(*sBf*), which may explain the above results. Because *κ* only contains information about a particular shelf, in particular the shelf break depth (i.e., *sB*), it could be used to compare the dynamics on different shelves.

### b. Reflected eddy

Most cases showed an isolated eddy as in the standard case during the reflection phase (Fig. 8). In general, short length scale, short time scale, strong friction, or gentle slope produced a more significant eddy in the reflection. The intensity of the reflected eddy was generally proportional to the coastal setup (Fig. 6). However, a longer time scale (say, 60 days) reduced the eddy intensity, as shown in Fig. 8d, although the coastal setup was similar to that with a time scale of 15 days. This could have been expected because the reflected eddy can propagate more freely than the incident wave, and therefore, given a long time scale, its energy radiated downshelf, thus reduced the eddy intensity. The amplitude of each mode was no longer decreasing with mode number (Fig. 5d). As in the results discussed in section 3c, in many (in fact all) cases, some modes were missing, but some others were associated with multiple eigenvalues. The cause for this phenomenon deserves further investigation. One possible reason was because the negative eigenvalue modes were squeezed in the offshore half of the shelf, thus some zero-crossings were degenerated or merged.

More interesting observations can be noted. For instance, increasing the time scale in general decreased the amplitude of the low-mode reflected waves (the thick red solid line, thick black solid line, and thin red solid line in Fig. 5d), which is consistent with Figs. 3b, 8c, and 8d, that the reflected eddy became weaker with a longer time scale. Another feature was that the mode number tended to be large for a shelf with low friction or steep topography (Fig. 5d). For high-friction or gentle slopes, i.e., large *κ*, the incidence of the oceanic signal moved strongly toward the coast, causing a large coastal setup. Moreover, in this case, the reflected eddy energy was concentrated in the low modes, producing a large eddy. On the other hand, for small *κ*, whether associated with low friction or a steep slope, the incidence angle of the oceanic signal was much smaller, and the reflected energy was concentrated in the high modes, producing a small eddy.

### c. The incidence angle

*roughly*understood through the dispersion relationship. Immediately after the oceanic signal penetrating the shelf break, it has a wavenumber in the

*x*direction and also propagates in −

*y*direction. Therefore, the wave can be roughly described as

*k*can then be solved. Then the wave vector [ℜ(

*k*),

*l*] signifies the wave direction, and the direction of the wave crest line, i.e., the incidence angle, can be determined as

*θ*approaches 90° for small time scale (1/

*σ*) or small wavelength (1/

*ϑ*), which is consistent with previous results.

*l*and

*ω*satisfy the dispersion relationship for a frictionless topographic Rossby wave,

*ϑ*and 1/

*σ*). It indicates that

*l*and

*ω*satisfying (45) generate a vanishing cross-shelf wavenumber

*k*and therefore an incidence angle close to 90°, leading to a strong coastal setup. This conclusion is similar to that of Huthnance (2004). Nevertheless, it should be noted that the above analysis was only a coarse interpretation of the study results, because the wave on the shelf is not freely propagating, and hence the actual incidence angle is not exactly the same as that predicted by the dispersion relationship.

## 5. Comparison with other theories

This section compares the theoretical solution with other theories, to further see the influences of relaxing the semigeostrophic approximation.

### Theories with semigeostrophic approximation

*η*term in (2) vanishes. For instance, in Csanady’s (1978) ATW theory, (2) was reduced to

_{yy}*β*-plane approximation, which was not considered here. Wise et al. provided a highly skillful solution to (47). Equation (47) results in a linear system similar to (18), but without the second-order term. It should be emphasized that both Csanady (1978) and Wise et al. (2020) had no intention to discuss finite-scale subinertial processes. However, because attention was not always paid to whether the scale of forcing would violate the semigeostrophic approximation, it is helpful to evaluate the scale dependencies of these theories.

Figures 10 and 11 show the results from Csanady’s ATW theory, Wise et al.’s theory, and this study. Only the incident phase is shown. Csanady’s and Wise et al.’s theories were both unable to produce the reflected eddy (not shown). The settings are listed in Table 1 and in the figure captions. For a long length scale (600 km) and a relatively long time scale (10 days), all three theories gave very similar results (Figs. 10a–c). In fact, given a long length scale and a long time period, Eqs. (4), (46), and (47) became the same. When the length scale approached the shelf width, the ATW and Wise theories still gave similar results, although the latter intruded farther into the shelf (Figs. 10e,f). For the cases with a period of 5 days, this difference became more apparent (Figs. 11e,f), even for long length scale (Figs. 10a,b). For a short length scale, the theory developed in this study gave a very different result (Figs. 10g,h and 11g,h), suggesting that the *η _{yy}* term is very important for shelf response.

*η*, which induced negative eigenvalue modes and the

_{yy}*η*term essentially caused the strong coastal setup and the associated wave reflections.

_{yy}## 6. Summary

The shelf response to a finite length scale and finite time scale subinertial offshore pressure signal was investigated. The general barotropic shelf wave equation was formulated and solved analytically with an integral transform method. The solution was a series associated with complex eigenvalues, half of which had a positive real part (positive eigenvalue for short) and half had negative real part (negative eigenvalue for short).

The general response of a wide shelf to an offshore pressure signal with length scale close to the shelf width is as follows. First, the imposed pressure signal at the shelf edge incident toward the coasts, causing a coastal setup at a place only slightly downshelf of the offshore forcing. For this reason, the large insulating region predicted by the ATW theory disappears. The incident wave is reflected back by the coast and the sloping bottom topography, forming an isolated eddy. The reflected eddy propagates both offshore and downshelf. Such a process is sketched in Fig. 2. This process could not be revealed by previous shelf wave models that adopted the semigeostrophic approximation.

Scaling analysis indicated that three nondimensional parameters are essential for the solution structure, i.e., the ratio between the shelf width and the forcing scale, *ϑ* ≡ *B*/*L*, the ratio between the inertial period and the forcing time scale, *σ* ≡ *ω*/*f*, and the “shelf conductivity” *κ* ≡ *r*/(*sfB*). The *ϑ* and *σ* are associated with forcing, whereas *κ* is solely related to the shelf property, which signifies the capability of friction in breaking the PV barrier and transmitting the pressure signal shoreward. The results indicated that coastal sea level setup increases for larger values of *ϑ* in a nearly linear manner, suggesting that the length scale of forcing is of essential importance. For a forcing time scale approaching 1/*f*, the shelf response behaves more or less as a plane Poincaré wave, say, tide, with strong coastal setup. For time scales longer than ∼7 days, the incident wave solutions are similar, but the reflected eddy becomes weaker. For a long length scale and a long time scale, the results of this study are similar to previous theories. The dispersion relationship can largely explain these dependencies.

Shelf conductivity, *κ*, was found to be essential. For high friction or a gentle slope, i.e., large *κ*, the incidence of the oceanic signal is strongly toward the coast, causing a large coastal setup. Moreover, in this case, the reflected wave energy is concentrated at low modes, producing a large eddy. In contrast, for small *κ*, associated either with low friction or steep slopes, the incidence angle of the oceanic signal is much smaller and toward the downshelf, and the reflected wave energy is concentrated in the high modes, producing a small eddy.

There are some limitations of the proposed theory. A periodic offshore signal was assumed to reduce the mathematical difficulty. As a consequence, the decaying downshelf propagation of reflected signal could not be well resolved. For an isolated signal at the shelf break, one solution could be to apply a Fourier expansion on *β* effect affecting the propagation of finite-scale offshore signal on a wide western boundary shelf deserves further investigations. The shelf was assumed to be barotropic in this study. Indeed, as pointed out by Chapman and Brink (1987), the shelf response is largely barotropic. However, later evidence has indicated that baroclinicity would enhance cross-shelf exchange (e.g., Yang et al. 2011; Zhang and Partida 2018), in particular the onshore intrusion (Cherian and Brink 2018). Hence, it is very interesting to see how a baroclinic shelf responds to offshore forcing. This study indicated that the shelf response to offshore forcing is highly scale dependent, which could be a starting point to further investigate more complicated processes in wide shelves.

## Acknowledgments.

The author appreciates the fruitful conversions with Prof. Hailong Yuan on the mathematical technique and Prof. Xiaopei Lin on interpreting the physics. The constructive comments received from Editor Joseph LaCasce, Editor Baylor Fox-Kemper, and three anonymous reviewers have substantially improved the manuscript. This study was financially supported by the National Key Research and Development Program of China (Grant 2022YFA1004401), the Innovation Program of Shanghai Municipal Education Commission (Grant 2021-01-07-00-08-E00102), and the Science and Technology Commission of Shanghai Municipality (21JC1402500).

## Data availability statement.

The sea level anomaly and absolute dynamic topography used in Fig. 1 are from the Archiving, Validation and Interpretation of Satellite Oceanographic (AVISO) data, which are now distributed by the Copernicus Marine Service (https://marine.copernicus.eu).

## APPENDIX A

### Derivation of (2)

*h*(1b)/∂

*x*] − [∂

*h*(1a)/∂

*y*] results in

*h*(1a)/∂

*x*] + [∂

*h*(1b)/∂

*y*] and using (1c) leads to

*f*× (A1) + (∂/∂

*t*)(A2) and rearranging the resulting equation, we have

*L*∼

*B*, and the elevation is scaled by the geostrophic current to the first order of accuracy. Substituting (A4) into (A3), using the notation

*σ*≡ 1/(

*fT*),

*α*≡

*r*/(

*fH*) yields

*σ*< 1,

*α*≪ 1, and

*α*/

*σ*can be of

*O*(1). Here we set

*σ*< 1 and

*σ*

^{2}≪ 1, which means the motion is subinertial, but its period can be only marginally longer than the inertial period 1/

*f*, thus the first-order time derivative in the lhs of (A5) or (A2) is retained. Hence, dropping terms in (A5) associated with

*α*, and

*σ*< 1 and

*α*≪ 1, (1a) and (1b) can be scaled to

*α*≪ 1 is questionable near the coast where the water depth is vanishing. However, because the current decays rapidly near the coast, the above approximation will not unduly affect the dynamics.

## APPENDIX B

### Derivation of (6b)

*u*vanishes, so (1a) and (1b) become

*h*→ 0 for

*x*→ 0, we have

## APPENDIX C

### Derivation of (32) and a Novel Treatment on the Upshelf Boundary Condition

*m*is an integer, then substitute (C1) into (30c)–(30e), after lines of algebra (mainly through integration by parts), we can have

*γ*and

_{j}*ε*are determined by the integral kernel function

_{j}*ψ*(

_{j}*x*) and the coefficient function

*a*(

*x*), which are related to the shelf geometry, roughness, and offshore forcing characteristics.

*α*, by contrary, is related to the upshelf boundary condition

_{j}*y*), see Eq. (30f). The

^{p}*y*) includes both

^{p}*y*) cannot be given arbitrarily. One way to relax this difficulty is to give one of these two, say

^{p}*α*, so the final solution is physically reasonable.

_{j}*M*,

*N*, and

*P*are all finite, (C6) turns to

*λ*

^{−}. In all of our calculations,

*α*is small (but not zero) for an offshore forced shelf wave, because the large

_{j}*α*} associated with

_{j}*λ*

^{−}are determined. Then, based on (30f),

*K*equations, but only

*K*unknowns, i.e.,

*α*} associated with

_{j}*λ*

^{−}(C8) can be used to solve for

*α*are determined. Then according to the constraint (C7), the sum of exponential terms associated with negative eigenvalue

_{j}*η*

_{0}and independent variables were dropped.

*λ*(

_{j}*y*−

*y*)], decaying exponentially for (

^{p}*y*−

*y*) → −∞. Hence, the imperfectness of

^{p}*m*in (C3) approach infinity, i.e., let

*y*→ +∞. Hence, for finite

^{p}*y*, the exp[

*λ*(

_{j}*y*−

*y*)] in (C11) vanishes, because ℜ (

^{p}*λ*) > 0. Therefore,

_{j}## APPENDIX D

### Derivation of (41)

*y*, the wavenumber in the

*y*direction is

*d*arctan(

*x*)/

*dx*= 1/(1 +

*x*

^{2}) and

*d*tan(

*x*)/

*dx*= sec

^{2}(

*x*).

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