SEARCH

SEARCH BY CITATION

Keywords:

  • internal tide;
  • internal wave;
  • physical oceanography;
  • tide

1. Introduction

  1. Top of page
  2. 1. Introduction
  3. 2. Momentum Equations and Pressure Decomposition
  4. 3. Interpretation of Tides in a Wedge Connected to a Flat Abyss
  5. 4. Summary
  6. References

[1] Gerkema [2011] objects to Kelly et al.'s [2010] conclusion that internal-tide pressure has zero depth average over a sloping bottom. This objection is based on the unfounded assumption that only internal-tide pressure can exist in a wedge under a rigid lid. Here, we refute this supposition, arguing that tides in a wedge contain mode 0 pressure because surface and internal tides are coupled through wave drag. Moreover, we note that Gerkema's [2011] definition of internal-tide pressure is incompatible with the accepted baroclinicity condition over a flat bottom, which requires the internal tide have zero depth average [Kunze et al., 2002; Gerkema and van Haren, 2007]. Specifically, the pressure field illustrated in Figure 1 of Gerkema [2011] cannot be matched to the known internal-tide solution over a flat abyss without producing unphysical discontinuities. In the following reply, we compare the internal-tide pressures and energy fluxes of Kelly et al. [2010] and Gerkema [2011] for tides in a wedge connected to a flat abyss [Wunsch, 1968]. We conclude that the internal-tide pressure proposed by Gerkema [2011] is incorrect and his objections to Kelly et al. [2010] are unfounded.

2. Momentum Equations and Pressure Decomposition

  1. Top of page
  2. 1. Introduction
  3. 2. Momentum Equations and Pressure Decomposition
  4. 3. Interpretation of Tides in a Wedge Connected to a Flat Abyss
  5. 4. Summary
  6. References

[2] Surface and internal-tide velocities are U = 〈u〉 and u′ = u − 〈u〉, respectively, where angle brackets indicate a depth average [Kunze et al., 2002; Gerkema and van Haren, 2007]. Depth averaging the momentum equation and calculating its residual produces the surface and internal-tide momentum equations

  • equation image
  • equation image

respectively, where D is wave drag [Bell, 1975], P depth-average pressure, and p′ residual pressure (i.e., p′ = p − 〈p〉 [Kelly et al., 2010]).

[3] Under a rigid lid, Kelly et al. [2010] identify p′ as internal-tide pressure [see also Kunze et al., 2002]. They motivate this definition by noting that p′ accounts for wave drag, is consistent with the baroclinicity condition over a flat bottom [Kunze et al., 2002; Gerkema and van Haren, 2007], and produces physically relevant energy fluxes [see Kelly et al., 2010, Figure 2].

[4] For the case of tides in a wedge, Gerkema [2011] begins by assuming that flow is “purely baroclinic,” trivializing the definition of internal-tide pressure, which must be known a priori to classify the flow. While Gerkema [2011] repeatedly states that only internal-tide pressure may exist in a wedge (for example, by referring to the flow as an “exact internal-wave solution” and stating that “the physical setting here precludes barotropic tides; the flow field is purely baroclinic”), these assertions are not supported by scientific reasoning. To illustrate this, we consider the physics of Gerkema's [2011] decomposition. For tides in a wedge, where U = 0, Gerkema [2011] implies that the internal-tide momentum equation is

  • equation image

where internal-tide pressure, p, is also total pressure. This equation is impractical for describing observations because it neglects wave drag, which is necessary for accurate tidal prediction [Jayne and St. Laurent, 2001; Arbic et al., 2010].

3. Interpretation of Tides in a Wedge Connected to a Flat Abyss

  1. Top of page
  2. 1. Introduction
  3. 2. Momentum Equations and Pressure Decomposition
  4. 3. Interpretation of Tides in a Wedge Connected to a Flat Abyss
  5. 4. Summary
  6. References

[5] To compare the internal-tide pressures and energy fluxes of Kelly et al. [2010] and Gerkema [2011], we extend Gerkema's [2011] discussion of tides in an infinite wedge to include a flat abyss [Wunsch, 1968]. The inclusion of the abyss serves the following three purposes: (1) it physically bounds the depth of the fluid, (2) it better represents the real ocean, and (3) it allows us to verify the compatibility of the solutions over the slope with the solution over the abyss, which is not in question [Kunze et al., 2002; Gerkema and van Haren, 2007].

[6] In the two-part domain, the stream function for upslope-propagating tides is matched at the wedge/abyss transition using Cauchy boundary conditions [Wunsch, 1968]. Following Gerkema [2011], we obtain velocity from the vertical derivative of the stream function, and pressure from the indefinite integral of the horizontal momentum equation [Gerkema, 2011, section 3.1]

  • equation image

so that pressure depends on a constant of integration, c0, which was not specified by Wunsch [1968]. Gerkema [2011] implicitly set c0 = 0, but here we consider other values, which are equally valid, and affect p but not p′ [Kelly et al., 2010]. Specifically, we use c0 ≠ 0 to match depth-average pressure over the wedge/abyss transition, ensuring that pressure and energy flux are continuous. The results of these calculations are shown in Figure 1.

image

Figure 1. Internal-tide pressure (at t = 0) and tidally averaged energy flux for tides in a wedge connected to a flat abyss (at x = 8.6) (c = 0.6, γ = 0.5, and n = 1 [Wunsch, 1968; Gerkema, 2011]). (a) Depth average of p′ is zero over the flat abyss. (b) Depth average of p (with c0 = 0) is nonzero over the flat abyss (highlighted in red), violating flat bottom tidal dynamics. (c) Depth average of p (with c0 ≠ 0). (d–f) Depth structure of internal-tide pressures. (g) Energy flux up′ indicates energy flux in the direction of wave propagation. (h) Energy flux up (with c0 = 0). (i) Energy flux up (with c0 ≠ 0) indicates some regions where energy flux opposes the direction of wave propagation (i.e., downslope at x = 2).

Download figure to PowerPoint

3.1. Application of Kelly et al. [2010]

[7] When the wedge is connected to an abyss, the depth average of p′ is zero over the flat abyss (Figures 1a and 1d), in accordance with the accepted baroclinicity condition [Kunze et al., 2002; Gerkema and van Haren, 2007]. A consequence of this decomposition is that all energy flux is in the direction of wave propagation (i.e., upslope, Figure 1g).

[8] Descriptively, the internal tide produces wave drag, which induces a surface-tide pressure gradient. Because surface-tide velocity is zero, this coupling does not transfer momentum or energy to the surface tide (as incorrectly suggested by Gerkema [2011]). Therefore, energy conversion (C = U · D) from the surface to internal tide is zero, and all energy associated with the internal tide propagates up the slope (Figure 1g).

3.2. Application of Gerkema [2011]

[9] In a wedge connected to a flat abyss, we illustrate two physical inconsistencies in the decomposition of Gerkema [2011], which arise from the requirement that pressure be continuous at the wedge/abyss transition. First, we reproduce the solution of Gerkema [2011] by setting c0 = 0 (Figures 1b, 1e, and 1h). In this case, p is identical to that of Gerkema [2011] within the sloped region, but has nonzero depth average over the flat abyss (Figure 1b), violating the accepted baroclinicity condition in this region [Kunze et al., 2002; Gerkema and van Haren, 2007]. Thus, defining p as internal-tide pressure when c0 = 0 is incompatible with flat bottom tidal dynamics.

[10] Next, we examine a solution where p is made to be consistent with flat bottom tidal dynamics. Here, c0 is chosen so that internal-tide pressure has zero depth average over the abyss (Figures 1c, 1f, and 1i). However, over the slope, this alternative solution produces regions of energy flux that are opposite the direction of wave propagation (i.e., downslope, Figure 1i at x = 2). The correct vertical distribution of energy flux must be upslope everywhere because the slope is subcritical. A nondivergent function cannot be used to correct up (as suggested by Gerkema [2011]), because it breaks the potential symmetry of the problem. As the abyssal boundary is open, the full domain can be taken to include a symmetric wave source and a second slope that mirrors the one considered here. In such a domain, adding a horizontally invariant depth profile of energy flux would produce unphysical asymmetry at the second slope. Thus, choosing p to be consistent with normal modes over a flat bottom produces unphysical energy flux in the wedge.

[11] Since c0 and the location of the abyss are unspecified parameters in Wunsch's [1968] solution, Gerkema's [2011] definition of internal-tide pressure should be physically relevant for all possible values. However, Gerkema's [2011] decomposition fails to meet this criteria because it sometimes produces p with nonzero depth average over the abyss and/or regions of energy flux that oppose the direction of wave propagation.

4. Summary

  1. Top of page
  2. 1. Introduction
  3. 2. Momentum Equations and Pressure Decomposition
  4. 3. Interpretation of Tides in a Wedge Connected to a Flat Abyss
  5. 4. Summary
  6. References

[12] Here we have examined Kelly et al.'s [2010] and Gerkema's [2011] definitions of internal-tide pressure in a wedge connected to a flat abyss. We have found that Gerkema's [2011] definition of internal-tide pressure: (1) does not allow for wave drag, (2) is incompatible with the accepted baroclinicity condition over a flat bottom, and (3) does not produce physically relevant energy flux. Conversely, Kelly et al.'s [2010] definition of internal-tide pressure is free from these problems. We therefore conclude that Gerkema's [2011] objections to Kelly et al. [2010] are unfounded, and internal-tide pressure is most accurately defined by Kelly et al. [2010]. Contrary to the conclusions of Gerkema and van Haren [2007] and Gerkema [2011], horizontal pressure gradients are not necessary for computing depth profiles of energy flux. The method for calculating energy fluxes proposed by Kunze et al. [2002] (and modified by Kelly et al. [2010]) remains the most accurate means of assessing energetics from conductivity-temperature-depth/lowered acoustic Doppler current profiler (CTD/lowered ADCP) data.

References

  1. Top of page
  2. 1. Introduction
  3. 2. Momentum Equations and Pressure Decomposition
  4. 3. Interpretation of Tides in a Wedge Connected to a Flat Abyss
  5. 4. Summary
  6. References