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  • View in gallery
    Fig. 1.

    (top) The Rossby number and base-10 logarithm of the Richardson number, Rog and log(Rig), in the Gulf Stream from a section taken during the winter as part of the CLIMODE experiment. (bottom) The ageostrophic shear from the same section with potential density (black contours) superimposed at increments of 0.2 kg m−3. Ageostrophic shears were calculated by assuming the cross-stream shear ∂υa/∂z was all ageostrophic and the along-stream shear was equivalent to the total shear minus the thermal wind shear, ∂ua/∂z = ∂u/∂z + f−1b/∂y. They show the coherent phase structure typical of a downward propagating internal wave, yet with slopes that run parallel to isopycnals. These shears coincide with regions of low Rig suggesting that baroclinic effects may be important in the wave dynamics. The wave shear occupies a relatively small part of the section, that is, about 30 km, so the velocities are not backrotated since they were observed over just several hours. These observations are from survey S1 in Inoue et al. (2010).

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    Fig. 2.

    The geostrophic velocity (shading) and potential density (contoured at 0.2 kg m−3 increments) for the idealized background flow given by (a) Eq. (16) and (b) the observations from the Gulf Stream.

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    Fig. 3.

    (top) log10 (Rig) and (bottom) Rog for the idealized background flow (16). Potential density contours with an increment of 0.2 kg m−3 are superimposed. These parameters are O(1) in the idealized jet similarly to the winter Gulf Stream as shown in Fig. 1.

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    Fig. 4.

    The frequency of oscillation of fluid parcels displaced at an angle θδ with respect to the horizontal, for a basic state without a background flow (Rog = 0, Rig = ∞, dashed), with a barotropic flow (Rog = −, Rig = ∞, solid) and with a baroclinic flow (Rog = 0, Rig = , dot-dashed). Baroclinicity both lowers the minimum frequency and shifts the angle θδ, where this occurs to coincide with the angle that isopycnals make with the horizontal θb.

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    Fig. 5.

    Rays, which run parallel to characteristics, illustrate the downward propagation of a wave packet with a frequency (a),(c) 0.95f and (b),(d) 0.98f in both (left) the idealized flow and (right) the Gulf Stream. Baroclinic effects expand the region where rays are confined from ωF (inside the green contour) to ωωmin (inside the magenta contour) and change the angle at which rays reflect from horizontal to the isopycnal angle. Note that there are regions where the contour marking ω = ωmin runs parallel to isopycnals. In these regions, the conditions for a slantwise critical layer are satisfied: multiple rays converge, the group velocity goes to zero, and waves are trapped. In the Gulf Stream observations, these slantwise critical layers are coincident with the region of high observed ageostrophic shear, as shown in Fig. 1.

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    Fig. 6.

    Schematic illustrating the physics of the minimum frequency for NIWs in geostrophic flows. Lines of constant geostrophic absolute momentum, Mg = ugfy (solid gray), and constant buoyancy bg (dashed black) are plotted in each frame. For (a) a flow with no relative vorticity or baroclinicity, (b) a barotropic flow with anticyclonic relative vorticity, (c) a baroclinic flow with only a vertical thermal wind shear, and (d) a flow with zero PV. The minimum frequency occurs when oscillations are along lines of constant buoyancy. When the gradient of Mg along bg is weaker than that shown in (a), the restoring force is reduced and the frequency is lower than f. In (d) this gradient is zero, hence there is no restoring force and ωmin = 0.

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    Fig. 7.

    The separation between the classical and anomalously low frequency regions for a wave with frequency ω = 0.95 in our idealized background flow and the slopes of the wave’s characteristics (top) λ+ and (bottom) λ. The checkerboard pattern indicates that the parcel oscillations along that characteristic are unstable with respect buoyancy. The wavy line pattern indicates that parcel oscillations are unstable with respect to momentum. The green contour is where ω = F; the magenta where ω = ωmin. The basic physics of parcel oscillations in these two quasi-unstable regimes is illustrated in Fig. 9.

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    Fig. 8.

    Schematic of characteristics ξ+ and ξ: There are three regimes depending on the frequency (left) above the effective inertial frequency F, (middle) at the effective inertial frequency, and (right) below the effective inertial frequency. Waves with ω < F are defined to exist in the anomalously low frequency range. If the sign of the isopycnal slope is switched, that is, θb < 0, the signs of the characteristic slopes are switched as shown in Fig. 7.

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    Fig. 9.

    Schematics of parcel displacements in a mean flow with baroclinicity and force diagrams illustrating how the components of the Coriolis force, , and buoyancy force, Fb = N2Y[tan(θb) − tan(θδ)], combine to make up the net force in the direction of displacement Fδ: (top) the classic range of parameter space where θb < θδ < θM; however, in a certain range of parameter space, parcel displacements may be (middle) shallower than buoyancy surfaces or (bottom) steeper than geostrophic absolute momentum surfaces—yet an oscillation will still occur. In these cases, although either Fb or Fc is destabilizing, Fδ is restoring.

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    Fig. 10.

    The imaginary part of the quantity ua/υa, given by (32), for a wave with ω = 0.95f in the idealized background flow for the two characteristics: (a) ξ+ and (b) ξ. The magnitude of this quantity indicates the local ellipticity of the hodograph, whereas the sign indicates the direction of rotation with either depth or frequency. Not only does the ellipticity differ for the two characteristics but the sense of rotation switches sign when parcel oscillations are unstable to momentum, that is, steeper than geostrophic absolute momentum surfaces (see Fig. 7). (c) The polarization relation at ωmin as a function of Rog and and (d) the ellipticity of the hodograph at ωmin as a function of .

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    Fig. 11.

    (top) Rays (parallel to characteristics) in the idealized background flow for waves with ω = 0.95f. The variation of the magnitude of the group velocity (normalized by its maximum along each ray) is colored on the rays. (middle) A numerical solution to Eq. (51) with ω = 0.95f and a point source at y ≈ 0.75 km, z ≈ −100 m. This is a snapshot of the streamfunction in time and the dashed cyan lines are the same rays shown in the top panel. (bottom) Energy density associated with the numerical solution for Ψ shown in the middle panel and computed with (40). These figures confirm ray tracing is qualitatively similar to the numerical solution. High energy densities, velocities, and shears occur near turning points and slantwise critical layers (defined in text) that run parallel to isopycnals.

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    Fig. 12.

    (a) The expansion parameter, , is plotted for the Gulf Stream section discussed throughout the paper and first displayed in Fig. 1. Values of |ε| > 0.25 are found within the magenta contours. The magnitudes of ε are typical for strong western boundary currents during winter and higher values may be observed. (b) A comparison between the formula for minimum frequency (19) and the first two terms in its Taylor series approximation (54) is plotted. (c) The relative error due to neglecting baroclinicity and approximating is plotted, whereas (d) the relative error due to approximating (54) is shown. The effects of neglecting baroclinicity are more severe than making the Taylor series approximation in this case. However, the error induced by assuming ε is small is nonnegligible here and may be worse in other cases where Rig or Rog are lower.

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Near-Inertial Waves in Strongly Baroclinic Currents

Daniel B. WhittEnvironmental Earth System Science Department, Stanford University, Stanford, California

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Leif N. ThomasEnvironmental Earth System Science Department, Stanford University, Stanford, California

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Abstract

An analysis and physical interpretation of near-inertial waves (NIWs) propagating perpendicular to a steady, two-dimensional, strongly baroclinic, geostrophic current are presented. The analysis is appropriate for geostrophic currents with order-one Richardson numbers such as those associated with fronts experiencing strong, wintertime atmospheric forcing. This work highlights the underlying physics behind the properties of the NIWs using parcel arguments and the principles of conservation of density and absolute momentum. Baroclinicity introduces lateral gradients in density and vertical gradients in absolute momentum that significantly modify the dispersion and polarization relations and propagation of NIWs relative to classical internal wave theory. In particular, oscillations at the minimum frequency are not horizontal but, instead, are slanted along isopycnals. Furthermore, the polarization of the horizontal velocity is not necessarily circular at the minimum frequency and the spiraling of the wave’s velocity vector with time and depth can be in the opposite direction from that predicted by classical theory. Ray tracing and numerical solutions illustrate the trapping and amplification of NIWs in regions of strong baroclinicity where the wave frequency is lower than the effective Coriolis frequency. The largest amplification is found at slantwise critical layers that align with the tilted isopycnals of the current. Such slantwise critical layers are seen in wintertime observations of the Gulf Stream and, consistent with the theory, coincide with regions of intensified ageostrophic shear characterized by a banded structure that is spatially coherent along isopycnals.

Corresponding author address: Daniel B. Whitt, Stanford University, 473 Via Ortega, Room 140, MC 4216, Stanford, CA 94305. E-mail: dwhitt@stanford.edu

Abstract

An analysis and physical interpretation of near-inertial waves (NIWs) propagating perpendicular to a steady, two-dimensional, strongly baroclinic, geostrophic current are presented. The analysis is appropriate for geostrophic currents with order-one Richardson numbers such as those associated with fronts experiencing strong, wintertime atmospheric forcing. This work highlights the underlying physics behind the properties of the NIWs using parcel arguments and the principles of conservation of density and absolute momentum. Baroclinicity introduces lateral gradients in density and vertical gradients in absolute momentum that significantly modify the dispersion and polarization relations and propagation of NIWs relative to classical internal wave theory. In particular, oscillations at the minimum frequency are not horizontal but, instead, are slanted along isopycnals. Furthermore, the polarization of the horizontal velocity is not necessarily circular at the minimum frequency and the spiraling of the wave’s velocity vector with time and depth can be in the opposite direction from that predicted by classical theory. Ray tracing and numerical solutions illustrate the trapping and amplification of NIWs in regions of strong baroclinicity where the wave frequency is lower than the effective Coriolis frequency. The largest amplification is found at slantwise critical layers that align with the tilted isopycnals of the current. Such slantwise critical layers are seen in wintertime observations of the Gulf Stream and, consistent with the theory, coincide with regions of intensified ageostrophic shear characterized by a banded structure that is spatially coherent along isopycnals.

Corresponding author address: Daniel B. Whitt, Stanford University, 473 Via Ortega, Room 140, MC 4216, Stanford, CA 94305. E-mail: dwhitt@stanford.edu

1. Introduction

Internal waves carry a substantial fraction of the kinetic energy in the ocean and have small vertical scales with large vertical shears (Ferrari and Wunsch 2009). Consequently they are thought to play a crucial role in vertical mixing (e.g., Munk and Wunsch 1998; Wunsch and Ferrari 2004; Ferrari and Wunsch 2009). Global maps of ocean mixing inferred from Argo profiles using a finescale parameterization for mixing by internal waves reveal that mixing is enhanced in regions of high eddy kinetic energy, particularly near separated western boundary currents (Whalen et al. 2012). Separated western boundary currents underlie the atmospheric storm tracks where time-variable winds efficiently inject kinetic energy into the near-inertial band of the internal wave spectrum (Alford 2003). This suggests that near-inertial waves (NIWs) could facilitate the enhanced mixing. The analysis of the Argo data by Whalen et al. (2012) provides further evidence of this, showing that mixing varies in concert with the seasonal changes in near-inertial energy in the mixed layer. However, how NIWs ultimately lead to enhanced turbulence is not well understood. One proposed mechanism is through the interaction of the waves with the circulation and its mesoscale eddies.

NIWs are known to interact strongly with mesoscale features because the vorticity of balanced flows significantly modifies their frequency (Mooers 1975). A variety of observations have been made that illustrate this interaction (e.g., Perkins 1976; Weller 1982; Kunze and Sanford 1984; Kunze et al. 1995). Kunze et al. (1995), in particular, show regions of high energy dissipation in the thermocline associated with anticyclonic vorticity, which lowers the frequency of the waves and allows for the existence of trapped subinertial waves with amplified energy density, consistent with the well-established theoretical predictions of, for example, Kunze (1985), Klein and Hua (1988), Klein and Treguier (1995), Young and Ben-Jelloul (1997), Balmforth et al. (1998), and Plougonven and Zeitlin (2005).

More recent observations collected in the Gulf Stream under wintertime forcing as part of the Clivar Mode Water Dynamics Experiment (CLIMODE) reveal strong NIWs with subinertial frequencies (Marshall et al. 2009). The NIWs were characterized by horizontally coherent structures in the ageostrophic shear that were aligned with the slanted isopycnals of the pycnocline (e.g., Fig. 1) and that were coincident with regions of enhanced dissipation (Inoue et al. 2010). The banded structure in ageostrophic shear was found in the north wall of the Gulf Stream where the balanced flow is cyclonic and strongly baroclinic, that is, where the relative vorticity, horizontal density gradient, and thermal wind shear are large. Similar observations of coherent wavelike shear bands along isopycnals have also been made in other strong frontal regions by, for example, Shcherbina et al. (2003) and Rainville and Pinkel (2004) but these observations were interpreted as waves constrained by the variation of relative vorticity. The effects of baroclinicity, which lead to amplification parallel to isopycnals, were not investigated and may have helped to explain some unexpected results in these observations.

Fig. 1.
Fig. 1.

(top) The Rossby number and base-10 logarithm of the Richardson number, Rog and log(Rig), in the Gulf Stream from a section taken during the winter as part of the CLIMODE experiment. (bottom) The ageostrophic shear from the same section with potential density (black contours) superimposed at increments of 0.2 kg m−3. Ageostrophic shears were calculated by assuming the cross-stream shear ∂υa/∂z was all ageostrophic and the along-stream shear was equivalent to the total shear minus the thermal wind shear, ∂ua/∂z = ∂u/∂z + f−1b/∂y. They show the coherent phase structure typical of a downward propagating internal wave, yet with slopes that run parallel to isopycnals. These shears coincide with regions of low Rig suggesting that baroclinic effects may be important in the wave dynamics. The wave shear occupies a relatively small part of the section, that is, about 30 km, so the velocities are not backrotated since they were observed over just several hours. These observations are from survey S1 in Inoue et al. (2010).

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

Two-dimensional primitive equation numerical simulations have also shown evidence of NIWs modified by the effects of baroclinicity in both a surface front and a coastal upwelling zone (Wang 1991; Federiuk and Allen 1996, respectively). However, the physical phenomena were not fully explored.

This body of recent observations and simulations suggests that the strong baroclinicity of frontal jets, not solely the vorticity, influences the dynamics of the NIWs that are propagating in them and can facilitate wave trapping, amplification, and breaking.1

Motivated to understand the physics behind the CLIMODE observations, in this article we use an appropriate analytical theory to explore how the properties, propagation, and energetics of NIWs are modified by strongly baroclinic geostrophic currents such as the Gulf Stream. By strongly baroclinic we mean that the gradient Richardson number of the geostrophic flow
e1
(where N is the buoyancy frequency and ug is the geostrophic velocity) is O(1). We will also consider mean flows with an order-one-magnitude vorticity Rossby number, (f is the Coriolis frequency, k is the vertical unit vector, and we keep the sign), as is appropriate for the Gulf Stream (e.g., Fig. 1) and other separated western boundary currents. The linearized governing equations for NIWs propagating in such currents are reviewed in section 2. These equations are similar to those used by Mooers (1975), except they are hydrostatic since we are studying only the waves of lowest frequency. The equations differ from those used in the theories of Kunze (1985) and Young and Ben-Jelloul (1997) in that they do not assume a mean flow with low Rog and high Rig and are, thus, appropriate for currents with strong baroclinicity and O(1) Rog and Rig. In sections 3 and 4, we discuss the mechanics, energetics, and propagation of waves under this governing system. Several of the unusual properties of these waves are elucidated using parcel arguments to understand the fundamental physics. In section 5, we present a numerical solution to the wave propagation problem in an idealized surface intensified baroclinic jet with properties similar to the winter Gulf Stream. We obtain monochromatic solutions and plot streamfunctions and energy densities. These results compare favorably with ray tracing. In section 6 we contrast our results to the theoretical results of Mooers (1975), Kunze (1985), and Young and Ben-Jelloul (1997). The article is concluded in section 7.

2. Governing equations

We begin with the inviscid, adiabatic, hydrostatic Boussinesq equations of motion on the f plane,
e2
e3
e4
e5
e6
where D/Dt denotes the material derivative, (u, υ, w) are the (x, y, z) components of the velocity, p is the pressure, ρ0 is the reference density, b = −/ρ0 is the buoyancy, and f = 2Ω sinΦ is the traditional Coriolis frequency, where Φ is latitude and Ω is the rotation rate of the earth.
Next we assume that the background flow is entirely in the x direction and, moreover, that all the motions are uniform in the x direction so that all derivatives with respect to x vanish. Furthermore, we assume that the wave perturbations are small compared to the background flow and hence nonlinear wave–wave interactions may be neglected. Moreover, we break the flow into steady geostrophically balanced and unsteady ageostrophic (wave) components,
eq1
where the geostrophic velocity ug and the steady part of the buoyancy bg are defined so that they are related by the thermal wind balance
e7
Thus, the governing equations of the ageostrophic wave motions become
e8
e9
e10
e11
e12
which are the same as those in Mooers (1975) except that we have made the hydrostatic approximation. We can make the hydrostatic approximation because we are interested in waves with frequencies much smaller than N and aspect ratios much smaller than 1.
For convenience, we now introduce a streamfunction ψ such that
eq2
and the system of governing equations reduces to a single partial differential equation for ψ
e13
where , , and . This is the well-known time-dependent Eliassen–Sawyer equation (Sawyer 1956; Eliassen 1962) that has since been derived in, for example, Hua et al. (1997), Plougonven and Zeitlin (2005), and Colin de Verdiere (2012).
Throughout the rest of the paper, we assume for simplicity that the time dependence can be represented by simple harmonic motion {i.e., ψ(y, z, t) = ℝ[Ψ(y, z)eiωt]} so (13) with the aforementioned simplifications becomes
e14
This is the governing equation that we will refer to throughout the paper. For the flows that we will consider, (14) can be hyperbolic, parabolic, and elliptic depending on the location. It is parabolic along the curve defined by S4N2(F2ω2) = 0 (which we will later define as the contour of the minimum frequency), elliptic outside this curve, and hyperbolic inside this curve. Since we are studying propagating waves in this system, we are primarily interested in the domain where this equation is hyperbolic. Moreover, we will always assume that the background flow is inertially stable, F2 > 0, statically stable, N2 > 0, and that N ≫ |f|. Furthermore, we assume that the potential vorticity (PV), q = F2N2S4, is positive, which ensures that ω is real and rules out symmetric instability (Hoskins 1974). However, we allow for the mean flow to be characterized by Rossby numbers, Rog = ζg/f (with sign), and geostrophic Richardson numbers, Rig = f2N2/S4, which have O(1) magnitudes, where is the vertical component of the relative vorticity. Put another way, this means that all three terms in this equation are important because we consider cases where both
e15
where S2/N2 is the isopycnal slope and (ηy, ηz) are characteristic horizontal and vertical wavelengths.

3. Basic wave properties

In a bounded domain with constant background properties, (14) has constant coefficients and can be converted into a Sturm–Liouville eigenvalue problem. In fact, even if N varies with depth, this procedure can still be followed (e.g., Gerkema and Shrira 2005a,b). However, we are interested in the case where all the coefficients in (14) vary in two spatial dimensions. We will ultimately have to obtain a numerical solution to this problem. However, we can glean physical insights from this equation without too many further assumptions about the solution. In particular, it can be shown that two fundamental properties of near-inertial waves (NIWs)—their minimum frequency and direction of propagation—are significantly modified by baroclinicity. However, first we introduce an idealized background flow in which we illustrate the various wave phenomena throughout the paper.

a. Idealized background flow

To cleanly illustrate the various properties of NIWs in a baroclinic flow, we construct an idealized baroclinic flow in geostrophic balance for use throughout the paper (see Fig. 2). The flow is constructed by assuming a steady, two-dimensional (y, z) density structure
e16
in which ρ0 = 1025 kg m−3 is the reference density, C1, n, and C2 are tuning constants, and Ly and Lz are the horizontal and vertical length of the domain. From this equation, we derive the buoyancy bg. We then assume a level of no-motion (ug = 0) at z = −Lz and integrate the thermal wind expression (7), from z = −Lz to z = 0 to obtain the geostrophic velocity ug and the geostrophic absolute momentum, which we define as
e17
where y is the jet-normal position, and Dy/Dt = υ, as usual. The horizontal (vertical) gradient of Mg surfaces tells you about the vertical (horizontal) component of the absolute vorticity in the geostrophic flow (see e.g., Eliassen 1962).
Fig. 2.
Fig. 2.

The geostrophic velocity (shading) and potential density (contoured at 0.2 kg m−3 increments) for the idealized background flow given by (a) Eq. (16) and (b) the observations from the Gulf Stream.

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

A flow with parameters similar to the wintertime Gulf Stream can be obtained from (16) by choosing (C1, n, C2, Ly, Lz) = (0.9 kg m−3, 2.5, −0.4 kg m−3, 120 km, 400 m) in (16) and letting f = 10−4 s−1. The Rossby number and Richardson number of this idealized geostrophic flow are displayed in Fig. 3 for comparison with the section crossing the Gulf Stream obtained during CLIMODE (in Fig. 1). The Gulf Stream observations are described in, for example, Marshall et al. (2009) and Inoue et al. (2010). With this choice of the tuning constants we have obtained a baroclinic flow that is relatively realistic, simple, and symmetric. It has slowly varying properties, but contains regions where Rog and Rig are O(1) similar to the observations.

Fig. 3.
Fig. 3.

(top) log10 (Rig) and (bottom) Rog for the idealized background flow (16). Potential density contours with an increment of 0.2 kg m−3 are superimposed. These parameters are O(1) in the idealized jet similarly to the winter Gulf Stream as shown in Fig. 1.

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

b. Minimum frequency

Wave solutions to (14) can only exist when the equation is hyperbolic. Therefore, to obtain the minimum frequency of NIWs governed by (14) in an arbitrary geostrophic mean flow, we find where (14) is parabolic, that is, where S4N2(F2ω2) = 0. In the barotropic limit (S2 = 0), waves must have ω > F and the minimum frequency is given by the effective Coriolis frequency,
e18
This is the well-known finding of Magaard (1968), Fomin (1973), Mooers (1975), and Kunze (1985) that anticyclonic (cyclonic) relative vorticity lowers (raises) the effective Coriolis frequency. Thus, mean-flow cyclonic (or anticyclonic) relative vorticity raises (or lowers) the minimum frequency that can be achieved by near-inertial waves. This result is important because waves with frequencies below f can exist for ζg < 0. For flows with laterally varying ζg the waves can be trapped and propagate rapidly in the vertical due to the “inertial chimney” effect (Lee and Niiler 1998).
For a baroclinic flow with S2 ≠ 0, the general expression for the minimum frequency is
e19
Figure 4 illustrates how the minimum frequency varies between a mean flow with no background variability (Rog = 0, ), one with barotropic variability (Rog = −, ), and one with baroclinic variability (Rog = 0, ). As we can see, the baroclinic effects, which are important for the vertical trapping of NIWs, enter through the Richardson number. As Fig. 5 shows, there is a substantial separation between the locations where ω = F (green contour) and ω = ωmin (magenta contour) in both our idealized flow and the Gulf Stream when ω = 0.95f and ω = 0.98f.2 This separation and the different shape of the dispersion relation when baroclinicity is present (see Fig. 4 and section 3d) foreshadow a significant difference in wave energy propagation due to baroclinic effects, which is not captured merely by the variation in F with depth. We will discuss this issue in more detail in sections 4 and 5.
Fig. 4.
Fig. 4.

The frequency of oscillation of fluid parcels displaced at an angle θδ with respect to the horizontal, for a basic state without a background flow (Rog = 0, Rig = ∞, dashed), with a barotropic flow (Rog = −, Rig = ∞, solid) and with a baroclinic flow (Rog = 0, Rig = , dot-dashed). Baroclinicity both lowers the minimum frequency and shifts the angle θδ, where this occurs to coincide with the angle that isopycnals make with the horizontal θb.

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

Fig. 5.
Fig. 5.

Rays, which run parallel to characteristics, illustrate the downward propagation of a wave packet with a frequency (a),(c) 0.95f and (b),(d) 0.98f in both (left) the idealized flow and (right) the Gulf Stream. Baroclinic effects expand the region where rays are confined from ωF (inside the green contour) to ωωmin (inside the magenta contour) and change the angle at which rays reflect from horizontal to the isopycnal angle. Note that there are regions where the contour marking ω = ωmin runs parallel to isopycnals. In these regions, the conditions for a slantwise critical layer are satisfied: multiple rays converge, the group velocity goes to zero, and waves are trapped. In the Gulf Stream observations, these slantwise critical layers are coincident with the region of high observed ageostrophic shear, as shown in Fig. 1.

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

With a little algebra, we may re-form (19) into several new expressions for ωmin that, although equivalent to (19), are especially useful for developing an intuition for the governing dynamics of NIWs in baroclinic currents. The first new form of (19) follows from the fact that, for a two-dimensional flow, the PV can be written in terms of the Jacobian of the background buoyancy and the geostrophic absolute momentum:
e20
Hence,
e21
The expression (21) tells us that, given a constant value of N2, the variation in the minimum frequency depends on the gradient of bg on Mg surfaces or equivalently the gradient of Mg on bg surfaces (see Fig. 6). This motivates us to write (19) in terms of the angles that surfaces of constant Mg and bg make with the horizontal, that is, θM = tan−1(F2/S2) and θb = tan−1(S2/N2). Then we may write the minimum frequency explicitly in terms of θM and θb:
e22
Fig. 6.
Fig. 6.

Schematic illustrating the physics of the minimum frequency for NIWs in geostrophic flows. Lines of constant geostrophic absolute momentum, Mg = ugfy (solid gray), and constant buoyancy bg (dashed black) are plotted in each frame. For (a) a flow with no relative vorticity or baroclinicity, (b) a barotropic flow with anticyclonic relative vorticity, (c) a baroclinic flow with only a vertical thermal wind shear, and (d) a flow with zero PV. The minimum frequency occurs when oscillations are along lines of constant buoyancy. When the gradient of Mg along bg is weaker than that shown in (a), the restoring force is reduced and the frequency is lower than f. In (d) this gradient is zero, hence there is no restoring force and ωmin = 0.

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

Before moving on, we highlight the important observation that (19) admits solutions where ωmin < F. In fact, for S2 ≠ 0, ωmin is always less than F. As discussed in Mooers (1975) and in more detail here, near-inertial waves have particularly unusual and counterintuitive behavior when ω < F. Therefore, it is useful to analyze the physics of NIWs in two separate frequency bands: the classical frequency range where ω > F and a range where ω < F that Mooers (1975) referred to as the anomalously low frequency band. Whether a wave is in one region or the other depends on its particular frequency and the background geostrophic flow that determines F and ωmin. As mentioned above, the separation between these two regions in both our idealized flow and the section from the Gulf Stream is illustrated in Fig. 5. The anomalously low frequency region is delineated by the contours where ω = F (green) and ω = ωmin (magenta) and is found where isopycnals steepen, the flow becomes more strongly baroclinic (i.e., Rig decreases), and the vertical vorticity and Rog increase.

c. Characteristics

The first step in analyzing the propagation of the NIWs in either the classical or the anomalously low frequency regime is a thorough investigation of the qualitative properties of the characteristics of (14) where it is hyperbolic (i.e., where ω > ωmin). In the hydrostatic limit, the slopes λ+,− of the two characteristics ξ+,− are
e23
As we will show in the next section, the characteristics indicate both the direction of energy propagation and wave-driven parcel displacements. The variations in characteristic slopes are plotted in Fig. 7 for a wave with ω = 0.95f in the idealized background flow. As Fig. 8 shows, the critical difference between the classical and anomalously low frequency range is that the two characteristics have slopes with the different signs in the classical case (ω > F) and the same sign in the anomalously low case (ω < F). When ω = F, one of the characteristics is flat but, as we will see in section 4, the horizontal component of the group velocity does not go to zero as it does in a barotropic mean flow. This result suggests that, as shown in Fig. 5, the slopes of characteristics are quite different if one neglects the effects of baroclinicity (i.e., sets S2 = 0) in a flow that is baroclinic. Again, this suggests that wave energy propagation, trapping, and amplification are significantly modified by baroclinicity.
Fig. 7.
Fig. 7.

The separation between the classical and anomalously low frequency regions for a wave with frequency ω = 0.95 in our idealized background flow and the slopes of the wave’s characteristics (top) λ+ and (bottom) λ. The checkerboard pattern indicates that the parcel oscillations along that characteristic are unstable with respect buoyancy. The wavy line pattern indicates that parcel oscillations are unstable with respect to momentum. The green contour is where ω = F; the magenta where ω = ωmin. The basic physics of parcel oscillations in these two quasi-unstable regimes is illustrated in Fig. 9.

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

Fig. 8.
Fig. 8.

Schematic of characteristics ξ+ and ξ: There are three regimes depending on the frequency (left) above the effective inertial frequency F, (middle) at the effective inertial frequency, and (right) below the effective inertial frequency. Waves with ω < F are defined to exist in the anomalously low frequency range. If the sign of the isopycnal slope is switched, that is, θb < 0, the signs of the characteristic slopes are switched as shown in Fig. 7.

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

d. Physical interpretation

To obtain a physical interpretation of these results, we now examine how an infinitesimal parcel of fluid would oscillate if it were given an instantaneous small ageostrophic velocity in a baroclinic background flow.

Before addressing this issue, however, it is useful to introduce the concept of total absolute momentum,
e24
The u-momentum Eq. (2) can then be rewritten,
e25
to show that total absolute momentum is conserved following a fluid parcel even for finite amplitude two-dimensional disturbances. Thus, Δua = −ΔMg, where Δ indicates the change over time following a fluid parcel. Without loss of generality, we can assume that ua(t = 0) = 0. Therefore, following a fluid parcel, in which is the parcel displacement vector and i, j, k the standard Cartesian unit vectors.
Now, consider a parcel perturbed a distance δ = Y cos(θδ) + Z sin(θδ) at an angle θδ in a baroclinic background flow, as shown in Fig. 9. We define the parcel initial position to be at (Y, Z) = (0, 0). Presuming the parcel is in a stable region (i.e., q > 0), this perturbation will induce an oscillation governed by
e26
where Fδ = FY cos(θδ) + FZ sin(θδ) is the force per unit mass on the parcel projected onto . The y and z components of the forces on the parcel may be obtained from the momentum equations and assuming that the parcel adjusts instantaneously to the pressure of the background flow. Thus we are left with
e27
e28
and Fc and Fb denote the Coriolis and buoyancy forces, respectively. The first of the above equations is an expression of absolute momentum conservation, while the second is an expression of buoyancy conservation. Thus we may rewrite (26) as
e29
which permits solutions of the form δ ~ exp(−iωt). This yields frequencies
e30
where the small angle approximation, consistent with hydrostasy, is used. Figure 9 contains a schematic of the force balance on a parcel for three different angles. Rearranging (30) to solve for θδ and again making use of the small angle approximation, we find that
e31
similarly to (23). Thus, to generate a wave with frequency ω > ωmin there are two possible parcel displacement directions, each of which is parallel to a characteristic ξ+,−. Since the ageostrophic velocity in the yz plane is parallel to parcel displacements, it follows that the energy flux, F = paua, is also aligned with the characteristics, confirming that ξ+,− indicate the directions of wave propagation.
Fig. 9.
Fig. 9.

Schematics of parcel displacements in a mean flow with baroclinicity and force diagrams illustrating how the components of the Coriolis force, , and buoyancy force, Fb = N2Y[tan(θb) − tan(θδ)], combine to make up the net force in the direction of displacement Fδ: (top) the classic range of parameter space where θb < θδ < θM; however, in a certain range of parameter space, parcel displacements may be (middle) shallower than buoyancy surfaces or (bottom) steeper than geostrophic absolute momentum surfaces—yet an oscillation will still occur. In these cases, although either Fb or Fc is destabilizing, Fδ is restoring.

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

Parcel arguments can also be used to understand the physics at the minimum frequency, illustrated in Fig. 6. It follows from (30) that the angle that minimizes the frequency corresponds to parcel displacements that run parallel to isopycnals, that, θδ = θbS2/N2. Consequently, for this frequency the buoyancy force (28) is zero and plays no role in dynamics of the oscillation. The restoring force of these oscillations arises solely from the Coriolis force and conservation of absolute momentum (27). The strength of this restoring force, and hence the frequency that it induces, is proportional to the change in geostrophic absolute momentum that a fluid parcel experiences as it moves along a density surface. The change in Mg thus scales with the along-isopycnal gradient in Mg, which is proportional to the PV, hence explaining the formulas for ωmin, (19) and (21).

As mentioned above, Fig. 4 illustrates this dispersion relation (30) for three different background conditions: one where there is no variation in the flow (F2 = f2 and S2 = 0, the classic case), one where there is only a horizontal gradient in momentum (i.e., ζg = −0.7f, S2 = 0 and the flow is barotropic), and one where baroclinicity is important and there is a wide anomalously low frequency range (F2 = f2 and S2 = 0.03N2). In the first and second cases, S2 = 0 and the minimum frequency coincides with purely horizontal displacements (θδ = 0) and all other frequencies correspond to two parcel displacements with equal but opposite angles. In the baroclinic case in contrast, where frequencies below F are anomalously low, the angle of parcel displacement at the minimum frequency is significantly different from horizontal; it is equal to the isopycnal slope.

What is also unusual is that for frequencies ωmin < ω < F the two parcel displacements that result in a given wave period have angles and of the same sign ( for θb positive as shown in Fig. 8), consistent with the analysis of the characteristics.

For the shallower characteristic, corresponding to parcel displacements at an angle for θb positive, parcel displacements are at a smaller angle than isopycnals, yet are in the same direction as the isopycnal tilt [i.e., and sgn() = sgn(θb)]. This phenomenon occurs in the checkered regions of our idealized flow as depicted in Fig. 7. A force diagram for this anomalous scenario, shown in Fig. 9, illustrates how parcel displacements at experience a destabilizing buoyancy force Fb. However, the Coriolis force Fc is strong enough so that the total force in the direction of the parcel displacement Fδ is stabilizing and thus results in oscillations. This unusual behavior is limited to the anomalously low frequency region and is purely a consequence of the baroclinicity of the flow.

On the steeper characteristic, corresponding to parcel displacements at an angle for θb positive, parcels move more closely to surfaces of constant geostrophic absolute momentum for all ω, as shown in Figs. 8 and 9. This causes the Coriolis force Fc to be reduced. However, the buoyancy force Fb is enhanced because the parcel oscillation has a large enough vertical component so that the net restoring force Fδ and thus frequency on this characteristic is the same as that for the shallower characteristic.

In certain regions of frequency space (occurring both when ω < F and ω > F but certainly not for all ω), the angle of parcel displacements on the steeper characteristic can be larger than the slope of Mg surfaces (i.e., for θb positive). In these regions, demarcated by superimposed wavy lines in Fig. 7, the Coriolis force is actually destabilizing. But, in this case the buoyancy force compensates for this destabilizing tendency and allows for an inertia–gravity wave rather than what would otherwise be symmetric instability. This is true because we are only considering background flow conditions for which |θM| is larger than |θb| and, hence, the PV is positive. Note that this phenomenon is not restricted to occur in the anomalously low frequency region as shown in Fig. 7. Nevertheless, slanted Mg surfaces, which in our model are purely a consequence of baroclinicity, are required for its occurrence.

Parcel arguments can also be used to derive the polarization relation of the waves. Noting that υa = −iωY, wa = −iωZ and wa/υa = λ = θδ, it follows from (27) that
e32
The imaginary part of the ratio of ua to υa is plotted throughout the idealized domain and for each characteristic ξ+,− in Fig. 10 assuming ω = 0.95f. The magnitude of this ratio tells you about the local ellipticity of the hodograph, whereas the sign tells you whether the velocity vector rotates clockwise (positive) or counterclockwise (negative) with time. Thus, unlike classical near-inertial waves, not all waves have a velocity vector that rotates clockwise with time when f > 0. As shown in Fig. 10, this could occur near the surface and thus has potential implications for the generation, resonant forcing conditions, and flux of near-inertial energy into the ocean. Except where S2 = 0 or ω = ωmin the polarization relation for the two characteristics is not the same. This is a markedly different behavior from classical inertia–gravity waves and is a consequence of the tilt in Mg surfaces and the dissimilar magnitudes of the characteristic slopes. The steeper characteristic has an angle that is closer to θM. Consequently the gradient in Mg in the direction of parcel displacements is smaller on this characteristic than on its shallower counterpart. Therefore, because absolute momentum is conserved for such displacements, ua and |ua|/|υa| will be correspondingly weaker. In the case that a characteristic slope is steeper than the absolute momentum surface, that is, |λ| > |θM|, the parcel becomes unstable to momentum and the sense of rotation of the parcel with time is reversed, as discussed above and shown in Fig. 10.
Fig. 10.
Fig. 10.

The imaginary part of the quantity ua/υa, given by (32), for a wave with ω = 0.95f in the idealized background flow for the two characteristics: (a) ξ+ and (b) ξ. The magnitude of this quantity indicates the local ellipticity of the hodograph, whereas the sign indicates the direction of rotation with either depth or frequency. Not only does the ellipticity differ for the two characteristics but the sense of rotation switches sign when parcel oscillations are unstable to momentum, that is, steeper than geostrophic absolute momentum surfaces (see Fig. 7). (c) The polarization relation at ωmin as a function of Rog and and (d) the ellipticity of the hodograph at ωmin as a function of .

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

At the minimum frequency the polarization relation becomes
e33
Figure 10c shows how the polarization of the horizontal velocity at the minimum frequency is modified as a function of mean-flow Rossby and Richardson number. In the absence of baroclinicity, |ua| > |υa| in cyclonic flow, whereas |ua| < |υa| in anticyclonic flow. However, in the presence of baroclinicity, |ua| < |υa| unless . The key observation is that for Rog ≠ 0, , and the waves with the lowest frequency are not circularly polarized—this in contrast to classical inertial motions. In particular, in regions where the minimum frequency is subinertial (ωmin < f) due to baroclinicity and/or anticyclonic vorticity, the waves’ horizontal flow is more rectilinear and aligned in the direction perpendicular to the geostrophic current (i.e., |υa| > |ua|). This is because under these conditions, the gradient in absolute momentum along isopycnals is reduced and hence the purely isopycnal displacements at ωmin generate weaker velocities in the direction of the background flow, as shown in Fig. 10d.

4. Energetics and propagation

In this section, we will characterize how near-inertial waves transport energy when propagating across baroclinic currents. To address this issue, we must either assume the solution takes the form of a plane wave or solve (14) numerically because the equation is nonseparable and the coefficients are nonconstant. We discuss our numerical solution to the problem in the next section. Here we develop an analytical expression for the wave-period-averaged energy density in terms of Ψ and then adopt the plane wave assumption, investigate the properties of the phase and group velocity, obtain a full set of polarization relations, and use these relations to derive the classical expression for the energy flux , where is the wave-period-averaged energy density, and cg is the group velocity vector.

a. Energy density

We begin by defining the energy density. We do this by first observing that there are pressure terms in (9) and (10) but not in (8) or (11). This symmetry suggests treating each of these pairs of equations in a similar way (Mooers 1970). Recalling that (υa, wa) = (∂Y/∂t, ∂Z/∂t) = (−iωY, −iωZ), we can integrate (8) and (11) and write
e34
e35
Then we use these results to eliminate ua from (9) and ba from (10) to obtain
e36
e37
Multiply (36) by , (37) by , where the asterisk denotes complex conjugate, and add to obtain
e38
where the right-hand side was obtained using the equation of continuity (12) and is the definition of the energy flux convergence. Moreover, we must be able to write the wave energy per unit volume in this model:
e39
We may then write the time-averaged wave energy per unit volume as
e40
Similar expressions appear in Mooers (1970), Mooers (1975), and Chuang and Wang (1981). The mean energy density (40) is equivalent to the usual hydrostatic form, . However, at any instant in a specific location . The equivalence in the time mean, assuming simple harmonic motion, comes about because the remaining terms in the equation, for example, the kinetic and potential energy production, each contain wave variables that are in quadrature with each other; that is, ua and ba are in quadrature with υa and wa (see Mooers 1975). In any case, the key result is that there is no net exchange of energy between the wave and the steady background flow in this system.
We will use (40) to obtain an energy density result when we numerically solve (14). However, we also compare our result with plane wave solutions and a ray theory where wave rays propagate along characteristics. In this case, it is useful to express the energy density as
e41
As in Mooers (1975), there is an energy equipartition in this system. However, in the nonhydrostatic case, (41) is .

b. Plane wave dispersion and polarization relations

We now make the Wentzel–Kramers–Brillouin (WKB) approximation of geometrical optics (i.e., the plane wave assumption). Strictly speaking, this approximation is only valid when the wavelength is much smaller than the length scale of variations in the background mean flow (i.e., spatial variation in the wavelength occurs over much larger scales than the wavelength) (Bender and Orszag 1978). This is not necessarily true for the flows of interest in this problem. Moreover, ray tracing cannot accurately describe tunneling problems, where there is a small barrier region in which ω < ωmin, or scattering problems. Furthermore, the ray tracing solution has grave errors at turning points (where the governing equation switches from hyperbolic to elliptic behavior). Thus, we check the results with a numerical solution, the accuracy of which does not depend on this scale separation.

First, we assume that the solutions to (8)(12) are all of the plane wave form. For example, , where (l, m) is the wavevector. Substituting these forms into (8)(12) yields a set of five algebraic equations that we can solve to obtain the dispersion relation and the polarization relations.

The dispersion relation obtained from this procedure is
e42
and the polarization relations in terms of are
e43
e44
e45
e46
where α = l/m is the aspect ratio of the wave. We anticipate that α+,− will be the negative of λ+,− because m/l = 1/α should be the slope of the wave vector that should be normal to the group velocity and parcel oscillations that are parallel to characteristics (i.e., k · u = 0). Our intuition is confirmed by inverting (42):
e47
where the last two equalities are obtained from (23) and (31). Thus, (42) is equivalent to the dispersion relation obtained from the parcel arguments (30) under the small angle approximation and the polarization relation for ua (43) is equivalent to (33). We also point out that, for waves satisfying the plane wave form, the horizontal velocity vector of downward (upward) propagating wave packets on characteristics steeper than Mg surfaces (where the imaginary part of ua/υa is less than 0 in Fig. 10) rotates counterclockwise (clockwise) with increasing depth—this in contrast with classical theory as described by, for example, Leaman and Sanford (1975).

c. Phase and group velocity

To obtain a first-order picture of near-inertial wave propagation in a baroclinic current, we obtain the phase and group velocity for plane wave solutions to our governing equation. As is generally done for plane waves, we define the phase velocity as the speed of propagation of a line of constant phase in the direction of the wave vector,
e48
The group velocity is defined as the gradient of ω in the wave vector space,
e49
We observe that cp · cg = 0. Thus, cp is orthogonal to cg as is typically the case for internal waves. As one would expect, the group velocity vectors have the same slope as the characteristics, discussed in section 3, shown in (47) and plotted in Fig. 7. We observe that, when |α| < |S2/N2| = |tan(θb)|, the sign of the vertical component of the group velocity and phase velocity can have the same sign, as shown in the checkered region of Fig. 7.

We also note that the group velocity goes to zero where ω = ωmin, as shown, for example, in Fig. 11. Consequently, for two-dimensional waves, we expect that most of the wave energy will be trapped inside the region where ω > ωmin.

Fig. 11.
Fig. 11.

(top) Rays (parallel to characteristics) in the idealized background flow for waves with ω = 0.95f. The variation of the magnitude of the group velocity (normalized by its maximum along each ray) is colored on the rays. (middle) A numerical solution to Eq. (51) with ω = 0.95f and a point source at y ≈ 0.75 km, z ≈ −100 m. This is a snapshot of the streamfunction in time and the dashed cyan lines are the same rays shown in the top panel. (bottom) Energy density associated with the numerical solution for Ψ shown in the middle panel and computed with (40). These figures confirm ray tracing is qualitatively similar to the numerical solution. High energy densities, velocities, and shears occur near turning points and slantwise critical layers (defined in text) that run parallel to isopycnals.

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

d. Energy flux

Given a background geostrophic flow and a solution Ψ, we can determine the energy density everywhere in the flow using (40). We would also like to be able to compute the energy density along a ray. Using the polarization relation for pa, (46), we find that
e50
Therefore, for steady conditions, that is, , is constant along a ray tube, where A is the ray tube area (Lighthill 1978). Consequently, we could compute the amplification in energy density along any ray path and compare these results with our numerical solution and (40). Nevertheless, we must be cautious as we expect that there may be some differences between the ray tracing and the numerical solution, especially near ω = ωmin as the plane wave solution is invalid near turning points (Bender and Orszag 1978).

5. Ray tracing and numerical solution

We now discuss two ways to develop a solution to the energy propagation and amplification problem for NIWs governed by (14). In both cases we solve the wave problem in our idealized test environment described in section 3 and plotted in Figs. 2, 3, and 5. We begin by describing and presenting results from ray tracing. Then we compare the ray tracing results with a fourth-order accurate finite difference solution to (14).

a. Ray tracing

In this section, we describe the ray-tracing solution to (14) where we use the WKB approximation and assume that the solution can be described by a plane wave as described in section 4. Since the medium varies, this is an approximation everywhere in the domain. Nevertheless, it allows for a great deal of theoretical progress and, as we will see, this solution qualitatively reproduces the features of the numerical solution. Therefore, a thorough understanding of the properties of this solution aids the understanding of NIWs in baroclinic currents.

The results of some ray-tracing computations are shown in Figs. 5 and 11. Figure 5 shows some rays, parallel to characteristics (23), computed with baroclinicity (i.e., with the S2 term included) and without baroclinicity (i.e., with the S2 term set to zero) so as to highlight the qualitative modification of the ray structure due to baroclinicity. Figure 11, on the other hand, shows that the ray-tracing results with baroclinicity compare favorably with the numerical solution described below. In the top panel of Fig. 11, the colored scatter lines are again rays, parallel to characteristics. The color indicates the magnitude of the group velocity |cg|, normalized by the maximum of |cg| along each ray (the group velocity magnitude is symmetric about y = 0). The background flow shown in both Figs. 5 and 11 is the same idealized flow described throughout the paper and the wave packets are monochromatic, with a frequency ω = 0.95f. The reflection rule, applied at vertical and horizontal boundaries and turning points requires that the characteristic and direction of energy propagation switch.

We make several observations. First, the rays are constrained to lie in the region where ωωmin. The group velocity (49) is not real outside this region and the governing equation is no longer hyperbolic as we have discussed. We also note that, as the ray approaches the magenta contour [where ω = ωmin and (14) is parabolic], the slope of the ray approaches the local slope of isopycnals, θb, as we would expect.

To further characterize the behavior of the solution near ωmin we use the terminology of “turning points” and “critical layers.” When we refer to behavior typical of a turning point, we mean that the behavior is qualitatively similar to the behavior at a turning point in a 1D Sturm–Liouville problem, defined rigorously in, for example, Bender and Orszag (1978). This is an internal reflection point in a ray theory where the group velocity goes to zero but multiple characteristics do not converge. More precisely, a turning point occurs where the slope of the ωmin contour is not parallel to characteristics. In contrast, a critical layer is defined as a region where many characteristic curves converge to the same point or line at ωmin. In this case, the characteristics are not only approaching the ωmin curve but their slopes are converging to the slope of this curve as well.

In general, the ωmin contour can act as both a turning point and a critical layer in the same background flow. To see this, consider our idealized domain. Except at two points in the legs of the magenta bowl (in Fig. 11), which bounds the region where ω > ωmin, the slope of this contour is not equal to the slope of characteristics or, equivalently, isopycnals. Therefore, away from these two points the ωmin contour is best characterized as a turning point because multiple characteristics do not converge. On the other hand, near these two points this contour is best characterized as a critical layer because all characteristics converge to one point here and the group velocity (parallel to characteristics) goes to zero.

Although the critical layer is localized to a point in the idealized flow (e.g., Fig. 5a), there is no particular requirement that a critical layer be a point. It merely requires that the ωmin boundary be locally parallel to characteristics or, equivalently, isopycnals. In the Gulf Stream flow, for example (in Fig. 5c), there are multiple critical layers where the ωmin contour is parallel to isopycnals. These regions coincide with the observed high ageostrophic shears, as shown in Fig. 1. We refer to these layers as slantwise critical layers because the isopycnals on which they are found are slanted. This phenomenon is analogous to the vertical critical layer described by Kunze (1985) where the relative vorticity of a mean flow increases with depth but buoyancy remains constant (see his Fig. 14). We adopt this new terminology to additionally distinguish slantwise critical layers from vertical critical layers that can form when inertia–gravity waves propagate parallel to a vertically sheared background flow (e.g., Booker and Bretherton 1967), in contrast to the case of normal incidence we consider here. The crucial difference between these two phenomena is that in a slantwise critical layer, as described here, the component of the wave’s phase velocity in the direction of the mean current is zero, so it can never equal the local velocity of the background flow. This means that a normally incident two-dimensional NIW in a steady two-dimensional background flow cannot transfer energy to the background flow by this process.

We use the fact that is constant along a ray tube to infer that energy density will increase along a ray as a consequence of two effects: (i) a reduction in the ray tube area, A, and (ii) a reduction in group velocity |cg|. The group velocity goes to zero as ωωmin (see the top panel of Fig. 11), which suggests that the energy density should go to infinity as ωωmin, particularly at critical layers where A → 0 as well. However, based on Bender and Orszag (1978), we might more reasonably expect the solution to be governed by Airy-function-like behavior near turning points. This suggests that we will see some amplification right at ωmin but also some energy leakage characterized by a rapid decay in energy density beyond turning points. Where ωmin is a critical layer, the ray tube area and the group velocity are both decaying to zero. Therefore, we might reasonably expect to see the largest amplification of energy density in these regions similar to Kunze (1985).

b. Numerical solution

We can resolve the energetics questions raised in the ray tracing analysis by solving (14) numerically. We use fourth-order central differences to discretize all spatial derivatives and assume Ψ = 0 at all four walls of our idealized test domain. We develop a discretized representation of (14):
e51
where = (F2Dzz + 2S2Dzy + N2Dyy) and ; Dzz, Dzy, and Dyy are fourth-order accurate central difference approximations to the second derivative operator with the boundary conditions included. The Laplacian frictional operator, = νυDzz + νhDyy, contains vertical and horizontal viscosities to damp motions near the grid scale. We set νυ = ωΔz2/π2 and νh = ωΔy2/π2 in which Δz and Δy are the vertical and horizontal grid spacings in meters. For the resolution presented here, νh ≈ 1 m2 s−1 and νυ ≈ 10−5 m2 s−1 in our idealized flow. Finally, b is a source function, which is zero except at a single point where it is one (labeled source point in Fig. 11). Thus, we are essentially numerically solving for the Green’s function of (14) for an oscillating forcing at a point.

We discretize the domain with 802 points evenly distributed in the horizontal and vertical directions. We present this numerical solution to (51) in Fig. 11 for ω = 0.95f and a point source at approximately 100-m depth in the middle of the domain. We plot the real part of the streamfunction Ψ in the middle panel and the wave-period-averaged energy density , given by (40), in the bottom panel. The real part of the streamfunction describes the wave when the phase is zero. Before discussing the results, some comments about the numerical solution are in order.

The computation (51) is not intensive; it takes only several seconds on a single core. Experiments with various levels of friction as well as nontraditional and nonhydrostatic effects were performed but are not presented. A certain amount of friction is required to damp very small-scale features. Without any friction, small-scale features with unrealistically large shears can exist at or near the grid scale. However, adding nonhydrostatic and nontraditional terms does not qualitatively change the results.

The results of the numerical computation show striking similarities to the ray tracing despite the fact that the medium varies relatively rapidly in space. Streamlines, high gradients in Ψ, and high energy densities occur along rays. Streamlines are nearly parallel to isopycnals near ωmin (the magenta contour) and, as expected, the numerical solution is amplified there and decays outside, similar to an Airy function at a turning point. Furthermore, the strongest amplification occurs near the critical layers (the corners of the bowl where ωmin is parallel to isopycnals in Fig. 11). The shear, which is by definition parallel to rays throughout the domain, and, in particular, isopycnals near ωmin, is strongest near critical layers where wave velocities, energy densities, gradients in energy density, and hence shears are largest.

In short, the results of the ray tracing and numerical solution are consistent. Both are useful tools for analyzing waves in baroclinic currents. Regardless of which method is used, however, it is important to include the effects of baroclinicity when the Richardson number of the background flow is less than ~10.

6. Discussion

We now compare the analysis presented here with others in the literature, in particular those of Kunze (1985) and Young and Ben-Jelloul (1997). The key result is that the dispersion relation obtained by Kunze (1985) and Young and Ben-Jelloul (1997), that is,
e52
is an approximation to (42) that is valid when the following parameter
e53
is small.3 Similar to the full expression for the frequency (42), the approximation (52) is minimized for waves with flow that is purely along isopycnals, that is, l/m = −S2/N2, yielding a minimum frequency
e54
In the limit of no baroclinicity, (54) returns the classic result of Kunze (1985): . Conversely, in the limit of a background flow with no lateral shear, (54) becomes , which is equivalent to Eq. (3.13) of Young and Ben-Jelloul (1997).
The approximation (54) is equal to the first two terms in a Taylor series expansion of the full expression for the minimum frequency (19), , where the small parameter of the expansion is
e55
Observations from the Gulf Stream reveal, however, that this parameter can be order one in magnitude (e.g., Fig. 12), suggesting that the approximate Eq. (54) may be inaccurate in such highly energetic background flows. We quantify the accuracy of this formula as a function of ε in Fig. 12 by comparing the full and approximate expressions for the minimum frequency. For |ε| < 0.25 the two expressions agree well and deviate from one another as |ε| → 1. There is a greater discrepancy between the two formulas for ε < 0. For example, the relative error in ωmin using (54) is 6% when ε = 1 but goes to infinity as ε → −1. Noting that the PV of the background flow can be written in terms of ε, q = f2N2(1 + ε), it follows that for nonzero stratification ε → −1 as the PV goes to zero. Observations of PV and stratification in western boundary currents suggest that the situation of zero PV and nonzero stratification, and hence ε → −1, may not be that unusual in these regions, especially during strong wintertime forcing (see e.g., Thomas and Lee 2005; Thomas and Joyce 2009; Thomas et al. 2013). Under these conditions, baroclinicity strongly modifies the physics of NIWs and the theory described here is most applicable. However, it is important to emphasize that, even if ε is not approaching −1 but Ri < 10, baroclinic effects should still be included, perhaps in approximate form. This is illustrated in Figs. 12c and 12d, which show the relative error in the dispersion relation due to using the approximate dispersion relation, (54), or the approximate effective Coriolis frequency, , instead of the full dispersion relation, (19).
Fig. 12.
Fig. 12.

(a) The expansion parameter, , is plotted for the Gulf Stream section discussed throughout the paper and first displayed in Fig. 1. Values of |ε| > 0.25 are found within the magenta contours. The magnitudes of ε are typical for strong western boundary currents during winter and higher values may be observed. (b) A comparison between the formula for minimum frequency (19) and the first two terms in its Taylor series approximation (54) is plotted. (c) The relative error due to neglecting baroclinicity and approximating is plotted, whereas (d) the relative error due to approximating (54) is shown. The effects of neglecting baroclinicity are more severe than making the Taylor series approximation in this case. However, the error induced by assuming ε is small is nonnegligible here and may be worse in other cases where Rig or Rog are lower.

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0132.1

As mentioned above, the theory presented originally by Mooers (1975) and more recently by Plougonven and Zeitlin (2005) is also appropriate in the strongly baroclinic regime. Our work is distinguished from that of both Mooers (1975) and Plougonven and Zeitlin (2005) in that we have emphasized the physical interpretation of NIWs rather than the mathematical properties of solutions to (14). We have also presented a numerical solution in an idealized jet with properties similar to the Gulf Stream to illustrate the results, highlight their applicability to real world flows and confirm the inferences derived from ray tracing and parcel arguments. We must also point out that the results of the hydrostatic theory presented in this paper and in Plougonven and Zeitlin are not entirely the same as those in Mooers. One particularly prominent difference is that the group velocity does not go to zero at ωmin if the nonhydrostatic equations are used (Mooers 1975). Nevertheless, this modification will have a negligible effect on the energy flux of NIWs in an environment like our model environment or the Gulf Stream. The group velocity still gets very small at this point and the hydrostatic approximation is appropriate for these low-aspect-ratio waves.

Plougonven and Zeitlin (2005) recently presented a similar hydrostatic formulation of the Eliassen–Sawyer equation. They used the theory to study geostrophic adjustment rather than wind-generated near-inertial waves, although they did touch upon the trapping of near-inertial waves in geostrophic jets. However, they analyzed the trapping phenomena using a barotropic form of (14); that is, they only considered variations in mean flow relative vorticity. Therefore, like Kunze (1985), they did not explore the baroclinic effects that we have shown are important.

Finally, we observe that the theory presented here is similar to the theory of NIWs on the nontraditional β plane (e.g., Gerkema and Shrira 2005b; Colin de Verdiere 2012). The horizontal variation in f due to changes in latitude on the nontraditional β plane is analogous to the horizontal variation in F due to variations in relative vorticity presented in this paper. As a result, on a β plane, NIWs have a turning latitude. However, because the horizontal component of the Coriolis frequency induces a vertical gradient in geostrophic absolute momentum by adding an extra to (17), the waves can propagate past this turning latitude and become trapped at depth as shown by Gerkema and Shrira. Because conservation of absolute momentum governs the physics of NIWs, the properties of these waves on the nontraditional β plane are analogous to the properties of NIWs in the anomalously low frequency region, illustrated in Figs. 5 and 11. For example, characteristic slopes of waves with frequencies, ω < F, have the same sign in both regimes (see, e.g., Gerkema and Shrira 2005a,b). However, the effects of baroclinicity overwhelm nontraditional effects at ocean fronts because the vertical shear and stratification are both much larger than the horizontal component of the Coriolis frequency so we neglect these effects in this work [see Colin de Verdiere (2012) for a careful parametric study of nontraditional effects using the nontraditional, nonhydrostatic, time-dependent Eliassen–Sawyer equation].

7. Conclusions

In this paper, we have analyzed two-dimensional near-inertial waves (NIWs) that propagate perpendicular to baroclinic geostrophic currents with O(1) Rossby and Richardson numbers. The work builds primarily off that of Mooers (1975), who derived the mathematical properties of nonhydrostatic inertia–gravity waves in this system, but emphasizes a new physical interpretation for the results using parcel arguments and conservation principles. In particular, conservation of absolute momentum MT = ufy is crucial for understanding the unusual properties of the waves in this system. One such unusual property of these waves is that their minimum frequency decreases with increasing baroclinicity. This can be understood geometrically. The waves of lowest frequency have parcel displacements along isopycnals and thus experience no buoyancy force. The restoring force for the oscillations in this case is provided solely by the Coriolis force, whose strength, by conservation of MT, depends on the along-isopycnal gradient of geostrophic absolute momentum, Mg = ugfy. This gradient is reduced as baroclinicity increased because isopycnals steepen while Mg surfaces flatten, thereby lowering the minimum frequency.

By lowering the intrinsic frequency, baroclinicity allows NIWs to exist in regions where the effective Coriolis frequency, , exceeds the wave’s frequency. Waves in this anomalously low frequency region have properties that differ dramatically from classical inertia–gravity waves. Like classical inertia–gravity waves, there are two characteristic slopes for a wave of a given frequency (as long as the frequency is not at its minimum). However, unlike classical inertia–gravity waves, the two characteristic slopes of these anomalously low frequency waves have the same sign, being centered about the isopycnal slope. In addition, on the shallower characteristic, vertical wavenumbers, phase velocities, and group velocities all have the same sign. This unusual property was first observed in primitive equation numerical simulations of a front by Wang (1991).

Near-inertial waves can be trapped and amplified in the anomalously low frequency regions. The amplification preferentially occurs near slantwise critical layers that run parallel to isopycnals. These are places where multiple characteristics converge and the magnitude of the group velocity decays to zero (see e.g., Fig. 11). At these critical layers, the waves attain their minimum frequency and hence parcel displacements, rays, characteristics, and lines of constant phase all run parallel to isopycnals. Thus, near a slantwise critical layer, the vertical shear of the NIW would be strongest and lines of constant shear would tend to align with isopycnals. This characterization is consistent with the enhanced ageostrophic shear in the Gulf Stream highlighted in Fig. 1, which occurs near a region with multiple critical layers, as identified by a ray-tracing calculation that accounts for the effects of baroclinicity (see e.g., Fig. 5).

Another property of the anomalously low frequency waves that differs from classical inertia–gravity waves is their polarization relation. In particular, the waves with the lowest frequency are characterized by horizontal velocities that are not circularly polarized, unlike classical NIWs. As baroclinicity is increased, the polarization relation of these waves becomes more rectilinear, with stronger velocities in the cross- versus alongfront direction. As shown by Thomas (2012) this change in polarization relation can lead to an efficient exchange of kinetic energy between the waves and balanced flows in regions of active frontogenesis. The idea being that, as the polarization relation shifts from circular to rectilinear, the waves induce a momentum flux that is in a particular direction. When this momentum flux is pointed down the gradient in momentum of the balanced flow, the waves act as an effective viscosity, extracting kinetic energy from the mean flow.

The polarization relation is modified in unexpected ways for waves of higher frequencies as well. In the presence of baroclinicity, the sense of rotation and the ellipticity of the hodograph traced out by the velocity vector over time for waves of the same frequency but different characteristics (i.e., different propagation directions) are not necessarily the same. For the case when one of the characteristics is steeper than surfaces of constant Mg, the velocity vector for a wave in the Northern (Southern) Hemisphere rotates counterclockwise (clockwise) with time, that is, opposite to what classical theory would predict. This implies that the resonance conditions for maximal wind work on the near-inertial motions no longer corresponds to an anticyclonic rotary wind oscillating at the inertial frequency, like the classical prediction of Pollard and Millard (1970). How much of an effect this has on the generation of NIWs in the proximity of the fronts, especially those associated with separated western boundary currents that underlie the midlatitude storm tracks, is an open question and one that will be the subject of future research on the generation process through which we hope to better understand the rates of wind generation, radiative decay, trapping, and dissipation of near-inertial energy in strongly baroclinic western boundary currents.

Another topic that we are pursuing is how three-dimensional NIWs (i.e., waves that have alongfront variability) in a strongly baroclinic current differ from the 2D waves described here. When the waves vary in the alongfront direction, a Doppler shift is present that can modify the conditions for wave trapping (e.g., Kunze 1985). Determining how 3D dynamics alters the key properties of NIWs, not just their trapping, is the subject of a follow-up study to this work.

Acknowledgments

We would like to thank four anonymous reviewers for their helpful suggestions. This work was supported by the Office of Naval Research Grant N00014-09-1-0202 and the National Science Foundation Grant OCE-0961714.

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1

Geostrophic flows with extremely high baroclinicity can develop enhanced shear and dissipation parallel to isopycnals even without significant inertial wave activity. Frontogenetic strain, in particular, can play an important role in this process and can also modify any internal waves in the front (Thomas 2012). However, frontogenesis does not appear to be important in the observations presented here and is not the focus of the present paper. See Winkel et al. (2002), Nagai et al. (2009), Thomas et al. (2010), and D’Asaro et al. (2011) for other examples of enhanced shear parallel to isopycnals due to waves and other phenomena.

2

The selected frequencies are somewhat arbitrary but, when ω is not too far from f and the flow is baroclinic, the observed separation between ωmin and F is typical. The qualitative nature of the results does not change for slightly different frequencies [i.e., (0.95 ± 0.05)f]. However, waves with much larger frequencies are not trapped whereas waves with much smaller frequencies cannot exist.

3

The theory of Young and Ben-Jelloul (1997) used in, for example, Balmforth et al. (1998) and Zeitlin et al. (2003) depends on an asymptotic expansion in small . Their dispersion relation, to first order, is the same as that in Kunze (1985). Thus, the Kunze results hold even when the WKB scaling assumption does not, so long as is small. In this paper, however, we are explicitly interested in conditions when is O(1). See (15).

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