## 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.

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}

*N*is the buoyancy frequency and

**u**

_{g}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 Ro

_{g}and high Ri

_{g}and are, thus, appropriate for currents with strong baroclinicity and

*O*(1) Ro

_{g}and Ri

_{g}. 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

*f*plane,

*D*/

*Dt*denotes the material derivative, (

*u*,

*υ*,

*w*) are the (

*x*,

*y*,

*z*) components of the velocity,

*p*is the pressure,

*ρ*

_{0}is the reference density,

*b*= −

*gρ*/

*ρ*

_{0}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.

*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,

*u*and the steady part of the buoyancy

_{g}*b*are defined so that they are related by the thermal wind balance

_{g}*N*and aspect ratios much smaller than 1.

*ψ*such that

*ψ*

*ψ*(

*y*,

*z*,

*t*) = ℝ[Ψ(

*y*,

*z*)

*e*

^{−iωt}]} so (13) with the aforementioned simplifications becomes

*S*

^{4}−

*N*

^{2}(

*F*

^{2}−

*ω*

^{2}) = 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,

*F*

^{2}> 0, statically stable,

*N*

^{2}> 0, and that

*N*≫ |

*f*|. Furthermore, we assume that the potential vorticity (PV),

*q*=

*F*

^{2}

*N*

^{2}−

*S*

^{4}, 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, Ro

_{g}=

*ζ*/

_{g}*f*(with sign), and geostrophic Richardson numbers, Ri

_{g}=

*f*

^{2}

*N*

^{2}/

*S*

^{4}, which have

*O*(1) magnitudes, where

*S*

^{2}/

*N*

^{2}is the isopycnal slope and (

*η*,

_{y}*η*) are characteristic horizontal and vertical wavelengths.

_{z}## 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

*y*,

*z*) density structure

*ρ*

_{0}= 1025 kg m

^{−3}is the reference density,

*C*

_{1},

*n*, and

*C*

_{2}are tuning constants, and

*L*and

_{y}*L*are the horizontal and vertical length of the domain. From this equation, we derive the buoyancy

_{z}*b*. We then assume a level of no-motion (

_{g}*u*= 0) at

_{g}*z*= −

*L*and integrate the thermal wind expression (7), from

_{z}*z*= −

*L*to

_{z}*z*= 0 to obtain the geostrophic velocity

*u*and the geostrophic absolute momentum, which we define as

_{g}*y*is the jet-normal position, and

*Dy*/

*Dt*=

*υ*, as usual. The horizontal (vertical) gradient of

*M*surfaces tells you about the vertical (horizontal) component of the absolute vorticity in the geostrophic flow (see e.g., Eliassen 1962).

_{g}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

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

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 (*C*_{1}, *n*, *C*_{2}, *L _{y}*,

*L*) = (0.9 kg m

_{z}^{−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 Ro

_{g}and Ri

_{g}are

*O*(1) similar to the observations.

(top) log_{10} (Ri_{g}) and (bottom) Ro* _{g}* 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

(top) log_{10} (Ri_{g}) and (bottom) Ro* _{g}* 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

(top) log_{10} (Ri_{g}) and (bottom) Ro* _{g}* 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

*S*

^{4}−

*N*

^{2}(

*F*

^{2}−

*ω*

^{2}) = 0. In the barotropic limit (

*S*

^{2}= 0), waves must have

*ω*>

*F*and the minimum frequency is given by the effective Coriolis frequency,

*f*can exist for

*ζ*< 0. For flows with laterally varying

_{g}*ζ*the waves can be trapped and propagate rapidly in the vertical due to the “inertial chimney” effect (Lee and Niiler 1998).

_{g}*S*

^{2}≠ 0, the general expression for the minimum frequency is

_{g}= 0,

_{g}= −

_{g}= 0,

*ω*=

*F*(green contour) and

*ω*=

*ω*

_{min}(magenta contour) in both our idealized flow and the Gulf Stream when

*ω*= 0.95

*f*and

*ω*= 0.98

*f*.

^{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.

The frequency of oscillation of fluid parcels displaced at an angle *θ _{δ}* with respect to the horizontal, for a basic state without a background flow (Ro

_{g}= 0, Ri

_{g}= ∞, dashed), with a barotropic flow (Ro

_{g}= −

_{g}= ∞, solid) and with a baroclinic flow (Ro

_{g}= 0, Ri

_{g}=

*θ*, where this occurs to coincide with the angle that isopycnals make with the horizontal

_{δ}*θ*.

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

The frequency of oscillation of fluid parcels displaced at an angle *θ _{δ}* with respect to the horizontal, for a basic state without a background flow (Ro

_{g}= 0, Ri

_{g}= ∞, dashed), with a barotropic flow (Ro

_{g}= −

_{g}= ∞, solid) and with a baroclinic flow (Ro

_{g}= 0, Ri

_{g}=

*θ*, where this occurs to coincide with the angle that isopycnals make with the horizontal

_{δ}*θ*.

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

The frequency of oscillation of fluid parcels displaced at an angle *θ _{δ}* with respect to the horizontal, for a basic state without a background flow (Ro

_{g}= 0, Ri

_{g}= ∞, dashed), with a barotropic flow (Ro

_{g}= −

_{g}= ∞, solid) and with a baroclinic flow (Ro

_{g}= 0, Ri

_{g}=

*θ*, where this occurs to coincide with the angle that isopycnals make with the horizontal

_{δ}*θ*.

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

Rays, which run parallel to characteristics, illustrate the downward propagation of a wave packet with a frequency (a),(c) 0.95*f* and (b),(d) 0.98*f* in both (left) the idealized flow and (right) the Gulf Stream. Baroclinic effects expand the region where rays are confined from *ω* ≥ *F* (inside the green contour) to *ω* ≥ *ω*_{min} (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* = *ω*_{min} 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

Rays, which run parallel to characteristics, illustrate the downward propagation of a wave packet with a frequency (a),(c) 0.95*f* and (b),(d) 0.98*f* in both (left) the idealized flow and (right) the Gulf Stream. Baroclinic effects expand the region where rays are confined from *ω* ≥ *F* (inside the green contour) to *ω* ≥ *ω*_{min} (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* = *ω*_{min} 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

Rays, which run parallel to characteristics, illustrate the downward propagation of a wave packet with a frequency (a),(c) 0.95*f* and (b),(d) 0.98*f* in both (left) the idealized flow and (right) the Gulf Stream. Baroclinic effects expand the region where rays are confined from *ω* ≥ *F* (inside the green contour) to *ω* ≥ *ω*_{min} (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* = *ω*_{min} 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

*ω*

_{min}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:

*N*

^{2}, the variation in the minimum frequency depends on the gradient of

*b*on

_{g}*M*surfaces or equivalently the gradient of

_{g}*M*on

_{g}*b*surfaces (see Fig. 6). This motivates us to write (19) in terms of the angles that surfaces of constant

_{g}*M*and

_{g}*b*make with the horizontal, that is,

_{g}*θ*= tan

_{M}^{−1}(

*F*

^{2}/

*S*

^{2}) and

*θ*= tan

_{b}^{−1}(

*S*

^{2}/

*N*

^{2}). Then we may write the minimum frequency explicitly in terms of

*θ*and

_{M}*θ*:

_{b}Schematic illustrating the physics of the minimum frequency for NIWs in geostrophic flows. Lines of constant geostrophic absolute momentum, *M _{g}* =

*u*−

_{g}*fy*(solid gray), and constant buoyancy

*b*(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

_{g}*M*along

_{g}*b*is weaker than that shown in (a), the restoring force is reduced and the frequency is lower than

_{g}*f*. In (d) this gradient is zero, hence there is no restoring force and

*ω*

_{min}= 0.

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

Schematic illustrating the physics of the minimum frequency for NIWs in geostrophic flows. Lines of constant geostrophic absolute momentum, *M _{g}* =

*u*−

_{g}*fy*(solid gray), and constant buoyancy

*b*(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

_{g}*M*along

_{g}*b*is weaker than that shown in (a), the restoring force is reduced and the frequency is lower than

_{g}*f*. In (d) this gradient is zero, hence there is no restoring force and

*ω*

_{min}= 0.

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

Schematic illustrating the physics of the minimum frequency for NIWs in geostrophic flows. Lines of constant geostrophic absolute momentum, *M _{g}* =

*u*−

_{g}*fy*(solid gray), and constant buoyancy

*b*(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

_{g}*M*along

_{g}*b*is weaker than that shown in (a), the restoring force is reduced and the frequency is lower than

_{g}*f*. In (d) this gradient is zero, hence there is no restoring force and

*ω*

_{min}= 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 *ω*_{min} < *F*. In fact, for *S*^{2} ≠ 0, *ω*_{min} 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 *ω*_{min}. 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* = *ω*_{min} (magenta) and is found where isopycnals steepen, the flow becomes more strongly baroclinic (i.e., Ri_{g} decreases), and the vertical vorticity and Ro_{g} increase.

### c. Characteristics

*ω*>

*ω*

_{min}). In the hydrostatic limit, the slopes

*λ*

_{+,−}of the two characteristics

*ξ*

_{+,−}are

*ω*= 0.95

*f*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

*S*

^{2}= 0) in a flow that is baroclinic. Again, this suggests that wave energy propagation, trapping, and amplification are significantly modified by baroclinicity.

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* = *ω*_{min}. 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

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* = *ω*_{min}. 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

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* = *ω*_{min}. 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

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, *θ _{b}* < 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

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, *θ _{b}* < 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

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, *θ _{b}* < 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.

*u*-momentum Eq. (2) can then be rewritten,

*u*= −Δ

_{a}*M*, where Δ indicates the change over time following a fluid parcel. Without loss of generality, we can assume that

_{g}*u*(

_{a}*t*= 0) = 0. Therefore, following a fluid parcel,

**i**,

**j**,

**k**the standard Cartesian unit vectors.

*δ*=

*Y*cos(

*θ*) +

_{δ}*Z*sin(

*θ*) at an angle

_{δ}*θ*in a baroclinic background flow, as shown in Fig. 9. We define the parcel initial position to be at (

_{δ}*Y*,

*Z*) = (0, 0). Presuming the parcel is in a stable region (i.e.,

*q*> 0), this perturbation will induce an oscillation governed by

*F*=

_{δ}*F*cos(

_{Y}*θ*) +

_{δ}*F*sin(

_{Z}*θ*) is the force per unit mass on the parcel projected onto

_{δ}*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

*F*and

_{c}*F*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

_{b}*δ*~ exp(−

*iωt*). This yields frequencies

*θ*and again making use of the small angle approximation, we find that

_{δ}*ω*>

*ω*

_{min}there are two possible parcel displacement directions, each of which is parallel to a characteristic

*ξ*

_{+,−}. Since the ageostrophic velocity in the

*y*–

*z*plane is parallel to parcel displacements, it follows that the energy flux,

**F**=

*p*

_{a}**u**

_{a}, is also aligned with the characteristics, confirming that

*ξ*

_{+,−}indicate the directions of wave propagation.

Schematics of parcel displacements in a mean flow with baroclinicity and force diagrams illustrating how the components of the Coriolis force, *F _{b}* =

*N*

^{2}

*Y*[tan(

*θ*) − tan(

_{b}*θ*)], combine to make up the net force in the direction of displacement

_{δ}*F*: (top) the classic range of parameter space where

_{δ}*θ*<

_{b}*θ*<

_{δ}*θ*; 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

_{M}*F*or

_{b}*F*is destabilizing,

_{c}*F*is restoring.

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

Schematics of parcel displacements in a mean flow with baroclinicity and force diagrams illustrating how the components of the Coriolis force, *F _{b}* =

*N*

^{2}

*Y*[tan(

*θ*) − tan(

_{b}*θ*)], combine to make up the net force in the direction of displacement

_{δ}*F*: (top) the classic range of parameter space where

_{δ}*θ*<

_{b}*θ*<

_{δ}*θ*; 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

_{M}*F*or

_{b}*F*is destabilizing,

_{c}*F*is restoring.

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

Schematics of parcel displacements in a mean flow with baroclinicity and force diagrams illustrating how the components of the Coriolis force, *F _{b}* =

*N*

^{2}

*Y*[tan(

*θ*) − tan(

_{b}*θ*)], combine to make up the net force in the direction of displacement

_{δ}*F*: (top) the classic range of parameter space where

_{δ}*θ*<

_{b}*θ*<

_{δ}*θ*; 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

_{M}*F*or

_{b}*F*is destabilizing,

_{c}*F*is restoring.

_{δ}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, *θ _{δ}* =

*θ*≈

_{b}*S*

^{2}/

*N*

^{2}. 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

*M*thus scales with the along-isopycnal gradient in

_{g}*M*, which is proportional to the PV, hence explaining the formulas for

_{g}*ω*

_{min}, (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 (*F*^{2} = *f*^{2} and *S*^{2} = 0, the classic case), one where there is only a horizontal gradient in momentum (i.e., *ζ _{g}* = −0.7

*f*,

*S*

^{2}= 0 and the flow is barotropic), and one where baroclinicity is important and there is a wide anomalously low frequency range (

*F*

^{2}=

*f*

^{2}and

*S*

^{2}= 0.03

*N*

^{2}). In the first and second cases,

*S*

^{2}= 0 and the minimum frequency coincides with purely horizontal displacements (

*θ*= 0) and all other frequencies correspond to two parcel displacements with equal but opposite angles. In the baroclinic case in contrast, where frequencies below

_{δ}*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 *ω*_{min} < *ω* < *F* the two parcel displacements that result in a given wave period have angles *θ _{b}* positive as shown in Fig. 8), consistent with the analysis of the characteristics.

For the shallower characteristic, corresponding to parcel displacements at an angle *θ _{b}* positive, parcel displacements are at a smaller angle than isopycnals, yet are in the same direction as the isopycnal tilt [i.e.,

*θ*)]. 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

_{b}*F*. However, the Coriolis force

_{b}*F*is strong enough so that the total force in the direction of the parcel displacement

_{c}*F*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.

_{δ}On the steeper characteristic, corresponding to parcel displacements at an angle *θ _{b}* 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

*F*to be reduced. However, the buoyancy force

_{c}*F*is enhanced because the parcel oscillation has a large enough vertical component so that the net restoring force

_{b}*F*and thus frequency on this characteristic is the same as that for the shallower characteristic.

_{δ}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 *M _{g}* surfaces (i.e.,

*θ*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 |

_{b}*θ*| is larger than |

_{M}*θ*| 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

_{b}*M*surfaces, which in our model are purely a consequence of baroclinicity, are required for its occurrence.

_{g}*υ*= −

_{a}*iωY*,

*w*= −

_{a}*iωZ*and

*w*/

_{a}*υ*=

_{a}*λ*=

*θ*, it follows from (27) that

_{δ}*u*to

_{a}*υ*is plotted throughout the idealized domain and for each characteristic

_{a}*ξ*

_{+,−}in Fig. 10 assuming

*ω*= 0.95

*f*. 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

*S*

^{2}= 0 or

*ω*=

*ω*

_{min}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

*M*surfaces and the dissimilar magnitudes of the characteristic slopes. The steeper characteristic has an angle that is closer to

_{g}*θ*. Consequently the gradient in

_{M}*M*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,

_{g}*u*and |

_{a}*u*|/|

_{a}*υ*| will be correspondingly weaker. In the case that a characteristic slope is steeper than the absolute momentum surface, that is, |

_{a}*λ*| > |

*θ*|, 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.

_{M}The imaginary part of the quantity *u _{a}*/

*υ*, given by (32), for a wave with

_{a}*ω*= 0.95

*f*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

*ω*

_{min}as a function of Ro

_{g}and

*ω*

_{min}as a function of

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

The imaginary part of the quantity *u _{a}*/

*υ*, given by (32), for a wave with

_{a}*ω*= 0.95

*f*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

*ω*

_{min}as a function of Ro

_{g}and

*ω*

_{min}as a function of

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

The imaginary part of the quantity *u _{a}*/

*υ*, given by (32), for a wave with

_{a}*ω*= 0.95

*f*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

*ω*

_{min}as a function of Ro

_{g}and

*ω*

_{min}as a function of

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

*u*| > |

_{a}*υ*| in cyclonic flow, whereas |

_{a}*u*| < |

_{a}*υ*| in anticyclonic flow. However, in the presence of baroclinicity, |

_{a}*u*| < |

_{a}*υ*| unless

_{a}_{g}≠ 0,

*ω*

_{min}<

*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., |

*υ*| > |

_{a}*u*|). This is because under these conditions, the gradient in absolute momentum along isopycnals is reduced and hence the purely isopycnal displacements at

_{a}*ω*

_{min}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 **c _{g}** is the group velocity vector.

### a. Energy density

*υ*,

_{a}*w*) = (∂

_{a}*Y*/∂

*t*, ∂

*Z*/∂

*t*) = (−

*iωY*, −

*iωZ*), we can integrate (8) and (11) and write

*u*from (9) and

_{a}*b*from (10) to obtain

_{a}*u*and

_{a}*b*are in quadrature with

_{a}*υ*and

_{a}*w*(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.

_{a}### 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* < *ω*_{min}, 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, *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.

*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):

*u*(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

_{a}*M*surfaces (where the imaginary part of

_{g}*u*/

_{a}*υ*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).

_{a}### c. Phase and group velocity

*ω*in the wave vector space,

**c**

_{p}·

**c**

*= 0. Thus,*

_{g}**c**

_{p}is orthogonal to

**c**

*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 |*

_{g}*α*| < |

*S*

^{2}/

*N*

^{2}| = |tan(

*θ*)|, 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.

_{b}We also note that the group velocity goes to zero where *ω* = *ω*_{min}, 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* > *ω*_{min}.

(top) Rays (parallel to characteristics) in the idealized background flow for waves with *ω* = 0.95*f*. 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* = 0.95*f* 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

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

(top) Rays (parallel to characteristics) in the idealized background flow for waves with *ω* = 0.95*f*. 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* = 0.95*f* 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

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

(top) Rays (parallel to characteristics) in the idealized background flow for waves with *ω* = 0.95*f*. 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* = 0.95*f* 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

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

### d. Energy flux

*p*, (46), we find that

_{a}*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*=

*ω*

_{min}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 *S*^{2} term included) and without baroclinicity (i.e., with the *S*^{2} 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 |**c**_{g}|, normalized by the maximum of |**c**_{g}| 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* = 0.95*f*. 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 *ω* ≥ *ω*_{min}. 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* = *ω*_{min} and (14) is parabolic], the slope of the ray approaches the local slope of isopycnals, *θ _{b}*, as we would expect.

To further characterize the behavior of the solution near *ω*_{min} 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 *ω*_{min} 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 *ω*_{min}. In this case, the characteristics are not only approaching the *ω*_{min} curve but their slopes are converging to the slope of this curve as well.

In general, the *ω*_{min} 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* > *ω*_{min}, the slope of this contour is not equal to the slope of characteristics or, equivalently, isopycnals. Therefore, away from these two points the *ω*_{min} 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 *ω*_{min} 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 *ω*_{min} 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 *A*, and (ii) a reduction in group velocity |**c**_{g}|. The group velocity goes to zero as *ω* → *ω*_{min} (see the top panel of Fig. 11), which suggests that the energy density should go to infinity as *ω* → *ω*_{min}, 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 *ω*_{min} but also some energy leakage characterized by a rapid decay in energy density beyond turning points. Where *ω*_{min} 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

*F*

^{2}

*D*+ 2

_{zz}*S*

^{2}

*D*+

_{zy}*N*

^{2}

*D*) and

_{yy}*D*,

_{zz}*D*, and

_{zy}*D*are fourth-order accurate central difference approximations to the second derivative operator with the boundary conditions included. The Laplacian frictional operator,

_{yy}*ν*+

_{υ}D_{zz}*ν*, contains vertical and horizontal viscosities to damp motions near the grid scale. We set

_{h}D_{yy}*ν*=

_{υ}*ω*Δ

*z*

^{2}/

*π*

^{2}and

*ν*=

_{h}*ω*Δ

*y*

^{2}/

*π*

^{2}in which Δ

*z*and Δ

*y*are the vertical and horizontal grid spacings in meters. For the resolution presented here,

*ν*≈ 1 m

_{h}^{2}s

^{−1}and

*ν*≈ 10

_{υ}^{−5}m

^{2}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 80^{2} points evenly distributed in the horizontal and vertical directions. We present this numerical solution to (51) in Fig. 11 for *ω* = 0.95*f* 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

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 *ω*_{min} (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 *ω*_{min} is parallel to isopycnals in Fig. 11). The shear, which is by definition parallel to rays throughout the domain, and, in particular, isopycnals near *ω*_{min}, 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

^{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*= −

*S*

^{2}/

*N*

^{2}, yielding a minimum frequency

*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

*ω*

_{min}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*=

*f*

^{2}

*N*

^{2}(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,

(a) The expansion parameter, *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 *ε* is small is nonnegligible here and may be worse in other cases where Ri_{g} or Ro_{g} are lower.

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

(a) The expansion parameter, *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 *ε* is small is nonnegligible here and may be worse in other cases where Ri_{g} or Ro_{g} are lower.

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

(a) The expansion parameter, *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 *ε* is small is nonnegligible here and may be worse in other cases where Ri_{g} or Ro_{g} 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 *ω*_{min} 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 *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 *M _{T}* =

*u*−

*fy*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

*M*, depends on the along-isopycnal gradient of geostrophic absolute momentum,

_{T}*M*=

_{g}*u*−

_{g}*fy*. This gradient is reduced as baroclinicity increased because isopycnals steepen while

*M*surfaces flatten, thereby lowering the minimum frequency.

_{g}By lowering the intrinsic frequency, baroclinicity allows NIWs to exist in regions where the effective Coriolis frequency,

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 *M _{g}*, 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 *ω*_{min} 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 *O*(1). See (15).