Simulating the Long-Range Swell of Internal Waves Generated by Ocean Storms

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to mix the upper ocean and to have the potential to mix the deep ocean enough to be important for climate simulations.NIWs are principally generated as a result of a resonant coupling between upper-ocean currents and mid-latitude atmospheric cyclones.Here, we report on simulated NIWs in an eddy-resolving general circulation model that is forced by a realistic atmosphere, and we make comparisons to NIWs observed from moored and shipboard measurements of currents.The picture that emerges is that as much as 16% of NIW energy (which is season dependent) radiates out of the mixed layer and equatorward in the form of low-mode, long-lived internal gravity waves; they transmit energy thousands of kilometers from their regions of generation.The large amount of energy in near-inertial motions at a given site is a combination of a local response to wind forcing and waves that have traveled far from where they were generated.

INtROductION motivation
Near-inertial waves (NIWs) are one of the most energetic modes of ocean variability, containing about half the ocean's kinetic energy at most sites (and even more beneath storm tracks).
Hence, they are of central importance to a variety of ocean processes, including mixed-layer deepening (Price et al., 1986); submesoscale processes, larval dispersion, and formation of thin layers (Franks, 1995); restratification of mixed layers and thermohaline intrusions (Hosegood et al., 2008); and mixing of both the upper and deep ocean (Hebert and Moum, 1994;Alford, 2003b).Yet, an understanding of NIW energy sources and redistribution by wave radiation is lacking, causing NIWs to be poorly represented in regional and global models.
Over the last decade, there has been significant community focus on internal tides, which produce large thermocline displacements, affect sound propagation, and control some hotspots of elevated turbulent mixing.Near-inertial internal gravity waves are probably of similar or greater importance, providing comparable kinetic energy and the vast majority of shear variance (Alford and Whitmont, 2007;Toole, 2007), and likely leading to a substantial amount of turbulent mixing.
Significant deficiencies remain in our understanding of the physical processes that determine their generation, evolution, and destruction.NIWs appear to be primarily forced by the wind (D' Asaro, 1985;Alford and Whitmont, 2007).The amount of energy input to the ocean in the form of NIWs has been estimated to be 0.5-0.7 TW (Watanabe and Hibiya, 2002;Alford, 2003a), based on forcing a slab model (Pollard and Millard, 1970) with reanalysis winds.However, Jiang et al. (2005) demonstrated sensitivity to the choice of wind product used, obtaining values ranging from 0.36-1.4TW.
Though uncertain, these calculations represent a large fraction of the 2 TW of energy demand required to maintain deep-ocean stratification (Munk and Wunsch, 1998;Wunsch and Ferrari, 2004;St. Laurent and Simmons, 2006).On the other hand, data from the Ocean Storms Experiment (D' Asaro, 1995;Levine and Zervakis, 1995) and historical moored current meters (Alford, 2003a) suggest that a substantial portion of NIW energy is radiated far from generation regions in fast-moving, low-mode waves.

Additionally, observations show energetic
NIWs at all depths (Silverthorne and Toole, 2006;Alford, 2010), with a greater percentage penetrating to depth (Alford et al., in press) than concluded by the Furuichi study.One possibility for these discrepancies is that mesoscale flows can affect generation (Weller, 1982), refract the waves once generated (Kunze, 1986), and amplify deep propagation (Lee and Niiler, 1998;Zhai et al., 2005).
Our study, unlike the Furuichi simulation, includes mesoscale eddies (Figure 1).It is intended to shed some light on these puzzles, and illustrate the r e l a t i v e v ortic it y r e l a t iv e vo r t i c i t y

Wind Generation of Near-Inertial Waves
The first part of NIW generation, the initial input of energy into the mixed layer by resonant forcing, is the best understood.Observed mixed-layer velocities quickly grow as great as 1 m s -1 (greater in hurricanes and typhoons).
A slab model by Pollard and Millard (1970) generally reproduces the initial amplitude and phase of observed mixed-layer oscillations.The skill of the slab model enables construction of the aforementioned global maps of wind work, using only reanalysis wind products such as those from the National Centers for Environmental Prediction (NCEP) and mixed-layer depth climatology.Note that recent work using high-resolution moored observations (Plueddemann and Farrar, 2006) pointed out that neglect of mixed-layer deepening by slab models likely causes these estimates to be high, although the degree of overestimation is uncertain and likely varies from place to place (Tom Farrar, Woods Hole Oceanographic Institution, pers. comm., 2012).Slab models predict strong inputs in the western portion of each basin at mid-latitudes, confirming that mid-latitude storms provide the bulk of the energy flux (with much of the remainder from hurricanes/typhoons).
The general distribution of energy flux in our model (Figure 2, gray scale) resembles the observed near-inertial energy in the mixed layer from previous slabmodel estimates and the observed distribution from drifters (Chaigneau et al., 2008), but the global total of 0.4 TW, as we will show, is on the low side of the range of estimates.

excitation of Interior motions
Once near-inertial motions in the mixed layer are generated, lateral variation leads to convergences and divergences that "pump" vertical motions at the base of the mixed layer, driving motions in the ocean interior (Figure 3).Anderson and Gill (1979) and Gill (1985) developed a modal formalism for describing this excitation of submixed-layer motions.
Much like the vibration of a guitar string, waves in the stratified ocean take the form of standing vertical oscillations of horizontal currents (with corresponding vertical displacements of isopycnals), which are termed the dynamic "modes." The lowest mode, or, by analogy, "harmonic, " of horizontal velocity corresponds to horizontal ocean currents that are uniform from top to bottom.This mode is commonly referred to as the "barotropic" mode or zeroth mode and is not an internal gravity wave.The first "baroclinic" or depth-dependent mode, indicated schematically in gray in Figure 3, is an internal gravity wave that takes the form of an oscillation, with flow going in one direction at the surface and in the opposite direction at depth, and with a "node" at the depth of the thermocline.Higher modes have higher numbers of nodes and hence a more complicated vertical structure.
An important feature of internal gravity waves is that the horizontal rate of propagation, or the "phase" speed, associated with each given baroclinic mode decreases with increasing mode number.
For typical ocean depths and stratification, the first baroclinic mode travels at about 3 m s -1 and the second at 1.5 m s -1 zeroth mode.The expected vertical structure of the modes can be computed from stratification alone, and profiles can be described by fitting the velocity profile to a linear combination of the modes.
To compute the ocean interior response, the slab velocity profile (motion in the mixed layer and none below) is projected onto the modes computed for pre-storm stratification.
As the modes propagate away, energy travels downward into the ocean interior, with the details controlled by the phasing between the various modes (see Figure 3).Far from the storm site, successive modes will arrive in the order of their phase speed so that we can expect to see the first mode arrive first, followed by the second, and so forth.Both high-and low-mode internal gravity waves are excited.High modes propagate along curving characteristics downward and equatorward.The higher modes have strong shear that results in mixing ("ε").Low-mode wave radiation, indicated in gray, takes the form of oscillations that propagate equatorward.upward characteristics and topographic scattering have been observed, but the processes involved are not completely understood.
relative and planetary vorticity; Kunze, 1985).Hence, in addition to "imprinting" mesoscales and submesoscales on NIWs (effectively reducing their wavelengths and increasing their propagation speeds), anticyclonic eddies can create regions known as "inertial chimneys" wherein a locally lowered effective inertial frequency allows rapid deep propagation.First proposed by Lee and Niiler (1998), inertial chimneys are also seen in numerical models (Zhai et al., 2005).and N (the buoyancy frequency).For this reason, near-inertial waves can only propagate slightly toward the pole on a beta plane, because they quickly reach a latitude where their frequency is equal to the local value of f, which is higher than their generation latitude, and then turn they toward the equator.Alford (2003b) first observationally demonstrated this tendency to propagate toward the equator.Note that this view contrasts with the original explanation of the near-inertial peak by Fu (1980) in terms of higherfrequency waves propagating poleward.

Local dissipation vs. Long-Range propagation
As a group of NIWs propagates down from the surface and toward the equator, it maintains the frequency associated with its generation latitude, which will exceed the local value of f.This phenomenon is commonly referred to as a blue shift, but it really arises from the reduction of f with latitude rather than an increase in the wave frequency.The terms "red" and "blue" refer to relatively lower and higher frequencies, analogous to the relative frequencies of red and blue visible light.The consequence is that at any given latitude and depth, the internal gravity wave spectrum (as measured by, for instance, horizontal velocity) will have a locally generated peak near f plus a higher-frequency (bluer) contribution by waves that originated shallower than and poleward of the given location where f was higher.The blue shift can be quantified by a ratio of the local Coriolis frequency f to the near-inertial peak so that an f peak / f local larger than one represents a blue shift and, conversely, a ratio smaller than one reveals a "red" shift.A blue shift of the near-inertial peak has been reported throughout the world ocean (e.g., Fu, 1981;D' Asaro and Perkins, 1984).

mOdeL deScRIptION
For this study, we used the Generalized Ocean Layer Dynamics (GOLD) model, an isopycnal-coordinate ocean model developed at the National Oceanic and Atmospheric Administration Geophysical Fluid Dynamics Laboratory.
Topography data were taken from the thirtieth-degree resolution ETOPO2 dataset (National Geophysical Data Center), bicubically interpolated to the target model grid resolution.Horizontal resolution is nominally one-eighth degree globally with resolution scaling from approximately 14 × 14 km at the equator to 7 × 7 km at 60°N (quasi-Mercator), transitioning to a quasiregular bipolar grid (Murray, 1996) poleward of 60°N where the resolution remains a nearly constant 7 × 7 km.
Isopycnal viscosity was set by two additive schemes.The first is a biharmonic scheme whereby a fixed parameter that has units of velocity set to 0.05 m s -1 is multiplied by the third power of the grid resolution.The second scheme is a Smagorinsky nonlinear eddy viscosity computed by multiplying a fixed parameter with units of velocity set to 0.001 m s -1 times the grid spacing.
The model time stepping uses a modesplitting scheme with a time step of 360 seconds and 7.5 seconds for the baroclinic and barotropic modes, respectively.
The nondimensional quadratic bottom drag coefficient is 0.003.Diapycnal mixing was calculated using the shear-driven entrainment scheme of Jackson et al. (2008).The mixed layer is modeled using a refined bulk mixed layer (Hallberg, 2003)  Near-inertial motions were isolated via band-pass filtering.

Re SuLtS
As is typical for a model of this resolu- "characteristics" (lines of characteristic slope), consistent with a frequency somewhat above f.Though collected in a different year, the general tendency to propagate downward and equatorward is seen in both model and observations.However, the observed shear is much greater, highlighting the model's limited ability to resolve the higher modes.A characteristic is drawn for a frequency (i.e., a 3% blue shift; Garrett, 2001).

Latitudinal Propagation and
Conservation of Frequency:

Blue Shifting
We examine the frequency response of the model in several ways.We compute horizontal velocity spectra every half-degree in latitude and every 2° in longitude, and average at each latitude to produce maps of spectral power as a function of frequency and latitude for the North Pacific.The resulting spectra of the modal, surface, and deep currents (Figure 6) reveal strong inertial peaks.
As internal waves propagate equatorward, they preserve their frequencies, which smear the spectra equatorward, as shown in the first three panels of Figure 6.The fourth panel shows the effect of this "painting" of frequency equatorward: the inertial peaks, which are combinations of local inertial response and "contamination" from signals that originated at higher latitudes where f is larger, has been blue-shifted to higher frequency.In the example given here, the shift is between 5 and 10% at 35°N for modes 1-4.The spectrum of the surface currents (green line) is nar- and is a factor of two stronger than in the Northern Hemisphere in all seasons.
The Northern Hemisphere varies from 0.06 to 0.18 TW over the year (variation of 300%).The low-mode signal takes roughly 20 days to propagate from 40°N to the equator, a distance of over 4,000 km (2.6 m s -1 , Figure 1D).

Ratio of Energy Input to Radiation
The average monthly wind work from

SummaRy aNd dIScuSSION
We have presented some aspects but we find that blue shifting frequently exceeds that predicted by Garrett (2001).
The net effect is that the inertial peak becomes increasingly broad and blue shifted equatorward of the storm tracks.
We note the absence of reliable observational estimates at latitudes equatorward of 25° precludes any definitive conclusion about blue shift at low latitudes.
The fact that NIW groups can radiate from subpolar latitudes across the equator (Figure 1D) suggests that in the midbasins (away from western boundary currents), NIW groups are able to pass relatively unscathed through a mesoscale eddy field (Figure 1A).This phenomenon is likely only true for the lower modes.Higher modes have smaller spatial scales and slower phase speeds and are expected to be more sensitive to refraction by mesoscale eddies.We caution that large-scale ocean models are limited in their ability to resolve the high modes, and, therefore, future effort should be directed toward refining our understanding of the radiative versus dissipative partitioning of the energy in near-inertial oceanic motions.
a R p e R L .S I m m O N S a N d m at t H e W H . a L f O R d S p e c I a L I S S u e O N I N t e R N a L WaV e S aBStR act.Near-inertial waves (NIWs) are a special class of internal gravity waves with periods set by planetary rotation and latitude (e.g., at 30° latitude, one cycle per 24 hours).They are notable because they contain most of the observed shear in the ocean and around half the kinetic energy.As such, they have been demonstrated These calculations give the "source" of NIWs, specifically, the lateral distribution of the power input.However, analogous to internal tides, NIWs appear to be able to travel far from their forcing regionsdownward into the ocean interior from their generation at the surface and also equatorward.Understanding their dissipation clearly requires understanding this lateral and vertical propagation.In a study using a global model similar to the one used here,Furiuchi et al. (2008) obtained a global wind work total of 0.35 TW, but without mesoscale flows.Though they obtained good agreement with some observational estimates of inertial energy in the mixed layer, they concluded that most NIW energy dissipates in the upper 150 m of the ocean.
figure 1.(a) Relative vorticity of sea surface currents, scaled by the coriolis frequency at 30°N.colormap saturates at 0.1.(B) Snapshot of east-west sea surface currents.colormap saturates at 25 cm s -1 .(c) Vertical velocity at 750 m depth.colormap saturates at 40 m per day.(d) time-latitude plot of vertical velocity at 750 m depth along the meridian indicated in (c).The data visualized in panels B-d have not been filtered or band passed in any way.
figure 3. processes associated with near-inertial generation, dissipation, and propagation.as storms move along the storm track, a local response occurs with frequencies near the local coriolis frequency.Both high-and low-mode internal gravity waves are excited.High modes propagate along curving characteristics downward and equatorward.The higher modes have strong shear that results in mixing ("ε").Low-mode wave radiation, indicated in gray, takes the form of oscillations that propagate equatorward.upward characteristics and topographic scattering have been observed, but the processes involved are not completely understood.
NIW evolution depends on vertical and horizontal wavelengths imparted onto interior motions by storms.The initial forcing excites all vertical modes in a red spectrum, with most energy going to the lowest modes.Horizontal scales are set through a combination of storm size and storm translation speed, plus the mesoscale vorticity field.Fundamental to understanding the energy cascade is determining the relative importance of (1) low-mode motions, which contain appreciable energy but quickly propagate away laterally, and (2) higher modes that remain in the forcing region owing to their slow vertical and meridional group velocities, and contain much of the vertical shear in the ocean.The distribution of the dissipation from a storm thus depends on the relative fraction of energy in each mode.Depending on the stratification, 30% or more of the energy may be contained in vertical modes one and two (D' Asaro, 1995;Alford, 2010;   Alford et al., in press).These low-mode disturbances propagate horizontally out of the generation region and may break as a result of topographic scattering, as do internal tides(Müller and Liu, 2000;Johnston and Merrifield, 2003), or nonlinear transfers within the internal wave field.The remainder of the energy is in higher wavenumber waves that have slow group velocities and greater shear, and are therefore likely to break relatively locally.The former are the focus of our study, in large part because our numerical model only resolves the first few vertical modes.Blue Shift of the Ocean current Spectrum Near the Inertial frequencyNear-inertial waves are special for a few reasons(Garrett, 2001), most notably for our purposes because of their nearness to the lower end of the range of possible frequencies for propagating internal gravity waves, which can only exist between f (the local Coriolis frequency) scheme(Griffies et al., 2009) using the atmospheric state and the evolving model ocean.The CORE atmospheric data have six-hourly variability for most atmospheric fields, obtained from the NCEP analysis.It repeats annually, with numerous corrections to address known biases and to balance the fluxes when integrated over the global ocean and over a year.Notably, our forcing differs from the CORE scheme described by Griffies et al. (2009) in one important regard: Figure 1D shows that these waves travel thousands of kilometers, apparently without disruption by the lower-frequency flows.For example, around yearday 17, a wave train originating at 40°N crosses through the equator into the Southern Hemisphere.It should be noted that these structures are also evident in the deep vertical velocity response in the simulations of Furiuchi et al. (2008, their Figure 2), and in Komori et al. (2008, their Figure 1).A "shadow" in the lee of the Hawaiian Islands is evident (Figure 4).Komori et al. (2008) also noted the shadowing behind the Hawaiian Islands.It should be stressed that Figure 1C,D has not been filtered or band passed in any way.The dominance of the near-inertial velocity signal emphasizes the fact that inertial motions have most of the ocean's energy.
figure 5. (top) model shear in a section north of Hawaii.(bottom) R/V Roger Revelle sonar shear along the same section, inertially back rotated.
figure 7. (a) Simulated blue shift as a function of depth and latitude.(b) Blue shift from mooring analysis by alford (2003a).(c) profile of modeled and observed blue shift with depth (panels a and b), averaged from 25° to 60° latitude.
of near-inertial wave physics in an eddy-resolving general circulation model (GCM) and made comparisons with moored estimates of peak shift (2,200 estimates at various depths and latitudes), moored estimates of energy flux (80 moorings, mode-1 and -2), and shipboard section(s).The model predicts that 0.36 TW of energy is transferred to near-inertial motions, which is on the low end of other estimates.This energy input occurs principally under storm tracks and has significant hemispheric and annual variability (Figure 2).The ratio of the spatially integrated wind work over an area to the wave radiation leaving the region is a measure of the amount of wind work that goes into stable, radiating low-mode internal waves, in other words, a measure of the fraction of the energy that dissipates large distances from the storm track.The general pattern is for the waves to radiate equatorward (southeast in the Northern Hemisphere).There is a marked anisotropy between wavenumber k (eastwest) and l (north-south), with l >> k.We find a mode-1 wavelength of about 500-600 km and a mode-3 wavelength of about 200-300 km.The east-west wavelength approaches basin scale.Westward propagation in the Southern Ocean has not been noted previously.Equatorward fluxes range from 3% to 16% of the energy input.This range of values varies with season and may reflect changes in seasonal mixed-layer depth between July and December.Note that the modeled fluxes are usually about 10-20% weaker than Alford (2003a) and the modeled wind work for the North Pacific is 10-20% higher than Alford (2003a).These two competing differences conspire to reduce the overall estimate of the percent of radiating energy (i.e., their ratio).Note, however, that previous experience with idealized GOLD suggests the simulated ratio can vary by as much as a factor of two, with sensitivity to model resolution and treatment of parameterized viscosity.Future work should examine the role of the mesoscale flows and other factors in determining this fraction on an event-by-event basis.Horizontal energy fluxes (Figure 2) agree well with mooring estimates.Looking at depth-latitude sections, energy propagates downward along characteristics (Figure 5).When compared to observed sections, slopes are qualitatively similar, but modeled wavenumbers are substantially lower, and vertical shear is 10 times weaker.Modeled and observed blue shift (Figure 6 and 7) increases from somewhat less than unity at high latitudes and near the surface to 1.3 at depth and near the equator (30% blue shift).This view of the ocean corroborates the Garrett (2001) view,