JUNE 2018 AN SONG ET AL . 1409
Geographical Distribution of Diurnal and Semidiurnal Parametric Subharmonic
Instability in a Global Ocean Circulation Model
J a,b a c d,eOSEPH K. ANSONG, BRIAN K. ARBIC, HARPER L. SIMMONS, MATTHEW H. ALFORD,
MAARTEN C. B f g h iUIJSMAN, PATRICK G. TIMKO, JAMES G. RICHMAN, JAY F. SHRIVER,
h
AND ALAN J. WALLCRAFT
aDepartment of Earth and Environmental Sciences, University of Michigan, Ann Arbor, Michigan
bDepartment of Mathematics, University of Ghana, Legon, Accra, Ghana
cCollege of Fisheries and Ocean Sciences, University of Alaska Fairbanks, Fairbanks, Alaska
dApplied Physics Laboratory, University of Washington, Seattle, Washington
e School of Oceanography, University of Washington, Seattle, Washington
fDivision of Marine Science, University of Southern Mississippi, Stennis Space Center, Mississippi
gCentre for Applied Marine Sciences, Bangor University, Anglesey, United Kingdom
hCenter for Ocean–Atmospheric Prediction Studies, Florida State University, Tallahassee, Florida
iOcean Dynamics and Prediction Branch, Naval Research Laboratory, Stennis Space Center, Mississippi
(Manuscript received 19 August 2017, in final form 2 February 2018)
ABSTRACT
The evidence for, baroclinic energetics of, and geographic distribution of parametric subharmonic
instability (PSI) arising from both diurnal and semidiurnal tides in a global ocean general circulation
model is investigated using 1/12.58 and 1/258 simulations that are forced by both atmospheric analysis
fields and the astronomical tidal potential. The paper examines whether PSI occurs in the model, and
whether it accounts for a significant fraction of the tidal baroclinic energy loss. Using energy transfer
calculations and bispectral analyses, evidence is found for PSI around the critical latitudes of the tides.
The intensity of both diurnal and semidiurnal PSI in the simulations is greatest in the upper ocean,
consistent with previous results from idealized simulations, and quickly drops off about 58 from the critical
latitudes. The sign of energy transfer depends on location; the transfer is positive (from the tides to
subharmonic waves) in some locations and negative in others. The net globally integrated energy transfer
is positive in all simulations and is 0.5%–10% of the amount of energy required to close the baroclinic
energy budget in themodel. The net amount of energy transfer is about an order of magnitude larger in the
1/258 semidiurnal simulation than the 1/12.58 one, implying the dependence of the rate of energy transfer
on model resolution.
1. Introduction There are several unanswered questions about the
fate of low-mode internal tides radiating away from to-
Internal tides are generated by barotropic tidal flow
pographic sources:
over rough topography in a stratified fluid. After
generation, vertical low-mode internal tides may ra- 1) What percentage of converted barotropic-to-baroclinic
diate into the interior of the ocean or break and mix in energy eventually goes into mixing the ocean? Mixing
the vicinity of rough topography. Internal tides that in the deep ocean, now thought to be primarily due to
escape away from their sources do not travel indefi- the baroclinic tides, wind-generated near-inertial
nitely since they are bound by their turning latitudes waves (NIWs), and the wind-driven general circula-
(the latitude at which their frequency is equal to the tion, has been argued to exert a strong influence on
local inertial frequency; Hendershott 1973; Wunsch the stratification and general circulation of the ocean
and Gill 1976). (Munk and Wunsch 1998; St. Laurent and Simmons
2006; MacKinnon et al. 2017).
Corresponding author: JosephK. Ansong, jkansong@umich.edu, 2) What are the dynamic processes befalling internal
jkansong@ug.edu.gh tides before they finally break?
DOI: 10.1175/JPO-D-17-0164.1

 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright
Policy (www.ametsoc.org/PUBSReuseLicenses).
1410 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48
3) What is the best way of parameterizing dissipation of near-inertial shear in the upper ocean around the North
internal waves in global ocean models? Pacific CL.Using idealized numerical simulations, Richet
et al. (2017) find that the presence of amean flow removes
Recent progress on the questions above and on other re-
the tidal dissipation peak at the PSI critical latitude,
lated physical oceanography problems has been reviewed
and they reported two weak peaks away from the CL.
by a Climate Process Team on internal-wave-driven ocean
They attribute the sensitivity of PSI to the mean cur-
mixing (MacKinnon et al. 2017).
rent that Doppler shifts the frequency of the primary
Parametric subharmonic instability (PSI) is one of
internal tide.
several potential processes responsible for removing
Simmons (2008) demonstrated that the PSI seen at
energy from internal tides on their journey from topo-
critical latitudes in the idealizedmodels run byMacKinnon
graphic sources. PSI is a particular resonant wave–wave
and Winters (2005) can also be seen in a realistic global
interaction in which a primary wave (the internal tide in
internal tide model. However, the model used in Simmons
this case) exchanges energy with daughter waves with
(2008) had several simplifications compared to the actual
smaller vertical scales and with nearly half the frequency
ocean. These include the use of a horizontally uniform
of the primary wave (McComas and Bretherton 1977).
stratification, the inclusion of only one tidal constituent
Observational studies (Alford et al. 2007; Qiu et al. 2012;
(the M2 semidiurnal tide), and the lack of atmospheric
MacKinnon et al. 2013a,b; Sun and Pinkel 2012, 2013;
forcing. The results presented here come from three
Alford et al. 2017), studies utilizing idealized numeri-
separate simulations set up to mimic realistic oceanic
cal models (Hibiya et al. 1998, 2002; MacKinnon and conditions through
Winters 2005; Furuichi et al. 2005; Gerkema et al. 2006;
Carter and Gregg 2006; Hazewinkel and Winters 2011; 1) forcing of the model with the three major semi-
Gayen and Sarkar 2013; Richet et al. 2017), and realistic diurnal tides in two simulations and with the three
global numerical models (Simmons 2008) continue to major diurnal tides in a third,
implicate PSI for shaping the internal wave dynamics. 2) inclusion of atmospheric forcing, and
Innovative laboratory experiments have also been de- 3) the resulting horizontally varying stratification.
signed to study PSI (see Sutherland 2013, and references The inclusion of atmospheric forcing ensures the pres-
therein; Bourget et al. 2014). The idealized numerical ence of a vigorous mesoscale eddy field as well as
study of MacKinnon and Winters (2005) found a sig- the generation of NIWs. Both eddies and NIOs may
nificant loss of energy from the M2 internal tide at its interact with low-mode internal tides, potentially
critical latitude (CL) of 28.88. Subsequent observational detuning them and altering the PSI behaviors. The in-
studies found that the energy loss at 28.88 was not as clusion of different tidal constituents is also important
catastrophic as predicted by the idealized study. Alford because the study by Hazewinkel and Winters (2011)
et al. (2007) suggest that one possible reason for this found that the growth rate of PSI is influenced by the
discrepancy is the presence of higher-mode waves in the spring-neap cycle. In addition, it is of interest to examine
real ocean. These waves move in different directions whether diurnal PSI can be detected in a global ocean
(see Zhao et al. 2010), thus detuning the perfect phase model, because observational studies (Alford 2008; Xie
locking that resulted in the exponential PSI growth in et al. 2016) and idealized numerical studies (Simmons
MacKinnon and Winters (2005). 2008) both suggest its occurrence.
An idealized numerical study by Hazewinkel and
Winters (2011) built upon the simulations of MacKinnon
and Winters (2005) by incorporating a vertically non- 2. The HYCOM model and methodology
uniform density stratification, the planetary beta effect, The simulations in this study were performed with the
and higher vertical resolution into their simulations. Hybrid Coordinate Ocean Model (HYCOM; Bleck
They observed PSI to occur in the upper ocean with 2002; Halliwell 2004; Chassignet et al. 2009), which is in
subsequent propagation of near-inertial oscillations use by the U.S. Navy as an operational model (Metzger
(NIOs) into the deep ocean. Importantly, they found et al. 2014). The simulations used here, however, are run
that the reduction of baroclinic energy flux due to PSI is in forward (non-data-assimilative) mode. Two simula-
sensitive to eddy viscosity. Using a vertical eddy vis- tions are forced by the three largest semidiurnal tidal
cosity of 5 3 1025m2 s21, they found a flux reduction of constituents (M2, S2,N2) at horizontal resolutions of
about 15%, consistent with the observations of Alford 1/12.58 and 1/258 at the equator. A third simulation is forced
et al. (2007) for beams crossing the critical latitude by the three largest diurnal constituents (Q1,O1,K1)
around Hawaii. Recent analyses of observational data by at a horizontal resolution of 1/12.58 at the equator.
Alford et al. (2017) further implicate PSI for heightened The model simulations use K-profile parameterization
JUNE 2018 AN SONG ET AL . 1411
(KPP; Large et al. 1994) as their mixed layer submodel.
We use a critical bulk Richardson number of 0.25. For
the shear instability term we set the maximum gradient
Richardson number to 0.7 with a cubic profile from 0 to
0.7 and a maximum value of 50 3 1024m2 s21 for vis-
cosity and diffusivity. The vertical eddy diffusivity for
unresolved background internal wave shear is pre-
scribed as 1.0 3 1025m2 s21 while the vertical eddy
viscosity due to unresolved background internal waves is
3.0 3 1025m2 s21 (Wallcraft et al. 2009), a value com-
parable to that used in Hazewinkel and Winters (2011).
The model has 41 hybrid vertical coordinate surfaces
with potential density referenced to 2000m and atmo-
spheric forcing from the Navy Global Environmental
Model (NAVGEM; Hogan et al. 2014).
We employ the parameterized topographic wave drag
FIG. 1. Periods of the diurnal (K ,O ,Q ;D collectively) and
scheme of Jayne and St Laurent (2001). Because global 1 1 1 1
semidiurnal (M2, S2, N2; D2 collectively) tides as well as periods
models are not able to resolve the breaking of internal of their half-frequency subharmonics [(1/2)D1 and (1/2)D2,
waves, we use a parameterized wave drag acting on the respectively]. The solid curve shows the local period of inertial
bottom flow to represent the generation and breaking of oscillations f as a function of latitude. Lines to the left of f are
unresolved high vertical modes by flow over topography forD2 andD1; those to the right are for (1/2)D2 and (1/2)D1.
(Arbic et al. 2004, 2010). Ansong et al. (2015) showed
that the barotropic and low-mode baroclinic tides in meter spectral densities; Ansong et al. (2017), who com-
simulations forced simultaneously by tides and atmo- pared the baroclinic tidal energy fluxes to fluxes computed
spheric fields compare more closely to satellite altimeter from current meter records; and Savage et al. (2017), who
observations when a parameterized internal wave drag compared the dynamic height frequency spectral densities
is applied to the bottom flow. Buijsman et al. (2016) with in situ depth profiling instruments.
showed that about 50% of the internal wave dissipation Figure 1 depicts the local period of inertial oscillations
can be attributed to bottom drag in 1/12.58 HYCOM. versus latitude. On the same plot we show the periods of
Because of storage limitations, the recorded hourly the diurnal and semidiurnal tides. We note that this is a
global three-dimensional model output is held to 60 days schematic that is only used to help in interpretation; in
for the runs at 1/12.58 resolution and to 30 days for the the real ocean the tidal periods could be Doppler shifted
1/258 simulation. This amount of data requires about by other motions. The CLs of the three diurnal tidal
48 terabytes of disk space from all simulations. Other constituents (Q1,O1,K1)—the location where half their
previous papers (Arbic et al. 2010, 2012; Timko et al. frequency equals the local inertial frequency—are 12.98,
2012, 2013; Richman et al. 2012; Shriver et al. 2012, 2014; 13.48, and 14.58, respectively. The CLs of the three
Ansong et al. 2015; Buijsman et al. 2015, 2016) can be semidiurnal constituents (N2,M2, S2) are 28.28, 28.88,
consulted for detailed discussions of the parameterized and 29.98, respectively. Because the semidiurnal CLs
topographic internal wave drag, self-attraction and occur near the turning latitudes of the diurnal tides,
loading, and other important details in our tide- spectral separation is impossible without very long time
resolving HYCOM simulations. Shriver et al. (2012) series data. This is the main reason that we forced the
demonstrated that the amplitudes of the barotropic and model with the diurnal and semidiurnal tides separately,
internal tide sea surface elevations inHYCOMcompare in contrast to the real ocean where the tidal constituents
well to altimeter-constrained models and along-track occur concurrently. The semidiurnal and diurnal signals
altimeters, respectively. The tidal currents in HYCOM are obtained by a Butterworth filter that bandpasses
have been compared to archived current meter records between 1.55 and 2.32 cpd and 0.80 and 1.20 cpd, re-
spanning about 40 years (Timko et al. 2012, 2013). On spectively. Their subharmonics are obtained by a band-
average over 5000 locations, Timko et al. (2013) find that pass between 0.80 and 1.20 cpd and 0.40 and 0.60 cpd,
the kinetic energy of the M2 tide is within a factor of respectively. We use HYCOM12S and HYCOM12D to
1.3 of observations. Other model–data comparisons represent the HYCOM simulations forced by the
performed with HYCOM include Müller et al. (2015), semidiurnal and diurnal tides, respectively, at 1/12.58
who compared the internal gravity wave (IGW) kinetic resolution, and HYCOM25S to represent the 1/258
energy (KE) frequency spectral densities to current simulation. The Rayleigh criterion that the differences
1412 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48
between resolved frequencies must be greater than the seawater, and C is a linear wave drag coefficient with
inverse of the data record length implies that the local units of meters per second. In Eqs. (3) and (4), (u, y) are
near-inertial oscillations cannot be separated from the horizontal velocity components in the zonal and meridi-
semidiurnal subharmonic waves between about latitudes onal directions, respectively, and velocities in parentheses
627.18 and 30.68 for 30–60 days. are bandpassed, whereas the term juaj is the total un-
filtered velocities. The variables ula and yla are the zonal
(u) andmeridional (y) velocity components, respectively,
3. Theory of the low-mode tide averaged over the bottom 10m.
a. Baroclinic energetics
b. PSI energy transfer
The energy balance analyses of the total barotropic
We next present the basic theory of energy transfer to
and baroclinic semidiurnal tides in a HYCOM simula-
subharmonic signals from the low-mode tide, following
tion run at 1/12.58 horizontal resolution are given in
previous studies (e.g., Frajka-Williams et al. 2006;
Buijsman et al. (2016). In this study, we analyze the
MacKinnon et al. 2013b; Sun and Pinkel 2013). In theory
baroclinic energy balance following the same approach
the wave vectors and frequencies (k1, v1), (k2, v2),
as discussed in Buijsman et al. (2016), namely,
(k3, v3), of the waves in a PSI triad interaction satisfy
f g f  g f g the resonant conditionsC 5 = F 1 D , (1)
l
v 1 v 5v , k 1 k 5k , (5)
where curly brackets indicate area and time averages, 1 2 3 1 2 3
C is the barotropic to baroclinic conversion rate, F is the with v1 ’ v2 5 v3/2. The variables in Eq. (5) with sub-
baroclinic energy flux vector, and Dl is the energy dissi- scripts 1 and 2 refer to the daughter waves of PSI, and
pation. Time-averaged and domain-integrated values of the third wave is the parent wave. Consider the mo-
the terms in Eq. (1) represent the amount of energy con- mentum equations for the inviscid Boussinesq system
verted from the barotropic to the baroclinic tide (con-
version); positive flux divergence represents generation ›u
1 u  =u5OT, (6)
of baroclinic energy while negative divergence in- ›t
dicates dissipation or energy conversion to barotropic
tides. The dissipation term represents all processes where OT refers to other terms not considered in our
removing energy from the internal tide including analysis, and let
wave–wave interactions and the parameterized topo-
u5 u 1 u 1u . (7)
graphic internal wave drag (see section 2). The energy 1 2 3
dissipation term is further separated into the dissipa- The velocities may be expressed in terms of Fourier
tion due to the topographic wave drag Dwl and qua- coefficients; for example, u 5 û ei(k1 x2v1t)1 1 1 c.c., where
dratic bottom friction Dbl, such that (Buijsman et al. c.c. refers to the complex conjugate. The energy equa-
2016) tion for one of the daughter waves, say u1, can be derived
f g f g f g f g by first substituting Eq. (7) into Eq. (6) and multiplyingD 5 D 1 D 1 R , (2)
l wl bl l through by u*1 (the complex conjugate of u1). By aver-
aging over large time and space scales, the oscillatory
where Rl is a residual term composed of discretization
terms vanish, resulting in (Sun and Pinkel 2013)
errors, viscous dissipation, and energy loss due to non-
linear wave–wave interactions of the tide and other ›E
1
motions. As in Buijsman et al. (2016), the wave drag and 1u*  (u 1 u 1 u )  =(u 1 u 1 u )5 0, (8)›t 1 1 2 3 1 2 3
quadratic bottom friction terms are computed using the
2
linear separation technique of Kang and Fringer (2012), where E1 5 (1/2)ku1k 5 (1/2)(u*1u1 1 y*1y1) and where
to arrive at we omit the ^ symbol in the above expression and
subsequent equations. The terms u*  (u2  =u3) and
fD g5 hr C ju jðu u 1 y y Þi 1, and (3) u*1  (u3  =u2) control the rate of energy transfer since
bl 0 D a a la a la
h ð i they drive u1 resonantly if the waves satisfy the resonantfD g5 r C u u 1 y y ) , (4)
wl 0 a la a la conditions. Thus, Eq. (8) reduces to the energy equation
for u1:
where the bottom drag coefficient is CD 5 0.0025; sub-
scripts a and a indicate averaging over the bottom ›E1 5 [2u*  (u  =u )2 u*  (u  =u )]1 c. c: (9)
10 and 500m, respectively; r0 is the average density of ›t
1 2 3 1 3 2
JUNE 2018 AN SONG ET AL . 1413
The equation governing energy transfers to u2 can wave propagation, and the divergence of the energy flux
be derived in a similar manner. Previous studies (e.g., quantifies energy sources and sinks (Nash et al. 2005). For
MacKinnon and Winters 2005; Young et al. 2008) the semidiurnal tides, the map of conversion rates from
suggest that, near the critical latitude, the term T5 our simulations (Figs. 2a,b) are comparable to previous
2u*  (u2  =u3) is the primary driver of PSI rather than studies (e.g., Egbert and Ray 2003; Simmons et al. 2004;1
the term 2u*  (u3  =u2). We will show that, indeed, Niwa and Hibiya 2011; Buijsman et al. 2016), showing1
the term 2u*  (u3  =u2) is negligible everywhere in the concentrated activity at hotspots such as midocean ridges1
model. We will compute energy transfers from term T, and shelf slopes. We find greater conversion in HY-
which m
ay be written inexpan
ded form as  COM25S than in HYCOM12S, especially in the AtlanticOcean. Because the internal tides are generated by flow
›u ›u ›y ›y
T52u* u 3 1 y 3 2 y* u 3 1 y 3 . (10) over topography, a higher-resolution model leads to
1 2 ›x 2 ›y 1 2 ›x 2 ›y better-resolved topographic features and hence greater
conversion rates. The conversion of the diurnal tides
In the equations above we have ignored the partial de-
largely lie within their turning latitudes (between latitudes
rivatives with respect to z because, close to the critical
308N and 308S), where theory predicts they can freely
latitudes where the PSI daughter waves have enhanced
propagate (Fig. 2c). However, diurnal tides may also exist
energy, the waves have vanishing vertical velocities and
poleward of their turning latitudes as coastally trapped
displacements (MacKinnon et al. 2013b), resulting in the
waves and as topographically generated waves in highly
relatively simpler expression in Eq. (10). To compute
constricted tidal straits (Niwa and Hibiya 2011), as seen,
the energy transfer terms, we first employ bandpass fil-
for example, around the Aleutian Islands. We find here
tering to obtain the horizontal velocities of the semi-
that most of the diurnal conversion is concentrated around
diurnal tide. The vertical wavenumber resonant condition
the western side of the Pacific and the Indian Oceans,
in Eq. (5) implies that the daughter waves in a PSI triad
consistent with Niwa and Hibiya (2011).
interaction with smaller vertical scales than the tide have
The globally integrated conversion rates in
oppositely signed vertical wavenumbers. Thus, to obtain
HYCOM12S and HYCOM25S are C 5 0.49TW and
the subharmonic signals at each grid point, we first
0.61TW, respectively (Figs. 2e,f). The amount of con-
bandpass for velocities with half the frequency of the tide,
version fromHYCOM12S is comparable to the 0.53TW
and then use rotary analysis to separate the signals into
obtained by Buijsman et al. (2016) using a HYCOM
clockwise (CW) and counterclockwise (CCW) compo-
simulation forced by the eight major diurnal and semi-
nents (Emery and Thomson 1997; MacKinnon et al.
diurnal tides. The parameterized wave drag used here
2013b). For linear internal waves, a sense of CW rotation
dissipates about Dwl 5 0.20TW of baroclinic energy in
with increasing depth is consistent with downward
HYCOM12S and about 0.28TW in HYCOM25S. We
(upward) energy (phase) propagation, while CCW ro-
note that the wave drag also dissipates about 0.71TW
tation is consistent with upward energy propagation
of barotropic energy in HYCOM12S and 0.61TW in
(Leaman and Sanford 1976; MacKinnon et al. 2013b).
HYCOM25S (not shown). The dissipation due to the
In addition, we will integrate over the tidal and sub-
low-mode tides, Dl, is about the same as the conversion
harmonic frequency bands to determine the amount
rate since the global integral of the flux divergence is
of spectral energy in each band. We will employ the
close to zero [see Eq. (1)]. As explained in Buijsman
subharmonic energy ratio (SER; Chou 2013; Chou
et al. (2016), the baroclinic dissipation by bottom fric-
et al. 2014) criterion, based on the power spectral
tion can be negative, as we see in Figs. 2e and 2f, as a
density (PSD), given by
result of the linear separation employed in the calcula-
PSD at subharmonic band tions. The linear separation method is used because of
SER5 , (11) its simplicity; a nonlinear approach yields additional
PSD at tidal band
cross terms. Figure A1 in Buijsman et al. (2016) com-
to quantify the fraction of energy in the subharmonic pares the two methods, showing that the nonlinear ap-
motions. In the rest of the paper, we present the results proach results in dissipative bottom friction.
of the different analyses. The globally integrated barotropic-to-baroclinic
diurnal conversion is 0.08TW, about 16% of the
HYCOM12S amount. Niwa and Hibiya (2011) obtain
4. Results of global energy balance analysis
approximately 0.12 (0.13) TW from the sum of K1 and
The barotropic to baroclinic conversion rates inform us O1 in a simulation run at 1/108 (1/158) horizontal reso-
about regions where internal tide energy is available to lution. Most of the diurnal energy conversion takes
potentially contribute to mixing, the energy flux identifies place in the western Pacific and Indian Oceans; very
1414 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48
FIG. 2. (left) Barotropic to baroclinic energy conversion (mWm22) for (a) HYCOM12S (b) HYCOM25S, and
(c) HYCOM12D. (right) Baroclinic energy balance analysis for corresponding resolutions in the left panels, with
barotropic to baroclinic conversionC, low-mode dissipationDl, topographic wave drag dissipationDwl, dissipation
due to bottom friction Dbl, and residual term Rl.
little takes place in the Atlantic. This is consistent with and semidiurnal internal waves into a broad contin-
previous studies (e.g., Egbert and Ray 2003; Niwa and uum of higher-frequency and higher-wavenumber
Hibiya 2011). The topographic wave drag in our diurnal internal waves. Here, one of our goals is to estimate
simulation dissipates approximately 0.05TW of energy, the amount of energy transferred from the low-mode
about 60% of the diurnal conversion. This diurnal wave internal tides to PSI subharmonic signals and to de-
drag to conversion ratio is larger than the 40% seen in termine whether this amount could account for a sig-
the semidiurnal tides. The larger diurnal ratio is likely nificant fraction of the residual term in the baroclinic
because the wave drag in HYCOM is tuned for the tidal energy budget.
semidiurnal tides and therefore may ‘‘overdamp’’ the
diurnal energy.
In both the semidiurnal and diurnal tides, a substantial 5. Subharmonic tidal signals
fraction of the baroclinic energy budget, about Rl 5 a. Example calculations
0.4TW for the semidiurnal tides and 0.03TW for the
diurnal tides, is unaccounted for. This residual energy Figure 3a shows an example time series of the band-
dissipation may be attributed to viscous and numerical passed northward velocity component of the sub-
dissipation in themodel (Buijsman et al. 2016), as well as harmonic NIWs, in the upper 2000m, at location MP3
loss to subharmonic signals via wave–wave interactions (28.938N, 163.448W; MacKinnon et al. 2013b) from HY-
such as the PSI mechanism. A previous study by Müller COM12S. After bandpassing for the subharmonic signal,
et al. (2015), which employedHYCOM simulations with we then employ rotary spectral analysis to decompose it
the same horizontal resolutions used here, showed into CW (Fig. 3b) and CCW (Fig. 3c) motions. At this
that energy is transferred out of the low-mode inertial location we see that there is subharmonic energy in both
JUNE 2018 AN SONG ET AL . 1415
FIG. 3. (a)–(c) Time series of northward velocity (m s21) fromHYCOM12S near locationMP3 (29.18N, 163.58W)
aroundHawaii showing the bandpassed signals at the (a) subharmonic near-inertial frequency band (NIW), and the
component of the subharmonic signal rotating (b) CW and (c) CCW. (d) The PSD of the signals in (b) and (c), and
(e) the PSD at location (20.58N, 163.58W). The green and magenta dashed lines indicate the frequencies of the
semidiurnal tidal constituents and their subharmonics, respectively, and the blue dashed line show the local inertial
frequency.
directions of motion but with greater energy in motions near-inertial waves from these two mechanisms at the
with CW rotation. This is shown quantitatively in Fig. 3d, critical latitude of the semidiurnal tides. That is the
where the PSD in themotionswithCWrotation is greater reason for further employing bispectral analysis (see
than the PSD for motions with CCW rotation. More- section 7 and appendix), to help identify interactions
over, the energy in the CW subharmonic band is about that are largely caused by the PSI mechanism. Thus,
the same as the energy in CW motions in the tidal the subharmonic signals considered in this section
band. This is likely due to the presence of local near- likely contain both the wind-generated NIW as well as
inertial oscillations (NIOs), which have the same those from PSI. The PSD at a location equatorward of
frequency as the subharmonic waves at this latitude the CL, where the tidal subharmonic signals and the
and could increase the energy in the subharmonic local near-inertial oscillations are easily separated, is
band. The numerical simulations of Hazewinkel and depicted in Fig. 3e; showing that the strengths of the
Winters (2011) show that the PSI mechanism appears upward and downward subharmonic motions are
to generate NIWs in the upper ocean with subsequent much closer to each other.
propagation into the ocean interior, similar to the
b. Evidence for PSI
locally generated NIOs, which are constrained to
propagate into the ocean interior. This implies that We present below a few example calculations showing
it is impossible to use only PSD to separate the that the characteristics of the subharmonic signals in
1416 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48
FIG. 4. (a) Snapshot of semidiurnal tidal signal of meridional baroclinic velocity from HYCOM12S close to
location MP3 (29.18N, 163.58W) around Hawaii. (b) As in (a), but after bandpassing around the subharmonic
frequency band. Observe the high vertical wavenumber structure in the upper ocean around the CL. (c) A vertical
profile through theM2 critical latitude (;28.88) from (b), showing the vertical scale of the subharmonic signal. (d)A
horizontal slice of (b) through 1000-m depth, showing the horizontal scale of the subharmonic signal in the critical
latitude band (258–308N).
certain locations are consistent with those of inertial PSI transfers energy to the smallest vertical scales al-
waves generated via the PSI mechanism. Figures 4a lowed by the formulation of the problem. The model
and 4b are a snapshot of baroclinic meridional velocity used here has thinner isopycnal layers in the upper
through location MP3 around Hawaii (28.88N, 163.58W) ocean, and somewhat thicker layers with depth. The
depicting both the tidal and subharmonic motions from average thickness of isopycnal layers between 100 and
HYCOM12S.We see a higher vertical wavenumber signal 1000m is about 100m. Thus, because PSI is large in the
in the upper ocean around the M2 CL. Outside the CL upper ocean where we have thinner layers, we expect to
region the subharmonic signals are weaker but they have have our best chance of resolving the daughter waves
comparable strength to the tide around theCL. TheNIWs there. On average, the subharmonic signals in Fig. 4c
from PSI have vanishing vertical velocities at the CL (e.g., have vertical wavelength lm of about 300m in the upper
MacKinnon et al. 2013b), and so we expect a concentra- ;2000m of the ocean. The vertical scales are smaller in
tion of subharmonic energy around the CL. Though the the upper ocean and increase in scale with depth. This
wind-generated NIWs have the same frequency as the is consistent with observations in MacKinnon et al.
subharmonic signals at the CL, they are free to propagate (2013b) around the same location, where they estimate a
away and so are not expected to have energy concen- vertical wavelength of about 200m (in the ;400–750-m
trated around the CL. We also note that the high vertical depth range) with larger scales at depth. The vertical
wavenumber disturbance seen in Fig. 4b is similar to dis- scales are also comparable to those in observations by
turbances found by Simmons (2008) in a global model run Alford et al. (2007), though the magnitudes of the ve-
without atmospheric wind forcing (see his Fig. 2). locities are much smaller in our model. Other observa-
We briefly discuss below the vertical and horizontal tional studies around Hawaii find the subharmonic
scales of the subharmonic signals around the CL. The waves from PSI to have vertical scales between 50 and
study by Hazewinkel and Winters (2011) suggests that 150m (Sun and Pinkel 2013).
JUNE 2018 AN SONG ET AL . 1417
FIG. 5. (a) Snapshot of the subharmonic baroclinic kinetic energy density (u2 1 y2) from the vertical section in
Fig. 4b, computed from HYCOM12S. (b) Temporal variation of vertically averaged subharmonic kinetic energy
density from the vertical section in (a).
Horizontal-wavenumber spectra computed from the near-inertial as expected. The subharmonic baroclinic
tidal and subharmonic bands at different depths in the KEdensity from the vertical section inFig. 4b is displayed
upper ocean show a range of lateral scales (not shown). in Fig. 5a, further emphasizing the upper-ocean in-
The tides peak at wavelengths of around 90–150km, and tensification of subharmonic KE density around the CL.
the subharmonics have wavelengths in the 60–100-km We find from Fig. 5b that the vertically averaged sub-
range. For example, we estimate a horizontal wave- harmonic KE densities roughly follow the spring-neap
length lk of about 100 km around the CL band at a depth cycle of the tidal KE, signifying that they are likely gen-
of about 1000m. This horizontal wavelength is compa- erated via the PSI mechanism (MacKinnon and Winters
rable to the lateral scales of near-inertial waves reported 2005; Alford et al. 2007; Hazewinkel and Winters 2011).
from observations (Alford et al. 2017). Figure 4d shows The bispectrum and bicoherence calculations (briefly
that the horizontal scale of the subharmonic signal is not explained in section 7 and the appendix) at this loca-
uniform across the CL. As a check, we can separately tion are displayed (in Fig. A1), showing significant
compute the frequency of the waves v associated with bicoherence at several vertical levels.
lm 5 300m and lk 5 100 km from the dispersion re-
lationship for internal gravity waves:
6. Global PSD calculations
m2
2
2 f
2 1k2N2 f 2 1 (k=m) N2 At each grid location, we integrate the PSD over the
v 5 5 , (12)
m2 1 k2 11 (k=m)2 tidal and subharmonic bands for both the CW and CCW
components. We emphasize that the PSD calculations
wherem5 2p/lm is the vertical wavenumber, k5 2p/lk from the subharmonic band alone are insufficient to
is the horizontal wavenumber, f is the Coriolis fre- distinguish PSI subharmonics from other oceanic mo-
quency, and N is the buoyancy frequency. Because tions of similar frequency in a model that includes at-
(k/m) ’ 3 3 1023 and N ’ 3 3 1025 s21, we see from mospheric forcing in addition to tidal forcing. The
Eq. (12) that v ’ f, indicating that the waves are calculations here present only an initial map of the
1418 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48
FIG. 6. Global distribution of bandpassed and vertically integrated PSD from HYCOM12S
for the (a) semidiurnal band, (b) subharmonic band, and (c) SER. Note that themaximumSER
in (c) is about 400.
distribution of variance in the total subharmonic band. and wind-generated near-inertial waves. The presence
Figure 6a is the band-integrated and vertically of atmospheric forcing in our high-resolution model
integrated PSD from the HYCOM12S tidal band. For ensures a vigorous mesoscale eddy field, comparable to
brevity, we display only the sum of the CW and CCW observations and yields large variabilities in the internal
components of the integrated PSD. The map reveals the tides in both the model and observations (Ansong et al.
hotspots of tidal activity similar to themap of conversion 2017). Figure 6c shows the SER [Eq. (11)], which has a
(see Fig. 2a). Figure 6b is similar to Fig. 6a but for sub- maximum value of about 400. Most large values of SER
harmonic signals with half the frequency of the tide. We are concentrated around the CL. We employ the SER
see a distribution of large subharmonic PSD around the later to filter out less energetic subharmonic signals
CL, as in the results of Simmons (2008, the bottom panel outside the critical latitude regions, especially in our
of his Fig. 3). However, in our results there is a greater bicoherence calculations. The vertically averaged PSDs
latitudinal spread around the CL as well as a continuous from HYCOM25S show larger kinetic energy variances
longitudinal distribution of these subharmonic signals, in both the tidal and subharmonic bands than in
in contrast to the patchy signals along the CL ob- HYCOM12S. Moreover, the SER from HYCOM25S
served in Simmons (2008). These differences in our re- (not shown) is also concentrated around the CL but
sults are probably caused by the interactions of the tides appears patchier than theHYCOM12S results in Fig. 6c.
with other motions and by the presence of atmo- The distribution of vertically integrated PSD from the
spherically forced motions such as mesoscale eddies diurnal frequency band is shown in Fig. 7a. We find that
JUNE 2018 AN SONG ET AL . 1419
FIG. 7. As in Fig. 6, but for HYCOM12D.
most of the diurnal kinetic energy variance occurs equa- difference frequencies of the primary waves. To quan-
torward of their turning latitudes (6308), as expected from tify nonlinear interactions, one must resort to higher-
theory, as well as in certain regions such as the Aleutian order spectral analysis techniques (Nikias and Petropulu
Arc and the Sea of Okhotsk, likely caused by topo- 1993). For signals arising from wave–wave interactions,
graphically trapped waves (Egbert and Ray 2003; Niwa the cumulant bicoherence spectrum can be used to
and Hibiya 2011) as mentioned earlier. The subharmonic measure the extent of the joint dependence of the
signals from the diurnal band are much weaker than their spectral components (Kim and Powers 1979). Unlike
semidiurnal counterparts (Fig. 7b), but they also have the power spectrum, the bicoherence spectrum may be
larger variances along their CL. The map of SER in this used to separate nonlinearly coupled waves (which ex-
case (Fig. 7c) reveals patchy locations of high subharmonic hibit phase coherence) from spontaneously excited in-
variance along the CL. We also find relatively high vari- dependent waves, without reference to the waves’
ance occurring poleward of theCL, in both the diurnal and amplitudes (Kim and Powers 1979). These authors also
semidiurnal PSD. This is likely caused, in part, by leakage show that the squared bicoherence can be used to
of mesoscale motions into the subharmonic band. measure the fraction of power in a given spectral band
due to the quadratic coupling. We have given a brief
summary of this statistical technique and the related
7. Results of bispectral analyses
concept of bispectrum in the appendix.
When different spectral components interact non- Other previous investigations of PSI in the ocean have
linearly, the resulting signal contains frequencies of the employed bispectral analysis (Hasselmann et al. 1963;
primary waves as well as daughter waves with sum and Furue 2003; Furuichi et al. 2005; Carter and Gregg 2006;
1420 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48
Frajka-Williams et al. 2006; MacKinnon et al. 2013b; effects of the background flow might partially explain
Sun 2010; Sun and Pinkel 2012, 2013; Chou et al. 2014; the presence of significant bicoherence values poleward
Xie et al. 2016). Because bispectral analysis was de- of the CL. The bispectral results for HYCOM25S are
veloped for stationary and random processes, its appli- displayed in Figs. 8c and 8d and illustrate a similar dis-
cation to nearly deterministic signals like the tides is tribution of significant bicoherence as in HYCOM12S.
challenging (Chou 2013). Another difficulty is the pos- The bicoherence takes values between 0 and 1. Here, we
sibility of obtaining significant bicoherence levels from find that most of the significant bicoherence values in
synthetic signals that are not interacting nonlinearly HYCOM25S generally lie closer to 1 (with a global av-
with each other (Chou et al. 2014). The work of Sun erage of about 0.73), whereas the values in HYCOM12S
(2010) and MacKinnon et al. (2013b) provides useful lie in the lower bound of the bicoherence range
information on the application of bispectral analysis to (with a global average of about 0.43). This implies
internal tides, as employed in this study. that the phase correlations between the tides and
In the following, we use the SER, as well as bispectrum the subharmonic signals in HYCOM25S are generally
and bicoherence calculated at each model layer, to de- stronger than in HYCOM12S. Thus, higher-resolution
termine the global patterns of bicoherence from our sim- simulations are likely to facilitate greater wave–wave
ulations. Using the CW and CCW components of the interactions between the tides and the subharmonics.
subharmonic complex velocity time series, we calculate the The bicoherence values in the diurnal case are generally
bicoherence at each grid point using the approach of Kim much smaller than in HYCOM12S, as displayed in
and Powers (1979). The calculation yields a value be- Fig. 9, and appear to be contaminated by the mesoscale
tween 0 and 1 at each grid location, with 1 indicating high motions, which are present in the bandpassed signals.
bicoherence (implying high phase correlation between the
b. Vertical distribution of bicoherence
tides and the subharmonic signals). We use significance
levels for the bicoherence developed by Elgar and Guza Figure 10a shows a zonally averaged meridional sec-
(1988). In our case values of 0.35 (for HYCOM12S tion of bicoherence driven by semidiurnal tides in the
and HYCOM12D) and 0.5 (for HYCOM25S) indicate Northern Hemisphere for HYCOM12S. Significant bi-
bicoherence at the 95%significant levels respectively (see coherence values occur throughout the water column
the appendix). We plot bicoherences at locations where around the CL with a peak just equatorward of the CL
both the peak bispectrum and significant bicoherence (Figs. 10a,b). We observe that even though PSI may
occur within the subharmonic frequency band. occur anywhere equatorward of the CL, its intensity
quickly falls off about 58 from the CL. This result is
a. Geographical distribution of bicoherence
consistent with the two-dimensional idealized results of
Figures 8a and 8b show the results of the vertically Furuichi et al. (2005, see their Fig. 5). Figures 10c and
averaged bicoherence calculations for HYCOM12S, 10d show that the significant bicoherence values occur-
while Figs. 8c and 8d show the same results for ring away from the CL region have smaller energy
HYCOM25S. It is likely that in some locations, espe- (based on the SER criterion). Figures 10c and 10d also
cially poleward of the CL, we will find subharmonic show that the interactions that are both energetic and
signals with significant bicoherence alongside small en- significant occur in the upper ocean and are concen-
ergy levels, as demonstrated by the SER calculations in trated around the CL. For instance, the significant
section 6. We use the SER here to filter out weak sub- bicoherence values occurring in the deep ocean in
harmonic signals that have significant bicoherence Fig. 10a have smaller averaged SER values than those in
values. For example, in Fig. 8a, locations with SER , 1 the upper ocean. Therefore, the prominence of the deep
are not plotted for clarity. Figure 8a displays a concen- ocean after the SER criteria has been applied is less, as
tration of significant bicoherence values around the CL. seen in Fig. 10d. As discussed in previous studies
Neglecting signals with SER , 2 further narrows the (MacKinnon and Winters 2005; Young et al. 2008;
distribution of the energetic signals with significant bi- Hazewinkel and Winters 2011), the subharmonic mo-
coherence to the CL (Fig. 8b). The observational study tions draw their energy from horizontal gradients of the
of Xie et al. (2016) suggests that near-inertial waves internal tides [see Eq. (9) and following discussion].
induced by PSI can be transported poleward beyond Thus, in a model with surface-intensified stratification
their CL by background geostrophic flow. Similarly, and consequent surface-intensified internal tides, PSI is
Richet et al. (2017) find two weak dissipation peaks of expected to be concentrated in the upper ocean as we
internal tides due to PSI at latitudes ;258 and ;358, find here. Because the rates of energy transfer in PSI are
from idealized numerical studies, and attributed this to proportional to the energy in the waves, transfer terms
the presence of a mean flow in their domain. Thus, the such as u›u/›x are amplified in highly stratified locations
JUNE 2018 AN SONG ET AL . 1421
FIG. 8. Global distribution of vertically averaged bicoherence for (a) HYCOM12S (SER.
1), (b)HYCOM12S (SER. 2), (c)HYCOM25S (SER. 1), and (d)HYCOM25S (SER. 2).
The plots display significant bicoherence values at the 95% significance level (b95%5 0.35 for
HYCOM12S and 0.50 for HYCOM25S), and every sixth grid point is plotted.
such as the thermocline (Frajka-Williams et al. 2006; the CL with significant bicoherence values is broader in
MacKinnon et al. 2013a). As further presented in Carter HYCOM25S (Fig. 10e) than in HYCOM12S and displays
and Gregg (2006), the ratio of nonlinear to linear terms two smaller peaks in the vertically averaged plot
in the equations of motion, such as u›u/›x/(›u/›tp), sffifficffiffiaffiffilffiffieffiffis (Fig. 10f). In contrast to HYCOM12S, we find significant
as the square root of the buoyancy frequency, N(z), bicoherence values deeper in the ocean in HYCOM25S
such that the nonlinear interactions could be expected to (Fig. 10h). Figure 10 (right panels) are the counterparts
be larger in regions of strong stratification. of Fig. 10 (left panels) but for HYCOM12D. Similar to
The center panels in Fig. 10 display the vertical distribu- the distribution in HYCOM12S, we find a concentration
tion of zonally averaged bicoherence for HYCOM25S. The of significant bicoherence values centered around the
pattern of bicoherence is similar to that of HYCOM12S diurnal CL over all depths. In this case, we also see
but generally shows higher bicoherence values, as men- significant bicoherence values poleward of the CL
tioned earlier. In addition, we see that the region around (Figs. 10i,j) as also reported by Xie et al. (2016).
1422 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48
FIG. 9. As in Fig. 8, but for HYCOM12D.
However, as shown in Figs. 10k and 10l, these poleward MP3, we find relatively large transfers between 300- and
bicoherence values are weaker than those equatorward 550-m depth (Fig. 11b, magenta curve). We note that the
of the CL. energy transfer rates (;10210Wkg21) from both profiles
are about an order of magnitude smaller than in observa-
tions around the same location (MacKinnon et al. 2013b,
8. Energy transfer rates
their Fig. 11). A horizontal slice of transfer rates around
We next present estimates of energy transfer rates MP3 at about 400-m depth is displayed in Fig. 11c. We see
computed using Eq. (9). We find that the term about the same amounts of positive and negative transfers
2u*1  (u3  =u2) is negligible everywhere compared to around MP3 with a maximum transfer rate of about 1.13
T52u*  (u  =u ) as a primary driver of PSI, as sug- 1029Wkg211 2 3 at (29.28N, 197.688E). The bispectral calcula-
gested by previous studies (not shown for brevity; see tions near MP3 are given in the appendix (Fig. A1), and
section 3).We thus compute the energy transfer rates using show significant bicoherence values in the 300–1000-m
the latter term in which u1 and u2 are the CW and CCW depth range. Figures 11d and 11e are the counterparts of
components of the bandpassed subharmonic velocities and Figs. 11a and 11b but for HCYOM25S. In this case, we find
u3 is the tidal velocity. Figure 11a displays a meridional more locations with relatively large transfer rates in the
section of energy transfer rates in the North Pacific region vertical section than inHYCOM12S. UnlikeHYCOM12S,
fromHYCOM12S.We see relatively large energy transfers the energy transfers also extendmuchdeeper into thewater
in the upper 100m of the ocean, the result of mixed layer column (;2500-m depth, not shown). The vertical profile
processes. Below 100m, we find patchy locations of posi- of energy transfer near MP3 (Fig. 11e, blue curve) shows
tive energy transfers (transfers from the tidal band to sub- largely negative transfers in HYCOM25S, unlike what we
harmonic signals) as well as negative/reverse transfers find in HYCOM12S (Fig. 11b, blue curve). Thus, the
(transfers from the subharmonics to the tidal band), con- dominant sign of energy transfer with depth appears to be
sistent with previously reported observational results influenced by local conditions and model resolution. En-
around Hawaii (MacKinnon et al. 2013b; Sun and Pinkel ergy transfers at the location just east of MP3 (Fig. 11e,
2013). The energy transfers in this case are largely located magenta curve) appear to have a similar vertical structure
in the upper 1000m of the ocean. A vertical profile of as in HYCOM12S, with largely positive energy transfers
transfer rates around location MP3 is displayed in Fig. 11b with depth.
(blue curve) showing positive transfers over most of the Figures 12a and 12b show vertically integrated glob-
profile below 200-m depth. A few degrees to the east of al energy transfer rates from the semidiurnal band.
JUNE 2018 AN SONG ET AL . 1423
FIG. 10. (a) Variation in depth and latitude of zonally averaged bicoherence for HYCOM12S. (b) Vertically averaged bicoherence
values in (a), showing a drop-off in intensity of bicoherence away from the critical latitude. The dashed vertical lines indicate the position
of theM2 critical latitude. (c),(d)As in (a), but with zonally averaged SER. 0.1 and SER. 0.5, respectively. (e)–(h)As in (a)–(d), but for
HYCOM25S. (i)–(l) As in (a)–(d), but for HYCOM12D.
The figures display a concentration of relatively large indicates that those subharmonic signals may have come
transfer rates around the CL after employing the SER from other motions rather than from PSI. Similar to the
criterion. Comparing Fig. 6b to the more rigorous cal- semidiurnal energy transfers, the diurnal transfer rates
culation of energy transfers in Fig. 12a, we see that some are both positive and negative, with a net posi-
of the signals appearing in Fig. 6b along the critical lat- tive transfer. In addition, the diurnal transfer rates are
itude (e.g., in the southeast Pacific region) vanish. This generally much smaller than the semidiurnal transfers
1424 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48
F . 11. Energy transfer rates from HYCOM12S (a) at a zonal section along latitude 29.148E (310210Wkg21IG ), (b) in a vertical profile
near location MP3 (blue curve) and just to the east of MP3 (magenta curve), and (c) at a depth of about 400m near MP3. (d),(e) As in
(a) and (b), but for HYCOM25S. The white dashed lines in (a) and (d) show the positions of profiles given by the blue andmagenta curves
in (b) and (e).
(Fig. 12c; note the different color scale used for clarity). (Fig. 12c), likely because diurnal tides are less energetic
As discussed in section 4, this is likely due to the fact that than semidiurnal tides.
the diurnal internal tides are generally weaker than the Figure 12d displays the zonally averaged semidiurnal
semidiurnals, due in part to the effect of the topographic energy transfer rates for the positive and negative
wave drag scheme that dissipates a comparatively larger components. We see that the transfers are almost
fraction of diurnal energy. We see relatively large di- equally partitioned between the positive and negative
urnal energy transfers in patchy locations in the east transfers with a net positive transfer over the globe. The
equatorial Pacific region and the southwestern Pacific globally integrated positive and negative energy trans-
region (to the northeast of Australia) around the diurnal fers are 10.197TW and 20.196TW, respectively, for
CL. These locations also show high diurnal subharmonic HYCOM12S, resulting in a net amount of ;0.001TW.
variance as seen in Figs. 7b and 7c. We also find that This amount is about 0.5% of the 0.40TW residual en-
the diurnal transfer rates appear contaminated by ergy transfer needed to close the HYCOM12S baro-
the mesoscale motions, especially around the Gulf clinic energy budget. In the case of HYCOM25S, the
Stream, Kuroshio, and Antarctic Circumpolar Current globally integrated amount of energy transfer is much
JUNE 2018 AN SONG ET AL . 1425
FIG. 12. Vertically integrated global distribution of energy transfer rates (Wkg21) be-
tween low-mode tides and subharmonic signals for (a) HYCOM12S, (b) HYCOM25S, and
(c) HYCOM12D. (d) Zonally averaged energy transfer rates from HYCOM12S (H12S) and
HYCOM25S (H25S), showing that the positive and negative transfers have about the same
magnitude, leading to a negligible net energy transfer. ‘‘T’’ is used to denote the sign of energy
transfer.
larger, with values of10.48TW and20.44TW, giving a integrated positive and negative amounts of energy
net amount of 0.04TW. This amount is about 10% of the for HYCOM12D are 10.086TW and 20.074TW, re-
residual energy transfer. Thus, increasing the horizon- spectively, giving a net value of 0.012TW. The net
tal model resolution increases the amount of energy amount of energy transfer in this case is about 4.0% of
transfer by about an order of magnitude. The globally the residual (;0.03TW).
1426 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48
We point out that the residual energy flux going into internal tides is dissipated by the wave drag. A signifi-
PSI is a small difference of large numbers and changes cant fraction of baroclinic energy dissipation of the
with model resolution, making it difficult to estimate the semidiurnal tides (;0.40TW) and the diurnal tides
uncertainty in the energy transfer values. For example, (;0.03TW) is not accounted for in our baroclinic energy
the globally integrated positive and negative transfers balance. This amount of residual energy dissipation is
from HYCOM25S are each about 0.5TW, which we attributed to viscous and numerical dissipation in the
assume to be a measure of the standard deviation. To model as well as loss to subharmonic signals via wave–
measure the standard error, we need an estimate of the wave interactions. A previous study using HYCOM
number of degrees of freedom (DOF). This is a difficult simulations, with the same horizontal resolutions em-
quantity to measure, so we give a crude estimate here. ployed here, showed that energy is transferred out of the
Let us take DOF 5 NT 3 NW 3 NV, where NT is the low-mode inertial and semidiurnal internal waves into a
number of degrees of freedom in time, NW is the broad continuum of higher-frequency and higher-
number of waves involved in a global estimate, and NV wavenumber internal waves (Müller et al. 2015). Thus,
is the number of degrees of freedom in the vertical di- one of the goals of the present study is to determine
rection. Because the internal tides are dominated by whether a significant percentage of the residual energy
mode-1 waves, let us take NV 5 1. We take NT to be dissipation in the model is due to energy transfer to
the number of M2 periods in the HYCOM25S record, subharmonic waves via the PSI mechanism.
namely, 30 3 24/12.42 ’ 58. For NW, we count the In this paper, we used power spectral density and
number of internal tide beams emanating from source energy transfer calculations, as well as bispectral anal-
regions in HYCOM to be about 90 (e.g., from Fig. 2a in ysis, to provide evidence for the occurrence of PSI in
Ansong et al. 2017). This yields DOFp5ffiffiffiffi5ffiffi2ffiffi2ffiffi0. So a rough HYCOM and to map out the geographic distribution of
estimate of the standard error is 0:5/ 52205 0.007TW, the hypothesized PSI. In contrast to Simmons (2008),
which is about 18%of themean value of 0.04TW.Given who focused on M2-only simulations, here we employed
the uncertainties in the estimate of DOF and the fact two simulations forced by three semidiurnal (N2,M2, S2)
that we are dealing with residuals of differences of large and one simulation forced by three diurnal (Q1,O1,K1)
numbers, we cannot state with confidence that the sign tidal constituents. All simulations also included atmo-
of the residuals is positive in the simulations. The large spheric forcing, thereby ensuring a vigorous mesoscale
increase in both positive and negative transfers with in- eddy field. Energy transfer computations using equa-
creases in resolution further increases our uncertainties tions similar to previous studies (e.g., MacKinnon et al.
and leaves open the possibility that still higher-resolution 2013b; Sun and Pinkel 2013) show that the largest energy
simulations will find somewhat different answers, thus transfers are near the CLs, with most large transfers
opening the door for future studies. confined to the upper ocean. However, we find that the
energy transfers are almost equally partitioned between
positive transfers (from low-mode tides to subharmonic
9. Summary and discussion
signals) and negative/reverse transfers (Fig. 12d), in
Diurnal and semidiurnal internal tide energy analyses contrast to previous idealized simulations (MacKinnon
are performed using three simulations of the HYCOM and Winters 2005). We compute a net positive energy
global ocean circulation model at 1/12.58 and 1/258 hori- transfer (energy loss from the tide due to PSI) in all
zontal resolutions. HYCOM12S and HYCOM12D are simulations with varying amounts. The net global energy
used to denote the HYCOM simulations forced by the loss is ;0.001TW in HYCOM12S, which is ;0.5% of
semidiurnal and diurnal tides, respectively, at 1/12.58 the residual energy needed to close the HYCOM12S
resolution, and HYCOM25S represents the 1/258 semi- baroclinic energy budget. The net global amount of
diurnal simulation. For the semidiurnal tides, the pa- energy transfer in HYCOM25S is relatively larger,
rameterized topographic wave drag in HYCOM12S about 0.04TW, representing approximately 10% of the
(HYCOM25S) dissipates about 0.20 (0.28) TW of baro- residual transfer. This shows that the rate of energy
clinic energy, representing about 41% (46%) of the transfer is sensitive to the model resolution, with the
globally integrated barotropic-to-baroclinic conversion finer resolution facilitating greater wave–wave interactions
[;0.49 (0.61) TW]. The semidiurnal amount of conver- and larger energy transfer rates. For the diurnal tides,
sion is comparable to previous computations (Egbert and we estimate a net global amount of energy transfer of
Ray 2003; Simmons et al. 2004; Niwa and Hibiya 2011; 0.012TW in HYCOM12 simulations, representing about
Buijsman et al. 2016). Compared to the semidiurnal tides, 4.0% of the residual transfer.
the globally integrated conversion in the diurnal tides is In addition to energy transfer calculations, we per-
much smaller (0.08TW); about 60% of the generated formed bispectral analysis to provide evidence for, and
JUNE 2018 AN SONG ET AL . 1427
map the geographic distribution of, PSI in the model. inertial (see our example calculations in section 5b). A
Direct application of the bicoherence method of Kim combination of all the reasons mentioned above gives us
and Powers (1979) shows evidence of PSI around the CL some confidence that the subharmonic waves are likely of
together with statistically significant bicoherence values PSI origin.
appearing in high-latitude regions. The geographic dis- Previous studies suggest that energy transfers to the
tribution of bicoherence is more localized to the CL daughter waves of PSI might be faster between the sub-
upon employing the criterion that the SER [Eq. (11)] is harmonics and higher-mode internal tides (mode $3)
at least 1. than between the subharmonics and the low mode-1
We have shown that, over all depths, the zonally waves (MacKinnon et al. 2013b). The HYCOM model
averaged bicoherence from both the diurnal and semi- presented here well resolves the mode-1 and mode-2
diurnal bands is concentrated around the critical lati- waves but barely resolves modes 3 and higher. This might
tudes, and that the intensity of the zonally averaged be a possible reason for our low estimate of the net global
bicoherence significantly decreases about 58 away from energy flux to the subharmonics. Thus, a suggestion
the CL. This is consistent with the previous idealized re- (O. Sun 2013, personal communication) for a future study
sults of Furuichi et al. (2005), who also found that the is to separate out the higher-mode tides and to use them to
intensity of PSI in the Pacific quickly drops over a dis- compute bicoherences and energy transfer rates. Another
tance of about 38 from the CL. The most energetic sub- future endeavor is to use harmonic analysis to separate
harmonic signals with significant bicoherence values also out individual tidal constituents, in contrast to band-
show high intensity in the upper ocean. This is consistent passed signals, to investigate PSI.
with the idealized simulations ofHazewinkel andWinters We emphasize that the idealized simulations in
(2011), who attributed the surface intensification of PSI Hazewinkel and Winters (2011) imply that results in
to the surface-intensified structure of the buoyancy fre- any model, including ours, are dependent upon the
quency. A reviewer pointed out that an alternative hy- model resolution and viscosities. They find a 15% tidal
pothesis for the observation that PSI daughter waves are energy loss to PSI using a vertical eddy viscosity of 5 3
enhanced at the critical latitudes is that they could be 1025m2 s21 and a 25% reduction by prescribing a zero
waves that are generated by other processes, but with an viscosity. We find here that most of the significant bi-
amplitude that is enhanced at the turning latitude. This coherence values in HYCOM25S generally lie closer to
alternative hypothesis cannot be nullified by our model the upper value of 1, whereas the significant values in
results. However, we think the hypothesis is unlikely for HYCOM12S lie in the lower range. This implies that the
several of the reasons presented in the paper. For in- phase correlations between the tides and the sub-
stance, if the waves were generated by other random harmonic signals in HYCOM25S are generally higher
processes in the model, they would not likely lead to an than in HYCOM12S. Thus, simulations with horizontal
enhancement of waves with significant bicoherence resolutions even higher than 1/258 are likely to facilitate
values and relatively higher energy transfers along the greater wave–wave interactions between the tides and
critical latitudes. Second, the theory of PSI energy the subharmonics. The simulations reported here are
transfers shows that near the critical latitude the primary too expensive to do sensitivity studies on different vis-
driver of PSI is the term T1 52u*1  (u2  =u3) rather than cosities and vertical resolutions. It is expected that the
the term T2 52u*1  (u3  =u2). Our results are consistent estimates of PSI energy transfer given here will be im-
with theory, because we find that indeed T2 is negligible proved upon in the future by running higher-resolution
everywhere on the globe whereas the term T1 produces simulations and by storing and processing longer-
transfers that are enhanced along the critical latitudes. duration records from the model, so that different mo-
The analysis ofT1 andT2 also shows that the subharmonic tions can be more accurately separated.
waves seen in our results draw their energy from the There are still open questions about PSI. One ques-
horizontal gradients of the tidal velocities, as found in tion is about the long-termbehavior of PSI. For instance,
observations (e.g.,MacKinnon et al. 2013a).A third piece how does the rate of energy transfer and geographical
of evidence is that the high vertical wavenumber distur- distribution of PSI change seasonally? This question
bance seen at the critical latitude in our analyses (e.g., could not be addressed here because of the short length
Fig. 4b) is qualitatively similar to that of Simmons (2008) of time series used. Studies have begun to explore the
in a model that was run without atmospheric wind forc- time-varying mixing signals of PSI origin (Qiu et al.
ing. Last, the estimated vertical scales of the subharmonic 2012). In addition, there are still challenges in isolating
waves are close to those seen in observations of PSI the subharmonic waves of PSI in the real ocean (and in a
daughter waves (e.g., Alford et al. 2007;MacKinnon et al. complex model like ours), where many frequencies are
2013a), and we have shown that they are indeed near present. The recent study byRichet et al. (2017) suggests
1428 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48
that the frequencies of the PSI subharmonic waves likely APPENDIX
depend upon both the frequency and horizontal wave-
number of the primary tide, as well as on mean currents, Bispectral Analysis
as a result of Doppler effects. Following previous work,
this study assumed that the frequencies of the PSI Given a zeromean signal x(t) with its complex Fourier
daughter waves are about half the tidal frequencies, and transform X(v), the bispectrum is defined as (Kim and
it would be interesting to investigate the effects of Powers 1979)
modifying the subharmonic frequencies, as suggested by
B(v ,v )5E[X(v )X(v )X*(v )] ,
Richet et al. (2017), in a global model. We hope to ad- 1 2 1 2 3
dress these and other interesting scientific questions in
where t is time, v is frequency, E[] is the expectation
future studies.
operator, and X* is the complex conjugate of X. The
implication of this definition is that the bispectrum is
Acknowledgments. This research represents a contri-
identically zero unless 1) the frequency components
bution to the Climate Process Team (CPT) project
v1, v2 and v3 5 v1 1 v2 are present in a given signal
‘‘Collaborative Research: Representing Internal-Wave
and 2) phase coherence (or phase consistency) is also
DrivenMixing in Global OceanModels,’’ which focuses
present among the three components. A quantitative
on improving estimates of mixing due to internal waves
measure of the bispectrum is the bicoherence defined by
in the ocean. The project is funded by the National
Kim and Powers (1979) as
Science Foundation and is led by Jennifer MacKinnon
of the Scripps Institution of Oceanography. We are
2 jB(v , 2v )j
grateful to Jennifer MacKinnon, Oliver Sun, Eric b (v ,v )5 h 1  i2 h  i, (A1)1 2 2 2
D’ASaro, Eric Kunze, and Sherry Chou for extremely E X(v )X(v ) E X(v )1 2 3
helpful discussions. We are also grateful to Tim Duda
for suggesting that we look at energy levels as well as with 0# b(v1, v2)# 1. Alternate normalizations of the
significance of the bispectra. Finally, we would like to bispectrum have also been presented in previous studies
acknowledge very helpful and thorough comments from (Elgar and Guza 1988; Hinich and Wolinsky 2005; Sun
two anonymous reviewers of this manuscript and an and Pinkel 2012).
earlier version of this manuscript, and from a third In practical applications, the bicoherence is esti-
anonymous reviewer of the earlier version. J.K.A. and mated as
B.K.A. acknowledge funding from the University of
~
Michigan Associate Professor Support Fund, supported b~5  B    , (A2)
by the Margaret and Herman Sokol Faculty Awards. E~ X(v )X(v ) E~ X(v )1 2 3
J.K.A. and B.K.A. gratefully acknowledge support from
National Science Foundation CPT Grant OCE-0968783 where
and Office of Naval Research Grant N00014-11-1-0487.  M
B.K.A. and P.G.T. acknowledge support from a B~5  1 
 (i) (i) X(v ) X(v ) X(v )*(i),
University of Texas Jackson School of Geosciences M 1 2 3i 
Development grant, Naval Research Laboratory   "   #1/2M(NRL) contract N000173-06-2-C003, Office of Naval ~ j 1 2( ) (i) (i)E X v X(v )j 5 
 X(v ) X(v )  ,
Research Grants N00014-09-1-1003 and N00014-11-1- 1 2 M 1 2i
0487, and National Science Foundation Grant OCE- "   #1/2M
0924481. H.L.S. was supported by NSF (CPT) Grant ~ j j 1 
  2E[ X(v ) ]5 X( )(i)v  .
OCE-0968838 and ONR Grant N00014-09-1-0399. 3 M 3i
M.H.A. was supported by Grant OCE-0968131. J.G.R.,
J.F.S., and A.J.W. were supported by the projects Here,M is the set of data records each of length, say,N.
‘‘Eddy resolving global ocean prediction including We divided our time series into 50% overlapping win-
tides’’ and ‘‘Ageostrophic vorticity dynamics’’ spon- dows, each of lengthN5 256h, and applied a Hamming
sored by the Office of Naval Research under Program window to each record as in previous studies (Nikias and
Element 0602435N. This work was supported in part Petropulu 1993; Kim and Powers 1979).
by a grant of computer time from the DOD High In practice, it is likely that a finite length time series,
Performance Computing Modernization Program at even with truly independent components, will have non-
the Navy DSRC. This is NRL contribution NRL/JA/ zero bispectrum. For these reasons, Elgar andGuza (1988)
7320-13-1693. have established significance levels of zero bicoherence
JUNE 2018 AN SONG ET AL . 1429
FIG. A1. Example bispectral calculation from HYCOM12S at location (29.148N, 163.58W), depicting the (a) bispectrum and
(b) bicoherence at about 500-m depth, and (c) significant bicoherence values at each vertical level. The vertical dashed lines in (c) show the
value at the 80%, 90%, and 95% significance levels.
to help determine if data are statistically consistent exactly at the location of the peak bispectrum because of
with a linear, random phase process. They find signifi- the normalization factor [the denominator in Eq. (A1)].
cance lepveffiffiffilffisffiffiffi pffiffiffiffiffiffiffiffiffioffiffi f b at 95%[ 6/nd (denoted by b95%) and For this reason, the bispectrum and the bicoherence
99%[ 9/nd, where nd is the number of degrees of need to be considered together to determine non-
freedom. Determining the number of degrees of free- linear interactions. Figure A1c depicts significant
dom in the case of internal waves is a problematic issue bicoherence values at each vertical level, showing
(Carter and Gregg 2006; Sun 2010; MacKinnon et al. two main depth ranges (300–1000m and 2100–2600m)
2013b; Sun and Pinkel 2012; Chou 2013). Following where the tides and the PSI subharmonics have sig-
MacKinnon et al. (2013b), we estimate nd5 23 (60/2.5)5 nificant bicoherence.
48 for the 60-day span of our HYCOM12S and
HYCOM12D time series, and nd 5 2 3 (30/2.5) 5 24
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