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Numerical Simulations of Baroclinic Tide Fields in the Global Ocean:

Estimation of Baroclinic Tide Energy Available for Deep Ocean Mixing
Yoshihiro Niwa

UTokyo Alliance, The University of Tokyo,
7‐3‐1 Hongo, Bunkyo‐ku, Tokyo, 113‐0033 Japan
Abstract

The gıobal distributions of the major semidiurnal ( M_{2} and S_{2} ) and diurnal ( K_{1} and O_{1} )baroclinic tide energy are investigated usin g a hydrostatic numerical model. A series ofnumerical simulations using various horizontal grid spacing of \Delta x=1/5^{\circ}-1/20^{\circ} shows thatgeneration of energetic baroclinic tides is restricted over prominent topographic features. It isaıso found that the energy conversion rate from barotropic to baroclinic tide is very sensitive
to horizontal grid spacing as well as the resolution of the model bottom topography.
Extrapolating the calculated results to the limit of zero horizontal grid spacin g(\Delta x=0^{\circ}) yields
the estimate of the global baroclinic conversion rate to be 1240 GW. The amount of baroclinic
tide energy dissipated in the open ocean below a depth of 1000m (the typical depth of themain thermocline) is estimated to be 560‐680 GW, which is comparable to the mixing energy
estimated by Webb and Sughinohara (2001) as needed to sustain the global thermohaline
circulation.

1. Introduction

Baroclinic tides are ubiquitous phenomena in the stratified ocean. They are generated bybarotropic tidal currents flowing over a rough ocean bottom such as mid‐ocean ridges andcontinentaı sheıf slopes; the stratified water is then verticaıly displaced up and down and partof the barotropic tide energy is converted into baroclinic tide energy.
Baroclinic tides have a strong influence on the global thermohaline circulation and henceon the world’s climate, because they are thought to contribute to turbulent mixin g in the deep
ocean; near the generation sites, the vertical shear of horizontal currents associated withbaroclinic tides can be strong enough to cause shear instability leading to locally enhanced
turbulent mixing. Turbulent mixing can also be induced even far from the generation sites,\cdot

because propagatming barocıim c tides nonımearly interact with the background internal waves
and/or mesoscale eddies and cascade part of their energy down to small dissipation scales
inducing shear instability. Thus induced turbulent mixing transfers heat from the surface

22
down to the dense deep water that originates from high ıatitudes, causing it to upweıı back tothe sea surface at low and mid latitudes. Actually, ocean general circulation models (e.g.,
Hasumi and Suginohara 1999) demonstrate that the rate and spatial distribution of turbulent
mixin g can strongıy controı the fundamentaı features of the global thermohaline circulationsuch as flow and density structures as well as the transport of heat and chemical tracers.
Accurate estimation of baroclinic tide energy is also crucial for understanding the energybalance of the globaı thermohaline circuıation. Munk and Wunsch (1998) estimated that 2100
GW (1 GW =10^{9}W) of mixing energy is required to pull up 30Sv (1Sv =10^{6}m^{3}/s ) of the dense

deep water below a depth of

1000m(the typical depth of the main thermocline) andpostulated that nearly half of that energy was supplied by barocıim c tides and the remaining
half by the wind‐induced intemal waves. On the other hand, Webb and Suginohara (2001)
suggested that the direct upwelling by wind stress (Ekman suction) in the Southem Ocean
around the Antarctic continent might pull up about half of the deep water formed in the North

Atlantic to the sea surface, reducing the required mixin g energy to as low as 600 GW. It still
remains a matter of debate as to which of these mixin g energy analyses is more convincin g.
An important step toward answerin g this question is to accurately estimate the amount ofbaroclinic tide energy available for deep ocean mixing.
In the present study, we investigate the spatial distribution of the major semidiurnal (M_{2}and S_{2}) and diurnal ( K_{1} and Oı) baroclinic tides usin g a fine‐grid three‐dimensional numericalmodel incorporatming realistic bottom topography, density stratification, and barotropic tidal
forcing. In order to accurately estimate the energy conversion rate from barotropic tides tobaroclinic tides, in particular, a series of numerical simulations with various horizontal grid
spacings is conducted to examine the sensitivity of the calculated results to the model grid
spacin g as well as the resolution ofthe model bottom topography.
2. Numerical simulations

Fig. 1 shows the model domain that covers the global ocean from 80^{\circ}S to 80^{\circ}N . The
governin g equations are the full three‐dimensional Navier‐Stokes equations under hydrostatic
and Boussinesq approximations. The governing equations are numerically integrated using
the Princeton Ocean Modeı (Blumberg and Melıor, 1987). To examine the sensitivity of themodel results to the horizontal grid spacin g as well as the resolution of the model bottom

topography, a set of numerical experiments is carried out usin g six different horizontal grid
spacings, namely, \Delta x=1/5^{\circ}, 1/6^{\circ}, 2/15^{\circ}, 1/10^{\circ}, 1/15^{\circ}, 1/20^{\circ} .

In all cases, 41 sigma levels are

23

80N
70N
60N
50N
40N
30N
2 ON
1 0N

E0

1 OS
20S

30S
40S
50 S

60S

70S
80S

20E

40E

60E

80E

100E

120E

140E

160E

80E

160W

t40W

20W

100W

80W

60W

40W

20VI

OW

)

Fig. 1. The whole model domain and the bathymetry. The regions including the prominent
topographic features are indicated by straight lined squares. The thick white ıine indicates theboundary between the open ocean and the coastal and marginaı seas.

assumed in the vertical. For details of the goveming equations and the numerical model,
readers are referred to Niwa and Hibiya (2004).
For each numerical experiment, the model topography is constructed by averaging the
1 ‐ \min(1/60^{\circ}) gridded global bathymetric data of ETOPOI within each model grid cell so as

to increase the resolution of the model topography with the decrease of the model grid
spacin g . The basic temperature and salinity fields are obtained from the annual mean data of
the National Oceanographic Data Center’s World Ocean Atlas. The model is forced by thetidal potentiaı forcin g of four major semidiurnal and diurnaı astronomicaı tidal constituents( M_{2}, S_{2}, K_{1} and Oı). The time periods of M_{2}, S_{2}, K_{1} and O_{1} tidal forcing are 12. 42hrs,
12. 00hrs, 23.93hrs and 25. 82hrs , respectively. Furthermore, in order to reproduce the realistic
barotropic tidal fieıd, we assimilate the barotropic surface tidal eıevation data from the tidal
prediction system of NAO. 99b (Matsumoto et al., 2000). The model is driven by the M_{2}, S_{2},K_{1} and O_{1} tidal forcin g for 60 days from an initial state of rest with time steps of 30s and3. 75s for the baroclinic and barotropic modes, respectively. The calculated time series data forthe last 20 days are harmonically analyzed to obtain the amplitude and phase of each tidal
constituent. In order to extract the information about baroclinic tides, we subtract thedepth‐averaged tidal currents from the calculated tidal currents.

24
3. Results

Figs. 2a and 2b show the snapshot of vertical displacement at the depth of 1000m in the
mid‐latitude North‐Western Pacific for semidiumal ( M_{2} and S_{2} ) and diurnal (Kı and Oı)
barocımic tides, respectively. The characteristic horizontal wavelength becomes ∼ 150km forsemidiurnal baroclinic tides and \sim 300km for diumal baroclinic tides. Energetic baroclinic
tides with an ampıitude up to severaı tens meters are excited over the Luzon ridge ıocatedbetween Taiwan and the Philippines and over the Izu‐Ogasawara ridge.

(a) Semidiurnal (M_{2}+S_{2})(b) Diurnal (K_{1}+0_{1})Fig. 2. (a) The snapshot of vertical displacement field for semidiumal ( M_{2} and S_{2} ) baroclimic
tides at depth 100m in the mid‐ıatitude North Westem Pacific. (b) same as (a) but for diumal( K_{1} and O_{1} ) baroclinic tides.

25
Fig. 2b shows that the propagation of diumaı baroclinic tides is limited equatorward of thelatitude about 30^{\circ} . This is because frequency of freely propagating linear internal waves must
be greater than the Coriolis frequency f(f=2\Omega\sin\theta where \Omega=1/24hrs^{-1} is the earth^{\dagger}srotation frequency and \theta is latitude), hence the diumal baroclinic tides cannot propagate freely
poıeward of the critical ıatitude of about 30^{\circ} where the diurnal tidaı frequency becomes equaı
to the Coriolis frequency (the exact critical latitudes for K_{1} and O_{1} baroclinic tides are 30. 01^{\circ}and 27.69^{\circ} , respectively).

To examine the generation of baroclinic tide energy more quantitatively, we calculate thedepth‐integrated energy conversion rate from the barotropic to baroclinic tides (Niwa andHibiya, 2004) defined as,
CNV

where

(x,y)= \int_{-H}^{0}g\overline{\rho'W_{bt} dz

H is the depth, g is the gravity acceleration,\rho' is the density deviation from the basic

density field due to baroclinic tide motions, W_{bt} is the vertical veıocity resultming from theinteraction of depth‐averaged tidal currents with bottom topographies and the overbar denotes
the temporaı average over one tidal period.
Thebaroclinic

tidal

energyconversion rate

integrated

over theglobalocean( \iiint_{g\iota oba\iota}CNV(x,y)dxdy) amounts to 9ı4 GW (673, 117, 83, 41 GW for M_{2}, S_{2}, K_{1} , and Oı
constituents, respectively) for the case of the fmest horizontal grid spacin g\Delta x=1/20^{\circ}.
However, the calculated baroclinic conversion rate is very sensitive to the choice of thehorizontal grid spacing (\Delta x) . Fig. 3 plots the globally integrated baroclinic conversion rates
obtained for each numerical simulation versus the horizontal grid spacin g(\Delta x) . The global
baroclinic conversion rate for the sum ofM_{2}, S_{2}, K_{1} and Oı tidal constituents increases by 512
GW (from 393 GW to 912 GW) as \Delta x is reduced from 1/5^{\circ} to 1/20^{\circ} . This significant increaseof the global baroclinic conversion rate is primarily attributable to the increase of the spatial
resolution of the model bottom topography rather than the improvement of the model grid
resolution itself. Actually, when \Delta x is reduced from 1/5^{\circ} to 1/20^{\circ} but with fixin g the model
topography resolution (i.e. the topography smoothed with averaging scale of 1/5^{\circ} is used inboth caıcuıations), the increase of the global baroclinic conversion rate is limited to 167 GW(from 393 GW to 560 GW).

Fig. 3 indicates that the logarithm of the global baroclinic conversion rate increases almost
l i n early with the decrease of \Delta x .

Hence, we extrapolate the least‐squares fitting in Fig. 3 to

26

1 0

\llcorner(Dag\subsetUJ\overline{\varpi}

5

\vee

\Phi

4^{\lrcorner}\varpi 2\nabla\propto\vdash-\subset\underli {\fac irle}{c \frac{O}\lcorne t\primel}1\overline{\fcircle}q>^{)}\lcorner\circ\subset

m

O

(o

\circ

Horizontal Grid Spacing ( x1/60 degree)
Fig. 3. The globally integrated baroclinic tidal energy conversion rates for the semidiumaı(solid circles) and diumal tidal constituents (solid triangles) and the sum of the four major
tidal constituents (solid squares) obtained usin g various horizontal grid spacing of 1/20^{\circ}-1/5^{0}.
Superposed on each pıot is the least squares fitting line.

the limit of zero grid spacin g(\Delta x=0^{\circ}) to estimate the global baroclinic conversion rate tobe 1238\pm 35 GW (937\pm 35,163\pm 3,92\pm 4,46\pm 3 GW for M_{2}, S_{2} , Kı and O_{1} tidal

constituents, respectively). Since the global barotropic tidal energy input from theastronomical tidal potential forcing amounts to 3500 GW (Munk and Wunsch, 1998), theabove estimate indicates that 35% of the total barotropic tidal energy is converted tobaroclinic tidal energy.
To examine the locations where the baroclinic tide energy is generated, the global
distribution of the baroclinic tidal energy conversion rate for the sum of M_{2}, S_{2}, K_{1} and O_{1}tidal constituents is shown in Fig.4, which is averaged over each 5^{\circ}\cross 5^{\circ} grid area. Note thatFig.4 shows the baroclinic conversion rate estimated by the extrapolation to the limit of \Delta x=0^{o} . The details of the extrapolation are explained in Niwa and Hibiya (2014). We can see thatthe generation of energetic baroclinic tides is concentrated in several specified regions.

27
SON

60N

40N

20N

EO

20S

40S

60S

80S
20E

40E

6

W 0W

10^{-3}W/m^{2})
0.16

0.23

0.33

0.46

0.66

0.93

1.33

1.88

2.65

3.75

5.30

7.50

10.6

15.0

Fig.4. The global distribution ofthe barocıinic tidal energy conversion rates for the sum of thefour major tidal constituents ( M_{2}, S_{2}, K_{1} and O_{1} ) averaged within each 5^{\circ}\cross 5^{\circ} grid area. The
baroclinic conversion rate estimated in the limit of zero horizontal grid spacin g(\Delta x=0^{\circ}) isshown.

Solomlslan
lndonesi

Archipela
Western‐lnd |Ridc‐

Mid‐Atlan

Ridc‐

East ChiS
lzu‐Ogasawc
Ridc‐
Polynesilslan

Andams
Hawai|

Ridc‐
Aleut|
lslan
K1
lslan

0

20

40

60

SO

100

120

140

160

Baroclinic Tidal Energy Conversion Rate (GW)
Fig.5. The integrated semidiurnal (M_{2}, S_{2}) and diumal (K_{1}, O_{1}) baroclinic tidal energy
conversion rates over each representative area (see Fig. 1) estimated in the limit of zero grid
spacin g(\Delta x=0^{\circ}) .

28
Fig.5 shows the barocımic conversion rate integrated over each of the significant
generation regions indicated in Fig.1 with the squares enclosed by straight lines. Over thesesignificant regions, about 70% of the globally integrated baroclinic tide energy is generated.
Especially, the Solomon Islands, the Indonesian Archipelago, the Westem Indian Ridges, andthe Mid‐Atlantic Ridges are the four most important generation regions, over which morethan 100 GW ofbaroclinic tide energy is generated.

Next, in order to estimate how much of the baroclinic tide energy is dissipated in the open

ocean below a depth of 1000m (the typical depth of the main thermocline) to induce thedeep‐ocean mixin g that sustains the thermohaline circulation, we first calculate thedepth‐integrated dissipation rate of baroclinic tide energy. Assuming nonlinear advection
effect is negligible, it is given by the difference between the energy conversion rate and theenergy flux divergence,

DIS(x,y)= \int_{-H}^{0}g\overline{p'W_{bt} dz-\int_{-H}^{0}[\frac{\partial} {\partial x}(\overline{p'u'})+\frac{\partial}{\partial y}(\overline{p'v'})]dz
where u', v' andp' are the eastward and northward velocities and the pressure perturbations

associated with baroclinic tide motions, respectively. The spatial distribution of DIS(x,y)(not shown here) is similar to that of CNV(x_{t}y) shown in Fig. 4, but extends more outward
from each prominent topographic feature as a consequence of baroclinic tide propagation. If

the boundary between the open ocean and the coastal and marginal seas is defmed as shownby the thick white line in Fig. 1, we can estimate the baroclinic dissipation within the open

ocean\iiint_{open}DIS(x,y)dxdy to be 833\pm 23 GW (405\pm 11 GW for the coastal and marginalseas) in the limit of \Delta x=0^{\circ}.
Recent turbulence observations have shown that the turbulent dissipation rate associatedwith breakin g of baroclinic tides decays upwards from the ocean bottom (z=-H) with ascale height ( \varsigma ) several tens to hundreds meters depending on the characteristic wavelengthof ocean bottom roughness (Decloedt and Luther, 2010). Taking this into account, we assume
that the depth‐integrated baroclimic tidal energy dissipation rateD(x,y) is vertically

distributed following a vertical structure function given by\phi(z_{t}\cdot H)=\frac{e^{-(z+H)/\varsigma} {\varsigma(1-e^{-H/\varsigma})
which satisfies\int_{-H}^{0}\phi(z;H)dz=1 . Then, the fraction of the baroclinic tide energy that isdissipated in the open ocean at depths greater than |z1 can be calculated as29

F_{open}(z)= \frac{\int\int_{open}[\int_{-H(x,y)}^{Z}DIS(x,y)\phi(z';H(x,y) dz']dxdy}{\int\int_{Global}CNV(x,y)dxdy}.

Fig. 6 shows the plots of F_{open}(z) (thick solid lines) for the scale height parameters \varsigma=10,
100, 500, 1000 m , respectively. The same function but for the coastal and marginal seas is

F_{coastal}(z) is also plotted in Fig. 6 (thin dotted line). Although the value of F_{open}(z) at eachzdecreases as the scale height parameter \varsigma increases, F_{open}(z) does not change appreciably

even if \varsigma is varied over two orders of magnitude. Fig. 6 indicates that 45‐55% of the global
baroclinic tide energy is dissipated in the open ocean below the depth of 1000 m(F_{open}(|z1=1000m)) . Multiplying this fraction by the global energy conversion rate estimated in the limitof zero horizontal grid spacin g (about

1240GW ), we can estimate that 560‐680 GW of thebaroclinic tide energy is supplied to the deep ocean mixin g which is available to sustain the
global overturning circulation.

100

200

\hat{\ve E}

L\tilde{a}^{300}
a

400

500

600
Relative Contribution to the Global

Baroclinic Conversion Rate (%)
Fig. 6. The cumulative depth distribution of the dissipation rate of baroclinic tide energy forthe four major tidal constituents ( M_{2}, S_{2}, K_{1} , and O_{1} ) within the open ocean (solid lines) andwithin the coastal and marginal seas (dotted lines) which are calculated assuming an
exponential vertical structure function with various scale height parameters \varsigma=10,100,500,
and 1000 m.

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3. Summary and Discussion

In the present study, using a three‐dimensional numerical model, we have investigated theglobal distribution and budget of the semidiurnal and diumal ( M_{2}, S_{2}, K_{1} , and Oı) baroclimic
tide energy. The sensitivity experiments have shown that the energy conversion rate from thebarotropic to baroclinic tides strongly depends on the horizontal grid spacing (\Delta x) as well as
the resolution of the model bottom topography; the energy conversion rate increases
exponentially as \Delta x is reduced from 1/5^{\circ} to 1/200.

This clearly indicates the necessity of very high resolution numerical simulations(horizontal grid spacin g much less than 1/100^{\circ} ) incorporating fine‐scaıe bathymetric data suchas those obtained by multibeam echo sounder observations from research ships or ROVs

(remotely operated underwater vehicle). However, such very high resolution numericalsimulations covering all the global ocean is impossible for the moment because of insufficientcomputer capacity as well as insufficient spatial coverage of multibeam bathymetric data.
In the present study, we have instead estimated the baroclinic energy conversion rate in thelimit of zero horizontal grid spacin g(\Delta x=0^{\circ}) by extrapolatming the exponential relationshipfound between the caıculated energy conversion rate and \Delta x . Fig. 7 shows the global tidal
energy budget obtained from the this study. We have found that 3500 GW of the global
barotropic tidal energy is converted into 1240 GW of baroclinic tide energy over theprominent topographic features in the global ocean, about 830 GW of which is dissipatedwithin the open ocean. We have also found that45‐55% of the global baroclinic tide energy is
dissipated below a depth of 1000 m (the typical depth of the main thermocıine) in the open
ocean so that 560‐680 GW of the baroclinic tide energy is estimated to be available for deep
ocean mixin g.

This value is just barely 30% of 2100 GW, namely, the energy required to sustain the gıobaı
overmrnin g circulation estimated by Munk and Wunsch (1998). They have argued that theremaining energy for deep ocean mixing could be supplied by the wind‐induced internal
waves. However, numerical simulations by Furuichi et al. (2008) demonstrated that the
globally integrated win d ‐induced near‐inertial internal wave energy available for deep ocean
mixin g was limited to 100 GW, because more than 70% of the wind‐induced near‐inertial

wave energy was dissipated within the top 200 m and hence not available for turbulent mixing

at greater depths. Even addin g the contribution from the win d‐induced internal waves,therefore, the estimated total energy available for deep ocean mixin g of 660‐780 GW is still31
31much insufficient to fulfill the mixin g energy estimated by Munk and Wunsch (1998). This
result seems to support the validity of the argument of Webb and Suginohara (2001) who
insisted that the required mixing energy can be reduced down to 600 GW by considering theexistence of the direct upweııing by wind stress (Ekman suction) in the Southem Ocean thatpulls up about half of the North Atlantic Deep Water (NADW) formed in North Atlantic.
Acknowledgements

I would like to thank Prof. Sunao Murashige and the participants to the workshop forgiving me the opportunity to present this work and to have useful discussions. Furthermore, I
would like to express the deepest respect to Prof. Mitsuhiro Tanaka for his significant
achievements in the research of water wave physics and for his dedicated efforts in the

promotion and education of this research field.

Global Barotropic Tide Energy

Fig. 7. The global tidal energy budget estimated in the present study

32
References

Blumberg A.\Gamma. , Mellor G.L. (1987), A description of a three‐dimensional coastal ocean
circulation model, In Heaps N. (ed.), Three‐dimensional coastal ocean models, AGU,
Washmington D.C..
Decloedt

T.,

Luthertopography‐catalyzed

D.S.

(2010),

diapycnaıOnasimplemixingintheempirical

abyssalparameterization

ocean,

of

Journal Physical

Oceanography, 40, 487‐508.

Furuichi N., Hibiya T., Niwa Y. (2008), Model‐predicted distribution of wind‐induced
intemal wave energy in the world’s oceans, Journal Geophysical Research, 113,
doi: 10. 1029/2008JC004768.

Hasumi H., Suginohara N. (1999), Effects oflocally enhanced vertical diffusivity over roughbathymetry on the world ocean circulation, Journal Geophysical Research, 104, 23367‐
23374.

Matsumoto K., Takanezawa T., Ooe M. (2000), Ocean tide models developed by assimilatming
Topex/Poseidon altimeter data into hydrodynamical model: a global model and a regionalmodel around Japan, Journal of Oceanography, 56, 567‐581.
Munk W.H., Wunsch C. (1998), Abyssal recipes II: energetics of tidal and wind mixing, Deep
Sea Res Part I, 45,1977−2010.

Niwa Y., Hibiya T. (2004), Three‐dimensional numerical simulation of M_{2} internal tides inthe East China Sea, Journal of Geophysical Research, 109, DOI: 10. 1029/2003JC001923.
Niwa Y., Hibiya T. (2014), Generation of baroclinic tide energy in a global three‐dimensional
numerical model with different spatial grid resolutions, Ocean Modelling, 80, 59‐73,
doi : 10. 1016/j . ocemod.2014.05.003.

Webb D.J., Suginohara N. (2001), Vertical mixing in the ocean, Nature, 409, 37.