Latitudinal Distribution of Mixing Rate Caused by the M

Latitudinal Distribution of Mixing Rate Caused by the M2 Internal Tide

JIWEI TIAN, LEI ZHOU, AND XIAOQIAN ZHANG

Physical Oceanography Laboratory, Ocean University of China, Qingdao, China

(Manuscript received 12 January 2004, in final form 18 April 2005)

ABSTRACT

Ten years of Ocean Topography Experiment (TOPEX)/Poseidon tidal data and an energy balancerelation are used to estimate the mixing rate caused by M2 internal tides in the upper ocean. The resultsindicate that latitudinal distribution of the mixing rate has a generally symmetrical structure with respect tothe equator. The maxima are distributed around 28.9°N and 28.9°S and can be as high as 3.8 � 10�5 m2 s�1

in the Pacific, 5 � 10�5 m2 s�1 in the Atlantic, and 3.7 � 10�5 m2 s�1 in the Indian Oceans. The minimum,which is only 10% of those at 28.9°, is located near the equator. The data imply that midlatitudes are thekey regions for internal tide mixing.

1. Introduction

The mixing rate in the ocean plays an important rolein the meridional transports of mass, momentum, andenergy, and exerts great influence on the global climatechange (Munk and Wunsch 1998). The M2 internal tideis one of the most important sources of mechanical en-ergy for mixing in the ocean. Thus, determination of thespatial and temporal distributions of the mixing ratecaused by M2 internal tides is of significance in gainingunderstanding on the meridional circulation processesand the dynamics of air–sea interactions. Such under-standing is of obvious importance in our continuouseffort toward the improvement of climate predictionmodels.

Jayne and Laurent (2001) included a parameteriza-tion of the internal tidal drag in the tide model, showingthe diffusivities generated by tidal flows over topogra-phy. Hasumi and Suginohara (1999) and Simmons et al.(2004) proved with numerical models that the oceangeneral circulations were significantly affected by ver-tical and horizontal distributions of eddy diffusivity. Sofar, all available ocean models are incapable of distin-guishing the mixing processes so that we are obliged totreat the mixing rate as a constant in most of the oceanmodels. However, studies have shown that the mixingrate changes greatly with rough bottom topography.

For example, in the microstructure observation andtracer experiments carried out for the deep Brazil Basinof the South Atlantic Ocean, Polzin et al. (1997) andLedwell et al. (2000) discovered that the mixing rate isless than or approximately equal to 1 � 10�5 m2 s�1 atall depths above the smooth abyssal plains and theSouth American continental rise but exceeds 1 � 10�4

m2 s�1 above the Mid-Atlantic Ridge. Heywood et al.(2002) estimated an average mixing rate of (39 � 10) �10�4 m2 s�1 in the Scotia Sea. Garrett (2003a) pointedout that there should be a significant change in themixing rate with the latitude. Gregg et al. (2003) gave amaximum mixing rate near 30°N in the North PacificOcean as 5 � 10�5 m2 s�1. Hibiya and Nagasawa (2004)used expendable current profiler data in the North Pa-cific and a turbulence parameterization model to com-pute the latitudinal distribution patterns of mixingrates, and discussed the effects of parametric subhar-monic instability on the latitudinal distributions of tur-bulent dissipation. However, it is costly to determinethe mixing rate by these traditional cruise observationsand, until now, there has been no reported attempt tolaunch large-scale investigations of these kinds in theopen ocean.

Fortunately, with the availability of satellite altimeterdata, Ray and Cartwright (2001) and Tian et al. (2003)estimated the energy fluxes of internal tides on a largescale. Egbert and Ray (2001) estimated the M2 tidalenergy dissipation from Ocean Topography Experi-ment (TOPEX)/Poseidon altimeter data. In this paper,considering the energy balance in the steady ocean that

Corresponding author address: Prof. Jiwei Tian, Ocean Univer-sity of China, No. 5 Yushan Road, Qingdao 266003, China.E-mail: [email protected]

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energy dissipation balances net internal tide flux in theinner ocean, we calculate the mixing rate caused byinternal tides in a suitably chosen control volume.Therefore, it is a meaningful attempt to estimate thedistribution of the mixing rate in the upper ocean usingthe altimeter data and dynamical relations of internalwaves.

This paper is organized into four topical sections.After the first introduction section, section 2 will de-scribe the method to calculate global energy flux of M2

internal tides and the mixing rate. Next, in section 3, theresults will be summarized, which is followed by discus-sions given in section 4.

2. Analysis method

a. Global distribution of M2 internal tide energyfluxes

In most parts of the ocean, M2 tides are the dominantcomponents of tides. They can be extracted accuratelyfrom the satellite altimeter data in which they show upas strong signals (Kantha and Tierney 1997; Ray andCartwright 2001; Tian et al. 2003). The method to cal-culate the energy flux of M2 internal tides was de-scribed in detail by Tian et al. (2003). Here we willdescribe the method only briefly along with an illustra-tive example.

Ten years of TOPEX/Poseidon altimeter tidal data(Benada 1997) from October 1992 to June 2002 consti-tute the baseline of this study. Tracks forming a dia-mond-shaped domain are chosen to obtain a relativelyaccurate 2D structure of the internal tide field.

First, to determine the main tidal components, wecompute the frequency spectra of TOPEX/Poseidontime series data. This is followed by harmonic analysisfrom which the tidal height of all main tidal compo-nents, especially that of M2 tide, is calculated. Second,the baroclinic components of the M2 tide are separatedfrom the barotropic ones according to their wavelengthdifference. Simple high-pass filtering (cutoff at approxi-mately 400 km) of tidal heights along the tracks canadequately remove the barotropic tides in the openocean (Ray and Mitchum 1997). Then, the baroclinictidal waves are treated as a superposition of severalplane waves, which propagate in different directions.Although the magnitude of the horizontal wavenumberof M2 internal tide is uniquely determined by the lineardispersion relationship, the wavenumbers in differentdirections, however, have different projections on aspecific altimeter track. Hence the corresponding peaksin the wavenumber spectrum along the altimeter trackare distinguishable. Thus, the number of the internalwaves in the superposition can be determined accord-

ing to the number of peaks in the wavenumber spec-trum, that is,

� � �i�1

n

�Ai cos��it � kix � liy� Bi sin��it � kix � liy�,

�1�

where � is the baroclinic tidal height, Ai and Bi are theamplitudes, �i is frequency, ki and li are x and y com-ponents of the wavenumbers respectively, and i � 1,2, . . . , n. Usually the peaks are obvious enough to bepicked out by eye, and hence a threshold spectral am-plitude is not assigned for the identification of indi-vidual peaks. Because every plane wave in Eq. (1) ishorizontally two-dimensional, the baroclinic tidalheights along all of the four altimeter tracks, which takethe form of a diamond-shaped domain, are used to de-termine ki and li by the least squares method. Third, thevelocity and energy flux of the internal tides are calcu-lated. That the Brunt–Väisälä frequency profiles havethe form of a Dirac function (Conkright et al. 1999)helps in the derivation of the analytical solutions to thedynamic equations of internal tides (Baines 1982). Herethe x coordinate is assigned to be the wave propagationdirection, letting Ki � k2

i l2i , i � 1, 2, . . . , n, where

the subscript i represents the index of the superpositioncomponent. Assuming that the velocity potential of thefirst baroclinic mode of internal tides has the form �i ��i exp(iKix), where �i satisfies

�i � ��Ci sinh�c0�iz

c2h �, �d � z � 0

sin�i�1 z

h�, �h � z � �d,�2�

and where

Ci �sin�i�1 � R�

sinhc0

c2�iR

, �i �Kih

c0, c0 � � �2 � f 2

N02 � �2�1�2

,

c2 � �1 � � f

��2�1�2

, R �d

h,

N0 is the Brunt–Väisälä frequency, d is the thermoclinedepth, and h is the water depth, the velocities ui and wi

of the first baroclinic mode of internal tides are calcu-lated by ui � ��i/�z and wi � ��i/�x. The totalvelocity is then u � �n

i�1(uini wiz), where ni is thehorizontal unit vector of the ith wave component, and zis the vertical unit vector. Last, the energy flux of theinternal tides is calculated by P(z) � p(z) · u(z), inwhich p(z) is the fluctuation pressure. Note that weonly obtain one energy flux of the internal tide for eachdiamond-shaped domain.

As an illustrative example given below, the energy

36 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 36


flux of M2 internal tide near the Ryukyu Trench hasbeen calculated using the above equations. A diamond-shaped calculation area is shown in Fig. 1. Ten years ofTOPEX/Poseidon tidal data at point A (27.8°N,132.8°W), marked in red circle in Fig. 1, are shown inFig. 2a. The spectrum at point A is shown in Fig. 3.Note that according to the folding frequency, the cor-responding spectrum peak of M2 tide should be at 62.1days (Ray and Mitchum 1997). The M2 tidal heightsafter the harmonic analysis are shown in Fig. 2b. Simplehigh-wavenumber-pass filtering along the tracks (cutoffis 400 km) is used to obtain baroclinic M2 tidal heights.The baroclinic tidal heights of the time series at point Aare shown in Fig. 2c. The high portion of the wavenum-ber spectrum along track B marked in red in Fig. 1 isshown in Fig. 4, which shows clearly the presence ofthree peaks. This is true also for the high portion of thewavenumber spectrum along other tracks in this calcu-lation area. Thus, the baroclinic tides in this area aretreated as a superposition of three plain waves. Apply-ing a least squares method, Ai, Bi, ki, and li can beaccurately determined by Eq. (1). Then, applying Eq.(2), the velocities of M2 internal tides can be calculated.Last, the vertically integrated energy flux of M2 internaltides in this area is calculated to be 3.9 kW m�1, asindicated by the black arrow in Fig. 1.

b. Estimation of mixing rate caused by M2 internaltides

From the viewpoint of energy balance, we can esti-mate the energy dissipation of M2 internal tides in the

upper ocean. Assuming that the ocean is at a steadystate, the energy balance of M2 tides is in the form of

� � S � � · P, �3�

where �, S, and P are the energy dissipation, energysource, and energy flux of M2 internal tides, respec-tively. The M2 internal tides derived from the TOPEX/Poseidon data are the first mode, which contains mostof the energy (Cummins et al. 2001; Ray and Cartwright2001). Thus, we constrain our estimations to the first

FIG. 3. Spectrum at point A in Fig. 1. According to the foldingfrequency, the peak of M2 tide is at 0.0161 day�1.

FIG. 1. A diamond-shaped calculation area encompassed byTOPEX/Poseidon tracks near Ryukyu Trench. Point A, markedin red, is the place where we obtain the results in Figs. 2 and 3.Track B, in red, is where we calculate the high portion of wave-number spectrum shown in Fig. 4. Energy flux of M2 internal tidescalculated in this area is represented by a black arrow.

FIG. 2. (a) TOPEX/Poseidon tidal data (Benada 1997) at pointA marked in red circle in Fig. 1; (b) M2 tidal heights at point Aafter harmonic analysis; and (c) baroclinic M2 tidal heights atpoint A after high-wavenumber filtering (cutoff is 400 km). Thehorizontal coordinate is MGDR cycle number. The interval ofeach two cycles is 9.92 days.


Fig 1 live 4/C


baroclinic mode. In the upper layer of the inner ocean,which is far from the coasts and with rough bottomtopography, the source term S can be omitted; that is,energy dissipation balances net internal tide flux in theinner ocean. Hence the energy balance Eq. (3) can besimplified as

� � �� · P. �4�

Integrating (4) in a suitable control volume �, the av-erage energy dissipation of M2 internal tides in the up-per ocean is

� � �1

|�| ��

� · P�x� d�, �5�

where the control volume comprises several adjacentdiamond-shaped areas, including horizontally theTOPEX/Poseidon orbits and vertically the upper layerfrom the ocean surface to the main thermocline bottom.Then the eddy diffusivity caused by the M2 internaltides is calculated based on an assumed relationshipwith average energy dissipation and stratification (Os-born 1980),

� � 0.2�

N2 , �6�

where the Brunt–Väisälä frequency N can be calculatedfrom ocean profile data (Conkright et al. 1999).

A flowchart showing the logistics of the method isgiven in Fig. 5.

3. Results and analyses

a. Global energy flux and latitudinal distribution ofmixing rate

Figure 6 shows the global distribution of the M2 in-ternal tide energy fluxes calculated from the TOPEX/Poseidon data. One can see that the energy flux is highnear the coasts and areas with abruptly changing bot-tom topography, as in the Ryukyu Trench, midoceanridges, and fracture zones, and propagates into the in-

FIG. 4. High portion of wavenumber spectrum (cutoff is 400km) along track B marked in red in Fig. 1.

FIG. 5. Procedures of the method to estimate the mixing rate caused by M2 internal tides.



ner ocean. Theoretically we can calculate a flux in eachTOPEX/Poseidon diamond. In real practice, however,it is a time-consuming task, and we do not have enoughcomputer time to accomplish the task. Therefore, the

energy fluxes shown in Fig. 6 are sparser than those inthe TOPEX/Poseidon diamonds.

Spatial distribution of the energy dissipation causedby M2 internal tides is shown in Fig. 7. Because the

FIG. 6. Global energy fluxes of M2 internal tides derived from the TOPEX/Poseidon data.

FIG. 7. The global energy dissipation of M2 internal tides.


Fig 6 7 live 4/C


control volume is composed of several adjacentTOPEX/Poseidon diamonds, one energy dissipation es-timate requires several energy fluxes. In addition, onecan see from the procedures of the method in Fig. 5 thatthere are not enough energy fluxes to estimate the en-ergy dissipation in a few areas and Eq. (4) is not appli-cable in a few other areas. Thus, the number of dots inFig. 7 is much less than that of the arrows shown in Fig.6. To illustrate by a real case how the energy dissipationdiagram as shown in Fig. 7 is constructed, a part of thearea in Fig. 6 south of the Aleutian Trench is extractedand displayed in Fig. 8. One can see that there are sixenergy fluxes (marked I–VI) in or near a control areacomposed of four TOPEX/Poseidon diamonds (encom-passed by red lines in Fig. 8) south of the AleutianTrench. In this domain, the average depth of the mainthermocline bottom is 300 m and the average Brunt–Väisälä frequency above the main thermocline bottomis 6 cph (Conkright et al. 1999), as shown by the em-bedded figure in Fig. 8. The fluxes I–IV flow into thecontrol volume, while fluxes V and VI flow beyond theedge of the domain. The energy dissipation of M2 in-ternal tides in this domain is assumed to be the sum-mation of fluxes I–IV. The calculated energy dissipa-tion in this domain is 1.29 � 10�8 W kg�1, and theestimated mixing rate caused by M2 internal tides is1.73 � 10�5 m2 s�1.

Figure 9 shows the respective latitudinal distributionsof the mixing rate in the Pacific, the Atlantic, and theIndian Oceans. One can see that the latitudinal distri-bution of the mixing rate exhibits a near-symmetricalstructure with respect to the equator. At 30°N and 35°S,

the mixing rate reaches a maximum value of 3.8 � 10�5

m2 s�1 in the North Pacific and 2.5 � 10�5 m2 s�1 in theSouth Pacific. At 30.5°N and 27.5°S, it reaches a maxi-mum value of 5 � 10�5 m2 s�1 in the North Atlantic and3.6 � 10�5 m2 s�1 in the South Atlantic. At 29.5°S, itreaches a maximum value of 3.7 � 10�5 m2 s�1 in thesouth Indian Ocean. The mixing rate decreases both atthe equator and toward the higher latitudes. At theequator it is only 10% of that at 28.9° in all of the threeoceans, at the same order as a molecular mixing coef-ficient. This indicates that midlatitudes are the key re-gions for mixing in the upper ocean. These results maybe related to parametric subharmonic instability (Mc-Comas and Müller 1981; Nagasawa et al. 2000; Hibiyaet al. 2002), which can enhance the mixing rate causedby M2 internal tides at 28.9°. Hence, we may concludefrom the discussion that the mixing rate should not beset as a constant, as practiced in many climate models,but rather as a parameter varying with latitudes.

b. Error analysis

The quality of the calculation of the M2 internaltides was tested using the ADCP data observed in theSouth China Sea from 20 August 2000 to 4 November2000. The ADCP was located at 20°34.8510�N,118°24.4610�E. Using the above method, we processedthe TOPEX/Poseidon (T/P) data concurrent with theADCP observations. The details of such comparisonshave been previously discussed by Tian et al. (2003).

The control volume is composed of several T/P dia-monds. The horizontal size of the control volume isdetermined by the size of the diamond. Because linearapproximations were applied to estimate the energydissipations, small control volumes are required. En-largement of the control volume would make the size ofthe control volume several times larger. This wouldchallenge the validity of applying Eq. (4) in a large area,and could lead to larger errors. The above consider-ation would therefore favor the choice of the smallestpossible control volume. On the other hand, if the con-trol volume is too small, there would be insufficientenergy flux to estimate the energy dissipation. Thelower boundary of the control volume is the bottom ofthe thermocline. The depth of the thermocline is from�300 to �900 m, as was measured by Conkright et al.(1999). We estimated that

� � O� |P|0ld�,

where �0 is the seawater density, l is the length of a legof the control volume, and d is the depth of the ther-mocline. One could see that, even if the error of d is as

FIG. 8. Energy flux of M2 internal tides derived from theTOPEX/Poseidon data south of the Aleutian Trench (extractedfrom Fig. 6). The vertical profile of the Brunt–Väisälä frequencyin this basin is shown in the embedded figure (Conkright et al.1999).


Fig 8 live 4/C


large as 50 m, the relative error of the dissipation rate� would still be no larger than 20%.

The calculations are based on 10 years of TOPEX/Poseidon altimeter tidal data. The nominal bias of tidaldata is 1.5 cm (Benada 1997). The wavenumber of baro-clinic M2 internal tides is of the order of 0.01 km�1, theaverage depth of the thermocline is assumed to be 500m, and the average water depth is about 4000 m. Theestimated bias of the energy flux in Fig. 6 is about 1000W m�1. We also estimate the error caused by the nomi-nal bias of tidal data, using the real parameters includ-ing the wavenumber, depth of the thermocline, andthe water depth in 50 random T/P diamonds. The esti-mated mean error of the M2 internal tidal energy fluxis 846 W m�1, the maximum error is 1142 W m�1, andthe minimum is 634 W m�1. If the average Brunt–Väisälä frequency is taken to be 3 cph in the innerocean, the estimated error of mixing rate is roughly 1 �10�5 m2 s�1.

There are some volumes in which the fluxes are in-sufficient to calculate the summation. There are also afew volumes near large topography, for example, nearthe east of Australia, in which the sum of fluxes is nega-tive. This means that Eq. (4) does not apply in suchareas and, as a consequence, not all energy fluxes in Fig.6 are used to calculate the energy dissipation shown inFig. 7.

The error is as large as one-quarter of the mixing ratecaused by M2 internal tides near 28.9°N/S and is evenlarger than that near the equator. However, our mainpurpose in this study is to show the quality of thelatitudinal structure of mixing rate from the viewpointof energy balance. The error as discussed above shouldtherefore have no effects on the latitudinal pattern thatwe concluded. Besides, the pattern of mixing rate va-riation with latitude in the North Pacific (Fig. 9a)is consistent with those shown in Fig. 2 of Gregg et al.(2003).

FIG. 9. Latitudinal distribution of the mixing rate caused by M2

internal tides in (a) the Pacific, (b) the Atlantic, and (c) the IndianOceans.



4. Discussion

Internal tides and wind-generated inertial waves aretwo important components of the internal waves forabyssal mixing (Garrett 2003b; Alford 2003). The lati-tudinal distribution of the mixing rate caused by M2

internal tides (Fig. 9a) is consistent with the field inves-tigation of Gregg et al. (2003) in the upper North Pa-cific. This indicates that mixing caused by M2 internaltides is so dominant that it can almost represent themixing in the upper ocean.

The available numerical models require an appropri-ate parameterization for the mixing rate. However, themixing rate changes both in space and in time. It istherefore hard to derive it from easily measurable pa-rameters such as temperature or salinity. Acquisition ofaccurate mixing rate thus requires enormous delicatefield observations, and such missions are cost-prohibi-tive at the present. On the other hand, if the distribu-tion pattern of the mixing rate shown in Fig. 9 could befurther supported by more field observations, we wouldthen be able to concentrate our investigation in theupper ocean only on two narrow belt areas around28.9°N and 28.9°S. This should reduce substantially thecost and workload in carrying out field observations.

Acknowledgments. This work was supported by theDepartment of Science and Technology, China,through State Key Basic Research Program (ProjectTG1999043800) and was partially supported by NOAANESDIS and NASA. The authors express their thanksto Prof. Rui Xin Huang for his helpful discussion andXinfeng Liang for his calculations.

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Latitudinal Distribution of Mixing Rate Caused by the M