Parameterization of ocean wave-induced mixing processes for finite water depth

Acta Oceanologica Sinica 2009, Vol.28, No.4, p.16-22http:/ /www .hyxb.org.cnE-mail: hyxbe@263.netParameterization of ocean wave- induced mixingprocesses for finite water depthYANG Yongzeng1,2*, ZHAN Run', TENG Yong'1 First ]Institute of Oceanography, State Oceanic Administration, Qingdao 266061, China2 Key Laboratory of Marine Science and Numerical Modeling, State Oceanic Administration,Qingdao 266061, ChinaReceived 3 March 2008; accepted 18 September 2008AbstractThree dimensional wave-induced mixing plays an important role in shallow water area. A quitedirect approach through the Reynolds average upon characteristic length scale is proposed to pa-rameterize the horizontal and vertical shallow water mixing. Comparison of finite depth case withinfinite depth results indicates that the difference of the wave-induced mixing strength is evident.In the shallow water condition, the infinite water depth approximation overestimates the mixingstrength in the lower layers. The nonzero horizontal wave-induced mixing presents anisotropic prop-erty near the shore. The Prandtl's mixing length theory underestimated the wave- induced mixingin the previous studies.Key words: Reynolds average, Characteristic length scale, Wave-induced mixing parameter1 Introductionessential for mixed layer depth simulation.However, the wave-induced mixing parameteriza-Overestimated sea surface temperature (SST) andtion of infinite water depth approximation is probablyunderestimated mixed layer depth in summer are com-unsuitable in nearshore region, especially in the re-mon problems in ocean circulation modeling (Martin,gion in which the water depth ranges from 0 to 20 m.1985; Kantha and Clayson, 1994). It is believed thatSo this study addresses finite water depth cases, andthese two problems are caused by insufficient surfacepresents a more reasonable expression of wave-inducedmixing. The mixed layer is caused by several mecha- horizontal and vertical mixing parameters (Section 2).nisms, such as vertical convection, solar heat penetra-In Section 3, these parameters are estimated primarilytion, and ocean wave-induced mixing. In the modelby integrating the LAGFD-WAM Wave Model. Com-of Ezer (2000), the overheating SST was turned down parison of finite depth case with infinite depth resultsby asumning that shortwave radiation penetrates to is analyzed. At the end of this section, we compare the .deep layers. In other models, Agrawal et al. (1992),vertical mixing with Prandtl's mixing length results.Craig and Banner (1994)， Terry et al. (1996, 1997,Some conclusions are summarized in Section 4.2000)，Mellor and Blumberg (2004) suggested thatsurface wave breaking could enhance mixing in up2 Model derivationper layers by turbulence. In cases when consider-ing three-dimensional wave motions for infinite waterWe consider the linear wave equations for finitedepth, Yuan et al. (1999) proposed a parameteriza-water depth,tion of wave-induced mixing for circulation modelingbased on Prandtl's mixing length theory. Yang et al. .Oφ= 0z≤0(1(2003) estimated the amplitude of wave-induced verti-cal mixing parameter Bv in the ECS (the East China{Uw1,Uw2,Uw3}= Vφ .(2Sea) by using the LAGFD-WAM Wave Model. By in-8ζ_ 0φz=0ot0zcorporating the wave-induced mixing and tidal mixing)dprocesses in the Huanghai Sea, Qiao et al. (2004) ana-)t"9ζ=0. z=0(4lyzed the temperature and salinity distributions along0φ36° N section and found that both mixing processes are=0z=-H(5中国煤化工This work is supported by the national young scientist fund of China under contractand special fundfor fundamental scientific research under contract (No. 2007G15).YHCNMHG*Corresponding author, E-mail: yangyz@fio.org.cn.YANG Yongzeng et al. Acta Oceanologica Sinica 2009, Vol. 28, No. 4, P.16-22where φ is velocity potential, 5, elevation of waves, and where the subscripts t, w, C represent turbulence, waveH water depth. The solutions of the above equations and current respectively. We first consider the wave-can be written ascurrent Reynolds stress which indicates the wavemixing intensity for current momentum. As givenζ=| | Aexp{i(R.r-wt)}dKby Yuan et al. (1999)， the wave-current Reynoldsstress isφ=[ [_igoshK(日+2)xsinhKHTwotij = -(Uwiltj) - (utilwj>.(11)exp{i(后.r- wt)}dh(7) The above characteristic length scale L,H,T forUw1 =[ wk1 gcoshK(H+z)、Reynolds average satisfyKAsinhKH xLt,Lw<< L<< Lcexp{i(后.r - wt)}dh .(8)Ht,Hw<< H<< HeUw2 =: [ (4kxcoshK(H+2》)xT,Tw< =2ZuusQryAT,(13)L2HTexp{i(后.p - wt}dk(10)where A(k) is the wave amplitude, and k: is the wave-where C△vj represent scale integral volume of fAuid el-number vector. Assuming that the wave field is sta-ements, N is the number of elements in volume L2H,titically stationary, that is (A()4(R) = 8(后 - and OT, the temporal interval.的)E(E).In the following, we adopt L = Lt = Lw,H =Ht= Hw, and T= Ti = Tw for simplicity. (13) can尸=ζbe rewritten as1z=0(ut1Uw3) =下2汇huu3Ov;AT; (14)L2 HwTwi=1j=1|Hwhere N1 is the number of elements in volumeL2Hw.For a given time t, one sort of fAuid elements has the.........V L,property uv3> 0, and the other, o3≤0 (Fig. 2).LSea floor .Fig.1. Sketch of the average scale.cn)●d1We now decompose Ui into three velocity com-ponents: a mean component ui, whose characteristicQu'<0scales are Te in time, Le, in horizontal and Hc in verti-u"'"=0cal, respectively; wave component Uwri, whose time andspace scales are Tw, and Lw(Hw); and a random tur-中国煤化工bulence component Uuti, whose time and space scalesare T, and L(Ht), such thatMYHCNMHGUi= ui+Uwi+ Uti. i= 1,2,3Fig.2. Sketch of sorts of fluid elements..18YANG Yongzeng et al. Acta Oceanologica Sinica 2009, Vol. 28, No.4, P.16-22For the first, there exist a height zj < z and time (19) can be also derived aszj= z+ A(z). Because thetj =-sinh2 KHu3△v;△T.(16)Given that uU'w3 and z - zj have the same sign for thesinh?K(H +z),。0u1first sort of elements,M当2S(z-z)%3Ao;OT;≥0.(17)For the wave-current Reynolds stress (ut3Uw1>, we. i=1 j=1can also obtainFor the second sort of ones, (ut1Uw3)2 has the same(ux3Uu)>=-忙E(R)Cosh*K(H +z)a .property. So<25(u1uw3) = (1unuw3)1+ (u1Uw3)2=-L.H..T.. xcosh2K(H + z)dk8u3(21)- zj|lu%sIOv;△T;. (18)i=1j=1Now we can formulate the other wave currentLet Hw= 2A(z) where A(z) is the wave amplitude at Reynolds stresses below. Letdepth z. For wave motion, the height zj = z- A(z)(or zj = z+A(z)) at which uw3 = 0. (18) can beE(R)sinh'^K(H+z2)dkB,expressed by the integration as{[+()sinh2K(H + z)(unus) =- δz 24()L2T ./=-a()J[.! u2E(的)2di(22)(z' -(z- A(z)) . dz1u3(z')|. dxdydT≈rz+A(z)|工E(后)cosh"K(H +2)ai{11K0z 2A(z)L2TwJz-A(z)Tw I(z' -(z- A()) . dz'uw3(z)| . drdydT =,黑E(R)cosh2K(H +2)dk//o0u1 18z 2A(2)( 2”-(z- A())cosh2K(H+z)|z'=z+A(z)Bn2={/1管E(问)|uw3(z)|. dxdydT|z'=z-A(z)。”Lw2,2号E(E:cosh2K(H+z)ail(24)= - 0u1AA(>)(luw3()>.(19)中国煤化工0zMHCNMH G.YANG Yongzeng et al. Acta Oceanologica Sinica 2009, Vol. 28, No.4, P.16-2219there are220- 35°N and 1189-133°E with horizontal resolution0u2of 19/4 by 19/4. The model temporal resolution isTwt11 = 2Bh:Twt22 = 2Bh2u15 minutes. The NCEP (National Centers for Envi-ronmental Prediction, USA) re-analyzed wind fieldsTw133= 2Bwith horizontal resolution of 1.25° by 1.0° and time0u1interval of 6 hours is interpolated into the model grid.Twt12= Twt21 = Bh1(25)8xFigure 3 shows the significant wave height distribution0u3,0u1on Aug.16, 2001, and the maximum reaches up to2 mTwt13 = Twt31 = Bh1+ Bin the ECS.Tut23= Twt32= Bra2 + B"8oy119.0° 121.0* 123.0* 125.0* 127.0° 129.0* 131.0°EWe can make decomposition for temperature and3.0" -salinity, θ= 0+0',S= 5+ S', where the quantities N1.8with over-bar and prime symbols denote mean values 31.0*1.6and fAuctuations. As described above, we can further29.1.obtain80)07.0°--(uw10)>= Bh1-(uw20)= Bn2y’0.80.6)百25.0"-(uw30'>= B,0..23.as8S-(uw1S')= Bn1 -(uw2S')= Bn2gn;0-(uw3S')=Buaz(26)Fig.3. Significant wave height distribution inthe ECS on 16 Aug., 2001 (Unit: m).3 Simulations .Figure 4 is the spatial wave- induced mixing pa-3.1 Comparison of finite water depth with in-rameters (Bn1 and Bn2) distributions at surface, andboth almost have the same peak value. But at thefinite depthshallow water site (31.5°N, 122°E)， Bh1 is about 5In this study, the 3rd generation LAGFD-WAM6 times larger than Bh2 from surface to bottom beWave Model is integrated in the ECS (East China cause waves propagate westward dominantly to theSea) in August, 2001 to compute the wave- numbershore (Fig. 5). Horizontal anisotropic wave inducedspectrum, then the wave- induced mixing parameters mixing is evident in the shallow water condition.are estimated tentatively according to the formulaeIn general, when the water depth is greater than(22)，(23) and (24). The area being computed covers 50 m, (22) can be reduced to119.0° 121.0° 123.0° 125.0" 127.0* 129.0* 131.0E119.0* 121.0" 123.0° 125.0* 127.0° 129.0* 131.0E0.180.1633.0* -33.0*- b31.0°-0.140.1229.0°0.1一0.10.0827.0" -27.0"0.0625.0° -0.0423.0°0.02 23.0中国煤化工0.02YHCNMHGFig.4. Distributions of Bn1 (a) and Bh2 (b) at surface (Unit: m2/s). ..2YANG Yongzeng et al. Acta Oceanologica Sinica 2009, Vol. 28, No. 4, P.16-220.0B, ={!E()exp{2K*z}dkw2E(丙)2.0- φ合4.0言-中exp{2Kz}dk(27)宫6.0-which is more practical for the application to the deep8.0water condition. Figure 6 is B。distribution at the sur-face and its maximum decreasing with depth according10.0-to (27). But for shallow water condition, this reduc-0.0x10° 4.0x10° 8.0x103 1.2x 1021.6x102tion of (27) can cause error. Figure 7 shows the differ-B.,B2 (m'/s)ence between finite and infinite water depth. And thelatter overestimates the mixing strength in the lowerFig.5. Vertical distributions of Bh1 (O) andlayers.Bn2 (O) at Point (31.5°N, 122°E).119.0° 121.0* 123.0° 125.0* 127.0° 129.0* 131.0°EI 0.30.2731.0" .0.2129.0°8- 0.1527.0°21225.0*一 0.09- 0.060.030.000.100.200.30Bv (m2/s)Fig.6. Bw distribution at surface(a) and vertical structure at Point(25°N, 131°E) (b) for infinitewater depth..0 Tet al. (1999) proposed the formulae of wave -inducedmixing parameters.0-B。=a | | E(k)exp{2kz}dk xi 4.0-”无(1-w2E(k)exp{2kz}dkBh1= 0(29)Bh2= 0(30)0.0x 10°4.0x108.0x10° 1.2x102under infinite water depth approximation.In theirmodel, a=1 and is still an undetermined coefficient.Fig.7. Comparison of Bv between finite (OFigure 8 is Bv distribution at surface and verticaland infinite (O) water depth at Point (31.5°N,structure of its maximum according to (28) where1229 E).a=1. So its ni 中国媒化工sale thanthatinFig. 6car the surface.3.2 Comparison to Prandtl's mixing lengthIt should beTYHCNMHGybyYuanettheoryal. (1999)， the Prandtl's mixing length l= A(z).Based on Prandtl's mixing length theory, Yuan If l= 2A(z), then a=4. Under this circumstance,.YANG Yongzeng et al. Acta Oceanologica Sinica 2009, Vol. 28, No. 4, P.16-222119.0°, 121.0* 123.0. 125.0°. 127.0°. 129.0*. 131.0E0.060.05431.0*-0.0480.04229.0°官80.0327.0.024宫1225.0"0.0180.0121623.0°0.006200.0x 10"2.0x 10-24.0x 10°6.0x 102Bv (m?/s)Fig.8. B。distribution at surface (a) and vertical structure at Point (25°N, 131°E) (b) according to Prandtl'smixing length.the numerical mixing strength agrees with ours apReferencesproximately. The vertical mixing strength decaysfaster with water depth in Fig. 8b than that in Fig.Agrawal Y C, Terry E A, Donelan M A, et al. 1992. En-hanced dissipation of kinetic energy beneath break-6b, which can be due to B, ~ e2Kz according to (27),ing waves. Nature, 359: 219 -220while B, ~ e3Kz according to (28).Craig P D, Banner M L.1994. Modeling wave -enhancedturbulence in the ocean surface layer. J Phys4 ConclusionsOceanogr, 24: 2546- -2559The three dimensional wave-induced mixing playsEzer T. 2000: On the seasonal mixed layer simulated bya basin-scale ocean model and the Mellor-' Yamadaan important role in shallow water area. In this study,turbulence scheme. J Geophys Res, 105(C7): 16843-a quite direct scheme through the Reynolds average16855upon characteristic length scale is proposed to pa-Kantha L H, Clayson C A. 1994. An improved mixedrameterize the shallow water mixing. The horizontallayer model for geophysical applications. J Geophysand vertical wave-induced mixing parameters are pre-Res, 99: 25235- 25266sented for finite water depth. The nonzero horizontalLedwell J R, Montgomery E T, Polzin K L, et al. 2000.wave-induced mixing in our model, which differs fromEvidence for enhanced mixing over rough topogra-the previous parameterization scheme, indicates thatphy in the abyssal ocean. Nature, 403(13): 179 - 182this mixing mechanism should be discussed in futureLevitus S.1982. Climatological Atlas of the World Ocean,studies.NOAA/ERL GFDL Professional Paper 13, Prince-Based on the integration of the LAGFD-WAMton N J, 173Wave Model in August 2001, we computed primarilyMartin P J. 1985. Simulation of the mixed layer at OWSthe wave-induced mixing strength for finite and infi-November and Papa with several models. J GeophysRes, 90: 581- -597nite water depth in the ECS. Comparison indicatesthat difference is evident, especially for shallow waterMellor G L, Blumberg A F. 2004. Wave Breaking andOcean Surface Layer Thermal Response. J Physcondition in which the infinite water depth approxima-Oceanogr, 34(3): 693- 698tion overestimates the mixing strength. The horizon-tal wave-induced mixing presents anisotropic propertyMellor G L, Yamada T.1982. 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