## F0AhmL

Where T is the time period over which the average is taken, assumed to be 1 year because of the seasonal cycle wtr is the vertical velocity along the 1-year trajectory and Ahm,L represents the mixed layer depth change accumulated over a 1-year trajectory in Lagrangian coordinates. Thus, this definition includes both the temporal average and the spatial average over a 1-year trajectory. The schematic diagram in Figure 5.27 illustrates this definition for a two-dimensional case. An instrument...

## Iff v iff pUnontidalVv jjj putidai vrdv 3105

The sign of the first term on the right-hand side of Eqn. (3.105) is unclear however, tidal flows are driven by tidal force, so that the last term in Eqn. (3.105) is positive. Thus, the last two terms in Eqn. (3.101) represent a positive contribution to the non-tidal motions, and they correspond to similar terms associated with the barotropic tides discussed in Section 3.5.3. 3.5.5 Interpretation of energy integral equations Wind energy sources for the kinetic energy In addition to the...

## Info

Fig. 4.81 a Idealized wind stress (in 0.1 N m2) and b Ekman pumping (in 10 6 m s) profiles used in the analytical model. lower layer thickness along the eastern boundary. In this section, we will simply assume that the upper layer outcrops along 18 N, and the reduced gravity for the upper and lower interfaces is 0.02 m s2. The only free parameter in this section is the lower layer thickness along the eastern boundary, h1e. For the case with h1e H0 150 m, the solution is shown in Figure 4.83....

## J f 1 fi J f 2 fi

Using the barotropic streamfunction f B, the nonlinear equations (4.154a) and (4.154b) can be rewritten as J (f 1, Py + F fB) wo + O(R) (4.156a) J (f 2, Py + F fB) O(R) (4.156b) These equations are linear because f B is a given function fromEqn. (4.155). These equations are first-order partial differential equations in the characteristic form, with a quantity that behaves like a barotropic potential vorticity q2 py + F fB as the characteristics. Under the assumption of infinitesimal friction,...

## P

Integrating this equation over ps, pb (where ps is the mixed layer density and pb is the unknown free boundary separating the moving water from the stagnant abyssal water) gives fpb 2 fpbe e2 fpb a2 2gpof2 fXe B2pdp + Bpdp + Ba dp Sf J wedx (4.325) From these equations, the calculation of the wind-driven gyre in the subpolar basin is reduced to solving the following free-boundary value problem in density coordinates Bp gh(x, y) at p ps (ps is unknown) (4.327) B Ba, Bp Bap (p), at p pb (pb is...

## U

Where U is the velocity scale, g is the gravity acceleration, and h is the depth of the water. The Froude number is the ratio of the advection velocity and the speed of the surface gravity waves. When the Froude number is equal to one, the advection velocity is equal to the phase velocity of the gravity waves, and the flow is called critical. When the Froude number is larger than one, the perturbation signals cannot propagate upstream, so the flow is called supercritical. gaATh3 gPASh3 Grasshof...

## V

Where U and L are the velocity and length scale of the motion, and v is the viscosity. The Reynolds number is the ratio of the inertial force to the viscosity force. When Re is small, the friction force is important, i.e., it is comparable with the inertial force, and the flow is laminar. When Re is large, the frictional force is less important, i.e., it is much smaller than the inertial force, so that the frictional effect is confined within very thin boundary layers near the solid boundary of...

## Ui u i ui p p p p p p3128

In addition, we assume that the Boussinesq approximations hold and the tidal force is omitted. Multiplying Eqn. (3.44) by ui and averaging leads to the equation for the turbulent kinetic energy 9,-9, - dp _ , .dp' d , , , , dut -K' + u-K> a> - - (a + a') + v- (u'ta - s - (ui j Scaling analysis indicates that for steady-state flows the basic balance is between the turbulence production, the dissipation, and the buoyancy work (Turner, 1973 Osborn, 1980) Defining the flux Richardson number as...

## L

Fig. 5.103 The Goldsbrough-Stommel circulation of the world's oceans, neglecting the inter-basin transport. Each arrow indicates the horizontal mass flux integrated over a 5 x 5 box, in Sv along the western boundary of each basin, there is a curve indicating the northward mass flux (in 106 m3 s) within the western boundary, which is required to close the circulation (Huang and Schmitt, 1993). evaporation and precipitation. The western boundary currents that perform the role of closing the...

## Description of the worlds oceans

The main focus of this book is the study of large-scale circulation in the world's oceans. As a dynamical system, the circulation in the world's oceans is controlled by the combined effects of external forcing, including wind stress, heat flux through the sea surface and seafloor, surface freshwater flux, tidal force, and gravitational force. In addition, the Coriolis force should be included, because all our theories and models are formulated in a rotating framework. In this chapter, I first...

## Hi P

From Eqns. (4.40, 4.49), the streamfunction within the boundary layer is (Fig. 4.5a) The global structure of the boundary layer solution depends on the choice of parameters used in the model, in particular the choice of R. From observations, the width of the Gulf Stream is about 50 km, thus we choose Ss 25 km. Assuming P 2 x 10-11 m s, hi 400 m, then the suitable choice is R Phi Ss 2 x 10-4 m s2. As an example, we choose a model mimicking the North Atlantic Ocean, with parameters he 300, g'...

## A

A Model with a rigid lid b Model with a free surface Fig. 4.2 Multi-layer models a with a rigid lid or b with a free surface. surface z 0, so the original problem of a moving boundary is reduced to one with a fixed boundary. Owing to the existence of a non-zero sea surface level Z 0 at the flat surface z 0, the equivalent hydrostatic pressure p pa is not constant. Using the hydrostatic relation, one can calculate the pressure in the layers beneath. Starting fromp pa at z 0 and integrating the...

## Hifi

Where the superscript II indicates that the definition applies to region II with two moving layers, and the subscripts indicate the individual layers. It is important to note that fi in these relations is not necessarily a constant. In fact, we can write it in the formfi (Hi), which indicates thatfi depends on the latitude of the outcrop line by tracing backward along the streamline Hi const. Therefore, in our discussion hereafter, we can write alternately fi fi + Sfi, where we imagine that Sfi...

## K

Where v is the viscosity and K is the temperature diffusivity. The Prandtl number is used to describe the ratio of momentum dissipation and tracer mixing. At the level of molecular dissipation and mixing, the Prandtl number for seawater is approximately equal to 8 however, owing to strong eddy and turbulent activity, the equivalent Prandtl number for the large-scale geostrophic turbulence may be quite different from this value.

## Ggw

Ks ,., , ,n + K (5.171) where K* 10-4 m2 s is the maximum diapycnal diffusivity for salt Angering Kis the constant diapycnal diffusivity due to other mixing processes unrelated to double diffusion. 0.3 x 10-4 m2 s was used in many studies (e.g., Zhang et al, 1998). Rc 1.6 is the critical density ratio above which the diapycnal mixing due to salt fingering drops dramatically, due to the absence of staircases n 6 is an index to control the decay of Kq, Ks with increasing Rp. For the case of...

## J f 1 qi wo y h Jf2 q2 Vh 2 DV2f2

Where J (g, h) gxhy - gyhx is the nonlinear Jacobian term, q1 and q2 are potential vorticity in the upper and lower layers where g' is the reduced gravity, H is the undisturbed layer thickness, and Xr is the Rossby radius of deformation. We have neglected relative vorticity in these equations because it is negligible for basin-scale motions. The F(fi - fi-1) terms are the contribution due to interface deformation, also called the stretching term, noting that the interface height is proportional...

## Pw psp

The corresponding diffusive mass flux, relative to the center of mass, is Jw pw (uw u) pwsm ps (1 s)m > 0 (3.39) Js Jw ps (1 s)m< 0 (3.40) Note that the center of mass moves upward with a velocity of u (1 s)m, slower than the rate of evaporation because the salt component is stagnant the diffusive salt flux defined here is the diffusive flux relative to the center of mass. The advective salt flux associated with the center is ups Jw, which is equal to the rate of diffusive salt flux, but...

## Hds fffv Tdv3183b

Is entropy production due to internal dissipation, Hthermal - jjj v (3.183c) is entropy flux and generation due to heat transport, including entropy fluxes through the air-sea interface and internal entropy generation through thermal mixing. In order to analyze this term we divide the ocean into two layers an upper layer (about 10-15 m, denoted as Vi) and a lower layer (denoted as V2) (Fig. 3.53). On the sea surface, there are five heat fluxes net short-wave radiation qsw, net long-wave...

## S 20s 10s Eq 10n 20n 30n 40n 50n 60n 70n

Fig. 5.179 Annual-mean northward freshwater transport in the Atlantic Ocean, in Sv. In particular, the overturning streamfunction in potential temperature coordinates is closely related to poleward heat flux therefore, for climate study, the overturning streamfunction in temperature coordinates may serve as the best diagnosis tool. The cores of the overturning streamfunction maps, labeled in red, represent the poleward transport of surface water and the return flow at depth. Since the model...

## S 40s 20s 0 20n 40n 60n

Fig. 2.12 Meridional distribution of potential density along 30.5 W (kg m3), using two different reference levels. unstable stratification is artificial. Were the equation of state linear, there would be no such artificial problems. Apparently, the vertical gradient of a0 changes signs at about 4 km depth. Thus, the water column at this depth range seems gravitationally unstable. A column of water taken from depth range 4-4.5 km would be unstable if it were adiabatically brought to the sea...

## Preface

With great progress being made in science and technology, we are becoming more interested in finding out how the climate system, including the oceanic general circulation, works on our planet. This book is written for the general reader who is searching for knowledge about oceanic circulation and its relevance to climate and the global environment on Earth. During the process of collecting the materials for this book, I have tried to achieve a sensible balance between the physical concepts...

## I I I I I I I I I I I I I I 012345678

Fig. 4.118 Layer thickness adjustment in response to wind stress perturbations, described in Fig. 4.117. he is the original layer thickness along the eastern boundary. Curves labeled with he indicate changes in the layer thickness along the eastern boundary hWW for changes of layer thickness at the western boundary and right at the equator hsw for changes of maximum layer thickness along the western boundary of the subtropical gyre and hw for changes of maximum layer thickness along the western...

## E 60e 90e 120e 150e 180 150w 120w 90w 60w 30w

30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W Fig. 1.33 Zonal section of salinity along 58.5 S (Southern Ocean). 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W Fig. 1.33 Zonal section of salinity along 58.5 S (Southern Ocean). 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W Fig. 1.34 Zonal section of potential density (kg m3) along 58.5 S (Southern Ocean). 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W Fig. 1.34 Zonal section of potential density (kg m3) along 58.5 S (Southern Ocean). the...

## W woo wobB511o

Where woo 1o 7m s, wo 5 x 1o 7m s. a No ridge, uniform upwelling b With ridge, uniform upwelling a No ridge, uniform upwelling b With ridge, uniform upwelling C c) No ridge, non-uniform upwelling d With ridge, non-uniform upwelling C c) No ridge, non-uniform upwelling d With ridge, non-uniform upwelling Fig. 5.62 Interfacial displacement (in units of m), the thickness of the abyssal layer is H 3,000 m, the uniform upwelling rate is W00 10-7 m s, the amplitude of the enhanced upwelling rate over...

## N 25n 30n 35n 40n 45n 50n

Fig. 4.113 The -3 C anomaly isotherms for 1977-81 (dashed line), 1982-6 (dotted line), and 1987-91 (solid line), superimposed upon the mean late-winter isopycnals (thin solid lines labeled with numbers, in From Figure 4.ii2, during the period from i977 to i989 the temperature anomaly moved down about 260 m, so the vertical velocity is slightly more than 20 m year, which is on the same order as the vertical velocity in this area of ocean, as inferred from the Ekman pumping rate on the order of...

## E P

FpBouss PoPS - P (E - P) PS (2.157) Since pS 0.02, the equivalent density flux in a Boussinesq model is 50 times smaller than the real density flux, and the sign is opposite to the real density flux. 2.7.4 Pitfalls of using the buoyancy flux to diagnose energetics of the oceanic circulation If the equation of state is linear in temperature, salinity, and pressure, then buoyancy is linearly proportional to density, so the balance (or the transport) of buoyancy, the temperature, the density, or...

## Jp2

I.e., the system can generate mechanical energy to sustain the circulation by itself. Thus, Sandstrom came to the conclusion that a closed steady circulation can be maintained in the ocean only if the heating source is situated at a level lower than the cooling source. Sandstrom also carried out laboratory experiments to demonstrate his postulation. In the first experiment, the heating source was put at a level lower than the cooling source. Strong circulation was observed between the levels of...

## W v cgf pg

Assume the following values for the basic parameters of the circulation f 10 4 s, P 2 x 10-11 s m, Lx 2.5 x 107 m, Ly 107 m, Ln Lsy 1.5 x 106 m, Aj 1,000 m2 s, c 0.16, g' 0.01 m s2, k 10-5 m2 s and the typical values for these depth scales are de 313m, dw 685m, and dK 368m. Comparing these three depth scales, it is clear that pycnocline depth associated with wind stress is the dominating scale thus, it is convenient to use this depth scale and introduce the nondimensional depth d D dw, for...

## TyBx Ahfxxxx457

Where ty B is the boundary layer solution. The scale width of the western boundary layer is the vorticity equation is reduced to This equation is subject to the following boundary conditions. First, the boundary layer solution should be finite and it should match the interior solution at infinity (the outer edge of the boundary layer) tyB tyi, n (4.61) Second, the western boundary is a streamline, tyB 0, at n 0 (4.62) In addition, two types of boundary condition may apply a) No-slip condition...

## S T

The simplest choice is a two-box model for a single-hemisphere ocean, in which temperature is relaxed toward the specified reference temperature of T To, T 0 and an air-sea freshwater flux p, p . To simulate the effect of the wind-driven gyre, a volume transport m between the two boxes is added (Huang and Stommel, 1992). Following Guan and Huang (2008), this volume transport is prescribed a priori, independent of the energy for mixing and temperature salinity (Fig. 5.131). Assuming that the...

## D

Sm I Wmb + Vmb hm,max (5.38) where hm,max is the annual maximal depth of the mixed layer Sf wmb + Vmbddyhm (5.39) where hm is the annual mean mixed layer depth. Note that both of these definitions treat the subduction in the local sense, so these two subduction rates do not include the average over the trajectory downstream. As a result, the rates calculated from these two equations are smaller than the rates calculated from the above-defined SE and SL, both of which include the contribution...

## E 60e 90e 120e 150e 180 150w 120w 90w 60w 30w 0

Fig. 1.11 Annual mean (NCEP-NCAR) rate of evaporation minus precipitation (cm yr). Fig. 1.13 Geothermal heat flux based on a semi-empirical formula, seafloor shallower than 2.6 km is excluded (mW m2). Fig. 1.13 Geothermal heat flux based on a semi-empirical formula, seafloor shallower than 2.6 km is excluded (mW m2). Fig. 1.19 Annual mean sea surface temperature anomaly ( C), deviation from the zonal mean. Fig. 1.19 Annual mean sea surface temperature anomaly ( C), deviation from the zonal...

## 2 hU

The potential vorticity balance for a closed streamline (or the whole basin) is between the frictional torque (mostly generated along the edge of the basin where the friction is non-negligible) and the wind stress torque imposed over the basin. Thus, no matter how small the frictional force is, it is essential for maintaining the basin-wide vorticity balance between wind stress input and frictional output. Furthermore, a purely inertial model is physically impossible, because vorticity input...

## Bpp Bxd p Bp Bxpb Bp Bx ps I Bp Bpxd p

Where the first term on the right-hand side is zero due to the matching boundary condition that horizontal velocity at the base of the wind-driven gyre vanishes. The last integral can be converted into another form d fpb 2 2 dpb 2 dps fpb dx I Bpdp Bp -Xp - Bp + 2i BpBxpdp (4.301) Combining equations (4.298, 4.299, 4.300, and 4.301), we obtain

## N 80n 70n 60n 50n 40n 30n 20n 10n

Fig. 5.55 a-d Transport of the western boundary current driven by a point sink specified at location 0q, in units of the total sink Sq. Fig. 5.55 a-d Transport of the western boundary current driven by a point sink specified at location 0q, in units of the total sink Sq. Fig. 5.56 Deep circulation in the world ocean postulated by Stommel (1958). Fig. 5.56 Deep circulation in the world ocean postulated by Stommel (1958). interior of each basin water uniformly moves upward, as assumed in the...

## McO Mg O Mt O

Since the vertical grids are uneven, in order to show the volume flux over a certain depth range, the volume flux in each layer is re-scaled as Gp Gpk-h , Gf Gth (5.380) where h0 100 m is the typical scale for most essential features associated with gyre-scale circulation. In comparison, the commonly used MOC streamfunction is defined as the vertical integration where m fk (Xf, 0j) is the meridional throughflow volume flux defined above. Thus, the MOC streamfunction includes both positive and...

## P dJpSp VdSJp

Equations (2.152) and (2.153) combined with Eqns. (2.144) and (2.147) constitute a complete set of basic equations of a dynamical system under the so-called Boussinesq approximations, i.e., the following simplifications are assumed Using the volume conservation to replace the mass conservation Retaining the buoyancy forcing associated with small density deviation Using volume conservation in the tracer (temperature and salinity) prognostic equations. 2.7.2 Potential problems associated with...

## S

Fig. 5.97 A finite difference grid box in the x z plane. dJL + (us)+ (us) + (wS)+ (wS) (S+ ) where (uS)+ and (uS) are the salt fluxes advected across the right and left boundaries, and S+ and S are the horizontal salinity gradients at the right and left boundaries similarly, (wS)+ and (wS) are the salt fluxes advected through the upper and lower boundaries, and Sf and (wS) are the vertical salinity fluxes at the upper and lower boundaries. In addition, vertical velocity is prescribed on the...

## N Wt

D) If the slippery boundary condition applies, fB,nn 0, at n 0 thus, C4 -03 3. The final solution is e) Structure of the western boundary current the layer thickness of the boundary current can be obtained from the semi-geostrophic relation h2 2f fi + hW, hW hi - 2f fi g' (4.67) The corresponding velocity field can be calculated accordingly as v . inertial western boundary current The existence of an inertial western boundary current was postulated first by Stommel (1954). His basic idea is...

## Thermohaline circulation

The balance of water masses in the world's oceans consists of two major processes water mass formation and erosion. Most water masses are formed near the upper surface and sink. Furthermore, through either transformation or erosion, water mass properties are continually transformed, so that a water mass gradually loses its identity. Therefore, some types of water mass are formed below the surface layers through the mixing of water masses originated from the sea surface however, in this chapter,...

## Ojm

Fig. 5.122 Three equilibriums for a two-box model of Stommel (1961) the stable note (indicated by a), the saddle (indicated by b), and the stable spiral (indicated by c) the arrows indicate the behavior of the model's solutions in the T-S diagram. density difference due to the temperature component. Such a large salinity difference is possible for a slow circulation only because of the long relaxation time scale. An unstable state that is thermally controlled, represented by the saddle point b...

## E 20e 30e 40e 50e 60e

Fig. 4.33 Isolines of potential vorticity in the bottom layer, with the heavy dashed line indicating the outcropping line of the upper layer. Although east of the thin dashed line potential vorticity contours are blocked by the eastern boundary, west of the thin dashed line potential vorticity contours disconnected from the eastern boundary are closed through the western boundary. boundary, along which geostrophic motion is forbidden. However, there are contours stemming from the western...

## E 150e 180 150w 120w 90w

Fig. 4.90 Climatological-mean circulation on four isopycnal surfaces a oq 23.2 b 24.2 c 25.0 and d 26.0 kg m3. Gray lines indicate depths (m) black and gray thick lines indicate pathways. The ventilation time for the pathways (in years) is indicated by a pair of numbers the left for the western boundary pathway and the right for the interior pathway. Gray thick lines near the eastern boundary in a and b indicate a narrow local pathway, a short-cut, from the subtropics to the equatorial regime...

## N 30n 45n 60n

Fig. 5.199 a Poleward heat flux in the 3-D ocean model (in 1015 W) b meridionally accumulated energy required for sustaining subsurface diapycnal mixing in the 3-D ocean model (in 109 W) (Huang et al., 2007). the ocean interior, which may enhance the MOC and poleward heat flux. On the other hand, more mechanical energy input can increase the mixed layer depth. This process also plays an important role in regulating the MOC and poleward heat flux. For example, tropical cyclones can input a large...

## Pp QB p

Bp -gh(x, y) at p ps (4.305) B Ba, Bp Bap, at p pb ( pb is unknown) (4.306) ' 2 rpb a2 2 g p f2 rxe B2dp + Ba dp wedx (4.307) Although this approach seems to be a simple extension of the early work by Welander (1971a), there are subtle differences between the new formulation and Welander's old formulation. In Welander's formulation, it was assumed that potential vorticity Q(B, p) in Eqn. (4.304) is a given function of B and p, and this equation is subject to boundary conditions specified at the...

## N 20n 30n 40n 50n 60n

Fig. 5.145 Meridional velocity distribution along the western boundary of the model basin, in The meridional overturning circulation constitutes the most important component of the thermal circulation (Fig. 5.146). In turn, this consists of the following four branches 1. a poleward flow of warm water in the upper ocean, as indicated by the relatively compact streamlines in the upper ocean 2. a narrow sinking branch near the northern boundary 3. a relatively broad and slow upwelling in the ocean...

## Is Pacific Ocean In Western Hemisphere

Where we is the Ekman pumping velocity calculated from wind stress. The total meridional volume flux involved in the wind-driven gyre across a latitudinal section of the basin interior is the sum of these two fluxes, i.e., MSv ME MG. The return flow into the western boundary is MSv. The bifurcation of the western boundary current is a nonlinear phenomenon however, here we assume that the bifurcation latitude of the western boundary current is the latitude where MSv vanishes this can be...

## N

Fig. 5.151 Sketch of the northward heat flux associated with the two equilibrium states redrawn from F. Bryan, 1986 . surface. Water has a much higher heat capacity than air thus, the ocean is the dominating factor influencing the climate, and climate on a planet without oceans would be extremely different from that on a planet with oceans. However, as shown in the numerical experiments by F. Bryan, this poleward heat flux can be changed substantially if the thermohaline circulation in the...