Interactive comment on “ Technical Note : Volume Transport Equations in Combined Sverdrup-Stommel-Munk Dynamics without Level of no Motion ”

I’m afraid I cannot recommend publication of this paper. First, much is made of the result that geostrophic transport vanishes if there is a level of no motion. This is not true. The simplest counter-example is the 1.5 layer ventilated thermocline where the lower layer is assumed at rest, and yet the interior flow maintains a net meridional flow via Sverdrup dynamics. Eq (14) is missing a term proportional to the integrated divergence of the geostrophic flow which, without a level of no motion, is a term involving


Introduction
The seminal papers by Sverdrup (1947), Stommel (1948), and Munk (1950) laid the foundation of wind-driven ocean circulation.Sverdrup balance (SB) represents the meridional volume 30 transports employing only the local wind-stress in a linear dynamical framework (Wunsch 2011).Stommel (1948) and Munk (1950) linear frictional ocean models are used to explain the existence of intensive western boundary currents.Their theories were established on the base of an assumption of level of no motion (H).What is the physics behind this assumption?What happens if level of no motion does not exist?35 Let the Cartesian coordinates be used with (x, y) the horizontal and z the vertical coordinates (upward positive) and (i, j, k) the corresponding unit vectors, and let the horizontal and vertical velocities be represented by V = (u, v, w).With low Rossby and Ekman numbers, steady state momentum and continuity equations are given by where f =2 sin  , is the Coriolis parameter,  the Earth rotation rate, and  the latitude; ρ is the density; p is the pressure; (A z , A h ) are the vertical and horizontal eddy viscosities; 45 , is the horizontal gradient operator.Vertical integration of the horizontal momentum equations ( 1a) and (1b) with respect to z from a depth z = -H(x, y) to the surface leads are the zonal and meridional transports, which satisfy 0 The vectors 55 represent wind stress, bottom stress due to vertical eddy viscosity, and bottom stress due to horizontal viscosity.Here, (u -H , v -H ) are the current velocities at depth z = -H; ρ 0 (= 1028 kg/m 3 ) is 60 the characteristic density.
If z = -H is a level of no motion, (u -H , v -H ) = 0 (also implying no stress at this level, i.e., τ (b)   = 0 if the drag law is used) and the P function in Sverdrup (1947) Ocean Sci. Discuss., doi:10.5194/os-2016-81, 2016 Manuscript under review for journal Ocean Sci.Published: 1 November 2016 c Author(s) 2016.CC-BY 3.0 License.
, , exists.65 Under the condition (8) (i.e., existence of the P function) cross differentiation of (3) and (4) will make disappearance of the horizontal pressure gradient terms and give the classical Sverdrup balance (SB) if no horizontal viscosity (A h = 0), curl( ) If z = -H is not a level of no motion, the P function does not exist.This is because 70 This leads to an impossible relationship ( ) ( ) Cross differentiation of (3) and (4) without a level of no motion leads to a revised Sverdrup transport equation, 75 Substitution of the geostrophic balance , Comparison between ( 9) and ( 14) leads to the fact that existence of a level of no motion is the same as vanish of meridional geostrophic transport in the system, (16) 85 However, verification of the accuracy of the SB theory is based on the comparison of the Sverdrup meridional transport (i.e., the surface wind stress curl) with the meridional transport calculated directly from the geostrophic currents based on hydrographic data (i.e., the meridional geostrophic transport).The first was that of Leetmaa et al. [1977], followed by the studies of Wunsch and Roemmich [1985], Böning et al. [1991], Schmitz et al. [1992], etc.Their results have 90 shown that the Sverdrup meridional transport is generally consistent with the meridional transport calculated directly from the geostrophic currents based on hydrographic data in the northeastern subtropical North Atlantic Ocean, but is inconsistent with the geostrophic transports in the northwestern subtropical North Atlantic Ocean.Meyers [1980] discussed the meridional transport of North Equatorial Countercurrent in the equatorial Pacific and found significant inconsistency 95 with the Sverdrup theory.Hautala et al. [1994] estimated the meridional transport of the North Pacific subtropical gyre along 24°N and noted that the Sverdrup balance is not valid in the western subtropical Pacific Ocean.Lately, Wunsch [2011] has evaluated the accuracy of the Sverdrup theory using an assimilated global ocean dataset.
A logical way to overcome such a mismatch between SB theory (no meridional geostrophic 100 transport) and verification (comparison of surface wind stress curl to meridional geostrophic transport) is to remove the level of no motion and instead to use a known level.The ocean bottom [i.e., z = -H(x, y)] is a reasonable choice.Therefore, H is referred to the ocean bottom depth here

Geostrophic Currents under Boussinesq Approximation
With the Boussinesq approximation, vertical differentiation of (13) and use of hydrostatic balance (1c) leads to the thermal wind relation, 115 Vertical integration of (17) from the ocean bottom [z = -H(x, y)] to any depth z leads to the calculation of the geostrophic currents from the density ρ, ) 120

Volume Transport Equations
Cross differentiation of (3) and (4) leads to the transport equation for the whole water column, where  is the volume transport streamfunction 0 0 , When the Rayleigh friction is used for the bottom stress (7b), Eq.( 20) becomes the extended Munk- When the Rayleigh friction is used for the bottom stress (7b) and horizontal viscosity vanishes (A h = 0), Eq.( 20) becomes the extended Stommel model, When A h  0, and the drag law is used for the bottom stress due to vertical viscosity, Eq(20) reduces to the extended Sverdrup transport equation, 135

Forcing Functions
Various transport equations ( 20), ( 23 and horizontal viscosity 140 The last three forcing functions depend on the bottom current velocities (u -H , v -H ) [see (7b), ( 7c) and ( 20)], horizontal diffusivity A h , and the bottom drag coefficient C D .The P-vector inverse method (Chu 1995, 2000, Chu et al. 1998a, b) is used to determine (u -H , v -H ) from hydrographic 145 data.The horizontal diffusivity (A h ) is taken the value of with the RMS of 3.32×10 -12 m/s 2 , which is almost two-orders of magnitude smaller than the density forcing.The computation shows the following relationship, 160 If wind forcing vanishes, the density driven Sverdrup transport streamfunction is calculated by If density forcing vanishes, the wind driven Sverdrup transport streamfunction is determined by which is the classical Sverdrup dynamics but the volume transport is for the whole water column rather than above a level of no motion.It is noted that in calculating the density forcing function  28)-( 30) are solved using the traditional method with Ψ = 0 at the east boudary and integrating westward to get the climatological annual mean density driven Sverdrup transport Ψ den (Fig. 4), wind driven Sverdrup transport Ψ w (Fig. 5), and density-wind driven Sverdrup transport Ψ (Fig. 6).180 Since the purpose of this note is to present the extended transport equations after removing level of no-motion rather than to simulate the volume transports, only major features on (Ψ den , Ψ w , Ocean Sci.Discuss., doi:10.5194/os-2016Discuss., doi:10.5194/os- -81, 2016 Manuscript under review for journal Ocean Sci.Published: 1 November 2016 c Author(s) 2016.CC-BY 3.0 License.Ψ) are discussed.Although the density and wind forcing functions are very different (Fig. 1 and Fig. 2) with low correlation coefficient (0.0699), the three Sverdrup transport streamfunctions (Ψ den , Ψ w , Ψ) show similar patterns in both hemispheres, i.e., subpolar gyre, subtropical gyre, 185 equatorial current, and equatorial counter current.The volume transport driven by density (Ψ den ) is stronger than driven by wind (Ψ w ).The correlation coefficient is 0.185 between Ψ den and Ψ w , 0.716 between Ψ den and Ψ, and 0.698 between Ψ w and Ψ.
Ocean Sci.Discuss., doi:10.5194/os-2016-81,2016 Manuscript under review for journal Ocean Sci.Published: 1 November 2016 c Author(s) 2016.CC-BY 3.0 License.after.Questions arise: How do the Sverdrup-Stommel-Munk equations change after H is changed from the level of no motion to the ocean bottom depth?What is new physics behind such a 105 change?This note will answer these questions.Following the same path as SB from (9) to (14), several new volume transport equations (called extended Sverdrup-Stommel-Munk equations) have been derived.The rest of the note is outlined as follows.Section 2 presents extended Sverdrup-Stommel-Munk transport equations.Section 3 depicts the world ocean climatological annual mean forcing functions.Section 4 shows the world ocean climatological annual mean density and wind 110 driven Sverdrup transport streamfunctions.Section 5 gives the summary.

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bottom drag coefficient C D is set as 0.0025 (seeChu and Fan, 2007).In this study, the five forcing functions are calculated from the three global datasets: (1) climatological annual mean hydrographic data downloaded from the NOAA 's World Ocean Atlas 2013 at the website: http://www.nodc.noaa.gov/OC5/woa13/for computing (U den , V den ) [see (17)], 150 (2) the NOAA 's ETOPO5 from the website: http://www.ngdc.noaa.gov/mgg/global/etopo5.HTML for bottom topography H(x, y), and (3) the Comprehensive Ocean-Atmosphere Data Set (COADS) at http://iridl.ldeo.columbia.edu/SOURCES/.DASILVA/.SMD94/.climatology/for computing climatological annual mean surface wind stress ( , The other three forcing functions are much smaller.Fig 3 shows the bottom stress forcing function due to horizontal viscosity [ 27)OceanSci.Discuss., doi:10.5194/os-2016-81,2016   Manuscript under review for journal Ocean Sci.Published: 1 November 2016 c Author(s) 2016.CC-BY 3.0 License.0699, which implies the independence between the meridional geostrophic transport for the whole water column and the surface wind tress curl.4.Density and Wind Driven Sverdrup Transport Streamfunctions165 The extended Sverdrup Transport equation due to density and wind forcing [vanish of other three forcing functions in (25)] is given by 0 and (29), the latitude ϕ is set as 15 o N for the zonal region of 0 o -15 o N, and as 15 o S for 175 the zonal region of 0 o -15 o S [see (26)].With the climatological annual mean density and wind forcing functions calculated in Scetion 3, the three Sverdrup transport euqations ( are established on the 190 assumption of level of no motion.This note shows that this assumption is equivalent to the assumption of no meridional geostrophic transport.To remove the level of no motion and instead to use bottom topography, extended Sverdrup-Stommel-Munk transport equations are derived in this note with adding four more forcing functions in addition to the surface wind stress: density, bottom meridional current, bottom stresses due to vertical and horizontal viscosities.The density 195 and wind forcing functions are dominant using the world ocean bathymetry, climatological annual mean hydrographic and surface wind stress data.The density and wind forcing functions are independent with very low correlation coefficient (0.0699).However, the Sverdrup transport streamfunctions under density, wind, and both forcing show similar patterns in both hemispheres, i.e., subpolar gyre, subtropical gyre, equatorial current, and equatorial counter current.The 200 correlation coefficient is 0.185 between density and wind forced Sverdrup transports; 0.716 between density and density-wind forced Sverdrup transports; and 0.698 between wind and density-wind forced Sverdrup transports.The author thanks Mr. Chenwu Fan's outstanding efforts on computational assistance, NOAA 205 National Centers for Environmental Information (NCEI) for World Ocean Atlas 2013, bathymetry data, and Atlas of Surface Marine Data.

Fig. 3 .
Fig. 3. Climatological annual mean bottom stress forcing due to horizontal viscosity [   / / h y x A Q x Q y     ] (unit: m/s 2 ) calculated from the NOAA/NCEI WOA13 data using the P-270 vector method and NOAA ETOPO5 data.

Fig. 4 .
Fig. 4. Climatological annual mean density driven Sverdrup transport streamfunction (unit: S V = 10 6 m 3 /s).It is noted that in calculating the density forcing function den V  , the latitude ϕ is set 275 as 15 o N for the zonal region of 0 o -15 o N, and as 15 o S for the zonal region of 0 o -15 o S.