Equilibrium surface elevation distribution in a baroclinic ocean with realistic bottom topography

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 08, NO. C0, 330, doi:0.029/2002jc00579, 2003 Equilibrium surface elevation distribution in a baroclinic ocean with realistic bottom topography Srdjan Dobricic Institute for Environment and Sustainability, Joint Research Centre, Ispra, Italy Received 2 August 2002; revised 28 June 2003; accepted 8 July 2003; published 2 October [] The objective of this study is to theoretically determine the equilibrium surface elevation distribution for the given density field in the ocean with the realistic bottom topography. It is first experimentally demonstrated in a barotropic ocean model experiment that during the process of the geostrophic adjustment in the ocean with the realistic bottom topography the equilibrium surface elevation rapidly adjusts to the most stable state independent of initial conditions. It follows from this result that in a baroclinic ocean the equilibrium gradient of the potential energy that can be converted into the kinetic energy of the vertically averaged flow is rapidly minimized, giving the theoretical solution for the equilibrium surface elevation field. This application of the minimum potential energy principle is experimentally validated in a numerical model simulation of the free surface geostrophic adjustment in the North Atlantic. INDEX TERMS: 4532 Oceanography: Physical: General circulation; 4255 Oceanography: General: Numerical modeling; 4556 Oceanography: Physical: Sea level variations; KEYWORDS: general circulation, potential energy, surface elevation Citation: Dobricic, S., Equilibrium surface elevation distribution in a baroclinic ocean with realistic bottom topography, J. Geophys. Res., 08(C0), 330, doi:0.029/2002jc00579, Introduction Copyright 2003 by the American Geophysical Union /03/2002JC00579 [2] It is well known from observations that in the ocean vertically integrated surface elevation and buoyancy gradients tend to partially compensate each other [e.g., Welander, 959]. The partial compensation between the two forces is the result of the adjustment of mass and velocity toward the geostrophic balance between the pressure gradient and the Coriolis force and the globally more stable state of the ocean in the gravity field. [3] For typical depths of the ocean it is clear that the surface elevation will redistribute faster than density variations can adjust by internal waves, advection, and diffusion. Formally, we can compare the speed of external gravity waves with the speed of internal gravity and Rossby and topographic waves, and with the speed of the advection. For typical depths of the ocean, external gravity waves represent by far the fastest mode, and also the external Rossby and topographic waves are much faster than internal waves. This means that the surface elevation distribution has time to reach the equilibrium state much more quickly than density variations can be redistributed. [4] In the special case of a flat-bottom barotropic ocean and constant Coriolis parameter, due to the conservation of the potential vorticity, the spatial distribution of the equilibrium surface elevation resulting from the geostrophic adjustment can be analytically calculated from the initial state [Rossby, 937, 938; Gill, 986]. However, no comparable theory exists for the case of a realistic ocean with a complex bottom topography: The theoretical studies for this case assumed that gravity waves were already dispersed and the solution was the state resulting from the adjustment by slower topographic and Rossby waves [e.g., LeBlond and Mysak, 978; Slørdal and Weber, 996]. Thus the objective of this study is to consider physical consequences of the adjustment by fast gravity-inertia waves and to define the equilibrium surface elevation distribution in the deep ocean with a realistic bottom topography. [5] In section 2 the Princeton Ocean Model (POM) [Blumberg and Mellor, 987; Mellor, 992] is applied in the diagnostic mode. This dynamical procedure to determine the equilibrium transport and surface elevation is well known and has been used for a long time in oceanography [e.g., Holland and Hirschman, 972; Holland, 973; Demin et al., 990; Ezer and Mellor, 994; Slørdal and Weber, 996]. These studies experimentally confirmed that the geostrophic adjustment by surface elevation and total transport is much faster than the internal adjustment which redistributes the density. For example, it was found that in midlatitudes the adjustment time for surface elevation and total transport is only a few days, while in the tropics it is less than a month [e.g., Holland, 973; Demin et al., 990]. Furthermore, including the prognostic calculation for temperature and salinity had a small influence on short-term simulations [e.g., Ezer and Mellor, 994; Slørdal and Weber, 996]. [6] In section 3 the barotropic configuration of POM is used to investigate experimentally evolution of the surface 8 -

2 8-2 DOBRICIC: EQUILIBRIUM SURFACE ELEVATION DISTRIBUTION elevation field and to what extent there is a memory of the initial conditions. Section 4 theoretically determines the equilibrium surface elevation field for a given density distribution giving a solution which is physically consistent with the result of the barotropic experiment from section 3. In section 5 the theoretical findings are experimentally validated in the North Atlantic with numerical simulations of the geostrophic adjustment process. 2. Numerical Model Setup [7] The POM model is chosen for the experimental validation, because it simulates explicitly the free surface adjustment process using the terrain following vertical coordinate, sigma, which provides more accurate representation of the bottom slope and calculation of the bottom vertical velocity from the kinematic boundary condition in comparison to models that have a box-like representation of the bottom topography with the horizontal bottom and zero bottom vertical velocity. The model code has a switch for the diagnostic mode in which fields of temperature, salinity, and wind stress are kept constant in time, and surface elevation, horizontal velocity components, and turbulent kinetic energy are freely adjusted in three-dimensional prognostic equations. As in previous studies [e.g., Holland, 973; Ezer and Mellor, 994], it is assumed that after month, vertically integrated velocity and surface elevation fields become steady. [8] Temperature and salinity fields are obtained from the Levitus climatology [Levitus, 982] and all experiments in this study are made using the original resolution of the Levitus data set ( ). In this way the model resolution in the meridional direction is constant (0 km) and in the zonal direction it varies from 0 km at the southern boundary to 40 km at the northern boundary. The use of the relatively coarse model resolution gives a possibility to make many numerical tests in the North Atlantic with available computers. Furthermore, it avoids the horizontal interpolation of temperature and salinity fields which might introduce artifacts. However, we will also perform several numerical tests to check the sensitivity of the experimental result to the horizontal resolution. [9] The model topography is made by averaging the 5-min horizontal resolution ETOPO5 data around each model point: there is no smoothing applied. Figure shows the model domain, land-sea mask with the position of open boundaries and the bottom topography. We can see that the model includes also small parts of the Pacific Ocean and Mediterranean Sea which are disconnected from the North Atlantic. We can expect errors in realistically simulating these areas due to the lack of accurate boundary conditions at open boundaries and the coarse model resolution, but we will require the validation of the general theory also in these areas, because it should be independent of model boundary conditions. Therefore these areas and their open boundaries will remain in all model experiments. The wind stress is horizontally interpolated from the climatology of Hellerman and Rosenstein [983] with the horizontal resolution of 2 2. [0] The external integration time step is set to 30 s, while the internal time step is set to 800 s. The constant in the calculation of the Smagorinsky horizontal viscosity is 0.2. Figure. Model domain, land-sea mask and bottom topography with the contour interval of 000 m. The land is shaded. At all open boundaries the radiation boundary condition is applied on the vertically integrated velocity component orthogonal to the boundary, while the parallel component is set to zero. [] The numerical experiment applied in this study is very similar to that made by Ezer and Mellor [994]. In their study they used the same model and the same climatological data sets, but experiments were performed with a variable horizontal resolution and over a smaller area. The horizontal resolution in their study [Ezer and Mellor, 994, Figure 2b] varies from 20 km close to Florida to 50 km at the northeast boundary to better represent the details of the circulation along the western coast. On the other hand, in this study the horizontal grid is a uniform latlon grid, because the numerical model will be only used to test the general theory, which must be independent of the model horizontal grid. Furthermore, the use of variable horizontal resolution could result in the artificial elimination of small-scale features in areas where the horizontal resolution is very coarse, while representing them fully in areas where it is very high. [2] In POM prognostic equations are based on Boussinesq and hydrostatic approximations. In the diagnostic mode the major part of the simulation is based on the solution of vertically integrated momentum and continuity equations. The integration limits are from the ocean bottom with the depth H to the free surface elevation þ f k V ¼ gh 0 þrv ¼ ð2þ z bdz 0 Adz þ t 0 þlðvþ r 0 In these equations, V = (U, V ) is vertically integrated momentum, b = gr/r 0 is buoyancy, f is the Coriolis parameter, r is density, r 0 is the constant reference density, ðþ

3 DOBRICIC: EQUILIBRIUM SURFACE ELEVATION DISTRIBUTION 8-3 T 0 is wind stress, and L(v) contains the bottom stress term and the vertical integral of advective and horizontal viscosity terms. In the diagnostic calculation, only the term L(v) directly depends on the three-dimensional structure of the velocity field. Consistently with the Boussinesq approximation, the vertical integral of the buoyancy gradient in equation () is very accurately approximated by the integral from the bottom to the zero reference level. [3] Although POM uses the sigma vertical coordinate, it can be noticed in equation () that in this study the buoyancy gradient, constant with time in the diagnostic simulation, was calculated on horizontal levels. This was done because in the first model experiments the direct calculation of the buoyancy gradient on sigma levels resulted in unrealistic model results (not shown). One could reduce the error in the calculation of the buoyancy gradient on sigma levels by increasing the horizontal resolution and by filtering the bottom topography, but in this study it is necessary to calculate the horizontal buoyancy gradient as accurately as possible without artificial corrections to the result. By calculating it on horizontal levels prior to the model simulation the numerical error due to the application of the sigma coordinate is completely eliminated. Furthermore, the model still has the exact bottom depth and slope, and the bottom vertical velocity is not artificially set to zero like in models that have the box-like representation of the bottom topography. [4] To test the model setup, a model simulation is performed with the full climatological buoyancy and wind stress forcing. Model outputs (Figure 2) show that, considering the relatively coarse horizontal resolution of the numerical model, major simulated features of surface elevation and total transport fields are in good agreement with observations and results of numerical simulations from other studies [e.g., Richardson, 985; Hogg et al., 986; Johns et al., 995; Ezer and Mellor, 994]. It is visible in both surface elevation and total transport fields that the Gulf Stream separates from the western coast at Cape Hatteras and that there is recirculation along the shallow western coastal shelf. [5] Especially close qualitative and quantitative agreement is obtained with the simulation result by [Ezer and Mellor, 994, Figure 3], even though they used a much higher horizontal resolution along the western coast. This shows that the spatial structure of the result is almost completely defined by the scales of the Levitus climatology, larger then the horizontal resolution in the model. For example, the surface elevation gradient along the Gulf Stream is much coarser then observed, but very similar to the Ezer and Mellor [994] result. They also reported that the change of the model resolution was not important for the result. [6] Large, probably unrealistic, total transport structures south of 25 N simulated by the model are similar to those obtained by Ezer and Mellor [994] and by some other diagnostic studies that used the same climatology for the diagnostic calculation in the North Atlantic [e.g., Mellor et al., 982; Greatbatch et al., 99]. It was shown by Ezer and Mellor [994] that those circulation structures are present either due to non-adjusted density field and bottom topography or due to errors in hydrographic data, because Figure 2. Model outputs of the diagnostic simulation with the full forcing by climatological density and wind stress. Equilibrium fields are obtained by averaging instantaneous fields from day 3 to day 40 of the model simulation. (a) Equilibrium surface elevation with the contour interval 0. m. (b) Equilibrium total transport stream function with the contour interval 0 Sv ( Sv = 0 6 m 3 s ). they were somewhat suppressed only when prognostic equations for salinity and temperature were included in their simulation. In this simulation, similar features exist at the same latitudes also in the part of the Pacific covered by the model domain. [7] The model simulation was then prolonged to year. After -year surface elevation and total transport fields became steady and at every instant were practically identical to the time averaged fields in Figure 2 (not shown). This experimental result verifies that fields shown in Figure 2, averaged from day 3 to day 40 of the model simulation, represent the stable equilibrium state. [8] In fact, even after a few days, the whole daily averaged surface elevation field stabilized at the state almost identical to the equilibrium in Figure 2. On the other hand, the adjustment of daily averaged fields of vertically integrated velocity toward the equilibrium steady state (Figure 2) took longer, especially in the subtropics, in agreement with the finding by Demin et al. [990]. The time-averaged total

4 8-4 DOBRICIC: EQUILIBRIUM SURFACE ELEVATION DISTRIBUTION Table. A Summary of the Main Differences Between Experiments a Experiment Description B a barotropic ocean with the realistic bottom topography B2 a barotropic ocean with the flat bottom E the reference experiment with the full buoyancy force E2 the forcing accounts only for the part that should be fully compensated by the surface elevation gradient E3 the forcing is only the part of the buoyancy force that should not be compensated even partially a The numerical setup description and results from experiments B and B2 are given in section 2, while section 5 contains the description and results from experiments E, E2, and E3. transport reached the state identical to the stable equilibrium only in the last 0 days of the simulation (days 3 40). 3. Geostrophic Adjustment in the Barotropic Model Setup 3.. Barotropic Model Setup [9] The next experiment is made to numerically simulate the geostrophic adjustment process in the barotropic ocean for a given initial surface elevation field. Table gives a short description of differences between the most important experiments performed in the study. The horizontal grid is the same as in the experiment from the previous section. There is no wind forcing and surface elevation, and total transport fields can freely adjust towards the equilibrium state which can depend only on the initial distribution of the surface elevation. Equation () þ f k V ¼ ghrh þl ðþ: [20] Kuo and Polvani [997] have reviewed a number of numerical simulations of geostrophic adjustment imposing an initial perturbation in the surface elevation. However, in contrast to the realistic bottom topography, they assume a flat bottom and constant Coriolis parameter (a case analytically solved by Rossby [937, 938]) Experiment B: Realistic Bottom Topography [2] To perform the experiment, the surface elevation field is initialized by the Gaussian hill distribution in the middle of the North Atlantic (Figure 3). The initial velocity is set to zero everywhere. The amplitude and the gradient of the Gaussian hill are chosen to be quantitatively comparable to the amplitude and the gradient of surface elevation simulated by numerical models in North Atlantic (e.g., Figure 2). However, qualitatively the amplitude of the initial disturbance is not important for the result as long as it is much smaller than the ocean depth H. [22] In the first simulation the term L(v) is set to zero. This eliminates the influence of advection, horizontal viscosity, and bottom stress on the result. Time-averaged fields of surface elevation and total transport are displayed in Figure 4. They are calculated as 0-day means from day 3 to day 40. Clearly, fields are practically flat, which means that the gradient of the potential energy of equilibrium surface elevation in the gravity field has become zero. However, the surface elevation is not steady after month of the simulation. As an example, instantaneous fields of surface elevation and total transport at the end of the day 40 are shown in Figure 5. ð3þ [23] To further analyze the result, the diagnostics were made using equations for the vorticity z and the divergence D of the vertically integrated velocity following from equation þrðfvþ ¼ gjðh; hþþ k ðraþ fflfflfflfflffl{zfflfflfflfflffl} fflfflfflfflffl{zfflfflfflfflffl} fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} k r ½ ðf VÞŠ ¼ rðghrhþþra fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflffl{zfflfflfflfflfflffl} ffl{zffl} ; ð5þ where the vector A represents the advection, horizontal viscosity and bottom stress terms, and J (a, b) =k (ra rb) is the Jacobian. [24] The experiment is repeated for year without nonlinear terms. Horizontal averages of absolute values of each of terms underlined by brackets in equations (4) and (5) are displayed in Figures 6a and 6b. Even after year the circulation did not become steady and the divergence did not vanish. Furthermore, tendency terms in momentum equations were significant in comparison to other terms. On the other hand, the 0-day averaged fields were almost flat throughout the simulation (not shown). [25] Therefore, the -year simulation was repeated with the inclusion of advection, horizontal viscosity, and bottom stress contained in A. In this case the flow became steady (Figures 6c and 6d). Practically, the ocean reached a state of the rest. However, it is evident that the adjustment to the Figure 3. Initial distribution of the surface elevation field in the barotropic experiment. The contour interval is 0. m.

5 DOBRICIC: EQUILIBRIUM SURFACE ELEVATION DISTRIBUTION 8-5 due to the requirement for the conservation of the potential vorticity [e.g., Gill, 986]. [28] Equilibrium surface elevation and total transport fields after month of the simulation are displayed in Figure 7. In agreement with theory, in this case the equilibrium surface elevation field is not flat and it is influenced by the initial surface elevation distribution. Its shape is slightly deformed in comparison to the initial shape, because of the boundary condition of no flow in the direction orthogonal to the coast. [29] Again, experiment B2 was prolonged in -year simulation, and again in the absence of the viscosity, the flow did not become steady. However, the 0-day averaged flow was steady and almost identical to that represented in Figure 7. When the viscosity and advection were included in the simulation, surface elevation and total transport became steady after several months of the simulation. Every instantaneous state was very similar to that represented as the equilibrium field in Figure 7 (not shown). Again, the adjustment to the steady state by viscosity takes months, but the balanced steady state is not flat due to the conservation of the potential vorticity. In the long term, dissipation of energy due to friction must lead to a flat surface elevation Figure 4. Experiment B equilibrium fields in the North Atlantic with the realistic bottom topography calculated as the mean from day 3 to day 40 of the simulation. (a) surface elevation with the contour interval 0. m. (b) Total transport with the contour interval 0 Sv. All isolines in the figure represent the zero value. steady state by viscosity takes months: much slower than the adjustment of equilibrium fields to the most stable state, which takes only days. [26] To further validate the finding that the equilibrium surface elevation distribution reaches the stable state independent of initial conditions, the experiment was repeated in several different configurations: using the original coarse resolution bottom topography with a two times higher horizontal model resolution, using a significantly smoothed bottom topography and by closing all open lateral boundaries. In every case the result was the same: The equilibrium surface elevation rapidly became flat, but in the absence of the viscosity the flow did not become steady (not shown) Experiment B2: Flat Bottom [27] To demonstrate that the model accurately simulates the geostrophic adjustment process, experiment B2 was performed in a special configuration with a flat bottom (4500 m) and constant Coriolis parameter (0.000 s ). According to theory, in this specially configured experiment the equilibrium surface elevation field cannot become flat Figure 5. Experiment B instantaneous fields at the end of the day 40 of the simulation in the North Atlantic with the realistic bottom topography. (a) Surface elevation with the contour interval 0. m. (b) Total transport with the contour interval 0 Sv.

6 8-6 DOBRICIC: EQUILIBRIUM SURFACE ELEVATION DISTRIBUTION Figure 6. Experiment B diagnostics of horizontal averages of absolute values of terms in the vorticity and divergence equations for year of simulation in the North Atlantic with the realistic bottom topography. Values are averaged over a -day period for the presentation. Units are m s 2. (a) Linearized vorticity equation terms. (b) Linearized divergence equation terms. (c) Vorticity equation with nonlinear terms. (d) Divergence equation with nonlinear terms. and zero total transport, but the experiment shows that this process takes much longer then a year. [30] Clearly, there can be some peculiar configurations of bottom topography, other then a flat bottom, that could make the equilibrium surface elevation field dependent on the initial state. Several special topographic configurations were tested with the numerical model. For example, the use of the bowl-shaped bottom topography and constant Coriolis parameter resulted in equilibrium fields qualitatively similar to those in Figure 7 (not shown). [3] On the other hand, the use of a variable Coriolis parameter resulted in flat equilibrium fields (not shown). The experimental result with variable Coriolis parameter is in very close qualitative agreement with results of basin mode calculations by [Liu et al., 999], [Cessi and Louazel, 200], [Cessi and Paparella, 200], [Cessi and Primeau, 200], and [Primeau, 2002]. They studied the first baroclinic mode, but as in the barotropic experiment performed in this study the variation of the Coriolis parameter resulted in the damping of the oscillations independent of viscosity. Furthermore, in this study it is found in experiments B and B2 that even without any viscosity the equilibrium fields rapidly reach the damped state, while the oscillation continues. The result obtained with the variable Coriolis parameter is in full agreement with the observed rapid damping due to the bottom topography, because the variation of the Coriolis parameter can be considered analogous to the variation of the bottom topography, getting more shallow towards the north [e.g., LeBlond and Mysak, 978]. [32] We can expect that in the general case of the geostrophic adjustment with the realistic bottom topography the equilibrium surface elevation field will most probably rapidly reach the most stable distribution independent of the initial state. Experiment B demonstrates that this applies to the North Atlantic. 4. Determination of Equilibrium Surface Elevation Distribution 4.. Splitting the Vertically Integrated Buoyancy Force [33] Results of the barotropic numerical experiment from the previous section can be used to determine the equilibrium surface elevation distribution corresponding to the density distribution in the ocean. In further analysis it is convenient to use the equivalent form of the vertically

7 DOBRICIC: EQUILIBRIUM SURFACE ELEVATION DISTRIBUTION 8-7 [34] To find the equilibrium state of surface elevation, equation (6) will be averaged in time. For a sufficiently long averaging period we will assume that the equilibrium velocity and surface elevation are steady. Then from equation (2) follows that the equilibrium vertically integrated velocity is fully rotational; that is, it can be expressed using only the stream function \Psi (V = k r\psi), and equation (6) becomes f r ¼ rp be r; ð8þ Figure 7. Experiment B2 equilibrium fields in the North Atlantic with the flat bottom (4500 m) and constant Coriolis parameter (0.000 s ) calculated as the mean from day 3 to day 40 of the simulation. (a) Surface elevation with the contour interval 0. m. (b) Total transport with the contour interval 0 Sv. integrated buoyancy force in equation () obtained after applying the Leibniz rule for the differentiation of an integral and the rule for the integration by parts. Also, the wind stress will be ignored, because it is not the subject of the study, and equations will be linearized assuming that the term L(v) is insignificant. Equation þ f k V ¼ rp b r; where p b is bottom pressure and r 0 is the potential energy of buoyancy per a unit surface [e.g., Mellor et al., 982], p b ¼ gh þ Z; Z ¼ bdz and ¼ zbdz: It should be noticed here that does not represent the full potential energy of the water column. The definition of the full potential energy of the water column, and the determination of the part which can be transformed into the kinetic energy of the vertically averaged flow, are given in Appendix A. ð6þ ð7þ where p be is the equilibrium bottom pressure which corresponds to the equilibrium surface elevation h e. Clearly, if there is only one steady state given by equation (8), it must be in the stable equilibrium. The existence of the unique solution to equation (8) is experimentally demonstrated in section 2 in the simulation with the full forcing by the wind stress and the vertically integrated buoyancy force and in barotropic ocean experiments in section 3. [35] Now the potential energy and bottom pressure will be split into two parts ( = c + n and p be = p bc + p bn ). The part c has a gradient which is exactly compensated by the vertically integrated gradient formed by the corresponding part of the bottom pressure p bc. Conversely, the remaining part n has a gradient which cannot be exactly compensated by the vertically integrated gradient of the remaining part of the bottom pressure p bn. It is important to notice that this splitting of and p be into as-yet unknown parts is defined in a way that there is no further possibility to split n in a part which can be fully compensated by the vertically integrated gradient of any part of p bn. [36] From our definition of the splitting, equation (8) can be rewritten as H rp bc þr c ¼ 0 ð9þ f r ¼ H rp bn þr n : ð0þ The theory does not depend on this splitting: We could also make the analysis by keeping only one equation following from equation (8). The splitting into two equations (equations (9) and (0)) merely simplifies the mathematical presentation: It does not restrict the possible solutions, because it allows any form for the partial balance between two terms on the right side of equation (0). [37] By taking the curl of equation (9), we find that both c and p bc must be constant along isobaths and only the gradient of the part of the potential energy of buoyancy that is constant along isobaths can be fully compensated by the surface elevation distribution. The two terms in equation (9) cancel each other and consequently cannot transform into kinetic energy of the vertically averaged flow. Only the two terms on the right side of equation (0) create the gradient of the potential energy in the gravity field which can be transformed into the kinetic energy. Therefore, the result of the geostrophic adjustment process is that the equilibrium surface elevation rapidly reaches the distribution which maximizes the compensated part of the potential energy gradient (i.e., the two terms in equation (9)) and minimizes the gradient of terms on the right side of equation (0). If the right side of equation (0) is zero, the total transport is

8 8-8 DOBRICIC: EQUILIBRIUM SURFACE ELEVATION DISTRIBUTION zero. As a result of the fast adjustment of the equilibrium surface elevation, the potential energy of buoyancy does not change, but the part of the potential energy which can be converted into the kinetic energy of the vertically averaged flow is rapidly reduced Assumption That is Constant Along Isobaths [38] If one were to assume that is constant along isobaths, it follows from equation (9) that its gradient can be rapidly and completely compensated by the bottom pressure gradient and c can become equal to. Therefore the fast adjustment of surface elevation will form a vertically integrated gradient of the bottom pressure which fully compensates the gradient of. [39] In this case the rest of the geostrophic adjustment becomes analogous to that simulated by the model in section 3.2 with h substituted by p bn and without any forcing term resulting from the buoyancy distribution. Clearly, in this case the only equilibrium solution in an ocean with the realistic bottom topography is zero gradient of p bn and zero total transport independent of the initial state Assumption That f is Constant [40] If the Coriolis parameter f is constant, equation (0) becomes rðf Þ ¼ H rp bn þr n : ðþ In this case the gradient of the non-compensated part of the potential energy represents the forcing term for topographic waves. However, in the steady state the first term on the right side of equation () is a pure gradient (no rotational part) because, as we can see in equation (), its right side must not have any rotational part. This would contradict the definition of p bc and p bn which says that this term must not be a pure gradient. [4] Therefore, in the stable equilibrium state the first term on the right side of equation () must be zero and only the second term remains. This means that the total transport is determined by the distribution of the noncompensated part of the potential energy n with the minimized gradient Variable f [42] In the general case of the variable Coriolis parameter, H rp bn cannot become zero, because in this case the right side of equation (0) also has a rotational part. After the fast adjustment by gravity-inertia waves the gradient of n represents the forcing term for Rossby and topographic waves. However, due to the adjustment to the most stable state, the first term on the right side of equation (0) will be minimized. [43] The remaining part of the bottom pressure gradient then consists only of the gradient of surface elevation h n, which minimizes the first term on the right side of equation (0) and which satisfies the equation obtained dividing equation (0) by f and taking the curl, gj H ; h f n ¼ J f ; n : ð2þ Furthermore, this term is the result of the dispersion of Rossby waves, so the variation of the surface elevation in equation (2) is damped close to the eastern boundary. The remaining part of the surface elevation h n can be calculated by integrating equation (2) from the zero value at the eastern boundary. [44] After finding n and h n we can define the distribution of h e, 2 rh e ¼ 4 gh 0 r@ z 3 bdz 0 Adz gh r n 5 þrh n : ð3þ [45] The first term on the right side of equation (3) surrounded by large brackets is the part of the equilibrium surface elevation gradient that forms the bottom pressure field which is constant along isobaths. The last term on the right side of equation (3) is the part of the equilibrium surface elevation gradient resulting from westward propagation of Rossby waves. It is defined by equation (2), and it is minimized in order to reach stable state. [46] One consequence of the adjustment of surface elevation to the stable equilibrium state is that the part of the geostrophic bottom flow, determined by buoyancy, is almost fully along isobaths, even when f is not constant. This is because the adjustment process minimizes r h n in equation (3), making it typically much smaller than the first term on the right side which determines the part of the bottom flow parallel to isobaths. On the other hand, if equilibrium surface elevation were not in the stable state, or even if the stable state were dependent on initial conditions, the last term could become comparable to the first term on the right side of equation (3). In that case the buoyancy determined part of the bottom geostrophic flow could deflect from the flow along isobaths. However, since h n is the result of the adjustment by Rossby waves, it is damped close to the eastern boundary. Consequently, the bottom flow close to the eastern boundary is almost fully along isobaths, while close to the western boundary it can diverge more significantly Nonlinear Momentum Equation [47] Experiment B showed that the use of the minimum potential energy principle to determine the equilibrium surface elevation distribution in the ocean with the realistic bottom topography does not depend on the linearization of the momentum equation; it is fully valid also for a nonlinear momentum equation. However, when it is nonlinear, the theoretical determination of the stable equilibrium surface elevation corresponding to a given density distribution becomes practically more difficult, because it involves more terms. Furthermore, slowly changing nonlinear terms can influence the dispersion properties of Rossby and topographic waves, further complicating the analysis. Therefore, in the experimental validation we must experimentally test the effect of neglecting nonlinear terms. However, it follows from section 4.2 that when is constant along isobaths, nonlinear terms cannot influence the solution, because the forcing term rapidly becomes zero Relation to Bottom Pressure Torque and JEBAR [48] Holland [973] showed that, in the case of variable Coriolis parameter, the non-zero bottom pressure torque

9 DOBRICIC: EQUILIBRIUM SURFACE ELEVATION DISTRIBUTION 8-9 relates buoyancy and surface elevation with the zonal change of the total transport. Also, Sarkisyan and Ivanov [97] directly related the change of the total transport along isolines of planetary vorticity to the Jacobian of bottom depth and potential energy of density, which they called the JEBAR term. JEBAR has been used as the forcing term in several theoretical studies [e.g., Huthnance, 984; Mertz and Wright, 992; Slørdal and Weber, 996] after the elimination of fast gravity waves. However, JEBAR does not uniquely define the circulation, because in the steady state it only determines the change of the stream function along isolines of planetary vorticity and it requires another condition for a unique solution [e.g., Sarkisyan and Ivanov, 97]. [49] After taking the curl of equations (9) and (0), terms in equation (9) vanish because p bc is constant along isobaths. The first term on the right side of equation (0) then gives a bottom torque formulation which is quantitatively identical to that with the full bottom pressure found by Holland ¼ J ð H; p bnþ: ð4þ Again, after taking the curl of equations (9) and (0) divided by H, terms in equation (9) vanish, this time because c is constant along isobaths. The second term on the right side of equation (0) is the JEBAR term which is found from n, J f H ; ¼J H ; n : ð5þ Equations (4) and (5) show that our theory is consistent with definitions of the JEBAR and bottom pressure torque. [50] Several studies used JEBAR to estimate the total transport in the North Atlantic. One method used the approximation, based on observations, that the total transport practically vanishes close to the eastern coast. Starting from the zero total transport boundary condition close to the eastern boundary, Mellor et al. [982] and Greatbatch et al. [99] calculated the total transport by integrating the JEBAR term along isolines of planetary vorticity. The resulting total transport field is similar to that explicitly simulated in section 2 (Figure 2) and in other numerical simulations by ocean models [e.g., Ezer and Mellor, 994]. [5] Bogden et al. [993] used another method. They assume that the vertical velocity at the bottom is zero and that the solution to equation (8) can be determined from an arbitrary condition named the dynamical free mode. By choosing this condition to be the minimization of density mixing at a certain depth, Bogden et al. [993] managed to obtain the major features of circulation patterns in the North Atlantic without spurious circulation structures in the subtropics. However, by explicitly simulating the geostrophic adjustment in the North Atlantic, Ezer and Mellor [994] showed that the spurious circulation structures in the subtropics are caused by non-adjusted density and bottom topography in the Levitus climatology. [52] Myers et al. [996] calculated the total transport in equation (8) from the zero condition at coasts, using an artificial resolution dependent horizontal viscosity term to connect the otherwise independent solution on different isolines of planetary vorticity. Consequently, their solution depends on the arbitrary chosen constant viscosity coefficient and the local change of the model resolution. [53] All the calculations using JEBAR described above are based on arbitrary assumptions. By contrast, in this study we merely look at the consequences of potential energy minimization: a universal process in nature. This limits how much of the potential energy can transform into the kinetic energy of the vertically integrated flow. The only assumption is that the surface elevation distribution can change much more quickly then the density distribution. Also, Rossby [937, 938] and Gill [986] implicitly used the minimum potential energy principle to analytically determine equilibrium fields resulting from the geostrophic adjustment in the gravity field. However, they only considered a barotropic flat-bottom ocean on the f-plane, in which special case, equilibrium fields depend on the initial state. [54] It is important to notice that physically, equation (8) describes the motion of particles which are not free, because their paths are in a spatially variable potential field. As a consequence, most mathematical solutions to equation (8) are not physically valid: The only correct one must also give a stable equilibrium, which means the minimizing the change of potential energy of the particles on their paths in the gravity field. The success in calculating the total transport in the North Atlantic in studies using JEBAR [Mellor et al., 982; Greatbatch et al., 99; Bogden et al., 993; Myers et al., 996] demonstrates that some of the assumptions used in those studies indirectly approximate the unique stable equilibrium solution to equation (8) presented here. [55] It can be noticed in section 4.4 that we also apply a zero boundary condition close to the eastern boundary, but this is only for h n (not for total transport). However, this assumption is physically justified by the evaluation of dispersion properties of Rossby waves. 5. Experimental Validation of Findings [56] In order to prove that the minimum potential energy principle really determines the equilibrium surface elevation field, POM was run using different forcing terms: the complete forcing term, only the part which should be compensated and the uncompensated part (Table ). To confirm the theory, the second experiment E2 should give the same surface elevation output as the first, but no total transport, whereas the third experiment E3 should give the same total transport output as the first, but with a flat surface elevation field. 5.. Experiment E: The Reference Experiment [57] In experiment E the POM model was run to acquire reference surface elevation and total transport outputs. It was forced only by the buoyancy force calculated from the Levitus climatology, and the wind stress was set to zero; all setup parameters were the same as in section 2. In this case the vertically integrated momentum equation in POM þ f k V ¼ gh 0 r@ z bdz 0 Adz þlðþ: v ð6þ

10 8-0 DOBRICIC: EQUILIBRIUM SURFACE ELEVATION DISTRIBUTION Figure 8 shows outputs of experiment E. We can see that the main difference from the initial model test with the full forcing by buoyancy and wind stress is in the total transport field. Nevertheless, like in the calculation by Greatbatch et al. [99], all major features that represent the total transport in the North Atlantic are well simulated. This confirms that they are mostly determined by the density distribution. There are large total transport features south of 25 N, another confirmation that they are the consequence of non-adjusted density and bottom topography. [58] It is important to notice that the validation of the theory presented in this section does not depend on the quality of the buoyancy data set, because the proof depends only on comparing one model result with another model result, not with the reality. Indeed, from this point of view the proof could be done with a completely invented climatology. [59] To test the influence of the horizontal resolution of the ocean model on the result, the experiment was repeated with the 4 times higher horizontal resolution. Now the meridional resolution is 28 km and the zonal resolution varies from 28 km at the southern boundary to 0 km at the northern boundary. The higher resolution gives a finer Figure 9. Experiment E equilibrium fields resulting from the Levitus climatology, but obtained using the high model resolution. They represent the mean from day 3 to day 40 of the simulation. (a) Surface elevation with contour interval 0.m. (b) Total transport stream function with contour interval 0 Sv. Figure 8. Experiment E equilibrium fields obtained from the Levitus climatology. They are obtained as the mean from day 3 to day 40 of the simulation. (a) Surface elevation with contour interval 0.m. (b) Total transport stream function with contour interval 0 Sv. bottom topography and coast (not shown). The temperature and salinity are horizontally linearly interpolated on points of the new grid. All model constants and numerical schemes are the same as in the coarse resolution experiment. Figure 9 shows that large-scale features of equilibrium fields are very similar to the coarse resolution result (Figure 8). [60] The only significant differences are visible in the total transport field close to the western coast where we could expect that nonlinear terms are relatively more important. With the higher resolution the recirculation along the shallow western coastal shelf and the circulation in the Gulf of Mexico are stronger, but there is no significant change in the main features of circulation direction and intensity, represented by the stream function gradient, in the rest of the Atlantic Ocean. Furthermore, the equilibrium surface elevation in the whole Atlantic Ocean has a very similar distribution to that resulting from the coarse resolution simulation (Figure 8). The high-resolution sensitivity experiment confirms that simulated equilibrium fields are mainly determined by the smallest spatial scales resolved by the Levitus climatological data set, which are well resolved already by the coarse resolution. Therefore the coarse

11 DOBRICIC: EQUILIBRIUM SURFACE ELEVATION DISTRIBUTION 8 - resolution model is sufficient for the validation of the theory Diagnostic Calculation of n and H n [6] To define the compensated part of the vertically integrated buoyancy force, it is convenient to start with the determination of terms on the right side of equation (0), using the finding that they must be minimized. The first step is to find the distribution of n which minimizes its gradient. [62] The gradient of is projected on the (s, n) orthogonal coordinate system where the axis s is parallel to isobaths. Using our definition that n = c, where cannot change and c must be constant along isobaths (i.e., c = c (n)), we can ¼ : ð7þ ð8þ The stable equilibrium surface elevation will be distributed in a way to globally n /@ n while keeping n /@ s. [63] At each iteration k the value of k c at a single isobath is obtained from k c ¼ lk c ; ð9þ where l = l(n). Starting from the first guess that c is the mean of along each isobath, equations (7) and (8) are iteratively solved by varying l. Whenever, for a variation of l at a single isobath, the global sum of the absolute value of r n is smaller then the value obtained before the variation, c is changed by l c and the resulting n is updated. The iterative procedure is continued until the sum of the absolute value of r n over the whole ocean stops to converge towards the minimum. Figure 0 shows the convergence of the sum of r n during 24 iterations of equations (7) (8) over the whole North Atlantic area. Although there is no formal guarantee, we assume that this solution to equations (7) (8) gives the absolute minimum distribution of the gradient of n. [64] The gradient of n found in this way (for a constant f ) is further used as the forcing term to find the corresponding distribution of h n for a variable f. Starting from the zero boundary condition close to the eastern boundary the distribution of h n is obtained by integrating equation (2) along isolines of planetary vorticity Experiment E2: Forcing With the Fully Compensated Part of Vertically Integrated Buoyancy Gradient [65] Now we subtract the gradients of n and h n that minimize terms on the right side of equation (0) from the full vertically integrated buoyancy force. In this experiment the vertically integrated momentum equation in POM is 0 þ f k V ¼ gh þðr n þ gh rh n ÞþLðÞ v 0 r@ z bdz 0 Adz ð20þ Figure 0. Reduction of the sum of jr n j in 24 global iterations over all isobaths. The zero iteration is the first guess in which c is assumed to be the mean of along each isobath. In 24 iterations the global sum is reduced from 2.8 m 2 s 2 in the first guess to 2.7 m 2 s 2 in the last iteration. After the last iteration, any further variation of l did not result in a reduction of the global sum. [66] Figure shows model outputs of equilibrium surface elevation and total transport fields after month of the simulation in experiment E2. We can see that the equilibrium surface elevation field is very similar to that obtained in the reference experiment E (Figure 8), whereas as expected, total transport is practically inexistent over the whole North Atlantic and, also, the two-gyre structure disappeared. [67] It is important to notice that according to theoretical findings from section 4.2, whenever the potential energy of buoyancy is constant along isobaths, the equilibrium total transport equals zero everywhere. This means that in this numerical experiment the total transport would become zero even if we do not approach the exact minimum of the gradient of the right side of equation (0). However, in that case the equilibrium surface elevation field from experiment E2 (Figure ) would significantly differ from the equilibrium surface elevation field from experiment E (Figure 8) Experiment E3: Forcing With the Non-Compensated Part of Vertically Integrated Buoyancy Gradient [68] In experiment E3, contrary to E2, only the remaining part of the vertically integrated buoyancy force is used for the forcing of vertically integrated momentum equations in POM. According to the theoretical findings, in this case the vertically integrated buoyancy force is only the driving force for the equilibrium total transport and the gradient of the surface elevation should be zero. The vertically integrated momentum equation in þ f k V ¼ gh rh ð r n þ gh rh n ÞþLðvÞ: ð2þ Figure 2 shows the model outputs of surface elevation and total transport fields. This time the surface elevation gradient is practically inexistent, and the two-gyre structure in the surface elevation field disappeared. However, as expected, the equilibrium total transport field is almost

12 8-2 DOBRICIC: EQUILIBRIUM SURFACE ELEVATION DISTRIBUTION small differences between equilibrium surface elevation and total transport fields between experiments could be due to nonlinear terms. In all experiments, nonlinear terms are calculated from the three-dimensional velocity which was forced by the same buoyancy force in all experiments. The only change of the forcing was in the vertically integrated momentum equation. As a consequence the three-dimensional velocity and the term L(v) are significantly different in experiment E3 in comparison to E and E2, because the equilibrium surface elevation in experiment E3 has zero gradient. For example, due to the absence of the surface elevation gradient in experiment E3, the equilibrium surface velocity is very small and the bottom velocity is very large in comparison to other experiments. Therefore, experiments were repeated using the switch for the two-dimensional mode available in POM, in which the nonlinear term L(v) is computed only from the vertically averaged velocity. As was expected, with this option, differences between equilibrium total transport fields in E and E3 were almost completely removed (not shown). Figure. Experiment E2 equilibrium fields obtained from the Levitus climatology. (a) Surface elevation with contour interval 0. m. (b) Total transport stream function with contour interval 0 Sv. According to the theory the equilibrium surface elevation distribution should completely compensate the buoyancy force. The fields are averages of simulated fields from day 3 to day 40 of the simulation. identical to that calculated in the reference experiment E (Figure 8) with the full buoyancy forcing Sensitivity Tests [69] To test the sensitivity of the result to the choice of n, several other simulations were performed by varying the value of n over the whole computational domain away from the value, which forms the minimum gradient determined by the iterative method. As expected, the equilibrium total transport in experiment E2 was always zero. On the other hand, the equilibrium surface elevation field in experiment E2 differed more significantly from that of experiment E, and gradients in the equilibrium surface elevation field in experiment E3 were more pronounced. The iteratively found solution evidently resulted in the most optimal splitting between compensated and non-compensated parts. It is also found that h n has a much smaller gradient then the rest of h e and that its variation does not significantly influence the experimental result (not shown). [70] In addition to the computational inaccuracy in numerically determining the most stable state of the ocean, Figure 2. Experiment E3 equilibrium fields obtained from the Levitus climatology. (a) Surface elevation with contour interval 0. m. (b) Total transport stream function with contour interval 0 Sv. According to the theory, in this experiment the surface elevation distribution cannot even partially compensate the buoyancy force. The fields are averages of simulated fields from day 3 to day 40 of the simulation.