Estimate of eddy energy generation/dissipation rate in the world ocean from altimetry data

Abstract

Assuming eddy kinetic energy is equally partitioned between the barotropic mode and the first baroclinic mode and using the weekly TOPEX/ERS merged data for the period of 1993~2007, the mean eddy kinetic energy and eddy available gravitational potential energy in the world oceans are estimated at 0.157 and 0.224 EJ; the annual mean generation/dissipation rate of eddy kinetic energy and available gravitational potential energy in the world oceans is estimated at 0.203 TW. Scaling and data analysis indicate that eddy available gravitational potential energy and its generation/dissipation rate are larger than those of eddy kinetic energy.

High rate of eddy energy generation/dissipation is primarily concentrated in eddy-rich regions, such as the Antarctic Circumpolar Current and the western boundary current extensions. Outside of these regimes of intense current, the energy generation/dissipation rate is two to four orders of magnitude lower than the peak values; however, along the eastern boundaries and in the region where complicated topography and current interact the eddy energy generation/dissipation rate is several times larger than those in background.

Appendices

Appendix 1: Formulation based on a two-layer model

A first-baroclinic-mode eddy can be examined in terms of a two-layer model, Fig. 9, where η is the sea level anomaly, h ₁ is the depth of the interface, d is the interfacial disturbance, (H ₁ and H ₂), (u ₁ and u ₂), and (ρ ₁ and ρ ₂) are the mean thickness, horizontal velocity, and density of the upper and lower layers. The corresponding pressure gradient in each layer is

$$ \nabla {p_1} = g{\rho_1}\nabla \eta, \nabla {p_2} = g{\rho_1}\nabla \eta - g\Delta \rho \nabla {h_1} = g{\rho_1}\nabla \eta - g\Delta \rho \nabla d, $$

(14)

where g is the gravitational acceleration, $\Delta \rho = {\rho_2} - {\rho_1}$ is the density difference between the upper and lower layers. These relations can be rewritten as

$$ \frac{{\nabla {p_1}}}{{{\rho_1}}} = g\nabla \eta, \frac{{\nabla {p_2}}}{{{\rho_2}}} \approx g\nabla \eta - g \prime \nabla d, $$

(15)

where $g \prime = g\Delta \rho /{\rho_2}$ is the reduced gravity. Geostrophic velocity in each layer is proportional to the pressure gradient, the right panel of Fig. 9. By definition, volumetric transport in each layer should satisfy the following constraint

$$ {u_1}{H_1} + {u_2}{H_2} = 0. $$

(16)

From these equations we obtain

$$ d = \frac{g}{{g\prime}}\left( {1 + \frac{{{H_1}}}{{{H_2}}}} \right)\eta . $$

(17)

Thus, the horizontal pressure term in the lower layer is reduced to

$$ \frac{{\nabla {p_2}}}{{{\rho_2}}} \approx - g\frac{{{H_1}}}{{{H_2}}}\nabla \eta = - \frac{{{H_1}}}{{{H_2}}}\frac{{\nabla {p_1}}}{{{\rho_1}}}. $$

(18)

Fig. 9

Sketch of the free surface and the layer interface in a two-layer model: left panel, free surface and interface of a two-layer model. Right panel, velocity pattern of the first baroclinic mode. The symbols are explained in the main text

When the lower layer is much thicker than the upper layer, velocity in the lower layer is much smaller than that of the upper layer; however, the volumetric transport in the lower layer is not negligible because it is exactly the same as that in the upper layer (with an opposite sign).

In the present two-layer model, if the lower layer is much thicker than the upper layer, the layer ratio term in Eq. 17 can be omitted, and the corresponding expression is reduced to

$$ d \approx \frac{g}{{g \prime}}\eta . $$

(19)

However, in our calculation, the exact expression 17 for our two-layer approximation of the stratification is used.

The AGPE for a two-layer model can be calculated as follows. Assume the undisturbed upper layer thickness is H ₁, the free surface elevation is η and the interface depression is d, Fig. 10. The reference state is defined as the state with minimal gravitational potential energy, which corresponds to a state with both the free surface and the interfacial surface leveled off, as shown by the dashed horizontal lines in Fig. 10. Since the vertical movement involved is very small, we assume that water density does not change with pressure. As a result, the only changes are as follows. Firstly, the free surface elevation anomaly is flatted out, as shown by the arrow in the upper part of Fig. 10. Secondly, the interface is flatted out, indicated by the solid arrow in the lower part of Fig. 10. However, other parts of upper and lower layer remain unchanged.

Fig. 10

Water parcel movement during the adjustment to a state of minimal gravitational potential energy

The calculation of AGPE is separated into two parts. For the upper part of the water column, we use the upper surface of the undisturbed upper layer as the reference state. The total gravitational potential energy of the water parcel before and after adjustment is

$$ \chi_{\rm{Top}}^0 = \frac{{g{\rho_1}b}}{2}{\eta^2}, $$

(20)

$$ \chi_{\rm{Top}}^1 = \frac{{g{\rho_1}}}{2}(B + b){a^2} = \frac{{g{\rho_1}}}{2}\frac{{{b^2}}}{{B + b}}{\eta^2}. $$

(21)

Thus, the corresponding available gravitational potential energy is

$$ \Delta {\chi_{\rm{Top}}} = \chi_{\rm{Top}}^0 - \chi_{\rm{Top}}^1 = \frac{{g{\rho_1}}}{2}\frac{{Bb}}{{B + b}}{\eta^2}. $$

(22)

For the lower part of the water column near the interface, there are two water parcels exchanging their positions. For simplicity, we use the non-disturbed interface as the reference level. Before the adjustment, the total gravitational potential energy for the upper layer parcel (on the lower left corner) and the lower layer (on the lower right corner) is

$$ \chi_{{{\rm{Bot}},1}}^0 = - \frac{{g{\rho_1}}}{2}b{(d - e)^2}, $$

(23)

$$ \chi_{{{\rm{Bot}},2}}^0 = \frac{{g{\rho_2}}}{2}B{e^2}. $$

(24)

The corresponding terms after adjustment have similar expressions,

$$ \chi_{{{\rm{Bot}},1}}^1 = \frac{{g{\rho_1}}}{2}B{e^2}, $$

(25)

$$ \chi_{{{\rm{Bot}},2}}^1 = - \frac{{g{\rho_2}}}{2}b{(d - e)^2}. $$

(26)

Thus, the available gravitational potential energy associated with the adjustment of these two water parcels are

$$ \Delta {\chi_{\rm{Bot}}} = \chi_{{{\rm{Bot}},1}}^0 + \chi_{{{\rm{Bot}},2}}^0 - \chi_{{{\rm{Bot}},1}}^1 - \chi_{{{\rm{Bot}},2}}^1 = \frac{{g\Delta \rho }}{2}\frac{{bB}}{{(B + b)}}{d^2}. $$

(27)

For an individual eddy, the width of the background stratification field is much larger than the width of the eddy, so that$B \gg b$, and the corresponding total available gravitational potential energy for the unit length, obtained by dividing the width of b, is

$$ \chi = \Delta {\chi_{\rm{Top}}} + \Delta {\chi_{\rm{Bot}}} \approx \frac{{g\Delta \rho }}{2}{d^2} + \frac{{g{\rho_1}}}{2}{\eta^2}. $$

(28)

Using Eq. 17, this is reduced to

$$ \chi \approx \frac{{g\Delta \rho }}{2}\left[ {1 + \frac{{g \prime}}{g}{{\left( {\frac{{{H_2}}}{{{H_1} + {H_2}}}} \right)}^2}} \right]{d^2}. $$

(29)

Since the reduced gravity is much smaller than gravity, the second term in Eq. 29 is negligible and the total available gravitational potential energy per unit length is

$$ \chi \approx \frac{{g\Delta \rho }}{2}{d^2}. $$

(30)

Our discussion above can be extended to the case of an eddy in a cylindrical coordinates. Assuming eddy dimension is much smaller than the dimension of the ocean, the results are the same.

The ratio of EAGPE and EKE for an eddy is estimated as follows. The geostrophic velocity of an eddy in the upper layer is estimated as

$$ {u_1} = g\left| {\nabla \eta } \right|/f \approx g{\eta_{{\max }}}/fr, $$

(31)

where f = 2Ωsinθ is the Coriolis parameter, Ω is the earth rotation rate, θ is the latitude, ${\eta_{{\max }}}$ is the maximal free surface elevation at the center of the eddy and r is the radius of the eddy. Therefore, the total amount of kinetic energy of an eddy integrated over the total area of the eddy, A, is estimated as

$$ {E_k} = \frac{1}{2}{\rho_1}{H_1}\left( {1 + \frac{{{H_1}}}{{{H_2}}}} \right)\iint_A {{u^2}{\hbox{d}}A} \approx \frac{1}{2}{\rho_1}{H_1}\left( {1 + \frac{{{H_1}}}{{{H_2}}}} \right){\left( {\frac{{g{\eta_{{\max }}}}}{{fr}}} \right)^2}A. $$

(32)

The corresponding total available gravitational potential energy of an eddy is estimated as

$$ {E_{\rm{agpe}}} \approx \frac{1}{2}\Delta \rho g{\left[ {\frac{g}{{g \prime}}\left( {1 + \frac{{{H_1}}}{{{H_2}}}} \right){\eta_{{\max }}}} \right]^2}A. $$

(33)

Thus, the ratio of these two types of energy for an eddy is

$$ R = \frac{{{E_{\rm{agpe}}}}}{{{E_k}}} \approx {\left( {\frac{r}{{{r_d}}}} \right)^2}, $$

(34)

where ${r_d} = \sqrt {{g \prime{H_1}{H_2}/({H_1} + {H_2})}} /f$ is the first radius of deformation. Thus, for eddy with radius close to the first deformation radius, the total energy is roughly equally partitioned between the EAGPE and EKE. However, most eddies identifiable from the oceanic datasets, especially from the altimetry, the horizontal length scale is much larger than the first radius of deformation (Chelton et al. 2007a; Stammer, 1997; Roemmich and Gilson, 2001). As a result, the eddy energy is mostly in the form of EAGPE.

Appendix 2: Inferring the two-layer model from a continuously stratified model

A vitally important step in formulating the two-layer model is to specify the equivalent depth of the mean interface and the density difference between the two layers. A simple approach is to use the depth of the main pycnocline and the associated density jump. In the following discussion this model will be called the thermocline model (TH-model). Such a model is, however, not suitable for the subpolar basin and the Southern Ocean where the main thermocline is poorly defined.

A better approach in parameterization of a two-layer model was described by Flierl (1978). Mesoscale eddy can be described in terms of the normal modes, and the standard formulation has been described in many previous literatures, e.g., Pedlosky (1987), Chelton et al. (1998), and Huang and Pedlosky (2002). Our notation here follows Flierl (1978). The normal modes can be defined as the following eigen value/function problem:

$$ \frac{\text{d}}{{{\hbox{d}}z}}\left( {\frac{{{f^2}}}{{{N^2}}}\frac{{{\text{d}}{F_n}}}{{{\hbox{d}}z}}} \right) + {\lambda_n}{F_n} = 0, $$

(35a)

$$ \frac{{{\text{d}}{F_n}}}{{{\hbox{d}}z}} = 0, z = 0, - H, $$

(35b)

where F _n (z) is the nth eigen mode,${\lambda_n}$ is the corresponding eigen value, N ² is the squared buoyancy frequency, and H is the depth of the sea floor. A normalization constraint is also applied to the eigen functions

$$ \int_{{ - H}}^0 {{F_i}{F_j}{\hbox{d}}z} = H{\delta_{{ij}}}. $$

(36)

Our study is focused on the first baroclinic mode. The choice of parameter for a two-layer model depends on the physical aspects of the problem as discussed by Flierl (1978). Unfortunately, no suitable formulation specifically designed for the study of the available potential energy is available at present time; thus, we will adapt the standard formulation for normal mode presented by Flierl (1978). Accordingly, the equivalent interface depth and the equivalent density step are

$$ {H_1} = \frac{H}{{1 + F_1^2(0)}}, $$

(37)

$$ \varepsilon = \frac{{f_0^2H}}{{{\lambda_1}g{H_1}\left( {H - {H_1}} \right)}}. $$

(38)

The equivalent reduced gravity is defined as

$$ g \prime = \varepsilon g. $$

(39)

This model will be called the equivalent two-layer model (EQ-model).

Appendix 3: Calculation of the annual mean generation/dissipation rate of mesoscale eddies

Through eddy identification and tracking, the time series of position and energy for an eddy were obtained and the total energy of an eddy at each moment during its lifetime were calculated as summation of EKE and EAGPE. The detailed algorithm of annual mean generation and dissipation rate of the mesoscale eddy is as follows.

Assume that we have a time series of an eddy, including its position and the SSHA at time $t = {t_1},{t_{{2,}}}...,{t_{{n - 1}}}$ with uniform time step of 1 week. In order to analyze the life cycle of an eddy, we need to define the beginning and end of the eddy. The beginning of an eddy is with zero energy, so that ${e_0} = 0$, and its time is defined as ${t_0} = - 2{t_1} + {t_2}$; its position is defined by a linear extrapolation from points 1 and 2: $({x_0},{y_0}) = \left( { - 2{x_1} + {x_2}, - 2{y_1} + {y_2}} \right)$. Similarly, the end of the eddy can be defined.

The energy source or sink within each pair of points $d{e_{{i,i + 1}}}$ is calculated as

$$ d{e_{{01}}} = {e_1} - {e_0} = {e_1}, $$

(41)

$$ d{e_{{i,i + 1}}} = {e_{{i + 1}}} - {e_i}, $$

(42)

The location of $d{e_{{i,i + 1}}}$ is in the middle of these two positions.

Gridded energy variation data set was required, so the 1 × 1° grid was chosen here. Suppose we have four grid points: (i, j), (i + 1, j), (i, j + 1), (i + 1, j + 1), the contributions to four grid points were calculated by the method of weighting. We assume there is a point source de locates (m and n) with a non-dimensional position (X and Y) within this grid net, X = m − i, Y = n − j. Thus, contribution of this source to the grid points at the four comers is:

$$ e\left( {i,j} \right) = {\hbox{d}}e\left( {1 - X} \right)\left( {1 - Y} \right), $$

(43)

$$ e\left( {i + 1,j} \right) = {\hbox{d}}e X\left( {1 - Y} \right), $$

(44)

$$ e\left( {i,j + 1} \right) = {\hbox{d}}e \left( {1 - X} \right)Y, $$

(45)

$$ e\left( {i + 1,j + 1} \right) = {\hbox{d}}e XY. $$

(46)

As a result, in 15-year accumulation the total contribution of these sources or sinks at those grid points is:

$$ E_{{i,j}}^{\rm{source}} = \sum\limits_{{n = 1,{\rm{for}} {e_n} > 0}}^N {{e_n}}, $$

(57)

$$ E_{{i,j}}^{\rm{sink}} = \sum\limits_{{n = 1,{\rm{for}} {e_n} < 0}}^N {{e_n}} . $$

(48)

The total contribution of these sources or sinks at each grid point divided by the 15-year time is the annual mean generation and dissipation rate of mesoscale eddies:

$$ w_{{i,j}}^{\rm{source}} = E_{{i,j}}^{\rm{source}}/T, $$

(49)

$$ w_{{i,j}}^{\rm{sink}} = E_{{i,j}}^{\rm{sink}}/T. $$

(50)

Appendix 4: Results in TH-model

Results from the TH-model are much smaller than the corresponding values obtained from the EQ-model, and the global sum of eddy energy generation rate is estimated at 0.113 TW (Table 3). In particular, the contribution from the ACC in the EQ-model is also much higher than that obtained from the TH-model. Such difference is due to the fact that the TH-model underestimates both the depth of the equivalent interface and the density jump across the interface, as shown in Figs. 2 and 3.

Accordingly, for the TH-model the EKE generation rate of cyclonic eddies is slightly lower than that of anticyclonic eddies. However, the EAGPE generation rate of cyclonic eddies is slightly higher than that of anticyclonic eddies. For the global sums, in the TH-model the EAGPE generation rate is 1.15 times larger than that of EKE (Table 4).

Since the interface depth in the TH-model is not suitable for the eddy in the subpolar basin and the Southern Ocean where the main thermocline is poorly defined, we present the results from the TH-model as a comparison and a sensitivity test.