A study of the omega equation for diagnosing vertical motions at ocean fronts

Journal of Marine Research, 54, 239-259,1996

A study of the omega equation for diagnosing vertical motions at ocean fronts

by Jean-Michel Pinot’, Joaquin Tintok and Dong-Ping Wang3

ABSTRACT Estimation of vertical velocity is a key issue for understanding ocean physics and transport

of biogeochemical tracers. We examine the accuracy of estimating vertical velocity in fronts with the omega equation. The diagnostic performance of the omega equation is evaluated by using vertical velocities obtained from simulation of frontal instabilities in a primitive equation model as a reference. We use two traditional quasigeostrophic methods in which the flow is either a geostrophic flow computed from density or a nondivergent flow derived from vorticity and also test two new formulations: a quasigeostrophic method using the total flow field and the semigeostrophic omega equation. Results show that all four formulations correctly diagnose the vertical velocity pattern. However, estimates provided by the traditional quasigeostrophic formulations have systematic bias. In contrast, the two new techniques, which are practically equivalent, produce unbiased vertical velocity diagnostic at fronts. These results point out the importance of including higher order dynamics than quasigeostrophy to take into account the ageostrophic advection in the front. Since adequate filtering of ADCP data is not yet available to obtain a suitable total flow, the semigeostrophic omega equation is proposed as the most valuable tool to compute vertical velocities from high resolution density measurements.

1. Introduction

Vertical velocity is a key factor for the exchanges of heat and salt between the surface and the ocean interior. It is also crucial for the vertical transport of all kinds of materials (plankton, suspended matter, pollutants) and for distributions of biogeochemical tracers. By bringing deep nutrients into upper levels, it is particularly relevant for the marine production in the euphotic layer (Woods, 1988). Mesoscale upwelling is considered as a major process producing patchy biomass patterns in ocean fronts (Strass, 1992). An accurate diagnostic of vertical velocity is thus crucial for understanding and predicting how the ocean responds to environmental changes.

1. Department de Fisica, Universitat de les Illes Balears, E-07071, Palma de Mallorca, Spain. 2. Institut Mediterrani d’Estudis Avancats (IMEDEA), Universitat de les Illes Balears, E-07071,

Palma de Malloria, Spain. 3. Marine Sciences Research Center, State University of New York, Stony Brook, New York, 11794,

U.S.A.

239

240 Journal of Marine Research [54,2

The difficulty of directly measuring vertical velocities in ocean fronts has made necessary the use of indirect methods to infer them. Several methods can be used, depending on the nature of the data set (Lindstrom and Watts, 1994). In the case of extensive high resolution ship surveys, the most widely used tool is the omega equation (o-equation). In a recent paper, Strass (1994) compared three different methods of estimating vertical motions based on: (1) the density equation, (2) the vorticity equation and (3) the omega equation. Based on the patterns of the forcing term (called the Q-vector), he showed that the omega equation gave the best estimates of vertical velocity. It should be noted that he did not perform a complete inversion of the omega equation to obtain actual vertical velocities.

There are several different formulations of the omega equation. The classical formulation of the omega equation based on the quasigeostrophic (QG) theory, was established by Hoskins et aZ. (1978). In practice, it allows computation of vertical velocity from geostrophic flow obtained from density measurements. Applications of this formulation to vertical velocity diagnostic in the ocean can be found in several recent studies (Tintore et al., 1991; Pollard and Regier, 1992; Allen and Smeed, 1996; Viudez et al., 1996a). However, the QG omega equation often has been applied to ocean fronts of large Rossby number; therefore violating the QG approximation, since in those cases higher order dynamics would be required.

In order to take into account the higher order dynamics, Hoskins and Draghici (1977) established a semigeostrophic (SG) omega equation, which also allows the diagnostic of vertical velocity from density measurements alone. Whereas the SG equations of motion have been used in prognostic frontogenesis simulations for atmospheric (Duffy, 1976; Snyder, 1991) and oceanic studies (Flagg and Beardsley, 1978; Macvean and Woods, 1980), there has not been much interest in using the SG omega equation as a diagnostic tool. To our knowledge, no application to an oceanic problem has been done. This might be related to the complexity of the application of this equation which requires transformation of space coordinates.

The wide use of ship-board ADCPs now allows high-resolution direct measurements of ocean currents. However, the vertical velocity component still remains below ADCP’s accuracy. Therefore, it is natural to think about constructing the Q-vector forcing term from direct velocity measurements. Leach (1987) believed that the total flow should not be used in the computation since it is contaminated by aliasing short-scale motions such as internal and inertial waves. He suggested that these short-scale motions can be filtered out by using only the nondivergent horizontal velocity field in constructing the Q-vector forcing term of the omega equation. Though the nondivergent flow can be assumed to be close to the actual geostrophic flow, it is not clear apriori that such approach is indeed valid.

The purpose of this study is to discriminate between different formulations of the omega equation by evaluating their respective performance. The QG omega equa-

19961 Pinot et al.: Study of omega equation 241

tion is computed using three different methods in which the Q-vector is obtained from the geostrophic, nondivergent and total flow respectively. The SG omega equation also is used to derive the vertical velocity. To investigate these different formulations of the omega equation, the data are obtained from a numerical primitive equation model simulation of the frontal circulation in an oceanic jet. Thus, the diagnostic results can be quantitatively evaluated with respect to the known vertical velocities. For this aspect, our methodology is similar to the one used by Strass (1994) for testing vertical velocity diagnostic procedures, except that we perform a complete inversion of the omega equation. Investigating the omega equation through diagnosing of a numerical model output in atmospheric examples can be found in Hoskins and Pedder (1980) for baroclinic waves and in Pauley and Nieman (1992) for a developing cyclone. It is worth noting that the main interest of using numerical simulations is that they provide self-consistent dynamical fields which can be used to discriminate between different diagnostic methods. There is no claim that those numerical studies actually represent the reality.

The three QG formulations are presented in Section 2 and the SG formulation in Section 3. The primitive equation model and frontal settings are described in Section 4. Results of the vertical velocity comparison between the various omega equation diagnostics and the primitive equation are presented in Sections 5 and 6. These results and their possible application to in situ density and velocity data are discussed in Section 7.

2. Quasigeostrophic omega equation

a. Q-vector computed from the geostrophic flow. At the level of approximation of QG theory, the equations for the horizontal velocity (u, v) and density (p) are:

i a a a C3P ,+u,~+v -up-f&=--

g aY 1

i $+u,$+v Lg+flu=-$

g dY i

i $+u$+v A+w-$Fo g ay i

(1)

(2)

(3)

where (z+, vg) is the geostrophic flow, w is vertical velocity, P is pressure andfo is the local Coriolis parameter. By combining (l), (2) and (3) the time derivatives can be eliminated and using the continuity equation:

f!F+?+g=o dY

(4)

242 Journal of Marine Research 15432

we obtain the classical formulation of the QG omega equation (Hoskins et al., 1978):

N2V;w + f 0” g = 2V.Q (5)

with

v+&+--$

and

(6)

(7)

where N is the buoyancy frequency. The QG omega equation requires previous knowledge of the density field p and also the choice of a reference level for computation of geostrophic velocities. Here, we use the model density to compute the dynamic height field at each level integrating from a reference level which coincides with the surface where the elevation is given by the model.

Both density and dynamic height fields’ first and second derivatives are contaminated by noise at grid resolution scale. We use a two-dimensional interpolation scheme based on successive corrections (Pedder, 1993) to smooth these fields. This scheme is based on an empirical influence function S(r) decreasing with distance r:

1 r2 S(r)=exp -2 z I 01

whose shape is controlled by an influence scale L chosen such that shortest scales be strongly damped. This algorithm approximates optimal interpolation for infinite iterations (Bratseth, 1986) and has proved to be useful for oceanic applications (Gomis et al., 1996). We perform horizontal interpolations independently at each level. The influence scale L is chosen as two times the horizontal grid spacing, say L = 4 km, in order to exclusively filter out noise at grid resolution scale and maintain a good agreement with the original model data at larger scales. The mean deviations between the analyzed and original fields are 0.1% and 1% for density and dynamic height respectively. Thus, the cross-frontal gradients, whose sharpness characterizes frontogenetic dynamics, are not smoothed by the analysis. Sensitivity of the results to the influence length scale is analyzed in Section 6.

b. Q-vector computedfrom a nondivergentflow. Leach (1987) proposed an alternative method to compute vertical velocity using direct velocity measurements (u, v). To


avoid aliasing by small-scale motions such as internal and inertial waves, he filtered the velocity field by removing the divergent part. He computed the relative vorticity 5 from the objective analysis of velocity measurements:

and then integrated the Poisson equation:

vi* = !g w-9

to calculate a stream function W. From q, a nondivergent field (u,,~, v,~) can be derived as:

allr a* %d = - - and v,d = z.

dY

The omega equation is still based on the QG formalism (Eq. Q-vector is now computed from nondivergent velocities:

(11)

5) except that the

(12)

To check this procedure, we use the vorticity given by the model and integrate at each level the Poisson equation (Eq. 10) using gradient (Neumann) boundary conditions (model velocity components) for the stream function as in Leach (1987).

c. Q-vector computedfrom the totalflow. It is also useful to check the performance of the QG omega equation (Eq. 5) by retaining the divergent part of the horizontal velocity field. The Q-vector is computed from the total velocities (utot, v,,,):

(13)

To test this alternative formalism, the Q-vector (Eq. 13) is directly computed from model’s total flow. The meaning of this formalism is analyzed in the Appendix.

3. Semigeostrophic omega equation

At the level of geostrophic momentum approximation, the SG equations for velocity and density (Hoskins, 1975) are:

i a a a a

1 l3P

,+u,+v-+w,u,-fov=--& 8Y

(14)


i ~+u~+v;+w-&+f,=-~

i ay

i

a a a a -g+u&+v-+wp=o.

aY i

(15)

06)

Note that SG equations, Eqs. 14, 15 and 16, include ageostrophic advections (both horizontal and vertical) which are neglected in QG equations, Eqs. 1, 2 and 3. To obtain a diagnostic equation for w, a transformation of coordinates is performed (Hoskins and Draghici, 1977) from physical space (x, y, z, t) to geostrophic space (X, Y, Z, T) defined as:

X = x + vg/fO, Y = y - ug/fO, Z = z, T = t.

The SG omega equation in geostrophic space can be written as follows:

(17)

a%* V;qgw* + f,” 7 = 2V.Q.

The Q-vector in the SG omega equation (Eq. 18) has the same expression as in QG except that it is computed in geostrophic space:

(19)

w* in (Eq. 18) is related to the vertical velocity w by:

W * = w/J (20)

where J is the Jacobian of the transformation of coordinates:

J = 1 + &/fo (21)

and & is the geostrophic relative vorticity:

Also, the square buoyancy frequency N2 that appeared in the QG omega equation (Eq. 5) is replaced by geostrophic potential vorticity qg in the SG omega equation (Eq. 18). Geostrophic potential vorticity is written as:

In practice, to compute the Q-vector in geostrophic space, model dynamic height (referenced to the surface model elevation) and density fields must be transformed from physical to geostrophic coordinates. At each level, for each model grid point


(.xj,yi), we compute (Xi, k) from (Eq. 17). (Xi, fi) is not a regular grid so that all the discrete fields have to be interpolated on a regular grid (Xi, x) to allow further calculations of derivatives by centered differencing. The interpolation scheme used is the same successive corrections algorithm (Pedder, 1993) used previously with the same influence scale L = 4 km to avoid excessive smoothing. After calculation of w in geostrophic space (Xi, x), the same methodology is applied to perform the inverse transformation and retrieve the w field in physical space.

4. Primitive equation model

The model used is a regional, high resolution, three-dimensional, primitive equation model (Wang, 1989). The methodology, the model geometry and parameter setting are based on a previous study by Wang (1993) and are briefly summa- rized here for completeness. The domain is a periodic channel 54 km long, 72 km wide and 200 m deep with a slip boundary condition at the coast and quadratic friction at the bottom. The resolution is 2 km in the horizontal and 20 m in the vertical. The Coriolis parameter is constant. Horizontal eddy viscosity and diffusivity are chosen as lo2 m2 s-i. The vertical eddy viscosity and diffusivity are based on the Munk-Anderson formula with neutral values of respectively lop2 m2 s-l and 10M4 m2 s-l. The initial state consists of a frontal jet in geostrophic balance, parallel to the coast, over which a slight disturbance is superimposed. The initial Rossby number of the jet (here 0.4) can be tuned by the adjustment of the setting parameter which controls the width of the initial surface density front. The model is run for five days.

In the primitive equation simulation, the initial perturbation evolves into growing frontal baroclinic waves which are well developed after five days (Fig. 1). At this time, instabilities are characterized by important ageostrophic motion. The Rossby number reaches a maximum value of 1.2 at the surface. As expected from theory, vertical velocity is related to streamline patterns (Figs. 1 and 2). Upwelling and downwelling are respectively found downstream and upstream of troughs. The results that are of particular interest are the strength of vertical velocity which reaches an order of magnitude of lop4 m s-l ( = 10 m day-‘), and the clear asymmetry pattern between upwelling and downwelling. Upwelling occurs in a broad area, in the anticyclonic region of the wave, with maximum velocities of 2 10m4 m s-i while downwelling is in a narrow zone, in the cyclonic region of the wave, with velocities up to 4 1O-4 m s-l. Wang (1993) showed that this asymmetry is related to the mesoscale convergence and divergence governed by ageostrophic processes.

5. Results

The omega equation is an elliptic partial differential equation which has a unique solution if boundary conditions are known on all the domain. Here we use model- derived boundary conditions at the surface, bottom and around four lateral bound-


Figure 1. Surface elevation in cm (model output at t = 5 day)

aries. In this way, the solutions only depend on the forcing term (Q-vector) accuracy. The sensitivity of relaxation to boundary conditions is further discussed in Section 6. An iterative relaxation scheme is used to perform the inversion of the omega equation. Note that in the case of the SG omega equation, the use of model-derived boundary conditions implies transforming the model vertical velocity at the boundaries to geostrophic coordinates. This is done by the same successive corrections interpolation technique as for the density and dynamic height fields.

As previously indicated, we use model output data as observations. Therefore, we apply the techniques described above to the model density and/or velocity outputs at five days, and compute the vertical velocity field Wdiag in the model region with the four different formulations of the omega equation, (l)-QG, (2)-SG, (3)-U”d and (4)-U,,,. Methods (1) and (2) rely on model density data and methods (3) and (4) on density and velocity data. Preliminary observations can be made from comparison between slices of Wdiag (Fig. 3) and W,,&el (Fig. 2) at 70 m. Largest w are found at about 70 m; hence this is a depth where the effect of the diagnostic technique is expected to be the strongest. Figure 2 and 3 show that the general w patterns are well reproduced with all four methods. The magnitude (10e4 m s-l) of vertical velocities and the asymmetry of upwelling/downwelling pattern are approximately reproduced. However, all diagnosed downwelling patterns are slightly broader than the model one. They indicate that the region of maximum ageostrophic motion (strong


42.0

Figure 2. Vertical velocities w,&l (10m4 m s-l) obtained at 70 m (model output at t = 5 day)

downwelling) is most sensitive to the approximations made in omega equation formulations. Method (1) apparently gives the largest error, as the diagnosed downwelling location is substantially shifted toward the upwelling zone (Fig. 3a). Method (2) which does not produce such a shift, gives generally much better results (Fig. 3b). Method (3) produces a w pattern much too smooth (Fig. 3~). Finally, method (4) well reproduces the downwelling/upwelling strength and location (Fig. 3d) and apparently gives the best diagnostic.

scatter plots of Wdiag versus W,o,+r illustrate more clearly the performance of the different diagnostic techniques (Fig. 4). The regression statistics (Table 1) also provide quantitative and systematic comparison. These plots and statistics are computed over more than 3000 points of the 3-D grid, excluding the boundary grid-points where model-derived boundary conditions are imposed for the inversion of the omega equation. Mean values are computed independently for upwelling (mean w+ j and downwelling (mean W-) regions using vertical velocities higher than 0.5 10m4 m s-l. Maximum upward/downward w values exhibit the same trend as mean values so that they are not presented here. Standard deviation (St. Dev.) and linear regression coefficient (Reg. Coeff.) also are calculated which helps to quantify the Consistency Of Wdiag with W,&r. The linear regression lines are included in Figure 4.

248

56.0

42.0

28.0

14.0

0.0

Journal of Marine Research 15472

-.5-

1 / , , , , , , , , ( , / , , / , , , , , , , I 1 1 1 1 1 1 1 1

0.0 18.0 36.0 54.0 72.0

Figure 3. Diagnostic of vertical velocities W&g (10e4 m SC’) at 70 m using: (I) Standard QG omega equation (method 1); (b) SG omega equation (method 2); (c) QCi omega equation with nondivergent flow (method 3); (d) QG omega equation with total flow (method 4).


56.0

42.0

28.0

14.0

0.0

56.0

42.0

28.0

1, ( , , , , , , , ( , , , , , , , , , I,, , , , , , , , , , , I,, 1 0.0 18.0 36.0 54.0 72.0

Figure 3. (Continued)


6 .-

5.-

.._. ” ‘. -6.- .:- :

.., .- _.

-7. I I I I I I I I I I I I -7. -6. -5. -4. -3. -2. -1. 0. 1. 2. 3. 4. 5. 6. 7.

Figure 4. Scatter plots of W&g versus W,,&l (10m4 m s-l) for all four diagnostic methods: (a) Standard QG omega equation (method 1); (b) SG omega equation (method 2); (c) QG omega equation with nondivergent flow (method 3); (d) QG omega equation with total flow (method 4).

Method (1) appears to bias both upwelling and downwelling patterns (Fig. 4a). Upwelling is overestimated while downwelling is more highly scattered. The same result also is revealed in the computed statistics (Table 1). The linear regression performed between QG and model results gives a rather good coefficient of 0.8, but this linear relationship is actually valid only for low w while it is not significant for high w values (Fig. 4a). Because of the presence of a systematic bias, the diagnostic value of the QG omega equation is quite poor. SG method (2) gives better results than QG method (1). The comparisons are also scattered (Fig. 4b), particularly for negative values, but they scatter uniformly from a linear relation with regression coefficient of 1.0. In other words, the SG omega equation produces no significant bias. The SG method however tends to overestimate downwelling (Table 1). Method (3) show that both mean upwelling and downwelling are underestimated (Fig. 4~). The w diagnostic is strongly biased with the regression coefficient less than 0.7. Finally, method (4) gives w values very close to the P.E. model reference. The scatter


7. I I I I I I

6.-

-5. ,_

._ : .: -, .

_: ‘:' ,_ . -. -6.

-7. -7. -6. -5. -4. -3. -2. -1.

I I I I I I . 1. 2. 3. 4. 5. 6. '7.


plot indicates consistency between the estimated and model w (Fig. 4d) with a regression coefficient of 0.8.

In order to quantify the accuracy of the different omega equation formulations, statistics of the E, error field (defined as E, = W&as - w modei) are given for the four diagnostics in Table 2. Maximum error and r.m.s. error decrease from method (1) to (4). Method (1) produces the largest error, up to 6.3 10m4 m s-l, and a large r.m.s. error of 1.1 low4 m s-l. Method (2) improves the diagnostic as the maximum error and r.m.s. error are reduced to 4.5 and 0.9 10m4 m s-i respectively. With method (3) the maximum error is 3.2 10e4 m s-l and r.m.s. error 0.6 10e4 m s-l. The best results are obtained with method (4), which gives a maximum error of 2.2 10e4 m s-l and r.m.s. error of 0.4 10e4 m s-l.

6. Sensitivity study

a. Analysis influence scale. Sensitivity of the vertical velocity diagnostic to the analysis scale L of the influence function (Eq. 8) is examined. Instead of using L = 4 km, we carried out an analysis with L = 10 km which is about the scale of the front width. In the case of the QG omega equation computed from geostrophic flow, the pattern


5. I 4 .-I

-6.

-7. -7. -6. -5. -4. -3. -2. -1.

I I I I I I I 1. 2. 3. 4. 5. 6. 7.


remains the same, but the vertical velocity magnitudes are substantially reduced by 50%. In contrast, in the case of the QG omega equation computed from total flow, the vertical velocity is not sensitive to L. In method (1) both density and dynamic height fields are analyzed before computation of the Q-vector while in method (4) only the density field is analyzed. This shows that the omega equation is most sensitive to the velocity field.

b. Boundary conditions. From examination of the structure of the w field in the vicinity of the boundaries, it appears that the solution of the omega equation is much more sensitive to relaxation on the surface boundary value. This is due to the fact that the vertical gradient of the vertical velocity (&V/&Z) is large in the nearsurface layer. Since realistically we have no knowledge of w value at the surface, we assume zero vertical velocity at the surface and test its validity. Statistics of the diagnostics obtained using this empirical boundary condition are compared with model results in Table 3. The quality of w estimates is decreased if compared with results obtained by using model-derived boundary conditions (Tables 1 and 2). Departures of w&as from w,&i in terms of mean upwelling and downwelling are increased for all four

19961 Pinot et al.: Study of omega equation

7.""""""" I

253

-3.

-4.

-5.

-6.

-7., , , I I I I I I I I I I I I -7. -6. -5. -4. -3. -2. -1. 0. 1. 2. 3. 4. 5. 6. 7.


Table 1. Statistics of the w field for the reference P.E. simulation and four diagnostic methods imposing model-derived boundary conditions.

w ( 10m4 m s-l) Mean w - Mean w + St. dev. Reg. coeff.

P.E. (1) QG (2) SG (3) Und (4) r/,0,

-1.6 -1.8

cg’

-2.0 1.1 1.4 1.0 -1.3 1.0 0.9 0.7 -1.4 1.1 1.1 0.8

Table 2. Statistics of the error field (E, = Wdiag - w,&r) for the four diagnostic methods imposing modei-derived boundary conditions.

E, (10-d m s-1) Mean Min MaX St. dev.

(1) QG -0.02 -6.3 3.6 1.1 (2) SG -0.16 -4.5 3.3 0.9 (3) und -0.03 -2.7 3.2 0.6 (4) utot 0.02 -1.8 2.2 0.4

2.54 Journal of Marine Research [54,2

Table 3. Statistics of the w field for the reference P.E. simulation and four diagnostic methods imposing w = 0 at the surface.

w ( 10e4 m s-l) Mean w -

P.E. -1.6 (1) QG -2.1 (2) SG -2.2 (3) umi -1.4 (4) uto, -1.6

Mean w + St. dev.

1.2 1.2 1.6 iTs 1.3 1.7 1.0 1.0 1.3 1.3

Reg. coeff. E, (st. dev.)

CL9 K!La 1.0 1.3 1.2 1.0 0.6 0.8 1.0 0.5

methods. Both the differences between standard deviations of W&,g and W,dei and the r.m.s. error become larger. However, the regression coefficients are not affected suggesting that the assumption of zero vertical velocity at the surface does not cause additional bias.

7. Discussion and conclusion

Methods (1) and (3) have been shown to be biased. Method (1) overestimates both downwelling and upwelling velocities by about 10%. This overestimation is consistent with the results of Viudez et al. (1996b) who compared estimates obtained by the QG omega equation and by density dynamical assimilation in a P.E. model applied to a hydrographic data set. Such a systematic bias indicates that methods (1) and (3) rely on assumptions that do not properly represent the ageostrophic frontal dynamics. Indeed, the reason for the bias is that these methods do not adequately take into account horizontal ageostrophic advection. This is clear in the case of method (1) where nonlinear terms are computed from geostrophic advection. In the case of method (3), the nondivergent field also filters out the horizontal ageostrophic advection. However, our results show that the nondivergent velocity field obtained from the integration of vorticity field is not equivalent to the geostrophic flow. Method (1) overestimates both downwelling and upwelling velocities while method (3) underestimates these quantities.

Results are significantly improved with methods (2) and (4) which give unbiased estimates. These two methods explicitly involve the ageostrophic terms. Methods (2) and (4) which both rely on the geostrophic momentum approximation are practically the same (see Appendix). This approximation specifically retains ageostrophic advection which plays a crucial role in mesoscale convergence/divergence. This accounts for the better performance of these two methods. Our findings are in agreement with those obtained by Pauley and Nieman (1992) about the importance of retaining ageostrophic terms for vertical velocity diagnostic. Pauley and Nieman (1992) compared the generalized omega equation (a diagnostic equation derived from primitive equations that involves total velocities) and the QG omega equation


performance for a developing atmospheric cyclone. They obtained poor QG vertical velocity estimates with bias in ascent/descent magnitude and location. By examining each term of the generalized omega equation, they find that the major correction to QG diagnostic is related to ageostrophic advection and diabatic heating. They concluded that forcing the QG omega equation with total flow and diabatic heating gives a result consistent with primitive equations.

In the application to real data where the absolute accuracy of vertical velocity estimate may be difficult to achieve, an essential criterion to discriminate between the diagnostic qualities of the different omega equation formulations relies on whether there exists a systematic bias. An unbiased estimate of the vertical velocity field is absolutely necessary to avoid major errors in computing the vertical transport of heat, salt, nutrients and geochemical tracers. We have shown that the standard QG omega equation has serious limitations for estimating the ageostrophic vertical velocity. Using a nondivergent flow field obtained from ship ADCP data to force the omega equation as done in Fiekas et al. (1994) also produces a systematic bias in the vertical velocity estimate. On the other hand, the omega equation forced by the Q-vector built from total velocities provides an accurate determination of vertical velocity. However, the success of this method will depend on an adequate removal of tides, internal and inertial waves from the velocity data. At present, the filtering methods are not yet reliable. Hence, the SG omega equation technique should be considered as the most valuable tool now available to compute vertical velocities in narrow fronts if density sampling resolution is adequate. Additional information is required to derive the absolute geostrophic flow. This information can be obtained by deriving a reference level from adequately analyzed ADCP velocity measurements if they are available (Pollard and Regier, 1992). These conclusions should be considered in the design of future in situ frontal experiments. The only difficulty of the SG method actually lies in the transformation of coordinates which requires a suitable interpolation scheme to avoid losing accuracy through successive interpolations.

Finally, our sensitivity study suggests that the accuracy (r.m.s. error) of the omega equation is slightly decreased due to the need in practice to assume zero vertical velocity at the surface. However, our results may overstate this error since isopycnals outcrop at the surface in our model. In the real ocean, isopycnals are flattened in the surface mixed layer, and this error may be smaller.

Acknowledgments. This work is a UIB contribution to the EUROMODEL project funded by the E.C. program MAST (MAS2-CT93-0066) and by the Spanish CICYT (AMB93-1046-CE). Jean-Michel PINOT acknowledges a doctoral E.C. fellowship (ref: MAST-913017) and a post-doctoral fellowship from the Spanish government (ref: CICYT-SB94-A29028891). We are very thankful to both anonymous referees whose thoughtful comments improved the quality of the manuscript.


APPENDIX Using the total flow in the quasigeostrophic omega equation

Following the work by Hoskins (1975) for atmospheric dynamics, the geostrophic momentum approximation ( 1 DvlDt 1 -=z 1 fv I) f or a Boussinesq, hydrostatic, friction- less ocean leads to the following set of semigeostrophic equations:

with

(Al)

642)

(A3)

(A41

where

(UP v) = (ug + %g, vg + bg).

Applying the time derivative (A4) to &.p gives:

D&p = &D,p - &u&p - d,vd,p - d,wd,p.

Using the third component of (Al), (A6) simplifies to:

a&P = -(Po/s)(QI - @ax4

bw

646)

(A71

where

a = kbo)6waxp + a,va,d (A@ and

N2 = -(gho)arP. 649)

Applying the time derivative (A4) to ap, gives:

D,a,v, = ap,v, - a$axvg - a,va,v, - apap,. (AlO)

Using the second component of (Al), (AlO) simplifies to:

Dd,v, = -fo4hg - a.+,a,v, - azvgayvg - ap,ap, - azVagayvg - a,wa,v,. (All)

The thermal wind relation can be derived from (A3):

(Pofoazvg~ Pofowg) = (-a~~ ga,p>. 6412)


Using thermal wind (A12) and continuity (A2), (All) can be written:

Ddzvg = -fo@, - WPofo)Q, -AI (A131

where

4 = -a,v,,a~, - axvagapg + atuagaxvg + azvagayvg. Combining (A7) and (A13) by using (A12), we obtain:

(A141

N2&w - f ;a24g = 2Q1 + fd,. (Al9

Taking the time derivatives (A4) of a,p and ap, and using thermal wind (A12) and continuity (A2) as above gives:

N2ap - f3,v, = 2Q2 + fdz 6416)

where

Q2 = ~~po)(ay~axp + a,va,p) (A171

and

A2 = -apagazvg - axu,ug + apu,axug + azvagayug w3)

Combining the x-derivative of (A15) with the y-derivative of (A16) and using continuity (A2) gives:

N2V&,w + f$$ = 2V.Q + fo(dxA, + ayA2) (A19)

where Q = (QI, Qd Now, for a frontal jet oriented along they-axis, we can assume ug x vg and a, +z a,

in (A14) and (A18) so that we can write:

Al = apu,a,vg and A, = 0

and as a consequence (A19) writes:

N*V&w + f$3,‘w = 2V.Q + fodr(d~agdxvg). WO)

(A20) is practically the same as the QG omega equation with total flow. The only difference is the last r.h.s. term in (A20) which adds to the Q-vector divergence. The results presented in the paper suggest that this extra term represents a minor correction to the vertical velocity diagnostic. Hence, the QG omega equation with total flow can be considered as consistent with the geostrophic momentum approximation in ocean fronts.


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Received: 5 December, 1994; revised: 31 October, 1995.

A study of the omega equation for diagnosing vertical motions at ocean fronts

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