WP 3.1.1 Improvement in geostrophic current determination.

Authors: Senior scientist Ole B. Andersen and State Geodesist Per Knudsen, KMS, Rentemestervej 8, DK-2400 Copenhagen NV, Denmark. (oa@kms.dk).

The ocean contains numerous dynamical regimes, each of which has to be considered separately. In this task we have limited the investigation into the regime representing most of the deep ocean of the world. This regime is also the most simple, representing the open ocean circulation at spatial scales larger than about 30 km and with time scales greater than one day.

This regime is characterised by motions that are nearly geostrophic and as such there is a balance between the Coriolis force and the pressure force, which means that the sea surface slope predicts the oceanic surface velocity or vice versa. The geostrophic balance can be written like

where u and v are eastward and northward currents respectively, g is the acceleration of gravity, (, ) are latitude and longitude, a is the Earth's mean radius, and h is the observed sea level height. The deflections of the vertical are called (, ) as north-south and east-west deflection usually given in seconds of arc. Finally f () is the Coriolis parameter f ()= 2 sin (), where is the earth rotation rate.

The geostrophic balance is only valid polewards of roughly 3 distance to the Equator. In the following figures global plots are shown excluding the Equator. As such the results and plots are not valid within roughly 3 distance of the Equator.

The error simulator (work package 2.1) was used to generate perturbations with given standard deviation per spherical harmonic degree of a fixed gravity field. The spherical harmonic field, EGM96 (Lemoine et al. 1997) completer to degree and order 360 was used as the "true" gravity field when such was needed. The simulated fields were perturbations of this field corresponding to GOCE and EGM96 standard deviations. The simulated perturbations of the geoid, the gravity field and the deflections did then have the contribution of the "true" gravity field removed, leaving errors for the subsequent study. A typical number of 50 perturbations of global 0.5 degree grids were used in the study. It is important to assume that the ocean surface topography is known from altimetry and can thus assumed to be as fixed. Consequently, the perturbations of the geoid can be considered in one to one agreement with perturbation in the dynamical topography.

The outcome of the perturbations of the deflection of the verticals were subsequently converted into geostrophic velocities and standard deviations of the pertubated fields were computed. The improvements in any quantity or more correctly, the reduction in error going from EGS96 to GOCE is subsequently calculated as the improvement in the standard deviation of the error-pertubations like:

Figure 3.1. The reduction in error for the east - u component of geostrophic current determination for GOCE versus EGM96. The mean value of the reduction is 56.1 percent. Maximum and minimum reductions are 35.6 and 64.1 percent, respectively. Equatorial regions have been masked.

Figure 3.2. The reduction in error for the north - v component of the geostrophic currents determination for GOCE versus EGM96. The mean value of the reduction is 64.0 percent. Maximum and minimum values are 40.7 and 66.9 percent, respectively. Equatorial regions have been masked.

Both the reduction for the east and north component of geostrophic currents shows clear zonal structures, corresponding to the modelled error statistics of the GOCE mission having largest errors and consequently smallest error reduction at high latitudes. However, the reduction for the east - u component is on average 10 percent smaller than the reduction in the north - v component, which is presently not fully understood. The small differences in the longitudinal direction are a consequence of the errors in the EGM96 gravity model as well as imperfections in the Monte Carlo simulator due to the number of simulations used in the investigation.

An evaluation of the improvements of the geoid as a function of wavelength and the reduction in errors in current velocities obtained using the geostrophic assumption are qualitatively investigated below, by making a schematic description of the different spatial scales of the mean topography of the WOCE Parallel Ocean Climate Model of Semtner and Chervin (1992) forced by the ECMWF winds and the model was provided by R. Tokmaknian. This model is also referred to as the POCM model. The model is only one out of many sophisticated global circulation models available today (e.g. OCCAM, MICOM) and although this model is not completely representable of the real ocean it clearly illustrates the importance of GOCE for ocean modelling.

The four year mean of the elevation field was investigated for spatial scales corresponding to what would be seen at wavelength where GOCE is superior to other gravity missions like GRACE and CHAMP. Namely the wavelength shorter than 500 km. A bell-shaped filter in the space domain was used to extract the corresponding part of the spectrum. At scales smaller than 500 km frontal signatures of the major currents are seen as well as many boundary signals. Dynamical processes at these short scales are important to the study for the ocean circulation and its heat and mass transport, which stresses the important contribution of GOCE at these spatial wavelength. The mean of the POCM model topography has a standard deviation of 56.8 cm with minimum and maximum values of -208 cm and 123 cm. The 500 km "high-pass"filtered POCM ocean topography has a standard deviation of 7.5 cm with minimum and maximum values of -96 and 84 cm. Consequently, the amplitude of the topography is clearly larger in many regions than the expected geoid errors of 2 cm for GOCE at these wavelength.

Figure 3.3. Ocean topography of the POCM model with spatial scales smaller than 500 km derived from four year mean value of the Semtner and Cherwin (1992) model. The standard deviation is 7.5 cm with minimum and maximum values of -96cm and 84 cm.

While variations of the sea level and thus of the ocean currents can be derived directly from satellite altimetry, the absolute elevation and hence absolute current can only be derived with respect to some quasi-stationary surface and not absolutely from altimetry. Consequently the permanent circulation and the mean flow can only be resolved by numerical modelling presently. The ocean transports most of its heat through its mean flow as well as variabilities (i.e. eddies), and through this it controls the climate. At any location eighter the mean flow or the variability about the mean flow will be the most dominant. Consequently, GOCE will be very important in resolving these issues together with GRACE and CHAMP.

In order to investigate the spatial scales corresponding to what would be seen at wavelength where GOCE is superior to other gravity missions the kinetic energy of the mean field was calculated using a geostrophic assumption. The geostrophic currents derived from the POCM mean elevation field were used and filtered for its contribution below 500 km wavelength. The global mean of the kinetic energy of the POCM model is of 12.6 (cm/s)2 and a standard deviation of 50.1 (cm/s)2. The 500 km "high-pass" filtered part of the kinetic energy has a mean value of 3.76 (cm/s)2 and a standard deviation of 19.6 (cm/s)2. Again the importance of GOCE in resolving frontal and boundary signals is obvious.

Figure 3.4 Weighted kinetic energy E of the mean flow having smaller spatial scales than 500 km. A geostrophic assumption was made and the geostrophic currents were filtered for their contribution with wavelength shorter than 500 km interval. The weighting by (sin(latitude))2 were performed to avoid singularities at the equator.

Under geostrophic conditions the mean surface meridional and zonal transports Tv and Tu can be determined using the sea surface height as the currents implies a massflux. In a simple approach the height h1, h2 at two positions can be used to determine the transport under barotropic conditions like

where is the density of sea water and d is the depth of the mixed layer. This depth was fixed at 200 meters for the following investigation.

It must be noticed, that the transport do not depend on the geostrophic currents and the improvement in transport determination is consequently not related to the improvement in the determination of the geostrophic currents. The transport is only depending on the surface height difference between the end points of the section, and the depth over the distance consequently the improvement in transport uncertainties are related to the determination of the geoid and not the deflections of the vertical.

Five sections were used for the investigation of the reduction of surface mean transport uncertainties. The five sections also shown in Figure 3.5 were

1) North Atlantic. Meridional currents though 59 N between 60 W and 10 W.

2) North Atlantic. Meridional currents through 48 N between 70W and 10W.

3) Gulf Stream near Cape Hatteras. Zonal transport through 60W between 38N and 42 N

4) Gulf Stream near Florida. Meridional transport through 30N between 85W and 75W

5) Drake Passage section. Zonal transport through 60W between 62S and 52 S

Figure 3.5 Location of five sections used to investigate the reduction of transport uncertainties.

The reduction in mean surface transport uncertainties are shown in Figure 3.6. The transport through the two high latitude zonal cross sections at 59N and 48 N section are only improved by a few percent (1% and 3%, respectively). This corresponds most probably with the fact GOCE is assumed to have has higher errors at high latitude. Similarly the lack of improvement in the north Atlantic ocean is most probably due to the error bars for the EGM96 being too optimistic as also noted by GOCE studies by Le Grand and Minster (1999), and by other authors like Wunch and Strammer (1998)

Figure 3.6 Reduction in surface transport uncertainties compared with EGM96 transport uncertainties (normalised to 100). The reduction are shown for the five sections in Figure 3.5.

Generally, the improvement are around 5-10 % with reduction of up to 22% in the Drake Passage. The fact that the improvement is greater in the Drake Passage most likely steams from the fact that the EGM96 has larger geoid errors in the southern ocean because of poorer coverage of gravity observations entering the model. (Lemoine et al, 1997).

The impact found here on uncertainties in mean surface transport are similar to what has been seem by other studies by Legran and Minster (1999), and the impact is comparable in magnitude to the impact on top-to-bottom transport uncertainties found by Ganachaud and co-workers (Ganachaud et al., 1997)


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Last change: 1999-06-17 by cct.