Hoskins (1982), using quasi-geostrophy, explains clearly (see especially p. 134–135) why an intensification of a front will result in a upwelling on the warm side, downwelling on the cold side, a warm-to-cold ageostrophic flow at the surface and cold-to-warm one at depth across the front, all in order to conserve the thermal wind balance. This model is however time-independent and it is not clear when the upwelling/downwelling starts and how the isopycnals (and their water mass) evolve with time. I need to understand the time-dependent case. See Bleck

*et al.*(1988) for a more physical and complete explanation (see below).Pollard and Regier (1992) is an impressive analysis of thermodynamical and dynamical data across a front in the Sargasso Sea, with, in particular, the calculation of potential vorticity, ageostrophic horizontal circulation and vertical one. Yet, it is not clear how I should analyze a model that has almost everything and may be easier and more obvious to understand. From this paper, I conclude that one idea would be to compute the ageostrophic motion and compare its direction with the direction of thermal front. Another idea may be to follow parcels along (using the isopycnal velocity field), check with salinity that the deduced transport is correct, comprehend the transport within and around eddies and conclude for nutrients.

Bleck

*et al.*(1988) uses a primitive equation model that conserves potential vorticity to study an idealized case of frontogenesis. Thanks to this paper, I finally understand what the upwelling and downwelling really mean. The following is my own description of frontogenesis:The initial condition is an isopycnal front, superimposed on which a cross-front convergent perturbation squeezes and intensifies the front. To stay simple, the perturbation is depth-independent so that in the frame that follows the perturbing motion, the layers stay stationary. This would be the only thing that occurs, would the dynamical equations (thermal balance and conservation of potential vorticity (PV) and mass) need not be satisfied. Due to the horizontal squeezing, the slope of the isopycnals and the horizontal gradient of the buoyancy become larger but the vertical shear of the flow does not changed, thus violating the thermal wind balance. Furthermore, the edges of the along-front geostrophic jet become steeper and the relative vorticity larger, violating the PV conservation.

To satisfy the dynamical constraints, the layers are responding in several aspects. First, a water column between two isopycnals that follows the perturbing motion changes its thickness to conserve PV and he increase in magnitude of its relative vorticity imposed by the squeezing motion: smaller thickness on the anticyclonic part of the front, larger on the cyclonic part of the front. Second, due to these changes in thickness and with no flow in the cross-flow direction at the vertical boundaries, the layers on the anticyclonic side spreads horizontally while those on the cyclonic side contract, shifting the center of the front away from the anticyclonic side toward the cyclonic side. There is thus in effect a cross-front ageostrophic flow from the anticyclonic to the cyclonic side of the front. In the model, the vertical shear of the jet and horizontal shear of the buoyancy decrease downward so that the horizontal shift also decreases downward: the front tilts toward the anticyclonic side downward. The level of tilting can be an indication of the maturity of the frontogenesis. Notice that the cross-front flow is

*relative to the convergent perturbing motion*: although on the cyclonic side of the front, water is being pulled toward the center of the domain by the perturbing motion, this pull is smaller than the one occurring on the anticyclonic side. Notice also that the tilting is a consequence of the boundary conditions: would it still occur if the ocean would have been considered infinite in the cross-front direction?Another consequence of these horizontal shift of mass is that the isopycnal layers up-well on the anticyclonic side and down-well on the cyclonic side, which tends to flatten the front. This flattening of the isopycnals corresponds to the transfer of potential energy to kinetic energy necessary to increase the surface jet and satisfy the thermal wind balance. The acceleration of the geostrophic jet is balanced by the Coriolis acceleration associated with the cross-front ageostrophic motion.

Secondary effects are: 1) “[t]he cyclonic and anticyclonic cells are not symmetric: the horizontal and vertical extent of the anticyclonic cell is larger than the extent of the cyclonic cell” and the cyclonic cell is stronger than the anticyclonic one, so that 2) the maximum positive relative vorticity is larger than the maximum negative relative vorticity and the cyclonic edge of the geostrophic jet is steeper than its anticyclonic edge, and 3) the maximum downwelling is stronger than the maximum upwelling. Why? One reason is that, for a convectively stable ocean (or equivalently the thickness h in the definition of PV stay positive), the relative vorticity has to stay larger than -f (f being the Coriolis parameter) so that magnitude of the anticyclonic relative vorticity is bounded, unlike that of the cyclonic relative vorticity.

One last point concerns the water columns that start initially at the center of the front. At these locations, the geostrophic jet is maximum and the relative vorticity is zero so that the squeezing does not affect the relative vorticity and PV of these water columns. The thickness of the layers at these locations does not change. The water columns, however, move with the ageostrophic flow as any other and keep defining what is the center of the front. Finally, notice that

*there is no exchange of water between the anticyclonic side and the cyclonic side of the front*, each side having a constant mass of water during the frontogenesis.**There is something wrong in my understanding concerning this last paragraph as the line showing the water columns that did not change their thickness is tilted in the other direction, toward the cyclonic side downward (Fig. 3)**.

- The introduction of Spall
*et al.*(1995) explains why frontogenesis can be as important as other wind-forced processes in the process of subduction.

- Bleck
*et al.*(1988), A two-dimensional model of mesoscale frontogenesis in the ocean,*Q. J. R. Meteorol. Soc.*,**114**, 347–371. - Hoskins (1982), The mathematical theory of frontogenesis,
*Ann. Rev. Fluid Mech.*,**14**, 131–151. - Pollard and Regier (1992), Vorticity and vertical circulation at an ocean front,
*J. Phys. Ocean.*,**22**, 609–625. - Spall
*et al*. (1995), Frontogenesis, subduction, and cross-front exchange at upper ocean fronts,*J. Geophys. Res.*,**100**, 2543–2557.