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The Stommel boundary layer

Stommel (1948) made the assumption that interfacial friction is in the form of a linear drag law. Thus, within the western boundary layer the downstream momentum equation is in an ageostrophic balance 

\(fhu=-g'hh_y-Rv\)

The dynamical balance of the circulation is best illustrated in terms of the potential vorticity change along closed streamlines. Since the relative vorticity is very small, the potential vorticity can be approximated by \((\Delta \rho /\rho_0)(f/h)\)

In the basin interior the wind-stress curl is a sink of potential vorticity; thus, potential vorticity declines downstream In the western boundary, the interfacial friction torque is a source of positive potential vorticity; thus, potential vorticity of a water parcel increases along the pathway.

In the basin interior, the interfrictional friction torque is negligible. Within the western boundary layer the wind-stress torque is negligible. 

Similarly, the energy balance along streamlines consits of two stages. First, in the basin interior, wind stress imposes mechanical energy into the circulation, primarily near both the northern and southern boundaries, where the zonal velocity and wind stress are large. This external mechanical energy input is balanced by an energy sink due to interfacial friction. 

 

 

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Munk boundary layer.

Munk (1950) postulated that the lateral friction works as the major dissipation mechanism. Lateral friction is parameterized in terms of \(\nabla ^2_h(h\vec{u})\)

Therefore, the basic momentum equations are as follows: 

\(-fhv=g'hh_x+A_h\nabla^2_h(hu)+\tau^x/\rho_0\\fhu=-g'hh_y+A_h\nabla_h^2(hv)\)

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Explain the meridional flow by Ekan pumping physical!

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  1. We find westerlies at mid latitudes with easterlies at low latitudes and polar regimes. 
  2. This wind stress pattern drives poleward Ekman flows at both low and high latitudes but drives an equatorward Ekman flow at mid latitudes. 
  3. The meridional convergence of Ekman flux in the upper ocean gives rise to the Ekman pumping and upwelling below the base of the Ekman layer. 
  4. In the basin interior, the relative vorticity is negligible, so potential vorticity for a water column is \(f/h\)
  5. Ekman pumping at the base of the Ekman layer compresses the water column height. In order to conserve potential vorticity \(f/h\), the individual water column moves toward the equator, where the Coriolis parameter \(f\) is smaller. 

Thus, Ekman pumping in the subtropical basin drives an equatorward flow in the ocean interior. Similarly, the Ekman upwelling in the subpolar basin drives a poleward flow in the ocean interior. 

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Explain the need of potential vorticity balance and why this can only be provided by a western boundary layer!

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The problem: the interior solution is valid all the way up to the eastern wall; thus, the solution in the interior can be obtained by starting the integration from the eastern boundary. On the other hand, there is a western boundary layer, so we would not be able to obtain the interior solution by integration from the western boundary. 

Explanation: 

We are in the subtropical basin: Water continously gains negative vorticity from the upper boundary. The circulation in the ocean interior is very slow, and the relative vorticity is negligible; thus, water parcels move southward to a place where the planetary vorticity is smaller. To close the ciruclation, water needs to move northward at the western or eastern wall and gain vorticity. 

-> No matter what kind of model we use, there should always be a place where positive vorticity is generated through either interfacial or lateral friction to counterbalance the vorticity input from wind-stress curl in the interior. 

As the graphic shows, only a wall to the left side can provide positive vorticity input through friction. Therefore, only a western boundary current can play the role of balancing potential vorticity in a closed basin. 

 

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Reciruclation: What is the mitovation for a closer look? 

Our discussion has been limited to models based on Sverdrup dynamic. The western boundary currents were just matched to the interior solution. Within this theoretical framework, the maximal streamfunction is totally determined by a zonal integration of the Ekman pumping velocity, started from the eastern boundary of the basin. 

Observations, however, indicate that the maximal volume flux in the Gulf Stream is about 150Sv, which is several times larger than the value calculated from the Sverdrup relation. 

The discrepancy between the linear theory and observations is due to two factors:

  1. Inertial terms in the horizontal momentum equations have been neglected for analyitical simplicity in the commonly used linear theories. 
  2. There is a strong interaction with stratified flow over topography, which is called bottom pressure torque or JEBAR (joint effect of baroclinicity and bottom relief)
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The JEBAR term

When the ocean bottom is not flat, the JEBAR term arises when the curl is taken of the vertically-averaged horizontal momentum equations. 

In fact, JEBAR comes directly from taking the curl of the veritically-averaged horizontal pressure gradient term. 

\(J(\Psi,\frac{f}{H})=J(\Phi,\frac{1}{H})+\frac{1}{\rho_0}[\frac{\partial }{\partial x}(\frac{\tau_s^y}{H})-\frac{\partial }{\partial y}(\frac{\tau_s^x}{H})]\)

The first term on the right-hand side is called the JEBAR term, and the second term on the right-hand side is the contribution due to wind-strss curl.

The equation states that changes of the barotropic streamfunction along the \(f/H\) contours are due to contributions from these two terms. 

The JEBAR term is associated with bottom pressure torque, and is the result of interaction between the stratification and bottom  topography. It can be readily seen that, for the case of a flat bottom, this terms vanishes. 

Greatbatch et al. (1991) analyzed the circulation on the North Atlantic Ocean. The most crucial contributor to the volume transport in the recirculation regime is the JEBAR term. 

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Explain the essence of the quasi-geostrophic theory! 

While geostrophic motion refers to the wind that would result from an exact balance between the Coriolis force and horizontal pressure-gradient forces,[1] quasi-geostrophic (QG) motion refers to flows where the Coriolis force and pressure gradient forces are almost in balance, but with inertia also having an effect.

This effect of inertia is reffered to the ageostrophic part of the velocity, which is under the following assumptions much smaller than the geostrophic part:

  1. The magnitude of accelartion is small compared to the magnitude of the Coriolis force:
    1. Then: \(|v_g|>>|v_a|\)
    2. This is the case if the Rossby number is small:
      1. \(R_0=\frac{U^2/L}{f_0U}=\frac{U}{f_0L}\)
      2. \(\frac{Du}{Dt}=fv-\frac{1}{\rho}\frac{\partial p}{\partial x}=f(v-v_g)\)
      3. if \(R_0\)is small -> \(\frac{Du}{Dt}\) is small -> \(v-v_g\)is small -> \(|v_g|>>|v_a|\)
  2. The \(\beta\)-plane approximation is valid. 
    1.  justifies letting the Coriolis parameter have a constant value  in the geostrophic approximation and approximating its variation in the Coriolis force term by .[4] However, because the acceleration following the motion, which is given in (1) as the difference between the Coriolis force and the pressure gradient force, depends on the departure of the actual wind from the geostrophic wind, it is not permissible to simply replace the velocity by its geostrophic velocity in the Coriolis term.

The most important implification is that vertical displacements are small compared to the background stratification! 

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Scale anaylsis for the \(1\frac{1}{2}\)-layer model 

At highest order of simplification the flow is purely geostrophic:This means also that on a f-plane the flow is non-divergent. 

To have vertical motion we need to consider the ageostrophic flow. 

The ageostrophic flow has 4 parts:

  1. \(\beta\)-part: \(u_a\)and \(v_a\) are parallel to \(u_g\)and \(v_g\), reinforcing (\(u_g,v_g\)) for y<0, weakening \((u_g,v_g)\) for y>0.
  2. Isallobaric part: is normal to the isallobars and flows towards the isallobaric low (where pressure is falling most rapidly) -> Convergence in regions where the pressure is falling most rapidly. 
  3. Non-linear part: 
    1. flow around the through is "subgeostrophic" (i.e. weaker than geostrophic), so that the pressure gradient force towards the centre of curvature is slightly stronger than the opposing Coriolis force, enabling the flow to take a curved path. 
    2. Likewise, around the ridge, the flow is slightly "supergeostrophic". 
    3. This leads to divergence ahead of the trough and convergene behind the trough (i.e. ahead of the ridge). Regions ahead of an upper level trough (i.e. ahead of the ridge).
    4. Regions ahead of an upper level trough are favourable for low level storm development. 
  4. Ekman part: ageostrophic flow to the right of the windstress.