isolines over these topographies are in the form of closed contours, and the circulation is in the regime of strongly nonlinear dynamics.
Since flow over steep topography can be quite strong, we reformulate the basic equations introduced in the subsection entitled "Inverse reduced-gravity model" in Section 5.2.2, including the nonlinear advection terms and the bottom friction terms. Assuming that the flow is steady, the momentum and continuity equations for an inverse reduced-gravity model are uux + vuy — f v = —g'Zx — Ru/h0. (5.111a)
where h and h0 are the layer thickness and its mean value, and R(u, v)/h0 is the bottom friction term. The specific form of bottom drag is used here in order to find a simple analytical expression of the solution, as will be discussed shortly. Cross-differentiating and subtracting a Z (m), Uniform upweling, no topography b Z (m), Uniform upweling, with topography
Eqns. (5.111a) and (5.111b) leads to a concise form of the potential vorticity equation
where Q = (f + vx - uy )/h is potential vorticity, including the contribution due to relative vorticity.
For the case with closed potential vorticity contours, we can calculate the integration of Eqn. (5.112) over the area Aq within a closed potential vorticity contour Cq. Note that, in the present case, there is a source/sink driving horizontal velocity; thus, there is no closed streamline as in the case discussed in Section 4.1.3. In fact, there is a mass flux crossing the potential vorticity contour Cq and entering area Aq . From continuity, the total mass flux across the boundary is equal to the total amount of upwelling inside area Aq . Furthermore, this is a model with friction, so that potential vorticity is not conservation along streamlines. Thus, strictly speaking, potential vorticity contours are different from the so-called geostrophic contours used in a similar situation with closed contours.
Integrating the left-hand side term of Eqn. (5.112) leads to
J Jaq JcQ J Jaq
In deriving this equation, we used the continuity equation (5.111c) and the fact that Q is constant along Cq. Thus, the integration of Eqn. (5.112) is
V k JJJaq k0 JcQ
Equation (5.114) regulates flows within closed potential vorticity contours. First, assuming that there is a net upwelling inside a closed contour cQ, the circulation must be cyclonic, as required by Eqn. (5.114), and this is true for both steep seamounts and trenches (Kawase and Straub, 1991; Johnson, 1998). Second, the circulation rate is inversely proportional to the frictional parameter - and the linearly proportional to total upwelling rate inside the closed contour. Third, since the relative vorticity is positive, vx - uy > 0, for cyclonic flow, the nonlinear term of the balance can further enhance the circulation.
Flow near deep trenches is one of the most interesting phenomena in the deep ocean because trenches are rather elongated features, with length scaling up to thousands of kilometers. Observations have indicated that there are indeed strong cyclonic circulations over steep trenches in the oceans. For example, there are strong deep boundary currents over the deep trenches in the northern and northwestern North Pacific Ocean, as shown in Figure 5.45 (Owens and Warren, 2001). Observations for other sites in the world's oceans were summarized by Johnson (1998).
In the discussion above the existence of the western boundary currents is required by mass conservation, but we have not discussed any dynamical constraints on the deep western boundary currents. To explore the dynamical structure of these currents we need a dynamical framework. We have discussed the inertial western boundary currents in Chapter 4. According to simple inertial theory, the width of the surface inertial western boundary currents is on the order of 30-50 km. However, observations indicate that the deep western boundary currents can be very wide, on the order of 500 km. To explain this phenomenon, Stommel and Arons (1972) postulated that the essential ingredient of such a broad western boundary current is the sloping bottom over which the inertial western boundary currents flow.
Consider an inverse reduced-gravity model with a sloping bottom (Fig. 5.66).
The current is semi-geostrophic
From these equations, we obtain the potential vorticity conservation and the Bernoulli conservation laws
where f is the streamfunction, and Q (f) = dBd(ff).
Stommel and Arons discussed the case of uniform potential vorticity. In this case, the solution for the inertial deep western boundary currents can be represented in terms of exponential functions. However, it is much easier to solve this problem using the stream-function coordinates transform, as we discussed for the surface western boundary currents in Section 4.1.
Introducing the nondimensional variables, x = Lx', h = Hh',f = fof', v = Vv', and f = tyf, then the basic equations (dropping the primes) are the following:
where s = aL/H is the nondimensional bottom slope, e = (kD/L)2 is the Burger number, and = (g'H)1/2/f0 is the internal radius of deformation.
Stommel and Arons (1972) showed that there are two types of boundary condition for the onshore side of the current: vanishing layer thickness or zero velocity. In the second case, f + vx h
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