11.3: Munk's Solution

Sverdrup’s and Stommel’s work suggested the dominant processes producing a basin-wide, wind-driven circulation. Munk (1950) built upon this foundation, adding information from Rossby (1936) on lateral eddy viscosity, to obtain a solution for the circulation within an ocean basin. Munk used Sverdrup’s idea of a vertically integrated mass transport flowing over a motionless deeper layer. This simplified the mathematical problem, and it is more realistic. The ocean currents are concentrated in the upper kilometer of the ocean, they are not barotropic and independent of depth. To include friction, Munk used lateral eddy friction with constant \(A_{H} = A_{x} = A_{y}\). Equations \((11.1.1-2)\) become:

\[\begin{align} \frac{1}{\rho} \frac{\partial p}{\partial x} &= \ fv + \frac{\partial}{\partial z}\left(A_{z} \frac{\partial u}{\partial z}\right) + A_{H} \frac{\partial^{2} u}{\partial x^{2}} + A_{H} \frac{\partial^{2} u}{\partial y^{2}} \\ \frac{1}{\rho} \frac{\partial p}{\partial y} &= -fu + \frac{\partial}{\partial z}\left(A_{z} \frac{\partial v}{\partial z}\right) + A_{H} \frac{\partial^{2} v}{\partial x^{2}} + A_{H} \frac{\partial^{2} v}{\partial y^{2}} \end{align} \nonumber \]

Munk integrated the equations from a depth \(-D\) to the surface at \(z = z_{0}\) which is similar to Sverdrup’s integration except that the surface is not at \(z = 0\). Munk assumed that currents at the depth \(D\) vanish, that \((11.1.5)\) apply at the horizontal boundaries at the top and bottom of the layer, and that \(A_{H}\) is constant.

To simplify the equations, Munk used the mass-transport stream function \((11.1.18)\), and he proceeded along the lines of Sverdrup. He eliminated the pressure term by taking the \(y\) derivative of \((\PageIndex{1})\) and the \(x\) derivative of \(\PageIndex{2}\) to obtain the equation for mass transport: \[\underbrace{A_{H} \nabla^{4} \Psi}_{\text{Friction}} - \underbrace{\beta \frac{\partial \Psi}{\partial x} = -\text{curl}_{z} T}_{\text{Sverdrup Balance}} \nonumber \]

where \[\nabla^{4} = \frac{\partial^{4}}{\partial x^{4}} + 2 \frac{\partial^{4}}{\partial x^{2} y^{2}} + \frac{\partial^{4}}{\partial y^{4}} \nonumber \]

is the biharmonic operator. Equation \((\PageIndex{3})\) is the same as \((11.1.9)\) with the addition of the lateral friction term \(A_{H}\). The friction term is large close to a lateral boundary where the horizontal derivatives of the velocity field are large, and it is small in the interior of the ocean basin. Thus in the interior, the balance of forces is the same as that in Sverdrup’s solution.

Equation \((\PageIndex{3})\) is a fourth-order partial differential equation, and four boundary conditions are needed. Munk assumed the flow at a boundary is parallel to a boundary and that there is no slip at the boundary: \[\Psi_{bdry} = 0, \quad\quad \left(\frac{\partial \Psi}{\partial n}\right)_{bdry} = 0 \nonumber \]

where n is normal to the boundary. Munk then solved \((\PageIndex{3})\) with \((\PageIndex{5})\) assuming the flow was in a rectangular basin extending from \(x = 0\) to \(x = r\), and from \(y = -s\) to \(y = +s\). He further assumed that the wind stress was zonal and in the form: \[\begin{align} T &= a \cos ny + b \sin ny + c \nonumber\\ n &= j \ \pi/s, \quad j = 1,2,\ldots \end{align} \nonumber \]

Munk’s solution (figure \(\PageIndex{1}\)) shows the dominant features of the gyre-scale circulation in an ocean basin. It has a circulation similar to Sverdrup’s in the eastern parts of the ocean basin and a strong western boundary current in the west. Using \(A_{H} = 5 \times 10^{3} \ \text{m}^{2}/\text{s}\) gives a boundary current roughly 225 km wide with a shape similar to the flow observed in the Gulf Stream and the Kuroshio.

Figure \(\PageIndex{1}\): Left: Mean annual wind stress \(T_{x}(y)\) over the Pacific and the curl of the wind stress. \(\varphi_{b}\) are the northern and southern boundaries of the gyres, where \(M_{y} = 0\) and \(\text{curl} \tau = 0\). \(\varphi_{0}\) is the center of the gyre. Upper Right: The mass transport stream function for a rectangular basin calculated by Munk (1950) using observed wind stress for the Pacific. Contour interval is 10 Sverdrups. The total transport between the coast and any point \(x, y\) is \(\psi(x, y)\).The transport in the relatively narrow northern section is greatly exaggerated. Lower Right: North-South component of the mass transport. After Munk (1950).

The transport in western boundary currents is independent of \(A_{H}\), and it depends only on \((11.1.9)\) integrated across the width of the ocean basin. Hence, it depends on the width of the ocean, the curl of the wind stress, and \(\beta\). Using the best available estimates of the wind stress, Munk calculated that the Gulf Stream should have a transport of \(36 \ \text{Sv}\) and that the Kuroshio should have a transport of \(39 \ \text{Sv}\). The values are about one-half of the measured values of the flow available to Munk. This is very good agreement considering the wind stress was not well known.

Recent recalculations show good agreement except for the region offshore of Cape Hatteras where there is a strong recirculation. Munk’s solution was based on wind stress averaged over \(5^{\circ}\) squares. This underestimated the curl of the stress. Leetmaa and Bunker (1978) used modern drag coefficient and \(2^{\circ} \times 5^{\circ}\) averages of stress to obtain \(32 \ \text{Sv}\) transport in the Gulf Stream, a value very close to that calculated by Munk.