oscillating_thermocline.ipynb

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     "source": "# A Wind-Driven Oscillating Thermocline\n$\\newcommand{\\ol}[1]{\\overline{#1}}$\n$\\newcommand{\\ab}[1]{\\langle #1 \\rangle}$\n$\\newcommand{\\pd}[2]{\\frac{\\partial #1}{\\partial #2}}$\n\n## Steady State\n\nConsider a wind driven channel in a statistically steady state, such as described by Marshall and Radko (2003). In the adiabatic ocean interior, the buoyancy equation satisfies\n\n$$ J(\\Psi_{res}, \\ab{\\ol{b}}) = 0 \\ . $$\n\nThe residual circulation is given by the sum of wind-driven (Ekman) and eddy components, such that\n\n$$ \\Psi_{res} = -\\frac{\\tau}{\\rho_0 f} + \\Psi^\\ast \\ .$$\n\nIn MR03, the eddy-induced transport $\\Psi^\\ast$ is specified using the QG transformed-eulerian-mean definition $ \\Psi^\\ast = \\overline{v'b'} / N^2$, where $N^2 = \\partial \\overline{b} / \\partial z$. An important limit is simply $\\Psi_{res} = 0$.\n\nThe domiant balance in the buoyancy equation is approximately 1D, between the vertical advection terms:\n\n$$ J(\\Psi_{res}, \\ab{\\ol{b}} ) \\simeq ( \\ab{\\ol{w}} + w ^\\ast) \\pd{\\ab{\\ol{b}}}{z}  $$\n\nwhere\n\n$$ \\overline{w} = - \\frac{1}{\\rho_0 f} \\pd{\\tau}{y} $$\n\nis the Ekman pumping velocity and\n\n$$ w^\\ast = \\pd{\\Psi^\\ast}{y}  $$\n\nis the eddy-induced vertical advection.\n\n## Time Dependent Response \n\nWe need to define an anomaly of the zonal average buoyancy field from the steady state. Define\n\n$$ b^+ = \\ab{b} - \\ab{\\ol{b}} \\ .$$\n\nWhen the system is not in a steady state, the buoyancy equation gains a time dependent term. The 1D approximation is \n\n$$ \\pd{\\ab{b}}{t}  = - ( \\ab{w} + w ^\\ast) \\pd{\\ab{b}}{z} \\ .$$\n\nHere we interpret the overbar as a zonal average but not necessarily a time average. Perhaps it is best to think of it as a low pass filter in time.\n\nWe impose time variability by making the winds, and the Ekman pumping, oscillate in time, such that\n\n$$ \\ab{w} = w_0 + w_1 \\sin(\\omega t) $$\n\nwhich means that $\\ab{\\ol{w}} = w_0$. We substitute this into the buoyancy equation\n\n$$ \\pd{\\ab{b}}{t}  = - ( w_0 + w_1 \\sin(\\omega t) + w ^\\ast) \\pd{\\ab{b}}{z} \\ .$$\n\n### No Eddy Response\n\nAssuming that the eddies do not respond at all, the steady state balance simply subtracts out of the equation for $b^+$, leaving\n\n$$ \\pd{b^+}{t}  = - w_1 \\sin(\\omega t)\\pd{\\ab{b}}{z} \\ . $$\n\nFurther assuming that the buoyancy anomaly is small, we can replace $\\pd{\\ab{b}}{z}$ with the time average, leading to a closed equation for the anomaly:\n\n$$ \\pd{b^+}{t}  = - w_1 \\sin(\\omega t)\\pd{\\ab{\\ol{b}}}{z} $$\n\ngiving\n \n$$ b^+ = \\frac{w_1}{\\omega} \\cos(\\omega t)\\pd{\\ab{\\ol{b}}}{z} \\ .$$\n\nThe RMS average of the buoyancy anomaly generated by the oscillations is therefore\n\n$$ \\ol{(b^+)^2}^{1/2} = \\sqrt{\\pi} \\frac{w_1}{\\omega} \\pd{\\ab{\\ol{b}}}{z} $$\n\n### Gent-McWilliams Eddy Response\n\nUnder the GM parameterization, the eddy-induced vertical advection is given by\n\n$$ w^\\ast = \\pd{\\Psi^\\ast}{y} = K_{GM} \\pd{s}{y} \\ .$$\n\nWe identify steady and oscillating components as above by\n\n$$ w^\\ast = w^\\ast_0 + w^{\\ast +} $$\n\nwhere\n\n$$ w^\\ast_0 = \\ab{\\ol{w^\\ast}} = K_{GM} \\pd{\\ol{s}}{y} \\ .$$\n\nWe can identify the oscillating component with thermocline oscillations:\n\n$$ w^{\\ast +} = K_{GM} \\pd{s^+}{y} $$\n\nwhere $s^+ = s - \\ol{s}$\nHow can we relate $s^+$ to $b^+$?\n\n$$ s = - \\frac{ \\partial_y \\ab{b}}{\\partial_z \\ab{b}} = \\frac{ \\partial_y \\ab{\\ol{b}} +\\partial_y b^+ }{\\partial_z \\ab{\\ol{b}} + \\partial_z b^+} \n= \\frac{ \\frac{\\partial_y \\ab{\\ol{b}}}{\\partial_z \\ab{\\ol{b}}} + \\frac{\\partial_y b^+}{\\partial_z \\ab{\\ol{b}}} }{1 + \\frac{\\partial_z b^+}{\\partial_z \\ab{\\ol{b}}}} \n$$\n\nApplying QG scaling and a binomial expansion, we find\n\n$$ s = \\ol{s} \\left ( 1 - \\frac{\\partial_z b^+}{\\partial_z \\ab{\\ol{b}}} \\right ) $$\n\ngiving\n\n$$ w^{\\ast +} = - K_{GM} \\pd{\\ol{s}}{y} \\frac{\\partial_z b^+}{\\partial_z \\ab{\\ol{b}}} $$\n\n"
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     "source": "### Kinematics\n\nThe above argument doesn't make much sense. Instead, let's consider kinematics. Let $\\eta$ be the displacement of the zonal-mean isopycnal from its reference position. We have\n\n$$ \\pd{\\eta}{t} = w_{res} $$\n\n$\\eta$ can be related to buoyancy anomalies through the taylor expansion\n\n$$ \\eta \\simeq -\\frac{b^+}{\\partial_z \\ab{b}} \\simeq -\\frac{b^+}{\\partial_z \\ab{\\ol{b}}} \\ .$$\n\nIt is still not possible to related $\\eta$ to changes in slope without considering the meridional stucture of the wind forcing. Otherwise the wind pumping just moves the isopcnals up and down uniformly, with no changes in slope.\n\n$$ s^+ = \\pd{\\eta}{y} \\simeq - \\frac{\\partial_y b^+}{\\partial_z \\ab{\\ol{b}}} $$\n"
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