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• Author or Editor: Stanley J. Jacobs
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# Upper Bounds for Forecast Errors I: Theory

Stanley J. Jacobs

## Abstract

A method is given for predicting errors in numerical weather forecasts due to imprecise knowledge of frictional constants, heating, and initial conditions. It is assumed that error bounds are known for these quantities. Given a measure of forecast error, it is shown that it is possible to find the maximum for this error for all admissible parameter settings by solving a two-point boundary value problem. The method yields the maximum error and the parameter setting which induces this error.

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# On Wind-Driven Lake Circulation

Stanley J. Jacobs

## Abstract

We consider here the flow induced by applying a wind stress at the surface of an initially quiescent lake. It is assumed that the Ekman number, based on an eddy viscosity, is small, and that the Rossby number is at most of the order of (Ekman number)Â½. Under these conditions, which are met in practice, a linear theory is applicable. The linear problem is solved using boundary layer methods. There are essentially five distinct regions: an outer region in which the horizontal velocity is independent of depth, Ekman layers at the upper and lower boundaries, a corner region at the edge of the lake at which the Ekman layers meet, and a shear layer adjacent to the corner region. Study of the Ekman layers provides the equations which hold in the outer and shear layer regions, and consideration of the corner region provides the boundary condition. The outer flow proves to he geostrophic and directed along curves of constant depth. The shear layer is needed to satisfy the boundary condition of zero net outward transport at the edge of the lake. If the wind stress is constant, or, more generally, has zero line integral around curves of constant depth, the transport is confined to the shear layer.

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# A Note on Predictability

Stanley J. Jacobs

## Abstract

It is assumed that in making a forecast the initial state of the system is known subject to an initial error of measurement. Given an upper bound for the initial error, rigorous bounds are determined for the error of the forecast and for the time interval in which the current error in the forecast is less than a pre-assigned values.

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# On The Stability of a Barotropic Basic Flow in a Stratified Atmosphere

Stanley J. Jacobs
and
Aksel Wiin-Nielsen

## Abstract

The stability of a barotropic, horizontally sheared, zonal current in a stratified atmosphere is investigated. It is found that several unstable internal modes may exist. These unstable disturbances have longer wave-lengths and somewhat smaller growth rates than unstable, two-dimensional, non-divergent disturbances. The energetics of the unstable internal modes are investigated and it is found that the energy source is the zonal kinetic energy while the disturbances convert eddy kinetic energy to eddy available potential energy. Several examples are shown for an atmosphere characterized by a constant lapse rate.

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