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# PATTERN ANALYSIS OF ONE SUMMER'S MULTILEVEL MAPS OF MONTREAL RAIN

J. S. MARSHALL and C. D. HOLTZ

## Abstract

For the summer of 1964, precipitation intensity in the vicinity of Montreal, observed by CPS-9 radar, was recorded on constant-altitude maps at six heights. To allow for attenuation (at λ = 3.2 cm) and other radar uncertainties, the distribution with rainfall rate R at 5,000 ft was matched to that for one rain gage at the surface. The extent (area × time) in excess of a given R decreased rapidly with increasing R and with increasing height: a factor 1000 from 0.4 to 100 mm hr–1, a factor 10 per 15,000 ft. The pattern in plan at any height (with resolution limited to about 2 mi in the recorded maps) can be described in terms of cells with intensity decreasing outward approximately exponentially. A logarithmic function of intensity was used: T = 1.66 log (R/R0) where R0 = 0.25 mm hr–1. The value of T for a contour bounding a central area A is given by T = Tc−x, where Tc is the maximum and x = (A/17.5 n.mi.2)1/3. (With this relation, one can estimate peak intensity from the area bounded by any given intensity.) The number of cells having Tc greater than a given value of T, per 108 n.mi.2 of map area (as for hourly maps of area 20,000 n.mi.2 throughout a 5,000-hr summer) is N0, where log N0 = 8.8−T−H and H = height/15,000 ft. As to shape, the maximum and minimum extents were always within a factor 6 of each other. Usually, the area in the horizontal section of a cell decreased with height. Occasionally it increased to a maximum, at about 20,000 ft, as much as four times greater than the area at 5,000 ft. These maxima aloft were generally identifiable with storms reaching 40,000 ft, which is tall for Montreal.

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# On the Relationship between Subduction Rates and Diabatic Forcing of the Mixed Layer

A. J. George Nurser and John C. Marshall

## Abstract

The transport of mass between a mixed layer, exposed to mechanical and thermodynamic forcing, and an adiabatic thermocline is studied for gyre-scale motions. It is shown that if the mixed layer can be represented by a vertically homogeneous layer, whose base velocity and potential density are continuous, then, at any instant, the rate at which fluid is subducted per unit area of the sloping mixed-layer base, S, is given bywhere h is the depth of the mixed layer, Qb = −fρ̄−1∂ρ/∂z|zh is th large-scale potential vorticity is the base, ℋnet is the heat input per unit area less that which warms the Ekman drift, αE, Cw, and ρ̄ are the volume expansion coefficient, heat capacity, and mean density of water, respectively. It is assumed that the mixed layer is convectively controlled and much deeper than the layer directly stirred by the wind. The field of S is studied in a steady thermocline model in which patterns of Ekman pumping and diabatic heating drive flow to and from a mixed layer overlying a stratified thermocline.

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# A Continuously Stratified Thermocline Model Incorporating a Mixed Layer of Variable Thickness and Density

John C. Marshall and A. J. George Nurser

## Abstract

A continuously stratified, steady thermocline model is formulated in which a mixed layer of variable depth and density overlies a stratified thermocline. Rather than prescribe the distribution of density and vertical velocity at the top of the permanent themocline, we explicitly represent the dynamics of the vertically homogeneous layer layer that overlies it; the density distribution at the sea surface, the depth of the mixed layer, and the structure of the thermocline are all found for prescribed patterns of Ekman pumping and surface buoyancy fluxes. If the potential vorticity of the thermocline is assumed to have a uniform value on isopycnal surfaces, it is shown that the problem can be reduced to one of finding the distribution of a single scalar field, the mixed-layer density, by the method of characteristics. Given this field and knowledge of the potential vorticity distribution in the thermocline, all other variables of the model can be found. The resulting model seems ideally suited to the study of the interaction of a mixed layer with a stratified thermocline, since it explicitly represents the lateral geostrophic flow through the sloping base of the mixed layer.

Idealized solutions are presented for both subtropical and subtrophical and in which, in response to patterns of wind and diabatic forcing, isopycnals outcrop into a mixed layer of variable thickness and density. The effect of both warming and cooling of the mixed layer on the structure of the gyre is investigated.

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# A Theoretical and Diagnostic Study of Solitary Waves and Atmospheric Blocking

N. Butchart, K. Haines, and J. C. Marshall

## Abstract

Theories which associate atmospheric blocking with isolated “free mode” solutions of the equations of motion are reviewed and the central role played by the potential function Λ ≡ dq/dψ (where q is the is the quasi-geostrophic potential vorticity and ψ is the streamfunction) is emphasized. This function provides the common dynamical link that draws together the weakly nonlinear (soliton) and fully nonlinear (modon) theories of isolated coherent structures.

A diagnostic study of the European blocking episode during October 1987 is presented and the relationship between q and ψ investigated by plotting scatter diagrams of quasi-geostrophic potential vorticity against the streamfunction on an isobaric surface. An approximate functional relationship is found allowing Λ to be defined. Over the blocking region, points on the scatter plot cluster around a straight line which is more steeply sloping than the straight line defined by points from nonblocking regions, demonstrating that the block exhibits a local minimum in Λ. Such a signature is characteristic of local fully nonlinear free mode structures, the prototype of which has been termed the “equivalent barottopic modon.” The data strongly suggest that blocking episodes can exhibit local free-mode dynamics and that their persistence may in part be attributed to the robustness and stationary nature of these local coherent structures.

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# Fluid Dynamics of Oceanic Thermocline Ventilation

John C. Marshall and A. J. George Nurser

## Abstract

A flux form of the Potential vorticity (PV) equation is applied to study the creation and transport of potential vorticity in an ocean gyre; generalized PV fluxes (J vectors) and the associated PV flux fines are used to map the creation, by buoyancy forcing, of PV in the mixed layer and its transport as fluid is subducted through the base of the mixed layer into the thermocline. The PV flux lines can either close on themselves (recirculation) or begin and end on the boundaries (ventilation). Idealized thermocline solutions are diagnosed using J vectors, which vividly illustrate the competing process of recirculation through western boundary currants and subduction from the surface.

Potential vorticity flux vectors are then used to quantify the flux of mass passing invisidly through a surface across which potential vorticity changes discontinuously but at which potential density and velocity are continuous. Such a surface might be the base of the oceanic mixed layer or, in a meteorological context, the tropopause. It is shown that, at any instant, the normal flux of fluid per unit area across such a surface is given, very generally, bywhere u is the velocity and n is the normal vector to the surface. Here ω is the absolute vorticity; B = −gDσ/Dt is the buoyancy forcing, with D/Dt the substantial derivative and σ the potential density; Q = −ρ−1ω·∇σ is the potential vorticity; ρ the in situ density., and g the gravitational acceleration. Square brackets denote the change in the enclosed quantity across the surface.

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# A New Treatment of the Coriolis Terms in C-Grid Models at Both High and Low Resolutions

A. J. Adcroft, C. N. Hill, and J. C. Marshall

## Abstract

Numerical models of the ocean typically employ gridpoint techniques in which the dynamical variables defining the state of the ocean are held on a staggered grid. One common arrangement of the variables, known as the Arakawa C-grid, is particularly prone to gridscale noise that is due to spatial averaging of Coriolis terms and that is manifest when the grid resolution is coarse with respect to the deformation radius. Here, the authors analyze the problem in the context of linear inertia–gravity waves and discuss the reason for the prevalence of noise. They suggest a solution to the problem in which the C-grid model variables are augmented with D-grid velocity variables. An analysis of the resulting C–D grid indicates favorable behavior and numerical results are presented to demonstrate this. Finally, they discuss the similarity in nature between the C–D grid and the Z-grid, to explain why the C–D grid works well at both high and low resolution.

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# Fifth International Winds Workshop

K. Holmlund, C. S. Velden, and J. Le Marshall

The International Winds Workshops interact with the Co-ordination Group for Meteorological Satellites with respect to important issues related to the derivation and utilization of wind information based on the imagery from geostationary satellites. It also provides a forum for data producers, data users, and the science community to exchange information on the characteristics of satellite-tracked winds and to optimize their use in several applications, including numerical weather prediction, nowcasting, and climate applications. The sequence of meetings began in Washington, D.C., in September 1991. Since then, meetings have been held in Tokyo, Japan, in December 1993; Ascona, Switzerland, in June 1996; and in Saanenmoser, Switzerland, in October 1998. This report describes the proceedings at the Fifth International Winds Workshop and includes the recommendations derived from the meeting.

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# Quasigeostrophic Potential Vorticity in Isentropic Coordinates

P. Berrisford, J. C. Marshall, and A. A. White

## Abstract

An isentropic coordinate form of quasigeostrophic potential vorticity is derived. Compared with well-known height and pressure coordinate versions, this isentropic form is more clearly related to Ertel's potential vorticity and the derivation from it is much simpler. A derivation from quasigeostrophic approximations to the governing equations is also given and the boundary conditions to be applied on isentropic surface are discussed. Analogous developments using isopycnal coordinates are given assuming an incompressible fluid model.

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# MEASUREMENT OF RAINFALL BY RADAR

J. S. Marshall, R. C. Langille, and W. Mc K. Palmer

## Abstract

Experiments at wavelength 10 cm have verified the theoretical finding that the power obtained from radiation reflected from rain is proportional to Z, the sum of the sixth powers of diameters of raindrops' contained in a representative volume. No value has been obtained for the constant of proportionality. Correlation of Z with, rainfall intensity R (both at ground level) on logarithmic scales shows that, within a factor of about two, Z α R 2. Vertical scans of rain storms by radar indicate that variation of rain content with height is moderate and accountable. It may be possible therefore to determine with useful accuracy the intensity of rainfall at a point quite distant (say 100 km) by the radar echo from that point.

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# THE DEVELOPMENT OF THE SIZE DISTRIBUTION OF RAINDROPS DURING THEIR FALL

E. C. Rigby, J. S. Marshall, and Walter Hitschfeld

## Abstract

Numerical methods are used to study the changes in the distribution of raindrops with size and in the radar echo, as rain falls. Changes brought about by collisions among the drops, by accretion of cloud, and by evaporation are considered. The distribution assumed aloft is that actually observed at the ground. This is justified, because the changes in the form of the distribution are found to be slight: an exponential type of distribution law would seem to be applicable at all heights. This result is taken to mean that the processes investigated cannot by themselves produce the distributions observed at the ground from distributions of a very different sort, or from the broad distributions of snow. A mechanism as yet unknown, probably involving drop break-up, would seem to be required. The work is done both for “continuous rain,” where conditions at all levels are assumed to be constant in time, and for showers, where they are assumed to be initially the same at all levels.

Unequal rates of fall of raindrops tend to increase the vertical extent of the radar echo in time. The calculated rates of motion of echo top and base depend critically on range and radar sensitivity; they are in reasonable agreement with those observed.

Where necessary, the rate of depletion of cloud due to rain falling through it is taken into account. A simple theory of this process, by Dr. K. L. S. Gunn, is described in an appendix.

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