J. Fluid Mech. (1986), vol. 173, p p . 431-471
Printed in Great Britain
43 1
Turbulent entrainment: the development of
the entrainment assumption, and its application
to geophysical flows
By J. S. TURNER
Research School of Earth Sciences, Australian National University, G.P.O. BOX 4,
Canberra, A.C.T. 2601, Australia
(Received 24 March 1986)
The entrainment assumption, relating the inflow velocity to the local mean velocity
of a turbulent flow, has been used successfully to describe natural phenomena over
a wide range of scales. Its first application was to plumes rising in stably stratified
surroundings, and it has been extended to inclined plumes (gravity currents) and
related problems by adding the effect of buoyancy forces, which inhibit mixing across
a density interface. More recently, the influence of viscosity differences between a
turbulent flow and its surroundings has been studied. This paper surveys the
background theory and the laboratory experiments that have been used to under-
stand and quantify each of these phenomena, and discusses their applications in the
atmosphere, the ocean and various geological contexts.
1. Introduction
The entrainment hypothesis was first introduced by Sir Geoffrey Taylor in a
wartime report on the dynamics of hot gases rising in air. He spoke about it later
at a Pacific Science Association meeting in 1949, but did not follow up that talk with
a published paper. The idea received a wider exposure through the review lecture by
Batchelor (1954)’ who at that time started two Cambridge students working on
theoretical and experimental aspects of the problem, and also revived G. I.’s interest
in the subject. This led to the joint publication (Morton, Taylor & Turner 1956,
referred to as I in the following sections) which is the more frequently quoted starting
point of this subject.
The entrainment hypothesis in its original form can be stated very simply: the
mean inflow velocity across the edge of a turbulent flow is assumed to be proportional
to a characteristic velocity, usually the local time-averaged maximum mean velocity
or the mean velocity over the cross-section at the level of inflow. The total inflow
at any position will depend also on the surface area and the geometry and dynamics
of the flow - whether it is axisymmetric or two-dimensional, a continuous jet or
plume or a suddenly released ‘thermal ’. In this form the assumption is deceptively
simple, and even obvious, since for a jet in uniform surroundings it can be deduced
from the similarity solution or justified by the most elementary dimensional
considerations. Its power was soon demonstrated, however, by the application to the
rise of plumes in stably stratified surroundings, a situation where similarity solutions
of the usual kind cannot be applied. It is of course another type of similarity
assumption, which implies the same kind of turbulent structure and balance of forces
at each height. It can be, and has been, criticized on the grounds that it is
demonstrably not exactly true in particular circumstances, and there have been

432 J . 8. Turner
debates about possible alternative formulations, but it has in fact been enormously
successful when applied to phenomena over a very wide range of scales. Though it
has been adapted to take account of additional physical processes, the basic
assumption remains an effective starting point for quantifying a wide variety of
mixing problems.
This paper has several related themes, all of them aimed at demonstrating the
power and practical success of the entrainment hypothesis, and reaching a better
understanding of why it works so well. The basic definitions and entrainment
equations will first be summarized and re-examined in the light of several recent
reviews of the fundamentals. The wide range of successful applications to convective
phenomena will then be described, with particular emphasis on geophysical examples.
These sections on convective flows are followed by a discussion of the extension to
mixing processes in stably stratified fluids, first with a mean flow, and then without.
Research on the mechanism of the entrainment process, particularly work which
sheds light on the role of the large eddies, will be discussed next. Finally, the effect
of large viscosity differences on the entrainment mechanism itself is examined. A
laboratory study of this aspect was motivated by a geological problem, the mixing
of magmas of different composition and physical properties, but it has much wider
implications for mixing processes in general.
2. Jets and plumes in a homogeneous environment
In summarizing the present state of the similarity theory, and the measurements
used to test it, we will follow the excellent reviews by Fischer et al. (1979) and
List (1982). They have treated both the axisymmetric and the two-dimensional cases,
but only the first will be described here, with emphasis on the comparison between
vertically directed turbulent jets (sources of momentum) and plumes (sources of
buoyancy) in a uniform environment. Only steady flows will be treated here, not the
stage immediately after the flow has been turned on (e.g. ‘the starting plume’,
Turner 1962). A two-dimensional inclined starting plume will, however, be discussed
in another context in $6.
The three basic integral properties of these flows (integrated across the cross-
sectional area at any level) are the fluxes of mass, momentum and buoyancy. These
are defined by:
pp = 2n pwrdr, (1) Jorn
where p is the ‘specific mass flux’ or volume flux, and w is the local mean vertical
velocity a t radius r from the vertical line above the
pm = 27~ pw2rdr, Sum
where m is the specific momentum flux; and
source ;
where /3 is the specific buoyancy flux and 9’ = gAp/po, the effective gravitational
acceleration, and po is the constant density of the environment.
The symbols Q , M and B are used to denote the initial values of p, m and /3. These
three are the primary variables governing the behaviour of axisymmetric turbulent
buoyant jets provided that the Reynolds number Mi/v exceeds a few thousand, as