# Pressure Problem of the Entraining Plume

Jun-Ichi Yano

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Jun-Ichi Yano. Pressure Problem of the Entraining Plume. 2020. hal-02911650

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# Pressure Problem of the Entraining Plume

by

Jun-Ichi Yano

CNRM UMR3589 (CNRS and Météo-France), 31057 Toulouse Cedex, France

## abstract

The standard similarity solution for the entraining plume is re-considered by adding a contribution of the pressure. By explicitly including temporal dependence as well as an equation for the radial velocity, a Poisson equation for solving the pressure is derived. When the top-hat approximation is maintained as in the original formulation, the pressure force perfectly cancels out both the buoyancy force and the entrainment effect. A remaining contribution of dynamic pressure works like a drag force. As a result, a growing plume solution becomes no longer available. It is concluded that an inclusion of a weak, but explicit radial dependence of the physical variables is crucial to avoid such a perfect cancellation. However, an explicit inclusion of a radial dependence no longer permits a similarity solution. An alternative, more straightforward remedy is to phenomenologically introduce effective mass factors to avoid a perfect cancellation. The present study provides a basic receipt for including a convective pressure in evaluating the convective velocity in model parameterizations.
# 1. Introduction

Convection is important phenomena in many geophysical flows. As pointed out in a textbook by Turner (1973), two major categories can be considered for convection. One is those arising under a homogeneous environment. The Rayleigh-Bénard problem is a prototype in this category. Another is those arising from local sources. Basic elements of convection in the latter category can be conceptually treated as plumes and thermals (Yano 2014).

Morton et al. (1956) performed a laboratory experiment of plumes generated by a localized, steady buoyancy source, and presented a similarity solution for describing these experimental plumes. However, as a major omission, a pressure term is not included in their momentum equation for their theoretical analysis. Though this approximation may be justifiable when a role of buoyancy is dominant in the dynamics, its contribution should still be quantified.

The present paper re-addresses Morton et al.'s similarity solution by adding a pressure term to the problem. Apart from the inclusion of this term, the formulation of the problem remains identical to that of Morton et al., including the top-hat approximation. The analysis is also exclusively focused on the case with no stratification, when a similarity solution is available. The adopted methodology has important applications to convection parameterizations, as discussed in the last section extensively (Sec. 6).

The formulation of the problem is presented in the next section. A key step is to consider the equation for that radial component of velocity explicitly so that a Poisson equation for solving the pressure can be derived in a straightforward manner. Inclusion of temporal tendency is also crucial, because the Poisson equation is derived under a condition of vanishing temporal tendency of the divergence of the flow.

A basic solution is presented in Sec. 3. As it turns out, under the top-hat approximation, the pressure term perfectly cancels out both the buoyancy force and the entrainment terms in the vertical momentum equation. A remaining dynamic pressure simply works as a drag force, and a growing plume solution is no longer available. As a remedy, in Sec. 4, effective mass factors are added to the formal pressure solution so that the perfect cancelation can be removed. General behaviour of this version of solution is discussed along with the implications of the result in Sec. 5.

# 2. Formulation

By following Morton et al., we assume that equations for the vertical mass flux, $\pi R^2 w$,
and the conservation of the total plume mass, defined over a horizontal section, $\pi R^2$, are given by

$$ \frac{\partial}{\partial t} R^2 w + \frac{\partial}{\partial z} R^2 w^2 = R^2 b - R^2 \frac{\partial p}{\partial z}, \qquad (1a) $$

$$ \frac{\partial}{\partial t} R^2 + \frac{\partial}{\partial z} R^2 w = \varepsilon R^2 w. \qquad (1b) $$

Here, we take a coordinate, $z$, upwards in vertical direction; $R$ is a height-dependent radius of the plume, $w$ the vertical velocity. By following Morton et al., a common factor, $\pi$, is already dropped from the equations. The buoyancy, $b$, is defined in Morton et al. as

$$ b = g \frac{\rho_0 - \rho}{\rho_0} $$

in terms of the densities $\rho$ and $\rho_0$ for the plume and the environment, respectively, and the acceleration of the gravity, $g$. Also by following Morton et al., the fractional entrainment rate, $\varepsilon$, is defined by

$$ \varepsilon = \frac{2\alpha}{R}, \qquad (2) $$

where $\alpha$ is a constant.

As modifications from Morton et al.'s original formulation, the pressure term (with $p$ the pressure) is added in Eq. (1a). Temporal tendencies (with $t$ the time) are also retained in both equations. Although a steady similarity solution is pursued in the following, as in Morton et al., this inclusion is crucial for deriving a Poisson problem for solving the pressure problem, as going to be seen below.

Another equation considered by Morton et al. is a conservation of vertical buoyancy flux:

$$ \frac{\partial}{\partial z} R^2 w b = 0. $$

This equation simply states that the vertical buoyancy flux

$$ H = R^2 w b \qquad (3) $$

is a constant with height. Thus, by using this constraint, one of the three variables is expressed in terms of the others. For example, the buoyancy is given by

$$ b = H/(R^2 w). \qquad (4) $$

By combining Eqs. (1a, b), we can derive a vertical momentum equation given by

$$ \frac{\partial w}{\partial t} + w \frac{\partial w}{\partial z} = b - \varepsilon w^2 - \frac{\partial p}{\partial z}. \qquad (1c) $$
Recall that a basic principle of deriving a Poisson equation for solving the pressure is to impose a condition that a temporal tendency of divergence vanishes by consistency with the mass continuity. For this purpose, we also need a momentum equation for the radial direction, $r$:

$$ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} = - \frac{\partial p}{\partial r}, \qquad (1d) $$

where the velocity in the radial direction is defined by $u$. The mass continuity (in terms of velocity) is defined by

$$ \frac{1}{r} \frac{\partial}{\partial r} r u + \frac{\partial w}{\partial z} = 0. \qquad (1e) $$

For deriving an equation for the pressure problem, we first apply $(1/r)(\partial/\partial r)r$ and $\partial/\partial z$ on Eqs. (1d) and (1c), respectively, and take a sum together:

$$ \begin{aligned} & \frac{\partial}{\partial t} \left[ \frac{1}{r} \frac{\partial}{\partial r} r u + \frac{\partial w}{\partial z} \right] + \frac{1}{r} \frac{\partial}{\partial r} r \left( u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} \right) + \frac{\partial}{\partial z} \left( w \frac{\partial w}{\partial z} \right) \\ & = \frac{\partial}{\partial z} \left( b - \varepsilon w^2 \right) - \left( \frac{1}{r} \frac{\partial}{\partial r} r \frac{\partial}{\partial r} + \frac{\partial^2}{\partial z^2} \right) p \end{aligned} $$

The first term with the time derivative vanishes by mass continuity (1e). Thus, after some re-arrangements, we obtain an equation for the pressure problem as

$$ \left( \frac{1}{r} \frac{\partial}{\partial r} r \frac{\partial}{\partial r} + \frac{\partial^2}{\partial z^2} \right) p = \frac{\partial}{\partial z} \left( b - \frac{2\alpha w^2}{R} \right) - \left[ \frac{1}{r} \frac{\partial}{\partial r} r \left( u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} \right) + \frac{\partial}{\partial z} \left( w \frac{\partial w}{\partial z} \right) \right]. \quad (5) $$

### 3. Basic Similarity Solution

We seek a similarity solution with the same $z$-dependence as considered in Morton et al. (1956). Thus,

$$ w = w_0 z^{-1/3}, \qquad (6a) $$

$$ b = b_0 z^{-5/3}, \qquad (6b) $$

$$ R = R_0 z, \qquad (6c) $$

where $w_0$, $b_0$, $R_0$ are constants to be determined. By direct substitution of Eqs. (6a) and (6c) into Eq. (1b), but dropping the time derivative, we immediately find

$$ R_0 = \frac{6}{5} \alpha \qquad (7) $$

as in Morton et al. (1956). Also note that $b_0$ can be defined from Eq. (4) under given $w_0$ and $R_0$. The solution for the radial velocity is obtained by substituting Eq. (6a) into the mass continuity (Eq. 1e):

$$ u = \frac{w_0}{6} r z^{-4/3}. \qquad (6d) $$
By substituting Eqs. (6a, b, d) into the right hand side of Eq. (5), we obtain a more specific form for the pressure problem:

$$ \frac{\partial^2}{\partial z^2} p = -\frac{5}{3} \left(\frac{\partial p}{\partial z}\right)_0 z^{-8/3} \qquad (8a) $$

with

$$ \left(\frac{\partial p}{\partial z}\right)_0 = b_0 - \left(\frac{2\alpha}{R_0} - \frac{1}{10}\right) w_0^2. \qquad (8b) $$

Here, the radial dependence drops out from the right hand side in the final form, thus also from the left hand side in Eq. (8a). This equation can be integrated once in vertical direction, and we obtain

$$ \frac{\partial p}{\partial z} = \left(\frac{\partial p}{\partial z}\right)_0 z^{-5/3}. \qquad (9) $$

This is the pressure force acts upon the vertical velocity in Eq. (1c).

By substituting this solution (9) into the right hand side of Eq. (1c), we find

$$ b - \varepsilon w^2 - \frac{\partial p}{\partial z} = -\frac{w_0^2}{10} z^{-5/3} \qquad (10) $$

with both the buoyancy force and the entrainment perfectly cancels out with a counterbalancing force from the pressure. The only contribution left behind is a dynamical pressure term that acts as a drag force. It is obvious that no steady solution is available under this configuration.

A specific situation involved with the given pressure solution (8a, b) can be seen by substituting it, along with Eqs. (6a, b), into Eq. (1a). We find that a solution must satisfy a condition:

$$ \left(\frac{2\alpha}{R_0} - \frac{43}{30}\right) w_0^2 = 0. $$

Thus, nontrivial solution with $w_0 \neq 0$ is possible only if

$$ R_0 = \frac{60}{43}\alpha $$

However, this constraint contradicts with the already obtained result (7).

## 4. Solution with Effective Mass Factors

Both the buoyancy force and the entrainment perfectly balances with the counterbalancing force arising from the pressure due to the top-hat approximation to the system. This point is more explicitly seen by retaining only the contributions from those two terms
in the right hand side of Eq. (5), and consider the resulting pressure solution, $p_b$ (i.e., buoyancy pressure). Thus, the problem reduces to:

$$ \left(\frac{1}{r}\frac{\partial}{\partial r}r\frac{\partial}{\partial r} + \frac{\partial^2}{\partial z^2}\right)p_b = \frac{\partial}{\partial z}\left(b - \frac{2\alpha w^2}{R}\right). \qquad (12) $$

Here, there is no dependence on $r$ in the right hand side, thus, $r$ derivatives also drops off from the left hand side. After integrating once in the vertical direction, we obtain

$$ \frac{\partial p_b}{\partial z} = b - \frac{2\alpha w^2}{R}. $$

That is exactly the perfect cancelation by pressure already obtained.

Note that a balance between the left hand side and the first term in the right hand side constitutes a hydrostatic balance. It also easily transpires that this balance is a direct consequence of horizontal homogeneity of the right hand side in Eq. (12). In reality, such a perfect horizontal homogeneity does not exist, thus the pressure, $p_b$, also presents a weak radial dependence. As a result, the perfect cancelation no longer realizes, but a weak imbalance remains.

A more formal manner to address this issue is to assume a certain radial dependence in variables inside the plume, and to try to obtain a revised solution for the pressure. Unfortunately, it is found that a similarity solution is no longer available under this modification. Some qualitative discussions from this perspective will be presented in the next section.

As for now, for making an analytical progress, rather artificially, we introduce factors, $\gamma_1$ and $\gamma_2$, representing relative effective masses to the formal solution (8b), thus

$$ \left(\frac{\partial p}{\partial z}\right)_0 = (1 - \gamma_1)b_0 - 2(1 - \gamma_2)\frac{\alpha w_0^2}{R_0} + \frac{w_0^2}{10}. \qquad (13) $$

Here, two separate factors, $\gamma_1$ and $\gamma_2$, are introduced so that contributions of the buoyancy force and the entrainment can easily be identified in the final result. Note that as a result, Eq. (13) is no longer a formal solution to Eq. (5), but as going to be suggested in the next section, this modification crudely incorporates an expected effect by including a horizontal inhomogeneity to the pressure problem. When these forcing terms are close to a horizontal homogeneity, $\gamma_1$ and $\gamma_2$ are close to zero. On the other hand, when they have sharp horizontal structures, $\gamma_1 \simeq \gamma_2 \simeq 1$, and the pressure does not counteract on those forces.

By substituting Eq. (13) into Eq. (1c), along with the other solutions, we obtain

$$ w_0 = \left[ \frac{12H}{5\alpha^2} \left( \frac{\gamma_1}{\gamma_2 - 7/50} \right) \right]^{1/3}. \qquad (14) $$
This solution may be compared with the original similarity solution obtained by Morton et al. (1956):

$$w_0 = \frac{5}{6} \left( \frac{9H}{10\alpha^2} \right)^{1/3}. \qquad (15)$$

Note that for the solution (14) to be physical, we require $\gamma_2 > 7/50$. This is considered the minimum degree of horizontal inhomogeneity required for the vertical velocity (within the plume). When we rather arbitrary set $\gamma_1 \simeq \gamma_2 \simeq 1/3$, we obtain $\gamma_1/(\gamma_2 - 7/50) \simeq 5/3$, and $w_0 \simeq (4H/\alpha^2)^{1/3}$. Furthermore, Fig. 1 plots $(\alpha^2/H)^{1/3}w_0$ as defined by Eqs. (14: solid) and (15: long dash) as functions of $\gamma = \gamma_1 = \gamma_2$. We fine that, rather intuitively, $w_0$ (Eq. 14) with the pressure effect is always larger than the original solution (Eq. 15) by Morton et al.

## 5. Simple Analysis on Horizontal Inhomogeneity

A full analysis of a solution with explicit horizontal dependence in entraining plume would be beyond the scope of the present paper, especially considering the fact that a similarity solution is no longer available. Nevertheless, a contribution of the horizontal dependence to the problem can be qualitatively estimated in the following manner, by following Bluestein (2013), Morrison (2016).

Let us consider only a contribution on buoyancy, $b$, on the pressure problem in Eq. (5). We also neglect the curvature effects in derivatives for simplicity. Thus, a problem to consider is:

$$(\frac{\partial^2}{\partial r^2} + \frac{\partial^2}{\partial z^2})p = \frac{\partial b}{\partial z}. \qquad (16)$$

We let the buoyancy to be

$$b = b_0 \cos \frac{\pi}{L} r \sin \frac{\pi}{D} z. \qquad (17)$$

Here, $L$ and $D$ are horizontal and vertical scales of buoyancy distribution. Note that an exact problem can be solved by using the 0-th order Bessel function in the radial direction. In this respect, the assumed form, $\cos \pi r/L$, may be considered an approximation to a Bessel function. This approximation provides the same conclusion as obtained by using an exact solution based on a Bessel function, but in a more straightforward manner (cf., Morrison 2016).

By substituting Eq. (17) into Eq. (16), we find that the resulting pressure-gradient force takes the form

$$\frac{\partial p}{\partial z} = (1 - \gamma)b \qquad (18)$$
with the effective mass factor, $\gamma$, defined by

$$ \gamma = 1 - \left[ 1 + \left( \frac{D}{L} \right)^2 \right]^{-1}, \qquad (19) $$

as shown by e.g., Jeevanjee and Romps (2016).

In general, with tall convection, $D/L \gg 1$, and $\gamma \simeq 1$, thus in this case, the pressure force has almost no counterbalance against the buoyancy. The laboratory experiment by Morton et al. (1956) may be considered to satisfy this scaling, thus their omission of the pressure term may also be justified.

However, it is important to note that the ratio $D/L$, is more precisely, that of local curvatures of the buoyancy distribution in vertical and horizontal directions. Especially, within a convective core, we expect that the buoyancy distribution is much homogeneous horizontally with a fairly large variation in the vertical direction. Thus, in this case, $D/L \ll 1$, $\gamma \simeq 0$, and the pressure force almost balances with the buoyancy.

Note that if we naively take typically adopted scales for deep convection, i.e., $D \sim 10$ km and $L \sim 1$ km, then $(D/L)^2 \simeq 10^2$, thus $\gamma \simeq 1$. However, this estimate could be very misleading for the reason just stated. On the other hand, as already remarked in the last section, we require $\gamma > 7/50$ to obtain a physically consistent solution. Thus there must be an upper bound for a possible homogeneity even within a convective core.

## 6. Implications to the Convection Parameterization Problem

The pressure problem of the entraining plume has been considered herein. Somehow, this problem has been left re-addressed for more than a half century since the problem was originally examined by Morton et al. (1956). However, the significance of this problem can hardly be overemphasized, because the mass-flux formulation currently adopted for convection parameterization in majority of operational models, both for weather and climate predictions, is built upon the entraining-plume model originally formulated by Morton et al. (1956). No doubt, there are various elaborations upon this original formulation. Inclusion of detrainment would probably be the most notable one. See Plant and Yano (2015) for full discussions.

Nevertheless, by inheriting this historical problem, today's convection parameterization still neglects a pressure contribution in evaluating the convective vertical velocity given in the form of Eq. (1c). It is widely known that neglect of the pressure contribution tends to over accelerate the convective velocity by buoyancy. A standard remedy, even today, is
to introduce an effective mass to curb this unfavourable tendency, as originally attempted
by Simpson *et al.* (1965): see de Rood *et al.* (2012) for a review of the current status.

The present study, effectively, provides a basic receipt for explicitly evaluating a pres-
sure contribution to the convective vertical velocity in mass-flux parameterizations. To
demonstrate this receipt, the entraining plume is taken as a prototype for a more general
problem in operational convection parameterizations. This prototype problem has already
revealed some basic issues in applying this receipt in operation. Most seriously, the top-
hat approximation adopted in mass flux formulation, if it is literally taken in solving the
pressure problem, leads to a prefect cancelation of both buoyancy force and the entrain-
ment effect by the pressure-gradient force. An only remaining force acting on the vertical
velocity stems from a dynamic pressure, which acts as a drag force.

It transpires that for solving the convective-pressure problem properly, the neglected
horizontal structure within convection must explicitly be included in the formulation. An
obvious remedy would be to introduce a prescribed radial dependence inside a plume.
This procedure would be relatively straightforward apart from some technical issues to
be taken care of, but not to be discussed herein. However, the obtained result would
sensitively depend on this arbitrary-prescribed radial structure.

A more straightforward way of counting this effect is to introduce a factor for effective
mass both in buoyancy and entrainment terms. This factor can qualitatively be estimated
by a simplified pressure problem as considered in Sec. 5. According to this analysis, when
a simple scaling for tall convection is adopted, the effective mass factor would be almost
unity, and the counter gradient effect of pressure on buoyancy and entrainment can be
neglected. On the other hand, within a horizontally homogeneous convective core, the
pressure is expected to perfectly cancel the buoyancy acceleration so that a quasi-steady
state may be maintained. A general case would be a middle of those two limits.

In the case of the entraining plume considered herein, a remaining contribution of the
dynamic pressure is relatively weak, and almost identical result as in Morton *et al.* (1956) is
recovered, when the effective mass factor is simply taken unity. However, this result is most
likely fortunate with a particular form of a similarity solution assumed. A similar analysis
as for the buoyancy pressure in Sec. 5 can easily show that the drag effect by the dynamic
pressure may be more substantial with a form of w²/D with a constant D measured by a
vertical scale of convection. Thus, effectively, the pressure drag doubles the entrainment
term in the momentum equation. However, keep in mind that this effect only takes a form
of entrainment in a virtual sense, and should not be confused with a physical entrainment
process. Especially, this process does not contribute to the plume-mass continuity (Eq. 1b).

It may be worthwhile to note that even a simple inclusion of a drag effect arising
from the dynamic pressure can substantially curb an excessive acceleration tendency of
convective vertical velocity by buoyancy in operational models. This could be a vital
option to be seriously considered. By addressing a prototype problem, the present paper
suggests various options for including the pressure effect in the convective-velocity problem
in operational models. These options are to be examined in forthcoming studies.

References

Bluestein, H. B., 2013: *Severe Convective Storms and Tornados*. Springer Berlin Heidelberg, 456pp.

de Roode, S. R., A. P. Siebesma, H. J. J. Jonker, and Y. de Voogd, 2012: Parameterization of the vertical velocity equation for shallow cumulus clouds. *Mon. Wea. Rev.*, **140**, 2424–2436.

Jeevanjee, N., and D. M. Romps, 2016: Effective buoyancy at the surface and aloft. *Quator. J. Roy. Meteor. Soc.*, **142**, 811–820.

Morton, B.R., G.I. Taylor, and J.S. Turner, 1956: Turbulent gravitational convection from
maintained and instantaneous sources. *Proc. Roy. Soc. London, A234*, 1–23.

Morrison, H., 2016: Impact of updraft size and dimensionality on the perturbation pressure and vertical velocity in cumulus convection. Part I: Simple, generalized analytic solutions. *J. Atmos. Sci.*, **73**, 1441–1454.

Plant, R. S., and J.-I. Yano (Eds), 2015: *Parameterization of Atmospheric Convection*, Volume I, World Scientific, Singapore, Imperial College Press, London, 515pp.
http://www.worldscientific.com/worldscibooks/10.1142/p1005

Simpson, J., R. H. Simpson, D. A. Andrews, and M. A. Eaton, 1965: Experimental cumulus dynamics. *Rev. Geophysics*, **3**, 387–431.

Turner, J. S., 1973: *Buoyancy Effects in Fluids*, Cambridge University Press.

Yano, J.-I., 2014: Basic Convective Element: Bubble or Plume?: A Historical Review.
*Atmos. Phys. Chem.*, **14**, 7019–7030. doi:10.5194/acp-14-7019-2014
<figure><img src="image_2.png" /><figcaption>Fig. 1: Plot of $(\alpha^2/H)^{1/3}w_0$ as defined by Eqs. (14: solid) and (15: long dash) as functions of $\gamma = \gamma_1 = \gamma_2$.</figcaption></figure>