overshot is hot quotation

And the book has some very neat tricks in it, most of which can be done on an 8-shaft loom. How to put a monk"s belt border on all four sides of a piece without losing your mind. How to float overshot motifs on a plain or twill background. The piece with motifs having different pattern weft colors in the same row by using one motif"s pattern weft as the other"s tabby was very cool. My personal favorite and the one that really excited me was the discussion of blended drafts. Basically it walks you through how to take any two drafts that have the same total thread count but different threadings, and combine them into a single new threading that will weave both. Any two 4-shaft overshot drafts can be combined on 8 shafts max, so I see a lot of potential for this. You can also combine different structures, say to put an overshot border on a textured towel.

In Overshot is Hot! Weaver’s magazine Editor Madelyn van der Hoogt presents the many articles and projects featuring overshot that appeared in Weaver’s and Prairie Wool Companion over a period of almost 20 years. During that time, overshot was transformed from a weave structure—used mostly for coverlets or placemat borders—into a contemporary, multicolor, multiuse pattern weave.

Innovative designers explore and apply variations and extensions of overshot drafting techniques and share their results on these pages. This book is your chance to have it all under one cover.

Our books are formatted in rich Portable Document Format (PDF)—which includes bookmarking, internal page links, and high-res images & charts so you can print pages if you wish. For best results, we recommend using Acrobat Reader (a free application from Adobe™) as a desktop application for viewing and printing. Acrobat Reader 6 or higher is required to take advantage of all the features listed above. This PDF can be transferred to your tablet with all of the same functionality when you use one of many apps (like iBooks in iOS).

With more than 30 of the best overshot projects compiled from 20 years of Weaver"s magazine and Prairie Wool Companion, this weaver’s reference combines step-by-step instructions with weaving theory. Such projects as heirloom linens, traditional coverlets, and colorful modern applications for scarves, table toppers, and wall hangings are presented, covering everything a crafter needs to know to design fabulous fabrics in overshot and its cousin star-and-diamond weave. Both beginners and advanced weavers working on at least four-shaft looms will delight in the projects provided.

The origin of the technique itself may have started in Persia and spread to other parts of the world, according to the author, Hans E. Wulff, of The Traditional Crafts of Persia. However, it is all relatively obscured by history. In The Key to Weavingby Mary E. Black, she mentioned that one weaver, who was unable to find a legitimate definition of the technique thought that the name “overshot” was a derivative of the idea that “the last thread of one pattern block overshoots the first thread of the next pattern block.” I personally think it is because the pattern weft overshoots the ground warp and weft webbing.

Overshot gained popularity and a place in history during the turn of the 19th century in North America for coverlets. Coverlets are woven bedcovers, often placed as the topmost covering on the bed. A quote that I feel strengthens the craftsmanship and labor that goes into weaving an overshot coverlet is from The National Museum of the American Coverlet:

Though, popular in many states during the early to mid 19th centuries, the extensive development of overshot weaving as a form of design and expression was fostered in rural southern Appalachia. It remained a staple of hand-weavers in the region until the early 20th century. In New England, around 1875, the invention of the Jacquard loom, the success of chemical dyes and the evolution of creating milled yarns, changed the look of coverlets entirely. The designs woven in New England textile mills were predominantly pictorial and curvilinear. So, while the weavers of New England set down their shuttles in favor of complex imagery in their textiles, the weavers of Southern Appalachia continued to weave for at least another hundred years using single strand, hand spun, irregular wool yarn that was dyed with vegetable matter, by choice.

And, due to the nature of design, overshot can be woven on simpler four harness looms. This was a means for many weavers to explore this technique who may not have the financial means to a more complicated loom. With this type of patterning a blanket could be woven in narrower strips and then hand sewn together to cover larger beds. This allowed weavers to create complex patterns that spanned the entirety of the bed.

What makes overshot so incredibly interesting that it was fundamentally a development of American weavers looking to express themselves. Many of the traditional patterns have mysterious names such as “Maltese Cross”, “Liley of the West”, “Blooming Leaf of Mexico” and “Lee’s Surrender”. Although the names are curious, the patterns that were developed from the variations of four simple blocks are incredibly intricate and luxurious.

This is only the tip of the iceberg with regard to the history of this woven structure. If you are interested in learning more about the culture and meaning of overshot, check out these resources!

The National Museum of the American Coverlet- a museum located in Bedford, Pennsylvania that has an extensive collection of traditional and jacquard overshot coverlets. Great information online and they have a “Coverlet College” which is a weekend series of lectures to learn everything about the American coverlet. Check out their website - coverletmuseum.org

Textile Art of Southern Appalachia: The Quiet Work of Women – This was an exhibit that traveled from Lowell, Massachusetts, Morehead, Kentucky, Knoxville, Tennessee, Raleigh, North Carolina, and ended at the Royal Museum in Edinburgh, Scotland. The exhibit contained a large number of overshot coverlets and the personal histories of those who wove them. I learned of this exhibit through an article written by Kathryn Liebowitz for the 2001, June/July edition of the magazine “Art New England”. The book that accompanied the exhibit, written by Kathleen Curtis Wilson, contains some of the rich history of these weavers and the cloth they created. I have not personally read the book, but it is now on the top of my wish list, so when I do, you will be the first to know about it! The book is called Textile Art of Southern Appalachia: The Quiet Work of Women and I look forward to reading it.

Weave structures often have specific threading and treadling patterns that are unique to that particular weave structure and not shared with others. This book takes you out of the traditional method of weaving overshot patterns by using different treadling techniques. This will include weaving overshot patterns as Summer/Winter, Italian manner, starburst, crackle, and petit point just to name a few. The basic image is maintained in each example but the design takes on a whole new look!

stitchips Overshot is Hot! is an electronic book delivered in a compact USB drive. Compatible with both Mac & Windows, stitchips are a great way to expand your craft library.

In Overshot is Hot! Weaver"s magazine Editor Madelyn van der Hoogt presents the many articles and projects featuring overshot that appeared Prairie Wool Companion over a period of almost 20 years. During that time, overshot was transformed from a weave structure—used mostly for coverlets or placemat borders—into a contemporary, multi-color, multi-use pattern weave. Innovative designers explore and apply variations and extensions of overshot drafting techniques and share their results on these pages. This book is your chance to have it all under one cover.

The digital content onstitchipsare Rich PDFs that are optimized to work with both Mac and Windows based products, providing you with high-res text and images that will print with crispness. Charts and images will easily enlarge for your comfort. Rich PDFs provide internal and external linking. Click on a link in the Table of Contents and you’ll be taken right to that section. Links to websites are also provided where appropriate. Bookmarks and search functions will help you navigate your stitchips quickly and easily.

For best results, we recommend usingAcrobat Reader(a free application from Adobe™) as a desktop application for viewing and printing. Acrobat Reader 6 or higher is required to take advantage of all the features listed above. This PDF can be transferred to your tablet with all of the same functionality when you use one of many apps (like iBooks in the iOS).

"Fool in the shower" is a metaphor attributed to Nobel laureate Milton Friedman, who likened a central bank that acted too forcefully to a fool in the shower. The notion is that changes or policies designed to alter the course of the economy should be done slowly, rather than all at once. This phrase describes a scenario where a central bank, such as the Federal Reserve, acts to stimulate or slow down an economy.

The expression is best summed up as the scenario when central banks or governments overreact to swings in the economic cycle and loosen monetary and fiscal policies too far and too fast, without waiting to gauge the impact of their initial actions. When the fool realizes that the water is too cold, they turn on the hot water. However, the hot water takes a while to arrive, so the fool simply turns the hot water up all the way, eventually scalding themself.

In the same way that it takes time for hot and cold water to work their way through home plumbing to the showerhead, so it takes time for monetary policy changes to work their way.

Friedman and other Monetarists have maintained that accounting for these lags between monetary policy and economic outcomes is an important part of wise monetary policymaking.

Any change made to stimulate a broad economy, especially one as large as the U.S., takes time to work its way through. In economic terms, Friedman described this by saying that there are long and variable lags between changes in monetary policy and changes in the economy.

The time between when a change in monetary policy is executed and changes in economic performance can be observed can be months or years, and the interval is not constant but can and does change over time. A move like lowering the fed funds rate can take anywhere from six months to two years to fully integrate into the economy and trickle down to changes in lending, investment, real output, and ultimately consumer prices.

These gaps happen because outside of idealized economic models, money is not neutral to the economy, and changes in the supply of money do not enter the economy uniformly distributed but at specific points and into the hands of specific market participants.

The point of where the new money enters the economy and the exact process by which it spreads through the economy is not fixed, but contingent on the specifics of monetary policy: who receives the new money first and in successive transactions, and general market conditions throughout the period of time that it takes to work through the economy.

For monetary policymakers, this poses a special problem if they are interested in achieving their publicly stated goals of stabilizing economic metrics such as unemployment and consumer inflation. They cannot observe the effects of any given change in monetary policy until some indefinite point in the future, and can’t be sure how long that will be.

Combined with the pressure to act to fix immediate problems in financial markets, this can lead a monetary policymaker to “overcorrect” monetary policy and create long term problems in reaction to short term demands. In light of this, many economists are often cautious about overreaching and prefer small consistent steps to enact change.

Friedman created the metaphor of the "fool in the shower" who is constantly tinkering with the hot and cold controls because they do not realize that there is a lag between the time they order up a temperature change and when such a change occurs.

Applied to the economy, the metaphor suggests that policymakers are prone to overshooting their target and making things worse rather than better. However, Freidman believed, as have most other Monetarists, such as Fed Chairmen Alan Greenspan and Ben Bernanke, that these lags can be approximated and accounted for by wise policymakers by making incremental changes in policy and tracking market conditions to model their effects.

However, given some of the extreme economic events, and monetary policy reaction to them, over the past few decades this may be more of a challenge than some believe. In an economy prone to financial crises, constant evolution of technology and economic relations, and subject to radical new nonstandard monetary policies, perhaps the impact of a fool in the shower will always be a lingering element to markets dominated as they are by central banks.

This comprehensive video workshop begins with an introduction to your instructors and personal remarks by Navajo weaver, Ann Walker Maloney. You"ll learn about looms, tools, yarn and equipment. They you will learn how to spin and ply edging cord (just in case you want to make your own), prepare the warp, apply the edging cord, secure the warp to the loom and make the heddles. As the weaving instruction begins, you will learn how to use the loom, create the sheds, and use the tools. The section on Planning a Rug Design will acquaint you with things you should think about before planning your own rug. This section includes specific information on weaving a design that reverses at the centre of the rug. As you begin weaving, you will learn the secret of layering in the first row, and how to weave from the top down. You will learn how to weave stripes, vertical lines (by interlocking wefts on the same row), squares or blocks, diagonals and diamonds. Once you learn these techniques you"ll be ready to plan and weave a rug design of your choice. You will learn many tips and solutions to potential problems throughout the video. And, Angie"s method of finishing a rug the Navajo way is magic! All this plus stories, traditions and rugs that will inspire.

Vulcan bomber at Wellesbourne airfield has just overshot the runway during a hot taxi run. Almost made it to the road. A Vulcan bomber has run off a runway at an airfield in Warwickshire, narrowly...

One of my favorite parts of working on my Ancient Rose Scarf for the March/April 2019 issue of Handwoven was taking the time to research overshot and how it fits into the history of American weaving. As a former historian, I enjoyed diving into old classics by Lou Tate, Eliza Calvert Hall, and Mary Meigs Atwater, as well as one of my new favorite books, _Ozark Coverlets, by Marty Benson and Laura Lyon Redford Here’s what I wrote in the issue about my design:_

“The Earliest weaving appears to have been limited to the capacity of the simple four-harness loom. Several weaves are possible on this loom, but the one that admits of the widest variations is the so-called ‘four harness overshot weave,’—and this is the foremost of the colonial weaves.” So wrote Mary Meigs Atwater in The Shuttle-Craft Book of American Hand-Weaving when speaking of the American coverlet and the draft s most loved by those early weavers.

Overshot, in my mind, is the most North American of yarn structures. Yes, I know that overshot is woven beyond the borders of North America, but American and Canadian weavers of old took this structure and ran with it. The coverlets woven by weavers north and south provided those individuals with a creative outlet. Coverlets needed to be functional and, ideally, look nice. With (usually) just four shaft s at their disposal, weavers gravitated toward overshot with its stars, roses, and other eye-catching patterns. Using drafts brought to North America from Scotland and Scandinavia, these early weavers devised nearly endless variations and drafts, giving them delightful names and ultimately making them their own.

When I first began designing my overshot scarf, I used the yarn color for inspiration and searched for a draft reminiscent of poppies. I found just what I was looking for in the Ancient Rose pattern in A Handweaver’s Pattern Book. When I look at the pattern, I see poppies; when Marguerite Porter Davison and other weavers looked at it, they saw roses. I found out later that the circular patterns—what looked so much to me like flowers—are also known as chariot wheels.

In A Book of Hand-Woven Coverlets, author Eliza Calvert Hall quotes the great William Shakespeare himself when writing of coverlet names: “What’s in a name? That which we call a rose by any other word would smell as sweet.” I, too, can’t help thinking of this quote as I survey my finished scarf. Would I love it as much if I saw chariot wheels instead of poppies and roses? Perhaps, but given the choice, I prefer the blossoms.

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Results. A smooth heat conduction profile (either fixed or through Kramers opacity law) leads to a relatively shallow power law with dos ∝ ℱn0.08 for low ℱn. A fixed step-profile of the heat conductivity at the bottom of the convection zone leads to a somewhat steeper dependency on dos ∝ ℱn0.12 in the same regime. Experiments with and without subgrid-scale entropy diffusion revealed a strong dependence on the effective Prandtl number, which is likely to explain the steep power laws as a function of ℱn reported in the literature. Furthermore, changing the heat conductivity artificially in the radiative and overshoot layers to speed up thermal saturation is shown to lead to a substantial underestimation of the overshooting depth.

Conclusions. Extrapolating from the results obtained with smooth heat conductivity profiles, which are the most realistic set-up we considered, suggest that the overshooting depth for the solar energy flux is about 20% of the pressure scale height at the base of the convection zone. This is two to four times higher than the estimates from helioseismology. However, the current simulations do not include rotation or magnetic fields, which are known to reduce convective overshooting.

Convective mixing in stars has important consequences, for example, in early and late phases of stellar evolution and for the diffusion of light elements. Furthermore, stellar differential rotation (e.g. Rüdiger 1989) and dynamos (e.g. Moffatt 1978; Krause & Rädler 1980; Brandenburg & Subramanian 2005) owe their existence to turbulent fluid motions. The efficiency of mixing in convective and radiative layers in stars differs greatly. The former are vigorously mixed on a timescale much shorter than the evolutionary timescale of the star. Thus it is of great interest to be able to predict where effective convective mixing occurs. The greatest uncertainty in this respect is the amount of overshooting from convection zones (CZ) to adjacent radiative layers.

Stellar structure and evolution models most often apply some variant of the mixing length (ML) model of Vitense (1953) to describe convection. These models are completely local and do not allow overshooting. Non-local extensions to ML models (e.g. Shaviv & Salpeter 1973; Schmitt et al. 1984; Skaley & Stix 1991) yield estimates of overshooting, but the validity of the ML approach has been questioned (e.g. Renzini 1987). More advanced closures of convection based on Reynolds stress (e.g. Xiong 1985; Deng et al. 2006; Garaud et al. 2010; Canuto 2011) are physically more consistent but challenging to implement (see, however, Zhang et al. 2012; Zhang 2013). Furthermore, testing and validation of the Reynolds stress models, for example, by comparison to three-dimensional numerical simulations, is still in its infancy (e.g. Kupka 1999; Snellman et al. 2015; Cai 2018).

A seemingly attractive option to study overshooting is to solve the governing equations directly by means of three-dimensional simulations. Numerical simulations of convection have been used to estimate the overshooting depth in numerous studies (e.g. Hurlburt et al. 1986, 1994; Roxburgh & Simmons 1993; Singh et al. 1995, 1998; Saikia et al. 2000; Brummell et al. 2002; Ziegler & Rüdiger 2003; Rogers et al. 2006; Tian et al. 2009; Pratt et al. 2017; Brun et al. 2017; Hotta 2017; Korre et al. 2019). Early studies indicated overshooting of the order of a pressure scale height at the base of the CZ, which is an order of magnitude more than typical estimates from helioseismology (e.g. Basu 1997). The difference between these studies and the Sun is that the energy flux imposed in the simulations is typically much greater than the corresponding solar flux. This leads to higher convective velocities in the simulations and to an overestimation of the overshooting. Scaling laws, based on the relation of convective velocities with the energy flux, arise in analytic models of overshooting (e.g. Zahn 1991; Rempel 2004) and predict a reduction of the overshooting depth as a function of decreasing flux. The primary aim of the current study is to establish these relations from carefully controlled numerical experiments.

Here these studies are revisited by a set-up where the heat conductivity is self-consistently computed using the Kramers opacity law. This set-up allows the depth of the CZ to dynamically adapt to changes in the thermodynamic state of the system (Käpylä et al. 2019a) and to produce a smooth transition between convective and radiative layers (Brandenburg et al. 2000; Käpylä et al. 2017). Furthermore, a significantly broader range of imposed flux values is covered than in any of the previous studies. Moreover, particular care is taken to isolate the effect of the input flux by performing models where the supercriticality of convection, degree of turbulence, and effective thermal Prandlt number are approximately constant as the flux varies. An effort is made to link to the earlier studies of Singh et al. (1998) and Tian et al. (2009) by targeted sets of simulations probing the influence of subgrid scale entropy diffusion on convection and the resulting overshooting depth. Finally, a critical assessment of some of the modeling choices of Hotta (2017) is presented.

An optically thick fully ionised gas is considered, where radiation is modelled through diffusion approximation. The ideal gas equation of state p = (cP − cV)ρT = ℛρT applies, where ℛ is the gas constant, and cV is the specific heat at constant volume. The radiative flux is given by

where K is the radiative heat conductivity. Two qualitatively different heat conductivity prescriptions are considered, where K either has a fixed profile K(z) or is a function of density and temperature, K(ρ, T). In the latter case, K is given by

Here a = 1 and b = −7/2 are used, which correspond to the Kramers opacity law for free-free and bound-free transitions (Weiss et al. 2004). Heat conductivity consistent with the Kramers law was first used in convection simulations by Brandenburg et al. (2000).

Owing to the strong depth dependence of the radiative diffusivity, χ = K/(cPρ), additional turbulent SGS diffusivity is used in the entropy equation to keep the simulations numerically feasible. Here the SGS flux is formulated as

where Γ0 is a cooling luminosity, T = e/cV is the temperature where e is the internal energy, and where Tcool = Ttop is a reference temperature corresponding to the fixed value at the top boundary.

The computational domain is a rectangular box where zbot ≤ z ≤ ztop is the vertical coordinate, where zbot/d = −0.45, ztop/d = 1.05, and where d is the depth of the initially isentropic layer (see below). In a few runs the domain extends to deeper layers such that zbot/d = −0.75 to accommodate deeper overshooting. The horizontal coordinates x and y run from −2d to 2d. The horizontal size of the box is thus LH/d = 4, and the vertical extent Lz/d is either 1.5 or 1.8.

where Fbot is the fixed input flux and Kbot(x, y, zbot) is the value of the heat conductivity at the bottom of the domain. On the upper boundary a constant temperature T = Ttop, coinciding with the initial value, is assumed.

where ρ0 is the initial value of the density at z = ztop. The models with Kramers heat conductivity are fully defined by choosing the value of the kinematic viscosity ν, the gravitational acceleration g, the values of a, b, K0, ρ0, T0 and the SGS Prandtl number

In addition to the explicit viscosity, SGS, and radiative diffusion, the advective terms in each of the Eqs. (2)–(4) are written in terms of a fifth-order upwinding derivative with a hyperdiffusive sixth-order correction where the diffusion coefficient depends locally on the flow, see Appendix B of Dobler et al. (2006).

Supercriticality of convection is roughly determined by min(RaRad, RaSGS). Both quantities vary as functions of height and are quoted near the surface at z/d = 0.85 for all models. Conventionally, the Rayleigh number in the hydrostatic non-convecting state is one of the control parameters. This is still true for the cases with fixed K profile in the current study, but in the runs with Kramers conductivity, the convectively unstable layer in the hydrostatic case is very thin and confined to the near-surface layers (Brandenburg 2016). Thus the Rayleigh numbers are quoted from the thermally saturated and statistically stationary states.

Here the primes denote fluctuations and overbars horizontal averages. The total convected flux (Cattaneo et al. 1991) is the sum of the enthalpy and kinetic energy fluxes:

where cP∇ad = cP(1 − 1/γ)=ℛ, and g = |g|. Above, ∇ is the actual temperature gradient realised in the system. Now it is possible to define the total flux and the flux transported by adiabatic stratification as

where ∇rad is a hypothetical radiative gradient in the absence of convection and Ftot = Fbot. The ratio of the total-to-adiabatic flux is the Nusselt number (e.g. Brandenburg 2016),

where ⟨⋯⟩t denotes a time average over the statistically stationary part of the time series. Error estimates for dos are obtained by dividing the time series into three equally long parts and considering their largest deviation from the time average over the whole time series as the error.

Typical flow patterns for Runs R4 and R7 are shown in Fig. 1. The flow structure observed in various studies of stratified non-rotating convection is recovered with connected downflows near the surface merging into isolated plumes at larger depths (e.g. Stein & Nordlund 1989, 1998). The downflows are surrounded by broader upflows. In the majority of the current simulations the flows are at best mildly turbulent with Re ≈ 20…40 such as in Run R4 in the left panel of Fig. 1. However, the qualitative large-scale structure of convection does not change at higher resolutions and Rayleigh and Reynolds numbers, see the right panel of Fig. 1 for Run R7.

Results regarding the energy fluxes and force balance from representative runs from Sets K, S, and DS are shown in Fig. 5. The left panels show the contributions to the energy flux and the superadiabatic temperature gradient ∇ − ∇ad. The fluxes for Runs K4 and S4 in Fig. 5a and c are qualitatively similar to those of Runs K and S of Käpylä et al. (2017), respectively. The main difference to the latter is the treatment of the near surface layers; cooling layer in the present runs as opposed to an imposed entropy gradient in Käpylä et al. (2017), and the somewhat different values of ℱn; 1.8 × 10−5 here versus 9 × 10−6 in Käpylä et al. (2017). The subadiabatic Deardorff zone encompasses roughly a quarter of the MZ in Run K4, whereas in Run S4, the DZ is almost absent. The runs in Set DS differ from those in Set S in that convection transports almost all of the energy due to the lower K2/K1 ratio. Assuming that the temperature gradient in the final statistically saturated state is nearly adiabatic, the fraction of convective transport can be estimated from (cf. Brandenburg et al. 2005)

Panels a, c, and e: total (black dash-dotted lines), convective (black), enthalpy (blue), kinetic energy (light purple), radiative (dark purple), and cooling (green) fluxes and the superadiabatic temperature gradient (red) from Runs K4, S4, and DS5, respectively. Panels b, d, and f: total averaged vertical forces (solid lines) and the power of the forces (dashed) on the upflows (red) and downflows (blue) from the same runs. The dotted lines in these panels show the corresponding viscous force and its power. The shaded areas indicate the BZ (darkest), DZ, and OZ (lightest) and the thick black line at the horizontal axis denotes the MZ.

Reaching the solar value of ℱn is currently not possible due to the prohibitive time-step constraint and the long thermal adjustment time involved (e.g. Brandenburg et al. 2000; Kupka & Muthsam 2017). However, it is reasonable to assume that the overshooting depth scales with a power law as a function of ℱn (e.g. Schmitt et al. 1984; Zahn 1991). Thus it is in principle possible to estimate the extent of overshooting in the Sun provided that a sufficiently broad range of higher flux values are probed and their results are extrapolated to the solar case. A few such studies can be found in the literature (e.g. Singh et al. 1998; Tian et al. 2009; Hotta 2017).

One of the most restrictive modelling choices in the past has been the use of a static heat conduction profile that effectively enforces the layer structure of the simulation. This can be seen from Fig. 6a where the vertical coordinates of the bottoms of the convection (zCZ) and overshoot (zOZ) zones are shown as functions of ℱn. The results for zCZ from Sets S and Sh show that the interface between the CZ and OZ stays at the initial position at z = 0 for all values of ℱn. The runs in Sets DS, DSS1, and DSS2 behave similarly, although the bottom of the CZ is shifted downward from its initial position. For Sets K and Kh the depth of the CZ is generally reduced in comparison to Sets S and Sh. In these runs the depth of the CZ increases as ℱn decreases. This is contrasted by the results from Sets P where a Kramers-like, but static, profile of K is used (see Fig. 2): here zCZ is practically fixed in the current range of ℱn.

Panel a: vertical (z) coordinates of the bottom of the CZ (zCZ, solid lines) and OZ (zOZ, dashed). The dotted red line indicates the bottom of the domain. Panel b: overshooting depth dos normalised by the pressure scale height Hp as a function of ℱn for Sets K (black), Kh (grey), P (blue), S (red), Sh (green), DS (purple), DSS1 (cyan), and DSS2 (orange). The purple diamond denotes Run DS5h. The dotted lines show approximate power laws from fits to simulation data; see Table 5. Panel c: comparison of Sets DS, DSS1, and DSS2 with the studies of Singh et al. (1998) (red) and Tian et al. (2009) (blue).

The drastic change from a steep power law in Set DS to the shallower power laws in Sets DSS1 and DSS2 is related to the difference in the diffusion of temperature fluctuations. Figure 7a shows that the effective Péclet numbers in OZ in all of these sets are comparable. However, as is shown in Sect. 4.3, the overshooting depth is insensitive to the Péclet number in this parameter range provided that the effective Prandtl number is not varied at the same time. The only difference between Sets DS and DSS1 and DSS2 is that in Set DS, the SGS entropy diffusion is omitted and thus the effective Rayleigh and Prandtl numbers vary as a function of ℱn, whereas in the latter two, they approach constant values for low ℱn, see Figs. 3 and 7b. Changing the effective Prandtl number leads to a dramatic change in the way convection transports energy in that the sound speed (temperature) fluctuations are enhanced over the velocity fluctuations with increasing Prandtl number, see Fig. 8a. This means that in Set DS the temperature fluctuations become increasingly more important in the enthalpy transport as ℱn (Pr) decreases (increases). This is immediately reflected in the overshooting depth as a smaller velocity fluctuation is required to carry the same flux. Figure 8b shows that in Set DSS1 the ratio of the temperature and velocity fluctuations remains practically constant as a function of ℱn. The break in the power law in Sets K and Kh can be understood similarly by the decreasing effective Prandtl number as ℱn increases.

Furthermore, the studies of Singh et al. (1998), Tian et al. (2009), and Hotta (2017) all considered the bottom of the CZ to be fixed and given by the initial non-convecting state. Although all of these studies used a fixed profile for the heat conductivity, which fixes zCZ, it does not necessarily stay at the same position as in the initial state; compare, for example, the solid blue and red and green curves in Fig. 6a. This is particularly clear for Sets DS, DSS1, and DSS2 which are similar to the set-up of Singh et al. (1998). However, it is hard to assess whether such a systematic error is present in the results of Singh et al. (1998). In the simulations of Hotta (2017) a similar issue is also possible but it appears that this effect may be small (e.g. his Fig. 8).

In an earlier study, Hotta (2017) concluded that the overshooting depth depends strongly on the resolution of the simulations. The numerical models of Hotta (2017) use a numerical diffusion scheme based on slope limiters where the effective Reynolds and Péclet numbers depend on the grid spacing. Here the explicit viscosity and entropy diffusion are varied to study this effect. Run K5 is taken as a reference run, and the diffusion coefficients were varied within current computational limits in Set R. Run K5 is referred to as Run R3 in Set R. The simulation strategy was such that two branches of runs were performed by taking a thermally saturated snapshot of Run K5 as a basis. In the low-Re branch the grid resolution was kept fixed and the diffusivities were increased (Runs R1 and R2). In the high-Re branch (Runs R4-7) the diffusivities were decreased, and if necessary, a snapshot from a previous simulation was re-meshed to a higher grid resolution (Runs R5 and R6). These two branches were ran consecutively such that the previous runs were first run to a thermally saturated state before changing the diffusivities for the next run to avoid long transients. The results for the normalised overshooting depth as a function of Re = PeSGS are shown in Fig. 9.

Overshooting depth normalised by the pressure scale height at zCZ as a function of Re for Set R. The inset shows dos/HP as a function of Ma = urms/(gd)1/2.

The current results suggest that overshooting is roughly constant as a function of Re. We note, however, that the data points with the highest values of Re (Runs R6 and R7) in Fig. 9 could not be run sufficiently long to establish that they are truly in a statistically stationary state. Thus the values of dos from these runs should be considered as upper limits. In any case, these results are at odds with those obtained by Hotta (2017), who found a steeply declining trend as a function of Re. This, however, is likely because Hotta (2017) modified the heat conductivity in the radiative layer to speed up thermal relaxation (see Sect. 4.5), and possibly exacerbated by the varying effective Prandtl number in his models (Sect. 4.2).

The inset of Fig. 9 shows dos/Hp as a function of Ma which quantifies the overall magnitude of the convective velocity. The current data suggest that the overshooting depth is independent of the overall velocity. This is at odds with Zahn (1991), for instance, who derived a Ma3/2 dependence. However, the range of values explored here is too narrow to draw definite conclusions.

Depth of the transition δtrans from nearly adiabatic to radiative zones as a function of input flux ℱn normalised by the pressure scale height at zCZ from Set K.

Profiles of K, normalised by Kbot, from Runs S7 (black) and S7m (red). The bottom of the initially unstable layer is indicated by the vertical dotted line at z = 0.

Time-averaged total (black dash-dotted), convective (black), enthalpy (blue), radiative (dark purple), kinetic energy (light purple), and cooling (green) fluxes from Runs S7 (solid) and S7m (dashed). The vertical dotted lines indicate the bottoms of the buoyancy (BZ) and Deardorff zones (DZ). The bottom of the overshoot zone (OZ) for Run S7 (S7m) is denoted by the dashed (dot-dashed) vertical line.

Furthermore, the current results indicate that modifying the heat conductivity in the layers below the CZ (e.g. Brun et al. 2017; Hotta 2017) leads to a substantial underestimation of the overshooting depth. The only way to extract reliable scaling of the overshooting depth as a function of ℱn currently is to run the simulations self-consistently to a thermally relaxed state without modifying the system parameters such as the heat conductivity. The present study also demonstrates the limits of this approach in that the runs with the lowest input flux require integration times of the order of several months even at a relatively low resolution of 2883. A more promising alternative to speeding up thermal saturation is to alter the thermodynamic quantities instead (e.g. Hurlburt et al. 1986; Anders et al. 2018). However, even this method has its limitations, and the applicability of this approach, for example, to rotating convection in spherical shells remains to be demonstrated.

The current results suggest that the overshooting depth in the Sun would be of the order of